Careful analysis of new field observations on the geometry of meandering tidal channels reveals that the values of meander wavelengths as well as curvatures, conveniently scaled by local channel width, fall within a fairly restricted range, suggesting the existence of some mechanistic process controlling meander formation. A three-dimensional model is then proposed to predict flow and bed topography in weakly meandering tidal channels. The analysis is developed for meander wavelengths much smaller than the tidal wavelength and assuming that sediment is transported both as bedload and as suspended load. Both conditions are typically met in nature. Due to the symmetry of the meander pattern investigated, the theory indicates that oscillations associated with the basic flow give rise to symmetric oscillations of the point bar–pool pattern around the locations of maximum curvature. However, no net migration in a tidal cycle is present, at least for periodic tides with zero mean. Suspended load leads both to an enhanced bottom deformation and to a downstream shifting of the position of the point bar. The model then provides the basis of a planimetric instability theory of the type developed for river meanders (Blondeaux & Seminara 1985). Though the available data do not yet allow a detailed quantitative comparison, it is shown that the wavelengths selected by the ‘bend mechanism’ are somewhat larger than those typically encountered in nature. The geomorphology of the process of meander formation in tidal environments is then discussed and, upon comparison with observational evidence, points out the need for various developments of the present model.

The form taken by a finite amplitude internal seiche in a shallow two-layer fluid within a long narrow container is akin to a shock wave in a gas-filled tube; an undular bore or surge is present which reflects back and forth along the length of the container. The internal surge in Loch Ness is sometimes observed to retain its amplitude over several seiche periods, and a resonance with the wind appears possible. This idea is explored by developing a theoretical model and by making a laboratory experiment, but difficulties are encountered in estimating the size of the parameters which characterize the natural phenomenon.

We introduce a new numerical method for the study of scalar mixing in two-dimensional advection fields. The position of an advected material strip is computed kinematically, and the associated convection–diffusion problem is solved using the computed local stretching rate along the strip, assuming that the diffusing strip thickness is smaller than its local radius of curvature. This widely legitimate assumption reduces the numerical problem to the computation of a single variable along the strip, thus making the method extremely fast and applicable to any large Péclet number. The method is then used to document the mixing properties of a chaotic sine flow, for which we relate the global quantities (spectra, concentration probability distribution functions (PDFs), increments) to the distributed stretching of the strip convoluted by the flow, possibly overlapping with itself. The numerical results indicate that the PDF of the strip elongation is log normal, a signature of random multiplicative processes. This property leads to exact analytical predictions for the spectrum of the field and for the PDF of the scalar concentration of a solitary strip. The present simulations offer a unique way of discovering the interaction rule for building complex mixtures which are made of a random superposition of overlapping strips leading to concentration PDFs stable by self-convolution.

Two-dimensional free surface flows without waves, produced by a submerged sink in a reservoir, are computed numerically for various configurations. For a sink above the horizontal bottom of a layer of fluid, there are solutions for all values of the Froude number F greater than some particular value. However, when the fluid is sufficiently deep, there is an additional solution for one special value of F < 1. The results for a sink at the vertex of a sloping bottom, treated by Craya and by Hocking, and for a sink in fluid of infinite depth, treated by Tuck & Vanden-Broeck, are confirmed and extended. In particular it is shown that as the bottom tends to the horizontal, the solution for a sink at the vertex of a sloping bottom approaches a solution for a horizontal bottom with F = 1. However solutions are found for all values of the Froude number F [ges ] 1 for a sink on a horizontal bottom.

We investigate experimentally the transition to turbulence of a yield stress shear-thinning fluid in Hagen–Poiseuille flow. By combining direct high-speed imaging of the flow structures with Laser Doppler Velocimetry (LDV), we provide a systematic description of the different flow regimes from laminar to fully turbulent. Each flow regime is characterized by measurements of the radial velocity, velocity fluctuations and turbulence intensity profiles. In addition we estimate the autocorrelation, the probability distribution and the structure functions in an attempt to further characterize transition. For all cases tested, our results indicate that transition occurs only when the Reynolds stresses of the flow equal or exceed the yield stress of the fluid, i.e. the plug is broken before transition commences. Once in transition and when turbulent, the behaviour of the yield stress fluid is somewhat similar to a (simpler) shear-thinning fluid. Finally, we have observed the shape of slugs during transition and found their leading edges to be highly elongated and located off the central axis of the pipe, for the non-Newtonian fluids examined.

A theory is developed for turbulence in a stably stratified fluid, for example in the experiments of Rouse & Dodu and of Turner where there is no shear and the turbulence is induced by a source of energy near the lower boundary of the fluid. A growing mixed layer of thickness D appears in the lower portion of the fluid and is separated from the non-turbulent fluid above, in which the buoyancy gradient is given, by an interfacial layer of thickness h. The lower mixed layer has a very weak buoyancy gradient and the large buoyancy difference across the interfacial layer is Δb.

As indicated by the experiments of Thompson & Turner and Hopfinger & Toly, and derived by the author in a recent paper, if u is the root-mean-square horizontal velocity and l is the integral length scale, the eddy viscosity ul is a constant in a homogeneous fluid agitated by a grid. When there is stratification, the theory indicates that the fluid motion is unaffected by buoyancy forces in the mixed layer, so that ul should again be constant in the lower portions of the mixed layer. Since l is proportional to distance, we may conveniently suppose that the source of the disturbances is at a level z = 0 where u is infinite in accordance with uz = K. Thus we may take K to be a fundamental parameter characterizing the turbulent energy source. Then z is distance above the plane of the virtual energy source. If the non-turbulent fluid has uniform buoyancy, DΔb = U2 may be shown to be constant. In general, whether constant or not, U may be taken to be a fundamental parameter expressing the stability. The quantity $\hat{R}i = U^{2}D^{2}/K^2$ is the most fundamental of the several Richardson numbers that have been introduced in this problem because, with its use, ‘constants’ of proportionality do not depend on the molecular coefficients of viscosity or diffusion (for high Reynolds number turbulence) or on the geometry of the grid.

The theory contains a number of results:

1. $u_e D/K \sim \hat{R}i^{-\frac{7}{4}},$ where $u_e = dD/dt$ is the entrainment velocity. Integration yields $D \infty t^{\frac{2}{11}}$ for a homogeneous upper fluid and $D \infty t^{\frac{2}{9}}$ for a linear upper density field. This $-\frac{7}{4}$ entrainment law compares with a $-\frac{3}{2}$ law suggested by several experimenters.

2. Turbulence in the interfacial layer is intermittent with intermittency factor $I_3 \sim \hat{R}i^{-\frac{3}{4}}$. The turbulent patches have dimension $\sigma_3 \sim D\hat{R}i^{-\frac{3}{4}}$.

3. If the (equal) root-mean-square velocities in an infinite homogeneous fluid at a distance D from the grid are denoted by u1 ∼ v1 ∼ w1, we find that the r.m.s. velocities near the interface are u2 ∼ u1, v2 ∼ u1 and $w_2 \sim u_1 \hat{R}i^{-\frac{1}{4}}$.

4. The buoyancy flux q2 near the interface may be expressed as $q_2 \sim w^3_2/D$.

5. h ∼ D, as observed in several experiments.

The stability of laminar axisymmetric jets and wakes, the two prominent examples of free shear layers, is investigated with respect to linear azimuthally periodic disturbances. The complete viscous disturbance equations are integrated numerically and the eigenvalues are obtained by matching the numerically advanced solutions to the asymptotic solutions at a large radius. Both spatial and temporal stability are examined for inviscid and viscid flows. It is found that the critical Reynolds number for the jet and the wake are not much different while the amplification rates for the wake become considerably greater than those for the jet as the Reynolds number increases. The axisymmetric shear-layer flows also seem to be more stable than the corresponding plane flows.

The goal of this paper is to derive expressions for the pressure forces and moments acting on an elongated body swimming in a quiescent fluid. The body is modelled as an inextensible and unshearable (Kirchhoff) beam, whose cross-sections are elliptic, undergoing prescribed deformations, consisting of yaw and pitch bending. The surrounding fluid is assumed to be inviscid, and irrotational everywhere, except in a thin vortical wake. The Laplace equation and the corresponding Neumann boundary conditions are first written in terms of the body coordinates of a beam treating the body as a fixed surface. They are then simplified according to the slenderness of the body and its kinematics. Because the equations are linear, the velocity potential is sought as a sum of two terms which are linked respectively to the axial movements of the beam and to its lateral movements. The lateral component of the velocity potential is decomposed further into two sub-components, in order to exhibit explicitly the role of the two-dimensional potential flow produced by the lateral motion of the cross-section, and the role played by the curvature effects of the beam on the cross-sectional flow. The pressure, which is given by Bernoulli's equation, is integrated along the body surface, and the expressions for the resultant and the moment are derived analytically. Thereafter, the validity of the force and moment obtained analytically is checked by comparisons with Navier–Stokes simulations (using Reynolds-averaged Navier–Stokes equations), and relatively good agreements are observed.

We report on experimental determinations of the temperature field in the interior (bulk) of turbulent Rayleigh–Bénard convection for a cylindrical sample with an aspect ratio (diameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ over height $L$) equal to 0.50, in both the classical and the ultimate state. The measurements are for Rayleigh numbers $\mathit{Ra}$ from $6\times 10^{11}$ to $10^{13}$ in the classical and $7\times 10^{14}$ to $1.1\times 10^{15}$ (our maximum accessible $\mathit{Ra}$) in the ultimate state. The Prandtl number was close to 0.8. Although to lowest order the bulk is often assumed to be isothermal in the time average, we found a ‘logarithmic layer’ (as reported briefly by Ahlers et al., Phys. Rev. Lett., vol. 109, 2012, 114501) in which the reduced temperature $\varTheta = [\langle T(z) \rangle - T_m]/\Delta T$ (with $T_m$ the mean temperature, $\Delta T$ the applied temperature difference and $\langle {\cdots } \rangle $ a time average) varies as $A \ln (z/L) + B$ or $A^{\prime } \ln (1-z/L) + B^{\prime }$ with the distance $z$ from the bottom plate of the sample. In the classical state, the amplitudes $-A$ and $A^{\prime }$ are equal within our resolution, while in the ultimate state there is a small difference, with $-A/A^{\prime } \simeq 0.95$. For the classical state, the width of the log layer is approximately $0.1L$, the same near the top and the bottom plate as expected for a system with reflection symmetry about its horizontal midplane. For the ultimate state, the log-layer width is larger, extending through most of the sample, and slightly asymmetric about the midplane. Both amplitudes $A$ and $A^{\prime }$ vary with radial position $r$, and this variation can be described well by $A = A_0 [(R - r)/R]^{-0.65}$, where $R$ is the radius of the sample. In the classical state, these results are in good agreement with direct numerical simulations (DNS) for $\mathit{Ra} = 2\times 10^{12}$; in the ultimate state there are as yet no DNS. The amplitudes $-A$ and $A^{\prime }$ varied as ${\mathit{Ra}}^{-\eta }$, with $\eta \simeq 0.12$ in the classical and $\eta \simeq 0.18$ in the ultimate state. A close analogy between the temperature field in the classical state and the ‘law of the wall’ for the time-averaged downstream velocity in shear flow is discussed. A two-sublayer mean-field model of the temperature profile in the classical state was analysed and yielded a logarithmic $z$ dependence of $\varTheta $. The $\mathit{Ra}$ dependence of the amplitude $A$ given by the model corresponds to an exponent $\eta _{th} = 0.106$, in good agreement with the experiment. In the ultimate state the experimental result $\eta \simeq 0.18$ differs from the prediction $\eta _{th} \simeq 0.043$ by Grossmann & Lohse (Phys. Fluids, vol. 24, 2012, 125103).

Longitudinal and transverse structure functions, $D_{ll}=\langle {\it\delta}u_{l}{\it\delta}u_{l}\rangle$ and $D_{tt}=\langle {\it\delta}u_{t}{\it\delta}u_{t}\rangle$, can be calculated from aircraft data. Here, ${\it\delta}$ denotes the increment between two points separated by a distance $r$, $u_{l}$ and $u_{t}$ the velocity components parallel and perpendicular to the aircraft track respectively and $\langle \,\rangle$ an average. Assuming statistical axisymmetry and making a Helmholtz decomposition of the horizontal velocity, $\boldsymbol{u}=\boldsymbol{u}_{r}+\boldsymbol{u}_{d}$, where $\boldsymbol{u}_{r}$ is the rotational and $\boldsymbol{u}_{d}$ the divergent component of the velocity, we derive expressions relating the structure functions $D_{rr}=\langle {\it\delta}\boldsymbol{u}_{r}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{r}\rangle$ and $D_{dd}=\langle {\it\delta}\boldsymbol{u}_{d}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{d}\rangle$ to $D_{ll}$ and $D_{tt}$. Corresponding expressions are also derived in spectral space. The decomposition is applied to structure functions calculated from aircraft data. In the lower stratosphere, $D_{rr}$ and $D_{dd}$ both show a nice $r^{2/3}$-dependence for $r\in [2,20]\ \text{km}$. In this range, the ratio between rotational and divergent energy is a little larger than unity, excluding gravity waves as the principal agent behind the observations. In the upper troposphere, $D_{rr}$ and $D_{dd}$ show no clean $r^{2/3}$-dependence, although the overall slope of $D_{dd}$ is close to $2/3$ for $r\in [2,400]\ \text{km}$. The ratio between rotational and divergent energy is approximately three for $r<100\ \text{km}$, excluding gravity waves also in this case. We argue that the possible errors in the decomposition at scales of the order of 10 km are marginal.

Surface pressure fluctuations induced by turbulent boundary-layer flow at ${\mathit{Re}}_{\theta } = 4755$ over small backward- and forward-facing steps are studied with large-eddy simulation. Four step heights that are 53, 13, 3.3 and 0.83 % of the boundary-layer thickness are considered to investigate the effects of step height on surface pressure characteristics and pressure-source mechanisms. The extent to which turbulent velocity fluctuations in the boundary layer and the separated shear layer contribute to the surface pressure fluctuations is examined with scaling of various pressure statistics and two-point correlations. For larger steps, vortical structures develop in the shear layer and the associated intense velocity fluctuations are the dominant source. Downstream of slightly less than one reattachment length from the step, the root-mean-square pressure is found to scale with the local maximum cross-stream Reynolds normal stress ${ \overline{{v}^{\ensuremath{\prime} \hspace{0.167em} 2} } }_{\mathit{max}} $. The pressure frequency spectrum at the maximum ${p}_{\mathit{rms}} $ location consists of an energy-containing range that scales with the mean reattachment length ${x}_{r} $ and a higher frequency range that rolls off with a slope close to $\ensuremath{-} 7/ 3$. As the step height decreases, the boundary-layer turbulent fluctuations become the dominant source, the ${ \overline{{v}^{\ensuremath{\prime} \hspace{0.167em} 2} } }_{\mathit{max}} $ scaling of ${p}_{\mathit{rms}} $ is no longer valid and the roll-off slope of the frequency spectrum becomes steeper. The downstream recovery of a step-perturbed boundary layer towards an equilibrium boundary layer is investigated from the point of view of surface pressure fluctuations. For steps with a strong separated shear layer, pressure fluctuations are found to decay rapidly for up to three reattachment lengths downstream of the step, within which approximately 60 % of the peak ${p}_{\mathit{rms}} $ is dissipated. Farther downstream, recovery is much slower. The pressure-recovery distances estimated for the largest backward and forward steps are 175 and 295 step heights, respectively.

Dynamic wetting is crucial to processes where liquid displaces another fluid along a solid surface, such as the deposition of a coating liquid onto a moving substrate. Numerous studies report the failure of dynamic wetting past some critical process speed. However, the hydrodynamic factors that influence the transition to wetting failure remain poorly understood from an empirical and theoretical perspective. The objective of this investigation is to determine the effect of meniscus confinement on the onset of dynamic wetting failure. A novel experimental system is designed to simultaneously view confined and unconfined wetting systems as they approach wetting failure. The experimental apparatus consists of a scraped steel roll that rotates into a bath of glycerol. Confinement is imposed via a gap formed between a coating die and the roll surface. Flow visualization is used to record the critical roll speed at which wetting failure occurs. Comparison of the confined and unconfined data shows a clear increase in the relative critical speed as the meniscus becomes more confined. A hydrodynamic model for wetting failure is developed and analysed with (i) lubrication theory and (ii) a two-dimensional finite-element method (FEM). Both approaches do a remarkable job of matching the observed confinement trend, but only the two-dimensional model yields accurate estimates of the absolute values of the critical speeds due to the highly two-dimensional nature of the stress field in the displacing liquid. The overall success of the hydrodynamic model suggests a wetting failure mechanism primarily related to viscous bending of the meniscus.

The early phase of high-speed liquid droplet impact on a rigid wall is characterized by compressibility effects through the creation of a shock wave attached to the contact area periphery. Initially, the area of compressed liquid is assumed to be bounded by the shock envelope, which propagates both laterally and upwardly into the bulk of the liquid. In this paper, an analytical model accounting for the lateral liquid motion in the compressed area is developed and compared to the axisymmetric numerical solution of the inviscid (Euler) flow equations. It is shown that the often employed assumption that the compressed area is separated from the liquid bulk by a single shock wave attached to the contact line breaks down and results in an anomaly. This anomaly emerges prior to the time when the shock wave departs from the contact line, initiating lateral liquid jetting. In order to remove this anomaly, the analytical model presented in this paper proposes the transition from a single to a multiple wave structure in the contact line region, prior to jetting eruption. The occurrence of this more complex multiple wave structure is also supported by the numerical results.

The unfolding of inertial instability in intially barotropic vortices in a uniformly rotating and stratified fluid is studied through numerical simulations. The vortex dynamics during the instability is examined in detail. We demonstrate that the instability is stabilized via redistribution of angular momentum in a way that produces a new equilibrated barotropic vortex with a stable velocity profile. Based on extrapolations from the results of a series of simulations in which the Reynolds number and strength of stratification are varied, we arrive at a construction based on angular momentum mixing that predicts the infinite-Reynolds-number form of the equilibrated vortex toward which inertial instability drives an unstable vortex. The essential constraint is conservation of total absolute angular momentum. The construction can be used to predict the total energy loss during the equilibration process. It also shows that the equilibration process can result in anticyclones that are more susceptible to horizontal shear instabilities than they were initially, a phenomenon previously observed in laboratory and numerical studies.

Kr, Xe and SiCI4 have been investigated for convective instabilities in closed vertical cylinders with conductive walls heated from below. Critical Rayleigh numbers NiRa for the onset of various convective modes (including the onset of marginally stable and periodic flow) have been determined with a high resolution differential temperature sensing method. Flow patterns were deduced from a multiple sensor arrangement. For the three lowest modes (i = 1, 2, 3) good quantitative agreement with linear stability theory is found. Stable oscillatory modes (periodic fluctuations of the mean flow) with a period of approximately 5 s are found for a relatively narrow range of NRa. The critical Rayleigh number NoscRa for the onset of oscillatory temperature fluctuations is 1348 ± 50 for an aspect ratio (height/radius) of 6.

Low- and high-frequency acoustic resonances are computed numerically via a high-order finite-element code for a generic two-dimensional high-lift configuration with a leading-edge slat. Zero mean flow is assumed, approximating the low-Mach-number situation at aircraft landing and approach. To avoid unphysical reflections at the boundaries of the truncated computational domain, perfectly matched layer absorbing boundary conditions are implemented in the form of the complex scaling method of atomic and molecular physics. It is shown that two types of resonance exist: resonances of surface waves which scale with the total airfoil length and longitudinal cavity-type resonances which scale with the slat cove length. Minima exist in the temporal decay rate which can be associated with the slat cove resonances and depend on the slat cove geometry. All resonances are damped owing to radiation losses. However, if coherent noise sources exist, as observed in low-Reynolds-number experiments, these sources can be enhanced acoustically by the above resonances if the source frequency is close to a resonant frequency.

Motivated by devices such as the atomic force microscope, we compute the drag experienced by a cylindrical body of circular or rectangular cross-section oscillating at small amplitude near a plane wall. The body lies parallel to the wall and oscillates normally to it; the body is assumed to be long enough for the dominant flow to be two-dimensional. The flow is parameterized by a frequency parameter $\gamma^2$ (a Strouhal number) and the wall–body separation $\Delta$ (scaled on body radius). Numerical solutions of the unsteady Stokes equations obtained using finite-difference computations in bipolar coordinates (for circular cross-sections) and boundary-element computations (for rectangular cross-sections) are used to determine the drag on the body. Numerical results are validated and extended using asymptotic predictions (for circular cylinders) obtained at all extremes of $(\gamma,\Delta)$-parameter space. Regions in parameter space for which the wall has a significant effect on drag are identified.

We consider a nominally uniform flow over a semi-infinite flat plate. Our analysis shows how a small streamwise disturbance in the otherwise uniform flow ahead of the plate is amplified by leading-edge bluntness effects and eventually leads to a small-amplitude but nonlinear spanwise motion far downstream from the leading edge of the plate. This spanwise motion is then imposed on the viscous boundary-layer flow at the surface of the plate – causing an order-one change in its profile shape. This ultimately reduces the wall shear stress to zero – causing the boundary layer to undergo a localized separation, which may be characterized as a kind of bursting phenomenon that could be related to the turbulent bursts observed in some flat-plate boundary-layer experiments.

The linear and weakly nonlinear stability of Poiseuille–Couette flow is considered for various values of the relative wall velocity 2uw, An account is given first of the asymptotic upper and lower branches of the linear neutral curve (s), followed by their disappearance, as uw is increased. Two main (and one minor) neutral curves are found to exist for smaller O(1) (or lesser) values of uw, then one for moderate O(1) values of uw, and none for larger O(1) values of uw. The cut-off velocity at which each main neutral curve disappears is determined, and in each case the whole neutral curve for uw just below the cut-off value is determined in closed form. Secondly, weakly nonlinear solutions are found to bifurcate subcritically from the neutral curve for uw just below cut-off, but to ‘bifurcate from infinity’ just above cut-off. This identifies a minimum threshold amplitude at the entry to the regime where no linear neutral curve exists.