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Rama Govindarajan Jawaharlal Nehru Centre Bangalore

Are the shallow-water equations a good description at Fr=1?. Hydrodynamic Instabilities (soon) AIM Workshop, JNCASR Jan 2011. Rama Govindarajan Jawaharlal Nehru Centre Bangalore. Work of Ratul Dasgupta and Gaurav Tomar. Shallow-water equations (SWE). Gradients of dynamic pressure.

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Rama Govindarajan Jawaharlal Nehru Centre Bangalore

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  1. Are the shallow-water equations a good description at Fr=1? Hydrodynamic Instabilities (soon) AIM Workshop, JNCASR Jan 2011 Rama Govindarajan Jawaharlal Nehru Centre Bangalore Work of Ratul Dasgupta and Gaurav Tomar

  2. Shallow-water equations (SWE) Gradients of dynamic pressure

  3. Inviscid shallow-water equations (SWE) Pressure: hydrostatic, since long wave Fr<1 h Fr >1 x

  4. Lord Rayleigh, 1914: Across Fr=1 Mass and momentum conserved Energy cannot be conserved If energy decreases, height MUST increase nndb.com U2,h2 dcwww.camd.dtu .dk/~tbohr/ U1,h1 Tomas Bohr and many, 2011 ...

  5. H2 U2 H1 U1 The inviscid description The transition from Fr > 1 to Fr < 1 cannot happen smoothlyThere has to be a shock at Fr = 1 Viscous SWE still used at Fr of O(1) and elsewhere In analytical work and simulations Can give realistic height profiles

  6. Viscous SWE (vertical averaging) closure problem h Fr=1 Singha et al. PRE 2005 similarity assumption parabolic no jump x Better model: Cubic Pohlhausen profile Watanabe et al. 2003, Bonn et al. 2009

  7. Planar BLSWE + Dasgupta and RG, Phys. Fluids 2010 In addition EXACT EQUATION: solved as o.d.e.

  8. Reynolds scales out h and f from same equation Similarity solutions for Fr >> 1 and Fr << 1 Downstream parabolic profile Upstream Watson, Gravity-free (1964)

  9. Velocity-profile does not admit a cubic term Drawback with the Pohlhausen model Although height profiles good

  10. BLSWE

  11. Velocity profiles Low Froude P solution Highly reversed. Very unstable

  12. Planar – Height Profile Velocity profile and h’: Functions only of Froude `Jump’ without downstream b.c.! Behaviour changes at Fr ~1 Upstream

  13. Circular – No fitting parameter Near-jump region: SWE not good? need simulations of full Navier-Stokes

  14. A circular hydraulic jump Simulations

  15. Tidal bores Arnside viaduct http://www .geograph.org .uk/photo/324581 http://ponce.sdsu.edu/pororoca_photos.html http://www.metro.co.uk/news/article.html?in_arti cle_id=45986&in_page_id=3 The pororoca: up to 4 m high on the Amazon Chanson, Euro. J. Mec B Fluids 2009

  16. Motivation: gravity-free hydraulic jumps (Phys. Rev. Lett., 2007, Mathur et al.)

  17. Navier-Stokes simulations – Circular and Planar GERRIS byStephane Popinet of NIWA, NewZealand Planar Geometry Note: very few earlier simulations Circular: Yokoi et al., Ferreira et al. 2002

  18. Elliptic??? Effect of domain size

  19. SWE always too gentle near jump

  20. PHJ - Computations Non-hydrostatic effects

  21. J, Fr ~ 1 N, Fr < 1 P, Fr > 1 Typical planar jump U, Fr < 1

  22. The story so far I - G + D + B + VS + VO = 0 BLSWE: I - G +VS = 0? Good when Fr > 1.5 Good (with new N solution) when Fr < 0.8 Fr ~1 I ~ G, singular behaviour as in Rayleigh equation KdV: I - G + D = 0

  23. Singular perturbation problem

  24. WKB ansatz take h’ large Lowest order equation O(1) Either e is O(R-1) or jump is less singular. With latter

  25. Only dispersive terms contribute at the lowest order {Subset of D} = 0 At order e {Different subset of D + Vo} = 0 No term from SWE at first two orders Gravity unimportant here!! (Except via asymptotic matching (many options)) h’ need not always be large In fact planar always very weak e ~ O(1) or bigger! No reduction of NS

  26. Undular region

  27. Model of Johnson: Adhoc introduction of a viscous-like term, I-G+D + V1 = 0. Our model for the undular region

  28. Conclusions Exact BLSWE works well upstream multiple solutions downstream, N solution works well Behaviour change at Fr=1 for ANY film flow Planar jump weak, undular Different balance of power in the near-jump region gravity unimportant Undular region complicated viscous version of KdV equation

  29. Always separates, separation causes jump? ..... Analytical: circular jump less likely to separate Circular jumps of Type 0 and Type II-prime Standard Type I Type ``II-prime’’ Type ``0’’

  30. Circular jump FrN=7.5 Increasing Reynolds, weaker jump

  31. Numerical solution: initial momentum flux matters

  32. Effect of surface tension

  33. Planar jumps – Effect of change of inlet Froude Wave - breaking Steeper jumps with decreasing Fr As in Avedesian et al. 2000, experiment Inviscid: as F increases, h2 increases

  34. Planar jumps – Effect of Reynolds 12.5 25 47 Steeper jumps with decreasing Reynolds

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