The viscous structure of a separated eddy is investigated for two cases of simplified geometry. I n 9 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier-Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in 9 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

A viscous-inviscid interaction is produced when a compressible laminar boundary layer encounters a corner. The correct mathematical structure for such interactions at large Reynolds number is given by the asymptotic triple-deck theory. In the present work the triple-deck equations for supersonic and hypersonic flows are solved for both compression and expansion corners. Results are presented for a range of corner angles, including separated cases, and are compared with experimental data and with finite Reynolds number calculations based on an interacting boundary-layer model.

An inverse problem in unsteady heat conduction is one for which boundary conditions are prescribed internally, the surface conditions being unknown. By specifying the boundary conditions at a single location, an exact solution is obtained as a rapidly convergent series with the well-known, lumped capacitance approximation as the leading term. Specific forms of the series are determined for sample inverse problems: solid slab, cylinder, sphere, and transpiration-cooled slab. The solution also is applied to direct problems, involving two-point boundary conditions. By truncating the series, approximate solutions of simple form result. The one-term and two-term approximations compare well with exact solutions.

A numerical computation of the laminar boundary layer on a fixed circular disk of radius a whose axis is concentric with that of a vortex having circulation Γ is described. The computations were started at the edge of the disk and continued inward toward the axis until the properties of the terminal flow became evident. A two-layer asymptotic expansion was formulated for the solution of the boundary-layer equations near the axis, and the terminal-flow properties revealed by the analysis are shown to be in excellent agreement with the numerical results. The structure of the terminal boundary layer consists of an inner layer next to the surface with thickness O[(ν/Γ)1/2r] in which the flow is primarily radial, and an outer layer with thickness O[(ν/Γ)1/2a] of predominantly inviscid nature in which the flow recovers to the external potential vortex. The mass flux in the outer layer does not vanish as r→0, indicating that the boundary layer must erupt from the surface at r=0in the manner envisioned by Moore.

Laboratory experiments on swirling flows through tubes often exhibit a phenomenon called vortex breakdown, in which a bubble of reversed flow forms on the axis of swirl. Mager has identified breakdown with a discontinuity in solutions of the quasicylindrical flow equations. In this study we define a tornado-like vortex as one for which the axial velocity falls to zero for sufficiently large radius, and seek to clarify the conditions under which the solution of the quasi-cylindrical flow equations can be continued indefinitely or breaks down at a finite height. Vortex breakdown occurs as a dynamical process. Hence latent-heat effects, though doubtless important to the overall structure and maintenance of the tornado, are neglected here on the scale of the breakdown process. The results show that breakdown occurs when the effective axial momentum flux (flow force) is less than a critical value; for higher values of the flow force, the solution continues indefinitely, with Long's (1962) similarity solution as the terminal state. When applied to the conditions of the 1957 Dallas tornado, the computed breakdown location is in agreement with Hoecker's analysis of the observations.

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The effect of surface topography on an otherwise two-dimensional boundary-layer flow is investigated. The flow is assumed to be steady, laminar and incompressible, and is described by triple-deck theory. The basic problem reduces to the solution of a form of the nonlinear three-dimensional boundary-layer equations, together with an interaction condition. The solutions are obtained by a spectral method, with the computations carried out iteratively in Fourier-transform space. Numerical results are presented for several cases including three-dimensional separation. Comparison is made with the predictions of linearized theory. The decay corridor observed by Smith is confirmed for one localized configuration, but not for another having a broader height distribution.

The Fourier transform method is applied to the problem of computing viscous flows involving boundary-layer separation, based on the triple-deck model of viscous-inviscid flow interaction. As used here, the method is pseudo-spectral in that the non-linear inertia terms are evaluated in physical variables, although the main computations are made in spectral variables. The Fast-Fourier-Transform algorithm is used to expedite the interated transformations.The method is much faster than conventional finite-difference