Integral methods for turbulent boundary layers in pressure gradients

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INTEGRAL METHODS FOR TURBULENT BOUNDARY LAYERS

IN PRESSURE GRADIENTS

Washlngton,DC. 20007

by

Paul S. Granville

APPROVED FOR _{PUBLIC RELEASE:}

DISTRIBUTION UNLIMITED

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(2)

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DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

WASHINGTON, D. C. 200b7

INTEGRAL METHODS FOR TURBULENT BOUNDARY LAYERS

IN PRESSURE GRADIENTS

by

Paul S. Granville

APPROVED FOR PUBLIC RELEASE:

DISTRIBUTION UNLIMITED

(4)

TABLE OF CONTENTS

ABSTRACT

Page.

ADMINISTRATIVE INFORMATION 1

INTRODUCTION 1

VELOCITY SIMILARITY LAWS FOR BOUNDARY LAYER. FLOW WITh PRESSURE

GRADIENTS - 2

INNER LAW OR LAW OF ThE WALL 2

OUTER LAW OR VELOCITY DEFECT LAW 2. LOGARITFflIC LAW

...

3

LAWOF THE WAKE 4

BOUNDARY LAYER PARAMETERS 5

GENERAL ....

-5

POWERLAW

VELOCITY SIMILARITY LAWS 6

WALL SHEARING STRESS 7

WALL SHEARING STRESS FOR FLAT PLATES . .

. 8

SHAPE PARAMETER FOR FLAT PLATES - 9 ENERGY THICKNESS AND SHAPE PARAMETER . . 10

ENTRALNMENT THICKNESS AND SHAPE PARAMETER 12

EQUILIBRIUM PRESSURE GRADIENTS . .

. 15

EFFECT OF PRESSURE-GRADIENT PARAMETER . 15

SHAPE P TER .

15

INTEGRAL METHODS .

17

GENERAL .

EQUILIBRIUM SHEAR-STRESS FACTORS ... 19

ENERGY METHOD 19

GENERAL . . .,

19

SHAPE PARAMETER EQUATION 20

POWER-LAW ENERGY THICKNESS 21

FERNHOLZ ENERGY THICKNESS 21

TWO-PARAMETER ENERGY THICKNESS . .

. 22

EXISTING RELATIONS FOR. DISSIPATION INTEGRAL 24

ENTRAINMENT METhOD .

. 24

GENERAL .

. 24

SHAPE PARAMETER EUATION . 26

POWER.LAW ENTRAINMENT ThICKNESS 27

HEAD ENTRAINMENT THICKNESS .

. 28

TWO-PARAMETER ENTRAINMENT ThICKNESS . .28

EXISTING RELATIONS FOR ENTRAINMENT FACTOR . .30

PARTIAL MOMENTUM METHODS . 30

GENERAL 30

POWERLAW VELOCITY PROFILE (s = m) . 33

POWER-LAW VELOCITY PROFILE (s 0) . . 34

(5)

Page

MOMENT OF MOMENTUM METHOD ₃₉

NONEQUILIBRIUM PRESSURE GRADIENTS ₄₀

REFERENCES ₄₃

LIST OF FIGURES

Page

Figure 1 - Comparison of Energy Shape Parameters 13

Figure 2 - Comparison of Entrainment Shape Parameters ₁₆ Figure 3 - Energy Method, M = f[H, R0]; Comparison of Various

Procedures ₂₃

Figure 4 - Energy Method, CD = f[H, R6]; Comparison of Various

Procedures ₂₅

Figure 5 - Entrainment Method, M = f[H, _{Re]; Comparison of Various}

Procedures ₂₉

Figure 6 Entrainment Method, E = f[H, R0]; Comparison of _Various

Procedures ₃₁

Figure 7 - Moment of Momentum Method, _{CT = f[H, R0]; Comparison of}

(6)

A

a

B1,B2, CD CD Cs Cs C T C1, C2 C3, C4 E e C * G C G H * H H H H 1.1 M m N n p p NOTATION

Slope of logarithmic velocity law

Factor in equation for flat-plate shape parameter, Equation (42) Interceptsof logarithmic velocity laws, Equations (6) and (9)

Factor in quation for flat-plate shape parameter, Equation (42) Dissipatio1 integral defined in Equation (89)

Dissipatio1 integral defined in Equation (94)

General shear-stress factor given in Equation (82)

Genetalshear-ste5S factor defined in Equation (84)

Shear-stres integral defined in Equation (192)

Constants in Equation (26) Constants in Equation (35)

Entrainment factor, Equation (115)

Subscript denoting equilibrium conditions

Rotta's shape parameter defined in Equation (21)

Velocity-defect energy shape parameter defined in Equaion (54)

Velocity-defect shape parameter defined in Equation (169)

Velocity-defect shape parameter defined in Equation (172)\

Shape parameter, H = 6 /0

Energy shape parameter defined in Equation (45)

Entrainment shape parameter defined in Equation (60)

Shape parameter defined in Equation (140)

Shape parameter defined in Equation (141)

Velocity-defect integral defined in Equation (32)

Pressure-gradient coefficient, Equation (82)

Relative position in boundary layer, m = s/6

Coefficient in Euation (82)

Power-law exponent, Equation (16) Coefficient in Equation (82)

(7)

R0 Momentum-thickness Reynolds number, R0 = US/v

s y-position in boundary layer

U Velocity at outer edge of boundary layer

u Mean tangential velocity component in boundary layer

Fluctuating tangential velocity

u u at position

y

S

Shear velocity, u =

yfT7

* u* General shear velocity, u t

v Mean normal velocity component

v' Fluctuating normal velocity w Coles' wake factor

x Streamwise distance

y Normal distance from wall

o Subscript denoting flat-plate conditions

0.3 _{Subscript denoting conditions at m = 0.,3}

Constants in Equation (28)

Clauser's pressure-gradient parameter defined in Equation (70)

y Velocity ratio, (u/U)0, Equation (25)

6 _{Boundary-layer thickness}

6 _{Displacement thickness}

C Factor in Equation (114)

o _{Momentum thickness}

0*

Energy thickness defined in Equation (44)

A _{Constant in Equation (202)}

v _{Kinematic viscosity}

p Density

-r Shear stress in boundary layer

taty=s

-r Wall shear stress

w

*

-t _{Characteristic shear stress, Equation (3)}

(8)

1References are listed on page 43,

ABSTRACT

Shape parameter differential equations are developed for turbulent boundary layers in pressure gradients incorporating

two-parameter velocity profiles Energy and entrainment

methods are included Shear stress factors are explicitly developed for equilibrium and quasi-equilibrium conditions.

ADMINISTRATIVE IF0RMATI0N

This work was funded by the Naval Ordnance Systems Command under

Subproject UR 109 01 03.

INTRODUCTION

The analytical prediction of turbulent boundary layers in pressure gradients has been the subject of intensive investigation not only because of the engineering applications but also because of the difficulties in developing methods suitable for all type of pres ute distributions. The

fundamental problem of the tubu1ent boundary layer is common to that of all turbulent flow: the lack of an adequate theory o the mechanics of turbu-lent motion.

Rtta3

has critically examined the varioUs predictive fnethods which

have appeared in the literature in recent years. The 1968 Stanford Con-ference4 tested the ability Of current methods to predict existing experi-mental re$ult from given pressure distributions.

Amoii'g the trends whicb may be ascertained from the Stanford Con-ference are:

The virtual abandonment of traditional 6-H fonfiulations (0 =

momentum thickness, H = shape parameter) when utilizing two-parameter velocity profiles such as the Coles law of the wake. _{However results are}

still given in terms of 0 and H.

The use of shear stress factors, (e.g.., the dissipation integral

for the energy equation) derived from equilibrium pressure gradients for use in non-equilibrium pressure gradients.

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It is now proposed to returrito the traditional -H methods even for two-parameter velocity profiles in deriving shape parameter equations for

the various integral methods. Analytical relations are obtained for the

various shape parameters for two-parameter velocity profiles. Analytical expressions are also derived for the various shear-stress factors under

equilibrium pressure gradients. Consideration is given tO applying

equilibrium shear-stress factors to non-e4uilibrium pressure gradients.

VELOCITY SIMI LARITY LAWS FOR BOUNDARY LAYER FLOW

WITH PRESSURE GRADIENTS

The velocity similarity laws, the law of tie wall and the velocity defect law, provide an analytical basis for pipe flow and boundary layer

flow on flat plates (i zero pressure gradient). The extension to boundary-layerflow with pressure gradients may be made to proceed as

follows:

INNER LAW OR LAW OF IE WALL

In addition to presure gradient dp/dx, the mean velocity component.

u parallel to the snooth wall (roughness and other effects may also be in-cluded) is consideréd to be dependent oi the usual local quantities: normal

where u =.

MT/P,

shear velocity. -OUTER LAW OR VELOCI'Y DEFECT LAW

In the boun4ry layer away from the wall, the velocity defect U - U

distance from the wll y, shearing kinematic viscosity v of the fluid

= £

Non-dimensional tatios may be fOrmCd

or

u_y

trêss ' at the dp wall ' density p and (1) (2) di V

3dxJ

pu

_I

(10)

the cumulative effect of the pressure gradients which may be represented by some appropriate characteristic value of the shearing stress T*. Mickley

et al.5 uses the value of shearing stress r at the inner edge of the outer layer as 1*; for adverse pressure gradients r becomes the maximum value of T and for zero pressure gradient

_Tw

Likewise McDonald and Stoddart6 use the maximum value of r as r for adverse pressure gradients. The density p

is an additional physical parameter. Analytically then

U - u = f[y/6,p,T*] (3)

where 6 is boundary layer thickness and U is value of u at y = 6. Non-dimensionally

- _F[L

6

where u is

}/t*/p

For flat plates (zero pressure gradient) r*

-and u

- u.

LOGARITHMIC LAW

Within the boundary layer the ranges of validity of the inner and outer laws are considered to overlap which results in logarithmic functions

for both the inner and outer laws for the common region. Equating the derivatives of u with respect to y for the inner and outer laws,

Experimentally Patel shows B1 = B10 for (IpuT) (dp/dx) < 0.01 where

B10 = B1[0] for flat plates (zero pressure gradient).

Also

(4)

Equations (2) and (4), produces

/ \ Btu/u i

(T.r

' T/

-lU-u . (5) A ') / (u y/v) - u1 \61

3(y/cS)

-Then integrating

uy

----=Ath_L_+B

T I '

_(6).

3dx

pU

(11)

boundary layers with

gradients B

LAW OF THE WAKE

- _{A in}

_{+ B20}

u*

_U.

I

T

?-

Ain,+--.B2,0

J!-

Ain-+B

U 2.

I

where B2 = .uT*/ut) B20 and B20 B2 for flat plates (zero pressure

gradient).

Thevalue of B12 then depends on the history of the effects of the pressure gradients up to the station being considered or B- f[x]. For

specially adjusted piessure gradients termed equilibrium pressure gradients, B2 can be held constnt with respect to x. Boun4a.ry layers on flat plates

in zero presstre gradient may be considered as a special case of equilibrium

B2 B In general even for equilibrium pressure Towards separation B2

8

It was observed by Coles that the experimental data for the outer

law outside the overlapping region had similarity in its deviation from

the logarithmic law

uch

that

UU_

_AinL+

U

I

(1

_w[tI)

where wy/6] is considered as a universal function teried the wake

function. The wake function given in tabular form by Coles8 was fitted

with a sigmoidal function by Hinze9

w = 1 + sin i -1 - cos

f

(11) (1) (10) or or

(12)

A polynomial fit is given by Moses1° as

and two-parameter if

U_flY

_H _R

15

u _L

_'

'

o

where :, momentum thickness

=

(1 - u/U) (u/U)dy;

displacement thickness = (1 - u/U)dy;

0 _*

H, shape parameter (due to Gruschwitz) = _{/0; and} R0, momentum thickness Reynolds number = US/v.

POWER LAW

An example of a one-parameter velocity profileis the familiar power law

which produces

[02

BOUNDARY LAYER PARAMETERS

GENERAL

Analytical models of the velocity profile are designated one-.

parameter if

(14)

(16)

w= 2

2(.)]

(12)

In an earlier analysis RottaU used as a first approximation a linear

function

w = 2j. ₍₁₃₎

1

The wake function is normalized so

_f

wd[y/ó] = 1. Also w[lj = 2.

(13)

Then

From the law of the wake

H= 2n+ 1

*

_H-1

T

H+l

OH-1

- H(H+l) H-1 \ (H - ₁₎₁ 2

I

H(H+l)

J

As shown in Figure 5 of Reference 12 the power law model prOvides a

sur-prisingly close fit to experimental velocity profiles in pressure gradients.

VELOCITY SIMILARITY LAWS

The velocity similarity laws provide a two-parameter velocity

pro-file.

A useful shape parameter is Rotta's shape parameter (also called

Clauser's shape parameter)

GE

d(.*)

(21)

If the velocity defect law is assumed to hold also to the wall, G is

constant for constant B2 of equilibrium boundary layers. Also from the

definitions 1 (H-l\ - (

Tw \

2 (H-1 = --'s-rn- / -(22)

(14)

4 A2 3.18 A + 0.75 B22

2 A + B2

WALL SHEARING STRESS

The coefficient of wall shearing stress or local skin friction is

expressed by

= _{f[H, Re]} (24)

1 13

As shown by Rotta and Patel, the law of the wake provides an

un-plicit relationship for (TW,pu2)[H,Re]. However an explicit relationship obtained in Reference 12 is a genera1zation of the procedure of Ludwieg

and Tillinann14 or 4

I

w

pU

T

it

'

H+1

w I w

\

fy

\ o

where

(iu2)

is the flat plate value for the same R8, (I/PU2

y = (u/U) and H are the flat plate values.

15 16

Empirically both Urani and Felsch express 1-as

y = c - C2 log H (Here log is taken as log10.)

Uram gives c1 = 0.9058 and c2 = 1.818 while Felsch gives c1 = 0.93 and

c2 = 1.95.

(23)

= f[R0].

At separation,

r/pU2

= 0, Uram's constants give H = 3.15 and

Felsch's constants give H = 3 which is closer to test data than H _{= 4 given}

17 . .

by the law of the wake. Nash uses H = 3 at separation in a modified

skin-friction fOrmula for pressure gradients. Also Equation (25) with

(15)

WALL SHEARING STRESS FOR FLAT PLATS

As show" in Reference 12, (T/PU2)0[R0] is derived from the Scho'enherr formul.a for the total drag of flat plates as

/

T

\ - 0.0146

- (log2R) (1/2 log[2 Re]

+ 0.4343)

Formulas of type

where

U

u

_t

From the law of the wake

IT

(w

'p 21 /

'U/

fc. R + o l 0 2

may also be deriedas follows:

Adding the et1apping inner and outer logarithmic laws, Equations

(6) and (9) p roduce

uô

T

=A2n

+B +B

v 1 2

Since u&/v = (u/U) (/0) R0 by definition

=-AZn--+AthR -Ath-+B +B

u 0 1 2

t

If the velocity defect law is assumed to hold to the wall

u / u \

ul

T1

_l

T

(16)

or where a .3026 A l

l-cA

and I = A + 4. B2

Then for f1at plates denoted by subscript o

AnR0+AZnH0-AinI10+B +B

T 1 2,0 2,n H c + C 0 3

4u

I

= a1 log R0 + a2 H -1

(/0

then from Equation (37)

1

(a1

lOg Re + 2) a B1 +

B20 + C3

_{A_n(A}

_{+ 1/2} 2

1-cA

SHAPE PARAMETER FOR FLAT PLATES

Since from Equation (22)

H 0 1-1 _a 0 1 ₂ H -1 = log Re + 0 0 ₀ (37) (38) (40) (41) If then

(17)

or

a1 a2

where = and ,=

0 0

ENERGY THICKNESS AD SHAPE PARAMETER

*,

The energy thickness e is defined as

E

6

(u)2]

The energy shape prameter H is defined as

* *

H e /e

For one-parameter velocity profiles

= f [H]

while for two-parameter velocity profiles

[1 (1a. log R0 (43)

= f [H. Re]

The simp1est relation is from the power-law velocity profile, a

one-parameter velopity profile, which is

4H

* _4H

H

3H1

-II

Closer empirical fits (one-parameter) have been obtained by various

18-22 investigators. (48) or H

H-i

=a log R0 (42)

(18)

From Weighardt:

A two-parameter relation H*

=

f[HR]

may be obtained from the

velocity similarity laws as follows. _{A velocity-defect law}

energy shape *

parameter G is defined as follows

Then, from Equation (22)

*

GE

From the appropriate definitions

1 2 2 (54) (55) * _{1.269 H} = H _{< H < 1.7} (49) H - 0.379 1.25 From Fernholz: *

_1272H

H H 1.35 < H < 2.8 ₍₅₀₎ H 0:37 +

5.4()

5 x 10 _{< R0 < 2.5 x 10}

From Moses et al.:

H* .02 + Q..S7 H + 0.095 H2

H (51)

From Goldberg:

*

- 2.78 H - 1 (52)

From Nicoll and Escudjer:

* -H = 1.431 0.0971 0.775 1.25 < H < 2.8 3 (53) H R0 < 8 x 1O4

(19)

Equation (56) was also obtained

= 3 -

H+(-)H

by Rotta.11

2 A + B2

G and G are then related through Equations (57) and (23) with B2

being the implicit parameter. At separatioT B2 -

and

GIG2

1.12. An

explicit nunier-ical fit gives closely

1.12

(

Then with Equation (56) and Equation (2)

- H + l.1 ± l..5 (H=l) (59)

p U.

where the variation

(25).

with R0 is obtained through T/PU given by Equation

The comparison in Figure 1 shows excellent agreement between the

two-parameter relation and the empirical one-parameter relations of

Feinho1z and of Nicoll and Escudier.

ENTRAINMENT THICKNESS AND SHAPE PARAMETER

*

The entrainment thickness is defined as - and the entrainmeflt shape parameter H a

(60)

For one-parameter velocity profiles

H = f [H] (61)

4.71 A B22 + 0.63 B2 From the law of the wake

* 12 A3 + 11.04 A2 B

(20)

* H 1.8 1 . 78 1.76 1.74 1.72 1. 70 1 .68 1 .66 1.64 1 .62 1 .60 1.58 1. 56 1.54 1.52 1 .50 1 .48 1.2 1.4 16 1.8 2.0 2.2 2.4 2.6 H 2.8 3.0

Figure 1 _{Comparison of Energy Shape Parameters}

R0=103

\_PARAMETER

2-PARAMETER R0=1

:oLLNDESCUDIER'

(21)

while for two-parameter velocity profiles

Au empirical fit by Head4 yields the one-parameter relation

H = 1.535

(H-0.7Y2715 + 3.3

(64)

A relation U =

fUR0J

may be obtained from the velocity similarity

laws. From the definition

(1

-LG)

Then from Equation

(22)

o i ( - 1

- G

_\

H2 and from Equation

(60)

=(frX-

)- H

(67)

This relation was also obtained by Mihel et al.4

and G may be related through EqUations

(33)

and

(23)

with B2

fi =

[H. R0]

(62)

The power law velocity profile gives

being the implicit parameter. At separation B2 -* and

G/l1

-close numerical fit gives

1.5.

A

G

_{- 1}

-5 (68)

G3'2

Then from Equation

(67)

H(H+2) H712

(\4

(69)

2(H-l)

_(H-l)5'2

_{\p u)}

2H

₍₆₃₎

H- 1

(22)

2

For H = 3,

T/PU

= 0 and H = 3.75.

The comparison in Figure 2 shows close _{agreement between the}

two-parameter values and the empirical fit of Head.

EQUILIBRIUM PRESSURE GRADIENTS

EFFECT OF PRESSURE GRADIENT PARA1'1ETER

It has been shown theoretically by Rotta and experimentally by Clauser that similarity is maintained if the pressure gradient parameter

13

- T dx (70)

w

is kept constant with respect to x or d13/dx = 0.

Then G and consequently B2 are constant. _{Empirically Nash23 obtains}

SHAPE PARAMETER

For equilibrium pressure gradients Equation (34) _{is stated as}

U = A n R0

2n H - A

2.n I

+ B1 + B2

T From Equation (22) H

1UA

_A _A B1 B2 = ₌ _{Qfl Re +}

_{n H}

--th I +

-. +

(71) O = 6.1

yL8l

+ 13 -1.7 Félsch16 obtains and Alber4'24 G = 6

l.8

- 13

-l..5

(72)

G 6.1 %'l.8l + 13 -0.40, 13

> 0

(73)

G=6.513+7.8067,

_13<0

(23)

2PARAMETER R0=b03 2PARAMETER 1O5 2PARAMETER -- - -.- - HEAD 18 20 2.2 24 26 2.8 10 9 7 H 3

(24)

In general A is constant and B1 is constant for smooth surfaces.

For equilibrium boundary layers G, '1 and B2 are constant with respect to

for a particular .

Differentiating Equation (75) with respect to x produces

A(H-l)2 [FI+(H+1)] '

dli

pU

0 -- - -

_dx

G H + A(H-l)

for equilibrium boundary layers.

An alternate form from Equation (22) is

.1

(_Tw

)2 dH '. p U2

GH

a n (Hl)B T w 2

PU

(76) (77) 1 G

H24fl.()

1

2.

where Tw/PU is given by Equation (25).

INTEGRAL METHODS

GENERAL

Integral methods for solving the turbulent boundary layers in pressure gradients refer to methods based on integrated forms of the equation of motion (momentum equation) and/or the equation of continuity, using various weighting factors which for incompressible two-dimensional

flow are au au

'dU

laT

u -

+

v -

=

U -

+

-ax ay dx

pay

au

_{+ - =}

ax

0 ax ay

(25)

du

= -P

where u'v' = Reynolds turbulent shear stress. The Reynolds turbulent

nor-mal stresses are not included though these may become quite significant

close to separation. Also close to separation p/y 0.

The classical integral form is von Krmn's momentum equation

ob-tained by integrating the equation of motion without using any weighting

factor T do

OdU

w

(H+2)U=

-pU

The purpose of variation of H with x

other integrated forms is to obtain eventually the

or dH/dx. The energy equation uses u as a weighting

factor. The entrainnient equation integrates the equation of continuity.

The moment-of-momentum equation uses y as the weighting factor. The partial

momentum equations partially integrate the equation of motion to differently

specified sublevels within the boundary layer. Details of these equations

follow. I

The shape parnieter equation may take the following forms

6 =- .M[H,R0] . !! + N[HR8] 2 - P[H1R0] c or where M, and Cs Cs TI P M[H,R]

(P[HR]

C -

N[HR])

TW2

P are coefficients, C is the generalized shear-stress factor

(26)

EQUILIBRIUM SHEAR-STRESS FACTORS

To obtain C5[H,R0] for equilibrium boundary layers, GdH/dx from Equation (83) is first converted to form

e P + N)

PU

and equated to Equation (76) to produce

= N A(H-l)2 [H+(H+l)8]

e HP P

[GH + A(H_l)2] P

where CSe represents the equilibrium value of C5.

e may be considered a function of H and Re if 8 is related to G by Equation (71) and G is a function of I-i and Re through Equation (22).

ENERGY METHOD

GENE RAL

The energy equation is obtained by multiplying the equation of motion Equation (78) by u and integrating over the whole boundary layer

1L (u3 e*) 2

dx _P

* *

In terms of energy shape parameter H ± S /0 and employing the momentum

equation for dS/dx * dEl

*edU

* T = (H1)H - H w _{+ CD}

PU

where 2 1.

CD=3

j

r-dy

0

called the dissipation integral, the shear-stress work integral or the production integral.

(27)

SHAPE PARAMETER EQUATION

and finally

where

and

Then from Equation (82)

-.

*

t

H BH dH BH

r

edu

;-

=';i

B Zn R I 2 - (H+l)

OLpU

0

=

[(H_l)H* + (H+l) B

I

:i

1*

BH - + B Zn Re DJ BH _U CD 2 1

f

B(u/U) dt''

J

B(y/) \S 0 -*

+ (H+l)

B

in

R3]

:

*

N (H* + .B in _Re)

(94)

- The objective Equation (88), to a

is to convert the energy shape parainetet equation,

shape parameter equation of forms Equations (82) or

(83).

For a two-parameter velocity profile

H* _{H* [H} _Zn Re]

Then expanding into partjal derivatives

*

dif

BH dH BH

dZnR

0

dx

-

BH 0 dx B 9n R,.,

(28)

Then M = H(H-1) (3H-1) N H(3H-1)

P=

(3H-1)2 4 [ (Hr- 1) +H] 4 _{- ])}2 _{[H+ (H+1)} _I D,e 311-1 (3111) [GH + A(H-1) ]

FERNHOLZ ENERGY THICKNESS

Here * ₁₂₇₂₁₁ _-4 ₄ H = H- 0.37 + 5.4 x 10 H dH 0.4706 -3 3 dH - 2 + 2.16 x 10 H (H-0.37) M (H-i) (H_0.37)[1.272 5.4 x 10 (H_O.37)H] 0.4706 - 2.16 x (H-0.37)2 H3 (50) (104)

P=-

* (97)

CDe is obtained from Equation (86) since CDe = CS,e for the. energy

equation.

POWER-LAW ENERGY THICKNESS

Here H = f[H] - 311-1 dH dH (3H-1)2 (48) (98) Then and

(29)

and dH * dH - +11 H J A(H-1)2 [H +(H+i)] dH* D,e _L H J _I GH + A(H-1)2 dH

TWO-PARAMETER ENERGY THICKNESS

Here

It

2'.

1.12(H-1) + 1.5 (H-i)

il

2 H

pU

With EqUation (25)

12(H2_i)

H2

and with Equations (37)

and

(42)

M N = H-i (k

_0.37)2

0.4706 -2.16

x 1O (H _0.37)2 H

It

i.5/---V pu

2 c2 (H-i) 2.31H +i\ HI %O

₎

2'

2a(H0-i) ( C2 +

2.3(H 1) \2.3H I

-

H +1

0

00

0 M, N, P, and CD e

are ob1tained from Equations (95), (96), (97),

and (86).

Figure 3 shows hw close the M values are for the various

formu-lations. (105) (106)

(107)

(59) (108) (109) 2

1.5(H-1)

0

(30)

/ / / /

//

_/

V--/I

//

/1

2-PARAMETER, R6=103 _FERNHOLZ /

/,

,,

//

POWER LAW

7'

R0=105 2-PARAMETER, 13 14 15 16 17

18

1.9 2.0 2.1 2.2 H

Figure 3 - Energy Method, M = f[H, R0]; Comparison of Various Procedures

14 13 12 11 10 9 8 7 6 4 3 2

(31)

EXISTING RELATIONS FOR DISSIPATION INTEGRAL TruckenbrOdt25 approximates 0.0112 D 1/6 Re - 11

from the analyses of Rotta.

Wa1z26 suggests for euilibrium boundary layers

Escudier et al.28 propose

where

*

₄₈₁

0.00962 + 0.1644 (H

-1.5)

R (0.231711 -0.2644)

e

where H is determined by Fernholz, Equation (50).

.A fit of experiflenta1. shear-stress data by Escudier and Spalding27 results in

1.094

CD (2

ENTRAINMENT METHOD

GENERAL

The entrainment equation was first proposed by i-lead29'4 on the basis

of physical reasoning iegarding tFie growth or entrainient of the developing

boi.iridary layer. it has since been found out that it can be derived y

integrating the equaticn of continuity, Equation (79) Michel et al 2

Figure 4 compares the dissipation integral for the various formulations

(.1.10) 0.004214 H -0.004572. (112) +

pU

+ .1) 2.715 0.0113 + (1 - (113)

*

2

*/2

*

-H

--H -1 (114)

(32)

9 8 7 6 3 2 8 7 6 5 3 22 2-PARAMETER C , R D,e 0

WALZ Coe R9103 2-PARAMETER CDe R0-10 WALZ C0 R0105

-- ESCUDIER "._... AND SPALDING ___________

ESCUDIER AND SPALDING CO3

I R0=103 C

-

-t1NIC0LLANDESCUDIER CD _e' ESCUDIER ET AL CO3 R0= 19 20 21 13 1.4 1 5 1.6 17 1 8 H

(33)

introduced a shear-stress factor by considering the equation of motion

Equation (78) at y = . The entrainment equation then resembles the energy equation and is given by

In terms of the entrainment shape parameter H ( - ó)/Ü and employing

the momentum equation for de/dx

SHAPE PARAMETER EQUATION

The objective is to convert the entrainment shape parameter

and finally (-BcI

\aHedu

H+

B 2n Re) E

(+1)Hftf

H E e dH dx H+1 I-., _B

\BH

Tw

-(H+

El--\

BflR

_"BHpU2

(115) (116) (120)

equation, Equation (116), to a shape parameter equation of forms Equations

(82) or (83).

For a two-parameter velocity profile

H

= fEH, 9n R0] (117)

Then expanding into partial derivatives

B dH B

0dxBH0dxBthR80

dH BH dH BH r Tw

e=-e+

dnR0

e dU (118) (119) dx (H+ 1

nR[

2 S

PU

U dx

(34)

where M =- (H+i) i + f. ai

\H

N=-IH+

-- .rw --

L(u/u),

2

PU

(122) (123) Here 2H (63) H- 1 Then (H_i)2 (125) M H(H2_l) (126) N = I-I(H-l) (127)

Hi2

and (128) E _{- (H-i)}2H (H+i) + 2A [H+(H+l)] (129) GH + A(H-l)2

P=-

(124)

and E is obtained from Equation (86) since E = C for the entrainment

e e S,e

method.

The actual evaluation of Ee and the accompanying shape parameter equation depends on the particular relation for entrainment thickness.

Some examples are now presented:

(35)

HEAD ENTRAINMENT THICKNESS

and

Here

TWO-PARAMETER ENTRAINMENT ThICKNESS

Here H(H+2) 3.8 H

(_

-

2(H-i) (H-i)5"2

'p

U2 With Equation (25) BH H2-2H-2 3.8 H7"2 = 2 ₊ (H-i)5,2 =

1'

+

ii

ç A(H-i)2 H+(H+1) L J

GH+A(H-1)

3/4 -''

I

2H-7 3c2 ) L'O-'-O 2.3(Hii)HY

and with quations

(7)

and (42)

57

H7"2

/_T\3"4[

fTw

\1/2

2aQ_i(

C2

2,fi R 572 1 2 I

IT

21

+ 2.3fH1\ \2..3H I H +1

0 (H-i) \Q U _/

_L

\p

U

°

₁

° °

°

M, N, P, an4 Ee are obtained from Equations (122), (123), (124), and (86).

Figure 5 shows large differences in M between the various formulations

(134) (69) (135) (136)

fl

1.535 (}O7)2715 +

3

₍₆₄₎ Then c dH - 4.168 (H-0.7 -3. 715

(130)

M= O.68(+1)(H-0.7) [

+ 2.15 W0.7)2.715] ₍₁₃₁₎ N 0.3683, (H0.7) [i 2..164(H0.7)2.715] (132) N 4.168

(133)

(36)

6 / / /

/

I

/

,

2-PARAMETER R0=103__\,'

'H

/<_HEAD

/

_/

/

H

/

,

2-PARAMETER R0=105 ' POWER LAW H H 1.3 14 1.5 1.6 17 1. 8 19 2.0 2.1 2.2 H

Figure 5 - Entrainment Method, M = f[H, R0}; Comparison of Various

14 13 12 11 10 9 4 3 2 1 0

(37)

where S W 2

pU

2

pU

U = U at y = s T = T at y

= S.

(139)

EXISTING RELATIONS FOR ENTRAINMENT FACTOR

Empirically Head4 obtains

E = 0.0306 [l.535(H_0.7)_2.715 + 0.3]_0.653 (137)

Also Nicoll and Ramaprian30 include Statford's data for separating flow and

obtain an empirical fit

E = 0.035 /l.25

(138)

Figure 6 compares the. various entrainment factors.

PARTIAL MOMENTUM THODS

GENERAL

Another group of integrated, equations which are transformable into

Thhape. parameter equations maybe obtained by integrating the equation of motion, Equation (78), to some intermediate value of y, say s[x]. s may be

S =

in where m is a constant which was used by Moses4 or s = 0 which was

used by Furuya and Nakamura.4 Another possibility is s[u/U = const] which

will not be treated here.

Integrating the equation of motion, Equation (78), to y = s produces

1

I u

IldU

j

ffdy

(38)

0.06 0.05 0.04 0.02 0.01 0 1.3 1.4 1.5 1 6 1 7 1 8 H

Figure 6 - Entrainment Method, E = f[H, R0]; Comparison of Various

Procedures

2-PARAMETER Ee R0=103

2-PARAMETER E ,, R =10

e o

NI COLL AND RAMAPRIAN E

-c

HEAD E

-E

0.03

(39)

and'

Then Equation (139) becomes

With

and

Then

Two new shape parameters are introduced.:

dx --

(fle)

-

+ H+(H+1)

HE

fdy

e S 2 I

(U

J

T S 2

pU

= fH,2,n _Re] fri

_f[Hth

Re] H b ozi dy T w 2

PU

- U RO (H +

1-afl

UH

H U H T S T w P. U dx (140) (141) (142) (143) (144) (143)

(40)

and for s = 1-1+1 U 2 m M

I

-

U HH+(H+1) [

n0

U a

-

3FI H + 2n R0

i\ + 3

2 R U s U 3H

H-U' S U 3R U.S BH 3H U 3H J:I U 3H U 3H

POWER-LAW VELOCITY PROFILE (s = m)

For 'power-law velocity' profiles Equation (16) H+i 2H

fs\2

H-I 4%cS) U S U

(S

H-i 2 S (146) (147) (148) (149) (150) (151) (153) (154) and H+ 1 =_ '[2 -, H(Hl) 'Zn m] (152) L" J Since H-i )+ I

(41)

and Also H-i u

r

5 I H-i ii _-

Lw+i

--

1 H-i H H

_LHH+1

For power-law velocity profiles

dH

P-H-i H(H+l) , (H+l\ s H-l) .6 and fOr s = th6

fH+l\ H

H

TrT)

'

(H-i)

[2 - (H2..l) n m]

Therefore from Equations (146), (147), and (148)

POWERLAW VELOCITY PROFILE (s = 0)

From Equations (150) and (19) H(H+i) (

mlH)

N g_1;) (l+mT') (H-.I) H m 9,nm 2H

iH-i

- (H-i) LFIIH+n

I

H-1

[H(H+l).

H+1 (1 55) (161) (162)

(42)

and dH dH Then Then for s = I H-i H(H+i) - (H-i) LH(H4i) M = (H-i) -

[H]

_}

N=

Hi{

-

n H-i [ H+i H

m[

H(H+i)]

p-

H-i

TWO-PAR.ANETER VELOCITY PROFILE (s =

m)

The objective is to obtain H[HRQ] and

shape parameter G defined as

s/cS 'U-u'

J

__)d(.)

0

'

tI

J

0 (__)d(..)\ -r/

From appropriate definitions

r

H-i 1H I H-i

LHH+iJ

9n H(H+I) H = (H+H)m - HG (171) Let us introduce (169) (170) H-i (H+i)]} (165) -H (i66) (167) (i68)

(43)

Likewise let us introduce shape parameter

defined as

Sf'5 ₂

() d(f)

From appropriate definitions

A

= (H-i)

.g.

+

()

-2 HG

= (H-i)

+

(A+H) - -2 HG

(173)

For s =

A

= (H-i) --

+ (fI+H)m -2 HG (174)

From the iaw of the wake, Equation (12)

A tn(1-2n m) + or from Equation (22) 1

G03

= 0.5541 + 0.4.10 (H-i)

(PT2)

G

L

(Yi)

d(f)

21

m

- m

1 -2

(177)

Unfortinateiy the condiftion s = 0 does not lend itself to this type of

analysis. Then Fl03 = .3(HH) -H [0.5541 + H-i

I

(172)

(175)

A+--G is related to A+--G throigh B.2 in Equation (23).

An empiricai fit for m = 0.3 (m = 0.3 was used by Moses) gives

(178)

(44)

consequently = A2 m(in2 rn-2 in

m+2)2

A B2 rn(in ml) + A B2 m3 [(2_rn) in m -1 2 3

94

5

46 rnl-2m +m

-2m

For m = 0.3 I

G03

= 1.7573 A2 + 1.2512 A B2 + 0.2571 B22 or

is related to G through B2 by Equation (23). An empirical fit gives for

m = 0.3

G03

G 2.04 - 0.686

+

-G2

G03

_2.04 _H2 Tw = 0.686 ₊ -) (H_1) p U (181)

(182)

Since '1

m2

d(*)

H03

= 0.3 H -0.122 H -0.686 H2

Vu2

_2.488V/')

(183)

From Equation (10) for m s/5

= 1

-

/-:;- [

A in in + B2 (1

4w[1)]

(184)

(45)

(46)

is the shear-stress integral.

32

while Nash and Macdonald suggest

MOMENT OF MOMENTUM METHOD

If the equation of motion, Equation (78), is multiplied by y and

then integrated from y = 0 to y = , a moment of momentum equation is

formed. Unfortunately the resulting equation is awkward to deal with on

the basis of the two-parameter velocity profile in order to obtain a shape

parameter equation. However a convenient form results ftom a power-law velocity profile which Tetervin and Lin3' originally obtained

From Equation (83) -M H(H+1)(H2-1) (193) 2 N = H H2-1 (194) p (H+1)(H2-1) (195)

2H 2A[H+(H+1) ](H-1)l

T

(196)

+ (H+1)2[GH+A

H_l]2J

U2

CT,e =

Nash and Hicks4 use

= 0.025 (i 1)2 (197) dH H(Hi-l)(H2-l) 0 dU + H (H2l) T _(H+l)(H2-l) CT (191) 0 2 U d where

PU

1 C T

E2

_J

T (192)

PU2(

0

(47)

Figure 7 compares the shear stress integrals of some of the various

formu-lations.

NONEQUILIBRIUM PRESSURE GRADIENTS

The use of equilibrium stress factors not only ensures agreement for equilibrium pressure gradients but also for quasi-equilibrium conditions where the G values do not remain constant but vary in accordance with the

equilibrium G- relatiOn, Relating equilibrium stress factors to H and R8 provides a built-in lag which is characteristic of the response of the

shear-stress distribution to sudden changes in pressure gradients

Lag-type equations have been proposed by Goldberg21 and Nash and Hicks4 of type

dC

- A (C

-

c)

where A is a constant adjusted to suit the experimental data.

(202)

Also McDona143 proposes

+ 1.16) (19 ) e p U CT,e = (O9 + 1.2) Tw (199)

Values of CT for strong adverse pressure gradients are given .ii

Reference 12 as (H-i where (200) \H+l 0.0378 )/52.9 log FL

_{_4.181(Tw}

(.201) =

(W1)

[1

_H21

j\p

(48)

10 9 8 7 6 0

13

14

15

16

17 1 8

19

20

H

Figure 7 Moment of Momentum Method, _CT = f[H,, R0];; Comparison of

Various Procedures 2.1'

22

POWER LAW R0=105 C , e POWER LAW R0= 1

-1Iu -___

(49)

Close to separation ordinary boundary layer conditions seem to fail. There are three-diiensional cross flows, normal-stress effects and normal

(50)

REFERENCES

Rotta, J.C., "Turbulent Boundary Layers in Incompressible Flow,"

in "Progress in Aeronautical Sciences," Vol. 2, A. Fern et al., eds., Pergamon Press, New York (1962).

Rotta, J.C., "Critical Review of Existing Methods for Calcu-lating the Development of Turbulent Layers" in "Fluid Mechanics Of Internal Flow," G. Sovran, ed., Elsevier Publishing Co;, Amsterdam (1967)

(Pro-ceedings of General Motors Symposium (1965).

Rotta, J.C., "Recent Developments in Calculation Methods for Turbulent Boundary Layers with Pressure Gradients and Heat Transfer," Transactions of ASME, J. Appi. Mech., Series E, Vol. 33, No. 2 (Jun 1966).

Kline, S.J. et al., eds., "Proceedings - Computation of

Turbulent Boun4ary Layers - 1968 AFOSR-IFP-Stanford Conference," Vol. 1, Thermosciences Div., Dept. Mech. Eng., Stanford University, Calif. (1968).

Mickley, H.S. et al., "Nonequilibrium Turbulent Boundary Layer," AIAA Journal, Vol. 5, No. 9 (Sep 1967).

McDonald, H. and Stoddart, J.A.P., "On the Development of the

Incompressible Turbulent Boundary Layer," British Aircraft Corp. Ae 225

(Mar 1965); also ARC RE,M 3484 (1967).

Patel, V.C., "Calibration of the Preston Tube and Limitations on its Use in Pressure Gradients," J. Fluid Mech., Vol. 23, Pt. 1 (Sep 1965).

Coles, D., "The Law of the Wake in the Turbulent Boundary Layer," J. Fluid Mech., Vol.1, Pt. 2 (Jul 1956).

Hinze, J.O., "Turbulence," McGraw-Hill, New York (1959).

Moses, H.L., !'The Behavior of Turbulent Boundary Layers in

Adverse Pressure Gradients," Gas Turbine Laboratory, Massachusetts

Institute of Technology Report 73 (Jan 1964).

:11. Rotta, J.., "On the Theory of the Turbulent Boundary Layer,"

Mitteilungen aus dem Max-Plank-Institut fr Strnungsforschung, No. 1

(51)

Granville, P.S., "A Method for the Calculation of the Turbulent Boundary Layer in a Pressure Gradient," David Taylor Model Basin Report 752

(May 1951).

Patel., R.P., "Afl Improved Law for the Skin Friction in an

Incompressible Turbulent Boundary Layer in any Pressure Gradient," Dept.

Of Mech. Eng., McGill University (May 1962).

Ludwieg, H. and Tillinann, W., "investigations of Surface Shearing

Stresses in Turbu1eit Boundary Layers," Ingenieur-Archiv, Vol. 17, No. 4,

p. 288 (1949); translated as NACA TM 1285 (May 1950).

Uram, E.M., "Skin-Friction Calculation for Turbulent Boundary.

Layers in Adverse Pkessu±e Distributions," J. AerO. Sci., Vol. 2.7, p. 75 (1960).

Felsch, K.O., "A Contribution to the Calculation of Turbulent Boundary Layers in Two_Dimensional Incompressible Flow," Deutsche Luft-und

Rauthfahrt Forschungsbericht 66-46 (July 1966), Royal Aircraft Establishment Library Translation 1219 (Ma.r 1967).

Nash, J.F., "A Note on Skin-Friction Laws for the Incompressible

Turbulent Boundary Layer," National Physical Laboratory Aerodynamics

Division Report 1135 (Dec 1964).

Wieghardt, K. and Tillmann, W., "On the Turbulent Friction

Layer for Rising Pressure," Kaiser Wilhelm - Institut ftfr Str'o4rnungsforschung

ZWB UM 6617 (Nov 1964); translated as NACA TM 1314 (Oct 1951).

Feritholz, H., "A New Empirical Relationship between the

Form-Parameters H32 and H12 in Boundary Layer Theory," J. Royal Aer. Soc.,

Vol. 66, No. 9 (Sep. 1962).

Moses, H.L. et al., "Boundary Layer Separation in Internal

Flow," Gas Turbine Laboratory, Massachusetts Institute of technology

Report 81 (Sep 1965). .

21! Goldberg, P., "Upstream History and Apparent Stress in Turbulent

Boundary Layers," Gas Turbine Laboratory, Massachuetts Institute of

(52)

Nicoll,' W.B. and Escudier, M.P., "Empirical Relationships be-tween the Siape Factors H32 and H12 for tJiiifoim-Density Turbilent Boundaty

Layers and Wall Jets," AIAA Journal., Vol. 4, No. 5 (May 1966).

Nash, J.F., "Turbulent-BoundaryLayer Behavior and the Auxiliary Equation," in "Recent Developments in oundaiy Layer Researth,"AGARDograph

97 (May 1965).

Alber, I.E., "Turbulent Boundary Layer Development," Dynamic

Science Co. TR-A68-101, Monrovia, Calif. (Jan 1968).

Truckenbrodt, E., "A Metho4 of Quadrature for Calculation of

the Laminar and Turbulent Boundary Layer in Case of Plane and Rotationally Symmetrical Flow," Ingenieur-Archiv, Vol. 20 (1952); translated as NACA

TM 1379 (May 1955).

Walz, A., "Uber Fortschritte in Nherungstheorie und Praxis der Berechnung Kompressibler laminaret und turbulentêr Grenzschichten mit

Warmeribergang," Zeitschrift ftr Flugwissenschaften, Vol. 13, No. 3 (Mar 1965).

Escudier, M.P. and Spalding, D.B., "A Note on the Turbulent Uniform-Property Hydrodynamic Boundary Layer on a Smooth Impermeable Wall; Comparisons of Theory with Experiment," (AD-805492) ARC Current Paper 815

(1966).

Escudier, M.P. êt al., "Decay of a Velocity Maximum in a Turbulent Boundary Layer," Aero. Quarterly, Vol. 18, Pt. 2 (May 1967).

Head, M.R., "Entrainment in the Turbulent Boundary Layer," ARC RM 3152 (Sep 1958).

Nicoll, W.B. and Ramaprian, B.R., "A Modified Entrainment Theory for the Prediction of Turbulent Boundary Layer Growth in Adverse

Pressure Gradients," ASME Transactions, J. Bas. Eng. Vol. 91, Series D,

No. 4 (Dec 1969).

Tetervin, N. and Lin, C.C., "A General Integral Form of the

Boundary-Layer Equation for Incompressible Flow with an Application to the Calculation of the Separation Point of Turbulent Boundary Layers," NACA

(53)

32. Nash, JJF. and Macdonald, A.G.J., "A Calculation Method for the Incompressible Turbulent Boundary Layer, Including the Effect of

Up-stream History of the Turbulent Shear Stress," National Physical. Laboratory Aero Report 1234 (May 1967).

33 McDonald, H., 'The Departure from Equilibtiuu of Turbulent Boundary Layers," Aero. Quarteily, Vol. 19, Pt. 1 (Feb 1968).

(54)

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Pasadena, Calif

4 Dept Mech Eng

Catholic Univ, Wash., D.C. 1 Prof M.J. Casarella 1 Prof P.K. Chang 1 Prof Keinhofer

1 Dr. C.S. Wells, Jr.

LTV Res Center, Dallas, Texas

2 St. Anthony Falls Hydr Lab

Univ of Minn., Minneapolis

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J. Levy, Hydrodynamics Dept Aerojet-Gen, Azusa, Calif

Westinghouse Electr Corp, Annapolis

Attn: M.S. Macovsky

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Ocean Systeips Op

North American Rockwell Corp Anaheim, Calif

Oceanics, In. Attn: A. Lehman Dr. R. Bernickêr

EssO Math E Systems, Inc. Florham

Park,

N.J.

Prof Douglas E. Aibott Fluid Mechanics Lab

School of Mechanical Engrg

Purdue Univ Lafayett, Indiana 47907 Mr. Irwin E. Alber Dynamic Science 1900 Walker Ave Monrovia, Calif 91019 Copies 2 Aerotherm 460 Calif AVe

Palo Alto., Calif

1 Mr. Robert N. Kendall 1 Mr. L. Anderson

Mr. Ivan Beckwith

NSAS Langley Research Center

Mail Drop=l6l Langley Station

Hampton, Va 23365

Mr. T. Cebeci

Douglas Aircraft Div

3855 Lakewood Blvd

Long Beach, Calif 90801

Prof Francis H. Clauser Vice Chancelor

Univ of Calif

Santa Cruz, Calif 95060 Prof Donald Coles, GALCIT

306 Karman Lab

CIT, Pasadena, Calif .91109 Prof G. Corcos

Dept of Mechanical Engrg

Div of Aeronautical Sci

Univ of Calif

Berkeley., Calif 94720

Dr. G. Deboy

Dept Mechanical Engr Purdue Univ Lafayette, Indiana 47907 Dr. George S. Deiwert Ames Lab Fluid Mechanics Br Bldg 229-1, Mail Stop N-229-4

Moffett Field, Calif

94035

Dr. C. Donaldson

Aeronautical Research Assoc.

of

Princeton, Inc

Princeton, N.J. 08540

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Aero Research Staff Mail Stop 55-38

(56)

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Rjn 308, Pierce Hall

Harvard Univ

Cambridge, Mass. 02138

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Cambridge, Mass. 02139

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Dept Mech Engrg Purdue Univ

Lafayette, Indiana 47907

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Pratt Whitney Aircraft Co

East Hartford, Conri 06100

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School of Engrg

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Tuskegee Institute Alabama 36088

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Div of Engrg Applied

Physics, Pierce Hall 324 Harvard Univ

Cambridge, Mass 02138 Prof H.W. Liepmann

Karman Lab

CIT, Pasadena, Calif 91109

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Aerospace Engrg Dept

153E Hammond Bldg

Penn State Univ Univ Park, Pa 16802 Mr. H. McDonald

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United Aircraft Research Ctr Silver Lane

East Hartford, Conn 06108 Prof George Mellor

Dept Aerospace Mechn Sci

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Princeton Univ

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Univ of Calif, Los Angeles

Dept Engrg

Los Angeles, Calif 90024

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Ill Institute of Technology Aerospace E Mech Engrg Dept

Chicago, Ill Dr. Hal L. Moses

Project Engr

Cornirg. Glass Works 3800 Electronics Drive

Raleigh, N.C. 27604

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Dept 72-14 Zone 403 Lockheed-Georgia Co

Marietta, Ga. 30060

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Univ of Notre Dame

Notre Dame, Tn4iana 46556

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Nielsen ngrg Research Co

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(57)

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Colorado State Univ Fort Collins, Colorado

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Renton, Washington Prof W.C. Reynolds

Mech Engrg Dept

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Gene±al Motors Techn Center

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Martin Marietta Corp

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do

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Baltimore, Md 21218 Prof H. Tennekes

Aerospace Engrg Dept

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Assistant Dean

Graduate School of Engrg Univ of Bridgeport

Bridgeport, Conn 06602

Dr. W.W. Willmarth

Dept Aerospace Engrg

Univ of Michigan

Ann Arbor, Michigan 48104

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Kirtland Air Force Base New Mexico 87117

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UNCLASSIFIED

DDFORM 1473

(PAGE 1)

( NOV 65 UNCLASSIFIED

---DOCUMENT CONTROL DATA - R & D

Security classification of rUle, body of abstract and indcxirtg annotation must be entered when the overall report Is classified) t. ORIGINATING ACTI VI TV (Co.porato author)

-Naval Ship Research and Development Center Washington, D.C. 20007

2a. REPORT SECURI TV CLASSIFICATION

UNCLASSIFIED

2b. GROUP

3- REPORT TITLE

INTEGRAL METHODS FOR TURBULENT BOUNDARY LAYERS IN PRESSURE GRADIENTS

4 DESCRIPTIVE NOTES (Type of report and incl.lsive dales)

Final Report

5. AU THOR(S) (First name, middle initial, last name)

- Paul S. Granville

6- REPORT DATE

April 1970

75. TOTAL NO. OF PAGES

55

7b. NO. OF REFS

33

Ba. CONTRACT OR GRANT NO.

-b. PROJECT NO. UR 109 01 03

C.

d.

Ba. OR)GINATORS REPORT NUMBER(S)

3308

9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report)

IS. DISTRIBUTION STATEMENT

-APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

It- SUPPLEMENTARY NOTES 2. SPONSORING MILITARY ACTIVITY

Naval Ordnance Systems Command Washington, D.C. 20360

13. ABSTRACT

-Shape parameter differential equations are developed for turbulent boundary layers in pressure gradients incorporating two-parameter velocity profiles. Energy and entrainment methods

are included. Shear stress factors are explicitly developed for equilibrium and quasi-equilibrium conditions.

(59)

UNCLASS I F lED

Security Classification

Id

-- -

-KEY WORDS - . -- LINK A - LINKS LINK C

ROLE WT ROLE WT ROLE WT

Boundary layer Pressure gradient Velocity profile