Using index matching and particle tracking, we measure the three-dimensional velocity field in an isotropic porous medium composed of randomly packed solid spheres. This high-resolution experimental dataset provides new insights into the dynamics of dispersion and stretching in porous media. Dynamic-range velocity measurements indicate that the distribution of the velocity magnitude, $U$, is flat at low velocity (probability density function $(U)\propto U^{0}$). While such a distribution should lead to a persistent anomalous dispersion process for advected non-diffusive point particles, we show that the dispersion of non-diffusive tracers nonetheless becomes Fickian beyond a time set by the smallest effective velocity of the tracers. We derive expressions for the onset time of the Fickian regime and the longitudinal and transverse dispersion coefficients as a function of the velocity field properties. The experimental velocity field is also used to study, by numerical advection, the stretching histories of fluid material lines. The mean and the variance of the line elongations are found to grow exponentially in time and the distribution of elongation is log-normal. These results confirm the chaotic nature of advection within three-dimensional porous media. By providing the laws of dispersion and stretching, the present study opens the way to a complete description of mixing in porous media.

We consider the instabilities of flows through a submerged canopy and show how the full governing equations of the fluid–structure interactions can be reduced to a compact framework that captures many key features of vegetative flow. First, by modelling the canopy as a collection of homogeneous elastic beams, we predict the steady configuration of the plants in response to a unidirectional flow. This treatment couples the beam equations in the canopy to the fluid momentum equations. Subsequently, a linear stability analysis suggests new insights into the development of instabilities at the surface of the vegetative region. In particular, we show that shear at the top of the canopy is a dominant factor in determining the onset of instabilities known as monami. Based on numerical and asymptotic analysis of the quadratic eigenvalue problem, the system is shown to be stable if the canopy is sufficiently sparse.

Buoyancy-driven exchange flow in a vertical duct is studied theoretically for a light pure fluid and a heavy fluid. The latter is a suspension composed of a Newtonian fluid and two populations of negatively buoyant particles of the same size but different densities (bidensity). In a previous study (Mirzaeian & Alba, J. Fluid Mech., vol. 847, 2018, pp. 134–160), the authors developed a lubrication model for monodensity suspension of particles of uniform size. The main observation of the monodensity study was the discovery of particle-enriched zones near heavy and light fluid fronts due to the relative motion of particles and the fluid. Distinct from the previous work, here, mismatched densities instigate a relative motion of lighter and heavier particles in addition to the movement of fluids. Other than the previously observed enrichment case, the bidensity case gives rise to a novel flow regime where there is enrichment of heavy particles but depletion of light particles near the interface. The transition to this regime is governed by a balance between the densities of heavy and light particles as well as those of light and carrying fluids for a given choice of initial volume fractions of particles. Such density balance is characterized by two dimensionless parameters comprising light and heavy particle-to-carrying-fluid density ratios. The transition mechanism is studied through additional simulations, revealing that the former increases with initial volume fraction of particles of either type, while the latter contrarily decreases. The effect of other parameters on the flow are discussed within the context of the paper.

Decaying turbulence in salt-stratified fluid with Schmidt number $700$ is investigated by direct numerical simulation. In the final period of decay, and after the Ozmidov scale becomes smaller than the Kolmogorov scale, potential-energy distribution due to salinity fluctuations shows large-scale clouds composed of structures smaller than the Kolmogorov scale. When these clouds appear, potential energy has a flat spectrum in the viscous-convective subrange, rather than a $k^{-1}$ spectrum observed initially before the stratification effect becomes significant. This transition occurs since the potential energy near the Kolmogorov scale or the primitive scale of stratified turbulence defined by $\sqrt{\unicode[STIX]{x1D708}^{\ast }/N^{\ast }}$, where $\unicode[STIX]{x1D708}^{\ast }$ is the kinematic viscosity and $N^{\ast }$ the Brunt–Väisälä frequency, decreases significantly due to the persistent conversion of potential energy into kinetic energy by the counter-gradient density flux.

We investigate the relationship between the metachronal wavelength of an array of beating cilia and the resulting fluid flow rate through numerical simulations. Our model is based on a hybrid immersed boundary lattice Boltzmann algorithm. Our results suggest that varying the metachronal wavelength of the cilium array affects the fluid flow rate by increasing or decreasing the spread of cilia during their active strokes. We quantify this behaviour by constructing an analytical model of the system and deriving an equation for free area within the cilium array that depends on the metachronal wavelength. We show that there is a strong correlation between free area and fluid flow rate that holds for different values of cilium spacing.

Second-order curved shock theory is developed and applied to planar and axisymmetric curved shock flow fields. Explicit equations are given in an influence coefficient format, relating the second-order gradients of pre-shock and post-shock flow parameters to shock curvature gradients. Two types of applications are demonstrated. First, the post-shock flow fields behind known curved shocks are solved using the second-order curved shock equations. Compared with the first-order curved shock equations, the second-order equations give better agreement with solutions obtained using the method of characteristics. Second, the second-order theory is applied to capture the curved shock shape with limited flow field information. In terms of the residual sum of squares of the curved shock, the second-order curved shock equations give a value one order of magnitude better than those given by the Rankine–Hugoniot equations and the first-order equations. This improved accuracy makes the second-order theory a good candidate for solving shock capture problems in computational fluid dynamics algorithms.

We present a new theoretical model describing gravity–capillary waves in orbitally shaken cylindrical containers. Our model can account for both one-layer free-surface and two-layer interfacial wave systems. A set of modal equations for irrotational waves is formulated that we complement with viscous damping rates to incorporate energy dissipation. This approach allows us to calculate explicit formulas for the phase shifts between wave and shaker which are practically important for the mixing efficiency in orbitally shaken bioreactors. Resonance dynamics is described using eight dimensionless numbers, revealing a variety of different effects influencing the forced wave amplitudes. As an unexpected result, the model predicts the formation of novel spiral wave patterns resulting from a damping-induced symmetry breaking mechanism. For validation, we compare theoretical amplitudes, fluid velocities and phase shifts with three different and independent experiments and, when using the slightly deviating experimental values of the resonance frequencies, find a good agreement within the theoretical limits.

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

Vortex interactions behind step cylinders with diameter ratio $D/d=2$ and 2.4 at Reynolds number ($Re_{D}$) 150 were investigated by directly solving the three-dimensional Navier–Stokes equations. In accordance with previous studies, three spanwise vortex cells were captured: S-, N- and L-cell vortices. In this paper, we focused on vortex interactions between the N- and L-cell vortices, especially the vortex dislocations and subsequent formations of vortex loop structures. The phase difference accumulation process of every pair of corresponding N- and L-cell vortices and its effects on the vortex dislocations were investigated. We revealed that the total phase difference between N- and L-cell vortices was accumulated by two physically independent mechanisms, namely different shedding frequencies and different convective velocities of these two cells. The second mechanism has never been reported before. The relative importance of these two mechanisms varied periodically in the phase difference accumulation process of every pair of corresponding N- and L-cell vortices. This variation caused the vortex dislocation process and the subsequent formation of the loop structures to change from one N-cell cycle to another. Our long-time observations also revealed an interruption of the conventional antisymmetric vortex interactions between two subsequent N-cell cycles in this wake. Moreover, the trigger value and the threshold value in the phase difference accumulation processes were identified and discussed. Both values contribute to better understanding of the vortex dislocations in this kind of wake flow. Finally, the universality of our discussions and conclusions was investigated.

Natural convection in porous media is a fundamental process for the long-term storage of CO2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Rayleigh number ($Ra$) for high values of $Ra$ ($Ra\gg 1300$). Here, we perform direct numerical simulations (DNS), fully resolving the flow field within the pores. We show that the boundary layer thickness is determined by the pore size instead of the Rayleigh number, as previously assumed. The mega- and proto-plume sizes increase with the pore size. Our DNS results exhibit a nonlinear scaling of the Sherwood number at high porosity, and for the same Rayleigh number, higher Sherwood numbers are predicted by DNS at lower porosities. It can be concluded that the scaling of the Sherwood number depends on the porosity and the pore-scale parameters, which is consistent with experimental studies.

We examine the scaling of the two-point correlation function for $\unicode[STIX]{x1D716}$, the energy dissipation rate, over a range of values of the separation $r$ between the two points and the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$. The correlation function is estimated from hot-wire measurements in grid turbulence, along the axes of wakes and jets, and along the centreline of a fully developed channel flow. When $Re_{\unicode[STIX]{x1D706}}$ exceeds a value of approximately 300, a condition which is achieved for both plane and circular jets, the correlation function collapses over nearly all values of $r$ when the normalization uses Kolmogorov scales. However, there is no collapse in either the power-law range or dissipative range when the normalization is on the integral (or external) length scale, which indicates that there is no self-similarity based on external scales. Although the maximum value of $Re_{\unicode[STIX]{x1D706}}$ is not much larger than $10^{3}$, the behaviour of the energy dissipation correlation function on the axes of plane and circular jets seems consistent with the first similarity hypothesis of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303) but not with the revised phenomenology of Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85).

We quantitatively test a dimensionless group (Blackwell et al., Phys. Fluids, vol. 27, 2015, 043101) for predicting the onset of splash behaviour for drop impacts of yield-stress fluids on horizontal surfaces pre-coated with the same fluid. Two classes of complex fluid are considered, with different chemistry and material microstructure: ‘gel’ suspensions of attractive hard colloidal clay particles (Laponite RD), and ‘glassy’ suspensions of crowded soft polymeric particles (Carbopol 940). High-speed imaging identifies drop impact regimes as a function of material composition, drop diameter, velocity and coating thickness. The high-dimensional parameter space collapses into the single dimensionless group, which we find successfully separates the impact results into distinct regimes. Moreover, it gives a constant critical value for the regime boundary as a function of dimensionless coating thickness, and remarkably this critical value is only a factor of 2 larger for Carbopol when compared to Laponite at several different concentrations. The results demonstrate insight that depends on macroscopic dynamic conditions and rheological properties largely independent of chemistry and material microstructure within the range of conditions studied.