An approximate calculation of the laminar heat transfer in the stagnation region of spheres and cylinders in high speed flows

TEl~"'~lrU:

HOGEscnoel MUI

"IueGTUlGI()UYI~

f!

fEB.

1973 < ;:-;: H

.

BlJUOTllEEK

von

KARMAN INSTITUTE

POR FLUID

DYNAMICS

TECHNICAL NOTE

84

AN APPROXIMATE CALCULATION OF THE LAMINAR HEAT TRANSFER IN THE STAGNATION REGION OF SPHERES

AND CYLINDERS IN HIGH SPEED FLOWS

by

Han s W. STOCK

RHODE-SAINT-GENESE, BELGIUM

von KAR MAN INSTITUTE FOR FLUID DYNAMICS

TECHNICAL NOTE

84

AN APPROXIMATE CALCULATION OF THE LAMINAR HEAT TRANSFER IN THE STAGNATION REGION OF SPHERES

AND CYLINDERS IN HIGH SPEED FLOWS

by

Hans W. STOCK

TABLE OF CONTENTS

LIST OF SYMBOLS

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ABSTRACT

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1. INTRODUCTION

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2. HEAT TRANSFER EQUATIONS •

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3. RESULTS AND DISCUSSION _•

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REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D

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Calculation of the temperature gradient at the wall in the two-dimensional case (cylinder) Calculation of temperature gradient at the wall in the axisymmetric case (sphere) Calculation of the velocity

gradient at the stagnation point Calculation of the Mach number dependence of the function F at the stagnation point for M + ~

~ i 1 1 2

6

8

- i

-LIST OF SYMBOLS.

A Constant defined by equation (8 A) a Velocity of sound

C Chapmants constant defined by equation

(6

A) c Pressure coefficient defined by equation (1 e)

p

c Specific heat at constant pressure

p

F Function defined by equation (13)

F

1 Function defined by equation

(7)

f Velocity ratio defined by equation ( 2 A)

h

h

H

k

Film or heat transfer èoefficient

statie enthalpy

2 Total enthalpy, H = h + u

2

Heat conductivity coefficient

L Reference length in equations (1 B)

M Mach number

m Quantity defined by equation (2 A)

m Exponent in the external velocity relationship defined

by equation

(8

A) Nu Nusselt number

p Pressure

Pr ?randtl number

q Heat flux per unit area and time

R Body radius, sphere or cylinder

ii

-S Entha1py function defined by equation (2 A)

T Temperature

u Velocity in the x- direction in the physical plane

U Velocity in the X- direction 1n the transformed plane

(Stewartson transformation)

x Streamwise distance in the physical plane .

X ~eamwise distance in the transformed plane (Stewartson

transformation)

y Distance normal to the wall in the physical plane

Y Distance normal to the wall 1n the transformed plane

(Stewartson transformation)

e

Pressure gradient parameter defined by equation (11 A)

y Specific heat ratio

n Variable defined by equation (

5

A)

e

Angle between stream direction and radius vector from the center of curvature of the nose

~ Dynamic viscosity

v Kinematic viscosity

ö Density

Subscripts.

~ Upstream infinity conditions, upstream of the shock

o Stagnation conditions

e ~nditions at the outer edge of the boundary layer

- i i i

-W Wall conditions

1 Quantities evaluated immediately downstream the normal shock

Quantities in the transformed plane (MangIer transf6r-mation)

- 1

-ABS TRA C T.

An approximate calculation method of the laminar

heat transfer in the stagnation region of isothermal spheres

and cylinders in high speed flows of a perfect gas is given.

A simplified expression for the Mach number dependenee of the

heat transfer in the stagnation point is developed. The

present method is compared to existing theories and

exper1-ments.

I. INTRODUCTION.

The heat transfer rates in the stagnation reg10n of

high speed vehicles are of interest for the design as they

are maximum in that region. Assuming that the blunt nose part

of the vehicles can be described by a sphere or a cylinder,for

an axisymmetric or a two dimensional configuration

respecti-vely,simple expressions for the heat transfer rates can be

developed.

In this note, the laminar heat transfer problem on

spheres and cylinders is treated using the laminar boundary

- 2

-2. HEAT TRANSFER~ EQUATIONS.

The dimensionless quantity used in heat transfer

calculations is the Nusselt number:

( 1 )

k

where L is a typical body length and the film coefficient h

is defined by :

h = qw

T - T

r

w

The recovery temperature T is equal to the stagnation

tem-r

perature T for a Trandtl number of unity o

Thus equat ion ( 2 ) gives for P

=

1

r

h

=

qw T

-

T

0 w

With Fourier's law

q

=

- k

-

aT

_ay

equation (1) leads to

{

l.! }

Nu = k.

ay

w· R GO k • { T - T } GO 0 W (4 )

3

-where R,being the radius of a sphere or a cylinder,is the typical body length.

The temperature gradient normal to the surface at the wall is for both, spheres and cylinders. following Cohen-Reshotko's method (Ref. 1) (For derivation. see Appendix A (Cylinders) and

B

(Spheres)) :

,

=

To·S _w du e

ëiX

1 Po du (6 )

The velocity gradient dxe• which can be assumed con-stant up to 0 ~ 800

away from the stagnation point following Lees (Ref. 2).is evaluated at the stagnation point assuming a pressure distribution which is described by the modified

Newtonian theory. for

M

_IX) > 2 (Ref. 2).

The resulting expression is Appendix C) du e

dX

with FI

=

=

u IX)

-

R

[2- .

_{Y ...} M2

Ta

p ...

r/

2 (1 - - )

Pol

T

_...

(for derivation see

(7.)

4

-,

Combining equations ( 5 ) • (6 ) and

( 7 )

gives

k ·R

~

J/2

Pw ₁ Po Nu = w

,

_Fl

_{( 8 )}

k (TO-T ) TO·S

-

.

-..,

_{w Po S·vO·C} Pe .., w

To rewrite equation (8) the following expressions will be used : T 1

-

w

T;

=

-k _IJ w w

k

_..,

= _IJ.., IJ_w T C w = _T

o

liO Pw

=

Pe Pw Pe T 0

=

--Po _{Po T w} S

w see equation (2A)

for Pr

=

1

with C being Chapman's constant (see equation

6A)

as

~

= 0 dy

equation of state

Combining equations

(8)

and

(9)

gives

Nu 00 with Re 00 S' w

=

_-s

w u • P • R .., 00

=

(10)

(11)

5

-Equation (10) gives Nu

_...

with Sf W

= -

S

w

~

POl

= ( - -

1) cos 20+1) p ...

TO

{..L

M2

T

y... ...

(12)

p ... l / J l / 2

(1- - )

POl Pe

using equation 2C for

-p .... which is va1id for M > 2.

In the ca1cu1ation of

S'

w

S

w

of reference 1 and a1so in equation

...

fo11owing the ana1ysis

3.

it was assumed that

the Prandtl number was equa1 to unity. To correct the ca1-culation for gases with Prandt1 numbers different from unity.

the fol1owing re1ation is proposed in Ref. 3 :

which is valid for Prandtl numbers from 0.6 to l.O.

Thus the fina1 expression is :

Nu

_...

S' w = -

S

w (Re ... ) 0.5

(14)

6

-3.

RESULTS AND DISCUSSION.

The pressure gradient parameter

a

for the stagna-tion point flow is equal to 0.5 or 1.0 for a sphere or a cy-linder respectively, (Ref.

4).

s'

_w

The ratio S is calculated uS1ng the analysis of

w

Ref. 1. for

a

=

0.5 and

a

=

1.0 respectively. The results which are valid for Pr

=

1.0 are shown in Fig. 1, plotted

ver-. T

sus the temperature rat10 w.

T

_o

The function F, which 1S valid for Mach numbers M > 2 is plotted in Fig. 2 versus the upstream Mach number

GD

Mand with

e

as a parameter. Fig. 3 shows that the function 00

F for

e

=

0° (stagnation point) depends nearly linearly on

M • For these conditions F can be approximated by the follo-00

wing equation, which isshown too in Fig. 3 for the Mach num-ber range2 < M «

f.

00

For

e

=

0°

F

=

0.48

+

0.774

M ₀₀

( '6)

In the limiting case, for M -+ 00 the function F 1n 00

the stagnation point (e = 0°) is evaluated in Appendix D.

For M -+00 and

e=

0° F

=

0.831 Moo 00

Using the expression (16) the equation (15) can be written for the stagnation point heat transfer for Mach numbers

2 < M <

7

00 Nu

-

=

S w S w ( 18 )

Finally, Figs.

4

and

5

show the comparison of the present calculation with experiments and different available

7

-theories, for spheres and cylinders respectively. It can be seen that the present calculation predicts the heat transfer coefficient in the whole stagnation region reasonably well, although the calculation is strictly only correct in the narrow vicinity of the stagnation point.

8

-REFERENCES

1. COHEN, C.B.

&

RESHOTKO. E.: Similar solutions for the compressible laminar boundary layer with heat transfer and pressure gradient.

NASA TN D 3325, Feb. 1955.

2. LEES, L.: Laminar heat transfer over blunt-nosed bodies at hypersonic flight speeds.

Jet Propulsion, April 1956, pp. 25~.

3. RESHOTKO, E.

&

COHEN, C.B.: Heat transfer at the forward stagnation point of blunt bodies. NASA TN D 3513, July 1955.

4.

SCULICHTING, H.: Grenzschichttheorie. Verlag G. Breun, Karlsruhe, 1951.

5.

BECKWITH, I.E.

&

GALLAGHER, J.J.: Local heat transfer and recovery temperatures on a yawed cylinder at a Hach number of 4.15 and high Reynolds numbers. NASA TR R 104, 1961.

6.

SIBULKIN, M.: Heat transfer near the forward stagnation point of a body of revolution.

J.A.S., Aug. 1952, pp. 570.

7.

VAN DRIEST, E.E.: The problem of aerodynamic heating. Aeron.Eng.Review. Oct. 1956, p. 26.

8.

KOROBKIN, I.: Local flow conditions. recovery factors and heat transfer coefficients on the nose of a

hemisphere-cylinder at a Mach number of 2.80. NAVORD R 2865.

9.

STEWARTSON, K.: Correlated incompressible and compressible boundary layers.

Proc. Roy. Soc., London, Ser.A, vol. 200, A 1060, Dec. 1949, p. 84.

9

-10. MANGLER. W.: Zusammenhang zvischen eb enen und rotations-symmetrischen Grenzschichten ~n kompressib1en

F1üssigkeiten.

A.l

-APPENDIX A CALCULATION OF TRE TEMPERATURE GRADIENT AT THE WALL IN THE TWO-DIMENSIONAL CASE

(Cylinder) .

From Cohen-Reshotko, Ref. 1

=

(l+S) - 1"2 _(IA) where S

=

_H

H

-

1 e 2 1

l.::.!

M2 m

=

+ e 2 e (2A) 1"

₌

-

u u e

Differentiating equation (IA) and for wall conditions gives

1-12 = T ((~) _

l.::.!

e

o

av

_{• w} 2

2

_m e 2 1" w (3A)

With 1"

=

0, as the velocity u at the wall is zero, one can

w

write

(4A)

To make use of the results of Ref. 1 to evaluate the independent variables x and y ~n the physical plane

have to be changed by the Stewartson transformation, Ref.

9,

to X and Y. Furthermore, the similarity variabIe n has to be introduced.

A.2

-The rollowing relationships are used

dX a dY =

..e....

-

e dy Po aO /m+l U Y e n =

_I

_--r

_V(il:

where C is the Chapman constant eva~uated at the wall and stagnation conditions

C =

~

To+102.5 /

~

T +102.5

w

(6A)

U is the velocity at the outer edge or the boundary layer e

in the transformed plane

ao U

=

u e e a e

(7A)

U u

U

=

-

u e e

and m is the exponent 1n the Falkner-Skan type cf velocity distribution outsiàe the boundary layer

(BA)

Thus equat ion

(4A)

can be expressed by

(~)

₌

_To

(ll

1.U.

ll)

=

T • S' Pw a

-

e

/~

-

U e ay _w an aY ay _w o v Po ao

_I

2 voX

Taking a

=

ao which is justified as the flow in the vicinity e

Pw

=

TOS' -_w

_Po

A.3

-U m+l e ~VQx (lOA)

The square root on the right hand side of equation 9A can be rewritten using equations

(5A). (7Ä).

and

(8A)

where 2m B

=

du e dxC

B

being the pressure gradient parameter. ao a e a _Pe e

- -

_{aa Po}

Combining equations (lOA) and (llA) gives

(l!) 3y w = To S'

~

/ dUe 1 PO w Po / dx

BoCovo

Pe (llA)

(12A)

B.l

-APPENDIX B CALCULATION OF TEMPERATURE GRADIENT AT THE WALL IN THE AXISYMMETRIC CASE

(Sphere)

It is possible to transform the axisymmetric bpundary layer flow to an equivalent two dimensional one using the Mangler transformation, Ref. 10.

The transformation formulas are

x

=

rex)

y

= -

_L

Y

The physical quant:ities are related by

u('X',y)

₌

u(x,y) T'(i',y)

₌

T{x,y) p(i',y)

₌

p (x,y) p (x ,y)

₌

p{x,y) ii'(i',y)

₌

lJ(x,y)

(lB)

(2B)

Equation (9A) can thus be written in the equivalent two dimensional plane

Taking a

=

ao and rewriting the square root term in equation e (3B) as ~n Appendix A gives

dü

_e

Pa

-dx

(4B)

B.2

-Going back to the coordinate system and the physical properties in the axisymmetric plane using e~uations

IB

and

2B

gives

as Pw

I

I dU e dx Po

To ( - ) - -

-a n p _w 0

a -

v 0 - C dx d-_x P _e

with

s

=

s

and dn

=

dn as can be Been easily from equatiens

(IB).

(2B). and (5A) equations (5B) gives :

du

I

_-

e

a-vO-c

dx (6B)

This leads finally to

=

st

Pw / I dU e

~

To

w

PO

a-vo·c

dx Pe

- C.I

-APPENDIX C CALCULATION OF THE VELOCITY GRADIENT AT THE STAGNATION POINT

The statie pressure distribution in the stagnation region of spheres and cylinders in supersonic flow can be described by the modified Newtonian theory :

(IC)

where c Pmax

~s the pressure coefficient at the stagnation point. Equation (IC) gives :

Pe POl

-- = --- -

~

I

.

J

cos2e + I

Pao Pao (2C)

As the flow in the vicinity of the stagnation point downstream of tne bormal shoek is ineompressible. Bernoulli's equation ean be used

or

Combining equations (2B) and (4B) g~ves

u

e

=

~

(Po

,(I.-cO. 2 9) -

P~

(1-co. 2 6)

l]

'/2

2 1/2

(p-

(POI-Poe») e

=

sine

- C.2

-du

From equation (5c) the velocity gradient dxe can easi1y be ca1cu1ated du _e

<rX

= with d0

₌₇

dx

(6c)

The velocity gradient at the stagnation point (0

=

0°) is then. using the equation of state and a

=

lyRT

du (~) dx 8=0 u..,

(_2 _Tc

p"")

1 /2

= - ( 1 - - ) R M2 T Po 1 Y "" ..,

D.l

-APPENDIX D CALCULATION OF THE MACH NUMBER DEPENDENCE OF THE FUNCTION FAT ..

THE STAGNATION POINT FOR M ~ m

oe

Rewriting equation (13) for

e

=

0° gives

~

~

1/2

POl 2 T

o

Pm 1/2

F(O.M~)

=

(---J(--r

~

(1 -

---»)

Pm yM m Po 1

m

Evaluating the different terms leads to

TO

r=

POl POl with for M _m for M

Combining equations (lD) - (3D) gives

- D.2 -F (0 r.1

"'CI»

: I M

(A .l:l)

• Cl> Cl> Y 1/2 And fina11y

F(O.M

"'CI»

=

0.831

M

co Cl>

(4D)

S~

-Sw

0.60

0.55

0.50

0.45

0.40

----

----~

~~

---

---~

~

~

I

~

_---

----

l.---

A

=

0

.

5 (SPHERE)

~

I

, ! i

I

I , I

I

t t I

COOLING - -. HEATING

- - - - ~

o

0.2

0.4

0.6

0.8

lO

1.2

1.4

1.6

T

_w

1.8

2D

'-T

_o

9=

6.0

~

--

-

-

t

-

T

La

19~

I

20°

5.0

I

-

--1---

-

I

30°

F

40

0 4.0 ~1

-

_50°

10

I

I

~

~:.;,;F:;F;»'

4 " " " ' = " " " - =

~

-160

0

2.0

I

~,....= 7'~

I

=-~

=-

~

I

=_---===

I

70°

1.0~:q:

J

::

oL.

--~----L---~--~--~~--W_

1D

2.0

3.0

4.0

5.0

6.0

7.0

Moo

FIG.2.

DEPENDENCE OF THE FUNCTION F ON THE MACH NUMBER M(X) WITH

e

6.0

/

/

5.0

Fe=oo

4.0

3.0

2.0

lO

o

lO

/

V

/

7

-/

/

/

---

EXACT EXPRESSION, EQUATION (13)

-

APPROXIMATE EXPRESSION, EQUATION (16)

20

3.0

4.0

5.0

sa

MCD

FIG.3.

DEPENDENCE OF THE FUNCTtON F ON THE MACH NUMBER MCI) IN THE

STAGNATION

POINT

18

NU

_eD

1.6

05

(ReeD)

1~

12

lO

0.8

0.6

0.4

02

o

~

~ l'~

-

H--I

I

.!?-. - -

--

CD

~

--=;;

---- LEES, REF 2

--

_""""

- -1-- .-...

"-~

THEORY

_

PRESENT

B "-

i ' ,

-,

_,

CALCULATION

,

i

,

~

I

",

,

I

"

I

'

..

"-~

"-~, B

"

_'

_..

h "-

"-~

,

'.

"

EXPERIMENT

o

ReQ)

=

Q.64 - 106

,

"

,

BECKWITH ET AL.

A ReCl)

=

1015 _10

6

,

_,,71.

I--

"-REF

5

al

ReeD

=

137

-10

6

~

-,

"

"'"

B

ReCl)

=

1915

_106

,

_,

~

"

,

,

_~

~.

"

i',

""'--"

_"

,

"

_"

r"- .... _...

---I...-~~---- - - --~ - -

-o

10

20

30

40

50

60

70

80

90

e

(deg)

FIG.4. COMPARISON OF THEORY AND EXPERIMENT OF THE HEAT TRANSFER FOR A

CYLINDER AT A MACH NUMBER OF M(X)

=

4.15

-1.8

- ---

----

--

-

-

--

+

----Nu

_CID

1.6

(Re

)05

CID

1.4

r---

I I

---

_~

I

---

_-

...

I ... ~ L'~ ...

~

... ...

,

,

--- 1 - - --- - " 8

~,

...

,

~

t I ...

.

J

...

.

'" ...

~

i ...

"

,

--1---

"

"

~

,

_,

,

_,

,

'"

...

---- LEES, REF 2

, ,

"

SIBULKIN, REF6,AND

,

---

THEORY

0

,

"

----VAN DRIEST, REF7

,

"

-

PRESENT CALCULATION

'"

... "r -...

12

lO

0.8

0.6

0.4

_,

EXPERIMENT

8

KOROBKIN, REF 8

'

... _...

-

... _-~

02

o

o

10

20

30

40

50

60

70

80

90

e

(deg)

FIG.5.

COMPARISON OF THEORY AND EXPERIMENT OF THE HEAT TRANSFER FOR A