A Mathematical Model Governing Tornado Dynamics: An Exact Solution of a Generalized Model

Abstract

We investigate in this paper the dynamics of tornadoes by considering that the real inflow radial velocity depends on both the radial and vertical coordinates. The formulation is based on the model for the radial velocity that has been deduced from an experimentally verified model of azimuthal velocity. We present an analytical model for steady, incompressible, and viscous fluids and try for exact solutions. Although all the three components depend on radial and axial coordinates, viscosity affects merely the azimuthal velocity and the pressure. It is observed that the magnitude of the radial velocity increases to the maximum at the core but reverses the trend beyond and vanishes as it reaches the centerline. The magnitude reduces linearly with axial distance as per the supposition. At the core, the larger the Reynolds number, the lower is the velocity for moderate Reynolds numbers. Insignificant impact is observed for very large Reynolds number. However, inside and outside the core, the trends are reversed. Radial pressure distributions for different axial positions are similar to theoretical, numerical, and experimental observations. As we move outward from the axis, pressure increases. The difference between the pressures at the axis and that in outward regions increases with height. Pressure falls with rising Reynolds number uniformly for all radial distances. This is an indication that quantitative difference in pressure is large between viscous and inviscid flows.

A generalized analytical viscous model for steady-state atmospheric vortices

“…The region above d is the convective zone. Pandey and Maurya [3] considered the linear vertical profile of radial velocity as u r z , 1…”

“…In the existing models of atmospheric vortices, there are often idealisations such as neither radial nor axial components of velocity (Lamb [1], Oseen [2], Pandey and Maurya [3], Rankine [4], Taylor [5]) or radial velocity depending only on the radial coordinate (Long [6], Rott [7]), or a special form of viscosity Kieu and Zhang [8]) and so on. Steady-state models (Lamb [1], Oseen [2], Rankine [4], Taylor [5], Long [6], Rott [7], Kieu and Zhang [8], Burgers [9]) are often used for validation of observed experimental data and numerical simulation.…”