The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.

The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocity u, rather than the spatial co-ordinate x, is used as the independent variable, and an equation for ∂f/∂u at a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.

The mixing layer bounding the exhaust plume associated with hypersonic highaltitude rockets is analysed as a laminar binary mixture of perfect gases with Lewis and Prandtl numbers of unity. A far-field approximation to the undisturbed jet core and a Newtonian pressure balance between the jet and ambient gases are used to construct the mixing-layer location. Longitudinal pressure variations are neglected and resultant errors are evaluated. Boundary conditions at the edge of the mixing layer are evaluated by streamline tracing to shock entry points. The sensitivity of properties in the mixing layer to variations in the plume angle of attack, engine nozzle efficiency and engine thrust are examined, and an approximate density and thrust scaling of mixing-layer overall properties is developed.

The results of an experimental investigation of the mean- and fluctuating-flow properties of a compressible turbulent boundary layer in a shock-wave-induced adverse pressure gradient are presented. The turbulent boundary layer developed on the wall of an axially symmetric nozzle and test section whose nominal free-stream Mach number and boundary-layer-thickness Reynolds number were 4 and 105, respectively. The adverse pressure gradient was induced by an externally generated, conical shock wave.

Mean and time-averaged fluctuating-flow data, including the experimental Reynolds shear stresses and experimental turbulent heat-transfer rates, are presented for the boundary layer and external flow, upstream, within and downstream of the pressure gradient. The turbulent mixing properties of the flow were determined experimentally with a hot-wire anemometer. The calibration of the wires and the interpretation of the data are discussed.

From the results of the investigation, it is concluded that the shock-wave/boundary-layer interaction significantly alters the shear-stress characteristics of the boundary layer.

We discuss the creation of a finite-element program suitable for solving incompressible, viscous free-surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. We conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related ‘stick-slip’ problem and the tube inlet problem, both of which also contain stress singularities, are also given.