# Study on SPH Viscosity Term Formulations
## 5 Results II--Model Application in Practical Viscous Flows
### Lid-Driven Flow with an Inside Cylinder  
To provide a further robust evaluation on the proposed ISPH viscosity force models, this section explores a more complex lid-driven flow case with a cylinder inside. The schematic set up of the model is shown in Figure 11a. There is a cylinder inside the cavity region, with its diameter being \(d=0.5\ L\). The 2D square cavity has a height \(H\) and width \(L\), with the dimensions of \(H=L=1.0\ \text{m}\). The center of the cylinder is located at \((0.5\ L,0.5\ H)\). The wall boundary of both the cylinder and the cavity region is set as the non-slip and non-penetration condition in the tangential and normal directions, respectively, where \(u_{r}\) and \(u_{n}\) indicate the velocity component in each direction. The velocity of the top wall of the cavity in the \(x\)-direction is given by \(U_{0}=1.0\ \text{m/s}\), which leads to a Reynolds number \(Re=1000\). The other computational parameters are kept unchanged as those used in Section 5.1. Based on the ISPH model simulations (with Visc-04), Figure 11b gives the result of velocity vector plots of the cavity flow at particle number \(N=10{,}000\) and simulation time \(t=40\ \text{s}\), when the steady results are achieved. Figure 12 shows the comparison of velocity amplitude patterns between the ISPH and OpenFOAM computations, and a qualitatively good agreement has been found.  
Figure 11: (**a**) Schematic setup of the lid-driven flow model with a cylinder inside; and (**b**) Velocity vector plot of the cavity flow at time \(t=40\ \text{s}\). Red, location of fluid particle; Black, velocity of flow field.  
Figure 12: Velocity amplitude patterns of the lid-driven flow with an inside cylinder computed by: (**a**) OpenFOAM; and (**b**) ISPH (Visc-04).  
To finally examine the performance of different viscosity force models, Figure 13 gives the comparisons of velocity \(ux\) distributions at \(x=0.5\) m and \(uy\) distributions at \(y=0.5\) m, computed by the ISPHs and FVM-based OpenFOAM. In this case, the computational parameters are: particle size \(dx=0.005\) m, time \(t=40\) s and Reynolds number \(Re=1000\). As shown in Figure 13, the results obtained by the four viscosity calculation methods are quite similar to each other, except the enlarged portion (where the velocity profiles change sharply) shows that the result obtained by VISC-04 is the most satisfactory compared with the OpenFOAM results. It is also noticeable that OpenFOAM generally gives higher values of the velocity than ISPH for both the lid-driven flow and the lid-driven flow with an inside cylinder.  
For convergence and error analysis, Figure 14 gives the numerical errors of minimum velocity \(ux\) of various ISPH viscosity force models under different particle resolutions, where \(Err\) is defined as the same error used in the previous section. Again all the ISPH errors decrease with the deceasing particle size. In addition, it is also disclosed that the error of VISC-04 is the smallest, which is consistent with the conclusion drawn from Section 5.1.  
Figure 13: Comparisons of four viscosity force models for the lid-driven flow with an inside cylinder: (**a**) \(ux\) velocity distribution; and (**b**) \(uy\) velocity distribution.