AD-A140 560 MIXANG-LENGTH FORMULAAIONS FOR TURR ENT BOUNDARY

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David W. Taylor Naval Ship Research &W& 6 0 ,4 fts" ft

and Development CenterBthesda. Maryland 20064 (See reverse side)

is Cof~OU010s oeumCu I @$aAN aO0fegs of *60O3? O*,5Naval Sea Systems Comand (SEA 0SR24) March 1984Hull Research and Technology Branch is moe ",V069 tVashington, D.C. 20362 44

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Roughmess Turbulent Dissipation RateMixing Length Turbulent Kinetic EnergyTurbulent Boundary LayerEddy Viscosity

ft AgsiOACT rCOM.m ea* if "tIear aw #~fto O 01.0Mixing lengths ere formulated for turbulent boundary layers over

arbitrarily rough surfaces from both the van Driest and from the Rottaprocedures. The associated eddy viscosities and the associated turbu-lent dissipation rates are also formulated. f,oP 147) 3 t@O uvS15~.t

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TABLE OF CONTENTS

PageLIST OF FIGURES ............................... Iv

NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ABSTRACT . .1

ADMINISTRATIVE INFORMATION . . . ... 1

INTRODUCTION ................ . . . . . . ... 1

CHARACTERIZATION OF ROUGHNESS DRAG o.*......... ......... 2

NI XING-LICG'T THEORY ............................ 4

EXISTING MIXING-LNGTH METHODS FOR ROUGH SURFACES .............. 7

ROTTA I .*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ROTTA II . . . . . . . . . ... 8

PROPOSED NEAR-WALL MIXING LENGTHS FOR ROUGH SURFACES ............ 10

GENERAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

INTmIEIATE ROUGHNESS REI .......... ... 10

FULL. ROUGH REGIME ............................ 11

GEN'ERALIZED ROTTA It .... o...................... 11

DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

xumnI CAL RESULTS ..... o........................ 13SUMMqA~tT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

APPENDIX A - CONVZRSION OF MIXING LENGTHS TO EDDYVISCOSITIES ..... ... . .......................... 23

APPENDIX I - EAR-WALl. VALUES OF TURBUIEN KINETICZNERGT (k) AND TURBULENTDISSIPATION RATE (c) .......... ...................... 27

APPENDIX C - HYPERBOLIC-TANGgRT MWOIFICATION FU CTION .............. 31

REUNoS... ... ..... ..... ..... ...... ...... 33

lit

LIST OF FIGURS

Page

1 - Variation of Integraud y(-) with . ... ................ . is

2 - Plot of A versus 51 for Intermediate

Roughness Regime ...... ..... ........................... 16

3 - Plot of Rotta I t versus 51 for Fully Rough Regime ............ ... 17

4 - Correlation of Rotta II Factor by with 5 forIntermediate Roughness Regime ...... ..... .................. 1

S - Correlation of Rotta II Factor by with 51 forFully Rough Regime ...... ..... .......................... 19

6 - Comparison of gotta I and Rotc. II Wall Mixing

Lengths 1. ......... ........................ .O.......20

7 -CAparlson of Mixing Lenghs. ...... .. ..................... 21

A.1 - Eddy Viscosities Over Rough Surfaces ...... .. ................. 25

3.l - Variation of Turbulent Kinetic Energy Near Wall ... ............ ... 29

B.2 - Variation of Turbulent Dissipation Rate Near Wall ............. .... 30

C.A - Comparison of y -) . . ....................... 32\dy/

ivi

NOTATION

A Slope of logarithmic velocity law, A I/IC

Intercept of logaritthnic velocity law in Ieynolds-numer mode(also drag characterization function)

81 Value of al for nmooth surfaces

Br Intercept of logarithmic velocity law in relative-roughneas mode

ar Value of Br for fully rough regime

lI12 Integrals defined by Equation (30)

k Roughness height

k1 ,k 2 .. Other roughness lengths

k Roughness Reynolds number, k * u k/v

Turbulent kinetic energy

I Mixing length

Pondimensionsl mixing length, t ° tlv

Value of t at the willw

K Modification function

T Roughness texture, T- k/k1 . kI/k 2,...

u Stremovise velocity component

u Nondimenuional u, u *uu

u Shear velocity, u t ___

w Subscript denoting conditions at the wall, y - 0

Normal distance from wall

Vi

* ay Nondimnuional y, y - u y/v

YL. laminar sublayer thickness

YL Nondimensional YL' YL " UTYL/V

as Drag characteraation. 5 AS - a

Ay gotta I shift

C Turbulent dissipation rate

Kvon Kirman constant

A Length factor In modification function

A Mondiuausional A, uI /v

* Ash A for hyperbolic-tangent modification function

X for van Driest modification functionV

A for -mooth surfacesV.2 V

Kinematic viscosity of fluid

Turbulent eddy viscosity

Density of fluid

t Shearing stress

Laminar shearing stress

T t Turbulent shear stress

IWall shearing stress

Vi

ASTRCT O

~' "Nitxlns length& are formulated for turbulent boundary

faces~ va aries th ehdo ankfromdSaig the m roeue.The afssoci aned mto

Its sbsqeddy moictie b nd ituhet assocand thrbuet kfinenitc

Furtermremienegie anublent red isosiptioraes arte wals mayberquateda

bThdar cordto tiot we* prafsorte eqatin for Dai .ror predlhip Rofth

ah Dvstan Cwaytfr.m t wasl authoize an iitnd ae by rheaeda whch sechmat-

ma b pedctd ymixing- nth form uareion ls otebudr al x~tn

calclatinmehod f9or smotha sufcsered the Prandtlva rs al mixinglethfr

lengh my beadstess to applyraoily rousefthe* Sucinh mueraodo smooth or

facs at telmtha o sublayer thans Splith I the rmehosdra fChactern Siati.1Of

Its ~ evr subsequent mi(ctobyMtublenc a ante mehod tfHag turl. ha

Wal siingeth a. orrther tonet 4no at ed~y vla ibae a iscarded.ar as

lentSchmdt ubrs e to p re amted thruen isco usblaye or beoth r a ce. t

devlomet o trblst onar refees. is In generl page niig3egh1.re

with distance swoy fom the wall until aI~mt au nrahdwihI opt

van Driest modification to the Prandtl wall mixing-length tormulatitan provided a

turbulent shear stress which started at the wall. Rottaa then adapted the

van Driest-Prandtl nixing length for mmoth surface. to rough surfaces by shifting

the position of the reference well an appropriate distance. otta presented tiw

americal results in graphical form for two well-k own rougthesses: the Nikuradew

sand-grain roughness and the Colebrook-White engineering roughneu. Later Cebec£

and Smith 2 provided an analytical fit to the well shift for the sand-grain tougha'9

which was then used by Cebect and Chang for calculations of turbulent boundary

layers over rough surfaces.for arbitrarily rough surfaces. the van Driest factor is crrelavd In this

paper vith the roughneso drag characterization functlon until a pit gn vitlui ot

zero t* reChed. This is conoidered to represent t*e bwing of tw tuily routhregime. Then. for the fuliy rough resite, the nixing 1.r4th i* constdored to enwultr

an initial value at the wall in accordance with Ilotte's oril4nal (195O) 4nalvmiw.

ror purposes t comparison. the *9cond lotta umwly%.1% of a wall ohift i

genraelized to produce numarical rtulto for the draA chAfractetIat1oi of arbitrary

rouhnes*. In Appendix A. an explicit converoion of mixing-lenaths, to edd viscoa-

krk1CM I* derived. In Appendix i, the a.ociated turbule nt kinetic tnwrgies, (f) and

turtulent dislpeticm rate () arc developed.

CHARACTIRIZATION Of AUMAiI OAE.

A brief review t presented of pertinent festute. of the ve1locitv s1imlatI1

Law-- an.d the assclated drag characteratt Ions ef t.agh aure tc.

Close to the ml. the losaritmic velocitii lea fat an arbitrarilv roolth

iurfacs. defined by a ufficient numbr of le-Ot factot% k. k 1 , ks, ... , I ILiven

in uhat may be c Iled the Roynolds-nember node a*

*A in a T a IT (3Mu t

end, in what may be called the relative-rmge~es *ode, as

- A kn I & f 1k ,Tj (2)u r

where

2

S Ba * A k (3)

mere u - strm tse velocity conponent

u - shear velocity, u - VI - moll shear *tress

W- density of fluid

y 0 uTy/v

y * normal distance from well

v * kinmatic viscosity of fluid

I * texture of rough"*ss configuratio. 1 ' kJk 1. kI/k .* S

h o rough.... teynoldo nuiber. k * uIkl"

Either al or a may be considered to be a roughpoo. drag rharactoritation

function. both a and B1, as* funrctifo of toughn.es Reynlds Wueber It MW

IIofAut 1..

The sueltoug'ests rg Res ar

both Il I nd 6 r very with h for, the 9~it 1

. K~ittaroug h

r * r * cnstant

lS *S i *Itl-A k

Another drag charecteriltation to St~m by .0t which repremmts sdtvition from

motth So ald rtof oth

). fulr roug

AS [k ,TI - 1 [k*.Tj - B1 (6)

A defined, IS is always negative for rough surfaces. Nikurad se1 used the P -

characterization while Hlasa11 preferred the .B (actutllv -.'.B) characterization.

ro expetimentally determine a characterization for a spici it: arhitraril, rough

Surf4ce, there are variou0 procedures available. The direct procedure requires

velockty ,nea ur int cloe to the wall to define a logaritlunic law as well as the

MC44kVUVinRC of the w411 4h)Vr stress. Simpler. indirect procedurs inav be used

Invuving nMV4urent ot thV average velocity of pipe flow.I the total drapi of a1)21

ft4C pl4E,. or hw torque of a rotating disk.13

StXl X;-LE.%CTH ThEORY

Accordind to the Prandtl mixing-length theor-i, the turbuslent .ht-ar t (

IS rVLlatVd to the I'l ,ttf graditent by

*t L2 (duj2 "7

twh r. I. la th. mixing lenitth which, In iteneral. li not 0, 1 at.|n * + i".i o t '-,'I

Stposiittion in tlw flow field.

r"; r,)t a!'.

ahve r C rct. - at a point in the mv : i'f tlow '. l r

'c-i turh 'nt QhcPr -2rteft ort

du 2 * ddv 'fv1

S- - the ewtonian Law of ViRcositv.

dv

fr A qhear laver with zero longitudinal prtesure zradient, C-- , ' t, "

wall. With this approxiWation, Equation (8) may be written In a nondi-.lnsin~l ,orm)

2 (d dA---" (Q)+d dv

II I II II . - - 'm - . .

where I ts a nondLmensional mixing length. t o u LIv, and u is nondinensional

velocity, u u/ut .

Solving for du dy as a quadratic expression and integrating from the vall.

where y * 0 and u 0. results in a velocity profite

u dy (10)

Close to the vall Prandcl proposed a linear variation of mixing length with

distance from the wall

;. * .- y(11)

where , is the von !irain constant. se of this relation, in Equation (10). lead%

to a logarithmic velocity law with A a l1-. However. an erroneous value of S

reult%. Furthermore, in so far as the logarittuic velocity law does not hold

right up to the wall. so the Prandtl vail mixing length does not hold. To remedy

thlq, a modification function ,(L %,w'y ) LA needed as a function of v and * where* *

is an additional nondlmensional length. - u .'/.. Furthermore. M should equal*b a

zero at y - 0 and equal unity at v - .

For smooth surfaces. van lDrieqt proposed the representation

X * 1 - exp (12)

V

where is the k associated with van Driest.v

Other modification functions may be formulated as, for example, the hyperbolic

tangent function alluded to by Patel

I- tanh ) (13)

h

where X h is the X used here.

' hy Hi should equal aero at y - 0 requires soe further discusstio. Theo-

retical Investiations5 indicate that the eddy viscosity , t should vary with y or*#4 * Iy at y a 0. Nov Equation (A.6) in Appendix A relates the eddy vis(-2bity to themixing length. A Haclaurin expansion gives

v t *

o2 1

Vt 2

A nonzero value of MI at y -0 such as M I would then result In

.2, ... (4)

which is not acceptable.A no laurtn expansion of the van Driest a g0ves

V

t 2 2 *

" ,-' y + .. 17)

v

which Is acceptable.

|6

Likaise., for the hyperbolic tangent M

V t .21( 2t" 2 .4 ' (IS)

EXISTING MIXINLEGTH MEMOS FOR ROUGH SURFACES

Some pertinent existing mixing-length methods for rough surfaces are now

critically examined.

ROTTA

An extension of the Prandtl well mixing length method to rough surfaces,

Initially proposed by Notta.7 seemed a laminar sublayer thickness that would de-

crease vith roughness or

t K(y -yL) (19)

* *k

vhere YL is the laminar sublayer thickness and yL a u YL/V. The introduction of E

in Equation (10) resulted in an integration in elementary functions and a relation

between YL and 51

* 1

7 L a1 + - (1- tn 4K) (20)

With increasing roughness, the laminar sublayer finally vanishes, yL " 0, and

a1 - - 11K (1-tn 4K). This is now considered to be the beginning of the fully

rough regime and the end of the intermediate roughness regime.*

For the fully rough regime, Rotta assumed an initial mixing length t at the

wall

t + Ky (21)

The introduction of this mixing length for the fully rough regime into the

velocity profile, Equation (10), results in an analytical solution.

7

o + tn(:I (22)

At higher values of this relation reduces to the usual log law, Equation (1), so

that

B - (tn4K-1) -- ~ ~~4 T

I / K Kt/ 2'Vv 4 ,

-I n (2t+2 ) (23)

i*

i~~~~ wl " 4 i ,-)- Vw"

This relates B and E:; therefore, the velocity profile, Equation (10), is a

function of y and B1.

ROTTA It

Rotta 8 seems to have abandoned the laminar sublayer approach in analyzing

rough surfaces after the whole concept of a purely laminar flow next to the wall

became untenable. Since there are turbulent stresses even in the laminar sublayer,

the sublayer has been renamed the viscous sublayer.

Rotta extended the van Driest formulation for smooth surfaces to rough surfaces

by adding a length Ay to the y coordinate such that

,(y +Ay ) -exp (24)

8

* *

where A is the value of A for smooth surfaces. When this mixing length is in-vhs V

corporated into the velocity profile, Equation (10), and related to the logarithmic

law, Equation (1), Ay becomes a function of ,[k ,T). Instead of solving the

general case of Ay M f(s], Rotta solved Ay - flk I for the Nlkuradse sand-grain

roughress and Ay * f(k I for the Colebrook-White engineering roughness and presented

the results graphically. Cebeci and Smith 2 fitted the results for the Nikuradse

sand-grain roughness to an empirical formula. Subsequently, Cebeci and Cang 9 used

this formula in boundary-layer calculations.

The Rotta formulation leads to an initial value of L , namely, t at the wall

(for y*-0) or

1W -CAy* 1- exp (25)

* **

Hence tw a f[B1 1 - f[k ,T] for all roughness regimes.

DAM

Dahm1 6 proposed a differential equation for the mixing lengths of rough sur-

faces based on empirical considerations, which, for zero mass Injection, becomes

di* 0.4 y - (Z -Z)-Z* 11.83 (26)dy

Examination indicates that this equation is a linear differential equation with a

solution given by

- 0.4 y + t* + (0.4) (11.83) -1 (27)

9y 11.83

L9

It may be noted that this formula resembles that of Rotta, Equation (21), for the

fully rough regime (K-0.4; also at y -0, and t -Lw ). In actual use, values of Ew

would have to be correlated with drag characterization (B I ) .

PROPOSED NEAR-WALL MIXING LENGTHS FOR ROUGH SURFACES

GENERAL

Equating the velocity profiles obtained from mixing lengths, Equation (10), and

the logarithmic law, Equation (1), produces

-a

yB- f2 dy* - .dy LEI* (28)

0 1 + /1+(2*)

y*

and with tnEI j (29)f Y

B1 f 2 dy + f2 - 1 dy (30)1 **

0 1 + /1+(2t* )2 y 1 + /l+(2t )2 IY

or

B I I + 12 (31)

where y is a sufficiently large value of y so that the second integrand of Equation

(30), dI2/dy , becomes negligible.

INTERMEDIATE ROUGHNESS REGIME

With the Prandtl-van Driest mixing length,

- c [ exp () (32)

10

Y*

In Equation (30), the van Driest factor A becomes a function of B1 and, hence, aV * I

function of roughness factors, k and T or A a f[B 1 - f1k ,T.v

Numerically then, A from the preceding equation decreases with decreasingv * * *

values of BI until the limit A v - 0 is reached. Then, t - Ky . This is the same

limit as Rotta I where the laminar sublayer disappears.

The range of A from its smooth value of A to its zero limit, may beV V's

considered to cover the intermediate roughness regime, at X. > A > 0.V,8 V

FULLY ROUGH REGIME

The original Rotta I proposal of an initial mixing length t at the wall forw

the fully rough regime will now be considered appropriate. Accordingly, Equations

(11). (22), and (23) apply.

GENERALIZED ROTTA II

In order to make comparisons, the Rotta II method of an additional length 6y

is now generalized. The Rotta II mixing length for rough surfaces, Equation (24),

is substituted into either Equation (28) or (30) for BI so that 67 becomes a

function of B, so that Ay - [I - f1k ,TJ.

DISCUSSION

Two methods have been described for the prediction of mixing lengths for

arbitrarily rough surfaces. There is the Rotta II method which is generalized in

terms of the roughness characterization function and applies to both the intermediate

and fully rough regimes. The RoLta I method extends the van Driest formula for

smooth surfaces to rough surfaces by means of a normal distance parameter.

The other method described consists of an extension of the van Driest formula

to the intermediate roughness regime; the van Driest factor is correlated to the

roughness characterization function. For the fully rough regime the Rotta I method

is to be used in which a wall mixing length is related to the roughness character-

ization function. The relative merits of each method are now discussed.

For the intermediate roughness regime, the Rotta 1I method leads to an initial

mixing length, Equation (25), at the wall while the van Driest factor method gives

11

a zero mixing length. A nonzero mixing length at the wall implies the existence of

a nonzero turbulent shear stress and a nonzero eddy viscosity at the wall as is

demonstrated next.

Now the turbulent shear stress near the well, Equation (7), nondimensionalized

to

w dy! " "*

becomes, by use of Equation (10) for du dy

2

t ______ j(34)

then, at the wall y - 0,

2 22 (--v 35)

Consequently, at the wall, the Rotta I method has a wall value of turbulent shear

stress, but the van Driest factor method has none.

Also for the eddy viscosity, Equation (A.6) gives

( w - 2-+(2t)2 (36)

Again, the Rotta I1 method has an initial value of eddy viscosity at the wall while

the van Driest factor method has none. In fact, the van Driest factor method

applied to the intermediate rough regime reLains the variation of eddy viscosity

with y*4 at the wall.

12

For the intermediate roughness regime, an initial value of mixing length,

turbulent shear stress, and eddy viscosity at the wall may all be considered

objectionable.

NUMERICAL RESULTS

In determing BI from an integration of mixing lengths by means of Equation

(30), it is necessary to specify a limiting value of y so that the integrand of 1

becomes practically zero. As an example, the integrand y (d12/dy) is plotted in

Figure I for the smooth case X a 26 and for the case of the beginning of the fullyv

rough regime n 0 with - 0.4 (the sae case as for ROTTA 1). To compensate for

the logarithmic abscissa, the ordinate is multiplied by y so that the area underthe curve represents the integral I2 Then. dl,/dy -y (d2/My) (d tny /dy )

Ac values of y above about 200 the integrand is close to zero.

In Figure 2, the van Driest factor X* is correlated with drag characterizationV

function B for the intermediate roughness regime by means of Equation (35). This

accommodates any arbitrary roughness. The limiting value is Bl - -1.325 for .v - 0.

Smaller values of B are then in the fully rough regime.*The wall mixing length 9 for the fully rough regime is plotted against the

wdrag characterization function B1 In Figure 3 In accordance with the Rotta I

relation, Equati-n (23).

In Figures 4 and 5 the Rotta factor .'y is plotted against the drag character-

ization function B1 , as a result of solving Equations (24) and (30). This corre-

lation applies to any arbitrary rough surface and, consequently, includes the

Nikuradse sand-grain roughness and the Colebrook-White roughness. *IA comparison is shown in Figure 6 for the mixing length at the wall as aw

function of the roughness characterization function B It should he noted that the

Rotta it formulation has values of . in the intermediate roughness regime in

accordance with Equation (25). On the other hand, the van Driest formulation has a

zero value. For the fully rough regime the Rotta I values are compared to the

Rotta It values of 2.w

A comparison of the variation of mixing lengths with normal distance y is

displayed in Figure 7 for various formulations on the basis of equal valties of B1.

The smooth case is shown for a van Driest factor of 26. As an example of the

intermediate roughness regime, a comparison i.; shown between the van Driest and

13

Rotta I1 formulations for - 2. It is to be noted that the Rotts IX value of the

mixing length is larger close to the wall. This is a result of having an initial

value t at the wall. Mixing lengths at the border between the intermediatew

roughness and fully rough regimes are also compared. Here A* - 0 and B- -I.325.

For the fully rough regime, an example comparison is shown for 51 - 6. It is to

be further noted that for large values of y all the values of t tend to converge

to the original Prandtl wall mixing length of I - as expected.

For arbitrarily rough sturfaces, two procedures have been developed. First is

the extension of the Prandtl-van Driest formulation, Equation (12), to the inter-

mediate roughness regime. Here the van Driest factor Av is correlated with dragv

characterization 81 in Figure 2. For the fully rough regime, the Rotta I procedure

miy be used as given by Equation (23) which is plotted in Figure 3 as t w aAalnst B.

Second, there is the Rotta II correlation, Equation (24), developed for arbitrarily

rough surfaces. Figure 4 shows the correlation of Ay with BI for the intermediate

roughness regime and Figure 5 shows that for the fully rough regime.

14

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21

APPENDIX A

CONVERSION OF MIXING LENGTHS TO EDDY VISCOSITIES

The concept of eddy viscosity Vt (more properly eddy kinematic viscosity) was

originally proposed during the last century by Boussinesq for turbulent flows as an

analogy to the usual viscosity for laminar flows or

Tt du

Equating this to the turbulent shear stress given in terms of mixing lengths,

Equation (7), produces the well known expression

V= z2du (A.2)

Note that eddy viscosity unlike laminar viscosity is not a property of the fluid,

but depends upon its position in the flow.

For various reasons it may be more desirable to relate eddy viscosity Gv ) to

mixing length without the presence of velocity gradient du/dy.

In general, shear stress (T) has laminar and turbulent contributions such that

T= (V+V- ) du (A.3)

Close to the wall r T w and then, nondimensionally,w

du = (A.4)

dy 1+V

Also from Equation (10)

du2du _ 2 (A.5)dy 1

d +1 l+(2*2

23

htozmp MLAwa.T rum

Equating the two expressions for du /dy results in

= 2 (2t*)2 - 1 (A.6)

This is the desired relation for converting mixing lengths to eddy viscosities.

At large values of 2., (V / PO- - Ky or V t -UZ- KUy.

For the van Driest formulation for arbitrarily rough surfaces, Figure A.1 shows

the variation of eddy viscosity ratio V t/V with normal distance y . The smooth case

is shown as well as the case for the boundary between the intermediate and fully

rough regimes. Also, as an example of the fully rough regime vt /v is shown for B, =

-6. At large values of y , Ct /V) - Ky as seen in Figure A.l.

It is to be noted that the increase of V /u with y presented here applies onlyt

to the boundary-layer region next to the wall. A limiting value is reached which2

corresponds to outer-region values such as those given by Cebeci and Smith.

24

UAU

Z CC >

e\o

CA-

25,

APPENDIX B

NEAR-WALL VALUES OF TURBULENT KINETIC ENERGY (k)AND TURBULENT DISSIPATION RATE (E)

A current modeling method for turbulence determines the turbulent shear stress

from the eddy viscosity which, in turn, is obtained from the turbulent kinetic

energy (k) and the turbulent dissipation rate (c) from

k2 (B.1)t iP e

where c is a constant.

Values of k and e are obtained from solutions of convection equations which are

partial differential equations. However, close to the wall it has been found that

values of k and C obtained from mixing lengths may be used as inputs to the partialdifferential equations for k and c with an improved accuracy in the solutions of

the equations.

By equating the prod. :tion and dissipation terms of the k-equation, Arora

et al. 6 related k and e to the mixing length close to the wall as follows,

k (du (B.2)/--dy)

and

2 Idu\ 2- ~ (B.3)

Elimination of the velocity gradient du/dy produces a direct relation between k and

c with t as follows. First the relation for k and c are nondimensionalized to

22duk - * (B.4)

u. 2 2 dy

27

and

4 *2 du(B.5)

u 4\dy/

Substitution of du /dy from Equation (10) produces

* 2

2 -- (B .6 )

and

2 -34 C k *2 (B.7)

uT L1+(29*),

Some general properties of k and e are now deduced. At higher values of t , j(Vc-/u2 )k approaches 1 and (V/U4 )c approaches i/Z*.

PT 4 TAlso (v/u )F reaches a maximum value of 1/4 at k r2= .

TThis maximum value is independent of the mixing-length model.

*

In Figures B.1 and B.2, the variation of k and E with y are shown for selected

roughness conditions.

28

I ,., /a m,, ,. (I

0 311

III0

I -4

4) LU00

N jz

I AU 013D-i11 II.

29)

l< I7. 44

0I'.

cc 0

At t4 d

3 30

wl-

APPENDIX C

HYPERBOLIC-TANGENT MODIFICATION FUNCTION

A hyperbolic-tangent function1 5 may be used as a modification function for

arbitrarily rough surfaces.

2M - tanh (y /xh) (C.1)

The factor Xh may be correlated with the roughness function B in accordance

with Equation (30). For Xh = 0, M - 1 and Z. Ky which marks the limit of the

intermediate roughness regime just as for the van Driest modifications function.

A comparison of the integrand y (dl2/dy) between the hyperbolic-tangent

function and the van Driest function is shown in Figure C.I.

31

lop

ZIP

322

REFERENCES

1. Patankar, S.V. and D.B. Spalding, "Heat and Mass Transfer in Boundary

Layers," C.R.C. Press, Cleveland. Ohio (1968).

2. Cebeci, T. and A.M.O. Smith. "Analysis of Turbulent Boundary Layers,"

Academic Press, N.Y. (1974).

3. Nituch, .J., S.S. Sjolander and M.R. Head. "An Improved Version of

Cebecl-Saith Eddy-Viscosity Model," Aeronautical Quarterly, Vol. 29, Part 3.

pp. 207-225 (Aug 1978).

4. Huang, T.T., N. Santelli and G. belt, "Stern Boundary-Lver on AXisVMmetric

Bodies," 12th Symposium on Naval Hydrodynamics, National Academy of Sciences.

Washington, D.C. (1979).

5. Sideman. S. and W.V. Plnczewski, "Turbulent Heat and .ass Transfer at

Interfaces: Transport Models and Mechanisms," in "Topics In Transport Phenomena."

C. Cutfinger, ed., pp. 47-207, Hemisphere Publir.hing orporation. Washington. D.C.

(1975).

6. Arora, R., K.K. Kuo and M.K. Razdan, "Near-Wall Treatment for Turbulent

Boundary-Layer Computations," AIAA Journal, Vol. 20, No. 11, pp. 1481-1482 (Nov

1982).

7. Rotta, J., "Dos in Wandnahe gultige Geschwindl gkeitsgesetz turbulenter

Stromungen," Ingenieur-Archiv, Vol. 18, pp. 277-280 (1950).

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

"Progress in Aeronautical Sciences," Vol. 2, A. Ferri, D. Kchemann and L.H.C.

Sterne, eds., pp. 1-219, Pergamon Press, N.Y. (1962).

9. Cebeci, T. and K.C. Chang, "Calculation of Incompressible Rough-Wall

Boundary-Layer Flows," AIAA Journal, Vol. 16, No. 7, pp. 730-735 (lul 1978).

10. Nikuradse, J., "Laws of Flow in Rough Pipes," .'ACA TW 1292 (Nov 1950)

(Translation from VDI-Forschungsheft 361, Jul/Aug 1933).

11. Hama, F.R., "Boundary-Layer Characteristics for Smooth and Rough Surfaces,"

Transactions, Society of Naval Architects and Marine Engineers, Vol. 62, pp. 333-

358 (1954).

33

L2. (Grativille, P.S. , "T'e Frictional Resiibtance and Turbulent !Boundarv 1.i'cr

ot lkgh Sur CLsv," Journal of Ship R -,%arcl, Vol. 2, !4,). 3, pp. 52-74 (Dec 1958).

r. ;r4nvtlI, P.S., "Drag-Charaterization Mthod for A.*.itrarilv Rou ,h

~~~b a. i . 4 4n% 'i Rotating Diaks, Journal oii Yluick FYnginver ug. * .1 . 104,

, p. 373-374 (Sep 1982),

1 .. van Dr 1sit * . K. . . "M Turbulent Flow Ne~ar A I-Li I . Journal o'f Avro,tuc icail

1. Vol. 23. pp. 1007-1011. 1036 (1956).

IS. Ptt' , V.C. "A Vn itLed 'iew of the Law of the WalI Us 11 !.ng Mixin,-.r,.th

Arronauticail 'uarterl..ol. 24, P.art 1, pp. 51-70 (Fvh 1973).

1 h'. t.ttw, T. . Pir ronalI Comin sicat Ion to T. C. 1. in and R . 1. -:wt r, "' ,3irbu I vnce

. .or fli-Speed Rough-Wall Iloundarv Layetr," AIAA lournal, ol. 20, No. 3,

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