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14 changes: 7 additions & 7 deletions _tutorials/incompressible_flow/Inc_Von_Karman/Inc_Von_Karman.md
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Expand Up @@ -33,11 +33,11 @@ The resources for this tutorial can be found in the [incompressible_flow/Inc_Von

### Background

When the Reynolds number $$Re=\rho \cdot V \cdot D / \mu$$ is low (Re < 40), the flow around a circular cylinder is laminar and steady. At around Re=49, the flow becomes unsteady and a periodic shedding of vortices forms in the wake of the cylinder, known as vortex shedding. The frequency of this vortex shedding is usually expressed in terms of the Strouhal number $$St= f \ cdot D / U_{\infty}$$, with f the shedding frequency, D the diameter of the cylinder and U the far-field velocity. Important experimental work can be found in the paper of Williamson, *Vortex Dynamics in the Cylinder Wake*, Annual Review of Fluid Mechanics (1996) [doi](https://doi.org/10.1146/annurev.fl.28.010196.002401). After around Re=180, a second frequency is observed experimentally, in the longitudinal direction. This frequency can only be observed in a 3D simulation. The increase in number of frequencies continues until after around Re=1000 the flow is considered fully turbulent.
When the Reynolds number $$Re=\rho \cdot V \cdot D / \mu$$ is low (Re < 40), the flow around a circular cylinder is laminar and steady. At around Re=49, the flow becomes unsteady and a periodic shedding of vortices forms in the wake of the cylinder, known as vortex shedding. The frequency of this vortex shedding is usually expressed in terms of the Strouhal number $$St= f \cdot D / U_{\infty}$$, with f the shedding frequency, D the diameter of the cylinder and U the far-field velocity. Important experimental work can be found in the paper of Williamson, *Vortex Dynamics in the Cylinder Wake*, Annual Review of Fluid Mechanics (1996) [doi](https://doi.org/10.1146/annurev.fl.28.010196.002401). After around Re=180, a second frequency is observed experimentally, in the longitudinal direction. This frequency can only be observed in a 3D simulation. The increase in number of frequencies continues until after around Re=1000 the flow is considered fully turbulent.

### Problem Setup

The configuration is a circular cylinder of 5 mm surrounded by a far field at $$L = 30 D$$ and a rectangular wake region of $$X = 150 D$$. The far-field velocity is $$U_{\infty} = 0.12 m/s$$. With a viscosity of $$\mu=1.0 \cdot 10^{-5}$$ and a density of $$\rho = 1 kg/m3$$, the Reynolds number is $$Re=120$$.
The configuration is a circular cylinder of diameter $$D=0.01 m$$ surrounded by a far field at $$L = 30 D$$ and a rectangular wake region of $$X = 150 D$$. The far-field velocity is $$U_{\infty} = 0.12 m/s$$. With a viscosity of $$\mu=1.0 \cdot 10^{-5}$$ and a density of $$\rho = 1 kg/m3$$, the Reynolds number is $$Re = 120$$.

![von_karman_mesh](../../../tutorials_files/incompressible_flow/Inc_Von_Karman/images/mesh.png)
Figure 2: Computational domain for the von Karman vortex shedding.
Expand Down Expand Up @@ -176,20 +176,20 @@ print("strouhal number = ",St)
```


When we compare the Strouhal number with the experimental data from Williamson, we see in Figure 4 that the frequency is underpredicted. We will vary some numerical settings to investigate if we can improve the prediction of the Strouhal number.
When we compare the Strouhal number with the experimental data from Williamson, we see in Figure 4 that the frequency is slightly overpredicted. We will vary some numerical settings to investigate the impact on the prediction of the Strouhal number.



### Numerical variations

![validation](../../../tutorials_files/incompressible_flow/Inc_Von_Karman/images/strouhal_cylinder_karman_variation.png)

Figure (5): Comparison of different numerical settings
Figure (5): Comparison of different numerical settings.

In Figure 5, we see the effect of different numerical settings on the prediction of the Strouhal number. The second order scheme predicts a Strouhal number of $$St = 0.1734$$, slightly over predicting the experimental value of $$St_{exp} = 0.170$$. Note that our predictions of the Strouhal frequency depends on the number of samples and sampling rate that we provide to the FFT. We took 2500 timesteps of 0.01 s which contains enough cycles for an accurate frequency prediction using an fft.
When switching from second order in time to first order, the Strouhal number is under predicted by 6 \% compared to the experimental value. Also, when increasing the time step from 0.01 s to 0.02 seconds, the St decreases by 2 \%. When increasing the time step even further to $$ \Delta t = 0.04 s%%, St is under predicted by 8 \%. The period of the dimensional frequency is $$f \approx 0.5 s$$, so with a timestep of 0.01 s we have 50 time steps per period, we have 25 time steps when $$\Delta t = 0.02 s$$, and only 12 time steps when $$\Delta t = 0.04 s$$. It is clear that 12 time steps per period is not sufficient.
When switching from second order in time to first order, the Strouhal number is under predicted by 6 % compared to the experimental value. Also, when increasing the time step from 0.01 s to 0.02 seconds, the St decreases by 2 \%. When increasing the time step even further to $$ \Delta t = 0.04 s $$, St is under predicted by 8 %. The period of the dimensional frequency is $$f \approx 0.5 s$$, so with a timestep of 0.01 s we have 50 time steps per period, we have 25 time steps when $$\Delta t = 0.02 s$$, and only 12 time steps when $$\Delta t = 0.04 s$$. It is clear that 12 time steps per period is not sufficient.

It is also known that the size of the computational domain influences the results, so we reduce the domain by half, $$L = 15 D$$ and $$X = 75 D$$. The Strouhal then increases to $$St = 0.1768$$, an increase of 2 \%. It seems that a far-field that is 15D away from the cylinder is sufficient.
It is also known that the size of the computational domain influences the results, so we reduce the domain by half, $$L = 15 D$$ and $$X = 75 D$$. The Strouhal then increases to $$St = 0.1768$$, an increase of 2 %. It seems that a far-field that is 15D away from the cylinder is sufficient.

As a final test, the testcase can be executed for varying Reynolds numbers, ranging from Re=60 to Re=180, giving the result in Figure (6).

Expand All @@ -204,4 +204,4 @@ We get a pretty good agreement compared to the experimentally measured values.
The paraview statefile to create the movie can be found here: [statefile_with_particles.pvsm](https://github.com/su2code/Tutorials/blob/master/incompressible_flow/Inc_Von_Karman/statefile_with_particles.pvsm)
and here:
[statefile_movablearrow_timeseries.pvsm](https://github.com/su2code/Tutorials/blob/master/incompressible_flow/Inc_Von_Karman/statefile_movablearrow_timeseries.pvsm)
Note that you have to select your own, local files when you load the statefile.
Note that you have to select your own, local files when you load the statefile.