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Overview

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

Case description

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

Mesh

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.

Rotational velocity as a function of radius Pressure as a function of radius