Dense particle suspensions are widely encountered in many applications and in environmental flows. While many previous studies investigate their rheological properties in laminar flows, little is known on the behaviour of these suspensions in the turbulent/inertial regime. The present study aims to fill this gap by investigating the turbulent flow of a Newtonian fluid laden with solid neutrally-buoyant spheres at relatively high volume fractions in a plane channel. Direct numerical simulation (DNS) are performed in the range of volume fractions ${\it\Phi}=0{-}0.2$ with an immersed boundary method (IBM) used to account for the dispersed phase. The results show that the mean velocity profiles are significantly altered by the presence of a solid phase with a decrease of the von Kármán constant in the log-law. The overall drag is found to increase with the volume fraction, more than one would expect if just considering the increase of the system viscosity due to the presence of the particles. At the highest volume fraction investigated here, ${\it\Phi}=0.2$, the velocity fluctuation intensities and the Reynolds shear stress are found to decrease. The analysis of the mean momentum balance shows that the particle-induced stresses govern the dynamics at high ${\it\Phi}$ and are the main responsible of the overall drag increase. In the dense limit, we therefore find a decrease of the turbulence activity and a growth of the particle induced stress, where the latter dominates for the Reynolds numbers considered here.

This paper investigates the electrohydrodynamical instability of two miscible flows in a micro-pipe subject to an axial electric field. There is an electrical conductivity stratification between the two layers. A weak shear flow arises from a constant axial pressure gradient. The three-dimensional linear stability analysis is studied under the assumption of a quasi-steady state. The influences of the conductivity ratio ${\it\eta}$, the interface location $a$, the interface thickness ${\it\delta}$, the Reynolds number $\mathit{Re}$ and the Schmidt number $\mathit{Sc}$ on the linear stability of the flows are investigated. The flow becomes more unstable for a larger conductivity contrast. When the conductivity in the inner layer is larger, the critical unstable mode can be dominated by either the corkscrew mode (the azimuthal wavenumber $m=1$) or the axisymmetric mode ($m=0$), which is dependent on the interface location $a$. It is observed that, when the interface is proximal to pipe’s wall, the critical unstable mode shifts from the corkscrew mode to the axisymmetric mode. When the conductivity is larger in the outer layer, the instability is dominated by the axisymmetric mode. A detailed parametric study shows that the flow is least stable when the interface between the two liquids is located at approximately $a=0.3$ and $a=0.2$ for conductivity ratios of ${\it\eta}=0.5$ and ${\it\eta}=2$ respectively. The flow becomes more stable as the interface becomes thicker, and the shear flow and ionic diffusion are found to have a stabilizing effect due to the enhancement of dissipation mechanisms.

High-Reynolds number gravity currents (GC) in a horizontal channel with circular/semicircular side walls are investigated by comparing experimental data and shallow-water (SW) theoretical results. We focus attention on a Boussinesq system (salt water in fresh water): the denser fluid, occupying part of the depth or the full depth of the ambient fluid which fills the remaining part of the channel, is initially at rest in a lock separated by a gate from the downstream channel. Upon the rapid removal of the gate (‘dam break’), the denser ‘current’ begins propagating into the downstream channel, while a significant adjustment motion propagates upstream in the lock as a bore or rarefaction wave. Using an experimental channel provided by a tube of 19 cm diameter and up to 615 cm length, which could be filled to various levels, we investigated both full-depth and part-depth releases, considered the various stages of inertial-buoyancy propagation (in particular, the initial ‘slumping’ with constant speed, and the transition to the late self-similar propagation with time to the power $3/4$), and detected the transition to the viscous-buoyancy regime. A first series of tests is focused on the motion in the lock while a second series of tests is focused on the evolution of the downstream current. The speed of propagation of the current in the slumping stage is overpredicted by the theory, by about the same amount (typically 15 %) as observed in the classical flat bottom case. The length of transition to viscous regime turns out to be ${\sim}[\mathit{Re}_{0}(h_{0}/x_{0})]^{{\it\alpha}}$ ($\mathit{Re}_{0}=(g^{\prime }h_{0})^{1/2}h_{0}/{\it\nu}_{c}$ is the initial Reynolds number, $g^{\prime }$ is the reduced gravity, ${\it\nu}_{c}$ is the kinematic viscosity of the denser fluid, $h_{0}$ and $x_{0}$ are the height of the denser current and the length of the lock, respectively), with the theoretical ${\it\alpha}=3/8$ and experimental ${\it\alpha}\approx 0.27$.

The purpose of this work is to investigate, for the first time, excitation of Faraday waves in small containers using two commensurate frequencies. This spatial restriction, which is encountered at low frequencies, leads to a wave composed primarily of one spatial eigenmode of the container. When two frequencies are used, the mode resonates primarily with one frequency, while the role of the second is to alter the instability threshold and the resulting nonlinear dynamics. As the parameter space expands greatly as a result of the introduction of three new degrees of freedom, viz. the frequency, amplitude and phase of the new component, the linear theory is first used as a guide to highlight basic two-frequency phenomena. These predictions and nonlinear phenomena are then studied experimentally with the system of Batson, Zoueshtiagh & Narayanan (J. Fluid Mech., vol. 729, 2013, pp. 496–523), who studied single-frequency excitation of different modes in a cylindrical cell. The two-frequency experiments of this work focus on excitation of the fundamental axisymmetric mode, and are quantitatively compared to the model via a posteriori Fourier decomposition of the parametric input. In doing so, experimental dependence of the instability on the new degrees of freedom is demonstrated, in accordance with the model predictions. This is done for a variety of frequency ratios, and overall agreement between the observed and predicted onset conditions is identical to that already reported for the single-frequency experiment. For each frequency ratio, the nonlinear behaviour is experimentally characterized by bifurcation and time series data, which is shown to differ significantly from comparable single-frequency excitations. Finally, we present and discuss a wave in which both temporal frequencies are used to simultaneously excite different spatial modes.

Transition to turbulence in shear flows is often subcritical, thus the dynamics of the flow strongly depends on the shape and amplitude of the perturbation of the laminar state. In the state space, initial perturbations which directly relaminarize are separated from those that go through a chaotic trajectory by a hypersurface having a very small number of unstable dimensions, known as the edge of chaos. Even for the simple case of plane Couette flow in a small domain, the edge of chaos is characterized by a fractal, folded structure. Thus, the problem of determining the threshold energy to trigger subcritical transition consists in finding the states on this complex hypersurface with minimal distance (in the energy norm) from the laminar state. In this work we have investigated the minimal-energy regions of the edge of chaos, by developing a minimization method looking for the minimal-energy perturbations capable of approaching the edge state (within a prescribed tolerance) in a finite target time $T$. For sufficiently small target times, the value of the minimal energy has been found to vary with $T$ following a power law, whose best fit is given by $E_{min}\propto T^{-1.75}$. For large values of $T$, the minimal energy achieves a constant value which corresponds to the energy of the minimal seed, namely the perturbation of minimal energy asymptotically approaching the edge state (Rabin et al., J. Fluid Mech., vol. 738, 2012, R1). For $T\geqslant 40$, all of the symmetries of the edge state are broken and the minimal perturbation appears to be localized in space with a basic structure composed of scattered patches of streamwise velocity with inclined streamwise vortices on their flanks. Finally, we have found that minimal perturbations originate in a small low-energy zone of the state space and follow very fast similar trajectories towards the edge state. Such trajectories are very different from those of linear optimal disturbances, which need much higher initial amplitudes to approach the edge state. The time evolution of these minimal perturbations represents the most efficient path to subcritical transition for Couette flow.

One of the simplest splashing scenarios results from the impact of a single drop on a deep pool. The traditional understanding of this process is that the impact generates an axisymmetric sheet-like jet that later breaks up into secondary droplets. Recently it was shown that even this simplest of scenarios is more complicated than expected because multiple jets can be generated from a single impact event and there are transitions in the multiplicity of jets as the experimental parameters are varied. Here, we use experiments and numerical simulations of a single drop impacting on a deep pool to examine the transition from impacts that produce a single jet to those that produce two jets. Using high-speed X-ray imaging methods we show that vortex separation within the drop leads to the formation of a second jet long after the formation of the ejecta sheet. Using numerical simulations we develop a phase diagram for this transition and show that the capillary number is the most appropriate order parameter for the transition.

In this numerical investigation we explore the possibility of applying free-stream vortices as a passive flow control method for delaying the transition to turbulence. The work is motivated by previous experimental studies demonstrating that stable streamwise boundary layer (BL) streaks can attenuate both two- and three-dimensional disturbances inside the BL, leading to transition delay, with the implication of reducing skin-friction drag. To date, successful control has been obtained using physical BL modulators mounted on the surface in order to generate stable streaks. However, surface mounted BL modulators are doomed to failure when the BL is subject to free-stream turbulence (FST), since a destructive interaction between the two is inevitable. In order to tackle free-stream disturbances, such as FST, a smooth surface is desired, which has motivated us to seek new methods to induce streamwise streaks inside the BL. A first step, in a systematic order, is taken in the present paper to prove the control idea of generating free-stream vortices for the attenuation of ordinary Tollmien–Schlichting waves inside the BL. In this proof-of-concept study we show that, by applying a spanwise array of counter-rotating free-stream vortices, inducing streamwise BL streaks further downstream, it is possible to alter the BL stability characteristics to such a degree that transition delay may be accomplished. For the demonstration we use direct numerical simulations along with stability analysis.

Graetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number $\mathit{Nu}_{x}$, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, $\mathit{Nu}_{x}$ scales with the ratio of position $\tilde{x}=x/L$ to Graetz number $\mathit{Gz}$, i.e. $\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$. Here, $L$ is the length of the heated or cooled tube section. The Graetz number $\mathit{Gz}$ corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent ${\it\beta}$ equals $1/3$. For no-shear flow, ${\it\beta}=1/2$. The results show that for partial slip, where the ratio of slip length $b$ to tube radius $R$ ranges from zero to infinity, ${\it\beta}$ transitions from $1/3$ to $1/2$ when $10^{-4}<b/R<10^{0}$. For partial slip, ${\it\beta}$ is a function of both position and slip length. The developed Nusselt number $\mathit{Nu}_{\infty }$ for $\tilde{x}/\mathit{Gz}>0.1$ transitions from 3.66 to 5.78, the classical limits, when $10^{-2}<b/R<10^{2}$. A mathematical and physical explanation is provided for the distinct transition points for ${\it\beta}$ and $\mathit{Nu}_{\infty }$.