From e18c6fbab950b41fb1dbfad7d4d8a356fe446c7d Mon Sep 17 00:00:00 2001 From: Nijso Date: Thu, 8 May 2025 14:16:53 +0200 Subject: [PATCH] Update Inc_Von_Karman.md fix typos --- .../Inc_Von_Karman/Inc_Von_Karman.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/_tutorials/incompressible_flow/Inc_Von_Karman/Inc_Von_Karman.md b/_tutorials/incompressible_flow/Inc_Von_Karman/Inc_Von_Karman.md index 7cf450b8..83a221f8 100644 --- a/_tutorials/incompressible_flow/Inc_Von_Karman/Inc_Von_Karman.md +++ b/_tutorials/incompressible_flow/Inc_Von_Karman/Inc_Von_Karman.md @@ -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. @@ -176,7 +176,7 @@ 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. @@ -184,12 +184,12 @@ When we compare the Strouhal number with the experimental data from Williamson, ![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). @@ -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. \ No newline at end of file +Note that you have to select your own, local files when you load the statefile.