# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 4 Conclusions  
This paper introduced the idea of multigrid initialization to overcome the drawbacks of solvers using single grids for viscoplastic flows. The proposed solver is easy to implement and significantly improves the convergence rate by reducing CPU time and number of iterations. The computational procedure employed in this work is based on the finite volume / SIMPLE algorithm in conjunction with the regularization scheme of Papanasatsuiou and is applied for the solution of the lid-driven cavity and pipe flow problems. The convection terms are discretized using the QUICK scheme in order to avoid any artificial diffusion that may be caused by interpolation schemes of low order of accuracy. The capability of the algorithm for solving viscoplastic flows was verified against a number of benchmark solutions for a range of Bingham numbers in the case of lid-driven cavity flow. A qualitative assessment was also conducted by monitoring the positioning of the principal vortex with regard to Bingham and Reynolds numbers. Results from the current study were accurate and compare favorably with the published results. In order to analyze the influence of the growth stress parameter \((m)\) of the regularization model on fluid motion, pipe flow tests for different values of \((m)\) and \(Bn\) were considered. Numerical simulations confirmed that a large value of \(m\) reproduced a large threshold-like region. The result also showed that as \(m\) increases, the yield surface converges to the true yield region, with no significant influence beyond a value of 400 in this particular case. Furthermore, the results clearly reveal that introducing the multigrid method as an initialization procedure yields better convergence for Navier-Stokes equations for viscoplastic flows compared to the single-grid method. A multi-grid initialization speed-up ratio as high as 6.45 was achieved for the finest grid size (320 \(\times\) 320). The present study can be extended to the numerical solution of non-Newtonian problems with a different definition of effective viscosity and accounting for complex geometries encountered in many industrial processes. This part is the subject of an ongoing investigation that is to be published in our future works. Conceptualization, S.M. and I.K. ; methodology, S.M., I.K. and F.B. ; validation, S.M., I.K. and F.B. ; formal analysis, S.M. and F.B. ; investigation, S.M., I.K., D.O. and F.B. ; data curation, S.M. ; writing--original draft preparation, S.M. ; writing--review and editing, S.M., I.K. and F.B. ; visualization, F.B. ; supervision, I.K., D.O. and F.B. ; project administration, D.O. and F.B. All authors have read and agreed to the published version of the manuscript. This research received no external funding. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Bingham Number**} & \multirow{2}{*}{**Number of Cells**} & \multicolumn{3}{c}{**CPU Time (s)**} & \multicolumn{3}{c}{**Number of Iterations**} \\ \cline{3-6}  & & **Single Grid** & \
\begin{tabular}{c} **MG** \\ **Initialization** \\ \end{tabular} & **Single Grid** &
\begin{tabular}{c} **MG** \\ **Initialization** \\ \end{tabular} & **Speed-Up Ratio** \\ \hline
0 & \(160\times 160\) & 88.2 & 35.8 & 6076 & 1910 & 2.46 \\  & \(320\times 320\) & 872.9 & 143.7 & 14,272 & 1964 & 6.07 \\
1 & \(160\times 160\) & 109.9 & 32.4 & 7643 & 1670 & 3.39 \\  & \(320\times 320\) & 990.1 & 153.4 & 16,337 & 1956 & 6.45 \\
10 & \(160\times 160\) & 81.6 & 29.8 & 5424 & 1569 & 2.73 \\  & \(320\times 320\) & 797.3 & 141.5 & 13,033 & 1856 & 5.63 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MG Initialization performance for different Bingham numbers. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The authors gratefully acknowledge the support and computing resources from the African Supercomputing Center (ASCC) at UM6P (Morocco). The authors declare no conflict of interest.