Arthur Rizzi

Convergence to Steady State

The time-dependent formulation is most often used to compute steady state solutions to the Euler equations. There are several mechanisms that drive the solution to a steady state. Here we shall concentrate on the dissipation effect due to the boundary conditions, and not to the effect of artificial viscosity. Therefore we shall study hyperbolic partial differential equations where the boundary effects are dominant. The results are also valid for more general classes of differential equations of essentially hyperbolic character, as for example the Navier-Stokes equations for high Reynolds numbers. The study is mathematical, much of it repeated from Ref. 1.

Modelling of Vortex Flows: Vorticity in Euler Solutions

In Chapter IX it was stated that wakes, respectively vortex sheets, and vortices can be handled in the frame of inviscid theory. This is done for a long time already in the frame of potential wing theory (see for instance Ref. 1). Panel methods, as discrete potential methods, can be seen as extension and generalization of — linear — potential wing theory to general and complicated aircraft geometries at subsonic and even supersonic speeds (see e.g. Ref. 2). With proper vortex sheet paneling, panel methods can also be employed on delta wings with leading-edge vortices, in this way allowing to compute non-linear lift problems with — linear — potential wing theory (see e.g. Ref. 3).

Fundamentals of Discrete Solution Methods

Wave propagation is an inherent feature of the solutions to the Euler equations. The first part of this chapter surveys the basic theory of wave propagation encountered in hyperbolic differential equations including the concept of haracteristics,1 discontinuities, and weak solutions.2 Boundary conditions and how they affect stability are also discussed.3 The second part introduces basic ideas on discrete methods, their accuracy and consistency, numerical stability and numerical boundary conditions.

Coupling of Euler Solutions to Viscous Models

Viscosity effects are present in every flow. If, however, the momentum forces acting in the flow are much larger than the viscous forces (large Reynolds number), the latter can be neglected. Flow with such properties — inviscid flows — can be described with the Euler equations, or even with the potential equation. The latter holds if the flow in addition is irrotational, and if compressibility effects are weak.

Historical Origins of the Inviscid Model

This book concerns the development and application of numerical methods to the solution of the Euler equations. A brief sketch of the evolution of the ideas that led to the formulation of these equations, the problems to which they were applied, and the efforts to solve them before computers were available then sets the proper perspective for an appreciation of the material that follows.

The Finite Volume Concept

The previous chapter presented fundamental concepts for the difference solution to hyperbolic problems in one space dimension. The situation becomes more complicated when the number of space dimensions increases, especially if the computational region does not conform naturally to a Cartesian mesh.

The Euler Equations

The Euler equations are an approximation of the Navier-Stokes equations with the viscous forces and the volume forces being neglected. The reason for doing so is a) because viscosity and heat conduction in a gas usually play a role only in a thin layer near solid surfaces, the thickness of which is much smaller than the characteristic length of the object being immersed in the fluid flow, b) because the volume forces are usually much smaller than the global forces generated by the dynamics of the flow.

Principles of Upwinding

Since the Euler equations do only contain first derivatives in space and time they obviously have no terms expressing the presence of damping in time and space. So if we construct a numerical scheme for their solution we have to keep in mind the fact that a numerical error creeping somehow into the iterative solution process might grow over all bounds leading to the blow up of the scheme since it is not damped. The art of the program des gner is to find a mean to incorporate numerical damping into the discrete approximation of the Euler equations which is small enough to reproduce the original equations as faithful as possible, but large enough to keep the course of iterations in a well ordered time evolution towards the steady state. One tool for this purpose is the addition of a higher derivative of the flow variables, multiplied by a suited coefficient, to each line of the Euler equations. This is called the artificial viscosity approach, and is the topic of the preceding Chapter V.

Methods in Practical Applications

In this chapter we present some results obtained from solutions computed to the Euler equations. They are a selection mainly from own work that covers a range of flow velocities, from very low to very high, and flow complexities, from simple academic cases to complicated industrial flow problems. These cases illustrate many different phenomena. Some are chosen to demonstrate the type and character of flow separation that is observed in Euler computations, other are chosen to demonstrate the practical utility of Euler solutions for industrial design.

A Note on the Use of Supercomputers

Computational fluid dynamics has had a truly dramatic development process in the last decade. The first broad attempts to solve the Euler equations came at the begin of the Eighties, see for example Ref. 1, the proceedings of one of the first GAMM-Workshops in the field of Numerical Fluid Mechanics in 1979. Now even Navier-Stokes solutions are feasible for complex configurations.