We analyse the stability of a thin film falling under the influence of gravity down a locally heated plate. Marangoni flow, due to local temperature changes influencing the surface tension, opposes the gravitationally driven Poiseuille flow and forms a horizontal band at the upper edge of the heater. The thickness of the band increases with the surface tension gradient, until an instability forms a rivulet structure periodic in the transverse direction. We study the dependence of the critical Marangoni number, a non-dimensional measure of the surface tension gradient at the onset of instability, on the associated Bond and Biot numbers, non-dimensional measures of the curvature pressure and heat-conductive properties of the film respectively. We develop a model based on long-wave theory to calculate base-state solutions and their linear stability. We obtain dispersion relations, which give us the wavelength and growth rate of the fastest growing mode. The calculated film profile and wavelength of the most unstable mode at the instability threshold are in quantitative agreement with the experimental results. We show via an energy analysis of the most unstable linear eigenmode that the instability is driven by gravity and an interaction between base-state curvature and the perturbation thickness. In the case of non-zero Biot number transverse variations of the temperature profile also contribute to destabilization.

This article presents an experimental study of magnetohydrodynamic convection in a tall vertical slot under the influence of a horizontal magnetic field. The test fluid is an eutectic sodium potassium Na22K78 alloy with a small Prandtl number of Pr ≈ 0:02. The experimental setup covers Rayleigh numbers in the range 103 [lsim ] Ra [lsim ] 8×104 and Hartmann numbers 0 < M < 1600. The effect of the magnetic field on the convective heat transport is determined not only by damping as expected from Joule dissipation but also, for magnetic fields not too strong, the convective heat transfer may be considerably enhanced compared to ordinary hydrodynamic (OHD) flow. Estimates of the isotropy properties of the flow by a four-element temperature probe demonstrate that the increase in convective heat transport accompanies the formation of strong local anisotropy of the turbulent eddies in the sense of an alignment of the main direction of vorticity with the magnetic field. The reduced three-dimensional nonlinearities in non-isotropic flow favour the formation of largescale vortex structures compared to OHD flow, which are more effective for convective heat transport. Along with the formation of quasi-two-dimensional vortex structures, temperature fluctuations may be considerably enhanced in a magnetic field that is not too strong. However, above Hartmann numbers M [gsim ] 400 the formerly strongly time-dependent flow suddenly becomes stationary with an extended region of high convective heat transport at stationary flow. Finally, for very high Hartmann numbers the convective motion is strongly suppressed and the heat transport is reduced to a state close to pure heat conduction.

In this paper, we study the interaction of two co-rotating trailing vortices. It is well-known that vortices of like-sign ultimately merge to form a single vortex, and there has been much work on measuring and predicting the initial conditions for the onset of merger, especially concerning the critical vortex core radius. However, the physical mechanism causing this merger has received little attention. In this work, we directly measure the structure of the antisymmetric vorticity field that causes the co-rotating vortices to be pushed towards each other during merger. We discover that the form of the antisymmetric vorticity comprises two counter-rotating vortex pairs, whose induced velocity field readily pushes the two centroids together. The merging velocity computed from the antisymmetric vorticity field agrees closely with the merging velocity measured directly from the total (original) flow field.

The co-rotating vortex pair evolves through four distinct phases. The initial stage comprises a diffusive growth, which can be either viscous or turbulent. In either case, the number of turns that they rotate around one another until the vortices start to merge increases with Reynolds number (Re). If one observes the streamlines in a rotating reference frame (moving with the vortices), then one finds an inner and outer recirculating region of the flow bounded by a separatrix streamline. When the vortices grow large enough in the first stage, diffusion across the separatrix places vorticity into the outer recirculating region of the flow, and this leads to the generation of the antisymmetric vorticity, causing convective merger. This second (convective) stage corresponds to the motion of the vortex centroids towards each other, and is a process which is almost independent of viscosity. During the late part of this stage, the antisymmetric vorticity is diminished by a symmetrization process, and the final merging into one vorticity structure is achieved by a second diffusive stage. The fourth and ultimate phase is one where the merged vortex core grows by diffusion.

Tsunami run-up and draw-down motions on a uniformly sloping beach are evaluated based on fully nonlinear shallow-water wave theory. The nonlinear equations of mass conservation and linear momentum are first transformed to a single linear hyperbolic equation. To solve the problem with arbitrary initial conditions, we apply the Fourier–Bessel transform, and inversion of the transform leads to the Green function representation. The solutions in the physical time and space domains are then obtained by numerical integration. With this semi-analytic solution technique, several examples of tsunami run-up and draw-down motions are presented. In particular, detailed shoreline motion, velocity field, and inundation depth on the shore are closely examined. It was found that the maximum flow velocity occurs at the moving shoreline and the maximum momentum flux occurs in the vicinity of the extreme draw-down location. The direction of both the maximum flow velocity and the maximum momentum flux depend on the initial waveform: it is in the inshore direction when the initial waveform is predominantly depression and in the offshore direction when the initial waves have a dominant elevation characteristic.

The flow over a cavity at a Mach number 0.8 is considered. The cavity is deep with an aspect ratio (length over depth) L/D = 0.42. This deep cavity flow exhibits several features that makes it different from shallower cavities. It is subjected to very regular self-sustained oscillations with a highly two-dimensional and periodic organization of the mixing layer over the cavity. This is revealed by means of a high-speed schlieren technique. Analysis of pressure signals shows that the first tone mode is the strongest, the others being close to harmonics. This departs from shallower cavity flows where the tones are usually predicted well by the standard Rossiter’s model. A two-component laser-Doppler velocimetry system is also used to characterize the phase-averaged properties of the flow. It is shown that the formation of coherent vortices in the region close to the boundary layer separation has some resemblance to the ‘collective interaction mechanism’ introduced by Ho & Huang (1982) to describe mixing layers subjected to strong sub-harmonic forcing. Otherwise, the conditional statistics show close similarities with those found in classical forced mixing layers except for the production of random perturbations, which reaches a maximum in the structure centres, not in the hyperbolic regions with which turbulence production is usually associated. An attempt is made to relate this difference to the elliptic instability that may be observed here thanks to the particularly well-organized nature of the flow.

The Richtmyer–Meshkov instability in an incompressible and compressible stratified two-dimensional ideal flow is studied analytically and numerically. For the incompressible problem, we initialize a single small-amplitude sinusoidal perturbation of wavelength λ, we compute a series expansion for the amplitude a in powers of t up to t(11) with the MuPAD computer algebra environment. This involves harmonics up to eleven. The simulations are performed with two codes: incompressible, a vortex-in- cell numerical technique which tracks a single discontinuous density interface; and compressible, PPM for a shock-accelerated case with a finite interfacial transition layer (ITL). We identify properties of the interface at time t = tM at which it first becomes ‘multivalued’. Here, we find the normalized width of the ‘spike’ is related to the Atwood number by (wm/λ)−0.5 = −0.33A. A high-order Pad approximation is applied to the analytical series during early time and gives excellent results for the interface growth rate a˙. However, at intermediate times, t > tM, the agreement between numerical results and different-order Padé approximants depends on the Atwood number. During this phase, our numerical solutions give a˙∝O(t−1) for small A and a˙∝O(t−0.4) for A = 0.9. Experimental data of Prasad et al. (2000) for SF6 (post shock Atwood number = 0.74) shows an exponent between −0.68 and −0.72 and we obtain −0.683 for the compressible simulation. For this case, we illustrate the important growth of vortex-accelerated (secondary) circulation deposition of both signs of vorticity and the complex nature of the roll-up region.

If a shear flow of a homogeneous fluid preserves the shape of its velocity profile, a standard formula for the condition for hydraulic control suggests that this is achieved when the depth-averaged flow speed is less than (gh)1/2. On the other hand, shallow-water waves have a speed relative to the mean flow of more than (gh)1/2, suggesting that information could propagate upstream. This apparent paradox is resolved by showing that the internal stress required to maintain a constant velocity profile depends on flow derivatives along the channel, thus altering the wave speed without introducing damping. By contrast, an inviscid shear flow does not maintain the same profile shape, but it can be shown that long waves are stationary at a position of hydraulic control.

Statistics of the mixed velocity–passive scalar field and its Reynolds number dependence are studied in quasi-isotropic decaying grid turbulence with an imposed mean temperature gradient. The turbulent Reynolds number (using the Taylor microscale as the length scale), Rλ, is varied over the range 85 [les ] Rλ [les ] 582. The passive scalar under consideration is temperature in air. The turbulence is generated by means of an active grid and the temperature fluctuations result from the action of the turbulence on the mean temperature gradient. The latter is created by differentially heating elements at the entrance to the wind tunnel plenum chamber. The mixed velocity–passive scalar field evolves slowly with Reynolds number. Inertial-range scaling exponents of the co-spectra of transverse velocity and temperature, Evθ(k1), and its real-space analogue, the ‘heat flux structure function,’ 〈Δv(r)Δθ(r)〉, show a slow evolution towards their theoretical predictions of −7/3 and 4/3, respectively. The sixth-order longitudinal mixed structure functions, 〈(Δu(r))2(Δθ(r))4〉, exhibit inertial-range structure function exponents of 1.36–1.52. However, discrepancies still exist with respect to the various methods used to estimate the scaling exponents, the value of the scalar intermittency exponent, μθ, and the effects of large-scale phenomena (namely shear, decay and turbulent production of 〈θ2〉) on 〈(Δu(r))2(Δθ(r))4〉. All the measured fine-scale statistics required to be zero in a locally isotropic flow are, or tend towards, zero in the limit of large Reynolds numbers. The probability density functions (PDFs) of Δv(r)Δθ(r) exhibit roughly exponential tails for large separations and super-exponential tails for small separations, thus displaying the effects of internal intermittency. As the Reynolds number increases, the PDFs become symmetric at the smallest scales – in accordance with local isotropy. The expectation of the transverse velocity fluctuation conditioned on the scalar fluctuation is linear for all Reynolds numbers, with slope equal to the correlation coefficient between v and θ. The expectation of (a surrogate of) the Laplacian of the scalar reveals a Reynolds number dependence when conditioned on the transverse velocity fluctuation (but displays no such dependence when conditioned on the scalar fluctuation). This former Reynolds number dependence is consistent with Taylor’s diffusivity independence hypothesis. Lastly, for the statistics measured, no violations of local isotropy were observed.

In this paper we consider, for modelling and simulation, a non-isothermal turbulent flow laden with non-evaporating spherical particles which exchange heat with the surrounding fluid and do not collide with each other during the course of their journey under the influence of the stochastic fluid drag force. In the modelling part of this study, a closed kinetic or probability density function (p.d.f.) equation is derived which describes the distribution of position x, velocity v, and temperature θ of the particles in the flow domain at time t. The p.d.f. equation represents the transport of the ensemble-average (denoted by 〈 〉) phase-space density 〈W(x, v, θ, t)〉. The process of ensemble averaging generates unknown terms, namely the phase-space diffusion current j = βv〈u′W〉 and the phase-space heat current h = βθ〈t′W〉, which pose closure problems in the kinetic equation. Here, u′ and t′ are the fluctuating parts of the velocity and temperature, respectively, of the fluid in the vicinity of the particle, and βv and βθ are inverse of the time constants for the particle velocity and temperature, respectively. The closure problems are first solved for the case of homogeneous turbulence with uniform mean velocity and temperature for the fluid phase by using Kraichnan’s Lagrangian history direct interaction (LHDI) approximation method and then the method is generalized to the case of inhomogeneous flows. Another method, which is due to Van Kampen, is used to solve the closure problems, resulting in a closed kinetic equation identical to the equation obtained by the LHDI method. Then, the closed equation is shown to be compatible with the transformation constraint that is proposed by extending the concept of random Galilean transformation invariance to non-isothermal flows and is referred to as the ‘extended random Galilean transformation’ (ERGT). The macroscopic equations for the particle phase describing the time evolution of statistical properties related to particle velocity and temperature are derived by taking various moments of the closed kinetic equation. These equations are in the form of transport equations in the Eulerian framework, and are computed for the case of two-phase homogeneous shear turbulent flows with uniform temperature gradients. The predictions are compared with the direct numerical simulation (DNS) data which are generated as another part of this study. The predictions for the particle phase require statistical properties of the fluid phase which are taken from the DNS data. In DNS, the continuity, Navier–Stokes, and energy equations are solved for homogeneous turbulent flows with uniform mean velocity and temperature gradients. For the mean velocity gradient along the x2- (cross-stream) axis, three different cases in which the mean temperature gradient is along the x1-, x2-, and x3-axes, respectively, are simulated. The statistical properties related to the particle phase are obtained by computing the velocity and temperature of a large number of particles along their Lagrangian trajectories and then averaging over these trajectories. The comparisons between the model predictions and DNS results show very encouraging agreement.

We study the effects of multiple scattering of slowly modulated water waves by a weakly random bathymetry. The combined effects of weak nonlinearity, dispersion and random irregularities are treated together to yield a nonlinear Schrödinger equation with a complex damping term. Implications for localization and side-band instability are discussed. Transmission and nonlinear evolution of a wave packet past a finite strip of disorder is examined.

Lighthill’s acoustic analogy is formulated for turbulent channel flow with pressure as the acoustic variable, and integrated over the channel width to produce a two-dimensional inhomogeneous wave equation. The equivalent sources consist of a dipole distribution related to the sum of the viscous shear stresses on the two walls, together with monopole and quadrupole distributions related to the unsteady turbulent dissipation and Reynolds stresses respectively. Using a rigid-boundary Green function, an expression is found for the power spectrum of the far-field pressure radiated per unit channel area. Direct numerical simulations (DNS) of turbulent plane Poiseuille and Couette flow have been performed in large computational domains in order to obtain good resolution of the low-wavenumber source behaviour. Analysis of the DNS databases for all sound radiation sources shows that their wavenumber–frequency spectra have non-zero limits at low wavenumber. The sound power per unit channel area radiated by the dipole distribution is proportional to Mach number squared, while the monopole and quadrupole contributions are proportional to the fourth power of Mach number. Below a particular Mach number determined by the frequency and radiation direction, the dipole radiation due to the wall shear stress dominates the far field. The quadrupole takes over at Mach numbers above about 0.1, while the monopole is always the smallest term. The resultant acoustic field at any point in the channel consists of a statistically diffuse assembly of plane waves, with spectrum limited by damping to a value that is independent of Mach number in the low-M limit.

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.

The three-dimensional particle paths due to a helical beat pattern of the flagellum of a sessile choanoflagellate, Salpingoeca Amphoridium (SA), are modelled and compared to the experimental observations of Pettitt (2001). The organism’s main components are a flagellum and a cell body which are situated above a substrate such that the interaction between these entities is crucial in determining the fluid flow around the choanoflagellate. This flow of fluid can be characterized as Stokes flow and a flow field analogous to one created by the flagellum is generated by a distribution of stokeslets and dipoles along a helical curve.

The model describing the flow considers interactions between a slender flagellum, an infinite flat plane (modelling the substrate) and a sphere (modelling the cell body). The use of image systems appropriate to Green’s functions for a sphere and plane boundary are described following the method of Higdon (1979a). The computations predict particle paths representing passive tracers from experiments and their motion illustrates overall flow patterns. Figures are presented comparing recorded experimental data with numerically generated results for a number of particle paths. The principal results show good qualitative agreement with the main characteristics of flows observed in the experimental study of Pettitt (2001).

Axisymmetric Rayleigh–Bénard convection in a cylinder with vertical axis is studied. The nonlinear behaviour is investigated near the onset of convection using an eigenfunction expansion. It is found that initially a steady target pattern develops; the number of rolls depends only on the aspect ratio of the box. At about 5% beyond onset, an oscillatory pattern develops, in which the number of rolls oscillates between two adjacent values. The transitions between the initial steady state and this oscillatory pattern are also investigated, and fall into two main categories. As the Rayleigh number is reduced to the transition point, either the period of the travelling wave tends to infinity, whilst its amplitude stays finite, or there is a sudden transition to a vascillating pattern, the amplitude of which becomes smaller and finally vanishes, whilst the period remains finite. The results are compared with experimental work.

We consider the motion of a liquid film falling down a heated planar substrate. Using the integral-boundary-layer approximation of the Navier–Stokes/energy equations and free-surface boundary conditions, it is shown that the problem is governed by two coupled nonlinear partial differential equations for the evolution of the local film height and temperature distribution in time and space. Two-dimensional steady-state solutions of these equations are reported for different values of the governing dimensionless groups. Our computations demonstrate that the free surface develops a bump in the region where the wall temperature gradient is positive. We analyse the linear stability of this bump with respect to disturbances in the spanwise direction. We show that the operator of the linearized system has both a discrete and an essential spectrum. The discrete spectrum bifurcates from resonance poles at certain values of the wavenumber for the disturbances in the transverse direction. The essential spectrum is always stable while part of the discrete spectrum becomes unstable for values of the Marangoni number larger than a critical value. Above this critical Marangoni number the growth rate curve as a function of wavenumber has a finite band of unstable modes which increases as the Marangoni number increases.