Grashof number

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The Grashof number is a dimensionless number in fluid dynamics and Heat Transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It is named after the German engineer Franz Grashof.

$Gr = \frac{g \beta (T_s - T_\infty ) L^3}{\nu ^2}\,$ For Flat Plates

$Gr_D = \frac{g \beta (T_s - T_\infty ) D^3}{\nu ^2}\,$ For Pipes

$Gr_D = \frac{g \beta (T_s - T_\infty ) D^3}{\nu ^2}\,$ For Bluff Bodies

where the D subscript indicates the length scale basis for the Grashof Number.

g = acceleration due to Earth's gravity

β = volumetric thermal expansion coefficient = 1/T; T in Kelvin

T_s = source temperature

T_∞ = film temperature

L = length

D = diameter

ν = kinematic viscosity

The Transition to turbulant flow occurs for $G r > 10 9$ for flat plates.

The product of the Grashof number and the Prandtl number gives the Rayleigh number, a dimensionless number that characterizes convection problems in heat transfer.

There is an analogous form of the Grashof number used in cases of natural convection mass transfer problems.

$Gr_c = \frac{g \beta^* (C_{a,s} - C_{a,a} ) L^3}{\nu^2}$