The vorticity interaction mechanisms governing miscible displacements in three-dimensional heterogeneous porous media are investigated by means of detailed simulations in the regimes of viscous fingering, dispersion, and resonant amplification. The computational results for spatially periodic and random permeability distributions are analysed in detail with respect to the characteristic wavenumbers and norms associated with the individual vorticity components. This allows the identification of the mechanisms dominating specific parameter regimes. Nominally axisymmetric displacements such as the present quarter five-spot configuration are particularly interesting in this respect, since some of the characteristic length scales grow in time as the front expands radially. This leads to displacement flows that can undergo resonant amplification during certain phases, while being dominated by fingering or dispersion at other times. The computational results also provide insight into the effects of density-driven gravity override on the interactions among these length scales. While this effect is known to play a dominant role in homogeneous flows, it is suppressed to some extent in heterogeneous displacements, even for relatively small values of the heterogeneity variance. This is a result of the coupling between the viscous and permeability vorticity fields in the viscous fingering and resonant amplification regimes. In the dispersive regime, gravity override is somewhat more effective because the coupling between the viscous and permeability vorticity fields is less pronounced, so that the large-scale structures become more responsive to buoyancy effects.

The development of a three-dimensional viscous incompressible flow generated behind an infinitely long circular cylinder, impulsively started into rectilinear motion and rotationally oscillating, is studied computationally. The numerical scheme, an hybrid vortex method, is used to integrate the velocity–vorticity formulation of the Navier–Stokes equations. The Reynolds number considered is $\Rey\,{=}\,400$, which is moderate though beyond the critical values $\Rey_2\,{\simeq}\,190$ and $\Rey_2'\,{\simeq}\,260$ for which the flow becomes spontaneously three-dimensional. The numerical method is explained and its main points are developed. This scheme is then applied to solve some two-dimensional problems, both in order to validate the method and to compute a nominal two-dimensional flow, required to measure the impact of three-dimensionality. The three-dimensional flow past a steady cylinder is also compared to benchmark simulations. Once the flow has become fully three-dimensional, beyond the transient regime and saturation of instabilities, the cylinder begins a rotary oscillation around its axis. Two kinds of rotations are considered: constant amplitude and several frequencies, and constant frequency and various amplitudes. When amplitude and frequency are high enough, the whole flow comes back to its two-dimensional state. This result gives a justification for two-dimensional computations in the literature related to rotating cylinders. For the first super-harmonic frequency of the flow, a parametric study is performed in order to find the impact of the amplitude on the topology of the flow. A bifurcation is clearly identified. Finally, the mechanisms involved in the return to a two-dimensional state are explained: the interaction between transverse instabilities and von Kármán streets is quantified by means of a correlation analysis.

For a fluid layer heated and cooled differentially at its surface, we use a variational approach to place bounds on the viscous dissipation rate and a horizontal Nusselt measure based on the entropy production. With a general temperature distribution imposed at the top of the layer and a variety of thermal boundary conditions at the base of the layer, the horizontal Nusselt number is bounded by $cR_H^{1/3}$ as the horizontal Rayleigh number $R_H \,{\rightarrow}\, \infty$, for some constant $c.$ The analysis suggests that the ultimate regime for this so-called ‘horizontal convection’ is one in which the temperature field develops a boundary layer of width $O(R_H^{-1/3})$ at the surface, but has no variation in the interior. Although this scenario resonates with results of dimensional scaling theory and numerical computations, the bounds differ in the dependence of the Nusselt measure on $R_H$. Numerical solutions for steady convection appear to confirm Rossby's result that the horizontal Nusselt number scales like $R_H^{1/5}$, suggesting either that the bound is not tight or that the numerics have yet to reach the asymptotic regime.

This paper presents part 2 of a study of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331–359). In the present paper we consider the particular range of parameters where the critical values of the scaled Rayleigh number $R$ for the onset of oscillatory and stationary convection are close to each other, and we develop and analyse a nonlinear theory in such a parameter regime which takes into account those mixed stationary and oscillatory modes of convection with common wavenumber vectors. Under a near-eutectic approximation and in the limit of large far-field temperature, we first determine a number of weakly nonlinear solutions, and then the stability of these solutions is investigated. The most interesting result is the preference for a mixed solution composed of standing and stationary hexagonal modes over a relatively wide range of the parameter values and for $R$ just above its lowest subcritical value where convection is possible. Such a preferred solution has properties mostly in agreement with the experimental results due to Tait et al. (Nature, vol. 359, 1992, pp. 406–408) in the sense that the flow is downward at the cell centres, upward at the cell boundaries and there is some tendency for channel formation at the cell nodes.

In this article we introduce a new two-phase model for compressible viscous flows of saturated mixtures consisting of a carrier fluid and a granular material. The mixture is treated as a multicomponent fluid, with a set of thermodynamic variables assigned to each of its constituents. The volume fraction occupied by the granular phase and its spatial gradient are introduced as additional degrees of freedom. Then, by applying the classical theory of irreversible processes we derive algebraic expressions for the viscous stresses and heat flux vectors, the momentum and energy exchanges between the two phases, as well as a parabolic partial differential equation for the volume fraction. In our model, thermal non-equilibrium between the two phases emerges as a source term of the evolution equation for the volume fraction, in contrast with earlier models.

Active control of flow over a sphere at $\hbox{\it Re}\,{=}\,10^5$, based on the free-stream velocity $U_{\infty}$ and sphere diameter $d$, is carried out for drag reduction using a time-periodic blowing and suction from a slit on the sphere surface. The forcing frequency range considered is one to thirty times the natural vortex-shedding frequency. With the forcing, the drag on the sphere significantly decreases by nearly 50% for the forcing frequencies larger than a critical frequency (about $2.85 U_\infty$/{\it d}$). For the forcing frequencies smaller than this critical frequency, the drag is either nearly the same as, or slightly smaller than, that without forcing. The critical forcing frequency is found to be closely associated with the onset of the boundary-layer instability. It is shown from the surface-pressure measurement, surface oil-flow visualization and near-wall streamwise velocity measurement that the disturbances from the high-frequency forcing grow inside the boundary layer and delay the first separation while maintaining laminar separation, and they grow further along the separated shear layer and high momentum in the free stream is entrained toward the sphere surface, resulting in the reattachment of the flow (thus forming a separation bubble above the sphere surface) and the delay of the main separation. The reverse flow region in the wake is significantly reduced and the motion in that region also becomes weak owing to the forcing. Finally, the variation of drag by the present forcing with respect to the Reynolds number is very similar to that by dimples on the surface, but is different from that by surface roughness.

We explore the spatial growth of disturbances developing on top of a base flow given by the Hagen–Poiseuille profile, which has been modified by a small axisymmetric and axially invariant distortion. Such deviations from the ideal parabolic profile may, for instance, occur in experiments as a result of experimental uncertainties. The optimal distortion (i.e. the distortion with a prescribed norm that induces the maximum growth rate) is computed by a variational technique. Unstable modes are found to exist for very small values of the norm of the deviation at low Reynolds numbers, and the instability is governed by an inviscid mechanism. The growth of these modes and the ensuing transition to turbulence is then studied by means of direct numerical simulations. Two possible paths of transition are found, one based on the exponential amplification of axisymmetric disturbances and the subsequent formation of $\uLambda$-vortices and the other based on the growth and breakdown of streamwise streaks.

The effect of high levels of free-stream turbulence on the transition in a Blasius boundary layer is studied by means of direct numerical simulations, where a synthetic turbulent inflow is obtained as superposition of modes of the continuous spectrum of the Orr–Sommerfeld and Squire operators. In the present bypass scenario the flow in the boundary layer develops streamwise elongated regions of high and low streamwise velocity and it is suggested that the breakdown into turbulent spots is related to local instabilities of the strong shear layers associated with these streaks. Flow structures typical of the spot precursors are presented and these show important similarities with the flow structures observed in previous studies on the secondary instability and breakdown of steady symmetric streaks.

Numerical experiments are performed by varying the energy spectrum of the incoming perturbation. It is shown that the transition location moves to lower Reynolds numbers by increasing the integral length scale of the free-stream turbulence. The receptivity to free-stream turbulence is also analysed and it is found that two distinct physical mechanisms are active depending on the energy content of the external disturbance. If low-frequency modes diffuse into the boundary layer, presumably at the leading edge, the streaks are induced by streamwise vorticity through the linear lift-up effect. If, conversely, the free-stream perturbations are mainly located above the boundary layer a nonlinear process is needed to create streamwise vortices inside the shear layer. The relevance of the two mechanisms is discussed.

We first study the impact of a liquid drop of low viscosity on a super-hydrophobic surface. Denoting the drop size and speed as $D_{0}$ and $U_{0}$, we find that the maximal spreading $D_{\hbox{\scriptsize\it max}}$ scales as $D_{0}\hbox{\it We}^{1/4}$ where We is the Weber number associated with the shock ($\hbox{\it We}\,{\equiv}\,\rho U_{0}^2 D_{0}/\sigma$, where $\rho$ and $\sigma$ are the liquid density and surface tension). This law is also observed to hold on partially wettable surfaces, provided that liquids of low viscosity (such as water) are used. The law is interpreted as resulting from the effective acceleration experienced by the drop during its impact. Viscous drops are also analysed, allowing us to propose a criterion for predicting if the spreading is limited by capillarity, or by viscosity.

Some model problems are considered in order to investigate wetting failure in liquid–liquid systems. Three geometries are considered, two-dimensional two-phase shear flow, two-dimensional driven capillary rise, and both two- and three-dimensional two-phase driven cavity flow. In the first two cases, the two fluids are made equiviscous. The driven cavity flow is investigated for both equi- and non-equiviscous fluids. Three methods of analysis are used for the equiviscous case, an essentially exact Fourier series method, a quasi-parallel approximation and a phase-field model. The Fourier series validates the phase-field method in that they both give nearly identical results for onset of instability. At relatively large slip length divided by channel width ($10^{-2}$), the capillary number at which onset of wetting failure (entrainment of the receding fluid in the advancing) occurs is highly dependent on the type of flow. This dependence, however, appears to diminish rapidly as the slip length becomes smaller. The capillary number for the onset of instability is moderately dependent on gravity level.

Three-dimensional phase-field calculations are then discussed that show wetting failure through tipstreaming and splitting instabilities. Spot checks indicate that the onset points of two- and three-dimensional instabilities are very close. It is hypothesized that tipstreaming can be understood in part as a quasi-two-dimensional phenomenon.

We use kinematic simulation (KS) to study the development of a material line immersed in a three-dimensional turbulent flow. We generalize this study to a material surface, cube and sphere. We find that the fractal dimension of the surface can be explained by the same mechanism as that proposed by Villermaux & Gagne (Phys. Rev. Lett. vol. 73, 1994, p. 252) for the line. The fractal dimension of the line or the surface is a linear function of time up to times of the order of the smallest characteristic time of turbulence (or Kolmogorov timescale). For volume objects we describe the respective role of the Reynolds number and of the object's characteristic size. Using the method of characteristics with KS we compute the evolution with time of a concentration field $C({\bm x},t)$ and measure the fractal dimension of the intersection of this scalar field with a given plane. For these objects, we retrieve the result of Villermaux & Innocenti (J. Fluid Mech. vol. 393, 1999, p. 123) that the Reynolds number does not affect the development of the fractal dimension of the iso-scalar surface and extend this result to volume geometries. We also find that for volume objects the characteristic time of development of the fractal dimension is the large scales' characteristic time and not the Kolmogorov timescale.

The influence of fluid thermal sensitivity on the centrifugal flow instabilities in pressure-driven (Dean) and drag-driven (Taylor–Couette) Newtonian shear flows is investigated. Thermal effects are caused by viscous heating or an externally imposed temperature difference between the outer and inner cylinders, $\uDelta T^{ \ast }$, or a combination of both. In all cases considered, the maximum temperature difference within the gap is small enough such that the base-state velocity profile and consequently the distribution of angular momentum are practically unchanged from those in the isothermal flow. The base-state temperature gradient can be approximated as a linear superposition of $\uDelta T^{ \ast }/d$, where $d$ is the gap width, and that caused by viscous heating. Numerical linear stability analysis shows that when $\uDelta T^{ \ast }\,{=}\, 0$, viscous heating causes the critical Reynolds number, Re$_{c}$, to be greatly reduced when the Nahme number, defined as the product of the Brinkman number, Br, and the dimensionless activation energy associated with the fluid viscosity, $\varepsilon $, is $O(\alpha^{2}/\hbox{\it Pr})$ where $\alpha $ and Pr denote the dimensionless critical axial wavenumber and Prandtl number respectively. Since $\alpha^{2}$ is $O(10)$ and typical Pr values for thermal sensitive liquids could be $O(10^{4})$, appreciable flow destabilization occurs even when Na is $O(10^{ - 3})$. In the absence of viscous heating, an externally imposed temperature gradient can lead to significant reduction in Re$_{c}$ when $S \,{ \equiv}\,(\varepsilon\uDelta T^{ \ast })/T_{1}^{\ast }\,{ <}\, 0$ and $\vert S\vert $ is $O(\alpha^{2}/\hbox{\it Pr})$, where $T_{1}^{\ast }$ denotes the temperature of the inner cylinder. The numerical linear stability analysis results are explained based on a simplified model derived from the linearized governing equations by invoking the narrow-gap approximation. This model shows that the thermo-mechanical coupling, arising from the convection of the base-state temperature gradient by radial velocity perturbation, amplifies the temperature fluctuations within the flow by a factor proportional to Pe/$\alpha^{2}$ where Pe denotes the Péclet number. This results in the reduction of local viscosity. Hence, the rate of dissipation of the velocity perturbations decreases causing the centrifugal instability to occur at lower values of the Reynolds number compared to the isothermal flow. Thermo-mechanical destabilization caused by vis- cous heating for $\uDelta T^{ \ast }\,{=}\, 0$ can be quantified by a scaling law of the form $\Lambda \,{=} [1+ \hbox{\it Pr}\,c_{1}$Na/$\alpha^{2}]^{ - 1 / 2}$ where $\Lambda $ is the ratio of the critical Reynolds number of the non-isothermal flow to that of the isothermal one and $c_{1}$ is a flow-dependent constant. Similarly, in the absence of viscous heating and for $\uDelta T^{ \ast }\,{<}\, 0$, $\Lambda\,{=}\, [1\,{+}\,\hbox{\it Pr}c_{2} S/\alpha^{2}]^{ - 1 / 2}$, where $c_{2}$ is a flow-dependent constant. When $\uDelta T^{ \ast }\,{>}\, 0$ and viscous heating are present, a numerical linear stability analysis shows that $\Lambda\,{\propto}\,\hbox{\it Na}^{k}$ where $k \,{<}\,0$ and it is dependent on $\uDelta T^{ \ast }$ and the flow type. Finally, we perform a nonlinear stability analysis for the Dean flow which shows that the bifurcation is supercritical for both stationary and time-dependent modes of instability.

We present numerical simulations of stably stratified, vortically forced turbulence at a wide range of Froude numbers. Large-scale vortical forcing was chosen to represent geophysical vortices which break down at small scales where Coriolis effects are weak. The resulting vortical energy spectra are much steeper in the horizontal direction and shallower in the vertical than typical observations in the atmosphere and ocean, as noted in previous studies. We interpret these spectra in terms of the vertical decoupling which emerges in the strongly stratified limit. We show that this decoupling breaks down at a vertical scale of $U/N$, where $N$ is the Brunt–Väisälä frequency and $U$ is a characteristic horizontal velocity, confirming previous scaling arguments. The transfer of vortical energy to wave energy is most efficient at this vertical scale; vertical spectra of wave energy are correspondingly peaked at small scales, as observed in past work. The equilibrium statistical mechanics of the inviscid unforced truncated problem qualitatively predicts the nature of the forced–dissipative solutions, and confirms the lack of an inverse cascade of vortical energy.

Separation of a supersonic boundary layer near a compression ramp is considered in the limit of large Reynolds numbers and for Mach numbers $O(1)$. When the ramp angle is small, the motion may be described by the well-known triple-deck theory describing viscous–inviscid interactions. For small values of the scaled ramp angle, steady stable solutions can be obtained. However, it is shown that when a recirculation zone is present and the ramp angle is sufficiently large, the flow in the recirculation zone is susceptible to convective instabilities when perturbations are introduced there. At still larger values of the scaled ramp angle, an absolute instability is shown to occur that leads to a violent local breakdown of the boundary layer. The calculated results are shown to be consistent with a theoretical criterion that is the necessary and sufficient condition for the onset of instability.

We present a combined analytical and numerical study of the instabilities of a pair of parallel unequal-strength vortices. We extend the analyses of a vortex in an external strain field (Crow, AIAA J. vol. 8, 1970, p. 2172; Widnall et al., J. Fluid Mech. vol. 66, 1974, p. 35) to include the orbital motion of the vortex pair. For counter-rotating pairs, the classic Crow-type periodic displacement perturbations are unstable for all vortex strength ratios, with fastest-growing wavelengths several times the vortex spacing. For co-rotating pairs, the orbital motion acts to suppress instability due to displacement perturbations. Instabilities in this case arise for elliptic perturbations at wavelengths that scale with the vortex core size. We also examine the influence of a second vortex pair by extending Crouch's (J. Fluid Mech. vol. 350, 1997, p. 311) analysis. Numerical results from a spectral initial-value code with subgrid-scale modelling agree with the growth rates from the theoretical models. Computations reveal the nonlinear evolution at late times, including wrapping and ring-rejection behaviour observed in experiments. A pair of co-rotating Gaussian vortices perturbed by noise develops elliptic instabilities, leading to the formation of vorticity bridges between the two vortices. The bridging is a prelude to vortex merger. Analytic, computational and experimental results agree well at circulation Reynolds numbers of order $10^5$.

The stability of a barotropic zonal jet aligned with zonal topography on the beta-plane is investigated. The topography is assumed to be spatially periodic, with a period much smaller than the width of the jet. The problem is examined both by linear normal-mode analysis and by direct numerical simulations.

The following results are obtained. If the topography is sufficiently weak, the growth-rate of the most unstable normal mode has two maxima. The long-wave maximum occurs at wavelengths comparable to the width of the jet, and is described by Benilov's (J. Phys. Oceanogr. vol. 30, 2000, p. 733) asymptotic theory. The short-wave maximum occurs at wavelengths comparable to the scale of the topography or at a shorter one. The nonlinear evolution of the flow is, in this case, similar to that in the case of a flat bottom, i.e. the jet begins to meander and breaks up into separate vortices.

For a stronger topography, long-wave disturbances are stable, as predicted by Benilov's (2000) asymptotic theory, whereas short-wave instabilities are still present. The instabilities are strongest near the lines of maximum shear. In nonlinear simulations, the flow becomes turbulent within narrow strips along these lines, and potential vorticity there homogenizes. As the strips grow wider, they begin to interact, and the subsequent evolution is again similar to that of a jet over a flat bottom: large-scale meandering and break-up of the jet into vortices. In the presence of topography, however, the vortices are ‘filled’ with short-wave turbulence, and break-up occurs later.