bluuebunny/crossref_metadata_2025_split huggingface dataset:
Flows in a rotating spherical shell: the equatorial case
Authors: de Verdière A. Colin , Schopp R. 
Published: April 2006

It is well known that the widely used powerful geostrophic equations that single out the vertical component of the Earth's rotation cease to be valid near the equator. Through a vorticity and an angular momentum analysis on the sphere, we show that if the flow varies on a horizontal scale L smaller than (Ha)1/2 (where H is a vertical scale of motion and a the Earth's radius), then equatorial dynamics must include the effect of the horizontal component of the Earth's rotation. In equatorial regions, where the horizontal plane aligns with the Earth's rotation axis, latitudinal variations of planetary angular momentum over such scales become small and approach the magnitude of its radial variations proscribing, therefore, vertical displacements to be freed from rotational constraints. When the zonal flow is strong compared to the meridional one, we show that the zonal component of the vorticity equation becomes (2Ω.Δ)u1 = g/ρ0)(∂ρ/a∂θ). This equation, where θ is latitude, expresses a balance between the buoyancy torque and the twisting of the full Earth's vorticity by the zonal flow u1. This generalization of the mid-latitude thermal wind relation to the equatorial case shows that u1 may be obtained up to a constant by integrating the ‘observed’ density field along the Earth's rotation axis and not along gravity as in common mid-latitude practice. The simplicity of this result valid in the finite-amplitude regime is not shared however by the other velocity components.Vorticity and momentum equations appropriate to low frequency and predominantly zonal flows are given on the equatorial β-plane. These equatorial results and the mid-latitude geostrophic approximation are shown to stem from an exact generalized relation that relates the variation of dynamic pressure along absolute vortex lines to the buoyancy field. The usual hydrostatic equation follows when the aspect ratio δ = H/L is such that tan θ/δ is much larger than one. Within a boundary-layer region of width (Ha)1/2 and centred at the equator, the analysis shows that the usually neglected Coriolis terms associated with the horizontal component of the Earth's rotation must be kept.Finally, some solutions of zonally homogeneous steady equatorial inertial jets are presented in which the Earth's vorticity is easily turned upside down by the shear flow and the correct angular momentum ‘Ωr2cos2(θ)+u1rCos(θ)’ contour lines close in the vertical–meridional plane.
Shear dispersion and anomalous diffusion by chaotic advection
Authors: Jones S. W. , Young W. R. 
Published: April 2006

The dispersion of passive scalars by the steady viscous flow through a twisted pipe is both a simple example of chaotic advection and an elaboration of Taylor's classic shear dispersion problem. In this article we study the statistics of the axial dispersion of both diffusive and perfect (non-diffusive) tracer in this system.For diffusive tracer chaotic advection assists molecular diffusion in transverse mixing and so diminishes the axial dispersion below that of integrable advection. As in many other studies of shear dispersion the axial distribution ultimately becomes Gaussian as t → ∞. Thus there is a diffusive regime, but with an effective diffusivity that is enhanced above molecular values. In contrast to the classic case, the effective diffusivity is not necessarily inversely proportional to the molecular diffusivity. For instance, in the irregular regime the effective diffusivity is proportional to the logarithm of the molecular diffusivity.For perfect tracer chaotic advection does not result in a diffusive process, even in the irregular regime in which streamlines wander throughout the cross-section of the pipe. Instead the variance of the axial position is proportional to t in t so that the measured diffusion coefficent diverges like In t. This faster than linear growth of variance is attributed to the trapping of tracer for long times near the solid boundary, where the no-slip condition ensures that the fluid moves slowly. Analogous logarithmic effects associated with the no-slip condition are well known in the context of porous media.A simple argument, based on Lagrangian statistics and a local analysis of the trajectories near the pipe wall, is used to calculate the constants of proportionality before the logarithmic terms in both the large- and infinite-Péclet-number limits.
The effects of thermal conditions on the cell sizes of two-dimensional convection
Authors: Ishiwatari Masaki , Takehiro Shin-Ichi , Hayashi Yoshi-Yuki 
Published: April 2006

The effects of thermal conditions on the patterns of two-dimensional Boussinesq convection are studied by numerical integration. The adopted thermal conditions are (i) the heat fluxes through both upper and lower boundaries are fixed, (ii) the same as (i) but with internal cooling, (iii) the temperature on the lower boundary and the heat flux through the upper boundary are fixed, (iv) the same as (iii) but with internal cooling, and (v) the temperatures on both upper and lower boundaries are fixed. The numerical integrations are performed with Ra = 104 and Pr = 1 over the region whose horizontal and vertical lengths are 8 and 1, respectively.The results confirm that convective cells with the larger horizontal sizes tend to form under the conditions where the temperature is not fixed on any boundaries. Regardless of the existence of internal cooling, one pair of cells spreading all over the region forms in the equilibrium states. On the other hand, three pairs of cells form and remain when the temperature on at least one boundary is fixed. The formation of single pairs of cells appearing under the fixed heat flux conditions shows different features with and without internal cooling. The difference emerges as the appearance of a phase change, whose existence can be suggested by the weak nonlinear equation derived by Chapman & Proctor (1980).
Gravity wave radiation from vortex trains in rotating shallow water
Authors: Ford Rupert 
Published: April 2006

Gravity wave radiation by vortical flows in the f-plane shallow-water equations is investigated by direct nonlinear numerical simulation. The flows considered are initially parallel flows, consisting of a single strip in which the potential vorticity differs from the background value. The flows are unstable to the barotropic instability mechanism, and roll up into a train of vortices. During the subsequent evolution of the vortex train, gravity waves are radiated. In the limit of small Froude number, the gravity wave radiation is compared with that predicted by an appropriately modified version of the Lighthill theory of aerodynamic sound generation. It is found that the gravity wave field agrees well with that predicted by the theory, provided typical lengthscales of vortical motions are well within one deformation radius.It is found that the nutation time for vortices in the train increases rapidly with increasing Froude number in cases where the potential vorticity in the vortices is of the same sign as the background value, whereas the nutation time is almost independent of Froude number in cases where the potential vorticity in the vortices is zero or of opposite sign to the background. Consequently, in the former cases, the unsteadiness of the flow decreases with increasing Froude number, so the effect of the inertial cutoff frequency is increased, leading to an optimal Froude number for gravity wave radiation, above which the intensity of the radiated waves decreases as the Froude number is further increased. It is proposed that the existence of a finite range of interaction within the vortices, for flows with positive vortex potential vorticity, may account for the strong dependence of nutation time on Froude number in those cases. The interaction scale within the vortices becomes infinite in the limit of zero vortex potential vorticity, and so the arguments do not apply in those cases.
Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet
Authors: Hussein Hussein J. , Capp Steven P. , George William K. 
Published: April 2006

The turbulent flow resulting from a top-hat jet exhausting into a large room was investigated. The Reynolds number based on exit conditions was approximately 105. Velocity moments to third order were obtained using flying and stationary hot-wire and burst-mode laser-Doppler anemometry (LDA) techniques. The entire room was fully seeded for the LDA measurements. The measurements are shown to satisfy the differential and integral momentum equations for a round jet in an infinite environment.The results differ substantially from those reported by some earlier investigators, both in the level and shape of the profiles. These differences are attributed to the smaller enclosures used in the earlier works and the recirculation within them. Also, the flying hot-wire and burst-mode LDA measurements made here differ from the stationary wire measurements, especially the higher moments and away from the flow centreline. These differences are attributed to the cross-flow and rectification errors on the latter at the high turbulence intensities present in this flow (30% minimum at centreline). The measurements are used, together with recent dissipation measurements, to compute the energy balance for the jet, and an attempt is made to estimate the pressure-velocity and pressure-strain rate correlations.
The effect of hydrodynamic interactions on the tracer and gradient diffusion of integral membrane proteins in lipid bilayers
Authors: Bussell Stuart J. , Hammer Daniel A. , Koch Donald L. 
Published: April 2006

Biological membranes can be considered two-dimensional fluids with suspended integral membrane proteins (IMPs). We have calculated the effect of hydrodynamic interactions on the various diffusion coefficients of IMPs in lipid bilayers. The IMPs are modelled as hard cylinders of radius a immersed in a thin sheet of viscosity μ and thickness h bounded by a fluid of low viscosity μ′. We have ensemble averaged the N-body Stokes equations to the pair level and have renormalized them following the methods of Batchelor (1972) and Hinch (1977). The lengthscale for the hydrodynamic interactions is λa = μh / μ′, Which is O (100a), and the slow decay of the interactions introduces new features in the renormalizations compared to the analogous analyses for three-dimensional suspensions of spheres.We have calculated the asymptotic limits for the short- and long-time tracer diffusivities, Ds and Dl, respectively, and for the gradient diffusivity, Dg, for ϕ [Lt ] 1 and λ [Gt ] 1, where ϕ is the IMP area fraction and λ = μh / (μ′a). The diffusivities are
\begin{eqnarray*}
D_s/D_0 &=& 1-2\phi[1-(1+\ln (2)-9/32)/(\ln(\lambda)-\gamma)],
D_l/D_0 &=& D_s/D_0 - 0.07/(\ln(\lambda)-\gamma),
D_g/D_0 &=& 1+\phi[-7+(6\ln(2)+7/16+0.37)/(\ln(\lambda)-\gamma)],
\end{eqnarray*}
where D0 is the diffusivity in the limit of zero area fraction, and γ = 0.577216 is Euler's constant. The results for Dl and Ds differ only slightly. The decrease in Dg/Do as ϕ increases contrasts with the result for spheres for which Dg/Do > 1.
A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity
Authors: Yang Joseph , Kubota Toshi , Zukoski Edward E. 
Published: April 2006

This work investigates the two-dimensional flow of a shock wave over a circular light-gas inhomogeneity in a channel with finite width. The pressure gradient from the shock wave interacts with the density gradient at the edge of the inhomogeneity to deposit vorticity around the perimeter, and the structure rolls up into a pair of counter-rotating vortices. The aim of this study is to develop an understanding of the scaling laws for the flow field produced by this interaction at times long after the passage of the shock across the inhomogeneity. Numerical simulations are performed for various initial conditions and the results are used to guide the development of relatively simple algebraic models that characterize the dynamics of the vortex pair, and that allow extrapolation of the numerical results to conditions more nearly of interest in practical situations. The models are not derived directly from the equations of motion but depend on these equations and on intuition guided by the numerical results. Agreement between simulations and models is generally good except for a vortex-spacing model which is less satisfactory.A practical application of this shock-induced vortical flow is rapid and efficient mixing of fuel and oxidizer in a SCRAMJET combustion chamber. One possible injector design uses the interaction of an oblique shock wave with a jet of light fuel to generate vorticity which stirs and mixes the two fluids and lifts the burning jet away from the combustor wall. Marble proposed an analogy between this three-dimensional steady flow and the two-dimensional unsteady problem of the present investigation. Comparison is made between closely corresponding three-dimensional steady and two-dimensional unsteady flows, and a mathematical description of Marble's analogy is proposed.
Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics
Authors: Gama S. , Vergassola M. , Frisch U. 
Published: April 2006

The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very weak large-scale perturbation of wavenumberkwill amplify it with a rate proportional tok2, independent of the direction.Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, possesses a centre of symmetry (parity-invariance) and has three- or six-fold rotational invariance to ensure isotropy of the eddy-viscosity tensor.The eddy viscosities emerging from the multiscale analysis are calculated by two different methods. First, there is an expansion in powers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyond the disk of convergence. Secondly, there is a spectral method. The two methods typically agree within a fraction of 1%.A simple example, the ‘decorated hexagonal flow’, of a time-independent flow with isotropic negative eddy viscosity is given. Flows with randomly chosen Fourier components and the required symmetry have typically a 30% chance of developing a negative eddy viscosity when the Reynolds number is increased.For basic flow driven by a prescribed external force and sufficiently strong largescale flow, the analysis is extended to the nonlinear régime. It is found that the largescale dynamics is governed by a Navier-Stokes or a Navier-Stokes-Kuramoto-Sivashinsky equation, depending on the sign and strength of the eddy viscosity. When the driving force is not mirror-symmetric, a new ‘chiral’ nonlinearity appears. In special cases, the large-scale equation reduces to the Burgers equation. With chiral forcing, circular vortex patches are strongly enhanced or attenuated, depending on their cyclonicity.
The stability of statically unstable layers
Authors: Thorpe S. A. 
Published: April 2006

We investigate the development of instability in a fluid with density locally of the form ρ0[1 −(N2 / g)z + A sin Kz], composed of an overall stable uniform gradient of buoyancy frequency, N, but with a superimposed sinusoidal variation of vertical wavenumber, K, and amplitude, A [Lt ] 1; g is the acceleration due to gravity and z is the upward vertical coordinate. Layers exist in which the fluid is statically unstable when the parameter r = N2 / gKA, is less than unity.When r is zero, the density is sinusoidal in z and the problem reduces to one studied by Batchelor & Nitsche (1991). Their solution, which finds a gravest mode of linear instability with terms having vertical motions independent of z and with horizontal scales large in comparison with K−1, is extended to non-zero r. An effect of a small, but finite, r is to stabilize the fluid, increasing the critical Rayleigh number and the corresponding non-dimensional horizontal wavenumber. The vertical scale of the mode which first becomes unstable is reduced as r increases. A small sinusoidal shear destabilizes the fluid.When r approaches unity, the density field contains regions of static instability which are of thickness small compared to K−1. The problem then approximates to one studied by Matthews (1988). Consistent solutions for the growth of disturbances are obtained by truncated series and, following Matthews, by the solution of a Fourier-transformed equation. A small uniform shear, characterized by a flow Reynolds number, Re > O, is found to stabilize the fluid, in that it increases the critical Rayleigh number of the onset of instability. This suggests that convective Rayleigh–Taylor instability, with constant phase lines parallel to the flow, is then the favoured mode of onset of instability. At very large Rayleigh numbers and at a Prandtl number of 700, however, the growth rate of the most rapidly growing linear disturbances may increase as Re increases from zero, and the form of the evolving flow is then less certain.The theory is used to estimate the scale and growth rates of instability in overturning internal gravity waves in the laboratory experiment described in a companion paper (Thorpe 1994).
Statically unstable layers produced by overturning internal gravity waves
Authors: Thorpe S. A. 
Published: April 2006

Internal waves in a uniformly stratified fluid of sufficiently large amplitude develop tilted layers in which the fluid is statically unstable. To investigate the evolution and subsequent development of this structure, experiments are made in which a horizontal rectangular tube containing a fluid of uniform density gradient is gently rocked at a selected frequency about a horizontal axis normal to the tube length. Large-amplitude standing internal gravity waves of the first mode are generated, and these steepen and overturn, the isopycnal surfaces folding to produce a vertically thin and horizontally extensive layer in which the fluid is statically unstable. In experiments with relatively small forcing, the layer persists for some 6 buoyancy periods, with no detected evidence of secondary instability, and static stability is re-established as the periodic flow reverses. The layer however breaks down, with consequent diapycnal mixing, when greater forcing is applied.The scale and growth rates of instability in the overturning internal gravity waves are estimated using the theory developed in a companion paper by Thorpe (1994a). For the parameters of the laboratory experiments with relatively small forcing, the growth rates are small, consistent with the absence of signs of secondary instability. Larger growth rates and disturbance amplification factors of about 70 are predicted for the conditions in the experiment in which mixing was observed to occur. The experimental observations are consistent with an instability having a longitudinal structure.We conclude that the form and development of breaking in internal gravity waves will vary according to the circumstances in which waves break, but depend on the Prandtl number of the fluid and, in particular, on the Rayleigh and Reynolds numbers of regions of static instability which develop as the waves overturn.
Expulsion of particles from a buoyant blob in a fluidized bed
Authors: Batchelor G. K. , Nitsche J. M. 
Published: April 2006

It is a significant feature of most gas-fluidized beds that they contain rising ‘bubbles’ of almost clear gas. The purpose of this paper is to account plausibly for this remarkable property first by supposing that primary and secondary instabilities of the fluidized bed generate compact regions of above-average or below-average particle concentration, and second by invoking a mechanism for the expulsion of particles from a buoyant compact blob of smaller particle concentration. We postulate that the rising of such an incipient bubble generates a toroidal circulation of the gas in the bubble, roughly like that in a drop of liquid rising through a second liquid of larger density, and that particles in the blob carried round by the fluid move on trajectories which ultimately cross the bubble boundary. Numerical calculations of particle trajectories for practical values of the relevant parameters show that a large percentage of particles, of such small concentration that they move independently, are expelled from a bubble in the time taken by it to rise through a distance of several bubble diameters.Similar calculations for a liquid-fluidized bed show that the expulsion mechanism is much weaker, as a consequence of the larger density and viscosity of a liquid, which is consistent with the absence of observations of relatively empty bubbles in liquid-fluidized beds.It is found to be possible, with the help of the Richardson-Zaki correlation, to adjust the results of these calculations so as to allow approximately for the effect of interaction of particles in a bubble in either a gas- or a liquid-fluidized bed. The interaction of particles at volume fractions of 20 or 30 % lengthens the expulsion times, although without changing the qualitative conclusions.
Unsteady flow near a moving cylinder
Authors: Dowling A. P. 
Published: April 2006

General representations are derived for both the velocity potential and the surface pressure fluctuations induced by an arbitrary distribution of vorticity near a manoeuvring cylinder. The cylinder is inextensible and in unsteady motion. Its axis may be slightly curved, with radius of curvature large in comparison with the cylinder radius.Two model problems are considered in detail to investigate the effect of lateral displacements of a cylinder with an established boundary layer. The boundary layer on the flexible cylinder is found to be shed once the lateral displacement of the cylinder axis exceeds the boundary-layer thickness. The unsteady pressures generated by this vortex shedding are investigated.