Weak viscous interaction on a spinning hollow cylindrical body in axial, adiabatic, supersonic flow.

< ~ H

von

KARMAN INSTITUTE

FOR FLUID

DYNAMICS

Technicai Note 74

.

..

,

WEAK VISCOUS INTERACTION ON A SPINNING HOLLOW CYLINDRICAL BODY IN

AXIAL, ADIABATIC, SUPERSONIC FLOW

by

Harry P. HORTON

RHODE-SAINT-GENESE, BELGIUM

von Karman Institute for F1uid Dynamics

Technica1 Note 74

WEAK VISCOUS INTERACTION ON A SPINNING HOLLOW CYLINDRICAL BODY IN

AXIAL, ADIABATIC, SUPERSONIC FLOW

by

Harry P. HORTON

ACKNOWLEDGEMENT

The author gratefully acknowledges the financial support of the Royal Society of London, under the European Fellowship Program, in conjunction with the Fonds National de la Recherche Scientifique of Belgium.

TABLE OF CONTENTS

SUMMARY

LIST OF FIGURES NOTATION

1. INTRODUCTION

2. WEAK VISCOUS INTERACTION ON A HOLLOW SPINNING CYLIND ER

2.1 General Analysis 2.2 The Hypersonic Limit

3. SMALL PERTURBATIONS TO THE ANALYTIC WEAK INTERACTION SOLUTION REFERENCES FIGURES page ~ ~~ i i i 1 3 3 10 13 17

i

-SUMMARY

Analytic expans10ns 1n ascending powers or the hypersonic interaction parameter, X, are derived for adiabatic weak viscous interactioh on a hollow circular cylinder spinning about its axis, mounted at zero incidence in a supersonic stream, the boundary layer being laminar.

It is shown that, 1n the hypersonic limit, the relation between the local to free-stream Mach number ratio and tbe inter-action parameter is independent of spin rate, if all vectorial quantities appearing in tbe detinition of the interaction parame-ter are resolved in the direct ion of the free-stream relative to an observer fixed to the rotating surtace.

A perturbation scheme is developed which generates exponentially-growing 'strong' interactions, starting from the weak interaction sOlutions, tor use as initial conditions for the calculation of interactions on hollow cylinder-flare configurations.

The analogybetween the type of flow considered and that on a yawed plate of infinite span is pointed out.

- ii

-LIST OF FIGURES

1. Flow configuration considered.

2. Effect of spin parameter, Kl

=

vo/u , on weak interaction ₀₀ pressure ratio.

- i i i

-NOTATION

aCD, a e Speeds of sound

a Axial velocity profile parameter

b Transverse velocity profile parameter C

=

(ll/ll }/(T/T ) "" CD

_ :E

[x.!!

m

2]

f

-

e + X-I l+m e f Y F

₌

F y G

=

h

=

:l

₌

't

Y "J,sy J

=

=

3x-1 K₂(2 -~ _y} X-I

1:

+ I + -I m e

=

K3(2

-1.

} y 21] sy

-

J_sy Rö-:-. M I l+m 1. _~ e

ï+m

c

M

_..,

m e CD .

..

6s ./ös. 1. 1.

*

=

6y./öy. 1. 1. ~ = 6 sy . / ö y. 1. 1.

..

...

6s . /ös. 1. 1.

..

..

=

6Sy./öy. 1. 1.

..

..

öy./ös. 1. 1. Chapman constant + 3x-1 X-I tan

e

- iv -m 2

...

K2

=

_l+m Kl

...

l+m K3 = _m e K2 e 3y-l m CD K4 = _y-l

-

_l+m CD

M

=

u/a Mach number

=

ul /a ... CD

=

ul _e

/a

_e } Mach numbers relative _{on the body surface}

to an observer rotating v-I 2 m =

2

M .... M

=

M /M _{e ...} p

=

ä{u/u )/ä{z/6f.) e 1. w

Axial wall shear parameter

Transverse wall shear parameter

p Statie pressure

Q Local perturbation amplitude

Re _u

=

u /v _...

Re

ul = ul ...

/v

GO

Unit Reynolds number of free-stream

Unit Reynolds number based on free-stream velocity relative to an observer rotating on the model surfac

r body radius w R _65.

*

1. R

=

*

=

Re .6 s . U 1. 2

-

_•

65. 1.

Transformed axial displacement thickness Reynolds number

Axial kinetic energy dissipation integr

s

,y,

z Orthogonal co-ordinates, see Fig. 1

- v

-s

= (S' - l)/E

Sl

=

s.sec

r

Distance along surface measured in direct ion of undisturbed stream relative to an observer

rotating on body surface

T Statie temperature Tl T2 U = U e u Cl) vQ = aX _sy = aJ _sy a _Cl) .u a _e a _Cl) = - . u a e = Or w a Z

=

....!:. a _Cl) e

[I

n •

_o

99 Cl

=

y

o,m,a,b

Transformed axial velocity component

Transformed external velocity

Velocity of undisturbed stream

Velocity of undisturbed stream relative to an observer rotating on the body surface

Circumferential velocity of body surface

P dz PCI)

Transformed ordinate normal to surface

(1 -

~e)dn

] -1

Logarithmic decrement of linearised pertur-bations, eq. (37)

Ratio of specific heats of gas at constant pressure and constant volume

Direction of undisturbed flow relative to an observer rotating on the body surface

...

6 s. 1

•

6y. 1 6 . 1 Ä =

6"

= E = n 9 s. 1

•

9s. 1 9y. 1 9sy. 1

*

9sy. 1 6. ( 1

=

J

(1 o R6

t.

----!. M3 C _GD X v Vo

u/u

)dZ e dz (U SO/V )1/26:./50 GD Cl) 1 (.I.:l) M3 00 Re 1 / 2 s 1 / 2 2 u 0 6 .

JO

1 U L)dZ

=

-

(1

-U U e e Ó .

JO

1 U U2 =

(1

-

-)dz U U2 e e 6 .

JO

1 V2 = dz V2 0 6.

JO

1 _U V

=

_U

dz Vo e 6.

f

l~

v 2 dz

=

V 2

o

e 0 vi

-Transformed axial displacement thickness

Transformed transverse displacement thickness

Transform~d boundary layer thickness,

defined by U/U

=

0.99

e

Normal ordinate used in similar solutions

Transformed axial momentum thickness

Transformed axial kinetic energy thickness

0 À lA v

₌

II / P P (J

!',.

9 9 v

=

_vo 0 M3C 1/2

_...

X

=

Re sl/2 u Ml!Cd2 Xl

=

Re s 1/2 u 1 1 Suffices B e i o OD WI dn vii

-Inclination of displacement surface to body surface

Exponent, eqs. (35)

Coefficient of viscosity of gas

Coefficient of kinematic viscosity of gas

Density of gas

Hypersonic interaction parameter

Hate of spin of body

Zero pressure gradient (Blasius) conditions

Local conditions in the external stream

Transformed quantity in the Stewartson plane

At start of interaction

Undisturbed free-stream conditions

- 1

-1. INTRODUCTION

The objective of this studY ~s to derive the initial

conditions required for the numerical solution by the method of Reference 1 of the problem of the boundary layer-shock wave

interaction provoked by a flare on a hollow cylindrical body spinning about its axis, this axis being aligned with a

super-sonic stream (Fig. 1). The case of laminar, adiabatic flow is

considered.

These initial conditions are derived in two stages. Firstly, an analytic expansion in ascending powers of X, the hypersonic interaction parameter, is derived for the weak

viscous interaction which exists on a spinning hollow cylinder of semi-infinite length, that is, in the absence of a flare. This problem is closely related to the classical one of weak viscous interaction on a flat plate at zero incidence exposed to a supersonic stream. 2 ,3

The presence of the flare causes upstream perturba-tions to this basic flow which, far upstream, die

exponential-ly with distance trom the flare. 4 The second stage of the

ana-lysis thus consists of deriving the form of perturbations to the weak interaction solution which grow exponentiallY with distance downstream.

We consider a perfect gas with constant ratio of

specitic heats, y, having a linear viscosity-temperature

dependence, and the Prandtlnumber is assumed to be equal to unity.

The analysis uses the system of boundary layer

equations in integral form developed in Ref. 1. This system

consists of the momentum and first moment-of-momentum (kine-tic energy) integral equations tor the longitudinal flow, the

momentum integral equation for- the transverse flow, and an

- 2

-A transformation of the Stewartson type is used to reduce the equations to incompressible form and relationships between transformed integral properties are derived in the form of polynomial fits from similar solutions of the boundary layer equations.

- 3

-2. WEAK VISCOUS INTERACTION ON A HOLLOW SPINNING CYLINDER

2.1 General Analysis

We consider here the analytic expanS1.on in ascending powers of X, the hypersonic viscous interaction parameter. for the weak viscous interaction induced by the boundary layer growing upon the hollow cylindrical portion of the spinning body, upstream of theflare-induced interaction.

The approach follows closely the analyses byfubota and Ko 3 , and Klineberg 5 , of the related two-dimensional problem.

The external flowproperties are assumed to be governed by the two-dimensional Prandtl-Meyer relationship, with the~me

justification and limitations given for the zero-spin case in Ref. 6.

The starting point for the analysis is the system of ordinary differential equations derived in Ref. 1, with dr Ids

w

put equal to zero since the body surface is parallel to the axis in the region considered. These are :

..-

....

_{dM M}

t;

_~+ dos .

•

dX + _{{(2~+l) K2(2-X )X}} os. 1. e =~ ex>

(1 )

ós . +

_<iS

P

ds 1. ds Y M _e R _os.

*

M _e 1.

•

dZs

•

M

"t

_----!. doy. _{+ Oy}of _{. ----:L +;t,} J<, os. dM 1.

-

e ₌ ~ ex> 1 _{( 2 )}

Py sy ds 1. ds sy M ds _{Rö!. M} ~ e e l.

..

..

dÖs.

•

dJ os. dM

_...l.f...

M ----!. + +{3J 2K2(2X' - J )Y<. --2. e ex> ( 3 ) J ós. + = R ds l. ds sy sy M ds _{R o:. M} e e l.

..

..

d~

..

dos . dOy·

..

_d:t' os. dM

~ M _ _ 1 + 1

..

_--Z + (f+f ~)~ e 00 F F - - + os. K3 0 y· = ds Y ds 1 ds 1 ds _{y e} ds _Rot. M e l. h ( 4 )

4

-where all quantities are defined in the Notation.

We recall fr om Ref. 1 that

~

=

~(a), :Uz a(a}.o(b), P

=

p(a},;l == Tl(a,b)/o(b), P

sy y

=

P (b) y

J = J(a}, J = T2(a,b)/a(b}, R == R(b),

sy ( 5 )

where a and bare the axial and transverse velocity profile parameters, and the functions are polynomials derived by curve-fitting discrete values calculated from 'similar sOlutions,l,7.

We firstly change the independent variable, s, to the

hypersonic interaction parameter

X

defined as :

x

=

Rel/2 1/2

u s

( 6 )

at the same time introducing a non-dimensional transformed streamwise displacement thickness, 6, and a normalised Mach

#0 number, M, defined by : 6

=

Rö* _s. ~ • X ,

M =

M /M _e IX>

The following set of equations results

6 dM

( 7 )

3y-l l+m 2(y-l) 1 P { } A ~~ + Ij dl? da + (2l+1}+K2(2-X )ao ~ - -

=

];6 -2(----=.}

fi

-

da _{y M dX} l+m dX dX X e

( 8 )

3y-l aTl Tl aTl db A l+m 2(y-l) d6 _{~}6~ +} l>. dM 1 T l 6-2(---=') Tl - - + ( - + -_{aa a aa -} - . 6 - + Tl·:;: _{ab -}

=

_l+m dX dX dX M dX X e

,

,

- 5 -1

=

-3y-l l+mG) 2(X-1 ) J à - 2 ( - )

x

(F+aoF

)~

+

{d~

+F 0

~}à ~

+ {Fya 00 y - da y oa - ob dX dX

...

à dM +(f+aof ) ..

-

=

Y _{M dX} where G _ 211 sy

[<F+aoFylA -

l+m 1 2(l+mG» X e l+m e d't } db - K3ao--Z à - + db -dX r+1 2(y-l) _M3tan0 U) m e X

(10)

(11)

(12)

We now assume, following Kubota and Ko 3 , expansions forllie inde-pendent variables

M,

à, a, b in ascending powers of X :

,

=

a

_o

+ a _{1 X} + •••

where the leading terms correspond to infinitesimal interaction,

X

~ 0, for which the pressure is constant, and the suffix 'WIl denotes 'weak interaction'.

The expanS10ns are here taken only up to thetirst order in

x,

since the values of X at the start of the

flare-induced interactions considered in Ref. 1 were always sufficiently

second 6 second

-order coefficient m2 ~s inc1uded because it fo11ows quite simp1y

from mi.

We now substitute the expansions, Eqns. 13, into the

system of equations 8 to 11, expressing ~, J,P, R,a by Tay10r

series in ai P , R,o by Tay10r series in bi and Tl, T2 by two-y

variab1e Tay10r series in both a and b. The displacement

sur-face slope, tan

e,

is re1ated to the externa1 Mach number M

by the Prandt1 Meyer re1ation

tan 0

_{• e}

₌

r

M e (_{M -1} 2 )1/2

c l

M2 1 + 2 dM

-

M • e

Expanding 0 in a Tay10r series ab out M

m, and using the first

member of Eqns. 13, yie1ds :

e

=

_{1 + m} m 1 {( 2 )-1/2 m2 + ₂ ( _l+m ) M -1 _m -m

(14)

Making these sUbstitutions, mu1tip1ying both sides of the

equa-tions by X, and equating terms of equa1 powers in X, we obtain :

To the order of zero in _~

2

°0

=

( 2P/~)B (16a)

2

(2R/J)B

7

-(2P _y 11</~ _sy )B

,

(16c) (l+m) { } mI

= -

( c l ) GD (t;tB + 1 + l/me) +

K~(2-Z'B)16B

.00 (16d) ---q- M (M2_1)l/2 rry CD GD

To the order of unity in X

(17a)

(17b)

(17c)

,

(17d)

where differentiation (partial or total as applicab1e) 1.S denoted by suffices a and b, and where

(3x-l) m CD m ex>

1.4.

K4

=

-

8 for X

=

x-l l+m co l+m CD

Elimination of 00 between Eqns. 16a and 16b leads to the re1ation (PJ)B

=

(~R)B' from which ao is found; re-substi-tution of this value of a into one of these two equations gives 00. Eqns. 16c and l6d then yield bo and ml respectively. Simul-taneous solution of Eqns. 17a and l7c determine ol and al' and Eq. l7b then gives blo Final1y, m2 is found from Eq. 17d. Before making these ca1culations, however, some simplifications may be

8

-introduced by using a particu1ar sOlution, for constant pressure and constant radius, of the partia1 differential boundary layer equations given in Ref, 1 :

(18)

Substitution of this re1ation into the definition of thewrious integral quantities results in the following identities :

16

_-

1

,

B

'X

_-

1

_Z

YB B

,

X

sYB

-

;eB 1

- - '2

RB •

Because of slight numerical inaccuracy resulting from the curve-fit procedure, these re1ations are not precisely satis-fied by the po1ynomial functions, but the errors are very smalle Equation 16c becomes redundant when these re1ations are used, and bo is taken to be exactly the zero-pressure gradient va1ue

(as opposed to the slightly different va1ue that would be given by solution of Eq. 16c).

The zero order coefficients are then

9

-The first order coefficients are given by

where al

=

where IS 1

=

where (y-l)(l+m ) GD (1+K 2 )all ml all

=

( 1-1 / X_B ) / (

~

[d11 (1+K 2 ) - Kit] mI dIl = 2 +

(1

~) R da B

ap

(1:...

- 1 )

_(1.!2.)

Py

ab

B cr

ab

B

,

dP

_1:.

_~) -3.630

ëi'ä'-

_{R da B}

=

,

all

=

2.111

.

,

°1+0.41al+ mI K".

2.294.

,

The coefficient m2 may be obtained directly from Eq. 17d.

(21)

(22)

( 2 4 )

The numerical values quoted above we re calculated using the pOlynomials for vanishingly small spin rate (Kl -+ O). However, as shown in Ref. 7, the polynomials are not significantly changed by spin, for values of Kl at least as great as unity, and the above relations may be considered universal.

As an example, Fig. 2 shows the variation of pressure ratio with X, for a range of values of the sp~n parameter Kl and a free-stream Mach number of

4,

calculated from the second-order Mach number expansion using the isentropic flow relations.

10

-2.2 The Hypersonic Limit

m

CD

In the hypersonic limit, when

v-l 2

~ M » 1

2 CD

,

the coefficients mI and m2 in the Mach number expansion tend to the limiting values

1

l..:.l

2 2

= -

2' (

2 ) (l + Kl) (m 1 1 +m 1 2 )

_•

(26)

This result may be more easily interpreted bymtro-ducing a new co-ordinate system, as shown in Fig. 3. and resol-ving quantities along the direction of the undisturbed flow relative to the body.

The undisturbed flow velocity and Mach number relative to earth are u • M • whilst relative to the body they are uI ,

-~

...

MI • Then if

_...

r

is the direction of the undisturbed flow relative to the spinning body, we have :

UI 2 vo 2 2

(--=.)

₌

1 +

( - )

- 1 + Kl = sec 2 r _• (27)

u u

CD 00

Also, M 00

=

MI ... cosr

Measuring sI along the relative wind direction,

SI

=

s.sec r (28) M 3 C1/ 2 _U ~ 00 Now _X

=

_{1/2 1/2} where Re

=

U v Re s

...

u

- 11

-Introducing now ~ hypersonic interaction p~r~meter, Xl, defined by flow properties measured in the sl-direction, such that Xl = , where Re ul = i t follows tha t Xl

=

(l+Kl)x 2

-,

•

Hence the Mach number expansion may be written ~n terms of Xl in the form Ml M 2 2 ~ _{- -.!:. =} 1 _{+ ml Xl + ml Xl + •••} Ml"" M co 1

_(cl)

2 where mi

₌

-

(mll+ml2) - 0.0479 2 2 (30)

Thus the Mach number expans~on, based onparameters measured along the relative wind direction, is not dependent upon the spin ratio Kl' and is identical with that for two-dimensional flow, in the hypersonic limit. This may be most easily shown by the corresponding expansion for pressure which, using the relation

~ M y-l

=

(--!:.) M v-I 2 ~ 14 » l 2 for

,

Cl) gives 1 + 0.335 Xl + 0.048 Xl - 2 •

This ~s ~n agreement with the two-dimensional result of Kubota and Ko3 •

This result may be more simply, but less rigorously, derived by an application of the 'principle of independence', since boundary layer parameters for constant pressure flow only enter into the coefficients mI and m2, for which special case the principle applies in compressible flow.

- 12

-Without entering into details, we may remark that the above result, and the preceding weak interaction analysis, are equally applicable to the~se of a flat plate of infinite span, yawed at an angle of

r

to a free stream of velocity U

...

and Mach number M .... In this case it I1By be shown that, in the hypersonic

limit, the weak interaction is independent of sweep angle if all quantities are measured in the direction of the undisturbed stream.

- 13

-3. SMALL PERTURBATIONS TO THE ANALYTIC WEAK INTERACTION SOLUTION

When the above weak interaction solution is used to

g~ve initial values for the computatioh of a flare-induced inter-action, small perturbations must be applied to these initial values in order to guide the numerical solution in the desired compressive sense. Ko and Kubota 8 have derived the consistent form for such perturbations in two-dimensional, adiabatic, hyper-sonic flow. We shall now make a similar analysis for the case of a sp inning hollow cylinder, again for adiabatic, hypersonic flow.

In practise, i t 1S found that this hypersonic perturbation scheme works well even at low supersonic speeds.

The starting point of the analysis lS again the system of equations 1 to

4.

Following Ko and Kubota, we introduce a non-dimensional strearnwise ordinate,

s,

and a non-dirnensional displacement thickness,

6,

defined by :

1/2

s

=

siso Ó = (u s 0 / v ) ó st / s 0

CD co 1

where sa lS the assurned position of the start of the 'strong' loc al interaction induced by the flare. We further introduce the small quantity E defined by :

E

=

,

so that E ~s closely related to the interaction parameter

x.

(32)

We are interested ~n the region where (s-so)/so

=

O(E);

and accordingly introduce :

....

s

=

s-l _E

14

-Now, again following Ko and Kubota, introduce pertur-bations of the form

- - - À'"'

o

= 0o(s) + €

6(s)

-where 00' ma, ao, bo are the unperturbed weak interaction solutions

at s

=

sa' and À is an unknown exponent greater than unity.

Substitution of the above relations into Eqns. 1 to

4,

together vith Taylor series expansions as used in the weakmter-action analysis, leads to the following set of equations for the

perturbations, for (X;1)Me 2 » 1 :

ra

dm

ma

di

=

0

'"'

Jo

:~

+

~(:~)o

:i+

(3+2K~)Jo !~ ~

=

0 , (36)

r-{) ,,-iJ j-db'"' 3X-l 2 "f)

ra ...

(2-4x) ...

-(l-ho)~o oo~+(----)(l+Kl)(l+~o)-- ~+mo

y-l ! -

=

0

15

-Here, Eqns. 19 have been used to simp1ify the resu1t, introducing a slight error because these re1ations are exact on1y at the

B1asius point.

This system of first-order ordinary differentia1 equa-tions is 1inear with respect to the derivatives of the pertur-bations, and the coefficients are constant. We therefore assume solutions of the exponentia1 type

L

=

A exp(als) , 6 0 ,.

a

=

B exp(als), m mo ,.

=

C exp(als), b

=

D exp(als),

which, on substitution into Eqns. 36, g1ve

(g) B 2

JoA + + (3+ 2K_{l)J O}C

=

0

da 0

,

02'

a1(, ) o'l' 01<,

~oA + (-2Z0 +X

_o

- 0 B + ,;eo C + (-2L + "2to

äb

_{0 )D}

=

0

aa oa ob 0

This set of 1inear simultaneous a1gebraic equations for A, B, C, D has rea1 solutions on1y if the determinant of the 1eft sides is zero, when A:B:C:D may be found from on1y three of the equations. Using the first three, we find that :

16

-A B - q, say, =

=

where _{P 2}

=

3J 0 - (2'l?0 + 1 ) ( dJ / d

-

t.h

P 3

=

Jo(l-to )f(d~/da)o

,

(40)

PI

=

~o(dJ/d~)O - JO

,

[

a

(X 1<)

1/

a

('l/ :/1,) p ..

=

- ~0(PI+P3)+P2 3a sy 0 3b sy 0 •

Putting Q

=

E À q exp(als), and combining Eqns. 37, 39 and

40,

we get

Me/Ma>

=

mo L1+QPI] a

=

aO~+QP3(1+K~

] b

=

bo~+QP

.. (l+Kf)]

,

(41)

Thus, i f arumerica1 perturbation, Q, is app1ied tothe weak interaction solution öo, mo, ao, bo at s

=

so, according to Eqns. 41, exponentia1 solutions will be generated by downstream integration which are of compressive type if Q < 0. and of&pansive type if Q > 0.

The logarithmic decrement, al , may be obtained by putting the determinant of the left sides of Eqns. 38 equal torero, but is complicated in form and not of interest 1n the present appli-cat ion of the ana1ysis.

17

-REFERENCES

1. Leblanc, R., Horton, H.P. and Ginoux, J.J., "The Calculation of Adiabatic Laminar Boundary Layer Shock Wave Inter-actions in Axi-Symmetric Flow : Part 11 - Ho11ow Cylinder-Flare Bodies with Spin", TN 73, 1971, von Karman 1nstitute for Fluid Dynamics.

2. Hayes, W.D. and Probstein, R.F., Hypersonic Flow Theory, Academic Press, 1959

3. Kubota, T. and Ko, D.R.S., "A Second-Order Weak 1nteraction Expansion for Moderately Hypersonic Flow past a Flat Plate", AIAA Journal, Vol. 5, No. 10, Oct. 1967~

pp. 1915-1917

4.

LighthilI, M.~., "On Boundary Layers and Upstream 1nf1uence. Part 11 - Supersonic Flows without Separation",

Proceedings of the Royal Society (London), Series A, Vol. 217, 1953, pp. 478-507

5. Klineberg, J.M., "Theory of Laminar Viscous-Inviscid Inter-actions in Supersonic Flow", Ph.D. Thesis, 1968, California Inst. of Techno1ogy, Pasadena, Calif.

6. Horton, H.P., "The Calculation of Adiabatic Laminar Boundary Layer Shock Wave 1nteractions in Axi-Symmetric Flow Part 1 - Hollow Cy1inder-F1are Bodies with Zero Spin", TN 60, 1970, von Karman 1nstitute for F1uid Dynamics 7. Horton, H.P., Leblanc, R. and Ginoux, J.J., "Integral

Relation-ships for Self-Similar Compressible Adiabatic Laminar Boundary Layers with Constant Transverse Velocity",

(to be published)

8. Ko, D.R.S. and Kubota, T., "Supersonic Laminar Boundary Layer along a Two-Dimensional Adiabatic Curved Ramp", AIAA Journal, Vol. 7, No. 2, Feb. 1969, pp. 298-304

FIG. 1. r w ",,--

-/ '

"

/ '

"

/ '

,

/ / \~ / '

~\

'"

_"

\

.~

~-~r)

o w _ - -

7--FlARE CIRCULAR CYLINDER

I

I

/

/

.-/

LONGTITUDINAL SECTION OF CYLINDER

4 3 2 1 o FIG. 2.

Ka-

4.0 ~ - 1.4 1 2

_x

EFFEcr OF SPIN PARAMETER. Kl- v _o Ju • ON l,VEAK

Cl)

INTERACTION PRESSURE RATIO

U_cD

RELATIVE Ta EARTH - ORIGINAL SYSTEM

/

v o

REUTIVE 1'0 BODY SURFACE - NEW SYSTEM

FIG. 3. CHANGE OF CO-ORDlNATES FROM EARTH-FIXED Ta