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

499

Appendix D: Optical Methods

in Compressible Flows

Introduction

A number of photographs showing various features of compressible ows, such as shock

waves, are given in the main body of this book. An example of such a photograph is shown

in Figure D.1. A very brief discussion of the methods used to obtain such photographs will

be presented in this appendix.

There are basically three such methods:

1. Shadowgraph

2. Schlieren

3. Interferometer

All of these methods utilize the fact that the speed of light through a gas varies with the

density of the gas, i.e., the fact that the refractive index, n, which is the ratio of the speed of

light in a vacuum to the speed of light in the gas, is a function of density, i.e.,

n = function (ρ) (D.1)

where

n

c

c

=

0

(D.2)

c

0

being the speed of light in a vacuum and c being the speed of light at some point in the

gas.

The relation between n and ρ is approximately linear and is usually written as

n

s

=+1 β

ρ

ρ

(D.3)

where ρ

s

is the density of the gas at 0°C and standard atmospheric pressure and β is a

constant that depends on the type of gas. Typical values of β are given in Table D.1. These

values strictly only apply at a particular wavelength of light.

Equation D.3 is sometimes written in terms of the Gladstone–Dale constant, K, such that

n = 1 + Kρ (D.4)

500 Appendix D

so

K = β/ρ

s

(D.5)

Because the speed of light depends on the density of the gas through which it is passing,

it follows that if the density changes in the gas, the speed of light will be different in dif-

ferent parts of the gas. However, there is another related effect produced by the change in

refractive index. If a beam of light passes through a gas in which there is a density gradient

normal to the direction of the beam, the light will be turned in the direction of increasing

density. This is shown schematically in Figure D.2.

The angle through which the light ray is turned is dependent on the gradient of density

normal to the direction of the light, i.e., for the situation shown in Figure D.2, on dρ/dy

which by virtue of Equation D.3 will be proportional to the gradient of the refractive index,

i.e., on dn/dy.

FIGURE D.1

Typical Schlieren photograph of supersonic ow over a body. (Courtesy of NASA.)

TABLE D.1

β Values for Various Gases

Gas β

Air 0.000292

Nitrogen 0.000297

Oxygen 0.000271

Water vapor 0.000254

Carbon dioxide 0.000451

501Appendix D

Shadowgraph System

Consider a series of light rays passing through a gas in which there is a vertical gradient of

density as indicated in Figure D.3.

Because of the deection of the light rays resulting from the density variations, if a screen

is placed in such a way that it intercepts the light rays that have passed through the gas,

the rays will be crowded together in some places and spread apart in other places as indi-

cated in Figure D.3. When the rays are crowded together, the screen appears lighter than

average, whereas when the rays are spread apart, the screen appears darker than average.

Hence, because of the density gradients in the gas, regions of light and dark will appear on

the screen as indicated in Figure D.4.

If the deection of the light rays shown in Figure D.3 is considered, it will be seen that if

there is a uniform vertical gradient of the density, all of the rays of light will be deected

Height

Density

Ray of

light

Glass

y

ε

FIGURE D.2

Bending of light beam in the presence of a density gradient.

Light

source

Light

source

Collimating

lens

Test section

Test section

Screen or

photographic

plate

Brightness

Average

Brighter

Darker

Average

Brighter

Darker

Average

Average

FIGURE D.3

Shadowgraph system.