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Layton, William J.; Rebholz, Leo G.

On relaxation times in the Navier-Stokes-Voigt model. (English) Zbl 07508633

Int. J. Comput. Fluid Dyn. 27, No. 3, 184-187 (2013).
Summary: We study analytically and numerically the relaxation time of flow evolution governed by the Navier-Stokes-Voigt (NSV) model. We first show that for the Taylor-Green vortex decay problem, NSV admits an exact solution which evolves slower than true fluid flow. Secondly, we show numerically for a channel flow test problem using standard discretisation methods that although NSV provides more regular solutions compared to usual Navier-Stokes solutions, NSV approximations take significantly longer to reach the steady state.

Cited in 24 Documents

MSC:

76-XX Fluid mechanics
35-XX Partial differential equations

Keywords:

Navier-Stokes-Voigt model; Navier-Stokes equations; relaxation times; Taylor-Green vortex; exact CFD solution
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References:

[1] Bowers, A. L.; Rebholz, L. G., Increasing Accuracy and Efficiency in FE Computations of the Leray-deconvolution Model, Numerical Methods for Partial Differential Equations, 28, 2, 720-736 (2012) · Zbl 1414.76035 · doi:10.1002/num.20653
[2] Brenner, S.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), New York: Springer-Verlag, New York · Zbl 0804.65101
[3] Bryan, K., Accelerating the Convergence to Equilibrium of Ocean-climate Models, Journal of Physical Oceanography, 14, 666-673 (1984) · doi:10.1175/1520-0485(1984)014<0666:ATCTEO>2.0.CO;2
[4] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics (2007), Berlin: Springer, Berlin · Zbl 1121.76001
[5] Cao, Y.; Lunasin, E.; Titi, E. S., Global Well-posedness of the Three-dimensional Viscous and Inviscid Simplified Bardina Turbulence Models, Communications in Mathematical Sciences, 4, 4, 823-848 (2006) · Zbl 1127.35034
[6] Chorin, A. J., Numerical Solution for the Navier-Stokes Equations, Mathematics of Computation, 22, 745-762 (1968) · Zbl 0198.50103 · doi:10.1090/S0025-5718-1968-0242392-2
[7] Douglas, J.; Dupont, T., Galerkin Methods for Parabolic Equations, SIAM Journal on Numerical Analysis, 7, 4, 575-626 (1970) · Zbl 0224.35048 · doi:10.1137/0707048
[8] Green, A. E.; Taylor, G. I., Mechanism of the Production of Small Eddies from Larger Ones, Proceedings of the Royal Society of London, Series A (Mathematical and Physical Sciences), 158, 499-521 (1937) · JFM 63.1358.03 · doi:10.1098/rspa.1937.0036
[9] John, V.; Layton, W. J., Analysis of Numerical Errors in Large Eddy Simulation, SIAM Journal on Numerical Analysis, 40, 3, 995-1020 (2002) · Zbl 1026.76028 · doi:10.1137/S0036142900375554
[10] Koseoglu, A., The Navier-Stokes Voight Model and Convergence to Equilibrium and Statistical Equilibrium (2011), Department of Mathematics, University of Pittsburgh
[11] Kuberry, P.; Larios, A.; Rebholz, L.; Wilson, N., Numerical Approximation of the Voigt Regularization of Incompressible NSE and MHD Flows, Computers and Mathematics with Applications (2012)
[12] Labovsky, A.; Layton, W.; Manica, C.; Neda, M.; Rebholz, L., The Stabilized Extrapolated Trapezoidal Finite Element Method for the Navier-Stokes Equations, Computer Methods in Applied Mechanics and Engineering, 198, 958-974 (2009) · Zbl 1229.76051 · doi:10.1016/j.cma.2008.11.004
[13] Layton, W., An Introduction to the Numerical Analysis of Viscous Incompressible Flows (2008), Philadelphia, PA: SIAM, Philadelphia, PA · Zbl 1153.76002
[14] Oskolkov, A. P., The Uniqueness and Solvability in the Large of Boundary Value Problems for the Equations of Motion of Aqueous Solutions of Polymers, 7 Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38, 98-136 (1973)
[15] Tafti, D., Comparison of Some Upwind-biased High-order Formulations With a Second Order Central-difference Scheme for Time Integration of the Incompressible Navier-Stokes Equations, Computers & Fluids, 25, 7, 647-665 (1996) · Zbl 0888.76061 · doi:10.1016/0045-7930(96)00015-1
[16] Taylor, G. I., On Decay of Vortices in a Viscous Fluid, Philosophical Magazine, 46, 671-674 (1923) · JFM 49.0607.02 · doi:10.1080/14786442308634295
[17] Voigt, W., Ueber Innere Reibung Fester Korper, Insbesondere der Metalle, Annalen der Physik, 283, 671-693 (1892) · JFM 24.0932.01 · doi:10.1002/andp.18922831210
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