applied mathematics and mechanics

520 Pages·2013·10.77 MB·English

APPLIED MATHEMATICS
AND MECHANICS

An International Scries of Monographs

EDITORS

F. N. FRENKIEL

University of Minnesota

Minneapolis,
Minnesota

G. TEMPLE

Mathematical
Institute

Oxford
University

Oxford,
England

Volume 1. K. OSWATITSCH: Gas Dynamics, English version by G.
Kuerti

(1956)

Volume 2. G. BIRKHOFF and Ε. H . ZARANTONELLO: Jet, Wakes, and Cavities

(1957)

Volume 3. R. VON MISES: Mathematical Theory of Compressible Fluid Flow,

Revised and completed by Hilda Geiringer and G. S. S. Ludford

(1958)

Volume 4. F. L. A L T : Electronic Digital Computers—Their Use in Science

and Engineering (1958)

Volume 5A. W. D. HAYES and R. F. PROBSTEIN: Hypersonic Flow Theory,

second edition, Volume I, Inviscid Flows (1966)

Volume 6. L. M. BREKHOVSKIKH: Waves in Layered Media, Translated from

the Russian by D. Lieberman (1960)

Volume 7. S. FRED SINGER (ed.): Torques and Attitude Sensing in Earth

Satellites (1964)

Volume 8. M I L T O N V A N DYKE: Perturbation Methods in Fluid Mechanics

(1964)

Volume 9. ANGELO MIELE (ed.): Theory of Optimum Aerodynamic Shapes
(1965)

MATHEMATICAL THEORY

OF COMPRESSIBLE FLUID FLOW

RICHARD VO N MISES

late Gordon McKay Professor of Aerodynamics

and Applied Mathematics, Harvard University

Completed by

HILDA GEIRINGER
G . S. S. LUDFORD

1958

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First Printing, 1958

Second Printing, 1966

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE

When Richard von Mises died suddenly in July, 1953, he left the first
three chapters (Arts. 1-15) of what was intended to be a comprehensive
work on compressible flow. By themselves these did not form a complete
book, and it was decided to augment them with the theory of steady plane
flow, which according to von Mises' plan was the next and last topic of
the first part.

This last work of Richard von Mises embodies his ideas on a central
branch of fluid mechanics. Characteristically, he devotes special care to
fundamentals, both conceptual and mathematical. The novel concept of
a specifying equation clarifies the role of thermodynamics in the mechanics
of compressible fluids. The general theory of characteristics is treated in an
unusually complete and simple manner, with detailed applications. The
theory of shocks as asymptotic phenomena is set within the context of
rational mechanics.

Chapters I V and V (Arts. 16-25) were written with the author's papers
and lecture notes as guide. A thorough presentation of the hodograph
method includes a discussion and comparison of the modern integration
theories. Shock theory once more receives special attention. The text ends
with a study of transonic flow, the last subject to engage von Mises' in­
terest.

In revising the existing three chapters great restraint was exercised, so
as not to impair the author's distinctive presentation; a few sections were
added (in Arts. 7, 9, 15). More than forty pages of Notes and Addenda,
partly bibliographical and historical, and partly in the nature of appendices,
follow the text. This is in line with von Mises' practice of keeping text free
from distraction, while at the same time providing a fuller background.
The text is, however, completely independent of the Notes.

Every facet of the work was studied jointly by us, in an attempt to con­
tinue in the author's spirit. Final responsibility for the text and the Notes
to Arts. 16-21, 25 and the Notes to Chapters I, I I lies, however, with Hilda
Geiringer (Mrs. R. v. Mises) and for the text and Notes to Arts. 22-24
and the Notes to Chapter I I I with G. S. S. Ludford.
The present book contains no extensive discussion of the approximation
theories, which have proved to be so fruitful. It was the author's intention
to discuss these in the second part of his work, along with various other

ν

vi
P R E F A C E

topics. The book has been written as an advanced text-book in the hope
that both graduate students and research workers will find it useful.

We are greatly indebted to many people for help given in various phases
of our task. Helen K. Nickerson, who was the much appreciated assistant
of von Mises in the writing of the first three chapters, helped in their
later revision and read Chapters IV, V with constructive criticism. The
whole manuscript was read by G. Kuerti, who suggested important im­
provements. S. Goldstein at times gave us the benefit of his unique in­
sight into the whole field of mechanics. M . Schiffer was always ready with
discussion and advice on delicate questions of a more mathematical nature.
The influence of C. TruesdelFs important contributions to the history of
mechanics is obvious in many Notes; he also readily provided more specific
information.

Very able assistance was rendered by W . Gibson and S. Schot, whom we
thank cordially for their dedicated interest and valuable help. Thanks are
also due to M . Murgai who prepared the subject index and to D. Rubenfeld
who made the final figures.

We are particularly grateful to F. N . Frenkiel as an understanding and
patient advisor. The work of Hilda Geiringer at the Division of Engineering
and Applied Physics, Harvard University, was generously supported by
the Office of Naval Research; that of G. S. S. Ludford, was carried out
under the sympathetic sponsorship of the Institute for Fluid Dynamics
and Applied Mathematics, University of Maryland, and its director Μ . H.
Martin.

Finally, much more than a formal acknowledgment is due to Garrett
BirkhofT, who enabled H. Geiringer to carry out her task under ideal work­
ing conditions. It is mainly due to his vision and understanding that this
last work of Richard von Mises has been preserved.

H I L D A
G E I R I N G E R

G.
S. S.
LUDFORD
Cambridge, Massachusetts
Fall 1957

The leitmotif, the ever recurring
melody, is that two things are

indispensable
in any reasoning, in any description
we shape

of a segment of reality:
to submit to experience
and to face

the language
that is used, with unceasing
logical
criticism.

from an unpublished paper of R. v. Mises

CHAPTER I

INTRODUCTION

Article 1

The Three Basic Equations

1. Newton's Principle

The theory of fluid flow (for an incompressible or compressible fluid,
whether liquid or gas) is based on the Newtonian mechanics of a small
solid body. The essential part of Newton's Principle can be formulated into
the following statements:

(a) To each small solid body can be assigned a positive number ra,
invariant in time and called its mass; and

(b) The body moves in such a way that at each moment the product
of its acceleration vector by ra is equal to the sum of certain other
vectors, called forces, which are determined by the circumstances
under which the motion takes place (Newton's Second Law). 1

For example, if a bullet moves through the atmosphere, one force is gravity
rag,* directed vertically downward (g = 32.17 ft/sec2 at latitude 45°N);
another is the air resistance, or drag, opposite in direction to the velocity
vector, with magnitude depending upon that of the velocity, etc.

By means of a limiting process, this principle can be adapted to the case
of a continuum in which a velocity vector q and an acceleration vector
dq/dt exist at each point. Let Ρ be a point with coordinates (x, y, z), or
position vector r, and dV a volume element in the neighborhood of P; to
this volume element will be assigned a mass pdV, where ρ is the density,
or mass per unit volume. Density will be measured in slugs per cubic foot.
For air under standard conditions (temperature 59°F, pressure 29.92 in. Hg,
or 2116 lb/ft 2), ρ = 0.002378 slug/ft3, as compared with ρ =
1.94 slug/ft3

for water. The forces acting upon this element are, the external force of

* Vectors will be identified by means of boldface type a, v, etc.; the same letter in
lightface italic denotes the absolute value ox the vector: a = | a |; components will
be indicated by subscripts, as ax , ay . The scalar and vector products of a and b will
be represented by a»b and a X b respectively. In the figures the bar notation
for vectors will be used namely a, g, etc.

1

2
I .
I N T R O D U C T I O N

gravity pgdV
and the internal forces resulting from interaction with ad­

jacent volume elements. Thus, after dividing by dV, the relation

(1)
ρ
= Pg + internal force per unit volume
at

is a first expression of statement (b).

T o formulate part (a) of Newton's Principle, note that the mass to be

assigned to any finite portion of the continuum is given by fpdV and there­

fore, since this mass is invariant with respect to time,

(2)

These two relations will be developed further in succeeding sections, but

first the meaning of the differentiation symbol d/dt occurring in Eqs. (1)

and (2) must be clarified.

The density ρ and the velocity vector q are each considered as functions

of the four variables x, y, z, and t, so that partial derivatives with respect

to time and with respect to the space coordinates may be taken, as well

as the directional derivative corresponding to any direction I, given by

d - = cos (/, x) ^- + cos (Z, y)
+ cos (Ζ, ζ) |- ,
θί
dx
dy
dz

where cos (Ϊ, x), cos (I, y), and cos (Z, z) are the direction cosines defining

the direction I. In particular, if I be taken as the direction of q, the direction

cosines may be expressed in terms of q, to give

d
d
,
d
,
d

where s is used in place of I to designate the direction of the line of flow; for

this direction ds = q dt. By d/dt in Eqs. (1) and (2) is meant, not partial

differentiation with respect to t at constant x, y, z, but rather differentiation

for a given particle, whose position changes according to Eq. (3):

(4)

d
=
^tdx_d_i_dyd^,dzd

dt
dt
dt dx
dt dy
dt dz

d
.
d
,
d
.
d
d
,
d

=
dt +
qxd~x
+
q " d y
+
q*dz
= dt +
qdi'

The acceleration vector dq/dt is the time rate of change of the velocity

vector q for a definite material particle which moves in the direction of q

at the rate q = ds/dt. The operation d/dt may be termed particle differen­

tiation, or material differentiation, as distinguished from partial differentia-

1.2
N E W T O N ' S E Q U A T I O N FOR A N I N V I S C I D F L U I D
3

tion with respect to time at a fixed position. An alternative form of Eq. (4)

is

( θ
d&~h +
(q-grad)-

Here, grad is considered as a symbolic vector with the components d/dx,

d/dy, and d/dz in accordance with the well-known notation of grad / for

the vector with components df/dx, df/dy, and df/dz, which is called the

gradient of f, and the scalar product q*grad means the product q times the

component of grad in the q-direction, i.e., q d/ds = qx d/dx + qy d/dy +

qz d/dz. Equations (4) or (4') will be referred to as the Euler rule of differ­

entiation?

When conditions are such that the partial derivative with respect to

time, d/dt, vanishes for each variable, the phenomenon is called steady, as

in steady flow, steady motion. Particle differentiation then reduces to

q d/ds.

2. Newton's equation for an inviscid fluid

The above equation (1) holds for any continuously distributed mass in

which the density ρ is defined at each point and at each moment of time.

Different types of continua may be characterized by the form of the term

which represents the forces arising from interaction of neighboring ele­

ments. An inviscid fluid is one in which it is assumed that the force acting

on any surface element d§>, at which two elements of the fluid are in con­

tact, acts in a direction normal to the surface element. It can be shown (see,

for example, [16],* p. 2) that at each point Ρ the stress, or force per unit

area, is independent of the orientation (direction of the normal) of dS. The

value of this stress is called the hydraulic pressure, or briefly pressure, p,

at the point P. The word "pressure" indicates that the force is a thrust,

directed toward the fluid element on which it acts: negative values of ρ

are not admitted.

For a small rectangular cell of fluid of volume dV = dx dy dz (see Fig.l),

two pressure forces act in the z-direction, on surface elements of area dy dz.

Taking the x-axis toward the right, the left-hand face experiences a force

ρ dy dz directed toward the right, while the right-hand face experiences a

force (p + dp) dy dz directed toward the left; here dp = (dp/dx) dx. The

resultant force in the rc-direction is thus

— dp dy dz = —^-dxdydz=
— ^
dV.

y
υ
dx
*
dx

Therefore the internal force per unit volume, appearing in Eq. (1), has

* Numbers in brackets refer to the bibliography of standard works, p. 502 if.

4
I. I N T R O D U C T I O N

ζ

~
Γ
^1
dz|- —

(P+dp)

dx

FIG. 1. Rectangular volume element with pressure forces in x-direction.

x-component — dp/dx; similarly, the remaining components are found to

be — dp/dy and — dp/dz. Hence

(I)
P~
= pg ~ grad ρ

expresses part (b) of Newton's Principle. This equation was first given by

Leonhard Euler (1755), but is usually known as Newton's
equation?

The vector Eq. ( I ) is equivalent to the three scalar equations

dqx
dp

dt
dx

(ΐ')

dqy

p l t
=
p
g *

dp

dqz
dp

Equations ( I ) and ( Γ ) are valid only for inviscid fluids. If viscosity is
present, additional terms must be included in the expression for the internal
force per unit volume, thus generalizing Eq. ( I ) . This will be discussed later
(Sec. 3.2).
The motion of the fluid may also be described by following the position
of each single particle of the fluid for all values of t. From this point of view
the so-called Lagrangian equations of motion are obtained, which determine
the position (x, y, z) of each particle as a function of t and of three parame­
ters identifying the particle, e.g., the position of the particle at t = to .4

Except in certain cases of one-dimensional flow, the Eulerian equations are
more manageable.
In either case the following basic concepts apply. The space curve de­
scribed by a moving particle is called its path or trajectory. The family of
such trajectories is defined by the differential equation dr/dt = q(r, t). On
the other hand, at each fixed time t = t0, there is a (two-parameter) family
οϊ streamlines: for t = U each streamline is tangent at each point to the
velocity vector at this point, its differential equation being dx X q(r, t0) = 0,

1.3
E Q U A T I O N OF C O N T I N U I T Y
5

or dx:dy:dz
= qx:qy:qz,
where qx = qx(x, y, z, to), etc. Thus in general the

family
of streamlines varies in time. Clearly the streamline through r at

t = to, is tangent to the trajectory of the particle passing through r at that

instant.

If, in particular, the motion is steady, i.e., at each point Ρ the same

thing happens at all times, then there is only one family of streamlines,

which then necessarily coincides with the trajectories.

In the general case it is convenient to consider, in addition, the particle

lines (or " world lines"). They are defined in four-dimensional x, y, z, i-space,

each line corresponding to one particle. They appear in Chapter I I I as the

x, Mines. A particle path is the projection onto x, y, 2-space of the respective

particle line.

3. Equation of continuity

In order to express part (a) of Newton's Principle—conservation of mass

—in the form of a differential equation, the differentiation indicated in Eq.

(2) could be carried out by transforming the integral suitably (see below,

Sec. 2.6). It is simpler, however, to consider again the rectangular cell of

Fig. 1. Fluid mass flows into the cell through the left-hand face at the rate

of pqx dy dz units of mass per second, and out of the right-hand face at rate

[pqx + d(pqx)] dy dz, where d(pqx) is the product of dx and the rate of change

of pqx in the ^-direction, or [d(pqx)/dx]
dx. Thus the net increase in the

amount of mass present in this volume element caused by flow across these

two faces is given by — [d(pqx)/dx]
dx dy dz, with analogous expressions for

the other pairs of faces. If we use the expression div v, divergence of the

vector v, for the sum (dvx/dx)
+
(dvy/dy)
+
(dvz/dz), the total change in

mass per unit time is —div (pq) dV. Now if the mass of each moving par­

ticle is invariant in time, the difference between the mass entering the cell

and that leaving the cell must be balanced by a change in the density of the

mass present in the cell. A t first, the mass in the cell is given by ρ dV, and

after time dt by (p + dp) dV, where dp = (dp/dt)dt. Thus the rate of change

of mass in the cell, per unit time, is given by (dp/dt) dV, so that

( I D
div(pq) =
- ^ .

This relation, valid for any type of continuously distributed mass, is known

as the equation of
continuity.5

A slightly different form of ( I I ) is obtained by carrying out the differ­

entiation of pq, giving

6
I. I N T R O D U C T I O N

or, using the Euler rule (4)

(ΙΙ')

Either form, ( I I ) or (II')> will be used in the sequel. In the case of a steady

flow, Eq. ( I I ) reduces to

while in the case ρ = constant, that of an incompressible fluid, Eq. ( I F )

gives

whether or not the flow is steady. The equation of continuity remains un­

altered when viscosity is admitted.

4. Specifying equation

The equations (I) and ( I I ) , which express Newton's Principle for the mo­
tion of an inviscid fluid and are usually referred to as Euler's equations, in­
clude one vector equation and one scalar equation, or four scalar equations.
There are, however, five unknowns: qx,
qy , qz,
p, and p, in these four
equations. It follows that one more equation is needed in order that a solu­
tion of the system of equations be uniquely determined for given "boundary
conditions ,\ Boundary conditions, in a general sense, are equations involv­
ing the same variables, holding, however, not in the four-dimensional x, y, z,
ί-space, but only in certain subspaces, as at some surface Φ(χ, y, z) = 0,
for all t (boundary conditions in the narrower sense), or at some time
t = to, for all x, y, ζ (initial conditions).

There exists no general physical principle which would supply a fifth
equation to hold in all cases of motion of an inviscid fluid, as do Eqs. ( I )
and ( I I ) . What can and must be added to ( I ) and ( I I ) is some assumption
that specifies the particular type of motion under consideration. This fifth
equation will be called the specifying equation. Its general form is

where it is understood that derivatives of p, p, and q may also enter into F. 6

The simplest form of specifying condition results from the assumption
that the density ρ has a constant value, independent of x, y, z, and t. It is
evident that if ρ is a constant, the number of unknowns reduces to four,
and ( I ) and ( I I ) are sufficient. This is the case of an incompressible fluid.
The most common form of a specifying equation consists in the assump-

div (pq) = 0,

div q = 0,

( I I I )
F(p, p, q> χ, y, *, t) = o,

1.4
S P E C I F Y I N G
E Q U A T I O N
7

tion that ρ and ρ are variable but connected at all times by a one-to-one
relation of the form

( I l i a )
F(p, p) = 0.

This means that if the pressure is the same at any two points, the density

is also the same at these two points, whether at the same or different mo­

ments in time. Examples of such (ρ, p)-relations are

(5a)
- = constant,
Ρ

(5b)
— = constant,
p"

(5c)
ρ = A - - ,
Β > 0,7
ρ

where κ, A, and Β are constants. In the first of the examples (5) pressure
and density are proportional; in general, it will be assumed that ρ increases
as ρ increases, and vice versa, so that dp/dp > 0. If the specifying equation
is of the form ( I l i a ) , the fluid is called an elastic fluid, because of the analogy
to the case of an elastic solid where the state of stress and the state of strain
determine each other. A large part of the results so far obtained in the
theory of compressible fluids holds for elastic fluids only.

The special assumption that the specifying equation is of the form ( I l i a )

is, however, too narrow to cover, for example, the conditions of the at­
mosphere in the large. It is well known from thermodynamics that for each
type of matter a certain relation exists among the three variables, pressure,
density, and temperature, the so-called equation of state. Thus the tempera­
ture can be computed when ρ and ρ are known. If the atmosphere were
assumed to be elastic, so that a specifying equation of the form ( I l i a ) held,
it would then be sufficient to measure ρ in order to know the temperature
as well. This is obviously not the case, so a specifying equation of the form
( I l i a ) cannot hold for the atmosphere in general. [The equation of state is
not a specifying equation, even of the more general type ( I I I ) , since it
implies temperature as a new variable.] In many aerodynamic problems,
only comparatively small portions of the atmosphere need be considered,
such as the vicinity of the airplane. In such cases, there is no objection in
principle
to the use of a specifying equation of the form ( I l i a ) , if this
brings the solution of the problem within reach.

The particular cases corresponding to the examples of (p, p)-relations
given in (5) can be interpreted in terms of certain concepts from thermo-

8
I. I N T R O D U C T I O N

1
F I G . 2. ρ versus ~ for (1) isothermal and (2) isentropic flow,
ρ

dynamics. For a so-called perfect gas* the equation of state is

(6)
V- =
gRT,
ρ

where Τ is the absolute temperature (°F + 459.7) and R is a constant de­
pending upon the particular gas. 8 For dry air, if considered as a perfect

gas, the value of R can be taken as 53.33 ft/°F. From Eq. (6) it follows
that for a perfect gas the condition (5a) implies a flow at constant tem­
perature, or isothermal flow.

The entropy S of a perfect gas is defined by

(7)
S =
g
R
. log — + constant,
7 - 1
py

where γ is a constant, having the value 1.40 for dry air. Thus the motion
of a perfect gas with the condition (5b) as specifying equation and κ = y
is isentropic. The motion of any fluid having the specifying equation (5b),
with κ > 1, will be termed
polytropic.

If ρ is plotted against 1/p, the specifying equation for an isothermal flow
is represented by (1), an equilateral hyperbola (see Fig. 2); the curve for an
isentropic flow is shown as the dotted line (2). Whenever the variation of ρ
and of p is confined to a small range of values, the relevant part of these, (or
other)* curves can be approximated by a straight line, giving the third type

* A perfect gas is not necessarily inviscid (see [16], p. 83).

1.5. A D I A B A T I C FLOW
9

of specifying condition (5c). This linearized form of the (p, p)-relation

(see Sec. 17.5) often facilitates the solution of a problem. Specifying equa­

tions which are not of the form ( I l i a ) will be discussed later (Arts. 2 and 3).

In the case of steady flow, which was defined by the added condition

d/dt = 0 for all five dependent variables, it would appear that there are

more differential equations than unknowns. Actually, if t does not occur

in the specifying equation, which is true a fortiori
when this equation-is of

the form ( I l i a ) , the system consisting of Eqs. ( I ) , ( I I ) , and ( I I I ) does not

include t explicitly. For such a system the assumption d/dt — 0 at t = 0

leads to a solution totally independent of t. Thus, the assumption of steady

motion is a boundary condition only, according to the definition at the

beginning of this section. The same is true in the case of a plane
motion,

which is defined by the conditions qz = 0 and d/dz = 0 for all other vari­

ables, provided ζ does not occur explicitly in the specifying equation.

5. Adiabatic flow

In many cases the specification of the type of flow is given in thermo­
dynamic terms. It is then necessary, in order to set up the specifying
equation ( I I I ) , to express these thermodynamic variables in terms of the
mechanical
variables. Two
examples
have
already
been
mentioned
above. If it is known (or assumed) that the temperature is equal at all
points and for all values of t, then the equation of state, which is a relation
bet\veen T, p, and p, supplies a relation of the form ( I l i a ) , F(p, p) = 0;
or, if the condition reads that the entropy has the same value everywhere
and for all times, the definition (7) supplies another relation of the type

The most common assumption in the study of compressible fluids is that
no heat output or input occurs for any particle. If this refers to heat trans­
fer by radiation and chemical processes only, the flow is called simply
adiabatic. If heat conduction between neighboring particles is also excluded
we speak of strictly adiabatic motion. In order to translate either assumption
into a specifying equation, the First Law of Thermodynamics must be
used, which gives the relation between heat input and the mechanical
variables. If Q' denotes the total heat input from all sources, per unit of
time and mass, the First Law for an inviscid fluid can be written

where cv is the specific heat of the fluid at constant' volume, and quantity
of heat is measured in mechanical units. The first term on the right repre­
sents the part of the heat input expended for the increase in temperature;
the second term corresponds to the work done by expansion. Equation (8)

( I l i a ) .

(8)