# A Lattice Boltzmann Method and Asynchronous Model Coupling for Viscoelastic Fluids
## 4 Numerical Results and Discussions
### Contraction Flows  
Viscoelastic fluids through contractions are important in the rapidly developed research field of micro fluidics, such as the polymer processing. As we know, the contraction flow has singularity at the reentrant corner, which makes calculations at high Weissenberg numbers challenging. The planar 4:1 contraction was seen as a benchmark [(28)] model in 1987 because of the geometrical simplicity and the known numerical difficulty. In this section, to further test the validity of the method presented in this paper, we simulate the Oldroyd-B fluids through a 4:1 contraction geometry at \(Wi=5\). Figure 6 shows the flow domain of which the ratio between the downstream and upstream height is 1 to 4. The characteristic length \(L\) and the characteristic velocity \(U\) are the half height and average velocity at the outlet, respectively. A parabolic Poiseuille flow is set at inflow by  
\[u_{x}=\frac{3}{128}(16-y^{2}),u_{y}=0, \tag{33}\]  
Figure 4: Comparison of the relative \(L_{2}\)-error of \(\tau_{xy}\) and \(\tau_{xx}\) for \(y\) at position \(x=5\) at \(Wi=15.0\) with \(\beta=0.5\) and Re = 1.0, respectively. Figure 5: The time development of the velocity components \(u\) at central point (5, 0.5) for \(Wi=15.0\) (right) with \(\beta=0.5\) and Re = 1.0, respectively. while the fully-developed conditions are given at the outlet, so the average velocity \(U\) at the out let is 1. No-slip boundary conditions are given along the stationary walls. Symmetry boundary conditions are imposed along the center-line. We take the dimensionless units \(U\) as 1, \(L\) as 1, parameter \(\beta\) as 1/9, and Re number as 1.0. To verify the mesh convergence, we carried out the calculations of a steady flow with \(Wi=1.0\) using three Cartesian meshes, M1, M2 and M3. The total number of cells and the grid size of each mesh are reported in Table 1. Figure 7 displays the temporal evolution of the local velocity at the intersection of the symmetry line and the contraction plane of the channel, for the different mesh. From Figure 7, we can see that the velocity tends to be meshing independent as the number of the nodes increased. To juggle the computational precision and efficiency, the numerical results are got in the next parts by utilizing the dense mesh of M3. For the generalized asynchronously coupled method, we set \(\delta t\approx 1.6~{}\times~{}10^{-5}\), \(\Delta t\approx 1~{}\times~{}10^{-3}\) and \(N=4\). Figure 8 displays the numerical results of velocity components along the horizontal center lines of the contraction with \(Wi=1.0\), comparing between the results of GAC and AC method, respectively. We can see that the difference of those two schemes is not obvious at low \(Wi\) number flows. That study of low \(Wi\) number flows is a benchmark example when considering it as a preview to the high \(Wi\) number flows in the following part. Figure 6: A diagram of the planar 4:1 contraction flow. Figure 7: Mesh dependence of the velocity components through the horizontal at \(Wi=1.0\) using mesh of M1, M2, and M3. To obtain some complex viscoelastic behaviors, we consider the dynamics of viscoelastic fluids at \(Wi=5.0\) using the GAC in this part. Figure 9 displays a set of instantaneous streamlines of flows with \(Wi=5.0\) on the finest mesh M3. Concerning the dynamics of viscoelastic fluids, we can find a series of different flow regimes, while increasing the \(Wi\) number to 5.0. From Figure 9, we observe that the elastic lip vortex grows in size as time goes by, eventually running up to the corner vortex region, and mixing with each other in a fairly complex dynamic process. The dynamical transitions are signed by a complex interaction between pulsating lip and corner vortices, which tend to approach and separate. In the literatures, through a full 4:1 planar contraction, [PERSON] et al. [5] and [PERSON] et al. [29], by carrying out numerical simulations of the Oldroyd-B model, also predicted the similar coalescence of lip and salient corner vortices into one big vortex with increasing elasticity.