In turbulent Rayleigh–Bénard (RB) convection, a transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number $Ra\approx 10^{13}{-}10^{14}$ a transition in the scaling of the global Nusselt number $Nu$ (the dimensionless heat transfer) and the Reynolds number with $Ra$ has been observed in experiments and very recently in direct numerical simulations (DNS) of two-dimensional (2D) RB convection. In this paper, we analyse the local scaling properties of the lateral temperature structure functions in the BLs of this simulation of 2D RB convection, employing extended self-similarity (ESS) (i.e., plotting the structure functions against each other, rather than only against the scale) in the spirit of the attached-eddy hypothesis, as we have recently introduced for velocity structure functions in wall turbulence (Krug et al., J. Fluid Mech., vol. 830, 2017, pp. 797–819). We find no ESS scaling at $Ra$ below the transition and in the near-wall region. However, beyond the transition and for large enough wall distance $z^{+}>100$, we find clear ESS behaviour, as expected for a scalar in a turbulent boundary layer. In striking correspondence to the $Nu$ scaling, the ESS scaling region is negligible at $Ra=10^{11}$ and well developed at $Ra=10^{14}$, thus providing strong evidence that the observed transition in the global Nusselt number at $Ra\approx 10^{13}$ indeed is the transition from a laminar type BL to a turbulent type BL. Our results further show that the relative slopes for scalar structure functions in the ESS scaling regime are the same as for their velocity counterparts, extending their previously established universality. The findings are confirmed by comparing to scalar structure functions in three-dimensional turbulent channel flow.

The linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but under certain conditions the flow may be linearly stable to axisymmetric disturbances but linearly unstable to non-axisymmetric disturbances.

Exact solutions for multipolar azimuthal-mode vortices in two-dimensional Euler flows are presented. Flow solutions with non-vanishing far-field velocity are provided for any set of azimuthal wavenumbers $m$ and arbitrary number $n$ of vorticity shells. For azimuthal wavenumbers $m=0$ and $m=1$, the far-field velocity is a rigid motion and unsteady flow solutions with vanishing far-field velocity are obtained by means of a time-dependent change of reference frame. Addition of these first two modes, in the case of $n=1$, results in a particular Chaplygin–Lamb (C–L) dipole, with continuous and vanishing vorticity at the vortex boundary. Numerical simulations suggest that this particular C–L dipole is stable.

Experimental results are presented for two turbulent boundary-layer experiments conducted at a free-stream Mach number of 4 with wall cooling. The first experiment examines a constant-temperature cold-wall boundary layer subjected to adverse and favourable pressure gradients. It is shown that the boundary-layer data display good agreement with Coles’ general composite boundary-layer profile using Van Driest's transformation. Further, the pressuregradient parameter βK found in previous studies to correlate adiabatic highspeed data with low-speed data also correlates the present cooled-wall high-speed data. The second experiment treats the response of a constant-pressure highspeed boundary layer to a near step change in wall temperature. It is found that the growth rate of the thermal boundary layer within the existing turbulent boundary layer varies considerably depending upon the direction of the wall temperature change. For the case of an initially cooled boundary layer flowing onto a wall near the recovery temperature, it is found that δT ∼ x whereas the case of an adiabatic boundary layer flowing onto a cooled wall gives δT ∼ x½. The apparent origin of the thermal boundary layer also changes considerably, which is accounted for by the variation in sublayer thicknesses and growth rates within the sublayer.

A class of ‘exact’ steady and unsteady solutions of the Navier—Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating in the (x, y)-plane with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x, — y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large Kármán's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation-point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Kármán's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite-time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial-value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

We consider non-geostrophic homogeneous baroclinic turbulence without solid boundaries, and we focus on its energetics and dynamics. The homogeneous turbulent flow is therefore submitted to both uniform vertical shear $S$ and stable vertical stratification, parametrized by the Brunt–Väisälä frequency $N$, and placed in a rotating frame with Coriolis frequency $f$. Direct numerical simulations show that the threshold of baroclinic instability growth depends mostly on two dimensionless numbers, the gradient Richardson number $\mathit{Ri}= {N}^{2} / {S}^{2} $ and the Rossby number $\mathit{Ro}= S/ f$, whereas linear theory predicts a threshold that depends only on $\mathit{Ri}$. At high Rossby numbers the nonlinear limit is found to be $\mathit{Ri}= 0. 2$, while in the limit of low $\mathit{Ro}$ the linear stability bound $\mathit{Ri}= 1$ is recovered. We also express the stability results in terms of background potential vorticity, which is an important quantity in baroclinic flows. We show that the linear symmetric instability occurs from the presence of negative background potential vorticity. The possibility of simultaneous existence of symmetric and baroclinic instabilities is also investigated. The dominance of symmetric instability over baroclinic instability for $\mathit{Ri}\ll 1$ is confirmed by our direct numerical simulations, and we provide an improved understanding of the dynamics of the flow by exploring the details of energy transfers for moderate Richardson numbers.

The injection of a two-dimensional jet into a uniform stream is considered, the fluids being assumed inviscid and incompressible. When the total head of the jet is much larger than that of the uniform flow, the motion is characterized by two disparate length scales, and uniformly valid asymptotic solutions can be found by the method of matched expansions. Inner and outer expansions are developed for the jet and the external flow. The first-order outer solution in the jet is the usual thin jet approximation which fails in the neighbourhood of the jet exit except for 90° injection, when it is uniformly valid. The basic non-linearity introduced by the pressure condition along the vortex sheet separating the jet from the external flow appears as a non-linear boundary condition for the first-order outer solution in the external flow. A novel feature of the analysis is the necessity of imposing a logarithmic singularity as an ‘inner’ boundary condition for the outer solution in the external flow. The first-order fluid speed and streamline deflection angle are shown to be given correctly to O(1) uniformly in the external flow (for all injection angles) by the first-order outer solution.

This paper presents an inviscid bluff-body wake model which correctly takes into account the displacement effect of the far wake by means of an appropriate source–sink system located behind the body. The separating streamlines, which in previous inviscid wake models have been regarded as the time-averaged shear layers emanating from the separation points within a small distance downstream of the body, can be interpreted as the displacement surface of the wake throughout the whole region of flow behind the body. The solutions for a normal flat plate, a circular cylinder and a 90° wedge are worked out and compared with experiments, where possible. The theoretical pressure distributions agree fairly well with the experimental ones. The shape of the separating streamlines obtained from the present theory is physically reasonable and compares well with experimental results for a normal plate and a 90° wedge.

We consider Marangoni convection in a heated layer of a binary liquid. The solute is a surfactant, which is present in both surface and bulk phases; the bulk gradient of the concentration is formed due to the Soret effect. Linear stability analysis demonstrates a well-pronounced stabilization of the layer due to the adsorption kinetics and advection of the surface phase. We derive nonlinear amplitude equations for longwave perturbations in the case of fast sorption kinetics (small Langmuir number) and demonstrate that with increase in the effect of the adsorption, subcritical excitation occurs. In the case of a finite Langmuir number, the weakly nonlinear problem is ill-posed. A physical mechanism of subcritical bifurcation is discussed.

Perturbation expansions are derived to second order in a wavenumber parameter for the unsteady lift induced on an aerofoil by disturbances convected past it at subsonic speeds. The results are used to discuss other approximate methods which have been used to predict the unsteady forces and noise generated by an aerofoil in turbulent flow.

Stably stratified and non-stratified flows past a touching pair of vortices with continuous velocity are considered. An asymptotic solution for the very long eddies is determined. Numerical results cover the whole range of subcritical stratification and eddy length.

The interaction of a weakly viscous solitary wave with a submerged semi-circular cylinder was examined using high-resolution two-dimensional numerical calculations. Two simulations were carried out: (a) as a baseline calculation, the propagation of a solitary wave over uniform depth; and (b) a solitary wave interacting with a submerged semi-circular cylinder. Large-scale simulations were performed to resolve the viscous boundary layers on the free surface, bottom and around the obstacle. Integral measures such as momentum and energy are analysed and compared against analytical approximations. For uniform depth, the loss in momentum and energy arises from the traction caused by the finite length of the domain bottom and the dissipation which is predominantly within the bottom boundary layer, respectively. The force on the cylinder is composed of (form) drag, inertial and hydrostatic components, the last factor arising from gradients in the height of the free surface. Morison’s semi-empirical equation is shown to provide a leading-order description of the force on the semi-circular cylinder. These elevated rates of change (momentum and energy) return to uniform depth values after a short period of time, indicating a localized effect of the obstacle. To interpret the flow field, vorticity, streamline and second invariant of the velocity gradient tensor plots were used to highlight relative thickness of boundary layers, vorticity distribution throughout the domain and stagnation points in the flow.

A new theoretical model is developed for the growth of a convecting fluid layer at the base of a stable, thermally stratified layer when heated from below. The imposed convective heat flux is taken to be comparable to the heat flux conducted down the background gradient so that diffusion ahead of the interface between the convecting and stable layers makes a significant contribution to the interfacial heat flux and to the rate of rise of the interface. Closure of the diffusion problem in the stable region requires the interfacial heat flux to be specified, and it is argued that this is determined by the ability of convective eddies to mix warmed fluid below the interface downwards. The interfacial velocity, which may be positive or negative, is then determined by the joint requirements of continuity of heat flux and temperature. A similarity solution is derived for the case of an initially linear temperature gradient and uniform heating. Solutions are also given for a heat flux that undergoes a step change and for a heat flux determined from a four-thirds power law with a fixed base temperature. Numerical calculations show that the predictions of the model are in good agreement with previously reported experimental measurements. Similar calculations are applicable to a wide range of geophysical problems in which the tendency for diffusive restratification is comparable to that for mixed-layer deepening by entrainment.

Prandtl's secondary mean motions of the second kind are driven by the variation of Reynolds stresses near resistive boundaries. In the flows considered here the turbulence is generated away from the boundary in the absence of a mean flow and then impacts onto a rigid surface placed into the flow at $t\,{=}\,0$. The initial development of the distorted flow is obtained using the linear approximation and the statistical analysis of rapid distortion theory, following Hunt & Graham (1978) assuming homogeneous stationary high-Reynolds-number turbulence with an integral length scale $L_\infty$ and r.m.s. velocity $v_\infty\prime$. First, the effects of axisymmetric anisotropy and of different forms of the spectra are analysed for turbulence impinging onto a plane surface lying at an angle $\alpha$ to the unit vector $\bf e$ of the axis of symmetry of the energy spectrum tensor $\Phi_{ij}(\mbox{{\boldmath k}})$. $R$ is defined as the ratio of the largest to smallest variances of the velocity components. The surface blocking leads to gradients of Reynolds shear stresses normal to the surface in the source layer $B^{(s)}$ with thickness of order $L_\infty$ and thence to a mean velocity $U(t) \,{\sim}\,{-}tv_\infty\prime^2 \sin 2\alpha (1\,{-}\,1/R) /L_\infty$ along the slope in the opposite direction of the projection of ${\boldmath e}$ onto the plane (i.e. in the direction ($\mbox{{\boldmath e}}\wedge\mbox{{\boldmath n}}$)$\wedge\mbox{{\boldmath n}}$ where ${\boldmath n}$ is the normal into the flow). $U$ is greatest near the surface where $y \,{\ll}\, L_\infty$. As a result of shear stresses being induced by the mean velocity gradient, a steady flow results over a time scale $T_L\,{=}\,L_\infty/v_\infty\prime$ – an order of magnitude estimate for the steady-state mean velocity is thence $U(t/T_L \,{\to}\, \infty) \,{\sim}\, v_\infty\prime(\sin 2\alpha (1\,{-}\,1/R))^{1/2}$. Secondly, the effect of a curved surface is studied by analysing isotropic turbulence near an undulating surface of wavelength $\Lambda$ and amplitude $H$, with a low slope so that $H \,{\ll}\, \Lambda$. The boundary condition of zero normal velocity at the curved surface generates larger irrotational fluctuations in the troughs, smaller fluctuations over the crest, and shear stresses over the slopes. The curl of the gradients of Reynolds normal and shear stresses within $B^{(s)}$ cause the growth of a mean vorticity which induces a mean velocity of order $-tv_\infty\prime^2/L_\infty$ within $B^{(s)}$ and a weaker recirculating velocity of order $-tv_\infty\prime^2/\Lambda$ in a deeper wave layer, $B^{(w)}$, with thickness of order $\Lambda$ outside $B^{(s)}$. The wavelength of the mean motion is $\Lambda$, with downward motions over the troughs and upward motion over the crest. As in the first case, a steady flow is predicted when $t/T_L \,{\gg}\, 1$. Anisotropic free-stream turbulence also induces mean motions on undulating surfaces with the same wavelength $\Lambda$ as that of the undulating surface, but the directions of these mean motions can be towards or away from the troughs/crests depending on the orientation of the anisotropy of the free stream. Flow visualization experiments conducted in a mixing box with oscillating anisotropic and isotropic grids demonstrated the existence of these mean flows and that they reach a steady state with an intensity and length scale comparable to those predicted. These results are also consistent with numerical simulation of Krettenauer & Schumann (1992) of convective turbulence over an undulating surface.

The focusing of a weak and nearly plane shock wave at an axisymmetric arête is examined. The initial-value problem governing the flow in the focal region is derived. A similitude is introduced which shows that the details of the flow depend on the initial shape and strength of the shock only through a simple rescaling of the variables. The dependence of the pressure coefficient on the physical parameters is given. It is seen that the maximum pressure coefficient is proportional to a parameter measuring the rate of focusing of the shock and to the square root of a parameter measuring the initial strength of the shock.