Numerische Mathematik - ljll.math.upmc.fr · Obstacles in incompressible viscous ﬂows 501 3.2. The H1 penalization Inthissectionwechoosetopenalizeinadditionthewholelinearpartofthe

Numer. Math. (1999) 81: 497–520 NumerischeMathematikc! Springer-Verlag 1999

A penalization method to take into account obstaclesin incompressible viscous flowsPhilippe Angot1, Charles-Henri Bruneau2, Pierre Fabrie21 IRPHE, CNRS UMR 6594, Equipe Mathematiques Numeriques pour la Modelisation, LaJetee – Technopole de Chateau-Gombert, F-13451 Marseille, France

2 Mathematiques Appliquees de Bordeaux, CNRS et Universite Bordeaux 1, 351, cours dela Liberation, F-33405 Talence, France

Received August 10, 1997

Summary. From the Navier-Stokes/Brinkman model, a penalizationmethod has been derived by several authors to compute incompressibleNavier-Stokes equations around obstacles. In this paper, convergence the-orems and error estimates are derived for two kinds of penalization. Thefirst one corresponds to a L2 penalization inducing a Darcy equation in thesolid body, the second one corresponds to a H1 penalization and induces aBrinkman equation in the body. Numerical tests are performed to confirmthe efficiency and accuracy of the method.

Mathematics Subject Classification (1991): 76D99; 76S05

1. Introduction and setting of the problem

About fifteen years ago, there were several attempts to penalize the no-slipboundary condition on the surface of an obstacle surrounded by a fluid.The aim was to avoid body-fitted unstructured meshes in order to use fastand efficient spectral, finite differences or finite volumes approximationson cartesian meshes. A way to do that is to add a penalized velocity termin the momentum equation of the incompressible Navier-Stokes equations.Following the former work of Peskin [12], [13], several authors (for instance[9]) add both a time integral of the velocity and a velocity penalization termonly at the points defining the surface of the obstacle. It appears that thepenalization has to be extended to the volume of the body to give correctphysical solutions at high Reynolds numbers [14]. In independent works,Arquis and Caltagirone [4] add a penalization term on the velocity definedon the volume of a porous body. This corresponds to a Brinkman type

498 P. Angot et al.

model with variable permeability where the fluid domain has a very largepermeability in front of the one of the porous medium. This model wasgeneralized later in [1] [2] to deal with fluid-porous-solid systems. In amore recent work [8], it is suggested that this model allows to compute thedrag and lift coefficients by integrating the penalization term inside the solidbody. Various works use the same methodology to compute incompressibleflows around a cylinder or behind a step [3], [5], [6], [10].The aim of this paper is to establish rigourous estimates of the error in-

duced by such penalizations and to show numerically the efficiency of themethod. In Sect. 3, we give the formal asymptotic expansion of the approx-imate solution with respect to the penalization coefficient. Then Sects. 4and 5 are devoted to the analysis of the proposed L2 and H1 penalizationrespectively. The L2 penalization consists in adding a damping term on thevelocity in the momentum equation whereas the H1 penalization includesin addition a perturbation of both the time derivative and the viscous term.The numerical tests to validate the mathematical modelling are presented inthe last section.

2. Preliminaries and notations

Let ! be a regular bounded connected open set in IR2, we assume that !contains I regular obstacles !i

s , 1 ! i ! I (see Fig. 1).We set

!s =I

!

i=1

!is , !f = !\!s

"is = #!i

s , $ = #!

where !f is the incompressible fluid domain in which the Navier-Stokesequations are prescribed. In the physical case, the motion is given by impos-ing a non homogeneous Dirichlet boundary condition on $ , whereas, forsake of simplicity in the mathematical study we assume that the motion, isimposed by an external source term. Consequently, we take an homogeneousDirichlet boundary condition for the velocity on $ .So we are looking for the solution of the following initial boundary value

problem:

#tuf " 1Re

%uf + uf · #uf + #pf = f in IR+ $!f

div uf = 0 in IR+ $!fuf(0, ·) = uf 0 in !fuf = 0 on #!f .

(2.1)

Obstacles in incompressible viscous flows 499

ΩΣ

Ω

Ω

Γf

s

i

s

i

Fig. 1. Domain

The first equation can be written in terms of the stress tensor &(u, p) =1

2 Re(#u + #ut) " pI) as:

#tuf + uf · #uf " div &(uf , pf) = f in IR+ $!fdiv uf = 0 in IR+ $!fuf(0, ·) = uf 0 in !fuf = 0 on #!f .

(2.2)

This form is used in Sect. 3 for the H1 penalization.Now, let us introduce the following functional spaces:

L2(!) = (L2(!))2H1(!) = {u % L2(!) ; #u % (L2(!))4}

H2(!) = {u % L2(!) ;#2u

#xi #xj% L2(!) , 1 ! i ! j ! 2}

H10(!) = {u % H1(!) ; u = 0 on #!}

H = {u % L2 ; div u = 0 ; u.n = 0 on #!}V = {u % H1

0(!) ; div u = 0}.

at last, we denote 1!s the function:

1!s(x) = 1 if x % !s1!s(x) = 0 if x % !f

3. Two models to penalize the Navier-Stokes equations

Instead of solving the problem (2.1) on!f , we solve an equivalent problemon thewhole domain! by penalizing the obstacles. The linear case is studiedin [3]

500 P. Angot et al.

3.1. The L2 penalization

The first idea is to force the velocity to be small in !s by solving:

#tu" " 1Re

%u" + u" · #u" +1'1!su" + #p" = f in IR+ $!

div u" = 0 in IR+ $!u"(0, ·) = u0 in !u" = 0 on $.

(3.1)

We set u" = u + 'u and p" = p + 'p to derive formally the equationssatisfied by u, p and u, p. Then we get from the first equation in (3.1) byidentifying the terms of same order:

1!su = 0(3.2)#tu " 1

Re%u + u · #u + 1!s u + #p = f(3.3)#tu " 1

Re%u + u · #u + u · #u + #p = 0.(3.4)

Thus by (3.2) u vanishes in !s. Consequently u satisfies equation (2.1) in!f and u checks:

us + #ps = 0 in !sdiv u = 0 in !s

(3.5)

and

#tuf " 1Re%uf + uf · #uf + uf · #uf + #pf = 0 in !f

div uf = 0 in !f .(3.6)

Let us remark that u, u and p, p are continuous on each "is. Hence ps is

given by:%ps = 0 in !i

s 1 ! i ! Ips = pf on "i

s

which yields us in !s by (3.5).Then uf is completly determined by adding:

uf = us on "is 1 ! i ! I

anduf(0, ·) = 0

to (3.6).

Remark 3.1 In !s, u verifies a Darcy type law associated to a Neumannboundary condition on the pressure. Thus the obstacles are associated toporous media.


3.2. The H1 penalization

In this section we choose to penalize in addition the whole linear part of theNavier-Stokes equations. So according to (2.2) we set:

"

1 +1'1!s

#

#tu" + u" · #u"

"div"

(1 + 1"1!s)&(u", p")

#

+ 1"1!su" = f in IR+ $!

div u" = 0 in IR+ $!u"(0, ·) = u0 in !u" = 0 on $

(3.7)

From the expansion form of u" and p", u" = u + 'u and p" = p + 'p, weget by identification of the terms of same order:

1!s#tu " div (1!s&(u, p)) + 1!su = 0,(3.8)

#tu + u#u " div &(u, p) + 1!s#tu

"div (1!s&(u, p)) + 1!s u = f,(3.9)

#tu + u · #u + u · #u " div &(u, p) = 0.(3.10)Closing equation (3.8) in !s with the natural boundary condition

&(us, ps) · nis = 0 on "i

s 1 ! i ! I

where nis denotes the unit normal vector pointing inside of !i

s, we find thatus & 0 on!s thanks to the damping term. Then by (3.9) u satisfies equation(2.2) in !f and by (3.9) and (3.10) u cheks:

#tus " div &(us, ps) + us = 0 in IR+ $!s

div us = 0 in IR+ $!s

(3.11)

and#tuf + uf · #uf + uf · #uf " div &(uf , pf) = 0 in IR+ $!f

div uf = 0 in IR+ $!f

(3.12)

Now, as in the previous subsection, the continuity of (u, u) and (p, p) oneach "i

s allows to close the problem by adding:

&(us, ps) · nis = &(uf , pf) · ni

s on IR+ $"is , 1 ! i ! I

us(0, ·) = 0 in !s

502 P. Angot et al.

to (3.11) anduf = us on IR+ $"i

s , 1 ! i ! I

uf(0, ·) = 0 in !f

to (3.12).Remark 3.2 In !s, u satisfies a Brinkman type equation associated to aNeumann type condition for the stress tensor. Once again the obstacles canbe viewed as sparse porous media.Remark 3.3 Instead of (3.7) we study a similar problem given by :

$

1 +1'1!s

%

#tu" " 1Re

div$

(1 +1'1!s)#u"

%

+u" · #u" +1'1!su" + #p" = f in IR+ $!

div u" = 0 in IR+ $!

u"(0, ·) = u0 in !

u" = 0 on $.

(3.13)

The mathematical analysis of (3.7) can be achieved through a mixed formu-lation as in [7]

4. The L2 penalization

In this section we study the behaviour of the solution u" of problem (3.1)when ' goes to zero.

#tu" " 1Re

%u" + u" · #u" +1'1!su" + #p" = f in IR+ $!

div u" = 0 in IR+ $!u"(0, ·) = u0 in !u" = 0 on $

(4.1)

where f is a given function in L!(IR+; L2(!)) which support is includedin !f . As it is well known, for ' given there exists an unique solution u" of(4.1) satisfying:

u" % C0(IR+;H) ' L2loc(IR

+; V )

#tu" % L2loc(IR

+; V ")


Remark 4.1 Note that this solution cannot be more regular in space as 1!s

is a discontinuous function.

As the results of this section are derived from energy estimates, we recallthe weak formulation of (4.1), for any ( % V

(#tu" , ()V !,V +1

Re

&

!#u" · #(dx +

&

!u" · #u"(dx

+1'

&

!1!su"(dx =

&

!f(dx.

u"(0, ·) = 0

(4.2)

Remark 4.2 It is allowed to extend (4.1) to test functions ( inW where:

W ='

( % L2(0, T ; V ) ; #t( % L2(0, T ; V ") ; ((T ) = 0(

4.1. Convergence result

In order to get a convergence result we need the following a priori estimates:

Lemma 4.1

supt#IR+

|u"|2 ! |u0|2 +Re2

)1sup

t#IR+*f*2

$1(4.3)

& t

0|#u"(*)|2d* ! |u0|2 + Re2 sup

t#IR+*f*2

$1 t(4.4)

where *f*$1 denotes the norm in H$1(!).

Proof. From (4.2) with ( = u" we get:

12

d

dt|u"(t)|2 +

1Re

|#u"(t)|2 +1'

&

!1!s(x)u2

"(t, x)dx ! *f*$1|#u"(t)|,

so,

d

dt|u"(t)|2 +

1Re

|#u"(t)|2 +2'

&

!1!s(x)u2

"(t, x)dx

! 1Re

*f*2$1,(4.5)

that gives after elementary computations (4.3), (4.4). !

From (4.5) we deduce directly:

504 P. Angot et al.

Corollary 4.1

2'

& t

0

&

!1!s(x)u2

"(*, x)dxd* ! |u0|2 + Re supt#IR+

*f(t)*2$1 t.(4.6)

As we are not able to derive an estimate on #tu" in V ", we choose toevaluate a fractional derivative in time of u" to obtain compactness.

Lemma 4.2 There exists a generic function g in C0(IR+) independent of 'such that:

|##t u"(t)|L2(0,t;!) ! g(t), ++ ! 1

4(4.7)

Sketch of proof: As the term 1"1!su" is a damping term, the ideas developed

in [11] , [16] can be applied (see also [7] ).

Following the idea of [7], we introduce the Hilbert space:

W ='

( % L2(0, T ; V ) ; #t( % L2(0, T ; V ") ; ((T ) = 0(

to obtain:

Lemma 4.3 There exists a generic function g in C0(IR+) depending on thedata such that:

1'

)

)

)

)

& t

0

&

!1!s(x)u"(*, x)((*, x)dxd*

)

)

)

)

! g(t) *(*W(4.8)

Proof. Integrating by part in time the first term of (4.2) we get for ( % W :

1'

& T

01!s(x)u"(t, x)((t, x)dxdt =

& T

0(u"(t), #t()V ,V!dt

" 1Re

& T

0

&

!#u"(t, x) · #((t, x)dxdt

"& T

0

&

!u"(t, x) · #u"(t, x)((t, x)dxdt

+& T

0

&

!f(t, x)((t, x)dxdt +

&

!u0(x)((0, x)dx.

The estimate (4.8) comes from straightforward majorations using (4.3) and(4.4). !

Finally, to show that the limit u of the sequence (u")" is solution ofhomogeneous Navier-Stokes equations in !f we need the last result:


Lemma 4.4 There exists a generic function g in C0(IR+) depending on thedata such that

|u"|L2(0,t;L2($is))

! g(t)'14 , +i 1 ! i ! I(4.9)

Proof. As u" belongs to L2(0, T ; V ) the trace of u" is well defined on eachboundary "i

s and we have the following inequality where c denotes alwaysa generic constant.

|u"(t)|L2(!is) ! c|u"(t)|

12L2(!i

s)*u"(t)*

12H1(!i

s).(4.10)

We first bound theH1 norm on !is by theH1 norm on ! ; then using (4.3)

and (4.4) we get:& t

0|u"(*)|2L2($i

s)d* ! g(t)

& t

0|u"(*)|L2($i

s)d*

! g(t)*

& t

0|u"(*)|2L2($i

s)d*

+

12

,

which gives with (4.10) and (4.6)& t

0|u"(*)|2L2($i

s)d* ! g(t)'

12 , +t % IR+,

where g is a generic function. !

Therefore we can show:Theorem 4.1 When ' goes to zero, the sequence (u")" converges to a limitu which satisfies:

u|!s = 0,

and u|!f is the unique weak solution of Navier-Stokes equation in !f .Moreover there exists h % W" such that:

1'1!su" , h in W " weakly

and

"& t

0(u(*), #t((*))V ,V!d* +

& t

0

&

!#u(*, x) · #((*, x)dxd*

+& t

0

&

!u(*, x) · #u(*, x)((*, x)dxd* + (h,()W ,W !

=& t

0

&

!f(*, x)((*, x)dxd* +

&

!u0(x)((0, x)dx

(4.11)

506 P. Angot et al.

Proof. From (4.3), (4.4) and (4.7) we get by a compactness result [16],

u" , u in L2(0, T ;H) strongu" , u in L2(0, T ;V ) weaku" , u in L!(0, T ;H) weak-

for a subsequence still denoted (u")" .Moreover from (4.8) we have

1'1!su" , h in W ".

At last, from (4.9)u = 0 on IR+ $"i

s.

As h is the weak limit of 1"1!su", one has for any ( in W such that

supp ((t, ·) - !f(h,() = 0.

Then (4.11) is straightforward and from the remark above u is the weaksolution of Navier-Stokes equation in!f . Finally, from Corollary 4.1 u = 0in!s and by uniqueness ofu|!f thewhole sequence (u"|!f )" converges. !

4.2. Error estimate

Now assuming that the solution uf of Navier-Stokes equations in !f is reg-ular enough, we can derive an error estimate.

So, for uf % L!(0, T, H10(!f)) ' L2(0, T ;H2(!f)) (which is true as

soon as uf 0 % H10(!f)), let us define u by:

u = uf in !fu = 0 in !s.

It is obvious that:

u % L!(0, T ;H10(!)) ' L2(0, T ; {H2(!f) ' H2(!s)})

and thatdiv u = 0 in IR+ $!.

So, we get for ( % V ,

0 =&

!f

$

#tu " 1Re

%u + u · #u + #p " f

%

(dx,


and after integration by parts

0 =&

!f

$

#tu(+1

Re#u · #(+ u · #u(" f(

%

dx

"I

,

i=1

&

$is

&(u, p)nif(d+(4.12)

where nif is the outward unit normal vector to !f on "i

s.In addition, as the 2Dweak solutionofNavier-Stokes equations is unique,

the weak limit u obtained in the previous theorem is identical to uf in!f assoon as u|!f belongs toH1

0(!f) and is divergence free in !f .

Theorem 4.2 Let

uf % L!(0, T ; H10(!f)) ' L2(0, T ;H2(!f))

be the solution of Navier-Stokes equations in !f and u defined as:

u = uf in !fu = 0 in !s.

Then there exists v" bounded in L!(0, T ;H) ' L2(0, T ; V ) such that:

u" = u + '14 v".

Moreover there exists a generic function g in C0(IR+) such that:

|v"|L2(0,t;L2(!s)) ! g(t)'12 .

Proof. To get the error estimate above we seek u" on the following a prioriform u" = u + 'w" and get from (4.2) as 1!su = 0, for ( % V

&

!

$

#tu(+1

Re#u · #(+ u · #u(" f(

%

dx +&

!1!sw"(dx

+'&

!

$

#tw"(+1

Re#w" · #(+ u · #w"(+ w" · #u(

%

dx

+'2&

!w" · #w"(dx = 0.

508 P. Angot et al.

According to (4.12) this reduce, after dividing by ' to:&

!

$

#tw"(+1

Re#w" · #(+ u · #w"(+ w" · #u(

%

dx

+1'

&

!1!sw"(dx + '

&

!wn · #w"(dx

=1'

I,

i=1

&

$is

&(u, p) · nifw"d+.

(4.13)

For fixed ' , w" is regular enough to take ( = w" in (4.13). So we get thefollowing energy estimate:

12

d

dt|w"|2 +

1Re

|#w"|2 +1'

&

!1!sw

2"dx

! 1'

I,

i=1

)

)

)

)

)

&

$is

&(u, p) · nifw"d+

)

)

)

)

)

+)

)

)

)

&

!w" · #uw"dx

)

)

)

)

! c

'

I,

i=1

|w"|L2($is)

+ c |w"| |#w"|

! c

'

I,

i=1

|w"|12L2(!i

s)|#w"|

12L2(!i

s)+ c |w"| |#w"| ,

as in the proof of Lemma 4.4.Then, using Young inequality to absorb the gradient terms, one gets:

d

dt|w"|2 +

1Re

|#w"|2 +2'

&

!1!sw

2"dx

! c |w"|2 + c

$

1'

|w"|12L2(!s)

%43,

and absorbing again |w"|L2(!s) by the third term of the left hand side, onehas:

d

dt|w"|2 +

1Re

|#w"|2 +2'

&

!1!sw

2"dx

! c |w"|2 + c'$ 32

Then, byGronwall lemma,we get forw"(0) = 0 aswe chooseu"(0) = u(0)


|w"(t)|2 ! g(t)'$ 32 ,

and

1Re

& t

0|#w"(*)|2 d* +

1'

& t

0

&

!1!s(x)w2

"(*, x)dxd* ! g(t)'$ 32 ,

where g is a generic continuous function on IR+. This gives the result forw" = u$u!

" . !

4.3. Interpretation of h

For weak solutions given by (4.11), we are not able to give a rigorous in-terpretation of the weak limit h of ( 1

"1!su")". Nevertheless, for regularsolution in time, we can derive an explicit formula.Following [15], we can show, under the compatibility condition:

" 1Re

%u"(0) + u"(0) · #u"(0) +1'1!su"(0)(4.14)

+#p"(0) " f(0) is bounded in L2(!)

and under the regularity assumption:

#tf % L!(IR+; L2(!))(4.15)

that #tu" is bounded in L!(0, T ;H) ' L2(0, T ; V ).Thus h belongs to L!(0, T ; V ").

Remark 4.3 The condition (4.14) is fulfilled as soon as u"(0)|!s = 0 andu"(0) % H2(!) which is physically relevant.

In this case, instead of (4.11) the limit u satisfies:&

!#tu(dx +

1Re

&

!#u · #(

+&

!u · #u(dx + (h(t),((t))V!,V =

&

!f(dx.(4.16)

Theorem 4.3 Under compatibility condition (4.14) and regularity condi-tion (4.15) h is uniquely determined by

(h(t),((t))V !,V = "I

,

i=1

&

$is

&(u, p)nif(d+ +( % V .(4.17)

510 P. Angot et al.

Proof. The result follows from (4.12) and (4.16). !

Remark 4.4 From (4.17) we can writeI

,

i=1

&

$is

&(u, p)nif(d+ = " lim

"%0

1'

&

!s

1!su"(dx.

When ( % V is such that ( & 1 in !is and ( & 0 in !j

s , j .= i, we get&

$is

&(u, p)nifd+ = " lim

"%0

1'

&

!s

1!isu"dx.

This last equation gives an explicit way to compute the lift and drag coeffi-cients through an integration over the volume of the obstacle [3], [8].

5. TheH1 penalization

We study in this section the behaviour of the solution (u")" of the penalizedproblem (3.7) presented in Sect. 3 when ' goes to zero.

$

1 +1'1!s

%

#tu" " 1Re

div$

(1 +1'1!s)#u"

%

+u" · #u" +1'1!su" + #p" = f in IR+ $!

div u" = 0 in IR+ $!

u"(0, ·) = u0 in !

u" = 0 on $.

(5.1)

As the aim of this section is to derive a better error estimate, wework directlywith regular solution in time, and so we take by example

f % L!(IR+; L2(!)) such that #tf % L!(IR+; L2(!)),(5.2)and we assume that the following compatibility solution is fulfilled.

" 1Re

div$

(1 +1'1!s)#u"(0)

%

+u"(0) · #u"(0) +1'1!su"(0) + #p"(0) " f(0)(5.3)

is bounded in L2(!).


Remark 5.1 The condition (5.3) is satisfied as soon as

u"(0) % H2(!) u"(0)|!s = 0 , div u"(0) = 0

By usual techniques, we can show that (5.1) has a unique solution underthe compatibility condition (5.3) and under the regularity assumption (5.2)which checks

(u")" is bounded in C0(0, T ; V )

(#tu")" is bouded in L!(0, T ;H) ' L2(0, T ;V ).

The variational formulation of (5.1) is, for any ( % V&

!

$

1 +1'1!s

%

#tu"dx +1

Re

&

!

$

1 +1'1!s

%

#u" · #(dx

+&

!u" · #u"(dx +

1'

&

!1!su"(dx =

&

!f(dx

u"(0, ·) = u0.

(5.4)

As pointed out in Sect. 3, we introduce u and u in C0(0, T ; V ) solution of:

u|!s = 0u|!f = uf ,

(5.5)

where uf is the solution of Navier-Stokes equations in !f with uf = 0 on$ /" and u defined as:

u|!s = us where

#tus " 1Re%us + us + #ps = 0 in IR+ $!s

div us = 0 in IR+ $!sus(0) = 0 in !s&(us, ps)ni

s = "&(uf , pf)nis on "i

s 1 ! i ! I

(5.6)

u|!f = uf where

#tuf " 1Re%uf + uf · #uf

+uf · #uf + #pf = 0 in IR+ $!f

div uf = 0 in IR+ $!fuf(0) = 0 in !fuf = us on "i

s 1 ! i ! I

(5.7)

512 P. Angot et al.

Remark 5.2 If the penalization is suppressed in the time derivative in (5.1),the resultingBrinkman equation in the solid (5.7) becomes a steady equation.

Remark 5.3 The existence and the regularity of uf is classical as&

$is

usnifd+ = 0, 1 ! i ! I.

The existence of us can be derived from [7] where such boundary conditionsare studied.

The regularity us % C0(0, T ;V ) ' L2(0, T ;H2(!s)) can be obtained bythe translation method of Nirenberg.

Then we set u" = u + 'u + 'w" and we establish energy estimates onw".

Theorem 5.1 Let u and u be defined by (5.5), (5.6) and (5.7), Then thereexists (v")" bounded in L!(0, t;H) ' L2(0, t;V) such that:

u" = u + 'u + '32 v".

Moreover there exists a generics function g in C0(IR+) such that:

|v"|L"(0,T ;L2(!s)) ! g(t)'12

|#v"|L"(0,T ;L2(!s)) ! g(t)'12

Proof. Replacing u" in (5.4) by its expression, we get as u|!s = 0:&

!1!s

$

#tu(+1

Re#u · #(+ u · #u(" f(

%

dx

+&

!1!s

$

#tu(+1

Re#u · #(+ u(

%

dx

+'&

!1!f

$

#tu(+1

Re#u · #(+ u · #u(+ u · #u(

%

dx

+'&

!

$

(1 +1'1!s)(#tw"(+

1Re

#w" · #() + u · #w"(

+1'1!sw"(+ w" · #u(

%

dx

+'2&

!(u · #u + w" · #u + u · #w" + w" · #w")(dx = 0 .

(5.8)


Using (5.5) , (5.6) and (5.7) the first three integrals reduce, after integratingby parts, to:

"'I

,

i=1

&

$Is

&(uf , pf)nif(d+.

Then (5.8) becomes when dividing by ':

+&

!

$

(1 +1'1!s)(#tw"(+

1Re

#w"#() + u · #w"(

+1'1!sw"(+ w" · #u(

%

dx

+'&

!(u · #u + w" · #u + u · #w" + w" · #w")(dx

=I

,

i=1

&

$Is

&(uf , pf)nif(d+.

Setting ( = w" we get by sobolev embeddings the following inequality:

12

d

dt

&

!

$

1 +1'1!s

%

w2"dx +

1Re

&

!

$

1 +1'1!s

%

|#w"|2 dx

+1'

&

!1!sw

2"dx

! c (|#u| + ' |#u|) |w"| |#w"| + c ' |u|12 |#u|

32 |w"|

12 |#w"|

12

+c

-

I,

i=1

|&(uf , pf |L2($is)

.

(|1!sw"| + |1!s#w"|) ,

as

|w"|L2($is)

! c/)

)1!isw"

)

) +)

)1!is#w"

)

)

0

.

where c always denotes a generic constant.Then, from the regularity of u and u, and from Young inequalities we have

514 P. Angot et al.

for ' ! 1:

d

dt

&

!

$

1 +1'1!s

%

w2"dx +

1Re

&

!

$

1 +1'1!s

%

|#w"|2 dx

+1'

&

!1!sw

2"dx

! c"

|#u|2 + '2 |#u|2#

|w"|2

+c '

-

|u| |#u|2 +I

,

i=1

|&(uf , pf |2L2($is)

.

which gives from Gronwall lemma as w"(0) = 0:

|w"(t)|2 +1'

|1!sw"(t)|2 ! 'g(t)

and& t

0

$

|#w"(*)|2 +1'

|1!sw"(*)|2%

d* ! 'g(t).

where g denotes a generic function in C0(IR+). !

6. Numerical validation

To validate the penalization we choose a numerical test easy to implementand for which the geometry is well fitted in order to avoid an additionalerror. So we compute the two-dimensional flow around a square cylinder ina channel, i.e. with a no-slip boundary condition on the horizontal walls of$ , as shown on Fig. 2.We use a cartesian mesh with 320 $ 80 cells on the whole domain ! =

!f / !s = (0, 4) $ (0, 1), in such a way that "s corresponds to meshlines. To get the reference solutions uref , we approximate the Navier-Stokesequations in !f associated to homogeneous Dirichlet boundary conditionson"s. Then we can compare for various values of ' the penalized solutionsto these reference solutions.

6.1. The L2 penalization

First, we investigate a steady case at the Reynolds number Re = 40 whereRe = UD

% with U = 1 defines the mean velocity of the Poiseuille flow atthe entrance section,D the size of the square and . the cinematic viscosity.


Ω Ω

Γ

fsΣ s

x

x1

2

Fig. 2. Computational domain

Table 1. Numerical measurement of the error estimate at Re = 40

! ||u!||L2("s) " for O(!#) ||u! " uref ||L2("f ) " for O(!#)10#2 3.81 10#2 9.69 10#2

10#3 5.80 10#3 0.82 1.65 10#2 0.7710#4 6.39 10#4 0.96 1.82 10#3 0.9610#5 6.46 10#5 0.99 2.01 10#4 0.9610#6 6.47 10#6 1.00 9.59 10#5 0.32

The isolines of pressure and vorticity are plotted on Fig. 3 for both the exactDirichlet boundary condition and the L2 penalization respectively. We canobserve that the pressure is continuous through the obstacle with the L2

penalization.Table 1 shows that the numerical error estimate varies asO(') in!s and

about O(') in !f . When the penalty parameter ' is going to zero, the errorof the solution in the fluid domain is still decreasing as shown on Table 1.However the gain in accuracy, which is in O(') for a range of ' valuesnot too small, becomes in fact limited by the discretization error as soonas ' is taken below a certain threshold. This is confirmed by the fact thatthe latter threshold is decreasing when the mesh step is smaller. Hence, thetheoretical error estimate given in Theorem 4.2 might not be optimal as ityields O('3/4) in !s and O('1/4) in !f .Then we compute unsteady solutions atRe = 80 to be sure that propagationeffects are not spoiled by the penalization method. We see on Fig. 4 and 5that a large value of ' induces a delay of the convection. Consequently, theStrouhal number St = fD

U , where f is the frequency of the vortex sheddingin the wake, is under estimated (Table 2). Let us note that the value of theStrouhal number for this internal flowdoes not correspond to thewell-knownvalue for such a Reynolds number in an open flow.

516 P. Angot et al.

Fig. 3. Comparison of the solution computed with Dirichlet boundary condition and L2

penalization with ! = 10#8 (pressure and vorticity fields at Re = 40)

Table 2. Strouhal number at Re = 80

uref ! = 10#8 ! = 10#6 ! = 10#4 ! = 10#2

St 0.2390 0.2390 0.2390 0.2387 0.2349


Fig. 4. Comparison of the solutions computed with Dirichlet boundary condition and L2

penalization with ! = 10#8 and ! = 10#2 at Re = 80 (pressure field)

For small values of ', we clearly observe on Figs. 4 and 5 that there isno difference between uref and u" when ' = 10$8. This means that theerror induced by the penalization method is much lower than the error ofapproximation.These numerical tests show the efficiency of the L2 penalization that

gives the same result thanDirichlet boundary condition as soon as the penaltyparameter ' is small enough. In addition, the pressure field is continuous inthe whole domain ! and the equation

1'u" + #p" = 0 in !s

is numerically satisfied, which confirms the asymptotic formal expansiongiven by (3.5). Hence, there is a Darcy flow in the solid at the order '. As itis carried out numerically in [10] and according to (4.17) and Remark 4.4,

518 P. Angot et al.

Fig. 5. Comparison of the solutions computed with Dirichlet boundary condition and L2

penalization with ! = 10#8 and ! = 10#2 at Re = 80 (vorticity field)

the lift and drag coefficients can be computed from

lim"%0

&

!s

#p dx = " lim"%0

1'

&

!s

u" dx =&

$s

&(u, p) · nf d+.

6.2. The H1 penalization

We perform the same numerical tests than in Sect. 6.1. The results plottedon Fig. 6 atRe = 40 show that there is a small discrepancy with the originalsolution. It is due to the discretization of the penalized viscous term. Indeed,the discretization of this term changes slightly the size of the obstacle andthus the recirculation zone in the wake is a little bit different.In the unsteady case, we observe also a discrepancy in the determination

of the Strouhal number even for ' = 10$8. This is still due to the varia-tion of volume of the body which is effectively taken into account by the


Fig. 6. Comparison of the solutions computed with Dirichlet boundary condition and H1

penalization with ! = 10#8 at Re = 40 (vorticity field)

discretization and induced by the approximation of the penalized viscousterm. Hence, theH1 penalization does not improve in practice the numericalresults as it is shown theoretically.

7. Conclusions

This paper gives a rigorous justification of the penalization method to takeinto account a solid body immersed in an incompressible viscous fluid inmotion. This method is a fictitious domain method which has been provedto be very easy to implement, robust and efficient. In addition, since thepenalization parameter ' can be taken as small as necessary, the penaltyerror is always negligeable in front of the error of approximation.

References

1. Angot, Ph., Caltagirone, J.P. (1990): New graphical and computational architectureconcept for numerical simulation on supercomputers. Proceedings 2ndWorld Congresson Computational Mechanics, Stuttgart 1, 973–976

2. Angot, Ph., Caltagirone, J.P. (1990):Natural convection through periodic porousmedia.Proceedings 9th Int. Heat Transfer Conf., Jerusalem, Hemisphere 5, 219–224

3. Angot, Ph.: Analysis of singular perturbations on the Brinkman problem for fictitiousdomain models of viscous flows. C. R. Acad. Sci. Paris, Serie I, 1997, (submitted)

520 P. Angot et al.

4. Arquis, E., Caltagirone, J.P. (1984): Sur les conditions hydrodynamiques au voisinaged’une interface milieu fluide - milieu poreux: application a la convection naturelle. C.R. Acad. Sci. Paris 299(1), Serie II, 1–4

5. Bruneau, Ch.-H.: Numerical Simulation and Analysis of the Transition to Turbulence.15th International Conference on Numerical Methods in Fluid Dynamics, 1996, toappear in Lecture Notes in Physics

6. Bruneau, Ch.-H., Fabrie, P. (1994): Effective downstream boundary conditions forincompressible Navier-Stokes equations. Int. J. Numer. Methods in Fluids 19, 693–705

7. Bruneau, Ch.-H., Fabrie, P. (1996): New efficient boundary conditions for incompress-ible Navier-Stokes equations: a well-posedness result. M2AN 30(7), 815–840

8. Caltagirone, J.P. (1994): Sur l’interaction fluide-milieu poreux: application au calculdes efforts exerces sur un obstacle par un fluide visqueux. C. R. Acad. Sci. Paris 318,Serie II, 571–577

9. Goldstein, D., Handler, R., Sirovich, L. (1993): Modeling a no-slip flow boundary withan external force field. J. Comput. Phys. 105(2), 354–366

10. Khadra, K., Parneix, S., Angot, Ph., Caltagirone, J.-P.: Fictitious domain approachfor numerical modelling of Navier-Stokes equations. Int. J. Numer. Meth. in Fluids(submitted).

11. Lions, J.L. (1969): Quelques methodes de resolution des problemes aux limites nonlineaires. Dunod, Paris

12. Peskin, C. (1982): The fluid dynamics of heart valves: Experimental, theoretical, andcomputational method. Annu. Rev. Fluid Mech. 14, 235–259

13. Peskin, C. (1977): Numerical Analysis of blood flow in the heart. J. Comput. Phys. 25,220–252

14. Saiki, E.M., Biringen, S. (1996): Numerical simulation of a cylinder in uniform flow:application of a virtual boundary method. J. Comput. Phys. 123, 450–465

15. Tartar, L. (1978): Topics on nonlinear Analysis, Preprint Orsay16. Temam, R. (1979): Navier-Stokes equations and numerical analysis, North-Holland,

Amsterdam

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Numerische Mathematik - ljll.math.upmc.fr · Obstacles in incompressible viscous ﬂows 501 3.2. The H1 penalization Inthissectionwechoosetopenalizeinadditionthewholelinearpartofthe