The problem of fluid motions in the form of inertial waves or inertial oscillations in an incompressible viscous fluid contained in a rotating spheroidal cavity was first formulated and studied by Poincaré (1885) and Bryan (1889). Upon realizing the limitation of Bryan's implicit solution using complicated modified spheroidal coordinates, Kudlick (1966) proposed a procedure that may be used to compute an explicit solution in spheroidal coordinates. However, the procedure requires an analytical expression for the $N$ real and distinct roots of a polynomial of degree $N$, where $N$ is a key parameter in the problem. When $0\,{\le}\,N\,{\le}\,2$, an explicit solution can be derived by using Kudlick's procedure. When $3\,{\le}\,N\,{\le}\,4$, the procedure cannot be practically used because the analytical expression for the $N$ distinct roots becomes too complicated. When $N\,{>}\,4$, Kudlick's procedure cannot be used because of the non-existence of an analytical expression for the $N$ distinct roots. For the inertial wave problem, Kudlick thus restricted his analysis to several modes for $1\,{\le}\,N\,{\le}\,2$ with the azimuthal wavenumbers $1\,{\le}\,m\,{\le}\,2$. We have found the first explicit general analytical solution of this classical problem valid for $0\,{\le}\,N\,{<}\,\infty$ and $0\,{\le}\,m\,{<}\,\infty$. The explicit general solution in spheroidal polar coordinates represents a possibly complete set of the inertial modes in an oblate spheroid of arbitrary eccentricity. The problem is solved by a perturbation analysis. In the first approximation, the effect of viscosity on inertial waves or oscillations is neglected and the corresponding inviscid solution, the pressure and the three velocity components in explicit spheroidal coordinates, is obtained. In the next approximation, the effect of viscous dissipation on the inviscid solution is examined. We have derived the first explicit general solution for the viscous spheroidal boundary layer valid for all inertial modes. The boundary-layer flux provides the solvability condition that is required to solve the higher-order interior problem, leading to an explicit general expression for the viscous correction of all inertial modes in a rapidly rotating, general spheroidal cavity. On the basis of the general explicit solution, some unusual and intriguing properties of the spheroidal inertial waves or oscillation are discovered. In particular, we are able to show that $\int_V ( {\bm u} \,{\bm \cdot}\, \nabla^2 {\bm u} )\,{\rm d}V \equiv 0$, where ${\bm u}$ is the velocity of any three-dimensional inviscid inertial waves or oscillations in an oblate spheroid of arbitrary eccentricity and $\int_V$ denotes three-dimensional integration over the volume of the spheroidal cavity.

The radial Navier–Stokes flow in a fracture bounded by impermeable corrugated rock surfaces is significantly different from the commonly used creeping flow model between two parallel surfaces, described by Darcy's law on the macroscale. Continuous variations in the Reynolds number along the radial coordinate determine the important role of the nonlinear inertial effects, which are reinforced by local oscillations of the velocity field caused by wall corrugation. The system behaviour is studied both analytically and numerically. A solution for the full system of Navier–Stokes equations in a thin cylindrical domain with oscillating walls is developed as a biparametric and two-scale asymptotic expansion with respect to fracture aperture and corrugation period. The numerical solution is derived based on the finite volume method. A new macroscale flow equation is obtained, which explicitly displays the relative roles of viscous dissipation caused by corrugation, local and global inertia, and cross inertia–viscous effects. The effective flow parameters are defined using analytical relationships.

We examine the phenomenon of polymer drag reduction in a turbulent flow through Brownian dynamics simulations. The dynamics of a large number of single polymer chains along their trajectories is investigated in a Newtonian turbulent channel flow. In particular, the FENE, FENE-P and multimode FENE models with realistic parameters are used to investigate the mechanisms of polymer stretching. A topological methodology is applied to characterize the ability of the flow to stretch the polymers. It is found using conditional statistics that at moderate Weissenberg number Wi the polymers, that are stretched to a large fraction of their maximum extensibility, have experienced a strong biaxial extensional flow. When Wi is increased other flow types can stretch the polymers but the few highly extended molecules again have, on average, experienced a biaxial extensional flow. Moreover, highly extended polymers are found in the near-wall regions around the quasi-streamwise vortices, essentially in regions of strong biaxial extensional flow.

A new approach to chemistry modelling for large-eddy simulation of turbulent reacting flows is developed. Instead of solving transport equations for all of the numerous species in a typical chemical mechanism and modelling the unclosed chemical source terms, the present study adopts an indirect mapping approach, whereby all of the detailed chemical processes are mapped to a reduced system of tracking scalars. Here, only two such scalars are considered: a mixture fraction variable, which tracks the mixing of fuel and oxidizer, and a progress variable, which tracks the global extent of reaction of the local mixture. The mapping functions, which describe all of the detailed chemical processes with respect to the tracking variables, are determined by solving quasi-steady diffusion-reaction equations with complex chemical kinetics and multicomponent mass diffusion. The performance of the new model is compared to fast-chemistry and steady-flamelet models for predicting velocity, species concentration, and temperature fields in a methane-fuelled coaxial jet combustor for which experimental data are available. The progress-variable approach is able to capture the unsteady, lifted flame dynamics observed in the experiment, and to obtain good agreement with the experimental data, while the fast-chemistry and steady-flamelet models both predict an attached flame.

The subject of the paper is the stability of the thermocapillary flow generated by an inclined temperature gradient. The investigation is carried out for water (Prandtl number $Pr = 7$) in the limit of a vanishing Biot number. By means of the linear stability analysis, the instability boundary of the return flow and the main instability modes have been found. The evolution of the convective regimes predicted by the linear stability theory and arising as a result of a nonlinear interaction has been studied by a numerical simulation of three-dimensional governing equations. Hexagonal cells differently oriented with respect to the main flow have been revealed as well as roll-like and quadrangular patterns. Transitions between different convective patterns occurring under changes of the governing parameters have been considered.

The dynamics and spectra of the quasi-diagonal direct interaction approximation (QDIA) closure for inhomogeneous two-dimensional turbulence over mean (single realization) topography are compared with results from direct numerical simulations (DNS). A more efficient version of the closure, termed the cumulant update QDIA (CUQDIA), has also been formulated and tested. Studies are performed for a range of resolutions, for large scale Reynolds numbers between very low ($R_{L} < 1$) and moderate ($R_{L} \approxeq 300$) and for wide ranges of topographic spectra and initial mean field and transient spectra. The QDIA-type closures are shown to be computationally tractable for general inhomogeneous flows, particularly in cumulant update form, and to perform extremely well when the turbulence is weak. At low ($R_{L} \approxeq 60$) to moderate ($R_{L} \approxeq 300$) Reynolds numbers the presence of significant amplitude small-scale mean fields and topography reduces the under-estimation of small-scale transient kinetic energy that is characteristic of the Eulerian direct interaction approximation (DIA). A regularized version of the CUQDIA closure (RCUQDIA) in which interactions are localized in wavenumber space, depending on specified cut-off ratios, has also been tested at moderate Reynolds number for cases when the small-scale mean fields and topography are weak. Excellent agreement has been found between the RCUQDIA closure and DNS results for turbulent flows with properties broadly similar to atmospheric spectra.

Hartmann layers are a common feature in magnetohydrodynamics, where they organize the electric current distribution in the flow and hence the characteristics of the velocity field. In spite of their importance their stability properties are not well understood, mainly because of the scarcity of experimental data. In this work we investigated experimentally the transition to turbulence in the Hartmann layers that arise in magnetohydrodynamic flows in ducts. From measurements of the friction factor a well-marked transition to turbulence was found at a critical Reynolds number, based on the laminar Hartmann layer thickness, of approximately 380, valid also for laminarization and for a wide range of intensities of the magnetic field. The sensitivity of this result to the roughness characteristics of the walls along which the Hartmann layers develop confirms that these layers are related to the transition observed and provides more information on its stability properties.

Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). The linear stability analysis of this flow provided unrealistically high critical Reynolds numbers, about two orders of magnitude higher than those observed in experiments. We propose an explanation based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes transient growth of two-dimensional optimal perturbations and the subsequent three-dimensional instability of the modulated streaky flow. In agreement with recent experimental investigations the calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number, where the transition occurs at realistic amplitudes of two- and three-dimensional perturbations.

A theoretical analysis is made of the transient Ekman layer of a rapidly rotating compressible fluid. In the initial state, both the fluid and the disk are in equilibrium in isothermal rigid-body rotation. Flow is initiated by imposing a mechanical/thermal disturbance on the rotating disk. By making detailed examinations of the energy balance in the Ekman layer, the energy transfer mechanisms are delineated. Two distinctive transient energy transfer mechanisms are identified: (i) the one-dimensional energy diffusion process in the axial direction, and (ii) the conventional Ekman layer flow which is similar to that of an incompressible fluid. The usefulness of a particular grouping of flow variables, which is termed the energy flux content, is emphasized. The major distinctions between compressible and incompressible fluids are ascertained. This analytical endeavour clarifies the features unique to a compressible rotating flow. A unified view is constructed to encompass the previously published theoretical findings which had been presented in a piecemeal fashion.

The evolution of pancake dipoles of different aspect ratio is studied in a stratified tank experiment. Two cases are reported here for values of the dipole initial aspect ratio $\alpha_0 = L_v/L_h$ (where $L_v$ and $L_h$ are vertical and horizontal length scales, respectively) of $\alpha_0 = 0.4$ (case I) and $\alpha_0 = 1.2$ (case II). In the first case, the usual decay scenario is observed where the dipole diffuses slowly with a growing thickness and a decaying circulation. In case II, we observed a regime where the thickness of the dipole decreases and the circulation in the horizontal mid-plane of the vortices remains constant. We show that this regime where the vertical length scale decreases can be explained by the shedding of two boundary layers at the top and bottom of the dipole that literally peel off vorticity layers. Horizontal advection and vertical diffusion cooperate in this regime and the decrease towards the viscous vertical length scale $\delta = L_h\hbox{\it Re}^{-1/2}$ occurs on a time scale $\alpha_0 \hbox{\it Re}^{1/2}T_A$, $T_A$ being the advection time $L_h/U$. From a scaling analysis of the equations for a stratified viscous fluid in the Boussinesq approximation, two dominant balances depending on the parameter $R = \hbox{\it Re}F_h^2$ are discussed, where $F_h = U/NL_h$ is the horizontal Froude number and $\hbox{\it Re} = UL_h/\nu$ is the Reynolds number, $U$, $N$ and $\nu$ being, respectively, the translation speed of the dipole, the Brunt–Väisälä frequency and the kinematic viscosity. When $R\gg 1$ the vertical length scale is determined by buoyancy effects to be of order $L_b = U/N$. The experiments presented in this paper pertain to the case of small $R$, where viscous effects govern the selection of the vertical length scale. We show that if initially $L_v \leq \delta$, the flow diffuses on the vertical (case I), while if $L_v \gg \delta$ (case II), vertically sheared horizontal advection decreases the vertical length scale down to $\delta$. This viscous regime may explain results from experiments and numerical simulations on the late evolution of stratified flows where the decay is observed to be independent of the buoyancy frequency $N$.

A comprehensive numerical and experimental study of normal shock waves in hypersonic axisymmetric jets of N$_2$ is presented. The numerical interpretation is based on the quasi-gasdynamic (QGD) approach, and its generalization (QGDR) for the breakdown of rotational–translational equilibrium. The experimental part, based on diagnostics by high-sensitivity Raman spectroscopy, provides absolute density and rotational temperatures along the expansion axis, including the wake beyond the shock. These quantities are used as a reference for the numerical work. The limits of applicability of the QGD approach in terms of the local Knudsen number, the influence of the computational grid on the numerical solution, the breakdown of rotation–translation equilibrium, and the possible formation of a recirculation vortex immediately downstream from the normal shock wave are the main topics considered.

Experiments and numerical simulations are carried out to verify the existence of the acoustic solitary wave in an air-filled tube with an array of Helmholtz resonators connected. Following up previous work (Sugimoto et al. 1999), the experiments are improved by using a newly designed piston driver to launch an initially plane pressure pulse and also by extending the tube length from 7.4 m to 10.6 m. To highlight the effect of the array of resonators, the case with no array is also examined in parallel. Direct and indirect checks are made to verify the existence of the solitary wave. The former compares the profiles and propagation speeds of pulses measured experimentally to the solitary-wave solution. The latter checks the validity of nonlinear wave equations in describing real wave evolution in the tube. Solving an initial-value problem numerically with weakly lossy effects of boundary layers and jet loss at the throat of the resonator, comparison is made between measured and simulated evolution. The validity of the equations in the lossy case is necessary to maintain the existence of the solitary wave in the lossless limit. It is revealed that nonlinear wave equations originally derived for unidirectional propagation in the tube can provide a good description of the real evolution, with some allowance for phase shifts on reflection at both ends of the tube. In particular, it turns out that the lossy effects are described quantitatively well. By establishing the validity of the equations, it is concluded that the acoustic solitary wave exists.

A theoretical investigation of the breakdown of the viscous wake downstream of a flat plate in supersonic flow is performed in this paper based on the large Reynolds number ($Re \to \infty $) asymptotic analysis of the Navier–Stokes equations. The breakdown is provoked by an oblique shock wave impinging on the wake a small distance $l_s$ downstream of the plate trailing edge. Two flow regimes are considered. In the first $l_s$ is assumed to be an $O(Re^{-3/8})$ quantity in which case the shock impinges on the wake within the region of viscous–inviscid interaction that is known to occupy a vicinity of the trailing edge with longitudinal extent of $O(Re^{-3/8})$. Under these conditions the interaction process may be described by the equations of the triple-deck theory. To obtain a numerical solution of these equations we used a rapid matrix Thomas technique in conjunction with Newton iterations. The results of the calculations not only predict the wake breakdown near the shock location but also reveal a hysteresis behaviour of the flow as the shock is moved downstream, giving rise to three solution branches. The second part of the paper is concerned with the flow regime when the shock interacts with the wake further downstream of the trailing edge triple-deck region: $l_s \gg Re^{-3/8}$. In this case the fluid motion proves to be inviscid to leading order not only in the upper deck of the interaction region but also everywhere inside the wake. Due to this simplification the interaction problem can be reduced to a single integro-differential equation governing the pressure distribution along the interaction region. With known pressure the Bernoulli equation may be used to find the velocity field. The Bernoulli equation also allows us to formulate a simple criterion which may be used to predict the onset of wake breakdown. We found that viscosity becomes important again in a smaller vicinity of the breakdown point where the flow reversal takes place. It is remarkable that the viscous–inviscid interaction problem governing the flow in this vicinity admits an analytical solution.

The experiments by Darbyshire & Mullin (1995) on the transition to turbulence in pipe flow show that there is no sharp border between initial conditions that trigger turbulence and those that do not. We here relate this behaviour to the possibility that the transition to turbulence is connected with the formation of a chaotic saddle in the phase space of the system. We quantify a sensitive dependence on initial conditions and find in a statistical analysis that in the transition region the distribution of turbulent lifetimes follows an exponential law. The characteristic mean lifetime of the distribution increases rapidly with Reynolds number and becomes inaccessibly large for Reynolds numbers exceeding about 2250. Suitable experiments to further probe this concept are proposed.

We report results of a series of detailed experiments designed to unveil the dynamics of a particle of radius $a$ moving in high-frequency, low-Reynolds-number oscillatory flow. The fundamental parameters in the problem are the Strouhal ($\hbox{\it Sl}$) and the particle Reynolds numbers ($\hbox{\it Re}_p$), as well as the fluid-to-particle density ratio $\alpha$. The experiments were designed to cover a range of $\hbox{\it Sl} \hbox{\it Re}_p$ from 0.015 to 5 while keeping $\hbox{\it Re}_p < 0.5$ and $\hbox{\it Sl} > 1$. The primary objective of the experiments is to investigate stationary history effects associated with the Basset drag, which are maximized when the viscous time scale $a^2/\nu$ is of the same order of the flow time scale $9/\Omega$, where $9$ is a geometrical factor for the sphere, $\nu$ is the kinematic viscosity and $\Omega$ is the angular frequency of the background flow. The theoretically determined behaviour of stationary history effects is confirmed unequivocally by the experiments, which also validate the fractional derivative behaviour (of order $1/2$) of the history drag for the range of parameters under study.

In this paper we report results from some investigations on the use of ultrasonics in controlling travelling bubble cavitation. Control of this type of cavitation, generated using a venturi device, has been achieved by manipulation of potential nuclei using a piezoelectric device, termed the Ultrasonic Nuclei Manipulator (UNM). The performance of the UNM, activated in continuous-wave (CW) and pulsed modes, has been studied over a range of dissolved gas concentration ($C$). The performance under CW-excitation is found to depend sensitively on $C$, with lack of control in near-saturated water samples. Failure to control cavitation at $C \approx 1$ under CW-excitation is suggested to be a result of bubble growth by rectified diffusion under these conditions. The pulsed mode of excitation of the UNM, in such cases, seems to be a very promising alternative. Further improvement is observed by using two piezoelectric crystals, one driven in the CW-mode and the second in pulsed mode, as the UNM. Through carefully designed experimentation, this has been traced to the movement of nuclei under the influence of Bjerknes forces. Besides reduction of noise, other measures of control have been identified and investigated. For example, it has been found that the maximum velocity achievable at the venturi throat can be increased from about 15 m s$^{-1}$ to about 22 m s$^{-1}$ with nuclei manipulation using ultrasonics.

A new and systematic approach is proposed to determine the migration of torque-free spherical bubbles immersed in a steady and non-uniform unbounded Stokes flow and subject to arbitrary capillary effects. The advocated procedure appeals to only a very few quantities on the surface of each bubble and is therefore suitable for a future numerical treatment of arbitrary clusters of bubbles. For a single bubble, the theory allows a straightforward analytical implementation and the predicted results agree well with Hetsroni & Haber (1970), Hetsroni et al. (1971) and Subramanian (1985). The thermocapillary motion of non-conducting spherical bubbles freely suspended in a quiescent liquid in the presence of an arbitrary ambient temperature $T_{\infty}$ is considered and it is shown that it is futile to determine the disturbed temperature field, whatever $T_{\infty},$ once bubbles are equivalent (i.e. experience the same velocity in a given uniform temperature gradient ${\bm \nabla}T_{\infty},$ as obtained by Acrivos et al. 1990 and Wang et al. 1994).

Journal of Fluid Mechanics, vol. 494 (2003), pp. 399–405