CALCULATION of COMPRESSIBLE, NONADIABATIC BOUNDARY LAYERS in 1 November 1971 LAMINAR, TRANSITIONAL and TURBULZNT FLOW by W METHOD of 6

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CALCULATION OF COMPRESSIBLE, NONADIABATICBOUNDARY LAYERS IN LAMINAR, TRANSITIONAL AND TURBULENT FLOW BY THE METHOD OF INTEGRAL RELATIONS

Prepared by NIELS EN: ENGINEERING AND RESEARCH, INC. Mountain View, Calif. 94040 for LangleyResearch Center

NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. NOVEMBER 1971

I TECH LIBRARY KAFB, NY

- -~~ ~ .~ 1. Report No. I 2. Government A-ion No. 3. Recipient's clltalog No.

CALCULATION OF COMPRESSIBLE, NONADIABATIC BOUNDARY LAYERS IN 1 November 1971 LAMINAR, TRANSITIONAL AND TURBULZNT FLOW BY W METHOD OF 6. PerformingOrgsnization Code INTEGRALRELATIONS

~" ~ .. .~ . ~~~ - I 7. Author(s) 8. Performing Orgsnization Report No. Gary D. Kuhn NEAR TR 25

. . ~ ~~ ~~" - . 10. Work Unit No. 9. MormingOrginization Name and Address 722-03-10-01 NielsenEngineering and Research,Inc. ~.~ 850 Maude Avenue 11. Contract or Grant No. Mountain View, 94040 .. Calif. NASl-9429 ..

" . 13. Type of Report andPeriod Covered 2. SponsoringAgency Name and Address . ContractorReport NationalAeronautics and SpaceAdministration 14. SponsoringAgency Code Washington, DC 20546

.- ...... ~ ~~ - -- - __ _~ - r 5. .Supplementary Notes

" .. " . ~~~ ~ . 6. Abstract The method of integral relations has been applied to the calculation ofcompressible, nonadiabaticlaminar, transitional, and turbulentboundary-layer characteristics. A two-layer eddy-viscosity model was employed for turbulent flow with transition produced by modulating the eddy viscosity by an intermittency function..--A computqr program was developed to do the calculations for two-dimensional or axisynrmetricconfigurations from low speeds to hypersonicspeeds witharbitrary Prandtl number. The program containsprovisions for tabular input of arbitrary configurationshapes and arbitrarystreamwise pressure, temperature, and Mach number distributions. Optionsare provided for obtaining initial conditions either from experimentalinformation or from a theoretical similarity solution.

The transition region can be described either by an arbitrary distribution of intermittency or by a functionbased on Emnons' probabilitytheory. Correlations were developedfor use in est- imatingthe parameters of thetheoretical intermittency function. Correlations obtained from other sourcesare used for estimating the transition point.

Comparisonswere made between calculated and measured boundary-layer quantities for laminar, transitional, and turbulentflows on flat plates, cones, cone-flares, and a waisted body ofrev- olution.Excellent comparisons were obtainedbetween the present theory and two other theories based on the method of finite differences.

The intermittency required to reproduce some experimental heat transfer results in hypersonic flow was found to bequite different from thetheoretical'function. It is suggestedthat the simple probability theory of mons may not be valid for representing the intermittency of hypersonic transitiohal boundary layers and that the program could be useful as a tool for detailed study of the intermittency of the transition region.

2 18. Distribution Statement Boundary layer Laminarboundary layer Unclassified - Unlimited Transitional boundary layer Turbulent boundary layer

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For sale by the NationalTechnical Information Service, Springfield, Virginia 22151 . . . ”...... - FOReWORD

The research described herein was sponsored by LangleyResearch Center, NASA, underContract NAS1-9429. Computer facilities were provided at LRC and at Ames ResearchCenter, NASA. The assistanceof the TechnicalMonitor, Mr. S. 2. Pinckney, incoordinating the research effort and providingtechnical information is gratefully acknowledged. The work was performed under the supervision of Dr. Jack N. Nielsen.

iii

TABLE OF CONTENTS Page Section -NO. SUMMARY 1 INTRODUCTION 2 LIST OF SYMBOLS 3 THEORETICAL BASIS 7 Assumptions 8 Boundary-Layer Equations €or Compressib1.e Turbulent Flow 8 Transformation of Axisymmetric Boundary-Layer Equations to Almost Two-Dimensional Form 13 Transformation of the Compressible Boundary-Layer Equations 15 Stewartson transformation 16 Dorodnitsyn transformation 19 Derivation of Ordinary Differential Equations 22 Development of integro-differential equations 22 Selection of approximating €unctions 26 Development of ordinary differential equations 29

Initial Conditions 36

Similarity solution 36 Direct €it of known velocity and temperature profiles 38 Known boundary-layer parameters 38 Least squares €it of velocity and temperature profiles 38

CALCULATION OF TRANSITIONAL BOUNDARY LAYERS 40

Prediction of the Transition Point 41 Turbulent Eddy-Viscosity Model 45 Modulation of the Eddy Viscosity in the Transition Region 50 Correlation €or the Extentof the Transition Zone 52

DETERMINATION OF ORDER OF APPROXIMATION FOR THE ENTHALPY FUNCTION 54

DESCRIPTION OF COMPUTER PROGRAM 58

V Page Section -No. EVALUATION OF THE METHOD 61 Incompressible Transitional Flow on a Flat Plate 61 Compressible Transitional Flow on a Sharp Flat Plate 62 Transitional Flowon a Sharp Cone 64 Adverse Pressure Gradient Flow on a Cone-Flare 65 Comparison with a Finite-Difference Method 66 Calculations with Specified Initial Conditions 68 Effect of Transverse Curvature in Laminar Flow 71 Turbulent Flow with Transverse Curvature 71 CONCLUDING REMARKS 73 REFERENCES 77 FIGURES 1 THROUGH20 a3

vi CALCULATION OF COMPRESSIBLE, NONADIABATIC BOUNDARY LAYERS IN LAMINAR, TRANSITIONALAND TURBULENT FLCW BY THE METHOD OF INTEGRAL RELATIONS By Gary D. Kuhn Nielsen Engineering& Research, Inc.

SUMMARY

The method of integral relations has been applied to the calculation of compressible, nonadiabatic laminar, transitional, and turbulent boundary-layer characteristics. A two-layer. eddy-viscosity model was employed for turbulent flow with transition produced by modulating the eddy viscosity by an intermittency function. A computer program was developed to do the calculations for two-dimensional or axisymmetric configurations from low speeds to hypersonic speeds with arbitrary Prandtl number. The program contains provisions for tabular input of arbitrary configuration shapes and arbitrary streamwise pressure, temperature, and Mach number distributions. Options are provided for obtaining initial conditions either from experimental information or from a theoretical similarity solution. The transition region can be described either by an arbitrary distribution of intermittency or by a function basedmons' on probability theory. Correlations were developed for use in estimating the parameters of the theoretical intermittency function. Correlations obtained from other sources are used for estimating the transition point. Comparisons were made between calculated and measured boundary-layer quantities for laminar, transitional, and turbulent flows on flat plates, cones, cone-flares, and a waisted body of revolution. Excellent comparisons were obtained between the present theory and two other theories based on the method of finite differences. The intermittency required to reproduce some experimental heat transfer results in hypersonic flow was found to be quite different from the theoretical function. It is suggested that the simple probability theory ofmons may notbe valid for representing the intermittencyof hypersonic transitional boundary layers and that the program could be useful as a tool for detailed study of the intermittency of the transition region. INTRODUCTION

Many empiricalmethods are available for estimating the properties of laminar and turbulentboundary layers. With the development of sophisticated electronic computers, many computerprograms have also been developedfor calculating the solutions of the boundary-layer equations in thelaminar and turbulentregions. At the same time, much researchhas been done on the nature of the boundary-layer flow in the transition regionbetween laminar and turbulentflow. Even now, however,the traLlsi- tionregion is notcompletely understood. The methodsof computing the boundarylayer in the transition region usually employ some empirical variationof the growth of the turbulent eddy viscosity, itself an empiricalquantity, along with a prediction of the starting point of the transitionregion developed from still other empirical correlations.

At thepresent time, there are two mathematicaltechniques for computingthe details of the boundary layer. Both techniques use the boundary-layerequations together with an empirical eddy-viscosity model forthe turbulent region. The techniquesdiffer essentially in how they solvethe boundary-layer equations. In thefirst method,an example of which is described. in reference 1, the solution is soughtby the method of finitedifferences. In the second method,described in references 2, 3, and 4, thesolution is bythe method ofintegral relations, in which the solution is represented to anydegree of accuracy by partial sums of infiniteseries.

In the work described in this report, the method of integral relations is adapted to developmentof a generalizedprogram for use as anengineer- ing tool for the calculation of boundary-layers on arbitrary two- dimensionalor axisymmetric bodies, with arbitrary flow conditions.

Themethod ofintegral relations was appliedin references 2, 3, and 4 to the calculation of laminar and turbulent,nonadiabatic, compressibleboundary layers. In the work describedherein, the method has been extended tocalculation of transitionalboundary layers. In addition,the theory has been extended to include arbitrary Prandtl number,and arbitrary Chapman-Rubesin constant,thus relaxing the assumption of a linear total temperature-velocity relationship as used in the previousapplications of the method.

2 The work presented in this reportwas accomplished under a 1-year effort sponsored by Langley Research Center, NASA, Contract No. NAS1-9429. The main effort was directed toward development of a computer program and the associated theory based on the method of infegral relations for calculation of transitional boundary layers. Dueto the general nature of the method, the program and theory are also applicable to the laminar and turbulent regions. The following sections contain the analyses which lead to the computer program and contain a substantial number of comparisons between prediction and experiment. An operating manual and a programmer's manual for the computer program are being issued as separate documents.

LIST OF SYMBOLS

a speed of sound elements of coefficient matrix defined by (88)eq.

elements of coefficient matrix defined by (89)eq.

parameter defined by eq.(26) parameters defined by eq. (150) elements of a coefficient matrix defined by (117) eq.

B parameter defined by eq.(27) right-hand side of eq.(86) , defined by eq. (90)

C specific heat for constant pressure P C Chapman-Rubesin constant defined by eq. (45) skin-friction coefficient,2.rJpeu: cf coefficients used in specifying the velocity gradient profile, eq. (77)

Ei coefficients used in specifying the enthalpy function, (82) eq. f frequency of Tollmien-Schlichting wave

3 weighting functions defined in eqs.(681, (69), and (70)

weighting function defined by eq.(83)

constants in transition correlation, eq.(121) integrand of integral defined by eq.(94)

static enthalpy total enthalpy transverse curvature index defined in eqs. , (1)(2) , and (3) constants in eddy viscosity, eq. (139)

constants defined in eq. (113)

reference length in two-dimensional coordinates or Probstein- Elliott coordinates (eq. (56))

L reference length in physical coordinate system m order of approximation of enthalpy function (eq. ) (82) m e (y - 1)MZ/2 M Mach number n order of approximation of velocity gradient profile,(77) eq. n' exponent of unit Reynolds number in transition correlation (eq. (121)1 reference Reynolds number defined by eq. (160)

Nusselt number pressure integrals definedby equations (94), (95), and (96)

integrals defined by eq. (119)

Prandtl number radius from axis of axisymmetric configuration or distance from datum planeof two-dimensional configuration laminar recovery factor rL

4 rT turbulent recovery factor Re transition Reynolds number, (ue/ve)xt Xt unit Reynolds number,u/v

Reynolds number,( ue/ve) 6*

Reynolds number of length of transition, (ue/ve)~x

reference Reynolds number, (ueo/veo)

enthalpy-Mach number parameter in transition correlations, eqs. (120), (121), and (122) S enthalpy function, (H/He) - 1

st Stanton number,kw (aT/ay) Jp u c [Taw - T"] I eep transverse curvature factor (eq.(28)) temperature velocity components in physical coordinates parallel and normal to surface, respectively

due

1 /2 U* reference friction velocity, (rJpw) + U u/u* u friction velocity, (r/p) 1/2 7 velocity components in Stewartson plane velocity parameter in eddy viscosity, defined by (138)eqs. and (143) normal velocity in Dorodnitsyn plane, (V/Ue)fio - v + iiq(be/Ue)

X,Y curvilinear coordinates in physical system

X' axial or reference coordinate (fig. 1) AX i integration step size

I coordinates ,of the Stewartson transformation,

Y+ P"U*Y/PW a angle of inclination of surface relative to referenceaxis (fig. 1) - a exponent in transition length correlation, eq.(150)

eddy-viscosity parameter,1 + E/F

Y ratio of specific heats r intermittency function intermittency function defined eq.by rB (161)

6 boundary-layer thickness

6* displacement thickness kinematic displacement thickness

E absolute eddy viscosity

E eddy conductivity H

E+ .. 0 momentum thickness .. enthalpy thickness h source function' parameter in intermittency (147))'(eq.

CL absolutemolecu,larviscosity ..

V kinematic viscosity, p/p coordinates in the Dorodnitsyn plane,

P density ul, uz, us error functions defined in eqs. (65), (66), and (67) -

6 functionsdefined by eqs. (98) through (103)

shear stress (mi)/aq streamfunction

Subscripts

aw adiabatic wall e refers to the boundary-layer edge

L refersto laminar flow

0 initialcondition at x.

stagnation condition

transition refersto turbulent flow condition at the wall free-stream condition

Special Notation

denotesa time averagequantity

denotesvariables in theProbstein-Elliott transformation

denotesdifferentiation with respect to 4 denotesa fluctuating quantity

THEOFU3TICAL BASIS

Themethod of integral relations used herein is due to Dorodnitsyn (ref. 5). Briefly,the mathematical procedure employed is that of representing the solution of a differential equation byan infinite series. An approximation to the solution is developed by taking a partial sum of theinfinite series. A discussion of some of thepertinent mathematical theorems associated with the application of the method to the partial

7 differential equations of the incompressibleboundary layer was presented by Murphy and Rose inreference 6. Inthe present analysis, the method is generalized to include compressible flow with arbitrary Prandtl number.

Since the present analysis is considerably more generalizedthan has beenpresented previously, the derivation of the important equations is presentedin detail. The basicanalysis is presentedin terms of a turbulentboundary layer since that is the most generalcase. The equations describing the transitional and laminarboundary layers differ from those of the turbulent case only in the definition of the effective, or eddy,viscosity.

Assumptions

The analysis is based on thefollowing assumptions:

(1) The governingequations are thosefor a compressibleturbulent boundarylayer. (2) The air behavesas an idealgas. (3) The molecularviscosity, F, is proportionalto the temperature. (4) The specificheat of the gas is constant. (5) The Prandtl number ofthe gas is constant, but arbitrary. (6) The conditions at the edgeof the boundary-layer are arbitrary. (7) The wall is either two-dimensionalor axisymmetric, but can have arbitrary variation in the direction of flow as longas the radius of curvatureof the wall is large compared tothe boundary layer. (8) The pressure is constantnormal to the wall. (9) The wall temperature is prescribed and may varyarbitrarily in thedirection of flow.

Boundary-Layer Equationsfor Compressible Turbulent Flow

Reference 1 presents the boundary-layerequations for compressible turbulentflow in a convenient form forboth axisymmetric and two- dimensionalflow. The basicnotation and coordinate scheme are shown infigure 1. A blunt body is shown; however, thepresent analysis applies equally well to a pointed body since the analysis is restricted to points downstream of a stagnationpoint or sharp leading edge or nose.Note also that the same symbols are used forthe physical coordinates of both

a two-dimensional and axisyrnmetric configurations. Thus, r denotes the distance of apoint from the axis of an axisymmetric configuration, or from the reference plane of a two-dimensionalconfiguration. The coordinates are a curvilinear system in which x is tne distance along the surface measured from the stagnation point or leadingedge. The dimension y is measured normal to the surface. In the differential equations, the transverse curvature terms are retained because of their potentialimport- ance in predicting the boundary-layer growth on long slenderbodies. The governing equations describingthe steady flow of a turbulent boundary layer are:

CONTINUITY

MOMENTUM

9 The boundary layer is assumedbe tothin and the terms such asp, u, H, and x are assumed to be-ofthe order of 1 and v and y are of the order of the boundary-layer thickness,6. This allows some of the correlation terms involving the fluctuation "- quantitiesto be neglected. - Specifically, the double correlation terms p'u', u'H', p'H', and p'v' are of the order of 6 at most, and the triple correlation terms p'v'u' and p'v'H' are of the order of- h2 at most.- Away from separation, the Reynolds normal stresses -puI2 . and -pv" are small and can be neglected. The Reynolds shear stress -pu'v' is eliminated through useof the definitions

and > B=l+-E c1 J The term pv'h' is eliminated through the use of an eddy-conductivity concept as follows: the total enthalpy is defined

1 H = h + h' + 7 [u' + v2 + 2uu' + 2w' + um2+ vI2]

As before, the terms ut* and v'' are assumed to be of at least second order so that the total enthalpy can be expressed

H = E+ H' where

- 12 H=h+-u2

and

H' = h' + uu' + vv'

Then

"- v'H' = v'h' + UU'V'

10 or

If aneddy conductivity is introduced so that

then

Or, defining a turbulentPrandtl nurrber as

- E PrT - -E H

gives

Laufer(ref. 7) haspointed out that some ofthe higher order correlation terms shouldprobably be retained in the energy equation for hypersonicflows. However, inthe absence of reliable experimentalor theoreticalinformation, an appropriate model forthose terms cannot be developed at the present time.

In thepresent analysis, the turbulent Prandtl number, PrT, has been assumed to be constantexcept in the transition region, where it is defined as

Prt = 1 - r (1 - PrT)

where r is a modulationfactor for the transition region and will be discussed in detail in a subsequentsection.

I Neglecting higher-order terms and incorporating the eddy-viscosity and eddy-conductivity concepts, the differential equations(1) to (3) become

CONTINUITY

& [rkpu] + aya [ rkpv] = o

MOMENTUM

ENERGY

aH PU + PV

Pr where the ten pv is a time-averaged quantity defined by

" pv = pv + p'v' (10)

In solving the differential equations,is it convenient to define a new variable

S=H-1 He with which the energy equation (eq.(9) ) becomes

where He is assumed to be a function of x.

12 Equations (7), (8), and (12) are easily applicable to laminar and transitional flow. In laminar flow, substitution of 6 = Prt - 1 reduces the equations to those for a laminar boundary layer. Further, suitable variation of the eddy viscosity and turbulent Prandtl number makesthe equations applicable to the transition region. The boundary conditions for this systemof equations are: At y = 0: u=v=o r = rw

s = s,

At y = m: u = ue(x)

au/ay = 0 v = ve (x)

s=o

At x = xo:

Transformation of Axisymmetric Boundary-Layer Equations to Almost Two-Dimensional Form In order to put the axisymmetric equations (k = 1) into a more convenient form, the Probstein-Elliott transformation (ref. 8) is applied. The coordinates of the axisymmetric bodyare shown in figure l(a). The Probstein-Elliott transformation is

dy = dy where rw (x' ) is specified by the body shapeand r (x,y) is given by

r(x,y) = rw(x') cos+ y a (15)

13 with

XI 1/2 x = [l + (drJd~')~] dx' 0

A stream functionthat satisfies the continuity equation, equation (7), is defined by

+rpuk

Let - - *=-A L u=u v-v

Then the continuity equation with -& pu = a? and

has the form

Applyingthe transformation to the momentum and energyequations (8) and (12) yieldsthe transformed equations

14 - p.(y)x

where

A=---1

B=P"@-+ "-1 1 Prt Prt Pr

and t is the transverse curvature factor

cos a t= 2L 2 xW

and

r - W cos a Y=LY+TY2 (29)

For flows in which the transverse curvature termsare negligible, letting k = 0 in equations (24) and (25) produces the equations of 8 two-dimensional boundary layer. The transverse curvature termsmay be negligible for an axisymmetric flow if the body radiusis lazge cornpard to the boundary-layer thickness. The Probstein-Elliott transformation is thus a first-order correction of the approximate equations forthe effect of transversecurvature.

Transformation of the Compressible Boundary-Layer Equations In order to further simplify theequations, two additional tranm- formations are applied. The Stewartson transformation (ref. 9) reduces the equations to a set of equations for an incompressibleflow. The Dorodnitsyn transformation (ref. 5) removes the explicit dependence on the molecular viscosity. The following analysis is presented in terms

15 - of the Probstein-Elliott coordinates,x- and G, with- the understanding that in the two-dimensional casex = x and y = y. Stewartson transformation.- In the Stewartson transformation the following variables areintroduced:

and

a e u=- OU a e

With these the boundary conditions become:

At = 0: u= v= 0 s=s W

At = m: U = ue = ae Me 0 au/ay = 0 s=o

At X = Xo: u = U0(Y)

s = S0(Y)

It is assumed that s, He, and the eddy-viscosity parameter, 6, transform directly. That is,

16 With the above definitions, the derivatives become

and

Then with the definitions

and

it is easily shown, usingrelations (31) through (39) and theperfect gas assumption along with the relations

and

(41) that the boundary-lay erequations in the Stewar t s on planeare:

CONTINUITY

17 u-+vg=ax&J (S+1)Ue=

a 1 +- (43) e

ENERGY

CVe & [(l + kt9)A - 0

'e 'e .a2 +c- - - He a 2e dYa [(l + ktF)BU ayau e 1 0

The derivatives dT /di? and dHe/dz are not transformed because He se and Tse are assumed to transform directly. Also, since He(?) is a boundary condition and He and are related by Tse nothing is gained- by transforming their derivatives. Likewise, the coordinate y is not transformed in the transverse curvature terms because integration of the equations across the boundary layer is anti- cipated and only corresponding values are needed in those terms. The Chapman-Rubesin constant,C, is assumed to be at most a function of x. In the work presented herein, C is evaluated at the wall temperature. That is,

c=- Tw + 198.6 1 (45) where Sutherland's lawis used to evaluate the viscosity.

18 Dorodnitsvn transformation.- The Dorodnitsyn transformation (ref. 5) converts the,viscousequations in the Stewartson variablesto a Set of non-viscous equations in variablesdefined as follows:

U 4 1’2 -v = (e) ‘e

where

At q = 00: G,,aq = o s=o

At 4 = 4,:

19 As inthe Stewartson transformation, it is assumed that S, He, and B transformas parameters, so that

The referenceconditions in relations (30) through (32) and (46) through (48) havebeen chosen in calculations made inthe present work as theconditions at the edge of the boundary layer at the station x. where thesolution is consideredto begin. This is a somewhat arbitrary choice,but is found to be mostconvenient. The referenceconditions could be those at any point related isentropically to the boundary-layer edge at xo. The referencelength 1 is completelyarbitrary. However, notethat for anaxisymmetric body, the reference length J? inthe equivalentplane must be related to thelength' L inthe physical system bythe transformation relation

From relations(46)

20 The application of relations(46) through (49) and (57) and (58) to equations (42), (43), and(44) is straightforward. The resultingequations are:

CONTINUITY aii a; ag+q--0 (59)

MOmNTUM

ENERCX

- K dHe " - (S + 1)U He dZ where

=y-l e 2 M2e

and

Me ' Peoae 0- 0 Pea, Me

and it hasbeen assumed that

-1 -=-dTS el -dHe TS dZ He dZ e by virtue of the assumption of constant specific heat.

21 Derivation of Ordinary Differential Equations Development of integro-differential eqgations.- The partial differential equations (59) , (60) , and (61) arereduced to a set ofordinary differential equations by the introduction of certain approximating functions, First, it is convenient to define a function

- Then, if @,S, and w areconsidered -to be representedby approximating functions in terms of thevariable u, equations (59), (60), and (61)can beexpressed as

- as - as A @u-+w--C--x aii

2Cme - 1 + me [(I + kt?) + -dHe - "1 He dx @(S + 1); = o3 (67) a; @ where oZy and u3 differ from zeroby virtue of theuse of approxi- al, - matingfunctions to represent @,S, and w. The exact formof the approximatingfunctions will be examined later. If the errors Y 02' and o3 arerequired to be orthogonalto three sets of linearly independentfunctions fiy gi, and hi' respectively,then

f fro, du = 0 i = 1,2,...,n (68) 0

22 dE- 0 i = 1,2, ....n (69) 0

{ his, dii 0 i = 1,2, ...,n 0

Requiring that

and

and addingequations (68) and (69) gives,after some preliminarymanipulation,

1 -2 a (f:ii) diZ + I a (fzv) d; - - @fz'(S + 1 - u ) du 'e 0 0 0 aii

Note that the semi-infiniteinterval in the independentvariable q has been transformedinto -the more convenientfinite interval [ 0,1] for the independentvariable u.

23 Following the same procedure as forequation (72) it is required that

fi = sf;

gi = Sfi (73)

hi = f * i

Then addingequations (68), (69), and (70)gives

du = 0 (74) 0

The integralsin equations (72) and(74) can be evaluated in a straight- forwardmanner ifthe condition fz(1) = 0 is consideredto be satisfied by requiring that

and fz ' is requiredto be bounded as E approaches 1 (7 approaches m) . Equations(72) and (74)then reduce to two integro-differentialequations inthe dependent variables @ (4,IT) and S (4,ii) , namely

24 and

2 k' - @Sf:= d; - -'e (S + 1 - ii )@Sfi du ue j 0 0

+ C [ (1 + kt?) ([sf;" + f;' "3 + 0 aii

- +-K dHe (S + l)@f:E - ($ - .'> Sf:'@ dE = 0 (76) He dx

Note thatequations (75) and (76) were obtainedby linear combinations ofthe equations (68) , (69), and (70)using particular forms for the weightingfunctions fiy giyand hi. It is clear thatthe conditions (68) , (69), and (70) are notnecessary conditions for equations (75) and (76) to be valid.In particular, this procedure offers no information * toaid in selecting either the weighting function - fi or the forms of the approximatingfunctions for @,S, and w. These functionsmust be determined on the basis of the boundaryconditions and other conditions the functions must satisfy.

25 Selection of approximating functions.- The choice of functionsto represent the variables 9 and S is not arbitrary. The functions must satisfy the boundary conditions as well as certain compatibility relations for boundary-layer flows. Specifically, the boundary conditions are:

At = 0: @ = finite s = sw

f: f: = bounded

At u = 1: @ = W' s=o * fi = 0

The Weierstrauss approximation theorem applied to (differentia the 1 equations (65), (66), and (67) alone, that is, without boundary conditions, - suggests that the functions @,S, and w could be approximated uniformly by polynomials on the closed interval [0,1]. However, caution mustbe used when applying the Weierstrauss theorem to cases with boundary conditions. In the present case the condition on @ at = 1 cannot be satisfied by a polynomial alone. Furthermore, the condition that S be - exactly equal to zero at u = 1 is a stronger condition than continuity as required by the Weierstrauss theorem,so that uniform convergence can1;o-L necessarily be guaranteed if the function S is represented by polynomials. This is especially important in the transitional and turbulent regions as willbe demonstrated subsequently. Velocity-gradient function, @:- To avoid the difficulty of the infinite condition on the function @ at u = 1, Dorodnitsyn (ref. 5) introduced a profile of the form

In references 2, 3, and 4, it was shown that a three-parameter(n = 3) function could adequately represent both laminar and turbulent profiles except in the vicinityof a separation point. At separation, the polynomial

26 formulationof equation (77) is inadequatefor any order. The three- parameterformulation was used in the present study and was found to

representadequately transitional profiles as well aslaminar and turbu- ' lent profiles.

Enthalpyfunction, S: Inprevious applications of the method of integralrelations to boundarylayers (refs. 2, 3, 4), the assumption was

made that the Prandtl number is unity. Under thatassumption, the energy , equationtakes the same form as the momentum equation and the total temperature distribution in the boundarylayer is a linear function of the normal- - izedvelocity, u, so thatthe dependent variable S can be representedby

However, as shown bySchlichting (ref. lo), fornon-unity Prandtl number, thevelocity and thermalboundary layers are not congruent. For a laminarboundary layer on a flat plate, with constant pressure along the plate,Schlichting shows thatthe temperature in the boundary layer is

This means that the enthalpyprofile parameter, S, is

r rr- -11 1 L - m E] e e

In a turbulentboundary layer, Pinckney (ref. 11) has shown thatthe temperatureprofile is a fourth-orderpolynomial. Converting Pinckney's static temperature function to the present notation yields

lme 2

+'Z l+me ue

where K is determinedby requiring the total energy deficiency of a boundarylayer, obtained by integration across the boundary layer, to be equal to the energy removed from theboundary layer upstream of the local station by radiation,surface heat transfer, or any other means. Inthe present work, theenthalpy function was representedby

27 r m 1 S = (1 - E) ISw + EiEil L J

This allowscontinuous calculation of the enthalpy profiles in laminar, transitional, and turbulentflows. For laminar flow on a cooledflat plate, the formulation was found to convergeuniformly so that an adequate approximation was obtained with a profile ofany order greaterthan rn = 2. Forturbulent flow, the combin.ation ofan approximate eddy-viscosity - function and thecondition at u = 1 wasfound to causepolynomials of ordergreater than m = 2 todevelop waveswhich rendered the polynomials inadequate to representthe variable S. The detailsof a studyof the best order €or thepolynomial in equation (82) will bepresented in a subsequentsection.

In summary, thefunctions chosen to represent the variables @ and S were

and

where the values n = 3 and m = 2 wereused for most of the work reported herein.

Sedectionof weighting functions: In order to assurethat the equa- tionsto be derived from equations (75) and (76) arelinearly independent, theweighting functions fi, gi, and hi must befamilies of linearly - independentfunctions. Furthermore, the €unctions must be bounded as u approaches 1. The functions f: chosen for thisanalysis were the same asthose used in reference 2, namely

i-1 ” f; = (1 - u)u

28 Develobent of ordinary differential equations.- Substitution of equations (77) , (82) , and(83) into equatio,;s (75) and (76) results in a set of six ordinary nonlinear differential equations in the six unknowns Ci(4) and Ei(E).' Since the Ci and Ei are functionsonly of the independentvariable, e, the resulting equations may be expressedin terms of the independentvariables, x and ii. Thus,with expressions (13), (361, and (571,

and

The differential equations to be solved are then

r 1 dC, and.

dE [aij] = B; where

i = 1,2,3,...yn a =- 1 iji+j j = 1,2y3,...,n

rn 'k i = 1,2,. ..,m a'ij (i+j+k)(i+ j+k+l) (89) j = 1,2, ...,m =xk= 1

29 l+me dH + 1 +[t dx 2 He dx and

- C[(i - l)Qi,l- iQi] - C(i - 1) [(i - 2)Ri-2 - iRi-l])

Ek + f [(i+j + k)(i + j + k+1) k=1 in which

30 Pi = J+ (1 + kt?) 6 -i-lU d~ = Fi du -@ [ (94) 0 0

Ri = f SFi dii 0

R, = L/ve where u e . 0 0

I (99)

I' Nusselt Number In this work, the Nusselt number is computed from the Stanton number as

Nu = (Pr)(Rex) St

Displacement Thickness

ee 0

Momentum Thickness

Enthalpy Thickness

m m m e

1= 1

(eq. (108) continued on next page)

34 Velocity Profile - In the two-dimensional system or the Probstein-Elliott system, the y coordinate is given by equation(97), and for axisymmetric flow, the physical coordinate is

1 /2 r W cos a cos a

Velocity Gradient

If k = 0, then L = 1.

Mach Number Profile

Temperature and Density Profiles

35 Initial Conditions

Inorder to solvethe differential equations (86) and (87), initial valuesof Ci and Ei must be specified. The requiredvalues can be determinedin several ways. The ways thathave been found to be most usefulare discussed in this section.

Similaritysolution.- For laminar flow on a flat plate at uniform temperaturewith constant pressure, similarity is known to exist. Solutionof equation (86) under those conditions yields

1/2 Ci = Ki (+)

where Ki is a constant. So theinitial values of Ci aresimply

1/2

‘i 0 = Ki (:)

For thelaminar similarity case, the equation (86) does not depend upon the Ei. Therefore, it is easyto show fromequation (87) that for the laminar similarity case

dEi

” dx -0

and equation(87) can be expressedas

where

n - Cf ‘k + 2 (0) Aij- 1 (i + j + k)(i+ j + k + 1) C,P~ 6jl k=1

and Bi = Sw cf2(0) 'k - CfZ' (0) - (i+ k) (i+ k + 1) f Cl k=1

where

1 for j = 1 { 0 for j > 1 and

Solutionof the equations (116) for the values of Ei is accomplishedby substitutingthe values

Ci = Ki and

intoequations (117) , (118) , and (119) . The values Ki havebeen determined by computing the solution to the equations(86) from arbitraryinitial conditions. The similarsolution is obtainedasymptotically.' The resultsfor n = 3 are

K, = 3.006443 (C) 1'2

- 37 K, = -3.041346(C) 1/2

K, = 0.7048994 (C) 1'2

Th,e same procedure as used toobtain Ki in thelaminar case can also be used to compute initialvalues of Ci and Ei for a fullyturbulent boundarylayer on a flat plate or for higher orders of approximation (n > 3) . Thisprocedure was discussedin reference 2. Direct fit of known velocity and temperature profiles.- If the ~~ velocityprofile, y versusu, - and thetotal temperature profiles can be known atthe initial station, the required values of Ei areeasily obtainedfrom equation (82) by substituting the values of S and ii at m pointsin the boundary layer at thatstation. Then, therequired valuesof Ci areobtained by substituting the n valuesof y corresponding tothe n valuesof ii intoequation (97) and solvingthe resultingset of n algebraicequations simultaneously. Note, however, thatfor anaxisymmetric configuration, the coordinate - y must be transformed tothe Probstein-Elliott value, y, before equation (97) is used. It must be notedthat this technique depends on thechoice of the pointsat whichvalues of ii and T areobtained in the boundary layer. Thus, ifall points are near the edge ofthe boundary layer, a different fit may be obtained than for points near the wall.

Known boundary-layerparameters.- A third methodof obtaining initial valuesof Ci and Ei €or a three-parameter(n = 3) velocitygradient profile depends upon knowledgeof theparameters Cf, St, 6*, and 9 plus - knowledgeof thetotal temperature at (m - 1) valuesof u. The skin- frictioncoefficient, Cf, is used toobtain C, from equation (104). Then theparameter E, is obtainedfrom equation (105) withthe known valueof the Stanton number. The (m - 1) valuesof Tt arethen used inequation (82) toobtain the remaining Ei. Finally,the equations (106) and (107)are used to obtain the remaining two parameters, C2 and C3.

This method canbe used for any orderof approximation on theenthalpy profile, but is limited to a three-parametervelocity-gradient profile.

Least squares fit ofvelocity and temperatureprofiles.- A fourth method of obtaininginitial conditions consists of solving the the required

38 - - parameters from known distributions of y versus u and S versus u by the method of least squares. This method was used in the of work reference 2. The method was not used in the present work, but is outlined here to generalize the presentation. Least squares fitting of the enthalpy function is straightforward. The residual equations are written inform the

and the coefficients Ei are obtained by solving the set of equations given by

where m' is the number of data points. Fitting of the velocity profiles is a more complicated process. First the- datamust be expressed in termsof a distribution of aii/aq versus u. That is,

1 - Gi = n

j=l

The residual equations are then written as

"3 -1 (%li 2 1 -ii = 3=1 - i Ri n n

39 An iteration procedure is then usedto compute the required values ofC 1- First it is assumed that

- for all ui and the C are computed by least squares. Then these C j j are used to compute the denominators of the residual equations and new values of C are obtained. This is repeated until successive iterations j yield the same solution within a desired degree of accuracy.

CALCULATION OF TRANSITIONAL BOUNDARY LAYERS

As discussed previously, the only thing that distinguishes between laminar, transitional, and turbulent flow in the present formulation is the parameter 0 in the dissipation integrals Pi, Qi, and Ri (eqs. (94) , (95), and (96)). Thus, the requirements of calculating transitional flow are (1) knowledge of the point where transition begins,(2) a function representing the turbulent eddy-viscosity 0, (3) a modulation function to cause the parameter 0 to go smoothly from its laminar value of unity to its fully turbulent distribution, and (4) knowledgeof where transition ends. In this section, the methods chosen for treating the requirements for calculating transitional boundary layers willbe discussed. For greater detail regarding the theoretical aspects of boundary-layer stability and transition, the reader is referred to the many excellent references on the subject. In particular, the report by M. V. Morkovin (ref. 12) is an excellent reviewof all aspects of the transition problem and contains an extensive list of references for the reader desiring detailed knowledge of specific topics. The report byL. Mack (ref. 13) is, to theauthor’s knowledge, the most current and comprehensive studyof laminar boundary- layer stability theory.

40 Prediction of the Transition Point At the present time, the transition point becannot predicted with confidence by purely theoretical methods. While sophisticated methodsof solving the equations describing the stability of the laminar boundary layer have been developed in recent years (refs.13, 14, 15) these methods are unableto predict the point where the growing linearized disturbances characteristic of boundary-layer stability theory become large enoughto trigger the breakdown which creates the transition region.A promising method of approach was taken by Donaldson 16)(ref. who devised a model of the transitional boundary-layer two-dimensional in the mean flow but three-dimensional in the disturbance flow. The model was apparently capable of predicting an onset of transition boundary for the case of fairly large disturbances introduced into a flat plate boundary layer. However, although the method appearsbe toan effective way of studying the transition process, it has not been developed into a general engineering tool. Purely empirical methods are also limited in application due to the myriad of factors affecting the transition. Some of these factors are Reynolds number, Mach number, unit Reynolds number, pressure gradients, nose bluntness, surface roughness, free-stream turbulence level, angle of attack, and radiated aerodynamic noise. In a recent paper, Morkovin (ref. 12) indicates that even thougha large amount of experimental transition data exists much of the informationhas not been recorded in sufficient detail to allow the separation of the effectsof all the different parameters. For this reason, the approach usually taken €or engineering applications is to try to correlate data with certain parameters or of groups parameters. Even though limited in scope depending upon the parameters chosen for the correlation, this method at leastcan provide a first approximation over a reasonably wide range of parameters. Definition of the transition point is difficult since experimentally transition takes place over a finite region and different kinds of data produce different indications of the point of transition. Optical techniques such as schlieren photographs indicate a point of transition which is somewhere near the middle of the transition region (ref. 17). Wall temperature or recovery factor determinations indicate a peak value which is also considered by Brinich (ref.18) to correspond approximately to the mean pointof the transition zone. The only quantities which yield well-defined points near the beginning of the transitionzone are the

41 Stanton nuniber, the skin-friction coefficient, and thepressure indicated by a pitotprobe placed near the surface (surface pitot pressure). Those quantitiesexhibit maxima andminima. The minimum ofany of those quantities occursnear the point of onset of transition. It is not clear, however, that the minimum values of each of the three quantities would necessarily occur at the same point.

It was found that the calculatedboundary-layer quantities, skin- frictioncoefficient, Cf, and Stanton number, St, tended to developthe same formof variation in the transition region as the imposed intermittencydistribution. A well-defined minimum is obtainedfor both quantities. On theother hand, the minimum ofthe surface pitot pressure distribution is less well defined. The surfacepitot pressure is a quantitywhich depends on the velocity profile in the boundary layer. Becauseof this, the surface pitot pressure distribution depends to some extent on the size ofthe probe relative to the boundary layer, andon theposition of the probe within the boundary layer. In addition, the possible effect of the probe on starting the transition cannot be dismissed.For these reasons, correlations based on measurements of sur- facequantities such as Cf or St arebelieved to be the most reliable €or predicting the onset of transition.

Inthe present work, twomethods areused for estimating the location ofthe transition point. These methods are (1) correlationsbased on collection of experimental data over a broadrange of experimental conditions and (2) useof the experimental location of transition. When the transition point is determinedby one of these methods, it is used as a directinput into the analytical solution. The remainder of thissection will be devoted to discussion of correlations for the transition Reynolds number.

In recentyears, several correlations have been developed for predictingtransition Reynolds nunibers. Some ofthese are describedin references19 through 23. Pate and Schueler(ref. 19) and Pate (ref. 20) showed that data on flat plates and cones in wind tunnelscould be correlated using parameters characteristic of the boundary layer on the wind-tunnel walls. However, thatkind of correlation is restrictedto use in wind tunnels and hencecannot be extended to predicting transition on models in a ballistic rangeor in freeflight. Deem et al.(ref. 21)

42 developed a correlation for sharp and cylindrically blunted flat plates at zero angle of attack based on Mach number, unit Reynolds number, leaaing-edge'bluntness, leading-edge sweep angle, and wall-to-adiabatic- wall temperature ratio. Their correlation, applicable onlyto wind tunnels, predicted the Reynolds numberof the end of transition withina factor of less than 2 for most of the data-used. This correlation was developed into a set of charts by Hopkins, ahd Jillie, Sorensen (ref. 22), who pointed out that the correlation can provide only a first approximation of the transition Reynolds number expected for flight vehicles. Correlation parameters developed atNASA Langley Research Center by Bertram and Beckwith (ref. 23) were to able predict the onset of transition Reynolds number for sharp-nosed cones within a factor of 2 when wind-tunnel, ballistics range, and flight data were considered separately. Researchers at Langley Research Center have recently developed an improved correlation for the onset of transition on sharp cones in wind tunnels and ballistics ranges. The specific relations for these correlations willbe presented subsequently. The correlation relations to be presented in this report will be directly applicable to sharp-nosed cones or sharp-leading-edged flat plates or hollow cylindezs. The relations may be used as first approximations for determining the transition point on other configurations with sharp noses or leading edges. The effects of nose or leading-edge bluntness and pressure gradients have been studied by a number of investi- gators (refs. 24, 25, 26). However, a completely general correlation accounting for all the various parameters affecting the transition point does not exist at this time. The correlations recommended for use with the present theory are as follows: Sharp Flat Plates

2c0.95 7- 0.167 s'J 1 0.6 10 Re =-R ~- Xt 6.094 MZ + 1.22 -

43 where

This relation was obtainedby dividing the expression given in reference23 for sharp cones in wind tunnelsby 3. A sample of the successof the relation is shown in figure2 where the predicted value of Re is Xt compared with experimental values from different sources (refs.18 and 27 through 32 and unpublished data from LRC).The prediction is within a factor of 2 €or most of the data. Sharp Cones Wind Tunnels and Ballistics Ranges*

n' [F + GS' + I(s')~] Rex = Re, 10 t where n', F, G, and I depend on the experimental environment as follows: Facility -n' -F -G -I Wind Tunnel 4.5158 0.4 -0.29861 0.027300 Ballistics Range 0.6 2.5955 -0.13680 0.014578

Free Flight (ref. 23)

3 0.6 102 [1-32 + 0.130 S'J = Rex -2 Re, t 6-09, MZ + 1.22 >Ie

For estimating the transition Reynolds number on blunt leading-edge flat platesor hollow cylinders or other two-dimensional or open-nosed axisymmetric configurations, the reader is referredto the charts of Hopkins, Jillie, and Sorensen (ref. 22).It must be remembered, however, that those charts referto the end of the transition region.A correlation for estimating the length of the transition region will be discussed

~~ ~. . . :c - - * The author gratefully acknowledges the cooperationof Mr. P. Calvin Stainback of LRC in providing this correlation relation.

44 in a subsequentsection of thisreport. Forblunted cones or other closed- noseaxisymmetric configurations, the work of Stetson andRushton (ref. 25) andMoeckel (ref. 26)and Zakkay and Krause (ref. 33) may be helpful. Some effects of nose bluntness on cones were studied experimentally by Stainback(ref. 34) and bySoftley, Graber and Zempel (ref.35) . However, the generalityof the results is uncertainsince each investigator examined flowover a limitedrange of Mach numbersand Reynolds numbers.

Turbulent Eddy-Viscosity Model Severalauthors have developed eddy-viscosity models for use in solvingthe turbulent boundary-layer equations. Some of these models are describedin references 1,.2, 36,37, and 38. Several models are compared inreferences 36 and 39. The criteria usedfor choosing a particular model for the present work were that the model yield results whichagreed reasonably well withthose of other models and that the model formulation beas simple as possible. The lattercriterion dictated that the best model for the present theory would beone which could be described in - termsof the independent variable u ratherthan the boundary-layer coordinatey. This makes calculation of theactual velocity profiles at eachintegration step unnecessary except at stations wherethey are desiredfor output, or in cases where transversecurvature is being accountedfor, thus maximizing the speed of the computer program.

The eddy-viscosity modelemployed in the present work is a two-layer model based on the work ofKleinstein (ref. 38) and of Clauser(ref. 40). Kleinsteinderived an expressionfor the eddy-viscosity of the inner layer ofan incompressible boundary layer (the laminar sublayer and the law ofthe wallregion). Clauser proposed that the eddy viscosityin the outer layer was constantat the value

6 = 0.0168 Reg* for anincompressible boundary layer.

45 Inner Layer Eddy-Viscosity Model Extension of Kleinstein's model for theinner layer to compressible flow is straightforward. With the definitions

PWU* y+ = - I, Y

theturbulent shear stress

8U -T = (w + PE) ay where it hasbeen assumed that

aU -pu'v' = ay becomes

V (1 + E+) 7au+ u*= r=PC 8Y from which

7=P'2- (l+€)-+ au+ Tw pw vw aY+

46 and

For incompressibleflow, Kleinstein found that Prandtl's mixing length theory gave

E+ = k: (y+2du+ ) dY+

For compressible flow, the assumption is made that the fluid properties do not change significantly in a distance dy. The major- contributions to the turbulent shear then come from the velocity correlation u'v'so that with the mixing length hypothesis E = k:y2 (du/dy) and the definitions (124) through (127), the expression (134) becomes

For large distances from the wall such thatE+ is large comparedto unity, equation (133) can be expressed

so that with equations (135) and (136)

+ + vw E = k,y uT y-

Retaining Rleinstein's definition, a new variable is introduced, namely + .=/u du'+ U 0 T where u+ can ba variable due to variationsof both the turbulent shear 7 and the density. Following Kleinstein, it is easily shown that the turbulent eddy-viscosity can be represented by the expression

47 where the values 0.41 and 7.7 areused for k, and k2, respectively. With the conditions

y = 0:

y = 6: T= 0

-= 0 aY

it canbe shown that the variation of the shear stress in a boundarylayer can be represented by

Foran equilibrium turbulent boundary layer in which dp/dx = 0 and

y -7 "-u- 6

it is easy to see that the shear stress can be assumed to be constant for theinner layer calculations, since values of y/6 at which significant - deviationsof r from rw occurrequire values of u nearunity, corresponding tothe outer layer of the boundary layer. Of course,flows with significantadverse pressure gradients may requirethe full polynomial expression.In the present work it was alwaysassumed that r = T W throughoutthe inner layer. This assumption is the"Prandtl hypothesis" which results in the classical lawof the wall.

The Kleinstein model of the inner layer eddy viscosity is conveniently - expressedin terms of u as follows. Since

48 it follows that

In the computerprogram written for the present analysis, the expression -(143) is evaluatedusing simple trapezoidalintegration for 11 valuesof U.

Outer Layer Eddy Viscosity Herring and Mellor(ref. 36)extended Clauser’s model tocompressible flowby defining the eddy viscosity in the outer layer as

where 6; is calledthe kinematic displacement thickness defined by

6; = [ (1 - E) dy 0

Otherauthors (refs. 1 and41) haveused an intermittency function in the outer layer to causethe eddy viscosity to decay to zero at the edge of the boundarylayer. The majoreffect of such an intermittencyfunction is confinedto the region ii > 0.9. The intermittencyfunction is thusof secondaryimportance in calculation of the boundary-layer solutions. For thisreason, no intermittencyfunction was usedin the outer layer in the present analysis.

A9 Modulation of the Eddy Viscosity in the Transition Region The functionalform of theea'dy viscosity in the transition zone is defined in the present work as

where r(x) is called the intermittency. The intermittency must satisfy =. = = the conditions that r 0 when x xt and r 1 when x is far downstream of xt. In the present work, the intermittency is specified in one of two ways: (1) an analytical expression derived from considera- tions of the probability of the flow being turbulenta given at statior or (2) an empirical distribution. The analytical function chosento represent r(x) is the Gaussian distribution derived by Emmons (ref. 42). In this analysis, the function is expressed as

-1 (x-x,) r=l-e

In the present analysis, the parameter h is determined by calculating the length of the transition region and requiring thatI' = 0.95 at the point of maximum skin friction or heat transfer. This process will be described in greater detail ain subsequent section concerned with correlation of the length of the transition region. Emmons derived the intermittency function after observing turbulent spots form ina water flow. He introduced a source-density function (related to the parameterh) to describe the production of turbulent spots, and showed that the probability of the flow being turbulenta given at point is the intermittency factor,r, given by equation (147). The existence of turbulent spotsa boundaryin layer has been confirmed ina water flow by Mitchner (ref.43), by Schubauer and Klebanogf (ref. 44) in low-speed air flows, and by James (ref. 45) in air flows at Mach numbers from2.7 to 10. Emmons' assumption that the spots grow independently of each other was confirmed by the experiments of Elder (ref. 46) . Dhawan and Narasimha (ref. 47) used theirown and Schubauer and Klebanoff's (ref. 44) experiments to show that Emmons' source density function couldbe best represented bya delta function inx, implying that the spots originate essentially alonga single line transverse to the

50 flow.Nagel (ref. 48), inextending the theory to hypersonic flow, showed that a key factor in the source density function is the frequency of spot formation. "hat frequency was found torequire the existence of a characteristic length which was identified as the average lateral spacing of theturbulent spots. This length was found to be related to free- stream disturbances which could be related to wind-tunnel size, model vibrations, or other external causes. The intermittency distribution across the boundary layer in the transition region is a function of the shape of the turbulent spots. Schubauerand Klebanoff (ref. 44) showed that at low speeds thespots have a nearly constant cross sectional area close to the surface and taper towardthe outer edge of the boundary layer. Owen (ref. 49) reported that the intermittency distributions across the boundary layer in the transition region were similar to those reported by Corrsin and Kistler (ref. 50) inturbulent boundary layers. That is, theintermittency varies from some maximum value near the wall to zero toward the outer edge of the boundarylayer. The intermittencydistribution in the direction normal to the wall is ofsecondary importance in relation to the streamwise distribution in determining the mean profiles and the heat transfer and shear stress at the wall. This is especiallytrue in the present analysis, since the intermittency is nearly constant across the inner part of the boundarylayer from which come the major contributions to the integrals involvedin the solution. For this reason, in this workno intermittency distribution was applied in the direction normal to the wall.

That Emmons' theory may not be generally applicable was discovered by Morkovin (ref. 12), who found that in plotting the distributions ofa number of surface variables from different experiments through the transition region, only about half of those examinedcould be fitted to the probabilitydistribution curves of Dhawan and Narasimha (ref. 47). Likewise, the presentauthor found that for some hypersonic flows the intermittency tends to increase faster toward the end of transition than the low-speed probabilitytheory predicts. One reason for this may be that some ofthe higher instability modes foundby Mack (ref. 13) become excited and contribute to the breakdown so that the source function is not adequately representedby a single line source, but should include new sources intro- duceddownstream of theoriginal transition point. In theabsence of a more eatisfactory theory of the spot distribution in hypersonic flow, the

51 simple probability theory of Emmons was chosen to be used in the computer program writtenfor the presenttheory. Provision was also made for the Ynput of an arbitrarydistribution of the function r(x). Inthis way, known deviations from the Emmons function can be accommodated in the computerprogram. In the test calculations to be discussedin this report,the experimental intermittency distributions were usedwhenever they could be clearly determined.

It is important to note at this point that the capability ofusing an arbitrarydistribution of r(x)allows the determination of theexperi- mentalintermittency distribution from data.Thus, the computer program canprovide a usefulengineering tool forstudying hypersonic transition or transition under other circumstances where the simple probability theory is inadequate.

Correlation for the Extentof the Transition Zone

Inthe present work, theparameter h inequation (147) is determined from relationsdescribing the length of the transition region. The length of theregion is definedas the distance between the minimum and the maximum ofthe heat transfer, skin-friction coefficient, or surface pitot pressure. A typicaldistribution is shown infigure 3. It was found that when theexpressions (139) and (146) wereused forthe eddy viscosity,the minimum of the calculated heat transfer and skin-friction coefficient occurred at approximately r = 0.01, and the maximum occurredat approximately r = 0.95.These values were used to determinethe point of onset of transition and theparameter h so thatif the Reynolds number of the lengthof the transition region is known, thelocation of the point of onset of transition is given by

Rex = Re - 0.061 Renx t Xmin and thefactor h is givenby

h = 2.66/(Ax) *

In most cases, due to the uncertainty involved with locating the transition point,the correction described in equation (148) is a minorone and may be neglected. The correction is necessaryif the minimum point is known .from specificdata.

52 In reference 47, it is shown with data for a limitedrange of Mach numbers thatthe quantity Re could be representedby the expression Ax

Re = 5 Rex Ax t

Potter andWhitfield (ref. 31) studiedthe transition zoneover a broad rangeof Mach numbers (2 M 2 8). They observed that theparameter Re Ax is independent of the unit Reynolds number andleading-edge geometry, and is basically only a function of the transition Reynolds number and the Mach number. That is,

Some insightinto the variation of ReAx with Mach number can be obtainedby examining the work ofNagel (ref.48), Lees and Reshotko (ref. 51), and Mack (ref. 13). Nagel showed thatthe source function in theintermittency (eq. (147)) could depend upon thefrequency, f, of the Tollmien-Schlichting wave most unstable with respect to the breakdown process.Nagel further showed thatthe Reynolds number based on the distance from the onset of transition to a givenpoint in the transition region was inverselyproportional to the square root of thatfrequency. That is,

ReAx = Rex - Rex t

Lees and Reshotko showed by boundary-layer stability theory that the most unstablefrequency was inverselyproportional to M2 forhigh Mach numbers. Thus, the length of transition Reynoldsnumber, Re Ax, would be linear with Mach number €orhigh supersonic and hypersonicflows.

Formulas for ReAx were developedby examination of experimental data. For flat plates and hollowcylinders, the data of Potter and Whitfield (ref. 31) were used.For cones, data were obtainedfrom references 20, 28, 34, and 52 through 61.The dataof Potter and Whitfield were found to suggest a form

U ReAx = f (Me)Rex t

53 The function f(Me) was assumed to be linear so that - Re = (x + EMe) Re: Ax t - The parameter a was determined as theslope of a plot of log (ReAx) -versus log (Rext) forfixed Mach number.Then - theparameters x and B were determinedfrom a plotof ReAx (Re ) -a versus Me. For cones, Xt this was a trial and error process,since the data exhibited a considerable amount ofscatter. Thus, theformula arrived at can be considered a first approximationfor arbitrary flow conditions. The valuesobtained forthe parameters are as follows: - - - Configuration -a -A -B Flat plates, hollow cylinders 0.575 237.5 61.8 Cones .6 20.0 34.5

Comparisonsof thepredicted value of Re withexperimental values from Ax severalsources are shown infigure 4. All data were obtainedin wind tunnelsexcept for one free-flightcase shown infigure 4(b). Some of the scatter of the data can be attributed to the quality and quantityof availabledata. In many instances,insufficient experimental data were obtainedto locate accurately the ends of the transition region. Also, the data shown in figure 4 wereobtained by several different methods. As for the location of the beginningof transition, as discussed previously, theobserved location of the endof transition depends upon the quantity beingobserved. Included in figure 4 aredata from heat-transfer measurements,surface pitot probe measurements, and walltemperature measurements. It is not clear that all three quantities will exhibit a maximum at the same point for a givenflow situation.

DETERMINATION OF ORDER OF APPROXIMATION FOR THE ENTHALPY FUNCTION

The selectionof the best order of the polynomial in equation (82) was doneby examining the solutions produced by various orders in both laminar and turbulentflow. The solutionswere compared withother theoriesas well as with experimental data.

54 First, itcan be shown that for a laminar boundary onlayer a flat plate, the Ei are functions only of the Prandtl number, Pr. For such a case, the velocity profiles are similar and the solution for the coeffi- cisnts of the approximating polynomial in the function@ (eq. (81)) is given by equation (113), and the solution for the Eiis obtained from equations (116). In order to evaluate the order of approximationof the enthalpy function, equations (116) were solved for values of m of 0, 1, 2, 3, 4, 5, and 6. The results areshown in figures 5 and6 for a Prandtl number of 0.72 and a wall-to-total temperature ratioof 0.3086. In figure 5 is shown the variation of total temperature as a functionof ii. "ie profiles appearto converge uniformlyso that there is only a slight difference between the curves for m= 2 and m = 3 and the higher- order curves are the same as that for= 3.m The variation of the static temperature is shown in figure6. The manner of convergence is more apparent for the static temperature than for the total temperature. The curve for m = 0 is relatively high, while that for m = 1 is low. The higher-order curves then appear to approach a limiting curve between those for m = 0 and 1 as m increases. The curve for the temperature profile discussed by Schlichting (eq. (79) ) is also shown in figure6. That curve is slightly higher than the apparent limiting curve for the present approximate profile except in t,he outer region(E greater than 0.7 in the figure) . Another test that canbe given the enthalpy function approximation in the laminar boundary layer is the value of the recovery factor. If El is set equal to Sw, then the heat transfer to the wall is zero, since

Then solution for Sw yields the adiabatic wall temperature. The recovery factor is then computed from the relation

r = 1 + Saw L (l Leme) (152)

55 where

T a .U ”-1 (153) ’aw T te

The variationof rL with the orderof the approximating funct ... ..n for S is shown infigure 7. The recoveryfactor appears to converge in approxi- matelythe same manner as thetemperature profiles shown in figures 5 and 6. As theorder of approximation, m, is increasedfrom 1, rL appears to approachthe theoretical value of (Pr)1’2 asymptotically.Fer m = 3, theerror in the recovery factor is approximately 4 percent,with the errordecreasing as m is increased.

Finally,the approximating function for S must be able torepresent theenthalpy profile in a turbulentboundary layer. In this case, the Ei cannot be obtainedas simply as for the laminar similarity case. Further- more, the Ei will depend to some extent upon theparticular model used to represent the eddy viscosity as well as upon theaccuracy with which thevelocity profiles are represented. In order to examine theeffect of theorder of approximation of the variable S inturbulent flow, the solutionfor a boundarylayer was calculatedfor m = 0, 1, 2, 3, and 4, startingin laminar flow with a similaritysolution, going through a transitionzone, and terminatingafter the boundary layer had become fullyturbulent. Both the eddy viscosity modelchosen forthe present workand that used inreference 2 were usedfor this study. Similar results were obtained with both models.

The variation of the total temperature profile at a station near the endof the transition zone forthe present eddy-viscosity model is shown infigure 8. The totaltemperature profiles for m = 3 and 4 arequite different from theprofiles for m = 0, 1, and 2. The profilesfor m = 3 and 4 display a wavinessnot indicated by the lower-order profiles. This phenomenon is explained by the fact that the equations used to calculate theprofiles (eqs. (86) and (87) ) containan approximate function, namely the eddy viscosity.Because of this, the function being represented by theapproximate series is notprecisely defined. This fact alone might onlyserve to require higher orders for convergence since the Weierstrauss approximationtheorem guarantees convergence if the functions are simply continuous. However, theimposition of the additional condition on the

56 function S at ii = 1 removes the guarantee of convergence and as the order of the approximation is increased the higher-order functions develop waves asthey try to satisfy both the boundary condition at ii = 1 and the differential equations. The importanceof the temperature profiJa approximation is shown in figure 9 where the distribution of the Stanton number in a transitional boundarylayer is shown forvalues of m of 0, 1, 2, 3, and 4. The presenteddy-viscosity model was used inthose calculations. The approxi- mationproduces a solution for the Stanton number whichappears to converge as does the temperatureprofile for m = 0, 1, and 2. Then for m = 3 and 4, a completely different solution is obtained. Convergence in the turbulent boundary layer will probablydepend on the velocityprofiles as well asthe eddy viscosity. It was found in previous investigations(refs; 2 and 6) thatthe three-parameter velocity gradient profilegiven by equation (77) is not the best for all conditions on a turbulentboundary layer. Specifically, that kind of profile was found in both references cited to be inadequate€or accurately representing the turbulentprofile close to separation. However, sincethe three-parameter profile is found to be accuratefor a widerange of conditions, its use represents a reasonableengineering compromise.

In viewof the results shown in figures 5, 6, 7, 8, and 9 and the precedingdiscussion, it is believedthat the value of m = 2 is a reasonableorder of approximation for the enthalpy function. The apparent discrepancybetween this series and theprofile ofPinckney (eq. (81)) is explainedby the fact that the present formulation is anapproximation to thetrue profile function. Some effort was expended to modifythe formu- lationby specifying the slope of the enthalpy function at ii: = 0 in terms ofPinckney's function (eq. (81)). However, such a device reduces the generality and predictivenature of the solution since knowledge of the recovery fact.or becomes a necessary part of the solution of the equations.Specification of the value of the parameter Ei and the recoveryfactor is tantamount tospecifying the Reynold's analogy factor since from equation (105)

57 It was reportedby Cary (ref. 62) that a comprehensive definition of Reynoldsanalogy is notavailable. In the absence of reliable information, fromwhich E, could be specified, it was decided to let E, vary independentlyalong with the other parameters, E,, E3,..., Em. A comprehensive search for more sophisticated approximating functions was beyond the scopeof the present effort and was notundertaken. It should be emphasized that the present formulation allows prediction of boundary- layerproperties for both adiabatic and nonadiabaticconditions. Some error may be encountered at adiabatic wall conditions since the recovery factor of thepresent theory for low ordersof approximation is about

4 percent lower thanthe theoretical value of (Pr)1’2. However, it will be shown subsequently that the theory yields predictions of heat transfer, skin-friction coefficient and other boundary-layer properties in excellent agreementwith experimental data over a wide rangeof conditions.

It is also believed that improvementof theeddy-viscosity model so that the equations being solved more exactly represent the true physical situation will help the present enthalpy profile formulation to converge for~turbulentflow as well as forlaminar flow. Until a more exactrepre- sentation of the eddy viscosity is found, it is recommended thatvalues of rn greaterthan 2 be usedwith caution.

DESCRIPTION OF COMPUTER PROGRAM

A generalizedcomputer program was written to calculate laminar, transitional, and turbulentboundary layers of a perfectgas on arbitrary two-dimensional oraxisymmetric bodies with arbitrary prescribed edge conditions and walltemperature. The programwas,based on equations (86) and (87) and their accompanying relations (88) through (103) as well as theauxiliary relations (104) through (112). The optionsfor initial conditions discussed in a previoussection were included in the program.

The basicinput data required for the computer program are as follows:

1. Referenceflow conditions, Relo, Meo, Tse , peo.These are most convenientlychosen as the conditions at the initis1 station although they canbe the conditions at any pointin a shock-freeflow. The edgepressure, peo,and theedge pressure distribution may beinput in arbitrary units since the distribution is normalizedwith respect to the reference value when calculationsare performed.

58 2. Prandtl nunibers PrL and PrT and recovery factors rL and rT.. The turbulent Prandtl number, PrT, is definedto be equal to 1.0 in laminar flow and is modulatedtb change continuously to the turbulent by value the expression

Recovery factors can be inputted arbitrarily or automatically computed as

In the transition region, the recovery factor is then computed as

rt = rL + r(rT - rL)

A. 3. Information for transition, ReXt’ Ax’ Re Optionsare included in the program for input of an arbitrary tabular distribution r ofversus x or calculating r from equation (147). If equation (147) is used, then the parameter A may be either inputted or computed from equation(149). If A is to be computed, then ReAx is required as input. 4. Distributions of the required edge conditions, wall temperature, and configuration coordinates are provided as input to the computer program in tabular form. A list of values of each quantity, pe,Tse, Me, and Tw can be input with a corresponding list of values ofx. For the configuration coordinates a subroutine is provided to compute aof tablevalues of x corresponding to the input values of x’ and rw. Interpolation for values of any quantity between the input values is accomplishedby second- order interpolation. Tables of values of derivatives of the quantities at the input stations are obtained by averaging the slopeson either side of a given input station computed using the first differences between stations. Subsequent values of derivatives at stations between input stations are computed by linear interpolation. In the present work, it was found that adequate smoothnessof the input data was easily achieved by obtaining the data from a smooth curve drawn to a scale from which three significant figures could be read. The most important requirement is that enough data

59 be input in regions of rapidchanges that the values obtained from the interpolation schemescan adequately represent the input distribution. An option is provided so that the edge conditions may be computed from isentropicflow relations. In that case, only the pressure distribution is necessaryas input. The Mach nuniber distribution is then computedfrom the relation

5. Initialconditions. Four options are provided: (1) inputof n valuesof Ci and rn valuesof Ei; (2) inputof n valuesof ii and - - y and m valuesof u and Tt; (3)input of values of Cf, St, 6*, 8, and (m - 1) valuesof ii and Tt; (4) calculationof Ci and Ei from a similaritysolution as described previously.

Outputfrom the computerprogram consists of values of x, XI, Rexy

addition,the profiles of E, y, y/6, aii/aq, dii/d(y/L), M/Me, T/Te, Ts/Tse, and f3 may be printed at theoutput stations if desired.

The integrationof equations (86) and(87) is accomplishedusing a fourth-order Adams-Moulton predictor-correctorintegration scheme (refs. 2, 3, and 63)with a fourth-order Runge-Kuttascheme used to obtain starting values. By doublingor halving the integration step size, the integration scheme is capableof some optimizationof the integration. Thus, numerical errors are kept within certain bounds by dividing the integration step by 2 whenever the error is too large and by multiplying the interval by 2 whenever lessaccuracy is necessarythan is beingattained. Because of this feature, the running time ofthe program is somewhat dependent upon the rate at whichquantities are changing. Typical run times forthe flat-plate and cone calculations to be discussedin the following sectioi, of thisreport were 1.0 to 2.0 minutesper run on the IBM 7094.The calculations described in this report were performed on the IBM 7094.

60 EVALUATION OF THE METHOD

The method of integral relations solution technique togetherwith the transitional flow structure and eddy-viscosity model discussed in previous sections of this report are applied in this sectionto a number of boundary-layer calculations. Included are comparisons with experimental measurements on various two-dimensional and axisymmetric configurations as well as comparisons with calculated solutions obtained by the method of finite differences. Cases were selected to demonstrate the ability of the method to compute boundary-layer solutions over a wide rangeof flow parameters as well as to demonstrate the flexibilityof the program for calculating boundary layers on arbitrary configurations. In all cases, air is assumed be to a perfect gas with a constant ratio of specific heats,y, equal to 1.4 and constant laminar and turbulent Prandtl numbers. Unless otherwise stated, the conditions at the edge of the boundary layer were computed assuming isentropic flow. Initial conditions fox most of the calculations were obtained assuminga similar solution from the leading edge or tonose the initial station of the calculations. In all calculations of transitional boundary layers, the velocity gradient and enthalpy function approximations employed three- parameter expressions (n = 3 and m = 2 in equations (77) and (82)) .

Incompressible Transitional Flow aon Flat Plate The first case to be examined is the data of Schubauer and Klebanoff (ref. 44). These data were shownby Dhawan and Narasimha (ref.47) to fit the probability theory of Emmons (ref.42) describing the intermittency of the transition region. Because of this, the case is an excellent one for demonstrating that the boundary-layer calculations have the correct response to the modulationof the eddy viscosityby the intermittency factor,r.

Also, the case demonstrates the applicability of the theory for low Mach D numbers.

61 The test conditions used for the calculations were:

Me = 0.072

Re, = 0.44(106) ft-'

Ts = 530°R

Tw = 530°R

= 2.31 ( Rext lo6)

h = 0.838 ft-*

Valuesof transition Reynolds number and thesource factor of the intermittencyfunction (eq. (147)) were determinedfrom the data. The configuration was a sharp-leading-edge flat plate of length 12 feet.

Infigure lO(a) is showna series ofcalculated velocity profiles through the transition region compared withthe experimental profiles. Excellentagreement is achievedfor the entire region. Likewise, the agreementbetween the experimental values of the skin-friction coefficient and thetheory shown infigure 10(b) is excellent.

CompressibleTransitional Flowon a Sharp Flat Plate

The nextcase to be examined is forcompressible flow on a flat plate. Some data were obtained from theLangley Research Center for the kat transfer to a sharp-leading-edge flatplate. The data were obtained on a model oflength 2.5 feet in the Continuous Flow Hypersonic Wind Tunnel. The tunnel Mach number was 10.39. Mach numbersfrom 6 to 10 on theplate were producedby varying the angle of attack of the plate. Due tothe rapid expansion required to produce a Mach number of 10 in the tunnel, an errorof 4 percentexisted in the static temperature and pressureat the edge of the boundary layer on theplate. This error in the temperature was acmunted for in the calculations by defining an effective totaltemperature so thatthe calculated local static temperature agreed with the data.

A characteristicof the experimental data was that all of it displayed anintermittency distribution in the transition region different from that

62 givenby Emmons' probabilityanalysis represented by equation (147). For this reason, the experimentalintermittency distribution was usedin calcu- 1at ions.

Recall that the intermittency represents the fraction of turbulence existingin the boundarylayer at a givenstation. The procedurefor determining the intermittency is somewhat iterative since the fraction of turbulencecannot be determineddirectly from the experimentaldata. To make a direct calculation of r, it would be necessary to know boththe laminar and thefully turbulent conditions at eachstation. This is clearly impossiblefor the turbulent conditions since the virtual origin of the turbulentboundary layer is unknown. Accordingly,the procedure followed was tofirst use equation (147) to compute theintermittency, then, by comparing the results of the boundary-layer calculation with the data the intermittencydistribution could be modified to fit the data. Two orthree trials were sufficient to produce good comparisonswith the data.

The results for three sets of dataare shown infigure 11 where the calculated distributions of the Stanton number are compared with the experimentaldata. Also shown inthe figures is theintermittency distri- butionused in the calculations as well as that given by equation(147). The conditionsused in the calculations were as follows:

Case 1 Case 2 Case 3

Me = 6.18 Me = 6.18 Me = 6.18

Re, = 1.32(106) ft-I Re, = 1.63 (lo6) ft-' Rel = 2 -03 ( lo6) ft-I

Ts = 1860°R Ts = 1815OR Ts = 1829OR

Tw = 549OR Tw = 552OR Tw = 553OR

PrL = 0.72 PrL = 0.72 PrL = 0.72

PrT = 1.0 PrT = 1.0 PrT = 1.0

Taw = 1526OR Taw = 1525OR Taw = 1527OR

In the reduction of the experimental heat-transfer data for presentation in terms of theStanton number, therecovery factor was assumed to be constantalong the plate at thelaminar value. In the theoretical

I calculations shown in figure 11, the same assumption was usedwith the values of rL and rT taken to be equal to thevalue required to make Taw correspond tothe value used for the experimental data.

Ineach case, it was discovered that the intermittency distribution required to fit the experimental data deviated from the distribution of equation (147) byincreasing more rapidly than that distribution near the end oftransition. A possiblereason for this was that the end ofthe transitionzone coincided with the endof theplate. The laminarheat transfer was predictedquite well without any adjustments. The solution generatedby the intermittency of equation (147) withthe parameter h determinedfrom the indicated Ax is shown in figures 11 (a) (b) and (c) forcomparison. The experimentaldata indicate that the intermittency at first increases more slowly than for the probability function but then rises to its ultimate value more rapidly than the probability function. Infigure ll(b), it is seenthat intermittency values greater than unity are required to match the data.

Transitional Flowon a Sharp Cone

Fischer(ref. 52) recentlyconducted a studyof boundary-layer transition on a loo half-anglesharp-nosed metal cone at a free-stream Mach number of7. The dataprovide an excellent case for comparison as well as fordemonstrating the flexibility of the computer program. Three cases were selectedat random fromreference 52 forcomparison. The testconditions used for the sample calculationswere as follows:

Case 1 Case 2 Case 3

Me = 5.54 Me = 5.54 Me = 5.54

Re, = 6.69(106) ft-l Re, = 7.01 (lo6) ft-’ Rel = 7.48(106)ft-l

TS = 1040°R TS = 1070°R TS = 1075OR e0 e0 e0

Tw = 551°R Tw = 556OR Tw = 548OR

PrL = 0.72 PrL = 0.72 PrL = 0.72

PrT = 1.0 PrT = 1.0 PrT = 1.0

64 In all three cases, thelaminar and turbulent recovery factors were taken to be (PrL) and (Pr,) 'I3, respectively. The experimentallocation of transition and theexperimental intermittency distribution determined in the same manner as for the previous examples were used in the numerical calculations. Comparisonsof the calculated results with the experimental Stanton number distributions are presentedin figure 12. The agreementbetween thenumerical results and theexperimental data is very good for all three test cases. Also shown withthe Stanton number distributionsare the intermittency distributions as determined to matchthe data and as given by equation (147) .

Adverse Pressure Gradient Flowon a Cone-Flare The previousexamples have demonstrated the ability of the theory and the computerprogram to calculate transitional boundary layers on flat plates and cones on which a constantpressure existed. In the next example, a case in which transition occurs in an adverse pressure gradient is considered. The data were obtainedby Zakkay, Bos, and Jensen(ref. 64)on a cone-flare modelon which a 7.5O half-anglecone was blendedsmoothly into an axisymmetric flare bodywith a constantradius of curvature. The data are an excellent case for demonstrating the ability of the theory and the computerprogram to calculate boundary layers under nonzero pressure gradients.In addition, since both the pressure distribution and the Mach number distributionare presented in reference 64, thecase is also a demonstration of the ability of the program to accept arbitrary boundary- layeredge conditions. A sketchof the configuration and thecoordinates used in the computerprogram is shown in figure 13. The referenceconditions used in the theoretical calculations corre- sponded to the conditions on thecone andwere as follows:

Re, = 4.86(106)ft-' 0

M = 8.0

Ts = 1800°R

Tw = 520°R

65 PrL = 0.70

PrT = 1.0

h = 42 ft-2

Rext = 6.88 (10')

Calculations were initialized on thecone 9 inchesfrom the nose. Initial conditions €or the calculations were obtainedfrom the similarity solution builtinto the computer program. The experimentaldistributions of the Mach number at the edgeof the boundary layer and the static pressure in the boundarylayer shown infigure 14 were usedfor the calculations. Transi- tion was assumed to begin at a. point estimated from the experimentaldata. The intermittency of the transition region was calculated by equation (147). Experimental data are presented in reference 64 for the heat transfer on the body in terms of a parameter Nu/NR where the Nusselt number and Reynolds number aredefined in terms oflocal stagnation conditions. That is ,

and

where Lz is a referencelength equal to 1 inch. The resultsof the theoretical calculations in terms of the heat-transfer parameter compare quite well with the data as shown in figure 15.

Comparison with a Finite-Difference Method

The technologyof finite-difference methods of solving the boundary- layer equations has recently been applied to solution of transitional boundarylayers by Harris (ref. 41).That technique apparently provides good comparisonswith experimental measurements. Therefore, it was

66 considered worthwhileto compare the results of a calculation made with that technique with the present theory. Accordingly, some calculations made by the techniqueof reference 41 were obtained and are compared with calculations from the present theory in figure16. The flow conditions of the calculations were as follows:

Re, = 1.706(107) ft-I

Me = 6.21

Ts = 1367OR

Tw = 584OR

PrL = 0.7

PrT = 0.9

Rex = 4.45 (lo6) t

h = 160.0

The configuration is looa half-angle sharp-nosed cone,2.5 feet in length. The calculations of the present theory we?- initialized at= x0.1 utiliz- ing the internally generated similarity solution of the computer program. The intermittency was computed from equation (147), using a value of the parameter h determined from the referenced calculations. In figures 16(a), (b), and (c), the initial velocity and temperature profiles of the two theories are compared. The initial velocity profiles show excellent agreement between the two theories. The temperature profiles show excellent agreement near the wall, but only fair agreement near the outer edge of the boundary layer. Comparison of turbulent profiles from a station downstream of the transition is presentedin figures 16(d) and (e). In this case, both the temperature profiles and the velocity profiles given by the two theories show good agreement over the entire boundary layer. In the referenced theory, the inner layer eddy viscosity was on based Prandtl's mixing-length hypothesis as described by Van Driest (ref. 65), while the outer layer was described by the same functionas used in the present theory (eq. (144)) except that an intermittency factor was used.

67 I

Cotnparisonof the two models at the turbulent station of figures 16(al and (e) is shown in figure 16 (f) . Skin-friction coefficient and Stanton number predictions for the two theories are compared infigures 16(g) and (h). The two theories are in excellentagreement in the laminar region for the skin-friction coefficient, displaying slight differences in the transitional and turbulentregions. Forthe Stanton number, a slight difference between the two theories is notedfor the entire length of the calculation. Finally, the two theories are in excellent agreement for the displacement and momentum thicknesses as shown in figure 16(i).

Calculations with Specified Initial Conditions

One of theuseful features of the present theory is that initial conditions may be obtainedin a number of ways. Fourmethods were discussed in a previoussection of this report. One ofthese methods is the similaritysolution used for initializing the previous test cases. The otherthree methods require known profiles. Two ofthese were tested by comparisonwith results of another theoretical calculation. The comparison calculations were produced by the finite-difference technique oE Bushnell and Beckwithdescribed in reference 66. Thoseauthors extezded the calculation technique to transitional flow by the use of an intermittency function likeequation (147). In addition, their intermittency function was multi- plied by a factorwhich was specifiedas a functionof r'. That is,

where r is givenby equation (147). Inthe calculations obtained from theauthors of reference 66 for comparison,the intermittency, rB(x) was specifiedat four points. These valuesof rB(x) were plotted and a smooth distribution of r foruse withthe present theory was obtainedgraphically.. The distribution is shown in figure17 (a) . It shouldbe noted that the distribution of r(x) shown in figure 17(a) was usedonly as an illustrative example to compare the presenttheory with another theory. No specialsignificance is implied byvalues of r greaterthan unity.

68 The configuration was a sharp-leading-edge flat plate. The reference conditionsused for the calculations were asfollows:

= 8.5 MeO

Rel = 2.295 (lo7) ft-’ 0

Ts = 640°R

Tw = 526OR

PrL = 0.688

PrT = 0.9

rL = r = T 0.89

The turbulentPrandtl number of thereference calculation varied across the boundarylayer from 0.9 to 1.5. The value 0.f 0.9 was chosenfor the presentcalculations. Two caseswere computed withthe present theory, corresponding to the two methods of obtaining initial conditions from known quantities. The initialconditions were asfollows:

Case 1: - -U y_ -Tt 0.300970.020068 570.0 -50396 .0336 590.0 - 804 28 -05361-80428

Case 2: Cf = 0.00019736

6* = 0.006595

e = 0.00014955

St = 0.00076354 - u = 0.50396

Ts = 590.0

I. The initial temperature profiles are compared in figure17(b). For Case 1, the fitted profile matches the reference profile very well over the entire profile while the profile for2 exactly Case matches the reference profile at the wall and at the specified(Ti point= 0.5) with good fit elsewhere. Comparison of the initial velocity profiles is presented in figure 17(c). As with the temperature profiles, an excellent fit was obtained for the conditions of Case1. Only at the outer edge of the boundary layer does the fitted profile deviate from the reference profile. For 2,Case with the integral and wall quantities matched, the fitted profile is slightly different from the reference profile in shape. Comparison of the skin-friction coefficient and the displacement and momentrun thickness variation along the plate is shown in figures 17(d) and 17(e). The reference calculations and the present calculations are in excellent agreement for both Case1 and Case 2. Note in particular that both calculative methods show the same response to the rather unusual intermittency variation. In figures17 (f), (9), and (h) are shown comparison of the velocity and temperature profiles at two stations downstream of the transition point. The first station was in the transition region where the intermittency factor, r, was approximately equal to3. The second station was near the end of the transition region, wherer = 1. At the first station, the temperatures within the boundary layer predicted by the present theory are slightly lower than thoseof the reference calculation. However, at the second station, the temperature profiles are in excellent agreement. Note also, that the temperature profiles corresponding to1 and Case Case 2 of the present theory are identical with each other at both of the stations shown in figures (9)17 and 17 (i). There is slightly more difference between the velocity profiles of the present theory and the reference theory than between the temperature profiles, as shown in figures 17 (f) and. 17At (h)both stations, the reference theory predicts a profile which has higher velocities a through greater portion of the boundary layer than does the present theory. Recall, however, that the wall and integral quantities predicted by both theoiries were in excellent agreement as shown in figures 17(d) and 17(e).

70 Effect of Transverse Curvature in Laminar Flow

On axisymmetric configurations, the effectsof transverse curvature can be important if the thickness of the boundarylayer is comparable to the radiusof the configuration. The effectof transverse curvature on a laminarboundary-layer calculation is illustratedin figure 18. The configuration being examined is the loo half-angle cone of reference 52 in a flow with a Mach number of 6.9. The pressure was assumed to be constant along the conesurface and unit Reynolds number lo4 ft-’ was chosen for illustrativepurposes. The effectsof transverse curvature (TVC) on the calculatedStanton number distributionare evident from figure18(a). The calculation in which transverse curvature was accounted for predicts a highervalue of St thanfor the calculationin which transversecurvature is neglected. However, the effectdiminishes as the calculation proceeds downstream, as evidencedby the fact that the error between the two predic- tionsdecreases from 29 percentat x = 0.2 foot to 14percent at x = 1.0foot. Similar results are found forthe displacement thickness of theboundary layer as shown infigure 18(b). The calculatedvelocity and temperatureprofiles are shown infigure 18(c) and (a). Sinceincreasing the Reynolds number has the effect of decreasing the boundary-layer thickness, it is expectedthat in mostcases of interest for transitional boundary-layercalculations transverse curvature can be neglected. This was indeedfound to be true for the data offigure 12, discussedpreviously. The inclusionof transverse curvature in the calculations for that figure hadno effect on the results.

The initial conditions for both cases shown in figure 18 were obtained from thesimilarity solution built into the computerprogram. This resulted in a.n initially transient solution for the case with transverse curvature included. However, thesolution approached a uniquevariation within a few integrationsteps (each step was approximately0.0001 foot).

Turbulent Flow with Transverse Curvature A goodexample of turbulent boundary-layer data for flows with variable pressure gradients and significant transverse curvature effects was presented byWinter, Smith, and Rotta(ref. 67). The data provide a particularly good case for comparison with the present theory because they allow the demonstration of many different capsbilitiss of the theory all in two calculations.

71 The configuration is an axisymmetric, piecewise continuous,waisted body (fig.19). The datapresented in reference 67 for M, = 1.4 were chosen for comparison. The transitionpoint was assumed to be at 1.5 inches from thenose of the body, corresponding to the location of a boundary-layer trip as reported in reference 67.

The test conditions for the caseconsidered were asfollows:

M, = 1.398

Re, = 2.02 (lo6) ft-I m

Ts = 536OR

= 524OR TW

Me = 1.06 0

= 0.252 Rext (10')

The reference conditions for the calculations were taken to be theconditions on theconical nose section of the body. The calculations were startedat x = 0.1 footusing the similarity solutionbuilt into the computerprogram. Transition fromlaminar toturbulent flow was computed automatically from thegiven value of Re with the intermittencygiven xt by equation (147). The sourcefunction in equation (147) was computed assuming the transition region to be equal in length to the length of the laminarboundary layer. The experimental Mach nuderdistribution is presentedin figure 19 alongwith the configuration. The isentropic pressure distribution corresponding to the given Mach number distribution was used asinput to the computerprogram. The conditionsbehind the nose shock wave wereobtained from reference 68.

Thiscase was a goodone fordemonstrating the smoothness requirements ofthe input distributions. The smoothnessof thedistribution describing theconfiguration is not critical since the body slopedoes not enter the calculationsdirectly. However, thedistributions ofpressure, Mach number, walltemperature, and totaltemperature must be smooth so that the

72 interpolationof the first derivatives required will be smooth. In the calculations discussed in this section, it was found to be sufficient to plot the distribution to a scale at which three significant figures could be obtainedfrom a smooth curve drawn through the data. The configuration being described has received considerable attention byauthors of theoretical methods. Calculations on thisconfiguration havebeen made by Cebeci, Smith, and Mosinskis(ref. 69) , Herring and Mellor (ref.36), and Harris(ref. 41). All ofthese authors used finite- difference methods. Until now, onlyHarris was able to compute a completely theoretical solution with nodependence on experimental profile or skin- frictioninformation for initial conditions. TO theauthor's knowledge, the solution presented here is the first to be obtained by the methodof integralrelations.

The calculated results are compared withthe experimental data for momentum thickness and skin-frictioncoefficient distributions in figure 20. The agreementbetween the theoretical and experimentaldistributions is very good. Inparticular, note the erratic behavior of the data and the theoryfor the skin-friction coefficient in the region x < 1 foot. Also, note that the inclusion of the transverse curvature terms in the solution had a significanteffect on the momentum thickness. The transversecurvature effect is the same as foundby the authorsof references 36, 41, and69. The skin friction is predicted quite well both with and without transverse curvature,particularly at the point of minimum skinfriction. Without transversecurvature, the momentum thickness is overpredictednear the downstream endof the model, but is in good agreementwith the data when transversecurvature is includedin the calculations.

CONCLUDING REMARKS

The method of integral relations has been applied to the calculation oflaminar, transitional, and turbulentboundary layers on arbitrary axisymmetric or two-dimensionalconfigurations. The technique employed is a new application of a method previously used successfully to predict the characteristics ofseparated laminar boundary layers and unseparated turbulent boundarylayers. An eddy-viscosity model hasbeen developed for use in the turbulent boundary-layer equations along with an intermittency function for the transition region. A computerprogram has been written to solve the

73 ...... _.._””

equations with arbitrarily prescribed boundary-layer edge conditions, walltemperature distributions, and Prandtl number for flows low from speeds to hypersonic speeds. A nuniber of methods of obtaining initial conditions are discussed. The solution canbe started from known information such as velocity and temperature profilesor knownvalues of skin friction, heat transfer, and other boundary-layer properties, or from a similarity solution which assumes constant pressure from the leading edge of the configurationhas no and dependence on experimental or empirical data. Other theoretical initial conditions could alsobe used but were not included in the work described here. Thus, the calculative method can eitherbe self-starting or rely on experimental information as the user chooses. The computer program canbe used to calculate the transition region between other laminar and turbulent boundary-layer calculations or it can be used to compute the entire solution from laminar through turbulent flow. Prediction of the transition point is performed by the user of the computer program using approximate formulas developedby correlating experimental data. The intermittency of the transition region be can computed using a function based on probability theory,or it can be provided as an arbitrary distribution. The computer program maybe used to determine the intermittency correspondingto experimental data. Good comparisons between the theory and experimental data for heat transfer and skin friction were made on variouskinds of configurations in both subsonic and supersonic flows. Transitional heat transfer on both sharp-leading-edged flat plates and sharp-nosed coneswas calculated by determining an intermittency distributionto fit the data. Comparisons of the present theory with calculations made using finite-difference theory indicated excellent agreement between thetwo theories. Calculation of the turbulent boundary layeron a body of revolution having both favorable and adverse pressure gradients yielded excellent agreement with the experimental skin-friction coefficient and momentum thickness, when the effects of transverse curvature were included. Most significantly, that calculation was made usingno experimental informationto obtain initial conditions. Finally, some comments are in order regarding the intermittency distribution in hypersonic boundary layers. The exact details of the transition

74 region flaw structure remain as one of the major unsolved problems of fluid mechanics. Data examined inthis research corroborated the finding of Morkovin (ref. 12) that the intermittency distributions of hypersonic boundary layers donot always fit the simple probability distribution of Emmons as used by Dhawan and Narasimha (ref. 47). Morkovin stated that of many hypersonic caseshe examined, only about half of then could be fitted to thelow-speed probability distribution curves of Dhawanand Narasimha. Cases were presented in this report in whichthe intermittency distributions found to fit the data were quite differentfrom that givenby Emmons' function. The cases presented hereinwere presented to demonstrate the accuracy and flexibility of the computer program and were not intended as evidence of the validity of the probability theory. However, the results suggest that the intermittency function should be a more complicated function than that given by the simple theory of Dhawanand Narasimha. The results also suggest an important use of the computer programas a research toolin the study of hypersonic transition, but such a studymust be accompanied by further experimental research before meaningful resultscan be obtained. Many experimental dataare available for heat transfer toflat plates and cones in wind tunnels. However, in order to study the intermittency in the transition region comprehensively, more dataare needed from free-flight experiments as well as from configurations with significant pressurevaria- tion. It is possible that the higher modesof instability described by Mack (ref. 13) could produce potential turbulent sources whichwould be triggered by the relatively high disturbance levelof a wind tunnel, but might not be triggered in free flight. However, available free-flight data are not adequate to determine the detailed distribution of theintermittency. Use of the computer programalong with experimental datato determine the intermittency in boundary layers over a wide rangeof parameters could lead to correlations with which the intermittencycould be predicted with greater confidence than withthe simple probability function of Dhawanand Narasimha. Moreover, such correlationscould lead to extension of the probability theory to more complicated source functions, than the delta-functionof Dhawan and Narasimha, with sources of turbulence being introduced downstreamof the original pgint of onset of transition.

75 Most available experimental data in hypersonic flow are for heat transferalone. Experiments in which both heat transfer and skin friction were measured for several Mach numberswould substantially improve the state of knowledgeof the transition region. These measurementscould be accompaniedby measurements of the intermittency as determinedfrom fluctuatingquantities in the boundary layer. Calculations could then be made to verify whether the intermittency determined to match transitional heat transfer data is the same as required to match skin-friction data. The experimentswould also provide information about Reynolds analogy in thetransition region. This might be usefulin improving the present formulation of the enthalpy profiles in the theory.

Another area in which re.liable experimentalinformation is needed is the effect ofpressure gradients on thetransition zone. This is needed fordevelopment of improved models of the eddy viscosity and turbulent Prandtl nuniber as well as for the improvementof the theory regarding the intermittency.

Inconclusion, the method hasbeen found to be a fast, flexible, and accurate method ofcomputing unseparated laminar, transitional, and turbulentboundary layers. As input,the computer program requires only very basic informationsuch as the flowconditions and data describing theconfiguration, plus information on the initialconditions. Use of the program as an engineering tool requires only a minimum amount of knowledgeof the theoretical methodsused in the calculations.

NielsenEngineering & Research,Inc. Mountain View, California October 5, 1970

76 REFERENCES

1. Smith, A. M. 0. and Cebeci, T.: NumericaiSolution of the Turbulent Boundary-LayerEquations. Douglas Aircraft Corp. Rep. No. DAC 33735, May 29,1967.

2. Lynes, L.L., Nielsen, J. N., and Kuhn, G. D.: Calculationof Compress- ible Turbulent Boundary Layers with Pressure Gradients and Heat Transfer. NASA CR-1303,Mar. 1969.

3. Goodwin, F. K., Nielsen, J. N., and Lynes, L.L.: Inhibitionof Flow Separationat HighSpeed. AFFDL-TR-68-119, Vols. I through 111, Mar. 1969.

4. Kuhn, G. D., Goodwin, F. K., and Nielsen, J. N.: Predictionof Super- sonic Laminar Separation by the Method of Integral Relations with Free Interaction. AFFDL-TR-69-87, Jan. 1970.

5. Dorodnitsyn, A. A.: General Method ofIntegral Relations and Its Application to BoundaryLayer Theory. Advances in Aeronautical Sciences,vol. 3, von Karmdn Ed., The MacMillanCo., New York, N. Y., 1962,pp. 207-219.

6. Murphy? J. D. and Rose, W. C.: Applicationof the Method ofIntegral Relations to the Calculation of Incompressible Turbulent Boundary Layers.Computation of Turbulent Boundary Layers, 1968 AFOSR-IFP- StanfordConference, Vol. I, 1969,pp. 54-75.

7. Laufer, J.: Thoughts on CompressibleTurbulent Boundary Layers. Rand Corp. Memorandum R"5946-PR, Mar. 1969.

8. Probstein, R. F. and Elliott, D.: The TransverseCurvature Effect in CompressibleAxially SymmetricLaminar Boundary Layer Flow. Jour. Aero. Sci.,vol. 23,p. 208, 1956.

9. Stewartson, K.: CorrelatedIncompressible and CompressibleBoundary Layers.Proc. of the Royal SOC., A, vol. 200,1949, pp. 85-100.

10. Schlichting, H.: BoundaryLayer Theory. Fourth Ed., McGraw-Hill Book Co., Inc., New York, N. Y., 1960, p. 346.

11. Pinckney, S. Z.: StaticTemperature Distribution in a Flat-Plate Compressible TurbulentBoundary Layer with Heat Transfer. NASA TN D-4611, June 1968.

12. Morkovin, M. V.: CriticalEvaluatlon of Trans'ition from Laminar to Turbulent Shear Layers withEmphasis on Hypersonically Traveling Bodies. AFFDL-TR-68-149, Mar. 1969.

13. Mack, L. M.: Boundary-Layer Stability Theory. Notes preparedfor the AIAA ProfessionalStudy Series High-speed Boundary-Layer Stability and Transition,San Francisco, Calif., June 14-15,1969.

77 14. Wazzan, A. .R., Okamura, T., andSmith, A. M. 0.: SpatialStability Study of Some Falknar-SkanSimilarity Profiles. Proc. Fifth U. S. Natl. Congress of Appl. Mech., A.S.M.E., Univ. of Minnesota, June 1966, p. 836.

15. Kaplan, R. E.: The Stabilityof Laminar Incompressible Boundary Layers in the Presence ofCompliant Boundaries. Mass. Inst. of Tech. Aero- elastic and Structures Research Lab., ASRL-TR-116-1.

16. Donaldson, C. duP.: A Computer Study of anAnalytical Model of Boundary-Layer Transition. AIAA Jour., vol. 7,no. 2, Feb. 1969, pp. 271-278.

17. Potter, J. Leith:Observations on the Influenceof Ambient Pressure onBoundary-Layer Transition. AEDC TR-68-36, Mar. 1968.

18. Brinich, P. F.: Effectof Leading-EdgeGeometry on Boundary Layer Transition at Mach 3.1. NACA TN 3659, Mar. 1956.

19. Pate, S. R. and Schueler, C. J.: RadiatedAerodynamic Noise Effects onBoundary-Layer Transition in Supersonic andHypersonic wind Tunnels. AIAA Jour., vol. 7, no. 3, Mar. 1969, pp. 450-457.

20. Pate. S. R.: Measurementsand Correlations ofTransition Reynolds Nunitjers onSharp Slender Cones at HighSpeeds. AEDC TR-69-i72, Dec. 1969.

21. Deem, R. E., Erickson, C. R., andMurphy, J. S.: Flat-Plate Boundary- LayerTransition at Hypersonic Speeds. AFFDL Rep. FDL-TDR-64-129, Oct. 1964.

22. Hopkins, E. J., Jillie, D. W., and Sorensen, V. L.: Chartsfor Esti- matingBoundary-Layer Transition on FlatPlates. NASA TN D-5846, June1970.

23. Bertram, M. H. and Beckwith, I. E.: NASA LangleyBoundary-Layer TransitionInvestigations Paper No. 18. Boundary-Layer Transition Study Group Meeting, Vol. 111, Session on Ground Test Transition Data and Correlations, William D. McCauley,Ed., Air Force Rep. BSD-TR-67-213, Vol. 111, Aerospace Rep. No. TR-0158(53816-63)-1, Vol. 111, Aug. 1967.

24. Van Driest, E. R. and Blumer, C. B.: Boundary-Layer Transition:Free- StreamTurbulence and PressureGradient Effects. AIAA Jour., vol. 1, no. 6,June 1963, pp. 1303-1306.

25. Stetson, K. F. radRushton, G. H.: k ShockTunnel Investigationof the Effects of Nose Bluntness,Angle of Attack andBoundary Layer Cooling on BoundaryLayer Transition at a Mach Number of5.5. ALAA Paper No. 66-495, presented at the4th Aerospace SciencesMeeting, Los Angeles, Callf.,June 27-29, 1966.

26. Moeckel, W. E.: Some Effectsof Bluntness on Boundary-Layer Transition and Heat Transfer at SupersonicSpeeds. NACA Rep. 1312,1957. Supersedes NACA TN 3653,1956.

78 27. Holloway, P. F. and Sterrett, J. R.: Effect of Controlled Surface Roughnesson Boundary-Layer Transition and Heat Transfer at Mach Numbers of 4.8and 6.0. NASA TN D-2054, Apr. 1964.

28. Brinich, P. F.: RecoveryTemperature, Transition, and Heat Transfer Measurements at Mach 2. NACA TN D-1047,Aug. 1961. 29. Brinich, P. F.: Effectof Bluntness on Transition for a Cone and a Hollow Cylinder at Mach 3.1. NACA TN 3979, May 1957.

30. Eaton, C. J., Pederson, J. R. C., Plane, C. G., andSouthgate,. A. C.: ExperimentalStudies at Mach Numbers 3,4, and5 of Turbulent BoundaryLayer Heat Transfer in Two-DimensionalFlows with Pressure Gradient. British Aircraft Corp. Rep. ST-2310, 1964. 31. Potter, J. Leith and Whitfield, J. D.: Boundary-Layer Transition Under HypersonicConditions. AEDC TR-65-99, May 1965,presented at AGARD SpecialistMeeting, Naples, May 10-14, 1965.

32. Cary, A. M., Jr.: TurbulentBoundary-Layer Heat Transfer and Transition Measurements with SurfaceCooling at Mach6. NASA TN D-5863, June1970.

33. Zakkay, V. and Krause, E.: BoundaryConditions atthe Outer Edge of the BoundaryLayer on BluntedConical Bodies. Air ForceOffice of AerospaceResearch, Aeronautical Research Lab., AFU 62-386, July 1962.

34. Stainback, P. C.: Effectof Unit Reynolds Number, NoseBluntness, Angle of Attack, andRoughness on Transition on a 5O Half-Angle Con. at Mach 8. NASA TN D-4961, Jan.1969.

35. Softley, E. J., Graber, B. C., and Zempel, R. E.: Experimental Observation of Transition of theHypersonic Boundary Layer. AIM Jour., vol. 7, no. 2, Feb.1969, pp. 257-263.

36. Herring, H. J. and Mellor, G. L.: A Method ofCalculating Compreesible TurbulentBoundary Layers. NASA CR-1144, Sept.1968.

37. Maise, G. andMcDonald, H.: MixingLength and Kinematic Eddy Viscosity in a CompressibleBoundary Layer. AIAA Paper No. 67-199, presented atthe 5th Aerospace Sciences Meeting, New York, N. Y., Jan. 23-26, 1967.

38. Kleinstein, G.: Generalized Law ofthe Wall and Eddy-Viscosity Model for Wall Boundary Layers. AIAA Jour.,vol. 5, no. 8, Aug. 1967, . pp.1402-1407.

39. Martelucci, A., Rie, H., and Sontowski, J. F.: Evaluation of Several Eddy Viscosity ModelsThrough Comparison with Measurements in Hypersonic Flows, AIAA Paper No. 69-688,,presented at AI- Fluid andPlasma Dynamics Conference, San Franclsco, Calif., June 16-18, 1969.

40. Clauser, F. H.: The TurbulentBoundary Layer. Advances in Appl. Mech., Vol. IV, Academic Press, Inc.,1956, pp.1-52.

79 41. Harris, J. E.: Numerical Solution of the Compressible Laminar, Transi- tional, and Turbulent Boundary'Layer Equations with Comparisons to Experimental Data. Ph.D.. Dissertation, Virginia Polytechnic Inst., May 1970. 42. Emmons, H. W.: The Laminar-Turbulent Transition in a Boundary Layer. Jour. Aero. Sci., vol. 18, 1951, p. 490.

43 * Mitchner, M.: Propagation of Turbulence from an Instantaneous Point Disturbance. Jour. Aero. Sci., vol. 27, 1954, pp. 350-351. 44. Schubauer, G. B. and Klebanoff, P. S.: Contributions on the Mechanics of Boundary-Layer Transition. NACA Rep. 1289, 1956. (Supersedes NACA TN 3489, Sept. 1955.) 45. James, C. S.: Observations of Turbulent-Burst Geometry and Growth in Supersonic Flow. NACA TN 4235, Apr. 1958.

46. Elder, J. W.: An Experimental Investigationof Turbulent Spots and Breakdown to Turbulence. Jour. Fluid Mech., vol. 9, no. 2, Oct. 1960, pp. 235-246.

47. Dhawan, S. and Narasimha, R.: Some Properties of Boundary Layer Flow During the Transition from Laminar to Turbulent Motion. Jour. Fluid Mech., vol. 3, 1958, p. 418. 48. Nagel, A. L.: Compressible Boundary Layer Stability by Time Integration of the Navier-Stokes Equations- and an Extensionof Emmons' Transition Theory to Hypersonic Flow. Boeing Flight Sci. Lab. Rep. No.. 119, Sept. 1967. 49. Owen, F. K.: Transition Experiments ona Flat Plate at Subsonic and Supersonic Speeds. AIAA Paper No.69-9, presented at the 7th Aero- space Sciences Meeting, New York,N. Y., Jan. 20-22, 1969. 50. Corrsin, S. and Kistler, A. L.: Free Stream Boundaries of Turbulent Flow. NACA Rep. 1244, 1955. (Supersedes NACA TN 3133, 1954.) 51. Lees, L. and Reshotko, E.: stability of the Compressible Laminar Boundary Layer. Jour. of Fluid Mech., vol. 12, Pt. 4, Apr. 1962. 52. Fischer, M. C.: An Experimental Investigation of Boundary-Layer Transition on a loo Half-Angle Cone at 6.9. Mach NASA TN D-5766, Apr. 1970. 53. DiCristina, V.: Three-Dimensional Laminar Boundary-Layer Transition on a Sharp8O Cone at Mach10. AIAA Jour., vol. 8, no. 5, May 1970, pp. 852-856. 54. Jones, J. J.: Experimental Investigation of the Heat-Transfer Rate to a Seriesof 20° Cones of Various Surface Finishesa Machat Number of 4.95. NASA Memo 6-10-59L, June 1959. 55. Softley, E. J.: Boundary Layer Transition on Hypersonic Blunt Slender Cones. AIAA Paper No. 69-705, presented at AIAA Fluid and Plasms Dynamics Conf., San Francisco,Calif., June 16-18, 1969.

80 56. Softley, E. J., Graber, B. C., and Zempel, R. E.: Transition of the Hypersonic Boundary Layer on a Cone. Part1 - Experimentsat = 12 and 15. General Electric Rep. No. 67 $D39, Contract No. AF 04 (694)772, NOV. 1967. 57. Mateer, G. G. and Larson, H. K.: Unusual Boundary-Layer Transition Results on Cones in Hypersonic Flow. AIAA Jour.,vol. 7, no. 5, Apr. 1969, pp. 660-664. 58. Rumsey, C. B. and Lee, D. B.: Measurements of Aerodynamic,Heat Transfer and Boundary-Layer Transition on15O a Cone in Free Flight at Supersonic Mach Numbers up 5.2.to NASA TN D-888, Aug. 1961. 59. Van Driest, E. R. and Boison, J. C.: Experiments on Boundary-Layer Transition at Supersonic Speeds. Jour. Aeronautical Sci., vol. 24, no. 12, Dec. 1957, pp. 885-899. 60. Merlet, C. F. and Rumsey, C. B.: Supersonic Free-Flight Measurement of Heat Transfer and Transition onIOo aCone Havinga Low Temperature Ratio. NASA TN D-951, Aug. 1961. 61. Rumsey, C. B. and Lee, D. B.: Measurements of Aerodynamic Heat Transfer and Boundary-Layer Transition onloo a Cone in Free Flight at Super- sonic Mach Numbers up to5.9. NASA TN D-745, May 1961. 62. Cary, A. M., Jr.: Summary of Available Information on Reynolds Analogy for Zero-Pressure-Gradient, Compressible Turbulent-Boundary-Layer Flow. NASA TN D-5560, Jan. 1970. 63. Hildebrand, F. B.: Introduction to Numerical Analysis. McGraw-Hill Book Co., New York, 1956, Chapter6. 64. Zakkay, V., BOS, A., and Jensen, P. F., Jr.: Laminar, Transitional, and Turbulent Flow with Adverse Pressure Gradient on a Cone-Flare at Mach 10. AIAA Paper No. 66-493,. presented at the 4th Aerospace Sciences Meeting, Los Angeles, Calif., June 27-29, 1966.

65. Van Driest, E. R.: On Turbulent Flow Near a Wall. Jour.of Aero. Sci., vol. 23, no. 11, Nov. 1956. 66. Bushnell, D. M. and Beckwith, I. E.: Calculation of Nonequilibrium Hypersonic Turbulent Boundary Layers.and Comparisons with Experimental Data. AIAA Paper No. 69-684, presented at AIAA Fluid and Plasma Dynamics Conf., San Francisco, Calif., June 16-18, 1969. 67. Winter, K. G. , Smith, K. G. , and Rotta, J. C.: Turbulent Boundary Layer Studies on a Waisted Body of Revolution in Subsonic and Supersonic Flow. AGARD - Recent Developments' in Boundary-Layer Research, proc. of a Specialist Meeting sponsored by AGARD Fluid Dynamics Panel, Naples, Italy, May 10-15, 1965,p. 933. 68. Sims, J. L.: Tables for Supersonic Flow Around Right Circular Cones at Zero Angle of Attack. NASA SP-3004, 1964.

81 69. Cebeci, T., Smith, A. M. O., and Mosinskis, G.: Calculation of Compress- ibleAdiabatic Turbulent Boundary Layers. AIAA Paper No. 69-687, presentedat AIAA Fluid and Plasma Dynamics Conf., San Francisco, Calif., June 16-18, 1969.

82 (a) Axisymmetricconfiguration.

Figure 1.- Boundary-layercoordinate System. (b) Two-dimensional configuration.

Figure 1.- Concluded.

84 Figure 2.- Comparison of experimentaldata withpredicted transition Reynolds numbers on flat plates.

85 *mox X- X-

Figure 3.- Typical distribution of a surface quantity in a transitional boundary layer.

86 I

(a) Flat rlates and hollowcylinders.

Figure 4.- Comparison of experimentaldata for length of transition region with predicted values.

87 . . ."

I SYMBOL REFERENCE 0 52 0 53 0 h n 0 0

0 0

(b) Cones.

Figure 4.- Concluded.

88 I .2 FLOW CONDITIONS I I 00

= 1779 OR 0.8

-Tt 0.6 - Tte

0.4 /=./ SYMBOL I m 0 0.2 ""_ I ...... e 2

I I I I 0- -~ 1 0 .2 .4 .6 .a I .o U

Figure 5.- Variation of total temperature in a laminar bound8ry layer for different orders of approximation.

89 T/

Figure 6.- Variation of static temperature in a laminar boundary layer for different orders of approximation.

90 I .o

0.9

‘L

0.8

I 2 3 4 5 6 m

Figure 7.- Variation of laminar recovery factorwith order of approximation of enthalpy function.

91 I.

I .2 FLOW CONDITIONS Me = 6.18 I.o Rs,/ft = 1.32(1O6) ft"

0.8

0.6

0.4 SYMBOL m 0 "- 2 0.2 "- - 3 ""- 4

0 .2 .8 1.0

Figure 8.- Variation of total temperature in a nearly turbulent boundary layerfor different orders of approximation- I .4 FLOW CONDITIONS I Me = 6.18 I .2 d Ret/ft = 1.32(1O6) ft"

= 1779 OR Tte I .o I-~""-=~

0.8

0.6

0.4 0 I "- 2 0.2 "" 3 - ""- 4

0 -1 - I I I I 2 I .4 I .6 I .8 2.o 2.2 2.4 X

Figure 9.- Stanton number distribution ina transitional boundary layer for various orders of approximation of the enthalpyprofile.

93 l=-.-LI 2.0' .05 r M = 0.072 h/ft = 0.44(1O6) ft-l .04 - 0 DATA (ref. 47) -THEORY .03 - 3

.02 - rcs .o I "-i *02D.o I .o I - OO .5 I.o OO .5 I .o OO .5 I.o u/ue due u/ue u/ue x = 5.25 ft. x = 5.75 ft. x = 6.75 ft. x = 8.0 ft.

(a) Velocity profiles.

Figure 10.- Comparison of theoreticaland experimental quantities in transitional flow ona flat plate. I

I I I I I ' I' M = 0.072 Re,/ft = 0.44 (IO6) ft" 0 DATA (ref. 47) -THEORY

(b) Skin-frictioncoefficient.

Figure 10.-Concluded.

95 2.0

0 DATA THEORY EXPERIMENTAL r I.o ”” THEORETICAL r ,,y 0.8 /‘ I .2 0.6 0.0 / ” / 0.4 - r - 0.8 -

02 I 0 0.5 I.o 1.5 2.0 2 a x, ft. (a) Me = 6.18, Rgl= 1.32(10°) ft”, Tt 1860°R, Tw 549OR.

Figure 11.- Comparison of theoretical and experimental Stanton number distributions ona sharp-leading-edge flat plate.

96 I "

~~ 2.0 I I

o DATA THEORY EXPER I .o

pr) 0.8 0- x 3j 0.6 L.! nQ, 0.4 c c 0 t I c .2 0 4- v, 0.2 0.8 r 0.4

(b) Me = 6.18, Re,= 1.63(10') ft-l, Tt = 1815OR, Tw = 552OR.

Figure 11.-Continued.

97 200

0 DATA THEORY EXPERIMENTAL r I .o

~JO 0.8 0- x 3j 0.6

L! Q -P 5c 0.4 c 0 tc 0 3j 0.2

0. I x, ft.

0 (c) Me - 7.45, Re,- 2.03(10') ft-', Tt 1829 R, Tw 5530R.

Figure 11.- Concluded.

I I

4.0 -

3.0 -

2.0 - / / / I .2 /

1.0 - - 0.8 - 0 DATA r - THEORY 0.4 0.6 - EXPERIMENTAL r - """THEORETICAL r /

- I 0.40- - ~ - ' I I 0 2 4 6 8 IO 12 x, in.

(a) Rq.l= 6.69(10°) ft-l, Tt -- 1040°R, Tw = 551°R.

Figure 12.- Comparison between theoreticaland experimental Stanton number distributionson a sharp-nosed cone at MiP = 7.

I c 0 cc 0 ti

(b) Re,= 7.01(106) ft-l, Tt = 1070°R, Tw = 556OR.

Figure 12.- Continued.

100 3.0

2.0 EXPERIMENTAL r "-" THEORETICAL r

I .o 0.8

0.6

0.4 0 2 4 6 8 IO 12 x, in,

Figure 12.- Concluded.

101 0 s x' s 1.46

'W = 0.1318 x'

'W 1.5 0. I977 1.5833 -2092 I .6667 .2258 I .75 .2458 I .8333 .2683 1.9 I67 .2933 2.0 .32 I7 2.0833 035 I7 2. I667 .3883 2.25 .4267 2.33 .4683

Figure 13.- Cone-flare configuration.

102 16

C P 19 21 23 25 27 x, in.

(a) Pressure distribution on theflare. 8

Me 6

19 21 23 25 27 29 x, in.

(b) Mach number on theflare.

Figure 14.- Conditions at the edge of the boundarylayer on an axisymmetric cone-flare.

103

I 1.0 -

0.8 -

0.6 -

0.4 r

IO 15 20 25 x, in.

Figure 15.- Comparison between the predicted and experimental heat transfer distributions on a cone-flare.

104 7.0 I I I I - PRESENTTHEORY "" FINITE DIFFERENCE THEORY 6.0 Me = 6.21 Re, = 1.706(107) ft" 5.0 Tt = 1367OR

c. Tw = 584OR 4.0 L en -0 v % 3.0

2.0

I .o

.2 .4 - .6 .8 I .o U

(a) Initialvelocity profile (x - 0.1 ft.) .

Figure 16.-Capariaon of two theoretical calculations of the boundary layer on a sharp cone.

1 05 7.0

6.0

5.0

4.0

3.0

2.o

I .o

I .o 2.o 3.0 4.0 5.0 T/ Te 0 .2 .4 .6 .8 I .o Tt /Tte

(b) Initial static and total temperature profiles inthe boundary layer (x = 0.1 ft.).

F j:gurc 16. - Continued .

106 (c) Initialtotal temperature profiles (x = 0.1 ft. ) .

Figure 16.-Continued.

107 5.0 I I I

4.0

3.O

2.0

I .o

(d) Velocity profile at end of transition (x - 0.5 ft.).

Figure 16.- Continued.

108 I .o

0.8

0.6

0.4

0.2

0 .2 .4 .6 .8

(e) Total temperature profile at end of transition (x - 0.5 ft.) .

Figure 16.- Continued.

1 09 I03

PRESENTTHEORY ”- REFERENCE 41 3

I o2

I 0.0

I .o

0 .2 .4 - .6 .8 1.0 U

(f) Eddy viscosity at end of transition (x - 0.5 ft.1.

Figure 16.- Continued.

110 I .6

I .4

I .2

I.o

0.8

0.6

0 x, ft.

(9) Skin-frictioncoefficient.

Figure 16.- Continued. I .o

0.8

0.6

0.4

0.2

0 I 1 I I I I 0.5 1.0 1.5 2.0 2.5 0 x, ft.

(h) Stanton number.

Figure 16. - Continued,. 6.0

5.0

4.0 lo3) 3.0

2.0

I .o

0.5 1.0 1.5 2.0 x, ft.

(i) Displacement and momentum thickness, ft.

Figure 16. - Concluded. o INTERMITTENCY GIVEN FOR FINITE DIFFERENCE CALCULATION 3.5 - INTERMITTENCY USED FOR PRESENTTHEORY

2.5

I .5

I .o

0.5

- d Jl'.2 .6 1.0 1.2 I 1.6 .4 .8 .4 x, ft.

(a) Distribution of intermittency.

Figure 17.- Comparison of finite-difference theory and present theory for different initial conditionson a flat plate.

114 I I I I

8 -T Te 6

(b) Initial static temperature profiles (x = 0.325 ft.).

Figure 17.- Continued.

115 .I2 - FINITE DIFFERENCE .I 0 THEORY

-08

-02

.2 .4 .6 .a 1.0 U

Figure 17.- Continued,

116 I .4 I I I I 1 I I

I.2

I .o

0.8

0.6

0.4 0 FINITE DIFFERENCE CALCULATEDPOINTS -

0.2 ""

I I I I I I I ~ 1 .2 .4 .6 .8 1.0 1.2 I.4 1.6 x, ft.

(d) Skin-frictioncoefficient.

Figure 17.- Continued.

117 .02 FINITE DIFFERENCEPRESENT THEORY I CALCULATIONS

8*/ L .o I OIL

.2 .4 .6 .8 1.0 1.2 I.4 1I .6 x/ L

(e) Displacement and momentum thickness.

Figure 17.- Continuedi

118 020

.I 8

.I6 FINITE DIFFERENCE .I4 THEORY

" - - "- .I2 E' " g .IO

.O&

.OE

.OL

.02

( .2 .4 - .6 .8 U

(f) Velocity profiles in transition region (x = 0.825 ft., I' - 3). Figure l7.- Continued.

119 12

IO d

Te 6 FINITE DIFFER- 4 PRESENT THEORY (both cases) 2

nl I I I t 1 "0 .2 .4 -6 .8 1.0

(9) Total temperature profile in transition region (X - 0.825 ft., r = 3). Figure 17.- Continued.

120 8.4 I I I I

-FIN ITE Dl FFERENCE THEORY

0.3 ""

E' " >r 0.2

0. I

(h) Velocity profiles in turbulent,region (X = 1.43, r = 1).

Figure 17.- Continued.

121 14

8 -T Te 6

FINITEDIFFER- ENCETHEORY "- PRESENT THEORY (both cases)

(i) Total temperature in turbulent region (X = 1-43, r = 1).

Figure 17.- Concluded.

122 .04 1 I I I NO TVC "- WITH TVC .O 3

St .02

.o I

.2 .4 .6 .8 I .o x/L (a)Stanton number.

.06 I 1 I 1 I I

I I 1 I 1 .2 .4 .6 .8 1.0 x/L

(b) Displacement thickness.

Figure 18.Calculation showing the effect of transverse curvature on a laminar boundary la er on acone; Me = 5.54;q = 104 ftJ.

123 4-0

NO TVC "- WITHTVC

3.0

2.0

I .o

.2 .4 .6 .8

Figure 18.- Continued.

124 4.0

NO TVC "- WITH TVC

3.O

2.0

I .c

C I .2 .4 .6 .8

0 I 2 3 4 5

(d) Static and total temperature profiles (x/L = 0.2).

Figure 18.- Concluded-

125 2 .o I I I I

I M, I .o - . -

0 I I I I 0 I 2 3 4 1 x', ft.

Figure 19.- Experimental Mach number distribution on a waisted body of revolution.

126 6.0 I I I I

0 DATA (ref. 67) 5.0 -PRESENT THEORY m- NO TVC 0 "" WITH TVC

2 I M, M, = 1.4 = 2.02(1O6) ft" 536OR \o Tw = 524OR

~~ 1.0 2.0 3.0 4.0 0 x', ft.

(a) Skin-friction coefficient.

Figure 20.- Comparison of theory and experimental data on a waistedbody of revolution.

127 x', ft.

(b) Momentum thickness.

Figure 20.- Concluded.

128 NASA-Langley, 1971 - 12 CR-1797