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In fluid mechanics, the Rayleigh number (Ra, after Lord Rayleigh) 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. For most engineering purposes, the Rayleigh number is large, somewhere around 106 to 108. The Rayleigh number is defined as the product of the Grashof number (Gr), which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number (Pr), which describes the relationship between momentum diffusivity and thermal diffusivity: Ra = Gr × Pr. Hence it may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities: Ra = B/μ × ν/α. It is closely related to the Nusselt number (Nu). 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 : This means the Rayleigh number is a type of Péclet number. For a volume of fluid of size in all three dimensions and mass density difference , the force due to gravity is of the order , where is acceleration due to gravity.

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