The small-amplitude wave modes inside a ducted inviscid compressible swirling flow are investigated. In order to avoid possible mathematical ambiguities arising from the use of an inviscid flow model, the wave modes are cast as the solution of an initial boundary value problem. Two families of propagating waves are found. The acoustic modes are supported by the compressibility effect of the flow. The rotational modes are sustained by the centrifugal force field associated with the mean flow rotation. Two cases, one with a free-vortex swirl and the other with a rigid-body swirl, are investigated in some depth. Numerical results are provided.

By using a Reynolds-averaged two-dimensional computation of a turbulent flow over an airfoil at post-stall angles of attack, we show that the massively separated and disordered unsteady flow can be effectively controlled by periodic blowing–suction near the leading edge with low-level power input. This unsteady forcing can modulate the evolution of the separated shear layer to promote the formation of concentrated lifting vortices, which in turn interact with trailing-edge vortices in a favourable manner and thereby alter the global deep-stall flow field. In a certain range of post-stall angles of attack and forcing frequencies, the unforced random separated flow can become periodic or quasi-periodic, associated with a significant lift enhancement. This opens a promising possibility for flight beyond the static stall to a much higher angle of attack. The same local control also leads, in some situations, to a reduction of drag. On a part of the airfoil the pressure fluctuation is suppressed as well, which would be beneficial for high-α buffet control. The computations are in qualitative agreement with several recent post-stall flow control experiments. The physical mechanisms responsible for post-stall flow control, as observed from the numerical data, are explored in terms of nonlinear mode competition and resonance, as well as vortex dynamics. The leading-edge shear layer and vortex shedding from the trailing edge are two basic constituents of unsteady post-stall flow and its control. Since the former has a rich spectrum of response to various disturbances, in a quite wide range the natural frequency of both constituents can shift and lock-in to the forcing frequency or its harmonics. Thus, most of the separated flow becomes resonant, associated with much more organized flow patterns. During this nonlinear process the coalescence of small vortices from the disturbed leading-edge shear layer is enhanced, causing a stronger entrainment and hence a stronger lifting vortex. Meanwhile, the unfavourable trailing-edge vortex is pushed downstream. The wake pattern also has a corresponding change: the shed vortices of alternate signs tend to be aligned, forming a train of close vortex couples with stronger downwash, rather than a Kármán street.

Trajectories of inertial spheroidal particles moving in a shear flow near a solid wall are calculated numerically from the Stokes flow equations by computing the hydrodynamic forces and torques acting on the particles. Near the wall these interactions cause coupling between the particle's rotational and translational motions. Due to this coupling an inertial spheroid is shown to move along an oscillatory trajectory, while simultaneously drifting towards the wall. This phenomenon occurs in the absence of gravity as a combined effect of three factors: particle non-spherical shape, its inertia and particle–wall hydrodynamic interactions. This drift is absent for inertialess spheroids, and also for inertial spherical particles which move along flow streamlines.

The drift velocity is calculated for various particle aspect ratios γ and relaxation times τ. An approximate solution, valid for small particle inertia is developed, which allows the contribution of various terms to the drift velocity to be elucidated. It was found that the maximum value of the drift velocity prevails for N(γ)γ2τs∼4, where s is the shear rate and N(γ) is a decreasing function of γ, related to the particle–wall hydrodynamic interactions. In the limiting cases of large and small inertia and also of very long and thin spheroids, the drift vanishes.

Possible applications of the results are discussed in the context of transport of micrometre particles in microgravity conditions. It is shown that the model used is applicable for analysis of the deposition of aerosol particles with sizes above 10 μm inhaled in the human respiratory tract in the absence of gravity.

Turbulent-shear-induced coagulation of monodisperse particles was examined experimentally in the nearly isotropic, spatially decaying turbulence generated by an oscillating grid. The 3.9 μm polystyrene microspheres used in the experiments were made neutrally buoyant and unstable by suspending them in a density-matched saline solution. In this way, particle settling, double-layer repulsion and particle inertia were negligible and the effect of turbulent shear was isolated. The coagulation rate was measured by monitoring the loss of singlet particles as a function of time and reactor turbulence intensity. By restricting consideration to experimental conditions where the singlet concentration was in excess, the effect of higher-order aggregate (i.e. triplet) formation was negligible and nonlinear regression using an integral rate expression that included terms for doublet formation and breakup was used to obtain the turbulent coagulation rate constant. The strength of the van der Waals attractions was characterized with the Hamaker constant obtained from Brownian coagulation experiments. Since particle bulk mixing was fast compared to the coagulation rate, the observed coagulation rate constants were averages over the local coagulation rates within the grid-stirred reactor. Knowledge of the spatial variation of turbulence within the reactor was necessary for quantitative prediction of the experiments because model predictions for the coagulation rate are nonlinear functions of shear rate. The investigation was conducted with particles smaller than the length scales of turbulence and since the smallest turbulent length scales, the Kolmogorov scales, have the highest shear rate they controlled the rate of particle aggregation. The distribution of the Kolmogorov shear rate at various grid oscillation frequencies was obtained by measuring the turbulent kinetic energy (E) using acoustic Doppler velocimetry and relating E to the Kolmogorov shear rate using scaling arguments. The experimentally measured turbulent coagulation rate constants were significantly lower than theoretical predictions that neglect interparticle interactions; however, simulations that included particle interactions showed excellent agreement with the experimental results. The favourable comparison provides evidence that the computer simulations capture the important physics of turbulent coagulation. That is, particle transport on length scales comparable to the particle radius controls the rate of turbulent shear coagulation and particle interactions are significant.

The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as ‘axisymmetrization’, is studied asymptotically and numerically. The vortex is perturbed at t=0 and differential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for infinite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity field is integrated against a smooth test function the result decays asymptotically as t−λ with λ=1+(n2+8)1/2, where n is the azimuthal wavenumber of the perturbation and n[ges ]1. The far-field stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).

Previous research has suggested that isolated, initially non-axisymmetric vortices in two-dimensional flows tend to become axisymmetric, in a coarse-grained sense, by purely inviscid mechanisms. That research, however, considered only vortices with broadly distributed vorticity. In this paper, it is shown that vortices with sufficiently steep edge gradients behave in a radically different way; in particular they can remain non-axisymmetric, apparently indefinitely. Such vortices, it is argued, are more typical in inviscid two-dimensional flows than the broadly distributed vortices previously considered, and hence the tendency for vortices to become axisymmetric is not generic to these flows.

Many wake flows exhibit self-excited flow oscillations which are sustained by the flow itself and are not caused by amplification of external noise. The archetypal example of a self-excited wake flow is the low Reynolds number flow past a circular cylinder. This flow exhibits self-sustained periodic vortex shedding above a critical Reynolds number. In general, control of such flows requires stabilization of many globally unstable modes; the present work describes a multiple-sensor control strategy for the cylinder wake which succeeds in controlling a simplified wake model at a Reynolds number above that at which single-sensor schemes fail.

Representation of the flow field by a finite set of coherent structures or modes, which are extracted by proper orthogonal decomposition and correspond to the large-scale wake components, allows the efficient design of a closed-loop control algorithm. A neural network is used to furnish an empirical prediction of the modal response of the wake to external control forcing. This model avoids the need for explicit representation of the control actuator–wake interaction. Additionally, the neural network structure of the model allows the design of a robust nonlinear control algorithm. Furthermore the controller does not necessarily require velocity field information, but can control the wake using other quantities (for example flow visualization pictures) which characterize the structure of the velocity field. Successful control of a simplified cylinder wake model is used to demonstrate the feasibility of the low-dimensional control strategy.

The average settling velocity of heavy particles under a body force field is studied numerically in stationary homogeneous isotropic turbulent flows generated by the direct numerical simulation and the large eddy simulation of the continuity and Navier–Stokes equations. The flow fields corresponding to different selected ranges of turbulent scales are obtained by filtering the results of the direct numerical simulation, and employed for calculating the particle motion. Wang & Maxey (1993) showed that as a consequence of the particle accumulation in the low vorticity region and the preferential sweeping phenomenon, the average settling rate in turbulence is greater than that in still fluid. In the present study, the phenomenon of particle accumulation in the low vorticity region is found to be controlled mainly by the small scales with wavenumber kω corresponding to the maximum of the dissipation (vorticity) spectrum. However, the increase of the average settling rate, 〈ΔvS〉, also depends strongly on the large energetic eddies. The small eddies with wavenumber greater than 2.5kω have essentially no effect on the particle accumulation and the average settling velocity. The large eddy simulation is thus adequate for the present study provided the smallest resolved scale is greater than 1/(2.5kω). Detailed calculations show that 〈ΔvS〉 is maximized and is of order u′/10 when τp/τk≈1 and vd/u′≈0.5 for Rλ=22.6–153, where τp is the particle's relaxation time, τk is the Kolmogorov timescale, vd is the settling rate in still fluid, u′ is the root mean square of the velocity fluctuation, and Rλ is the Reynolds number based on the Taylor microscale.

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

A detailed two-part computational investigation is conducted into the dynamical evolution of two-dimensional miscible porous media flows in the quarter five-spot arrangement of injection and production wells. High-accuracy direct numerical simulations are performed that reproduce all dynamically relevant length scales in solving the vorticity–streamfunction formulation of Darcy's law. The accuracy of the method is demonstrated by a comparison of simulation data with linear stability results for radial source flow.

Within this part, Part 1 of the present investigation, a series of simulations is discussed that demonstrate how the mobility ratio and the dimensionless flow rate denoted by the Péclet number Pe affect both local and integral features of homogeneous displacement processes. Mobility ratios up to 150 and Pe-values up to 2000 are investigated. For sufficiently large Pe-values, the flow near the injection well gives rise to a vigorous viscous fingering instability. As the unstable concentration front approaches the central region of the domain, nonlinear interactions between the fingers similar to those known from unidirectional flows are observed, such as merging, partial merging, and shielding, along with secondary tip-splitting and side-branching instabilities. At large Pe-values, several of these fingers compete for long times, before one of them accelerates ahead of the others and leads to the breakthrough of the front.

In contrast to unidirectional flows, the quarter five-spot geometry imposes both an external length scale and a time scale on the flow. The resulting spatial non-uniformity of the potential base flow is observed to lead to a clear separation in space and time of large and small scales in the flow. Small scales occur predominantly during the early stages near the injection well, and at late times near the production well. The central domain is dominated by larger scales.

Taken together, the results of the simulations demonstrate that both the mobility ratio and Pe strongly affect the dynamics of the flow. While some integral measures, such as the recovery at breakthrough, may show only a weak dependence on Pe for large Pe-values, the local fingering dynamics continue to change with Pe.

The increased susceptibility of the flow to perturbations during the early stages provides the motivation to formulate an optimization problem that attempts to maximize recovery, for a constant overall process time, by employing a time-dependent flow rate. Within the present framework, which accounts for molecular diffusion but not for velocity-dependent dispersion, simulation results indeed indicate the potential to increase recovery by reducing the flow rate at early times, and then increasing it during the later stages.

Direct numerical simulations are employed to investigate the coupling between the viscous fingering instability and permeability heterogeneities for miscible displacements in quarter five-spot flows. Even moderate inhomogeneities are seen to have a strong effect on the flow, which can result in a complete bypass of the linear growth phase of the viscous fingering instability. In contrast to their homogeneous counterparts (cf. Part 1, Chen & Meiburg 1998), heterogeneous quarter five-spot flows are seen to exhibit a more uniform dominant length scale throughout the entire flow domain. In line with earlier findings for unidirectional displacements, an optimal interaction of the mobility and permeability related vorticity modes can occur when the viscous length scale is of the same order as the correlation length of the heterogeneities. This resonance mechanism results in a minimal breakthrough recovery for intermediate correlation lengths, at fixed dimensionless flow rates in the form of a Péclet number Pe. However, for a constant correlation length, the recovery does not show a minimum as Pe is varied.

Confirming earlier observations, the simulations show a more rapid breakthrough as the variance of the permeability variations increases. However, this tendency is far more noticeable in some parameter regimes than in others. It is furthermore observed that relatively low variances usually cannot change the tendency for a dominant finger to evolve along the inherently preferred diagonal direction, especially for relatively small correlation lengths. Only for higher variances, and for larger correlation lengths, are situations observed in which an off-diagonal finger can become dominant. Due to the nonlinear nature of the selection mechanisms at work, a change in the variance of the heterogeneities can result in the formation of dominant fingers along entirely different channels.

In this paper the two-layer flow past a cylinder in a rotating frame is modelled numerically. This study follows on from the many theoretical and experimental examinations of the one-layer flow past a cylinder. The main aim of this study is to investigate the geophysically relevant effect on the flow of varying the vertical shear, while keeping the other parameters the same in both layers. This will include an examination of the contour plots of vorticity and streamfunction and measuring the size of the separated region and the position of separation as a function of the vertical shear. The most interesting result is that the separated region in the slower layer grows much larger than that in the faster layer as the vertical shear is increased.

Two free waves propagating in a parallel shear flow generate a critical layer when their nonlinear interaction induces a perturbation whose phase velocity matches the basic-state velocity somewhere in the flow domain. The condition necessary for this to occur may be interpreted as a resonance condition for a triad formed by the two waves and a (singular) mode of the continuous spectrum associated with the shear. The formation of the critical layer is investigated in the case of freely propagating Rossby waves in a two-dimensional inviscid flow in a β-channel.

A weakly nonlinear analysis based on a normal-mode expansion in terms of Rossby waves and modes of the continuous spectrum is developed; it leads to a system of amplitude equations describing the evolution of the two Rossby waves and of the modes of the continuous spectrum excited during the interaction. The assumption of weak nonlinearity is not however self-consistent: it breaks down because nonlinearity always becomes strong within the critical layer, however small the initial amplitudes of the Rossby waves. This demonstrates the relevance of nonlinear critical layers to monotonic, stable, unforced shear flows which sustain wave propagation.

A nonlinear critical-layer theory is developed that is analogous to the well-known theory for forced critical layers. Differences arise because of the presence of the Rossby waves: the vorticity in the critical layer is advected in the cross-stream direction by the oscillatory velocity field due to the Rossby waves. An equation is derived which governs the modification of the Rossby waves that results from their interaction; it indicates that the two Rossby waves are undisturbed at leading order. An analogue of the Stewartson–Warn–Warn analytical solution is also considered.

We study the stability of the interface between (a) two adjacent viscous layers flowing due to gravity through an inclined or vertical channel that is confined between two parallel plane walls, and (b) two superimposed liquid films flowing down an inclined or vertical plane wall, in the limit of Stokes flow. In the case of channel flow, linear stability analysis predicts that, when the fluids are stably stratified, the flow is neutrally stable when the surface tension vanishes and the channel is vertical, and stable otherwise. This behaviour contrasts with that of the gravity-driven flow of two superimposed films flowing down an inclined plane, where an instability has been identified when the viscosity of the fluid next to the plane is less than that of the top fluid, even in the absence of fluid inertia. We investigate the nonlinear stages of the motion subject to finite-amplitude two-dimensional perturbations by numerical simulations based on boundary-integral methods. In both cases of channel and film flow, the mathematical formulation results in integral equations for the unknown interface and free-surface velocity. The properties of the integral equation for multi-film flow are investigated with reference to the feasibility of computing a solution by the method of successive substitutions, and a deflation strategy that allows an iterative procedure is developed. In the case of channel flow, the numerical simulations show that disturbances of sufficiently large amplitude may cause permanent deformation in which the interface folds or develops elongated fingers. The ratio of the viscosities and densities of the two fluids plays an important role in determining the morphology of the emerging interfacial patterns. Comparing the numerical results with the predictions of a model based on the lubrication approximation shows that the simplified approach can only describe a limited range of motions. In the case of film flow down an inclined plane, we develop a method for extracting the properties of the normal modes, including the ratio of the amplitudes of the free-surface and interfacial waves and their relative phase lag, from the results of a numerical simulation for small deformations. The numerical procedure employs an adaptation of Prony's method for fitting a signal described by a time series to a sum of complex exponentials; in the present case, the signal is identified with the cosine or sine Fourier coefficients of the interface and free-surface waves. Numerical simulations of the nonlinear motion confirm that the deformability of the free surface is necessary for the growth of small-amplitude perturbations, and show that the morphology of the interfacial patterns developing subject to finite-amplitude perturbations is qualitatively similar to that for channel flow.

Journal of Fluid Mechanics, vol. 268 (1994), pp. 231–265

It has recently come to our attention that our paper, which describes Marangoni-driven flow near a contact line, overlooks solutions involving a general thermal boundary condition on the free surface (private communication, S. J. Tavener 1997). These new solutions are applicable for non-isothermal flows in a corner region where one boundary is a rigid plane (and either perfectly insulating or perfectly conducting) and the other is a free surface upon which a general thermal boundary condition is applied. We describe these additional solutions below.