scholarly journals Linear biglobal analysis of Rayleigh–Bénard instabilities in binary fluids with and without throughflow

2012 ◽  
Vol 713 ◽  
pp. 216-242 ◽  
Author(s):  
Jun Hu ◽  
Daniel Henry ◽  
Xie-Yuan Yin ◽  
Hamda BenHadid
Keyword(s):  
Reynolds Number ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Critical Rayleigh Number ◽  
Two Dimensional ◽  
Binary Fluids ◽  
Transverse Modes ◽  
Aspect Ratios ◽  
Rayleigh Benard ◽  
Mode A

AbstractThree-dimensional Rayleigh–Bénard instabilities in binary fluids with Soret effect are studied by linear biglobal stability analysis. The fluid is confined transversally in a duct and a longitudinal throughflow may exist or not. A negative separation factor $\psi = \ensuremath{-} 0. 01$, giving rise to oscillatory transitions, has been considered. The numerical dispersion relation associated with this stability problem is obtained with a two-dimensional Chebyshev collocation method. Symmetry considerations are used in the analysis of the results, which allow the classification of the perturbation modes as ${S}_{l} $ modes (those which keep the left–right symmetry) or ${R}_{x} $ modes (those which keep the symmetry of rotation of $\lrm{\pi} $ about the longitudinal mid-axis). Without throughflow, four dominant pairs of travelling transverse modes with finite wavenumbers $k$ have been found. Each pair corresponds to two symmetry degenerate left and right travelling modes which have the same critical Rayleigh number ${\mathit{Ra}}_{c} $. With the increase of the duct aspect ratio $A$, the critical Rayleigh numbers for these four pairs of modes decrease and closely approach the critical value ${\mathit{Ra}}_{c} = 1743. 894$ obtained in a two-dimensional situation, one of the mode (a ${S}_{l} $ mode called mode A) always remaining the dominant mode. Oscillatory longitudinal instabilities ($k\approx 0$) corresponding to either ${S}_{l} $ or ${R}_{x} $ modes have also been found. Their critical curves, globally decreasing, present oscillatory variations when the duct aspect ratio $A$ is increased, associated with an increasing number of longitudinal rolls. When a throughflow is applied, the symmetry degeneracy of the pairs of travelling transverse modes is broken, giving distinct upstream and downstream modes. For small and moderate aspect ratios $A$, the overall critical Rayleigh number in the small Reynolds number range studied is only determined by the upstream transverse mode A. In contrast, for larger aspect ratios as $A= 7$, different modes are successively dominant as the Reynolds number is increased, involving both upstream and downstream transverse modes A and even the longitudinal mode.

scholarly journals Flow states in two-dimensional Rayleigh-Bénard convection as a function of aspect-ratio and Rayleigh number

2012 ◽  
Vol 24 (8) ◽  
pp. 085104 ◽  
Author(s):  
Erwin P. van der Poel ◽  
Richard J. A. M. Stevens ◽  
Kazuyasu Sugiyama ◽  
Detlef Lohse
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Flow States

Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate

2014 ◽  
Vol 751 ◽  
pp. 570-600 ◽  
Author(s):  
Mark C. Thompson ◽  
Alexander Radi ◽  
Anirudh Rao ◽  
John Sheridan ◽  
Kerry Hourigan
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Flat Plate ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Near Wake ◽  
Aspect Ratios ◽  
Von Karman ◽  
Von Kármán ◽  
Mode A

AbstractWhile the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Bénard–von Kármán state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Bénard–von Kármán-like near wake into a two-layered wake, through to a secondary Bénard–von Kármán-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.

Convection in a rotating cylinder. Part 1 Linear theory for moderate Prandtl numbers

1993 ◽  
Vol 248 ◽  
pp. 583-604 ◽  
Author(s):  
H. F. Goldstein ◽  
E. Knobloch ◽  
I. Mercader ◽  
M. Net
Keyword(s):  
Rayleigh Number ◽  
Aspect Ratio ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Rotation Rate ◽  
Critical Rayleigh Number ◽  
Onset Of Convection ◽  
Aspect Ratios ◽  
Prandtl Numbers ◽  
Rayleigh Numbers

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

Roll-pattern evolution in finite-amplitude Rayleigh–Beńard convection in a two-dimensional fluid layer bounded by distant sidewalls

1984 ◽  
Vol 143 ◽  
pp. 125-152 ◽  
Author(s):  
P. G. Daniels
Keyword(s):  
Rayleigh Number ◽  
Temporal Evolution ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This paper considers the temporal evolution of two-dimensional Rayleigh–Bénard convection in a shallow fluid layer of aspect ratio 2L ([Gt ] 1) confined laterally by rigid sidewalls. Recent studies by Cross et al. (1980, 1983) have shown that for Rayleigh numbers in the range R = R0 + O(L−1) (where R0 is the critical Rayleigh number for the corresponding infinite layer) there exists a class of finite-amplitude steady-state ‘phase-winding’ solutions which correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of the Rayleigh number is varied. It has been shown (Daniels 1981) that in the temporal evolution of the system the final lateral positioning of the rolls occurs on the long timescale t = O(L2) when the phase function which determines the number of rolls in the system satisfies a one-dimensional diffusion equation but with novel boundary conditions that represent the effect of the sidewalls. In the present paper this system is solved numerically in order to determine the precise way in which the roll pattern adjusts after a change in the Rayleigh number of the system. There is an interesting balance between, on the one hand, a tendency for the number of rolls to change by the least number possible and, on the other, a tendency for the even or odd nature of the initial configuration to be preserved during the transition. In some cases this second property renders the natural evolution susceptible to arbitrarily small external disturbances, which then dictate the form of the final roll pattern.The complete transition involves an analysis of the motion on three timescales, a conductive scale t = O(1), a convective growth scale t = O(L) and a convective diffusion scale t = O(L2).

Onset of Rayleigh–Bénard convection in a rigid channel

1989 ◽  
Vol 199 ◽  
pp. 257-279 ◽  
Author(s):  
M. S. Chana ◽  
P. G. Daniels
Keyword(s):  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Two Dimensional ◽  
Asymptotic Results ◽  
Onset Of Convection ◽  
Aspect Ratios ◽  
Channel Aspect Ratio ◽  
Vertical Walls ◽  
Rayleigh Benard

A two-dimensional Galerkin formulation of the three-dimensional Oberbeck-Boussinesq equations is used to describe the onset of convection in an infinite rigid horizontal channel uniformly heated from below. The dependence of the critical Rayleigh number on the channel aspect ratio is determined and results are compared with those of an idealized model studied by Davies-Jones (1970). Asymptotic results are derived for both narrow and wide channels, corresponding to limits of small and large aspect ratios respectively. In the latter case the main core flow, consisting of two-dimensional rolls with axes perpendicular to the vertical walls of the channel, can be represented by the solution of an amplitude equation. Close to the walls, however, the motion remains fully three-dimensional and a reversal of the vertical flow is associated with a local subdivision of each main roll into a pair of co-rotating rolls.

scholarly journals Aspect ratio effects in Rayleigh–Bénard convection of Herschel–Bulkley fluids

2017 ◽  
Vol 34 (5) ◽  
pp. 1658-1676 ◽  
Author(s):  
Mohammad Saeid Aghighi ◽  
Amine Ammar
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Rayleigh Number ◽  
Two Dimensional ◽  
Content Type ◽  
Bingham Number ◽  
Aspect Ratios ◽  
Power Law Index ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Purpose The purpose of this paper is to analyze two-dimensional steady-state Rayleigh–Bénard convection within rectangular enclosures in different aspect ratios filled with yield stress fluids obeying the Herschel–Bulkley model. Design/methodology/approach In this study, a numerical method based on the finite element has been developed for analyzing two-dimensional natural convection of a Herschel–Bulkley fluid. The effects of Bingham number Bn and power law index n on heat and momentum transport have been investigated for a nominal Rayleigh number range (5 × 103 < Ra < 105), three different aspect ratios (ratio of enclosure length:height AR = 1, 2, 3) and a single representative value of nominal Prandtl number (Pr = 10). Findings Results show that the mean Nusselt number Nu¯ increases with increasing Rayleigh number due to strengthening of convective transport. However, with the same nominal value of Ra, the values of Nu¯ for shear thinning fluids n < 1 are greater than shear thickening fluids n > 1. The values of Nu¯ decrease with Bingham number and for large values of Bn, Nu¯ rapidly approaches unity, which indicates that heat transfer takes place principally by thermal conduction. The effects of aspect ratios have also been investigated and results show that Nu¯ increases with increasing AR due to stronger convection effects. Originality/value This paper presents a numerical study of Rayleigh–Bérnard flows involving Herschel–Bulkley fluids for a wide range of Rayleigh numbers, Bingham numbers and power law index based on finite element method. The effects of aspect ratio on flow and heat transfer of Herschel–Bulkley fluids are also studied.

Heat Transfer for High Aspect Ratio Rectangular Channels in a Stationary Serpentine Passage With Turbulated and Smooth Surfaces

2013 ◽  
Author(s):  
Matthew A. Smith ◽  
Randall M. Mathison ◽  
Michael G. Dunn
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Aspect Ratio ◽  
Flow Behavior ◽  
Reynolds Numbers ◽  
Hydraulic Diameter ◽  
Internal Cooling ◽  
Number Range ◽  
Aspect Ratios ◽  
Parallel Configuration

Heat transfer distributions are presented for a stationary three passage serpentine internal cooling channel for a range of engine representative Reynolds numbers. The spacing between the sidewalls of the serpentine passage is fixed and the aspect ratio (AR) is adjusted to 1:1, 1:2, and 1:6 by changing the distance between the top and bottom walls. Data are presented for aspect ratios of 1:1 and 1:6 for smooth passage walls and for aspect ratios of 1:1, 1:2, and 1:6 for passages with two surfaces turbulated. For the turbulated cases, turbulators skewed 45° to the flow are installed on the top and bottom walls. The square turbulators are arranged in an offset parallel configuration with a fixed rib pitch-to-height ratio (P/e) of 10 and a rib height-to-hydraulic diameter ratio (e/Dh) range of 0.100 to 0.058 for AR 1:1 to 1:6, respectively. The experiments span a Reynolds number range of 4,000 to 130,000 based on the passage hydraulic diameter. While this experiment utilizes a basic layout similar to previous research, it is the first to run an aspect ratio as large as 1:6, and it also pushes the Reynolds number to higher values than were previously available for the 1:2 aspect ratio. The results demonstrate that while the normalized Nusselt number for the AR 1:2 configuration changes linearly with Reynolds number up to 130,000, there is a significant change in flow behavior between Re = 25,000 and Re = 50,000 for the aspect ratio 1:6 case. This suggests that while it may be possible to interpolate between points for different flow conditions, each geometric configuration must be investigated independently. The results show the highest heat transfer and the greatest heat transfer enhancement are obtained with the AR 1:6 configuration due to greater secondary flow development for both the smooth and turbulated cases. This enhancement was particularly notable for the AR 1:6 case for Reynolds numbers at or above 50,000.

Numerical Prediction of Contaminant Removal from Cavity in Horizontal Channel by Constrained Interpolated Profile Method

2014 ◽  
Vol 695 ◽  
pp. 384-388
Author(s):  
Nor Azwadi Che Sidik ◽  
A.S. Ahmad Sofianuddin ◽  
K.Y. Ahmat Rajab
Keyword(s):  
Reynolds Number ◽  
Aspect Ratio ◽  
Removal Rate ◽  
Contaminant Removal ◽  
Square Cavity ◽  
Profile Method ◽  
Aspect Ratios ◽  
Contaminants Removal ◽  
Parabolic Flow ◽  
High Removal Rate

In this paper, Constrained Interpolated Profile Method (CIP) was used to simulate contaminants removal from square cavity in channel flow. Predictions were conducted for the range of aspect ratios from 0.25 to 4.0. The inlet parabolic flow with various Reynolds number from 50 to 1000 was used for the whole presentation with the same properties of contaminants and fluid. The obtained results indicated that the percentage of removal increased at high aspect ratio of cavity and higher Reynolds number of flow but it shows more significant changes as increasing aspect ratio rather than increasing Reynolds number. High removal rate was found at the beginning of the removal process.

Power Augmentation of a HAWT by Mie-type Tip Vanes, considering Wind Tunnel Flow Visualisation, Blade-Aspect Ratios and Reynolds Number

2003 ◽  
Vol 27 (3) ◽  
pp. 183-194 ◽  
Author(s):  
Yukimaru Shimizu ◽  
Edmond Ismaili ◽  
Yasunari Kamada ◽  
Takao Maeda
Keyword(s):  
Reynolds Number ◽  
Wind Tunnel ◽  
Aspect Ratio ◽  
Reynolds Numbers ◽  
Flow Visualisation ◽  
Velocity Measurements ◽  
Horizontal Axis Wind Turbine ◽  
Aspect Ratios ◽  
Detailed Surface ◽  
Blade Tips

Wind tunnel results are reported concerning the effects of blade aspect ratio and Reynolds number on the performance of a horizontal axis wind turbine (HAWT) with Mie-type1 tip attachments. The flow behaviour around the blade tips and the Mie-type tip vanes is presented. Detailed surface oil film visualization and velocity measurements around the blade tips, with and without Mie vanes, were obtained with the two-dimensional, Laser-Doppler Velocimetry method. Experiments were performed with rotors having blades with different aspect ratio and operating at different Reynolds numbers. The properties of the vortices generated by the Mie vanes and the blade tips were carefully studied. It was found that increased power augmentation by Mie vanes is achieved with blades having smaller aspect ratio and smaller Reynolds number.

The Influence of Real Gas Radiation On the Stability and Development of Benard Convection in a Two-Dimensional Layer

2021 ◽  
Author(s):  
Brent W. Webb ◽  
Vladimir Solovjov
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Radiative Heating ◽  
Two Dimensional ◽  
Gas Temperature ◽  
Layer Depth ◽  
Real Gas ◽  
Gas Radiation ◽  
Induced Flow ◽  
Buoyancy Induced Flow

Abstract The influence of real gas radiation on the thermal and hydrodynamic stability is investigated in a two-dimensional layer of radiatively participating H2O and/or CO2 heated from below. The non-gray radiation effects of the two species are treated rigorously using a global spectral approach, the Spectral Line Weighted-sum-of-gray-gases model. The phenomena are explored by solving the full coupled laminar equations of motion, energy, and radiative transfer from the low-Rayleigh number, pure conduction-radiation regime through the onset of buoyancy-induced flow to the developed Bénard convection regime. The evolution of the thermal, velocity, and radiative heating fields is studied, and the critical Rayleigh number is characterized as a function of species mole fraction, average layer gas temperature, layer depth, wall emissivity, and the total gas pressure. It is found that participating radiation in the medium has the effect of stabilizing the layer, delaying transition to buoyancy-induced flow. The development of buoyancy-induced flow and temperature, along with the radiative heating are presented. It is found that the critical Rayleigh number in the radiatively participating gas layer can be more than an order of magnitude higher than the classical convection-only scenario. The onset of instability is found to depend on the species mole fractions, average gas temperature in the layer, wall emissivity, layer depth, and total pressure. Generally, all other variables being the same, H2O has a greater stabilizing influence on the layer than CO2.

Sign in / Sign up

Export Citation Format

Share Document