The increased thickness of the boundary layer contributes to two types of interaction of the boundary layer with the inviscid flow field

346 C . . "VATH

Variables in dimensionless form

X xJL

~ yJL

fl eJL

1. Introduction

The contribution of skin friction on the drag of slender missile at super- sonic or hypersonic Mach number is more than that of the pressure distribu- tion. One of the main factors in missile design is the extreme surface tempera- ture produced by the frictional heating of the missile. Knowledge of the bound- ary layer properties at high Mach number is essential to the missile designer because friction drag and heat transfer can then be determined.

At hypersonic speed the deceleration of the fluid by a shock wave or by viscous processes in a boundary layer generally produces very high tempera- tures. Due to high temperature, there is an increase in the thickness of the boundary layer over the thickness encountered at the same free stream Rey- nolds number at lower speeds, also various physical phenomena, like dissocia- tion and ionization, diffusion of atoms and ions occur, which cause the air to depart from a perfect gas behaviour. The increased thickness of the boundary layer contributes to two types of interaction of the boundary layer with the inviscid flow field; (a) pressure interaction and (b) vorticity interaction. "Pres- sure" interaction is important for slender bodies, and "vorticity" intcraction is important both for thc slender as well as for blunt bodies.

The effect due to these interactions can be neglected and the air can be considered as a perfect gas if the Mach number is not very high (~C

<

^10).

An approximate method for the calculation of the laminar compressible boundary layer on three dimensional bodies in axial flow, based on the momen- tum and the energy integral equations of the boundary layer, has been devel- oped by ROTT and CRABTREE [1] under the following assumptions:

(a) The surface is heat insulated (no heat transfer).

(b) The Prandtlnumber, P, of the gas is unity.

(c) The viscosity,,u, varies proportionally to the absolute temperature, T.

Further ROTT [2] developed a method which is applicable to adiabatic walls when It ."-' T across the boundary layer, but ,u ^{r - J} Tⁿ outside the boundary layer, and therc is no restriction concerning the Prandtl number. But they did not take the effect due to the presence of shock wave into account.

HAKTZSCHE and WKNDT [3] have shown that the equations for a thin laminar boundary layer on a circular cone at zero angle of incidence in a super- sonic stream with attached shock can be reduced by means of a simple trans- formation to equations of the same form as those for a flat plate.

They have obtained the characteristics of the boundary layer in terms of the flo'w condition just outside the boundary layer of the cone and not in front of fthe attached shock wave.

CO.lfPRESSIBLE LAJ[[SAR BO[;SDARY LA YER 347 The present author applies the approximate method of ROTT for the cal- culation of the characteristics of the laminar compressible boundary layer on supersonic or hypersonic flow past unya'wed infinite solid circular cone.

The basic assumptions are:

(a) The cone angle, cc, and the free stream Mach number, M=, are such that the non viscous flow is conical and irrotatiollal and the shock wave is attached to the vertex of the cone.

(b) There is no restriction on the Prandtl number, P.

( c) The viscosity, 11, is proportional to T across the boundary layer i. e.,

II T n

-'- = - - , otherwise fl ~ T . TlV

(d) Boundary layer is thin compared to the shock layer and considered as separate from the inviscid flo'w field. "Pressure" interaction and "vorticity"

interaction have been neglected.

(e) The air behind the shock is considered to be pcrfect.

The present method is simple in operation and gives momentum thick- ness as a simple quadrature. All the characteristics have been analytically expressed in terms of free stream quantities.

2. Characteristics of the inviscid flow in the shock layer

. \

The non viscous flow outside the boundary layer is assumed to be un- affected by the presence of the thin layer and is conical. All physical quantities are constant along rays from the apex and along generators of the cone. There- fore, physical quantities at the outer edge of the boundary layer depend on cr., hut not on x. The various characteristics of the flow at the surface of the cone have been obtained as in [4.] by assuming that the density in the shock layer is constant. The results are given below:

cosec² y.

J( (1)

_ I

sin )

K = B

+

^{In I - sec lfJ}

\ 1 - cos lfJ '

B = _ In (' _~~) _ _ c_o_s_cc_

. 1 - cos cc sin~ cc

--- +

2 ^{(Y - 1)} NI:' sin² If'

=E

⁽²⁾

Y 1

1*

348 G.NATH

B

+

In - - - ' - -

+

^{cosec2 11' [1}

+

^{Y (1 - E)}

^M!

^{sin! 11']}

l

^{1 -sin cos 11' 11'}

⁾²

(3)

1 - - - - K2

1 _ cosec⁴ IX

K2 (

sin 11'

)2

B

+

^In

+

^{cosec2 11'}

1 - cos 11' 1

B

+

^In

+

^{cosec2 11'}

[E{l

+

^{Y (1 - E) M~ sin 21p}]}

(4)

[ ₁₊ Y-I _{-2-M~ (I-e) sin}21p ]

l '

^{sin 11'}

⁾²

1 - cos 11'

1 - - - (5)

(1.0 20

1'If", 6

!po 24.5

121 3.3193

/2",

El- 7.4063

p",

Tl _2.2313

T;::

M1 3.6759

cosec4 IX

M~--­

M i = - - - - -K2 Tl T",

Table I

20 30

7.5 6

23.6 35.1

3.8592 4.2251 10.7003 14.0925 2.7727 3.3354 4.1432 2.7084

30 7.5 34.25

4.6852 21.0376 4.4902 2.9413

COMPRESSIBLE LAMINAR BOUNDARY LA YER 349 Mach number, M~, and shock wave angle, "P, are related by:

[

21(

AI

^{oo = cosec '1jJ (Y}

+

1)

{B +

In ( sin '1jJ

j +

1 - cos 1jJ

c~s

"P } _ (Y _ 1) 1(

]~

sm²1f!

(7)

they are given in Table 1.

3. Velocity, temperature, density etc. in the bonndary layer

Knowing the characteristics of the inviscid flow at the outer edge of the boundary layer, the various characteristics of the boundary layer can be easi- ly obtained. According to the simplifying assumption of the von Karman- Pohlhausen method, any velocity profile in the boundary layer may be repre- sented as a member of a one-parameter family of profiles. Thus, the velocity profile may be written as:

u

y

^{1 (}

y')2

-;;; =

^{li (f3)}

e ⁺

^L2

^mdp) e

dUI 8²

(T

^j3-2Y

where

p

is a paramerer and mi

= - - - -- __

^{1_ Y-1}

dx 1'0 To,

(8)

(9) For the present case mi = 0, as u₁ is independent of X.

p

can be made equal to mi' According to the modification proposed by TrrwAITEs [5], the universal function li(mi) can be calculated fr<¥ll Fig. 1 of [5], when mi

=

^O.

Since

e

can be obtained from [18], -U can be known.

Ul

Hence, ^{II =}

r

⁽⁰⁾

L

cosec² cc

I 8 1(

As in [2],

T ^w 'ip Y - 1 71,1'0 cosec,l cc

- =

^V - - j y 1 . : ' - - -

+

Too 2 1(2

1 cosec⁴ cc 1(2

1 _

(B +

In sin 1f! )2 1 - cos 1jJ

• [E {I

+

Y

M:'

(1 - E) sin2"P}]

(10)

(11)

(12)

+

^cosec2'1jJ

(13)

350 G.IVATH

1 ^I Y - 1 71"'~

' - 2 -^1Y.L1

V -

Y - 1 1+ P - - M i

2

cosec·¹o::

1 - - - - K2

(14)

(15)

l

^E

⁺

^{In sin 1p}

⁾²

1 _ 1 - cos 1p

cosec²1p

The pressure inside the boundary layer is the samc as the pressure on thc sur- face in the inviscid flow.

T IT w. ^{l ' -I?w} etc. are given in T hIlI a e . 1?1

aO

M", P hG> T"

T,=-;y:

12", 12,

~ 12""

~ Too

To

T;.

H

20 20

6 6

0.737 1 3.3199 3.7026 0.3012 0.2701 0.9998 0.8965 7.4076 8.2615 1.1153

10.9230 12.2974

Table II 20

7.5 0.737

: 20 130 . 0.737

I1I

~11.5

^{11 6}

3.9471 4.4332 I 2.2593

0.2533 ^I

1

0.2255 I 0.4426

!

0.9777 0.8705 I 1.8701

I

10.9439 12.2919 i 7.5357 1.1232 1 1.0919 13.1756 11.9215 7.1141

13~ 13~.5

1

I

^0.737

2.4670 2.4853 30

7.5 I

2.7303 0.4053 0.4024 0.3662 1. 7126 1.8852 1.7160 8.2286 111.1595 12.2596 1

7.8602

1.0986 I 1 7.9257 ¹ 8.8056

i

COMPRESSIBLE LA,yIINAR BOU1YDARY LAYER

4. Momentum thickness and skin friction coefficient

351 --...I

The momentum thickness, 0, of the compressible boundary layer is given by:

where

Hence,

If

- 2

^{l'P . T}

·1-11

(_o)

^y2

^Uf dx

⁽¹⁷⁾

. Tw

y = x since

_

0.15E(~)IlI·~II-1l

02 = Too. T^tv x

Re ~osec2 cc K

P = l , - = 1 To Ttll

l ^E (

^{_ 1}

^{T tl (T}

^{_ 0}

^{)1- 11}

J

^{= 315}

e

^{= 8.5135 Too I . Ttv}

37 Re cosec² cc

K

(18)

(19)

H=(i;)H

ⁱ

+ i:

^-1=[1+

^lip

^{y 2}

111Ii]IHi+l1-1

⁽²⁰⁾

where

Hi

can be calculated from Table I of [5].

Co^{= - - -}

!o _{!! ... txl}

U:.,

⁽²¹⁾

1'u Ta

where - can be obtained from [4],-from [14], Ml from [6] and li from

Too

7;v

Fig.l of [5]. 0, ^Co etc. are given in Table Ill.

352 ^{G. NATH}

Table

m

a' 20 20 20 20 20 20 20 20

M", 6 6 6 6 7.5 7.5 7.5 7.5

P 0.737 1 0.737 1 0.737 1 0.737 ,1

n 0.5 0.5 0.8 0.8 0.5 0.5 0.8 0.8

9 (Re

2.8191 2.745 3.379 3.339 3.714 3.599 4.476 4.425

Yx

CD~ ^{0.307 0.315 0.256 0.258 0.255 0.262 0.207 0.209}

0*

V ^~e

^{30.792 33.755 36.909 41.058 48.932 53.700 58.971 66.025}

bV ^~e

^{23.998 23.368 28.765 28.424 31.607 30.638 I 38.104 37.670}

aO 30 30 30 30 30 30 30 30

Mo> 6 6 6 6 7.5 7.5 7.5 7.5

P 0.737 1 0.737 1 0.737 1 0.737 1

n 0.5 0.5 0.8 0.8 0.5 0.5 0.8 0.8

-If-¥

^{1.527 1.494 1.161 1.716 1.788 1.747 2.048 2.029}

o -x

CDVRex 0.462 0.471 0.410 0.412 0.417 0.426 0.363 0.366

*

VRe

^{10.863 11.743 12.314 13.487 14.171 15.382 16.232 17.865}

e -

_x

-If-¥

^{17.434 17.273}

0 - 12.999 12.718 14.736 14·.608 15.221 14.872 x

5. Results of the numerical calculations

The numerical results are given in Tables I to Ill. From the results, it is possible to know the effects of Prandtl number, P, Mach number, ~M~=, semi- vertical angle of the cone, 'X, Reynolds number, Re and index of viscosity tem- perature law, n, on the various characteristics of the boundary layer.

(a) Effect of Prandtl number: Temperature at the wall, TIV , H, Co and 0* increase, but

e,

^{band Ibv} decrease as P increases.

(b) Effect of Mach number: Temperature, Tw , H,

e,

0 and 0* increase, but QIV and Co decrease as lVI= increases.

(c) Effect of Reynolds number: Reyuolds number affects only

0, b,

^Co

and

b*.

They decrease as Re increases.

COMPRESSIBLE LAMINAR BOUNDARY LAYER 3.'13

(d) Effect of the index of the viscosity-temperature law: (j, CO'

J*

and J depend upon n, others are independent of it. They increase as n increases.

(e) Effect of semi-vertical angle of the cone: (j,

J,

H, 15* and P_w decrease, but Co increases as IX increases.

6. Conclusions

The momentum thickness,

e,

and houndary layer thickness,

J,

the dis- placement thickness,

J*,

H, and T w increase considerably as Mach number is increased but Co decreases as NI

=

increases. The effect of Prandtl numher on the momentum thickness,

e,

the skin friction coefficient, Cff, and the boundary layer thickness, ;5, is small, ·whereas its effect on wall temperature, Tw, wall density, fiw, Hand b* is considerably large. n has little effect on

0,

Cf, "(5*

and ;5, others are independent of it. The effect of IX on the boundary layer char- acteristics is rather appreciable.

7. Summary

In the present paper, the approximate method of Rott has been applied for the cal- culation of thc steady compressible laminar boundary Jayer characteristics under certain assumptions when the flow past an unyawed semi-infinite solid circular cone is supersonic or hypersonic. The assumptions are: (a) The cone semivertical angle, ()(, and the free-stream Mach number, .0/1 =, are such that the non viscous flow is conical and isentropic, the shock wave is straight and attached to the vortex of the cone. (b) The boundary layer is thin compared to the shock layer and considered as separate from the inviscid flow field. Pressure interaction and vorticity interaction have been neglected. (c) The air behind the shock is considered as perfect. (d) There is no restriction on Prandtl number, P, and is considered to be invariant with temperature. (e) The viscosity, p, is proportional to T across the boundary layer. otherwise f.l ~ Tⁿ and the body is insulated i.e., there is no heat transfer. The present method is simple in operation. The characteristics of the boundary layer have been analytically expressed in terms of free stream quantities.

References

1. ROTT, N. and CRABTREE, L. F.: Simplified laminar boundary layer calculations for bodies of revolution and for yawed wings. Jour. Aero. Sci. 19, 553-558 (1952).

2. ROTT, N.: Compressible laminar boundary layer on a heat insulated body. Readers' Forum.

Jour. Aero. Sci. 20, 67-68 (1953).

HANTZSCHE, W. and WENDT, H.: The laminar boundary layer on a circular cone at zero incidence in a supersonic stream. Rep. and Trans. No. 276, British IlL A. P., Aug. 1946.

NATH, G.,: Hypersonic flow past unyawed circular cones. Journal of the Indian Mathemati- cal Society 28, 7-24 (1964).

5. THWAITES, B.: Approximate calculation of the laminar boundary layer. Aero. Quar. 1, 245- 280 (1949).

Girishwar NATH, P. and T. Colony, Qr. No. 66, Kidwai Puri, Patna-l, India