The instability of a bed of particles sheared by a viscous fluid is investigated theoretically. The viscous flow over the wavy bed is first calculated, and the bed shear stress is derived. The particle transport rate induced by this bed shear stress is calculated from the viscous resuspension theory of Leighton & Acrivos (1986). Mass conservation of the particles then gives explicit expressions for the wave velocity and growth rate, which depend on four dimensionless parameters: the wavenumber, the fluid thickness, a viscous length and the shear stress. The mechanism of the instability is given. It appears that for high enough fluid-layer thickness, long-wave instability arises as soon as grains move, while short waves are stabilized by gravity. For smaller fluid thickness, the destabilizing effect of fluid inertia is reduced, so that the moving at bed is stable for small shear stress, and unstable for high shear stress. The most amplified wavelength scales with the viscous length, in agreement with the few available experiments for small particle Reynolds numbers. The results are also compared with related studies for turbulent flow.

In this paper we investigate the large-eddy simulation (LES) of the interaction between a turbulent shear flow and a free surface at low Froude numbers. The benchmark flow field is first solved by using direct numerical simulations (DNS) of the Navier–Stokes equations at fine (1282 × 192 grid) resolution, while the LES is performed at coarse resolution. Analysis of the ensemble of 25 DNS datasets shows that the amount of energy transferred from the grid scales to the subgrid scales (SGS) reduces significantly as the free surface is approached. This is a result of energy backscatter associated with the fluid vertical motions. Conditional averaging reveals that the energy backscatter occurs at the splat regions of coherent hairpin vortex structures as they connect to the free surface. The free-surface region is highly anisotropic at all length scales while the energy backscatter is carried out by the horizontal components of the SGS stress only. The physical insights obtained here are essential to the efficacious SGS modelling of LES for free-surface turbulence. In the LES, the SGS contribution to the Dirichlet pressure free-surface boundary condition is modelled with a dynamic form of the Yoshizawa (1986) expression, while the SGS flux that appears in the kinematic boundary condition is modelled by a dynamic scale-similarity model. For the SGS stress, we first examine the existing dynamic Smagorinsky model (DSM), which is found to capture the free-surface turbulence structure only roughly. Based on the special physics of free-surface turbulence, we propose two new SGS models: a dynamic free-surface function model (DFFM) and a dynamic anisotropic selective model (DASM). The DFFM correctly represents the reduction of the Smagorinsky coefficient near the surface and is found to capture the surface layer more accurately. The DASM takes into account both the anisotropy nature of free-surface turbulence and the dependence of energy backscatter on specific coherent vorticity mechanisms, and is found to produce substantially better surface signature statistics. Finally, we show that the combination of the new DFFM and DASM with a dynamic scale-similarity model further improves the results.

The complex flow features that arise from the impact of a shock wave on a concave cavity are determined by means of high-speed video photography. Besides additional information on features that have previously been encountered in specific studies, such as those relating to shock wave reflection from a cylindrical wall and those associated with shock wave focusing, a number of new features become apparent when the interaction is studied over longer times using time-resolved imaging. The most notable of these new features occurs when two strong shear layers meet that have been generated earlier in the motion. Two jets can be formed, one facing forward and the other backward, with the first one folding back on itself. The shear layers themselves develop a Kelvin–Helmholtz instability which can be triggered by interaction with weak shear layers developed earlier in the motion. Movies are available with the online version of the paper.

The initial growth of a disturbance induced by a near-wall circular cylinder in a flat-plate boundary layer is experimentally investigated using both particle image velocimetry and hydrogen bubble visualization techniques. The secondary spanwise vortices appear in the near-wall region as a direct response to the outside passing wake vortices, consistent with previous studies on similar models (Pan et al., J. Fluid Mech., vol. 603, 2008, pp. 367–389; Mandal & Dey, J. Fluid Mech., vol. 684, 2011, pp. 60–84). The streamwise variation of the total disturbance energy within the boundary layer shows a two-stage growth, which characterizes the initial transition process: the first exponential growth stage, followed by a region with slower growth rate. It is revealed that these two stages of growth are related to the formation and the destabilization of the secondary vortex in the near-wall region. The technique of dynamic mode decomposition is used to decompose the total disturbance into temporally orthogonalized modes, and it shows that the first growth stage largely results from the increased disturbance at the same frequency as that of the wake vortex shedding, while the second growth stage comprises the disturbance growth in a number of frequencies, especially the lower ones.

Two coupled problems are investigated: a complete description of long-wave vortex ring oscillations in an ideal incompressible fluid, and an examination of sound radiation by these oscillations in a weakly compressible fluid.

The first part of the paper relates to the problem of eigen-oscillations of a thin vortex ring (μ[Lt ]1) in an ideal incompressible fluid. The solution of the problem is obtained in the form of an asymptotic expansion in the small parameter μ. The complete set of three-dimensional eigen-oscillations and axisymmetric modes (two-dimensional oscillations) is obtained. It is shown that, unlike the vortex column oscillations which have the form of simple angular harmonics, the majority of eigen-oscillations of a thin vortex ring have a more complex form which is a combination of two harmonics in the leading approximation. This leads to dramatic changes in the efficiency of sound radiation produced by modes of the vortex ring in comparison with the corresponding modes of the vortex column.

In the second part of the paper the solution obtained is used to investigate the process of sound radiation by vortex perturbations in a weakly compressible fluid. The vortex ring eigen-oscillations are classified according to their sound radiation efficiency. It is shown that the modes with the dimensionless frequency ω≈1/2 radiate sound most efficiently. They are two isolated modes, two infinite families of Bessel modes and a set of axisymmetric modes. The frequencies of these modes are in the interval Δω=O(μ).

The results obtained are compared with known experimental data on acoustic radiation of a turbulent vortex ring. Within the limits of the theory derived an explanation of the main characteristics of sound radiation is presented.

Equations for the development of streamwise (or secondary) vorticity, for stationary and rotating systems, are derived directly from the vector equation for vorticity, using intrinsic co-ordinates. This approach places emphasis on the coupled equations for the vorticity components parallel and normal to the streamline. The equations derived are more general than those hitherto available, being valid for compressible, stratified and viscous flow. Some interpretation is given of the physical mechanism causing secondary flow.

Both a weakly nonlinear analytic theory and direct numerical simulation are used to document processes involved during the spin-up of a rotating stratified fluid driven by wind-stress forcing for time periods less than a homogeneous spin-up time. The strength of the wind forcing, characterized by the Rossby number ε, is small enough (i.e. ε[Lt ]1) that a regular perturbation expansion in ε can be performed yet large enough (more specifically, ε∝E1/2, where E is the Ekman number) that higher-order effects of vertical diffusion and horizontal advection of momentum/density are comparable in magnitude. Cases of strong stratification, where the Burger number S is equal to one, with zero heat flux at the upper boundary are considered. The Ekman transport calculated to O(ε) decreases with increasing absolute vorticity. In contrast to nonlinear barotropic spin-up, vortex stretching in the interior is predominantly linear, as vertical advection negates stretching of interior relative vorticity, yet is driven by Ekman pumping modified by nonlinearity. As vertical vorticity is generated during the spin-up of the fluid, the vertical vorticity feeds back on the Ekman pumping/suction, enhancing pumping and vortex squashing while reducing suction and vortex stretching. This feedback mechanism causes anticyclonic vorticity to grow more rapidly than cyclonic vorticity. Strict application of the zero-heat-flux boundary condition leads to the growth of a diffusive thermal boundary layer E−1/4 times thicker than the Ekman layer embedded within it. In the Ekman layer, vertical diffusion of heat balances horizontal advection of temperature by extracting heat from the thermal boundary layer beneath. The flux of heat extracted from the top of the thermal boundary layer by this mechanism is proportional to the product of the Ekman transport and the horizontal gradient of the temperature at the surface. The cooling caused by this heat flux generates density inversions and intensifies lateral density gradients where the wind-stress curl is negative. These thermal gradients make the potential vorticity strongly negative, conditioning the fluid for ensuing symmetric instability which greatly modifies the spin-up process.

In this paper we study the stability of long, steady, two-dimensional salt fingers. It is already known that salt fingers carrying a large enough density flux are unstable to long-wavelength internal-wave perturbations. Stern (1969) studied the mechanism of this, the collective instability, and it was studied in more detail by Holyer (1981). We extend the earlier work to include perturbations of all wavelengths, as well as long-wavelength perturbations. By applying the methods of Floquet theory to the periodic salt fingers, the growth rates of perturbations are found. For both heat-salt and salt-sugar systems the collective instability, which can be recognized by its frequency of oscillation, does not have the largest growth rate. There is a new, non-oscillatory instability, which, according to linear theory, grows faster than the collective instability. We study the instabilities that arise by using a combination of analytical and numerical methods. Further work will be necessary in order to assess the importance of these instabilities in different physical situations and to examine their development as their amplitude increases.

Disturbance interactions in wave triads and multiwave systems of various configurations are investigated to reveal the mechanism of laminar-turbulent transition in Blasius and pressure-gradient boundary layers. The averaging method of weakly nonlinear instability theory in quasi-parallel flows is applied. Tollmien-Schlichting-wave resonant interaction is shown to be the only leading mechanism of subharmonic (S)-type transition. The mechanism universally dominates in boundary layers excited by sufficiently small initial disturbances. The role of any other mode is inefficient. Weakly nonlinear models are concluded not to explain the K-type transition scenario. The results of the study are employed to interpret physical and numerical experimental data.

We have measured the probability density functions (PDFs) of density fluctuations and of density-gradient fluctuations in decaying stratified turbulence, using a thermally stratified wind tunnel. The turbulence was generated by passing the flow through a biplanar grid at the entrance to the test section. The linear mean vertical temperature gradient could be adjusted to produce different stratification strengths. The PDFs of the density-gradient fluctuations exhibit extended exponential tails, while those for the density fluctuations are nearly Gaussian. As the turbulence decays away from the grid the exponential tails of the density gradient PDFs become steeper and the central rounded part of the distribution widens. The tail steepness scales approximately as $Re \lambda{-\frac{1}{2}}$. Buoyancy forces are not the cause of the exponential tails, since when normalized in r.m.s. units the behaviour of the tails is independent of stratification strength. The vertical temperature gradients ∂θ/∂z (measured using two cold wires) show a strong positive skewness close to the grid where the turbulence is most vigorous. This skewness is not caused by non-Boussinesq effects and is present for all stratification strengths. We propose a simple phenomenological model (similar to that of Budwig et al. 1985), based on stirring of fluid parcels advected in the mean gradient, to explain the presence of this skewness. The skewness observed by other researchers and their interpretations are discussed in the context of this model. The buoyancy flux PDF also shows strong exponential tails and is very strongly skewed. Both of these properties are consistent with joint-Gaussian statistics of the vertical velocity and temperature fluctuations.

Progressive short waves with a narrow frequency band are known to be accompanied by long set-down waves travelling with the groups. In finite depth, scattering of short waves by a large structure or a varying coastline can radiate free long waves which propagate faster than the incident set-down. In a partially enclosed harbour attacked by short waves through the entrance, such free long waves can further resonate the natural modes of the harbour basin. In this paper an asymptotic theory is presented for a harbour whose horizontal dimensions are much greater than the entrance width, which is in turn much wider than the short wavelength.

We study the formation, growth and disintegration of jets following the impact of a drop on a thin film of the same liquid for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{We}<1000$ and $\mathit{Re}<2000$ using a combination of numerical simulations and linear stability theory (Agbaglah, Josserand & Zaleski, Phys. Fluids, vol. 25, 2013, 022103). Our simulations faithfully capture this phenomena and are in good agreement with experimental profiles obtained from high-speed X-ray imaging. We obtain scaling relations from our simulations and use these as inputs to our stability analysis. The resulting predictions for the most unstable wavelength are in excellent agreement with experimental data. Our calculations show that the dominant destabilizing mechanism is a competition between capillarity and inertia but that deceleration of the rim provides an additional boost to growth. We also predict over the entire parameter range of our study the number and timescale for formation of secondary droplets formed during a splash, based on the assumption that the most unstable mode sets the droplet number.

Aeroacoustic measurements and analysis have been made for an unshrouded rotor partially immersed in a planar equilibrium turbulent boundary layer at low Mach number. This configuration provides an idealized model of inflow distortion effects seen when a rotor is mounted adjacent to the hull or fuselage of a vehicle. At low and moderate thrust conditions, the rotor produces broadband noise organized into haystacks produced by large eddies of the ingested turbulence being cut multiple times by successive rotor blades. At high thrust, however, the acoustic signature changes and becomes louder and more tonal. This change is accompanied by separation of the boundary layer from the wall in the vicinity of the rotor blade disk. The separation region is highly unsteady and populated by intense vortex structures. Acoustic analysis suggests that blade–vortex interactions with these structures are the source of the additional tonal noise at high thrust.

Low-momentum releases of buoyant material from area sources are investigated in the context of the model of Morton, Taylor & Turner (1956) (MTT). The general solution of the model equations is shown to include the case of a converging–diverging flow which we have used to model plumes from the area sources. These solutions have as asymptotes certain power solutions in terms of height, i.e. the MTT solution at large heights and a non-entraining solution near the source. The new solutions exhibit distinct and interesting flow features such as a neck (point of minimum cross-section) and a velocity peak somewhat above. The new results have been compared with the few known experimental data sources and reasonable agreement is demonstrated. In the process we have also examined the importance of the Boussinesq approximation and find results recently published not to be valid except in one special case.

In this article, a biglobal stability approach is used in conjunction with direct numerical simulation (DNS) to identify the instability mode coupling that may be responsible for triggering large thrust oscillations in segmented solid rocket motors (SRMs). These motors are idealized as long porous cylinders in which a Taylor–Culick type of motion may be engendered. In addition to the analytically available steady-state solution, a computed mean flow is obtained that is capable of securing all of the boundary conditions in this problem, most notably, the no-slip requirement at the chamber headwall. Two sets of unsteady simulations are performed, static and dynamic, in which the injection velocity at the chamber sidewall is either held fixed or permitted to vary with time. In these runs, both DNS and biglobal stability solutions converge in predicting the same modal dependence on the size of the domain. We find that increasing the chamber length gives rise to less stable eigenmodes. We also realize that introducing an eigenmode whose frequency is sufficiently spaced from the acoustic modes leads to a conventional linear evolution of disturbances that can be accurately predicted by the biglobal stability framework. While undergoing spatial amplification in the streamwise direction, these disturbances will tend to decay as time elapses so long as their temporal growth rate remains negative. By seeding the computations with the real part of a specific eigenfunction, the DNS outcome reproduces not only the imaginary part of the disturbance, but also the circular frequency and temporal growth rate associated with its eigenmode. For radial fluctuations in which the vorticoacoustic wave contribution is negligible in relation to the hydrodynamic stability part, excellent agreement between DNS and biglobal stability predictions is ubiquitously achieved. For axial fluctuations, however, the DNS velocity will match the corresponding stability eigenfunction only when properly augmented by the vorticoacoustic solution for axially travelling waves associated with the Taylor–Culick profile. This analytical approximation of the vorticoacoustic mode is found to be quite accurate, especially when modified using a viscous dissipation function that captures the decaying envelope of the inviscid acoustic wave amplitude. In contrast, pursuant to both static and dynamic test cases, we find that when the frequency of the introduced eigenmode falls close to (or crosses over) an acoustic mode, a nonlinear mechanism is triggered that leads to the emergence of a secondary eigenmode. Unlike the original eigenmode, the latter materializes naturally in the computed flow without being artificially seeded. This natural occurrence may be ascribed to a nonlinear modal interplay in the form of internal, eigenmode-to-eigenmode coupling instead of an external, eigenmode pairing with acoustic modes. As a result of these interactions, large amplitude oscillations are induced.

This paper presents a study of steady gravity currents entering a two-layer system, with the current travelling either along the boundary to form a boundary current, or between the two different layers to form an intrusion. It is shown that, at the front of an intrusion, the streamlines meet at angles of 120° at a stagnation point. For an energy-conserving current the volume inflow rate to the current, the velocity of propagation and the downstream depths are determined. In contrast to the pioneering study of Benjamin (1968), it is found that the depth of the current is not always uniquely determined and it is necessary to use some principle additional to the conservation relationships to determine which solution occurs. An appropriate principle is obtained by considering dissipative currents. In general, if the volume inflow rate to a current is prescribed, the current loses energy in order to maintain a momentum balance. We thus suggest the criterion that the energy dissipation is a maximum for a fixed volume inflow rate. It is postulated that the energy which is lost will go to form a stationary wave train behind the current. A nonlinear calculation is carried out to determine the amplitude and wavelength of these waves for intrusions. Such waves have been observed on intrusions in laboratory experiments and the results of the calculation are found to agree well with the experiments. Similar waves have not been observed on boundary currents because the resulting waves have too much energy and break.

This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.

An experimental study, investigating the mean flow and turbulence in the wind drift layer formed beneath short wind waves was conducted. The degree to which these flows resemble the flows that occur in boundary layers adjacent to solid walls (i.e. wall-layers) was examined. Simultaneous DPIV (digital particle image velocimetry) and infrared imagery were used to investigate these near-surface flows at a fetch of 5.5 m and wind speeds from 4.5 to 11 m s−1. These conditions produced short steep waves with dominant wavelengths from 6 cm to 18 cm. The mean velocity profiles in the wind drift layer were found to be logarithmic and the flow was hydrodynamically smooth at all wind speeds. The rate of dissipation of turbulent kinetic energy was determined to be significantly greater in magnitude than would occur in a comparable wall-layer. Microscale breaking waves were detected using the DPIV data and the characteristics of breaking and non-breaking waves were compared. The percentage of microscale breaking waves increased abruptly from 11% to 80% as the wind speed increased from 4.5 to 7.4 m s− and then gradually increased to 90% as the wind speed increased to 11 m s−. At a depth of 1 mm, the rate of dissipation was 1.7 to 3.2 times greater beneath microscale breaking waves compared to non-breaking waves. In the crest–trough region beneath microscale breaking waves, 40% to 50% of the dissipation was associated with wave breaking. These results demonstrated that the enhanced near-surface turbulence in the wind drift layer was the result of microscale wave breaking. It was determined that the rate of dissipation of turbulent kinetic energy due to wave breaking is a function of depth, friction velocity, wave height and phase speed as proposed by Terray et al. (1996). Vertical profiles of the rate of dissipation showed that beneath microscale breaking waves there were two distinct layers. Immediately beneath the surface, the dissipation decayed as ζ−0.7 and below this in the second layer it decayed as ζ−2. The enhanced turbulence associated with microscale wave breaking was found to extend to a depth of approximately one significant wave height. The only similarity between the flows in these wind drift layers and wall-layers is that in both cases the mean velocity profiles are logarithmic. The fact that microscale breaking waves were responsible for 40%–50% of the near-surface turbulence supports the premise that microscale breaking waves play a significant role in enhancing the transfer of gas and heat across the air–sea interface.

Reflection of an obliquely incident solitary wave by a vertical wall is studied numerically by applying the ‘high-order spectral method’ developed by Dommermuth & Yue (1987). According to the analysis by Miles (1977a, b) which is valid when ai [Lt ] 1, the regular type of reflection gives way to ‘Mach reflection’ when ai/(3ai)½ ≤ 1, Where ai is the amplitude of the incident wave divided by the quiescent water depth d and ψi is the angle of incidence. In Mach reflection, the apex of the incident and the reflected waves moves away from the wall at a constant angle (ψ*, say), and is joined to the wall by a third solitary wave called ‘Mach stem’. Miles model predicts that the amplitude of Mach stem, and so the run-up at the wall, is 4ai when ψi = (3ai)½.

Our numerical results shows, however, that the effect of large amplitude tends to prevent the Mach reflection to occur. Even when the Mach reflection occurs, it is ‘contaminated’ by regular reflection in the sense that all the important quantities that characterize the reflection pattern, such as the stem angle ψ*, the angle of reflection ψr, and the amplitude of the reflected wave ar, are all shifted from the values predicted by Miles’ theory toward those corresponding to the regular reflection, i.e. ψ* = 0, ψr = ψi, and ar = ai. According to our calculations for ai = 0.3, the changeover from Mach reflection to regular reflection happens at ψi ≈ 37.8°, which is much smaller than (3ai)½ = 54.4°, and the highest Mach stem is observed for ψi = 35° (ψi/(3ai)½ = 0.644). Although the ‘four-fold amplification’ is not observed for any value of ψi considered here, it is found that the Mach stem can become higher than the highest two-dimensional steady solitary wave for the prescribed water depth. The numerical result is also compared with the analysis by Johnson (1982) for the oblique interaction between one large and one small solitary wave, which shows much better agreement with the numerical result than the Miles’ analysis does when ψi is sufficiently small and the Mach reflection occurs.

This paper describes the results of experiments conducted on a circular jet simultaneously excited by two different acoustic tones. By varying the phase between two signals at harmonically related frequencies, control can be exercised on the process of harmonic generation – sometimes the process being virtually destroyed. This is shown to be so for both harmonic and subharmonic generation, but the latter is more difficult to control.