Rotating cylinders

Overview

• Solver: simpleFoam
• Incompressible
• Steady
• Laminar
• Multiple Reference Frame (MRF)
• TODO: tutorial path

The case comprises two cylinders, with inner radius $R_1$ rotating with angular velocity $\Omega_1$, and outer radius $R_2$ rotating with angular velocity $\Omega_2$.

The laminar case corresponds to a Reynolds number of 100. where the Reynolds number os defined as:

$Re = \frac{|\u_0| d}{\nu}$

Where $\u_0$ is the angular velocity of the inner cylinder, i.e.

$\u_0 = \Omega_1 R_1$

and $d$ is the distance between the cylinders, i.e. $R_2 - R_1$. Using an inner and outer radii of 1 and 2, respectively, and setting the kinematic viscosity to $1$, the angular velocity of the inner cylinder is 100 rad/s.

Analytical solution

Taylor 77 shows that the velocity, $\u_{\theta}$ is described by:

$$\u_{\theta} = A r + \frac{B}{r}$$

where $A$ and $B$ are constants defined as:

$$A = \frac{\Omega_1 \left( 1 - \mu \frac{R_2^2}{R_1^2} \right)}{1 - \frac{R_2^2}{R_1^2}}$$ $$B = \frac{R_1^2 \Omega_1 (1 - \mu)}{1 - \frac{R_1^2}{R_2^2}}$$

Here, $\Omega_1$ and $\Omega_2$ are the rotational speeds of the inner and outer cylinders, and

$$\mu = \frac{\Omega_2}{\Omega_1}$$

The steady flow equations for this case, in cylindrical co-ordinates reduces to

$$\frac{1}{\rho}\frac{\partial p}{\partial r} - \frac{\u_{\theta}^2}{r} = 0$$

On integrating with respect to radius an expression for the pressure is recovered:

$$p = \frac{A^2 r^2}{2} + 2 A B \ln (r) + \frac{B^2}{2 r^2} + C$$

Mesh

• 2D structured mesh created using blockMesh

Boundary conditions

• Outer cylinder fixed
• Inner cylinder rotates at a fixed angular velocity

Results

The rotational velocity, $\u_\theta$ can be directly reported during the calculation using a fieldCoordinateSystemTransform function object.