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# 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:

## Analysis Type and Mesh

Tool Type: OPENFOAM®

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:

Find below the standard mesh used for this case: Figure 2: Standard mesh, with region refinements applied to the area around the inner cylinder wall.

## 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:

## 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.