Rayleigh-Bénard Convection

Rayleigh-Bénard convection describes the flow in a liquid that is heated from below. Since we use a nondimensional formulation, temperatures $T_A$ and $T_B$ are only used when rescaling back to dimensional quantities.

Governing Equations

In a domain of dimensions $L \times B \times D$, the governing equations for Rayleigh-Bénard convection are given by

\[\begin{split}\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= -\frac{1}{\rho_0}\nabla p + \nu \nabla^2\mathbf{u} - \frac{\rho g}{\rho_0}e_z\\ \nabla \cdot \mathbf{u} &= 0\\ \frac{\partial T}{\partial t} + (\mathbf{u} \cdot \nabla) T &= \kappa_T \nabla^2T\end{split}\]

with boundary conditions

\[\begin{split}x &= 0, L &:~& u = v = w = \frac{\partial T}{\partial x} = 0\\ y &= 0, B &:~& u = v = w = \frac{\partial T}{\partial y} = 0\\ z &= 0 &:~& u = v = w = 0, T = T_B\\ z &= D &:~& u = v = w = 0, k\frac{\partial T}{\partial z} = h(T_A-T)\end{split}\]

where $T_A$ is the temperature of the gas just above the interface. See Symbols for the definition of all quantities used here.

Nondimensional formulation

In TransiFlow, we implement the following nondimensionalized equations

\[\begin{split}\frac{\partial \tilde{\mathbf{u}}}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde{\mathbf{u}} &= -\tilde\nabla \tilde p +\frac{1}{\mathrm{Gr}^{1/2}}\tilde\nabla^2\tilde{\mathbf{u}} + \tilde Te_z\\ \tilde\nabla \cdot \tilde{\mathbf{u}} &= 0\\ \frac{\partial \tilde T}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde T &= \frac{1}{\mathrm{Pr}\mathrm{Gr}^{1/2}}\tilde\nabla^2\tilde T\end{split}\]

with boundary conditions

\[\begin{split}\tilde x &= 0, A_x &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde x} = 0\\ \tilde y &= 0, A_y &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde y} = 0\\ \tilde z &= 0 &:~& \tilde u = \tilde v = \tilde w = 0, \tilde T = 1\\ \tilde z &= 1 &:~& \frac{\partial \tilde u}{\partial \tilde z} = \frac{\partial \tilde v}{\partial \tilde z} = \tilde w = 0, \frac{\partial \tilde T}{\partial \tilde z} = \mathrm{Bi} \tilde T\end{split}\]

Here $x$, $y$ and $z$ are scaled by $D$, and hence $A_x = L / D$ and $A_y = B / D$. The other quantities are scaled using $u = \hat u \nu / D$, $t = \hat t D^2 / \nu$, $p = \hat p (\mu \nu / D^2) - \rho_0 g z$, $T = (T_B - T_A) \hat T + T_A$ and additionally $\hat u = \tilde u \mathrm{Gr}^{1/2}$, $\hat t = \tilde t \mathrm{Gr}^{-1/2}$, $\hat p = \tilde p \mathrm{Gr}$, $\hat T = \tilde T$. Moreover, the Prandtl number is given by $\mathrm{Pr} = \nu / \kappa_T$, the Rayleigh number by $\mathrm{Ra} = (\alpha_T g \Delta T D^3) / (\nu \kappa_T)$, the Grashof number by $\mathrm{Gr} = \mathrm{Ra} / \mathrm{Pr}$ and the Biot number is given by $\mathrm{Bi} = h D / k$. All other quantities are defined in Symbols.

Perturbation formulation

Alternatively, we can also look at the perturbation from the motionless solution

\[\bar T(\tilde z) = 1 - \frac{\mathrm{Bi}}{\mathrm{Bi} + 1}\tilde z\]

which is less prone to numerical errors. This results in the following nondimensional formulation

\[\begin{split}\frac{\partial \tilde{\mathbf{u}}}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde{\mathbf{u}} &= -\tilde\nabla \tilde p +\frac{1}{\mathrm{Gr}^{1/2}}\tilde\nabla^2\tilde{\mathbf{u}} + \tilde Te_z\\ \tilde\nabla \cdot \tilde{\mathbf{u}} &= 0\\ \frac{\partial \tilde T}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde T &= \frac{1}{\mathrm{Pr}\mathrm{Gr}^{1/2}}\tilde\nabla^2\tilde T + \frac{\mathrm{Bi}}{\mathrm{Bi + 1}}\tilde w\end{split}\]

with boundary conditions

\[\begin{split}\tilde x &= 0, A_x &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde x} = 0\\ \tilde y &= 0, A_y &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde y} = 0\\ \tilde z &= 0 &:~& \tilde u = \tilde v = \tilde w = \tilde T = 0\\ \tilde z &= 1 &:~& \frac{\partial \tilde u}{\partial \tilde z} = \frac{\partial \tilde v}{\partial \tilde z} = \tilde w = 0, \frac{\partial \tilde T}{\partial \tilde z} = \mathrm{Bi} \tilde T\end{split}\]

Parameters

These are the relevant parameters in the parameters dictionary for this problem type.

Parameter name	Default value	Notes
'Problem Type'		Set to 'Rayleigh-Benard' or 'Rayleigh-Benard Perturbation'
'Rayleigh Number'	1.0	Unused if Gr is defined
'Prandtl Number'	1.0
'Grashof Number'	Ra / Pr	Overrides Ra if defined
'Biot Number'	0.0
'Asymmetry Parameter'	0.0	Used to switch branches