This paper is a theoretical and experimental investigation of the onset of the buoyancydriven longitudinal roll cells that occur when a liquid layer flows over a heated horizontal plate. Linear stability theory is applied under the assumption that the spatially developing temperature profile can be treated locally, that is at each axial position, as being ‘frozen’. Using the film thickness as the length scaling factor, the critical Rayleigh numbers associated with the onset of longitudinal rolls are found to be considerably lower than measured values. A modified local stability analysis using the thermal boundary-layer thickness as the scaling factor is shown to agree with experiments. Predicted wavenumbers and the position of the onset of cellular convection are in agreement with wavenumbers measured by flow-visualization techniques. The position of the onset of cellular convection is also obtained from heat-transfer measurements at the heated surface. In the asymptotic limit of a linear undisturbed temperature profile the classical solutions of the Rayleigh-Bénard problem for the critical Rayleigh number and wavenumber are recovered, the only effect of the flow being the structure of the secondary flow that occurs when the system is unstable. Amplification theory is also compared with experimental data for the position at which the thermal effects of the convection are detectable and for the wavenumbers measured by flow visualization. The thermal amplification ratio $\overline{Nu}$ and a velocity-disturbance amplification ratio $\overline{w}$ are used to interpret the onset of discernible cellular convection. The data are not consistent with any single amplification ratio over the range of Rayleigh numbers studied (103 < Ra < 3 × 105), and the theoretical and experimental results suggest that a band of wavenumbers rather than a single wavenumber is encountered when cellular convection occurs.

This paper attempts to determine the optimum profile of a two-dimensional plate that produces the maximum hydrodynamic lift while planing on a water surface, under the condition of no spray formation and no gravitational effect, the latter assumption serving as a good approximation for operations at large Froude numbers. The lift of the sprayless planing surface is maximized under the isoperimetric constraints of fixed chord length and fixed wetted arc-length of the plate. Consideration of the extremization yields, as the Euler equation, a pair of coupled nonlinear singular integral equations of the Cauchy type. These equations are subsequently linearized to facilitate further analysis. The analytical solution of the linearized problem has a branch-type singularity, in both pressure and flow angle, at the two ends of plate. In a special limit, this singularity changes its type, emerging into a logarithmic one, which is the weakest type possible. Guided by this analytic solution of the linearized problem, approximate solutions have been calculated for the nonlinear problem using the Rayleigh-Ritz method and the numerical results compared with the linearized theory.

Using the Hamiltonian approach of Krasitskii (J. Fluid Mech., vol. 272, 1994, pp. 1–20), we derive a variant of the Zakharov equation in which the wave-induced mean surface elevation and the surface potential of the wave-induced mean flow are represented as separate variables governed by separate evolution equations. The kernel function of this new variant is simpler, and in particular also well defined in the uniform-wave-train limit for waves on finite depth. This form of the Zakharov equation may be advantageous in some applications. One example is the derivation of nonlinear Schrödinger equations in the narrow-band limit, where the handling of the mean flow and mean surface is significantly simpler than when starting from the original Zakharov equation. In this paper we have used the alternative form of the Zakharov equation to derive a Hamiltonian nonlinear Schrödinger equation for directional waves on arbitrary depth, valid to one order higher in bandwidth than the Hamiltonian equation recently presented by Craig, Guyenne and Sulem (Wave Motion, vol. 47, 2010, pp. 552–563).

A discrepancy between the enhancement in overall burning rate and the enhancement in flame surface area measured for high-intensity turbulence is addressed. In order to reconcile the two quantities, an additional contribution from the effective turbulent diffusivity is considered. This contribution is expected to arise in sufficiently intense turbulence from eddies smaller than the flamelet thickness. In the present work, the enhancement in diffusivity arising from these eddies is estimated based on a model energy spectrum; individual contributions from all turbulence length scales smaller the flamelet thickness are integrated over the corresponding portion of the spectrum. It is shown that diffusivity enhancement, estimated in this manner, is able to account for the measured discrepancy between the overall burning rate enhancement and flame surface area enhancement. The factor quantifying this discrepancy is formalized as a closed-form function of the Karlovitz number.

The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier–Stokes equations, with a view to assessing rigorously the consequences of the scale invariance (an exact property of the Navier–Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for infinitely large Reynolds number, is applied to the transport equations for the second- and third-order moments of the longitudinal velocity increment, $(\unicode[STIX]{x1D6FF}u)$. Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the fluid viscosity, $\unicode[STIX]{x1D708}$, and the mean kinetic energy dissipation rate, $\overline{\unicode[STIX]{x1D716}}$ (the overbar denotes spatial and/or temporal averaging), are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the flatness factor of $(\unicode[STIX]{x1D6FF}u)$ and shows that these distributions must exhibit plateaus (of different magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ increases indefinitely. Also, the skewness and flatness factor of the longitudinal velocity derivative become constant as $Re_{\unicode[STIX]{x1D706}}$ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}$ with $n=2$, 3 and 4 cannot be represented by a power law of the form $r^{\unicode[STIX]{x1D701}_{n}}$ when the Reynolds number is finite. It is shown that only when $Re_{\unicode[STIX]{x1D706}}\rightarrow \infty$ can an $n$-thirds law (i.e. $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}\sim r^{\unicode[STIX]{x1D701}_{n}}$, with $\unicode[STIX]{x1D701}_{n}=n/3$) emerge, which is consistent with the onset of a scaling range.

An analytical dispersion relation describing the linear stability of a plane detonation wave to low-frequency two-dimensional disturbances with arbitrary wavenumbers is derived using a normal mode approach and a combination of high activation energy and Newtonian limit asymptotics, where the ratio of specific heats γ→1. The reaction chemistry is characterized by one-step Arrhenius kinetics. The analysis assumes a large activation energy in the plane steady-state detonation wave and a characteristic linear disturbance wavelength which is longer than the fire-zone thickness. Newtonian limit asymptotics are employed to obtain a complete analytical description of the disturbance behaviour in the induction zone of the detonation wave. The analytical dispersion relation that is derived depends on the activation energy and exhibits favourable agreement with numerical solutions of the full linear stability problem for low-frequency one- and two-dimensional disturbances, even when the activation energy is only moderate. Moreover, the dispersion relation retains vitally important characteristics of the full problem such as the one-dimensional stability of the detonation wave to low-frequency disturbances for decreasing activation energies or increasing overdrives. When two-dimensional oscillatory disturbances are considered, the analytical dispersion relation predicts a monotonic increase in the disturbance growth rate with increasing wavenumber, until a maximum growth rate is reached at a finite wavenumber. Subsequently the growth rate decays with further increases in wavenumber until the detonation becomes stable to the two-dimensional disturbance. In addition, through a new detailed analysis of the behaviour of the perturbations near the fire front, the present analysis is found to be equally valid for detonation waves travelling at the Chapman–Jouguet velocity and for detonation waves which are overdriven. It is found that in contrast to the standard imposition of a radiation or piston condition on acoustic disturbances in the equilibrium zone for overdriven waves, a compatibility condition on the perturbation jump conditions across the fire zone must be satisfied for detonation waves propagating at the Chapman–Jouguet detonation velocity. An insight into the physical mechanisms of the one- and two-dimensional linear instability is also gained, and is found to involve an intricate coupling of acoustic and entropy wave propagation within the detonation wave.

A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, . Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate can occur for arbitrary values of , viscous Reynolds number , and random-force correlation time and orientation angle in the shearing plane. The value of is independent of the domain size. The shear dynamo is ‘fast’, with finite in the limit of . Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small and , the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying () and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.

We investigate the deformation of a linear viscoelastic compliant coating in a turbulent flow for a wide range of coating parameters. A one-way coupling model is proposed in which the turbulent surface stresses are expressed as a sum of streamwise-travelling waves with amplitudes determined from the stress spectra of the corresponding flow over a rigid wall. The analytically calculated coating deformation is analysed in terms of the root-mean-square (r.m.s.) surface displacement and the corresponding point frequency spectra. The present study systematically investigates the influence of five coating properties namely density, stiffness, thickness, viscoelasticity and compressibility. The surface displacements increase linearly with the fluid/solid density ratio. They are linearly proportional to the coating thickness for thin coatings, while they become independent of the thickness for thick coatings. Very soft coatings show resonant behaviour, but the displacement for stiffer coatings is proportional to the inverse of the shear modulus. The viscoelastic loss angle has only a significant influence when resonances occur in the coating response, while Poisson’s ratio has a minor effect for most cases. The modelled surface displacement is qualitatively compared with recent measurements on the deformation of three different coatings in a turbulent boundary-layer flow. The model predicts the order of magnitude of the surface displacement, and it captures the increase of the coating displacement with the Reynolds number and the coating softness. Finally, we propose a scaling that collapses all the experimental data for the r.m.s. of the vertical surface displacement onto a single curve.

A combined approach using system identification and feed-forward control design has been applied to experimental laminar channel flow in an effort to reduce the naturally occurring disturbance level. A simple blowing/suction strategy was capable of reducing the standard deviation of the measured sensor signal by 45 %, which markedly exceeds previously obtained results under comparable conditions. A comparable reduction could be verified over a significant streamwise extent, implying an improvement over previous, more localized disturbance control. The technique is effective, flexible, and robust, and the obtained results encourage further explorations of experimental control of convection-dominated flows.

We investigate the conditions that determine the detachment of a water drop from different vibrating textured plates by using vertical vibrations. The plate surfaces were patterned by a lattice of pillars of different shapes with different geometrical arrangements. The acceleration threshold for the water droplet to bounce off the surfaces was measured as a function of the excitation frequency. In each case, the acceleration threshold presents a minimum at the natural frequency of the droplet. The minimum acceleration required for the take-off is larger for small droplets than for large droplets. Namely, one finds that the value of the threshold depends on the size of the droplet and on the maximum apparent contact area between the droplet and the substrate. The theoretical model takes into account the energy necessary to break the capillary bridges between the droplet and the pillars of the surface. This model captures the main ingredients explaining the drop size dependence of the acceleration threshold for the take-off.

It is shown that in Stokes's flow the perturbation field, due to the addition of one more sphere to a shear flow of a fluid containing a number of non-interacting spheres, has the property that the total additional shearing force, acting on any plane normal to the direction of velocity change, is zero. However, the perturbation velocity, integrated over such a plane, takes a constant value, positive if the plane lies on one side of the sphere and negative if it lies on the other side. It follows that the effect of all the spheres is not to alter the shearing stress at all, but to reduce the mean shear by a factor 1 – 2·5c, where c is the concentration. This suggests that Einstein's viscosity law should be altered to η = η0/(1 – 2·5c) when c is not small.

Direct numerical simulations of temporally evolving reacting compressible mixing layers have been performed to study the viscous superlayer (VSL) at the outer edge of the interface layer. Budgets of the transport equation of enstrophy conditioned on the normal distance from the turbulent/non-turbulent interface are used to examine the features of the VSL. A new method is introduced to detect and to quantify the thickness of the VSL in reacting compressible flows. It involves finding the correlation coefficient of the viscous diffusion term with the viscous dissipation term. It is shown that, while compressibility seems to have little effect on the thickness of the VSL, as the level of heat release increases, the thickness of this layer decreases. Furthermore, it is observed that a thinner VSL propagates slower, resulting in a decrease of the rate of entrainment into the mixing layer.

Simultaneous measurements of streamwise velocity ( $\tilde{U}$ ) and concentration ( $\tilde{C}$ ) for a horizontal plume released at eight different vertical locations within a turbulent boundary layer are discussed in this paper. These are supplemented by limited simultaneous three-component velocity and concentration measurements. Results of the integral time scale ( $\unicode[STIX]{x1D70F}_{c}$ ) of concentration fluctuations across the width of the plume are presented here for the first time. It is found that $\unicode[STIX]{x1D70F}_{c}$ has two distinct peaks: one closer to the plume centreline and the other at a vertical distance of plume half-width above the centreline. The time-averaged streamwise concentration flux is found to be positive and negative, respectively, below and above the plume centreline. This behaviour is a resultant of wall-normal velocity fluctuations ( $w$ ) and Reynolds shear stress ( $\overline{uw}$ ). Confirmation of these observations is found in the results of joint probability density functions of $u$ (streamwise velocity fluctuations) and $\tilde{C}$ as well as that of $w$ and $\tilde{C}$ . Results of cross-correlation coefficient show that high- and low-momentum regions have a distinctive role in the transport of passive scalar. Above the plume centreline, low-speed structures have a lead over the meandering plume, while high-momentum regions are seen to lag behind the plume below its centreline. Further examination of the phase relationship between time-varying $u$ and $c$ (concentration fluctuations) via cross-spectrum analysis is consistent with this observation. Based on these observations, a phenomenological model is presented for the relative arrangement of a passive scalar plume with respect to large-scale velocity structures in the flow.

In isothermal non-coalescence behaviours of a droplet against a wall, an air film of micrometre thickness plays a crucial role. We experimentally study this phenomenon by letting a droplet levitate over a moving glass wall. The three-dimensional shape of the air film is measured using an interferometric method. The mean curvature distribution of the deformed free surface and the distributions of the lubrication pressure are derived from the experimental measurements. We vary experimental parameters, namely wall velocity, droplet diameter and viscosity of the droplets, over a wide range; for example, the droplet viscosity is varied over two orders of magnitude. For the same wall velocity, the air film of low-viscosity droplets shows little shape oscillation with constant film thickness (defined as the steady state), while that of highly viscous droplets shows a significant shape oscillation with varying film thickness (defined as the unsteady state). The droplet viscosity also affects the surface velocity of a droplet. Under our experimental conditions, where the air film shape can be assumed to be steady, we present experimental evidence showing that the lift force generated inside the air film balances with the droplet’s weight. We also verify that the lubrication pressure locally balances with the surface tension and hydrostatic pressures. This indicates that lubrication pressure and the shape of the free surface are mutually determined. Based on the local pressure balance, we discuss a process of determining the steady shape of an air film that has two areas of minimum thickness in the vicinity of the downstream rim.

We present a variational data assimilation method in order to improve the accuracy of velocity fields $\tilde{\boldsymbol{v}}$, that are measured using particle image velocimetry (PIV). The method minimises the space–time integral of the difference between the reconstruction $\boldsymbol{u}$ and $\tilde{\boldsymbol{v}}$, under the constraint, that $\boldsymbol{u}$ satisfies conservation of mass and momentum. We apply the method to synthetic velocimetry data, in a two-dimensional turbulent flow, where realistic PIV noise is generated by computationally mimicking the PIV measurement process. The method performs optimally when the assimilation integration time is of the order of the flow correlation time. We interpret these results by comparing them to one-dimensional diffusion and advection problems, for which we derive analytical expressions for the reconstruction error.