1. The Fluid Dynamics of Tornadoes Richard Rotunno NCAR NCAR is funded by the National Science Foundation Lecture 3: Numerical Simulations & Dynamics of The Two-Celled Vortex

2. Numerical Model of Ward Vortex Chamber …inadequate top boundary condition…  …difficult to model top boundary condition… Ward (1972, JAS)

3. Numerical Model of Ward Vortex Chamber …inadequate top boundary condition…  …difficult to model top boundary condition… Rotunno (1977, JAS)

4. Numerical Simulations of Ward Chamber S = 0.1S = 0.4S =1.0 Rotunno (1979, JAS)

5. Fiedler Chamber Fiedler (1994, Atmos-Ocn) Swirl RatioReynolds # Boundary condition

6. Nondimensional Governing Equations in Cylindrical Polar Coordinates velocity scale length scale buoyancy scale

7. Fiedler Chamber Fiedler (1994, Atmos-Ocn) Boundary condition Buoyancy force

8. Imposed Buoyancy Force and Grid r

9. r

10. Steady Solution v. Swirl Ratio

15. Two-Celled Vortex End-Wall Vortex End-Wall Boundary Layer Corner Flow Outer Flow

16. The Two-Celled Vortex

17. Fiedler Chamber Fiedler (1994, Atmos-Ocn) Boundary condition Boundary condition @ Eliminate lower boundary layer / zero-stress condition 

18. Free-Slip @ z=0 Two-Celled Vortex Steady Solution v. Swirl Ratio

19. Free-Slip @ z=0 Two-Celled Vortex Steady Solution v. Swirl Ratio

20. Free-Slip @ z=0 Two-Celled Vortex Steady Solution v. Swirl Ratio

21. Free-Slip @ z=0 Two-Celled Vortex Steady Solution v. Swirl Ratio

22. Free-Slip @ z=0 Two-Celled Vortex Steady Solution v. Swirl Ratio

23. r z Circulation Around the Domain Two-Celled Vortex At steady state and applying boundary conditions 

24. r z Conservation of Angular Momentum Thermodynamic Speed Core size Increases with Swirl Ratio Two-Celled Vortex: Balance of Buoyancy and Centrifugal Tendencies (Rankine Vortex)

25. Core Size and Max Velocity from Free-Slip Two-Celled Vortex Experiments

26. Imposed Buoyancy Force and Grid r

27. Two-Celled Vortex End-Wall Vortex End-Wall Boundary Layer Corner Flow Outer Flow

28. Summary 1.Fiedler Chamber also produces a realistic range of vortex structures 2.Free-slip conditions eliminate the surface boundary layer and corner flow  3.Two-celled vortex has its own distinct dynamics, essentially the classic mechanism of conservation of angular momentum