A Simple Theory for Fire Whirls

1. A Simple Theory for Fire Whirls Nilton O. Renno Associate Professor University of Michigan nrenno@umich.edu

2. Background • Fire whirls can be catastrophic because: • Rotation inhibits mixing producing tall and extremely hot flames • Tangential winds produced by fire whirls can easily reach tornadic intensity • Intense tangential winds and high temperatures can produce fire blow-ups • Fire whirls can produce erratic and unpredictable propagation of forest fires • Fire whirls can be extremely dangerous to firefighters as evidenced by the 2001 Fish Fire Environmental Fluid Dynamics

3. Where are fire whirls more likely to form? Environmental Fluid Dynamics

4. …Near a fire boundary Environmental Fluid Dynamics

5. Fire whirls and fire fingers Environmental Fluid Dynamics

6. Fire whirl theory • A simple theory for fire whirls can be derived starting with the: • Energy equation • First law of thermodynamics • Ideal gas law Environmental Fluid Dynamics

7. Energy equation Taking the dot product of the velocity vector with the equation of motion and assuming steady state, we get (1) Environmental Fluid Dynamics

8. First law of thermodynamics The first law of thermodynamics applied to moist air states that (2) Environmental Fluid Dynamics

9. Ideal gas law Environmental Fluid Dynamics

10. Fire whirl and fire plume Environmental Fluid Dynamics

11. The idealized circulation Environmental Fluid Dynamics

12. Integrating the first law of thermodynamics across the steady-state circulation, we get (3) which shows that the net heat input is equal to the work done by the circulation. Environmental Fluid Dynamics

13. Integrating the energy equation over the idealized circulation and plugging equation (3), we get (4) which shows that in steady-state, the net heat input balances the work by friction. Environmental Fluid Dynamics

14. Assuming flat bottom Integrating the energy equation (1) from the radius of influence to a stagnation point at the center of the vortex, we get (5) Environmental Fluid Dynamics

15. Defining Equation (5) can be written as (6) Environmental Fluid Dynamics

16. Defining the thermodynamic efficiency of a steady-state convective heat engine as (7) Environmental Fluid Dynamics

17. Equation (6) can be written as (8) Environmental Fluid Dynamics

18. Using the ideal gas law, we get Environmental Fluid Dynamics

19. where Thus, the pressure drop across a fire whirl is Environmental Fluid Dynamics

20. Thermodynamic efficiency It follows from equation (7) that the thermodynamic efficiency of the fire whirl can be written as where H is the fire plume depth. This indicates that fire whirls (and fire blow-ups) are more likely to occur in regions of deep convective layer, even when the ambient wind is calm. Environmental Fluid Dynamics

21. Thus, there is a positive feedback between heat input and the fire whirl intensity via its tangential windspeed. Assuming cyclostrophic balance at the radius of maximum wind, we can write Environmental Fluid Dynamics

22. Angular momentum budget The angular momentum per unit mass is Neglecting the Coriolis acceleration term at the radius of maximum wind, we get Environmental Fluid Dynamics

23. Vortex radius Finally, we get Environmental Fluid Dynamics

24. A large fire whirl Environmental Fluid Dynamics

25. A storm forced by a fire Environmental Fluid Dynamics

26. Conclusions • Our theory predicts that: • Fire whirls are more like to form at fire boundaries • Fire whirls are more intense and blow-ups more likely to occur when the convective layer is deep, even if the ambient wind is calm • A positive feedback between fire whirls and heat flux that can produce extremely hot flames • A quantitative test of our theory is underway Environmental Fluid Dynamics