Towards 2D overland flow simulations Olivier Delestre Laboratory - - PowerPoint PPT Presentation

Towards 2D overland flow simulations

Olivier Delestre

Laboratory J.A. Dieudonn´ e & Polytech Nice Sophia University of Nice – Sophia Antipolis

CEMRACS 2013

Problem context

Preventing overland flow and erosion From upstream...

(Photos : Yves Le Bissonnais, INRA)

...to downstream.

Downstream zones modifications (watersheds)

◮ Where is the water coming from ? ◮ Where is it flowing ?

Use of physical models is required to :

◮ simulate flow (volumes and location) ◮ suggest changes (grass strip).

to carry out improvements

Shallow Water (Saint-Venant) system

y x z h u v z, z + h O

• u

Data : topography z, rain P, infiltration I Unknowns : velocities u, v, water height h    ∂th + ∂x (hu) + ∂y (hv) = P − I ∂t (hu) + ∂x

• hu2 + gh2/2
• + ∂y (huv) = gh(−∂xz − Sf x)

∂t (hv) + ∂x (huv) + ∂y

• hv 2 + gh2/2
• = gh(−∂yz − Sf y)

Strategy

◮ Properties of the 1D Shallow Water system ◮ Choice of the method depending on the properties ◮ Validation : analytical solutions and laboratory experiment ◮ Application : field data

1D Shallow Water system

a b x O q(t, a) q(t, b) P(t, x) I(t, x) z, z + h

Data : topography z, rain P, infiltration I Unknowns : velocities u, water height h A system of conservation laws ∂th + ∂x(hu) = P − I ∂t(hu) + ∂x(hu2 + gh2/2) = gh (−∂xz − Sf ) (1)

System properties (I) : Hyperbolicity

Setting q = hu U = h q

• , F(U) =
• q

q2/h + gh2/2

• , B =
• P − I

gh (−∂xz − Sf )

• ,

compact form ∂tU + ∂xF(U) = ∂tU + F ′(U)∂xU = B, Hyperbolicity if h > 0 : λ−(U) = u −

• gh,

λ+(U) = u +

• gh

Saint-Venant gaz dynamic Froude number Fr = |u| c Mach number |u| c c = √gh free surface waves celerity c =

• p′(ρ) sound speed 1

subcritical Fr < 1 subsonic supercritical Fr > 1 supersonic

• 1. p(ρ) = ρRT perfect gaz

System properties (II) : Conservation laws

Integral of equation (1) in x ∂th + ∂xq = P − I, gives d dt b

a

h(t, x) dx = q(t, a) − q(t, b) + b

a

P(t, x) − I(t, x)dx, Mass conservation of water. Second equation : momentum equation

System properties (III) : Steady states

System properties (III) : Steady state

∂th + ∂x(hu) = P − I ∂t(hu) + ∂x(hu2 + gh2/2) = gh (−∂xz − Sf ) (2) ∂th = ∂tu = ∂tq = 0 ∂xhu = P − I ∂x(hu2 + gh2/2) = gh (−∂xz − Sf ) .

System properties (III) : Steady states

Lac at rest equilibrium u = 0 g(h + z) = Cst . z, z + h O x Hsur = z + h = Cte

Numerical method (I)

Finite volume method

xi−1 xi−1/2 xi xi+1/2 x ∆x tn+1 O t tn ∆tn xi+1

We integrate ∂tU + ∂xF(U) = 0

• n the volume

[tn, tn+1[×]xi−1/2, xi+1/2[, and we set Un

i =

1 ∆x xi+1/2

xi−1/2

U(tn, x) dx we get Un+1

i

= Un

i − ∆t

∆x

• F n

i+1/2 − F n i−1/2

• ,

with the interface flux approximation F n

i+1/2 = F(Un i , Un i+1) ∼

1 ∆t tn+1

tn

F(U(t, xi+1/2)) dt.

Numerical method (I)

◮ For each choice of F(UG, UD) we have a different finite volume scheme :

HLL, kinetic, Rusanov, VFRoe-ncv, suliciu, ...

◮ second Order

◮ in space : MUSCL, ENO, modified ENO ◮ in time : Heun

Numerical method (I)

◮ For each choice of F(UG, UD) we have a different finite volume scheme :

HLL, kinetic, Rusanov, VFRoe-ncv, suliciu, ...

◮ second Order

◮ in space : MUSCL, ENO, modified ENO ◮ in time : Heun

◮ Coupling with the source term (topography ∂xz)

Necessity : compatibility with steady states

Steady states (II)

∂th + ∂x(hu) = 0 ∂t(hu) + ∂x(hu2 + gh2/2) = −gh∂xz (3) ∂th = ∂tu = ∂tq = 0 hu = Cst u2/2 + g(h + z) = Cst . We consider u = Cst g(h + z) = Cst .

Hydrostatic reconstruction (II) [Audusse et al., 2004]

We define z∗ = max(zG, zD) and    U∗

G = (h∗ G, h∗ GuG), U∗ D = (h∗ D, h∗ DuD)

h∗

G = max(hG + zG − z∗, 0)

h∗

D = max(hD + zD − z∗, 0)

. Thus, we have        FG(UG, UD, ∆Z) = F(U∗

G, U∗ D) +

• g(hG

2 − (h∗ G)2)/2

• FD(UG, UD, ∆Z) = F(U∗

G, U∗ D) +

• g(hD

2 − (h∗ D)2)/2

• ,

where F(UG, UD) is the numerical flux.

Friction treatment

Shallow Water system with friction f ∂th + ∂x(hu) = 0, ∂t(hu) + ∂x(hu2 + gh2/2) + h∂xz = −hf , (4) f = f (h, u) friction force (on the bottom) Several friction laws possible

◮ Manning :

f = n2 u|u| h4/3

◮ Darcy-Weisbach :

f = F u|u| 8gh

Friction treatment

◮ Apparent topography [Bouchut, 2004]

We consider : zapp = z + bn with ∂xbn = Sn

f

Friction treatment

◮ Apparent topography [Bouchut, 2004]

We consider : zapp = z + bn with ∂xbn = Sn

f ◮ Semi-implicit [Bristeau and Coussin, 2001]

qn+1

i

+ F |qn

i |qn+1 i

8hn

i hn+1 i

∆t = qn

i + ∆t

∆x

• Fi+1/2 − Fi−1/2
• with qn+1∗

i

for the right part, we have qn+1

i

= qn+1∗

i

1 + ∆t F|un

i |

8hn+1

i

Validation on analytical solutions – SWASHES

New test cases :

◮ Saint-Venant/shallow water :

◮ data z ◮ unknowns h et u (and so q)

Validation on analytical solutions – SWASHES

New test cases :

◮ Saint-Venant/shallow water :

◮ data z ◮ unknowns h et u (and so q)

◮ test cases

◮ data h and q (and so u) ◮ unknown z

Validation on analytical solutions – SWASHES

New test cases :

◮ Saint-Venant/shallow water :

◮ data z ◮ unknowns h et u (and so q)

◮ test cases

◮ data h and q (and so u) ◮ unknown z

◮ Several possibilities

◮ several friction laws ◮ diffusion source term [Delestre and Marche, 2010] ◮ rain source term

• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

• 16
• 14
• 12
• 10
• 8
• 6
• 4
• 2

2 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

Validation on analytical solutions – SWASHES

Apparent topography (subcritical-subcritical)

• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

Validation on analytical solutions – SWASHES

Semi-implicit (subcritical-subcritical)

• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

Validation on analytical solutions – SWASHES

Semi-implicit (subcritical-supercritical)

• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

Validation on analytical solutions – SWASHES

Semi-implicit (supercritical-subcritical)

• 6
• 5
• 4
• 3
• 2
• 1

1 200 400 600 800 1000 z, z+h (m) x (m) topographie surface libre niveau critique

Summary of the chosen numerical method

◮ Numerical flux : HLL ◮ Second order scheme : MUSCL ◮ Friction : semi-implicit treatment ◮ Shallow Water system with rain P

∂th + ∂x(hu) = P ∂t(hu) + ∂x(hu2 + gh2/2) + h∂xz = −hf (5) time splitting/explicit treatment

Validation on experiments – INRA rain simulator

Settings of the experiment

x O z L q(t, L) Lc = 4 m P(t) 5 cm

0 ≤ t ≤ 250s R(x, t) = 50 mm/h if (x, t) ∈ [0, L] × [5, 125] 0 else

Analytical solutions and simulations

1 2 3 4 5 6 7 50 100 150 200 250 q (g/s) Temps de simulation (s) f=0.12, numerique f=0.12, cinematique f=0.12, exact f=0.34, numerique f=0.34, cinematique f=0.34, exact

Water height and velocity at equilibrium

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.5 1 1.5 2 2.5 3 3.5 4 h (m) x (m) f=0.12, numerique f=0.12, cinematique f=0.12, exact f=0.34, numerique f=0.34, cinematique f=0.34, exact 0.02 0.04 0.06 0.08 0.1 0.12 0.5 1 1.5 2 2.5 3 3.5 4 u (m/s) x (m) f=0.12, numerique f=0.12, cinematique f=0.12, exact f=0.34, numerique f=0.34, cinematique f=0.34, exact

What about reality ?

t (s)

9 8 7 6 5 4 3 2 1 0 0 50 100 150 200 250

q(.,L) (g/s)

”Calibration”

Darcy-Weisbach Manning

1 2 3 4 5 6 7 8 9 50 100 150 200 250 q (g/s) Temps de simulation (s) Mesures f=0.1 f=0.11 f=0.12 f=0.13 f=0.14 f=0.15 f=0.16 f=0.17 f=0.18 1 2 3 4 5 6 7 8 9 50 100 150 200 250 q (g/s) Temps de simulation (s) Mesures n=0.008 n=0.009 n=0.01 n=0.011 n=0.012 n=0.013 n=0.014 n=0.015 n=0.016

A simulation result (Manning)

1 2 3 4 5 6 7 8 9 50 100 150 200 250 q (g/s) Temps de simulation (s) Mesures n=0.013

Parcels in Niger ([Esteves et al., 2000], IRD)

Darcy-Weisbach

20 40 60 80 100 120 140 1000 2000 3000 4000 5000 P (mm/h) t (s) Hyetogramme 20 40 60 80 100 120 140 1000 2000 3000 4000 5000 q (mm/h) t (s) Hydrogramme Full SWOF Mesure ERO

Manning

20 40 60 80 100 120 140 1000 2000 3000 4000 5000 P (mm/h) t (s) Hyetogramme 20 40 60 80 100 120 140 1000 2000 3000 4000 5000 q (mm/h) t (s) Hydrogramme Full SWOF Mesure ERO

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Velocity measures by SVG [Planchon et al., 2005] Number of cells : 40 × 100 (4 m × 10 m)

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

20 40 60 80 100 120 140 1000 2000 3000 4000 5000 6000 7000 P (mm/h) t (s) Hyetogramme 10 20 30 40 50 60 70 80 1000 2000 3000 4000 5000 6000 7000 q (mm/h) t (s) Hydrogramme Full SWOF Mesure Thies

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Number of cells : 160 × 200

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Homogeneous coefficients

Coefficients homogenes : h (m) 1 2 3 4 x (m) 2 4 6 8 10 y (m) 0.001 0.002 0.003 0.004 0.005 0.006 Coefficients homogenes : h (m) 1 2 3 4 x (m) 2 4 6 8 10 y (m) 0.002 0.004 0.006 0.008 0.01 0.012

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Homogeneous coefficients

Coefficients homogenes : Fr 1 2 3 4 x (m) 2 4 6 8 10 y (m) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Coefficients homogenes : Fr 1 2 3 4 x (m) 2 4 6 8 10 y (m) 0.5 1 1.5 2 2.5 3

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Velocities

5 10 15 20 25 30 5 10 15 20 25 30 v simulees (cm/s) v mesurees (cm/s) FullSWOF 2D PSEM 2D

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Heterogeneous coefficients

Coefficients heterogenes : h (m) 1 2 3 4 x (m) 2 4 6 8 10 y (m) 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005 Coefficients heterogenes : Fr 1 2 3 4 x (m) 2 4 6 8 10 y (m) 0.5 1 1.5 2 2.5 3

Thies parcel – Senegal ([Tatard et al., 2008], IRD)

Velocities

5 10 15 20 25 30 5 10 15 20 25 30 v simulees (cm/s) v mesurees (cm/s) FullSWOF 2D PSEM 2D

FullSWOF

◮ object and inheritance ◮ variables encapsulation ◮ vector class (2d) ◮ objects ”distributor” ◮ fixed CFL and fixed ∆t ◮ Doxygen documentation ◮ Free open source software

Malpasset dam break simulation

(Cordier et al., CEMRACS 2012)

Malpasset dam break simulation

Malpasset dam break simulation

10 20 30 40 50 60 70 80 90 6 8 10 12 14 h+z (m) Gauge number Maximum water elevation FullSWOF 2D Telemac-2D (Hervouet00)

Thank You !

HLL flux F(UG, UD) =      F(UG) if 0 < c1 F(UD) if c2 < 0 c2F(UG) − c1F(UD) c2 − c1 + c1c2(UD − UG) c2 − c1 else , with two parameters c1 < c2. For c1 and c2, we take c1 = inf

U=UG ,UD( inf j∈{1,2}λj(U)) and c2 =

sup

U=UG ,UD

( sup

i∈{1,2}

λi(U)). with λ1(U) = u − √gh and λ2(U) = u + √gh.

retour

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., and Perthame, B. (2004). A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput., 25(6) :2050–2065. Bouchut, F. (2004). Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, volume 2/2004. Birkh¨ auser Basel. Bristeau, M.-O. and Coussin, B. (2001). Boundary conditions for the shallow water equations solved by kinetic schemes. Technical Report 4282, INRIA. Delestre, O. and Marche, F. (2010). A numerical scheme for a viscous shallow water model with friction.

• J. Sci. Comput., DOI 10.1007/s10915-010-9393-y.

Esteves, M., Faucher, X., Galle, S., and Vauclin, M. (2000). Overland flow and infiltration modelling for small plots during unsteady rain : numerical results versus observed values. Journal of Hydrology, 228 :265–282.

Planchon, O., Silvera, N., Gimenez, R., Favis-Mortlock, D., Wainwright, J., Le Bissonnais, Y., and Govers, G. (2005). An automated salt-tracing gauge for flow-velocity measurement. Earth Surface Processes and Landforms, 30(7) :833–844. Tatard, L., Planchon, O., Wainwright, J., Nord, G., Favis-Mortlock, D., Silvera, N., Ribolzi, O., Esteves, M., and Huang, C.-h. (2008). Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil. Journal of Hydrology, 348(1-2) :1–12.