● ✌

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i NATIONALADVISORYCOMMImEEFOR AERONAUTICS

TECHNICAL NOTE 3888

COMPRESSIBLE IAMINAR

i!“pJ

HIMT TRANSFER WITH PRESSURE GRADIENT

By Eli Reshotko

Lewis Flight Propulsion Laboratory

Cleveland, Ohio

Washington

December 1956

AFMPCf .-. *. *. .

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TECN LIBRARY KAF6, NM

I!lllllullllllllillllllll!llull“--uclh7DqaNATIONAL ADVISORY COMMITTEE FOR AERONAUTl~fi

TECHNICAL NOTE 3888

ISIMPLIFIEDM@THOD FOR EE?TIMATINGCOMPRESSIBLE LAMI.NARHEAT

TRANSFER WITH PRESSURE GRADIENT

By Eli Reshotko

SUMMARY

In the present report an approximation is made in the method of Tech-nical Note 3326, which simplifies the calculation of heat transfer. Goodagreement with the method of Technical Note 3326 is expected for isother-mal surfaces with adverse and small favorable pressure gradients regard-less of surface temperature and also for flow with large favorable pres-sure gradient over highly cooled surfaces.

●

INTRODUCTION●

A major factor influencing the designhypersonic flight is aerodynamic heating.

of vehicles for supersonic andWhile for simple shapes such

as the flat plate, the cone, and stagnation points the amount of la&inarheat transfer maybe estimated using sirgpleformulas, such is not general-ly the case for a calculation with arbitrary pressure gradient.

An integral method for calculating compressible laminar boundsrylayers with heat transfer and pressure gradient is given in reference 1.This method is an adaptation of the correlation technique of Thwaites(ref. 2) to the compressible boundary layer using the exact solutions ofreference 3. The present report points out an approximation which sim-plifies the calculation of laminar heat transfer. Examples are presentedto illustrate the approximate technique, and numerical.comparison is madewith the complete method of reference 1. Someslender bodies introduced by Jack and Diaconisconjunction with the examples.

SYM80LS

A coefficient in eqs. (2) and (3)

approximate techniques for(ref. 4) sre described in

i B exponent of Mach number in eqs. (2) and (3)(fig. 1)

2“

Cf

CfR~

Nu

CP

‘P

D

h

K

k

L

2

M

Nu

local skin-friction coefficient,4

1~ %&

Re~olds analogy parameter (ref. 1)

P -Pmpressure coefficient, ~

~ ~#:

specific heat at constant

throat diameter of rocket

heat-transfer coeffl.cient

-~(K=4 for y=

thermal conductivity

arbitrary length

pressure

nozzle (example

1.4) “

()e tw audimensionless shear parameter (ref. 1), ~~ee GW

Mach number

Nusselt number, ~%

-(~)e2(#@n correlation number (ref. 1),

w

/

dpep, dimensionless pressure gradient, ~ ~TM~e

NACA TN 3888

Pr Pratidtlnuuber, ~ (evaluated at wall temperature)

P

R

Rew

pressure

radius of axially symnetric body

p#exReynolds number, — —.

4

a

..—L..”

*

● �

NACA m

P

J r/Sw

t

%

x

Y

a

r

e

● P

vu

P

‘rW

3888

n6tw[bY(t/teflw

tw - to

temperature

external velocity

coordinate slong surface

coordinate normal to surface

heat-transfer

co*r-J

exponent of Prandtl number for

ratio of specific heats

boundary-layer momentum

absolute viscosity

kinematic viscosity

density

thickness

au()

shear streSS at wd_~J ~ W w

3

modification

Subscripts:

Cyl cylinder

e local flow outside boundary layer (external)

w wall or surface value

m free-stream static value

o free-stream stagnation value

CALCULATION

The calculation of laminar heat

PROCEDURE

trsusfer over isothermal surfaces byk the method of reference 1 is first summarized and then the approximation-

introduced.

dThe heat-transfer parameter Nu/! for Prandtl nuniber1 from

equations (36) and (37) of reference 1 is

4 NACA TN 3888

(1)

I to

The factor n P’ ~j which iS ~rect~ ProP~tfon~ to the square of ~m-e

entum thickness, is obtained in reference 1 by quadrature from the momen-tum integral equation. For two-dimensional flow

P=

while for exially symmetric flow

E

●

‘iJ

(2)

n A

j (0)

L >K@-1R2d(~

~=t K t e (3)

plr(); #-1# o

e Oe

Reference 1 suggests that for moderate press~e gradients the coefficientA be set equal to 0.44 and the coefficient B be chosen from a figurewhich is presented herein as figure 1. If an expected favorable pressuregradient is consistently that corresponding to stagnation-pointflow orstronger, the coefficients A and B should be evaluated from the ap-pendix. Such large pressure gradients might be encountered, for example,on the nose portions of blunt bodies or in choked nozzles.

Thus, all quantities in equation (1) are determined directly by theMach number and static-temperaturedistributionswith the exception ofthe shear or skin-frictionparameter 2 and the Reynolds analogy param-

()CfR+eter —Nu

These parameters are presented in reference 1 as func-Pr=l“

tions ’ofthe correlation number n. In order”to complete the calculationof heat transfer according to reference 1} the parameter n must be eval- S

/

‘ouated from the quantity n P’—t ~ which requires differentiation of the

e *

NACA TN 3888 5

.

()C#lewpressure distrib~tionj the parameters Z and ~ are then ob-

Pr=ltained and substituted into equation (l). However, as can be seen from

.figure 2, the ratio

()

C$;, which is the desired combination of

Nu Pr=lparameters, is relatively insensit~ to n for many cases of interest.

Hence, it is suggested that~

be taken as 0.11 for all values

offorFor

the

“

s

\~)Prdn.1 This assumption gives an error of less than 10 percent exceptflows with large favorable pressure gradients or very near separation.large favorable pressure gradients, a further suggestion is made in

appendix with regard to the quantityz

()

C#ew.

Nu Pr=l

The expression for heat transfer upon setting&&

equal to

WY)=0.11 is

(4)

/

‘owhere nP~ ~ is obtained from either equation (2) or (3). The quantity

ePr” acc~unts for the effect of ?randtl number on heat transfer. Valuesof a. suggested by Tifford and Chu (ref. 6) are, for small pressuregradients, 1/3; for large adverse pressure gradients, 1/4; and for ex-treme favorable pressure gradients, 1/2.

It should be noted and emphasized here, that if skin-friction in-formation is desired, the parameter 2 must be evaluated, and, therefore,the complete method of reference 1 must be used.

k

%his same approximation can be obtained from the almost linear vari-ation of a heat-transfer parameter r/~ with correlation number n

* shown in fig. 6 of Crocco and Cohen (ref. 5). The present approximationcorresponds to r/n ~= -0.22.

6 NACA TN 3888

●

EXAMPLES

Three examples are given. The first two sre a cone cylinder and a ‘+1#

parabolAc-nose cylinder for which Jack and Diaconis (ref. 4] recentlypresented experimental information. The numerical calculations for thesetwo examples are for a free-stream Mach number of 3.12, auibientstatictemperature of 180° R, and for the surface cooled to ambient static tem-perature. The third example is that of the rocket nozzle calculated inreference 1.

I. Cone Cylinder

Because the cone cylinder is composed of two simple shapes, it lendsitself to a somewhat generalized trea&ent. The len~h of the cone sur-face will be designated L as shown in the following sketch:

*

Over the conical portion of the body the surface Mach number is constantso that from equation (3)

()n

‘oP’ ~

e cone

and from equations (4) sad (5)

0.44() t

3

(4-)Nu = 0.575 Pr=13‘% cone.

in which the coefficient 0.575 is identically that indicated by theexact boundary-l~er solution for a Prandtl number of 1.

.

v

NACA TN

on

3888

the cylindrical portion of the tidy from equation (3),

0 [()#-l ‘eK

n 44econe ? cone

to = K 3p, ~

w-1 ‘e

e ~

and the heat-transfer parsmeter Nu/~

equation (7) into equation [4).

.

.

(7)

is obtainedby substituting

The heat-transfer rate on the cylinder side of the cone-cylinderJunction canbe readily evaluated. From equations (4) and (7),

b

\

()MM >K

(r)J!Ju (’d-’)Nu ‘Cyl Cyl=

R% cone‘% cyl at junction

‘!Equations (5) to (8) are used to calculate

.0

tKMB-l

()

e

‘cone r cone

(8)

the heat transfer to the

% -included-angle cone cylinder reported in reference 4. The pressure

distribution for this cone cylinder, shown in figure 3, was obtained usingthe theory of VanDyke (ref. 7). An examination of the calculated valuesof Nu/@w shows that the presented approximate technique gives results

tithin 2 percent of those using the complete method of reference 1. Theabrupt change in the heat-transfer psrsmeter at the cone-cylinder junctionis due to the abrupt bresk in the pressure and Mach nuuiberdistribution atthis station.

Jack and Diaconis (ref. 4) have proposed that the heat transfer onthe cylindrical portion of the body be estimated by a formula whichmatches the cylinder vslue at the cone-cylinder junction and the flat-plate value at infinity. This relation takes the form (for x>L)

~’)cone (9)

8

where

NACATN 3888

●

✃

[Al-)Nu

Rew CyI at junctionc=

()k coneas evaluated from equation (8). The use of equation (9) avoids the neces-sity of performing the integration of equation (3) over the cylindrical *portion of the body, and it is seen from figure 3 that equation (9) agrees Gclosely with the othed estimations of heat transfer on the cylindrical

m

portion of the body. All three curves in figure 3 represent the data of —reference 4 to within 15 percent.

11. Parabolic

A sketch of the parabolic nose

Nose Cylinder

cylinder together with its pressure “

distribution as calculated using the theory of reference 7 is shown infigure 4. The distribution of heat-transfer parameter Nu/fi as cal-culated by the present approximate technique using equations (3) and (4)

;

is seen to be within 3 percent of that obtained using the complete methodof reference 1.

A further simplification for this slender psrabollc-nosed body isproposed in reference 4. If it is assumed that the Mach number over thebody is fairly constant and that the heat-transfer variation is due prim-=ily to the geometric variation of the body from conical to cylindricalshape, then from equations (1) and (3)

* ‘w =. F (10}

The curve of equation (10) is seen fromment with those of the complete (ref. 1) andthree curves represent the experimental datapercent.

111. Rocket Nozzle

The third example is that of the rocketan example in reference 1. It is an example

R2 dx

figure 4 to be in good agree-simplified techniques. Allof reference 4 within 12

h

nozzle (fig. 5) also used as “Ninvolving large favorable

NACA TN 3888 9

.

pressure gradients with heat transfer. The combustion-chamber stagnation

-4pressure is assumedtion temperature is0.78, and the ratiocooled to a uniformt~to = 0.2.

The solid line

Ml parameter Hu/&

to be 500 pounds per square inch absolute, the stagna-taken as 4~0 R, the Prandtl number is assumed to beof specific heats is 1.3. The nozzle wall is assumedtemperature of 800° R, which corresponds to

in figure 5 is the variation of the heat-transferfrom the complete method of reference 1. The siut-

plified technique was applied in two ways: first, as proposed in theteti of this report and second, as proposed for large favorable pressuregradients in the appendix. The results are shown in figure 5. Whileboth applications of the simplified technique result in an overestimationof the heating lead, the techntque for large pressure gradient matches theresult of reference 1 at the stagnation point and overestimates the heat-transfer rate at the throat onl.yby about 13 percent.

CONCLUDING REKARKS

An approximation is made in the method of Technical Note 3326 whichsimplifies the calculation of heat transfer.

The simplified method is expected to be adequate for isothermal sur-faces w3th adverse and small favorable pressure gradients regardless ofsurface temperature level and also for flow with large favorable pressuregadient over cooled surfaces. .

Good agreement is obtained in these ranges between the simplifiedtechnique and the complete method oftrative examples.

Lewis Flight Propulsion LaboratoryNational Advisory Committee for

Cleveland, Ohio, September

Technical Note 3326 for the il.lus-

Aeronautics13, 1956

.

w-

10 NACA TN 3888

*APPENDIX - CALCUIJYTIONPROCEDURE FOR LARGE FAVORABLE PRESSURE GRADIENTS

The simplified technique as proposed in the text of this report is %

set up in such a way as to match the exact boundsry-layer solution forzero pressure gradient. For large favorable pressure gradients (corres-ponding to stagnation-pointflow or stronger), the use of the values of

A, B, andL

CfRew

()

proposed in the text may greatly overestimate the

Nu Pml ~amount of aerodynamic heating. - .- Oa

The values of A and B in the following table correspond to linestangent to the curves of figure 3 of reference 1 at the plane stagnation

point. ‘itilarW) ‘he values ‘f /CfR~\and a are for plane stag-

F7’r-nation point flow.

tw z~ A B

()

CfR~

Nu Pr=l

o 0.380 2.35 0.102.2 .372 2.53 .099.6 .362 3.20 .093

1.0 .410 4.94 .083

a

0.4.4.4.4

.

w

The simplified technique presented in this report is not recommendedfor use with Isrge favoralle pressure gradients over heated surfaces.

—

While the numbers presented in this appendix are for matching atplane stagnation-pointflow, a similar match could be made at any desiredcondition using the information in figure 3 of reference 1 and in figure2 of this report.

REFERENCES

1. Cohen, Clarence B., and Reshotko, Eli: The Compressible Laminar Buund-ary Layer with Heat Transfer..andArbitrary PressureTN 3326, 1955.

2. ThWaites, B.: Approximate Calculation of the LaminsrAero. Quarterly, vol. 1, Nov. 1949, pp. 245-280.

Gradient. NACA

●

Boundary Layer.

-d

NACATN 3888

.

3. Cohen, Clerence B., and Reshotko, Eli:

Jpressible LE&mzr Boundary Layer withGradient. NACATN 3325, 1955.

I-1

SinUar Soltiions for the Com-Heat Transfer and Pressure

4. Jack, John R., and Diaconis, Nick S.: Heat-Transfer Measurements onTWO EUdies of Revolution at a Mach Ihmiberof 3.12. I?ACATN 3776,1956.

5. Crocco, L., and Cohenj C. B.:co Compressible Laminar Ikmndary Layer with

2 Heat Transfer and Pressure Gradient. 50 Jahre Grenzschichtforschung,* Eine Festschrift in OrigineXbeitragen. Verlag Friedr. Vieweg & Sohn,

Braunschweig, 1955, pp. 280-293.

6. M.fford, Arthur N., and MU, S. T.: Heat Transfer in Lmninsr EoundaryLayers Subject to Surface Pressure smd Temperature Distributions.Proc. Second Midwestern Ccmf. onFltid Mech., Ohio State Univ.,

Z Mar. 17-19, 1952, pP. 363-377.

2 7. VanDyke, Milton: Practical Calculation of Second-Order Supersonic-y Flow Past IionllftingBodies of Revolution. NACA TN 274A, 1952.E

●

u NACA TN 3888

.

“.1#

14

Two-dimensional

suregrsdient

E

AxialJ.js~etricclosed-bodyfavor-ablepi’essure

10gradiegt~

.

/

B8 /

Smallpressure gradien’:

6

pressure

4

WalJtemperatureratio,-tW/tO

.

?

—

Figure 1. - Valuesof exponentB to be used in equations(2)or(3). (Fkomref.1.)

.14

.lz

.10

mm

.W

.04

.02

Oarrelationnumber, n

Figure 2. - Ratio of cheer p?uemk.er to ReynoI& analogy pemmeter frm SOMMOIM repcdedinreference1.

G

.

ml

H-i.lO_ -

t- ~“

“2 4

f.04 /

.02

0

/

-.02

/ f

/ ‘

-.04 ,.

.6— Canpleternethcdofref.1——-sllmpllfledteclmiqle

.5

.4

%

.—

.“

0 5 10 I-6Mal tistancelfn.

Figure3.-_le 1: C@e cyllnderlx@.

.-

NACA TN 3888 15

.6

.5

.30 5 10 ls

Axialtistance,in.

Figure 4. - Ekample II: Parabolic-nose-cyllder body.

16 NACA ~ 3888

.

.

.60

.36

Ron le th=t> Ccsfpleteztethodof ref. 1h8 \

Shpllfied technique

/ \ AB a

.52 I \

\,()

C$;

\ / -iiir E&d# \ ———0.44 3.ciJ O.1.1 0.337

I f\

\ —-— .372 2.s5 .099 .4

.46\ ,’

\

\\

\//‘ \

\

k “u\

In , \

\ \\

\.40 Y \

\F A b \r I

1

.36 \

.32

\

.2801 2 3 a 5 6

Distenceelong nozzleWSIL, x/D

.

.

-. ~

%-

NACA - Langley Field, V..