In order to develop a vortex model able to describe the flow conditions and characteristics of a translating vortex, potential flow theory is used. Whilst it is acknowledged that this is a highly simplified approach, it does at least provide a scientifically robust framework in which to undertake the analysis. Additionally, within a tornado the inertial effects are likely to be significantly larger than the viscous effects and as such this approximation is likely to be sufficient (particularly for the purposes used below). The configuration considered is shown in Figure 1, which illustrates the 2D plan view of the ground plane of a fixed reference frame. The tornado is represented using a free vortex circulating in the anti-clockwise direction with the circulation strength of Γ ( = 2 π r c V θ m a x , where V θ m a x is the maximum tangential velocity of the vortex and r c is the radial distance which V θ m a x occurs. NOTE: r c is an analytical convenience for this study and not actually a core radius. Since potential vortices do not have a finite core, an r c is merely used to define a circulation strength). The vortex translates in the direction parallel to the x-axis from − x to + x with the translating speed of V t r a n s . A body represented using a point, B , is placed at the origin at a distance of r from the translating path of the vortex (Section 2.2 details further analysis using an airfoil to represent the body). Since the motion is assumed to be a free vortex flow (where tangential velocity is inversely proportional to the distance from the rotation axis), the tangential velocity at point B will increase with a decrease in r . Additionally, if the body (point B , hereafter referred to as case 1) is placed at a distance of + r , the velocity induced by the vortex at point B is greater (as V θ m a x + V t r a n s due to the direction of vortex circulation), whereas, if placed at a distance of − r , the velocity at point B is lower (as V θ m a x − V t r a n s ). The potential function ( φ v ) of the vortex can be expressed as φ v = Γ 2 π θ = Γ 2 π tan − 1 y p   x p − V t r a n s t   (1) where θ is the azimuth angle, y p is the location of any point in the y-direction, x p is the location of any point in the x-direction (e.g., at point B, x p = 0 and y p = 0) and t is the time step (as shown in Figure 1).

The velocity components of the vortex can be obtained by taking the derivatives of φ in the x and y-direction, i.e.: V x   =   Γ 2 π   − y p ( x p − α V θ m a x   t ) 2 + ( y p ) 2 (2) V y =   Γ 2 π   ( x p − V t r a n s t ) ( x p − α V θ m a x   t ) 2 + ( y p ) 2 (3) where α is the dimensionless translating velocity (the ratio between the wind velocity and the translating velocity, α =   V t r a n s V w i n d   or α = V t r a n s V θ m a x   ) and V θ m a x   is the reference velocity (maximum tangential velocity), taken as 1 in what follows. Using Eqs 2, 3 in conjunction with the unsteady Bernoulli equation where the time derivative of the potential function, ∂ φ ∂ t can be shown to be: ∂ φ ∂ t   =   Γ 2 π   y p V t r a n s ( x p − V t r a n s t ) 2 + ( y p ) 2 (4) And after some manipulation (see Supplementary Appendix SA), the overall pressure coefficient can be expressed as: C P = C P s + C P a d j (5) where C P s is the stationary pressure coefficient ( C P s = − V t o t a l 2 V θ m a x 2 ), C P a d j is the adjustment term for the pressure coefficient ( C P a d j = 2 ∂ φ ∂ t V θ m a x 2 ) and V t o t a l is the resultant velocity. After further manipulation the adjusted overall pressure coefficient can be expressed as (Supplementary Appendix SA): C P = − r ( x p − α V θ m a x   t ) 2 + y p 2 [ − 2 α − r ] (6) where r is the distance from vortex translation path (set as positive or negative depending on the vortex’s relative location to the x-axis). Thus, Eq. 6 represents the overall pressure coefficient as a function of vortex translating speed at any point. It should be noted that if the translating velocity is zero ( α = 0), then Eq. 6 at the radial distance of r c will give the same static pressure as the Rankine vortex model (1882) (see Supplementary Appendix SA). As noted in Eq. 6, the overall pressure coefficient consists of two terms. The adjustment term ( C p a d j ) includes a time derivative of the potential function, which is a function of the distance from the vortex centre and the velocity of vortex translation. Thus, C p a d j represents the effects arising from the movement of the vortex while C P s represents the effects due to the presence of the vortex. As a result, the magnitude of C p a d j is dependent on the speed of the vortex translation, whereas if the vortex has no translation (stationary vortex), the overall pressure is solely the magnitude of C P s .

The impact of vortex translation on the pressure field at point B is shown in Figure 2, where the vortex translates from x = −4 to 4. The translating velocity of α = 0.1 and 0.3 were used for comparison and values of r corresponding to +1 and −1. The black line in Figure 2 represents C P s , i.e., the pressure coefficient without the adjustment term induced by vortex translating motion. It is evident from Figure 2 that the contribution from C p a d j can be considerable and is a function of the relative translation speed of the vortex. For example, when α = 0.1 and t = 0 (where circulation centre is at x = 0 and C P is maximum), Eq. 6 reduces to + / − 2 α . Thus, the overall value of C P increases (decreases) by 20% (−20%) depending on the value (and sign) of r . However, it is significant to point out that when α = 0.3, the changes are observed to be ± 60%. Such drastic changes in static pressure magnitude could have important implications from a design perspective.

Results presented in this section have shown that vortex translation is important and can have significant impact on the static pressure field. It was demonstrated that the pressure adjustment, C p a d j , is dependent on the translating speed of the vortex; However, this only represents the static pressure at a particular point in space. Understandably, the superposition principle, which is valid in potential flow, could be somewhat questionable in real flows. Nevertheless, in what follows it will be shown that the potential flow framework is useful in terms of providing insight into the effects of vortex translation.

The previous section illustrated the potentially large changes in pressure which could occur at a point due to vortex translation. In order to conduct an extensive evaluation of the impact of tornado translation on the pressure field and overall force on a body, the flow around a two-dimensional body is considered. A thin symmetrical airfoil was chosen to represent the body as the streamlined geometric configuration should minimise flow separation. Additionally, the particular airfoil cases examined are well documented, and there are reliable analytical methods which have proven to be accurate for calculating the lift force on this shape (Mittal and Saxena, 2002; Akbari and Price, 2003; Hoarau et al., 2006; Hu and Yang, 2008; Liu et al., 2012; Al Mutairi et al., 2017); such comparisons need to be made with care as implicit in our analysis is the assumption that the flow is two-dimensional which is clearly not the case for low-rise structures. However, for the purposes of the current paper this assumption is considered reasonable. The configuration of the simulation is shown in Figure 3, which illustrates the plan view of the 2D ground plane of a fixed reference frame. Similar to the configuration considered in Section 2.1, the airfoil is placed at a distance of r from the translating path of the vortex with the leading edge of the airfoil positioned at the origin. The vortex rotates in the anti-clockwise direction with the circulation strength of Γ and translates in the direction parallel to the x-axis from − x to + x with the dimensionless translating speed α . The lift force in this context can be considered to be analogous to the side force of a 2-D body (e.g., low-rise structures) when viewed in plan; where positive lift denotes the force towards the circulation centre of the vortex while negative lift denotes the force away the circulation centre of the vortex. Given the symmetrical nature of the system, only the results relating to positive values of r will be discussed in what follows as it yields greater magnitude of lift force.

Several cases were devised, details of all cases explored in this study are listed in Table 1. Case 1 represents the vortex translation past a point in the flow field (as presented in Section 2.1). Case 2 investigates the impact of varying thickness of the airfoil by considering three airfoils with varying thickness, h / c = 0.12, 0.24 and 0.06 corresponding to Case 2a, Case 2b and Case 2c respectively. The chord length, c of the airfoil in each case is c = 1 with the distance to the vortex translating path centre of r = 2, resulting in a relative scale of c / r = 0.5. Case 3 explores the effects of scaling where varying airfoil chord lengths and distance to the vortex translating path are simulated and compared. For this case, three different distances from the vortex translating path where examined: r = 1, 2, and 4. In addition, three airfoils with varying chord lengths of c = 1, 0.5, and 0.25 denoted as Case 3a, Case 3b and Case 3c respectively were simulated. This resulted in relative scales of c / r = 1, 0.5 and 0.25 for Case 3a, . = 0.5, 0.25 and 0.125 for Case 3b and c / r = 0.25, 0.125, and 0.0625 for Case 3c.

The number of tornado occurrences based on the F scale (from 1950 to 2007) and EF scale (from 2007 to 2020) are shown in Figures 10A,B. Attributed to each section of Figures 10A,B are two different percentage values: the first percentage indicates the total number of records in each dataset corresponding to that scale, whilst the second percentage (the figure in brackets) corresponds to the amount of data at a particular scale which is complete and hence used in the analysis. For example, in Figure 10A, F1 scale data accounts for 34% of all of the F scale tornadoes in the data set but only 12% of these are complete and used in the analysis. In total, 4,830 and 16,967 tornadoes corresponding to F scale and EF scale tornadoes respectively have been considered below. Figure 10 indicates that in general, F0 and EF0 scale activity accounts for majority of the tornado occurrences (39 and 52% respectively), while tornado occurrences corresponding to the either F4/5 or EF4/5 are considerably less frequent (2% or less in both cases).

The mean translating velocity ( V t r a n s ) of each tornado is based on the simple calculation of translated distance/duration. Translated distance in this context is calculated using the start and end longitudes and latitudes and assumes a straight-line translation. Thus, the calculated distance travelled is rather arbitrary but for the purposes of the current analysis is considered suitable. The duration parameter is simply obtained from the difference between the end time of the tornado and the start time–both data are given in the database. Similar to the distance calculation, this may not be wholly accurate but, for the purposes of the current analysis, it has been assumed that the data in the database is sufficiently accurate. Figures 10C,D illustrates the calculated translating velocity against scale and suggest that, on average, there is relatively little variation in the translating velocity regardless of scale, i.e., average values ∼19 m/s (see Table 3). However, as indicated in Figures 10E,F significant variations exist within each scale with the possible exception of F/EF 5 data, which is likely due to an artifact from the lack of data (at high return periods); further, the trend as shown in Figures 10A,B suggests that this could simply be due to there being insufficient records in these categories making the drawing of meaningful conclusions rather difficult. Thus, while the actual translating speed appears to be independent of the scale of the tornado, it should be interpreted with care.

Understandably, with the improvement of tornado reporting and measurement, the more recent data (EF scale data) are significantly more comprehensive and contains less missing information. Thus, further analysis are conducted only using the EF scales. The typical ranges of wind speed that each EF scale covers were used to convert EF scales to actual wind speeds. For each scale, a simple average velocity ( V w i n d   ) has been obtained using the maximum and minimum values of the range; however, for EF5, V w i n d   is simply taken to be equivalent to the lowest 3 s gust velocity. Thus, for EF scales of 0, 1, 2, 3, 4, and 5, the average wind velocities are 33.5 m/s, 43.8 m/s, 54.9 m/s, 67.3 m/s 81.8 m/s, and 89.4 m/s respectively. Whilst this is highly inconsistent, it has already been noted above that the data in these categories are too sparse to enable any meaningful conclusions to be drawn. The normalising of the data in this way introduces an additional unknown, namely the difference between the mean values in a particular range and the two extremes of that range, where the uncertainties for the EF scales of 0, 1, 2, 3, 4, and 5 are 13, 12, 10, 10 and 9% respectively. Given the variability within each scale as shown in Figure 10, what follows should be interpreted with caution.

Figure 11A shows the normalised translating velocity, α   (where = V t r a n s     / V w i n d   ) of tornadoes of EF scale whilst Figure 11B illustrates the variation in the data. The translating velocity, α for EF scale 0, 1, 2, 3, 4, and 5 are 0.37, 0.39, 0.34, 0.28, 0.25, and 0.25 respectively, while the standard deviation for the corresponding EF scales are 0.3, 0.22, 0.17, 0.13 0.08, and 0.07 respectively. Figure 11A indicates a minor decrease in α as the scale increases, i.e., in relative terms the translation speed varies slightly with respect to tornado scale. However, this is expected as the average wind velocity increases with the increase in scale.

Notwithstanding the additional variability that has been introduced by non-dimenisonalising the data, with the similarity in the mean translating velocities (both the F and EF scales as presented in Figure 10), it is evident that the translating velocities of tornadoes are independent of scales, thus permitting all data to be combined. As a result, all data of the EF scale are combined (denoted as “combined scale”) and further analysed. Additionally, due to the lack of sufficient data for the scales with higher uncertainties (EF0, EF1, and EF 5) these data have been removed for the purpose of this analysis.

The mean translating velocity of the combined scale data is 18.8 m/s with the standard deviation of 10.57 m/s and the skewness and Kurtosis of 3.45 and 24.26 respectively, while the normalised mean translating velocity is 0.32 with the standard deviation of 0.17 and the skewness and Kurtosis of 2.68 and 19.48 respectively. In comparison, the dimensional results have marginally higher skewness as well as kurtosis in comparison with the normalised data. This difference can be attributed to the normalising process as tornadoes at each scales were normalised with the respective mean wind velocity, thus resulting in the difference in skewness and kurtosis. Additionally, an analysis was conducted using the 3-parameter Weibull probability density function. It was found that by using the shape parameter, β of 3, scale parameter, η of 31, and location parameter γ of 10, the Weibull probability density function fits the combined scaled data very well.

Overall, results presented in this section demonstrated that while the average wind velocity of tornadoes varies drastically depending on the magnitude of the respective scales (F and EF scale), the actual translating speed of tornadoes are independent of the scales. The mean translating velocity of tornadoes is 18.8 m/s with a normalised mean translating velocity of 0.32, but with the possibility of translating up to the normalised velocity of 0.37. Having established the relative range of translating speeds which occur in naturally occurring tornadoes, it is now possible to explore the effect that such a range may have on the generated pressure field and force coefficient.

Results presented in Section 2.2.3 have shown that by considering the flow angles at both the leading edge and the trailing edge, the maximum forces on an airfoil subjected to a moving vortex can be predicted using uniform flow with the appropriate range of inflow angles. However (as presented in Section 2.2.3), varying the chord lengths and distances from the vortex translating path will result in a range of flow angles; thus, an expression summarising the flow angle and the relative scale of the airfoil to the vortex is presented below.

Based on Figure 9, it can be observed that all three cases show similar trends, where the range of angles decreases with the decrease in scale. Thus, it is possible to fit a curve to obtain an expression containing the chord length and distance from the vortex translating path: Θ =   − 15.7   ( c / r ) 2   + 61.03 ( c / r ) − 0.05 (9) where Θ is the maximum difference in inflow angle between the leading edge and the trailing edge. It should be acknowledged that Eq. 9 does not contain any variable regarding the AoA in order to obtain the maximum lift on an airfoil. As noted above, the current analysis does not consider flow separation and as such Eq. 9 should be considered as conservative.

Initially, Case 1 which represents vortex translation past a point with no body present in the flow is explored. The maximum adjustments to the pressure coefficient, C p a d j at the point x = 0 and y = r with the vortex translating velocities of α = 0.1, 0.2, 0.3, 0.4 and 0.5 are shown in Figure 12A. Based on this figure, it can be observed that C p a d j increases linearly with the increase in translating velocity with a gradient of 2. Thus, in the overall pressure coefficient Eq. 5, the adjustment term, C p a d j is replaced with α , as a function of vortex translating velocity, as: C P =   C P s + 2 α (10) where C P s can be obtained using equation [A7]. The resulting expression as presented in Eq. 10 can be used to estimate the maximum overall pressure coefficient at point under varying vortex translating velocities. Next, in order to quantify the impact of vortex translation on the surface pressure coefficient on the airfoil, the mean value of C P a d j over the surface for all cases of Case 2 (Case 2a, 2b, and 2c) and Case 3a is calculated. It should be noted that to retain consistency, Case 3a is employed as the chord lengths are identical. Figure 12B shows the mean adjustments to the pressure coefficient on the airfoil when the vortex circulation centre is located where the maximum lift coefficient for the airfoil occurs, with varying airfoil thickness (Case 2) and varying relative scale (Case 3a). Figure 12B outlines the mean C P a d j of airfoil with varying thickness of h / c = 0.06, 0.12 and 0.24 with the vortex circulation centre at x / c = −1 and translating velocities of α = 0.1, 0.2, 0.3, 0.4, and 0.5. Figure 12C shows the mean C P a d j of airfoil with varying relative scales of c / r = 1, 0.5 and 0.25 with the vortex circulation centre at x / c = −2.2, −1, and 0 respectively (as shown in Figure 6A) and translating velocities of α = 0.1, 0.2, 0.3, 0.4, and 0.5.

As illustrated in Figure 12B, whilst increasing the thickness of the airfoil results in the increase in C P a d j , the translating velocity is the primary factor affecting C P a d j . All three cases show linear increase with α , with gradients of 1.32, 1.36, and 1.38 for h / c = 0.06, 0.12, and 0.24, respectively. Similarly, the adjustment term, C p a d j is replaced with α , and thus, the overall pressure coefficient can be rearranged as a function of airfoil thickness and vortex translating velocity as: C P =   C P s + ( 0.244   h / c   + 1.32 ) α (11)

Equation 11 illustrates that the overall pressure coefficient at a point is a function of translation speed and relative scale. This is an important finding since it suggests that adding the translation speed of a tornado to the maximum wind speed generated by the flow field will not result in an appropriate value for the pressure coefficient. Eq. 11 can be used to estimate the maximum overall pressure coefficient at point under varying vortex translating velocities. Additionally, the comparison of varying scales showed that the increase in relative airfoil size results in a considerable increase in C P a d j ; however, the major factor in the increase in adjustments can be attributed to the translating velocity of the vortex. All three scales show a linear increase with the α , with gradients of 1.64, 1.36, and 1.25 for the scales c / r = 1, 0.5 and 0.25, respectively. The overall pressure coefficient can be rearranged as a function of the relative airfoil scale to the vortex translating path and vortex translating velocity as: C P =   C P s + ( 0.526   c / r   + 1.1 ) α (12)

By combining Eqs 11, 12, it is possible to obtain a multi-parameter expression containing both the scale and airfoil thickness summarising the adjustment on an airfoil as: C P =   C P s + ( 0.244   h / c +   0.489   c / r   + 1.114 ) α (13)

Equation 13 can be used to estimate the maximum overall pressure coefficient on the airfoil under varying vortex translating velocities, where the distribution of C P s on the surface of the airfoil when the maximum lift coefficient occurs can be obtained using the range of AoA of Θ as discussed in Section 4.1. With that being said, these generalised equations are unable to summarise every geometrical configuration and is limited to only the symmetrical airfoil. This may be unrealistic for bluff bodies for the reasons outlined above. Moreover, by neglecting the wake effects, additional forces induced by drag are ignored; as demonstrated in Section 2.2.3, the high relative inflow angle as experienced by the airfoil far exceeds the critical angle of attack, which is undoubtably unrealistic and would result in the stalling of the airfoil. Therefore, further physical or numerical simulations should be conducted in order to validate these generalised equations; physical vortex generators with the capability to effectively model the translation of tornadoes, although restricted to lower translating speeds (approximately 0.5–2 m/s) (Razavi and Sarkar, 2018; Refan and Hangan, 2018) could validate the effects of tornado translation on pressure on the airfoil at lower translating speeds; concurrently, CFD simulations could also provide as an alternative via using sliding mesh to simulate the translating movement, thus simulating vortex translation at higher speeds. Notwithstanding the limitations, the findings presented in this study demonstrate the significant importance of considering vortex translation on the overall pressure coefficient.

Procedures to apply the proposed flow angles and pressure adjustments in Sections 4.1, 4.2 can be used in a number of ways:

• As a framework to reproduce the flow conditions of a moving tornado in which physical or numerical methods could be based upon–the force exerted on a (Civil Engineering) structure, and thus, wind load can be estimated using a typical boundary layer wind tunnel with the suggested range of flow angles, without the need to employ a vortex generator.

Identifying the relative size of the airfoil is, of course, the first step in calculating the flow angles, which includes the specification of the reference length (chord length) of the airfoil, c , and the distance to the tornado translating path, r . By using the relative scale of the airfoil to the vortex, c / r , a range of angle, Θ , representing the maximum difference in flow angle between the leading and trailing edge can be obtained using Eq. 9 (e.g., an airfoil with the chord length of c = 1 and the distance to the vortex translating path of r = 2, yielding with the relative scale of c / r = 0.25, the maximum difference in flow angle is approximately 14.2^o).

By employing a numerical or physical boundary layer wind tunnel, the angle of attack at which the maximum lift occurs can be obtained. The lift force in this context can be considered to be equivalent to the side force of traditional (Civil Engineering) structure when viewed in plan. It would be advisable also to simulate the additional range of +/- Θ ; for example, for a symmetrical airfoil at Reynolds number of 179,000, the maximum C L s of approximately 1.6 occurs at the AoA of approximately 10^o (Drela, 1989). Using the relative scale of c / r = 0.25, the additional AoA of −4.2^o and 24.2^o with the C L s of approximately −0.5 and 0.9 respectively, should also be considered. Whilst the magnitude of C L s simulated at the additional AoA are (expectedly) lower than the maximum C L s of 1.6, the corresponding drag coefficient, C D changes drastically, with the magnitude of approximately 0.007, 0.013 and >0.03 for −4.2^o, 10^o and 24.2^o respectively. Thus, highlighting the associated increase and decrease in force (lift or drag) that should be considered to fully account for the flow conditions as induced by a moving tornado. It is acknowledged that due to the limitations of the current methodology, the wake effects are neglected. However, this simple example demonstrates the significance of considering the effects induced by the additional angles of Θ . Additionally, for the calculation of force coefficient on the airfoil, the effects caused by the translating velocity is not considered as α was found to have very minimal effects on the overall force (as discussed in Section 2.2.2).

By conducting the simulation using physical/numerical boundary layer wind tunnel, the surface pressure distribution on the airfoil can be measured, which can be used as input to the variable of C P s in Eq. 13 as well as the specification of the airfoil thickness, h . The increase in surface pressure distribution was observed to increase considerably with the increase in translating velocity, α . As discussed in Section 3, it is evident that the translating velocity of tornadoes appears to be independent of the F and EF scales, with the average α of 0.32. Thus, using these input parameters, an airfoil with the relative scale of c / r = 0.25 and thickness of h / c = 0.12 will yield the adjustment of C P a d j = 0.4 to be added to every measured pressure point in order to account for the effects induced by the tornado translating movement, but can increase up to C P a d j = 0.52 with the translating velocity of α = 0.37. The resulting overall surface pressure distribution on the surface can be used as a reference to determine where large increase in pressure could occur, which may lead to the failure of cladding (or roof tiles) and require structural reinforcement.

Admittedly, this procedure of application as described, is a simplified approach; whilst the superposition principle is valid in potential flow, in real flow this could be somewhat questionable. Additionally, the flow is also assumed to be 2-D which is clearly not the case for a naturally occurring tornado or a physically simulated vortex; physical and numerical studies (Natarajan and Hangan, 2012; Gillmeier et al., 2017; Refan and Hangan, 2018) have shown that while the dominant flow component of tornado-like vortices is the tangential velocity, the vertical and radial velocity component could be up to the magnitude of approximately 35 and 58% of the tangential velocity respectively, therefore, the inclusion of the vertical and radial component of velocity could potentially affect the inflow angle drastically. With that being said, the findings presented in this study have shown that the framework is useful in terms of providing insight into the effects of vortex translation on the pressure field and the force on a body and providing a foundation upon which future simulation methods could be based.