Rayleigh number

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection. In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free or natural convection. It characterises the fluid's flow regime: a value in a certain lower range denotes laminar flow; a value in a higher range, turbulent flow. Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection. The Rayleigh number is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Hence it may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities. It is closely related to the Nusselt number. For most engineering purposes, the Rayleigh number is large, somewhere around 106 to 108. It is named after Lord Rayleigh, who described the property's relationship with fluid behaviour. The Rayleigh number describes the behaviour of fluids (such as water or air) when the mass density of the fluid is non-uniform. The mass density differences are usually caused by temperature differences. Typically a fluid expands and becomes less dense as it is heated. Gravity causes denser parts of the fluid to sink, which is called convection. Lord Rayleigh studied the case of Rayleigh-Bénard convection. When the Rayleigh number, Ra, is below a critical value for a fluid, there is no flow and heat transfer is purely by conduction; when it exceeds that value, heat is transferred by natural convection. When the mass density difference is caused by temperature difference, Ra is, by definition, the ratio of the time scale for diffusive thermal transport to the time scale for convective thermal transport at speed u {displaystyle u} : This means the Rayleigh number is a type of Péclet number. For a volume of fluid of size l {displaystyle l} in all three dimensions and mass density difference Δ ρ {displaystyle Delta ho } , the force due to gravity is of the order Δ ρ l 3 g {displaystyle Delta ho l^{3}g} , where g {displaystyle g} is acceleration due to gravity. From the Stokes equation, when the volume of fluid is sinking, viscous drag is of the order η l u {displaystyle eta lu} , where η {displaystyle eta } is the viscosity of the fluid. When these two forces are equated, the speed u ∼ Δ ρ l 2 g / η {displaystyle usim Delta ho l^{2}g/eta } . Thus the time scale for transport via flow is l / u ∼ η / Δ ρ l g {displaystyle l/usim eta /Delta ho lg} . The time scale for thermal diffusion across a distance l {displaystyle l} is l 2 / α {displaystyle l^{2}/alpha } , where α {displaystyle alpha } is the thermal diffusivity. Thus the Rayleigh number Ra is where we approximated the density difference Δ ρ = ρ β Δ T {displaystyle Delta ho = ho eta Delta T} for a fluid of average mass density ρ {displaystyle ho } , thermal expansion coefficient β {displaystyle eta } and a temperature difference Δ T {displaystyle Delta T} across distance l {displaystyle l} . The Rayleigh number can be written as the product of the Grashof number and the Prandtl number: For free convection near a vertical wall, the Rayleigh number is defined as: