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

305

10

Flow with Heat Transfer

Introduction

Most of the previous chapters in this book have been devoted to a discussion of ows in

which the effects of viscosity could be neglected over most of the ow eld and to ows that

could be assumed to be adiabatic. However, the effects of viscosity and heat transfer can

be of dominant importance in some ow situations. In this chapter, consideration will be

given to some illustrative situations involving compressible ows in which heat transfer has

a signicant effect. This heat transfer to the ow may be the result of heat transfer from the

walls over which the uid is owing or it may be the result of chemical reactions in the ow.

This chapter is basically broken down into three sections. The rst section is predomi-

nantly concerned with external ows, i.e., ows over the surface of a body, and deals with

so-called aerodynamic heating and with the factors that affect the heat transfer rate in

such situations. Only a brief introduction to the topic is given. The second section is con-

cerned with heat transfer effects in internal ows of the type dealt with in the previous

chapter except that here the ows considered are not adiabatic. The third section deals

very briey with two types of shock wave in which heat generation or release plays a very

important role.

Aerodynamic Heating

When a gas ows over a surface, the gas in contact with the surface is brought to rest as

a result of viscosity. A rise in temperature is associated with this decrease in velocity at a

surface. If the gas velocity over the surface is high, this temperature rise, associated with the

slowing of the ow near a surface, can become quite large. This, basically, is what is referred

to as the “aerodynamic heating” of a surface. The phenomena is illustrated in Figure 10.1.

The velocity is zero at the surface because of the action of viscosity, and the temperature

rise across the boundary layer is the result of the work done on the ow by the viscous

forces, i.e., the temperature rise is produced by the dissipation of kinetic energy into heat

as the result of the work done by the viscous forces. The temperature rise, i.e., the aerody-

namic heating, is therefore said to be the result of “viscous dissipation.”

Aerodynamic heating is particularly important in hypersonic ows. However, in such

ows, as will be discussed in Chapter 12, changes in the specic heats and the chemi-

cal nature of the gas can occur due to the very high temperatures arising in such ows.

Further, because the temperature rises that occur at the surface are so high in hypersonic

306 Introduction to Compressible Fluid Flow

ow, radiation heat transfer can become important. For these reasons, the analysis of aero-

dynamic heating in hypersonic ow can be very complex. Attention in this chapter there-

fore will be restricted to ows at high subsonic and at supersonic velocities. Radiation

effects will be ignored in this chapter.

The Adiabatic Surface Temperature

Consider ow over a nearly at surface at a Mach number M as shown in Figure 10.2. It is

assumed that the surface is adiabatic, i.e., that there is no heat transfer to or from the sur-

face. In this case, the surface temperature is denoted by T

wad

. Consider two points A and B

as shown in Figure 10.2.

Point A is in the freestream outside the boundary layer, whereas point B is on the sur-

face. If the ow between points A and B is adiabatic, then, since M at point B on the surface

of the plate is zero, it follows as shown before using the energy equation that

T

T

M

T

T

wad

∞

∞

∞

=+

−

=1

1

2

2

0

γ

Flow

Boundary

layer

Temperature

profile

Temperature

increase

Adiabatic wall

T

1

T

1

T

wad

>

FIGURE 10.1

Temperature rise near the surface of a body.

M

A

T

B

Adiabatic

surface

∞

FIGURE 10.2

Flow situation considered.