In this talk I will present a method to evolve arbitrary sets. These sets may be open or closed curves/surfaces, and may not be regular (e.g. fractal). Obvious applications are interface tracking in multi-phase/multi-component fluid flows. The method relies on the gradient-augmented level set framework, and a two-grid approach. I will discuss efficiency, accuracy, stability and end with various applications.

Classical Riemann problem solutions for two-phase, multi-component flows in porous media are based on the assumption of constant flow rates. When the pressures at the inlet and outlet boundaries are constant, the flow rate will vary as a function of time, and hence, existing analytical solutions are no longer valid.

In this presentation, existing Riemann problem solutions for constant flow rates are generalized to constant pressure boundaries. This generalization is achieved through the determination of an analytical closed form solution for the flow rate and pressure distribution within the porous medium as a function of time.

The new analytical solution is valid for arbitrary flow geometries in the sense that the cross sectional area of the flow medium between injection boundary and producer boundary is arbitrary. In particular, it is shown that the flow velocity varies not only because of widening/narrowing of the flow channel, but also because the geometry has a nontrivial effect on the structure of the Riemann solution itself.

As examples on arbitrary flow geometries, the constant pressure boundary Riemann solutions for radial flow, spherical flow in addition to standard linear flow will be presented.

The new solutions have numerous applications to flow calculations between injectors and producers in oil reservoirs, for example as combined with streamline modeling with arbitrarily shaped stream tubes. Such applications will be discussed for water flooding and gas flooding of oil reservoirs.

Finally, the new analytical solutions will be compared to standard numerical approximations for the purpose of grid sensitivity analysis.

In this talk I will discuss efforts to numerically model various physical processes in lakes. One aspect of the talk will consider internal wave motions on the basin scale, while the other will consider smaller scale modelling involving the interaction of internal waves with the lake bottom. In each case the typical, publicly available numerical model is under-resolved or somehow misrepresents the physics that occurs in the field. Sometimes, this is by design, with convenient mathematical approximations (e.g. the hydrostatic approximation) and properties of numerical schemes (dissipation) resulting from the a priori decision to under-resolve the physical processes in question. At other times this is by necessity, since the physical phenomenon in question spans too many scales to be adequately resolved. I will discuss efforts to circumvent under-resolution with either improved physical parametrizations (e.g. weakly non hydrostatic effects), improved numerical methods (high order discontinuous Galerkin methods), or high resolution 'scale-up' numerical experiments using direct numerical simulations.

Direct numerical simulations of potential enstrophy in strongly stratified, weakly rotating turbulence will be presented. When rotation and stratification are strong - as in large-scale atmospheric and oceanic flows - the potential vorticity is approximately linear in the flow variables, and so the potential enstrophy is a quadratic invariant. At smaller scales, however, the quadratic approximation becomes questionable. We will show that both quadratic and higher-order potential enstrophy are possible in this regime, depending on the effects of viscosity. As a result, laboratory and geophysical scale stratified turbulence have very different potential enstrophy dynamics.