NASATechnical

Paper2914

1989

National Aeronautics andSpace Administration

Office of Management

Scientific and TechnicalInformation Division

A Review of High-Speed,Convective, Heat-Transfer

Computation Methods

Michael E. Tauber

Ames Research Center

Moffett Field, California

https://ntrs.nasa.gov/search.jsp?R=19890017745 2020-05-03T05:09:53+00:00Z

NOMENCLATURE

B

C

C1

cf

Cp

D

F

f

g

H

H

h

h'

J

k

k

L

Le

M

fil

m

Nu

Pr

P

pp

Ps

_kv

r

rn

mass-addition parameter (Eq. 70) r

mass concentration R

constant in Eq. 37 Re

skin-friction coefficient S t

specific heat at constant pressure t

binary diffusion coefficientU,V,W

diffusion function (Eq. 6)

similarity stream function (Eq. 13) V,,o

total enthalpy ratio, H/H e x,y,z

total enthalpy, h + (u 2 + w2)/2 Z

cavity depth (Eq. 65) ct

static enthalpy 13

reference enthalpy (Eq. 53) F

exponent (j = 0 for yawed, infinite cylinder; j = 1 yfor axisymmetric body)

3

coefficient of thermal conductivity

roughness element height (Eq. 50)

cavity length (Eq. 65) 0

Lewis number, pcpD/kf A

Mach number

IXmass addition rate

gOmolecular weight of gas

Nusselt number,/twCpx/k(haw - hw)P

Prandtl number, lacp/kcp

pressure

peak pressure (Eq. 68)

stagnation pressure

heat transfer rate to the body surface

radius of body of revolution

nose radius of hemisphere

PRECEDING PAGE

t •

cnthalpy recovery factor

gas constant

Reynolds number, pux/la

Stanton number,/tw/peue(haw - hw)

static enthalpy normalized by total enthalpy atboundary layer edge, h/H e

velocity components in the x,y,z directions,

respectively

free-stream velocity

chordwise, normal, and spanwise coordinates

Compressibility factor

angle of attack

pressure gradient parameter (Eq. 21)

heat-transfer function ratio (Eq. 43)

specific heat ratio

cone half angle, wedge angle, or flat-plate anglewith respect to free stream

inverse density ratio across shock, 9,WPe

boundary layer momentum thickness

sweep, or yaw, angle of wing leading edge or

cylinder

coefficient of viscosity

reference value of coefficient of viscosity

similarity variables (Eq. (9-10))

density of gas

density and viscosity product ratio (Eq. 17)

stream function (Eq. 11)

ratio of heat transfer at surface with mass addition

to value without mass addition (coldwall value)(Eq. 69)

spanwise velocity ratio, w/w e

BLANK NOT FILMED

iii

Subscripts

A,M

aw

bt

C

cyl

e

atoms and molecules in binary mixture

adiabatic wall (no heat transfer at wall)

beginning of boundary layer transition

cone

cylinder

value at boundary layer edge

f

FP

S

SL

W

O0

chemically frozen value

flat plate

stagnation point value

stagnation line

wall, or local body surface, value

free-stream conditions

iv

SUMMARY

The objective of this report is to provide useful engineering formulations and to instill a modest degree of

physical understanding of the phenomena governing convective aerodynamic heating at high flight speeds.Some physical insight is not only essential to the application of the information presented here, but alsoto the effective use of computer codes which may be available to the reader. The paper begins with adiscussion of cold-wall, laminar boundary layer heating. A brief presentation of the complex boundarylayer transition phenomenon follows. Next, cold-wall turbulent boundary layer heating is discussed. Thistopic is followed by a brief coverage of separated flow-region and shock-interaction heating. A review ofheat protection methods follows, including the influence of mass addition on laminar and turbulentboundary layers. The paper concludes with a discussion of finite-difference computer codes and acomparison of some results from these codes. An extensive list of references is also provided fromsources such as the various AIAA journals and NASA reports which are available in the open literature.

1. INTRODUCTION

The original impetus for developing means of pre-dicting the aerodynamic heating during high-speed atmo-spheric flight was the advent of the long-range missile.Only a modest extension of technology was required fromthe missile atmospheric entry velocity of about 6 km/sec tothe 7.5-kin/see entry speed of vehicles returning from satel-lite orbit. It is the fortuitous behavior of air molecules at

high temperatures that made the early formulations of aero-dynamic heating applicable to a speed of about 9 kin/see,thereby covering both the missile and satellite entry flightregime.

The pioneering papers of Lees (ref. 1) and Fay andRiddell (ref. 2), published in 1956 and 1958, respectively,provided the basic physical and mathematical modeling forhypervelocity convective aerodynamic heating. However, theearly work (refs. I and 2) was hampered by a lack of infor-mation of the high-temperature thermodynamic and transportproperties of air. Succeeding computations by Beckwith andCohen (ref. 3) and Cohen (ref. 4), published in 1961, usedthe high-temperature thermodynamic properties of Moeckeland Weston (ref. 5) and the transport properties of Hansen(ref. 6), both published in 1958. The state of the art in con-vective heat transfer at hypersonic speeds in the early 1960swas well presented by Dorrance (ref. 7). Possibly the bestcurrent source of information for the working engineer is the

chapter on forced convection in external flows by Rubesinet al. (ref. 8).

In a high-velocity flow, the physical mechanism of heattransfer differs from the molecular heat-conduction processoccurring at lower speeds. At sufficiently high speeds, theair that has passed through the strong shock wave sur-rounding the front of the body becomes so hot that themolecules dissociate into atoms. When the atoms diffuse

through the boundary layer into the cooler region near the

wall, the heat of recombination that is released as moleculesform can contribute significantly to the heating of the wail.Therefore, at high speeds convective heating consists ofboth conduction and diffusion (due to atomic recombination)

of energy through the boundary layer. The ratio of energytransported by diffusion to that by conduction is known asthe Lewis number and is an important parameter in high-speed, laminar, boundary layer calculations. The chemicalreactions in the laminar boundary layer are frequently frozenat altitudes in excess of about 45 km according to Fay(ref. 9). Therefore, the heating can be potentially reduced(ref. 2) if the wall is made of a noncatalytic, glassy, orceramic material which inhibits atom recombination. For

example, Rakich et al. (ref. 10) have shown that the Shuttleorbiter tiles are noncatalytic. Metals and many ablative heat-shield materials that are widely used on missiles arecatalytic; therefore, finite-rate catalytic effects will not beconsidered in the following discussions. Wall catalysiseffects are included in the finite-difference boundary layer andsurface material interactions computer codes which will bementioned later.

The objective of this review is to provide usefulengineering formulations and to instill a modest degree ofphysical understanding of the phenomena governingconvective aerodynamic heating. Some physical insight isnot only essential to the application of the information pre-sented here, but also to the effective use of computer codes

which may be available to the reader. (In many codes, thephysics are obscured by the mathematical procedure requiredto solve the differential equations.) An extensive list of ref-erences is also provided from sources such as the variousAIAA journals and NASA reports which are available in theopen literature.

The paper begins with a discussion of cold-wall,laminar boundary layer heating. (The term "cold wall"means that no mass addition occurs at the body surface andthat the influence of the wall affects the heating only

throughthetemperatureofthewall.)A briefpresentationofthecomplexboundarylayertransitionphenomenonfollows.Next,cold-wallturbulentboundarylayerheatingis dis-cussed.Thistopicis followedbyabriefcoverageof sepa-ratedflow-regionheating.A reviewof heatprotectionmethodsfollows,includingtheinfluenceof massadditiononlaminarandturbulentboundarylayers.Thereportcon-cludeswithadiscussionof comparisonsandsomeresultsfromfinite-differencecomputercodes.

Withinthetextof thereport,theMKSmetricsystemof unitsis used.However,mostof thefiguresthatareshownoriginatedinotherpapersandarenotreplottedifEnglishunitswereoriginallyused.Therefore,conversionfactorsarepresentedintheAppendix.

TheauthorisdeeplygratefultoMrs.Lily Yangforherassistanceinthepreparationofthismanuscript.

2. LAMINAR HEAT TRANSFER--COLDWALL

In this section, analytic solutions are derived for thelaminar boundary layer heating of surfaces with and withoutpressure gradients. Air will be treated as a real gas inequilibrium and continuum flow is assumed. However, thechanges in heating that occur in low-density flows will bediscussed.

2.1 Boundary Layer Equations

The boundary layer equations are written in the usualform, which assumes that the thickness of the layer is smallcompared to the body's radius of curvature and thatcentrifugal forces can be neglected. Following the historicdevelopment of references 1 and 2, the equations are furthersimplified by assuming a binary mixture of "air molecules"(instead of nilxogen and oxygen) and "air atoms" in thermo-dynamic equilibrium. (The binary gas mixture assumptiongreatly simplified the computation and was initially justifiedbecause the final results agreed well with experiments. Sub-sequently, it was computationally validated by Moss(ref. 11).) The expressions for the conservation of mass,momentum in the x and y directions, and energy are(ref. 4)

vurJ) + pvrJ)=0 (1)

3u 3u due 3 3 uon _x'x+ PV _yy= peue "_-" + _yy (_ _"_-y) (2)

"_x3W 3w c) ()3y 3y dwoy 3p_pu + pv - !Lt _ and - 0 (3)

3y - 0 (4)

aH aH a [_ff aH IXpu-_x+pv _y -_y (1 + F) _y + Prf

x(Prf-l-F) u _yy+ w (5)

The coordinate systems are shown in figure 1. For athree-dimensional body of revolution, j = 1 and the velocitycomponent w = 0. For a two-dimensional, infinite, yawedcylinder, j = 0, w = w e = constant and 3w/3z = 0 in theboundary layer.

(a) COORDINATES FOR THE YAWED INFINITE CYLINDER

y, v

--tP,-

{b) COORDINATES FOIl THE BODY OF REVOLUTION

Figure 1.- Boundary-layer coordinate systems.

For a binary gas mixture in thermodynamic equi-librium, the continuity equation does not have to be writtenfor the individual species because the species concentrationsare uniquely determined by the pressure and enthalpy, or anyother two slate properties. In the energy equation, the diffu-sion function, F, is defined as

F(p,h) = (Le- 1)(hA - hM) (_hA) p=const"(6)

The boundary conditions for the conservation relations,equation (1-5), are

at y=0u=v=w=0

H = Hw(x)(7)

and at the boundary layer edge

u = Ue(X) and w = w e = const.

H = H e = const.(8)

The momentum and cnergy equations, equations (3)and (5), respectively, are now transformed to modified Leessimilarity coordinates (refs. 1 and 2) by letting

and

;0=" 19dy

_oq 2(_ - _) o

permits transforming equations (2), (3) and (5) into thesimilarity coordinate systems. Now the momentum andenergy equations become, respectively,

,9, (an2) 1- d::

where bt0 is a reference value of the coefficient of viscos-

ity, and _ is a function of x, as yet undetermined. To

satisfy continuity, equation (1), the stream function is

introduced

0x - -or (1 lb)

Combining equations (10) and (1 la) gives

q (v)an - uo 2(_ - _-) (12)

Now a new dependent variable can be defined as

f(x,h) - g (13)

bt0 q 2(_-_-)

where the velocity profile is given by

Of u(14)

an - Ue

Writing the nondimensional spanwise velocity and enthalpyratio, respectively, as

;(_,n)- w (15)we

H

g(_JI) = _ (16)

and the ratio of the products of density and viscosity as

qo= Pla (17)PePe

(18)

_9-11"(q0 _0-_q) + (1 - dd£_) (Orl_' )

_ o.L= 2(_- {) (°f a.L or,,an a_ _ an/ (19)

=_- (1 - Prf + F)

X (ts-te)3-_-_ q ( an.] _'(_ )

+ 2(_- _) (of ag _ af a_zAvan ag a¢ an/ (20)

where the pressure gradient parameter, 13,is defined as

13 2(_ - _,) ts due= Ue 7"_" (21)

and the ratio of static to total enthalpy is

h (©t=_'=g-(1-ts)_ 2-(t s-re) Of 2 (22)

For the body of revolution at a=0, ts=l and _=0.On the infinite yawed cylinder, ts is constant and given by

ts=l -

2 V 2 sin2 AW e

- 1 (23)2He 2He

whichisthestaticenthalpyratioattheboundarylayeredgealongthestagnationline.Theboundaryconditionsin thesimilaritycoordinatesystemforequations(18)-(20)areatrI =0 (wall)

f(0,_)=_fqf(0,_)= _(0,_,)= 0 (24a)

g(0,_,)= gw(_)=tw(_) (24b)

as 1"1-->oo (boundary layer edge)

of (_,.)_- g = _,(_,,_) = g(_,_) = 1 (25)

2.2 Similar Solutions

Partial differential equations were traditionally difficultto solve numerically. However, when written in the local

similarity coordinates, the equations can be readily solvednumerically. Exact solutions can also be found for a(limited) number of conditions where the flow external tothe boundary layer and the wall temperature becameindependent of the streamline coordinate _,(x).Practicalexamples of such exact similar solutions are the flow at athree-dimensional stagnation point, at the stagnation line ona two-dimensional infinite cylinder, and on an inclined sharpflat plate or a sharp cone having a constant wall tempera-ture. For the preceding conditions, the partial differentialequations become ordinary differential equations, which aremuch easier to solve.

Most practical problems of interest, however, do not

have exact solutions. A widely used approximation is to

assume that such flows can be treated as if they were locally

similar. The procedure consists of assuming that the deriva-tives with respect to _(x) vary relatively slowly and can be

neglected, while the local values are used for the termswhich depend on _(x) through the external flow or the wall

conditions. Again, a set of ordinary differential equations

results which can be solved for the required boundary layerprofiles. The solutions thus found are then applied at each

value of _(x) in the nonsimilar flow by using the approxi-

mate values of ts and the local values of te, gw, and 13.

Under the local similarity conditions, the value of d_/d_

becomes arbitrary and may be set equal to zero (ref. 3). The

momentum and energy equations, equations (18)-(20), can

now be written as ordinary differential equations, whereprimes denote differentiation with respect to rI, to yield

(q°f")'+fP'=_ (f')2-g+ (l-ts'_22tsIv'--_s J _ -te t_ --- _)1

(26)

(q_(')' + f_'= 0 (27)

T

= {'-_"(l - Prf + F)Prf

(28)

The boundary conditions are

f(0) = f'(0) = ((0) = 0 (2%)

g(0) = gw (29b)

while the parameters [3, ts, re, and gw are evaluated locallyalong the body. The similar boundary layer equations can be

numerically integrated in a straightforward manner for high-speed flight conditions if the high-temperature thermody-namic and transport properties are known.

2.3 Thermodynamic and Transport Properties

In 1958, the first set of high-temperature thermo-dynamic (ref. 5) and transport (ref. 6) properties for airbecame widely available and were extensively used for thefollowing two decades. Although more precisely calculatedvalues for the transport properties were published in theearly 1960s by Peng and Pindroh (ref. 12) and Yos (rcf. 13),the reports were not widely circulated. In references 14-16,the transport properties from references 6, 12, and 13 werecompared and it was shown that Hansen's coefficients ofviscosity and thermal conductivity were too low at tempera-tures above 2000 K. While such inaccuracies should not be

ignored, it was shown by Howe and Sheaffer (ref. 17) thatvarying the coefficient of thermal conductivity by a factor of10 resulted in a 40% increase in the stagnation point heatingrate at a speed of about 18 km/sec.

For use with computer codes, the analytic approx-imations for high-temperature air properties developed byWorbs and Bolster (ref. 18) are useful. When fast computersare available, the program of Gordon and McBride (ref. 19)can be used as a subroutine and gives thermodynamic prop-ertics and species concentrations to 6000 K that compareclosely with those of reference 5.

2.4 Heat Transfer Rate at a Stagnation Pointor Line

The general expression for the heat transfer rate to thewall can be written (ref. 4)

3h= I.tw (1 + Fw) (_-Y)w (30a)_w Prwf

or, in the similar coordinate system

(Ow_twt s(30b)

The dimensionless heat transfer parameter consisting ofNusselt number and Reynolds number, evaluated usingstatic wall conditions, can be written as

NUw g'w

(Rew) 0-5 - gaw - gw(31)

where

gaw = te + r(1 - to)

and _ is the recovery factor. Two exact solutions to equa-tions (26)-(28) consisting of the axisymmetric stagnationpoint and the stagnation line on a yawed, infinite cylinderwill be presented next. The equations were numericallyintegrated using references 5 and 6. The calculations wereperformed to temperatures up to the onset of ionization,which corresponds to a flight speed of almost 9 km/sec. Aunit Lewis number was used so that F = 0; this assump-tion can introduce errors of up to 8% in the heating rate(rcf. 4).

For the axisymmetric stagnation point j = 1, [3 = 0.5,ts = te = 1 and due/dx = ue/x, so that equation (31) gives

Nuw _ g'w

(Rew)0.5- 1 - gw

For a recovery factor, r, of 0.85 and

(I+F)

Reference 4 gives the expression

Nuw 0.57Pr_4 (. pe)le ] 0"45 (36a)

and for

PoPc _ >

Nuw 0.57Pr_4 ( 9elUe "]0"67to-TCZwJs(36b)

where equation (36b) is applicable for large yaw angles andhighly cooled walls. Note that for

Correlation of the numerical calculations yields (ref. 4)

Nuw _ 0.767Pr_4 ¢ Pelae _]0'43 ew)0.5 t - TW)s(32)

which can be compared to the expression from rcfcrence 2of

Nuw _ 0.768Pr_4 ( pel.te "_040(Row)0.5 tp-"_WJs

(33)

Using equation (32) yields the heat transfer rate

OtN S -- 07 7 (½o.5Pr 0"4 (He - hw)s(Pe_e)0"43(pwPw)0"07 s

wf (34)

For the yawed, infinite cylinder j = 0, 13 = 1,ts=te< l, and due/dx=ue/x so that

NHw g'w

(Rew) 0.5 - gaw - gw(1 + Fw) (35)

the ratio of equation (32) to equation (36a) disagrees byonly 5% from the theoretical value of 2 -0"5, which relates

the stagnation heating rates on an axisymmetric body and acylinder.

The usefulness of the correlation equations are furtherimproved by writing the relations directly in terms of flightcondition. Examples of such correlations are shown infigure 2 for the laminar heat transfer parameter for the stag-nation point on a hemisphere, an unswept circular cylinder,sharp cones, and a sharp flat plate. (A wall temperature of300 K was used for the calculations illustrated in figure 2to facilitate comparison with shock tube data.) Note that themaximum difference between the four hemispherical stagna-tion point calculations (refs. 2 and 20-22) is about 15% andcan, partly, be attributed to using different high-temperaturetransport properties. The sharp cone values (rcfs. 23 and 24)differ by 8% at low speeds and much less at high speeds.

The calculations shown as straight lines in figure 2 canbe used to derive simple analytic approximations for theheat transfcr rate. Noting that

Nuw C1T_

(Rew) 0.5- V a(37)

1.0

.9

.8

.7

.6

.5

.4

Z ,3

.2

I I I I I

2 3 4 5 6 7 8 9 10 15 20

FLIGHT VELOCITY, km/sec

I I I I I 1 I _ I I ]

3-D STAGNATION POINT

2-D STAGNATION LINE

..... SHARP CONE

MARVIN & DEIWERT ....... FLAT PLATE

IDDEL

"''"_'-_ / HOSHIZAK|

_"---.- _""-'-_ __ _- DeRIENZO

MARVIN/'_" "_" _ ..-_-" _-',-,-_..._'''"__"'_ & PALLONE

"'_ G CHAPMAN/

I l I I I

Figure 2.- Comparison of laminar convective heat-transfer parameter calculations.

where the exponent "a" varies from 0.3 to 0A, "b" variesfrom 0.10 to 0.20, and CI is a constant, and substitutingthe expressions for Nusselt and Reynolds number, yields theaxisymmetric stagnation point heat transfer rate

)051( LRTwZw)jv2 + h_o-

°-'× PsS

(38)

where Ps is the stagnation pressure. The velocity gradientat a hemispherical stagnation point can be evaluated usingNewtonian theory (ref. 2), which gives

(39a)

where

PS: p_V2o_( 1- 2) + Poo

and e = P_o/Ps. At high speeds e << 1, and

('_)s = V_° (2_:)0"5rn(39b)

An empirical expression for e which is valid within about+10% in the speed range from 103 m/see to 104 m/see, is

e = 11.4V_ 0"57 (39c)

For wall temperatures of less than about 4000 K, thePrandtl number variation is small and the coefficient of vis-

cosity can be approximated by

T0.71law- w

If the exponent, b, in equation (37) is set equal to 0.145,then the bracket in equation (38) becomes independent ofwall temperature. Using a value for the exponent, a, of

0.3575, agd if the flight speed is high enough so thath_ << V-/2, one can now write equation (38) in the sim-ple and useful form

:( / wbtws --- 1.83(10 -4) k rn ) V 1 - _s m"-_"

where the nose radius, rn, is in m, the free-stream density isin kg/m 3, and the flight velocity is in m/see. The correla-

tion equation for air given by Marvin and Deiwert (ref. 21)is nearly the same as equation (40) (_mgnadon point heat-ing correlations in gases other than air are given in refs. 21and 25.) The analogous expression for the swept, infinitecylinder is

_tWcy1= 1.29(10-4)/r-'_yl)0"5(1- 0.18sin2A)V3

_w) w_. (41)x 1 - os A m2

where

haw = hoo + 0.5V2(1 - 0.18 sin 2 A)

and A is the sweepback angle. In writing equation (41) ithas been implicitly assumed that the stagnation pointvelocity gradients are identical for the sphere and cylinder.According to reference 8, the Newtonian value, equa-tion (39), is accurate for the circular cylinder, but 7.7% toolow for the sphere. If the velocity gradient value of refer-ence 8 were used in equation (40), the constant wouldincrease from 1.83 to 1.90.

A comparison with experimental data is shown infigure 3 for the four hemispherical stagnation point correla-tions of figure 2. Up to a speed of 14 km/sec, all four cor-relations, from references 2 and 20-22, lie within the data

spread, which is about +25%. In addition to the shock tubedata, five measurements based on observations of the onsetof melting of projectiles fired in a ballistic range byCompton (ref. 26) are also shown. The ballistic range dataconfirm the shock tube measurements and theories of refer-

ences 20 and 21 in the low ionization velocity regime. Atspeeds above 14 km/sec where ionization effects are verystrong, Hoshizaki's formulation (ref. 20) appears to give thebest agreement with the data of Rose and Stankevicsfief. 27). The strong ionization regime is also characterized

by intense radiative heating, which can be much greater thanthe convective heating.

Newtonian theory, equation (39), can be used to predictthe velocity gradients reasonably well on circular cylindersand hemispheres. However, for blunter configurations moreprecise numerical methods, or experimentally derived values,must be used. In figure 4, a comparison betweenexperimentally determined velocity gradients and valuescalculated using Newtonian theory shows the limitations ofthe Newtonian approximation for determining stagnationpoint velocity gradients on blunt bodies.

2.5 Blunt-Body Heating-Rate Distributions

The distribution of the heat-transfer rate around a blunt-

nosed axisymmetric body, or a yawed cylinder, can bedetermined using the local boundary layer similarityapproximation. Following reference 4, the ratio of the localheat transfer to the stagnation value is written as

dUe_sl05

F (42)

where the velocity gradient parameter, 13, is

2ts(due/dx) fx 2j]3=pwl'twu2te(r/L)2jF2 0 pwbtwUe(_) F2dx

(43)

20(106 )

16

12

A I

"5 8

O

[]

1 i i _ r i i i

BALLISTIC RANGE - COMPTON (REF26) HOSHIZAKI

I I I ! I I I I t

2 4 6 8 10 12 14 16 1B

FLIGHT SPEED, km/sec

Figure 3.- Stagnation point convective heating comparisons.

o

ii

x

¢_ x .8 --

.7 _i

I _ .6

.4

.3

CC ,2

Ck

I,,-

Z

"J .1 --

n,-

_- ,2

o

p--Xs /'/

FROM / /

MEASURED _ / //

PRESSURES / / _

/

//

/

I f I I I I

-,1 0 .1 .2 .3 .4

BLUNTNESS PARAMETER, Xs/r s

Figure 4.- Stagnation-point velocity gradient variation withbody bluntness at Mach 4.76.

I1 + Fw 'w g'w sF = Prw Prws

The gradient of the enthalpy ratio at the wall, g'w, is deter-mined by soh, ing the locally similar boundary layer equa-tions (26)-(28). The full solution of Eqs. (42) and (43)requires iterating, because F is inside the integral inEq. (43). However, it is consistent with the local similarityassumption to set F = 1 in Eq. (43), thus eliminating theiteration. The result is essentially the same as that of Kempet al. (ref. 28) and is shown in figure 5 for a hemisphereand figure 6 for a circular cylinder. Also illustrated in fig-ures 5 and 6 is the solution of reference 1, which results

when F = 1 in both Eq. (42) and (43). For the axisym-metric body, numerical solutions were correlated to yield

m ,du e. -0.5

qw s s pw_twuer

1 + 0.096(13te) 0.5

1/)68(44)

for a Lewis number of one and gw << I.

1.0

.9

.8

.7

.4

."_ .3

O

%O

O

KEMP ET AL (REF. 28) O""

LEES (REF. 1)

COS (x/R)

I 1 I I I 1 I I _ i

0 10 20 30 40 50 60 70 80 90

x/R BODY ANGLE, deg

Figure 5.- Heat-transfer distribution on hemisphere cylin-der. (From ref. 28; reprinted with permission of TheAmerican Institute of Aeronautics and Astronautics.)

Also shown in figures 5 and 6 are heating solutionsfollowing a cosine variation of the body angle measuredfrom the stagnation point. Although it is approximate, thecosine variation of heating has a theoretical basis (ref. 1) atleast to the body sonic point (near 45 °) and appears to givereasonable results to about 70 ° beyond the stagnation point.

1,0 _^COSINE

I "_ KEMP ET AL

I _ (REF. 28)6.d

.2 "_"5, _

0 20 40 60 80 100 120

180 fi, degrees

Figure 6.- Comparison of predicted and measured heat fluxdistributions on a circular cylinder normal to a stream,

The local similarity solution procedure has also beenapplied to a blunt-faced axisymmetric body with a smallcorner radius. The results of using the methods of ref-erences 1 and 28 to calculate the heating distribution areillustrated in figure 7. Again, the method of reference 28 is

2.0

1.0

.8

.6

"_ .4

.2

LEES (REF. 1) \k?_

KEMP ET AL (REF. 2..8) _,

R oo'X

I I 1 I 1 I I

.2 .4 .6 .8 1.0 1.2 1.4

x/R

Figure 7.- Heat-transfer distribution on flat-nosed body.(From ref. 28; reprinted with permission of TheAmerican Institute of Aeronautics and Astronautics.)

and 15c is the cone half angle. In practice, all cones havesome tip bluntness. Usually, the nose blunting will reducethe heating en the cone surface aft of the nose. In an analy-sis made for a perfect gas of 7 = 1.4, Vallcrani (ref. 29)showed that the local heating is reduced at a flight Machnumber of 5, or greater, for 5° < _5c < 25 °, but increasedfor larger half-cone angles.

The sharp wedge, or inclined flat plate, heating rates canbe found from equation (45) by using the Manglertransformation which is

qwFp = _twc/(3) 0"5 (47)

However, the Newtonian approximation for peue is not asaccurate for a wedge as for a cone. After adjusting theNewtonian values, an approximation for the inclined flatplate which is equivalent to equation (46) is

qw -- COJx FP)°5

seen to give somewhat better results than that of refer-ence I, although there is much scatter in the dam.

2.6 Cone and Flat Plate Heating Rates

The laminar boundary layer heating of bodies withoutpressure gradients (_ = ducJdx = 0) such as sharp cones andflat plates at angle of attack will now be considered. Theexpression for the heating rate on a sharp cone in air is,from reference 23, after noting that the total enthalpy term

should be replaced by the recovery enthalpy,

0.018 (peue'_0'5h0.85(1 hw ) WilWc- prw \-'x'-] aw _ - h-"_w _ (45)

where Pc is in atm, Ue is in m/see, x is_in m, and h is inm2/sec 2. At high speeds, where h,_ << V_,]2, and using theNewtonian flow approximation

peue = p,,_V3 sin 2 6c cos 8c

and the transport properties of reference 15, equation (45)becomes

x

where

x V3.2 sin 6c (1 hh_w/ Wm 2

haw = h,,_ + 0.40V 2o,o

where 6FPplate.

hw ) _ (48)×v22sin 1 - m2

is the wedge angle, or angle of attack of the

2.7 Wing Leading-Edge tleating

The heating along a cylindrical leading edge of a finite-length wing can be approximately calculated using theexpression of Rubesin (ref. 30)

.2i:lWLE = qwcy 1 • 2 sin2A) 0"5+ qwFp cos ot (49)

where Clwcvi is given by equation (41) and qFP by equa-tion (48). For highly swept leading edges, equation (49) islimited to small angles of attack, since the stagnation linemoves from the cylindrical leading edge onto the lower sur-face of the wing as the angle of attack increases.

(46)

2.8 Low-Density Flow Heating

The peak hypersonic aerodynamic heating experiencedby missiles or manned vehicles always occurs in thecontinuum flow regime. However, sometimes vehicles havesmall nose or wing leading-edge radii, and these regions of

the vehicle can experience low-density flow phenomenawhich alter the heating. For example, if a vehicle flying athigh altitude has a small nose radius, the Reynolds numberon the nose ceases to be large. In that case, the basicassumptions of the Prandtl boundary layer theory are vio-lated as the boundary layer becomes thick and slip can occurat the body surface. In addition, shock waves can become

thick,ordiffuse,andoccupya significantfractionof theshock layer. The low-density flow field and heating are dis-cussed by Shorenstein and Probstein (ref. 31) andShorenstein (ref. 32) for flat plates and cones. Axisymmetricstagnation point heating measurements are presented andcompared with theory by Boylan (ref. 33). One parameterthat characterizes low-density flows is the Knudsen number,which is the ratio of molecular mean free path to body size.The increase in stagnation point heat transfer coefficientwith Knudsen number and altitude was calculated by Mosset al. (ref. 34) and is shown in figure 8. For a nose radiusof 2.54 cm, the heat-transfer coefficient, predicted by thedirect simulation Monte Carlo (DSMC) method, increasestenfold in the altitude range of 60-100 km. Of course, theheat-transfer rate decreases by two orders of magnitudebecause of the atmospheric density decrease with altitude.From figure 8, it is also evident that neglecting slip at thewall can lead to a 130% overprediction of the heating at aKnudsen number of 0.1. (The results from DSMC methodshave been verified by comparison with experimental data byNomura (ref. 35).)

1.0

.8

.6

Ch

.4

.2

.001

/

.01

I

50

,38

I / 0 DSMC

//--_ [] VSL, NO SLIP

//_ Eix VSL, WITH SLIP

.10 1.0 10 100

,x_/r n

' 9'070 110

ALT, km

Figure 8.- Stagnation-point heat-transfer coefficient versusKnudsen number. V_ = 7.5 km/sec, rn = 2.54 cm.(From ref. 34.)

This concludes the discussion of cold-wall laminar heat

transfer. Next, boundary layer transition will be brieflycovered.

3. BOUNDARY LAYER TRANSITION

The transition of the boundary layer from laminar toturbulent flow remains the most complex problem in fluidmechanics. At hypersonic speeds, turbulent boundary layerheating can be several times greater than laminar heating.Therefore, it is essential that some form of reliable means of

predicting transition be available to avoid the penalties thatresult from overly conservative design. Because transition isinfluenced by many factors, the engineer must rely onempirical relations derived from test data. However, none ofthe ground test facilities can simulate most of theparameters of interest; in fact, the operating characteristicsof many test facilities have been found to strongly influencethe data. In the following sections, some examples of testdata will be discussed and a few correlation charts and

formulas from the open literature will be presented.

It has long been known that among the importantparameters influencing transition are the boundary layer edgeReynolds number and Mach number. The results of plottingmeasurements of the Reynolds number calculated at thebeginning of transition against Mach number are illustratedin figure 9 for flow on cones. At first glance, figure 9appears to be a shotgun pattern of data points, with transi-tion Reynolds numbers varying from about 1 million to30 million. However, even within this jumble of data thereare some definite trends. First, note that the flight data givethe highest transition Reynolds numbers and the wind tun-nel data give the lowest values. The ballistic range pointsfall, more or less, in the middle. The tendency to predictearly transition in many wind tunnels has been widelyobserved (refs. 36 and 37) and correlated by Dougherty andFisher (ref. 37) with the intensity and, to some extent, thefrequency of the disturbances in the facilities. The problemsencountered in wind tunnel test measurements of transition

have encouraged the use of ballistic ranges (ref. 38 and 39).More recently, the development and use by Beckwith(ref. 40) of a low-disturbance, high Reynolds number,supersonic wind tunnel has yielded valuable data (refs. 41and 42). For example, Chen et al. (ref. 42) measured similartransition Reynolds numbers at a Mach number of 3.5, onflat plates and slender cones with adiabatic walls. In high-noise wind tunnel tests, the uansition Reynolds numbers on

fiat plates were only half as large as the values measured oncones (ref. 42), implying that the flat-plate boundary layerwas more sensitive to free-stream disturbances.

60 x 106

40

20

10

86

n,-

4

2

1.8

0

Aa__ 0

A FREE FLIGHT

I 0 _ I I I I I l

2 4 6 8 10 12 14 16

M e

Figure 9.- Typical transition dam---cones.

10

Someempiricalcorrelationsfor thebeginningoftransitionfoundin theopen literature will now be discussed.Despite the data scatter in figure 9, there is a discernibletrend of increasing transition Reynolds number with risingMach number. The same trend was observed by Softleyet al. (ref. 43) and is shown in figure 10 (taken fromref. 43).The data in figure 10 are for sharp cones, having

Reynolds numbers based on nose radii of less than 200. Thehigh stability of the laminar boundary layer to disturbancesat hypersonic edge Mach numbers has been observed byother researchers (ref. 44) and lends credence to the trendshown in figure 10. However, the influence of walltemperature on hypersonic transition appears to be an openissue, with some tests showing strong effects and other testsindicating little effect (rcf. 43).

4x107

3

10798

..=" 7× 6

y-5

0 A

O

106 J J i i 1 1 t I I I l J0 5 10

M e

for surface-roughness-induced transition is presented byAmirkabirian et al. (ref. 49) for the Shuttle orbiter and isshown in figure 11. Although the Shuttle tiles are verysmooth, the "roughness" results from misaligned tries andthe gaps between tiles. Again, most of the flight data pointsare well above the shaded band, which is based on wind-tunnel tests. An approximate correlation is (ref. 49)

Re0/Me = const. (51)

600

50O

400

Rej)__ 300

M e

200

100

FLIGHT DATA WIND TUNNEL DATA

O- STS-1 ..... SMOOTH MODEL

[3 - STS-2 CORRELATION

0 - STS-3 _ DATA RANGE FOR

A- STS-4 _ ROUGH MODELS

[]

8o [] o oA ¢

A Ao £ oA 0

I I I 1 I

0 ,2 .4 .6 .8 1.0

×/L

J I Figure 11.- Flight values of the transition parameters.15 (From ref. 49; reprinted with permission of The

American Institute of Aeronautics and Astronautics.)

Figure 10.- Effect of local Mach number on transitionReynolds number--sharp slender cones--uniform walltemperature. (From ref. 43; reprinted with permission ofThe American Institute of Aeronautics and Astronautics.)

Unlike the hypersonic boundary layer, the subsonic oneexisting on the blunt noses of high-speed flight vehicles iseasily tripped by surface roughness. The mechanism hasbeen extensively studied and correlations have been pub-lished (refs. 45-47). Since strong pressure gradients exist onthe blunt noses, the correlations are for Reynolds numbersbased on boundary layer momentum thickness rather than

the body lengths used for sharp cones and flat plates. A for-mula suggested by Laderman (ref. 46) is

T._e_0"7(5O)

where k/0 is the ratio of roughness element height to localmomentum thickness. (Charts of momentum thickness inhigh-speed flight for bodies with various amounLs of nosebluntness can be found in ref. 48.) Another correlation

where the constant varies from 150 to 350, depending on theratio of roughness height to momentum thickness, etc.Additional transition data are available for configurations ofpractical interest such as blunted cones at angle of attack(ref. 50) and on swept-wing leading edges (ref. 51). Despitethe high-transition Reynolds numbers which can be expectedat hypersonic speeds, full-sized vehicles will still experienceturbulent boundary layers over much of their surface. The

heat transfer resulting from turbulent boundary layers willbe discussed next.

4. TURBULENT HEAT TRANSFER--COLDWALL

For the laminar boundary layer, it was shown that exactsolutions could be found which model the viscous flow over

important regions of a vehicle. In contrast with the laminarboundary layer, there is no complete theory for modeling theturbulent transport of mass, momentum, and energy. Theequations for the turbulent boundary layer can be formulatedonly with the aid of experimental data such as mixinglengths, etc. Therefore, all turbulent "theories" are semi-empirical and care must be taken when these formulations

11

areappliedbeyondtherangeof conditionsforwhichtheywerederived.

4.1 Comparison of Approximate Methods

A number of approximate methods have been widelyused to calculate the turbulent boundary layer heating. All ofthe methods rely on Reynolds analogy which relates skinfriction and heat transfer (ref. 8). In addition, all methods usesome form of coordinate transformation for extending thewell-verified incompressible skin-friction and heat-transferformulas to the compressible, high-speed conditions. Thetransformations therefore are functions of Mach number,

Reynolds numbers, and temperatures at the boundary layeredge and at the wall. Among the most widely usedapproaches are the reference enthalpy, or reference tem-perature, Spalding and Chi, Van Driest II, and Coles. Thesemethods have been extensively compared and their formula-tions explained in the literature by Hopkins and Inouye(ref. 52) and by Cary and Bertram (ref. 53); therefore, onlybrief descriptions will be presented here.

The reference temperature concept was originated byRubesin and Johnson (ref. 54) and the coefficients modifiedby Sommer and Short (ref. 55). The basic assumption isthat a temperature within the boundary layer can be calcu-lated, using an expression with empirically determinedcoefficients, which will yield the correct skin friction in acompressible, supersonic boundary layer when the incom-pressible relations are used. Although the method was firstformulated for laminar boundary layers, it was found to givegood results for turbulent boundary layers also. Eckert(ref. 56) extended the method to high-speed flows in

equilibrium having real gas effects by using enthalpy inplace of temperature. Van Driest II refers to Van Driest'ssecond method which uses the von Karman mixing length(ref. 57). In addition, the Crocco temperature profile isassumed through the boundary layer. The Spalding and Chimethod (ref. 58) uses Van Driest's formulation to relate theincompressible and compressible boundary layer skin fric-tion. Empirical expressions are used containing the bound-ary layer edge, wall, and recovery temperatures. The cor-relation functions were empirically determined using a largebody of data. However, some of the data contained sys-tematic errors, yet all data were weighed equally. Themethod of Coles (ref. 59) is based on the concept of a con-stant mean temperature of a layer, called the substructure,which is located near the wall but extends beyond the lami-nar sublayer. The substructure temperature is based on theboundary layer edge Mach number and temperature and thewall temperature and is used to evaluate a Reynolds number.Additional mathematical details on the above four methodscan also be found in refercnce 8.

The ability of the four methods to predict turbulent heattransfer has been compared by several authors. Thecomparisons of Hopkins and Inouye (rcf. 52) are shown infigure 12 for the Mach number range 4.9 to 7.4 and forratio of wall temperature to recovery temperature from 0.1to 0.82. The conclusion drawn from figure 12 and compar-isons in reference 52 is that the Van Driest II method gave

12

the best overall results while Spalding-Chi consistentlyunderpredicted the heating. However, note that disagreementbetween the prediction and the data of 20% to 30% is com-mon. The conclusions of reference 52 are corroborated bythe results of Chien (ref. 60) in figure 13 for a wall tem-perature ratio of 0.2 or greater. Chien's tests were conductedat Mach 7.9. For a very cold wall temperature ratio of 0.11,the Spalding-Chi method is shown (fig. 13) to be best.However, shock tube experiments (refs. 61 and 62) at walltemperature ratios from 0.01 to 0.24 and at Mach numbersof 1.5 to 3.1 indicate the best agreement with Van Driest IIin reference 61, while Ref. 62 supports using Spalding-Chi. In fact, reference 53 and Cary (ref. 63) concluded thatthe Spalding-Chi method yielded the best results over theentire temperature ratio range 0.1 to 0.7 at Mach numbers of4 to 13. Charts of turbulent heat-transfer coefficients based

on the Spalding-Chi correlations were made by Heal andBertram.(ref. 64) Zoby and Graves (ref. 65) statisticallystudied the effect of using two different forms of Reynoldsanalogy on the Van Driest II, Spalding-Chi, and referenceenthalpy methods. Data from ground-based facilities andfrom flight tests were analyzed. It was concluded theColbum modification of Reynolds analogy, which is

cfSt - 2Pr0.667 (52)

usually resulted in somewhat better agreement with test datathan using von Karman's version of Reynolds analogy. Acomparison of the three heat-transfer prediction methodswith the data (ref. 65) typically yielded rms errors of 15% to18%.

ZldJ

rr

THEORY80i i _ l i i , i _ l

601-SOMMER AND SHORT /, ]4ol-- ¢ ,4

-40 t i x i i J i 1 1 i

80 ' ' I I I I I I

6ol-SPaLD,nG ' Jk ,Oo% ,,,× 60 / r I I I I I I J I 1

40bVAN DRIEST II I' 20l- ............ z_" ¢':__ A0 ,4

×[ ._ 0 ,--,_ _7 " ' _7/ "

= "' -20 ]- _ _ .... v

_/-=IS "= 40 I I , , , , l , , ,

' ' ' ' ' ' ' 42ol-- . A<> ,401_ . • . f_'

-40 _ I I t I I I I I

0 .2 .4 .6 .8 1.0

Tw/Taw

Figure 12.- Effect of wall-temperature ratio on predictedheat transfer. M e = 4.9 to 7.4; 2Ch/Cf = 1.0. (Fromref. 52; reprinted with permission of The American

Institute of Aeronautics and Astronautics.)

lxl0 -3

St .6

.4

O

(a)L

x

t W

-- : 0.35T o

1 L I I I

lx10 -3

St .6

00". 0

"o.%°

°o

{b)I

\ o°°

T Wm = 0.2T

O

I I I t I

lx10 -3

.8

.6

St

.4

.2

°°

(c)

o

o".9O "°%

o'-_o......_ o0°o o o

T w-- : 0.tlT o

VAN DRIEST t.... SOMMER-SHORT TURB

----- SPALDING CHI [ THEORY1CLARK-CREEL

........ BLASIUS-LAMINAR THEORY

I 1 I 1 I

8 12 16 20 24x106

Re x

Figure 13.- Stanton number variation with Reynoldsnumber and wall temperature ratio. (From ref. 60;

reprinted with permission of The American Institute ofAeronautics and Astronautics.)

The conclusion is that all the semiempirical methods

give only approximate results that can be off by 20%, orsometimes more. However, for wall temperature ratios

below 0.2, Spalding-Chi does seem to predict the heatingbetter than the other three methods. It should also be notedthat finite-difference codes for calculating perfect gas turbu-lent boundary layer properties have been developed byWilcox (refs. 66-68) and, no doubt, others.

4.2 Reference Enthalpy Method

The reference enthalpy method has been widely used forthree decades. The most common expression for thereference value of the enthalpy (ref. 56) is

h' = 0.22haw + 0.28he + 0.5hw (53a)

although a number of variations have been used. At leastone other version (ref. 8) given by

h'= 0.18haw + 0.32he + 0.5hw (53b)

yields slightly better results. The Eckert version, equa-tion (53a), was used by Arthur et al. (ref. 69) to derive

simple, closed-form correlation equations for the turbulentflat-plate heating. The Blasius incompressible, turbulentskin-friction relation was used in reference 69 for Reynoldsnumbers to 10 million in the form

cf 0.0296

2 - (Re')0.2(54)

and the Schul_-Grunow equation

cf 0.185

_'= (logl0 Re') 2584 (55)

for Reynolds numbers above 10 million. In equations (54)and (55), the prime denotes that the quantity is evaluated atthe conditions corresponding to the reference enthalpy. Thethermodynamic and transport properties of references 5and 6 were used to calculate the heat-transfer rate at altitudesfrom 3 to 60 km and at velocities from about 1 to

11 km/sec. The resulting correlation, which is valid to+15% for Voo > 1500 m/see, but less than 3960 m/see

_q,,vFp =3.72(10-4)(p_ sin 2 8 cos 2"22 8) 0.8

(x - Xbt)0-2(Tw/555) 0-25

:( wx V 37 0.9 - m2 (56a)

and for Voo > 3960 m/sec

itwFp =2.45(10-5)(p,,_ sin 2 8 cos 2'62 8) 0.8

(x - Xbt) 0"2

x V 7 0.9 - m2 (56b)

In equation (56) the Newtonian approximation for pressureand velocity were used in the form

Pc = 9_v2 sin2 8 (57)

13

Ue=V_ cos_5 (58)

Therefore, the heat transfer is underpredicted at small valuesof 8. (For example, at 5 = 10°, the heating will be about6.5% too low, which is well within the uncertainty of the

reference enthalpy method.) Note that for the beginning ofthe turbulent boundary layer, the location at which transi-tion begins, Xbt, was chosen. While this choice is some-what conservative, it is supported by the correlations of ref-erences 53 and 65. The turbulent heat transfer on a sharp

cone of half angle, _, would be given, approximately, by(ref. 8)

qwc= 1.15qwFr, (59)

The relations derived so far are valid if the boundarylayer is either laminar or turbulent. In locating the effectiveorigin of the turbulent boundary layer in equation (56), itwas assumed that the transitional flow region is equal in

length to the preceding laminar flow distance. For lack of abetter model, it is assumed here that the heating rate in thetransitional region varies linearly with distance, having thelaminar value at the beginning of transition and the fullyturbulent value at the end of transition. This approximationgives, roughly, the correct average skin friction according toDhawan and Narasimha (ref. 70), and, therefore, the correctheating rate.

It is now possible to calculate the heat-transfer rate on asimple configuration. The example chosen here is theheating along the fuselage windward centerline of the Shut-de orbiter during the first flight (see fig. 14, from Tauberand Adelman (ref. 71)). The calculations were made usingequations (48) and (56) by assuming that the bottom sur-face was a wedge at the local surface slope and in radiativeequilibrium. The agreement between the measured valuesand the calculations is fairly good. The turbulent calcula-tions are definitely too high, possibly because of the coldwall conditions or the conservative choice of the origin. The

error bars on the data represent the effect of uncertainties inthe surface emissivity (ref. 72). At the flight conditionsshown in figure 14, molecular dissociation was sufficientlysmall to make catalytic wall effects unimportant in thelaminar boundary layer. Finite-rate wall catalysis has notbeen observed in turbulent boundary layers, probablybecause of the intense mixing.

4.3 Cross-Flow and Surface RoughnessEffects

Three-dimensional shapes, or axisymmetric bodies atangle of attack, all experience cross flows. The heating ofsuch complex flows must, generally, be computed usingfinite-difference methods (refs. 73 and 74). However, for a

few geometrically simple shapes, or in the plane ofsymmetry, approximate methods can be used. One suchexample is the calculation of the turbulent heating on theleading edge of a swept wing. The cross flow induced by theleading-edge sweep promotes boundary layer transition(ref. 51). Also, a turbulent boundary layer at the wing-fuselage junction can cause transition at the leading edge. Toapproximately account for the turbulent flow along the

14

0

6

_'4

<r_

F-

w

-'i'-0

I

,,,,,_ _/DATA

V = 3536 m/sec

ALTITUDE : 52.9 km

c_= 39.2I I I I I I

m

_ - 34.1

1 I 1 I 1 I

_, - 30.8

1 I I I I

.1 .2 .3 .4 .5 .6X/L

Figure 14.- Comparison of centerline heating calculationswith STS-1 flight data. (From ref. 71.)

leading edge, equation (56) must be used for the flat-platecontribution in equation (49).

Another example where the heating in a complex flowcan be treated approximately is the sharp cone at angle ofattack. For this case, the tangent cone approximation(ref. 8) yields reasonably good results when the angle ofattack is less than the cone half angle. In the tangent-conemethod, an imaginary cone is used having a half anglewhich equals the slope of the local ray on the cone at angleof attack. However, when the cone also has a blunt nose,

the heating distribution becomes more complex (seefig. 15, from Widhopf (ref. 75)) and must be calculatednumerically (ref. 74).

Vehicles designed to fly at hypervelocities are usuallyprotected by ablative heat shields. Frequently, ablationcauses a rough surface which can trigger early boundarylayer transition. Roughness-induced transition on bluntnoses of bodies, where the boundary layer is thin and theflow subsonic, has been widely observed and studied(refs. 45-47). In addition to triggering premature transition,the surface roughness can increase the turbulent heating byover 100%. The turbulent heating caused by roughnesselements equal in size to about 1% of the nose radius isillustrated in figure 16 (from Chen (ref. 76)). The peakrough-wall turbulent heating formulation of Chen predicts

..#oO"

1

.8

.6

.4

M_ = 10.6

Re_/ft = 12(10) 6

DATA FAIRING

c_ = 20

15

20 '¸

I I 1 I I I I 1 I I1 2 3 4 5 6 7 8 9 10

Sfr n

Figure 15.- Turbulent heat-transfer distributions along thewindward and leeward rays for various angles of attack.(From ref. 75; reprinted with permission of TheAmerican Institute of Aeronautics and Astronautics.)

"G

%

c0

°o"

500

450

400

350

30O

250

200

150

100

50

O •

G.E NsMETHOOOI N°°FROUGHNESS

• (SMOOTH WALL) I

[] \ I

• TS-- 1400 R, Re./FT-- 2 x lO' i ; _mSMOOTH ROUGH ( K = 0.04 in)

I [][] 0 o=0'90' I

• • c_= 0', 90' II •I

1 I I I 1 I I I I

10 20 30 40 50 60 70

O, deg

Figure 16.- Heat transfer to hemispheres having very roughand smooth walls in a typical tunnel environment.(From ref. 76; reprinted with permission of TheAmerican Institute of Aeronautics and Astronautics.)

more than twice the smooth-wall value and agreesreasonably well with the heavily scattered data. Note, also,that the peak heating rate occurs about 30 ° off the stagnationpoint, compared to 35 ° for the smooth surface value. Thecomputation is performed by numerically integrating a setof modified turbulent boundary layer equations which arebased on the assumption that the effect of roughness on themixing is restricted to a relatively thin layer near the wall.

Next, the heating associated with regions of separatedflow will be discussed. This will be followed by a brief

coverage of shock-interaction heating.

5. SEPARATED FLOW AND SHOCK-INTERACTION HEATING

Although regions of separated flow occur frequentlywhen control surfaces are deflected to large angles on hyper-sonic vehicles, the computation of such viscous flow fieldsis complex and relies heavily on finite-difference, numericalmethods. Another mechanism that can cause flow separa-tion, and severe local increases in heating near the flowreattachment point, is the presence of gaps, or cavities, andsteps in the surface. The discussion of heating in cavitieswill be followed by a brief coverage of the heating expe-rienced in the base regions of bodies. The extremely high

local heating that can occur when shocks intersect is coveredlast.

5.1 Shock-Induced Separated-Flow Heating

In a supersonic or hypersonic flow, a shock is generatedwhen the local surface slope increases abruptly. Forexample, such a change in slope occurs when a controlsurface is deflected. Since the static pressure increases

through the shock, the higher pressure is fed upstreamthrough the subsonic part of the boundary layer, whichcauses the flow to separate (fig. 17). For a given pressureincrease, the extent of the separation region in a laminarboundary layer is greater than in a turbulent one since theformer has a greater fraction of subsonic flow. A region ofrecirculating flow is formed in the comer with the fluidadjacent to the wall moving upstream. The heating in theseparated-flow region is increased over the value ahead of theseparated region and peaks at, or slightly beyond, where theboundary layer reattaches on the compression surface. The

heating in separated and reattached flows has recently beenreviewed by Merzkirch et al. (ref. 77).

Extensive heat-transfer measurements were performedby Holloway et al. (ref. 78) in laminar, transitional, andturbulent separated flows. The experiments were performedat Mach 6 and at maximum length Reynolds numbers from0.9 million to 7.3 million. An example of the heat transferin a turbulent boundary layer at Mach 6 with attached andseparated flow on a 75 ° sweptback, sharp delta wing is

15

oo

D ,'\\\\\\"\\\\\_" \\\_\\\\\\ ""<,

Figure 17.- Supersonic compression corner flow.

shown in figure 18 (from Keyes et al. (ref. 79)). The wingis at zero angle of attack. The flap deflection of 20 ° does notcause flow separation, whereas the deflection of 40 ° producesstrong separation. (The problem of trying to calculate theheating on the flap is evident from the variety of methodsused.) Various researchers have correlated the ratio ofturbulent heating rates ahead of the separated region to thepeak rates on the flap with the corresponding pressure ratio.Back and Cuffel (80) give the empirical relation

Ch2/Chl = (P2/Pl)0"85 (60)

RESULTANT SHOCK .._ /

INDUCED SHOCK -_

LEADING EDGE SHOCK \ _...._..._...--.-_ REGION

NECK../

X _ X COMPRESSION

BOUNDARY-LAYER EDGE

Figure 18.- Stanton number variation on 75 ° delta wingwith trailing-edge flap, Re/m ---2.7 x 107;c_ = 0 °.

(From ref. 79; reprinted with permission of TheAmerican Institute of Aeronautics and Astronautics.)

where their (dimensional) heat transfer coefficient is definedits

Ch =/t/(haw - hw) (61)

Nestler (ref. 81) suggests an exponent of 0.80 in equa-tion (60) and Alzner and Zakay (ref. 82) find values from0.725 to 0.815 for turbulent flows.

For hypersonic laminar boundary layers, Mathews andGinoux (ref. 83) suggest a relation in terms of Stanton

number at peak heating, Stmax, which is approximately

Stmax - (Rel) -0'5 (62)

where Rel is the Reynolds number ahead of the separatedregion on the flat plate. The constant of proportionality

appears to be a strong function of wall temperature ratio.

The problem of determining the heating when flowseparation occurs has been attacked successfully numeri-cally, Shang and Hankey (ref. 84) calculated the supersonicflow field for the turbulent boundary layer case, while Hungand MacCormack (ref. 85) did the hypersonic, laminar case.More recently, Lawrence et al. (ref. 86) used a perfect gasparabolized Navier-Stokes (PNS) code to calculate thehypersonic flow on a sharp flat plate at zero angle of attackwith a flap (fig. 19). (The flow field shown in fig. 19 couldsimulate the hypersonic flow over a sharp-nosed airfoil witha large flap.) A comparison of calculated laminar heat trans-fer with experimental data from reference 86 is shown infigure 20. At a Mach number of 14, the 15° deflection angledid not cause significant separation. A newer version of thecode, incorporating real gas effects, was used to perform thesame calculation as above, but at twice the previousReynolds number and at a Mach number of 20 (ref. 87). Theresult is shown in figure 21. (Since the calculations wereperformed on different computers, no direct comparison ofcomputation time is possible.) The heat-transfercoefficient of references 86 and 87 is defined as

Ch = Pr St(peue/pooVoo).

Shock-induced boundary layer separation and increasedlocal heating also occur when a shock wave impinges on theflow over a flat plate. Experiments performed by Back andCuffel (ref. 88) in a turbulent boundary layer at Mach 3.5confirmed the empirical relation given by equation (60), butgave an exponent of about 0.73. Skebe et al. (ref. 89)

performed detailed boundary layer measurements of theeffects of shock interactions on a flat plate having initiallaminar, transitional, and turbulent flow. The Machnumbers ranged from 2 to 4 and Reynolds numbers permeter varied from about 4.7 million to 30 million.

St

MODIFIED SPALDING-CHI--C LWING

..... FLAP (METHOD 1)

_- FLAP (METHOD 3)

_-- FLAP (METHOD 4)

.... SEPARATED REGION

.O2 [ I

.01 __-- y/c

O O

[] .09

<) .18

.001 -

.0004 -tICX_

-.6 -.4 -.2 0

s/c

ATTACHED FLOW, & -

l I 12}_ u,':_ '

_REATTACH- / I_MENT FROM

OIL FLOW _C)''_

3

--'-4

-.6 -.4 -.2 0 .2s/c

20 SEPARATED FLOW, _ - 40

Figure 19.- Hypersonic compression comer flow

16

JF- 10-2ZuJ

LI_

10-3

z<

I---

<

_ 10-4

: 1.o4x1o5v

BEAM-WARMING SCHEME

2ND ORDER ROE'S SCHEME

O HOLDEN AND MOSELLE DATA

I J L J

.8 1.2 1.6 2.0

x/_

Figure 20.- Comparison of heat-transfer coefficients onsharp flat plate with 15" compression comer. (Fromref. 80; reprinted with permission of The AmericanInstitute of Aeronautics and Astronautics.)

1.0

.8O

.60

x

.40

.20

0

/ Tw:100o o

--PRABHU ET AL

OTANNEHILL ET AL

I 1 I I I I

.50 1.0 1.5 2.0 2.5 3.0

XA

Figure 21.- Comparison of heat-transfer coefficients onsharp flat plate with 15° compression comer. (Fromref. 87; reprinted with permission of The AmericanInstitute of Aeronautics and Astronautics.)

Dot,ailed boundary layer surveys were made, but heat transferwas not measured. Hedge (rcf. 90) compared numerical cal-culations using MacCorrnack's code (ref. 85) with pressuresmeasured for laminar boundary layer shock interactions atMach 8. He found good agreement except near the begin-ning of the induced separation region. Hung et al. (ref. 91)present correlations for the peak heating from shockimpingement on a flat plate at hypersonic speeds. The fol-lowing expressions for the Stanton numbers at peak heatingarc given in reference 91:

for laminar/laminar flow

Stma x = 0.664(Re'x)-0-5(pr') _)-667 (63)

for laminar/turbulent and turbulent/turbulent flow

Stmax = 0.0576(Re'x)-0"2(pr') -0"667 (64)

where the prime denotes that the parameter is based on thereference enthalpy value (eq. 53a) at the peak heating loca-tion, Viegas and Coakley (ref. 92) studied turbulence modelsthat were used in the Navier-Stokes equations to calculateshock-separated turbulent boundary layers. They concludedthat the one-equation model proposed by Glushko andRubesin gave superior results, including more accurate skin-friction values, than the zero equation model.

5.2 Heating in Cavities

Cavities occur on all vehicles and can cause largeincreases in local heating at high speeds. First, cavities cantrigger boundary layer transition. Second, the heating withinthe cavity can be much higher than the heating of the sur-face just ahead of the cavity. Experiments of heating withincavities in turbulent flows were reported by Nestler et ai.(ref. 93) and some of their results are shown in figure 22.The ratio of the flat-plate heating rate to that measuredwithin the cavity is shown in figure 22; C h is defined byequation (61). (The curves have been drawn through theoriginal data points, which are deleted.) The Mach numberon the plate was 6.3 and the local Reynolds number wasabout 6 million. The turbulent boundary layer thicknessjust ahead of the cavity location was 1 cm. For the cavitylength-to-depth ratio (L/H) used in figure 22 of 5, 15, 30,the cavity depth was 2.5 cm, 2 cm, and 1 cm, respectively.Note that for L/H = 5, the flow passes over the cavity andthe heating on the cavity floor is decreased from the undis-turbed flat-plate value. When L/H = 11, the nature of theflow in the cavity changes according to Gortyshov et al.(ref, 94). For values of L/H larger than 11, the flowexpands into the cavity, attaches on the floor, and separatesnear the downstream corner, where the heating near reat-tachment reaches three times the undisturbed flat-plate value.

For supersonic turbulent flow, the average heat transferin the cavity is (ref. 94)

Stav e = 0.48(ReL)-0.4(L/H) 0.2 (65)

where the Stanton and Reynolds number properties arc eval-uated using the undisturbed flow conditions and the cavitylength (ref. 77). For cavities in supersonic and hypersoniclaminar boundary layers, Lamb (ref. 95) gives the followingexpression:

St = 2.07(Re6) -1 (66)

In equation (66), the Reynolds number is, again, evaluatedat the flow conditions upstream of separation and is basedon the boundary layer thickness, fi, ahead of the cavity,while the Stanton number is related to the heating on theleeward surface (ref. 77).

17

2.5

2.0

1.5o

1.0

I0 1.0

AMe = 6.3 I!

Ch ° = VALUE OF C hFOR NO CAVITY/]

r j _ I I I

,2 .4 ,6 .8

X/L

Figure 22.- Relative heat-transfer coefficient measuredalong cavity floor. (From ref. 93; reprinted withpermission of The American Institute of Aeronautics andAstronautics.)

intersection of strong shock waves can produce free shearlayers, jets, or expansions. Different shock interactions andthe resultant flows were delineated and studied by Edney(ref. 99). In reference 99 it was found that large increases inheating result from free shear layer attachment and super-sonic jet impingement. Keyes and Morris (ref. 100) per-formed experiments at a free-stream Mach number of 6.Turbulent shear layer attachment on a hemisphere was foundto produce local heating rates which were up to 14 timeshigher than the undisturbed stagnation point value. Anapproximate correlation for the ratio of peak local heattransfer to the (undisturbed) stagnation point value (ref. 100)can, again, be written using the corresponding pressureratio, in the form

Chp/Ch s = C(pp/Ps)1.35 (68)

where C = 1.58 when the shear layer is turbulent andC = 0.63 when it is laminar. The dependence of the heatingon the shear layer state (laminar, turbulent, or transitional)is, again, emphasized by Keyes (ref. 101). It is shown thatthe initial state of the shear layer can have a major influenceon the heating.

6. HEAT-PROTECTION METHODS

5.3 Heating in Base Regions

Since the heating in the base regions of hypersonicvehicles is one to two orders of magnitude less than on thewindward portions of most vehicles, the topic will be dis-cussed only briefly. Francis (ref. 96) presents flight data forablating very slender (4.5 ° half-angle) cones with turbulentboundary layers. For a blunted cone, the heating in the baseregion was, approximately, 6% of the cold wall value with-out ablation on the cone afterbody. Bulmer (refs. 97 and 98)gives a correlation for turbulent flow base heating on conesin hypersonic flow in the form

(Nu/Pr)b - 35.5(Reb/Rec) 2.2 (67)(Nu/Pr)c

In equation (67), the subscript b refers to the evaluation atconditions where the flow has expanded isentropically to thebase pressure and the reference length is the base radius. Thesubscript c refers to the flow conditions on the cone justahead of the base (ref. 77). For base heating where a laminarboundary layer exists, reference 77 suggests using equa-tion (66).

5.4 Shock-Interference Heating

Shock-interference flows occur when, for example, thebow shock from the blunt nose of a vehicle impinges on theleading edges of wings, fins, and inlet cowl lips. Theanalysis of shock-interference heating is fundamentallycomplex, but cannot be ignored because heating rate

increases of one order of magnitude have becn observed. The

The preceding discussion has been limited to methodsof calculating the heating rates at the body's wall in theabsence of interactions with the surface, i.e., the cold wallheating. The following discussion will deal with methods ofprotecting the body surface from the aerodynamic heating.

Several heat-protection methods are widely used. If theheating rate and duration of the pulse is modest, a high-heatcapacity metal heat sink, such as Beryllium oxide, canabsorb about 6.3 MJ/kg (about 2700 Btu/lb). For manypurposes, a more effective mechanism of heat absorption isto transfer mass from the surface to the boundary layer. Thethree main methods of mass-addition cooling are describedbelow.

Film cooling: fluid is injected into the boundary layernear the stagnation point, from where it spreads back overthe body.

Transpiration cooling: the fluid passes through a porouswall into the boundary layer. This differs from film coolingin that fluid injection can occur all along the wall.

Ablation: the surface material is allowed to melt and

vaporize, and enters the boundary layer in a liquid or gaseousstate. The mass addition of fluid into the boundary layerreduces heat transfer by (I) absorbing heat through phasechanges of the material from solid to liquid to gas and(2) thickening the boundary layer and altering the velocityprofile. Ablative heat shields are frequently impregnatedwith carbon fibers. The fibers provide structural reinforce-ment and, at high temperatures, form a char layer. The charis porous and permits percolation of gases and reradiates

18

heat.For example,at a temperatureof 3000K, over400W/cm2canbereradiated.

Toprotectthestructureandtheinteriorcomponentsofthevehicle,insulationmaterialisrequiredbetweentheheatshieldandthestructure.Sincetheheattransferoccursbyconduction,materialshavinglow conductivityandlowdensityareused.

t,.1 Mass-Addition Cooling

The most widely used method of protecting vehiclesagainst the high heating encountered in hypervelocity flightis mass-addition cooling. An early study of (stagnationre_ion) film cooling was made by Howe and Mersman (ref.

102). Low (ref. i93) calculated the effect of transpirationcooling on the iaminar boundary layer on a flat plate insupersonic flow where the injectant was air. Subsequently,Pappas and Okuno (refs. 104-106) measured the influence ofinjecting various gases, which were both lighter and heavierthan air, on the heat transfer on cones, having laminar andturbulent boundary layers, in supersonic flow. Kaattari (ref.107) and others perlbrmed mass-injection experiments onblunt-nosed bodies, including hemispheres. The results ofthe fluid-injection experiments were correlated for laminarboundary layers in reference 23. For the turbulent case, thecoefficients were found empirically. The correlation is,using an expression of the form (ref. 7)

= ttw/£1WB=0 = 1 - a(mc/mw)nB + b(mc/mw)2nB 2 (69)

in figure 24 for laminar flow. Again, the agreement isreasonably good. The beneficial effect of injecting gaseswhich are lighter than air is evident. Since lighter gasesthicken the boundary more than heavy ones, the velocity

_gradient in the boundary layer is reduced more. (For Freon,mw = 120 was used.) The influence of molecular weightratio is more pronounced for the turbulent boundary layerthan for the laminar one, as is evident in figure 25. Forgases that are heavier than air, equation (69) underpredictsthe wall cooling. Note also in figure 25 that in theturbulent flow case, the mass-injection efficiency decreaseswith increasing edge Mach number. This observation isconfirmed by Jeromin (rcf. 109) in his lengthy

1.0

_'_'" I_.-.,. \ . O LAM

_'"_ [] TURB

_. STAG PT

_'_Z" _ TURB CONE

LAM(__ STAG P_ "_ "_ x _'I_E]

I I

.5 1.0 1.5 2.0 2.5 3.0B

where mc and mw are the molecular weights of the shock-layer gas and the injection gas, respectively. Thedimensionless mass-addition parameter, B, is defined as

Figure 23.- Heat-transfer decrease at supersonic speeds forinjection of air into air. (Laminar theory from rcf. 23;data from rcfs. 104-107.)

Stagnation point (lam.)Cone, or flat plate (lam.)Cone, or flat plate (turb.)

ri3 WB- (He- hw) (70) 71, DATA BY PAPPAS AND

l\kX FREON OKUNO (REF. 104)

I\\\/ I-A,R Me_:4

.8t __HELILIM EQ, 69

.4

o,_,.2 "_'_-",_

bOXB=O

where flaw is the mass-addition rate. The constants a, b,and n in equation (69) have the following values:

a b

0.72 0.13 0.250.795 0.11 0.250.29 0.015 0.5

The results from equation (69) are compared with themeasuremenks of references 104-107 by plotting the heatingreduction as a function of the mass-addition parameter. Infigure 23, the heating reduction is shown when air isinjected into an air boundm'y layer, mw = file, at the stagna-tion point and on cones, or flat plates, having laminar andturbulent flow. The agreement is good. It is also apparentthat a much higher injection rate is required to cool the wallwhen turbulent flow exists, as confirmed by Dershin et al.(ref. 108). The effect of molecular weight ratio on theagreement between equation (69) and the data is illustrated

0 ,5 1 1.5 2.0 2.5 3.0

B

Figure 24.- Laminar hcat-transfer decrease at supersonicspeeds on sharp cone for three injected gases. (Theoryfrom ref. 23; data from ref. 104.)

19

1.0(

.8

.6

,4

DATA BY PAPPAS AND

OKUNO IREF. 106)

M e = 3.67 O

oQ_L I UM O _ O"

_ - EQ. 69

\, , \ _ t l !

.5 1.0 1.5 2.0 2.5 3.0B

Figure 25.- Turbulent heat-transfer decrease at supersonicspeeds on sharp cone for three injected gases. (Data fromref. 106.)

.2 3

,1 2

0

0 1 2 3

B

Figure 26.- Reduction of Stanton number for air injection

(experimental data) on flat plates and cones withturbulent boundary layers. (From ref. 109.)

survey paper and is shown more dramatically in figure 26(based on ref. 109).

Expressions such as equation (69) can be used toapproximate the behavior of relatively low-temperaturevaporizing ablators, for example, Teflon. For high-temperature, carbonaceous ablators, the presence of chemicalreactions between the injected and resident species in theboundary layer must be included in addition to the surface

20

radiation. Neither of these phenomena is accounted for inequation (69). The performance of carbon ablators wascalculated by Pu_ and Bartlett (ref. 110), who dcrived

empirical relations, similar to equation (69), for sevcralmaterials. A stagnation point correlation for graphite, fromnumerical solutions, is (ref. 110)

_'=l-aB+bB 2+cB 3-dB 4 (71)

where a=0.6563, b=0.01794, c=0.06365, and

d = 0.01125. The previously defined heat-transfer ratio, _,is related to equation (71) by

(72)

where the enthalpy ratio AHc/H e accounts for gas-phasechemical reactions and diffusion effects for injected species.The variation of _ with the mass-addition parameter, equa-tion (71), is shown in figure 27 and the enthalpy ratio fromchemical reactions is givcn in figure 28. (Both figures arefrom ref. 110.)

1.0

,8 i

.6

,4

,2

0

B

.2 .4 .6 .8 1.0 1.2 1,4

I_,% E I ! l , vQ. 71 PT2 0,01 TO 150 atrn

H e 500 TO 13,000 Btu/Ib

PT 0.2 TO 1.0 atm _ l UPPER

' 2 _ _LEGEND

H e 40,000 TO 60,000 Btu/Ib

EQ. 71

LEGENDI I 1 t_"""""'l_ I

1.6 1.8 2.0 2.2 2.4 2,6 2.8 3.0

B

Figure 27.- Correlation of blowing parameter with graphiteablation boundary layer solutions. (From ref. 110;reprinted with permission of The American Institute ofAeronautics and Astronautics.)

Slender vehicles can experience very high heating at thenose in hypervelocity flight, causing surface recession andshape change at the nose. If the nose ablation is asym-

metric, an undesirable pitching moment results. The slender-vehicle nose-ablation problem is formulated and the resultsof numerical solutions are presented by Chin (ref. 11 l). The

roughness of the ablating surface can also cause prematureboundary layer transition, which further intensifies the heat-ing, as previously mentioned. The effects of nose surfaceroughness on transition and heating are discussed by Grabowand White (ref. 112). Slender-body nose-shape changes asdetermined from wind tunnel tests using low-temperature

z_

1-"1

1.2

1.0

.8

.6

.4

.2

I

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8

B

Figure 28.- Variation of AHc/H e with blowing rate andtotal enthalpy. (From ref. 110; reprinted withpermission of The American Institute of Aeronautics andAstronautics.)

ablators were measured by Kobayashi and Saperstein

(ref. 113). Using film cooling to reduce the surface reces-sion and shape change of slender noses was studied by Goldet al. (ref. 114). The heat protection by mass injection wasalso theoretically studied for other geometric shapes and willbe briefly covered next.

The effect of mass injection on the laminar heatingalong the stagnation line of an infinite, swept cylinder wascalculated by Libby and Kassoy (ref. 115). For massiveblowing, it was found that the spanwise flow separated, butthe chordwise flow remained attached. Wortman analyzed theeffect of mass addition on several blunt shapes ranging fromhemispheres to cylinders (refs. 116 and 117). In refer-ences 116 and 117 the laminar boundary layer equations aresolved for a real gas in equilibrium and for a variety of lightand heavy (compared to air) injected gases. The results arepresented in the form of correlation functions. In refer-ence 118, Wortman calculates the reduction of laminar

heating by mass injection on sharp cones at angle of attack.It was found that normalizing the heat transfer by the zeroangle of attack values eliminated the influence of injectedgas properties. Inger and Gaitatzes (ref. 119) computed theeffect of strong blowing on high Reynolds number laminarflow, at supersonic speeds, about slender bodies, includinginduced pressure effects. Some of the effects computed inreferences 118 and 119 are open to question, however. Forexample, it is well known that cross flow on cones at largeangles of attack (rcf. 118) can induce boundary layer transi-tion. It has also been observed that massive blowing at highReynolds numbers can cause transition very close to thestagnation point of a blunt body, as will be discussed next.

6.2 Injection-Induced Transition

Demetriades et al. (ref. 120) and Kaattari (ref. 107)

observed boundary layer transition caused by mass injectionon blunt bodies in hypersonic flows. The effect wasobserved at Reynolds numbers based on body diameter aslow as about 1 million. Similar results were observed by

Wimberly et al. (ref. 121) on a slender cone tested at a frec-stream Mach number of 12.2 and a flee-stream Reynoldsnumber of 4 million. In most of the cases observed, theoccurrence of transition raised the heating above the laminarvalue without mass addition. Park (ref. 122) presented

calculations to explain the injection-induced early transitionon blunt bodies. However, no general method is availablefor predicting the injection rates that will lead to prematuretransition.

6.3 Ablation Material Response

To calculate the response of an ablation material tohigh heating rates, it is necessary to know the high-temperature properties of the material. Eighty-five differentgraphitic materials were tested by Lundell and Dickey(refs. 123 and 124). The pressures ranged up to 4.5 arm andtemperatures up to 4000 K. The materials that were studiedincluded ATJ graphite, both two- and three-dimensional car-bon-carbon composites, pyrolytic graphite, mesophasegraphite, glassy carbon, and natural graphite. It was foundthat ATJ graphite performed slightly better than the single-phase materials (pyrolytic graphite, mesophase graphite, andglassy carbon). However, at temperatures above 3600 K theATJ mass loss increased exponentially with temperature.The mechanism responsible for the greatly increased massloss was particulate removal, or spallation, probably caused

by compressive thermal stress.

A different class of ablation materials consisting ofceramics was studied by Ziering (ref. 125). Metallic andceramic materials have much greater mechanical strengththan graphite, for example, and would experience much lesserosion when traversing dust or rain clouds. Tests of thefollowing materials were reported in reference 125: siliconnitride, silicon carbide, tantalum carbide, and tungsten. Sili-con nitride's mass loss, surface recession, and energyabsorbed per unit mass were found to be by far the best ofthe group, with silicon carbide rating a distant second. Theenergy absorbed per unit mass ablated was 6 MJ/kg(2600 Btu/lb) for silicon nitride, while silicon carbide'svalue was 0.72 MJ/kg (310 Btu/lb).

For additional information on the application of mass-transfer cooling, Harmett (ref. 126) is recommended.

7. FINITE-DIFFERENCE CODES

A large number of computer codes exist which can beused to calculate heat transfer at high velocities. Most ofthese codes were not intended for widespread use and havenever been documented. By necessity the following discus-sion will be limited to a small, but representative, samplingof codes that are being used to calculate cold-wall aerody-namic heating. The discussion concludes with a descriptionof some codes that can be used to compute surface ablation.

Thompson et al. (ref. 127) compared the heating on

blunted slender cones at angles of attack as calculated by

21

fourdifferentcodeswithbothwindtunnelandflightdata.The codes consisted of a three-dimensional, viscous flow,

finite-difference method called VSL3D, and three approxi-mate, engineering-type codes known as MINIVER, AERO-HEAT, and INCHES. As expected, VSL3D gave the bestagreement with the data. Of the approximate methods,MINIVER predicted the nose bluntness effects poorly andAEROHEAT underpredicted the laminar flow heating atangle of attack by as much as 30%. The INCHES code gavegood results at zero angle of attack, but at small angles ofattack the windward ray heating was underpredicted by up to40%. However, the engineering-type codes all used smallamounts of computer time and their utility must be evalu-ated in that light.

7.1 Space-Marching Codes--PNS

The most widely used methods to calculate hypersonicflow fields and heat transfer are the downstream marching

solutions of the parabolized Navier-Stokes equations knownas PNS codes. The parabolized approximation is valid if thesteady-state shock-layer flow, which must be supersonic,and the subsonic portion of the viscous-layer flow are bothin the streamwise direction. Therefore, flows with largeamounts of separation, and the attendant reverse flow,cannot be treated; however, cross-flow separation ispermitted. When the above limitations are imposed, theNavier-Stokes equations becomes parabolic in the stream-wise direction, thus permitting the downstream marchingfrom an initial, supersonic data plane. Chaussee (ref. 128)evaluated the ability of a perfect gas PNS code to predictheat transfer on a blunted bicone having laminar and turbu-lent boundary layers. Comparisons with Mach 8 wind tun-nel data gave reasonably good results when the dampingcoefficients (artificial viscosity) were held at the smallestvalue giving stable solutions. Chaussee and Rizk (ref. 129)used the same PINS code to make calculations of the flow

over deflected control surfaces on several body shapes. Theflap deflections were kept small enough to avoid separatingthe turbulent boundary layer. Although heating rates werenot calculated, pressure distributions and other flow-fieldinformation were presented. Rizk et al. (ref. 130) used thePNS code to calculate heating on a bicone at angle of attackon the windward, leeward, and 90 ° plane. Their results werecompared with wind tunnel measurements made at Mach 10and are shown in figure 29 (from ref. 130). The comparisonshows some disagreement near the nose of the vehicle. Thedifferences on the aft part of the lee side are due to boundarylayer transition which the code cannot predict.

The PNS code calculations of references 128-130 were

made assuming a perfect gas. It would be surprising if theaerodynamic heating in a Mach 10 flow could be accuratelypredicted using the perfect gas assumption. In fact,Balakrishnan and Chaussee (rcf. 131) showed that using

4

2

0z

I-<t,t,i'I-

i11>

I--<..i

WINDWARD PLANE

II • EXPERIMENT

CALCULATION

i i 1 l t 1 1 1 1 1 i

• EXPERIMENT, _ = 90 PLANE

• EXPERIMENT, LEEWARD PLANE

CALCULATION

2.011_ • • •

0_2 6 10 14 18 22

X-STATION

Figure 29.- Axial variation of the heat-transfer rate.Red = 8.3 x 104; ct = 10°. (From ref. 130.)

perfect gas calculations caused the stagnation point heatingto be significantly underpredicted at high speeds. Efforts toexplain Shuttle orbiter surface-temperature measurementsprovided one impetus to include real gas effects in PNScodes. Prabhu performed this task and added the capability tocompute chemical nonequilibrium effects (refs. 132 and133). Subsequently, Prabhu et al. (ref. 134) extended thePNS code to permit three-dimensional flow computations.Examples of Prabhu's calculations are shown in figure 30,for a real gas in chemical equilibrium, and in figure 31 forthe chemically reacting flow. Both figures 30 and 31 are for

sharp cones having a 10° half angle, flying at a speed of8100 m/sec at an altitude of 61 km. The small cone angleresults in an equilibrium shock layer temperature of 3100 Kat zero angle of attack. The relatively low temperature limitsthe nonequilibrium effects. Tannehill et al. (ref. 135) modi-fied a PNS code (ref. 86) to include chemical reactions andalso added an upwind-differencing algorithm. The upwind-differencing method improves shock capturing and avoidsthe instabilities inherent in central-differencing schemescaused by flow-field discontinuities such as shocks. Theamount of artificial viscosity that is needed to control theoscillations occurring in central-differencing schemes canlead to inaccuracies in the computed results, as was observedin reference 128.

22

.05

.04

NONEQUILIBRIUM AIR

O IDEAL GAS ('_ = 1.4)

• EQUILIBRIUM AIR

.02

.01

0 1 2 3

DISTANCE ALONG BODY AXIS, X (m)

Figure 30.- Axial variation of Stanton number on 10°sharp cone for various gas models, (_ = 0. (From ref. 132.)

.05

.04

.03

St

.02

.01

_ = 0 deg

O _ = 10 deg (LEESIDE)

• c_ = 10 deg (WINDSIDE)

_• • • •

O

oE) E) _ C) IO I_ I . •

1 2 3

DISTANCE ALONG BODY AXIS, X (m)

Figure 31.- Effect of angle of attack on 10 ° sharp cone heattransfer. (From ref. 134; reprinted with permission ofThe American Institute of Aeronautics and Astronautics.)

7.2 Time-Iterative Codes

To overcome the inherent limitations of the PNS codes,new methods are being developed to calculate thehypervelocity, real gas flow about three-dimensional bodies.The new codes use a time-iterative procedure which elimi-nates the need for artificial viscosity. However, thecomputation time for the time-dependent methods is nearlytwo orders of magnitude greater than for the downstreammarching codes.

One example of a recently developed time-iterative codeis the work of Palmer (ref. 136). Palmer's code employs animplicit, flux-splitting, shock-capturing method andincludes equilibrium, real-gas effects. The code is beingextended to treat chemical reactions (ref. 137).

Edwards et al. (ref. 138) compared four computer codesby using each to calculate hypersonic flow over twodifferent bodies. The four codes consisted of an upwind PNSsolver, UPS (ref. 86), and three time-iterative methodswhich solved the three-dimensional, thin-layer Navier-Stokes equations. The time-ilerative codes consisted of acentral-differencing, time-variational, diminishing codeknown as VAIR3D/TVD, a partial flux-split code calledF3D, and an upwind solver, UWIN. A comparison of windtunnel measured heating rates with calculations using theabove codes is shown in figure 32 (from ref. 138). Thecomparison is for a bicone model at Mach 10 and a 10°angle of attack. All the codes give similar results withminor exceptions. Near the body's nose, the heating in the90 ° meridional plane is underpredicted. Also, the occurrenceof boundary layer transition on the leeward side increases themeasured heating well above the calculated laminar flowvalues. (None of the codes can predict transition.) A secondcomparison of the codes with each other is illustrated infigure 33. A generic, hypersonic body shape was used; thebody is to be tested in the NASA Ames 3.5-ft hypersonicwind tunnel. Note that the heat-transfer coefficients, corn-puled at Mach 7.4 and _ = 10°, vary by over a factor of 2.Apparently the strong cross flow occurring on the body atangle of attack poses difficulties for all four codes, althoughabout 70,000 grid points were used. Unfortunately, noexperimental data are available yet to determine which of thefour codes gives the best results under these conditions. The

CPU time for the three time-iterative codes ranged from37 to 90 min on the CRAY X-MP, compared to about

1 min for the UPS, space-marching code. Although greatstrides have been made using finite-difference methods tocompute the (cold-wall) heat transfer on three-dimensionalbodies, complex viscous flow fields still pose difficulties.

23

7

<rr"

t,.9Z1,-<LU"l"

T--<_1

w2

(a) WINDWARD SYMMETRY PLANE

[313

3 (b) ,; = 90; MERIDIANAL PLANE

AND LEEWARD SYMMETRY

[-1

n EXPT. 90 deg

-- VAI R3D/TVD

.......... UWlN1313

[] 13 ....... UPS...Fa --'-- F3D

.. . O EXPT. LEE

o o__---_/_._.-:-_

0 4 8 12 16 20 24 28

X

Figure 32.- Comparisons of heat-transfer rate for bicone.Moo = 10, 0t = 10 °. (From ref. 138.)

mr

¢y

zIdd

GkLU-

U.l

oo

t,t-

z

)-I-<LU-r

_.__';s_'C_c_r,o__2 L --_-----_3

= L = 0.941m

.3 7.. (a) WINDWARD SYMMETRY PLANE

,,_,-. ,'k",,:-.L....,,_J/ \

._ -,.....:........... \

........ UWIN ' '" .... .. ",

::::::: \::. \ ....

.2 - (b) LEEWARD SYMMETRY PLANE

--.2:::

l I t I I I

.25 .35 .45 .55 .65 .75 .85X. m

Figure 33.- Comparison of heat-transfer coefficients.Moo = 7.4, (x = 10°. (From ref. 138.)

24

7.3 Ablation Material Interaction Codes

Compared to the calculation of cold-wall heating, theinteraction between the boundary layer and an ablatingsurface can be much more complex. Several codes which canbe used to calculate mass-addition effects with chemicalreactions will be discussed. The codes can be used to calcu-

late ablative heat-shielding requirements.

The boundary layer integral matrix procedure (BLIMP)code (ref. 139) was developed to calculate the effects ofchemical nonequilibrium in laminar and turbulent boundarylayers over surfaces of arbitrary eatalycities. The procedure isapplicable to nonsimilar, multicomponent, laminarboundary layers with thermal diffusion and second-order,transverse curvature effects. The surface boundary conditionsinclude coupling with transient ablation energy and massbalances. The BLIMP code has been used to analyze thesurface catalytic effects experienced on the Shuttle orbiter(ref. 10). A relatively simple analysis for noncharringablators was developed by Matting and Chapman (ref. 140).The method of reference 140 is applicable to melting,vaporizing, or subliming surfaces, but treats only the stag-nation point. Moss (ref. 11) developed a code for solving thefully viscous, ablating wall problem at and downstream ofthe stagnation point. Both convective and radiative heatingcan be treated. The coupling between the chemically reacting

boundary layer and charring ablators was formulated byKendall et al. (ref. 141). The code is known as CMA. Thecomputation is one-dimensional; although various bodyshapes can be used, the upstream history of the boundarylayer is ignored. The surface ablation in gases other than aircan also be analyzed; for example, rocket nozzle flows, orflight through the atmospheres of other planets.

In concluding this discussion of convective heatingcalculation methods, it should be emphasized that theoreticalmethods alone are insufficient to accurately predict the high-speed heating of a vehicle's surface. Codes are not yet pre-cise enough to be relied upon solely to compute heating incomplex viscous or three-dimensional flows. However,

experiments are costly, time-consuming, and frequentlyincapable of simulating flight conditions. The judicious useof verified codes and other theoretical procedures can greatlyreduce the amount of experimental testing that is required.Only by using a combination of reliable theoretical methodssupplemented by carefully conducted experiments can theheating of a vehicle be determined with a high level ofconfidence.

Ames Research Center

National Aeronautics and Space AdministrationMoffett Field, California 94035March 15, 1989

25

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139. Murray, A. L.: Facilitation of the BLIMP ComputerCode and User's Guide. Prep. by Acurex Corp.,Aerotherm Div., Huntsville, AL, for FlightDynamics Laboratory, USAF Systems Command,Wright-Patterson AFB, OH, AFWAL-TR-86-3101,vols. 1 and 2, Jan. 1987.

140. Matting, F. W.; and Chapman, D. R.: Analysis ofSurface Ablation of Noncharring Materials withDescription of Associated Computing Program.NASA TN D-3758, 1966.

141. Kendall, R. M.; Bartlett, E. P.; Rindal, R. A.; andMoyer, C. B.: An Analysis of the CoupledChemically Reacting Boundary Layer and CharringAblator (Parts I-VI). NASA CR-1060-1065, 1968.

31

APPENDIX

Listed below are some useful conversion factors.

1 Btu = 1054.8 J

1 Btu/lb = 2325.4 J/kg

1 ft = 0.3048 m

1 lb = 0.4536 kg

1 slug/ft 3 = 515.2 kg/m 3

1 W = 9.4805(10 -4) Btu/sec

I W/cm 2 = 0.88076 Btu/ft 2 sec

1 W/m 2 = 8807.6 Btu/ft 2 sec

PRECEDING PAGE BLANK NOT FILMED

33

Report Documentation Page

1. Report No.

NASA TP-2914

2, Government Accession No.

4. Title and Subtitle

A Review of High-Speed, Convective, Heat-Transfer

Computation Methods

7. Author(s)

Michael E. Tauber

9, Performing Organization Name and Address

Ames Research Center

Moffett Field, CA 94035

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546-0001

3. Recipient's Catalog No.

5. Report Date

July 1989

6. Performing Organization Code

8. Performing Organization Report No.

A-89042

10. Work Unit No.

506-40-11

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Paper

14. Sponsoring Agency Code

15. Supplementary Notes

Point of Contact: Michael E. Tauber, Ames Research Center, MS 230-2, Moffett Field, CA 94035

(415) 694-6086 or FTS 464-4126

16. Abstract

The objective of this report is to provide useful engineering formulations and to instill a modest

degree of physical understanding of the phenomena governing convective aerodynamic heating at

high flight speeds. Some physical insight is not only essential to the application of the information

presented here, but also to the effective use of computer codes which may be available to the

reader. The paper begins with a discussion of cold-wall, laminar boundary layer heating. A brief

presentation of the complex boundary layer transition phenomenon follows. Next, cold-wall

turbulent boundary layer heating is discussed. This topic is followed by a brief coverage of sepa-

rated flow-region and shock-interaction heating. A review of heat protection methods follows,

including the influence of mass addition on laminar and turbulent boundary layers. The paper con-

cludes with a discussion of finite-difference computer codes and a comparison of some results

from these codes. An extensive list of references is also provided from sources such as the various

AIAA journals and NASA reports which are available in the open literature.

17. Key Words )Suggested by Author(s))

Aerodynamic heating

High-speed aerodynamic heatingHeat-transfer computations

19, Security Classif. (of this reportl

Unclassified

18. Distribution Statement

Unclassified - Unlimited

20. Security Classif. (of this page)

Unclassified

Subject category: 34

21. No. of pages

36

22. Price

A03

NASA FORM 1626 OCT 86 l'or sale by the National Technical Information Service, Springfield, Virginia 22161 NASA-Langley, 1989