Grashof number#
Named after: Franz Grashof (1826-1893).
$$\text{Gr} \stackrel{\text{def}}{=} \frac{g \beta (T_s - T_\infty) L^{3}}{\nu^{2}} \sim \frac{\text{buoyancy}}{\text{viscous damping}}$$
Description#
Measures buoyancy forcing relative to viscous damping. It is the natural-convection analogue of a Reynolds-number scale based on temperature-induced density differences.
Quantities#
| Name | Symbol | SI units | Dimension |
|---|---|---|---|
| gravitational acceleration | \(g\) | \(\mathrm{m}\,\mathrm{s}^{-2}\) | \(\text L\,\text T^{-2}\) |
| thermal expansion coefficient | \(\beta\) | \(\mathrm{K}^{-1}\) | \(\Theta^{-1}\) |
| surface-ambient temperature difference | \(T_s - T_\infty\) | \(\mathrm{K}\) | \(\Theta\) |
| characteristic length | \(L\) | \(\mathrm{m}\) | \(\text L\) |
| kinematic viscosity | \(\nu\) | \(\mathrm{m}^{2}\,\mathrm{s}^{-1}\) | \(\text L^{2}\,\text T^{-1}\) |
Regimes#
Natural convection
| Range | Regime | Description |
|---|---|---|
| 0 – 1e+08 | laminar | Viscous damping keeps buoyancy-driven motion smooth and ordered. |
| 1e+08 – 1e+09 | transitional | Buoyancy and inertia are strong enough for unsteady plumes and mixed laminar-turbulent behavior to appear. |
| 1e+09 – ∞ | turbulent | Buoyancy-driven inertia dominates viscous damping. Natural-convection boundary layers and plumes become turbulent. |