A residual distribution approach to the Euler equations that preserves potential flow.

Abstract

The main goal of this work is to solve the steady Euler equations governing inviscid compressible fluid flow as cleanly and accurately as possible, focusing on the limiting cases where the physical behavior of the flow changes. Incompressible flow, where either the free stream Mach number or the local Mach number approach zero, and potential flow which is appropriate to shock-free flow originating in a uniform stream both present difficulties for conventional Euler schemes and the symptoms are generally excessive numerical entropy generation and loss of accuracy. However these cases of zero Mach number or zero vorticity should simply be stepping stones towards the design of an accurate Euler code with shock capturing capability. Following such a line of reasoning, this dissertation will present a numerical implementation of an Euler solver based on the unique decomposition of the steady equations into their simplest homogeneous parts. In the supersonic regime, the decomposition takes the form of scalar advection for entropy, enthalpy and two acoustic variables. In subsonic flow, the acoustic part stays coupled and forms a 2 x 2 system of Cauchy-Riemann type. To further ensure strict decoupling between hyperbolic and elliptic parts at the discrete level, the advective and acoustic equations must be integrated into completely independent cell residuals. Residual-distribution schemes are well suited for solving Euler equations as expressed in the decomposed form above. In the fluctuation-splitting approach, the maximally decoupled hyperbolic and elliptic residuals are split to nodes according to a physically sound distribution method, until reaching convergence. For those fluctuations representing scalar advection, multidimensional upwinding is the optimal choice whereas for residuals originating from the 2 x 2 elliptic subsystem, an accurate approach is least-squares (LS). Within the LS minimization process, updates to the variables underlying the elliptic part, namely p and theta, are constrained such that entropy and enthalpy are unaffected. Computational results reveal high accuracy and robustness for a large range of Mach numbers, with very little sensitivity on mesh orientation or irregularity. However, the convergence rate of the least-squares scheme is currently a hindrance and it should be considerably improved to make the whole method attractive in applications. In regions of incompressible flow, the scheme behaves very well, quite an encouraging fact when knowing that <italic> M</italic> &rarr; 0 is a difficult limit for Euler solvers. In the case of uniform entropy and enthalpy inflow and if no shocks occur, potential flow is recovered with extremely low levels of numerical entropy generation.