The general equivalence principle of Tacina & Dahm (2000) (Part 1) that extends scaling laws for non-reacting flows to account for density changes due to reaction heat release is applied to turbulent mixing layers to develop physically based scaling laws for heat release effects in exothermic reacting mixing layers. This leads to an ‘extended density ratio’ $s^+$ based on the equivalent elevated temperature for one of the two free-stream fluids that accounts for the density variations within the layer due to exothermic reaction. When used in place of the isothermal density ratio $s$ in scaling laws for growth rate and entrainment ratio in non-reacting mixing layers, resulting predicted effects of heat release show good agreement with measured values, and reveal subtle effects of stoichiometry previously unnoticed in experiments. Results also suggest ways to achieve increased growth rates and entrainment ratios due to heat release in turbulent mixing layers. These results for heat release effects in mixing layers, and earlier results for heat release effects in the near and far fields of planar and axisymmetric jets, support the validity and utility of the equivalence principle between exothermic reacting turbulent shear flows and a corresponding equivalent non-reacting flow under otherwise identical conditions.

Viscous heating can play an important role in the dynamics of fluids with a strongly temperature-dependent viscosity because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local increase in temperature near the tube walls with a consequent decrease of the viscosity and a strong stratification in the viscosity profile which can trigger instabilities and a transition to secondary flows.

In this paper we present two separate theoretical models: a linear stability analysis and a direct numerical simulation (DNS) of a plane channel flow. In particular DNS shows that, in certain regimes, viscous heating can trigger and sustain a particular class of secondary rotational flows which appear organized into coherent structures similar to roller vortices. This phenomenon can play a very important role in the dynamics of magma flows and, to our knowledge, it is the first time that it has been investigated by a direct numerical simulation.

The variability of the physical thickness of fully developed turbulent interfaces is examined using scalar measurements in the outer far-field regions of round jets at a Reynolds number of $\Re\,{\sim}\,20\,000$ and Schmidt number of $\Sc\,{\sim}\,2000$. The interfacial thickness is considered in terms of the inverse magnitude of the scalar gradient across the interface and its relation to the scalar dissipation rate. The thickness variations and their conditional statistics are examined on outer interfaces at a resolution of ${\sim}\,1000^3$ with data that capture the full transverse extent of the flow. At the resolution of the present measurements, the interfaces are observed to exhibit highly intermittent thickness variations that consist of striation patterns, or undulations, along the interfacial surfaces. The conditional probability density of the interfacial thickness is found to be nearly lognormal, in agreement with previous studies. A new scale-local density measure of the interfacial thickness is formulated to examine the effects of coarse graining and the dependence of the thickness on resolution scale. The scale-local thickness density, conditionally averaged on the outer interfaces, is found to exhibit self-similarity in a range of resolved scales. This observation of self-similar behaviour, in conjunction with intermittency, provides a physical ingredient useful for studies of phenomena sensitive to turbulent interfaces.

Circulation driven by horizontal differential heating is studied, using a double-walled Plexiglas tank $(20 \,{\times}\, 15\,{\times}\, 2.5$cm$^3$) filled with salt water. For instances of heating/cooling from above and below, results indicate that there is always quasi-equilibrium circulation. In contrast to most previous results from experimental/numerical studies, circulation in our experiments appears in the form of a shallow cell adjacent to the boundary of thermal forcing. The non-dimensional stream-function maximum confirms the 1/5-power law of Rossby, $\Psi\,{\sim}\,Ra^{1/5}_L$. Dissipation rate measured in the experiments appears to be consistent with theory.

For cases of heating/cooling from a sloping bottom, circulation is similar to cases with a flat bottom; circulation is strong if heating is below cooling, but it is rather weak if heating is above cooling. Nevertheless, circulation in all cases is visible to the naked eye.

The unstable structure of the near wake of a vertical cylinder, in a fully developed laminar free-surface layer, is characterized in relation to the unsteadiness of the horseshoe (necklace) vortex system about the upstream surface of the cylinder. A cinema technique of high-image-density particle image velocimetry allows space–time imaging of the critical regions of the flow and thereby wholefield representations of patterns of the flow structure, in conjunction with spectra and cross-spectra at a large number of points over the flow domain.

The unsteadiness of the near wake was examined over a range of wake stability parameter $S \,{=}\, c_{f}{D/h}_{w}$, in which $c_{f}$ is the bed friction coefficient, $D$ is the cylinder diameter, and $h_{w}$ is water depth; this range of $S$ was selected such that the classical Káarmán mode of vortex formation remained completely suppressed. Within this range, increase of the Reynolds number, based on depth $h_{w}$ of the shallow layer and $D$ of the cylinder, yielded the onset and development of an instability mode that takes the form of a varicose, as opposed to a sinuous, pattern of vortices. It is related to the unsteadiness of the horseshoe (necklace) vortex system on the upstream side of the cylinder. The process of vortex formation in the near wake is interpreted in terms of multiple, coexisting layers of vorticity due to both the horseshoe vortices and the vorticity layer associated with separation from the cylinder.

Furthermore, it is demonstrated that when the near wake is stable at sufficiently low values of the Reynolds number, based on depth $h_{w}$ and cylinder diameter $D$, application of external perturbations via small-amplitude rotational oscillations of the cylinder, at the most unstable frequency of the separating shear layers, can lead to destabilization of the near wake in a sinuous mode of small-scale vortical structures. Moreover, this type of rotational perturbation of the cylinder, applied at the expected frequency of large-scale Kármán vortex formation, can also yield destabilization of the near wake in this mode. These types of perturbations lead to substantial alterations of the patterns of vorticity and streamline topology, as well as Reynolds stresses and entrainment velocities of the separating shear layers, along the bed, relative to patterns above the bed.

Polymer stretching in random smooth flows is investigated within the framework of the FENE dumbbell model. The advecting flow is Gaussian and short-correlated in time. The stationary probability density function of polymer extension is derived exactly. The characteristic time needed for the system to attain the stationary regime is computed as a function of the Weissenberg number and the maximum length of polymers. The transient relaxation to the stationary regime is predicted to be exceptionally slow in the proximity of the coil–stretch transition.

The cross-stream migration of a single neutrally buoyant rigid sphere in tube flow is simulated by two packages, one (ALE) based on a moving and adaptive grid and another (DLM) using distributed Lagrange multipliers on a fixed grid. The two packages give results in good agreement with each other and with experiments. A lift law $L \,{=}\, CU_s (\Omega_s-\Omega_{\hbox{\scriptsize{\it se}}})$ analogous to $L \,{=}\, \rho U\Gamma$ which was proposed and validated in two dimensions is validated in three dimensions here; $C$ is a constant depending on material and geometric parameters, $U_s$ is the slip velocity and it is positive, $\Omega_s$ is the slip angular velocity and $\Omega_{\hbox{\scriptsize{\it se}}}$ is the slip angular velocity when the sphere is in equilibrium at the Segré–Silberberg radius. The slip angular velocity discrepancy $\Omega_s-\Omega_{\hbox{\scriptsize{\it se}}acute;$ is the circulation for the free particle and it changes sign with the lift. A method of constrained simulation is used to generate data which is processed for correlation formulas for the lift force, slip velocity, and equilibrium position. Our formulae predict the change of sign of the lift force which is necessary in the Segré–Silberberg effect. Our correlation formula is compared with analytical lift formulae in the literature and with the results of two-dimensional simulations. Our work establishes a general procedure for obtaining correlation formulae from numerical experiments. This procedure forms a link between numerical simulation and engineering practice.

A fundamental relation is derived governing the Lagrangian kinematics of fluid particles on the surface of nonlinear ocean waves which may be known only stochastically. The horizontal trajectories of fluid particles on the free surface are shown to obey a pair of coupled nonlinear Ricatti-type ordinary differential equations driven by the temporal and spatial gradients of the free-surface elevation defined relative to an Eulerian frame. This equation is explicit in that it does not require the solution of a fully nonlinear potential flow free-surface problem and may be viewed as a deterministic or stochastic equation depending on the interpretation of the definition of the free-surface elevation. It is free of empirical corrections often used to estimate the particle kinematics above the calm water surface, is valid in potential flow and for waves of large steepness in two and three dimensions and in waters of all depths and may be used for the evaluation of the extreme unsteady loads exerted on surface piercing vertical circular cylinders by steep random waves.

The rotational and translational motions of fibres in fully developed isotropic turbulence are simulated for a range of turbulence Reynolds numbers. Equations for fibre motion based on the leading-order slender-body theory relate the fibre's translational and rotational velocities to zeroth and first moments of the fluid velocity along the fibre length. The translational and rotational motions of fibres with lengths that exceed the size of the smallest eddies are attenuated by the filtering associated with these spatial averages. The translational diffusivity of the fibres can be predicted using a simple theory that neglects any coupling between fibre orientation and the local direction of the fluid velocity. However, the coupling of fibre orientation with the axes of extension and rotation is found to greatly reduce the amplitude of the rotary motions and the rotational dispersion coefficient. The rotary dispersion coefficient is found to be on the order of the inverse integral time scale. However, its variation with Reynolds number suggests that the rotary dispersion is influenced by all the scales of turbulence over the limited range of Reynolds numbers explored in our simulations.

In this paper the problem of the jet deflector of optimum shape has been solved. The deflector divides a jet that effuses from a semi-infinite channel of finite width. The goal of the investigation is to define the shape of the deflector that provides either its minimum wetted arclength under the given deflection angle or (which is equivalent) the deflection of the jet through the maximum angle under the given arclength of the deflector. An exact analytical solution of the problem has been found and it has been shown that the solution realizes a global extreme. A series of optimum deflectors is constructed for a variety of deflection angles and contraction jet coefficients.

In this paper we consider the diffraction of waves by a sharp edge in three-dimensional flow with non-zero mean vorticity. This is an extension of the famous Sommerfeld problem of the diffraction of waves by a sharp edge in quiescent conditions. The precise problem concerns an infinitely long annular circular cylinder, which contains a concentric semi-infinite circular cylinder which acts as a splitter. The mean flow has both axial and swirl components, and cases in which the splitter is arranged with either a leading edge or a trailing edge relative to the axial flow are considered. This is a model of a number of practical situations in the aeroengine context. We treat both sonic and nearly-convected incident disturbances, and two regimes are considered; one in which the azimuthal order, $m$, of the incident waves is $O(1)$, and a second in which $m\,{\gg}\,1$. A solution for $m\,{=}\,O(1)$ in the case of rigid-body swirl is found using the Wiener–Hopf technique, and special care is needed to handle the infinite accumulation of scattered nearly-convected modes which results from the presence of the mean vorticity. Simplification in the limit $m\,{\gg}\,1$ then allows us to consider more general swirl distributions. A number of effects arise due to the presence of mean vorticity. This includes the generation of sound at a trailing edge due to the scattering of a nearly-convected disturbance, which is to be contrasted with the way in which a convected gust passes a trailing edge silently in uniform mean flow.

A study was performed using direct numerical simulation to examine the interaction of external turbulence with a nominally columnar, large-scale vortex at a vortex Reynolds number $\hbox{\it Re}_V \,{\equiv}\, \Gamma / \nu \,{=}\, 3000$. A multi-step procedure is used to generate initial conditions in which the external turbulence has the wrapped, nearly azimuthal form characteristic of turbulence around a large-scale vortex structure. The proper-orthogonal decomposition method is used to extract specific modes of the vortex turbulence that dominate the kinetic energy and enstrophy fields. The effect of turbulence initial intensity and length scale on the turbulence structure and its influence on the large-scale vortex are examined. It is observed that the external turbulence wraps around the large-scale vortex and advects radially inward toward the vortex core. The dominant axial length scale of the external turbulence appears to scale with the vortex core diameter, with the mode with the largest enstrophy having a wavelength of about twice the core diameter. The turbulence induces a bending wave on the vortex core with axial wavelength approximately equal to the dominant wavelength of the external turbulence. The turbulent enstrophy decays according to a power-law expression for cases with moderate initial turbulence intensity. For sufficiently strong initial turbulence intensity, the turbulence breaks up the large-scale vortex core, creating strong turbulence within the vortex core.

In this work the electro-osmotic flow in a rectangular channel such that the channel height is comparable to its width is examined. Almost all previous work on the electro-osmotic flow in a channel has been for the case where the channel width is much greater than the channel height and the flow is essentially one-dimensional and depends only on channel height. We consider a mixture of water or another neutral solvent and a salt compound such as sodium chloride for which the ionic species are entirely dissociated. Results are produced for the case where the channel height is much greater than the electric double layer (EDL) (microchannel) and for the case where the channel height is of the order of the width of the EDL (nanochannel). Both symmetric and asymmetric velocity, potential and mole fraction distributions are considered, unlike previous work on this problem. In the symmetric case where all quantities are symmetric about the centreline, the velocity field and the potential are identical as in the parallel-plate one-dimensional case. In the asymmetric case corresponding to different wall potentials, the velocity and potential can be vastly different and reversed flow can occur. The results indicate that the Debye layer thickness is not a good measure of the actual width of the electric double layer. The binary results are shown to compare well with experiment and asymptotic solutions are also obtained for the case of a three-component mixture which may be applied to biomolecular transport.

We report on the results of a combined experimental and numerical study on mode interactions of rotating waves in Taylor–Couette flow. Our work shows that rotating waves which originate at a Hopf bifurcation from the steady axisymmetric Taylor vortex flow interact with this axisymmetric flow in a codimension-two fold-Hopf bifurcation. This interaction gives rise to an (unstable) low-frequency modulated wave via a subcritical Neimark–Sacker bifurcation from the rotating wave. At higher Reynolds numbers, a complicated mode interation between stable modulated waves originating at a different Neimark–Sacker bifurcation and a pair of symmetrically related rotating waves that originate at a cyclic pitchfork bifurcation is found to organize complex $Z_2$-symmetry breaking of rotating waves via global bifurcations. In addition to symmetry breaking of rotating waves via a (local) cyclic pitchfork bifurcation, we found symmetry breaking of modulated waves via a saddle-node-infinite-period (SNIP) global bifurcation. Tracing these local and global bifurcation curves in Reynolds number/aspect ratio parameter space toward their apparant merging point, unexpected complexity arises in the bifurcation structure involving non-symmetric two-tori undergoing saddle-loop homoclinic bifurcations. The close agreement between the numerics and the experiment is indicative of the robustness of the observed complex dynamics.

Receptivity of compressible mixing layers to general source distributions is examined by a combined theoretical/computational approach. The properties of solutions to the adjoint Navier–Stokes equations are exploited to derive expressions for receptivity in terms of the local value of the adjoint solution. The result is a description of receptivity for arbitrary small-amplitude mass, momentum, and heat sources in the vicinity of a mixing-layer flow, including the edge-scattering effects due to the presence of a splitter plate of finite width. The adjoint solutions are examined in detail for a Mach 1.2 mixing-layer flow. The near field of the adjoint solution reveals regions of relatively high receptivity to direct forcing within the mixing layer, with receptivity to nearby acoustic sources depending on the source type and position. Receptivity ‘nodes’ are present at certain locations near the splitter plate edge where the flow is not sensitive to forcing. The presence of the nodes is explained by interpretation of the adjoint solution as the superposition of incident and scattered fields. The adjoint solution within the boundary layer upstream of the splitter-plate trailing edge reveals a mechanism for transfer of energy from boundary-layer stability modes to Kelvin–Helmholtz modes. Extension of the adjoint solution to the far field using a Kirchhoff surface gives the receptivity of the mixing layer to incident sound from distant sources.

Two-dimensional (plane) solitary waves on the surface of water are known to bifurcate from linear sinusoidal wavetrains at specific wavenumbers $k\,{=}\,k_{0}$ where the phase speed $c(k)$ attains an extremum $(\left. \hbox{d}c/\hbox{d}k \right |_{0}\,{=}\,0)$ and equals the group speed. In particular, such an extremum occurs in the long-wave limit $k_{0}\,{=}\,0$, furnishing the familiar solitary waves of the Korteweg–de Vries (KdV) type in shallow water. In addition, when surface tension is included and the Bond number $B\,{=}\,T/(\rho gh^2)\,{<}\,1/3$ ($T$ is the coefficient of surface tension, $\rho$ the fluid density, $g$ the gravitational acceleration and $h$ the water depth), $c(k)$ features a minimum at a finite wavenumber from which gravity–capillary solitary waves, in the form of wavepackets governed by the nonlinear Schrödinger (NLS) equation to leading order, bifurcate in water of finite or infinite depth. Here, it is pointed out that an entirely analogous scenario is valid for the bifurcation of three-dimensional solitary waves, commonly referred to as ‘lumps’, that are locally confined in all directions. Apart from the known lump solutions of the Kadomtsev–Petviashvili I equation for $B\,{>}\,1/3$ in shallow water, gravity–capillary lumps, in the form of locally confined wavepackets, are found for $B\,{<}\,1/3$ in water of finite or infinite depth; like their two-dimensional counterparts, they bifurcate at the minimum phase speed and are governed, to leading order, by an elliptic–elliptic Davey–Stewartson equation system in finite depth and an elliptic two-dimensional NLS equation in deep water. In either case, these lumps feature algebraically decaying tails owing to the induced mean flow.

We study vibrational instabilities of the thermocapillary return flow driven by a constant temperature gradient along the free surface of an infinite layer that vibrates in its normal direction with acceleration of amplitude $g_1 $ and frequency $\omega _1 $. The layer is unstable to hydrothermal waves in the absence of vibrations beyond a critical Marangoni number $M$. Modulated gravitational instabilities with $M\,{=}\,0$ are also possible beyond a critical Rayleigh number $R$ based on $g_1 $. We employ two-time-scale high-frequency asymptotics to derive the equations governing the mean field. The influence of vibrations on the hydrothermal waves is found to be characterized by a dimensionless parameter $G$ that is proportional to $R^2.$ The return flow at $G\,{=}\,0$ is also a mean field basic flow and we study its linear instability at different Prandtl numbers $P$. The hydrothermal waves are stabilized with increasing $G$ and reverse their direction of propagation at particular values of $G$ that decrease with increasing $P$. At finite frequencies, a time-periodic base state exists and we study its linear instability by calculating the Floquet exponents. The stability boundaries in the $(R,M)$-plane are found to be composed of two intersecting branches emanating from the points of pure thermocapillary or buoyant instabilities. Three-dimensional modes are always preferred and the region of stability, while anchored at the point of hydrothermal waves corresponding to $R\,{=}\,0$, is found to grow without bound along the $R$-axis with increasing frequencies. Results from the two approaches are shown to be in asymptotic agreement at large frequencies.

We calculate a rigorous dual bound on the long-time-averaged mechanical energy dissipation rate $\varepsilon$ within a channel of an incompressible viscous fluid of constant kinematic viscosity $\nu$, depth $h$ and rotation rate $f$, driven by a constant surface stress ${\bm\tau}\,{=}\,\rho u^2_\star\xvec$, where $u_\star$ is the friction velocity. It is well known that $\varepsilon \,{\leq}\, \varepsilon_{\rm Stokes}\,{=}\,u^4_\star/\nu$, i.e. the dissipation is bounded above by the dissipation associated with the Stokes flow.

Using an approach similar to the variational ‘background method’ (due to Constantin, Doering & Hopf), we generate a rigorous dual bound, subject to the constraints of total power balance and mean horizontal momentum balance, in the inviscid limit $\nu \,{\rightarrow}\, 0$ for fixed values of the friction Rossby number $Ro_\star\,{=}\,u_\star/(fh)\,{=}\sqrt{G}E$, where $G\,{=}\,\tau h^2/(\rho \nu^2)$ is the Grashof number, and $E\,{=}\,\nu/fh^2$ is the Ekman number. By assuming that the horizontal dimensions are much larger than the vertical dimension of the channel, and restricting our attention to particular, analytically tractable, classes of Lagrange multipliers imposing mean horizontal momentum balance analogous to the ones used in Tang, Caulfield & Young (2004), we show that $\varepsilon \,{\leq}\, \varepsilon_{\max}\,{=} u^4_\star/\nu-2.93 u_\star^2 f$, an improved upper bound from the Stokes dissipation, and $\varepsilon \,{\geq}\, \varepsilon_{\min}\,{=} 2.795 u_\star^3/h$, a lower bound which is independent of the kinematic viscosity $\nu$.

In this study we consider the unsteady separated flow of an inviscid fluid (density $\rho_{f}$) around a falling flat plate (thickness $T$, half-chord $L$, width $W$, and density $\rho_{s}$) of small thickness and high aspect ratio ($T \ll L \ll W$). The motion of the plate, which is initially released from rest, is unknown in advance and is determined as part of the solution. The flow solution is assumed two-dimensional and to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. Throughout its motion, the plate continually sheds vorticity from each of its two sharp edges and the unsteady Kutta condition, which states the fluid velocity must be bounded everywhere, is applied at each edge. The coupled equations of motion for the plate and its trailing vortex wake are derived (the unsteady aerodynamic loads on the plate are included) and are shown to depend only on the modified Froude number $\Fr = T\rho_{s}/L\rho_{f}$. Crucially, the unsteady aerodynamic loads are shown to depend on not only the usual acceleration reactions, which lead to the effect known as added mass, but also on novel unsteady vortical loads, which arise due to relative motion between the plate and its wake. Exact expressions for these loads are derived.

An asymptotic solution to the full system of governing equations is developed for small times $t > 0$ and the initial motion of the plate is shown to depend only on the gravitational field strength and the acceleration reaction of the fluid; effects due to the unsteady shedding of vorticity remain of higher order at small times.

At larger times, a desingularized numerical treatment of the full problem is proposed and implemented. Several example solutions are presented for a range of modified Froude numbers $\Fr$ and small initial inclinations $\theta_{0} <\pi/32$. All of the cases considered were found to be unstable to oscillations of growing amplitude. The non-dimensional frequency of the oscillations is shown to scale in direct proportion with the inverse square root of the modified Froude number $1/\sqrt{\Fr}$. Importantly, the novel unsteady vortical loads are shown to dominate the evolution of the plate's trajectory in at least one example. Throughout the study, the possibility of including a general time-dependent external force (in place of gravity) is retained.

This paper applies a diffuse-interface model to simulate the deformation of single drops in steady shear flows when one of the components is viscoelastic, represented by an Oldroyd-B model. In Newtonian fluids, drop deformation is dominated by the competition between interfacial tension and viscous forces due to flow. A fundamental question is how viscoelasticity in the drop or matrix phase influences drop deformation in shear. To answer this question, one has to deal with the dual complexity of non- Newtonian rheology and interfacial dynamics. Recently, we developed a diffuse-inter-face formulation that incorporates complex rheology and interfacial dynamics in a unified framework. Using a two-dimensional spectral implementation, our simulations show that, in agreement with observations, a viscoelastic drop deforms less than a comparable Newtonian drop. When the matrix is viscoelastic, however, the drop deformation is suppressed when the Deborah number $De$ is small, but increases with $De$ for larger $De$. This non-monotonic dependence on matrix viscoelasticity resolves an apparent contradiction in previous experiments. By analysing the flow and stress fields near the interface, we trace the effects to the normal stress in the viscoelastic phase and its modification of the flow field. These results, along with prior experimental observations, form a coherent picture of viscoelastic effects on steady-state drop deformation in shear.