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

Shallow-water equations (SWE) Gradients of dynamic pressure

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

Lord Rayleigh, 1914: Across Fr=1 Mass and momentum conserved Energy cannot be conserved If energy decreases, height MUST increase nndb.com dcwww.camd.dtu.dk/~tbohr/ Tomas Bohr and many, U 1,h 1 U 2,h 2

5 H1H1 H2H2 U1U1 U2U2 The inviscid description The transition from Fr > 1 to Fr < 1 cannot happen smoothly There 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

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

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

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

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

BLSWE

Velocity profiles Low Froude P solution Highly reversed. Very unstable

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

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

Simulations A circular hydraulic jump

u.edu/pororoca_ photos.html 81 Tidal bores Arnside viaduct Chanson, Euro. J. Mec B Fluids cle_id=45986&in_page_id=3 The pororoca: up to 4 m high on the Amazon

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

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

Effect of domain size Elliptic???

SWE always too gentle near jump

23 PHJ - Computations Non-hydrostatic effects

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

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

Singular perturbation problem

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

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

Undular region

Model of Johnson: Adhoc introduction of a viscous-like term, I-G+D + V 1 = 0. Our model for the undular region

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

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’’

Circular jump Fr N =7.5 Increasing Reynolds, weaker jump

Numerical solution: initial momentum flux matters

Effect of surface tension

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

Planar jumps – Effect of Reynolds Steeper jumps with decreasing Reynolds