We compare two-dimensional vorticity and passive scalar cascades seen as a gradient enhancement process. Our criteria are based on conditional averages of the first and second Lagrangian derivatives of vorticity and passive scalar gradients in relation to the local flow geometry. In order to interpret these criteria, transient properties are derived for random vorticity and scalar fields, showing that the second-order Lagrangian derivatives of vorticity and passive scalar gradients may behave differently. Cascades obtained in numerical simulations of decaying and forced incompressible turbulence are analysed. First-order analysis reveals that the direct cascade in elliptic domains is more efficient than previously suspected. While several first-order diagnostics collapse to a single curve for vorticity and passive scalars, second-order diagnostics consistently show that the vorticity gradient exhibits faster temporal fluctuations than the passive scalar gradient, a property which we anticipate qualitatively in the study of random fields.

The method of characteristics is used to calculate the flow ahead of an impulsively started piston moving at constant velocity. Particular attention is paid to the development of weak shock waves which are either fully or partly dispersed at very large distances from the piston. It is found that the global features of the flows may be represented in similarity form, and the graphs obtained allow extrapolation to very weak waves.

The present investigation is concerned with natural convection in a wedge-shaped domain induced by constant isothermal heating at the water surface. Complementary to the study of daytime heating by solar radiation relevant to nearshore regions of lakes and reservoirs previously reported by the same authors, this study focuses on sensible heating imposed by the atmosphere when it is warmer than the water body. A semi-analytical approach coupled with scaling analysis and numerical simulation is adopted to resolve the problem. Two flow regimes are identified depending on the comparison between the Rayleigh number and the inverse of the square of the bottom slope. For the lower Rayleigh number regime, the entire flow domain eventually becomes isothermal and stationary. For the higher Rayleigh number regime, the flow domain is composed of two distinct subregions, a conductive subregion near the shore and a convective subregion offshore. Within the conductive subregion, the maximum local flow velocity occurs when the thermal boundary layer reaches the local bottom, and the subregion eventually becomes isothermal and stationary. In the offshore convective subregion, a steady state is reached with a distinct thermal boundary layer below the surface and a steady flow velocity. The dividing position between the two subregions and the major time and velocity scales governing the flow development in both subregions are proposed by the scaling analysis and validated by corresponding numerical simulation.

The current work analyses the onset characteristics of Rayleigh–Bénard convection in confined two-dimensional two-layer systems. Owing to the interfacial interactions and the possibilities of convection onset in the individual layers, the two-layer systems typically exhibit diverse excitation modes. While the attributes of these modes range from the non-oscillatory mechanical/thermal couplings to the oscillatory standing/travelling waves, their regimes of occurrence are determined by the numerous system parameters and property ratios. In this regard, the current work aims at characterising their respective influence via methodical linear and fully nonlinear analyses, carried out on fluid systems that have been selected using the concept of balanced contrasts. Consequently, the occurrence of oscillatory modes is found to be associated with certain favourable fluid combinations and interfacial heights. The further branching of oscillatory modes into standing and travelling waves seems to additionally rely on the aspect ratio of the confined cavity. Specifically, the modulated travelling waves have been observed to occur (amidst standing wave modes) at discrete aspect ratios for which the onset of oscillatory convection happens at unequal fluid heights. This behaviour corresponds to the typical $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$:$n$ resonance where the critical wavenumbers of convection onset in the layers are dissimilar. Based on all of these observations, an attempt has been made in the present work to identify the oscillatory excitation modes with a reduced number of non-dimensional parameters.

This paper describes an investigation of the acoustic properties of a rigid plate with a periodic pattern of holes, in a compressible, ideal, inviscid fluid in the absence of mean flow. Leppington & Levine (J. Fluid Mech., vol. 61, 1973, pp. 109–127) obtained an approximation of the reflection and transmission coefficients of a plane wave incident on an infinitely thin plate with a rectangular array of perforations, assuming that a characteristic size of the perforations is negligible relative to that of the unit cell of the grating, itself assumed to be negligible relative to the wavelength. One part of the present study is of methodological interest. It establishes that it is possible to extend their approach to thick plates with a skew grating of perforations, thus confirming recent results of Bendali et al. (SIAM J. Appl. Math., vol. 73 (1), 2013, pp. 438–459), but in a much simpler way without using complex matched asymptotic expansions of the full wave or to a grating of multipoles. As is well-known, effective compliances for the plate can then be derived from the corresponding approximations of the reflection and transmission coefficients. These compliances are expressed in terms of the Rayleigh conductivity of an isolated perforation. Consequently, in one other part of the present study, the methodology recently introduced by Laurens et al. (ESAIM, Math. Model. Numer. Anal., vol. 47 (6), 2013, pp. 1691–1712) to obtain sharp bounds for the Rayleigh conductivity has been extended to include the case for which the openings of the perforations on the upper and lower sides of the plate are elliptical in shape. This not only enables the determination of these bounds and of the associated reflection and transmission coefficients for actual plates with tilted perforations but also yields single expressions covering all usual cases of perforations: straight or tilted with a circular or an elliptical cross-section.

Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

In a recent article (Nouar, Bottaro & Brancher, J. Fluid. Mech., vol. 592, 2007, pp. 177–194), a linear stability analysis of plane Poiseuille flow of shear-thinning fluids has been performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturbation. The current paper focuses on the first-principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity with the shear rate on the flow stability with respect to a finite-amplitude perturbation. A weakly nonlinear analysis, using the amplitude expansion method, is adopted as a first approach to study nonlinear effects. The bifurcation to two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning behaviour is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that (i) the nonlinearity of the viscous terms tends to reduce the viscous dissipation and to accelerate the flow, (ii) the harmonic generated by the nonlinearity is smaller and in opposite phase to that generated by the quadratic nonlinear inertial terms and (iii) with increasing shear-thinning effects, the bifurcation becomes highly subcritical. Consequently, the magnitude of the threshold amplitude of the perturbation, beyond which the flow is nonlinearly unstable, decreases. This result is confirmed by computing higher order-Landau constants.

The time-harmonic three-dimensional finite-depth floating-body problem is reformulated as a boundary integral equation. Using the elementary fundamental solution that satisfies the boundary condition on the sea bottom but not the linearized free surface condition, the integral equation extends over both the ship hull and the free surface. It is shown that this integral equation is free of irregular frequencies, that is, it has at most one solution.

The nonlinear evolution of a continuous spectrum of travelling waves resulting from the growth of unstable disturbances in circular Couette flow has been investigated. Numerical solution of the governing integro-differential equations for different initial conditions shows that the equilibrium states of Taylor-vortex, wavy-vortex or spiralvortex flows are not unique, but depend on the initial disturbance. The presence of multiple solutions at a fixed Reynolds number for a given Taylor–Couette geometry has been known since Coles’ seminal contribution in 1965. The current study indicates that the equilibrium state of flows on a stable bifurcation branch is a natural consequence of nonlinear wave resonance and is dependent on the initial conditions. The resulting wavenumber can take any value within an accessible finite band. Since such multiple solutions have also been found numerically for mixed-convection flows and experimentally for several other flows, there is evidence to support the conclusion that a non-uniqueness in the sense of Coles is a generic property for all fluid flows.

An asymptotic theory describes the stationary instability of convective flow between differentially heated vertical planes at large Prandtl numbers. The theory is concerned with the structure for A [Gt ] 1 where A is a Rayleigh number based on the horizontal temperature difference and the distance between the planes. As such it is relevant to the instability of flow in a vertical slot of aspect ratio h [Gt ] 1 where the convective regime corresponds to order-one values of a non-dimensional parameter γ which partly depends on the vertical temperature gradient generated in the slot and can be approximated by γ4 = A/8h. Instability is shown to set in at a critical value of γ that compares well with experimental observation. The lower branch of the neutral curve conforms to a boundary-layer type approximation while the upper branch has a critical-layer structure midway between the planes which becomes fully developed as the first reversal of the vertical velocity of the base flow is encountered near the centreline.

The scattering at a rectilinear step change in depth of a shallow-water vortex pair consisting of two patches of equal but opposite-signed vorticity is studied. Using the constants of motion, an explicit relationship is derived relating the angle of incidence to the refracted angle after crossing. A pair colliding with a step from deep water crosses the escarpment and subsequently propagates in shallow water refracted towards the normal to the escarpment. A pair colliding with a step from shallow water either crosses and propagates in deep water refracted away from the normal or, does not cross the step and is instead totally internally reflected by the escarpment. For large depth changes, numerical computations show that the coherence of the vortex pair is lost on encountering the escarpment.

Thermal convection in a horizontal water layer being cooled from above and absorbing solar radiation is simulated numerically at the Prandtl number Pr = 7. Three different regimes arising are investigated here. The first is characterized by intermittent convection, the second by steady-state convection, and the third is convection free. The transitions occur at different values of J0/Q, the ratio of downward solar-radiation flux just below the surface to heat flux through the interface (assumed to be constant), but at almost the same Rayleigh number. The generalized heat-conduction law is found to be valid.

Using water–salt water laboratory experiments, we investigate the mechanism of erosion by a turbulent jet impinging onto a density interface, for moderate Reynolds and Froude numbers. The Froude number is defined by $Fr_{i}=u_{i}/\sqrt{b_{i}g^{\prime }}$, with $u_{i}$ and $b_{i}$, the typical velocity and width of the jet at the interface, and $g^{\prime }$ the reduced gravitational acceleration. The Froude number $Fr_{i}$ characterizes the competition between inertial forces against the restoring buoyancy force. Contrary to previous observations reporting baroclinic destabilization of the interface, we show that the entrainment, in the range of parameters explored here, is driven by interfacial gravity waves. The waves are generated by the barotropic excitation coming from the turbulent fluctuations of the jet; they are then amplified by a mechanism of wave-induced stress; and they finally break and induce entrainment and mixing. Based on those physical observations, we introduce a scaling model for the entrainment rate, which varies continuously from $Fr_{i}^{3}$ to an $Fr_{i}$ power law from small to large Froude numbers, in agreement with the present and some of the previous laboratory data.

The two-dimensional problem of the free sloshing of an inviscid fluid in a vertically walled tank with an arbitrary bed shape is solved at both first and second order in the Stokes expansion of the velocity potential. The approach employed at both orders uses Green's functions for a flat bed in conjunction with the Cauchy–Riemann equations to derive integral equations for the tangential flux along the varying bed. The first- and second-order potentials everywhere in the fluid may then be related to these fluxes. Significant analytic progress is made with the calculation of various contributions to the integral equations at second order. The equations at first and second order are ultimately solved using a variational principle equivalent to the Galerkin method, giving efficient and accurate results. In particular, the work involved in determining the second-order solution is no more intensive than in solving the first-order problem. The first-order solution is shown to reproduce known results for specific bed shapes. The method is applied to a range of bed shapes and the second-order correction to the free-surface elevation is illustrated.

An asymptotic analysis of Rayleigh–Taylor unstable ablation fronts encountered in inertial confinement fusion is performed in the case of a strong temperature dependence of the thermal conductivity. At leading order the nonlinear analysis leads to a free boundary problem which is an extension of the classical Rayleigh–Taylor instability with unity Atwood number and an additional potential flow of negligible density expelled perpendicular to the front. The nonlinear evolution of the front is analysed in two-dimensional geometry by a boundary integral method. The shape of the front develops a curvature singularity within a finite time, as for the Birkhoff–Rott equation for the Kelvin–Helmholtz instability.