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Helmholtz decomposition code

VERSION>=2021.09.01

Introduction

Analysis of turbulence and vortices requires knowledge of the kinetic energy, is conservation and its modes. This tool performs the Helmholtz Decomposition, in order to extract the compressive and rotational components of kinetic energy of the flow as shown in Tsubota, Fujimoto, Yui (2017) Numerical Studies of Quantum Turbulence..

We define the effective velocity field as $\omega(\textbf{r},t)=\sqrt{\rho(\textbf{r},t)}v(\textbf{r},t)$ . From this, we can calculate the kinetic energy as:

$E_k=\frac{m}{2V}\int \omega(\textbf{r},t)^2d\textbf{r}=\frac{m}{2}\sum_{\textbf{k}}|\widetilde{\omega}(\textbf{k},t)|^2$ ,

where $\widetilde{\omega}(\textbf{k},t)=\mathfrak{F}[\omega(\textbf{r},t)]=\int \omega(\textbf{r},t) \frac{e^{-i\textbf{k}\cdot\textbf{r}}}{V}d\textbf{r}$ is the Fourier Transform of the effective velocity.

From this, we can apply the Helmholtz Decomposition theorem to obtain the compressive and rotational components of the effective velocity field:

$\omega(\textbf{r},t)= \omega_0(t)+\omega_c(\textbf{r},t)+\omega_i(\textbf{r},t)$

$\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{} \frac{\textbf{k}\cdot \widetilde{\omega}(\textbf{k},t)}{k^2}\; \textbf{k}e^{i\textbf{k}\cdot\textbf{r}}$

$\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{} \left \{ \widetilde{\omega}(\textbf{k},t) - \frac{\textbf{k}\cdot \widetilde{\omega}(\textbf{k},t)}{k^2}\; \textbf{k}\right \}e^{i\textbf{k}\cdot\textbf{r}}$

Note that in our specific case, we assume the component $\omega_0(t)$ to be zero across the whole lattice.

Usage

TODO: explain how to use... The cast line consists of few specific entries. After the executable name, call the .wtxt file that needs to be analyzed, together with a number indicating the spin component you wish to inspect:

0 is for spin a (up)
1 is for spin b (down)
2 is for spin a+b

Here is an example of how it should look like:

[gabrielw@node2066 tmp]$ mpirun -np 5 wslda-hdc-3d test2.wtxt 0          

         Adding new variables [wc_a, wi_a] to w-data set.
         Number of measurements = 10
         Starting time          = -1 
         Lattice size   = 80 x 80 x 16 
         Ending time            = 9.000 
         Component = a 
         Variables being produced = wi_a & wc_a 
         Creating file: test2_hh_a.txt
         Creating file: test2_hhspec_a.txt

         Reading frame [1], currently at 1%
         ....
        Completed in    4s ( 0m 4s)!

For large lattices, the code might take longer than it shows due to MPI memory cleaning routines.

The code outputs four files:

Two of them are .wdat, and they contain data about $\omega_0(t)$ and $\omega_0(t)$ in binary form. These can be read via VisIt by selecting the corresponding quantity from the VisIt HUD (wc*/wi*);
One .txt file, containing time, compressive energy, rotational energy, compressive energy calculated via spectral decomposition and rotational energy calculated via spectral decomposition in this order;

TODO: Explain what are output files... their meaning and content, in what units are columns

Limitations

Presently the code supports only 3d data. However, you can apply it to 2D as well. Before that, you need to increase data dimensionality of the datasets you wish to analyze. You can do it using wdata tool wdata-datadim-up. For example:

[gabrielw@node2066 tmp]$ wdata-datadim-up test1.wtxt test2 3
# WDATA TOOL: DATADIM MODIFIER
# READING INPUT DATA: `test1.wtxt`
# ************************ LATTICE ***************************
# LATTICE: 80 x 80 x 16
# SPACING: 1.000000 x 1.000000 x 1.000000
# VOLUME : 80.000000 x 80.000000 x 16.000000
# DIM-IN : 2
# DIM-OUT: 3
# RESIZING `rho_a`...
# RESIZING `rho_b`...
# RESIZING `delta`...
# RESIZING `j_a`...
# RESIZING `j_b`...
# RESIZING `nu`...
# RESIZING `tau_a`...
# RESIZING `tau_b`...
# RESIZING `V_a`...
# RESIZING `V_b`...
# RESIZING `V_ext_a`...
# RESIZING `V_ext_b`...
# RESIZING `delta_ext`...
# RESIZING `velocity_ext_a`...
# RESIZING `velocity_ext_b`...
# RESIZING `alpha_a`...
# RESIZING `alpha_b`...
# RESIZING `A_a`...
# RESIZING `A_b`...
# WRITING `test2.wtxt`
# DONE.

Developers

Andrea Barresi, Warsaw University of Technology