This test case solves the 2D Poiseuille flow in a vertical cylinder at \( Re=100 (stationary flow test)\).

The aims of this test case is to validate the discretization of the Stokes / Navier-Stokes equations in a 2D axisymmetric coordinate \((r,y)\).

Figure 1: Axisymmetric coordinate system

Configurations

The radius and the height of cylinder are denoted by R and H respectively which are equal to \(0.5\) and \(0.5\). Therefore the axisymmetric domain is square, its opposite corner coordinates being \((0,0)\) and \((0.5,0.5)\).

We solve the Navier-Stokes equations on the whole domain.

The two following basic configurations are used:

• flow in a straight circular pipe, where a slip boundary condition is set on the left boundary and a wall is set on the right one. If \( v_{mean} \) is the mean velocity, the velocity profile is given by:

  \[ v(r) = 2 \ v_{mean} \ \left[ 1 -\left( \frac{r}{R} \right)^2 \right] \]

• flow between two concentric circular pipes, where wall is set on the left and right boundary conditions and the velocity profile is given by:

  \[ v(r) = 2 \ v_{mean} \ \left( ln(r/R_{1}) \frac{R^2-R_{1}^2}{ln(R/R_{1})} - r^2 + R_{1}^2 \right)/ \left( R_{1}^2+R^2-\frac{R^2-R_{1}^2}{ln(R/R_{1})} \right) \]

In both cases, the problem can be solved with periodic boundary conditions (the flow is forced with a pressure gradient put on the right-hand-side of \( v \) component of the Navier-stokes equation), or with an inlet bottom boundary condition and a Neumann one on the top boundary.

If periodic boundary conditions are used, the source term is set to:

• for the flow in straight circular pipe:

  \[ f = 8 \mu v_{mean}\ / R^2 \]

• for the flow between two concentric circular pipes:

  \[ f = 8 \ \mu v_{mean} \ / \left( R_{1}^2+R^2-\frac{R^2-R_{1}^2}{ln(R/R_{1})} \right) \]

Runtime parameters

Navier-Stokes equation is used with HYPRE or MUMPS solvers.

Comments

Second order spatial convergence is expected.

The advection term could not be discretized in this test case since it is equal to zero. Nevertheless, its activation allows to check its validity.

Spatial convergence is done thanks to a .json file.

Results

Second order convergence is observed with \(L_\infty\), \(L_1\) or \(L_2\) norms and regular meshes. With periodic boundary condition, pressure is verified to be equal to \(0\) up to the computer precision as well as the components of the velocity \(v\).

2D results with inlet

• Case: The straight circular pipe

• Case: The concentric circular pipes

2D results with periodicity

• Case: The straight circular pipe

• Case: The concentric circular pipes