2.1 Introduction

In this chapter we consider the Navier-Stokes equations for a compressible fluid and show how they can be put into a form more convenient for turbulent flows. We follow the procedure first introduced by Reynolds in incompressible flows: we regard the turbulent motion as consisting of the sum of the mean part and a fluctuating part, introduce the sum into the Navier–Stokes equations, and time1 average the resulting expressions. The equations thus obtained give considerable insight into the character of turbulent motions and serve as a basis for attacking mean-flow problems, as well as for analyzing the turbulence to find its harmonic components. However, before these governing conservation equations for compressible turbulent flows are obtained, it is appropriate to write down the conservation equations for mass, momentum, and energy.

In the following sections we shall discuss the conservation equations and their reduced forms in terms of rectangular coordinates, and for convenience we shall use the summation notation. For a discussion of the conservation equations in terms of another coordinate system, the reader is referred to [1].

2.2 The Navier–Stokes Equations

The well-known Navier–Stokes equations of motion for a compressible, viscous, heat-conducting, perfect gas may be written in the following form [2]:Continuity

(2.2.1)

Momentum

(2.2.2)

Energy

(2.2.3)

wherethe stress tensor τij, heat-flux vector qj, and total enthalpy H are given by

(2.2.4)

(2.2.5)

(2.2.6)

Inthese equations, Λ is the bulk viscosity (= −2/3μ), μ the dynamic viscosity, k the thermal conductivity, and h the static enthalpy. In Eq. (2.2.4), δij is the Kronecker delta, having the value 1 for i = j and 0 for ij. A summation is understood for repeated indices.

Sometimes it is more convenient to express the energy equation in terms of static enthalpy h, rather than total enthalpy H. Then using Eqs. (2.2.1) and (2.2.2)(2.2.3) becomes

(2.2.7)

whereis the dissipation function.

Equations (2.2.1)–(2.2.7) apply to laminar as well as to turbulent flows. For the latter, however, the values of the dependent variables are to be replaced by their instantaneous values. A direct approach to the turbulence problem, namely the solution of the full time-dependent Navier–Stokes equations, then consists in solving the equations for a given set of boundary or initial values and computing mean values over the ensemble for solutions, as discussed in Section 1.10. Even for the most restricted problem – turbulence of an incompressible fluid that appears to be a hopeless undertaking, because of the nonlinear terms in the equations. Thus, the standard procedure is to average over the equations rather than over the solutions. The averaging can be done either by the conventional time-averaging procedure or by the mass-weighted averaging procedure. Both are discussed in the next section.

2.3 Conventional Time-Averaging and Mass-Weighted-Averaging Procedures

In order to obtain the governing conservation equations for turbulent flows, it is convenient to replace the instantaneous quantities in the equations of Section 2.2 by their mean and their fluctuating quantities. In the conventional time-averaging procedure, for example, the velocity and pressure are usually written in the following forms:2

(2.3.1)

(2.3.2)

whereand are the time averages of the bulk velocity and pressure, respectively, and (xi, t) and p″(xi, t) the superimposed velocity and pressure fluctuations, respectively. The time average or “mean” of any quantity q(t) is defined by

(2.3.3)

Inpractice, is taken to mean a time that is long compared to the reciprocal of the predominant frequencies in the spectrum of q; in wind-tunnel experiments, averaging times of a few seconds to a minute are usual. Clearly, the time average is useful only if it is independent of t0; a random process whose time averages are all independent of t0 is called “statistically stationary.” A nonstationary process (e.g., an air jet from a high-pressure reservoir of finite size) must be analyzed by means of “ensemble averages.” The ensemble average q is the average of a large number of instantaneous samples of q, of which one sample is taken during each run of the process at time t0 after the process starts. A process with periodicity imposed on turbulence, like the flow in a turbomachine, can be analyzed by phase averages – take averages at a given point in space over many events in which a blade is at a given position relative to that point. For further discussions, see [3].

For a fluctuating quantity q″(t), the average, , is zero, that is,

(2.3.4)

Averagevalues similar to Eqs. (2.3.1) and (2.3.2) can be written for the other flow quantities, such as density, temperature, and enthalpy, as follows:

(2.3.5)

(2.3.6)

(2.3.7)

(2.3.8)

where

As an example, let us consider the continuity and the momentum equations in the forms given by Eqs. (2.2.1) and (2.2.2), respectively, and show how they can be obtained by using the conventional time-averaging procedure for compressible turbulent flows. If we substitute the expressions given by Eqs. (2.3.1), (2.3.2), and (2.3.5) into (2.2.1) and (2.2.2) and take the time average of the terms appearing in the resulting equations, we obtain the mean continuity and the mean momentum equations in the following forms:

(2.3.9)

(2.3.10)

Forincompressible flows, d = 0. As a result Eqs. (2.3.9) and (2.3.10) can be simplified considerably; they become

(2.3.11)

(2.3.12)

We see from Eqs. (2.3.9)–(2.3.12) that the continuity and the momentum equations obtained by this procedure contain mean terms that have the same form as the corresponding terms in the instantaneous equations. However, they also have terms representing the mean effects of turbulence, which are additional unknown quantities. For that reason, the resulting conservation equations are undetermined. Consequently, the governing equations in this case, continuity and momentum, do not form a closed set. They require additional relations, which have to come from statistical or similarity considerations. The additional terms enter the governing equation as turbulent-transport terms such as and as density-generated terms such as and . In incompressible flows, the density-generated terms disappear, as is shown in Eqs. (2.3.11) and (2.3.12). In compressible flows, the continuity equation (2.3.9) has a source term, , which indicates that a mean mass interchange occurs across the mean streamlines defined in terms of ūi. It also indicates that the splitting of ui according to Eq. (2.3.1) is not convenient; it is not consistent with the usual concept of a streamline. For that reason, we shall replace the conventional time-averaging procedure by another procedure that is well known in the studies of gas mixtures, the mass-weighted-averaging procedure, which was used by Van Driest [4], Favre [5], and Laufer and Ludloff [6]. Mass-weighted averaging eliminates the mean-mass term and some of the momentum transport terms such as and across mean streamlines. We define a mass-weighted mean velocity

(2.3.13)

wherethe bar denotes conventional time averaging and the tilde denotes mass-weighted averaging. The velocity may then be written as

(2.3.14)

where(xi, t) is the superimposed velocity fluctuation. Multiplying Eq. (2.3.14) by the expression for (xi, t) given by Eq. (2.3.5) gives

Timeaveraging and noting the definition of (xi, t), we get

Fromthe definition of ūi, given by Eq. (2.3.13) it follows that

(2.3.15)

Notethe important differences between the two averaging procedures. In the conventional time averaging, in the mass-weighted averaging,

Similarly, we can define the static enthalpy, static temperature, and total enthalpy thus:

(2.3.16)

(2.3.17)

(2.3.18)

where

(2.3.19)

Also,

(2.3.20)

The expressions...