This test case solves the annular Couette flow with immersed boundary conditions.

This test case solves the annular Couette flow with immersed boundary conditions.

This test case suite focuses on the second-order immersed boundary methods applied to the incompressible Navier-Stokes equation with Dirichlet boundary condition. The objectives of this test case are:

1. verify the core of the immersed boundary methods.
2. verify the order of spatial convergence of the methods.

For more details, see [1] and [2].

Configurations

Physical domain.

We study the flow between two cylinder of radii \(R_i=0.01\) and \(R_e=0.2\). Angular velocity if the inner clinder \(w_i=0.5\) whereas angular velocity of the outer cylinder is equal to 0.

Solutions

The analytical solution of annular Couette flow is given by:

\begin{align} u_\theta(r)=\frac{Ar}{2}+\frac{B}{r} \end{align}

and

\begin{align} p(r)=p_0 - \frac{(Ar)^2}{8} + ABln(r)-\frac{1}{2}(B/r)^2 \end{align}

where \(r\) is the distance from the center of inner cylinder, \(p_0=0\), and the constants A and B follow,

\begin{align} A=2 \frac{w_oR_o^2 - w_iR_i^2}{R_o^2 - R_i^2}, B=(w_i - w_o)\frac{(R_iR_o)^2}{R_o^2 - R_i^2} \end{align}

Boundary condition on velocity are given by the exact solution.

Runtime parameters

The following common settings are used:

• Second ordered centered scheme are used
• Several immersed boundary methods are verified on this test case: direct, linear, LGS, LIS, Q, QO, QGS, QGSO, QIS, QISS1 (see [2] for details).

Results

Second-order convergence is obtained for the L2 error norms for u and pressure. The unexpected convergence (i.e. stair-like behavior) seen for the \(L∞\) error norm of u for the LGS, Q, and QGS methods is characteristic to many of the methods based on the standard linear method for the current test case. As seen from the \(L∞\) error norm of pressure, the standard linear method reaches an error saturation which is significantly reduced with the LGS, Q, and QGS methods. The LGS, Q, and QGS methods also locally increase the order of convergence of the L∞ error norm of pressure between grids \(N^2=64\) and \(N^2=256\).

Linear

LGS

LIS

Q

QGS

QISS1

References

[1] J. Picot, S. Glockner, Discretization stencil reduction of direct forcing immersed boundary methods on rectangular cells: the Ghost Node Shifting Method, Journal of Computational Physics, 364, pp18-48, 2018.

[2] A. M. D. Jost and S. Glockner, Direct forcing immersed boundary methods: Improvements to the Ghost Node Method, Journal of Computational Physics, volume 438, 110371, 2021.