This chapter is concerned with the main topic of the monograph, namely, the solution of the GRP for quasi-1-D, inviscid, compressible, nonisentropic, time-dependent flow. In Section 5.1 we formulate the problem and study its solution in the Lagrangian and Eulerian frames. In particular, we state and prove the main ingredient in the GRP method, Theorem 5.7. A weaker form of this theorem leads to the “acoustic approximation” (Proposition 5.9). Summary 5.24 gives a step-by-step description of the GRP analysis. In Section 5.2 we present the GRP methodology for the construction of second-order, high-resolution finite-difference (or finite-volume) schemes. Starting out from the (first-order) Godunov scheme, we present the basic (E1) GRP scheme. It is based on the acoustic approximation and constitutes the simplest second-order extension of Godunov's scheme. This is followed by a presentation of the full array of GRP schemes (as well as MUSCL). Generally speaking, the presentation in this chapter follows closely the GRP papers [7] and [10].

The GRP for Quasi-1-D, Compressible, Inviscid Flow

In Section 4.2 we studied the Euler equations (4.45) governing the quasi-1-D flow in a duct of variable cross section. We emphasized in particular the role of the Riemann problem (“shock tube problem”), namely, the IVP subject to initial data (4.100). As we shall see in this chapter, the solution to the Riemann problem is a basic ingredient in the numerical resolution of the flow.

In Definition 2.15 we gave the most practical version of the entropy condition. It limits admissible shocks to those obtained by the intersection of “forward-moving” characteristics. These are therefore discontinuities that “cannot be avoided” or replaced by a rarefaction wave. In this Appendix we give some further insight into this concept of an “entropy satisfying” weak solution to (2.1), (2.2).

Our starting point is the physical notion of a “vanishing viscosity solution.” In general terms, an equation leading to discontinuous solutions [such as (2.1)] is supplemented by “dissipative terms” (also referred to as “viscous terms”). In analogy to the physical situation, such terms have a “smoothing effect” on solutions with large gradients, thus replacing discontinuities by “transition zones” where the solution varies smoothly, albeit rapidly. As the viscous effects are diminished, those transition zones shrink to surfaces of zero width, across which the solution has a sharp jump. Mathematically speaking, the additional viscous terms are often represented by second-order derivatives with a small (“vanishing”) coefficient.

To illustrate the situation, we consider the “moving step” problem for Burgers' equation (Example 2.12).

The phenomenological theory discussed in the previous chapter did not permit the parameterization of the energy dissipation. In this chapter spectral turbulence theory will be presented to the extent that we appreciate the connections among the turbulent exchange coefficient, the energy dissipation, and the turbulent kinetic energy. In the spectral representation we think of the longer waves as the averaged quantities and the short waves as the turbulent fluctuations. Since the system of atmospheric prediction equations is very complicated we will be compelled to apply some simplifications.

Fourier representation of the continuity equation and the equation of motion

Before we begin with the actual transformation it may be useful to briefly review some basic concepts. For this reason let us consider the function a(x) which has been defined on the interval L only. In order to represent the function by a Fourier series, we extend it by assuming spatial periodicity. Using Cartesian coordinates we obtain a plot as exemplified in Figure 12.1. The period L is taken to be large enough that averaged quantities within L may vary, i.e. the averaging interval Δx ≪≪ L.

Certain conditions must be imposed on a(x) in order to make the expansion valid. The function a(x) must be a bounded periodic function that in any one period has at most a finite number of local maxima and minima and a finite number of points of discontinuity.

Introduction

The vertical structure of the atmospheric boundary layer is depicted in Figure 13.1. The lowest atmospheric layer is known as the laminar sublayer and has a thickness of only a few millimeters. It is difficult to verify the existence of this layer because of its small vertical extent. Within the laminar sublayer all physical processes such as the transport of momentum and heat are regulated by molecular motion. In most boundary layer models the existence of this layer is not explicitly treated. It stands to reason that there also exists some type of a transitional layer between the laminar sublayer and the so-called Prandtl layer where turbulence is fully developed.

The lower boundary of the Prandtl or surface layer is the roughness height z0 where the mean wind is assumed to vanish. The vertical extent of the Prandtl layer is regulated by the thermal stratification of the air and may vary from about 20 to 100 m. In this layer all turbulent fluxes are approximately constant with height. The influence of the Coriolis force may be ignored this close to the earth's surface, so the turning of the wind within the Prandtl layer may be ignored. The wind speed, however, increases very strongly in this layer, reaching a value of more than half the wind speed at the top of the boundary layer.

This chapter addresses one of the most central issues of computational fluid dynamics, namely, the simulation of flows under complex geometric settings. The diversity of these issues is briefly outlined in Section 8.1, which points out the role played by the present extensions: the (1-D) “singularity tracking” and the (2-D) “moving boundary tracking” (MBT) schemes. Section 8.2 deals with the first extension, and Section 8.3 is devoted to an outline of the second one. In the former we present the scheme methodology and refer to GRP papers for examples. In the latter, the basic principles of the method are presented, and we refer to [39] for more algorithmic details. Finally, an illustrative example of the MBT method shows how an oval disk is “kicked-off” by a shock wave.

Grids That Move in Time

In Part I of this monograph we dealt with finite-difference approximations to the quasi-1-D hydrodynamic conservation laws, where the underlying grid was fixed and equally spaced in the majority of cases. In our two-dimensional numerical extension (Section 7.3) we restricted the treatment to a Cartesian (rectangular) grid. Naturally, finite-difference approximations assume their simplest form on such grids, and the motivation for seeking geometric extensions comes primarily from physical applications.

Introduction

The representation of atmospheric flow fields by means of spherical functions has a long history. Haurwitz (1940) represented the movement of Rossby waves by means of spherical functions. The development of the spectral method for the numerical integration of the equations of atmospheric motion goes back to Silberman (1954), who integrated the barotropic vorticity equation in spherical geometry. The spectral method attracted the attention of others and studies were performed, for example, by Lorenz (1960), Platzman (1960), Kubota et al. (1961), Baer and Platzman (1961), and Elsaesser (1966). Lorenz demonstrated that, for nondivergent barotropic flow, the truncated spectral equations have some important properties. Just like the exact differential equations, they preserve the mean squared vorticity, called enstrophy, and the mean kinetic energy. Platzman pointed out that this very desirable property automatically eliminated nonlinear instability, which at that time was a substantial difficulty in grid-point models. The early work made use of the so-called interaction coefficients to handle nonlinearity. This cumbersome procedure was replaced by the efficient transform technique for solving the spectral equations, which was devised independently by Orszag (1970) and by Eliasen et al. (1970). In compressed form the essential information on spectral modeling is given by Haltiner and Williams (1980). Much valuable information about spectral techniques – which is usually not readily available – can be extracted from the “gray” literature. We refer to an excellent report by Eliasen et al. (1970). Finally, we refer the reader to an excellent article on “Global modelling of atmospheric flow by spectral methods”, by Bourke et al. (1977).

The numerical investigation of specific meteorological problems requires the selection of a suitable coordinate system. In many cases the best choice is quite obvious. Attempts to use the same coordinate system for entirely different geometries usually introduce additional mathematical complexities, which should be avoided. For example, it is immediately apparent that the rectangular Cartesian system is not well suited for the treatment of problems with spherical symmetry. The inspection of the metric fundamental quantities gij or gij and their derivatives helps to decide which coordinate system is best suited for the solution of a particular problem. The study of the motion in irregular terrain may require a terrain-following coordinate system. However, it is not clear from the beginning whether the motion is best described in terms of covariant or contravariant measure numbers.

From the thermo-hydrodynamic system of equations, consisting of the dynamic equations, the continuity equation, the heat equation, and the equation of state, we will direct our attention mostly to the equation of motion using covariant and contravariant measure numbers. We will also briefly derive the continuity equation in general coordinates. In addition we will derive the equation of motion using physical measure numbers of the velocity components if the curvilinear coordinate lines are orthogonal.

In order to proceed efficiently, it is best to extend the tensor-analytical treatment presented in the previous chapters by introducing the method of covariant differentiation.

Computational fluid dynamics (CFD) is a relatively young branch of fluid dynamics, the other two being the experimental and the theoretical disciplines. Its rapid development was enabled by the spectacular progress in high power computers, as well as by a matching progress in numerical schemes.

The starting point for the formulation of CFD schemes is the governing equations. In fact, the term “fluid dynamical equations” is much too general and indeed ambivalent. In practice there exist numerous models of such equations. They reflect a variety of stipulations on the nature of the flow, such as compressibility, viscosity, or elasticity. They also involve various effects such as heat conduction or chemical reactions. A large portion of these models do not fall, mathematically speaking, under the category of “hyperbolic conservation laws,” which is the subject matter of this monograph. We refer the reader to the book by Landau and Lifshitz [75] for a general survey of fluid dynamical models.

In this monograph we are concerned with time-dependent, inviscid, compressible flow, which is studied primarily in the “quasi-one-dimensional” geometric setting. This leads to a system of partial differential equations expressing the conservation of mass, momentum, and energy. There are various approaches to the numerical resolution of this system, such as the classical method of characteristics or the “artificial viscosity” scheme.