Grass yard number

Physical key figure
Surname	Grass yard number
Formula symbol	${\displaystyle {\mathit {Gr}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Gr}}={\frac {g\,\beta \,(T_{\mathrm {s} }-T_{\infty })L^{3}}{\nu ^{2}}}}$
${\displaystyle g}$ Acceleration due to gravity ${\displaystyle \beta }$ Coefficient of thermal expansion ${\displaystyle T_{\mathrm {s} }}$ temperature ${\displaystyle T_{\infty }}$ Resting temperature ${\displaystyle L}$ Characteristic length ${\displaystyle \nu }$ kinematic viscosity
Named after	Franz Grashof
scope of application	viscous currents

The Grashof number (named after Franz Grashof , 1826–1893) is a dimensionless 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 through viscosity , multiplied by the ratio of the inertial force to the viscous force: ${\displaystyle Gr}$

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

With

• ${\displaystyle g}$ Gravitational acceleration ( ) ${\displaystyle \approx 9{,}81\;\mathrm {\tfrac {m}{s^{2}}} }$
• ${\displaystyle \beta }$ Coefficient of thermal expansion
• ${\displaystyle T_{\mathrm {s} }}$ temperature
• ${\displaystyle T_{\infty }}$ Resting temperature
• ${\displaystyle L}$ Characteristic length
• ${\displaystyle \nu }$ kinematic viscosity .

The reformulation of the Navier-Stokes equations in the dimensionless form results in the form equivalent to the definition given above

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

With

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

The Grashof number can also be converted into an equivalent Reynolds number in order to then be able to apply the formulas of free convection to forced convection :

${\displaystyle Re_{\mathrm {eq} }={\sqrt {0{,}4\cdot {\mathit {Gr}}}}}$

See also

• Hagen number

Web links

• Further information