Numerical analysis and computational testing of a high accuracy Leray-deconvolution model of turbulence. (English) Zbl 1191.76061

Summary: We study a computationally attractive algorithm (based on an extrapolated Crank-Nicolson method) for a recently proposed family of high accuracy turbulence models, the Leray-deconvolution family. First we prove convergence of the algorithm to the solution of the Navier-Stokes equations and delineate its (optimal) accuracy. Numerical experiments are presented which confirm the convergence theory. Our 3d experiments also give a careful comparison of various related approaches. They show the combination of the Leray-deconvolution regularization with the extrapolated Crank-Nicolson method can be more accurate at higher Reynolds number that the classical extrapolated trapezoidal method of Baker (Report, Harvard University, 1976). We also show the higher order Leray-deconvolution models (e.g. \(N = 1,2,3\)) have greater accuracy than the \(N = 0\) case of the Leray-\(\alpha \) model. Numerical experiments for the \(2d\) step problem are also successfully investigated. Around the critical Reynolds number, the low order models inhibit vortex shedding behind the step. The higher order models, correctly, do not. To estimate the complexity of using Leray-deconvolution models for turbulent flow simulations we estimate the models’ microscale.


76F65 Direct numerical and large eddy simulation of turbulence
76F02 Fundamentals of turbulence
35Q35 PDEs in connection with fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI Link


[1] Leray, J Math Pur Appl Paris Ser IX 13 pp 331– (1934)
[2] Leray, Acta Math 63 pp 193– (1934)
[3] Geurts, Phys Fluids 15 pp l13– (2003)
[4] Geurts, J Turbulence 7 pp 1– (2006)
[5] Optimal design of fluid flow using subproblems reduced by large eddy simulation, Preprint ANL/MCS-p1117-1003 ( 2003).
[6] , and , On the convergence of the Leray-{\(\alpha\)} model to the trajectory attractor of the 3d Navier-Stokes system, 2005.
[7] Cheskidov, R Soc London, Proc Ser A Math Phys Eng Sci 461 pp 629– (2005) · Zbl 1145.76386
[8] Ilyin, Nonlinearity 19 pp 879– (2006)
[9] Vishik, Russian Math Dokladi 71 pp 91– (2005)
[10] and , Introduction to inverse problems in imaging, IOP Publishing, Bristol, 1998. · Zbl 0914.65060
[11] Layton, Analysis and Applications · Zbl 1153.76002
[12] Guermond, Phys D Nonlinear Phenom 177 pp 23– (2003)
[13] Galerkin approximation for the Navier-Stokes equations, Harvard University, 1976.
[14] and , Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern simulation strategies for turbulent flow, editor, Edwards, Ann Arbor, MI, 2001, p. 21.
[15] Adams, Phys Fluids 13 pp 997– (2001)
[16] Adams, J C P 178 pp 391– (2002)
[17] Adams, Phys Fluids 11 pp 1699– (1999)
[18] Adams, Phys Fluids 13 pp 2985– (2001)
[19] Finite element methods for viscous incompressible flows: A guide to theory, practices, and algorithms, Academic Press, Boston, 1989.
[20] and , The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
[21] Introduction to the Numerical Analysis of Incompressible, Viscous Flows, SIAM Publications, Philadelphia, 2007.
[22] Germano, Phys Fluids 29 pp 1755– (1986)
[23] Germano, Phys Fluids 29 pp 1757– (1986)
[24] Galdi, Math Models Methods Appl Sci 10 pp 343– (2000)
[25] and , Convergence analysis of the finite element method for a fundamental model in turbulence, Technical report, University of Pittsburgh, 2006. In revision, SIAM J Numer Anal.
[26] Dunca, SIAM J Math Anal 37 pp 1890– (2006)
[27] , and , Mathematics of large eddy simulation of turbulent flows, Springer, Berlin, 2006. · Zbl 1089.76002
[28] A remark on regularity of elliptic-elliptic singular perturbation problem, Technical report, available at, http://www.math.pitt.edu/techreports.html, 2007.
[29] Heywood, SIAM J Numer Anal 2 pp 353– (1990)
[30] Applied functional analysis: applications to mathematical physics, Springer-Verlag, New York, 1995.
[31] John, Int J Numer Methods Fluids 50 pp 713– (2006)
[32] Large eddy simulation of turbulent incompressible flows, Analytical and numerical results for a class of LES models, Lecture notes in computational science and engineering, Vol. 34, , , , , and , editors, Springer-Verlag, Berlin, Heidelberg, New York, 2004.
[33] Large eddy simulation for incompressible flows, Springer-Verlag, Berlin, 2001.
[34] Street, Phys Fluids A 5 pp 3186– (1993)
[35] and , FreeFem++ webpage: http://www.freefem.org.
[36] Muschinsky, JFM 325 pp 239– (1996)
[37] Layton, J Math Anal Appl 325 pp 788– (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.