In this paper, linear stability analysis on a Newtonian fluid film flowing under the effect of gravity over an inclined porous medium saturated with the same fluid in isothermal condition is carried out. The focus is placed on the effect of the anisotropic and inhomogeneous variations in the permeability of the porous medium on the shear mode and surface mode instabilities. The fluid–porous system is modelled by a coupled two-dimensional Navier–Stokes/Darcy problem. The perturbation equations are solved numerically using the Chebyshev collocation method. Detailed stability characteristics as a function of the depth ratio (the ratio of the depth of the fluid layer to that of the porous layer), the anisotropic parameter (the ratio of the permeability in the direction of the basic flow to that in the direction transverse to the basic flow) and the inhomogeneity functions are presented.

The free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.

The lowest drag shape of fixed volume in Stokes flow has been known for some 40 years. It is front–back symmetric and similar to an American football with ends tangent to a cone of $60^{\circ }$. The analogous convex axisymmetric shape of fixed surface area, which may be of interest for particle design in chemistry and colloidal science, is characterised in this paper. This ‘other’ optimal shape has a surface vorticity proportional to the mean surface curvature, which is used with a local analysis of the flow near the tip to show that the front and rear ends are tangent to a cone of angle $30.8^{\circ }$. Using the boundary element method, we numerically represent the shape by expanding its tangent angle in terms decaying odd Legendre modes, and show that it has 11.3 % lower drag than a sphere of equal surface area, significantly more pronounced than for the fixed-volume optimal.

A single, simply stated approximation transforms the equations for a two-dimensional perfect fluid into a form that is closely analogous to Maxwell’s equations in classical electrodynamics. All the fluid conservation laws are retained in some form. Waves in the fluid interact only with vorticity and not with themselves. The vorticity is analogous to electric charge density, and point vortices are the analogues of point charges. The dynamics is equivalent to an action principle in which a set of fields and the locations of the point vortices are varied independently. We recover classical, incompressible, point vortex dynamics as a limiting case. Our full formulation represents the generalization of point vortex dynamics to the case of compressible flow.

A new theory for the emergence of dispersion in shallow water hydrodynamics in two horizontal space dimensions is presented. Starting with the key properties of uniform flow in open-channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux and mass density. The theory gives a new perspective – from the viewpoint of hydraulics – on how and why key shallow water models like the Korteweg–de Vries equation and Kadomtsev–Petviashvili equations arise in the theory of water waves.

Experimental observations of the flow of a suspension of solid fraction ${\it\phi}\approx 0.084$ over a circular cylindrical post in a shallow microchannel (depth smaller than the cylinder radius) find that the recirculating wake behind the obstacle at moderate Reynolds numbers is depleted or devoid of particles. Particles injected into the wake exit to regain the depleted state. By numerical simulation of the discrete particle motion, the basis for the depletion behind the cylinder is studied; rather than a shallow channel, the numerical simulations consider a periodic domain, mimicking the flow past an infinite cylinder. The Reynolds number is defined, using the average axial velocity ${\bar{U}}$, diameter of the obstacle $D$ and the kinematic viscosity of the suspension ${\it\nu}$, as $Re={\bar{U}}D/{\it\nu}$, and is studied for $Re<30$ in the simulation – conditions for which the pure fluid exhibits an extended steady closed-streamline (recirculating) wake behind the cylinder; unsteadiness is found to be suppressed by the channel walls in the experiments, allowing steady flow at a larger $Re$ than expected for an infinite cylinder (up to at least $Re=300$). The simulations use the lattice-Boltzmann method to determine the motion of the fluid and neutrally buoyant particles. The trajectory of a single particle (small relative to the cylinder) shows migration to a limit cycle inside the wake. With an increase of the number of particles in the wake alone (no particles in the free stream), particles can escape the wake due to velocity fluctuations. Simulation of the flow of suspensions of ${\it\phi}=0.04,0.06$ and 0.08 demonstrates that there is particle exchange between the wake and the free stream; the net flux of particles out of the wake leads to a particle-depleted wake, qualitatively very similar to the experimental observation.

We present new families of gravity–capillary solitary waves propagating on the surface of a two-dimensional deep fluid. These spatially localised travelling-wave solutions are non-symmetric in the wave propagation direction. Our computation reveals that these waves appear from a spontaneous symmetry-breaking bifurcation, and connect two branches of multi-packet symmetric solitary waves. The speed–energy bifurcation curve of asymmetric solitary waves features a zigzag behaviour with one or more turning points.

The physics of the roughness sublayer are studied by direct numerical simulations (DNS) of an open-channel flow with sandgrain roughness. A double-averaging (DA) approach is used to separate the spatial variations of the time-averaged quantities and the turbulent fluctuations. The spatial inhomogeneity of velocity and Reynolds stresses results in an additional production term for the turbulent kinetic energy (TKE) – the ‘wake production’; it is the excess wake kinetic energy (WKE), generated from the work of mean flow against the form drag, that is not directly dissipated into heat, but instead converted into turbulence. The wake production promotes wall-normal turbulent fluctuations and increases the pressure work, which ultimately leads to more homogeneous turbulence in the roughness sublayer, and to the increase of Reynolds shear stress and the drag on the rough wall. In the fully rough regime, roughness directly affects the generation of the wall-normal fluctuations, while in the transitionally rough regime, the region affected by roughness is separated from the region of intense generation of these fluctuations. The budget of the WKE and the connection between the wake and the turbulence suggest strong interactions between the roughness sublayer and the outer layer that are insensitive to the variation of the outer-layer conditions. Furthermore, the present results may have implications for the relationship between the roughness geometry and the flow dynamics in the region directly affected by roughness.

We investigate the settling speeds and root mean square (r.m.s.) velocities of inertial particles in isotropic turbulence with gravity using experiments with water droplets in air turbulence from 32 loudspeaker jets and direct numerical simulations (DNS). The dependence on particle inertia, gravity and the scales of both the smallest and largest turbulent eddies is investigated. We isolate the mechanisms of turbulence settling modification and find that the reduced settling speeds of large particles in experiments are due to nonlinear drag effects. We demonstrate using DNS that reduced settling speeds with linear drag (e.g. see Nielsen, J. Sedim. Petrol., vol. 63, 1993, pp. 835–838) only arise in artificial flows that, by design, eliminate preferential sweeping by the eddies. Gravity and inertia both reduce the particle r.m.s. velocities and falling particles are more responsive to vertical than to horizontal fluctuations. The model by Wang & Stock (J. Atmos. Sci., vol. 50, 1993, pp. 1897–1913) captures these trends.

We investigate the response of a laboratory aquifer submitted to artificial rainfall, with an emphasis on the early stage of a rain event. In this almost two-dimensional experiment, the infiltrating rainwater forms a groundwater reservoir which exits the aquifer through one side. The resulting outflow resembles a typical stream hydrograph: the water discharge increases rapidly during rainfall and decays slowly after the rain has stopped. The Dupuit–Boussinesq theory, based on Darcy’s law and the shallow-water approximation, quantifies these two asymptotic regimes. At the early stage of a rainfall event, the discharge increases linearly with time, at a rate proportional to the rainfall rate to the power of ${\textstyle \frac{3}{2}}$. Long after the rain has stopped, it decreases as the squared inverse of time (Boussinesq, C. R. Acad. Sci., vol. 137, 1903, pp. 5–11). We compare these predictions with our experimental data.

The effect of the Reynolds number on the behaviour of inertial particles in wall-bounded turbulent flows is investigated through large-scale direct numerical simulations (DNS) of particle-laden canonical channel flow spanning almost a decade in the friction Reynolds number, from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau } = 150$ to $\mathit{Re}_{\tau } = 1000$. Lagrangian particle tracking is used to study the motion of six different particle sets, described by a Stokes number in the range $\mathit{St} = 1\text {--}1000$. At all Reynolds numbers a strong segregation in the near-wall region is observed for particles characterized by intermediate Stokes number, in the range $\mathit{St} =10\text {--}100$. The wall-normal concentration profiles of such particles collapse in inner scaling, thus suggesting the independence of the turbophoretic drift from the large-scale outer motions. This observation is also supported by the spatial organization of the suspended phase in the inner layer, which is found to be universal with the Reynolds number. The deposition rate coefficient increases with $\mathit{Re}_{\tau }$ for a given $\mathit{St}$. Suitable inner and outer scalings are proposed to collapse the deposition curves across the available ranges of Reynolds and Stokes numbers for the different deposition regimes.

We investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

Numerical simulations of the transverse dam-break waves (TDWs) produced by the sudden removal of a gate on the side of a waterway are conducted based on the shallow-water equations to find solutions to a family of water-diversion problems. The Froude numbers in the main flow identify the members of the family. The depth and discharge profiles are analysed in terms of Ritter’s similarity variable. For subcritical main flow, the waves are comprised of a supercritical flow expansion followed by a subcritical outflow. For supercritical main flow, on the other hand, the waves are analogous to the Prandtl–Meyer expansion in gas dynamics. The diversion flow rate of two-dimensional TDWs on a flat bed is 55 % greater than the one-dimensional flow rate of Ritter in the limiting case of zero main flow, and approaches the rate of Ritter in the other limit when the value of the Froude number in the main flow approaches infinity. The diversion flow rate over a weir is generally higher than the rate over a flat bed depending on the Froude number of the main flow. These numerical simulation results are consistent with laboratory observations.

The swimming behaviour of micro-organisms can be strongly influenced by the rheology of their fluid environment. In this article, we experimentally investigate the effects of shear-thinning (ST) viscosity on the swimming behaviour of an undulatory swimmer, the nematode Caenorhabditis elegans. Tracking methods are used to measure the swimmer’s kinematic data (including propulsion speed) and velocity fields. We find that ST viscosity modifies the velocity fields produced by the swimming nematode but does not modify the nematode’s speed and beating kinematics. Velocimetry data show significant enhancement in local vorticity and circulation and an increase in fluid velocity near the nematode’s tail. These findings are compared with recent theoretical and numerical results.

For inhomogeneous turbulent flows at high Reynolds number, it is shown that the redistribution term in Reynolds-stress turbulence models can be determined from the velocity–acceleration correlation. It is further shown that the drift coefficient in the generalized Langevin model (which is used in probability density function (PDF) methods) can be determined from the Reynolds stresses and the velocity–acceleration correlation. These observations are valuable, since the second moments of velocity and acceleration can be measured in experiments, in direct numerical simulations and in well-resolved large-eddy simulations (LES), and hence these turbulence-model quantities can be determined. The redistribution is closely related to the pressure–rate-of-strain, and the unknown in the PDF equation is closely related to the conditional mean pressure gradient (conditional on velocity). In contrast to the velocity–acceleration moments, these pressure statistics are much more difficult to obtain, and our knowledge of them is quite limited. It is also shown that the generalized Langevin model can be re-expressed to provide a direct connection between the drift term and the fluid acceleration. All of these results are first obtained using the constant-property Navier–Stokes equations, but it is then shown that the results are simply extended to variable-density flows.

We examine the effect of large bulk viscosity on the classical problem of two-dimensional shock–boundary-layer interaction. The flow is taken to be steady and supersonic over a flat adiabatic plate. The boundary layer is taken to be laminar and the fluid is modelled as a perfect gas with a bulk viscosity that is large compared with its shear viscosity. The flow details are computed using a fifth-order weighted essentially non-oscillatory finite difference scheme and a third-order Runge–Kutta scheme for the spatial and temporal discretizations. The primary result of interest is the suppression of separation when the ratio of bulk to shear viscosity is sufficiently large.

The interaction between vortical isotropic turbulence (IT) and a normal shock wave is studied using direct numerical simulation (DNS) and linear interaction analysis (LIA). In previous studies, agreement between the simulation results and the LIA predictions has been limited and, thus, the significance of LIA has been underestimated. In this paper, we present high-resolution simulations which accurately solve all flow scales (including the shock-wave structure) and extensively cover the parameter space (the shock Mach number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M_s$, ranges from 1.1 to 2.2 and the Taylor Reynolds number, ${\mathit{Re}}_{\lambda }$, ranges from 10 to 45). The results show, for the first time, that the turbulence quantities from DNS converge to the LIA solutions as the turbulent Mach number, $M_t$, becomes small, even at low upstream Reynolds numbers. The classical LIA formulae are extended to compute the complete post-shock flow fields using an IT database. The solutions, consistent with the DNS results, show that the shock wave significantly changes the topology of the turbulent structures, with a symmetrization of the third invariant of the velocity gradient tensor and ($M_s$-mediated) of the probability density function (PDF) of the longitudinal velocity derivatives, and an $M_s$-dependent increase in the correlation between strain and rotation.

It has recently been demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic realm. The walker, consisting of a droplet plus its guiding wavefield, is a spatially extended object. We here examine the dependence of the walker mass and momentum on its velocity. Doing so indicates that, when the walker’s time scale of acceleration is long relative to the wave decay time, its dynamics may be described in terms of the mechanics of a particle with a speed-dependent mass and a nonlinear drag force that drives it towards a fixed speed. Drawing an analogy with relativistic mechanics, we define a hydrodynamic boost factor for the walkers. This perspective provides a new rationale for the anomalous orbital radii reported in recent studies.

The spectral eddy viscosity (SEV) concept is a handy tool for the derivation of large-eddy simulation (LES) turbulence models and for the evaluation of their performance in predicting the spectral energy transfer. We compute this quantity by filtering and truncating fully resolved turbulence data from direct numerical simulations (DNS) of neutrally and stably stratified homogeneous turbulence. The results qualitatively confirm the plateau–cusp shape, which is often assumed to be universal, but show a strong dependence on the test filter size. Increasing stable stratification not only breaks the isotropy of the SEV but also modifies its basic shape, which poses a great challenge for implicit and explicit LES methods. We find indications that for stably stratified turbulence it is necessary to use different subgrid-scale (SGS) models for the horizontal and vertical velocity components. Our data disprove models that assume a constant positive effective turbulent Prandtl number.

In Haller & Beron-Vera (J. Fluid Mech., vol. 731, 2013, R4) we developed a variational principle for the detection of coherent Lagrangian vortex boundaries. The solutions of this variational principle turn out to be closed null geodesics of the Lorentzian metric induced by a generalized Green–Lagrange strain tensor family. This metric interpretation implies a mathematical analogy between coherent Lagrangian vortex boundaries and photon spheres in general relativity. Here, we give an improved discussion of this analogy.