THE INFLUENCE OF PRANDTL NUMBER AND SURFACE ROUGHNESS ON THE

RESISTANCE OF THE LAMINAR SUB-LAYER TO

MOMENTUM AND HEAT TRANSFER.

Chandra Lakshman Vaidyaratna Jayatilleke

1966

Imperial College of Science and Technology

SUMMARY

It is shown that, the development of a general theory

for the calculation of momentum- heat- and mass-transfer in

2-dimensional flows past surfaces, necessitates expressions

of the effects of Prandtl/Schmidt number variation and surface

rolighness on the laminar sub-layer.

A comparison of drag, and heat- and mass-transfer (at low

rates) data of flow in smootivpipes, and theoretical formulae a

based on a Couette flow anilysis, permits the recommendation

of simple yet accurate formulae for the evaluation of the

effect of Prandtl/Schmidt number variation on the laminar

sub-layer.

The examination of velocity profile and drag data indi-

cate the nature of the information on surface 'roughness

effects which need be incorporated in the theory. A model

is proposed, of the flow close to the surface; and this

provides a basis for the formulae.

An experimental study of the hydrodynamics and heat

transfer in a radial wall-jet is reported; and comparisons

made of the the predictions of jet behaviour with the data.

Useful details of the theoretical prediction procedures

are given in the form of appendices.

3

ACKNOWLEDGEMENTS

The work described in this thesis was carried out during

the tenure of a Ceylon Government University Scholarship.

I am grateful to Professor D. B. Spalding, of Imperial

College, for the suggestion of the problem, advice at all

stages of the work, encouragement and understanding.

Thanks are also due to Professor J. C. V. Chinnappa, of

the University of Ceylon, for his kind advice and understand-

ing. The granting of leave and provision of partial financial

support, by the University of Ceylon is gratefully acknow-

ledged.

I also thank: my colleagues E. Baker, S. V. Patankar,

M. Wolfshtein, M. P. Escudier and G. N. Pustintsev, for

help and useful discussions; Dr. D. F. Dipprey, of the Cali-

fornia Institute of Technology, for providing unpublished

data from his work on surface roughness; and the members of

the Technical Staff of the Mechanical Engineering Department,

Imperial College,-, helped in the speedy execution of the

experimental arrangements.

CONTENTS

Introduction 5

1. The basic model 10

2. Some features of turbulent flows near walls 17

3. The P-expression for flows past smooth 26

surfaces

4: Hydrodynamic effects of surface roughness 48

5. Couette-flow analysis of heat transfer from 68

rough surfaces

6. Experimental investicration of a radial 77

wall-jet

7. Application of the theory 90

Concluding remarks 105

Nomenclature 109

List of references 118

Tables 1 - 8 130

Appendices 1 - 10 175

Figures 209

INTRODUCTION

It is not necessary here to stress the importance of

being able to predict the quantitative aspects of the hydro-

dynamics and heat- or mass-transfer processes occuring in

turbulent fluid flows past solid boundaries.

The problem has been stated and the important studies

of many of its aspects have been enumerated in the paper by

SpaldingEb21, in which is expounded the "Unified Theory of

Friction, Heat transfer and mass transfer in the turbulent

boundary-layer and wall-jet".

This general theory in its present state of development

has been applied successfully to many cases of hydrodynamics

of flow past flat plates with or without mass-transfer,

boundary layers recovering down-stream of a disturbance on

the surface, flow over surfaces in the presence of pressure

gradients, and of heat transfer in flat-plate boundary-layers

with many interesting boundary conditions. The general the-

ory at present can be seen as the initial stage of an attempt

at understanding many important physically-controlled solid -

fluid interactions, as shown in figure (i).

In the theory although we are not directly involved with

the micro-structure of the turbulence, we can yet express its

grosser manifestations in the form of auxiliary functions and

empirically determined constants which have to be incorpora-

ted, in order to make the eouations soluble and the solutions

realistic.

••• 0

The development of our understandin7 of the interactions

shown in figure (i) involves a two-fold task: firstly, the

collection and codification of discrete bits of information

confined to a particular region; secondly, the devising; of

a mathematical model which has manifold aspects but yet cali

incorporate localised information, thus helping to unify our

knowlede and dissolve boundaries sept:rating the recrions,

The present work belon7s to the first category and arises

from the consideration that the presence of the wall has an

influence on the turbulence pattern within the fluid. Thus

in the case of a smooth wall it is found that turbulent

velocity fluctuations become uncorrelpted in its immediate

vicinity and decreasinq,ly so as we move away from it; the

molecular transport properties become important in this re—

glen near the wall, which is refer7ed to as the "laminar or

viscous sub-layer", A rudimentary Couette-flow analysis

shows how this influence can be specified in ouantitative

descriptions of the flow; for example, in the familiar

41 velocity profile expression,

u+ 1 /, )

the term r is a parameter into which the sub-layer effect

can be lumped. The sub-layer also affords an extra resic--

tance to heat transfer, which depends on the laminar Prandt1

The meanings of the symbols used are (riven in the section

entitled NGMI1NCLATURE

7 number when the wall is smooth; this extra resistance which

we shall refer to as P, will be discussed in detail later.

In the context of figure (i) the present work may be

seen as stepping out in two directions; delving into some

details of the situations coming within the fields A and B,

and then making the paths meet in the region C. The follo-

wing objectives were set:

(a) the collection of heat transfer and drag data for

smooth pipes and extraction from them of a relationship bet-

ween the laminar Prandtl number and P;

(b) the collection of all available information on

flows past rough surfaces and devising a means of estimat-

ing the influence of roughness on the parameter E mentioned

earlier and also on P;

(c) the conduction of experiments to detect whether

the influence of roughness is confined to the region very

close to the surface; and

(d) the application of the general theory in its pre-

sent state development to the calculation of hydrodynamics

and heat-transfer in the conditions of the present experi-

ment and of any other available flows past rough walls.

The body of the work is sub-divided into four sections.

In section I the theoretical framework into which the four

aspects of the present work fits, is outlined. The sub-

sequent sections II, III and IV are concerned respectively

with: the influence of Prandtl number variation on the sub-

layer in flows past smooth walls; the effect of wall rough-

8

ness on hydrodynamics and heat transfer; and an experimental

investigation of the hydrodynamics and heat transfer in a

radial wall-jet on a rough surface.

9

SUCTION I

10 CHAPTER 1

THE BASIC LODEL

1.1 The physical system

The system which we direct attention to is the turbu-

lent boundary layer formed by the flow of a fluid past a

surface, as shown in figure 1.1. The surface may be either

smooth or rough. There can be slots discharfing fluid along

the surfE.ce parallel to the main-stream; the main-stream can

either have a finite velocity or be at rest. Heat transfer

may occur at the surface; and the fluid from the slot differ

in temperature from the ambient.

In order to simplify the mathematical problem, the

follovAng restrictions are placed on the system:

10 steady flow;

2. fluid homogeneous in phase, having uniform molecu-

lar properties and turbulent;

body forces, such as those due to gravitation, ab-

sent;

4e two-dimensional flow.

By two-dimensional flows we mean those in which local

quantities such as velocity and temperature depend on only

two space variables. Under this category would be flows past

plane surfaces, general cylindrical surfaces having their

axes normal to the flow and stationary axi-symmetrical bodies

with axes lying-, in the general direction of the flow.

11

1.2 System of co-ordinates

The general system of co-ordinates in respect of which

the equations are written is .shown in figure 1.2; that

illustrated is the general case of an axi-symmetrical body.

R is the distance of a general point S on the surface,

from the axis of symmetry.

The distance x is measured along the trace of the given

surface on the plane passing through its axis and S; and y is

measured along the local normal.

1.3 The differential equations

Our ai.11 here will be to present the important equations

devoid of much of the details of their derivations so as to

enable the present work to be placed in the proper context.

The partial differential equations g)overning fluid flow

in the system shown, are as follows:-

Mass conservation:

71(pRu) + A(oRv) = 0 ... (1.3-1)

Momentum:

au au pu dx + pv 77: dua

ay dx (1.3-2)

The Prandtl boundary layer assumptions and the momentum

equation for inviscid flow outside the boundary layer, viz:

LIE o(1 .3-3) -dx- dx

are implicit in the above equations.

As shown, for example, by Spalding[0], equation (1.3-3)

12

can be integrated across the boundary layer to yield:

1 a r 1-Y6 dyc, pu dy= mS PvG PUG dx ... (1.3-24)

L-,0

which is referred to as the "integral mass conservation

equation". The suffices S and G refer to the conditions at

the surface and' at the outer".edge'of*the baanslary layer res-

pectively. 1% is the mass flux from the surface into the

fluid stream and vG the fluid velocity in the y-direction at

the edge of the boundary layer. The thicknes::of the boun-

dary layer at a given section is denoted by VT

Combination of equations (1.3-1) and (1.3-2), with

suitable manipulation and integration yields the "inteo.ral

momentum deficit equation".

1 d R

1 f- YG 1. -!R ou(uG - u) dy

Multiplication of (1.3-2) by u and subseaucnt integra-

tion with respect to y results in the 'interal kinetic-

enerqy-deficit eouation":

1 a R da

F ' Yri 2 2 --, i u u ,

/ G 2 IR puk-2- - --) dy, = • r

L 0 J

au dy ti ay

l'ollowing the practice of Spaidinp:D21 we introduce

the folio wing quantities which acre to be used in rewriting

the equations in dimensionless form:

dy

(1.3-5)

15.

m

a T/(Pu,g)

11V(PG)

u/uG

Y/YG

_ (dyG/dx)

-1

(1.577) • 0 0 ( 1 . 3 8 ) 0 0 ,C ( . 3 - 9 ) 0 0 0 (1.3-10) C 0 ( 1.3-11)

Il z ... (1.3-12) -o r

12 I z 2,. ucf; ... (1.3-13) JO 1

13 = z3d ... (1.3-14) -0 r

J ( 1 . 3-15) I slidF.,

0

H12 F..-- (1 - 21) /(I1 - 12) ... (1.3-16)

H32 ..,... (Ii - I3)/(11 - 12) ... (1.3-17)

RG --z- pYGuGill ... (1.3-18)

Rm .7.. 11%. ... (1.3-19)

R2 = (Ii - I2)RG ... (1.3-20)

R3 -.z (Ii - I3)RG

x

, . . ( 1.3-21)

Rx (P/P-) fo

u dx ... (1.3-2 2)

With the aid of these definitions, the differential

equations can be written in the following forms:-

14

Mass conservation:

dRm d(ln R)

dR Rm dR - m- mG ... (1.3-23) x x

Momentum deficit:

dR + (1 + 1112)R2 x + R2 dRx

Kineti -energy deficit:

dR3

d(ln R) d(ln

dRx + R3 dRx + 2R3 dRx

dR2 d(ln R) d(ln uG)

dRx - + sS

O 60 (1.3-24)

m + • (1.3-25)

If we are to calculate heat- or mass-transfer, the differ-

ential equation governing the conservation of a property p has

to be added to the abvea collection. In the case of heat trans-

fer,

.(

the conserved property would be enthalpy; or in the case

of mass-transfer without chemical reaction, the fractional

concentration by mass of a fluid component. For a detailed

discussion of conserved properties, one may refer to Spaldinp-

C8Q] . The equation governing the conservation of p within an

x-wise element of the boundary layer is:

... (1.3-26)

where ji t is the flux corresponding to the property (P•

YG il tRr pu(c - cps ) dy.} JS

0

We can rewrite (1.3-26) in terms of:

Rpll = (9S - (PG,hGI 0,1

where, 5q)

... (1.3-27)

Az d ... (1.3-28) 0,1

G 0 0

(1.3-31)

with,

(P )/( PS - (PG) and

Scp,S P' ((PS - (PG )

so that, (1..3-26) becomes,

dR9,1 d(ln R) dRx 11 ,1 dRx

ScD ,S ( (PS - (PG)

According to Spaldingr82j, 183], we can generate solu-

tions to the hydrodynamic problem by taking equations (1.3-23)

and (1.3-20 into consideration. It is also possible to

effect the same by solving equations (1.3-20 and (1.3-25);

the difference between the two methods lying in the fact that

the former requires the use of the law rfoverning entrainment

of fluid into the boundary layer and the latter requires

instead, the dissipation integral. We shall, however, keep

in touch %:ith both methods as each-one has its own limitations

and may offer advantages in particular situations.

If we examine the differential equations and the defi-

nitions which preceded them, it becomes clear that the problem

resolves into a mathematical part, of workin7 out a numerical

solution, and a physical part, of finding auxiliary relations

connecting some variables with others and laying down initial

and boundary conditions.

It is the physical aspect of the problem which demands

more attention since the mathematical solution would usually

be tractable when auxiliary relations and other conditions

-r11 S

16

are available.

he equations and definitions written so far arc valid

whether the flow is laminar or turbulent. The features of

turbulence will be incorporated in our model via the aaxiliaI,

relations, and so ire turn to some turbulent flop phenomena

for these relations.

17

CHAPTER 2

SOME FEATURFS OF TURBULITT Ii1JOWS NEAR WALLS

2:1 Introduction

We shall confine our attention to flows where heat- or

mass-transfer rates arc not large enough to have an affect

on 'the hydrodynamics; lame heat-transfer rates would bring

about temperature differences which can cause appreciable

fluid property variations across the stream, and large mass-

transfer rates would bring about higher momentum-transfer

rates, due to the transverse mean motion, than would obtain

due to eddy diffusivity alone.

The information presented in this chapter pertain main-

ly to turbulent flows past smooth walls; and most of them are

presented in greater detail in the work of SpaldingD2, 83],

Nicoll and Tscudier[52q1 and Escudier[22j.

The origins of our information are measurements of

velocity-profiles, temperature-profiles, shear-stress-pro-

files, surface friction and heat-transfer rates. The auxi-

liary relationships have either been deduced from these data

or so arranged that they generate results which have some

degree of conformity with observations.

2.2 The velocity profile and local drag law

Spalding [82] used a velocity-profile of the form:

z = s1/2u+ (1 z )(1 - cos v0/2 • 0 • (2.2-1)

It is readily seen to be a superposition of two components:

sS 1/2 u which + is called the 'wall component'; and (1 - zE)(1 -

cos 7r4/2 which is called the 'wake component'.

The dependence of u+ on the non-dimensional

from the wall, which is defined as,

Y+ = Ydr(TsP)/11

V 0 0

18

distance y

(2.2-2)

is given by the universal relation:

u+ = 1 ln(Ey+) ... (2.2-3)

in the case of zero mass-transfer from the surface. IC is

known to be a universal constant and E is a number whose

value is taken as constant for smooth surfaces. The relation.

ship between and y+ is modified by surface variations such

as permeability and roughness; we shall not concern ourselves

with the effects of the former and shall leave those of the

latter for consideration later.

By the substitution in (2.2-1) of the main-stream

conditions:

z = 1

E, = 1 1/2

RGsS

(2,2-4) and.

together with u+ from (2.2-3) , we obtain the local drab- law:

c,1/2 0 0 0 (2.2

where,

e H ln(ERA12)

000 (2.2-6)

The velocity-profile expression can now be rewritten as:

19

zE(1 + + (1 - zE) (1 - cos 72- )/2

... (2.2-7)

The role of the parameteii zE can be understood more

clearly at this stage: it determines the relative proportions

of wake and wall components present in a riven velocity

profile. The way in which the two components add up to form

a composite profile is shown in figure 2.1. zE is related

to the other parameter -r by the drag. law (2.2-5).

Instead of the wake function:

TIL" (1 - cos WW2 ... (2.2-8)

other forms have becn surTgested, notably, the linear one:

rz = ( 2 2 -9)

In addition to simplifying the algebra, the profile with a

linear wake is seen to fit well the data for turbulent jets

near smooth wall, as shown in figures 2.2a and2.2b.

While developing our programmes for computation, Some

flexibility in the choice of profile is to be maintained by

writing the velocity profile expression as:

zE (1 + lnE) + L(1 - A)(1 - cos 77-4)/2 + Aq(1 - zE) .e1

000 (2.2-10)

so that A = 1 rives the linear wake and A = 0 the cosine form.

Finally, it must be remarked that the profile suggested

assumes that the wake and wall influences to be co-extensive,

and that we may expect discrepancies where this condition

is invalidated in any way.

20

2.3 The entrainment law

The form in which equation (1.3-25) is written, J,

attributes to the boundary layer an ability to 'entrain'

fluid from the free stream outside it.

SpaldingL82] hypothesised that the dimensionless rate

of entrainment, represented by -90_, is dependent on the

parameter zE. 2,fter examining a considerable amount of data

from boundary layers and walijets with finite main-streams

over smooth plane surface, Nicoll and Tscudier[52] have

recommended a relationship of the form:

-mG = C1(1 - zE) , for 0 < au _5 1

... (2.3-1) and 0 = C2zE - C3'

for 1 < zE

In the case of wall-jets with no mairiLstream flow, the

entrainment law becomes:

-r./zE = C2 ... (2.3-2)

The values of constants recommended by the above authors

are: C1 = 0.075; C2 = 0003; and C3 = 0.02.

Entrainment rates deduced from some velocity traverses

in a radial wall-jet over a smooth surface, during the

present experiment, indicate that,

C2 = 0.04 (2.3-3)

is more appropriate for thie conficr,uration. This has bar,-,

confirmed later by datafrom other radial wall-jets.

The above suo-gestions for the entrainment law are,

however, tentative, and have to be improved by a thorough

experimental investiation of various flows.

21

2.4 The dissipation integral

The value of the dissipation integral which is denoted

by s can be derived experimentally from: shear stress profiles

measured using hot-wires; shear-stress profiles deduced from

velocity-profile data by the application of the integral

momentum equation to parts of the boundary layer; and velocity

traverse data via the integral kinetic-ener'7y-deficit equa-

tion, (1.5-25).

Spalding[W, has suggested two forms of functional

relationships: one, expressing -g as a function of m, zE and e;

and the other, as a function of H12, H32, 97, and m.

EscudierD2J states that a satisfactory 4zE' ,C)- which

is valid for zE > 0.6, can be generated from a mixing-length

distribution of the form:

a e/yG for z!. < ... (2.L-1)

and A = A, for ,ki./K- <

where A is the ratio of the local mixing-length 4! to the

local boundary layer thickness yG. The values of constants

recommended are: Al = 0.075 and k.= 0.41. Graphically, this

mixing-length distribution is as shown in firlTure 2.3. Using

this mixing length distribution and a velocity-profile

assumption, local shear stresses are calculated by means of

the formula:

S = Al az1 az (3V.

— . (2.4-2)

from which .6 follows according to equation (1.3-16).

22

The auxiliary relations riven so far are concerned with

the hydrodynamic problem; the collection is incomplete due

to the absence of expressions of the I-interrrals in terms of

z and y, obtained from the velocity profile.

The p- (heat- or mass-) transfer problem reauires the

solution of eouation (1.3-31), which acrain requires further

auxiliary relations, We find these in the form of a o-proff'_e

assumption and a c-flux law at the wall.

2.5 The p-profile

We write down the Sr-profile expression by analogy bet-

ween the velocity- .and p-fields; thus,

1/2+

ouoss ( 9 -9E) W i;)- ... (2.5-1)

Here, p+ is a dimensionless value of the coserved property

as dictated by a wall-law; and its variation is obtained from

a Couette-flow analysis which will be given in detail in

subsequent section of this work. It suffices to ouote here:;

the result:

= 00

( + + P) 0 0 0 (2.5-2; where o0 is the ratio of the eddy diffusivities for p-trann-

fer and momentum transfer in the turbulent fluid where

laminar transport properties are neglia-ible, and P is the

resistance of the laminar sub-layer to p-transfer on accour

of the laminar Prandtl or Schmidt number of the fluid being different from oo.

0.5(1 + z ) I1

121 1 A[-3 + zE(-3

+ (1 — A) [ + z (-1 E 4

23

The second 7roup on the R.H.S. of (2.5-1) is a wake-

component with taking the form given by either (2.2-7)

or (2.2-8). The expression in (2.5-1) can be recast in the

form:

1 1 - g —P (1 - gE) (1 + In 8, ) e O 0 0 (2.5-3)

with,

(C TG) /(TS 9G) O00 ( 2 .5-14)

and

... (2.5-5)

Also implicit in (2.5-3) are: the definition p+, which is

given in (3.2-12); and the statement

j" E(1 - OE)

T'S = Puc(Ts - a 0 •t'co ... (2.5-6)

which expresses the 'local p-transfer law at the wall'.

The profile assumptions enable us to evaluate the

1-integrals which follow.

2.6 The 1-integrals

As mentioned before the I-intcgrals contain terms which

are derived from the cosine-wake as well as the linear-wake;

putting A equal to zero gives the former while A equal to

unity gives the latter. The relevant algebraic forms are:

,.. (2.6-1)

2(1 1.5 2 ci zEk-3 +

0.411) 4.. , (3 1.589 2 ?, / L'E` 8 ,t/ + e, 2

(2.6-2)

(1 - A) (0.2055 + 0.7948zE

zE) 1 (2.6-Q 0 0 •

3

r 1 10.25 4. (0.25 °•333) -e/

24

(0.25 + 0.833 + z7, 0.75N 21

)

+ ( ‘1 - A) .3125 + (0.1875 - 0.326% izE

and

I091 = gE[A(T r i z7 ,

+ -3"

- (0.1875

+ (0.3125

1.5z, (

0.5795, 0.461)z E

6) + z3

t/2

2.0945 5.539 +

1 - A) ( 3zE

t,3 E

(2,6-3)

0.8943zE ') - 4), + 8- + 8 - /1

(1 - 0E) 1A(0.25 + 0.75zE - 2zE)

L ,Ci '

On the foregoing Dag.es we have presented briefly the

differential equations governing hydrodynamics and heat-

transfer in turbulent flows near walls, and most of the aux-

iliary relations required in solving them. Some points

which have been treated briefly, will be discussed below

greater detail. One such item is the relationship linkj

P and e the laminar Prandtl/Schmidt number of the fluid.

The term P appears in the local (p-transfer law as expressed

in (2.5-5). Section II which f-12_ows is devoted to a survc:

of experimental and theoretical information, which has been

carried out in order to find the best form of the P-functjr-

for smooth c'17Thces.

SECTION II

25

26 CHAPTER 3

THE P-EXPRESSION FOR FLOWS PAST SMOOTH SURFACES

3.1 Introduction

In section 2.5 we introduced a term, P, to account for

the enhanced resistance to heat- or mass-transfer offered by

a layer of effectively laminar fluid near the surface and

having a fraction of the thickness of the boundary layer. I:

is associated with the 'wall' component of the temperature- be

or concentration-profile which is assumed tmmalogous to the

'wall' component of the velocity profile, and to be deter-

mined by universal laws, likewise.

The universality of the wall component of the velocity

profile, i.e. its independence of the main-stream conditions,

has been demonstrated by the analysis of velocity-profile

data and by the satisfactoriness of the drag-laws derived on

the basis of such an asc,.umption, as reported by Schubauer and

Tchen 7l•jo Althourrh no direct comparison with data has been

carried out to demonstrate the universality of the T-profile,

the consequences of the analogy between temperature and velo-

city profiles have been shown to be valid in the case of

pipe-flow; for example, by the investirrations of Deissler

1131.

Equation (2.5-1) is a statement of the generalisation

by Spalding [82] of the concept of the analogy between the (p-

and velocity-fields so as to form a basis for the calculation

of heat- or mass-transfer in complex 2-dimensional turbulrnJ

27

flows; and it is a part of the present task to recommend a

suitable me; of evaluating P.

The basis for our recommendation will be: the analyses

of heat-transfer in turbulent Couctte-flow by various authors

starting from Prandt1[59] and Taylor 85] who were the first

to recognise that a major portion of the resistance of boun-

dary layer to heat-transfer resided in the laminar sub-layer;

and a comparison of the P-functions derived from these ana-

lyses with experimental data collected from the literature.

It turns out that the satisfactory P-functions are far more

complicfted than their purpose demands; hence the opportunity

has been taken, to recommend a few simple yet sufficiently

accurate formulae.

3.2 Heat- or mass-transfer in turbulent Couette-flow

The characteristic feature of a turbulent Couette-flow

is the dependence of velocity u and property 9 on y only.

It is also specified that the shear stress T and the flux Jg

corresponding to the property 9 do not change with y.

It follows from dimensional analysis that the velocity-

profile is expressible by a unique relationship of the form:

where,

and

u+

= 114-(-y4-} ... (3.2-1)

u+ a u/Nr( Ts/p) ... ( 3.2-2)

Y+ ::,* YtT(TsP)/11 ... (3.2-3)

28

A total viscosity, µt, is defined by:

t Ts/( du/dy) 00 0 ( 3 .2- 4)

and a total exchange coefficient, of the property p, by:

j,,/( d9/d3r) (3.2-5)

In the case of heat-transfer, for example,

t = BOO (3.2-6)

k being the thermal conductivity and c the specific heat at

constant pressure, of the fluid.

The total Prandtl or Schmidt number, of is defined by:

0 [1. t - = t/r 0.. (3.2-7) Dimensional analysis leads to the result:

at = at(-y+' a} (3.2-8;

where a is the laminar Prandtl or Schmidt number of the fluid;

and also that:

= Zt0.74)- (3.2-9

By virtue of the constancy of shear-stress and flux, we de-

rive that:

and

where,

1-it/11 = —t = dy4-/de

Et/at = aY+/'?+

(2 (2,.., ) tit 0) Ai- tt

Schmidt number:Mass-transfer::Prandt1 number:Heut-transfer

29

Et can be eliminated between (3.2-10) and (3.2-11) to give

the important result:

... (3.2-13)

In principle, co+ can be evaluated, since of can be related

to u+ by means of (3.2-1) and (3.2-8).

In spite of the fact that in real pipe-flows the shear-

stress variea over the cross-section because of the pressure

gradient necessary to overcome pipe friction, the velocity-

profile resembles that of a Couette-flow remarkably; hence

the justification for the use of Couette-flow analysis on

pipe-flows.

The velocity profile which is usually taken is of the

form:

u+

= • In y+ + const. O 0 0 (3.2-14) which is seen to fit the data over a large part of the pipe

cross section (fissure 3.1).

The drag coefficient, s2

In pracitice, one of the measurable variables in pipe-

flow is the bulk velocity, a, of the fluid; and it is custo-

mary to calculate drag coefficient on its basis. It is

defined by,

r'R

Ra ▪ 271-1 u(R - y) dy • (3.2-15)

which may be rewritten as,

+ +

r -u_ = 1

0 t du+

Jo

30

• YR +

= 2 u'(1 - y-474) d(37-4-/y) ... (3.2-16) .!0

in ter is of dimensionless variables, yilz- beinc, the dimension-

less ipe-radius. On substitution for u+ p from (3.2-10 cnd

evaluation of the integral, .0 r'et the relationship between

bulk velocity u+ and the centre-line velocity u+ as:

a+ 1 . 5,/k;- ( 3 . 2-17)

The dra coefficient s is defined as

sp „to ?I:3 2) = ( l/u+) 2

(3.2-13)

Then it follows from (3.2-17) and (3.2-18) that:

5l/2 - ... (3.2-19) -

The Stanton number S, p 2

For gr-tr.nnsfer in P flo the Stanton number• is

defined by:

... (3.2-20)

c,t (93 -

which can be reduced to :

= 1/Wa+) (3.2-21)

c 91) where is the mixed-mean-value of cp over the pipe cross-

section and 4+ .the correspondin- non-dimensional value. It

is possible to derive a relation between Ff.,+ and 41z on the

basis of the q: y+ relation being lor.arithmic over most

of the stream as a consequence of at having- a constmat value

oo within the turbulent core; this beinq,

31

-+ , (1.5 1.25\ = L'O‘ - 2 -

U.

+/ 0 0 0 (3.2-22)

By suitable manipulation of equations (3.2-17), (3.2-13),

(3.2-18), (3.2-22) and (3.2-21) , we obtain:

s1/2

T,P (a t - 00) au+ 4. 1.25 s \

772 (14_

‘ 1,e2 Pi

... (3.2-23)

which relates the Stanton number and the drag coefficient.

Many previous authors have either,(i) given formulae

which can be reduced to the above form, or, (ii) proposed

+ relations which enable the quadrature in (3.2-23) to

be evaluated. The followng comments can be made about the

terms in (3.2-23).

(a) The term on the L.H.S., being. proportional to the inver-

se of the Stanton number, represents the total resistance to

the transfer of

(b) We have contrived to express it as the sum of two

resistances in series appearin7 on the R.H.S.; of which the

the lati;er is what would remain if 0t was made equal to o

over the whole cross-section of the pipe.

(c) The first term of the R.H.S., then represents the extra

resistance which arises solely due to any differences between

ot and oo.

Let this extra resistance be represented by o0P; so that,

P - du+ ( 3.2-24)

32

Consideration . of (3.2-8) and also the form of the integral

above, leads L.s to expect that:

P = $0, ... (3.2-25) 00 It is empirical knowledge that at differs significantly

from 00 only in the region very close to the wall (0 < <

i.e. in the region which is usually called the laminar sub-

layer. An alternative state7cnt of this fact has already been

used in the derivation of (3.2-22). 0P therfore represents

the extra resistance to -transfer offered by the laminar

sub-layer on account of the total Prandtl or Schmidt number

in it being different from that in the turbulent core. A

further consequence of this fact is that the upper limit of

the integral in (3.2-23) can be extended to infinity withau

the value of the integral being affected appreciably.

Equation (3.2-23) can now be written as,

1/2

S 001P + (1 + 2 1.25 ,

°P)1'1,3 ,1/2 (3.2-23

(Pyr

3.3 Summary and comparison of previous theories

The forms of equation (3.2-23) which can be attributed

to various authors differ mainly in the following respects

N (1) assumptions regarding the at.(-0, u+)- relation; (2) the value of 00' if indeed this is taken as constant;

(3) the treatment of the quantity,

1.25 1 + s 2 P

33

Accordingly, in Table 1, their theories are classified with

respect to these features. Another, and perhaps secondary,

feature is the way in which the quadrature for P has been

evaluated after a of variation has been specified; some of

the ()Jo, 114.} functions permit this to be worked out in closed

form, whilst others necessitate numerical integration. When

a closed form exists it is entered in the appropriate column

in the Table.

Even when there is no closed form, it is posstblp to

find an asymptotic expression for the numerical solution at

high Prandtl or Schmidt numbers, and this too is included as

it is an important point of comparison.

Instead of starting from a of variation and devising a

P-function, it is possible to fit a curve to experimental

data, directly; the expressions for P obtained in this way

would be accompanied by the word 'empirical'. The recommen-

dation made in the present work is of this character.

The distribution of total Prandtl/Schmidt number, lot

Inspection of Table 1 shows that various forms have been

suggested for the of distribution. Even so, they do not

give widely differing values of of corresponding to given o

and u+. Many of

1

00

the (3t

1

a t

variations can be cast in the form:

f(.11+, ... (3.3-1) 1 00

1 o

3L

where 'f' stands for some function; or such a relationship

can be generated from:

at = ot 5/-.1- 9 RD' 1

and u „. (303-2) 4-

= 1.14-(-y+)- I 1 .-, which are presented in some references.

In the'fluid layers where both the molecular and eddy

transport processes are effective, the function 'f' is the

same as t which vies defined in (3.2-9) and hence can be

related directly to the velocity-profile (see Appendix 1).

All the authors with exception of Prandtl, Taylor,

Hoffmann and Bilhne accept the presence of such a mixed mode

of transport, at 1;;ast within a buffer region which is inter-

posed between the turbulent core and the laminar sub-layer;

of these Murphree, Rannie, Reichardt, Lin et al, Deissler,

Petukhov and Kirillov, Mills, Gowariker and Garner, Wasan and

Viilke, and Rasmussen and Karamcheti accept the existence of

an eddy transport effect right up to the wall. The existence

of a solely turbulent core is postulated by all except Marti-

nelli, Rannie, Reichardt, Petukhov and Kirillov, Mills , Go7/-ariker and Garner, and Rasmussen and Karamcheti.

Prandtl, Taylor, Hoffmann, and Rehne, all of whom sug-

gested two layer models of the flow chose the location of the

discontinuity so as to obtain agreement of the resulting

(p-transfer laws with experimental data available. to them, and

not corresponding to the points of discontinuity in the

velocity profiles they had assumed.

35

A plot of 'f' against (figure 3.2), enables the com-

parison of the important features of the at distributions;

they are all seen to follow the same general trend, having

differences with regard to details such as the number of sub-

divisions and the placing of their lines of demarcation.

Values of a0

All the authors, except Reichardt,_ have chosen Go coual

to unity, although many of them recognised the possibility

of its value lying between 0.5 and 2. This is reasonable on

account of the fact that the effect of the difference of a 0

from unity would become implicit in any adjustments made in

order -CO fit the p-transfer law to experimental data.

The value of o0, however, would become important, accord-

ing to our analysis, in the ease Prandtl or Schmidt numbers

smaller than unity° But in such a case the Couette-flow

assumptions become ouestionable; so that we shall limit our

analysis to fluids with Prandtl/Schmidt numbers not much less

than unity.

Transformation of boundary layer relationships for ap-olicatior

to pipe-flows

In the case of boundary layer flows, the reference state

is that of the main-stream, and for pipe-flows it is the bulk

state which is defined in terms of average flow rates and

mixed mean values of q

The factor 1 1.25 sP has been introduced to account Ag 2

for the fact that the bulk states with respect to the velocity

and 9-distributions respectively are different from each

other and from that at the pipe-axis, which, crudely, corres-

ponds to the main-stream state of a boundary-layer. It is

clear that most authors did not make such a correction.

Generalisation: of the intec;=al_for P

The complexity of most of the 0t distributions does not

permit closed form expressions for P. During the evaluation

of P starting from a given at t relationship, the result

can be generalised for a value of ao other than the one which

has been specified by the author, by the substitution of ot/00

and a/00 in places where at and a appear respectively, as

indicated by a combination of (3.3-1) and (3.2-4); namely, Poo _

P = j(55 - 1)1 11 + 2 1 -1

(E, - 1) du' ... (3.3-2) 0 JO _ GO j

In the case of already tabulated values of P, this

generalisation can be effected by taking the tabulated value;

of a to mean cs/c50.

Asymptotic expressions for P

The Et-, u+ relationships can he approximated by:

Et = 1 + alu+ + a2(u+)2 + eeee e e c

for small values of u+, when the sub-layer is not hypothesised

to be solely laminar. . _

Theoretical reasons have been given 1 21a:,86j for the

lowest exponent in this series being not less than 3. If we

denote this lowest expcnent by b, then, (3.2-2) reduces to:

1)11 Jo

+.go

ab

-b ab(u+)i

sin77./b)

-1

1

d e 0 0

0 •

I

0

0

37

(3.3-)4)

(3.3-5) L b

p

which gives,

P -> (2 ) a0

as o/a0 ->

Hence it is possible to have an asymptotic expression

for P, for large a, irrespective of whether a closed form

exists or not, for the integral in (3.3-2). Such asymptotic

expressions are entered in Table 10

3d1. Choice of the P-function

Experimental data

Overriding the appeal of various hypotheses and mathe-

matical maniptlations, the criterion for the suitability of a

suggested P-function is its agreement with experimental data.

Hence a part of the present task has been the collection of

all avilahle experimental results for heat- and mass-transfer

in smooth pipes, covering a large range of Prandtl/Schmidt

numbers, so as to enable this.

The usual method of comparison has been to use plots of a

Nusselt or Stanton number aglimst Reynolds number with Prandtl

number as a parameter. Such a method of comparison would

either restrict the comparison to data obtained for specific

Reynolds and Prandtl/Schmidt numbers only, or necessitate

interpolation so that an unwarranted sense of exactness may

be created.

38

In the present work the testing ground is the (o0P, a)

space. Experimental values of o0P are obtained, from values

of Stanton number and drag coefficient via:

s1/2 0 1.25 0P exp = S

(1 sP) p,P sp

which is simply (3.2-23) , rewritten. When drag coefficients

are not given alon7 with heat- or mass-transfer data they are

calculated by the use of the Prandtl-von Karman formula:

0 a a (3.4-1)

= 2.46 ln(RDsp/2 ) + 0.292 • • •

is taken as 0.40. The highest value

given in the literature is 0.41; therefore, in its

position it cannot cause significant inaccuracies

s -1/2 P

TIE value of 14.

(3.4-2)

of /s

present

owing to

uncertainity of its value. Two values of 0 have been tried:;

namely, 0.9 and 1. The values of S(P3P' sP

and a which have

been extracted from the literature are given in Table 2.

The results have been plotted in figures 3.3 and 3.4,

actually in terms of ooP + 9 against a, on logarithmic co-

ordinates; the first figure, for ao = 1 and the second, for

a0 = 0.9. Logarithmic co-ordinates have been used on account

of the large range of each variable involved; and 9 has been

added so as to enable the plotting of negative values. Dots

and crosses have been used because the use of various symbols

would be only confusing.

The following points regarding these figures may be noted

(a) The differences between the figures are very slight and

are noticeable only at the low values of a, thus indicatir

39

the relative unimportance of changing the value of from

unity, at high o. There is, however, a slight decrease of

scatter in the latter figure which can be attributed to the

change of oo to 0.9.

(b) The data points fall on a single band of Although they

been obtained by diverse experimental techniques; the

o0P values increasing steadily with increase of the Prandtl/

Schmidt number. The best curve fitting the points can be made

to pass through P = 0 at 0 at the lower end and the clus-

ter of points derived from the data.. of rdeL0] at a values

around 8. But a degree of uncertainity is introduced to the

slope at high a because the data, of Lin,. Denton, Gaskill,.

Putnamrial appearing at that end and indicated by crosses,

were obtained in an annular flow configuration; thus making

the values of s be of doubtful applicability. The data for

a values slightly lower than for the above set show a large

amount of scatter as they were probably affected by surface

to bulk temperature differences, the fluids being some heavy

and light oils.

The large scatter at low o values can be attributed

primarily to the fact that the calculation of o0 P for thew

involves taking the difference of two terms which of the same

order of magnitude.

(c) Two curves are shown on each figure. Of these the bro-

ken lines represent equations of the form:

)40

°OP = A1f(6/60)

3/4 _ 0 0 0 (3. 4- 3

and the full lines:

rt 3/4 - 1 A GOP 1L 0

+ A2exp(-A3a

(3

The values

A2' and A3 al-e as follows:

CO A2 A

3

0.9 8.32

0.28 0.007

1.0 9.00

0.28 0.007

The second term in seuare-brackets on the R.H.S. of (3.4-24

is a correction factor introduced so as to obtain an improved

fit at moderate a, than with the simpler fprm

Comparison of tbnics with experiment

The P-expressions of various authors as given in Table

are represented on figures 3.5 and 3.6; two figures being

used for the sake of clarity. The area occupied by the ex-

perimental data is shown in outline on each figure; and a

curve representing equation ( 3 .14-4) with appropriate constants

is given as the i mean-line' through the data. A a0 valu::, of

unity has been used in 7eeping with the view of most authors

The following remarks can be made regarding the various

curves.

(a) The curve of Fannie is seen to lie well below the data,

and obviously to have the wrong slope at high values of oc.

(b) On the other extreme are the curves of Prandtl, and von

Darman, which seem to have the wroni asymptotic slope and

141

also to deviate from the data for a greater than 15

approximately,

(c) Wirth an increasing degree of closeness to the mean-line,

at high a, come the curves of Gowariker and Garner, Hoffmann,

Reichardt, Mills, Petukhov and Kirillov, Lin et al, Rasmussen

and Karamcheti, Kutatladze, Wasan and Wilke and of Deissler.

The coves of Lin et al deviate the most at values of a

between 1 and 10.

(d) It is difficult to choose between the exponents 2/3 and

3/4 for the asymptotic form.

(e) If we consider the data represented by crosses to be

reliable then we may say that the curves of Deissler, Wasan

and Wilke, and Kutateladze have an acceptable trend at high

falues of 0.

3.5 An examination of the limitations of Couette-flow

analysis

The Couette-flow analysis is a very restricted solution

of the partial-differential equation governing cp-tnansfer in

pipe-flows. We have neglected the axial convection and con-

duction effects and assumed invariant cp - ps profiles, in

order to simplify the analysis. As a result our analysis

would be restricted to cases where the flux j" is uniform.

The experimental data we have used; have come mainly fro7.

systems with electrically heated tubes, wetted-wall columns

or counter-current two-fluid heat exchangers; so that this

boundary condition is at least approximated.

L2

It would be interesting, if not necessary, to examine

whether there is any significant change in the solution when

we change boundary condition to one of unifcrm Ts, and also

whether the exact solutions indicate a dependence of P on the

Reynolds number, an effect which is assumed to be absent in

the Couette-flo;I model.

An examination of some 'exact' solutions

The differential equation overn:_ng the T-field in a

Pipe-flow, when ax.ial conduction has been neglected but axial

convection taken into account, is:

u = 1 -:- ) 1 ... (3.5-1) .

ax r p 0 00 ari

where r is the distance from the pipe-axis and u the eddy-

viscosity. The axial conduction term is said to he negligi-

ble under the present circumtances. Its inclusion also

complicates the solution very much; therefore, it has been

the practice to leave that term out.

Equation (3.5-1) has been solved for the two boundary

conditions of uniform T-flux and uniform T-potential at

the surface, by the followng authors:

Sleicher and Tribus[77]

Siegel and. Sparrow [751 Kays and Leung [35]

each pair of authors choosing a particular velocity, eddy-

viscosity and total Prandtl number distributions which arii

shown in Table 3 .

43

The possibility was examined, of adding a term to the

expression in (3.2-2), so as to account for the convection

term in (3.5-l) on the Insis of its exact solutions. The

additional term was expected to depend on the Reynolds number

and also on a parameter which specifies the boundary condition

at the surface. A suitable definition for this parameter

which we shall refer to as BP' was considered to be:

(dcps/dx) 0 00

(dF,6 /d2)

the value of Bp being zero for the case of constant cps, and

unity for that of uniform flux at the surface.

The exact solutions which are presented in terms of

Nusselt number, were transformed to P by the use of drag coef-

ficients calculated with formula (5.4-2). These P-values are

shown plotted, on figure 3.7, against s1/2 with a as para-

meter. The computations of Kays and Leung for the uniform

cps case were not available. Those of Siegel and Sparrow are

few in number and show only small differences between the two

boundary conditions; therfore the percentage differences bet-

ween P values for the two boundary conditions are shown in

Table Li..

The following remarks can be made regarding the compari-

son:

(a) The lack of smoothness in the distribution of points of

Kays and Leung may be attributed to round-off errors in the

tabulated Nusselt numbers; and in the case of Sleicher and

Tribus, to truncation errors, their solution being given as

BP (3.5-2)

a series.

(b) The solutions of Kays and Leung, which have been made by

the use of an eddy viscosity hypothesis: of the form suggested

by Deissler[13], do not differ significantly from his Couette-

flow solution,

(c) The exact solutions fall within the spread of experimen-

tal data for each c

(d) The solutions of Sleicher and Tribus do not come within

15 percent. of the. mean-line through the data or within 25

percent. of the predictions of Kays and Leun7. They also do

not show any systemmatic difference between the two boundary

conditions.

(e) The percentage differences of P-values for the two-boun-

dary conditions, derived from the computaticns of Siegel and

Sparrow are about 15 percent. at the Prandtl number of 0.7.

This however does not justify the specification of a compli-

cated correction for differing boundary conditions, because

the experimental data with which any final comparison has to

be made show a scatter of about -I- 35 percent.

3.6 Recommendation cf a calculation procedure

It has been observed tl,:at all except three of the theo-

ries follow the data 1-easonably well. Therefore, our choice

of a formula for the evaluation of P has to be based on th-,

criterion of simplicity, especially in view of the applica-

tion in our general theory of flows near surfaces.

Hence, we can eliminate all except. the closed form

45

expressions. Such closed form expressions are but a few, and

even these are seen to lack the simplicity of (3.4L4)

In the light of the finding7s of Rotta[5] and Ludwieg[40]

a value of 0.90 for oo seems quite appropriate for our applf •

cations.

Therefore, the following formula for the calculation of

P can be recommended:

.24 (0/(30 ) 3/4. - 1

(̂j + 0.28 exp(-0.0070/00) 1

... (3.6.-1)

with o0 = 0.9.

Where a simpler formula is required, especially at Fran,

dtl/Schmidt numbers greater than about 50, the formula,

P = 9.24(c/60 )3/4 - i

(3.6-2)

can be used.

Stanton numbers for heat- or mass-transfer may be cal-

culated by the use of (3.2-23) and the recommended P-expres-

sion; a value of 0.4 is appropriate for K, and the drag-. co-

efficient obtainable via the formula (3.4-2) or any other, one

which is appropriate for the given flow configuration.

In the final analysis, P would depend on the mean velo-

city-profile nnd the turbulent fluctuations close to the

surface (i.e. within about 1/5 the boundary layer thickness

from it.at low Reynolds numbers, and less at higher ones) and

of course, the laminar fluid properties. The distributions

of these velocity fluctuations near the surface in the case

46

of pipe-flows show a great resemblance to those of boundary

layer flows; typical velocity distributions for the two sys-

tems being shown in figure 3.8. Hence it would be reasonable

to apply the P-function derived from pipe-flow data to boun-

dary layer flows, and with some reservations, to wall-jet

flows.

3.7 Clobure

Formulae (3.5-1 and (3.5-2) are recommended for the

evaluation of P which appears in the local heat flux laws

the general Q--profile expression (2.5-3) of turbulent boun-

dary layers and wall-jets.

The validity of the formulae are subject to the following

conditions:

1. in the case of mass transfer processes, the mass-transfer

driving force, B, is small (i.e. -0.1 < B

2. heat transfer rates are not high enough to bring about

large temperature differences which would influence the flow

due to property variations across the stream, and introduce

ambiguity in the choice of fluid properties;

3. the surface is hydrodynamically smooth.

The last condition aptly leads us to the next aspect of

the present work; namely, a study of the effects of surface

roughness on the laminar sub-layer.

SECTION III

47

48 CHAPTER 4

HYDRODYNAMIC EFFECTS OF SURFACE ROUGHNESS

4.1 Introduction

In the context of the general theory introduced in Sec-

tion I, it is necesnary to devise a means of incorporating a of

cuantitative descriptionA the effects of surface roughness into

the model so as to widen the scope of its applications.

The interest in roughness has arisen among various wor-

kers for two basic reasons: on one hand the desire to avoid

it, shown by shipbuilaers; on the other hand the deliberate

introduction of it to improve heat transfer, done by nuclear-

reactor designers and others. Between these two extremes we

stand, attempting to learn how a fluid stream behaves in the

presence of surface roug-hness.

Our primary recuirement is a drag; correlation, for this

is the statement of the direct result of the interaction bet-

ween a surface and a fluid-stream. It is also necessary to

examine whether the influence of roughness extends sufficient-

ly far from the surface so as to affect the dissipation and

entrainment processes.

4.2 Classification of roughness types

The problem of row7hness is complicated by the possi-

bility of having a Freat variety of shapes, sizes and distri-

butions of roughness elements. Nevertheless, it will be

seen that these fall into groups, each having a characteris-

tic behaviour pattern; so it behoves us to commence with an

49

attempt to separate the various roughnesses according to

their appearance.

Roughness may be broadly classified into two types: ir-

regular; and regular. By regular we mean that all the ele-

ments are identical in shape and size,.and are distributed

according to a definite -.pattern. For example, in the first

category we have the roughness formed of sand grains having

various sizes; in the second, those produced by knurling and

the machining of threads.

It is also possible to make a distinction between two

dimensional roughness, i.e. one formed of elements which are

ridges of grooves having uniform cross-section and placed at

right angles to the flow; and three-dimensional roughness,

i.e0 one formed of discrete lumps or cavities.

Another aspect is, whether the elements are packed

together as closely as possible or distributed.

We may also notice subtle similarities of behaviour

depending the superficial features of the elements.

4.3 Applicability of pipe-flow results to boundary-layer

flows

From a comparison of velocity profiles obtaining in pipes

and boundary layers, HamaN drew the conclusion that the

roughness effect on the wall-law of the velocity-profile was

universal irrespective of the external flow conditions. He

found that the velocity defect law was universal; whilst the

50

wall-law was affected to an extent dependent on the magnitude

of a 'roughne,3s parameter'.

Perry and Joubert[561 extracted the 'wake component' of

the velocity-profiles in a boundary layer along a rough sur-

face in the presence of an adverse pressure gradient acting

on the stream. They found that the wake-profile could be des-

cribed by Coles' wake function. Another obervation was that

the law of the wall was affected in the same way by the

roughness as in the case of a constant pressure boundary lay-

er.

:6ettermannT] too showed the validity of Coles' wake

profile and the wall-law modification from further studies

of constant pressure boundary layers.

4.4 Drag-law for maximum-density, uniform roughness

We shall first devote our attention to the simplest type

of roughness: that formed of elements which are uniform in

size and packed as closely as -. ossible; they may be regular

or not, but the important thing is that there is uniformity

at least in a statistical sense, i.e. they have a very narrow

distriution of sizes. In this case the rourrhness size can

be characterised by one particular dimension, usually the

heio-ht of elements. The uniform sand-grain roughness of

Nikuradse[5A, pyramidal roughness of Cope [10 and Stamford

[814] and the v-groove roughness of Kolar 1361 belong to this

category.

1

51

Kikuradse (531 found that the ratio yr/D, where yr is the

mean height of roughness and D the pipe diameter, is an

adequate parameter in the correlation of pipe-flow drag data;

i.e. all the roughnesses with the same yr/D values fell on

a single curve. The drag curves he obtained are shown in

figure 4.1.

Among other things, he established that AN, defined by,

AN = u - ln(Y/Srr) .0. .L-1) was a function of the uuantity,

Rr = YrigTsp)/P. noo ( .4--2)

forthis particular rou7hness; so, the drag law is,

2.83 t/(81-,7 2.5 ln(D/yr) - 3.75 = AN

with,

AN = AN(-Rr}

0.0 (4.4-3)

• (4.4-4) If we write the velocity-profile expression in the

familiar way,

u • ln(Ey+) • ( 2 . 2- _3)

then,

Equations

exp(i.f_Ar) E = R (4.4-5)

r

(4-4-5) and (4.4-3) together enable us to derive

an empirical relationship between E and Rr from pipe-flow

drag data. The values so derived are shown plotted in figure

4-2, on logarithmic co-ordinates, Hama r29, writes the

logarithmic velocity-profile in the form:

52

L\ u = 1 In y au - 11 K. Ts/p

... (4.4-6)

BH being the additive constant in the smooth wall case. If

write EM as the value of E appropriate for this, then,

B = E- H - /4, In jui 0 • 5 ( • 4- 7) so that,

u fl(Ts4)

The E 1'2,, relation is a suitable form of the drag cor-

relation for insertion into the c-eneral theory calculation

procedures, and permits evaluation of the drag coefficient

for a rough surface under given hydrodynamic conditions (see

Appendix 2).

4.5 The E w Rr relation for uniform sand-grain roughness

A close examination of the E Rr curve (figure 4.3)

for Nikuradse's sand—rain roughness data, shows that it can

be divided into 3 sections:(i) that to the , left of A, where

it is parallel to the Rr axis and E has the value Eli"; (ii) the

portion betwocn A and B, which is referred to as the transi-

tion between the hydrodynamically smooth and fully rough

conditions; (iii) the fully rough conditions being represen-

ted by the portion to the right of B, which has an equation

of the form:

E = /Rr • ( 4.5-1) being a constant of the order 30.

- in(E, /E) . (4.4-8)

53

Although Nikuradse recognised the presence of the tran-

sition region, he did not delve deeper into it, but fitted

a piecewise linear charactefistic when the need arose. It

was usual for subsequent workers on flows past rough surfaces

to work with roughnesses which they considered to be large

enough to ensure fully rou.--h operation.

Nedderman and Shearer [521 attempted to construct a model

of this transition flow. They envisaged a condition where

only the portion of the roughness element which protruded

above a hypothetical sub-layer was resoonsible for producing

a drag component which increased as the square of the velo-

city. This led them to derive a drag formula which can be

represented by the curve in figure 4.4.

Morris [49.] too engaged in a curve-fitting exercise

which took him only a part of the way in the transition re-

-ion.

By taking into account a statistical description of the

roughness, a satisfactory E characteristic for the transition

zone can be generated as shown in the remainder of this sub;,

section.

Sand-grain size distribution

A photograph of the sand covered surface, appearing in

Nikuradse's paper shows that there were sand-grains of a

rang.e of sizes in a roughness of given nominal size; the

smallest Frain being about half the size of the largest. Of

course this is to be expected since the method of obtaining

54

grains of approximately uniform size would have been to col-

lect those which passed though a sieve of suitable size and

were stopped by one with a slightly smaller mesh; and sand.•

grains being of irregular shape, this would result in a range

of sizes which would perhaps have had a distribution as shorn

in figure 4.5.

We may also boar in mind that the total number, T, of

sand-grains having a size greater than a given size yr2c'

is

given by a laterally inverted cumulative freauency distribu-

tion; and 1 y r,u I N dyr,g ••• (4.5-2)

j Yr c

Some hydrodynamic considerations

From experiments on flows past spheres and cylinders,

has been shown that the ability of a body to shed vortices

depends on whether its characteristic Reynolds number is

above a critical value, Re c (Schlichting[70]. ).

This is seen to be valid also for bodies placed in

contact with a smooth surface so that the flow past it is

sheared, as indicated by the experiments of Sacks[7]. There

is some indication that the value of critical Reynolds number' ,

based on the friction velocity, (gTs/p), lies between 0 and

50.

As the nominal roughness Reynolds number increases, more

of the elements will become capable of vortex generation; we

55

will call this becoming 'active'. It is possible to think of

a critical size yr c all the trains having a size larger ,

than this being active; then,

r,c = y rR e,c/R r (4.5-3)

Looking back on the distribution curve, figure 4.5, we

can see the possibility of deriving a relation between the

total number, Ta, of active elements and Rr, of the form

shown in figure 4.6. (Note that the Ta /'J Rr relation is not

anti-symmetrical due to the nature of the yr c ̂a Rr relation.)

There is reason to think that the onset of activity of

an element is delayed by the increase of local turbulence

level. This turbulence level will in turn depend on the

number of elements which are already active; so that Re,c

will increase witn increase of Rr

Drag on the surface

Here we make the further assumption that the areas

occupied by the active and inactive elements are in the ratio

of the numbers in the two groups. Hence, if we define a as

the fractional projected area of the active elements, then

a can be related to Ta and thence to Rr

The drag coefficient of the surface can be considered as

made up of two parts, i.e.,

s = asE + (1 - sal

where sE

and sM are the drag coefficients which would obtain

if the same main-stream flow existed over fully roucrh and

56

entirely smooth surfaces respectively. (Details of various

steps in the acrivation of (4.5-4) are given in Appendix 3) -

For fully rouc-h surfaces, the drag law is,

E = POtr ( 4 . 5- 5)

and for a smooth one,

.13

▪ (a constant) • ( 24.5-6) Hence it is pos3ible to derive that (Appendix 3) ,

/ 2 \ 2 1 -1/2 E = !a(Rr/(1) + (1 - a) ... ( 4 • 5- 7) i Eia

1 1_ __J

Application to Hikuradse's data

If we assume a simple auadratic distribution of sand-

grain sizes, i.e.,

N = .A.... 1 - X) ... ( 4 . 5-8) IX

with,

X = (Yr,g Yr,1)/( Yr,u — Yr, 1) (4.5-9)

then,

2 c = 1+ 2X3 - 3Xc where,

Xc = (Rr,l/Rr)n(Rr,u - Rr)/(Rr,u - Rr,l)

... (4.5-10)

— (4.5-11)

Equations (4.5-11), (4.4-10) and (4.4-7) together

the E(--Rr and have to be chosen to 7ive a R r,u Rr,1

satisfactory fit to the data; for sand grain roughness,

Xc = 0 .0 2248( 100 — R) /R° .584 • .. ( 4.5-12) is found to give a satisfactory fit for Rr lying between

and 100.

57

Some consequences of the validity of the model

1. Roug'Inesses which can be controled to a greater

degree, in size and shape would show a narrower range of

transition.

2. Rough surfaces, thourrh nominally similar, need not

have the same transition characteristic.

Indeed, this is found to be the case: a close examina-

tion of Nikuradse's data show a distinct curve for each rough

pipe. They all, however, form a narrow band of points be-

cause the basic shape of the size distribution function would

be governed by the laws of crystal structure, (or perhaps

the type of mesh of the sieve '.) and other unknown, but common,

factors which we lump into the statistical description.

In this respect we find the man-made rouhnesses showing

a tendency to have more scattered transition data than

natural roughness elements.

3. Roughneses which have areas of smooth surface

interspersed with the elements will not have a tending to

unity as nr increases. Therefore their operation will never

be fully rough, and the characteristics will be curves having

slope greater than -1 on the E Rr plot. This point will

be discussed later.

L.6 Drag measurements of Dipprey and Sabersky

Dipprey and Sabersky[15] conducted drag measurements in

pipes having a roughness which was meant to simulate uniform

sand-grain roughness. The rough surfaces were prepared by

58

clectrodeposition of metal on sand-coated mandrels and then

dissolving the pond away to expose a rough metallic surface

which had the 'general apearance of an array of close packud

sand-renains l .

Strictly speakinp:, this rouchness should exhibit differ-

ences of behaviour at least in the tansitiop roughness 7one:

for the elements in this case are negatives' of sand-grains.

There would be a wider distribution of sizes of projections;

and pernaps an effect more of cavities rather than protrusionu

at lower Rr values.

E values deduced from their data[itil are shown in figure

L1.8. The moan line of Hikuradse's data is also shown. The

following points are evident:

1. the two rou5:hnesses behave alike when the surfaces are

fully rough;

2. the data of Direy, Sabersky show consistently lower

values of E in the transition region.

L.7 TAEr} for pyramidal roughness

Pyramidal-type rou7hness has been studied by Cope r12] and

Stamford[84]; the former using whole pyramids whilst the

latter used truncated ones. Cope's drag data are shown on

figure 4.9a and those of 1.;tamford on figure 4.9b. E

calculated from the two sets of data are shown on figures 4.1u

and 4.11 respectively.

The followinp• features are noteworthy:

1. The points for both rou7hnesses fall on* the same curve

59

Rr > 50 approximately.

2. Cope's data give E values which rise rapidly, as Rr drops

below 50, to values above the smooth limit; this being unreal-

istic as it indicates a rough pipe which is 'smoother' than

a smooth one. This discrepancy is probably due to buoyancy

effects which become significant as Reynolds nuiaber decreases

in his vertical-tube heat-exchanger configuration.

3. A feature of Stamford's data is that although the rough-

ness elements of his three pipes respectively were not exactly

similar, all the points fall on a single curve, suggesting

that: if the elements on one surface are not exactly similar

to those on another but have tie same basic shape, then the

suifaces would have a common E --Rr cu-ove; only the size of

the elements being important. This lends further support to

the statistical basis for the derivation of E.(1/r given in

sub-section 4.5.

4. A comparison with the mean line of Nikuradse's data, also

shown on figure 4.11, shows that for the same nominal height,

pyramidal roughness..is 'rougher' than sand roughness.

5. Stamford's data indicate a very short transition region zs

whichLexpected of machined roughnesses.

4.8 V-groove roughness used by Kolar

The closely packed roughnesses considered above were of

a 3-dimensional type; in contrast, that of Kolar:M is 2-dim-

ensional. E values calculated from his data are shown in

60

fiFure 4.12. He used three similar v-groove roughnesses of

relative roughness height, yr/D, of 0.0545, 0.0371 and 0.0189

respectively. The points are seen to lie on a curve, which

does not show any systematic difference between the three

rouFhnesses.

As in the case of Cope's data, a transition re,j_on

the points deviate significantly from the fully-rough line,

can be noticed; unlike for Stamford's data. It is interesting

to note the following point in this connexion: the rough pipes

of Stamford were produced by indenting plane sheets with the

required roughness pattern and then forming them into pipes,

whereas, Kolar and Cope used available pipes which were mach-

ined internally to produce the roughness. Hence there is the

likelihood that Stamford had better control over the uniform-

ity of elements.

For application to boundary-layer computations the TkRr)-

is approximated by a continuous piecewise smooth function

represented by the 3-segmented curve shown on figure 4.12.

E. for natural rouhness

Commercial steel, wrought iron and galvanised iron pipes

are found to have drag coefficients differing from those of

copper, brass or, glass pipes under the same flow.conditionse

This is attributed to slight waviness and surface irregulari-

ties left by manufacturing processes and any other unknown

causes.

This roughness has been referred to as 'commercial rough--

(31

ness' by Schlichtino. [70], but we consider it -oreferable to

use the term 'natural rour:hness', as opposed to 'machined'

or any other 'artificial' rourrhnesses.

The roughness hei9..ht is not definable in the same way as

we define the nominal height of sand roughness; and it is

usual to Quote an 'egrAvalent sand roughness hei7ht' vdlich is

defined as the height of the uniform sand-roughness that rives

the same drag coefficient under identical hydrodynamic con-

ditions. The eczuivalent sand nrain roughnesses of some

surfaces are listed in Table 5. It has been shown by Colebrookal], Smith and Lpstein[78],

and Muller and Stratmann[50], that the drat correlation which

has been proposed by Colebrook for naturally rouch pipes is

is satisfactory for new or corroded metal pipes and some non-

metallic pipes as well.

The E4Rr} derived from Colebrook's formula is,

34.02ARr + 3.305) • • • (4.9-1)

and is shown graphically in fiffure 4-14 together with those

of other roughncsses.

The accuracy in the use of this formula is limited by

the fact that,th,e—choice of ccAivalent sand-r:raiti roughness

cannot be done with certainity. An extra word of caution is

necessary when one applies the above formula to boundary

layer flows, due to the fact that, equivalent sand roughness

heicrhts indicated in Table 5 are for the texture produced

under conditions of pipe manufacture, which may not be the

S2

same as that produced when the material is rolled to form

sheets.

4.10 Distributed roughness

Significant departure from the E"---,Rr curve for closely

packed roughness arises when elements are distributed over

the surface with areas of smooth surface interspersed.

Owing to the complex nature of the distributed rouFhness

problem no particular attempts were made during the present

study to derive generalised relationships between E and the

geometrical parameters of the roughness.

The E ,--Rr characteristics for a tvoical roughness of

this type - formed by wires stretching almost at right angles

to the flow , - are given in figure 4.14. The parameter

in this plot is pr/yr, pr bein the spacinr of the wires. The

magihitude of the slope is seen to be lower than that of the

line for 'fully rough' sand-grain roughness. The same trends

are shown by roughnesses composed of distributed pyramids and

triangular ridges respectively. The data in figure 4.14,

which are from the work of -Ialherbe[44], have been fitted by a

curve having the equation,

38.0(0.0362/Ar) 1/exp(0.123 + 0.0082pr/Yr)

... (4.10-1)

which is valid for pr /yr > 6.25 and 20 < R < 200.

From the drag data correlation of Bettermann[51 for

rides with square cross-section, we can derive the E

Rr

63

relation

SRO . E = aR expl7).94 - 4.90 ln(pr/yr)1] e.. (4.10-2) r L which is seen: to be valid for 180 < Rr <750 and 2.7 < Pr/Yr

< 4, according to his data.

It may have been observed that equations (4.10-1) and

(4.10-2) indicate op;site trends of E for changes of pr/Yr.

This is the implication of the phenomenon of there being a

particular value of pr/yr for which the drag is a maximum

with respect to pr/yr, at a constant Reynolds number, for

roughness elements of given shape and size. The 'optimum'

value of pr/yr and the maximum value of drag coeffient expres-

sed in terms of the smooth wall drag cofficient under iden-

tical hydrodynamic conditions appear to depend on the height

of the roughness relative to the duct dimension, D, or the

boudary-layer thickness,

Ealherbe[44] shows that, for a given Reynolds number,

irrespective of the cross-sectional shape of elements, the

value of pr/yr for maximum drag lies between 6.5 and 10;

roughnesses with larr'cr yr/b values usually having larger

pr /yr at maxima. This lack of dependence on shape, however,

is not corroborated by the data of Savage and Myers[691.

We should also note that Malherbe's data show a depen-

dence of the r relation on yr/b whereas Detterman,7'sj5

do not. The latter were obtained in a boundary-layer flow

whilst the former are from .a duct flow. This is perhaps an

indication of an important point: that, in thc,.case of

6L.

distributed roughness it would not be eau to generalise duct

flow results for the purpose of application to boundary layer

calculations.

Morris[0] recognises a distributed type roughness formed

of grooves where conditions are suitable for the formation of

standin7 eddies in the grooves; when this obtains, he calls

the surface 'quasi-smooth'. The drag coefficient can be writ-

ten as the sum of that of a smooth surface under the same

hydrodynamic conditions and a constant. Morris states • t

all flows having this behaviour, encountered by him had groove

width to depth ratios slightly greater than unity. One of the

roughnesses used. by Sams 0661 exhibited this feature; his

data for square thread type roughness are summarised in figure

4.14.

Some data on flows past surfaces with 3-dimensional

distributed roughnesses are to be found in the work of

Ambrose IT1, and Doenecke[i,5]. The former used pipes roughened

with small circular cylindrical projections and with cylindri-

cal cavities. The latter made measurements in boundary layer

flows on plates roughened with short cylindrical projecting

elements.

4.11 Other hydrodynamic considerations

Up to now-..we have been preoccupied with the relationship

between E and the roughness parameters, which would be re-

quired in the process of generating solutions to the hydro-

dynamic problem starting from a velocity profile assumption.

65

If the differential equations given in sub-section 1.3

are written in terms of shape factors H32 and H12, then an

auxiliary relation involving these become necessary. Nicoll

and Escudi'-r [52aj have recommendedt,

0.0971 0.775 H32 = 1.431 - GOO (4.11-1) H 2 12 H12

which is shown compared with rough-surface boundary layer data

of Brunello[7] and Bettermann[5] in figure 4.15. Although the

detailed disposition of the data leave much to be desired, the

curve can be considered representative of the data,. Also

shownn in the same figure is the curve representing the equa-

tion,

H32 0.25(H12 + 3)2/H12 • • • (4.11-2)

which is derived from a simple linear velocity profile assump-

tion.

A comparison, on figure 4.16, of the H32 J zE relation

derived by Spalding[83], on the basis of the linear velocity

profile, with rough wall data shows good agreement.

Spalding [83] has proposed,

z7ss + 0.008(1 - zB)3

(4.11-3)

for the aalculation of '6 for smooth surfaces. Enetty thick-

ness data shown on figure 4.17 enable the calculation of

values which are seen, on figure 4.18, to agree very well

with the theoretical values. The agreement, however, is Irok

found to be due to the predominance of the first term on the

R.H.S. iione the less, this does not depreciate the formula

66

(4.11-3); we only cannot make a pronouncement on the validity

of the constant eddy viscosity hypothesis. For these data

ilvalues are around 5 to 6; such low values 1.eing a feature

of flows past rough surfaces. A point of difference from the

way smooth surface data agree with the above relation should

be noted: for smooth surfaces the experimental data seem to

deviate increasingly from the theory when .g values are higher

than about 0.0015, but in the case of the rough surface the

agreement seems to improve or at least remain satisfactory a

as b. increases aP)ove this value.

4.12 Closure

1. Experimental E ti Rr curves are summarised on figure

4.14. Irrespective of the value of yr/D all data for maximum

density roughnesses having elements of a given shape have a

unique DOZr}.

2. Comparison of Nikuradse's data[53] with those of

Dipprey and saberskyD-4, 151 and of Stamford's data with Cope's

[l21 indicate that roughnesses with the same general appearance

have a common 'fully rough' E r,, Rr characteristic.

3. A model based on a rudimentary statistical descrip-

tion of the elements, is seen to produce a satisfactory curve

fit of the transition portion of Tikuradse's sand-grain E AiRr

characteristic.

In general the width of the transition should depend

directly on the degree of uncertainity regarding the size of

roughness elements.

67

4. In the case of diStributed roughness elements, two

more geometri:al paraTeters have to be brought into the pic-

ture in addition to yr/D; namely, those describing the longi-

tudinal and lateral spacings of the elements. Usually one

5 of these haw& been eliminated either by making, the elements

2-dimensional or making the spacings in the two directions

equal.

5. E correlations for roughnesses formed by circular

wires, triangular ridges and square ridges have been obtained

from available data. Those derived from the data of fialherbe

44 are seen to be very limited in application on account of

the dependence of E on yr/D.

6. It has been shown that for boundary layers on rough

flat plates, the relations between H32 and H12, and between

H32 and z recommended for:smooth surfaces can be applied. A

simple and widely applicable s(-z -e')- is found to cover flows

past rough surfaces as well, according to the limited amount

of data available.

58

CH T722 5

COUETTE-FLU:i ANALYSIS OF HI-TAT T=TSII7R rROM ROUrl-H .3CTRI110ES

5.1 Introduction

The influence of roughness on heat transfer has to be

introduced through a modification of v', 'which gives it a

dependence on the roughness.

Diprcy and Sabersky[l5], and Owen and Thomson r1.551 11,7,ve

used P as a means of corelatinir their heat-transfer and

(sublimation) mass-transfer data respectively. They proposed

model flows which are satisfactory in the 'fully rough' flog:

In this chapter we present an improved model flow which

behaves in a satisfactory manner even in the '14drodynamically

smooth' and 'transition' regimes; together with more P Rr or P rte. E data which have been extracted from the literature.

5.2 Heat transfer from a surface with maximum density

uniform roucl:nness

Dipprey and Saborsky(151, and Owen and Thomson[55] con-

cerned themselves with uniform, maximum density type rough-

nesses. The former authors used a roughness which could be

described as sand-indentations and the latter, tl.o rourshnesses,

one composed of pyramidal elements and other of ridges having

a triangular cross-soction.

Dipprey and Sabersky proposed a model floe.: where standing;

eddies which acted as intermediaries in the heat-transfer

process between surface and main stream, existed in the

69

cavities which formed the roucliness. The scouring action of

the eddies extracted heat from the cavity walls according; to

a law of the form:

Sc yr - 0 - ( 5 2-1)

where 3c is an appropriately defined Stanton numbel, for the

flow within the cavities; with the result,,

3 - 1 - = cl

73-- (P'S S ( 5 . 2- 2)

where, / A = 1 - lnkER r)

according. to our notation. Since the model is found.to be

valid for fully rourrh flow only, A is taken as 8.48, the

value appropriate for this. The authors find that,

n = 0.2, q = 0.44, and a = 5.19

give satisfactory fit of the data for Rr > 65.

Owen and Thomson, on the other hand suppose that around

-irojecting. rou-thness elements used by themselves, are wrapped

horse-shoe shaped eddies which acc,ur the surface and act in

the same way as those of the previous authors. They derive

the result:

sS .1 0 h5 0.8 S - C tiss = 0.52 Rr°' 0 ( 5 . 2- 5)

for the rou7hness type they used; with,

17.8, for pipe-flows

12.6, for flow between parallel plates

and C = 0, for boundary layer flows.

70

This formula too is valid for fully rough flows only.

The diffnrences in values of the coefficients and the

indices in (5.2-2) and (5.2-3) have be-n ascribed by 0,: on and

Thomson to differences in tie third term on the L.H.S. and

the dificrences of roughness element shape.

However, the liriitationo of the applicability of both

theories indicate the incompleteness of the models, altheuch

the picture presented of the 'fully rouh' flow scorns to be

adequate. 'co have explored the possibility of devising an

improved model, successfully.

An examination of Dipprey and Sabor,skji s data

The data of Dipproy and Saber-sky D-149 15] expressed in

terms of P and Rr are shown plotted in figure 5.1 and exhibit

the following features:

(a) the curves for each Prandtl number start off from the

value appropriate to smooth pipe flow, at Rr = 0;

(b) the curves deviate froTfl constant P linos as Rr increases

from zero;

(c)- then the curves dip to pass through a minimum after which

they rise monotonically. The curve for the lowest Prandtl

number does not show a minimum at all.

These points prompt us to look for two mechanisms rather

than one as proposed b r the above mentioned authors. These

two should be mechanisms which oppose each other and vary in

relative strength as the roughness Reynolds number Rr incrpses.

71

Without much difficulty we can find one mechanism in the

scouring action of the eddies as sup7ested previously. This

would predominate at high Rr because all the elements will be

exposed to these eddies which may either be stabilised in

cavities of wrapped around projecting elements. Their effect

would be to increase the sub-layer resistance P as Rr increa-

ses, as demonstrated by the previous authcres. One may raise

the question as to how the sub-layer resistance P can increase

under these circumstances, at all; because the effect of

increased scourine' should be to decrease the resistance to

heat-transfer The increase of P, however, is not anomalous

because it is actually the ratio of the resistance to heat-

transfer to the resistance to momentum transfer. Tn the case

of a flow 'past a smooth surface, the increase of Reynolds

number has no significant effect on this ratio. On the other

hand, for a change of Reynolds number in the case of flow

past a rough surface, the chan7es of the two resistances need

not necessarily be proportional. The transfer of heat between

the solid and fluid takes place by a molecular process, very

close to the interface whether eddies increase the mixinr

above it or not; whereas, hydrodynamic resistance is not only

due to the momentum transferred to the surface but also due

to the extraction of momentum by the eddies.

At the low Rr end of the scale we can envisage the

following mechanism, especially in the liE.ht of the model

proposed in sub-section 4.5. As Rr increases above the limit

72

of the hydraulically smooth re7ion,the active elements promote

turbulence at the outer ed7c of the laminar sub-layer, which

results in a reduction of F below tho smooth-pipe value.

Since the eddy producingmechanism is directly linked to the

deviation of E from that of a smooth surface, E and not flr is

the variable to which this effect would be directly related.

Even the first mechanism should be a function of E because

here too the scourin^ action begin to be effective only

when El diff ers from the smooth surface value. a1 a9 a-7,

P = b (1/F 1/E,) + (171E„,) pM

So we write,

... (5.2-3)

the suffix H denoting smooth surface conditions. The forms

of the component terms have been laid down from a considera-

tion of the behaviour reellired of them. The first term is

written on the lines of the proposal of the previous authors

the Quantity within brackets becomes proportional to at

high Rr, and in the hydraulically 'mooth regime the scouring

action would be altogether absent.

The values of constants have b::-n determined so as to

fit the data, and the resulting expression is,

3.15 o0.695 , 0.359 0./E - 0.116) + 0.274 PIZ° .0

... (5.2-0

which is shown ploted on figure 5.2 along with the data.

In applications it has to be used in conjunction with rela-

tionship aivine E in terms of Rr, which, for Dipprey and

73

Sabersky's data is,

-7,1/2 0.00-2.1089 a(Rr + 3.4_8)

2 + 0.013327(1 - I E=

.,. (5.2-5) v!ith, a -3 . 1 + c - 3X2

and c = 0.02586(70 - R,)/(Rr + 3.48)0°475

This is not a disadvantae since J has to be determined durin7

the solution of the hydrodynamic problem.

5.3 Data of Stamford and Cope

The other set of data available for 3-dimensional rour.h-

ness elements are those of Stamford1841 and Coper121. P

values deduced from these, are plotted in figure 5.3. They

indicate the same exponent of 1/E at small E, as Dipprey and

Sabersky's data. It is also interesting to note that both

sets of data can be represented by the formula:

00.695 E-0.359 = 62.1 ... (5.3-1)

which is also shown on the figure. This lends support to our

idea that roun-hnesses havinT elements of the same creneral

shape, in this case pyramids, behave in the same manner in

the fully rough rcgime.

5.4 Data of Kolar

F values deduced from the data of Kolar are shown in

figure 5.4.

They show a downward trend at lom- Rr values and appear

to be asymptotic to a line havin,7 the same gradient as for

the other data.

5.5 P-values for natural rou7hness

F-values derived from the data of Smith and Epstein[78]

for r, alvanised iron, resin bonded graphite and standard steel

pies do not show a systematic differences boteen the three

materials.

The trend of the data sureests the same asymptotic

behaviour as other data.

5..S Other types of roughness

L.10,hour-h heat-transfer experiments with cone distributed

roughness types have 'eeen reported in the literature by &came-

lauri[26], rd arils and Sheriff [21] and Droycott and Lawther

[1], to quote a few names; their methods of presentation

have the following disadvantages where our method of correla

tion is concerned:

l. unusual channel e.-eolf,etry;

2. absence of drat data;

3. only local values -iven.

These data have therfore not been analysed this presenta-

tion.

5.7 Closure

The amount of data directly useful in formulatine

functions is seen to be lielited, althour;;11 there are many

reports on heat-transfer from rough surfaces in the litera-

ture.

The scheme of correlating heat-transfer data in temas of

P re(Taires good control of Prandt1 numer, or of Rourlmcas

Reynolds number, Rr, during; experiments

75

SECTION IV

76

77

CHAFTM 6

EXPLRIMEAL INVESTIGATION (M A RADIAL WAIL-ET

6.1 Introduction

In keeping with the general exploratory nature of the

present investir.ation it was thourrht fit to conduct an experi-

ment on the flow development and heat transfer in a radial

wall-jet on a rough surface.

This was ce);idered an interesting problem since the

effect of wall roughness on this type of flow has not been

investigated.

The wall-jet on a smooth surface has been studied

analytically by Glauert[251; and Eakke 121 has made measurements

of jet growth and velocity decay.

The radial configuration has particular appeal since the

problem of achieving 2-dimensionality does not arise. The

wall-jet thickness increases and the velocities decay with

increase of the distance, xi from the slot; and in the case

of a surface with uniform hei;iht of rour-hnes this would

amount to a ch=r.e in the rouThnese, Reynolds number' Rr' viith

x, which is an interestin situation for the application of

the general theory outlined in section I.

In the present Chapter we shall deal with the important

experimental details and in the next, the application of our

prediction methods to the present experiment and also to the

case of a boundary layer on a rough surface, reported in the

literature.

76

6.1 Basic pFeces of ecuipment

The arranrrement of the main items of ETeuipment is shown

schematically in fic.ure 6.1, and wo:-hin section is shown in

firure 6.2.

Air from the fan is iraected radially alonr the plate,

through the uniform rap formed betwe-n the flancre at the end

of the delivery pipe and the plate. 'de shall refer to the

gap throuch which air is blown as the ? slot'.

A smooth plate and two rourhened plates 7e7,-e used. The

smooth plate and one rourhenea plate were heavily insulated

at the back and the other rouffh plate was uniformly heated

by means of an electrical 'Aeatinr pad'.

Fan and delivery system

A centrifugal fan was used for providing; the air. Air

leaving; it was led through a short converrin7.duct into the

delivery pipe, Interposed between the flanre of the fan and

duct .cas a fine wire mesh screen. Connexion between delivery

pipe and duct was mar2,,, Via a metal bellows, so as to minimise

the vibrations which were tr=s7tited ?Thom the fan to the

workin- section. The 3--inches-internal-diameter delivery

piue was of P.V.C. and had a lcncrth of 10 feet which was

considered sufficient to produce a reasona-ply develoT3ed

turbulent pipe. flo,- profile at the noszle end. In addition

to reducinp: vibration, the bellows piece served the addition-

al purpose of accomodatinF the profT,ressiv expansion of the

pipe, which to,-)k place as the apparatus warmed up; it

79

therfore helped to reduce the T waring up' time.

Nozzle

The nozzle in this case was the passage formed between

the face of the flange on the delivery pipe and the flat

:plate. The flancl:e face was shaped as shown in figure 6.3.

The shape eras determined by trial; the aims in shaping it

being:

(a) to avoid expansion of air flowing through;

(o) to obtain;a velocity distribution as close to

uniform as possible; and

(c) to avoid separation of the flow at the inner portion

of the flange face.

The same flange was used throughout the tests since it would

have been impracticable to look for flange shapes which

satisfied condition (b) for each slot height. 7;ith the

Present noz::le it was possible to use a maximum slot height

of 0.405 inch without flow being separated from the face of

the flange.

Typical slot velocity-profiles with the fan at full

power are shown in figure 6.4. They show that for a.slot

height of about 0.1 inch9 the velocitydistribution is reason-

ably uniform; but the distributj.on develops an increasin-

slant as the slot is made larrer.

The choice of roughness

The following factors influenced the choice of roughness

type used in this experiment.

80

1. For the prediction of hydrodynamic aspects, the Ellr)-

characteristi-, should be available.

2. The knowledge of the P variation is not necessary

for the adiabatic mall .temperature_ predictions.

3. The heat-transfer predictions necessitate the know-

ledge of the P-function for air floing past the riven sur-

face.

L. The rouFhness should be capable of being made axi-

s:vmmetrical whilst keeping the same distribution throughout.

Therefore an emery covered surface was used for hydro-

dynamic and adiabatic wall-temperature measurements, since

the ERI,} for sand rou7hneEs could be useda The v-groove

roughness of _Kolar was found suitable for the heat-transfer

runs ',ecause it satisfied the conditions. In addition it

was easy to produce and control during the making.

The emery roughncss used had an average height of 0.0082

inch, and v-grosve rou,hness a depth of 0.014 inch.

The plate assembly

Li) Adiabatic plates

(a) Smooth adiabatic plate; .2, 3 ft. souare iperPex T

plate was used in the smosth-surface runs.

PresE,ur tappin7s and therraocouple junctions were fixed

at points along two radial linos as shown in figure 6.5.

The spacings of the pressure ta-spinr,s '-nd the thermocouple

junctions are riven in Apenddx

83.

(b) A 'pers-eex' sheet covered with 'emery' cloth was

used as the roug,h adiabatic plate. The plate had only

thermocouple junctions. A 5.9 inch diameter circular portion

in the centre was smoothed by filling. with 'Araldite' so as

to produce identical injection conditions for both smooth

and rough plates.

Both plates were insulated by a layer of glass-wool

applied to the back of the plate and held in place by a

sheet of expanded polystyrene. The glass-wool layer was about

1.5 incches thick and loosely packed; and the polystyrene

sheet was about 1 inch thick.

(2) Heat-transfer plate

The 2 ft. 11 in. diameter rough plate was of 'hard alu-

minium' and had a spiral v-groove machined on it leaving; a

5.9 inches diameter smooth area in the contre. Thermocouples

were embedded in the plate along a radial line at regular

intervals.

An 'Iso--pad' heating element was used for heating the

plate. The heating pad was designed to give unifonu heat

flux.

The rough plate was bolted to a rectangular °Sindanyo'

plate with the 'Iso-.Lad' sandwiched between them. The back

of the 'Sindanyo' plate Was heavily lagged with glass wool

held in place by a shallow rectangular metal casing. A 1

inch. `;.]..,k sheet of expanded polystyrene *as taped to the

back of the metal casing to provide additional insulation.

82

(3) Positioni-ir, of the plates

The plates er,e held in position by a jack which was

mounted on a sioted-anr:le framework. This framework had

foot--scr&vis which permitted adjustment of the f sauareness t

of the plate withh respect to the delivery pipe. Sauareness

was tested by means to be described in section 6.3.

The slot height was varied by moving, the plate assembly

by means of the jack.

6.2 Instrumentation

Velocity profiles were mf-asured with a flattened Pitot

probe of height 0.0042 inch. The accuracy of the probe was

checked by comparison of measured dynamic heads with those

indicated by a large (0.08 inch) circular Pitot probe when

placed in the same stream Zo significant difference was

observed in the range of velocities that were to be measured

in the wall-jet. Although this check was made in pipe-flow,

no check was possible in the wall-jetflow. Although various

corrections for displacement effects have been suggested in

the literature, as enumerated by Bradshaw and CeeL6. , no

corrections were made to the readings as an insufficient

correction would be worse than no correction at all.

The dynamic pr,,- sures greater than 4.5 inches of paraffin

were measured with. a vertical U-tube manometer. An inclined

manometer was u4ed for dynamic pressures in the ranF'e of 2

to 4.5 inches of paraffin and a micro-manometer was used for

heads less than about 2 inches of paraffin.

83

The manometers were filled with paraffin. The possible in-

accuracies in the riding of these manometers are listed in

Appendix 5.

The traverse unit shown in fltrure 5.6 was used to hold

the Pitot probe in -any' required position. The probe was

actuated by means of a micrometer head; it could be located

with an aceurac:y better than 0,005 inch. The distance of

the measurin7'station from the slot was measured using a

steel-rule graduated to read 0,02 inch.

Calibrated copper-constantan thermocouple wire was used.

In the case of smooth and emery rough plates., the thermocoup-

les were placed in holes drilled in the plates_, so that they

were flush with the working faces of the plates and held in

place with 'Araldite r .

The aluminium plate had the thermocouples inserted into

holes drilled into it from the reverse side to within 1/64

inch of the troughs of the roughening grooves, and held in

place by wedges of copper wire. The wires from the junctions

were led out between the heater pad and plate at ri,Tht angles

to the radius passing through the thermocouple wells so that

the wires remained isothermal for some length from the junc-

tions. The main series of thermocouples was placed alonc: one

radius of the plate whilst others were placed at known points

so as to enable the symmetry to be checked.

A thermocou;-de junction mounted at the end of a tube

fixed parallel to the one carrying the Pitot probe in the

811.

traversing unit shown in figure 6.6, was used to measure

temperature profiles.

La7ooratory standard instruments were used in making

measurements of thermocouple c. m. f.'s and power input to

the heater.

Details regarding all the items of equipment are given

in appendix 5.

S.3 Operating procedure

Setting up of plates

3efore the actual running of the tests an impotant phase

of the work was the setting up of the plate. The aim was to

obtain a required slot height and have it as uniform as

possible, over the whole periphery of the nozzle.

For the applicability of the theory, it was necessary

for the flow to be axioymmetrical.

In the first instance, the lip of the nozzle flange and

the impingement region were made free froR unevenness.

To put the plate in position the frame which was to carry

it was detached from that carrying the'u-elivry pipe 'After

the plate was mountad, the frames were brought together but

not bolted. The plate; was centred laterally with respect to

the flange; meanwhile, care was taken to see that the slot

height at the ends of the horizontal diameter of the flange.

were the same. Then the foot-screws of the frame carrying

the plate were adjusted to centre the plate finally. Unifor:1-

ity of the slot heic:ht was checked by the use of slip gauges.

85

The frames were then bolted together and they did maintain a

uniform slot height satisfactorily. Subsecuent variations

of the slot hei,7-ht wore effected by means of the jack.

The symmetry of the flow was checked by measurin- the

velocity at many stations around the slot and equidistant

from the axis, and was found to be satisfactory.

Velocity profiles

The fan was started and the jet was allowed to work for

about 1 to 2 hours for the apparatus to reach steady tempera-

tures; longer times being allowed for smaller slots. This

was necessitatecU firstly, by the desire to avoid any uncer-

tain thermal expansions from havinc- any sin-nificant effects

on the measurements; and secondly, by use of the warmth of

the air to produce the aiabatic wall temperature rise.

When the temperatures had become steady, the velocity-

profile measurements were comi,enced, the first station beine.

at the slot. The next two stations were chosen at roughly 5

and 8 slot heir hts downstream, and the remaining' stations at

progressively increasing spacins between each other.

Altogether about 8 profiles were done for each slot settinc7;

and each profile required about l2 hours for completion.

every profile the Pi tot aerobe was progressively moved

away from the position of contact with the surface. Break

of contact of the probe with the surface was established as

follows. To start with, the probe was pressed against the

su-rfacedthe micrometer head rotated backwards slicrhtly,

E;S

to remove any back-lash. Then the micrometer head was rotated

to displace the prbbe by 0.001 inch each time and thc mano-

meter reading noted. During this the reading would decrease

slightly and stay at the minimum value until there was a

sharp rise which indicated that the probe was on longer in

contact with the surface, The 0.001 inch movement was conti-

nued for about 3 more readinrrs; thc actual zero readin to

the nearest 0.0005 could be found by plotting the manometer

reading- and taking the meeting point of the horizontal line

through the points during cotact and a straight line drawn

through the few points obtained after 'lift-off'.

The profile was completed by taking readings with thc

probe at various positions with smaller spacings near the

surface and increased spacines once the maximum velocity

point was passed.

Adiabatic wall temperatures.

Adiabatic wall temperatures were measured on the smooth

and rough walls -qith the warm air injection through the slot.

The warmth was produced by the action of the fan and flow

through the pipe.

The temperature readinys wore t -cn during the last

velocity profile measurement of each run to ensure that

the temperatures were as steady as possible, allov,ihr for

ambient temperature fluctuations.

heat transfer from v-grooved

For a given slot heir:ht, the surface temperature distri-

87

butions corresponding to various heat inputs 'Tere obtained,

starting from zero input to about 750 watts.

In this case a limited number of temperature profiles

were lAeasured in addition to velocity profiles.

6.4 Data reduction

The readinas obtained -;ere reduced to velocities, tem-

peratures and distances etc. Ly means of fromulac listed in

appendix 6.

6.5 Review of the reduced data

Some interesting observations that should be made, altho-

ugh no specific use has been made of them, are:

l. The variation of velocity profiles at the slot due to

changes in slot height, shown in figure 6.4.

Since it was not feasible to change the nozzle-flange

shape to suit every slot height in a regular way, the same

nozle was used so that any influence on the flow development

would -be systematic.

2. A static pressure distribution on the smooth surface; a

typical distribution being as shown in figure 6.7.

All the differences in static pressure were ignored in

the reduction of data and in the making of theoretical

predictions.

3. The indication of negative Fitot heads as the probe was

moved away from the surface and near the 'edge' of the jet,

the reference pressure being atmospheric.

This effect had been noticed by Bradshaw and C6C

88

as Tell.

VelocityEpofiles

A set of measured. velocity profiles on the smooth surface

is shown in fic;ure 6.8, and a set for the rough surface on

figure 6.9. They both have the same rreneral a-opearance.

Adiabatic wall temperature

DurinL; the measurements it noted that the tempera-

tures of the. air in the pie and at the sta-nation pint were

higher than the surface temperature readin,i;s just inside the

slot. The surface temperatures remained apprecialy constant

for some distance do-,:nstream of the slot as shown by the

adiabatic wall temperature distribution 7iven in fiFure 6.10.

Since the drop in temperature between nozzle eatry and exit

would be due to some uncertain heat transfer mechanism within

the nozzle, the slot temperature used in normElisin!7; the a

adiabatic wall temperature rise was the mean of the gs

on either side of the slot.

Temeraturc-prpfiles on heat transfer surface

h set of measured temperature profiles is shown in fi-

gure 6.11. The followinr;7 observations can be made:

There is a region where the jet temperature is higher

than that of the surface, and heat transfer occurs to the

surface. Downstream of this region, the direction of the

heat flow is reversed. Due to the fact that the metal plate

heavily insulated at the back this reversal of flux is

observed even in the case of n7) electrical heating.

The surface temperature distribution correspondinc to the

profiles in figure 6.11 are shown in figure 6.12. Two curves

are shown, one corresponding to the temperatures indicated

by the probe thermocouple whilst in contact with the surface,

and the other to the temperatures indicated 1c7 thermocouples

embedded in the wall. The two curves cros each other, nd

the paint of concurrence corresponds to the reversal of wall

heat flux.

90

CHAPTER 7

APPLICATION OF THE THEORY

Tel Introduction

The experimental programme reported in the previous

chapter provides many interesting rlications of the general

theory outlined in Chapter 2

The theory has ben applied 20 far to systems with

parallel flow; but the present experiment provides an appli-

cation to radial flow

i icon and Escudier [52a] report on the successful pre-

diction of the hydrodynamics of a parallel wall-jet with a

finite main-stream; whereas, the wall-jet reported of here

is in stagnant surroundings.

Something entirely new in the present work is the attem-

pt to incorporate roughness effects into the hydrodynamics

and heat transfer calculations. Schlichting reports of a.

procedure to estimate the local- and total-skin-friction of

rough surfaces, devised on the hypothesis that all rouryhnesses

have E(..Rr characteristics of the same form; but with the aid

of the present theory ve are able to allow for different

roughness typos, and variations of the flow itself. In the

present experiment we have the interestirw feature of the

variation. df the roughness effect; the velocities are gene-

rally decreasing and the layer thickness increasing with the

increase of x, which would result in a more rapid decrease

of Rr than in the case of simple boundary-layer flow.

91

7.2 The hydrodynamic problem

A prerequisite of any successful application of the

theory is the knowledge of the values of the constants, in

the auxiliary relations, which are valid for a <oiven system.

The entrainment constant

The entrainment conot'nt is :Thrived from the velocity

profile. data accordinr to the procedure outlined in appendix

7. Its value for the case of rourh walls is found to be

approximately the same as that for smooth wails. Like typical

smooth wall data, as shown in the paper Tby Nicoll and Dscudier

:52a1 there is wide variation in values. This is probably

because of the uncertainity in the velocity intorals and the

necessity to find their derivatives.

In any case the test for suitability would be the

predictions of jot behaviour.

The mixingr-lenrrth constant

The mixinc7-lenrth constant .1 has been defined in sub-

section 2.4.

From the definition 6, the velocity profile assumption,

the mixine length assumption (2.4-1) and the definition of

mixinr length (2.4-2) it follows that,

1, zE,

(7.2-1)

For the usual values of Al• 0.08), over approximately

4/5 of the boundary layer, has the value of 11.. Thus,

92

s i11 f4. x ... (7.2-2)

where 'f' means 'some function of

For a wall-jet, one of Spalding's statements [831 can be

modified to give,

2 47,E 4

( 7 . 2- 3) z

A comparison of (7.2,2) and (7.2-3) shows that, if the

entrainment constant for radial flow is different from that

for parallel flow, then, to bring about a corresponding change

in VzE3 9 we should change the value of ..X1 in the proportion,

1 C2 for radial flow 1/2

L C2 for parallel flow

Nevertheless, the value we obtain for .1,1 by the above

procedure would only be a rough estimate; for the entrainment

constant is approximate to begin with, and the equations

such as (7.2-3) too are approximate.

Initial values

The region of af,plicability of the theory is apparently

that downstream of the station where the mixi -layer origin-

ating at the uper lip of the slot and the layer sheared by

the surface, join up. Profile 2 on figure 6.9 is one ob-

taininF7, very close to the initial station.

The following initial values have to be supplied to the

computer programme ,Jhoch is used to solve the differential

equations by a Runge-Kutta procedural

93

1. maximum velocity, umax;

2. distance from surface to where the velocity is

half the maximum value, y1/2;

3. zE; 4 --e'.

The first two are experimental values and the other two arc

generated from these using the velocity-profile assumption

and the E(-Rr} for the given roughness, by the procedure shown

in appendix 8.

Basically the same programme is used as for a boundary

layer with a finite mainstream velocity, the modification

being effected by inserting a very small value for this velo-

city (--10-6 x slot velocity)

Both methods of solution, namely, the 'entrainment meth-

od' which involves the solution of equations (1.3-20 and

(1.3-25), and the 1-6 - method' which involves the solution

of (1.3-25) and (1.3-26), were applied. Details of the method

of solution arc given in appendix 9 and a graphical compari-

son of the methods in figure 7.1.

It is clear from figure 7.1 that both the 'E - method'

and 'entrainment method' can give comparable predictions.

During the development of the general theory many points

such as the choice of various constants were left open so that

the predictions could be manoeuvred to give a reasonable fit.

The important choices which had to be made were those of the

94

mixing-length constant Al' the entrainment constant C2 and the

form of E(.1R.r}. Decisions regarding the best wake-profile and

the procedures for calculating -6, or the best set of initial

values (appendix 8) were trivial because reasonable changes

of these caused only negligible changes in predictions.

Figure 7.1a shows the effect of changing Al, and 7.1b

that of changing C2' on the predictions. Some experimental

data are also shown on these figures. These comparisons

enable us to pick out,

Ai = 0.139

and.

C2 = 0.039

as being slightly better than other values; they are higher

than those operating in a parallel flow situation.

Figure 7.1c shows that the incorporation of the rough

surface E variation is an improvement, because the prediction

of y1/2 is substantially better.

(7,rowth and velocity decay of wall-jets on smooth, emery

covered and v-grooved surfaces respectively are shown in

figures 7.2, 7.3 and 7.4.. The full lines are predictions

made by the use of -6 - method; the circles and crosses are

experimental values of umax/uc and y1/2/yc respectively. The

entrainment method also gave predictions of comparable accu-

racy; these are not shown. Other pertinent details are that

umax and y1/2 were used as initial values and that the E(.R1}

used was that for uniform sand grain roughness.

95

Agreement between theory and experiment are satisfactory,

except for the smallest slot height of 0.031 inch of the

'emery roughness series' (figure 7.3b). With a slot as small

as this the slight non-uniformities which are negligible in

the case of larger slots, would become prominent.

7.3 Comparison of velocit-L_profiles for smooth and rough

surfaces, and their development

The velocity profiles obtaining at the same distance

downstream of the slot with identical slot conditions are

shown in figure 7.5, plotted on co-ordinates u/umax and

Y/Y1/2*

A basic difference is revealed in that, although the

smooth surface data are well fitted by a 'log + linear' pro-

file, the rough surface data seem to favour the 'log + cosine'

profile rather than the closest 'log + linear' profile, both

of which have been drawn on that figure. All other cases

verified this finding; figures 7.6a and b showing two more

cases

This4perhaps an indication of an influence of the surface

roughness on the wake component. Whether the.roughness impo-

ses a radically different turbulence pattern has to be resol-

ved by a detailed invostigaLiuh using hot-wire probes.

The development of profiles on smooth and rough surfaces

are indicated by figures 7.6a and 7.6b respectively. It can

be seen that the value of e' of the fittin,r profiles varies

over a wider range in the case of the rough surface than in

96

the case of the smooth one.

A relation between shape factors derived by Nicoll and

Escudier[520 can be simplified for the case of wall-jets in

stagnant surroundings to,

H32.H12 = 1.10 (7.3-1)

The shape factor data from the present experiment Fives the

value of the above product as,

H32.H12 = 1.096 0 06 (7.3-2)

The value of the product for profiles close to the slot,

say within about 10 slot heicrhts, is sirrnificantly below 1.1;

and this appears to be characteristic of the undeveloped

velocity profile, i.e. one which has not reached a shape that

can be represented ell by the assumed velosity profile.

7.4 Estimation of

The variations of the intevIal,

! G

R: u3 dy

such as thoes shown in figure 7.7 enable the estimation of

the dissipation inte;7ral by the application of equation

(1.3-6) in a manner like that shown in appendix 7.

For the smooth surface we obtain the values,

97

Vz3 yo in.

.0077 .20

.0078 .22

.00S6 .135

.0070 .065

and for the emery-rough surface,

s/z3 YC in.

.0076 .20

.0080 .12

.0078 .29

.0063 .03

.111 the above values are higher than those for parallel

jets quoted elsewhere; this is consistent with the high 1,- 2..1ue

of mixing-length constant used in the theoretical solutions.

7.5 Adiabatic wall temperature

When the wall is adiabatic, the integral 9 conservation

equation, (1.3-30), simplifies to,

R R(p,1 constant ... (7.5-1)

This equation can be simplified after substitution of express-

ions for R9- and Ig,i, together with initial conditions, 1

as shown in appendix 10, to give,

98

= (Ps - Pc)/(Pc - PG)

RoRa,ozE,0{(1/3 - 1.5/r) + (1 - 1)(3/8 - .8945/4V1

R RG zE rA(1/3 - 1.5/e) ( 1 - ( 3/8 - .8945/e )_.] ... (7.5-2)

where subscript 0 denotes initial conditions.

P is referred to as the 'thermal effectiveness of the

surface'.

Out of the quantities R, product RG.zE and the term in

sauare brackets, the radius R has the widest variation; there-

fore the choice of the initial value R0 is very important.

For making the predictions shown on figures 7.9 and 7.10, the

initial stations were found by trial. The distances of the

initial stations from the slot, in terms of slot height, are

shown in figure 7.11.

A theory developed 3y Cole FlOal for predicting the ini-

tial region lenf7ths of parallel wall-jots in mainstreams with

finite velocities, indicates an increase of initial region

length with increase of slot Reynolds number. In the present

experiment, increase of slot Reynolds number corresponds to

increase of slot height and therefore we should have the

opposite trend to that shown by our data. This contradiction

should not, of course, be taken too seriously, because the

slot in the present case is circular and the flow is spreading

radially. Also reported in Cole's paper are the experimental

data of Kuethe, for a parallel wall-jet in still-air; his

99

value of x0 /yC is approximately 12.5. If we extrapolate our

values to zero) slot height, then we would be approaching,

mathematically, the case of a straight slot; and then we ob-

tain a value of approximately 12.8, for xo/yc, which is remar::-

ably close to that from Kuethe.

The initial region lengths on the emery-rough surface

also show a systematic variation with slot height. The

smooth surface is seen to have larger initial regions than

the rough surface, for the same slot conditions.

7.6 Heat transfer from a rough surface into a wall-jet

The heat transfer problem was a rather complex one. ri'be

complexity was mainly brow -ht about by the fact that the .2:11f.f...

plate was a thick (3/8 in.) metallic one, in which radial

conduction effects were Quite important, as the theoretica,

predictions confirmed later. There was the further comT-)1:.cF__-

tion that the air in the jet was warmer than the ambient.

addition the surface was rough.

The heat-transfer system is shown schematically in

figure 7.12.

Heat-transfer from the plate is governed by equation

(1.3-29). The heat flux into the jet is not equal to the

amount supplied by the heater-pad because of heat conduction

along the plate; therfore an additional equation which

governs the conduction in the plate has to be solved simul-

taneously with equation (1.3-29). The heat-balance equatien

100

for an x-wise clement of the plate is,

kmt ar d( 4'S i - 0..

gh) J`; = 0 711 + R 1 ( 7.6-1) - S E Rc

P dx dui

whore J" is the heat-flux from the heater-pad to the plate. Jii

Initial values

As discussed in sub-section 6.5, the jet is warmer than

the plate in the region close to the slot and heat-transfer

occurs into the plate. Further downstream the heat-flux

reverses direction. The point at which the heat flux rever-

sal occurs is taken as the initial 1;oint for the inter-ration

of the equations. The temperature profile hare, corresponds

to adiabatic conditions; the. thermal boundary layer definitely

has a thickness equal to that of the jet, as shown in figure

6.11. Tho assumption of the theoretical temperature profile

enables the evaluation of the initial value of I8,1 (appen-

dix 10).

The other initial values required are those of the

surface temperature above the ambient, and the radial temper-

ature gradient of the surface.

Execution

Details of the solution are given in appendix 10. The

equations governing heat-transfer are solved simultaneously

with those of the hydrodynamicl problem, since the values of

zE, and RR are required. The s - method is used for the

hydrodynalHic solution, the details of which are 'riven in

appendix 9.

101

Predictions have been made of surface temperatures of

the plate for a given heat-flux from the heater-pad.

:omparison of predictions with measurements

The predicted and measured values of surface tempera-uul-.

arc shown in figures 7.13a, 7.13b, 743c and 7.13d.

Altogether, 12 luns each having given values of slot heig1-1

and heater input are shown; and the following, observations

can be made.

1. The predictions follow the data closely for some distcnc..:

downstream and then begin to deviate.

2. The deviations are so as to under-estimate the surface

temperature.

3. The range of agreement between predictions and the dat

increases of -,p7:C2 in nto

L PreCctions which are not included in the figures, made

without taking conduction in the plate into account

incorrect trends and had larger deviations from the data

those presented.

some rema:ks

In view of the fact that the heat-transfer system war

complex, we can say that the predictions are quite satisfee.

tory.

The inferiority of the predictions at the outer regio

of the plate may be attributed to the inexactness of the

heat balani:e c -i:lation of the plate, (75-1) which has bee71.

102

made one-dimensional, in order tc simplify the computation.

In the present form of this equation and the_mcan6 of solutic

applied, is implicit, that the plate is infinite in extent:

whereas the actual plate was finite. The finite size of the

plate may have introduced an amount of 2-diMensionality into

the conduction process, especially, close to the periphery.

Nevertheless, the inclusion of this over-simplified equation.

at least, did cause a si7nificant improvement. A further

indication of the inadequecy of the heat balance equation iL;

the fact that the predictions improve as the heat input is

increased; this increase would have the effect of minimisin

the importance of the y-derivatives of e within the plate,

which have been omitted.

7.7 An application of the calculation procedures to a

boundary layer flow

Details of the flow

Perry and Joubcrt[561 report an investicratien of the

hydrodynamic aspects of a flow over a roux-h surface in the

presence of an adverse pressure 7-fadient. They have present -

values of dran• coefficient deduced by a method which is an

extension of that proposed h ClauserE16] for flows past

past smooth surfaces. Also reported in their work are main-

stream velocity variation with x, shown in ficrurc 7.14 and

non-dimensionalised velocity profile data in a 7raphical for

103

The rouThness used by them was formed of rid:,-es of

square cross-section, placed at ri-ht angles to the flow

The roughness was made geometrically similar to that usea

thoore.

1-)etails of computation

The s - method of prediction was used. The E-function

was that derived from the drag data of Moore (rported in

1_10i)

Mainstream velocity and its fradient at a given valu

of x were provided by a sub-routine which interpolated from

a table of u values.

Initial value of zE was estimated from the velocity

profilerrraph,andobtainedfromthiszE and the initial

value of s. taken from the tabulation of data.

Comparison of prediction with erucriment

The predicted drag coefficients arc seen compared with

the data, in figure 7.14. Llso the figure is the

prediction made by taking the same initial values, but

assuning the surface to be smooth. The aarocmcnt of the pre-

dictions made usin the rouoh. surface E-function are much

rlo 6 1-,atisfactory than those with the assumption of a s000th

surface.

This exercise indicates to some extent the validity of.

the modification 1.:Iade by us to the theory, so c-.s to enable

the calculation of flows past rour,h surfaces. Interestingly

1014.

enough, the application has been to a boundary layer flow

with a pressul'e gradient.

105

C 0 NOLUD I E.G REIslARKS

Achievements

The achievements of the work which has been described,

may be recapitulated as follows:

1. An empirical relationship which links the resistance of

the laminar sub-layer to heat- or mass-transfer with the c.

Prandtl or hmidt number respectively, has been derived.

Chapter 2 contains the details of the Couette-flow analy-

sis as applied to smooth-pipe-flows from which the above rela-

tionship follows. Previous theories have been summarised and

compared with experimental data. This has enabled the deriva-

tion of simple yet accurate expressions which describe the

effect of the Prandtl/Schmidt number variation on the laminar

sub-layer. This is in the:form of the P(.0/60 } for smooth

surfaces.

2. A survey of investigations of flow in rough pipes, has

led to the recognition of some unifying features of the

interaction of rough surfaces with fluid streams.

The most important feature is the possibility of describe

in7 the fully doveloped velocity profile in steady flows past

rough surfaces by a two component expression as done by

Spalding in the case of flows past smooth surfaces.

3. Means of specifying the quantitative effects of roughness

on the processes of momentum-transfer and heat-transfer

through the laminar sub-layer. have been derived„eppeetially

with a view to its incorporation in a general prediction

106

method for flows past rough surfaces.

The influences of surface roughness on the wall-law

velocity-profile, can be described by the function EOR.1)-. No

attempt has been made to make a complete inventor of E‘Rd-

for all types of roughness. E(-1=tr} have been given for uniform

sand grain roughness and v-groove roughness in view of the

applications made later in the course of the work.

Yore notewor.thylpaints - areithat:

(a) the demonstration that strict geometrical similarity of

the roughness elements is not necessary for them to have the

same ..Ft.r)- characteristic in the 'fully rough' flow regime;

(b) the distribution of sizes of elements on a given rough

surface plays an important part in deter ining the behaviour in

the transition region between 'hydrodynamically smooth' and

fully rough operation;

(c) the mechanism of heat transfer for a rough surface can

be described more fully by the recognition of the gradual

transition to fully rough operation on account of the distri-

bution of element size.

4. The successful application of the general method of pre-

diction, to tit radial wall, jets on two types of rough surfaces

and also to a boundary layer flow reported in the literature.

These applications bear out the relevance of the modifi-

cations to E, for rough walls. Although the information has

been taken from one extreme, of pipe-flow, to the other, of

wall-jet flow, its incorporation has resulted in a marked

107

improvement of the -1;redictiol.s.

The SUCC3SS of the heat transfer predictions is indica-

tive of the fact that the present theoretical framework can

easily be built upon; in this case to take wall conduction

and jet temperature into account.

Limitations and further developments

The conditions of validity of the P-expression for smooth

surfaces, are given at the end of chapter 2. Improvements

of that expression can be done on the lines of making allowan-

ces fo large temperature differences between the surface and

fluid stream, and high diffusive mass-transfer rates.

Further work which should be done regarding flows past 1.5

rough surfaces Am as follows:

1. Collection of more drag data on flows past pyramidal and

other controlled roughnesses, to enable us to draw further.^.

inferences on the nature of transition from hydrodynamically

smooth to fully rough flow. Perhaps there may be no hydro-

dynamically smooth flow at all; but in this case there is

further need for drat data with transition form laminar to

trubulent flow.

2. Attention has bern drawn to the fact that it would be

difficult to interpret data from pipe flows with distributed

roughness, for use in boundary layer calculations. In such

cases direct experiments on boundary layers are necessary.

3. The flow past an abrupt change of surface roughness is

108

one that needs further investigation. Logen and Jones 1-01

report an expriment on such a change in a pipe-flow, and

present measurements of the variation in turbulence intensity

and velocity distribution. However, such measurements would

not be directly applicable to boundary layer flows due to the

differences in the way the tow flows are confined.

The values of constants X1 and C, have been determined

so as to make the predictions of velocity decay and jet-p-rowth.

agree with data,. Since our model is an ap-Qroximate one, there

is the possibility these values of constants are not suffi-

coently accurate for making predictions of other quantities

associated with the flow. This detail was not examined tho-

roughly in the course of. the•present work.

(3.4-3)

(3.4-4)

(3.3-3)

109

NOMENCLATURE

Symbol Meaning Eauc.tion

of

occurcne

, 111 9 '-2- 21- 3

-b

BIi

BP

Parameter specifying the wake profile (2.2-9)

(A = 1: linear wake, A = 0 cosine wake)

Exponents in the general expression (5a2-

for the rough surface P-cxpression

Coefficients in the recommended smooth

surface P-expression

Coefficient of (u+)b in the series ex-

pension or

Coefficient in roughness clement ='.izc

distribution

uantity used by Nikuradse in the

correlation of velocity profiles of

flow in rough pipes

Exponent of II+ in the first term con- (3.3-3)

taming u in the series expansion

for Et

Term used by Hama in the correlation (4.4-6)

of velocity data of flows past rough

surfaces.

-- transfer boundary condition Dare- (3.5-2)

meter

110

C2 Entrainment constant (2.571)

cp Sipcific heat of fluid at constant ( 3 . 2-6) pressure

D Pipe diameter ( . 4- 3) E Term used in Couette-floT: velocity-- (2.2-3)

profile expression

Ee value of E for flov past a fully- (A.3-5)

rough surface

E-

Value of F. for flow past a hydro- (4.4-7)

dynamically smooth surface

h12 Shape factor (1.5716)

H32 Ratio of kinctic-encr7y-thickness to (1.3-17)

to momentum thickness

11,12,13 Inte2,rals associated with the velo- (1.5-12,13

city profile and

0,1 Inte!Jral associated with the Sr and (1.5-28)

velocity profiles

J" Flux from surface into fluid stream, (1.5-26)

associated with property 9

jE

Electrical power input to the heater

pad

Heat-flux from heater pad to wall (7.6-1)

(2.4-1)

Abbreviation for a logarithm (2.2-6)

Quantity analogous to efor c---transfer (2.5-5)

111

m Nor-dimonsionalised mass-flux into (1.3-1C,

flid streaT from the surface

-rib -ion-dimonsionalised rate of entrain- (1.3-11)

ment into the boundary layer from the

mainstream

Mass-flux into fluid stream from the (1.3-4)

surface

Humber of rouHaness elements of a (4-5-2)

particular size in a riven sample of

elements

pp quid pressure in lbm-ft-s units

P Dimensionless measure of the addi-

tional resistance to 9-transfer due

to the laminar Prandtl/Schmidt num-

ber beinp: dif-ierent from that of the

turbulent fluid and the presence of

surface roughness

PM Talue of P of a smooth surface with (5.2-3;

fluid of the came Prandtl/Schmidt

number

Pr Pitch of two-dimensiinal rouc-Phness (4.10-1)

elements

R Distance of a -point on an axi-symmetri- (1.3-1)

cal body from the axis

R2 Reynolds number based on momentum (1.3-2c

thickness and mainstream velocity

112

R3 Reynolds number based on energy thick- (l.3-21)

ne3s and mainstream velocity

-D Reynolds number of pipe-flow based on (304-2

pipe-diameter and bulk velocity of the

fluid

Critical Reynolds number of rouqhness c,c

(405-3)

elements, based on friction velocity,

(s Af( Ts/p

RG

Reynolds number based on boundary - (1..5-1a)

layer thickness and mainstream velo-

city

Rm Non-dimensionalised mass-flux within (1.3-19)

tne boundary layer

Rr Roughness Reynolds number based on (4.4-2)

element height and friction velocity

Rrl'Rr,u Roucrhness Reynolds number correspon- (4.5-11)

,

dine to the lower and upper limits of

roughness heights respectively

tt Reynolds number based on distance (1.3-22)

along the surface

R Non-dimcnsionaliscd c-flux within the (1.3-26) 9 1

boundary layer in the flow direction

s non-dirilensionalised shear-stres in (1.3-9)

the boundary layer

-6 Avcracrc value of s on a velocity (1.3-15)

basis

sc Non-dimensionslised shear stress on

the. surface if it were wholly covered

with active roFaness elements

sM Non-dimensionalised shear stress on (405-4)

en effectively smoth surface unfter

the: same h7drodynamic conditions as

for s

Eon-dimension-Used shcai C.6ress on (3,2-15)

pipe due to fluid

ss Value of s at. the surface (F. cf/2) (2.274)

Stanton number for c-transfer (2.3-28) 8S9c

Number c. rourrhnoss elements having a (4.5-2)

size greater than a given value in a

sample of rourrhnoss elements

Number of active elements in a riven -a

u ;'luid velocity in the direction of the (103-

mainstream

u Dui veloeit:: of the fluid in pipe

uC Velocity of air injection at the slot (

Mainstream velocity in a boundary la- (1.3-2)

yer flow

Dimensionless mca—lre of velocity in

a Coustte-flow analysis

Dimensionless measure of "culL (3.2-]6

in a pipe

114

uR

vC

x

xC

X

y YG

y

Yr,l'Yr9u

YR

zE

Dimensionless measure of centre-line (3.2-17)

vC.ocity in a

Velocity component in a direction ( 1 . 3- 1)

normal to the surface

Value of v at the 'edge' of the boun- (1.3-4)

dary layer

Distance along the surface in the ( 1 . 3-1)

mainstream direction

Height of slot opening

Normalised value of roughness height

(4.5-11)

Distance measured from surface

(1.372)

Thickness of boundary layer

(1.3-4)

Dimensionless measure of distance from ( 2.2-2)

the surface in a Couette-flow analysis

Nominal height of roughness

(4.4-1)

reneral value of roughness height in

(4.5-2)

a given sample of elements

Lower and upper limits, respectively, (4.5-2) of roughness height in a given sample

of elements

Dimensionless measure of pipe radius (3.2-1S)

in Couette-flo,:: analysis

Velocity in mainstream direction (1.3-7)

Parameter in the assumed velocity

(2.2-6)

profile

115

a Fractional arca of rough surface, (4.5-0

oc-mpied by 'active' elements

Coefficient in E relation for (4.5-1)

fully-rough flour

'Total' exchange coefficient pertain- (3.2-5)

ing to the property cp in a turbulent

fluid

.A u Term used by Hama in correlation of (4.4-6) tATs/p)

of velocity profiles in flows past

rourrh surfaces

Thermal effectiveness of a surface

((Ps - %)/(QC

t Ratio of 'total' viscosity of turbu:r.

lent fluid to the lauinar viscosity

Cu'- h Eddy diffusivities for momentum- and

heat-transfer respectively

ivTormalised measure of conserved pro-

perty, E (P - 93,)/(Ps - 9G)

Parameter in the Q-profile

Prandtl's length constant

Non-simensionalised mixing-lenr,th

Value of in ouier part of boundary

layer

Laminar viscosity

Total viscosity in turbulent flow

(7.5-2)

(3 . 2-10)

(3.5-1)

(2.5-3)

(2.5,3)

(2.2-3)

(2.471)

(2.4-1)

(1.3718)

(302-4)

1.16

Distance from surface, normalised by (1.3-28)

di-ision with yr

Density of fluid (1.371)

Laminar Prandtl/Scmidt number of (3.2-29)

fluid

of Total Prandtl/Schmidt number (3.277)

00 Value of of in the fully turbulent (2.5-2)

reqion of the fluid

ti Shear-stress in fluid in lbm-ft-s (1.3-2)

units

tiS Shear stress on surface (1,375)

L. conserved property (1.3-26)

Mixed mean value of (2 over the cross- (3.2-20)

section of a pipe

9 Hypothetical value of c corl-cspond- (2.5-1)

inF to 0 = GE

9G Value of c in the main-stream (1.3-26)

Dimensionless measure of c in Couette- (2.5-1)

flow analysis

Dimensionless measure of c in (2.5-1)

Couette-flows

(Ps Value of (2 at surface (2.5-1)

Normalised wake-function (2.2-7)

117

Subscripts

State which would exist at surface if the free

mixing layer component existed by itself

c The conditions which would obtain if all the

roughness elements were 'active

exp Values determined experimentally

Mainstream state

M The conditions obtaininp- if all the i,oughness

elements were 'inactive'

max Appertaining to the point where the velocity

profile has a maximum

State of fluid adjacent the surface of a pipe

State of fluid at the centreline of of a pipe

S State of fluid in a boundary layer adjacent the

surface

1/2 Appertainin,, to the point where the velocity

difference u - uG has half its maximum value

118

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47 Moody, L.F.: Friction factors for pipe flow; Trans. A.S.

H.E., vol. 66 (1944)1005-1006.

48 Morris, HAI.: A new concept of flow in rough conduits;

Croce A.S.C.E. Separate No. 350 (1953).

L.9 Hor.ris, F.H. and W.G.; Heat transfer to oils

and water in pipes; Ind. Emg. Chem., vol. 20 (1928) 23L1-

240.

50 huller, W. and Stratmann, H.: Friction losses in pen-

stocks; Sulzer Tech, Rev., No. 3 (1964) 111-120.

51 Murphree, E.V. Relation between he: .t transfer and f11 4;

friction; Ind. Eng. Chem., vol. 24 (1932) 726-736.

52 Neddermann, R.M. and Sherer, C.J.: Correlations for

the friction and velocity profile in the transition re-

gion for flow in sand rouhened pipes; Chem. Eng. Sci—

vol. 19 (1964) 423-425.

15

52a Nicoll, W.B. and Escudier, M.P.: Empirical relationships

between shape factors H32 and H12 for uniform-density

turbulent boundary layer and wall-jets; TWF/TN/3, Imp-

erial College, Mech. Eng. Dept., July 1965.

53 Nikuradse, J.c Stromungsgesetze in rauhen Rohren; VDI

Forschungsheft,.361.(19S3).

54 Nunner, W.: Warmeubergang und Druckabf all in rauhen

Rohren; VDI Forschungsheft, 455 (1956)

55 Owen, P.R. and Thomson, Heat transfer across

rough surfaces; J. Fluid uec,I., vol. 15 (1963) 321-334.

56 Perry, A.E. and Joubert, P.N.: Rough wall boundary layers

in adverse pressure gradients; J. Fluid Mech., vol. 17

(1964) 193-211.

57 Petukhov, B.S. and Kirillov, V.V.: The problem of heat

exchange in the turbulent flow of a liquid in tubes;

(Eng. Translation of Russian original),C.T.S. No. 613,

May 1959.

58 Pohl, W.: Einfluso dor Pandrauhicrkcit auf den Warmouber-

gang an Wasser; 1Jorsch. Ing.-wes., vol. 4 (1933) 230-237.

59 Pranati, L.: Eine Beziehung zwischen Warmeaustauch und

Str3mungswiderstand der Flussirkeit; z. Physik, vol. 11

(1910) 1072-1078.

60 Rannie, W.Do: Heat transfer in turbulent shear flow;

J. Aero. Sci., vol. 23 (1956) 485-489.

126

61 Rasmussen, M. and Karamcheti, K.: On the viscous sub-

layer of an incompressible turbulent boundary layer;

3UDAER(Feb.1965), Dept. of Aeronautics and Astronautic • .

Stanford University.

62 Reichardt, Die WarmeiibertraFunF in turbulenten Rei•

burmsehichten; z, anpcw. Math. u, Mech., vol.20 (1940,

297-528. (EnF. Trans.: Heat transfer throucrh turbulent

friction layers; KACA TM 1047, (1943) )

63 Reichardt, H.: Die Grundlagen des turbulentes WarmeUber-

gangs; Arch. Mesa -iiarmetechnik, vol. 2 (1951) 129-142.

(Thn . Trans.: The principles of turbulent heat trans-

fer; - in Recent Advances in Heat and Mass Transfer,

pulished by Mc (1961) 223-252.)

Reynolds, 0.: On the extent and action of heating sur-

faces in steam boilers; Sci, Papers, vol. 1, Camb. Univ.

Press (1901).

65 Rotta, J.C.: Temperaturverteilungen in der turbulenten

(renzschicht an der ebencn Platte; Int. J. Heat Mass

Transfer, vol. 7 (1964) 215-227.

66 Rouse, H.: Advance Mechanics of Fluids; -published by

John Wiley (1959).

67 Sacks, Shin friction experiments on rough wall

Proc. A.S.C.E., J. of the Hydraulics Div., HY5 - Paper

1664, vol. 84 (1958)

127

68 Sams, E.W.: Experimental investigation of average heat

transfer and friction coefficients for air flowing in

circular tubes having square thread type roughness;

ACA RM E52D17 (1952).

69 Savage, DX, and Myers, J.E.: The effect of artificial

surface roughness on heat momentum transfer; J.A.I.Ch.E._

vol. 9 (1963) 694-702,

70 Schlichting, H.: Boundary Layer Theory, published by

Mc Graw-Hill (1960).

71 Shubauer, G.B. and Tchen, C,Li.: Section B - Turbulent

Flow; in - Turbulent Flows and Heat Transfer, edited by

C.C.Lin, published by Oxford Univ. Press (1959).

72 Schwartz, W.H. and Cosart, Two dimensional turbu-

lent wall-jet; J. Fluid Mech., vol. 10 (1961) 481-495.

73 Sheriff, N., Gumley, P. and France, J.: Heat transfer

characteristics of roughened surfaces; Chem. Proc. Eng.,

vol. 45 (1964) 624-679.

74 Sherwood, T.K.: Heat transfer, mass transfer and fluid

friction relationships in turbulent flow; Ind. Eng. Chem-

vol. 42 (1950) 2077-2084.

75 Siegel, R. and Sparrow, E.M.: Comparison of heat trans-

fer results for uniform heat flux and uniform wall

temperature; J. of Heat Transfer, Trans. A.S.M.E.,

Series C, vol. 82 (1960) 152-153.

128

76 Sigalla, A.: Lcasurements of skin friction in a plane

turbulent wall-jet; J. Roy. Acro. Soc., vol. 62 (1958)

873-877.

77 Bleicher, C.A. Jr. and Tribus, Heat transfer in a

pipe with turbulent flow and arbitrary wall-temperature

distribution; Trans. A.S.M.E., vol. 79 (1957) 789-797,

78 Smith, J.W. and Epstein, H.: Effects of wall roughness

on convective heat transfer in commercial pipes; J.A.I,

Ch.E., vol, 3 (1957) 242-284.

79 Spalding, D.B.: A new formula for the law of the wall;

J. App. Mech., vol, 28 (1961) 455-458.

80 Spalding, D.B.: Convective Mass Transfer, published,-

Edward Arnold, (1963)

81 Spalding, D.B.: Contribution to the theory of heat tr

fer across a turbulent boundary layer; Int. J. Heat

Transfer, vol. 7 (19644) 745 -761.

82 Spalding, D.B.: A unified theory of friction heat tra_],:

fer and mass transfer in the turbulent boundary layer

and wall-jet; ARC Rep. No. 25,925 (1964).

83 Spalding, D.B.: New light on the kinetic energy dcfl-

ccip.ation of the turbulent layer; Imperial College, MccT1

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energy-deficit equation of the turbulent boundary layer.

AGARDoraph 97, Part 1, (1965).

129

84 Stamford, S.: The effect of roughness on the heat trans-

fer from a pipe to a moving fluid; Ph. D. Thesis, Univer-

sity of London, (1958).

85 Taylor, G.I.: Conditions at the surface of a hot body

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86 Townsend, A.A.: The Structure of Turbulent Shear Flow,

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87 von Karman, T.: The analogy between fluid friction and

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88 Wasan, D.T. and Wilke, C.R.: Turbulent exchange of mo-

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Int. J. Heat Mass Transfer, vol. 7 (1964) 87-94.

130

1. AUTHOR (S)

TABLE 1

SU-MARY OF

THEORIES

In this table the

theories are

=anzrd in

chronological

order.

P - EXPRESSION

7 ASYMPTOTIC FORM

8 REiiARKS

2. R.L2ERENCE AND YYEAR

0t'

4 •

00

5.TRL'ILAENT OF .1 + 5sp/(4K-?-)

6. TVALUATION OF

131

PRANDTL, L. 1 j REYN oL CS 0.

1,4910 FG4-1 1901

0- 3

011 e

4

P • 9 ) 1 )

p

In these and some other early works thia expressions

attributed to the authors were implicit.

132

TAYLOR, 0 I .

u.i4" : t

k.t IA If (-5- .

I / 2 „ — -3 "1- — • • P

44. 1

- • -

P " .7

Taylor assured that for iph Reynolds numbers

and larTe pipe diameters:

sp, olt).ta.n1

Hence the, value 1253 in coluan. 6

133

.A/1 i../ J- P.;-! k) EE-• V L - V .

2 L.51] •.., /932 •

3

, +Yr : G+ = ___ _ jA =

1_ _4_ .1_ lky 1-- 3

wPieie F -(

-----)-- F -(-y + 6 ci) '"

, y , L,G ) _ -1

.,.... 4—AF3 L

i 4- (//

, ? _1 [ 2 - '-'./ - 1- '2--'11 Cd_ Y)

i —. ':-.., AT

- 0 ) 17 AP-7-, fr, • r)

i ,,,,eci epi, p ir, C., (/i .

'-- ,

I - t Y ( t 1 Lt ..., , - , F

., .1.- _ 07:.

p, 6., in 01 ,,,i) ai e i:v of I- ( 605,. 0-f PL) , 72..1.71-.

!4- 1

Lz- J

I ' 1 (5 IA /- c + fzi r)(._ ta i" I )

C civA kl VI 0 L 6 ..i.:- vi r i -I- t ,...: ,-t i'v,

-,:••q :A a -1:- lo vl ( :i : :2 - '

(-_-,---- Ives 61. toli{e (..) 4: 1\1,„ LA cz !..,

6 h c i,,,v,,, ok.

i-Lii..) • h (31,i 0 r

7 f) ,15.. o„, 2 /13.

/01„. ,,,- pic);-e e if./ 6, ) 1_ I-) <- . ,f - ;:t(- -1- o

Ex 100 fr, s , 0 .,,, r 0 r- .-1-- ; n(-- ''''' cf,:i ..• . i ( y ) -- :,- i. = ! --I- co ,-• , •• • - t

p r- ,)1:-_-,o :•-_, e a -.,. v,..;

134

, i Hoi:mi\NN1-7-, IT; t..1 1-1 M E VJ.

2 L3I] 4940 F? 1 1 9 3 8 - )

7

7 6-

...•

> 4J1 1 • qk..

hiiner e i,t+i -L.- A{ 6-4- -1/()

= 7 5- z.:, C

s vy-, 1 1.6. Vs t 0

1--) c..) 1: vy, o r,.r,s V,/ l t 11

A( a- > 9 ‘ v ),-,

ph

c; (j v- e e vv ; t in

'A L4 4,1 ti) 0 v S

expev,vv)evit—.,

4- 4

..) --....

-1/6 P ----- 7.54 C (-_1)

P -::

.......

5 03 ,

10 1)

A '

57,0'..00 12,...)01

— i

I

1 .A 1•Itirl. 0-i --: C;-.', 0 74-d,

? P ---, 7.54

C t, in ,-... -L Co vl Ac.)

y.

135

4 v 0 VI 1 < Gt v- v-y-p) V\ 7.

2 1 27j , '1939.

-3 5 y+< 30 .. (5:t -___ - V 7!-.7

• [Al.._ r.,- 1 fr , v _

a.- - ,--1

4- 1

,. i ti

6 P z.- 5 (1.5 -- - 1) IA [1 + —,;---; (cr. -

7 P --,7 5 G-.

8

136

1 REICHARDT i—f .

-,-, [621 19A a

3

(-)< t.A -̀• 2. ' 0- — 6- —

2 <1 1:.:,.. 1 — 1 I.. 5 — !4- - T. t __ j__

,)-- 0-0 L I S ...7., -- 1

1 S •

5 ‹... i A .. (5-t

4 i -cc, , 10 c k 6 I- 6 e I-- t cf vo I t-,, E

5 R.-

; r,;:. — '...._,.._ • (,,,. :- coe r.Q:_i. 1 .0v-, :—..-01-ca) _ , _.2...._ iAo, r? , 4-• I ' t A ----. +. -

•P I: ( 6- 1 " 6-- — I? T) 9 0 ( (:-•- I ) I ;

1 C a Irvk t_DCA V P (-A

1/, i 0 le, 1 T.;

-- , '

PC p .=_ ,c \

vE- v ‘, ;1,1( V Li v i a -I: I 0 vi OC 12c. \A/ 1 17 1,-, IR D

1: i:, 2; 1-.1 a t ‘A/ tA It.' (7' Q S 2 ( / 4,-, t

7 P—' 2C .1

P

137

4 MARTINELL), R . C .

2 Lif-5 1 ' 1947

3 = t _.. ,i-

,

+ /

+ ..-

3o < ./1- :

04 _ 2-

I

F-- l

c, 1, ,,- _-

)

5 Y I + :`--;

3.

Y \(1 - i' i—

,.2.

•0-" :- 2 • 5

05 \ '.‘_/

4

,-- _, , Ni.)....

P .:-_. p )--

I, 1 V) :...1 14., ';' Y.' l el. 7(

,,

7 _

e VO vl i< 0 v e-v1 0 in ' 5

4-_-1.ev-,doped -f-'0 ,,-

4-. .--

-lc t-i . 0 o i/ y ,),

1 (1. t o \,../

c

-C I 0 ,,,/,,

138

R .6,NNIE '././ . D .

7 1-6::-) . 1 , 1-, 19 C- 1

0< .y --'• < 27.

CA V‘ d ll+- -:,.

0.4-

s • _ I+ si r) In 2 ( l' -1-- 4/ iK i )

ci- - 1 , 1 ,..,• 1 2/ -frii

..-

UL) / ±. a/ 0 ,e1 (4 k 4 y -f ) Ai i<1 , w ( 1 k

_, AA' / 1,.. 0. 0

= 1 1 m v-l'H-- 3 ) J--- = 2 11< V-47

)

14 52 4/^1k,ci = .

- S

4

--.-.-- 1

/ 2_ /C 1./ i< •,1- ,

_ c, y Os' > 1

(2-

)<, ( 6-- 1 )

..-'

VT:::

1?̀ = 5. • -- .) y — _27

4% i< , ( 1 - (.'")

7 p ----7. 2 2 0 4177-6.-. ___ 4 c:, . A

139

I RE 'C. 1-1Al2 .DT , H

2 [631 ) • 1951

3

y+< 6 ', 6:t = 0 +'2•7x1,5. y-'43 ] ir L 4. / I 0-

1 Y+ y# >6 , 6- _ 1 ,.4_ 0.4cy +__11 kor,. v,,--7 1,c/---_-- _ k i -

1- ... -.1 ( 1:1 ) " A V '- q

c-, ___7 ____< .1- S 1

6-o

C I ' L`C./ 4 it trAok-/_) ( (p

(4, L- — 1.i )

711)2 IA+ =

J k — , ,'1

\ + 0 . 4 \-- y -4- — it - 0.tiltn -I 0

0 1nd v: ...--,-,. R D 15-p /2

4 vavi e S f r o vv; ( (It W 0 ti k 0 (1 , S /.7c. a x r ;. ) p-y, &C in va 1,,, ci 0.7!,9

S -;,•-; 1

17o v TR—, > .1.,

;_',"

r2113°°

1:• f I

b

A 0 lot) 100,)

k 1 1 .1 4 1 9 tc) . • , -) ' .56 -.• 117 - ' .- e)! ` '- ' i

II- / S. p —Y 5 8 6

140

I LIN , C, ,.,) Mw.iLTON Q.IN , 4 PUTNAM , G- • L .

2 - _ Lz . 2 ', . 19 5 3 .

''.

.- • , .- + '7, ' / 1

i --",,,, 4- 1 Y-2-- `t j

-

t --•

( -- l ;4 • 17, i - / L .) (7.6 1 4 • S )

2 y 4.

t - J_ I 11 . +0 v, .14 5 Uk ---- • ,. , , _.... ,) 4_ ,f•• _,- ,

3 L '2- 1 - 2...__ I . _ - 43 I _ ____Y 14•S" l.I.L ,s 4 ., _..< 30 . (7, _ 1 + .9..„--1 /[1_ ,_ t . t 1-1- - o • c.,' 59) 1_. ___ - . . , j ,.,- !T ,.

-,-..-L- c.-, -1- ()A 4 \

L4 -'- ,___ 4. ' 774- (v\ -4-- ÷ rj ' - ."- } ....

+ _ :: S 4 - S

(,,i _ -,

4

5 ^-' 1

; F 6-‘). 4_ ci, / 0 4.77 ,o_2/ _ — it,

": :.--i 1+ L7,. .L.4 (5-

p = 6, („4 (1 --1- o , 04 1 ci-

r_.- ) , 1 +---2--- J

Li 144 ' '; _I.- 2 - k i 's -• C, '/3 ' 2

', th.•:- I ,/4. • L... _

1 0 6_, I/1 1 71,,tJ; - i 14.' 5

—I- -4— fir3 -t a 4/1 /•— b , ..;

7 1 / 3

P ____. 1 7. 53 a-

8

m 0 ci ,4, ,o; -t .0 ,-, 0 -t vo,,, I <'0..., pi, O. vo 's ±(,' € 0 v y b•,,

i illif o ci v r, t I . 0 v't !di' ,2,:i ci ,-, vi• ..s: (...):..i . , - / A, ;,1t 0 tine

(60,-,',.-1,:,,i- s.rd, - to?ev

141

I DEI.,'"3LP-R , P. u.

2 [1:Si ' 19 SS

3

0 <-4-1 _ 7.- 4- 4 r

I - e X ri t A ̀ f L P (-t1-11, 1 4. 2- 6 ' • " , 41-1.

l'...:- • 12 4

t I 7 . 4 - - - -4- i'l CA-- -91

- '

V + 0 d ÷

l) -i- y ,..,

; -(- vi 7.- IA +' y 4. [1 - e x ('.:,

, -I- /_., , ) , u -+ , 2 , 778 uo ky / 2 b i -f-

1.4 4- = u+ C i y + , Z6'

-4 -- - ) )

( - v. ) LA .

, , ,/ 1

4 i

5 I ( ii,y, ID i i c ik )

6 vi LA vy\ e v't c 0 (

7 P - -- /4 8 • ::: ( - ---- 9

8

1/42

P E T i< V, 1-3 • 5 • 4 L a , v. s

I-57j 495z

+ • ; 0 Ly i; t 1)1 ; t I 4. Or4 ,, _ I,1:9y, (,, "Vf f) 0

- `, - I + 2 C

+ 7 + e -Y/t1 I I (---j

0 • 3

4

-z

IA - v‘ (! 4- ') • /4 y I . ') 1

# 2

/ 1

c (

e p

Aprie. oxivykOk- e(y P cc, be ext---)Keli7Ser_71 a5

-73 07_, 2 rs

7

1.3 (1:7 (-1 ov, VY) 0 CI .c a E. 0 o~ C t:

V ;Jo, .1 Q (1) ,riCo v irecte

cii~t ✓ tb Lm 0 v

143

M I L A F

1962

F_ , - 2 / S

I < i< V _i. ( i< IA ‘ it_ LA ) 1 -4-- -- Fe _._i _k- LA _ ___— — E — G 2 _

L 0.- -t. \

E 1

. 1 2 „,. „ + k, t 4.- ( ) — ) ( ic. !A j , —1 - .1/4: IA — -----

2 G

u+ >0

6'

+

= 0 • 4- = 9.025

11 !_,4 tin e :` ico I

7

Qc e E

( 1,1 5 [79 J

144

4 6- 0 Vt/ A 1? i /: Er!{ V . P . j Ci A r2 N E I? F .

2 L2 7 i 1 9 ( 2

-- _

4- JV 1011.1Ce

1 4 - , b

2 1 >: 1 0 7

U

0- - ..1 _

R ,.

I. l 2 1 -x k) CA f.71:-

/

4 1

4-- — 1 ti

7 P -1,- 6•07 3 _. ,

145

i< u TA T r-- i___ A D 7 t::--- S, cs:) ,

2 11_37j ; 19 6 2

--4,

0< y 1-< 6 ; 04. = 6- , ilo r G-<, 1 0 1-1- r_

6:4: 7: .00014g , +

/r • 1

,...•__ i

(,000148y+

(3- "o 4- 11 = y+

II .4_ , 0 2 yl-- 2) 6\./ 4<lo ', 6-i_ -_-_(0 20y 4-1)/

/L a- o-

tif _-_- C iv, ft-4- C1

:30 < \j+ : 6-74, 7-- tro

tA-1- -.:, 2 . 5 to y

4 —1)

\ -1

4 1

S 1

1-2.1 6 - 2 AI T. 6. '4 4_ -'i 2 Tr

1 +I 5.; 13-' - I -I-• 0 • 5 () 1,11

1 4 0

7

• .7__ 0-

p • I 6 1 4 ______-> .10

2

1146

1 W A.S 1-: N D. T. it w 1 L.),<F c • R .

[8E] , 1,6 .3 .

0 < Y+‘ 2a '

W;tk

+ ii

-4 +3 . 4..•)(.,xio y +15.1Sx(C6 y-1-4

1 -4- ! — ./.1. • 16)00 4 _y + 7:I- I IS% td by+ 4

1- 4. • 16)<to -4..-,,/- _,.. it5 . 1 ,.: Xis, -- '-',J-+- 4 — -4- - -1 0- a" _7-6 +

0 1-4.16,00-11- y4.3+-15.15X13 V 2.1

—4,i-4 , - 6 —i• 04A to

,, -I- 3. 07, )= 10 j-+ 5

I

--,•-: I

6 1,-, L.4 vv, e r 1 c.. '.i. I

---7 / P --?- i 7 - 2 7:7 ( 7 - - 2/3

8

147

1 RASiviUSEN, M • 1 l< A.RAMC !-tE 7 I i<

Pi2ESE N T PgroMAAE:NO TioN

2 . \ G 1 i ) 1965 ..._ 1,--)6 b

-I- 1 I -I- --- ' - 1 1,-117) Le- !J _-i-Z. ?...t-t- i

/-- — — 1 _

.

— — 2/c (-: ok fr. (Al- — 1 ,'• L 4 ' ) ol

4- 2 fi. . -f-

L_(1<!/- ) - --)

4 t 0.9 aind !, 0

S I I -4- 7 554 5p .

b . I vv., ..! Y l ( CA (

empirica (

1-7 cr N3izt. 11 _ 0

j _0.007.Lii

+ 0 ' 2 9 e ,c3

7 . 2/3

P —> 12. 7 C7 0_ , :-.S /Li

P *> 9,24 (-- 1 0--:,)

6.02 6.487 6.89 4.913 5.76 9.94 5.89

7.040 7.367

7.07 7.26

5.323 5.649

5.75 6.03

6.42 8.119 7.83 6.245 6.39 6.62 4.996 6.02 6.75 4.542 5.87 6.38 3.534 5.81 6.77 3.847 5.77

data for a = 0.60

TABLE 2

COLLECTED DATA

Barnet, W.I. and Kobe, K.A. [3]

1)48

s 3 A* so? S.t) (7), P lc, X12 Scpx103 igpx10

7.942 6.56 5.715 6.63 7.216 6.64 6.364 6.16 7.,432 6,87 6.472 6.18 7.515 7.06 5.451 6.48 8.141 7.38 6.412 6.67

7.180 6.62 5.895 6.78 7.036 6.67 5.040 6.92 6.532 6.63 6.208 7.06 8.237 6.62 6.664 7.32 5.943 6.78 6.700 7.65 6.268 6.39 5.787 5.38 6.328 6.18 5.763 6,63 10.494 6.02 6.736 6.22 7.156 6.32 6.112 6.30 6.172 6.32 5.691 6.49 5.580 6.17 5.799 6.30 5.823 6.28 6.172 6.13 5.542 6.52 7.108 6.07

above data for

6.04 6.06 6.13 6.25 6.34 6.92 5.91 6.12 6.12 6.34 6.49 6.73 6.02 5.94 5.89 6 .20 6.42 6.62

a = 0.76

6.413 6.75 5.813 6.38 6.545 6.77 6.858 6.89 7.290 7.07 6.65o 7.26 8.299 7.51 9.044 7.83 5.200 6.02 5.693 5.87 5.441 5.81 4.708 5.77 4.853 5.76 5.429 5.76 6.245 5.75 5.753 6.03 6.065 6.64 6.209 6.39

kripX10 Si" x 101 41cp ici0

7.492 5.332 5.344 5.747 5.669 6.077 5.080 5.236 5-441 5.621 5.837 5.921 5.320 5.594 5.897 5.356 5.633 5.969

4.613 6.06 1 5.238 4.414 6.13 I 5.72o 4.984 6.25 6.273 5.465 6.62 5.564 7.196 6.13 6.302 6.813 6.34 6.515 7.493 6.49 5.706 5.493 6.73 6.387

above

149

Scpx103

Bernado, E. and Eian, C.5. [14]

50,11k103 XpX)02- 550,,,x I x 02 a-

1.51 5.39 3.5 1.00 4.92 4.5 .52 5.90 25.2 1.58 5.32 3.1 16.66 5.76 3.7 .57 5.46 25.1 1.65 5.24 2.6 1.66 5.72 3.4 .46 5.62 25.7 1.53 5.33 2.5 1.76 5.62 2.9 -45 5.47 25.8 1.72 5.18 2.3 1.81 5.69 2.7 .50 5.92 25.5

1.66 5.08 2.1 2.01 5.69 2.3 .33 5.38 41.3 1.65 5.26 2.7 2.00 5.51 2.2 .33 5.46 41.9 1.62 5.27 2.8 1.50 5.94 4.9 .36 5.52 41.3 1.71 5.83 3.9 1.72 5.75 3.7 .34 5.64 41.3 1.75 5.73 3.3 1.77 5.69 3.3 .35 5.77 41.9 1.84 5.67 2.9 1.81 5.6o 2.9 .38 5.93 41.1 1.73 5..57 2.6 1.82 5.89 2.6 .38 5.75 40.9 1.79 5.51 2.3 2.08 5.52 2.2 .38 5.85 41.5 1.81 5.45 1.42 4.45 1.4 .38 5.95 41.1 1,60 5.77 3,3 1.30 4.56 1.6 .37 6.13 42.1 1..75 5.69 3.2 1.27 4.58 1.9 .0 6.29 40.7 1.85 5.63 2.8 1.20 4.65 2.3 .41 6.62 40.9 1.84 5.22 2.5 1.06 4.74 2.7 .65 5.29 9.8 1.91 5.52 2.3 1.09 4.86 3.5 .61 5.43 11.8 1.83 5.50 2.0 1.00 5.01 4.9 -57 5.59 14.6 1.99 5.48 2.3 1.30 4.68 1.9 .51 5.78 18.7 1.79 5.63 2.9 1.20 4.76 2.3 .45 6.03 24.8 1.53 5.78 3.9 1.22 4.77 2.3 .37 6.33 34.5 1.25 5.04 3.4 1.14 4.86 2.8 .66 6.56 18.3 1.18 5.16 4.4 1.11 4.39 3.6 .63 6.62 18.5 1.24 5.09 3.7 1.06 5.10 4.6 .59 6.01 18.6 1.28 5.02 3.3 .31 6.84 59.9 .54 5.78 18a7 1.31 4.95 2.9 .40 6.43 41.8 .52 5.63 18.6 1.36 4.89 2.5 -46 6.17 32.3 .54 5.43 18.5 1.44 4.85 2.3 .48 5.92 25.5 .51 5.41 18.6 1.45 4.81 2.1 .51 5.60 25.2 .48 5.34 18.6 1.49 4.77 1.9 .47 5.66 26.9 1.00 5.19 6.2 1.23 4.55 1.9 .43 5.76 29.8 1.12 4.88 3.6 1.20 4.69 2.1 .43 5.83 32.h 1.01 4.96 4.2 1.21 4.63 2.3 .41 5.92 35.6 .98 5.08 5.0 1.17 4.68 2.6 .37 5.04 40.1 .88 5.23 6.6 1.12 4.13 2.9 .35 5.18 47.0 .82 5.38 8.4 1.10 4.81 3.4 .32 5.32 53.0 .77 5.56 11.0 .95 5.20 6.2 .95 5.64 6.3 .72 4.83 6.4

1.09 4.87 3.6 .85 5.39 6.3 1.15 5.89 6.6

150 Bernado and Lian (contd.)

1.01 4.97 4.2 .77 5.21 6.3 .52 4.92 11.7 .94 5.08 5.1 .74 5.08 6.4 .49 5.04 13.8 .84 5.37 3.3 .78 4.98 6.4 .46 5.17 16.6 .76 5.59 1_5 .73 4.90 6.4 .46 5.28 19.1 .go 5.22 6.4 .71 4.82 6.2 .43 5.45 23.4

1.09 5.83 6.3 .90 5.21 6.4 .56 5.77 18.o 5.3 5.5o 17.8 .47 5.18 17.6 .43 5.02 17.8 4.9 5.27 17.5 .46 5.10 17.7

Chilton, T.H. and Colburn, A.P.f91

S 41),13 a S 0 ,-- 2 jsp)00

.00732 0.61 .007Z 0.72 7.03 681 st 711 il 6.92 641 ii ci 6.55 613 if 640 if 6.47

Svp04 „Fp x102.

Ede, A.J. [20]

Sg)F,X1104 krgpx tc? cs S(7.)0X f 04 ,rsp x 102 25.87 5.06 .706 33.3 5.94 .705 27.5 5.13 .708 24.95 5.07 .706 38.1 6.38 .705 29.3 5.54 .708 26.4 5.08 .706 42.8 6.61 .706 32.9 6.02 .708 29.5 5.48 .702 40.7 6.84 .706 33.1 6.24 .708 28.4 5.54 .706 42.1 6.86 .706 33.o 6.34 .708

42.6 7.24 .705 41.9 6.94 .706 36 .0 6.38 .708 22.8 4.91 .703 43.0 7.00 .707 35.1 6.76 .708 26.4 5.09 .703 27.1 5.21 .705 31.9 6.98 .708 30.4 5.54 .706 29.9 5.28 .705 26.2 7.03 .708 28.9 5.49 .703 31.9 5.35 .705 32.1 7.12 .708

34.4 6.03 .706 35.1 6.02 .705 40.8 6.83 .708 24.6 4.93 .705 37.7 6.54 .705 42.5 6.91 .708 26.5 5.13 .705 38.9 6.74 .705 31.4 5.44 .784 26.0 5.13 .706 41.4 6.94 .705 31.3 5.47 .704 29.9 5.52 .705 41.6 7.23 .705 32.1 5.47 .704 30.4 5.56 .706 23.6 4.85 .708 27.1 5.07 .704 31.3 5.58 .706 23.9 4.88 .708 30.3 5.53 .704 8.89 5.32 8.09 13.7 5.58 5.98 11.49 7.17 7.57 6.29 5.30 7.85 10,36 7.43 6.52 12.50 7.29 6.80 8.90 5.35 8.09 11.91 7.14 6.71 11.52 7.17 7.32

151 Ede (contd.)

9.32 5.53 7.84 12.49 6.74 6.69 13.89 7.26 6.25 9.23 5.49 7.92 12.01 6.17 6.85 11.41 6.66 7.70 9.52 5.41 7.71 11.82 6.03 6.79 12.40 6.77 6.81 10.00 5.73 7.82 10.66 5.54 6.95 12.31 1.03 7.01 9.94 5.72 7.56 10.04 5.51 7.10 14.83 7.07 5.63

10.94 6.09 8.81 8.07 5.63 10.5 12.02 6.63 7.56 10.85 6.06 7.53 8.48 5.61 10.1 13.26 6.71 6.45 9.43 5.55 8.31 10.47 5.54 6.99 11.07 6.27 8.19 9.53 5.53 8.03 10.71 6.39 6.39 11.69 6.42 7.73 9.46 5.54 8.23 12.48 6.43 6.91 11.30 6.24 7.84

9.70 5.53 7.96 12.92 6.37 6.27 12.30 6.33 6.90 9.73 5.53 8.19 8.46 5..13 7.33 11.55 6.22 7.46 9.68 5.49 7.68 9.63 5.47 7.01 12.60 6.28 6.36 9.74 5.52 7.75 10.47 5.98 6.68 8.00 5.13 9.47

10.01 5.46 7.05 11.06 6.32 6.75 8.60 5.23 8.95

9.80 5.50 7.80 8.61 5.12 7.57 7.79 5.16 9.87 9.93 5.54 7.13 9.72 5.49 8.07 8.39 5.28 9.60 9.49 5.53 8.26 10.25 5.61 7.76 7.66 5.36 9.99 9.60 5.54 7.28 9.60 5.41 8.09 =7.69 5.20 9.83 9.64 5.50 8.04 10.19 5.61 7.77 8.43 5.31 9.42

9.92 5.42 6.89 9.78 5.47 7.97 8.14 5.17 9.15 9.51 5.49 8.18 10.50 5.58 7.43 8.71 5.31 9.36 9.37 5.40 7.40 9.57 5.41 8.06 8.58 5.16 9.01 9.73 6.78 8.87 10.10 5.60 7.66 8.73 5.75 9.75 9.94 6.73 8.41 9.83 5.45 7.74 7.41 5.70 9.48

8.78 6.37 9.54 10.74 5.54 6.96 9.60 5.70 9.09 8.98 6.32 8.99 10.81 5.98 7.86 9.38 5.54 8.79 9.20 6.05 9.70 11.44 6.12 7.36 9.49 5.96 8.84 9147 6.03 9.15 10.73 5.97 7.74 9.68 5.51 8.34 10.61 6.78 7.56 11.50 6.08 7.13 9.07 5.80 9.79

11.36 6.73 7.53 10.54 6.00 8.11 8.89 5.63 9.50 11.59 6.62 7.55 11.07 6.15 7.88 9.88 6.15 9.30 12.29 6.52 6.81 11.51 6.69 7.74 10.70 5.96 8.91 12.86 6.62 6.40 11.75 6.82 7.29 9.22 6.30 9.99 14.21 6.41 5.05 12.13 740 7.04 9.12 6.11 9.90

12.57 6.72 7.01 14.31 7.15 5.79 9.82 6.30 9.47 10.23 6.10 9.10 10980 5.67 7.46 9.16 5.21 7.50 7.66 5.37 9.99 12.03 6.13 7.05 10.55 6.11 7.88 6.16 5.23 9.98 12.27 6.85 6.64 9.09 5.26 7.69 10.73 S.92 8.77 13.31 6.58 6.19 9.70 6.10 8.12

11.08 6.64 8.44 9.30 5.75 9.34 11.62 6.72 7.78 10.03 6.92 9.48 7.72 4.99 9.15 10,65 5.72 7.61 10.80 6.67 9.26 9.60 5.67 8,58 11.08 6.72 8.12 10.84 5.53 7.11 7.93 4.92 8.34 9.65 5.76 8.03 10.47 5.76 7.79 9.54 5.68 8.80 11.73 6.64 7.15

Ede (contd.) 152

10.51 5.59 7.58 8.14 4.94 8.60 10.69 5.97 6.91 10.06 5.70 7.77 9.56 5.70 8.86 6.64 7.19 9.88 5.53 7.53 8.22 4.96 8.64 10.45 f.76 6.90 12.61 6.41 6.53 10.4L 6.13 8.44 11.25 6.55 6.48 13.27 6.22 6.03 9.12 5.28 8.10 10.97 5.58 6.11

11.61 6.57 7.32 10.15 6.18 8.81 11.90 6.91 6.75 12.53 6.38 7.00 8.85 5.32 8.59 10.40 5.61 6.48 12.66 6.58 6.63 10.69 6.05 7.28 15.36 5.95 6.51 13.34 6.32 6.19 9.27 5.32 7.63 9.86 5.10 6.26 11.83 5.35 3.30 10.82 6.03 7.67

Gilliland, E.R. and Sherwood, T.K. 24

a 1.80 stp,rmo3 ispx1o2

1.875 Sy,pX103 ta-pX102

1.85 519,PX)03 Affp X102

2.16 pX 103 A& X 10

4.02 6.62 6.65 7.24 4.08 6.99 4.16 7.42 3.69 6.22 3.75 7.05 3.90 6.54 3.55 7.50 3.40 5.93 3.95 7.03 3.49 6.26 3.67 6.88 )4.62 7.33 4.03 6.33 3.37 6.04 4,34 7,63 4,62 7.59 3.65 6.27 4.12 6.99 3.79 7.07

3.95 6.78 3.68 6.40 4.24 7.56 3.56 6.69 3.47 6.18 3.16 5.92 3.51 6.35 6.17 6.38 4.35 7.34 5.07 7.73 4.43 7.16 3.86 7.10 4,35 7.34 5.07 7.73 4.43 7.16 3.86 7.10 4.31 7.33 4.20 7.33 3.88 6.94 3.43 6.35

3.90 6.80 4.35 7.34 4.22 7.00 3.21 6.16 3.78 6.8o 4.16 6.57 4.16 7.03 3.93 7.54

3.91 6.56 4.54 7.33 4.36 7.62 4.59 7.30 3.86 6.84

a 0.60 1.83 2,26 2.17

6.75 6.15 3.08 6.13 3.86 6.92 3.17 6.16 6.47 6.19 3.97 6.85 3.36 6.06 2.93 5.87 7.53 6.8o 3.47 6.33 2.79 5.83 3.17 6.33 6.94 6.51 3.22 6.00 3.53 7.41 3.33 6.63 0.86 6.62 3.08 5.77 3.75 7.68 3.57 7.C5 7.67 4.54 7.24 3.51 7.18 4.41 7.73 6.18 6.28 4.57 6.63 3.44 6.87 5.44 7.35 6.54 5.95 3.98 7.17 3.18 6.38 3.34 6.46 6.03 6.27 3.68 6.52 3.90 7.35 3.15 6.16 6.74 6.53 3.35 6.15 3.48 7.39 4.13 7.43 7.23 6.92 4.06 .21 3.67 7.42 4.07 7.44 4.47

153

Grele, M.D. and. Gideon, L. 28

5 xio ,r5-P xi 0 3

SAP x10 apx 102

O- 1

S mos Zpxio

99,? 1.60 5.49 4.104 1.68 6.01 5.066 2.18 5.60 4.739 1.56 5.48 3.818 1.81 6.13 4.900 1.47 5.49 3.998 1.76 5.48 3.818 1.70 5.86 4.840 1.69 5.71 4.833 1.55 5.44 3.518 1.64 0.35 6.599 1.32 5.70 4.)1)10 1..86 5.62 3.956 1.57 6.66 7.279 1.84- 5.84 5.065 1.63 5.62 4.157 1.58 6.35 5.426 1.45 5.62 4.030 1.61 5.19 5.395 1.57 6.54 5.351 1.37 6.00 4.941 1.59 5.13 5.646 1.99 5.50 3.958 1.48 5.85 3.880 1.50 6.00 4.887 2.25 5.49 3;872 1.44 6.13 4.888 1.56 6.22 5.208 1.46 5.41 3.316 1.21 6.34 6,414 1.44 6.43 5.897 1.23 5.61 3.080 1.31 6.57 6.994 1.60 6.43 5,707 1.50 5.63 4.020 1.58 6.63 4.553 1.26 5.90 5.288 1.62 5.52 4.214 1.32 6.11 5.509 1.63 5.60 4.931 1.40 5.81 4.577 1.90 5.72 4.777 2.22 6.00 4,697 1.55 5.72 4.632 1.53 6.19 5.003 1.8o 5,85 5.084 1.27 6.42 5.667 1.63 5.74 4,246 1.74 6.39 5,383

a = 1.6o

Jackson, M.L. and Ceaglske, N.H. 32

p -1032.87 3.11 3.06 2.52 2.82 2.565 2.49 .2:385 xidl 7.26 6.58 6.32 7.04 6.61 5.88 5.93 5.69

2.925 3.18 asp 6 x102 3.25 2.465 2.66

if§ X 102 6.24

o = 1.06

6.4o 7.18 7.26 6.80

StP,1"103 2.625 2.365 2.485 1.995 2.73 2.105 2.33 2.845,- Vg-ip x102 5.95 5.74 6.03 5.67 6.15 5.70 5.80 6.10

S4,P'r103 2.312 2.58 2.45 2.565 2.60 2.405 2.44 2.24 4i-e xIo2 6.96 6.87 6.96 6.52 7.12 5.95 5.83 5.62

154

Kaufman, S.J. and Iseley, P.D. DA

3 2 5,,, x IQ lipX I 0 cr

3 ?__ .px S io 4i-gpxio cr 3 .2

S pxo 4r.-57xio P, cr

2.36 6.22

1..94 6.22

1.35 6.22

2.84 4.35 9.82

1.86 5.54

1.54 5.54

1.24 5.54

3.40 5.25 9.15

1.27 1.16 1.02

5.11 5.11 5.11

6.35 8.35 11.6

Lin, C.S., Denton, E.B., Gaskill, H.S. and Putnam, G.L.

59,p x10 2

it, --p); 10 6 2

Sc, 100 ,.A.F,X 10 (5". 5 So,pX 41-s- px 4D2

cr

9.00 6.22 308 3.05 6.22 1380 5.19 5.54 462 7.05 6.22 L40 2.19 6.22 2170 3.82 5.54 900 5.47 6.22 615 1.69 6.22 3110 2.76 5.54 1380 4.14 6.22 890 7.82 5.54 325

Morris, W.G. and :.Whitman, P.H. 11.4.(A S,pX AgpX d Cr SpipNi04 ,,X-p X I 02 C%

x 04 118-p 102

25.2 6.22 5.53 5.66 28.1 5.45 5 -.21 21.0 24.6 6.19 3.19 5.99 29.3 4.17 5.52 31.9 21.4 5.92 2.84 4.88 5.64 20.2 2.49 6.06 67.o 20.3 5.72 2.93 4.82 5.48 19.5 3.27 5.70 45.7 18.6 5.70 2.985 5.79 5.69 19.0 2.42 6.03 71.0 17.5 5.56 3.02 4.99 5.25 18.5 0.72 7.44 126 16.7 5.46 3.03 4.54 5.27 20.1 1.20 6.98 108 16.6 5,45 3.02 5.29 5.18 19.3 o.73 6.59 83 16.2 5.35 3.04 5.10 5.17 19.3 1.49 7.12 87 17.5 5.32 3.09 3.73 5.40 36.4 0.94 6.38 150 15.7 5.31 3.19 4.54 6.08 17.6 1.5o 6.27 93 18.3 5.24 3.00 5.34 5.67 10.9 1.46 6.7o 97 4.13 6.91 41.3 5.12 5.50 12.6 1.00 6.72 165 4.22 6.72 40.6 4.54 5.7o 12.8 :1.17 7.08 132 4.37 6.43 38.4 4.28 5.92 16.6 2.10 7.06 59 .5 4.53 6.38 37.4 2.38 7.08 57.o 3.23 6.20 60.0 4.44 6.15 36.8 3.59 6.29 25.1 3.37 6.21 112 4.44 5.94 35.4 2.10 7.34 67.o 2.34 6.83 110 4.37 5.80 35.0 1.88 7.23 87.0 2.41 6.58 63.5 4.42 5.66 33.9 2.02 7.05 86.0 3.13 6.00 62.5 4.24 5.52 36.1 1.84 6.88 96.0 3.33 6.03 75.o 4.16 5.41 35.8 3.97 5.79 32.2 3.02 6.12 79.0 1.41 6.98 215 1-81 6.73 loo 2.88 6.21 100 4.50 7.14 26.7 5.61 5.15 15.5 2,49 6.35 96

Siege!

Sparrow

[7 5]

4 so -1

100 ,7 10 ,

&LA {u) Y4). AUTHORS y o y) 6-0 -E ; rainge

0-;) o'o-j--( 0-) .(-s f;"-Tiatr

C767- as ve r) by Teri Cr: s [13] (ros /610) be in9 a correct l'ov, -9-ac tow based o Sig erk m easu rerne,ils

— 7. 5

✓ovr expelfiv ental velociEy

profiles via vovx Karmun's relation

Velocity mec, s revvieinl-s of Sleicher

Slerchev TrI1,c)s

[77]

Tan smi.tayer 1- . .12.4 61- y+D -ex p(:124. u-4-y+)7 Y+ d y -1- (At A) v. 10‘. ')+ 4 26

-4- •124tA4y+b" -- ext., (124 ufy+)] -(.4 = • 36 y1-6 - Y f/Yit ) -I : yt;-3 A J1 N- -Foe 0 -'C Y1-‹. 26

tA1-1__ tin -Y-1.1- 12, 9491 3 2G y÷ ys.2-4- = novi-diyyiensiovIolisecl ec,c,filA S

--36 26 of P1.13 e .

1v stAbiayer

14+ 0 1 -F - exP(rAu+y1- )]

Ih -tu v btAlenE Cov e

U-4-= 2..S. yt + 5.5

6;=-- 1.2 21 Terikins W;l\1

y +6_142x 4-2 yl 'CM-Y+ ig5 a vy, viti ply fit,

-for y-4->42 ctor o-P 1.2

3 --10

Kays

LetA

[35]

(eE°/P)( Vito ev

y

1-t (A

- ex p (11 v11" )j •ol 5'4 -For y-÷<42

ay -F.

TABLE 3 : DETAILS o'F EXACT SOLUTIONS

Percentage difference PF - P

P1 100 PF

156

TABLE 4.

Percentage differences between P values for boundary

conditions of uniform wall-flux and uniform wall potential,

deduced from the data of Siegel. and Sparrow 751

0 0.7 10 100

RD

104 17 - -

5 x 104 22 -5 -.2

105 20 0 .2

5 x 105 21 5 .6

PF P - value at uniform wall flux

P - value at uniform wall potential

157

TABLE 5

Equivalent sand grain roughness height, of some naturally

rough surfaces

The equivalent sand grain heights of many common surfaces

are to be found in the literature; e.g. in the works of

Moody [47] and Schlichting

It is clear that there can be wide variation in the

values quoted; even so, their application would be an improve-

ment in the case of rough surfaces.

Typical equivalent sand grain roughness values of some

naturally rough surfaces are given in this able.

Material yr inches source reference

Galvanised iron 0.002 - .004

Steel (7.8 + 0.9) 10-4 i

Rerdn bonded graphite 0.0025 - .0027 1

Liquid film exposed to 0.0037 exp RY:112:F

gas stream range of data:

yp = film thickness P.m ry 12800 - 19 200 uE = mean velocity of

fluid in film

Welded steel

(a) new 0.0015 - .004

(b) uniformly rusted 0.005 - .016

(c) cleaned after 0.004 - .008

long use

[78]

and yEuvpE/117,^-1 127 - 508

;

• 'VELOCITY PROFILE DATA

Table 6a Run No: SH-D1

Slot height = 0.223 in. Smooth surface:,

X= 0. IN. X= 2.15CIN. X= 3.6031N. X= 5.62CIN. x= 7.620IN.

Y IN. U FT/S Y IN.- U FT/S r TN, U FT/S se 1K. U FT/S Y T. U FT/S C.0021 157.33 0.C326 191.55 0.0026 143.37 0.0021 89.64 :.3021 64.06 C.Cn31 161.83 G..".^36 201.57 Z.0036 149.42 0.,;(131 91.27 n.0131 66.12 0.:1,1 41 166.74 0.0056 230.56 v.0056 170.48 0.0041 97.54 0.Cn4/ 72.12

' ...0i61 211.76 n.c"76:248.50' C.C.076 182.52 0.051 133.26 0.Ce".71 85.80 C.0081 275.13 c.0396 257.69 0.0096 187.32 "..0^71 115.94 0.0111 93.08 0.6101 313.81 0.0116 264.88 G.0116 191.52 c.2n91 121.79 3.'151 96.47 0.0121 327.24 •3.0136 269.24 G.0135 193.82 C.0111 125.4: 7.1191 98.89 0.2141 332.68 0.0196 272.21 6.0195 195.64 0.3131 127.72 7,.;231 100.41 0.0171 334.56 3.0176 275.81 C.0176 197.18 3.0151 129.52 C.0271 101.82 -.9201 335.63 .1.o196 278.07 c.0216 231.31 0.5171 131.13 •-.'311 103.26 0.1231 335.63 :.3216 280.31 ;.0256. 202.76 3.0191 132.28 .-.351 1n3.91 C.C271 335.63 r.1236 281.59 :..3296 234.24 C.0231 134.21 "..%391 104;42 C.0371 335.09 ;.0256 282.22 :.0336 204.54 :.0271 135.63 0.0431 105.10 r.0521 332.95 - 0.276 283.80 1.0376 205.37 r..1.311 137.43 -.'.471 105.95 0.1!,21. 326.14 :.3296 285.06 0.0426 205.71 135.08 ,0.0351 .-.".;511 116.03 C.1521 319.75 -.0336 285.69 0.2476 2115.37 0.3391 138.40 :..:591 106.79 0.1771 314.39 0.-.376 286.31 0.1576 203.63 ':.n431. 139.36 2.0671 107.58 0.2021 281.25 .r:416 236.31 C.0776 2:0.56 :.f471 140.89 7.7751 137.37 0.2121 257.62 0.0466 285,:69 :.1026 190.13 0.0511 r41.26 :.0871 134.58 0.2221 119.04 :.0545 231.90 3.1526 169.95 3.0551 141.52 .:.1371 1'16.45 0.2371 -0. ,-..0616 277.42 0..21'26 140.83 7.;1591 141.77 0.1271 104.03

-..3756 265.56 0..2275 126.29 3.0671 141.26 p.2271 94.22 0.1256 214.06 0.2376 122.20 :.n751 140.76 0.3271 76.45 r.15C6 19r,.38 C.2776 19.61 1.0331 139.74 -..4271 65.10 0.17.36 162.76 0.2776 38.35 ..G951 134.02 3.5271 43.62 -.10- 6 143.31 :.2P76 95.6 3.1371 132.81 "..5471 49.62 0.2:5 13109 :.1^76 84.19 k.;1331 121.97• -,.5771 38.14 0.2c,:-.;, 74.49 „1,,/, 65.'4 9 -.25S1 10'+.42 -.4791 55.r,S 6.275:, 48.23 0.3676' 49.36 -.1151 88.04 "..4871 C.

. :'.21.26 22.18 f.317t, 32.50 ',.3111 .85.47 C.295 -0. ' L.4776 21.41 ::.3401 30.64.

.4126 C. C.3731 73.41 . 0.3c,-31* 66.87

::.4191 56.39 • 4191 39.25 .

• C.5391 31.-V-

__.... .1.5511 !".

p

Table 6b

Run No: SH-D2 Skit height = 0.20 in.

X= 0. J!:... X=L2.468IN. X= 3.40:1IN. T X= 4.480IN. X= 5.480IN. X= 7.422IN. X= 9.468IN.

Y IN. 1 U FT/S y IN. U FT/S y im, U FT/S y ul. U FT/S Y IN. U FT/S Y IN. UFI/S Y IN. U FT/S C.i.0O21 179.32 0.0021 179.23 3023 142.59 L..0321 109.31 0.0021 89.48 0.0021 65.43 (..,.;021 49.15 0.0031 205.92 L.....231 160.37 :...'..033 151.65 ;..'.:031 112.09 C.0031 92.61 0.0031 68.48 0.0031 51.48 0.8341 239.25 0.0041 182.33 x.0043 161.63 0.2051 127.03 0.0041- 97.45 0.0041 73.78 0.0041 51.48 %'.h."..201 262.83 2.;.351 138.09 '2.4053 169.74 .......361 135.80 0.0051 104.09 0.0':151 78.51 -%-0(151 59.25 U.0161. 273.45 0.0061 195.05 ,,;.0.953 169.74 C.0091 145.14 0.0061 110.15 0.0061 82.61 0.. 0061 63.22 . 0.031 311.44 1:.9281 219.:9 ....U073 181.15 ..-2121 150.14 0.0081 117.61 0.0101 91.14 Z.0111 .72.33 0.0.:.:1 324.77 C...:2121 231.71 2.0282 184.39 0.0151 153.39 0.0111 122.83 0.0141 95.32 0.0161 75.44 t..0121 337.31 0....:121 240.30 .1.0113 189.18 ....d31 155.48 0.0141 126.58 0.0181 98.14 C...1111 72.38 1.).4'141, 339.92 1U141 245.13 O.0133 191.43 ...i:211 157.16 0.0171 V28.71 0.0221 100.37 0.0161 75.44 .;..161 339.4'.: x..161 249.79 .......153 194.21 ,.....241' 158.72 0.0201 130.17 0.0261 111.49 0.0211 77.99 :.t..191 339.92 ..0.1.81 251.93 ..2.0 1 73 195-.56 ;...:.'271 159.38 0.0231 132.08 0.0301 102.31 0.0261 79.37 ...241 339.43 :.....:2'1. 254.39' ..0193 197.35 .....3.,1 161.93 0.0261 132.87 0.0341 103.13 6.0311 30.35 :.0291 338.88 .,' -:2231 256.47 :.::213 199.56 ..2331 151.34 0.0291 133.66 0.0391 103.94 0.0361 30.99 ,...341 338.36 ::.C2 S1 257-86 . ...13) 220.'0 4.2331 161.41 0.0321 134.90 0 3.441 104.74 0.0411 ' 81.79 ...'...441 337.10 ....k..331 238.39. .1.1.,?53 220.87 -.2361 161.98 0.0351 136.14 3.0491 105.34- 0.0461 82.32 ‘,..'54i 335.47 C ...381. 258.54 ....1773 201.74 -"I411 162..17 0.0381 136.29 0.0541 105.66 0.0511 32.69 . , (61 333.52 ....,431 257.51 :._3:3 222.61 L...:461 162.36 0.0431 137.06 3.0391 106.01 0.0561 83.22 :.:841 337.',.97. .....c31 253.69 ...:.353 2:2.96 .....1511 162.48 0.0481 137.57 3.0641 106.21 0.0611 83.37 ..1'.91 322.61' ,...631. 247..44 :,...4:'3 2'23.39 2961 162.21 0.0531. 137.73 0.0591 106.13 0.0711 83.74 1.1141 325.13 ,......131' 234.38 - 1..-.3 232.16 :....61' 161.79 C.C581 137.67 C.0791 116.01 C.0111 84.10 .:.1401 323.68 L,.1'81 203.17 :.:5'3 199.12 ....711 16C.,27 0.0581 137.67 0.0891 *105.85 0.0911 84.20 -1511 315.:6 2.1331 149.97 ..47.:3 196.'.1 .....311 158.52 0.0681 137.82 0.0991 105.34 0.10.11 13.79 ...1::91 317.::6 2.1531 171.79 ,.'_9.3 187.69 ..':911 155.98 0.0731 136.75 0.1241 103.36 0.1211 83.58 .16=)1. 21'7.23 .,.1631 162.21 ..11.3 176.71 ..1'11. 153.79 0-0881 135.83 8.1491 101.37 0.1711 81.26

...- 1791 283.21 :,..1131 '157.C. ,.14..3 162.65 2.1261 146.92 0.0931 133.97 0.1741 98.92 0.2211 77.50 -.1941 272.73 ..2^C1 122.53 '..19.3 !:r.9,7 :..1511 138.18 0.1231 130.17 0.2241 91.33 0.2711 73.63 :.1891 249.31 ;..2231 124.95 _.24:3'106.62 0.1701 1,9.53 0.1481 124.58 0.2741 84.62 0.3211 58.75 .1911 186.55 2.20). 85.12 .25.-3 1;22.33 ..i"11 12;.72 0.1931 112.40 0.3241 76.73 3.3711 63.22 "1021 157.2? 1..278: 65.E2 ..2,4;. 3 78.10 ,:...8:+61 11:,..1Q 0.2431 102.88 1..3141 75.13 0.3961 54.54 -.1131 1.=!..33 ..291 49.64 ...31.:1 63.47 ...2511 1,:.1.4.,: :1.29:11 85.32 3.:E' 741 63.36 7,.3 1 59.62 _.lax',. L 3.74 ,.-Y22.). .33.47 ..34., 49.79 :.;311 e2.E'l c.34n1 71.44 -..4741 60.35 i .4211 5.,..21 -19_1 54.'-' ..3231 3. -36,33 33.4'. ..-.311 61.9.1 0.3561 68.45 -.:4741 51.16 '7.4711 53.,.7 -.2-.t.1 -.3. I. ..3703 25.35 .7.311 5:..32 6.3931 57.49 0.5241 43.43 0.5211 50.02

---1 t1')2 2. -.: . 1,21 1 34.2,-; 0.4431 42.'70 C.5401 33.28 :.5711 44..21 .,.,111 32.77 0.4731 36.01 0.5741 2,1.37 0.6211- 35.12

. ; •

.4511 „ 0.40)31 -.5131

27-.74 22.40

0.6491 0-6751

'",-,..:5 12.94

2.7211 23.77 2.- 7011 16.77

. 7.51A1 D. C-6131 9.15 3.3211 10.04 0.5951 0. 0.8431 ...c,.

Table 6c

Run No: SH-D3

Slot height = 0.065 in: -1

X= C. IN. X= 0..5.150IN. X= 1.1CCIN. X= 1.5901N. X= 2.C6CIN. X= 3.070IN. X= 6.54718. X = 9"121"

1Y IN. U FT/S Y IN. 11 FT/S Y IN. U FT/5 Y IN. UFT/S Y IN. UFT/S Y IN. U FT/5 Y IN. UFT/5 Y IN. U FT/S

0.C121. 218.19 0.002r. 221.66 0.0021 179.46 0.0O21 139.67 0.0026 119.43 0.0021 80.59 3.0026 38.28 0.0028 23.49

0.0031 232.82 C.303L 222.47 C.0031 185.34 0.3031 143.47 0.0036 123.11 0.0031 83.77 C.0036 40.09 0..0038 24.51 0.0041 259.27 0.004L .225.26 0.0041 195.67 0.0041 151.95 0.3146 132.19 0.0041 88.28 0.0066 48.76 0.0048 25.78 0.0051 286.08 3.0:5L 235.73 C.C151 213.18 0.0051 160.54 0.0956 140.68 r.0151 94.99 3.0096 52.80 0.0178 30.77

0.0(171 341.48 0.306L .251.57 0.0.161 220.60 0.:061 170.26 0.0066 146.88 0.3071 104.60 0.0126 54.67 0.0108 34.17 0.0091 352.28 C.306L 281.66 0.0071 228.18 C.01171 176.98 3.0076 148.69 C.0191 110.95 0.0156 56.17 3.3138 36.28 r.0111 352.78 0.012I 301.57 3.0081 211.29 0.0161 185.38 3.0086 152.83 0.0111 113.59 0.3166 57.09 0.017 8 37.61 3.0141 353.29 '7.112/ 311.19 0.0091 235.13 0.3101 189.20 • 3.0r96 155.72 3.0131 116.13 0.3216 57.84 C.021 8 38.69 0.0191 353.2.9 0.0141- 315.19 0.01C1 237.40 0.0111 191.55 0.0106 156.86 0..3151 117.73 0.3246 58.61 3.0256 39.51

0.3291 353.29 (3.016L 318.35 C.0111 239.65 3.0121 192.95 :.0116 158.56 7.0171 118.55 0.0276 61.64 -,.0298 40.32 34391 354.30 0.018L 318.85 C.C121 241.51 0.5131 194.33 :.1126 159.68 0.0191 116.88 0.:3e.6 62.51 0.0338 40.97 0.0441 354.55 3.021L 318.85 ri.3131 242.99 1.0151 196.62 0.r136 161.35 0.0211 120.08 r.3336 63.09 •:.:.378 41.45 0.0491 355.35 0.0241 -515.47 1..1141 246.64 0.0171 197.53 3.0146 162.34 0.,.)231 120.66 0.0386 63.63 3.0428 42.00

0.0541 353.24 0.n28L 309.18.1.0151 248.09 3.0191 198.44 0.0156 163.54 0.0251 121.27 0.0436 64.21 3.0478 42.38 3.05431. 346.15 3.033L -299.19 3.0171 248.09 0.0211 199.34 0.0176 164.63 7.0271 121.95 0.0486 64.52 3.0'52 8 42.84 0.0581 316.55 3.053L .Z43.92 :'•7 191 247.36 0.0231 199.69 - .1196 165.71 3-1291 122.10 0.1536 64.94 1.3578 43.01 0.0591 286.39 0.1681 198.71 7.0221 245.91 0.3271 199.34 0.0216 166.78 0.0321 122.27 0.0586 65.14 . 0.0628 43.28

0.3631 226.21 7.:"731 275.81 .0.0271 242.25 0.0311 197.53 0..0236 167.85 1.0361 121.95 0.0636 65.25 0.0728 43.72 0.0611 165.17 0.078L a67.49 0.0371 229.74 0.0351 195.71 °.0256 168.06 0.0401 121.88 0.0686 65.28 0.0828 43.72 3.0621 99.03 .3831. Z51.23 C.'521 208.94 5.3431 190.14 (.0296 168.27 0 .0471 121.24 0.0736 65.10 3.1928 43.90 ,- .1641 26.70 - .393L 120.30 C.3771 169.72 C.:631 173.4C 3.0336 167.85 0.1571 121.16 0.0836 64.82 0.1028 44.28

:.1641 0. 3.1331. 93.66 0.3071 149.55 0.r.831 153.71 r.0186 165.71 :.)771 113.74 :.1C86 63.38 0.1128 44.7.6

. 3.1131 3.1161

59.77 0.

03971 r.1171

134.67 119.19

3.1331 - .1131

130.81 120.57

0.0436 (.0506

164.33 161.35

0.1021 0.1271

105.06 94.99

0.1586 3.2186

59.24 54.25

.0.1228 0.1428

44..'2 43.43

0.1271 86.78 1.1231 109.76 0.0636 154.57 3.1521 85.06 0.2586 48.69 3.1628 43.15 0.1521 47.63 ..1131 99.76 3.0806 141.95 3.1771 75.27 0.3086 43.48 1•.2123 41.45 0.1621 31.54 0.1531 78.45 0.106 125.99 0.2021 66.02 0.3586 37.37 3.3128 36.63

• - .1671 17.47 0.1731 57.22 0.1216 111.37 7..2121 61.65 0.1886 33.80 0.4123 31.17 3.1771 0. 0.1931 36.18 3.1406 96.47 0.2221 56.79 0.4086 31.60 3.5128 25.82

0.2031 23.82 0.15:6 81.14 1.2121 53.23 C.4336 29.21 3.5623 22.97 3.2281 0. 1.1716 73.43 3.2521

. 46.12 0.4836 22.77 3.6128 20.80

3.2-6 51.08 0.2771 35.79 0.5336 17.83 3.6678 15.94 0.2156 39.34 0.212.1 28.01 0.5586 14.13 3.5878 21.66 0.2256 30.93 ,- .3071 22.48 C.6336 0. 0.6123 20.89 0.21;6 22.46 0.3271 O. 0.6878 15.94

241.6 16.15 0.7628. 0. _ 6.2756 0,

Table 6d

Run No EH-D1 Slot height = 0-405in. Emery surface

x= 0.. In. X= 1.630IN. X= 2.400IN. X= 3.690IN. X= 6.080IN. X=10.060IN.

Y. IN. uFris y IN. U FT/SY IN. U FT/S Y IN. U FT/S Y IN. UFT/S Y IN. _pFT/S 0.0037 125.t8 0.0037 118.11 0.0037 126.91 0.0037 92.83 0.0037 69.29 0.0037 40.52 0.0047 129.96 0.0047 124.02 0.0047 130.39 0.0047 94.541 0.0047 71.06 0.0047 42.94 0.0057 136.63 0.0067 131.03 0.0067 136.43' 0.0057 96.76. 0.0077 74.14 0.0067 44.56 0.0087 190.45 C.0217 178.42 0.0087 140.94 0.0107 110.77. 0.0277 87.12 0.0427 55.22 D.0137 234.01 0.0417 217.70 0.0287 177.48 0.0307 134.49' 0.0477 93.59 0.1177 60.44 0.0187 247.31 0.0667 245.50 0.0457 195.70 0.0507 144.76 3.0677 97.13 0.1427 61.02 0.0237 254.•40 0.0317 250.91 0.0637 204.21 0.0707 149.62' 3.0927 98.93 0.1677 61.02 0.0337 259.24 0.0917 251.05. 0.0787 208.54 C.0907 152.00. 0.1227 99.83 0.1927 61.02 0.0437 259.93 0.1017 249.48 0.0987 210.25' 0.1057 151.29 0.1477 99'.47 0.2177 60.90 0.0637 257.57 0.1167 245.50 0.1187 206.39 0.1207 150.22 0.1727 97.78 0.2427 60.14 0.1137 254.40 0.1917 199.26 0.1387 201.12 0.1357 149.62 0.272.7 89.13 0:2927 58.35 0.2137 244.42 0.200 128.97 0.1787 181.47 0.1557 145.99 0.3727 74.38 0.4427 52.59 0.2637 237.03 0.2917 103.59 0.2787 111.05 0.2057 133.83 0.4727 55.54 0.5427 45.12 0.3137 213.34 0.3167 76.04 0.3037 90.67 0.2807 86.54 0.4977 50.53 0.6427 37.66 0.3387 183.50 0.3567 23.80 0.3287 -70.59 0.3807 69.70 0.5227 46.51 0.6927 34.52 0.3637 146.48 0.3917 0. 0.3537 50.16 0.4307 45.74 0.5727 34.72 0.5427 G. 0.3737 134.66 0.3787 22.27 0.4557 33.15 0.5977 0. - 0.3837 122.21 0.3987 0. 0.4007 8.42 0.4037 77.75 3.3057 0. . 0.10:%.07 0. .

Table 6e

Run No: EH-D2

Slot height = 0.29 in.

X= 0. IN. X= 1.780IN. X= 2.500IN. X= 3.600IN. X= 4.460IN. X= 5.680IN. X= 6.953IN. X= 8.641IN.

Y IN. U FT/S y IN. U FT/S -? IN U FT/S y IN. uFT/S Y IN. U FT/S y Uti. U FT/S Y IR. UFT/S Y IN. UFT/S 0.3037 147.53 0.0037 145.50 0.0037 153.59 0.0037 114.10 0.0037 98.33 0.0037• 76.29 0.0037 65.18 0.0037 44.76 0.0047 162.16 0.0047 149.30 0.0057 155.93 0.0047 115.15 0.0047 99.87 0.0057 78.94 C.0057 67.78 0.0057 47.79 0.0057 194.08 0.0057 153.24 0.0077 162.21 0.3997 128.02 0.0057 102.32 0.0087 82.65 0.0087 71.00 0.0117 56.77 0.0077 223.04 0.0037 164.44 0.0097 168.25 0.0197 146.63 C.0077 106.57 C.0337 105.89 0.0537 93.07 0.0567 74.98 3.0127 296.86 3.0137 183.36 0.0157 183.72 0.0397 196.10 0.0127 114.91 0.0587 114.89 0.1037 98.63 C.1067 79.11. 0.1177 317.43 0.0237 213.41 0.0257 202.96 0.0497 171.41 0.0327 133.21 0.0787 118.39 0.1287 99.16 0.1567 80.10 0.0227 316.56 0.0287 227.14 0.0407 220.11 0.3597 174.52 0.0427 139.78 C.0937 120.62 0.1537 99.27 0.1817 80.08 0.0327 318.52 0.0337 240.08 0.0557 229.79 C.0697 176.56 0.0527 143.57 0.1087 121.35 0.1787 98.55 0.2067 79.25 3.0577 315.43 0.0357 249.53 0.0657 233.32 0.0797 177.57 0.0627 146.35 0.1337 121.21 0.2037 97.17 0.2567 77.47 0.1077 335.60 0.0437 258.63 3.0757 234.87 6.0947 178.58 0.0727 147.27 0.1587 119.73 0.3037 86.55 0.3567 69.55 0.1577 299.24 0.0537 269.08 0.0857 235.26 0.1097 176.56 0.0827 148.48 0.2087 114.28 8.4037 71.44 0.4567 60.70 C.1827 295.36 0.0637 277.22 0.1057 231.37 0.1297 173.49 0.0977 149.08 4.3097 96.79 G.5037 54.49 0.5567 49.94 0.2327 279.96 0.0737 283.57 0.1557 202.24 6.1547 167.18- 0.1127 149.08 0.3837 80.14 0.6037 36.79 G.6567 37.72 C.2577 245.78 0.3937 287.56 0.2057 166.08 n.2047 150.25 0.1327 148.48 0.4337 66.87 0.6537 27.90 0.7067 32.18 0.277.7 207.36 6.1037 276.58 .2317 145.-.8 0.2297 139.10 0.1577 144.20 0.4837 54.77 0.7257 12.21 9.7777 24.53 0.2877 144.67 0.1187 272.69 0.2557 123.46 0.2547 128.38 0.2077 133.88 0.5587 33.23 0.7367. 0. 0.8547 15.64 0.2917 -0. 0.1437 254.22 3.2559 123.22 0.3047 104.87 0.2577 118.11 0.5837 27.19 0.9377 4.47

3.2437 132.71 0.26;7 114.99 0.3297 91.69 3.3077 101.63 C.6087 16.95 . 3.9827 0. 0.2437 130.84 3.3957 80.64 0.3547 79.46 0.3577, 82.76 0.6477 0. 0.2537 0.2687

121.63 102.28

0.3307 6.3557

59.94 29.12

0.3797 0.4297

65.62 36.52

0.3827 0.4327

72.48 52.91

•

0.31E7 39.46 0.3657 16.30 0.4547 16.84 0.4527 32.56 0.3477 0. 0.3917 C. 0.4977 0. C.5077 17.68 0.5317 10.05 . . 0.5627 C.

Table 6f

Run f\lo:EH-D3

Slot height = 0.270 in.

X= C. IN. X= 1.580IN. X= 2.6001N. X= 3.530IN. x= 4.470IN. X= 5.5601N. .X= 7.000IN. x= 8.7C0IN.

Y IN. U FT/S Y IN. U FT/S Y IN. uFr/S- y IN. UFT/5 Y IN. U FT/S Y IN. UFT/5 Y IN. U FT/S Y IN. U FT/S 0.003T 42.29 0.0037 61.9 3 0.3037 52.44 6.0037 44.53 0.0037 36.68 0.0037 29.30 0.0037 20.93 0.0037 15.73 0.0047 49.71 0.0047 64.95-.0.00047 53.56 0.0047 46.21 0.0047 37.01 0.0047 30.15 0.0057 22.51 0.0047 16.58 0.0067 70.30 0.0057 66.54 0.0057 55.03 0.0057 47.06 0.0067 38.21 0.0057 31.18 0.0077 23.64 0.0217 22.91 0.0117 102.52 0:0077 69.21 0.0067 55.97 0.0167 54.59 0.0087 39.69 0.0157 35.66 0.0237 29.63 0.0467 25.44 0.0217 116.51 0.0097 72.31 0.0087 58.23 0.0367 60.78 0.0237 45.92 0.0657 41.87 0.0787 33.08 0.0717 26.26 0.0267 117.10 0.0197 85.16 0.0187 66.54 0.0417 61`.67 0.0737 51.58 0.0757 42.20 0.1037 33.63 0.0967 26.31 0.0317 117.64 0.0247 90.32 0.0287 72.26 0.0467 62.41 0.0787 52.46 0.0857 42.39. 0.1137 33.78 0.1217 27.04 0.0417 117.55 0.0297 94.84 0.0337 74.63 0.0517 63.33 0.0837 52.69 C.0957 42.77 0.1237 33.91 0.1467 27.06 0.0517 117.40 0.0347 98.80 0.0387 76.24 0.0567 64.18 0.0887 52.76 0.1057 42.37 A.1337 33.90 0.1717 25.86 0.0717 117.10 0.0397 102.61 0.0437 77.60 0.0617 64.65 0.0987 52.74 0.1157 42.96 0.1437 33.85 0.1967 26.78 0.0867 116.36 C.0447 105.29 0.0487 78.63 0.0667 64.92 0.1237 52.46 0.1307 42.68 0.1537 33.79 0.2967 24.77 0.1367 114.94 0.0497 107.26 0.0537 79.60 0.0767 65.12 0.2237 45.65 0.1457 42.43 0.1787 33.29 0.3967 22.82 0.1867 110.21 0.0597 109.36 0.0587 80.03 0.0867 65.25 0.3237 33.88 0.1657 41.95 0.2707 30.38 0.4967 20.02 0.2117 103.37 0.0697 110.94 0.0637 80.68 0.0967 65.09 0.3487 30.26 0.2657 36.95 0.3787 26.07 0.5967 15.27 0.2367 93.47 0.0797 110.31 0.0737 81.33 0.1167 64.12 0.3737 26.84 0.3657 29.55 0.4537 22.45 0.6217 14.51 0.2467 88.49 0.0897 108.39 0.0837 81.11 0.1667 58.96 0.3987. 24.17 0.4157 24.33 0.5°37 19.31 0.6467 13.66 0.2567 79.59 0.1047 104.56 0.0987 30.47 0.2667 41.83 0.4587 15.20 0.4407 21.66 0.5287 17.65 0.6717 11.80 0.2667 23.59 0.1297 94.84 0.1237 77.15 0.2917 35.65 0.4737 12.38 0.4657 18.21 0.5337 15.90 0.6967 16.411 0.2677 15.25 0.1797 70.36 0.1737 66.49 0.3167. 32.01 0.5677 0. 0.5257 11.18 0.6037 12.47 0.7967 6.33 0.2687 0. 0.2047 58.45 0.2237 53.10 0.3417 26.50 0.5657 7.S4 0.6537 7.28 0.9967 ,C.

0.2147 52.81 0.2487 45.73 0.3917 14.41 0.6357 0. 0.7037 4.03 0.2297 44.97 0.2587 42.10 C.4017 11.49 0.7437 0. -0.2447 36.67 0.2687 39.22 0.4547 C. . C.2597 23.39 0.2987 31.74 . 0.2797 18.70 0.3687 10.52 0.2997 7.17 0.3767 7.51 0.3197 0. 0.4037 0.

Table 6g

- c Run No: EH-D4 Slot height = 0•.21 in.

X= 0. IN. X= 1.330IN. X= 1.870IN. X= 2.640IN. X= 4.07Q1N. Y IN. U FT/S V IN. UFT/S Y IN. uvris Y IN. U FT/S Y IN. tuFT/5 0.0037 170.43 0.0037'199.62 0.0037 175.40 0.0037 115.60 0.0037 99.07 0.0047 177.66 0.0047 193.29 0.0047' 179.02 0.0047 123.79 0.0047 100.80 0.0057 198.41 0.0057 204.90 0.0057 182.46 0.0057 129.42 0.0057 103.56 0.0067 237.47 0.0067 209.62 0.0067 185.37 0.0077 135.48 0.0077 107.57 0.0077 264.32 0.0087 215.48 0.0087 193.82 0.0127 150.43 0.0107 113.63 0.0127 325,.10 0.0117 225.96 0.0137 208.43 0.0177 164.03 0.0157 121.64 0.0147 329.70 0.0167 246.66 0.0187 220.06 0.0227 176.05 0.0207 128.30 0.0167 332.65 0.0217 262.39 0.0237 229.95 0.0277 185.55 0.0257 134.81 0..0187 •333.18 0.0267 275.29 0.0287 239.06 0.0327 192.52 0.0307 138.72 0.0227 334.51 0.0317 287.62 0.0337 246.40 0.0377 198.45 0.0357 142.02 0.0277 333.98 0.0367 294.04 0.0387 252.82 0.0427 202.45 0.0407 145.00 0.0427 332.65 0.0417 302.40 0.0437 257.70 0.0477 205.50 0.0457 148.04 0.0677 329.43 0.0467 307.93 0.0487 261.47 0.0527 210.64 0.0507 149.23 0.1.177 322.36 0.0517.311.38 '0.0537.263.84 0.0577 212.33 0.0557 151.60 0.0677 315.13 0.0567.313.37 0.0587'266.53 0.0627 215.25 0.0607 252.89 0.1777 305.59 0.0617 313.88 0.0637 260.86 0.0677 213.17 0.0657 153.94 0.1677 294.77 0.0667 313.09 0.0687 267.86 0.0727 213.58 0.0707 154.51 0.1977 122.02 0.0717 309.66 0.0737 267.53 0.0777 213.-58 0.0807 155.66 0.2077 245.18 0.0817 305.03 0.0837 264.86 0.0877 213.17 0.0907 155.66 0.2117- 159.24 0.0967 290.08 0.0937 259.08 0.0977 210.22 0.1007 155.66 0.2137 66.58 0.1217 254.47 0.1187 237.19 001077 206.37 0.1107 155.20 0.2147 0. 0.1467 212.99 0.1437 208.43 0.1177 202.45 0.1207 154.51

0.1717 170.61 0.1687 177.52 0.1427 138.78 0.1307 152.19 0.1967 127.16 0.1937 146.77 0.1927 154.52 0.1557 146.83 0.2217 88.03 0.2187 115.13 0.2177 134.82 0.2057 130.13 0.2467 46.35 0.2437 87.50 0.2427 115.73 0.2557 113.94 0.2567 21.40 0.2837 36.60 C.2677 95.04 0.2807 104.41 0.2717 0. 0.2937 14.54 0.3177 56.35 0.3057 94.29

0.3137 0. 0.3427 32.53 0.3557 73.03 • 0.3527 21.58 0.4057 52.4

0.4127 0. 0.5297 25.20 0.6177 0.

.-.

Run No:EH-D5

......;,...

: . - .._--_ .. -'--"'--'.-•..-., '-"'-- ' ..-.- .

Table 6h

Slot' height = 0.21 in.

-./

x= O. IN. x= 7.b7blN. X= 4.b20IN. X= 3.1COIN. X= 1.770IN.Y HJ. "TIl: V .IN UFT I S YIN. U FT/S y IN. U FT IS Y IN . U FT IS0.0037 82.14 0.OC37 21.09 C.0037 49.5b- 0.0037· 12b.77 0.0037 91.950.0047 93.27 0.C041 25.90 0.0047 50.40 0.0047 i27.61 0.0057 95.150.0057 109.88 0.00b7 26.57 0.0057 51.24 C.0057· 128.85 0.0077 98.9b0.0107 162.11 c.coa7 27.87 0.0077 53.13- 0.00b7 131.07 0.0127 107.620.0157 175.77 0.0137 31.44 0.0227 63.59 0.0087 13b.87 0.0277 lzq.770.07.07 178.78 0.0237 3b.b2 0.0477 n.28 0.0137 145.90, 0.0377· 139.890.0257 179.17 C.033? 39.23 0.0727 73.24 0.0337 Ib6.99 0.0477 -14b.11:1.0307 179.77 0.0587 42.01 0.0827 73.95 0.0537 176.29 Q.0577 150.310.0357 133.91 0.0837 43.b6 0.0927 74.19 0.0737 178.69 0.0677 150.790.0437 '178.97 0.0937 44.46 0.1027 73.72 0.0837 1..78.b9 0.0777 150.310.0501 1:78.78 0.1037 44.62 0.1127 73.48 0.0937 177 .49 0.1027 139.890.0757 178.38 0.1137 45.01 0.1227 73.09 0.1187 171.08 0.1527 105.84-0.1257 .17b.78 0.1237 45.09 0.1477 71.28 0.1687 150.51 0.1777 88.230·.1507 175.37 0.1331 45.33 0.1977 b5:34 (I.218"t 122.(;1 0.-2027 iO.230.1757 165.3b 0.1437 45.48 0.2477. 57.77 0.2687 93.42 0.2277 53.080.1857 156.53 C.1587 45.25 0.2977 49.13 0.2937 . 77.89 0.2427 43.210.1957 141.52 O.lR37 44.4b 0.3227 44.85 0.318'7 b3.59 0.2527 3b.l00.2057 72.43 0.203i 43.6b 0.3477 40.11 0.3437 4b.13 0.2777 18.7b0.2Q77 26.51 0.2587 42.2b 0.3727 35.b4 0.3587 35.73 0.3127 O.0.2257 O. 0.3087 39.86 0.4227 25.89 0.3b37 27.29

0.3587 35.29 0.5177 20.57 0.3777 0.0.3331 34.94 0.5277 0 •.0./.337. 32.070.4837 28.61. O.5R37 22.180.&337 18.320.7137 !I.31C.9 /. 7 7 O•..

. I

...-

.' l. r."

!

'1-.I. f

J......0\\.11

Table 6j

Run No:EH-D6 ..Slot height = 0.20 in.

x= a" IN. X= 1.080IN. X= 2.110IN. X= 3.1001N. X= 4.660IN. X= 6.7121N. X= 9.437IN.

Y IN- U FT/S y IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. UFT/S Y IN. U FT/S Y IN. U FT/S 0.00335 188.61 0.0035 177.99 0.0035 158.06 0.0035 126.75 0.0035 90.56 C.0055 62.35 0.0035 39.15 0.00415 190.02 0.0045 179.98 0.0045 161.41 0.0045 128.84 0.0045 93.25 0.0345 64.03 0.3055 39.51 0.00T5'254.45 0.0055 186.29 0.0055 166.31 0.0055 131.58 0.0055 94.57 0.0055 65.13 0.0065 43.13 0.010'.5 293.41 0.0065 191.94 0.0065 170.C2 0.0065 134.27 0.0075 98.05 0.0105 70.35 0.0975 41.85 0.01lE5 333.16 0.0075 197.43 0.0085 177.22 0.0085 138.20 0.0125 105.52 0.0205 77.97 0.0115 46.42 0.07T5 337.14 0.0125 226.25 0.0135 191.73 0.0135 149.37 0.0225 114.98 0.0305 83.24 0.0315 57.00 0.0275 337.66 0.0175 250.30 0.0185 205.13 0.0235 164.18 0.0325 120.97 0.0405 86.97 C.0515 61.59 0.0305 337.14 0.0225 273.C8 0.0235 214.57 0.0335 173.17 0.0375 123.29 C.0605 91.53 C.0715 64.01 0.04(P5 336.61 0.0275 291.37 C.0285 221.52 0.0385 176.74 0.0425 125.56 0.0705 92.68 0.0915 65.54 0.05W5 334.49 0.0325 305.10 0.0335 227.88 0.0435 178.75 0.0475 126.96 0.0805 93.82 0.1115 65.91 0.101.5 333.96 0.0375 315.70 C.0385 233.29 0.0485 181.72 0.0525 128.35 0.0855 94.38 0.1315 66.87 0.13015 325.60 0.0425 322.95 0.0435 237.76 0.0535 183.09 0.0575 129.72 0.0905 94.75 0.1515 67.24 0.15103 322.58 0.0475 327.60 0.0485 240.82 0.0585 .184.16 0.0625 130.23 0.0955 95.13 0.1715 67.24 0.16'5 317.30 0.0525 329.77 0.0535 243.77 0.0635 185.13 0.0675 131.59 0.1055 95.13 0.1915 66.98 0.1705 305.00 0.0575 330.58 0.0585 245.23 0.0685 186.09 0.0725 132.27 0.1155 95.39 0.2165 66.45 0.185 284.17 0.0625 330.04 0.0635 245.95 0.0.735 186.09 0.0775 132.67 0.1255 95.42 0.2665 64.17 0.1905 270.38 0.0675 328.69 0.0685 246.68 0.0785 186.09 0.0825 132.94 0.1355 95.31 0.3165 61.76 0.16,,85 249.51 0.0775 323.78 0.0735 245.95 0.0885 185.13 0.0875 132.94 0.1505 94.75 0.3915 57.00 0.19w.5 210.87 0.0075 315.99 0.0785 245.23 0.1035 182.21 0.0925 132.94 0.1655 94.01 0.4915 49..72 0.20(05 113.16 0.1025 299.21 0.1035 233'.68 0.1285 174.20 0.1025 132.67 C.1705 92.49 0.5665 43.67 0.201_5 44.23 0.1275 258.69 0.1265 214.15 0.1785 151.16 0.1175 131.59 0.2405 87.78 0.6165 38.70 0.20225 0. 0.1525 210.94 0.1535 190.33 0.2035 136.90 0.1425 128.62 0.2905 81.74 0.6665 34.65

0.1775 158.97 0.1785 165.23 0.2285 122.42 0.1925 119.50 0.3405 73.78 0.6915 32.55 0.2025 109.20 0.2035 139.46 0.2535 107.84 0.2425 106.19 0.3905 66.20 0.7415 27.87 0.2175 78.75 0.2285 112.47 0.2785 93.44 0.2925 91.91 C.4405 58.07 0.7915 22.23

. 0.2275 55.47 0.2535 88.13 0.3035 78.42 0.3175 83.45 0.4655 53.18 0.8415 16.16 0.2325 46.23 0.2785 62.91 0.3285 62.07 0.3425 76.14 0.49C5 48.15 1.0475 0. 0.2425 22.21 0.2935 47.33 0.3585 44.49 0.3675 67.79 0.5155 43.69 0.2625 0. 0.3035 34.14 0.3685 33.63 0.3925 59.45 0.5655 34.66 .

Q.3135 19.71 0.3765 25.91 0.4175 55.45 0.5905 28.51 0.3385 0. 0.3935 15.04 0.4675 34.66 0.6155 21.43

0.4235 0. 0.4925 24.94 0.7345 0. --__ 0.5375 D.

Table 6k

Run No: EH-D7. Slot height 0.12 in.

X. 0. IN. X= 0.560IN. X= 1.200IN. X= 1.8001N. X- 2.880IN. X= 4.8504N.

Y IN. , U FT/S Y IN. U FT/S y IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. UFT/S 0.0037 179.61 0.0037. 200.38 0.0042. 173.88 0.0037 142.71 0.0037 98.89 0.0042 73.72' 0.0047 187.31 0.0047. 207.36 0.0052 178.98 0.0057 150.63 0.0057 102.52 0.0052 74.27 0.0057'230.60 0.0077. 237.70 0.0062 185.40 0.0107. 166.41 0.0107 113.37 0.0062 75.36 0.0077 312.80 0.0127. 277.05 0.0072 191.61 0.0157 180.82 0.0156 123.85 0.0162 82.90 0.0117 345.65 0.0157. 297.76 0.0102 206.96 0.0187 187.13 0.0157 124.18 0.0262 88.46 0.0167'347.18 0.0187, 318.80 0.0132 219.95 0.0217 192.77 0.0207 131.88 3.0312 90.65 0.0217 346.67 0.0217 332.17 0.0162 231.98 0.0247 197.80 0.0257 13/.86 0.0362 92.32 0.0317 345.39 0.0247 340.88 0.0192 243.33 0.0277 202.27 0.0307 142.60 0.0412• 93.54 0.0417 344.62 0.0277 345.28 0.0222 252.05 0.0307 206.64 0.0407 149.11 0.0462 94.90 0.0617 343.59 0.0327 348.36 0.0252 258.40 0.0357 211.76 6.0507 153.85 0.0512 95.50 0.0867 344.62 0.0377 348.10 0.0282 264.25 0.0407 216.35 0.0607 155.70 0.0562 96.54 0.0917 344.38 0.0427 347.08 0.0312 269.65 0.0457 217.92 0.0707 156.16 0.0662 97.64 0.1017 296.51 0.0527 339.58 0.0352 275.26 0.0507 219.63 0.0807 154.54 0.0762 98.04 0.1067 188.28 0.0627 323.50 0.0402 278.84 0.0557 220.04 0.0907 152.68 0.0862 98.18 0.1117 173.59 0.0877- 257.44' 0.0452 280.77-0.0607 219.22 0.1157 144.22 0.0962 98.00 0.1137 158.65 0.1127 174.30 0.0502 279.16 0.0657 217.59 0.1407 132.56 0.1062 97.71 0.1147 156.40 0.1227 141.69 0.0602 270.32 0.0707 214.70 0.1657 119.32 0.1262 96.17 0.1157 151.80 001327 107.46 0.0752 248.10 0.0807 208.36 0.1907 106.02 0.1762 88.46 0.1167 0. 0.1377 92.27 0.1002 203.90 0.1057 185.21 0.2157 91.54 0.2262 78.46

0.1477 60.49 0.1252 159.30 0.1307 158.15 0.2407 76.28 0.2762 67.18 0.1527 43.58 0.1502 113.89 0.1557 129.73 0.2657 61.72 0.3262 55.10 0.1577 20.54 0.1652" 88.95 0.1657 119.11 0.2907 46.49 0.3512 49.16 0.1677 0. 0.1752 71.88 0.1807 103.02 0.3157 31.65 0.3762 41.89.

0.2002 22.49 0.2057• 73.31 0.3407 10.02 0.4262 29.14 0.2182 0. 0.2157 .63.78 0.3707 0. 0.4762 16.58

0.2307• 45.29 0.5472 0. 0.2457 22.57 0.2557 10.30 0.2727 0.

;Table 6I

Run No: EH-D 8 Slot height = 0.0305 in.

X= C. IN. X= 0.2301N. X= 0.5251N. X=-0.9501Ns X= 1.4101N. X= 2.4801N. X= 4.2001N.

Y IN. U FT/S i-IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. UFT/S Y IN. UFT/5 0.0037 255.80 0.0037 241.25 0.0037 139.54 0.0037 87.31 0.0037- 64.22 0.0037 39.14 0.0047. 25764 0.0047 258.54 0.0047 251.98 0.0047 143.30 0.0047 90.59 0.0057 67.58 0.0047 41.96 0.0057. 20.67 0.0057 274.38 0.0057 257.51 0.0057 148.75' 0.0057 93.52 0.0067 69.22 0.0067 44.42 0.0067. 22.01 0.0077 328.10 0.0077 269.56 0.0087 154.02 0.0067 96.30 0.6097 71.61 0.0097 47.01 0.0117 27.07 0.0097 347.37 0.0107 281.72 0.0107 161.86 0.0117 107.83 C.0197 83.82 0.0247 53.88 0.0317 .33.63 0.0117 351.40 0.0137. 282.34 0.0127 166.18 0.0137 112.46 0.0247 88.86 0.0397 56.46 6.0567 35.88 0.0137 353.40 0.0167. 275.39 0.0147 168.30 0.0157 117.14 0,0297. 90.86 G.0497 57.40 0.0817 36.39 0.0157 354.89 0.0227. 251.27 0.0167 169.87 0.0187 120.30 0.0347 91.82 6.0597 57.74 0.0917 36.75 0.0177 355.33 0.0277. 222.22 0.0187 170.91 0.0217 123.33 0.0447 92.31 0.0697 57.11 0.1017 36.67 0.0207 355.38 0.0377. 153.73 0.0237 172.45 0.0267 124.32 0.0547 90.66 0.0847 55.67 0.1117 36.23 0.0237 352.90 0.0407 99.38 0.0287 166.18 0.0317 123.42 0.0647 86.50 0.1247 48.20 0.1317 35.63 0.0257 319.96 0.0427 83.99 0.0387 .148.16 0.0367 122.47 0.0797 79.38 0.1747 36.87 0.1817 32.73 0.0277 185.68 0.0477 53.51 0.0482 123.38 -0.0517 111.83 0.1547 33.52 0.1997 31.41 0.3317 19.69 0.0287 194.02 0.0587 0. 0.0487 123.37 0.0767 84.65 0.1797 18.74 0.2747 10.81 0.3567 16.63 0.0297 32.48 ' --- ' - 0.0637 91.48 0.0867 74.87 0.2097 .0. 0.3247 O. 0.4067 10.20 0.0307 0. 0.0737 67.12 0.0967 64.00 . 0.4317 5.76

0.0887 31.07 0.1067 53.80 0.5287 0. 0.0937 15.82 0.1267 30.79 0.0997 0. 041417 13.25 . .

0.1567 Os

Table 6m.

Run No: Sbt height .• 0.050 in. V- grooved surface

X= 0. IN. X= 0.300IM. X= 0.7201N. X. 1.520IN. X= 2.7101N. X= 4.3201N, X= 6.2651N. ... X= 8.656IN.

.- IN. . U FT/S Y IN. • U Pus Y IN. , U FT/5 Y IN. U FT/S Y IN. US •Irl Y IN. LI FTtS, -y IN. UFT/S :.0054 279.45 0.0049. 182.83 0.0049 144.31 0.0049 75.99 0.0059 470.0049. 30.36 0.0049 20.74' b.0049 12.10 '.0064 301.46 0.0059. 186.12 C.0059 145.51 0.0059 76.66 0.006 9 48.00 0.0069 30.52 0.0069 20.81 D.0099 14.14 1.0114 349.26 0.0069. 195.20 0.0069 147.87 0.0069 78.52 0.0089 50.10 0.0099 31.86 0.0099 21.26 '.0.0999 21.12 :.0164 350.74 0.0089 214.24 0.0089 151.47 0.0089 82.91 0.0119 52.77 0.0249 36.73 0.0249 24.344 0.1999 22.93 ).0214 350.74 0.0139. 255.89 0.0109 162.41 0.0189 97.22' 0.0219 57.93 0.0399 40.10 0.0749' 30.47' 0.2249 23.44 D.0314 352.21 0.01698 276.06 0.0159 186.30 0.0239 102.22 0.0319 61.44 0.0549 41.53 0.1249 31.42 0.2499 23.62 ?.0364 353.68' 0.0189 286.53 0.0209 188.16 C.0289 105.85 C.0419 63.75 0.0699 42.67 0.1749 31.37 0.2749 23.46 ).0414 354.02 0.0209 293.'99 0.0239 192.71 0.0339 108.10 0.0469 64.77 0.0849 42.98 0.1999 31.04 0.3249 22.69 ).0444 346.28 0.0229 300..40 0.0269 195.4U 0.0389 109.68 0.0519 65.03 0.0949 42.97 0.2499 29..25 0.4499 19.33 ..0464 321.97 0.0249 305.54 0.0299 196.72 0.0439 110.52 0.0569 65.40 0.1049 42.86 0.3499 24.18. 0.5499 16.33 1.0474 285.57 0.0269 30T.30 0.0329 197.52 0.0489 110.80 0.0619 65.35 0.1249 41.87 0.3999 20.79 0.5999 14.63 . :.0484 189.57 0.0289. 308..08 0.0359 197.34 0.0539 110.24 0.0719 65.08 0.1499 40.15 0.4499 17.19) 0.6499 12.86 1.0494 109.97 0.0309 307.52 0.0409 194.51 0.0639 108.51 0.0969 62.55 0.1999 35.27 0.4999 14.45 0.7499 10.61 ).0504 25.62 0.0339 303.55 0.0459 188.62 0.0839 100.86 0.1219 58.93 0.2499 30.35 0.5499 11.70 1.1159 0. 3.0514 0. 0.0369 295.46 0.0559 175.76 0.1039 90.06 0.1469 53.29 0.2999 .22.94 0.7159 0.

0.0439 266.48 0.0759 137.53 0.1289 75.76 0.1719 47.13 0.3149 21.61 0.0539. 205.57 0.0859 117.09 0.1489 62.48 0.1869 43.09 0.3249 20.24 b.0589 166.47 0.0959 96.68 0.1589 53.98 0.1969 40.54 0.3749 14.48 C.C639 122.94 0.1009 58.20 0.1689 48.34 0.2119 37.08 0.4499 3.33 0.0689 91.25 0.1059 80.37 0.2039 25.42 0.2269 32.75, 0.5499 • 0. ' 0.0739 60-56 0.1309 35.85 0.2539 0.' 0.2469 .28.00 0.0789 24-,84 0.1409 5.86 0.2719 22.86 - 0.0839 0.. . 0.1559 0.1559 0. 0.3719 0.

Table. 6n

Run No: VH-D2 Slot height = 0.15 in.

X= 0. IN. X= 1.230IN. X= 1.850IN. X= 2.800IN. X= 4.120IN. X= 6.094IN. X= 9.220IN.

Y IN. U FT/S Y I. U FT/S Y IN. U FT/S Y IN. u_FT/c Y IN. U FT's_ y IN. uFT_Je Y IN. _U FT/S 0.0049 278.31 0.0049' 168.81 0.0049 152.11 0.0049' 117.74 0.0049 85.20 0.0049 55.25* 0.0049 33.01. 0.0049 281.43 0.0059. 170.88 C.0059 157.24 0.0059 119.16 0.0059 86.63 q.0119 55.44 0.0979 51.48 0.0059 283.91 0.0069 173.44 0.0069 161.24 0.0079 120.77 0.0079 88.44 0.0129 56.19 0.1229 52.15 0.0079 295.38 0.0089. 179.91 0.0119 174.29 0.0099. 124.35 C.0109 92.13 0.0149 58.51 0.1479 52.66 0.0149 341.50 0.0119 189.44 0.0219 195.32 0.0199 140.82 0.0259 105.61 0.0209 62.68 0.1729 53.01 0.0199 340.99 0.0219 223.91 0.0319 210.13 0.0299 153.46 0.0409 113.92 0.0309 68.56 0.1979 52.87 0.0299 339.44 0.0319 254.43 0.0419 218.78 0.0399 157.44 C.0509 116.95 0.0559 78.13 0.2229 52.52 0.0599 335.30 0.0369 265.93 0.0469 221.98 C.0449 163.51 0.0609 119.33 0.0709 81.64 0.5479 36.61 0.1099 331.10 0.0419 275.03 0.0519 223.57 0.0499 165.66 0.0709 121.31 0.0859 83.76 0.6479 29.55 0.1249 326.85 0.0469 283.22 0.0569 225.14 0.0549 167.79 0.0809 122.23 0.1009 84.85 0.6979 26.01 0.1349 315.14 0.0519 288.46 0.0619 226.32 0.0599 168.84 0.0509 122.52 0.1159 85.22 1.0589 0. 0.1449 281.12 0.0569 292.39 C.0719 225.93 0.0649 169.88 0.1309 122.39 0.1259 85.63 0.1499 89.68 0.0619 .293.59 0.1969 111.E2 C.0699 170:92 0.1109 122.23 0.1359 85.59 0.1509 30.57 0.1419 179.91 0.2069 100.25 0.0799 170.92 0.1259 121.08 0.1559 85.01 0.1519 0. 0.1569 151.22 0.2319 73.39 0.0899 170.92 0.1759 110.79 0.2059' 81.64

0.1669 132.63 0.2669 28.97 0.0999 167.79 0.2259 97.67 0-.2559 76.77 0.1919 87.97 0.2719 21.32 0.1149 162.97 0.2759 82.06 3.3059 70.48 0.2169 44.16 0.2919 0. 0.1299 157.44 0.3259 64.87 0.3559 62.63 0.2269 11.80 0.1549 145.16 G.3409 58.92 0.4059 54.09 0.2369 0. 0.2049 115.18 0.3559. 52.83 0.4559 45.88

0.2449 89.60 0.4009 36.50 0.4309 39.82 ' 0.2549 83.32 0.4379 26.48 0.4809 39.79

0.2799 67.41 0.5159 0. 0.5059 37.74 0.3299 29.84 0.7809 0. E.3799 0.

• ;

Table , 6 p

Run No: VH-D3

Slot height = 0.25 in .

X=-0. IN. X= 1.260IN. X= 3.120IN. X= 4.520IN. X= 6.5001N. . k= 2.070IN.

Y IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. U FT/S Y IN. U FT/S 0.0051 214.99 8.6052 173.31 0.0049 131.71 0.0049 94.77 0.0049 64.14 0.0049 148.19 0.0071 248.48 0.0072 176.35 0.0059 133.58 0.0079 99.29 0.0069 65.71 3.0059 150.94 0.0101 291.60 0.0092 180.31 0.0069 135.68 0.0229 116.07 0.0099 67.45 0.0079 154.08 0.0151 319.32 0.0132 187.99 0.0089 160.76 C.0379 126.48 0.0149 70.97 0.0129 166.43 0.0291 323.93 0.0232 221.63 0.0489 183.88 0.0479 129.77 0.0399 85.59 0.0229 190.45 0.0251 324.80 0.0332. 258.44 0.0539 186.27 0.0579 131.09 0.0649 92.55 0.0329 210.20 0.0301 324.25 0.0382 273.71 0.0589 187.22 0.0679 137.04 0.0899 96.32 0.0379 185.18 G.0401 323.16 0.0432 286.33 0.0689 189.56 0.0779 137.69 0.0999 97.19 0.0429 224.61 0.0601 319.87 0.0482 296.64 0.0789 190.31 C.0879 138.58 0.1099 97.51 0.0479 230.41 0.0851 315.71 0.0532 304.87 0.0889'189.75 0.0979 138.97 0.1199 97.87 0.0529 235.71 0.1351 306.93 8.0582 310.04 0.0989 188.63 0.1079 138.97 0.1299 97.87 0.0579 239.18 0.1851 299.36 0.0632 313.44 0.1439 162.43 0.1229 137.69 0.1399 97,98 0.0629 242:61 0.2101 286.10 0.0682 315.41 0.1689 164.68 0.1379 136.39 0.1499 97.69 0.0679 244.64 0.2301 254.10 0.0732 315.97 0.2189"138.27 0.1879 126.27 0.1649 97.33 0.0729 246.32 0.2401 233.85 0:0782 315.41 0.2439 122.05 0.2379 112.69 0.1899 96.06 0.0779 246.98 0.2451 213.75 0.0882 312.31 0.3139 76.96 0.2879 97.68 0.2649 88.42 0.0829 246.98 0.2481 170.09 0.1132 295.45 C.3939 20.50 0.3379 81.08 0.5149 45.66 0.0879 246.98 0.2501 60:57 C.1632 230.24 0.4539 0. 0.3629 71.21 0.5399 40.94 0.0979 244.30 0.2511 0. 0.1882 187.84 0.3879 62.96 0.5649 35.60 0.1129 237.46

0.1982 169.71 0.4129 53.63 0.7699 0. 0.1629 193.88 0.21-32 143.16 0.4629 33.19 0.2129 140.65 0.2382 99.47 0.5529 0. 0.2279 125.15 0.2632 55.12 0.2379 113.98 C.3012 0. 0.2629 87.00

0.2879 60.22 0.3129 28.56 0.3429 . 0.

172

TABLE 7

Thermal effectiveness data of radial wall-jet

SMOOTH SURFACE

in.yC 0.0.5 0.125 0.223

x/yc E Yo x/Yo

E

0 1.000 0 1.000 0 1.000 4.77 .912 9.40 .978 1.345 ..()29

12.30 .8975 6.4 .953 3.59 .900 20.0 .772 10.4 .957 5.83 .919 27.7 .667 14.4 .8625 8.08 .906

35.4 .560 18.4 .738 10.32 .771 50.75 .476 26.4 .6265 14.8 .679 66.15 .405 34%4 .524 19.27 .553 81.5 .350 42.4 .4465 23.97 .473 96.9 .314 50.4 .404 28.27 .425

120.0 .267 62.4 .354 35.0 .351 150.7 .214 78.4 .294 43.9 .303 196.9 .124 102.4 .210 57.4 .152

EMERY SURFACE

yc in.

0.41 0.29 0.21 0.11

x/y C a x/yc E x/Yc E x/yc a

34.8 .265 48.9 .252 67.7 .230 126.8 .213 27.4 .331 38.5 .318 53.3 .288 99.8 .261 21.2 .404 29.8 .393 41.2 .352 77.3 .311 16.2 .501 22.8 .481 31.6 .4,1 59.3 .384 12.5 .590 17.6 .585 24.4 .557 45.8 .450

10.1 .715 14.2 .671 19.6 .606 36.78 .525 8.84 .798 12.4 .744 17.2 .672 32.2 .560 7.61 .858 10.7 .815 14.8 .743 27.7 .609 6.37 .933 8.96 .892 12.4 .838 23,2 .692 5.14 .978 7.22 .957 10.0 .927 18.74 .777

3.90 .994 5.49 .990 7.60 .990 14.23 .870 2.67 .998 3.75 1.002 5.19 1.007 9.73 .956 2.05 1.007 2.88 .990 3.99 .998 7.48 .965 1.432 1.004 2.01 1.012 2.79 1.018 5.23 1.000 .. 3 .988 1.15 1.000 1.-9 .991 2.97 1.000

.198 1.006 .28 1.013 .

.J9 ,.

:994 .72 .988

173

TABLE 3

Heated plate surface temperatures

V - GROOVE ROUGHJJESS

Yo = slot heir7.ht, j = power input to heater.

y = 0.25 in.

FJ --r-- - 0 w 150 w 300 w 450 w 750 vr

x in. T -T o17 S ('- o, " -T 1

j-g G Ts-TG o- T -T r G G

d m m , i -1. i S LT

14.1 10.22 10.75 12.58 14.39 17.48 13.1 10.21 10.96 12.58 14.54 17.59 12.1 10.70 12.34 13.37 14.51 18.05 11.1 11.20 12.89 13.26 15.34 17.80 10.1 11.29 13.40 14.10 16.07 18.46 9.1 12.62 13.97 14.45 16.30 18.79 8.1 13.57 14.71 15.18 15.73 19.04 7.1 14.75 15.34 16.22 1-7.63 19.91 6.1 16.18 17.23 17.56 19.03 21.24 5.1 18.91 18.97 19.12 20.48 22.87 4.1 20.33 21.30 21.58 22.85 24-93 3.1 22.96 23.72 23.94 25.25 27.15 2.1 25.83 25.78 25.87 26.83 28.26 1.1 27.10 27.28 27.58 28.66 30.09 .1 28":06 28.05 28.35 29.30 30.56

-0.1 29.21 28.35 28.69 29.82 31.19

Yo = 0.15 J ---10- 0 w 300 w 450 w E x in. - T$ TG °F T -T °E1 S G . °IP Tr. -T 6 G

14.1 0 H

0-\

Op if LiTh 11 -\

rR

.0 r) C

O 01H

rrl

N N

Pr) c0

N N -\

•

a .

0 • 0 0

• 0 • .• o 0 0

0 0

NC

O O

'N 0 0

N re 1

•Co cX)

H

0-) H

HH

HI—

IHH

NN

NN

N

1.:!..51 15.14 13.1 11.93 15.18 12..1 12.31 14.51 11.1 12.40 15.58 10.1 11.95 16.26 9.1 13.21 16.62 8.1 1)4.20 16.83 7..1 15.43 17.79 6.1 18.12 19.00 5.1 19.59 20.30 4'.1 21.93 22.67 3..1 24.52 25.34 2..1 26.80 27.49 1..1 29.40 30.23 •0y.1 30.66 31.42 - .1 31.25 32.19

ze'PC 6T'ilc Lc*3c

"(=)c 02°63 T5'2 55'Lz OT°22 L9°9z C9'53 56° 9z oLoLz 99°9'd 0L.° 9z C9*W 9g*9z

Pc°Pc t7L°55 t6,00c 5'9Z

99°a T5'9-6 T9*gz -17T°5 a*Pz Z9-1-73 -179-17 Lz ° 5z 39-43 9C°P3 zC°P 95*PE

Lg*CC 69' z5 95*oc 05°a oL'gz

2..*5 20°zz Og'T T5°03 22'6T oL°6-E 59` 6T P-u6T 00° 61 1-79°2T TL°9T

c000c 6o° 63 cPosz o5°C3 0°T -[9°6T CT*9T 9T'LT 9t7' ST OT*gT 59*PT gC'VE OecT u4o5T Circi gc*CT

99°05 9z' 05 59*Lz 63*Pz CI*TZ 6L°2-E TL'9T OZ•gT 96*CT 00'ci 5°3-[ t79° TT 56' OT T9° 0T

9Z*OT 9t100T

T* - T'O-'UT T°3 -U.0 T*P T'S r.9 T',L T°2 i*.6 T0 OT "PIT "UZI

T°PT

0 i0 I-°,1 0 1-21 ,; -si

_g a00 1 w

i. ci doI-"I *TIT x

arT A 009 =A Ogq M 005 IA OgT PA C -•-.4--- 1.

*uT g0'0 = o.g

PLT

175

APPENDIX 1

In a turbulent Couette-flow,

du_ du ... (A.1-1) ti 77z 't S P - -ry and

J"= J's' s (r + peh)tT r

From (A.1-1), and the definitions,

11M T s 4)) , y+ = ri(rg)

we obtain:

(1 uP\ P't du+ + ---) = = ,_ z-7-, Et ... (A.1-3) il P. dy'

(A.1-1) and (A.1-2) when combined, give:

,tm l't + P6u du p,, du _ = ...., ...._ ... (A.1-L1.) ' ,5-g ri + peh dp rt thP

Substitution of:

= _R.. • r , co = Cu ; CD+ = ( - (Ps)kiKs'sp)Aig

and. a = t - ilt/rt in

results in: Et

+ (Et - 1)-1 a0

1 1 00 6t

1 1 -

d(p+

du+

and

1 Et

at ... (A.1-5)

... (A.1-6)

176

APPENDIX 2

The use of r} for the calculation of local,(1us.

We have,

vrith,

E = ERr). ... (A.2-1)

Rr y r hi(T S p) / 0 0 0 (A.2-2)

In a boundary layer problem where 2 parameters, such as

RG and zE' determine the local conditions, the application of

E4.Rr} for the calculation of local drag coefficient is as

follows.

and (2.2-5) is,

with,

Ss E -u/(4)

/2 ss = i<zE/-e'

= In (ERlas,1/2)

... (A.2-3)

... (A.2-4)

... (A.2-5)

... (A.2-6)

. . (A.2-2) gives,

YrueP zE Rr

The steps in the evaluation of .Fare,

(1) Assume a value of 8)

(2) Calculate Rr by the use of (A.2-6)

(3) Obtain E from ERI,}

(Li) Use (A.2-4) and (A.2-5) for finding a new value of E

based on E form step (3)

177

(5) Repeat (2) , (3) and (L!.) until _e• becomes sufficiently

accurate.

When eis known, (A.2-0 enables the calculation of the drag

coefficient.

If Rm is specified instead of RG, then the steps in

determining _ei are the same with (A.2-5) rewritten as,

R In E 1 zr '

L Il _e/ ... .2-7)

and 1 , , + zEk-2 — ) II

Pipe-flows may be accomodated if we write,

(Y6)b.-layer (D/2) pipe

... (A.2-9)

(uG)b-layer

F. (UR) pipe

which, together with the relation

uR 3 ,,rsp

1 + 2 /•<-

for a pipe-flow

and the empirical fact

ILE/uR 1

lead to,

3 'I-s -ei = K (1 + P) + in( 2) ,.,, (A.2-10) ti 2 /- sp

_e - . ln(M.Dsp1/2) • . • i (A.2-n)

and

178

with

and

where

RD z. pDa/p,

.m. ,z,/(pri 2) s P

U = bulk velocity of fluid in pipe

Rr is defined in this case by

Yr , 1/2 Per :--:. D — RD sP eo. (L02-12)

179

APPENDIX 3

Derivation of (4.5-7)

The total drag coefficient on the surface has been

resolved into two components: that due to the 'active' ele-

ments which are shedding vortices and that due to the rest

of the surface, which is effectively smooth.

i.e. s = a se + (1 - s.1Y. ... (A.3-1)

a is the projected area of the 'active' elements per unit area

of a n equivalent smooth pipe surface; se the effective drag

coefficient of the surface if all the elements were active,

and sm that of the surface if it was smooth.

For a fully rough surface:

E = t /12r ... (A.3-2)

where f3 is a constant, and for a smooth one:

E = Em ( a constant) SOC (Zi'03-3)

Fr'2 pipe flci in general, we have the drag- law riven by

(A.2-10) and (A.2-11); viz.,

1/2 1/2, k- 3sP ln(E Dsp ) = —7-775-(1 + ) + ln(2)

s;'' 2 A=

i.e. in E = - ln(RDP s1/2) + + (-3 + In 2) --- sp

Then, in particular, we have,

t 1% In Em = - 1nLRDsm/2) + 1/2 + a sm 3-4)

- s-1/2\m P I ...

s-1/2)

(A.3-6)

... (A.3-7)

... (A.3-8)

C C 0 (A.3-9)

0 0 0 (A.3-10)

ln(E/E ) = 0.5 ln(sM/sp) (s-1/2

ln(ERr41) = 0.5 ln(se/sp) s-1/2

( e

In the transition zone we may write

se s P

so that

s.e/sp (ERr/)2

and sm/sp (F/EM) 2

180

where a = 2 + In 2

for the effectively smooth areas. Similarly, for the portions

where the elements are 'active',

in Ee = ln(P/Rr) - ln(RDse1/2 ) + + a

se

0 0 • (A.3-5) By a combination of (A.3-3), (A.3-0 and. (A...3-5) , we obtain:

which when subsituted in (A.3-1) rive

[a(R / )2 + (1 - a)/qi

This is the same as equation (4.5-7)

Details of application of the theory to Nikuradse's E Rr

data

We assume a quadratic distribution for N,

i.e. N .43)X(1 - .. (A.3-11)

where

181

X (y r, yr,1 )/(yr,u - yr,1 ... (A.3-12)

g

Yr, , = general value of roughness height similar

If all sand grain roughnesses were geometricallyAto each

other, then

Yr,1

Yr 1

.., (A.3-13)

r,u n Yr J yr being the nominal height of rouFhness.

It has been stated in the text that the critical Reynolds

number Re,c for the onset of activity of roughness elements

should increase with Rr. Let us hypothesise a variation:

Re ,c = a+bRr ( A . 3- 14)

as a first approximation.

(4.5-3) wives

r yr(a c b Rr)/Rr ... (A.3-15) ,

According to (4.5-2), the number Ta of active elements at a

liven value roughness Reynolds number is riven by:

c Ta

= AX( 1 - X) dX = AD( 1 + _ 3x)/6 0

... (A.3-16)

where

Xc - Yr,u Yr,1 ( n - yr

a + bRr - mRr B CRr

(n m)Rr Rr

(say) • (A.3-17)

a + bRr Rr m Yr Yr,c Yr,1

that

a = 0 at Rr =

and a = 1 at Rr

=

Thus,

C = Rr,i/(Rr,a.

and

so that,

B = - R C r,u

Rr,1 1 ... (A.3-19)

riyu.j

- Rr,u) . (A.3-20)

182

Since a is the fractional number of 'active' elements,

therefore, a = Ta/(AD/6)

= 1 + 2X3 - 3X2 000 (A0318) c

Constants Is and C can be determined from the condition

Rr 1 Rr,1 7

Rr Xc - Rr Rr,1 - Rr,u

... (A.3-21)

It is found, however, that this form for Xc dines not

Ove sufficient flexibility when fitting, the shape of the

curve it brings about being entirely determined by the values

of Rr u and Rr l' • this is only to be expected if we remember , ,

the restrictions tlaced on the distribution function and the

critical Reynolds number variation. To remedy this inflexi-

bility we modify Xc

Xc

to:

[-Rr J711-1t Rr,1 - Rr. .. (A.3-22) - r - Rr,1 - Rr,u

From Nikuradse's data:

Em = 8.12; p = 30.03; Rr u = 100. 9

183

It is seen from figure 4.2 that the lower limit of the

transition zone is hardly distin7uishable from the intersec-

tion of the lines representing

E =and E = 30.03/11r

Therefore,

Rr,1 = 307

l'he value of n which gives a satisfactory fit is about

0.546; The resulting set of equations which can be used fOr

calculating E are:

0.02248(100 - Rr)

c 0,584 Rr

1 + 2X3 - 3X2 (A.J-23)

and

with

= = r- I a(R r/p) 2 +

/2 — ct)/E-1 _ I

= 30.03 and EM = 8.12

• 0 C.

184.

APPENDIX 4

Fortran IV Subroutines used for evaluating E and P for

V-groove roughness

SUBROUTINE EFUNC (RR,AE,DERR) RR2= ( 175.6/23.7) 1./ . 409) RR1=( 175.6/7.5) **( ./1 . 09) LF(RR-RR1)198,199,200

198 AE=7.5 DERR=0. RETURN

199 AE=7.5 DERR=.00',6 RETURN

200 IF(RR.GT.RR2)G0 TO 201 AE=175.6/R141(1.409 DERR=-1.4094.AE/RR RETURN

201 AE=23.7/RR DERR=-AE/RR RETURN END

SUBROUTINE PFUNC(RR,P) IF(RR-47.)101,101,102

101 P=-1.925+.12064RR RETURN

102 P.-1.925+.12064:47.+.0193*(RR-47.) RETURN END

185

APPY.L.MIX 5

Details of Apparatus

Fan:- 'Sturtevant' Monoirram No.5; Capacity 2500 cfm against

a head of 23 inches of water gauge; driven by a 15 h.p. motor•

Delivery pipe:- Inner diameter 3 inches; lenFth lOffeet;

material P.V.C.

Flange(forminrr nozzle) :- 5.95 inches diameter; material 'Pers-

pex'.

Smooth plate:- 'Perspex' sheet, 3 feet square. 3/8 inch thick.

Location of thermocouples: (distances from slot) -.2, .3, .8,

1.3, 1.8, 2.3, 3.3, 4.3, 5.3, 6.3, 7.8, 9.8, 12.8 inches resp-

ectively.

Static pressure holes: 14 holes at 1 inch intervals, first

one at .85 inch from the slot.

Emery covered plate:- 'Perspex' plate as in the case of smooM

one but covered with (Trade 1 1/2 emery cloth; average height

of rourrhness 0.0082 inch.

Location of thermocouples; (distances from slot) -.95, -.46,

.04, .29, .54, .79, 1.04, 1.54, 2.04, 2.54, 3.05, 3.55, 4.05,

5.05, 6.55, 8.55, 11.05, 14.05 inches respetively.

Plate with v-groove roughness:- Material: hard aluminium.

Eroove 60° V, depth 0.014 inch; in the form of a spiral If

0.022 inch pitch,.

186

Location of thermocouples: (distances from slot) -.90, .11,

1.09, 2.11, 3.09, 4.11, 5.11, 6.1, 7.1, 8.1, 9.1, 10,1, 11,1,

12.1, 13.1, 14.1. inches respectively.

Heater for rough plate:- 'Iso-pad' 800 w, flat circular

heater, overall diameter 2ft llin., with bin. diameter hole

in the centre.

Power to heater: control - 'Variac'

measurement - Cambridge A.C. Test set, No.

L356579.

Thermocouple wire:- 'Honeywell' type 9B105.

Measurement of thermocouple e.m.f.'s:- Selector switch:'Cro-

pico t , Type SP1 No. 7262. Potentiometer: made by Cambridge

Instruments.

Pitot probe:- Made of Stainless-steel hypodermic tubing,

having a rectangular aperture 0.0042 x 0.050 inch.

Manoteters:- Fluid: paraffin, having a specific gravity of

0.787 at 6o°p.

Vertical Manometer: U-tube type; maximum reading 40 in-

ches of paraffin gaurTe; least count ofsscale 0.1 inch,

redable to an accruacy of 0.05 inch; likely error in

velocity calculated from measured head < 0.5 percent.

ii. Inclined hanometer: N.P.L. type; 40 tubes; variable

inclination, can be set to nearest 1/2 of angle with

the aid of a clinometer; least count of scale 0.1 inch;

187

smallest inclination used 1L1.50; likely error < 0,6

percent

Micromarnmeter: U-tube type; liquid level determined

by brinFing a pointer which is attached to a micrometer

head, in contact with liquid surface; least count of

micrometer 0.0001 inch; likely error in velocity < 1

percent.

188

APPENDIX 6

Data Reduction

1. Velocity data:-

The pitot-head readings in inches of paraffin (gauge)

were converted to velocities in feet per second by means of

the formula:

u = 18 .9104 kf( TG h/hB)

where,

hB = barometric height in inches of mercury

TG

= absolutetchiperature of air,.in °Y.

h = pitot head in inches of paraffin.

2. Pitot-probe position:-

The distance of the pitot-probe from the surface is

determined from the reading of the traverse unit micrometer.

Allowance has to be made for the height of the probe

opening; and a fraction of the roughness height has been added

in keeping with the recommendation of Perry and Joubert [56-1.

The formula used is:

y = y - y 0.5 y + 0.2 yr Pm Po

where,

y is the height of the centre of the probe opening from

the datum surface,

the reading of the traverse unit micrometer for the Pm

given position of the probe,

189

y Jo the value of y Pm m when the pitot-probe is touching

the tips of the roughness elements,

yp the overall height of the Pitot-probe opening

yr average height of roughness elements.

3. Temperature measurements:-

The thermocouple e.m-f.'s were reduced to temperatures

in °P by the useeof the calibration formula (derived by G:E.

lms)

- 4c( a - 2 c

where,

e = Thermocouple e.m.f., measured in millivolts,

a = -0.6704468

0.02e 5O

and c = 1.372913 x 10-5

4. Integrals associated with the velocity profile:--

The values o-P,

00

Di

u dy

J 0

Pp == -- u2dy

cm3

Jo

3d u y

were obtained from the reduced velocity profile data by the

use of Simpson's rule with variable step-length.

T op 9

190

APPENDIX 7

The entrainmeLt constant

The means of estimating the entrainment constant is

based on the integral mass conservation equation (1.3-2-1).

For the case of a flow with zero mass-transfer at the

surface and with a velocity profile having a maximum, equation

(1.3-4) and the definition of m0 -7iven in equation (1.3-11)

lead to,

1 r, - - j- _

Rumax dx L Rj-

u dy 111G/'max ... (A.7-1)

The experimental data enable us to compute values of the

integral on the L.H.S. at many stations downstream of the slat.

Hence a graph of the Quantity in curly brackets on the L.E.S.

against x can be drawn and its gradient obtined.

This can be substituted in (A.7-1) along with the values

of umax and R to give mG/zmax° frowl

The value of ,C can be estimated f -cm a procedure given

in appendix 8; and this enables to evaluate zmax/z2.

Then the entrainment constant follows from

C 2 E MG/ZE = (MG/zmax)(ZMaX/ZE)

C 0 ( A .7-2)

It must be stressed that the value of C2 obtained is a

rough estimate only because it'is very sensitive to the errors

in the graphically determined derivatives. Errors can be

191

magnified about 5 times. Hence any recommendation that is

made has to be based on a comparison of the predictions with

the data.

A 'Flample calculation:

rata of wal-,jet on a smooth surface with y0 = 0.223 in.

A plot of R f u dy vs. x qives a p-radient of 0.314 ft3/in0,3, • 0

at x = 8 in.

Cori,esponding value of R.0max = 64.97 ft2/s.

° • • -

G m /zmax

0.314 x 1264.97 = 0,0581

have 1 ti • •••• 10, hence z x/z., ma .57

- m G = 0.039 = C2

More values of C2 deduced as above, are:

YC rou7hness C2

O.15 v-groove 0.033

O.25 v-F(foove o.o37

O.035 emery 0.026

O.15 emery 0 .040

192

APPENDIX 8

Procedure for finding the initial values of and uE for a

wall-jet in stagnant surroundings

In order to start the integration of the differential

equations by the Runge-Kutta procedure, the initial values of

X' and u,b have to be known.

The initial values usually known are those of Rm' umax,

R2 etc. For a 7iven rourrhness type the EkRr relation Y1/2'

can be specified.

The following procedures can be used for finding and

uE They are based on a velocity profile assumption.

General: All the procesures have to make use of a sub-routine

for finding the values of z/z/11 and / when especified. r,lx -1/2

"ae assume a velocity profile of the form:

z/zF = 1 + —1 In 8, -t_s3(-P- ... 0..8-1) -e'

which is simply a restatement of the velocity-profile given

in sub-section 2.2, for a case of zE--- 00.

At the velocity maximum,

d(z/zE) = 1 - W -Ir'd.5“)-1 = 0 ... (A.8-2)

d-, ?"4 .:

This equation can be solved for the value of P' max by an

iterative procedure. The value ofmax can then be substitu-

ted in (A.8-1) to give zma/zE; so that we have

max/zE = fm"'" ... (A.8-3)

where 'fm' means 'some function oft.

193

is the value at which 41/2 2-; izE = 0.5 (zmax/zE)

i.e. 0.5 ( zmaizE) = 1 + -,11n 3./2 - td 1/2). . . (A.8-14) 4„

This equation can also be solved by iteration for the value

Thus we have,

YG,1/ 2 = C/2(- / ... (A.8-5)

Procedure when umax, Y1/2 and E(B.r are given:

By the combination of (2.2-6) and (2.2-5) we have:

of 8,1/20

= 111( ERG k zE4r)

(A.2-1) and (A.2-6)

E = Ecar}

Br = pYrkuC ZE/( p, f t )

(A.8-6) can be rewritten as,

= In E z 1 py1/2umax

'max 41/2

and (A.8-7) as,

z ). max

with,

FY,/ E(.

1-1,

.. L (A.8-6)

1 .,. (A.8-7)

... (A.8-8)

(A.8-9)

(A.8-8) and (A.8-7) can be solved by the following procedure,

when the relationship (A.8-9) is specified:

1. Assume a value of ..e / 8) ;

2. Find zmax/zE using (A.8-3) ;

3. F 1/2 using (A.8-5) ; , 4. Calculate E using (A.8-9).

194.

5. Obtain a new value of -9'by substitution of terms into

the R.H.&. of equation (A.8-8);

6. Repeat the steps, using the value of-e'from 5, until

the required accuracy is obtained.

When €' is known, zmax/zE follows from (A.8-3) ; then,

= umax/zmaxE /z) . .. uE (A.8-9)

Procedure when R m , u max and E(-Rr)- are given

For this case we write (.A.8-6) as,

R, tez, -8' ln E

11 ti

.. (A.8-10)

when z is infinite (2.6-1) --->

Il 1 1 ... (A.8_11) zE (A.8-10) and (A.8-11) give,

e = LERm 2ie/(-e' — 2)] ..c (A.8-12)

As before (A.8-12) can be solved in conjunction with (A.8-9)

for ?'

Procedure when Rm, y1/2 and umax are given

We have,

i.e.

Il

1

,d0

1

(U

j

0

z

g zE

i.e. 1 = pu dy ZE uE yG

0

Rm (zmax/zE)1/2

1.95

Y1/2 umax

• • 0 ( A .8-13) Since zmax/zE 1/2 and are functions of r, and Rm y1/2

and umax are known, (A.8-13) can be solved by iteration to

give X' as follows:

1. Assume

2. Find zmax/T,E and g1/2 ;

3. Calculate new value of -e' by the use of,

- Rm(zmal E)P.;1/211/(pumaxyl/2)

8 0 0 (A.8-1)0

4n Repeat procedure until the desired accuracy is obtained.

1 2

196

APPENDIX 9

Procedure for integration of the hydrodynamic equations

(a) Entrainment method

Differential equations:

( 1 .3- 23) dRm d(in R)

Rm - m - m_ ... (A.9-1)

dRx (1.3-24)

dR2 d(ln R) a( in tb.) + R2 + (1 + H12 )R2 dRx dRx dRx

= 111 + Sn 000 (A.9-2)

We have also the relation,

Ro = Rm(I1 - 12)/11 . . (A.9-3)

which results from a combination of (1.3-19) and (1.5-20).

The term containing R and also R2 can be eliminated bet-

ween (A.9-1) , (A.9-2) and (A.9-3) , to give,

d

x [11

1 - / IiRm --aTE = 11ss + 97,(I, - 12) + 12m 1 - d(ln lb) ... (A.9...w

- (1 - I„)R m dRx

In general, we have,

I.1 = ii(- zz,,, -e'). 0000 i = 1, 2

/ and ii = -ek.Rm, zE).-

197

• 0 = z_ a e aRin )

(8Ii an. m)

... (A.9-5)

azEiRm oI1 (a -e'

azril .e.a pzE Rm

The L.H.S. of (A.9-4) can be simplified to

L.H.S. - I R —(1 ) m dR 2 1

- I1Rm dzE a I2\ aR. a 12 dRx az..E4 dRx aRm1

J

It can be shown that, in (A.9-5)

tal i ai' L3.tlz \azEI

aIi

e >

and also ( az.

E e' > >

(aV ORm) zE

Hence,

dz, oI2 - Rm dR OZE

all

az .E

L .H.S

(A09-4)

dzE (1 - I2) Rm d(ln -uc.

(II i2) Ills - - 11b. - -dR x Rm aRx ai2 121 all

ozz) ,e,

C C • (A.9-6)

Equations (A1.9-l) and (A.9-6) are to be solved for

Rm and zE' by a Runge-Kutta procedure.

198

Auxiliary relationships:

all ai2 I I --- and E

can be expressed in terms of zs I19 2' as az

and €' by means of (2.6-1) and (2.6-2).

93 is related to zs by the entrainment law; and ss to z

and by the drag-law.

Combination of (2.2-6), (2,2-5), (1.3-19) and (2.6-1),

gives

l - zEq _ k]

ln m . (A.9-7)

(A.2-1) is,

and (A.2-6) is,

=

Rr PYrue-zE/(P'-11

O 0 0 (A.9-8)

If Rm and zE are known, then -e' can be obtained by the simul-taneous solution of (1,9-7) and (A.9-8) with a procedure

similar to that used for solving (A.2-1) , (A.2-5) and (A.2-6)

for t

(b) :S. - method

Differential equations:

(1.3-224)

2 d(ln R) d( ln + R2 + (1 + 11,2)R,

dRx dRx dRx

= m + ss ... (A.9-9)

di i a if a zE aRc zE dI. N al i dzE

ozE p, dRx 4 0 dRx C 0 • i =

1, 2, 3

199

(1.3-25)

dR3 + R + 2R, d(ln R) d(ln uG )

(A.9-10)

If we make substitusions for R , R3 and 1112 in terms of

I2 , and RG by means of (1.3-20), (1.3-21) and. (1.3-16),

then(A.9-9) and. (A.9-10) become, reppeCtively,

d( ln R) d( In uG (Ii - I2)RG + (Ii - I2)RG + (1 - I2)R0

dRx dRx dRx

= m + ss (A.9-11)

d[ - ln R) d( ln uG ) (Si I3)RG] + (I1 - I3)RG + 2(Ii - I3)RG

x dRx dRx =m + (A.9-12)

m + 2E dRx

3 dRx -) dRx

ae now have,

=

and -e = -ei (-R.0 zE)-

As in (A.9-5) , we again find that,

aIi al . az, z 67'7. R E G

and.

• C • i 1, 2, 3

• 0 0

(A.9-13)

If we substitute this in (A.9-11) and (A.9-12) , and solve the dz

Nand

dRE resulting: equations for and. —dRG

9 then we obtain, x x

200

dRG 1 dR D Q1 az - 13)

- Q2 az8(1 - 1 I2) (A. 9-1 4)

and dzE 1 dRx

::---- ii• Q2(Ii - 12) - Q 1 - 13) ... (b.9-15)

where,

d(ln u) _ d( In R) Q1 -• m + ss 1 - I2)RG RG(II - I2)

dRx dRx

d( lnun.) d( ln R) Q2 4- M 2-6 - 2(11 - 13)R, - R(I1 - I3)

dRx dRx and

- az (Ii - 13) - (Ii az (II - 12)

Equations (L.9-14) and (1....9-15) are to be solved for RG and zE.

Auxiliary relationships:

611 812 813 I1' I29 I --- =--,and a 1' 2' 3' a zE' azE z,

of z and -el by means of (2.6-1) , (2.6-2) and (2.6-3)

Instead of which ap:.:ears in the entrainment method,

here we have 73, the dimensionless value of the dissipation

integral. -6 is expressed as a function of zE and -e' by the

procedure which is recommended by Escudier [23] and is outlined

in sub-section 2.4. evaluations

which is required in the course of theA is obtained in

terms of RG and z, by the solution of (L.8-6) and (A.8-7)

can be expressed in terms

201

General remarks

Other information reauired be both entrainment and

methods are specifications of,

(1) main stream velocity, uc , variation with respect to x,

(2) R variation with respect to x.

From geometrical and/or kinematic considerations we can write the

di the specific forms of 1)1(.4 and R(-14, and hence 75p16) and

dR dx*

These can be transformed into functions of Rx by means

of the transformation

dRx dx

UGP

Notes on the application to the radial wall-jet in stagnant

surroundings

The equations for a flow with a finite main-stream velo-

city can be used for the case of sta7nant surroundings by the

expedient of puttin7

lo 6x(slot velocity)

The R variation is given by

R = x x C

the distance x being measured fron the slot; x0 is the radius

of the slot.

262

Initial values

The procedures outlined above reouire the initial values

of zE' RG

and Rm.

Experimental initial values can be conver-

ted to values of -'and uE by the procedures outlined in appen-

dix 8. Then zE' RG

and Rm can be obtained by the use of

ZE = uE/11G

and

with

G frd exp(e)

E z,E

Rm

I1 = 0.5 + zE(0.5 - 1/t)

Integration step-length

For convenience in plotting and the saving of computer

time, the integration step length is varied as integration

proceeds.

Output

The numerical integration procedure generates values of

z.5, and Rm (or V, together with the corresponding value of

These can be converted to output relevant to a wall-jet by the

following steps:,

1. calculate the value of zmax/zE corresponding to

2. then, (umax(11C) = (zmax/zE)zE(uc/y

3. calculate value of c/2 corresponding to -e' 4. (y1/2/y-c) = ]./2 RGII/(ueg) °

203

Details of computer prorrrams

The computer programs are composed of the following sub -

routines:

1. MAIN:- This is the routine which reads in the initial

(experimental) values and computes the initial values of

zE

and Rm (or RG) using procedures given in appendix 8. It

also does the output of generated values in the required form.

The step-length is chan7ed in,a specified fashion as the

integration proceeds, by this routine.

2. Subroutine DFQ:- This is library subroutine which

effects the Runge-Kutta numerical integration.

3. Subroutine DER:- DEQ calls on this subroutine for the

computation of derivatives of the dependent variables at a

given station.

4. Subroutine SP:- This subroutine computes the value of

-S. for specified values of 11,-e,and zs It can use either a

cosine or a linear wake profile.

5. Subroutines VT , and WIDTH:- The former provides the

values of uc and clu_/dx; and the latter, values of R and

dR/dx.

6. subroutine LIThE:- This vives the values of zmax and F / '1/ 2

corresponding to a given value of -I?:

6. Subroutine EFUNC:- This computes the value of :E and

dE/dRr corresponding to a specified value of Rr.

2 04.

APPIMIX 10

Solution of tae cp-transfer problem

(a) Thermal Effectiveness of the surface

Equation (1.3-31) can be written, for the condition

Scp,s = 0, as:

mod (R R ) = T1 0 x

This can be integrated with respect to Rx to give,

R R(1)21= const.

i.e. (Ts - TORGIgo. = const. (L..10-2)

Under the condition of no transfer corresponding to p

at the wall and equality of the p and hydrodynamic boundary

layers,

(2.5-6) OE 1

and (2,6-4)

zTo 1.5zE

IQ21 = + 3 + ( 1 - (- +

-r

For some distance downstream of the slot the value of (Ps

is the same as that of stuff injected from the slot,

i.e. (PS20 = (PC (1..10-4)

If R = R0 at the last station at which this condition obtains

and suffix 0 denotes tho corresponding values of other quan-

tities, then

R0(93,0 - (PG)%-,0I0,1,0 = R(TS- cG)RGIQ,l

... (L.10-5)

3zE 0 . 8945 zE, 8

205

and aF,.ain, because of (A.10-0

R0(cp0 cf&RG90I091,0 = R((ps - TG)1IQ,1 (A.10-6)

93 (PG _ RG90 IG9190 To - 9G

0 •

u R RG IQ -

In the present application cp stands for enthalpy, and

E. is referred to as the thermal effectiveness of the surface.

In the present

I0,1 zE[iL(-3

Hence, RoRa,ozE,o

case:

1

t Fn3

zE >>

1.5)

1;

1 -

(1 (

therefore,

0 . ft.3? 45)1 _

. 10- 7)

+ (

_ .,5\ L.) (i £0, 0 X0 45)1

•-• 1 R RG zE 1±.(-3 1 . 5) • (1 - -t- ( .8?1-1-5)1

. (

(b) Heat transfer from the surface in the present experiment

Important geometrical and other details of the system

are shown schematically in figure 7.12.

The differential equation governing heat transfer from

the surface into the jet is,

d c0,1 + R d( dR R) = Js/(pi) ... (A.10-8) dRx 9,1 x

which is a combination of (1.3-31) and (1.3-30). This can be

rewritten with the aid of (1.3-27), as

dR Pc's - TG)RGI(;),]j = JAR/(pu) ... (A.10-9)

206

(2.5-6) gives

JS (1-9E) zE = (Ts - 9G)

iftp 0

If the conduction in the plate is considered onc-dimen- 2

sional, i.e. !3_2 arc negligible compared to the x-wise ay' a 9 ay2

derivatives; and the heater supplies a flux of juE' then,

kmt d d( rns - PG 51 E c R J = J" + — dx dx

is the heat balance equatidn for an element of the plate.

Here, km = thermal conductivity of the plate material

cp = specific heat cif air, and

thickness of plate.

Also,

R = x + xC ... (A.10-12)

Temperature traverses made within the jet show that a

linear-wale is suitable for the temperature-profile; therefore,

IQ,l F1 F29 (A.10-13)

a

a3(1 zE) 1zE

'

(A.10-10 '

and a2zE alz,

F2 2 _e , - ( 1 - zE) a3 • (1 - Zr) /6

Fl

000 (A.10-15)

with

(A.lo-lo)

where,

a1 a2 - f- 1.5 ; a3 0.25 -e,

207

For solution by computer, using a Runge-Kutta procedure,

the equations arc recast as'fbllows:

Let D

PS - PG ( . 10 -17)

F

DR I 0,1 O 0 0 (L.10-18) and (dD/dx)

O 0 • (L.10-19) We have also the transformation relation,

d dalx

p, d PuG. dx ... (1,.10-20)

(A.10-19), (A.10-20), (A.10-10), (A.10-8) and (A.10-12)

together give,

\ I! dF pD(1 - Q

E) z-k2 u

G F _ _ dx _e e 01J, X -

9

(A.10-11), (1.10-19) and (A.10-10) give,

[7-93 ( 1 - Qs) DzEK2 c

jfl _ dx 00 4.f' E kmt x

x•C ▪ — ( 11.. 10 - 21)

xC • • • (A.10-22)

To summarise: differential equations (A.10-19), (A.10-21)

and (1.10-22) have to be solved simultaneously with the hydro-

dynamic equations.

Initial conditions:

In addition to the usual hydrodynamic initial conditions

the following thermal initial values are required: D; ; 1(41; 9

E and i. Of these, D and arc obtained from the experimen-

tal9

va-Mues of Ts- TG and dTs/dx at the initial station.

If we choose thefinitihl station at the point where the

flux interchanged between the jet and the surface reverses

208

direction; then at this station the surface is adiabatic and

(1.10-10) gives

G = 1

Hence,

IG,1

1 F2 '

where F1 and F2 are as defined by (2.10-1/4 and (1.10-15) , and

depend on hydrodynamic conditions only.

41 is evaluated by the use of the value of P found from

the specified P-expression for the rough surface, corresponding

to the initial value of Rr.

Execution:

the Runge-Kutta procedure advances the integration

by a stop, we would have new values of P ZE, F, D and LS .

Then the corresponding value of 1(41 can be obtained by means 9

of (A.10-18); and F1 and F2 can be evaluated from the values

of -rand e which correspond to the new values of E.6 and zu.

Then the new value of GE follows from the substitution of

the required quantities into (‘1.10-13).

Now the integration can be advanced through another step

and so on.

.se

.cal •ity

e•ss-;y

able 1•'.-

3ure Lents

. . . • —

• • • . . .

. •

:is

- .v.•‘..‘

,

,•-• -

1g ..... ,-, •-..,"'

in Dr A ...

•• -• , •••.... -•

- • • .. •

term- R.. P. -

, /

- .- -

.,.•

Is' / ,,

_ ,

, ... ,,-

.,,-

Smooth surface

Rough i

Blowing 'uStion

Surface activity

2-Ph flow

Chem acti

Comp ibil

Vari prop ties

Pres ()Tad

Mixi Jaye

Fran numb

Isot al;

VA

RIA

TIO

NS

M

AIN

S-T

RE

AM

2o9

SURFACE VARIATIONS

Firfure

*-Reynolds Fluid: fluid in which Reynolds Analogy is valid.

2 10

figure 1.1: General f low - configuration

I_

figure 1.2 : System of co-ordinates

composite profile

wall component,

zE T

wake 1- zE component

• 5

• 5

1

z

wake component

1 —zE

_L

z wal I

component

composite profile

• 5

zE

Boundary layer , zE < 1

Wall jet , zE >1

figure 2.1. Assumed velocity profile schematic

2 1 1

, ..01 2 5 A 7 B 9 .1 6 7 a 9 1.

Y/ Y1/2

figure 2.2a: Smooth wall jet velocity profile

.1 .01

--- profile of Bradshaw and Gee [6] .

logarithmic wall law + linear wake

figure 2.2b: Smoth wall jet velocity profile

umax 1

Al

A. vE

1 ,

figure 2.3: Assumed mixing- length distribution

2/4

30

20

10

0 1 10 1• 10 104 y+

figure 3.1: Couette-flow velocity profile comparison with pipe-flow data

..„ -Taylor von Korman

Martinel I i -Reichardt (1940)

Murphree Rannie

Prandtl .--'

i 10 20

figure 3.2a 30 40

+ LJ

Rasmussen and Karamcheti ,.Lin et al.

Reichardt (1951) ,,Wasan and Wilke

Gowariker and Garner (high Deissl er

.-Petukhov and Kirllov Lin et al. Spalding and Mills

RD)

10 20

figure 3.2b 40 aZ

104

103

1

102

7

r • . . .

; A 0

. S.

. • . .

II

.. ,

. •I • .. •

•

P=9.0 [(6/0-o-

P..- 9-0go/cfc)75

-1r ).75 - 1 j + -28 exp(-

- 1 -1j

-007o-/o- )]

/ Z

2 Z1

Z .

.

.

.

. .

figure 3.3: Variation of P with a- ; d-o =

1 10

1 1

1 •1 10 103 4 10 102

.6 • •

• • •

•

>"

• • •• • •

•

,,

•

V

6 .

• r

•

•

P.9.27Ro-/a0 P.9.2 7[(a-/

175 1

51--/7=,

-.1]

-1][i+ •27exp(-0070r/cr0 )]

/

7 figure 3.4: Variation of P with a- ; ao = .9

10

1

10

00

10

10

Prandtl- v9n Korman Hofmann

ReiOardt

''Present recoinmendation

fi

10

102 1

figure 3.3 . Comparison of theories with experiment

103

10

Gowariker &.GarRer Mills

Kutateladze

Present reco

Wasan & Deissler_--1

,- mmendation ;

Wilke„-

-:, • Rasmussen Petukhov

& Karamcheti & Kiri llov

r,1

; I I

Present sim Dl if ied recomrnendction

. ,

i I I

I I I

•

1 10

102

103 104

fiaure 3.6 : Comparison of theories with experiment

N O

01 0

• -

in

n

:11111r1: :1:

. 1 :

:

• •

.

• : :

- r t,- •

•

: : i

Ul

:(f3 a)

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.....

• 11.1.

• .

... .

• , .. ..„;l: .

1: II;

" •

. ...; li• ..... • ::-1:11 • : :

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tr,-.::•••1:4%Ti.::;_, ,. .

tr

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it,i: " "

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t,:t: -1' ' '

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11 ill: t

pt-. " ..11:'!: 111:11,111 "., I::

titt;;i:: T''• •111._:1-2-1-I-

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..... ••,1

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.„. =1:11

. ::;;:•' . .....

:It ..... :77.7

1 1

7 -I

• 7

0 •1 • 2 .3 .4 .5 .6 . .7 z

figure 3.8: Comparison of turbulent intensities in pipe and boundary layer flows (from [66], [70] )

pipe • -- boundary layer

.2 103 104 105

RD 106

o.8 0

.6

.4

fig.4.1: Drag data of Nikuradse [53]

10

5

5

5

5

•001 1

2 3 4 5 6 8 1

10 102 2

2 3 4 5 6 8

Rr

2 3 4 5 6 3 4 5 6 8

224

figure 4.2: E-(Rr-} from Nikuradse's data

• Nikuradse (Expt,)

Rr Rr figure 4.4 figure 4.3

YrI Yr Y ru Y rg lower upper

I limit I limit nominal general

figure 4.5

T

i

R ru Rr

E

frequency

I Jr

figure 4.6

i

1 ' Nedderman & %.,,,, \ Shearer

(Theory)

225

3 4 5 6 2

Rr 8 9 I

10 4 10 6 7 8 9 2 3

figure 4.7: Curve-fit of sand-grain E-data in transition region

10

•

: I .

1 • • • . • - . • . • • • , '•

•:::1:::

..•

: 1..

r:

_ _ •

• .. • • • • . •

• ' • !"'. •?

4 5 6 7 8 9 I 2 10

-

Sok

2 3 4 5 6 8 1(:) 2 ii

4 c 3 10 0 103 2 3 4 5 6 8

Rr

227

r o

P ,

I

o I

i --,

TH 1

J 1 7 I -

1 i i r 11

L -H i 1

4_ 1 i 1

Li r i 1 . i

-I- r

4L --,--- mean curve o sald rough Bess Lao

4 4.4

H

Tll

i i

figure 4.8: E{Rr} from data of Dipprey [15]

2

.001

10

6 5

E 4 3

2

1

6

4

a

6 5

4

3

•O) a

6 5

4

3

I r t INN m 6 5 .01 iC 6.:1 a 41.••wa.aest.iito.P.4.•acriosili.4 6 64"..4'Ilialiffic14111.4111rdranrIll Iltr:7.611Mesinummgravr.d...idderdommodammipun Nino

D D 41104101A .01.1./ 1. • 14110101119014•44.4,1111U

G d us.0.1"

103 104 .001

' I 101 .41. 1,1, 11.i .14]

rtt

105

figure 4.9b: Drag data of Stamford

---"-.- - ..• ": 1•'1':4 i.

T 4:511i -:- 4. =--". 4--,t7.,11,-._-__,_-+ - _1F-1 4-•-: ,--.,--:- .. c ' .=•• ti= "112:: gill: • t „ .i.,

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i,

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s• . 1. -

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104 RD 10

figure 4.9a: Drag data of Cope [12]

2 sS

.01

2ss

.01

•001 103

218

103 102 2 3 4 5 1( 2 3 4 5 6 8 10

229

.. fE

. 1 , , LI:

, .-•- ,

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-"- . • T 1 ii

4 -4 ' foiliNt_ rough'; sand rain

through- mean U de

Stamford ,. •-/ t lipcteri8tic

.1-- ' ' 1

i , , r ., , , , . , „ r. aia~ a _ fi • •'f1 , ini••• sehmoi ini 1 t ___.: . , IIIIM I , I ' 't-----7 - -t - , - 1 III

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i- 1 ; : 1 -I- I

t r.

. ,--

L.._,__1_i ! _________1

_.._ _.; .. . . . ; . 1 ;

-t-; -1, 1

-1 i . , 1 ; ' 1 , , 1

i

Rr

figure 4.10: EE Rd- from data of Cope [12]

•001

10 8

E 6 5

4

3

2

1 8

6

4

3

2

.1 8

6

5

4

.01 8

6

4

3

2

figure 4.11: E(Rd from data of Stamford [84]

10

Rr

230

2 3 1 5 i02

2 3 4 5 6 103

.001

.91

5

4

11111

1111111111111,111

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nd g air tic

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14

3 5 6

2 3 4 5 6 2 3 4 5 10 2 • 4 5 8 1, 0 2

23!

r r T

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Rr

figure 4.12: EfiRr-} from Kolar's data

•Q1 .1

1 a

6

5 4

3

2

1 B

5

3

2

1

6

5

4

3

2

•1 8

6

5

4

3

2

equa ation tion

pi Yr 5 17 5

232

1

•-• .0

6.25 25 55

10 .

I ' I

ti

2

/D

6 5

4 ( 4.10 -1a) (4.10 -lb)

(4)

2 50 70

3

2 3 4 5 6 3 4 5 6 8 102 8 3 4 5 & 4 1 8 10 2 '103 2 Rr

figure 4.13: EfIR } for 2-d distributed roughness formed of wires

.001

1 8

6 5

4

3

.1 e

.01 e 6 5

3

2

6 5

4

3

2

AJowains aff15!1

U 9 S t c 9 S V E

100' 8 9 S b E Z 8 9 S r E

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•

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S

LO •

S

S

3 S 9

9

OL

2— 0

• data from Bettermann [5]

relation derived from linear velocity profile [83]

I I I I I 1 I I I I .4 .6 •8 1.0 1.2

figure 4.16: ZE

data of: Brunel I o •

Bettermann 0

proposal of Nicoll and Escudier; from [23]

234

1

0

4

H12 figure 4.15:

r

4

400 500 700 800 900 1000 1100 1200 V rr

236

- _ 71 -

_ h ,

...

_ • ,

. ...

. • __ , _ _

: _ _.

.

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1_

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:

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I I .

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I

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it

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.

, c ornpqison-of___fropl _-

:la

heory--of—Spaldiri eAp

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_

w ith ;dicta of Better reianh

1

I

!

1

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1 10

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ri,

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ftgyre 5.1: N Rr , c3 } 1at° i for

tri .

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• :

4 4 I • .1, '•

102

Frtuation_

• • • ; : : I , S.-1:• ••- •

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1 . :

: Data. •

.tpi 2 3 4 5 6 7 8 9 .1 2 3 4 5 6 7 a 9 1 2 3 4 5 6 7 8 9 10 E

figure 5.2 : P-( E 3 for sand- indentation roughness

:It• ; 1 i r

2 :001 3 3 4 5 6

a 9801 7 2 5 3 6 4

E

41 7111177:17 1 o

4q 11111111111 1111111111111 111111' . INWHIMMIERNIMIN

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figure 5.3: PE) - for pyramidal roughness

100 • : :

1 I : I

1101 3 2 2 3 2 4 5 6 7 8 9 1 4 S 6 7 89 .11 3 4 5 6 B 9 10

• " '

: i--f-

10

........

•

r 1.1 0

•oYe

: I :

I •

• 1

figure 5.4 : PEE} for v-groove and natural roughnesses; a = .7

11

Principal dimensions delivery pipe: 3 in. i. d., 10 ft. long nozzle: 5.9 in. d. adiabatic plate: 3 ft. sq. heated plate 2 ft. 11 in. d.

adiabatic plate

heated plate

bel lows nozzle

61bil

11110 delivery pipe -

perspex' - glass-wool

slot --/

expanded polystyrene

aluminium plate

heating pad

sindanyo' metal casing

figure 6.1 : General arrangement of Apparatus

traverse unit,

242

figure 6.2: Working section

3in. d.

243

5.90 in. d.

\ \ \ \ \ \ \ \ \ \

figure 6.3 : Sectional view of nozzle

u f t

300

244

200

100

y -,12 in.

22 in. 29 in.

I I I I 0 1 2 3 4

y in.

figure 6.4: Velocity profiles at slot

thermocouples

1 4444444444W dt

static pressure tappings

figure 6.5 Smooth plate - underside

temperature probe

pitot probe

•

micrometer head

figure 6.6: Traverse unit

9 fl

Static 6

pressure at wall

inches of paraffin

2-47

figure 6.7 : Static pressure along wal

x inches from slot

u ft/s.

360

320

figure6•8 : Set of meciurecll velocity _1(?rotiles : Radial wall-het on smooth: walli

_ 280

240 yc .= •065 in.

distance frOrn slot (inches)

200

45'

A 1.10

1.69 ■ 2:06

160

at slot • 3.07

4.56 • -- 6.34 A 9.81

120

80

' • if

I i t • 1 i.

•

: 11 • i - . % T 1

T-•

! ! • figurTe Q.9: Set Of rneaspred! velOcity profiiles: -.

Radial w011-jet :on emery sui:face . 1 , -4-- •

i . ' !

I • 1

- 1-- I I r • 1---- ----- . , i !

. :

rC-

4-

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:

I • i ' . I

1

! ;

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1 ' 1 1

-

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-

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L 4 , 1 ,,

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1 ice.) ! , : i — • 1 :-- ••••a i . 1 i --:

1:: •

"59 '‘ • \ •-•94 • •

100e-

• 7

yc = .15 in.

R = 2.55x104

Heater input = 450 w. V-groove roughness

4* 60 90-

6.625

9•78

80- •

0 •1 • 2 -3 • 4 • 5 • 6 -7 •B .9 y in. ts,

V1̀ figure 6 11. Temperature profiles on heated rough -wall .

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figure 7.14: Rough wall boundary layer : prediction

data of: Perry and Joubert [56]