Considering the barotropic instability problem of the mean westerly current in the atmosphere we have performed a series of experiments in a rotating vessel (using water and a barotropic model) to study the behaviour of a zonal asymmetric basic current with respect to small perturbations. In the centre of a rotating cylindrical vessel (of large diameter and rotation rate ω) a smaller cylinder was installed, the rotation of which relative to the vessel, at a rate Δω, generates a nearly two-dimensional field of mean relative motion within a sharply defined region. The dominant zonal velocity component $\overline{v}$ shows monotonic radial decrease within this so-called friction zone. Now what happens if the relative rotation of the inner cylinder, the source of momentum, suddenly vanishes, i.e. δΩ = 0? The main result is that the basic zonal current $\overline{v}$, which now has an asymmetric radial profile ($\overline{v} = 0 $ at the inner cylinder and the outer edge of the friction zone), breaks down into vortices, the number of which, the integer wavenumber n, is a function of the parameter ε = δω/ω alone: n = n(ε); increasing ε eff effects a decrease of n. For a theoretical discussion of the experimental results we assume this to be a problem of barotropic instability and base our analytical considerations on the two-dimensional non-divergent vorticity equation, frictional forces being neglected. By applying a perturbation method and prescribing a realistic asymmetric basic current we can derive the relation ν = {[¾π/ln (ε + 1)½]2 + 1}½, which yields the real wavenumber v as a function of the parameter ε = δω/ω. The analytical results are in good agreement with the experiments.

Negative or rarefaction shock waves may exist in single-phase fluids under certain conditions. It is necessary that a particular fluid thermodynamic quantity Γ ≡ −½δ In (δP/δν)s/δ In ν be negative: this condition appears to be met for sufficiently large specific heat, corresponding to a sufficient level of molecular complexity. The dynamic formation and evolution of a negative shock is treated, as well as its properties. Such shocks satisfy stability conditions and have a positive, though small, entropy jump. The viscous shock structure is found from an approximate continuum model. Possible experimental difficulties in the laboratory production of negative shocks are briefly discussed.

An asymptotic description is proposed for supersonic laminar flow over a wedge or a backward-facing step, for large Reynolds number and for a base or step height which is small compared with the boundary-layer length. The analysis is carried out for adiabatic wall conditions and a viscosity coefficient proportional to temperature. In a particular limit corresponding to a very thick boundary layer, a similarity law is obtained for the base pressure $\overline{p}_b$. For a thinner boundary layer an asymptotic form for $\overline{p}_b$ is obtained which shows the dependence on the parameters explicitly and which permits good agreement with experiment. This latter result is based on an inviscid-flow approximation for the corner expansion and for reattachment, with viscous forces important primarily in a thin sublayer about the dividing streamline. A prediction of the pressure distribution at reattachment is given and the result is compared with experimental pressure distributions.

Second-order mean fields of motion and density are calculated for the two-dimensional problem of an internal gravity wave packet (the waves are predominantly of a single frequency o and wavenumber k) propagating as a wave-guide mode in an inviscid, diffusionless Boussinesq fluid of constant buoyancy frequency N, confined between horizontal boundaries. (The same mathematical analysis applies to the formally identical problem for inertia waves in a homogeneous rotating fluid.)

To leading order the mean motions turn out to be zero outside the wave packet, which consequently possesses a well-defined fluid impulse [Iscr ]. This is directed horizontally, and is given in magnitude and sense by \[ {\cal I} = \alpha{\cal M};\quad\alpha = \frac{2c_{\rm g}(c-c_{\rm g})(c+2c_{\rm g})}{c^3-4c^3_{\rm g}}. \] Here [Mscr ] is the so-called ‘wave momentum’, defined as wave energy divided by horizontal phase velocity c ≡ ω/k, and cg = c(N2–ω2)/N2, the group velocity.

If the wave packet is supposed generated by a horizontally towed obstacle, [Mscr ] appears as the total fluid impulse, but of this a portion [Mscr ]-[Iscr ] in general propagates independently away from the wave packet in the form of long waves. When the wave packet itself is totally reflected by a vertical barrier immersed in the fluid, the time-integrated horizontal force on the barrier equals 2 [Iscr ] (and not 2 [Mscr ] as might have been expected from a naive analogy with the radiation pressure of electromagnetic waves.)

Experiments are described in which the vortex shedding from a bluff body and the base pressure coefficient have been measured in a shear flow. It is shown that the shedding breaks down into a number of spanwise cells in each of which the frequency is constant. The division between the cells is thought to be marked by a longitudinal vortex in the stream direction and this is supported by evidence from experiments where a longitudinal vortex was generated in an otherwise uniform flow.

The variable-coefficient Korteweg–de Vries equation \[ H_X + {\textstyle\frac{3}{2}}d^{-\frac{7}{4}}HH_{\xi} + {\textstyle\frac{1}{6}}\kappa d^{\frac{1}{2}}H_{\xi\xi\xi} = 0 \] with d = d(εX) is discussed for solitary-wave initial profiles. A straightforward asymptotic solution for ε → 0 is constructed and is shown to be non-uniform both ahead of and behind the solitary wave. The behaviour ahead is rectified by matching to the appropriate exponential form and, together with the use of conservation laws for the equation, the nature of the solution behind the solitary wave is discussed. This leads to the formulation of the solution in the oscillatory ‘tail’, which is again matched directly.

The results are applied to the development of the solitary wave into variable-depth water, and the predictions are compared with those obtained, for example, by Grimshaw (1970, 1971). Finally, the asymptotic behaviour of both the solitary wave and the oscillatory tail are assessed in the light of some numerical integrations of the equation.