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| Grashof number | |
|---|---|
| Name | Grashof number |
| Dimension | L^3 T^-2 |
| Quantity | Dimensionless number |
| Related | Reynolds number, Prandtl number |
Grashof number
The Grashof number is a dimensionless quantity used in fluid dynamics and heat transfer to characterize buoyancy-driven flow; it compares buoyant to viscous forces and governs the onset of natural convection. Developed in the context of 19th and 20th century studies of heat and fluid motion, the Grashof number appears in scaling arguments, stability analyses, and empirical correlations for heat transfer in enclosures, vertical plates, and porous media.
The Grashof number is defined as the ratio of buoyancy to viscous forces for a fluid element in a gravitational field; it plays a role analogous to the Reynolds number for forced convection and the Rayleigh number when combined with the Prandtl number. In problems involving thermal buoyancy near surfaces such as vertical plates, cylinders, and spheres studied by researchers in the tradition of Ludwig Prandtl, Osborne Reynolds, and Joseph Boussinesq, the Grashof number determines whether laminar or turbulent free convection regimes occur.
Starting from the momentum balance in the Boussinesq approximation used by practitioners following Joseph Boussinesq and Lord Rayleigh, buoyant acceleration arises from density variations driven by temperature differences in a gravitational field like that characterized by Isaac Newton’s laws. Comparing the order of magnitude of buoyancy (proportional to gβΔTL) to viscous diffusion (proportional to ν^2) yields the Grashof number; this scaling emerges in boundary-layer analyses pioneered by Ludwig Prandtl and in linear stability studies such as those by G. I. Taylor and Lord Rayleigh. Physically, large Grashof numbers indicate buoyancy-dominated flows that can transition to turbulence similarly to high Reynolds number shear flows examined by Osborne Reynolds.
The classical form of the Grashof number for thermal buoyancy is Gr = g β (ΔT) L^3 / ν^2, where g is gravitational acceleration as measured relative to standards like experiments under Galileo Galilei’s laws, β is the volumetric thermal expansion coefficient referenced in materials treated by Robert Boyle and Jacques Charles, ΔT is the characteristic temperature difference used in analyses by John von Neumann and Ludwig Prandtl, L is a characteristic length consistent with geometries studied by Henri Bénard and Lord Rayleigh, and ν is kinematic viscosity catalogued in compilations by National Institute of Standards and Technology. Variants include a solutal Grashof number in mass-transfer analogs used in studies by Thomas Young and concentration-driven convection problems explored by Joseph Plateau.
The Grashof number is central to predicting natural convection heat transfer for heated vertical plates in classical experiments such as those by Henri Bénard, for enclosures in studies influenced by Sadi Carnot’s thermodynamic lineage, and in geophysical flows considered by John Dalton and Milutin Milanković. Engineers use Grashof-based correlations for Nusselt numbers in electronic cooling designs by firms and laboratories following methods from AT&T Bell Laboratories and NASA research. In environmental applications, Grashof scaling informs analyses of buoyant plumes in the atmosphere influenced by James Joule-type thermodynamic processes and in oceanographic contexts studied by Fridtjof Nansen.
Closely related numbers include the Rayleigh number (Ra = Gr·Pr), which combines Grashof and Prandtl effects in convection stability problems examined by Lord Rayleigh; the Prandtl number (Pr), important in thermal boundary-layer similarity researched by Ludwig Prandtl; and the Schmidt number (Sc), used in mass-transfer analogies developed in the tradition of Thomas Graham. The Richardson number links buoyancy to shear in stratified flows studied by Lewis Fry Richardson and is used alongside Grashof scaling in atmospheric sciences as pioneered by Vilhelm Bjerknes.
Experimental determination of Grashof-number-dependent correlations for heat transfer—such as Nusselt versus Grashof relations—was advanced in laboratory programs at institutions like Massachusetts Institute of Technology, Imperial College London, and Wright-Patterson Air Force Base. Standard test setups include vertical plates, horizontal cylinders, and enclosed cavities whose boundary conditions were catalogued by researchers at National Bureau of Standards. Empirical correlations often take the form Nu = C·Gr^m·Pr^n and are tabulated for specific geometries and materials in handbooks used by practitioners at General Electric and Siemens.
Use of the Grashof number typically assumes the validity of the Boussinesq approximation, small density variations as in classic analyses by Joseph Boussinesq, and a clear separation of scales for L and flow features as in boundary-layer theory from Ludwig Prandtl. It may be invalid in high-compressibility regimes examined in aerodynamics by Theodore von Kármán or in chemically reacting flows studied by Stanislav Ulam-era research groups, and caution is required when complex geometries, strong stratification, or transient buoyancy forcing—as in volcano plume studies associated with Charles Darwin-era observations—dominate.
Category:Dimensionless numbers