Numerical method for Saint-Venant equations and related modelsberstel/Mps/Afides/AFIDES2014/Gunawan.pdf · Outline Introduction Saint-Venant equation Numerical simulation Discussion

OutlineIntroduction

Saint-Venant equationNumerical simulation

Discussion and Remarks

Numerical method for Saint-Venant equations andrelated models

Putu Harry Gunawan

Encadre par Robert Eymard et David Doyen au

Laboratoire d’Analyse et de Mathematiques Appliquees, UPEM

23 Octobre 2013

Putu Harry Gunawan Numerical method for Saint-Venant equations and related models

OutlineIntroduction



1 Outline2 Introduction3 Saint-Venant equation

2D Saint-Venant equations1D Model of Saint-Venant1D Saint-Venant equationsDiscretization flows

4 Numerical simulationDam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation

5 Discussion and RemarksPutu Harry Gunawan Numerical method for Saint-Venant equations and related models

OutlineIntroduction



Ocean wave

Surfing man (Source : http ://foundwalls.com)


OutlineIntroduction



Coastal wave

Sanur Beach Bali (Source :

http ://everythingspossible.wordpress.com/2010/10/24/sanur-bali/)


OutlineIntroduction



River

The Ayung river Bali (Source : http ://tamanbebek.com)


OutlineIntroduction



Open channel flow

Irigasi system (Source : http ://www.tender-indonesia.com)


OutlineIntroduction



Atmosphere phenomena

El Nino wave (Source :http ://www.futura-sciences.com)


OutlineIntroduction



Tsunami, december 26, 2004 Sumatra, Indonesia(http ://tsun.sscc.ru)


OutlineIntroduction




2D Shallow water system : Equations

ht + (hu)x + (hv)y = 0

(hu)t +

(hu2 +

1

2gh2

)x

+ (huv)y = −ghzx

(hv)t + (huv)x +

(hu2 +

1

2gh2

)y

= −ghzy

where h is the height of watersurface, g is a gravitationalaceleration, z is topographyand u,v are the velocities indirection x and y repectively.


OutlineIntroduction




h + z sea level elevation

lateral velocity

x

u

h

z

Figure: The 1D model of SWE with undisturbed water depth d(x).


OutlineIntroduction




1D Shallow water system : Equations

ht + (uh)x = 0

(hu)t + (hu2)x +1

2gh2

x = −ghzx ,

where h is water hight, u is the velocity of water, z istopography/bathymetry, and g is gravitational aceleration.


OutlineIntroduction




Discretization


OutlineIntroduction



Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation

Dam break

Figure: Comparison all flux numerics in dam break wet bed with 1000 grid


OutlineIntroduction




Dam break

Figure: Comparison all flux numerics in dam break wet bed with 1000 gridzoomed


OutlineIntroduction




Dam break

Figure: Comparison the velocity of all flux numerics in dam break dry bedwith 1000 grid


OutlineIntroduction




Dam break

Figure: Comparison the velocity of all flux numerics in dam break dry bedwith 1000 grid zoomed


OutlineIntroduction




Transcritical flow experiment

Figure: The experiment of Transcritical flow with shok by Emriver geomodelssource : http ://serc.carleton.edu/details/files/19076.html


OutlineIntroduction




Transcritical flow

Figure: The simulation of Transcritical flow with shok in Staggered grid andSuliciu by Bouchut, 2004.


OutlineIntroduction




Model system of multi-layer SWE

Figure: The model of two layers


OutlineIntroduction




The system

By Bouchut & Zeitlin (2009)

∂thj + ∂x(hjuj) = 0

∂t(hjuj) + ∂x(hju2j + gh2

j /2) + ghj∂x

z +∑k>j

hk +∑k<j

ρkρj

hk

= 0

where hj ≥ 0, j = 1, · · · ,m are the fluid depth with m layers, uj arethe velocities, and z(x) is the topography. The constants g ,

0 < ρ1 ≤ · · · ≤ ρmare the gravity and the densities of the fluids respectively.


OutlineIntroduction




Simulation 1 by Bouchut & Zeitlin (2009)

Figure: The initial condition


OutlineIntroduction




Simulation 1 by Bouchut & Zeitlin (2009)

Figure: The interface at t = 0.5


OutlineIntroduction




Simulation 2

Figure: Simulation using 10 layers with 1700kg/m2 diff density for eachlayers


OutlineIntroduction




Mangroves

(photo : Bahama Bob Leonard)


OutlineIntroduction




Model system of mangroves

Assuming mangrove as a rigid porous media. In a porous media, dynamicequation are :

ht + (uh)x = 0

(hu)t + (hu2)x +1

2gh2

x = gh(Sf − zx),

where the term Sf denotes the drag force per unit mass due to mangrovefriction. In this equations, we use the Manning’s friction law as the drag forceper unit mass due to mangrove friction. The Manning’s friction law is given as :

Sf = −Cf u|u|h4/3

= −Cf q|q|h10/3

with Cf = n2m is drag coefficient, nm is denote Manning coefficient and q = hu

is the discharge.


OutlineIntroduction




SimulationThis simulation using genarate wave paddle given as,

h(0,0) =

∣∣∣∣A0 sin

(π

Tp

)∣∣∣∣ ,and

u(0,0) = 2

∣∣∣∣A0 sin

(π

Tp

)∣∣∣∣ .

Figure: Tsunami model in Mangrove area


OutlineIntroduction




Simulation using Suliciu flux

Figure: The result of effect mangrove in estuarine with Suliciu relaxation scheme


OutlineIntroduction




Simulation using Staggered grid

Figure: The result of effect mangrove in estuarine with staggerd grid scheme


OutlineIntroduction




Droplet on bassin flat bottom

Figure: The flat botom simulation


OutlineIntroduction




Dam break in 2D flat bottom

Figure: The dry dam break (left column) and wet dambreak (rightcolumn) Putu Harry Gunawan Numerical method for Saint-Venant equations and related models

OutlineIntroduction




Dam break in 2D with topography

The collapse of the St. Francis dam in California back in the1920’s. (http ://rivercityrevolution.wordpress.com)


OutlineIntroduction




Dam break in 2D with topography

The skecth for partial dam breach or instataneous opening of sluicegates by Fennema 1990.


OutlineIntroduction




The simulation

Figure: (a). Simulation by Fennema 1990, (b). Simulation usingStaggered grid

(a) (b)


OutlineIntroduction




Planar Surface in a parabolid

(http ://www.consumerwarningnetwork.com)

Figure: Wine in a glass


OutlineIntroduction




Planar Surface in a parabolid eqs

The topography in a paraboloid shape is defined by,

z(x ,y) = −h0

(1− (x − L/2)2 + (y − L/2)2

a2

)for each (x ,y) in [0; L]× [0; L], where h0 is the water depth at central point and a isthe distance from the central point to the zero elevation of the shoreline. Theanalytical periodic solution of this test is given by,

h(x ,y ,t) = max(

0, ηh0

a2

(2(x − L

2

)cos(ωt) + 2

(y − L

2

)sin(ωt)− η

)− z(x ,y)

)u(x ,y ,t) = −ηω sin(ωt)

v(x ,y ,t) = ηω cos(ωt)

where the frequency ω is defined as ω =√

2gh0/a. In thissimulation, we take t = 0 at the analytical solution as initialcondition and use the parameters a = 1, h0 = 0.1, η = 0.5, L = 4and T = 3 2π

ω .


OutlineIntroduction




Numerical simulation 2D

Figure: Planar surface in parabolid


OutlineIntroduction




Slice view in 1D

Figure: Planar surface in parabolid in y = 2


OutlineIntroduction




Dry wet 2D with topography

Figure: Simulation with topography


OutlineIntroduction




Coriolis force


OutlineIntroduction




Equatorial wave (El nino, Rosbby, Kelvin wave)


OutlineIntroduction




In this presentation, we introduced the implementations of descretizationof staggered grid scheme in fluid dynamics area.The applicability of staggered grid scheme to rapidly varied flow is robust(guarantee mass conservation, non-negative water level, and correctmomentum balance).Staggered grid scheme has more advantages than collocated grid scheme(i.e simple implementation with various large-scale simulation and complexsystems, Not need solving using Riemann solver, More accuracy, etc see(Stelling 2006)).Staggered grid scheme might need more unknown variables, but with themodern programming language technique, it is not a big problem.


OutlineIntroduction




Documents

Numerical method for Saint-Venant equations and related modelsberstel/Mps/Afides/AFIDES2014/Gunawan.pdf · Outline Introduction Saint-Venant equation Numerical simulation Discussion