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Rama GovindarajanJawaharlal Nehru CentreBangalore
Work of Ratul Dasguptaand Gaurav Tomar
Are the shallow-water equations a good description atFr=1?
Hydrodynamic Instabilities (soon)AIM Workshop, JNCASR Jan 2011
Inviscid shallow-water equations (SWE)
Pressure: hydrostatic, since long wave
h
x
Fr<1
Fr >1
gh
uFr
Quh
Cghu
22
2
2
dx
dhguu
puu
x
)(
1
Lord Rayleigh, 1914: Across Fr=1
Mass and momentum conservedEnergy cannot be conservedIf energy decreases, height MUST increase
nndb.comdcwww.camd.dtu.dk/~tbohr/
Tomas Bohr and many, 2011 ...
U1,h1
U2,h2
5
H1
H2
U1
U2
The inviscid description
11
1
1
gH
UFr 1
2
22
gH
UFr
The transition from Fr > 1 to Fr < 1 cannot happen smoothlyThere has to be a shock at Fr = 1
1gHc Viscous SWE still used at Fr of O(1) and elsewhereIn analytical work and simulationsCan give realistic height profiles
Fr=1
x
h
Singha et al. PRE 2005 similarity assumptionparabolicno jump
Viscous SWE (vertical averaging) closure problem
32 )()()()(
cbaU
u
Watanabe et al. 2003, Bonn et al. 2009
Better model:Cubic Pohlhausen profile
yyx udx
dhguu )(
Re)(,)( xh
dxd
xh
y
Qf
Planar BLSWE
)/(,/Re
)(
0
2/32/1 hgQFrQ
udx
dhguu
u
yyx
+
EXACT EQUATION: solved as o.d.e.
fffffFr
hf
Re1
Re' 22
011100 ),(,),(,),( fff
11 ),( fIn addition
Dasgupta and RG,
Phys. Fluids 2010
Reynolds scales out
Downstream parabolic profile Upstream Watson, Gravity-free (1964)
h and f from same equationSimilarity solutions for Fr >> 1 and Fr << 1
0Re' 2 fhf
9
Drawback with the Pohlhausen model
Although height profiles good
Velocity-profile does not admit a cubic term
Planar – Height Profile
Velocity profile and h’: Functions only of Froude
`Jump’ without downstream b.c.!
Behaviour changes at Fr ~1
8141.Re' hUpstream
Circular – No fitting parameter
Near-jump region: SWE not good? need simulations of full Navier-Stokes
http://ponce.sdsu.edu/pororoca_photos.html
http://www.geograph.org.uk/photo/324581
Tidal bores Arnside viaduct
Chanson, Euro. J. Mec B Fluids 2009
http://www.metro.co.uk/news/article.html?in_article_id=45986&in_page_id=3
The pororoca: up to 4 m high on the Amazon
Navier-Stokes simulations – Circular and Planar
GERRIS by Stephane Popinet of NIWA, NewZealand
Circular: Yokoi et al., Ferreira et al. 2002
Planar Geometry
Note: very few earlier simulations
I - G + D + B + VS + VO = 0
BLSWE: I - G +VS = 0?Good when Fr > 1.5Good (with new N solution) when Fr < 0.8
KdV: I - G + D = 0
Fr ~1I ~ G, singular behaviour as in Rayleigh equation
The story so far
?0'')'')(( 2 UcU
take h’ large
hh /'/1
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 + Vo} = 0
No term from SWE at first two ordersGravity unimportant here!! (Except via asymptotic matching (many options))
Model of Johnson: Adhoc introduction of a viscous-liketerm, I-G+D + V1 = 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 separateCircular jumps of Type 0 and Type II-prime
Standard Type I
Type ``II-prime’’
Type ``0’’
Planar jumps – Effect of change of inlet Froude
Wave - breaking
Steeper jumps with decreasing Fr
As in Avedesian et al. 2000, experimentInviscid: as F increases, h2 increases
2i
Frh'