In this paper, we revisit the eddy viscosity formulation to highlight a number of important issues that have direct implications for the prediction of near-wall turbulence. For steady wall-bounded turbulent flows, we make the equilibrium assumption between rates of production ($P$) and dissipation ($\epsilon $) of turbulent kinetic energy ($k$) in the near-wall region to propose that the eddy viscosity should be given by ${\nu }_{t} \approx \epsilon / {S}^{2} $, where $S$ is the mean shear rate. We then argue that the appropriate velocity scale is given by $\mathop{(S{T}_{L} )}\nolimits ^{- 1/ 2} {k}^{1/ 2} $ where ${T}_{L} = k/ \epsilon $ is the turbulence (decay) time scale. The difference between this velocity scale and the commonly assumed velocity scale of ${k}^{1/ 2} $ is subtle but the consequences are significant for near-wall effects. We then extend our discussion to show that the fundamental length and time scales that capture the near-wall behaviour in wall-bounded shear flows are the shear mixing length scale ${L}_{S} = \mathop{(\epsilon / {S}^{3} )}\nolimits ^{1/ 2} $ and the mean shear time scale $1/ S$, respectively. With these appropriate length and time scales (or equivalently velocity and time scales), the eddy viscosity can be rewritten in the familiar form of the $k$–$\epsilon $ model as ${\nu }_{t} = \mathop{(1/ S{T}_{L} )}\nolimits ^{2} {k}^{2} / \epsilon $. We use the direct numerical simulation (DNS) data of turbulent channel flow of Hoyas & Jiménez (Phys. Fluids, vol. 18, 2006, 011702) and the turbulent boundary layer flow of Jiménez et al. (J. Fluid Mech. vol. 657, 2010, pp. 335–360) to perform ‘a priori’ tests to check the validity of the revised eddy viscosity formulation. The comparisons with the exact computations from the DNS data are remarkable and highlight how well the equilibrium assumption holds in the near-wall region. These findings could prove to be useful in near-wall modelling of turbulent flows.

The two-dimensional vortices characterizing the bottom boundary layer of both progressive and solitary waves, recently discovered by experimental flow visualizations and referred to as vortex tubes, are studied by numerical solution of the governing equations. In the case of progressive waves, the Reynolds numbers investigated belong to the subcritical range, according to Floquet linear stability theory. In such a range the periodic generation of strictly two-dimensional vortex structures is not a self-sustaining phenomenon, being the presence of appropriate ambient disturbances necessary to excite certain modes through a receptivity mechanism. In a physical experiment such disturbances may arise from several coexisting sources, among which the most likely is roughness. Therefore, in the present numerical simulations, wall imperfections of small amplitude are introduced as a source of disturbances for both types of wave, but from a macroscopic point of view the wall can be regarded as flat. The simulations show that even wall imperfections of small amplitude may cause flow instability and lead to the appearance of vortex tubes. These vortices, in turn, interact with a vortex layer adjacent to the wall and characterized by vorticity opposite to that of the vortex tubes. In a first stage such interaction gives rise to corrugation of the vortex layer and this affects the spatial distribution of the wall shear stress. In a second stage the vortex layer rolls up and pairs of counter-rotating vortices are generated, which leave the bottom because of the self-induced velocity.

We examine pulse interaction and bound-state formation in interfacial turbulence using the problem of a falling liquid film as a model system. We perform direct numerical simulations (DNSs) of the full Navier–Stokes equations and associated wall and free-surface boundary conditions and we examine both analytically and numerically a low-dimensional (LD) model for the film. For a two-pulse system, DNSs reveal the existence of very rich and complex pulse interactions, characterized by either overdamped, underdamped or self-sustained oscillating behaviours, all of them found to be in excellent agreement with LD results. Having demonstrated the reliability of the LD model for two-pulse systems/smaller domains, we use it to investigate larger domains with many interacting pulses, where DNSs are computationally expensive. We demonstrate that such systems are likely to be dominated by a self-sustained oscillatory dynamics.

The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random multiplicative model of turbulence and calculating subdominant terms, we explain here why such corrections cannot be present.

The dynamics of inertial particles in turbulence is modelled and investigated by means of direct numerical simulation of an axisymmetrically expanding homogeneous turbulent strained flow. This flow can mimic the dynamics of particles close to stagnation points. The influence of mean straining flow is explored by varying the dimensionless strain rate parameter $Sk_{0}/{\it\epsilon}_{0}$ from 0.2 to 20, where $S$ is the mean strain rate, $k_{0}$ and ${\it\epsilon}_{0}$ are the turbulent kinetic energy and energy dissipation rate at the onset of straining. We report results relative to the acceleration variances and probability density functions for both passive and inertial particles. A high mean strain is found to have a significant effect on the acceleration variance both directly by an increase in the frequency of the turbulence and indirectly through the coupling of the fluctuating velocity and the mean flow field. The influence of the strain on the normalized particle acceleration probability distribution functions is more subtle. For the case of a passive particle we can approximate the acceleration variance with the aid of rapid-distortion theory and obtain good agreement with simulation data. For the case of inertial particles we can write a formal expression for the accelerations. The magnitude changes in the inertial particle acceleration variance and the effect on the probability density function are then discussed in a wider context for comparable flows, where the effects of the mean flow geometry and of the anisotropy at small scales are present.

A small region (called the surface tension region) where pressure differences across the oil–water and oil–air interfaces are important is shown to exist between the gravity viscous and monolayer regions in a spreading oil slick (Di Pietro, Huh & Cox 1978). The importance of this new region is that (i) it is necessary in order to connect the gravity–viscous and monolayer regions and (ii) it is a region where slopes of interfaces are large. This idea is used to find the thickness profile of an oil layer contained upstream of a barrier (an oil boom) placed across a channel in which water is flowing at a constant velocity. The assumption is also made that the velocity difference across the oil layer is small compared with the water velocity. The general conditions for the validity of the results are then discussed together with the modifications to the theory which are necessary if the boundary layer in the water below the oil should be turbulent rather than laminar. Good agreement is found to exist between experimental results for unsteady spreading on quiescent water in a channel and the results of the theory applied to this situation assuming quasi-steady spreading.

An analytical solution is presented for steady inviscid separated flows modelled by hollow vortices, that is, by closed vortex sheets bounding a region with fluid at rest. Steady flows past arbitrary obstacles protruding from an infinite wall are considered. The solution is similar to that of the vortex patch model; it depends on two free parameters that define the size of the hollow vortex and the location of the separation point. When a sharp edge constrains the separation point (Kutta condition), the solution depends on a single parameter. As with the vortex patch model, families of growing vortices exist, which represent the continuation of desingularized point vortices. Numerical results are presented for the flows past a semicircular bump, a Ringleb snow cornice and a normal flat plate. The differences from the previous results found in the literature are analysed and discussed with the present solutions for the flow past a normal flat plate.

The flow field produced by a shock wave reflecting from a helium gas-liquid interface was investigated using a cryogenic shock tube. Incident and reflected shock waves were observed in the gas; transmitted first- and second-sound shocks were observed in the liquid. Wave diagrams are constructed to compare the data with theoretical wave trajectories. Qualitative agreement between data and theory is shown. Quantitative differences between data and theory indicate a need for further analysis of both the gas-liquid interface and the propagation of nonlinear waves in liquid helium.

This work was a first step in the experimental investigation of a complex non-equilibrium state. The results demonstrate clearly the usefulness of the cryogenic shock tube as a research tool. The well-controlled jump in temperature and pressure across the incident shock wave provides unique initial conditions for the study of dynamic phenomena in superfluid helium.

Efforts to model accretion disks in the laboratory using Taylor–Couette flow apparatus are plagued with problems due to the substantial impact the end plates have on the flow. We explore the possibility of mitigating the influence of these end plates by imposing stable stratification in their vicinity. Numerical computations and experiments confirm the effectiveness of this strategy for restoring the axially homogeneous quasi-Keplerian solution in the unstratified equatorial part of the flow for sufficiently strong stratification and moderate layer thickness. If the rotation ratio is too large, however (e.g. ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{3/2}$, where ${\it\Omega}_{o}/{\it\Omega}_{i}$ is the angular velocity at the outer/inner boundary and $r_{i}/r_{o}$ is the inner/outer radius), the presence of stratification can make the quasi-Keplerian flow susceptible to the stratorotational instability. Otherwise (e.g. for ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{1/2}$), our control strategy is successful in reinstating a linearly stable quasi-Keplerian flow away from the end plates. Experiments probing the nonlinear stability of this flow show only decay of initial finite-amplitude disturbances at a Reynolds number $Re=O(10^{4})$. This observation is consistent with most recent computational (Ostilla-Mónico, et al.J. Fluid Mech., vol. 748, 2014, R3) and experimental results (Edlund & Ji, Phys. Rev. E, vol. 89, 2014, 021004) at high $Re$, and reinforces the growing consensus that turbulence in cold accretion disks must rely on additional physics beyond that of incompressible hydrodynamics.

We calculate the force on a periodic array of spheres in a viscous flow at small Reynolds number and for small volume fraction. This generalizes the known results for the force on a periodic array due to Stokes flow (zero Reynolds number) and the Oseen correction to the Stokes formula for the force on a single sphere (zero volume fraction). We use a generalization of Hasimoto's approach that is based on an analysis of periodic Green's functions. We compare our results to the phenomenological ones of Kaneda for viscous flow past a random array of spheres.

We investigate stability with respect to two-dimensional (independent of z) disturbances of plane-parallel shear flows with a velocity profile Vx=u(y) of a rather general form, monotonically growing upwards from zero at the bottom (y=0) to U0 as y → ∞ and having no inflection points, in an ideal incompressible fluid stably stratified in density in a layer of thickness ℓ, small as compared to the scale L of velocity variation. In terms of the ‘wavenumber k – bulk Richardson number J’ variables, the upper and lower (in J) boundaries of instability domains are found for each oscillation mode. It is shown that the total instability domain has a lower boundary which is convex downwards and is separated from the abscissa (k) axis by a strip of stability 0 < J < J0(−)(k) with minimum width J*=O(ℓ2/L2) at kL=O(1). In other words, the instability domain configuration is such that three-dimensional (oblique) disturbances are first to lose their stability when the density difference across the layer increases. Hence, in the class of flows under consideration, it is a three- not two-dimensional turbulence that develops as a result of primary instability.

The stability of a two-dimensional inviscid flow in an annulus between two permeable cylinders is examined. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It is shown that, independent of the direction of the radial flow, the basic flow is unstable to small two-dimensional perturbations provided that the ratio of the azimuthal component of the velocity to the radial one is sufficiently large. The instability is oscillatory and persists if the viscosity of the fluid is taken into consideration.

Two-dimensional steady thermocapillary flows in a liquid layer over a substrate, which has a uniform temperature and sinusoidal topography, are investigated by asymptotic theory. Here, the buoyancy effect is negligible and the interface is not significantly disturbed under low Marangoni number and low capillary number. A temperature gradient along the gas/liquid interface causes recirculating flows. For a small aspect ratio, which yields a sinusoidal topography with a long wavelength relative to the mean depth of the liquid layer, the second-order solutions are obtained analytically. The basic solutions show vertical diffusion of heat and vorticity from the substrate and interface, respectively. In the second corrections, the horizontal diffusion of heat weakens the overall flow and the convection of heat intensifies it.

A state of permanent, isothermal non-wetting of a solid surface by a normally wetting liquid may be achieved if the surface moves tangentially to a liquid drop that is pressed against it. Surrounding gas is swept into the space between the liquid and solid creating a lubricating film that prevents wetting. The length scales of the drop and the film are typically three or more orders of magnitude different, making numerical simulation difficult from a resolution standpoint. The present paper focuses on a hybrid approach employing lubrication theory for the thinnest portions of the gas film and numerical simulation for the liquid and outer gas phases.

In this paper, we consider a viscous instability of a stratified boundary layer that is a form of the familiar Tollmien–Schlichting (T-S) waves modified by a stable density stratification. As with the usual T-S waves, the triple-deck formalism was employed to provide a self-consistent description of linear and nonlinear instability properties at asymptotically large Reynolds numbers. The effect of stratification on the temporal and spatial linear growth rates is studied. It is found that stratification reduces the maximum spatial growth rate, but enhances the maximum temporal growth rate. This viscous instability may offer a possible alternative explanation for the origin of certain long atmospheric waves, whose characteristics are not well predicted by inviscid instabilities. In the high-frequency limit, the nonlinear evolution of the disturbances is shown to be governed by a nonlinear amplitude equation, which is an extension of the well-known Benjamin–Davis–Ono equation. Numerical solutions indicate that as a spatially isolated disturbance evolves, it radiates a beam of long gravity waves, and meanwhile small-scale ripples develop on its front to form a well-defined wavepacket. It is also shown that for jet-like velocity profiles, the standard triple-deck theory must be adjusted to account for both the displacement and transverse pressure variation induced by the inviscid flow in the main layer. The nonlinear evolution of high-frequency disturbances is governed by a mixed KdV–Benjamin–Davis–Ono equation.

The wave resistance of a two-dimensional pressure distribution which moves steadily over water of finite depth is computed with the aid of four approximate methods: (i) consistent small-amplitude perturbation expansion up to third order; (ii) continuous mapping by Guilloton's displacements; (iii) small-Froude-number Baba & Takekuma's approximation; and (iv) Ursell's theory of wave propagation as applied by Inui & Kajitani (1977). The results are compared, for three fixed Froude numbers, with the numerical computations of von Kerczek & Salvesen for a given smooth pressure patch. Nonlinear effects are quite large and it is found that (i) yields accurate results, that (ii) acts in the right direction, but quantitatively is not entirely satisfactory, that (iii) yields poor results and (iv) is quite accurate. The wave resistance is subsequently computed by (i)-(iv) for a broad range of Froude numbers. The perturbation theory is shown to break down at low Froude numbers for a blunter pressure profile. The Inui-Kajitani method is shown to be equivalent to a continuous mapping with a horizontal displacement roughly twice Guilloton's. The free-surface nonlinear effect results in an apparent shift of the first-order resistance curve, i.e. in a systematic change of the effective Froude number.

Circular hydraulic jumps are familiar in single layers. Here we report the discovery of similar jumps in two-layer flows. A thin jet of fluid impinging vertically onto a rigid horizontal plane surface submerged in a deep layer of less-dense miscible fluid spreads radially, and a near-circular internal jump forms within a few centimetres from the point of impact with the plane surface. A jump is similarly formed as a jet of relatively less-dense fluid rises to the surface of a deep layer of fluid, but it appears less stable or permanent in form. Several experiments are made to examine the case of a downward jet onto a horizontal plate, the base of a square or circular container. The inlet Reynolds numbers, Re, of the jet range from 112 to 1790. Initially jumps have an undular, laminar form with typically 2–4 stationary waves on the interface between the dense and less-dense layers but, as the depth of the dense layer beyond the jump increases, the transitions become more abrupt and turbulent, resulting in mixing between the two layers. During the transition to a turbulent regime, single and sometimes moving multiple cusps are observed around the periphery of jumps. A semi-empirical model is devised that relates the parameters of the laboratory experiment, i.e. flow rate, inlet nozzle radius, kinematic viscosity and reduced gravity, to the layer depth beyond the jump and the radius at which an undular jump occurs. The experiments imply that surface tension is not an essential ingredient in the formation of circular hydraulic jumps and demonstrate that stationary jumps can exist in stratified shear flows which can be represented as two discrete layers. No stationary circular undular jumps are found, however, in the case of a downward jet of dense fluid when the overlying, less-dense, fluid is stratified, but a stationary turbulent transition is observed. This has implications for the existence of stationary jumps in continuously stratified geophysical flows: results based on two-layer models may be misleading. It is shown that the Froude number at which a transition of finite width occurs in a radially diverging flow may be less than unity.

The steady flow of a stream emerging from a nozzle, hitting a horizontal plate and falling under gravity is considered. Depending on the length of the plate L and the Froude number F, the plate can either divert the stream or lead to its detachment. First, the problem is reformulated using conformal mappings. The resulting problem is then solved by a collocation Galerkin method; a particular form is assumed for the solution, and certain coefficients in that representation are then found numerically by satisfying Bernoulli's equation on the free surfaces at certain discrete points. The resulting equations are solved by Newton's method, yielding various configurations of the solution based on the values of F and L. The lift exerted on the plate is computed and discussed. If the plate is long enough, physically meaningful solutions are found to exist only for values of F greater than or equal to a certain critical value F0, which is to be determined. Results are presented, both for F > F0 where the detachment is horizontal and for F = F0 where the detachment point is a stagnation point at a 120° corner. Related asymmetric flows where the rising stream is inclined are also studied.

An analytical theory is developed for a class of stable quasi-geostrophic vortices propagating either in the westward direction with supercritical velocities (c<−βR2d) or eastward, where Rd is the radius of deformation and β is the gradient of the Coriolis parameter. The numerical spectral calculations, initiated by the theoretical solutions, indicate that the supercritical vortices move initially with the predicted velocity, but later slow down to the speed of the long planetary waves. The period of time during which an eddy is propagating with its initial velocity is analysed as a function of its strength and (initial) speed. The theoretical solution has a distinctive streamline signature (‘alpha gyre’), which also appears in the final state of the numerical calculations, including those for an initially completely symmetric vortex placed on the β-plane.

We analyse the stability of plane Poiseuille flow with a streamwise system rotation. It is found that the instability due to two-dimensional perturbations, which sets in at the well-known critical Reynolds number, Rc = 5772.2, for the non-rotating case, is delayed as the rotation is increased from zero, showing a stabilizing effect of rotation. As the rotation is increased further, however, the laminar flow becomes most unstable to perturbations which are three-dimensional. The critical Reynolds number due to three-dimensional perturbations at this higher rotation case is many orders of magnitude less than the corresponding value due to two-dimensional perturbations. We also perform a nonlinear analysis on a bifurcating three-dimensional secondary flow. The secondary flow exhibits a spiral vortex structure propagating in the streamwise direction. It is confirmed that an antisymmetric mean flow in the spanwise direction is generated in the secondary flow.