Rayleigh number Benard flow

At low Rayleigh numbers, Wragg (W6) found a smaller Ra dependence, resembling more the dependence in laminar free convection. In this range of Ra numbers, a cellular flow pattern is believed to exist, analogous to that of thermal and surface tension-driven cellular convection (Benard cells F3). In the range where the convection is turbulent, the Ra1/3 dependence has been confirmed over seven powers of Ra by Ravoo (R9), who used a centrifuge to vary the body force at constant bulk composition. 

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. 

Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid. When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value A 7. The flow has a stationary cellular character with a spatial periodicity of about 2d. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T in temperature creates warmer and cooler regions, and due lO buoyancy effects the former tends to move upwards and the latter downwards. When AT < AT, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T and the velocity (normal to the layer) vary as exp (i j,y) with x ji/rf, the threshold is given by the dimensionless Rayleigh number... 

The instability of liquid films subjected to surface tension and/or density gradients causes convection flow forming vortex cells known as Benard cells. The Marangoni (Ma) and the Rayleigh (Ra) numbers are often used for determining the conditions for the onset of the formation of convection cells caused by the surface tension gradient and density gradient, respectively. 

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