A theory and simulation code are developed to study non-steady sources as means to control sonic booms of supersonic aircraft. A key result is that the source of sonic boom pressure is not confined to the length of the aircraft but occupies an extensive segment of the flight path. An aircraft in non-steady flight functions as a synthetic aperture antenna, generating complex acoustic waves with no simple relation to instantaneous volume or lift distributions.

The theory applies linear acoustics to slender non-steady sources but requires no far-field approximation. The solution for pressure contains a term not seen in Whitham's theory for sonic booms of distant supersonic aircraft. The term describes a pressure field that decays algebraically behind the Mach cone and, in the case of steady flight, integrates to a ground load equal to the weight of the aircraft. The algebraic term is separate from those that describe the sonic boom.

Two non-steady source phenomena are evaluated: periodic velocity changes (surge), and periodic longitudinal lift redistribution (slosh). Surge can attenuate a sonic boom and covert it into prolonged weak reverberation, but accelerations needed to produce the phenomenon seem too large for practical use. Slosh may be practical and can alter sonic booms but does not, on average, result in boom attenuation. The conclusion is that active sonic boom abatement is possible in theory but maybe not practical.

In this study, the linear stability of mixed-convection flow in a vertical channel is investigated for both buoyancy-assisted and -opposed conditions. The disturbance momentum and energy equations were solved by the Galerkin method. In addition to the case with a zero heat flux perturbation boundary condition, we also examined the zero temperature perturbation boundary condition. In general, the mixed-convection flow is strongly destabilized by the heat transfer and therefore the fully developed heated flow is very unstable and very difficult to maintain in nature. For buoyancy-assisted flow, the two-dimensional disturbances dominate, while for buoyancy-opposed flow, the Rayleigh–Taylor instability prevails for zero heat flux perturbation boundary condition, and for the zero temperature perturbation on the boundaries the two-dimensional disturbances dominate except at lower Reynolds numbers where the Rayleigh–Taylor instability dominates again. The instability characteristics of buoyancy-assisted flow are found to be strongly dependent on the Prandtl number whereas the Prandtl number is a weak parameter for buoyancy-opposed flow. Also the least-stable disturbances are nearly one-dimensional for liquids and heavy oils at high Reynolds numbers in buoyancy-assisted flows.

From an energy budget analysis, we found that the thermal–buoyant instability is the dominant type for buoyancy-assisted flow. In buoyancy-opposed flow, under the zero temperature perturbation boundary condition the Rayleigh–Taylor instability dominates for low-Reynolds-number flow and then the thermal–shear instability takes over for the higher Reynolds numbers whereas the Rayleigh–Taylor instability dominates solely for the zero heat flux perturbation boundary condition. It is found that the instability characteristics for some cases of channel flow in this study are significantly different from previous results for heated annulus and pipe flows. Based on the distinctly different wave speed characteristics and disturbance amplification rates, we offer some suggestions regarding the totally different laminar–turbulent transition patterns for buoyancy-assisted and -opposed flows.

In this work, we investigate the suitability of models based solely on continuum hydrodynamics for Stokesian Dynamics simulations of sheared suspensions of non-Brownian hard spheres. The suspensions of interest consist of monolayers of uniform rigid spheres subjected to a linear shear field. Areal fractions ranged from ϕa = 0.2 to 0.6. For these suspensions, two sets of Stokesian Dynamics simulations were performed. For the first set, particle interactions were assumed to be strictly hydrodynamic in nature. These simulations are analogous to those of Brady & Bossis (1985) and Chang & Powell (1993). For the second set of simulations, particles were subjected to both hydrodynamic and short-range repulsive forces. The repulsion serves as a qualitative model of non-hydrodynamic effects important when particle separation distances are small. Results from both sets of simulations were found to be within the range of established experimental data for viscosities of suspensions. However, simulations employing the pure hydrodynamic model lead to very small separation distances between particles. These small separations give rise to particle overlaps that could not be eliminated by time-step refinement. The instantaneous number of overlaps increased with density and typically exceeded the number of particles at the highest densities considered. More critically, for very dense suspensions the simulations failed to approach a long-time asymptotic state. For simulations employing a short-range repulsive force, these problems were eliminated. The repulsion had the effect of preventing extremely small separations, thereby eliminating particle overlaps. At high concentrations, viscosities computed using the two methods are significantly different. This suggests that the dynamics of particles near contact have a significant impact on bulk properties. Furthermore, the results suggest that a critical aspect of the physics important at small particle separation distances is missing from the pure hydrodynamic model, making it unusable for computing microstructures of dense suspensions. In contrast, simulations employing a short-range repulsive force appear to produce more realistic microstructures, and can be performed even at very high densities.

An asymptotic analysis for the long-time unsteady laminar far wake of a bluff body due to a step change in its travelling velocity from U1 to U2 is presented. For U1 [ges ] 0 and U2 > 0, the laminar wake consists of a new wake of volume flux Q2 corresponding to the current velocity U2, an old wake of volume flux Q1 corresponding to the original velocity U1, and a transition zone that connects these two wakes. The transition zone acts as a sink (or a source) of volume flux (Q2 – Q1) and is moving away from the body at speed U2. Streamwise diffusion is negligible in the new and old wakes but a matched asymptotic expansion that retains the streamwise diffusion is required to determine the vorticity transport in the transition zone. A source of volume flux Q2 located near the body needs to be superposed on the unsteady wake to form the global flow field around the body. The asymptotic predictions for the unsteady wake velocity, unsteady wake vorticity, and the global flow field around the body agree well with finite difference solutions for flow over a sphere at finite Reynolds numbers. The long-time unsteady flow structures due to a sudden stop (U2 = 0) and an impulsive reverse (U1U2 < 0) of the body are analysed in detail based on the asymptotic solutions for the unsteady wakes and the finite difference solutions. The elucidation of the long-time behaviour of such unsteady flows provides a framework for understanding the long-time particle dynamics at finite Reynolds number.

We consider a model of a freely expanding gas cloud of tri-axial ellipsoidal shape, Gaussian density profile, proposed by Dyson (1968). The ellipsoids are deformable, but are further constrained here to have principal axes maintaining a fixed orientation in space: the study of the more general rotating flows is deferred to a future work. Our main result is that, when the fluid is a monatomic gas with adiabatic index γ = 5/3, the model is completely integrable by quadratures. Solutions starting from a state of rest are describable by elliptic functions; the generic solution however is a more general transcendent that cannot be reduced to elliptic type.

The complete integrability of Dyson's model may be ascribed to the fact that it possesses the Painlevé property (Ince 1956; Ablowitz & Segur 1977), meaning, essentially, that the solutions are meromorphic functions of the independent variable, admitting only pole singularities. However, the correct choice of independent variable here is not just the physical time t: rather, it is the thermasy (van Danzig 1939) u = ∫ Tdt, which is one of the potentials occurring in the Clebsch transformation.

Further investigation will be required to test Dyson's full ‘spinning gas cloud’ model for an eventual Painlevé property.

Following the study in Part 1 of cross-flow and other non-symmetric effects on vortex/wave interactions in boundary layers, the present Part 2 applies the ideas of Part 1 and related works to an incident axisymmetric flow supplemented by a small swirl or azimuthal velocity. This is with a view to possibly increasing understanding of vortex breakdown. The wave components involved are predominantly inviscid Rayleigh-like ones. The presence of the swirl leads to extra features and complications associated mainly with extra logarithmic contributions but for the dominant interactions essentially the same equations as in Part 1 are found. These dominant nonlinear interactions must be based on azimuthal wavenumbers of ± 1 in the case of the Squire jet with swirl. In contrast to Part 1, which consisted mainly of an analysis of the quasi-bounded solutions, a representative set of numerical solutions of the full integro-differential amplitude equations is presented, for realistic axial and swirl velocity profiles. The work points also to the influence of further increases in the incident swirl.

Surface wave instabilities in a two-dimensional thin draining film are studied by a direct numerical simulation of the full nonlinear system. A finite element method is used with an arbitrary Lagrangian–Eulerian formulation to handle the moving boundary problem. Both temporal and spatial stability analysis of the finite-amplitude nonlinear wave regimes are done. As the wavenumber is decreased below the linear cut-off wavenumber, supercritical sinusoidal waves occur as reported earlier from weakly nonlinear analysis and experiments. Further reduction in wavenumber makes the Fourier spectrum broad-banded resulting in solitary humps. This transition from nearly sinusoidal permanent waveforms to solitary humps is found to go through a quasi-periodic regime. The phase boundaries for this quasi-periodic regime have been determined through extensive numerical parametric search. Complex wave interaction processes such as wave merging and wave splitting are discussed. In the exhaustive numerical simulations performed in this paper, no wave-breaking tendency was observed, and it is speculated that the complex wave-interaction processes such as wave merging and wave splitting curb the tendency of the film to break.

In this paper, the oscillatory and steady streaming velocities over a permeable bed are studied both theoretically and experimentally. Three different sizes of glass beads are used to construct permeable beds in laboratory experiments: the diameters of the glass beads are 0.5 mm, 1.5 mm, and 3.0 mm, respectively. Several experiments are performed using different wave parameters. A one-component laser-doppler velocimeter (LDV) is used to measure the horizontal velocity component inside the Stokes boundary layer above the solid and permeable surfaces. It is observed that neither oscillatory nor steady velocity components vanish on the permeable surface. The ‘slip velocities’ increase with increasing permeability. Based on the laminar flow assumption and the order of magnitude of the parameters used in the experiments, a perturbation theory is developed for the oscillatory velocity and the steady wave-induced streaming in the boundary layers above and inside the permeable bed. The theory confirms many experimental observations. The theory also provides the damping rate and the phase changes caused by the permeable bed.

Generalized Taylor dispersion theory is used to study the chaotic laminar transport of a reactive solute between eccentric rotating cylinders in the presence of an inhomogeneous chemical reaction. The circumstance considered is that of laminar axial ‘Poiseuille’ flow in the annular region between the two non-concentric cylinders, accompanied by a secondary, generally chaotic, flow induced via alternate rotation of the cylinders. A Brownian tracer introduced into the flow is assumed to undergo an instantaneous, irreversible reaction on the surface of the outer cylinder. The resulting effective transversely and time-averaged reaction rate, axial solute velocity, and axial convective dispersivity are computed. When chaos is present, the effective reaction rate is increased to a value several times larger than occurs in the absence of chaotic transport. It is found that an optimum alternation frequency exists, and that this frequency decreases with increasing transverse Péclet number (Peq). It is also observed that the maximum achievable reaction rate increases with (Peq). The effect of laminar chaotic mixing on the mean axial solute/solvent velocity ratio is to drive its value towards the perfectly mixed value of 1.0, despite the removal of solute from the slower-moving axial streamlines near the outer (reactive) cylinder wall. Lastly, in the presence of transverse chaotic transport, the convective Taylor contribution to the axial solute dispersivity acquires a value up to several orders of magnitude smaller than that achievable by means of non-chaotic convection.

A Kolmogorov-type similarity theory of locally homogeneous and isotropic turbulence generated by a Smagorinsky-type large-eddy simulation (LES) at very large LES Reynolds numbers is developed and discussed. The underlying concept is that the LES equations may be considered equations of motion of specific hypothetical fully turbulent non-Newtonian fluids, called ‘LES fluids’. It is shown that the length scale ls = csδ, which scales the magnitude of the variable viscosity in a Smagorinsky-type LES, is the ‘Smagorinsky-fluid’ counterpart of Kolmogorov's dissipation length $\eta = v^{3/4}\epsilon^{-1/4}$ for a Newtonian fluid where ν is the kinematic viscosity and ε is the energy dissipation rate. While in a Newtonian fluid the viscosity is a material parameter and the length ν depends on ε, in a Smagorinsky fluid the length ls is a material parameter and the viscosity depends on ε. The Smagorinsky coefficient cs may be considered the reciprocal of a ‘microstructure Knudsen number’ of a Smagorinsky fluid. A combination of Lilly's (1967) cut-off model with two well-known spectral models for dissipation-range turbulence (Heisenberg 1948; Pao 1965) leads to models for the LES-generated Kolmogorov coefficient αLES as a function of cs. Both models predict an intrinsic overestimation of αLES for finite values of cs. For cs = 0.2 Heisenberg's and Pao's models provide αLES = 1.74 (16% overestimation) and αLES = 2.14 (43% overestimation), respectively, if limcs → (αLES) = 1.5 is ad hoc assumed. The predicted overestimation becomes negligible beyond about cs = 0.5. The requirement cs > 0.5 is equivalent to Δ < 2ls. A similar requirement, L < 2η where L is the wire length of hot-wire anemometers, has been recommended by experimentalists. The value of limcs → (αLES) for a Smagorinsky-type LES at very large LES Reynolds numbers is not predicted by the models and remains unknown. Two critical values of cs are identified. The first critical cs is Lilly's (1967) value, which indicates the cs below which finite-difference-approximation errors become important; the second critical cs is the value beyond which the Reynolds number similarity is violated.

In this paper, we analyse the development of initially small, periodic, voidage disturbances in gas-fluidized beds. The one-dimensional model was proposed by Needham & Merkin (1983), and Crighton (1991) showed that weakly nonlinear waves satisfied a perturbed Korteweg–de Vries or KdV equation. Here, we take periodic cnoidal wave solutions of the KdV equation and follow their evolution when the perturbation terms are amplifying. Initially, all such waves grow, but at a later stage a rescaling shows that shorter wavelengths are stabilized in a weakly nonlinear state. Longer wavelengths continue to develop and eventually strongly nonlinear solutions are required. Necessary conditions for periodic waves are found and matching back onto the growing cnoidal waves is possible. It is shown further that these fully nonlinear waves also reach an equilibrium state. A comparison with numerical results from Needham & Merkin (1986) and Anderson, Sundaresan & Jackson (1995) is then carried out.

The long-wave instability of unidirectional spatially periodic flows is investigated by means of asymptotic expansions. It is shown that the wavevector of the most dangerous disturbances is generally inclined to the direction of the basic stream. A new type of long-wave oscillatory instability is discovered, and a comparison with results of previous investigations is performed.

We report here an experimental study of a round vertical liquid jet that, after achieving a self-preserving state, is subjected to volumetric heating between two diametral stations. The heat injection is achieved by applying a voltage across the stations, the jet fluid having been rendered electrically conducting by the addition of acid. Using laser-induced fluorescence, digital image processing and laser-Doppler anemometry, the flow properties of the jet have been studied in detail. It is found that, with sufficient heating, the jet no longer grows linearly with height, and the decay of both centreline velocity and turbulence intensity is arrested, and may even be reversed just beyond the zone of heat addition; nevertheless the entrainment decreases, which is at variance with the hypotheses often made for modelling it. This behaviour is here attributed to the disruptive influence that, as the present experiments show, the volumetric heating has on the large-scale vortical structures in the jet, which are known to be largely responsible for the engulfment of ambient fluid that is the first step in the entrainment process. It is shown that a new non-dimensional heat release number correlates the observed data on changes in jet width. An integral model that would describe the effect of local heating is proposed, and implications for cloud development in the atmosphere are discussed.

Longitudinal mass transport in branched oscillatory flows is greater than in non-branched oscillatory flows. Here a derivation is given of a longitudinal diffusion equation which governs the long-term mass transport when there is perfect synchronism of the flow in adjacent branches. An explicit formula is obtained for the shear dispersion coefficient (effective longitudinal diffusion) when a sinusoidal flow excursion crosses a junction in geometrically self-similar flows with negligible secondary flow. A single junction crossing can be sufficient to double the shear dispersion as compared to an unbranched flow at the same frequency.

The trapping of linear water waves over two-dimensional topography is investigated by using the mild-slope approximation. Two types of bed profile are considered: a local irregularity in a horizontal bed and a shelf joining two horizontal bed sections at different depths. A number of results are derived concerning the existence of trapped modes and their multiplicity. It is found, for example, that the maximum number of modes which can exist depends only on the gross properties of the topography and not on its precise shape. A range of problems is solved numerically, to inform and illustrate the analysis, using both the mild-slope equation and the recently derived modified mild-slope equation.

The effects of entrained air in cushioning water impact on a wall are estimated by using a flow which has many similarities to the severe flip-through impacts that have been identified for water waves hitting a vertical wall. This is a filling flow which rapidly fills a confined region, such as a crack between blocks, or the space beneath a deck projecting from the coast (Peregrine & Kalliadasis 1996). The main properties of the filling flow are easily calculated, including the high-pressure peak which corresponds to the pressure peak of a flip-through. This work extends the study of filling flows to the case where the filling liquid is an air–water mixture, thus giving explicit results for the reduction of peak pressure due to the compressibility of entrained air. The behaviour of a bubbly liquid subject to substantial pressure changes is considered. Expressions are derived for an air–water mixture treated as a compressible fluid. The reduction in pressure from the incompressible case is found to be large even for relatively small air content, and depends more on the reduction in fluid volume than any other feature of the pressure–density relation. Results are presented in such a way that they may be used to estimate compressibility corrections to both the maximum and background pressures in a flip-through wave impact if corresponding incompressible pressure values are available.

The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics.

The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg–de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg–de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg–de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant.

Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.

Nonlinear stability theorems analogous to Arnol'd's second stability theorem are established for continuously stratified quasi-geostrophic flow with general nonlinear boundary conditions in a vertically and horizontally confined domain. Both the standard quasi-geostrophic model and the modified quasi-geostrophic model (incorporating effects of hydrostatic compressibility) are treated. The results establish explicit upper bounds on the disturbance energy, the disturbance potential enstrophy, and the disturbance available potential energy on the horizontal boundaries, in terms of the initial disturbance fields. Nonlinear stability in the sense of Liapunov is also established.