Grashof number

The Grashof number (named after Franz Grashof, 1826-1893) is a dimensionless characteristic number in fluid mechanics that is suitable for estimating flows in thermal convection. It indicates the ratio of the static buoyancy of a fluid to the force acting on the fluid due to viscosity, multiplied by the ratio of the inertial force to the viscous force:

${\begin{aligned}Gr&={\frac {F_{Auft}}{F_{viskos}}}\cdot {\frac {F_{traeg}}{F_{viskos}}}\\&={\frac {g\cdot \gamma \cdot (T_{\mathrm {s} }-T_{\infty })\cdot L^{3}}{\nu ^{2}}}\end{aligned}}$

with

acceleration due to gravity ( $\approx 9{,}81\;{\mathrm {{\tfrac {m}{s^{2}}}}}$ )
$\gamma$ coefficient of thermal expansion by volume
$T_{{\mathrm s}}$ temperature
$T_{{\infty }}$ rest temperature
Characteristic length
$\nu$ kinematic viscosity.

Reformulating the Navier-Stokes equations into the dimensionless form yields the form equivalent to the definition given above

$\Leftrightarrow Gr={\frac {\left|\rho -\rho _{0}\right|}{\rho _{0}}}\cdot {\frac {g\cdot L^{3}}{\nu ^{2}}}$

with

$\rho$ Density
$\rho _{0}$ Density in the undisturbed fluid.

One can also convert the Grashof number into an equivalent Reynolds number in order to subsequently apply the formulae of forced convection to free convection:

Grashof number

See also

Search by letter