Lid-driven Cavity

In the lid-driven cavity a fluid contained in a box is driven by a lid located at the top of the box that moves at a constant velocity $U$.

Governing Equations

In a domain of dimensions $L \times B \times D$, the governing incompressive Navier-Stokes equations are given by

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

with boundary conditions

\[\begin{split}x &= 0, L &:~& u = v = w = 0\\ y &= 0, B &:~& u = v = w = 0\\ z &= 0 &:~& u = v = w = 0\\ z &= D &:~& u = U, v = w = 0\end{split}\]

See Symbols for the definition of all quantities used here.

Nondimensional formulation

In TransiFlow, we implement the following nondimensionalized equations

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

with boundary conditions

\[\begin{split}\tilde x &= 0, 1 &:~& \tilde u = \tilde v = \tilde w = 0\\ \tilde y &= 0, A_y &:~& \tilde u = \tilde v = \tilde w = 0\\ \tilde z &= 0 &:~& \tilde u = \tilde v = \tilde w = 0\\ \tilde z &= A_z &:~& \tilde u = 1, \tilde v = \tilde w = 0\end{split}\]

Here $x$, $y$ and $z$ are scaled by $L$, and hence $A_y = B / L$ and $A_z = D / L$. The other quantities are scaled using $u = \tilde u U$, $t = \tilde t L / U$, $p = \tilde p \rho_0 U^2 - \rho_0gz$. Moreover, the Reynolds number is given by $\mathrm{Re} = UL / \nu$. 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 'Lid-driven Cavity'
'Lid velocity'	1.0
'Reynolds Number'	1.0