Computational Fluid Dynamics-First

8/13/2019 Computational Fluid Dynamics-First

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Computational Fluid Dynamics

Fluent Modeling CourseFirst: An Introduction to CFD

Lecturer: Ehsan.A.Saadati

Sharif Uniersity of !echnology

"#$ %roup&!ehran: Second Edition Fall '(()

ehsan.saadati*gmail.com

###.petrodanesh.ir


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Contents

+. Mathematical Modeling

+. %oerningE,uations'. !he %eneral Scalar !ransport E,uation

'. -umerical Methods

. Mesh !erminology and !ypes

. Discretiation Methods

.

Solution of Discretied E,uations. Accuracy/ Consistency/ Sta0ility and Conergence


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Contents

1. Fluid Flo#+. Discretiation of the Momentum E,uation

'. Discretiation of the Continuity E,uation

1. !he SIM2LE Algorithm

3. !he SIM2LE4 Algorithm

5. !he SIM2LEC Algorithm

6. Direct s. Iteratie Methods

7. Multigrid Methods

3. Solution 4esiduls


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Mathematical Modeling

%oerningE,uations

!he Momentum E,uation 8 Considered in 9&Direction

!he Energy E,uation


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The differential momentum equation for a Newtonian fluid with constant

density and viscosity

+

+

+

=

+

+

+

2

2

2

2

2

2

z

u

y

u

x

u

x

p

z

uw

y

uv

x

uu

t

u

+

+

+

=

+

+

+

2

2

2

2

2

2

z

v

y

v

x

v

y

p

z

vw

y

vv

x

vu

t

v

( ) ( ) ( )0=

+

+

+

z

w

y

v

x

u

t

+

+

+

=

+

+

+

2

2

2

2

2

2

z

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

w

Convection Piezometric pressure gradient Viscous termsLocal acceleration

Continuity Equation ( ) ( ) ( )

0=

+

+

z

w

y

v

x

u


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%oerningE,uations

!he Species E,uation

The equation for the conservation of mass for a chemical specie i may "e written in

terms of its mass fraction# Yi# where Yi is defined as the mass of species i per mass ofmi$ture! %f &lic'(s law is assumed valid# the governing conservation equation is

Giis the diffusion coefficient for Yiin the mi$ture and Riis the rate of

formation of Yithrough chemical reactions!


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Mathematical Classification of

2artial Differential E,uations

Elliptic2artial Differential E,uations

Let us consider steady heat conduction in a *ne+

,imensional sla"# as shown in &igure! The governing equation

and "oundary conditions are given "y

This simple pro"lem illustrates important properties of elliptic

P,Es! These are-

.+The temperature at any pointx in the domain is influencedby the temperatures on both boundaries.

/+%n the a"sence of source terms# T(x)is "ounded "y the

temperatures on the "oundaries! %t cannot "e either higher or

lower than the "oundary temperatures!


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;yper0olic2artial Differential E,uations

consider the one+dimensional flow of a fluid in a channel# as shown in &igure! The velocity of the

fluid# 1# is a constant2 also 13! &or t35# the fluid upstream of the channel entrance is held at

temperature T! The properties 6 and Cp are constant and '5! The governing equations and "oundary

conditions are given "y-


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2artial Differential E,uationsThe solution to this pro"lem is-

The solution is essentially a step in T, traveling in the positive x direction with a velocity U, as

shown in igure!

8e should note the following a"out the solution-

.! The upstream boundary condition (x"9 affects the solution in the domain! Conditions

downstream of the domain do not affect the solution in the domain!

/! The inlet "oundary condition propagates with a finite speed# U.

:! The inlet "oundary condition is not felt at pointx until t" x#U


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!emperature


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2ara0olic2artial Differential E,uations

Consider unsteady conduction in the sla" in &igure! %f k,

and Cpare constant, $nergy Equation may "e written in terms

of the temperature Tas!

%t is clear from this pro"lem that the varia"le tbehaves

very differently from the variablex. The variation in tadmits

only one%way influences, whereas the variablexadmits two%

way influences. tis sometimes referred to as the marching or

parabolic direction.


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Some More Specification a0out 2ara0olicE,uations Are:

.+ The "oundary temperature T& influences the temperature T(x,t) at every point in

the domain# =ust as with elliptic P,E>s!

/+ *nly initial conditions are re'uired (i.e., conditions at t"9! No final conditions are

required# for e$ample conditions at tinfinite! 8e do not need to 'now the future to solve

this pro"lem?

4+ The initial conditions only affect future temperatures, not past temperatures.

+ The initial conditions influence the temperature at every point in the domain for all

future times! The amount of influence decreases with time, and may affect different

spatial points to different degrees!)+ @ steady state is reached for t infinite ,Aere# the solution "ecomes

independent of Ti(x,&).%t also recovers its elliptic spatial "ehavior!

0+ The temperature is "ounded "y its initial and "oundary conditions in the a"sence

of source terms!


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=ehaior of the Scalar !ransport E,uation:

The general scalar transport equation we derived earlier Bmentioned "elow9 has much

in common with the partial differential equations we have seen here!

The elliptic diffusion equation is recovered if we assume steady state and there is noflow!

The same pro"lem solved for unsteady state e$hi"its para"olic "ehavior!

The convection side of the scalar transport equation e$hi"its hyper"olic "ehavior!

%n most engineering situations# the equation e$hi"its mi$ed "ehavior# with the

diffusion terms tending to "ring in elliptic influences# and the unsteady and convection

terms "ringing in para"olic or hyper"olic influences!


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Mesh !erminology and !ypes

The physical domain is discretized "y meshing or gridding it! The fundamental unit of

the mesh is the cell Bsometimes called the element9! @ssociated with each cell is the

cell centroid. cell is surrounded "y faces, which meet at nodes or vertices. n three

dimensions, the face is a surface surrounded "y edges. %n two dimensions# faces

and edges are the same!


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-ode&=ased and Cell&=ased Schemes

Numerical methods which store their primary un'nowns at the node or verte$

locations are called node%based or vertex%based schemes. Those which store them

at the cell centroid# or associate them with the cell# are called cell%based schemes.

inite element methods are typically node+"ased schemes# and many finite volume

methods are cell+"ased! &or structured and "loc'+structured meshes composed of

quadrilaterals or he$ahedra# the num"er of cells is appro$imately equal to thenum"er of nodes# and the spatial resolution of "oth storage schemes is similar for

the same mesh! &or other cell shapes# there may "e quite a "ig difference in the

num"er of nodes and cells in the mesh! &or triangles# for e$ample# there are twice as

many cells as nodes# on average! This fact must "e ta'en into account in deciding

whether a given mesh provides adequate resolution for a given pro"lem! &rom the

point of view of developing numerical methods# "oth schemes have advantages and

disadvantages# and the choice will depend!


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4egular and =ody&fitted Meshes

%n many cases# our interest lies in analyzing domains which are regular in shape-

rectangles# cu"es# cylinders# spheres! These shapes can "e meshed "y regular

grids# as shown in &igure! The grid lines are orthogonal to each other# and conform

to the "oundaries of the domain! These meshes are also sometimes called

orthogonal meshes!

a-egular and


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&or many practical pro"lems# however# the domains of interest are irregularly shaped

and regular meshes may not suffice! @n e$ample is shown ! Aere# grid lines are not

necessarily orthogonal to each other# and curve to conform to the irregular geometry!

%f regular grids are used in these geometries# stair stepping occurs at domain

"oundaries# as shown! 8hen the physics at the "oundary are important indetermining the solution# e!g!# in flows dominated "y wall shear# such an

appro$imation of the "oundary may not "e accepta"le!


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Structured/ =loc> Structured/ and Unstructured Meshes

The meshes shown in &igure aare e$amples of structured meshes. *ere, every

interior verte$ in the domain is connected to the same num"er of neigh"or vertices!

&igure 0shows a "loc'+structured mesh! Aere# the mesh is divided into "loc's# and

the mesh within each "loc' is structured! Aowever# the arrangement of the "loc's

themselves is not necessarily structured!

&igure a

&igure "


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Structured %rid Created 0y an Elliptic %enerator for a -ACA Airfoil


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Structured/ =loc> Structured/ and Unstructured Meshes

&igure c shows an unstructured mesh! Aere# each verte$ is connected to an ar"itrary

num"er of neigh"or vertices! 1nstructured meshes impose fewer topological

restrictions on the user# and as a result# ma'e it easier to mesh very comple$

geometries!

&igure c


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Conformal and -on&Conformal Meshes

@n e$ample of a non+conformal mesh is shown in &igure d! Aere# the vertices of a

cell or element may fall on the faces of neigh"oring cells or elements! %n contrast# the

meshes in &igures a# 0and care conformal meshes!

&igure d


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Cell


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Cell Shapes Deshes may "e constructed using a variety of cell shapes! The most widely used are

quadrilaterals and he$ahedra! Dethods for generating good+quality structured

meshes for quadrilaterals and he$ahedra have e$isted for some time now! Though

mesh structure imposes restrictions# structured quadrilaterals and he$ahedra are

well+suited for flows with a dominant direction# such as "oundary+layer flows! Dore

recently# as computational fluid dynamics is "ecoming more widely used foranalyzing industrial flows# unstructured meshes are "ecoming necessary to handle

comple$ geometries! Aere# triangles and tetrahedral are increasingly "eing used#

and mesh generation techniques for their generation are rapidly reaching maturity!

@nother recent trend is the use of hy"rid meshes! &or e$ample# prisms are used in

"oundary layers# transitioning to tetrahedral in the free+stream!


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@re


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@re


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Discretiation Methods

Finite Difference Methods The finite difference method is "ased on the Taylor series e$pansion

a"out a point#x

( )xxuu

x

x

ux

x

uuu

i

ii

ii

+

+

+

+=

+

+

asdefinediswhere

H.O.T.2

1

2

2

2

1

( )xxuu

x

x

ux

x

uuu

i

ii

ii

+

+

=

asdefinediswhere

H.O.T.2

1

2

2

2

1


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Finite Difference Methods

&inite difference methods appro$imate the derivatives in the governing differential

equation using truncated Taylor series e$pansions!

y including the diffusion coefficient and dropping terms of *BDI/9

&inite difference methods do not e$plicitly e$ploit the conservation principle in deriving

discrete equations! Though they yield discrete equations that loo' similar to other

methods for simple cases# they are not guaranteed to do so in more complicated cases#

for e$ample on unstructured meshes!


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Finite Element Methods

8e consider again the one+dimensional diffusion equation! There are different 'inds of finite

element methods! Let us loo' at a popular variant# the Faler'infinite element method! Let F "e

an appro$imation to F!


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Thus# we require

The weight functions +i are typically local in that they are non%ero over element i, "ut are

zero everywhere else in the domain! &urther# we assume a shape function for F# i!e!#

assume how Fvaries "etween nodes! Typically this variation is also local! &or e$ample we

may assume that Fassumes a piece+wise linear profile "etween points . and / and

"etween points / and : in &igure /!.! The Faler'in finite element method requires that

the weight and shape functions "e the same! Performing the integration in Equation /!..

results in a set of alge"raic equations in the nodal values of F which may "e solved "y a

variety of methods! 8e should note here that "ecause the Faler'in finite element methodonly requires the residual to "e zero in some weighted sense# it does not enforce the

conservation principle in its original form! 8e now turn to a method which employs

conservation as a tool for developing discrete equations!


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Finite


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Discretiation Methods Finite


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Solution of Discretiation E,uations


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Solution of Discretiation E,uations?hat are pro0lems of direct solution method@

@ solution for fis guaranteed if A&+ can "e found! Aowever# the operation count for the inversion of an N$N matri$ is of *BN/9! Consequently# inversion is almost never employed in practical pro"lems!

Dore efficient methods for linear systems are availa"le! &or the discretization methods of interest here# A is sparse# and for structured meshes it is "anded!

,irect methods are not widely used in computational fluid dynamics "ecause of large computational and storage requirements! Dost industrial C&, pro"lems today involve hundreds of thousands of cells# with +. un'nowns per cell even for simple pro"lems! Thus

the matri$ A is usually very large# and most direct methods "ecome impractical for these large pro"lems! &urthermore# the matri$ A is usually nonlinear# so that the direct method must "e em"edded within an iterative loop to update nonlinearities in A. Thus# the direct

method is applied over and over again# ma'ing it all the more time+consuming!


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Solution of Discretiation E,uations

Iteratie Methods %terative methods are the most widely used solution methods in computational fluid dynamics! These methods employ a guess+and+correctphilosophy which progressively improves the guessed solution "y repeated application

of the discrete equations!

Let us consider an e$tremely simple iterative method# the Fauss+


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Solution of Discretiation E,uations The neigh"or values# f$and f+ are re'uired for the update of f2. These are assumed 'nown at prevailing values! Thus# points which have already "een visited will have recently updated values of f and those that have not will have old values!

:!


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Accuracy @ccuracy refers to the correctness of a numerical solution when compared to an e$act solution! %t is more useful to tal' of the truncation error of a discretiation method. The truncation error associated with the diffusion term using the finite difference method is 6((DI9 /9 as shown "y Equation! This simply says that if

diffusion term is represented by the right hand side# the terms that are neglected are of 6((DI9 /9!

Thus# if we refine the mesh# we e$pect the truncation error to decrease as ((DI9 /9! %f we dou"le the $+direction mesh# we e$pect the truncation error to decrease "y a factor of four! The truncation error of a discretization scheme is the largest truncation error of each of the individual terms in the equation "eingdiscretized! The order of a discretiation method is n if its truncation error is 6((DI9 n9!

Dethods of very high order may yield inaccurate results on a given mesh! Aowever# we are guaranteed that the error will decrease more rapidly with mesh refinement than with a discretization method of lower order!


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Conergence

8e distinguish "etween two popular usages of the term convergence! 8e may say that an iterative method has converged to a solution# or that we have o"tained convergence using a particular method! y this we mean that our iterative method has successfullyo"tained a solution to our discrete alge"raic equation set ! 8e may also spea' of convergence to mesh independence! y this# we mean the process of mesh refinement# and its use in o"taining solutions that are essentially invariant with further refinement!


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Discretiation of the Momentum E,uation

Consider the two+dimensional rectangular domain shown in "elow &igure! Let us

assume for the moment that the velocity vector < and the pressure p are stored at

the cell centroids! Let us also assume steady state! The momentum equations in the

x and y directions may be written as!


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8e see that Equations )!. and )!/ have the same form as the general scalar transport equation#

and as such we 'now how to discretize most of the terms in the equation! Each momentumequation contains a pressure gradient term# which we have written separately# as well as a

source term B4u or 4v) which contains the body force term# as well as remnants of the stress

tensor term! Let us consider the pressure gradient term! %n deriving discrete equations# we

integrate the governing equations over the cell volume! This results in the integration of the

pressure gradient over the control volume! @pplying the gradient theorem# we get

@ssuming that the pressure at the face centroid represents the mean value on the face# we write


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Completing the discretization# the discrete u+ and v+momentum equations may "e

written as

Fiven a pressure field# we thus 'now how to discretize the momentum equations!

Aowever# the pressure field must "e computed# and the e$tra equation we need for its

computation is the continuity equation! Let us e$amine its discretization ne$t!


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Discretiation of the Continuity E,uation

&or steady flow# the continuity equation# which ta'es the form-

%ntegrating over the cell 2 and applying the divergence theorem, we get

@ssuming that the 6


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Determining Density and 2ressure

&or practical C&, pro"lems# sequential iterative solution procedures are frequently adopted

"ecause of low storage requirements and reasona"le convergence rate! Aowever there is a difficultyassociated with the sequential solution of the continuity and momentum equations for incompressi"le

flows! %n order to solve a set of discrete equations iteratively# it is necessary to associate the discrete

set with a particular varia"le! &or e$ample# we use the discrete energy equation to solve for the

temperature!


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Density =ased for Compressi0le Flo#s

2ressure =ased for Incompressi0le flo#s

There are a num"er of methods in the literatures which use the density as a primary

varia"le rather than pressure! This practice is especially popular in the compressi"le flow

community! &or compressi"le flows# pressure and density are related through an equation of

state! %t is possi"le to find the density using the continuity equation# and to deduce the pressure

from it for use in the momentum equations!


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Direct Solution is "ut of 4ich

Se,uential Iteratie Method ?ill =e Used

%t is important to realize that the necessity for pressure+ and density+"ased

schemes is directly tied to our decision to solve our governing equations sequentially

and iteratively! %t is this choice that forces us to associate each governing differential

equation with a solution varia"le! %f we were to use a direct method# and solve for thediscrete velocities and pressure using a domain+wide matri$ of size :1x:1, (three

variables % u#v#p + over N cells9# no such association is necessary! Aowever# even with

the power of today>s computers# direct solutions of this type are still out of reach for most

pro"lems of practical interest! *ther methods# including local direct solutions at each

cell# coupled to an iterative sweep# have "een proposed "ut are not pursued here!


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!he SIM2LE Algorithm

The SIMPLEBSemi+Implicit Method for 2ressure Lin'ed Equations9 algorithm and

its variants are a set of pressure+"ased methods widely used in the incompressi"le flow

community! The primary idea "ehind


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%f we su"tract equation of momentum which using initial pressures from the momentum

equation uses real pressure valuesBsatisfies continuity equation9 we can drive an

equation for relation "etween pressure correction and velocities correction values!


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8e now ma'e an important simplification! 8e appro$imate resultant Equations as


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earranging the last equation we ta'e "elow one for pressure correction which can

"e solved iteratively


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!he SIM2LE4 Algorithm

The SIMPLERalgorithm has "een shown to perform "etter than SIMPLE! This is primarily "ecause the


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!he SIM2LEC Algorithm

The


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Direct s. Iteratie Methods

Linear solution methods can "roadly "e classified into two categories# direct or iterative! ,irectmethods# such as Fauss elimination# L1 decomposition etc!# typically do not ta'e advantage of matri$

sparsity and involve a fi$ed num"er of operations to o"tain the final solution which is determined to

machine accuracy! They also do not ta'e advantage of any initial guess of the solution! Fiven the

characteristics of the linear systems outlined a"ove# it is easy to see why they are rarely used in C&,

applications! %terative methods on the other hand# can easily "e formulated to ta'e advantage of the

matri$ sparsity!


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Multigrid

8e saw "efore that the reason for slow convergence of Gauss-Seidelmethod is

that it is only effective at removing high frequency errors! 8e also o"served that low

frequency modes appear more oscillatory on coarser grids and then the Fauss+


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Multigrid

8e 'now that in general the accuracy of the solution depends on the discretization2therefore we would require that our final solution "e determined only "y the finest

gridthat we are employing! This means that the coarse grids can only provide us

with corrections or guesses to the fine grid solution and as the fine grid residuals

approach zero

*ne strategy for involving coarse levels might "e to solve the original differential

equation on a coarse grid! *nce we have a converged solution on this coarse grid#

we could interpolate it to a finer grid! *f course# the interpolated solution will not in

general satisfy the discrete equations at the fine level "ut it would pro"a"ly "e a

"etter appro$imation than an ar"itrary initial guess! 8e can repeat the process

recursively on even finer grids till we reach the desired grid!


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Multigrid

Coarse %rid Correction

@ more useful strategy# 'nown as coarse grid correction is based on the error e'uationecall that the error esatisfiesthesamesetof equations as our solutionif we replace the source vector "y the residual-

we can already see that the strategy outlined a"ove has the desired properties! &irst of all# note that if the fine

grid solution is e$act# the residual will "e zero and thus the solution of the coarse level equation will also "e zero!

Thus we are guaranteed that the final solution is only determined "y the finest level discretization! %n addition#

since we start with a zero initial guess for the coarse level error# we will achieve convergence right away and not

waste any time on further coarse level iterations! @nother useful characteristic of this approach is that we use

coarse level to only estimate fine level errors! Thus any appro$imations we ma'e in the coarse level pro"lem only

effect the convergence rate and not the final finest grid solution!


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Multigrid


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Solution 4esiduals


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Solution 4esiduals


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Solution 4esiduals

@t the end of each solver iteration# the residual sum for each of the conserved varia"les is computed and stored! *n a

computer with infinite precision# these residuals will go to zero as the solution converges! *n an actual computer# theresiduals decay to some small value Bround+off9 and then stop changing Blevel out9! &or single+precision computations

Bthe default for wor'stations and most computers9# residuals can drop as many as si$ orders of magnitude "efore hitting

round+off! ,ou"le+precision residuals can drop up to twelve orders of magnitude!

Definition of 4esiduals for the 2ressure&=ased Soler

@fter discretization# the conservation equation for a general varia"le at a cell P can "e written as

Aere aP is the center coefficient# an" are the influence coefficients for the neigh"oring cells# and " is the contri"ution of

the constant part of the source term


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Solution 4esiduals

This is referred to as the Mun+scaled residual! %t may "e written as

This Mscaled residual is defined as

The scaled residual is a more appropriate indicator of convergence for most pro"lems


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Solution 4esiduals

The pressure+"ased solver(s scaled residual for the continuity equation is defined as

The denominator is the largest a"solute value of the continuity residual in the first five iterations!

%t is sometimes useful to determine how much a residual has decreased during calculations as an

additional measure of convergence! &or this purpose# &L1ENT allows you to normalize the residual

Beither scaled or un+scaled9 "y dividing "y the ma$imum residual value after D iterations# where D is set

"y you in the esidual Donitors panel in the Iterationsfield under -ormaliation!


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4eferences

Purdue 1niversity+ C&, oo'let

&luent )!: ,ocumentation


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!he End

y- Ehsan

Documents

Computational Fluid Dynamics-First