Finite Element Methods for Fluids

FINITEELEnvIENT

METFIODS FORFLUIDS

Olivier Pironneau

Universitd Pierre-et-Marie-Curie & INRIA, F rance

JOHN WILEY & SONSChichester New York

Brisbane Toronto SingaPore

MASSONParis Milan

Barcelone Mexico

I 9 8 9

Table of contents

Foreword. Notation

Introduction

1. partial d.ifferential equations of fluid mechanics.

1 . 1 O r i e n t a t i o n " " " 1 1

L.Z The general equations of Newtonian fluids: conservation of ma^ss,

momenturrr, "n.rgy and the state equation " 15

1.3 /nurs cid, f lows " " ' 19

!.4 Incomprt,ssible and weakly compressible flows " "20

1 . 5 l r r o t a t i o n a l f l o w s " " " ' 2 2

1.6 The Stokes Problem " " ' 23

L.7 Choice of equations " " ' 24

1 . 8 C o n c l u s i o n . " " " 2 5

weakly irrotational flows2 .

2.3

2.42.52.6

3. : Convection - diffusion phenomena

B.l Introduction: boundary layers, instabil ity of centered schemes . ...57

3.2 Stationary conaections: introduction, schemes based on the method

of characteristics, upwind schemes, the method of streamline diffusion,

,;.;l;;;; rh;.#Uprvinding by discontinuities, upwinding by cells'

3.3 Conuection diffusion 1: introduction, theoretical

vection - diffusion equation, time approximation' " " "75

g.4 Conuection diffusion 2: discretisation of total derivatives, the Lax -

Wendroff/Taylor-Galerkin scheme, streamline upwinding (SUPG) 84

4. : The Stokes problem

4.1 Slalement of the Problem ' ' ' 95

4 .2 Func t iona lse t t ing " " " '96

4.J Disqetisation; the Plbubble/P1 element, the PlisoP2/P1 element,

the P2/P1 element, the non conforming PL I P0 element , the non

conformin g P2 bubble / Pl element, comparison ' " " 98

5.

FINITE ELEMENT METHODS FOR FLUIDS

4.4 Numerical solution of the linear systems .' genera,lities, solution ofthe saddle point problem by conjugate gradient . 109

4.5 Ermr estimates: abstract*setting, general theorems, verification ofthe inf-sup condition. . .113

4.6 Other haundary conditions: slip boundary conditions, boundary con-ditions on the pressure and on the rotational. .. i. . ... 120

The Navier-Stokes equations5 . t In t rodue t ion . . . . . . L255.2 Eristence, uniqueness, regularity: the variational problem, existence,

uniqueness, regularity, behavior at infinity, The case v =O . ..,.t275.3 Discretisatian in sTtace: generalities, error estimates . . 1335.4 Discretisation in tim.e: semi-explicit discretisation, implicit and semi-

implicit discretisations, solution of the nonlinear systern . . . L375.5 Discretisation af the total deriuatiae .. . .1425 . 6 O t h e r m e t h o d s . . . . . 1 4 95.7 Simulation of turbulent flows . . . . .151

Euler, compressible Navier-Stokes and the shallow water equa-tions.6.L In t roduct ion. . . . . 1616.2 The compressible Euler equations.' statement of the problem, 2D tests,

existence, a few centered schemes, some upwind schemes. . .1616.3 The cornprcssible Naaier-Stokes equalions: introduction, an exam*

ple, an implicit scheme for weakly compressible flows, Euler-basedschemes . . . . . . 172

6.4 The Shallow water equations: introduction, a velocity-depth scheme,A flux-depth scheme, comparison. . .. ..I77

6.5 Conclusion

6.

References

Appendix

fndex

: A finite elernent program for fluids on the Macintosh 197

206

The Unseen grew aisible to sludent eyes,,

Erplained was the irnntense Inconscienl''s scheme,

Aailacious lines were traced upon the ooid;

The Infinite was reduced to square and cube'

Sri Aurobindo Saaitri. Book 2, Canto XI

Foreword

This book is written from the notes of a course given by the author at the

Universit6 Piene et Marie Curie (Paris 6) in 1985, 86 and 87 at the Master

Ievel. This course addresses students having a good knowledge of basic nu-

merical analysis. a general idea about variational techniques and finite element

methods for partial differential equations and if possible a little knowledge of

fluid mechanics; its purpose is to prepare them to do research in numericai

analysis applied to problems in fluid mechanics. Such research starts, very

often nith practical training in a laboratory; this book is therefore chiefly ori-

ented towards the production of programs; in other words its aim is to give

the reader the basic knowledge about the formulation, the methods to analyze

and to resolve the problems of fluid mechanics with a view to simulating them

numerically on computer; at the same time the most important error estimates

available are given.

Unfortunately, the field of Computational Fluid Dynamics has become a

vast area and each chapter of this book alone could be made a separate lVIasters

course: so it became necessary to restrict the discussion to the techniques

which are used in the laboratories known to the author, i.e. INRIA, Dassault

Industries, LNH/EDF-Chatou, ONERA...Moreover, the selected methods are

areflection of his career (andhis antiquated notions ?). Briefly, in aword, this

book is not a review of all existing methods.

In spite of this incompleteness the arrthor wishes to thank his close collabo-

rators rvhose work has been presenteci in this book and hopes that it would not

8 FINITE ELEMENT METHODS FOR FLUIDS

hurt them to have their results appear in so partial a work: MM C. Bernardi,J.A. D6siddri, F. Eldabaghi, S. Gallic, V. Girault, R. Glowinski, F. Hecht, C.ParEs, J. Pdriaux, P.A. R"avia,rt, P-h- Rostand, and-J.II. Saiac. The authorwishes to thanks his colleagues too, whose figures are reproduced here: MrsC. Begue and M"O. Bristeau, MM B. Cardot, J.M. Dupuy, F. Eldabaghl, J.Hasbani, F. Hecht, B. Mantel, C. ParEs, Ph. Rostand, J.H. Saiac,V. Schoenand B. Stoufilet as well as the following institutions AMD-BA, PSA, SNCF andSTCAN with special thanks to AMD-BA for some of the color pictures. Theauthor'also wishes to thank warmly MM P.G. Ciarlet and J.L" Lions for theirguidance for the publication of this work, G. Arumugam and R. Knowles fcrtheir assistance in the translation into English and finally Mrs C. Demars fortyping the script on MacWriteTM. This book has been typeset with Tfif,; thetranslation from MacWrite to TeXture?M has been made with EasyTeX"M.

MacDraw, MacWrite and Macintosh are trade marks of Apple ComputerCo.

Tff,is a trade mark of the American lt{ath. Society.T$tures is trade mark of Blue Sky Research Co.EasyTbX and MacFEM are copywrite.softrvares of Numerica Co. (23 Bd

de Brandebourg BP 215, 94203 lvry/Seine, France)

Notation

References:A number in brackets, ex (1), refers to the equation (1) of the current

chapter . The notation (1.1) refers to the equation (1) of the chapter 1.

The numbers in square brackets like [2] indicate a reference in the bibliog-

raphy.

Vectors, matrices, scalar products ...

Unless stated otherwise the repeated indices are summed. Thus,

u .a = u ia i

denotes a scalar product of u with o.For all vectors u, u and for all tensors of order 2 A, B we denote :

(u 8 u);; - uiaj; trA - A;i; A: B = tr(ABr), lAl = (tr '+er1*

Differential operators :

The partial derivatives with respect to t or r j are denoted respectively

0a 7tt; 0au,t=

Ei ui , i =

6t a, i - -

6

The classical grad, div, curl and laplacian operators are denoted respectively

Yp, V .u , V x u , Au

10 FINITE ELEMENT

The operator V (nabla) applies also

V.A= A; i , ; (Vu) ; i -

METHODS FOR FLUIDS

to vectors and tensors ... for example

, i , i ( and thus u.Yu - u ;u, i )

Little functions:O(r): Any function obeving l lo( l l ) l l S.t; , , , , , , ,,1r) i Any f,rr, . t ion obeving l im"*o l io(") l l / l l " l l =0'

GeometrY :

The domain of a PDE is in general denoted by o, which is a bounded

open set in R' (n-2 or 3); its volime element by dx and the boundary by 0o

or often I. . The domain is, in general suflftciently regular (locally on the one

side of the boundary which is Lipschitz continuous) f9r ys to. define a normal

n(x) for almost all x of the bo.rndary l, but we admit domains with corners'

The tangents are denoted by r in 2D and r:,r2 in 3D; the curvilinear abscissa

is noted by s and the element length (or area) is denoted by d7'

[Ol stands for the area or volume of Q'

diam(A) is its diameter '

Function SPaces

Pk = space of polynomials of degree iess thân

Co(Q) - space of continuous functions on O'

,'('Oj = space of square integrable functions'

o f L 2 ' b y ( , ) a n d t h e n o r m b v l l o o r l l s ' n :

or equal to k.

We note the scalar Product

(o, b) = a ( c ) . b ( c ) ; l l o - @ , a ) *

Ht(O) = Sobolev space of order l .We note i ts norm Uv l l l l r or l l l l t , t '

;i';igi = spaces of ^[/1(O) functions wit]r zero trace on I'

;1;icli" = Sobolev space'of order k wittr functions having range in R''

Finite Elements:A triangulationof Q is a covering by disjoint tria'ngles (tetrahedra in 3D)

such that the vertices of .flflr,, the boundary of the union of elements' are on

the boundary of o. The singular points of 0Q must be vertices of don'A

triangulation is regular if no angle tends !" 9 or zr when the element size h

tends to 0. A triangulation is uniform if all triangles are equal' -An interpolati of a function rp in a polynomial space is the polynomial

function 91 which is equal to g onsome foints of the domain (called nodes)'

A finite element is conforming if the space of approrimation is included in

I{1 (in practice continu.ous functions). The most fiequently used finite element

in this book is the pl conforming element on triangles (tetrahedra). An

element is said to be lagrangian ( othert Tut be Hermite") if it uses only

values of functions at nodes and no derivatives.

T

Introduction

computational fluid dynamics (cFD) is in a fair way to becoming an rm-

portant engineering tool like wind tunnels. For Dassault industries, 1986 was

the year when the -numerical

budget overtook the budget for experimentation

in wind tunnels. In other domains, like nuclear security and aetospace' exper-

iments are difficult if not impossible to make. Besides, the liquid state being

one of the four fundamenstal states of matter, it is evident that the practical

applications are uncountable and range from micro-biology to the formation of

stars. At the time of writing, CFD is ihe privilege of a few but it is not hard to

foresee the days when multipurpose fluid mechanics software rvill be available

to non-numerical engineers on workstations'

Simulations of fluid flows began in the early 60's with potential flows'

first incompressible or compressible hypersonic, then compressible transonic (cf'

Ritchmeyer - Morton [1]). The calculations were done using finite difference or

panel ,rr.thod, 1.i nt.fUiatZ]). Aeronautic and nuclear industries have been the

principa,l users of numerical ialculations. The 70's saw the first implementation

of the finite element method for the potential equation and the Navier-stokes

equations (Chung[,Temam[3], Thomasset[a]); also during tliat time the de-

velopment of finiie difference methods for complex problems like compressible

Navier-Stokes equations continued (see Fe

years have seen the development of far

flows (multigrid, (Brandt[6], Hackbus]

[7]), vectorization (Woodward et al[8])

to reach certain objectives (spectral n

(Chorin [10]) methods of lattice ga^s (

T2 FINITE ELEMENT METHODS FOR FLUIDS

treatment of problems which are more and more complex like the compressible

Navier-Stokes equations with interaction of shock-boundary layers, Knutsen

boundary layers (rarefied gases, see Brun[ ]), free surfaces "'And yet in spite

of the apparent ,t.."r, the day to day problems like flow in a pipe, in a glass

of water with ice, in a river, around a car still remain unsolvable with today's

computers. simulation of turbulence, Rayley-Benard instabilities ." lie stiil

further in the future. suffice it to say that the subject wili have to be studied

for a good number of years yet to come!

Why restrict ourselves only to finite element methods ? many reasons; the

first because one can not explain everything in 200 pages' the next because the

author has more experience with this method than with other methods but

most importantly because if one needs to know only one method then this is

the best one to krro* ; in fact, this is the only technique (in 1987) which can

be applied to all equations without restriction on the domain occupied by the

fluid.We have also chosen to study low degree finite elements because practical

flows are often singular (shock, turbulence, boundary layer) and to do better

with higher order Llements one has to do a lot more than just doubling the

degree of approximation (Babuskafi, Paterafl)

The chapters of this book more or less reflect the historical discovery of

the methods. But since this book addresses itself to specialists in numerical

analysis, chapler / recalls the main equations of fluid mechanics and their

derivations.

Chapter g treats of irrotational flow, i.e. the velocity of the flow is given

by the relation

u = V p

where g , the flow potential, is a real valued function '

In,compressibli potential flows provide an occasion for us to recall the iter-

ative methods used to solve Neumann problems by the finite element method:

a 6 n n; - = u o n ron

treated in Chapter 1 :

( 1 )

- L 9 = 0 i n O ;

The transonic potential equation is also

v. (1 - lvp l ' ) ; -ve l = 0 in Q; #=0 on I

and solved b.v an optimization method.

This chapter ends with the description of a potential correction

for weakly irrotational flows. So it deals rvith a method of solution for

stalionary syslem:

(2)

(3)

mcthodEuler 's

V.pu = 0, V.[p*ou] + Vp = 0, v. tp"(r* i* i* l l = 0 in Q, (4)

INTRODUCTION 13i

where u.n is given on all of I and p,u and p are given on parts of f according

to the signs oi rr.lr *}f,)tl'.

chapter 3 deals with convection - diffusion phenomena, i.e. equations of

the type :

X *v.e' - k\p- / in Q,

with g given on the boundary and at an initial time. The methods of upwinding

and artificial viscosity are introduced'

ln chapter y' we give some finite element methods to solve the generalized

Stokes problem :

o u - A u * V p = f , V . * = 0 i n Q ,

with u given on the boundary. This problem which is in itself useful for low

ReynolJs number flows is also utilized as an intermediate step for the resolution

of Navier-Stokes equations.

The chapler ishows how one solves the Navier-Stokes equations using the

methods giuen in chapters 3 and 4. Some models of turbulence as well as the

methods for solving them are given.

Finally, in chapter 6 we give a few methods to solve the compressible Euler

equations (tZltgl with p = C = t = 0) and the Navier-Stokes equations :

H.v .pu = o

W .V.(p,r 8') * vp - qAu - (3 + €)v(v.u) = 0

*r** j , ' i t+v{pur+.#iu (e)

= v.{ ioot+ lrr(vu + vu") + (€ - f a)w 'Ju] + f 'u- n p

with u,p given on the boundary, p given on part of the boundary and at an

initial time.In this same chapter, we also consider the case of Saint-Venant's shallow

water equations because they form an impottant bidimensional approximation

to the incompressible Navier-Stokes equations and they are of the same type

as the cornpressible Navier-Stokes equations'

The book ends with an appendix describing a computer program, written

by the author to illustrate the course and used to solve some of the problems

studied. Unfortunately this software makes extensive use of the 'Toolbox' of

the Appte Macintosh"M and it is not portable to other machines

(5)

(6)

(7)

(8)

14 FINITE ELEMENT METHODS FOR FTUIDS

A FE\M BOOKS r.OR, FURTHER, READING

t1] A. Baker:Finite element computational flaid mechanics' McGraw Hill'

1985.

t2l J.P. Benque, A. Ilaugel, P.L. Viotlet:Engineering applications of hyd'raulics

I/. Pitman, 1982.

t3l G. Chavent, G. Jaffrey: Mathematical methods and f'nite elements for

relen)orr stmulations. North-Holland, 1986'

[4] T. Chung: Finite element analysis in fluid mechanic's' McGraw Hill' 1978

iui c Crrueli.r, A. Seigal, A. van Steenhoven: Finit'e element methods and

Naaier-Stokes equa{ionr. Mathematics and its applications series, D' Rei-

del Publishing co, 1986.

t6] G. Dhatt, G.-Touzot: Une pr6sentation de la m6thode des t4l6ments finis'

Editions Maloine, 1984.

t?] v. Girault, P.A. Raviart:Finit e etetnent method for Nauier-stokes equ&'

tions. Springer Series SCM, 5, 1985'

t8] R. Gto*irrrili, Numerical methods for nonlinear aariational problems'

Springer Series in comp. physics' 1984'

[g] R Grirbe., J. Rappaz:-Finite element methods in linear magnetohydrody-

namics. Springer series in ccmp' physics, 1985'

[10] C. Hirschi Nimerical compulation of intemal and external flows' Wiley,

1988.T.J.R. Hughes: The finite elernent method. Prentice Hall, 1987.

c. Johnson:Numerical solul,ion of PDE by the finite element method' cam-

bridge university Press, 1987.R. ieyr"t, T. iuytot' Computational methods for fluid flows. Springer

series in computational physics. 1985'

F. Thomasset: Implementation of the finite element method for the Naaier-

stokes equalions. springer series in comp. physics. 1981.

[1 1][12]

[13]

[14]

Chapter 1

Some equations of fluid mechanics

1. ORTENTATION.

The aim of this chaPter is to recall

as well as the approximation principles

simulations. For more details, see for

Landau-Lifchitz [141], Panton [187].

the basic equations of fluid mechanics

which we propose for their nume.rical

example Anderson [4], Bachelor [10],

2. GENERAL EQUATIONS OF NEWTONIAN FLUIDS

2.1 Equation of conservation of ,rn&ss '

Let p(x,t) be the density of a fluid at a point c at time t ;let u(c,t) be

its velocity.Let O be the domain occupied by the fluid and O a regular subdomain of O.

To conserve the mass, the rate of change of mass of the fluid in o,-O(Jo dl at,

has to be equal to the mass flux across the boundary 0o ofo, - fuo Pu.n,'( n

denotes the exterior normal at 0O and the surface element)


Figure 1.1 : A volume elernent of fluid O.

Using Stokes formula

l"arbitrary,

f

V . u = | , . n ( 1 )J A O

we get imrnediately the equation of conserva-and the fact that O istion of rnass

P,t*Y.(Pu) = 0

2,2 Equation of conservation of momentum

Now let us write Newton's equations for a volume element O of the fluid.A part ic le of the f lu id at s at an instant t wi l l be at x*u(x,r)e+o(&) at

instant t + k ; its acceleration is therefore

l t * i fu(x + u(x, t )k, t + e) - u(r , t ) l = u,1*uiu,5 (g)

The external forces on O are :

- The external forces on a unit volume (electromagnetism, Coriolis, grav-i tv.. .) , fo f

- The pressure forces: I uop,-The viscous constraints due to deformation of the fluid (a' is a tensor):

II o 'n .

Jao

Newton's laws are. therefore. written

f f f

Jon@l*uvu) =

Jof -

Juo(pn-o 'n) ,or, by using Stokes formula

(2)

SOME EQUATIONS OF FLUID h{ECHANICS

V.o ' ) ,

which gives :

p ( u , t * u V u ) * V P - Y - o ' = f '

To proceed further we need a hypothesis to relate the stress tensol a/ with

the velocity of the fluid. BV defrnition of the word "fluid" ' ilY force' even

infinitesimal, applied to o should produce a displacement, thus a' being the

effect due to the viscosity of the fluid, it must depend on the derivatives of the

velocity and tend to zero with them'

The hypothesis of Newtonian fl,ow is a linear law relating s' ta vu ' (I

being the unit tensor) :

o' = rt(Yu +vur) + (€ * l lro."

4 and { are i,he first and second viscosities of the fluid'

Frcm (1), (3) and (4) we derive the equation of conseruation of momentum

p(u,t* uvu) * vp - v.h(vu + vu") + (€ - !) 'o'") - f ' (5)

It is easy to verify that the equations can also be written as

p(u,t*uVu) - q\u- (3 + €)V(V u) + Yp = f (6)

because v.Vu - Au and V.vuT - v(v.u). Taking into account the co'tinuit3'

equation (2), rne could also rewrite (6) in a conservative form :

W .Y.(puo u) + vp - qLu- (3 + ()v(v 'u) = f ' (7)

- Pasty fluids (near to a state of solidification)

- micro.fib.r, ir fluids (some biomechanics fluids with long molecules),

17

loo@,,r uVu) -

Io,t - vP +

(4)

18 FINITE ELEMENT METIIODS FOR FLUIDS

Mixture of fluid-particles (blood, for example),Rarefied ga^ses.

2.3 Equation of conservation of energy and the state equation .

Lastly, an equation called conseruation of energy can be obtained by writ-

ing the total energy of a volume element O(t) moving with the fluid-

We note that the eners/ E is the sum of the work done by the forces and

the amount of heat received. The energy of a volume element O is the sum of

potential energy pe and the kinetic energy put /2 , fo p( u212 *t) . The work

done by the forces is: Jo u. f + fuouo' .n -

fuopn.u-If there is no sourie of heat (combustion...) then the amount of heat

received (lost) is proportional to the gradient flux of the'temperature 0 :

fuo rcY?.n. We therefore get the equation :

* I,uJ'k ++)) - l"{Lpk + {)t' '+ v rup( { + dt

- l.u.f

- lro[u(o'

-pr) * nYg)n

With (1) and Stokes'formula, we obtain :

*ru * *n + v.(ufpt* *e) +rl) = Y.(uo' + nvfi + f.u

For an ideal fl,uid, C, and Co being the physical constants, we have

(8)

e = C r ?

and the equation of state

! = R ep

where R is an ideal gas constant. With 7 = ColC, = RlCr+(9) * follows :

, _ p" -

p 0 - 1 ) '

With (4), (8) becomes :

0 . u 2 p , , , 2

#vT*#r +v t " tpT*h6+f "

= V.{rcV g +lrt(Yu* Vu") + (€ - fn)rv.rJr}

(e)

(10)

1 we can write

( 1 1 )

( 12)

SOME EQUATIONS OF FLUID MECHANICS

This equation can be rewritten as a function of temperature

g,t * uYr- * (7 = l)ov ." - *d'ePUo

= *t tv.r l2(€ -2rr /3)* lvu *Yurf i lPUu

By introducing entroPY :

R , ps =

F t o s /

-(12) can be rewritten in the form (cf. Landau-Lifchitz [141 p' 236]):

N(Tr* uvs) - \ lvu* Vurl ' + (( - f nltv' ulz + nLl (15)

Remark 2 :

Some values of the physical constants

19

( 13)

(14)

p(g lr*3) q(s lcm.s)a i r 1 .210-3 1.810-4

utater 1 0.01

'r1 .41

€= 0= 0

n(cm2 ls)0.2

1.410-3

R(cm2 f sz .o C)2.871060.2410

conditions :- u,g given on the boundary, p calculated from (10)

- p given on the part of the boundary in which u'n <�0'

A nunrber of numerical simulations have been made (mostly using finite

differences) but they are extremely onerous and limited to quasi non-stationary

laminar flows.For these reasons and also with a view towards getting partia,l analytical

solutions, the fluid mechanicists have proposed some approximations to the

complete s1'stem ; it is, in general, these approximated systems that we solve

numericalll'.

3. INVISCID FLOWS.

If we neglect the loss of heat by thermal diffusion (rc - 0) and the viscous

effects (rl = € = 0) the simplified equations are knowrr as the cornpressible Euler

equations. So equation (15) becomes :

2A FINITE ELEMENT METHODS FOR FTUIDS

Y, .uvs = o

hence s is constant on the lines tangent at each

fact a stream line is a solution of the equation :

( 16)

point to u (streane lines). In

and so

P , t * V ' ( P u ) = 0

p(u, t *uVu) +Yp - f

p - C f ( C = u ' o # ) .

The problems (2X5X12) and (19)-(21) will be studiedalgorithm for finding the stationary solutions of (19)-(21)Chapter 2.

4. INCOMPRESSIBLE OR WEAKLYFLOWS.

, ' ( r ) - u(x( r ) , r )

fts@g),r) = #*.X =s,r *uvs = o.

(17)

( 18)

If s is constant and equal to s0 constant at time 0 and if s is also equal to s0 on

the part of f whereu,.n ( 0, then from (17) we see that (18) has an analytical

solution s = s0. Note however that (18) has no meaning if u ha.s a shock. In

any case, experiments show that s decreages through shocks.

Finally there remains a system of two equations with two unknowns :

where

(1e)

(20)

(2 1)

in chapter 6 but anwill also be given in

COMPRESSIBLE

In the case when p is practically constant (water for example or air withIow velocity) we can neglect its derivatives. Then (2), (6) become the incom-pressible Naaier Stokes equations :

V.u = Q (22)

u , t * u V u * Y p - v A u , - f ( 2 3 )

where u=r l lp is the reduceduiscosi tyof thef lu id andp/pandf lp havebeenreplaced by p and J.

The same approximation can be applied to equations (19)-(20) and weobtain (22)-(23) with v = 0, known as the incompressible Euler equations.

V . u = 0 (24)

Remark 3.With (22X23) or (2a)(25) one can no longer use (21) because the svs-

tem becomes over'determined. This ambiguity can be explained by srudf ing

(19)(21)when the velocity of sound (dp/dfirl2 tends to inflnity. one ca:i shorv

iXiui""r*an -Majda [fOf]) that if C tends to infinity in (21) then p gtrs ro in-

finity but Yp r"muins bounded. Thus the pressure variation continues ttr have

a physical interpretation in (22)(23) or (24X25) but p no loniler represeirls the

physical pressure.

An equation for the temperature 0 can be obtained from (13) by a-r'uning

p constant; if / = 0 we have :

X.uYo-ho t__h tou+Yur l2

SOME EQUATIONS OF FLUID MECHANICS

rtr,t * uYu *Yp - f

The Navier-stokes and Euler's equations will be studied in chapter 5.

If also u and p - po are srriall, then (27) is close to

u , t *Cpo7- lvp = oP, t *PoY 'u = 0 ;

which gives

P,tt -cPo'Lp - 0.

V . u = 0

u , 1 l u Y u * V P - v L u - - g 9 e g

g, t *uv0 - rc ' L0 - ht ru

+ Yur l2

2 T

(25)

( 26)

An intermediate approximation, called weakly cornpressible , in betn-een

the complete systet" (2), (5), (12) (compressible Navier-Stokes ) and iucom-

pressible Navier-Stotces 1ZZ;-(23), is interesting because it exhibi'us tlie hyper-

bolic character of the underlying acoustics in (2), (5), (12) and (19)(21)'

Assumethat { =q= rc = 0 , wh ich imp l iesscons tan t and sopp- t co t ls tan t ;

under the assumpiion that p oscillates around a value p0 one gets :

p , t * u Y P * P o V . u = 0

u,t *uYu * Cpot- tVp - 0. (27)

(28)

(2e)

(30)


where e3 is the unit vector in the vertical direction.

5. IRROTATIONAL FLOWS :

We could try to determine if, for suitable boundary conditions, there exist

solutions of equations satisfying

V x u = O ; ( 3 1 )

these solutions are called itrotational

But as (22) impties the existence of rp (x,t) such that

u = Ys (32)

these solutions are also called potential'

Using the identities :

A u = - V x V x u * V ( V . u ) ( 3 3 )

uVu = *ux (V x u ) * " t ; l (34 )

we say that (24)-(25) have such solutions for / = 0 if rp is a solution of the

Laplace equalion :

Ap = 0, (35)

with (32) and

1 ^p = k -

, lvvl ' . (36)

This type of flow is the simplest of atl. It will be studied in chapter 2.

In the same way, with (32)' (27) implies

P , I * Y P Y P * P a L 9 - A ( 3 7 )

v(p , ,+ | l v , r l ' *ycpot - tp ) =o (38)

If we neglect the convection t.i* YpY p this system simplifies to a nonlinear

wa,ae equation :

p,tt -c/.p + f,lv vl' ,, = d,(t) (39)

where c = 7C po? is related to the velocity of the sound in the fluid.

Finally, we show that there exist stationary potential solrrtions of (tg)'

(20), (21) with.f = 0. Using (34), (19) could be rewritten as follows :

SOME EQUATIONS OF FTUID MECHANICS

- p u x V x u + p v + . v P = o '

Taking the scalar product with u' we obtain

" bo+* Vpl = o

Also, on taking into account (21)

' .Ov* * {-'c1v p1 = s

Or

_ , u 2 - + C 1 ; p . t - \ = o ( 4 3 )uP'v\n .y - L

So, the quantity between the pa.renthesis is constant along the stream lines,

that is we have

o 1 2

p - po(k- i l#Indeedasys tern t ikeuV(=0 is in tegra tedexac t ly l i ke(16)andg ives ' {cons tan tif it is constant on the part of the bo-undary where u.n 10. Thus if p0 and & are

constant at the entrance boundary of Q (u.ni 0), if all the streamlines intersect

the entrance boundary of O and if u is non zero' then (a4) holds, then (40)

implies that v x u is parallel to u and, at least in 2 dimensions , this implies

that u derives from a potential (i.e. (31)). Flom (2) and (4a) we deduce the

trans onic potential f l,ow equation.

v. t (k- lvPl ' ) ;hv 'P1 =o

This problem will be studied in chapter 2'

Remark 4:Equation (35) is consistent with (45) since we can return to it by assuming

p constant (Cf (44)).

6. THE STOKES PROBLEM.

Let us come back to the system (19)-(21) and let us rewrite it in non

dimensional form.Let U be a characteristic velocity of the flow under study (for example one

of the non homogeneous boundary conditions )'

23

(40)

(41)

(42)

(44)

(45)


Let L be the characteristic length (for example the diameter of Q) and ?

a characteristic time (which is a priori equal to LIU).Let us put

u ' = # ; t ' - i t , ' = *

Then (15) and (16) can be rewrit ten as

V6' 'ul - 0

, L \ , , - - t r t ^ . 1 t r r - 2 ;7 - ^ ( - " f '

\V6)u' ,t ' {utY ' 'u' + U-'v x'P - (fu)a"'u' - f frSo, if we put T - Ll(J , P' = P/Uz

, '=hthen (g8)-(Bg) is the same as (15)-(16) with "prime" variables.The inverse of

z' is calle d Reynolds number. Let us give some examples :

(46)

(47)

(48)

(4e)

(unitsM I{ S) Umicro - organism 10-4

glider 1sailboat(shell) 0, 1

ca,r 3airplane 30t,anker I

L v R e - W10-4 10-6 10-2

1 0 . 1 5 x 1 0 - a 7 x 1 0 41 10-6 1053 0 . 1 5 x 1 0 - a 6 x 1 0 51 0 0 . 1 5 x 1 0 - 4 2 x 1 0 7

2 0 0 1 0 - 6 2 x 1 0 8

When v' >> !, v'Lu' dominates u'Vu' and u!r, in (a8); it becomes the

Stokes problert :

(50)

(51)

7. CHOTCD OF EQUATTONS.

Given a fluid system to simulate numerically,chosen? Unfortunately, there is no automatic rule

simplest system compatible with tle phenomenon'

what equations need to be

; one seeks, in general the

-vA,u *Yp - f

V . u = 0

Experience shows that (50)-(51) is indeed an excellent approximation of (15)-

(tO) for "low Reynolds number fiows" in dimension three ; in dimension two the

Stotes problem might not have solution in an unbounded domain. We recall

in this context that a two dimensional flow is not an infinitely thin ffow, but

a flow invariant under translation in a spatial direction and whose velocity is

perpendicular to the invariant direction' .

The Stokes problem will be studied in chapter 4'

S O M E E Q U A T I O N S o F F L U I D M E C I I A N I C S 2 S

Let us take the case of an aircraft wing. The first thing that an engineer

wishes to know is evidently the resulta.nt of th" fluid forces on the wing ; that

is the lift and the drag. The lift being- sornewhat unrelated to the viscosity

we can integrate the transonic equatioi to calculate it (cf Chapter 2)' But if

there is a lot of vorticity then *" iru,',r" to take the Euler equation, compressible

if the velocity if large, incompressible if not. on the other hand, there is no

chance to calculate ihe drag with Euler's equations; we have to integrate the

Navier-Stokes equations (in ttre boundary layer or in the whole domain)' The

problem b..omo more complicated if the flow is "detached"; though the iift

is not a viscous phenomenon, the generation of eddies neal the trailing edge

is fundamentally a phenomenon riated to the viscosity' The oscillation of

the lifi around it, u.r"ruge value cannot be approximated correctly unless the

Navier-stokes equations are solved'

To conclud", th" choice of a system to solve is beyond the scope of this

book and this chapter gives orrly ,ome examples. A thorough discussion with

a physicist or with an engineer is therefore indispensable before embarking on

numlricat simulations which are often long and expensive'

8 C O N C L U S I O N :

ThenomenaPDE's :

Etliptic: Stokes equations,

Parabolic ; TemPerature

ible Navier-Stokes equatio'n.

incompressible Potential flow '

propagation, convection-diffusion' Incompress-

compressible Navier stokes equations describe a large number of phe-

which from the mathematical point of view belong to 3 categories of

Hyperbolic: Euler's equations, wave equations of the acoustics of a fluid,

convectron' +hor rharo ;nes not exist a uni , approximate

We state also, that there does not exist a universal scheme to

numerically these three types of equations, which explains why each subsystem

enumerated above and its numerical integration is treated in a separate chapter

of this book.

Chapter 2

Irrotational and weakly irrotational flows

1. OruENTATION

W e h a v e s e e n i n c h a p t e r l t h a t i t i s p o s s i b l e ( w i t h s u i t a b l e } o u n d a r y c o n -ditions) to find irrotational solutions (v x , = 0) to the general flr'rid mechanics

equations under the following hypotheses :

- inviscid flow- no vorticity generated by discontinuities (shocks) cr by the boundaries'

In this chapter, we will study finite element approximations of those equa-

tions. The first p".i duuls with the Neumann problem, where we recall also the

finite element method. In the ,..orrd part, *e d"al with subsonic compressible

flows, which will be solved by an optimization method' we then extend the

method to transonic flows. Finally in the thir-d-and fourth parts we study the

resorution of the probrem in terms bf stream function and a rotationar correc-

ii". -"tt od b-ased on Helmoltz decomposition of vector fields'

z. INCOMPRESSIBLE POTENTIAL FLOWS

2.1 Generalities.If the density p is constant, the viscosity and thermal diffusion are negligi-

ble and if the flow does.not depend on time then the velocity u(r) and pressure

p(c) are given for all points s € O of the fluid by

V . u = 0 ( 1 )


V x u = 0

. 1 ,p = k - i u '

where & is a constant if the flow is uniform at infinity.

Figure 2,t . A diuergent nozzle

Example 2Calculation of the flow around a symmetric wing (the non-symmetric

lifting case will be studied later).

Figure 2.? A syrnrnetric aifoil

(2)

(3)

On the boundary | = 0O in general the normal component of the velocity,

u.n, is given by :

u . n = g o n f ( 4 )

We remark that (2) implies the existence of a potential p such that :

u = V p ( 5 )

So the complete system is simply rewritten as a Neumann problem.

A P = 0 i n A

0p* _ g o n fon

Example 1.Calculation of low velocity laminar flow through a nozzle.

(6)

(7)

l .e.

g4

IRROTATIONAT AND QUASI IRROTATIONAT FLOWS 2:S

R.emarkwe could have used (1) to say that there exists r/ such that

u = V x r h ( 9 )

Then, the comPlete sYstem becomes

V x V x / = 0 i n O ( 1 0 )

Y x { t . n - s o n f ( 1 1 )

If the flow is bidimensional (invariant by translation in one direction) then

(10)-(11) become :

A?, - 0 (12)

r t$ (x ( s \ - Jg@O)h vo ( s )e r ( 13 )

We study this method at the end of the chapter. Note that in 3 dimensions,

the operator in (10)-(11) is not strongly elliptic and that r/ is a vector though

rp is a scala.r.

2.2. yariational formulation and discretisation of (6)-(7)

Proposition 1Lety e irp1rith non zero &aerage onA. If g is rvgular,(g e gtlzlf))and

tf

f o o (14 )J r

then problern (6)-(7) is equiualent to the following uariational problem :

f fI vev, - | s. Yw €.W (15)

J n J r

? eW = {w€ / {1 (o) ; ln

,= 0 } (16)

Proof :We see, in multiplying (6) bV ur, and integrating on Q with the help of

Green's formula that :

f f 0o f -

Jn_o,n, * [#, =

JnYevw. (1i)

, If we qse (6) and (7) in (17) we get


I ooo* - [ n- vtu € H'(o) (18)J n J f

Putting w = L we get (14). As (1S) doesn't change if we replace p by p +

constant and 1.r by , * constanl, we can restrict ourselves to W. The converse

can be proved in the same way ; from (15) with the assumption that rp is regular

(in "F/2(O)) ro as to use (17), we get :

l r -Oo) ,* l r r# , -g) to=o Yw€w (1e)

By taking tu to be zero on the boundary, we deduce that :

[ -Oolu., = 0 Vto such that [-r* - O,Ja Jn

and so, from the theory of Lagrange multipliers, there exists a constant I such

that

- h,9 = )X in O. (20)

Note also that (19) implies

* = n o n r ( 2 1 )on

Finally, (17) with w = L (20), (21) and (14) imply that the constant in (20) iszero.

Remarkw given by (16) is equivalent to HL(qln one could have set up the

problem in //1(A) lR at the start but this presentation allows a natural con-struction for the discretisation of I{'. Besides, it is interesting to note that if(1a) is not satisfied, p is the solution of (20) instead of (6).

Proposition 2If A is a bounded d,omain with Lipsch,itz boundary and if g is in Ht/z(f)

then (15) has a unique solul ion.

Proof :In W and with bounded Q, Poincar6's inequality is satisfied:

f r ' s c f l v p l ' ( 2 2 )J n ' J o '

So the bilincar form associated with (15) is W - elliptic :

I lor l '2 ] min{ c- ' , r } l le l l? (23)J n '

- z

As W is a closed subspace of I/t(O) the linear map

IRROTATIONAL AND QUASI IRROTATIoNAL FIJows 31"i'r

':lr$t

t!) + [ o.atJ r

is continuous. The resurt foilows from Lax-Milgram theorem (cf ciarlet [55]

Strang-Fix [223 ], Lions-Magenes [155])'

Proposition 3Ler {wnl be a sequence of internal approdmations (wn c w) of w' such

that

v w € W 3 { ' , , n } n t ! ) h l - W n s a c h t h o l l l . o - @ l [ * 0 w h e n h - + 0 ( 2 4 )

Then the sclution 9n e W6 of

I o roott)h = [-n*n Ywn € wn (25)Jn Jf

conaerges strongly in f{1(o) to g, the solution of (15)-(16).

Proof :Usini (15) with u.r6 and subtracting from (24)' we obtain

I O(rn- rp)Vu,';, = 0 Ywn €Wn (26)J N

which shows that tPr, is a solution of the problem

ft,^{ I*lV(,n - dl2| (27)

and so lf ,bn is an approximation of p in the sense of (2a) we have

I lo@n - ,P)l' < f-tot'ta - e)12 (28)J o J o

and hence the Proof (Cf' (22))'

Proposition 4If tiere erist q and C(A) such Lhat in aildition to (21) we haae

l l rn - r l [ < Ch" l l , l l "+ '

then l lPo - Pl lr ( C'ho l lPl l .+t

RemarkIf O is convex' we have also

(2e)

(30)

lpn -Plo < Ctt ha+rl lPl l "+t ' (31)


Proof :The error estimate (30) can easily be deduced from( 28) (29) and (22). To

prove (31) one has to use the duality argument of Aubin-Nitsche (see Ciarlet

[56])

Lagrangian triangular finite elements (also called P1 conforming):

O is divided into triangles (tetrahedra in 3p) {n}t...t such that

- Tx fiTt = @, or 1 vertex, or I whole side (resp. side or face) when k * I- The vertices of the boundary of UTr are on f- The singular points of I (corners) are on the boundary |6 of UTn.

We note that Qr, = U?r, h -- 0 U Tx , {qt}il are the vertices of the

triangles, and h is the longest side of a triangle :

o = ,',r,rf'stt'y lqt - q'l

Hn= { rue Co (on ) t *n l r r €P ' }

W n - { r o € H o ' t X u n = 0 }Jor

(32)

(33)

(34)

where Po denotes the space of polynomials in n variables of degree less than

or equal to o (O C n) and C0 the space of continuous functions.If Q is a polygon then I,[, C W and proposition 3 applies. If moreover, we

assume that all the angles of the triangles are bounded by 0r ) 0 and 02 1 rwhen h * 0 then the proposition 4 can be applied.

With (33) Wh is of f inite dimension, say M-1. Let {tu' i}r..y-1 be a basisof W1 ; if we write gn in this basis, we have

M - L

p,,(r)= t giw'i(o)1

and if we replace tu'1, by w'i in (25) we get a (M -L)x (M - 1) linear system.

(35)

(36)

(3i)with iD = {pt . .pu-r } , vw' iYw' i , Gj - l,o,'i

A O - G

f

A i j = |J N

Example IWith a = 1 a basis of Wn can easily be constructed. Let {}f (c)h-r...,,+r

denote the barycentric coordinates of s in ?r with respect to its n * L verticesthen the number of basis functions is,N - 1 (i.e. M = N) that is one

Iess than the total number of vertices {gh=r...1y. Let u,'i be the canonical basisfunction of Lagrangian elements of order 1:

IRROTATIONAL AND QUASI IRROTATIONAL FLOWS

*i(r) = Af (c) if k is such that o €Tx, (q' € Tt)

- 0 otherwise

33

(38)

then

*,, (r)- ui(o) - 1"1',f) i - L..t/ - 1Jnx

is a basis of Wn.

Figure 2.3 : Erample of namerical singularity obtained by pull ing

a basisjunction ; we renxark in the top middle a distortion of isoualue lines '

the sides (edges).

2.3. Resolution of the linear system by the conjugate gradient

method.

(3e)

FOR FLUIDS

. This algorithm aPPlies also

an construct the sYstem (36),

obtained is more regular (cf"

34 FINITE ELEMENT METHODS

speed the solution when properly preconditioned

to positive semi-definite linear systems; thus we c

(37) with all the basis functions uri. The solution

figures 3, 4).

Figure 2.4: Result obtained by the canjwgate gradient rnethod

withoat rernoaing the basis function'

Let us recall the conjugate gradient method briefly'

Notation:L , e f t , x € R N b e a r u n k n o w n s u c h t h a t A r - b , w h e r e A e f f x N , A T - A ,

b e f t N .

Let C be a positive definite matrix and < , )c the scalar product associ-

ated with C:

1 a ,b )c= o r Cb ; l l o l l c - (a r Ca) * (40)

Algorithrn 10. initialization:Choose C g RNXN positive definite (preconditioning matrix ), e small

positive, a}such that C;a € ImA (0 for example), and set g0 = 7ro =

- C - ' ( A * o - b ) , n = 0 .

1 Calculate :

< g " , h n ) c

hnr Ahn

= rn * p"h"

gn - p"C-L Ah"

p n =

gn* t

^n+L -v -

^ln

hn+t

(41)

(42)

(43)

(44)

(45)

l lg"+ ' l l3= lteEiz-= gn+r * f h"

2If ll yn+l llc < e , stop else incement n by 1 and go to 1'

IRROTATIONAL AND QUASI IRROTATIONAL FLOWS 35

Note that C is neuer inaer'ted and that 9 = C-L z is a short hand' notation

for "solue Cy = z"

Remark 1This algorithm can be viewed as a particular case of a more general algo-

rithm used lo find the minimum of a function (see below). Here it is applied

to the computation of the minimum of E(r) - xT A, f 2 - br c ' One can verify

thati) p" given bV (1) is also the minimum of E('^ + ph")

ii;'-ni+t givln by (aB) is also the gradient of E with iespect to the scalar

product associated with C, that is :

g n + 1 _ _ C - 1 1 A c " + 1 _ b ; (46)

Proof :

nn*r = rn * p" hn <__> Atn*r = Asn * pn Ah"

<+ C-t (Ax"+l - b) _ C- t (Arn - b) + p"C-L Ah"

Remark 2The only divisions in the algorithm are by 6nTn1rn and llg"llc' The first

one could be zero if the kernel of A is non empty whereas the second is non zero

by construction; so to prove that the algorithm is applicable even if det(, ) - 0

we have to prove that hnr Ah ts never zero. But before that let us show

convergence.

Lemma

h i r A h k - 0 , v j < f t

< g k , g j ) c = 0 , v j < k

< gt ' ,hi )c- o, Yi < k

Proof :Let us proceed by the method of induction. Assuming that the property is

truefor j <k ( n, let us prove that (a7)(4SX49) are t ruefor al l j < k < n* 1

i) Multiplying (a3) by and using (47) and (49):

< g n + t , h i > c = < g h , h i > c - p n < C - r A h " , h i > c = 0 - h i r A h n = 0 , ( 5 0 )

i f j < n ; I f i =n then i t i s ze ro bY (41) .

ii) Using (ab):

(47)

(48)

(4e)

6i


< gn+L,gi )c=1 gn+1,,hi - 7i 6i ' t )c= 0 (51)

bV i) above.iii)Finally, again from (ab) we have, if j < n :

6n*tT,E6i - (gn+, + Tnhn), eni = go*tT Ani +0 = g'*tt ,ki+'-: gi) - ,r

(52)where we have used (a?) to get the second equality and (43) for the last one.

If j - n, we have to use the definition of 7". Let us show that 7n is also

equal to -gn+rr Ah'o /hT Al , . . We have :

gn* tTAh, < g '+1 - gn ,gn* t )c _ - l lg "+ t l la

f f i - f f i - f f i ( 5 3 )We have used (43) for the first equation and (a8) and (49) for the second.

We leave to the reader the task of showing that {47),(49) are true for & = 1.

We end the proof by showing that 6nT tr6n is never zero-

Indeed if n is the first time it is zero then we have:

,0 = hnT Ah', h" e IrnA + /rt = 0 =+ gn = -7n-16n-1

= gn-L - ,n-t 6-r tr1n-t

so by (41)

_'n-Llh" l , =< gn-thn-t > _pn-L6n-f f t r6"-1 = 0

but by hypothesis h"-1 is not zero.

CorollatyThe algorithm conuerges in N iterations at the most.

Proof :Since the g" are orthogonal there cannot be more than N of them non

z,ero. So at iteration n = N, if not before, the algorithm produces gn = 0 that

is C-t(Ar" - b) = O.

Choice of CHowever it is out of question to do N iterations because N is a very big

number. One can prove the following result :

Proposition 5:If x* is the solution and xb is the computed solution at the lcth iteration,

then

rRROT/ffIONAt AND QUASI IRROTATIONAL FLOWS &7

( : a

< r k - x ' , A ( r r - o * ) ) c ( 4 ( l t A ' . : ) ' * < c 0 - x * , A ( x o - r * ) ) c' ' p u A + r '

where pcA it the conditian namber of A (r.u,tia between the largest and smallest

eigenuolues of Az = AC z ) in the netric inlroduced by C.

Proof .' see Lascaux-Theodor [9], for example.

One can also prove superconvergence results, that is the sequence {rb}converges faster than all the geometric progressions (faster than r& for all r)

but this result assumes that the number of iterations is large with respect to

N .We can easily verify that if x0 = 0 and C - A we get the solution in one

iteration. So this is an indication that one should choose C 'near' to A. When

we do not have any information about ,4., experience shows that the following

choices are good in increasing order of complexity and performance:

s;i = o7j',=r!'i,li, it < zC;j = 0, otherwise'

C - incompletetely factorized matrix of A.

(54)

(55)

(56)

We recall the principle of incomplete factorization (Meijerink-VanderVorst

ll72l, Glowinski et al [97]):We construct the Choleski factorization .L' of A (- L'L'r) and put

L;i - 0 if -4ii = 0, L;i = L'; i otherwise'

One can also construct directly .L instead of .L'by putting to 0 all the elementsof Ct which correspond to a zero element of A, during the factorization (cf [10])but then the final matrix may not be positive definite.

Proposition 6If A is positiae semi definite, b is in the image of A, and Cso is in the

image of A, then the conjugate grvdient algorilhm conaerges towards the unique

solution z' of the lineor system Ar = b; which uerifies Cx' € Im A.

Proof :From (a2) and (45) we see that

Crn ,Cgn,Ch" e ImA => Cxn* t ,Cgn* ' ,Chn+t e ImA

The property is thus proved by induction.


2.4 Cornputation of nozzles

If Q is a nozzle we take, in general, $ zerc on the walls of the nozzle and

g = 1r.o.fl, ua constant at the entrance and exit of the nozzle.

The engineer is interested in the pressure on the wall and the velocity field.

One sornetime solves the same problem with different boundary conditions at

the entrance f 1 and at the exit f2 of the nozzle :

-Ap - 0 in Q, 9lr, = 0, 9lr, = constant,

The sam" *"i"1*'01.1::;., u:,qE2

o g ,

frlr"-rrufz = 0. (57)

Figure 2.5 : Computation of f low in a nozzle . The trtangulalionand the leuel l ines of the potenl ia l are shown.

2.5. Computation of the lift of a wing profileThe flow around a wing profile S conesponds, in principle, to flow in

an unbounded exterior domain, but we approximate infinity numerically by u

boundary l* at a finite distance ; so Q is a two dimensional domain with

b o u n d a r y f = f * U S .

One often takes u* constant and

g l r - = ' t t ,oo . r t r t 9 l ; = 0

Unfortunately, the numerical results show that with these boundary conditions,the flow generally goes around the trailing edge P. As P is a singular point off , lVg(c)l tends to infinity when x - P and the viscosity effects (4 and () are

(58)

l \ .

1l - .

I\

IRROTATIONAL AND QUASI IRROTATIONAL FLOWS 39'

no longer negligible in the neighborhood of P (see figure 6)' The modeling of

the floiv UV (f )-12) is not valid and (2) has to be replaced by (c..' constant) :

V x u - u 6 n

where 6> is the Dirac function on the stream line E which passes through P.

Figure 2.6 FIow around an airfoih D is the wake'

One takes then O - t as the domain of computation' So we have

a boundary condition on D. Since D is a stream line : '

o g , n[rr lE = u '

But as u is continuous along E we also have :

oP, oP 'f i lx ,+

= Et

- .

by integrating this equation on E we obtain :

to add

(5e)

4 , ,fr@|"*

- Plx,-) = o i 'e' for

where p \s a constant which is to

on P, or better bY :

l vp (P+) l '= l vp(P- ) l '

which comes out to be the same but is interpreted as a continuity condition on

the pressure.It can be proved using conformal mappings that with (60)-(61) the solution

<p does not defend on the position of D (which is not known a priori) but with

(fO; tftir is not the case: the sclution depends on the position of E. So we solve

- A P = 0 i n O - t

P l n + - P l x , - = 0

ivp(P+) l ' = lvp(P-) l '

some constant g Plz* - Plx,- = 0 (60)

be determined with the help of (59) written

(6 1)

(62)

(63)

(64)

40 FINITE ELEMET{T METHODS FOR FLUIDS

0 e ,A;;lan

- s;

equation (64) which is calied the Joukowski condition . lt deals with

tinuity of the velocity and also of the presEure .

To solve (62)-(65) a simple method is to note that the solution

(63)-(65) is l inear in P :

p(x) - ea b) + 0@'( ' ) - ,Po( ' ) )

(65)

the con-

of (62)-

(66)

- p0 is the solution of (62),(65) and (53) with f = 0, i.e.

Apo = o, ffV

= g, g continuous across x

.1,=pt -p0 is thesolut ion of (62),(65) withg= 0 and (63) with p- L, i .e.

L.t = 0, ffi i, - o, rfb* -,hll.- = L, Vdlr* = Vdb-

The variational formulation of this second problem is: find g e Wl such that

I v 4 , v * = o Y w e W { ;Ja

Wou = { rh e I / t (Q) : d l r* - t l , l>- = 9, V/ [* = Vdlp- ]

We find fl bV solving (6a) with (66) : it is an equation in one variable B.We can show tl,at the Llft Cy (the vertical component of the resultant of theforce applied by the fluid on S) is proportional to B :

c1 - ?Plu*l

where p is the density of the fluid.

Practical Implementation :

In practice we can use Pl finite elements though the Joukowski condition

requires that we know Y tp at the trailing edge ; with Pt , Y g is piecewise

constant and so the triangles should be suffieiently small near the trailing edge.

In [11] a rule for refining the triangles near the trailing edge (to keep the

error O(h'i :can be found the size of the triangles should decrease to zero

as a geometric progression as they approach P and the rate of the geometric

progression is a function of the angle of the trailing edge. Experience shows

that one can apply the condition (61) by replacing P+ and P- by the triangles

which are on S and have P as a vertex.

(67)

TRROTATIONAL AND QUASI IRROT/$IONAL FLOWS 4t

Figure 2.7 configuration of the triangles near the trail ing edge'

==-'

Figure 2.8 : Erample of numerical results : a) equipotentials

b) rriangulalion arcund the neighborhood of the profi le

c) strvamlines with lift d) and withoul lift'

Another method, which is no doubt better from the theoretical point of

view, but very costly in terms of computer programming time is to include a

,p".iul basis function in place of the one associated with P, to represent the

singularity. It can be shown that rp(c) is like ls-pl"l(zr-F) in the neighborhood

of ihe traiiing edge, where B is the angle of the trailing edge. One can find the

complete p.oof iriirisvard [102], and we give here only an intuitive justification'

Let us work in polar coordinates with origin at P and 0 - 0 corresponding

to one edge of S ani Q = 2tr - p for the other. Let / b. the l imit of 9(r,0)1,^

(rn not necessarily an intcger), i.e.

p ( r , 0 ) - r ^ f ( 0 ) + o ( r - )

The equations for P are:

ffitr- =r-'H?,0) =o =+ /'(o) = o


E P , - r 0 g ,f f i ls*

- r- ' f f i ( r ,2r- B) = 0=+ f ' (2o- B)=0

Lp = . � ' f . ,# + r -2# - , * * , f (o) t f t , (o) - o

The general solution of the last equation being f (0) = osin(rn(d-do)) thefirst two conditions give :

d o - 0 , m ( 2 r - � g ) = o

that is m - r/(2" * P).We have therefore in the neighborhood of P:

p(r, 0) -- arf i"l4#,01 + ogfu 1

which suggests taking a basis function associated with P as

." (r, q - rf i"tO*EllIoQ)

where IoU) is a smooth function which is equal to unity near P and zero farfrom P.

Results using this technique can be found in Dupuy [70], for example.

3.POTENTIAL SUBSONIC STATIONARY FLOWS :

3.1 Variational formulation

One still assumes that the effects of viscosity and thermal diffusion arenegligible but does not assume that the fluid is incompressible. If the flow isstationary, inotational at the boundary and behind all shocks if any, then onecan solve the lrcnsonic polential equation

1v.[(k-r lvr l ' )*vp] =o in 0 ' (os)

The velocity is still given by

the density by

and the pressure by

u - Y g

o-po(k-] i ; torf l=,

p - po(frr

(6e)

(70)

IRROTATIOITAL AND QUASI IRROTATIONAL FLOWS +3

The constants Pg , Po are usually known at the outer boundary. The boundary

conditions are Srven on the normal flux pu'n' rather than on the normal velocity:

1 . - , , t 2 \ 3 b _ o o n f ( Z l )( k - ; lYv , , 0n

r

To simplifY the notation, let us Put

p(vp)= po(& - |tvrt ' l=' t72)

Proposition 7proilr* (1il-{15) is equiualent to a aariational equation. Find

F eW = {u € f { l (0 ) ; f - * *= 0}J A

(where $is ong giaen function with non zero nxean on Q ) such that

. a f

[Â -

f tvrt'l;kv'Pv' = Jrn'

Yw € w; (73). / l t

M o r e o a e r , o n | s o l u t i a n o f ( 4 q i s a s t a t i o n a r y p o i n t o f t h e f u n c t i o n a l

E(p) - - f ^ tn - f , t vv t \# - l ,o (74)

J N

Proof :To prove the equivalence between (68) and (73)-(74) we follow the proce-

dure of proposition 1- Let us consider now

E(v +} r ) -

[^w- ] toto *, \ tu) l ' ) t - # l rs@*lu)J a A

We have :

E',x(p+ ) tu) - * [^fr - ] tot t+ ]r l2)*o(, * ' \ t r )vtu -

h l rn*7 - r J n

So all the solutions of (49) are such that

E ' , x (p * ) tu ) l r=o = 0

and ncw t,he result follPws.

Proposition 8

If b < (2k0 -t)10 + 1) ) tlz then E defined' ov (70) is conuer in

Wb = {p eW : lYe l < b } (75)


Proof :Let us use (71) to calculate E,AA in the direction 1l, :

7 - L d 2 E1 4tr2 lr=o =

Irp(ve)lv *12 -Fb

fno9dz-t(ve'Yw)2

- l,p(v p)z-r1vu,l2(fr - f,vrrtl + *tffil'r

, Ip(ve)2-^'lvrol2(r -f,lvol'(l + #))

> (e -*ur*:+(r -+f{lrro,rra

Corollary 1If b < (2k (z - t)/(t + 1) )

tl2 the problem

f 1 . a . L ' Y f

,#i#p, -

Jr&- ulvvl'�)# -

F Jrn, (76)

admits a unique solution.

Proof :Wb is closed convex, E is W - elliptic,convex, continuous.

Corollary 2If the solation of (76) is such that lVpl { b at oll points, then it is also

the solution of (69).

3.2 , Discretisation

Proposition 9Let b < (2k (t - t) l (T + 1) )

Llz and let

Wl = {pn e Wn : lYpnl S a} $7)

Assame thaf. W!, anil Wb satisfy (21) with strong conaergence in WL'a i kt usapprodmate (75) by

*P#; - l.a-f,vrol')#o

-hl,no (78)

If {pnln or" solutions inWl'*then gy -* I the solution of (76).

Proof :As lVrpl,l is bounded by b, one can extract a subsequence which converges

weakly in Wl'*(Q) weak *. Let d be its limit.

IRROTATIONAL AND QUASI IRROTATIONAL FLOWS 45

Let g be the solution of (?6) and fl6rp the interpolate of-g in the sense of

(24).Ui n be the functional of problem (78). Then since Wl CWb we have :

E ( d < E ( p n ) < E ( T I 1 P ) .

The weali semi-continuity of. E and the fact that Pn is the solution of (78)

implies that any element to of WD is the limit of {tr,'6|, wn e Wl in Wi'*

( l r rnh*s l l , - un l l i , * = 0)

E(d S E(rl,) ( liminf E(tp1) S lim E(TInp) - E(p)

but <p is the minimum and so

E(d - E(Ir).

The strong convergence of gn towards g in I/t(O) is proved

convexity and the l4rb -ellipticity of E.

(7e)

(80)

by using the

3,3. Resolution by conjugate gradients :

To treat the constraint " lVpl < b " the simplest method is penalization.

We then solve :

,ftt#^ E'(Pn)

E,(pn) = - [ A -* lornl,)* - + [ n, + p I Kb' - lVpr, l ' )*] ''

J n ' z ' ] - t J r J n

where p is the penalization parameter which should be large. The penalization

is only to avoid the divergence of the algorithm if in an intermediate step the

bound 6 is violated. If the solution reaches the bound 6 in a small zone then

Corollarl' 2 doesn't apply any more; (experience shows that outside of this zone

the calculated solution is still reasonable); but in this case it is better to use

augmented Lagrangian methods (cf Glowinski [95]).

(8 1)

m i n : l 0 t E - 5 - x - - x - : l 2 0 E - 3 n o x : 2 7 5 E - 3

Figure 2.9: Compute,l resull of a lransonic nozzleThe fgurc rcpresents the isoualues of lvpl, obtained by soluing (81).

Taking into account the fact that Wn is of finite dimension, (81) is anoptimization problem without constraint with respect to coeffi.cients of gn ona basis of W6 :


N - 1

pn(r) = I piwi{x)I

t * ,T*- tE ' (Po)

To solve (81), we use the conjugate gradient method with a preconditioning

constructed from a Laplace operator with a Neumann condition. Let us re-

call the preconditioned conjugate gradient algorithm for the minimization of a

functional.

Algorithm 2 (Preconditioned conjugate Gradients) :

Problem to be solved: "?F

E(t) (82)

0. choose a preconditioning positive definite matrix c; choose e > 0 small, M

a large integer. Choose an initial guess z0' Put n = 0'

1. Calculate the gradient af E with respect to the scalar product defined

by C; that is the solution of

Cgn = -V,E(r") (83)

I f l l g " l l c < e s t o P.. else if n = 0 put ho = go else put

* - llg"ll'c (84)7 = Tle;ill5

h n = g n + 1 h " - l ( 8 5 )

3 . Calculate the minimizer p' solution of

min{E(2" + ph")} (86)

Put

zn*t = zn + pn hn (87)

If n ( M increment n by 1 and go to 1 otherwise stop'

Proposition 10

If E is strictly conaer and twice d,ifferentiable, algorithm 2 generates a

seque-nce {t"} which conaerges (. = A, M = +oo) towards the solution of

problem (82).

The proof can be found in Lascaux-Theodor [1a2] or Polak [193] for exam-

ple. Let us recall that convergence is superlinear (the error is squared every N

iterations) and that practical experiments show that ./i{ iterations are enough

IRROTATIONAL AND QUASI IRROTATIONAT FLOWS 47

or even more than adequate when we have found a good preconditioner C.

With a good preconditioner, the number of iterations is independent of N.

This is the case with problem (81) when C is the matrix A constructed in (37).

The linear systems (83) could be solved by algorithm 1. As in the linear case'

the algorithm works even if the constraint on the mean of gn is active provided

that the gradient g' is projected in that space.

Gelder's Algorithm [s6]:The fixed point algorithm, introduced by Gelder [86], when it converges

gives the result very fast :

v.[(k -f,lro"l')o5 ven+'] - o in o

09n*r

0n

(88)

(8e)1 ^ r(k -; lvp^ l ' ) ;=' ' = g o n f

It is conceptually very simple and easy to program. Once discretised by finite

elements (88), (89) gives a Iinear system at each iteration, but the convergence

is not guaranteed; horvever experience shows that it works well when the flow

is everywhere subsonic. The coefficients depend on the iteration number n ; so

one should avoid direct methods of resolution. If one solves (88) by algorithm

1 (conjugate gradient) then it becomes a method which from a practical point

of vierv is very near to algorithm 2.

4 . TRANSONIC POTENTIAL FLOWS :

4.1 Generalit ies :If the solution of (?5) is such that lVgl = b almost everywhere on I C O,

measure(I) > 0 then rve cannot solve (68) by (75).

From physics, u'e know that when the modulus of the velocity lV,pl is

superior to [2k(7 - t)/0 + 1)] t lz the flow is supersonic. If we linearize (68)

in the neighborhood of p., we get :

Y.[pY6p- p2-1YpV69] -0 (e0)

One can show by the same calculation as (74) that this equation is localiy

ell iptic in the subsonic zone and locally hyperbolic in the supersonic zone. Fur-

thermore numerical experience shows that (68) has many solutions in general

C0 but not C1 (lVpl is discontinuous), the discontinuities are where lV,pl is

equal to the speed of sound. We have to impose an additional condition to

avoid the jumps in the velocities (shocks) going from subsonic to suoersonic in

the flow direction.We have seen in Chapter 1 that the entropy s is constant if q , (, rc are

zero ; but this is no longer valid when the velocities are discontinuous ; when

4 is small, for example, 4Au becomes big. It is shown in Landau-Lifchitz lIaI)that the entropy created by the shock is proportional to the third power of the


jumps in the velocity across the shock in the flow direction : the entropy being

increased through shocks it can only be produced by supersonic to subsonic

shocks. An original method for impmiqg this condition (called an entropy

condition) h* been proposed by Glowinski [95] and studied by Festauer et al

[77], Necas [180] :Given 2 positive constants b and M,Let

u, € I{l(Q), lVpl < b,

The variational inequality in (91) rneans that

Lp<M in O (92)

which effectively avoids all positive jumps of dg/dz in the x-direction if / issmooth in the y-direction.

Results leading tciwards the existence of solutions for (68)(71X92) can befound in Kohn-Morawetz [135]. The following property is a key to show thatsubsequences converge in Wil.

Proposition 11The spac" Wlr is compaet conuet in fJ1(A).

Proof (Festauer et al [?fl)The convexity of Wla is clear ; let us show that it is compact. Let us

define

G, ( . ) - [Fo" .Vr .u* M* ) (93 )J N

G(w)- [1v rv**Mw) (e4)J N

If g" + I in .F/l(O) weak* then Gn - G in I/t(Q) weak* and as G"(tr,') >0 Vur we have, from a lemma of N{urat [178], Gn * G strongly in I4zt'*(O)weak*

But

lV (p " -d l |= (G ' -G) (p " -p ) *0 (95 )

The functional used for the computation of subsonic flows is no longerconvex in the supersonic zones; one must find another functional. A possibilityis to minimize the square of the norm of the equation:

<MI * * V rp )0 ( e1 )

In'o= o)

w ! * = { p e r l ' ( Q ) , - [ v e Y wJ N

min llv.t(fr -'rtoot\ rrrvplllllP-er€IIf (o)nwL

(e6)

IRROTATIONAL AND QUASI IRROTATIONAL FLOWS q9

The choice of a suitable norm is very important to insure existence of solution

and a fast algorithm. Ilere A-1 is .l"urly a good choice but it requires glr

to be known (Diri.t l"t boundary conditions). with Neumann conditions (also

with mixed conditions) one can solve:

''Jfl.4{l-lYelzdx:

. (Ve , Vtr.,) = (p(Yp)Vp, V') - Iro*

Vur € I/t(O)

Indeed if we let e be defined bY

-Ae = v.[(e -f,Vrn #ve], € € //3(0)

then (96) can be written equivalently as

(ei)

lv. l ' ) .

A finite element

lVe612 : (e8)

(V.r, , Vr6) - (p(Ypn)Ypn, V'r) - [ô*n Ywn € Wn]

J T

It should be noted that this problem always has a solution since it is the min-

imization of a positive differentiable function in a finite dimensional bounded

space. But 96 will be an approximation tc the solution of the'transonic prob-

lem only if e1 r 0; and the set of points on which the constraints of TYly arc

active must tend to a set of measure zero'

In Glowinski [g5] details of the conjugate gradient solutions of this problem

can be found. w" r"frr to Lions [154] for a thorough study of optimal control

problems.

4.2 , Numerical considerations

Practically, experience shows that the constraint (92) is always saturated

and so the calculated solution depends on M (cf. Bristeau et d [aa]) and that

the choice of M is thus critical.

In the actual computation one may prefer to use "upwinding" or " artificial

viscosity" to take care of the entropy condition'

Upwinding will be studied in the following chapter. Let us give a simple

version due to Hughes [115].Equation (49) is aPProximated bY

mlne-er€I{f ({t')nwfu

discretisation of (97) is

rm i n { I

vn€WIv Jn

{1.


nrx: ?t9F.3min:58083 - x--x-:603F3

Figure 2.10: Transonic flow computed by Gelder's algorithm

with upwinding.

f 1 r - a r r - |

I tn - *tooi l?)F vpl* 'Vur - l^wn Ywn eWn en+1 € Wn (99)

J o ' z J r

where W;, is, for example, a Pl confor*ing triangular finite element approxi-

mation of W and where (Vpr,)a is calculated on the triangle f'1r), nearest to

?3 " upwind" of the flow instead of calculating it on the triangle ?3

(Yel)alr. = VP"lrr(t)

Numerical results with this method can be found in Hughes [115] or Dupuy

[70] for example.

Artificial viscosity will also be studied in chapter 3 ; one can add viscous

terms to the equation (they were there in the Navier-stokes equations and

because of thernthe entropy can only grow). Following Jameson [121] (see also

Bristeau et al.[aa]), we consider

[^rr -

f,vrr l')o. Yei+tvwn * h lrr..f-;l(u'n'

- ")+]v ei*'Y p'ilu1'd'xJ o z J t L " "

( 1 0 0 )

= [ n * nJ r

where the utn is an upwind approximation of ivg[[, I il the speed of sound

and, pf is an upwind approximation of (k- (l12) lv ei1z1 ttt.-t-; 0f l0s stairds

for (u'llul)V/,1he detirrative in the direction u ; this upwind approximation

can be constructed as above or bY

(Vei)air* = (1 - or)Vpllr* + uYg>iltrr*r

where c.r is a relaxation parameter'

Finally, notice that (99) and (100) are Gelder-type algorithms to solve

underlying nonlineu. "q,rutions; other methods can be used such as the I/-1

TRROTATIONAL AND QUASI IRROTATIONAL FLOWS '61

least-square method presented above ((96)-(98)) ot GMRES, a quasi Newton

algorithm rvhich will be studied in chapter 5 ; results for these can be found in

Bristeau et al [44].

5. VECTOR POTENTIALS :

Let us return to (1), (2) and study the possibi l i ty of calculat ing u by

u = V x r,'. For simpiicity, we shall assume that Q is simply connected. Let f

be piecewise twice differentiable and f; be its simply connected components.

Proposi t ion 12 :

Le t i € I2 (0) such lha tY.u € ,2 (O) . Le t $ € HL(Q) / I? the so lu t ion o f

a .a a . l c . t c . l a ,

a r * , r *

Figure Z.LL: Potential transonic fl,ow around a M6 wing;

,o*prid by J.M. Dupuy: the figure shows the Mach numbers on

the wings and presure coefficienls on seclions of the wing.

COUHBE DES CP AILE WINC M6Cotlttl CO|IlllU(t t O(xrrt$ xUrarPU.'ror|lt DI rarll t CL rCDIrCt *nll

oil,rttl DFCoalllult 3 Colm$ tptrlltxlrutn||lt 0c lacll t c.|ll tclocxct | 0!a

ro.ltlt 0C @UC8l3 l|| L lll I llto.rut 03 iorlt grl u llgrt. tt


(Vd, Vru) = (u, Vu,') Vu € I/n(O)/n

ond $ e lH' (O)/Rlt the soluti on of

x u )

I,Vu e .V

t r .n = 0 ) ,

(101)

(102)

(103)

Proof :The proof being rather long, we shall give only the main ideas and the

details can be seen in Bernardi [30], Dominguez et al [6?], Eldabaghi et al [73].First let us note that (101) et (102) are well posed so that S and $ are unique.

Then we see that the solution of (102) satisfies Y ''h = 0' In fact' by taking v

= Vg in (102) with g zero on | (because of (103))'we see :

(V * 4,V x u) + (V.r / ,V'r) - (u ' V

,h eV = {u € f / l (O)s : u x nlp = 0,

then

u * Y d * V x {

(V.rl,,Aq) = 0 Vq € If'(Q) n H;(O) such that

that is we almost have

we see that (101) imPlies

{ = V x t + y 6 - u

V . { = 0 { . n = n . Y x $

and (102) implies

V x { = Q

because of the following identitY :

(104)

(105)

(106)

l ,#=o

(V . r l , , s )=0 Vg€ L ' (A )

Nevertheless, one can prove, without much difficulty, that (105) implies Y.t =

0. Now let

(107)

(108)

(1oe)

(oV x b) = (V x o, b) * lro.(n

x b) (110)

Besides, if 1, 12 &te two vectors tangential to f and orthogonal to each other

then

a r

TRROTATIONAL AND QUASI IRROTATIONAT FLOWS 5$

$ x n - g * r ! . r ; - [ + b l t . r i ) , ' r = 0 ( 1 1 1 )

so that

n.V x ,1, = -jtt.rt),r" + (rlt.rz),rr = 0i (112)

the proof is completed by using the following property :

V x { = Q , V . € = 0 , { . t l r = 0 + { = 0 . ( 1 1 3 )

Proposition t3 :With the same conditions as in proposilion 12, we can also use the fol[ow-

ing decomposilion :Let g be the solution in Hl&)lR of

(Vd, Vu,') = -(V.u, u) Vu € HL@)lR

ond $ € I{i(0)3 the solution of

(1 14)

(116)

(1 17)

( V x r l , , Y x u ) + ( V . r / , V . r ) = ( u , V x u ) Y a e V ( 1 1 5 )

, 1 , x n - V q o n r , / r h . n = 0J l i

where q is the solution of the Beltrami equation on l.

t t yy - [u .n . v tu€r / ' ( r )t?rJr or; or; Jr-'

where r; ar€, two orthogonal tangents to T. Then

u = V x t * V p

Proof :The proof is the same as that of proposition 12, except for (111) onward.

Instead of (112), we have:

n.Y x r l t = 'o'q o2q "= -(fr * W) = u,.n (118)

where the last equality is due to (117). Finally, we also have €.n = 0 and so€ = 0 .

We have therefore two decompositions of u. In the first one, 0$fdn = u.nwhereas in the second n.V x ,t, = u.n and 0$/0n = 0.

If V.u = 0 in order that rf = 0, one has to use the second decomposition. Sowe see that the potential vector resolution of incompressible 3D flow requires,in general,


- the solution of a Beltrami equation on the boundary (117),

- the solution of a vectorial elliptic equation (115)'

RemarkAn alternative form for (117) is

f - - 0 q 0 u f -I v ( tvu - -? - = I u .nw,Vr l € I / t ( f )

J r ' O n O n J r

6. ROTATIONAL CORRECTIONS:

We can use proposition 12 and the transonic equation (64) to construct a

method of solving the stationaxy Euler equations :

V.(pu) = 0 (11e)

Y . ( p u 8 ' ) * V p = 0 ( 1 2 0 )

v.tp. ,( | , ,2*J- l ) l =o (121)' 2 " r - L p "

As in chapter 1, we deducefrom (121) thatpp-tll0 - 1) + u212 is constant

on the streamlines, that is, if coo is the upstream intersection of the stream

line with f, we have

+' - ** - I / ( r " " ) on {c i x ' = u(r ) , c(0) = r " " } (L22)t - L p ' 2

\

By definition of the reduced entropy .9, we have

4 = s (123)il

and ,S is constant on the streamlines, except actoss the shocks. In fact, r.r = Vx

u satisfies

O = V . ( p u 8 u ) * V p = - ' p u x V x u * V p + p V + 0 2 4 )

= -pux c,u * pVH - -{-'gg' J - t

and so pu Y S - 0, but this calculation is not valid if u is discontinuous.

However we can deduce frorrr (12a) that

n'l- -L

u = (u x V// - ?T

x VS)lrl- ' + \pu (125)

where ,\ is adjusted in such a way that V.u = 0, i.e.

IRROTATIONAL AND QUASI IRROTATIONAL FLOWS

Figure 2.L2: Transonic flow with entropy correction; corn-

puted by F. El-Dabaghi using the Helmollz decomposil ion. Thp

Mach numbers are shown with and without entropy corection; the

oth.er f igun shows lhe entropg.

PuYA= -V.[(u - ')(rx Vf/ - P't-t ' Y V1

;1

Let us now use proposition 12 ; we get the following algorithm :

$*.

ot'lo

qg

(126)

AlgorithmO. Choose u continuous and calculate the streamlines

1. Calculate a nerV I/ by transporting /{(r-) on the streamlines and a

new S by transporting S(r*) on the streamlines and if necessary adding a

jump [S] calculated from the Rankine Hugoniot conditions across the shocks

(conservation of [.pu],lpu' + pl,[u'12 + tpl0 - t)p])

H I N = 0 . t 1 2 8l ' lFX = 1 .36 ' {


2. calculate c.r by (125)with p calculated by (122) and (tz3), i'e'

p(u) =#r, - i"'ltu. (127)

and .\ calculated bY (126).3 . calculate d'bv resolving (102) (we shall apply (110) to calculate the

right hand side).4 . Calculate { by solving (119), i 'e' with (127) :

v.[p(vd+vx{)J =o off i l r - t (128)

5 . Put u = Vd + V x r/ and return to 1'

We remark that when w 111, r/' is small so all that remains is (12p) i'e'

the transonic equation. This algorithm is therefore a rotational correction to

the transoni. eqoution. The convergence of the algorithm is an open problem

but it seems to be quite stable (see nt Dabaghi et al [3J) and it gives good

results when , i, not too large. F o* a practical point of view, the main

difficulty remains in the identification of shocks and applying the Rankine-

Hugoniot condition in step 1 (see llafez et al [105J or Luo [161] for an alternative

method).Finally, we see that it requires the integration of 3 equations of the type :

uV{ - . f , { lp given' (129)

this will be treated in the next chapter'

Other decompositions have been used to solve the Euler equations. An

example is the clebsch decomposition used. by Aker et al. [ 71] and zil L2421

,, = Vp * sVl where g, E, $ arc scalar valued functions. Extension of these

methods to the Navier-stokes equations can be found in El Dabaghi et al [73],

Ecer et. al. [71] , Ziil1242 ).

Chapter 3

Convection diffusion Phenomena

1. INTRODUCTIONThe following PDE for ,f is called a linear convection-diffusion equation:

6 , t * a 4 + V . ( " d ) - V . ( z V i l = f ( 1 )

we have: ,{

("o'd),, - e"t (d,t + od)

so with such a change of function one can come back to the case o = 0; secondly

the diffusion terms and the dissipative ternrs have similar effects physically.

These equations arise often in fluid mechanics. Here are some examples :

- Temperature equation for 0 for incompressible flows,

of convection-diffusion.


In general, z is small compared to U L ( characteristic velocity x charac-

teristic length ) so we must face two difficulties:- boundary layers,- instability of centered schemes.

On equation (1) in the stationary regime when u = 0 we shall study first

three types of methods:- characteristic methods,- streamline upwinding methods,- upwinding-by-discontinuity.Then we will study the full equation (1) beginning with the centered

schemes and finally fior the full equation, the three above mentioned methods

plus the Taylor-Galerkin / Lax-Wendroff scheme.

1.1. Boundary layers

To demonstrate the boundary layer problem, let us consider a one dimen-

sional stationary version of (1) :

"i;, ="i*:;; =', Iwhere u and u are constants. The analytic solution is

6@\ _ (L, , , e) l@Llv)( '5] (4). \ / \ ; ) L T - @ t

When u luL '-* 0,

(x - r )d + \ -

- / i f u ( 0 , e x c e p t i n c = 0 ,

u

d * l i f u > 0 , e x c e p t t n x = L .u

But the solution of (2) with z =0 is

4 _ @ - r ' ) ( b )

if we take the second boundary "onditiorlin (3) and

, t6 - - ( 6 )

if we take the first one.-vVe see that when vfuL * 0, the solution of (2) tends to the solution

with u =0 and one boundary condition is lost; at that boundary,for u << 1,there is a boundary layer and the solution is very steep and so it is difficult tocalculate with a few discretisation points.

CONVECTION DIFFUSION PHENOMENA 59

Figure 3.1 : According to the

with a boundary laYer tosign of u, the solution is giaen by (5) or (6)

catch the aiolated boundary condition'

1.2, Instability of centered schemes :

Finite element methods correspond to centered finite difierence schemes

when the mesh is uniform and these schemes do not distinguish the direction

of flow.Let us approximate equation (2) by Pt conforming finite elements' The

variational formulation of (2)

1L 1L

J, u4 ''u '

Jo 6 ''tr ''

is approximated bY

f L

l- @On,rwn * vdh,tlrh,, - u' '1) = 0 Vt'o' l continuous Pl piecewise (8)

J O

and zero on the boundarY.

If ]0,L[ is divided into intervals of length h, it

be written as

is easy to veriiY that (8) can

f , {0 , * r -d i - ' l - f r tO i+ l - 26 i +d i - ) 1 ' i = t ' " ' 'N - 1 (e )

t r 'here 6 j = {n( ih) , j =0, . . . ,N - L lh i 4o = drv = 0 '

Let us find the ,ig.nuu,lues of the linear system associated rvith (9), that

is, let us solve

1 L=

Jo tD, Vtu € I{;00,t[) (7)

/"()) - det

2 c r - \ L - c r 0- 1 - a 2 a - \ l - a

- L - a 2 o - \ L - a

0 - 1 - a Z c t - A


where we have put a = 2v/(hu). One can easily find the recurrence relationfor .I,. :

In = (2o - t )1"-r * (1 - oz)In-z

and so when \ = 2a then lzp+r - 0 Vp. This eigenvalue is proportional to v,and tends to 0 when v - 0. So one could foresee that the system is unstablewhen v/hu is aery small.

There are several solutions to there difficulties :1. One can try to remove the spurious modes corresponding to null or

v€ry small eigenvalues by putting constraints on the finite element space (seeStenberg 1220) for example).

2. One can solve the linear systems by methods which work even for non-definite systems. For example Wornom- Hafez [239] have shown that a blockrelaxation method with a sweep in the direction of the flow allows (9) to besolved without upwinding.

3. Finally one can modify the equation or the numerical scheme so as toobtain non-singular well posed linear systems: This is the purpose of upwindingand artificial diffusion.

Many upwinding schemes have been proposed (Lesaint-Raviart [149J, Hen-rich et "I. [110], Fortin-Thomasset [83], Baba-Tabata [7], Hughes [i15], Benqueet al. [20], Pironneau [190] ... see Thomassetlzzglfor example). We shall studysome of them. For clarity we begin with the stationary version of (1).

2. STATIONARY CONVECTION:

2.L. Generalities :

In this section, we shall study the problem

V . ( " d ) = f i n O d l r = d rwhere u is given in I4Z1'*(O), / is given in .Ll(A) and

( 10)

(11 )E = {c € f : n(x) .u(r ) < 0}

E

Figure 3.2 : Eis the part of I wherc, the fluid enters ( u.n < 0 ).

f-E

CONVECTION DIFFUSION PHENOMENA 6.1

A variational formulation for this problem is to search for 6 e fi1q such

that

(g,,uYw) - [ *1ru.n -

[ f* vur € Ht(o) with ur-p = 0.J E J n

Here dr € Ht/z(E),,r, V.u € I,r(O) , f e fi(n) is enough but with more regu-larity there is an explicit solution:

Proposition 1 :Let Xft;s) be the solution of

Xt = tr(X), X(0) = c. (12)

Let Xy(x) - X(c; sp) De the interseclion of {X(x;s) : s < 0} wilh E.Then

6@) =dr(Xr(r))r-. / : : V'u(X(r;r))ds

!,o,r{*{r; r ; ;r / , '

-v u(X(c;t))dr 4,

( 13 )is solut ion of (10).

Proof :

u.vd - x 'vO - {O{x( ' ; r ) ) - f (x(x;s)) - 6v.ud,s

on integration this equation gives (13).

Ccrollary 1 :If atl the streamlines inlersectE, then (10) has a unique salution.

Remark :The solution of (10) evidently has the following properties :- posit ivity : f ) 0, dr > 0 =+ 6 >0- c o n s e r v a i i v i t y : V ' u = 0 , f = 0 + [ r u ' n $ d ' s - 0 V S c l o s e d c o n t o u r C Q- stabi l i ty : luQl (") < l / l 4Cnl - u.n(Xy ( ' ) ) ldr(xr( ' ) ) lWe seek, if possible, numerical schemes which preserve the above proper-

ties.A priori the simple Pl triangular conforming finite elements and

(rr1, V.(udi,)) = (f ,*n) V.r, €Wn tor,lr = 0; 6n e Wn di, l> = 0 (14)

do not preserve the above properties. In fact, if Q is a square divided into trian-

gles with the sides parallel to the axes and if u = (1,0) we obtain the centered

finite difference scheme, which we know is unstable, when / is irregular:

62 FINITE ELEMBNT METHODS FOR FTUIDS

2 ( 4 i + t , i - 4 ; - 1 , i ) + d i + r , i + t - 4 ; , i + t ' * 6 ; , i - t - 6 ; - 1 , i - 1 - 6 h f i , i

where 6;,5 is the value of { at vertex rj. The solution of (14) cannot be unique

when V.u = 0 since it is a skew symmetric linear system '

j+1

j

j -1

i -1

Figure 3.3tA uniform lriangulalion

2.2. Scheme 1 (characteristics)

i i+l

and the corresponding numbering.

One could propose the follorving scheme :

1) u is approximated by un , Pr conforming, from the values of u at the

Proposition 2 :

l d - d n l * < 7 ' g

u ,here C depend .s on l {p11 , * , l f l r , * , l r l r , - , l / l * and on d iam(A) .

(15)

Proof :To simplify, let us assume that V.u = 0; the reader can build a similar

proof in the general cabe.Let us evaluate the error along the characteristics; let X;,(r; .) be the

solution of

X ' = r n ( X ) , X ( 0 ) = c ,

CONVECTION DIFFUSION PHENOMENA

approximated as in step 2 above ; then

( l ru - r l * * lVul* lxn - Xl

Using the Belman-Gronwall lemma :

lXn-Xl < luu- � u l * (e lv" l - r ' - t ,

where .[ is the length of the longest characteristic. Finally,

63

lx| - X'l = l"r(Xr) - "(X)l < l(u,, - "XXr,)l + lu(Xr,) - "(x)l

( 16)

:l60(q')- d(qr)l s f (x)dsl

< lVdrl* l(Xn, - XrXq') l+ t lV f l* lXn - Xl* * l"r^ - sr l l /1"" (17)

As lsp" -r"l is the diference in length between the discrete and the continuouscharacteristics , so long as the boundary is regular enough with the help of (16)we obtain the result:

l6ok\ - 6k') l - < c l" -ui , loo

Numerical considerations :

This scheme is accurate but rather costly because one has to calculateall the characteristics starting from all the vertices. For example for a squaredomain with N2 vertices, one must cross 1,"(lf - 1) triangles in total when the

velocity is horizontal, so the cost of this computation is 0(/Vr3i z;

if //, denotesthe total number of vertices.

One could reduce the cost by first calculating them for the gt which are farfrom E in the direction u then using a linear interpolation for all the nodesfor which there exists two opposite triangles already crossed by previouslycomputed characteristics.

ldr(Xrr,(qt)) - dr(Xr(qi))l+ | [" ' ,(*n)d., - f"J o J o

Remark (Saiac)I f u = Y x t a n d { i s k n o w n

,/,(Xrn(qi)) = tk\, Xrr(qt) estarting from q'.

then Xp6 can be computed from the equationl; so there is no need to cross the domain

64 FINITB ELEMENT METHODS FOR FLUIDS

Figurc 3.4tThe characteristics are broken lines joining uertices ('. If two

characteris{ics go around a aer"tet q, then lhe ch.aracteristic which starts fromq rnay be skiPqted.

To calculate the {{*}* a good algorithm is to

nates t l f * l ) ; - r . .*+r of €t and decompose t l ( {*)

D' lo '= "0(€&)l

l r ! - oi

then 6r+r = Di=r..' lf *t qi (' = 2 or 3) with

lf*t - pp! + )f where p is such that

)f*t t o, f l such that )f+t - 6.

This calculat,ion is explained in detail in (120).

Z.g. ScJreme 2 (upwinding by discretisation of the total deriva-

tive)

A weak formulation of (10) is written as:

Find qt in I2(O) such that

-($,uVu,) * [ 6r*u.n - ( f ,*) Vtu € I / t (Q), such that ur lp-E = 0.J E

Let f (Q) tre a translation of Q by 6 in the direction *u

r ; * t O l = [ s * u ( c ) 6 : s € O ]

To construct, t,he scheme, we consider the following two approximations :

uYw(x)= |tr t '+

u(c)6) - ,( ' ) l

use the barycentric coordi-i" pl such that

(18)

( 1e)

CONVECTION DIFFUSION PHENOMENA :65

t s - 6 [ s u . n + o ( 6 ) - ' ' r :

Jn-on"6-(o) Jr-r

With these approximations (20) can be approximated in the space Wn of. PL

functions continuous on the triangulation o by the following problem :

(dn,wn) - ln*rrn

On(r)*n(' * u(c)d) - 6(f ,rn) - a lrOr*h,f, 'Tr,

V . l € W n ; * n l r - r = 0

or, with the convention that u6(c) = 0 if x ( Q :

Find 4neWn such thatfor all u. '6 e Wn with tnlr-> = 0:

(6n,wn) - [ O^@)w6(x+u(x)6)dr = 6( f , -n) - a fÔr 'h t t r ' t ,J a J E

One may prefer schemes where the boundary conditions are satisfied in

the strong sense. For this one should start with the following variational for-

mulation of (10):Find { such that {[s

- dr and

-{O,uvu) * f Qwu"n - ( f , r ) Vu. ' € Ht(o), u[ : 0 ' (20)J f - E

Then by the same argument one finds the following discrete problem:

Find {1 with 6n - dro € I'% and

with Wn = { .n eCo(O) , rn l r* e Pt Ylc, w1, l r = 0 } .

Comments :1. We could have performed the same construction by using the formula :

v.( rd) - uYQ + l .v .u= l (O( ' ) - 6@- u(r)d)) + d@ - u(e)d)v .u(r )d '

and could have obtained, instead of (21)'

f(6n, .n)=

JnLn(s - u( r )6) t r r , ( tX l - 6V.u(c) )dr+ 6( f , *n) , V n e W6'

(22)


This is also a good scheme, but as will become clearer the scheme (21) is

conservative whereas (22) is not.

Z, If we use mass lumping ( a special quadrature formula with the Gauss

points at the vertices), that is, {qt}t being the vertices of ?:

where n = 2 (resp.3), lTl the area (resp. volume) of a triangle (resp. tetrahe-

dron) (this formula is exact if f is affine) then the linear system from (21) and

(22) becomes quasi-tridiagonal, in particulu (22) becomes :

dr,(qi) = dn(qt - ulqi)dxr - 6v.u(q')) + f(q')6

[28]).We also see how one can construct schemes of higher order by using a

better approximation of uVto'

Proposition 3 :

If u is constant, and if all the streamlines cutE, problem (21) has a unique

solution

Proof :If there were 2 solutions, their difference €7, would be zero on X and would

satisfy:

(25)

Taking u)h = €6, \{€ would obtain :

I r = , | T ' [ I f k ' )J r '

- (n + 1 ) l . . n * r

l.o l3,o S l.r, lo,oler,o(/ * ud)lo,rrn"r-(o)

le6o(r * ud)lfsrnr;(o) = ln^r;rn

e1,(v)2detf v(c +

= lei,l3,nnrr*(n)

(23)

t24)

I r n * o - t e 6 u 1 , o ( I * u 6 ) - oJa Jon"d-(o)

(26)

but

So

u6\-rldy (27)


(28)lenlo,n-onrr+io1 = 0

which means that e1 is zero on all the vertices of the triangles which have a

non-empty intersection with O - O nTt'(Q) .

Applying similar reasoning to Q n ry(A) and proceeding step by step, we

can show that e1 is zero everywhere.

Remark on the convergence :

At least for the case u constant, using the same technique ,we would get

an error estimate of the type

l6n - dlo,o-o.rz"+(o) S C(6' + h')

and so the final error is 0(d +h2 /6). The diffusivity of the schemes is a problem;

Bristeau-Dervieux [40] have studied a 2nd order interpolation of uYw which

gives better results, i.e. less dissipations, (but which is less stable).

Generalization of the previous schemes :

The first scheme corresponds to an integration along the whole length

of the characteristics whereas the previous scheme corresponds to a repeated

integration on a length 6. If one wants an intermediate method, one could

proceed as follows (Benque et al. [20]) :Let D be a subset of O and ,!.@) the solution of

uY lb- - 0 in D, ,h . la -D =7t )

where

A - D - { c e 0 D : u ( r ) . n ( r ) < 0 }

As for (13), it is easy to see that

, !*@) - w(X6p(x))

where X is thesolutionof (12) and X6p the intersection with A-D. From (10)we deduce

f t r fI f r l t . - l v ( " i l$ ,= | wgu.n* I , / , -du .n

J D J D J A - D J A + D

where AD+ is the part of 0D where u.n 2 0. By choosing an approximationspace Wn, a method to calculate X6p and a quadrature forrrrula, one coulddefine a family of schemes of the type :

f (x)w(Xpp(n))dx Ywn € Wn

and this family contains the above two schemes.

t Q7,u.nut1,oXnan* t w11$1,u.n- tJ A + D J A - D J D


2.4. Scheme 3 (Streamline upwinfing SUPG) '

(V . ( rd i , ) - t , t un *hV. (uu ; ' ) ) =0 Yun €Wn (2e)

Cornments :

The idea is the following :

The finit" elem"rrt method leads to centered finite difference schemes be-

cause the usual basis functions tud (the "hat function") are symmetric (on a

uniform triangulation) with respect to vertices gi. so replacing the Galerkin

formulation

(V.(rdr,), *d) = (f,,t) Vf , 6n = d* * L4r' ' (30)

by a nonsynunetric "Petrov-Galerkin" formulation (cf. Christie et al. [5a])

(V. ( "dr ) , * ' ' ) = ( f ,w" )

where the to,i have more weight upstream than downstream ; that is the case

of the function x -+ u,i + hV.(utui) (see figure 3.5). We note also that one

could work either with u,'i or ui in the second member.

Figure 3.5 :

The pt basis functions and the Pelrou-Galerkin functions of Hughes.

Another interprctation could also be giuen ;

Equation (29) is also an approximation of

V.( rd) -huV(V. ( "d) ) - f -huYf

that is, when u is constant

(32)

(3 1)


v . ( "d ) -hv . [ (u8u )v / ] - f - huv f (33 )

So we have in fact added a tensorial viscosity hu I u; now this tensor has itsprincipal axis in the direction u, that is we have added viscosity only in thedirection of the flow. Furthermore what is added on the left is also added onthe right because (10) diferentiated gives uV(V.[u{]) - uV/.

Proposition 4 :i) The scheme (29) has a unique solutionii) ry u €. Wt,* the solution satisfies the following inequalitg :

l6n _dlo,n 1ch*ldlr , r , (A4)

Proof when V.u ) o ) 0 :If (29) has two solutions, their difference ep satisfies

hlv.(ue;)I3+ f, l -ro ")€i+ * Ir-ru.nef;dl-0 (35)

because

(v . (ue^) , . r , ) = | { to . " ) ro , rn ) * ,

l r r . r r l (36)

So el, is zero.Let us find an error estimate :Subtracting the exact equation from the approximated equation, we ob-

tain:

(v.(u(d - dr , ) ) ,wn * hY.(uw6)) = o Ywn € Wn (37)

By taking u'A = tbn- {7, where di, is the interpolation function of 4, we get :

(V.(uu1,), ur * hY .(uw1,)) = (V.(u(rltn - d)) ,wn * hV.(utu1,)) (38

= ((V.u)( tn - 6),*n) - (V.("wn), tbn - d - hv.(u(fu- d)))

t u.nw6(1t11 - 4)J r - t

or using (36) and the error estimate of interpolation (ld*rbnlo S ,hz-o, a = 0,a n d f / - , l r n l o , r ( , h 3 1 2 ) ,

r f . _ . o r f 6

; JnN q"o+;

Jr_ru'nwz' ,+ hlv.(u*o) l1o

< Ch2lV.(utu6) lo + h* lu.ry*olo,r-> + Chzl(V.u)u,, ls;


frorrr which we deduce that : luVto;lo,o ( Ch,l(u.n)t/ '*nlo,r-r S Ch3l2 and

l(v.ulttzu,r,lo S Chslz, which in turn gives

lrr,lo,o = ldr, - ,l,nlo,n ( Chi (39)

frorrr which the result follows since

16 - |fl S cn* + ldr, - 4l S C'hi

Remark:If V.u is not positive the proof uses the following argument:

I.,et {.t - e-h(r),p(r) where }(r) is any smooth function. Then with u =

ek(')u, tlt is a solution of

V.(url) = f '

but V.u - eb(V .u * uYk), so one can choose /c so has to have V.u > 0'

2.5. Scherne a (Upwinding by discontinuity) :

If { is approximated by polynomial functions which are discontinuous on

the sides (f*.") of the elements, one can introduce an upwinding scheme via

the integrals on the sides (faces), using the values of dl at the right or the left

of tlrr: sides depending on the flow direction'

l,esaint t148] has intXoduced one of the first methods of this type and has

givc, error estimates for Friedrich systems. In the case of equation (10) let us

J"ur,....................................................................................,l, dn in the space of functions which are piecewise polynomials but not

nec<:$sarily continuous on a triangulation UQ of Q :

W n - { r o , w n l r € P u }

We ;r,pproximate (10) bY

u.nl$fiw1 f *n Ywn € Wn YT (42)

whcrc dr, is taken \nWn, n is the external normal to 0T,l6nl denotes the jump

of (t1, across 0T' from upstream of. 0T :

[dr ] ( t ) - I im,*s+(6n(x * eu(c)) - 6n(e - eu(r))) , Yx € 0T (43)

ancl 0-? is the part of d? where u.n ( 0.If f intersects f - the convention is

that, </1, = 6ron the other side of f- i.e. 6n(r- eu(r)) = dr(") when c € l- '

ll.emark 1:

lf V.u = 0 and /c = 0 (/;, piecewise constant ) (42) becomes

(40)

(41)

l,.nv.(udn) -

lr-,fI- t ,


f l- | u.nlg1,l- I t vT i .e.J A - T J T

dnlr - t I , in lr ; t lu.nl * f -ru [ - tu.n1J J a-rnar; Jr J a-T

7 l

(44)

(45)

(46)

where 4 i. the triangle upstream of T which shares A-T. In one dimensionwith u = L and diam(f) = h it reduces to 6; = 6;-t * hf, for all f.

One can prove that the scheme is positive if all the streamlines cut E :

. f > 0 , d r > 0 + d o > 0

Remark 2 :If the solution ,/ is continuous, we expect that dl will tend to a continuous

function also in @2) ; so we have added to the standard weak formulation aterm which is small (the integral on A-f).

Proposition 5 :If Y .u - 0, if all the streamlines cut E and if h is small, (/2) defines 6n

tn a unrque manner

Proof :We suppose for simplicity that /p e WnLet el be the difference between the 2 solutions. Let ? be a triangle such

that d-?is in D. Then (a2) wi th u)h= e1l7 on 7,0 otherwise, gives

r rl u V e 6 l - I " . " [ e 6 ] e 1 - 0

J T J A _ T

I f n f e ? f- - ; I u .ne"1,* | u . "++ | u .nen l r rn l r -

L J A - T J A T _ A - T 4 J A _ T

where ?- is the triangle upstream of T tangent to 0T- . rc AT- is in E then(46) becomes:

t u.n! - o.J ar-a-r 2

AT - A-7. Thus step by step we canS o e 6 = 0 o n cover the domain O.

Convergence :Johnson -Pitkd,ranta ll27l have shown that this scheme is of O(ht+t/z) for

the error in L2 -�norrn and 0(ht+1) if the triangulation is uniform;this resultwas extended by Ritcher [2a] who showed that the scheme is of O(hr+t) if thetriangulation is uniform in the direction of the flow only. The proofs are ratherinvolved so we give only the general argument used in [127] and we shall onlysHow the following partial result:

T 2 F I N I T E E L E M E N T M E T H O D S F o R F T U I D S

ProPosition 6

1f *'= 0 (piecewise canstant elernents), g is regular, Y'u = A, and f=Q

then

l le -Pr , l l " ScJh

where

As in {Lzi), we define I+ = F- ! and we introduce the following notation:

4 o ' b > r I / l u ' n l o ' b 'I J ar -rnar

1 a,,b )s= f fu.nlo.u.J S

with this notation the discretb problem can be written in two ways:

< 9t - 9 ; , to ! ) l r * 19, l , 'urh ) t=Iqt 'ur r ' ) l Vun €Wn

- <,l f -utf , ,pf >r, * ( pn,url )p+ =1 Pttult )r Ywn e Wn

so if we add the two forms we see ttrat gn is also a solution of:

h * 19n '? / ' r ) r

- 2 < g r , t r l ) > Y w n € W n

By taking urrr = th we find that the scheme is stable:

l lpol l" s clPrlo,r '

The reader can establish the consistency of the scheme in the same way and

obtain :

<p+ -g- , ' , , f , )n - <. r f -w^,9- )n* q � � � � � � � � � � � � � � � - ' ,u ' l r ) r= 2<-pr 'wn)n

So for all r/n in I'7r,

l l P o - e l l " < l l { i , - P l l "

lf ,bn is a piecewise constant interpolation of p one h* lp -,l,nl = ll'�Plho(h),

so

(f /, lu nlo(h)')i s c'/h

lloll, = (I !urrr.nl(o+

- o-)' * Irlu.nla2)uz


and the result follows.

2.6 Scheme 5 (UPwinding bY cell)

The drarvback of the previous method is that there are many degrees of

freedom in discontinuous finite elements (in 3D there are on the average 5 times

more tetrahedra, at least, than nodes).

To rectify this, Baba-Tabata [7] associated to all conforming Pl function

a function Po on cells centered around the vertices and covering Q.

To each node qd we associate a cell oi defined by the medians not origi-

nating from gi, of the triangles having qd a.s one of the vertices.

Figure 3.6zThe cell oi associaled to aerl.ex q'

Let $1, be a function P1 - piecewise on a triangulation and continuous.

To dn we associate a function d'r,, pi"..wise P0 on the cells od by the formula

73

d ' n l o , = : [ 6 n @ )l q ' l J o i

Conversell', knowin1 $'n, P0 on the ot' one could associate

in Pl by

d'uo,= # f, ont,)

(47)

to it a function ry'6

,hn&i) = {)L|,, (48)

Then we apply the scheme 4 to //n but we store in the computer memory only

6 n ,This means that our problem is now to find 6n,, PL continuous on a trian-

gulation, equal to dri, on E and satisfying (in the ca-se where V.u = 0 )

r f- l u . n f g ' n f - l t v i ( 4 e )

J a - o , J o ;

where

(50)


To show that this scheme is well posed (i.e. that {6 is uniquely defined) one

must show that (50) has a unique solution when di, is known because we have

already shown that (a9) has a unique solution when all the streamlines cut E.

This is difficult in general but easy to show if all the triangles are equilateral.

Then an elementary computation of integrals gives

l,,rn = I l.dn*i + f,!,' lo'and multiplying by di and summing, we see that

(5 1)

(52)

(53)

(54)

(55)

2.7 Cornparison of the schemes

So far we have presented three classes of methods for the stationary con-

vection diffusion equation. It is not possible to rank the methods since each

has its advantages and drawbacks: for instance, upwinding by characteristics

gives good results but is conceptually more difficult and harder to program

while SUPG is conceptually simple but perhaps too diffusive if an adjustment

parameter is not added. Experience has shown to the author that personal pref-

erences (such as scientific background or previous experience with one method)

are the determining factors in the choice of a particular method; beyond that

we can only make the follon'ing broad remarks:1. Upwinding by the discretisation of total derivatives has been very suc-

cessful with the incompressible Nar-ier-Stokes equations where conservativity

is not critical.2. Streamline diffusion (SUPG) has been very successful in the engineering

environment because it is conceptually simple.3. Upwinding by discontinuitl' has been remarkably successful with the

compressible Euler equaticns because it is the right framework to generalize

finite difference upwinding schemes into finite elements. (It has even received

the name of Finite Volumes in this context when used with quadrangles).

3. CONVECTION.DIFFUSION 1 :

3.1. Generalities :In this section, we.consider the equation

To' l,,ro= 0 =+ t l.nl* ; t v' l.e') '= 0 =+ 6n = 0

6, t * V. (ud) - v \d. = f in Oxl0, ? [

d ( t ,0 ) = 6o( r ) i n O ;

olr = dr

CONVECTION DIFFUSION PI{ENOMENA -{rb

and knowing that in g"rr".rl for fluid applications v is small, we search for

schemes which work "ln"t with u = 0. Evidently, if v - 0, (55) should be

relaxed to

6lt = 6r (56)

rf.v =0, (53), (54), (55) is a particular case of stationary convection (eq (10))

for the variables tt, i) on Ox]0,T[ with velocity {t ' , 1}'

Thus orr" .orrtd ,i*ply rid . difiusion term to the previous schemes and

get a satisfying theory. -g"t

irwe use the "cylindrical" structure of Qx]0,T['

we can devise two new methods, one implicit in time without upwinding and

the other semi-implicit but unconditionally stable. We begin by recalling an

existence and uniqueness result for (53) and a good test problem to compare

the numerical schemes: the rotating hump'

Test Problem.

Consider the case where O is the square ]- 1,1.[2 and where u is a velocitl'

field of rotation around the origin

u ( x , t ) = { Y , - r } '

We take 6r =0, f = 0 and

do(o) = e( l r - ro l ' - r ' ) -" Vr such that lc - r0 l < r ,

do(r) = 0 elsewhere.

When v = A,d0 i , convected by, so i f we look at t - ' { , ,y,6( ' ,y, t ) } we wiI I

see the region {r,y r 6(r,d * 0} rotate without deformation; the solution is

periodic in timl and we can overl ap 6 and /0 after a time corresponding to

one turn. It v { 0 the phenomenon is similar but the hump flattens due to

diffusion; we may also get boundary layers near the boundary

3.2.Some theoretical results on the convection-diffusion equation'

Proposition 7:

Let e be an open bounded. set wilh boandary | Lipschit 'z; we denote by n

i|s ederior normal ; Ihe sgstern

d , t * u v 6 * a 6 = f i n Q = Q x ] 0 , ? [ ( 5 7 )

d ( ' , 0 ) = d o ( c ) V r e O ( 5 8 )

4 @ , t ) - s ( x , t ) V ( c , t ) € E = ( ( t , t ) : u ( x , t ) ' n ( c ) < 0 ) ( 5 9 )

has a unique solut ion in Co(0, T; , ' (O)) -when 40 e L'(A), I € Co(0, f ; , f 2(f ))

and, a,u e. L*(Q), Lipschitz in x, f e L'(Q)' r

76 FINITE ETEMENT METHODS FOR FLUIDS

Proof :We complete the proof by constructing the solution. Let X(") the solution

of

,1

ix(") - u(x(r), r) if x(r) e o (60)dr

= o otherwise

with the boundary condition

X( t ) = c . (61)

If u is the velocity of the fluid, then X is the trajectory of the fluid particle that

passes s at time t. With u e L@(Q) problem (60)-(61) has a unique solution.

As X depends on the parameters c,t, we denote the solution X(c,t;r) ; it is

also the "characteristic" of the hyperbolic equation (57).

We remark that :

I

f rb{xt . , t ; r ) , r ) , r ) | ,=t = 6, t (x , t ) * uv6(t , t ) (62)

So (57) can be rewritten as :

6 , , * a 6 = f ( 6 3 )

and integrating

d{ r , t ) = e f c (x ( r ) ' ' )d ' [ . \ * [ ' f ( *@) ,op [ "

a (X( r ) ' t )d r 6o1 (64)

J O

we determine ) frorn (58) and (59).I f X ( c , l ; 0 ) € l , t h e n

t = s (X(c , t ;0 ) , r ( r ) ) (65)

where r(o) is the t ime (< t) for X(c,t;r) to reach f.I f X(x, t ;0) € O, then

\ = 6 " (X(e , t ;0 ) ) (66 )

Corollary (o = V.u) :With the hypothesis of the proposit icn 7 and i f u € L'(0,7;H(diu,A)) n

L*(Q), Lipschitzinx,problem (53), (58), (59) with u = 0 has a uniqae solutionin Co(o,r ; r2(o)) .

-(rt recal l that H(dir,Q) = {u € .[2(O)" : V.u e L2(0)}/.

Proposition 8 :Let Q be a bounded open set in R" with I Lipschitz . Then problem


.:':' I

b,t *v.(" i l - V.(rcVO) - f i " Q = OxJ0,?[ (67)

d(r ,0) - do(c) in o (68)

d = s on f xlO, ?[ (69)

has a unique solut ion in L2(0,?; Ht(O)) i f rc i , i , u i € r ' " (0) , u i , i € t r*(Q) '

f e Lz(Q) , 4o € rz (a) , 9 e L ' (0 ,7 ;nU2( t ) ) , and i f there er is ts a ) 0 such

thot

77

n;iZ;Zi > olzl ' VZ e R".

Proof :See Ladyzhenskaya [138].

(70)

(7 1)

(72)

3.3. Approximation of the Convection-Diffusion

equation by discretisation first in time then in space.

In this paragraph we shall analyze some schemes obtained by using finite

difference methods to discretise 0Sl0f and the usual variational methods for

the remainder of the equation. Let us consider equation (67) with (70) and

assume, for simplicity from now on and throughout the chapter that rc;i =

v 6 ; i , v ) 0 , u e L * ( Q ) a n d

V.u = 0 in Q 7 r .n =0 on f x ]O,? [

4 = 0 o n f x ] 0 , ? [

As before, Q is also assumed regular.The reader can extend the results with no difif iculty to the case u.n t ' 0,,

with the help of paragraph 2.As usual, we divide ]0,T[ into equal intervals of length & and denote by

d"(r) an approximat ion of S@,nk).

3.3.1. Implicit Euler scheme :

We search d;*t € Hoh, the space of polynomial functions of degree p on

a triangulation of Q, continuous and zero on I such that for all tur, € f/sh we

have

1

i fo;*L - 6t ,wh) * (u"+ly oi* t ,wh) + u(v$i+t , vro) - ( . f "+ ' ,*n) (73)

This problem has a unique solution because this is an N x N linear system, N

being the dimension of Ilon :


(A + frB;O'�"+t - kF + /O (74)

where A; i - ( r i , * , t ) + vk(Vwi,Vur) , B; i = (un*lVr i , * t ) , Ft = ( f ,wi) ,I - (wi,u,-r) and where {rt} is a basis of Jloa and di the coefficients of {6 onthis basis. This system has a unique solution because the kernel of A + &A isempty :

0 -- 4,r (A+ kB){, - {r Ar/, + { = o. (75)

Proposition 9 :I f 4 e L'(0,T; I{n+t(a\ and 6,t € L"(0,7; Hp(a)), we haae

wt - 4@k,.)13 + uklv($i - 6(nk,.))13)\ < c@p + r,) (76)where p is the degree of the polynonrial approrimalion for 6l.

Proof :a) Estimate of the en'or in timeLet $n+t be the solution of the problem discretised in time only :

1

i ( 4 " * ' - 6 " ) + u ' * 1 Y 6 " + r - u \ f n + L - T n r r ; $ " + L l r = 0 ; Q o = a g T )

the error."(t) - 6"(u) - $(nk,c) satisf ier." lr = 0, €o = 0 and

f{ ."*t -€') q11,n+rven+t - ua.en*L =r

lot( l -g)

f f iO,(r* 0)k)d0 (28)

where the right hand side is the result of a Taylor expansion of $(nk * fr,c).Multiplying (78) by 6*+r and integrating on Q makes the convection termdisappear and we deduce that

l lr"+t l l, = (1.'+t 13 + ttrlv."+t 13) * < kzr'ldl'rlo,r*10,"[ + ll." l l,. (i9)

So we have, for all n:

ll."ll' S kr/rld','rrlo,e (80)

b) Estimation of the ercor in spaeeLet {n - di * 6n ; by subtracting (77) from (73) we see

! , r "* t - € ' ,wh)* (un+rv{n+r ,wh)* z(v( '+ l , v tu l , ) - 0 (g1)& \ s

Let rli be the interpolation of /" ; then f' = {i + ,ltt - g" and €fr satisfies(we take u)h = €l+t)

CONVECTION DIFFUSION PIIENOMENA 79

l l€f+'n7 S Kf,l,l{l*t l" - U,i+r - 6n*L - vf, + 6",€f+')- (82)

- f r (un+tv( , i ;+ t - 6n*r ) ,€ ;* t ) - zk(v ( . t i * t - 6n*1) , v( ;+t )

lE[+' l l , < lKr l l , + cklhp l ld i , l lp,o + hP ( l l r l l * ,e * u) l ld l lp+r,n]

because l.l, < ll.l l, , ,l,i*'- ,/t i. the interpolation of 4"+r - 4" which is an

approximation of k$',r.Finally we obtain (76) by noting that

(83)

(84)l ld; - dfuk)ll, < l l(fr11, +nhf - 4" l l, + l ld("e) - 6"11,

Comments :We note that the error estimate (?6) is valid even with u = 0. Thus, we

do not need any upwinding in space. This is because the Euler scheme itself

is upwinded in ti; (it is not symmetric in n and n * 1). The constant c in

(76) depends on l ldllu+t,o and that if y is small (67) has-a boundary layer and

lldllo+1,p tends to infinity (if dr is arbitrary ) * ,-p+1/2. This requires the

moiif ic.t ion of dr or the imposition of: h 11 vt-Ll2p.

A third method consists of replacing z by ,n in such a way that h <<

,I-rl2p is always satisfied. This is the artificial viscosity method.

3.3.2. Leap frog scheme :

The previous scheme requires the solution of a non-symmetric matrix for

each iteration. This is a scheme which does not require that kind of operation

but which works only when b is O(h).

fifofr | - 6i-t , *o) * (u"Y $f;, *n) + ;1Nt6il*' + 6f-'1, vrr,) (85)

= (.f"+1 ,wn) Vrn € Hon, di* ' € Hon

To start (85), we could use the previous scheme.

Proposi t ion L0The sch.eme (85) ts marginallg stable , i.e. lhere erists a C such that

16|,l" < Clfla,e(l - CluU"f l-l (86)

for a l l k such that

* .# (87)


Proof in the case where u is independent of t and f = 0.

We use the energy method (Ritchmeyer-Morton [197], Saiac [211] for ex*

ample).Let

s" - Wf*' t3 + lAillo + zt'("v 6T, 4i+')then

^9" - sn-r - l4i+'13 - Wf-1|3 + zt'("v41,6;*t + d;-1) (89)

but from (Sb), with un = 67*'+ df-l and if / = g

ldl+t l', - ldi-'13 + 2k(uvgi,Ll+' + 6i-r) + rr;v(d;+' + 6;-')13 = 0; (e0)

so Sr, 5 .S"-1 ( ... S Sr. On the other hand, using an inverse inequality

(uvdi,d[* ') > -f tr t"" 1; l ;+' lr l4tr lo ) - lr l *$t ld;+'13 + ldi lS) (e1)

we get

s, > (r - cklTl* )1dl+'13 + ldf, l3l

(88)

(e2)

Convergence :As for the Buler scheme, one can show using (86) that (85) is O(hP + k2)

for the -L2 norm error

3.3.3. Adams-Bashforth scheme :

It is important to make t,hc dissipation terms implicit as we have done forA{ because the leap-frog scht-'tne is only marginally stable (cf. Richtmeyer-Morton [197]). In the same n'ay, if D * O (u.n # 0), it is necessary to makeimplicit the integral on f -D. [|or this reason, we consider the Adams-Bashforthscheme of order 3 which is explicit u'hen we use mass lumping and which hasa better stability than the leap-frog scheme.

' t . . 2 : 1 1 6 . - , 5

,{Oi*' - dl i ,rn) =

froto,iv trr6) -

iuWf-',?Dh) + tu1af,- ' ,*n)

V.r € 1161, where

b(6n,r r ) = - [ (uV/ ; , r ur ; , ) * v(Y$6,Vwn) - ( / , rn) ]

The stability and convergence of the scheme can be analysed as in $3.3.5.


3.3.4 The d' schemes :

In a general way, let ,4 and B be two operalors and the equation i.r tirne

be

u , t * A u * B u = / ; " ( 0 ) = u o

We consider a scheme with three steps

81


1 . ^

#@"*t - rr") * Aun+e * Bun - f"+s

&(un*l-a - un+t) * Aun+e * Bun+r-l - y*Fe

1

h @ " * , - u n * L - r ) + A u n * L * B u ' + r - e - | n * L

We will use these for the Navier-Stokes equations.

3.3.5. Adaptation of the finite difference techniques :

Alt the methods presented up to now in this section have been largely

studied on regular grids in a finite difference framework. The same techniques

can be used in a finite element context to estimate stability and errors' with

the following restrictions :- constant coefficients (u and u constants),- uniform triangulation- influence of boundary conditions is difficult to take into account

The analysis of finite difference schemes is based on the following funda-

mental property:Stability * consistency :* convergence.

To give an example, let us study the Crank-Nicolson scheme for (67) in

the ca^se where u is constant :

!ro1*' - di,wh)+ |t"vt 67*' + 4i),r,,) * itvtof*t + 4i),v,n)

= (.f"+ i ,*n) Vri , Q Hon

Although the L2 stability is simple to prove by taking u)h = 6i*t + 41, we will

consider the other methods at our disposal.By choosing a basis trt) of ifsr, we rould write explicitly the linear system

corresponding to the case ,f = 0 and {nlr = 0.

B(O'+r - O') +f;eg"+l + Q') = o


where Bii = (rt ,uf ) Ati = v(Ywi,Vnl ) + (uVut ,rrt) .-,, '

If we know the eigenvalues and eigenvectors of A in thp metric B, i.e. the

solutions {)i,li} of \Br! - A$, then by decomposing (D' on this basis, we get

(Bo)l+r - (Bo)l*; ?i?1r + fr ; ;

By asking the amplification factor F = (1-k )i/2)/(t + k îlz) to be smaller

than 1 , we ca,n deduce arl interval of stability of the method.

Evidently, the smallest real part of the eigenvalues is not known but could

be numerically determined in the beginning of the calculation (as in Maday et

al. [164] for the Stokes problem).- On the other hand, when //on is constructed with pt - continuous func-

tions, the Crank-Nicolson scheme on a uniform triangulation is identical to the

following finite difference scheme (exercise) :

S 2

ttoo?,|'+ dlii,*' + dliii + 6?,t:' + dl-ti-1 + 6!jl,i + dl,#'l

+$ftolfir+1 - $?jl,i-')(', 1u,) * (2u1 - u')(6?ll,i - 6?!r',)

*(2u2 - ")(d?,ii, - diil,)l + ukl46r,i' - d?{l,i - d?,1}' - d?jl,i - d?,|:r)= idem in d" by changing & to -,t.

If there exists solutions of the form :

4?'n = 'Pn '2it{ Ix+mv)

they have to satisfy

tn+Ifp * cos((l + m)h) * cos(lh) * cos(rnh)l

. l c h .*z

u [sf n((l + m)h)(q * u2) * srn(lh)(2q - uz) + sin(mh)

(2u, - "r) l * 2uk[2 - cos(lh) - cos(rnh)] l

=,r!"lsame factor but fr * -&]

So we have a formula for the amplification factor.Finally, it is easy to see that the above finite difference scheme is consistent

to order 2. So we have a presumption of convergence of 0(h2 + k') for themethod on a general triangulation .

4.CONVECTION DIFFUSION 2.

In this section we analyze schemes for the convection-diffusion equation

(53)-(55) rvhich still converge when u = 0 without generating oscillations.

+


This classification is somewhat arbitra"ry because the previous schemes canbe made to work when u = 0 or when g is irregular. But we have put in thissection schemes which have been generalized i;o nonlinear equations (Navier-

Stokes and Euler equations for example). Let us list the desirable properties

for a scheme to work on nonsmooth functions d :- convergence in .L* notm,- positivity: g > 0 + gh ) 0,- convergence to the stationary solution rvhen | -+ oor- localization of the solution if v = 0 ( that is to say that the solution

should not depend upon whaiever is downstream of the characteristic whenu -0)-

4.L. Discretisation of the total derivative:

4.L.L Discretisation in time.We have seen that if X(c, t; r) denotes the solution of

I Y?(" ) - u (X( r ) , r ) ; X( t ) - oCIT

-

a4,t * uV 4 - +6(X (o , t ; r ) , t ) l '=,, O T

(e3)

then

Thus, taking into account the fact that X(" , (" * 1) f t ; (n + 1)e) = o, wecan write:

(4,t * uv 6)n+L = f [0"*t( ' ) - d^(xn('))]

where X"(r) is an approximation of X(c, (n + 1)&;n,t).We shall denote bV Xi an approximation 0(fr') of Xn(c) and by Xi an

approximation O(et) (the differences between the indices of Xand the exponentsof & are due to the fact that X' is an approximation of X obtained by anintegration over a time & ; thus a scheme 0(e") gives a precision 0(ka+t1;.

For exampie

Xi@) = s - u"(t)k (Euler scheme for (93))

(e4)

(e5)

(e6)

Xt @) = t - un* * (t - un @lllf ( Second order Runge-Kurta) (97)

modified near the boundary so as to get X;'(A) C Q. To obtain this inclusionone can use (96) or (97) inside the elements so that one passes from xto X" (x)by u broken line rather than a straight line (see (18X19)).

This yields two schemes for (53) :


1 . .

;(d"*t - 6" oxi) - u\dn+L - 1n*1

1

i (d"* ' -6"ox;)- io( f "+ l * 4")= f "++

Lemma 1 :I f u is wgular and i fx i (O) C Q, the schemes (98) and (99) are L2-stable

and conuerge in 0(&) cnd 0(&') rcspectiuely.

Proof :Let us show consider (98). We multiply by 6"+t

'

l ,6"+Ll2 + r f |V6"+11, < ( l / "+t l f + ld"oXt l) ld"+t l (100)

But the map o + X preserves the volume when u' is solenoidal (V.u = 0). So

from (96) :

s5

' 'ri:ti

(e8)

(ee)

lQ" oxillo - t 6" (y)'drrlvxll* 'dy < ld" l3,o(1 + "k')Jxi (n)

( 1 0 1 )

(103)

Hence {t verifies

l ld" l l" < ' i l / lo,q * ld"lo,nl (102)

To get an error estimate one proceeds as in the beginning of the proof ofproposition 9 by using (95).

Remark :Xi(Q) C O is necessary because u.n = 0. Otherwise one only needs

Xi(o) f't dfz c D.

4.L.2 Approximation in space.Now if we use the previous schemes to approximate the total derivative

(scheme (98) of order 1 , scheme (99) of order 2) and if we discretise in spaceby a conforming polynomial finite element we obtain a family of methods forwhich no additional upwinding is necessary and for which the linear systemsare symmetric and time independent .

Take for example the case of (98) :

t,df,*t *n *n, lnV{[+1Vtr7, =

Vtur, € IIon

* lror,Gi@))w1,(x)+ L w nn In,"

d. i* ,€ Hon


where I{or, i, the space of continuous polynomial approximation of order 1 on

a triangulation of O, and zero on the boundaries'

ProPosition 11 :

I/x;(o) c Q, the scheme (103) is Iz(Q) stable eaen if u = 0.

Proof :

one simply replaces uh by ox+t in (103) and derives upper bounds :

ld;*' l:, s l*Ldl*'l' + k ln'orr*'vd;+' (104)

= n lnyn+td;*'

* !*6i(xi.:))dl+'( ')< (,t l f"+' lo + ldioxi( ') lo)ldl+' lo

S (ldtlo(1 + ir ' l* &l;n+tlo)ldl+' lo

The last inequality is a consequence of(101)'

FinallY bY induction one obtains

ldf lo,n s (1 + f,n')"(ld[ lo,a + I ftl/" lo,n) (105)

(105) but theRemark :

By the same technique similar estimates can be found for

norms on 6f & {f; will b. ll.ll, (cf' (79))'

Proposition 12 .If Hon is a pr confonnins approdmation of Ht(Q) then !!:

t"(tt) 1t9r*of the error berween gi solution o7 (103) and, $" ,iluti,on of (95) is 0(h2 lk +

h).Thus the scheme is 0(h2 l& + & + h) .

Proof :One subtracts (98) from (103) to obtain an equation for the projected

error:

where fIn 6"*1

(106)

(107)

l.r

t f* t = 6i* ' - I lnd"*t ,

is an interpolation in I{oir of 6"+t. One gets

[ ,T+'wn * *, I vef;+1vtu6 - [ eioxi?t)h =Jn-" Jn ' " Ja

4n*t - Ilr,d'+r)rn * vk I o(f"*t -�fIn6"*t)Vrr;nJ{t

- [ t.." -rrnd\oxi.oJn


trlom (10?), with u)h = ef+1 we obtain

l luf*t ll? sfll.il l, + lld"+1 - frnd"+t ll, + ld" fln6" lo)ll.l+t ll,

therefore

l l . l * t l l , < l l . i l l , +c(hz *ukh1

Remark :By comparing with (103), we see that e6 and ,/1, are solutions of the same

problem but for et, .f is rePlaced bY :

1 . , . . ^ 1 1 , 6 - L t I

i(d"*t - trrd"+t) - rao(d"+t - Ilrd"+t) - itO"

-frn6")oxi,

where Ar, is an approximation of A. We can bound independently the first and

the last terms. By-working a little harder (Dougtas-Russell [69]) one can shown

that the error is, in fact, 0(h + & + min(h2/k' h)).

Proposition 13 :

With lhe second order scheme in time (99) and a similar approxirnation

in space one c&n buitd schemes O(hz + k2 + min(h3/ft,h')) with respect to lhe

.L2 norm:

(di+t, @r) + j l tvt4i* ' + 4"), vrr) - ($ioXt ,*n) + kU"+* ,*n) (10s)

Vrr .€ Honi6 i* te Hon

where I{sh is a P2 conforming approdmation o/ff$(O)

Proof :The proof is left es an exercise.

T h e c a s e v = A :We notice that (103) becomes

f . - | f - _ , r

lO i * t t nh= | O iox t *n+k I f n * t *n Ywne Hon (109 )J " J n J o

6i*t € Hon

That is to say

dl*t = rln (4toxil + krlhf"+L (110) '

8?

iir:


where IIl is a .02 projection operator in Woi,. Scheme (108) becomes :

6;*' - r lr (6toxt)+ f no (f"+' + /") (111)

If / = 0 the only difference between the schemes are in the integration formula

for the characteristics. Notice also that the numerical diffusion comes from the.L2 projection at each time step. Thus it is better to use a precise integration

scheme for the characteristics and use larger time steps. Experience shows that

k x L\h/u is a good choice.Notice that when v and / are zero one solves

6 , t * u Y $ = Q d @ , 0 ) - 4 o ( r ) ( 1 1 2 )

-(since we have assurned 7t.n= 0, no other boundary condition is needed).Since V.u = 0, we deduce from (112) that (conservativity)

I ou,e) = [ oo@) v, (11s)Jn JN

On the other hand, from (109), with trr, = 1

f , . f II r;*' (') - | 6ioxi - | at(fidetlvxl l-'dv (114)

Jn Ja Jxi.(o)

So if detlY Xi | = 1 (which requires Xi(O) - O), one has

I o ; * ' ( x )d " x - [ +T@)dv - [ do . ; ) a , ( 11b )JA Jf l JTI

We say then that the scheme is conservative. It is an important property inpractice.

4.L.3. Nurnerical implementation problems :

Two points need to be discussed further.- How to compute X"(c),- How to compute In :

In = lndooxi*o

(116)

Computat ion of (116) :As in the stationary case one uses a quadrature formula:

In * D'r 4o(xr,(€&)) 'r ,(€r) (11?)

For example with P1 elements one can takeu) {{}} = the middles of the sides, uk = ox/3 in 2D, orl4 in 3D, where

or is the area (volume) of the elements which contain (e. ":

Figure 3.8 : Numerical simulation of a stationary conaection equa-

tion by characterisft lcs and PI conforming elemenls (top) and compari-

son with the leap-frog *P1 elemenl scheme (bo[tom); in the second case

the stationary solution is compuled as the l imit of the transient problem.

(Courtesy of J.H. Saiac).

b) The 3 (a in 3D) point quadrature formula (Zienkiewicz l24ll, Stroud

12241) or any other more sophisticated formula ; but experiments show that

qnudrature formula rvith negative weights c.rft should not be chosen. Numericai

tests with a 4 points quadrature formulae can be found in Bercovier et al. [27].Finaliy, another method (referred as dual because it seems that the basis

functions are convected forward) is found by introducing the following change

of variable:


4 n (y) * n(X ; t (y)),IetlY X ;

I ldy

89

(118)lnonoxnu)h

= l",rn,


Then the quadrature formula are used on the new integral. If V.u is zero one

can even take

r = I ,k 4n(€t)rr,(X; '((u))rf (11e)

It is easy to check that this method, (109), (119), is conservative while (109),

(117) is not. Numerical tests for this method can be found in Benque et al.

[20].The stability and error estimate when a quadrature formula is used is an

important open problem ; it seems that they have a bad effect in the zones

where u is small (Suli 12261, Morton et al.[177]).

Computation of X"(r) :Formula (96) and (9?) can be used directly. However it should be pointed

out that in order to apply (117), (same problem with (119)) one needs to know

the number I of the element such that

x"({e) € rrThis problem is far from being simple. A good method is to store all the

numbers of the neighboring (by a side) elements of each element and compute

the intersections {€},X'((&)} with all the edges (faces in 3D) between {} and

X"(€k); but then one can immediately improve the scheme for X6 by updating

u with its local value on each element when the next intersection is searched;

then a similar computation for (18)-(19) is made:Let {u;} be such that u = Di u;Qi, Drr = 0, whet. tqd} are the vert ices

of the triangle (tetrahedron):Find p such that

(120)

(121)) l = \ ; * p u ; + flr; = o, )i > o.f

This is done by trial and error; \ile assume that it is l* which is zero, so:

Arr.P = - - ,

Fm(r22)

and we check that ); ) 0, Vt . If it is not se we change rn until it works.Most of the work goes into the determination of the u;. Notice that it may

be practically difficult to find which is the next triangle to cross when X"(C&)

is a vertex, for example. This requires careful programming.

When lu - unloo is 0(hr; the scheme is 0(hr). If ut is piecewise constant

one must check that ul,.n is continuous across the sides (faces) of the elements(when V.u = 0) otherwise (121) may not have solutions other than p = 0. IfIrh = V " do and r/1 is Pl and continuous then ut.n is continuous.

4.2. The Lax-Wendroff/ Taylor-Galerkin scheme[25]

Consider again the convection equation


6, t *Y. (uS) - f in Ox l0 ' " [

For simplicity assume that u.nlr = 0 and that u and f ate

A Taylor expansion in time of / gives:

t26"+1 = 6" + k6i * ?0:rr+

0(fr3)

If one computes d,r and {,11 from (123)' one finds

9 l

(123)

independent of l.

(124)

4n*t = 6' �+ k[ / -Y.(u6\ l - f ;v.@lf -Y.(u6"1])+0(e')

or again

(125)

L2

6n+1 = On + k[/ - V.(rd"n + ;t-V.("/) * V.[uV ,(u6^)]l + o(t3) (126)

This is the scheme of Lax-Wendroff [144]. In the finite element world this

scheme is known as the Taylor-Galerkin method (Donea [68]). Note that the

last term is a numerical diffusion 0(k) in the direction u' because it is the

tensor ut 8 ut.Let us discretise (126) with If6 , the space of Pl continuous function on a

triangulation of O :

(dl*t , .n) -- ((6f, ,*n) + k(f - V .("6i l , 'o)

-f;9 fuf),wnl - *AYwn,

v.('di)) Ywn e Hn

The scheme should be O(h, + &') in the .L2 norm but it has a CFL stability

condition (Courant-Friedrischs-Lewy) (see Angrand-Dervieux [2] for a result of

this type on an O(h)-regular triangulation for scheme (I27) with mass lumping)

k < C , h ,lu l -

An implicit version can also be obtain by changing n into

in (126)

(6i*', trn) + ft(v.(u 6i*\,.n) * f;to.f"O;+t) ,uYw1,) (129)

- k( f , ro) * (6 i , rn) .+(V.(u/) , tu6) , Ywn e Hn

In the particular case when V.u = 0 and ,.rlr = 0,we have the following

result :

(r27)

(128)

n * 1 a n d & i n t o - f r


ProPosition 14 :

If u-and f are independent of t and V.u = 0, u.n1r = 0, then scheme (129)

has a unique solution which satisfies

ldf lo sltilo + Tlllo *Trlv'(uf)lo (stabilitv)

l6i - 6@t )lo < C(h' + k') (convergence)

(6i+' ,wh)-1- /c(v.(u di*\,wnl + $fv .@6T+\, uvtu;, ) + v(v 67+r

- k(f ,wn) + (6i , .n)+ f ; tv t uf) ,wn), Ywn e Hon

(130)

(131)

t 2 1 " 2

l6i*rlr, + iluy6i*'l '� sl4ilold[+'lo + elf lold"+'lo + ] lv.("/)loldl+'lo(132)

thus if one divides by ld;+tlo and adds all the inequalities, (130) is found.

To obtain (131) one should subtract (126) from (i29) :

1n2

(u"*t ,un) + ! ("V. '* t ,

uYwlr)* f r (uVe'*r , ur) = (e" + 0(fr3) ,*n) (133)

where e = 6n-4.Let en = 6n- r/1 where ?l,r, is the projection of { in /{n with

rhe norm l l . l l * = (1.13 + f t? l2luV.Wfl ' . Then €A = e + 0(h2) and from (133),we get

l l . l * ' lh S l l . f l l +c$h2 +f r ' )

The result follows.

Remarks :

1. The;rrevious results can be extended without difficulty to the case u t'0 with the scheme:

(134)

, Vrn)

(135)

Higher or.der schemes :

Donea also'report that very good higher order schemes can obtained easily

by pushing t,he Taylor expansion further; for instance a third order scheme

would b" (/ -= 0):


r,iirl

t 2 1 r2

4n+t' - t6" -ky.(uS") * ?V.luv.(ug")J + u v.[uv.(u(@n*1 - d"))] i = 0

4.3.The streamline upwinding method (SUPG)'

The streamline upwinding method studied in $2.4 can be applied to (123)

without distinction between t and c but it would then yield very large linear

systems. But there are other ways to introduce streamline diffusion in a time

dependent convection-diffusion equation.

The simplest (Hughes [116]) is to do it in space only; so consider

@i*' ,Ixh *rv.(uu.'7,)l + f tv @ldi+ dl+tl) ,wn * rv'(utu,,))

k u . - . , ,+;F@,[+' + di), V,r, ) - ryT lr,(^(d;+' + 6i)rv'(uu"1,))

- k ( f "+ i ,wn * rV . (u tu l ) ) + @T, ,un * rV . (uu1, ) ) , Ywn e Hon

where fi is an element of the triangulation and u is evaluated at time (n* ll2)k

if it is time dependant; r is a parameter which should be of order h but has

the dimension of a time.With first order elements, Adr, = 0 and it was noticed by Tezduyar 1227)

that r could be chosen so as to get symmetric linear systems when Y.u = 0;

an elementary computation shows that the right choice is r = klz.Thus in

that case the metho is quite competitive, even though the matrix of the linear

system has to be rebuilt at each time step when u is time dependant.

The error analysis of Johnson [125] suggests the use of elements discontin-

uous in time, continuous in space and a mixture upwinding by discontinuity in

time and streamline upwinding in space.'Io this enci space-time is iriangulated with prisms. Let Q" = Qx]n&, (t *

1),b[ , let Wiube the space of funct ions fn{o, t } which are zero on f x]n&,( t*

l)fr[ continuous and piecewise affine in r and in I separately on a triangulation

by prisms of Q" ;We search /[ with 6i - 6rn € I'7f;6 solution of

g3

t- Jq'

lo^*1,

ldi,, +V(rdi)l[rn * h(w1,,1* V(uu.,6 ))d,xdt + [^ uVdi.Vwndxdt (136)J Q O

6TQ, (n - l )k * 0)u7 ' (c , ( , - L)k)da f ( .n * h(wnl* V,(utup,))dxdt

94 FINITE ETEMENT METHODS FOR FTUIDS

Yw6 € Wi6

Note that if N is the number of vertices in the triangulation of Q" , equation

(136) is an ^lf x JV linear system, positive definite but non symmetric''

One can show (Johnson et al. [127]) the following :

* I*4Ttb,(n

- l)k - o)' ' '6(c, (n t)k)dc,

l T ^ l

tlr- wr - 6l1o,n&li s c(h* k8)lldlln'�(e)'

4.4, Upwinding by discontinuity on cells:

(d;+',ud) - f,, l, ,,u'd!",6i+Laxla^t

+ vk(vdi+'v't) -

= (d;,ri) + &(f, rd), vi, 6i*' - 6rn e Hon

where u,i is the continuous piecewise affine function a.ssociated with the fth

vertex of the triangulation of Q.

This scheme is used in Dervieux et at [64][65J for the Euler equations (see

Chapter 6).

(137)

Chapter 4

The stokes problem

1. POSITION OF THE PROBLEM

The generalized stokes problem is to find u(r) e R" and p(c) € R such

that

e u - i ' , A , u + V P - f Y . u : 0 i n Q , ( 1 )

n = Itrr on | - AO. (1')

where q,y ate given positive constants and / is afunction from Cl into -R'".

It can come from at least two sources :

1) an approximation of the fluid mechanics equations such as that seen in

Chapfer 1, ihat is when the Reynolds number is small (microscopic flow, for

example), then o = 0, and in general f = 0;il L time discretisation of the Navier-Stokes equations. Then a is the

inverse of the time step size and f un approximation of -uVu.

In this chapter, we shall deal with some finite element approximations of

(1). More details regarding the properties of existence, unicity and regularity

can be found in Ladyzh"ortuyu [138], Lions [153] and Temam [228] and regard-

ing the numerical approximations itt Girault-Raviart [93], Thomasset [229]'Girault-Raviart [93], Glowinski [95], Hughes [116] '

Test Problem :The most classic test problem is the cavity problem :

96 FINITE ELEME}IT METHODS FOR FLUIDS

The domain Q is asquare ]0 , t l ' ,o=0,u - L , f = 0 and the boundarycondition:

u = 0 on all the boundary except the upper boundary ]0, l[x{1} whereu - ( 1 . , 0 ) .

max:ffi

Figure 4.L : Stokes fiow in a cauily

This problem does not have much physical signifigance but it is easy tocompare with solutions computed by finite differences. We note that the solu-tion has singularities at the two corners where u is discontinuous (the solutionis not in HL).We can regularize this test problem by considering on the upperboundary 10 , l [ x {1 } '

u = x ( l - r )

2. FUNCTIONAL SETTING.

Let n be the dimension of the physical space (A C r?") and

/(A) = {u e Ht (O)" ' V.u = Q}

J , ( 0 ) = { u € l ( 0 ) ' , 1 " - 0 }

Let us put

(2)

(3)

and consider the

a , b € L ' ( Q ) "

A , B e L z ( Q ) x "

= (/, u) Vu € J,(O)

€ J,(O)

( o , b ) - [ o , u ,Jn

r(A, 81 = | A,i B,i

J N

problem

a(u,u) * z(Vu, Vu)

(4)

(5)

u - u ' t

(6o)

(6b)

THE STOKES PROBLEM

where "'r is an extension in J(O) of u1. '*'ii''

frgrnrna 1

tf ", € Hrl2 (I)" ond if [ruy.n =0 1hen there edsts an exteasr.o*r u'p €

/(O) of up.

Remarks1. For notational convenience we assume that up always admits an exten-

sion with zero divergence and we identify up and utp '

2. Taking account of this condition and without loss of generaiity' we

arrive by translation (u = u - "'r) at the case ur - 0 '

Theorem 1If f e Lr(Q)" and uf- € J(fr) , problem (6) has a unique solution-

Proof :/"(o) is a non empty closed subspace of f/.l(o)' and the bilinear form

{u,r } -+ a(u,o) + z(Vu,Va)

is l{j(a) e[iptic. with the hypothesis u -(f ,u) continuous' the theorem is a

direct consequence of the Lax-Milgram theorem'

97

Remarkwe also have a variational principle : (6) is equivalent to

. 1 / \ 1,ft}"r'l ,o("'u) + |u(vu'v")

- (f 'u)

Theorem 2If the solution of (6) is c2 then it satisfies (1). Conuersely, vnder the

hypot-hesis of theore* i, i1 {u,p} e HL(o)" x r2(o) is a solution of (t) then u

is a solution of (6).

Proof :V{e apply Green's formula to (6) :

[ @ , - u a u ) . u = [ f , v u € / " ( o ) ( 8 )J n J a

Now we use the follorving theorem (Cf Girault-Raviart [92] for example)

f r t . o - o v u € r ' ( o ) w i t h v . u = 0 u ' n l p - 0 ( 9 )

There exists q e Lz(Q) such that I - Vq (10)

(7)

- We recall that (Vp,u) = 0 for all v € Jr(o)'


So (8) implies the existence of p e ,2(O) such that

u u - u \ u = f - Y P ( 1 1 )

and the result follows. Conversely, by multiplying (1) with v € Jo(O) and

integrating we obtain (6) since ?r = 0 :

r r f f- I A , u a = / V u V u a n d l p Y . o - l o Y P

Ja Ja Jn Jn

Remark1. It is not necessary that u be C2 for p to exist, but then the proof is

based on a more abstract version of (9)(10). Indeed it is enough to notice that

the linear map :

L$t 'S : , * ( f ,u) - a(u, u) - z(Vu, Vu)

is zero on Jo(o) because the space orthogonal to J(O) is the sp_ace of gradients,

thus there exists a q in ,'(0) such thai L(a) = (vq, u) for all u in I/3(o)" ;hence (1) is true in the distribution sense.

2. Proble;n (1) can also be directly studied in the form of a saddle point

problem (Cf. Girault-Raviart [92] for example),

a(u,a) $ v(Yu,Vr) - (V.r, p) = (f ,o) Vu € I{ot(O)"

(V . r ,q ) =0 Vq€ L28) /R

We shall use later this formulation in the error analysis of the methods.

B. The uniqueness of the pressure, p (up to a constant) is not given by

theorem 1 but it can be shown by studying the saddle point problem'

3. DISCRETISATION

Let {J1}1, be a sequence in a finite dimensional space , Joh = Jp, n /{(Q)",

such that

vu € J"(O) there exists un € Jon such that llro - rllr "* 0 when h -* 0 (L2)

We consider the approximated problem :

Find u1 such that

a(un, on) * v(Yu1,, Vun) = (.f, un) Yan € Joni uh - u!v^ e Jon' (13)

THE STOKES PROBLEM

where u'1.. is an approximation of ut in Jr,'

Theorem 3If Jon is non empty the problem (13) has a unique solution'

ProofLet us consider the Problem

1r4in - *{"("n, ur,) * v(Yu1,Yur,) - (/, "r')}

f . - U ' f r e J o h L

Since Jon is of finite dimension ff, (1a) is an optimization of a strictly quadratic

function in N variables ; thus it u,imitt a unique solution' By writing the first

order optimality conditions for (14), we find (13)'

A counter examPle '

Let Q be a quadrilateral and ft be the triangulation of 4 triangles formed

with the diagonals of O. Let

Jon = {on e Co(o) ' 7)hlr . € Pt ul l r - l = 0 '

( V . , l , g ) = 0 V g c o n t i n u o u s , p i e c e w i s e a f f i n e o . T r . }

There are only two degrees of freedom for u1 (its values at the center) but there

are 4 constraints of which 3 are independent, so./on is reduced to {0'0}'

F i g u r e 4 . 2 : a a l u e s o f t h e a e l o c i t y a l t h e u e r . t i c e scf the first countererample and the 'pressure tor the second'

Remarks1. One might think that the problem with the above counter example

stems from the fact that there are not enough vertices. Although Joir would

not, in general, be empty when there are more vertices, this approximation is

not good, as we shall ,.", b..uuse there are too many independent constraints'

For example, let Q be a square triangulated into 2(N - 1)2 triangles formed by

the straight rines parailel to the r uid y axis and the lines at 45o (figure 4.'2)'

\ve have also 2N2 constraints and 2(lgz- -4N) = 2N2 degrees of freedom when

N is large. .,

99

(14)

o

- l

100 FINITE ELEMENT METIIODS FOR FTUIDS

2. The pressure cannot be unique in that case. Indeed with Js6 defined

as above there exists {po } piecewise affine continuous such that

(V.ri,,Pn) = 0 Vut Pl continuous on ?h

These are the functions py, which have values alternatively 0, L and -1 on the

vertices of each triangle (each triangle having exactly these three values at the

three vertices.) Then (V.rr,,pn) is zero because

(V.rr , ,pn) =f{v.ro) l r , ( i I pn(qtr))r , u

i=Lr2,3

So the constraints in Jo1 are not independent and the numerical pressure asso-

ciated with the solution of (1a) (the Lagrangian multiplier of the constraints)cannot be unique.

To solve numerically (13) (or (1a)) we can try to construct a basis of Jon.Let {ui}f be a basis of Jo1.. By writ ing u1 on this basis,

, r r (r)= I u ir i l r )*ul- ,"(r) , (15)1 . . N

i t is easy to see by putting (15) in (13) with u6 = ui that (13) reduces to alinear system

A U = G ( 1 6 )

with

A; i = o ( r i ,d ) + u (Yu i , Vr j ) , (17)

Gi - $ ,o i ) - o (u ! r r ,u i ) - u (yu !p^ , v , r ' ) . (18)

However there are two problems

1) It is not easy to find an internal approximation"..'of /(O) ; that is ingeneral we don't have Jon C /o(Q), which leads to considerable complicationsin the study of convergence.

2) Even if we succeed in constructing a non empty Jor, which satisfies (12)and for which we could show convergence, the construction of the basis {ui} is :

in general difficult or unfeasible which makes (15X16) inapplicable in practice.

Before solving these problems, let us give some examples of discretisationfor Jon in increasing order of difficulty. These examples are all convergent andfeasible as we shall see later. They are all of the type

Jn = { rn eVn , (V . rn , { )n = 0 Vq € 8n }

J a n = { r n e J n , u i , l r = 0 }

THE STOKES PROBLEM 101

As usual, t?1& ) denotes a triangulation of Q, Qr, the union of T*, {gi }; = r -.-rv ,

the vertices of the triangulation, ilttl the j - th barycentric coordinate of s

with respect to vertice, {q*r}i=r.,r, ir of the element ?t'We denote respectively

by

Ns, the number of vertices,

Ne, the number of elements,

Nb, the number of vertices on l,

Na, thenumbero fs idesof the t r iangu la t ion ,andNf , the number of faces of the tetrahedra in 3D'

P* denotes the space of polynomials (of degree ( m) in n variables' We

recall Euler's geometric identitY :

l fe - No * Ns = 1 in 2D , Ne - N/ * No - Ns - -1 in 3D

3.1. P1 bubblelPl element (Arnold-Brezzi-Fortin [5])

Let f(r) be the bubble function associated with an element Tn defined

b y :

pk(r) - tI rf (r) on ft and o otherwisej = 1 . . n | 1

The function plk is zero outside of T1, and on the boundary and positive in the

interior of Tr. Let rlli(t) be a function defined by

*i(r) = )f (c) on all the Tr, which contain the vertex i and 0 otherwise

Let us put :

Vn ={ I r ' rd ( r )+ t b r p r ( r ) : Vo ' ,bk € R" }j k

that is the set of (continuous) functions having values in -R" and being the sum

of a continuous pi"."*ir" uffin" function and a linear combination of bubbles.

We shall remark that dimVn - (Ns * Ne)n'

Let

( 1e)

r02

Qn

that is a set of piecewise

We have : dim Q1' = Ns


= {I f ur(x) : vy' € E} (20)i

affine and continuous functions (usual Pl element).

Figure 4.SzPosition af the degrees of freedom in the Pr - bubble/Pr element

Let us put

Jn = {rr, g fir : (Y .u;,, Qn) = 0 Vqh € Qh}

Jon = {ro g J1 : ul lr^ = 0}.

(21)

(22)

It is easy to show that all the constraints (except one) in (21) are independent.

In fact if N is the dimension of Vn the constraints which define J6 are of the

form BV - 0 where V are the values of u; at the nodes. So if we show that

{ o n , V q n ) = 0 Y a n e V n + Q n = c o n s t a n t

that proves that Ker BT is of dimension 1 and so the ImB is of dimension N-1,

i.e.B is of rank N - 1 which is to say that all the constraints are independent

except one (the pressure is defined up to a constant).Let ei be the itlr vector in the cartesian system. Let us take be = 6*,;ei,

ai - 0 and u;, in (19). Then

Q = ' � a ' 7 q o ' a r e a ( T ; )

= (on, Yqn) - Arr l r ' 1r , * f

+ 8n -- constant

So we have (n - 2 or 3):

d i m J n = ( n - 1 ) ( N s * N e ) n - A r s * 1 = N s * n N e * 1 ( 2 3 )

We shall prove that this element leads to an error of 0(h) in the HLnorm for

the velocity. Whereas, without knowing a basis (local in r) one cannot solve(13) and one has to use duality for the constraints. A basis, local in c, for .1,

THE STOKES PROBLEM TO3;

is not known (by local in c, we mean that each basis function t,' has a suppor$.

around qi and it is zero far frorn qd).

Remark1. We can replace the bubble by a function zero on ETx piecewise affine

on B internal triangles of each triangle obtained by dividing it at its center ; we

obtain the same convergence result.

Z. The bubbles are vety easily eliminated in practice because

they are orthogonal to the pt -basis functions ((Vrt, V1,') = 0).

As pointed out by Lohner and Pierre, they are equivalent to an ar-

rif iciat dissipation -V.(c(h, r)vp) added to (lb) where ("(h, t)l"b =

(-o, t) ' l [(V.u, V wb)ar ea("6)].

Figure 4.4zPosition of lhe i legrees of freedom in lhe element P|,lPt.

3.2. P1 iso P2/P1 elernent (Bercovier-Pironneau [26])We construct a triangulation T:7,12 from the triangle ft in the following

manner :

Each element is subdivided into s sub-elements (s = 4 for triangles and 8

for tetrahedra) by the midpoints of the sides-

We put

Vn - {ro e Co(Oo)" : a6lT1,€ (Pt)" VTx e T +}

Qn -- {qn e co(ou) : qnlTx € Pl YTr eTn}'

(24)

(25)

We construct ,Iot by (21)-(22)

Knowing that the number of sides of a triangulation is equal to lf e * Ns - 1

if n=2 (Ns*Nf-Ne-l if n=3) we get, in 2-D :

d i m J n = 2 N s * 2 ( N s * N e - 1 ) - N s * 1 = 3 - l { s * 2 N e - t ( 2 6 )


Figure 4.5 : Degrees of freedom in the PtisoPzf Pt element

The independence of the constraints is shown in the following way :

We take urr € I/r, with 0 on all nodes (vertices and midpoints of sides)except at the middle node ql of the side a;, intersection of the triangles ?r, and

ft,. Then :

0 = (u6, Vgr,) - uhtelX f Vqnlr*iarea(Tr,)li -L,2

Let g1, and q4 be the values of gn at the two ends of a1; if we take u;(q') =

Qt, - {r, then the above constraint gives gr, = grr.So the gy, which satisfy theconstraints for all uy. are constant.

This elemcnt also gives an error of 0(h) for the Hrnorm. We know how toconstruct a basis with zero divergence usable for this element (Hecht [109]).

3.3. P[ZIPL element (Hood-Taylor [113])

vn = {ri, e Co(ou)" 1 ahlr* € (P') vfr}

Qn -- {qo e Co(oo) : qnlr* € Pt v/r}

anrl of course (2I)-(22).

(27)

(28)


Figure 4.6 : Position of the degrees of freedom in the P'l Pt element '

This element is 0(h2) for the I/1 error and has the same dimension (26)

but it gives matrices with bandwidth larger than (24)-(25)' The proof of the

independence constraints is the same as above. The dimension of Jn is there-

fore:

d i m J n _ 2 ( N s * N e _ 1 ) + 2 N s - ^ ^ [ s * 1 = 3 N s * 2 N e _ I

Construction of a zero divergence basis :

We know how to construct a basis with zero divergence usable for this

element (Hecht t10g]) ; figilre 4.7 gives the non zero directions at the nodes of

each basis funct'ion *ittt 1"ro divergence, as well as its support (set of points

where it is non zero).

Figure 4.72 Basis wilh zero

For the 3 families of functions,a side, ? a triangle and g a vertex.

diuergence for the P2 IPL element .

the notations ere the following : o denotes

We have dra,wn the triangles on which'the

2Ns functions Ne functions Na functions


function is non zero and the directions of the vectors, values of these functions

at the nodes, when they a,re non zero.

Since all constraints but one in "Ir, are independent ; the dimension of "I1,is

dimJl - d imVl - d imQl= 2. f fs * 2Na- Ns * 1 = 2Ns * l Ie * Na

the last equality being a consequence of Euler's identity.

We shall define three families of vectors :- 2N s vectors, each with 0 on all t,he nodes (middle of the sides and vertices)

except a vertex g where the vector is arbitrary .- Ne vectors u&, each having value 0 on all the nodes except on the 3

mid-points of the sides of atriangle where nk = Dr..r.\;oi with, for example :

where o1 denotes a unitary vector tangent to the side a and urd the basis functionP2 associated to the vertex gi.

- N a vectors u non-zero only on a side a and on a side o1 and a2 of the2 triangles which contain a. If an denotes the unitary vector normal to a wecould take, for example u = Aan + pol * va! with :

Ar=r , )z=f f i# , ls=

,\ _ _ (Vu.,l, o,.) .. _ _(Vtr3, orr) (Ywa, ao)-@' F=-p . , r *5 ' u= - (Vum

We leave it to the reader to verify that the vectors are independent and withzero divergence in the sense of (2t)(27X28).

3.4. P1 nonconforminglP0 element (Crouzeix-Raviart [60])

Vn - {u7, continuous at the middle of the sides (faces) ,nlr* € (.Pt)"} (29)

Qn = {qo, qol r* € Po} (30)

and (21)-(.22).


Figure 4.StPosition of degrtes of f\edom for the element Pt nonconform'

ins lPo .

Q = ( V . u r , , g r , ) = t /u J";d"q1,u1,.nd,1 = I length(sid") (qnlr*, - Qnlrr,)

:+ en = constant.

In .R2 we have

d , i m J n - N e * 2 N s - 1 ( 3 1 )

we know how to construct a local basis with zero divergence for this element'

In dimension 2 we d.efine two families of basis vectors of Jn' one with indices

on the middle of the sides qdr '

, , , j ( g u , ) = ( g d q J ) m V g } l m i d d I e o f t h e s i d e { q u , q , } e r ( 3 2 )

and the other with indices on the internal vertices ql

o,,(q, i) = 6ri f f i

vsi j middle of {qtq'}

where n is normal to qdqi itt the direct sense'

(33)

Ns functionsNlfunctions

Figure 4.92 Zero i l iuergence basis for the Pr nonconforming Po element


We remark that for the two families we have :

because each integral is zero. Moreover if Na denotes the number of sides thereare No * Ns basis functions in this construction ; that number is also equalto 2Ns * Ne - 1 which is exactly the dimension of Jn kf . (31)); so we have abasis of Jn.

In 3-D a similar construction is possible (Hecht [109]) with the middleof the faces and the vertices but the functions thus obtained are no longer

independent; one must remove all the functions v'I of a maximal tree withoutcycle made of edges.

3.5. P2bubblelPL discontinuous element . (Fortin [80])

(V.r ' ,en) =! to l r - t u t .nd,x = o Yqn e Qnx JT*

vn - {ro e co(Qo)" : u6(c) - u)h+t btpt@), .n lr ,I

(34)

(36)

€ ( P k + t ) " , h € R " \

(35)Qn - {qo , qnlT* € Pe} (qn is not continuous),

& -0 or 1. With (21)-(22) this element is 0(h') in norm HL if & - 1 and

0(h) if & - 0. Since the constraints are independent (except one as usual) wecan easily deduce the dimension of Jon. The construction of a basis with zerodivergence for this element can also be found in [15]. We note that if we removethe bubbles and if & - 1 the element becomes 0(h).

f igure 4.10: Position of degrees of freedom for the elemehtP2bubblef PL disconlinuous .

RemarkOne can construct without

als by replacing in the formulaeI m with respect to eocl of its

difficulty a family of elements with quadrilater-P* by Q* , the space of polynomials of degreevariables.

THE STOKES PROBLEM

v (r(m)) - A(m) * u (l (m)) + v1r1-;;

109 "

3.6 ComParison of elements:

The list below is only a small subset of possible elements tfor the solution

of the Stokes problem. 14l" huu" only give" ih" elements which are most ofLen

used. The choi.u or an element is based on the three usual criteria :

- memory caPacitY,- precision,- progranrming facilitY'

Even with recent developments in computer technolcgy the memory ca-

pacity is still the deterrninini factor and so we have given the elements in the

order of increasing memory requirement in 3D for the tetrahedra ; but that

could change for computers *iit, 256 Mwords like the CRAY 2 which could

handle elements which are now considered to be too costly !

4. RESOLUTION OF THE LINEAR SYSTEMS

If we know a basis of Jon and of '/t

construct I' and G by (1?)-(18) and sc

method but relatively costly in memor;

To oPtimize the memoty we store <

dimensional array again denoted by /

indices of rows una lot,r*ns of non zero elements are stored'

m = 0LooP on i and j i, \I f a ; i + 0 , i l o ' m = m * 1 , A ( * ) - a i i ' I ( * ) = i ' J ( m ) = I

end of IooP .

fTL1,1 o, = fn.

This method is well suited particularly to the solution of the linear system

A U - G

by the conjugate gradient method (see chapter 2) because it needs only the

following oPerations :

g iven U calculateV = AU;

now this operation is done easily with array A' I and J :

To get V(i) one Proceeds as follows :

Init ialize V - 0.

For rn - 1.. rr"714as do


4.L. Resolution of a saddle point problem by the conjugate gra-

dient method.

All the examples given for Jol are of the type

Jon = {rr, e Von : (V.rn, {n) = 0 Vqh € 8i,} (36)

where Von is the space of functions of V6 zero on the boundaries. An equivalent

saddle point problem associated with (1a) is

min max{ lo1',o,uh)+ f,rNtn,

vun)-(f,uo)*I_pt(v.uh,qi)} (3i)ur -u!^ €vor, p; ' 2 z l . .M

where {qt|* is a basis of Qn.

Theorem 4When Jon is giaen b! (36) and is non etnply with dimVn - N ,dimQn - M and von c I4(0)" f O is polygonal), the discrelised Stokes

problem (13) is equiualent to (37). In addition, (37) is equiualent. to

a(un or,) * v(Vu6, Vur,) + (Vpr, , un) = ( f ,on) Yv6 € Vol (38)

(V.rr , , { r ) = 0 Yqn € Qn (39)

uh - u'y^ e Vo6, pn € Qn

These arc, the necessary and sufficient conditions for u1 to be a solution of

(13) because they are the first order optimality conditions for {1/r).

ProofProblem (13) is equivalent to (14) which is a problem with quadratic cri-

teria and linear constraints. The min-max theorern applies ancl the conditions

for the qualification of the constraints are verified (Cf. Ciarlet [56], Luenberger

[162] Polak [193] for example). The gt are the M Lagrangian multipliers withM constraiuts in (36). The equivalence of (37) with (38)-(39) follows from the

argument that if L is the criteria in (37)

OL AL# = 0 , 6 = o

( 4 0 )

at the optimum, where ui are the components bf un expressed in ttre basis of

%1,. These are the necessary and sufficient conditions for {un,po} to be thesolution of (37) becaus6 L is strictly convex on I/o6 in ul and linear in p'

Remarks .1. (37X38) can be found easily from (1); this formulation was used in

engineering well before it was justified.

THE STOKES PROBLEM T1'I'

2. without the inf-sup condition (?9), there is no unicity of {pi}l'

Proposition 1If {ri}tl, o basis of Vonand {qi}M a basis of Qn' problem (35)-(39) is

eqaivolenl Io o linear sYstem

(4i)

sis {ui } and P being the

A ; i o ( r i , r l ) + v ( Y u i , V ' j ) , f , j = l " N ( 4 2 )

B ; i = (Vq ' , d ) i - L "M, j - l "N (43)

G i = U , r l ) - o ( r r ^ ,d ) - z (Vur^ ,V ' j ) i = 1 "N (44)

( e B \ / u \ -\a ' o / \ P )

-

with U 6eing the coeficient of un erpressed on

coefi,cienls-of p;rerplss'ed' on the basis {qt}'

(f)the ba

ProofWe note that if u;'

Then we decomPose uh

(38) and (3e).

Proposition 2

L . Solae

is zero on f6 we have

(V.rr, , { t) = -( '0, Vgr,)

and p6 on their basis and take t)h = ry' and g7'

s o l u e A u o = G - B P o

put ga = Btyo (- Bt A-LG) n = 0'

(46)

- q ' i n

Problem (11) is equiaalent lo

(8, A- t B)P = Bt A-rG, u = -A-L BP + A-LG

ProofIt is suffi.cient to elimin ate (J f,rom (a1) with the first equations' This can

be done because A is invertible as shown in (a8) :

U t A U - a ( u t , u n ) * v { Y u y , , V u l ) = 0 * u n = 0 * ' t J - 0 ( 4 8 )

(47)

(4e)

, (50)

t12

2. Put


Au = Bq"

and C z = Bt i 1- At A-r Bq")

set o _ ls"lL

P n + 1 = p n - p q . "

un*l : un - pu (so that Aun*t - -Bp"+t + G)

g t + 1 = g n * p z

3. IF ( fg '+ l1< e)or(n > nMan) THEN s lop ELSEPnI

(57)

(58)

and Return to I wi th n:=n*l

Choice of the preconditioning.Evidently the steps (51) and (56) are costly. We note that (51) can be

decomposed into n Dirichlet sub-problems for each of the n components of uy.This is a fundamental advantage of the algorithm because all the manipulationis done only on a matrix not bigger than a scalar Laplacian matrix.

(56) is also a discrete linear system. A gooci choice for C (Benque and al

[20]) can be obtained from the Neumann problem on Q for the operator -A ,when we use for Qn an admissible approximation of HL(o) (Cf. 3.3, 3.2 and3 . 1 ) :

(Vdn,Vqn) - (1, qo) Vsr e QnlR =+ (b9)

Cii = -An;j = (Vqt ,Yqi ) Vi, j (boundary points included) . (60)

This choice is based on the following observation : from (1) we deduce

-A,p -- V.f in Q,frn+ - f . n + v L , u . no n

So in the case v K t there underlies a Neumann problem.This preconditio:ring can be improved as shorvn by Cahouet-Chabard [47]

by taking instead of -An :

C : ( u I o ' o A i ' ) _ 1 w i t h N e u m a n n c o n d i t i o n s o n t h e b o u n d a i y � � �

(5 1)

(52)

(53)

(54)

(55)

(56)

^' - lg"+'l?, - wq n * t = g n + t * l q "

THE STOKES PROBLEU 1.13

where f1, is the operator associated with the macrlr trri,u)J) where toi is the

canonical basis function associated with the verte: {. In fact, the operator

Br A-18 is a discretisation of V.(a/ - vA)- 1V. Ier* f o = 0, it is the identity

and if v = 0,, it is -An.

4.2. Resolution of the saddle point proble-ur by penalization

The matrix of problem (41) is not positive def;n-;e.

Rather iet us consider

(* -:,) (s )We can eliminate P with the last equations

( ,4+ l"u ') t r -G

We solve this system by a standard method since the matrix A * e-r B Bt rs

symmetric pooitive definite, (but is still a big matrlx)- This method is easy to

program because it can be shown that (63) is equivalent to

/ c \- { ; } ( 6 2 )\ l l t\ " , /

and there remains :

a(ui , , , r , ) * u(Y u; , Vul) * f { v . "n ,q t ) ( v . . ' , i . s i ) - ( f , on )L . . M

1e

(63)

(64)

Yu1, € Vo6 u1, - ulrn € l"on

It is useful to replace e by €10i, B; being the area of the support of q'; this

corresponds to a penalization with the coefficients 0i in the diagonal of / in

(62). Or"her penalizations have been proposed. A clever one (Hughes et al

[118]) is to penalize by the equation for u with u: Vg :

(V ' ' r , , qn) = 0

is replaced b3'

( v . r r , , qn ) * e l ( vp r ,Vq r ) - (A "o ,vg r , ) - ( f , vq6 ) ] = 0

but the term (Arr , ,Vgn) can be dropped because V.u7, = 0 (see (129) below).

5. ERROR ESTIMATION

(The presentation of this part follows closely Girault-Raviart [93] to which

the reader is referred for a more detailed description of the convergence of finite

element schemes for the Stokes problem.)

5.1. The abstract set uP

Let us consider the following abstract problem

LL4 FINITE ELEMENT METHODS FOR FLUIDS

Find un €..1L where

Jn = {t* e I/6 : }(u6, Qi,) = 0 Vqn-e Qnl' (65)

such that

a(un,ro) = (f ,on) Vur € Jt (66)

with the following notations :- I/6 sub-space of V ,I/ Hilbert space, dim I/1' ( *oo- Qn sub-space of 8, O Hilbert space, dim 8i, ( *oo- a( , ) continuous bilinear of V xV'+ R with

a(u,a) > rllrl l?, Vu € Y (V-ellipticity) (67)

- b( , ) continuous bilinear of V x Q -+ R such that

0 < 9 : - ' ' b ( ' ' q )!;36:Egffi (68)

- ( , ) duality product between I/ and (its dual) V'.We remark that these conditions are satisfied by the Stokes problem with

a(u,u) = a(u, u) + u(Vu, Vu)

b(r , q) - - (V.r , q)

v = H ] ( o ) " Q = L 2 ( a ) l nA - i n f { a , u ) .

(, ) = scalar product of .f2(n).

P - L/C where C is such that for all g there is a u such that V.u - g and

lvr lo S Clqlo

Remark:The notation in this formulation is somewhat m:sleading ; in fact Jon and

V6 &te rrow denoted .lr and I/1,1

The following proposition compares (66) with the continuous problem :

a (u ,u ) = ( f ,u ) Yu € J (69) :

u € J = { u € V : D ( o , s ) = 0 V q € 8 } ( 7 0 )

5.2. General theorems

Proposition 3If un and, u are solutions of (65)-(66) and (69)-(7A) respectiuely, andif Vn

is non empty, then

THE STOKES PROBTEM

l l" - urllv < C()X"l$^ ll" - urllv * o^?6^ llp - q,,lleJ

Proafwe can show (Brezzi [39]) thanks to (68) that (69)-(70) is

the following problem : find {t , p} E V x Q such that

a(u,a) * b(o, P) = ( f , r ) Ya € V

b ( u , s ) = 0 Y q e Q

By replacirrg, by rr e Jn in (72) and by subtracting (66) from

a(u - uh,ah) = -b(op, ,P - Pn) Yun e Jn

Or, by taking into account (65)

a(u - uh, un) - -6(rn ,P - Qn) Yan € Jn Vqh € Qi'

and for all u6 € Jn

a(wn - u6 uh) = e(utn - u,uh) - b( ' r , ,P - qn)

So (cf (6?) and the continuity of a and b)

l l l . t - uhl lv S l lo l l l l 'o - " l lv + l lb l l l lp- qol lo

which implies

l l " - ur , l lv < l l " - whl lv + l l .u - 'n l lu <

(1 + l lcl lA- ') l l rr , 'o - ul lv + t- ' l lbl l l lp - ql, l lq Ywn € Jn Yqn

> p ' > 0

1 1 5

(71)

(75)

(i6)

(77)

(7e)

(80)

equivalent to

(72)

(73)

(72) we get

(74)

e Q (78)

Proposition

(Brezzi [39])

if o^116^ ,::g^

then ,"tt!^ l l" - ,r,l lt

b(,r,, gr,)

l lun l lv l lqhl lq

( (1 + Sl "jtl^ ll, - ,i,l lu

ProofLet us show first that if zn is in the orthogonal space of Jn tn Vn (denoted

Jf ) we have

p ' < b(rn, qn)Yqn € Qn Yzn e J* (&1)

llrr,llllqr,ll

1 1 6 FINITE ELEMENT METHODS FOR FLUIDS

This will be a direct consequence of (?9) if we show the following :

zn € Jf * fqo such that en solution of ' b('n'41) 't

, :89^t141;tIn fact, the optimality conditions of the problem"Sup"are :

b ( y o , q i ) * A ( t o , U i , ) = 0 Y Y n € V n

with ,\ - -ll"oll-u; since the conditions are necessary and sufficient for z1 to

be a solution, it is sufficient to define gt as a solution of

b (y^ , , { r ) = ( tn ,yn) Yyn eVn

which is possible because if B the operator defined by the above equation is

not of maximal rank then by taking {r, in the kernel, we see that 9' = 0 in (79).

Now let an €Vn be arbitrary. \Ye can rvrite nh = zn * tu6 where wn € Jn

and z1 e J*. Let us show that l lzl l l S Cll" - ,ull for all u in J :

Since zn e J* we have

b( rn ,8n) - b ( ro ,9 i , ) = -b { " - uh , qh) Yqn € Qn

Then we have from(8l)

t h ( 2 , , n , \

l l"ollu s g-iff i s l lbll l '-L l l" - unllv (82)

s o V t u 6 € J n , 1 r n Q V n s u c h t h a t

l l, - wnllv S ll, - ur,ilv +llrnllv S (r $lll,, - unll, (83)

Theorem 5Let u7, and u be the solutions of (65)-(66) and (69)-(70)

I f we haae (67), (68) and (79) then

l l" - u^llv s ct, l2l^ l l" - unllv + o#6^ l lp - qollel (84)

ProofThe proof follows directly from the 2 previous propositions.

Theorem 6If p and ph are the pressures associated with u and u6 (Lagrange multiplier

of the conslraints),we also haue

llp - pnllq S c(l)t,ltf"^ ll, - uhllv * o^'35" llp - q,,llql (85)

TEE STOKES PROBTEM

its corresponding discrete equation, we deduce that

TT7

Proof :From (7a) and

b(on,Pn - {n) = a(u - uh, ur) * b(tr1 ,P 8n) Yun € Vn Vqn e Qn (86)

So from (79)

l lpo - i^ t te < Pt- l sup,r . { [o( r , - uh,ah) * b(ur , ,P- Qh)] / l l ' r , l lv ]

< Pt - ' ( l l " l l l l " - un l l v + l lb l l l l p - qn l le ) '

We obtain :

l lp - pnl le < P'- ' l lol l l lu - uol lv + ( l lbl lB'- ' + t) l lp - q' , l la (87)

To study the generalized problem (64), we consider the following abstract

problem where c(p,q) = (Cp,g) is a symmetric Q- elliptic continuous bilinear

form :

Find ui e

where C6 and

(88)

{8e)(e0)

of h, we haue

(e 1)

{CnPn, {n) = c(Pn, qn) YPn, qn e Qn

(BT"n, q; ) = b(r i , , gr , ) Yun € Vn, Vgn e

Theorem 7[Inder the hypathesis of theorern 5 and if I is independent

l l , i - unilv + l lpi - pnlle S .cl l / l lo

ProofWe createan asymptotic expansion of uf, and of pi,

uf, =,rl + ,ut1, + '. '

p ' h - p o n * r p t n * . . .

that

(C;'B'n'uon,BTro) = o

o(uf l , ur) + (C;t gT"tn,BToo) = ( f ,un),

o("'n, ur) * (C;' nT"7, 4 ou) = 0...Von e Vn

I/6 such that

o(" i ,o r , ) * u- t (C; t RT" i ,BToo) - ( f ,un)

.B1 are defined by the relations

Qn

So (88) implies

(e2)

(e3)

(e4)

(gr)


By putting {n = Clr B{,un, p?, - C;L BTu},,,we see that ufl is solution of the

,,orrp"rrulized probrlm 1'oo;. we easiiy show that the ui6 are bounded indepen-

dently of h, and hence the theorem holds'

Remark:If the inf sup condition is not verified (88) is still sensibles but the error

due to penalization is 0(./.) (cf. Bercovier [25])'

5.3. Verification of the inf-sup condition (79)'

Finally for all the examples in section 3 if remains only to show the suffi-

cient condition (?9); we follow Nicolaides et al ltqtl'We shall do this for the element PrbubbtelPt. We recall the notation :

I/ represents.[f61(O)l', Q = Lz(fj4l&, I/lr and Q6 area'pproximations of

the spaces (Prbubbli,-zero on the boundary, for 1z6 and Pl for 8t, both are

assumed conforming). The lemma 1 below does not depend on the particular

form of Vn and Q1 :

Lemma 1The inf-sap condition (79) is eqaiualent lo the eristence of o IIn

IIrr : I/ - Vn (96)

such that (C is indePendent of h):

( u - t l l u , V g r , ) = 0 Y q n € Q n , Y u € V ( 9 7 )

l l r r , r l [ 5 Cl l r l l r (e8)

ProofIf III exists, we have

,::e^ffi >:Eeqffi#=;:vffi (ee)>c-t;Eylf f i >c- 'glqol" (g,-c-tB) (100) :

Conversely, let IIl be an element in the orthogonal of J1, which satisfies

(I I i , r , Vqh) = (r , Vqh) Yqn € Qn (101)

then (cf. (81))

l lr lnrl l, 1r'LJlfrpJ <P'-' l l ' l l , (102)

THE STOKES PROBLEM 1,19

,n:l'i.11.

(103)l ( r , Vq) l - [ (v .u , q ) l < lV . r lo lq lo S C l l r l l r lq lo .

Theorem 8If the lriangulation is regular (no angle tends to 0 or r when h tends to

0), then the pt - bubble/PL element satisfies the inf-sup condition with P'

independent of h. so we hate the following en'or eslimate :

because

Then we have

( I Io , - a ,Y qn)

and we easily show that

l l " - 'n l l t + lp - pn lo I Ch l lu l l2 .

ProofTo simplify we assume n = 2. and we shali apply lemma 1. Let u be

arbitrary in I/$(0)"Let IInu be the interpolate defined by its values of the vertices g' of the

triangulation.

i l r ,u(q') = u(q')

(104)

( i06)

(10 i )

(108)

(10e)

- Here there is a technical difficulty because ?/ may not be continuous and one

must give a meaning to u(qi). We refer the reader to [12, lemma II.4.1] .

Nevertheless the values at the centre of the elements are chosen such that

f f

I rrpdn - | udx YTrJTu JTx

=+'1,.( I Io r -u )dc ]Vgr , l r * =0

l l rrr ,r l l ' s cl lr l l ' .Remark

1. This proof can be applied immediately to the case where the bubbla is

replaced by the element cut into three by the centre of a triangle.

2. It is not always easy to find a II1, with (96X98); another method is to

prove (79) directly on a small domain made of a few elements and then shot'

(Boland-Nicolaides [33]) that it implies (79) on the whole domain.

6. OTHER BOUNDARY CONDITIONS

6.1 Boundary Conditions without frictionThe friction on the surface is given by v(Yu * Vu")n - pn. A no-friction

.-t

condition on lr is written as

r20 FINITE ELEMENT METIIODS FOR FLUIDS

v(Yu+ Vu")n - Pn = 0 on lr C 0Q; (110)

and

This condition may be useful if rr is an artificial boundary (such as the exit of

a canal) or a free surface'

Let us consider the Stokes problem with (110) and

u = 0 o n 0 0 - f r ( 1 1 1 )

To inc lude(110) in thevar ia t iona l fo rmula t ionof theStokesprob lem,wehaveto consider :

a(u,a) + itv" + Yur, vu * v'r) = (f' u) vu € Jor(a)

u - u r€ J01(O) = tu € J (O) "u =0 on 0Q - f r )

When V.u = 0 we have :

( V . V u T ) ; = u i , i i = 0

(Vu * vur ,vr") : (v, * vu", v')

So (112) imPlies

o(u ,u ) - z (Au , o ) * t v (Vu1 Vu" )n u - ( f ' ' ) Vu € Jo r (A) (116).

J f r

one can prove without difficulty rhar (116) implies (1) an-d (111)'

The finite element approximation "f (ffZ;-1113) and the solution of the

linear system is done as before. Howevet'in inl saddle point algorithm the

knowledge of the pressure do., ,rol d'ecouple the velocity components (cf (51)

in Algorithm 2).

6.2 Slipping Boundary Conditions'

It is also interesting to solve the stokes problem without tangential friction

and a slip condition on the velocity :

r . lv(yu*Yur)n-pn] = 0, ' r .n =0 on f r C 0O; l t = t l ' r on aO-f 1 (117)

where r is a vector tangential to 0o (in 3D there are2 conditions and 2 tangent

vectors) .Wetreat the-probleminthesamemannerbysolv ing.

o(u, u) + itv"+ Vu", Vu * v") = (/, ') Vo € Jo"(o) (118)

(112)

(113)

(114)

(1 15)

THE STOKES PROBLEM 12r

u- ut€ Js, .1(Q) = { , € / (Cl) i t ) =0 on 0Q - f1 i u 'n l r , = 0} (11'g)

Here also the previous techniques can be applied ; however there are some

additional problems in the approximation of the normal (cf. verfiirth [233])'

Some turbulence models replace the physical no slip condition on the sur-

face of a solid by the slip condition :

. 0 u .a u . r * ( - ) . r = b , u . n = 0 o n 1 1 C A 0- O n '

where r is a tangent vector to f1 (two vectors in 3D)

We can eith-er rvork wiih the standard variational formulation, that is :

a(u,o) * z(Vu, Vu) * f a1tr.7) = (f , ,) + [^ br 'u Vu € ' /o"r(Q) (121)'

J r r J l r

u - u p , € J a n t ( A ) = { u e J ( O ) : u = 0 o n 0 Q - f r ; u ' n l r , = 0 } ( 1 2 2 )

or we can also reduce it to the previous case by noting that n'du f 0r - -u'r f R'

so that (120) is equivalent to ;-(at - a+rl Rwhere ,R is the radius of curvature):

a 'u . r 1 r (Vu *Yur )n = b , u "n = 0 on 11 C A0

a condition which can be obtained with the variational formulation

a (u ,u )+ iw " *Yu r bu . r (123)

V o € / o ' r ( O ) u - u r € J o " r ( Q ) '

Finally, rve note that

( V q , u ) - 0 V q € I / t ( Q )

and so rather than working with ..Io',r

work with :

( i20)

, Vu * Vrt) * lr ,ar.tr-t)

= (f ,r) * l r ,

<+ V.u = 0 and u.nlp = 0

which is difficult to discretise' we can

J6" , (A) = {u €Ht (A) , (Vq,u) - 0 Vq € t ' (g t ) ; t r lan- r , = 0 }

where the normals do not appear explicitly; but condition (120) is satisfied in

the weak sense only. parbs [rgs] has shown that this is nevertheless a good

method.

6.3. Boundary conditions on the pressure and on the vorticity

To simplify the presentation, let us assume that O and 0Q are simpll'

connected. Let us consider the Stokes problem with velocity equal to up on a

part of the boundary f 2 and with pr"r.rib.d pressure p0 on the complemenfirl'

L22 FINITE ELEMENT METHODS FOR FLUIDS

part 11 of 0Q. If the tangential components of the velocity are prescribed onlllllr (r x n = ur x n) then the problem is well posed and we can solve it byconsidering the following variational formulation :

a ( u , u ) * z ( V x u , V x u ) * u ( Y . u , V . r ) - ( f , 4 + [ p o a . n V u € J o * ^ r ( O )J| ,

(r24)u-ur € Jox,,r(O) = {u e J(O) : u = 0 on 12 = 00-lr; ?xnlp. = 0} (125)

The finite element approximation of this space is carried out exactly as inchapter 2,$5

If we solve

a(u,a)*z(V x u, V x u)* u(Y.u,V.r ) = ( f ,q+ [ d , .a Vu € Jo"r (O) (126)J f r

u * u r € J o n t ( O ) = { u € l ( O ) t r - 0 o n 1 2 = 0 O - f 1 ; u . n l r , = 0 } ( 1 2 7 )

then we compute the solution of the Stokes problem with, on 11 :

where up and;::;.;,*",T;;i ,ft-- r n- v d'

Finally, V. Girault [91] has shown that the following problem

{

(128)

a(u,u) * z (V x u ,V x u) + v(V.u ,V. r ) = ( f ,a ) Vu € J6" (O)

u-ut € /6" (0) = {u € / (O) : u .n lp = 0 ;n .V " r l " - 0 } ,

yields

L t .n = I l t . f l t 2 .V x u = 0 , y = / .n on I' d n

Naturally, we can 'mix' (125),(127) and the above condition on parts of I.To sum up, we can treat a large number of bcundary conditions but there

are certain compatibilities needed. Thus rt does not seem possible to have onthe same boundary u.n and p prescribed. Also it is not clear whether it ispermissible to give all the components of V x u on a part of the boundary in3D. For more details, see Begue et al.[18].

6.4 Mixing different boundary conditions.

If we want to mix the boundary condition studied in $6.2 with those of$6.3 we are confronted with the choice of the variational formulation:

we have to use (Vr, vo) near to a boundary and (V x u, V x u) on otherparts of the boundary!

'

THE STOKES PROBLEM

The following formulae are satisfied for all regular u,n

polygonal boundarY :

(V. r , V . r ) + (V

d u - u \ , u * Y P = f , V . u = 0 i n Q

ulp, = u1'

0uu.n = ur.ni au.r + r.fr - b on fz

u x n = u r x n ; p = p s o n l s

L t . n = u r . n i n x v x 7 t = d o n l a

Then we could solve :

123

and all O having4

x u, V x u) - (V,r, Vr) - f;rOun-

u'nV'ul

o(u, u) t v(Yu,Vr) - i hro"'n - u'nY'u * uVtr 'n - u'nY 'af

L J I

(12e)

f to r*Vr?, Vu*Vr" ) - (V. r r , V . r ) * (Vu, Vu)+ ! ;oou.n-n .nY.u l

(130)

(131)

= (/ , u) * f dl Vu € Jo*r(O) u - ur € /o"r(O)J l r

But the replacement of (Vu * Vur,Vu * Vo") by (Vu,Vu) produces an ad-

ditional teim (V.rr, V.r), which is zero in the continuous case and non zero in

the discrete case.

There are further simplifications. suppose, for example, that we have

to solve the following problem in O with the simply connected polynomial

boundary | = fr U lz U le U I+'

(132)

(133)

(134)

(135)

(136)

(137)o(un, ur,) * v(Yu6, Vrr ,) * o l r"un.uh

= (f ,rr,) * Ir,rro, * lr"poon.n

- lr,o.rn

*.f,r".(ulvup .n - ah.nv.up) Yan e JL


uy, -url, € JL ={ro g J6 : uslrr = 0, trl X nl1, = 0, un'nlrrur. = 0}

In fact, the extra integral is a function of up and not of u because it contains

only ta.ngential derivatives of u.n on la and on ls it contains only tangential

derivatives of tangential components of u (in fact V.u - n}u/0n ).

RemarkSince these formulations apply only to polygonal domains one should be

cautious of the "Babuska paradox" for curved boundaries;

the solutions obtained on polygonal domains may not converge to the right

solution when ttie polygon converges to the curved boundary.

Figure 4.LL : Iso aalues of pressure on a sphere in a Stokes flow.On the first fi,gure (luft) the computation was done with the PI/PI inrsalid

element (oscillations can be seen) on the second figun (right) the P[isoP2/P1element was used (Coartesy of AMD-BA).

Il r

Chapter 5

Incompressible Navier- Stokes equations

1. INTRODUCTION

The Navier-Stokes equations :

u 1 * u Y u * V P - u \ , u = f ( 1 )

V.u = 0 (2)

govern Newtonian incompressible flows ; u and p are the velocity and Pressure.These equations are to be integrated over a domain Q occupied by the fluid,during an interval of time l0,T[. The data are :

- the external forces /,- the viscosity z (or the Reynolds number Re),- the initial conditions at t - 0: uo,- the boundary conditions : uF, for example.

These equations are particularly diffrcult to integrate for typical applica-rion (high Reynolds numbers) (small z here) because of boundary layers andturbulence. Even the mathematical study of (1)-(2) is not complete ; thguniqueness of the solution is still an open problem in 3-dimensions.

126 FINITE ELEMENT METI{ODS FOR PLUIDS

There any many applications of the incompressible Navier-stokes equa-

tions ; for examPle :

- heat transfer problems (reactors,boilers"')

- aerodynamics of vehicles (cars, trains, airplanes)

- aerodynamics inside motors (nozzles, combustion chamber"')

- meteorology, marine currents and hydrology'

Many tests f,robtems have been devised to evaluate and compare numerical

methods. Let us mention a few:

a ) The cauitg ProblernThe fluid is driven horizontally with velocity u by the upper surface of a

cavity of size o (cf. figure 1). The heynolds number is defined by : R - uaf u'

(cf. ihomasset izzo1;. The domain of computation Q is bidimensional'

-

Figure 5.1

b ) The backward steP :

Th" purabolic velocity profile (Poiseuille flow) at the entrance and at the

exit section are given, such that the flux at the entrance is equal to the flux at

the exit. The domain is bidimensional and the Reynolds number is defined as

u*l'f v where l'is the height of the step and uoo the velocity at the entrance

J1h" centre of the parabolic profile (cf. Morgan et al [174]):

l = 3 L - 2 2 o = 1 b = 1 . 5 .

Figure 5.2 The backward step problem (l'=1);

I' t

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS 127

c ) The cylinder problem.

The problem is two dimensional until the Reynolds number rises above

approximately 200 at which point it becomes tridimensional. The domain Q is

a periodic system of parallel cylinders of infinite length (for 3D computations

one could. take a cylinder length equal to 10 times their diameter ). The integral,

Q, of u.n at the entrance of the domain is given and is equal to the total flux

of the flow. The Reynolds number is defined by 3Q l2u (ct. Ronquist-Patera

[202]).

L

Figure 5.3

In this chapter, we give some finite element methods for the Navier-Stokes

equations which are a synthesis of the methods in chapters 3 and 4. We begin

by recalling the known main theoretical results. More mathematical details can

be found in Lions [153], Ladyzhenskaya [138], Temam 12281; details related to

numerical questions can be found in Thomasset L2291, Girault-Raviart [93],Temarn [228) and Glowinski [95].

2. EXISTENCE, UNIQUENESS, REGULARITY.

2.L. The variational problem

We consider the equations (1)-(2), itt Q - Qx]O,T[ where O is a regular

open bounded set in R', tr = 2 or 3, with the following boundary conditions :

I'

u( r ,0 ) - r ro ( r ) e € Q

u ( x , t ) - u r ( r ) r € f , t € ] 0 , ? [

We embed the variational form, as in the Stokes problem, in the space /r(fl) :

/ ( C l ) = { t , € H l ( O ) " : V . u = 0 } , ( 5 )

J"(o) - {u € J(o) : u lp = Q}. (6)

The problem is to f ind u e L2(o,?, /(o)) n tr* (o,T, r '(a)) such that

(u , , ,u ) + z (Vu , Vu) * (uVu ,u ) - ( f , u ) , Vu € J , (A ) (7 )

(3)

(4)

<)

Ott

r28

In (?) (, ) denotes the

is in the .L2 sense in t.

We search for u in

exist.

FINITE ELBIUENT METHODS FOR FLUIDS

u(0) = uo (8)

u - ur e L2(o,T, J"(0)) (9)

scalar product 'n

L2 in s as in chapter 4 ; the equality

We tale f in L2 (Q) and uo, ur in J(O)'

,* in order to ensure that the integrals containing uVu

2.2. Existence of a solution'

Theorem 1The problern (7)-(9)

Proof (outline)We follow Lions [153];

let us a^ssume ur = o' Let0) in .FI' with s = nf 2; thrs

< * i , ! ) = . \ ; ( t u i , u ) V u € V r .

-(the .l are arranged in increasing order, the notation ( ) represents a scalar

pioduct in % "tri f1n) the seiof C* functions with compact support ; if""

-i ,1rt]r-are the eigenvectors of the Stokes problem with q = v = 1)'

Let um be the approximated solution of (?)-(9) in the space generated by

the first nr solutions of (10) :

has al least ane solut' ion.

the outline of the proof only is given' To simplify,

{rt}, be a dense basis % of t, € D(O)" : V'u =

basis is defined bY :

(10)

(uT ,u/) + (u*Yu^ , ' f ) = (f ,"") i - L'"rn

u^ -- t 'i'id = 1 . . m

,^(0) = uo* the comPonent of uo on u)*

On multiplying (11) by u' and summing we get

( 11)

( 12)

( 13)

arf,lr^l? + rlVu^l? = (f ,u^) because (uVu, u) = 0 Vu € J,(O)' (14)

From (14) we deduce easilY that

which allows us to assert

lution on [O,T] and that

L* (O ,7, L2 (Ct)) .

1 t

I lv "^ (o)12 do 1 luo^17 + Clf lo,e (15)J "

that the differential system in {ui} (11) has a so-

solution is in a bounded set of L2(0,T,Jr(CI)) n

lr^l 'o * 2v

INCOMPRESSIBLB.NAVIER.ST0KESEQUATIONS

Let P^ be the projection from Jr(o)' into the sub-space generated

{*1 , . .w^} . F rom (1) we deduce tha t

uT = P^f -uP^Lu* - P^(u^Yu^)

r29

bv

P* being a self-adjoint projector, its norm in time as an operator from vr' in

frr, i, ler-s thun or-.q.ruf to 1. The mapp_ing p * Lg being continuous from

"rr(nl inro J6(O),, Au is bounded in L'(O,f, /o(0)'). Finally, by Soboiev

embeddings, one shows that

( 16 )

( 17)l luvul l(ro), S Cll t l l ro(o)^ with o = &

Which demonstrates that

uf bounded in L ' (O,T, Js(Q) ' ) V-. (18)

By taking a subsequence such that u^ converges to u in L2(O,T, /"(O))

weakly, i*(O,T,rt(A)) weakly* , L'(Q) strongly and such that uf" ' Ltt

in L2(b,ft io(0)') *.u(ly, one can pass to the l imit in (11)'

2.3. IJniqueness

Theorem 2

In lwo il imensions problem (l)-(0) has a unique solution.

ProofThe following relies on an inequality which holds if Q is an open bounded

s e t i n R ' , ( l d l r d e n o t e s t h e s e m i r r o r r n i n I / l : 4 * ( V d , V d � ) t l ' ) '

l lp lk . rn l S c lp lo,o* lp l r ,n i v ' € H8(o) (19)

because (trVu, tu) is then bounded by l '1, l ' l t l ' l rLet there be two solutions for (7)-(9) and let tr be their difference.

We deduce from (7) with u = u) i

So ur is zero.

Theorem 3

I! luoll is srnol I or if u is smaolh (u e L* (O'T ; '4(o)) then the solution

of (l)-(0) is uniqae in three dimensions'

Proof : See Lions [153]. l

|t,t,l l' +, l"' lo*l'dt - - l"'@o,)wdt (20)

, " I"l,l,l,l, lulldt ,, I"' fulldt * t I" fulllulld'o

130 FINITE ELEME}:T METHODS FOR FLUIDS

RemarkIt should be emphasized that theorem 3 says that if u is smooth, it is

unique ; but one cannot in general prove this type of regularity for the solution

constructed in theorem 1 ; so its uniqueness is an open problem. There are

important reports which deal with the study of the possible singularities of the

solutions. For instance, it is known that the singularities are "local " (Cafarelli

et al. [46], Constantin et al. [59]) and we know their Hausdorff dimension is

less than 1.

2.4. Regularity of the solution.

There are two important reasons for studying the regularity of the solutions

of (7)-(e) :

1. As we have seen, the study of uniqueness is based on this regularity

2. The error estimates between the calculated solution and exact solution

depend on the f/p norm of the latter. Unfortunately it is not unusual to have

u € H2 but it is rather difficult to find initial conditions which give u € H3

(Heyrvood-Ranacher [111]) so it would appear useless from the point of view

of classical error analysis, to approximate the Navier-Stokes equations with

finite elements of degree 2 because they will not be more accurate than finite

elements of degree 1; however even if there is some truth to what was said, this

conclusion is too hasty and in fact one can show, on linear problems, that the

error decreases with higher order elements even when there are singularities

(but it does not decrease as much as it would if there were no singularities)..

In the case of two dimensions, there exists general theorems on regularity,

for example :

Theorem 4 (Lions [153])I f f and f I are ;n t ' ' (91' , f ( . ,0) in L2(Q)2, uo in Jo(o) n H2(a)2 then the

solut ion u of (7)-(9) is in L2(0,r ; f2(Q)2).

As in 3 dimensions, as one cannot show uniqueness, the theorems are more

delicate. Let us give an example of a result which, together with theorem 3

il lustrates point 1.

The Hausdorff dimension d of a set O is defined as a limit (if it exists) of

log A'(e)/log(Lle) when e * 0, where N(.) is the minimum number of cubes

having ihe length of their sides less than or equal to e and which cover O.

(d = 0 if O is a point, d - Lif O is a curve of f inite length, d - Zfor a surface...

non integer numbers can be found in fractals (Feder [76])

Theorem 5 .(Cafarelli et al.[a6])

Suppose lhat f is in Lc(Q)3, I ) 512 , thatY.f = 0 and that uo.n and

ur &re zero. Let u be a solution of (7)-(9) and S be the set of {",1} such that

INCOMPRESSIBLENAVIER-STOKEStrQUATIONS

u e Ln"(D) for aII neighborhoods D of {,,t}. Then the Hausdorff dimension

in space-time of S is less than 1'

2,5. Behavior at i:rfinitY.

In general we are interested in the solution of (t)-(a) for large time because

in practlce the flow does not seem highly dependent upon initial conditions :

the flow around a car for instance does not really depend on its acceieration

history.There are many ways "to forget" these initial conditions for a flow ; here

are two examPles :

1. the flow converges to a steady state independent of t ;

2. the flow becomes periodic in time'

3. quasi-periodic florv : the Fourier transform f *-+

arbitrary point of the domain) has a discrete spectrum.lu ( r , t ) l (where r I s an

4. Chaotic florvs with strange attractors: t -"+ lu (c , t ) l

1 3 1

continuous spectrum and the Poincar6 sections (for example the

{rr(", nk),,u2(x,nlc)}' for'a given r ) have dense regions of points

complete zone of spuce (in case 1, the Poincar6 sections are reduced to

*h"r n is large, in-cases of 2 and 3, the points are on a curve).

has apoints

filling aa point

In fact, experience shows that the flows pass through the 4 regimes in the

order 1 to 4 when the Rel,nolds number Re increases and the change of regime

takes place at the bifurcation points of the mapping Ll --'+ u, where u is the

stationary solution of ( 1)- (a). As in practice Re is very large (for us z is small) '

4 dominates. The following points are under study :

a) Whether there exists attractors, and if so, can we characterize any of

their properties ? (Hausdorff dimension, inertial manifold,...cf. Ghidaglia [89],

Foias et at.[zs] cr Berg6 et al [29] and the bibliography therein).

b) Does u(x,t) behur'. in-astochastic way and if so, by which }aw ? can

we deduce some equations for average quantit ies such ff i u, lt, l l ' , lV x ul2"'

this is the problem of turbulence modeling and we shall cover it in a little

more detail at the end of the chapter . (cf Lesieur [150], for example and the

bibliography for more details).

Here are the main results relating to point a)'


Consider (7)-(9)with up = 0, f independent of t, and O a subset of .r?2.This system has an attractor whose Hausdorff dimension is betwen cfuealsand CRez where Re = t/f diam{A)/v (cf. Constantin and a1.[59], Ruelle

[206][207]). These results are interesting because they give an upper bound forthe number of points needed to calculate such flows (this number is proportionalb v-sla).

In three dimensions, we do not know how to prove that (7)-(9) with thesame boundary conditions has an attractor but we know that if an attractorexists and is (roughly) bounded by M in W1'- then its dimension is less thanCMzlau-sla (cf . [10]) .

2.6. Euler Equations (z = 0)

As v is small in practical applications, it is important to study the limitu * 0.If. u - 0 the equations (7)-(9) become Euler's equations. For the problemto be meaningful we have to change the boundary conditions ; so we considerthe following problem :

u t * u V u * V P - f , V . u = 0

u(0) = uo

u.nlr - I

"(r) - ur(c), Vs € X = {u € | : u(c).n(x) < 0}.

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

if x e x. (2e)

Proposition 1If Q is bidirnensional and simply connected, if f e L'(Q), u0 € ff1(O)2,

g = ut.n and (uy); are in Htlz (t) and the integrat of g on f ds zero, then (21)- (2 i l has a unique solut ion in L2(O,T ; Hl(O),)n L*(O,T,L2$t)2).

Proof (sketch )To simplify, we assume f = 0,g = 0. Since V.u = 0 there exists a r/ such

that :

U l = $ r Z r U Z = - t r t

By taking the curl of (21)-(22) and by putting

uL,z -rtrz.,1 = -Al

We see that

,(o) ::*:::;, -j' . ,,,(r) = c.rp(r) (derived from g and up I here c..,s, is zelo)


If u is continuous, Lipschitz and g is zero, (27)-(23) can be integrated by thn

method of characi"rirti." (cf. chapter 3) and o; has the same smoothness as

ii", "f V x u0. So we have an estirnate which relates llrll' to the Hoiderian

norm C of u (Kato [129])

l l , l t S c ( l l " l l + l l v "o l l )

Moreover, from (25)-(26), (23) with I = 0 gives

dl r = o ,

Thus from (26), we have

l l r l l '+ , S Cl l r l l ' .

(30)

(3 1)

(32)

So it is sufficient to construct a sequence in the following way :

u' given * cr,rn*l calculated by (27)-(29) .c.,n+1 gl,o"n 9 ,n*1 calculated by (25),(26):(3q)

and to use the eJtimates (30) and (32) to pass to the limit'

A complete proof can be ,""r, itt Kato [129], McGrath[170], Bardosflo]'

Remark 1In three dimensions we only have an existence theorem in the interval

lO,rl,r small. In fact, equation (27) becomes

w 1 luYw - r ' rVu = 0

which can give terms in elt where .\ are the eigenvalues of vu'

A number of numerical experiments have shown the formation of singular-

ities after a finite time but the results have not yet been confirmed theoretically

(Frisch [84], Chorin [52], Sulem [225])'

Remark 2\Ye knorv little about convergence of the solution of the Navier-Stokes

equation t,owards the solution of buler's equation when t/ -1 0' In station-

arl ''laminar" cases, one can prove existence of boundary layers of the form

,:t ti; ti b"i1g the distance to the boundary) in the neighborhood of the walls

(Landau-Lifschitz [141], Rosenhead [203]) ; so convergence is in 12 at most in

those cases.

3. SPATIAL DISCRDTISATION

3.1. Generalit ies.

The idea is simple: we discretise in space by replacing J"(O) by J,1, in

(7)-(e):

L s 4 F I N I T E E L E M E N T M E T H O D S F o R F L U I D S

Find un € Jl, such that

(uo,r , r r , ) + (u6Vu6, un) * u(Vu6, Von) = ( / , ' i , ) Yun € Jon (33)

"n(0) = u?, uh - ur^ € Jon (34)

lf Jon is of dimension N and if we construct a basis for Jon then (33)-(34)

becomes a nonlinear differential system of N equations:

A{J' + B(tl,U) + uCU - G. (35)

One could use existing library prograuls (tINPAK[152] for example) to solve

(3b) but they are not efficient, in general, because the special structure of the

matrices is not used. This method is known as the "Method of Lines"

So we shall give two appropriate methods, taking into account the following

two remarks :

1" ) If u )) 1 (33)-(34) tends towards the Stokes problem studied in Chap-

ter 4.2") If u = 0 (33)-(34) is a non-linear convection problem rn Jorr. In par-

ticular in 2-D the convection equation (2?) is underlying the system so the

techniques of chapter 3 are relevant.

As we need a method which could adapt to all the values of v we shalllo

take lhe finite elements in the famity studied in the Chapter I and the method

for time discretisation adaptetl to conuection sludied in Chapter 9. But before

that, let us see a convergence theorem for the approximation (33)' (35).

3.2. Error Estinate .

Let us take the framework of Chapter 4: J(A) is approximated by

Jn = {u1, € V1, : (V.rr, , qn) = 0 Yqn € Qn.| (36)

and "Io(Q) is approximated bY Jon - Jn fl I/ot(Q)". We put :

Voh = {rn e Vn t ,nlr^ = 0}. (3i)

Assume that {Vr1' ,Qn} is such that- there exists Tl1, : V] = H2(O)" n Hl(A)" + Voh such that

(sr , , V. ( , - t l6u)) - 0 Yqn e Qn l l , - I l r ,u l l S Chl l r l l r ,n . (38)

- there exists 11 : HL(Q) * Qr, such that

lq - or ,q lo,o ( Chl lq l l t ,n (39)

INCOMPRESSIBLE NAVIER.STO KES EQ UATIO NS

inf sup -(9'o'qn) \ aqr,Eex vxevoh lqn*1ffi

t '

nll 'nllt S f l 'r,lo V'r, € I/q,, (inverse inequality)

135

(40)

(41)

Remark :These conditions are satisfied by the Pl + bubbte/Pr element and among

others, the element Pr iso P2 f PL.

To simplify, Iet us assume that

u o = 0 a n d u f = 0 ,

Theorem 5. (Bernardi-Raugel [31])a)n=2. If the iolution o1 fl 'p)-is i i ,n L'(A,f ; H&(A)')nC0(0, T; Le(g)'),

q > Z, then problern (33)-(3/) has only one solutisn for h small satisfying

l l " - ul l l7zp,r;ai(o)) ! C(v)hl[uVull6,q

b) n=3. I f the nornl of v-ru in co(O,r ; , r3(a)3) is smal l and i f u €

Lr (0 ,7 ;H2(A)3) n .A1(O,T;L2(Q)3) then (3 i l -06) has a un ique so lu t ion w i th

lhe same errvr est imate as in a)-

Proof (sketch)The proof is technical, but it is based on a very interesting idea (introduced

in Brezzi et al [38])We define the operator

G ; u - . G ( u ) - u Y u - f . ( 4 2 )

\\'e observe that the non stationary Stokes problem

u,1 -u ! 'u * Yp - g , V .u = 0 , u (0 ) = Q, u r = 0 (43 )

defines a linear map .L : g + u and that the solution of (7)-(9) is nothing but

the zero of u -* u * LG(u) :

u solution of (Z)'(9) <+ .F(") = u + LG(u) - 0' (14)

Similarly, for the discrete problem \f Ln is the nonstationary discrete Stokes

operaior underlying (33)-(3a):

u solution of (33)-(3a) <+ Fn(") - uh * L6G(ur,) = 0' (45)

To prove the existence of a solution for the discrete problem, we note that

the derivative of Fl, with respect to u, F'1r(u), is a linear map (because F;' is


quadratic) and that it is an isomorphism if u6 is near to u. Then we see that

the solution is also a fixed Point of

p + 9 1 1 ( a ) = o - F i ( r u ) - t F r ( r ) ( 4 6 )

for all ?r; rr€&r to u. For rp'^ - I - F'(r)-tF'(r) in the neighborhood of 0

is a contraction and from a fixed point theorem we have the existence and

uniqueness of the solution.

To obtain an estimate, we calculate the following identity :

l l" - ,rr,l l = llr;tlri.(" - 'o) - (rn(") - Fn(rr,)) + r'11u;111 (47)

7 L

S llr;-tt/ lr;t,ro) - Fi,@n * 0(u - "a))1.(" - u1)d0l+ rr,(u)ll

S llri-'l l[ 'up llri("a) - Fi,(r)lll lu - u6ll + llrr,(")ll]

So. for small h we have :

l l " - rn l l <C l l r r , (u ) l l

Moreover

Fr, (") = u + L1,G(u) - u * LG(u) * (Ln - L)G(u) = (Ln - L)G(u) (49)

but f l(Ln-f,)G(u)ll is precisely the error in the resolution of the non-stationary

Stokes problem'wiltt g - G(u). Elence the result. For more details, see [ ][5J

[16,rV$3]

Remark 1The proof of theorem 5 suggests that the usual inf sup condition on the

conforming finite elements in the error estimate for the non-stationary Stokes

problem is necessary for Navier-Stokes equations.

Remark 2 .The error estimate of theorem 5 shows that h should be small when v is

small. We note that lluVull depends also on v. For example in a boundary

layer in r-v/{ ',

(48)

l rluVuls = (

J: ,-r "- # aill, = o1r- 11 (50)

since c(") ^, 0(z), we should take h << urla. The same reasoning leads to

h << vs'18'ifwe want lu--uolr,e (( 1. We note that the numbers are less pes-

simistic than those given by the attractor argument. This is not contradictory

as there must exist attractor modes of modulus less than u-trla

4. TIME DISCRETISATION

4.1 Semi-explicit discretisation in t

r37

Let us take the simplest explicit scheme studied in Chapter 3, that is the

Euler scheme.

Seheme 1 :Find ul+ t with ul+t - urt € Jor, such that

1 , .

i { " ;* t , , r ) = s(uh) vun € Jor, (51)

g(ui ) - -u(Yqi, Vrr ,) + ( /" , ur) * i , " t , un) - (u[Vu[, u6) (52)

Bvidently, /c has to satisfy a stability condition of the type

IN COMPRESSIBLE N AVIER-STO KES EQUATIONS

k < C ( u , h , u )

As for the Stokes problem (51) is equivalent to

f t" l*t, un) * (Vpo, ?,r) = g(uh) vur, € voi,

( V . r l + t , 9 n ) = 0 Y q n € Q n ;

?tri*' + (Vpi , rd) = e(rd) * 7r": -

l{ui,") vi

( V . r l * t , g r ) = 0 Y q n € Q n " ; + t ( t ) - D " t ' i ( " ) '

We can now substitute u1 in (57) and we obtain

(53)

(54)

(55)

(56)

(5 i )

E L

f I(vph , ui)(v qh,ai) - G(qn) vt.- o i

(58)

that is a linear system of a Laplacian type.

The main drawback of these types of methods is the stability (53)-

There exist schemes which are ahnost unconditionally stable such as the

rational Runge-Kutta used by Satofuka [212]:

To integrate

V,t = F(V) (be)


we use

Scheme 2

vn+t = vn + [2s1(g1,93) - g3(g1 , 91) ] (g3 , 9 t ) - t (60)

where (.,.) represents the scalar product in the space of lz(t) and

91 = kF(v" )

9 2 = k F ( V " - c a 1 )

s 3 = b g t + ( L - b ) g '

This method is of order 2if Zbc - -1 and of order 1if it is not. For a linear

equation with constant coefficients, the method is stable when 2bc 1-1 (for the

proof, it is sufficient to calculate the amplification coefficient when it is applied

to (59) and F(Y) - FV where F is a diagonal matrix). For our problem, we

must add a step of spatial projection on zero divergence functions of Jon. That

is, we solve :

#"?*'* (vpr, ui) =

#ui*' Yan € von (61)

(V . r l * t , Qr ) = 0 Yqn € Qn (62)

Numerical experiments using this method for the Navier-Stokes equations dis-

cretised with the PrisoP2 fPl finite elements can be found in Singh [215].

4.2. Semi-Implicit and Implicit discretisations

A semi-implicit discretisation of O(k) for (3a)-(36) is

Scheme 3:

1 1

, { "T* t ,an ) * u (Yu i+ r , vu r )+ ( r [ vu [+ t ,a r , ) = ( f "+L , r r , ) *

7 fu t ,on ) (63 )

A version O(k') is easy to construct from the Crank-Nicolson scheme. Thisscheme is very popular because it is conceptually simple and almost fully im-plicit : yet it is not unconditionally stable and each iteration requires thesolution of a non symmetric linear system. Thus on the cavity problem with10 x 10 mesh and k = 0.1 the method works well until u x L1500, beyond whichoscillations develop in the flow.

Let us consider the following implicit Euler scheme of order O(k) \

Scheme / :

1 . .

i { " ; * t , uh)+v(v t ;+ t

Vrr, € "Iol. This scheme

system.

INCOMPRESSIBLENAVIER-STOKESEQUATIONS 139

, von)+(,r f ,* tvr f* t , un) = ( f^+t, un)* i , " t , o1) (64)

is unconditionally stable but we must solve a non-linear

4.3 Solution of the non-linear system (59) '

This non-linear system can be solved by Newton's method, by a least

square rnethod in I/-1 or by the GMRES algorithm:

4.3.1 Newton's Method

The main loop of the algorithm is as follows :

F o r p - L . . p M a c d o :1. Find 6ur wi th

(hun, r^ ) ; t u (Y6un,Yrn)+ ( r l * t 'pY6u7, * 6u1,Yu|^* t '0 , ? ' ,h ) = (65)

- { ( r l * t 'P ,un) l * , r1vu;* t ' ' , V,n)

- ( f ' + t , uh ) - 1 r i , r n ) f l Vun

2. Put

+ ( r l * t ' oVu l+1 'P , 'n )

e Jon 6un e Jon (66)

uf,*r,n+r -u;+t 'P +6un (67)

But here also, experience shows that a condition between k, h and v is neces-

sary for the stability of the scheme becaus e if u is very small rvith respect to

h, problem (bg) has many br-anch of solutions or because the convergence con-

ditions for Newton's method (hessien ) 0) are not verified. Again fcr a cavity

with a 10 x l0 and /c = 0.1 we can use this scheme ti l l Re-1000 approximately.

If the term ufluis upwind.ed, one can get a method which is unconditionally

stable for ali mesh ,t ; this is the object of a few methods studied beloH'.

4.3.2 Abstract least-squares and 0 scheme'

This methcd has been irrtroduced in Chap,ter 2, Paragraph 4.2. It consists

here of taking for uf;+l the solution of the problem

min - { [ fuo - whl2dc : uh -ur € Joh,w 1 - u y Q , J o l J A

{68)

L40 TINITE ELEMENT METIIODS FOR FLUIDS

Figure 5.4 : Streamline of a bidimensional f low in a caaity. The compu'

tatian ias done with the PlisoP2/Pl element at Reynolds 50A with a uniform

triangulation of /A0 :riangles and abslract least squares wilhoul upwinding.

(Courtesy of AMD-BA) .

1 . , - a - r - l 1

l@n,u r , ) * v (Vu1 , ,V r r ) * (u rVu . ' r ,un ) - ( . f "+1 ,on) * n@' i , ' n )

Von € ' / o r , ]k '

The solution of this problem by a preconditioned conjugate gradient method,

solving three Stokes problems per iteration (calculation of the step size in the

descent directions is analytic because the optimized quantity is of degree 4).

This method has good stability qualities. Only symmetric linear systems are

solved but it is more costly than other methods ; in Bristeau et al [aa] a set of

numerical experiments can be found.

To decrease the cost Giowinski [96] suggested the use of a d scheme because

it is second order accurate for some values of the parameters and because it

allorvs a d.ecoupling between the convection and the diffusion operators.

If B is the convection operator ulV and A is the Stckes operatol the d

scheme is:

h@"*t - 'u,n) + Au"+o + Bun = fn*t

1 1r,n*r-0 - un*e) + Au"+t * Bun*r-e - / .n*t- |

@7;;1'\*

h@"*t - un*L- l ) + Au"+, + Bun*1-0 = y+r

r; :OMPRESSIBLE NAVIER-STOKES EQUATIONS

Figure 5.5 : streamlines for the backward slep problem with the P!;'Pl

elemeni, air.racl leasl squares without upwitt,ding at Reynolds 191. (Coun'sy

of AMD-8.t,:

So the least .quare method will be used only for tire central step while the rwo

other steps ?-:€ generalized Stokes problems. Alternatively, the convection step

can be soli ' : i by other methods such as GMRES'

4.3.3 The GMRES algorithm.

The GlrfRES algorithm (Generalized lMinimal RESidual method) is a

quasi-Newrr-,ri3n method deviced by Saad [210]. The aim of quasi-Newtonian

methods is t', solve non-linear systems of equations :

F ( u ) = 0 , u € E N

The iteratir-e procedure used is of the type :

un*1 - t l ,n - ( / " ) - t F(rr ' )

where Jn ts an approximation of F'(un). To avoid the calculation of F'the

following approximation may be used:

' r ' l

. = F ( , * 6 r ) - F ( r )D 6 F ( u ; a ) -

6

As in the conjugate gradient method to find tlte

considers the Krilov spaces :

I {n = SP{ 'o ' J ro " ' ' J"-"0}

where r0 = -F - Jao, io is an approximation of the solution to find. In

GMRES the approximated solution o' used in (69) is the solution of

,.fti" llro' - J (, - ,o) ll.

= F ' (u)u. (69)

solution u of Ju - -F otre

T42 FtrNITE ETEMENT METHODS FOR FLUIDS

The algorithm below generates a quasi-Newton sequeace which {t "} hopefully

(it is only certain in the linear case) will converge to the solution of .F (u) = g

in .RN :

Algorithm (GMRES) :

0 . Initialisa,tion: Choose the dimension & of the Krylov space; choose

u6. Choose a tolerance e and an increment 6 , choose a preconditioning neatrix

^ 9 € E N * N . P r r t n = 0 .

1. a. Compute with (69) r| - -S-t(f; + Jnun), *rn = ,ll ll""ll, where

Fn - F(u") and /,,u - DtF(u"ia)-b. For i = 2...,,t compute r{ and ud from

j - 1

4 - s-t lD6l(u,;,4-') - Ihi;- 'rLli = l

uf^ - -"r .l l / " l l

where h?,i = *f S-t DaF{uniu)r^)2. Find un*t the solution of

min l lS- 'F(r ) l l 'u=u'+fi c;u, ' i

k

O t r O 2 r . . . A k J t = 1

3. I f l lF(""+t) l l < t stop else change n into n*1.

The implementation of Y. Saad also includes a back-tracking procedure{or

the case where u,, departs too far away from the solution. The result is a black

box for the solution of system of equations which needs only a subprogram to

calculate r'(") given u; there is no need to calculate F'. Experiments have

shown that this implementation is very efficient and robust.

5. DISCRETISATION OF THE TOTAL DERIVATIVE.

5.1. Gcneralities

Let us apply the techniques developed in chapter 3:

u ,1 luYu- l (u " * r - unoXn)k '

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS 143i

we obtain the following scheme :

Seheme 5:

1 . . - r r I

f t " ;* t , r r ) + z(Vu[+l , Vrr ,) - ( f " , ur) * 7@ioXi,rn) Yan € Jon (70)

where Xt (r) is an approximation of the foot of the characteristic at time n,b

which p*t.t through a at time (" * 1)& (Xf (t) = r - uf,fr)'

We note that the linear system in (70) is still symmetric and independent

of n. If Xf is well chosen, this scheme is unconditionally stable, and convergent

in 0(e + h) .

Theorem 6 :rf

xf (0) c Q, ldet(Vxf )-' l < t + Ck,

then scheme 5 is unconditionally stable.

Proof :This depends upon the change of variable y - Xi@) in the integral

I owtx))dx - t -.$@)d"t(v xi)-t dv (72)Jn Jx;(n)

From this we obtain

luioxilo,n ( lder(Vxfi )- t l*,t,l,rl lo,n (73)

In getting a bound from (70) with vh =,r;*t and by using (73) and hypothesis

(71), we obtain

l lr l* ' l l ' , = lr l* '13 + r"lv ui+tl3 S (kl/" lo + (1+ Ct)l" i lo)l" l* ' lo (74)

Since lr i , lo < l l rol l , , we obtain

l l r l * ' l l , s * l / " lo + (1+ cf t) l luf l l " ; (75)

or, after summation,

l lr[+' l l" s kt l / t lo(r + ck)"-i + l l"f l l l ,(1 + ck)" (76)i ( n

and hence the result follows with n Sf /k.

Remark :If u[ - V x t6 with $n Pr piecewise and continuous and if Xi is the

exact solution of

(71)

.,ttr

t44 FINITE ETEMENT METHODS FOR FTUIDS

t t , t , r , , '

I

,7

It

v

P

t

L

1 - a - a 4 a a a -

The aortet behing the car'

Figure 5.6 : 3D f"ow around a car

with the Plbubble/P1 element al Reynolds

50A. The f gures show the trace of the mesh

on lhe ear, the uelocil ies an a plune per'

pandicular to the fl,ow behing half the car

and the prvssare on the car at the plane of

sgmetry. (Computed bY F. Eecht).

frx^(') = u[(-x1(r)) (77)

then (71) is satisfied with C = 0. It is theoretically possible to integrate (77)

exactiy ,irr." X1 is then a piecervise straight line but, in practice, the exact

calctrlation of the integral (uio Xt,un) is unnecessarily costly ;we use adirect

Gauss formula (Pironneau [190])

(u ioX 1, , uh) = f "n(X"({ t ) ) ,n(4t )oot

or a dual formula (Benqu6 et al. [22])

(u iox7 , ,uh) = - f "n ( { t ) ro (x f

- t (€ t ) ) t t t

5.z.Emor analysis wi th "1 OtOOle/P1 '

Definition of the method :

To simphfY, we assume that u; - 0 ; we choose

Joh = t r r , g Von: (V.r r , ,Qn) = 0,Vgi , gQn\ (80)

voh = {ro e C'(a) : anlr € P1 ,YT eT|} (81)

Qn = {qn e C"(o) .qh l r € P1,vr e Tr , } ' (82)

where Tr, is a regular triangulation of Q ( assumed to be polygonal) and,Tn17

is the triangulatLn obtained by dividing eacb triangle (resp. tetrahedron) in 3

(resp. 4) bt joining the center of gravity to the vertices (fig. 4.2).

The pressure on the car

at the mid-plane.

(78)

(7e)

t I I I / / ' r r t

l:Ti'W1i

INCOMPRESSIBLE NAVIER.STOKES EQUATIONS L15

we defin " xt(r) * the extremity of the broken line [xo - x,Xt,...,X**r

- Xtr(r)i such thai ihe xi arcthe intersections with the sides (faces), Xj-1 is

calculated,from xj in_the direction -"T(xi) and the time covered from c to

,m*r is b :

For each i l ,p > 0 such that 1j+t = xi * p"i(xi) l t , e ATi,

l x i + t - x i l _ ,lu iGm--

(83)

(84)

In practice, we use (78) or (?9) with the 3 (resp. 4) Gauss points inside the

triangles, but with this integral approximation, the error estimation is an open

probil*; furthermore (71.bJ ir ntl easily satisfied, so we shall define another

xt .

Definition of X[ satisfYing (71) :

Lef {ti be the solution of

(V * , l t t ,V * ,o) + (V.r / i ,Y.wn) = ( " f ; ,V X tu l ) ,Yw6 €Wn, (86)

,! t e Wn - {*n e V6 | 1t)1x nlP = 0' w7.nd7 = 0) (8i)

where f; are the connected components of f . Let us assume for simplicity that

Q is simply connected. This problem has already been studied in paragraph

5 of chapter 2 (here however wt X n is zero on I instead of tll x nn ; Wn rs

,ron "-jty because I is a polygon . We have seen that (86)-(87) has one and

only onl solution and that V * r/ti = ui. Since y x tpt.l.gt::..*ise constant

and with zero divergence in the ..ttt. of dittt ibutions, (83)(84) enables us to

compute an analyticlsolution of (77) if we replace ,3 by v x /l ; then (71) is

satisfied.So we shall define x'i u a solution of (77) with vx {f;, and we will

assume that (uf; " X't, o;,) is calculated exactly'

Theorem 7 :

Assume Cl conl)er, bound,ed, polygonal and, u € wr-'cF(flx]0,?'[)t n

C" (0 ,7 ; I { ' (0 ) t ) , p € Co(0 , T ;H2(O)) . Le t {u i ( , ) } " be the so lu t ions inVon

of

1 . - r l I ' n t r l n \ L J -

I@"*16,un) * u(vui+t, Vrn) - ( f", ur,) * i t" ;Pxi ,un), Yun e von (88)

where x,i is calculated by (77) with ui replaced by Y x ,hl and tli solution of

(s6), (57).' , Tt rn if u is the eracl salxtion of the incompressible Nauier'Stskes equa-

lions, (1)(/,) with uv = 0,we haae

t,,


Figure 5.7 : 3D fl,ow below a lrain between the wheel aiis; computed withthe PI bubble/Pl element at Reynolds 1000. The r$gures show lhe lrace of themesh on the solid boundaries and lhe aelocities in a plan section parallel to thetrack. (Computed by C. Paris)

r t t t l t r t t l t t t t

l l l l l l

-u ( . , , ,e )13) * sc l++h+e l (8e)

of u, affine in llullr,llpll" and erponential

(l" i - u(.,nk\lf l+ unppT

where the conslant C is independentin llullr,oo.

Remark 1 :Note that (89) is true even with v -* 0 (Euler's equation) but it is necessary

that the solution of (1)-(+) remain in I,T/1,* when u -, 0 (which assumes that

/ depends on z). Numerically, we observe that the method remains stable evenwith u - 0 so iong as the solution of the continuous problem is regular.

Remark 2 :The hypothesis 'O convex' can be relaxed. It is only necessary in order

that (r/ - ' lV x ,lt l2 + lV .nl, l ')1/2 be a norm.

To prove (89), we shall follow these three steps :-es t ima teV x r l t t - " i- estimate Xt - Xn- estimate ui - u' where u' is the value of the exact solution at time n/c.Before starting, we note that from (1)-(4), we can deduce

IIiCOMPRESSIBLE NAVIER-STOKES EQUATIONS L4J'

lun*, -u, \ ,un*l yon*l = fn +!r"oxn +lc lu,rr l*0(1), V.r '* l = 0 (90)k

uo given, un*Llr = 0 (91)

where 0(1) is a function bounded independently of & and z.

Lemma 1:

lxr - X"lo < keklVr"l* lv x $t - ," lo (92)

Proof :From the definitions of Xn and X, we have

* r * ^ -X ) = V x tn (Xn) -u ( r ) = Vx ,hn (Xn) -u ( r r , ) *u ( r1 ) - " ( ' ) ( 93 )

and so

t

filx^ - Xl,(") S l(V x $n - u)oX1,l" + lu(r1) "(')1, (e4)

S lV x Qn - u lo* lVu l " " lXr - X l , ( t )

Since xt(t) = x(t) = s, we deduce (92) from the Bellman-Gronwall inequalit l '

(by the integration of (9a) on the interval ]nk,(n+1)k['

Lemma 2 :

lV x t / r - , lo S Cl" i - r ' lo * hlVu"ls (95)

Proof:Let t" be a solution of

( V x , h " , Y x u ) + ( V . t / " , V ' t ) = ( ' " , V X w ) , Y w € W ( 9 6 )

* '€ W =t r € I { t (O) " : u x n l r = O, I r rw 'nd 'y=

0} (9 i )

By subtracting (96) from (86)' we see

(V x (r l ' i , -r | ,"),V >< rrr) + (V.(r/ [ - ' l t") ,Y'wn) = ( ' f ; - u'n,V x u' ' ;) (98)

Let {;, be a projection -of $" in wn, that is, the solution fit wh of i


(V x €i,, V x tur,) + (V.{n, V.rh ) = (V x rt)", V X .r, ) * (V .rlr", V.tun), Ywn € Wn(ee)

By takingun = €n-r/1t in (99) and by subtracting (86), we see that

lV x ({i - t l , t lo < Clun - u*lo (100)

Finally, we obtain the result because u is equal to V x r/ and

lv x (/ [ -d)lo S lV x (d,f -€n)lo + lV x ({r, - r / ,) lo < C(l"n -ulo* hlVu"ls)(101)

the last inequality comes from (100) and from the error estimate between (1,

and rl (the error of projection is smaller than the interpolation error).

To have existence and uniqueness for (86)*(99), the bilinear form should

be coercive, which we can prove only for convex polygonal domains (Girault et

al [e3]).

Corollary 1 :

lxf - x"lo s ec(lv"l""xl"i - Ltrnlo + hl (102)

Proof of Theorem 7 :We subtract (90) from the variational form of (88) :

( r l * t - un*t , rn) * uk(V(u[* t - , ' * t ) , Vur,) + e(V(pl* t . pn+1), rn) (103)

- (u iox t - u ,n oXn, un) * k ' lu , r l " " (0 (1) , u1)

Then we proceed as for the error analysis of the Stokes problem (cf. (4.74)-

(4.78)). From (103), we deduce that for all wn € Jon and all e4, € Qn, we

have

("1*t - wh,on) + vk(Y(ui+ ' - *o), Vrr , ) + &(v(p[+l - en),a6) (104)

= (un*1 - wh,un) * vk (v (u"+1 - ro ) ,Yan) + t1v1pn*1 - Qn) ,a l )

+(uioXi - un oXn, or,) * k' lu,rl*101t;, u;),Ya1, € Jon

So (see (7a) for the definition of ll.ll,)

l l" l*t -uil l , < l lr"+t -*nll,+klv(p"+l -gr,) lo+luioXi un oXnlo+k2lir.]-( 1q5)

which gives if we choose Q6 eQual to the interpolation of Pn+1

INCOMPRESSIBLE NAVIER-STOKES EQUATIO I\S 149

tii.

l l r [ * , -u . * I l l , < l l r l * t - rnh l l , + l l * r , -un*L l l , S 2 [ [un+t -whl l ,+k l ip"+t l l '(106)

+lui oXn - unoXnlo * &21' ,r l*

Taking uh as equal to the interpolation of un*t ' tlle see that

l l r l* t - un*111, < cl(h'* vhk)l lr"+t l l ,

+hkllp"+tll, + lr' lu,rl* * luioxi - u" oX"lo

it remains to estimate the last term ;

(107)

luioxi - LtnoX'[o s l"f, oxt - u"oxf,lo * u'o6r - X")lo (108)

S l ,r i - tr ,nl"0 + lvr" l- lxf - Xnlo S (1 + Ck)l l" i - '" l l ' +C'kh

where C and C' depend upon lVrl- (cf. (102))' BV using a recurrence argu-

ment, the proof is completed because (107) and (108) give

l l r l * t - un* t l l , S (1 +Cr f ) l l " i - ? tn l l , + C r (h '+hk+k2 ) (10e)

6. OTHER METHODS.

In practice, all the methods given in chapter 3 for the convection-diffusion

equation can be extended to thJNavier-stokes equations' Paragraph 5 is an

"*u*pl" of such an extension. one can also use the following :

- Lar-\\tendrofi artificial viscosity method : horvever it cannot be com-

pletely explicit (sanne problem as in $4.1) unless one adds Pt in the divergence

"qnution (T.*u- t228], Kawahara [131]);- the uprvinding *.thod by disctntinuity ; this could be superimposed with

the other methojs oi s+.r und of $4.2. In this way we can obtain methods which

converge for all " 1.f. Fortin-Thlmusr"t [83], for an example of this type);

- slreamline upwinding method (SUPG)'

6.1 suPG and AIE (Adaptive Implicit/Explicit scheme)

As in paragraph 4.3 in chapter 3 a Petroa-Galerkin variational formulation

for the Navier-stokes is used with the test functions un * ru1,vu1 instead of

a h i


(uo,r* unvun * vpr, ,un * ru6Yu1,) * v(Yuv,,Yun) - ut, [^ ru11va6a,u1,I " t i

- ( f ,ay { ru lVu6) Vr l € Von(Y.un, { t ) = g Yqn € Qn.

Here r is a parameter of order h, Tt is an element; we have used the simplestSUPG method where the viscosity is added in space only. Johnson [125] righttysuggest in their error analysis the use of space-time discretisation (see 3.4.3) .

A semi-implicit time discretisation gives the following scheme:

(ui* ' r - uT

* uivuf+r, urr * ruf ;Ya.,)+ (vpl+' , ur,) * (vp[ , ruf ,Yu')t f r

*u(Vui+t, Vun) - r+ lr,ruftVulAuf

= (,f"+1 ,at * ruf;Yu6) Vur, € Vor

( V . r l * t , 9 r ) = o Y q n € Q n .

As noted in Tezduy ar 1227), it is possible to choose r so as to have symmetriclinear systems because the non-symmetric part comes from:

^ , n * L

(h , rufv 'n) * (uf ;Vul+t , 'o)

Now this would be zero if r = h and if uiVui*l was equal to V.(u[ g "l+t).So this suggests the follorving modified scheme:

Find ,i+t - ur € Vor, and pf+l e Qn such that

,uf,+t - u?(# * uiV'l*t ,an * kuftYul) + (v'uf; ,uf,+t ''n)

+(vpl+t,ur,) * (vpi,kuftyu1) + z(vuf,*t,vro) - r+ lr,nurou1,auft

= ( f "+t ,0n * ku iYul r ) Yan €Von

( Y . u l + t , Q n ) = g V q n € Q n .

Naturally a Crank-Nicolson O(k') scheme can be derived in the same way.Similarly a semi-explicit first order scheme couid be:

( I * ' - 'uT * u iYui ,uh * ku iYal , ) + (vp[+ ' ,oo)+ (vp i ,kuf ;vr r , )' k

*u(Yui , Vr t ) - rU, I kuf ,Yul ,Aut = ( fn , rn * ku iYul , ) Yan € VonT JT,

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS 1 5 1

(v . r l * t , gn) =-o Yqn e Qn.

Still following Tezduyar [227] to improv'e the computing time we can use

the explicit scherie in regiorrr *h..e the local courant number is large and

the implicit scheme when it is small; the decision is made element by eiement

based on the local Courant number for convection, l"i ln kf fu and for diffusion

vk fhl, and on a measure of the gradient per element size of uf; ; here hr is the

average size of element l- So we define n numbers by

pl(q- max{t,itr, * - t,(ffi) F twr - .}

where Uj (resp ul) is the maximum (resp minimum) of u1;(c) on element n

and similarly for U!,"? on Q instead of fi'

The constant c is lhosen for each geometry; then lt Bt;(c) ) 0 the contri-

bution to the linear system of the convection terms on element I in the i - lh

momenrurn equation is taken explicit (with uf, instead of u[+1) and otherwise

implicit. similarly lf vklh! is less than one, the contribution to the matrix

of the l inear system from v frryun,Yan is taken with uT, - uf, (explicit) and

with ui+1 otherwise.

7. TURBULENT FLOW SIMULATIONS.

7.L. Reynoldst Stress Tensor.

As we have said in the beginning of the chapter, there are reasons to

b e l i e v e t h a t w h e n t . + o o , t h e s o l u t i o n o f ( 1 ) - ( + ) e v o l v e s i n a s p a c e o f d i m e n s i o nproportional to v-sl+.

For practical applications z (= l/Re) is extretnell 'small and so it is nec-

essary to have " lur!. number of points to capture the limiting solutions in

t ime.we formulate the following problem (the Reynolds problem) :

Let ui, be a (random) solution of

u , t * uYu* Vp - v \u = 0 , V 'u = 0 in Ox ]0 ,? [ (110)

u (c ,0 ) - uo (x ) I w( r ,u ) , u lp = u r ( 1 1 1 )

n,here *(r,.) is a random variable having zero average.

Let ( ) be the dverage operator with respect to the law of u introduced

by r .u . Can we ca lcu la te 1u ) , 1u I u > " ' ?

This problem corresponds closely to the goals of numerical simulation of

the Navier-stokes equations at large Reynolds number because, when v 1{ L',

u is unstable with respect to initial conditions and so the details of the fibw

152 FINITE ELEMBNT METHODS FOR FLUIDS

cannot be reproduced from one trial to the next. Thus, it is more interesting

t o f i n d . o n e m a y a l s o b e i n t e r e s t e d i n .Only heuristic sslutions of the Reynolds problem are known (see Lesieur

[150] for example) but let us do the following reasoning :

If we continue to denote by ,, the average < u'. > and by u' the difference

u'*- < u!* ), then (110) becomes :

u,t * uY u* Vp - u L,u+ V. u' @ u' = - (ut,r + ut Y u * uv ut + vp' - v a'u' ) ( 1 12)

V . u ' + V . u = 0 ( 1 1 3 )

because uVu = V.(u @ u) when V.u = 0.If we apply the operator (.) to (112) and (113), we see that

u , t * uVu* Vp - u ! ,u+V. = 0 ,V 'u = 0 (114)

which is the Reynold,s equation and

R = ( u t @ u t >

is the Reynolds tensor. As it is not possible to find an equation for .R as a

function of u, we will use an hypothesis (called a closure assumption) to relate

.R to Vu.It is quite reasonable to relate .R to Vu because the turbulent zones are

often in the strong gradient zones of the flow. But then R(Vu) cannot be

arbitrary chosen because we must keep (114) invariant under changes of coor-

dinate systems. In fact, it would be absurd to propose an equation for u which

is independent upon the reference frame. One can prove (Chacon-Pironneau

[b0]) ttiat in this case the only form possible for .R is (see also Speziale [218]).

R = a(lV u + Yrf l ')/ + b(lvu * Yur l 'xv" * vu").

in 2D. In 3D

(115)

(116)

(118)

(11e)

R - aI * b(vu * Vu") * c(vu * Yur)2 (117)

where a, b and c are functions of the 2 nontrivial invariants of (Vu * Yu').

So we get

V.rt, = Va * V.[b(Vu + Vur)] + V.[c(Vu + Vu")2]

but Vo is absorbeC by the pressure (we change.p to p * o) and so a law of the

type

1Q = b(Vu * Vu") * c(Vu + Vu")2

has the samc effect. In 2D, it is enough to specify a function in one variable

b(s) and in 3D two functions in two variables b(s, s'), c(s, s') where s and s' are

the two invariants of Vu * Vu".

INCOMPRESSIBLENAVIER-STOKBSEQUATIONS153

- -:-i

- - _

--

Figure 5.8 : Turbulant flnwsly's turbulent m'odel (ComPuled

a sphere with Smagorin-

7.2. The Smagorinsky hypothesis [216] :

Smagorinsky proposes that b - ch2lVu * Yur l, that is

R- -ch ' lvr+ vu" l (v"+ vur) , c = o '01

where h is the average mesh size used for (114)' This hypothesis is compatible

with the symmetry aird a bidimensional analysis of R ; it is reasonable in 2D but

not sufficient in # lso.riale [21g]). The fact that h is involved is justified by an

ergodic irypothesis which u*o,rnt, to the identification of ( ) with an average

operator in a space on a ball of radius h. Numerical experiments show that

we obtain satisfying results (Moin-Kim [173]) when we have sufficient points to

cause u, to .o.r"rplrrd to the beginning of the inertial range of Kolmogorov'

7.3. The k - e hvpothesis (Launder-Spalding [1a3]):

The sc-called k-* e model was introduced and studied by Launder-Spalding

[143], Rodi [200J anrong others'

Define the kinetic.nergy of the turbulence k and average rate of dissipation

of energy of the turbulence e bY:

< lu ' l ' > (121)

(r22)f Vu' * Yu'T l ' >;

then R, &, € are modeled bY

simulation around

by C. Paris)-

(120)

h - t' = t {


, 1n2n -

'ttct - "u?Fu + Vur), (ru = o.og)

n h2 1r2

f r , t * uYk - +r lvu * vu' l ' - v.( tu;o* l * e = o

. E c l , - , r T r 2 a / & ' . - t e 26, r * uVe -

1 ; lV" *Vur l ' - V . ( r , ;O. ) + cz7 = g

(123)

(124)

(125)

I2

with c1 - 0.1256 t c2 = 1.92, ce = 0.07.A rough justification for this set of equations is as foilows. First one notes

that k2f e has the dirnension of a length bquare so it make$ sense to use (123).To obtain an equation for &; (i12) is multiplied by u' and averaged (we

recall that .4 : B = A;18;i) :

 + V . 

* lo < u'2r. t ' t - o;

That is to say with (123)

t 2 n , t2

k,t -cr , | (v"+v ur) : Vu*uV/c -u 1 u ' Lu'

The last 3 terms cannot be expressed in terms of u, & and €, so they must bemodelled. For the first we use an ergodicity hypothesis and replace an ensembleaverage by a space average on a ball of centre c and radius r, B(x,r):

- < u'L,u/ >=< lYu'12 > + [ u' .{ar.J an6, r1 on

By symmetry (quasi-homogeneous turbulence) the boundary integral is small.The second term is modelled by a diffusion:

< u,v* >= -v. < u, g u, > yk;2

If u'2 and u' were stochastically independent and if the equation for u'2 waslinear, it would be exact up to a multiplicative constant, the characteristic timeof u'. The third term is treated by " similar argument as the first one so it isconjectured to be small. So the equation for & is tbund to be :

&,r * uV& - ++lvu* Yurl' - v.(. rYorrr e = 0 (126)

To obtain an equation for € one may take the curl of (112),multiply it by Y x u'and use an identity for homogeneous turbulence:

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS t c o

e = u < l V x , r ' 1 2 >

Letting LD' -- Y x u' , one obtains:

0 : 2 u l - r ' . ( r ' , r * ( u * u ' ) V ( r . ' + u ' ) - ( ' * a r ' ) V ( u + u ' ) - v A ' u ' ) >

= € , r1 uVe* -2 , 

-2r(< ut @ u' >; Vu * V. < (r ' I , ' )u ' >) + 2v2 < lv ' ' l ' >

because

V x (u' x a,r) = wVu' - u'Vu'

the term I utV x (r' x t.l)-> is neglqcted for symmetry rea'sons ; < u'Vuw'2 >

is modelled by a diffusion just as in the & equation; by frame invariance, (

ut &Lot > should also depend only on Yu * Vu", /c and e therefore in 2D it can

only be proportional to Vu+YuT and by a dimension argument it must be

proportional to &. The third terrn is neglected because it has a small spatial

*"u.r, and the last term is modelled by reasons of dimension by a quantity

proportional to e2lk. Finally one obtains :

c t , - T , t - . f t ' - r e 2€,t * uYe - f i tv"* vurl ' - v.(.,;o.) * czi - 0

The constants are adjusted so that the model makes good predictions for

a few simple flows such as turbulence decay behind a grid.

Natural boundary conditions could be

k , e g i v e n a t t = 0 ; I : l p - 0 , € [ P = e 1 ' ( 1 2 8 )

however an attempt can be made to remove the boundary layers from the

computation-l domain by considering " wall conditions" (see Viollet [235] for

further details)

k l r= u* ' r l , ' , .1 .=#

u .n=0 , r ru . i +g f f= t

where /i is the Von Karman constant (/i = 0.41), 6 the boundary layer thick-

ness, u* the friction velocity, g = c* k'lr, d = cttk2lle6(B +Io9(6lD))] where

D is a roughness constant and B is a constant such that u.r matches approx-

imately the viscous sublayer. To compute u*, Reichard's law may be inverted

(by Newton's method for examPle) :

(r27)

(12e)

(130)

156 FINITE BLEMENT METHODS FOR FLUIDS

Figure 5.9 : Computation of tarbulent f i.ow in a cauity with the k - e

model. The QL/P|-discontinuoas quadrilateral element has been used. The

figures show the streamlines and the iso-k lines. (Computed by B. Cardot).

u*f2.5tos(1 + 0.4 \ l+ 7.8( | - e-i# -

Hro aa$ )) - u-r

So in reality o and B in (3a) are nonlinear functions of u.r,e,k.

Positiveness of e and k.

For physical and mathematical reasons it is essential that the system yields

positive values for /c and e. At least in some cases it is possible to argue that

if the system has a smooth solution for given positive initial data and positive

Dirichlet conditions on the boundaries then & and e must stay positive at later

times.For this purpose we looks at

^ kH - -

lf.4 denotes the total derivative operator, 0l7t+uY and E denotes (Uz)lVu*

YuT 12, then

D t o - t - o r r - * r r ,e e

* a2E(cu - . r ) - 1

- 02 E(c,- , ') * ?o t 1", ,,5r.(lr4 - L * cz' v

* cz + crY .k|Y0 + 2crY0.v( l, * Gu - ,)alv .t i ,v',

Because ct, 1c1 and c2 ) !, 0 will stay positive and bounded when there are

no difusion terms because d is a solution of a stable autonomous ODE along

the streamlines:

4 0 - a 2 E Q r - r t ) - L * c z

Also 0 cannot become negative when ct, = c. because the moment the minimum

of 0 is zero at (r,t) we will have:

V g = 0 , d = 0

By writing the g equation at this point, we obtain :

0, t = c2 - 1 ) 0

which is impossible because I cannot become negative unless 0,t 3 0'

similarly we may rewrite the equation for & in terms of 0 :

b

Dtk - lne1vr, * vurl ' - v.(" t kovk)+ | = o2 | ' | \ r a

Here it is seen that a minimum of fr with ,b = 0 is possible only Lf D& = 0 at

that point, which means tfrat ft will not become negative.

Note however that ,L may have an exponential growth rf crgz lVu*Vu'l' >

2; this will be a valuable criterion to reduce the time step size when it happens.

7.4. Numerical Methods.

Let us consider the following model :

' t t r , t *uYu*Vp- v . [ z ( l vu+vu r lXV" *vu r ) ] = 0 , V 'u = 0

u . n = 0 , a u . r * r w - b

where a,b aregiven and v is an increasing positive function of lVu*Vu?l .

This problem is well posed ( there is even a uniqueness of solution in 3D

under reasonable irypotheses on v (cf. Lions U53])).The variational formulation can be written in the space of functions having

zero divergence and having normal trace zero :

(r , r , r ) + (uVu, u) * (133)

INCOMPRESSIBLE NAVIER.STOKES EQUATIONS

V o € J , " ( Q ) ; u € J

L57

( 1 3 1 )

(132)

1 ^

i t r lvu * vu" l)(v u + Yur ), V, * Vur)

f+

Jrlau.u - bu)dl - Q

,,(Q) = {u e Ht (o)t r V.u = o, u.nlp = o} (134)

1

158 FINITE ETEMENT IfETHODS FOR FLUIDS

By discretising the total derivative, n"e can consider a semi-implicit scheme,

f tr;*' - uioxi, vr) * ir*(vul*' + (vu[+')t), vur, + (Vrn)t) (135)

+ [ gr7- [)und-r = 0, Yan e JoonJ I

u ; + t € J o n n = { a n e V n : ( V . u ; , , { i , } = 0 Y q n € Q n ; a n . n n l r = 0 } ( 1 3 6 )

where V1 ar-d Q6 arc as in chapter 4 and where :

vi - v(lvuf + (vuf)"1) (13?)

There is a difficulty with Janl and the choice of the approximated normal n6,

especially if Q has corners.

We can also replace Jorrl by

J'onn = {rn € Vn , ( t ,7 , ,Vg; r ) = 0 Yqn eQnj (138)

(141)

(r42)

because

0 = (u, Vq) = - (V.r , a) + [ u .nqdî , Vq +J r

V.u = 0 , u .n lp - 0 .

(13e)With J'onothe slip boundary conditions are satisfied in a weak sense only but

the normal nn does not now appear .The techniques developed for Navier-Stokes equations can be adapted to

this framework, in particular the solution of the linear system (134) can be car-

ried out wiih the conjugate gradient algorithm developed in chapter 4. However

we note that the matrices would have to be reconstructed at each iteration be-

cause u depends on n.

To solve (114), (123) (the equations k - r), we can use the same method ;we add to (133)-(134) (& is replaced b1's)

+(1@+t>xJ n k

^ n '

lr i - "rhlvr?+t * Vui+ll2l1x1t;;at, ton) = 0, Ywn € Won

kI*' - qtoXt, un) * rrrr|{1rqt*', Vrn)

^ 2

(.f*t - eioXi, tun) * trc,()vrf,+' ,V.f,)c 6


*(:."r -llof lvrl*'* vu[+l,l, + uftXxfq)dt,un) = 0, Ywn € won

The integrals from n& to (n + l)fr are carried out along the streamlines in order

to stabilize the numerical method (Goussebaile-Jacomy [99]).

So, at each iteration, we must- solve a Reichard's law at each point on the walls,

- solve (134),- solve the l inear systems (141)-(142)'

The algorithm is not very stable and converges slowly in some cases but

it may be modified as follows.

As in Goussebaile et al.[99] the equations fiot q-€ are solved by a multistep

algorithm involving one step of convection and one step of diffusion. However

in this case the convection step is performed on q,0 rather than on g' €'

The equation for g is integrated as follows:

(q[*',,r)- (qioxt, ,n)+(ql *' I^:*"r ,TfiV"f*vuf f -#r,uh) (i43)

n n 2

+kc t (SVO;* l ,V r6 ) - 0 Vr r , € Qon = { ro e Qn t *n l r = 0 }c 6

sn - &rn € Qon (144)

But the equation for e is treated in two steps via a convection of 0 - qf e tbaL

does not include the viscous terms

(d;* tr,*n) (d;*, l^:-"r i t lv,[ *yuirl '(", - cr,),w) (14b)

- (| ioxf ,tor,) + (", - I,w1,)k

where 0t = qt let Then el+tlz is found as

.?*i = Xi:i (146)"h _ e ; * ,

and a diffusion step can be applied to find ul*t '

( .1* ' , toh)+&c. ( * r r i * ' ,Vrn) = ( .1*L , tn) $47)c t

V r r , € Q o h i u ; + t - e r n { - Q o n

Notice that this scheme cannot produce negative values for qfr+1 and e[+1 when

{{

. f i


because ct ) cu and

the coef;ficient of gl*t

the production terms

(?rr , lvu i*Vu[ f - i l r>-1

cz ) 1 so (145) generates only positive d

is positive. Note that (148) is a stability

are greater than the dissipation terms'

Ir,,*ro,(148)

while in (143)condition when

Chapter 6

Euler, Navier- Stokesand the shallow 'water equations

1. ORIENTATION

In this chapter, we have grouped together the three principal problems of

fluids mechanics wherc the nonlinear hyperbolic tendency dominates.

First, ne shall recall some general results on nonlinear hyperbolic equa-

tions. Then we shall give some finite element methods for the Euler equations

and finally we shall "xt.nd the results to the compressible Navier-Stokes equa-

tions.In the second part of this chapter, we shall present the necessary modifica-

tions needed to apply the previous schemes to the incompressible Navier-stokes

equations averaged in e3 : saint-venant's shallow water equations.

2. COMPRESSIBLE EULER EQUATIONS

2.L. Position of the Problem

Tle general equations of a perfect fluid in 3 dimensions can be written as

(c f . (1 .2 ) , (1 .7 ) , (1 .12 ) )

dF'


w , t * v . F ( w ) = 0 ( 1 )

and, in relation to (1.12)

E - p(e+f,Wp)

and

W - IP, Put, Prt2, pus, Efr

F(W) - ln(w), Fr(w), F (w)lr

F;(W) = lpu;, PUtui * 6uP, Puzu; * 6z;P, Putu; * 6s;p,ut(E + p)]" (4)

p-(7- lxE' - f ,a"n

w, t *Do,Wl#=o

The determination of a set of Dirichlet boundary conditions for (1) to be well

defined is a difficult problem. Evidently we need initial conditions

W(r,O) = W'(r) Vs e f, l

and to deterrnine the boundary conditions, we write (1) as :"

(2)

(3)

(5)

(8)

where A;(W) is the 5 x 5 matrix whose elements are the derivative of 4 with :

respect to Wi.Let n;(x) be a component of the external normal to I at c. One can show

that

B(W,n) = \dt1w1";is diagonalizable with eigenvalues

) r ( t ) = u , . n - c , I z ( " ) - ) a ( " ) - ) n ( " ) - u . n , ) s ( " ) = u . n * c ( 9 )

where c - (tplil l/2 is t,he velocity of sound. Finally, by analogy with the linearproblem, we see that it is necessary to give Dirichlet conditions at each points of I with negative eigenvalues. We will then have 0, 1, 4 or 5 conditions onthe components of W according to different cases. The important cases are:

- supersonic entering flow (u.n < 0,lu.nl ) c) : 5 conditions for example, ;

P , Pu; , P- supersonic exit flow : 0 conditions.- subsonic entering flow : 4 conditions, independent combinations of lrV.li

where ld is the left eigenvector associated to the eigenvalue ); ( 0.- subsonic exit flon' : 1 condition (in general, on pressure)- f low slipping at the surface (u., - 0): 1 condition, i.e.: u.n = 0.

EULER. NAVIER STOKES AND THE SHALLOW WATER EQ"' 163

2.2. Bidimensional test Problemsa) FIow through a canal with an obstacle in the form of an arc of circle

(ftizzi-Viriand[199])

1

Figure 6.1 :5iot ionary fiow aroanil an obstacle in the form of an arc of

circle. The picturc shows the geometry, the type of boundary conditions and

the position of sftocfts.

The height of the hump is 8% of its length. The flow is uniform and

subsonic at the entrance of the canal, such that the Mach number is 0'85' \\ie

have slipping flow along the horizontal boundary and along the arc (u'n - 0)l

the qualitl' of the result can be seen, for example, from the distribution of

entropy created bY the shocks'

b) F/oti, in a backwaril step canal (woodward-colella[238])

<_-.+0.6

Figure 6.2 zShockwaaes conf,guration in the fl,ow oaer a.^step'

u

P

p

u

P

p


This flow is supersonic at the exit and we give only conditions at the

entrance p = L, p = L.4,u = (3,0). The quality of the results can be seen in the

position of the shocks and of the .[ contact discontinuity lines.

c)Symmetric f low with Mach I around a cglinder. (Angrand-Dervieux[2])

Figure 6,3:Positions of shocks at Mach 8.

In this example, the diffrculties arise because the Mach number is large,

the domain is unbounded and because there is a quasi rarefaction zone behind

the cylinder.

2.3. Existence

One should interpret (1) in the distribution sense because there could be

discontinuities : to have uniqueness for the probiem, one must add an entropy

condition. The physical (reduced) entropy, s,

S - b s L" p'v

should decrease in time when we follow the fluid particles (t -' S(X(r)'t))). In

certain cases. (Kruzkov[136], Lions[153]) one can show that if the solution of

(1) is perturbed by adding a viscosity -eLW then when € + 0 it converges to

ihe entropy solution of (1). So the existence and uniqueness of the solution of

(1) can be studied as a limit of the viscosity solution.

For systern (1) with the above prescribed conditions there exists the fol-

Iowing partial results :- an existence theorem for a'local solution by a Cauchy-Kowalevska argu-

ment (cf. Lax [145])- a globai L*iri"n.* theorem in one dimension (cf. Majda [165])

( 10)

- a global existence

(scalar equation) and in

[1 53J) .

EULER, NAVIER STOKES AND THE SHATLOW WATER EQ"' 165

theorem in the case wherc W has only one component""

arbitrary dimension (Kruzkov [136],Glimm [94]' Lions'

2.4, Some centered schemes

To obtain finite element methods for equations (1)' we can use the method

developed in chaPter 3:- artificial viscositY methods,- semi-Lagrangian methods,- streamline viscositY methods'

a )Ar t i f i c ia lv iscos i ty rne thods(Jameson[121] )we take for (1) u.r .*pli.it discretisation in time and a Pl conforming

finite element method for sPace :

1 . .

|{wi*' - wr ,vn) - vwt),evn)

-(v .LG(Wi)Vwfl,Vn) = a YVn € Un;

Vy, .F(Wi) .n6 (11)

t * t - w f n e u o n

*1,w

where G is a viscosity tensor chosen carefully [15] and where Usl, is the space of

vector valued functions , piecewise - Pr continuous, on the trianguiation and

*hor" components corresponding to those where I'7r is known, are zero on f ;

i,ff6 is obtained by replacing in Wf the known.values of Wr '

As viscous and convection ternl are taken into account explicitly in tilne'

there is a stability condition on fr; if one is interested only in stationary solu-

tions, mass lrr*pi.rg can be used to make the scheme really explicit; in addition

a preconditioner.un b" added to accelerate it. The viscosity chosen by Jameson

is approximatelY :

v (c(iv) vw) = -*rlfiivn"# Y -tb-ah2# t.u#) (12)

rvhere z is the flow direction and o and b are constants of O(1); however it is

written as a nodal scherne in the sense that mass lumping is applied to the first

term of (11) and the contribution of the diffusion term is something like

I ( t ( w i - w ^ ) - I ( w * - w r ) )i , keT ( i ) me N( r ) r€N(k )

where .M(f) is the set of rieighbor nodes of node f and T(f) the triangles which

contain the ith vertex.

b) Lax-wendroff/Taylor-Galerkin Method (Donea [68]' Lohner et

al . [160])


We start with a TaYlor exPansion

wn+t = wn + kwi * **3r+ o(e2)

From (1), we deduce that

(13)

- f;rrtw1)

" - [email protected](wf)n1.v1,, YVn

wxl, = l?l-1 (rw - otc lurrr*rrnnt

-tJ I

w,tt =-v.( Wr= -v.(r; (w)wn) = v.(r,i(w)v.r(I4l)) (14)

So (13) can be discretise<i as

(w;;;;;*,, vi,) = (wt, vr,) + k(F (wn, vvr,) - T*t(14# )v.r (wil,vvr,) (15)

€ Uon

wn+r - wpfl+t * cD* M;t(I,I - Mr)w" ,

c ) P r e d i c t o r - c o r r e c t o r v i s c o s i t y m e t h o d ( D e t v i e u x [ 6 a . ] . ) � � �This is another form of method b. We start with a step (predictor) in

which Wf is piecewise constant on each element T and calculated from

with o = (t+ tfr)12 (cf. Lerat and Peyret [189]). Then, we

by solving

( 16)

compute WX*t

lWi*' - wt,vn)n+ ((1 - Or(wil + Or(wf), vvr,) (17)

V1,.F(W\) t * (G(l'vf )VWi,YV6) = 0, YVn € Un

EULER, NAVIER STOKES AND TIIE SI{ALIOW \ryATER EQ"' 167

where p - ll2aand where (, )r, denotes the mass lumping quadrature f91mu!.3.i

G is chosen to be proportional to h2 and to the first derivatives of Wf; (to give

more viscosity where W is nonsmooth)'

d ) S t r e a r n l i n e u p w i n d i n g m e t h o d s ( S U P G ) . ( H u g h e s - M a l i e t [ 1 1 9 ] 'J ohnson-SzepessY [128])

Thebas ic idea is toaddtheequat ionsof theprob lemto thebas is func t ions ;we consider the scheme

!:.'r (Wn,, +V.F(Wh), Vr, * h(Vn,, * A;(Wn)Vh,i))dt

a(A'oW6,1+ I A';W1,,,,) #i

' ' " l o + l v W n l )

I+, I lwt+ - wf- t(I wh,,,vn,,,)

J a i

( 18)

+(Wi+ -wt-,V{- ) = 0

where [/'t is the left or right limit of u(t) when | --+ nk. we discretise in time

and in space in a coupled manner, (cf cirapter 3) for.examl]e with continuous

p1 approximation i' spu"" and Pi discontinuous in time' Though the method

is efficient numerically, (Mallet [166]) the coxvergence study of the algorithm

suggests 3 modifications (Johnson-szepessy [128]):- use of entropy formulation of Euler's equations'

- replacing t;; hM in (1g), where M is a matrix function of wn,

. .ddi,,g another viscosity called ,'shock capturing,,

The final scheme is

l^:.'r(A'own,r+t A!rwn,,,,vn rh(I A;)- * lA'ovn,t+ t A';v1', ',))dt ( 19)i i ) o I

+(wf+ - Wi- ,Vt*) - hd VV^ e

where

Vf = {{Jn,PL continuous in c, Pl discontinuous in t,

such that (Jn; = 0 on the parts of I where Wn; is given ),

\Are choose s= 1 , b << ! , c= 1 , we pu t VU = l [ J , r ,u , r , ] " and rve take

r l nY h

(20)

(2 i )d - tJnx1 " t , ( n {1 )k I

The matrices .4,; are comPuted from

A| = ( ,S,r*)- t , A! , = A;A'o.- The scheme is imPlicit in time

Burger's equation, (F' (W) = W'12,- R, result which converges towards

the entropy function S(W) (cf (10)) bv

like the Crank-Nicolson scheme' For

W having onlY one comPonent) on Q

the continuous solution when h -* 0 can

168 FINITE ELEMBNT METHODS FOR FLUIDS

be found in [17] ; also It W has only one component Johnson and Szepessy haveshown that the method converges to the entropy solution.

In the general case one can also show that if Ir[, converges to W, then Wsatisfies the entropy condition

s(w),, + I s,- F;(w),", ( o

e) General Remarks on the above schemes

Due to the complexity of the problem, we still do not know how to proveconvergence of the scheme in a general case (except maybe SUPG). So theanalysis often reduces to the study of finite difference schemes on regular gridsand of a linear equation (like that of chapter 3).

The schemes with artificial viscosity have the advantage of being simple butto be efficient, the regions where the viscosity plays a role must be small: thisis possible with an adaptatiae mesh where we subdivide the elements if lvlazl islarge (Lohner [158], Palm6rio and al. [186], Bank [12], Kikuchi et at. [1BB]). Butthe stability condition forces us to reduce k if h diminishes so more iterationsare needed than with implicit schemes; this requires a careful vectorizationof the program and/or the use of fast quadrature formulae (Jameson [121]).Finally, the main drawback as the choice of viscosity coefficient and the timestep size, one may prefer the schemes which use upwinding methods, is notbeing more robust (but not necessarily more accurate). On the theoreticalside, it is sometimes possible to show that the schemes satisify the entropyconditions in the case of convergence (Leroux [1aZ]).

2.5. Upwinding by discontinuity

As in chapter 3, upwinding is introduced through the discontinuities ofF(Wn) at the inter-element boundaries, either because Wn is a discontinuousapproximatton of W , or because for a\l Wn continuous, we know how to asso-ciate a discontinuous value at right and at left, at the inter-element boundary.

a) General flamework (Dervieux [64j, Fezoui 178], Stouffiet et al. t2220.As in chapter 3, for a given triangulation, we associate to each vertex qd

a cell at obtained by dividing triangles (tetraedra) by the medians (by medianplanes). Thus we could associate to each piecewise continuous function in atriangulation a function P' (piecewise constant ) in ai by the formula

(22)

By multiplying (1) bV a characteristic function of a' and by integration (Petrov-Galerkin weak formulation) we obtain, after an explicit time discretisation, thefollowing scheme :

wxl", = fr [,,*oo,

EULER, NAVIER STOKES AND THE SHAbLOW WATER EQ"' 169

wi*'(q') = wtk'). fr lu, oor*rr." vt (23)

On at O l, we take fa(W) - F(W) and we take into account the known

components of Wr ;rrrd "lr.*here Fa(W) is a piecewise constant approximation

ot F(W) verifYing

I u,, rotwxl.n =

Y,*r*'^1o,, w\4n) f u,, n,,'

O(Y, V) = F(V), for all Iz

b) Definition of the flux O :

Let B(W,n) € 65x5 (ax 4 in 2D) be such that

F(W) .n - B(W," ) .W YW Yn (26)

Note that B is the same in (8) because F is homogeneous of degree I in w

i;'iiti - ^F(W)) and ro B is nothing_but F!(W)n;. As ri'e have seen that

b il aiusonalizabl., th"t" exists T € 'R5x5 such that

B - T-L I\T

where A is the <iiagonal matrix of eigenvalues'We denote

A* = diag(*mor (*A; , 0) ) , B* - T- r Lr T

l B l = B + - B - , B = B * + B -

We can chocse for Q one of the following formulae :

Ostv (Vi ,Vi) = B+(Vt)Vt + B- (Vi)yi (Steger-Waiming)

* v s 1 7 i , V j ) _ g + ( r y ) v ' + , _ ( u , t , u ' ) v j ( v i 3 a y a s u n d a r a m ) ( 3 1 )

, v r ( V i , V j ) = f ; t , v , ) + r 1 t r ' i ) + l B ( f f l K V j _ v ' ) ] 1 v a n L e e r ) ( 3 2 )

oo'(v' ,vi) - f,trfvd) + r(v t, - l:, Wy)ld.rrl (osher) (33)

(24)

(25)

(27)

(28)

(2e)

(30)


the guiding idea being to get Or(Vt ,Vi) = F(Vt), if ,\; is positive and F(Vi),

i f ) r < 0.So iDsw , for instance, can be rewritten as follows :

osw (vi,vi) = t[F(v') + F(vi)f + ] i lB(v') lvi

- lB(vi)lvi l ,

because

F(V') + r lVi ) = (B+(yt) + B- (VtDVt + (B+( Vi) + B- (Vi))Vi ;

The first term, if alone, would yield an upwind approximation. The second,

after summation on all the neighbors of i, is an artificial viscosity term. The

Van Leer and Osher schemes rely also on such an identification ; the flux of

Osher is built from an integral so that it is Cl continuous ; the path in R5

from Vi to yj is chosen in a precise manner along the characteristics so as to

capture exactly singularities like the sonic points.

c) Integration in time.The previous schemes are stable up to CFL of order | (clwlk lh < 1 where c

depends on the geometry);if one wants only the stationa.ry solution, the scheme

can be speeded up by a preconditioning in front of 0W/Et ;these schemes are

quite robust ; 3D flows at up to Mach 20 can be computed ; but they are not

precise. Schemes of order 2 arc being studied.

d) Spatial approximations of order 2.A clever way (due to Van Leer) to make the previous schemes second order

is to repl ace Vi and Vi by the interpolates 7i- and 7'+ defined on the edges

where they must be computed by :

v i - - ! , i + ( vy ) r @ ; { )

v i+ - v i - ( yy ) i ( ( ; { )2

An upwinding is also introduced to compute the gradients (cf. Stouffiet et al.

12220. An alternative is to use discontinuous elements of order 2 or higher

(Chavent-Jaffr6 [51]).

2.6. Convergence : the scalar conservatiol equation case

The convergence of the above scheme is an open problem except perhaps

in lD or when W has only one component (instead of 5!); -this is the case of

the nonlinear scalar conservation equations which govern the water-oil concen-

tration in a porous media; let us take this a^s an exbmple. Let f be in Ct(R)

and /(c,r) € E be a solution of

EULER, NAVIER STOKES AND THE SHALLOW WATER EQ." T7L

Figure 6.4: simulation of an inaisciil cornpressible flow around a sp&ce

shuttle by soluing the Euler equalions. (coartesy of AMD-BA)'

Q,t*Y /(d) = 0 in Q =10, TlxR', d(0) = do in R" (34)

with {o having comPact suPPort

One can rto* (k*rkou [136]) that for 6o e L*(R") there exists a unique

solution (34) satisfying the entropy inequalities

O..t * V.f" 5 0 in Q for all c

6?*' = n({dl});

(35)

where

o.(d) = ld - cl, F"(il = (/(d) - /('))'g "(6 - ") (36)

M o r e o v e r , d i s a l s o t h e l i m i t o f 6 ' , € + 0 , t h e o n l y s o l u t i o n o f

Oi +V'( / (d ' )) - e 4 '6 ' = 0 in Q, d '(0) = ' Oo ' (3i)

Finally, if BV(R") denotes the space of functions of bounded variations) \\'e

have :

4o eL r (R" )nBV(R" ) + Q eL* (0 ,T ; r t (n " ) ) nBV(R" ) (38 )

In the finite difference w<.,rld, the following definitions are introduced :

Let {{i} be the values at the vertices ot 6f@) @i - 6i(qt)); an explicit

scheme

,(39)


is monotone if I{ is a non-decreasing function in each of its arguments. A

scheme ts TVD (Totat Variation Diminishing) if

l ld;+' l l ru < l ldf l ls" where l ldi l lsv = ffi (40)q i , q j n e i g h b o r s

These notations can be generalized to finite element schemes at least in the

case of uniforrn meshes. In [10] a quasi-rnonotone finite element scheme is

constructed with upwinded fluxes and a viscosity of O(h), and one gets a class

of methods for which the convergence is proven'

Implicit methods with viscosity have the advantage of being automatically

stable and in the case of convergence (where all the diffficulties lie) convergence

occurs towards an entropy solution.

For example, let us consider the following scheme :

1 . .IfW;*L -Qi, tui,)*(V .f @f+t), @r) *eh" (Y$[+t, VtI/r) = 0, Ywn € Vn (41)

where 71, is a space of conforming finite elements'

By takin g un - |i*t, we get the stability

l6i*' lz s l{il , ld; +' | , (42)

because (V./(dn),dn) is equal to a boundary integral on E which is positive

plus c [V{6 l ]Under iather strong convergence hypotheses, by taking u'r6 eQual to the

interpolation of sgn(d -: c) one can show in the limit of (a1)

(6 , t sgn(6 - ' ) ) + (V . f (d ) , ' gn (6 -c ) ) *e l im h " (Y{n ,Yssn(Q- " ) ) = 0 (43 )

but

(vd, yssn(g- c)) = lror_"=0,* 0 r , ,

on

so (a3) implies ultimately (35).' As mentionned above, SUPG*shock capturing is one case where conver-

gence can be proved by u similar argument (and many more ingredients [128])

3. COMPRESSIBLE NAVIER.STOKES EQUATIONS

3.1. Generalit ies

The equations (1.2), (1 7), (1.12) can also be written as a vectorial system

tn W. With the notations (2)-(5), we have

EULER, NAVIER STOKES AND THE SHALLOW WATER EQ"' 1Zi3

W,t * V.r(14/) -V.K(W''VW) = 0

where K(W,VW) is a l inear 2nd order tensor in VI4z such that

Ki,r = o, K.,2,s,4 - qYu+ (( + f )rv'",

K ,s = 4(vu + Yur)u+ (C - ld'v ' ' * .o(X - $lThe term v.K(w,vw) is seemingly a diffusion term if p,! > 0 since it

responds to 4Au + (rtl3+ ()v(v."; i" th9 2"d, 3"d and 4rh equations of

.r,i to K(co lilan i it pc"[lv.rl'(( - 2q13)+ lvu * v$ lqlz) in the last(c f . (1 .13)) :

{ 44\

(45)

(46)

cor-(44)one

(47)- l*"wv.K(w,vw)

= - I*J,tLu* (3 + C)v(v.,)1,,

- l*"(*c,t;on *f,r,[lv.,['(( -r|,)* lVu + vu"1]11

, l^"(rlv.,l '* (3 + C)lv-,I ') + .* l^"l ' 'f

*yc, l."l'o .ul'(c - r+) +f,:v" +vf l'l > oPmax J R3

So we can think that the explicit schemes in step 1 to integrate (1) can be

applied to ( a) if the stability condition on the time step k is modified :

klcmidh,ft; (48)

where /{ is a function of rc,1, (. This modification is favorable to the schemes

obtained by artificial viscosiiy but still when the boundary conditions are dif-

ferent boundary layers arise boundary layers in the neighborhood of the ivalis'

The boundary conditions can be treated in the strong sense (they appear

in the variational space Vor, instead of V6) or in the weak sense (they appear

as a boundary term in the variational formulation)'

Flow around a NACA0A12 airfoil

A test problem is given in Bristeau et al.[43] which concerns a 2D flo$'

around a rving profile NACA0012. Taking the origin of the reference frame at

the leadii,g .Jg. the equation of the upper surface of the profile is :

y - 0.I7735t/c-0.075597s -0.2I2836x2+0.17363s3 - 0.06254x4, 0 ( r ( 1

The boundary conditions at infinity are such that the Mach number is 0.85,

the Reynolds number (uplri is 500 and the angle of attack is 00 or 100. The

temperatures at infinity and on the surface are given

i

174 FINITE ELEMENT METHODS FOR TLUIDS

Figure 6.* FIow around a NACAATIL. The figurc shous theposition of shocks and the bounilary lagers.

3.2. An example

As an example, we present the results obtained by Angrand et al. [2] andRostand [204].

The algorithm used is a modification of (16)-(17) and the boundary con-ditions are treated in the weak sense. The problem treated was the modelingof rarefied g* layers in the neighborhood of the surface ( Knudsen layers) andso a Rcbin type boundary condition is applied:

A ( W ) W + K ( W , V W l n - s o n f (4e)where,4 is a 2nd order tensor and g € R5. (If .,4 -.* *oo we obtain the Dirichletconditions and 1f A -* 0, the Neumann conditions).

Let Un be the space of functions having range in B5 continuous and piece-wise P1 on a triangulation of Q. Problem (aa) is approximaied by

1

iWX+' - wf ,vn)n - (Fn(wt), vl/r.) + (/{h( wt ,vwi),yvo) (b0)

f _+ l [F (wt )von+(A(Wt)Wt -e )v r ] =0 YVn€t1n

J r

K7, possibly contains also the artificial viscosity if needed, F6 the upwinded fluxand (, )1, the ma^ss lumping. This scheme is mostly used to calculate stationarystates.

In Rostand [204], the flow around a wing profile (2D) NACA0012 has beencalculated with the following boundary conditions :

u.n =0, n.D(u).r * Apu.r= 0 u.D(u)."+ f i f f i - a (b1)

where D(") = Vu * YuT - 2/3 V.u I, .4 is a coefficient to be chosen and Pr isthe Prandtl number.

we point out the following problems which are yet to be solved :

EULER, NAVIER STOKESAND THE SHALLOW WATBR 8Q... t75

i \..\Figure 6.6: simulation of the compressible Naaier'stokes equalions

around, a NACA0012 by method (60) with Pr conforming elements' In the

figurv which has a shoik, A - | of (61) is and in lhe olher one A - 100'

(Computed bY Ph. Rostand)'

- find non reflecting boundary conditions at infinity towards downstream

- find a fast scheme (implicit ?) and more accurate (order 2 ?)

- remove the pressure oscillations near the walls and the leading edges in

particular; it seems +,hat one has to use a different approximation for p and u

when u (( c, which makes sense from what we know about incompressible the

Navier-Stokes equations.

- make the scheme a feasible approximation for small Mach number also,

(something which does not exist at the moment)


3.3 An extension of incompressible methods to

the compressible case.

Since the incompressible Navier-Stokes equations studied in Chapter 5 are

an approximation of the compressible equations when p tends to a constant, it

should be possible to use the first one as an auxiliary solver for the second one.

56 we shall try to associate the continuity equation and the pressure. The

complete system (44) can be written as a funct,ion of a = logp us shown below,

Bristeau et al.[al]. Here the method is presented with ( = 0) :

o,t *uYo * V.u = 0

u ,1 {uYu+ ( r - l ) (gvo + Yg) - , te -o (Lu + iV (V .u ) )

0,t * uV| * (7 - 1)0V .u = e-o (rcL,A * F(Vu)) (53)

where F(Vu) - r l lVu+Yur1z12-2q1V." | '2f t (and where A and rc have been

normalized).An implicit time discretisation of total derivatives gives rise to a general-

ized Stokes system if we treat the ternperature explicitly :

1 - , 1 - - i t 1

Eo"*t * V.u"*l = io"

oXn ,

!u**t - ne-o'(ar, ,* , + 1v(v. t f+1)) + (z - L)e"von*rK J

1 _- iu"oX"

- ( t - l )VP",

1 , . - n - , r 1 - n

(1 + +(r - l )v ,un)Tn+r - e-oo rca,o"* t =;0"oxn * e-o ' ' F(Vu")U k \ '

, ' l c

A nrethod for solving th'iJ system is to multiply the last two equations by "o

so as to obtain

(52)

(54)

(55)o q * Y ' u = 9

B u - o A u * b Y o - f

6 0 - c A ' ? = h

where o and c are consta:rt. So following the method of chapter 4; we take the

divergence of the second equation, substitute it in the first and obtain :

qo*V . tW -oA) - ' ( f - bVa) l = 0

an equation that we solve by using the conjugate gradient method.

This method has the advantage of leading back to the methods stqdied

in chapter 5 when the flow is quasi-incompressible, but has the drawback of

E U L E R , N A V I E R S T O K E S A N D T I I E S H A I I 0 \ \ r W A T E R E Q . . . L 7 7

't,

.r/"' _'-----.1

.: \--'-1t---------. .r , t

Figure 6.7:a NACA0012 bY

giving a methodtend to 0.

Solution of the comPressible

4 ,SAINT.VENANT'SSHALLOWW.ATEREQUATIONS

4.L. Generalities

Let us go back to the incompressible Navier-5t-r:kes equations with gravitl'

(e = -9.81)

the rnethod of ParagraPh 3'3

which is non-conservative for

l i.;r ier- Slokes equalions around

(C',;rnputed bY M'O' Bristeau)'

L' '- ' .er's equations when 4 and x

t r , t * V .u @ a *YP - vL 'u - - 7ez

and assume that the domain occupied by the fluid

is true in the case of lakes and seas '

Figure 6.8 : Vertical section t'hrough I'he fluid domain

the bed,'of a lake and' the water surface'

Let zr(rt,,z',t) be the height of the surface and z1("'"

thebed .Thecon t inu i t yequa t ion in tegra ted in rs=zg ives

V.u = 0 (56)

has a small thickness, which

showing

, t) the height of

178

but by the definition of u3 we have d,z, f dt - us(z r) so, taking into account the

"*tfip condition at the bottom and defining u' z by

(58)

FINITE ELEMENT METHODS FOit FLUIDS

[" , .udz - us(2,) - u"( ' i l * Vrz '( [" ' udz) = 0J r ,

- u \ " r - \ " J t t

f z t

I ud t = ( r , - z )a (x1 ,sz , t ) , z = zs - z l

J z t

one can rewrite (57) as

rn (b6)(a) we negrecr arl th. r":;Ji:"::,1" ,n,,0 equation sives

0p- = 9oz

(57)

(5e)

(60)

By putting p calculated from (60) in the two first equations of (56)(a) integrated

in cg we see

(za),t* V.(zu A u) + gzY z - vA'(za) = 0

The system (59), (61) constitutes the shallow water equations of saint-venant

(for more details, s.. B.nq,r6 et al. [22]' fo1 example)' For large regions of

water (seas), the coriolis forces, propottional to u x v where c'r is the rotation

vector of the earth must be added. other source terms, -f , itt the right hand

side of (61) could come from- the modeling of the wind effect (f constant)- the modelini of the friction at the bottom zl U - clululz)'

If we change scales in (59), (61) as in (1'39)'we obtain

(61)

2 t , 1 , * Y / ' ( z t o ' ) = 0

(z'a') ,1, + V' .(r ' , ' €) u') + F-2 z'Y' z ' - R-t L' (z ' ' ' ) = 0

with ft - lalLf v and the Froude number

F '= l ' l' @

lf we expand (61) and divide by z,we get

u , t *uVu * gY z - uz-L A, (zu) = 0

If F << 1 and R << 1, ,hen gvz dorninates the convection term oVu and the

difiusion term vz-t Lzu ;*t t* approximate (65) (59) by'

z . t * V . ( z t r ) = S u , t * g Y z = 0 (66)

(62)

(63)

(64)

(65)

EULER, NAVIER sToKEs AND THE SHALTOW WATER 8Q... L7,g

wbich is a nonlinear hyperbolic system with a propagation velocity equai to

rF.If R >> 1 so that we can neglect the viscosity term, we Eemark that (59)'

(61) is similar to the Euler equations with 7 = 2, p = z and an adiabatic

approximation plf = constant ; on the other hand if in (65) we set u = z1)

then it becomes ri*itut to the incompressible Navier-Stokes equations with an

artificial compressibility such as studied by Temam [228]:

u , t * Y . ( r * 1 u O u ) + * V r t - z A ( u ) = 0 z , t * V ' u = 0 ;2 \ /

this problem is well pose,cl with initial data on u, z and Dirichlet boundary data

on u only.But if we make the following approximation:

z-r L,(za) 3 Au,

then (bg)(61) becomes similar to the compressible Navier-Stokes equations with

7 = L :

u , t * u V u * g Y z - v A , u = 0

z , t * V . ( z u ) = 0 ;

An existence theorem of the type found by Matsumura-Nishida [169] may be

obtained for these equations ; this means that the problem (59), (61) is well

posed with the following boundary conditions :

z , u g i v e n a t t = 0 $ 7 a )

ulp given I and z given on all points of I where u..n < 0 (67b)

with the con{ition that z stays strictly positive. This fact is confirmed by an

analysis of the following problem (Pironneau et al [192]):

g Y z - u A ' o = A V . ( z u ) - g ;

with (6?b). This reminds us that certain seemingly innocent approximations

have a dramatic effect on the inathematical properties of such systems.

Bcundary conditions for these shailow water equations is a serious problem;

here are some difficult examples found in practical problems:- on the banks v;here z * 0- in deep sea where one needs non reflecting boundary conditions if the

computational domain is not to be the wholesurface of the earth.- if the water level goes down and islands appear ( z : 0)'

'-fest Problem :A simple non stationary test problem is to

metric and Gaussian in form with 2m height in

side returning to rest.

study a wave which is axis-vm-

a square of 5m depth and 10rn


Figure 6.9 : Computational domain (lrft) and aertical seclion through the

physical domain .

4.2. Numerical scheme in height-velocity formulation

When the Froude number is small, we shall use the methods applicable

to incompressible Navier-Stokes equations rather than the methods related to

Euler's equations.We shall neglect the product term V z Yu (small compared to uVu) in the

development of A,(za) in the formulation (59), (65), that is we shall consider

the system

z , t * u V z * z Y . a = 0 ( 6 8 )

u l * u V u * g Y z - u A , a = 0 ( 6 9 )

z (0) = zo ,a(0) - uo in Q; u lp = u r r z l2 , = z r (70)

For clarity, w€ assume that u1 = 0. Recall that D is that part of f where the

flux enters (u.n < 0). If D/Dt is the total derivative, (68)-(69) can also be

written

D z

V + z Y . a - 0

DuD t + g Y z - u \ a = 0 '

With the notation of chapter 3, we can discretise the systemimplicit Euler scheme

(7 1)

(72)

with the semi-

(73)

(74)

for thewe find

I r -^+t -r,E ' \ .

- o n o X ^ ) + z ^ v . u * * 1 = o

1 . -

; ( r - * t - uô - ) + gYzm*t - uL 'a*+ l = o

For spatial discretisation, we shall pse the methods given in chapter 4

Stokes problem : we choose QL * ,2(Q) and Von = I{i(Q)" and

ItT*t,of+\ sblution of

EULER, NAVIER STOKES AND THE SHALLOW WATER EQ"'

where

181

l{,f*' , frt+ (v.uf,+', sn) - lt'r oxr ,#) Yq1' € Qn (75)

' l . - r 1 r - t n J � i t t

t

I t r f * t ,?rh) +v(Yr$+t ,vrn) +s(vzf*1,ur) -

r t 'Tox{ ' tn) v" €voh

(76)

we recall that xi(") is an approximation of X(*k,c), which is asolution of

# = uf;(X(r),r), X((** 1)k) = c, FT)

thus the bounda,ry condition on z6;p appears in (75) in the calculation of

zf;oXff(x), c € E.At each iteration in time, we have to solve a linear system of the type

( e _ B \ f y \ = f l ) ( i 8 )\ s ' - D ) \ . . 2 ) - \ G /

Bii (vq' ,d), Gi - -l4f "'f ,#,

Fi -- irf oxf ,d), A;i hru. ,"r)

+ i{o*' ,vu") (7e)

As with the Stokes problem, the matrix of the linear systems is symmetric but

here it is nonsingulu, whatever -may be the chosen element {,v.n,Qnl' In fact,

by using (?6), to'"ti.rri' ate u[+r, we find an equation for tT+' '

1 n r e 4 B + D ) Z * B r A - r F - G

the matrix of this linear system is always positive definite (cf. (79))' It-can

be solved by the conjugate gradient method gxactly as in the stokes problem'

A preconditioner .un ulro be constructed in the same manner ; Goutal [100]

suggests :

c = ? - A ntcg

where D is given in (79) and -A6 is a Laplacian matrix with Neumann condi-

tion (this preconditioning conesponds to a discretisation of the operator on z

in the continuous case when ' = 0)'

Stability and convergence are open problems but the numerical perfor-

mance of the method is good[l3]'

4.3. A numerical scheme in height-flux formulation

W e p u t D _ z u s o t h a t ( 5 9 ) - ( 6 1 ) c a n b e r e w r i t t e n a s

I f s 'qrn . . - - I --'t)

k Jn zf '

(80)

(8 1)


Figure 6.10: Res'ults for the test prcblem at time t = 0'2s for the shal'

low water equution with conuection terms in the msmenlutn equations' The

contputation has been ilone with the forn'alation using height 'and aelocity' Pr

,rnjorrning elements and, at Reynolds namber 20; lhere art' 100 equal triangles'

The uelocities anil the leuel lines of the surface haae been plot'ted- (Computed

by V. Schoen).

z , t * V 'D = 0

1D, t *V . ( ;D I D) * szY z - vL 'D = 0

Let us take as a model proUtJ* the case where

z(0) - z' , D(0) - Do in O (84)

D = Dr on lxl0,T[ (85)

In [13] the following scheme is studied : with the same finite element space I/61

,rri Oon, the functiorr in the space Q1 which are zero on l, one solves

!Qf*' - 2f;, qn) + (v'Df+t'{r') = 0 Yqn e Qn (86)

! PE - uf; oxf,, tur,) * (y ,T*t, uh) *'--g nil*t, vH) (87)k g ' , t r

v h v " n ' * I ' / ' \ ' ' - t l ' " r 9 '

" ' Z h

i 1 -i , + i ( r t V . u f , t u l ) = g l w n e V o n

where 't)* =,DT lrT and wher e Xf is as i.n- the previous paragraph' that 'is' it

is calculated with i1, : Xf (c) = I - af (t)k '

(82)

(83)

EULER, NAVIER STOKES AND THE SHALLOW WATER EQ.. . 183

Each iteration requires the resolution of a linear system of the type (78)

but with

1 . .Dij lt{d,ai) A;j

6 J"W * uvw'o ,r)

(88)

We note that ,4 is no Ionger symmetric except when u = 0. The uniqueness

of the solution oi (86)-(87) is no longer guaranteed except when u 1{ 1 rvhich

makes it symmetric again. When v *0, we can also use the following approx-

imation

FT,v,i)Another method can be devised by working withreplacing (r|*t - 2f;, gr,) in (SO) by (ri^*' * ri ,of this algorithm is also an open problem.

f Df . ' -m(7,J"ffiv'TVun

(8e)

z' = z2 instea d of z and byqn I QtT)). The convergence

= l*#"rf*lvton -

4.4. Comparison of the two schemes

The main difference between the height-velocity and height-flux formula-

tions is in the treatment of the convection term in the continuity equation and

in the boundary conditions. We expect the first formulation to be more stable

for flows with high velocity. The other factors which should be considered in

selecting a method are :- the boundary conditions,- the choice of variable to be conserved- the presence of shocks (torrential regime, u = 0) ; (86) ii in conservative

form whereas (75) is not.

5 . CONCLUSION

In this chapter we have briefly surveyed some recent developments in the

f;eld of compressible fluid dynamics and the shallow water equations. We have

omiited reacting flows (fluid mechanics * chemistry), the Rayleigh Benard

problem and flows with free surfaces because a whole book would have been

necessary in place of just one short chapter; these subjects are also evolving

quite rapidly so it is difficult to write on them without becoming rapidly ob-

solete; even for compressible flows interactions between the viscous terms and

the hyperbolic terms is not well understood at the time of writing.

The architectures of computers are also changing fast; dedicated hardrvare

for fluid mechanic problems is already appearing on the market and this are

likely to influence deeply the numerical algorithms'

Finally compressible turbulence is, with hypersonic combustion, one of the

challenge for the 90's.

References

t1] F. Angrand: Viscous perturbation for the compressible Euler equations,application to the numerical simulation of compressible viscous flows.M.O. Bristeau et al. ed. Numerical simulalion of conl,pressible Naaier-Stokes flows. Notes'on Numerical f luid mechanics, 18 , Vieweg, 1987.

[2] F. Angrand, A. Dervieux: Some explicit triangular finite elements schemesfor the Euler equations. J. for Num. Meth. in Fluids , 4 ,749-764, L984.

[3] H. Akay, A. Ecer: Compressible viscous flows with Clebsch variables.in Numerical methods in laminar and turbulent ftows . C. Taylor, W.Habashi, M. Hafez ed. Pineridge press 1989-2001, 1987.

[4] J.D. Anderson, Jr.: h'Iodern compressible flow with historical perspective.McGraw Hill. 1982. see also Fundamental Aerodynamics lt i lc-Graw Hill1984.

[5] D. Arnold,F. Brezzi, M. Fortin: A stable finite element for the Stokesequations . Calcolo 2I(4 ) 337-344,1984.

t6] O. Axelson, V. Barker: Finite elemenl solulion af boundary aalue problems.Academic Press 1984.

[7] K. Baba, M. Tabata: On a conservative upwind finite element schem forconvection diffusion equations. ̂ R,4/r?O numer Anal . 15 ,1 ,3-25,1981.

[8] I Babuska, W. Rheinboldt: Error estimates for adaptive finite element

, computations. ^9/AI[ J. Numer Comp. 15. 1978.\i [9] I. Babuska: the p and h - p v'ersion of the finite elernent method: the

state of the art. in tn Finite element, theary and applicalion. D. Dwoyer,M Hussaini, R. Voigt.eds. Springer ICASE/NASA LaRC series. 199-2391988.

[10] G.K. Bachelor: An introdu,ction to Fluid Dynamics. Cambridge UniversityPress, 1970 .

[11] A. Baker: Finite element computational f luid, mechanics . McGraw-Hill,1985.

[12] R. Bank: PTLMG User's guide. Dept of lt{ath. UCSD tech. report. Jan1988.

REFERENCES 185

[13] R. Bank, T. Dupont, H. Yserentant: The Hierarchical basis multigrid

method. Konrad zuse zentram Pruprint sc-87-1. 1987.

[14] R. Bank, L.R. Scott: on the conditioning of Finite Element equations

with highly refined. meshes. Math Dept. Pennstate reEearch report 87013.

1987.

t15] R. Bank, T. Dupont, H. Yserenlant : The Hierarchichal muitigrid

method. tICLA math preprint SC87-2' 19E7'

[16] C. Bardos: On the two dimensional Euler equations for

f lu ids . J . MaIh . Ana l . App l . ,4O , 769-790,1972 '

incompressible

[12] T.J. Beale, A. Majda: Rate of convergence for viscous splitt ing of the

Navier-stokes equations. Math. comp . 37 24T260, 1981.

[18] C. Bcgue, C. Conca, F. Murat, O. Pironneau: A nouveau sur les 6quations

de Navier-Stokes avec conditions aux limites sur la pression. Note CRAS

t 304 Serie I t .,23-28 , 1987

[19] C. Begue, C. Conca, F. Murat, O. Pironneau: Equations de Navier-Stokes

avec conditions aux limites sur la pression. Nonlinear Partial Differential

equations and their applications. Collbge de France Seininar I , (H. Brezis

and J.L. Lions ed) Pitman 1988.

[20] J.P. Benqu6, O.Daubert , J. Goussebai le, H. Haugel : Spl i t t ing up tech-

niques for computations of industrial f lows. InVistas in applied mathe'

matics . A.V. Balakrishnan ed. Optimization Software inc. Springer,

1986.

[21] J.P. Benqu6, A. Haugel , P.L. Viol let , : Engineer ing appl ical ions of hy-

draulics I/. Pitman, 1982.

L2ZJ J.p. Benque - B. Ibler - A. Keraimsi - G. Labadie: A finite element method

for the Navier-Stokes eq. In Norries ed. 1110-1120, Pentech Press' 1980.

[23] J.P. Benqu6, B. Ibler, A. Keramsi, G. Labadie: A new finite element

method for Navier-Stokes equations coupled with a temperature equation.

In T. Kawai, ed. Proc. 4th Int . Syrnp. on Fini te elements in f low problems

North-Hol land 295-301, 1982.

124) J.P. Benque, G. Labadie, B. Latteux: Une methode d'6l6ments finis pour

Ies 6quations de Saint-Venant, 2nd Conf. On numerical methods in laminar

and turbulent f lows. Venice, 1981.

t2b] M. Bercovier : Perturbation of mixed variational problems. RAIRO s6'rie

rouse 12(3 ) 2ll-236 ,1978.

[20] M. Bercovier - O. Pironneau: Error Estirnates for Finite Element solution

of the Stokes problem in the primitive Variables. Numer. Math . 33,

2rr-224, 1979.[2 i ] M. Bercovier, O.Pironneau, V.Sastr i : Fini te elements and character ist ics

for some parabolic-hyperbolic problems:Appl. Math. Modell ing, 7 , 89-96,

1983.

t28] A. Bermudez, J. Durani: La m6thode des caract6ristiques pour les

problbrnes de convection-diffusion stationnaires. MA2N RAIRO 2L 7-26,

1987.

[29] P. Berg6, Y. Pomeau, Ch. Vidal: De I 'ordre dans Ie chaos . Hermann

1984.

186 FINITE ELEMENT I{ETHODS FOR FLUIDS

[80] C. Bernardi. Thbse de doctorat Sime cycie, Universit6 Paris 6' 1979.

[at1 C. Bernardi, G. Raugel: A conforming finite element method for the time-

dependant Navier-Stokes equations. SIAI{ J. Numer. Anal. 22 ,455-473,1985.

[32] S. Boivin: A numerical method for solving the compressible Navier-Stokes

equations. (to appear in IMPACT 1989).

[43] J. Boland, R. Nicolai<ies: Stability of finite elements under divergence

constraints.SlAM J. Numet'. Anal . 2A ,4.722-73I. 1983.

[34] J. Boris, D.L. Book: Flux corrected transport. J. Comp. Phys- 20 397-

431. 1976.

[35] A. Brandt: Multi-level adaptive solution to boundary value problems.

Malh. of comp.31, 333-39L, 1977.

[36] C. Brebbia: The Boundary Element Method for Engineers. Pentech, 1978.

[37] Y. Brenier: The transport-collapse algorithm, SIAM J. Numer Anal . 2t

p1013, 1984

t38] F. Brezzi, J. Douglas: Stabilized mixed methods for the Stokes problem.

(to appear).

t39] F. Brezzi: On the existence, uniqueness and approximation of saddle-point

problems arising from Lagrange multiplier-.. RAIRO Anal Numer. 8 , 129-

1 5 1 , 1 9 7 4 .

[40] M.O. Bristeau, A. Dervieux: Computation of directional derivatives at the

nodes: $2.4 in F. Thomasset: Implementation of f.nite element methods forNaaier-Stolces equations, Springer series in comp. Physics, 4&50, 1981.

t41] M.O. Bristeau, R. Glowinski, B. \{antel. J. Periaux, C. Pouletty: So-

lution of the compressible Navier-Stokes equations by least-squares and

finite elements. In Bristeau et ai. ed:Name rical simulation of compressible

Nauier-Stokes fl,ows. Notes on i\umerical f luid mechanics, 18 , Vieweg,

1987.

l42l M.O. Bristeau , R. Glowinski, B. lfantel, J. Periaux, G. Rog6: Adaptive

finite element methods for three dimensional compressible viscous flow

simulation in aerospace engineering. Proc. 11th Int. Conf. on Numerical

methods in fluid mechanics. Will iamsburg. \ ' irginia (1988) ( Springer, to

appear )[43] M.O. Bristeau, R. Glowinski, J. Periaux. I1. Viviand: Numerical sim'

ulation of compressible Nauier-Stokes fiou's . Notes on Numerical fluid

rnechanics, L8 , Vieweg, 1987.

l44l M.O. Bristeau, R. Glowinski, J. Periaux, O. Pircnneau, P. Perrier: On the

numerical solution of nonlinear problems of fluid mech. by least squares.

Comp. Meth. in Appl . Mech . . L7 lL& March 1979.

[45] R. Brun: Transport et relaration dans les i.coulements gazeux: Masson

1986.'[46]

L. Cafarell i , R. Kohn, L. Nirenberg: On the regularity of the solutions

of the Navier-stokes equations. Comm. Pure and Appl. Math. , 35 ,

771-831, 1982.

[47] J. Cahouet, J.P. Chabart: Some fast 3D finite element solvers for the

generalized Stokes problem. Int. J. Numer. Melhods. in Fluids . 8,

REFERENCES 187

869-895. 1988. : :

[ag] Cahouet, Chabart: Some fast 3-D Finite Element solvers for Generalized

Stokes problem. Rapport EDF HBl4ll87'03' 198

[49] T. Chacon: Oscillaiions due to the transport of cro-structures' SIAM

J. App l . Math .48 5 p112&1146 ' 1988 '

[50] T. ch".or,, o. Pironneau: on the mathematical foundations of the k - e

model. In Vistas in applieil mathematics . A.V. Balakrishnan ed. Opti-

mization Software inc. Springer, 1986'

[51] G. Chavent, G. Jaffr6: Mathematical methods and f,roite elements for rcser'

uoir sh,rulations. North-Holland, 1986'

[52] A.J. chorin: Numericalstudy of slightly viscous flow- J. Fluid Mech. 57

785-796. 1973.

[bg] A. J. Chorin: A numerical tr{ethod for Solving Incompressible viscous flow

problems. J. Comp. Physics ' 2, L2-26, 1967'

tb4] i. Christie, D.F. Griff iths, A.R. Mitchell, O.C. Zienkiewicz: Finite ele-

ment methods for second order differential equation-s with significant first

derivatives. rnt. J. Nurner. Meth. Eng ,10 , 1389-1396, 1976.

[bb] ph. Ciarlet: The Finite Element Method. for Ell iptic problems. North-

Hol land, 1978.

[56] P.G. Ciarlet Elasticitl trid,imensiannelle. Masson Ed.' 198S.

ifZi g. Cockburn: The quasi-monotone schemes for scalar conservation laws.

IMA preprint 277 , University of Minnesota, oct. 1986.

[b8] G. Comte-Bellot: Ecoulement turbulent entre deux parois planes. Publi-

cations scientifiques et techniques du ministEre de I'air' 1980'

[59] P. Constantin, c. Fioas, o. Manley, R. Temam: Determining modes and

fractal dimension of turbulent flows. J. Fluid Mech - l-50 , 427-440, 1985'

t60] M. Crouzeix - P.A. Raviart: Conforming and non-conforming finite ele-

ment element methods for the stationary Stokes eq. RAIRA . R3. 33-76,

1973 .

[61] C. Cuvelier, A. Segal, A.A. van Steenhoven: Finite Element Methods and

Nauier-Slokes equations. N{athematics and its applications series, D. Rei-

del Publishing Cc. 1986.

[62] R. Dautray, J.L. L ions: Analyses math| .mat iques et calculs num4'r iques.

Masson, 1987.

[63] J.W. Deardorff: A numerical study of 3-d turbulent channel flow at large

Reynolds numbers. J. Fluid Mech. 4l ,2,453-480, 1970'

[04] A. Dervieux: Steady Euler simulations using unslrtL,ctured meshes . Von

Karman Lecture notes series 188404. Mars 1985'

[6b] J.A. Desideri, A. Dervieux: Compressible Fluid Dynamics; cornprc,ssible

flo, solaers using unslruclurvd grids . Von Karman lecture notes series.

March 1988.

t6O] Q.Dinh, R. Glowinski, J. Periaux: Domain decomposition for elliptic prob-

lems . In Finite element in fluids. 5 R. Gallager, J. Oden, O. Zienkiewicz,

T. Kawai, M. Kawahara eds' p45-106. Wiley' 1984

[OT] J.M. Dominguez, A. Bendali, S. Gallic: A variational approach to the

vector potential of the Stokes probiem . J. Math. Anal. and Appl. I07,2,


537-560, 1987.

[68] J. Donea: A Taylor-Galerkin method for convective transport problems.

J. Nurner. Meth. Eng,2A, 101-120, 1984. (see also J. Donea, L. Quar-

tapelle, V. Selmin: An analysis of time discretization in the Finite Element

SJution of Hypeibolic problems. ,I. Comp. Phgsics, T0 2 1987-

[69] J. Douglas, T.F. Russell: Numerical methods for convection dominated

diffusion problems based on combining the method of characteristics with

finite elernent methods or finite difference method. SIAM J. Numer Anal.

1 9 , 5 8 7 1 - 8 8 5 , 1 9 8 2 .

[70] J. M. Dupuy: Ca]culs d'dcoulements potentiels transsoniques; rapport in'

terne Ecole PolYtechnique 1986.

[71] A. Ecer, H. Akai: Computation of steady Euler equations using finite

element metho d. AIAA . 21 Nx3, p343-350' 1983'

[72] F. El-Dabaghi, J. Periaux, O. Pironneau, G. Poirier: 2d/3d finite element

solution of ihe steady Euler equations . Int. i. Numer. Meth- in Fluid .

7 , p1191-1209. 1987.

[Zg] f . El-Dabaghi, O. Pironneau: Stream vectors in 3D aerodynamics. Numer.

Math. , 48 363-394, 1986.

F4] F. El-Dabaghi: Steady incompressible and compressibie solution of Navier-

Stok., "quutions by rotational correction. in Numerical met. for fluid dy-

namics. K.W.Morton and M.J. Baines. Clarendon press.p27&281. 1988.

[Tb] J. Essers: Computational methods for turbulenl transonic and aiscous

flows. Springer, 1983.

F6] J. Feder: Frac|als. Plenum Press. 1988.

lZZj nn.f"stauer, J.Mandei, J.Necas: Entropy regularization of the tansonic

Potential f low problem. Commentaliones Mathemalicae Uniuersit iatis

Carolinae. Prague, 1984. See also C. Necas: transonic flow Masson Mas-

son 1988.

t?8] F. Fezoui: Resolution des equations d'Euler par un sch6ma de Van Leer

en 6l6ments finis. Rapport Inria 358' 1985.

[Tg] C. Foias, R. Temam: Determination of the solution of the Navier-Stokes

equations by a set of nodal values. Math. Comput ., 43 , 167, 117-133,

1984.

[80] M.. Fortin: Calcul num6rique des 6coulements par la mdthode des 6l6ments.

Ph.D. Thesis, Universi te Par is 6, 1976.

[81] M. Fortin, R. Glowinski: Augmenled lagranqian methods . North-Holiand

1983.

[82] M. Fortin, A. Soulaimani: Finite element approximation of compressible

viscous flows. Proc. computational methods in flow analysis, vol 2, II.

Niki and M. Kawahara ed. Okayama [Jniuersily Science press ' (1988)

t83] M. Fortin, F. Thomasset: Mixed finite element methods for incompressible' f low problems. J. Comp. Physics . 31 113-i45' 1979.

[84] Ur Frisch, B. Hasslacher, Y. Pomeau: Lattice-gas automata for the Navier-- -

Stokes equation.Physical Reuiew letters. F6 , 14, 1505-1508, 1986^. .

[85] G.P. Galbi: Weighted residual energy methods in fluid dynamics ' Springer

1985.

REFERENCES 189

[86] D. Gelder: solution of the compressible flow equations. InL. J. Nume"7.

Meth. Ens. (3) 35-43, 1987'

[87] W. Gentzsch: vectorization of computer programs wilh application to cFD

. Notes on numerical fluid mechanics' 8 Vieweg 1984'

[88] p. Geymonat, P. Leyland: Analysis of the linearized compressible Stokes

fields. Tth ISABE conf . Beijing. 1985.

[g1] V. Girault: Finite Element Method for the Navier-Stokes equations

with non-standard boundary conditions in R3. Rapport du Laboratoire

d'Anall,se Num6rique de l 'Universit6 Paris 6 n0 R87036, 1987.

v.Girault, P.A. Raviart: Finite Element Approilmations of the Nauier-

.gloles Eq . Lecture Notes in Math' Springer ,1979'

V. Girault, P.A. Raviart: Finite Elements methods for Naaier-Slokes equa- '/

[e2]

[e3]l ions. Springer series SCM vol 5, 1986.

[g4 ] J . G l imm: Comm. Pure and App l . Math . 18 . p697. 1965.

igSj R. Glowinski: Numerical methods for nonlinear uariationnal problems.

springer Lectures Notes in computationnal Physics, 1985.

196] R. Glorvinski: Le 0 sch6ma. Dans M.O. Bristeau, R. Glowinski, J. Periaux.

Numerical methods for the Navier-Stokes equations. Comp. Phy. report

6 , 7 3 - i 8 7 , 1 9 8 7 .

[gT] R. Glowinski, J. Periaux, O. Pironneau: An efficient precondition-

irrg scheme for iterative numerical solution of PDE.Malhematical Mod-

el l ing,L979.

[g8] R. Glon,inski, O. Pironneau: Numerical methods for the first biharmonic

problem. SIAM Reuiew 2t 2 pl67-2I2. 1979.

[gg] J. Goussebaile, A. Jacomy: Application e la thermo-hydrolique des

m6thodes d'6clatement d'op6rateur dans le cadre 6l6ments finis: traite-

ment du modble ft - e . Rapport EDF-LNH HEl4Il85.Ll, 1985.

[100] N. Goutal: Rlsolution des 6quations de Saint-Venant en r6gime transcri-

t ique par une m6thode d'6l6ments finis. Thise, Universit6 Paris 6, 1987.

[101] P.M. Gresho.R.L. Sani: On pressure conditions for the incompressible

Navier-Stokes equations. in Finite Elements in Fluids 7 R. Gallager

et al ed. Wiley 1988.

[102] P. Gr isvard: El t ipt ic problems in non-smooth domains. Pi tman, 1985.

[103] M. Gunzburger: Ir{athematical aspects of f inite element methods for in-

compressible viscous flows. tn Finite elemenl, theory and application. D.

Dwoyer, \'[ Hussaini, R. Voigt eds. Springer ICASE/NASA LaRC series.

r24-t50 1988.

[104] L. Hackbush: The Multigrid Method: lheory and applications. Springer ',

series SCII, 1986.

[105] M. Hafez, W. Habashi, P. Kotiuga: conservative calculation of non isen-

tropic transor'ic flows. AIAA . 84 L929. 1983.


[106] L. Halpern: Approximations paraxiales et conditions absorbantes. ThEse

d'6tat. Universit6 Paris 6. 1986.

[107] J. Happel, H. Brenner: Low Reynolds namber hydrod,gnamics. Prentice

IIall 1965.

[108] F.H. Harlow, J.E. Welsh: The Ma,rker and cell method.P/r,ys. Fluids 8

,2182-2189, (1965).

[109] F. Hecht: A non-conforming P1 basisserie analyse numerique . 1983.

with free divergence in ,R3. RAIRO

[110] J.C. Henrich , P.s. Huyakorn, o.c. Zienkiewicz, A.R. Mitchell: An upwind

finite element scheme for the two dimensional convective equation. Int. J.

Num. Meth. Eng . 11- , 1831-1844, 1981'

[111] J. Ileywood-R. Rannacher: Finite element approximation of the nonsta-

tionnary Navier-stokes equations (I) SIAM J. Numer. Anal . 19 p275.

1982.

[112] M. Holt: Numerical methods in fluid dynamics. Springer, 1984'

iffg] p. Hood - G. Taylor: Navier-Stokes eq. using mixed interpolation. in

Finite element in fl,ow problera oden ed. UAH Press,L974.

114 K. Horiuti: Large eddy simulation of turbulent channel fl'ow by one-

equation modeling. J. Phys. Soc. of Japan. 54 , 8, 2855-2865, 1985.

[1lb] t.f .n. Hughes:A simple finite element scheme for developing upwind finite

elements. .Inl. J. Num. itteth. Eng .L2 , 1359-1365, 1978.

[116] T. J.R. Hughes: The f ,n i te elemenl method. Prent ice Hal l , 1987'

[f fZ] T.J. R. Hughes, A. Brooks: A theoretical framework for Petrov-Galerkin

methods with discontinuous weighting functions: application to the

streamline upwind procedure. In Finite Elemenls in Fluids, R. Gallagher

ed. Vol 4., Wiley, 1983.

[118] T.J.R. Hughes, L.P. Franca, M. Mallet: A new finite element formulation

for compubational fluid dynamics . Comp. Metl,. in Appl. Mech' and Eng'

63 97-112 (1987) .

[11g] T.J.R. Hughes, M. Mallet: A new finite element formulation for compu-

tational f luid dynamics.Computer Meth. in Appl. Mech. and Eng' 54

,341-355, 1986.

[120] M. Hussaini, T. Zang: Spectral methods in fluid mechanics. Icase report

8 6 - 2 5 . 1 9 8 6 .

[121] A. Jameson: T]ansonic flow calculations. In Numerical meihods in fluid

mec[. H. Wirz, J. smolderen eds. p1-87. McGraw-Hi i l . 1978.

lL22l A. Jameson, T.J. Baker: Improvements to the Aircraft Euler Method.

AIAA 25th aerospace meeting. Reno Jan 1987.

[12g] A. Jameson, J. Baker, N. Weatherhill: Calculation of the inviscid transonic

flow over a complete aircraft . AIAA paper 86-0103, 1986.

lLZ4) C. Johnson: Numerical solution of PDE by the f,nite elemenl' meihod ''

Cambridge university press, 1987.

[125] C. Johnson: Streamline diffusion methods for problems in fluid mechanics.

Finite elements in fluids, vol 6. R. Gallagher ed. wiley, 1985.

[120] C. Johnson, U. Nd,vert, J. Pitkiranta: Finite element methods for linear

hyperbolic problerns. Computer Meth. In Appl. Mech. Eng' 45 ,285-312,

REFERENCES

1985.

for compressible flows, T.E. Tedzuyar ed. AMD-78 , 1987.

[129] Kato: \on stationary flows of viscous and ideal fluids in 'R3 'J' Func' Anal

p261-287. 1979.

[132] M. Karvahara: Periodic finite element method of unsteady flow' Int' J'

Meth. in Eng. LL 7, P1093-1105. 1977.

[133] N. Kikuchi, T. Torigaki: Adaptive finite element methods in computed

aided engineering. Danmarles Tekniske HOJSKOLE, Matematisk rcport

i 9 1

198&09.

[134] S. Klainerman, A. Majda: Compressible andPurv Appl . Math.35, 629-651' 1982.

[135] R. Kohn, C. Morawetz: To appear. See also

solution for a transonic flow problem-Comm.

incompressible fluids. Comm

C. Morawetz: On a weak

pure and dppl . math. 38 ,

797-818,1985.

[130] S.N. Kruzkov: First order quasi-l inear equations in several independent

variables. Math USSR Sbornik, 10 , 2L7-243, 1970'

[132] Y. A. Kuznetsov: Matrix computational processes in subspaces. in Com-

pul ing methods in appl ied sciences and engineer ing VI .H. Glowinski , J 'L.

Lions ed. North-Holland 1984.

[1g8] O. Lad-r-zhenskaya: The Mathematical Theory of uiscous incomprcssrble

flows . Gordon-Breach, 1963.

[139] O.A. Ladyzhenskaya, v.A. Solonnikov, N.N. ouraltseva:paraboliques lin€aires el quasi-lin{.aires . MIR 1967.

Equot ions

[140] H. Lamb: I{ydrodEnamics. Cambridge University Press, 1932.

[tat1 1,. Landau- E. Lifschrtz: Mecanique des Fluides . MIR Moscou, 1953'

if+Zj e. Lascaux, R. Theodor: Analyse num|,rique malricielle appliqu|'e d I 'arl

de I ' inginieur . Masson, 1986.

[ i43] B.E. Launder, D.B. Spalding: Mathematical models of turbulence Aca-

demic press, 1972.

[i44] P.D. Lax: On the stability of difference approximations to solutions of

hyperbolic equations with variable coefficients. Comm. Pure Appl. Math .

I p267, 1961.

[145] P. Lar: Hyperbolic systems of conseruation laws and lhe mathematical

theory of shock waues, CMBS Regional conference series in applied math.

1"1 , slAli, 1972.[146] J. Leray, C. Schauder: in H. Berestycki, These d'6,tat, Universit6 Paris 6,

1982.

192 FINITB ELEMtrNT METHODS FOR FLUIDS

[147] A. Leroux: sur les systbmes hyperboliques. Thh,se d'6,tat. Paris 1979.

ir+Aj p. Lesaint: Sur la r6solution des systbrnes hyperboliques du premier ordre

par des m6thodes d'6ltlments finis. ThEse de doctorat d'6tat, Univ. Paris

6 1975.

[14g] P. Lesaint, P.A. Raviart: On a finite element method for solving the neu-

tron transport equation. In Mathernatical aspect of finite elemenls in PDE

. C. de Boor ed. Academic Press, 89-L23' 1974'

[1b0] M. Lesieur: Tturbulence in fl.uids . Martinus Nijhoff publishers, 1987.

frs11 J. Lighthil l : waues in fl,uids cambridge university Press,1978.

[ r rz ] L INfACK, [ rser ' � s gu ide . J .J . Dongara c .B . Mo ler , J .R. Bunch, G.W.

Steward. SIAM MonograPh, 1979.

[153] J.L. L ions: Quetrques Methodes de Resolut ion des

Nonlineaires . Dunod, 1968.

[154] J.L. Lions. controle des systemes gouaern',s par des

part ie l les Dunod, 1969.

[155] J.L. Lions, E. Magenes: Problemes aur l imites non

Problimes aur l imites

6qu ations aur d6.riu6. es

homogenes et applica-

t ions . Dunod 1968.

[lb6] J.L. Lions A. Lichnewsky: Super-ordinateurs: 6volutions et tendances,

la uie d,es sciences, comptes rcndus de I 'acad,6mie des sciences, s6rie

gdn6rale,1, Octobre 1984.

[1bi] P.L. Liorrr, Solution viscosit6 d'EDP hyperbolique nonlineaire scalaire.

S6minaire au Collbge de France. Pitman 1989.

[158] R. Lohner: 3D grid generation by the advancing front method. In Laminar

ond turbulent f lows. C. Taylor, W.G. Habashi, H. Hafez eds. Pinneridge

press, 1987.

[15g] R. Lohner, K. Morgan, J. Peraire, O.C. Zienkiewicz: Finite element meth-

ods for high speed flows. AIAA paper. 85 1531.

[160] R. Lohner, K. Morgan, J. Peraire, M. Vahdati: Finite element flux-

cogected transport (FEM-FCT) for the Euler and Navier-Stokes equa-

tions. in Finite Elements in Fluids 7 . R. Gallager et al ed. Wiley 1988.

[161] Luo-Hong: R6soluti6n des 6quations d'Euler en !I ' - u en r6gime

transsonique. Thise Universitd Paris 6. 1988.

[162] D. Luenberger: Optimization by aector space methods. Wiley 1969.

ifOal n. MacCormack: The influence of CFD on experimental aerospace facil-

ities; a fifteen years projection. Appendfu C . p79-93. National academic

press, Washington D.C. 1983.

[ld4] Y. Maday, A. Patera: Spectral element methods for the incompressible

Navier-Stokes equations. ASME state of the art surtsey in comp. mech. E.

Noor ed. 1987.

[165] A. Majda: Comprvssible fl,uid flows and systems of conseraation laws

Applied math sciences series, Springer Vol 53, 1984.

[i66] ],t. Mallet: A finite element method for computational fluid dynamics'

Doctoral thesis, University of Stanford, 1985.

[167] M. Mallet, J. Periaux, B. Stouffiet: On fast Euler and Navier-Stokes

solvers. Proc. 7th GAMM conf. on Numerical Methods in Fluid Me'

chanics. Louvain. 1987.

REFERENCES

[16g] A. Matsumura, T. Nishida: The initial value problem for the equation_s of

motion of viscous and heat conductive ga"ses. J. Math. Kyoto uniu . 20 ,

67-104, 1980.

[169] A. Matsumura, T. Nishida: Initial boundary value probiems for the equa-

tions of motion in general fluids, in Computer Melhods in applied sciences

and Engineering. R. Glowinski et al. eds; North-Holland, 1982.

[1?0] F.J. McGrath: Nonstationnary plane flows of viscous and ideal fluids'

Arch.Rat. Mech. Anal . 27 p328-348, 1968'

[171] D. Mclaughin, G. Papanicolaou, O. Pironneau: Convection of microstruc-

tures. Siam J. Numer. Anal - 45 ,5, p780-796' 1982'

l11zl J.A. Meijerink- H.A. Van der Vorst: An iterative solution method for linear

systems of which the coefficient matrix is a symmetric M-matrix. Math.

of Comp . 31, 148-162 ,L977.

[1?B] p. Moin, J. Kim: Large eddy simulation of turbulent channel flow. J.

F lu id . Mech. L18 , P341, 1982,

[1?4] K. Morgan, J. Per iaux, F.Thomasset: Analysis of laminar f low ouer a

bockward facing slep : Vieweg Vol 9: Notes on numerical fluid mechanics.

1985.

[1?5] K. IVlorgan, J.Peraire, R. Lohner: Adaptive finite element flux corrected V

transport techniques for CFD. rn Finite element, lheory and applicalion.

D. Dwoyer, M Hussaini, R. voigt eds. Springer ICASE/NASA LaRC se-

r ies. 165-174 1988.

[1?6] K.W. Morton: FEM for hyperbolic equations. In Finite Elemenls in

phgsics: computer physics reports vol 6 No 1-6, R. Gruber ed. North

Holland. Aout 1987.

ILTT) K.lV. Morton, A. Priestley, E. Suli: Stability of the Lagrange-Galetkin

method with non-exact integration. RAIRO-M2AN 22 (4) 1988, p625-

654.

[17S] f. NIurat: l ' injection du cone positif de l{-Ldans W-t'e est pacte

p o u r t o u t q < 2 ' J . M a t h p u r e s e t A p p l ' 6 0 , 3 0 9 - 3 2 1 , 1 9 8 1 '

[1ig] i.C. N.d.I"., A new family of f inite element in ,R3 . Numer m 50 ,

57-82. 1986.

[180] J. Necas: Ecoulemenls de fl,uides; compacit|, par entropie. Masson' 1989'

if3f] R.A. Nicolaides: Existence uniqueness and approximation for generalized

saddie-point problems. ; IAM J. Numer. Anal . 19 , 2,349-357 (1981).

[182] J . Nitsche, A. S.hutz: On ]ocal approximation propertie s of L2 -pro j ection

on spline-subspaces. Applicable analysis ,2 , 161-168 (L972)

[183] J.T. Oden, G. Carey: Fini te elements; mathemal ical aspects . Prent ice

Hall, 1983.

[184] J.T. Oden, T. Strouboulis, Ph. Devloo:for high speed compressible flows. inGallager et al ed. WileY 1988.

Adaptive Finite element methods

Finite Elements in Fluids 7 R.

193

[185] S. Orszag: Statistical theory of turbulence.(1973). in

Balian-J.L.Peule ed. Gordon-Breach, 234-374, Lg77 '

[186] B. Palmerio, V. Bil let, A.Dervieux, J. Periaux: Self adaptive mesh refine-

ments and finite element methods for solving the Euler equations. Pro-

Fluid dynamics . R.

194 FINITE ELEMENT METBODS FOR FLUIDS

ceeding of the ICFD conf . Reading 1985.

[18?] R.L. Panton: Incomprvssible flow. wiley Interscience,1984.

[188] C. Pa^res: Un traitement faible par elements finis de la condition de glisse-

ment sur une paroi pour les equations de Navier-Stokes. Note CRAS. 307

I p101-106. 1988.

[189] R. Peyret, T. Taylor: Computalional methoils for fluid flows. Springer

series in computational physics, 1985.

i190] O. Pironneau: On the transport-diffusion algorithm and its applications

to the Navier-Stokes equations. Numer. I{ath. 38 , 309-332' 1982.

[191] O. Pironneau: Conditions aux limites sur la pression pour les equations

de Navier-Stokes. Note CLAS.303,i , p403-406. 1986.

[192] O. Pironneau, J. Rappaz: Numerical analy-sis for compressible isentropic

viscous flows. (to appear in IMPACT' 1989)

[193] E. Polak: Compulational methods in optimization. Academic Press, 1971.

[194] B. Ramaswami, N. Kikuchi: Adaptive finite element method in numerical

fluid dynamics. in Computational methads in flnw analysis. H. Niki, M.

Kawahara ed. Okayama University press 52-62. 1988.

[195] G. Raugel: These de Seme cycle, Universitd de Rennes, 1978.

[196] G. Ritcher: An optimal order error estimate for discontinuous Galerkinmethods. Rutgers University, conxputer science reporl Fev. 1987.

[197] R. Ritchmeyer, K. Morton: Differvnce melhods for init ial ualue problems.

Wiley 1967.

[198] L. Ptizzi, H.Bailey: Split space-marching finite volume method for chemi-

cally reacting supersonic flow. AIAA Journal, L4 ,5,62L-628' 1976.

[199] A. Rizzi, H. Viviand eds: Numerical methods for the computation of in-

uiscid transonic flows with shock waaes. 3 , Vieweg, 1981.

[200] W. Rodi: T\rrbulence models and their applications in hydrolics. Inst.

Ass. for Hydrolics. State of the art paper. Delft. 1980..

[201] G. Rog6: Approximation mixte et acceleration de la convergence pour

les 6quations de Navier-Stokes compressible en elements finis. Thise ,Universit6 Paris-6 (1989).

12021 E. Ronquist, A. Patera: Spectral element methods for the unsteady Navier-

Stokes equation. Proc. Tth GAMM conference on Numerical methods in

fl,uid mechanics . Louvain la neuve, 1987.

[203] L. Roaenhead ed.: Laminar Boundary layers. Oxford University Press.

1963,

[204] Ph. Rostand: Kinetic boundary layers, numerical simulations. Inria repttrt

728,L987.[205] Ph. Rostand, B. Stouffiet: TVD schemes to compute compressible viscous

flows on unstructure meshes. Proc. 2nd Int Conf. on hyperbolic problems,

Aachen (FRG) (To appear in Viewes ) (1988).

[206] D. Ruelle: Les attradteurs 6tranges'La recherche, 108, pL32 1980'

[207] D. Ruelle: Characteristic exponents for a viscous fluid subjected to time

dependant forces.Com. Math. Phys. 93 , 285-300' 1984'

[208] I. Ryhming: Dgnamique d,es fl,uides Presses Polytechniques Romandes,

1985.

REFERENCES

B. Saramito: Existence de solutions stationnaires pour un problbme de'

fluide compressible. DPH-PFC 1224 (1982)'

Y. Saad: Krylo\ subspace methods for solving unsymmetric linear systems'

Math of CornP. 37 105-126, 1981'

J.H. Saiac: On the numerical solution of the time-dependant Euler equa-

tions for incompressible flow. Int. J. for Num. Meth. in Fluids ' 5 ,

195

[2oe]

[210]

[21 1]

637-656 , 1985.

1212) N. Satofuka: Unconditionally stable

tions of compressible viscous flows'

f lu id mech. 7 , V ieweg. 1987 '

explicit method for solving the equa-

Proc. 5th Conf. on nulner. meth' in

difference simulations of tur-

Comp. Physics- 18 376-404'[213] V. Schumann: subgrid scale model for finite

bulent flows in plane channel and annuli' J'

1975.

[214] L.R. Scott, M. Vogelius: Normdivergence oPerator in sPaces of

estimates for a maximal inverse of the

piecewise polynomials. RAIRO L{LAN '

19 111-143. 1985.

[215] V.P. Singh: Rdsolution des 6quations de Navier-Stokes en 6l6ments finis

par des -6thod.r de d6composition d'op6rateurs et des sch6mas de Runge-

kutta stables. Thise de Sime cycle, universit6 Pars 6, 1985'

[216] J.S. Smagorinsky: General circulation model of the atrnosphere. Mon'

Weather Reu . 91 99-164, 1963'

[217] A. Soulaimani, M. Fortin, Y. Oue]let, G. Dhatt, F. Bertrand: simpie

continuous pressure element for 2D and 3D incompressible flows. Comp'

Meth. in Appl. Mech. and Ens ' 62 47-69 (1987)'

[21g] C. Spezialer -On

non-linear K-l and /{ - e models of turbulence. J. Fluid

Mech. 178 P459. 1987.

[219] C. Speziale:T\rrbulence modeling in non-inertial frames of references'

ICASE rePort 88-18.

[220] R. Stenberg: Analysis of mixed finite element methods for the Stokes

problem; a unified approach. Math of comp .42 9-23. 1984.

t221] b. Stoufiflet: Implicite finite element methods for the Euler equations. in

Numerical method for the Euler equations of Fluid ilynamics. P. Angrand

ed. SIAil{ series 1985.

12221 B.stouffiet, J. Periaux, L. Fezoui, A. Dervieux: Numerical simulations

of BD hypersonic Euler flows around space vehicles using adaptive finite

eiements AIAA paper 8705660' 1987'

l22B) G. Strang, G. Fix: An analysis of the finite element method . Prentice

Hall 1973

l214l A.H. Stroudl: Approdmcte calculation of multiple integrals . Prentice-Hall

1971 .

[2?b] C. Su]em, P.L Sulem, C. Bardos, U. Frisch: Finite time analycity for the

3D and 3D Kelvin-Helmholtz instability. comm. Math. Phys. , 80 ,4 8 5 - 5 1 6 , 1 9 8 1 .

12261 E. Suli: Convergence and non-linear stability of the Lagrange-Galerkin

method for the Navier-Stokes equations. Numer. Math. 53 p459-483'

1988.

196 FINITE EI,EMENT METHODS FoR FLUIDS

1227) T.E. Tezduyar, Y.J. Park, H.A. Dear$llfj Finite element procedures for

time dependent convection-diffusion-rea.cl,ion systems. in Finite Elements

in Fluids 7 . R. Gallager et al ed. Wiley 1988'

l22S) R. Temam :TheorE ond Nu*erical Analysi.s of the Nauier-Stokes eg . North-

Holland 1977.

[22g] F. Thomasset: Implementation of Finite Elernent Methods for Naaier-

Stokes eq . Springer series in Comp' Phyriics 1981'

[2g0] N. Ukeguchi, tl. Sataka, T. Adachi: On the numerical analysis of com-

pressible flow problems by the modified 'f l ic'method. Comput- f luids. 8,

2 5 1 , 1 9 8 0 .

[2g1] A. Valli: An existence theorem for cotnf)ressible viscous flow. Boll Un.

Mat . I t . Ana l . Funz . Appt . 1&C, 3L7- i l2 ! t ,1981 '

l2ZZl B. van Leer: Towards the ultimate consr:rvation difference scheme III. /.

Comp. Phgsics . 23 ,263-275,1977.

[2Bg] R. Verfurth: Finite element approximaLion of incompressible Navier-Stokes

equations with slip boundary conditions. Numer Math 50 ,697-721'1987.

[284] R. Verfurth: A preconditioned conjugal,r: gradient residual algorithm for

the Stoke, probi"*. in R. Braess, W. l lackbush, U. T[ottenberg (eds)

Deaelopments in multigrid method,s, Wilr:y, 1985'

[2gb] p.L. Viollet: On the modelling of turbult:nt heat and mass transfers for

computations of buoyancy affected flows. Proc. Int. Conf. Num. Meth.

for Laminar and Tltrbulent Flows - Yertcnia, 1981'

[230] A. Wambecq: Rational Runge-Kutta mr:t,ltods for solving systems of ODE,

Comput ing ,2A , 333-342, 1978.

[237] L. Wigton, N.Yu, D. Young: GMRES :rc<:r:leration of computational fluid

dynamic codes. AIAA paper 85-1494, 'p67-74' 1985'

[238] P. Woodward, Ph. Colella: The numerir:;r'l eimulation of 2D fluid

flows with strong shocks. J. comp. Pltusnt:s ,54 , LI5-173, 1984.

[23g] S.F.Wornom M. M. Hafez: Implicit corrslrvative schemes for the Euler

equat ions . AIAA Journai 24,2, 215-223, l { ) lJ6 '

[240] N. Yanenko: The method of froctional sl,ryn. Springer' 1971.

iz+ti O.C. Zienkiewicz:Thi f inite element rttrlhod in engineering science.

McGraw-Hill 1977, Third edition.

1242) W. Zil j: Generalized potential f low theor.y nrrtl direct calculation of veloci-

ties applied to the numerical solution of l,lrc Navier-Stokes equations. Inl.

J . Numer . Meth . in F lu id 8 ,5 ,599-612 ' l { )88 '

Appendix

MacFEM:A finite element software for the Macintosh

1. INTRODUCTION

example).io i"u.h this aiin, we have used fully the graphic interface of modern

workstations but since we wanted inexpensive equipment we have used Apple's

MacintoshTM. The rlrawback is of course that N{acFEI\I is not portable to

other machines; it runs only on any 'Mac' with at least 512K bytes of memory

(display of results is in colour on a Mac II).

2 . FUNCTICNS

MacFEM is an application program to help the input of data for the finite

element methods on triangles in dimension 2. It is menu and mouse driven and

allows the user to define graphically the domain O and the data of the PDE

(right hand side and boundary conditions for example)

198 FINITB ELEMENT METHODS FOR FLUIDS

- The domain of the PDB Q is defined by its triangulation which is inputentirely with the mouse by selecting functions in the menu .

Figure A.1 : MacFEM 's n1,enu, bar with a selection being made (itemnew ) from the Menu File .

-Boundary conditions are input analytically (ex. dlr = x*!-1).A numberi can be attached to any part of the boundary to allow different formulas ondifferent segments. Similarly Q can be divided into subdomains and a differentnumber j can be attached to each part. The analytical functions that definethe right hand side and the boundary condition can also refer to i or j.

This part of MacFEM is written in Pascal and cannot be modified (5000instructions for the Mac expert because it calls the Toolbor .).But the l ibraryof programs for the solutions of the PDE by the finite element method is writtenin FORIRAN and the sources are available to the users (it is necessary to usethe Microsoft or Absoft cornpilers to make modifications to this part).

At the time of writing the following modules are available:1. Laplace equation with any O, f , gr,g :

Plr, = er, fflr, - n.Applications :Incompressible inviscid ffows (Chapter 2).

Methods: The P1 finite elemert method. the l inear svstem is solved bv aCholesky factorization.

2. The transonic equation with any Q, gr,g in a quasi subsonic regime:

lVel < 1.2,

v.(1 - f,VrfY=vp= 0 in o, elr, = pr, U*lr, = n.

-4,9 - / in Q, ( 1 )

(2)

Applications : compressible potential subsonic and weakly transonic flows (Cup-ture 2)

Method: The Pl f inite eleirrent method, the l inear system is solved by uCholesky factorization. Gelder's algorithm with upwinding ir, la Hughes in thesupersonic zones .

3. The convection diffusion equation with any Q, f , go,gr,g i

Edi t Modi fg 0uner Rld Enter Dota Create Macror

APPENDIX 199

t ':;;:i:

p , t *uYg -uLg= / i n Q ,p ( t , 0 )= po ( r ) , P l r , =P r , f f b " -n .

( 3 )

Applicationl : Convection-Diffusion (chapter 3).

Methotls: The P1 finite element method and discretisation of the total

derivative (z = 0 is allowed).

4. The Stokes problem in rlt - c.r with mY O, f ,rlrr,g ,

L'4, - / in o , rb l r = , l ' r , H

=g on | . (4)

Applications : The Stokes problem(chapter 4)

Methods: The Pl f inite element method on a mixed formulation in tls,u;

the boundary operator rlr -* 0t!/0n is built and solved by a Choleski factor-

ization (see Glowinski et al.[98J et al. for more details ).

5. Solution of the general 2nd order scalar PDE

ae -�V.(AV,p) - / in O elr, = er, kp * ffib, - n, (5)

where O, c , An,Atz - Azr ,Azz , f ,9 r ,9 ,& are any func t ions so long as the

problem remains strongly elliptic.Application : Tool to solve other problems (such a^s time dependant prob-

lems)Methods: The Pl f init 'e element method, the l inear system is solved by u

Cholesky factorization.

6. The GMRES subroutine, a Stok-es equation in P1/Pl*bubble and an

incompressible Navier-Stokes equations solver are under development.

3. INPUT OF DATA.

3.1 TriangulationTo create a triatrgulation one

use the following functions- Add nodes or triangles- Remove nodes, individually- Move nodes and glue them- Symmetries.

can start from pre-triangulated domains and

or by regionto other nodes.

Triangulations are created in two steps; first a sketch is created with few

triangles, then it is refined, the scales are defined and the nodes are adjusted

to their exact positions as necessary.

200 FIIITE ELBMENT METHODS FOR FLUIDS

Figure A.2

Figure A.3

In the second steP one can also:-refine the mesh b1' dividing all the triangles of a region into

triangles.-regularize a regicn by having each node moved to the center

bours.-Changediagonals. move, add or remove nodes'

Figure A.4

four smaller

of its neigh-

F l l e Ed i t Rid Enter Dota Create Macroe

u , u a t - t : =Rdd Nodef temoue NodeMoue NodeMoue I6 l uef hotrgr' 0iugonn{s0iuit l t l In ZoomSrrroc{h 2oom He5; ior t

fn ter 0atarquare 2H2Square 5n5Square 4u4Squore 416Squsre 6H4

Circle 2x6Circle 5xBCirc le 4u l0

Neru Trlanqulat lon

Square uxhCircle uxh

a t i le Ed i t Rid Enter 0ata Create Macros

Hdd NodeBemoue NodeMoue NodeVDtrt l l i Flt teChange D lagona l sI l iu ide in ZoomSmoo th zoom Reg ion

APPENDIX

#triangulation#

201

different on each part of the boundary

Figure 5: ExamPle; This

angles 7nd Port of a ring and

hmn. Number 2 is assigneil la

3.2 Definition of data'

In most practical ca"ses' the data are

so it is necessary to number these parts'

lriangulalion was obtained by glueing two rect-

lhen refi'ning he result; cost of lhe operalions:

the region defined by the clased curue

Figure A.6: To each part of Lhe boundary an inleger is assigned by speci-

fying the first oni tort poini of thi part of the bound.ary in lhe counter clockwise

d,irection.

Then a compiler of expressions allows the input of 9a9!r. function' Prede-

fined dialogue templates are used for this' The compiler builds allays of values


of the user's functions at the vertices or at the center of gravity of the triangles;

these values are stored in a text file for future use.

t f J t t tvrurr ' nIr| 'g ut lr l l rr l

Solue: -Az-d in O, z'ei sr dzlon'ei on aQi. CBUTION:urite onlg on the highl lghted (black)l ine.

x+U

Boundarg l iz-et; enter el here

Boundarg 2:z-e?; enter e2 here

Boundorg 5:z-e51 enter e5 here

Eoundarg 4 :z-e4; enter e4 here

Boundarg 5:z-e5; enter e5 here

Boundorg 6:az/an -e6; enter e6 Boundary 7 :oz/on =e7; enter e?

Figure A.7: The pred.efined dialogue template to build the lwo array fi les

f and e made of the aalues of the right. hand side of a Laplace equation (f ) and

of the aalues of the boundary conditians on the boundary uertices (e). Here

lhe user has entered f = x * A and the boundary cond"i t ions are about to be

specified. This takes less than 30 sec.

Figure A.8: Another dialogue template; in lhis one the user specif,es his

data by a sftprt Pascal-l ike inslruction.

3.3 Solutions of the PDEsThe solution part is a separate application (shown below as the first icon)

that can be initiated from I\{acFEM or from the finder. Here, once initiated,

PDDSolver will read the data array e, d prepared by MacFEM and produce

an output file array, here Laplacian which can be plotted by MacFtrM. Note

that the solution can be also carried out on another computer also, if the

files e and d are transferred; this is easy because they are text files and they

Should Uou use another or rou of ssme r ize ber ide H,g, l enter nome:

l f l - f t hen e : -0 e l se i t t - 2 t hen e : - l e l se i f i ' 6 t hen e : -

AppENDIX 203

can be transferred by any communication program' The file tube contains the

triangulation data; for Macintosh use it is compacted but it can be stored

unpacked if it is to be sent to another computer'

f f i H HTrS. o d

Figure 9: Various icons of f'Ies generated by MacFEM

Trianqulat ion

6 Fortran sgstem

t ibMACFEM

O MS TOBTBRNC P0E-SoluerC] Source FileeD Subroutinee

Figure A"tO: Selection of lhe resolution module'

S o l u t l o n Z : Znox- 'l .O0O Zrrin' .0OOO press <CRr

GPD€-Solver

ffiMrcfEM

HLpbcir

o lUarp 9 SCSI

fTs{T-lt-ffi-l

;"** .,[ 0pen I

ftance-l

Figure A 11: Results.MacPlus it to,ok 6 sec.

In the present case lhere were 161 nodes and on

Q TronsonicS aplE U t i l i t i e s


3.4 Visualization of results.The results of the computation are stored in a fiIe; the contours can be

plotted (including a zoom on a part of the domain); a plot with colour patchescan be made if on a MacII. The results can also be visualized as a curve alonga user defined segment.

4. EXAMPLES.

4.1 Compressible flow in a curved pipeEqdation (2)must be solved with p = 0 at the inlet, p)g/0n = 0.4 on the

lateral walls and 0pl0n = A.25 at the outlet (the lower vertical part of theboundary). The flow has a supersonic pocket and the shock can be seen.

R0 (lloch t ir 0.6339: Zrax- .9680 Znin: .668? prcss <C-Rr

Figure A.LZ :Transonic flow in a curtted channel.

4.2 Tlansonic flow with shocks.Again (2) must be solved with the same boundary conditions except

the outlet where g = 0.25 and on the following geometry (nozzle):

Figure A.13 : Triangulation; symmetry is used to reduce time.

APPENDIX 205

RO (llqctt I lr 0.6339: Znqx' .8?09 Zrln- '4599 pmsr <CR>

Figure A.14 :Resutts; the supersonic zone and the shock can be seen'

Upwinding is cames out with values on the upstream triangles. Ilowever

Gelder,s algorithm is not very stable in this case and there seem to be two

subsequences each converging to a different limit' We plan to improve this

computatio

Index

AAbstract least square 49

Accoustics 22Adam-Bashforth 82Artificial viscosity 80

Attractor 131

BBBL condi t ion 115,181(B abuska- Br ezzi-Lad y zhe nskay a)

Barycentric coordinates 33,64

Bellman-Gronwall lemma 63

Beltrami (Problem of ) 52

Bernoulli larv 28

Boundary layer 58Boundary element method 11

Boussinesq approximation 21

cCavity problem 96,126

Centered scheme 59

CFL condition(Courant- Friedrisch-Levy) 92,140

CFD 11Characteristics 62Choleski 37Clebsch variables 56Coercivity 30Compressible flow 20,42,167,I7 3

Condensation of mass 66'137

Conforming (element) 10

Conjugate gradient 34' 1 1 1

Conjugate gradient (noniinear) 46

Conservation law 162,171

Conservative form 171

Conservative scheme 88

Convection 60,76Convection-diffusion 75,84

Crank-Nicolson 166

DDiffusion 75Diffusion (streamline) 68

Dirichlet (problem) 29,49

Drag 17

EElement, Pl bubble IPL 101,144- P1 iso P2lP1 104- Pl nonconforming/P0 106- P 1 1 0- P k 1 0- P2lP1 105Ellipticity 28Enthalpy 123Entropy 19,123,169Buler eq. (incompressible) 132- eq. (compressible) 53'161- scheme 78Explicit schemes 137

FFCT 167Finite differences 82,171- elements 10- volume 14,73,169Flow, compressible 42, 173- incompressible 27,125- irrotational27,42- potential27,42- transonic 42,47- viscous 95,125FIux Corrected tansPort 167Flee boundary L3,L77Froude number 178Fractional step 83

GGalerkin method 31,128Gelder's algorithm 47GI\{RES 142Green's formula 30

HHausdorff dimension 130Helmoltz' decomposition 51,52Hermite element 10Hyperbolicity 21,25

I tJrKImplicit schemes 79Incomplete factorization 37Incompressible flow 27 .I25Inertial manifold 130Inf-sup condition 115,181Interpolate 10Internal approximation 10,31Inviscid flow 2I,27, 131, 167Joukowski condition 39k - e modelKnuCsen layer I2,I75

LLagrangian element 10Laplace equation 22Lax scheme 66Lax-Milgram lemma 31Lax-Wendroff scheme I 1.165Leap-Flog 82Lifr 40

M , NMach number 170Mass lumping 66,137,167NACAOO 12 173Navier-Stokes eq.- incompressible 125- compressible 172Neumann problem 28Newton algorithm 139Newtonian flow 17Node 10Non-conforming element 10Nozzle 28

PPanel method 11Parabolic 25Parallel computing 12Particle method 12Petrov-Galerkin 68Penalization 45Poincard inequality 30Potential (function) 22Potential (vector) 51Preconditioning 34,46,II2Predictor-corrector 166

RRankine-Hugoniot conditions 54Rayley-Benard problem 21Reichard law 155Reynolds number 24Reynolds tensor 152Runge-Kutta scheme 85

SSaddle point 115Saint Venant's problem L3,I77semi-implicit schemes 137Shallow water eq. 13,177Shocks 47Slip boundary cond. L2I,I55Smagorinsky model 153Sound speed 47Stokes problem 24Stokes theorem 98Stream function 29Streamlines 20Streamline upwinding 68Stress tensor 17suPG 69,93,167

TTaylor-Galerkin scheme 91, 165Temperature equation 210 scheme 83,139Total derivative 84,L42Tlansonic 23,42tiangulation regular 10Triangulation uniform 10Turbulence 151

U , V , . wUpwinding 64Upwinding by cell 73,168Upwinding by discontinuity 94Uzawa algorithm 111Variational formulation 29Variational inequality 48Viscositv 17Viscosity (artificial) 80, 165Von Karman constant 155Wave 21Wing 39

APPENDIX II

Figure 1 : Computation of a compressible inuiscid flow around a 2Dcross section of a space shuttle by a f,nite element method on the compressihleEuler eqaalions. This picture shows the power of the finite element methodfor automatic adaptiue rnesh ref,nement within the iteratiae algorithm for thesolution of the PDE; the mesh is rc,fined. when the gradient of lhe aelocity islarge. The width of the shoc improues as the the mesh becomes small. Thenum,ber of uertices does not incrc.ase so much since it goes from 70/ to 1109and finally to 1896. (Courtesy of Dassault industries, AMD,BA).

Figure 2 : Mach numbers on the surface of an airVlane obtained by solaingthe potential lransonic equalion by a finite elemenl melhod of degree 1. (TheMach at infinity is 0.85 and lhe angle of attack is L0" ). (Courtesg of Dassaultindustries, AMD-BA).

II

IiI

. tI

Figure 3 : Computalion of the conuection of a passiue spot by an irro-

tationnal f low in a nozzle; the method used is the discretization of the total

deriuatiue. (computation done with MacFEM on a MacII).

Figure 4:Zoom on a part of the domain of computat ion for the s imulat ian

of the incompressible Naaier-Stokes equalions in a nozzle modeling an air ' intake

ind discretized with the P1 iso P2 / P1 element and a method of abstract least'

square. (Re=750, angle of incidence 40o ). (Courtesy of Dassoult industries,

AMD-BA).

(Next pages) :

Figure 5 : Mesh on the boundary and pressure n-L&p for a simplif 'ed model

of a ci and, its plane of symetry. The results haae been obtained with lhe

Pl-bubble-P1 element and a discrelization of the total deriuatiue for the in-

corn[)ressible Naaier-Stolces equations. (Computed by F. Hecht and C. Paris:

INnIA * cooperation with PSA).

Figure 6 : Solution of the conxpressible Euler equations around' a space

shuttle by |he finite elemenl method of degree I and a MUSCL type upwind'

ing (Osher). The lrfach number ot infinily is 25 and 40o of angle of allack.

(Courtesy of Dassault industries, AMD-BA).

Figure 7 : Viscous compressible fl,ow around a wing profi le at Mach 2 and

Reynolds number 2000. The fi,nite element method, is also of degree I for the

pressure (top) the density (bottom) and the uelocity (the middle picture shous

the Mach number). (courtesy of Dassault industries, AMD-BA).

Figure 8 : Compressible uiscous fl,ow around an isolated air intake con'L'

puled by the sarne rnethod as in figure 7. (Re=200, Mach at inf,nity=A.9,

incidence 40" ).(Courtesy of Dassault industries, AMD-BA).

.:. tlj',tii,

6l:

Documents

Finite Element Methods for Fluids