High-resolution direct numerical simulations of isotropic homogeneous turbulence are used to understand the differences between the effects of spatial intermittency on the energy dissipation rate and on surrogates for the dissipation rate that are based on measurements of a subset of the strain rate tensor. In particular, the one-dimensional longitudinal and transverse surrogates, as well as a surrogate based on the asymmetric part of the strain rate tensor, are considered. The instantaneous surrogates are studied locally, locally averaged in space and conditionally averaged to see what statistics of the dissipation rate might accurately be inferred given measurements of the surrogates. The simulations with the Reynolds numbers based on the Taylor microscale of 102–235 are highly resolved for accurate evaluation of higher-order statistics. The probability densities of the local and locally averaged surrogates are significantly different from the corresponding statistics for the dissipation rate itself. All of the surrogates are more intermittent than the dissipation rate, the transverse surrogate is more intermittent than the longitudinal and these trends are still prominent even when the fields are spatially averaged at length scales close to the integral length scale. As a consequence, the intermittency exponent computed from the moments of the locally averaged longitudinal and transverse surrogates is approximately 1.5 and 2.2 times higher, respectively, than that computed by the same method from the dissipation rate field. In addition, while different methods of computing intermittency exponent from the dissipation rate field yield the same result, different methods applied to a surrogate are inconsistent.

The problem of a vortex pair, rising obliquely at an angle of 45° toward a deformable free surface in a viscous, incompressible fluid, is solved with the aid of the Navier—Stokes equations. The full nonlinear boundary conditions at the free surface are applied. The oblique interaction of the vortex pair with the free surface results in a number of novel features that have not been observed for the special case of a vertical rise, reported earlier. These features include the directional change of trajectories near the free surface and the occurrence of waves driven by the vortex pair. Moreover, surface tension can completely change the flow characteristics such as the direction of the trajectories and the generation of secondary vortices. Numerical solutions are presented for selected Reynolds, Froude, and Weber numbers.

Langmuir circulations (LCs) generated by the interaction between wind-driven currents and surface waves can engulf the whole water column in neutrally stratified shallow water and interact with the turbulence in the bottom boundary layer. In this study, we perform a mechanistic study using wall-resolved large-eddy simulations (LES) based on the Craik–Leibovich equations to investigate the effects of LCs on turbulence statistics in the bottom half of shallow water. The highest Reynolds number considered in this paper, $Re_{\unicode[STIX]{x1D70F}}=1000$, is larger than the values considered in wall-resolved LES studies of shallow-water Langmuir turbulence reported in literature. The logarithmic layer is diagnosed based on a plateau region in the profile of a diagnostic function. It is found that the logarithmic layer disrupted at $Re_{\unicode[STIX]{x1D70F}}=395$ reappears at $Re_{\unicode[STIX]{x1D70F}}=1000$, but the von Kármán constant is slightly different from the traditional value $0.41$. To study the effects of LCs on turbulence statistics, LCs are extracted using streamwise averaging. The velocity fluctuations $u_{i}^{\prime }$ are decomposed into a LC-coherent part $u_{i}^{L}$ and a residual turbulence part $u_{i}^{T}$. It is found that the profiles of LC-coherent Reynolds shear stress $-\langle u^{L}v^{L}\rangle$ obtained at various Reynolds numbers are close to each other in the water-column coordinate $y/h$, with $h$ being the half-water depth. As the Reynolds number (or, by definition, the ratio between the outer and inner length scales) increases, the influence of LCs on the near-bottom momentum transfer is reduced, which is responsible for the reappearance of the logarithmic layer. At all of the Reynolds numbers under investigation, the peaks of $\langle u^{L}u^{L}\rangle$ are collocated in the water-column coordinate $y/h$, while those of $\langle u^{T}u^{T}\rangle$ are collocated in the inner-scale coordinate $y/(\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}})$. Due to the increase in the distance between the peaks of $\langle u^{L}u^{L}\rangle$ and $\langle u^{T}u^{T}\rangle$ with the Reynolds number, the profile of $\langle u^{\prime }u^{\prime }\rangle$ forms a bimodal shape at $Re_{\unicode[STIX]{x1D70F}}=700$ and $1000$.

The dynamics of helicity in homogeneous skew-isotropic freely decaying turbulence is investigated, at very high Reynolds numbers, thanks to a classical eddy-damped quasi-normal Markovian (EDQNM) closure. In agreement with previous direct numerical simulations, a $k^{-5/3}$ inertial range is obtained for both the kinetic energy and helical spectra. In the early stage of the decay, when kinetic energy, initially only present at large scales cascades towards small scales, it is found that helicity slightly slows down the nonlinear transfers. Then, when the turbulence is fully developed, theoretical decay exponents are derived and assessed numerically for helicity. Furthermore, it is found that the presence of helicity does not modify the decay rate of the kinetic energy with respect to purely isotropic turbulence, except in Batchelor turbulence where the kinetic energy decays slightly more rapidly. In this case, non-local expansions are used to show analytically that the permanence of the large eddies hypothesis is verified for the helical spectrum, unlike the kinetic energy one. Moreover, the $4/3$ law for the two-point helical structure function is assessed numerically at very large Reynolds numbers. Afterwards, the evolution equation of the helicity dissipation rate is investigated analytically, which provides significant simplifications and leads notably to the definition of a helical derivative skewness and of a helical Taylor scale, which is numerically very close to the classical Taylor longitudinal scale at large Reynolds numbers. Finally, when both a mean scalar gradient and helicity are combined, the quadrature spectrum, linked to the antisymmetric part of the scalar flux, appears and scales like $k^{-7/3}$ and then like $k^{-5/3}$ in the inertial range.

A ship, towing a heavier-than-water cable with a neutrally buoyant slender cylinder attached to the downstream end of the cable, is considered. The neutrally buoyant element contains a sonar array. Linear changes in the ship's velocity cause perturbations of both the cable and cylinder. The form of transverse vibrations of the neutrally buoyant cylinder was determined in Part 1. The propagation of disturbances along the cable is investigated in this paper. In particular, the effectiveness of the cable at isolating the sonar array from forcing due to unsteady ship motion is examined.

The cable and cylinder are found to be stable under constant towing conditions. It is therefore appropriate to investigate their response to forcing. Meanderings in the ship's track produce transverse displacements of both the cable and the cylinder. These transverse oscillations entirely decouple from any in-plane motion. The propagation of disturbances of frequency ω along the cable depends strongly on the value of the non-dimensional frequency ωlC/U, where lC is the cable length and U is the towing speed, and only weakly on the other cable parameters. The cable acts as an effective low-pass filter to transverse oscillations, the amplitude of disturbances with non-dimensional frequency greater than 10 being reduced by at least 90% as they propagate along the cable.

Unsteadiness in the ship's speed can result in in-plane deflections of the cable, and vertical oscillations of the cylinder containing the sonar array. In contrast to the transverse oscillations a significant proportion of the in-plane disturbances at the ship travels to the sonar array at all values of the frequency. Low- and high-frequency analytical forms are derived to explain why this occurs. Perturbations in the ship's position are most effectively transformed into vertical oscillations of the array at a frequency of 2.8U/lC. The effect of cable properties on transmission along the cable is investigated. The transmission again depends on the value of the non-dimensional frequency ωlC/U. Parameter changes, which increase the cable critical angle, increase the proportion of the disturbance at the towing point that is transformed into vertical array motion, for a fixed value of ωlC/U. This is explained by reference to the low- and high-frequency analytical solutions.

The stability of an incompressible parallel flow with a linear wall slip – where the amount of wall slip is proportional to the velocity gradient – is analysed. Numerical computations are performed for microchannel flows in the slip-flow region. The wall slip causes short-wave instability while the slip-flow model is stable for long waves. This instability disappears once the slip wall condition is replaced by the no-slip condition.

An experimental study of the interaction of weak shock waves in a liquid with bubbles and solid particles has been conducted. Cavities were punched, and solid particles were cast, into a thin sheet of gelatine clamped between two transparent blocks. A shock of pressure 0.3 GPa was introduced by impacting the gelatine layer with a flyer plate. The subsequent collapse of the cavities was photographed using high-speed framing cameras, and waves in the gelatine were visualized using schlieren optics. Assorted cavity/particle geometries were studied. In the first, cavity and particle were aligned on an axis parallel to the incident shock front. The jet crossing the cavity was found to deviate from the perpendicular to the shock front. This deviation was towards the solid particle when separations were small and away from the particle when separations were increased. When a cavity was placed upstream of a solid particle the collapse time was reduced. Conversely, when a cavity was placed downstream of a solid particle, collapse time was increased and the closure was more symmetrical. These observations were explained in terms of wave reflections. Collapses where the cavity/particle axis was inclined to the incident shock showed features of each of the geometries described above.

The symmetric near wake of a flat plate with a circular trailing edge, exhibiting pronounced shedding of wake vortices, is investigated via a direct numerical simulation. The separating plate boundary layers are turbulent and statistically identical. The present study focuses on the vortical structures, the distribution of phase-averaged velocity statistics and vorticity fluctuations in the near wake. Braid and vortex core regions are investigated via contours of instantaneous vorticity and pressure with an emphasis on the three-dimensionality of near-wake structures. Rib vortices are observed in the braids. The vortex cores are populated with both ribs that wrap themselves around the cores and spanwise-elongated regions of spanwise vorticity near the centres of the cores. Production terms in the transport equations for phase-averaged fluctuating vorticity show that stretching of rib vortices via the phase-averaged strain rate produces significantly less turbulent vorticity than turbulent stretching in the braids and cores.

We investigate the effect of intergranular cohesive forces on the properties of self-diffusion in dense granular flows. The study is based on a series of simulated plane shear flows at different inertial and cohesion numbers, in which transverse diffusivities are measured. Results evidence an increase in diffusivity by up to two orders of magnitude when introducing cohesion. This strong effect is analysed using the Green–Kubo framework, expressing the diffusivity in terms of instantaneous grain velocity fluctuations and their time correlation. This analysis shows that cohesion, by forming enduring clusters in the flow, enhances the velocity fluctuations and their time persistence, which both contribute to enhancing grain mixing and self-diffusion.

In this paper a turbulent plane channel flow modified by a distributed transpiration at the wall, with zero net mass flux, is studied through direct numerical simulation (DNS) using the incompressible Navier–Stokes equations. The transpiration is steady, uniform in the spanwise direction, and varies sinusoidally along the streamwise coordinate. The transpiration wavelength is found to dramatically affect the turbulent flow, and in particular the frictional drag. Long wavelengths produce large drag increases even with relatively small transpiration intensities, thus providing an efficient means for improved turbulent mixing. Shorter wavelengths, on the other hand, yield an unexpected decrease of turbulent friction. These opposite effects are separated by a threshold of transpiration wavelength, shown to scale in viscous units, related to a longitudinal length scale typical of the near-wall turbulence cycle. Transpiration is shown to affect the flow via two distinct mechanisms: steady streaming and direct interaction with turbulence. They modify the turbulent friction in two opposite ways, with streaming being equivalent to an additional pressure gradient needed to drive the same flow rate (drag increase) and direct interaction causing reduced turbulent activity owing to the injection of fluctuationless fluid. The latter effect overwhelms the former at small wavelengths, and results in a (small) net drag reduction. The possibility of observing large-scale streamwise-oriented vortical structures as a consequence of a centrifugal instability mechanism is also discussed. Our results do not demonstrate the presence of such vortices, and the same conclusion can be arrived at through a stability analysis of the mean velocity profile, even though it is possible that a higher value of the Reynolds number is needed to observe the vortices.

An experimental study of fully developed uniform-density turbulent flow in a circular pipe containing one or two rods located off-centre is described. The friction factor in both cases was found to be approximately 5 % higher than the simple pipe friction factor. The shear stress distribution on the rod surface was determined using calibrated boundary-layer fences. The normalized shear stress distributions were independent of Reynolds number in the range 3·7 × 104 to 2·15 × 105. Mean-velocity measurements were obtained to check the validity of the universal law of the wall near the rod surface. Secondary-flow velocities were measured by a hot-wire anemometer and integrated to yield the secondary-flow stream function. Secondary-flow velocities of the order of 1 % of the mean velocity were observed. In the gap between the two pins, however, the secondary-flow velocities were only ½% of the mean velocity. It is demonstrated that the secondary flow cannot be neglected if a force balance is used to determine the shear stress distribution on the rod surface.

The relaxation and breakup of an elongated droplet in a viscous and initially quiescent fluid is studied by solving the full Navier–Stokes equations using a three-dimensional finite volume method coupled with a moving mesh interface tracking (MMIT) scheme to locate the interface. The two fluids are assumed incompressible and immiscible. The interface is represented as a surface triangle mesh with zero thickness that moves with the fluid. Therefore, the jump and continuity conditions across the interface are implemented directly, without any smoothing of the fluid properties. Mesh adaptations on a tetrahedral mesh are employed to permit large deformation and to capture the changing curvature. Mesh separation is implemented to allow pinch-off. The detailed investigations of the relaxation and breakup process are presented in a more general flow regime compared to the previous works by Stone & Leal (J. Fluid Mech., vol. 198, 1989, p. 399) and Tong & Wang (Phys. Fluids, vol. 19, 2007, 092101), including the flow field of the both phases. The simulation results reveal that the vortex rings due to the interface motion and the conservation of mass play an important role in the relaxation and pinch-off process. The vortex rings are created and collapsed during the process. The effects of viscosity ratio, density ratio and length ratio on the relaxation and breakup are studied. The simulations indicate that the fluid velocity field and the neck shape are distinctly different for viscosity ratios larger and smaller than O(1), and thus a different end-pinching mechanism is observed for each regime. The length ratio also significantly affects the relaxation process and the velocity distributions, but not the neck shape. The influence of the density ratio on the relaxation and breakup process is minimal. However, the droplet evolution is retarded due to the large density of the suspending flow. The formation of a satellite droplet is observed, and the volume of the satellite droplet depends strongly on the length ratio and the viscosity ratio.

Bioconvection is an important phenomenon in aquatic environments, affecting the spatial distribution of motile micro-organisms and enhancing mixing within the fluid. However, stratification arising from thermal or solutal gradients can play a pivotal role in suppressing the bioconvective flows, leading to the aggregation of micro-organisms and growth of their patchiness. We investigate the combined effects by considering gyrotactic motility where the up-swimming cells are directed by the balance of the viscous and gravitational torques. To study this system, we employ a continuum model consisting of Navier–Stokes equations with the Boussinesq approximation coupled with two conservation equations for the concentration of cells and stratification agent. We present a linear stability analysis to determine the onset of bioconvection for different flow parameters. Also, using large-scale numerical simulations, we explore different regimes of the flow by varying the corresponding boundary conditions and dimensionless variables such as Rayleigh number and Lewis number ($\mathit{Le}$) and we show that the cell distribution can be characterized using the ratio of the buoyancy forces as the determinant parameter when $\mathit{Le}\lt 1$ and the boundaries are insulated. But, in thermally stratified fluids corresponding to $\mathit{Le}\gt 1$, temperature gradients are demonstrated to have little impact on the bioconvective plumes provided that the walls are thermally insulated. In addition, we analyse the dynamical behaviour of the system in the case of persistent pycnoclines corresponding to constant salinity boundary conditions and we discuss the associated inhibition threshold of bioconvection in the light of the stability of linearized solutions.

A fluid contained between two parallel walls, one of which is at rest and the other moving in the longitudinal direction with a constant velocity, is examined when a standing sound wave is imposed in the transverse direction. Vortical acoustic streaming appears in the region between the walls. The streaming is not affected by the main flow. A qualitative analysis is presented for the Navier–Stokes equations governing the steady-streaming component of the motion. The study considers the case of flow with high streaming Reynolds number and makes an explicit determination of the vorticity in the inviscid core region. The effect of the streaming upon the shear flow in the longitudinal direction is then analysed asymptotically. A periodic structure of the wall shear stress in the transverse direction is detected in which vast areas of vanishing wall shear stress alternate with narrow regions where it increases significantly. A relation expressing the mean wall shear stress in terms of the streaming Reynolds number is derived. Results obtained show that acoustic streaming results in a marked enhancement of the mean wall shear stress at the walls.