scholarly journals Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers

2016 ◽  
Vol 791 ◽  
Author(s):  
Juan Sánchez ◽  
Ferran Garcia ◽  
Marta Net
Keyword(s):  
Numerical Study ◽  
Very High Frequency ◽  
Onset Of Convection ◽  
Single Vortex ◽  
Conduction State ◽  
Prandtl Numbers ◽  
The Stability ◽  
Rayleigh Numbers ◽  
Very High ◽  
Fluid Spheres

A numerical study of the onset of convection in rotating internally heated self-gravitating fluid spheres is presented. The exploration of the stability of the conduction state versus the Taylor and Prandtl numbers supplies a detailed idea of the laws that fulfil the four types of solutions obtained at low Prandtl numbers. The main result found is that axisymmetric (torsional) modes of convection are preferred at high Taylor numbers in the zero-Prandtl-number limit. This instability appears at low Rayleigh numbers and gives rise to an oscillating single vortex of very high frequency.

On the analogy between thermal and rotational hydrodynamic stability

1966 ◽  
Vol 24 (1) ◽  
pp. 165-176 ◽  
Author(s):  
Walter R. Debler
Keyword(s):  
Temperature Distribution ◽  
Initial Temperature ◽  
Rotating Cylinder ◽  
Temperature Profiles ◽  
Convection Pattern ◽  
Onset Of Convection ◽  
Single Vortex ◽  
Angular Velocities ◽  
The Stability ◽  
Rayleigh Numbers

The correspondence between the eigenvalues for the problem of the onset of convection in a fluid confined between two horizontal plates and for the stability of viscous flow between two cylinders rotating at almost the same angular velocity has been known for some time. The recent work of Chandrasekhar (1961) has prompted the extension of the analogy to a larger group of rotating cylinder problems and their associated convection cases in which the primary temperature distribution is parabolic. This paper shows the analogy between these two problems and presents data which give the corresponding temperature distribution for a given ratio of angular velocities between the two cylinders. The equivalent Rayleigh numbers are listed for the Taylor numbers given by Chandrasekhar (1954). The eigenfunctions for several of the parabolic temperature profiles are determined. These results show that the single vortex convection pattern becomes a double vortex for certain initial temperature distributions. The critical Rayleigh numbers for the stability of a layer of water which is near 4 °C is also found by analogy.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

The instability of a layer of fluid heated below and subject to Coriolis forces

1953 ◽  
Vol 217 (1130) ◽  
pp. 306-327 ◽  
Keyword(s):  
Coriolis Force ◽  
Thermal Instability ◽  
Horizontal Layer ◽  
Onset Of Convection ◽  
Dimensional Parameter ◽  
Coriolis Forces ◽  
Effective Gravity ◽  
The Stability ◽  
Bounding Surfaces ◽  
Rayleigh Numbers

This paper is devoted to examining the stability of a horizontal layer of fluid heated below, subject to an effective gravity ( g ) acting (approximately) in the direction of the vertical and the Coriolis force resulting from a rotation of angular velocity Ω about a direction making an angle ϑ with the vertical. It is shown that the effect of the Coriolis force is to inhibit the onset of convection, the extent of the inhibition depending on the value of the non-dimensional parameter T = 4 d 4 Ω 2 cos 2 ϑ/ v 2 , where d denotes the depth of the layer and v is the kinematic viscosity. Tables of the critical Rayleigh numbers ( R c ) for the onset of convection are provided for the three cases ( a ) both bounding surfaces free, ( b ) both bounding surfaces rigid and ( c ) one bounding surface free and the other rigid. In all three cases R c →constant x T 2/3 as T →∞ ; the corresponding dependence of the critical temperature gradient (— β c ) for the onset of convection, on v and d , is gαβ c = constant x ĸ (Ω 4 cos 4 ϑ/ d 4 v ) 1/2 ( ĸ is the coefficient of thermometric conductivity and α is the coefficient of volume expansion). The question whether thermal instability can set in as oscillations of increasing amplitude (i.e. as 'overstability’) is examined for case ( a ), and it is shown that if ĸ/v <1.478, this possibility does not arise; but if ĸ/v >1.478, over-stability is the first type of instability to arise for all T greater than a certain determinate value. It further appears that these latter possibilities should be considered in meteorological and astrophysical applications of the theory.

Linear stability analysis of mixed-convection flow in a vertical pipe

2000 ◽  
Vol 422 ◽  
pp. 141-166 ◽  
Author(s):  
YI-CHUNG SU ◽  
JACOB N. CHUNG
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Mixed Convection ◽  
Shear Instability ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Vertical Pipe ◽  
Prandtl Numbers ◽  
The Stability ◽  
Rayleigh Numbers

A comprehensive numerical study on the linear stability of mixed-convection flow in a vertical pipe with constant heat flux is presented with particular emphasis on the instability mechanism and the Prandtl number effect. Three Prandtl numbers representative of different regimes in the Prandtl number spectrum are employed to simulate the stability characteristics of liquid mercury, water and oil. The results suggest that mixed-convection flow in a vertical pipe can become unstable at low Reynolds number and Rayleigh numbers irrespective of the Prandtl number, in contrast to the isothermal case. For water, the calculation predicts critical Rayleigh numbers of 80 and −120 for assisted and opposed flows, which agree very well with experimental values of Rac = 76 and −118 (Scheele & Hanratty 1962). It is found that the first azimuthal mode is always the most unstable, which also agrees with the experimental observation that the unstable pattern is a double spiral flow. Scheele & Hanratty's speculation that the instability in assisted and opposed flows can be attributed to the appearance of inflection points and separation is true only for fluids with O(1) Prandtl number. Our study on the effect of the Prandtl number discloses that it plays an active role in buoyancy-assisted flow and is an indication of the viability of kinematic or thermal disturbances. It profoundly affects the stability of assisted flow and changes the instability mechanism as well. For assisted flow with Prandtl numbers less than 0.3, the thermal–shear instability is dominant. With Prandtl numbers higher than 0.3, the assisted-thermal–buoyant instability becomes responsible. In buoyancy-opposed flow, the effect of the Prandtl number is less significant since the flow is unstably stratified. There are three distinct instability mechanisms at work independent of the Prandtl number. The Rayleigh–Taylor instability is operative when the Reynolds number is extremely low. The opposed-thermal–buoyant instability takes over when the Reynolds number becomes higher. A still higher Reynolds number eventually leads the thermal–shear instability to dominate. While the thermal–buoyant instability is present in both assisted and opposed flows, the mechanism by which it destabilizes the flow is completely different.

scholarly journals A theoretical and numerical study of thermosolutal convection: stability of a salinity gradient solar pond

2011 ◽  
Vol 15 (1) ◽  
pp. 67-80 ◽  
Author(s):  
Dalila Akrour ◽  
Mouloud Tribeche ◽  
Djamel Kalache
Keyword(s):  
Stability Analysis ◽  
Boundary Conditions ◽  
Numerical Study ◽  
Salinity Gradient ◽  
Thermosolutal Convection ◽  
Separation Coefficient ◽  
Onset Of Convection ◽  
Solar Pond ◽  
Negative Case ◽  
The Stability

A theoretical and numerical study of the effect of thermodiffusion on the stability of a gradient layer is presented. It intends to clarify the mechanisms of fluid dynamics and the processes which occur in a salinity gradient solar pond. A mathematical modelling is developed to describe the thermodiffusion contribution on the solar pond where thermal, radiative, and massive fluxes are coupled in the double diffusion. More realistic boundary conditions for temperature and concentration profiles are used. Our results are compared with those obtained experimentally by authors without extracting the heat flux from the storage zone. We have considered the stability analysis of the equilibrium solution. We assumed that the perturbation of quantities such as velocity, temperature, and concentration are infinitesimal. Linearized equations satisfying appropriate prescribed boundary conditions are then obtained and expanded into polynomials form. The Galerkin method along with a symbolic algebra code (Maple) are used to solve these equations. The effect of the separation coefficient y is analyzed in the positive and negative case. We have also numerically compared the critical Rayleigh numbers for the onset of convection with those obtained by the linear stability analysis for Le = 100, ?a = 0.8, and f = 0.5.

Convection in a rotating cylindrical annulus. Part 4. Modulations and transition to chaos at low Prandtl numbers

1997 ◽  
Vol 350 ◽  
pp. 209-229 ◽  
Author(s):  
J. HERRMANN ◽  
F. H. BUSSE
Keyword(s):  
Rossby Waves ◽  
Landau Equation ◽  
Neutral Curve ◽  
Onset Of Convection ◽  
Ginzburg Landau ◽  
Cylindrical Annulus ◽  
Chaotic Convection ◽  
Finite Distance ◽  
Prandtl Numbers ◽  
Rayleigh Numbers

Thermal Rossby waves driven by centrifugal buoyancy in a rotating cylindrical fluid gap become unstable right at the onset of convection when the Prandtl number is small. The Benjamin–Feir–Newell instability leads to modulated thermal Rossby waves which can also be described by a generalized Ginzburg–Landau equation. A resonance instability occurs at a finite distance in Rayleigh number from the neutral curve. It leads to two independent wave patterns propagating past each other and finally gives rise to vacillations of the amplitude of convection. Most of these features can be described to a good approximation by a system of three coupled amplitude equations. Time integrations based on a Galerkin expansion show transitions to chaotic convection at higher Rayleigh numbers.

scholarly journals Strong downdrafts preceding rapid tropopause ascent and their potential to identify cross-tropopause stratospheric intrusions

2018 ◽  
Vol 36 (5) ◽  
pp. 1403-1417 ◽  
Author(s):  
Feilong Chen ◽  
Gang Chen ◽  
Chunhua Shi ◽  
Yufang Tian ◽  
Shaodong Zhang ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Very High Frequency ◽  
Vhf Radar ◽  
Stratospheric Intrusion ◽  
Different Seasons ◽  
Cutoff Low ◽  
The Stability ◽  
Stratospheric Intrusions ◽  
Global Reanalysis ◽  
Very High

Abstract. The capability of measuring three-dimensional wind and tropopause structure with relatively high time and vertical resolution makes very-high-frequency (VHF) radars a potentially important tool for studying various processes of the atmosphere. However, at present several unanswered questions remain regarding the use of VHF radars to identify possible stratospheric intrusions. Here the potential detection of stratospheric intrusion events is discussed using the Beijing MST (mesosphere–stratosphere–troposphere) radar located at Xianghe (39.75∘ N, 116.96∘ E). During the passage of a cutoff low in late November 2014, a deep V-shaped tropopause structure and strong downdrafts (> 0.8 m s−1) immediately preceding the rapid tropopause ascent (> 0.2 km h−1) were observed. Within the height region of the downdrafts, the stability of the radar tropopause seems to be weakened. Analysis results from global reanalysis and the satellite data, as well as the trajectory model, have shown clear evidence of downward stratospheric intrusions (dry ozone-rich and depleted methane air) associated with the strong downdrafts. A total of 20 typical cases of such strong downdrafts, occurring during various synoptic processes in different seasons, have been presented, and 15 of them are exactly associated with some form of stratospheric intrusions. Four years (2012–2015) of such downdrafts are further discussed. The observations reveal that the strong downdrafts preceding the rapid tropopause ascent can be a valuable diagnostic for monitoring intrusion events, which helps us to gain a better understanding of stratospheric intrusions in VHF radar observations.

Onset of multicellular convection in a shallow laterally heated cavity

1987 ◽  
Vol 411 (1841) ◽  
pp. 327-350 ◽  
Keyword(s):  
Stability Boundary ◽  
Thermal Gradients ◽  
Cavity Flows ◽  
Flow Core ◽  
End Effects ◽  
Shallow Cavity ◽  
Prandtl Numbers ◽  
The Stability ◽  
Hadley Cells ◽  
Rayleigh Numbers

Shallow cavity flows, driven by horizontal thermal gradients, are analysed over a range of Rayleigh numbers R and Prandtl numbers σ. Emphasis is placed on the limit in which R is comparable in size to the cavity aspect ratio L . Departures from two-dimensional steady Hadley cells, defined by the limit L → ∞ at fixed R and σ, are shown to be associated with nonlinear end effects. Eigenvalue calculations indicate the existence of a critical Prandtl number σ c , below which the core structure is not necessarily parallel with the horizontal boundaries. For σ < σ c the parallel flow core is destroyed at Rayleigh numbers R > R c (σ). Results for the stability boundary R c (σ) are presented.

scholarly journals On the linear convective stability of a TiO2 particle–based nanofluid layer heated from below

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401982894
Author(s):  
Ildebrando Pérez-Reyes ◽  
René Osvaldo Vargas-Aguilar
Keyword(s):  
Volume Fraction ◽  
Fluid Layer ◽  
Tio2 Particle ◽  
Oscillatory Convection ◽  
Implicit Equation ◽  
Rayleigh Convection ◽  
Prandtl Numbers ◽  
Model Equations ◽  
The Stability ◽  
Rayleigh Numbers

The linear hydrodynamics of Rayleigh convection in a horizontal nanofluid layer heated from below was studied. The hydrodynamic stability of the fluid layer bounded by two horizontal perfect thermal conducting walls was extended to analyze steady and oscillatory convection, and the role played by thermophoresis. Experimental data of TiO2 particle–based nanofluid was used to discuss the stability of the fluid layer. Results on the relationship between thermal and volume faction Rayleigh numbers are used to discuss experiments in nanofluid Rayleigh convection, while the absence of thermophoresis in the model equations was also considered. For this nanofluid, steady convection sets in at critical wavenumber ac = 3.12, but thermal RT and nanoparticle volume fraction RV Rayleigh numbers are given by an implicit relationship. For the onset of oscillatory convection, the wavenumber is also obtained from an implicit equation involving RT and RV. Results are discussed in terms of physical dimensionless parameters of the system like the Lewis and Prandtl numbers. This work complements the earlier efforts of Tzou and more recently by Nield and Kuznetsov.

Effect of Imposed Wall Temperature Oscillations on the Stability of Natural Convection in a Square Enclosure

1995 ◽  
Vol 117 (1) ◽  
pp. 113-120 ◽  
Author(s):  
Q. Xia ◽  
K. T. Yang ◽  
D. Mukutmoni
Keyword(s):  
Numerical Study ◽  
Vertical Wall ◽  
Laminar Flows ◽  
Time Dependent ◽  
Square Enclosure ◽  
Temperature Oscillations ◽  
Vertical Walls ◽  
Flow Transitions ◽  
The Stability ◽  
Rayleigh Numbers

The present numerical study is directed toward buoyancy-driven laminar flows in a two-dimensional square enclosure with differential heating at the vertical walls. The top and bottom walls are insulated. A time-dependent temperature varying sinusoidal perturbation is imposed on the hot vertical wall. The cold vertical wall is maintained at a constant temperature. The fluid is air with a Prandtl number of 0.72. Computations were carried out at one imposed frequency, which is of the same order as the first natural frequency of the system. It was found that the perturbations destabilized the flow in that higher amplitudes lead to lower critical Rayleigh numbers for the flow transitions. Computations spanned four regimes: periodic, quasi-periodic with two frequencies, quasi-periodic with three-frequencies, and chaotic.

Sign in / Sign up

Export Citation Format

Share Document