Governing Equations for Rayleigh-Bénard Convection (NOB) - Part 1

03 Dec, 2024

Basic Laws

The three major dynamical law governing Rayleigh-Bénard convection are, namely, the

continuity equation
momentum equation
energy equation

These laws describe the conservation of mass (or volume for incompressible flows), momentum and energy in continuum mechanics.

The relevant dimensional non-Oberbeck-Bousinessq Rayleigh-Bénard is then formulated by assuming:

incompressible, viscous flow augmented by
- buoyancy force
- Coriolis force

Non-dimensionalisation of the non-Oberbeck Rayleigh-Bénard equations will be detailed in Part 2.

Also see sources for detailed derivation.

Continuity Equation

For incompressible flows,

\nabla \cdot u = 0

Equations of Motion

Incompressible Navier-Stokes in conservative form

\frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} = - \frac{1}{ρ} \frac{\partial P}{\partial x_{i}} + ν (T) \frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}} + f_{i}

More succinctly:

D_{t} u = - \frac{1}{ρ} \nabla P + \nabla \cdot [ν (\nabla u + (\nabla u)^{T})] + F

A multitude of body forces can be included depending on the nature of the problem. For a simple, rotating NOB Rayleigh-Bénard problem (see ¹):
1. Buoyancy force: $- \rho \boldsymbol{g} \alpha \Delta $
2. Coriolis force: $2 Ω \times u$

Energy equation

Internal energy equation (see ²):

ρ C_{p} D_{t} T = - \nabla \cdot (k \nabla T) + β T D_{t} P + τ : \nabla u + Q

where:

τ : \nabla u = [\begin{matrix} τ_{11} & τ_{12} & τ_{13} \\ τ_{21} & τ_{22} & τ_{23} \\ τ_{31} & τ_{32} & τ_{33} \end{matrix}] \nabla u

For isobaric flows: $k = C_{p} ρ κ$ and $D_{t} P = 0$
We also assume viscous dissipation term $τ : \nabla u$ and internal energy production $Q$ to be neglibible

Equation simplifies to:

ρ C_{p} D_{t} T = - \nabla \cdot (C_{p} ρ κ \nabla T)

And thus:

D_{t} = - \nabla \cdot [κ \nabla T]

References

P. A. Davidson, Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge: Cambridge University Press, 2013.↩
Ronald L. Panton. Incompressible Flow. John Wiley Sons, Ltd, 2013.↩

#NOB #RBC