## 2 Issues with Wind Tunnel Modelling of Flow and Dispersion Processes in NVLBs
### Finding of ABL Parameters  
To find the desirable roughness length, \(z_{0}\), and displacement height, \(d_{0}\), of the modelled ABL, the common procedure of fitting the mean velocity (\(U\)) profile to the logarithmic law-of-the-wall is used:  
\[U=\frac{u_{*}}{\kappa}ln\left(\frac{z-d_{0}}{z_{0}}\right), \tag{1}\]  
where \(\kappa\) is the von Karman constant (usually taken as 0.4) and \(z\) is the measured height. The velocity profile should be measured at the domain entrance (i.e., at the end of the development section of the wind tunnel). For the estimation of friction velocity, \(u_{*}\), one can use the assumption that the measured mean vertical momentum flux (\(<u^{\prime}w^{\prime}>\), where \(u^{\prime}\) and \(w^{\prime}\) are the fluctuation components of the longitudinal and vertical velocities, respectively, and the angular brackets represent time averaging) is nearly equal to \(u_{*}^{2}\) in the region just above the roughness height [37]. Equation (1) can be solved for \(z_{0}\) in cases when \(d_{0}\) is equal to zero (see Table 1) or when \(d_{0}\) is obtained from fitting (using the least squares method) of the measured velocity profile of the modelled ABL to the power (\(\alpha\)) law:  
\[\frac{U(z)}{U_{ref}}=\left(\frac{z-d_{0}}{z_{ref}-d_{0}}\right)^{\alpha}, \tag{2}\]  
Figure 4: The most used devices for development of atmospheric boundary layer (ABL) in wind tunnels including Irwin’s spires and roughness elements (adapted from [30]—Copyright 1981, Elsevier). where \(U_{ref}\) is the velocity at the reference height. The longitudinal integral length scale, \(L_{ux}\), is an essential measure of the dominant eddy size in a modelled ABL and is usually determined from the autocorrelation of the time series, assuming that [PERSON]'s hypothesis about frozen turbulence is valid. The modelled \(L_{ux}\) is then compared with that observed in full-scale measurements for different roughness lengths and heights. Figure 5 reflects the following characteristics of \(L_{ux}\) in a full-scale ABL, which were observed by [PERSON] [28]: \(L_{ux}\) is many times bigger than the height at which it is observed; \(L_{ux}\) increases with increasing height, while it decreases with increasing roughness; and from a height of around 300 m above the terrain roughness, \(L_{ux}\) decreases with height, regardless of terrain roughness. These characteristics are fundamental, as they facilitate the first estimation of the dimensions of the roughness elements and vortex generators used to model a required ABL. The last important characteristic of the simulated ABL is the spectrum of turbulence (or, more precisely, the spectral distribution of the kinetic energy of turbulence). As in the case of the integral length scale of turbulence, the spectrum of turbulence (obtained from velocity time series using the discrete Fourier transform) should be tested at different heights of the modelled ABL and should be in good agreement with that observed in full-scale measurements. The evaluation of the agreement entails a qualitative comparison (matching) of the curves of the turbulence spectra between the wind tunnel and full \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Roughness Class** & **Slightly Rough** & **Moderately Rough** & **Rough** & **Very Rough** \\ \hline Type of terrain & Snow, water surface & Grassland, farmland & Park, suburban area & Forest, inner-city area \\ \(z_{0}\) (m) & 10\({}^{-5}\)–5 \(\times\) 10\({}^{-3}\) & 5 \(\times\) 10\({}^{-3}\)–0.1 & 0.1–0.5 & 0.5–2 \\ \(\alpha\) & 0.08–0.12 & 0.12–0.18 & 0.18–0.24 & 0.24–0.40 \\ \(d_{0}\) (m) & \(\approx\)0 & \(\approx\)0 & \(\approx\)0.75 \(\times\)\(h\) & \(\approx\)0.75 \(\times\)\(h\) \\ \hline \hline \end{tabular} \(h\) is the mean height of vegetation and buildings (in m).  
\end{table} 
Table 1: Roughness lengths (\(z_{0}\)), profile exponents (\(\alpha\)), and zero plane displacements (\(d_{0}\)), according to [32].