# American Institute of Mathematical Sciences

March  2021, 14(3): 785-801. doi: 10.3934/dcdss.2020333

## Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows

 1 Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Karlovo Náměstí 13,121 35 Prague 2, Czech Republic 2 Institute of Mathematics, Czech Academy of Sciences, Žitná 25,115 67 Prague 1, Czech Republic 3 Mediterranean Institute of Oceanography - MIO, UM 110 USTV - AMU - CNRS/INSU 7294 - IRD 235, Université de Toulon, BP 20132 F-83957 La Garde cedex, France 4 Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,186 75 Prague 8, Czech Republic

* Corresponding author: Petr Knobloch

Received  January 2019 Revised  November 2019 Published  March 2021 Early access  April 2020

The paper presents a numerical study of the efficiency of the newly proposed far-field boundary simulations of wall-bounded, stably stratified flows. The comparison of numerical solutions obtained on large and truncated computational domain demonstrates how the solution is affected by the adopted far-field conditions. The mathematical model is based on Boussinesq approximation for stably stratified viscous variable density incompressible fluid. The three-dimensional numerical simulations of the steady flow over an isolated hill were performed using a high-resolution compact finite difference code, with artificial compressibility method used for pressure computation. The mutual comparison of the full domain reference solution and the truncated domain solution is provided and the influence of the newly proposed far-field boundary condition is discussed.

Citation: Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333
##### References:
 [1] T. Bodnár and L. Beneš, On some high resolution schemes for stably stratified fluid flows, Finite Volumes for Complex Applications VI, Problems & Perspectives, Volume 1, 2, Springer Proc. Math., Springer, Heidelberg, 4 (2011), 145-153. doi: 10.1007/978-3-642-20671-9_16. [2] T. Bodnár, L. Beneš, P. Fraunié and K. Kozel, Application of compact finite-difference schemes to simulations of stably stratified fluid flows, Applied Mathematics and Computation, 219 (2012), 3336-3353.  doi: 10.1016/j.amc.2011.08.058. [3] T. Bodnár and P. Fraunié, On the boundary conditions in the numerical simulation of stably stratified fluids flows, Topical Problems of Fluid Mechanics 2017, Institute of Thermomechanics CAS, Prague, (2017), 45-52. [4] T. Bodnár and P. Fraunié, Artificial far-field pressure boundary conditions for wall-bounded stratified flows, Topical Problems of Fluid Mechanics 2018, Institute of Thermomechanics CAS, Prague, (2018), 7-14. [5] T. Bodnár, P. Fraunié and H. Řezníček, Numerical tests of far-field boundary conditions for stably stratified stratified flows, Topical Problems of Fluid Mechanics 2019, Institute of Thermomechanics CAS, Prague, (2019), 1-8. [6] M. Braack and P. Mucha, Directional do-nothing condition for the Navier-Stokes equations, Journal of Computational Mathematics, 32 (2014), 507-521.  doi: 10.4208/jcm.1405-m4347. [7] L. Ding, R. J. Calhoun and R. L. Street, Numerical simulation of strongly stratified flow over a three-dimensional hill, Boundary-Layer Meteorology, 107 (2003), 81-114.  doi: 10.1023/A:1021578315844. [8] J. G. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 22 (1996), 325-352.  doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. [9] J. C. R. Hunt and W. H. Snyder, Experiments on stably and neutrally stratified flow over a model three-dimensional hill, Journal of Fluid Mechanics, 96 (1980), 671-704.  doi: 10.1017/S0022112080002303. [10] P. Marchesiello, J. C. McWilliams and A. Shchepetkin, Open boundary conditions for long-term integration of regional oceanic models, Ocean Modelling, 3 (2001), 1-20.  doi: 10.1016/S1463-5003(00)00013-5. [11] I. Orlanski, A simple boundary condition for unbounded hyperbolic flows, Journal of Computational Physics, 21 (1976), 251-269.  doi: 10.1016/0021-9991(76)90023-1.

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##### References:
 [1] T. Bodnár and L. Beneš, On some high resolution schemes for stably stratified fluid flows, Finite Volumes for Complex Applications VI, Problems & Perspectives, Volume 1, 2, Springer Proc. Math., Springer, Heidelberg, 4 (2011), 145-153. doi: 10.1007/978-3-642-20671-9_16. [2] T. Bodnár, L. Beneš, P. Fraunié and K. Kozel, Application of compact finite-difference schemes to simulations of stably stratified fluid flows, Applied Mathematics and Computation, 219 (2012), 3336-3353.  doi: 10.1016/j.amc.2011.08.058. [3] T. Bodnár and P. Fraunié, On the boundary conditions in the numerical simulation of stably stratified fluids flows, Topical Problems of Fluid Mechanics 2017, Institute of Thermomechanics CAS, Prague, (2017), 45-52. [4] T. Bodnár and P. Fraunié, Artificial far-field pressure boundary conditions for wall-bounded stratified flows, Topical Problems of Fluid Mechanics 2018, Institute of Thermomechanics CAS, Prague, (2018), 7-14. [5] T. Bodnár, P. Fraunié and H. Řezníček, Numerical tests of far-field boundary conditions for stably stratified stratified flows, Topical Problems of Fluid Mechanics 2019, Institute of Thermomechanics CAS, Prague, (2019), 1-8. [6] M. Braack and P. Mucha, Directional do-nothing condition for the Navier-Stokes equations, Journal of Computational Mathematics, 32 (2014), 507-521.  doi: 10.4208/jcm.1405-m4347. [7] L. Ding, R. J. Calhoun and R. L. Street, Numerical simulation of strongly stratified flow over a three-dimensional hill, Boundary-Layer Meteorology, 107 (2003), 81-114.  doi: 10.1023/A:1021578315844. [8] J. G. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 22 (1996), 325-352.  doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. [9] J. C. R. Hunt and W. H. Snyder, Experiments on stably and neutrally stratified flow over a model three-dimensional hill, Journal of Fluid Mechanics, 96 (1980), 671-704.  doi: 10.1017/S0022112080002303. [10] P. Marchesiello, J. C. McWilliams and A. Shchepetkin, Open boundary conditions for long-term integration of regional oceanic models, Ocean Modelling, 3 (2001), 1-20.  doi: 10.1016/S1463-5003(00)00013-5. [11] I. Orlanski, A simple boundary condition for unbounded hyperbolic flows, Journal of Computational Physics, 21 (1976), 251-269.  doi: 10.1016/0021-9991(76)90023-1.
Vertical velocity contours and flow streamlines in the plane of symmetry
Vertical velocity isosurfaces
Vertical velocity contours in the plane of symmetry - truncated solution
Vertical velocity contours in the plane of symmetry - truncated domain - $\frac{\partial p}{\partial n} = 0$
Contours of the transversal velocity component $v$ and flow streamlines
Contours of the vertical velocity component $w$ and flow streamlines
Isosurfaces of the transversal velocity component $v$
Isosurfaces of the vertical velocity component $w$
Computational domain and its extension
Inlet velocity profile setup
Contours of the transversal velocity component $v$ - nondimensionalized $\widetilde{v} = v/U_{*}$
Contours of the vertical velocity component $w$ - nondimensionalized $\widetilde{w} = w/U_{*}$
Isosurfaces of the transversal velocity component $v$ - nondimensionalized $\widetilde{v} = v/U_{*}$
Isosurfaces of the vertical velocity component $w$ - nondimensionalized $\widetilde{w} = w/U_{*}$
Pressure contours in the plane of symmetry
Longitudinal velocity contours in the plane of symmetry
Vertical velocity contours in the plane of symmetry
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