NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS ANDANALYSIS OF THE SOLUTIONS

Charles-Henri Bruneau 1

Abstract : The aim of this survey isto discuss some of the difficulties one canencounter both when solving Navier-Stokesequations for incompressible flows by an ob-stacle and analysing the approximate solu-tions. Far to be exhaustive, some main as-pects of the numerical simulation are delib-erately pointed out, in addition to the waythe obstacle is taken into account and to thefar field boundary conditions. Then, usingone of the robust methods it is possible tosimulate the transition to turbulence for in-creasing Reynolds numbers. That means tocompute transient solutions which need tobe analyzed and here is the second topic ofthis paper. Indeed, the classical tools likeFourier analysis are very efficient as long asthe solution is periodic but useless when thesolution is more complex. Despite the de-velopment of wavelets and new algorithmsit seems still difficult to distinguish quasi-periodic and chaotic solutions.

1 Introduction

It is nowadays quite impossible to reviewall the ways the researchers have found outall around the world and for thirty yearsto solve the Navier-Stokes equations for in-

compressible flows. There are now classicalbooks devoted to these equations and theirapproximations [6, 11, 12, 18, 25, 30, 31].There are also international conferences fo-cusing globally or partially on this topic[17, 29, 23, 9, 16]. This shows the suc-cess of Navier-Stokes equations among thecomputational fluid dynamics community.Success that gives rise to a tremendous re-search activity and to so many papers thereader is overwhelmed. Therefore this paperdoes not pretend to give an exhaustive re-view of the field but only some commentson some aspects of the formulations, theboundary conditions, the approximations,the solving methods and also the analysisof the solutions. Indeed, using a method ro-bust enough on a fine mesh at least in theboundary layer area, it is now possible tocompute transient solutions quite easily in2D and even in 3D when making the bestof the new computers and the new compu-tational techniques. That means that onehas to use appropriate tools of analysis toqualify the computed solutions. As long asthe solutions are periodic this is very easyby Fourier analysis but when they are morecomplex it is very difficult to analyze pre-cisely the solution even in laminar cases.

1Mathematiques Appliquees de Bordeaux, Universite Bordeaux 1351 cours de la Liberation, 33405 Talence cedex, France.bruneau@math.u-bordeaux.fr

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2 Navier-Stokes models

2.1 The equations

From the mass and momentum conservationlaws, it is easy to derive Navier-Stokes equa-tions for an incompressible Newtonian vis-cous fluid in a domain Ω ⊂ IRN with N ≤ 3

∂tU + (U · ∇)U − 1

Re∆U +∇p = F

in ΩT = Ω × (0, T ) (2.1)

div U = 0 in ΩT . (2.2)

The first equation can be rewritten as

∂tU + (U · ∇)U − div σ(U, p) = F in ΩT

(2.3)

or

∂tU + (U · ∇)U − div σ(U, p) = F in ΩT

(2.4)

where σ(U, p) and σ(U, p) are respectivelythe pseudostress tensor and the stress tensordefined by :

σ(U, p) =1

Re∇U − p I (2.5)

σ(U, p) =2

ReD(U) − p I

with D(U)ij =1

2

(∂ui∂xj

+∂uj∂xi

)(2.6)

with U = (ui)i the velocity vector, p thepressure, Re the dimensionless Reynoldsnumber and F the external forces. Most of-ten F = 0 and the motion is given througha non homogeneous Dirichlet boundary con-dition imposed on a part ΓD of the bound-ary ∂ Ω.

These equations for the primitive vari-ables can be transformed by introducing thevorticity ω = ∇ ∧ U . The general form in3D of the velocity-vorticity equations is

∂t ω + (U · ∇)ω − 1

Re∆ω = (ω · ∇)U

+∇∧ F in ΩT

∇∧ U = ω in ΩT

divU = 0 in ΩT

(2.7)

where the second part of the non linearterm is treated as a source term in the firstequation. In 2D this term vanishes. Thereare other models like the stream function-vorticity model which is valid only in 2D(see [14] for more details).

2.2 The initial datum

The evolution problem (2.1) (2.2) requiresan initial condition

U(x, 0) = U0(x) in Ω (2.8)

and from a mathematical point of viewthis initial datum must belong to the rightspace, in particular U0 must a priori satisfyboth the divergence-free condition and theboundary conditions. In practice it is notso easy to check the divergence-free condi-tion and the numerical experience shows itis not compulsory. The first time steps willproduce the good initial solution.An other question related to the ini-tial condition is the use of a high-orderscheme in time requiring several initialdata U−j(x), 0 ≤ j ≤ J for the first timesteps. This can be solved either by settingU−j(x) = U0(x) for 0 ≤ j ≤ J or by usingan Euler scheme for these first time steps.

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2.3 The boundary conditions

It is well-known that the boundary condi-tions constitute one of the main difficul-ties we can encounter. Here there are threetypes : inflow, no-slip and outflow or openboundary conditions. The first two corre-spond to non homogeneous and homoge-neous Dirichlet conditions, the last one ismuch more difficult to find out in order toget a well-posed problem and a realistic ap-proximate solution. These conditions aregathered for instance when computing theflow behind an obstacle in a channel (figure1) where the domain Ω has for boundary∂Ω = ΓD ∪ Γ0 ∪ Γ1 ∪ ΓN .

On ΓD the flow at infinity (U∞, p∞) isimposed, that is a Poiseuille flow is set atthe entrance section. On Γ1 there is a no-slip condition U = 0 as well as on Γ0 if themesh is adapted to the limit of the obsta-cle. We shall see in the next section thatthere are other ways to take into accountthe obstacle. But the condition to set onΓN is far to be so easy. Indeed if ΓD isnot too close to the obstacle Ω0 , the Dirich-let condition is relevant at the entrance sec-tion and does not produce any perturba-tion. On the contrary, even when ΓN isnot so close to Ω0 some boundary condi-tions can produce strong reflections whenvortices are convected through the artificiallimit. The treatment of the open boundaryconditions for Navier-Stokes equations is it-self a large field of research as nothing tellus what to do to get on the truncated do-main the restriction of the solution on theinfinite domain. There are essentially twoways of dealing with this difficulty whichare either to use a buffer region outside ofΩ in wich the equations are modified [8, 27]or to impose the best condition known onΓN . Many researchers have find out good

open boundary conditions and we refer to[28] and references therein for more details.One of the most used is probably the zero-stress boundary condition σ(U, p)n = 0 wegeneralize in [3] by

σ(U, p)n = σ(U∞, p∞)n

orσ(U, p)n = σ(U∞, p∞)n

for Stokes flow and by for instance

σ(U, p)n+1

2(U · n)−(U − U∞)

= σ(U∞, p∞)n (2.9)

or

σ(U, p)n+1

2(U · n)−(U − U∞)

= σ(U∞, p∞)n (2.10)

for Navier-Stokes flows with the notationa = a+− a−. On ΓN , the new term is equalto zero except if U.n is negative to easethe convection of vortices and avoid reflec-tions. From a mixed formulation we canshow by energy estimates that conditions(2.9) or (2.10) yield a well-posed problem[2].

2.4 The obstacle

To take into account the obstacle there areessentially two ways, either the mesh is con-structed so that Γ0 is approximated by thesides of some cells or an immersion methodis used. In the first case a no-slip boundarycondition U = 0 is imposed on Γ0 and thecomputation is done on the unstructuredmesh via a finite elements or a finite vol-umes approximation. In the second case acartesian mesh is applied on D = Ω∪Γ0∪Ω0

and the approximation is achieved by meansof finite differences or spectral methods. Of

3

Ω ΩΓΓ Γ

Γ

Γ0 x

x

1

2

1

1

ND0

0

Figure 1: Computing domain

course, this needs an additional tool to rep-resent the obstacle. One is to force the no-slip condition at the surface of the body byadding a feedback forcing function to themomentun equations. The points definingthe surface are chosen by the user, theycan be either the closest vertices or the in-tersection points between the body surfaceand the mesh. In this second case severalinterpolations can be used [13, 27]. An-other tool is to consider the obstacle as aporous medium with a very small perme-ability coefficient K. This yields a fluid-solid formulation in the domain D by solv-ing Navier-Stokes equations in the fluid andDarcy equations in the solid. One way todo that is to set K = 1 at every point in Ωand to set K = KΩ0 1 at every point inΩ0. Then equations (2.1),(2.2) are replacedby

∂tU + (U · ∇)U − 1

Re∆U +

U

ReDaK3

+1

K∇p = F in DT = D × (0, T ) (2.11)

divU = 0 in DT (2.12)

where Darcy number is given by

Da =1

ReKΩ0

. It is clear that this adds

an extra work as it requires to solve equa-tions (2.11), (2.12) in Ω0. But the cartesianmesh simplifies the computation and allowsto use the spectral or multigrid methods.Moreover, the pressure in Ω0 permits tocompute the drag and lift forces [5]. Finallythe velocity in Ω0 is of the same order thanKΩ0 .

3 Numerical simulation

3.1 The approximation

It is obvious that we can not give here evenan outline of the numerous types of approx-imations used to solve the Navier-Stokesequations. Indeed, this needs several books[6, 11, 12, 18, 25, 30, 31]. But , we can givea taste of the main difficulties which lay onone hand on the equilibrium between theconvection and the diffusion terms and onthe other hand on the divergence-free con-dition. It is now clear that we have to treat

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the convection term explicitely to avoid ar-tificial numerical diffusion in time. Let ussay that this term must be expressed attime nδt when computing the solution attime (n + 1)δt. On the contrary the otherterms can be discretized implicitely at time(n+1)δt. The discretization in space is sub-jected to the mesh and thus to the way theobstacle is taken into account. The moreused is probably the finite volumes approx-imation on unstructured meshes [24]. Butwith one of the immersion procedures it ispossible to benefit of the spectral or finitedifferences methods ([19, 27] or [4]). Asthe convection term is put in the secondmember of the momentum equation (2.1) or(2.11), a centered discretisation of the otherterms yields a well-conditioned matrix easyto invert.

But the convection term needs somemore work. A good discretization is neededto guarantee the success of the simulation.Indeed, one has to be very careful deal-ing with this term as every extra diffu-sion brought up by the scheme is added to− 1Re

∆U and changes the real value of theReynolds number. For instance, the dis-cretization of

u∂u

∂x

at point j in one dimension by a first-orderupwind scheme

unj (unj − unj−1)/δx (3.1)

if unj is positive corresponds to a second-order approximation of

u∂u

∂x− δx

2

∂

∂x(|u|∂u

∂x)

and thus adds a viscosity term that altersthe Reynolds number. Consequently thesimulation can be qualitatively correct butnot quantitatively. It is well-known that the

critical Reynolds number corresponding tothe first Hopf bifurcation for the driven cav-ity problem is not yet determined for sure.Because, for this problem, this first bifur-cation occurs at high Reynolds number andtherefore it is not easy to achieve a good ac-curacy. Then it is necessary to use a lessdiffusive scheme ; a possible choice is to re-place (3.1) by

unj−1/2 (4unj − 5unj−1 + unj−2)/3δx

− unj+1/2 (4unj − 5unj+1 + unj+2)/3δx (3.2)

if unj−1/2 is positive and unj+1/2 is negative.The results presented in this paper are ob-tained with such a scheme. For the drivencavity problem, it yields the hopf bifurca-tion for Re around 7500 which appears inrecent work to be a good value of the criticalReynolds number for this problem [23, 16].Other results are generally obtained withhigh order compact schemes.

Another key to the success is the approx-imation of the divergence-free condition andhere again there are numerous methods todo it. For a finite elements approximation,many ways were developped to find a goodapproximate space and often the equation(2.2) is satisfied only in a weak sense oneach element [30]. With finite differences,a easy way to approximate (2.2) is to use acentered discretization on a staggered grid(figure 2) so that divU = 0 can be writtenat the pressure point directly without inter-polation [4].

But probably the most famous way toforce the incompressibility is given by theduality method as described in the next sub-section.

3.2 The convergence methods

The last point is to find the whole method ofresolution which insures a good performance

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u2

u1 p u1

u2

Figure 2: A staggered cell

necessary to observe the long time be-haviour of the solution. As already pointedout, the duality method is one choice. Then,the pressure plays the role of the Lagrangemultiplier and is computed by Uzawa’s al-gorithm. Coupling this to a good gradienttype method to invert the linear system agood performance can be achieved.

Another choice is to use the decomposi-tion of the solution in its different scales.The new and now well-known nonlinearGalerkin method consists in cutting thesescales into two parts, the large and thesmall ones. Then, the original Navier-Stokes equations are splitted into two partsto better represent the relative behaviour ofthe two types of scale. The result is animprovement of the performance obtainedwith a classical method whatever the ap-proximation is [21]. Linked also to the dif-ferent scales, the multigrid method is a verystrong tool [1, 15]. Indeed, by using succes-sive grids it is possible to capture very fastthe scales related to each grid. On one handthe solution is computed on a really coarsemesh to get the large scales and on the otherhand the finer the grid is the smaller thescales can be reached [4, 33].

This choice of method is decisive. In-deed to make a direct simulation of the tran-sition to turbulence it is necessary both touse a very fine mesh in the boundary layer

and to compute the solution for a long time.Even in 2D, this can require several days ofcomputing time on the best work stations.

4 Numerical tests and

analysis of solutions

The numerical tests presented here corre-spond to the domain Ω of figure 1. Thechannel is the rectangle (0, L) × (0, 1) withL = 3 or 4 and the obstacle is a circle ofradius 0.2 which center is located at point(1, 0.5). We recall Re is a dimensionlessReynolds number. To get a meaningfulReynolds number, one has to multiply Reby the diameter of Ω0 which is here d =0.4. So a solution at Re = 100 correspondsto a solution at real Reynolds number 40.For low Reynolds numbers the numericalexperiments are performed on a uniformgrid of 256 × 64 cells which is fine enoughto describe the solution. For instance, atRe = 100 there is a symmetric steady solu-tion with a recirculating bubble behind thecylinder as we can see on the stream func-tion isolines (figure 3).

Then, as Re increases, the steady so-lution looses its stability to the benefit ofa purely periodic solution very stable forhigher values of Re. The recirculation zonealternates from the top to the bottom of the

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Figure 3: Solution at Re = 100

cylinder. We can detect quite accuratelythe critical Reynolds number correspondingto the first Hopf bifurcation. Indeed, forthis simple geometry it corresponds to theloss of symmetry. The question is : Are wesure it is quantitatively correct ? To answerthis question we can make another numer-ical test on the same geometry with d =0.2 by applying an open boundary condi-tion on Γ1 instead of the no-slip condition.In this case a constant flow U = (1, 0) is im-posed on ΓD instead of the Poiseuille flow offlowrate 1.

We then have a very well-known physicaltest and can compare the results with physi-cal experiments. The values of the Strouhalnumber St for various real Reynolds num-bers are in very good agreement with thephysics [32] and assert the accuracy of themethod (table 1). Nevertheless this compar-ison is possible only for low Reynolds num-bers.

Coming back to the initial problem andusing a finer mesh, we increase Re to reachother regimes. For Re = 1000, for instance,there is still a periodic solution but this timethere are strong alternate vortices convectedthrough the domain. We can then controlthat the open conditions (2.10) do not affectthe solution computed on a shorter domainas it can be seen on the isolines of the vor-ticity field (figure 4). Indeed both solutions

are computed with exactly the same param-eters. The only difference is the length ofthe domain L = 4 and L = 3. We seein particular that there is no reflections in-duced by the artificial boundary and thatthe computed solution on the shorter do-main corresponds to the restriction of thesolution computed on the larger domain atthe same time (figure 4).

Then, increasing Re, there is still a peri-odic solution until Re = 3700 but with var-ious behaviours. Indeed, a classical Fourieranalysis reveals that approximately fromRe = 200 to Re = 3700 the flow is peri-odic and exhibits the same main frequencyfm ' 1 (this value depends on the vari-ous parameters). But from Re = 2200 toRe = 3600 it appears two subharmonicscorresponding to f ' fm

3and 2fm

3and for

Re = 3700 it appears seven subharmonicscorresponding to f ' fm

8and its multiples

(figure 5). Let us note that this qualita-tive behaviour, in particular the number ofsubharmonics, changes with the geometry ofthe obstacle. For instance, a square of sidelength 0.4 does not give the same subhar-monics.

Until now, the Fourier analysis is a veryefficient tool that gives very accurately thefrequencies of a time signal corresponding tothe value of one component of the velocityat a chosen point of the domain behind the

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Re Reynolds number Computed St Value of St in [32]300 60 0.130 0.136500 100 0.164 0.164800 160 0.188 0.186

Table 1: Comparison of the Strouhal number

Figure 4: Comparison of the solutions obtained on the domain Ω with L = 4 and L = 3

cylinder. Of course, the behaviour of thesolution and thus the spectrum does notdepend of the point. We can complete theanalysis with a phase portrait that corrob-orates the presence of subharmonics as thesame curve is drawn several times. We seeon figure 6 that a wavelets analysis is not soaccurate even if the main frequency fm andits subharmonics are detected at Re = 3700.

At Re = 3800, a long time simula-tion shows us the solution alternates be-tween two states. It is well-known that thewavelets are very efficient to detect a dis-

continuity [7, 22, 26] but here there is aslow transition between the two states muchmore difficult to analyze as it contains alarge part of the spectrum. On figure 7 arerepresented the time-frequency analysis ob-tained with both a windowed Fourier trans-form and an adapted wavelets transform.We see the two methods provide about thesame informations but as soon as there is atransition the analysis is spoiled.

When the Reynolds number increases weget more complex solutions. We can seethe field becomes more complex and lookschaotic (figure 8). Nevertheless the different

8

A

F 0 . 1 0 1 E + 0 1

U2

U1

A

F 0 . 9 9 5 E + 0 0

U2

U1

A

F 0 . 9 7 8 E + 0 0

U2

U1

Figure 5: Spectrum and phase portrait for Re = 1000, 2500 and 3700

analysis give very few informations exceptthe presence of the main frequency whichchanges with time. We think that a match-ing pursuit procedure [20] can give somemore informations providing a good dictio-nary adapted to the kind of signal we ana-lyze is used.

Another question is to understand thebehaviour of the vortices, how they are

convected through the domain for highReynolds numbers, if they can merge or bedivided into two parts ? Here again, somework has been done by means of waveletsessentially to compress two-dimensional tur-bulent flows [10]. The result shows that a2D field can be analyzed with this tool andwell represented. It is now necessary to seeif such an analysis from time to time can

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0 .

0 . 2

0 . 4

0 . 6

0 . 8

1 .

1 . 2

1 . 4

1 . 6

1 . 8

2 .

f r e q u e n c y

0 . 5 . 1 0 . 1 5 . 2 0 . 2 5 . 3 0 . 3 5 . 4 0 . 4 5 . 5 0 . t i m e

Figure 6: Wavelets analysis of the signal at Re = 3700

give enough informations to understand thebehaviour of the whole field.

5 Conclusion

With the improvement of both the numeri-cal techniques and the computing power itis now possible to compute directly from theNavier-Stokes equations good transient so-lutions in route to turbulence. However itis still difficult to qualify the computed so-lutions with the existing tools of analysis.

6 Acknowledgements

We would like to thank Roger Gay, BrunoTorresani and Jean-Francois Delage forfruitful discussions and the numerical testson wavelets analysis.

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