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Foundations of Computational Fluid Mechanics

1.1 Basic Concepts of Fluid Mechanics

Computational fluid dynamics attempts to solve the dynamical structures of systems within the fluid approximation, namely when the behaviour of the fluid is dominated by collisions between the constituent molecules. This implies that the mean free path is everywhere smaller than the characteristic dimensions of the problem, and that the mean collision interval short compared to the characteristic time so that the system may be treated as a continuum and individual molecular motions are treated by averaging. These averaging techniques may be applied to plasma when the Debye length is small and the averaging leads to the magneto-hydrodynamic (MHD) approximation (Pert, 2021). The derivation and validity of the dynamical MHD equations are developed in Pert (2021). Only plasma within the MHD approximation will be treated here. The collision-free model of plasma are fully treated in Section 8.1 and in more detail by Birdsall and Langdon (1991) and Hockney and Eastwood (1988).

The key underlying equation of motion in the MHD equations are those of compressible fluid mechanics: MHD requiring the introduction of additional effects. Making use of the concept of time splitting discussed earlier these additional behaviours may be treated by separate additional routines. The key fluid routines thus form a basic core to which additional physics is tied. It is therefore appropriate to examine with some care the treatment of the compressible fluid initially and subsequently the additional terms.

Compressible fluid is described by its thermodynamic state and by the velocity of flow. These values must be specified at each point within the flow field. The thermodynamic state is specified by two state variables (e.g. pressure and density ), the remainder being then known from the equation of state. The velocity is defined by the three-vector components . There are thus five independent quantities, which vary in space and time , and must be determined subject to the constraints on the system. The equations of fluid mechanics contain five equations, which are simply the basic conservation laws of physics applied to moving fluid, whose derivation is well known (Pert, 2013).

Typical problems investigated using these basic fluid techniques include aerodynamics (e.g. aeronautical design or car design), meteorology, oceanography and plasma dynamics.

Let us now examine the characteristic structure of compressible fluid mechanics which, as we shall see, are hyperbolic in form, which leads to problems in formulating numerical approximations.

1.2 The Basic Equations of Fluid Mechanics

1.2.1 Eulerian Frame

The simplest co-ordinate system, and in many cases the simplest, in which to define the flow of fluid is the stationary one which is relative to the laboratory, known as the Eulerian frame.

In the laboratory (or Eulerian) frame, the general equations of fluid mechanics can be written in several forms, of which the general conservation form is most appropriate for a general discussion (Pert, 2013). In Cartesian tensor form, these are:

where is the specific internal energy of the fluid, the specific enthalpy and the heat flux vector. is the momentum flux tensor

is the total stress tensor

where is the viscous stress tensor and the Kronecker delta (1 if and 0 otherwise).

The heat flow vector may be written in terms of the temperature gradient and the thermal conductivity ()

The temperature is determined from the equation of state in the form .

In the laboratory (or Eulerian) frame, the general equations of fluid mechanics can be written in several forms, of which the general conservation form is most appropriate for a general discussion (Pert, 2013).

where is the specific internal energy of the fluid, the specific enthalpy, and the heat flux vector. is the momentum flux tensor

is the total stress tensor

where is the viscous stress tensor and the Kronecker delta

These five equations contain six unknown quantities, namely , to which we must include an additional term, e.g. entropy to obtain closure. This is normally the equation of state : for example for a polytropic gas , where is the adiabatic index of the gas. The equation of state normally includes coefficients like viscosity and thermal conductivity, which allow the viscous stress and thermal conduction flux to be calculated knowing the spatial gradients of velocity and internal energy.

Neglecting the viscous stress , the momentum balance equation may be cast in a more familiar form, Euler's equation, by subtracting the first equation from the second

where is the appropriate sound speed, (namely Conservation Law form.

The set of equations are of the general conservative form

where is the time derivative moving with fluid, is a general conserved quantity and the flux associated with it.

Clearly, for any volume V enclosed by a bounding surface S,

which expresses the same result in an integral form expressing global balance within the system.

1.2.2 Lagrangian Frame

Alternatively, it is frequently convenient to consider the fluid particles stationary within their own frame, known as the Lagrangian frame. The transformation from one frame to the other is accomplished by the advection operator. Thus, if the particle velocity is , it can be easily seen that if the time derivative in the Lagrangian frame, namely with respect to the stationary particle, is , then the transformation from Lagrangian to Eulerian systems for the general quantity is

which we call the advection equation.

1.3 Ideal (Dissipationless) Flow - Hyperbolic Equations

In many fluids, viscosity and thermal conduction are weak (in particular in gases), i.e. the fluid behaves as an ideal dissipationless continuum. To a first approximation in plasma, viscosity can also be neglected and the medium treated as a two-component fluid (electrons and ions), but thermal conduction is an important element and dissipational, which may be treated separately within our numerical scheme. We will therefore seek to develop the numerical approximations in an ideal inviscid fluid and subsequently introduce corrections to the flow model necessary to take account of dissipative effects. In these circumstances, the equations of fluid dynamics (1.1) form a set of quasi-linear hyperbolic equations.

1.3.1 One-Dimensional Isentropic Flow

One-dimensional isentropic flow

Let us consider the simple one-dimensional case of the time-dependent Eulerian non-dissipative system, which can be expressed in the general form

where and are general 'vector sets' (column matrix) of the fluid variables

and is the matrix . This set of three equations represents the general conservation equation in one dimension, where is explicitly a function of the variables alone. It therefore has the form of three first-order quasi-linear differential equations in two variables and .

Since the mathematics is messy if the full set of Eqs. (1.54) are used, it is convenient to reduce them to a simpler form by explicitly forming Euler's equation (1.14) from the mass (1.1a) and momentum conservation (1.1b) equations using the pressure/density relation for the adiabatic sound speed , and taking account of the entropy constancy. We correspondingly replace the energy conservation Eq. (1.1c) by the isentropic condition, such that the entropy of a fluid particle remains constant, Eq. (1.14b).

where is the entropy and is the sound speed squared. For this set

The array is given by substitution

The fluid equations are hyperbolic if the eigenvalues of are real. In this case of an ideal fluid, they can be shown to be , where is the isentropic sound speed: . To find the eigenvalues , solve the determinantal equation

to give . The eigenvectors are

respectively. is the identity matrix: (.

Since the eigenvalues are distinct we may diagonalise by a similarity transformation so that

where is diagonal and is a function of the variable only. Hence, substituting for

for the eigenvalues . The functions , obtained from , are constant on the lines , and are known as Riemann invariants. For this case, they take the form

for the ideal fluid equations.1

In principle, a knowledge of the initial values of these quantities will allow...