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Flow Reversal in a Simple Dynamical Model of Turbulence

Flow Reversal in a Simple Dynamical Model of Turbulence. Kees Kuijpers, 0632305 Mariska van Rijsbergen, 0636290. Content. Introduction Shell models Pitchfork bifurcation GOY model Conclusions and further research. Introduction. Focus of the article: Rayleigh- Benard convection

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Flow Reversal in a Simple Dynamical Model of Turbulence

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  1. Flow Reversal in a Simple Dynamical Model of Turbulence Kees Kuijpers, 0632305 Mariska van Rijsbergen, 0636290

  2. Content Introduction Shell models Pitchfork bifurcation GOY model Conclusions and further research Flow Reversal in a Simple Dynamical Model of Turbulence

  3. Introduction Focus of the article: Rayleigh-Benard convection Research also useful for the magnetic polarity of the Earth Flow Reversal in a Simple Dynamical Model of Turbulence

  4. Shell models Simple model of turbulence Find a simplified version of the Navier-Stokes equations Shell models Flow Reversal in a Simple Dynamical Model of Turbulence

  5. From Navier-Stokes to the GOY-model The Navier-Stokes equation: The Fourier transformed Navier-Stokes equation is The GOY model is given by Flow Reversal in a Simple Dynamical Model of Turbulence

  6. Conservative quantities (1) Conservation of these quantities require f=0 and ν=0. The GOY model reduces to: Theorem of Noether: For each symmetry there is a corresponding conservation law Flow Reversal in a Simple Dynamical Model of Turbulence

  7. Conservative quantities (2) Conservation of energy Conservation of helicity Conservation of enstrophy Flow Reversal in a Simple Dynamical Model of Turbulence

  8. Conservation of energy The energy is given by: Conservation of energy requires: Taking a=1, and b=-eps then c=-(1+eps) Flow Reversal in a Simple Dynamical Model of Turbulence

  9. Conservation of enstrophy The enstrophy is given by: Conservation of enstrophy requires: Take bn=-eps=-5/4, and kn=2n k0then the enstrophy is conserved Flow Reversal in a Simple Dynamical Model of Turbulence

  10. Conservation of helicity The helicity is given by: Conservation of helicity requires: Take bn=-eps=-1/2, and kn=2n k0then the helicity is conserved Flow Reversal in a Simple Dynamical Model of Turbulence

  11. GOY shell model vs. Sabra shell model GOY shell model: a=1 and c=-(1+b) Sabra shell model: a=1 and c=(1+b) Flow Reversal in a Simple Dynamical Model of Turbulence

  12. Flow reversal in the model GOY shell model: Instead of a force, we use a different approach Flow Reversal in a Simple Dynamical Model of Turbulence

  13. Pitchfork bifurcation • Two possible standard solutions: • Subcritical bifurcation: • Supercritical bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  14. Our problem: Goy-model Imperfect parameter In our case we have a special form of the supercritical pitchfork bifurcation: Imperfect supercritical pitchfork bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  15. Our problem: Goy-model Introduce Rewriting We get the standard form with Flow Reversal in a Simple Dynamical Model of Turbulence

  16. Solutions imperfect supercritical bifurcation • For h=0 we get the same solution as in the normal supercritical bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  17. Solutions imperfect supercritical bifurcation • For h≠0 we get solutions depending on h: Flow Reversal in a Simple Dynamical Model of Turbulence

  18. Solutions imperfect supercritical bifurcation, with r constant Flow Reversal in a Simple Dynamical Model of Turbulence

  19. Calculation GOY model with bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  20. Calculation GOY model with bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  21. Calculation GOY model with bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  22. Calculation GOY model with bifurcation Flow Reversal in a Simple Dynamical Model of Turbulence

  23. Two state model In order to get a two state model, we introduced a pitchfork bifurcation: The quantity real(Φu1*) is the amount of energy transferred by mode u1 to smaller scales (u2, u3) Flow Reversal in a Simple Dynamical Model of Turbulence

  24. Find the average values for the upper and lower state (1) Assume: where Now, we can compute β: Our bifurcation formula: Combination of the last two equations: Flow Reversal in a Simple Dynamical Model of Turbulence

  25. Find the average values for the upper and lower state (2) Flow Reversal in a Simple Dynamical Model of Turbulence

  26. Conclusion • We can work with a simple model for the fluid motion • We can simulate: • A system with two stable solutions • The reversal of the flow Flow Reversal in a Simple Dynamical Model of Turbulence

  27. Further research • The influence of k0 needs to be investigated • The influence and physical meaning of b (-eps) • The reliability of the model • Is the energy really conserved? • Why do our solutions sometimes explode? Flow Reversal in a Simple Dynamical Model of Turbulence

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