# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 3 Numerical Results
### Algebraic Convergence of the SIMPLE/Regularization Procedure  
In viscoplastic flows, derivatives of the flow variables take significantly higher values across the yield surfaces in the limit between the unyielded and yielded regions. This disparity is due to the fact that these derivatives are zero inside the rigid zones and non-zero outside. In addition, increasing the Bingham number and growth stress parameter would further widen this gap. As a result, solution errors are found to be much larger there than the rest of the domain, causing inefficiencies and deterioration in convergence rates of iterative solvers. In such cases, using fine grids can considerably reduce the numerical errors and address more efficiently the high values of the derivatives. However, it is well known that the number of iterations required for convergence increases almost linearly with the number of control volumes for the single grid set ([PERSON] et al., 2019; [PERSON] et al., 2019). To address this limitation, one may notice that using an appropriate initial solution is very important for fine grids and contributes to the enhancement of the convergence of iterative solvers. To this end, a steady lid driven cavity flow simulation is carried out for different Bingham numbers 0, 1, and 10 at \(Re=10^{2}\) using single-grid (SG) and multi-grid (MG) initialization methods. Figure 14 demonstrates the convergence of the \(L^{\infty}\) norm of \(u\) and \(v\) residuals as a function of the number of SIMPLE iterations on the finer grids. It shows that the performance of SIMPLE, as a single-grid solver, deteriorates, and more iterations are needed as the number of control volumes increases for all cases. In contrast, computations with MG initialization exhibit near constant number for both 160 \(\times\) 160 and 320 \(\times\) 320 grids. In all cases, the number of iterations has drastically been reduced compared to the single grid computation. One may also observe that convergence becomes faster in the case of \(Bn=10\), but it also becomes more unstable and oscillatory. It should be noted that for higher Bingham numbers, convergence difficulties may be encountered. In order to overcome this difficulty, low under-relaxation factors for pressure and velocity field, denoted by \(\alpha_{p}\), \(\alpha_{u}\), and \(\alpha_{v}\) can be considered within the numerical solver ([PERSON], 2018). This technique was found to be efficient in making the solver more robust, but we observed that it leads to a slow down in convergence  
Figure 13: Apparent viscosity profiles along the vertical section \(x=0.5L\) calculated with \(m=100\) (solid lines) and \(m=800\) ( red dotted lines), \(Re=10^{2}\) and \(Bn=10\). rate. Another remedy to this problem could be the suggestion of [PERSON] and [PERSON] [19] in a multigrid context. They recommend updating the viscosity on the finer grid only and holding its value constant on coarse grids within the multigrid cycle. The performance of multi-grid (MG) initialization is based on the speedup characteristics, and a comparison between the number of iterations and computation times, as summarized in Table 1. In all cases, CPU time is recorded until the convergence of residuals is achieved. In the case of MG Initialization, the total simulation time includes computation time on both multilevel grids during initialization, and the finer grid, which is the true representation of the simulation time. The simulations are performed on a Core i7-6700 HQ CPU @ 2.60 GHz \(\times\) 8, 8 GB RAM-based computer. According to our numerical experiments, the speed-up ratio between single grid and multigrid initialization methods is around 6 for  
Figure 14: The \(L^{\infty}\) norm of \(u\) and \(v\) residuals as a function of the number of SIMPLE iterations on the fine grid, for \(Bn=0\), 1, and 10 (\(Re=10^{2}\)). Results are shown for simulations with single-grid (SG) in red lines and multi-grid (MG) initialization in blue lines. 320 \(\times\) 320 grids and up to 3 for 160 \(\times\) 160 grids.