Shape Sensitivity and Design for Fluids with Shocks

Shape Sensitivity and Design for Fluids with Shocks

OLIVIER PIRONNEAU*

University of Paris VI & IUF, Laboratoire Jacques-Louis Lions, 175 rue du Chevaleret, F-75013 Paris, France

(Received March 2003)

Dedicated to Professor Mutsuto Kawahara on the occasion of his 60th birthday For many problems of compressible fluid dynamics it is desirable to find the sensitivity of the shock position with respect to the shape of the domain occupied by the fluid. One application is for the minimization of the sonic boom of airplanes; another is for the stability of the stream in fast-flowing rivers or canals. Classical calculus of variation is not valid for these cases because of the presence of Dirac functions appearing when a discontinuous function is differentiated, but we show here on the compressible potential flow equation how to find the equations of the derivatives and what are the linearized problems. Some numerical test cases are given for illustration.

Keywords: Partial differential equations; Nozzle flow; Sensitivity; Transonic equation; Shallow water equations; Optimal shape design

INTRODUCTION

There are two pioneering papers on the linearized Euler equations for potential flow: one by Majda (1983) and one by Godlewskiet al.(1998). There are also many studies directed to the Euler equation (see Giles and Pierce, 2001;

Alekseev and Navon, 2002; Homescu and Navon, 2003 for example). In Bardos and Pironneau (2002a,b) we have reinterpreted the results of Godlewski et al. (1998) and introduced a formalism which we shall apply here to find the linearized transonic equation in cases where the position of the shock is sensitive to the parameter of the linearization.

Following di Cesare and Pironneau (2000) and extending Pironneau (2002) we analyze the stationary problem for the transonic equation, we formally derive an equation for the derivative of the potential of the flow with respect to parameters in the data, including the shape of the domain, we discuss the well posedness of the problem and we present some numerical tests to confirm the theory.

This is done first in one dimension of space for stationary and transient cases and then for two-dimensional cases.

We also adapt the results to the shallow water equations for the stability of streams in rivers and canals.

COMPRESSIBLE POTENTIAL FLOWS Isentropic Flow

Consider the Euler equations for compressible perfect isentropic flows in a domainVof boundaryG.

›trþ7·ðruÞ ¼0; ›tðruÞ þ7·ðru^uÞ þ7r^g¼0: ð1Þ For aerodynamics g¼1:4 and for shallow waters g¼1:

Initial and boundary conditions must be given such as r¼r⁰;u¼u⁰ at time zero inVandu·n¼uG·n onG plusr¼rG;onG²¼{x[G:uG·n,0}:If in addition the inflow velocity is supercritical thenu·s¼uG·smust be given too. The vectors n, sare the outer normal and tangent vector(s) toG.

Irrotational Flow

The following holds for any vector fieldu 7·ðru^uÞ ¼ru·7uþ7·ðruÞu

¼r7u²

2 2ru£7£uþ7·ðruÞu: ð2Þ

ISSN 1061-8562 print/ISSN 1029-0257 onlineq2003 Taylor & Francis Ltd DOI: 10.1080/1061856031000113617

*E-mail: pironneau@ann.jussieu.fr

The Caseg>1

So the irrotational stationary solutions to Eq. (1) satisfy (recall that7r^g¼ ðgr=g21Þ7r^g21):

7£u¼0 ›trþ7·ðruÞ ¼0;

›tuþ7 u²

2 þ g

g21r^g21

¼0:

The first equation tells us that u derives from a potential f, i.e. u¼7f: The third equation gives an algebraic relation betweenrandu²,2gr^g21þ ðg21Þu²

¼K2›tf;where the constantKis fixed by the boundary conditions. After renormalization it can be written as

›tfþ1

2j7fj²þr^1=b¼1 ›trþ7·ðr7fÞ ¼0: ð3Þ Stationary solutions of this system must satisfy the transonic equation:

7· 12j7fj²b

7f

¼0 in V;

12j7fj² b›f

›njG_N ¼g;

f¼fG on GDUG\GN;

ð4Þ

with g¼1:4; b¼1=ðg21Þ ¼2:5 in air. Boundary conditions come from the knowledge of ru·n on G.

On G², ›f/›s is also known, or equivalently f up to a constant k_j on each connected component G²_j of G².

There are also integral quantities which are prescribed and which come from an integration over time of Eq. (1), the conservation of mass and the conservation of momentum which may put a constraint of compatibility on the data above:

ð

V

rþ ð

G

ru·n¼ ð

V

r⁰þ ð

G

r⁰u⁰·n;

ð

V

ruþ ð

G

ruðu·nÞ ¼ ð

V

r⁰u⁰þ ð

G

r⁰u⁰ðu⁰·nÞ:

For instance, from Eq. (4) we see that it is necessary that the integral onGofgbe zero, therefore the integral ofron Vis prescribed.

An entropy inequality must be added for well posedness (see Glowinski, 1984; Necˇas, 1989):

Df. 21:

It is automatically satisfied whenu is continuous and also when uis discontinuous with a decreasing jump in the direction of the flow u.

A time-dependent, simple (yet physically meaningful) model between Eqs. (3) and (4) is

›ttf27· 12j7fj²b

7f

¼0 in V£ð0;TÞ

f¼f⁰; ›tf¼f¹ on V£{0}

12j7fj² b›f

›n ¼g on G_N£ð0;TÞ;

f¼fG on GD£ð0;TÞ:

ð5Þ

The Caseg51

For shallow water flows,g¼1and the trick used, namely 7r^g¼ g

g21r7r^g21

must be replaced by7r¼r7logr:Then Eq. (3) becomes

›tfþ1

2j7fj²þlogr¼1 ›trþ7·ðr7fÞ ¼0; ð6Þ and the stationary potential flow equation for shallow waters is:

7·e^212j7fj27f¼0 in V;

e^212j7fj2›f

›nj_GN ¼g; fj_GD ¼fG: ð7Þ The time-dependent approximation of Eq. (6) is

›ttf27·e^212j7fj27f¼0 in V£ð0;TÞ f¼f⁰; ›tf¼f¹ on V£{0}

e^212j7fj2›f

›n ¼g on G_N£ð0;TÞ;

f¼fG on G_D£ð0;TÞ:

ð8Þ

Slender Shock Tube

We can compound the caseg¼1with the caseg.1in one framework by defining

rðuÞ ¼ ð12u²Þ^b with b¼ 1

g21 if g.1 and

rðuÞ ¼e^212u2 if g¼1:

WhenVis slender of lengthLon thexaxis with cross- section of surface S(x) then, following Landau and Lifschitz (1956), the solution to Eq. (4) is

rðuÞuðxÞSðxÞ ¼K;

ðL 0

rðuÞSðxÞdx¼a ð9Þ

where K is fixed by the inflow/outflow boundary conditions and a is the prescribed total mass of gas.

Equation (9a) has two solutions at eachx, one subcritical and one supercritical. Only one discontinuous switch from super to subcritical is allowed by the entropy condition and the position of the switch (shock) is determined by Eq. (9b).

Derivative with Respect to the Total Mass

Let u⁰ denote the derivative of u with respect to a.

By differentiation of Eq. (9a) we find that u⁰¼0 almost everywhere. However, the position of the shockxsvaries withaand since

u¼u2þ ðu_þ2u2ÞHðx2xsÞ;

whereHis the Heavyside function, we have

u⁰¼2x⁰_s½udðx2xsÞ where ½uUuþ2u2: By the same principle

d

darðuÞ ¼2x⁰_s½rðuÞdðx2xsÞ:

Putting this information into Eq. (9b) differentiated gives the shock displacement:

x⁰_s¼2 1

½rðuÞSðx_sÞ:

Derivative with Respect to the Inflow/Outflow Condition The derivative u⁰ of u with respect to K will have two parts, a regular part denoted by u⁰_K and a singular part which as before is2x⁰_s½udðx2x_sÞ:After differentiation, Eq. (9) is

MðuÞu⁰_K¼1

S with MðuÞ

¼

ð12u²Þ^b 122b_12u^u22

if g.1 ð12u²Þe^212u2 if g¼1

½rðuÞSðx_sÞx⁰_s¼2 ðL

0

r_ðuÞu⁰_KSdx where

r_ðuÞUdrðuÞ

du giving x⁰_s¼ 21

½rðuÞSðxsÞ ðL

0

r_ðuÞ MðuÞdx:

ð10Þ

Derivative with Respect to Shape

Assume now that S( · ) depends on a parameter b and denote S⁰ its derivative in b. Differentiation of

Eq. (9) yields

MðuÞu⁰_S¼2KS⁰ S² ;

½rðuÞSðxsÞx⁰_s¼ ðL

0

ðr_ðuÞu⁰_SSþrðuÞS⁰Þdx: ð11Þ

Here toox_s⁰can be computed explicitly:

x⁰_s¼2 1

½rðuÞSðx_sÞ ðL

0

Kr_ðuÞ MðuÞS2rðuÞ

S⁰dx:

Remark 1 In all cases the shock position is unstable when½u!0(weak shocks).

The Time-dependent Slender Case

Integrated over the cross-section of the slender domain {ðx;yÞ:2S_mðxÞ,y,SMðxÞ; x[ð0;LÞ}; with S¼S_M2S_m;Eq. (5) leads to

›ttf21

S›xðSrð›xfÞ›xfÞ ¼0: ð12Þ As before when S depends on a parameter a, the derivativef⁰with respect toawill be discontinuous and will satisfy in the sense of distribution theory:

›ttf⁰2›xð›urðuÞ›xfþrðuÞ›xf⁰Þ2›xS S MðuÞu⁰

¼ ›xS S

0rðuÞu: ð13Þ

As an illustration assume thatS⁰is zero before the shock and that stability is studied at a stationary equilibrium state.

ThenrðuÞu¼Kbefore the shock and sof⁰¼0 before the shock and the Rankine – Hugoniot conditions at the shock imply›xf⁰¼0: Hence, Eq. (13) can be integrated after the shock only for x[ðS;LÞ with the boundary conditions

›xf⁰¼0 at the shock s; f⁰ðLÞ ¼0 or ›xf⁰ðLÞ ¼0:

Numerical Simulation

A simple numerical test was made. A geometry is chosen:

SðxÞ ¼1þaðbÞ* sinðpðx=LÞ; x[ð0;LÞ with að0Þ ¼0:2 andL¼1:Stability is studied around the stationary state, i.e. u(x) satisfying ue^2ð1=2Þu2¼0:35=SðxÞ: Equation (13) is integrated up toT ¼10 by an explicit scheme with all

“primes” being derivatives with respect tobwitha⁰such that0:35ð›xS=SÞ⁰¼sinð2pðx=LÞÞ:

Figure 1 showst!f⁰ð0;tÞ;which is proportional to the displacement of the shock, andx!f⁰ðx;TÞ:This is a case where the displacement of the shock can be computed quite easily.

THE TWO-DIMENSIONAL CASE

Assume now thatf, solution of Eq. (4) is a function of a scalar parameter a via the data of the partial differential equation and that it has a shock S(a). We wish to differentiatefwith respect toa. For clarity we will seek the result ata¼0:

Denote bya(x)athe distance in the directionn_S, normal to the shock S pointing inside Vþ, between S(a) and SUSð0Þ;i.e.

SðaÞ ¼{xþaaðxÞn_SðxÞ: x[S}:

Denote by V^ the region before the shock and after the shock. Then,I_Dbeing the characteristic function of a setD,

u¼u2þ ðu_þ2u2ÞI_Vþ

whereu^are smooth functions. The derivative ofIV_þis a Dirac mass onS(see Bardos and Pironneau, 2002). So if f⁰,u⁰denotes the derivative off,uwith respect toa, we have

u⁰¼u⁰₂þ ðu⁰_þ2u⁰₂ÞIVþ2½uadS

FIGURE 1 Top: Variation of the shock position (downstream to its initial value) versus time. Bottom: Water level before (smoother curve) and after the change in geometry, at final time.

and therefore if u⁰ has a Dirac mass on S, f⁰ must be discontinuousacrossS, and

½f⁰_S¼2½u_S·n_Sa: ð14Þ

Away from the shock the transonic equation can be differentiated, giving:

7·ðr⁰7f·7wþr7f⁰Þ

;7· r 122bu^u 12juj²

7f⁰

¼0:

ð15Þ

But Eq. (15) makes no sense at the shock because a Dirac mass is multiplied by a discontinuous function.

In Bardos and Pironneau (2002) a formalism has been introduced to differentiate such functions. Suppose that u andvare discontinuous at x(a). Then the following is expected:

u⁰¼u⁰_a2½ux⁰_sdðx2x_sÞ v⁰¼v⁰_a2½vx⁰_sdðx2x_sÞ ðuvÞ⁰¼ ðuvÞ⁰_a2½uvx⁰_sdðx2x_sÞ:

The trick is to define

u¼

1

2ðu^þþu²Þ at the shock

u elsewhere

and then notice that “(uv)⁰¼{u}¯v⁰þu⁰{v}¯v¯” contains the identity “[uv]¼{u¯}[v]þ[u]{v}¯”.

When b¼5/2, r²¼(12u²)⁵¼AB withA¼(12u²), B¼C²;C¼A²;

2rr ⁰¼A⁰BþAB ⁰¼22uu⁰Bþ2ACC ⁰

¼22uu ⁰ð12u²Þ⁴þ4ð12u²Þ ð12u²Þ²AA ⁰

¼22uu ⁰ðð12u²Þ⁴þ4ðð12u²ÞÞ²ð12u²Þ²Þ: ð16Þ

Therefore, the linearized transonic equation is 7·ðM7 f⁰Þ ¼0 with M

¼ ð12u²Þ^2:5 122b^{u^u} 12u²

inV\S ð17Þ

withu¼ j7fjand, at the shockS M ¼ ð12u²Þ^2:5

£ I2ð12u²Þ⁴þ4ð12u²Þ²ð12u²Þ² ð12u²Þ^2:52

u^u

!

: ð18Þ

Notice that both formulae forM agree inV\S.

FIGURE 2 Level lines offwith a changing potential at the outflow boundary going from 0.4 (left) to 0.44 (right). Bottom: Plot ofx!fðxÞj_Gw:

In variational form, Eq. (4) is

;w[H¹ðVÞ ð

V

r7f·7w¼ ð

G

gw:

WhenV¼VðaÞ;assuming thatg¼0on the parts ofG which depend ona, the derivative of this formulation is (see Pironneau, 1983)

;w[H¹ðVÞ ð

V

M7f⁰·7w¼2 ð

G

ðr7f·7wÞa

which is also Proposition 1.

Proposition 1 The differentiated transonic equation with respect to boundary variations a is a system for ðf⁰_S;½f⁰;x⁰_sÞ;its regular part, its jump across the shockS and the variation of the shock position:

;w[H¹ðVÞ ð

V\S

M7 f⁰·7w2 ð

S

M½7f·n_S›w

›nx⁰_s

¼2 ð

G

ðr7f·7wÞa: ð19Þ

Remark 2 (19) contains the Rankine – Hugoniot con- dition½M7f⁰·n_S¼0. It is interpreted as

7·ðM7f⁰Þ ¼0; ½M7 f⁰·nj_S¼0;

M7 f⁰·nj_G¼2r›f

›s

›a

›s; ½f⁰ ¼2½u·n_Sx⁰_S: ð20Þ

Comparison with the one-dimensional case shows that an additional integral condition is needed, like the total mass of fluid specified.

NUMERICAL SIMULATION

Consider first a very simple case, that of a transonic divergent nozzle with supersonic inflow conditions and a difference of potential so that there is a shock in the flow. This example has been investigated in Bardos and Pironneau (2003) for sensitivity with respect to changes in the inflow or outflow boundary condition.

We recall the results obtained. Upstream of the shock the PDE is hyperbolic and so the perturbation f⁰ is zero because it is entirely defined by the inflow conditions, which are zero, integrated on the streamlines.

Suppose we want to move the shock by a known quantity x_s⁰, then the value off⁰is known on Sand we must find the boundary conditionfwhich gives^›f_›n⁰¼0on S, for instance by solving

minf

ð

S

ðf⁰2f⁰_SÞ²:7·ðM7f⁰Þ ¼0; in V^þ

›f⁰

›n j_S¼0; ›f⁰

›n jG ¼f

:

FIGURE 3 Mach lines with a changing Neumann outflow condition at the outflow. Bottom: Plot ofx!fðxÞj_Gw andx!ð›f=›xÞðxÞj_Gw on the symmetry lineGw.

Numerical Algorithm

As the transonic equation is nonlinear we used a fixed point algorithm with a small under-relaxation parameter (for instance 0.01); convergence is obtained with 50 – 200 iterations:

7·ðr^m7f^mþ1Þ ¼0:

To computer^mþ1we compute the two rootsu^of ð12u²Þ^bu¼12j7f^mþ1j²b

j7f^mþ1j and set r^mþ1¼ ð12u²₂Þ^b if u decreases or if both u andrugrow on the streamline and setr^mþ1¼ ð12u²_þÞ^b otherwise.

The Divergent Nozzle

Using freefemþ(http://www.freefem.org) we com- puted the solution of the transonic equation in a symmetric nozzle of equation

Gw¼ ðyðxÞ;xÞ:yðxÞ ¼1þ1

8ð3x²22x³Þ; x[ð0;1Þ

: We performed two computations withui¼0:4;and a potential difference of 0.4 or 0.44. The level curves of

›f

›x for these are reported on Fig. 3.2. They show that only the region after the shock changes, as predicted by the theory. Finally, we have solved numerically the PDE of f⁰ in the domain right to the shock with Neumann homogeneous conditions except on the outflow bound- ary where we have a Dirichlet condition equal to 0.04.

The level lines are shown in Fig. 3.3. It predicts a shock displacement parallel and of distance 0:04=ð0:62 0:2Þ ¼0:1 which is compatible with the experiments of Fig. 3.2.

The Convergent Divergent Nozzle The nozzle’s equation is now

Gw¼{ðyðxÞ;xÞ:yðxÞ ¼1þ1

8ð3x²22x³Þ;

x[ð20:9;0:9Þ}:

The inflow and outflow boundary conditions are now ru·n¼M*ð12M²Þ^bþk

3 1 162y

M¼0:4; k¼0 or 1:

Next we solved the equation for f⁰ with Neumann condition equal to zero at the shock boundary and equal to

1

3ð₁₆¹ 2yÞ at the outflow boundary. The level lines are shown in Figs. 3.3 and 4. This test is not very conclusive, it is at the border of numerical noise. The results on f⁰ predict a displacement of the shock at the symmetry line equal to 0.003/0.4, i.e. a little less than one per cent, which is about what we obtain numerically. However, there should not be any changes left of the shock and the numerical scheme does not show that.

Perspective

The numerical strategy for nozzle flows may not work for wing profiles because the elliptic zone surrounds the hyperbolic zone and so there is no independent decomposition of the zones. Another problem with the method we have used here is that it is necessary to “track”

the shock and remesh the domain with the shock as one boundary.

It would be convenient to put the linearized equation, the Rankine – Hugoniot conditions and the equation for the shock displacement into one single variational

FIGURE 4 Level lines off⁰for the first and second test.

formulation. To do so we observed in Bernardi and Pironneau (2002) that

ð

V

7f·7w¼ ð

›D

›w

›nD

,f¼I_D:

This leads us to try, for an appropriateg ð

V

M7f⁰7w¼ ð

S

g›w

›n

;w[H¼{w[H¹ðVÞ: wjG_i<Go ¼0}

with M¼r I22bu^u 12juj²

: ð21Þ

The solution would satisfy (15), and its Rankine – Hugoniot conditions, but it is hard to see that Eq. (14) would be satisfied because an integration by part in the neighborhood of Sgives a term which has no meaning, MdS. However, in mixed form consider a similar problem with

U⁰;M7f⁰¼r I22bu^u 12juj²

7f⁰;

that is (subject to non-homogeneous boundary con- ditions):

ð

V

ðM²¹U⁰·Wþf⁰7·WÞ þ ð

S

gW·n¼0

;W [Hðdiv;VÞ ð

V

7·U⁰w¼0 ;w[L²ðVÞ

ð22Þ

for which the unknowns are U⁰,f⁰,g. Then f⁰ is also discontinuous and there is no ambiguity when integrated by part becauseU⁰does not jump acrossS.

Two problems remain:

1. Justify Eq. (22) and show that it definesgindeed (ifM was elliptic everywhere g would be a data of the problem, but we have shown that it is not so for transonic flows, at least in the case of nozzles).

2. Solve numerically Eq. (22) with automatic numerical detection ofS.

References

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paper185.pdf.

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