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
Saint-Venant equationNumerical simulation
Discussion and Remarks
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
Saint-Venant equationNumerical simulation
Discussion and Remarks
Ocean wave
Surfing man (Source : http ://foundwalls.com)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Coastal wave
Sanur Beach Bali (Source :
http ://everythingspossible.wordpress.com/2010/10/24/sanur-bali/)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
River
The Ayung river Bali (Source : http ://tamanbebek.com)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Open channel flow
Irigasi system (Source : http ://www.tender-indonesia.com)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Atmosphere phenomena
El Nino wave (Source :http ://www.futura-sciences.com)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Tsunami, december 26, 2004 Sumatra, Indonesia(http ://tsun.sscc.ru)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
2D Saint-Venant equations1D Model of Saint-Venant1D Saint-Venant equationsDiscretization flows
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.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
2D Saint-Venant equations1D Model of Saint-Venant1D Saint-Venant equationsDiscretization flows
h + z sea level elevation
lateral velocity
x
u
h
z
Figure: The 1D model of SWE with undisturbed water depth d(x).
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
2D Saint-Venant equations1D Model of Saint-Venant1D Saint-Venant equationsDiscretization flows
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.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
2D Saint-Venant equations1D Model of Saint-Venant1D Saint-Venant equationsDiscretization flows
Discretization
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
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
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
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 gridzoomed
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Dam break
Figure: Comparison the velocity of all flux numerics in dam break dry bedwith 1000 grid
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Dam break
Figure: Comparison the velocity of all flux numerics in dam break dry bedwith 1000 grid zoomed
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Transcritical flow experiment
Figure: The experiment of Transcritical flow with shok by Emriver geomodelssource : http ://serc.carleton.edu/details/files/19076.html
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Transcritical flow
Figure: The simulation of Transcritical flow with shok in Staggered grid andSuliciu by Bouchut, 2004.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Model system of multi-layer SWE
Figure: The model of two layers
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
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.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Simulation 1 by Bouchut & Zeitlin (2009)
Figure: The initial condition
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Simulation 1 by Bouchut & Zeitlin (2009)
Figure: The interface at t = 0.5
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Simulation 2
Figure: Simulation using 10 layers with 1700kg/m2 diff density for eachlayers
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Mangroves
(photo : Bahama Bob Leonard)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
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.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
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
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Simulation using Suliciu flux
Figure: The result of effect mangrove in estuarine with Suliciu relaxation scheme
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Simulation using Staggered grid
Figure: The result of effect mangrove in estuarine with staggerd grid scheme
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Droplet on bassin flat bottom
Figure: The flat botom simulation
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
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
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Dam break in 2D with topography
The collapse of the St. Francis dam in California back in the1920’s. (http ://rivercityrevolution.wordpress.com)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Dam break in 2D with topography
The skecth for partial dam breach or instataneous opening of sluicegates by Fennema 1990.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
The simulation
Figure: (a). Simulation by Fennema 1990, (b). Simulation usingStaggered grid
(a) (b)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Planar Surface in a parabolid
(http ://www.consumerwarningnetwork.com)
Figure: Wine in a glass
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
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 .
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Numerical simulation 2D
Figure: Planar surface in parabolid
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Slice view in 1D
Figure: Planar surface in parabolid in y = 2
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Dry wet 2D with topography
Figure: Simulation with topography
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Coriolis force
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Dam breakTranscritical flowMulti-layer EqsMangroves frictionDroplet on bassin flat bottomDam break in 2DPlanar Surface in a parabolidCoastal wave simulationAtmosphere simulation
Equatorial wave (El nino, Rosbby, Kelvin wave)
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models
OutlineIntroduction
Saint-Venant equationNumerical simulation
Discussion and Remarks
Discussion and Remarks
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.
Putu Harry Gunawan Numerical method for Saint-Venant equations and related models