A Semi-Implicit Multiscale Scheme for Shallow Water Flows at Low Froude Number

A new large time step semi-implicit multiscale method is presented for the solution of low Froude-number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite-volume discretization is based on a Cartesian grid and is second order accurate. The basic properties of the method are validated by numerical tests. This development is a further step in the development of asymptotically adaptive numerical methods for the computation of large scale atmospheric flows

An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface Omega(t) in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic 7 × 7 system that is highly nonlinear. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features

Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance. We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs for particular examples in one spatial dimension

On a Cartesian cut-cell methodology for simulating atmospheric ice accretion on aircraft

Atmospheric in-flight ice accretion has been a significant operational hazard in aviation for decades. Super-cooled water droplets impinge on exposed surfaces such as wings and rotor blades. These droplets may freeze on the surface thereby changing lift characteristics and disturb weight and aerodynamic balances. The multiple length scales involved prevent designing dynamically similar flows making traditional aeronautical engineering tools such as wind tunnel experiments not suitable. Therefore, computational fluid dynamics (CFD) methods have proved an attractive alternative to study atmospheric icing effects. However, most approaches are based on simple incompressible models and are only suited for small ice heights due to the difficulty of dynamically tracking the ice accretion. This thesis aims to develop novel mathematical models to capture more relevant phenomena and to improve the numerical methods to allow dynamic tracking of the air-ice interface. The initial chapter presents an augmented air and droplet model which tracks droplet temperatures thereby producing more accurate heat fluxes for the phase transition calculation. Firstly, we validate our novel model for common ice accretion test cases and find excellent agreement with literature. The advantage of the augmented system is demonstrated by applying it to an experimental setup that studies the heat exchange between water droplets and air for various flow conditions. We find excellent agreement between our model and the experiment for all presented cases whereas constant-temperature approaches match only for short interaction times. Finally, we apply the new system to study the droplet temperatures around various aerofoil and find significant temperature differences compared with conventional models. The following chapter studies the freezing process on the wing geometry. Presently, the most advanced model is based on lubrication theory, however, linear terms are truncated. We extend the series expansion to include first order terms and demonstrate that the additional order is necessary to accurately capture the thin film flow on a cylinder. Furthermore, we extend the lubrication-theory- based approach which was limited to simple geometries. The extended model is valid on arbitrary wing shapes making it more relevant for engineers studying real-world problems. The penultimate chapter combines the previous two to give a simulation of the full icing process. We integrate it with a Cartesian cut-cell method which can cope dynamically with moving interfaces. The robustness and performance of the cut-cell techniques allow us to simulate ice growth on real-world geometries. We demonstrate this capability by presenting results of the dynamic ice growth on a NACA 0012 aerofoil - making this the first such numerical experiment.EPSR