Grashof number

The Grashof number Gr 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 frequently arises in the study of situations involving natural convection. It is named after the German engineer Franz Grashof.

for vertical flat plates

for pipes

for bluff bodies

where the L and D subscripts indicates the length scale basis for the Grashof Number.

g = acceleration due to Earth's gravity

β = volumetric thermal expansion coefficient (equal to approximately 1/T, for ideal fluids, where T is absolute temperature)

T_s = source temperature

T_∞ = film temperature

L = length

D = diameter

ν = kinematic viscosity

The transition to turbulent flow occurs in the range 10⁸ < Gr_L < 10⁹ for natural convection from vertical flat plates. At higher Grashof numbers, the boundary layer is turbulent; at lower Grashof numbers, the boundary layer is laminar.

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.

where

and

g = acceleration due to Earth's gravity

C_a,s = concentration of species a at surface

C_a,a = concentration of species a in ambient medium

L = characteristic length

ν = kinematic viscosity

ρ = fluid density

C_a = concentration of species a

T = constant temperature

p = constant pressure

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