Validation Case: Taylor-Couette Flow

The Taylor-Couette flow validation case belongs to fluid dynamics. This test case aims to validate the following parameters:

• Rotating wall
• Velocity profile
• Pressure profile

SimScale’s simulation results were compared to analytical results obtained from methods elucidated in the Scholarpedia article on Taylor-Couette flow\(^1\).

Geometry

The so-called Taylor-Couette flow occurs in the gap between two infinitely long concentric cylinders, when at least one of them is rotating. Therefore, the geometry for this project consists of a slice of an annulus between two cylinders, as seen in Figure 1:

The dimensions of the geometry are given in Table 1:

Geometry parameters	Dimension \([m]\)
Outer radius (a)	1
Inner radius (b)	0.35
Thickness of the slice (c)	0.1

Analysis Type and Mesh

Tool Type: OPENFOAM®

Analysis Type: Steady-state incompressible flow

Turbulence Model: Laminar

Mesh and Element Types: The mesh used in this case was created in SimScale with the standard algorithm.

Find in Table 2 an overview of the resulting mesh:

Case	Mesh Type	Cells	Element Type
Taylor-Couette flow	Standard	488457	3D tetrahedral/hexahedral

Find below the standard mesh used for this case:

Simulation Setup

Material:

• Viscosity model: Newtonian;
• \((\nu)\) Kinematic viscosity: 1e-5 \(m²/s\);
• \((\rho)\) Density: 1 \(kg/m^3\).

Boundary Conditions:

Before defining the boundary conditions, the current nomenclature will be used for the rest of this documentation:

In the table below, the configuration for both velocity and pressure are given at each of the boundaries:

Nomenclature	Boundary Type	Velocity	Pressure
Inner wall	Custom	Rotating wall: 0.001 \(rad/s\) around the positive y-axis	Zero gradient
Outer wall	Custom	Fixed value: 0 (no-slip condition)	Zero gradient
Sides	Custom	Symmetry	Symmetry

Reference Solution

The analytical solution\(^1\) for Taylor-Couette flow is computed from the simplified Navier-Stokes in cylindrical coordinates. Before calculating the velocity and pressure profiles, we need to calculate two constants, \(A\) and \(B\):

$$A = \frac {\omega_{out}R_{out}^2-\omega_{in}R_{in}^2}{R_{out}^2-R_{in}^2} \tag {1}$$

$$B = (\omega_{in} – \omega_{out})R_{out}^2\frac {R_{in}^2}{R_{out}^2-R_{in}^2} \tag{2}$$

where:

• \(\omega_{in}\) and \(\omega_{out}\) \([rad/s]\) are the rotational velocities of the inner and outer walls, respectively;
• \(R_{in}\) and \(R_{out}\) \([m]\) are the inner and outer walls’ radius.

The resulting velocity profile \(U\) is a function of radius \(r\). The equation is given below:

$$U(r) = Ar + \frac {B}{r} \tag {3}$$

Similarly, for pressure \(P\):

$$P(r) = A^2\frac {r^2}{2}+2ABln(r)-\frac {B^2}{2r^2} \tag {4}$$

Result Comparison

The velocity and pressure variation in the radial direction obtained with SimScale are compared to the analytical solution.

In Figure 5, we can see the velocity profile due to the rotation of the inner wall.

Last updated: June 23rd, 2021