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Rotating Zones: Taylor-Couette Flow

Overview

The aim of this test case is to validate the following parameters of incompressible laminar couette flow in the annular region of two infinite cylinders (one stationary, one rotating):

  • Velocity profile
  • Pressure profile

The presence of both stationary and rotating components in the geometry necessitates the need for distinct representative zones. SimScale provides two options to approach this problem: the Multi-Reference Frame (MRF) method and the Arbitrary Mesh Interface (AMI) method. Briefly stated, MRF uses source terms for coriolis and centrifugal forces to model the rotation, while AMI explicitly ‘rotates’ the appropriate mesh, interpolating values at each iteration over the interface between the fixed and rotating zones. AMI is inherently transient, but MRF allows for cheaper and quicker results. Thus, MRF uses the steady-state simpleFoam solver in contrast to the pimpleDyMFoam solver used by AMI.

The geometry was uploaded on to the SimScale platform, and appropriate meshing techniques used to generate stationary and rotating zones. Simulation results of SimScale were compared to analytical results obtained from methods elucidated in [1].

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Geometry

The infinite cylinders were modeled as a a slice of their annulus (see Fig.1.). The dimensions are shown in Table 1. The outer cylinder remains stationary, while the inner rotates with an angular velocity of \(0.001\ rad.s^{-1}\).

RotatingZones-geometry

Fig.1. Geometry of the annular slice

Analysis type and Domain

The geometry was uploaded on the SimScale platform, and appropriate meshing techniques were employed to generate fixed and rotating zones in the mesh.

Tool Type : OPENFOAM®

Analysis Type : Incompressible (Laminar)

Mesh and Element types :

Table 2: Mesh Metrics
Mesh type Cells in x Cells in y Cells in z Number of nodes Type
snappyHexMesh 100 10 100 100000 3D hex
RotatingZones-mesh

Fig.2. Mesh used for the SimScale case

Simulation Setup

Fluid:

Kinematic viscosity: \(10^{-5}\ m^2s^{-1}\)

Boundary Conditions:

For both MRF and AMI the rotating mesh zone is assigned an angular velocity of \(0.001 rad.s^{-1}\) about the y-axis, although it is interpreted differently by the two cases. The velocity and pressure bounary conditions for MRF and AMI have been stated in Table 3.

Table 3: Boundary Conditions for MRF and AMI Cases
Parameter Inner Wall (A) Outer Wall (B) Sides (C)
Velocity \(0.0\ ms^{-1}\) (Moving wall velocity) \(0.0\ ms^{-1}\) Symmetry
Pressure Zero Gradient Zero Gradient Symmetry

Results

The analytical solution for Taylor-Couette flow is computed from the simplified Navier-Stokes in cylindrical coordinates [1]. The velocity profile is given by:

\[u(r) = Ar + \frac{B}{r}\]

Here,

\[A = \frac{u_{out}r_{out} - u_{in}r_{in}}{{r_{out}}^2 - {r_{in}}^2}\]

and

\[B = u_{in}r_{in} - A{r_{in}}^2\]

The pressure distribution is given by:

\[p = \frac{A^2r^2}{2} + 2AB\log{r} - \frac{B^2}{2r^2} + C\]

A comparison of the velocity and pressure variation in the radial direction obtained with SimScale (using MRF and AMI) with analytical results is given in Fig.3A and 3B.

Vel Prs

Fig.3. Visualization of velocity and pressure (A, B) in the radial direction for MRF and AMI

Disclaimer

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.