Rayleigh number, critical

For shapes which have an appreciable fraction of area over which the normal component of the buoyancy force is directed away from the surface (e.g., a heated horizontal plate facing upward) the turbulent regime where SF oc (Ray occurs at low Ra. For example, the horizontal plates of Fig. 10.3 have only one side exposed, the side for which the normal component of buoyancy is directed away from the plate. Here the power relation applies at Ra > 10. For horizontal cylinders, on the other hand, such a relationship is exhibited for Ra > 5 X 10. For spheres in Fig. 10.2, there is no indication of this transition even at Ra = 10. The critical Rayleigh number, Ra/, above which the j-power relationship applies is correlated by... 

The increasing temperatures with depth in a sedimentary basin cause a thermal expansion and hence a decrease in density of the groundwater with depth. This vertical density stratification may induce free convection of the groundwater if the critical Rayleigh number (R = 40) is exceeded (e.g. Wood and Hewett, 1982). 

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...

For a given wave number, the least stable mode corresponds to n = 1 (this yields the smallest value of Ra ). If we plot Ra versus a lor n = 1, as in Fig. 12-5, we obtain the so-called neutral stability curve. For a given a, any value of Ra that exceeds Ra (a) corresponds to an unstable system, whereas any smaller value is stable. The critical Rayleigh number, Ra, for linear instability is the minimum value of Ra for all possible values of a, and the corresponding value of a = am, is known as the critical wave number. 

We have noted earlier that the case of a fluid layer with two free surfaces is qualitatively representative of the general nature of the solution for more realistic boundary conditions. Of course, the change from a slip to no-slip boundary condition does tend to stabilize the flow somewhat so that the critical Rayleigh number increases in the order... 

Plot your results for the critical Rayleigh number, Rac, as a function of k/d2. How small does k/d2 have to be before we reach the Darcy limit, and how large before the result reduces to the classical Rayleigh-Benard case ... 

Critical Rayleigh Numbers for Horizontal Cavities Restricted in the Horizontal Direction. 

TABLE 4.8 Critical Rayleigh Numbers Racp and Rac, for Horizontal Rectangular Parallelepiped Cavities Having a Perfectly Conducting Wall (Ra,) or an Adiabatic Wall (Ra ) [31-33]... 

TABLE 4.9 Critical Rayleigh Numbers Ra and Ra at Different Values of DIL for Horizontal Circulary Cavities ... 

Horizontal Fluid Layers. A uniform volumetric heat production q " in a horizontal layer bounded above by an isothermal surface and on the sides and bottom by adiabatic surfaces is depicted in Fig. 4.40. For a stationary fluid, the Nusselt number defined in the figure is Nu = 2, and the temperature difference used to construct the Rayleigh number is T0 - 7] = q" L l2k. As Ra increases from zero, the layer remains stable and heat flow is by conduction until a critical Rayleigh number of 1386 is reached [167]. Thereafter convection promotes a monotonic increase in Nu with Ra. For water (2.5 < Pr < 7), and for Ra < 10 2, the heat transfer data of Kulacki et al. [166-168] are accurately represented by... 

Rac critical Rayleigh number governing the initiation of small eddy convective... 

Raci, Racp critical Rayleigh numbers for cavities with adiabatic and perfectly conducting walls, respectively... 

On purely qualitative grounds Low and Brunt (L8), in 1925, proposed that the presence of a solid wall in place of a free surface should double Rayleigh s criterion for instability and that solid walls at both top and bottom should quadruple the critical Rayleigh number. 

In 1926-27, Jeffreys (J3, J4) attempted to extend Rayleigh s result to a more realistic set of boundary conditions, first using finite differences to obtain successive approximations to the solution of Eq. (31) and later using a method of undetermined coefficients for the case corresponding to two solid conducting boundaries. In the latter manner, he computed a critical Rayleigh number of 1709.5. 

The Critical Rayleigh Number as Calculated by Various Investigators for Different Boundary Conditions... 

As noted above, these authors also proved the general validity, for the Rayleigh problem, of the principle of exchange of stabilities. Further, by formulating the problem in terms of a variational principle, Pellew and South-well devised a technique which led to a very rapid and accurate approximation for the critical Rayleigh number. Later, a second variational principle was presented by Chandrasekhar (C3). A review by Reid and Harris (R2) also includes other approximate methods for handling the Benard problem with solid boundaries. 

In a recent paper. Sparrow, Goldstein, and Jonsson (Sll) presented the solution to the Rayleigh problem with a radiation boundary condition of the type of Eq. (38) at the upper free surface. Some of their results, for both a constant flux and a constant temperature bottom, are shown in Table IV. Of special interest is their solution for the limiting case of a constant-flux upper surface and a constant-flux bottom, for which the critical Rayleigh number was found to be only 320. 

In 1936, Avsec (A2) confirmed Rayleigh s theory in a rough way for air layers confined between metal plates spaced from 1.1 to 6.3 cm apart. The onset of convection was detected visually by means of smoke within the chamber. In 1938, Schmidt and Saunders (S5) repeated the experiments of Schmidt and Milverton, using air as well as water, and obtained an average critical R of 1750. In 1958, an extensive study by Silveston (S9) using the Schmidt-Milverton technique confirmed the Rayleigh theory for four liquids in addition to water, for depths from 1.45 to 13 mm. Silveston s results yielded a critical Rayleigh number of 1700 51. 

A refined, heavy, viscous hydrocarbon oil. Volkovisky reports viscosity data but no other physical properties. The critical Rayleigh number, however, is reported. 

Actually, in Pearson s calculation, the critical Rayleigh number was taken equal to 571, the value obtained by Jeffreys using incorrect boundary conditions. The correct value, 1108, was arrived at by Low. 

Figure 9.2 (Left) Convective fluid flow and effect on the waveform, slightly above the critical Rayleigh number. (Right) Fluid flow at Rayleigh numbers greatly exceeding the critical value, where axisymmetric flow is expected. The waveform has a parabolic shape.

What about ascending fronts If a front were to propagate upward, then the hot polymer-monomer solution in the reaction zone could rise because of buoyancy, removing enough heat at the polymer-monomer interface to quench the front. With a front that produces a solid product, the onset of convection is more complicated than the cases that we considered in Chapter 9, because the critical Rayleigh number is a function of the velocity (Volpert et al., 1996). Bowden et al. (1997) studied ascending fronts of acrylamide polymerization in dimethyl sulfoxide. As in the iodate-arsenous acid fronts, the first unstable mode is an antisymmetric one followed by an axisymmetric one. Unlike that system, in the polymerization front the stability of the front depends on both the solution viscosity and the front velocity. The faster the front, the lower the viscosity necessary to sustain a stable front. 

The flux between two lateral walls caused by the nonuniformity of the ion concentration profiles in the layers adjacent to the electrodes is of the same nature as the heat convection arising while the bottom wall is heated [2]. In the latter case a disturbance of the steady state occurs if the Rayleigh number reachs a certain (critical) value (Ra = gPd AT/vx, where P is the coefficient of bulk heat expansion, d is the distance between the walls, AT the increment of temperature, v the dynamic viscosity, and X the thermal diffusivity) the liquid transforms into a new state with a periodic cell structure in such a way that the circulation in the interior of each cell has an opposite direction compared to that of the adjacent one. According to previous evaluations [53] the critical Rayleigh number in the case of lateral rigid walls is about 1700. 

---