Abstract
Numerical solutions of the problem of steady incompressible viscous fluid flow in a plane channel with a backward-facing step have been obtained by the grid method. The fluid motion is described by the Navier–Stokes equations in velocity-pressure variables. The main computations were performed on a uniform 6001 × 301 grid. The control-volume method of the second order in space was used for the difference approximation of the original equations. The results were validated for the range of Reynolds numbers (100 ≤ Re ≤ 3000) by comparing them with the experimental and theoretical data found in the literature. The stability of the computational algorithm at high Reynolds numbers was achieved by using a fine difference grid (a small grid step). The study has been carried out for a short channel at Reynolds numbers from 1000 to 10 000 with a step of 1000. A nonstandard structure of the primary vortex behind the step—the presence of numerous centers of rotation both inside the vortex and in the near-wall region under it—has been revealed. The number of centers of rotation in the primary recirculation zone is shown to grow with increasing Reynolds number. The profiles of the coefficients of friction and hydrodynamic resistance to the flow as a function of Reynolds number have also been analyzed. The results obtained can be useful for comparison and validation of the solutions of problems of such a type.
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References
Gartling, D.K., A test problem for outflow boundary conditions-flow over a backward-facing step, Int. J. Numer. Meth. Fluids, 1990, vol. 11, no. 7, pp. 953–967. https://doi.org/10.1002/fld.1650110704
Rogers, S.E. and Kwak, D., An upwind differencing scheme for the incompressible Navier–Stokes equations, Appl. Numer. Math., 1991, vol. 8, no. 1, pp. 43–64. https://doi.org/10.1016/0168-9274(91)90097-J
Durst, F., Pereira, J.C.F., and Tropea, C., The plane symmetric sudden-expansion flow at low Reynolds numbers, J. Fluid Mech., 1993, vol. 248, pp. 567–581. https://doi.org/10.1017/S0022112093000916
Valencia, A. and Hinojosa, L., Numerical solutions of pulsating flow and heat transfer characteristics in a channel with a backward-facing step, Heat Mass Transfer, 1997, vol. 32, no. 3, pp. 143–148. https://doi.org/10.1007/s002310050104
Batenko, S.R. and Terekhov, V.I., Effect of dynamic prehistory on aerodynamics of a laminar separated flow in a channel behind a rectangular backward-facing step, J. Appl. Mech. Tech. Phys., 2002, vol. 43, no. 6, pp. 854–860. https://doi.org/10.1023/A:1020712520195
Biswas, G., Breuer, M., and Durst, F., Backward-facing step flows for various expansion ratios at low and moderate Reynolds numbers, J. Fluids Eng., 2004, vol. 126, pp. 362–374. https://doi.org/10.1115/1.1760532
Batenko, S.R. and Terekhov, V.I., Friction and heat transfer in a laminar separated flow behind a rectangular step with porous injection or suction, J. Appl. Mech. Tech. Phys., 2006, vol. 47, no. 1, pp. 12–21. https://doi.org/10.1007/s10808-006-0002-7
Abu-Nada, E., Al-Sarkhi, A., Akash, B., and Al-Hinti, I., Heat transfer and fluid flow characteristics of separated flows encountered in a backward-facing step under the effect of suction and blowing, J. Heat Transfer, 2007, vol. 129, no. 11, pp. 1517–1528. https://doi.org/10.1115/1.2759973
Bubenchikov, A.M., Firsov, D.K., and Kotovshchikova, M.A., Numerical solution of 2D viscous fluid dynamics problems using finite volume method (FVM) on triangular grid, Mat. Model., 2007, vol. 19, no. 6, pp. 71–86.
Bruyatskii, E.V. and Kostin, A.G., Direct numerical simulation of flow in a plane channel with sudden expansion based on the Navier–Stokes equations, Prikl. Gidromekh., 2010, vol. 12, no. 1, pp. 11–27.
Poponin, V.S., Kosheutov, A.V., Grigor’ev, V.P., and Mel’nikova, V.N., A method of spectral elements for solving plane problems of viscous fluid dynamics on unshifted unstructured grids, Izv. Tomsk. Politekh. Univ., 2010, vol. 317, no. 2, pp. 31–36.
Fomin, A.A. and Fomina, L.N., Numerical simulation of the viscous incompressible fluid flow and heat transfer in a plane channel with backward-facing step, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, vol. 25, no. 2, pp. 280–294. https://doi.org/10.20537/vm150212
Aleksin, V.A. and Manaenkova, T.A., Calculation of incompressible separated fluid flows, taking into account heat transfer, Izv. Mosk. Industr. Univ., 2011, no. 4 (24), pp. 28–38.
Borzenko, E.I. and Khegai, E.I., Numerical simulation of the steady-state Herschel–Bulkley fluid flow in a channel with sudden expansion, Vestn. Tomsk Univ., Mat. Mekh., 2016, no. 1 (39), pp. 68–81. https://doi.org/10.17223/19988621/39/8
Armaly, B.F., Durst, F., Pereira, J.C.F., and Schönung, B., Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech., 1983, vol. 127, pp. 473–496. https://doi.org/10.1017/S0022112083002839
Cruchaga, M.A., A study of the backward-facing step problem using a generalized streamline formulation, Commun. Numer. Meth. Eng., 1998, vol. 14, no. 8, pp. 697–708. https://doi.org/10.1002/(SICI)1099-0887(199808)14:8%3c697::AID-CNM155%3e3.0.CO;2-0
Lee, T. and Mateescu, D., Experimental and numerical investigation of 2D backward-facing step flow, J. Fluid. Struct., 1998, vol. 12, no. 6, pp. 703–716. https://doi.org/10.1006/jfls.1998.0166
Chiang, T.P., Sheu, T.W.H., and Fang, C.C., Numerical investigation of vortical evolution in a backward-facing step expansion flow, Appl. Math. Model., 1999, vol. 23, no. 12, pp. 915–932. https://doi.org/10.1016/S0307-904X(99)00019-0
Mouza, A.A., Pantzali, M.N., Paras, S.V., and Tihon, J., Experimental and numerical study of backward-facing step flow, in Proceedings of the 5th National Chemical Engineering Conference, Thessaloniki, Greece, 2005. https://doi.org/philon.cheng.auth.gr/philon/site/sdocs/paper%205PESXM_Tihon.pdf.
Erturk, E., Numerical solutions of 2D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions, Comput. Fluids, 2008, vol. 37, no. 6, pp. 633–655. https://doi.org/10.1016/j.compfluid.2007.09.003
Tihon, J., Pénkavová, V., Havlica, J., and Šimčík, M., The transitional backward-facing step flow in a water channel with variable expansion geometry, Exp. Therm. Fluid Sci., 2012, vol. 40, pp. 112–125. https://doi.org/10.1016/j.expthermflusci.2012.02.006
Saleel, C.A., Shaija, A., and Jayaraj, S., On simulation of backward facing step flow using immersed boundary method, Am. J. Fluid Dyn., 2013, vol. 3, no. 2, pp. 9–19. https://doi.org/10.5923/j.ajfd.20130302.01
Teruel, F.E. and Fogliatto, E., Numerical simulations of flow, heat transfer and conjugate heat transfer in the backward-facing step geometry, Mec. Comput., 2013, vol. 32, no. 39, pp. 3265–3278. https://doi.org/www.cimec.org.ar/ojs/index.php/mc/article/viewFile/4551/4480.
Moffatt, H.K., Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 1964, vol. 18, no. 1, pp. 1–18. https://doi.org/10.1017/S0022112064000015
Sinha, S.N., Gupta, A.K., and Oberai, M., Laminar separating flow over backsteps and cavities, Part I: Backsteps, AIAA J., 1981, vol. 19, no. 12, pp. 1527–1530. https://doi.org/10.2514/3.7885
Erturk, E., Discussions on driven cavity flow, Int. J. Numer. Meth. Fluids, 2009, vol. 60, no. 3, pp. 275–294. https://doi.org/10.1002/fld.1887
Fomin, A.A. and Fomina, L.N., On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers, Mat. Model. Chisl. Metody, 2015, no. 4 (8), pp. 92–109. https://doi.org/mmcm.bmstu.ru/articles/65/.
Fomin, A.A. and Fomina, L.N., Numerical simulation of viscous incompressible fluid in a short plane channel with backward-facing step, Mat. Model., 2016, vol. 28, no. 5, pp. 32–46.
Roache, P.J., Computational Fluid Dynamics, Albuquerque, NM: Hermosa, 1976.
Kosma, Z., The method of lines for the computation of incompressible viscous flows, Proc. Appl. Math. Mech., 2009, vol. 9, no. 1, pp. 483–484. https://doi.org/10.1002/pamm.200910214
Loitsyanskii, L.G., Mechanics of Liquids and Gases, New York: Begell House, 1995.
Belotserkovskii, O.M., Gushchin, V.A., and Shchennikov, V.V., Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid, USSR Comput. Math. Math. Phys., 1975, vol. 15, no. 1, pp. 190–200. https://doi.org/10.1016/0041-5553(75)90146-9
Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Washington: Hemisphere, 1980.
Fomin, A.A. and Fomina, L.N., Acceleration of the line-by-line recurrent method in Krylov subspaces, Vestn. Tomsk Univ., Math. Mekh., 2011, no. 2, pp. 45–54.
Lashkin, S.V., Kozelkov, A.S., Yalozo, A.V., Gerasimov, V.Yu., and Zelensky, D.K., Efficiency analysis of parallel implementation of simple algorithm on multi-processor computers, Vychisl. Mekh. Splosh. Sred, 2016, vol. 9, no. 3, pp. 298–315. https://doi.org/10.7242/1999-6691/2016.9.3.25
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Original Russian Text © A.A. Fomin, L.N. Fomina, 2018, published in Vychislitel’naya Mekhanika Sploshnykh Sred, 2017, Vol. 10, No. 3, pp. 260–275.
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Fomin, A.A., Fomina, L.N. Numerical Solution of the Problem of Incompressible Fluid Flow in a Plane Channel with a Backward-Facing Step at High Reynolds Numbers. J Appl Mech Tech Phy 59, 1211–1226 (2018). https://doi.org/10.1134/S0021894418070052
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DOI: https://doi.org/10.1134/S0021894418070052