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

427

13

Low-Density Flows

Introduction

It has been assumed in all of the preceding discussion in this book that the gas behaves as a

continuum, i.e., that the molecular nature of the gas does not have to be considered in ana-

lyzing the ow of the gas. However, it may not be possible to use this continuum assumption

in the analysis of the ow when the density of the gas is very low. Flows in which the density

is so low that noncontinuum effects become important are often termed “rareed gas ows.”

The conditions under which noncontinuum effects become important and the nature

of the changes in the ow produced by these effects is the subject of this chapter.

Noncontinuum effects can have an important inuence on the ow over craft operating

at high altitudes at high Mach numbers. They can also have an important inuence on

the ow in high vacuum systems. However, because this book is intended to give a broad

introduction to compressible uid ows, no more than a very brief introduction to the

topic of noncontinuum effects will be given here despite their signicant practical impor-

tance in a number of situations.

Knudsen Number

A gas can be assumed to behave as a continuum if the mean free path, i.e., the average

distance that a molecule moves before colliding with another molecule, λ, is small com-

pared with the signicant characteristic length, L, of the ow system. The ratio of λ/L is, of

course, dimensionless and is called the Knudsen number, Kn, i.e.,

Kn

L

=

λ

(13.1)

To relate the Knudsen number to the dimensionless parameters used elsewhere in the

study of compressible ows, it is convenient to be able to relate the coefcient of viscos-

ity to the mean free path. To do this, consider three layers distance λ apart in the ow as

shown in Figure 13.1.

Because molecules arriving at plane A shown in Figure 13.1 from plane B have not col-

lided with any other molecules over the distance λ, they arrive with an excess mean veloc-

ity of λ ∂u/∂y. Similarly, molecules arriving at plane A from plane C arrive with a mean

velocity decit of λ ∂u/∂y. When the molecules from planes B and C arrive at plane A, they

428 Introduction to Compressible Fluid Flow

collide with the molecules on this plane and attain the mean velocity on this plane. Because

of the change in mean momentum that therefore occurs at plane A, there is effectively a shear

force acting on plane A that is proportional to the excess or decit of momentum with which

the molecules arrive, i.e., proportional to λ ∂u/∂y. The net force per unit area will then be

proportional to the number of molecules arriving per unit area per unit time multiplied by

λ ∂u/∂y. Now the number of molecules arriving per unit area per unit time on plane A will

depend on the number of molecules per unit volume, i.e., on the density, and on the mean

speed of the molecules, c

m

, i.e., the force per unit area, the shear stress, τ, will be given by an

equation that has the form

τρ

∝λ

c

u

y

m

∂

∂

(13.2)

However, by denition, the coefcient of viscosity, μ, is given by

τµ=

∂

∂

u

y

(13.3)

Comparing these two equations indicates that

μ ∝ ρc

m

λ (13.4)

However, the mean molecular speed is proportional to the speed of sound, a, since a

sound wave is propagated as a result of molecular collisions. Hence, Equation 13.4 gives

μ ∝ ρaλ (13.5)

from which it follows that

λ∝

µ

ρa

(13.6)

A more complete analysis gives

λγ

= 126. µρ/

a

.

Using Equation 13.6, it will be seen from the denition of the Knudsen number given in

Equation 13.1 that

Kn

aL VL

V

a

∝

µ

ρ

µ

ρ

=

B

A

C

u

u

u

u

u

+

–

λ

y

y

∂

∂

∂

∂

λ

λ

λ

FIGURE 13.1

Layers considered in the analysis of viscosity.