Elements of Gasdynamics Liepman

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38 ELEMENTS OF G A S D Y N A M I C S

c, c, T), because the contribution to c, from the vibrational modes

depends on the temperature.

3) At still higher temperature, 1 and h h(p, T) because of dis-

sociation and ionization, i.e., because of processes tha t change the number

of particles.

4) A gas is both thermally and calorically perfect if T, T 0 ,

and p p,.

5) For monatomic gases the effects due to vibrational modes and dis-

sociation are absent.

2 1 Introduction

C H A P T E R

One-Dimensional Gasdynamics

We shall begin our study of the motion of compressible fluids with the

case of one-dimensionalow.

This definition applies to flow in a channel ortube, such as that illustrated in Fig. 2.1, which may be described by specify-

ing the variation of the cross-sectional area along its axis, A A(x), and

in which the flow properties are uni-

form over each cross-section, th at is,

p p(x), p p(x), etc. Similarly,

the velocity u, which is normal to

the cross-section, should be uniform

over each section, u x ) . These

quantities may also be functions of

time t, i the flow is nonstationary, orFIG. 1 One-dimensional flow in a stream

nonsteady. tube.

These conditions are not as restric-

tive as they may appear. For instance, if there are sections over which the

flow conditions are not uniform it is still possible to apply th e results be-

tween sections where they are uniform, tha t is, one-dimensional. Even at

nonuniform sections the results may often be applied to suitable mean

values.Furthermore, the one-dimensional results are applicable to the individual

stream tubes of a general, three-dimensional flow, x being the coordinate

along the stream tube.? We shall see in Chapter 7 what additional relations

are needed for this application.

For a n incompressible fluid, practically all the information about a one-

dimensional flow is contained in the kinematic relation, u is inversely

proportional to A ; the pressure is obtained from the (independent)

Bernoulli equation. In compressible flow, on the other hand, the variation

of the density makes the continuity and momentum equations inter-

dependent, and the relation between velocity and area is then not SO simple.

t I n Chapter 7 we uses fo r the streamline coordinate but x appears to he more convenient

here

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56 E LE M E NTS OF GA S DYNA M I C SONE -DI M E NS I ONA L GA S DYNA M ZC S 57

2.12 Flow at Constant Area apply locally to conditions on either side of a shock, provided it is normal

to the streamline (Fig. 2.9~).Consider adiabatic, constant-area flow (Fig. 2 . 9 ~ )hrough a nonequilib- Of course, the shock relations may be applied to equilibrium sections of

rium region (shown shaded). If sections 1 and 2 are outside this region, real constant-area ducts, such as the one shown in Fig. 2.9a, but it is neces-then the equations of continuity, momentum, and energy are sary that the friction forces on the walls be negligible, since there are no

/xu1 = ~ 2 ~ 2 (2. 41~ ) friction terms in the momentum equation. An example is the constant-area

(2.41b)supersonic diffuser, in which an adverse pressure gradient reduces the wall

p1+ plu12 = p2 + ~ 2 ~ 2 ~friction to negligible values. The diffusion occurs through a complicated,

hl + $u12= h2 + 3 ~ 2 ~ (2.416) three-dimensional process involving interactions between shock waves and

boundary layer. For equilibrium to be attained the diffuser must be long, inThe solution of these gives the relations tha t must exist between the flow

curious contrast to the normal shock, for which equilibrium is reached in aparameters at the two sections; it will be worked out presently.

very short distance.

Figure 2.10~hows an example of compression in a constant-area duct;

mijJfbFig. 2.10b shows, in contrast , an example of a normal shock wave.

2 13 The Normal Shock Relations for a Perfect Gas

0Equations 2.41 are the general equations for a normal shock wave. I t

will usually be necessary to solve them numerically (see Exercise 3 . 6 ) .a) b ) c) However, for a gas that is thermally and calorically perfect i t is possible

FIG. 2.9 Illustrating a change of equilibrium conditions in constant area flow. a) Uni- to obtain explicit solutions in terms of the Mach number M1 ahead ofform conditions on either side of a region of nonuniformity or dissipation; b ) normal the shock.

shock wave; c) shock wave normal t o flow on st:eamline a-b. Dividing the two sides of the momentum equation (2.41b), respectively,

by plul and p2u2, which are equal from the continuity equation, givesThere is no restriction on the size or details of the dissipation region so

long as the reference sections are outside it. In particular, i t may be ideal- U 1 - U 2 = - - - = - - -2 PI az2 a12ized by the vanishingly thin region, shown in Fig. 2.9b, across which the p2u2 plul yu2 YUI

flow parameters are said to jump. The control sections 1 and 2 may thenbe brought arbitrarily close to it. Such a discontinuity is called a shock Here the perfect gas relation a2 = yplp has been used. Then a12 and

wave.t Of course, a real fluid cannot have an actual discontinuity, and thisa22 may be replaced by using the energy equation for a perfect gas,

is only an idealization of the very high gradients tha t actually occur in a u12 a12 uz2 az2 Y 1 a 2- - = - - - - -

shock wave, in the transition from state 1 to 2. These severe gradients 2 7 - 1 2 7 - 1 2 7 - 1produce viscous stress and heat transfer, i.e., nonequilibrium conditions,

inside the shock. After some rearrangement there is obtained the simple relationThe mechanism of shock-wave formation, as well as some details of condi-

ulu2 =tions inside the dissipation region, will be discussed later. For application b(2.42)

to most aerodynamic problems, it is sufficient to calculate the jumps in the This is known as the Prandtl or Meyer relation.equilibrium values, and to represent the shock as a discontinuity. Since In terms of the speed ratio M* = ula*, this equation isthe reference sections may be brought arbitrarily near to the shock, the

device of a constant area duct is no longer needed, tha t is, the results always M*2 = l/M*l (2.43)

?In this book a shock wave will always be represented by a double line, as in Fig. 2.9b Now M* 1 corresponds to M 1, and thus the Prandtl relation showsand c No implication about its structure is intended. On shadowgraphs e.g., Fig.2.10b)

tha t the velocity change across a normal shock must be from supersonic toa shock wave appears as a dark line followed by a bright line, for reasons explained in

Article 6.13. subsonic, or vice versa. I t will be shown later t hat only the former is



















































































































































SMALL PERTURBA TION THEORY 203

Small Perturbation

Theory

8.1 Introduction

I n a great nu mber of aerodynamic problems one is interested in the

perturbation of a known fluid motion. The most common and obvious

case is th at of a uniform, stea dy flow (Fig. 8.1) . Le t U denote the uniform

velocity, and choose a coordinate system in which U is parallel to the

a) Uniform flow b ) erturbed flow

FIG.8 1 Perturbation of a uniform flow by a thin body

XI-axis. Density, pressure, an d tem perature are also uniform in this basicmotion and will be denoted by om p Tm, respectively. Th e corresponding

velocity of sound is am, nd the M ach number is U / a M,. Th e velocity

field of t his ba sic flow is given by

Assume now that a solid body, for example, an airfoil, is placed in this

uniform stream. The body disturbs the basic motion, and changes itsvelocity field, which, in the presence of the body, may be written,

U I w are called induced or pert urbation velocity components.

202

Th e object of this chap ter is to study th e case for which these perturba-

tion velocities are small compared with the mean velocity U . We shall

assume that

and shall simplify the equations of motion by neglecting small terms in the

perturbation velocities. In this way we shall be able to arrive a t equations

which, though not always linear, are still much simpler than the full equa-

tions, and which form the basis for by far the largest p art of airfoil theory,

wing theory, flow past slender bodies, wind tu nnel inte rference problems,

transonic flow, etc.

According to convenience, we shall use the notation X I , x2, x3 or x, y zfor the coordinate system. xl or x will usually be in the direction of the

undisturbed flow; in two-dimensional problems, it is customary to use y as

the normal coordinate, whereas, in problems involving wing-like (planar)

bodies, it is conventional to let z be the coordinate normal to the wing plane,

and y the spanwise coordinate.

8 2 Derivation of the Perturbation Equations

Th e equations of motion for s tead y frictionless flow were obtained in

Article 7.12, in the form

Writing this out in full, and substituting the velocity field defined in Eq.

8.1, we obtain the equ ation in terms of per turbation velocities

a2 ma y be obtained in term s of th e perturbation velocities, from the energy

eq u a t io n (7 .5 0 ~)or a perfect gas,





















































































































































































































































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Elements of Gasdynamics Liepman