# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 3 Numerical Results
### Verification of the Numerical Method: Lid Driven Cavity  
We now turn our attention to validating the numerical results obtained using the Simple/regularization procedure of our code. To this end, we consider the test case of lid-cavity flow, which is the prototypical recirculation flow and has long been used as a standard problem for testing and evaluating Navier-Stokes solvers. Many authors provided high quality benchmark results for this particular problem, most famous of which are [PERSON] et al. (2016) who used a coupled strongly implicit multigrid (CSI-MG) in a uniform mesh of \(257\times 257\). They presented a second-order accurate results that have served as \"The\" result for Newtonian flows to compare against ever since. All the results presented in this section are obtained on Cartesian grids consisting of \(320\times 320\) square control volumes. To validate the present algorithm, first, a Newtonian flow \((Bn=0)\) in the \(Re=10^{3}\) case is calculated. Figure 6a,b show the evolution of the horizontal and vertical velocities along the mid-planes \(x=0.5\) and \(y=0.5\), with the corresponding numerical result of [PERSON] et al. (2016). The present results are in good agreement with those numerical data. Now we consider the steady viscoplastic flow at various Bingham numbers (Bn) in the case of \(Re=10^{3}\). For this particular flow, many of the previous studies ([PERSON] et al. (2012); [PERSON] et al. (2017); [PERSON] et al. (2018); [PERSON] et al. (2013)) formulated the main  
Figure 6: Sections of velocity along the vertical (**a**) and horizontal (**b**) mid-planes, \(Re=10^{3}\)([PERSON] et al., 2016). quantitative results in terms of the features of the flow vortices. To this end, we need to calculate the stream function on the whole domain through integration of the velocity field (\(u=\partial\psi/\partial y,v=-\partial y/\partial x\)). The location of the primary vortex formed in lid-driven cavity flow is significantly influenced by both Bingham and Reynolds numbers. This shifting of location is easily noticeable, which allows us to further assess the qualitative aspect of the simulation results. The evolution of the center position of the principal vortex is compared to the results of the predictions performed by [PERSON] et al. (2011) and [PERSON] et al. (2013), with a regularized constitutive law on Figure 7. The obtained values once again compare fairly well with the literature, indicating that the treatment of the shear stress and viscosity in the non-Newtonian flow prediction is feasible and the steady SIMPLE solver for Bingham fluid is correct. The evolution of the principal vortex location as a function of the yield stress drawn in Figure 7 shows that the primary vortex goes up toward the moving lid as \(Bn\) increases. Increasing the yield stress leads to restricting the flow region closer to the upper moving boundary due to the resistance to lid motion, which gets higher. In viscoplastic flows, rigid zones are located at the bottom of the cavity where the shear stresses are very low as the fluid in contact is motionless due to the no-slip boundary condition. These unyielded regions expand significantly with the increase of the Bingham number, leading the main vortex to move toward the upper boundary as there is less space for flow to happen. The streamline contour evolution shown in Figures 8-10 highlights the progressive growth of the unyielded zones. At constant \(Re\) number, as the value of the Bingham number increases, the main vortex center gets closer to the upper cavity side and the zone sheared by the upper boundary becomes thinner. Figures 8-10 display the flow field as the bingham number increases, for \(Re=10\) and \(Re=10^{3}\), respectively. When \(Re=10\), the streamline contours are symmetric for all Bingham numbers, 2, 10, and 100. However, the unyielded zone is proportional to the yield stress magnitude.