Dispersion of small particles emitted from an area source at the surface into a fully developed high-Reynolds-number boundary layer flow is studied as a theoretical model for pollen dispersion in the neutral atmospheric boundary layer. The particle plume above the area source is assumed to behave as a particle concentration boundary layer. Boundary layer scaling and the assumption of self-preservation lead to an analytical solution in the form of a similarity function that has an additional dependence on the ratio of gravitational settling and turbulent diffusion velocities. Similar arguments are used to predict patterns of deposition onto the surface downstream of the source. Theoretical predictions are tested using a suite of large-eddy-simulation numerical experiments, with good agreement. The combined analysis of theoretical and numerical results reveals interesting features in the patterns of downstream deposition, such as non-monotonic trends in isolation distance with particle settling velocity and surprisingly large isolation distances for practically relevant parameter ranges. Possible effects of turbulence on effective settling velocity are highlighted.

The nonlinear dynamics of a viscous filament surrounded by a second viscous fluid arranged in a core-annular configuration when a radial electric field acts in the annular region, are studied analytically and computationally using boundary element methods. The flow is characterized by the viscosity ratio, an electric Weber number measuring the strength of the electric field, a geometrical parameter measuring the thickness of the undisturbed annular region, as well as a computational parameter that fixes the wavenumber of the undulations. Axisymmetric solutions are computed by direct numerical simulations in the Stokes limit for general values of the parameters when the two fluids have equal viscosities, and an asymptotic theory is carried out to produce a novel evolution equation for thin film dynamics valid when the undisturbed annular thickness is small and the viscosity ratio is of order one. It is established (in agreement with previous computations in the absence of electric fields) that a sufficiently thick annulus enables thread breakup while a sufficiently thin one (approximately one fifth of the undisturbed thread radius for the case of equal viscosities, for instance) suppresses pinching and drives the interface to approach the tube wall asymptotically without actually touching it. The present simulations show that the electric field affects the dynamics drastically in several ways. First, it promotes interfacial wall touchdown in finite time and a comparison between direct simulations and the asymptotic solutions are in fair agreement. Second, the electric field acts to suppress pinching in the sense that solutions that lead to jet breakup due to a thick enough viscous annulus are driven to wall touchdown. When pinching takes place we find that the ultimate pinching solutions are self-similar and recover the non-electrified ones to leading order for the range of parameters studied.

A wide range of applicability of the Reynolds analogy between turbulent momentum and heat transport implies inherent difficulty in diminishing or enhancing skin friction and heat transfer independently. In the present study, we introduce suboptimal control theory for achieving a dissimilar control of enhancing heat transfer, while keeping the skin friction not increased considerably in a fully developed channel flow. The Fréchet differentials clearly show that the responses of velocity and temperature fields to wall blowing/suction are quite different, due to the fact that the velocity is a divergence-free vector field while the temperature is a conservative scalar field. This essential difference allows us to achieve dissimilar control even in flows where the averaged momentum and energy transport equations have an identical form. It is also found that the resultant optimized mode of control input exhibits a streamwise travelling-wave-like property. By exploring the phase relationship between the travelling-wave-like control input and the velocity and thermal fields, we reveal that such control input contributes to dissimilar heat transfer enhancement via two different mechanisms, i.e. direct modification of the coherent components of the Reynolds shear stress and the turbulent heat flux, and indirect effects on the incoherent components, through modification of the mean velocity and temperature profiles. Based on these results, a simple open-loop strategy for dissimilar control is proposed and assessed.

We report an experimental investigation of the longitudinal space–time cross-correlation function of the velocity field, , in a cylindrical turbulent Rayleigh–Bénard convection cell using the particle image velocimetry (PIV) technique. We show that while Taylor’s frozen-flow hypothesis does not hold in turbulent thermal convection, the recent elliptic model advanced for turbulent shear flows (He & Zhang, Phys. Rev. E, vol. 73, 055303) is valid for the present velocity field for all over the cell, i.e. the isocorrelation contours of the measured have an elliptical curve shape and hence can be related to via with and being two characteristic velocities. We further show that the fitted is proportional to the mean velocity of the flow, but the values of are larger than the theoretical predictions. Specifically, we focus on two representative regions in the cell: the region near the cell sidewall and the cell’s central region. It is found that and are approximately the same near the sidewall, while at the cell centre.

We study the stability of a thin, Newtonian viscous sheet floating on a bath of denser fluid. We first derive a general set of equations governing the evolution of a nearly flat sheet, accounting for geometrical nonlinearities associated with moderate rotations. We extend two classical models by considering arbitrary external body and surface forces; these two models follow from different scaling assumptions, and are derived in a unified way. The equations capture two modes of deformation, namely viscous bending and stretching, and describe the evolution of thickness, mid-surface and in-plane velocity as functions of two-dimensional coordinates. These general equations are applied to a floating viscous sheet, considering gravity, buoyancy and surface tension. We investigate the stability of the flat configuration when subjected to arbitrary in-plane strain. Two unstable modes can be found in the presence of compression. The first one combines undulations of the centre-surface and modulations of the thickness, with a wavevector perpendicular to the direction of maximum applied compression. The second one is a buckling mode; it is purely undulatory and has a wavevector along the direction of maximum compression. A nonlinear analysis yields the long-time evolution of the undulatory mode.

Deformation of a viscous drop in shear at finite inertia and the streamlines around it are numerically investigated. Inertia destroys the closed streamlines found in Stokes flow. It creates reversed streamlines and streamlines spiralling around the vorticity axis. Spiralling streamlines spiral either towards the central shear plane or away from it depending on the viscosity ratio and the inertia. The zones of open or reversed streamlines as well as streamlines spiralling towards or away from the central shear plane are delineated for varying viscosity ratio and Reynolds number. In contrast to the infinite extent of the closed Stokes streamlines around a rigid sphere in shear, the region of the spiralling streamlines in the vorticity direction both for a rigid sphere and a drop shrinks with inertia. Inertia increases deformation, and introduces oscillations in drop shape. An approximate analysis explains the scaling of oscillation frequency and damping with Reynolds and capillary numbers. The steady-state drop inclination angle with the flow axis increases with increasing Reynolds number for small Reynolds number. But it decreases at higher Reynolds number, especially for larger capillary numbers. For smaller capillary numbers, drop inclination reaches higher than (the direction of maximum extension), critically affecting the interfacial stresses due to the drop. It changes the sign of first and second normal interfacial stress differences (and thereby these components of the effective stresses of an emulsion of such drops). Increasing viscosity ratio orients the drop towards the flow axis, which increases the critical Reynolds number above which the drop inclination reaches more than .

A computational large eddy simulation (LES) study is presented of the interaction between a turbulent boundary layer separating from a rounded ramp in a duct and a pair of spanwise-periodic, round synthetic jets, actuated upstream of the nominal separation line. Several scenarios are considered, for different injection angles and velocity ratios. In all cases, the actuation frequency corresponds to the shedding-instability mode of the separated shear layer. Experimental data, available for both the baseline flow and one actuated configuration, are used to verify the validity of the computational solutions. The analysis includes a separation of coherent and stochastic contributions to the time-averaged statistics of the auto- and cross-correlations of the fluctuations. The control authority is examined by reference to the effects of the actuation on the size of the separated zone, the momentum thickness of the boundary layer, the velocity field, various turbulence quantities and phase-averaged properties. The study demonstrates that the principal aspect of the interaction, at mean-flow level, is an increase in mixing provoked by the formation of strong streamwise vortices away from the wall, the induction of much weaker streamwise vortices close to the wall, and the extra production of stochastic turbulence caused by unsteady straining. The coherent stresses arising from the periodic perturbations are high – typically 5 times the levels of the unperturbed flow – but only within about 5–7 diameters of the jet orifice, and 2 orifice diameters on each side of the jet, and these are dominant primarily in the outer parts of the boundary layer. Stochastic turbulence is also elevated, but more modestly. The global effect of the actuation is a reduction of 10–20 % in the length of the separated region and 20–40 % in the thickness of the reverse-flow layer, depending on the actuation scheme, counter-flow actuation being the most effective. This reduction is mainly associated with a delay in separation. These results highlight the need for synthetic jets to be placed close to the separation zone and for the inter-jet distance to be of order 5 or lower to achieve a high level of separation-control authority.

We investigate the classical problem of the extension of an axisymmetric viscous thread by a fixed applied force with small initial inertia and small initial surface tension forces. We show that inertia is fundamental in controlling the dynamics of the stretching process. Under a long-wavelength approximation, we derive leading-order asymptotic expressions for the solution of the full initial-boundary value problem for arbitrary initial shape. If inertia is completely neglected, the total extension of the thread tends to infinity as the time of pinching is approached. On the other hand, the solution exhibits pinching with finite extension for any non-zero Reynolds number. The solution also has the property that inertia eventually must become important, and pinching must occur at the pulled end. In particular, pinching cannot occur in the interior as can happen when inertia is neglected. Moreover, we derive an asymptotic expression for the extension.

Motivated by complex multi-fluid geometries currently being explored in fibre-device manufacturing, we study capillary instabilities in concentric cylindrical flows of fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier–Stokes problem. Generalizing previous work by Tomotika (), Stone & Brenner (, equal viscosities) and others, we present a full linear stability analysis of the growth modes and rates, reducing the system to a linear generalized eigenproblem in the Stokes case. Furthermore, we demonstrate by Plateau-style geometrical arguments that only axisymmetric instabilities need be considered. We show that the case is already sufficient to obtain several interesting phenomena: limiting cases of thin shells or low shell viscosity that reduce to problems, and a system with competing breakup processes at very different length scales. The latter is demonstrated with full three-dimensional Stokes-flow simulations. Many cases remain to be explored, and as a first step we discuss two illustrative cases, an alternating-layer structure and a geometry with a continuously varying viscosity.

A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number () and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely , is not a sufficient condition for instability. It is also shown that the critical value of required for instability, where is the length of the region in a wave in which and is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that (where is the growth rate of the instability averaged over , the period in which parcels of fluid are subjected to ) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely and , are shown to be valid only for a limited parameter range.

Velocity-gradient evolution in compressible turbulence is modelled with an autonomous dynamical system of equations that are able to explain important non-isentropic, Mach-number and viscous effects. This enhanced homogenized Euler equation (EHEE) model is validated against the Burgers equation and direct numerical simulation (DNS) computations at the appropriate limits of Mach number. The EHEE model qualitatively captures crucial nonlinear physical features – especially various dilatational aspects of the flow field – observed in DNS over a range of Mach, Reynolds and Prandtl numbers. The model can serve as the basis of more quantitative statistical and stochastic closure models.

Three-dimensional transitional flow of a rarefied monatomic gas over a spinning sphere is studied numerically by the direct simulation Monte Carlo method. Gas molecules interact with each other as hard spheres. The Maxwell model of specular–diffuse scattering is used to describe the interaction between gas molecules and the sphere surface. The effect of all dimensionless governing parameters of the problem on the flow structure, distributions of stresses and heat flux density on the sphere surface, aerodynamic force and torque exerted on the sphere, and heat flux on the sphere surface is analysed. Simulations are conducted at Mach numbers raging from to and Knudsen numbers raging from to . Two qualitatively different streamline patterns are observed around the sphere depending on its dimensionless rotational velocity. The direction and magnitude of the transverse Magnus force in transitional flow depend on the Knudsen and Mach numbers. The torque and heat flux coefficients are found to be functions of the Mach number and dimensionless rotational velocity. The effect of rotation on the sphere aerodynamics weakens with an increase in the sphere temperature with respect to the gas temperature in the free stream. A complete set of equations is developed to fit the computed values of the aerodynamic and heat flux coefficients in a case when the sphere temperature is equal to the temperature of the free stream.

Classical stability theory for swept leading-edge boundary layers predicts eigenmodes in the free stream with algebraic decay far from the leading edge. In this article, we extend the classical base flow solution by Hiemenz to a uniformly valid solution for the flow upstream of a bluff body, which includes a three-dimensional boundary layer, an inviscid stagnation-point flow and an outer parallel flow. This extended, uniformly valid base flow additionally supports modes which diverge algebraically outside the boundary layer. The theory of wave packet pseudomodes is employed to derive analytical results for the growth rates and for the eigenvalue spectra of this type of mode. The complete spectral analysis of the flow, including the algebraically diverging modes, will give a more appropriate basis for receptivity studies and will more accurately describe the interaction of perturbations in the free stream with disturbances in the boundary layer.

The mechanisms by which turbulent shear flow causes waves on a gas–liquid interface are studied analytically, with a critical assessment of the possible role played by wave-induced Reynolds stresses (WIRSs). First, turbulent flow past a corrugated surface of a small slope is analysed; the surface can either be stationary or support a travelling wave. This problem serves as a useful model because direct numerical simulation (DNS) and experimental data are available to test the analysis, and because this picture is itself a model for the fully coupled two-layer problem. It is demonstrated that the WIRSs play no significant role in shear-driven turbulent flow past a moving wavy wall, and that they alter the structure of the flow only in a quantitative fashion in the pressure-driven case. In the shear-driven case in particular, excellent agreement is obtained with previously reported DNS results. Two closure assumptions are made in our model: the first concerns the wave-induced dissipation of turbulent kinetic energy; the second concerns the importance of rapid distortion. The results of our calculations are sensitive to the assumptions used to close the wave-induced dissipation but are insensitive to the details of the rapid-distortion modelling. Finally, the fully coupled two-layer problem is addressed in the setting of waves of small amplitude, where it is demonstrated that the WIRSs do not play a significant role in the growth of interfacial waves, even at relatively high Reynolds numbers. Again, good agreement is obtained between data from experiments and DNS.

The present study is motivated by a need to produce stability modes to assist in the understanding and control of unsteady separated flows. The flow configuration is a NACA 0015 aerofoil with laminar leading-edge separation and turbulent recirculation. In previous water tunnel experiments, this flow configuration was measured in an unperturbed (uncontrolled) separated state, and a harmonically perturbed (controlled) reattached state. This study presents numerical data of the unperturbed case, and recovers stability modes to describe the evolution of perturbations in this environment. The unperturbed flow is numerically generated using large eddy simulation. Its temporal properties are quantified via a Fourier analysis of the velocity time history at selected points in space. The leading-edge shear layer instability is characterized by instantaneous vortex structures, and the bluff body shedding is illustrated by proper orthogonal decomposition modes. Statistical measures of the velocity field agree well with the water tunnel measurements. Finally a stability analysis is undertaken using a triple decomposition to distinguish between the time averaged field, the unsteady scales of motion, and a coherent wave (perturbation). This analysis identifies that perturbations in the region immediately downstream of the separated shear layer have the highest spatial growth rates. The associated frequency is of the order of the sub-harmonic of the shear layer instability.

The pattern in an image of flow visualizations using reflective flakes stems from their non-uniform orientation rather than their spatial accumulation. It is shown, based on the assumption that flakes are infinitely thin elliptic discs without inertia, that the temporal evolution of their orientations is identical to that for infinitesimal material surface elements. In general, bright regions in a visualized image are the superposition of those where the flake (i.e. the material surface element) orientation is isotropic and those where flakes tend to align in the direction for which the incident rays are reflected into the line of sight. A non-trivial example of the visualization of a steady flow in a precessing sphere is given to verify these conclusions.

Experimental observations of vortex breakdown in a rotating lid-driven cavity are presented. The results show that vortex breakdown for cavities with high aspect ratios is associated with the appearance of stable helical vortex multiplets. By using results from stability theory generalizing Kelvin’s problem on vortex polygon stability, and systematically exploring the cavity flow, we succeeded in identifying two new stable vortex breakdown states consisting of triple and quadruple helical multiplets.