Tornado Debris Flight

What is modelled

The wind of a tornado is modelled as a Rankine vortex: solid-body rotation inside the core and a speed falling as 1/r outside it, with a slow inflow toward the axis and an updraft in the core. A piece of debris starting at rest feels the air drag from the wind it moves through, and gravity. Light pieces with a big area follow the wind quickly and can rise; heavy compact pieces lag behind.

Equations

• Wind vθ = V r/R inside, V R/r outside; inflow ur = −f vθ; updraft w = fw V e−(r/R)2.
• Motion m dv/dt = ½ ρ Cd A |vrel| vrel − m g.
• Response time τ = 2m / (ρ Cd A U), terminal speed √(2 m g / (ρ Cd A)).
• Cyclostrophic pressure drop in the core ρ V2.

Checks run in the page

• The circulation and pressure drop of the vortex.
• The speed of a body in a steady wind against its closed form.
• The same flight for pieces with the same m / (C_d A).
• The terminal speed in still air.

Honest simplifications

The vortex is steady and axisymmetric with no turbulence or translation, the debris is a point with one fixed drag coefficient (a tumbling board has a drag that changes with its orientation), air density is constant, the debris starts free and is not first torn loose from a building, and it stops on first contact with the ground. This is an illustration of the physics, not a damage or safety prediction.

• Press Timber board 2x4 and Roof sheet and compare how high each goes.
• Press Strong tornado, plywood and see the speed approach the peak wind.
• Raise the debris mass in the graph on the right and see the highest point fall.