Residual-based turbulence models and arbitrary Lagrangian-Eulerian framework for free surface flows

We present a residual-based turbulence model for problems with free surfaces. The method is derived based on variational multiscale ideas that assume a decomposition of the solution fields into overlapping scales that are termed as coarse and fine scales. The fine scales are further split hierarchically into fine-scales level-I and fine-scales level-II. The hierarchical variational problems that govern the two fine-scale components are modeled employing bubble functions approach. The model for level-II scales is variationally embedded in the mixed field level-I problem to yield a stable level-I formulation. Subsequently, the model for level-I scales that in fact constitutes the fine-scale turbulence model is then variationally injected in the coarse-scale variational form. A significant feature of the method is that it does not contain any embedded tunable parameters. To accommodate the moving boundaries we cast the formulation in an arbitrary Lagrangian-Eulerian frame of reference. The free surface boundary condition is imposed weakly which results in a formulation that conserves the volume of the fluid. A variety of benchmark problems show the accuracy and range of applicability of the proposed formulation and results are compared with published data. A wavy bed problem is investigated to show the interaction of turbulence generated at the bottom surface with the free surface thereby leading to irregular free surface elevations.