Hexagonal convection cells

Fig. 7. Contour lines of the type obtained by Benard for the free surface of a hexagonal convection cell using the Fabry-P6rot interferometer.

Benard in the year 1900 observed that hexagonal convection cells are formed within thin film of molten spermaceti of about 0.5-1 mm depth that were heated from below, with the cell spacing somewhat more than three times the liquid depth. These cells are now referred as Benard cells (Figure 5.31). Bdnard initially assumed that surface tension at the free surface of... 

These observations consummated in a growth model that confers on the millions of aligned zone 1 nanotubes the role of field emitters, a role they play so effectively that they are the dominant source of electron injection into the plasma. In response, the plasma structure, in which current flow becomes concentrated above zone 1, enhances and sustains the growth of the field emission source —that is, zone 1 nanotubes. A convection cell is set up in order to allow the inert helium gas, which is swept down by collisions with carbon ions toward zone 1, to return to the plasma. The helium flow carries unreacted carbon feedstock out of zone 1, where it can add to the growing zone 2 nanotubes. In the model, it is the size and spacing of these convection cells in the plasma that determine the spacing of the zone 1 columns in a hexagonal lattice. 

Benard convection cells [27, 28] a liquid with an inverse temperature gradient (hot below and cool on top) may exhibit thermal convection. Less dense parts of the liquid well upward whereas denser parts show down-welling. The convection cells may arrange in hexagonal order in which the center of each cell wells downwards and the rim wells upwards. The cells stem from the concerted movement of many molecules and cease when the temperature gradient is below a threshold at which the thermal equilibrium canbe reached solely bythermalconductionandnotconvection. 

BENARD CONVECTION CELLS. When a layer of liquid is heated from below, the onset of convection is marked by the appearance of a regular array of hexagonal cells, the liquid rising in the center and falling near the wall of each cell. The criterion for the appearance of the cells is that the Rayleigh number should exceed 1700 (for rigid boundaries). 

Figure 10.6.1 Plan photograph taken by Henri Benard of hexagonal cells in a thin film of molten spermaceti from his original experiments on convection cells induced by surface tension gradients. [Courtesy of Prof. Simon Ostrach. From Benard 1900.1...

Since the diameter of the convective cells is approximately equal to the depth of liquid, then for a depth/diameter ratio of 0.5, it is possible to envisage up to six convective cells with six thermals in a hexagonal pattern carrying heat from the tank floor to the surface, in addition to the much stronger convective flows up the wall of the tank and down the central jet from the surface (see Fig. 3.4). 

In conventional Benard convection [14], flie liquid surface is stationary in the laboratory frame of reference, and hence the cells are stationary, and are observed as a regular array of hexagonal convection or Benard cells. In each cell, hot fluid rises in the centre and cold fluid falls along its boundary (see Fig. 4.7). 

When the Rayleigh number exceeds the critical value, fluid motion develops. Initially, this consists of a series of parallel two-dimensional vortices as indicated in Fig. 8.35a. However at higher Rayleigh numbers a three-dimensional cellular flow of the type indicated in Fig. 8.35b develops. These three-dimensional cells have a hexagonal shape as indicated in the figure. This type of flow is termed Benard cells or Benard convection. 

Heat transfer in horizontal enclosed spaces involves two distinct situations. If the upper plate is maintained at a higher temperature than the lower plate, the lower-density fluid is above the higher-density fluid and no convection currents will be experienced. In this case the heat transfer across the space will be by conduction alone and Nus = 1.0, where 8 is still the separation distance between the plates. The second, and more interesting, case is experienced when the lower plate has a higher temperature than the upper plate. For values of Grs below about 1700, pure conduction is still observed and Nu = 1.0. As convection begins, a pattern of hexagonal cells is formed as shown in Fig. 7-12. These patterns are called Benard cells [33]. Turbulence begins at about Gr6 = 50,000 and destroys the cellular pattern. 

Figure 3.9 The Benard cells in a flat vessel of liquid (A) a schematic of the cell formation due to the self-organized convection of a heated liquid, and (B) the top view of the cells. The convective vertical downstream is centered in the hexagons.

The characteristics of heat transfer through a horizontal enclosure depend ou whether the hotter plate is at the top or at the bottom, as shown in Fig. 9-22. When the hotter plate is at the top, no convection currents develop in the enclosure, since Ihe lighter fluid is always on top of the heavier fluid. Heat transfer in tlris case is by pure conduction, and we have Nu - 1. When the hotter plate is at the bottom, the heavier fluid will be on top of the lighter fluid, and there will be a tendency for Ihe lighter fluid to topple the heavier fluid and rise to the top, where it comes in contact with the cooler plate and cools down. Until that happens, however, heat transfer is still by pure couduc-tion and Nu — I. When Ra > 1708, the buoyant force overcomes the fluid resistance and initiates natural convection currents, which are observed to be in the form of hexagonal cells called BSnard cells. For Ra > 3 X 10, the cells break down and the fluid motion becomes turbulent. 

The analysis with disturbance quantities of the form of Eqs. (10.6.15) indicates a periodic structure in the x, z plane but the shape of the cells associated with the solution is not specified and higher order nonlinear theory is required to define a particular cellular structure. Palm (1960) has shown that in the parallel Rayleigh problem for steady buoyancy driven convection of a liquid film heated from below, the cells approach a hexagonal form as a consequence of the variation of the kinematic viscosity with temperature. 

PALM, E. 1960. On the tendency towards hexagonal cells in steady convection.. Fluid Mech. 8, 183-192. 

As described earlier, Benard performed a number of careful interferometric measurements of the deformation of the liquid surface during convection, and correctly attributed the observed relief to the forces of surface tension. Benard s hexagonal cells were concave and were divided hy ridges, whose level was 1.7 p above the < 11 center when the layer of spermaceti was 1.20 mm thick and at a temperature of lOO C. Smaller thicknesses and lower temperatures yielded less pronounced ridges. Benard s thermal measurements placed the lateral temperature difference between the cell center and the cell partitions at approximately 1°C. In 1939, Hershey (HI) obtained an expression for the steady-state surface elevation due to surface tension differences caused by temperature differences on the surface. This expression reduces to... 

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