Math 2.0

Fluid mechanics Definition

Rayleigh–Bénard convection

Builds on Navier–Stokes equationsBoussinesq approximation for buoyancyRayleigh numberPrandtl numberHydrodynamic stability

Definition

Consider a fluid layer 0<z<d0 < z < d, unbounded in the horizontal directions, under gravity gg in the direction −z-z. Let ν\nu be its kinematic viscosity, κ\kappa its thermal diffusivity and α\alpha its thermal expansion coefficient. The bottom plate is held at temperature T0+ΔTT_0 + \Delta T and the top plate at T0T_0, with ΔT>0\Delta T > 0. Let ρ0\rho_0 be a reference density and 𝐞z\mathbf{e}_z the upward unit vector. In the Boussinesq approximation, the velocity 𝐮\mathbf{u}, pressure pp and temperature TT satisfy ∂t𝐮+(𝐮⋅∇)𝐮=−1ρ0∇p+νΔ𝐮+gα(T−T0)𝐞z,∇⋅𝐮=0,\partial_t \mathbf{u} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho_0} \nabla p + \nu \Delta \mathbf{u} + g \alpha (T - T_0) \mathbf{e}_z, \qquad \nabla \cdot \mathbf{u} = 0, ∂tT+𝐮⋅∇T=κΔT.\partial_t T + \mathbf{u} \cdot \nabla T = \kappa \Delta T. These equations have the conduction solution 𝐮=0\mathbf{u} = 0, T=T0+ΔT(1−z/d)T = T_0 + \Delta T (1 - z/d). Rayleigh–Bénard convection is the flow with 𝐮≠0\mathbf{u} \neq 0 that this system develops when the conduction solution is unstable.

Remarks

The control parameters are the Rayleigh number Ra=gαΔTd3/(νκ)\mathrm{Ra} = g \alpha \Delta T d^3 / (\nu \kappa) and the Prandtl number Pr=ν/κ\mathrm{Pr} = \nu / \kappa. Linear stability theory shows that the conduction state is unstable exactly when Ra\mathrm{Ra} exceeds a critical value Rac\mathrm{Ra}_c that does not depend on Pr\mathrm{Pr}. For two rigid no-slip plates, Rac≈1708\mathrm{Ra}_c \approx 1708 with critical wavenumber kcd≈3.12k_c d \approx 3.12. For two stress-free plates, Rayleigh's case, Rac=27π4/4≈657.5\mathrm{Ra}_c = 27 \pi^4 / 4 \approx 657.5 with kcd=π/2k_c d = \pi / \sqrt{2}.

Example: just above Rac\mathrm{Ra}_c, with rigid plates, the observed flow consists of steady parallel convection rolls with wavelength close to 2πd/kc≈2d2\pi d / k_c \approx 2d. Non-example: a layer heated from above, ΔT<0\Delta T < 0, is stably stratified, and the conduction solution is stable for every value of ∣ΔT∣\abs{\Delta T}. The hexagonal cells in Bénard's original experiments with a free upper surface are now attributed mainly to surface tension gradients, the Marangoni effect, rather than to buoyancy.

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Boussinesq approximation for buoyancy, concept, no evidence levelBoussinesq approximat…conceptHydrodynamic stability, concept, no evidence levelHydrodynamic stabilityconceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptPrandtl number, concept, no evidence levelPrandtl numberconceptRayleigh number, concept, no evidence levelRayleigh numberconceptRayleigh–Bénard convection, definition, no evidence levelRayleigh–Bénard conve…definition
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  1. Boussinesq approximation for buoyancy concept
  2. Hydrodynamic stability concept
  3. Navier–Stokes equations concept
  4. Prandtl number concept
  5. Rayleigh number concept
  6. Rayleigh–Bénard convection definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rayleigh–Bénard_convection
About this page
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URIm2://7PNDRDMHJ7
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MSC76E06, 76R10
DefinesRayleigh–Bénard convection
UsesNavier–Stokes equations, Boussinesq approximation for buoyancy, Rayleigh number, Prandtl number, Hydrodynamic stability
Changed3 November 2026

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