ad-757 871 a simplified approach to the analysis ofad-757 871 a simplified approach to the analysis...
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AD-757 871
A SIMPLIFIED APPROACH TO THE ANALYSIS OFTURBULENT BONDARY LAYERS IN TWO ANDTHREE DIMENSIONS
F. M. White, et al
Rhode Island University
Pr:epared for:
Air Force Flight Dynamics Laboratory
November 1972
DISTRIBUTED 3Y:
National Techi cal Information Service
U. S. DEPARTAdENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151
AFFDL-TR-72- 36
. SIMPLIFIED APPROACH. TO THEANALYSIS OF TURBULENT BOUNDARY
SlLAYERS IN TWO AND THREE DIMENSIONS
F. hi. WHITE
RJ. C. LESSMANN
G. H. CHRISTOPH
DEPARTMENT OF MECHANICAL ENGINEERINGAND A.PPLIED MECHANICS7
UNIVERSITY OF RHODE ISLANDKINGSTON, RHODE ISLAND
TECHNIC A1L REPORT AFFDL-T'•- 36 D L
NOVOMBER 1972 Ujrý~. ~ E
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Kingston, Rhode Island 02881 N/A3, REPORT TITLE
A SIMPLIFIED APPROACH TO THE ANALYSIS OF TURBULENT BOUNDARY LAYERS INTWO AND THREE DIMENSIONS
4. DESCRIPTIVE NOTES (7ype of report and inclusive datee)
Final Report - June 1971 through Jul- 1972S. AITHOOR(S) (Last name. first name, initial)
White, F.M.Lessmann, R.C.Christoph, G.H.S. REPORT DATE 7a. TOTAl. NO. OF PAOIeR 7b. NO. Op RErS
November 1972 t I 71ema CONTRACT OR GRANT NO. 9a 0RISINATOR'S RRPORT MUMMEQR(S)
I F33615-71-C-1585
b. PROJECT W.t. 1426
Task No. O 96. & Err-•e @oRT NO(S) (An?-**,rn-•,m Matmabe aelE
d.Work Unit 17 AFFDL-TR-72-13610. AVA ILABILITY/LIMITATION NOTICES
Approved for public release, distribution unlimited.
I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITYDepartment of the Air Force
* IAF Flight Dynamics Laboratory(APSC).•,' Wright-Patterson APB, Ohio 45433
13 AnSTRACT..cfts report deals with a new method of calculation of coupled
skin friction andheat transfer in two-dimensional turbulent boundarylayers and also with the calculation of axisymmetric and three-
dimensional turbulent skin friction. It is based upon an integr1lapproach which assumes analytic law-.,.-the-vall velocity and temper-ature profiles through the boundary layer.N In the two-dimensional anaxisymmetric cases, this results in ordinary differential equationswhich have the skin friction and/or wall heat transfer as the onlydependent variables. No shape factors or integral thicknesses appear
in this theory. Flow separation is predicted by the explicit criter-
ion Cf = 0. Th:, three-dimensional boundary layer analysis results in
two coupled partial differential equations for 1) the freestream com-
ponent of'skin friction; and 2) the tangent of the angle between thetotal surface shear stress and the freestream direction.
'The new theory is compared to several important experimentalstudies and the general agreement is .exre1Wff. .. AI16', even it%'"Its mo tcomplicated, the present theory is co'siderably simpler than othercurrently available calculation sch-me. This is particularly true
of the three-dimensional analysis.>
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AFFDL-TR-72-136
FOREWORD
This final technical report wss prepared by F.M. White,R.C. Lessmann, and G.H. Christoph of the Department of Mechan-ical Engineering and Applied Mechanics of the University ofRhode Island under Contrait F33615-71-C-1585, "Analysis of theTurbulent Boundary Layer in Axisymmetri- and Three-DimensionalFlows."
The contract was initiated under Project No. J426, "Ex-perimental Simulation of Flight Mechanics,' Task No. 142604:"Theory of Dynamic Simulation of Flight Environment." Thework was administered by the Air Force Flight Dynanics Labor-atory, Wright-Patterson Air Force Base, Ohio, Dr. Jaese T. VanKuren (FX), Project Engineer.
The work was accomplished during the period 1 June 1971through 30 June 1972.
The report was submitted by the authors in Jrly 3972.
A supplement containing an outline of the theoreticalmethod for engineering use is available under the title "RapidEngineering Calculation of Two-Dimensional Turbulent SkinFriction."
This technical report has been reviewed and is approved.
P PP. #J!TOIATOSChAef, Flight Mechanics DivisionAir Force Flight DynamicsLaboratory
li
TABLE OF CONTENTS
Chapter
INTRODUCTION 1
ShEAT TRANSFER AINLYSIS 4
1.1 Introduction 4
1.2 Development of thbi New Method 4
1.3 Ccp3psrison with Beat TransferIxpe -iments 21
1.3.1 Flat Plate rata 21
1.3.2 Data of Mretti & Kays 24
1.,; Conclusion 30
2 AXISYMJWTIiC CONPFXSSUI&Z FLOW 32
2.1 Introduc tion 32
2.2 The AxLsymietric Law-of-the-Wall 33
2.3 Dr2,,4Az' of the BasicDiffere.ntial Equation 38
2 C.4 C.arlsor. *ith Experiwent 45
. . 1 Comprezsible i'.-n. along aSlender CýIlnder 47
•.a.2 Sipersonic Attacled FlowPast 6. Cone 53
2. '-.3 Supersontc Flow with PressureGradient 59
2.5 Sumary 62
3 THE PREDIICTION Or Tw'HR-DDNwSIOiwAs FPCTION 63
3.1 Introduction 63
3.2 Specification of Velocity Correlations 69
3.3 -Ae Methcd of Solution 75
II
TABLE OF CONTENTS (Continued)
Chap ter E__t
3 3.4 Compari3on wi .h Experiment 80.. 4.1 The 1tating Dirc Pivblem 80
3.4.2 , tons Swept, Forward-F&ing Step 92
3-4.3 Data of Francis and Pierce 97
3.4.4 Data of Ifinksiek and Pierce 104
3.5 Sumary 107
Apperdix A INTEGRAL COEFFICIENTS 109
RF•ER•ES 118
4
I.
~~a lip,.... . . ... ii • f mI
LIST OF ILLUSTRATIONS
Figure fPage
1.1 Illustration of tne Effect of VariousParameters on the Velocity Law-of-the-Wall for Compressible Turbu!-ent Flow 11
1.2 Illustration of the Effect cf ;VariousParameters on the Temperature Law-of-the-Wall for Compressible Turbulent Flow 12
1.3 Comparison of the Present Hear-Wall Theorywith the Actual Law-of-the-Wate 20
1.4 Comparison of Theory with Zero PressureGradient Data of Moretti and Kays (19E5) 26
1.5 Comparison of Theory with Mild FavorablePressure Gradient Data of Mcretti andKaYs (1965) 27
1.6 Comparison of Theory with Severe FavorablePressure Gradient Data of Aoretti andKays (1965) 28
1.7 Comparison of Theory with Adverse PressureGradient Data of Moretti and Kays (1965) 29
2.1 Definition Sketch Illus;rating Coordinatesfor Axisyrmetric Bounfary Layer Flow 34
2.2 Some Examples of the Axisymmetric Law-of-the-Wall, Computed From Equationz (2-10)and (2-12) 39
.3 The Compressibility and Wall TemperatureParameter A, from Equation (2-32)
2.4 Skin Friction on an Adiabatic Cylinder ata Frees tream Mach Number of 5 -'orVarious Values of Ra = U r /v , Computed
o Eqn.(2-34). 50
2.5 Skin Frictlon Detervination for SupersonicFlow along a Slender Cylinder by a Modified j
Axis-mwmetrric Clauser-Plot, Eqn.(2-40) 52
vi
LIST OF ILLUSTRATIONS (Continued)
Figure e
.ilustratoion cf 'he Turbulenz Cone Rale
2.- Comparison of Theory and Experiment forSupersonic Flow past a Bodv ofRevolution, After Allen (1970) .i
3.1 .Three-bD-iensional Boundary Layer CoordinateSvs.:e=, .ith Skewing in One Direction Only
3.2 Three-Dimensional Boutndar, Layer withBila-e -al Skewing
3.-3 Sk.etch uf the :riangular Hodographa.pprox*rmation, after johnston (1957) 73
3.4 Verification of the Streamwise Law-of-the-Wall f)r Turbulent Flow on a Rtating Disc 83
3.5 7eriMication of Mager's Hodograph forRotat.ig Uisc Flow..-
3.6 Comparison of Theory" and Experiment forStre Lmwse Skin Fric tion ovi a Rotating Disc E7
3.7 Comps-rion of Theory and Experniment for,hc" Wall Streamlire Tangent on aRotating Disc 88
3.8- Comparison of Rotating Disc CrossflowVelocity Profiles with the Hoiographof Von Karman (19a,6) 91
3-9 Comparison of Present 7-Theory for ThrePressure Distribution Assumptions withthe Streamwise Skin Friction Eata ofjohnsýon (197C) for Flow near a 450Swept Step 9:
3.10 Comparison of Present Theory for ThreePressure Di.lrillution Assumptions withthe Wall Streamline Tangent Data ofJomnston (1970) for Flow Near a SweptStep 9':
:ii
LIST OF ILWUSTPATIONS (Con:inued)
Figure ae
3.11 Comparison of Present Theory With theCurved Duct Experiment of Francis and'Pierce (196'), Series : 99
3.12 Comparison of Theory With Wall Streamlineiangen- Data of Francis and Pierce (1967),Series 1 100
3.13 Comparison of Present Theory WithExperizmen--al Skin Friction as Estimatedin a Curved Duct by Francis and Pierce(19i7), Series 2 10i
3.14 Comparison of "'eory With Wall StreamlineTangent Data of Francis and Pierce'19,-7", Series 2 102
3.15 Comparison of Present Theory WithEstimated Skin F-Friction in the Cur-iedDuct Experiment of Klinksiek andPierce (1970) 105
3.16 Comparison of Present Ti.eory With WallStreamline Tangen, Data of 12inksiekand Pierce (19-1)
LIST OF TABLES
Table ,&
1-1 Comparison of Six Theories with Flat PlateFriction Data 23
2-1 Comparison of Theory with the Data ofRichmond (1957) for Supersonic FlowAlong a Slender Cylinder at He = 5.8 58
"viii
LIST OF SYMBOLS
K-gish
A van Driest factor defined by Eqn. (2-32)
Ai Integral coefficients defined by Eqns. (2-26,27,28)
C D DraG coefficient
cf Local skin friction coefficient, cf = 2VwPeUe2
c h Local Stanton number, ch = q/PeCp e(Taw-Tw)
c Specific heat at constant pressure
Cp Pressure coefficient
D i Integral coefficients defined by Eqns. (3-53) to (3-58)
Fc,F x Transformation functions defined by Eqn. (1-38)
G i integral coefficients defined by Eqns. (3-38,39)
ho Stagnation enthalpy, h =- h + u2 /2
h S Direction along an external stremal'le
h~n Direction normal to an external streamline
Hi Integral coefficients defined by Eqns. (1-29,30)
k Specific heat ratio, k = c p/cV
kw Fluid thermal conductivity at the wall
L Reference length
M Mach number
P Fressure
P Turbulent Pranrtl number, ' 0.9T
q Heat flux
r° Body radius++Dimensionless body radius, z roU*/w
r0 U 0 w
ixI
LIST OF SYMBOLS (Continued)
r Defined in ?igure 2.1
r Recovery factor, = '0.89
R Perfect gas constant
Ra Radius Reynolds number, Ra = Ue ro/-e
R Local Reynolds number, • = U ex/Ve
RL Effective Reynolds number defined by Eqn. (1-31)
t Dimensionless wall temperature, t = T/Two
Absolute temperature
TWO Reference wall temperature
T+ Temperature ratio, T+--T/Tw
u,v,w Velocities in the h 1 s,y, h2 n directions respectively
u* Friction velocity, u* = ( ws/pw)1/2+ +
Dimensionless veloci.y, u = u/u*
U Freestream velocitye
U0 Reference velocity
v* Friction velocity, v* = (rwn/Pw)I/2
V Dimensionless freestream velocity, V = Ue, 1U0+ +
w Dimensionless velocity, w = w/u*
x,y Coordinates parallel and normal to the wall
X* Dimensionless cocrdinate, x* = x/L+ +
y Dimensionless coordinate, y = yu*/vw
Y Coordinate, Y = ro in(r/ro)
1y ~~Dimensionless coordinate, Y~ = u/
x
IST OF S.0MBOLS (Continued)
CL Limensionless pressure gradient parameter, Eqn. (1-12)
1 Dimensionless heat transfer parameter, Eqn. (1-16)
Y Dimensicnless compressibility parameter, Eqn. (1-16)
5 Boundary layer thickness
+ Dimensionless boundary layer thickness, 5+=6u* DVi
C Eddy viscosity
0 Tangent of the angle between the total surfaceshear stress vector and the freestream direction
K von Karman's constant, K - 0.4
Dimensionless skin friction variables, Xl=(C/cf)
Dimensionless heat transfer variable, X2 _ T,w e w
I Absolute viscosity
v Kinematic viscosity
p Density
1 Shear stress
* Stream function
* Viscosty power-law exponent, a 0.67
SAngular velocity of rotating disc
Subscripts
aw Adiabatic wall
e Freestream
n Normal
o Initial value
s Streamwise
w Wall
_____xi
INTROrUCTION
This report summarizes work to cate on a new method,
simple yet accura e, for analyzing the turbulent boundary
layer for the varl.ation of sarface shear stress and heat
transfer under fairly arbitrary conditions. Earlier work
was concerned with vwo-dimensional compressible flow and
was summarlzed in the report by :.-dite -nd Christoph (1970)
and a subsequent published peper, White and Zhristoph
(1972). Another, more comprehensive paper, including such
effects as roughness and wall transpiration, has also been
published: White and Christoph (1971). The latter paper
is also confined to two-dimensional flow.
The present report extends this new theory to ÷wo new
types of conditions: 1) a skin friction theory for
axisymmetric and three-dimensional flow conditions; and
2) a new analysis for two-dimensional surface heat
transfer. The approach retains the earlier philosophy of
using only wall-related temperature and velocity profiles
to develop integral relations for skin friction and heat
transfer which are devoid of shape factors or integral
thicknesses. Thus, for examplL; the axisymmetric theory
results in a single differential equation for wall skin
friction, with no other variables eppearing.
It is convenient to di.ida the report into tnree
chapters. Th_ first chapter develops a two-dimensional thec.
for surface heat transfer (or Stanton namber) for compress-
ible flow with arbitrary wall tempe-ature. This extcnds the
14
A
earlier report., i,.hite and Thristoph (1970), which consider-
ed -kin friction only and elirminated the temperature through
a Crocco approximation. The new theory results in two first-
order coupled differential equations for skin friction
coefficient and Stanton number. The theory is compared with
variable wall tempersvure End pressure gradient experiments,
anr an approximate correlation is found which relates the
Reymolds analogy factor (ch/cf) to the local pressure
gradient.
The qecond chapter develops an analysis for compress-
ible axisyrwietric flow with pressure gradients. T-he boundary
layer is assrTaed tc be thick, so that transverse curvature
is ir•ortant, and temperature is eliminated through the
Crocco &pproximation. The resalt is a single first-order
differential equatiop for t"he skin friction coefficient.
Comparison is made with experiments for supersonic flow
along thin zylinders and bodies of revolution. The transverse
curvature effect can be quite large, causing skin friction
increases of 1004 or more in both subsonic and supersonic
flows. Some earlier work on this type of analys.s was
given in a paper by White (1972).
The third and final chapter is concerned with three-
dimensional incompressible turbulent skin friction. The
distinguishing characteristic of a three-dimensioneCL
boundary layer is a crossflow normal to the freestream
direction, resulting in a vector wall shear stress which has
both streamwise and crossflow components. Since both shear
2
components vary with bosh surface co~rdlnares, partial
differential eqdations are ine-.-itable. The present analysis
uses both a streamwise ana a crossflow law-of-the-wall to
deriv2 two first order coupled part:Lal differential
equrations for the two surface shear :omponencs. As with
previous work, there are no shape factors o r integral thick-
ness present. Also, the partlal d'Lffe--.ti•a' equat-ons
are of such simple fPn.i a .... , *-ý .. y - l" to -, v'
simple finite difference techniqae. .'-' " -i-rs
applied to several experimental flows.. ant' ,'Ie %sx-eement is
very promi.inj, particularly considering t. lative
inaccuracy of present three-dimensisn!.f). flcow :asurements.
i
:I
3 1•
ThaDter One
H.at Trarsfe;• naryi:is
introduction
The purpose of this chapter is to develop a method for
calculating beat trarsfer coefficients for arbitrarily high
Mach numbers, variable wall temperatures, and variable
streamwise p-.essure gradients. The present analysis is
restricted to steady, two-dimensional flow of a perfect gas
in a compressible turbulent boundary layer. The lateral
pressure gradient will be neglected since, although the
lateral gradient strongly influences the momentum thickness
in high Mach number fi1sws, it appears to have negligible
effect on the approach proposed here. Twv coupled, first
order, ordinary differential equations are derived which
have for dependent variables -only the skin friction coeffi-
cient and the Stanton number. Stanton numbers are calculat-
ed and compared to the data of Moretti and Kays (1965). A
relationship among skin friction, heat transfer, and pressure
gradient is presented.
1.2 Development of the New Method
The equations governing compressible, two-dimensional
turbulent boundary layer mean flow analyses were apparently
first given by Young (1953) in the following form:
4
a) he ctntinnuity equation:
P U) + (pv) = 0
b,) The momentum equation:
ýu ýU = +u _+
e) _The energy equation:
6ho Oh C y
Pu - + v -+(q-u¶) 4--
d) The perfect gas law:
p = RT, or: T/T. = p/p)
Here h = c T + u2,'2 is the stagnation enthalpy, and theo p
symools q and T represent the heat flux and shear
stress, respectively. That is,
Z)T P 'T 'T=t u0
q k - p-U; = i - p 'V-
Tnere are six unknowns (p ,u,v,h,,q,T) and only four
equations, so that further relations are needed. The
finite difference methods model the eddy viscosity and eddy
conductivity, thus correlating the variables q and T with
5
local conditions, andi then attack the complete governing
equations. The so-called von Karman integral methodseliminate p v from equations (1-2) and (1-3) by equation
(1-1) and then integrate with respect to y across the entire
boundary layer, thus obtaining the vyn hriian integral
relations. Additional relations for these integral methods
are then brought in as correlations between integral param-
eters.
The compressible flow analysis of White and Christoph
(1970, 1972) was based on Crocco's approximation for the
energy equation - cf. Schllchting (1968):
Prandtl number = unity; T = a - bu + cu, (1-6)
where ab,= are determined from boundary conditions.
Basically, the analysis of White and Christoph (1970,1972)
non-dimensionalizes the velocity u with respect to wall
quantities and then integrates equations (1-1) and (1-2)
with respect to the law-of-the-wall variable y" = y-u*/ Vw,
where u* = (Tw/pw)1/2 is the friction velocity. With
Crocco's approximation one can compute local heat transfer
only through a Reynolds analogy. Crocco's approximation
will not be made here. Instead, the energy equation, as
given by equation (1-3), will be used. 7-he velocity u
and the temperature T are non-dimensionallzed as follows:
u k= ux,y)/u*(x) and T = T(x,y)/Tw(X) . -7
Quantitati.ve expressions for - an,. T+ are needed.
The simple incompressible formula
? n(y) 5 (18)
and a similar ex:presslun for T are not adequate, Decause
we need to include compressibility, heat transfer, and
pressure gradient effects in t)- velocity and temperature
correlations. An analy-tcal approach of finding expressions
for u+ and T+ was desired, so thalt as little empiricism as
possible would be introduced into the present method. The
scheme adopted was based on the eddy viscosity approach of
Delssier (1959), which assumed that the Prandtl mixing
length approximation could be extended to the variable
density case with no further changes, Let us first derive
an expression for u . The total shear stress is related
to an eddy viscosity and mixing length as follows:
S: , (1-9)
where E = P ,y I
and s = 0.4 is von Kgrman's constant. Also, near the wall,
7
the boundary layer is approx:iately a Couette flow with
negligible acceleration, so that equation (1-2) becomes
¶- -•+• (I-10)
Non-dimensionali7.ing equations (1-9) and (1-1D) with respect
to wall vzribbles, we neglect the laminar portion of the
total shear in equation (1-9) (This step greatly simplifies
the final form of u" but has a negligible effect on the
numerical values of u+.). Equating equations (1-9) and
(1-10) results in
_u (T+(! + a y+))1/2 (`-i!)ky + K Y +
where
a d (1-12)
This is the same parameter a , termed the "pressure
gradient" parameter, which appeared in White's (1969)
analysis, and which was suggested by the work of Mellor
(1966). It is an ideal parameter for this method because
it is directly related to the shear stress without any
8
integral thizknesses. ie now need an expression for T+.
Following the same approach, the total heat flux is relatei
to an eddy conductivity as follows:
C
q = -( I + C) Z-=
where P T is the turbulent Prarndtl nu;._.er. Near the wall,
one obtains from equation (1-3)
qW UT + q (1-14;
Again, non-di-mensionalizing equations (1-13) and (1-14),
neglecting the laminar portion of the total heat flux inequ-:.tion (1-13), and equating equaticns (i-i3• and (i-14)
results in
.T+ = (0-2y u+(l + a y+))T+ 1/2
;3y + KY + (1 + y+) 1/2
where
PT andPTU*2
Pw p OW 2PTW
The parameters 1 and y will be referred to as the "heat
9
transfer' parameter and zhe "compressibility" parameter,
respectively. The effectb of the parameters a , a ,
and N on u+ are shown in Figure (1-1). Positive a
(adverse pressure gradient) and positive a (cold wall
heat transfer) raise a + above the Incompressibl. logarith-
mic law, and negative a (favorable pressure gradient)
and negative f (hot wall heat transfer) have the opposite
effect. The parameter y , which is always positive,
lowers u+ celow the incompressible result. If more than
one of these parameters are nonzero, the effect is roughly
an additive one. Precisely these iffects have been found
experimentally, e.g., Kepler and O'Brien (1959), Lee et al
(1969), and Brott el al (1969). The "heat transfer"and
"compressibility" parameters have the same qualitative
effect on T as they do on u+. The "pressure gradient"
parameter alone does not affect T+. However, a positive
a combined with either a positive 1 or a negatLve
8 depresses the 0 effect on T+. A negative a has
the opposite effect. The compressibility effect is
increased with a positive a and decreased with a nega-
tive 0 . These results are shown in Figure (1-2).
Equations (1-11) and (1-15) give, for zero pressure
gradient,
6+ ay+ + +- -- u1 (1-17)
8u+ 8+ 8u+
10
/
Effect of positive a (adversepressure gradient) and /positive 6 (cold wall /
+ heat transfer). x
Effect of y (compressibility),negative m (favorable pressuregradient), and negative B(hot wall heat transfer).
cannot be negative)
log(y+)
Figure 1.1. ILLUSTRATION OF THE EFFECT OF VARIOUS
PARAMETERS ON THE VELOCITY LAW-OF-IHE-WALL
FOR COMPRESSIBLE TURBULENT FLOW.
11 1
f /T/
//
+log(y ;
Figure 1.2. ILLUSTRATION OF THE EFFE-CT OF VARIOUS
PARAMETERS ON :,TyE TEMPERATURE LAW-OF-THE-WALL
FOR COMPRESSIBLE TURBULENT FLOW.
1'2
Integrating equation (1-!7) f'ives
T = 1 + u - u+ (-)
which is the form of Crocco's relation used by White rnd
Christoph (1970, 1972). If a does not equal zero, one
then obtains
S•T - u+(1+ ( y) (1-19)zmu+ a y +
which is not a Crocco relation.
Let us continue the method of solution, by first
defining a compressible stream function * as follows
T - Pu; T - - v (1-20)
Eliminating p v from equations (1-2) and (1-3) by
equation (1-20), and non-dimensi nalizing u and T as in
equation (1-7) gives
Pu'u + (u-u+) - *- 2-- (u*U+)7x Tx - VWCy +
Su! (1-21)
13
qw
and
( u* + 5 u* _ (C TT+ +
Pw 2TY+
u* • (uT + q) (1-22)Vw ýy+
The pressure gradient in equation (1-21) is related to the
freestream conditions by the inviscid Bernoulli relation:
d " e 0dUe (1-23)
The y +-derivatives in equations (1-?1) and (1-22) are no
problem because these equations are going to be integrated
with respect to y , but the x-derivatives must be calculated.
The functional relationships of u+, ±+, and A has
dimensions of viscosity) are
U+ IT+, # ALw = fcny+ W. 2 1, ) 11-24)
The Y-derivatives are calculated by using the chain rule:
y+ a 84
14
For skin friction ano heat transfer calculations, it is
ccnvenient to define further non-dimensionalizations:
x= x/L ; UeUo = (0/C2)/2
(1-26)T = /T q = /( ) 3
- ,Pr w' e0 wewp~~~
where L, Uo, and Two are constant reference values. Now+
use equations (1-26) in the definitions for y a, O, , id
y and carry out the partial derivatives in equation (1-25).
In so doing, derivatives of vw, P., and Te will arise.
Eliminate such derivatives in 7avor of (dIdx*) and
(dV/dx*) through eqation (1-4) and a viscosity power-law
ass'umtion U TTx where T is taken here as 0.67 for air.
Also use the relation between velocity, pressure, and
temperature in the freestream:
- dV dv -k d7ek Me V - -- I-, (k = c/C) / (1-27)e V e
The net result is that equation (1-25) can now be written as:
i. 1 dX 1 - + -.._+ - 3-•-AX*= TT -Y -+ 3a 1 6a + V 2v '>-V
V'I v I r- + (I/ V )"+- 23 + 2V
(1-28)
+- -y ( 11 - 1/2) ;-- + z + 1/2) -
1 _+ (k+l)M2 Vr
ap
15 5e
where the primes denote differentiation with reEpect to x*.
Equation (1-28) is now substituted into equations (1-21)
and (1-22) and the resulting equations irlgrated from
+ .4,
y =0 ( - w) to y = t (T = 0). The result is
•X-*" (m-3a(•) -) i,2 '
+ XI -7V' - 1 -H- (2l) "e (.•H 2-H,-H) 5 -29)
ti -
- l • a(+l H2•H = RLV
and
- 3xaH ) (-6HSH) )_
Tx LL 2dx*H
- t (+l/2),, +{ =
where .RL is an effective Rieynolds number defined as:
UoL ie Te 1/2RL - .2 ([S-) (•-) .(1-31)
e w w
1,~~~ ~ ~ ~ 4 .+)!2-;4 ;L
The Hi coefficients are Integrals in-Xo12ring a - n
law-of-the-wall parameters; they o're li'ted in Appendix A.
It is possible to numerically evaluate these H's onceu+ I u+ fu+ +T+ •T+ -
expressions for the derivatives umb dei- e , ,
16
) _mmm • • • ml m • • mmm • • U 0 L • ( LL e m mm (r 112Im • lm • l III I
- and are found. Expressions for th-se derivatives
are obtained by differentiating, with respect to a, 0, and+ +
y, the previously derived integrals for u , T +, and t,
Eqns.(1-11,15,20) respectively, at a fixed y Integral
expressions for these partial derivatives are given in
Appendix A; the change of variables du+ = (+/Ay+ )dy+ has
been made. The coefficients H are then evaluated by+
integrating, the relations in Appendix A from u = 0 to
u+ = ue+ = )I(Te/Tw)l/2for various values of a, j3, and
Y. The turbulent Prandtl number PT, is taken as 0.9
The starting value of T is 1.0, and the initial value of
y+ is taken to be 0.1108 = exp(-5.5/2.5) to match with the
incompressible logarithmic law (Eqn.l-8) in the limit. Curve
fit expressions for the coefficients Hi are also given in
Appendix A.
The parameters a, (3, and y are related to the dependent
variables A1 nd x 2 as follows:
2 - W (1/v)/RL
Yue2 PT (2 M2e (To/Tw) (1-32)e T 2 e+ 2
U = P K X M2 (k-l)(Te/Twe e 1
P 0.9 was found experimentally by Simpson, Whitten andTMoffat (1970). It is also the value assumed in most
finite-difference calculations, e.g. Cebeci (19F'l).
17 4
Equations (1-29) and (1-30) are, then, two coupled,
first order, nonlinear, ordinary differential equations in
Xi and x 2 only. The freestream Mach number and wall tempera-
ture distributions must be known, plus the values of X1 and
)2 at some initial x*. Unfortunately, 5+ appears in
equation (1-29) and must be calculated. However, 5+ need
not be known precisely for accurate calculations of ii and
12. An approximate way of calculating 6+, and the one
recommended, is to first assume a Crocco law of the form in
equation (1-18). With this expression for T+, equaticn
(1-11) may be integrated analytically as
-1 2Yu+ -'I 2yu+-Q
(1-33)
+ ' 2 (P-Po + l ( =oIT
where Q = (f2+4)1I/2 and P = (l+ay+)1/2
Equation (1-33) needs to be initialized. By plotting u++
versus y , as in Figure 1-I, it is seen that the curves for
various values of a, 0 aad Y converge at y+ slightly less
than ten. The initial conditions uo = 0 and y+ = 0.1108+
were taken. Unfortunately, y is an. implicit function of
u so that an iteration scheme must be used.
The dependent variable X2 is directly related to the
Stanton number
ch er/T --a (1-34)
18
where the Stanton nmiber has been defined as
ch ew (-35)e o e aw
Since XI is defined as (2/c 1 ) ., equations (1-29) and
(1-30) give c, and ch directly - no auxiliary information
is needed about the behavior of integral thickness or shape
factors. Another imnportant and unique property of equations
(1-29) and (1-30) is an explicit flow separation criterion.
As separation is approached, dX 1 dx* approaches infinity
and hence c, approaches zero - which is precisely the
definition of two-dimensional separation. One need not
worry about attempting to predict separation according to
a particular value of the slape factor.
one basic assumption is made in the development of the
present method; tha:. is, that the velocity and the tempera-
ture profiles, correlated by inner variables only, are valid
all the way to the edge of the boundary layer. This
assumption leads to a slight miscalculation in 6+, as seen
by Figure 1-3. The outer wake is not predicted accurately.
But, as was pointed out earlier, it is not terribly
imrortant to know b+ precisely in order to obtain accurate
skin friction and heat tranrsfer coefficients. Also, in a
supersonic turbulent boundary layer, the wake almost
disappears. Of course, there will be some error in the
momentum and the displacement thicknesses, if one wishes to
calculate them. In passing, it should be noted that, from
19
mm •Nmmm m • nmmm • • • m mm m • . '*1m
+ 5+
U ,
PRESENT THiEORY:/
+ E . 3/ TRUE "WAKE"
INCOMPTRiSSIBLELOGARITHMICLAW, EQN. (1-8)
log(y+)
Figure 1-3. COMPARISON OF THE PRESENT NEAR-WALL
THEORY WITH THE ACTUAL LAW-OF-THE-WAKE.
0
m m • i • ~ mm • nl • /20 anii• m i ml n • • m si ~ l
equation (1-11), if ay+ is greater than minus one, then
' I I < 0. Thus the velocity gradient is negative, or
tne velocity insice thc botzndary -3yer has become greater
than the freestream velci'.yv. -o avoid this difficulty,
it is suggested that one take Fla- = -I/a in the case of
a favorable pressure gradient. This is not believed to
handicap the present methoc.
1.3 Comparison With Heat 'ramnsfer Experiments
1.3.1 Flat Plate rita
First, it seems appropriate to consider a special case,
namely when da- dq p = = G. For this case, thenaey hndx -•x dx
present analysis reduces to the analysis given by White and
Christoph (1970, 19"). Equacion (1-29) reduces to:
d).!n! ' = , (1-36)
which integrates to:
0.455
-f A-Li 0.06 (TT /
where A is the van Lriest flat plate parameter, given by
equation (2-32) and plotted in Figure 2-3.
3I4
21l
This result is expressed in terms of a flat plate
compressibility transformation for skin friction as follows
(in the .notation of Spalding and Chi (1964)):
1 A2
Cf = •Cf (Rn.Rx) where F =Af C f inc.
(1-38)
and F.Rx = (Te/Tw) +.A
The present theory and five other theories, the five most
popular and accurate knc;a4n, were compared to 427 adiabatic
and 230 cold wall heat tntsfer data points. The five
other theories were cast in the same form as equation (1-38),
but of course F and F were differtnt for each theory.
All theories were computed with the incompressible skin
friction formula of Spalding and Chi (1964), which apparent-
ly gives the best agreement with incompressible friction
data. Both the root mean square error and the mean abso-
lute error were computed. The results are shown in Table
1-1. For adiabatic wall, the methods of Spalding and Chi
(1964), van Driest (1956b) and the present theory are
equally accurate. For cold wall heat transfer, the present
theory has the lowest percent error, with !Mcore (1962)
second. The present method is thus the most accurate flat
plate theory available at the presený time, with the added
advantage that it can be extended to more general freestream
and wall conditions (White and Zhristoph 1971).
22
TABLE 1i-i
COMPARISON OF SIX THEORIES WITH FIAT PLATE FRICTION DATA
ADIABATJI: -;27 POINTS COLD WALL: 230 POINTS
AUTHOR RMS ERROR ABS % ERROR RMS % ERROR ABS % ERROR
Eckert (1955) 12.4; 9.06 29.45 25.56
Moore (1962) 8.87 6.54 17.69 13.08
Sommer and 9- l0 7.77 23-55 20.14Short (1955)
Spalding & 7.59 5-46 21.13 16.94Cýi (194)
Van Eriest#2 (1956b) 7.55 5.46 17.49 13.81
PresentTheory, 7.80 5-26 14.31 11.28Eqn. (1-37)
Note: P.MS Error = 1 12)
MEAN ABS Error Y lei1
Data Source: Wnite dnd Christoph (1970).
23
1.3.2 Data of Moretti and Kays
Next, the present method was compared to the data of
Moretti and Kays (1965). Their data were taken in a
rectangular duct which followed a system of rectangular
nozzles. This test section was designed to provide various
desired distributions of stream velocity and surface tempera-
ture. The velocities and heat transfer rates for these
experiments were low, so the boundary layers were essentially
constant property layers. However, the important effects
of wall temperature variation and freestream velocity
variation were present. In fact, predicting heat transfer
under varying freestreAm conditions, especially accelerat-
ing flows, has become of practical concern in such problems
as cooling of gas turbine blades and rocket nozzles.
Figure 1-4 shows the Stanton number as a function of
position along the rectangula- duct for a constant free-
stream velocity and for a sharp Cecrease in the wall :,empera-
ture. No data was given by More .ti ano Keys (1965) over the
first one-third of the flow field. Since the Stanton
number must be known at the starting point of integration of
equations (1-29) and (1-30), and since it was iesired to
predict the effect of the wall temperature decrease which
occurred in the firsk one-third of the flow field, equation-.
(1-29) and (1-30) we-e integrated forward and backward from
the first data poinc. Skin friction coefficients were not
given by Moretti and Kays, so the starting value of k was
24
chosen such -hat - 0.< c•. The curve-fit wall tempera-
ture distribution was chosen to be of the form
= - a - b cos -(y(x*-x*)/c (1-39)
where (a,b,c) were fitted constants and x* was the x*
position where the wall temperature change started. The
present theory and the digital computer finite difference
method of Herring and Mellor (19%) are compared to the
data in Figure 1-4. Agreement by both methods is excellent.
The second Stanton number distribution measured by
Moretti and Kays consisted of a wall temperature change plus
an accelerating freestream velocity. Again, the wall
temperature distribution was fitted by equation (1-39),
but the freestream velocity distribution was fitted by a
polynomial. The present method and the method of Herring
and Mellor are compared with the data in Figure 1-5. Once
again both methods accurately predict the data. A more
severe test of the present method, and of the method of
Herring and Mellor, is the steep favorable pressure
gradient as given by Moretti and Kays' data in Figure 1-6.
If one believes in a constant Reynolds analogy factor, one
would cxpect an increase in the Stanton number or at most a
c')nstant or slightly decreasing Stanton number. As seen in
Figure 1-6, the Stanton number decreases considerably.
Both the present theory and the method of Herring and Mellor
254
30()11200 5
0 R = 3.8 >r 10 5/footU (ft/sece
i1 1i I I -A0 10 20 30 40 50 60 70
X - in.
30 ooooookocooox(T -T ) !0e 20
0F 10
0 10 20 20 40 50 E0 70X - in.
0. 004
0.003
C h
0.002
-- PRESENT THEORY, EQNS. (1-29,30),--- ------ HERRING & MELLOR (1968)
0 I I ii0 10 20 30 40 50 00 70
X - in.
Figure 1.4. COMPARISON OF THEORY WITH ZERO PRESSURE
GRADIENT DATA OF MORETTI AND KAYS ,1965).
26
U (ft/sec)e300 T II2O00
200 DATP, MORETTI & KAYS (1965)
100 0 0 0 00 000000000
o I I I I I0 10 20 30 40 50 60 70
X - inches
30
010 I- 0
(-T 1000000 ix 0000• 0
0 10 20 30 40 50 60 70
X - inches
0.004
0.003
Ch
0.002
PRESENT THEWRV. EQNS. (1-29,30)0.00!I
[ ------HERZTNG AND MELLOR (1968)
01 I I I i I0 10 20 30 40 50 60 70
X - inches
Figure 1.5. COMPARISON OF THEORY WITH MILD FAVORABLE
PRESSURE GRADIFNT DATA OF ?&RETTI & KAYS (1965).
27
U (ft/sec)
300 I
20010 DATA, MORETTI & KAYS (1965
1000 01 00 0c 0
ol 0 1 0 1 0 1 0 1 1 1
0 10 20 30 40 50 60 70
X - incheýs
30 I I I I
20 0ua~pg:i 006(TeT) 0
10 800F oFr
0 10 23 30 40 50 60 70X- inches
0.004
0.003
Ch
0.002
PRESENT THEORY, EQNS.(1-29,30) "0- -
0-00 1HERRING AND MELLOR (1968) Oý.•
0 _ I I l I0 10 20 30 40 50 60 70
X - inches
Figure 1.6. COMPARISON OF THEORY WITH SEVERE FAVORABLE
PRESSURE GRADIENT DATA OF MORETTI & KAYS (1965).
28
U (ft/sec)300 !I
200
100 0
0 I I 1 I - I I
0 10 20 30 40 150 60 70x - inches
30
(Te -T ) I I I I I
0 DATA, MORETTI & KAYS (1965) _•.,= 0•y•Oooooo
10 - 0
0 oo ooqoooooopooooooooo6....0 10 20 30 40 50 60 70
X -- inches
0.004
0.003 K
0.002
0.001 PRESENT THEORY, EQNS. (1-29,30)
------.HERRING & MELLOR (1968)
oI i I I I I ,0 10 20 30 40 50 60 70
X - inches
Figure 1.7. COMPARISON OF THEORY WITH ADVERSE PRESSURE
GRADIENT DATA OF MORETTI & KAYS (1965).
29
qualitatively predict this decrease. However, neither the
Dresent method nor the method of Herring and Mellor are
able to fully recover after the favorable pressure gradient
has ended - the present method doing somewhat better than
the method of Herring and Mellor.
The effect of an adverse pressure gradient on the
Stanton number is shoun in Figure 1-7. There is a notice-
able difference between the present method and the method
of Herring and Mellor toward the end of the flow region.
The present method shows a slight increase in the Stanton
number, whereas Herring and Mellor show a slight decrease.
The Jata of Moretti and Kays remain approximately constant
and are in reasonable agreement with either theory.
l.LL Conclusion
The purpose of this chapter was to develop a simple
and accurate method for analyzing heat transfer and skin fric-
tion in two-dimensional, compressible turbulent boundary
layers. This method uses law-of-the-wall velocity and
temperature correlations which include pressure graident,
heat transfer, and compressibility effects. After integrat-
ing the continuity, momentum, and energy equations across
the boundary layer, two coupled differential equations
result which have as their dependent variables the skin
friction coefficient and the Stanton number. Contrary to
the von Karman technique, integral thicknesses anJ shape
factors are not used. The present heat transfer analysis,
which is for arbitrary freestream velocity and wall
temperature distributions, compares favorably with the data
of Moretti and Kays (1965) and is as accurate as the
sophisticated finite difference technique of Herring and
Mellor (1968).
31
Chapter Two
Axisymmetric Compressible Flow
2.1 Introduction
In this chapter, our basic approach is applied to
derive a method for computing skir friction in a thick
axisymmetric, compressible turbulent boundary layer. The
analysis is restricted t,. steady flow and a perfect gas is
assumed for convenience. The freestream Mach number and
pressure gradient may be arbitrarily large and variable.
Nonadiabatic wall temperature is allowable, but the
variations should be "modest", that is, the streamwise
derivative of wall temper-cure is neglected. The tempera-
ture is eliminated in favor of velocity through the Crocco
approximation. It has already been shown that this
simplification has little effect on the accuracy of skin
friction computation (White and Christoph 1972).
The transverse curvature effect is introduced through
a coordinate change suggested by Rao (1967). The resulting
differential equation for skin friction (Eq. 2-25) is thus
valid for arbitrarily large ratios of the boundary layer
thickness to the body radius.
Rao's transformation was used by White (1972) to
compute the skin friction in turbulent incompressible flow
past a long cylinder of constant radius. The basic results
of that paper can be summarized as two formulas for turbu-
32
lent incompressible skin friction on very long cylinders:r 04 R /
0.0015 + 0.20 + 0.016 (x/ro) 0 x (2-1)
C = 0.0015 + 0.30 + 0.015 (L/ro) 0) 4 0 1/3
Here ro in the cylinder radius, x is the axial distance
from the leading edge, and ?x = U x/!, is the local Reynolds
number for an assumed constant freestream velocity Uo. 0 Me
quantity CD is the total friction drag coefficient on a
cylinder of length L. EquatVins (2-1) are accurate to + 5%over the entire turbulent flow range (106<R. < 109) and for
cylinder lengths up to (L/r) 0. Similar results will
be obtained here for compressible flow along a cylinder.
2.2 The Axisymmetric Law-of-the-Wall
In order to use the rresent me-hod, it is necessary to
have a realistic expression for the law-of-the-wall in a
thick axisymmetric boundary layer. This law differs
considerably from the two-dimensional law because of the
effect of the radial variable in the moment.um equation.
With reference to Figure 2-1, let the body radius be r o(x).
Within the sublayer, both the convective acceleration and
the pressure gradient terms in the momentum equation are
negligible. The equation reduces to:
33
=(2-2)or: r = ro Sw constantA r %w •"
With T thus approximated by the viscous stress only, we may
integrate Eqn. (2-2) to obtain an approximate velocity
profile in the sublayer:
Lw u" ro0 w ln(r/r 0 ) (2-3)
If we now define the shear velocity ue such that rw = Pw u*2
Eqn. (2-3) may be rewritten in law-of-the-wail form:
+ r+ ln(r/rt) (SUBLAYER) (2-4)0 +
where u =u/u* and rU pwu*ro0 /w. Equation (2-4) is in
marked contrast to the two-dimensional sublayer, for which
Figure 2.1. DEFINITION SKEITCH ILLUSTRATING COORDINATES
FOR AXISY.METRIC BOUNDARY LAYER FLOW.
34
+ + +u simply equals y , as in Chapter 1. Since y correlates
the entire wall region in two-dimensional flow, it was
suggested by Rao (1967) that the variable ro0 ln(r/ro) in
Eqn. (2-4) would similarly correlate an axisymmetric flow.
Thus Rao's hypothesis for incompressible axisymmetric flow
is that
u fcn(Y), where Y = r0 in(r/ro) ) (2-5)
Equation (2-4) would hold in the sublayer, while the loga-
ritIhmic overlap layer would be characterized by the relation
U i in(Y+) + B ( - 0.40, B 5.5) (2-6)
Rao (1967) and later White (1972) showed that Eqn. (2-6)
is an excellent approximation for P wide variety of thick
axisymmetric boundary layers. As the boundary layer becomes
thicker, this axisymmetric wall law becomes valid across
the entire boundary layer, that is, the outer or "wake"
layer becomes vanishingly small.
Note that Eqn. (2-6) is consistent with a turbulent
eddy viscosity c and mixing length L given by
Sp2 Y (r/r) - cos(a) , (2-7)
or t = •Y (r/r)1/2r0
where a = tan-1 (dro/dx). For this report, we shall take
35
So 00, that is, we neglect the slope of the bod;' and take,
approximately, r = r0 + y. :he error is negligible except
near the stagnation point, which does not concern us. Since
Y as defined by Eqn. "2-5) is esaller than y, k as defined
by Eqn. (2-7) is smaller than its two-dimensional equivalent
I = Ky, the physical explanation being that a cylinder has
less ability to create 'urbulent shear than a plane surface.
Equation (2-7) may be used to derive a law-of-the-wall
for compressible flow with pressure gradient. As in the
two-dimensional case (White and Christoph 1971), we neglect
the convective a*celeration, and the axisymmetric streamwi:ie
momentum equation becomes
dP e_r r.j - 0 (2-8)
Integrating from the wall (r=r ) outward, we have:
o de (r2 ro) (2-9)
This may be integrated again by assuming that viscous shear
is negligiblc and that T = £ (Au/Ay), where c is given by
Eqn. (2-7). Substituting for . and rearranging in terms of
law-cf-the-wall variables, we obtain:
du 1 1()I/2 " + a 2Y +r-'+ y+ o(e 0 1)
(2-10)
where a = dPe
w
36
The pressure gradient parameter a is the same quantity
which appeared in previous analyses for two-dimensional
flew (Chapter I and Wh. te 1969).
Equation (2-10) can be integrated for the velocity
profile u +(Y+) if we make an assumption about the density
variation. Since we are concerned only with predicting
skin friction, we assume a perfect gas,
p = p R T, or: pw/p-T/T . (2-11)
2plus a Crocco approximation, T 1- a + b u + c u , which may
be rewritten in terms of wall variables as follows:
+ +2
T/T 1+ u -+ , (u-12)
where 5 and y are the same parameters used in previous
work (White and Christoph 1970, 1972):
13 = qwu/(TwkwU*) = heat transfer parameter;
(2-13)
Y = r u, 2 /(2c pTw ) = compressibility parameter.
Equations (2-10) and (2-12) may now be combined and integrat-
ed to obtain the desired lpw-of-the-wall u+(Y+,a,'3,y). Of
particular interest is a closed form which can be obtained
for the special case (a=O,B=O):
1 si "ln(Y+/y+) (2-14)u(Y+'O0'0'Y) = in-
4
I
where we take = 0.1108 so that the formula reduces in the
limit of large radius to the two-dimensional law-of-the-wall.
373
Equation (2-14) corresponds to supersonic adiabatic flow
along a cylinder, for which experimental thick boundary
layer data is available (Richmond 1957).
Some velocity profiles obtained from integrating
Eqn. (2-10) are shown in Figure 2-2. The effect is generally
the same as in the two-dimensional flow study (White and
Christoph 1970), with positive a (adverse pressure gradient)
raising the curves above the incompressible log law and y
(the compressibility effect) tending to lower the curves.
Note that the parameter r+ has no effect on the curves
unless a is finite.
2.3 Derivation of the Besie Differential Equation
We now assume that the compressible law-of-the-wall
for a thick axisymaetric turbulent boundary layer is known
as the integrated combination of Equations (2-10) and (2-12):
++ + 1 1 2
0 (2-15)+ + +.-:-F1 -ro (1- e2Y /ro) +
This relation, if accepted as a reasonable approximation
across the entire boundary layer, provides closure to the
problem of computing the wall shear stress on a body of
revolution. That is, we can now derive a single different-
ial equation for Cf(x) under arbitrary flow conditions.
The boundary layer equations for compressible, axi-
symmetric, turbulent boundary layer flow are given by, for
38
50 1oII IIII I I1111111
: = 10 r+ = 100:0 0
40 (.02,0,0 02,0,0) I2(.02,0,.001(.02,0,.001 .o,,oo)-
Sr+ = 100:30 (803,0,0) -
30 -, 6
20
10
I0
10 100 1000 3000
y r tn(r/r )0
Figure 2-2. SOME EXAMPLES OF THE AXISYNMETRIC LAW-OF-THE-WALL,
COMPUTED FRMM EQUATIONS (2-10) and (2-12).
339 I
example, Herring and Mellor (1968):
a) Continuity:
I(pur) + =(pvr) 0 (2-16)
b) Momentum:
pur • - - r e +
(2-17)Y(rT)
c) Energy:hh
"h° + pvr = r(q + (2-18)
where the notation is the same as in Figure 2-1. As in
Chapter 1, the quantity ho = h + u 2/2 is the stagnation
enthalpy. Now in fact Eqn. (2-16), although included for
completeness, is not needed here, since the temperature relates
directly to velocity through the Crocco approximation,
Eqn. (2-12). As mentioned, the effect of this simplification
upon skin friction is slight, but knowledge of the ýtanton
number distribution is lost. If desired, one could extend
the temperature wall law technique of Chapter 1 to this axi-
symmetric case for computing wall heat transfer.
The continuity relation (2-16) is satisfied identically
by defining an axisymmetric stream function i. such that:=pur ; = - pv r (2-19)
Y p u r p vr
[ 40
Since p, u, r, and y are each related to wall variables, it
fellows from the first of Eqns. (2-19) that # itself is a
law-of-the-wall xariable:
2 U"+ +
t/v-wro =.L P dY+ fcnFY ,a,.,13r v (2-20)0 Pw ro
This fact is important in computing the velocity v from the
second of Eqns. (2-19). Utilizing wall law variables
wherever possible, the momentum relation, Eqn. (2-17), now
becomes:
p u+ Ž•(v•u+) - •.k -.,, Ž•( xu+) = - r -A 'Iw- ~y+
(2-21)
Vw AV
where ui (Tp, is the wall friction velocity. The y+
derivatives are related to Rao's variable v+ through the
relation
A % • (2-22)•y+ r o 0 ;y+•
which follows from Eqn. (2-5). The x derivatives must be
handled by the chain rule, since each of the parameters
(a,,.Y, ro) in the law-of-tne-wall is a function of x.
Thus we substitute
2týj, + 2-a ._ + Z1 XL%x ;x + + x (2-23 x
+ (2-23)+ •r0
•xrO
i m mm • • i m N mm ~ • mmnM • maM mm • mm iI i I .41•
In carrying out the partial derivatives in Eqn. (2-23), we
rctain our previous approximation (White and Christoph 1970,
?972) of neglecting the derivatives with respect to x of
j -w ,LL'0, and qw wherever they appear in ,5, -y , etc. It
is felt that these terms have only a small effect on wall
skin friction.
It remains only to carry out our basic integral
procedure, namely, combining Eqns. (2.-21, 22, 23) and inte-
grating the entire equation with respect to Y+ across the
boundary layer from (Y_ = 0, ' = rw) to (Y+ = Y +,r = 0).
The result is a single first order differential equation for
the wall friction velocity u*(x). It is convenient to
define dimensionless variables as in Chapter 1:
= "2- f ; x* = x/L ; V = Ue(x)/Uo , (2-24)
where Cf = 2¶/(PeeU), and L and Uo are suitable reference
values. In terms of these variables, the final integral
momentum equation becomes
d"x*(A - 3aA2 ) + V VX(F - A1 ) - .1 A2 (1!V)''
1 (2-25)
0
1 12where RL = (%L/ve)("r a)(T/T) and F =
2 PY-+/r+r0(e e -
42
Equation (2-25) is the central result of this chapter - a
first order ordinary differential equation for X(x*),
subject only to some assumed initial condition X(X*0) and
known flow conditions U (x), T (x), Tw(x), ro(x), and Me(X).
The functions (AV, A2 , A.) arise from the integration
across the boundary layer and are defined as follows:
Ye = (T e/TlXp(2Y /ro) r u+2 - Ou + 6u-+ 2Y u + ;)A~R, 2/w Y
+ au • + r ayaY+ dY+ (2-26)11wr--' Ii "w - wro ay •Y+
Ye ++ + + al1A- t exp(2:+/ro) U+ (127)2 T . ; a W r ( 2- 2 7 )
+Y+ +
1 (u+/Ti)exp'2Y+/r+) r(u+) jdY$3 o Vwo NO(2-28)
where r()= (Y+ -+ r) -y+ + r0
Is
243;1
In order for Eqn. (2-25) to be viable, it is necessary to
have analytic approximations for these coefficient functions.
After extensive numerical computation of Eqns. (2-.26, 27,
28), using Eqn. (2-15) for the wall law, we offer the
following curve-fit approximations:
A T A3 E=•.0 expf3 (I4 -+ 0.2 S+001 2'A1 2 " (2-29)
where S exp(O.4A /A)/r+ ,0
+ =,ro RL%/A L)
= 1 r+++r 2-- 0 {exp(2Ye/ro) -+ )
A- 0.066' expl 0.84 x + 0.2 S + 0.0018 s 2 (2-30)
A- A (i+0.2 Q)
and where
Ye 0.1108 exp( 0.4C - 2.2 Q2 /3) , (2-31)
where I = sin-l1 •(2vu+- 1)/(t2+ 4-)1/2}
Equation (2-31) is a curve-.'it to the law-of-the-wall itself,
Eqn.(2-15), and Y+ is needed to evaluate the pressure gradiente
parameter Q above. Note that, in the limit of very large+r° Q approaches (rt+), which is the incompressible two-
dimensional parameter originally used by White (1969).
44
The parameter "A" in these formulas accounts for the
combined effect of well temperature and Mach number, as
was done in White and Christoph (1972). It is the same
parameter defined in a flat plate analysis by van Lriest
(1956):
A = (TaT - 1) /Fsin- (a/c) + sin-l(b/c)1 , (2-32)
where a = (Taw+Tw)/Te - 2 ; b = (Taw -Tw)/Te ;
SF(Ta+Tw)/Tel' - 4Tw/Te
Values of A for various Mach numbers and wall temperature
ratios are shown in Figure 2-3. Note that the incompressible
adiabatic limit is A = 1.
2.4 Comparison with Experiment
Equation (2-25) may be used to compute the turbulent
skin friction distribution Cf(x) = 2/i2 (x) along a body of
revolution in arbitrary subsonic or supersonic flow, provid-
ing only that variations in wall temperature are not too
great. A starting value )>o(x=Xo) is needed; if the
computation is started at the transition point, the value
S= 20 is recommended. The solution of (2-25) may beo
continued downstream until the separation point (if any) is
predicted to occur when the coefficient (Ai-3aA2 ) vanishes.
We now consider three applications: a) flow along a
slender cylinder; b) supersonic attached flow past a cone;
45
ADIABATIC WALL:
5
4 4
3 2
A
2
010 5 10 15 20
MACH NUMBER - Me
Figure 2.3. THE COMPRESSIBILITY AND WALL
TEMPERATURE PARAMETER A, FROM EQN.(2-32).
46
and c) supersonir- :-ow pa!t slender body of revolution.
2.4.1 Compressible Flow Along a Slender ýylinder
Tne application of the p.resent theory is to consider
the effect of compressibility and transverse curvature with
no pressure gradient. A cylind!r has r° = constant and, to
excellent approximation, Ue = U = constant. Equation (2-25)
reduces to:
Ad,A1 d-* = . , (2-33)
which may be separated and integrated for a direct relation
between skin friction and local Reynolds number:
1/2 1"" ' R Al(),O,F,,v,ro) di(2-34)
x0
With zero pressure gradient, -he curve-fit approximation
to A1 is:
A 1- 8.0 exp. (0.4.8 + 0.0018 Z')1 , (2-35)L J
where Z = exr(O.; 0.+,A/r+0
Further, ro is directly related to A and the radius Reynolds
number Ra = U ro/Ve, as follows:R. eo e
+ _112r h/ \(A/e)(TTe) 1 (2-36)
Since R is constant for the cylinder, Eqrs. (2-35,36. ma-,
be substituted into Fan. (2-3'4) and the Integration ccr-4ed
.7
out for the desired relation Rx (X), or better, its inverse
Cf = Cf(RxRa). As an example, the computed skin friction
on an adiabatic cylinder at a Mach number Me = 5 is shown
in Figure 2-4. It is seen that there are substantial
transverse curvature effects which raise the skin friction
over its flat plate value. As Ra becomes very large, which
is equivalent to the cylinder being very short, the flat
plate skin friction is approached from above.
Since the integral in Eqn. (2-34) cannot be found in
closed foi.i. it is appropriate to find a curve-fit formula
for the results, in the spirit of the correlations (2-1)
proposed for incompressible flow. It was found that the
flat plate formula of White rind Christoph (1972) could be
generalized to include the effect of x/ro = Rx IR as
follows:1
Cf o.455/1[A (b R 'ew (Te/T A) (2-37)
where b C 0.06/fl + O.025(x/ro )6/7]L~ 0
This formula is valid with accuracy to +iOC, over the entire
turbulent Reynolds number range (105-109) and for Mach
numbers from zero to ten. The slenderness ratio (x/ro) may
be as high as 10; note that the formula predicts a roughly
5increase in skin friction for (x/rt) as low as ten. For
very small (x/r 0 ), the formula reduces to the compressible
flat plate formula, Equation (24) of White and Christoph
(1972).
48
Equations (2-•) or (2-37) may be compared with
experimental data of Richmond k1957) for supersonic
turbulent flow past long cylinders. Richmond generated a
supersonic flow at M 5.8 past three cylinders: ro = 0.012,
0.032, and 0.125 inches, respectively. For the largest
cylinder (0.125"), skin friction was measured with a floating
element balance for one condition (Ra = 20,400) and was
estimated from the velocity profiles for the other two
cylinders. The authors attempted an alternate skin
friction determination, using a sort of axisymetric
"Clauser-plot" which may be inferred from our postulated
law-of-the-wall, Eqn. (2-15). Since the cylinders are
assumed adiabatic and the pressure gradient is zero (a=f=O),
Eqn. (2-15) may be integrated in closed form to give the
following expression:
u+(Y+,0,0,v,'o - 1 d ii In(Y+/Y) , (2-38)
where = 0.1108 -o match with the incompressible log law
in the limit. By introducing -he definition of v from
Eqn. (2-13) and rearranging, we obtain an equivalent
logarithmic law:
i lrrYiO.1N) Y where a = U sin-1 (U/U•/u¢ •- 'r (/r)(2-39)
and where U = (2c 'Tr )I/2
49
0.000
0.00000
10 610O 7 10O 1
R'C
Figure 2-L4. SKIN FRICTION ON Ag !DIABATIC CYLINDER AT
A FREESTREAM MACHl NUMBER OF 5 FOR VARIOUS VALUES
OF R a= U r /v ,COMPUTED~ FROM EQUATION (2-34).
50
This relation shows that a plot of the "reduced" velocity
u versus Rao's variable Y4 should be semi-logarithmic and
follo-i the incompressible law. Further, if the velocity
profile u(y) is known, one may use Eqn. (2-39) to infer the
friction velocity u* and h-ence C f itself. That is, if we
eliminate uL in favor of Cf. Eqn. (2-39) may be rewritten
as follows:
rc 7= (C-4o
where r = (5/Ue)(Te/Tw) and R =a tn(r/ro)
(%e/Uw)(Te/Tw)1/2
Equation (2-40) may be plotted with C, as a par i.eter, as
shown in Figure 2-5. When data for r versus T is placed or.
this chart, it should possess a logarithmic overlap layer
from which one can Infer C, by interpolating between the
plotted parametric lines. This has been done in Figure 2-5
for four supersonic cylindrical profiles given by Richmond
(1957). Note that all profiles demonstrate the proper
slope and behavior and thus substantiate the present law-of-
the-wall approach. In all four cases, the inferred Cf is
lower than that reported by Richmond (1957). Table 2.1
summarizes the skin friction data and compares with two
theories: the more "exact' integral computation of Eqn. (2-34,
and the curve-fit formul of Eqn. (2-37). Also included for
comparison is a flit plate computation for Rx in the same
51
0.6
0.5 /• 'r'o
U e /T
/30.4
0.3
0.2 -EXPERIMENT OF RICHMOND (1957)
o ....... 2510 .. 0.012
0.1 V ....... 6730 0.032-* ...... 23300 ... 0.125
A ...... 26400 ... 0.125
0 1 V I Ii iI . I 1 1 111111 I4 4
100 1000 10 "xlO1
Ra(ul/e/)(T /T) Tn(r/r)
Figure 2.5. SKIN FRICTION DETERMINATION FOR SUPERSONIC FLOW
ALONG A SLENDER CYLINDER BY A MODIFIED AXISYMMETRIC
CLAUSER-PLOT (EQ. 2-40).
52
range. Note tnat transverse curvature has increased cf
by at least 40% to as much as 100%. The present theory
seems in good agreement, falling between the two skin
friction estimates (Richmorid and the "'Clauser-plot"). To
the author's knowledge, no other theory, whether integral
or finite difference In nature, has been applied to these
important thick axisymmetric supersonic boundary layer
experiments. -4ichmond (1957) also reported low speed
friction and velocity profile measurements along slender
cylinders; these were analyzed in the incompressible theory
of White (1972).
2.4.2 Supersonic Attached Flow Past a Cone
A classic problem in the literature is that of super-
sonic flow past i cone at zero incidence. If the resAiting
shock wave is attached to the cone vertex, the freestream
flow along the cone surface is approximately constant
velocity (ax = 0). This is somewhat analogous to the
cylinder case of the previous example, except that here the
surface radius r0 varies. If x is along the cone generators,
we have
ro x sin , (2-41)
where 0 is the cone half-angle. It is of interest to try
and relate the cone skin friction to an eaulvalent flat
plate c- cylinder at the same Reynolds number RX. Accord.a-'r
53
i
to the classic turbulent analysis by van Driest (1952),
C f at position Rx on the cone is equal to -f on a flat
plate at position (Rx/2). Thus, for equal Rx, the cone
skin friction is 10-15% higher than the plPte C f. This
analysis of course does not account for boundary layei
thickness effects, which the present theory includes as+
the (r ) effect on tha law-of-the-wall. With U, = constant,0
Eqn. (2-25) reduces forcone flow to:
drA1 dx* +A 3 - 7 (-2
and, since T is constant, (/ro )(dr 0/dX*) = "/x* from
Eon. (2- 2 1). If transverse curvature is important, i.e.
if the boundary layer is thick compared to the cone radius,
then A and A depend upon r+ (x,x*) and the variables are1 3 o
not separable in Eqn. (2-42). However, if we assume a
thin boundary layer (large r+), then our curve-fit
approximations give:
A1 , 1 - a e (a -. O, b ... 48.A)
(2-43)
:'or this case, Eqn. (2-42) has a closed form solution:
i _b R, x
which may be compa-ed to the analogous solution for flat
plate flow, Eqn. (2-37) or (1-36):
54
-w -
Inx x,*/a (2-46)
By comparing (2-44) and (2-45), one may deduce a first-
order turbulent "c .ne rule":
C (at RX) = Of (at (/FI+0.96/A) (2-46)fcone plate
This may be compared with the classic cone rule developed
by van Driest (1952):
C, (Rx) = Cf ( RX/2) (47)Scone plate
From Figure 2-3, A - 1.0 at low Mach numbers, so that the
correction factor for Reynolds number equals 1.96 approxi-
mately, or very close to van Driest's factor of 2.0. A
later analysis by Tetervin (1969) gives a factor of
(2.0 + 1/N), where N is the exponent in an assumed (,/Nth)
Dower-law relation between Cif and momentum thickness
[ Reynolds number. Si~ice N is of the order of 5 to 9,
increasing with -Reyn~olds number and Mach number, Tetervin's
factor is also about two. Figure 2-e-.a sh.ows values of
the ratio of cone to plate friction coefficient as
calculated from Eqn. (2-46). Thne predicted increase in .
cone fricýýIon is of the order of 5 to 15%, being higher
at low Reynolds numbers.
in fact, howeer, neither Eqn. (2-46) nor the van
Driest or Tetervin analysis is accurate for small cone
55
1.2
Cf (cone)
C (plate) M e 2.0f 5.0
10.01.i __20.0_
1.1010 10o6 10o7 10o8 19R
x
a) THIN BOUNDARY LAYER ASSUMPTION: EQN. (2-46).1..4 I -
1. 1 CONE H ALF-ANGLE:
1.31320°
C f (cone)000
Cf (plate) o 0f 451.2
12.2.0
M 2.0 THIN LAYER -e
"
ASSUMPTION
1.0 I I A
105 106 107 108 109Rx
b) EXACT CALCULATIONS FOR Me = 2.0, EQN,(2-42).
Figure 2.6. ILLUSTRATION OF THE TURBULENT CONE RULE.
564
angles, because of the transverse curvature effect
(large 6/ro). That is, a numerical or graphical solution
of Eqn.(2-42), using the actual formulas fcr A and A.3 frcm
Eqns.(2-29,31), shows a further increase in cone skin friction
at the smaller cone angles. These numerical solutions are
illustrated in Figure 2.6.b for a freestream Mach number of
2.0 along the cone surface. Thus transverse curvature adds
an additional 10 to 30% to the skin friction, and only the
thicker cones (20° to 450 half angle) approximate the first
order cone rule above (2-46,47). By analogy with the cylinder
analysis (Eqn.2-37), the relevant transverse curvature
parameter is (x!r 0 ) = csc(cf) for a cone. Hence the numerical
solutions in Figure 2.6.b may be correlat-d into the following
formula for constant pressure (attached) flow along a cone:
Cf(cone) 2 24 .455 (i -- l (2-48)A 2n11b F i( T/T" (I4e/9w )/A
e -l
where b = 0.06(1 + 0.96/A)- (1 + 0.03 csc T)-i.
This formula ir- accurate to about +_% over the complete
supersonic turbulent cone flow range, including constant
nonediabatic wall temperature, which is accounted for by
the van Driest parameter A in the formula (Eon.2-32).
57
rvY)
000 0 0 00000 0 0
U.' e, 0
CA Jý v*0 0.
LI'
HH
000 0 0
9z 0
z
wx E- U
E-1~c o '
0) 00 0 0 00 0 0 0 00
E- 4-7
00
000
C.H co C% -1
0
w0 0 0 0 Iy" H
Lr Cu u C\.
-0 0
1 .00 0 0 0
58
2.4.3 Supersonic Flow with Pressure Gradient:
For 11ow past a general body of revolution, the complete
differential equation for skin friction, Eqn.(2-25), should
be used, since a and r 0= r (x). In one sense, these
effects are mutually exclusive, because if pressure gradients
are important, transverse curvature is probably not important,
and vice versa. That, is, only a slender body can have large
(6/ro), but a slender body cannot have large pressure gradients.
However, the shape effect term (A3/r )(dro/dx*) is nearly
always important.
We give here a tyvpical solution of the general theory
by comparing Eqn.(2-25) with the skin friction measurements
of Allen (1970) for supersonic flow past a so-called Haack-
Adams body of revolution, shown to scale in Figure 2.7. The
shape of the body is given by the following formula:
r /rmax = 0.70(1_t2) o.16934 cos- t + t(l-t 2 )/2 1/2
where t = (1 - 2Z/L). (2-49)
Note that Z is the axial, not the surface, coordinate (see
Figure 2.1). Allen (1970) tes4eid a wind tunnel model with
L = 36 inches and r 1.3- !. inches. Skin friction at seven
stations was measured with a Preston tube and also estimated
by 1) the 3aronti-Libby law; and 2) the Fent-r-Stalmach law.
The upstream MYcl number was varied from 2.5 to 4.5, and
the example shown (M. = 2.96) is typical of both the data
and the theory. With L = 36 inches, the Reynolds number RL
6in Eqn.(2-25) was equal to 3.OxlO6. The freestream Mach
number distribution Me (x) is shown, and the velocity
distribution is of a similar shape and was curve-fit by
the expression
V(x*) -" 1 + 0.156(t - t 2 ) , t = x* - 0.139. (2-50)
Figure 2.7 shows three different analytical estimates:
1) the exact theory, Eqn.(2-25); 2) a flat plate theory
based on local Rx, Eqn.(1-36); and 3) a long cylinder
theory, Eqn.(2-37), based on local Rx and r 0 (x). It is
seen that the cylinder and flat plate formulas give an
estimate of the average skin friction on the body. The
difference between these two is about 5%, which representsthe transverse curvature effect. The exact theory is in
excellent agreement with the data over the entire body.
The higher values of Cf (compared to a flat plate) at the
front of the body are due to roughly equal contributions
from I) the term (A3 /ro)(dro/dx*); and 2) the favorable
pressure gradient. Similarly, the lower values at the rear
are due to 1) a change in sign of (dr 0 /dx*); and 2) the
adverse pressure gradient. The same excellent agreement
with experiment w•s found for the other three conditions
(M = 2.5, 3.95, 4.5) investigated by Allen (1970).
60
3.2
M DATA, ALLEN (1970) 0 0
3.0 0- 0
2.8 0 I I I I0 0.2 0.4 0.6 0.8 1.0
x*= x/L
0.003 -- DATA, ALLEN (1970):
O = BARONTI-LIBBY LAW
f3 = FENTER-STALMACH LAW
f = PRESTON TUBE
0.002--
THEORY:EXACT, EQN. (2-2 5)CYLINDER, EQN. (2-37)
-------- FLAT PLATE, EQN. (1-36)
0.001-
m= 2.96
HHAAACKK-ADDAMSS BODY Y
f I I III
0 0.2 0.4 0.6 0.8 1.0x*= x/L
Figure 2.7. COMPARISON OF THEORY AND EXPERIMENT FOR
SUPERSONIC FLOW PAST A BODY OF REVOLUTION AFTER ALLEN (1970).
61
2.5 Summary
This chapter has presented a complete theory for the
computation of turbulent skin friction in compressible
axisymmetric flow past arbitrary bodies of revolution. The
approach leads to a single first order ordinary differential
equation, Eqn. (2-25), for C.,.(; ) and accounts for supersonic±
flow, nonadiabatic wall temperatuie, mad extremely thick
boundary layers, where a transverse curvature correction is
necessary. Applircations to cylinder flow, cone flow, and
flow past a pointed body of revolution all show good
agreement with experiment. It is therefore thought that
the present theory has no significant deficiencies. Siince
this theory is also apparently the simplest comparable
analysis to be found in the literature, we therefore
recommend it to engineers for general usage.
62
Thapter Three
The Prediction of Three Dimensional Skin Friction
3.1 Introduction
The most common type of boundary layer encountered in
practice, and the most intractable from the standpoint of
analysis, is the three-dimensional turbulent boundary layer.
Reviews of the state of the art with respect to this impor-
tant class of flows have been given by Cooke and Hall (1962),
Joubert, Perry and Brown (1967) and more recently by Nash
and Patel (1972). Existing theories are few and seem unduly
complicated while available experimental data are usually
unreliable. Historically, two approaches have been used to
analyze three dimensional turbulent boundary layers; the
integral approach and the differential approach.
Typical of the integral approach is the work of Smith
(1966), and Cumpsty and Head (1967). These authors used two
momentum integral equations plus an auxiliary equation to
account for variations in the streamwise shape factor. They
extended Head's (1958) two-dimensional entrainment relation
to three dimensions but retained his two-dimensional empirical
functions.
Differential methods have been given by Bradshaw (1971)
and Nash (1969); both are extensions of the two-dimensional
method of Bradshaw et al (1967) which utilizes the turbulent
energy equation. The major difficulty with differential
63
S.............................................. .••, , , , - •, ,, , , -
methods is in the specification of boundary conditions. For
a given domain, boundary conditions must be specified along
all surfaces where fluid is entering from the outside.
In this chapter a new integral method is developed
which greatly reduces the computational difficulties inherent
in the calculation of three dimensional skin friction. The
analysis results in two coupled, non linear partial differ-
ential equations with the skin friction coefficient and the
tangent of the angle between the total surface shear stress
vector and the shear stress vector in the freestream direction
as the only unknowns. This is accomplished by assuming
suitable "law-of-the-wall" velocity zorrelations for both
the freestream and cross flow directions and then inte-
grating the governing equations with respect to the law-of-
the-wall coordinate y+. The resulting partial differential
equations ray be easily solved by finite difference tech-
niques since they require initial data along only a single
curve.
Thus the new method is a compromise between the "classi-
cal" integral and differential techniques. It is much more
straight-forward and contains considerably less empirical
content than the former, and is computationally much
simpler than the latter.
At the present time this method is limited to incompress-
ible flows; however, extending it to account for compress-
ibility, heat transfer, and variable fluid properties should
b4 possible in view of chapter 1 of this report and the past
64
efforts of White and Christoph (1970).
In order to discuss three-dimensional bounJary layer
flows it is most appropriate to work in a curvilinear
coordinate system composed, in part, of the projection of
the free streamlines onto the boundary surface. The
coordinate along :hese free streamline curves is designat-
ed as s, while the coordinate along the orthogonal
trajectoriez of the free streamlines, in the surface, is
designated as n, and the coordinate normal to the surface
is designated as y. Corresponding to these three coordinate
directions are the metric coefficients hl, h 2, and h 3, and
the mean velocity components u, w, and v respectively. For
simplicity, it shall be assumed that the curvature of the
bounding surface does not change abruptly and that the
boundary layer thickness is small compared with the principal
radii of curvature of the bounding surface, so that h3 may
be taken equal to unity and y becomes a simple distance
normal to the surface. As a consequence, h1 and h2 are
known functions of s anId n provided that the inviscid flow
over the surface is known. This coordinate system is shown
in Figure ?.J.
(h + (h W) + 0 (3-1)
the s component of the momentum equation
65
Y
w
h2 5n,w
CROSSFLOW TOTAL SURFACE
SHEAR VECTOR
hls,u
FREESTREAM
Figure 3.1. THREE-DIMENSIONAL BOUNDARY LAYER
COORDINATE SYSTEM WITH SKEWING
IN ONE DIRECTION ONLY.
66
u • u w - +ua u + u w ý h l w h
hAP + u(3-2)
ph1 'A~ S P sAphI •s p •y
and the n component of the momentum equation
U4+ + ? w + uw >h2 u2 h IK1s 2 :n AY 17 5s E nh 1 12 1 2
-1 p + .1 n (3-3)p--2 A n p -Ay
In these equations rS and Tn represent total shearing
stresses a-nd include the effects of viscosity and Reynolds
szress, Since only incompressible flow is to be considered,
the energy eoua~ion is not needed.
The distinguishing feature of a three dimensional
boundary layer is crossflow (the w component of velocity)
and the resulting vector character of the surface shear
-,tress. Crossflow occurs when the freestream streamlines
are curved, which gives rise to an unbalanced centrifugal
pressure gradient in the boundary layer. Such crossflow
imposes a shear stress on the surface, normal to the free-
stream direction ('n), which skews the to-cal surface stress
vector. If the direction of curvature of the freestream
streamline is reversed, skewing ir two lateral directions
can occur in a single velocity profile. Typical velocity
profiles showing skewing in one and two directions are shown
y
W=0 ee
tan-e
h 1 ,ls.u
h 2 1,n,w TOTAL SURFACE FREESTREAM
CROSSFLOW SHFAR VECTOR
Figure 3.2, THREE-DIMENSIONAL BOUNDARY LAYER
WITH BILATERA. SKEWING.
68
tn Figures 3.1 and 3.2.
Since tht present method depends upon an apriori
specifica-ion of -velocity correlations in both the free-
stream direction and the crossflow direction it would seem
appropriate to discuss these in some detail.
-.2 Specification of Velocity Correlations
Ln general, integral techniques are insensitive to
de~ails of the assumed velocity correlations. However,
this does not mean that one can be completely cavalier in
their specification- In order to insure that the chosen
correlations incorporate the correct physical parameters,
consider that near the wall the force balances given by
equations (z-2) and (3-3) reduce to
and= + 1 (3-5
"n nw K Y (3Fn)2
Assuming tha; the shear stress in the freestream
direction in a three-dimensional boundary layer may be
represented by the same eddy viscosity relation as for a
two dimensional boundary layer gives
_2 2,auauPK = Y I's-yF (3-6)
with von K~rman's constalnt K equal to 0.4. Nondimensional-
69
izing equations (3-4) and (3-6) with respect to wall
quantities and solving for the velocity gradient gives
+ + 112.; u = ( I + a s y ) 3 -7 )
The pressure gradient parameter a. is defined by
where u* = (•swp)1/2 is the shear velocity in the free-
stream direction. Equations (3-7) and (3-8) are exactly
the same ,'esults as have been previously obtained by White
and Christoph (1972) for the two-dimensional case; except
that here a. includes the curvilinear scale factor h
Therefore, it will be assamed that the law-of-the-wall
co.relation for mlie streamwise vel( city ccmponent, in a
three-dimensional boundary layer (Eqn. z-7) does not differ
from the two-dimensional correlation '•is is a reasonable
assumption as has been pcinted out '- -, (1969) and others.
it would bp extremely attractive to carry through
3imilar arguments for the c-ossflow correlation. However,
this immediately leads to excessive comriication. The
reason is that ,thle simple eddy viscosity approximation of
equa+lon (3-6) does not appear adequate to desc-ibe the
crossflow profile. Instead one must invoke a coupled edey
viscosiLy expression such as suggested by Cousteix, Quemard
and Michel (1971).
70
F F2-1 2, a•u, ;Bw(3
where F is a sublayer damping factor suggested by van Driest
(1956)
F = 1-exp _ 3 (, 2. (3-1
and1 = 0.085 6 Tanh (3.-1)
Nondimensionaiizing and combining (3-5) and (3-9)
leads to
.•+ ( 3-, , , +
nw 2wi•;h
a i n 3- 1 ;•n =n--ur F7Fn'
The sym.&ol stands for the tangent of the angle between
the zotal surface shear vector and the shear stress vector
in the freestream dtrection.
F.xpression (3-12) is obv! '-sly much too complicated t,,
be tractable as a crossflow ve-ocity correlation. Fortuna'--
an alternative exists in the simple hodograph models, which
often give a better representation of the crossflow velocitv
profile than does (3-12).
The earliest hodograph model was proposed by Prandtl
71
(1946) who speculated that
w = u9(•) (3-14)
where g was a function such that g(O) = 1 and g(1) = 0. He
took g to be
g,6) - 1- 6 (3-15)
Mager (1951) suggested a more accurate form in which
g(Y) = (1- 2(3-16)
Recently, several more sophisticated hodograph models have
been proposed. Johnston (1957) represented the crossflow
velocity by a triangular hodograph such that
-e = 0 2ý-' U- < (3-17)
we P (1- -) ;-u > (IL) ;(3-e e p
where (u/Ue)p corresponds to the apex of the triangie and is
given by
( (1 + ) (3-18)
Up Tan f
and Ue is the freestream velocity. Johnston's model is
sketched in Figure 3-3. This model has several shortcomings.
There are situations in which the outer region is not
adequately represented by a straight line and there are
situations ii which the apex of the triangle is not well
72
W
Ue INNER I OUTER
REGION REGION
!U
tan-l
0 U U
0 1.0
Figure 3.3. SKETCH OF THE TRIANGULAR HODOGRAPH
APPROXIMATION, AFTER JOHNSTON (J 957). 2*
7
73"-
defined. Also, he experimental determination of 0 is
very difficult due to the fact that the inner region of the
model actually corresponds to the viscous sub layer.
Eichelbrenner and Peube (1966) modeled the crossflow velocity
by a polynomial of the form
w = U P I ----,, I +u blu + ,u + e•)4u_eF e- e IT e "e Ue
(3-Z-9)
where a,b,c,d,e are evaluated by using beiundary conditions
at the wall and at the outer edge of the boundary layer.
These constants are also extremely difficult to verify
experimentally. This model does, however, hare the
advantage of being able to predict the "S-shaped' crossflows
which pz-e discussed in Section 3.1 and observed experi-
mentally by Klinksiek and ?lerce (1970).
With these considerations in mind it was decided by
the authors to employ the hodograph model as proposed by
Mager (1951) in the form
U+ t+)(3-20)
This choice seemed a good compromise between complexity and
relative accuracy. In fact this correlation compares well
with some experimental data as shown in Figure 3-5. Until
such time as more reliable three-dimensional data become
available, and in particular profile data, the use of a more
sophisticated hodograph model does not seem warranted.
74
•.3 The Method of Solution
In order to solve equations (3-1), (3-2), and (3-3)
they are first nondimensionalized using the shear velocity
in the freestream direction. The result is
a + U*w+uu (u*u) t u*u+) + v )- 21 (U* +)
+ 4* W - 2 +2 ah, _ _+UC u w* -I u*w+ W = ?1-p + * s
h17 h~i h . p. 1).s pV-
and
u+ + u+) + Z +
( ) (u•.+) + v - .+ (u*w")
u*2 u+w+ %h2 u* 2 u+2 'h1 ýj nhl--• As hhh2 n Ph 62,
where the velocity v is determined from the continuity
equation:6 +
v r a 1 , 1 u (hlUW)' +, ~ ~ ~ u _• u"+) +•7 u4+ dy+~1 1
(3-23)
Expressions for the partial derivatives with respect to both
s and n are needed. Note that
+u fcn(y , aS) (3-24)
and
+ + +Iw =fcn(y, as 8, 6+) (3-25)
Therefore,
75
+ ?k u •s +
'U+ S a
AI += = + 2s ai e a-+ 26 +wA6 s ilk + W- -Y-+ 'F- s (3-27)
a~s s
plus similar expressions for the n derivatives. For the
purpose of skin friction calculations, it is convenient to
make the following non-dimensionalizations:S+ ( 1 /f~i2" e UoL
= e U+s= ) , hL/L, h* = h/L RL =-
(2-28)
V = Ue(s,n)/Uo
where L is a reference length, and U is a refereince speed.
Then, from the definition of a
s + 1 (l/V) (3-29)
and
:as - 3 . ý I + 1 3 a2 (i/v) (3-30)Anl _ s A1n ~ l~
The partial derivatives of 6+ with respect to s and n are
more difficult to compute. Equation (3-7) integrates
analytically to
+ =1 r)P n(j o+l~u . 2(o-p + in ( ) .
(3-31'.i r2(p-l) + ln ((3-31)).
76
where+ 1/2
At the edge of the boundary layer, equation (3-31) becomes
+- 1) + 1n(pr- UL )1 (3-32)e 1 a(PY
where
Pb= (I + as
Differentiating equation (3-32) with respect to s and n
results in:
"6+- 1)1 r- p-Pp3 . (3-33)=s ! 6as - 6 P6;s (-3
- asp6 a s 2p6 2 -
and a similar expression for the n derivative. Combining
equations (3-26), (3-27), (3-29), (3-30), and (3-33) gives:
+ UI + I1u 1 • +u1
++
+u + 1 '1.1 1 'u+
vn -1n i (+3-3)V • ~ •V77S
77i
S ...... z •:=• • - • ..... ...... = •• z• • - -•:- - . ..•: - == :J : •• - ..... -:-• J- •• _-._. ...... •• ....
11 I ! • , [ !1I I I <1)
•, + 8-"+ + -- + K(z (p- AR-s"= 'i•sL y 1 3• + iaP
+23a
- a(•P P +fl 6 J 1 •-X *Ly ay (3-36)Ss S Ps L a y1J
+ -,:..+ ( 2P + 3+pn + . 2()
s__-. a 5 'p V = ••-•=p
•nW t1 cnr _~ 4
S S P5
a•. (s 1+PB _6 3p 2 ?n k S L •2
+ + 2 3 3 +
F" al•" 6, Dns•-
+ b5 -P6 +P6) ')w 1 AVf 1' (337)%
LCas s 2 P62
After seubztitutinZ the partie! derivatives, equations (3-34)
throu/h (3-37), into equnI.~ons (3-21) and (3-g2) and inte-Arwtinj the r.zulting equrtiorns from y+ = 0 (+ = 'rw) to+
42m = + (' 0). L ne obtr ins
""- a
a. "- 62#+ VýW yW
1' kl V 2' ::V G(G1 -3aO 2 ) + G\T (x 25+-G1 ) -1
2 -)+,
+ ( -3a G - XGG 2
(3-38)
x 4 4GL 2 1 1 1
E*R (-7) • G( RV i+-n) 7
x i ýI4t 5
12 12 ( 7Si (X 1+1 1. (G 33 a G~G 4 -x~ G+ ~ l~
'. ý n '11 s--12 1 13' L
(3-39)
Equa*ti nTs- (3-38)' an (-39 are twh, 7n pd 15n
22
+ l 3 G I GT2 i172
The G coefficients in equations (3-38) and (3-39) are
deflned ~in terms of quadratures over the velocity correlations
and various gradients of thc' velocity correlations. These
are ieb'ulated and evaluated ia appendix A.
Equations (3-38) an~d (3-39) are two, coupled, non-
linear, partial differential equations which have for
dependent variables only the skin friction coefficient and
79
the tangent of the angle between the total surface shear
stress vector and the shear stress in the freestream
direction - no integral thicknesses need be calculated.
The inviscid flow solution is assumed known, a may bes
computed fr.. its definition, and 6+ may be computed from
equation (3-32). As should be the case, equation (3-38)
reduces to the incompressible, two-dimensional analysis8 + 8
of White (1969) when - = 0, w = 0, and = h. .
If 1
3.4 Comparison with Experiment
-.4.3 The Rotating Disk Problem
A convenient starting point to test the validity of
any three-dimensional theory is the problem of a smooth
plane disk rotattng in an otherwise stationary fluid.
While axisymmetric, this problem exhibits the essential
feature of a three-dimensional flow, namely crossflow and
thb resulting veclor character of the surface shear stress.
The presenc theory has been applied to the rotating disk
problem in a paper by Lessmann and Christoph (1972) and
is discussed below.
The boundary layer equations, equations (3-1), (3-2)
and (3-3), may be reduced to those governing the boundary
layer flow on a rotating disk by identifying s with the
coordinate el and n with the coordinate r, both fixed in
space. The metric coefficients become h. = r and h2 = 1
and the equations reduce to
80
r •- (rW) -- 0 (3-40)()y
ýu iu uw_1w •7 + v ý- + r p•gl(-l
and
;5W ;3 W ~ rW r.gu + v T•_T (--421wr• v y r F ;%r
All derivatives with respect to 9l have been eliminated
because of symmetry.
The boundary conditions on the equations are that at
the rotating plate v=w=O and u=-rlV. Here f, is the angular
velocity of the rotating disk, assumed counterclockwise
about the y axis. In the freestream u, v, and w as well as
the sh,-aring stresses are zero. As a consequence of this
(3-43)
For what follows, it will be convenient to transform
equations (3-41) and (3-42) so as to obtain more conven-
tional boundary conditions. Defining a new circumferential
velocity as
ut = u + rO (3-4Ls-)
the boundary conditions at the plate become u' tv=w=O and
.n the freestream v=w=-O while u' = rt. Using equation (3-44)
equations (C-41) vnd (3-42) become
813
and
W 6- + ,, + 2Cu' - 1-2 _ (3-46)
Mager's crossflow velocity correlation, equation (3-20),
will be used and since a = 0,
+ y 55 (3-
will be used ror the streamwise velocity. Comparison of
these two velociz. correlations with the data of Chain and
Head (1969) is shown in Figures 3-4 and 3-5. The velocity
v is eliminated from equations (3-45) and (3-46) by the
continuity equation, and the resulting equations are non-
dimensionalized with respect to wall variables. For
convenience u* is related to the circumferential skin
friction coefficient by
;,,)1/2 rn0
4=(2; s =UT (3-48)
and r is non-dimensionalized by the relation
'n = r (n/ V) 1/2 (3f49)
Note that v is the square root of the radial Reynolds
nuwmber. in integrating the two momentum equations from
-I-" + +y = to y = &', the shear velocity in the radial
direction, v* =(•/P) will appear. The shear velocity
82
0 0-4 4 ý
F~ 0 H
[zz
E- 0-4
0E4 - z-~ 10
40H
rq.
00
00
0. 0-l0 C:U H
40 ~
0 00
0 -0 a z+ T, tr L.
ý4 e4 -4
Z 0 tKr0 + C
CC aL0 -
84
• is elimIrnated in -'evor of u* b- the expression
o "(3-so)
The result is
D- +. .-
n d d2 I
where
•2=• }•+ + +
S;-NY t6;-+ ) y )dy0
(3-5-3q5+ (0.61 5 239 x + 9.84)
+ +
D2 d--)dy+o 0
C(7154)"
(3-55)= 86 1 (153 X -9.5)
85
(31-55)
4-- . c - i 1II"I
+'r- = - - +-. - v+ d;+5 ' ". •," "C -
-( >,
4- +
(=)' w •'.i'-u - -4-- w~c*~ *-*S-a. -
C 0d
r6~~- X,,..,-U d
This problem redu.:es to the consideratlcn of coupledord.nary differentla2 equations as a result of the fact that
the rotating disk is a twc-coorcinate pD-eblem.
The results of the r,:-esent ana.vsis cons-st o; Dre-
dictions, using equations (3-1) and (3-52), for the
circumferential skin friction coefficient c, and thc ratio
of the crc)ssflow shearing .- tress to the cir-cmferentia'
shear4ng str.ss, S. These are compared with experimentai
results in Figures 3-6 and 3-7. Fig•re 1-6 shows that the
present celculstion of c-- compares very favorably with the
86
wJ
C. I ,-4
[ Ii:
Ile-
z• z
F <
UZ
rzu
0d
nii
4-4*4c
e rz87
0
CIOI
>x
<l Z
Ii"[ I
•z
I19
88 -'
ii I,, F
I I
I ] i ! I - ! I
• o
experimental cata and falls approximately halfway between
the rezults of 7ooper (1971) and von Karman (1946).
Cooper UZsEc a finite diffe'ence scheme based on the work
of Sm-ih and 'ebeci (19t6,197l), Figure 3-7 is reproduced
from the w-rk of Cha= ane Head (1969) and compares several
predic:ions for - with one measured value. Unfortunate -v,
'he zicer.ain-v asscciated with this single data point is
relqtively large. With this in mind, it can be seen that
th-r.t. theope,, as well as the entrainment method of
-hMr anu Head (i960), and the prediction of Banks and
3add (l•92) gi-'e plausible results. Both the present
ca3-culatlon and that of Cham and Head show a tendency tc
fall -with incregsing radial Reynolds number, while the
prediction of Banks and Gadd (1962) is for a constant value
of. -,oo.er .1-oi1) in, not concern himself with predicting
to-al she3r and hence no comparison with his work is possibl-
on This poin..
Since the rresent approach and the analyt.€ of von
KYr-an (-.ma) are intezýral methods it is informative to
con:i-der 'Vbo,!h thIe simi-arities and impo-tant differences
between these -wo works. The velocity correlations used
by von Kia"en are fin the present notation)
u = ,-y/I) (3-59)
and
w - eu(l-y,/) (3-6o)
a9
Y -
These are to be compared with equations (3-47) ana (3-20).
It is well known that the 1/7 power law gives a good
representation of fully turbulent jelocity profiles; at
leas-. for a limited range of Reynolds numaer. This fact
accounts for the relatively good agreement with the
circumferentia-t skin friction date. At the same time, the
crossflow correlation used by -...n Karman does not give. a
good representation of the radial velocity profile as is
shown in Fig,.re (3-8). Also, in order to get an exact
solution of his -quations. von Ka.can assumed P to be
con:-;tant and the bo-ndary layer thickness 5 proportional to
r These facts accowit for th.ý relatively poor pre-
diction of q = 0.162. In order to carry throug, the
calculation fcllowing von Karman, one must express the wall
shear stresses T and rrw as functions of P and 6. This
is in contrast to the piesent analysis in which the
dependent variables are a and the circumfe-ential skin
friction coefficient. No shear stress correlations are
necessary. One can carry cut von Kxrman's analysis retain-
ing the r dependence of @ and 6 and also consider the effect
cf replacing (3-60) with Mager's correlation. As is shown
in Figure 3-6, both of these lead to essentially the same
skir friction curve, which is somewhat low. The effects of
these changes on the prediction of 9 is more drastic as
shown in Figure 3-T. When one keeps the original velocity
correlatien but allows fo" r variation, the result is a
90
w -o
QD 0
-E--~ ZZZ
~0 (
4(l -1 40EO0 1
-f,14 0 -0 C;
00
00
0 N
C c.~c
0 01
00
91
curve wnich falls rapidly to an asy.p-0o-'c benavior
approaching e = 0.162. Using Mager's correlation with
von Karman's approach gives ;- prediction for O which agrees
closely with the present method for small Reynolds number
and becomes asymptotic to P = 0.217 for hige-er Reynolds
number. Apparently this difference is dae to the fact
that the logarithmic law-cf-t :e-wall properly accounts for
Reynolds number variation, while the 1/7 power law does not.
It should be pointed ou' that all calculations of o
in the present study assumed an initial xalue of a 0.228.
This initial value was chosen so as to generate a monatonic
curve for @ as a f-nction of r. it was found that perturbing
this initial condition caused the calculaticn to rapidly
converge to the curve shown in Figure 3-7.
3.4.2 Johnston's Swept, Forward-Facing Step
Johnston (1970) experimentally investigated the flow
over a two inch high rectengular step, swept at forty five
degrees to th-- main flow direction. While an example of
a fully three-dimensional situation, this flow is not
strictly a boundary layer due to the large variation of
pressure in the normal direction. For the purpose of the
present analysis the ;'.reestream" v± It, d1ist-ibition was
dzterzined by fitting a polynomial co Ž• ..... con's pressure
coefficient data. Three differ-rnt Luch distributions were
92
considered, one corresponding to the wall pressure coefficient,
o:ne to rhe vertically averaged pressure coefficient, and
one assuming a constant "freestream" velocity. As expected,
use of the average pressure coefficient gave the best
results but none of the calculations agreed well with the
Bradshnw (1971) also analyzed Johnston's data using a
finite difference scheme. His calculations, using the wall
pressure clefficient, showed similar behavior to the present
Pesults. In irder to overcome this difficulty, Bradshaw
determined the local value of the "freestream" pressure
gradient from tte two-dimensional pressure coefficient
data given by" Johnston. With this additional complication
Bradshaw's final results agreed fairly well with the data
except in :rie region close to separation. In this context,
seDaration refers to the condition where the surface flow
in this "bounda-y layer" has become parallel to the step.
The present analysis follows directly from equations
(3-3d) and (3-39). Making the assumption that the swap
is infinite, and hence that all v&riations parallel to the
s tep are zero, relater derivatives in the s and n directions
by
SA-2.- ---- (1-61)
Also the metric coefficients h1 and h 2 are equal to unity.
These simplifications result in two ordinary differential
93
equations for X and 9 as functions of s* = s/L. The
characteristic length L is taken to be the distance from
the step to the initial point of the calculation, along
the "freestream line".
The results of this analysis are shown on Figures
3-9 and 3-10. Also, Bradshaw's finite difference
calculation is displayed for comparison. On Figure 3-9,
it can be seen that use of te vertically averaged pressure
coefficient gives good agreement with the ,kin friction
up to about s* = 0.8. The calculation using the wall
pressure coefficient is low while use of a constant free
stream velocity gives results which are much too high.
Bradshaw's results are also in good agreement bLt extend
as far as s* = 0.85. None of these predictions agree with
the measured separation, w. -ch occurs at about s* - 0.93.
Figure 3-10 shows a comparison between the present
theory and Johnston's data for P. Again the average
pressure gradient case gives the best agreement but
predicts separation too early. Bradshaw's calculation is
somewhat better but also shows early separation.
These results should not be construed as indicating
some intrinsic deficiency in the present theory, since this
flow is not truely a bcundary layer flow. Actuallj, in
order to accurately predict the swept step flow field one
should use the full Navier-Stokes equations and not the
boundary layer equations.
i -----4
0.004
C =0.p
0.003 •.
C (WALL)
s pp
0.002
BRADSHA%
(1971) O
0.002 \
0 = DATA, JOHNSTON (1970)
0
0.4 0.5 0.6 0.1 0.8 0.9 1.0
s*= s/L
Figure 3.9. COMPARISON OF PRESENT THEORY FOR
THREE PRESSURE DISTRIBUTION ASSUMPTIONS WITH
T3E STRFANWISE SKTN FRICTION DATA OF JOIMNSTON
(1970) FOR FLOW NEAR A 450 SWEPT STEP.
95
1.0 I I
0.8 0
0.6
PRESENT THEORY:8 C (AVG) 0O
0.4
o/
C p (WALL /XBRADSHAW
0.2 P 0 (1971)1If
p -
0.4 0.5 0.6 0.7 0.8 0.9 1.0
s* = s/L
Figure 3.10. COMPARISON OF PRESENT THEORY FOR
THREE PRESSURE DISTRIBUTION ASSUMPTIONS WITH
THE WALL STREAMLINE 7ANGENT DATA OF JOHNSTON
(1970) FOR FLOW NEAR A SWEPT STEP.
96
4u--- W- -v- - - -VIIP
3.4.ý -ata of Francis and Pie-..e
Francis and Pierce (1967) meas;,: ad P, the tengent of
the angle between the total surface shear stre's vector and
the shear stress in the freestream ',=e.tion, in a rectan-
gular channel. T:h 10 inch wide chainnei consisted of a !E
inch straigh' inlet section followed by a curved section
of either a 55 inch centerline radius af curvature (referred
to as series 5) or a 25 inch centerline radius of curvature
(referred to as series 2) and a 48 inch straight discharge
section. Most of tle measurements were taken on the center-
line but a few of the measurements were taken 2 inches
either side of the centerline. The RevnoLds numbcr was
1.07+O.03xi0'J per foot for both series 5 and series 2.
The freestream velocity variation was about 0.3 percent per
foot, while the crossfioe velocity variation was from 30
to 54 percent per foot in the radial direction. This
experiment .•-ovides the possibility of examining effects
produced by a crossflow pressure gradient without the
influence of a strong pressure gradient in the freestream
flow direction.
Fdlja-ýions (3-38) and (3-39) are solved with the
following simt ifications:
a•s = 0, .*, -s=•-• '"2•-' -Y " W •' = '
(3-611 ,%aO-l.o - _ _
1 1 1-:Y -T 0 5W hU ;%n R*
97
where x*, y*, and I' are respectively the streamwis-e flow
direction, the direction normal to the streamwise flow
direction, and the radius of curvat-ure, al- =on-di-ension•.!-
ized with respect to the length of the channel. This
results in two coupled partial difLferential eauations for
2I and o as functions of x* and y*. 1- is assumed that
and a are known along some initial line x* - no sideC
boundary conditions need be known. The partial derivatives
occurring in these equations may be numerically approxixated
in a n'rmber of ways. For illustrative purposes, they were
calculated by forward di'ferences. Eackward or central
differt..ces couid have been used 4ust as earilv. Also,
if more accuracy is desired it Is possible to keep
correcting - and -w-ith their average values iteratively.
T-he results of these caluclations for the center streami-
line are compared to the measurec values of a end to the
skin friction as computed by Francis and Pierce uslrZ the
Ludwieg and Tillmann (1950) shear law, as shown in Figures
3-11, 3-12, 3-13, and 3-1,4. Agreement with tne data over
about the first half of both charonels is qui,.e good, but
then the theory and the data dive,-ge. The skian friction as
computed by the present theory is ,wer than that computed
by the IAdwieg and Tillmann law, and tha theoretical values
of P are considerably higher than the meas'ired valuerz. One
possible reason for these discrepancies is that Mager's
profile may not adequately represent Francis and Pierce's
data. Shanebrook and Hatch (1970) suggested a -more
98
PRE.SEN~T THEORY:
EQNS., (3-38) AND (3-39)
0.0023
o.~oI 0 DATA: FRANCIS & PIERCE (1967)
C 1- 20 30 40 50
Arc- Length S finthes)
Figure 3- -. COMPARISON OF PRESENT THEORY
WITH THE CURVED DUCT EXPERIMENT OF
FRANCIS AND PIERCE (1967), SERIES 5.
99
0.7
C.6 VON KARMAN METHOD f
USINZZ MAGER PROFILE,AFTER SHANEB ROOK
0.5 AND HATCH (1971)
0.4
"0. PRESENTTHEOJRY
0.2
/ KARMAN INTEGRAL METHODUSING SHANEBROOK ANDHATCH (p2=9,q2=2) PROFILE
0 i I - I I
0 10 20 30 40 50 60
ARC LENGTH S - Inches
Figure 3-12. COMPARISON OF THEORY WITH WALL
STREAMLINE TANGENT DATA OF FRANCIS AND
PIERCE (1967), SERIES 5.
100
CSC
:• .," 005
0 0 00cc 000
o.0o,3 ~-
PRESENT THEORY
0. 0021 a
1 i0 2P. 30 40 50 60
ARC LENIGTH - S - Inches
Fiaure 3.13. CC-M.i'ARISON OF PRESENINT THEORY WITH
KXPERIMENT.lAI SKIN FRICTION AS F.STI.AT.D IN
A C.•RVED D-.CT BY FRANCIS & PIERCE (1967),
SERIES 2.
101
0.8
KARMAN METHOD
USING MAGER"S
0.7 HOD(rRAPH
/,,0/
0.4 -
e I C O\ \ R
0.3
0 WIH SHANEBROOK & 0
HATCH (p. =9,q2,=2 )
.Ug)DOGRAPH PIRDFIL-F 0
0.3 1 Q
0 !0 2-" 3 40 50 60
ARC LENG_!7" S - Inches-
Figure 3.14. CODPARISON OF TifDRf WITH WALL
STREAMLINE TANGE&4T 0ATA OF FP.!CIS
AND PIERCE %1967). SERIES 2.0
102
sophisticated family of hodograph models for crossflow
profiles. Their method extended Eichelbrenner's (1966)
polynomial representation by forcing consecutive higher
order derivatives to be zero at the wall and at the
boundary layer edge. This family is called the (pi,qj)
family of hodograph models where pi is the number of
consecutive zero derivatives at the wall beginning with
the ith derivative and qj is the number of consecutive
zero derivatives at the boundary layer edge beginning
with the Jth derivative. Besides containing many terms,
Shanebrook and Hatch's hodograph models also contain the
angle 0 that is used in Johnston's (1957) triangular
model, Figure 3-3. By the use of the momentun integral
equations and an entrainment equation, Shanebrook 'and
Hatch compared their hodograph model with Mager's velocity
correlation for Francis and Pierce's data. Shanebrook
and Hatch's model shows better agreement with the data,
but the same trends as the present theory, as seen in
Figures 3-12 and 2-14. However, Nash and Patel (1972)
showed that Mager's correlation was in excellent agreement
with the crossflow velocity data for Francis and Pierce's
experiment.
Therefore, it is felt that the disagreement between
the present theory and Francis and Pierce's data is not the
fault of the assumed crossflow velocity correlation. Rather,
it is believed, as Nash and Patel pointed out, that the
103
boundar• layer in Francis and Pierce's curved channel was
dominated by the effects of comer flow at the 'Junction of
the side wails and the floor of the channel. These effects
were not accounted for in the present analysis.
3.4.4 Data of Klinksiek and Pierce
As is pointea out in the introduction to this chapter,
M-ger's velocity correlation is not capable of represent-
ing the so-called S-shaped crossflow velocity profiles,
Figure 3-2. However, it is .telieved that the present theory
will give. accurate results even if the crossflow velocity
profile is only grossly approximated; as long as 9 is well
predicted, and apparently Mager's correlation predicts 9
accurately. KLinksiek and Pierce (1970) conducted an
experiment in which they obtained S-shaped crossflow
velocity profiles. Their experiments were conducted in a
doubly curved channel of rectangular cross section. An
initial straig ) section of 78.7 ince :., was followed by a
60 degree bend with a 25 inch centerline radius of curvature.
This in turn was followed by a 12 inch straight section and
a 60 degree bend in the opposite direction with a 55 inch
centerline radius of curvature. The same equations (3-38)
and (3-39) tha* were used to compare with the data of
Francis and Pierce apply to tICis situation. The results are
shown in Figures 3-15 and 3-16. The theoretical values of
p are considerably above the data, presumably again because
of side wall influence. The S-shaped profiles were observed
104
.4
0
0 00
f0 0
0 PRESENT
rTHEORY
0.002 0
STRAIGHTI 25 INCH STR 55 INCH NEGATIVESECqION .RADIUS SEC RADIUF OF CURVATURE
0.001
0 2C 4u 60 80 100 120 140
ARC LENGTH S - Inches
Figure 3. 15. COMPARISON OF PRESENT THEORY WITH
ESTIMU TED SKIN FRICTION IN THE CURVED DUCT
EXPERIMEiNT OFKLINKSIEK & PIERCE (1970).
105
1.2 -
1.0
0
0.8
0000
0.6 Q PRESENTTHEORY
0.4 -0
00.2 -
0 0
0 0
0
-0.2 - ccP 0 0 00
0o 00-0.4
STRAIGHT 25 INCH STR 55 INCH NEGATIVE-0.6 [SECTION 1 RADIUS I SEC I RADIUS OF CURVATURE
-0.81 1 1 1 - --A0 20 40 60 s0 100 120 140
ARC LENCTH S - Inches
Figure 3.16. COMPARISON OF PRESENT THEORY WITH WALL
STREAMLINE TANGENT DATA or KLINKSIEK & PIERCE (1971).
,o6
w.
in the lat'i' cur:ed portion of the channel with the
inch centerline radius of curvature (starting at &.
inches in Figu-es 3-15 and 3-16). By this point in the
calcula;ion the theory considerably diverges from the data.
T--hnut no conclusion can be drawn about the general use of
Mager's cc;'relat.!on in situations where bilateral skewing
of tne crcssflow p0rofile is p "esent.
-.5 Summa.Y
Tne aDDroach to calculating three-dimensional skin
friction presented in this chapter seems to offer great
p-omise. it. is an integral method which utilizes "'law-
of-the-wall velocity correlations and results in two,
coupled, partial differential equations having the stream-
wise skin friction and the angle between the total surface
shear stret:s vector and the shear stress in the freestream
direction as -he aependent variables. This analysis
contains significartly less empiricism than "classical"
integral methods and is computationally much simpler than
currently availacle differential approaches.
An area requiring more study is the problem of
specifying a suitable crossflov, velocity correlation.
Mager's hodograph model was used here as it would seem to
offer the Gez- current compromise between accuracy and
complication. However, it is not capable of predicting
c•ossfl:w profiles vith reversed skewing; and this must
107
be regarded as a deficiency for any generally applicable
theory. it is felt that the development of a more
accurate crossflow model must wait upon the avaiLlability of
more accurate profile data.
The approach developed here compares extremely well
with the rotating disk data of Chan and Head and with other
predictions of this flow. Th,! relatively poor performance
of the present method when compared to the data of -Johnston,
Francis and Pierce, and Klinksiek ana Plerce should not
be taken as an indication of any serious deficiency in the
theory. Johnston's swept step flow was not truly a
boundary layer due to the presence of a large vertical
pressure gradient and the Francis/Fierce and Klink-siek/
Pierce data sets we'e strongly influenced by channel c:rner
flows. Thus, serious evaluation of the present method must
await the availability of a clean set of experimental data
for a three-dimensional boundary layer flow. Such a set of
data should include direct measurements of skin friction in
both the freestream and crossflow directions as well an
accurate measurements of the pressure gradient along the
freestream line and velocity profile data.
108
Nw
APPENDIX A
A~l. Tne coe-fficients H i for use in the heat transfer
differerntisl equations, Eons.(1-29,30), bre deflined as:
hi u+ g-;' +2YUa p au* t 2 a+3
G +T w w
r J2 ¶U 1 aU at )J +
o T+ -
6+ +2 U+a+ S au t
42 T+ T+a
__~ ~ uPaTf +T~r
T __ R- T+ ~u 2
4. yf - kT at ,S H! u - 22u
+y 4 ?t ;.+ R ? yu-;u y&'v
6 - 2-r+ ;a 21 jw c -T+~ -3 U
109
6* + 1pT u+ aT+' 8PT W+ a* u2au
0 w
u - dy+
1 + + Bu+BT+ OPTr ZTT+ ao u+ 2 au +H -- f0 -1PTu - PT • '* r-' + 211T ay÷ V" Y R ;F -S-o)
+ u+ Y u a* ) d
I•ne various partial derivatives which are needed in the above
expressions may be evaluated as follows:
+ ÷u + 8 y+ du+tzH •f- [T~y + Cry f - T+ 2
S=0 0 11+ CE) 2] 2(I ÷ y+)
I ÷S 0 2 T+ 0 4+ a+
e us+ my dua+ U+- [ f 2u + du+
+r+ Ue - du+
0 + GY+
110
du4 .
dii
-lm I w Imp - WWI-"-
DT+ u+
f e du
0 l+ay+
+ u +3T fe 2 u+ du+ u 2
o e
D o U e + + +u(u u+ . )+ .•.•=f K Y du -- + au =--u _{By++ 2ayy+ a+I.+ld,(W 0 (I+/y T+51 2 5/2 0 u(I + ay+)2
I" e y+ du + _ + •: + + au • + •+f C d -ru +. t u ft -2y( 1+0y i lw o (l+ay+.) 1 12 _T 5•25 4 -u fC. (1 ÷ a,+)
due Y d ++ .1+ _ _ _ __+ +_ au_+f + a -+u+ + )(I.+ ) -Sduo4 O-Y + 1 2T 5 /2 7s• u f
+I a
Ane following are curve--it expressions for the Coefficients Hi
0.475 u+ (1 + 0.22 y u+2)H1 8.5 expf e- e[1 + 0.i(ct6 /]( + 0.3jSu÷) }
e
111
.4
0.4 tu+ (I + 0.22 y u÷2)
H2 0.062 exD{ e 1 e21 + 0.12(a6+) ] (1 + 3.3 Ru÷)
e
0.52 u (1 + 0.22 yU+ý)
H3 1C.O exD{ e e[. + 0.08(a6+)1/2] (1 t 0.22
0.5 1+ (I + 0.32 yu- 2 )
Hl 4 2.5 exp{ e e
[I + 0.09(at6+) 2 ] (I + 0.3 8u+)e
2 (0.7 + uOO y) u (0.492-3.78) UH -0.2 y exp{ C } S-1 . ex{e }
(1 + 0.02(a6+) 1 / 2 ] 1-158 (1 + 0.(a+6 )1/2]
7 exp (._.6.3 + 400y) U
1 + (2000 y)7.6 rl + 0.03(e+)1 /21
0.435 u+ (1 + 35 vi )
H 0.051 exp{ e e6[1 + 0.25(a6÷) X2]
(0.8 - 5 S) U+
- 0.14 8 exp{ e
[1 + 0.25(aS+)1/2]
112
(0. q82-3.78) u+H 4 3.5 8 exp{ e
7 1 - 158 [1 + 0.1(06+)1/2)
(0.46+260y- 36) u+
+ 10000. Y 7. exp{1 e1 + e(3200 y) [1 + 0.05(ct+)1/2]
0.,,6 u +2 .33 2
H = 0.33 exp{ e e(1 + (I + 0.25 Oue)
In order to use these curve-fit expressions, one first
evaluates a, 13, and y from their definitions, EqnE.(l-32),
and then evaluates 6 + from the two-dimt.'sional law-of-
the-wall, Eqn.(1-33). The only other parameter in the
formulas for H, is U+. which is directly related to
skin friction by the relation u-- X(Te/T )/ 2.
1U3
A.2. The coefficients Gi for use in the three-dimensional
theory, Eqns.(3-3 8 , 39), are defined as follows:
G1 = +2 y
G 3 z U W+dyt0
6 + u
G = f [+ - f dy+ +d
44-
G 2 I f [ - wi'9" f0 s 0 s
+2 3
%6P6 P6 + P6 au Y V d 4. d 4
+ --:L f3 -+ I u1/
6 P6G• - 1(+ - + dy4] d+
wher P6 U+a /+ 2 +6 P+ y+ a,++ Y +
G 5 IF a d6yi ] dyýI y
6P • f f .5
0 0
-1412
mw -- - -U- -
+ + + ;u+ + /
Ty+ dy. dy
ou + +f[ LW+ + f u dv I dv+
U+ 3 - -, au dy]3
o v 0 3s
-2 3+ I-- +.-(S ').~ dv4-
2 2 +
+
G f +_____ +10 0 m v i d
GS d
5+ +
+ + +-r +~ ~ -
S - 0 s
.+ 2 3 +CI0 D. -DA + D .+ + V +
5 6 +w + u ;w f v-v-~ 2 2 3T - -j' 0 dvT+ 1rd
s 6
I t.1) +I~ 4 3w+ y ý + + +013 TV+ f ;A+ tly 4
o s 0
= f u 4 -n &
;E).
w
1I5 a [ aw o Y +
G 152 + , W wf dy ] dy0 0
6 + + +
167 = j [u cly I d- yu0 0
G + + 4 w + +Gu 17- = f yýJ U dy 3 dy+
Using the law-cf-the-wall and the Mager hodograph, these
integrals may be evaluated numerically as functions of the4-
parameters 1 , s Op, and 5 . The following curve-fits
are suggested for the numerical results:
G 1 "8.5 expt 0.475 XI /(l + O.l(a 6+)1f2) )
G_ 0.062 exp[ 0.8S4 X A(l + 0.12(a 6+ 1/2z 1 S
G ()6+ (0.33 X 2 -3.06X + 9.84 - 6(0.078X1 1.50 a 6)+3 1 1 . s
G• -S32500. G exp(0.285 X1/(1 + O.3(us 6+)1/2
S5 1 06+ (0.33 X1 - a s6+(0.1? 1 - 0.3 as 6)]
G 6 (1.53 X - 9.84 - aS 6+(0. 1 - 0.2 s 6+))
G7 e6 [0.33X2 - 4.59X1 + 19.65 - a 6 +(.6 X1 - 1.1 +s6*))7 iS
G at *[ 2.50 1 - 12.5 - S(0.08 1 - 0.15 a 6+ )6 +
+ 26 +(0.2 112 _ 2.28 X, + 8.33)
91 1r.9 " -0.71 0 exp[0.83 X 1/A l + 0..12(ULs 6+)I1/2)
M+ 2 _+_r [0.07 _ 0.28 X + 0.42 - a 6 (0.92 A - 2.0 a 6+)]10O 1 [00 1i s 1 s
Gil 1 26+[0.2X 2.28A + 8.33 - a 5+(0.481 0.82a
G12 - 5.50 0 exp(0.63 ii+/(i 0.1(a s6+)I2)]
G )2S+10.08 A2 _ 0.58 A + 0.83 - a 6+(0.45 A - 0.9U C S+)]13 1 1 s 1 s
G 14 G3
GI " W +(0.40 X 2 5.0 A + 18.7 - 0.30 a 6 +A )151 1 9
S16 +[X2 - 5.0 A + 12.5 - a 6+(4.0 I - 4.0 a 6+)]21 1 s I s
- ei6+(o.2 ^ - 1.5 X1 I+ 6.95)
G 05 + [O. A 2 _ 6.11 A + 19.65 - a 6 (0.87 A+ -17 1 1 . (1 s 1 s
It is these curve-fits which w-re used to compare the
present theorv,- Eqns.13-38.39), with the three-dimensional
flow experiments of Figures 1.9 through 3.16.
117
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"Now
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