A discussion of approximate theories for inviscid hypersonic flow on concave surfaces

June,

1970.

A DISCUSSION OF APPROXIMATE

THEORIES FOR INvISCID HYPERSONIC

FLOW ON CONCAVE SURFACES

by P. A. Sullivan

ftttti i:". -

t"(:tSC: '0('1

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Vl.I'" ~ . • ;"1: f I .. G I U/(;TJOU\A -, uNOl:

BIBLIOTHEEK

29 SEp,

1910

.

.,

..

A DISCUSSION OF APPROXIMATE

THEORIES FOR INVISCID HYPERSONIC FLOW ON CONCAVE SURFACES

by

P. A. Sullivan

Manuscript received December, 1969 •

ACKNOWLEDGEMENT

Much of the work described in this report was completed while the author was a Beit Fellow at the Imperial College of Science and Technology, London, England. Recent interest in a number of problems for which this is relevant led him to revise and extend the work during the past year. He wishes to express his appreciation of the encouragement and advice given by

Mr.

J. L. Stollery [Imperial College] ,

Mr.

J. G. Church assisted with the calculations completed at UTIAS.

AC KNOWLEDG EMENT

Much of the work described in this report was completed while the author was a Beit Fellow at the Imperial College of Science and Technology, London, England. Recent interest in a number of problems for which this is relevant led him to revise and extend the work during the past year. He wishes to express

his appreciation of the encouragement and advice given by

Mr.

J. L. Stollery [Imperial College] ,

Mr.

J. G. Church assisted with the calculations completed at urIAS.

J

SUMMARY

This work examines the applicability of the well known approximate in-viscid theories for estimating surface pressure on sharp leading edged bodies in hypersonic flow to concave surfaces, for the particular case when there is only a single shock wave attached to the leading edge. It is shown that the various methods give widely differing estimates. Some exact limiting

self-similar solutions are used to show that the waves reflected off entropy lines and the shock are very important . The thin shock layer theory, for which the Newtonian plus centrifugal theory is the first approximation, is shown to give a fairly good estimate of the centrifugal pressure rise across the layer. A combined tangent wedge plus centrifugal rule is proposed to overcome the major objection to the thin shock layer theory, which is seriously inaccurate near the loading edge because of the failure of the strong shock approximation.

1. 2. .J

3.

..

4.

'

.

TABLE OF CONTENTS INTRODUCTION

APPROXIMATE THEORIES IN INVISCID HYPERSONLC SLENDER BODY FLOW

2.1 The Theory of Thin Shock Layers

2.2 Shock Expansion or Simp~e Wave Theory

2.3

Tangent Wedge and Newtonian Rules

2.4

Application to Concave Surfaces

DESCRIPTION OF THE INVISCID SIMILAR SOLUTIONS

3.1

The Hypersonic Small Disturbance Equations

3.2

The Similar Solution

3.3

Integration of the Similar Solutions

3

.4

A Class of Non-Similar Surfaces

3

.5

Accuracy of Computation RESULTS ANI) COMMENTS

4.1

Power Law Surfaces

4.2

Non-Similar Surfaces

4.3

The Newtonian Plus Centrifugal Rule

4.4

Applicability to Non-Slender SUIfaces ,cOi'JC LUS IONS

APPENDICES: A) An Example of an Overfocussed Flow

REFERENCES FIGURES

B)

Resume of Dimensional Arguments for Similar Solutions

C) Singular Behaviour of Similar Solutions

D)

Tabula~ed Inviscid Similar Solutions E) Exact and Approximate Values of Zand R

1 3 3

5

8

8

9 9 11 12

14

15

15

15

16

17

18

19

20

Arabie: C P j K k .J M n p p R R ,R s v s T t U 00 u, v u V x, y Z SYMBOLS pressure eoeffieient .lp

if

2 0 0 0 0

j

=

0: two dimensional flows

hypersonie similarity parameter: K

= M

~ 00

exponent in power law body shape equation: Yb Maeh number; M is free stream Maeh number

00

Exponent in pressure distribution along power law body n

=

2(k-l)

defined by Eq.

3.17

statie pressure defined by Eq.

3.17

refleetion eoeffieients

distanee along the body surfaee statie temperature

time in equivalent one dimensional unsteady problem: x =

free stream veloeity

components of velocity in the x,y directions respectively

u - U

00

defined by Eq.

3.17

U t

00

co-ordinates parallel and perpendicular to the flow direction respectively

G:reek € p Suoscripts b 00 e s w L

specific heat ratio, c

Ic

p v ()'-l)

I

()'+1)

À/Às

stream function

body slope; dYb/dx

=

tan~

K(X)

is meridional curvature of surface defined by Eq.

3.18

viscosity

defined by Eq.

3.25

density

shock slope; dy

Idx

=

tan~

s

nondimensional stream function fpr nonsimi1ar surfaoes defined by

Eq.

(3.41)

denotes conditions at the body surface predicted by inviscid theory

free stream conditions far ahead of body edge of boundary layer

conditions just downstream of the shockwave

denotes conditions at the body surface predicted by boundary layer theory

1. INTRODUCTION

The equations of i~viscid hypersonic flow, that is the Euler equations together with the oblique shock relations, form a complex system which in general can only be solved by the use of numerical techniques. Consequent2y considerable attention has been given to the development of approximate methods that wil~ allow rapid estimation of fluid properties at the surface of airfoils and bodies in high Mach number flow. A survey of these methods is available in the book by Hayes and Probstein (1966), and a more elementary account is given by Chernyi

(1961.) •

Both of these texts give detailed treatments of flows in which the lead-ing edge of the body is sharp, and the field is everywhere supersonic. However, the discussions of ~he usefulness of these methods are limited almost completely

to convex geometries,- such as airfoils and fuselages. Recently interest has developed in flows for w~ich the inviscid or outer flow would be that over a con-cave surface • Two examples are the ramjet i~take and the reattaching boundary layer. Accordingly it becomes important in any complete treatment of this branch of hypersonic aerodynamics to evaluate the usefulness of the various approximate methods when applied to concave surfaces.

Inviscid concave surface flows can have a complex wave structure. In addition to yhe leading edge shock wave, the compression waves generated by the surface can themselves coalesce and form shock waves. In this process strong shear layers and finite reflected waves can be generated, and the intersection of this shock wave with the leading edge shock will also generate shear iayers and strong refLected waves. This problem has been discussed by Bird (196~) and

Sullivan (1963 and 1964). The type of flow field in which shock formation occurs may be calred over-focussed. In general such fields cannot be described simply

and will require direc\ application of the method of characteristics for rota-tional flows in order to obtain realistic surface distributions. They may be contrasted with flows in which the compression waves intersect the leading edge shock wave before they themselves can form shock waves. The latter type of flow may be called underfocussed. A typical examp~e of an overfocussed flow obtained in the Imperial College hypersonic gun tunnel is discussed in Appendix A. This report is concerned only with underfocussed flows.

The discussion and ~alysis is restricted to slender bodies, primarily because of the considerable analytical simplification that is introduced. A comment on the relevance of the results to nonslender surfaces is given in Section 4.4. The basic slender body theory for high Mach number flight is des-cribed in some detail by Hayes and Probstein (1966, pp.32-54). It is sufficient to note here that two major simplifications are introduced by the theory into the analysis of hypersonic inviscid flows. One is the hypersonic similarity rule; and the other is the equivalence principle, by which the equations of a two dimensional slender body can be reduced to the equations of one-dimensional unsteady flow. Although the latter set of equations are simpler in some res-pects than the two dimensional steady flow equations, they are still nonlinear and generally require numerical solution. This may be contrasted with the small disturbance theory for supersonic and subsonic flight, where the equations

are reduced to those for the acoustic problem. However, unlike the linear theory, the hypersonic small disturbance theory can give a full description of the wave structure of a rotational flow. It is also very accurate, since the error it imposes is of second order in the slenderness ratio whereas the error incurred by the use of the linear theory is of first order. In the general case

to make further progress with the hypersonic small disturbance theory it is neces-sary to introduce additional assumptions.

A number of other approximate methods have been developed for the solution

of locally supersonic flows. The approximate methods which might be applied to

underfocussed flows

are:-(i) tangent wedge rule

(ii) shock-expansion or simple wave theory (iii) Newton-Busemann theory

(iv) Newtonian impact theory

These methods are described in some detail by Hayes and Probstein (1966) and Chernyi (1961); they are summarized in Section 2. They can be applied to both the Euler equations and the hypersonic small disturbance equations, and in the latter case they can then be regarded as approximate schemes for solving the hypersonic small disturbance equations. It is the applicability of the approxi-mate theories (i) tb (iv) above to concave surface flows which is to be examined in this report.

The method of assessment will be to use a set of exact limiting solutions to the hypersonic small disturbance equations which are available through the use

of the equivalence principle. This principle allows the well known self-similar

solutions to the piston problem to be applied to hypersonic slender body flows. These

solutions are described in considerable detail by Sedov (1959, Chapter IV), and

their application to hypersonic flows is discussed by Hayes and Probstein (1966

pp.55-92). In terms of the hypersonic flow, the solutions exist for bodies of

the form Yb

=

Dbxk and if the gas is calorically perfect. Further, the fre~

stream Mach number Moo and the local body slope dYb/dx must be such that

Moo dYb/ dx

»

1 with dYb/dx

«

1, that is, the flow must be Mach independent.

When k < 1 the surface is a blunted slender ogive, and when k

>

1 the surface is

cusped and concave. The solutions have been applied to hypersonic flows quite

extensively for k <1, but the present work appears to have been the first attemp~

to exploit them for concave surface flows. The practical utility of these

solu-tions is limited since they are restricted to certain body shapes; however, they

are essentially exact limiting solutions to the hypersonic small d1sturbance

equa-tions, so that they may in certain circumstances be used as a standard against

which the simpler approximate methods listed above can be compared. In this

report, some numerical results for the similar solutions are presented and the approximate methods are compared with the results from the similar solutions. Comments on the accuracy of the approximate solutions are made.

A difficulty wi~h the interpretation of the similar solutions has been

their singular behaviour at the surface. For the convex surface case there has been considerable discussion in the literature; and much of this is summarized

by Freeman (1965), to which the reader is referred for a systematic account. For

the concave surface case, a systematic investigation of the singular b~haviour

was given by Sullivan (1966). The singularity in the density and surface

tempera-ture was related to the behaviour of the flow at the leading edge. For k

>

1,

assump-tion Moo dYb/dx

»

1 is violated there and a low-entropy layer appears in the flow. This layer is presented as a singularity in the similar solutions. However, it is shown by Sullivan

(1966)

that for the pressure, which is the quantity of primary interest here, the similar solutions give the correct value at the surface. It is worth noting that in contrast to the behaviour for k

<

1, the solutions for k

>

1 are the exact limiting solutions to the hypersonic small disturbance equa-tions, for M ~oo. This occurs because it is the st rong shock approximation

00

which is violated near the leading edge, and the domain in which the approximation is not sufficiently accurate becomes vanishingly smal~ in the limit. Further, an exact solution to the full Euler equations will yield an infinity in pip at the surface of a cusped body at M

=

00, since the streamlines immediately adjacent to

00

the surface undergo a pure isentropic compression. This may be contrasted with the behaviour for k

<

1, where the singularity is associated with the failure of the slender body approximation which is not affected by the limit M ~oo.

00 2. APPHOXIMATE THEORIES IN INVISCJ;D HYPERSONIC SLENDER BODY FLOW

2.1 The Theory of Thin Shock Layers

A characteristic feature of hypersonic flows is the tendency of the leading edge shock wave to wr.ap itself around the body as M increases. For

00

sufficiently large M ,the shock shape in the neighbourhood of the forepart

00

of a body approaches the body shape. The region between the shock wave and the body, known as the shock layer, is thin. The fluid in it is dense when compared with the free stream density, and is travelling at speeds which are not much less than the free stream speed. The shapes of the streamlines in the shock layer are not much different from the body shape.

These observations have been used to develop a successive approximation scheme for constructing solutions (e.g. Freeman,

1956).

The density ratio across an oblique shock wave in a perfect gas is given by

( / -_{/ +} 1 ) ₁ + 2 (2.1)

When Moo

=

00, P~Ps = (/-1//+1). Hence as / ~ 1 at M

=

00 ,P ~ 00 and the shock 00 s

wave must collapse into the body. Therefore method is developed by applying the limit

E

=

to the equations.

/ - 1

/ + 1 ~ 0

in a perfect gas the thin shock layer

(2.2)

If the body is slender, the first approximation is particularly simple. This analysis was given by Cole

(1957)

and Cherny[

(1961).

Their results are quoted here. Let the shape of the body be given by Yb

=

B(x) where x and y are coordinates parallel and perpendicular to the free stream. The stream is moving at velocity U , and has density p and pressure p •

Let ~ be the value of x for which a streamline ~ passing through any point (x,y) crosses the shock wave.

Then

+

°

(E)

(2.3)

The pressure at any point in the layer is given by :e(xzn _B2 B (x) {

B~;~P)

} =: (x) +

l~;

B(x) Poo Lfoo x

(2.4)

The pressure on the surface of the body is p(x,O) 2 B (x) B(x)

=

B (x) + xx

ifoo x 1 + j

Poo

This is the slender body form of the well known Newtonian plus Centrifugal formula. The first term on the right hand side of Eq. 2.5 is the shock pressure risee The

second term is the difference in pressure across the shock layer associated with its curvature; it is identically zero if the body profile curvature (i.e.,

B _xx ) is zero. The density can be calculated from the entropy conservation law, which becomes

~

(x,

~)

=

~

{ :

if

+

B~ (~)

}

(2.6)

00

The streamline shape is given by

y (x,Ö =: B(x) + ~ I _{-2 2}

}Bj(~) B~

d~

-

jJf

1

Moo + E B ~

(I

+

(2.7) 0 (1+j)B2_x ;1-B _xx {B _x _

B~+j(~)

} BJ(x)

The shock position is found by calculating y(x,x) from Eq. 2.7. Note that in Equations 2.5, to 2.7 it is implicit that B(x

L)

=

0. If it is not, then B(x) must be replaced by {B(x) - B(X

L)} to ensure that the formulae correctly esti-mate the mass flow onto the surface.

The application of this method to power law shapes (Yb

=

Dxk) is of particular interest. The surface pressure on such bodies is

~\. Pb

Poo

u!

~~

(k - 1) }

k (1 + j)

where tl b

when k> 1.

k-l

is the local body slope.

kD x P E

~ tl~

{ (k - 1)

}

pb

=

1 + Poo k( 1 +

j)

According to this formula, Pb

The density at the surface is given by

-7 00 as M -7 00 00

When applied to convex surfaces, Eq. 2.5 is found to considerably under-estimate the surface pressure. This occurs because the shock layer is rarely sufficiently thin, particularly if the body is slender. Usually the shock layer is thin only in the neighbourhood of the stagnation region; but in this reg ion a different analysis is required becapse the shape of the streamlines differs greatly from the body shape.

There are three other situations in which the thin shock layer theory exhibits a physically unsatisfactory behaviour. According to Eq. 2.5., on a convex surface, for which B < 0, at some point sufficiently far downstream

xx

from the leading edge, the pressure difference across the shock layer will balance the shock pressure rise. At this point (for which B > 0) the surface pressure becomes identically zero. According to Eq. 2.7, y(x; ~) -700 when this occurs,

so that theory is not uniformly valid. The theory also fails downstream of points of discontinuity of profile curvature and slope. At discontinuities in

curvature, the theory predicts a discontinuity in the surface pressure, when

it is known fr om the theory of characteristics that discontinuities in pressure gradient will occur. The behaviour downstream of discontinuities in slope is of interest when tl

b increases through the discontinuity. The theory fails in this case because it does not take account of the shock pattern.

2.2 Shock-Expansion or Simple Wave Theory

The method of characteristics provides a technique for the exact solution of the~supersonic regions of an inviscid flow field. However, it usually requires the use of numerical procedures, so that in practice digital computers are used.

Epstein (1931) suggested a method, known as the shock-expansion or simple

wave theory, which is a simplification of the method of characteristics. The Mach waves in a two dimensional field can be divided into ~wo gnoups. One carries the disturbances generated by the body away from the body; these are here called the principal waves. The second group, called the reflected waves, comprise those which carry the disturbances generated at the shock waves and streamlines in the field toward the body. According to the simple wave theory, the properties on the surface of a two-dimensional aerofoil immersed in a flow which is everywhere supersonic are calculated by neglecting the effect of a reflected Mach wave. Hence the pressure dis tri but ion on the surface of a convex aerofoil is obtained by calculating conditions at the leading edge by the oblique shock relations, and then matching a Prandtl~Meyer expansion to the body geometry. It is simple to apply, ~~d it yields an estimate of the pressure which depends only on the local body shape. The method has been extended by Eggers and Syvertson (1952) to permit the calculation of the shock shape and the field, and has been adapted to the condtruction ofaxially symmetric flows.

Consider its extension to concave surfaces. As on a convex body, at the leading edge and at positive discontinuities in profile slope, the oblique

shock relations are used to obtain the pressure just downstream. Over the remainder of the profile isentropic simple wave theory is used. The geometry of the body is an important factor in determining the accuracy of the methode If the body has a wedge-shaped forebody then the first principal wave is generated well down-stream of the leading edge. The reflected waves then intersect the body muc~

further downstream, or they may even miss it entirely. The simple wave method is then exact or very accurate when it is used to estimate the surface pressure. To justify its use on bodies on which the reflected characteristics intersect the surface well forward, it must be shown tha~ the disturbances propagated along the reflected charac~eristicsare_much weaker than those propagated along the principal characteristics. This is particularly important at hypersonic Mach numbers, since as M increases, the leading edge shock closes down onto the body

()()

at a faster rate than do the Mach waves. This causes more of the reflected characteristics to hit the body.

~here are 'two sources of reflected Mach waves. These are the shock waves

and the sliplines. Consider the reflection process at the shock wave. A re-flection coefficient R is defined as the ratio of the pressure difference across the reflected characte~istic to the difference across the incident principal characteristic, that is

R

s (2.10)

where the pressures are defined in figure 2.1. Hayes and Probstein (1966), giving an analysis originally due to Lighthill (1949) and Chu (1952), show that for

M

= ()()

and if the shock angle ~

«

1, then

00 R s

2JÊ -

.Jl(l-E)

2JÊ

+

.Jl

(1-

E)

as f:.:,~ -7 0

(2.11)

This limiting value of R depends only on l .

s R s is negative and is small if 1 is not close to one. Hence

sion will be reflected as a so that a principal wave is

on a convex surface where f:.:,~

<

0, the weak compression. However, as 1 -7 1

reflected undiminished in strength.

incident

expan-; R -7 (-1),

s

The reflection process at the slip-lines is depicted in Figure 2.2. A refleètion coefficient

R

is defined as before:

By matching reflection,

v R

v

the pressures on either side of i t can be shown that if MA

»

1

""

~

- MA

Rv

~

+ MA

(2.12)

the slip-line just downstream of the and ~

»

1

(2.13)

When the slip-line is weak, ~ ~ MA and R ~ O. The analysis of the effect of R on the reflected wave system given by ~ayes and Probstein (1966, p.504)

v

si~ce R is positive and the incident principal wave is an expansion, the

re-flectedVwave generated at the slip-lines is an expansion. Therefore the reflected waves generated at the shock and at the slip-lines in a convex surface flow tend to cancel, and can be much weaker than consideration of either mechanism alone indicates. Consider now the behaviour of R in the flow over a concave surface, for which the principal waves are compressi6ns. As R remains negative when ~~>O, the reflected waves generated at the shock will be ex~ansions. But as the shock is concave, in Eq. 2.13, ~

<

MA and Rv is now negative. Hence the reflected waves generated at the slip-lines are expansions. In contrast to the behaviour on a convex surface, the disturbances carried by the two types of reflected wave generated on a concave surface add. The strength of the reflected waves will therefore increase with increase in the strength of the s~ip-lines, or v.(p/pl).

Numerical comparisons with exact solutions obtained by the method of

characteristics* have shown that if I

>

1.3, the simple wave method gives excellent results on slender convex bodies with sharp leading edges. As I ~ 1 the error increases rapidly. It is expected that the method will overestimate the surface pressure when it is applied to an underfocussed flow on a concave surface. The

error should increase with i~crease in the curvature of the profile.

For slender bodies the hypersonic smal~ disturbance approximations to the shock and simple wave relations can be used. The application of the method is then part~cularly simple, since explicit formulae are available. ~he angle of shock wave at the ~eading edge (~L) is given by

~ 2 1/2

+ {( 2(1-E)) + l} K = M ~ =

SL CXl L (2.14 )

where ~

=

M CXl ~L' and ~L is the leading edge slope. The pressure and Mach

number behind the shock are then given by

PL (1 + E) K2 SL E Poo (2.15)

{

2 K SL

M~

_{{ ( 1 +E)}

_K~L

_{- E} {} 2 + (l-E) } E K SL (2.16)

The Mach number and pressure distribution over the rest of the profile are then given by jL M 1 M L

=

C_l-E E 1+ E E (2.17)

where M is the Mach number, p is the pressure and ~ is the local body slope.

* see for example the results quoted in

(i)

Hayes and Probstein (1966) p. 506,and (ii) Chernyi (1961) p.193, Fig. 4.15.

2.3 Tangent Wedge and Newtonian Rules

According to the tangent wedge rule, the pressure at a given point on the surface of a two-dimensional body is estimated by calculating the pressure generated on an "equivalent" wedge of angle equal to the local slope of the body, and which is immersed in the same free stream. The method is based on the physical idea that, if the shock 1ayer is thin, then the properties on the surface cannot differ greatly from those just behind the shock wave. This is discussed by Lees (1955). The advantage of this method is its simplicity. When the body is slender, the equivalent wedge pressure may be obtained by using the hypersonic small disturbance equations (Eq. 2.15) and the surface pressure is then given by a simple explicit expression. The estimate obtained depends only on the 10cal surface slope. The rule has been found useful on slender convex surfaces, particularly when the

simple wave method breaks down; such as when l is close to one, or when a blunt leading edge generates astrong shear layer adjacent to the surface. However, it neglects the wave structure of the field completely, and therefore is totally inadequate in regions where strong or finite waves of either famifY are generated. In contrast to the simple wave ~heory, the tangent wedge rule cannot be applied to those regions of the slender body having a negative slope. The agreement mentioned above should therefore be regarded as fortuitous.

A method which is ofteI\ '.: l used in problems such as flight dynamics calculations because of its extreme simplicity, is based on one of the concepts of Newton's gas dynamics. The individual gas particles in the free stream are assumed to move in straight lines. When they hit the body they lose only their momentum normal to the surface and then slide along the surface in an extremely thin layer. The momentum law then requires that

p

=

b

. 2 _a

slnu

b (2.18)

Examination of the oblique shock relations shows that this expression is exact in the thin shock layer limitgiven in Eq. 2.2 for the flow over a wedge. Hence i t can be considered as an approximation to the tangent wedge rule.

The thin shock layer t.heory shows that the centrifugal pressure difference across the shock layer is of the same order of magnitude as the shock pressure rise, so that the use of the Newtonian formula without the centrifugal term is from this point of view, inconsistent. In spite of this, Eq. 2.18 gives a. better estimate of the surface pressure on convex surfaces than does the Newtonian plus centrifu~al formula. This occurs because the Nevrtonian formula underestimates the shock pressure rise, and the centrifugal term subtracts from the shock pressure risee

2.4

Application to Concave Surfaces

The approximate methods discussed above are applied to an underfocussed flow on a concave surface • The forward part of the body has the form:

y

=

il (x + x

3 )

L (2.19)

where ilL is the leading edge angle. To demonstrate the effect of discontinuity in curvature~ the surface is continued as a ramp (that is at constant slope il

hypersonic similarity rule applies (Hayes and Probstein

1966,

p.41) and

c

-E-

=

f(M ~ , ,) ~2 00 L oL (2.20)

The surface pressure distributions for Moo ~L = 1.0 and ; , = 1.4 are given in figure 2.3. The theories give widely differing estimates. The tangent wedge rule gives the correct pressure at the leading edge, and far upstream on the ramp. All of the theories predict a constant pressure on the ramp. The NewUünian plus centrifugal formula gives a discontinuity at the junction. To demonstrate the accuracy of the small disturbance assumption, the pressure on the ramp, estimated by the simple wave theory, using the exact (nonslender) shock and Prandtl-Meyer wave relations is given. The error is less than~2%. In Fig. 2.4, the effect of variation of Moo and ~L on the estimates of ramp pressure is given. The differences between the estimates increase as Moo~L increases.

It is obvious that the discrepancies observed in the predicted surface pressures indicate the need for an exact solution to act as a basis for comparison of the various approximate methods. The small disturbanee approximation itse\f appears to be very accurate, but the additional approximations and assumptions introduced remain open to question.

3. SUMMARY OF THE SIMILAR SOLUTIONS

3.1 Hypersonic Small Disturbance Equations

For a two dimensional or axially symmetrie flow the hypersonic small disturbance equations are:

contin ui ty: U dP/₀₀ dX +

_dy

d (pv) + j pv/y 0 (3.1)

normal momentum: U dV/dX + v êlv/dy +-1 dp/dy 0 (3.2)

00 _P

streamwise momentum: U ₀₀

_di'

dÜ + v

_dy

dÜ ₊

_P

1

~

₌

₀ (3.3)

entropy: U d

~)+

_v d

(~)

=

0 (3.4 )

00

di

dy

Here x and y are the co-ordinates parallel and perpendicular to the free stream direction respectively, U is the free stream velocity, Ü and v are the pertur-bations in velocity parallel to x and y respec~ively, p is the pressure and P is thedensity. Also j

=

0 and j

=

1 for two dimensional and axially symmetrie flow respectively. The boundary conditions required are the tangency condition at the body described by y

=

Yb(x):

_1_ + _2_ { CJ2

if

l + 1 )' 00

~

₀₀

}

ps/poo = ()' - l)/(l + 1) +

~~1)-~CJ2

2 v/U _s = -00 l+l -1 { 2 CJ

CJ-ti

/U

= -

2/)'+1 { CJ2 -

!...}

s 00 M2 00 00 (3.6) (3.8)

lt is assumed that only one shock is present in the field. lts shape, which is

part of the solution to the problem, is given by y

=

Y (x); so that the local

s

shock slope CJ is given by

dy s

dx

=

tanCJ CJ (3.10)

The subscript "00" denotes quantities in the free stream and "s" denotes quantities

immediately downstream of the shock. The free stream Mach number is denoted by

M

00

The equation for

ti

is decoupled from the equations for p,p and v so

that it is possible to solve for the latter quantities independently. Then the

transformation

x = uoot (3.11)

reduces equations 3.1 and 3.2, 3.4 to 3.8~ a.l1d 3.10 to the equations of one

di-mensional unsteady flow. The quantity

ü

can then be found by integration of

(3.4), or by using the hypersonic small disturbance approximation to Bernoulli!s eq~ation, which is:

Uu

+ v2 + ..L. E =..L. poo (3.12)

00

2""

l-l P l-l Poo

The problem for

ti

will not be considered further. Proof of these equations is

given by Hayes and Probstein (l966, p. 32) and by Chernyi (1961, p.55).

If the Mach number M is so large that CJ2» l/Mf , then the oblique

00 co

shock rela tions can be approximated by their Mach independent or "strong" form:

Ps _{2 2}

7

-

)'+1 CJ

P 00 .:.0

v

ju

=

s 00 2 1+1 2 1+1 cr 2 cr

These are the form required for the similar solutions.

3.2 Similar Solutions

The arguments used to demonstrate the existence

are summarized in Appendix B. They are an adaptation o~

given by Sedov (1959). For a body of the form Yb

=

DbX

and vare replaced by

~ px and if y is replaced by R

V

=

v x U Y (3.16 )

of self similar solutions

the argument, originally

, if the depeudent ~~r~ables,p,p

(3.17)

(3.18)

then the differential equations for p, p and v reduce to the form

continuity: À

{1:

i!llR (V-k) + dV}+ (1 + j) V

R dÀ dÀ

o

(3.19)

momentum: À {(V-k) dVjdÀ +

1:

dP} + V(V-l) 2P

R

dÀ

R

o

(3.20 )

entropy: À {

v -

k }

{~ ~~

-

~ ~}

+ 2 (V -1)

=

0 (3.21)

The equations contain only À as independent variable.

The boundary condition a:tiFtlte)Ll?p~ctreiC.ames

V

=

k at À

=

1 (3.22)

The shock boundary conditions have to be cast into a form which is independent of

x or y. The dimensional arguments show that this can be achieved only when the

flow is Mach independent, so that the strong form of the oblique shock relations

are required. Then the shock shape is given by

y

=

(3.23)

and where À is a constant to be determined as part of the solution to the

prob-lem. The oBlique shock relations then become

P s

2k

1+1 (3.24 )

convenient to use it in place of À as independent variable. It is used here in the form

v

= ~ + k ~

= -

₍

-

/-1 ) k

s 1+1

Further, by intraducing Z which is defined by

Z

=

_R

lP

=

(3.25 )

(3.26 )

the equations can be decoupled. Af ter same algebra the following equations are obtained: dZ d~ where A

c

E Z

iA~3

+

B~2

+

C~

- 2Z

(~

+ D)

1

-"[

{

~3

+

E~2

+

F~

_-

Z{(l+j)~+

_{G } }}

=

d~tn À2 ₌ d~ ~ d[tn _d~

R]

=

k(k-l) _{(1- / )} 2 k-l Z

HH+

I

cp Z,~ Z - ~2 cp -

{(l+j)(~+k) dà~nÀ2

+ 1 } B D F (k-l) (3-/) + j (/-1) k

(k-l)/I

k(k-l) G

2(k-l)/1

+ (j+l)k (3.27) (3.28 )

The equation for Z can be solved independently and those for À and Rare reduced to quadratures.

3.3 Integration of the Similar Solution

There are nine singularities of equation 3.27. Those relevant to flows with k

<

1 are discussed by Lees and Kubota (1957). For k> 1 only one is of interest here, that which occurs at ~ ~ O.

In the limit ~ ~ 0 equation 3.27 approaches the form

dZ ~ Z

It can be shown that Z/S ~oo as S ~ 0, so that [see Appendix B] dZ 2ZD ds ~

W-

(3.31) and * _sa Z _Zb as S ~ 0 (3.32) where 2D 2~k-l) Ct G 2(k-l) + )'k(l+j) Alsm R - R* _b

s-a

It can be seen that when k

<

1, Z ~ 00 and R ~ 0 as S ~ 0; whereas when k

>

1, Z ~ 0 and R ~ 00 as s ~

o.

In both cases P remains fini te and tends to Pb' where

The method of integration adopted in the present calculations was as follows. The variable À was replaced by

so that equation (3.28) is replaced by

d($n~) =

Z - s2

ds cp

Then equations (3.27) to (3.29) were integrated simultaneouslyon a digital com-puter using a fourth order Runge-Kutta technique. The computation was started at s , where the shock jump conditions give

s Z s 2 2)'()' -l)k 2 ()'+l) R s 9'+1 )'-1 ' ~ s = 1 (3.38)

At some suitable point near s

=

sb

=

0, the calculations were matched to series solutions which were valid in the neighbourhood of s

=

O. The series for each variable contained a free constant Z~ , R~ , ~b which was determined by matching with the computed numerical values at the matcn point. The surface pressure was then determined by using equation 3.35. This procedure is illustrated ilin figure 3.1. It should be noted that the series were powers of S, where the exponents were initially unknown functions of the parameters )', k and j. The series and their method of derivation are given in Appendix C.

The series solutions valid near s

=

0 also gave a simple check on the accuracy of the numerical somputations. This is discussed in Section 3.4.

3.4 A Class of Non Self-Similar Surfaces

By constructing the equations for the streamlines in the field the solu-tions for the similar surfaces may be extended to a class of surfaces which have non-zero leading edge angles. For these surfaces , the similar solutions predict the correct values of the density and temperature as well as the surface pressure, provided that the leading edge angle is not so small that the surface is effecti-vely in the low entropy layer.

Figure 3.2 depicts the field for a self-similar flow when k

>

1. The curve ABC is the body and ADGE is the shock. The shape of a streamline ~ ==

constant which does not pass through the origin is depicted by GHJ. In the hyper-sonic flow, GHJ may be replaced by a surface of the same shape. If x

1 is the value

of x for which the derived surface has the required leading edge angle, then

ijL == (1 -

E) CT

(1-E)

\

This expres sion is given by

By continuity

obtained from Eq. 2.14

k Db x l+j YS1 Ys == \ 1+' Y J s D kx k-1 1 by letting M -7 ()() ()()

u

()() dy o

This equation may be non-dimensiona1ized to give

1

l+j

the shock shape is

(3.40)

~ is a non-dimensiona1 stream function. Equation 3.41 can be written as

(3.42)

Note that ~b == ~(o) == -l/(l+j) since for a power 1aw surface x

1 == O. Since ~ is a function of ~ on1y, it may be eva1uated simultaneously with Eqs. 3.27.

(i)

(ii) (iii)

To construct a non-similar surface, the fo1lowing procedure is adopted: Given ?'~_ k~ D and 1')1; xL is determined from Eq. 3.39,

Given an x, ~ is determined from Eq. 3.41,

-The values of V,R,P and ~ corresponding to ~ are obtained from the tabu-lated solution. -- J_

(iv) The ordinate of the surface is given by k Y 2L D x T]b

(v)

The slope is (J .44 ) 1'3 v

Y.x

U x co

(iv) The pressure is

p _Pco

11

y2/x2 P (~) co through leading surface it then

Since the streamline chosen to be the non-similar surface does not pass

the origin, then for M sufficiently large, the solution is valid at the

co

edge and near the surface of the derived body. When k

<

1, the derived

also has a sharp leading edge. If xL is sufficiently far from the origin

satisfies the equivalence at the leaaing edge.

3.5 Accuracy of Co~utation

The functions ~ and the series solutions valid near

S

=

0 can be used to

check the accuracy of the numerical computations. The series solution for d~/ds

can be formed from Eq. (3.42) and the series for Zl$nR and $nÀ. This may be

integrated to give ~=~ o fl +(l+j)T] e 0 0 G l-ex

s

1 - ex (3.46 )

The quantity ~ is determined in the same way as Z , that is, from the computed

o 0 ~

value of ~ at the match point, S. In the limit s ~ 0, ~ ~~. But by definition

n 0

~(s) ~ -1 as

S

~ O. Hence 1 + ~ is given in the tables in Appendix D. In all o

cases this error estimate is less than 0.3%.

4 •

rus

ULT S AND COMMENT S

4.1 Power Law Surfaces

The value of ViRiP_i and ~ for certain values of k and

r

in the ranges

115 ~ k

<

4.0 and 1.15 ~

r

~ 1.67 for j

=

0 are tabulated in Appendix D. The

distributions of Zi R and P within the shock layer for k

=

4.0 and

r

=

1.4 .are

given in Fig. 4.1.

In Fig. 4.2 the quantity

(4.1)

wedge estimate for M

=

00 (Eqs. 2.14 and 2.15), the Newtonian rule (Eq. 2.18) and the Newtonian plus c~ntrifugal (Eq. 2.5) theories. The Newtonian plus centrifugal estimate agrees moderately well with the similar solution, except near k

=

1. As M ~oo, the simple wave estimate (Eq. 2.17) for a cusped body becomes

00

2/)'-1

)'-1 (M 1'l )

2 00 b (4.2)

Hence, Pb ~ 00 as M ~ 00 and the simple wave solution cannot be readily compared 00 2

here. In Fig. 4.3'~b/k is plotted against )' for k

=

4, and is compared with the approximate estimates used in Fig. _I 4.2. The similar solution depends only weakly on:)' and tends to the estimate given by Newtonian plus centrifugal theory as )' ~l.

Although the series solutions for Zl R and À were originally derived to

allow the numerical solutions to be carried through to ~ = 0, it was subsequently found that they gave a good representation of the solutions for equations (3.27) to (3.29) across the entire shock layer. To obtain this representation it is only necessary to determine the constants Z , ~ and R by requiring the solutions

0 0 0

to give the correct values of Z, R and ~ at the shock. The tabulated values of Zand R for k

=

2 and )'

=

1.40 as determined by the third order series solutions are given in Appendix E. They are compared with the numerical values. Th~

approximate solutions for

Z,

R

and Pare also plotted in Fig. 4.1. The agreement is good; the error is less than

6%.

It appears that the series solutions given in this report all provide a very good approximate solution to the complete problem. It is worth noting that these solutions are valid for the case k

<

1 so that they should also be useful for these flows.

4.2 Non-similar Surfaces:

The similar solutions for

)'=

1.40, k

=

3, and D

=

1/150 have been used to construct solutions for non-similar surfaces. The prinoQpal parameters of the derived surfaces are given in Table 4.1. In figure 4.4 the surface pressure computed for case B from the similar solution is compared with the values computed by the approximate theories discussed in Sect. 2 and a new suggested rule to be discussed in Section 4.3.

TABLE 4.1

Principal Features of Derived No~siIDilar Surfaces. The Basic Similar Surface is Yb

= x3/150 with

~

=

1.40 Case

M 1'lL

00

T

.L

1'l/-aL

Comments

A 0.75 3033 These can be regarded as results for a B 1.50 3.33 surface wi th ~L= 8.60 0 3.00 3.33 at

M =

5,10 and 20 C 00 respectively D 0.50 8.00 Shows effect of

1'l/1'lL

The good agreement of the Newtonian plus centrifugal theory with the similar

solu-tion towards the rear of the surface demonstrates the importance of the

centri-fugal term, while the simple wave theory grossly overestimates the pressure at

the end of the surface. This clearly shows that the reflected disturbances

can-not be neglected.

The pressure distributions for case D are given in Figure

4.5.

Here, to

emppasize the details in the neighbourhood of the leading edge, C has been plotted

on a logarithmic scale. The value of Moo~L has been deliberately Pchosen to

il+ustrate the consequences of the failure of the strong shock approximation.

The surface depicted in Fig. 4~5 corresponds to M

=

10. At the leading edge,

00

the tangent wedge theory should give an accurate estimate of the inviscid surface

pressure. In the limit x ~ xL' the exact inviscid solution is the wedge solution

so that equation 2.15, which is based on the hypersonic small disturbance

approxi-mations and not on the strong shock approximation, should give very accurate

estimate for ~L

=

2.900• Both the similar solution and the Newtonian plus

centri-fugal theory are seriously in error in the neighbourhood of the leading edge.

Note however that for case D, Moo~R

=

4.00 so that the strong shock assumption

M ~»l is satisfied near the rear of the surface so tha~ the similar solution and

00

the Newtonian plus centrifugal theory should give fairly accurate values of p

in this neighbourhood. At the end of the surface, the tangent wedge estimateW

of p is 0.732 of the similar solutîon estimate.

w

In Fig.

4.6

the various theories are compared for Case C. Here the

sur-face pressure is plotted in the form P(x)/PTW(x) where PTW(x) is the tangent wedge

estimate at the given value of x. Since Moo~L

=

3.00 and Moo~R

=

10, the strong

shock approximation should apply over the entire surface. At the leading edge,

the similar solution, which is simply the wedge solution for M ~oo lies below the

tangent wedge estimate by

6.7%.

At the end of the surface, fo~ M· = 20,

00

~R

=

28.70 and the approximation sin~ ~ ~ is in error by

4%.

This is a good

indicator of the errors incurred by the slenderness approximation, so that taken

together, the two criteria suggest that for Moo

=

20 ~he similar solution should

be in error relative to the exact inviscid solution by about

7%

at most. The

results show the same general trend as those give~ in Figures

4.5

and

4.6

,

Finally in Figure

4.7

the results for case A are plotted. For the surface

drawn in figure

4.4

these solutions correspond to M

=

5.0. The strong shock

00

approximation is again clearly inaccurate the leading edge. The simple wave

theory here gives results near the end of the surface which do not differ greatly

from the value predicted by the similar solution; this behaviour may be

con-trasted with that in Figures

4.4

and

4.6

and indicates the strong dependence

on M of the wave reflection process at the sliplines.

00

4.3 The Tangent Wedge plus Centrifugal Rule:

The results given in Sections 4.1 and 4.2 have demonstrated the

im-portance of centrifugal effects in concave surface flows. In those regions where the similar solutions should be reasonably accurate the Newtonian plus

centri-fugal theory gives remarkably good agreement. On the ot her hand the Newtonian

plus centrifugal theory suffers the same limitation as the similar solution,

namely the Mach independence assumption. Because in concave surface flows the

theory cannot be used with any degree of confidence as a simple method for cal-culating p . If however the shock pressure rise term in equaticn (2.5) is re-placed by rhe tangent wedge estimate then this obstacle is overcome. The expres-sion for surface pressure p becomes

w C P

2(1-E)

~

₀₀ [K 2 _ 1] + 2B(x) s

(4.3)

where K is given by equation

2.14.

Note that, since the centrifugal term vanishes as the ïeading edge is approached, this composite rule gives the tangent wedge estimate at the leading edge. On the other hand it incorporates the centrifugal term in the regions where it is most useful, namely near the trailing edge. Consequently, it appears that this may be a useful simple rule for estimating p

in underfocussed flows on concave surfaces. w

This composite estimate, which may be conveniently called the "tangent wedge plus centrifugal" rule is plotted with the other estimates in figures

4.4,

4.5

and

4.6.

It tends to overestimate the value given by the similar solution.

In case C [Figure

4.6 ]

at the end of the surface the p as estimated by the rule is in excess of the similar solution by 11%. However, ït should be remembered that the exact solution for finite M should lie somewhat above the similar

00

solution estimate; this is suggested by the fact that the value of C on wedge p

in an inviscid flow monotonically decreases towards its asymptotic limit as

M ~oo. Hence the error in use of the rule should be less than that suggested

00

by the similar solution.

4.4

Applicability to Nonslender Surfaces

Although for simplicity the present analysis and the relevant discussion has been developed using the hypersonic small disturbance theory, the general conclusions drawn apply when the slenderness approximation is insufficiently accurate. Each of the approximate theories discussed in Section 2 has its coun-terpart for nonslender bodies; except tha~ the algebra is now more complicated. The tangent wedge theory requires solution of the implicit oblique shock equa-tions, simple wave theory requires use of the tabulated Prandtl-Meyer function, while the Newtonian-plus centrifugal formula becomes[see for examp1e Hayes and Probstein

(1966)]

C P o

dJ

+ 20t(x)

J

~p

.

costl dx x U.J,

(4.4)

where, here tantl

=

dYb/dx. The tangent wedge plus centrifugal rule can also be applied; except that there is now no exp1icit formula. The oblique shock equa-tions have to be solved and the centrifugal correction is given by the last term"in equation

4.4.

Clearly, since one of the important conclusions of the present work rests on an analysis of the wave structure of the flow one important limitation is that the flow remain supersonic in the shock layer. However, the condition for applicability is more stringent than this. If the analysis of the inter-action of the principal waves with the sliplines is reformulated without the restriction MA

»

1, it can be shown that

~

M2 _

₁

_rÇ

_~-

₁

R _v A

(4.5)

~

I>f-

_A

1

+

_{

_~

-

1

Put ~

1\

+ 5 and expand in a Taylor series to obtain to first order

(~

- 2) }

R

-

-

tl\

(MÎ-l)

5

( 4.6)

v

The sign of Rv depends both 5 and~. As ~ decreases, Rv changes sign at

~ =

J2

so that as M

-7..r

2 the waves ref1ected. vanish.

5.

CONCLUSIQNS

The justification for the conclusions in Section

4

of this report rest primarilyon the use of the similar soLutions as a reference. Although they are exact solutio~s to the hypersonic small disturbance equations, the question remains: how useful are they at large~ fipite M? This answer can be obtained

00

either by reference to exact numerical solutions to the full Euler equations

or to experiment. The experiments reported by Sullivan

(1966)

suggest that at

least the surface pressure is given reasonably accurately for Mach numbers as

low as Moo =

7.5

and for slopes ~R z 280; however, further work is required,

especially to check the usefulness of the tangent wedge plus centrifugal rule. Since one of the complications associated with experimental work on this type of

surface is boundary layer separation, and since the present work is concerned with the utility of approximations to the exact Euler equations, it would seem that a comparison with numerical solutions for the four cases used in Section

4

is desirable. Since computer programs which can do programs which can be used to give the exact inviscid solution are available (see for example Shih,

1964)

Bird, G. A. Chernyi, G. G. Chu, B.T. Cole, J. D. Eggers, A. J. Jr., Syverston, C. A. Epstein, P. S. Freeman, N. C. Freeman, N. C. Hayes, W. D. Probstein, R. F. Lees, L. Lees, L. Kubota, T. Sedov, L. I. Shih, L. Y. Sullivan, P. A. SUllivan, P. A. REFERENCES

Effect of Wave Interactions on Pressure Distributions in Supersonic and Hypersonic Flow, AIAA J. Vol. 1, p. 634 (1963)

Introduction to Hypersonic Flow. Academic Press, New York (1961)

On the Weak Interaction of Strong Shock and Mach Waves Generated Downstream of the Shock. J. Aero Sci. Vol. 19, p.433 (1952)

Newtonian Flow Theory for Slender Bodies, J. Aero. Sci. Vol. 24, p.448 (19~7)

Inviscid Flow About Airfoils at High Supersonic Speeds.NACA Tech. Note 2646, (1952)

On the Air Resistance of Projectiles, Proc. Nat. Acad. Sc. USA, Vol.17, p.532 (1931).

Qn the Theory of Hypersonic Flow Past Plane and Axially Symmetric Bluff Bodies, J. Fluid Mech. Vol. 1, p.366 (1956)

Asymptotic Solutions in Hypersonic Flow. Research Frontiers in Fluid Dynamics, Chapter 10, Seeger and Temple (ed) Interscience, New York (1965)

Hypersonic Flow Theory. 2nd Edition; Vol.l, Inviscid Flows, Academic Press, New York (1966).

Hypersonic Flow. Proc. 5th International Aero. Conference, Los Angeles, 241-276 (1955)

Inviscid Hypersonic Flow Over Blunt Nosed Slender Bodies. J. Aero. Sci. (24), p.195 (1957)

Similarity and Dimensional Methods in Mechanics, Moscow; Gostekhizdat. English Translation (M. Holt ed) New York and London (1959) Academic Press

A Generalized Method for the Calculation of Super-sonic and HyperSuper-sonic Flow Behind Curved Shock Waves. McGill University, Mech. Eng. Dept. Report 64-2, Montreal, Canada, January 1964.

An Investigation of Hypersonic Flow Over Concave Surfaces and Corners. University of London, Ph.D. Thesis, Faculty of Engineering (1964)

Inviscid Hypersonic Flow on Cusped Concave Surfaces.

APPENDIX A: DISCUSSION OF AN OVERFOCUSSED FLOW

Overfocussed flows can be analytically rather complicated since the process of shock formation and the interaction with the leading edge shock wave tend to generate strong shear layers and reflected waves. This is illustrated by the

following example reported by Sullivan (1964). A surface comprising of a 100 wedge joined to a 300 _{ramp by a smooth curve was tested at free stream Mach numbers of}

M

=

7.5

and 10 respectively in the Imperial College gun tunnel. Surface pressure

00

measurements and ~chlieren photographs were obtained. Figure A.l is a ~chlieren photograph of the flow field at M

=

7.5,

while Figure A.2 gives the surface

00

pressure measurements. The theoretical estimates given in Figure A.2 are based on the full nonslender tangent wedge rule and the simple wave theory. The simple wave theory gives the ramp pressure if no wave reflection occurs, while the tan-gent wedge rule gives the pressure that should be generated far downstream.

The interpretation of the field generated at M

=

7.5

is given in Figure

00

A.3. The compression waves CF formed by the curved surface, degenerate into a shock EDB, and form a slipline DR and reflected waves EG, DH. This slipline is the inner of the two sliplines visib~e in Figure A.l. The reflected waves DG, BH cause the first fall-off in the ramp pressure (A.2). The shock EDB interacts with the leading edge shock AB to generate the shock BK, the slipline BL, and the reflected wave BM. This reflected wave causes second fall-off in the ramp pressure. Both BM and BL can be seen in Figure A.l.

The field generated at

Mw

=

10 is similar to that at

Mw

the two groups of reflected waves have moved together.

APPENDIX B: DlMENSIONAL AEGUMENTS FOR THE EXISTENCE OF SIMILAR SOLUTIONS The following argument was given originally by Sedov (1959, p.146). It is repeated and extended here for clarity. In the one-dimensional unsteady flow, that is the piston problem, the dependent variables, p, pand v are functions of

the independent variables y and tand of the parameters which entèr into the shock and boundary conditions. Thus, in general, we might write

p, p, v fns

(y,

t, a, b, c, d ••.• )

where a,b,c, etc., are parameters. If the initial conditions are two of the parameters are the initial density and pressure pand

00 other will be constants which describe the motion of the piston.

piston motion might be given in the form

y p

i=O in which the c. are the parameters.

l (B.l) uniform then p and the 00 For example, (B02) the

Now the dimensions of the above quantities can be expressed in terms of certain fundamental dimensions; and these are of ten taken to be those of mass length and time M, Land T. Typically, the dimensions of pressure may be written

[pJ = ML -1 T -2

(B.3)

where the square brackets indicates dimensions. Since a functional expression such as (Bol) must have dimensional equality and since the dimensions of pand p contain that of mass, then at least one of the parameters must have dimensions which contain that of mass o Without loss of generality we may assume it to be "a" and that its:dimensions are

(B.4 )

We can now apply the II-theorem to the dimensional relations(Bl) using x, tand a as the reference quantities having independent dimensions, and we find

P, R, V - fns (B, C, Do ••. ) (B. 5)

where P, R, Vare the non-dimensionalII's replacing p,p, v respectively. They are defined by CX+l t3+2 Y t a R P p, CX+t3 t3 y t a P, V tv y (B 06)

The number of independent variables and parameters is reduced by three, and B,C, D are the non-dimensional lI's replacing b,c,d etc. Now "a" cannot be the only parameter present in the formulation for if this was the case then

(B.5)

would require

Thus there must be at least one other parameter present in the formulation, and

its dimensions must be independent of those of "a". Assume that b is this

para-meter. If the dimensions of b were not independent of those of a, then

[b]

[a]5

and any solutions would have to be of the farm (B07)0 We can assume that

eb]

=

~Ir\rn

~

(BoS)

Agakn, this is not restrictive, since if[b]

bja to satisfy (30S)0

~~n

then we can replace b by

Now if "a" and "b" are the only parameters present in the formulation,

then

P,

R, V

Hence, P,R,V dep end on y and t only through the combination ~

=

ymtn so that there

is only one independent variabIe ~, replacing y and to Usually, there will be

other constants "c", "d" etc., in the formulation and generally their dimensions

will be such that they form 11' s which contain y and t in the form

k

CC]

= (ymtn) x

t~

~ so that there are two independent variables, ~ and t • the remaining constants are such that they depend on a

typically

then by (BoS) typically

But if the dimensions of

and b only, that is, if

CB

010) so that again there is only one independent variable. To SelID up: If among the

parameters in the formulation of a one-dimensional unsteady flow there are only two which have independent dimensions then the problem can be reduced to one

in which one independent variabIe ~ replaces y and to Such flows are called

"self-similar" •

The above arguments can be applied to the piston problem in the following way. In the general case the motion cannot be self similar since the constants Poo' Poe and the C

i have tlrree independent dimensions among thema Even if we take

the simplest motion Y C tk the quantities p ,P and C have independent

p 0 CY.l 00 0

dimensions o However, it is physicalJ,y meaningful to set p :::::< 0; this correspands

00

to the strong shock limit. Then since there are only two constants, pand C

00 0

this motion must be self-similar and the independent variable is

~

=

yjtk• It

is straightforward ta show that the shock shape must also be a power law, that

is Y ~ t k • Now Y = Y (p , C , t) but it is not possible to replace this by

s s s 00 a

a non-dimensional relation. But if Y = Y

Cc

,t) then a non-dimensional r~­

s s O k

lation can be formed o Since [C]

=

[Y ]j[tJk , Y~t 0 For the hypersonic slender

o s s

APPENDIX C: BEHAVIOUR OF THE SIMILAR SOLUTIONS NEAR THE BODY In the limit ç ~ 0 equation 3.27 approaches the form

dZ

dç

I

Z [Cç [Fç -- 2ZD ZG

1

(C.l)

where initially, the behaviour of Z/ç in the limit ç ~ 0 is unknown. There are three alternatives. Assume Z ~ Z çCl: as ç ~ 0, then on substituting in (B.l)

o we find Cl: =

[

If Cl: >1 then

(B.2)

becomes C F (l-y) (C.2) (C.3 )

which, since y

>

1, contradicts the original assumption. Putting Cl:

=

1 determines

Z

which cannot be allowed, since

Z

must be determined by the shock boundary

o 0

conditions. Try Cl:

<

1. Then from (2)

Cl: = 2D

G

2(k-l)

2(k-l) + y(l+j)k

(c.4)

which is consistent provided k

>

O. A similar analysis of equation (3.29) shows

that R ~ R* ~

-

a

as ~ ~ 00

b

Third order series solutions valid near ç = 0 were derived for all dep en-dent variables to ensure accurate representation of the solutions near the body. It was found that the series were in powers of ç where the exponents were initial-ly unknown functions of the parameters. Consequently, an unusual procedure had to be adopted to find the series. Consider for example the series for z. It

was assumed that

~L

Cl:.

Z Z. ç 1

1 (C.5)

i=O where we specify only that,

Cl:i+l

>

Cl:. 1 etc. ,

(c.6)

For clar.±ty, the first three terms of (C.5) are written

Z

~

LçCl: +

Mç~

+ Nç6 (C.7)

The order arguments given above have shown that Cl: = 2D/G. In the following de-velopment it is necessary to retain all terms arising from products of the trun-cated expression (C.7) for z since it is not initially known which powers are

of higher order, that is for example if

cx+~>5 or cx+t3.<5

_(C.S)

From (C. 7)

~

~

exL~CX-l

+

~M~~-l

+ 5

N~5-1

+ ••. (C.9)

Inserting (C.7) and (C.9) into equation (3.27) gives, af ter some algebra

cxL~ex+3 + ~M~~+3 + 5N~5+3 + aEL~CX+2 + ~EM~~+2 + 5EN~5+2

+

aFL~CX+l

+

~FM~~+l

+ 5FN

~5+1

_ a(1+j)L2

~2CX+l_ ~(l+j)~ ~2~+1_5(1+j)N2~5+1

- (cx+5)(1+j)

LN~CX+5+1_(

CX~

)(1+j)LMsa+~+1_(~+5)(1+j)MN~~+5+1

_ ex GL2s 2a _ ~G~s2~ _ 5GN2s25

-

(

cx+~

)

GLMsa~

- (a+5)

GLN~CX+5

-

(~+5)

GMNs~+5

= ALs3+a + AMs3+~ + ANs 3+5

+ BL ~2+a + BMs2~ + BN~2+5 + CLsl+a + CMsl+~ + CNs l +5 _ 2L2 s; ~+l _ 2~ s2~+1 _ 2N2 s 25 +1 _ 4LMsa+~+1 _ 4LN~a+5+1 _ 4MNs~+5+1 _ 2DL2s~ _ 2D~s2~ _ 2~N2~25 -

4DLMsa~

- 4DLNsa +5 -

4DMNs5~

(C.10)

Since CX

<

1, the lowest power in (10) is s2CX, which gives as before;

Gcx-2D=0; cx= 2D

G (C.ll)

By virtue of (B.6) the only possibilities for the next lowest power of

S

in (B.10)

are sCX+l or sCX~. There are three alternatives, either ~ >1, ~ = 1 or ~ < 1. The assumption that ~ > 1 or ~ < 1 leads to contradictions. For if ~ >1 then we

must equate the coefficients of sa+l to zero so that we must have

or

- (O:+b) GLM + 4DLM = 0

a

+ ~ 4D

G ~=CX < 1

A similar argument eliminates ~ < 1. Then we must have ~

coefficients of sCX+l and sCX~

aFL - (a + 1) GLM + CL + 4DLM = 0

1 so that equating