Nonlinear mode theory of shear flow development

Abstract

The major part of this thesis is concerned with the description of a new technique for analysing the development of shear layers (particularly in free flows). A principal feature of this method is the expansion of the velocity distribution at each streamwise station x in a series of orthogonal functions (in the coordinate normal to the main flow). The Fourier coefficients in such an expansion (into modes) are now functions of x; the equations of motion when written in terms of these coefficients form an infinite system of coupled quasilinear ordinary differential equations for the modal amplitudes. A variety of examples are then considered to study how effectively this system of equations can be solved in various commonly occurring flows. The examples include: the non?linear wake subjected to favourable and adverse pressure gradients (including the extreme case of interaction with shock waves where comparisons with experiment are provided), jets exhausting into fluid at rest or in motion, and mixing layers. In all cases the initial conditions (in the form of a velocity distribution) can be considered to be essentially arbitrary. These studies show that a single term approximation for the velocity profile often gives results of acceptable accuracy; much higher accuracies can be easily obtained by including several modes and by solving the system of equations on a computer using a straightforward Runge–Kutta procedure. Compared to finite?difference schemes for solving the equations of motion the present method shows enormous savings in computing effort in all the examples considered, with (in some cases at least) even improved accuracies over the best available results. In all these examples the flow is laminar and two?dimensional, although occasionally the axisymmetric case is also considered. Extension to turbulent flow is possible if satisfactory models for the Reynolds stress are available. Furthermore the fluid is either incompressible or a model perfect gas (Prandtl number unity, viscosity proportional to temperature) for which the well?known Howarth–Dorodnitsyn transformation enables reduction to the incompressible problem. Several loosely connected studies investigating the effect of relaxing assumptions on Prandtl number and viscosity are therefore included in the thesis. These concern, first, similarity solutions for linearised wakes, which are here presented in a much more general form than before; second, the boundary layer on a flat plate. Flow at very high Prandtl number is analysed by the method of matched asymptotic expansions; at moderate Prandtl numbers (and with power?law viscosity?temperature relations), it is concluded that deviations from the properties assumed for the model fluid do not have a large effect on the final results. Incidentally Meksyn’s asymptotic method is given a new interpretation which sheds light on the conditions under which the method is likely to be successful, and enables us to make quantitative estimates of the effect of fluid properties.