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1 SIAM J. MATH. ANAL. Vol. 37, No. 6, pp c 26 Society for Industrial and Applied Mathematics Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see ON THE STOLZ ADAMS DECONVOLUTION MODEL FOR THE LARGE-EDDY SIMULATION OF TURBULENT FLOWS A. DUNCA AND Y. EPSHTEYN Abstract. We consider a family of large-eddy simulation (LES) models with an arbitrarily high consistency error O(δ 2N+2 ) for N =, 2, 3,... that are based on the van Cittert deconvolution procedure. This family of models has been proposed and tested for LES with success by Adams and Stolz in a series of papers, e.g., [Deconvolution methods for subgrid-scale approximation in largeeddy simulation, in Modern Simulation Strategies for Turbulent Flow, R. T. Edwards, Philadelphia, 2, pp. 2 4], [Phys. Fluids, (999), pp ]. We show that these models have an interesting and quite strong stability property. Using this property we prove an energy equality, existence, uniqueness, and regularity of strong solutions and give a rigorous bound on the modeling error u w, where w is the model s solution and u is the true flow averages. Key words. large-eddy simulation, scale similarity models, deconvolution, approximate deconvolution models AMS subject classifications. 76F65, 76D3 DOI..37/S Introduction. We consider the problem of modeling the motion of large structures in a turbulent fluid. This involves the interaction of many complex decisions made in the simulation. To isolate some effects, we study herein the correctness of the approximate deconvolution modeling (ADM) approach to closure pioneered by Adams and Stolz; see, e.g., [], [9]. The pointwise velocity and pressure, u,p, in an incompressible viscous flow satisfy the Navier Stokes equations () (2) u t + (uu T ) Δu + p = f, u =, u(x, ) = u (x). We study () subject to periodic boundary conditions (with zero mean) u(x + Le, t) =u(x, t) for x R 3, <t T. Periodic boundary conditions separate the hard problem of closure for the interior equations from another hard problem of wall laws and near wall models in turbulence. Let an overbar denote a local spacial averaging associated with a length scale δ which commutes with differentiation. Averaging the Navier Stokes equations gives the nonclosed equations for u, p, (3) u t + (uu T ) Δu + p = f, u =. Received by the editors October 2, 23; accepted for publication (in revised form) June, 24; published electronically March 3, 26. This research was partially supported by NSF grant DMS Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 526 (ardst2@math. pitt.edu, yee@pitt.edu). 89

2 LARGE-EDDY SIMULATION OF TURBULENT FLOWS 89 Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see Let the averaging operation u u be denoted formally by G so u =Gu. In the most interesting cases G is not invertible. Nevertheless, the closure problem (of replacing uu T by a tensor depending only on u) is solved once the approximate deconvolution problem (of approximating the action of G ) is solved. The van Cittert approximation to G can be developed in various ways (see [2] and section 2 for a precise definition of it). The simplest is to find an approximation to u by extrapolating from the resolved scales of u to those of u. The first three examples are (4) u G u := u u G u := u u u G 2 u := 3u 3u + u (constant extrapolation in δ), (linear extrapolation in δ), (quadratic extrapolation in δ). Let G N u denote the analogous Nth degree accurate approximate inverse (section 2). Calling (w,q) the approximations that result when this is used in (3) to treat the closure problem, we are inevitably led to the fundamentally important question of how well the solution w of the resulting model, (5) w t + (G N w(g N w) T ) Δw + q = f, w =, matches the behavior of the true flow averages u. This question has obvious theoretical and experimental components. We consider herein the theoretical parts of the question for the whole family of models. Our analysis is based on a delicate skew symmetry property that the model s nonlinear interaction terms have when the averaging operator is the differential filter ϕ ϕ (as studied by Germano [4]). Here for given ϕ L 2 (Q), ϕ is defined to be the unique periodic solution of (6) δ 2 Δϕ + ϕ = ϕ in Q, where Q denotes the d-dimensional cube of size L>,Q=(,L) d. Our analysis is for periodic boundary conditions. We believe that many of the results presented in this work can be extended to nonperiodic boundary conditions with further research. Indeed, the basic model (5) does not increase the order of the differential operator, so the model makes perfect sense coupled with any of the well-posed boundary conditions used for the Navier Stokes equations. Remark.. The model (5) using G was considered recently in [6] and [7]. On the other hand, practical calculations of Adams and Stolz in [] and [9] have stressed the superiority of models of order 4, 5 and higher in practical tests. Herein we show that a single, unified mathematical theory is possible for the entire family of models building on the analysis in [6] and [7]. 2. Deconvolution models. It has been pointed out by Germano (presented well in [5]) that with the differential filter ϕ := ( δ 2 Δ+I) ϕ it seems that no deconvolution is necessary; one can write exactly ϕ := ( δ 2 Δ+I)ϕ. This leads to the exact model for u given by (7) u t + (( δ 2 Δ+I)u[( δ 2 Δ+I)u] T ) Δu + p = f subject to the periodic boundary conditions. One criticism with using the exact deconvolution model (7) to predict u is that going from the Navier Stokes equations to (7), no information is lost.

3 892 A. DUNCA AND Y. EPSHTEYN Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see Thus there is no reason to believe that (7) can be approximated with fewer degrees of freedom than the Navier Stokes equation itself. Another difficulty with (7) is that any model that increases the order of the differential equation must be supplied with extra boundary conditions. Thus for nonperiodic problems, models such as (7) shift the essential difficulty from interior closure to the harder problem of specifying as boundary conditions the higher derivatives of turbulent velocities at walls. Thus approximate deconvolution which will lose information is necessary. The van Cittert method of approximate deconvolution (see [2]) constructs a family G N of inverses to G as follows: writing G = I (I G), a formal inverse to G can be written as the nonconvergent power series, Truncating this series gives (8) G = G N = (I G) n. n= N (I G) n. n= The first three approximations are given in (4). Lemma 2.. The operator G N : L 2 (Q) L 2 (Q) is compact, self-adjoint, and positive. Proof. The operator G : L 2 (Q) L 2 (Q) is compact and self-adjoint. Multiplying (6) by ϕ and integrating over Q gives δ 2 ϕ 2 + ϕ 2 =(ϕ, ϕ) 2 ϕ ϕ 2. It follows that G is positive and G. Let h N (x) = N k= ( x)k. By the definition of G N, G N = h N (G) and, consequently, G N is also a compact self-adjoint operator. Because h N is positive on [, ], which contains the spectrum of G, it also follows that G N is positive. Remark 2.. The operators {G N } N satisfy the following recursion: (9) (I δ 2 Δ)G N u = δ 2 ΔG N u +(I δ 2 Δ)u. The following lemma, which is a consequence of the identity G N G = I (I G) N+, was proved in []. Lemma 2.2. For smooth u the approximate deconvolution (8) has the consistency error O(δ 2N+2 ), () locally in Q and also u G N u =( ) N+ δ 2N+2 Δ N+ G N+ u, u G N u δ 2N+2 u H 2N+2 (Q).

4 LARGE-EDDY SIMULATION OF TURBULENT FLOWS 893 Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see Lemma 2.2 shows that G N u gives an approximation to u to the accuracy O(δ 2N+2 ) in the smooth flow regions. Thus it is justified to use it for the closure approximation (uu T ) (G N u(g N u) T )+O(δ 2N+2 ). If μ denotes the usual subfilter scale stress tensor μ(u, u) :=uu T ūū T, then the closure approximation is equivalent to the closure model () μ(u, u) μ N (u, u) :=G N u(g N u) T ūū T. The true subgrid stress tensor μ(u, u) is both reversible and Galilean invariant (Sagaut [8]). Thus many feel that appropriate closure models should at least, to leading order effects, share these two properties. We next show that the model () is both reversible and Galilean invariant. Lemma 2.3. For each N =,, 2,... the closure model () is reversible and Galilean invariant. Proof. Reversibility is immediate. Galilean invariance also follows easily once it is noted that Uw T =Uw T so G N (Uu T =UG N (u) T. Using these and other analogous properties gives μ(u +U, u +U)= [G N (u)g N (u) T +UG N (u) T + G N (u)u T + UU T (u + U)(u +U) T ] = [G N (u)g N (u) T ūū T ]+ G N (u)u +U (G N (u) (u)u U (u) = [G N (u)g N (u) T ūū T ]= μ N (u, u), since u = G N (u) = G N (u) =anduu T =UU T. Let 3. Variational spaces. Q denotes a d-dimensional cube of size L >, Q =(,L) d. H m (Q) ={u H m loc(r n ) u periodic with period Q} and { } H m (Q) = u H m (Q) u dx =. Q For the variational formulation of the scale similarity model with periodic boundary conditions, we consider the spaces of divergence-free functions and V = {u H (Q), u =inr d } H = {u L 2 (Q), u =inr d } as in Temam [2].

5 894 A. DUNCA AND Y. EPSHTEYN D(Q) is defined as Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see and (2) D(Q) ={ψ C (R d ) ψ is periodic with period Q} D(Q T )={ψ C ([,T) R d ) for t [,T),ψ(,t) is periodic with period Q and ψ has compact support in variable t [,T)}. The space of vector valued functions D(Q) is defined as D(Q) =D(Q) d. The other spaces D(Q T ), H, H p (Q), and V, L 2 (Q) are defined accordingly. Remark 3.. Because the inclusion H 2 (Q) H is compact, the inverse of the Laplacian operator ( Δ) : H H 2 (Q) H is a bounded, self-adjoint, and compact operator. This implies that there exists an orthonormal basis (w j ) j N of H consisting of eigenfunctions of the Laplacian operator. 4. The models and the existence of weak solutions. Definition 4.. The strong form of the Stolz Adams model that we analyze is as follows: Find (w,q) such that (3) and (4) w (H 2 (Q) H) d for a.e. t [.T ], w (H (,T)) d for a.e. x Q, q H (Q) L 2 (Q) if t (,T] w t Δw + ((G N w)(g N w) T )+ q = f in (,T) Q, w = in (,T] Q, w t= = u in Q, Q q dx = in (,T]. Definition 4.2. Let f L 2 (,T; V ) and w H 2 (Q). A measurable function w :[,T] Q R d is a weak solution of (4) if (5) and (6) w L 2 (,T,H (Q)) L (,T; H) [( w, ϕ ) ] ( w, φ) ( ((G N w)(g N w) t T ),ϕ) dt = (f,ϕ)dt (w,ϕ()) for all ϕ D(Q). The following lemma gives an energy inequality satisfied by the strong solutions of the Stolz Adams models. We mention here that the same argument is used to derive an energy inequality for the approximate solutions in the proof of existence of weak solutions to the Stolz Adams models.

6 LARGE-EDDY SIMULATION OF TURBULENT FLOWS 895 Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see Lemma 4.. If w is a strong solution of (4) as in Definition 4., then w satisfies the following energy inequality: (7) 2 ( w(t) 2 + δ 2 w(t) 2 )+ w(s) 2 + δ 2 Δw(s) 2 ds 2 ( T ) K f(s) 2 V ds + w 2 + δ 2 w 2 for all t [,T] with K = max{ 2 G N 2 L 2 (Q), 2 δ2, 2, δ2 2 G N L 2 (Q)}. Proof. We multiply (4) by the test function ϕ =( δ 2 Δ + I)G N w and integrate on Q. Because the weak form of the nonlinear term will vanish, (8) (9) ( ((G N w)(g N w) T ), ( δ 2 Δ + I)G N w) =( ((G N w)(g N w) T ), ( δ 2 Δ + I)G N w)=( ((G N w)(g N w) T ), G N w)=, we obtain the following energy equality: d (2) 2 dt (w, ( δ2 Δ+I)G N w)+(δw, ( δ 2 Δ+I)G N w)=(f, ( δ 2 Δ+I)G N w). In the above equality all terms ( δ 2 Δ+I)G N w are replaced using Remark 2., leading to d 2 dt (w, ( δ2 Δ+I)w)+ d 2 dt (w, δ2 ΔG N w) (Δw, ( δ 2 Δ+I)w) + δ 2 (Δw,δ 2 ΔG N w)=(f, G N w). Using integration by parts and the commutation property of the operator G N with differentiation gives (2) d 2 dt w 2 + d 2 δ2 dt w 2 + δ2 d 2 dt ( w, G N w) + w 2 + δ 2 Δw + δ 4 (Δw, G N Δw) =(f, G N w). We then integrate on [,t] and obtain 2 w(t) δ2 w(t) 2 + δ2 2 ( w(t), G N w(t)) + + δ 2 Δw(s) 2 ds + δ 4 (Δw(s), G N Δw(s)) ds = w(s) 2 ds (f(s), G N w(s)) ds + 2 w δ2 w 2 + δ2 2 ( w, G N w ). We use the positivity of the operators (G N ) N in the above inequality to get (22) 2 w(t) δ2 w(t) 2 + w(s) 2 ds + δ 2 Δw(s) 2 ds (f(s), G N w(s)) ds + 2 δ2 w w 2 + δ2 2 ( w, G N w ).

7 896 A. DUNCA AND Y. EPSHTEYN Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see An application of Cauchy s inequality on the first term on the right-hand side above gives (f(s), G N w(s)) ds 2 G N 2 L 2 (Q) f(s) 2 V ds + 2 We use this inequality in (22) to obtain 2 w(t) δ2 w(t) 2 + δ 2 f(s) V G N L 2 (Q) w(s) L 2 (Q) ds T w(s) 2 L 2 (Q) ds. Δw(s) 2 ds + w(s) 2 ds 2 G N 2 L 2 (Q) f(s) 2 V 2 ds + 2 δ2 w w 2 + δ2 2 G N L 2 w 2 L. 2 Remark 4.. The turbulence model based on the approximate deconvolution procedure introduced by Stolz and Adams in [9] and tested by Stolz, Adams, and Kleizer in [] and [] contains a relaxation term added to the right-hand side of the first equation in (4) to drain energy near cutoff length scale. This term takes the form (23) χ ω (I G N G)w, where χ ω > is a function of space and time. If χ ω is smooth and bounded in space and time, the addition of the relaxation term (23) does not change the mathematical results proved in this paper. For this model one can derive an energy estimate like Lemma 4. by treating the weak form of the relaxation term in the following way: ( χ ω (I G N G)w, ( δ 2 Δ+I)G N w) =( χ ω (I G N G)w,G N w)+(χ ω (I G N G)w,δ 2 G N Δw) C(ε, χ ω,n,δ,, G ) w 2 + ε Δw 2 for given ε> One can then pick ε = δ2 δ2 4, which results in the cancellation of the term 4 Δw on the right-hand side (see (2)) and apply the Gronwall lemma to get an energy inequality similar to (7). Based on this energy inequality all other results proved here for the Stolz Adams model without the relaxation term can be extended to the case where the relaxation term is incorporated into the equations. For less regular functions χ ω > the same results cannot be proved with the same arguments; this case requires further investigation. Proposition 4.. Let T>. Then for w H 2 (Q) H and f L 2 (,T; V ), there exists a weak solution w of (4) in the sense of Definition 4.2. This solution w belongs to the space L 2 (,T,H 2 (Q)) L (,T; V); it is L 2 -weakly continuous and satisfies the following energy inequality: (24) 2 ( w(t) 2 + δ 2 w(t) 2 )+δ 2 Δw(s) 2 ds ( ) K f(s) 2 V ds + w 2 + w 2

8 LARGE-EDDY SIMULATION OF TURBULENT FLOWS 897 Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see for all t [,T] with K = max{ 2 G N 2 L 2 (Q), 2 δ2, 2, δ2 2 G N L 2 (Q)}. Proof. The proof uses the Faedo Galerkin method. We will use Galdi [3] as a reference and only point out the differences between the proof of existence of the weak solution of the Navier Stokes equations and the proof of existence for our models. We pick an orthonormal basis {ψ j } j D(Q) ofh consisting of eigenfunctions of the Laplacian operator as in Remark (3.). Let (25) w k (x, t) = k η kr (t)ψ r (x) for k N be the solution of the following ODE system: ( ) wk (26) t,ψ r + ( w k, ψ r )+( (G N w k )(G N w k ) T,ψ r )=(f,ψ r ) r= for all r =,...,k with the initial condition (w k (),ψ r )=(w,ψ r ) for all r =,...,k. It follows that the coefficients η kr satisfy the following ODE system: (27) dη kr dt + k a ir η ki + i= for all r =,...,k with the initial condition η kr () = C r k a ijr η ki η kj = f r i,j= for all r =,...,k, where a ir = ( ψ i, ψ r ),a ijr =( ((G N ψ i )(G N ψ j ) T ),ψ r ),f r =(f,ψ r ), and C r = (bw,ψ r ). The function f r belongs to L 2 [,T) for any r, and consequently (27) has a unique solution near, η kr W,2 (,T k ), where T k T. Because w H 2 (Q) H there exists u H such that u = w.for the ODE defined above we have (w k,ψ r )=(w,ψ r ) for all r =,...,k. This gives (28) (w k,ψ r )=(u,ψ r ) for all r =,...,k. But w k, G k = span{ψ j } j=,...,k and G k is an invariant subspace of the Laplacian operator. Consequently, we can replace ψ r in formula (28) with (I δ 2 Δ)w k, to get (29) (w k, (I δ 2 Δ)w k, )=(u, (I δ 2 Δ)w k, )=(u, w k, ). Integrating by parts the first term above and using Cauchy s inequality in the second, we get (3) w k 2 + δ 2 w k 2 =(u, w k, ) 2 ( u 2 + w k 2 ),

9 898 A. DUNCA AND Y. EPSHTEYN which gives the following estimate: Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see (3) 2 w k 2 + δ 2 w k 2 2 u 2. We want to prove that we can pick T k = T. In (26) we replace ψ r with (I δ 2 Δ)G N w k. We can do this since (I δ 2 Δ)G N w k (t) G k = span{ψ j } j=,...,k for any t [,T). In the same way in which the energy inequality (7) for strong solutions was derived, we obtain (32) t 2 ( w k(t) 2 + δ 2 w k (t) 2 )+δ 2 Δw k (s) 2 ds + Δw k (s) 2 ds M, 2 where ( ) (33) M := K f(s) 2 V ds + w k 2 + δ 2 w k 2 with K = max{ 2 G N 2 L 2 (Q), 2 δ2, 2, δ2 2 G N L 2 (Q)}. M does not depend on t and using (3) M also does not depend on k. Due to orthonormality of the family {ψ j } j in H we get that a priori the coefficients η kr satisfy η kr 2 2M 2 for any t [,T),r =,...,k, and k N. This implies that for any k there exists a global solution (that is, on [,T)) η kr W,2 [,T), r =,...,k, of the ODE system (26). In the same way as in Galdi [3] one can show, using estimate (32), that there exists a subsequence of w k (which is redenoted by w k ) which converges weakly in V uniformly in t to a function w L (,T,V). From estimate (32) we infer that the sequence w k is bounded in L 2 (,T,H 2 (Q)); consequently, it contains a subsequence (which is redenoted by w k ) which is weakly convergent to a function w L 2 (,T,H 2 (Q)). One can show, taking limits of w k in the space L 2 (,T,L 2 (Q)), that w = w. It follows that w (H 2 (Q) H) d. We can show that w satisfies the variational equality (6) in the same way as in Galdi [3] taking the limits of w k in equality (26). In the case of Stolz Adams models, when taking limits, the nonlinear term is handled in the following way: one needs to show that for a given eigenfunction ψ r, (G N w k G N w k, (I δ 2 Δ) ψ r ) (G N w G N w, (I δ 2 Δ) ψ r )ds. However, (G N w k G N w k, (I δ 2 Δ) ψ r ) (G N w G N w, (I δ 2 Δ) ψ r )ds = (G N w k G N w k,ψ r ) (G N w G N w,ψ r )ds

10 Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see LARGE-EDDY SIMULATION OF TURBULENT FLOWS 899 t (G N (w k w) G N w k,ψ r )ds + (G N w G N (w k w),ψ r )ds G N 2 L 2 (Q) w k w L 2 (,T,L 2 ) ψ r w k L 2 (,T,L 2 ) + (G N w G N ( (w k w)),ψ r )ds. The first term on the right-hand side above converges to since w k w in L 2 (,T,L 2 (Q)), and the second converges to because w k w weakly in L 2 (,T,L 2 (Q)) and the operator G N is self-adjoint. The energy inequality (24) is obtained in the same way as in the case of the Navier Stokes equations taking limits in (32). Lemma 4.2. The weak solution w that was constructed in the previous theorem is also a strong solution of (4). Proof. This follows directly from definition (6), the regularity proven for the solution, and an integration by parts. Lemma 4.3. The weak solution w of (4) constructed in Proposition 4. is the unique weak solution of (4). Proof. This is a consequence of the regularity of w. The proof is the same as in the case of the Navier Stokes equations. 5. An a priori estimate of the modeling error. Our goal here is to give an a priori estimate of the modeling error u w. In this direction there are several fundamental problems. First, in three dimensions there is no proof of uniqueness of weak solutions u of the Navier Stokes equations. Thus for u a general weak solution of the Navier Stokes equations, the best result attainable in the usual norms with the present technique seems to be the following. Proposition 5.. Let w = w(δ) be the unique strong solution of the model (4). Then there is a subsequence δ j as j and a weak solution u of the Navier Stokes equations such that w(δ j ) u in L (,T,L 2 (Q)) L 2 (,T,H (Q)). Proof. This proof follows that of Theorem 3. of Layton and Lewandowski [6]. The second question concerns the right norm. Obviously if we are restricting our attention to general weak solutions, the right norm must be a very weak norm for which the modeling residual uu T G N u(g N u) T is not only well defined but also vanishes as δ. The answer to this question is still unknown; see, e.g., Layton and Lewandowski [6] for first steps. The third question concerns extracting a rate of convergence for u w which gives some insight into the model s accuracy on the laminar regions. This problem is much simpler. It reduces to proving the highest possible rate of convergence for u w for very smooth solution u. In the remainder of this subsection we give the answer: the modeling error is a priori O(δ 2N+2 ) for smooth u. Proposition 5.2. Assume u is a weak solution of the Navier Stokes equations and u L 4 (,T,L 2 (Q)). For w L 2 (,T,H 2 (Q)) L (,T; V) aweak solution of (4) and τ := uu T G N u(g N u) T there exists a positive constant P = P (, N, u L 4 (,T,L 2 )) such that (34) u w 2 L (,T,L 2 ) + (u w) 2 L 2 (,T,L 2 ) P (, N, u L 4 (,T,L 2 ) ) τ 2 L 2 (,T,L 2 ).

11 9 A. DUNCA AND Y. EPSHTEYN Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see Proof. To begin we derive an equation for φ := u w. First we note that w is a unique strong solution of the model and under stated regularity assumptions on u, u is a unique strong solution of the Navier Stokes equations; see [2, Remark 3.3]. Thus there are no subtleties in the derivation of the error equation. Equality (3) can be rewritten as (35) u t + (G N ug N u T ) Δu + p = f + (G N ug N u T uu T ), u =. Subtraction gives the equation for ϕ := u w, ϕ t + (G N ug N u T G N wg N w T ) Δϕ + (p q) = τ in (,T) R d, ϕ = in (,T] R d, (36) ϕ t= = inr d, p q dx = in (,T]. We multiply the first equation in (36) by (I δ 2 Δ) G N ϕ and then integrate on Q. Following exactly the same computations as in Lemma 4. gives (37) d 2 dt ϕ 2 + d 2 δ2 dt ϕ 2 + δ2 d 2 dt ( ϕ, G N ϕ)+ ϕ 2 + δ 2 Δϕ + δ 4 (Δφ, G N Δϕ) = ( τ,g N ϕ)+b(g N ϕ, G N u, G N ϕ), where b is the standard trilinear form Q b(u, v, w) =((u )v, w). The first term on the right-hand side is bounded as follows: ( τ,g N ϕ) = (τ,g N ϕ) τ G N L 2 ϕ 2 G N 2 L 2 To bound the second term we use Young s inequality ab ɛa (4ɛ) 3 b 4 3, together with the standard estimate for the trilinear form to obtain that for any ɛ>, Plugging ɛ = b(g N ϕ, G N u, G N ϕ) C(Q) u ϕ 2 ϕ 3 2, τ ϕ 2. b(g N ϕ, G N u, G N ϕ) ɛ G N 2 ϕ (4ɛ) 3 GN u 4 G N ϕ 2. 2 G N 2 into the above inequality, we get that b(g N ϕ, G N u, G N ϕ) 2 ϕ ( ) 2 3 G N 2 GN 4 u 4 ϕ 2.

12 LARGE-EDDY SIMULATION OF TURBULENT FLOWS 9 Using the last two inequalities in (37) gives Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see d 2 dt ϕ 2 + d 2 δ2 dt ϕ 2 + δ2 d 2 dt ( ϕ, G N ϕ)+δ 2 Δϕ + δ 4 (Δφ, G N Δϕ) 2 G N 2 L 2 τ (2) 3 GN 3 u 4 w 2. Gronwall s inequality and positivity of the operators (G N ) N give ( ) ϕ 2 3 exp 2 s 4 (2) 3 GN 3 u 4 2 ds GN 2 L 2 τ 2 L ds. 2 For fixed N we have that G N +(+ G )+(+ G ) 2 + +(+ G ) N, and since for every δ, G it follows that G N 2 N+ uniformly in δ. Under the assumption that u L 4 (,T,L 2 ) we infer the existence of a constant M = M(, N, u L 4 (,T,L 2 )) such that (38) ϕ 2 L (,T,L 2 ) M( ) T, N, u L 4 (,T,L 2 ) τ 2,T,L. 2 To estimate ϕ 2 L 2 (,T,L 2 ) we integrate (37) from to t and, using inequality (38), we obtain φ 2 L 2 (,T,L 2 ) R(, N, u L 4 (,T,L 2 ) ) T τ 2,T,L 2 for positive constant R = R(, N, u L4 (,T,L 2 )). Consequently, there exists a constant P = P (, N, u L 4 (,T,L 2 )) such that (39) ϕ 2 L (,T,L 2 ) + φ 2 L 2 (L 2 ) P (, N, u L 4 (,T,L 2 ) ) τ 2 L 2 (,T,L 2 ). Proposition 5.3. Under the conditions of the previous theorem, if u H N+ (Q), there exists P = P (, N, u) such that (4) u w 2 L (,T,L 2 ) + (u w) 2 L 2 (,T,L 2 ) P (, N, u)δ2n+2. Proof. An application of Lemma 2.2 gives τ 2 L 2 (,T,L 2 ) C(u)δ2N+2 ; (4) will then follow from (39).

13 92 A. DUNCA AND Y. EPSHTEYN Downloaded 8/2/4 to Redistribution subject to SIAM license or copyright; see 6. Conclusions. The Stolz Adams deconvolution models analyzed herein are shown to have very good mathematical properties, better than any other large-eddy simulation model for turbulent flows that is currently used. There exists a weak solution of these models; that solution is unique, and further it is shown that it belongs to higher order Sobolev spaces and that it is also the strong solution of the models. We proved that the Stolz Adams models give a good description of the local spatial averages of fluid velocities, the modeling error converges to, and the rate of convergence is also derived. This paper provides the mathematical foundations of the Stolz Adams models, giving guidance for practical computations with these models. REFERENCES [] N.A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in largeeddy simulation, in Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, Philadelphia, 2, pp [2] M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing, Bristol, UK, 998. [3] G.P. Galdi, Lectures in Mathematical Fluid Dynamics, Birkhäuser-Verlag, Basel, Switzerland, 2. [4] M. Germano, Differential filters of elliptic type, Phys. Fluids, 29 (986), pp [5] J.L. Guermond, J. Tinsley Oden, and S. Prudhomme, Mathematical perspectives on large eddy simulation models for turbulent flows, J. Math. Fluid Mech., 6 (24), pp [6] W. Layton and R. Lewandowski, Analysis of the Zero th order model for Large Eddy Simulation of Turbulence, technical report, 23. [7] W. Layton and R. Lewandowski, A simple and stable scale similarity model for large eddy simulation: Energy balance and existence of weak solutions, Appl. Math. Lett., 6 (23), pp [8] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Springer-Verlag, Berlin, Heidelberg, New York, 2. [9] S. Stolz and N.A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids, (999), pp [] S. Stolz, N.A. Adams, and D. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Phys. Fluids, 3 (2), pp [] S. Stolz, N.A. Adams, and D. Kleiser, The approximate deconvolution model for large-eddy simulations of compressible flows and its applications to shock-turbulent-boundary-layer interaction, Phys. Fluids, 3 (2), pp [2] R. Temam, Navier Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 995.