Table of contents

A new method of approximation for the small-gap Taylor problem is described, in which non-axisymmetric modes appear as a perturbation of axisymmetric modes. A detailed study of the dominant-mode pattern is made in a neighbourhood of (δ, μ) = (0, −0.7834) and the exact linearized equations are used to extend this. Here δ and μ are the dimensionless gap width and the ratio of the angular velocity of the outer cylinder to that of the inner cylinder. We find for −0.7834 > μ > −1.5 that as δ decreases the wave number of the most unstable mode increases.

Parametrically excited waves are usually modelled with a nonlinear amplitude equation. It has recently been demonstrated that the behaviour of these waves depends critically upon the coefficient of the cubic damping term in the nonlinear amplitude equation, and especially upon the sign of this coefficient (see Decent and Craik [J. Fluid Mech. 293 (1995) 237]. However, very little work has been carried out on theoretically determining the value of this coefficient. This paper derives the coefficient of cubic damping for the single-mode nonlinear amplitude equation which models two-dimensional gravity waves in a narrow rectangular container. Energy dissipation in the main body of the fluid and in boundary layers at the sidewalls and at the surface is considered. Theoretical results agree fairly well with an experiment carried out by Decent and Craik (1995).

Volume integrals of the second and third invariants, i.e. Q_A and R_A, respectively, of the velocity gradient tensor A_ij over an incompressible flow domain are shown to vanish for certain combinations of boundary conditions used in a large variety of direct numerical simulations of turbulent flows. For these turbulent flows, the dissipation of total kinetic energy is directly proportional to the total enstrophy. Betchov [J. Fluid Mech. 1 (1956) 497] showed that for homogeneous flows an increase in mean enstrophy implies that it is more likely that the intermediate rate-of-strain is positive rather than negative. This paper shows that using the framework of the invariants of the velocity gradient tensor and a volume integral formulation an analogous implication can be derived for more general classes of flows. Specifically, in addition to the case of homogeneous flows, this paper shows that in the cases of unbounded inhomogeneous flows (i.e. free shear flows) and for wall-bounded semi-infinite domain flows with zero tangential pressure gradient an increase in total Q_W, the second invariant of the rate of rotation tensor, implies that it is more likely that the intermediate rate-of-strain is positive rather than negative.

A new similarity solution is obtained for flow of a uniform stream past an aligned, semi-infinite flat plate. The fluid is incompressible, of constant density, and constant absolute viscosity. A detailed examination is made of flows involving steady rates of accretion and of ablation at the leading edge. Both the classical Blasius boundary layer and the Rayleigh-Stokes shear layer are encompassed, each representing a different extreme case of the similarity solution. In contrast to the accretion case, only relatively small rates of ablation can be tolerated and there is a qualitative change in the boundary-layer solutions.