Benchmark
The Flag 2D example is used to benchmark PVade. For more detail on the Flag 2D example, you can refer to Flag 2D. Benchmarking of PVade was done through validation of lift, drag and position of point A as described in Turek and Hron [1], and proceeds through three steps: (1) validation of the structure solve only (CSM3), validation of the fluid solve only (CFD3), and validation of the coupled fluid-structure interaction (FSI2).
CSM3 benchmark
For benchmarking of the structural dynamics, we used CSM3 in Turek and Hron [1] and compared the x- and y-position of point A (see Flag 2D) The table shows the results obtained for CSM3.
ndof |
ux of A [×10−3] |
uy of A [×10−3] |
|
|---|---|---|---|
PVADE |
6522 |
-15.368 ± 15.369 [1.0956] |
-65.624 ± 67.515 [1.0956] |
Truth (CSM3) |
6468 |
−14.279 ± 14.280 [1.0995] |
−63.541 ± 65.094 [1.0995] |
A plot for x- and y-position of point A is shown below, the blue line represents the time series tip displacement obtained from PVade and the black line from the reference paper.
CFD3 benchmark
For benchmarking of the fluid analysis, we used CFD3 in Turek and Hron [1] and compared the Lift and Drag over the time of the simulation. Additionally, rather than matching as closely as possible the number of degrees of freedom as in CSM3, we show convergence to the finest values reported by Turek and Hron as a function of increasing mesh resolution, i.e., decreasing values of \(\Delta x\), the spacing between finite elements at the flag’s surface.
Drag |
Lift |
|
|---|---|---|
PVade \(\Delta x = 0.0125\) |
387.109 ± 0.000 [0.100] |
-11.713 ± 0.000 [0.100] |
PVade \(\Delta x = 0.0100\) |
414.202 ± 3.364 [4.400] |
-23.958 ± 314.375 [4.400] |
PVade \(\Delta x = 0.0075\) |
432.383 ± 5.289 [4.400] |
-16.137 ± 424.951 [4.400] |
PVade \(\Delta x = 0.0050\) |
435.956 ± 5.600 [4.400] |
-12.505 ± 437.862 [4.400] |
PVade \(\Delta x = 0.0040\) |
437.246 ± 5.749 [4.400] |
-12.891 ± 439.845 [4.400] |
Truth (CFD3) |
439.45 ± 5.6183 [4.3956] |
−11.893 ± 437.81 [4.3956] |
The lift and Drag versus time are shown below
FSI2 benchmark
For the coupled FSI, we use FSI2 from Turek and Hron [1] and compared the position of point A as well as lift and drag over the duration of the simulation.
ux of A [×10−3] |
uy of A [×10−3] |
Drag |
Lift |
|
|---|---|---|---|---|
PVade |
-14.78 ± 13.23 [3.8] |
1.28 ± 80.62 [1.9] |
204.81 ± 69.60 [3.8] |
1.77 ± 232.09 [1.9] |
Truth (FSI2) |
−14.58 ± 12.44 [3.8] |
1.23 ± 80.6 [2.0] |
208.83 ± 73.75 [3.8] |
0.88 ± 234.2 [2.0] |
The lift, drag, and position of point A (the flag tip) over time are shown below:
The simulation of FSI2 can be visualized below. The top half of the visualization shows the vorticity of the fluid as it responds to and drives the flexible flag. The bottom half shows the displacement of the flag and the adapatation induced in the surrounding fluid mesh points.
