×

Residual stress of approximate deconvolution models of turbulence. (English) Zbl 1273.76206

Summary: We consider the case of a homogeneous, isotropic, fully developed, turbulent flow. We show analytically by using the \(-5/3\) Kolmogorov law that the time-averaged consistency error of the \(N\)th approximate deconvolution LES (large eddy simulation) model converges to zero following a law as the cube root of the averaging radius, independently of the Reynolds number. The consistency error is measured by the residual stress. The filter under consideration is a second-order differential filter, but the 1/3 law is still valid in the case of the Gaussian filter and a large class of filters used in LES. We also show how the \(1/3\) error law can be derived by a dimensional analysis.

MSC:

76F65 Direct numerical and large eddy simulation of turbulence
76F05 Isotropic turbulence; homogeneous turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams N. A., Modern Simulation Strategies for Turbulent Flow (2001)
[2] Bardina J., Technical Report No (1983)
[3] Bernardi C., Automatic insertion of a turbulence model in the finite element discretization of the Navier-Stokes equations, in preparation (2006)
[4] Berselli L. C., Large Eddy Simulation (2004) · Zbl 1299.76096
[5] Bertero M., Introduction to Inverse Problems in Imaging (1998) · Zbl 0914.65060 · doi:10.1887/0750304359
[6] Brossier F., Mathematical Modelling and Numerical Analysis 36 pp 345– (2002) · Zbl 1040.35057 · doi:10.1051/m2an:2002016
[7] Chen S., Physica D 133 pp 49– (1999) · Zbl 1194.76069 · doi:10.1016/S0167-2789(99)00098-6
[8] Chen S., Physics of Fluid 11 pp 2343– (1999) · Zbl 1147.76357 · doi:10.1063/1.870096
[9] Chen S., Physical Review Letter 81 pp 5338– (1998) · Zbl 1042.76525 · doi:10.1103/PhysRevLett.81.5338
[10] Constantin P., Physical Review Letters 69 pp 1648– (1992) · doi:10.1103/PhysRevLett.69.1648
[11] Doering C. R., Physics of Fluids 87 pp 359– (1995)
[12] Dunca A., SIAM Journal of Mathematical Analysis (2006)
[13] Foias C., Contemporary Mathematics 208 pp 151– (1997)
[14] Foias C., Physica D pp 152– (2001)
[15] Foias C., Journal of Functional Analysis 2 pp 1384– (1989)
[16] Frisch U., Turbulences (1995)
[17] Galdi G. P., Mathematical Models and Methods in the Applied Sciences 10 pp 343– (2000)
[18] Germano M., Physics of Fluids 29 pp 1757– (1986) · Zbl 0647.76042 · doi:10.1063/1.865650
[19] Layton W., Journal of Differential Equations 55 pp 151– (1984) · Zbl 0497.34055 · doi:10.1016/0022-0396(84)90079-2
[20] Layton W., Applied Mathematical Letters 16 pp 1205– (2003) · Zbl 1039.76027 · doi:10.1016/S0893-9659(03)90118-2
[21] Layton W., Un filtre pour la SGE Étudié à la lumière du K41 (2004)
[22] Layton W., Discrete and Continuous Dynamical Systems Series B 6 pp 111– (2006)
[23] Leray J., Acta Mathematica 63 pp 193– (1934) · JFM 60.0726.05 · doi:10.1007/BF02547354
[24] Lesieur M., Turbulence in Fluids (1997) · Zbl 0876.76002
[25] Lewandowski R., Analyse Mathématique et Océanographie (1997)
[26] Lewandowski R., Nonlinear Analysis TMA 28 pp 393– (1997) · Zbl 0863.35077 · doi:10.1016/0362-546X(95)00149-P
[27] Lewandowski R., Journal of Mathematical Fluid Dynamics (2006)
[28] Lewandowski R., Some consequences of a regularity assumption on the solutions of the Navier-Stokes equations, in preparation (2006)
[29] Lilly D. K., Proceedings IBM Scientific Computing Symposium on Environmental Sciences (1967)
[30] Lions P. L., Mathematical Topics in Fluid Mechanics (1996) · Zbl 0866.76002
[31] Mohammadi B., Analysis of the k–{\(\epsilon\)} Model (1994)
[32] Olson E., Journal of Statistical Physics 113 pp 799– (2003) · Zbl 1137.76402 · doi:10.1023/A:1027312703252
[33] Pope S., Turbulent Flows (2000)
[34] Reynolds O., Philosophical Transaction of Royal Society of London A 186 pp 123– (1895) · JFM 26.0872.02 · doi:10.1098/rsta.1895.0004
[35] Sagaut P., Large Eddy Simulation for Incompressible Flow (2001) · Zbl 0964.76002
[36] Stolz S., Physics of Fluids, II pp 1699– (1999)
[37] Stolz S., Physics of Fluids 13 pp 997– (2001) · Zbl 1184.76530 · doi:10.1063/1.1350896
[38] Temam R., Navier Stokes Equations (1984) · Zbl 0568.35002
[39] Wang X., Physica D 99 pp 555– (1997) · Zbl 0897.76019 · doi:10.1016/S0167-2789(96)00161-3
[40] Zaidman S., Almost-Periodic Functions in Abstract Spaces (1985) · Zbl 0648.42006
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.