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

.OCTOBER 1972

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

OCTOBER 1972

TABLE OF CONTENTS

LIST OF SYMBOLS • • • • • • • • • • • • • • • ABSTRACT • • • • • • • • • • • • • • • • • •

1. INTRODUCTION • • • • • • • • • • • • • • • • 2. HEAT TRANSFER EQUATIONS • • • • • • • • • • • 3. RESULTS AND DISCUSSION • • • • • • • • • • •

REFERENCES

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

• • • • • • • • • • • • • • • • • 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)

F1

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

Re Reynold's number

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

r .~covery conditions

- iii -

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

layer theory.

- 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 tem­r

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 dx e • which can be assumed con-

stant up to 0 ~ 80 0 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 > 2 (Ref. 2). IX)

The resulting expression is

Appendix C)

du e

dX

with

FI =

= u

IX) -R

[2- . M

2 Y ...

Ta r/ 2 p ...

(1 - -) Pol T ...

(for derivation see

(7.)

for M > 2 ...

- 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

= To 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/Jl/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.

To

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.

Z.A.M.M •• Band 28. April 1948. Heft 4. p. 97.

- 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 1 = H -

e

2 1 l.::.! M

2 m = + e 2 e

(2A)

1" u = -u e

Differentiating equation (IA) and for wall conditions gives

1-12

= T ((~) _ l.::.! e o av 2 2 • w 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

(~) (ll 1.U. ll) Pw a /~ U To T • S' e e = = - -ay an aY ay o v Po ao I 2 voX w w

Taking a = ao which is justified as the flow in the vicinity e of the stagnation point is incompressible leads to :

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 -

APPE NDIX 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 = - Y L

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 Pa e --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 (-) - - --

anp 0 a -v 0 - C dx d- P w x 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 dUe ~ To w PO a-vo·c dx Pe

which is equivalent te equation (12A).

- 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 cos 2 e + I ~ .J 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. 29) - P~ (1-co. 2 6) l] '/2

2 1/2 (p- (POI-Poe»)

e = sine

(4c)

- C.2 -

du From equation (5c) the velocity gradient dx e can easi1y be

ca1cu1ated

du e <rX =

with

d0 dx

=7

(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

(lD)

- 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 Tw

1.8 2D '-

To

FIG.1. VARIATION OF S'w/Sw WITH THE DIMENSIONLESS WALL TEMPERATURE

9=

6.0 ~-- - -t -T La 19~ I 20°

5.0 I ---1------ I 30°

F 400

4.0 ~1 - 50°

10 I I ~ 4"""'="""-= -1600 ~:.;,;F:;F;»' ~

2.0 I ~,....= 7'~ I =-~ =-~ I =_--------=== I 70°

1.0~:q: J:: oL. --~----L---~--~--~~--W_

1D 2.0 4.0 6.0 7.0 5.0 3.0 Moo

FIG.2. DEPENDENCE OF THE FUNCTION F ON THE MACH NUMBER M(X) WITH e AS A PARAMETER

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

7.0

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 _106

, ,,71. I--

"-

REF 5 al ReeD = 137 -106 ~-

, "

"'" 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 SPHERE AT A MACH NUMBER OF MCX)= 2.80