Overview

The Grashof number (Gr) is a dimensionless parameter used in fluid mechanics and heat transfer to quantify the relative importance of buoyancy-driven forces compared with viscous forces in a fluid. It is a principal criterion for assessing natural (free) convection: larger values indicate stronger buoyancy effects and a greater tendency for convective motion driven by temperature or concentration differences.

Definition and formula

In thermal convection the common form of the Grashof number is written as Gr = g · β · ΔT · L^3 / ν^2, where g is the acceleration of gravity, β the thermal expansion coefficient of the fluid, ΔT the characteristic temperature difference, L a characteristic length, and ν the kinematic viscosity. The number is dimensionless and can be defined locally (Gr_x) or with different choices of L depending on geometry.

Physical interpretation

Grashof compares buoyancy forces (which drive flow when density gradients exist) to viscous damping. When Gr is small, viscous forces dominate and motion is suppressed; when Gr is large, buoyancy overcomes viscous resistance and vigorous convective motion develops. The Boussinesq approximation is usually assumed when applying Gr in thermal problems (small relative density changes).

  • Rayleigh number: Ra = Gr·Pr, combining buoyancy and thermal diffusion effects (Pr is the Prandtl number).
  • Richardson number: Ri ≈ Gr/Re^2, used to compare natural and forced convection effects when a bulk flow (Reynolds number Re) is present.
  • Solutal Grashof: an analogous form for concentration-driven (mass transfer) buoyancy using a solutal expansion coefficient and concentration difference.

Applications and examples

The Grashof number guides the design and analysis of many systems: natural ventilation in buildings, cooling of electronic enclosures, free convection along heated plates and cylinders, enclosure heat transfer, solar collectors, and geophysical flows. Engineers use Gr (often with Ra and Pr) to select correlations for heat transfer coefficients and to predict whether flow will be laminar or transitional to turbulence; the specific critical ranges depend on geometry and boundary conditions.

History and limitations

Named after the 19th-century engineer Franz Grashof, the number is a classical tool in convective heat transfer. Its use assumes continuum behavior and usually small density variations (Boussinesq). For very large temperature differences, compressibility, radiation, or variable properties may invalidate simple Gr-based correlations. For more detailed treatments and correlations consult standard heat transfer texts or technical references: see reference.