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.

In this chapter we consider the system of equations governing compressible reacting flow. The fluid is a homogeneous mixture of two species. The evolution of the flow under the mechanical conservation laws of mass, momentum and energy is coupled to the (continuous or abrupt) conversion of the “unburnt” species to the “burnt” one. We take the simplest model of such a reaction, namely, an irreversible exothermic process. The equation of state of the fluid depends on its chemical composition. The resulting (augmented) system is still nonlinear hyperbolic (in the sense of Chapter 4) and is amenable to the GRP methodology. The basic hypotheses are presented in Section 9.1, leading to the derivation of the characteristic relations and jump conditions. In Section 9.2 we describe the classical Chapman—Jouguet model of deflagrations and detonations, and the Zeldovich—von Neumann—Döring (Z—N—D) solution is presented in Section 9.3. In Section 9.4 we study the generalized Riemann problem for the system of reacting flow. The treatment here is close to that of the basic GRP case (Section 5.1), but there are significant differences because of the reaction equation. In Section 9.5 we outline briefly the resulting GRP numerical scheme and study a physical problem of ozone decomposition.

Here the fluid dynamical theory and GRP schemes of Chapters 4 and 5 are applied to one-dimensional test cases. The problems are aimed primarily at demonstrating the capabilities of the scheme, but they are also revealing of nontrivial fluid dynamical phenomena that arise even at the relatively simple one-dimensional settings considered here. In Section 6.1 we treat a shock tube problem, using several scheme options to solve it. An interesting class of fluid dynamical problems is that of wave interactions, to which Section 6.2 is devoted. We selected four different cases in this class, shock—shock, shock—contact, shock—rarefaction, and rarefaction—contact interactions. In each case the GRP solution is compared to either an exact one or to a solution of a Riemann problem that approximates the exact one in some “asymptotic” sense. In the remainder of the chapter we employ the quasi-one-dimensional (“duct flow”) scheme, solving three different problems, comparing each numerical solution to the corresponding exact one. Section 6.3 treats a spherically converging flow of cold gas, and Section 6.4 is devoted to the flow induced by an expanding sphere. Finally, in Section 6.5 we present a detailed treatment of the steady flow in a converging—diverging nozzle, obtained numerically as a large-time solution by the GRP scheme.

The notation for the fluid dynamical variables here is identical to that of Chapters 4 and 5.

Introduction

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 over 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. We will discuss this situation later.

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 toward 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 and assuming that the curvilinear coordinate lines are orthogonal.

The starting point of the analysis is the equation of motion in the absolute coordinate system. The description of the motion in a rotating and time-dependent coordinate system, in general, requires knowledge of the rotational velocity of a point in the atmosphere as well as the deformation velocity of the material surface on which the point is located.

This chapter introduces the GRP method in the context of the scalar conservation law ut + f(u)x = 0. We start in Section 3.1 with the classical first-order (conservative) “Godunov Scheme,” which leads naturally to its second-order GRP extension. Section 3.2 contains a number of numerical (one-dimensional) examples, for linear and nonlinear equations, illustrating the improved resolution obtained by the GRP method. In Section 3.3 we extend the GRP methodology to the two-dimensional scalar conservation law ut + f(u)x + g(u)y = 0. Analytical and numerical results are compared for simple as well as complex wave interactions.

From Godunov to the GRP Method

In this section we discuss the GRP method, aimed at a high-resolution numerical approximation of the solution to a conservation law of the form (2.1). We always assume that f(u) is strictly convex:f″(u) ≥ μ > 0. It is shown that this method is a natural analytic extension to the Godunov (upwind) scheme. This latter scheme has been extensively studied in Section 2.2, in the context of the linear convection equation. We start here by studying this scheme in the nonlinear case.

As in Section 2.2, we take a uniform spatial grid xj = jΔx, − ∞ < j < ∞, and uniformly spaced time levels tn+1 = tn+k, t0 = 0.