A Discrete Kato Type Theorem on Inviscid Limit of Navier-Stokes Flows

PROOF COPY 210706JMP

Discrete Kato-type theorem on inviscid limit of Navier-Stokes flows

Wenfang共Wendy兲Cheng

School of Computational Sciences, Florida State University, Tallahassee, Florida 32306 Xiaoming Wang^a兲

School of Mathematics, Fudan University, Shanghai, China and Department of Mathematics, Florida State University, Tallahassee, Florida 32306

共Received 25 July 2006; accepted 24 October 2006兲

The inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics. Here we present a somewhat surprising result related to numerical approximation of the problem. More precisely, we show that numerical solutions of the incompressible Navier-Stokes equations converge to the exact solution of the Euler equations at vanishing viscosity and vanishing mesh size provided that small scales of the order of␯^/Uin the directions tangential to the boundary are not resolved in the scheme. Here␯ is the kinematic viscosity of the fluid andUis the typical velocity taken to be the maximum of the shear velocity at the boundary for the inviscid flow. Such a result is somewhat counterintuitive since the convergence is ensured even in the case that small scales predicted by the conventional theory of turbulence and boundary layer are not resolved since under- resolution共which is allowed in our theorem兲in advection dominated problem usu- ally leads to oscillation which inhibits convergence in general. The result also indicates possible difficulty in terms of numerical investigation of the vanishing viscosity problem if rigorous fidelity of the numerics is desired since we have to resolve at least small scales of the order of␯^/Uwhich is much smaller than any small scales predicted by the conventional theory of turbulence. On the other hand, such a result can be viewed as a discrete version of our result关X. Wang, Indiana Univ. Math. J. 50, 223 共2001兲兴 which generalized earlier the result of Kato 关in Seminar on PDE, edited by S. S. Chern共Springer, NY, 1984兲兴where the relevance of a scale proportional to the kinematic viscosity to the problem of vanishing viscosity is first discovered. ©2007 American Institute of Physics.

关DOI:10.1063/1.2399752兴

I. INTRODUCTION

One of the central and most useful systems in fluid dynamics is the Navier-Stokes system for incompressible homogeneous Newtonian fluids which governs the motion of fluids like air and water under normal conditions

⳵^u␯

⳵^{t +共u␯·ⵜ兲u␯−}␯⌬u^␯+ⵜp^␯=f, in⍀, 共1兲

divu^␯= 0, in⍀, 共2兲

u^␯=b,on⌫, 共3兲

a兲Author to whom all correspondence should be addressed: [email protected] 1

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PROOF COPY 210706JMP

u^␯=u₀, att= 0, 共4兲

whereu^␯=共u₁^␯, u₂^␯, u₃^␯兲is the velocity field in the Eulerian coordinates,p^␯is the kinematic pres- sure,f=共f₁, f₂, f₃兲 is the external body force, and the positive constant ␯ is the kinematic vis- cosity. The velocityb at the boundary satisfies the no-penetration condition

b·n= 0, 共5兲

where n is the unit outward normal to the boundary ⌫=⳵⍀. This includes the case of Taylor- Couette-type flows among others. The boundary condition sometimes is referred to as a charac- teristic boundary condition since the boundary consists of stream lines all the time.

There is abundant literature on the Navier-Stokes systems. The interested reader may consult the books byConstantin and Foias共1988兲, Doering and Gibbon共1995兲,Ladyzhenskaya 共1969兲, Majda and Bertozzi共2001兲, orTemam共2001兲for the mathematical theories of the Navier-Stokes equations.

For realistic fluids like air and water, the kinematic viscosity is very small and hence we may formally set it to zero and arrive at the Euler system for incompressible inviscid共dry兲fluids

⳵^u0

⳵^{t +共u0·ⵜ兲u0+ⵜp0=f, in⍀, 共6兲}

divu⁰= 0, in⍀, 共7兲

u⁰·n= 0, on⌫, 共8兲

u⁰=u0, att= 0. 共9兲

More importantly, if the characteristic flow speedUis large or the characteristic length scaleLis large, the Reynolds number which is defined as

Re =LU

␯ ^共10兲

is large and the nondimensionalized Navier-Stokes system takes the same form except the kine- matic viscosity is replaced by the reciprocal of the Reynolds number which is very small. This provides another scenario where inviscid approximation is needed.

Such an approximation has been utilized in many applications. A physically important ques- tion is then whether such an approximation can be justified via the zero viscosity limit of the Navier-Stokes equations.

The mathematical investigation of such a problem is extremely difficult due to the singular nature of the problem which involves a boundary layer and the nonlinear nonlocal nature of the systems involved. Extensive efforts have been made to resolve the inviscid limit problem which lead to many partial results 关see, for instance, Prandtl 共1905兲, von Karman 共1956兲,Schlichting 共1979兲, etc., from the physical perspective, and Bona and Wu 共2002兲, Weinan and Engquist 共1997兲,Kato共1984兲,Ladyzhenskaya共1969兲,Oleinik共1963兲,Oleinik and Samokhin共1999兲,Mat- sui共1994兲,Sammartino and Caflisch共1996兲,Temam and Wang共1997,1998兲,Wang共2001兲, and Xin and Zhang共2004兲for some of the mathematical results兴.

Confronted with such a difficult problem, we naturally resort to numerical methods, especially with today’s powerful computer and efficient and accurate numerical schemes. A natural question to ask is the fidelity of the numerical results. More precisely, let

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PROOF COPY 210706JMP

u^k=u_h

k

␯k

be a sequence of numerical solutions of an appropriate numerical scheme with kinematic viscosity

␯kand mesh sizehksatisfying the vanishing viscosity and mesh size assumption

␯k→0,h_k→0, ask→⬁ our questions are as follows:

Does lim

k→⬁u_h

k

␯k=u⁰imply lim

k→⬁u^␯k=u⁰? 共11兲

Does lim

k→⬁u_h

k

␯k⫽u⁰imply lim

k→⬁u^␯k⫽u⁰? 共12兲

What we will demonstrate below is that numerical solutions to the Navier-Stokes system always converges to the exact solution to the Euler system at vanishing viscosity if small scale of the order of ␯^/U tangential to the boundary is not resolved in the scheme. Therefore, numerically observed convergence at vanishing viscosity may have nothing to do with the convergence of the continuous solutions 关solutions of the Navier-Stokes system 共1兲兴 at vanishing viscosity to the solution of the Euler system共6兲. This indicates the difficulty in studying such an inviscid limit problem. Such a result is eluded to inWang共2001兲and is somewhat surprising since convergence is ensured even in the case that small scales predicted by the conventional theory of turbulence and boundary layer theories are not resolved in the scheme共recall that under-resolution in an advection dominated problem usually leads to oscillation which inhibits convergence in general兲.

The rest of the manuscript is organized as follows. In the next section we introduce the notion of appropriate truncation of the Navier-Stokes system and formulate our main result. We then compare the small scale in our theorem with other small scales predicted by conventional theory of turbulence and boundary layer theory. A sketch of the proof of the main result is presented in the third section. Some numerical results will be presented in Sec. IV, and we offer our concluding remarks in the last section.

II. MAIN RESULT AND REMARKS

It is apparent that the convergence of numerical solutions of the Navier-Stokes system to that of the Euler system should not be expected for arbitrary truncation, but for suitable approxima- tions of the Navier-Stokes system. Thus we need to introduce the notion of appropriate truncation.

Also the problem involves several limits: time step; spatial scale; and viscosity. The essential ingredients of an appropriate truncation are the consistency共as required by all convergent numeri- cal schemes兲and a bound on the truncated time averaged energy dissipation rate that is indepen- dent of the kinematic viscosity 关as is consistent with the Kolmogorov theory, see, for instance, Doering and Gibbon共1995兲,Foiaset al. 共2001兲, andFrisch共1995兲兴.

In order to focus on the main issue and for the sake of exposition, we consider flow in a two-dimensional共2D兲channel. Moreover, we consider discretization in the direction tangential to the boundary only 共no time discretization or spatial discretization in the direction normal to the wall兲. This allows us to concentrate on the phenomena related to tangential共to the wall兲 spatial discretization only as it is the focus of our main result. The result stated here remains valid for 3D general domain with discretization in the directions tangential to the wall in a boundary layer done using local curvilinear coordinates, and the additional assumption that the Euler system possesses a smooth enough solution on that fixed time interval under consideration.

For the channel geometry with periodicity in the horizontal direction, it is natural to use Fourier spectral truncation in the horizontal direction and thus a natural 共suitable兲 truncation would be the following Galerkin truncation:

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PROOF COPY 210706JMP

⳵^uk

⳵^{t +Pk共共u}

k·ⵜ兲u^k兲−␯k⌬u^k+ⵜp^k=Pkf, 共13兲

divu^k= 0, 共14兲

兩u^k兩z=0,h=Pkb, 共15兲

兩u^k兩t=0=Pku₀, 共16兲

wherePkis the projection onto the firstKkmodes inx, i.e., Pku=

兺

兩j兩ⱕK_k

e^2␲ijx/Luˆ^j,

冉

^兺

冊

The consistency of such a truncation is obvious. An appropriate bound on the energy dissipation rate will be derived later in the next section.

We also introduce the following quantity as a typical velocity:

U= sup

k

关0,Tmax兴⫻⌫兵兩P_k共b₁−u₁⁰兲兩其. 共18兲

Our main result is as follows.

Theorem 1: Suppose that we have a smooth solutionu⁰ of the Euler system共6兲 on the time interval 关0 ,T兴. [This is guaranteed in the 2D case with smooth enough data satisfying certain compatibility condition, seeTemam共1975兲.] Letu^kbe the solution of the truncated Navier-Stokes system共13兲with kinematic viscosity␯k.Assume that the following conditions are satisfied:

K_k→⬁共consistency兲, 共19兲

␯k→0共vanishing viscosity兲, 共20兲

Kk

␯k

LU→0共under-resolved condition兲. 共21兲

Then

u^k→u⁰. 共22兲

More precisely, there exists a generic constant␬independent of k such that

储u^k−u⁰储L⬁共O,T;L²兲ⱕ␬共共K_k␯k兲¹⁵+储u⁰−P_ku⁰储L²共O,T;H¹兲+储u⁰−P_ku⁰储L⬁共O,T;L²兲兲. 共23兲 The under-resolved condition 共21兲 can be written in terms of the smallest scale, denoted ls, resolved by the numerical method in the direction tangential to the boundary. Indeed, since Kkls

=L, the under-resolved condition is equivalent to

␯k/U ls

→0. 共24兲

This means that scales of the order␯/Uare not resolved in the scheme.

The appearance of this small scale is a little bit surprising since it is smaller than any of the known scales predicted by conventional theory of turbulence and boundary layer theory. Here we recall a few well-known small scales 共Foias et al., 2001; Doering and Gibbon, 1995; Prandtl, 1905;Frisch, 1995; among others兲:

• Prandtl boundary layer thickness:

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PROOF COPY 210706JMP

冑

• Kolmogorov dissipation length共3D兲:

冉

冊

where ␧ is the energy dissipation rate per unit volume and is presumably independent of the kinematic viscosity. 关The energy dissipation rate per unit volume scales as U³/h. In the case of boundary driven flow, the typical velocity is specified by the boundary value and thus is indepen- dent of the viscosity共see, for instance,Doering and Gibbon, 1995; Foiaset al., 2001;Wang, 1997;

among others兲. In the case of body force driven flow, it is easy to derive that ␧共⬃U³/h兲 ⱕ␬兩f兩L²Uh^−3/2, where f is the external body force, the typical velocityU is defined as the root- mean-square 共space and time averaged兲 velocity. It is also know 共see Doering and Foias, 2002, among others兲 that Re共=hU/␯兲ⱖ␬Gr^1/2共=兩f兩_L2

1/2h^3/4␯兲. Therefore, U⬃兩f兩_L2

1/2h^−1/4. Hence the ␧ is independent of viscosity for fixed body force and for turbulent flow which saturates the Kolmog- orov scaling. Similar arguments can be made for the entropy dissipation rate. The Taylor micro- scale can be estimated in the same fashion.兴

• Kraichnan dissipation length共2D兲:

冉

冊

where␩is the enstrophy dissipation rate per unit volume which is presumably independent of the kinematic viscosity.

• Taylor microlength:

冉

冊

Notice that these small length scales are all much bigger than␯^/U. Even the thickness of a viscous sublayer 关共␯^/U兲log Re兴 predicted by some boundary layer theory is bigger than ␯^{/U at large} Reynolds number共and the thickness of viscous layer is a small scale in the direction normal to wall only兲. Thus, if one follows the conventional wisdom, one would just resolve the small scales predicted by conventional theory and thus the numerical results would indicate convergence of numerical solutions to that of the Euler system共see Sec. IV below兲.

Of course,␯^/Uappear as the natural small scale in certain circumstances such as the boundary layer thickness in the presence of suction at the boundary. The appearance of the thickness␯^/Uis directly related to the suction which makes the boundary layer thinner and stable共seeTemam and Wang, 2000,2002兲. Even in that case, the scale of␯^{/Uappearsonly}in the directionnormalto the boundary in the boundary layer.

The relevance of small scales of the order of ␯^/U to the inviscid limit problem was first discovered byKato共1984兲and was improved to the case of small scale of the order of␯^{/Uin the} directions tangential to the boundary in an appropriate boundary layer byTemam and Wang共1998兲 andWang共2001兲. The main result here is essentially a discrete version of the main result stated in Wang共2001兲and thus a discrete Kato-type result.

Notice that the main result implies convergence even if small scales predicted by the conven- tional theory of turbulence and boundary layer theory listed above are not resolved in the scheme.

For instance, we may choose

Kk=

冉

冊

冉

冊

This is what we mean by under-resolution. The convergence of numerical solutions under the under-resolved condition is puzzling since the small scale resolved here can be much bigger than

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PROOF COPY 210706JMP

any of the small scales predicted by the conventional wisdom as we discussed in the previous paragraphs. It is well-known that we usually expect oscillation共Gibbs-type phenomena兲in con- vection dominated systems if small scales are not well-resolved共see, for instance, Gottlieb and Orszag, 1977;Fletcher, 1988; Morton, 1996;Cheng and Temam, 2002;Cheng, Temam, and Wang 2000; among others兲. Oscillation usually inhibits convergence which is contradictory to our main result.

III. SKETCH OF THE PROOF

Throughout this section, ␬ will denote a generic constant independent of the kinematic vis- cosity␯or truncation wave number K_k.

Our proof is along the line ofKato共1984兲andTemam and Wang共1998兲with some modifi- cation. The basic idea is to construct a so-calledbackgroundflow关Hopf- type technique 共Hopf, 1955兲兴with a free parameter␣which interpolates between the viscous sublayer共Kato-type result兲 and laminar boundary layer共Prandtl theory兲.

For simplicity we consider channel flow共flat boundary兲and two-dimensional case only. The case with curved boundary can be treated in the same way as in our previous work共Temam and Wang, 1997;Wang, 1997兲using curvilinear coordinates. The three-dimensional case is very simi- lar to our work on energy dissipation rate共Wang, 2000兲.

Our approach is close to the idea ofVishik and Lyusternik共1957兲共see alsoLions, 1973兲in the sense that we seek a corrector which approximates the difference between the viscous and inviscid solution. Hence it is slightly different from Kato’s共Kato, 1984兲approach.

Since we are interested in the asymptotic behavior of the solutionu^kto the Galerkin truncated Navier-Stokes system共13兲, we naturally compareu^kto the spectral truncation of the solution to the Euler equation, namely,Pku⁰. Notice thatPku⁰satisfies the system

⳵

⳵^{tPku0+Pk共共Pku0·ⵜ兲Pku0兲+ⵜPkp0=Pkf+gk, 共30兲}

divPku⁰= 0, 共31兲

兩P_ku⁰·n兩z= 0, h= 0, 共32兲

兩Pku⁰兩t=0=Pku₀, 共33兲

where

gk= −Pk共共共I−Pk兲u⁰·ⵜ兲u⁰兲−Pk共共Pku⁰·ⵜ兲共I−Pk兲u⁰兲. 共34兲 It is easy to see thatgkis small for largekdue to the consistency assumption and the smoothness assumption on the inviscid solutionu⁰. Indeed

储g_k储L²ⱕ␬共^储ⵜu^0储L⬁储共I−P_k兲u^0储L²+储P_ku⁰储L⬁储ⵜ共I−P_k兲u^0储L²ⱕ␬^储ⵜu^0储L⬁储共I−P_k兲u^0储H¹. 共35兲 We now follow the strategy of the continuous case and compare u^k to Pku⁰ with the aid of a corrector共background flow兲. For this purpose we need to first establish the upper bound on the energy dissipation rate independent of the kinematic viscosity for the truncated Navier-Stokes system共13兲just as in the continuous case.

Let␾be a fixed共smooth兲incompressible flow that matchesbon the boundary of the domain.

The existence of such flows is classical共see, for instance,Temam, 2001andWang, 2001兲. Consider

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PROOF COPY 210706JMP

v^k=u^k−Pk␸^. We then deduce thatv^ksatisfies the following system:

⳵^vk

⳵^{t +Pk共共v}

k·ⵜ兲v^k兲+Pk共共v^k·ⵜ兲Pk␾兲+Pk共共Pk␾^·ⵜ兲v^k兲−␯k⌬v^k+ⵜp^k

=P_kf− ⳵

⳵^tPk␾^−Pk共共P_k␾^·ⵜ兲P_k␾兲+␯k⌬P_k␾^,

divv^k= 0, 兩v^k兩z=0,h= 0, 兩v^k兩t=0=Pk共u0−␾共0兲兲.

Multiplying both sides byv^k, integrating over⍀, we have 1

2 d dt兩v^k兩_L22

+␯k兩ⵜv^k兩_L22ⱕ兩ⵜPk␾兩L⬁兩v^k兩_L22

+␬

冉

^冏

^冏

冊

which implies

储v^k储L⬁共0,T;L²兲ⱕ␬^, which further implies

␯k

冕

兩ⵜv^k兩_L2 2 dtⱕ␬^,

where␬ is a constant independent ofk共or␯k兲. Sinceu^kandv^kdiffer by P_k␾, we also have

␯k

冕

兩ⵜu^k兩_L2

2 dtⱕ␬. 共36兲

Next we move on to the issue of convergence ofu^ktou⁰under the under-resolved condition. We first introduce a corrector 共background flow兲 just as in the continuous case. The key idea, in addition to the ones that we had for the continuous case, is a reverse Poincaré inequality which implies the smallness of energy dissipation rate due to the tangential derivative of the flow.

Define the stream function

␺^k共x,z,t兲=Pk共b1共x,0,t兲−u₁⁰共x,0,t兲兲

冕

␳

冉

冊

where the cutoff function␳ satisfies the following properties:

␳僆C^⬁关0,⬁兲,

␳共0兲= 1,

␳⬘共0兲= 0, supp␳傺关0, 1兲,

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PROOF COPY 210706JMP

冕

␳^{= 0,}

兩␳兩L⬁ⱕ1, 兩␳⬘^兩L⬁ⱕ2, and the typical velocityUis defined as in Eq.共18兲.

The corresponding velocity field is

␪^k共x,z,t兲= curl␺^k共x,z,t兲=

冉

⳵␺^k

⳵^x

冊

The typical velocity defined is a natural generalization of the continuous one 共max_关0,T兴⫻⌫兩b1

−u₁⁰兩兲to this truncated case. This new typical velocity dominates the continuous version since we have, for smooth enoughb₁−u₁⁰,

lim

k→⬁Pk共b1−u₁⁰兲=b₁−u₁⁰. Next, we consider the adjusted differences

w^k=u^k−P_ku⁰−␪^k. 共39兲 Our goal is to prove w^k→0 which implies our final result since␪^k→0 inL^⬁共0 ,T;L²兲and Pku⁰

→u⁰inL^⬁共0 ,T;L²兲as kapproaches infinity.

It is easy to verify that w^ksatisfies

⳵^wk

⳵^{t +Pk共共u}

k·ⵜ兲w^k兲−␯k⌬w^k+ⵜq^k= −⳵␪^k

⳵^{t +}␯k⌬u⁰+␯k⌬␪^k−Pk共共␪^k·ⵜ兲␪^k兲−Pk共共w^k·ⵜ兲␪^k兲

−P_k共共u⁰·ⵜ兲␪^k兲−P_k共共w^k·ⵜ兲P_ku⁰兲−P_k共共␪^k·ⵜ兲P_ku⁰兲+g_k, 共40兲

divw^k= 0, 共41兲

兩w^k兩z=0,h= 0, 共42兲

兩w^k兩t=0= 0. 共43兲

Thanks to the explicit construction of our␪^k, we have

冏

冏

ⱕU_t²L␯k

␣^U+Utx2 L␯k3

␣^3U3,

兩ⵜ␪^k兩L²

2 ⱕ2U_x²L␯k

␣^U+U2 L␣^U

␯k

+U_xx² L␯k 3

␣^3U3,

兩Pk共␪^k·ⵜ兲␪^k兩L²

2 ⱕ5U²U_x²L␯k

␣^U+U2Uxx2 L␯k3

␣^3U3+Ux4 L␯k3

␣^3U3,

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PROOF COPY 210706JMP

兩Pk共Pku⁰·ⵜ兲␪^k兩_L22ⱕ2

冉

^冏

^冏

^冏

^冏

^冏

^冏

冊

^冉

^冊

兩Pk共w^k·ⵜ兲Pku⁰兩L²ⱕ兩ⵜPku⁰兩L⬁兩w^k兩L²,

兩Pk共␪^k·ⵜ兲Pku⁰艋兩L²

2 兩ⵜPku⁰兩_L2⬁

冉

␣^3U3

冊

冉

␣^3U3

冊

where we have used the impermeable wall boundary condition共32兲, and

Ut= sup

k

关0,Tmax兴⫻⌫

再 ^冏

^冉

^{冊 冏} 冎

Ux= sup

k max

关0,T兴⫻⌫

再 ^冏

^冉

^{冊 冏} 冎

Utx= sup

k max

关0,T兴⫻⌫

再 ^冏

^冉

^{冊 冏} 冎

Uxx= sup

k max

关0,T兴⫻⌫

再 ^冏

^冉

^{冊 冏} 冎

We then deduce, via the standard energy method, 1

2 d dt兩w^k兩L²

2 +␯k兩ⵜw^k兩L²

2 ⱕ␯k

冑

␯k

+U_xx² L␯k 3

␣^{3U3兩ⵜwk兩L2+}␯k2兩⌬u⁰兩L²

2 +␬兩w^k兩L² 2 +␬兩u⁰

−Pku⁰兩_H1

2 +␬

冉

␯k3

␣³

冊

冕

Notice the last共nonlinear兲term can be rewritten as

冕

冕

w₁^k⳵^w1 k

⳵^x ␪1

k+

冕

k+

冕

k+

冕

k, 共45兲 and hence we have the following estimates on the nonlinear term, thanks to the explicit construc- tion of the corrector共seeWang, 2001兲:

2

冕

k=

冕

k兲²␪1

k= −

冕

k兩L² 2 ,

2

冕

kⱕ2U兩w₃^k兩L²共⌫_␦兲

冏

冏

冏

冏

冏

冏

= 2␯

␣

冏

冏

冏

冏

4

冏

冏

+4␯

␣²

冏

冏

,

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