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An arbitrary Lagrangian-Eulerian $$\sigma$$ (ALES) model with non-hydrostatic pressure for shallow water flows. (English) Zbl 0967.76065
From the summary: We develop an arbitrary Lagrangian-Eulerian model in the $$\sigma$$ coordinate system (ALES) for shallow water flows, based on the unsteady Reynolds averaged Navier-Stokes equations. Unlike the conventional $$\sigma$$ coordinate system, non-hydrostatic pressure is incorporated together with the effect of moving free surface, giving a general scheme. The standard $$k-\varepsilon$$ turbulence model is used to calculate the eddy viscosity. Wave flows over bars are appropriate test cases for which experimental data are available, and comparisons are favourable. The model is also applied to the problem of wave and wave/current flows over a trench.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 76D33 Waves for incompressible viscous fluids
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##### References:
 [1] Lamb, H., Hydrodynamics, (1945), Dover New York · JFM 26.0868.02 [2] C.W. Hirt, An arbitrary Lagrangian-Eulerian computing technique, In: Proceedings of the second International Conference on Numerical Methods in Fluid Dynamics, Berkeley, USA, 1970 · Zbl 0271.76021 [3] Hirt, C.W.; Amsden, A.A.; Cook, J.L., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. comp. phys., 14, 227-253, (1974) · Zbl 0292.76018 [4] Ramaswamy, B., Numerical simulation of unsteady viscous free surface flow, J. comp. phys., 90, 396-430, (1990) · Zbl 0701.76036 [5] Kennedy, J.M.; Belytschko, T.B., Theory and application of a finite element method for arbitrary Lagrangian-Eulerian fluids and structures, Nucl. eng. design, 68, 129-146, (1981) [6] Ghosh, S.; Kikuchi, N., An arbitrary Lagrangian-Eulerian finite element method for large deformation analysis of elastic-viscoeplastic solids, Comput. meth. appl. mech. eng., 86, 127-188, (1991) · Zbl 0825.73687 [7] Chan, R.K.-C., A generalized arbitrary Lagrangian-Eulerian method for incompressible flows with sharp interfaces, J. comp. phys., 17, 311-331, (1975) · Zbl 0301.76009 [8] Pracht, W.E., Calculating three-dimensional fluid flows at all speeds with an eulerian-Lagrangian computing mesh, J. comp. phys., 17, 132-159, (1975) · Zbl 0294.76016 [9] Demirdžić, I.; Perić, M., Finite volume method for prediction of fluid flow in arbitrary shaped domains with moving boundaries, Int. J. num. meth. fluids, 10, 771-790, (1990) · Zbl 0697.76038 [10] Thé, J.L.; Raithby, G.D.; Stubley, G.D., Surface-adaptive finite volume method for solving free surface flows, Num. heat trans., part B, 26, 367-380, (1994) [11] Kelkar, K.M.; Patankar, S.V., Numerical method for the prediction of free surface flows in domains with moving boundaries, Num. heat trans, 31, 387-399, (1997) [12] Nomura, T.; Hughes, T.J.R., An arbitrary Lagrangian-Eulerian finite-element method for interaction of fluid and a rigid body, Comput. meth. appl. mech. eng. part B, 95, 115-138, (1992) · Zbl 0756.76047 [13] Navti, S.E.; Ravindran, K.; Taylor, C.; Lewis, R.W., Finite element modelling of surface tension effects using a Lagrangian-Eulerian kinematic description, Comput. meth. appl. mech. eng., 147, 41-60, (1997) · Zbl 0901.76034 [14] Phillips, N.A., A coordinate system having some special advantages for numerical forecasting, J. meteorol., 14, 184-185, (1957) [15] Haney, R.L., On the pressure gradient force over steep topography in sigma coordinate Ocean models, J. comput. phys., 21, 610-619, (1991) [16] Szabo, P.; Hassager, O., Simulation of free surfaces in 3-d with the arbitrary Lagrange-Euler method, Int. J. num. meth. eng., 38, 717-734, (1995) · Zbl 0823.76048 [17] W. Rodi, Turbulence models and their applications in hydraulics, 3rd ed., 1993 [18] Stelling, G.S.; van Kester, J.A.T.M., On the approximation of horizontal gradients in sigma co-ordinates for bathymetry with steep bottom slopes, Int. J. num. meth. fluids, 18, 915-935, (1994) · Zbl 0807.76062 [19] Stansby, P.K.; Zhou, J.G., Shallow-water flow solver with non-hydrostatic pressure: 2D vertical plane problems, Int. J. num. meth. fluids, 28, 541-563, (1998) · Zbl 0931.76064 [20] Casulli, V.; Cattani, E., Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. math. applic., 27, 99-112, (1994) · Zbl 0796.76052 [21] Stansby, P.K., Semi-implicit finite volume shallow-water flow and solute transport solver with k−ϵ turbulence model, Int. J. num. meth. fluids, 25, 285-313, (1997) · Zbl 0947.76059 [22] Rahman, M.M.; Faghri, A.; Hankey, W.L., Computation of turbulent flow in a thin liquid layer of fluid involving a hydraulic jump, J. fluids eng., 113, 411-418, (1991) · Zbl 0728.76033 [23] S. Beji, T. Ohyama, J.A. Battjes, K. Nadaoka, Transformation of nonbreaking waves over a bar, In: Coastal Engineering 1992, Proceedings of the 23rd International Conference vol. 1, Venice, Italy, 1992 pp. 51-61 [24] Ohyama, T.; Kioka, W.; Tada, A., Applicability of numerical models to nonlinear dispersive waves, Coastal eng., 24, 297-313, (1995) [25] Alfrink, B.J.; van Rijn, L.C., Two-equation turbulence model for flow in trench, J hydraulic engrg ASCE, 109, 941-958, (1983) [26] Basara, B.; Younis, B.A., Predictions of turbulent flows in dredged trenches, Journal of hydraulic research, 33, 6, 813-824, (1995)
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