×

Conservation laws of turbulence models. (English) Zbl 1110.76028

Summary: The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of Navier-Stokes equations. We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and \(N\)th order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.

MSC:

76F99 Turbulence
76F65 Direct numerical and large eddy simulation of turbulence
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ditlevson, P.; Guiliani, P., Cascades in helical turbulence, Phys. rev. E, 63, (2001)
[2] Chen, Q.; Chen, S.; Eyink, G., The joint cascade of energy and helicity in three dimensional turbulence, Phys. fluids, 15, 2, 361-374, (2003) · Zbl 1185.76081
[3] Chen, Q.; Chen, S.; Eyink, G.; Holm, D., Intermittency in the joint cascade of energy and helicity, Phys. rev. lett., 90, 214503, (2003)
[4] Liu, J.; Wang, W., Energy and helicity preserving schemes for hydro and magnetohydro-dynamics flows with symmetry, J. comput. phys., 200, 8-33, (2004) · Zbl 1288.76093
[5] Frisch, U., Turbulence, (1995), Cambridge Univ. Press · Zbl 0727.76064
[6] Gresho, P.; Sani, R., Incompressible flow and the finite element method, vol. 2, (1998), Wiley · Zbl 0941.76002
[7] Foias, C.; Holm, D.; Titi, E., The navier – stokes-alpha model of fluid turbulence, Phys. D, 152-153, 505-519, (2001) · Zbl 1037.76022
[8] Dunca, A.; Epshteyn, Y., On the stolz – adams deconvolution model for the large-eddy simulation of turbulent flows, SIAM J. math. anal., 37, 6, 1890-1902, (2005-2006) · Zbl 1128.76029
[9] Berselli, L.; Iliesu, T.; Layton, W., Mathematics of large eddy simulation of turbulent flows, (2005), Springer Berlin
[10] Adams, N.A.; Stolz, S., Deconvolution methods for subgrid-scale approximation in large eddy simulation, () · Zbl 1147.76506
[11] Stolz, S.; Adams, N.A., On the approximate deconvolution procedure for LES, Phys. fluids, 11, 1699-1701, (1999) · Zbl 1147.76506
[12] Moffatt, H.; Tsoniber, A., Helicity in laminar and turbulent flow, Ann. rev. fluid mech., 24, 281-312, (1992) · Zbl 0751.76018
[13] Moreau, J.J., Constantes d’unilot tourbilloinnaire en fluide parfait barotrope, C. R. acad. sci. Paris, 252, 2810-2812, (1961) · Zbl 0151.41703
[14] W. Layton, R. Lewandowski, On the Leray deconvolution model, Technical Report, University of Pittsburgh, 2005 · Zbl 1210.76084
[15] Arakawa, A., Computational design for long-term numerical integration of the equations of fluid motion: two dimensional incompressible flow. part I, J. comput. phys., 1, 119-143, (1966) · Zbl 0147.44202
[16] Cheskidov, A.; Holm, D.D.; Olson, E.; Titi, E.S., On a Leray-α model of turbulence, Proc. R. soc. lond. ser. A math. phys. eng. sci., 461, 629-649, (2005) · Zbl 1145.76386
[17] J.J. Bardina, H. Ferziger, W.C. Reynolds, Improved subgrid scale models for large eddy simulation, AIAA Pap. 80-1357
[18] Girault, V.; Raviart, P., Finite element methods for the navier – stokes equations, (1986), Springer Berlin · Zbl 0413.65081
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.