The effects of thermally insulating boundaries on rapidly and almost rigidly rotating gas flows are examined. It is shown that, on a thermally insulating boundary, all boundary layers disappear to zeroth order and that the geostrophic flow alone satisfies the kinematical boundary condition on such a boundary. The temperature gradient of the geostrophic flow is on a horizontal thermally insulating boundary corrected by a weak Ekman layer of strength E½ where E is the Ekman number. On a vertical thermally insulating boundary, the temperature gradient of the geostrophic flow is in the general case corrected by E¼ and $E^{\frac{1}{3}}$ Stewartson layers of strengths E¼ and $E^{\frac{1}{3}}$ respectively.

This paper considers two-dimensional flow generated in a stably stratified porous medium by monotonic differential heating of the upper surface. For a rectangular cavity with thermally insulated sides and a constant-temperature base, the flow near the upper surface in the high-Darcy–Rayleigh-number limit is shown to consist of a double horizontal boundary layer structure with descending motion confined to the vicinity of the colder sidewall. Here there is a vertical boundary layer structure that terminates at a finite depth on the scale of the outer horizontal layer. Below the horizontal boundary layers the motion consists of a series of weak, uniformly stratified counter-rotating convection cells. Asymptotic results are compared with numerical solutions for the cavity flow at finite values of the Darcy–Rayleigh number.

An earlier paper (Howe 1967) considered a non-linear theory of open-channel steady flow of deep water past a slowly modulated wavy wall. The wave pattern on the free surface of the water was obtained as the solution of a stably posed elliptic Cauchy problem, the main feature of the solution being the appearance of a ‘shock’ across which there is an abrupt change of phase. Such phase jumps can occur in a wide range of similar problems, but the advantage of the present case is that it is rather well suited to experimental investigation. This paper is therefore a lead-in to the more general problem of phase jumps, and uses the principle of conservation of energy in conjunction with the earlier solution to predict the possible position of the discontinuity on the free surface of the water. The possible nature of the free surface in the vicinity of the phase jump is also discussed (figure 4). This is a region where the width of the wave troughs becomes dramatically shorter than that of the neighbouring troughs. An approximate method of determining the line along which the phase jump occurs, not depending on a knowledge of the solution of the Cauchy problem, is also presented.

The problem of gas motion inside a resonance tube, closed at one end by a plug and fitted at the other with an oscillating piston is treated analytically and numerically. An analytical model is derived for arbitrary piston motion and gas oscillations about the first resonance frequency, where the gas flow is characterized by a shock wave travelling periodically back and forth in the tube. The model is obtained by a perturbation analysis in terms of a small-amplitude parameter $\varepsilon$. All the hydrodynamic properties of the gas are predicted with accuracy up to the second-order terms of $\varepsilon$. Isentropic and adiabatic problem formulations are addressed. Expressions for spatial distributions of the time-averaged hydrodynamic gas properties are derived for any frequency within the resonance band. It is shown that they are determined by the gas adiabatic exponent $\gamma$ and the law that governs the motion of the piston. The analytical model is verified by comparison with a numerical solution, showing good agreement.

To account for our previous experimental results, the flow past a flat plate set parallel to a uniform stream is investigated on the basis of Oseen's equations of motion. We are mainly concerned with the wake region and calculate the velocity field in detail for the range of Reynolds numbers 20–3000, corresponding to the experiments. The calculation shows that there is a velocity overshoot in the velocity distribution near the trailing edge, with the same magnitude as in the experiments. Furthermore, we confirm the experimental fact that the velocity on the centre-line of wake recovers in proportion to the square root of the distance from the trailing edge in the near wake. The pressure field is also examined to see the streamwise pressure gradient in the near wake.

Tornadoes are one type of violent flow phenomenon and occur in many places in the world. There are many research methods that aim to reduce the loss of human lives and material damage caused by tornadoes. One effective method is numerical simulation such as that in Ishihara et al. (J. Wind Engng Ind. Aerodyn., vol. 99, 2011, pp. 239–248). The swirling structure of the Navier–Stokes flow is significant for both the mathematical analysis and numerical simulations of tornadoes. In this paper, we try to clarify the swirling structure. More precisely, we performed numerical computations on axisymmetric Navier–Stokes flows with a no-slip flat boundary. We compared a hyperbolic flow with swirl and one without swirl, and observed that the following phenomenon occurs only in the swirl case: the distance between the point with the maximum magnitude of velocity $|\boldsymbol{v}|$ and the $z$-axis changed drastically at a specific time (which we call the turning point). Besides, an ‘increasing velocity phenomenon’ occurred near the boundary, and the maximum value of $|\boldsymbol{v}|$ was obtained near the axis of symmetry and the boundary when the time was close to the turning point in the swirl case.

The problem of supersonic flow over an inclined flat-plate delta wing with supersonic edges is solved to second order in incidence. This solution for surface pressure is uniform and fully analytic. The approach utilizes a reverse-flow integral method previously developed for second-order problems. This method is augmented by a number of techniques appropriate to its framework. The simplification over standard techniques achieved by using these reverse-flow methods is quite substantial and makes the problem tractable.

Reverse-flow procedures give a volume-surface integral relation that connects the second-order forward flow over the body of interest with the linearized reverse-flow over a related body. A singular integral equation is generated from the integral relation by introducing the edge sweep of the reverse-flow wing as a free parameter. An inversion is available which gives the second-order solution on the surface of the wing. The solution is then made uniformly valid using techniques previously developed.

The concept of available energy in a stratified fluid is revisited from the point of view of non-canonical Hamiltonian systems. We show that the concept of available energy arises when we minimize the energy subject to the constraints associated with the existence of Lagrangian invariants. The non-canonical structure implies that there exists a class of dynamically equivalent Hamiltonians, related by a local (in phase space) gauge symmetry. A local diagnostic energy can be defined via the Hamiltonian density chosen imposing a specific gauge-fixing condition on the class of dynamically similar Hamiltonians. The gauge-fixing condition that we introduce selects a specific local diagnostic energy which is well suited to study the effect of diabatic processes on the evolution of the available energy. Non-inertial effects, which are notoriously elusive to capture within an energetic framework, are naturally included via conservation of potential vorticity. We apply the framework to stratified flows in inertial and non-inertial frames. For stratified Boussinesq flows, when the initial distribution of potential vorticity is even around the origin, our framework recovers the available potential energy introduced by Holliday & McIntyre (J. Fluid Mech., vol. 107, 1981, pp. 221–225), and as such, depends only on the mass distribution of the flow. In rotating flows, the isopycnals of the ground state are generally not flat, and the ground state may have kinetic energy. We finally demonstrate that flows in non-inertial frames characterized by a low Rossby number ($Ro$), the local diagnostic energy has, to lowest order in $Ro$, a universal character.

The small-perturbation theory for steady, inviscid, non-equilibrium flow is extended to include the transonic speed range. The resulting transonic small-perturbation equation for the velocity potential is solved for flow inside a wavy cylinder. It is shown that this solution gives a representation of transition through sonic speed at the narrowest sections of the cylinder. This extension of the theory is applied to non-equilibrium nozzle flow.

The problem of determining the threshold of motion of a sediment particle resting on the bed of an open channel has historically been dominated by an approach based on the time–space-averaged bed shear stress (i.e. Shields criterion). Recently, experimental studies have promoted an alternative approach to predict the dislodgement threshold, which is based on the impulse of the flow-induced force. Nonetheless, theoretical analyses accompanying these studies result in complex expressions that fail to provide a direct estimate of said impulse threshold. We employ the work–energy principle to derive a prediction of the fundamental impulse threshold that the destabilising hydrodynamic force must overcome in order to achieve full particle dislodgement. For the bed configuration studied, which is composed of spheres, the proposed expression depends on the mobile particle’s size and mass, and shows excellent agreement with experimental observations previously published. The derivation presented in this paper may thus represent a robust theoretical framework that aids in the reinterpretation of existing data, as well as in the design of future experiments aimed at analysing the importance of hydrodynamic impulse as a criterion for prediction of particle dislodgement.

The transition from steady to oscillatory three-dimensional convection in a cube of saturated porous material is calculated to occur at Rayleigh number R = 584 due to seven pairs of thermal blobs which circulate around the cube. This travelling wave instability is shown to be closely related, first as regards structural characteristics and then as regards mechanism of instability, to an analogous instability in two dimensions. The correspondence with the two-dimensional flow is established via a correspondence with a nonlinear base flow in a box of square planform of a different aspect ratio (l/√2) and ultimately derives from the symmetries of the base flow in the cube.

A class of constants of motion is obtained for the time-dependent form of the Rayleigh stability equation, which provides stability criteria for velocity profiles with multiple inflexion points.

The possibility of acoustic control of the two-dimensional instabilities of a lossless plane shear layer of vanishing thickness is studied. The shear layer is formed from a body of incompressible fluid sliding over another fluid at rest. It is unstable through the generation of Kelvin–Helmholtz waves. We consider the possibility of adding to this linearly unstable flow a simple source, driven in such a way that its field interferes destructively with the instability to render the flow stable. The required strength of the unsteady control source is determined in terms of the fluctuating velocity at some fixed position in the moving fluid. We show that no unstable Kelvin–Helmholtz wave could survive the action of such a source. Next, we examine the scope for constructing the control signal from a measurement of the flow velocity at some fixed position. The source is a linear functional of the monitored velocity and we give the transfer function that would be required for the instabilities to be controlled. We prove that such control action would completely stabilize the otherwise unstable vortex sheet, and that other alternative sensor/actuator arrangements could also be effective. We go on to show that our particular very simple arrangement could not in fact be realized because, if required to work at all frequencies, it would not be causal. If we insisted on causality the vortex sheet would then only be stabilized over most frequencies. That would of course make the controlled flow completely different from the vortex sheet whose instabilities are so well known – and troublesome. We conjecture that there will exist some variations of the basic control arrangement described here that are both physically realizable and effective over the required frequency range. From our study of the initial value problem we have concluded that short perturbations would be attenuated very rapidly.

A numerical solution is presented for predicting the flow and heat transfer due to free convection in the entry region of a cooled vertical pipe, open at both ends. Following Lighthill (1953) the Kármán-Pohlhausen method is used in this analysis. Velocity and temperature profiles are assumed satisfying the physical boundary conditions, and the integrated forms of the equations of motion and the equations themselves at the axis and the walls of the pipe are used to calculate the various parameters involved in the profiles assumed. Two cases of constant wall temperature and linearly decreasing temperature with height were considered. Cooling decelerates the fluid in the boundary layer that forms at the wall and the fluid in the core is accelerated by virtue of continuity. As cooling progresses, at Rayleigh number O(103) the fluid in the boundary layer starts reversing at first and then assumes progressively increasing velocities. Graphs are presented for the development of the predicted velocity and temperature profiles and the other parameters involved, for a situation when the boundary layer fills the whole pipe.

We consider the finite-amplitude instability of incompressible spherical Couette flow between two concentric spheres of radii $R_1$ and $R_2$$({>}R_1)$ in the narrow-gap limit, $\varepsilon\,{\equiv}\,(R_2-R_1)/R_1\,{\ll}\,1$, caused by rotating them both about a common axis with distinct angular velocities $\Omega_1$ and $\Omega_2$ respectively. In this limit it is well-known that the onset of (global) linear instability is manifested by Taylor vortices of roughly square cross-section close to the equator. According to linear theory this occurs at a critical Taylor number $T_{\rm crit}$ which, remarkably, exceeds the local value $T_c$ obtained by approximating the spheres as cylinders in the vicinity of the equator even as $\varepsilon\,{\downarrow}\,0$. Previous theoretical work on this problem has concentrated on the case of almost co-rotation with $\delta\,{\approx}\,(\Omega_1\,{-}\,\Omega_2)/\Omega_1\,{=}\,\OR(\varepsilon^{1/2})$ for which $T_{\rm crit}\,{=}\,T_c\,{+}\,\OR(\delta^2)\,{+}\,\OR(\varepsilon)$. In this limit the amplitude equation that governs the spatio-temporal modulation of the vortices on the latitudinal extent $\OR(\varepsilon^{1/2}R_1)$ gives rise to an interesting bifurcation sequence. In particular, the appearance of global bifurcations heralds the onset of complicated subcritical time-dependent finite-amplitude solutions.

Here we switch attention to the case when $\varepsilon^{1/2}\,{\ll}\,\delta\,{\le}\, 1$. We show that for Taylor numbers $T\,{=}\,T_c+\OR((\delta\varepsilon)^{2/3})$ there exists a locally unstable region of width $\OR((\delta\varepsilon)^{1/3}R_1)$ within which the amplitude equation admits solutions in the form of pulse-trains. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and consists of a wave propagating towards the equator under an envelope. The pulse drifts at a slow speed (relative to the wave velocity) proportional to its distance (and away) from the equator. Both the wavelength and the envelope width possess the same relatively short length scale $\OR((\varepsilon^{2}/\delta)^{1/3}R_1)$. The appropriate theory of spatially periodic pulse-trains is developed and numerical solutions found. Significantly, these solutions are strongly subcritical and have the property that $T\to T_c$ as $\varepsilon\,{\downarrow}\,0$.

Two particular limits of our theory are examined. In the first, $\varepsilon^{1/2}\,{\ll}\,\delta\,{\ll}\,1$, the spheres almost co-rotate and the pulse drift velocity is negligible. A comparison is made of the pulse-train predictions with previously obtained numerical results pertaining to large (but finite) values of $\delta/\varepsilon^{1/2}$. The agreement is excellent, despite the complicated long-time behaviour caused by inhomogeneity across the relatively wide unstable region.

Our second special case $\delta\,{=}\,1$ relates to the situation when the outer sphere is at rest. Now the poleward drift of the pulses leads to a slow but exponential increase of their separation with time. This systematic pulse movement, over and above the spatial inhomogeneity just mentioned, necessarily leads to complicated and presumably chaotic spatio-temporal behaviour across the wide unstable region of width $\OR(\varepsilon^{1/3}R_1)$ on its associated time scale, which is $\OR(\varepsilon^{-1/3})$ longer than the wave period. In view of the several length and time scales involved only qualitative comparison with experimental results is feasible. Nevertheless, the pulse-train structure is robust and likely to provide the building block of the ensuing complex dynamics.

Stratified, inviscid channel flow over a thin barrier or into an abrupt contraction is considered on the hypotheses that the upstream dynamic pressure and density gradient are constant (Long's model) for those parametric régimes in which the hypotheses are tenable for finite-amplitude disturbances, namely k < 2 for the barrier and k < 1 for the contraction, where k = NH/πU is an inverse Froude number based on the Vaisälä frequency N, the channel height H, and the upstream velocity U. Reverse flow in the neighbourhood of the forward stagnation point, which implies the formation of an upstream separation bubble, is found for certain critical ranges of k. The maximum barrier height for which the dominant lee-wave mode can exist without reversed flow either upstream or downstream of the barrier is 0·34H. The limiting case of a half space is considered briefly, and forward separation is found for κ = Nh/U > κs, where κs = 2·05 for a thin barrier and 1·8 for a semi-circular barrier. The corresponding values for reverse flow in the lee-wave field are κc = 1·73 and 1·3, respectively.

The steady flow of an ideal gas with finite conductivity past a thin wedge section in an aligned magnetic field is considered within the linearized theory. The wedge is assumed to be an insulator or a perfect conductor. The flow field around the obstacle is found by numerical integration of the Fourier transform for various values of the Mach number, the Alfven Mach number and the magnetic Reynolds number in the main stream. The outer flow is estimated by the method of asymptotic expansions. These results disclose how the inner flow including gas-dynamic features changes to the magnetohydrodynamic outer flow.

The application of slender-body theory to the evaluation of the three-dimensional surface velocities induced by a boundary layer on an elliptic cylinder is considered. The method is applicable when the Reynolds number is sufficiently large so that the thin-boundary-layer approximation is valid. The resulting potential problem is reduced to a two-dimensional consideration of the flow over an expanding cylinder with porous boundary conditions. The limiting solutions for a flat plate of finite span and a nearly circular cross-section are obtained in a simple analytic form. In the former case, within the limitations of slender-body theory, the results are in exact agreement with the complete three-dimensional solution for this geometry.