Flow behind curved shock waves

September, 1979

FLOW BEHIND CURVED SHOCK WAVES

by

S. Mo1der

UTIAS Report No. 217 CN ISSN 0082-5255

•

..

•

FLOW BEHIND CURVED SHOCK WAVES

by

S. Molder

Ryerson Polytechnical Institute Toronto

and

Institute for Aerospace Studies Toronto

Submitted July,

1979

September,

1979

UTIAS Report No.

217

CN ISSN

0082-5255

•

Acknowledgements

It is a pleasure to acknowledge Dr. I. I. Glass for hi~ constructive

suggestions and review of this report.

The financial support received from the National Science and Engineering Council of Canada and the U.S. Air Force under Grant AF-AFOSR-77-3303 is

"

"

Summary

Equations are presented for the gradients of pressure, density and velocity behind curved shock waves in a uniform upstream flow. Formulas are gi ven for

the inclination of constant property lines behind two-dimensional, conical and

dotibly curved shocks. Graphical results are presented for

r

=

1.4

and a

free-stream Mach number of 3.0. An equation is derived which relates the inc~ination

of bhe isobars, isopycnics and isotachs. A gener al relationship is derived for dotibly curved shocks which connects the two shock curvatures and the streamline

curvature just behind the shock wave. Curved shock theory is applied to the

calculation of flow curvature and pressure gradients in the vicinity of a normal

shock as well as to finding the orientation of the sonic line behind ~ curved

1. 2.

3.

4.

5.

6.

7.

8.

9·

10. CONTENTS Acknowledgements Summary Notation INTRODUCTION SHOCK-WAVE GEOMETRY CONSERVATION EQUATIONS

3.1 Flow Equations tor a Stationary Curved Shock-Wave

Surface

3.2 Equations Governing Continuous Flow

3.3 Lines of Constant Pressure, Density, Velocity and

Flow Inclination

COMPATIBILITY EQUATIONS FOR FLOW GRADIENTS AT SHOCK WAVES FLOW wrrH PLANAR SYMMETRY

CONICAL FLOW

DOUBLY-CURVED SHOCK WAVE IN UNIFORM FLOW NORMAL SHOCK WAVES

8.1

Normal Shock Waves in Non-Uniform Flow

8.2

Normal Shock Waves in Uniform Flow

THE SONIC LINE CONCLUSIONS REFERENCES FIGURES

APPENDIX A - DERIVATION OF EQ. (5.16) APPENDIX B - RELATIONSHIP FOR

d5jdn

APPENDIX C - VORTICITY EQUATION

ii iii v 1 2 2 3

4

4

8

10

14

15

19

20 21 23

24

25

A

a

[AB]

B

[BC]

C

[CA]

Cp' C

v D E G K M n p p q R r s Notation

coefficient in the compatibility equation sJ;>eed of sound

coefficient· in the compatibility equation B C' _{2 -} B'C ₂

coefficient in the compatibility equation

CA2 -

C'~

sp~cifiè heats at constant press.ure and temperatUJ'e streamline curvatur~ (= d5/ds)

coefficient in the compatibility equation coefficient in the campatibility equation

the vorticity functi.on at a curved shock wave

[=

r~/Sa

=

~2/(V2·Sa) for an irrotational upstream]

Mach number

stre~ine coordinate measured perpendicular to the streamline non-dimensional pressure gradient

[=

(qp/dS)/PV2]

pressure

2 dynamic pressure (= pV /2) gas cons tan t in p

=

pRT

ratio of shock-wave curvatures (= Sa/Sb)

shock-wave radius of curvature correspond~ng to the curvature Sa shock-wave radius of curvature correspondin~ to the curvature Sb distance measured from an axis of symmetry; also the transverse curvature of the flow

streamline coordinate measured along the streamline

shock-wave curvature in the plane containing the up- and down-stream velocity vectors

v

Greek

ex

'Y

r

5 5 1, 52 6 7T p

e

~ Symbols

shock-wave curvature in plane perpendicular to the shock and the plane containing S

a

magnitude of flow velocity

.angle fr.om streamline to isorudc

ratio of specific heats (= C _p

Ic )

v

flow deflection through the shock wave

inclination of the velocity vector on the up- and downstrerum side of a sh'ock wave

denotes a difference

di stance measured along an isobar fluid density

inclination of the shock with respect to the velocity vector upstream of the shock

vorticity

Subscripts and Superscripts

1 2

*

t p p v 5,

denotes conditions upstream of a shock wave; also coefficients on the left-hand-side of the compatibility equations

denotes conditions downstream of a shock wave; also conditions on the right-hand-side of the compatibility equations

refers to conditions when the flow velocity equals the sound speed total conditions refers to isobar refers to isopycnic refers to isotach refers to isoclinics

refers to coefficients in the second compatibility equations

1. INTRODUCTION

Crocco (Ref. 1) first considered the flowbehind a single curved ~hock wave

and pointed to the existence of a shock angle, s~ch that the flow behind the

shock could be straight even though the shock was curved. This angle of the shock wave (which varies with upstream Mach number for shocks in uniform flow) has been called the Crocco point. Thomas (Ref. 2) derived an equation relating

shoc~ and streamline curvature for two-dimensional flow and provided numeri cal

~esults for curved shocks in uniform flow (Ref.

3).

In Ref.

4

Thomas provided consistency relations for higher derivative$ of shock and streamline curvature

and used these (Ref. 5) to give the first three approximations to the presswe

behind a curved shock on a curved two-dimensional bo~. Lin" and Rubinov (Ref.

6) used theequation of Thomas (Ref. 2) to show that, for an irrotational

up-stream flow, a normal shock at a continuously curving convex wall ispo~sible

only for Mach ntunbers above a certain critical value. They also derived à

number of interesting relationships for flow behind a single curved spock "

attached to a curved body.

Shock curvature relations and flow variabie gradients behind the shock ware

derived by Drebinger (Ref. 7) for both two-dimensional and axisymmetric flow,

in both cases under the assumption of a uniform and irrotational freestream.

Flow variabie gradients we re derived also by Gerber and Bartos (Ref. 8) who

then used these to find directions of constant Mach number and constant density contours behind two-dimensional and axisymmetric shock waves.

It is the purpose of the present report to present equations which relate

f~ow curvature and pressure gradients on the up and downstream sides of curved shocks. We permit the flow to be rotational on both sides of the shock, and allow

the shock itself to be dotibly curved. In this way the theories described above

are extended by one more level of generality so as to permit considera~ion of

doubly curved shocks facing rotational flow.

The theory will then be applied to finding property graclients and lines of

constant property values behind two-dimensional, conical and doubly curved "

shocks in a uniform freestream.

The theories developed relate to flow gradients near discontinuities. On

one hand this makes the theories useful in that they provide a de eper insig~t

into flow behaviour near discontinuities as weIl as the behavio~ of the

dis-continuities themselves. This is especially useful in attempts to give an

understanding of the ve~y complicated phenomena associated with"the various

forms of shock interaction (Refs. 11, 12). On the other hand one must keep in

mind the increasingly more and more approximate nature of the th~ories as one

p"roceeds away from the discontinuity surfaces or the interaction points. Neve"

r-theless, when the laternative is a long and tedious finite difference calculation, we are tempted to boldly extend this first order theory into regions where the tpeory is aQmittedly wanting in accuracy. Such approaches can be justified only

a pOEJteriori by comparisons against other theories or experimental results •

In speaking of numerical calculations we should notice here that the present calculation of gradients downstream of a shock produces a set of higher order boundary conditions which may then be used in the numerical calculation. This

should help to eliminate many of the problems presently associated with bound~ry

Just as in differential calculus the derivative of a function leads to an insight into the shape of the function so this theory furnishes a deeper under-standing of the nature and behaviour of shock waves as weIl as the surrounding flow fields.

2. SHOCK-WAVE GEOMETRY

A doubly curved shock-wave element is shown in Fig. 1. At the point x,

onthe shock wave, the upstream ve+ocity vector is VI and the downstream vector

is V2.. These vectors are at angles 51 and

52

respecU .. \'ely to a coordinate axis

of xyz. Flow deflection through the shock at x is 5 =

52 -

5J.

and the angle of

the shock wave with respect to the vector VI is 9. Twomutually orthogonal

planes PLI and PL2, intersecting along the line L1L2, both intersect the shock

at x in such a way that LLL2 is normal to the shock at x. Two traces axa . and

bxb .' are formed by the intersection of the shock and the two planes. At the

point x these traces have the radii· of curvature Ra and

Rb,

respectively. For

the spoon-shaped shock element shown, both centres of curvature are upstream of

the shock (in both dire'ctions the shock is concave towards the upstream-flow) ,

and for this case we define the radii to be negative so that the curvatures,

which are the negative reciprocals ofthe radii· of curvature, are positive:

S

a

1

~

The other three possible shock shapes are shown in Fig. 2. This is a plot

of the two shock curvatures Sa and Sb. In the first quadrant both Sa and Sb .

are positive and we get the spoon shock just discussed above. In the second quadrant Sa is negative and Sb is positive and we get a shoe-horn shaped shock wave. Such waves are associated with internal flow of the type discussed in Refs. 17, 19, 20 and 21. In the third quadrant we get the familiar type of

shock which appears over projectiles, shells and more-or-less axisymmetric blunt bodies in supersonic flight. References in this area are too numerous to list comprehensively. For numerical calculations of this flow the reader is referred to Refs. 22 and 24. The saddle-type shock shown in the fourth quad-rant would appear on flared cones such as the "aero-shell" configu.ratiQn. The

positive Sa-axis (where Sb = 0) represents a snow-shovel shaped shock wave such

as would appear on a two-dimensional compression ramp. Along the negative

Sa-axis we would find a shock with planar symmetry (two-dimensional) such.as would

appear on an unswept blunt 'leading edge of a wing. On the positive Sb-axiS we

have internal conical flow such as found in same types of internal compression inlets (Refs. 17, 18) and on the negative Sb-axis we have the familiar flow

over a right-circular cone at zero angle of attack. The origin, where Sa

=

Sb

=

0,

represents flow through a plane shock wave. This type of diagram then represents

all conceivable shock wave surfaces for both uniform and non-uniform upstream

con-ditions. · A few representative radial lines through the origin denote constant

values of the ratio Sa/Sb, which we denote by script 1( and refer to later in the

theoretical development. Some values of

tJl

are shown around the periphery of

Fig. 2. .

3. CONSERVATION EQUATIONS

These two sets of equations are then combined in Section

4

,to yield two eq~a­

tions which relate gradients of flow properties and streamline curvatures on the upstream and downstream side of the shock wave in terms of parameters

specifying shock orientation and curvature. We assume that the flow is steady, adiabatic, inviscid and that the gas is therma11y and ca1orica11y perfect. Pis-continuities, such as shocks and slip 1ayers, are infinitesima11y thin.

In this section we wi11 deve10p also an equation which re1ates the inclina-tions of the 1ines of constant pressure, density and velocity.

3.1 Flow Equations for a Stationary Curved Shock-Wav~ Surface

Consider a stationary shock-wave element with conditions denoted by the subscript 1 in the front and subscript 2 behind it. The shock wave is at an acute ang1e e to the oncoming velocity vector V1 and produces a flow def1ection 5 and ~ velocity V2 downstream. In the p1ane which coritains both VI and V2 the shock has a curvature Sa and a curvature Sb in the p1ane which is norma1 toboth the shock and the first plane. Both these curvatures are de:fined as posi tive when the shock surface is concave, as viewed from the upstream direction; and as a consequence the radii of curvature, Ra

=

-l/Sa and Rb

=

-I/Sb are' both negative for such a shock.

The usua1 continuity, momentum and energy equations can then be written (Refs.

16,

21) for conditions immediate1y in front of and behind the- shock-wave element. Continuity: (3.1) Moment-um: Tangential: (3.2) Normal: Energy:

(3.4)

In these equations p is the pressure , p the densi ty and

7

=

_{C /Cv is} the ratio of specific heats. The quantity a* is the speed of

soun~

at the

sonic condition, which is an invariant for adiabatic flow and can be related to the tota1 temperature of the flow by a*2

=

2yRTt/( y+1), R being the gas constant.

The vorticity jump across a curved shock is (Ref. 9)

:**

, 2

t.g

(1 - P1! ,P2) cose . S (3.5)

=

Pl!P2 V 1 a

**

We use a right-hand coordinate system, in contrast to the 1eft-hand system used in Ref.

9;

consequent1y our vorticity jump as given by Eq. 3.5 is the negative of that in Ref.

9.

Thus, if we know the conditions immediately upstream of' the shoclcand the shock

angle a, then we can,'from Eqs. 3.1 to 3.4, calculate the downstream conditions

P2' V2, P2 and B. If, in addition, the curvature Sa is given,then we can

cal-cula te the vortici ty j ump from Eq. 3.5.

3.2 Equations Governing Continuous Flow

In the regions immediately in front of and behind the shock, the flow is governed by the equations of inviscid compressible flow (Euler equations) .

These can be expressed in streamline coordinates (s, n) (Ref.

9).

Continuity: d _{pVr +} db ₀

ds

pVr

dIi

=

(3.6) Momentum: N' dV p

ds

+®

_ds

=

0 (3.7) N'2 dB p

ds

+~

=

0 (3.8) Energy: 7

(N'2

'

dB + oE dP ) _ V (V dB + !: )

=

0 7-1 P

ds

Pdll P

ds

s

where, s is measured along the streamline,

n is measured normal to the streamline in the same plane that contains the curvature Sa'

r is the distanee fram an axis of symmetry (if such an axis exists)

and can, at the shoclt, 'he related to the '.'transversell

curvature Sb

by r = -cosa/Sb

~ is the vorticity defined by:

!:

=

dV + V dB

=

~ -Ön

ds

(3.l0)

3.3 Lines of Constant Pressure, Densitl' Velocity and Flow Inclination

The theme of this report centres around second order quantities ofp~ysical

flow variables (i.e., gradients of) pressure, density, velocity and flow inclina-tion. The paths along which these gradients are zero are called:

isobars isopycnics' isotachs constant pressure constant densi ty constant velocity

constant flow inclination

We will attempt to establish equations for the orientation of these lines in the region immediately bebind a curved shock wave in steady supersonic flow.

Since we consider adiabatic flow,

holds everywhere, then along a line of constant velocity there must be constant

temperature • Since

Cp

is a constant i t follows then that these are also lines

of constant enthalpy, and since a2 = yRT then they are also the lines where the

speed of sound is constant. If the velo city and the speed of sound are constant then the Mach number must be also. In summary then, the lines of constant tem-perature, enthalpy, sound speed and Mach number are all collinear with the iso-tachs. In the work that follows we will refer to the isotachs only while

remembering that all the other lines just mentioned also are identically included. The direction of development for the orientation of the various lines of constant properties (hereinafter collectively referred to as isoaxics*) follows similar paths. To illustrate we will give the complete mathematical development only for the isobars.

Suppose the distance along the isobar is measured by TT and the isobar is

inclined at an angle apto the streamline. Then

~TT

= cosCi

~

+ sinx

dIl

011 P os p

diî

But the isobar is the line for which

dIl/dTT

=

O. Thus,

Using Eq.

(3.8)

we can write this as

tanx

p =

dIl/dS

2 pV

d5/dS

For the isopycnic - the constant density line - we have

From the energy equation

(3.9),

Z

R.2E

= pvg y - l p d n 1 I' - 1 )V2

d5

p

ds

(3.11) (3.12) (3.13) (3.14 )

*From the Greek lOO - meaning "constant" and a.E;la. - meaning "value"; this

From the momentum equation, Eq. (3.7), and tbe definition for the speed of sound a2

=

'1P/p, we can write (see Appendix A)

(3.16)

Substituting (3.15) and (3.16) in (3.14) gives

dp 2

- 'S

pV

taro

=

_.--;o~o::; _ _ __

p

dB

(1-1 ) g/V -

dS

wher'e op is the ang1e between the streamline and the 1ine of constant densi ty and

s

is the vorticity as defined by Eq. (3.10).

In a very simi1ar waj we can find the 1ine of constant velocity - the isotach::

taro _v

= -

§Vit

_~

The isoc1ines, or lines of constant flow inc1ination are (see Appendix BOfor oB/cm) :

(3.19)

The above equations for the inc1ination of'isoaxics are quite general in that they app1y to steady, inviscid, adiabatic flow of a ca1orica11y perfect gas. Note that Eqs. (3.13), (3.17) ~d (3.18) on their right hand sides contain only the three parameters (dp/os) / pV2, g/V and oB/ds. E1iminati..og these -para-meters fram Eqs. (3.13), (3.17) and (3.18) we end up with a single equation re1ating

op,

Op and Oy:

2 - 7

=

1 t ara tan::x p p

_ .L:...l

tano: v (3.20)

This equation is p10tted in Fig. 3; it permits us to find one of

ap,

op or

ay

having previous1y found two others. Various possib1e combinations of the orientation of the isoaxics withrespect to the streamline s, are shown in Fig. 3. Fram the Eq. (3.20) we see tha t if

av

and

op

are posi ti ve then

°

p is positive also. The sma11 sketches in Fig. 3 show the various possib1e cambinations of directions of the isoaxics with respect to the streamline.

'.

All sketches showav as negative; this of course is not always so. When

av

is positive we can still use Fig. 3 to obtain the correct va1ues (and their signs) of ap and a_p by simp1y flipping the isobars and the isopycniçs about the streamline. Note that Eq. (3.20) is identical1y satisfied whenever CXp

=

a

p = avo

For the special case of irrotationa1 flow (s/v

=

0) Eqs. (3.13), (3.17) and (3.18) bhen became,

which then yie1d

2

t~p = (~!pV

tana _v

=

(dp/os) /pv

2

dB/d

s a = a = a p p v (3.22) (3.33)

This means that, in irrotational flow, the isobaxs and the isopycnics and iso-tachs are all collinear.

It is of periphera1 interest to note that, for the case of a hydraulic analogue where hydraulic jumps and surface waves on water behave as a gas with

r

::0_{2, we can readi1y deduce from Eq. (3.20) that a p}

=

avo

See sketch below.

Of simi1ar peripheral interest is the limiting hypersonic flow where

r

~1 (Ref.

9).

In this case ~

=

_{a p' and the isotachs are undefined since}

r

=

1 implies constant velocity flow. See sketch on page 8.

The dashed 1ines in Fig. 3, with various va1ues of

e

spaeed alongside, is the locus of CXp, a_p and

avIS

behind a curved shock i:n uniform flow at M1

=

3. This curve w~ll be further ref~rred to in Section

5.

1--r

f~+'

4. COMPATIBILITY EQUATIONS FOR FLOW GRADIENTS AT SHOCK WAVES

We now take derivatives of Eqs. (3.1) to (3.4) along the shock wave and relate these derivatives, through geometrie relations, to flow variable gradients along and normal to the streamlines. Some of these gradients ean then be eliminated through Eqs. (3·.6) to (3.9). This derivation is quite tedious and will not be repeated here. An outline for the case

ot:

planar symmetrie flow is given in Ref. 10.

The resulting equations are as follows:

(4.la)

A'P _{1 1} + B'D _{1 1} + E''''' ₁ = A'P + B'D + C'S + G'S

"ol 1 2 2 2 2 a b (4.lb) Equations (4.la) and (4.lb) are two linear algebraie equations and ean be solved for any two variables from the set of seven,

provided that we know the values of the remaining five variables.

For a uniform upstream flow Pl = Dl

=

Il

=

0 and Eqs. (4.la) and (4.lb) beeome

(4.2a)

o

=

A~P

=

B'D + C'S + G'~

-""2 2 2 2 a -b (4.2b)

where

P_l

=

(dI> los ) 1/ Pl V 1 2 is the upstream pressure gradient, 2

P₂

=

_{(dI> los ) 2/ P2 V 2} is the downstream pressure gradient.

Dl

=

(05/os)1 is the strea.mline eurvature in front of the shock,

is the streamline curvature bebind the shock,

are the shock curvatures,

is the vorticity in front of the shock.

, ,

,

, , , ,

The coefficients Al B1 El A2 B2 C G and Al B1

EJ.

A2 B2 C G are all. functions of the specific heat ratio

r,

the upstream Mach number M1' and the shock ang1e 8, and the ang1e, 51, that the upstream flow makes with the x-axis. Tbis last ang1e, 51, introduces effects of upstream flow divergence and~ecames signifi-cant on1y when Sb

f

O.

The coefficients are:

A

=

2cos8 {1 - r _ (3_{NL 2 _ 4)sin}2 8 } 1

r

+ 1 2 -""1 B

=

2sin8 { Z -

5

+. (4 _ NL 2)sin2 8 } 1

r

+ 1 I 2 -~ E

=

2sin38 {

(r _

l)NL 2 + 2} 1

r

+ 1 -~ -B 2 C G sin28 - - "::::'2-co-s~(-8";;'---:5~) 2sin28 -

-r

+ 1 4 2

-

-

_r

+ 1 sin 8

=

sin8 cos8

=

sin~8 sine 8 - 5) 2sin( 8 - 5)

sin5₁

, 2 2 2

Ai

·

= M:!.

cos5 cos 8 -

(M:!. -

1)cos(28 + 5)

(4.3) (4.4) (4.5) (4.6) (4.7)

(4.8)

(4.10) (4.11) (4.12)

where

BI

=

-sin2e

2

Cl-

=

-sin25/2cos(e - 5)

G I

=

-tane sin5_I sine e - 5) + sine tan( e - 5) sïn5₂

(r -

1)~2sin2e

+ 2 (r +

1)~2sin2(e

- 5) = sine cose sine e - 5) cos ( e -

5)

(4.13) (4.14)

(4.15)

(4.16)

(4.l7a) (4 .• l7b)

The coefficients are plotted in Figs. 4 and

5

for Ml

=

3 and

r

=

~.4. Note that

G

=

0 for a unifor.m upstream flow. Various useful deter.minants of the

coeffi-cients are plotted in Fig.

6,

and the ratios of these determinants are shown in

Fig.

7.

All of these figures are for a representative Mach number of 3.0 and

r

=

1.4.

Having specified 7, MI and e, we can calculate all the coefficients A2.,

B2, C, G,

A2'

B2'

Cl and GI • Knowing the two shock curvatures Sa and Sb leaves

us with only P2 and

D2

as unknowns to be solved from Eqs. (4.2a) and (4.2b).

These are then the gener al campatibility equations for a doubly curved shock in unifor.m flow.

The equations derived in this section will be applied first to a number of classical and weIl-understood supersonic flows. This is done in order to gain insight into the meaning of the equations and to verify their correctness against known limiting and exact cases.

5 • FLOW wrrH PLANAR SYMMETRY

I

This is the flow associated with, for example, the leading edge of an unswept wing. It is commonly also referred to as two-dimensional flow. For

this type of flow the transverse curvature is zero, i.e., Sb

=

0 and G

=

O.

Equations (4.2a) and (4.2b) can then be solved for the pressure gradient and the streamline curvature behind the shock interms of the shock curvature Sa.

C A2 - c'~

D2

=

~B2

_{- A2B 2}

If the shock is not cuxved (Sa

=

0) then both the pressure gradient and

stream-1ine curvature vanish for any and all va1ues of 1"

M+

and 9. Numerical

calcula-tion of the coefficients in the denominator A2B2 - A2 B2, of Eqs. (5.1) and (5.2)

has shown that, for realistic values of 1', the value of the denominator never

becomes zero nor infini te. This means that the gradients cannot become infini te1y large for finite va1ues of the shock curvature. Or converse1y, if the shock can

be made to have a kink (Sa ~

00)

then the downstream pressure gradient and the

flow streamline curvature become very large. If the numerator of Eq. (5.1) is zero, i. e. ,

then the pressure gradient is zero whatever the shock curvature. Similarly, if

[CA]

=

_{CA2 - C'A2}

=

0 (5.4)

then the streamline curvature is zero whatever the value of Sa. Both of these

equations re1ate M1, 8 and 1" Their solutions have been discussed in Refs. 1,

7, 8

and 10. The particular value of 8 = 9c where D2 = 0, irrespective of Sa,

is called the Crocco point. This is the point on a curved two-dimensional shock wave where the curvature of the streamline behind the shock is zero irrespective of the value of the shock curvature.

Note that the slope of streamlines in the p-5 po1ar p1ane (Refs.- 11, 12, 13)

can be readily calculated from

(5.5) It is interesting to note that the polar streamline slope is independent of the

shock curvature for shocks in uniform flow. The implication of this to Mach reflection of shock waves is discussed in Ref. 14, where it has been shown that shock waves, at the triple point of Mach reflection, can so adjust their curva-tures that the polar streamline directions on either side of the slipstream are equal. This eliminates the need to invoke singular behaviour of the shock waves at the triple point and discourages the consequent appeal to viscous effects as explanations for paradoxes and discrepancies associated with triple-shock behaviour.

The vorticity jump across a curved shock wave is given in Ref. 9 as

(5.6)

Since the upstream vorticity is zero this then becomes the vorticity downstream of the shock and we can write (Appendix C):

or

~2

M

sin(e - 5) [ t a n e J2 Sa r 2 == V 2

=

V2

=

tane tante - 5)

r

=

KS 2 a (5.7a) A graph of' K

=

sin( e - 5)

r

tane - 1 ] 2 tane tan( e -

5)

(5.8)

is plotted in Fig. 8, for y = 1.4 and MI

=

3. The vorticity function, K, is a factor, which depends on Mach number and shock angle, and which multipIies the aurvature to gi vethe vortici ty behind the shock. We notef'rom Fig. 8 that the highest vorticity is produced by a highly curved shock at a shock 'angle ·of about 65 degrees; this f'or a Machnumber of 3.0. Also note ·that f'or a normal shock,

e = 90°', and a Mach wave e'= 1-1

=

19.5°, no vorticity is produced no matter what value the shock-.curva.ture .has.

The particular conditions P2, D2 and r2' behind the 'shock wave, wil1 now be applied j.n the gen.e.r.al equat.i.ans ·f'or the inclinati.an of' the isoaxics.. The results will then yield the isoaxic inclinations behind the shock wave. Using Eqs. (3.13), (3.17) 'and -<3.·18) and the def'i.nitions for P2' D2 and r2 and their expressions in planarly symmetrie f'low gives:

tara v tara p

=

_P_2-..",--

=

[BC J

-r

₂ + D 2 -[AB]K + (CA] (5.9) (5.10) (5.11)

In all of' these expressions the shock curvature has divided out of the numerátor and the denominator , so that ctp, (Xp and CXv are not f'unctions of the shock curva-ture Sa but vary with y, e and Ml only. The angles

CXp,

(Xp and CJ.v are plotted in Fig. 9 against the shock angle e for y

=

1.4 and MI

=

3.

It is noted that,

(a) The isotach is perpendicular to the streamline (C1.v = 90°) when

e

=

49,

(b) the isopycnic is perpendicular to the streamline (CJ._p = 90°) when

(c) the isobar is perpendicular to the streamline (CXp

=

90°) when

e

=

65.

The last of these conditions is seen to lie between the sonic point (8 = 8*)

and the maximum deflection angle point (8

=

8ma.x). In fact the isobar is per-pendicular to the streamline at the Crocco point. We note also that at the

sonic point behind the shock (8

=

8*) the value of av is about +25°. Since

av

denotes the angle of the isotach, as well as the constant Mach numberline, then at the sonic point

av

is the inclination of the sonic line, and we can conclude further that at a point behind the shock the sonic line lies above the streamline for two-dimensional flow at Ml

=

3. At 8

=

750

av = ap = ap

=

0. This means that all the isoaxics (except the isocline) and the streamline are collinear at this point, so that, moving downstream behind the shock we find not only constant pressure but also constant density and flow speed. The isoclines or lines of constant flow inclination are,

-

-

---~~ ₂

sin52

(1 - ~ )P2 - r

-which for two-dimensional flow becames,

=

_---...[...;;.;:CA~JL--­ (1 -

~

2)[BC]

(5.l2a)

From this equation we see that the isocline is perpendicular to the streamline when the flow is sanic (Me

=

1). Also if the streamline curvature is zero

(D2

=

0), then the streamline must proceed at a constant angle, i.e., it is collinear with the isocline; hence a5 = 0, which is in fact as predicted by Eq. (5.l2b). The equation also shows that if the pressure gradient along the

-streamline is zero (P2

=

0), then the isocline is perpendicular to the stream-line in two-dimensional flow. Using (5.9) and (5.l2b) we can eliminate D2/P2 to get, for planarly symmetric flow:

-1 1 _ ~2

This expression holds not just behind the shock but throughout the flow field. For low Mach number flow Me ~ 0, giving

This means that in the incompressible limit, for two-dimensional flow, the isobars and isoclines are orthogonale

Tbis means that a sonic line is intersected at right angles either by an isobar or an isocline.

The orientations of the isoaxics leads us to.a more detailed understandirig of the flow behaviour in the region immediately downstream of a curved shock.

6 . CONICAL FLOW

Conical flow appears when there is no variationof flow quantities along

rays emanating from a common ovigin. Supersonic flow over conical bodies generally produces conical flows. Typical of such flows is that found over an arrow-head shaped delta-wing, and a cone at an angle of attack. A classical simple case is

the flow over a circular cone at zero angle of attack. Less 'familiar examples are two flows associated with ducted bodies described in Refs. 17 and 18. ~he last three flows are all axisymmetric versions of conical flow. An important

charac-teristic of the conical supersonic flows is that the associated shock wave is also conical. This means that Sa

=

O. As we are still dealing with a uniform free-stream we can write the general compatibility equations, Eqs. (4.la) and (4.lb), in the reduced formats for conical flow:

( 6.la)

(6.lb)

We note that G

=

0 since 51

=

O. The two equations can be solved for the pressure gradient and the streamline curvature bebind the shock,

(6.2)

(6.3)

As we approach the apex of the conical flow Sb -+

±

00 and both

P2

and D2 -HO.

Tbis is evidence of the first-order singularity at the apex of conical flows. If we write Sb

=

-cose/r, where e is the shock angle, then r is the radius of curvature of the shock wave measured in aplane perpendicular to the freestream direction. In fact it is the local radius of curvature of the shock wave cross-section, when this cross~section is taken at right ang1es to the freestream direction. Since neither ~2G' nor A2G' are generally zero then we conc1ude that there are no points where we would expect either the pressure gradient or the strea.mline curvature to be zero as was the case with planar symmetric flow.

For conica1 flows the po1ar strea.mline slope is

A significanee, similar to that described for two-dimensional flow in Ref. 14,

can be ascribed to the polar streamline slope in coni~al flow.

The angle that the isobar makes with the streamline immediately behind a conical shock is found from,

tanx

p (6.5)

Tbis angle is plotted in Fig. 10. As with two-dimensional flow, the direction of the isobar is independent of shock curvature. Of course this must be so

since otherwise the flow would not be conically symmetrie. In contrast to

two-dimensional flow the isobar in conical flow is always inclined positively to the streamline (compare Figs. 9and 10).

The flow behind a conical shock is irrotational; thus r2

=

0 and we find

the isopycnics and isotachs from Eqs. (5.10), (5.11), (6.2) and (6.4):

(6.6)

(6.7)

As was stated previously, for general irrotational flow, the isotach, isobar

and isopycnic for conical flow are cOllinear, as confirmed here by Eqs. (6.5), (6.6) and (6.7),

ex

_p

=

ex

_p

=

ex

_v (6.8)

Therefore, Fig. 10 is good also for

ex

_{p and}

ay

for conical flow at Ml = 3 and

7

=

1.4. We note from Eqs. (4.6) and (4.7) for

A2

and B2 that

tara _p,p,v

= -

B2/A2

=

tan(

e -

5)

that is,

ex

= 8 - 5

p,p,v

Tbis means that in conical flow the line of constant properties bebind the shock lies in the back surface of the shock. This is as we would expect, since in conical flow, properties are constant along rays and the shock is just a sheet of such rays.

7. ,DOUBLY-CURVED SHOCK WAVE IN UNIFORM FLOW

For a doUbly-curved shock wave in a uniform stream the compatibility equa-tions, Eqs. (4.la) and (4.lb), became,

From these we can wri te, D

= _

_A2(C'Sa + G'Sb) - A

2

_{CS a}

=

2 [AB] [BC]Sa + _{B2G 'Sb} [AB] (7.la) (7.lb)

These relations give the pressure gradient and the streamline curvature behind

a doubly-curved shock wave in uniform flow. Note that both the pressure gradient

and the streamline curvature bebind the shock depend only on '1, M, 9, and the

two shock curvatures Sa and Sb. These equations allow us to find the pressure gradient and the streamline curvature behind any of the shocks depicted in Fig. 2. In terms of the coefficients shown in Figs. 4, 5 and 6 we can write Eqs.

(7.2) and (T~3) as

(7.4)

(7.5)

In order to appreciate the usefUlness of these equations let us consider

a

numerical example of a shoehorn shaped shock, shown in the second quadrant in

Fig. 2. The radii of curvature of the shock are Ra

=

2m and Rb

=

-lIn. This

gives Sa = -liRa

=

-1/2 and Sb

=

-l/Rb

=

1. Suppose the upstream Mach number

is 3.0, and the local shock angle is 40°. From Fig.

4

we get A2

=

.950,

B2

=

-.520, G'

=

-.080; and from Fig. 5 we get [AB]

=

.335, [BC]

=

-.620 and

[CA]

=

-.595. Substituting these values into Eqs. (7.4) and (7.5) gives,

P2

=

-.620

(_l)

+ -.520 x -.080 (1)

=

-0.63

-.560 2 -.560

-1.0 (

1)

D2

=

-.560 -

2

+

The first of these values means'that just behind the shock the pressure is

and the strèa.m1ine itself has a radius of curvature of -1/D2

=

-1/- .87

=

1.15m (i.e., i t is bending downward in the flow direction) .

,

From Eqs. (7.4) and (7.5) we see that P2

=

0 when

S

_Ib

BG'

R.

_p = - =jf" a

=

_{- (BC]} 2 . Sb _a

(7.6)

and D2

=

0 when S

_Ib

A G' ~

_-

a 2 Sb R

=

[CAj

c a (7.7)

These two conditions are generalizations of the two conditions discussed in Section

5

(Eqs. (5.3) . and (5.4)]. The first one states that Rb/Ra must be in the ratio -B2G'/(BC] in order for P2

=

O. The secondcondition states that Rb/Ra must have the value A2G' /(CA) in order for D2 =

o.

This last condition is

a generalization of the Crocco point in two-dimensional flow. Physically i t means that if the shock is curved in such a way that the ratio of the shock curvatures in each case takes on the two specific values given above, then .in one case the pressure remains constant along the streamline behind the shock, and in the

other case the streaml.ine is straight behindthe shock.

Values of -B2G'/[BCJand A2G'/[CAJ are plotted in Fig .• 11. From this we.

see that it is always possible, at least for 'Y

=

1.4 and MI

=

3,. to obtain a Crocco point and an isobaric point behind a doubly curved shock provided one can impose the necessary ratio of shock curvatures, and the required ratios are those given in Fig. 11. The curve for ~c

=

Rb/Ra in fact shows what ratios

Rb

and Ra must be .in, in order to have a straight streamline behind the shock. Similarly the curve for

Olp

gives the ratio Rb/Ra required to produce zero pressure change along the streamline immediately behind the shock at any given shock angle 9.

Note th at when 9

-+ 90

1J _{then ÎR-p}

=' -1, which implies that for a normal shock the pressure along the streamline behind the shock is constant if the shock

curva-tures are equal and apposite in sign. Referring to Fig. 2 this means that we must have either a shoehorn or saddle shock at this condition. Some

considera-tion wi~ reveal that these two types of shock waves are really the same for 9

=

90

9 • A8ain some reflection regarding flow symmetry leads us to conclude

that the flow behind a normal shock wave is straight, no matter how the shock is curved. This is in contradistinction to Fig. 11 where ~c

-+ -.69

as 9

-+90·,

indicàting that the flow is straight only when the ratio of cUrvature radii approaches -

.69.

A closer examination of the expres sion for ~c shows however that ~ c ;::: % at 9 =

90°,

indicating that the flow behind the. nonnal shock is straight whatevër be the shock curvature.

Note that Eq. (7.3) can be written as

(7.8)

This is a generalization of the shock-to-streamline-curvature equations for

two-dimensional flow found in Refs.

3,

5 and

6.

We note that if any two of these curvatures are ze:r;-o then the third must be zero also.

where

From Eqs. (7.4) and (7,.5') we see that the po1ar stream:Lines1ope is

P2 (BC]Sa + _B2G'Sb

D2 - (CA)Sa ~ ~G'Sb

IR

== S /S _{a b}

=

R _--b

/R

_a

(7.9)

This equation shows that the slope of the pólar streamlines is dependent not on1y

on

r,

Mi' and e, as was the case for two-dimensiona1 and conicalflows, but also

on the ratio of shock curvatures

R.

The general equations for> the 1ines of

constant property values (isoaxics) remain as given by Eqs. (5.12a), (5.17) end

(5.20a) and (5.19) for r2' but the va1ues of P2 and D2 are now give'n by Eqs.

(7.4) and (7.5). Thus the isobar's inc1ination to the streamline is given by,

where

(BC]Sa + _B2G'Sb

(CA)Sa - ~G'Sb

=

R

= S _a

/8..

_-è = R _-ö /R _a The ang1e of the iS9Pycnic to the streamline is,

ta.r:a p

(BC

1&l

+ _B2G t (CA)~ - ~Gt

ta.r:ap

=

((r -

l)K(AB) - (CA))R +

~Gf

In a simi1ar fashion the isotachs are inc1ined to the streamlines at

oy,

and the isoclinics are inc1ined at,

t

ara

= _

05!Os

5

(1 -

Mf)[dp/os]/PV~

+ -sin5 S cose b (7.10) (7.11) (7.12)

.

.

(CA]

=

~G'

-

-

---~---~~

(1 -

~2)((BC]

+ _B2G') + [AB]

~;~~

Note that CX₅

=

0 at the generalized Crocco point, where

Ki.

=

~c

=

A2G' /[CA].

This would of course have to be so since the streamline behind the Crocco point is straight, i .e., it is a 1ine of constant inc1ination (isoclinic) and consequent1y the ang1e between the streamline and the isoclinic, cx5

=

O. Also cx5

=

90~ when

6(=

(AB]sin5/(~2

- l)cosS)

(BC]

- B G'

2

A number of observations are now appropriate for the dotib1y-curved shocks shown in Fig. 2. The first is that the values of po1ar streamline slope and

inc1inations of various constant property 1ines depend on the ratio of shock curva-tures rather than the curvacurva-tures themse1ves. There is thus a certain geometrie simi1arity between the spoon-shock and the she11-shock on·one hand and the shoe-horn-shock and sadd1e-shock on the other. In the first case ~ is positive for both shocks and in the second case it is negative for both shocks. Even though the shapes of these shocks are different the ang1es that the isoaxics make with the streamlines are the same provided y, M1' S and~ are the same.

Note that for the 1imiting cases of two-dimensiona1 flow (R -+

\Xl)

and conical flow (~

=

0), Eqs. (7.11), (7.12) and (7.13) reduce to Eqs. (5.9), (5.10) and (5.11) or (6.5), (6.6) and (6.7) respective1y.

8.

NORMAL SHOCK WAVES

Brief mention was made in the previous section of norma1 shock w~ves in re1ation to their shape at the constant pressure and Crocco points. We wi11 now examine the normal shock wave in more detail; first, for a uniform upstream flow, and then for the more gener al case of a non-uniform upstream flow.

In the 1imiting case of a norma1 shock wave we have that 5

=

O,S

=

7r/2, sins

=

1 and coss

=

O. With these values the coefficients of the campatibility equations, (4.3) to (4.16), become:

A = 0

1

C

=

0 G = 0 BI

=

0 1 B'

=

0 2

C'

=

-2(Ml2 1)/CMi2(r +

I)}

G'

=

-2(M 1 2 - 1)/{Mi2

(r

+

I)}

8.1 Normal Shock Waves in Non-Uniform Flow

For the above conditions~ the compatibility equations (4.la) and (4.lb) become,

(8.la)

A'P

=

A'P + C'S + G'S

1 1 22 a b (8.lb)

The fundamental fact that these two equations are uncoupled (Le., no terms of one appear in the other) leads to some interesting consequences.

First we note that if the freestream is irrotational then fl ~ 0, and the ratio of streamline curvatures becomes,

(8.2)

This means that the ratio of s trea.m1.ine curvatures, in front of and behind a normal shock, has a unique value at a given Mach nuniber Ml' As was concluded in the previous discussion (Section 7) we must necessarily have D2

=

0 if Dl

=

0, i.e., a normal shock in a uniform upstream flow will produce no flow

curvature even if the shock is curved. However it is possible to have D2

=

0 for a finite value of Dl and this occurs at Ml

=

1.483 when the right hand side

of Eq. (8.2) equals zero. This is the condition at which a normal shock will straighten out a curved upstream flow. Another interesting point is at Ml

=

1.662 where the streamline curvature remains unchanged through the normal shock (i.e., D2/Dl

=

1). In other words, for this Mach number only is it possible for a normal shock to sit on a surface of constant curvature. These and intermediate values of D2/Dl are plotted in Fig. 12 for both convex (solid) and concave (dashed)

streamlines.

The second compatibility condition (8.1b) can be written

(8.3)

This shows that if the normal shock is plane (Ra,

Rb

~.oo), or if the curvatures are equal but opposi te in sign, then

(8.4)

Since M12/M22 is larger than one it means that aplane normal shock will amplify and change the sign of the upstream pressure gradient.

8.2 Normal Shock Waves in Uniform Flow

In uniform flow Dl

=

Pi

=

Pl

=

O. For these conditions Eq. (8.1a) gives D2

= 0

wbich means that the flow curvature behind a normal shock in uniform flow is zero no matter what the shock curvature may be. From Eq. (8.1b) we can write

2 P

2 = ----~2

('Y+l)~ (8.5)

Tbis gives the pressure gradient bebind the normal shock in terms of the radii of curvature of the shock, and it shows that the pressure gradient behind an axisymmetric shock, for which R

=

Ra = Rb' is twice as large as it is bebind the two-dimensional counterpart (R

=

Ra, Rb = 0). For blunt-body shock waves the radii of curvature are positive, thus P2 is positive indicating that the pressure increases behind the shock. Tbis we know to be so since from behind the shock the pressure must eventually reach a higher stagnation point value on the blunt body surf ace . The pressure gradient being larger for axisymmetric bodies gives qualitative confirmation to thefact that the shock detachment distance is larger for two-dimensional than for axisymmetric flows.

It is difficult to find experimental corroboration of the relationship between shock curvature and pressure gradient immediately behind a curved normal

shock. However, there exist many numerical calculations of the flow between a curved shock and a blunt body. We compare our curved shock theory against the finite difference results of Salas (Ref. 25). These results are calculated by a time-asymptotic method originally developed by Moretti and we campare in the table below the pressure gradient along the stagnation streamline immediately bebind the shock for a spherical body as given by the calculations of Salas and

the present theory. Fram the finite difference calculations we use the first two points bebind the bow shock, which 1ie on the stagnation streSJllline ,and the calculated radius of curvature of the shock Ro' The pressures at these points and their spatia1 ooordinates are used to find the finite difference approximation of the pressure gradient on the stagnation s treamline at the,

shock (t:sp/t:sx). The comparab1e pressure gradient is also calculated analytic8J.:1y;, fr am

where

2

P2 =""'---"':=:'2

(r+ 1

)M.z

Since the flow in question is ~isymmetric 2

The values of !:sp/!:sx and (Op/os) 2 are compared in the tab1e below.

M R /:p/~ _{(èpjds )2} 0 1·5 3.178 1.8118 1.8049 1.4 3.802 1.3083 1.3011 1.3 4.943 0.8700 0.8526 1.2 7.437 0.4896 0.4748 1.15 9·970 0.3262 0.3221' 1.10 16.395 0.1825 0.1772 1.08 21.945 0.1324 0.1270

The last two columns of the above tab1e show a close correspondence for the pressure gradient as calculated by a finite difference scheme and our analytical expression. The agreement of these two tota11y differing approaches 1ends cre-dence to both and provides conficre-dence in app1ying them to other flow situations as weIl.

These straightforward app1ications of the curved shock theory have yie1ded numerical answers for streamline curvature and pressure gradients about normal shock waves as weIl as provided qualitative confirmation of known bow-shock behaviour and shown close correspondence with finite difference ca1culations.

•

9 .

THE SONIC LINE

Generally the flow bebind a curved shock wave is divided into a supersonic and a subsonic region by a sonic line. The importance of the orientation of the sonic line is discussed in Ref.

9,

where it is shown that for two-dimensional flow the angle between the sonic line and the stream1.ine at a point immediately bebind the shock is given by

t~*

v

3

' 2

tan (9* - 5*)[3(7 + l)tan (9* - 5*) + 5 - 71

2 2

[1 -

tan

(e* -

5*)][(r

+ l)tan (9* - 5*) + 2}

where

e*

is the shock angle for which the flow behind the shock is sonic and 5* is the flow deflection through the shock at this condition. The angle ~ is the angle between the isotacn and the streamline - in this case the sonic line and the streamline. Using this fact we can, from our previous work, quite readily write down the generalization to the above equation, from Eq. (7.13), as

[BC]R + B 2G'

=

'"7.(K=['":"!_AB:::""I]r--_ "";I't=CA':""I]~}=R--:-+ -:'A--: 2G::"":"'

This expression gives the inclination of the sonic line as a function of free-stream Mach number, shock angle at the sonic point, and the ratio of the two radii of curvature of the shock at the sonic point; and it allows us to cal-culate this inclination for all the possible shock shapes shown in Fig. 2. All coefficients in the RHS of Eq. (9.2) are evaluated at the sonic condition, and R. is the ratio of shock curvatures Rb/Ra. The equation has been plotted for various values of

R

in Fig. 13. In this figure &L -+ ± 00 corresponds to flow with plan ar symmetry and it is seen that the sonic surface is inclined negatively with respect to the streamline except at Mach numbers below about

1.8

where the angle is positive. We see also that for spoon and shell shocks which have a positive ~, the value of ~ lies closer to zero than saddle or shoehorn shocks for which R is negative. Tbis means that for the first two shocks the sonic surface lies closer to the stre.amline than it does for the latter two shock~

wave types. For conical shocks, cR..

=

0, the angle ~

=

9* - 5*'"". From Fig. 13 we see that all graphs -+± 90 as Ml -+1. Also we have calculated that as Ml-+ 00 the graph for ~

=

+.5

approaches 0, and the graph for

fZ-

== -.2 approaches +90. Other graphs have intermediate asymptotes at Ml -+ 0 0 . This allows us to

con-clud~ that in the range

+.5

~ ~ ~

+1,

a;

is always negative; i.e., the sonic line lies below the streamline. In the range -.2 ~

R

~ +.2,

cÇ

is always positive so that the sonic line lies above the streamline. For other values of

tl

the behaviour is more complex, which prevents the drawing of similar general conclusions. Nevertheless Fig. 13 gives a comprehensive picture of sonic line behaviour for all possible shock shapes.

This application of the curved shock theory to the sonic line has illus-trated its simplicity of application in deriving exact numerical results for a complex transonic problem in gasdynamics.

Parallel approaches can be used to derive the orientations of other lines of constant property at any point behind a doubly curved shock wave. In parti-cular .ane can think of using Eq. (7.12),

tana

p

[BC]~ + B 2G'

= -[("-7--~1"""') K~( AB""""""'] --...,t.,.."C~A

...

J},...R..-+....,.~...,G~'

or its two-dimensional reduction,

tana

=

(BC]

P

(7 - l)K[AB] - [CA]

to calculate the lines of constant density bebind curved shock waves. One could

then compare th~se calculated results against interferometric pictures of

super-sonic .flow over spheres and cylinders.

10. CONCLUSIONS

Equations were derived for finding the gradients of flow variables bebind

curved shock waves in uniform flow. Graphical results are presented for

r

=

1.4

and a freestream Mach number of Ml =

3.

Relations are given for inclinations

of all constant property lines behind two-dimensional, conical and dotibly-curved

shocks. An equation was derived which relates the inclination of the isobars,

isopycnics and isotachs. Tbis equation ~plies anywhere in the flow field and

not just bebind the shock wave. A general relationship was derived for

dotibly-curved shocks which relates the two shock curvatures to the curvature of the streamline just downstream of the shock.

,In particular it is found that for curved shocks the flow property

gradients depend nonlinearly on 7, Ml and 8, and linearly on the shock

curva-tures Sa' and Sb. The inclinations of the constant property lines are

indepen-dent of the shock curvature for both two-dimensional and conical flow, but do dep end on the ratio of shock curvatures for flow behind a doubly curved shock.

Straightforward applications of the curved shock theory have provided exact numerical resUlts for flow behaviour near a normal shock as well as at the sonic

point bebind an oblique shock. Normal shock results, are in agreement with finite

1. Crocco, L. 2. Thomas, T. Y. 3. Thomas, T. Y. 4. Tnomas, T. Y. 5. Thomas, T. Y. 6. Lin,

c.

C. Rubinov, S. I. 7. Drebinger, J. W. 8. Gerber, N. Bartos, J. 9. Hayes,

w.

D. Prob stein, R. F. 10. Molder, S. 11. Ben-Dor, G. G1ass,

r.r.

12. Henderson, L. F. 13. Guder1ey, K. G. 14. Mö1der, S. 15 • Rao, P.O. 16. Chernyi, G. G. REFERENCES

Singo1arita De11e Corrente Gassosa Iperacustica

Ne11 , Intorno di Una Prora a Diedro; L'Aerotechnica, 17 (1937), pp. 519-536.

OnCurvedShockWaves, J. Math. &Phys., Vol. 26, 1947, p. 62.

Calculation of the Curvatures of Attached Shock Waves, J. Math & Phys., Vol. 27, 1948, p. 279. The Consistency Re1ations for Shock Waves, J. Math. & Phys., Vol. 28, 1949, p. 62.

The Determination of Pressure on Curved Bodies

Bebind Shocks, Comm. Pure & Appl. Math., Vol. 3, 1950, p. 103.

On the Flow Behind Curved Shocks, J. Math & Phys., Vol. 27, 1948, p. 105.

Detached Shock Waves, Ph.D. Thesis, Harvard Univ., May 1950.

Calculation of Flow Variab1e Gradients Behind Curved Shock Waves, J. Aerospace Sci., Vol. 28, 1960, p. 958.

Hypersonic Flow Theory, 2nd Ed., Academic Press, 1966.

Ref1ection of Curved Shock Waves in Steady Supersonic Flow, CASI J., Vol. 4, No. 2, Sept. 1971.

Domains and Boundaries of Non-Stationary Oblique Shock-Wave Ref1ections, I - Diatomic Gas, J. F1uid

Mech., 1979, Vol. 91, Part 4, p. 459-496.

On the Conf1uence of Three Shock Waves in a Perfect Gas, Aeronautical Quarter1y, Vol. XV, May 1964, p. 181.

The Theory of Transonic Flow, Pergamon Press, 1962. Po1ar Streamline Directions at the Triple Point of Mach Interaction of Shock Waves, CASI Trans., Vol.

5,

No. 2, Sept. 1972.

Streamline Curvature and Velocity Behind Curved Shocks, AIAA

J.,

Vol. 11, No. 9, Sept. 1973. Introduction to Hypersonic Flow, Academic Press, 1961.

.'

17. Molder, S. 18. Molder, S. 19. Minassian, L. B. 20. Lee, B. H.

K.

21. Lin, T. C. Rub in , S. G. 22. Collar, A. R. Tinkler, J. 23. Pai, S-1. 24. Holt, M. 25. Salas, M. D.

Internal, Axisynnnetric, Conical Flow, AlM J., Vol.

5,

No. 7, July 1967.

Busemann Inlet for Hypersonic Speeds, Jour. Space-craft

&

Rockets, Vol. 3, No. 8, Aug. 1966.

1ntegral Method for Internal Hypersonic Flows,

AIAA

J., Vol.

8,

June 1970, p. 1151.

The Aeronautical Quarterly, Vol. 22, 1971, pp. 233-256.

Internal Hypersonic Flow, Paper No. 73-APM DDD, Jour. Appl. Mech., Trans. of the ASME. Also PIBAL Report No. 73-4.

Hypersonic Flow, Colston Symposium Papers, Butter-wor th t s, 1960.

1ntroduction to the Theory of Conwressible Flow, Van Nostrand.

Numerical Methods in Fluid Dynamics, Springer-Verlag, 1977.

Flow Properties for a Spherical Body at Low Super-sonic Speeds, Paper presented at the 25th Anniversary Symposium of the Aerodynamic Laboratories, Polytechnic Institute of New York, 1979.

x

FIG. 1 DOUBLY CURVED SHOCK WAVE ELEMENT IN STEADY FLOW. Vl AND V2 ARE FLOW VELOCITIES IN FRONT OF AND BEHIND THE SHOCK INCLINED AT ÖJ. AND Ö2

RESPECTIVELY TO TEE X-AXIR. .

-'

ct

Z IJ::

~

_-_CD _ _ _ _ W_'_NG_S_H_OC_K_ 0 ~ SNOW SHOVEL SHOCK

+1 ~ u o :J: en -' ct U Z o u -' ct Z IJ:: W ~ W

o

0

!JP

~

=

-1

80

60

40

--20

-40

-60

_{- - =}

2-Y

1

Y-1

-15

tonap

ton

ap

ton av

-/

-80

I

-75

/~Q

-8

FIG.

4

COMPATIBILITY COEFFICIENTS FOR

r

=

1.4,

Ml

=

3 .

. 6 .4 .2 0 10

e

-.2 B2 -.4 -.6 -:8

.6 .4 CAB] - A2 B2 - A'2 B2 [BC] - B2 C' - B2 C [CA] - CA'2 - C' A2 MI-3 Y-I.4 [BCl

FIG.

6

DRrERMrnANTS OF COMPATIBn.rry COEFFICIENTS FOR

r

=

1.4 AND

MJ.

=

3.

8 6 4 [BCl CAB] 2 0 -2 -4 -6 -8

K

1.2

LO

.8

.6

.4

.2

00102030

405060708090

e

FIG. 8 THE VORTICITY FUNCTION K

=

sin( 8-5) tan8

tan8 1

[

2 tan( 9-5) - ]

80 60 40 20 Op

op

0 e*8max 10 20 30 40 50 60 70 80 90

8

°v

-20 -40 -60 -80

FIG. 9 ANGLES BEl'WEEN STREAMLINE AND V ARIOUS ISOAXICS BEHIND A

TWO-DIMENSIONAL CURVED SHOCK,?, = 1.4, M1 =

3.

90~---~ 80 70 60 Op 50

op

40

°v

₃₀ 20 10

o

0 10 20 30 40 50 60 70 80 90

e

SHOCK ANGLE

2 ~p ~c 0'1--...l....--'--~=::.J....----L--'4---.L...!...I...-....l....-....

e

90 -I -2

FIG. 11 RATIO OF CURVATURES OF DOUBLY CURVED SHOCK WAVE AT TEE CONSTANT

PRESSuRE AND CROCCO CONDrrIONS; i

=

1.4,

_M1

=

3,

IR..

=

Sa/~'

6 5 02 M~O'+3-2M~) 4 Di' (Y-I)M~+2 3 O₂ NS D,

>l<

2 '-, ... ... ... " TRANSONIC O~---~L---~ M, '1.483 -I L-OO:::::::=---.l _ _ _ _ ---.l.. _ _ _ _ - - L _ _ _ _ ...L-_ _ - - l I 1.2 1.4 1.6 1.8 2.0

o

~

__ + . 2

-20

~

__

---~05---40

~---.I---60

_-.2

!

~=

Rb/Ra

80

/4

*

5

6

Ov

-80

-60

-40

+.5

-20

o

MI

APPENDIX A

DERIVATION OF EQ. (5.16)

The mamentum equation, Eq.

(3.7),

is given by

oV dI> pV

ds

+

ds

= 0

Since the flow along streamlines is isentropic we can write the definition for

the sound speed, a2

=

(dI>/oP)s in the equivalent form,

or

1.2e_ 1 dI>

pos-~~

v a p

Using the momentum equation above to eliminate

dI>/Os

and

2:

dI>

=

2-.

(-pv;

~

)

p

ds

2 ' Os a p or

!2e=_if

~

p OS

v

OS (5.16)

APPENDIX B

RELATIONSHIP FOR d5/On

The continuity equation is given by,

Expanding thi s gi ve s Divide by pVr, dr oV op 05 pV

ds

+ pr

ds

+ Vr

ds

+ pVr dÏÏ = 0 1 or + 1 oV + 1 op + Ö~ - 0

rds

Vds Pds

ön-Note that dr/os = sin5 and (l/p)OpjOs= -Mfov/os/V (Appendix

A).

With these

stibstitutions the continuity equation becomes,

1 -

if

oV + 05 _ sin5 V

ds

dii---r-From Eq. (3.7), Thus d5 _2 dp / 2

dii

= (1 - ~)

ds

pV

- -

sin5 _r

APPENDIX C

VORTICITY EQUATION

The kinematic vorticity at any point in the flow is defined by

\

With this definition (note the sign difference fram Ref. 9) the jump in

vor-ti city across the shock is

So that downstream of a shock, with an irrotational upstream, we would have

or

r

E

~

=

V1 P2 ( 1 _ P1)2 cose • S

2 _{V2 V2 P1} P₂ a

Using Eqs.

(3.1)

and

(3.2)

this can be written as

=

sin(e - 5)

r

tane