Aerodynamics

Abstract

Computational methods are now pervasive in the science of aerodynamics. Because previously existing numerical methods proved inadequate for fluid flow simulations, the emergence of computational fluid dynamics (CFD) as a distinct discipline has sparked the development of an entirely new class of algorithms and a supporting body of theory, which are the main theme of this article. After a review of mathematical models of fluid flow, methods for solving the transonic potential flow equation (of mixed type) are examined. The central part of the article discusses the formulation and implementation of shock‐capturing schemes for the Euler and Navier–Stokes equations. The article next discusses the merits of the contrasting approaches of finite difference, finite volume, and finite element methods for the treatment of flows in complex geometric domains, together with the treatment of boundary conditions. A detailed presentation of explicit and implicit time‐stepping schemes for both steady and unsteady flows precedes a discussion of preconditioning and multigrid procedures for accelerating the convergence of steady state calculations. Dual time‐stepping schemes are recommended for the simulation of unsteady flow. In order to realize the potential benefits of CFD, it is essential to move beyond simulation to aerodynamic (and ultimately multidisciplinary) optimization. The article concludes with a discussion of aerodynamic shape optimization via control theory.

Efficient Relaxation Methods for High-Order Discretization of Steady Problems

“…For steady problems however, temporal order of accuracy is immaterial, and the use of this scheme is merely justified by the fact that simpler schemes, such as forward Euler or the 2-stage TVD RK scheme [Shu and Osher (1988)] are not linearly stable with DG or Spectral Difference methods, which may lead to overactive limiters in the TVD discretization, and hence to compromised accuracy. An alternative are low order but high-CFL number schemes, such as TVD / SSP schmes [Shu (1988); Gottlieb et al (2001)] or Jameson's high-CFL number multistage schemes [Jameson (1983[Jameson ( , 1993[Jameson ( , 2004] , which have been very popular in standard finite-volume CFD computations. These latter schemes have been designed using Fourier analysis for a linear model equation with the aim to maximize the stability region and at the same time provide good high-frequency error damping properties, which improves performance within multigrid algorithms.…”

“…Good results are usually obtained using W cycles, following standard nomenclature, see e.g. [Jameson (2004)]. These are defined by allowing transfer to the next higher level only if the solution has been advanced twice on the current mesh.…”

On nonlinear preconditioners in Newton–Krylov methods for unsteady flows

“…This is done using the nonlinear multigrid method for the computation of steady flows of Jameson et al [4]. There, two special Runge-Kutta schemes for the convective and the dissipative fluxes, which have large stability regions, are used as a smoother.…”

Blending Dual Time Stepping and Newton‐Krylov Methods for Unsteady Flows

“…This is done using the nonlinear multigrid method for the computation of steady flows of Jameson et al [1]. For Euler flows, this needs only three to five multigrid steps per time step, whereas for Navier-Stokes flows, this is significantly slower and sometimes more than a hundred steps are needed for convergence.…”