Taylor-Couette Flow

Taylor-Couette flow describes the flow of a liquid enclodes by two rotating cylinders. The inner cylinder of radius $r_i$ rotates with angular frequency $\omega_i$, the outer cylinder of radius $r_o$ rotates with angular frequency $\omega_o$.

Governing Equations

In a cylindrical domain of dimensions $d \times 2\pi \times L$, where $d = r_o - r_i$, the governing equations for Taylor-Couette flow 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

\[\begin{split}r &= r_i &:~& u = w = 0, v = \omega_i r_i\\ r &= r_o &:~& u = w = 0, v = \omega_o r_o\\ z &= 0, L &:~& u = v = w = 0\end{split}\]

where $\omega_o$ and $\omega_i$ are the angular frequencies of respectively the outer and inner cylinders. In the $\theta$-direction we apply periodic conditions, i.e. $u(r, 0, z, t) = u(r, 2\pi, z, t)$ and similar for other quantities. See Symbols for the definition of all quantities used here.

Nondimensional formulation

In TransiFlow, we implement the following nondimensionalized equations

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

with boundary conditions

\[\begin{split}\tilde r &= 1 &:~& \tilde u = \tilde w = 0, \tilde v = \tilde \omega_i\\ \tilde r &= \eta^{-1} &:~& \tilde u = \tilde w = 0, \tilde v = \tilde \omega_o \eta\\ \tilde z &= 0, L / r_i &:~& \tilde u = \tilde v = \tilde w = 0\end{split}\]

Here length is scaled by $r_i$, $u = \tilde u v_i$, $t = \tilde t r_i / v_i$, $p = \tilde p \rho_0 v_i^2 - \rho_0gz$, $\omega = \tilde \omega v_i / r_i$. Moreover, the Taylor number is given by $\mathrm{Ta} = v_i r_i / \nu = \mathrm{Re}_i r_i / d = \mathrm{Re}_i (\eta^{-1} - 1)^{-1}$ with $\eta = r_i / r_o$. 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 'Taylor-Couette'
'Reynolds Number'	1.0	Unused if Ta is defined
'Taylor Number'	Re / (1 / eta - 1)