Rayleigh-Benard

The same / lias also boon used as a model for spatiotemporal interniittency in Rayleigh-Benard convection ([cili88], [davl89]). 

Both type-I intermittency [berge80] and type-II intermittency [duboisS] has been observed in a Rayleigh-Benard experiment, Generally speaking, however, only a relatively small subset of experiments appear to be consistent with this particular route,... 

Figure 2.9.9(a) shows a schematic representation of a thermal convection cell in Rayleigh-Benard configuration [8]. With a downward temperature gradient one expects convection rolls that are more or less distorted by the tortuosity of the fluid filled pore space. In the absence of any flow obstacles one expects symmetrical convection rolls, such as illustrated by the numerical simulation in Figure 2.9.9(b). 

Fig. 2.9.9 (a) Schematic cross section of a compartments at the top and bottom, respec-convection cell in Rayleigh-Benard configura- tively. (b) Velocity contour plot of typical tion. In the version examined in Refs. [8, 44], a convection rolls expected in the absence of any fluid filled porous model object of section flow obstacles (numerical simulation). 

Fig. 2.9.10 Maps of the temperature and of the experimental data. The right-hand column convection flow velocity in a convection cell in refers to numerical simulations and is marked Rayleigh-Benard configuration (compare with with an index 2. The plots in the first row, (al) Figure 2.9.9). The medium consisted of a and (a2), are temperature maps. All other random-site percolation object of porosity maps refer to flow velocities induced by p = 0.7 filled with ethylene glycol (temperature thermal convection velocity components vx maps) or silicon oil (velocity maps). The left- (bl) and (b2) and vy (cl) and (c2), and the hand column marked with an index 1 represents velocity magnitude (dl) and (d2).

W. J. Goux, L. A. Verkruyse, S. J. Salter 1990, (The impact of Rayleigh-Benard convection on NMR pulsed-field-gra-dient diffusion measurements), J. Mag. Reson. 88, 609. 

Eidson, T. M. (1985). Numerical simulation of the turbulent Rayleigh-Benard problem using subgrid modelling. Journal of Fluid Mechanics 158, 245-268. 

Previous computations (189) show that the critical value of Rat for non-Boussinesq conditions is approximately the same as that for a Boussinesq fluid in a box heated from below, at least when H2 is the carrier gas. Thus, results from the stability analysis of the classical Rayleigh-Benard problem of a two-dimensional fluid layer heated from below (see reference 190 for a review) may be used to indicate the type of behavior to be expected in a horizontal reactor with insulated side walls. As anticipated from this analysis, an increase in the reactor height from 2 to 4 cm raises the value of Rat to 4768, which is beyond the stability limit, Rat critical = 2056, for a box of aspect ratio 2 (188). The trajectories show the development of buoyancy-driven axial rolls that are symmetric about the midplane and rotating inward. For larger values of Rat (>6000), transitions to three-dimensional or time-de-... 

Before starting to discuss (116), we make an observation. The fast time evolution (116) is also observed in driven systems that cannot be described on the level Ath- For example, let us consider the Rayleigh-Benard system (i.e., a horizontal layer of a fluid heated from below). It is well established experimentally that this externally driven system does not reach thermodynamic equilibrium states but its behavior is well described on the level of fluid mechanics (by Boussinesq equations). This means that if we describe it on a more microscopic level, say the level of kinetic theory, then we shall observe the approach to the level of fluid mechanics. Consequently, the comments that we shall make below about (116) apply also to driven systems and to other types of systems that are prevented from reaching thermodynamical equilibrium states (as, e.g., glasses where internal constraints prevent the approach to Ath)-... 

HYDRODYNAMIC AND HYDROMAGNETIC STABILITY. S. Chandrasekhar. Lucid examination of the Rayleigh-Benard problem clear coverage of the theory of instabilities causing convection. 704pp. 5b x 8b. 64071-X Pa. 12.95... 

V. Croquette, P. Le Gal, A. Pocheau, and R. Guglielmetti, Large-scale characterization in a Rayleigh-Benard convective pattern, Europhys. Lett., 1, 393-399 (1986). 

In this framework an interesting example is the Lagrangian motion in velocity field given by a simple model for Rayleigh-Benard convection [31], which is given by the stream function ... 

Plapp B. R, Egolf D. A., Bodenschatz E. and Pesch W., Dynamics and Selection of Giant Spirals in Rayleigh-Benard Convection, Phys. Rev. Lett. 81, 5334 (1998). 

The first problem considered is the classic problem of Rayleigh-Benard convection -namely the instability that is due to buoyancy forces in a quiescent fluid layer that is heated... 

Figure 12-4. The Rayleigh-Benard configuration d for buoyancy-driven convection in a horizontal fluid...

F. NATURAL CONVECTION IN A HORIZONTAL FLUID LAYER HEATED FROM BELOW-THE RAYLEIGH-BENARD PROBLEM... 

The Rayleigh Benard configuration is sketched in Fig. 12 4. We assume that the fluid is between two infinite plane surfaces that are separated by a distance d. The lower surface is at a constant temperature i and the upper one is at a lower temperature T0. Our starting point is the time-dependent Boussinesq equations, (12-160), (12 169), and (12-170), but now in dimensionless form,... 

The boundary conditions depend, of course, on the nature of the flat surfaces at z = 0 and 1. We shall return to conditions for the velocity components shortly. It may be noted that the Rayleigh Benard problem has historically been stated in a slightly different form, which is equivalent to rescaling 0 to 0 = O/Pr. The difference is that (12 181) and (12 184) then become... 

Figure 12-5. Stability criteria for the Rayleigh-Benard problem. The two curves shown are the neutral stability curves for the modes n = 1 and n = 2. The region above the curve for n = I is unstable, whereas that below is stable. The critical Rayleigh number is 657.511 at a critical wave number of 2.221...

It is worthwhile illustrating the proof of the principle of exchange of stabilities for the Rayleigh-Benard problem. Not only will this allow us to discuss the derivation of instability criteria for the case of no-slip boundaries, but the approach to proving this principle can also be applied to other problems. 

As in the case of the Rayleigh-Benard problem, there is a steady-state solution of these equations. 

As noted earlier in conjunction with the Rayleigh-Benard analysis, this is equivalent to redefining a dimensionless temperature and concentration... 

As in the Rayleigh Benard problem, the boundary conditions for two free surfaces are... 

In this section, we consider the classic problem of a fluid layer of depth d, with an upper surface that is an interface with air that is maintained at an ambient temperature 7o. The fluid layer is heated from below, and we shall assume that the lower fluid boundary is isothermal with temperature T (> To). This problem sounds exactly like the Rayleigh-Benard problem with a free upper surface. However, we consider the fluid layer to be very thin (i.e., d small) so that the Rayleigh number, which depends on d3, is less than the critical value for this configuration. Nevertheless, as previously suggested, the fluid layer may still undergo a convective motion that is due to Marangoni instability. 

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