The role of acoustics in flame/vortex interactions is examined via asymptotic analysis and numerical simulation. The model consists of a one-step, irreversible Arrhenius reaction between initially unmixed species occupying adjacent half-planes which are allowed to mix and react by convection and diffusion in the presence of an acoustic field or a time-varying pressure field of small amplitude. The main emphasis is on the influence of the acoustics on the ignition time and flame structure as a function of vortex Reynolds number and initial temperature differences of the reactants.

In this paper we consider the interaction of two unequal solitary waves travelling in opposite directions. The interaction equations are derived from the perturbation expansion of the Euler equations, which lead to the Boussinesq equation at lowest order. These equations are recast to obtain two weakly coupled KdV equations which are solved by the method of inverse scattering. We show that the amplitude of each solitary wave after interaction is reduced. This change in the amplitudes is shown to be fifth order in ε, the order of magnitude of the amplitude of the waves. This is in agreement with the change of amplitude produced by the interaction of two equal waves which arises in the reflection of a wave by a vertical wall.

Unsteady gasdynamic concepts are used to model the piston-driven compression of a confined gas. Perturbation methods, based on the limit of small piston Mach number, are used to construct solutions. The piston Mach number increases smoothly from zero to a maximum value, Mp = O(10−2) during an acoustic time period ta* = O(10−4 s). A linear a coustic field is generated and is represented in terms of an infinite series of Fourier spatial modes. During the longer piston time period tp* = O(10−2 s) the piston moves at constant speed. A multiple-timescale formulation is used to separate the instantaneous acoustic field from the accumulated bulk response of the gas to piston compression. The latter is found to be identical to the classical quasi-static results from equilibrium thermodynamic calculations. Nonlinear effects become important on the piston timescale. Modal interactions are represented by a system of coupled, nonlinear ordinary differential equations for the time-dependent Fourier coefficients. A numerical solution for this system describes the wavefront steepening to form a weak shock and its propagation back and forth repeatedly inside the cylinder.

Spin-stabilized projectiles with liquid payloads can experience a severe flight instability characterized by a rapid yaw-angle growth and a simultaneous loss in spin rate. Laboratory experiments and field tests have shown that this instability originates from the internal fluid motion in the range of high viscosity. After evaluation of the experimental data and analysis of the equations for the fluid motion in a spinning and nutating cylinder, we have developed a simple model of this flow. Disregarding the finite length of the cylinder, this model provides the flow field and the viscous contribution to the liquid moments in analytical form. At low Reynolds number, the flow field agrees well with computational results for the centre section of a cylinder of aspect ratio 4.3. The roll moment caused by this flow largely agrees with experimental data for a wide range of Reynolds numbers. Estimates of the temperature variation indicate that discrepancies at very low Reynolds numbers may originate from associated changes of the viscosity during the experiments.

The energy of internal waves tends to be propagated along certain characteristic paths inclined at fixed angles to the vertical direction; the angle of inclination depending only on the wave frequency and the density stratification (not on the wavelength). The reflexion of such waves by smooth plane surfaces has been discussed recently by Sandstrom (1966).

In the present paper the role of surface roughness is examined. Surprisingly, it appears that quite small-scale irregularities can completely alter the reflecting properties of a surface; the tangential scale of the roughness elements may be much smaller than the wavelength of the incident or reflected waves. All scales of roughness are relevant, down to those comparable in magnitude to the thickness of the oscillatory boundary layer. For tidal waves in the ocean this thickness is of the order of 1 m.

The behaviour of the coefficient of transmission as a function of the angle of incidence appears at first sight to be extremely complicated. Some simple examples of periodic surface roughnesses are discussed and elucidated: a sawtooth, a square-topped wave and a simple sine-wave. The transmission coefficient T for a sine-wave, for example, is shown in figure 9. An approximate expression for T is also derived in the case of a slowly modulated sine-wave (figure 10). These results are for a non-viscous fluid. The effects of viscosity are also considered qualitatively.

Convection in a rotating rectangular basin with differential thermal forcing at one horizontal boundary is examined using laboratory experiments. The experiments have an imposed heat flux boundary condition, are at large values of the flux Rayleigh number ($Ra_{F}\sim O(10^{13}{-}10^{14})$ based on the box length $L$), use water with Prandtl number $Pr\approx 4$ and have a small depth to length aspect ratio. The results show the conditions for transition from non-rotating horizontal convection governed by an inertial–buoyancy balance in the thermal boundary layer, to circulation governed by geostrophic flow in the boundary layer. The geostrophic balance constrains mean flow and reduces the heat transport as Nusselt number $Nu\sim (Ra_{F}Ro)^{1/6}$, where $Ro=B^{1/2}/f^{3/2}L$ is the convective Rossby number, $B$ is the imposed buoyancy flux and $f$ is the Coriolis parameter. Thus flow in the geostrophic boundary layer regime is governed by the relative roles of horizontal convective accelerations and Coriolis accelerations, or buoyancy and rotation, in the boundary layer. Experimental evidence suggests that for more rapid rotation there is another transition to a regime in which the momentum budget is dominated by fluctuating vertical accelerations in a region of vortical plumes, which we refer to as a ‘chimney’ following related discussion of regions of deep convection in the ocean. Coupling of the chimney convection in the region of destabilising boundary flux to the diffusive boundary layer of horizontal convection in the region of stabilising boundary flux gives heat transport independent of rotation in this ‘inertial chimney’ regime, and the new scaling $Nu\sim Ra_{F}^{1/4}$. Scaling analysis predicts the transition conditions observed in the experiments, as well as a further ‘geostrophic chimney’ regime in which the vertical plumes are controlled by local geostrophy. When $Ro<10^{-1}$, the convection is also observed to produce a set of large basin-scale gyres at all depths in the time-averaged flow.

The classical point-vortex model for a Kármán vortex street is linearly stable only for a single, isolated, marginally stable, case. This property has been shown numerically to hold for streets formed by symmetric rows of uniform vortices of equal area. That result is extended here to the case in which the areas of the vortices in the two rows are not necessarily equal. The method used is an analytic perturbation valid when the vortex areas are small, and applied using an automatic symbolic manipulator.

The near and very near wake of a flat plate with a circular trailing edge, with vigorous vortex shedding, is investigated with data from direct numerical simulations (DNS). Computations were performed for four different combinations of the Reynolds numbers based on plate thickness ($D$) and momentum thickness near the trailing edge (${\it\theta}$). Unlike the case of the cylinder, these Reynolds numbers are independent parameters for the flat plate. The objectives of the study are twofold, to investigate the entrainment process when the separating boundary layers are turbulent and to better understand the instability of the detached shear layers (DSLs). A visualization of the entrainment process, the effect of changing the ratio ${\it\theta}/D$ on entrainment and wake-velocity statistics, and a way of understanding entrainment in a phase-averaged sense via distributions of the turbulent transport rate are provided here. The discussion on shear-layer instability focuses on the role of log-layer eddies in the destabilization process, the effect of high-speed streaks in the turbulent boundary layer in the vicinity of the trailing edge on shear-layer vortex generation rates, and a relationship between the prevalence of shear-layer vortex generation and shedding phase that is a result of an interaction between the shedding process and the shear-layer instability mechanism. A power-law relationship between the ratio of shear-layer and shedding frequencies and the Reynolds numbers mentioned above is obtained. A discussion of the relative magnitudes of the exponents is provided. A second power-law relationship between shed-vortex strength and these two Reynolds numbers is also proposed.

We study by a fully nonlinear three-dimensional pseudospectral time-splitting simulation, the feedback control of a layer of fluid heated from below. The initial condition corresponds to a steady large-amplitude preferred convection state obtained at Prandtl number of 7.0 and Rayleigh number of 104, which is about six times the Rayleigh critical value. A robust controller based on the LQG (linear–quadratic–Gaussian) synthesis method is used. Both sensors and actuator are thermal-based, planar, andassumed to be continuously distributed. The simulated results show that large-amplitude steady-state convection rolls can be suppressed by the linear LQG controller action. The Green's function of the controller gives the shape of the control action corresponding to a point measurement. In addition, for Rayleigh numbers below the proportional feedback control stability limit, this controller appeared to be effective in damping out steady-state convection rolls as well. However, in a region very near the proportional control stability limit, proportional control action induces subcritical g-type hexagonal convection, which is obtained here through direct simulations. Note that well above this proportional control limit, the LQG still damps out all convection. The nonlinear plant model is validated by comparing check cases with published results.

We have performed linearized direct numerical simulations (DNS) of flow induced by radial wall injection forced by white-noise Gaussian forcings. We have shown that the frequency spectrum of the flow exhibits low-frequency discrete peaks in the case of a spatial structure of the forcing that is large scale. On the other hand, we observed that the spectrum becomes smooth (with no discrete peaks) if the spatial structure of the forcing is of a smaller extent. We have then tried to analyse these results in the light of global stability analyses. We have first computed the eigenvalue spectrum of the Jacobian and shown that the computed eigenvalues in the frequency range of interest were strongly damped and extremely sensitive to numerical discretization choices if large domains in the axial direction were considered. If shorter domains are used, then the eigenvalues are more robust but still extremely sensitive to the location of the upstream and downstream boundaries. Analysis of the $\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}}\epsilon $-pseudo-spectrum showed that eigenvalues located in a region displaying ‘background’ values of $ \epsilon $ below $10^{-12}$ were extremely sensitive and confirmed that all values in this region of the spectrum were actually quasi-eigenvalues. The eigenvalues are therefore ill-behaved and cannot be invoked to explain the observed discrete frequency selection mechanism. We have then performed a singular value decomposition of the global resolvent matrix to compute the leading optimal gains, optimal forcings and optimal responses, which are robust quantities, insensitive to numerical discretization details. We showed that the frequency response of the flow with the large-scale forcing can accurately be reproduced by an approximation based on the leading optimal gain/forcing/response. Analysis of this approximation showed that it is the projection coefficient of the forcing onto the leading optimal forcing that is responsible for the discrete frequency selection mechanism in the case of the large-scale forcing. From a more physical point of view, such a discrete behaviour stems from the streamwise oscillations of the leading optimal forcings, whose wavelengths vary with frequency, in combination with finite extent forcings (which start or end at locations where the leading optimal forcings are strong). Experimental results in the literature are finally discussed in light of these findings.

The jet flow of a shear-thinning power-law fluid is examined theoretically as it emerges from a channel at moderate Reynolds number. Poiseuille flow conditions are assumed to prevail far upstream from the exit. The problem is solved using the method of matched asymptotic expansions. A similarity solution is obtained in the inner layer near the free surface, with the outer layer extending to the jet centreline. An inner thin viscous sublayer is introduced to smooth out the singularity in viscosity at the free surface, allowing the inner algebraically decaying solutions to be matched smoothly with the solution near the free surface. A Newtonian jet is found to contract more than a shear-thinning jet. While both the inner-layer thickness and the free-surface height are $O(\mathit{Re}^{-1/3})$, and grow with downstream distance, the sublayer thickness is smaller, $O(\mathit{Re}^{-(1+n)/3})$, growing with distance for $n < 0.5$, and decaying for $n > 0.5$. The relaxation downstream distance for the jet is found to grow logarithmically with $\mathit{Re}$.

A direct numerical simulation is performed on the full time-dependent three-dimensional Navier–Stokes equations in a spatially developing plane-channel flow at a Reynolds number of 10 000. Two-dimensional eigenfunctions based on the solution of the Orr–Sommerfeld equation are introduced at the inflow with random noise added to simulate a vibrating ribbon transition experiment. The flow is allowed to choose a natural path to secondary instability, either K-type (after Klebanoff) or H-type (after Herbert), depending on the amplitude of the two-dimensional disturbance. For low-amplitude two-dimensional disturbances (1 % of the centreline velocity), H-type modes are found to dominate, while a doubling of the amplitude (2 % of the centreline velocity) produces a mixed H-type/K-type disturbance field with explosive growth of the secondary modes. In addition, the use of a suction/blowing slot that is phase lagged with respect to a fixed wall pressure signal is demonstrated to significantly reduce the energy in the primary mode owing to the destruction of phase between the streamwise and wall-normal velocity components. The use of forward finite-time Lyapunov exponents to generate Lagrangian coherent structures as a means of flow visualization is also presented, showing qualitative agreement with previous experimental visualizations, and represents a viable means of identifying characteristic vortical flow structures.

A numerical investigation of the velocity, pressure and vorticity fields very near the injection of flat and thin two-dimensional gas jets or liquid sheets between two parallel high-speed gas coflows is performed. The motivation of this research is to uncover some basic physical mechanisms underlying twin-fluid atomization. Conservation equations and boundary and initial conditions are presented for both single-phase jets and two-phase liquid sheet/gas-stream systems. Both infinitely thin and thick solid walls are considered. Apart from the gas Strouhal and Reynolds numbers appearing in the dimensionless single-phase flow equations, the liquid Reynolds number, the momentum flux ratio, the gas/liquid velocity ratio and the Weber number enter the two-phase flow dimensionless formulation. The classical numerical techniques for single-phase jets are supplemented with the volume-of-fluid (VOF) method for interface tracking and the continuum surface force (CSF) method to include surface tension in two-phase flow systems. Ad hoc convection algorithms in combination with a developed version of the fractional-step scheme allows a significant reduction of the numerical diffusion, maintaining localized and sharp interfaces. The action of the surface tension is correctly found via the CSF with a smoothed scalar-field approximation.

Results for single-phase jets with thin-wall injectors indicate qualitatively correct features and trends when varying the Reynolds number and the coflow/jet ratios: thick-wall injectors significantly modify the vorticity and pressure near fields; increasing the Reynolds number leads to larger flow disturbances; larger coflow/jet velocity ratios yield more perturbed near flow fields. For single-phase jets the Strouhal number as a function of the Reynolds number follows the usual trends of flows behind a circular cylinder.

For two-phase flows, increasing the gas Reynolds number leads to larger liquid-sheet deformations and to a reduction of the breakup length; a plot of the gas Strouhal number, in the presence of a liquid sheet, as a function of the gas Reynolds number displays a monotonically decreasing curve, contrary to that for a gas jet. This observation strongly suggests that the gas vortex shedding mechanism is modified by the liquid-sheet motion. The gas vortex shedding frequency as a function of the liquid-sheet oscillation frequency follows a straight line with a slope of approximately $45^{\circ}$ for momentum flux ratios greater than roughly 0.45; for values below 0.45 the gas vortex shedding frequency remains constant while the liquid sheet varies its oscillation frequency. Increasing the surface tension leads to a larger breakup length. Thin trailing edges almost double the sheet oscillation frequency and more than halve the perturbation wavelength compared to thick trailing edges.

This paper follows a recent article of Nambiar et al. (J. Fluid Mech., vol. 812, 2017, pp. 41–64) on the linear rheological response of a dilute bacterial suspension (e.g. E. coli) to impulsive starting and stopping of simple shear flow. Here, we analyse the time dependent nonlinear rheology for a pair of linear flows – simple shear (a canonical weak flow) and uniaxial extension (a canonical strong flow), again in response to impulsive initiation and cessation. The rheology is governed by the bacterium orientation distribution which satisfies a kinetic equation that includes rotation by the imposed flow, and relaxation to isotropy via rotary diffusion and tumbling. The relevant dimensionless parameters are the Péclet number $Pe\equiv \dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D70F}$, which dictates the importance of flow-induced orientation anisotropy, and $\unicode[STIX]{x1D70F}D_{r}$, which quantifies the relative importance of the two intrinsic orientation decorrelation mechanisms (tumbling and rotary diffusion). Here, $\unicode[STIX]{x1D70F}$ is the mean run duration of a bacterium that exhibits a run-and-tumble dynamics, $D_{r}$ is the intrinsic rotary diffusivity of the bacterium and $\dot{\unicode[STIX]{x1D6FE}}$ is the characteristic magnitude of the imposed velocity gradient. The solution of the kinetic equation is obtained numerically using a spectral Galerkin method, that yields the rheological properties (the shear viscosity, the first and second normal stress differences for simple shear, and the extensional viscosity for uniaxial extension) over the entire range of $Pe$. For simple shear, we find that the stress relaxation predicted by our analysis at small $Pe$ is in good agreement with the experimental observations of Lopez et al. (Phys. Rev. Lett., vol. 115, 2015, 028301). However, the analysis at large $Pe$ yields relaxations that are qualitatively different. Upon step initiation of shear, the rheological response in the experiments corresponds to a transition from a nearly isotropic suspension of active swimmers at small $Pe$, to an apparently (nearly) isotropic suspension of passive rods at large $Pe$. In contrast, the computations yield the expected transition to a nearly flow-aligned suspension of passive rigid rods at high $Pe$. We probe this active–passive transition systematically, complementing the numerical solution with analytical solutions obtained from perturbation expansions about appropriate base states. Our study suggests courses for future experimental and analytical studies that will help understand relaxation phenomena in active suspensions.