Abstract
We study numerically four regularization models with deconvolution for density currents, namely, Boussinesq-α, Boussinesq-ω, Boussinesq–Leray and Modified-Boussinesq–Leray. A Crank–Nicolson in time and finite element in space algorithm is proposed and proved to be unconditionally stable and optimally convergent, which is verified through convergence rates in simulations. Lastly, the regularized models are compared through the two-dimensional Marsigli’s flow benchmark for Re = 2000 and Re = 5000. We found that Boussinesq-α and Boussinesq–Leray models produced the most accurate solutions in the low Reynolds number test and, as expected, all regularized models had their solutions improved when deconvolution order was increased. On the other hand, in the high Reynolds number test the Boussinesq–Leray provided the best solution. Besides, the Boussinesq–Leray model is also more advantageous from the computational point of view because its momentum and filter equations are decoupled enabling to increase the deconvolution order with no significant increase in the computational cost.
Funding
Igor O. Monteiro is partially supported by PRH-PB16. Carolina C. Manica is partially supported by CNPQ, 4805225/2011-0.
Acknowledgment
The authors are very grateful to Prof. Leo G. Rebholz for his useful advices in the development of this article.
References
[1] G. A. Baker, V. A. Dougalis, and O. A. Karakashian, On a higher order accurate fully discrete Galerkin approximation to the Navier–Stokes equations, Math. Comp. 39 (1982), 339–375.10.1090/S0025-5718-1982-0669634-0Search in Google Scholar
[2] L. Berselli, P. Fischer, T. Iliescu, and T. Özgökmen, Horizontal large eddy simulation of stratified mixing in a lock-exchange system, J. Sci. Comput. (2011).10.1007/s10915-011-9464-8Search in Google Scholar
[3] L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations, Math. Methods Appl. Sci. 38 (2015), 1437–1450.10.1002/mma.3160Search in Google Scholar
[4] V. P. Bongolan-Walsh, J. Duan, P. Fischer, T. Özgökmen, and T. Iliescu, Impact of boundary conditions on entrainment and transport in gravity currents, App. Math. Model. 31 (2007), 1338–1350.10.1016/j.apm.2006.03.033Search in Google Scholar
[5] A. L. Bowers and L. G. Rebholz, Increasing accuracy and eflciency in FE computations of the Leray-deconvolution model, Numer. Methods Part. Differ. Equ. 28 (2012), 720–736.10.1002/num.20653Search in Google Scholar
[6] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15, Springer, 2008.10.1007/978-0-387-75934-0Search in Google Scholar
[7] M. I. Cantero, S. Balachandar, and M. Garcia, High-resolution simulations of cylindrical density currents, J. Fluid Mech. 590 (2007), 437–469.10.1017/S0022112007008166Search in Google Scholar
[8] Y. S. Chang, X. Xu, T. M. Özgökmen, E. Chassignet, H. Peters, and P. Fischer, Comparison of gravity current mixing parameterizations and calibration using a high-resolution 3D nonhydrostatic spectral element model, Ocean Model. 10 (2005), 342–368.10.1016/j.ocemod.2004.11.002Search in Google Scholar
[9] J. Connors, Convergence analysis and computational testing of the finite element discretization of the Navier–Stokes alpha model, Numer. Methods Part. Differ. Equ. 26 (2010), 1328–1350.10.1002/num.20493Search in Google Scholar
[10] M. Germano, Differential filters of elliptic type, Physics Fluids (1958-1988)29 (1986), 1757–1758.10.1063/1.865650Search in Google Scholar
[11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, 1986.10.1007/978-3-642-61623-5Search in Google Scholar
[12] M. Gunzburger, Finite Elment Methods For Viscous Incompressible Flows. A Guide to Theory, Practice, and Algorithms, Academic Press, 1989.10.1016/B978-0-12-307350-1.50011-9Search in Google Scholar
[13] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), 251–265.10.1515/jnum-2012-0013Search in Google Scholar
[14] H. R. Hiester, M. D. Piggott, P. E. Farrell, and P. A. Allison, Assessment of spurious mixing in adaptive mesh simulations of the two-dimensional lock-exchange, Ocean Model. 73 (2014), 30–44.10.1016/j.ocemod.2013.10.003Search in Google Scholar
[15] M. Ilıcak, S. Legg, A. Adcroft, and R. Hallberg, Dynamics of a dense gravity current flowing over a corrugation, Ocean Model. 38 (2011), 71–84.10.1016/j.ocemod.2011.02.004Search in Google Scholar
[16] M. Ilıcak, T. M. Özgökmen, E. Özsoy, and P. Fischer, Non-hydrostatic modeling of exchange flows across complex geometries, Ocean Model. 29 (2009), 159–175.10.1016/j.ocemod.2009.04.002Search in Google Scholar
[17] M. Ilıcak, T. M. Özgökmen, H. Peters, H. Z. Baumert, and M. Iskandarani, Performance of two-equation turbulence closures in three-dimensional simulations of the Red Sea overflow, Ocean Model. 24 (2008), 122–139.10.1016/j.ocemod.2008.06.001Search in Google Scholar
[18] A. A. Ilyin, E. M. Lunasin, and E. S. Titi, A modified-Leray-α subgrid scale model of turbulence, Nonlinearity19 (2006), 879.10.1088/0951-7715/19/4/006Search in Google Scholar
[19] R. Ingram, A new linearly extrapolated Crank–Nicolson time-stepping scheme for the Navier–Stokes equations, Math. Comp. 82 (2013), 1953–1973.10.1090/S0025-5718-2013-02678-6Search in Google Scholar
[20] W. Layton, An Introduction to the Numerical Analysis of Viscous Incompressible Flows, SIAM, Philadelphia, 2008.10.1137/1.9780898718904Search in Google Scholar
[21] W. Layton, C. C. Manica, M. Neda, M. Olshanskii, and L. G. Rebholz, On the accuracy of the rotation form in simulations of the Navier–Stokes equations, J. Comput. Phys. 228 (2009), 3433–3447.10.1016/j.jcp.2009.01.027Search in Google Scholar
[22] W. Layton, C. C. Manica, M. Neda, and L. Rebholz, Numerical analysis and computational testing of a high accuracy Leraydeconvolution model of turbulence, Numer. Methods Part. Differ. Equ. 24 (2008), 555–582.10.1002/num.20281Search in Google Scholar
[23] W. Layton, C. C. Manica, M. Neda, and L. G. Rebholz, Numerical analysis and computational comparisons of the NS-alpha and NS-omega regularizations, Comput. Methods Appl. Mech. Engrg199 (2010), 916.10.1016/j.cma.2009.01.011Search in Google Scholar
[24] J.-G. Liu, C. Wang, and H. Johnston, A fourth order scheme for incompressible Boussinesq equations, J. Sci. Comput. 18 (2003), 253–285.10.1023/A:1021168924020Search in Google Scholar
[25] Z. Luo, Proper orthogonal decomposition-based reduced-order stabilized mixed finite volume element extrapolating model for the nonstationary incompressible Boussinesq equations, J. Math. Anal. Appl. 425 (2015), 259–280.10.1016/j.jmaa.2014.12.011Search in Google Scholar
[26] C. Manica, M. Neda, M. Olshanskii, L. Rebholz, and N. Wilson, On an eflcient finite element method for Navier–Stokes-ω with strong mass conservation, Comput. Methods Appl. Math. 11 (2011), 3–22.10.2478/cmam-2011-0001Search in Google Scholar
[27] C. C. Manica, M. Neda, M. Olshanskii, and L. G. Rebholz, Enabling numerical accuracy of Navier–Stokes-α through deconvolution and enhanced stability, ESAIM: Math. Model. Numer. Anal. 45 (2011), 277–307.10.1051/m2an/2010042Search in Google Scholar
[28] W. W. Miles and L. G. Rebholz, An enhanced-physics-based scheme for the NS-α turbulence model, Numer. Methods Part. Differ. Equ. 26 (2010), 1530–1555.10.1002/num.20509Search in Google Scholar
[29] T. M. Özgökmen and E. Chassignet, Dynamics of two-dimensional turbulent bottom gravity currents, J. Phys. Oceanogr. 32 (2002), 1460–1478.10.1175/1520-0485(2002)032<1460:DOTDTB>2.0.CO;2Search in Google Scholar
[30] T. M. Özgökmen, P. F. Fischer, J. Duan, and T. Iliescu, Entrainment in bottom gravity currents over complex topography from three-dimensional nonhydrostatic simulations, Geophys. Research Letters31 (2004).10.1029/2004GL020186Search in Google Scholar
[31] T. M. Özgökmen, P. F. Fischer, J. Duan, and T. Iliescu, Three-dimensional turbulent bottom density currents from a high-order nonhydrostatic spectral element model, J. Phys. Oceanogr. 34 (2004), 2006–2026.10.1175/1520-0485(2004)034<2006:TTBDCF>2.0.CO;2Search in Google Scholar
[32] T. M. Özgökmen, T. Iliescu, and P. F. Fischer, Large eddy simulation of stratified mixing in a three-dimensional lockexchange system, Ocean Modelling26 (2009), 134–155.10.1016/j.ocemod.2008.09.006Search in Google Scholar
[33] T. M. Özgökmen, T. Iliescu, and P. F. Fischer, Reynolds number dependence of mixing in a lock-exchange system from direct numerical and large eddy simulations, Ocean Model. 30 (2009), 190–206.10.1016/j.ocemod.2009.06.013Search in Google Scholar
[34] T. M. Özgökmen, W. E. Johns, H. Peters, and S. Matt, Turbulent mixing in the Red Sea outflow plume from a high-resolution nonhydrostatic model, J. Phys. Oceanogr. 33 (2003), 1846–1869.10.1175/2401.1Search in Google Scholar
[35] L. G. Rebholz, A family of new, high order NS-α models arising from helicity correction in Leray turbulence models, J. Math. Anal. Appl. 342 (2008), 246–254.10.1016/j.jmaa.2007.11.031Search in Google Scholar
[36] L. G. Rebholz and M. M. Sussman, On the high accuracy NS-alpha-deconvolution turbulence model, Math. Models Methods Appl. Sci. 20 (2010), 611–633.10.1142/S0218202510004362Search in Google Scholar
[37] O. San and J. Borggaard, Principal interval decomposition framework for POD reduced-order modeling of convective Boussinesq flows, Int. J. Numer. Methods Fluids (2015), 37–52.10.1002/fld.4006Search in Google Scholar
[38] S. Stolz and N. A. Adams, An approximate deconvolution procedure for large-eddy simulation, Physics Fluids11 (1999), 1699.10.1063/1.869867Search in Google Scholar
[39] S. Stolz, N. Adams, and L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Physics of Fluids13 (2001), 997–1015.10.1063/1.1350896Search in Google Scholar
© 2017 by Walter de Gruyter Berlin/Boston
