Transport Phenomena

In order to ease the discussion we consider in general steady compressible and two-dimensional flow in Cartesian coordinates. The nomenclature is given in Fig. 4.1. The coordinate x is the stream-wise coordinate, tangential to the surface. The coordinate y is normal to the surface. The components of the velocity vector V in these two directions are и and v, the magnitude of the speed is V = V_ = /u[26] [27] + v2.1

v

u

x

The governing equations of fluid flow will be discussed as differential equa­tions in the classical formulation per unit volume, see, e. g., [1, 2]. The fluid is assumed to be a mixture of thermally perfect gases, Chapter 2. However, for the discussion we usually consider it only as a binary mixture with the two species A and B.

Fluid flow transports the three entities

— momentum (vector entity),

— energy (scalar entity),

— mass (scalar entity).

The transport mechanisms of interest in this book are

1. convective transport,

2. molecular transport,2

3. turbulent transport,

4. radiative transport.[28]

For a detailed discussion the reader is referred to the literature, for in­stance [2].

Under “convective” transport we understand the transport of the entity under consideration by the bulk motion of the fluid. In steady flow this hap­pens along streamlines (in unsteady flow we have individual fluid particle
“path lines”, and also “streak lines”, which represent the current locus of particles, which have passed previously through a fixed point). Under which circumstances a flow can be considered as steady will be discussed at the end of this section. If convective transport is the dominant transport mechanism in a flow, we call it “inviscid” flow.

“Molecular” transport happens by molecular motion relative to the bulk motion. It is caused by non-uniformities, i. e., gradients, in the flow field [3]. The phenomena of interest for us are viscosity, heat conduction, and mass diffusion, Table 4.1.

Table 4.1. Molecular transport phenomena. Phenomenon Molecular transport of Macroscopic cause Viscosity momentum non-uniform flow velocity Heat conduction energy non-uniform temperature Diffusion mass non-uniform concentration of species, pressure and temperature gradients

Molecular transport occurs in all directions of the flow field. Dominant directions can be present, for instance in boundary layers or, generally, in shear layers.

If molecular transport plays a dominant role in a flow field, we call the flow summarily “viscous flow” .[29] In fact, as we also will see later, all fluid flow is viscous flow. It is a matter of dominance of the different transport mechanisms, whether we speak about inviscid or viscous flow.

“Turbulent” transport is an apparent transport due to the fluctuations in turbulent flow. In fluid mechanics we treat turbulent transport of momentum, energy and mass usually in full analogy to the molecular transport of these entities, however with apparent turbulent transport properties, see, e. g., [4]. In general the “effective” transport of an entity Фef f is defined by adding the laminar (molecular) and the turbulent (apparent) part of it:

Фeff = Ф1ат + ФЫтЬ – (4.1)

“Radiative” transport in the frame of this book is solely the transport of heat away from the surface for the purpose of radiation cooling (Chapter 3). In principle it occurs in all directions and includes emission and absorption processes in the gas, see, e. g., [2, 3, 5], which, however, are neglected in our considerations.

In Table 4.2 we summarize the above discussion.

Item	Momentum	Energy	Mass
Convective transport	X	X	X
Molecular transport	X	X	X
Turbulent transport	X	X	X
Radiative transport		X

Table 4.2. Schematic presentation of transport phenomena.

The three entities considered here, loosely called the flow properties, change in a flow field in space and in time. This is expressed in general by the substantial time derivative, also called convective derivative [2, 6]

This is the derivative, which follows the motion of the fluid, i. e., an ob­server of the flow would simply float with the fluid.

We note in this context that the aerothermodynamic flow problems we are dealing with in general are Galilean invariant [1]. Therefore we can con­sider a flight vehicle in our mathematical models and in ground simulation (computational simulation, ground-facility simulation) in a fixed frame with the air-stream flowing past it. In reality the vehicle flies through the—quasi­uniform—atmosphere.

The first term on the right-hand side, д/dt, is the derivative with respect to time, the partial time derivative, which describes the change of an entity in time at a fixed locus x, y, z in the flow field.

The substantial time derivative D/Dt itself is a specialization of the total time derivative d/dt

where the observer moves with arbitrary velocity with the Cartesian velocity components dx/dt, dy/dt, and dz/dt.

In our applications we speak about steady, quasi-steady, and unsteady flow problems. The measure for the distinction of these three flow modes is the Strouhal number Sr. We find it by means of a normalization with proper reference values, and hence a non-dimensionalization, of the constituents q of the substantial time derivative eq. (4.2)

q = qref q*■ (4.4)

Here q = t, u, v, w, x, y, z, while qref are the respective normalization parameters. q and qref have the same dimension, whereas q* are the corres­ponding non-dimensional parameters, which are of the order one.

If the residence time is small compared to the reference time, in which a change of flow parameters happens, we consider the flow as quasi-steady, because Sr ^ 0. Steady flow is characterized by Sr = 0, i. e., it takes infinitely long for the flow to change in time (tref ^ж).

Unsteady flow is present if Sr = 0(1). For practical purposes the assump­tion of quasi-steady flow, and hence the treatment as steady flow, is permitted for S’r ^ 0.2, that is, if the residence time tres is at least five times shorter than the reference time tref. This means that a fluid particle “travels” in the reference time tref five times past the body with the reference length Lref.

Again we must be careful with our considerations. For example, the move­ment of a flight vehicle may be permitted to be considered as at least quasi­steady, Section 1.5, while at the same time truly unsteady movements of a control surface may occur. In addition there might be configuration details, where highly unsteady vortex shedding is present.

The flows treated in this book are considered to be steady flows. In the following sections, however, we present and discuss sometimes the governing equations also in the general formulation for unsteady flows.