This test case solves the Poiseuille flow with immersed boundary conditions.
The global boundaries are defined by the intevales:
\begin{align} [0, 2] &\quad\text{in the flow direction,} & [-1, +1] &\quad\text{otherwise.} \end{align}
The immersed boundary is either a flatchannel of a cylinder:
\begin{align} y = \pm h &\quad\text{(flatchannel),} & x^2 + y^2 = h^2 &\quad\text{(cylinder),} \end{align}
where \(h = \frac{1}{2} \frac{262144}{177147} \), which is half the wolf interval of the Pythagorean scale. This has been chosed to avoid the boundary to match any grid.
The velocity has the following parabolic shape: solution:
\begin{align} u(y) = \hat{u} \left(1 - \frac{y^2}{h^2}\right) &\quad\text{(flatchannel),} & u(r) = \hat{u} \left(1 - \frac{r^2}{h^2}\right) &\quad\text{(cylinder).} \end{align}
The flow is pushed by the navier source term, which should be equal to:
\begin{align} f_x = 2 \frac{\mu\hat{u}}{h^2} &\quad\text{(flatchannel),} & f_y = 4 \frac{\mu\hat{u}}{h^2} &\quad\text{(cylinder),} \end{align}
where \( \mu \) is the dynamic viscosity of the fluid. Using the navier source term yields a constant pressure gradient, which allows the domain to be periodic in the flow direction.
The average velocity is computed from the integral:
\begin{align} V \bar{u} = \int_{-h}^{+h}{u(y)\mathrm{d}y} = 2 h \hat{u} \int_0^1 {(1 - z^2) \mathrm{d}z} &\quad\text{(flatchannel),} \\ V \bar{u} = \int_{0}^{2\pi}\int_{0}^{+h}{u(r)r\mathrm{d}r\mathrm{d}\theta} = 2 \pi h^2 \hat{u} \int_0^1 {(1 - z^2) z \mathrm{d}z} &\quad\text{(cylinder),} \end{align}
giving:
\begin{align} \bar{u} = \frac{2}{3}\hat{u} &\quad\text{(flatchannel),} & \bar{u} = \frac{1}{2}\hat{u} &\quad\text{(cylinder).} \end{align}
In the actual test case, the dynamic viscosity of the fluid is set to match a given Reynolds number, which is defined by:
\begin{align} \mathit{Re} = \frac {2h \times \bar{u}} {\mu} \end{align}
We choose to set \(\bar{u}\) and \(\mu\) as follows:
\begin{align} \bar{u} &= \frac {1} {2h} & \mu = \frac {1} {\mathit{Re}} \end{align}
Values are given in the table below.
Geometry | \( \bar{u} \) |
---|---|
flatchannel | \( \tfrac{59049}{131072} = 0.45050811767578125\ldots \) |
cylinder | \( \tfrac{177147}{524288} = 0.33788108825683594\ldots \) |
and \( \mathit{Re} = 20\).
Boundary | Condition |
---|---|
left | Neumann |
right | Neumann |
bottom | Wall |
top | Wall |
back | Wall |
front | Wall |
Label | D's | Size |
---|---|---|
2D | 32×32 | |
3D | 16×16×16 |
Label |
---|
poiseuille |
poiseuille_3D |
The following common settings are used:
Mesh | \(L^\infty\) error | Order | \(L^2\) error | Order |
---|---|---|---|---|
16 | 4.69593096e-03 | n/a | 7.95143694e-03 | n/a |
32 | 1.07964363e-03 | +2.1209 | 1.84280242e-03 | +2.1093 |
64 | 1.75571802e-04 | +2.6204 | 3.00863636e-04 | +2.6147 |
128 | 3.75877241e-05 | +2.2237 | 6.42961656e-05 | +2.2263 |
256 | 1.24056177e-05 | +1.5993 | 2.13212113e-05 | +1.5924 |
512 | 1.45314236e-06 | +3.0937 | 2.49594936e-06 | +3.0946 |