Heat and mass transfer in turbulent boundary layers

Abstract

The foundation is laid of a general theory of heat and mass transfer in turbulent boundary layers. Both the mathematical and the physical aspects of the theory are dealt with.The mathematical contribution mainly consists of an implicit, finitedifference procedure for solving the boundary-layer equations. The novel features of the procedure include : the employment of a grid which adjusts its width so as to conform to the thickness of the layer in which the dependent variables vary significantly ; and the use, near a wall, of algebraic relationships expressing the once-for-all integrations obtained by the neglect of the longitudinal convection there. Calculations are presented to demonstrate that the solution procedure is accurate, economical and widely applicable.It is pointed out that a good mathematical tool prepares the road for research into physical hypotheses.On the physical side, van Driest's hypothesis is used to illustrate the algebraic relationships for the region near a wall ; their validity is tested by reference to experimental data including some of the present author. 3. 4.This sequence of events has also influenced the layout of this thesis.The contribution made by the new calculation procedure appeared so significant that I have given it the pride of place in this thesis, the earlier methods being only briefly mentioned. The work (including my experiments) relating to the physical aspects has been presented as illustrative material.It is true that, in the course of time, the calculation procedure described here is also likely to be superseded. What will be of lasting value to me is the training I received from Professor Spalding. The A finite-difference solution procedurePage 48 Choice of co-ordinate system 48 Conservation equations in x--u co-ordinates 50 2.3The entrainment rates 51 2.4The difference equation 54 Slip values at boundaries 602.5-1 What are slip values 60 2.5-2 Slip-value relations for a wall boundary 62 2.5-3 Slip-value relations for a free boundary 65 2.5-4Slip-value relations for a symmetry-line boundary 66 2.6Treatment of special boundary conditions 68 2.6-1The case of prescribed total flux through the wall 68 2.6-2The case of a symmetry-line boundary 69 Solution of the difference equations 71 Miscellaneous matters 72 2.8-1Grid-control formulae 72 2.8-2Calculation of the normal distance 73 2.8-3A characteristic thickness of the layer 75 2.8-4Choice of the forward step 76 3.Capabilities of the solution procedure: demonstration and appreciation 77 Purpose of the chapter 77 Some boundary-layer calculations 77 3.2-1 Plane turbulent mixing layer 77 3.2-2Compressible laminar boundary layer on flat plate 80 3.2-3Compressible turbulent boundary layer on flat plate 82 3.2-4 General remarks 86 Achievements and shortcomings of the present method 87 Suggestions for future work 888. Illustrations Examination of validity of the relationships 103 Further developments 108 Summary of the relationships for turbulent flow 109 Implications of the mixing-length hypothesis 112 Purpose of...