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The vorticity equation and its applications

The vorticity equation and its applications. Felix KAPLANSKI Tallinn University of Technology feliks.kaplanski@ttu.ee Tallinn University of Technology, Estonia. Examples of vortex flows. Examples of vortex flows. Examples of vortex flows. VORTEX BREAKDOWN IN THE LABORATORY

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The vorticity equation and its applications

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  1. The vorticity equation and its applications Felix KAPLANSKI Tallinn University of Technology feliks.kaplanski@ttu.ee Tallinn University of Technology, Estonia

  2. Examples of vortex flows

  3. Examples of vortex flows

  4. Examples of vortex flows VORTEX BREAKDOWN IN THE LABORATORY The photo at the right is of a laboratory vortex breakdown provided by Professor Sarpkaya at the Naval Postgraduate School in Monterey, California. Under these highly controlled conditions the bubble-like or B-mode breakdown is nicely illustrated. It is seen in the enlarged version that it is followed by an S-mode breakdown.

  5. Examples of vortex flows VOLCANIC VORTEX RING The image at the right depicts a vortex ring generated in the crater of Mt. Etna. Apparently these rings are quite rare. The generation mechanism is bound to be the escape of high pressure gases through a vent in the crater. If the venting is sufficiently rapid and the edges of the vent are relatively sharp, a nice vortex ring ought to form.

  6. Examples of vortex flows

  7. Vortex ring flow

  8. Virtual image of a vortex ring flow www.applied-scientific.com/ MAIN/PROJECTS/NSF00/FAT_RING/Fat_Ring.html - Force acts impulsively

  9. Overview: • Derivation of the equation of transport of vorticity • Describing of the 2D flow motion on the basis of vorticity w and streamfunction y instead of the more popular (u,v,p)-system • Well-known solutions of the system (w, y )

  10. NSE

  11. Vorticity

  12. Vorticity transport equation

  13. Helmholtz equation

  14. Vorticity equation on plane 1) 2) 1 3 1 4 2)-1)= 3 2 4 2

  15. Taking into account: Continuity equation

  16. Vorticity equation on plane 4 3 3 4

  17. Cylindrical coordinate system In cylindrical coordinates (r , q ,z ) with -axisymmetric case

  18. Vorticity equation: axisymmetric case 1) 2) 1 1 4 3 2 1 2)-1)= 3 2 4 2 Proof with Mathematica

  19. Taking into account: Continuity equation

  20. Vorticity equation: axisymmetric case 3 4 3 4

  21. Vorticity transport equation for 2D : q=1- axisymmetric vortices, q=0 – plane vortices The Stokes stream function can be introduced as follows and gives second equation

  22. For 3D problem: generalized Helmholtz equation , , , where For 3D problem we can not introduce streamfuction Y like for 2D problem.

  23. For 3D problem: generalized Helmholtz equation in cylindrical coordinates , , , where

  24. For 2D problem: (u,v,p) (w, y) Winning: two variables instead of three Losses: difficulties with boundary conditions for streamfunction

  25. Vortex flow 2-D plane

  26. Streamfunction and (u, v) through vorticity and u, v are given by

  27. Solutions, which contain vorticity expressed through delta-functions

  28. Vortex flow.

  29. Vortex flow.

  30. The Biot-Savart Law

  31. Solutions, which contain vorticity expressed through delta-functions

  32. Vortex flow.

  33. Vortex flow.

  34. Other solutions

  35. Vortex flows. Hill’s ring

  36. We use polar coordinates (r, q)and assume symmetry

  37. Solution Further we define constant c and find solution Proof with Mathematica

  38. Appropriate tangent velocity

  39. Burgers vortex (a viscous vortex with swirl)

  40. Vorticity

  41. Irrotational Flow Approximation • Irrotational approximation: vorticity is negligibly small • In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation (Ex. 10-3)

  42. Irrotational Flow Approximation2D Flows • For 2D flows, we can also use the streamfunction • Recall the definition of streamfunction for planar (x-y) flows • Since vorticity is zero, • This proves that the Laplace equation holds for the streamfunction and the velocity potential

  43. Elementary Planar Irrotational FlowsUniform Stream • In Cartesian coordinates • Conversion to cylindrical coordinates can be achieved using the transformation Proof with Mathematica

  44. Elementary Planar Irrotational FlowsLine Source/Sink • Potential and streamfunction are derived by observing that volume flow rate across any circle is • This gives velocity components

  45. Elementary Planar Irrotational FlowsLine Source/Sink • Using definition of (Ur, U) • These can be integrated to give  and  Equations are for a source/sink at the origin Proof with Mathematica

  46. Elementary Planar Irrotational FlowsLine Source/Sink • If source/sink is moved to (x,y) = (a,b)

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