## Test 03

2-D Driven cavity without gravity

(B.D. Rogers)

The following test case is based upon a widely used validation test case used for other numerical codes, e.g. finite volume, finite difference, spectral element, etc. This makes the problem ideal for assessing the ability of different SPH techniques in comparison with conventional methods where the results are well known. The problem consists of a 2-D rectangular cavity filled with fluid that is covered by a horizontal moving lid as shown in the schematic in Figure 1. The case has been studied among others including Ghia et al. 1982, Ku et al. 1987 & Chern et al. 2005.

For cases with Re = UL/nu less than 1000, the flow is distinguished by a main vortex in the centre, and two secondary vortices in the bottom corners (see Figure 2). For Re = 10,000, there is an extra vortex in the upper left corner. Numerical results are commonly assessed by plotting the steady-state streamlines and velocity profiles down the centerline of the tank. This is shown excellently in Figures 2-5 from Chern et al. (2005) which are Republished with permission, Emerald Group Publishing Limited.

**Tasks**

In order to perform a meaningful comparison, people are requested to perform the following simulations for Re = 1000, and 10000, presenting the information detailed.
For three resolutions, i.e. initial particle arrangements of 50x50, 70x70 & 100x100 within the cavity (not including the walls), please produce:

A streamline plot at steady state (see Figure 2)

Comparison of central profile u-velocity along y-direction at steady state (see Figure 3)

A plot of total KE versus time

Some measure of convergence of the simulation, e.g. global relative error of density:

where **x** is a position vector to the original particle positions, n denotes time level and is the SPH interpolated value at position **x**.

**Publications using this test case as a benchmark**

Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach, Journal of Computational Physics, 228(18), 6703-6725, DOI: 10.1016/j.jcp.2009.05.032.

Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby PK (2008) Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method, Journal of Computational Physics, 227(18), 8417-8436, DOI: 10.1016/j.jcp.2008.06.005.

Basa M, Quinlan NJ, Lastiwka M. (2009) Robustness and accuracy of SPH formulations for viscous flow, Int. J. Numer. Meth. Fluids, 60, 1127–1148 DOI:10.1002/fld.1927 10.1002/fld.1927.

de Leffe M, Le Touzé D, Alessandrini B (2011) "A modified no-slip condition in weakly compressible SPH", Proc. 6th International SPHERIC Workshop, Ed. T Rung & C Ulrich, Hamburg University of Technology (TUHH), 8-9 June 2011, ISBN: 978-3-89220-658-3, p. 291-298.

Khorasanizade, S., Sousa, J.M.M. (2014) A detailed study of lid-driven cavity flow at moderate Reynolds numbers using Incompressible SPH, Int. J. Num. Meth. Fluids, DOI: http://dx.doi.org/10.1002/fld.3949.

**References**

Ghia, U., Ghia, K. and Shin, C.A. (1982) High-Resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method”, Journal of Computational Physics, 48, 387-411, DOI: http://dx.doi.org/10.1016/0021-9991(82)90058-4.

Ku, H.C., Hirsch, R.S. and Taylor, T.D. (1987) A pseudospectral method for solution of the three dimensional incompressible Navier-Stokes equations, Journal of Computational Physics, 13, 99-113.

Chern, M.J., Borthwick, A.G.L. and Eatock Taylor, R. (2005) Pseudospectral element model for free surface viscous flows, Int. J. Num. Meth. For Heat & Fluid Flow, 15(6), 517 – 554.