- Email: [email protected]
Eddy solutions of approximate deconvolution models of turbulence
Applied Mathematics Letters 24 (2011) 23–26
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On Taylor/Eddy solutions of approximate deconvolution models of turbulence William Layton ∗ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
article
info
Article history: Received 11 August 2009 Received in revised form 22 July 2010 Accepted 1 August 2010 Keywords: Green–Taylor vortex Eddy solution Approximate deconvolution model Large eddy simulation
abstract This article shows that so called general Green–Taylor solutions, also called Taylor solutions or eddy solutions, of the Navier–Stokes equations are also exact solutions to approximate deconvolution models of turbulence. Thus, these special structures in flows exist as exact features in the models studied and their persistence/transient behavior is exactly determined by their stability, not by the effects of modelling or truncation errors. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Consider the 2d Navier–Stokes equations1 (NSE) under 2π periodic boundary conditions: for Ω = (0, 2π )2 , t > 0 ut + u · ∇ u − ν△u + ∇ p = 0
and ∇ · u = 0.
(1)
One fundamental exact solution of (1.1) is the Green–Taylor vortex. In the simplest case, it is an array of signed vortices which decay in place as t increases. These have been used as a fundamental test problem in CFD, e.g., [1–4] (and many others since) and to explore the analytic structure of some turbulence models by Berselli [5] (a remarkable and interesting paper), Barbatto et al. [6] and others. In this note we follow this latter approach and use general Green–Taylor vortices of the NSE, also called eddy solutions, Walsh [7], and Taylor solutions, Berselli [5], to explore similarly when a relatively new approach to turbulence models can replicate these fundamental vortex structures. First we recall from [7] (see also [8–10]) the definition of Green–Taylor/Taylor/eddy solutions of the NSE. Definition 1. Let φ(x) be 2π periodic and satisfy
− △φ = λφ,
∇ · φ = 0.
(2)
Then u(x, t ) = e−νλt φ(x) with pressure p satisfying
∇ p = −u · ∇ u is a Taylor solution/eddy solution of the NSE.
∗
Tel.: +1 4126248312; fax: +1 4126248397. E-mail address: [email protected]. URL: http://www.math.pitt.edu/∼wjl.
1 The usual abbreviations: NSE = Navier–Stokes Equations, CFD = Computational Fluid Dynamics, LES = Large Eddy Simulation, ADM = Approximate Deconvolution Model, and SFNSE = Space Filtered NSE. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.08.003
24
W. Layton / Applied Mathematics Letters 24 (2011) 23–26
For such u, ∇ × (u · ∇ u) = 0 and thus such a p exists. Eddy solutions are exact solutions of the (linear) vector heat equation for which the NSE nonlinearity u · ∇ u can be exactly cancelled by a pressure term ∇ p. The vorticity of an eddy solution exactly satisfies a linear heat equation which is the reason for its simple exponential decay in time. We note that 3d generalizations, corresponding to viscous extensions of abc (Arnold–Beltrami–Childress) flows, have been given by Ross Ethier and Steinman [11]. The eigenvalues of the 2d Stokes problem are given by λ = λm,n = m2 + n2 , where m, n are positive integers. The construction of Taylor/eddy solutions can also begin with scalar eigenfunctions of the Laplacian, [7]. If ψ is 2π periodic and satisfies
−△ψ = λψ then φ = (ψx2 , −ψx1 ) satisfies (2). We prove in Section 2 that Taylor/eddy solutions of the NSE are also exact solutions of the general Approximate Deconvolution Model (ADM) of turbulence, (3) below, proving that ADMs’ structure allows persistence of a large class of special solutions of the NSE. The spacial structure of the ADM solutions is exactly the same as for Taylor/eddy solutions of the NSE and the decay exponents are slightly larger for the ADM Taylor/eddy solution than for the corresponding NSE Taylor/eddy solution. 2. Approximation deconvolution models of turbulence Simulation of the pointwise velocity in turbulent flows by solving the NSE (1.1) down to the last persistent scale of motion is not feasible within the time and resource constraints in many important applications. The normal approach is instead to derive (approximate) equations for local spacial averages (denoted by u(x, t )) and to solve these for the (approximate) averaged velocities (denoted herein by w(x, t )). The class of models we study herein, Approximate Deconvolution Models (ADMs) were pioneered by Geurts [12] and Stolz, et al. [13–15]. A sound theoretical foundation exists supporting their practical effectiveness, e.g., [16–18]. In large eddy simulation (LES) the averages are defined by a local spacial filter (with filter radius denoted by δ ); see [16,19] for details and examples of models and filters. In most turbulent flow problems, run time and computer resource limitations are such that a fully resolved direct numerical simulation of the flow is not feasible. Thus, LES typically begins with an estimate of the finest feasible meshwidth △x. The averaging radius δ is then selected by the user typically to be O(△x). Herein we select a differential filter (an idea of Germano [20]) denoted by overbar or the action of the filter operator G given by u = Gu,
G := (−δ 2 △ + 1)−1 .
Averaging (1.1) leads to the non-closed Space Filtered NSE (SFNSE) ut + ∇ · uu − ν△u + ∇ p = 0,
∇ · u = 0.
In ADMs an approximate filter inverse/deconvolution operator D is constructed and used to close the SFNSE by uu = (formally) G−1 (u)G−1 (u) ≃ D(u)D(u); there are several used. We assume the chosen deconvolution operator D can be expressed as D = f (G), where f : R → R. Significant examples of deconvolution operators where D = f (G) include modified Tikhonov, [18], with D = ((1 − µ)G + ∑ µI )−1 , for µ > 0, van Cittert, [21], with DN = Nn=0 (I − G)n , and iterated and optimized Tikhonov regularization. An added time / secondary regularization term χ (w − D(w)) is typically added (see [17,22] for background on this term). This yields the general ADM which is
wt + ∇ · D(w)D(w) − ν△w + ∇ q + χ (w − D(w)) = 0,
∇ · w = 0.
(3)
Here w denotes the resulting approximation of u. By analogy to spectral methods for periodic problems (which are currently the ‘‘gold standard’’ for problems to which they apply), the secondary regularization term χ (w − D(w)) is comparable to spectral vanishing viscosity in damping scales near the cutoff length. The closure model of the nonlinear term ∇ · D(w)D(w) is comparable to the 3/2 rule in providing a more accurate representation of the restriction of the nonlinear term to the resolved scales. An abstract theory of the general ADM is now in place, [18]. We prove next that Taylor/eddy solutions of the NSE are also exact solutions of the above general ADM. The spacial structure is exactly the same as for Taylor/eddy solutions of the NSE and the decay exponents are slightly larger for the ADM Taylor/eddy solution than for the corresponding NSE Taylor/eddy solution. Theorem 2. If u(x, t ) = e−νλt φ(x) is a Taylor/eddy solution of the NSE then (w, q) where
w(x, t ) = e
−α t
φ(x),
∇ q = −D(w) · ∇ D(w)
[
α = − νλ + χ 1 − f
1
δ2 λ + 1
1
δ2 λ + 1
]
,
W. Layton / Applied Mathematics Letters 24 (2011) 23–26
25
is a Taylor/eddy solution of the ADM. Further ∇ × D(w) · ∇ D(w) = 0 so q exists. The energy and enstrophy of Taylor/eddy solutions decay exponentially at the same rates: Energy(t ) = e−2λν t Energy(0),
and
Enstrophy(t ) = λe−2λν t Enstrophy(0).
Proof. The proof is an extension of the construction in the NSE case. Indeed, we will show that ∇ × D(φ) · ∇ D(φ) = 0 so the ADM pressure q exists and then it is simple to verify by direct substitution that w(x, t ) satisfies
wt − ν△w + ∇ q + χ (w − D(w)) = 0,
∇ · w = 0.
First we recall from [7] that ∇ × (φ · ∇φ) = 0. Indeed, let φ = (u, v). It is easy to verify by a direct calculate that
∂ ∂ (uux + v ux2 ) − (uvx1 + vvx2 ) = 0 ∂ x2 ∂ x1 using −△(u, v) = λ(u, v) and ux1 +vx2 = 0. Since differential and convolution operators commute under periodic boundary
conditions, it follows that ∇ × (φ · ∇φ) = 0 and ∇ × (φ · ∇φ) = 0 are equivalent. Consider ∇ × (D(w) · ∇ D(w)). Thus, this is zero provided ∇ × D(w) · ∇ D(w) = 0. Let w = D(w). we claim that −△ w = λ w, ∇ · w = 0 so by the (above) same argument as the NSE case ∇ × (φ · ∇φ) and thus (going backwards) ∇ × (D(w) · ∇ D(w)) = 0. Indeed, in the absence of boundaries all the operators involved commute. Thus
∇ ·w = ∇ · D(w) = e−αt ∇ · D(φ) = e−αt ∇ · f (G)φ = e−αt f (G)∇ · φ = 0, −△ w = e−αt D(−△φ) = e−αt λD(φ) = λe−αt D(φ) = λ w. The claimed decay of energy and enstrophy is directly calculated from the exact solution.
A close look at the proof shows that the following also holds. Corollary 3. If u(x, t ) is a 2π periodic solution of the 2d or 3d NSE (1.1) with ∇ × (u · ∇ u) = 0, then this same u(x, t ) is an exact solution of the ADM (3) with χ = 0. For most families of deconvolution operators the zeroth order member is D = I (i.e., approximate u by u). For this simplest case we have the rate constant
α0 = −λ(ν + χ δ 2 /(λδ 2 + 1)). Thus the effect of time relaxation/secondary regularization here is (i) to increase the viscosity coefficient slightly from ν to roughly ν + χ δ 2 , and (ii) due to the denominator in the second term, the additional damping acts slightly more strongly on larger spacial scales (smaller λ). This seems paradoxical since the intent of time relaxation is to damp smaller scales exponentially in time. We note however that in Taylor/eddy solutions all scales are damped exponentially so a clear conclusion cannot be drawn from this point. For Nth order van Cittert deconvolution we directly calculate that
δ2 λ αN = − νλ + χ λδ 2 + 1 [
] N +1
.
Thus the effect is similar but as N increases the rate constant αN decreases to the NSE value. Acknowledgement The research presented herein was partially supported by NSF grant DMS 0810385. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A.J. Chorin, Numerical solution of the Navier–Stokes equations, Math. Comp. 22 (1968) 745–762. J. Kim, P. Moin, Application of a fractional-step method to incompressible Navier–Stokes equations, J. Comput. Phys. 59 (1985) 308–323. M.E. Brachet, Direct simulation of three dimensional turbulence in the Taylor–Green vortex, Fluid Dynam. Res. 8 (1991) 1–8. S.A. Orszag, Numerical simulation of the Taylor Green vortex, in: Comput. Meth. in Appl. Sci. and Eng. Part 2, Springer, Berlin, 1974, pp. 50–64. L.C. Berselli, On the large eddy simulation of the Taylor–Green vortex, J. Math. Fluid Mech. 7 (2005) S164–S191. D. Barbato, L.C. Berselli, C.R. Grisanti, Analytical and numerical results for the rational large eddy simulation model, J. Math. Fluid Mech. 9 (2007) 44–74. O. Walsh, Eddy solutions of the Navier–Stokes equations, in: J.G. Heywood, K. Masuda, R. Rautmann, V.A. Solonikkov (Eds.), The NSE II-Theory and Numerical Methods, in: LNM, vol. 1530, Springer, 1992. R. Berker, Integration des equations d’un fluide visquex incompressible, in: Handbuch der Physik, vol. 8/2, Springer, Berlin, 1963, pp. 1–384. A.E. Green, G.I. Taylor, Mechanism for the production of small eddies from larger ones, Proc. Roy. Soc. A 158 (1937) 400–521. G.I. Taylor, On the decay of vortices in a viscous fluid, Phil. Mag. 6 (46) (1923) 671–674. C. Ross Ethier, D.A. Steinman, Exact full 3D Navier–Stokes solutions for benchmarking, Internat. J. Numer. Methods Fluids 19 (1994) 369–375. B.J. Geurts, Inverse modeling for large eddy simulation, Phys. Fluids 9 (1997) 3585. N.A. Adams, S. Stolz, A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys. 178 (2002) 391–426.
26
W. Layton / Applied Mathematics Letters 24 (2011) 23–26
[14] S. Stolz, N.A. Adams, On the approximate deconvolution procedure for LES, Phys. Fluids II (1999) 1699–1701. [15] S. Stolz, N.A. Adams, L. Kleiser, An approximate deconvolution model for large eddy simulation with application to wall-bounded flows, Phys. Fluids 13 (2001) 997. [16] L.C. Berselli, T. Iliescu, W. Layton, Large Eddy Simulation, Springer, Berlin, 2004. [17] W. Layton, M. Neda, Truncation of scales by time relaxation, J. Math. Anal. Appl. 325 (2007) 788–807. [18] I. Stanculescu, Existence theory of abstract approximate deconvolution models of turbulence, Annali dell’Univ. di Ferrara 54 (2008) 145–168. [19] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Springer, Berlin, 2001. [20] M. Germano, Differential filters of elliptic type, Phys. Fluids 29 (1986) 1757–1758. [21] M. Bertero, B. Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing Ltd., 1998. [22] S. Schochet, E. Tadmor, The regularized Chapman–Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal. 119 (1992) 95.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On Taylor/Eddy solutions of approximate deconvolution models of turbulence William Layton ∗ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
article
info
Article history: Received 11 August 2009 Received in revised form 22 July 2010 Accepted 1 August 2010 Keywords: Green–Taylor vortex Eddy solution Approximate deconvolution model Large eddy simulation
abstract This article shows that so called general Green–Taylor solutions, also called Taylor solutions or eddy solutions, of the Navier–Stokes equations are also exact solutions to approximate deconvolution models of turbulence. Thus, these special structures in flows exist as exact features in the models studied and their persistence/transient behavior is exactly determined by their stability, not by the effects of modelling or truncation errors. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Consider the 2d Navier–Stokes equations1 (NSE) under 2π periodic boundary conditions: for Ω = (0, 2π )2 , t > 0 ut + u · ∇ u − ν△u + ∇ p = 0
and ∇ · u = 0.
(1)
One fundamental exact solution of (1.1) is the Green–Taylor vortex. In the simplest case, it is an array of signed vortices which decay in place as t increases. These have been used as a fundamental test problem in CFD, e.g., [1–4] (and many others since) and to explore the analytic structure of some turbulence models by Berselli [5] (a remarkable and interesting paper), Barbatto et al. [6] and others. In this note we follow this latter approach and use general Green–Taylor vortices of the NSE, also called eddy solutions, Walsh [7], and Taylor solutions, Berselli [5], to explore similarly when a relatively new approach to turbulence models can replicate these fundamental vortex structures. First we recall from [7] (see also [8–10]) the definition of Green–Taylor/Taylor/eddy solutions of the NSE. Definition 1. Let φ(x) be 2π periodic and satisfy
− △φ = λφ,
∇ · φ = 0.
(2)
Then u(x, t ) = e−νλt φ(x) with pressure p satisfying
∇ p = −u · ∇ u is a Taylor solution/eddy solution of the NSE.
∗
Tel.: +1 4126248312; fax: +1 4126248397. E-mail address: [email protected]. URL: http://www.math.pitt.edu/∼wjl.
1 The usual abbreviations: NSE = Navier–Stokes Equations, CFD = Computational Fluid Dynamics, LES = Large Eddy Simulation, ADM = Approximate Deconvolution Model, and SFNSE = Space Filtered NSE. 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.08.003
24
W. Layton / Applied Mathematics Letters 24 (2011) 23–26
For such u, ∇ × (u · ∇ u) = 0 and thus such a p exists. Eddy solutions are exact solutions of the (linear) vector heat equation for which the NSE nonlinearity u · ∇ u can be exactly cancelled by a pressure term ∇ p. The vorticity of an eddy solution exactly satisfies a linear heat equation which is the reason for its simple exponential decay in time. We note that 3d generalizations, corresponding to viscous extensions of abc (Arnold–Beltrami–Childress) flows, have been given by Ross Ethier and Steinman [11]. The eigenvalues of the 2d Stokes problem are given by λ = λm,n = m2 + n2 , where m, n are positive integers. The construction of Taylor/eddy solutions can also begin with scalar eigenfunctions of the Laplacian, [7]. If ψ is 2π periodic and satisfies
−△ψ = λψ then φ = (ψx2 , −ψx1 ) satisfies (2). We prove in Section 2 that Taylor/eddy solutions of the NSE are also exact solutions of the general Approximate Deconvolution Model (ADM) of turbulence, (3) below, proving that ADMs’ structure allows persistence of a large class of special solutions of the NSE. The spacial structure of the ADM solutions is exactly the same as for Taylor/eddy solutions of the NSE and the decay exponents are slightly larger for the ADM Taylor/eddy solution than for the corresponding NSE Taylor/eddy solution. 2. Approximation deconvolution models of turbulence Simulation of the pointwise velocity in turbulent flows by solving the NSE (1.1) down to the last persistent scale of motion is not feasible within the time and resource constraints in many important applications. The normal approach is instead to derive (approximate) equations for local spacial averages (denoted by u(x, t )) and to solve these for the (approximate) averaged velocities (denoted herein by w(x, t )). The class of models we study herein, Approximate Deconvolution Models (ADMs) were pioneered by Geurts [12] and Stolz, et al. [13–15]. A sound theoretical foundation exists supporting their practical effectiveness, e.g., [16–18]. In large eddy simulation (LES) the averages are defined by a local spacial filter (with filter radius denoted by δ ); see [16,19] for details and examples of models and filters. In most turbulent flow problems, run time and computer resource limitations are such that a fully resolved direct numerical simulation of the flow is not feasible. Thus, LES typically begins with an estimate of the finest feasible meshwidth △x. The averaging radius δ is then selected by the user typically to be O(△x). Herein we select a differential filter (an idea of Germano [20]) denoted by overbar or the action of the filter operator G given by u = Gu,
G := (−δ 2 △ + 1)−1 .
Averaging (1.1) leads to the non-closed Space Filtered NSE (SFNSE) ut + ∇ · uu − ν△u + ∇ p = 0,
∇ · u = 0.
In ADMs an approximate filter inverse/deconvolution operator D is constructed and used to close the SFNSE by uu = (formally) G−1 (u)G−1 (u) ≃ D(u)D(u); there are several used. We assume the chosen deconvolution operator D can be expressed as D = f (G), where f : R → R. Significant examples of deconvolution operators where D = f (G) include modified Tikhonov, [18], with D = ((1 − µ)G + ∑ µI )−1 , for µ > 0, van Cittert, [21], with DN = Nn=0 (I − G)n , and iterated and optimized Tikhonov regularization. An added time / secondary regularization term χ (w − D(w)) is typically added (see [17,22] for background on this term). This yields the general ADM which is
wt + ∇ · D(w)D(w) − ν△w + ∇ q + χ (w − D(w)) = 0,
∇ · w = 0.
(3)
Here w denotes the resulting approximation of u. By analogy to spectral methods for periodic problems (which are currently the ‘‘gold standard’’ for problems to which they apply), the secondary regularization term χ (w − D(w)) is comparable to spectral vanishing viscosity in damping scales near the cutoff length. The closure model of the nonlinear term ∇ · D(w)D(w) is comparable to the 3/2 rule in providing a more accurate representation of the restriction of the nonlinear term to the resolved scales. An abstract theory of the general ADM is now in place, [18]. We prove next that Taylor/eddy solutions of the NSE are also exact solutions of the above general ADM. The spacial structure is exactly the same as for Taylor/eddy solutions of the NSE and the decay exponents are slightly larger for the ADM Taylor/eddy solution than for the corresponding NSE Taylor/eddy solution. Theorem 2. If u(x, t ) = e−νλt φ(x) is a Taylor/eddy solution of the NSE then (w, q) where
w(x, t ) = e
−α t
φ(x),
∇ q = −D(w) · ∇ D(w)
[
α = − νλ + χ 1 − f
1
δ2 λ + 1
1
δ2 λ + 1
]
,
W. Layton / Applied Mathematics Letters 24 (2011) 23–26
25
is a Taylor/eddy solution of the ADM. Further ∇ × D(w) · ∇ D(w) = 0 so q exists. The energy and enstrophy of Taylor/eddy solutions decay exponentially at the same rates: Energy(t ) = e−2λν t Energy(0),
and
Enstrophy(t ) = λe−2λν t Enstrophy(0).
Proof. The proof is an extension of the construction in the NSE case. Indeed, we will show that ∇ × D(φ) · ∇ D(φ) = 0 so the ADM pressure q exists and then it is simple to verify by direct substitution that w(x, t ) satisfies
wt − ν△w + ∇ q + χ (w − D(w)) = 0,
∇ · w = 0.
First we recall from [7] that ∇ × (φ · ∇φ) = 0. Indeed, let φ = (u, v). It is easy to verify by a direct calculate that
∂ ∂ (uux + v ux2 ) − (uvx1 + vvx2 ) = 0 ∂ x2 ∂ x1 using −△(u, v) = λ(u, v) and ux1 +vx2 = 0. Since differential and convolution operators commute under periodic boundary
conditions, it follows that ∇ × (φ · ∇φ) = 0 and ∇ × (φ · ∇φ) = 0 are equivalent. Consider ∇ × (D(w) · ∇ D(w)). Thus, this is zero provided ∇ × D(w) · ∇ D(w) = 0. Let w = D(w). we claim that −△ w = λ w, ∇ · w = 0 so by the (above) same argument as the NSE case ∇ × (φ · ∇φ) and thus (going backwards) ∇ × (D(w) · ∇ D(w)) = 0. Indeed, in the absence of boundaries all the operators involved commute. Thus
∇ ·w = ∇ · D(w) = e−αt ∇ · D(φ) = e−αt ∇ · f (G)φ = e−αt f (G)∇ · φ = 0, −△ w = e−αt D(−△φ) = e−αt λD(φ) = λe−αt D(φ) = λ w. The claimed decay of energy and enstrophy is directly calculated from the exact solution.
A close look at the proof shows that the following also holds. Corollary 3. If u(x, t ) is a 2π periodic solution of the 2d or 3d NSE (1.1) with ∇ × (u · ∇ u) = 0, then this same u(x, t ) is an exact solution of the ADM (3) with χ = 0. For most families of deconvolution operators the zeroth order member is D = I (i.e., approximate u by u). For this simplest case we have the rate constant
α0 = −λ(ν + χ δ 2 /(λδ 2 + 1)). Thus the effect of time relaxation/secondary regularization here is (i) to increase the viscosity coefficient slightly from ν to roughly ν + χ δ 2 , and (ii) due to the denominator in the second term, the additional damping acts slightly more strongly on larger spacial scales (smaller λ). This seems paradoxical since the intent of time relaxation is to damp smaller scales exponentially in time. We note however that in Taylor/eddy solutions all scales are damped exponentially so a clear conclusion cannot be drawn from this point. For Nth order van Cittert deconvolution we directly calculate that
δ2 λ αN = − νλ + χ λδ 2 + 1 [
] N +1
.
Thus the effect is similar but as N increases the rate constant αN decreases to the NSE value. Acknowledgement The research presented herein was partially supported by NSF grant DMS 0810385. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A.J. Chorin, Numerical solution of the Navier–Stokes equations, Math. Comp. 22 (1968) 745–762. J. Kim, P. Moin, Application of a fractional-step method to incompressible Navier–Stokes equations, J. Comput. Phys. 59 (1985) 308–323. M.E. Brachet, Direct simulation of three dimensional turbulence in the Taylor–Green vortex, Fluid Dynam. Res. 8 (1991) 1–8. S.A. Orszag, Numerical simulation of the Taylor Green vortex, in: Comput. Meth. in Appl. Sci. and Eng. Part 2, Springer, Berlin, 1974, pp. 50–64. L.C. Berselli, On the large eddy simulation of the Taylor–Green vortex, J. Math. Fluid Mech. 7 (2005) S164–S191. D. Barbato, L.C. Berselli, C.R. Grisanti, Analytical and numerical results for the rational large eddy simulation model, J. Math. Fluid Mech. 9 (2007) 44–74. O. Walsh, Eddy solutions of the Navier–Stokes equations, in: J.G. Heywood, K. Masuda, R. Rautmann, V.A. Solonikkov (Eds.), The NSE II-Theory and Numerical Methods, in: LNM, vol. 1530, Springer, 1992. R. Berker, Integration des equations d’un fluide visquex incompressible, in: Handbuch der Physik, vol. 8/2, Springer, Berlin, 1963, pp. 1–384. A.E. Green, G.I. Taylor, Mechanism for the production of small eddies from larger ones, Proc. Roy. Soc. A 158 (1937) 400–521. G.I. Taylor, On the decay of vortices in a viscous fluid, Phil. Mag. 6 (46) (1923) 671–674. C. Ross Ethier, D.A. Steinman, Exact full 3D Navier–Stokes solutions for benchmarking, Internat. J. Numer. Methods Fluids 19 (1994) 369–375. B.J. Geurts, Inverse modeling for large eddy simulation, Phys. Fluids 9 (1997) 3585. N.A. Adams, S. Stolz, A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys. 178 (2002) 391–426.
26
W. Layton / Applied Mathematics Letters 24 (2011) 23–26
[14] S. Stolz, N.A. Adams, On the approximate deconvolution procedure for LES, Phys. Fluids II (1999) 1699–1701. [15] S. Stolz, N.A. Adams, L. Kleiser, An approximate deconvolution model for large eddy simulation with application to wall-bounded flows, Phys. Fluids 13 (2001) 997. [16] L.C. Berselli, T. Iliescu, W. Layton, Large Eddy Simulation, Springer, Berlin, 2004. [17] W. Layton, M. Neda, Truncation of scales by time relaxation, J. Math. Anal. Appl. 325 (2007) 788–807. [18] I. Stanculescu, Existence theory of abstract approximate deconvolution models of turbulence, Annali dell’Univ. di Ferrara 54 (2008) 145–168. [19] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Springer, Berlin, 2001. [20] M. Germano, Differential filters of elliptic type, Phys. Fluids 29 (1986) 1757–1758. [21] M. Bertero, B. Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing Ltd., 1998. [22] S. Schochet, E. Tadmor, The regularized Chapman–Enskog expansion for scalar conservation laws, Arch. Ration. Mech. Anal. 119 (1992) 95.