Rotating cylinders

- Solver: simpleFoam
- Incompressible
- Steady
- Laminar
- Multiple Reference Frame (MRF)
- $FOAM_TUTORIALS/incompressible/simpleFoam/rotatingCylinders

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.

Taylor **[taylor_stability_1922]** 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:

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

- 2D structured mesh created using blockMesh

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

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

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