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

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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 indi- cate the nature of the information on surface 'roughness effects which need be incorporated in the theory. A model is proposed, of the flow close to the surface; and this provides a basis for the formulae. An experimental study of the hydrodynamics and heat transfer in a radial wall-jet is reported; and comparisons made of the the predictions of jet behaviour with the data. Useful details of the theoretical prediction procedures are given in the form of appendices. 3 ACKNOWLEDGEMENTS The work described in this thesis was carried out during the tenure of a Ceylon Government University Scholarship. I am grateful to Professor D. B. Spalding, of Imperial College, for the suggestion of the problem, advice at all stages of the work, encouragement and understanding. Thanks are also due to Professor J. C. V. Chinnappa, of the University of Ceylon, for his kind advice and understand- ing. The granting of leave and provision of partial financial support, by the University of Ceylon is gratefully acknow- ledged. I also thank: my colleagues E. Baker, S. V. Patankar, M. Wolfshtein, M. P. Escudier and G. N. Pustintsev, for help and useful discussions; Dr. D. F. Dipprey, of the Cali- fornia Institute of Technology, for providing unpublished data from his work on surface roughness; and the members of the Technical Staff of the Mechanical Engineering Department, Imperial College,-, helped in the speedy execution of the experimental arrangements. CONTENTS Introduction 5 1.The basic model 10 2.Some features of turbulent flows near walls 17 3. The P-expression for flows past smooth 26 surfaces 4: Hydrodynamic effects of surface roughness 48 5.Couette-flow analysis of heat transfer from 68 rough surfaces 6. Experimental investicration of a radial 77 wall-jet 7. Application of the theory 90 Concluding remarks 105 Nomenclature 109 List of references 118 Tables 1 - 8 130 Appendices 1 - 10 175 Figures 209 INTRODUCTION It is not necessary here to stress the importance of being able to predict the quantitative aspects of the hydro- dynamics and heat- or mass-transfer processes occuring in turbulent fluid flows past solid boundaries. The problem has been stated and the important studies of many of its aspects have been enumerated in the paper by SpaldingEb21, in which is expounded the "Unified Theory of Friction, Heat transfer and mass transfer in the turbulent boundary-layer and wall-jet". This general theory in its present state of development has been applied successfully to many cases of hydrodynamics of flow past flat plates with or without mass-transfer, boundary layers recovering down-stream of a disturbance on the surface, flow over surfaces in the presence of pressure gradients, and of heat transfer in flat-plate boundary-layers with many interesting boundary conditions. The general the- ory at present can be seen as the initial stage of an attempt at understanding many important physically-controlled solid - fluid interactions, as shown in figure (i). In the theory although we are not directly involved with the micro-structure of the turbulence, we can yet express its grosser manifestations in the form of auxiliary functions and empirically determined constants which have to be incorpora- ted, in order to make the eouations soluble and the solutions realistic. ••• 0 The development of our understandin7 of the interactions shown in figure (i) involves a two-fold task: firstly, the collection and codification of discrete bits of information confined to a particular region; secondly, the devising; of a mathematical model which has manifold aspects but yet cali incorporate localised information, thus helping to unify our knowlede and dissolve boundaries sept:rating the recrions, The present work belon7s to the first category and arises from the consideration that the presence of the wall has an influence on the turbulence pattern within the fluid. Thus in the case of a smooth wall it is found that turbulent velocity fluctuations become uncorrelpted in its immediate vicinity and decreasinq,ly so as we move away from it; the molecular transport properties become important in this re— glen near the wall, which is refer7ed to as the "laminar or viscous sub-layer", A rudimentary Couette-flow analysis shows how this influence can be specified in ouantitative descriptions of the flow; for example, in the familiar 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 follo- wing objectives were set: (a) the collection of heat transfer and drag data for smooth pipes and extraction from them of a relationship bet- ween the laminar Prandtl number and P; (b) the collection of all available information on flows past rough surfaces and devising a means of estimat- ing the influence of roughness on the parameter E mentioned earlier and also on P; (c) the conduction of experiments to detect whether the influence of roughness is confined to the region very close to the surface; and (d) the application of the general theory in its pre- sent state development to the calculation of hydrodynamics and heat-transfer in the conditions of the present experi- ment and of any other available flows past rough walls. The body of the work is sub-divided into four sections. In section I the theoretical framework into which the four aspects of the present work fits, is outlined. The sub- sequent sections II, III and IV are concerned respectively with: the influence of Prandtl number variation on the sub- layer in flows past smooth walls; the effect of wall rough- 8 ness on hydrodynamics and heat transfer; and an experimental investigation of the hydrodynamics and heat transfer in a radial wall-jet on a rough surface. 9 SUCTION I 10 CHAPTER 1 THE BASIC LODEL 1.1 The physical system The system which we direct attention to is the turbu- lent boundary layer formed by the flow of a fluid past a surface, as shown in figure 1.1. The surface may be either smooth or rough. There can be slots discharfing fluid along the surfE.ce parallel to the main-stream; the main-stream can either have a finite velocity or be at rest. Heat transfer may occur at the surface; and the fluid from the slot differ in temperature from the ambient. In order to simplify the mathematical problem, the follovAng restrictions are placed on the system: 10 steady flow; 2. fluid homogeneous in phase, having uniform molecu- lar properties and turbulent; body forces, such as those due to gravitation, ab- sent; 4e two-dimensional flow. By two-dimensional flows we mean those in which local quantities such as velocity and temperature depend on only two space variables. Under this category would be flows past plane surfaces, general cylindrical surfaces having their axes normal to the flow and stationary axi-symmetrical bodies with axes lying,- in the general direction of the flow. 11 1.2 System of co-ordinates The general system of co-ordinates in respect of which the equations are written is .shown in figure 1.2; that illustrated is the general case of an axi-symmetrical body. R is the distance of a general point S on the surface, from the axis of symmetry. The distance x is measured along the trace of the given surface on the plane passing through its axis and S; and y is measured along the local normal. 1.3 The differential equations Our ai.11 here will be to present the important equations devoid of much of the details of their derivations so as to enable the present work to be placed in the proper context. The partial differential equations g)overning fluid flow in the system shown, are as follows:- Mass conservation: 71(pRu) + A(oRv) = 0 ... (1.3-1) Momentum: 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".
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