Proper orthogonal decomposition (the Karhunen-Loève expansion) is applied to convective flows in a tall differentially heated cavity. Empirical spatial eigenfunctions are computed from a multicellular solution at supercritical conditions beyond the first Hopf bifurcation. No assumption of periodicity is made, and the computed velocity and temperature eigenfunctions are found to be centro-symmetric. A lowdimensional model for the dynamical behaviour is then constructed using Galerkin projection. The reduced model successfully predicts the first Hopf bifurcation of the multicellular flow. Results determined from the low-order model are found to be in qualitative agreement with known properties of the full system even at conditions far from criticality.

An outline of the classical far-and near-field solutions for small-amplitude one-dimensional unsteady flows in a general inviscid relaxing gas is given. The structure of the complete flow field, including a non-linear near-frozen (high frequency) region at the front, is obtained by matching techniques when the relaxation time is ‘large’.If the energy in the relaxing mode is small compared with the total internal energy, the solution in the far field is, in general, more complex than that predicted by classical theory. In this case the rate process is not necessarily able to diffuse all convective steepening. An equation valid in this limit is derived and discussed. In particular, a sufficient condition for the flow to be shock-free is established. For an impulsively withdrawn piston it is shown that the solution is single-valued both within and downstream of the fan. Some useful similarity rules are pointed out.The corresponding formulation for two-dimensional steady flows is also noted in the small energy limit.

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity w( IJI) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (l) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form wayn, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the 68:103-188 (1983) Copyright © 1983 by the Massachusetts Institute of Technology Published by Elsevier Science Publishing Co., Inc. boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar. STUDIES IN APPLIED MATHEMATICS

Supersonic nozzle flows of a condensable vapour are considered in the high activation limit for homogeneous nucleation. Conditions are determined under which the final collapse of the supersaturated state is described by a condensation shock. It is shown that the shock zone is associated with droplet growth: droplet production occurs in a thin layer upstream of the growth region. Some new scaling laws are obtained for the structure of the production layer.