# Experimental Study of Downburst Wind Flow over a Typical Three-Dimensional Hill
## 4 Empirical Equations for Wind Profiles  
### Horizontal Wind  
As the foregoing analyses show that the most significant speed-up occurs when \(r_{\rm h}=1.0D_{\rm jet}\), the corresponding cases were used to inform the determination of the design wind loads. Five positions, including the crest, the hill foot, and the locations with altitudes of \(0.25H\), \(0.5H\), and \(0.75H\), respectively, are considered in this section. From the perspective of design, a conservative but reasonable wind load is preferred. \(U_{\rm max}\) denotes the maximum horizontal wind velocity of the profile. Thus, for each position at a given altitude in each case, the envelope values of horizontal wind velocity, in which the value at a relative height is the maximum among the corresponding horizontal velocities measured at all measuring locations, are utilized to enable a conservative design, as illustrated in Figure 25. In the figure, the normalized envelope values are compared with the curve proposed by [PERSON] et al. [8] in the form of  
\[\frac{U(Z)}{U_{\rm max}}=1.55(\frac{Z}{\delta})^{\frac{1}{\delta}}\left[1-{\rm erf }(0.7(\frac{Z}{\delta}))\right] \tag{5}\]  
where \(\delta\) is the wall jet thickness equal to the relative height from the ground to where the horizontal wind velocity is \(0.5U_{\rm max}\)[8]. As shown in Figure 25, except for the values  
Figure 24: \(M_{\rm t,max}\) at crest. measured at the hill foot, the test results are in good agreement with Equation (5). Furthermore, based on Equation (5), a modified phenomenological model is summarized herein to characterize the test results measured at the hill foot, and has the form of  
\[\frac{U(Z)}{U_{\text{max}}}=7.311(\frac{Z}{\delta})^{0.528}[1-\text{erf}(-1.267( \frac{Z}{\delta}))]-15.062\bigg{(}\frac{Z}{\delta}\bigg{)}+1.475\bigg{(}\frac{Z }{\delta}\bigg{)}^{3} \tag{6}\]  
The comparison of Equation (6) with the test results for the hill foot shows a satisfactory agreement, as shown in Figure 25b. In Equations (5) and (6), \(U_{\text{max}}\) and \(\delta\) can be estimated using the formulae  
\[\delta/D_{\text{jet}}=a_{1}+a_{2}\phi+a_{3}\phi^{2}+a_{4}\phi^{3}+a_{5}\phi^{4 }+a_{6}/\eta+a_{7}\eta+a_{8}\eta^{2}+a_{9}\eta^{3} \tag{7a}\] \[U_{\text{max}}/V_{\text{jet}}=b_{1}+b_{2}\phi+b_{3}\phi^{2}+b_{4}\phi^{3}+b_{ 5}/\eta^{2}+b_{6}/\eta+b_{7}\eta+b_{8}\eta^{2}+b_{9}\eta^{3}+b_{10}\eta^{4} \tag{7b}\]  
where \(\eta=H/D_{\text{jet}}\). Moreover, \(a_{i}\) (\(i=1,\,\ldots,\,9\)) and \(b_{i}\) (\(i=1,\,\ldots,\,10\)) are the coefficients, and can be obtained by fitting the test results. The values of \(a_{i}\) and \(b_{i}\) are tabulated in Tables 5 and 6, respectively. Equation (7) accounts for the effects of the slope of the hill and the hill height on the wind profiles. Actually, considering that the maximum horizontal velocity observed in an undisturbed downburst wind field is approximately \(1.0V_{\text{jet}}\), Equation (7b) provides a relatively accurate estimation of \(M_{\text{tmax}}\), which would enable a rational and economic structure design. The comparison of the test results with the estimated results is illustrated in Figure 26. In the figure, selected hill models with \(L=0.5D_{\text{jet}}\) and selected hill models with \(L=1.0D_{\text{jet}}\) are used for the comparison. A satisfactory agreement is observed. The value \(\gamma_{\text{U}}\) can be calculated as the ratio of the measured horizontal wind velocity, \(U_{\text{Exp}}\), to the estimated horizontal wind velocity, \(U_{\text{E}}\), as follows:  
\[\gamma_{\text{U}}=U_{\text{Exp}}/U_{\text{E}} \tag{8}\]  
Figure 25: Normalized envelope values of horizontal wind velocity: (**a**) on hill; (**b**) hill foot.