The Burgers equation, in spherical and cylindrical symmetries, is studied numerically using pseudospectral and implicit finite difference methods, starting from discontinuous initial (N wave) conditions. The study spans long and varied regimes–embryonic shock, Taylor shock, thick evolutionary shock, and (linear) old age. The initial steep-shock regime is covered by the more accurate pseudospectral approach, while the later smooth regime is conveniently handled by the (relatively inexpensive) implicit scheme. We also give some analytic results for both spherically and cylindrically symmetric cases. The analytic forms of the Reynolds number are found. These give results in close agreement with those found from the numerical solutions. The terminal (old age) solutions are also completely determined. Our analysis supplements that of Crighton & Scott (1979) who used a matched asymptotic approach. They found analytic solutions in the embryonic-shock and the Taylor-shock regions for all geometries, and in the evolutionary-shock region, leading to old age, for the spherically symmetric case. The numerical solution of Sachdev & Seebass (1973) is updated in a comprehensive manner; in particular, the embryonic-shock regime and the old-age solution missed by their study are given in detail. We also study numerically the non-planar equation in the form for which the viscous term has a variable coefficient. It is shown that the numerical methods used in the present study are sufficiently versatile to tackle initial-value problems for generalized Burgers equations.

In this paper the solitary-wave solutions of the Euler equations of motion are treated as a perturbation of the KdV equaton. We show analytically, that the amplitudes to two solitary waves are altered after interaction. This change in amplitude is calculated, showing that the smaller wave dereases in amplitude and the larger wave increases in amplitude.

We determine the circumstances under which baroclinic instability in the Charney model subjected to localized time-periodic forcing manifests itself as a wavetrain that oscillates at the source frequency and amplifies in space with distance from the source; analytical and numerical results describing the salient characteristics of such waves are presented. The spatially amplifying disturbance is a hitherto unsuspected part of the response to a pulsating source, and coexists with the more familiar neutral Rossby wavetrains; it is likely to play a role in a wide range of atmospheric and oceanic phenomena.

The central results rely on a careful application of a causality criterion due to Briggs. These results illustrate a practical means of attacking spatial instability problems, which can be applied to a broad class of systems besides the one at hand. We have found that the Charney problem with positive vertical shear is not absolutely unstable, so long as the wind at the ground is non-negative. This implies that spatial instability and forced stationary-wave problems are well posed in an open domain under typical atmospheric circumstances.

The amplifying waves appear on the downstream side of the source, have eastward (downstream) phase propagation and have wavelengths that increase monotonically with decreasing frequency, becoming infinite at zero frequency. When the surface wind is not too large, the spatial amplification rate has a single maximum near the frequency ωm = (f/N)Uz, where f is the Coriolis parameter, N is the stability frequency and Uz is the vertical shear; the rate approaches zero at zero frequency and asymptotes algebraically to zero at large frequency for any positive surface wind. Distinct Charney and Green modes do not appear until the surface wind is made very large. The amplification rate at ωm becomes infinite as surface wind approaches zero, suggesting a mechanism for the concentration of eddy activity.

We also discuss the relationship of these results to the structure of low- and high-frequency atmospheric variability.

In a previous paper we considered the nonlinear stability of a cylindrical mixing layer in an incompressible fluid at large Reynolds numbers. Nonlinear evolution results in the formation of vortex structures in the vicinity of the corotation radius rc. This paper considers the same model but in a compressible fluid. A fundamental difference implied by the presence of compressibility is the possibility of the generation of disturbances which are no longer localized near the shear layer but embrace the entire region. These are acoustic waves generated in the region of corotation resonance and emitted into the periphery. In the r > rc region lines of equal density are trailing spirals. The nonlinear evolution of such disturbances is determined by redistribution of the mean flow inside the critical layer (CL). It is shown that only two possible types of CL, viscous and unsteady, can be realized here. For both types of these regimes, evolution equations describing the dynamics of a spiral density wave amplitude are obtained and their solutions analysed. It appears that at any values (provided that they are small enough) of initial supercriticality of the flow, an explosive growth of amplitude occurs which continues as long as values comparable with background ones are reached.

A cylindrical duct absorbing energy from incident surface waves is considered. The asymptotic properties of the scattering and radiation potentials are determined, to yield the hydrodynamic quantities on which the energy absorption characteristics of the duct can be shown to depend. It is shown that it is possible to tune the resonant response of the duct to absorb the maximum theoretical energy at a given frequency. Curves are presented showing the variation of energy absorption and the amplitude of the duct response with frequency for various depths of submergence and various tuning frequencies.

Previous proposed modifications in the respective constitutive forms of the Newtonian deviatoric stress tensor and the no-slip boundary condition imposed upon viscous fluids at solid surfaces, wherein the fluid's mass velocity is replaced by its volume velocity, furnishes a complete continuum-hydrodynamic description of thermal transpiration phenomena occurring in a closed capillary tube filled with a single-component gas or liquid, the former at negligibly small Knudsen numbers. The resulting expression for the steady-state thermomolecular pressure difference $\Delta p$ existing between the two ends of the capillary, the latter maintained at different temperatures, is free of empirical parameters, such as Maxwell's thermal-slip coefficient, upon which current non-continuum theories of the phenomenon are based. The predicted $\Delta p$ (with the pressure highest at the hotter end) is shown to agree well with experimental data for gases in the near-continuum limit of vanishingly small Knudsen number. Also discussed is the experimentally observed lack of dependence of $\Delta p$ upon the physicochemical properties of the capillary walls, an observation which accords with the predictions of our theory. Our proposed volume velocity-based rationalization of the phenomenon of thermal transpiration offers a strictly continuum no-slip alternative to Maxwell's widely-accepted thermal creep explanation thereof, involving slip of the fluid's mass velocity at a non-isothermal surface. The agreement of our theoretical predictions of the thermomolecular pressure difference with experimental data, which is essentially indistinguishable in accuracy from that provided by Maxwell's thermal creep theory, provides further support for the viability of the generic volume velocity-based framework underlying our theory, the latter having recently been used to also rationalize related thermophoretic and diffusiophoretic phenomena in gases, as well as thermal diffusion in liquids.

Conventional boundary-layer theory cannot be applied when the fluid velocity outside the layer changes direction, and the leading edge of a finite body changes ends. In this paper an approximate method for examining the details of the boundary layer during a single flow reversal (occurring at t = 0) is described. It is based on the expectation that (a) long before reversal (t < -t1), there will be a quasi-steady boundary layer appropriate to flow in one direction; (b) long after reversal (t > t2) there will be a quasi-steady boundary layer appropriate to flow in the opposite direction, and (c) in between there will be a period of pure diffusion. The method is applied to a simple heat-transfer problem, in which a fluid of thermal diffusivity D flows with uniform velocity U = At over the plane y = 0; the strip 0 < x < L of the plane is maintained at temperature T1, while the restof the plane and the fluid far away have temperature T0. The approximate solution is compared with an exact solution of the boundary-layer equation, and is shown to give an accurate prediction of the heat transfer as a function of time. The boundary-layer approximation itself breaks down in regions of length O(D2/3A−1/3) near the ends of the heated strip, as usual; it also breaks down in the neighbourhood of the point x = 1/2At2, t > 0, at which the influence of the new leading edge is first felt after flow reversal. In a solution of the full equation, this region is examined in detail, and boundary-layer theory is shown to be sufficiently accurate for the calculation of heat transfer.

The spatial stability of the plane, two-dimensional jet flow to infinitesimal disturbances is investigated by taking into account the effects of transverse velocity component and the streamwise variations of the basic flow and of the disturbance amplitude, wave-number and spatial growth rate. This renders the growth rate dependent on the flow variable as well as on the transverse and streamwise co-ordinates. Growth rates for the energy density of the disturbance and the associated neutral curves are provided as a function of the streamwise co-ordinate. Variation of growth rate of the disturbance stream function and streamwise component of velocity with the transverse co-ordinate is also given for different disturbance frequencies and streamwise locations. Results are compared with those for the parallel-flow stability analysis, and also with those for an analysis that accounts for only some of the non-parallel effects. It is found that the critical Reynolds number based on the growth of energy density of the disturbance depends on the streamwise co-ordinate and lies within the range (around 20) found experimentally, while the parallel-flow theory yields a rather low value of 4·0.

The irrotational flow past two slender bodies of revolution at angles of yaw, translating in parallel paths in very close proximity, is analysed by extending the classical slender body theory. The flow far away from the two bodies is shown to be a direct problem, which is represented in terms of two line sources along their longitudinal axes, at the strengths of the variation rates of their cross-section areas. The inner flow near the two bodies is reduced to the plane flow problem of the expanding (contracting) and lateral translations of two parallel circular cylinders with different radii, which is then solved analytically using conformal mapping. Consequently, an analytical flow solution has been obtained for two arbitrary slender bodies of revolution at angles of yaw translating in close proximity. The lateral forces and yaw moments acting on the two bodies are obtained in terms of integrals along the body lengths. A comparison is made among the present model for two slender bodies in close proximity, Tuck & Newman's (1974) model for two slender bodies far apart, and VSAERO (AMI)–commercial software based on potential flow theory and the boundary element method (BEM). The attraction force of the present model agrees well with the BEM result, when the clearance, h0, is within 20% of the body length, whereas the attraction force of Tuck & Newman is much smaller than the BEM result when h0 is within 30% of the body length, but approaches the latter when h0 is about half the body length. Numerical simulations are performed for the three typical manoeuvres of two bodies: (i) a body passing a stationary body, (ii) two bodies in a meeting manoeuvre (translating in opposite directions), and (iii) two bodies in a passing manoeuvre (translating in the same direction). The analysis reveals the orders of the lateral forces and yaw moments, as well as their variation trends in terms of the manoeuvre type, velocities, sizes, angles of yaw of the two bodies, and their proximity, etc. These irrotational dynamic features are expected to provide a basic understanding of this problem and will be beneficial to further numerical and experimental studies involving additional physical effects.

The effects of randomly irregular bathymetry on the propagation of interfacial gravity waves are studied. Modelling sea water by a two-layer fluid of different densities, weakly nonlinear waves much longer than the sea depth but comparable to the horizontal scale of bathymetry are treated by Boussinesq approximation and multiple-scale analysis. For transient wave pulses, the governing equation for the pulse profile is shown to be an integro-differential equation combining KdV and Burgers terms. Quantitative and qualitative effect of disorder on the attenuation of wave amplitude, reduction of wave speed and change of wave profile are examined numerically and analytically based on the asymptotic approximation. For time-harmonic waves, mode-coupling equations are derived and examined for the competition between diffusion by random scattering, steepening by nonlinearityand frequency dispersion for a broad range of depth ratios.

An experimental study is presented on the spatial structure of the internal wave field emitted by a horizontally oscillating sphere in a uniformly stratified fluid. The limits of linear theory and the nonlinear features of the waves are considered as functions of oscillation amplitude. Fourier decomposition is applied to separate first harmonic waves at the fundamental frequency and higher harmonic waves at multiples of this frequency. For low oscillation amplitude, of 10% of the sphere radius, only the first harmonic is significant and the agreement between linear theory and experiment is excellent. As the oscillation amplitude increases up to 30% of the radius, the first harmonic becomes slightly smaller than its linear theoretical prediction and the second and third harmonics become detectable. Two distinct cases emerge depending on the ratio Ω between the oscillation frequency and the buoyancy frequency. When Ω > 0.5, the second harmonic is evanescent and localized near the sphere in the plane through its centre perpendicular to the direction of oscillation, while the third harmonic is negligible. When Ω < 0.5, the second harmonic is propagative and appears to have an amplitude that exceeds the amplitude of the first harmonic, while the third harmonic is evanescent and localized near the sphere on either side of the plane through its centre perpendicular to the direction of oscillation. Moreover, the propagative first and second harmonics have radically different horizontal radiation patterns and are of dipole and quadrupole types, respectively.

The effects of plasma heating and thermal non-equilibrium on the statistical properties of a low-Reynolds-number ($Re_{\tau } = 49$) turbulent channel flow were experimentally quantified using particle image velocimetry, two-line planar laser-induced fluorescence, coherent anti-Stokes Raman spectroscopy and emission spectroscopy. Tests were conducted at two radiofrequency plasma settings. The nitrogen, in air, was vibrationally excited to $T_{vib} \sim 1240\ \mathrm{K}$ and 1550 K for 150 W and 300 W plasma settings, respectively, while the vibrational temperature of the oxygen and the rotational/translational temperatures of all species remained near room temperature. The peak axial turbulence intensities in the shear layers were reduced by 15 and 30 % in moving across the plasma for the 150 and 300 W cases, respectively. The plasma did not alter the transverse intensities. The Reynolds shear stresses were reduced by 30 and 50 % for the 150 and 300 W cases. The corresponding Reynolds shear stress correlation coefficient was also reduced, which indicates that the large-scale structures were diminished. Finally, the plasma enhanced the turbulence decay in the zero-shear regions, where the power law decay $t^{-1/n}$ exponential factor $n$ decreased from 1.0 to 0.8.

The slow motion of a body through a stratified fluid bounded laterally by insulating walls is studied for both large and small Peclet number. The Taylor column and its associated boundary and shear layers are very different from the analogous problem in a rotating fluid. In particular, the large Peclet number problem is non-linear and exhibits mixing of statically unstable fluid layers, and hence the drag is order one; whereas the small Peclet number flow is everywhere stable, and the drag is of the order of the Peclet number.

The evolution of barotropic vortices over a topographic, axisymmetric mountain in a homogeneous rotating fluid is studied experimentally. The aim is to identify the main physical processes observed in (i) a horizontal plane of motion, perpendicular to the rotation axis of the system, and (ii) a vertical plane across the diameter of the mountain. The vortices are monopolar cyclones initially generated near or over the topography. Initially, the vortices drift towards the mountain due to the -effect associated with the topographic slope. On arriving, they turn around the obstacle in an anticyclonic direction, whilst anticyclonic vorticity is generated over the summit. The long-term vorticity distribution is dominated by the original cyclone elongated around the topographic contours and the generated anticyclone over the tip of the topography. In the vertical plane an oscillatory uphill–downhill flow is generated, which is directly related to the drift of the cyclone around the mountain. Depending on the vortex characteristics, the period of the oscillation ranges from 4 to 10 times the rotation period of the system. The horizontal and vertical flow fields are reproduced numerically by using a shallow-water formulation, which allows a detailed view of the vertical motions, hence facilitating the interpretation of the experimental results. In addition, the cyclone–anticyclone pair over the mountain is compared with analytical solutions of topographically trapped waves. A general conclusion is that vertical motions persist for several days (or rotation periods), which implies that this mechanism might be potentially important for the vertical transport over seamounts.

We model the behaviour of isolated sources of finite radius and volume flux which experience a sudden drop in buoyancy flux, generalizing the previous theory presented in Scase et al. (J. Fluid Mech., vol. 563, 2006, p. 443). In particular, we consider the problem of the source of an established plume suddenly increasing in area to provide a much wider plume source. Our calculations predict that, while our model remains applicable, the plume never fully pinches off into individual rising thermals.

We report the results of a large number of experiments, which provide an ensemble to compare to theoretical predictions. We find that provided the source conditions are weakened in such a way that the well-known entrainment assumption remains valid, the established plume is not observed to pinch off into individual thermals. Further, not only is pinch-off not observed in the ensemble of experiments, it cannot be observed in any of the individual experiments. We consider both the temporal evolution of the plume profile and a concentration of passive tracer, and show that our model predictions compare well with our experimental observations.

Direct numerical simulation is performed with a focus on the characterization of nonlinear dynamics during reflection of a plane internal wave at a sloping bottom. The effect of incoming wave amplitude is assessed by varying the incoming Froude number, $Fr$, and the effect of off-criticality is assessed by varying the slope angle in a range of near-critical values. At low $\mathit{Fr}$, the numerical results agree well with linear inviscid theory of near-critical internal wave reflection. With increasing $\mathit{Fr}$, the reflection process becomes nonlinear with the formation of higher harmonics and the initiation of fine-scale turbulence during the evolution of the reflected wave. Later in time, the wave response becomes quasi-steady with a systematic dependence of turbulence on the temporal and spatial phase. Convective instabilities are found to play a crucial role in the formation of turbulence during each cycle. The cycle evolution of flow statistics is studied in detail and qualitative differences between off-critical and critical reflection are identified. The parametric dependence of turbulence levels on Froude number and slope angle is calculated. Interestingly, at a given value of $\mathit{Fr}$, the turbulent kinetic energy (TKE) can be higher for somewhat off-critical reflection compared to exactly critical reflection. For a fixed slope angle, as the Froude number increases in the simulated cases, the fraction of the input wave energy converted into the turbulent kinetic energy and the fraction of the input wave power dissipated by turbulence also increase.

Numerical computations are employed to study the phenomenon of oscillatory forcing of flow through porous media. The Galerkin finite element method is used to solve the time-dependent Navier–Stokes equations to determine the unsteady velocity field and the mean flow rate subject to the combined action of a mean pressure gradient and an oscillatory body force. With strong forcing in the form of sinusoidal oscillations, the mean flow rate may be reduced to 40% of its unforced steady-state value. The effectiveness of the oscillatory forcing is a strong function of the dimensionless forcing level, which is inversely proportional to the square of the fluid viscosity. For a porous medium occupied by two fluids with disparate viscosities, oscillatory forcing may be used to reduce the flow rate of the less viscous fluid, with negligible effect on the more viscous fluid. The temporal waveform of the oscillatory forcing function has a significant impact on the effectiveness of this technique. A spike/plateau waveform is found to be much more efficient than a simple sinusoidal profile. With strong forcing, the spike waveform can induce a mean axial flow in the absence of a mean pressure gradient. In the presence of a mean pressure gradient, the spike waveform may be employed to reverse the direction of flow and drive a fluid against the direction of the mean pressure gradient. Owing to the viscosity dependence of the dimensionless forcing level, this mechanism may be employed as an oscillatory filter to separate two fluids of different viscosities, driving them in opposite directions in the porous medium. Possible applications of these mechanisms in enhanced oil recovery processes are discussed.

Townsend (1954) has shown that turbulent vorticity may rotate and strain a diffusion wake, thereby increasing the contribution of molecular diffusion to the total mean dispersion over short diffusion times. To test whether any such effect occurs at longer diffusion times, the lateral dispersion of both helium and of carbon dioxide in air were measured downstream from a continuous point source in the turbulence produced by a grid in a wind tunnel. The data show that, for long diffusion times, accelerated molecular diffusion is negligible, so that molecular diffusion makes only an independent contribution to the total dispersion.

A theory is constructed for rotating plane Couette flow of ferrofluid that is subject to the field generated by a periodic array of magnets. The system that is analysed contains a substantial lateral magnetic buoyancy, or magnetic gravity, allowing the configuration to be used in experimental studies of stratified shear flows in a connected geometry.

However, the spatial variation of the magnetic vector field of the magnet stack leads to magnetically generated wavy flows via the action of flow vorticity on the particle orientation in the suspension. The basic rotating Couette flow instabilities may also be affected by the same mechanism, which is sometimes referred to as rotational or ‘magnetic viscosity’. Theoretical calculations show that the directly excited wavy flows are generally small, for anticipated experimental conditions, except when they resonate with the natural linear instabilities of the Couette flow. A weakly nonlinear analysis is carried out in order to predict the behaviour in these cases. Magnetic effects stabilize the fundamental roll instability of rotating Couette flow by about 10% for a typical laboratory realization.

The flow of an incompressible elastico-viscous fluid between two parallel, infinite disks is investigated when one disk is held at rest and the other performs rotary oscillations about their common axis. It is found that the purely periodic primary motion has associated with it a secondary steady velocity distribution, as well as a secondary periodic motion with twice the frequency of the primary. The steady component of the secondary flow is discussed in detail.