The symmetries include periodicity, bilateral symmetry, rotational symmetry and permutational symmetry of the velocity components. More specifically, we considered the following one-parameter family of the initial conditions,

$$\begin{aligned}
 u_{0,x} & = \sin x  \bigl( \cos (3y) \cos z - \cos y \cos (3z)\bigr) U_0\\
 u_{0,y} & = \sin y  \bigl( \cos (3z) \cos x - \cos z \cos (3x)\bigr) U_0\\
 u_{0,z} & = \sin z  \bigl( \cos (3x) \cos y - \cos x \cos (3y)\bigr) U_0\end{aligned}$$

(cf. [@BCK15]). The Reynolds number components were chosen as follows, $U_0 = 0.01,  L = 2 \pi$ ($2 \pi$-periodic box), $\nu = \frac{1}{3} 10^{-4}$, resulting in the Reynolds number $Re = \frac{U_0 L}{\nu}$ of approximately $2000$.

The data set was created utilizing a mixed spectral finite elements-Fourier method. The $XY$-plane was discretized with a Galerkin pseudospectral discretization and extended in the z-direction with a Fourier discretization. This allows for a scalable solver and also exploits the speed of the Fast Fourier Transform similarly as in [@HKB7]. The Navier-Stokes equation was solved using an operator splitting method (Velocity Correction Scheme) leading to several systems of equations of reduced complexity to be solved instead of one large matrix system. That is while maintaining a splitting error 