Differentially Heated Cavity

In the differentially heated cavity, the fluid is driven by two opposite vertical walls that have different temperatures $T_A$ and $T_B$. Since we use a nondimensional formulation, $T_A$ 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 are given by

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

with boundary conditions

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

where $T_A$ is the temperature of the left wall and $T_B$ is the temperature of the right wall. See Symbols for the definition of all quantities used here.

Nondimensional formulation

In TransiFlow, we implement the following nondimensionalized equations

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

with boundary conditions

\[\begin{split}\tilde x &= 0 &:~& \tilde u = \tilde v = \tilde w = 0, \tilde T = \frac{1}{2}\\ \tilde x &= A_x &:~& \tilde u = \tilde v = \tilde w = 0, \tilde T = -\frac{1}{2}\\ \tilde y &= 0, A_y &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde y} = 0\\ \tilde z &= 0, 1 &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde z} = 0\end{split}\]

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 $u = \hat u \nu / D$, $t = \hat t D^2 / \nu$, $p = \hat p (\mu \nu / D^2) - \rho_0 g z$, $T = (T_B - T_A) \hat T + 1 / 2 (T_A + T_B)$ 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 $\mathrm{Ra} = (\alpha_T g \Delta T D^3) / (\nu \kappa_T)$ and the Grashof number by $\mathrm{Gr} = \mathrm{Ra} / \mathrm{Pr}$. All other quantities are defined in Symbols.

Parameters

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

Parameter name	Default value	Notes
'Problem Type'		Set to 'Differentially Heated Cavity'
'Rayleigh Number'	1.0	Unused if Gr is defined
'Prandtl Number'	1.0
'Grashof Number'	Ra / Pr	Overrides Ra if defined