scholarly journals A nonlinear model for rotationally constrained convection with Ekman pumping

2016 ◽  
Vol 798 ◽  
pp. 50-87 ◽  
Author(s):  
Keith Julien ◽  
Jonathan M. Aurnou ◽  
Michael A. Calkins ◽  
Edgar Knobloch ◽  
Philippe Marti ◽  
...  
Keyword(s):  
Heat Transport ◽  
Boundary Layers ◽  
Fluid Layer ◽  
Plane Layer ◽  
Linear Stability Theory ◽  
Reduced Model ◽  
Ekman Pumping ◽  
Free Boundaries ◽  
Nonlinear Feedback ◽  
Thermal Wind

A reduced model is developed for low-Rossby-number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominantly aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need to resolve the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer, the nonlinear feedback from Ekman pumping would be able to generate heat transport that diverges to infinity at finite Rayleigh number. This layer arrests this blowup, resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing, the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting that this behaviour may be prevalent in geophysical and astrophysical settings. As the rotation rate increases, the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.

Rapidly rotating turbulent Rayleigh-Bénard convection

1996 ◽  
Vol 322 ◽  
pp. 243-273 ◽  
Author(s):  
K. Julien ◽  
S. Legg ◽  
J. Mcwilliams ◽  
J. Werne
Keyword(s):  
Heat Transport ◽  
Fluid Layer ◽  
Limited Range ◽  
Momentum Conservation ◽  
Transition To Turbulence ◽  
Ekman Pumping ◽  
Vortex Interactions ◽  
Slip Surfaces ◽  
Background Temperature ◽  
Convection Rolls

Turbulent Boussinesq convection under the influence of rapid rotation (i.e. with comparable characteristic rotation and convection timescales) is studied. The transition to turbulence proceeds through a relatively simple bifurcation sequence, starting with unstable convection rolls at moderate Rayleigh (Ra) and Taylor numbers (Ta) and culminating in a state dominated by coherent plume structures at high Ra and Ta. Like non-rotating turbulent convection, the rapidly rotating state exhibits a simple power-law dependence on Ra for all statistical properties of the flow. When the fluid layer is bounded by no-slip surfaces, the convective heat transport (Nu − 1, where Nu is the Nusselt number) exhibits scaling with Ra2/7 similar to non-rotating laboratory experiments. When the boundaries are stress free, the heat transport obeys ‘classical’ scaling (Ra1/3) for a limited range in Ra, then appears to undergo a transition to a different law at Ra ≈ 4 × 107. Important dynamical differences between rotating and non-rotating convection are observed: aside from the (expected) differences in the boundary layers due to Ekman pumping effects, angular momentum conservation forces all plume structures created at flow-convergent sites of the heated and cooled boundaries to spin-up cyclonically; the resulting plume/cyclones undergo strong vortex-vortex interactions which dramatically alter the mean state of the flow and result in a finite background temperature gradient as Ra → ∞, holding Ra/Ta fixed.

Stabilization of the no-motion state in the Rayleigh–Bénard problem

1994 ◽  
Vol 447 (1931) ◽  
pp. 587-607 ◽  
Keyword(s):  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Control Flow ◽  
Feedback Controller ◽  
Linear Stability Theory ◽  
Motion State ◽  
Conduction State ◽  
Bénard Problem ◽  
Benard Problem ◽  
Rayleigh Benard

Using linear stability theory and numerical simulations, we demonstrate that the critical Rayleigh number for bifurcation from the no-motion (conduction) state to the motion state in the Rayleigh–Bénard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of a feedback controller effectuating small perturbations in the boundary data. The controller consists of sensors which detect deviations in the fluid’s temperature from the motionless, conductive values and then direct actuators to respond to these deviations in such a way as to suppress the naturally occurring flow instabilities. Actuators which modify the boundary’s temperature or velocity are considered. The feedback controller can also be used to control flow patterns and generate complex dynamic behaviour at relatively low Rayleigh numbers.

Stabilization of the No-Motion State of a Horizontal Fluid Layer Heated From Below With Joule Heating

1995 ◽  
Vol 117 (2) ◽  
pp. 329-333 ◽  
Author(s):  
J. Tang ◽  
H. H. Bau
Keyword(s):  
Joule Heating ◽  
Fluid Layer ◽  
Control Strategies ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Small Perturbations ◽  
Motion State ◽  
Conduction State ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

Using linear stability theory and numerical simulations, we demonstrate that the critical Rayleigh number for bifurcation from the no-motion (conduction) state to the motion state in the Rayleigh–Be´nard problem of an infinite fluid layer heated from below with Joule heating and cooled from above can be significantly increased through the use of feedback control strategies effecting small perturbations in the boundary data. The bottom of the layer is heated by a network of heaters whose power supply is modulated in proportion to the deviations of the temperatures at various locations in the fluid from the conductive, no-motion temperatures. Similar control strategies can also be used to induce complicated, time-dependent flows at relatively low Rayleigh numbers.

scholarly journals The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems

2015 ◽  
Vol 471 (2175) ◽  
pp. 20140689 ◽  
Author(s):  
Michael A. Calkins ◽  
Keith Julien ◽  
Philippe Marti
Keyword(s):  
Prandtl Number ◽  
Scaling Laws ◽  
Fluid Layer ◽  
Time Derivative ◽  
Density Perturbation ◽  
Plane Layer ◽  
Relative Magnitude ◽  
Critical Parameters ◽  
Anelastic Equations ◽  
Prandtl Numbers

The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.

Linear amplification factor transport equation for stationary crossflow instabilities in supersonic boundary layers

2020 ◽  
pp. 095441002095499
Author(s):  
Jiakuan Xu
Keyword(s):  
Transport Equation ◽  
Boundary Layers ◽  
Amplification Factor ◽  
Reynolds Numbers ◽  
Linear Stability Theory ◽  
Local Flow ◽  
Flow Parameters ◽  
Compressible Boundary Layers ◽  
Non Local ◽  
Flow Variables

Based on the database from linear stability theory (LST) analysis, a local amplification factor transport equation for stationary crossflow (CF) waves in low-speed boundary layers was developed in 2019. In this paper, the authors try to extend this transport equation to compressible boundary layers based on local flow variables. The similarity equations for compressible boundary layers are introduced to build the function relations between non-local variables and local flow parameters. Then, compressibility corrections are taken into account to modify the source term of the transport equation. Through verifications of different sweep angles, Reynolds numbers, angles of attack, Mach numbers, and different cross-section geometric shapes, the rationality and correctness of the new transport equation established in this paper are illustrated.

scholarly journals Rigorous bounds on the heat transport of rotating convection with Ekman pumping

2020 ◽  
Vol 61 (2) ◽  
pp. 023101
Author(s):  
B. Pachev ◽  
J. P. Whitehead ◽  
G. Fantuzzi ◽  
I. Grooms
Keyword(s):  
Heat Transport ◽  
Ekman Pumping ◽  
Rotating Convection ◽  
Rigorous Bounds

Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3

2011 ◽  
Vol 674 ◽  
pp. 5-42 ◽  
Author(s):  
CHRISTIAN S. J. MAYER ◽  
DOMINIC A. VON TERZI ◽  
HERMANN F. FASEL
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Skin Friction ◽  
Boundary Layers ◽  
Wave Spectrum ◽  
Linear Stability Theory ◽  
Transition To Turbulence ◽  
Transitional Region ◽  
Mean Flow ◽  
Mach 3

A pair of oblique waves at low amplitudes is introduced in a supersonic flat-plate boundary layer at Mach 3. Its downstream development and the concomitant process of laminar to turbulent transition is then investigated numerically using linear-stability theory, parabolized stability equations and direct numerical simulations (DNS). In the present paper, the linear regime is studied first in great detail. The focus of the second part is the early and late nonlinear regimes. It is shown how the disturbance wave spectrum is filled up by nonlinear interactions and which flow structures arise and how these structures locally break down to small scales. Finally, the study answers the question whether a fully developed turbulent boundary layer can be reached by oblique breakdown. It is shown that the skin friction develops such as is typical of transitional and turbulent boundary layers. Initially, the skin friction coefficient increases in the streamwise direction in the transitional region and finally decays when the early turbulent state is reached. Downstream of the maximum in the skin friction, the flow loses its periodicity in time and possesses characteristic mean-flow and spectral properties of a turbulent boundary layer. The DNS data clearly demonstrate that oblique breakdown can lead to a fully developed turbulent boundary layer and therefore it is a relevant mechanism for transition in two-dimensional supersonic boundary layers.

Sign in / Sign up

Export Citation Format

Share Document