On the effects of thermally insulating boundaries on geostrophic flows in rapidly rotating gases

1979 ◽  
Vol 95 (1) ◽  
pp. 97-118 ◽  
Author(s):  
Fritz H. Bark ◽  
Lennart S. Hultgren
Keyword(s):  
Boundary Condition ◽  
Temperature Gradient ◽  
Boundary Layers ◽  
Ekman Layer ◽  
Gas Flows ◽  
Zeroth Order ◽  
Ekman Number ◽  
Geostrophic Flow ◽  
Geostrophic Flows

The effects of thermally insulating boundaries on rapidly and almost rigidly rotating gas flows are examined. It is shown that, on a thermally insulating boundary, all boundary layers disappear to zeroth order and that the geostrophic flow alone satisfies the kinematical boundary condition on such a boundary. The temperature gradient of the geostrophic flow is on a horizontal thermally insulating boundary corrected by a weak Ekman layer of strength E½ where E is the Ekman number. On a vertical thermally insulating boundary, the temperature gradient of the geostrophic flow is in the general case corrected by E¼ and $E^{\frac{1}{3}}$ Stewartson layers of strengths E¼ and $E^{\frac{1}{3}}$ respectively.

Coriolis-force attenuation of blocking in a stratified flow

1979 ◽  
Vol 94 (4) ◽  
pp. 711-727 ◽  
Author(s):  
M. R. Foster
Keyword(s):  
Boundary Layers ◽  
Detailed Examination ◽  
Ekman Layer ◽  
Ekman Pumping ◽  
Ekman Number ◽  
Coriolis Forces ◽  
Ekman Layers ◽  
The Face ◽  
Vorticity Generation ◽  
Upstream Flow

Even very small Coriolis forces are shown to alter significantly the nature of the upstream wake of an object in slow (small Froude number) translation through a non-diffusive stratified fluid. If the Ekman number is of order one, the far upstream extent of the wake is reduced. If the fluid rotation is sufficient to make the Ekman number small, the contraction of the wake is much greater. We study a particular case in detail; the Ekman number is small enough to make horizontal boundary layers Ekman layers. In this case, the wake is confined to the vicinity of the object, the upstream flow arising from a combination of Ekman pumping and baroclinic vorticity generation. The upstream flow is described by an eigenfunction whose amplitude is dependent on object geometry. If the object is a semi-infinite rectangular parallelepiped, that amplitude is determined by detailed examination of the shear layer at the face of the parallelepiped and its interaction with the Ekman layer on the top surface of the object

scholarly journals Three-Dimensional Electro-Thermal Analysis of a New Type Current Transformer Design for Power Distribution Networks

Energies ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 1792
Author(s):  
Bingbing Dong ◽  
Yu Gu ◽  
Changsheng Gao ◽  
Zhu Zhang ◽  
Tao Wen ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Temperature Gradient ◽  
Field Distribution ◽  
Distribution Network ◽  
Current Transformer ◽  
Internal Temperature ◽  
Temperature Field Distribution ◽  
Type Design ◽  
New Type

In recent years, the new type design of current transformer with bushing structure has been widely used in the distribution network system due to its advantages of miniaturization, high mechanical strength, maintenance-free, safety and environmental protection. The internal temperature field distribution is an important characteristic parameter to characterize the thermal insulation and aging performance of the transformer, and the internal temperature field distribution is mainly derived from the joule heat generated by the primary side guide rod after flowing through the current. Since the electric environment is a transient field and the thermal environment changes slowly with time as a steady field under the actual conditions, it is more complex and necessary to study the electrothermal coupling field of current transformer (CT). In this paper, a 3D simulation model of a new type design of current transformer for distribution network based on electric-thermal coupling is established by using finite element method (FEM) software. Considering that the actual thermal conduction process of CT is mainly by conduction, convection and radiation, three different kinds of boundary conditions such as solid heat transfer boundary condition, heat convection boundary condition and surface radiation boundary condition are applied to the CT. Through the model created above, the temperature rise process and the distribution characteristics of temperature gradient of the inner conductor under different current, different ambient temperatures and different core diameters conditions are studied. Meanwhile, the hottest temperature and the maximum temperature gradient difference are calculated. According to this, the position of weak insulation of the transformer is determined. The research results can provide a reference for the factory production of new type design of current transformer.

Experimental and numerical investigation of turbulent convection in a rotating cylinder

2009 ◽  
Vol 642 ◽  
pp. 445-476 ◽  
Author(s):  
R. P. J. KUNNEN ◽  
B. J. GEURTS ◽  
H. J. H. CLERCX
Keyword(s):  
Temperature Gradient ◽  
Boundary Layers ◽  
Power Law ◽  
Vertical Velocity ◽  
Stereoscopic Particle Image Velocimetry ◽  
Turbulent Heat Transfer ◽  
Bulk Temperature ◽  
Rossby Number ◽  
Cyclonic Vorticity ◽  
Vertical Vorticity

The effects of an axial rotation on the turbulent convective flow because of an adverse temperature gradient in a water-filled upright cylindrical vessel are investigated. Both direct numerical simulations and experiments applying stereoscopic particle image velocimetry are performed. The focus is on the gathering of turbulence statistics that describe the effects of rotation on turbulent Rayleigh–Bénard convection. Rotation is an important addition, which is relevant in many geophysical and astrophysical flow phenomena.A constant Rayleigh number (dimensionless strength of the destabilizing temperature gradient) Ra = 109 and Prandtl number (describing the diffusive fluid properties) σ = 6.4 are applied. The rotation rate, given by the convective Rossby number Ro (ratio of buoyancy and Coriolis force), takes values in the range 0.045 ≤ Ro ≤ ∞, i.e. between rotation-dominated flow and zero rotation. Generally, rotation attenuates the intensity of the turbulence and promotes the formation of slender vertical tube-like vortices rather than the global circulation cell observed without rotation. Above Ro ≈ 3 there is hardly any effect of the rotation on the flow. The root-mean-square (r.m.s.) values of vertical velocity and vertical vorticity show an increase when Ro is lowered below Ro ≈ 3, which may be an indication of the activation of the Ekman pumping mechanism in the boundary layers at the bottom and top plates. The r.m.s. fluctuations of horizontal and vertical velocity, in both experiment and simulation, decrease with decreasing Ro and show an approximate power-law behaviour of the shape Ro0.2 in the range 0.1 ≲ Ro ≲ 2. In the same Ro range the temperature r.m.s. fluctuations show an opposite trend, with an approximate negative power-law exponent Ro−0.32. In this Rossby number range the r.m.s. vorticity has hardly any dependence on Ro, apart from an increase close to the plates for Ro approaching 0.1. Below Ro ≈ 0.1 there is strong damping of turbulence by rotation, as the r.m.s. velocities and vorticities as well as the turbulent heat transfer are strongly diminished. The active Ekman boundary layers near the bottom and top plates cause a bias towards cyclonic vorticity in the flow, as is shown with probability density functions of vorticity. Rotation induces a correlation between vertical vorticity and vertical velocity close to the top and bottom plates: near the top plate downward velocity is correlated with positive/cyclonic vorticity and vice versa (close to the bottom plate upward velocity is correlated with positive vorticity), pointing to the vortical plumes. In contrast with the well-mixed mean isothermal bulk of non-rotating convection, rotation causes a mean bulk temperature gradient. The viscous boundary layers scale as the theoretical Ekman and Stewartson layers with rotation, while the thermal boundary layer is unaffected by rotation. Rotation enhances differences in local anisotropy, quantified using the invariants of the anisotropy tensor: under rotation there is strong turbulence anisotropy in the centre, while near the plates a near-isotropic state is found.

scholarly journals Interaction of Ekman Layers and Islands

2013 ◽  
Vol 43 (5) ◽  
pp. 1028-1041 ◽  
Author(s):  
Michael A. Spall ◽  
Joseph Pedlosky
Keyword(s):  
Boundary Layers ◽  
Numerical Models ◽  
Phase Changes ◽  
Ekman Transport ◽  
Return Flow ◽  
Kelvin Wave ◽  
Density Anomaly ◽  
Depth Direction ◽  
Ekman Layers ◽  
Geostrophic Flows

Abstract The circulation induced by the interaction of surface Ekman transport with an island is considered using both numerical models and linear theory. The basic response is similar to that found for the interaction of Ekman layers and an infinite boundary, namely downwelling (upwelling) in narrow boundary layers and deformation-scale baroclinic boundary layers with associated strong geostrophic flows. The presence of the island boundary, however, allows the pressure signal to propagate around the island so that the regions of upwelling and downwelling are dynamically connected. In the absence of stratification the island acts as an effective barrier to the Ekman transport. The presence of stratification supports baroclinic boundary currents that provide an advective pathway from one side of the island to the other. The resulting steady circulation is quite complex. Near the island, both geostrophic and ageostrophic velocity components are typically large. The density anomaly is maximum below the surface and, for positive wind stress, exhibits an anticyclonic phase rotation with depth (direction of Kelvin wave propagation) such that anomalously warm water can lie below regions of Ekman upwelling. The horizontal and vertical velocities exhibit similar phase changes with depth. The addition of a sloping bottom can act to shield the deep return flow from interacting with the island and providing mass transport into/out of the surface Ekman layer. In these cases, the required transport is provided by a pair of recirculation gyres that connect the narrow upwelling/downwelling boundary layers on the eastern and western sides of the island, thus directly connecting the Ekman transport across the island.

Structure and energetics of optimal Ekman layer perturbations

1997 ◽  
Vol 333 ◽  
pp. 97-123 ◽  
Author(s):  
RALPH C. FOSTER
Keyword(s):  
Boundary Layer ◽  
Normal Mode ◽  
Geostrophic Wind ◽  
Ekman Layer ◽  
Acute Angle ◽  
Small Scale ◽  
Mean Flow ◽  
Near Surface ◽  
Geostrophic Flow ◽  
Neutrally Stratified

The optimal non-modal perturbations for the neutrally stratified boundary layer in a rotating frame of reference (Ekman layer) are found for a Reynolds number characteristic of the planetary boundary layer (PBL). Two classes of non-modal instabilities are found: evanescent perturbations, with lifetimes up to about one hour, and growing instabilities. The important difference between these types of perturbations is whether or not the optimal non-modal perturbation projects onto an unstable normal mode. The evanescent instabilities are of smaller scale and are oriented at larger angles to the surface isobars compared to either the growing perturbations or normal-mode instabilities. The optimal perturbations take the form of vortices at an acute angle to the geostrophic flow that rapidly transform into streaks with associated overturning motion. The energetics of the optimal perturbations are investigated in detail to clarify the instability mechanism throughout its evolution.Nonlinear stability analyses of the neutrally stratified Ekman layer have shown that the normal-mode instability will equilibrate with the mean flow to form boundary-layer-scale equilibrium roll eddies aligned closely with the geostrophic flow. However, numerical simulations do not generate these rolls in neutral stratification although they often realize small-scale near-surface streaks oriented at large angles to the geostrophic wind. The evanescent optimal perturbations bear a close resemblance to the simulated streaks. It is proposed that the non-model perturbation mechanism is associated with the streaks.

On the precession of a resonant cylinder

1970 ◽  
Vol 41 (4) ◽  
pp. 865-872 ◽  
Author(s):  
Roger F. Gans
Keyword(s):  
Boundary Layers ◽  
Solvability Condition ◽  
Rotating Cylinder ◽  
Resonant Mode ◽  
Ekman Layer ◽  
Resonant Response ◽  
Order Problem ◽  
Precession Rate ◽  
Resonance Phenomena ◽  
Strong Amplitude

A fluid contained in a rotating cylinder has an inertial mode which is excited by forced precession of the container. Wood's (1965, 1966) early work specifically excluded resonance phenomena. Recently McEwan (1970) has discussed resonance phenomena for strong amplitude of excitation, corresponding to rapid precession in this work.In this paper the magnitude of the resonant response for small precession rate is precisely calculated by matching the Ekman layer suction to the precessional forces. The procedure is to find the resonant mode v1, compute its boundary layers, $\tilde{{\bf v}}_1$, and the associated Ekman layer suction. The second-order problem has a solvability condition which is satisfied by matching the Ekman layer suction to the precession.

The influence of surface topography on rotating convection

1996 ◽  
Vol 313 ◽  
pp. 147-180 ◽  
Author(s):  
Peter I. Bell ◽  
Andrew M. Soward
Keyword(s):  
Surface Topography ◽  
Rossby Waves ◽  
Finite Amplitude ◽  
Length Scale ◽  
Mean Flow ◽  
Radial Temperature ◽  
Static State ◽  
Geostrophic Flow ◽  
Bump Height ◽  
Geostrophic Flows

Busse's annulus is considered as a model of thermal convection inside the Earth's liquid core. The conventional tilted base and top are modified by azimuthal sinusoidal corrugations so that the effects of surface topography can be investigated. The annulus rotates rapidly about its axis of symmetry with gravity directed radially inwards towards the rotation axis. An unstable radial temperature gradient is maintained and the resulting Boussinesq convection is considered at small Ekman number. Since the corrugations on the boundaries cause the geostrophic contours to be no longer circular, strong geostrophic flows may be driven by buoyancy forces and damped by Ekman suction. When the bumps are sufficiently large, instability of the static state is dominated by steady geostrophic flow with the convection pattern locked to the bumps. As the bump size is decreased, oscillatory geostrophic flow is possible but the preferred mode is modulated on a long azimuthal length scale and propagates as a wave eastwards. This mode only exists in the presence of bumps and is not to be confused with the thermal Rossby waves which are eventually preferred as the bump height tends to zero. Like thermal Rossby waves, the new modes prefer to occupy the longest available radial length scale. In this long-length-scale limit, two finite-amplitude states characterized by uniform geostrophic flows can be determined. The small-amplitude state resembles Or & Busse's (1987) mean flow instability. On losing stability the solution jumps to the more robust large-amplitude state. Eventually, for sufficiently large Rayleigh number and bump height, it becomes unstable to a long-azimuthal-length-scale travelling wave. The ensuing finite-amplitude wave and the mean flow, upon which it rides, are characterized by a geostrophic flow, which is everywhere westward.

Numerical Study of Flow and Heat in Long Micro and Nano Channels

2008 ◽  
Author(s):  
Masoud Darbandi ◽  
Shidvash Vakilipour
Keyword(s):  
Boundary Condition ◽  
Numerical Study ◽  
Slip Flow ◽  
Constant Heat Flux ◽  
Gas Flows ◽  
Transfer Rates ◽  
Compressibility Effect ◽  
Momentum And Heat Transfer ◽  
Compressibility Effects ◽  
Early Transition

In this work, we extend a numerical tool capable of solving compressible and incompressible gas flows to study the momentum and heat transfer rates in micro/nano channels with high aspect ratio (L/H = 8000), where the compressibility effect is dominant. The constant heat flux thermal boundary condition is firstly applied at the wall. Next, the flow regime is extended to the early transition regime employing a high order slip velocity boundary condition based on the kinetic theory assumptions. The accuracy of the present results in the slip flow regimes is evaluated against other available theoretical and experimental results. The thermal and compressibility effects on the pressure and Knudsen number distribution are extensively studied along the channel at early transition regimes up to Kn = 0.5. Likely, this Knudsen is the highest one to be reached via applying the foregoing boundary conditions.

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