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The Rayleigh number is defined as the product of the Grashof number (Gr), which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number (Pr), which describes the relationship between momentum diffusivity and thermal diffusivity: Ra = Gr × Pr.
A critical value exists for the Rayleigh number, above which fluid motion occurs. [3] = The ratio of the Grashof number to the square of the Reynolds number may be used to determine if forced or free convection may be neglected for a system, or if there's a combination of the two.
Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. [1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed.
The relative magnitudes of the Grashof number and the square of the Reynolds number determine which form of convection dominates. If G r / R e 2 ≫ 1 {\displaystyle {\rm {Gr/Re^{2}\gg 1}}} , forced convection may be neglected, whereas if G r / R e 2 ≪ 1 {\displaystyle {\rm {Gr/Re^{2}\ll 1}}} , natural convection may be neglected.
where S T, T T denote the salinity and temperature at top layer, S B, T B denote the salinity and temperature at bottom layer, Ra is the Rayleigh Number, and Pr is the Prandtl Number. The sign of Ra S and Ra T will change depending on whether it stabilizes or destabilizes the system.
Note that whereas the Reynolds number and Grashof number are subscripted with a scale variable, the Prandtl number contains no such length scale and is dependent only on the fluid and the fluid state. The Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity.
[3] [4] k is the thermal conductivity of the fluid, L is the characteristic length with respect to the direction of gravity, Ra L is the Rayleigh number with respect to this length and Pr is the Prandtl number (the Rayleigh number can be written as the product of the Grashof number and the Prandtl number).
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