This paper describes an experimental and numerical investigation of concentrated vortex flow past a sphere in a constant-diameter pipe. As the swirl was increased at a fixed sphere Reynolds number of approximately 1100, the length of the mean downstream separation bubble decreased. For a small range of swirl intensity, an almost stagnant separation bubble formed on the upstream hemisphere. A further increase in swirl caused the bubble to become unstable and develop into an unsteady spiral disturbance. At very high swirl the downstream separation bubble was eliminated and an unsteady separation zone extended far upstream. Calculations of the vorticity field from surface fits to azimuthal and axial velocity data suggest that upstream separation is caused by the distortion of vortex filaments in the diverging flow approaching the sphere. Numerical solutions of steady inviscid axisymmetric flow past a sphere exhibit a fold in the vicinity of upstream separation. It is suggested that this accounts for the extreme sensitivity encountered in the experiments.

Axisymmetric gravity currents released from behind a lock over a solid horizontal bottom in a system rotating about a vertical axis are investigated. The major parameters are $\C$, the ratio of Coriolis to inertia forces, and $H$, the ratio of the total height of the ambient fluid into which propagation occurs to the initial height of the dense fluid inside the lock. We focus attention on the small $\C$, almost inviscid case in which a significant radius of propagation is obtained. A two-layer shallow-water inviscid approximation is developed and it is shown that the angular velocities in the upper and lower layers are coupled by simple relationships which can be used to simplify both the time-dependent and the steady-state lens solutions. However, the major features predicted by the two-layer model differ only slightly from these obtained with a one-layer model. Finite-difference solutions of the full Navier–Stokes equations are also presented and compared with the shallow-water approximations.

We search channel flow for unsteady solutions for different Reynolds numbers and configurations by extending a shooting method which was previously used to obtain a travelling-wave solution. A general initial condition is considered. A periodic-like solution to the incompressible Navier–Stokes equations in a minimal flow unit is found. One cycle of the solution consists of two typical intervals: a single-streak period and a double-streak period. The solution seems to be periodic; however, it cannot be distinguished from a heteroclinic cycle which consists of two heteroclinic orbits connecting two single-streak solutions, because the solution is tracked only for one and half periods.

A new method is introduced for calculating time-dependent, non-Newtonian flows of fluids described by integral constitutive equations. The starting point for the method is the integral form of the solution to the equation sof motion, valid in the limit of low Reynolds number. Because of the non-Newtonian nature of the fluid, this solution includes an integral over the domain of the flow, which is not present in boundary integral methods. This integral over the fluid volume (in three dimensions) or area (in two dimensions) is converted to a Lagrangian reference frame, and discretized for numerical evaluation. Because points in the integrand move with the fluid velocity, values of the non-Newtonian portion of the stress can be found by integrating the deformation at those points in conjunction with a suitable integral constitutive equation. The contribution to the total velocity field of the non-Newtonian stress at each fluid element is that of a point dipole, and the method bears many similarities to the point-vortex method for calculating inviscid flows. Like the point-vortex method, it is necessary to introduce cutoff functions that remove the singular nature of the dipole–dipole interactions. In addition, to render the method computationally feasible, the interactions between the dipoles must be calculated by the fast-multipole method or some comparable approach. Methods for calculating cutoff functions and implementing the fast multipole method are discussed, and simulation results are presented for one- and two-phase time-dependent flows of viscoelastic fluids between eccentric and concentric rotating cylinders.

The focus of this paper is the effect of spatial non-uniformity in the ambient flow on the forces acting on a rigid sphere when the sphere Reynolds number, $Re$, is in the range 10 to 300. Direct numerical simulations (DNS) based on a pseudospectral methodology are carried out to solve for the unsteady three-dimensional flow field around a sphere which is either held stationary or allowed to translate freely under the hydrodynamic forces. The various components of the total force, namely the inertial, steady viscous, and history forces, are systematically estimated in the context of linearly varying straining flows. The inertial forces are isolated by computing the rapid changes in the drag and lift forces in response to a rapid acceleration of the ambient flow. It is shown that the inertial forces arising due to convective acceleration at moderate Reynolds numbers follow the inviscid flow result. While the effect of temporal acceleration depends only on the sign and magnitude of the acceleration, the effect of convective acceleration is shown to depend also on the initial state of the ambient flow. A simple theoretical argument is presented to support the numerical observations. It is also shown that the effect of convective acceleration on the steady viscous force can be realized on a slower time scale. The results show that the history kernels currently available in the literature are not adequate to represent the effect of non-uniformity on the history force.

Modern linear control theory has recently been established as a viable tool for developing effective, spatially localized convolution kernels for the feedback control and estimation of linearized Navier–Stokes systems. In the present paper, the effectiveness of these kernels for significantly expanding the basin of attraction of the laminar state in a subcritical nonlinear channel flow system is quantified using direct numerical simulations for a range of Reynolds numbers ($Re_{\CL}=2000, 3000 {\rm and} 5000$) and for a variety of initial conditions of physical interest. This is done by quantifying the change in the transition thresholds (see Reddy et al. 1998) when feedback control is applied. Such transition thresholds provide a relevant measure of performance for transition control strategies even in the nonlinear regime. Initial flow perturbations with streamwise vortices, oblique waves, and random excitations over an array of several Fourier modes are considered. It is shown that the minimum amplitude of these initial flow perturbations that is sufficient to excite nonlinear instability, and thereby promote transition to turbulence, is significantly increased by application of the control feedback. The kernels used to apply the feedback are found to decay exponentially with distance far from the origin, as predicted by the analysis of Bamieh, Paganini & Dahleh (2002). In the present paper, it is demonstrated via numerical simulation that truncation of these spatially localized convolution kernels to spatially compact kernels with finite non-zero support does not significantly degrade the effectiveness of the control feedback. In addition to the new state-feedback control results, exponential convergence of a localized physical-space state estimator with wall measurements is also demonstrated. The estimator and the full-state feedback controller are then combined to obtain a wall-information-based linear compensator. The compensator performance is also quantified, and key issues related to improving the performance of this compensator, which is degraded compared with the full-state feedback controller, are discussed.

Torsional Couette flow between a rotating disk and a stationary wall is studied experimentally. The surface of the disk is either rigid or covered with a compliant coating. The influence of wall compliance on characteristic flow instabilities and on the laminar–turbulent flow transition is investigated. Data obtained from analysing flow visualizations are discussed. It is found that wall compliance favours two of the three characteristic wave patterns associated with the transition process and broadens the parameter regime in which these patterns are observed. The results for the effects of wall compliance on the third pattern are inconclusive. However, the experiments indicate that the third pattern is not a primary constituent of the laminar–turbulent transition process of torsional Couette flow.

We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose radius is changing in a prescribed manner. We construct a class of exact solutions of the Navier–Stokes equations in the case when the vessel radius is a function of time alone so that the cross-section is circular and uniform along the pipe axis. The Navier–Stokes equations admit solutions which are governed by nonlinear partial differential equations depending on the radial coordinate and time alone, and forced by the wall motion. These solutions correspond to a wide class of bounded radial stagnation-point flows and are of practical importance. In dimensionless terms, the flow is characterized by two parameters: $\D$, the amplitude of the oscillation, and $R$, the Reynolds number for the flow. We study flows driven by a time-periodic wall motion, and find that at small $R$ the flow is synchronous with the forcing and as $R$ increases a Hopf bifurcation takes place. Subsequent dynamics, as $R$ increases, depend on the value of $\Delta$. For small $\Delta$ the Hopf bifurcation leads to quasi-periodic solutions in time, with no further bifurcations occurring – this is supported by an asymptotic high-Reynolds-number boundary layer theory. At intermediate $\Delta$, the Hopf bifurcation is either quasi-periodic (for the smaller $\Delta$) or subharmonic (for larger $\Delta$), and the solutions tend to a chaotic attractor at sufficiently large $R$; the route to chaos is found not to follow a Feigenbaum scenario. At larger values of $\Delta$, we find that the solution remains time periodic as $R$ increases, with solution branches supporting periods of successive integer multiples of the driving period emerging. On a given branch the flow exhibits a self-similarity in both time and space and these features are elucidated by careful numerics and an asymptotic analysis. In contrast to the two-dimensional case (see Hall & Papageorgiou 1999) chaos is not found at either small or comparatively large $\Delta$.

Trapped-mode frequencies are computed for submerged horizontal circular cylinders in a two-layer fluid consisting of a finite layer bounded above by a free surface and below by an infinite layer of greater density. Two separate cases are examined. First we consider waves which propagate along the top of a single cylinder at frequencies below the cut-off for wave propagation away from the cylinder, and find modes where the motion is concentrated around the interface, modes which are concentrated around the free surface, and modes which involve significant motion of both the interface and the free surface. Secondly, we compute modes above the cut-off for a pair of submerged cylinders in the lower fluid.

High-Reynolds-number asymptotics and numerical simulations are used to describe two-dimensional, unsteady, pressure-driven flow in a finite-length channel, one wall of which contains a section of membrane under longitudinal tension. Asymptotic predictions of stability boundaries for small-amplitude, high-frequency, self-excited oscillations are derived in the limit of large membrane tension. The oscillations are closely related to normal modes of the system, which have a frequency set by a balance between membrane tension and the inertia of the fluid in the entire channel. Oscillations can grow by extracting kinetic energy from the mean Poiseuille flow faster than it is lost to viscous dissipation. Direct numerical simulations, based on a fully coupled finite-element discretization of the equations of large-displacement elasticity and the Navier–Stokes equations, support the predicted stability boundaries, and are used to explore larger-amplitude oscillations at lower tensions. These are characterized by vigorous axial sloshing motions superimposed on the mean flow, with transient secondary instabilities being generated both upstream and downstream of the collapsible segment.

The problem of nonlinear adjustment of localized front-like perturbations to a state of geostrophic equilibrium (balanced state) is studied in the framework of rotating shallow-water equations with no dependence on the along-front coordinate. We work in Lagrangian coordinates, which turns out to be conceptually and technically advantageous. First, a perturbation approach in the cross-front Rossby number is developed and splitting of the motion into slow and fast components is demonstrated for non-negative potential vorticities. We then give a non-perturbative proof of existence and uniqueness of the adjusted state, again for configurations with non-negative initial potential vorticities. We prove that wave trapping is impossible within this adjusted state and, hence, adjustment is always complete for small enough departures from balance. However, we show that retarded adjustment occurs if the adjusted state admits quasi-stationary states decaying via tunnelling across a potential barrier. A description of finite-amplitude periodic nonlinear waves known to exist in configurations with constant potential vorticity in this model is given in terms of Lagrangian variables. Finally, shock formation is analysed and semi-quantitative criteria based on the values of initial gradients and the relative vorticity of initial states are established for wave breaking showing, again, essential differences between the regions of positive and negative vorticity.

Mixing of a conserved scalar representing the mixture fraction, of primary importance in modelling non-premixed turbulent combustion, is studied by direct numerical simulation (DNS) in strongly turbulent planar shear layers both with and without heat release at a reaction sheet. For high heat release, typical of hydrocarbon combustion, the mixing is found to be substantially different than without heat release. The probability density function of the scalar and the conditional rate of scalar dissipation are affected by the heat release in such a way that the heat release substantially decreases the overall reaction rate. To help clarify implications of the assumptions underlying popular models for interaction between turbulence and chemistry, the local structure of the scalar dissipation rate at the reaction sheet is extracted from the DNS database. The applicability of flamelet models for the rate of scalar dissipation is examined. To assist in modelling, a characteristic length scale is defined, representing the distance around the reaction sheet over which the scalar field is locally linear, and statistical properties of this length scale are investigated. This length scale can be used in studying values of the rate of scalar dissipation that mark the boundary between flames that feel a constant scalar dissipation field and those that do not.

In this article we investigate time-periodic shear flows in the context of the two-dimensional vorticity equation, which may be applied to describe certain large-scale atmospheric and oceanic flows. The linear stability analyses of both discrete and continuous profiles demonstrate that parametric instability can arise even in this simple model: the oscillations can stabilize (destabilize) an otherwise unstable (stable) shear flow, as in Mathieu's equation (Stoker 1950). Nonlinear simulations of the continuous oscillatory basic state support the predictions from linear theory and, in addition, illustrate the evolution of the instability process and thereby show the structure of the vortices that emerge. The discovery of parametric instability in this model suggests that this mechanism can occur in geophysical shear flows and provides an additional means through which turbulent mixing can be generated in large-scale flows.

New measurements of the Nusselt number have been made in turbulent thermal convection confined in a cylindrical container of aspect ratio unity. The apparatus is essentially the same as that used by Niemela et al. (2000), except that the height was halved. The measurement techniques were also identical but the mean temperature of the flow was held fixed for all Rayleigh numbers. The highest Rayleigh number was $2 \times 10^{15}$. Together with existing data, the new measurements are analysed with the purpose of understanding the relation between the Nusselt number and the Rayleigh number, when the latter is large. In particular, the roles played by Prandtl number, aspect ratio, mean wind, boundary layers, sidewalls, and non-Boussinesq effects are discussed. Nusselt numbers, measured at the highest Rayleigh numbers for which Boussinesq conditions hold and sidewall forcing is negligible, are shown to vary approximately as a 1/3-power of the Rayleigh number. Much of the complexity in interpreting experimental data appears to arise from aspects of the mean flow, including complex coupling of its dynamics to sidewall boundary conditions of the container. Despite the obvious practical difficulties, we conclude that the next generation of experiments will be considerably more useful if they focus on large aspect ratios.

Dispersed two-phase flows often involve interfacial activities such as chemical reaction and phase change, which couple the fluid dynamics with heat and mass transfer. As a step toward understanding such problems, we numerically simulate the sedimentation of solid bodies in a Newtonian fluid with convection heat transfer. The two-dimensional Navier–Stokes and energy equations are solved at moderate Reynolds numbers by a finite-element method, and the motion of solid particles is tracked using an arbitrary Lagrangian–Eulerian scheme. Results show that thermal convection may fundamentally change the way that particles move and interact. For a single particle settling in a channel, various Grashof-number regimes are identified, where the particle may settle straight down or migrate toward a wall or oscillate laterally. A pair of particles tend to separate if they are colder than the fluid and aggregate if they are hotter. These effects are analysed in terms of the competition between the thermal convection and the external flow relative to the particle. The mechanisms thus revealed have interesting implications for the formation of microstructures in interfacially active two-phase flows.

Journal of Fluid Mechanics, vol. 448 (2001), pp. 289–313