We measured the correlation spectrum of pressure fluctuations in a driving mixing layer with a Taylor-scale Reynolds number up to by a newly developed pressure probe with spatial and temporal resolutions that are sufficient to analyse inertial-subrange statistics. The influence of the mean velocity gradient tensor in the mixing layer, which is almost constant near its centreline, is studied using an idea similar to that underlying the linear response theory developed in statistical mechanics for systems at or near thermal equilibrium. If we write the spectrum as , where is the isotropic Kolmogorov spectrum in the absence of mean shear, then for small the deviation due to the shear is approximately linear and is determined by a few non-dimensional universal constants in addition to , and the mean energy dissipation rate. We also measured the pressure–velocity and velocity–velocity correlation spectra. Deviations from isotropy due to shear are shown to be approximately proportional to at large .

The paper presents a comprehensive analysis of gas motion created by a strong explosion in an atmosphere whose density is an exponential function of altitude. For the near field (i.e. short times after initiation), an exact analytical solution of the equations of motion is obtained by means of a perturbation technique. For the far field (i.e. long times after initiation), a similarity solution associated with a logarithmic front trajectory is derived. The two are shown to be well matched with each other. Finally, a fully-algebraic approximate solution is given that qualitatively reproduces all the salient features of the exact and asymptotic solutions, while quantitatively it is in fair agreement with their results.

This paper deals with the definition and description of optimal streaky (S) perturbations in a Blasius boundary layer. First, the asymptotic behaviours of S-perturbations near the free stream and the leading edge are studied to conclude that the former is slaved to the solution inside the boundary layer. Based on these results, a quite precise numerical scheme is constructed that allows concluding that S-perturbations produced inside the boundary layer, near the leading edge, can be defined in terms of just one streamwise-evolving solution of the linearized equations, associated with the first eigenmode of an eigenvalue problem first formulated by Luchini (J. Fluid Mech., vol. 327, 1996, p. 101). Such solution may be seen as an internal unstable streaky mode of the boundary layer, similar to eigenmodes of linearized stability problems. The remaining modes decay streamwise. Thus, the definition of streaks in terms of an optimization problem that is used nowadays is not necessary.

A higher order theory for two-dimensional turbulent boundary-layer flow of a compressible fluid past a plane wall is formulated, for moderately large values of the Reynolds number, by the method of matched asymptotic expansions. The parameters (γ − 1) M2∞ and the molecular Prandtl number are assumed to be of order unity. The analysis deals with the set of Reynolds equations of mean motion (which are underdetermined without an additional set of closure hypotheses) and assumes that the non-dimensional fluctuations in velocity, temperature and density are of order U*, (friction velocity divided by free-stream velocity a t some designation point), while fluctuations in pressure are of order U2*.The first-order results of the present study lead to asymptotic laws for velocity and temperature distributions which correspond to the law of the wall, logarithmic law and defect law, and also to skin friction and heat-transfer laws. It turns out that the first-order defect law depends upon the gradient of entropy and stagnation enthalpy and the law of the wall is independent of viscous dissipation. The second-order terms of the present work (accounting for mean convection due to turbulent mass flux, viscous dissipation in the inner flow and displacement effects in the outer flow) describe the necessary corrections to first-order terms due to low Reynolds number effects. In the overlap region the second-order results, for the law of the wall and the defect law, show bilogarithmic terms along with logarithmic terms.

In this study, we address the modification of sheared turbulence by dispersed inertial particles. The preferential sampling of the straining regions of the flow by inertial particles in turbulence leads to an inhomogeneous distribution of particles. The strong gravitational loading exerted by the highly concentrated regions results in anisotropic alteration of turbulence at small scales in the direction of gravity. These effects are investigated in a rapid distortion theory (RDT) extended for two-way coupled particle-laden flows. To make the analysis tractable, we assume that particles have small but non-zero inertia. In the classical results for single-phase flows, the RDT assumption of fast shearing compared to the turbulence time scales leads to the distortion and shear-induced production of turbulence. In particle-laden turbulence, the coupling between the two phases under rapid shearing induces number density fluctuations that convert gravitational potential energy to turbulent kinetic energy and modulate the turbulence spectrum in a manner that increases with mass loading. Turbulence statistics obtained from RDT are compared with Euler–Lagrange simulations of homogeneously sheared particle-laden turbulence.

The properties of artificially initiated turbulent spots over a heated plate were investigated in a water channel. The instantaneous velocity field and surface Stanton number were simultaneously established using a technique that combines particle image velocimetry and thermochromic liquid crystal thermography. Several characteristics of a spot are found to be similar to those of a turbulent boundary layer. The spacing of the surface heat transfer streak patterns within the middle or ‘body’ of a turbulent spot are comparable to the low-speed streak spacing within a turbulent boundary layer. Additionally, the surface shear stress in the same region of a spot is also found to be comparable to a turbulent boundary layer. However, despite these similarities, the heat transfer within the spot body is found to be markedly less than the heat transfer for a turbulent boundary layer. In fact, the highest surface heat transfer occurs at the trailing or calmed region of a turbulent spot, regardless of maturity. Using a modified set of similarity coordinates, instantaneous two-dimensional streamlines suggest that turbulent spots entrain and subsequently recirculate warm surface fluid, thereby reducing the effective heat transfer within the majority of the spot. It is proposed that energetic vortices next to the wall, near the trailing edge of the spot body, are able to generate the highest surface heat transfer because they have the nearest access to cooler free-stream fluid.

When a layer of liquid is heated from below at a rate which exceeds a certain critical value, a two- or three-dimensional motion is generated. This motion arises from the action of buoyancy and surface tension forces, the latter being due to variations in the temperature of the liquid surface.

The two-dimensional form of the flow has been studied by a numerical method. It consists of a series of rolls, rotating alternately clockwise and anticlockwise, which are shown to be symmetrical about the dividing streamlines. As well as a detailed description of the motion and temperature of the liquid, and of the effects on these characteristics of variations in the Rayleigh, Marangoni, Prandtl and Biot numbers, a study has been made of the conditions under which the motion first starts, the wavelength of the rolls and the rate of heat transfer across the liquid layer.

This study concentrates on a relatively simple model of a mushy layer originally proposed by Amberg & Homsy (1993) and later studied in further detail by Anderson & Worster (1995). We extend this model to the case in which the system is in a state of uniform rotation about the vertical. Of particular interest is to determine how the rotation of the system controls the bifurcating convection with both the oblique-roll planform and the planform of hexagonal symmetry. We find that two-dimensional oblique rolls can be either subcritically or supercritically bifurcating, depending on a pair of parameters (K1/CS, [Tscr ]), where K1 measures how the permeability linearly varies with the local solid fraction, CS relates the compositional difference between the liquid and solid phases to the variation of composition throughout the mushy layer, and the Taylor number [Tscr ] gives a measure of the local Coriolis acceleration relative to the viscous dissipation in a porous medium. The three-dimensional convection with hexagonal symmetry is found to be transcritical. Furthermore, distorted hexagons with upflow at the centres can be either subcritical or supercritical, depending on the value of the Taylor number [Tscr ].

We analyse the disturbed flow in the subsonic laminar boundary layer, disturbances being generated by local heating elements, which are placed on the surface. It is exhibited that these flows are described in terms of free interaction theory for specific sizes of thermal sources. We construct the numerical solution for the case of a flat subsonic stream in the viscous asymptotic layer, in which the flow is described by nonlinear equations for vorticity, temperature and an interaction condition which provides the influence of perturbations to the pressure in the main order. The obtained solutions are compared with those for corresponding linear problems with small perturbations. It is demonstrated that strong temperature perturbations in some situations allow us to obtain the flow close to the separated flow.

Linear stability analysis is used to investigate instability mechanisms for a horizontally oriented hyperbolic tangent mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important parameters governing inviscid dynamics are maximum shear , buoyancy frequency , angular velocity of rotation and characteristic shear thickness . Growth rates associated with the most unstable modes are explored as a function of stratification strength and rotation strength . In the case of strong stratification, growth rates exhibit self-similarity of the form . In the case of rapid rotation we also observe self-similar scaling of growth rates with respect to the vertical wavenumber and rotation rate. The unstratified cases show dependence while the strongly stratified cases show dependence where represents the difference between the angular velocity of rotation and least stable anticyclonic angular velocity, . Stratification was found to stabilize the inertial instability for weak anticyclonic rotation rates. Near the zero absolute vorticity state, stratification and rotation couple in a destabilizing manner increasing the range of unstable vertical wavenumbers associated with barotropic instability. In the case of rapid rotation, stratification prevents the stabilization of low , high modes that occurs in a homogeneous fluid. The structure of certain unstable eigenmodes and the coupling between horizontal vorticity and density fluctuations are explored to explain how buoyancy stabilizes or destabilizes inertial and barotropic modes.

We present an analytical theory of scattering of tide-generated internal gravity waves in a continuously stratified ocean with a randomly rough seabed. Based on a linearized approximation, the idealized case of constant mean sea depth and Brunt–Väisälä frequency is considered. The depth fluctuation is assumed to be a stationary random function of space, characterized by small amplitude and a correlation length comparable to the typical wavelength. For both one- and two-dimensional topographies the effects of scattering on the wave phase over long distances are derived explicitly by the method of multiple scales. For one-dimensional topography, numerical results are compared with Bühler & Holmes-Cerfon (J. Fluid Mech., vol. 678, 2011, pp. 271–293), computed by the method of characteristics. For two-dimensional topography, new results are presented for both statistically isotropic and anisotropic cases.

The problem of near-trapping of linear water waves in the time domain for rigid bodies or variations in bathymetry is considered. The singularity expansion method (SEM) is used to give an approximation of the solution as a projection onto a basis of modes. This requires a modification of the method so that the modes, which grow towards infinity, can be correctly normalized. A time-dependent solution, which allows for possible trapped modes, is introduced through the generalized eigenfunction method. The expression for the trapped mode and the expression for the near-trapped mode given by the SEM are shown to be closely connected. A numerical method that allows the SEM to be implemented is also presented. This method combines the boundary element method with an eigenfunction expansion, which allows the solution to be extended analytically to complex frequencies. The technique is illustrated by numerical simulations for geometries that support near-trapping.

The roll-up of an initially spherical vortex sheet into a vortex ring is computed using the vortex blob method. The ring sheds about 30% of its circulation into a tail which, in turn, rolls up into a ring that sheds circulation. The process repeats itself at smaller and smaller scales in a self-similar manner. The relation between the vortex shedding and the energy of the vortices is investigated. In contrast, an initially cylindrical vortex sheet rolls up into a vortex pair that sheds essentially no circulation.

For established axisymmetric turbulent miscible Boussinesq fountains in quiescent uniform environments, expressions are developed for the fluxes of volume, momentum and buoyancy at the outflow from the fountain: the outflow referring to the counterflow at the horizontal plane of the source. The fluxes are expressed in terms of the fountain source conditions and two dimensionless functions of the source Froude number, ${\rm Fr}_{0}$: a radial function (relating a horizontal scale of the outflow to the source radius) and a volume flux function (relating the outflow and source volume fluxes). The forms taken by these two functions at low ${\rm Fr}_{0}$ and high ${\rm Fr}_{0}$ are deduced, thereby providing the outflow fluxes and outflow Froude number solely in terms of the source conditions. For high ${\rm Fr}_{0}$, the outflow Froude number, ${\rm Fr}_{out}$, is shown to be invariant, indicating (by analogy with plumes for which the ‘far-field’ Froude number is invariant with source Froude number) that the outflow may be regarded as ‘far-field’ since the fluxes within the fountain have adjusted to attain a balance which is independent of the source conditions. Based on ${\rm Fr}_{out}$, the fluxes in the plume that forms beyond the fountain outflow are deduced. Finally, from the results of previously published studies, we show that the scalings deduced for fountains are valid for $0.0025 \lesssim {\rm Fr}_{0} \lesssim 1.0 $ for low ${\rm Fr}_{0}$ and $ {\rm Fr}_{0} \gtrsim 3.0 $ for high ${\rm Fr}_{0}$.

Double-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers – thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the ‘point-of-failure’ scale ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H_{pof}$) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that $H_{pof} $ can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport – the flux-gradient-aberrancy law – is proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier–Stokes equations. Compressibility is quantified by the number of density scale heights $N_{\unicode[STIX]{x1D70C}}$, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index $n=1.499<n_{a}=1.5$ is considered, where $n_{a}$ is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number $Pr$ and to $N_{\unicode[STIX]{x1D70C}}$. For large $Pr$ and small $N_{\unicode[STIX]{x1D70C}}$, the quasi-geostrophic columnar mode loses stability first; while for relatively small $Pr$ a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small $Pr$ and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers $Ra$ for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical $Ra$ for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative $Ra$. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-$Ra$ and large-$N_{\unicode[STIX]{x1D70C}}$ regime.