A detailed investigation of fully developed transient flow in a pipe has been undertaken using water as the working fluid. Linearly increasing or decreasing excursions of flow rate were imposed between steady initial and final values. A three-beam, two-component laser Doppler anemometer was used to make simultaneous measurements of either axial and radial, or axial and circumferential, components of local velocity. Values of ensemble-averaged mean velocity, root-mean-square velocity fluctuation and turbulent shear stress were found from the measurements.

Being the first really detailed study of ramp-type transient turbulent flow, the present investigation has yielded new information and valuable insight into certain fundamental aspects of turbulence dynamics. Some striking features are evident in the response of the turbulence field to the imposed excursions of flow rate. Three different delays have been identified: a delay in the response of turbulence production; a delay in turbulence energy redistribution among its three components; and a delay associated with the propagation of turbulence radially. The last of these is the most pronounced under the conditions of the present study. A dimensionless delay parameter τ+[= √2τUτ0/D] is proposed to describe it. The first response of turbulence is found to occur in the region near the wall where turbulence production peaks. The axial component of turbulence responds earlier than the other two components and builds up faster. The response propagates towards the centre of the pipe through the action of turbulent diffusion at a speed which depends on the Reynolds number at the start of the excursion. In the core region, the three components of turbulence energy respond in a similar manner. Turbulence intensity is reduced in the case of accelerating flow and increased in decelerating flow. This is mainly as a result of the delayed response of turbulence. A dimensionless ramp rate parameter γ[= (dUb/dt) (1/Ub0)(D/Uτ0] is proposed, which determines the extent to which the turbulence energy differs from that of pseudo-steady flow as a result of the delay in the propagation of turbulence.

The effects of unstable thermal stratification and mean shear on chemical reaction and turbulent mixing were experimentally investigated in reacting and non-reacting liquid mixing-layer flows downstream of a turbulence-generating grid. Experiments were carried out under three conditions: unsheared neutrally stratified, unsheared unstably stratified and sheared neutrally stratified. Instantaneous velocity and concentration were simultaneously measured using the combination of a laser-Doppler velocimeter and a laser-induced fluorescence technique. The results show that the turbulent mixing is enhanced at both large and small scales by buoyancy under unstably stratified conditions and therefore the chemical reaction is strongly promoted. The mean shear acts to enhance the turbulent mixing mainly at large scales. However, the chemical reaction rate in the sheared flow is not as large as in the unstably stratified case with the same turbulence level, since the mixing at small scales in the sheared neutrally stratified flow is weaker than that in the unsheared unstably stratified flow. The unstable stratification is regarded as a better tool to attain unsheared mixing since the shearing stress acting on the fluid is much weaker in the unstably stratified flow than in the sheared flow.

A weakly nonlinear analysis is employed to investigate the onset of plume convection in the mushy layer of a binary solution directionally solidified from below. An improved mathematical model including the constant pressure condition at the melt/mush interface is applied to analytically analyse the nonlinear behaviour of the convection. Results show that, due to the consideration of the constant pressure condition at the interface, the present analytical results are much better in comparison with the experiments than those obtained by previous studies, in which the no-vertical-flow condition at the interface was considered. It is also shown that the bifurcation to three-dimensional hexagon convection, corresponding to the onset of plume convection, is subcritical. For the case of small concentration ratio [Cscr ] (equation (5b)) the channel of the plume is generated at the top of the mush and grows downwards into the mush. For the case of large [Cscr ], the channel may be generated at the interior of the mush and grow upwards to the top of the mush. The possible parameter regime in which the flow is of a stable down-centre hexagon is discussed.

The dynamics of subgrid-scale energy transfer in turbulence is investigated in a database of a planar turbulent jet at Reλ ≈ 110, obtained by direct numerical simulation. In agreement with analytical predictions (Kraichnan 1976), subgrid-scale energy transfer is found to arise from two effects: one involving non-local interactions between the resolved scales and disparate subgrid scales, the other involving local interactions between the resolved and subgrid scales near the cutoff. The former gives rise to a positive, wavenumber-independent eddy-viscosity distribution in the spectral space, and is manifested as low-intensity, forward transfers of energy in the physical space. The latter gives rise to positive and negative cusps in the spectral eddy-viscosity distribution near the cutoff, and appears as intense and coherent regions of forward and reverse transfer of energy in the physical space. Only a narrow band of subgrid wavenumbers, on the order of a fraction of an octave, make the dominant contributions to the latter. A dynamic two-component subgrid-scale model (DTM), incorporating these effects, is proposed. In this model, the non-local forward transfers of energy are parameterized using an eddy-viscosity term, while the local interactions are modelled using the dynamics of the resolved scales near the cutoff. The model naturally accounts for backscatter and correctly predicts the breakdown of the net transfer into forward and reverse contributions in a priori tests. The inclusion of the local-interactions term in DTM significantly reduces the variability of the model coefficient compared to that in pure eddy-viscosity models. This eliminates the need for averaging the model coefficient, making DTM well-suited to computations of complex-geometry flows. The proposed model is evaluated in LES of transitional and turbulent jet and channel flows. The results show DTM provides more accurate predictions of the statistics, structure, and spectra than dynamic eddy-viscosity models and remains robust at marginal LES resolutions.

Rayleigh–Bénard convection in horizontal layers of binary fluid mixtures heated from below with realistic horizontal boundary conditions is studied theoretically using multi-mode Galerkin expansions. For positive separation ratios the main difference between the mixtures and pure fluids lies in the existence of stable three-dimensional patterns near onset in a wide range of the parameter space. We evaluated the stationary solutions of roll, crossroll, and square convection and we determined the location of the stability boundaries for many parameter combinations thereby obtaining the Busse balloon for roll and square patterns.

The baroclinic stability characteristics of axisymmetric gravity currents in a rotating system with a sloping bottom are determined. Laboratory studies have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer model derived from the shallow-water equations to study the stability characteristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading order, but not quasi-geostrophic, since the height deflections of that layer are not small with respect to its scale height. The upper-layer dynamics are quasi-geostrophic, with the Eulerian velocity field principally driven by baroclinic stretching and a background topographic vorticity gradient.

Necessary conditions for instability, a semicircle-like theorem for unstable modes, bounds on the growth rate and phase velocity, and a sufficient condition for the existence of a high-wavenumber cutoff are presented. The linear stability equations are solved exactly for the case where the gravity current initially corresponds to an annulus flow with parabolic height profile with two incroppings, i.e. a coupled front. The dispersion relation for such a current is solved numerically, and the characteristics of the unstable modes are described. A distinguishing feature of the spatial structure of the perturbations is that the perturbations to the downslope incropping are preferentially amplified compared to the upslope incropping. Predictions of the model are compared with recent laboratory data, and good agreement is seen in the parameter regime for which the model is valid. Direct numerical simulations of the full model are employed to investigate the nonlinear regime. In the initial stage, the numerical simulations agree closely with the linear stability characteristics. As the instability develops into the finite-amplitude regime, the perturbations to the downslope incropping continue to preferentially amplify and eventually evolve into downslope propagating plumes. These finally reach the deepest part of the topography, at which point no more potential energy can be released.

Previous polynya flux models have specified a constant value for the collection thickness of frazil ice, H, at the polynya edge. In certain circumstances, this approach can cause the frazil ice depth, h, within the polynya, to exceed H, a result which violates assumptions made in the formulation of the ice flux balance equations at the polynya edge. To overcome this problem, a parameterization for H is derived in terms of the depth of frazil ice arriving at the polynya edge and the component, normal to the polynya edge, of the frazil ice velocity relative to the velocity of the consolidated ice pack. Thus, H is coupled to the unknown polynya edge. Using the new parameterization for H, an analysis of the unsteady one-dimensional opening of a coastal polynya is presented. Analytical solutions are also derived, using the new parameterization for H, for steady-state two-dimensional polynyas adjacent to a semi-infinite and finite-length coastal barrier, the latter case representing a prototype island. In all cases, the solutions show close qualitative and quantitative agreement with those derived using a constant value for H. However, the steady-state two-dimensional polynya edge can, in certain circumstances, exhibit a corner at the point where the offshore equilibrium width is reached. Precise conditions for the existence of a corner are derived in terms of the orientation of the frazil ice velocity (u) and the consolidated ice velocity (U). Upper and lower bounds are also obtained for the area of the steady-state island polynya, and it is shown that over a large range of orientations of u and U, the area exceeds that associated with the island polynya with constant H. Finally, two simulations of the St. Lawrence Island Polynya are presented using the new parameterization for H, and the results are compared with the H-constant theory.

The ability of the helicity decomposition to describe compactly the dynamics of three-dimensional incompressible fluids is invoked to obtain new descriptions of both homogeneous and inhomogeneous turbulence. We first use this decomposition to derive four coupled nonlinear equations that describe an arbitrary three-dimensional turbulence, whether anisotropic and/or non-mirror-symmetric. We then use the decomposition to treat the inhomogeneous turbulence of a channel flow bounded by two parallel free-slip boundaries with almost the ease with which the homogeneous case has heretofore received treatment. However, this ease arises from the foundation of a random-phase hypothesis, which we introduce and motivate, that supersedes the translational invariance of a turbulence that is hypothesized to be homogeneous. For the description of this channel turbulence, we find that the three-dimensional modes and the two-dimensional modes having wave vectors parallel to the boundaries each couple precisely as in a homogeneous turbulence of the corresponding dimension. The anisotropy and inhomogeneity is in large part a feature incorporated into the solenoidal basis vectors used to describe an arbitrary solenoidal free-slip flow within the channel. We invoke the random-phase hypothesis, a feature of the dynamics, with closures, such as Kraichnan's direct-interaction approximation and his test-field model, in addition to the one most utilized in this manuscript, the eddy-damped quasi-normal Markovian (EDQNM) closure.

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric magnetic field. Slow steady non-axisymmetric motion is driven by a prescribed non-axisymmetric body force; both rigid and stress-free boundary conditions are considered. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E limit analytically for values of the Elsasser number Λ less than order unity and they are compared with new numerical results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Though the ageostrophic layers correspond to those previously isolated by Kleeorin et al. (1997) for axisymmetric flows, the quasi-geostrophic layers have a new structure resulting from the asymmetry of the motion.

In the absence of magnetic field, the inviscid limit exhibits a strong shear singularity on the tangent cylinder only removeable by the addition of viscous forces. With the inclusion of magnetic field, large viscous forces remain whose strength [Zscr ] was measured indirectly by Hollerbach (1994b). For magnetic fields with dipole parity, cf. Kleeorin et al. (1997), [Zscr ] increases throughout the range Λ [Lt ] 1; whereas, for quadrupole parity, cf. Hollerbach (1994b), [Zscr ] only increases for Λ [Lt ] E1/5.

The essential difference between the dipole and quadrupole fields is the magnitude of their radial components in the neighbourhood of the equator of the inner sphere. Its finite value for the quadrupole parity causes the internal shear layer – the Hartmann–Stewartson layer stump – to collapse and merge with the equatorial Ekman layer when Λ = O(E1/5). Subsequently the layer becomes an equatorial Hartmann layer, which thins and spreads polewards about the inner sphere surface as Λ increases over the range E1/5 [Lt ] Λ [Lt ] 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined through matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by [Zscr ] decreases with increasing Λ. Since [Zscr ] depends sensitively on the detailed boundary layer structure, it provides a sharp diagnostic of new numerical results for Hollerbach's model; the realized [Zscr ]-values compare favourably with the asymptotic theory presented.

A systematic numerical analysis is performed for the quasi-periodicity in the wake where a circular cylinder is rotationally oscillated in time. The main emphasis is placed on the identification of frequency selection subjected to the controlled perturbations in the vicinity of lock-on. The frequency responses are scrutinized by measuring the lift coefficient (CL). A direct numerical simulation is made to portray the unsteady dynamics of wake flows at Re = 110. It is found that, after the shedding frequency is bifurcated at the boundary of lock-on, one frequency follows the forcing frequency and the other gradually converges to the natural shedding frequency. The asymptotic convergence phenomena are observed by solving the Van der Pol equation and the circle map. A new frequency selection formula is proposed. The quasi-periodic states are interpreted in terms of the forcing frequency, shedding frequency and modulated frequencies by employing the torus concept and the CL(t) diagram. In the quasi-periodic state, the variation of magnitudes and relevant phase changes of CL with forcing phase are examined.

We study small-scale two-dimensional non-local turbulence, where interaction of small scales with large vortices dominates in the small-scale dynamics, by using a semi-classical approach developed in Dyachenko, Nazarenko & Zakharov (1992), Nazarenko, Zabusky & Scheidegger (1995), Dubrulle & Nazarenko (1997) and Nazarenko, Kevlahan & Dubrulle (1999). Also, we consider a closely related problem of passive scalars in Batchelor's regime, when the Schmidt number is much greater than unity. In our approach, we do not perform any statistical averaging, and most of our results are valid for any form of the large-scale advection. A new invariant is found in this paper for passive scalars when their initial spectrum is isotropic. It is shown, analytically, numerically and using a dimensional argument, that there is a spectrum corresponding to an inverse cascade of the new invariant, which scales like k−1 for turbulent energy and k1 for passive scalars. For passive scalars, the k1-spectrum was first found by Kraichnan (1974) in the special case of advection δ-correlated in time, and until now it was believed to correspond to an absolute thermodynamic equilibrium and not a cascade. We also obtain, both analytically and numerically, power-law spectra of decaying two-dimensional turbulence, k−2, and passive scalar, k0.

This paper examines changes in the mean velocity profiles of turbulent boundary layers subjected to system rotation. Analysis of the data from several studies conducted in the large rotating wind tunnel at the University of Melbourne shows the existence of a universal linear correction to the velocity profile in the logarithmic region. The appropriate parameters relevant to rotation are derived and correlations are found between the parameters. Flows with adverse pressure gradients, zero pressure gradients and secondary flows are examined and all appear to exhibit the universal linear correction, suggesting that it is robust.