Numerical Schemes Comparison in Solving the Problem of Laminar Flow in the Suddenly Expanding Channel

Authors: Madaliev M.E. Published: 04.03.2023
Published in issue: #1(106)/2023  
DOI: 10.18698/1812-3368-2023-1-4-22

Category: Mathematics and Mechanics | Chapter: Computational Mathematics  
Keywords: flat channel with reverse step, separated flow, control volume method, Navier --- Stokes equations


The paper studies fluid flow in a two-dimensional channel with sudden expansion (x/h = 2). Calculations were made for the laminar flow mode based on numerical integration of the nonstationary Navier --- Stokes equations. Various flow characteristics were determined at Re = 100--800. Results were obtained for the longitudinal speed profiles in various channel sections and lengths of the primary and secondary vortices at various values of the Reynolds number after the step. The friction coefficient distribution on the channel lower side along its length is presented for various values of the Reynolds number. For the difference approximation of the initial equations, the control volume method was applied; relationship between speed and pressure was found using the SIMPLE procedure. For numerical solution of the problem, the following QUICK schemes were introduced: McCormack's second-order accuracy, Warming --- Cutler --- Lomax's third-order accuracy and Abarbanel --- Gotlieb --- Turkel's fourth-order accuracy. To confirm correctness, numerical results were compared with the experimental data taken from the literature sources

Please cite this article in English as:

Madaliev M.E. Numerical schemes comparison in solving the problem of laminar flow in the suddenly expanding channel. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2023, no. 1 (106), pp. 4--22 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2023-1-4-22


[1] Blasius Н. Laminare Stromung in Kanalen Wechselnder Breite. Zeitschrift fur Math. und Phys., 1910, vol. 58, no. 10, pp. 225--233.

[2] Honji H. The starting flow down a step. J. Fluid Mech., 1975, vol. 69, no. 2, pp. 229--240. DOI: https://doi.org/10.1017/S0022112075001413

[3] Sinkha S.P., Gupta A.K., Oberay M.M. Laminar separated flotation of steps and caverns. P. 1. Flow behind step. Raketnaya tekhnika i kosmonavtika, 1981, vol. 19, no. 12, pp. 33--37 (in Russ.).

[4] Armaly В.F., Durst F., Pereira J.C.F., et al. Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mech., 1983, vol. 127, pp. 473--496. DOI: https://doi.org/10.1017/S0022112083002839

[5] Chang P.K. Separation of flow. Elsevier, 1970.

[6] Gogish L.V., Stepanov G.Yu. Turbulentnye otryvnye techeniya [Turbulent separation of flow]. Moscow, Nauka Publ., 1979.

[7] Le H., Moin P., Kim J. Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech., 1997, vol. 330, pp. 349--374. DOI: https://doi.org/10.1017/S0022112096003941

[8] Durst F., Melling A., Whitelow J.H. Low Reynolds number flow over a plane symmetric sudden expansion. J. Fluid Mech., 1974, vol. 64, iss. 1, pp. 111--118. DOI: https://doi.org/10.1017/S0022112074002035

[9] Cherdron W., Durst F., Whitelow J.H. Asymmetric flows and instabilities in symmetric ducts with sudden expansions. J. Fluid Mech., 1978, vol. 84, iss. 1, pp. 13--31. DOI: https://doi.org/10.1017/S0022112078000026

[10] Macadno E.O., Hung T.-K. Computational and experimental study of a captive annular eddy. J. Fluid Mech., 1967, vol. 28, iss. 1, pp. 43--64. DOI: https://doi.org/10.1017/S0022112067001892

[11] Kumar A., Yajnik K.S. Internal separated flows at large Reynolds number. J. Fluid Mech., 1980, vol. 97, iss. 1, pp. 27--51. DOI: https://doi.org/10.1017/S0022112080002418

[12] Plotkin A. Calculations by the spectral method of some separated laminar flows in channels. Aerokosmicheskaya tekhnika, 1983, no. 7, pp. 75--85 (in Russ.).

[13] Acrivos A., Schrader M.L. Steady flow in a sudden expansion at high Reynolds numbers. Phys. Fluids, 1982, vol. 25, iss. 6, pp. 923--930. DOI: https://doi.org/10.1063/1.863844

[14] Kuon O., Pletcher R., Lyuis Dzh. Calculation of flows with sudden expansion using the boundary layer equations. Teor. osnovy inzh. rasch., 1984, vol. 106, no. 3, pp. 116--123 (in Russ.).

[15] Pletcher L. Limits of applicability of the boundary layer equations for calculating laminar flows with symmetric sudden expansion. Teor. osnovy inzh. rasch., 1986, no. 2, pp. 284--294 (in Russ.).

[16] Malikov Z.M., Madaliev M.E. New two-fluid turbulence model based numerical simulation of flow in a flat suddenly expanding channel. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2021, no. 4 (97), pp. 24--39 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2021-4-24-39

[17] Lee Y.S., Smith L.C. Analysis of power-law viscous materials using complex stream, potential and stress functions. In: Encyclopedia of Fluid Mechanics. Vol. 1. Flow Phenomena and Measurement. Gulf, 1986, pp. 1105--1154.

[18] Roache P.J. Computational fluid dynamics. Hermosa, 1972.

[19] Taylor T.D., Ndefo E. Computation of viscous flow in a channel by the method of splitting. In: Holt M. (eds). Proc. of the Second Int. Conf. on Num. Methods in Fluid Dynamics. Lecture Notes in Physics, vol. 8. Berlin, Heidelberg, Springer, 1971, pp. 356--364. DOI: https://doi.org/10.1007/3-540-05407-3_51

[20] Durst F., Peireira J.C.F. Time-dependent laminar backward-facing step flow in a two-dimensional duct. J. Fluids Eng., 1988, vol. 110, iss. 3, pp. 289--296. DOI: https://doi.org/10.1115/1.3243547

[21] Alleborn N., Nandakumar K., Raszillier H., et al. Further contributions on the two-dimensional flow in a sudden expansion. J. Fluid Mech., 1997, vol. 330, pp. 169--188. DOI: https://doi.org/10.1017/S0022112096003382

[22] Brandt A., Jr. Dendy J.E., Ruppel H. The multigrid method for semi-implicit hydrodynamic codes. J. Comput. Phys., 1980, vol. 34, iss. 3, pp. 348--370. DOI: https://doi.org/10.1016/0021-9991(80)90094-7

[23] Hackbusch W. Multigrid methods for applications. Springer, 1985.

[24] Lange C.F., Schafer M., Durst F. Local block refinement with a multigrid flow solver. Int. J. Numer. Methods Fluids, 2002, vol. 38, iss. 1, pp. 21--41. DOI: https://doi.org/10.1002/fld.202

[25] Kim J., Moin P. Application of a fractional-step method to incompressible Navier --- Stokes equations. J. Comput. Phys., 1985, vol. 59, iss. 2, pp. 308--323. DOI: https://doi.org/10.1016/0021-9991(85)90148-2

[26] Durst F., Peireira J.C.F., Tropea C. The plane symmetric sudden-expansion flow at low Reynolds numbers. J. Fluid Mech., 1993, vol. 248, pp. 567--581. DOI: https://doi.org/10.1017/S0022112093000916

[27] Kaiktsis L., Karniadakis G.E., Orszag S.A. Unsteadiness and convective instabilities in a two-dimensional flow over a backward-facing step. J. Fluid Mech., 1996, vol. 321, pp. 157--187. DOI: https://doi.org/10.1017/S0022112096007689

[28] Leonard B.P. A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Eng., 1979, vol. 19, iss. 1, pp. 59--98. DOI: https://doi.org/10.1016/0045-7825(79)90034-3

[29] MacCormack R.W. The effect of viscosity in hypervelocity impact cratering. AIAA Paper, 1969, no. 354. DOI: https://doi.org/10.2514/6.1969-354

[30] Warming R.F., Kutler P., Lomax H. Second- and third-order noncentered difference schemes for nonlinear hyperbolic equations. AIAA J., 1973, vol. 11, no. 2, pp. 189--196. DOI: https://doi.org/10.2514/3.50449

[31] Abarbanel S., Gottlieb D., Turkel E. Difference schemes with fourth order accuracy for hyperbolic equations. J. Appl. Math., 1975, vol. 29, iss. 2, pp. 329--351. DOI: https://doi.org/10.1137/0129029

[32] Loytsyanskiy L.G. Mekhanika zhidkosti i gaza [Fluid mechanics]. Moscow, Nauka Publ., 1987.

[33] Patankar S.V. Numerical heat transfer and fluid flow. CRC Press, 1980.

[34] Malikov Z.M., Madaliev M.E. Numerical simulation of two-phase flow in a centrifugal separator. Fluid Dyn., 2020, vol. 55, no. 8, pp. 1012--1028. DOI: https://doi.org/10.1134/S0015462820080066

[35] Erkinjon son M.M. Numerical calculation of an air centrifugal separator based on the SARC turbulence model. J. Appl. Comput. Mech., 2020, vol. 6, no. S, pp. 1133--1140.

[36] Malikov Z.M., Nazarov F.Kh. Study of an immersed axisymmetric turbulent jet in comparative analysis of turbulence models. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 2 (101), pp. 22--35 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2022-2-22-35

[37] Nazarov F.Kh. Comparing turbulence models for swirling flows. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2021, no. 2 (95), pp. 25--36 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2021-2-25-36