# A Backwards-Tracking Lagrangian-Eulerian Method for Viscoelastic Two-Fluid Flows
## 4 Results
### Jet Buckling  
The simulation of jet buckling is a common flow case for testing the capabilities of numerical methods for free surface flow. The phenomenon has been numerically investigated for different viscoelastic constitutive models by several authors [14; 15; 16; 17; 18; 20; 26]. The phenomenon occurs for a fluid jet that flows onto a rigid plate under certain conditions. [PERSON] [50] proposed a condition that is based on experimental and theoretical observations, stating that a planar Newtonian jet will buckle if \(H/D>3\pi\) and \(\mathrm{Re}<0.56\), where \(H\) is the distance from the inlet to the plate, \(D\) the inlet width, and \(\mathrm{Re}=\rho UD/\mu\), where \(U\) is the inlet velocity. A modified yet approximate condition, based on numerical investigation, was later proposed by [PERSON] and [PERSON], stating that buckling should occur if  
\[\mathrm{Re}^{2}\leq\frac{1}{\pi}\frac{(H/D)^{2.6}-8.8^{2.6}}{(H/D)^{2.6}}. \tag{34}\]  
It is remarked that, in simulations, numerical round-off errors are responsible for triggering the buckling, since the flow setup itself is actually symmetric. Figure 15 shows a schematic of the fluid jet buckling domain that is used for simulation with the Lagrangian-Eulerian method proposed in this work. The domain is \(50\,\mathrm{mm}\) wide and \(100\,\mathrm{mm}\) high. An inlet of width \(D=5\,\mathrm{mm}\) is located at the center of the upper boundary. At the inlet, a uniform velocity \(U=0.5\,\mathrm{m/s}\) is imposed. The upper boundaries to the left and right, respectively, are outlets with the Dirichlet pressure condition \(p=0\). The remaining domain boundaries are treated as walls by imposing the no-slip condition. Gravity is acting in the negative \(y\)-direction with \(g=9.81\,\mathrm{m/s^{2}}\). Figure 14: Simulated swell ratio compared to data from the literature. The viscoelastic fluid is a single-mode Oldroyd-B fluid with the viscosity ratio \(\beta=0.1\), density \(\rho=1000\,\mathrm{kg}/\mathrm{m}^{3}\), and the total viscosity \(\eta_{t}=10\,\mathrm{Pa}\cdot\mathrm{s}\). The surrounding Newtonian viscosity is air with viscosity \(\mu_{\mathrm{air}}=1.8205\times 10^{-5}\,\mathrm{Pa}\cdot\mathrm{s}\) and density \(\rho_{\mathrm{air}}=1.204\,\mathrm{kg}/\mathrm{m}^{3}\). The Reynolds number for the flow is defined as \(\mathrm{Re}=\rho UD/(\mu+\eta)\), which, for the chosen parameters, yields \(\mathrm{Re}=0.25\). A Weissenberg number is defined as \(\mathrm{Wi}=\lambda U/D\). The flow is simulated for the Oldroyd-B fluid with \(\lambda=0.1\,\mathrm{s}\), corresponding \(\mathrm{Wi}=10\). As reference, a simulation is also performed with a Newtonian fluid, for which \(\beta=\mathrm{Wi}=0\). In the viscoelastic jet buckling simulations that were performed in this work, the octree grid is adaptively refined with two levels around the viscoelastic fluid phase. As the simulation progresses, the grid is therefore updated and the cells are refined and coarsened, where needed. A grid dependence study is performed while using three adaptive grids of increasing resolution. The grids have the respective base cell sizes \(\Delta x_{\mathrm{base}}=D/2,D/4,D/8\), which correspond to the finest cell sizes \(\Delta x_{\mathrm{min}}=D/8,D/16,D/32\). The simulations are performed with a constant time step \(\Delta t\) satisfying \(\Delta tU/\Delta x_{\mathrm{min}}=0.08\). The grid dependence is evaluated with respect to two simulation properties. Firstly, the free surface \(\alpha=0.5\) is compared for the three adaptive grids. The first normal stress difference \(N_{1}=\tau_{yy}-\tau_{xx}\) along the jet at \(x=5D\) is also compared. Because the buckling itself is triggered by numerical round-off errors, the exact conditions at which the buckling starts in the simulations may vary between grids. Therefore, the grid study comparison is made at a simulation time before buckling is initiated. In Figure 16, the free surface is compared for the three grids at the simulation times \(t=130\,\mathrm{ms}\) and \(t=140\,\mathrm{ms}\).