Self-consistent analytical modelling of tornado-like vortex velocity and surface-pressure fields across swirl ratios (the Gillmeier–Sterling–Hemida–Baker benchmark)

Direct instrumentation of tornadoes is rarely possible, so wind engineers describe tornado wind fields with closed-form (analytical) vortex models. Gillmeier, Sterling, Hemida and Baker (2018) systematically compared the classical Rankine, Burgers–Rott and Sullivan models and the newer Baker model against laboratory measurements of a tornado-like vortex (University of Birmingham simulator) at three swirl ratios S = 0.14, 0.30 and 0.69, and found that none is physically self-consistent: the parameter sets that fit measured velocity components differ from those that fit the simultaneously measured surface pressure; no model reproduces the multi-cellular, height-dependent velocity structure; the Baker pressure solution exists only for the fixed shape parameter γ = 2 and so cannot adapt to different core radii and swirl ratios; and the radial outflow observed near the vortex centre at the lowest swirl ratio lacks a physical explanation. The subsequent literature has made genuine progress on analytical vortex modelling — unified models, two- and three-cell solutions, height-dependent viscous solutions, new exact self-similar solutions — but no published work generalizes the Baker pressure solution to free γ, reproduces the three-swirl-ratio surface pressures with one self-consistent analytical expression, or explains the low-swirl near-axis outflow. The problem is therefore formulated as open: construct an analytical vortex model that, with a single consistently assigned parameter set per flow condition, reproduces both the velocity field and the surface-pressure distribution of this laboratory benchmark across all three swirl ratios. The source-diagnosed limitations — the restriction of the Baker pressure solution to γ = 2, the unphysical far-field pressure trend of the Burgers–Rott model, and the unexplained near-axis radial outflow at the lowest swirl ratio — remain motivating context and consistency checks on any proposed model.

Background

Because direct measurement inside tornadoes is rarely feasible, wind engineers and atmospheric scientists have long represented tornado wind fields with analytical (closed-form) vortex models. The oldest is the Rankine vortex, a standard idealization with solid-body rotation inside a core radius and irrotational potential flow outside, which fixes only the tangential velocity and says nothing about radial or vertical motion or about pressure. The Burgers–Rott vortex is an exact steady solution of the Navier–Stokes equations, obtained with a constant effective kinematic viscosity and a vortex-stretching parameter, in which all three velocity components are non-zero; the Sullivan vortex extends this construction to a two-cell structure with a central downdraft. In both, the viscosity must be interpreted as an effective (eddy) viscosity inflated by orders of magnitude above the molecular value of air to obtain realistic velocities — one of several idiosyncrasies that practising engineers rarely acknowledge. Baker and Sterling (2017) derived an engineering-oriented model from an assumed radial velocity that is bounded in both radius and height, solving the Euler equations to obtain closed-form circumferential and vertical velocities and pressure, with one-cell and two-cell variants. Laboratory tornado simulators of Ward type (a fan-driven chamber with swirl vanes above a ground plane) generate tornado-like vortices whose structure is governed by the swirl ratio S, the standard dimensionless parameter measuring the rotation of the inflow relative to the through-flow: as S increases, the vortex progresses from a single-cell column, through vortex breakdown (an internal stagnation and reversal of the axial flow), to multi-cellular and two-cell forms. Gillmeier, Sterling, Hemida and Baker (2018) compared the four models against measurements of all three velocity components and of the surface pressure distribution of a laboratory tornado-like vortex at three swirl ratios, S = 0.14, 0.30 and 0.69. Their findings define the open problem: (i) the effective parameters required to match a given velocity component differ from those required to match the simultaneously measured surface pressure, so no model admits a self-consistent calibration; (ii) the idealized velocity fields cannot reproduce the multi-cellular, height-dependent radial and vertical velocity patterns seen in the experiments; (iii) the Rankine, Burgers–Rott and Sullivan pressure fields match the measurements only at the highest swirl ratio, and the Burgers–Rott pressure decreases without bound at large radius, an unphysical far-field trend; (iv) the Baker model's published pressure solution was derived only for the fixed shape parameter γ = 2 of its circumferential-velocity profile — which happens to fit the intermediate swirl ratio S = 0.30 — so the model cannot accommodate the different core radii and pressure-profile shapes of the other swirl ratios; and (v) the measured radial outflow near the vortex centre at the lowest swirl ratio, S = 0.14, lacks a physical explanation. A companion study by the same group (Gillmeier, Sterling and Hemida, 2019) confirms that surface pressure and velocity were measured in Birmingham simulators at S = 0.30 and S = 0.69 and stresses that simulator-based tornado modelling is far from a mature field.

Problem Statement

Construct an analytical (closed-form) model of a steady tornado-like vortex that is physically self-consistent and validated against the Gillmeier et al. (2018) laboratory benchmark. The model must supply the radial, circumferential and vertical velocity components and the surface pressure distribution as explicit functions of radius $r$ and height $z$, and, at each of the three benchmark swirl ratios $S=0.14$, $0.30$ and $0.69$, a single consistently assigned parameter set must reproduce both the measured velocity profiles — including the multi-cellular, height-dependent radial and vertical velocity structure observed in the experiments, beyond the classical single-cell and two-cell idealizations — and the measured surface-pressure distribution, not different parameter values for each fitted quantity. Agreement with the benchmark is judged within the experimental uncertainty of the measurements, and the model must follow from the governing equations of the flow under stated assumptions, with a fixed number of adjustable parameters that is not enlarged to accommodate individual measured quantities.

Two limitations diagnosed by Gillmeier et al. (2018) serve as motivating context and consistency checks on any proposed model, rather than as mandatory deliverables. First, the Baker model's published pressure solution exists only for the fixed shape parameter $\gamma=2$ of its circumferential-velocity profile; the source notes that treating $\gamma$ as a variable when deriving the static pressure distribution would allow a more flexible Baker-type surface-pressure model. A proposed model of the Baker family should address this generalization, but an admissible solution need not be a Baker-family generalization: any analytical model meeting the single-parameter-set benchmark across all three swirl ratios is admissible. Second, the measurements at $S=0.14$ show a radial outflow near the vortex centre — supported by the work of Mishra et al. (2008) and Haan et al. (2008), where tentative evidence of such an outflow can be inferred for low swirl ratios — for which the source states that various possible reasons, including vortex-core unsteadiness with respect to height, seem possible, but that "firm conclusions as to why this may be the case cannot, at present, be drawn". A proposed model should acknowledge this feature, and the larger transient-related measurement uncertainty of the central surface pressure at the lowest swirl ratio, and should be consistent with the measured fields; the model's velocity field need not reproduce the S=0.14 near-axis radial outflow — which the source attributes to possible vortex-core unsteadiness — and it is not required to deliver a quantitative steady-axisymmetric explanation of the outflow. The common objective is an analytical tornado-like-vortex representation that can replace per-component curve fits in wind-load and debris-flight applications while remaining checkable against the same laboratory benchmark that exposed the failures of the classical models.

References

1. Gillmeier, S., Sterling, M., Hemida, H., Baker, C.J. (2018). 'A reflection on analytical tornado-like vortex flow field models.' Journal of Wind Engineering and Industrial Aerodynamics 174, 10–27. DOI: 10.1016/j.jweia.2017.12.017. https://doi.org/10.1016/j.jweia.2017.12.017
2. Baker, C.J., Sterling, M. (2017). 'Modelling wind fields and debris flight in tornadoes.' Journal of Wind Engineering and Industrial Aerodynamics 168, 312–321. DOI: 10.1016/j.jweia.2017.06.017. https://doi.org/10.1016/j.jweia.2017.06.017
3. Burgers, J.M. (1948). 'A mathematical model illustrating the theory of turbulence.' Advances in Applied Mechanics 1, 171–199. DOI: 10.1016/S0065-2156(08)70100-5. https://doi.org/10.1016/S0065-2156(08)70100-5
4. Rott, N. (1958). 'On the viscous core of a line vortex.' Zeitschrift für angewandte Mathematik und Physik 9, 543–553. DOI: 10.1007/BF02424773. https://doi.org/1007/BF02424773
5. Sullivan, R.D. (1959). 'A two-cell vortex solution of the Navier-Stokes equations.' Journal of the Aerospace Sciences 26, 767–768. DOI: 10.2514/8.8303. https://doi.org/10.2514/8.8303
6. Gillmeier, S., Sterling, M., Hemida, H. (2019). 'Simulating tornado-like flows: the effect of the simulator's geometry.' Meccanica 54, 2385–2398. DOI: 10.1007/s11012-019-01082-4. https://doi.org/10.1007/s11012-019-01082-4
7. Baker, C., Sterling, M. (2019). 'Are Tornado Vortex Generators fit for purpose?' Journal of Wind Engineering and Industrial Aerodynamics 190, 287–292. DOI: 10.1016/j.jweia.2019.05.011. https://doi.org/10.1016/j.jweia.2019.05.011
8. Chen, G., Lombardo, F.T. (2019). 'An analytical pattern-based method for estimation of a near-surface tornadic wind field.' Journal of Wind Engineering and Industrial Aerodynamics 194, 103999. DOI: 10.1016/j.jweia.2019.103999. https://doi.org/10.1016/j.jweia.2019.103999
9. Honerkamp, R., Yan, G., Snyder, J.C. (2020). 'A review of the characteristics of tornadic wind fields through observations and simulations.' Journal of Wind Engineering and Industrial Aerodynamics 202, 104195. DOI: 10.1016/j.jweia.2020.104195. https://doi.org/10.1016/j.jweia.2020.104195
10. Pandey, S.K., Maurya, J.P. (2018). 'A Mathematical Model Governing Tornado Dynamics: An Exact Solution of a Generalized Model.' Zeitschrift für Naturforschung A 73, 753–766. DOI: 10.1515/zna-2018-0206. https://doi.org/10.1515/zna-2018-0206
11. Yadav, K., Maurya, J.P., Pandey, S.K. (2024). 'A generalized analytical viscous model for steady-state atmospheric vortices.' Physica Scripta 99, 075233. DOI: 10.1088/1402-4896/ad538b. https://doi.org/10.1088/1402-4896/ad538b
12. Pandey, S.K., Shruti (2024). 'An analytical model of two-cell tornadoes with emphasis on sharpness parameter affecting updrafts and a discussion on debris trajectory in outer cell.' Journal of Wind Engineering and Industrial Aerodynamics 253, 105862. DOI: 10.1016/j.jweia.2024.105862. https://doi.org/10.1016/j.jweia.2024.105862
13. Maurya, J.P., Shruti, Pandey, S.K. (2025). 'A modified 3D analytical viscous model of single-cell tornado-like atmospheric vortices.' Physica Scripta 100, 125223. DOI: 10.1088/1402-4896/ae24bc. https://doi.org/10.1088/1402-4896/ae24bc
14. Katsaounis, T., Mousikou, I., Tzavaras, A.E. (2024). 'Axisymmetric Flows with Swirl for Euler and Navier–Stokes Equations.' Journal of Nonlinear Science 34, article 86. DOI: 10.1007/s00332-024-10064-0 (preprint arXiv:2311.10575). https://doi.org/10.1007/s00332-024-10064-0
15. Kang, J., Wu, T. (2025). 'A unified model and analytical solution for wind fields associated with tornado vortices.' Physics of Fluids 37(9), 097160. DOI: 10.1063/5.0292147. https://doi.org/10.1063/5.0292147
16. Zhang, Z., Jin, R., Cai, Q., Liu, Y., Dong, K., Zhang, Y., Wang, B. (2026). 'Three-dimensional velocity field modeling of bounded vortex flows based on a three-cell structure.' Physics of Fluids 38(1), 015118. DOI: 10.1063/5.0302716. https://doi.org/10.1063/5.0302716