The Effect of Methods of Calculating Velocity on Cell Faces in Sub-critical Open Channel Flow Simulation

Document Type : Research Article


1 Master of Science, Department of Civil Engineering, Yazd University, Yazd, Iran

2 Assistant Professor, Department of Civil Engineering, Yazd University, Yazd, Iran

3 Professor, Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran


A 2D vertical numerical model for solving unsteady Navier-Stokes equations with capability of calculating dynamic pressures in free surface flows is used in the present study. In this model, the projection method is applied to solve the equations in non-orthogonal curvilinear coordinate system with collocated grid arrangement. The velocities (fluxes) on cell faces are calculated using a “linear interpolation” method and some “momentum interpolation” methods. These methods are evaluated using some test cases including flow in a trench, flow over a sill and gradually varied flow (M2 profile). The results show that for mild free surface slope, all the momentum interpolation methods implemented in the model have the same accuracy. Also, the linear interpolation of velocity results in an acceptable accuracy. In other words, the checker board pressure fluctuation does not occur in flow domain. For sharp free surface slope, the linear interpolation of velocity causes nonphysical pressure fluctuation and results in divergence. Therefore, the momentum interpolation methods are inevitable. Comparing the required computational time of the methods shows that momentum interpolation methods needs less time in comparison with velocity linear interpolation method. The tests show that evaluating the cell face velocities are more important in longitudinal direction than the vertical ones. Therefore, if the momentum interpolation method is used for east and west faces of cells and the linear interpolation of velocity is used for top and bottom faces, the model runs in less time without reducing the accuracy.


دهقان، ب. ز.؛ هادیان، م. ر.و زراتی، ا. ر. (1390). "توسعه یک مدل عددی برای پیش بینی فشارهای دینامیک در جریان کانال‌های باز با شبکه منحنی‌الخط غیرمتعامد"، مجله هیدرولیک، دوره 6، شماره 4، ص.ص. 43-58.
هادیان، م. ر. و زراتی، ا. ر. (1388). مدل‌های عددی آب‌های کم‌عمق و کاربرد آنها در مهندسی رودخانه و سواحل (معادلات حاکم و روش‌های حل)، انتشارات دانشگاه صنعتی امیرکبیر (پلی‌تکنیک تهران).
Ahmadi, A., Badiei, P., and Namin, M. M. (2007). "An implicit two-dimensional non-hydrostatic model for free-surface flows." International Journal for Numerical Methods in Fluids, 54(9), 1055-1074.
Alfrink, B. J., and van Rijn, L. C. (1983). Two-equation turbulence model for flow in trenches, Delft Hydraulics Laboratory.
Blom, P., and Booij, R. (1995). "Turbulent free-surface flow over sills." Journal ofHydraulic Research, 33(5), 663-682.
Casulli, V., and Stelling, G. S. (1998). "Numerical simulation of 3D quasi-hydrostatic, free-surface flows." Journal of Hydraulic Engineering, 124(7), 678-686.
Casulli, V. (1999). "A semi-implicit finite difference method for non-hydrostatic, free-surface flows." International Journal for Numerical Methods in Fluids, 30(4), 425-440.
Christian, C., and Corney, P. (2004). "Three dimensional model of flow over a shallow trench." Journal of Hydraulic Research, 42(1), 71-80.
Choi, S. K. (1999). "Note on the use of momentum interpolation method for unsteady flows." Numerical Heat Transfer: Part A: Applications, 36(5), 545-550.
Choi, S. K., Nam, H. Y., and Cho, M. (1994). "Systematic comparison of finite-volume calculation methods with staggered and non-staggered grid arrangements." Numerical Heat Transfer, Part B Fundamentals, 25(2), 205-221.
Cubero, A., and Fueyo, N. (2007). "A compact momentum interpolation procedure for unsteady flows and relaxation." Numerical Heat Transfer, Part B: Fundamentals, 52(6), 507-529.
Ferziger, J. H., and Peric, M. (1997). Computational methods for fluid dynamics., Pub.: Springer.
Hadian, M., Zarrati, A., and Eftekhari, M. (2005). "Development of an implicit numerical model for calculation of sub and super-critical flows." International Journal of Engineering, 18(1), 1.
Hoffman, G. J. C. M. (1992). "Two-dimensional mathematical modelling of local-scour holes." Communcations on Hydraulic and Geotecnical Engineering, Faculty of Civil Engineering Delft University of Technology.
Kobayashi, M., and Pereira, J. (1991). "Numerical comparison of momentum interpolation methods and pressure-velocity algorithms using non-staggered grids." Communications in Applied Numerical Methods, 7(3), 173-186.
Koçyigit, M. B., Falconer, R. A., and Lin, B. (2002). "Three-dimensional numerical modelling of free surface flows with non-hydrostatic pressure." International Journal for Numerical Methods in Fluids, 40(9), 1145-1162.
Lee, J., Teubner, M. D., Nixon, J., and Gill, P. M. (2006). "A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows." International Journal for Numerical Methods in Fluids, 50(6), 649-672.
Li, B., and Fleming, C. A. (2001). "Three-dimensional model of Navier-Stokes equations for water waves." Journal of Waterway, Port, Coastal, and Ocean Engineering, 127 (1), 16-25.
Lien, F., and Leschziner, M. (1994). "A general non-orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, Part 1: Computational implementation." Computer Methods in Applied Mechanics and Engineering, 114(1), 123-148.
Majumdar, S. (1988). "Role of underrelaxation in momentum interpolation for calculation of flow with non-staggered grids." Numerical Heat Transfer, 13(1), 125-132.
Miller, T., and Schmidt, F. (1988). "Use of a pressure-weighted interpolation method for the solution of the incompressible Navier-Stokes equations on a non-staggered grid system." Numerical Heat Transfer, Part A: Applications, 14(2), 213-233.
Olsen, N. R. B. (2000). "CFD algorithms for hydraulic engineering." Class notes , ( nilsol/cfd/ cfdalgo .pdf).
Papageorgakopoulos, J., Arampatzis, G., Assimacopoulos, D., and Markatos, N. (2000). "Enhancement of the momentum interpolation method on non-staggered grids." International Journal for Numerical Methods in Fluids, 33(1), 1-22.
Rhie, C., and Chow, W. (1983). "Numerical study of the turbulent flow past an airfoil with trailing edge separation." AIAA journal, 21(11), 1525-1532.
Shyy, W., Udaykumar, H. S., Roa, M. M., and Smith, R. W. (1996). Computational fluid dynamics with moving boundaries, Taylor & Francis Publisher.
Stansby, P. K., and Zhou, J. G. (1998). "Shallow‐water flow solver with non‐hydrostatic
pressure: 2D vertical plane problems." International Journal for Numerical Methods in Fluids, 28(3), 541-563.
Versteeg, H. K., and Malalasekara, W. (1995). An introduction to computational fluid dynamics, the finite volume method., Longman Book Publisher.
Wu, C. H., and Yuan, H. (2007). "Efficient non-hydrostatic modelling of surface waves interacting with structures." Applied Mathematical Modelling, 31(4), 687-699.
Xia, C., and Jin, Y. C. (2006). "Multilayer averaged and moment equations for one-dimensional open-channel flows." Journal of Hydraulic Engineering, 132(8), 839-849.
Xu, H., and Zhang, C. (1998). "Study of the effect of the non-orthogonality for non-staggered grids—the theory." International Journal for Numerical Methods in Fluids, 28(9), 1265-1280.
Yu, B., Kawaguchi, Y., Tao, W. Q., and Ozoe, H. (2002). "Checkerboard pressure predictions due to the underrelaxation factor and time step size for a non-staggered grid with momentum interpolation method." Numerical Heat Transfer: Part B: Fundamentals, 41(1), 85-94.
Yu, B., Tao, W. Q., Wei, J. J., Kawaguchi, Y., Tagawa, T., and Ozoe, H. (2002). "Discussion on momentum interpolation method for collocated grids of incompressible flow." Numerical Heat Transfer: Part B: Fundamentals, 42(2), 141-166.
Yuan, H., and Wu, C. H. (2004 (a)). "A two-dimensional vertical non-hydrostatic σ model with an implicit method for free-surface flows." International Journal for Numerical Methods in Fluids, 44(8), 811-835.
Zang, Y., Street, R. L., and Koseff, J. R. (1994). "A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates." Journal of Computational Physics, 114(1), 18-33
  • Receive Date: 20 September 2015
  • Revise Date: 10 January 2017
  • Accept Date: 13 February 2017
  • First Publish Date: 13 February 2017