# Application of parallel computing technologies for modeling the flow separation process behind the backward facing step in the channel with the buoyancy forces

## DOI:

https://doi.org/10.26577/jmmcs-2018-1-493## Keywords:

domain decomposition method, backward facing step, projection method, separation and reunion of flows, separated flow, buoyancy forces## Abstract

The paper presents numerical solutions of the two-dimensional laminar flow behind the backward

facing step in the channel with the buoyancy forces. A two-dimensional incompressible Navier-

Stokes equation is used to describe this process. This system is solved numerically by the projection

method, which is approximated by the control volume method. The resulting Poisson equation

satisfying the discrete equation of continuity is solved by the Jacobi iterative method at each time

step. The numerical solutions of the laminar flow behind the backward facing step are compared

with the numerical results of other authors. This numerical algorithm is completely parallelized

using various geometric decompositions (1D, 2D and 3D). Preliminary theoretical analysis of the

various decomposition methods effectiveness of the computational domain and real computational

experiments for this problem were made and the best method of domain decomposition was

determined. In the future, a proven mathematical model and numerical algorithm with the best

decomposition method can be applied for various complex flows with the buoyancy forces.

## References

steps."J. Basic Engng. 84 (1962): p. 317.

[2] Armaly, B. F. and Durst, F. "Reattachment length and recirculation regions downstream of two dimensional single

backward facing step In Momentum and Heat Transfer Process in Recirculating Flows."ASME HTDVol. 13 (1980): l-7.

ASME, New York.

[3] Aung, W. "An experimental study of laminar heat transfer downstream of backsteps."J. Heat Transfer. 105 (1983):

23-829.

[4] Aung, W. "Separated forced convection."Proc. ASMEIJSME Thermal Enana Joint Conf. 2(1983): 499-515. ASME. New

York.

[5] Aung, W., Baron, A. and Tsou, F. K. "Wall independency and effect of initial shear-layer thickness in separated flow and

heat transfer."Int. J. Hear Muss Transfer. 28(1985): 1757-1771.

[6] Aung, W. and Worku, G. "Theory of fully developed. combined convection including flow reversal."J. Hear Transfer. 108

(1986): 485-488.

[7] Chiang, T.P., Tony, W.H. and Sheu Fang, C.C. "Numerical investigation of vortical evolution in backward-facing step

expansion flow."Appl. Math. 23(1999): 915-932.

[8] Chorin, A.J. "Numerical solution of the Navier-Stokes equations."Math. Comp. 22 (1968):745-762.

[9] Chung, T.J. "Computational fluid dynamics."2002. 1034 p.

[10] Durst, F. and Whitelawj, H. "Aerodynamic properties of separated gas flows: existing measurements techniques and new

optical geometry for the laser-Doppler anemometer."Prog. Heat Mass Transfer. 4(1971): 311.

[11] Eaton, J. K. and Johnson, J. P. "A review of research on subsonic turbulent flow reattachment."AIAA J. 19 (1981):

1093-1100.

[12] Fletcher, C.A.J. "Computational techniques for fluid dynamics 2."Springer-Verlag New York. 1 (1988): 387.

[13] Goldsteinr, J., Eriksenv, L., Olsonr, M. and Eckerte, R.G. "Laminar separation, reattachment and transition of flow over

a downstream-facing step."J. Basic Engng. 92 (1970):732.

[14] Gosmana, D. and Punw, M. "Lecture notes for course entitled: ’Calculation of recirculating flow’."Heat Transfer Rep.

74 (1974):2.

[15] Kumara, Yajnikk S. "Internal separated flows at large Reynolds number."J. Fluid Mech. 97 (1980):27.

[16] Issakhov, A. "Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal power

plant."International Journal of Nonlinear Science and Numerical Simulation. 16(5) (2015): 229-238. doi:10.1515/ijnsns-

2015-0047.

[17] Issakhov, A. "Mathematical modeling of the discharged heat water effect on the aquatic environment from thermal

power plant under various operational capacities."Applied Mathematical Modelling. 40(2) (2016): 1082-1096.

http://dx.doi.org/10.1016/j.apm.2015.06.024.

[18] Issakhov, A. "Large eddy simulation of turbulent mixing by using 3D decomposition method."J. Phys.: Conf. Ser. 318(4)

(2011): 1282-1288. doi:10.1088/1742-6596/318/4/042051.

[19] Karniadakis, G. E. and Kirby II, R. M. "Parallel Scientific Computing in C++ and MPI: A Seamless Approach to

Parallel Algorithms and their Implementation."Cambridge University Press, 2000. 630 p.

[20] Lin, J.T., Armaly, B.F. and Chen, T.S. "Mixed convection in buoyancy-assisting, vertical backward-facing step

flows."International Journal of Heat and Mass Transfer. 33(10) (1990): 2121-2132.

[21] Ngo, I. and Byon, C. "Effects of heater location and heater size on the natural convection heat transfer in a square cavity

using finite element method."J. Mech. Sci. Technol. 29 (7) (2015): 2995.

[22] Oztop, H. F. and Abu-Nada, E. "Numerical study of natural convection in partially heated rectangular enclosures filled

with nanofluids"Int. J. Heat. Fluid Fl. 29(5) (2008):1326-1336.

[23] Sebanr, A. "Heat transfer to the turbulent separated flows of air downstream of a step in the surface of a plate,"J. Heat

Transfer. 86 (1964):259.

[24] Simpson, R. L. "A review of some phenomena in turbulent flow separation,"J. Fluid Engng. 103 (1981): 520-533.

[25] Sparrow, E. M., Chrysler, G. M. and Azevedo, L. F. "Observed flow reversals and measured-predicted Nusselt numbers

for natural convection in a one-sided heated vertical channel."J. Heat Transfer. 106 (1984): 325-332.

[26] Sparrow, E. M., Kang, S. S. and Chuck, W. "Relation between the points of flow reattachment and maximum heat transfer

for regions of flow separation."Int. J. Heat Mass Transfer. 30 (1987): 1237-1246.

[27] Sparrow, E. M. and Chuck, W. "PC solutions for heat transfer and fluid flow downstream of an abrupt, asymmetric

enlargement in a channel,"Numer. Hear Transfer. 12(1987):1940.

## Downloads

## Published

## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*97*(1), 143–158. https://doi.org/10.26577/jmmcs-2018-1-493