Principles of Computational Fluid Dynamics

Fluid dynamics is a classic discipline. The physical principles governing the flow of simple fluids and gases, such as water and air, have been understood since the times of Newton. Sect. IX of the second book of Newton’s Principia starts with that came to be known as the Newtionian strees hypothesis: “The resistance arising from the want of lubricity in the parts of a fluid, is, other things being equal, proportional to the velocity in the parts of the fluid are separated from another”. This hypothesis is followed by Proposition LI, in which the flow generated by a rotating cylinder in an unbounded medium is considered. The period of the orbit of a fluid particle is found to be proportional to the distance r from the cylinder axis. This is not correct. This is not correct. The source of the error is that master balances force instead of torque; this may be of some consolation to beginning students who find mechanics difficult. The closing remark “All of this can be tested in deep stagnant water” must be taken with a grain of salt. Newton was more interested in celestial mechanics than in fluid dynamics. His aim was to test Descartes's vortex theory of planetary motion, which would gain credibility if the period of the orbit of a particle in this flow would be proportional to r ^3/2; in fact, it is proportional to r ². The mathematical formulation of the laws that govern the dynamics of fluids has been complete for a century and a half. In the nineteenth century and the beginning of the twentieth, eminent scientists and engineers were drawn to the subject, and gave it clarity, unificaiton and elegance, as exemplified in the classic work fo Lamb (1945), that first appeared in 1879. In the perface to the 1932 edition Lamb writes, that the subject has in recent years received considerable developments, classic fluid dynamics having a widening field of practical applications. This has ramained true ever since, especially because in the last forty years or so classic fluid dynamics finds itself in the company of computational fluid dynamics. This new discipline still lacks the elegance and unification of its classic counterpart, and is in a state of rapid development, so that we can do no more than give a glimpse of its current status. But first, we take a look at classic fluid dynamics.

As seen in the preceding chapter, fluid dynamics in governed by partial differential equations. Therefore knowledge of the numerical analysis of partial differential equation is indispensable in computational fluid dynamics. Introductions to this subject of varying degree of difficulty are Fletcher (1988), Hackbush (1986), Hall and Porching (1990), Hirsch (1988), Lapidus and Pinder (1982), Mitchell and Griffiths (1994), Morton and Mayers (1994), Grossmann and Roos (1994), Strikwerda (1989), Quarteroni and Valli (1994), Richtmyer and Morton (1967); the last book in particular is useful for practitioners of computational fluid dynamics.

Nonuniform grids are often used to obtain accuracy in regious where the solution varies rapidly. We will see that on nonuniform grids, finite volume and finite difference discretization are not equivalent. On arbitrary nonuniform grids the local truncation error is usually larger than on uniform grids, or grids on which the mesh size varies smoothly. This has sometimes led to confusion. Cell-centered finite volume discretization is sometimes advised against, because the local truncation error is larger than for vertex-centered finite volumes, and is in fact of the same order even as the term that is approximated. Nevertheless, this is a good discretization method that is popular in reservoir engineering and porous media flow computation, and in computational fluid dynamics in general. The source of the confusion is that the relation between the local and global truncation error is complicated. Surprisingly, the global truncation error is small, as we will see. Of course, it is the global truncation error that counts.

The convection-diffusion equation has been derived in Sect. 1.11. We take ρ = 1, and obtain For the physical significanece of the terms in this equation, see Sect. 1.11. The equation is assumed to be linear, with φ the only unkown. A number of important aspects of the numerical analysis of the equations of fluid dynamics show up in this simple equation. Its simplicity allows a thorough analysis of these aspects, which will be given in this chapter. Readers who have some experience in computational fluid dynamics man think at first sight that our treatmen of such a simple linear equation is too detailed, but it is a fact that sometimes cotroversial and not always well understood important issues, notably the occurrence of numerical ‘wiggles’, the specification of outflow boundary coditions, singular perturbation aspects (the occurrence of boundary layers when D << 1, in a sense to be made precise shortly, a common situation in fluid dynamics and the role of false (numerical) viscosity, can be brought out and clarified completely in the context of this simple equation.

The equation to be studied in this chapter is the nonstationary convection diffusion equation: See Sect. 2.3 for initial and boundary conditions and further comments, and Theorem 2.4.3 for the maximum principle satisfied by (5.1). Discretization of the spatial derivatives has been discussed in the preceding chapters. In the present chapter, attention will be focussed on temporal discretization. As a consequence, numerical stability will loom large. As in the preceding chapter, the aim is not merely to discuss the numerical solution of the relatively simple equation (5.1), but to prepare the ground for the numerical solution of the equations of fluid dynamics. This will guide our selection of numerical methods to be discussed, from the plethora of schemes available. Further introducation to the subject may be found in Morton (1996), Strikwerda (1989).

In this chapter, the incompressible Navier-Stokes equations in Cartesian co-ordinates discretized on uniform grids will be considered, discussing most of the basic numerical principles in a simple setting. In practical applications, of course, nonuniform grids and general coordinate systems are prevalent; these will be studied in later chapters, as will be the compressible case. We can be relatively brief in discussing discretization of the Navier-Stokes equations, because we prepared the ground in our extensive discussion of the convection-diffusion equation in Chapter 3, 4 and 5. Therefore it will not be necessary to discuss again the various possiblities for discretizing convection, or von Neumann stability conditions.

An in-depth treatment of iterative methods in linear algebra will not be given, because this is a huge subject, that would require a seprate volume. Instead, only a bird’s eye view of the subject will be given, providing ample references to the literature, paying particular attention to difficulties that are peculiar to the incompressible Navier-Stokes equations. The uninitiated reader who wants to use these methods is advised to consult the reference that we will quote. The algebraic systems that arise by finite volume discretization are extremely sparse, and also very large, beacause many gird points are required for accuracy. Therefore iterative methods are more efficient and demand far less storage than direct methods, especially in three dimensions. In two dimensions, direct methods using sparse matrix techniques can still be useful. Hence, we will confine ourselves to iterative methods, with one exception: the equation for the pressure correction can often be solved very efficiently by so-called fast Poission solvers, based on Fourier transformation and/or cyclic reduction.

We now enter the realm of hyperbolic equations. Instead of starting with an introduction to numerical methods for hyperbolic equations in general, we prefer to begin by plunging into the particular case of the shallow-water equations. Other hyperbolic equations will be considered in Chapters 9 and 10. The derivation of the shallow-water equations from the Navier-Stokes equations is given in Sect. 1.16. The resulting equations (1.121) and (1.124) govern to a good approximation flows in channles, lakes, rivers and seas, under the shallowness assumptions mentioned in Sect. 1.16. The core of mathematical models for flows on a planetary scale in the atmosphere and in the oceans of the shallow-water equations, but the computing methods used are specialized, and a sufficient introduction would exceed the confines of this book. We will therefore restrict ourselves mainly to applications in hydraulics, and not go into geophysical fluid dynamics. More extensive introductions to computing methods for the shallow-water equations are given in Abbott (1979), Kowalik and Murty (1993), Vreugdenhil (1994).

In Chap. 8 we have intoduced systems of hyperbolic differential equations by discussing the particular case of the shallow-water equations. Especially in the linerarized case, these form a very simple example of an hyperbolic system, permitting an easy introduction of basic concepts as characteristics, wave fronts and proper boundary conditions. In the nonlinear case, new phenomena occur, notably shocks and other types of discontinuity. These occur when the local flow velocity is larger than the phase velocity of waves. The discretizations discussed in Chap. 8 break down in this case. In order to prepare for the treatment of compressible flows, we give an introduction to scalar conservation laws in the present chapter.

Since almost all computing methods for the Euler equations in two and three space dimensions rely heavily on techniques developed for the one-dimensional case, we devote a full chapter to the one-dimensional Euler equations. In the one-dimensional case many intersting analytic aspects can be brought to light, and this we do first. The shock tube problem is a very useful test problem for discretization schemes; we will present its analytic solution. Then we turn to discretization methods. More extentive introductions to numerical methods for the Euler equationsare given by Godlewski and Raviart (1996), Kröner (1997), Laney (1998), Majda (1984), Toro (1997), Smoller (1983), Hirsch (1990).

In the foregoing chapters we have restricted ourselves to Cartesian computational grids, in order to avoid unnecessary technicalities in discussing the basic principles of computational fluid dynamics. In the present and in the following tow chapters, we will go into the complications brought about by geometric complexity of the domain in which the flow takes place. These complications are mainly of a technical nature, and no new basic principles will emerge. Nevertheless, for successful computation of flows in general domains, the technicalities involved in computation on general grids must be considered carefully.

The purpose of this chapter is to generalize the methods described in Chap. 10 for computing inviscid compressible flows in one dimension to more dimensions and general domains. Although no new principles will emerge, this generalization is not completely straightforward, except for the JST scheme (which is its great charm), which will therefore not be discussed here.

We will mostly discuss the three-dimensional case, leaving the two-dimensional case for the reader to work out. The domain \(\Omega \subset \mathbb{R}^{3}\) is arbitrary, and is assumed to contain a single block boundary-fifted structured grid with hexahedrons as cells, as discussed in Chap. 11. In practice the shape of the domain is often so complicated, that domain decomposition (cf. Sect. 11.3 has to be used. Domain decomposition is also frequently used to create sub-tasks for parallel processing. Conditions to couple subdomain solutions and iterative methods to iterate to global convergence have to be devised.)

Most of the introductory remarks made in Sect. 12.1 apply to the persent chapter as well, and will not be repeated. First, the compressible case will be discussed. The inviscid terms can be discretized as in Chap. 12, so that only the spatial discreization of the viscous terms needs to be given. Then the incompressible Navier-Stokes equations will be considered. The methods discussed in Chap. 6 will be generalized from Cartesian grids to structured boundary-fitted grids. A unified method for both the compressible and the incompressible case will be persented in the following chapter.

If there are regions in the flow domain where the Mach number M is not small, the incompressible Navier-Stocks or Euler equations cannot be applied. In principle, the compressible equations of motion are uniformly valid as the Mach number ranges from zero to supersonic (until real gas effects set in). Therefor it suffices to forego the simplifications that incompressibility brings, and all one has to do is to employ the compressible equations of motion. However, as will be discussed below, the standard numerical methods that have been developed for compressible flow (discussed in Chap. 10 and 12) break down or do not function properly when M ≲ 0.2. Methods to remedy this will be discussed in the present chapter. It would be ideal if a unified method could be found that is accurate and efficient for both compressible and incompressible flows. Such unified methods with accuracy and efficiency more or less uniform in the Mach number have indeed been proposed, and will be discussed below.