We present a nonlinear unsteady Darcy's equation which includes inertial effects for flows in a porous medium or Hele-Shaw cell and discuss the conditions under which it reduces to the classical Darcy's law. In the absence of surface tension we derive a generalized Polubarinova–Galin equation in a circular geometry, using the method of conformal mapping. The linear stability of the base-flow state is examined by perturbing the corresponding conformal map. We show that inertia always has a tendency to stabilize the interface, regardless of whether a less viscous fluid is displacing a more viscous fluid or vice versa.

In many turbulent flows near obstacles or in ducts, the turbulence is inhomogeneous in two directions perpendicular to the main flow direction. In convective flows, there may initially be no mean motion. In both types of flow the gradients of Reynolds stresses drive mean motions in directions of inhomogeneity. Using the method of rapid distortion theory developed by Hunt & Graham (J. Fluid Mech., vol. 84, 1978, p. 209), we analyse these gradients where homogeneous isotropic turbulence is impinging onto two semi-infinite flat rigid surfaces intersecting at right angles. The mean velocity is assumed to be uniform (i.e. the surfaces move at free-stream velocity, or the boundary layers are very thin). The inhomogeneous spectra, variances and Reynolds-stress gradients are evaluated. For isotropic free-stream turbulence with mean square velocity u2∞1, the mean square velocity fluctuation at a high Reynolds number in the corner is u21(X, 0, 0) = 2.121u2∞1, independent of the form of the spectrum. This is explained by estimating how the free-stream eddies are blocked by the two walls. The gradients of Reynolds stresses force a mean secondary flow to develop; its direction is into the corner, and its magnitude at time t is of order tu2∞1/L∞, where L∞ is the integral scale. These results are tested in a wind tunnel experiment. A turbulence-generating grid installed at the entrance to the test section generates nearly isotropic, grid-generated turbulence. A corner plate with faces parallel to the mean flow and sharp edges is placed downstream of the grid so that shear-free turbulence impinges onto the corner plate. The turbulent Reynolds number based on , is 1400 at the leading edge of the plate. A hot-wire anemometry is used to measure instantaneous velocities. The experimental results are consistent with the rapid distortion theory estimates for the variances and the secondary mean motion, which is in the same direction and has the same order of magnitude as Prandtl's analysis of shear-driven secondary flow (of the second kind). We conclude that the blocking mechanism adds to the shear effects and has a significant and sometimes dominant contribution to the crossflows wherever it acts in two non-parallel directions, such as convection in a corner. Consequently, mean transport into corners occurs for most kinds of distorted flow with weak viscous stresses, which has many engineering and environmental implications. There are also implications for the chaotic nature of many confined flows near corners.

We present a numerical study of the reversals and reorientations of the large-scale circulation (LSC) of convective fluid in a cylindrical container of aspect ratio one. We take Prandtl number to be 0.7 and Rayleigh numbers in the range from 6 × 105 to 3 × 107. It is observed that the reversals of the LSC are induced by its reorientation along the azimuthal direction, which are quantified using the phases of the first Fourier mode of the vertical velocity measured near the lateral surface in the midplane. During a ‘complete reversal’, the above phase changes by around 180°, leading to reversals of the vertical velocity at all the probes. On the contrary, the vertical velocity reverses only at some of the probes during a ‘partial reversal’ with phase change other than 180°. Numerically, we observe rotation-led and cessation-led reorientations, in agreement with earlier experimental results. The ratio of the amplitude of the second Fourier mode and the first Fourier mode rises sharply during the cessation-led reorientations. This observation is consistent with the quadrupolar dominant temperature profile observed during the cessations. We also observe reorientations involving double cessation.

The thermogravitational instability in a fluid layer of a non-Newtonian medium heated from below is investigated. Linear and weakly nonlinear analyses are successively presented. The fluid is assumed to obey the Carreau–Bird model. Although the critical threshold is the same as for a Newtonian fluid, it is found that non-Newtonian fluids can convect in the form of rolls, squares or hexagons, depending on the shear-thinning level. Similar to Newtonian fluids, shear-thickening fluids convect only in the form of rolls. The stability of the convective steady branches is carried out to determine under which specific conditions a pattern is preferred. The influence of the rheological and physical parameters is examined and discussed in detail.

In this paper, unsteady thin aerofoil theory is extended to the case of two interacting aerofoils that oscillate harmonically perpendicular to the direction of the incident flow. The two aerofoils represent the headsail and mainsail of a yacht that sails upwind in waves. The developed theory is validated against theoretical data from the literature and results from wind tunnel tests with rigid, high-aspect ratio sail models oscillating at reduced frequencies from k = 0 to k = 0.68. Good agreement is found between the predicted and measured chordwise pressure distributions. An application of the theory to the case of an International America's Cup Class yacht reveals that the time-varying components of the aerodynamic forces are small and that the thrust gain is minimal, i.e. only very little energy can be extracted from the unsteady flow about the sails. No attempt is made to investigate the influence of the flexibility of the sails, three-dimensional effects or phenomena related to dynamic stall.

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.