Mean Azimuthal Flow Research Articles

Simultaneous PIV and thermography measurements of partially blocked flow in a differentially heated rotating annulus

A radial barrier has been mounted in a differentially heated rotating annulus that partially blocks the azimuthal flow component. The experiment can be seen as an analog to geophysical flows with constrictions, e.g., the Antarctic Circumpolar Current. However, the experiment has been carried out without a particular natural flow in mind. The main interest was to observe a baroclinic annulus flow that does not become saturated. Hence, in contrast to the annulus flow without a barrier, the partially blocked flow remains transient and surface heat fluxes associated with baroclinic life cycles can be studied. The annulus can be subdivided into the upstream half of the barrier, where waves amplify, and the downstream half of the barrier, where waves decay. In the upstream half, the azimuthal mean flow is moderate but with a significant positive eddy radial heat flux. In the downstream half, we find a strong jet in the mean azimuthal flow and furthermore an increased radial mean temperature gradient. The latter points to a weakened or even reversed radial eddy heat flux in the lee side of the barrier. Temperature anomalies appear as large bulges in the outer part of the annulus. Moreover, an outward shift of vortex centers can be observed with respect to centers of temperature anomalies. This phase shift between pressure and temperature anomalies differs from that of classical Eady modes of baroclinic instability.

Non-invasive turbulent mixing across a density interface in a turbulent Taylor–Couette flow

In this paper, we present new experimental measurements of the turbulent transport of salt across an interface between two layers of fluid of equal depth but different salinities. The fluid is confined to a cylindrical annulus with a vertical axis. The outer cylinder is stationary and the inner cylinder rotates to produce a turbulent flow field consisting of an approximately irrotational mean azimuthal flow, with narrow boundary layers on the inner and outer cylinders. We focus on the limit of high-Richardson-number flow, defined as Ri = gΔρH/(ρ0u2rms), where ρ0 is a reference density, Δρ is the time-dependent difference of the layers' mean densities, urms is the root mean square of the turbulent velocity fluctuations and H is the layer depth. The mean flow has Reynolds number of the order of 104−105, and the turbulent fluctuations in the azimuthal and radial directions have root-mean-square speed of order 10% of the mean azimuthal flow. Measurements based on our experimental system show that when the Richardson number is in the range 7 < Ri < 200, the interface between the two layers remains sharp, each layer remains well mixed, and the vertical flux of salt between the layers, Fs ~(1.15 ± 0.15)Ri−1𝒜(H/ΔR)urmsΔS, where ΔS is the spatially-averaged time-dependent salinity difference between the layers and in general 𝒜(H/ΔR) is a dimensionless function of the tank aspect ratio, here taken to be unity, with ΔR being the gap width of the annulus. The salt transport appears to be caused by turbulent eddies scouring and sharpening the interface and implies a constant rate of conversion of the turbulent kinetic energy to potential energy, independent of the density contrast between the layers. For smaller values of Ri, the flow regime changes qualitatively, with eddies penetrating the interface, causing fluid in the two layers to co-mingle and rapidly homogenize.

Mode coupling analysis and differential rotation in a flow driven by a precessing cylindrical container

We present a theoretical weakly nonlinear analysis of the dynamics of an inviscid flow submitted to both rotation and precession of an unbounded cylindrical container, by considering the coupling of two Kelvin (inertial) waves. The parametric centrifugal instability known for this system is shown to saturate when one expands the Navier–Stokes equation to higher order in the assumed small precession parameter (ratio of precession to rotation frequencies) with the derivation of two coupled Landau equations suitable to describe the dynamics of the modes. It is shown that an azimuthal mean flow with differential rotation is generated by this modes coupling. The time evolution of the associated dynamical system is studied. These theoretical results can be compared with water experiments and also to some numerical simulations where viscosity and finite length effects cannot be neglected.

Interacting oscillatory boundary layers and wall modes in modulated rotating convection

Thermal convection in a rotating cylinder near onset is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime dominated by the Coriolis force. For thermal driving too small to support convection throughout the entire cell, convection sets in as alternating pairs of hot and cold plumes in the sidewall boundary layer, the so-called wall modes of rotating convection. We subject the wall modes to small amplitude harmonic modulations of the rotation rate over a wide range of frequencies. The modulations produce harmonic Ekman boundary layers at the top and bottom lids as well as a Stokes boundary layer at the sidewall. These boundary layers drive a time-periodic large-scale circulation that interacts with the wall-localized thermal plumes in a non-trivial manner. The resultant phenomena include a substantial shift in the onset of wall-mode convection to higher temperature differences for a broad band of frequencies, as well as a significant alteration of the precession rate of the wall mode at very high modulation frequencies due to the mean azimuthal streaming flow resulting from the modulations.

Numerical and experimental study of the time-dependent states and the slow dynamics in a von Kármán swirling flow

The characteristics of time-dependent swirling flows in a von Kármán configuration are investigated numerically up to Reynolds number Re = 3000 (based on the angular velocity and the radius of the cylinder), and experimentally within turbulent regimes. Experimental results are analyzed together with the periodic and aperiodic flow obtained numerically. In the present configuration, the fluid is contained in a cylindrical cavity with aspect ratio (height to radius) Γ = 2 and the motion is driven by the exact counter rotation of the end walls while the sidewall is at rest. Spectral direct numerical simulations show that for this geometry the axisymmetric base flow becomes unstable to non-axisymmetric perturbations with azimuthal wavenumber m = 1 through a subcritical bifurcation, and the corresponding flow exhibits a pattern with one cat's eye in the axial-azimuthal planes. Increasing the Reynolds number the flow becomes unstable to non-axisymmetric steady perturbations with even azimuthal wavenumbers, and the corresponding flows exhibits a two cat's eyes pattern. The occurrence of cat's eye pattern in radial-azimuthal surfaces was observed in this and other aspect ratio cavities and is associated with vortices in 3D steady flows with characteristic azimuthal modes [Nore, C., Tuckerman, L.S., Daube, O. and Xin, S., The 1 : 2 mode interaction in exactly counter-rotating von Kármán swirling flow, J. Fluid Mech., 2003, 477, 51–88; Lackey, T.C. and Sotiropoulos, F., Relationship between stirring rate and Reynolds number in the chaotically advected steady flow in a container with exactly counter-rotating lids, Phys. Fluids, 2006, 18, 1–14]. Time-dependent regimes are obtained numerically when the value of the Reynolds number is Re ≥ 1500. The time dependency is associated with a pulsation of the two vortices found in the steady regime. Experimental visualizations and measurements show that in turbulent regimes the flow also exhibits two vortices, but in this case they travel in the azimuthal direction with a frequency compatible with the frequency obtained in the numerical simulations at much lower Reynolds number. The azimuthal drift of these vortices is associated with the asymmetry of the mean azimuthal flow with respect to the equatorial plane.

Direction reversal of a rotating wave in Taylor–Couette flow

In Taylor–Couette systems, waves, e.g. spirals and wavy vortex flow, typically rotate in the same direction as the azimuthal mean flow of the basic flow which is mainly determined by the rotation of the inner cylinder. In a combined experimental and numerical study we analysed a rotating wave of a one-vortex state in small-aspect-ratio Taylor–Couette flow which propagates either progradely or retrogradely in the inertial (laboratory) frame, i.e. in the same or opposite direction as the inner cylinder. The direction reversal from prograde to retrograde can occur at a distinct parameter value where the propagation speed vanishes. Owing to small imperfections of the rotational invariance, the curves of vanishing rotation speed can broaden to ribbons caused by coupling between the end plates and the rotating wave. The bifurcation event underlying the direction reversal is of higher codimension and is unfolded experimentally by three control parameters, i.e. the Reynolds number, the aspect ratio, and the rotation rate of the end plates.

Rotation effect on near-wall turbulence statistics and flow structures

Large eddy simulation (LES) of turbulent flow through an axially rotating pipe, coupled with nonlinear dynamic subgrid-scale (SGS) model, is carried out to investigate rotation effect on the near-wall turbulence characteristics and flow structures. In the rotating turbulent pipe flow, it is found that the tendency towards a relaminarized flow appears and the axial velocity fluctuation is suppressed; however, the azimuthal fluctuation is enhanced due to the presence of the pipe wall rotation. The joint probability density function (joint PDF) of the velocity fluctuations and the probability density function (PDF) of the helicity fluctuation are analyzed in detail. It is revealed that the resolved Reynolds stress and helicity fluctuation in the wall region are closely related to the correlation between the velocity and vorticity fluctuations and affected significantly by the rotation-induced azimuthal mean flow. Further, the budgets of resolved Reynolds stresses indicate that the rotation effect is responsible for the more active turbulent energy redistribution and the production of the azimuthal turbulence fluctuation. The near-wall inclined streaky structures with respect to the axial direction are ascribed to the spiral motion of the fluid induced by the rotating pipe. The turbulence characteristics revealed in this study are of great help for the understanding of physical fundamentals in the rotating turbulent flows and for the development of reliable turbulence model.

A laboratory study of mean flow generation in rotating fluids by Reynolds stress gradients

Laboratory experiments were conducted that demonstrate that a mean azimuthal flow can be produced by introducing Reynolds stress gradients to a rotating fluid with zero initial mean flow. This mechanism may play a role in the generation of mean currents in coastal regions. The experiments entail the establishment of turbulence in a thin annular‐shaped region centered within a cylindrical test cell through the use of a vertically oscillating grid. This region rests in a horizontal plane perpendicular to the vertical axis of the tank, and the entire system is placed on a turntable to simulate background rotation. Flow visualization techniques are used to depict qualitative features of the resulting flow field. Measurements of the mean and turbulent velocity fields are performed using a two‐component laser‐Doppler velocimeter. The results show how rectified currents (mean flows) can be generated via Reynolds stress gradients induced by periodic forcing of the grid. In the absence of background rotation, rectified flow is observed in the radial and vertical directions only. The presence of background rotation tends to organize these motions in that the flow tends to move parallel to the turbulent source, i.e., in the azimuthal direction, with the source (strong turbulence) located to the right, facing downstream. The influence of rotation on the Reynolds stresses and their gradients as well as on the ensuing mean flow is evaluated, and the observations are examined by considering individual contributions of the terms in the Reynolds‐averaged momentum equations.

Three-dimensional numerical simulation of thermocapillary flows in cylindrical liquid bridges

The dynamics of thermocapillary flows in differentially heated cylindrical liquid bridges is investigated numerically using a mixed finite volume/pseudo-spectral method to solve the Navier–Stokes equations in the Boussinesq approximation. For large Prandtl numbers (Pr = 4 and 7) and sufficiently high Reynolds numbers, the axisymmetric basic flow is unstable to three-dimensional hydrothermal waves. Finite-amplitude azimuthally standing waves are found to decay to travelling waves. Close to the critical Reynolds number, the former may persist for long times. Representative results are explained by computing the coefficients in the Ginzburg–Landau equations for the nonlinear evolution of these waves for a specific set of parameters. We investigate the nonlinear phenomena characteristic of standing and pure travelling waves, including azimuthal mean flow and time-dependent convective heat transport. For Pr [Lt ] 1 the first transition from the two-dimensional basic flow to the three-dimensional stationary flow is inertial in nature. Particular attention is paid to the secondary transition leading to oscillatory three-dimensional flow, and this mechanism is likewise independent of Pr. The spatial and temporal structure of the perturbation flow is analysed in detail and an instability mechanism is proposed based on energy balance calculations and the vorticity distribution.

Magnetostrophic magnetoconvection

Perhaps the most fundamental problem in dynamo theory is the smallness of the Ekman number E which is O(10 −15) based on estimates of the molecular diffusivity of the Earth's core. It is difficult to achieve values much lower than O(10 −4) in numerical models. The “headline” value of E can be reduced using hyperviscosity but it is becoming clear that this is not helping our aim of moving towards the geophysically relevant parameter regime of low viscosity, and several factors indicate that E= O(10 −4) is not small enough. The magnetostrophic approximation is an alternative approach that neglects viscous effects altogether ( E=0) in the interior of the core. A consequence of this is Taylor's constraint [Taylor, J.B., 1963. The magnetohydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc. R. Soc. London, Ser. A 274, 274–283]. So far, fundamental problems have prevented the successful working of a convection-driven dynamo model using this approach [Walker, M.R., Barenghi, C.F., Jones, C.A., 1998. A note on dynamo action at asymptotically small Ekman number. Geophys. Astrophys. Fluid Dyn. 88, 261–275]. Nevertheless, the difficulties with hyperviscosity mean that it is important to continue to learn as much as possible about the E→0 limit through the ( E=0) magnetostrophic approximation. If viscous effects are retained only in Ekman boundary layers, a standard analysis modifies Taylor's constraint to give a prescription for the mean azimuthal flow. This so-called “geostrophic flow” V G( s, t)1 φ is the dominant nonlinear effect when the magnetic field is weak and can act to equilibrate an otherwise exponentially growing linear solution in an “Ekman state”. Two variants of this approach have been studied: mean-field dynamos and magnetoconvection. Here, we consider the latter. We use a cylindrical geometry [adopting cylindrical polars ( s, φ, z)]. In the presence of an imposed magnetic field B 0, we time-step the governing equations to find the dependence of the nonlinear (Ekman state) solutions as a function of the Rayleigh number Ra. We investigate the role of alternative magnetic boundary conditions and of whether there is an inner cylindrical boundary or not. These introduce modest quantitative differences. The major qualitative difference arises from the orientation of gravity g and the imposed temperature gradient ∇ T. When in the axial ( z) direction, a transition towards a Taylor state (along the lines envisaged by Malkus and Proctor [Malkus, W.V.R., Proctor, M.R.E., 1975. The macrodynamics of α-effect dynamos in rotating fluids. J. Fluid Mech. 67, 417–443.]) is found at Ra close to Ra c. When g and ∇ T 0 are aligned in the radial ( s) direction, supercritical Ekman states are found for a large range of values of Ra, and there is no evidence of the system moving towards a Taylor state. These results are consistent with the earlier work of Skinner and Soward [Skinner, P.H., Soward, A.M., 1990. Convection in a rotating magnetic system and Taylor's constraint II. Geophys. Astrophys. Fluid Dyn. 60, 335–356.] and Jones and Roberts [Jones, C.A., Roberts, P.H., 1990. Magnetoconvection in rapidly rotating Boussinesq and compressible fluids. Geophys. Astrophys. Fluid Dyn. 55, 263–308]. The expected cancellation in the Taylor integral is clearly observed as a Taylor state is approached. This emphasises the inability of the standard prescription for V G to be useful other than for an Ekman state.

Trapped‐wave modification and critical surface formation by mean flow at a seamount with application at Fieberling Guyot

Effects of radially and vertically sheared mean azimuthal flow on trapped waves at a seamount are shown to include changes to resonance frequency and spatial structure and the formation of two types of critical surface. In waves calculated using bathymetry, stratification strength, and frequency (0.66ƒ) appropriate to amplified diurnal currents observed at Cobb Seamount (46.8°N), mean flow based on measurements there does not cause critical surfaces. Even though clockwise mean flow vorticity is a significant fraction of ƒ, effects on the wave are limited to weakly reduced spatial scales and an increase in resonance frequency much weaker than estimates of the Doppler shift. The Doppler shift is opposed by variations in mean horizontal shear that cause a downslope potential vorticity gradient opposite that of the bottom slope supporting the wave; mean vertical shear is unimportant. In the general case, mean flow can cause (1) a stratified seamount‐trapped wave critical surface where the intrinsic frequency vanishes, causing both horizontal and vertical scales of the wave to diminish and (2) an internal wave critical surface where the intrinsic frequency rises to the effective Coriolis frequency, or low‐frequency bound for superinertial internal waves. An internal wave critical surface bounds a “superinertial cap” within which subinertial currents are effectively superinertial. Mean flow at Fieberling Guyot (32.4°N) reduces the effective Coriolis frequency by 0.43ƒ and causes a superinertial cap having an internal wave critical surface for diurnal currents (0.93ƒ) that coincides with measured turbulence maxima extending several kilometers away laterally above the seamount.

Application of the Eliassen Balanced Model to Real-Data Tropical Cyclones

Abstract The Eliassen balanced vortex model assumes gradient balance of the azimuthal mean flow. This assumption was tested by calculating mean and eddy terms in the radial momentum equation in the synoptic-scale environments of two tropical cyclones. The azimuthally averaged gradient balance was accurate to within 15%–25% in the free atmosphere outside the core, even in the asymmetric outflow layer. Balanced secondary circulations correlated well with circulations that included gradient thermal wind imbalance terms. Although the balanced model lacks Galilean invariance, balanced circulations were largely insensitive to use of a fixed coordinate or a coordinate moving with the storm. This occurred because changes in eddy heat and angular momentum fluxes largely offset one another. The two-dimensional balanced solutions provide a reasonably robust measure of circulations induced by azimuthal eddy processes in the tropical cyclone environment. Nevertheless, individual forcing functions, such as the commonly...

Ringdown of inertial waves in a spheroidal shell of rotating fluid

The role of inertial waves in the dynamies of the Earth's fluid core has been investigated through laboratory experiments on a spheroidal shell of rotating fluid. In these experiments inertial waves of azimuthal wavenumber 1, Ekman number 0(10 −5), Rossby number 0(10 −1) were excited by precession of an inner spheroidal body. Proximity to resonance was achieved by adjusting the ratio of the frequency of precession of the inner body to the rotational speed of the container to be near the eigenfrequency of the inertial wave mode being studied. Once the system was near resonance the perturbation was stopped and ringdown records were obtained. Amplitude, eigenfrequency and decay rate were recovered simultaneously for the principal and neighbouring modes excited using an iterative linearized least squares procedure. The recovery of complex eigenfrequencies for non-axisymmetric inertial waves in this shell geometry has given experimental verification of their existence. For those waves of azimuthal wavenumber one, a significant nonlinear interaction among modes is inferred from the simultaneous recovery of neighbouring modes. Other non linear effects include a mean azimuthal flow which appears to be stable for the low spatial order modes studied. These results contrast with highly unstable mean flow found experimentally in similar experiments carried out in cylindrical geometry.