Random Vortex Method for Geometries with Unsolvable Schwarz-Christoffel Formula

Authors

1 Department of Mechanical Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

2 Department of Mechanical Engineering, Institute of Energy & Hydro Technology (IEHT), Mashhad, Iran

Abstract

In this research we have implemented the Random Vortex Method to calculate velocity fields of fluids inside open cavities in both turbulent and laminar flows. the Random Vortex Method is a CFD method (in both turbulent and laminar fields) which needs the Schwarz-Christoffel transformation formula to map the physical geometry into the upper half plane. In some complex geometries like the flow inside cavity, the Schwarz-Christoffel mapping which transfers the cavity into the upper half plane cannot be achieved easily. In this paper, the mentioned mapping function for a square cavity is obtained numerically. Then, the instantaneous and the average velocity fields are calculated inside the cavity using the RVM. Reynolds numbers for laminar and turbulent flows are 50 and 50000, respectively. In both cases, the velocity distribution of the model is compared with the FLUENT results that the results are very satisfactory. Also, for aspect ratio the cavity (α) equal 2, the same calculation was done for Re=50 and 50000. The advantage of this modelling is that for calculation of velocity at any point of the geometry, there is no need to use meshing in all of the flow field and the velocity in a special point can be obtained directly and with no need to the other points.

Keywords


1.     Wolfshtein, M., "Some comments on turbulence modelling", International Journal of Heat and Mass Transfer,  Vol. 52, No. 17, (2009), 4103-4107.
2.     Chorin, A.J., "Numerical study of slightly viscous flow", Journal of Fluid Mechanics,  Vol. 57, No. 4, (1973), 785-796.
3.     Chorin, A.J., "Vortex sheet approximation of boundary layers", Journal of Computational Physics,  Vol. 27, No. 3, (1978), 428-442.
4.     Chorin, A.J., "Vortex models and boundary layer instability", SIAM Journal on Scientific and Statistical Computing,  Vol. 1, No. 1, (1980), 1-21.
5.     Sethian, J.A. and Ghoniem, A.F., "Validation study of vortex methods", Journal of Computational Physics,  Vol. 74, No. 2, (1988), 283-317.
6.     Beale, J.T., "A convergent 3-d vortex method with grid-free stretching", Mathematics of Computation,  Vol. 46, No. 174, (1986), 401-424, S415.
7.     Cottet, G.-H., "A new approach for the analysis of vortex methods in two and three dimensions", in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Elsevier. Vol. 5, (1988), 227-285.
8.     Gharakhani, A. and Ghoniem, A.F., "Three-dimensional vortex simulation of time dependent incompressible internal viscous flows", Journal of Computational Physics,  Vol. 134, No. 1, (1997), 75-95.
9.     Ghoniem, A.F. and Cagnon, Y., "Vortex simulation of laminar recirculating flow", Journal of Computational Physics,  Vol. 68, No. 2, (1987), 346-377.
10.   Knio, O.M. and Ghoniem, A.F., "Three-dimensional vortex simulation of rollup and entrainment in a shear layer", Journal of Computational Physics,  Vol. 97, No. 1, (1991), 172-223.
11.   McCracken, M. and Peskin, C., "A vortex method for blood flow through heart valves", Journal of Computational Physics,  Vol. 35, No. 2, (1980), 183-205.
12.   Summers, D., Hanson, T. and Wilson, C., "A random vortex simulation of wind‐flow over a building", International Journal for Numerical Methods in Fluids,  Vol. 5, No. 10, (1985), 849-871.
13.   Blot, F., Giovannini, A., Hebrard, P. and Strzelecki, A., "Flow analysis in a vortex flowmeter-an experimental and numerical approach", in 7th Symposium on Turbulent Shear Flows, Volume 1. Vol. 1, (1989), 10.13. 11-10.13. 15.
14.   Ghoniem, A., Chorin, A. and Oppenheim, A., Numerical modeling of turbulent flow in a combustion tunnel. 1980, Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, CA (US).
15.   Giovannini, A., Mortazavi, I. and Tinel, Y., "Numerical flow visualization in high reynolds numbers using vortex method computational results", ASME-PUBLICATIONS-FED,  Vol. 218, No., (1995), 37-44.
16.   Gagnon, Y., Giovannini, A. and Hébrard, P., "Numerical simulation and physical analysis of high reynolds number recirculating flows behind sudden expansions", Physics of Fluids A: Fluid Dynamics,  Vol. 5, No. 10, (1993), 2377-2389.
17.   Martins, L.-F. and Ghoniem, A.F., "Simulation of the nonreacting flow in a bluff-body burner; effect of the diameter ratio", Journal of Fluids Engineering,  Vol. 115, No. 3, (1993), 474-484.
18.   Shokouhmand, H. and Sayehvand, H., "Numerical study of flow and heat transfer in a square driven cavity", International Journal of Engineering Transactions A: Basics,  Vol. 17, No. 3, (2004), 301-317.
19.   Taghilou, M. and Rahimian, M., "Simulation of lid driven cavity flow at different aspect ratios using single relaxation time lattice boltzmann method",  (2013).
20.   Zafarmand, B., Souhar, M. and Nezhad, A.H., "Analysis of the characteristics, physical concepts and entropy generation in a turbulent channel flow using vortex blob method", International Journal of Engineering, TRANSACTIONSA: Basics Vol. 29, No. 7, (2016), 985-994.
21.   De, S. and Bathe, K.-J., "Towards an efficient meshless computational technique: The method of finite spheres", Engineering Computations,  Vol. 18, No. 1/2, (2001), 170-192.
22.   Oppenheim, A.K., "Dynamics of combustion systems, Springer,  (2008).