The tangential velocity (Vt ) profile in the field can be obtained from Doppler radar measurements. Doppler radars have been used to collect the data of over 200 individual tornadoes as reported in Wurman et al. (2013). Wurman et al. (2007) asserted that due to the beam limits, the radar measurements are limited to about 20m above the ground. On the other hand, the engineers are interested at the elevations less than 10m above ground level (AGL), where the typical buildings are located. Mathematical technique of Ground-Based Velocity Track Display (GBVTD) uses data of the Doppler radar measurements to find the Vt close to the ground. Kosiba andWurman (2010, 2013) and Refan et al. (2017) used this technique to find the tornado features of actual tornadoes. However, they reported the vertical location of the maximum Vt (zmax) occurs between 30m to 200m AGL. In addition, Nolan (2013) claimed that the close to ground Vt profile of the GBVTDs is affected by debris and thus close to ground, Vt measurements by the GBVTD are biased.
To better understand the tornadic flows, the laboratory simulators or tornado vortex chambers (TVCs) are employed. In these simulators, Vt is influenced by the following parameters as reported by Davies-Jones (1973): Reynolds number (Re), the aspect ratio (AR), and swirl ratio (S), as defined below:
Where, Ho is the inlet height of the chamber and the reference length as shown in Figure 1, Vro is the radial velocity at Ho and ν is the kinematic viscosity of air. Using Re≥4.5x10 4 in the TVC models makes the tornado simulations independent of the Re as reported by Refan et al. (2017). In addition, aspect ratio (AR) is defined as:
Where, ro is the radius of the tornado or tornado simulator and is equal to half of its width. Also,
Here Vto is the tangential velocity at the inlet height Ho. Equation (3) implies that S, AR, Vro and ro influence the Vto. That is:
Vto=2SHo/(roVro )
In our entire write up, variables with a * like (Vt*) is the non-dimensionalized variable and without * is dimensionalized variable.
Figure 1. Schematic of a simulator and its parameters
The major objective of this paper is to know the tornadic wind field around 10m from the ground.
This will help to design low rise buildings much better and with lower susceptibility to tornadic wind hazard. The relatively small size of the laboratory simulators results in large geometric scaling ratios (Refan et al., 2017) and those simulators cannot evaluate close-to-ground Vt. In addition, the scale ratios reported by different researchers are based on either length scale or velocity scale. The length scale is calculated either using core radius rc or location of the maximum tangential velocity zmax and the velocity scale is based on the maximum tangential velocity. In this work, none of the scale ratios is introduced. The detailed study conducted by Refan (2014) reports wind speed from 20 m to 80 m from the ground from both field measurements and experimental tornado simulator. Hence, it is difficult to collect wind speed around 10m from the ground using the existing data.
Well-refined computational fluid dynamics (CFD) models can compute the Vt at less than 10m
AGL. Dominguez and Selvam (2017) proposed an axisymmetric CFD model to simulate a tornado chamber of 1.0km x 2.0km, where ro=1.0km, Ho=1.0km and total height (h) =2Ho=2.0km. They used a minimum grid spacing (MGS) of 0.001Ho in the vertical axis which amounts to 1.0m from the ground for Ho=1.0km. They reported the maximum Vt occurring at less than 10m AGL.
However, their study was limited to ro=1.0km, whereas in actual tornadoes the ro may vary. From observations of different tornadoes by National Weather Service (NWS), it can be inferred that the significant tornadoes have ro in the range of 0.7km to 2.3km (Kashefizadeh, 2018). Therefore, the specific objectives of this research are:
1. To vary the ro and study its influence on the maximum Vt with respect to Vro and its location. Hangan and Kim (2008) and Refan (2014) showed that Vt is dependent on the S parameter and thus in order to investigate the effect of ro on the maximum Vt, it is necessary to investigate effect of variation of S on the maximum Vt.
2. To investigate effect of ro on less than 10m-AGL velocity profile. Typical buildings are located at elevation of z=3.3m. Therefore, the maximum Vt will be investigated at z=3.3m. Results centered on these objectives will be highly valuable to develop recommendations for safer design of buildings.
Governing equations: In this study, non-dimensional Navier-Stokes equation in cylindrical coordinate system is employed using Large Eddy Simulation (LES) for an axisymmetric model.
This reduces the 3D problem to 2D problem and thus reduces the computational time. Details of the equations are reported in Kashefizadeh (2018). The governing equations are nondimensionalized using Vro and Ho as the reference values. The reference value for Ho and Vro are considered to be 1km and 60m/s, respectively. For these reference values, the Re will be greater than 1x10 8 .
The computational domain in this study is similar to the computational domain of Dominguez and Selvam (2017). Their non-dimensional computational domain is 1x2 (ro=Ho & h=2Ho). In this study since ro is varied from 0.7km to 2.3 km, the non-dimensional ro* varies from 0.7 to 2.3. The increment of ro* is 0.1, which means that ro* will be 0.7, 0.8, 0.9 and so on.
The boundary conditions of the axisymmetric model are similar to study of Wilson and Rotunno (1986) as shown in Figure 2. For cells close to the ground, law of the wall is used as proposed by Neale et al. (2006).
Dominguez and Selvam (2017) used MGS=0.001Ho alongside the r-and z-axes. The present study also uses the same MGS along the r-and z-axis in the vicinity of the axisymmetric line (z-axis).
Then the grid is exponentially increased by a factor of 1.1 and the maximum spacing is considered to be 0.1Ho. Figure 3 shows the computational domains for non-dimensional ro* of 0.8 and 2. The grid sizes ranged from 46x60 to 63x60 in the r and z direction respectively. In Equation 6, the Vr*(z*) and Ho* are constant in this work, and the two parameters S and ro * will be varied to determine the Vt*(z*).
The CFD model uses SOLA-Yaqui type algorithm to solve the equations (Hirt et al, 1975). In this method, a staggered grid is used where velocities are stored at the nodes and the pressure at the middle of the cell. In the momentum equation, the diffusion and convection terms are respectively implicit and explicit. All terms other than convection in the NS equations are approximated using second order finite volume method (FVM). The QUICK scheme is used for convection term. At this time, the pressure is solved using SOLA type pressure correction. The advantage of using the Yaqui-type configuration is to avoid the problem of pressure-velocity decoupling (Harlow and Welch, 1965;Selvam, 1992). The computer model is run for 5 or 10 time units with a time step of 0.1 to satisfy the CFL condition.
For each ro in the range of 0.7km to 2.3km, various S parameters in the range of 0.2 to 1.5 are used and their tornado wind-fields are investigated to determine the swirl ratio that produces the maximum Vt.
The swirl ratios affect the structure of tornadoes. Hangan and Kim (2008) and Tari et al. (2010) showed before the touchdown, S is small and tornado has a single-cell structure as shown in The S for maximum Vt for each ro is determined and plotted with S for touchdown in Figure 8. It can be seen in Figure 8 that the touchdown S increases by increasing ro. The touchdown S is in the range of 0.40≤S≤0.9 for 0.7km≤ro≤2.3km. Similarly, Figure 8 shows that the swirl ratio S of the maximum Vt increases by increasing ro and is in the range of 0.50≤S≤1.2. This finding is in agreement with the previous studies where Lewellen et al (1997) suggested that by increase of ro, the S producing maximum Vt is likely to increase. Moreover, it can be seen that the S value corresponding to that of the maximum Vt is always greater than the S value corresponding to touchdown S, which implies that the maximum Vt occurs beyond the touchdown. Therefore, in the investigation, only swirl ratios that produce tornadoes beyond touchdown are considered because these are the ones, which may affect the buildings close to the ground. Therefore, it can be concluded that for all radii, the maximum Vt occurs beyond the touchdown stage. Vt for various tornado radii at z=3.3m, which is the height of a typical low rise building and the maximum Vt gradually reduces from 2.5Vro to almost 0.6Vro for ro from 0.7km to 2.3km. Similarly, Figure 9(c) shows that minimum zmax is 21m AGL for ro=0.7km, and by increasing ro, the zmax will also increase. However, for ro≥2.0km, the zmax is constant at 64m. These simulations show that the zmax is in the range of 21m to 64m, whereas radar measurements report zmax in the range of 30m to 200m. Figure 9(d) presents the rc for different ro where rc, is the radial distance of the location of the maximum Vt from the tornado center. It can be seen that rc is in the range of 100m≤rc≤460m for 0.7km≤ro≤2.3km. Table 1 presents a summary of the results.
Table 1 shows that for ro=1.0km, the highest peak is Vt=4.99Vro at S=0.60 and zmax=28.0m. Wilson and Rotunno (1986) reported the maximum Vt=5.0Vro for ro=1.0km. The reported value is for a single study of S=0.28 and zmax=1.016km. Lewellen et al. (1997) reported that for ro=1.0km and S=0.94, the maximum Vt is 6.6Vro at zmax= 27m AGL. Tari et al (2010) used a laboratory simulator and suggested that for ro=1.0km, S=0.68 produces the maximum Vt at a height of 0.34ro.
The difference of the results from the present study to that of Tari et al. (2010) can be due to differences in the geometry of the simulator chamber. The tornado simulator in this work is a based on Ward type, whereas the Tari et al (2010) simulator is similar to Iowa State University. The difference in the tornado chamber to touch down condition and other issues needs to be investigated further. Likewise, increase in core radius rc with increase in chamber radius ro is in agreement with studies of Ward (1972), Davies-Jones (1973), Jischke andParang (1974), Church et al (1979), Church and Snow (1993), Baker and Church (1979)
In this section, the effect of changing the swirl ratio on radial Vt profile is investigated. Figures 10 through 13 show the radial Vt profiles for different ro at heights z=4.5m, z=9.5m, z=18.5m and z=51m. Figure 10 shows the radial Vt profiles at z=4.5m for ro=0.8km, 1.5km, 1.7km, and ro=2.0km. This figure shows that for all radii at z=4.5m, the radial Vt profile has two peaks and does not resemble the Rankine Combined Vortex Model (RCVM) profile. Also, one can see that the double curvature slowly decreases as ro increases. Figure 11 shows the radial Vt profiles for ro=0.8km, 1.5km, 1.7km, and 2.0km at z=9.5m. Here, for ro greater than 0.8km, double peaks in the radial profile is distinctly observed. For ro=0.8km, there is a slight kink close to the center. For z=18.5m and 51m, radial velocity profiles are also plotted in Figures 12 and13. In Figure 12, slight kink is observed for higher ro and in Figure 13 there is no double curvature at all for all radius. However, Refan (2014) did not make any observation. Also the peaks appeared close to the ground and away from the center in their case. These differences may be due to the way vortex chamber is built and further detailed studies are warranted. Similarly, Church et al. (1979) showed occurrence of two peaks on the velocity profile, but did not report the elevation of occurrence of the double-peak. Church et al. (1979) stated that occurrence of the secondary peak on the profile is due to the strong shear force close to the ground. It is an important observation which implies increased intensity of tornadoes close to ground. Several conclusions are made from this section:
1. For lower elevation, there are double peaks observed close to the ground for all radius ro considered in this work. When the elevation increases, the double peaks slowly disappear from smaller ro. Therefore, wider tornadoes have higher intensity due to strong shear forces.
2. Alternatively, these observations imply that RCVM model applies for higher elevation and lower ro.
3. For all elevations, it is noted that when the ro decreases the maximum Vt increases or when In this section, the simulation results will be compared against the radar measurements of actual tornadoes. For this purpose, the ro of actual tornadoes, taken from radar measurements, are used in the simulation; the resulting tornado structure, rc, and zmax are then compared to the data collected from actual tornadoes. This comparison is done for 6 tornadoes as shown in Table 2. Comparison of the structure of the tornadoes shows that for all 6 cases, the computational values are in the range with radar measurements. Also, comparison of rc shows that the radar measurements report a fairly higher value than the simulations. This discrepancy is due to the debris effect in the radar measurements (Kosiba and Wurman, 2010) which causes the radars measure higher values for the rc. The computed rc values have error varying from 9% to 37% with respect to field observation.
The error is far more for higher rc compared to lower ones. Comparison of the zmax of the radar measurements to the simulations is possible for three actual tornadoes of Spencer, Manchester, and Goshen Wyoming tornadoes. Table 2 shows that for these three tornadoes, the zmax of simulations comply well with the actual tornadoes. Refan et al. (2017) stated that if two scaling criteria match in comparison of the simulations to the radar measurements, then the simulations are reliable. Therefore, simulation results in the present study are in close range with field measurements.
A numerical tornado simulator was proposed in order to investigate effect of the tornado radius ro on the maximum tangential velocity Vt of tornadoes. The following conclusions are made from the simulations:
1. Increasing ro increases the touchdown swirl ratio and the swirl ratio for maximum Vt in the range of ro considered for simulation.
2. Increasing ro increases zmax. For 0.7km≤ro≤2.3km, zmax occurs in the range of 20m<zmax<64m, whereas the radar measurements reported zmax in the range of 30m<zmax<200m.
3. Investigating the maximum Vt at different elevations above and below 10m shows that an increase of ro causes the maximum Vt to decrease with respect to Vro.
4. For all ro, at z<10m AGL, the radial Vt profile has two peaks. For higher z, the double peaks in the radial profile occurs for larger ro. In addition, these peaks appear close to the center of the chamber. This radial profile is different from RCVM flow and the detailed CFD study helped to visualize this phenomenon. However, the effect of this on force exerted on buildings is yet to be investigated. Similar double peaks were also observed by Refan (2014) but the double peaks appear away from the center and this may be due to different type of vortex chamber. Church et al. (1979) stated that occurrence of the secondary peak on the profile is due to the strong shear force close the ground. More detailed study on the effect of different vortex chamber on double peak occurrence needs to be conducted.
The first author acknowledges the support received from Womble Professorship and the second and third authors acknowledges the support received from National Science Foundation, under award number CMMI-1762999. The authors acknowledge one of the anonymous reviewer comments which helped to improve the paper extensively.