Second Approximation to Supersonic Conical Flows

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SECOND APPROXIMATION TO SUPERSONIC CONICAL FLOArS

A Thesis Presented to the Faculty of the Graduate School of Cornell University for the Degree of Doctor of Philosophy

by

Franklin K. Moore

ProQuest N um ber: 10834647

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•BIOGRAPHICAL SKETCH OF THE AUTHOR The author was born in Milton, Massachusetts, on August 24, 1922, and attended public schools in Milton and in Glen Kook, New Jersey*

He was graduated in 1940 from Ridgewood High School,

Ridgewood, New Jersey, and then entered the School of Mechanical Engineering at Cornell, receiving the degree BS in ME in February, 1944# He was inducted into the Army as a private in May, 194S, and was commissioned a second lieutenant in July, 1944*

He

served as an ordnance supply officer and adjutant on Adak, in the Aleutian Islands. After separation from the service in October, 1946, he entered the Graduate School at Cornell, majoring in Engineering Mechanics.

Since the summer of 1947 he has been studying for

the degree of PhD, major in Aeronautical Engineering. He was married in September, 1946, to the former Anne K. Smyth of Ridgewood, New Jersey. in March, 1948.

A son, David Elliott, was born

The author wishes to acknowledge his indebtedness to the faculty of the Graduate School of Aeronautical Engineering, and in particulot to Drs* W*R, Sears and Y.H. Kuo, for their generous and invaluable advice in conneetion with the development of this thesis.

SECOND APPROXIMATION TO SUPERSONIC CONICAL FLOWS

I.

SUMMARY The method of expansion of the velocity potential in powers

of a small geometrical parameter is applied with a view to finding a second order correction to the linear theory for steady, Isentropio, supersonic flow of a non-viscous perfect fluid about a body having conical symmetry. 1.

Two oases are considered*

Body entirely within the tip Maeh cone— for this case it

is shown that the assumption of continuity of velocities at the forward Each cone applies to the second approximation if to the first.

A method is developed for formulating at a mean plane

boundary conditions at a thin body* and the particular integrals of the differential equations applicable to the second-order velocity perturbations are .developed in integral form.

The second

order approximation to the pressure coefficient on an "arrow-head11 airfoil at aero angles of attack and yaw, and infinite in downstream extent, is computed by this technique. 2.

Body in part beyond the Mach cone— it is shown that the

above method breaks down for this case, and reasons are adduced for its failure*

-1-

II.

INTRODUCTION The linearised theory of supersonio flow in a non-viscous

perfect fluid has reached a high state of development.

Because

of its approximate nature, however, this theory is subject to error when applied to airfoils of finite thickness or angle of attack.

There is no simple way of estimating this error, except

in those few cases where higher approximations or exact solutions are known.

The results of the linear theory for the pitching

moment of airfoils are particularly suspect. It would be valuable, therefore, to attempt an improvement of the linear theory for three-dimensional cases.

This problem

,is approached in this paper via the method of expansion of the velocity potential in powers of a small geometrical parameter (thickness, say).

This method will be applied to steady, isen-

tropic problems having conical symmetry; that is, when the free stream is disturbed by a body generated by rays from a common focal point. If a velocity potential is expanded in powers of a thickness parameter, and the resulting series is introduced into the isentropic, compressible equation of motion, a linear hyperbolic differential equation similar to the wave equation is obtained from terms of first order in thickness, for supersonic velocities. The solution of this equation, satisfying the boundary conditions

an the flow obstacle is called the first approximation*

If terms

of second order are collected, a differential equation results which

involves the first approximation, and whose solution repre­

sents the second approximation* Certain aspects of this problem require discussion prior to any attempt to obtain a seoond approximation! a) Isentropy and Irrotatlonalityi We expect to be dealing with a flow character!zed by a uniform stream, which, by passage through'an attached conical bow wave, develops into a conical flow extending to infinity downstream*

The fluid is to be considered

ta compressible, inviscid continuum, except possibly at the bow wave itself.

Thus, the only dissipative phenomena occur within the

bow wave, and only then if this wave is a shock wave.

Therefore,

the only entropy changes which can take place in this flow are discontinuous ones occurring at the bow wave, and hence, the rest of the flow may be considered isentropic* >& . If one expands the entropy discontinuity at the shook (if such occurs) in powers of the Maoh number change across the shock, the leading term turns out to be proportional to the cube of this vach number change, or, to the cube of the thickness parameter. Therefore, since the purpose here is to develop a second order theory, we may neglect the entropy change across a bow shock, and conclude that the entire flow is isentropic, to the second order in thickness# We may then apply the theorem which states that, for a fluid obeying a barotropic (e.g*, isentropic) equation of state, a fluid particle once having zero rotation will continue to have zero

border of magnitude, eaoh term of the expansion must be oonlc&l. o) Justification of the expansion of the Telocity potential in powers of a thickness parameteri

This justification, to be rig­

orous, would require a proof that the velocity potential is, at each point in space, a function of the parameter differentiable to all orders-

This proof appears impossible, and we must rely upon

intuition.

We would expect a solution to a regular differential

Equation, in the neighborhood of regular able to this treatment.

boundaries, to be amen­

Of course, in the neighborhoods of stag­

nation points, or of discontinuities in body curvature, we expect a breakdown. Perhaps a more compelling argument is the following:

We know

that a first approximation exists— its justification need not rest in the applicability of Taylor’s theorem, but merely in the validity of a sma 11-perturbation theory.

Then we may say that to suppose

the existence of a second approximation is merely to say that there exists a correction to the small-perturbation theory, of the order of the square of the thickness parameter, and that succeeding cor­ rections are of higher order. We must further note that the ultimate justification for such suppositions lies in the agreement of results with experiment, d) Boundary conditions: (1)

On the body— this presents no difficulty.

prescribes that, to the highest order of approximation contemplated, there is no flow through the body surface.

One merely

(2 )

On the Waoh cone— here we are on shaky ground, even

in the first approximation*

Our trouble lies in the fact that in

an isentropic theory, we must treat the tip -Maoh eone as a flow boundary, which in reality it is not* since conditions there are a consequence of the shape of the obstacle to the flow*

In gen­

eral, a shock wave results from the presence of such an obstacle, n-nd we cannot treat a shock wave with an isentropic theory* Recourse must then be had to considerations outside the frame­ work of the theory in order to describe conditions at the Eaeh eone (or shock wave), and this description must appear as a boundary condition.

There are two possibilities— either the shock is of

aero strength (to the order of approximation), and hence is coin­ cident with the Mach cone (to the same order)} or a shock of non­ zero strength exists, and hence must appear at some conical location other than that of the Mach. cone. We must assume that the shook is attached, as it surely will be, for thin bodies.

In either case,

we must insure that the Rahkine-Hugoniot relations are satisfied to the proper order of approximation. In the first approximation to conical flows, the assumption of the former condition is customarily made, thus automatically satis­ fying the Rahkine-Hugoniot relations.

When the body is a cone at

zero yaw, for which the exact solutions of Teylor-Macoll, et al., are available, this is known to be the case. as the delta wing, one must use

an

In other oases, such

argument similar tothe following:

The shock wave near the apex is indicated by the envelope of

disturbances emanating from a very small region (vanishingly small, in fact, as the apex is ,more olosely approached), whereas, in eases where the linear theory provides discontinuous velocities at the Jfceh surface (e.g., the two-dimensional wedge), the shock wave is indicated by the envelope of disturbances emanating from a lateral line distribution of such small regions.

Thus we expect that a con­

ical shook proceeding from the apex of a delta wing would have a strength of smaller order of magnitude than that appearing in the case of the two-dimensional wedge.

We may also note that the

"supersonic source” method*, which accurately predicts the strength (to the first order) of the shock in the wedge case, predicts aero strength in the case of the deltawing. In the second approximation, we have the-same problem.

We mast

decide, by some method alien to isentropic theory, upon the nature of the second approximation to the shock wave.

The argument which

follows leads to the conclusion that, if the no-shook condition applies in the first approximation, it also applies in the seobnd, and again, the Rahkine-Hugoniot relations are automatically satisfied. This result is not restricted to conical flows. Consider any one of the velocity components to be expressed as follows: u; r where U;

is the free stream value and £

-

-(I)

is the thickness parameter.

1* Puckett, A.S., "Supersonic Wave Drag of Thin Airfoils", Journal of the Institute of Aeronautical Sciences, 13 (No. 9), pp 475-484, 1946. .

^IJow, consider the meaning of the sign of

£ — to change the sign

of € means to change the sign of the vertical flow velocity at the bodyj i.e*, to change the obstacle trom one producing com­ pression to one of the same curvature and inclination to the wind, but producing expansion, or vice versa* The eqxiation (l) above shows that €U**is an odd function of 6, that is, if there is a velocity discontinuity at the shook wave, it will represent compression or rarefaction, depending on the sign of

€ •

, however, is an even function of

6 , that is, any

discontinuity in the second approximation will be compression (say), irrespective of the sign of

€. •

This situation is represented in

Fig* (l) for the case of the two-dimensional wedge*

£ U‘

M

clcK

Wedqe

Fig. (1)

Uote that for and for

• i

£ > Q a the addition of €

€ - ^ 4 u i f r z )^-'"j

•- - e ' l u - i f ™ *f">

=o

(>.5)

^epllecting terms in Eq* (1*5) of order €. * '

r^v^T-

whence,

|where

-***?.- * % - * X - o

1

/3 r 1/7%^ - i

_________________________ (lfc;

J

§F~ This is the equation of the linear-perturbation theory. Collecting terms in Eq. (1.5) of order

= ^

* iv1?yt'x * 4 m-

= *u

£ ;

^

m

+ ^

-15-

;*

4 IV

+ ^ .o^

* Z

^

+ ^ o ^

, y jo ^

_ _(l,7

i^j^gerting Eq* (1*6)- into Eq* (1*7)#

+ W\]--(,.8)

=2U{('^ whence,

V

I

t

ll= - * £ { ( > ♦ ^ K S ) 1? * « & ♦ v v ^ x ^ i ] - - {L3)

Thus, the equations may be written*

-y"

=o

-

= -h* ^ 2.

-

{

o

- CI.W)

*

- (S,co) = ^71-c^r ( s , « > = - f i ^ ^ s ( A s 4 S( , - S ^ x ss] ------

-

V I V'«(S,**0 ■*• j§- ^ !rj7jf ^ ('+ ^S>-54)ttfir:,cc'Xs +S(|-S4) COT40 X ss -0-sU'— V x ^rLX',LsIui) -

rrr^j3 |((4+5t-s4)

These equations are developed in Par. 2.a, App* A*

-16-

4

«J X Sto]-

&(i-s4)^J«

3*

Development of solutions for

Conparing Eqs.

ULll\

V a\ and u r -»• tli

u t,)l +-

+ u r ‘'H ^

■fi =

_

_ _

(3.1)

_

(3 .^

where

i ! V

“

t/p

~

“

T

-

-

-

-

(3.3)

J

Whence, Eqs. (2.4), (2.5), (2.6) become, respectively*

v '*lX'~ p v

-< *•« )

V/C3> - - ^—

^3 £0 +4S*- 54)««puj & s '♦•SO+S'^^s'Oio^to ^.55, - 0 ~ 5 Z) \ u o

V a k J ' c* U

- f 7 T £ ^ £ o * - 4 S x- 5 * ) ^

^ SuJ^

+ s O + s 1X ' ' ^ z)

t - 0 ' S 1)*1

^

(3.5) t^ ^ s s

cO Jk S u J ^ - ~ ~(3.fe)

We next introduce the transformation

2

=

s e :u )

7

v

• C s' s e " ’“> ]

.

~

and find the following relations:

%=*L H*k= a1 -

1 a*ac

■&-

-

L 5 e lt^ A

- c s e ~ : u J i.

■11 = - s V * “£ , 4 s ’Jia w ’£*’ a*3?

CcrotO -

s

---- (3,8)

+ _£l_ = — [ * % £ r C li! + 2r^-^l 1 ?

«=

J

1

»tef

c =o. Performing the double integration indicated in Eq. (3.17), we \

find that

- __ (3,i8)

5 v | {

'-'■’h ; 1 - 1 * / W

^=flil8 equation is developed

-m

in Par.3, App.

A.

^ ---- To obtain a similar expressionfor the particular integral -

>

of

(3.6), we first simplify the equation as follows:

It we write

i+ < Sl~5* ^ -(‘"Sl)a + £ (t + s*)

# Eq- (3.6) becomes

V W *> =

s ( '- s 'l) ^ 5 i^

V ^noting that ::

•

o-sJ

5

--(3 .IO )

* T | c ( S ■—•^ 1 ) “ s ^ S S 6 (^i-s') / (,-s-1)1

—

- -

—

,

_

\

> -jjP8 may write Eq. (3.20) as

+

to

or, p

~^SS (i-s )

, .■

It

- r = > &

\

" "**•

" S is [-*— "

V-uy-^K

-

ss

..

1 Noting that

we write Eq.(3.22)

n

0 “ Sl)x

/

h

^

i

(3.33)

“ (gjCyj'^s

"

as follows: 1

(wcj

iuJ I

As

? - -L S_T ^ to^ i i l 1 .o 7 +

i- S 'J

l i H

0 - s ') 1

sS (3 K >

+

gpafcrodueing the relation (3.14), and m^THng use of the relations (S.8) to (3.12), and assuming that uX^is of the fona cur«.>a J [ p where

satisfies

«-

- —

--------------------------

Eq. (3.24), andV ^ ^ - O ,

-19-

we have

0 , 2 S’)

i ;

2: i ,

-

L{- u - 0

t (1* £ a ) f / f

J

»-£*

^ f L ' ^ e f *fr»V_fe 7 J t-C2z

- (i + c z ) %

f ^frz-^i:

'

/-C6

- 2/^'^2-fjV£c -/ 1-Pi

+j V z ^ f ± ^

j _

_ (3.30

The details of this are to be found in Par. 4, App. A. An expression for the particular integral of Eq. (3.5) for cpuld be derived in a similar manner. 4.

Boundary conditions— a) On the body? Vie prescribe simply that the flow through

the body surface be zero to the second order of approximation.

It

would be convenient to satisfy these conditions at some plane, as is

«

customarily done in first order solutions.

This can be done, as

will be shown in Sect* V. Qb the Mach cone: The argument presented in Sect. II indicates that we nay take the shook strength to be zero to the second approximation, so long as we are satisfied with the same condition as applied to the first approximation.

r t a MPLE—

1#

1 "ARBOff-HEAD" SWEPT BEHIND MASH CONE

Statement of problem and formulation of boundary conditions— N a c K Cone.

ZW

J_

Fig, (4)

v}

2A

X

Fig. (5)

Fig. (6)

Fig. (4) shows the configuration of the body— -an "arrow-head" at aero angles of attack and yew with respect to a free stream of ■velocity U and Mach number H

•

£

is the thickness parameter.

As shown in Fig. (5), 2.€.* is the angle formed by the intersections of the body surfaces with a plane passed perpendicular to the

leading edge* 'Fig. (6) defines

a new set of

( Z remains unchanged*) where

lies in the

coordinates, leading edge,and

X 7

is perpendicular to the leading edge* From Fig* (5) we have:

6 ' = £ C«-C A

----------

(|.l)

and from Fig. (S) we hare: X' -

X

A - ^ aero A

X Caro A *t

lx'=

^

a e c

A

— — —

—

-- —

—

— —

— 0*2.)

—

- (I -3)

—

—

- “ ° a I * ‘------------------------------o.-o

The boundary condition on the body is: /

\ _

- £, &4-C. A

~ —

—

—

— -------7-------- — (l-5)

Introducing Eq* (1*4), this becomes (%.)b = 6 ( ^ * ) b - € erf A ( % ) b where the subscript (

----------------------- (l.t)

means evaluation on the body.

In accordance with Eq* (1*5), Sect*II, we write: ^

: u x

+e

4-e 7- ^ ^ + *. *

If we assume that the value of ^

—

—

—

—

0*7)

3)

The boundary condition on the Maoh cone 1st «

’) . -

f

^

v

The solutions for

- w

?i -

* « > ;a

= °

- - - ( * • ,+>

and ^ C^are to be found by solving the

differential equations (1.10) and (l.ll) of Sect. II, subject to the boundary conditions given by Eqs. (l.ll), (1.12), (1.15), and (l.ld) above, and those arising from the irrotationality conditions. It is now convenient to restate the problem as it appears after application of the Tschaplygin transformation which was introduced in Par. 2, Sect. II.

We first take note of the following!

a) Eq. (2.1), Sect. II, shows that on the 5-a.ch cone (1 6 G5'=5C)r 7 "-1

;

5 -1

b) Also from Eq. (2.1), Sect. II, we find that and to find the ^

-

* w

5 -coordinate of the leading edge, we Yrrites a

>

-h Cfihf-A = 1±$2

ft

^

xS

-23-

»

Define JL as the

5 -coordinate of the leading edge -

-

-

-

(1*15)

whence:

£ * i {z j

l =

s

A *

I

3 _ _ _ _ _ ----- |6)

C©-f A [| _ /

o)

5, and henee

i # Is & dimensionless quantity*

Fig. (7) represents the plane in which the problem will be solved* The boundary conditions are as follows t Eq* (1*1) becomes: (ux (0)o = ZJ ,

uJ ^ c > s S - \

-

—

—

-

----*(3.a,Z) —_

(3 .0 ,3)

-

^

^

^

-

^

-1

‘-it* I:fuiiy** ^ U+z)x *

(c.a.3)

Using the previous conventions as to branches,we write:

^ 2

+ -JZ~ U-l)-lTT^ -

4* +

S 3

“TT

-—

---- (4,4.1)

ViTe now consider -the following: ( ^ o )0

of **^e31