# Experimental Simulation of Thunderstorm Profiles in an Atmospheric Boundary Layer Wind Tunnel
## 2 Literature Review
### Mean Wind Velocity  
Thunderstorm velocity can be described by the classical decomposition rule as the sum of a slowly varying mean wind velocity and rapidly varying turbulent fluctuation [28; 29; 30]. In general, the slowly varying mean wind velocity radial component \(\overline{U}_{r}(z,r,t)\), is a function of the height (\(z\)), radial distance from the center of the downdraft (\(r\)) and time (\(t\)) [31; 32; 33]. The variation with the height of downburst outflow is characterized by a maximum value in the interval of 50-120 m above the ground level, showing a particular nose-like shape [2; 5]. In the following paragraphs, some of the models proposed in the literature are reviewed, reporting the expressions of the vertical profile at time and radial distance (\(r=r_{m}\)), where the maximum velocity appears over the whole field. [PERSON] and [PERSON] [34] developed an analytical model for the velocity field induced by thunderstorms based on the large-scale numerical weather model TASS. Focusing on the time and radial distance where the maximum velocity appears, the vertical profile is expressed as follows:  
\[\overline{U}(z)=1.35\cdot\overline{U}_{m}\Big{(}e^{-z/z^{*}}-e^{-z/\epsilon} \Big{)} \tag{1}\]where, \(\overline{U}_{m}\) is the maximum mean wind speed, \(z^{*}\) is the height at which \(\overline{U}(z^{*})=0.5\cdot\overline{U}_{m}\), and \(\varepsilon\) is a characteristic height in the boundary layer. The authors established the relationships \(z_{m}=0.22\cdot z^{*}\), \(z^{*}=12.5\cdot\varepsilon\), with \(z_{m}\) being the height at which the maximum wind speed \(\overline{U}_{m}\) occurs. [PERSON] [35] modified the previous empirical model through the definition of a new radial dependency, since large discrepancies with full-scale measurements were noted. He proposed the following expression:  
\[\overline{U}(z)=1,22\cdot\overline{U}_{m}\Big{\{}e^{[z_{1}(\frac{z}{2m})]}-e^ {[z_{2}(\frac{z}{2m})]}\Big{\}} \tag{2}\]  
where \(c_{1}=-0.15\) and \(c_{2}=-3.2175\)[35]. [PERSON] and [PERSON] [36] proposed an empirical expression to predict the downburst wind velocity vertical profile, calibrated on laboratory tests and Computational Fluid Dynamics (CFD) simulations. After some experiments varying the distance from the impinging jet, experimental data were fitted through the following expression:  
\[\overline{U}(z)=A\Big{(}\frac{z}{z^{*}}\Big{)}^{B}\Big{[}1-\text{erf}\Big{(}C \cdot\frac{z}{z^{*}}\Big{)}\Big{]}\cdot\overline{U}_{m} \tag{3}\]  
where \(A=1.55\), \(B=1/6\), \(C=0.7\), erf is the error function, \(z^{*}\approx 5.5z_{m}\); this relationship is obtained by fixing \(\overline{U}(z^{*})=0.5\cdot\overline{U}_{m}\) in Equation (3). [PERSON] and [PERSON] [37] performed impinging jet tests and measured the profiles using hotwire anemometers, pressure rakes and particle image velocimetry techniques, finding the best fit of the data for the Wood and Kwok expression, Equation (3), to be \(A=1.52\), \(B=1/6.5\), and \(C=0.68\), which lead to \(z^{*}\approx 7.7z_{m}\). In addition, good agreement was found between the empirical formulation and the full-scale measurements from the NIMROD Project and CFD numerical simulations carried out in the same work. [PERSON] et al. [38] proposed an empirical model to obtain the vertical profile that was fitted and verified by a 3D CFD simulation. The model considers the effects of nonlinear growth of boundary layer thickness by incorporating varying characteristic length scales \(z_{m}\) and \(r_{m}\) as functions of the independent variables \(r\) and \(z\), respectively. According to their model, the vertical profile of the maximum radial velocity is given by:  
\[\overline{U}(z)=\overline{U}_{m}\bigg{(}\frac{z}{z_{m}}\bigg{)}^{\gamma}e^{ \gamma(1-\frac{z}{z_{m}})} \tag{4}\]  
where \(\gamma=0.159\) and \(z_{m}=0.0393D\), \(D\) being the downburst diameter. [PERSON] et al. [39] developed an empirical model based on the collection of numerical and experimental simulation results and recorded field data.