A four-way coupled Euler - Lagrange approach using a variational multiscale method for simulating cavitation

An Euler-Lagrange model is developed to simulate bubbly flow around an obstacle with the aim to resolve large and meso-scales of cavitation phenomena. The volume averaged Navier-Stokes equations are discretized using finite elements on an unstructured grid with a variational multiscale method. The trajectory of each bubble is tracked using Newton's second law. Furthermore, bubble interaction is modeled with a soft sphere contact model to obtain a four-way coupled approach. The new features presented in this work, besides using a variational multiscale method in an Euler-Lagrange framework, is an improved computation of the void fraction. A second order polynomial is used as filtering function and the volume integral is transformed by applying the divergence theorem twice, leading to line integrals which can be integrated analytically. Therefore, accuracy of void fraction computation is increased and discontinuities are avoided as is the case when the kernel touches a Gauss point across time steps. This integration technique is not limited to the chosen spatial discretization. The numerical test case considers flow in a channel with a cylindrical obstacle. Bubbles are released close to the inflow boundary and void fractions up to 30% occur at the stagnation point of the obstacle.