J.-L. Guermond 1 FOR TURBULENT FLOWS LARGE EDDY SIMULATION MODEL A HYPERVISCOSITY SPECTRAL

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1 A HYPERVISCOSITY SPECTRAL LARGE EDDY SIMULATION MODEL FOR TURBULENT FLOWS J.-L. Guermond 1 Collaborator: S. Prudhomme 2 Mathematical Aspects of Computational Fluid Dynamics Oberwolfach November 9th to 15th, LIMSI-CNRS, Orsay, France 2 Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin

2 Part Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 THE NAVIER STOKES EQUATIONS Overview Part I: Basic facts about the Navier Stokes equations Part II: Proposition of a paradigm for LES Part III: The hyperviscosity model Part IV: The spectral hyperviscosity approximation 1

3 Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 BASIC FACTS ABOUT THE NAVIER STOKES EQUATIONS 2

4 Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 THE NAVIER STOKES EQUATIONS u: velocity, p: pressure Ω is the fluid domain tu + u u + p ν 2 u = f in Ω u = 0 in Ω, u Γ = 0 or u is periodic, u t=0 = u0, u0 is the initial data f a source term. ρ is chosen equal to unity. 3

5 Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 EXISTENCE AND UNIQUENESS J. Leray (1934): introduces the notion of turbulent solution. A turbulent solution is a weak solution in L 2 (0, T ; H 1 (Ω)) L (0, T ; L 2 (Ω)). J. Leray uses mollification to prove existence: ψ D(R 3 ), ψ 0, R 3 ψ = 1, ψɛ(x) = 1 ɛ ψ ( x ɛ). tuɛ + (ψɛ uɛ) uɛ + pɛ ν 2 uɛ = f E. Hopf (1951) et al. uses the Galerkin technique to prove existence. The question of the existence of classical solutions for long times is open. This question is linked to the question of uniqueness of turbulent solutions. Clay Institute 1M$ prize. 4

6 Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 SUITABLE WEAK SOLUTION V. Sheffer (1976) introduces suitable weak solutions: DEFINITION: A NS weak solution is said to be suitable (or dissipative) iff t( 1 2 u2 ) + (u( 1 2 u2 + p)) ν 2 ( 1 2 u2 ) + ν( u) 2 f u 0. THEOREM: (Caffarelli-Kohn-Nirenberg (1982)) The dimension of the set of singular points of suitable weak solutions in Ω [0, + ] is less than 1. Best partial regularity theorem to date. 5

7 6 Caffarelli-Kohn-Nirenberg P 1 (S) = 0 S = {(x, t) Ω ]0, T [, u L (V ), V s.t. (x, t) V } Singular set: S P k (X) = lim δ 0 + inf{ r k i, X Q(M i, ri), ri < δ} Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003

8 Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 THEOREM: (Duchon-Robert (2000)) The limits (as ε 0 + ) of Leray-regularized NS solutions are dissipative. The question is open for Galerkin weak NS solutions. Proof: uɛ u in L 2 (H 1 ) (up to subsequences) and uɛ u in L p (L q ), 1 p 4q 3(q 2), 2 q < 6. Test with φuɛ, φ D(Q) (OK, since uɛ is smooth!). Q 1 2 u2 ɛ ( tφ + ν 2 φ) (ψɛ uɛu 2 ɛ + p ɛuɛ) φ + ν( uɛ) 2 φ fuɛφ = 0 u 2 ɛ u2 in L p (L 1 ), p 1 } ψɛ uɛ u in L 4 (L 3 ) u 2 ɛ u2 in L 4 2(L 3 ψɛ uɛu 2 ɛ 2) uu2 in L 4 3(L 1 ) 7

9 Part I Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 Proof (continued): Periodic domain and f = 0 2 pɛ = (ψɛ uɛ uɛ) pɛ L 2 c ψɛ uɛ uɛ L 2 c uɛ L 4 pɛ p in L 4 } 3(L 2 ). pɛ p in L 4 3(L 2 ) u u in L 4 (L 2 ) pɛuɛ pu in L 1 (L 1 ) Q ( u ɛ) 2 φ 2 Q φ (u ɛ u) u + Q φ( u)2 lim inf ɛ 0 Q ( u ɛ) 2 φ Q ( u)2 φ 8

10 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 LARGE EDDY SIMULATIONS 9

11 10 Almost no reasonable mathematical theory for LES. The objective is to modify the NS model so that the new model is amenable to numerical simulations. Concept introduced by Leonard (1974) OBJECTIVES OF LES Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003

12 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 THE FILTERING-EVERYTHING PARADIGM Let ( ) : w w be a regularizing operator acting on spaceand time-dependent functions (we call it a filter). Assume that the filter is linear and commutes with differential operators. Apply the filter operator to the Navier Stokes equations: tu + u u + p ν 2 u = f T, u = 0 u Γ = 0, or u is periodic, u t=0 = u0, where have introduced the so-called subgrid-scale tensor: T = u u u u 11

13 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 THE FILTERING PARADOX THE CLOSURE PROBLEM: The Goal of LES (Leonard (1974)) is to model T in terms of u only, without resorting to u. THEOREM: (Germano (1986)) Exact closure is possible, e.g., take v := (I ε 2 2 ) 1 v PARADOX: (Guermond-Oden-Prudhomme (2001)) If the filter induces an isomorphism, the solution set of NS is isomorphic to the solution set of the filtered equations. The two sets of weak solutions being isomorphic, one should expect to use the same number of degrees of freedom for approximating the NS solution as for approximating the filtered solution. 12

14 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 THE FILTERING PARADOX CONCLUSION: Filtering and achieving exact closure may not reduce the number of degrees of freedom. Filtering the Navier Stokes equations is an efficient approach only if inexact closure is performed. The filtering-everything paradigm is probably nonsensical. 13

15 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 PROPOSITION FOR A NEW PARADIGM FOR LES DEFINITION: A sequence (uɛ, pɛ)ɛ>0 is said to be a LES solution of NS if For all ɛ > 0, (uɛ, pɛ) is uniquely defined for all times (i.e. regularization technique s.t there is existence and uniqueness for uɛ and pɛ for all times). uɛ u, pɛ p (up to subsequences) and (u, p) is a suitable weak solution of NS. EXAMPLE 0: Galerkin approximations are regularizing techniques (so-called DNS). Does a Galerkin solution converge to a suitable weak solution? (500k$ question?). 14

16 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 EXAMPLES OF LES EXAMPLE 1: Leray s regularization is a LES technique (1934)! However the model is not frame invariant (is it important?). EXAMPLE 2: NS-α model (Holmes Marsden Ratius (1998), Holmes et al. (1999)). Theoretical mechanics tuɛ + (uɛ) uɛ ( uɛ) T uɛ + pɛ ν 2 uɛ = f Alternative interpretation tuɛ + ( uɛ) uɛ + πɛ ν 2 uɛ = f. It is a Leray regularization where the nonlinear term is ( u) u + ( u2 2 ). The equation is frame-indifferent. uɛ u (up to subsequences) and u is suitable 15

17 Part II Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 EXAMPLES OF LES EXAMPLE 3: Ladyzenskaja (1967) proposed tuɛ + uɛ uɛ + pɛ (ν uɛ + εt(d)) = f, uɛ = 0 uɛ Γ = 0, or u is periodic, uɛ t=0 = u0. where operator T is nonlinear, and D = 1 2 ( u ɛ + ( uɛ) T ). For instance T(ξ) = β( ξ 2 )ξ with cτ µ β(τ ) c τ µ, with µ 1 4 is possible. β(τ ) = τ 1/2 yields Smagorinsky s model. THEOREM: (Ladyzenskaja (1967)) The modified NS equations have a unique weak solution for all t > 0. Moreover uɛ u (up to subsequences) and u is suitable. 16

18 Part III Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 EXAMPLE 4: HYPERVISCOSITY 17

19 Part III Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 HYPERVISCOSITY EXAMPLE 4: Lions (1959) proposed to use hyperviscosity. Ω is the d-torus, where d is the space dimension. tuɛ + uɛ uɛ + pɛ ν 2 uɛ + ε( 2 ) α uɛ = f, uɛ = 0 uɛ is periodic, uɛ t=0 = u0. THEOREM: ( Lions (1959)) The modified NS equations have a unique weak solution for all t > 0 if α > d+2 4. Moreover uɛ u (up to subsequences) and u is suitable. QUESTION: How this technique can be implemented numerically? 18

20 Part III Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 HYPERVISCOSITY: A PRIORI ESTIMATES Testing the momentum equation by uɛ yields 1 d 2 dt u ɛ 2 L 2 + ν uɛ 2 L 2 + ɛ ( 2 ) α 2 uɛ 2 L 2 = (f, uɛ). Integration in time, gives the following estimate: uɛ 2 L 2 + T 0 u ɛ 2 H α dt u0 2 L 2 + f 2 L 2 (L 2 ). In a similar manner, testing by uɛ,t yields uɛ,t 2 L 2 (L 2 ) + u ɛ 2 H α u0 2 H α + T 0 dt Ω u ɛ 2 uɛ 2 dx+ f 2 L 2 (L 2 ). Using Hölder s inequality, we write Ω u ɛ 2 uɛ 2 dx uɛ 2 uɛ 2, L 2p L 2p 1 p + 1 = 1. p 19

21 Part III Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 HYPERVISCOSITY: A PRIORI ESTIMATES Owing to Sobolev inequalities we also have uɛ L 2p uɛ L 2p if 1 2p 1 1 2p d, uɛ L 2p u ɛ H α 1 if 1 2p 2 α 1 d. These two conditions yield and α d+2 4 and p 2d d+2, uɛ,t 2 L 2 (L 2 ) + u ɛ 2 H α u0 2 H α + T 0 u ɛ 2 H α uɛ 2 H α dt+ f 2 L 2 (L 2 ) and since T 0 u ɛ 2 H α dt is bounded, Gronwall s lemma yields uɛ,t L 2 (L 2 ) + uɛ L (H α ) c(ν, u0, f, ɛ). Existence of solutions in L (0, T ; H α ), T > 0, is proved by means of the Galerkin technique using the a priori estimates. 20

22 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 SPECTRAL HYPERVISCOSITY 21

23 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 FOURIER APPROXIMATION DEFINITION: H s (Ω) = {u = k Z 3 uke ik x, uk = u k, k Z 3 (1+ k 2 ) s uk 2 < + }. PN = (u, v) = (2π) 3 Ω uv = k Z 3 ukvk. { p(x) = k N c ke ik x, ck = c k }, Velocity space XN = Ṗ 3 N, Pressure space MN = ṖN. 22

24 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 FOURIER APPROXIMATION H s (Ω) k Z 3 vke ik x = v PNv = k N vke ik x PN LEMMA: PN satisfies the following properties: 1) PN is the restriction on H s (Ω) of the L 2 projection onto PN. 2) s 0, PN L(H s (Ω);H s (Ω)) 1. 3) PN commutes with differentiation operators. 4) v H s (Ω), µ, 0 µ s, v PNv H µ N µ s v H s. 5) v PN, µ, s, s µ, PNv H µ N µ s v H s. 23

25 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 A NAIVE HYPERVISCOSITY MODEL DEFINITION: Q(x) = 1 (2π) 3 1 k N k 2α e ik x. Q un(x) = Ω un(y)q(x y)dy = If α is an integer Q un(x) = ( 2 ) α un 1 k N k 2α uke ik x Let ɛ > 0 Find un C 1 ([0, T ]; XN) and pn C 0 ([0, T ]; MN) such that ( tun, v) + (un un, v) (pn, v) + ν( un, v) + ɛ (Q un, v) = (f, v), v XN, t (0, T ], ( un, q) = 0, q MN, t (0, T ], un t=0 = PNu0. 24

26 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 A NAIVE HYPERVISCOSITY MODEL Consistency error in L 2 : ɛ Q un L 2 ɛ un H 2α = O(ɛ 1 2). But ɛ cannot be too small to play the regularizing effect we expect. Consistency error cannot be arbitrarily small. But if u H s, where s may be arbitrarily large if u is smooth, u PNu L 2 cn s u H s = O(N s ). The interpolation error can be arbitrarily small. The hyperviscosity regularization spoils the consistency. 25

27 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 A SPECTRAL HYPERVISCOSITY MODEL KEY: Actually it is not necessary to stabilize the low wave numbers since they should be controlled by means of the L 2 a priori estimate. DEFINITION: Let α > 5 4 ɛn = N β, with 0 < β < and β > 0 (two parameters). { } 4α 5 If α 3 2 2, 4α(α 1) Otherwise. 2α+3 < 2α Ni = N θ, with θ = β 2α, Q(x) = (2π) 3 k 2α e ik x Ni k N 26

28 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 A SPECTRAL HYPERVISCOSITY MODEL One can also define the hyperviscosity kernel as follows: Q(x) = 1 ˆQ k k 2α e ik x, (2π) 3 Ni k N where the viscosity coefficients ˆQ k are such that 1 ˆQ k N i 2α k 2α, k Ni. This definition has the practical advantage of ensuring a smooth transition of the viscosity coefficients across the threshold Ni. All the results stated hereafter hold also with this definition. 27

29 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 A SPECTRAL HYPERVISCOSITY MODEL We d like β large so that ɛn = N β is small. But the condition un u where u is dissipative enforces β < { } 4α 5 If α 3 2 2, 4α(α 1) Otherwise. 2α+3 < 2α α β < 1 2 < 8 7 < 8 3 < < θ < 1 6 < 2 7 < 4 9 < 6 11 < 8 13 Admissible values of the parameters α, β, and θ. 28

30 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 A SPECTRAL HYPERVISCOSITY MODEL Note that ɛn = N 2α i Find un C 1 ([0, T ]; XN) and pn C 0 ([0, T ]; MN) such that ( tun, v) + (un un, v) (pn, v) + ν( un, v) + ɛn(q un, v) = (f, v), v XN, t (0, T ], ( un, q) = 0, q MN, t (0, T ], un t=0 = PNu0. Note that un = 0 since un MN, i.e. the approximate velocity field is solenoidal. 29

31 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 PROPOSITION: The hyperviscosity perturbation is spectrally small in the sense that ɛn Q vn L 2 c N θs vn H s, vn H s (Ω), s 2α. PROOF: Note that Q un 2 L 2 = k 4α uk 2. Ni k N Using the fact that 2α s and that Ni k k, we have: k 4α = N i 4α ( k Ni ) 4α N i 4α ( k Ni ) 2s = N 4α 2s i k 2s Moreover, from the definition of Ni, ɛn, and θ, we have N 4α 2s i = ɛ 2 N N 2θs so that ɛ 2 N Q u N 2 L 2 ɛ 2 N ɛ 2 N N 2θs Ni k N k 2s uk 2 N 2θs un 2 H s 30

32 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 PROOF: Observe that ɛn un 2 H α = ɛn un 2 = ɛn(q un, un)+ Ḣ α ɛn k 2α u k 2 1 k <N i un L (L 2 ) + ν 1/2 un L 2 (H 1 ) + ɛ 1/2 N u N L 2 (H α ) c. LEMMA: We have the a priori estimates To estimate the last term in the above inequality. Use the fact that ɛn = N 2α i and that k 3 k. ɛn k 2α u k 2 3 α N 2α i N 2α i 1 k <N i Since un is solenoidal and using the above bound, we obtain un 2 L 2 + ν un 2 L 2 (H 1 ) + ɛ N un 2 L 2 (H α ) 1 k <N i uk 2 3 α un 2 L 2. u0 2 L f 2 L 2 (L 2 ) + (3α ) u N 2 L 2 (L 2 ). The result is a consequence of Gronwall s lemma. 31

33 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 LEMMA: We have un L p (H 2/p ) + un L p (L q ) c, with 1 p 4q 3(q 2), 2 q 6, pn L 4/3 (L 2 ) c. PROOF: First, we observe that 2 : MN MN is bijective. Then, we multiply the momentum equation by ( 2 ) 1 pn (note that ( 2 ) 1 pn XN is an admissible test function). By using several integrations by parts, we obtain pn 2 L 2 = ( pn, ( 2 ) 1 pn) = ( tun + ν 2 un un un + f, ( 2 ) 1 pn) = (un un, ( 2 ) 1 pn), since un and f are solenoidal = ( (un un), ( 2 ) 1 pn) = (un un, ( 2 ) 1 pn) un 2 L 4 pn L 2, i.e. pn L 2 un 2 L 4. Use the bound on un, for q = 4, p =

34 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 COROLLARY: Let s be a real number such that s 2α 1 > 3, we have 2 tun L 4/3 (H s ) c. THEOREM: un u and u is a weak suitable solution to NS. PROOF: Test the momentum equation by PN(uNφ): ( tun, PN(uNφ))+ (un un, PN(uNφ)) (pn, PN(uNφ)) +ν( un, PN(uNφ))+ɛN(Q un, PN(uNφ)) = (f, PN(uNφ)) Define R = (un un, PN(uNφ) unφ) and because un is solenoidal, we have (un un, PN(uNφ)) = (un un, unφ)+r = ( 1 2 u N 2 un, φ) + R. 33

35 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 For the remainder R we have R = (un un, PN(uNφ) unφ), un 2 L 4 (PN(uNφ) unφ) L 2, N 1 α un 2 L 4 unφ H α, Approximation property, N 1 α un 2 L 4 un H α φ H s. Leibniz-like rule, s > α To bound un L 4 we proceed as follows. un 2 L 4 un 2 H r, Sobolev emb., 1 4 = 1 2 r 3, r = 3 4, un 2(1 γ) un 2γ L 2 H α, Interp. inq., γα = r, γ = 3 4α, 2α un 3 H α. At this point, there are two possibilities: either 3 2α 1 or 3 2α > 1. 34

36 Part IV Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 If α < 3 2, then un 2 L 4 un H α un 3 2α 1 H α, N 3 2 α un H α un 3 2α 1 L 2, Inverse inequality. Then, owing to ɛn = N β R N 5 2 2α un 2 H α φ H s = N 5 2 2α+β ɛn un 2 H α φ H s. That is to say, owing to the hypothesis β 4α 5 we have 2 T 0 R 0 as N +. 35

37 Part V Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 CONCLUSIONS 36

38 Part V Mathematical Aspects of Computational Fluid Dynamics, Oberwolfach November 9-15, 2003 CONCLUSIONS The new LES paradigm is constructive: i.e. enforcing the regularized solution to converge to a suitable weak solution implies strong constraints on the numerical methods. Extension to non periodic domains and finite elements is likely to be highly technical. There is no easy a priori estimate on the pressure. There is no easy a priori estimate on tuh (Fourier analysis gives uh H 3 8 ɛ (L 2 ), this is not enough!) The nonlinear term does not pose difficulties! The trouble maker is the product phuh! 37