A novel fully implicit finite volume method applied to the lid‐driven cavity problem—Part I: High Reynolds number flow calculations

Abstract

SUMMARYA novel implicit cell-vertex ÿnite volume method is described for the solution of the Navier-Stokes equations at high Reynolds numbers. The key idea is the elimination of the pressure term from the momentum equation by multiplying the momentum equation with the unit normal vector to a control volume boundary and integrating thereafter around this boundary. The resulting equations are expressed solely in terms of the velocity components. Thus any di culties with pressure or vorticity boundary conditions are circumvented and the number of primary variables that need to be determined equals the number of space dimensions. The method is applied to both the steady and unsteady two-dimensional lid-driven cavity problem at Reynolds numbers up to 10000. Results are compared with those in the literature and show excellent agreement.

Using the Lid-Driven Cavity Flow to Validate Moment-Based Boundary Conditions for the Lattice Boltzmann Equation

“…We note that there is excellent agreement between the present work and other methods. In particular, the results are very similar to those obtained by Sahin and Owens [23], who used a stylised and nonuniform finite volume discretisation and Botella and Peyret [24] who used a spectral method. Table 3.…”

“…4 confirms that our method is second order accurate. The results in this figure not only show the expected convergence properties of the LBM but a comparison with the assumed highly accurate data of Botella and Peyret [24] and Sahin and Owens [23] shows excellent agreement, giving confidence to the predictive capabilities of the LBM with moment-based boundary conditions. Furthermore, we have not attempted to optimise the MRT collision operator.…”

“…The minimum values of the primary stream function are also inserted in the tables. Moreover, we compare the results with other methods which are the finite volume [23,34], the finite difference [22,32], the spectral method [24], and LBMs [27,33] at different Reynolds numbers. Our data in this table was found using 257 × 257 grid points, the same spatial resolution Table 2.…”

“…The lid-driven cavity problem has been used to assess the accuracy of many numerical methods, including finite difference and volume methods [22,23], spectral methods [24], and lattice Boltzmann [9,[25][26][27]. In Section 2 we introduce the discrete velocity Boltzmann equation before showing that this model …”

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

“…More numerical experiments on this type of nonlinear solution method are carrying out now and a comparison between it and the continuation method will be reported. Figure 1-2 plot the profiles of two components of the computed velocity, and these plots are close to the results given in [29]. There are many references testing the three preconditioners and their variants on the LDC problem, c.f.…”