Introduction to Compressible Fluid Flow, Second Edition, 2nd Edition

21

2

Equations for Steady One-Dimensional

Compressible Fluid Flow

Introduction

Many of the compressible ows that occur in engineering practice can be adequately mod-

eled as a ow through a duct or streamtube whose cross-sectional area is changing rela-

tively slowly in the ow direction. A duct is here taken to mean a solid-walled channel,

whereas a streamtube is dened by considering a closed curve drawn in a uid ow. A

series of streamlines will pass through this curve as shown in Figure 2.1. Further down-

stream, these streamlines can be joined by another curve as shown in Figure 2.1.

Since there is no ow normal to a streamline, in steady ow, the rate at which uid

crosses the area dened by the rst curve is equal to the rate at which uid crosses the

area dened by the second curve. The streamlines passing through the curves effectively

therefore dene the “walls” of a duct and this “duct” is called a streamtube. Of course, in

the case of a duct with solid walls, streamlines lie along the walls and the duct is effec-

tively also a streamtube.

In the case of both ow through a streamtube and ow through a solid-walled duct,

there can be no ow through the “walls” of the system, there being no ow through a solid

wall and, by denition, no ow normal to a streamline. The two types of “duct” are shown

in Figure 2.2.

Examples of the type of ow being considered in this chapter are those through the

blade passages in a turbine and the ow through a nozzle tted to a rocket engine, these

being shown in Figure 2.2. In many such practical situations, it is adequate to assume that

the ow is steady and one-dimensional. As discussed in the previous chapter, steady ow

implies that none of the properties of the ow are varying with time. In most real ows

that are steady on the average, the instantaneous values of the ow properties in fact uc-

tuate about the mean values. However, an analysis of such ows based on the assumption

of steady ow usually gives a good description of the mean values of the ow variables.

One-dimensional ow is strictly a ow in which the reference axes can be so chosen that

the velocity vector has only one component over the portion of the ow eld considered,

i.e., if u, v, and w are the x, y, and z components of the velocity vector then strictly for the

ow to be one-dimensional, it is necessary that it be possible for the x direction to be so

chosen that the velocity components v and w are zero (see Figure 2.4).

In a one-dimensional ow, the velocity at a section of the duct will here be given the

symbol V, as indicated in Figure 2.5.

Strictly speaking, the equations of one-dimensional ow are only applicable to ow in

a straight pipe or streamtube of constant area. However, in many practical situations, the

equations of one-dimensional ow can be applied with acceptable accuracy to ows with

22 Introduction to Compressible Fluid Flow

G

H

F

D

A

B

C

E

FIGURE 2.1

Denition of a streamtube.

D

D

A

B

A

B

C

C

Streamlines

FIGURE 2.2

Solid-walled channel and streamtube.

Quasi-one-

dimensional flow

Quasi-one-

dimensional flow

FIGURE 2.3

Typical duct ows.

x

y

v

u

w

z

FIGURE 2.4

One-dimensional ow.

23Equations for Steady One-Dimensional Compressible Fluid Flow

a variable area provided that the rate of change of area and the curvature of the system

are small enough for one component of the velocity vector to remain dominant over the

other two components. For example, although the ow through a nozzle of the type shown

in Figure 2.6 is not strictly one-dimensional, because v remains very much less than u

the ow can be calculated with sufcient accuracy for most purposes by ignoring v and

assuming that the ow is one-dimensional, i.e., by only considering the variation of u with

x. Such ows in which the ow area is changing but in which the ow at any section can be

treated as one-dimensional, are commonly referred to as “quasi-one-dimensional” ows.

Control Volume

The concept of a control volume is used in the derivation and application of many equa-

tions of compressible uid ow. As discussed in the previous chapter, a control volume

is an arbitrary imaginary volume xed relative to the coordinate system being used (the

coordinate system can be moving) and bounded by a control surface through which uid

may pass as shown in Figure 2.7.

V

1

V

2

1

2

FIGURE 2.5

Denition of velocity V.

v

u

x

FIGURE 2.6

Flow situation that can be modeled as one-dimensional ow.

Flo

w

Control volume

FIGURE 2.7

Control volume in a general two-dimensional ow.