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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:

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

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

Usage

TODO: explain how to use...

[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)!

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, but before that, you need to increase data dimensionality. 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