A quantitative shallow water simulation technique for aerodynamic noise study is investigated. The technique is employed for the case of a free jet issuing from a nozzle with variable wall roughness.

The experimental results obtained are in remarkably good agreement with the theory. It is concluded that there is great potential for further work in even less well understood areas of aerodynamic noise research.

The propagation of internal Alfvén-ácoustic-gravity waves in a compressible, stratified, inviscid, perfectly conducting, isothermal atmosphere in the presence of a horizontal magnetic field is investigated by considering both the horizontal and the vertical component of the group velocity. The vertical component of the group velocity is important because it determines the speed at which energy travels upwards and becomes available for heating the upper regions. The regions of propagation and no propagation of waves are delineated for different magnetic Mach numbers, in a refractive-index domain. The horizontal and vertical group velocities are compared with the corresponding phase velocity of the wave motion. It is found that the horizontal group velocity of the internal waves is always less than the horizontal phase velocity for small magnetic fields and vice versa for large magnetic fields, whereas the vertical group velocity is always opposite in direction to the vertical phase velocity for small magnetic fields and vice versa for large magnetic fields. We have also drawn the reflexion condition in a wave-number-frequency domain for different Mach numbers.

When water is slightly stratified, internal gravity waves are considerably shorter than surface waves of comparable frequency. Here, this fact is exploited in demonstrating that an internal wave is unstable when it forms part of a resonant triad with a surface wave and another internal wave whose wave number is approximately equal to that of the original internal wave. It is suggested that in a system where there are two classes of waves of comparable frequencies but greatly differing wavelengths the short waves may be expected to generate long waves by this mechanism.

The assumptions of the shock-expansion method are re-examined. Although the shock-expansion method and Whitham's rule are used to treat two different classes of problems, certain similarities between these two methods are noted. It is suggested that a single procedure leads to solutions for the entire flow field for both classes of problems.

One-dimensional planar premixed flames propagating in a uniform flow are susceptible to hydrodynamic instabilities known (generically) as Darrieus–Landau instabilities. Here, we extend that hydrodynamic linear stability analysis to include a lateral shear. This generalization is a situation of interest for laminar and turbulent flames when they travel into a region of shear (such as a jet or shear layer). It is shown that the problem can be formulated and solved analytically and a dispersion relation can be determined. The solution depends on a shear parameter in addition to the wavenumber, thermal expansion ratio, and Markstein lengths. The study of the dispersion relation shows that perturbations have two types of behaviour as wavenumber increases. First, for small shear, we recover the Darrieus–Landau results except for a region at small wavenumbers, large wavelengths, that is stable. Initially, increasing shear has a stabilizing effect. But, for sufficiently high shear, the flame becomes unstable again and its most unstable wavelength can be much smaller than the Markstein length of the zero-shear flame. Finally, the stabilizing effect of low shear can make flames with negative Markstein numbers stable within a band of wavenumbers.

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies of barotropic $\unicode[STIX]{x1D6FD}$-plane turbulence have shown that coherent flows emerge from a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases. The emergence of large-scale flows has been attributed to a new type of collective, symmetry-breaking instability of the statistical state dynamics of the turbulent flow. In this work, we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid without any imposed mean north–south thermal gradient and with turbulence supported by an external random stirring. We use a second-order closure of the statistical state dynamics, that is termed S3T, with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising zonal jets and/or large-scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. Our theory predicts that there is a large bias towards the emergence of barotropic flows. If the scale of excitation is of the order of (or larger than) the deformation radius, the large-scale structures are barotropic. Mixed barotropic–baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and insignificant for higher energy input rates. The predictions of the S3T theory are compared with nonlinear simulations. The theory is found to accurately predict both the critical transition parameters and the scales of the emergent structures but underestimates their amplitude.

In steady axisymmetric flows in a closed swirl chamber one can distinguish between the swirl flow proper, with components normal to the meridian plane, and a secondary flow whose components lie in the meridian plane. One can trace the motion of a particle within the meridian plane. The closed path so obtained will be called a streamline, to be parametrized by a stream function ϕ. One can distinguish between the flow in a boundary layer, where the velocity gradient is large, and a core flow, where the velocity and temperature gradients are relatively small. The present article is concerned with only the core flow. In high Reynolds number flows in which the streamlines are not closed there are three quantities which are constant along the streamlines: the total enthalpy (the right-hand side of Bernoulli's equation), the entropy and the moment of momentum of the particles with respect to the axis of symmetry. These are determined by conditions in the entrance cross-section. In flows with closed streamlines these quantities are ultimately determined by the cumulative effects of viscosity and heat conductivity. Conditions expressing cumulative effects enter the analysis as integrabiliby conditions necessary for the existence of a second approximation in a development of the flow field with respect to the reciprocal of the Reynolds number. They are re-expressed as the balance equations for energy, entropy and angular momentum which are to be satisfied on all surfaces ϕ = constant. One thus obtains an algorithm which leads from expressions for the total enthalpy, the entropy and the angular momentum as functions of ϕ to the residuals in the balance equations, also computed as functions of ϕ. The functions with which this algorithm starts must be chosen such that the residuals in the balance equations become zero. The secondary flow can be arbitrarily slow only if the Prandtl number is ½. At the centre of the secondary motion the balance equations are linearly dependent. This fact introduces an additional free parameter which allows one to compute secondary flows with different speeds. The linearized algorithm has the character of a Fredholm integral equation. This suggests an iterative solution similar to a Neumann series. The particles experience periodic changes of state which can be discussed as thermodynamic cycles. Such an analysis shows that the heat inputs occur on average at lower temperatures than the heat outputs. Besides the work that maintains the swirl motion, which is provided by shear-force components normal to the meridian plane, one therefore needs additional work, provided by the shear-force components within the meridian plane, to maintain a secondary motion. Responsible for this state of affairs is the fact that particles which do not quite maintain adiabatic temperatures move within a field with a large pressure gradient caused by the swirl component. This makes the flow sensitive to disturbances in the energy balance.

The gravitational settling of a homogeneous suspension of Brownian particles on an inclined plate is considered. The hindered settling towards the wall and the viscous, buoyancy-driven bulk motion of the sediment are considered assuming steady conditions and accounting for the effects of Brownian diffusion, shear-induced diffusion and migration of particles due to a gradient in shear stress. Generally, the results show the development of a sediment boundary layer where the settling towards the wall is balanced by Brownian diffusion at the beginning of the plate and by shear-induced diffusion further downstream. Compared to previous results in the literature, the present theory allows steady-state solutions for extended values of the plate inclination and particle volume fraction above the sediment; upon reconsidering the case with non-Brownian particles, a new similarity solution, with a stable shock in particle density, is developed.

The frictional torque on a dumbbell rotating with time-dependent angular velocity is calculated from the hydrodynamic interaction between the two ends of the dumbbell. This leads to the correlation function for the random torque in rotatory Brownian motion. Although the motion of each dumbbell end has the characteristics of a translational motion, the correlation function at large times decays like t−5/2, as in the case of a solid sphere. The correlation function may be calculated for the limiting case of very small angular displacements. The results for displacements of arbitrary magnitude are the same provided that terms quadratic in the angular velocity are negligible.

Long-term marginal stability of a new family of isolated oceanic vortices is analysed. Sign reversal of the radial gradient of the potential vorticity anomaly, as implied by the isolation requirement, leads to vortex unsteadiness but does not break the coherence of the vortex, which remains marginally stable even for high absolute Rossby numbers $Ro\simeq 0.8$. The marginally stable vortices are characterized by a zero amount of potential vorticity anomaly on every isopycnal. The marginally stable final state is an unsteady vortex whose inner one-signed potential vorticity anomaly experiences revolution, rotation, precession and nutation.

A flowing granular material can behave like a collection of individual interacting grains or like a continuum fluid, depending in large part on the energy imparted to the grains. As yet, however, we have no general understanding of how or under what conditions the fluid limit is reached. Marston, Li & Thoroddsen (J. Fluid Mech., this issue, vol. 704, 2012, pp. 5–36) use high-speed imaging to investigate the ejection of grains from a granular bed due to the impact of a spherical projectile. Their high temporal resolution allows them to study the very fast processes that take place immediately following the impact. They demonstrate that for very fine grains and high impact energies, the dynamics of the ejecta is both qualitatively and quantitatively similar to what is seen in analogous experiments with fluid targets.

The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented.

The transient adjustment process of a compressible fluid in a rapidly rotating pipe is studied. The system Ekman number E is small, and the assumptions of small Mach number and the heavy-gas limit (γ = 1.0) are invoked. Fluid motion is generated by imposing a step-change perturbation in the temperature at the pipe wall Tw. Comprehensive analytical solutions are obtained by deploying the matched asymptotic technique with proper timescales O(E−1/2) and O(E−1). These analytical solutions are shown to be consistent with corresponding full numerical solutions. The detailed profiles of major variables are delineated, and evolution of velocity and temperature fields is portrayed. At moderate times, the entire flow field can be divided into two regions. In the inner inviscid region, thermo-acoustic compression takes place, and the process is isothermal–isentropic with the angular momentum being conserved. In the outer viscous region, diffusion of angular momentum occurs. The principal dynamic mechanisms are discussed, and physical rationalizations are offered. The essential differences between the responses of a compressible and an incompressible fluid are highlighted.

The issue of stability of the analytically obtained flow is addressed by undertaking a formal stability analysis. It is illustrated that, within the range of parameters of present concern, the flow is stable when ε ∼ O(E).

The paper is devoted to an investigation of convective turbulence. A simplified approach is used for this purpose. It considers an isolated turbulent pulsation as the eigensolution to the corresponding equations of thermohydrodynamics. Turbulence is generated by nonlinear interaction of pulsations: not all interactions, but only the most probable of them are investigated. It is assumed that during convection these are interactions of cells located along the gravity vector, i.e. lying in a vertical line, and lateral interaction of the cells is ignored. This assumption allows one to consider the process of the evolution and interaction of cells as axially symmetric. It is also assumed that the vertical scales of convective cells are larger than their horizontal scales. Therefore, the Boussinesq equations simplified in accordance with the theory of vertical boundary layers can be used. The fact that buoyancy forces, in addition to diffusion, influence the increase of the vertical scales, serves as a basis for this assumption. These assumptions make it possible to obtain the analytical and numerical–analytical solutions, which qualitatively describe the evolution and interaction of convective cells of two essentially different scales: (i) centimetre-scale convective pulsations and (ii) thermals and convective clouds, and to reduce the problem to the solution of nonlinear equations (equations of the Burgers type). Two opposite tendencies are revealed, manifested in the interaction of convective cells. First, there is coagulation of cells and fine nonlinear effects associated with it, which are known from observations and supported by the theory. Secondly, there is destruction of a strong rising cell through its collision with a weak descending ‘cold’ cell. It is assumed that the destruction of cells corresponds to the absence of solutions, when some parameters reach their critical values. A numerical solution to a more accurate problem without simplifications of the vertical boundary layer serves as a basis for this hypothesis. It shows that at critical values of the parameters the process of ‘wave turnover’ begins. It is accompanied by entrainment of the motions of the cold surrounding air into a system of convection and fast dissipation of a cell. In the simplified model, this dissipation is considered to be instantaneous and is called destruction. When the cells are sufficiently strong vertically, weak random fluctuations in the fields of meteorological elements cause their destruction. These results make it possible to propose a hypothesis which relates the degree of instability of cells with the probability of their existence, and to construct functions of cell distributions.

Three-dimensional shock wave reflection comprises flow physics that is significantly different from the well-documented two-dimensional cases in a number of aspects. The most important differentiating factor is the sweep of the shock system. In particular, this work examines the nature of flow fields in which there is a transition of shock reflection configuration in three-dimensional space. The flow fields investigated have been made to exist in the absence of edge effects influencing the shock interaction and transition, as found previously to exist in conventional double-wedge studies. In general, the shock configurations are those with central regular and peripheral Mach reflection portions. It is shown that the sweep angle of the portions on either side of the transition point is subject to a cusp, as per an analytical model that is developed. This is confirmed with the use of numerical models with additional evidence provided by experimental results using oblique shadow photography. Further application of the principles of three-dimensional shock analysis and those pertaining to the sweep cusp model yield important insights regarding the overall shock geometry and that at transition.

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r−5/3, while the y component (in the direction of vorticity) decays like r−4/3. The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as yw ~ zw ~ |x|1/3. The maximum x and z components of the velocity are located in the wake cores; they scale like |x|−2/3 and |x|−1 respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.