Fluid flows in a librating cylinder

We study swirling electrovortex flows in a cylinder filled with GaInSn metal using axisymmetric and large-scale three-dimensional numerical simulations. In our set-up electrical currents enter and exit the cell symmetrically through wires and the result is a von Kármán-like flow. Three inductionless and an inductive flow regimes are identified. Scaling laws for the magnitude of the velocity in each of these regimes are obtained both numerically and explained theoretically. We study how the aspect ratio of the cell affects the flow and how symmetrically wired cells are different from asymmetrical wired cells. We vary the radius of the connecting wires and propose a simple model that captures how the flow's intensity varies with the wire radius.

Barotropic flow over bottom topography— experiments and nonlinear theory

Motivated by the desire to understand the rich dynamics of precessionally driven flow in a liquid planetary core, we investigate, through numerical simulations, the precessing fluid motion in a rotating cylindrical annulus, which simultaneously possesses slow precession. The same problemhas been studied extensively in cylinders, where the precessing flow is characterized by three key parameters: the Ekman number E, the Poincaré number Po and the radius-height aspect ratio Γ. While in an annulus, there is another parameter, the inner-radius-height aspect ratio ϒ, which also plays an important role in controlling the structure and evolution of the flow. By decomposing the nonlinear solution into a set of inertial modes, we demonstrate the properties of both weakly and moderately precessing flows. It is found that, when the precessional force is weak, the flow is stable with a constant amplitude of kinetic energy. As the precessional force increases, our simulation suggests that the nonlinear interaction between the boundary effects and the inertial modes can trigger more turbulence, introducing a transitional regime of rich dynamics to disordered flow. The inertial mode u111, followed by u113 or u112, always dominates the precessing flow when 0.001 ≤ Po ≤ 0.05, ranging from weak to moderate precession. Moreover, the precessing flow in an annulus shows more stability than in a cylinder which is likely to be caused by the effect of the inner boundary that restricts the growth of resonant and non-resonant inertial modes. Furthermore, the mechanism of triadic resonance is not found in the transitional regime from a laminar to disordered flow.

Several planetary bodies in our solar system undergo a forced libration owing to gravitational interactions with their orbital companions, leading to complex fluid motions in their metallic liquid cores or subsurface oceans. In this study, we numerically investigate flows in longitudinally librating spherical shells. We focus on the Ekman number dependencies of several shear layers when the libration frequency is less than twice of the rotation frequency and the libration amplitude is small. Time-dependent flows mainly consist of inertial waves excited at the critical latitudes due to the Ekman pumping singularities, forming conical shear layers. In particular, previous theoretical studies have proposed different scalings for the conical shear layers spawned from the critical latitudes at the inner boundary. Our numerical results favour the velocity amplitude scaling predicted by Le Dizès & Le Bars (J. Fluid Mech. 2017, 826, 653) over the scaling initially proposed by Kerswell (J. Fluid Mech. 1995, 298, 311), though the Ekman numbers in our calculations are not sufficiently small to pin down this scaling. Non-linear interactions in the boundary layers drive a mean zonal flow with several geostrophic shears. Our numerical results show that geostrophic shears associated with the critical latitudes at the inner and outer boundaries exhibit the same scalings, i.e. an amplitude of over a width of . Apart from the geostrophic shear associated with the critical latitude, our numerical results show that the reflection of inertial waves can induce a geostrophic shear with an amplitude of over a width of . As the amplitude of the geostrophic shears increases as reducing the Ekman number, the geostrophic shears in the mean flows may be significant in planetary cores and subsurface oceans given small Ekman numbers of these systems.

An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the (x, y)-plane, themrigid side walls and the two rigid end walls being in contact with the fluid. Here (x, y, z) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container atQcm3/sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocityu= −∇ϕ − ∇ ×A, is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects onuof uniform rotation of the whole system with angular velocity Ω about thez-axis when the normal component ofuon the side walls is independent ofz.In the simplest conceivable case,D≡zu−zlis infinite (butD/Qremains finite). End effects are then negligible anduis everywhere independent ofz.The solenoidal component ofu, − ∇ ×A, corresponds tojgyres, one for each of thejirreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within them-ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇2ϕ = 0, depends onQbut not on Ω,jandv(the coefficient of kinematic viscosity), ∇ ×Adepends on all these quantities but vanishes identically whenjΩ = 0. WhenjΩ ≠ 0 butv→ 0, ∇2A+ 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form.End effects cannot be ignored whenDis finite. WhenDis independent ofxandyand equal toD0(say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3(v/Ω)½(δ [Lt ]D0), the end effects that then arise are only confined to these boundary layers whenj= 0. Whenj≠ 0 boundary-layer suction influences the flow everywhere; thus ∇2Aand ∇ϕ (but not ∇ ×A) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇2Abeing the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δs(typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δs. WhenDis not everywhere independent ofxandy,non-viscous end effects arise which produce relative vorticity in the main body of the fluid even whenj= 0.Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.

The flow in a split cylinder with each half in exact counter rotation is studied numerically. The exact counter rotation, quantified by a Reynolds number $\mathit{Re}$ based on the rotation rate and radius, imparts the system with an $O(2)$ symmetry (invariance to azimuthal rotations as well as to an involution consisting of a reflection about the mid-plane composed with a reflection about any meridional plane). The $O(2)$ symmetric basic state is dominated by a shear layer at the mid-plane separating the two counter-rotating bodies of fluid, created by the opposite-signed vortex lines emanating from the two endwalls being bent to meet at the split in the sidewall. With the exact counter rotation, the additional involution symmetry allows for steady non-axisymmetric states, that exist as a group orbit. Different members of the group simply correspond to different azimuthal orientations of the same flow structure. Steady states with azimuthal wavenumber $m$ (the value of $m$ depending on the cylinder aspect ratio $\unicode[STIX]{x1D6E4}$) are the primary modes of instability as $\mathit{Re}$ and $\unicode[STIX]{x1D6E4}$ are varied. Mode competition between different steady states ensues, and further bifurcations lead to a variety of different time-dependent states, including rotating waves, direction-reversing waves, as well as a number of slow–fast pulse waves with a variety of spatio-temporal symmetries. Further from the primary instabilities, the competition between the vortex lines from each half-cylinder settles on either a $m=2$ steady state or a limit cycle state with a half-period-flip spatio-temporal symmetry. By computing in symmetric subspaces as well as in the full space, we are able to unravel many details of the dynamics involved.

In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate $\unicode[STIX]{x1D6FA}^{\ast }$ using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency $\unicode[STIX]{x1D714}^{\ast }$ within the inertial wave interval $(0,2\unicode[STIX]{x1D6FA}^{\ast })$ such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle $\unicode[STIX]{x1D703}_{c}=\text{acos}(\unicode[STIX]{x1D714}^{\ast }/(2\unicode[STIX]{x1D6FA}^{\ast }))$ which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singularities of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman (Phil. Trans. R. Soc. Lond. A, vol. 264, 1969, pp. 597–634). These solutions are characterized by two parameters only: a real parameter $m$, which measures the strength of the underlying singularity, and a complex amplitude coefficient $C_{0}$. We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a solid plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a solid plane are both well described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter $m=1$ corresponding to a Dirac source at the edge of the disk and a coefficient $C_{0}\propto E^{1/6}$ where $E$ is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter $m$ for the similarity solution is $m=5/4$, leading to a coefficient $C_{0}\propto E^{1/12}$, that is larger than for the case of a disk for small Ekman numbers. A simple general expression for $C_{0}$ valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.

We investigate viscous dissipation in linear flows driven by small-amplitude longitudinal librations in rotating fluid spheres focusing on the rapid rotation regime applicable to planets. Viscous coupling can resonate with inertial modes in the bulk of the fluid when the frequency of the forcing is within the range $(0,2\Omega _0)$ , where $\Omega _0$ is the mean angular velocity of the sphere. We solve the linearised equations of motion with a semi-spectral numerical method and with an asymptotic expansion exploiting the small Ekman number, $E$ , which quantifies the strength of viscous forces relative to the Coriolis force. Our results confirm that the dominant contribution to the dissipation occurs in the Ekman boundary layer with leading-order scaling $E^{1/2}$ . When the forcing frequency coincides with that of an inertial mode, dissipation is reduced by as much as 9 % compared with boundary layer theory alone. The percentage-wise reduction is independent of $E$ and the frequency width of the reduction envelope scales as $E^{1/2}$ . At non-resonant frequencies conic shear layers develop in the bulk interior and, together with the Ekman layer bulge at critical latitude, slightly enhance dissipation. We confirm critical latitude bulge and shear layer contributions to the overall dissipation scale as $E^{4/5}$ and $E^{6/5}$ respectively, becoming negligible compared with dissipation in the main boundary layer as $E\rightarrow 0$ . The frequencies at which the dissipation enhancement from critical latitude effects is maximised are displaced from the inviscid limit periodic orbit frequencies by a factor that scales with $E^{0.23}$ .

Physical phenomena fundamental to rotating, baroclinically driven flows are studied with reference to results of numerical simulation of rotating annulus flows, using a modified Galerkin Model. Both local and global effects of sources, sinks, and transports of heat and momentum are discussed. A convenient ‘energy exchange” diagram reveals detailed information that is used to analyze nonlinear equilibration and amplitude vacillation of quasi-geostrophic baroclinic eddies. Transient inertial oscillations, sidewall boundary layers, and internal boundary layers are also discussed. A detailed study of symmetric flows is made, eleven of which are tested numerically for stability with respect to three-dimensional disturbances of a given zonal wave number. Two of the four unstable cases are integrated to a numerical steady state with finite-amplitude, quasi-geostrophic baroclinic waves. With the ‘rigid-lid’ geometry assumed, the average zonal velocity is zero, resulting in zero phase velocity of the waves. The structure of the thermal wave is nearly coherent in the vertical. These numerical results are consistent with laboratory observations. The eddy flow is quasi-geostrophic except in horizontal boundary layers, where the flow is driven toward low pressure. A small cross-isotherm advection is sufficient to maintain the temperature wave against diffusion and vertical advection. The eddy flow adjusts spontaneously toward the form of the fastest growing or slowest decaying disturbance representable by the truncated space resolution. The eddy flow feeds energy into the mean zonal flow in ‘barotropic-type’ interactions reflected mainly by the familiar ‘tilted trough’. During equilibration, the eddy flow alters the mean zonal flow in such a way that eddy energy sources are reduced relative to energy sinks. However, this adjustment is small compared to the change of total flow, which reflects a relatively large change of eddy amplitude. This suggests that small errors in the mean zonal flow representation can lead to relatively large errors in total flow representation. In most flows studied the kinetic energy dissipation is concentrated in thin boundary layers. In spite of this thinness, the basically laminar character of these dissipative boundary layers allows accurate and economical numerical simulation through the use of characteristic functions, which is a natural refinernent of the basic Galerkin method used. (In this prototype study, only “moderately characteristic’ functions are used, thus sacrificing numerical economy while simplifying the programming.) Similarly, the generation of potential energy, which is transformed into the kinetic energy of the flow, is accurately simulated. In most cases studied, this generation is also concentrated in thin boundary layers where thermal energy is extracted from cold fluid and added to warm fluid.

: Negligible wall shear stress provides the criterion that minimizes the distance over which boundary layer can decelerate without separating from the surface. Over the past few years, we have been carefully investigating the boundary layer that is continuously maintained on the verge of separation, without actually being allowed to separate 1. The experimental facility that was used consists of a contraction, a constant area throat, and a two-dimensional variable geometry expansion inserted into the 2' by 3' test section of a low speed wind tunnel (figure 1). The expansion is corrected of Lexan supported by electric actuators mounted on gimbals and capable of distorting the surface under computer control. The surface geometry can vary over a wide range of pressure gradients and streamline curvature. The transition between the throat and the test surface is accomplished by means of a circular cylinder to which the lexan surface is attached, thereby establishing the initial slope of the test surface. The side-wall boundary layers were removed by using suction and the measurements were made on the boundary layer developing on the curved surface.

5 - VORTEX FLOWS IN DUCTING AND FAN

Steady, linear hydromagnetic boundary layer at the free surface of a rotating baroclinic fluid is analysed under Boussinesq approximation. The flow is driven by the buoyancy forces which exist due to the density variations associated with baroclinicity. Meridional circulations of mass flux and electric current flux occur due to the boundary layer suction as well as due to the hydromagnetic effects. The boundary layer suction dominates and the dynamics of the flow reduce to that of nonmagnetic case for α2<O(E), whereE is the Ekman number and α2 (=σB 0 2 /ϱΩ) is the magnetic interaction parameter, which represents the ratio of hydromagnetic to Coriolis forces. When α2>O(E), meridional circulation induced in the interior due to hydromagnetic effects is stronger than that can be supported by the boundary layer. The electric current flux through the interior is stronger for any value of α than that can be supported by the boundary layer. Further it is argued that hydromagnetic effects suppress the laminated flow (∂v/∂z≠0) in the interior. As a result, the meridional circulations are strong when α=O(1).

The connection between the heat transfer and characteristic flow velocities of planetary core-style convection remains poorly understood. To address this, we present novel laboratory models of rotating Rayleigh–Bénard convection in which heat and momentum transfer are simultaneously measured. Using water (Prandtl number, Pr≃6) and cylindrical containers of diameter-to-height aspect ratios of Γ≃3,1.5,0.75, the non-dimensional rotation period (Ekman number, E) is varied between 10−7≲E≲3×10−5 and the non-dimensional convective forcing (Rayleigh number, Ra) ranges from 107≲Ra≲1012. Our heat transfer data agree with those of previous studies and are largely controlled by boundary layer dynamics. We utilize laser Doppler velocimetry (LDV) to obtain experimental point measurements of bulk axial velocities, resulting in estimates of the non-dimensional momentum transfer (Reynolds number, Re) with values between 4×102≲Re≲5×104. Behavioral transitions in the velocity data do not exist where transitions in heat transfer behaviors occur, indicating that bulk dynamics are not controlled by the boundary layers of the system. Instead, the LDV data agree well with the diffusion-free Coriolis–Inertia–Archimedian (CIA) scaling over the range of Ra explored. Furthermore, the CIA scaling approximately co-scales with the Viscous–Archimedian–Coriolis (VAC) scaling over the parameter space studied. We explain this observation by demonstrating that the VAC and CIA relations will co-scale when the local Reynolds number in the fluid bulk is of order unity. We conclude that in our experiments and similar laboratory and numerical investigations with E≳10−7, Ra≲1012, Pr≃7, heat transfer is controlled by boundary layer physics while quasi-geostrophically turbulent dynamics relevant to core flows robustly exist in the fluid bulk.

The flow in an enclosed completely filled rapidly rotating cylinder that is driven by the differential corotation of the top lid is studied numerically. Although the flow is in a very simple geometry, the fast background rotation and large differential rotation of the lid lead to very thin boundary layers with a variety of instability modes with very fine spatial scales as well as inertial waves that are sustained in the fast rotating interior flow and that interact with the viscous modes in the sidewall boundary layer, leading to complex spatiotemporal dynamics. The numerical simulations are compared and contrasted to experimental visualizations of the sidewall boundary layer instabilities reported by Hart and Kittelman [“Instabilities of the sidewall boundary layer in a differentially driven rotating cylinder,” Phys. Fluids 8, 692 (1996)]. The experiments report observing axisymmetric rolls propagating down the sidewall layer for small differential corotation of the top lid. As the differential rotation was increased, backward tilted diagonal rolls that precess slightly retrograde with respect to the rotating sidewall and forward tilted rolls with prograde precession significantly faster than the sidewall rotation were observed. For still larger differential rotation, a wavy turbulent state that has backward tilted structures erupting from deep within the sidewall layer into the interior and is riding on the forward tilted diagonal rolls in the deep layer was observed. Our simulations capture all of these states and strongly suggest that the observed axisymmetric rolls are unstable and were only transiently observed due to the slow and continuous increase in the differential rotation employed in the experiments. The influence of inertial waves driven by the sidewall instabilities on the three-dimensional wavy turbulent state is discussed.

To mitigate vortex shedding for flow and noise control of a circular cylinder, an experimental approach combining air blowing and porous coating was implemented simultaneously as a hybrid method. Localized air blowing was symmetrically applied through structured porous media at four angles, corresponding to different regions of the flow field: boundary layers, shear layers on the cylinder, separated shear layers, and the cylinder's base. The study involved synchronizing near-field pressure fluctuation and far-field noise measurements with flow field measurements obtained via particle image velocimetry. Near-field pressure measurements were taken around the cylinder's circumference using a remote sensing method. This comprehensive investigation revealed that vortex shedding primarily induced pressure fluctuations at the cylinder's shoulders, resulting in the propagation of acoustic waves to the far field. The hybrid method, alongside the separate application of porous coating and local blowing, showcased substantial efficacy in mitigating near-field pressure, consequently leading to a reduction in far-field noise. These techniques achieved this by strategically shifting the vortex formation region further downstream and expanding the wake region compared to the baseline. Notably, the hybrid method, particularly when local blowing was applied at the base of the porous coated cylinder, exhibited a significantly enhanced impact in this regard, resembling the behavior observed with the individual application of porous coating.