Influence of swirl ratio and radial Reynolds number on wind characteristics of multi-vortex tornadoes

Abstract

In this study, systematic numerical simulations are conducted to investigate how swirl ratio and radial Reynolds number affect the wind characteristics of multi-vortex tornadoes. By properly controlling boundary conditions, multi-vortex tornadoes are produced in a cylindrical computational domain. Six cases with different swirl ratios are studied to examine the influence of swirl ratio, while five cases with different radial Reynolds number are studied to investigate the influence of radial Reynolds number. To facilitate the characterization, the core size and rotational speed of subvortices, as well as the relative distance between the subvortex and the core radius of the main vortex, are defined. The results demonstrate that the increase in swirl ratio leads to the increase in the number of subvortices. For the overall vortex, the increase in swirl ratio decreases the maximum tangential velocity but increases the core radius of the overall flow. For subvortices, for the case where four subvortices are produced, the increase in swirl ratio increases the core size of subvortices but decreases the rotational speed of subvortices. While the increase in radial Reynolds number does not change the number of subvortices produced, it decreases the core size of subvortices, but increases the rotational speed of each subvortex.

Keywords

multi-vortex tornado swirl ratio radial Reynolds number subvortex CFD

Introduction

A multi-vortex tornado is a tornado that contains two or more than two small vortices (called subvortices in the following) swirling around a central point (Edwards 2017). Multi-vortex tornadoes have been observed and recorded in the past 50 years (Fujita 1970; Agee et at. 1975, 1977; Ipauley and Snow 1988; Wurman 2002; Kosiba and Wurman 2008; Wurman et al., 2014). The recent two devastating tornadoes, the Joplin, MO tornado of 22 May 2011 and the El Reno, OK tornado of 31 May 2013, which had been reported to contain multiple vortices inside, are categorized as multi-vortex tornadoes (Kuligowski et al., 2014; Wakimoto et al., 2016). The Joplin tornado was claimed to be the costliest tornado on record, which killed 161 people, injured more than 1000 and caused $2.8B of property damage (NOAA 2011). The El Reno tornado was claimed to be the largest tornado ever documented and three well-trained, experienced tornado researchers were killed by the subvortex of this tornado (Wurman et al., 2014; Bluestein et al., 2015). Due to the presence of multiple subvortices, this type of tornado has more complicated inner flow structure. In addition, inside these tornadoes, the width of extreme winds was found to be very narrow and almost located inside or around the subvortices (Fujita, 1981). Therefore, if some buildings were damaged by the multi-vortex tornadoes, the swirling subvortices could carry wind-born debris and attacked surrounding buildings, resulting in secondary damage.

In order to study tornadoes and their flow characteristics, researchers developed Tornado Vortex Chambers (TVCs) to simulate tornadoes in the lab environment (Wan and Chang 1972; Ward 1972; Davies-Jones 1973; Church et al. 1977, 1979; Baker and Church 1979; Rotunno 1979; Lund and Snow 1993; Wang et al., 2001; Sarkar et al., 2005; Haan et al., 2008). Through testing in these TVCs, researchers found that tornadic wind characteristics were controlled by the following three non-dimensional parameters: swirl ratio, radial Reynolds Number (Rer) and aspect ratio. These three parameters were actually first introduced by Lewellen (1962) in a solution for three-dimensional vortex flow with strong circulation. Among these three parameters, swirl ratio was considered as the dominant one and its influence has been discussed (Church et al., 1979; Ward, 1972; Davies-Jones 1973; Lewenllen et al., 2000). As swirl ratio increased, a jet-like flow turned into a single-celled vortex and then formed a double-celled vortex by following the formation of a stagnation point and vortex breakdown; Finally, the increase in swirl ratio led to the development of multiple subvortices due to the instability of cylindrical shear layers. Davies-Jones (1973) studied the influence of swirl ratio on core radius of a single axisymmetric vortex by using experimental simulation and found that core radius increased with swirl ratio and that the volume flow rate was an important factor in producing intense vortices. Church et al. (1979) pointed out that the transitions between the different types of core flow structure is independent of radial Reynolds number at very large radial Reynolds numbers.

Although previous studies have discussed the relationship between swirl ratio and tornado vortex, only a few experimental studies and numerical simulations in small scale (e.g., numerical simulations of tornado vortex produced in laboratory tornado simulators) reported that secondary vortices were found in the main tornado vortex (Natarajan and Hangan 2012; Nasir et al., 2014; Refan and Hangan 2018). For a multi-vortex tornado, how swirl ratio affects the properties of subvortices (e.g., the rotational speed and the core size of each subvortex, and so on) has not been systemically studied. In addition, it is worth noting that most studies in laboratory experiments produced low Reynolds number and nearly laminar flow (Rotunno, 2013). The Reynolds number in most laboratory experiments and numerical simulation is between 10⁴ and 10⁶, compared to 10⁹ or even higher in natural tornadic wind flows (Monji, 1985; Refan and Hangan, 2016 & 2018; Tang et al., 2018). Therefore, how does radial Reynolds number (10⁹) affects subvortices is unknown. Meanwhile, based on the radar observation of a multi-vortex tornado (Wurman 2002), although several subvortices are present at the same time in the wind field, they swirl with the overall vortex. Therefore, whether the increase in swirl ratio or radial Reynolds number results in the change in the relationship between subvortices and the overall vortex is not clear.

To bridge these knowledge gaps, the objective of this study is to investigate how swirl ratio and radial Reynolds number affect wind characteristics of multi-vortex tornadoes using systematic CFD simulations. The remainder of this paper is organized as follows. First, the definitions of swirl ratio and radial Reynolds numbers are reviewed and the properties of subvortices in a multi-vortex tornado (i.e., rotational speed of each subvortex, core size of each subvortex, as well as the relative distance between the subvortex and the core radius of the overall vortex) are defined. Then, the CFD simulation setup is described. To be specific, six cases with different swirl ratios are simulated to investigate the influence of swirl ratio; and five cases with different radial Reynolds numbers are simulated to examine the influence of radial Reynolds number. Next, the simulation results of tornadic wind field are extracted to illustrate the influence of different swirl ratio and radial Reynolds number on properties of subvortices. Finally, the force balance in radial direction is conducted to reveal the essential reason of changes in wind characteristics of multi-vortex tornadoes compared to single-vortex tornadoes.

Review on swirl ratio and radial Reynolds number, and definitions on properties of individual subvortices in a multi-vortex tornado

In this section, after reviewing the definitions of Swirl Ratio (S) and Radial Reynolds Number (Re_r), the properties of individual subvortices in a multi-vortex tornado (i.e., the rotational speed and the core size of each subvortex, as well as the relative location of each subvortex in the overall vortex) are defined.

Physically, swirl ratio (S) is a measure of the ratio of the circulation strength around the periphery of the vortex to the updraft strength (Davies-Jones, 1981). It was first introduced to measure the properties of the tornado vortex generated in the Ward-type laboratory tornado simulator (see Figure 1), as shown in equation (1).

S = \frac{r_{0} M}{2 Q} = \frac{\tan θ}{2 a}

(1)

where

θ

is the inflow angle,

r_{0}

is the radius of the flow outlet,

2 π Q

is the volume flow rate through the apparatus,

2 π M

is the circulation at

r_{0}

, and

a

is the aspect ratio, as expressed in equation (2).

a = \frac{h}{r_{0}}

(2)

where h is the height of the flow inlet. All these parameters used in equation (1) are either physical dimensions or properties that can be adjusted by changing the initial experiment setup. Therefore, swirl ratio can reflect the initial setup of the simulated tornadoes. Besides the expressions of swirl ratio in equation (1), Haan et al. (2008) proposed a modified definition of swirl ratio, as expressed in equation (3).

S = \frac{π r_{c}^{2} V_{c}}{Q} = S_{c}

(3)

where

r_{c}

and

V_{c}

are the core radius and the maximum tangential velocity in the wind field (at the core radius), respectively. For the same simulated tornado, the two expressions in equations (1) and (3) may give different swirl ratio values (Zhao et al., 2021). As the parameters in the expression in equation (1) can be determined at the beginning of each simulation, facilitating the generation of different simulation cases for parametric study, the definition of swirl ratio specified in equation (1) is used to calculate the swirl ratio in each case in this study.

Figure 1.

Ward’s laboratory tornado simulator.

Radial Reynolds number is another parameter that controls the dynamics of the air flow. It is defined as

R e_{r} = \frac{ρ Q}{μ h}

(4)

where

ρ

is the air density,

μ

is the dynamic viscosity, and

h

is the height of the flow inlet.

To investigate the characteristics of individual subvortices, the rotational speed and the core size of each subvortex, as well as the relative location of each subvortex in the main flow, are defined, as illustrated in the schematic diagram of a multi-vortex tornado in Figure 2. The large circle represents the core radius of the overall vortex, with $r_{c}$ denoting the core radius and $ω_{c}$ denoting the maximum tangential velocity of the overall vortex; the small circle represents one of its subvortices, with $ω_{s}$ denoting the rotational speed of this subvortex. This subvortex not only rotates around their own center but also swirls with the main vortex near the core radius ( $r_{c}$ ).

Figure 2.

Schematic diagram of a multi-vortex tornado (only one subvortex shown).

If it is assumed that the distribution of rotational speed (tangential velocity) of the main vortex along the radius follows Rankine Vortex model, the tangential velocity of the main vortex at Point A is linearly proportional to $ω_{c}$ , which is the same case for the tangential velocity of the main vortex at Point B, as expressed in

ω_{A} = \frac{r_{A}}{r_{c}} ω_{c}

(5)

ω_{B} = \frac{r_{B}}{r_{c}} ω_{c}

(6)

where

r_{A}

and

r_{B}

denote the radii where Points A and B are located. In fact, this assumption has been validated by the present authors, as shall be shown in Figure 7.

At Point A, the air flow of the overall vortex and the subvortex rotates in the same direction, the total velocity is the sum of $ω_{A}$ and $ω_{s}$ , which is actually the maximum tangential velocity along that radius and designated as $V_{m a x}$ . At Point B, the air flow of the overall vortex and the subvortex rotates in opposite directions, the total velocity is the difference between $ω_{B}$ and $ω_{s}$ , which is the minimum tangential velocity along that radius and designated as $V_{m i n}$ . The above analysis can be expressed using the following two equations.

V_{m a x} = ω_{s} + ω_{A}

(7)

V_{m i n} = - ω_{s} + ω_{B}

(8)

By substituting equations (5) and (6) into equations (7) and (8), $ω_{s}$ , the rotational speed of the subvortex ( $ω_{s}$ ) can be solved from equations (7) and (8) as

ω_{s} = \frac{r_{B} V_{m a x} - r_{A} V_{m i n}}{r_{A} + r_{B}}

(9)

When applying equation (9) to obtain $ω_{s}$ of a specific subvortex in a simulated tornadic wind field, the tangential velocity distribution along the radius that is through the center of that specific subvortex should be extracted first; then $V_{m a x}$ and $V_{m i n}$ can be easily obtained by looking for the peak and trough values on the extracted tangential velocity distribution along the radial distance; $r_{A}$ and $r_{B}$ are identified as the radii at which $V_{m a x}$ and $V_{m i n}$ are found.

Let us use $r_{s}$ to represent the core radius of this subvortex, which can be calculated by

r_{s} = \frac{r_{A} - r_{B}}{2}

(10)

In addition, the relative location of the subvortex in the main vortex can be represented by the relative distance (RD) between the center of the subvortex and the core radius of the main vortex, which can be calculated as

R D = r_{s} + (r_{c} - r_{A})

(11)

The normalized expression of RD is designated as normalized relative distance (NRD) and calculated as

N R D = \frac{R D}{r_{c}}

(12)

CFD simulation setup

In the authors’ previous work, the Spencer, SD. Tornado of 30 May 1998, which is a double-celled single-vortex tornado, has been successfully simulated and validated based on the radar-measured velocity data. Based on this model, the boundary conditions of the computational domain were modified to produce a multi-vortex tornado and major differences between multi-vortex tornadoes and single-vortex tornadoes were identified (Zhao et al., 2021). In this study, to investigate the influence of swirl ratio (S) and radial Reynolds number (Re_r) on the wind characteristics of multi-vortex tornadoes, upon the previous model, the boundary conditions are further modified to produce multi-vortex tornadoes with different S and Re_r.

Similar to the authors’ previous work, a cylindrical computational domain is applied to simulate tornadic wind field, as shown in Figure 3. The radius of the computational domain is $R = 800 m$ and the total height is $(h + h_{0}) = 1100 m$ . The inflow is simulated by a velocity-inlet on the side wall with a height of $h$ . The outflow is set up on the top wall, which is considered as a pressure-outlet. Except the velocity inlet and pressure outlet, all other boundary surfaces are set up as no-slip walls. In the region near the ground surface, where boundary layers are present, the grid inflation technique is applied to generate fine mesh. To be specific, 50 grid layers are applied, and the thickness of the first grid layer is 0.01 m, with a 1.2 ratio of thickness growth to avoid sudden change in grid size. The y plus value is around 20. The total number of cells in each simulation is approximately 3.5 million. Standard sea-level atmospheric conditions (pressure = 101,325 N/m², temperature = 298 K, and dynamic viscosity = 1.715 × 10^-5 kg/(s*m) are adopted at the inlet and outlet.

Figure 3.

Cylindrical computational domain to simulate a multi-vortex tornado.

A transient, incompressible CFD simulation is conducted to obtain the velocity and pressure data in wind field. To be specific, large eddy simulation (LES) with the WALE (Cwale = 0.325) subgrid model (Nicoud and Ducros 1999) is applied.It assumes that momentum and mass are mainly transported by large eddies, while small eddies are numerically modeled by subgrid model. The governing equations are filtered time-dependent Navier-Stokes equations, which are solved by a segregated implicit solver, with a SIMPLEC (Semi-Implicit Method for Pressure Linked Equation-Consistent) method for Pressure-velocity Coupling, as the SIMPLEC scheme usually has a better convergence than PISO (Pressure–Implicit with Splitting of Operators) (Van Doormaal and Raithby 1984; Hangan and Kim 2008). In addition, the simulation applies the Least Squares Cell Based scheme for Gradient, which is used to discretize the convection and diffusion terms in the flow conservation equations, the second-order discretization scheme for the pressure equation, and the bounded central differencing scheme for momentum convection-diffusion equation (Anderson and Bonhaus 1994; Barth and Jespersen 1989; Leonard 1991). The time step size is set as Δt = 0.02 s and the total simulation duration is 500 s.

The velocity input at the velocity inlet are indicated by equations 13–15, which were obtained from the radar-measured velocity data, as detailed in (Zhao et al., 2017). The radial, tangential, and vertical velocities applied on the velocity-inlet surface are input using User Define Function (UDF).

v = 21.97 {(\frac{z}{20})}^{0.12}

(13)

ω = {\begin{matrix} 49.26 {(\frac{z}{20})}^{0.1736} - 81.37, z \geq 20 \\ - 32.11 {(\frac{z}{20})}^{0.1429}, z < 20 \end{matrix}

(14)

w = {\begin{matrix} - 7.652 {(\frac{z}{20})}^{0.0908} - 1.049 {(\frac{z}{20})}^{1.1736} + 0.1236 z + 6.7923, z \geq 20 \\ - 0.1384 {(\frac{z}{20})}^{1.8541} + 0.7017 {(\frac{z}{20})}^{1.1429}, z < 20 \end{matrix}

(15)

To investigate the influence of S on properties of individual subvortices, six cases are simulated, as listed in Table 1. To separate the influence of S from the influence of Re_r (to ensure S to be the single variable in the parametric study), only the radius of pressure-outlet

r_{0}

is adjusted from 465 m to 800 m, which leads S to increase from 0.735 to 1.264, while Re_r remains the same.

Table 1.

Setup of simulated cases for investigating influence of swirl ratio

Case #	h (m)	$r_{0}$ (m)	a	v (m/s)	$ω$ (m/s)	$\tan θ$	S	Re _r
1	200	465	0.43	20.3	32.11	0.632	0.735	1.02 × 10⁹
2	200	500	0.40	20.3	32.11	0.632	0.790	1.02 × 10⁹
3	200	550	0.36	20.3	32.11	0.632	0.878	1.02 × 10⁹
4	200	600	0.33	20.3	32.11	0.632	0.948	1.02 × 10⁹
5	200	700	0.29	20.3	32.11	0.632	1.090	1.02 × 10⁹
6	200	800	0.25	20.3	32.11	0.632	1.264	1.02 × 10⁹

h- height of velocity inlet; $r_{0}$ - radius of pressure outlet; a-aspect ratio; v-tangential velocity at velocity inlet; $ω$ -radial velocity at velocity inlet; $θ$ -inflow angle; S-swirl ratio; Re_r-radial Reynolds number.

Similarly, to investigate the influence of Re_r on properties of individual subvortices, five cases are simulated, as listed in Table 2. To ensure Re_r to be the single variable in this group of cases, the radial velocity and tangential velocity at the velocity inlet are both scaled in 1.25 times, 1.5 times, 1.75 times and 2 times in each case. This adjustment leads Re_r to increase from 1.02×10⁹ to 2.04×10⁹, where S remains the same. It is noted that the tangential and radial velocities listed in Tables 1 and 2 are the velocities at the elevation of 20 m. These values are used to calculate S and Re_r.

Table 2.

Setup of simulated cases for investigating influence of radial Reynolds number.

Case #	h(m)	$r_{0}$ (m)	a	v (m/s)	$ω$ (m/s)	$\tan θ$	S	Re _r
1	200	800	0.25	20.3	32.11	0.632	1.264	1.02 × 10⁹
2	200	800	0.25	20.3 ×1.25	32.11 × 1.25	0.632	1.264	1.28 × 10⁹
3	200	800	0.25	20.3 × 1.5	32.11 × 1.5	0.632	1.264	1.53 × 10⁹
4	200	800	0.25	20.3 × 1.75	32.11 × 1.75	0.632	1.264	1.76 × 10⁹
5	200	800	0.25	20.3 × 2	32.11 × 2	0.632	1.264	2.04 × 10⁹

Simulation results and discussion: influence of swirl ratio on wind characteristics of overall vortex and properties of individual subvortices

Inside a multi-vortex tornado, the maximum negative pressure does not occur at the center of the overall vortex. Instead, it occurs at the center of each subvortex, which is very close to the core radius of the overall vortex. This has been observed in the authors’ previous study (Zhao et al., 2018). Meanwhile, the subvortices present their own circulation (rotation) inside the overall tornado vortex. Therefore, one convenient way to find out how many subvortices are produced in the domain is to plot 3D flow of tornadic wind field. Another approach is to draw the static pressure contour with the associated streamlines on a horizontal plane. Figure 4 presents the 3D view of the generated tornadic wind field starting from the horizontal plane at the elevation of 5 m, and the contour of static pressure on the same horizontal plane. Figure 5 presents the contour of static pressure and the streamlines on the horizontal plane at the elevation of 80 m. Both Figures 4 and 5 are drawn based on the instantaneous pressure data at t = 500 s. As shown in Figure 4(a)–4(c), three swirling subvortices are produced inside the overall vortex near the ground. These subvortices are tilted. In addition, some vortices, which are much smaller and shorter than these three subvortices, are also found near the ground. These vortices are considered as local vortices, not tornado subvortex, due to their small sizes. From Figure 4(d) to Figure 4(e), four subvortices are produced in the overall vortex and they present wider rotating air columns than the ones in Figure 4(a)–4(c). In these three cases, although more than four subvortices were observed at an early stage (before the tornado is relatively stable over time), they only last for a few seconds; eventually the number of subvortices becomes four. Furthermore, small local vortices are not found in these three tornadoes.

Figure 4.

3D flow structure starting from the horizontal plane at Z = 5 m, with contour of static pressure on the same horizontal plane presented, to study the influence of S. (a) S = 0.735, (b) S = 0.790, (c) S = 0.878, (d) S = 0.948, (e) S = 1.090, (f) S = 1.264.

As shown in Figure 5, the maximum negative pressure occurs at the center of each subvortex instead of the domain center, and small intense circular streamlines are present inside the core of the overall vortex, which verify the presence of multiple subvortices in the computational domain. Same as observed in Figure 4, from Figure 5(a) to 5(c), three subvortices are produced; In Figure 5(d) to Figure 5(f), four subvortices are produced. This phenomenon suggests that when swirl ratio (S) increases, the number of subvortices increases (from 3 to 4), which shows the critical S should be between 0.878 and 0.948. In addition, when S increases, the core size of the overall vortex also increases, based on the gradual enlargement of the intense circular streamlines of the overall vortex.

Figure 5.

Contour of static pressure and streamlines on the horizontal plane at Z = 80 m to study the influence of S. (a) S = 0.735, (b) S = 0.790, (c) S = 0.878, (d) S = 0.948, (e) S = 1.090, (f) S = 1.264.

When the increased range in S is low, the increase in S does not increase the number of subvortices. This can be observed in the first three cases (from S=0.735 to S= 0.878), as shown in Figure 4(a)-4(c) and Figure 5(a)–5(c). It indicates that the increase in circulation strength is not high enough to break the flow into more subvortices. A similar observation can be found for Cases 4–6 (from S = 0.948 to S = 1.264). However, in any case, the increase in S does increase the size of subvortices. In addition, for Cases 1–3, when only three subvortices are produced, the sizes of the three subvortices are significantly different and some small local vortices are produced. In Cases 4–6, when four subvortices are produced, the sizes of the four subvortices are similar and no small local vortices are observed, which indicates that the layout of four subvortices is more stable.

To investigate how S affects the overall vortex, the azimuthally averaged tangential velocity along the radial distance is extracted and presented in Figure 6. It shows that the increase in S increases the core radius of the overall vortex (consistent with the observation in Figures 4 and 5), while it decreases the magnitude of the maximum tangential velocity.

Figure 6.

Comparison of azimuthally averaged tangential velocity distribution along radial distance under different values of S.

To characterize the properties of individual subvortices, equations (9)–(12) are applied to calculate the rotational speed, the core size and the relative location of each subvortex. It is noted that the derivation of these equations is based on the following two assumptions: (1) the rotational speed of the overall vortex follows the Rankine Vortex model (linear relationship from tornado center to core radius of the overall vortex); and (2) the subvortex is located inside the core of the overall vortex. Before describing the results for each case, these two assumptions are verified using Case 6 (S = 1.264). To verify the first assumption, the azimuthally averaged tangential velocity profile in the area without subvortices, which is between Line 1 and Line 4 (-43° 16°, as shown in Figure 7(a)), is extracted and presented in Figure 7(b). Obviously, the tangential velocity profile of the overall vortex still follows the trend of the Rankine Vortex model. In addition, as shown in Figure 5, the subvortex is always located inside the core region of the overall vortex, which verifies the second assumption.

Figure 7.

Validation of the tangential velocity profile of the overall vortex still following Rankine Vortex model. (a) Only the data between the two dashed lines are used for obtaining the azimuthally averaged tangential velocity profile, (b) Comparison of the obtained tangential velocity distribution along the radial distance with Rankine model.

To examine the influence of increased S on subvortices, for each simulated case, the instantaneous tangential velocity distribution along the radius, which are through the center of each subvortex (Lines 1–4, as shown in Figure 7(a)), is averaged over 1 s and the mean values are presented in Figure 8. In addition, all properties of a subvortex defined in Section 2 are calculated based on equations (9)–(12) for each subvortex, and the maximum negative pressure at the center of each subvortex is also extracted, as listed in Tables 3. For comparison, the azimuthally averaged tangential velocity profile is also provided in Figure 8 and Figure 13.

Figure 8.

1-s averaged tangential velocity along radii on the horizontal plane at Z = 80 m. (a) S = 0.735, (b) S = 0.790, (c) S = 0.878, (d) S = 0.948, (e) S = 1.090, (f) S = 1.264.

As shown in Figure 8 and Table 3, when S is relatively small (S=0.735), all three subvortices have relatively large rotational speeds (around 44 m/s), while one subvortex has significantly smaller core size than the other two, but the rotational speed of this subvortex is much larger than the other two subvortices. It shows that the tangential velocity distribution along the radius through all three subvortices is very close each other and also very close to the azimuthally averaged tangential velocity profile, which means that the subvortices do not affect the overall vortex too much. This may be due to the multi-vortex tornado just transiting from a single-vortex tornado. As S increases, three subvortices start to show strong diversity. For example, in Case 3 (S = 0.878), the largest rotational speed is 1.6 times the smallest one. In addition, in almost all cases, the subvortex with the largest rotational speed also possesses the largest maximum negative pressure among all the generated subvortices. For Cases 2 and 3, the instantaneous values show significant deviation from the azimuthally averaged value, which suggests that the flow of the overall vortex is significantly changed in the area where the subvortex is involved.

Table 3.

Properties of subvortices for all six cases simulated to investigate influence of S

Case #	Subvortex #					Subvortex #
	SV#1	SV#2	SV#3	SV#4	Mean	SV#1	SV#2	SV#3	SV#4	Mean
	Rotational speed (m/s)					Core size (m)
1	38.7	43.2	50.4		44.1	33.8	46.3	20.0		33.3
2	57.9	39.1	51.3		49.4	51.3	32.5	55.0		46.3
3	49.8	42.8	31.9		41.5	46.3	46.3	47.5		46.7
4	42.0	41.9	39.6	42.7	41.5	55.0	48.8	53.8	55.0	53.1
5	35.9	38.7	34.6	34.7	36.0	75.0	78.8	70.0	80.0	75.9
6	29.2	37.4	32.7	32.1	32.8	91.3	71.3	67.5	68.8	74.7
	Relative distance (m)					Radius of maximum tangential velocity (m)
1	39.0	79.0	50.0		56.0	223.0	205.7	213.5		214.1
2	89.0	75.0	90.0		84.7	211.7	202.3	211.8		208.6
3	129.0	129.0	115.0		124.3	215.5	236.8	253.9		235.4
4	103.0	126.0	124.0	105.0	114.5	296.0	304.2	274.6	286.8	290.4
5	70.0	89.0	103.0	135.0	99.3	390.6	389.6	372.6	344.7	374.4
6	141.0	91.0	78.0	91.0	100.3	379.8	381.5	401.1	390.8	388.3
	Maximum tangential velocity (m/s)					Lowest negative pressure (x-10³ Pa)
1	71.3	69.6	76.8		72.6	5.61	3.81	3.79		4.40
2	90.7	67.7	62.7		73.7	5.85	4.30	2.98		4.38
3	59.7	67.2	52.6		59.8	1.81	3.06	2.18		2.35
4	61.2	62.8	66.2	59.1	62.3	2.53	2.20	2.34	2.13	2.30
5	55.9	62.2	57.6	61.3	59.3	1.76	1.73	1.47	1.65	1.65
6	45.5	62.5	52.3	53.4	53.4	1.27	2.17	2.06	1.60	1.78

Among these three cases (Cases 1–3), as S increases, the relative distance between subvortex and the core radius of the overall vortex gradually increases, which means that the subvortices get closer to the center of the overall vortex when S increases. On the contrary, the maximum negative pressure in subvortices decreases as S increases.

From Cases 4–6 (S = 0.948, 1.090 and 1.264), as S increases, the core sizes of subvortices mainly increase, while the rotational speeds decrease. This trend is the same as the change of the azimuthally averaged tangential velocity and core radius of the overall vortex with the increase in S. Meanwhile, as S increases, the relative distance decreases, which means that the subvortices get closer to the core radius in higher S. The subvortex possessing the largest rotational speed is always the one presenting the largest maximum negative pressure, which is the same as Cases 2 and 3.

Figure 9 presents the variation of the averaged values of properties of subvortices with S. As shown in Figure 9(b), the increase in S leads to the increase in the core size of subvortices no matter whether three or four subvortices are present. As shown in Figure 9(c) and 9(d), the increase in S results in the subvortices leaving the core radius of the overall vortex when the number of subvortices is three, but results in the subvortices approaching to the core radius of the overall vortex when the number of subvortices is four. As shown in Figure 9(a), when three subvortices are present, no clear trend is found on the rotational speed of subvortices as S increases. However, when four subvortices are present, the increase in S decreases the rotational speed of subvortices. Overall, the rotational speed mainly decreases as S increases.

Figure 9.

Variation of mean rotational speed, core size, relative distance and normalized relative distance of subvortices with swirl ratio.

To illustrate the variance of subvortices in one tornado, the boxplots of the properties of subvortices are presented in Figure 10. In the generated multi-vortex tornadoes with three subvortices, the variance of subvortices in terms of rotational speed and core size is more significant than the multi-vortex tornado with four subvortices, which may suggest that the tornadoes with four subvortices is more stable than the one with three subvortices. However, no matter whether three or four subvortices are present, the increase in S results in the increase in the variance of rotational speed and core size. No clear trend is observed on the variance of both relative distance and normalized relative distance.

Figure 10.

Boxplot of rotational speed, radius, relative distance and normalized relative distance of subvortices with different swirl ratios.

Simulation results and discussion: influence of radial Reynolds number on wind characteristics of overall vortex and properties of individual subvortices

For all the cases listed in Table 2, the streamlines of the wind flow and the contour of static pressure on the horizontal plane at Z = 80 m are presented in Figure 11. All these figures are based on the instantaneous data at t = 500 s. In all the simulated cases, four subvortices are produced in the domain. The layout of tornado core and subvortices is very similar among all the five cases. The difference mainly lies in that the core region and subvortices become less symmetric when Re_r increases. Figure 12 presents the azimuthally averaged tangential velocity along the radial distance. It shows that the increase in Re_r does not change the core radius of the overall vortex (as S remains the same), but it increases the maximum tangential velocity of the overall vortex.

Figure 11.

Streamlines and contour of static pressure on a horizontal plane at Z = 80 m to study influence of Re_r. (a) Re = 1.02 × 10⁹, (b) Re = 1.28 × 10⁹, (c) Re = 1.53 × 10⁹, (d) Re = 1.76 × 10⁹, (e) Re = 2.04 × 10⁹.

Figure 12.

Comparison of azimuthally averaged tangential velocity distribution along the radial distance under different Re_r.

To explore the influence of Re_r on properties of individual subvortices, for each simulated case, the 1-s averaged tangential velocity profile along the radius that is through the center of each subvortex is extracted and presented in Figure 13. For comparison, the azimuthally averaged tangential velocity profile of the overall vortex is also included in Figure 13. The properties of individual subvortices are calculated based on equations (9)–(12) and listed in Table 4. The averaged value of each property is provided in each table and plotted in Figure 14. As Re_r increases, the averaged rotational speed of subvortices mainly increases, while the averaged core size of subvortices decreases. In addition, in each case, the largest rotational speed and the largest maximum negative pressure are always found in the same subvortex. This trend is also observed in the cases simulated to study the influence of S. No clear trend is observed on the relative location of subvortices.

Figure 13.

Tangential velocity along radii on the horizontal plane at Z = 80 m. (a) Re = 1.02 × 10⁹, (b) Re = 1.28 × 10⁹, (c) Re = 1.53 × 10⁹, (d) Re = 1.76 × 10⁹, (e) Re = 2.04 × 10⁹.

Table 4.

Properties of subvortices for all five cases simulated to investigate influence of Re_r.

Case #	Subvortex #					Subvortex #
	SV#1	SV#2	SV#3	SV#4	Mean	SV#1	SV#2	SV#3	SV#4	Mean
	Rotational speed (m/s)					Core size (m)
1	29.2	37.4	32.7	32.1	32.8	91.3	71.3	67.5	68.8	74.7
2	40.7	39.8	55.7	48.4	46.2	52.5	90.0	43.8	73.8	65.0
3	48.8	61.9	50.0	55.8	54.1	71.3	58.8	47.5	71.3	62.2
4	52.7	70.0	62.2	72.1	64.3	52.5	46.3	56.3	52.5	51.9
5	54.9	64.1	67.9	65.4	63.1	73.8	51.3	60.0	50.0	58.8
	Relative distance (m)					Radius of maximum tangential velocity (m)
1	91.3	71.3	67.5	68.8	74.7	379.8	381.5	401.1	390.8	388.3
2	52.5	90.0	43.8	73.8	65.0	336.4	392.3	386.7	376.8	373.0
3	71.3	58.8	47.5	71.3	62.2	369.0	365.4	354.6	388.3	369.3
4	52.5	46.3	56.3	52.5	51.9	357.5	346.7	391.3	360.1	363.9
5	73.8	51.3	60.0	50.0	58.8	344.0	313.2	364.3	389.7	352.8
	Maximum tangential velocity (m/s)					Lowest negative pressure (x-10³ Pa)
1	45.5	62.5	52.3	53.4	53.4	1.27	2.17	2.06	1.60	1.78
2	65.4	68.2	82.4	64.6	70.2	1.53	2.96	5.03	3.51	3.26
3	80.8	91.1	90.7	85.2	86.9	4.01	7.29	4.29	5.23	5.21
4	91.9	96.9	99.4	114.2	100.6	10.46	10.27	8.47	8.45	9.41
5	140.0	90.9	107.4	109.0	111.8	16.63	8.01	9.60	8.34	10.64

Figure 14.

Variation of 1-s averaged rotational speed, core size, relative distance and normalized relative distance of subvortices with radial Reynolds number.

In addition, the subvortices show a large variance in rotational speed when a higher Re_r is reached, as shown in Figure 15. As Re_r increases, no significant change is observed on the variance of core sizes of subvortices. However, the variance itself is rather large in all cases. As to the variance of relative distance, it does not show clear trend when Re_r increases.

Figure 15.

Boxplot of rotational speed, radius, relative distance and normalized relative distance of subvortices with different radial Reynolds numbers.

Force balance analysis

For any civil structures subjected to straight-line winds, if the winds exhibit as a steady and incompressible flow, the wind pressure acting on civil structures can be predicted using Bernoulli’s Equation. However, the complicated flow characteristics of tornadoes make Bernoulli’s equation unsuitable for predicting the pressure acting on civil structures induced by tornadoes. Therefore, to capture the tornado-induced wind pressure on civil structures, the Bernoulli’s equation needs to be tracked back to its original form, the Navier-Strokes equation (equation (16)). The Navier-Strokes equation in cylindrical polar coordinates (r, ⍬, z) along the radial direction with three velocity components (u, v, w) is shown in equation (16)

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} - \frac{v^{2}}{r} - 2 Ω v = - \frac{1}{ρ} \frac{\partial p}{\partial r} + υ (\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{u}{r^{2}} + \frac{\partial^{2} u}{\partial z^{2}})

(16)

where

\frac{\partial u}{\partial t}

is the variation of the radial velocity with time,

u \frac{\partial u}{\partial r}

and

w \frac{\partial u}{\partial z}

are the radial advection term and the vertical advection term,

- \frac{v^{2}}{r}

is the centrifugal force, and

- \frac{1}{ρ} \frac{\partial p}{\partial r}

is the radial pressure gradient force. The external source term (

- 2 Ω v < 0.1)

and the diffusion term (

υ (\frac{\partial^{2} u}{\partial r^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{u}{r^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) < 10^{- 4}

) in this case are small enough to be negligible.

To demonstrate the contribution of each term in the force balance, all these terms are calculated along the radius through the center of one subvortex in the multi-vortex tornadoes at the height of 80 m and presented in Figure 17. $\frac{\partial u}{\partial t}$ is calculated from the instantaneous values of the last two time steps of simulation results. All other terms are calculated from the instantaneous values of the last time step of simulation results (t = 500 s).

For comparison, all these terms are also calculated along a radius in the related single-vortex tornado, as shown in Figure 16. From Figure 16, the pressure gradient force and centrifugal force are the only dominated forces acting in the tornadic wind field and they balance each other. Therefore, the Navier-Strokes equation (equation (16)) can be simplified into the cyclostrophic balance, as expressed in equation (17)

\frac{1}{ρ} \frac{\partial p}{\partial r} = \frac{v^{2}}{r}

(17)

Figure 16.

Forces in radial direction of a single-vortex tornado along radial distance.

Figure 17.

Forces in radial direction of four multi-vortex tornadoes. (a) S = 0.735, Re_r = 1.02 × 10⁹, (b) S = 0.878, Re_r = 1.02 × 10⁹, (c) S = 1.264, Re_r = 1.02 × 10⁹, (d) S = 1.264, Re_r = 2.04 × 10⁹.

Between a multi-vortex tornado and a single-vortex tornado, three major differences in the force balances are observed: (1) in a multi-vortex tornado, the pressure gradient force exhibits one large negative spike and one large positive spike, verifying the pressure change along radius around the subvortex. Along the radius, before the subvortex center, pressure changes from positive to negative resulting a positive pressure gradient force, while after subvortex center, pressure changes from negative to positive resulting a negative pressure gradient force; (2) in a multi-vortex tornado, besides the centrifugal force and the pressure gradient force, the variation of the radial velocity with time, the radial advection term and the vertical advection term are also significant, which suggests that the variation of radial velocity with respect to time and space is significant, demonstrating the wind flow in a multi-vortex tornado is more transient. In addition, the increase in S makes the radial advection term less significant, while the increase in Re_r makes the radial advection term more significant; and (3) in a multi-vortex tornado, the large forces occur far away from the domain center, around the radius where subvortices revolve, where the velocity is larger and velocity gradient and pressure gradient are larger. In the multi-vortex tornado, at the tornado center (the center of the overall vortex), the pressure is uniform and not that low. On the contrary, in a single-vortex tornado, at the tornado center, the pressure is very low due to atmospheric pressure drop and pressure gradient is high.

Conclusions

In this study, systematic CFD simulations are conducted to investigate how swirl ratio and radial Reynolds number affect the wind characteristics of multi-vortex tornadoes. To facilitate the characterization, the rotational speed and the core size of each subvortex, as well as the relative location of each subvortex in the overall vortex are defined. In terms of the influence of swirl ratio, the following conclusions are obtained. First, the increase in swirl ratio (from 0.70 to 1.25) leads to the increase in the number of subvortices (from 3 to 4). Second, as swirl ratio increases, the core radius of the overall vortex increases, while the maximum tangential velocity decreases. Third, the increase in swirl ratio decreases the rotational speed of subvortices when four subvortices are present and increases the core size of subvortices no matter whether three or four subvortices are present. Fourth, the increase in swirl ratio results in the subvortices leaving the core radius of the overall vortex when three subvortices are present, but results in the vortices approaching to the core radius of the overall vortex when four subvortices are present. Fifth, the fact that the variance of subvortices in terms of rotational speed and core size is more significant in the multi-vortex tornado with three subvortices may suggest that the multi-vortex tornado with four subvortices is more stable. The increase in radial Reynolds number does not change the layout of multi-vortex tornado (the number of subvortices), but it significantly affects the characteristics of subvortices. To be specific, the increase in radial Reynolds number increases the rotational speed of each subvortex, but decreases the size of subvortices.

Through the force balance analysis, it demonstrates that: (1) in a single-vortex tornado, the pressure gradient force and centrifugal force are the only dominated forces acting in the tornadic wind field and they balance each other. Therefore, the Navier-Strokes equation (equation (16)) can be simplified into the cyclostrophic balance; and (2) in a multi-vortex tornado, the pressure gradient force exhibits one large negative spike and one large positive spike. In addition, the variation of the radial velocity with time, the radial advection term and the vertical advection term are also significant. Furthermore, the large forces occur around the core of the subvortex, which is far away from the domain center.

In future studies, civil structures with different archetypes will be placed in the simulated wind fields of multi-vortex tornadoes to characterize the wind effects of this type of tornado on civil structures. The present study assumes that the tornado does not translate. In the future, more simulations will be conducted with the translation of tornado vortex included. In addition, structural dynamic analysis will be conducted to find the structural responses of civil structures under the wind loads induced by multi-vortex tornadoes, which will indicate the failure mechanisms of civil structures under multi-vortex tornadoes. Deformation accumulation and damage accumulation may have to be considered in structural analysis, as a civil structure may be attacked by different subvortices sequentially. Based on the obtained wind effects and failure mechanisms of civil structures, the pressure equation specified in ASCE 7 can be properly modified for a proper tornado-resistant design.

Footnotes

Acknowledgements

The authors greatly appreciate the financial support from the VORTEX-SE Program within the NOAA/OAR Office of Weather and Air Quality under Grant No. NA20OAR4590452. The authors also greatly appreciate the financial support from National Science Foundation, through the project, “Damage and Instability Detection of Civil Large-scale Space Structures under Operational and Multi-hazard Environments” (Award No.: 1455709), and two other projects (#1940192 and #2044013).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by National Science Foundation (1455709, 1940192, 2044013), and National Oceanic and Atmospheric Administration (NA20OAR4590452).

ORCID iDs

Yi Zhao

Guirong Yan

Ruoqiang Feng

Houjun Kang

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