vortex.md

June 7, 2025 · View on GitHub

Simulation of co-rotating and counter-rotating vortices

We will initialize our vortex with a Lamb-Oseen vortex, which is defined in cylindrical coordinates as:

U_{\theta}(r) = \frac{\Gamma}{2 \pi r} \left[1 - e^{-\large \frac{r^2}{r_c^2}} \right]

The vorticity is thus written as:

\large \omega = \frac{\Gamma}{\pi r_c^2} \: e^{-\large \frac{r^2}{r_c^2}}

With $\Gamma$ the circulation of the flow, $r$ the radial coordinate, and $r_c$ the mean radius defined as $r_c=\sqrt{4\nu t+r_0^2}$ .

Non-dimensionalization

We need to non-dimensionalize the initial condition. To do this, we introduce the same characteristic quantities as for the non-dimensionalization of the Navier-Stokes equations:

$L$ : characteristic length
$U$ : characteristic velocity
$T = L/U$ : characteristic time

The non-dimensional variables are defined by:

\begin{aligned} \tilde{r} &= \frac{1}{L}r \\ \tilde{r}_0 &= \frac{1}{L}r_0 \\ \tilde{r}_c &= \frac{1}{L}r_c \\ \tilde{t} &= \frac{U}{L}t \\ \tilde{\omega} &= \frac{L}{U}\omega \end{aligned}

Thus, the non-dimensional form of the initial condition is written as:

\begin{aligned} \begin{cases} \tilde{r_c} = \sqrt{4 Re^{-1} \tilde{t} + \tilde{r_0}^2}\\ \large \tilde{\omega} = \pm A e^{-\large \frac{\tilde{r}^2}{\tilde{r_c}^2}} \\ \tilde{r} = \sqrt{(\tilde{x} - \tilde{x_0})^2+(\tilde{y} - \tilde{y_0})^2} \end{cases} \end{aligned}

The characteristic velocity is defined as: $U=\frac{\Gamma}{A \pi \tilde{r_c}^2 L}$ and the characteristic length: $L=2\pi$ . The Reynolds number is thus: $Re = \frac{UL}{\nu} = \frac{\Gamma}{A \pi \nu \tilde{r_c}^2}$ . The choice of sign determines the direction of rotation of the vortex.

Merger of two co-rotating vortices

We initialize the simulation with two vortices rotating in the same direction (co-rotating), with the same intensity $A=10$ and the same size $\sigma = 0.5$ , separated by a distance $R=1$ . The simulation is performed over a duration of $30$ seconds at a Reynolds number of 2000.

vorticity

https://github.com/user-attachments/assets/a47447f4-31ed-460e-a302-e4a0b335e0c5

Other physical quantities

Merger of three co-rotating vortices

We perform the same simulation with three vortices.

vorticity

https://github.com/user-attachments/assets/e8318755-a6eb-42c1-871e-d4fe016f48aa

Other physical quantities

stream function

https://github.com/user-attachments/assets/b507bd66-dbef-41b8-a679-c6c6921d8874

velocity magnitude

https://github.com/user-attachments/assets/5139d689-f107-486a-9785-092694d22da4

velocity in x

https://github.com/user-attachments/assets/f9b37764-c4fe-4e0e-b063-c08fecbf9803

velocity in y

https://github.com/user-attachments/assets/8995cb6d-4412-49bc-8902-060bb2b5aeaf

Vortex dipole

We perform the same simulation but by choosing one of the two vortices to rotate in the opposite direction in order to obtain two counter-rotating vortices and thus create a dipole.