In this paper, the effects of free-stream turbulence on stagnation-point flow and heat transfer are investigated through large eddy simulation (LES) of homogeneous isotropic turbulence impinging upon an isothermal elliptical leading edge. Turbulent mean flow and Reynolds stress profiles along the stagnation streamline, where the mean flow is strain dominant, and at different downstream locations, where the mean flow gradually becomes shear-dominated, are used to characterize evolution of the free-stream turbulence. The Reynolds stress budgets are also obtained, and the turbulence anisotropy is analysed through the balance between the mean flow strain and the velocity pressure gradient correlation. In the presence of free-stream turbulence, intense quasi-streamwise vortices develop near the leading edge with a typical diameter of the order of the local boundary-layer thickness. These strong vortices cause the thermal fluxes to peak at a location much closer to the wall than that of the Reynolds stresses, resulting a greater sensitivity to free-stream turbulence for the heat transfer than the momentum transfer. The heat transfer enhancement obtained by the present LES agrees quantitatively with available experimental measurements. The present LES results are also used to examine the eddy viscosity and pressure-strain correlations in Reynolds stress turbulence models.

From analytical investigations it is well known that the roll-up of an inviscid plane vortex sheet which separates at the edge of a body is a self-similar process which can be described by scaling laws. Unlike plane vortices, ring vortices have a curved rotational axis. For this special vortex type experimental investigations as well as calculations in the literature suggest that the scaling laws are only partially valid. The main goal of this work is to clarify how far these similarity or scaling laws are also valid for the formation of viscid laminar vortex rings. Therefore, the formation process of laminar vortex rings was investigated numerically using a CFD (computational-fluid-dynamics) code. The calculations refer to an experimental setup for which detailed experimental data are available in the literature. In this setup, laminar ring vortices are generated by ejecting water from a circular tube into a quiescent environment by means of a piston. First, a case based on a constant piston velocity was investigated. Comparing calculated and measured data yields a very good agreement. Further calculations were made when forcing the velocity of the piston by three different time-dependent functions. The results of these calculations show that the formation laws for inviscid plane vortices are also valid for the formation process of viscid ring vortices. This applies to the normalized axial and radial position of the vortex centre as well as the normalized diameter of the vortex spiral. However, the similarity laws are valid only if the process is considered in a special frame of reference which moves in conjunction with the front of the jet and if the starting time of the formation process with respect to the starting time of the ejection is taken into account. Additionally, the formation of a ring vortex, which occurs during the start-up process of a free jet flow, was calculated. The results confirm a dependence for the motion of the jet front, which is known from analytical considerations and allows some interesting features to be identified.

The weakly nonlinear dynamics of large-scale perturbations in a viscoelastic flow is investigated both analytically, via asymptotic methods, and numerically. For sufficiently small elasticities, dynamics is ruled by a Cahn–Hilliard equation with a quartic potential. Physically, this amounts to saying that, for small elasticities, polymers do not alter the purely hydrodynamical mechanisms responsible for the nonlinear dynamics in the Newtonian case (i.e. without polymers). The approach to the steady state is quantitatively similar to the Newtonian case as well, the dynamics being ruled by the same kink–antikink interactions as in the Newtonian limit. The above scenario does not extend to large elasticities. We found a critical value above which polymers drastically affect the dynamics of large-scale perturbations. In this latter case, a new dynamics not observed in the Newtonian case emerges. The most evident fingerprint of the new dynamics is the slowing down of the annihilation processes which lead to the steady states via weaker kink–antikink interactions. In conclusion, polymers strongly affect the large-scale dynamics. This takes place via a reduction of drag forces we were able to quantify from the asymptotic analysis. This suggests a possible relation of this phenomenon with the dramatic drag-reduction effect taking place in the far turbulent regime.

The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of non-dimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a non-dimensional parameter B (inverse Burger number). The transcritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is close to unity is considered in the shallow-water (small-aspect-ratio) limit. For weakly rotating flow over a small isolated obstacle (M → 0) a similarity theory is developed in which the behaviour is shown to depend on the parameters Γ = (F−1)M−2/3 and ν = B1/2M−1/3. The flow pattern in this regime is determined by a nonlinear equation in which Γ and ν appear explicitly, termed here the ‘rotating transcritical small-disturbance equation’ (rTSD equation, following the analogy with compressible gasdynamics). The rTSD equation is forced by ‘equivalent aerofoil’ boundary conditions specific to each obstacle. Several qualitatively new flow behaviours are exhibited, and the parameter reduction afforded by the theory allows a (Γ, ν) regime diagram describing these behaviours to be constructed numerically. One important result is that, in a supercritical oncoming flow in the presence of sufficient rotation (ν ≳ 2), hydraulic jumps can appear downstream of the obstacle even in the absence of an upstream jump. Rotation is found to have the general effect of increasing the amplitude of any existing downstream hydraulic jumps and reducing the lateral extent and amplitude of upstream jumps. Numerical results are compared with results from a shock-capturing shallow-water model, and the (Γ, ν) regime diagram is found to give good qualitative and quantitative predictions of flow patterns at finite obstacle height (at least for M ≲ 0.4). Results are compared and contrasted with those for a two-dimensional obstacle or ridge, for which rotation also causes hydraulic jumps to form downstream of the obstacle and acts to attenuate upstream jumps.

Consideration of near-wall turbulence in the functional space that emphasizes the level of anisotropy of the velocity fluctuations not only provides an understanding of th causative physics behind remarkable effects of turbulent drag reduction, but also lead to the logical design of a surface topology which is shown experimentally to be capable o producing a significant reduction of viscous drag which far exceeds what has been achieved so far.

We examine the effects of the variation in dissipation-range resolution on the accuracy of inertial-range statistics and intermittency in terms of the direct numerical simulations of homogeneous turbulence and passive-scalar turbulence by changing the spatial resolution up to 20483 grid points while maintaining a constant Reynolds number at Rλ ≃ 180 or ≃ 420 and Schmidt number at Sc = 1. Although large fluctuations of the derivative fields depended strongly on Kmaxη and were underestimated when Kmaxη≃1, where Kmax is the maximum wavenumber in the computations and η is the mean Kolmogorov length, the behaviour of the spectra and the scaling exponents of the structure functions up to the eighth order in the range of scales greater than 10η was insensitive to variations in Kmaxη, even when Kmaxη≃1. The relationship between the spatial resolution and asymptotic tail of the probability density functions of the energy dissipation fields was studied using the multifractal model for dissipation, and the results were confirmed by comparison to the simulation data. Degradation of the statistics arises from modifications to the flow dynamics due to the finite wavenumber cutoff and the use of a coarser filter width for the data, which is obtained using a reasonable accuracy criterion for the flow dynamics. The effect of the former was less than that of the latter for the low-to-moderate-order statistics when Kmaxη≥1. We also discuss the universality of the inertial-range statistics with respect to variations in the dissipation-range characteristics.

Vortical flows with an axial (z-axis) swirl and a toroidal circulation (in the (rho,z)-plane) can be observed in a wide range of fluid mechanical phenomena such as flow around rotary machines or natural vortices like tornadoes and hurricanes. In this paper, we obtain exact analytical solutions for a general class of steady systems with such three-dimensional circulating structures. Assuming incompressible ideal fluid, a general single-variable equation, known as the Squire–Long equation, can be constructed which can uniquely describe the velocity fields with steady axial and toroidal circulations. In this paper, we consider the case where this type of flow can be analysed by solving a linear homogeneous partial differential equation. The derived equation resembles the governing equation of the hydrogen problem. As a result, we obtain a quantization relation which is similar to the expression for the quantized energy states in a hydrogen atom.

For circulating flows, this formalism provides a complete set of orthogonal basis functions which are regular and localized. Hence, each of the basis solutions can be used as a simplified model for a realistic phenomenon. Moreover, an arbitrary circulating field can be expanded in terms of these orthogonal functions. Such an expansion can be potentially useful in the study of more general vortices. As illustrations, we present a few examples where we solve the linear homogeneous equation to analyse fluid mechanical systems which can be models for circulating flow in confined geometry. First, we consider three-dimensional vortices confined between two parallel planar walls. Our examples include flows between two infinite planar walls, inside and outside a vertical cylinder bounded at the ends by horizontal plates, and in an axially confined annular region. Then we describe the special way in which the basis functions should be superposed so that a complicated steady velocity-field with three-dimensional vortical structures can be constructed. Two such cases are discussed to indicate that the derived solutions can be used for complicated fluid mechanical modelling.

In this theoretical study, a linear spatio-temporal analysis is performed on unconfined and confined inviscid jet/wake flows in order to determine whether they are absolutely or convectively unstable. The impulse response is evaluated in the entire outer fluid, rather than just at the point of impulse, over a wide range of density ratios. This confirms that the dominant saddle point can validly migrate into the plane of diverging eigenfunctions. This reveals that, at certain density ratios and shear numbers, the response can grow upstream in some directions with a cross-stream component, even though it decays directly upstream of, and at, the point of impulse. This type of flow is convectively unstable when unconfined, but becomes absolutely unstable when confined. Other effects of confinement are described in a previous paper. Together, these articles have important implications for the design of fuel injectors, which often employ confined shear flows at high Reynolds number and large density ratios to generate strong mixing in combustion chambers.

A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus–Benjamin–Feir instability, the most basic features of which had been captured by a complex Ginzburg–Landau equation model. Here, we analyse this instability in rotating convection using direct numerical simulations of the Navier–Stokes equations in the Boussinesq approximation. Several properties of the wall modes are computed, extending the results to far beyond the onset of convection. Extensive favourable comparisons between our numerical results and previous experimental observations and complex Ginzburg–Landau model results are made.

Hypersonic flows involving two or more bodies travelling in close proximity to one another are encountered in several important situations. The present work seeks to explore one aspect of the resulting flow problem by investigating the forces experienced by a secondary body when it is within the domain of influence of a primary body travelling at hypersonic speeds.

An analytical methodology based on the blast wave analogy is developed and used to predict the secondary force coefficients for simple geometries in both two and three dimensions. When the secondary body is entirely inside the primary shocked region, the nature of the lateral force coefficient is found to depend strongly on the relative size of the two bodies. For two spheres, the methodology predicts that the secondary body will experience an exclusively attractive lateral force if the secondary diameter is larger than one-sixth of the primary diameter. The analytical results are compared with those from numerical simulations and reasonable agreement is observed if an appropriate normalization for the relative lateral displacement of the two bodies is used.

Results from a series of experiments in the T5 hypervelocity shock tunnel are also presented and compared with perfect-gas numerical simulations, with good agreement. A new force-measurement technique for short-duration hypersonic facilities, enabling the experimental simulation of the proximal bodies problem, is described. This technique provides two independent means of measurement, and the agreement observed between the two gives a further degree of confidence in the results obtained.

We consider a neutrally buoyant and initially uncharged drop in a second liquid subjected to a uniform electric field. Both liquids are taken to be leaky dielectrics. The jump in electrical properties creates an electric stress balanced by hydrodynamic and capillary stresses. Assuming creeping flow conditions and axisymmetry of the problem, the electric and flow fields are solved numerically withboundary integral techniques. The system is characterized by the physical property ratios R (resistivities), Q (permitivities) and λ (dynamic viscosities). Depending on these parameters, the drop deforms into a prolate or an oblate spheroid. The relative importance of the electric stress and of the drop/medium interfacial tension is measured by the dimensionless electric capillary number, Cae. For λ = 1, we present a survey of the various behaviours obtained for a wide range of R and Q. We delineate regions in the (R,Q)-plane in which the drop either attains a steady shape under any field strength or reaches a fold-point instability past a critical Cae. We identify the latter with linear instability of the steady shape to axisymmetric disturbances. Various break-up modes are identified, as well as more complex behaviours such as bifurcations and transition from unstable to stable solution branches. We also show how the viscosity contrast can stabilize the drop or advance break-up in the different situations encountered for λ = 1.

This paper is concerned with the nonlinear interaction between a planar and a pair of oblique Tollmien–Schlichting (T-S) waves which are phase-locked in that they travel with (nearly) the same phase speed. The evolution of such a disturbance is described using a high-Reynolds-number asymptotic approach in the so-called ‘upper--branch’ scaling regime. It follows that there exists a well-defined common critical layer (i.e. a thin region surrounding the level at which the basic flow velocity equals the phase speed of the waves to leading order) and the dominant interactions take place there. The disturbance is shown to evolve through several distinctive stages. In the first of these, the critical layer is in equilibrium and viscosity dominated. If a small mismatching exists in the phase speeds, the interaction between the planar and oblique waves leads directly to super-exponential growth/decay of the oblique modes. However, if the modes are perfectly phase-locked, the interaction in the first instance affects only the phase of the amplitude function of the oblique modes (so causing rapid wavelength shortening), while the modulus of the amplitude still evolves exponentially until the wavelength shortening produces a back reaction on the modulus (which then induces a super-exponential growth). Whether or not there is a small mismatch or a perfect match in the phase speeds, once the growth rate of the oblique modes becomes sufficiently large, the disturbance enters a second stage, in which the critical layer becomes both non-equilibrium and viscous in nature. The oblique modes continue to experience super-exponential growth, albeit of a different form from that in the previous stages, until the self-interaction between them, as well as their back effect on the planar mode, becomes important. At that point, the disturbance enters a third, fully interactive stage, during which the development of the disturbance is governed by the amplitude equations with the same nonlinear terms as previously derived for the phase-locked interaction of Rayleigh instability waves. The solution develops a singularity, leading to the final stage where the flow is governed by fully nonlinear three-dimensional inviscid triple-deck equations. The present work indicates that seeding a planar T-S wave can enhance the amplification of all oblique modes which share approximately its phase speed.

We consider the dynamics of a hollow cylindrical shell that is filled with viscous fluid and another, nested solid cylinder, and allowed to roll down an inclined plane. A mathematical model is compared to simple experiments. Two types of behaviour are observed experimentally: on steeper slopes, the device accelerates; on shallower inclines, the cylinders rock and roll unsteadily downhill, with a speed that is constant on average. The theory also predicts runaway and unsteady rolling motions. For the rolling solutions, however, the inner cylinder cannot be suspended in the fluid by the motion of the outer cylinder, and instead falls inexorably toward the outer cylinder. Whilst ‘contact’ only occurs after an infinite time, the system slows progressively as the gap between the cylinders narrows, owing to heightened viscous dissipation. Such a deceleration is not observed in the experiments, suggesting that some mechanism limits the approach to contact. Coating the surface of the inner cylinder with sandpaper of different grades changes the rolling speed, consistent with the notion that surface roughness is responsible for limiting the acceleration.

The Stokes formula for the resistance force exerted on a sphere moving with constant velocity in a fluid is extended to the case of micropolar fluids. A non-homogeneous boundary condition for the micro-rotation vector is used: the micro-rotation on the boundary of the sphere is assumed proportional to the rotation rate of the velocity field on the boundary.

Internal gravity waves excited by the turbulent motions in a bottom Ekman layer are examined using large-eddy simulation. The outer flow is steady and uniformly stratified while the density gradient is set to zero at the flat lower wall. After initializing with a linear density profile, a mixed layer forms near the wall separated from the ambient stratification by a pycnocline. Two types of internal wave are observed. Waves with frequencies larger than the free-stream buoyancy frequency are seen in the pycnocline, and vertically propagating internal waves are observed in the outer layer with characteristic frequency and wavenumber spectra. Since a signature of the pycnocline waves is observed in the frequency spectrum of the mixed layer, these waves may affect the boundary-layer turbulence. The dominant outer-layer waves have a group velocity directed 35-60° from the vertical axis, which is consistent with previous laboratory studies. The energy flux associated with the radiated waves is small compared to the integrated dissipation in the boundary layer, but is of the same order as the integrated buoyancy flux. A linear model is proposed to estimate the decay in wave amplitude owing to viscous effects. Starting from the observed wave amplitudes at the bottom of the pycnocline, the model prediction for the spectral distribution of the outer layer wave amplitude compares favourably with the simulation results.

The flow field due to two normal impinging liquid jets is different from the flow field associated with a single normal impinging liquid jet, and even from the flow field around two normal impinging compressible fluid jets. Depending on the spacing between the two jets and their relative strengths, different kinds of hydraulic jump interactions are possible, resulting in a variety of flow patterns. The present study experimentally elucidates the jump--jump interactions formed in such cases, for different values of inter-jet spacings and for different strengths of the individual jets. Analogous flow fields associated with the interactions between a single impinging jet and a fence are also studied to allow convenient experimental flow vizualizations.

A new result is derived for the mass flow rate per unit volume through a scalar iso-surface – called here the ‘iso-surface mass flow density’. The relationship of the surface mass flow density to the local entrainment rate per unit volume in scalar mixing and to the local reaction rate in turbulent premixed combustion is considered. In inhomogeneous flows, integration of the surface mass flow density across the layer in the direction of the mean scalar inhomogeneity yields the mean entrainment velocity in scalar mixing and the turbulent burning velocity in premixed combustion. For non-premixed turbulent reacting flow, this new result is shown to be consistent with the classical result of Bilger (Combust. Sci. Technol. vol. 13, 1976, p. 155) for fast one-step irreversible chemical reactions. Direct numerical simulation data for conserved scalar mixing, isothermal reaction front propagation and turbulent premixed flames are analysed. It is found that the entrainment velocity in the conserved scalar mixing case is sensitive to a threshold value. This suggests that the entrainment velocity is not a well-defined concept in temporally developing mixing layers and that scaling laws for the viscous superlayer warrant further investigation. In the isothermal reaction fronts problem, the characteristics of iso-surface propagation in a low Damköhler number regime are investigated. In premixed flames, the effects of non-stationarity on the turbulent burning velocity are addressed. The difference from the existing methods for determining turbulent burning velocity, and the implications of the present results for flames with multi-dimensional complex geometry are discussed. It is also shown that the surface mass flow density is related to the turbulent scalar flux in statistically stationary one-dimensional premixed flames. Variations of the local propagation characteristics due to departure from an unstretched laminar flame structure are shown to decrease the tendency to counter-gradient transport in turbulent premixed flames.

Electrowetting has recently been explored as a mechanism for moving small amounts of fluids in confined spaces. We propose a diffuse-interface model for drop motion, due to electrowetting, in a Hele-Shaw geometry. In the limit of small interface thickness, asymptotic analysis shows that the model is equivalent to Hele-Shaw flow with a voltage-modified Young–Laplace boundary condition on the free surface. We show that details of the contact angle significantly affect the time scale of motion in the model. We measure receding and advancing contact angles in the experiments and derive their influence through a reduced-order model. These measurements suggest a range of time scales in the Hele-Shaw model which include those observed in the experiment. The shape dynamics and topology changes in the model agree well with the experiment, down to the length scale of the diffuse-interface thickness.

Three-dimensional highly resolved simulations are presented for cylindrical density currents using the Boussinesq approximation for small density difference. Three Reynolds numbers (Re) are investigated (895, 3450 and 8950, which correspond to values of the Grashof number of 105, 1.5 × 106 and 107, respectively) in order to identify differences in the flow structure and dynamics. The simulations are performed using a fully de-aliased pseudospectral code that captures the complete range of time and length scales of the flow. The simulated flows present the main features observed in experiments at large Re. As the current develops, it transitions through different phases of spreading, namely acceleration, slumping, inertial and viscous Soon after release the interface between light and heavy fluids rolls up forming Kelvin–Helmholtz vortices. The formation of the first vortex sets the transition between acceleration and slumping phases. Vortex formation continues only during the slumping phase and the formation of the last Kelvin–Helmholtz vortex signals the departure from the slumping phase. The coherent Kelvin–Helmholtz vortices undergo azimuthal instabilities and eventually break up into small-scale turbulence. In the case of planar currents this turbulent region extends over the entire body of the current, while in the cylindrical case it only extends to the regions of Kelvin–Helmholtz vortex breakup. The flow develops three-dimensionality right from the beginning with incipient lobes and clefts forming at the lower frontal region. These instabilities grow in size and extend to the upper part of the front. Lobes and clefts continuously merge and split and result in a complex pattern that evolves very dynamically. The wavelength of the lobes grows as the flow spreads, while the local Re of the flow decreases. However, the number of lobes is maintained over time. Owing to the high resolution of the simulations, we have been able to link the lobe and cleft structure to local flow patterns and vortical structures. In the near-front region and body of the current several hairpin vortices populate the flow. Laboratory experiments have been performed at the higher Re and compared to the simulation results showing good agreement. Movies are available with the online version of the paper.

Harmonic axial motion of the inner cylinder in the Taylor–Couette system can efficiently shift the onset of instability to larger inner cylinder rotation rates. However, once instability has set in, a rapid sequence of symmetry-breaking bifurcations results in complex spatio-temporal dynamics even for very low post-critical values of the rotation rate. Using spectral computations, we present a detailed study of this sudden transition, shedding light on the nature of the complex flows observed in recent laboratory experiments. In particular, it is shown that these bifurcations are responsible for some of the experimentally observed frequencies which had been attributed to background noise. Movies are available with the online version of the paper.