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

Shallow-water equations (SWE)

2/''

/'

hh

hh

h

hg

puuut

u

12

Gradients of dynamic pressure

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

fhFr

fffffhRe

1'

1'

22

SVGI

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

8141.Re' hUpstream

Circular – No fitting parameter

Near-jump region: SWE not good? need simulations of full Navier-Stokes

Simulations

A circular hydraulic jump

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

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. 2002

Planar Geometry

Note: very few earlier simulations

Effect of domain size

Elliptic???

SWE always too gentle near jump

23

PHJ - ComputationsNon-hydrostatic effects

P, Fr > 1

N, Fr < 1J, Fr ~ 1

Typical planar jump

U, Fr < 1

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

Singular perturbation problem

....''''Re

1 hDGI

/' hh

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))

Undular region

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’’

Circular jump FrN=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, experimentInviscid: as F increases, h2 increases

2i

Frh'

Planar jumps – Effect of Reynolds

Steeper jumps with decreasing Reynolds

12.5

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