The vertical temperature structure of homogeneous stratified shear turbulence is investigated using new rapid vertical temperature measurements in a thermally stratified wind tunnel. Six cases of gradient Richardson number, Rig = N2/(dŪ/dz)2, where N is the Brunt–Väisälä frequency (N2 = (g/t¯)dt¯/dz), are studied, spanning a range 0.015 [les ] Rig [les ] 0.5. Three- to five-hundred high-resolution temperature profiles are made for several streamwise stations for each case of Rig. These measurements are supplemented with standard fixed-point, Eulerian measurements of streamwise and vertical velocity fluctuations and temperature fluctuations and with an eight-point vertical rake of temperature probes using standard hot-wire and cold-wire techniques. Vertical profiles uniquely enable the computation of available potential energy (APE), Thorpe scales (LTh), and the diapycnal flux (ϕd), as well as one-dimensional vertical wavenumber temperature spectra. These quantities are compared with Eulerian measurements of turbulent kinetic energy (KE), potential energy (PE), and buoyancy flux. It is found that the one-dimensional vertical wavenumber temperature spectrum contains more energy at smaller scales compared to the horizontal spectrum, owing in part to shear distortion, which leads to larger mean square vertical gradients of fluctuating temperature as compared to mean square horizontal gradients. The combination of shear and stratification, especially for cases where the turbulence decays with evolution, accelerates the evolution toward small-scale anisotropy compared to just shear or just stratification. It is found that in highly stratified cases, the diapycnal flux can persist after buoyancy flux has collapsed to negligible values, indicating enhanced heat transfer without turbulent mixing. For low Rig, large-scale vertical advection creates both high local temperature gradients and regions of static instability. Associated with the regions of instability is APE, which grows relative to KE for the least stratified cases. For high Rig, the turbulence evolves to a wavelike state, containing some counter gradient fluxes and unstable patches. This wavelike state has higher heat flux efficiency than the more turbulent states owing to the low dissipation but relatively high diapycnal flux.

The motion of fluid droplets in capillary tubes subject to the action of a mean pressure gradient and an oscillatory body force is studied via numerical computations. The effects of the oscillatory forcing on the bulk flow rate and on the droplet velocity are evaluated, and results are presented for a range of forcing conditions, fluid properties and drop sizes. For large droplets (whose undeformed diameter exceeds that of the capillary tube), significant enhancement in the bulk flow rate is observed when the drop capillary number is small and the oscillatory forcing is strong. The enhancement is associated with increased droplet deformation in the presence of oscillatory forcing. The dependence of the flow enhancement on the amplitude, frequency and waveform of the oscillatory body force is evaluated for a range of fluid properties.

The motion of fluid droplets in constricted capillary tubes is investigated for flows subject to the combined action of a mean pressure gradient and an oscillatory body force. Numerical computations are employed to determine the effect of the oscillatory forcing on the mean flow rate and the mean droplet velocity. In the absence of oscillatory forcing, a critical pressure gradient for droplet propagation exists, below which droplets become plugged in the narrow constrictions of the tube. For mean pressure gradients below this threshold, oscillatory forcing is shown to be an effective means for unplugging the constrictions and remobilizing the droplets. For this remobilization process to occur, the oscillatory forcing level must exceed a specified value, and the oscillatory frequency must remain below a critical frequency. Quasi-steady models are shown to give effective predictions of the unsteady dynamics over a wide range of conditions.

The steady two-dimensional laminar flow of an air stream, flowing past a solid surface at high Reynolds number, is examined in the presence of rainfall. As raindrops sediment on the surface they coalesce and form a continuous water film that flows due to shear, pressure drop and gravity, in general. In the limit as the boundary layer and film thickness remain smaller than the radius of curvature of the surface a simplified lubrication-type formulation describes the flow field in the film, whereas the usual boundary layer formulation is applied in the gas phase. In the case of a flat plate and close to the leading edge, x → 0, a piecewise-self-similar solution is obtained, according to which creeping flow conditions prevail in the film and its thickness grows like x3/4, whereas the Blasius solution is recovered in the air stream. Numerical solution of the governing equations in the two phases and for the entire range of distances from the leading edge, x = O(1), shows that the film thickness increases as the rainfall rate, r˙, increases or as the free-stream velocity, U∞, decreases and that the region of validity of the asymptotic result covers a wide range of the relevant problem parameters. In the case of flow past a NACA-0008 airfoil at zero angle of attack a Goldstein singularity may appear far downstream on the airfoil surface due to adverse pressure gradients, indicating flow reversal and eddy formation inside the liquid film, and, possibly, flow separation. However, when the effect of gravity becomes evident in the film flow, as the Froude number decreases, and provided gravity acts in such a way as to negate the effect of the adverse pressure gradient, the location of the singularity is displaced towards the trailing edge of the airfoil and the flow pattern resembles that for flow past a flat plate. The opposite happens when gravity is aligned with the adverse pressure gradient. In addition it was found that there exists a critical water film thickness beyond which the film has a lubricating effect delaying the appearance of the singularity. Below this threshold the presence of the liquid film actually enhances the formation of the singularity.

The compression wave generated by a high-speed train entering a tunnel is studied theoretically and experimentally. It is shown that the pressure rise across the wavefront is given approximately by

formula here

where ρo, U, M, [Ascr ]o and [Ascr ] respectively denote the mean air density, train speed, train Mach number, and the cross-sectional areas of the train and the uniform section of the tunnel. A monopole source representing the displacement of air by the train is responsible for the main pressure rise across the wave, but second-order dipole sources must also be invoked to render theoretical predictions compatible with experiment. The principal dipole is produced by the compression wave drag acting on the nose of the train. A second dipole of comparable strength, but probably less significant in practice, is attributed to ‘vortex sound’ sources in the shear layers of the back-flow out of the tunnel of the air displaced by the train.

Experiments are performed that confirm the efficacy of an ‘optimally flared’ portal whose cross-sectional area S(x) varies according to the formula

formula here

where x is distance increasing negatively into the tunnel, [lscr ] is the prescribed length of the flared section, and [Ascr ]E is the tunnel entrance cross-sectional area, given by

formula here

This portal is predicted theoretically to cause the pressure to increase linearly with distance across a compression wavefront of thickness ∼ [lscr ]/M, which is very much larger than in the absence of flaring. The increased wave thickness and linear pressure variation counteract the effect of nonlinear steepening of the wave in a long tunnel, and tend to suppress the environmentally harmful ‘micro-pressure wave’ radiated from the far end of the tunnel when the compression wave arrives. Our experiments are conducted at model scale using axisymmetric ‘trains’ projected at U ∼ 300 k.p.h. (M ≈ 0.25) along the axis of a cylindrical tunnel fitted with a flared portal. The blockage [Ascr ]o/[Ascr ] = 0.2, which is comparable to the larger values encountered in high-speed rail operations.

We compute numerically the amplitude of long thin fingers that form in a liquid stratified with sugar S* and salt T* (measured in buoyancy units), for which τ = kS/kT = 1/3 is the ratio of the two diffusivities and the Prandtl number is Pr = v/kT ∼ 103, where v is the viscosity. The finger layer in our model is bounded by rigid and slippery horizontal surfaces with constant T*, S* (the setup is similar to the classical Rayleigh convection problem). The numerically computed steady fluxes compare well with laboratory experiments in which the fingers are sandwiched between two deep (convectively mixed) reservoirs with given concentration differences ΔT*, ΔS*. The model results, discussed in terms of a combination of asymptotic analysis and numerical simulations over a range of density ratio R = ΔT*/ΔS*, are consistent with the (ΔS*)4/3 similarity law for the fluxes. The dimensional interfacial height (H*) in the reservoir experiments (unlike that in our rigid lid model) is not an independent parameter, but it adjusts to a statistically steady value proportional to (ΔS*)−1/3. This similarity law is also explained by our model when it is supplemented by a consideration of the stability of the very thin horizontal boundary layers with large gradients (∂S*/∂z) which form near the rigid surfaces. The preference for three-dimensional salt fingers is also explained by a combination of analytical and numerical considerations.

The stirring and mixing properties of one-phase coaxial jets, with large outer (annular) to inner velocity ratio ru = u2/u1 are investigated. Mixing is contemplated according to its geometrical, statistical and spectral facets with particular attention paid to determining the relevant timescales of the evolution of, for example, the interface area generation between the streams, the emergence of its scale-dependent (fractal) properties and of the mixture composition after the mixing transition. The two key quantities are the vorticity thickness of the outer, fast stream velocity profile which determines the primary shear instability wavelength and the initial size of the lamellar structures peeled-off from the slow jet, and the elongation rate γ = (u2 − u1)/e constructed with the velocity difference between the streams and the gap thickness e of the annular jet. The kinetics of evolution of the interface corrugations, and the rate at which the mixture evolves from the initial segregation towards uniformity is prescribed by γ−1. The mixing time ts, that is the time needed to bring the initial scalar lamellae down to a transverse size where molecular diffusion becomes effective, and the corresponding dissipation scale s(ts) are

formula here

where Re and Sc denote the gap Reynolds number and the Schmidt number, respectively. The persistence of the large-scale straining motion is also apparent from the spectra of the scalar fluctuations which exhibit a k−1 shape on the inertial range of scales.

A new mechanism of internal wave breaking in the subsurface ocean layer is considered. The breaking is due to the ‘resonant’ interaction of shoaling long internal gravity waves with the subsurface shear current occurring in a resonance zone. Provided the wind-induced shear current is oriented onshore, there exists a wide resonance zone, where internal wave celerity is close to the current velocity at the water surface and a particularly strong resonant interaction of shoaling internal waves with the current takes place. A model to describe the coupled dynamics of the current perturbations treated as ‘vorticity waves’ and internal waves propagating over a sloping bottom is derived by asymptotic methods. The model generalizes the earlier one by Voronovich, Pelinovsky & Shrira (1998) by taking into account the mild bottom slope typical of the oceanic shelf. The focus of the work is upon the effects on wave evolution due to the presence of the bottom slope. If the bottom is flat, the model admits a set of stationary solutions, both periodic and of solitary wave type, their amplitude being limited from above. The limiting waves are sharp crested. Space–time evolution of the waves propagating over a sloping bottom is studied both by the adiabatic Whitham method for comparatively mild slopes and numerically for an arbitrary one. The principal result is that all onshore propagating waves, however small their initial amplitudes are, inevitably reach the limiting amplitude within the resonance zone and break. From the mathematical viewpoint the unique peculiarity of the problem lies in the fact that the wave evolution remains weakly nonlinear up to breaking. To address the situations when the subsurface current becomes strongly turbulent due to particularly intense wind-wave breaking, the effect of turbulent viscosity on the wave evolution is also investigated. The damping due to the turbulence results in a threshold in the initial amplitudes of perturbations: the ‘subcritical’ perturbations are damped, the ‘supercritical’ ones inevitably break. As the breaking events occur mainly in the subsurface layer, they may contribute significantly to the mixing and exchange processes at the air/sea interface and in creating significant surface signatures.

In this study we have theoretically investigated the effect of parallel superposition of modulation on the stability of single-layer Newtonian and viscoelastic flows down an inclined plane. Specifically, a spectrally based numerical technique in conjunction with Floquet theory has been used to investigate the linear stability of this class of flows. Based on these analyses we have demonstrated that parallel superposition of modulation can be used to stabilize or destabilize flow of Newtonian and viscoelastic fluids down an inclined plane. In general at low Reynolds number Re (i.e. O(1)) and in the limit of long and O(1) waves the effect of dynamic modulation on the stability of viscoelastic flows is much more pronounced; however, relatively large modulation amplitudes are required to achieve significant stabilization/destabilization. In addition, the dependence of the most dominant modulation frequencies on Re and the Weissenberg number We have been identified. Specifically, it has been shown that for Newtonian flows low-frequency modulations are destabilizing and the most dominant modulation frequency scales with 1/Re. On the other hand, for viscoelastic flows in the absence of fluid inertia low-frequency modulations are stabilizing and the most dominant modulation frequency scales with 1/We. In finite-Re viscoelastic flows the most dominant destabilizing modulation frequency scales with 1/Re while the most stabilizing modulation frequency scales with 1/WeRe. Finally, it has been demonstrated that the mechanism of both purely elastic and inertial instabilities in flows down an inclined plane is unchanged in the presence of dynamic modulation.

Motivated by the non-Newtonian properties of mucus and the bilayer nature of fluid lining in the pulmonary airways, we investigate surfactant transport on both single and bilayer fluid systems; the aim is twofold. First, we explore the influence of two principle rheological properties of mucus, yield stress and shear thinning, on the surfactant spreading behaviour. Secondly, in these airways, mucus, which has substantial non-Newtonian properties, overlies the periciliary liquid layer (PCL) which is primarily Newtonian, and we incorporate this bilayer structure into the analysis. This consists of the derivation of coupled spatio-temporal evolution equations describing the layer thicknesses and surfactant concentration. Subsequent analytical methods examine limiting cases where similarity variables can be usefully employed, and more generally numerical simulations are performed.

This paper studies the nonlinear development of two-dimensional Tollmien–Schlichting waves in an incompressible flat-plate boundary layer at asymptotically large values of the Reynolds number. Attention is restricted to the ‘far-downstream lower-branch’ régime where a multiple-scales analysis is possible. It is supposed that to leading-order the waves are inviscid and neutral, and governed by the [Davis–Acrivos–]Benjamin–Ono equation. This has a three-parameter family of periodic solutions, the large-amplitude (soliton) limit of which bears a qualitative resemblance to the ‘spikes’ observed in certain ‘K-type’ transition experiments. The variation of the parameters over slow length- and timescales is controlled by a viscous sublayer. For the case of a purely temporal evolution, it is shown that a solution for this sublayer ceases to exist when the amplitude reaches a certain finite value. For a purely spatial evolution, it appears that an initially linear disturbance does not evolve to a fully nonlinear stage of the envisaged form. The implications of these results for the ‘soliton’ theory of spike formation are discussed.

The collision of a strong vortex with a surface is an important problem because significant impulsive loads may be generated. Prediction of helicopter fatigue lifetime may be limited by an inability to predict these loads accurately. Experimental results for the impingement of a helicopter rotor-tip vortex on a cylindrical airframe show a suction peak on the top of the airframe that strengthens and then weakens within milliseconds. A simple line-vortex model can predict the experimental results if the vortex is at least two vortex-core radii away from the airframe. After this, the model predicts continually deepening rather than lessening suction as the vortex stretches. Experimental results suggest that axial flow within the core of a tip vortex has an impact on the airframe pressure distribution upon close approach. The mechanism for this is hypothesized to be the inviscid redistribution of the vorticity field within the vortex as the axial velocity stagnates. Two models of a tip vortex with axial flow are considered. First, a classical axisymmetric line vortex with a cutoff parameter is superimposed with vortex ringlets suitably placed to represent the helically wound vortex shed by the rotor tip. Thus, inclusion of axial flow is found to advect vortex core thinning away from the point of closest interaction as the vortex stretches around the cylindrical surface during the collision process. With less local thinning, vorticity in the cutoff parameter model significantly overlaps the solid cylinder in an unphysical manner, highlighting the fact that the vortex core must deform from its original cylindrical shape. A second model is then developed in which axial and azimuthal vorticity are confined within a rectangular-section vortex. Area and aspect ratio of this vortex can be varied independently to simulate deformation of the vortex core. Both axial velocity and core deformation are shown to be important to calculate the local induced pressure loads properly. The computational results are compared with experiments conducted at the Georgia Institute of Technology.

Theoretical and computational aspects of the self-induced motion of closed and periodic three-dimensional vortex sheets situated at the interfaces between two inviscid uids with generally different densities in the presence of surface tension are considered. In the mathematical formulation, the vortex sheet is described by a continuous distribution of marker points that move with the velocity of the fluid normal to the vortex sheet while executing an arbitrary tangential motion. Evolution equations for the vectorial jump in the velocity across the vortex sheet, the vectorial strength of the vortex sheet, and the scalar circulation field or strength of the effective dipole field following the marker points are derived. The computation of the self-induced motion of the vortex sheet requires the accurate evaluation of the strongly singular Biot-Savart integral whose existence requires that the normal vector varies in a continuous fashion over the vortex sheet. Two methods of computing the principal value of the Biot-Savart integral are implemented. The first method involves computing the vector potential and the principal value of the harmonic potential over the vortex sheet, and then differentiating them in tangential directions to produce the normal or tangential component of the velocity, in the spirit of generalized vortex methods developed by Baker (1983). The second method involves subtracting off the dominant singularity of the Biot-Savart kernel and then accounting for its contribution by use of vector identities. Evaluating the strongly singular Biot-Savart integral is thus reduced to computing a weakly singular integral involving the mean curvature of the vortex sheet, and this allows the routine discretization of the vortex sheet into curved elements whose normal vector is not necessarily continuous across the edges, and the computation of the self-induced velocity without kernel desingularization. Numerical simulations of the motion of a closed or periodic vortex sheet immersed in a homogeneous fluid confirm the effectiveness of the numerical methods for a limited time of evolution. Numerical instabilities arise after a certain evolution time due to the ill-posedness of vortex sheet dynamics. The motion may be regularized by desingularizing the Biot-Savart kernel using either Krasny's (1986b) method or spectrum truncation. Depending, however, on the physical mechanism that drives the motion, the instabilities may persevere.

A new theory for the turbulent plane wall jet without external stream is proposed based on a similarity analysis of the governing equations. The asymptotic invariance principle (AIP) is used to require that properly scaled profiles reduce to similarity solutions of the inner and outer equations separately in the limit of infinite Reynolds number. Application to the inner equations shows that the appropriate velocity scale is the friction velocity, u∗, and the length scale is v/u∗. For finite Reynolds numbers, the profiles retain a dependence on the length-scale ratio, y+1/2 = u∗y1/2/v, where y1/2 is the distance from the wall at which the mean velocity has dropped to 1/2 its maximum value. In the limit as y+1/2 → ∞, the familiar law of the wall is obtained. Application of the AIP to the outer equations shows the appropriate velocity scale to be Um, the velocity maximum, and the length scale y1/2; but again the profiles retain a dependence on y+1/2 for finite values of it. The Reynolds shear stress in the outer layer scales with u2*, while the normal stresses scale with U2m. Also Um ∼ yn1/2 where n < −1/2 and must be determined from the data. The theory cannot rule out the possibility that the outer flow may retain a dependence on the source conditions, even asymptotically.

The fact that both these profiles describe the entire wall jet for finite values of y+1/2, but reduce to inner and outer profiles in the limit, is used to determine their functional forms in the ‘overlap’ region which both retain. The result from near asymptotics is that the velocity profiles in the overlap region must be power laws, but with parameters which depend on Reynolds number y+1/2 and are only asymptotically constant. The theoretical friction law is also a power law depending on the velocity parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly, although the difference from linear growth is small.

It is hypothesized that the inner part of the wall jet and the inner part of the zero-pressure-gradient boundary layer are the same. It follows immediately that all of the wall jet and boundary layer parameters should be the same, except for two in the outer flow which can differ only by a constant scale factor. The theory is shown to be in excellent agreement with the experimental data which show that source conditions may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter, B1 = (Umv/Mo)/ (y+1/2Mo/v2)n = constant where n ≈ −0.528, appears to be required to determine the entire flow for a given source.