## 2 Homogeneous Turbulence  
For a given flow, the instantaneous velocity vector is referred to as \(u_{i}\) whose components in the streamwise, vertical and spanwise directions \((x,y,z)\) are \(u_{i}=(u,v,w)\). The Reynolds decomposition in the streamwise direction is defined as \(u=\left\langle U\right\rangle+u^{\prime}\), where \"\(\left\langle\,\cdot\,\right\rangle\)\" denotes time-average (capital letters are also used to emphasize averaged magnitudes). The characteristic size of the largest eddies is identified as the distance \(L\) required to nullify the correlation function \(\mathcal{R}_{ij}(\mathbf{r},t)\). With this assumption, the integral lengthscale  
\[L_{ij}^{(d)}=\int_{0}^{\infty}\mathcal{R}_{ij}(\mathbf{e}_{d}r,t)dr \tag{1}\]  
in the direction \(d\) (set by the unitary vector \(\mathbf{e}\)) is defined [37]. From all scales defined by this expression, those most commonly used are the longitudinal integral lengthscale \(L_{1}=L_{11}^{(1)}\) as well as the transversal one \(L_{2}=L_{22}^{(1)}\). Similarly, the longitudinal Taylor lengthscale (or micro-scale) \(\lambda_{1}=\lambda_{11}^{(1)}\) is defined by the osculating parabola to the correlation function \(\mathcal{R}_{ij}(\mathbf{r},t)\). If isotropy is assumed (or at least between the 1 and 2 directions) the equivalences \(L_{22}^{(1)}=L_{11}^{(2)}\) and \(\lambda_{22}^{(1)}=\lambda_{11}^{(2)}\) are also valid. In the absence of shear, the [PERSON] hypothesis of frozen turbulence can be adopted more comfortably. This is, it is assumed that the turbulence field does not change as it is convected by the mean wind at \(\left\langle U\right\rangle\), which yields the equivalence between the spatial and temporal correlations. In this way, correlations can be made from the time series of each velocity component. In particular, the autocorrelation will provide the integral time scales \(\mathcal{T}_{11}\) and \(\mathcal{T}_{22}\) from where the integral lengthscales can be computed by means of \(L_{1}=\left\langle U\right\rangle\mathcal{T}_{11}\) and \(L_{2}=\left\langle U\right\rangle\mathcal{T}_{22}\). Likewise, the longitudinal Taylor lengthscale can be calculated from the expression  
\[\frac{1}{\lambda_{1}^{2}}=\frac{\left\langle U\right\rangle^{-2}}{2}\left\langle \left(\frac{\partial u_{1}^{\prime}}{\partial t}\right)^{2}\right\rangle \tag{2}\]  
as seen in [38] (note that Equation (2) differs from the one presented in that report by a factor of \(\sqrt{2}\)). Considering the turbulent kinetic energy of a field \(k=\frac{1}{2}\left(\left\langle u_{1}^{\prime 2}\right\rangle+\left\langle v_{2}^{ \prime 2}\right\rangle+\left\langle w_{3}^{\prime 2}\right\rangle\right)\) and the assumption of isotropy, \(\lambda_{2}\) is related to the amount of dissipation of \(k\) by  
\[\varepsilon=\frac{15\
u\left\langle{u_{1}^{\prime}}^{2}\right\rangle}{\lambda _{2}^{2}}=\frac{30\
u\left\langle{u_{1}^{\prime}}^{2}\right\rangle}{\lambda_{ 1}^{2}}\ . \tag{3}\]  
When grid turbulence is used to approximate the theoretical case of decaying isotropic turbulence, the characteristics observed at different positions downstream from the grid correspond to the time evolution of isotropic turbulence with zero mean velocity. Thus, a decay during the interval \(\Delta t\) is approximated by that occurring within \(\Delta x\) in a wind tunnel. In this manner, it has been observed that the decay of \(k\) follows the expression  
\[\frac{k}{\left\langle U\right\rangle^{2}}=c_{A}\left(\frac{x-x_{0}}{M}\right) ^{-n}\ \, \tag{4}\]  
where \(M\) is the turbulence grid size, \(c_{A}\) (also written as \(1/A\)) is a fitting parameter, \(n\) is the decay exponent and \(x_{0}\) a virtual origin. Equation (4) is commonly employed to track the streamwiseturbulence intensity decay, replacing \(k\) for \(\left<{u_{1}^{\prime}}^{2}\right>\). While Ref. [37] mentions \(1.1\leq n\leq 1.3\) and \(c_{A}\simeq 1/30\), Ref. [39] reports to have observed \(n=1.25\) with \(x_{0}=0\), whereas Ref.