The Influence of Prandtl Number and Surface Roughness on the Resistance of the Laminar Sub-Layer to Momentum and Heat Transfer

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 indicate 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 understanding. 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 D. F. Dipprey, of the Cali- fornia Institute of Technology, for providing unpublished data from his work on surface roughness; and the members of the Technical Staff of the Mechanical Engineering Department, Imperial College,-, helped in the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 hydrodynamics 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 theory 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 incorporated, 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 velocity profile expression,41 + 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 following objectives were set: (a) the collection of heat transfer and drag data for smooth pipes and extraction from them of a relationship between the laminar Prandtl number and P; (b) the collection of all available information on flows past rough surfaces and devising a means of estimating 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 present state development to the calculation of hydrodynamics and heat-transfer in the conditions of the present experiment 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 subsequent 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 turbulent 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 molecular properties and turbulent; body forces, such as those due to gravitation, absent; 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: dua au au: pu + pv 77 ay (1.3-2) dx dx 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 1-Y6 dyc, r ... (1.3-24) pu dy= mS PvG PUG dx 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 respectively. 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 boundary 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 f- YG 1. 1 d ou(uG - u) dy dy R -!R (1.3-5)

Multiplication of (1.3-2) by u and subseaucnt integration with respect to y results in the 'interal kinetic- enerqy-deficit eouation": F ' Yri 2 2 --, i u u , au 1 a puk-/ 2-G - --) dy, = ti dy R da IR 2 • r ay L 0 J

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: 15. u/uG (1.577)

Y/YG • 0 0 ( 1 .3 8 )

a T/(Pu,g) 0 0 ,C ( . 3- 9 ) m 11V(PG) 0 0 0 (1.3-10)

_ (dyG/dx) C 0 ( 1.3-11)

-1 Il z ... (1.3-12) -o r I 2,. 12 z ucf; ... (1.3-13) JO 1 13 = z3d ... (1.3-14) -0 r I slidF., ( 1. 3-15) J 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) R .z- 3 (Ii - I3)RG , . . ( 1.3-21) x R (P/P-) f u dx ... (1.3-2 2) x o

With the aid of these definitions, the differential equations can be written in the following forms:- 14

Mass conservation: dR d(ln R) m R - m- mG ... (1.3-23) dRx m dR x Momentum deficit:

dR2 d(ln R) d(ln uG) + R + (1 + 1112)R2 - + sS dR x 2 dRx dRx Kineti -energy deficit: O 60 (1.3-24) dR d(ln R) d(ln 3 + R + 2R m + • (1.3-25) dRx 3 dRx 3 dRx If we are to calculate heat- or mass-transfer, the differential equation governing the conservation of a property p has a to be added to the abve.( collection. In the case of heat transfer, 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: YG il ... (1.3-26) tRr pu(c - cps) dy.} JS 0 where jit is the flux corresponding to the property (P• We can rewrite (1.3-26) in terms of: I ... Rpl l = (9S - (PG,hG 0,1 (1.3-27) where, 5q) 0,1 Az d ... (1.3-28) with,

(P )/( PS - (PG) and -r11 S Scp,S P' ((PS - (PG) so that, (1..3-26) becomes,

dR 9,1 d(ln R) ScD ,S ( G 0 0 (1.3-31) dRx 11 ,1 dRx (PS - (PG) According to Spaldingr82j, 183], we can generate solutions 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 definitions 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 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 mainly 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-profiles, surface friction and heat-transfer rates. The auxiliary 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: s1/2u+ (1 z = z )(1 - cos v0/2 • 0 • (2.2-1)

It is readily seen to be a superposition of two components: s1/2 +u which is called the 'wall component'; and (1 - zE)(1 - S

18 cos 7r4/2 which is called the 'wake component'. The dependence of u+ on the non-dimensional distance y from the wall, which is defined as,

Y+ = Ydr(TsP)/11 V 0 0 (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 (2,2-4) and. 1/2 RGsS + 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: z (1 + lnE) + L(1 - A)(1 - cos 77-4)/2 + Aq(1 - zE) E .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: = C1(1 - zE) , for 0 < au _5 1 -mG ... (2.3-1) 0 for 1 < z and = C2zE - C3' E 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, (2.3-3) C2 = 0.04 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 equation, (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 can be generated from a mixing-length is valid for zE > 0.6, 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 — . (2.4-2) (3V.

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

+ 1 2 ( 9 -9E) W i;)- ... (2.5-1) ouoss/ + 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: + + P) 0 0 0 = 00( (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. 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 e O 0 0 (2.5-3) 1 - g (1 - gE) (1 + P— In 8, ) with,

/(TS O00 ( 2 .5-14) (C TG) 9G) 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" (1 - O ) E E ... (2.5-6) T'S = Puc(Ts - a 0 •t'co 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: I 0.5(1 + z ) ,.. (2.6-1) 1 1 1 2(1 1.5 2 ci 12 A[-3 + zE(-3 zEk-3 +

+ (1 — A) [ + z (-1 0.411) 4.. , (3 1.589 2 E 4 ?, / L'E` 8 ,t/ + e, 2 (2.6-2) 24 r 1 (0.25 + 0.833 21 10.25 4. (0.25 °•333) 0.75N 3 -e/ + )z 7, 0.326% + ‘( 1 - A) .3125 + (0.1875 - izE

0.5795, 0.461)z - (0.1875 t/2 E

+ 2.0945 + 5.539 + 6)z 3 (0.3125 t,3 E (2,6-3) and E r i z7, 1.5z, 3zE 0.8943z I 9 = - 4), ') + ( 1 - A) ( 8- + 0 1 gE[A(T + -3" 8 - /1 2zE (1 - 0E) 1A(0.25 + 0.75z - ) L E ,Ci '

(1 - A) (0.2055 + 0.7948zE

0zE) 01 • (2.6-Q 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 auxiliary 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. 25

SECTION II 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 determined 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 velocity 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 boundary layer to heat-transfer resided in the laminar sub-layer; and a comparison of the P-functions derived from these analyses 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: + u = 114-(-y4-} ... (3.2-1) where, + u a u/Nr( Ts/p) ... ( 3.2-2) and + Y ::,* YtT(TsP)/11 ... (3.2-3)

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,

= BOO (3.2-6) t 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: [ 0t -= 1. t/r 0.. (3.2-7) Dimensional analysis leads to the result: + (3.2-8; at = at(-y ' a} 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 derive that:

1-it/11 = —t = dy4-/de and Et/at = aY+/'?+ where, (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: r -u_+ 1 + = 0 + t du ... (3.2-13) Jo 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,

•YR + = 2 u'(1 - y-474) d(37-4-/y) ... (3.2-16) .!0 in ter is of dimensionless variables, yilz- beinc, the dimensionless pipe-radius. On substitution for u+ 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

) = l/u+) 2 sp „to?I:3 2 ( (3.2-13)

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

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

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,

,( 1.5 1.25\ -+ 0 0 0 (3.2-22) = L'O‘ - 2 -+/ U. By suitable manipulation of equations (3.2-17), (3.2-13), (3.2-18), (3.2-22) and (3.2-21) , we obtain:

1/2 s 4. 4_ 1.25 s \ (a t - 00) au+ 772 ‘(1 1,e2 Pi T,P ... (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 0 P + (1 + 1.25 , ,1/2 S 01 2 °P)1'1,3 (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 + 2 sP 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 variation has been specified; some of evaluated after a 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. variation and devising a Instead of starting from a of 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 recommendation 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 variations can be cast in the form: and u+. Many of the (3t 1 1 0 a 0 t ... (3.3-1) 1 1 f(.11+, 0 0 o 3L where 'f' stands for some function; or such a relationship can be generated from:

at = ot 5/-.1- 9 RD'1 4- „. (303-2) and u = 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 interposed 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 suggested 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 comparison 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 a0 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, according 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 s has been introduced to account Ag 2 P 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, corresponds 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 _ 1 -1 P = j(55 - 1)1 11 + 2 (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: 37

-1 p 1)11 +.go ab(u+)i-b de 0 0 0 (3.3-)4) Jo which gives, sin77./b) 1 P -> 0 • 0 (3.3-5) (2a ) ab 0 L b I -> 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 mathematical 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: 1/2 s 0 (1 1.25 0 a a (3.4-1) 0P exp = S p,P sp 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: -1/2 /2 = • • • sP 2.46 ln(RDsp ) + 0.292 (3.4-2)

TIE value of 14. is taken as 0.40. The highest value of /s given in the literature is 0.41; therefore, in its present position it cannot cause significant inaccuracies owing to

uncertainity of its value. Two values of 0 have been tried:; namely, 0.9 and 1. The values of and a which have S(P3P' sP 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 at the lower end and the clus- 0 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 o P for thew 0 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 3/4 _ 0 0 0 (3.4- 3 °OP = A1f(6/60) and the full lines:

A rt 3/4 - 1 + A2exp(-A3a GOP 1L 0 ( 3

The values A2' and A3 al-e as follows:

C A A O 2 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 experimental 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 conduction 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:

1 ... (3.5-1) u = -:-. ) 1 ax r p 0 00 ari the eddy- where r is the distance from the pipe-axis and u viscosity. The axial conduction term is said to he negligible 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) B 0 00 (3.5-2) P (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 coefficients calculated with formula (5.4-2). These P-values are shown plotted, on figure 3.7, against s1/2 with a as parameter. 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 between P values for the two boundary conditions are shown in Table Li.. The following remarks can be made regarding the comparison: (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 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 experimental 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-boundary 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 complicated 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 theories 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 application 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 ^(j + 0.28 exp(-0.0070/00) 1 ) 3/4. - 1 ... (3.6.-1) = 0.9. with o0 Where a simpler formula is required, especially at Fran, dtl/Schmidt numbers greater than about 50, the formula, = (3.6-2) P 9.24(c/60 )3/4 - i can be used. Stanton numbers for heat- or mass-transfer may be calculated by the use of (3.2-23) and the recommended P-expression; a value of 0.4 is appropriate for K, and the drag.- coefficient 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 velocity-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 systems being shown in figure 3.8. Hence it would be reasonable to apply the P-function derived from pipe-flow data to boundary 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 boundary 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. 47

SECTION III 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 workers 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 between a surface and a fluid-stream. It is also necessary to examine whether the influence of roughness extends sufficiently 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 possibility of having a Freat variety of shapes, sizes and distributions of roughness elements. Nevertheless, it will be seen that these fall into groups, each having a characteristic 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: irregular; and regular. By regular we mean that all the elements 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 surface in the presence of an adverse pressure gradient acting on the stream. They found that the wake-profile could be described 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 layer. :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 0.0 (4.4-3) with, } AN = AN(-Rr • (4.4-4) If we write the velocity-profile expression in the familiar way,

u • ln(Ey+) • ( 2 . 2- _3) then, exp(i.f_Ar) E = R (4.4-5) r Equations (4-4-5) and (4.4-3) together enable us to derive from pipe-flow an empirical relationship between E and Rr 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

u = 1 In y au - L\ 11 ... (4.4-6) K. Ts/p

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

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

BH -= /4, In E-jui 0 • 5 ( • 4- 7) so that, u - in(E, /E) . (4.4-8) fl(Ts4) The E 1'2,, relation is a suitable form of the drag correlation 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 transition between the hydrodynamically smooth and fully rough conditions; (iii) the fully rough conditions being represented 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. 53

Although Nikuradse recognised the presence of the transition region, he did not delve deeper into it, but fitted a piecewise linear charactef istic 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 velocity. 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 is yr2c' given by a laterally inverted cumulative freauency distribution; and 1 y r,u I N dyr,g ••• (4.5-2) jY r 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

/'J R shown in figure 4.6. (Note that the Ta r 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 s and s E M are the drag coefficients which would obtain if the same main-stream flow existed over fully roucrh and

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(R / ) + (1 - a)Eia ... ( 4 • i r (1 1 5- 7) 1_ __J Application to Hikuradse's data If we assume a simple auadratic distribution of sand- grain sizes, i.e., N = .A....IX 1 - X) ... (4 . 5-8) with,

X = (Yr,g Yr,1)/( Yr,u — Yr, 1) (4.5-9) then, c = 1+ 2X 2 (4.5-10) 3 - 3Xc ... where,

Xc = (Rr,l/Rr)n(Rr,u - Rr)/(Rr,u - Rr,l) (4.5-11) Equations (4.5-11), (4.4-10) and (4.4-7) —toge ther

the E(--Rr R r,u Rr,1 and have to be chosen to 7ive a 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 examination of Nikuradse's data show a distinct curve for each rough pipe. They all, however, form a narrow band of points because 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 differences 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 roughness 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 --R cu-ove; only the size of r 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-dimensional. 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 machined internally to produce the roughness. Hence there is the likelihood that Stamford had better control over the uniformity 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 conditions. 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,

34.02AR r + 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,

) 1/exp(0.123 + 0.0082pr/Yr) 38.0(0.0362/Ar ... (4.10-1) which is valid for p r /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 E = SaR. O expl7).94 - 4.90 ln(p /y )1] e.. (4.10-2) r L r r 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 expressed in terms of the smooth wall drag cofficient under identical 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 dependence 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 written 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 cylindrical 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 required in the process of generating solutions to the hydrodynamic problem starting from a velocity profile assumption. 65

If the differential equations given in sub-section 1.3 then an are written in terms of shape factors H32 and H12, auxiliary relation involving these become necessary. Nicoll and Escudi'-r [52aj have recommendedt, 0.0971 0.775 H32 = 1.431 - H 2 GOO (4.11-1) 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 equation, 2 H32 0.25(H12 + 3) /H12 • • • (4.11-2) which is derived from a simple linear velocity profile assumption. 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 thickness 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 description 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 picture in addition to yr/D; namely, those describing the longi- tudinal and lateral spacings of the elements. Usually one of these haw&5 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 and z recommended for:smooth surfaces can be applied. A H32 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] concerned themselves with uniform, maximum density type roughnesses. 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 ridg es 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

cavities which formed the roucliness. The scouring action of the eddies extracted heat from the cavity walls according; to a law of the form:

S - 0 - 2-1) c yr ( 5 where 3 is an appropriately defined Stanton numbel, for the c flow within the cavities; with the result,,

3 - 1 cl ( 5 . 2- 2) (P'S 73--S - = where, 1 / A = - 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:

s S .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 increases, 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 R r 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. So we write, a a9 a-7, 1 (1 + (171E„,) p P = b /F 1/E,) M ... (5.2-3) the suffix H denoting smooth surface conditions. The forms of the component terms have been laid down from a consideration 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, 0.695 , 0.359 .0 3.15 o 0./E - 0.116) + 0.274 PIZ°

... (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 relationship aivine E in terms of Rr, which, for Dipprey and

Sabersky's data is, -7,1/2 2 E= 0.00-2.1089 a(Rr + 3.4_8) + 0.013327(1 - I .,. (5.2-5) a -3 v!ith, . 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 presentation. 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 literature. The scheme of correlating heat-transfer data in temas of

P re(Taires good control of Prandt1 numer, or of Rour lmcas 75

Reynolds number, Rr, during; experiments 76

SECTION IV 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 experiment 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 viith amount to a ch=r.e in the rouThnese, Reynolds number' Rr' 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 additional 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 knowledge of the P-function for air floing past the riven surface. 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 hydrodynamic 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 aluminium' 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) Positioniir,- 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 inaccuracies 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 thermocouples 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 'Aralditer . 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 junctions. 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 temperatures; longer times being allowed for smaller slots. This was necessitatecU firstly, by the desire to avoid any uncertain 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 manometer 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, temperatures 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, although 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 temperatures 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 figure 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 application to radial flow i icon and Escudier [52a] report on the successful prediction 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 attempt 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 characteristics of the same form; but with the aid have E(..Rr 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 than in the case of simple boundary-layer flow. of Rr 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

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,

(7.2-1) 1, zE, For the usual values of Al• 0.08), over approximately . Thus, 4/5 of the boundary layer, has the value of 11.

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 ( 7 . 2- 3) z 47,E 4 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 Vz3 we should change the value of the proportion, E9 ..X1 in 1 1/2 C2 for radial flow 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 velocity (--10-6 x slot velocity) Both methods of solution, namely, the 'entrainment method' 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 comparison 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 accuracy; these are not shown. Other pertinent details are that 1} umax and y1/2 were used as initial values and that the E(.R 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 velocitL_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' profile, 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 resolved 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,

H .H 0 06 32 12 = 1.096 (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 expressions for R and Ig,i, together with initial conditions, 9- 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; therefore 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 initial 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 x 0 /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 obtain 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 simultaneously with equation (1.3-29). The heat-balance equatien 100 for an x-wise clement of the plate is, kmt ar d( - gh)i 711 4'S J`; = 0 + - R 0.. ( 7.6-1) S E Rc dx 1 P dui whore J"Jii is the heat-flux from the heater-pad to the plate. 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 reversal 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 (appendix 10). The other initial values required are those of the surface temperature above the ambient, and the radial temperature 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 mainstream 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 analysis as applied to smooth-pipe-flows from which the above relationship follows. Previous theories have been summarised and compared with experimental data. This has enabled the derivation 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 distribution 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 hydrodynamically 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 X and C, have been determined 1 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.

109

NOMENCLATURE

Symbol Meaning Eauc.tion of occurcne 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

, 111 9 Coefficients in the recommended smooth '-2- 21- 3 (3.4-3) surface P-expression (3.4-4) + b -b Coefficient of (u ) in the series ex- (3.3-3) 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

BIi Term used by Hama in the correlation (4.4-6) of velocity data of flows past rough surfaces. B -- transfer boundary condition Dare- (3.5-2) P meter

110

(2.571) C2 Entrainment constant 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 value of E for flov past a fully- (A.3-5) Ee rough surface

E- Value of F. for flow past a hydro- (4.4-7) dynamically smooth surface Shape factor (1.5716) h12 (1.3-17) H32 Ratio of kinctic-encr7y-thickness to to momentum thickness 11,12,13 Inte2,rals associated with the velo- (1.5-12,13 city profile and Inte!Jral associated with the Sr and (1.5-28) 0,1 velocity profiles J" Flux from surface into fluid stream, (1.5-26) associated with property 9 j Electrical power input to the heater E 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 additional resistance to 9-transfer due to the laminar Prandtl/Schmidt number 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 Reynolds number of pipe-flow based on (304- -D 2 pipe-diameter and bulk velocity of the fluid Critical Reynolds number of rouqhness (405-3) c,c elements, based on friction velocity,

(s Af( Ts/p

R Reynolds number based on boundary - (1..5-1a) G layer thickness and mainstream velocity Non-dimensionalised mass-flux within (1.3-19) Rm tne boundary layer Roughness Reynolds number based on 2) Rr (4.4- element height and friction velocity Roucrhness Reynolds number correspon- (4.5-11) Rrl', Rr,u 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 s a c Non-dimensionslised she r stress on the. surface if it were wholly covered with active roFaness elements s Non-dimensionalised M 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 Dimensionless measure of centre-line (3.2-17) vC.ocity in a

Velocity component in a direction ( 1 . 3- 1) normal to the surface vC Value of v at the 'edge' of the boun- (1.3-4) dary layer x Distance along the surface in the ( 1 . 3-1) mainstream direction Height of slot opening xC

X Normalised value of roughness height (4.5-11) y Distance measured from surface (1.372)

Thickness of boundary layer YG (1.3-4) y 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

Yr,l'Yr9 u Lower and upper limits, respectively, (4.5-2) of roughness height in a given sample of elements (3.2-1S) YR Dimensionless measure of pipe radius in Couette-flo,:: analysis

Velocity in mainstream direction (1.3-7) z Parameter in the assumed velocity (2.2-6) E 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 (7.5-2)

((Ps - %)/(QC

t Ratio of 'total' viscosity of turbu:r. (3 . 2-10) lent fluid to the lauinar viscosity Eddy diffusivities for momentum- and (3.5-1) Cu'- h heat-transfer respectively ivTormalised measure of conserved pro- (2.5-3)

perty, E (P - 93,)/(Ps - 9G) Parameter in the Q-profile (2.5,3) Prandtl's length constant (2.2-3) Non-simensionalised mixing-lenr,th (2.471) Value of in ouier part of boundary (2.4-1) layer Laminar viscosity (1.3718) Total viscosity in turbulent flow (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 77) of Total Prandtl/Schmidt number (3.2 (2.5-2) 00 Value of of in the fully turbulent 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

LIST OF REFERENCES

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3 Barnet, W.I. and Kobc, K.A.: Heat and vapour transfer in a wetted all tower; Ind. Eng. Chem., vol. 33 (1941) 436-442.

4 Bernado, E. and Eian, C.S.: Heat transfer tests of aqueous ethylene glycol solutions in an electrically heated tube; NACi VR E-136, (1945) ; Supercedes NACA ARR E5807.

5 Bettermann, D.: Contribution u l' etude de la couche limite turbulente 14 long de plaques rugueuses; Centre National de la Recherche Scientifique, Faculte des Sciences de Paris, Laboratoire Aerothermique, Rapport No. 65-6

6 Bradshaw, P. and Gee, M.T.: Turbulent wall-jets with or without external stream; A.R.C. 22,008, FM 2971, 1960. 119

7 Bruncllo, G.: Contribution a li ctudc dc la covcction forces; Publ. Solent. et Tochn. No. 332, 1957.

8 Bdhne, W.: Die VArractibcrtrac-unp. in nhen Fldssigkeiten bei turbulenten Str6mung; Jahrbuch 1937 der deutschen Luftfahrtforschung.

9 Chilton, T.H. and Colburn, A.P.: Lass transfer (Absorp- tion) coefficients; Ind. Eng. Chem., vol. 26 (1934) 1183-1187.

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10a Cole, E.H.: A procedure for predicting the adiabatic wall temperature in the film-cooling process, based on the unified theory; Report No. IC/HJR/26, Imperial C College, Ilech. Eng. Dept., Feb. 1965.

11 Colebrook, C.F.: Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws; J. Inst. Civil rngineers, vol. 11, (1938/39) 133-156.

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13 Deissler, R,G.: Analysis of Turbulent heat transfer and friction in smooth pipes at high Prandtl or Schmidt numbers, NACA TN 3145 (1950. 120

1L4 Dipprey, D.F.: Personal Communication, (1965).

15 Dipprey, D.F. and Sabershy, R.H.: heat and momentum transfer in smooth and rough tubes at various Prandti numbers; Int. J. Beat Kass Transfer, vol. 5, No, 5 (196'

329-353.

Doenecke, J.: Contribution a l'etude de la convection forcee turbulente le long de plaques rucueuses; These, Faculte des Sciences, Paris, (juin 1953),

17 Downie-bpith, heat transfer and pressure drop data for an oil in a coper pipe; Trans, A.I.Chem.E., vol. LXXI (1934) 83-95.

18 Draycott, A. and Lawther, h.R.: Improvement of fuel element heat transfer by use of roughened surfaces and the aplication to a 7--rod cluster; Mt. Developments in. Heat Transfer, A.S.U.T., Part III, Paper 64 (1951) 545- 552.

19 Eagle, A.E. and Ferguson, R.L.: On the coefficient of heat transfer from the internal surface of tube walls; Proc. Roy. Soc., St-ries A, vol. 127 (1930) 540-556.

20 T:de, A.j.: The heat transfer coefficient for flow in a pipe; Int. J. Heat mass Transfer, vol. 4 (1961) 105-11'

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21a Elrod, H.G.Jr.: Note on the turbulent shear stress near a wall; o Aero. Sci., vol. 24 (1957) 08.

22 Fscudier, M.P.: The distribution of the mixing-length in turbulent flows near walls; TWP/TN/1, Imperial Coll-

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23 Lscudier, H,P. and Spalding, D.B.: A note on the turbulent uniform property hydrodynamic boundary layer on a smooth impermeable wall; comparison of theory with experiment; A.R.C. 27,302, I 3462, Aug. 1965.

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25 Clauert, The wall jet; J. Fluid Mech., vol. 1 (1956) 625-643.

25 C-omelauri, V.: The influence of artificial roughness on convective heat transfer; Int. J. Heat Mass Transfer, vol. 7 (1964) 653-663,

27 Cowariker, V.R. and earner, F.H.: Analysis of heat and -ass transfer between solid wall and fluid streams at both low and high Reynolds numbers; A.E.R.E., R 4197 (1962).

28 Grele, iA.D and uideon, L.: Forced convection heat transfer chararacteristics of molten sodium hydroxide; ACA RN E53L09, 1953. 122

29 llama, P.R.: Boundary layer characteristics for smooth and roue: surfaces; Trans. Soc. of. Naval Arch. and fmrir.-. En7rs., vol. 62 (1954) 338-358.

30 Lanratty, T.J. and Enp;en, Interaction between a turbulent air stream and a moving water surface;

9 vol. 3, No. 3 (Sept. 1957) 299-304.

31 Hofmann, E.: Uber die Eesetzmassigkeiten der Warme and der Stoffiibertragunz7 in zlatten and rar_hen Rohren; For- schun7 auf dem C;e1Diete des In7inieurwesens, vol. 11 (194SI 159-169,

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33 Jenkins, R.: Variation of ed4 coLuctivity with Prandt:. odulus and its use in thr . predietions .of turbulent hea',

transfer coefficients; Heat transfer and Fluid Mechani Institute, 147, Stanford University (1951)

34 Kaufmann., S.J. and iseley, F.D.: Preliminary investiation of heat transfer to water flowinrf. in as electrical_ heated Inconel tul.)e; RM E 500, (1950).

35 Kays, W.M, and Leunrf, E-v.: Heat transfer in annular passages - Hydrodynamically developed tubulent flow wit'_71. arbitrarily preselibed heat flux; Int. J. Heat Mass fer, vol. 6 (1963) 537-557. 123

36 Kolar, V.: Heat and mass transfer in a turbulent medium; Collection Czechoslov. Chem. Commune, vol. 26 (1951) 335- 545.

37 Kutateladze, S.S.: Fundamentals of heat transfer, Mash7iz 1962; Eng. Trans. published by Perammon Press, 1955.

38 Lancet, R.T.: The efiect of surface roup-hness on the convection heat transfer coefficient for fully developed turbulent flow in ducts with uniform heat flux; Trans. A.S.M.E., Series C, vol. 81, 7:1-o. 2 (1959) 168-174-

39 Laufer, J.: The structure of turbulence in fully developed pipe flow; NACA Rept 1033 (1951).

40 Ludwieg, H.: Bestimmung des Verhaltnisses der Austausch,. koeffizienten fur -vfarme and Imruls bei turbulentenGreirz- schichten; z. fur Flu.9,wiss, vol. 4 (1956) 73-81-

41 Lin, C.S., Denton, E.B., Caskill, H.S. and Putnam, G.L. Difftsion controlled electrode reactions; Ind. Eng. Chem, vol. 43 (1951) 2136-2143.

42 Lin, C.S., Moulton, R.W., and Putnam, G.L.: Mass transf- ':Detween solid wall and -Fluid stream; Ind. Eng. Chem., vol. 45 .(1953) 636-640.

43 Logan, E. and Jones, J,D.1 Flow in a pipe following an abrupt increase in surface rouoJaness; Trans. A.S,M.13., Paper No. 62 - Hyd - 5 (1962) 124

44 Halherbe, J-H.: Influence des rugosite de paroi sur les coeffidints d'echange thermique at de perte de charge; Centre d'Etudes Lucleaires de Saclay, Rapport C.E.A. Yo. 2283 (1963).

45 Martinelli, R.C.: Heat transfer to molten metals; Trans A,S,.E., vol. 69 (1947) 947-959.

)46 Mills, A.F.; An analysis of heat transfer and friction for turbulent flow in. a tube; D.I.C. Dissertation, Imperial Colle7e, Mech. Eng. Dept.,(1962).

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 region 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 transfer; - in Recent Advances in Heat and Mass Transfer, pulished by Mc (1961) 223-252.)

Reynolds, 0.: On the extent and action of heating surfaces 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).

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76 Sigalla, A.: Lcasurements of skin friction in a plane turbulent wall-jet; J. Roy. Acro. Soc., vol. 62 (1958) 873-877.

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1. AUTHOR (S)

2. R.L2ERENCE AND YYEAR

0 t' TABLE 1

SU-MARY OF THEORIES

In this table the theories are

=anzrd in 4 • 00 chronological 5.TRL'ILAENT OF .1 + 5sp/(4K-?-) order. 6. TVALUATION OF

P - EXPRESSION

7 ASYMPTOTIC FORM

8 REiiARKS 131

1 j REYN oL CS 0. PRANDTL, L.

FG4-1 1901 1,4910

0- 3 011 e

P • 9 ) 1 )

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 fI . (-5-

I / 2 „ — • • -3 P "1-

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-• L - V .

L.51] 2 ..,• /932 • Y , + r : G+ = ___ _ jA = -----)-- F -(-y + 1_ _4_ .1_ lky 1-- 3 6 ci) '"

wPieie , F -( y , L,G ) _ -1

3 , ? .,.... _1 [ 2 - Cd_ Y) '-'./ - 1- '2--'11 4—AF3 L ATi —. ':-.., i 4- (// , I - t Y 1 ., .1.- _Lt ..., , '--- (, t - fr0, ) 17• r) AP-7-, F 07:.

p, 6., in 01 ,,,i) ai e i:v of I- ( 605,. 0-f PL) , 72..1.71-. i ,,,,eci epi, p ir,C., (/i .

!4- 1

Lz- I ' J 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, 6 h c i,,,v,,, ok.

-,:••q :A a -1:- lo vl ( :i : :2 - '

(-_-,---- Ives 61. toli{e (..) 4: 1\1,„ LA cz !.., i-Lii..) • h (31,i 0 r

f) ,15.. o„, 2 /1 7 3.

/01„. ,,,- pic);-e e if./ 6, ) 1_ I-) <- . ,f- ;:t(- -1- o p r- ,)1:-_-,o :•-_, e a -.,. v,..; Ex 100 fr, s , 0 .,,, r 0 r- .-1-- ; n(-- ''''' cf,:i ..• . i ( y t ) -- :,- i. = ! --I- co ,-• , •• • -

134

,i Hoi:mi\NN1-7-, IT; t..11-1 M E VJ.

2 L3I] 4940 F? 1- ) 1 9 3 8

7 6- s vy-, 1 1.6. Vs t 0 ...• 7 1--) c..) 1: vy, o r,.r,s V,/ l t 11 > 4J11 • qk.. A( a- > 9 ‘ v ),-, e i,t+i -L.- A{ 6-4- hiiner ph -1/() = 7 5- z.:, C c; (j v- e e vv ; t in 'A L4 4,1 ti) 0 v S expev,vv)evit—.,

4- 4

..) --....

-1/6 P -:: 5 03 A ' — i P ----- 7.54 C (-_1) , I 10 1) 5,0'7 .0. 0 12,...)01 ...... 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.

- V 7!-.7 5 y+< 30 .. (5:t -___ • [Al.._ r.,- 1 fr , v _ -3

a.- - ,--1

4- 1

,. i ti

P z.- 5 (1.5-- - 1) IA [1 + —,;---; (cr. - 6

7 P --,7 5 G-.

8 136

1 REICHARDT i—f .

-,-, [621 19A a

(-)< t.A-`• 2. ' 0- — 6-

1 I.. 5 — !4- - 2 <1 1:.:,.. 1 T. — t __ j__ 3 ,)-- 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

R.- _ , _.2...._ ; ,;:r . — '...._,.._ • (,,,. :- coe r.Q:_i. 1 .0v-, 5 :—..-01-ca) , 4-• I ' t A ----. +. - r? iAo,

— •P -- I? T) 9 ' 0 ( (:-•- I ) I ; , I: ( 6- 1 " 6--

PC .=_p , c \

,-, IR D 1 C a Irvk t_DCA V P (-A vE- v ‘, ;1,1( V Li v i a -I: I 0 vi OC 12c. \A/ 1 17 1

1: i:, 2; 1-.1 a t ‘A/ tA It.' (7' Q S 2 ( / 4,-, t 1/, i 0 le, 1 T.;

7 P—' 2C .1

137

4 MARTINELL), R . C .

2 Lif- 5 1 ' 1947

,i- t _.. , = 3 I l Y \ F-- (1 - i',.2. i — + ..- c, 1, ,,- _- 3. 05

3o < ./1- : ) 0-" :- 2 • 5 \ '.‘_/ + /

04 _ 2- 5 Y I + :`--;

_, ,,--Ni.)....

P .:-_. p )--

1 ,, I, V) :...1 14., ';' Y.' l el. 7(

_ 7

4-. .-- c

e VO vl i< 0 v e-v1 0 in ' 5 -lc t-i . 0 o i/ y ,), 1 -C I 0 ,,,/,, 4-_-1.ev-,doped -f-'0,,- t o \,../ (1. 138

R .6,NNIE '././ . D .

7 11-,- 6:: -) . 1 , 19 C- 1

0< .y --'• < 27. s • _ I+ si r) In 2 ( l' -1-- 4/ iK i ) ci- - 1 , 1 ,..,• 1 2/ -frii ..- ) UL) / w ( 1 k 4/^1k,ci = 14. 52 CA V‘ d ll+- -:,. ±. a/ 0 ,e1 (4 k 4 y -f ) Ai i<1 ,

_, AA' / 1,.. 0. 0

0.4- = 1 1 m v-l'H-- 3 ) J--- = 2 - S 11< V-47

--.-.-- 1

/ 2_ /C 1./ i< •,1- , )<, ( 6-- 1 ) VT::: _ c, y Os' > 1

..-'

4% i< , ( 1 - (.'") (2-

1?` = 5. • -- .) y — _27

0 7 p ----7. 2 2 41767-.-. ___ 4 c:, . A 139

I RE 'C. 1-1Al2 .DT , H

2 [631 )• 1951 c-, = 0 +'2•7x1,5. y-'43 ] ir L 4. 7___ < ____ .1- S 1 y+< 6 ', 6:t / I 0- 6-o 1 Y+ C I ' L`C./ 4 it trAok-/_) 3 y# 6- _ 1 ,.4_ 0.4cy +__11 kor,. v,,--7 1,c/---_-- (4, L- — 1.i ) >6 , _ k i - ( (p

1- ... -.1 ( 1:1q ) " A V '- + , ,'1 IA = k — 711)2 0 . 4 \-- y -4- — it - 0.tiltn -I J0 \ +

01nd v: ...--,-,. R D 15-p /2

a x r ;. p-y, in va 1,,, ci 0.7!,9 4 vavi e S f r o vv; ( (It W 0 ti k 0 (1 , S /.7c. ) &C

S -;,•-; 1

17o v TR—,.1., > r2113°° b

;_'," 1:• f I A 0 lot) 100,)

-' .- e)! k 11 41 .1 9. • tc), '-) .56 -.• 117 ` '- ' i

II- / S. p —Y 5 8 6

140

I LIN , C, ,.,) Mw.iLTON Q.IN , 4 PUTNAM , G- • L .

- _ 3 2 Lz .. 2 ', . 19 5

.- • , + '7, .- i ' / 1 --",,,, 4- 1 ( Y-2-- j -- l ;4 • 17, i - / L .) (76. 1 4 • S `t) 4. 2 y t - -

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1962

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RASiviUSEN, M • 1 1 Pi2ESE N T PgroMAAE:NO TioN l< A.RAMC !-tE 7 I i< . 2 \..._ G 1 i ) 1965 1,--)6 b

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empiric a (

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TABLE 2 COLLECTED DATA

Barnet, W.I. and Kobe, K.A. [3] 3 kripX10 S.t) lc, X12 cpx10 3 i" x 101 41cpici0 s A* (7), P S igpx10 S so? 7.942 6.56 5.715 6.63 7.492 6.04 6.413 6.75 7.216 6.64 6.364 6.16 5.332 6.06 5.813 6.38 7.,432 6,87 6.472 6.18 5.344 6.13 6.545 6.77 7.515 7.06 5.451 6.48 5.747 6.25 6.858 6.89 8.141 7.38 6.412 6.67 5.669 6.34 7.290 7.07 7.180 6.62 5.895 6.78 6.077 6.92 6.65o 7.26 7.036 6.67 5.040 6.92 5.080 5.91 8.299 7.51 6.532 6.63 6.208 7.06 5.236 6.12 9.044 7.83 8.237 6.62 6.664 7.32 5-441 6.12 5.200 6.02 5.943 6.78 6.700 7.65 5.621 6.34 5.693 5.87 6.268 6.39 5.787 5.38 5.837 6.49 5.441 5.81 6.328 6.18 5.763 6,63 5.921 6.73 4.708 5.77 10.494 6.02 6.736 6.22 5.320 6.02 4.853 5.76 7.156 6.32 6.112 6.30 5.594 5.94 5.429 5.76 6.172 6.32 5.691 6.49 5.897 5.89 6.245 5.75 5.580 6.17 5.799 6.30 5.356 6 .20 5.753 6.03 5.823 6.28 6.172 6.13 5.633 6.42 6.065 6.64 5.542 6.52 7.108 6.07 5.969 6.62 6.209 6.39 above data for a = 0.76 4.613 6.06 1 5.238 6.02 6.487 6.89 4.913 5.76 4.414 6.13 I 5.72o 9.94 7.040 7.07 5.323 5.75 4.984 6.25 6.273 5.89 7.367 7.26 5.649 6.03 5.465 6.62 5.564 6.42 8.119 7.83 6.245 6.39 7.196 6.13 6.302 6.62 4.996 6.02 6.813 6.34 6.515 6.75 4.542 5.87 7.493 6.49 5.706 6.38 3.534 5.81 5.493 6.73 6.387 6.77 3.847 5.77 above data for a = 0.60 149

Bernado, E. and Eian, C.5. [14]

Scp x103 XpX)02- 550,,,x I x 02 a- 50,11k103 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 ,-- 2 S a S 0 jsp)00 41),13 .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

Ede, A.J. [20]

Svp04 „Fp x102. S(7.)0X f 04 ,rsp x 102 Sg)F,X1104 krgpx tc? cs 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 7.15 152 Ede (contd.)

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 1.875 1.85 2.16 stp,rmo3 ispx1o2 Sy,pX103 ta-pX102 519,PX)03 Affp X102 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

3 2 s 1 5 xio ,r5- xi 0 apx 10 O- S mo Zpxio P SAPx 10 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

Jackson, M.L. and Ceaglske, N.H. 32

a = 1.6o 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 asp 6 x102 3.25 2.925 3.18 2.465 2.66 if§ X 102 6.24 6.4o 7.18 7.26 6.80 o = 1.06 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 ?__ 3 .2 5,,, x IQ lipX I 0 3 cr S .px io 4igpxio- cr SP, pxo 4r.5-7 xio cr 2.36 6.22 2.84 1.86 5.54 3.40 1.27 5.11 6.35 1..94 6.22 4.35 1.54 5.54 5.25 1.16 5.11 8.35 1.35 6.22 9.82 1.24 5.54 9.15 1.02 5.11 11.6

Lin, C.S., Denton, E.B., Gaskill, H.S. and Putnam, G.L.

2 6 2 2 (5". 5 59,p x10 it, --p); 10 Sc, 100 ,.A.F,X 10 So,pX 41-s- px4D 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 x 04 118-p 102 S,pX AgpX d Cr SpipNi04 ,,X-p X I 02 C% 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 TABLE 3 : DETAILS o'F EXACT SOLUTIONS

AUTHORS y o y) &LA {u) Y4). 6-0 -E ; rainge .(-s Slerchev 0-;) o'o-j--( 0-) Velocity mec, s revvieinl-s ovr expelfiv ental veloci Ey ✓ f;"-Tiatr TrI1,c)s of Sleicher profiles via vovx Karmun's C767- as ve r) by [77] relation Teri Cr: s [13] (ros /610) be in9 a correct l'ov, (eE°/P)( Vito 1-t -9-ac tow based o ev erk m easu rerne,ils (A Sig y — 7. 5

12.4 +D -ex p(:124. u-4-y+)7 Siege! Tan smi.tayer 1- . . 61- y so -1 4 Y+ d y -1- A) v. 10‘. ')+ 4 26 Sparrow (At 100 1-6 - Y f/Yit ) -I : yt;3- A ,7 10 , J1 -4- •124tA4y+b" -- ext., (124 ufy+)] -(.4N- = • 36 y [7 5] -Foe 0 'C- Y1-‹. 26 ys.-4-2 = novi-diyyiensiovIolisecl ec,c,filA S tA1-1__ tin -Y-1.1- 12 --36 26 , 9491 3 2G y÷ of P1.13 e . 1v stAbiayer Kays - ex p (11 v11" )j ay-F. 6;=-- 1.2 LetA 14+ •ol 5'4 -For y-÷<42 21 Terikins W;l\1 0 1 -F - exP(rAu+y1- )] [35] y+6_ 142x 4-2 yl 'CM-Y+ ig5 a vy, viti ply fit, Ih -tu v btAlenE Cov e ctor o-P 1.2 -for y-4->42 3 U-4-= 2..S. yt + 5.5 --10 156

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 R D 104 17 - - 5 x 104 22 -5 -.2 105 20 0 .2 5 x 105 21 5 .6

PF - P Percentage difference P1 100 PF

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 improvement 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 [78] 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 and yEuvpE/117,^-1 127 - 508 fluid in film Welded steel (a) new 0.0015 - .004 (b) uniformly rusted 0.005 - .016

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: 7.1 191 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 ,0.0351 135.08 .-.".;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 ::.3401 C.295 -0. ' L.4776 21.41 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 !". 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 ....d 31 155.48 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 3.4410 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 33.'7,.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 0.2241 -.1941 272.73 ..2^C1 122.53 '..19.3 !:r.9,7 :..1511 138.18 0.1231 130.17 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 p .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 3 -.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,. 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 ---1t1')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 „ 0.4)31 27-.74 0.6491 '", 2.7211 . ; .4511 0 -,..:5 23.77 • -.5131 22.40 0-6751 12.94 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= 9 X= C. IN. X= 0..5.150IN. X= 1.1CCIN. X= 1.5901N. X= 2.C6CIN. X= 3.070IN. 6.54718. X = "121 " UFT/5 Y IN. U FT/S 1Y IN. U FT/S Y IN. Y IN. U FT/5 Y IN. UFT/S Y IN. UFT/S Y IN. U FT/5 Y IN. 11 FT/S 0.0028 23.49 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 40.09 0..0038 24.51 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 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.C1 51 213.18 0.0051 160.54 0.0956 140.68 r.0151 94.99 3.0096 52.80 0.0178 30.77 34.17 71 341.48 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 0.0(1 3.3138 36.28 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 r.0111 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 37.61 352.78 38.69 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 0.3246 3.0256 39.51 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 58.61 0.0276 -,.0298 40.32 0.3291 353.29 (3.016L 318.35 C.0111 239.65 3.0121 192.95 :.0116 158.56 7.0171 118.55 61.64 40.97 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 :.:.378 41.45 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 • 0.0386 3.0428 42.00 0.0491 355.35 0.0241 -515.47 1..1141 246.64 0.0171 197.53 3.0146 162.34 0.,.)231 120.66 63.63 64.21 478 42.38 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 3.0 3.05431. 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 42.84 346.15 3.033L - 1.3578 0.0581 316.55 7 191 247.36 0.0231 199.69 - .1196 165.71 3-1291 122.10 0.1536 64.94 43.01 3.053L .Z43.92 :'• 0.0628 0.0591 0.1681 198.71 7.0221 245.91 0.3271 199.34 0.0216 166.78 0.0321 122.27 0.0586 65.14 . 43.28 286.39 0.0728 43.72 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 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 0.0611 165.17 0.078L 3.1928 C.'521 208.94 5.3431 190.14 (.0296 168.27 0.0471 121.24 0.0736 65.10 43.90 3.0621 99.03 .3831. Z51.23 0.1028 44.28 ,- .1641 .393L 120.30 C.3771 169.72 C.:631 173.4C 3.0336 167.85 0.1571 121.16 0.0836 64.82 26.70 - 0.1128 44.7.6 :.1641 0. 3.1331. 93.66 0.30 71 149.55 0.r.831 153.71 r.0186 165.71 :.)771 113.74 :.1C86 63.38 3.1131 03971 134.67 3.1331 130.81 0.0436 164.33 0.1021 105.06 0.1586 59.24 .0.1228 44..'2 59.77 0.1428 . 3.1161 0. r.1171 119.19 - .1131 120.57 (.0506 161.35 0.1271 94.99 3.2186 54.25 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 0.1886 0.4123 31.17 • - .1671 17.47 0.1731 57.22 0.1216 111.37 7..2121 61.65 33.80 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 121.63 0.3307 59.94 0.3797 65.62 0.3827 72.48 • 0.2687 102.28 6.3557 29.12 0.4297 36.52 0.4327 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...... ;,... .-

: . - . ._--_ .. -'--"'--'.-•..-., '-"'-- ' ..-.- .

-./

Table 6h

Run No:EH-D5 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 IS 0.0037 82.14 0.OC37 21.09 C.0037 49.5b- 0.0037· 12b.77 0.0037 91.95 .I 0.0047 93.27 0.C041 25.90 0.0047 50.40 0.0047 i27.61 0.0057 95.15 0.0057 .. 109.88 0.00b7 26.57 0.0057 51.24 C.0057· 128.85 0.0077 98.9b .- 0.0107 162.11 c.coa7 27.87 0.0077 53.13- 0.00b7 131.07 0.0127 107.62 0.0157 175.77 0.0137 31.44 0.0227 63.59 0.0087 13b.87 0.0277 lzq.77 0.07.07 178.78 0.0237 3b.b2 0.0477 n.28 0.0137 145.90, 0.0377· 139.89 .' l 0.0257 179.17 C.033? 39.23 0.0727 73.24 0.0337 Ib6.99 0.0477 -14b.11 . r :1.0307 179.77 0.0587 42.01 0.0827 73.95 0.0537 176.29 Q.0577 150.31 ." 0.0357 133.91 0.0837 43.b6 0.0927 74.19 0.0737 178.69 0.0677 150.79 ! 0.0437 '178.97 0.0937 44.46 0.1027 73.72 0.0837 1..78.b9 0.0777 150.31 0.0501 1:78.78 0.1037 44.62 0.1127 73.48 0.0937 177 .49 0.1027 139.89 '1- 0.0757 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 88.23 0·.1507 175.37 0.1331 45.33 0.1977 b5:34 (I.218"t 122.(;1 0.-2027 iO.23 0.1757 165.3b 0.1437 45.48 0.2477. 57.77 .I 0.2687 93.42 0.2277 53.08 0.1857 156.53 C.1587 45.25 0.2977 49.13 0.2937 . 77.89 0.2427 43.21 0.1957 141.52 O.lR37 44.4b 0.3227 44.85 0.318'7 b3.59 0.2527 3b.l0 0.2057 72.43 0.203i 43.6b 0.3477 40.11 0.3437 4b.13 0.2777 18.7b 0.2Q77 26.51 0.2587 42.2b 0.3727 35.b4 0.3587 35.73 0.3127 O. . f 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. 32.07 0.4837 28.61 J. . O.5R37 22.18 0.&337 ..... 18.32 0\ 0.7137 !I.31 \.11 / C.9 . 7 7 ..O• 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= 0.300IM. X= 0. IN. 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. USrl •I 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. 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 yC in. 0.0.5 0.125 0.223 x/y E c E Yo x/Yo 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 y 0.41 0.29 0.21 0.11 c in. 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 o- m m d, x in. T -T o17 " -T o,1 Ts-T T -T r i -1. i S ('- j-g G G G G 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 0 w 300 w 450 w J E ---10- x in. T$- TG ° F TS -T G ° E1 . Tr.6 -T G ° IP rr l 1.:!..51 15.14 NC N 14.1 0 a 13.1 H 11.93 15.18 O . 0 12.31 14.51

12..1 O' 0 - \ N 0 12.40 15.58 Pr

11.1 HH O p • 0 ) 0 11.95 16.26 c0 10.1 if 0 • N 13.21 16.62 N 9.1 HH

re 1)4.20 16.83 8.1 N LiTh - 1 \ 17.79 1

I—IHHNNN 15.43 7..1 1 - \ 6.1 • 18.12 19.00 •C o 19.59 20.30 cX 5.1 rR .• o ) 21.93 22.67 4'.1 H .0

24.52 25.34 3..1 r 0 ) 2..1 CO 26.80 27.49 29.40 30.23 NN 01H 1..1 0 - ) •0y.1 0 30.66 31.42

- .1 31.25 32.19 T* - Pc°Pc Lg*CC c00 0c 99°05 ze'PC T'O- 6T'ilc t7L°55 69' z5 6o° 63 9z' 05 t6 0c 95*oc cPosz 59*Lz 'UT Lc*3c ,0 T°3 5'9Z 05°a o5°C3 63*Pz "(=)c CI*TZ -U.0 02°63 99°a oL'gz 0°T 6L°2-E T*P T5'2 T5'9-6 2..*5 -[9°6T 20°zz CT*9T TL'9T T'S 55'Lz T9*gz r.9 7T°5 Og'T 9T'LT OZ•gT OT°22 -1 L a*Pz T5°03 9t7' ST 96*CT T', L9°9z 00'ci T°2 C9'53 Z9-1-73 22'6T OT*gT oL°6-E 59*PT 5°3-[ i*.6 56° 9z -179-17 t79° TT Lz 5z 59` 6T gC'VE T0 OT oLoLz ° "PIT 39-43 P-u6T OecT 56' OT 99°9'd T9° 0T "UZI 0L.° 9z 9C°P3 00° 61 u4o5T 79°2T Circi 9t10 0T C9*W zC°P 1- T°PT 9g*9z 95 *PE TL°9T gc*CT 9Z*OT i. ci 0 _g I-"I *TIT x 0 1 ,; -si a0 1 w do i0 I-°,1 0 1-2 arT OgT PA C -•-.4--- 1. A 009 =A Ogq M 005 IA .g *uT g0'0 = o

PLT

175

APPENDIX 1

In a turbulent Couette-flow,

ti du_ du ... (A.1-1) 77z 't S P - -ry and J"= J's' (r + peh)tT r From (A.1-1), and the definitions,

11M T s 4)) , y+ = ri(rg) we obtain: uP\ P't du+ (1 + ---) = = ,_ 7-z-, E ... (A.1-3) il P. dy' t (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. at =- ilt/rt in results in: E d(p+ t a ... (A.1-5) du+ t + (Et - 1)-1a0 and 1 1 0 1 0 6 t 1 1 ... (A.1-6) Et - 176

APPENDIX 2

The use of r} for the calculation of local,(1us. We have, E = ERr). ... (A.2-1) vrith,

Rr y rhi(T S p) / 0 0 0 (A.2-2) In a boundary layer problem where 2 parameters, such as determine the local conditions, the application of RG and zE' E4.Rr} for the calculation of local drag coefficient is as follows. Ss E -u/(4) ... (A.2-3) and (2.2-5) is, /2 ss = i

. . (A.2-2) gives, YrueP zE Rr ... (A.2-6)

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.

is specified instead of RG, then the steps in If Rm determining _ei are the same with (A.2-5) rewritten as,

R In E I1l zr' ... .2-7) L _e/ and 1 , , II + zEk-2 — )

Pipe-flows may be accomodated if we write,

(D/2) pipe (Y6) b.-layer ... (A.2-9) (uG) F. (UR) b-layer pipe

which, together with the relation

3 ,,rsp u R 1 + for a pipe-flow 2 /•<- and the empirical fact

ILE/uR 1 lead to,

-ei = K (1 + 3 'I-s P) + in( 2) ,.,, (A.2-10) tis p 2 /- and _e- . ln(M.Dsp1/2) • . • i (A.2-n) 178 with RD z. pDa/p, and s .m. ,z,/( pri 2) P where U = bulk velocity of fluid in pipe

Rr is defined in this case by

Yr , 1/2 eo. Per :--:. D — RD sP (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' elements which are shedding vortices and that due to the rest of the surface, which is effectively smooth. i.e. s = a s + (1 - s.. ... (A.3-1) e 1Y a is the projected area of the 'active' elements per unit area the effective drag of a n equivalent smooth pipe surface; se 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 k- 3sP ln(E Ds1/2,p ) = —7-775-(1 + ) + ln(2) s;'' 2 A= i.e. in E = - ln(R s1/2) + + (-3 + In 2) DP ---sp

Then, in particular, we have,

In Em = - 1nLRDsmt 1%/2) + + a 3-4) s1/2m

180 where a = 2 + In 2 for the effectively smooth areas. Similarly, for the portions where the elements are 'active',

e 1/2 in E = ln(P/Rr) - ln(RDse ) + + a se

0 0 • (A.3-5) By a combination of (A.3-3), (A.3-0 and. (A...3-5) , we obtain:

ln(sM/sp) (s-1/2 -1/2\ ln(E/E ) = 0.5 m - sP I ... (A.3-6) ln(ERr41) = 0.5 ln(se/sp) -1/2 s-1/2) (se ... (A.3-7) In the transition zone we may write

se sP ... (A.3-8) so that

s.e/sp (ERr/)2 C C 0 (A.3-9) and sm/sp (F/EM) 2 0 0 0 (A.3-10) 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) 181

y ... (A.3-12) where X (y r,g r,1 )/(yr,u - yr,1 Yr, , = general value of roughness height similar If all sand grain roughnesses were geometricallyAto each other, then Y r,1 Yr 1 .., (A.3-13) r,u n Yr J y being the nominal height of rouFhness. r 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: ( A . 3- Re ,c = a+bRr 14) as a first approximation. (4.5-3) wives

r ,c yr(a 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 T = AX( 1 - X) dX = AD( 1 + _ 3x)/6 a 0 ... (A.3-16) where a + bRr Yr,c Yr,1 Rr m Yr Xc - Yr,u Yr,1 ( n - yr

a + bR - mR B CR r r r (say) • (A.3-17) (n m)R R r r 182

Since a is the fractional number of 'active' elements, therefore, a = Ta/(AD/6)

= 1 + 2X3 - 3Xc2 000 (A0318) Constants Is and C can be determined from the condition that a = 0 at R = Rr,1 1 r ... (A.3-19) and a = 1 at R = r riyu. j

Thus, C = Rr,i/(Rr,a. - Rr,u) . (A.3-20) and B = - Rr,u C so that, R R R X - r 1 r,1 7 r ... (A.3-21) c Rr Rr,1 - Rr,u

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 to: [-Rr J711-1t Rr,1 - Rr. .. (A.3-22) X - R c - r - Rr,1 r,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 • 0 C. (A.J-23) and r- /2 == 2 I a(R r/p) + — ct)/E-1_ I with = 30.03 and EM = 8.12 184.

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 respectively. 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 inches 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 T = absolutetchiperature of air,.in °Y. G 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 y Pm Po r 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 m when the pitot-probe is touching Pm 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 - op T 9 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

2 Pp --== u dy

cm3 3 u dy Jo were obtained from the reduced velocity profile data by the use of Simpson's rule with variable step-length. 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- - j- _ - u dy / Rumax dx L R 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

C2 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. 2/s. Cori,esponding value of R.0max = 64.97 ft

° • • - m 0.314 x 1264.97 = 0,0581 G /zmax ti 1 ••••• .57 have 10, hence zma x /z.,

= 0.039 = C - mG 2

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, relation Y1/2' R2 etc. For a 7iven rourrhness type the EkRr can be specified. The following procedures can be used for finding and u They are based on a velocity profile assumption. E General: All the procesures have to make use of a sub-routine

for finding the values ofr ,lxz/z /11 and -1/2/ when especified. "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 - -Ir'd.5“)1- = 0 ... (A.8-2) d-, ?"4 W.: This equation can be solved for the value of P'max by an can then be substitu- iterative procedure. The value ofmax ted in (A.8-1) to give zma/zE; so that we have

... (A.8-3) max/zE = fm"'" where 'fm' means 'some function oft. 193

41/2 is the value at which 2-; izE = 0.5 (zmax/zE) i.e. 0.5 1 + -,11n 3./2 - td 1/2). . . (A.8-14) ( zmaizE) = 4„ This equation can also be solved by iteration for the value of 8,1/20 Thus we have, YG, ... (A.8-5) 1// 2 = C/2(- , and E(B. are given: Procedure when umax Y1/2 r By the combination of (2.2-6) and (2.2-5) we have:

= 111( ERG k zE4r) .. L (A.8-6) (A.2-1) and (A.2-6) E = Ecar} 1 .,. (A.8-7) with, Br = pYrkuC ZE/( p, f t ) (A.8-6) can be rewritten as,

E z 1 py1/2umax ... (A.8-8) = In 'max 41/2 and (A.8-7) as, FY,/ E(. (A.8-9) 1-1, z max ).

(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,

(A.8-9) uE = umax/zmaxE/z) ...

r) Procedure when R m , u max and E(-R - are given

For this case we write (.A.8-6) as, R, tez, -8' ln E .. (A.8-10) 11 ti 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, 1 Il z ,d0

(U 1 i.e. g z j E 0

1.95 i.e. 1 = pu dy ZE uE yG 0

Rm (zmax/zE) 1/2 Y1/2 umax

• • 0 ( A .8-13) Since zmax/zE and 1/2 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,

1 - Rm(zmal E 2 )P.;1/211/(pumaxyl/2)

8 0 0 (A.8-1)0

4n Repeat procedure until the desired accuracy is obtained.

196

APPENDIX 9

Procedure for integration of the hydrodynamic equations

(a) Entrainment method

Differential equations: ( 1 .3- 23) dR m d(in R) - m - m_ ... (A.9-1) dRx Rm (1.3-24) dR a( in tb.) 2 d(ln R) + (1 + H dR + R2 dRx 12)R 2 x 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 between (A.9-1) , (A.9-2) and (A.9-3) , to give,

IiRm --aTEd [11 - / = 11ss + 97,(I, - 12) + 12m x 1 1 - d(ln lb) ... (A.9...w - (1 - I„)R m dRx In general, we have,

ii(- zz,,, 0000 i = 1, 2 I.1 = -e'). and ii = -ek.R/ m, zE).-

197

(8Ii a e • 0 an. m) z_ = aRin) ... (A.9-5) oI1 (a -e' azEiRm azril .e.a pzE Rm

The L.H.S. of (A.9-4) can be simplified to

L.H.S. - I Rm —dR(1 2 1) dzE a I2\ aR. a 12 - I R dR 1 m dRx az..E4 x aRm1 J

It can be shown that, in (A.9-5)

aIi tali ai' > e L3.tlz \azEI and also ( az. (aV > > OR E e ' m) zE

Hence, dz, oI2 all L .H.S - R m dR OZE az.E

(A09-4)

d(ln -uc. - (II - i2) 11b. - Ills dz (1 - I2) Rm -dR x R E m aRx ai2 121 all ozz),e,

C C • (A.9-6) Equations (A1.9-l) and (A.9-6) are to be solved for R and z by a Runge-Kutta procedure. m E'

198

Auxiliary relationships: all ai I I --- and 2 can be expressed in terms of zs I19 2' as azE 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 . (A.9- 7) ln m ]

(A.2-1) is, =

O 0 0 (A.9-8) and (A.2-6) is, Rr PYrue-zE/(P'-11 -e' can be obtained by the simul- If Rm and zE are known, then 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) d( ln 2 d(ln R) + R2 + (1 + 11,2)R, dR dRx dRx x = m + ss ... (A.9-9) 199

(1.3-25) d(ln R) d(ln uG ) dR3 + R 3 + 2R, m + 2E dRx dRx -) dRx (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 dR dRx dRx x = m + ss (A.9-11) d[ - ln R) d( ln uG ) (Si I3)RG] + (I1 - I3)RG + 2(Ii - I3)RG dR x dRx x =m + (A.9-12) ae now have,

= • C • i 1, 2, 3 and -e = -ei (-R.0 zE)- As in (A.9-5) , we again find that, aIi al . az, z E 67'7. RG and. dii a if a zE aR c zE dI. N al i dz E 4 0 C 0 • i = 1, 2, 3 dRx ozE p, dRx

• 0 0 (A.9-13) If we substitute this in (A.9-11) andNand (A.9-12) , and solve the dz resulting: equations for E and. dR—G then we obtain, dRx x 9

200

dRG 1 2 az8(1 - dR D Q1 az - 13) - Q 1 I2 ) (A. 9-1 4) and dz 1 E ::---- ii• Q2(Ii - 12) - Q -1 1 ) ... (b.9-15) dRx 3 where, d(ln u) _ d( In R) R (I - I ) Q1 -• m + ss 1 - I2)RG G I 2 dRx dRx d( lnun.) d( ln R) M 4- 2-6 - 2(11 - 13)R, - R(I - I ) Q2 1 3 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: 61 81 81 1 =--,and2 3 can be expressed in terms I1'1' I22'9 3'I ---a zE' azE a 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)

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* by means These can be transformed into functions of Rx of the transformation dR x UGP dx

Notes on the application to the radial wall-jet in stagnant surroundings The equations for a flow with a finite main-stream velocity 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 xC 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 R and Rm. of zE' G Experimental initial values can be converted to values of -'and uE by the procedures outlined in appendix 8. Then z R and R can be obtained by the use of E' G m

ZE = uE/11G frd exp(e) G E z,E and Rm E(0.5 - 1/t) with I1 = 0.5 + z

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 z and R (or G) using procedures given in appendix 8. It E m R 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. and F / 6. subroutine LIThE:- This vives the values of zmax '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 ) = 0 x T1 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 3z 0 . 8945 zE, + + ( 1 - (- + E IQ21 = 3 -r 8

For some distance downstream of the slot the value of (Ps is the same as that of stuff injected from the slot, i.e. (PS 20 = (PC (1..10-4) at the last station at which this condition obtains If R = R0 and suffix 0 denotes tho corresponding values of other quantities, then

R0(93,0 - (PG)%-,0I0,1,0 = R(TS- cG)RGIQ,l ... (L.10-5)

205

and aF,.ain, because of (A.10-0

R0(cp0 cf&RG90I091,0 = R((ps - TG)1IQ,1 (A.10-6)

RG90 IG9190 93 (PG _ 0 • - To - 9G Ru 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 case: zE >> 1; therefore,

1 1.5) _ 0 .t.f 3? 45)1 I0,1 zE[iL(-3 + ( 1 -

Hence, Ft _ .,5\ ( L.) (i X0 45)1 RoRa,ozE,o n3 £0, (1 0

•-• 1 1 5 • ( .8?1-1-5)1 R RG zE 1±.(-3 - . ) -t- (1

. ( .10- 7)

(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( R) = Js/(pi) ... (A.10-8) dRx 9,1 dRx 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 (A.lo-lo) = (Ts - 9G) iftp 0 If the conduction in the plate is considered onc-dimen- 2 3_2 a 9 sional, i.e. !ay' 2 arc negligible compared to the x-wise ay derivatives; and the heater supplies a flux of jE'u then, d( rns kmt d - PG5 1 J = J"E +c —R 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, F (A.10-13) IQ,l 1 F29 where, a z 1 E (A.10-10 Fl ' a3(1 zE) and a2zE alz, F2 2 '_e, - ( 1 - zE) a3 •(1 - Zr) /6

000 (A.10-15) with 0.25 a1 a2 - f- 1.5 ; a3 -e,

207

For solution by computer, using a Runge-Kutta procedure, the equations arc recast as'fbllows: Let D ( . 10 -17) PS - PG

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 p, d ... (1,.10-20) dalx PuG. dx (A.10-19), (A.10-20), (A.10-10), (A.10-8) and (A.10-12) together give, pD(1 - Q \) z-k2 u dF _ E I! G _ F ▪ — ( 11.. 10 - 21) dx _e e01J, X - 9 x•C (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 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 hydrodynamic equations.

Initial conditions: In addition to the usual hydrodynamic initial conditions the following thermal initial values are required: D; ; 1(419 ; and i. Of these, D and arc obtained from the experimen- E 9 tal 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. 2o9

2-Ph .se flow

Chem .cal acti •ity ONS Comp e•ss-

ATI ibil ;y RI A V Vari able prop 1•'.- ties

AM . . . Pres 3ure • —

-TRE ()Tad Lents • • •

S . . . .

N • - .v .•‘..‘ ,-, •-..,"' AI Mixi ..... M 1g Jaye :is , ,•-• -

Fran in A ... numb Dr •• , -• • -• •••.... -• .. •

, / Isot term- - .- - Is' / _ al; R.. P. ,, , , - .,.• ... ,,- .,,- Smooth Rough Blowing Surface surface i 'uStion activity

SURFACE VARIATIONS

Firfure

*-Reynolds Fluid: fluid in which Reynolds Analogy is valid. 2 10

figure 1.1: General f low - configuration

figure 1.2 : System of co-ordinates 2 1 1 1

z composite profile

wall •5 component, z E T 1- z wake E component

•5

< 1 Boundary layer , zE

z wal I component z E composite profile

•5 1 —z wake E component _L

z >1 Wall jet , E figure 2.1. Assumed velocity profile schematic , ..01 5 A 7 B 9 .1 2 6 7 a 9 1.

Y/ Y1/2 figure 2.2a: Smooth wall jet velocity profile u max 1

--- profile of Bradshaw and Gee [6] .

logarithmic wall law + linear wake

.01 .1

figure 2.2b: Smoth wall jet velocity profile 2/4

A. vE

1 , figure 2.3: Assumed mixing- length distribution

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 30 40 + figure 3.2a LJ Rasmussen and Karamcheti ,.Lin et al. Reichardt (1951) ,,Wasan and Wilke R ) Gowariker and Garner (high D Deissl er .-Petukhov and Kirllov Lin et al. Spalding and Mills

10 20 40 aZ figure 3.2b 104

103

. • r . . • . . . ; A 0

2 . S. 10

.. , .75 -1r P=9.0 [(6/0-o- ) -1 j + -28 exp(- -007o-/o- )] . • I • -75 1 •.. P..- 9-0go/cfc) -1j II • 1

2 . Z1 figure 3.3: Variation of P with a- ; do- = Z . . Z . / . .

1 10 1 1 10

10 • • •• .6 • • • • • • •

• 10 • >" • r

• P.9.27Ro-/a 175-1][i+ •27exp(-0070r/cr0 )] • .6 0 • 1 7=, P.9.2 7[(a-/ 51--/-.1] V 10 ,,

7 figure 3.4: Variation of P with a- ; ao = .9 / 00 1 •1 1 10 102 103 104 Prandtl- v9n Korman Hofmann ReiOardt

''Present recoinmendation fi

2 10 10 1 figure 3.3 . Comparison of theories with experiment Gowariker &.GarRer Mills Kutateladze

103 Wasan & Wilke„- Deissler_--1 Rasmussen & Karamcheti Present reco mmendation ;,- -:, • Petukhov & Kiri llov

r,1 Present sim Dl if ied recomrnendction

;I I

. , i I I

I I N I O

• 3 1 10 102 10 104 fiaure 3.6 : Comparison of theories with experiment

n 01 0 •- in :11111r1: :1: . 1 : • : : - r t,- • • : : i : • • .

_ . _ ,, . 1, ,. . ,. ..,,, ,,, ..,, 1. ., . , .• .,.., ..,... „...... Itl -.:' -7. -`1,::,-1- tr,-.::•••1:4%Ti.::;_, H t:-.; ti ii ii . : 1..-: : ., it,i: t.:11!...:1 t,:t: It !! 1,,,jit, . e l' ::-;-; ' : : : : 0 t tt- - - • ••• ..., ii:' :: : ''''' ::',...... "::: :::.-:, :. .-: .:' -.- : : ..: i _„: _:_-- ‘.._- _4- - !...,:,; !...., I ,-„,-„. :.! ...... : i , . . , . _. _ . . .'-'-' ' ""' ' t r*- ' ' 1-1- : I. I ..; 1 ' . • .' . ' ' .• • ' ' " " "" 1 .... -1 ' ' ' .- -.-t. .1.1-,...... !...... ---.-...--.—.--...... - — -..— — -,...--., - . , . , . . . . . , . .

, ... . . I 14: 1;1 -;,. 1 r .-.1 I, • .1.,1 I,',.: .1;tt.I.: 1., I. . , . . , , . • . ! , . ..,. ,.!: i:: ... • r ... • ,-. ,,,,,,,..11;...... 1 • . ; ; • ; „ . • ...... • , • ... ••,1 •;.0, " ..11:'!: 111:11,111 "., I:: •I'; :• •;,:1--,., t,._.pt-. titt;;i:: T''• •111._:1-2-1-I- 4:11-,-,--:11".-:::::-:, ;: • :-: ; : 1 : ..... 1.:::41.? : 1 :',.• 14:1:::::11:111:. ;:•;;‘,1)- ; ..,;!: 11,..,„ ...-4-• • 11: .„. .. .. :11,"...... , t.,11..1, •.•1,;,•. •-., t.. • • 1 1 "": • ' ; :1 7 -I =1:11 •. - . , .,. .., ...... r .. . . : '•:.;:, X i I.:: , ::';:i::ii: ...,,,...... ::;;:•' 11 ill: • 7 ..... t ...... LU 0., :It ..... :17.4, :ti: r....,,:s. :•:-, ,.,:" :77.7 ..,— . -,.....1.:,...... ,...c, , : 7:11 - ..„ ..... '%•••••;-4

4-: . t(1 : .. • ;.; •":1_ :77 ID • 0 'iJ . 117:: z}:::. • 71: 7'77" • . ' 7 ...... :. . 7 ' T .... . : . . : • • LUl L. . 4- :(f3 .::1 .1_;... 77 . : a) • ..... „ \\,...... ::" .... 77:7 I. : I 11.1. 7 7 - ..... • — •'': • 0' HO; : ' . • • . : :0 . • ...... • , .. ..„;l: . 1: II; ; : ! • .• n " • . ...; 1:77 :777;117! li• ..... • ::-1:11 • : : b ro . • • t • • 0 (Y) 0 0 0

0 6 + d 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 o.8 0

.2 3 4 10 10 105 106 R D fig.4.1: Drag data of Nikuradse [53] 224

•001 2 3 4 5 6 8 1 2 3 4 5 6 3 4 5 6 8 2 3 4 5 6 8 2 1 10 102 Rr

figure 4.2: E-(Rr-} from Nikuradse's data 1 225 '%.,,,, Nedderman & E \ Shearer (Theory)

• Nikuradse (Expt,)

R Rr r figure 4.3 figure 4.4

frequency

I Jr i i Y Y R YrI Yr ru rg ru lower upper Rr limit I limit I nominal general figure 4.5 figure 4.6 10

: I .

1 • • • . • - . • . • • • , '•

•:::1::: ..•

: 1..

_ _ •

• .. • • • • . •

- • ' • !"'. •?

• Sok 2 3 4 6 7 8 9 4 5 6 7 8 9 I 2 2 3 4 5 6 8 9 I 10 10 Rr 10 figure 4.7: Curve-fit of sand-grain E-data in transition region 227

10 r

6 o 5 P ,

E 4 3

2 I I o i 1 --, TH 1 6 J 1 7 4 I - 1 i i r 11 2 L- H i 1

a 4_ 1 6 i 5 r Li1 i 4 . i1

-I- r --,--- mean curve o sald rough Bess Lao •O) 4L 4 a 4.4 6 H 5 4 Tll 3

i .001 i 2 3 4 5 6 8 2 3 4 5 6 8 1(:) 2 4 c 3 ii 10 0 103 Rr

figure 4.8: E{Rr} from data of Dipprey [15]

I r t INN m 6 218 2 s 5 .01 S iC •acriosili.4 6 64"..4'Ilialiffic14111.4111rdranrIll 6.:1 a 41.••wa.aest.iito.P.4. .idderdommodammipun Nino Iltr:7.611Mesinummgravr.d.. • D D 41104101A .01.1./ 1. 14110101119014•44.,14 111U d us.0.1" .01 G

---"-.- i. -:- 4. 4-•-: ,--.,--:- t - 4 T 4--,t7.,11,-._-__,_-+- _1F-1 .. c ' .=•• ti= "112:: „ ..• .i., ":1•'1': 4:511i =--". gill: • .I.{'. a-..-1 ._, ,_,,-_- II 11111111111•11111 I 11 it -It, :I:. 4` . n • ...... 2 rh,-,-4-- -4-4-'1 7+ ' ,t, 111' . • i 1 ... —,-,- 'T-,----4 =, -.4,-:,' -,-,-, ,,,, :, , . , ,- f i,

' I 101 .41. 1,1, 11.i .14] rtt .001 3 10 104 105 figure 4.9b: Drag data of Stamford

_: _-i_ zifi-tT " ' •- i -! •• -:.• _i-i-_1_:t H-1-I _I; ! .i ;t,, i'.k Ili • T --' _ • • . ..-:_---- I_-E..:_t-'_,t's,,--_: 7 = - .r.--. -_-0;fi.- :-Ti'lli,..='-- 'fit= :-. 1. " iT-Y , 1.::+:-.1 1 1-. 4 A -1- fi-1,-,. ii-ii -... 7.1, s• - -==i____ _-- '. ------;--T=E: - Z--q-1 '-•=:-- '-'1,?; .,'-'-- . zip :''di ,ia..-iiihrzi.-1_1-_-j_ji.,..:-, . .s 1,, ••..:mg..i..--r•l' 4H, - .- -; • sw ..x.— .r .... L.-4-- ._.... rr...... , t.„i„of„.7_i_-_,:iFEr-,..', ,.. •:=•,, ,_ :.::,..Ffins11.-_;.lipi pr.—. . — ,7. --.\-4,- -,-.11 I ''"'- ___, . ,.. 4 v .: -6, , -I ' "*. 4-r- -1-.4,- ,,trl,, = .:::+-- - .1-=.==-4-1---1-- 7 t--4-," Iii.::::. -_,..- / , ,t,11„ i 1 ..-'"-I-'',h.- *. 1111 ..... :a 2ss -,-- .,-_1-4 .I _,...._,.+ 4 J I to dot 4.-, ..'':,--.-..t..-. 1---ta ti.--. ; -, ---.- i, ..ib_i ---1- t .--1----tv-AF ' irs ---1.---t- --, ,-,, 1-7-4-, , - -,-, .1, ;-,:, : -, '..1-1 -1-• ,lir44,_ '-• ''-i 1 it ,,, ._ , ., ,., , -_-_ -_,",--r--4-_-_"- "---' . •' tIt*.ir t'.7...it if-tit '41144- 1 INI l ey i -1-,11. ,1---14...;+ ..41: .14.4 t,,,.4-. Lt__-:±1 ±-1 -1-_,,... 1-f . I 414 kniumn i,■-.T ' 4- -4-1--;- : 4_, '4 '11 +14' am• imill ' ' ' 1 1 1 - ---..--i-',---,--w, -r-i-H. ;Ill- --' '4.1.- -1-• 4" H. it 7,7' --f +-H- ' ' I I -t 1 — 't 1 till 4 I. .:4.J.-'41.. '11. L 4:1 1 I1 , III I 1 i -1 I4 , . ti "4 t.4. 'fil- 4. ' -f- I _i_1_,_1_ ' ' ! I.-. .t I I ' ' 1 1 1 I t " " I il 1/ 40'.', 4 ' II lift 1 , ., dm m • 1. I ' / I- ' 1 !--I-- 'I-47-Iq t; ' • 1,, :11:_i:i4,-., '4___. ' i . ,444 ., ,ii„ Le. I ' 6 to 4 II .-+1.M+ _4_1______. I _Li. ai l_LLi: 414. li .4', .il. ,,,...,,, . , ' I I _l_i.,_ 1IQ •1 ',, , t li_h l 1 IT .(I , II , W I Hi :Iil, I I 1 1 jili j I ii illij.1 f .01 -tHil 'ff.: •- f-t-- ; _,_ 4V-“,..„ :-Li. ..'11.1 ii- ,H gr.- 3 fi'-'0'1 • -:fft:-117,-:'7,I::=1- .... .:11"t,, --El' -i-,-..4. 11 ._-_.2. .- f.-..-27I•U- 4-L'...1-4., .-.3- 1••' F-:,,t4. 4.f...,4 ..t....'•-n-'•--=ti' 74tR -.- , TA«V. .41-t 'T: „ '.-- •-,r -1_---.1---• .. ----: f.— . ,;-:...1 _1-. '11::_,W47-iii':TEi. -:,IiiP-Z:--4- 1.:,--1----1F------A.-4-,, ._ ,., ... -: ii : -t -r------4- ---11-.0 -:P „,---.-'-:;Lrt---1 -trt•-- -• "--„,• '-.1; .=--• -----4-- . -IL,. 1 ' • Hli 9 "• .. r ...... -----, „ .,„ FIMOMININII /111•111•• • n• •1•111••• -4-.,,,,, y. . , •-•”. _._ 33...111•14.44.4114, r... .__._._I_, • •••141111110•1••••••••• •• .t .- 4 Etf---- MI •IIIIIMPIIIL II •• • -.--- '•. - -''lial, - I 1 1H+.- ii _r-t 4--,- --s- _ .-1-•--,--.•- i''• ,-: '_ Mil -7T7 1... .6..9:V.. ifi1,1r. 1 i1 i ----,u -- ,... ___!_,_i__-_____,. __43,--...... -41-1, -...... i--,3-4...,:f4 .-.±... _ -.,-. "" • « " 1 4.:•:" r • lir - 1„, 4 - __....-4__. 4_ _t_-:-_.. ---t -:-_...t; =4,t- 4, _v. ....t._ „_..., L --,-,_ .L.E..„-r....4_ r_tti._,,÷ :1 ...m...... • T, ...nct, 1 i1 -- - . 1_, I+ _#'aLr...1 -4.....11-,. ::-Ii ...1_4!,....-4,--4-4-__.'------r '. I t , lir 1 . 1 hth ,---11------,------4-r 2.1- •,,- t-- 4-4rt 411 f, ;j--i-. $ -- ---i I i_i --.1 -t. NI =Is ra cariril --_-_-.;--_L- t:T_-,T=_-41-_4 t.4.4 .;:t-,,.;:,,..-irli t:_,! 4-4., ..., : i--4- 4 ,---74----7 -. 1-h---- , ri T it ' '" u ' I da." , -.... --1 L-_-_i---ti-t_:1 4,1- u_tl.-' "'14. . 7:71: !----4-_,_-.4-1--sild. 1.11f.'- --•''',' 4,--.." 't , ,,t--___r _l_i_t , i i 441, 1.1.1 it W. :.„ 1_4:_i_ -- - ' ' ' '' 4,- , ' ,.„1: 114, I -.. .ti LI:.,:: _ 47 tii ill , 1.' lit} .___L-_L_:_ ___.1-__I__41;, -- 4L.11.1± -. 41, 414i ii..: I-- ',“.. i F.,. . ,H--,-. ii ti--t4A-1. ,' L± I'L. 'i.' I, ' , 1 , . , , , , , , . ,14 , 4-1 I . 1 4.41 .111 1 ' • , . .1._, •i_l_i 11'' ,' '4,1 ,, .11;- . tti •1 I'" -t . i' • r , '4 ---. , 4 "-----4-4- i --' --+- '1:-- -.1,,:i, • IIII 1 . 4-1,.- -4, I IkI4 -f,44....- --I- „ I- --H---,- -1,--1 4-4 i 4 4 +-1-.,1 1 ,It 4 i.1: .0.! ',III..4L2i_t: iii.L44:::_n-__:. :-_,---4- 1--:1-« .-H.I -J.: i l ' 144 .1.- --I-, {4 1 1 .1i. U. --,----4 -- -1 --..--- .:. 1 1 i .,,,i 1 i-- ' " 4,', '' 141 44 14,441441i _I i_ --.-4441.'l 41 li.1.144• I II' Its •001 1-171.--riTrit, i lilt OH iv: [ri l'oi ii, lit, ,1.' ,,, I I 1- 1 1 Ili Oil , il 1 0! ill': 711 1 I I I , 3 4 10 10 RD 10 figure 4.9a: Drag data of Cope [12] 229

10 . .. 8 1fE , , LI: E 6 5 .-•- , ,

1 .,, ,i., -r-- -1- . , . j_ i 1 1 , , . , 8 , • , ' 1

6 . 1 I T -"- 1 . • 4 ii

3 -4 4 ' foiliNt_ rough'; sand rain 2 .1-- mean U de •-/ tlipcteri8tic ' ' 1 , through- Stamford . i .1 r , ., , aia~ a _ fi • •'f1 , , . , „ r. 8 ini••• sehmoii ni 1 t ___.: . , I , I ' 6 IIIIM 't-----7 - -t - , - 1 5 III 4 . i- --- t 1 ; : l ---- 1 -I- I

r. .01 t 8 . ,-- 6 1 L.._,__1_i ! ______.._ _.; .. 4 1 . ; . . 1. ; 3 -t-,; ;1 -1, i . , , 1 2 -1 ' 1 i

•001 2 3 4 5 6 8 10 2 3 4 5 102 103 1(

figure 4.10: EE Rd- from data of Cope [12]

230

5 4 11111

1111111111111,111 11111111111111111111111111111111111111111111MEMINIMIMIII IMIE11111 111111111111111111111111111 1C111111111M11111111 101111111111111111M1111 riliM11111111 111111.11110111 =MUM 1111111 1111111111111M11 1111111111111 11111111111211MEMilli• nd g air tic

1111111IM111111111111.111111111111.MMI 11111111 1.111111111111M11111111MIT NE11111011111MINiimMINIBIRMII NESIMINM1111111111mw 1 ',11111011111011111111111111111 agal= • -11111111111Elni ■

1111111 I III 14 .91 111111111 P •H [ MEUNIER.. EN III El IIIIMMENNI OEM mimmoomi• mum 1.1 '11

.001 3 5 6 2 3 1 5 2 3 4 5 6 i02 103 R r

figure 4.11: E(Rd from data of Stamford [84] 23!

1 -, r r T a —1— 11=11111111=1111111111111111111111111111 iiiki 6 5 III I 4 im REIM MIMI

2 11

B MAN imp!ili. Mk 5 MIIN Id IIIN

3 n11 2 iIII , i L 1 i I . 6 5 ®111111 1 4 IN _ li 3 1 in ■

2 ill

•1 IMIIMINIMIIIIIIIIIIIMMINI111011111111111111111111 1111111111=1 IIIINIMI 8 11•111•11111111MIN111111111111111111111111M1111111111111111111111111111MM. MI IIIIINIMEINIIIMIIIIIIIIIIIIIIIIII 1111111111101111•1111111 ______4 6 MIMI RS 1 5 NMI 4 III 11 MI 3 . 111 i \

2 111

•Q1 2 3 4 5 6 2 3 4 5 2 • 4 5 8 1, 0 .1 10 2 Rr

figure 4.12: EfiRr-} from Kolar's data 232

10 .

6 5 4

2 I ' I

1 1 ti 8

6 5 4

2 /D .1 e •-• .0

6 5 ation ( 4.10-1a) 4 equation (4.10 -lb) 3

piYr 5 .01 17 5 e 2 6 50 5 70 3 (4) 6.25 2 25 55 .001 2 3 4 5 6 3 4 5 6 8 102 3 4 5 & 4 1 8 10 2 8 '103 2 Rr figure 4.13: EfIR } for 2-d distributed roughness formed of wires AJowains aff15!1

U 9 S t c 9 S V E 8 9 S b E Z 8 9 S r E 100'

I IS 1111111111111111111111111111 !Mints d(gft-'7 ILLIMIIIIIII 1111111151111111411 S I weimmumagamWill supw•••••••minimmmanumummumun umpowanwpmetposonsow2mirvio •Is ...... LO • I • IMIL INNIMMINIF MI • a fj. ilkimminimmillima iOMIIIIIISIMMEIIIMIMAINalall 1111111111111111111 S MI .1MIMI VINEIMMNI1101111131111111111111101115111111111111 11111111111111111111111 NOE inriunisbamouirtnowimmi 1111111111111111111111111111

MEM InNEOMMINNIEFAMIPIINEM 11111411191100\41 111

S 1111111111M111111111111 IdaIIIUHIILVIIIIILIILIIIWIIIMEIHUIIIIIIIOIIIHIIIIIII INN MIMI Jim I MN 11111111111111111101111111111111111111111111111111111MMINIMILIIIIIMOMMIIMONUMIEHMINION 41 oliniliillimi iiimiiiiiiimmint.imisiimmoiniumimaimmieumesamixortanpumonemmommo : . up MIMI 11111111E111111• '":":: 1111111110111111P11 am 11111111 3 S 9 IM 11111111 111111111111111611L; 9 • OL 234

2— 0

1 • data from Bettermann [5]

relation derived from linear velocity profile [83] 0 I I I I I 1 I I I I .4 .6 •8 1.0 1.2 Z figure 4.16: E

data of: Brunel I o • Bettermann 0

proposal of Nicoll and Escudier; from [23]

4 H12 figure 4.15: r

400 500 700 800 900 1000 1100 1200

V rr 236

_ - 71 -

, _ h ...

_ • ,

. ...

, _ _ . • __

: _ _.

. " :

4--- • -i . ; 1

-- -F-71 . I . I I I - 3 ; :la eAp - , it , , ------.. .41 .184_ cornpqison-of___fropl _- heory--of—Spaldiri ! . : r w ith ;dicta of Better reianh

I 1

! 1 1_ _

• • •

4 . I 4 i • . i I 1 1 i 1

_ . • . 1 _. -Or-'-'1-..... :. 1 I : I 1 i 10 . -? - H - i - --', " } t - -- (- i 1 . i iI 1 . 4 ,-- •! - • - ri, f i I • L K -1.----i----..- 4, , t 1, --' I ,I : - -I - i ir--____4:----ffer • T t - 7 ; --v; ' i

6Q 100. . tri ftgyre 5.1: N Rr , c3 } 1at° i for roughn 1 2 10

Frtuation_ • : :

• 1 . •• • ; : : I , S.-1:• ••- • . : • :: •••••.:11:••••. ":', . : : ; :-::: . 1::• :::* .

: • : : • • ::: •:::

::_• .:::

• . : : : : :•:::::: . ....

• .••• • :• • • •

:.- j J • j :.*: • :.• ::'• • ••• -.• ••••i 410 : 1 . : ..„ " •••• 4 : : : : • • : Data. :of • • . . I j • • ••• • : •••I !!•. . I , • ,; j j ;:,, :..; • " ' I . . .1, '• •;:. •-• ••"' I • 4 4 '1!" •i•,` J.1 • :j

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 .tpi E figure 5.2 : P-( E 3 for sand- indentation roughness :It• ; 1 i r MIN11111 111.11=1111111111111111E11

41 7111177:17o 1 4q 11111111111

• h 111111' 1111111111111 INWHIMMIERNIMIN '3411111111111111111Milli MINEINMEMIN. : : ...... : . EhHI 11111 11111111E111111111. ibis 311111 11111111111111111111 : I 11110111;1011111111111111111111 II di 11111:

• • . -t immuniI • ' • • I .1 1 •.. i• f4I• ,•• I ." ;..,

2 3 4 5 6 3 4 5 6 2 3 4 5 6 7 :001 a 98 01 7 8 a il E figure 5.3: PE)- for pyramidal roughness 100 • : : : I : I • 1 I : I

• " ' : i--f-

......

•

r •oYe 1.1 0 • 1

1101 2 3 4 5 6 7 8 9 1 2 3 4 5 6 2 3 4 S 6 7 89 .11 B 9 10 figure 5.4 : PEE} for v-groove and natural roughnesses; a = .7 Principal dimensions delivery pipe: 3 in. i. d., 10 ft. long adiabatic heated nozzle: 5.9 in. d. plate plate adiabatic plate: 3 ft. sq. heated plate 2 ft. 11 in. d.

bel lows nozzle

11110 aluminium delivery pipe - plate slot--/ heating pad 61bil 11 expanded polystyrene perspex' - metal glass-wool sindanyo' casing

figure 6.1 : General arrangement of Apparatus 242

traverse unit,

figure 6.2: Working section 243

3in. d.

5.90 in. d.

\ \ \ \ \ \ \ \ \ \

figure 6.3 : Sectional view of nozzle 244

u ft

300

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 2-47

figure 6.7 : Static pressure along wal

9 fl Static 6 pressure at wall inches of paraffin

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 at slot

45' • 3.07

A 1.10 4.56 160 1.69 • -- 6.34

■ 2:06 A 9.81

120

80 T-• ' • f I i • i t • 1 i.

: 11 • i - . % T 1 ----3 ! ! • 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 !

. : I .16 int rC- ---1------7- -I------i-- :

: 4- I • i ' . I

! ;

H---;--i-- -E- ' ' - i i

1 ' 1 1

• -1 1; ---. I ' 1 t ; 1___. ; , , ; I } •1 .3 .4 11.0 • In. , I I i : 1 i. . 1-- -1 • • -1a0u-ia; 4,1104v1-- 1!lnclia!PD paInso61^1 :i„9.4in!i- -- 2it 01. ( 9

•

T , 1 ,, 4

L ; 4-! ---t--- . ! 1 1 ice.) ! , : i — • 1 :-- ••••a i . 1 i --: 1:: • "59 '‘ • \ •-•94 • yc = .15 in. • R = 2.55x104 100e- Heater input = 450 w. • 7 V-groove roughness

4* 60 90- 6.625 9•78

80-

•

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I • • • ;• 1r Isis : IV' •

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surface • Emery rough

o Smooth

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1 •1 .2 .3 .J inches figure 711: Variation of initial- region length with slot- height AIR

CIRCULAR METAL PLATE • I,

,3 E H EAT 1E R. -PAD

--F;sure 7.12: Rawliat w•all-jeE : Hea+ transfer system

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271 ft/s

120

110

100

90 2 4 6 g 10 12 x ft. data : main stream velocity variation

• ss drag data — prediction using rough wall •005 E- expression prediction with same initial values, and E =7-5

.003

.001

4 6 8 10 12 x ft.

figure 7.14: Rough wall boundary layer : prediction data of: Perry and Joubert [56]