Part-I … Comparative Study and Improvement in Shallow Water Model

Part-I …Comparative Study and

Improvement in Shallow Water Model

Collaborators: Prof. Kim Dan Nguyen & Dr. Yu-e Shi

Speaker: Dr. Rajendra K. Ray Date: 16. 09. 2014

Assistant Professor,School of Basic Sciences,

Indian Institute of Technology Mandi,Mandi-175001, H.P., India

Dr. Rajendra K. Ray

2

Outlines

Introduction

Governing Equations and projection method

Wetting and drying treatment

Numerical ValidationParabolic Bowl

Application to Malpasset dam-break problem

Conclusion

Dr. Rajendra K. Ray 16.09.2014

3

Introduction

Free-surface water flows occur in many real life flow situations

These types of flow behaviours can be modelled mathematically by Shallow-Water Equations (SWE)

The unstructured finite-volume methods (UFVMs) not

only ensure local mass conservation but also the best possible fitting of computing meshes into the studied domain boundaries

The present work extends the unstructured finite volumes method for moving boundary problems

Many of these flows involve irregular flow domains

with moving boundaries


4

Governing Equations and projection method

Shallow Water Equations:

Continuity Equation

)1(0

y

hv

x

hu

t

Z s

Momentam Equations

)2(

2

o

bxHH

s

y

huA

yx

huA

xx

Zghhvf

y

huv

x

hu

t

hu

)3(

2

o

byHH

s

y

hvA

yx

hvA

xy

Zghhuf

y

hv

x

huv

t

hv


5

Governing Equations and projection method …

Projection Method:

Convection-diffusion step

)4(

2

*2

2

*2***

2

*2

2

*2***

y

q

x

qA

y

vq

x

uq

t

q

y

q

x

qA

y

vq

x

uq

t

q

yyH

yyy

xxH

xxx

Wave propagation step

)5(1 12

2

2

22ns

nsss ZZZwhere

A

BZ

yxA

gh

)6(

,,,

1

22

2211

**

2

2

2

2

wyy

wxx

ny

nx

h

ny

nxyxyx

ns

ns

LLqqhC

gFFdtdtA

y

q

x

q

dty

L

x

L

y

q

x

q

dty

Z

x

ZghB


6

Governing Equations and projection method …

Velocity correction step

)7(11)1( 11

*1

FdtdtFqdtL

x

Zghdt

x

Zghdtqq n

xx

ns

ns

xnx

Equations (4)-(8) have been integrated by a technique based on Green’s theorem and then discretised by an Unstructured Finite-Volume Method (UFVM).

)8(11)1( 11

*1

FdtdtFqdtL

y

Zghdt

y

Zghdtqq n

yy

ns

ns

yny

The convection terms are handled by a 2nd order Upwind Least Square Scheme (ULSS) along with the Local Extremum Diminishing (LED) technique to preserve the monotonicity of the scalar veriable

The linear equation system issued from the wave propagation step is implicitly solved by a Successive Over Relaxation (SOR) technique.


7

Steady wetting/drying fronts over adverse steep slopes in real and discrete representations


8

Modification of the bed slope in steady wetting/drying fronts over adverse steep slopes in real and discrete representations


9

Wetting and drying treatment

The main idea is to find out the partially drying or flooding cells in each time step and then add or subtract hypothetical fluid mass to fill the cell or to make the cell totally dry respectively, and then subtract or add the same amount of fluid mass to the neighbouring wet cells in the computational domain [Brufau et. al. (2002)].

To consider a cell to be wet or dry in an particular time step, we use the threshold value as the minimum water depth (h) )10( 3O

If the cell will be considered as dry and the water depth for that cell set to be fixed as for that time step

,hh


10

Conservative Property

Definition: If a numerical scheme can produce the exact solution to the still

water case: )(,0, IVHZ s

then the scheme is said to satisfy the Conservative Property (C-property)

[Bermudez and Vázquez 1994].

Proposition 1. The present numerical scheme satisfies the C-property.

Proof. The details of the proof can be found in Shi et at. 2013 (Comp & Fluids).


11

Numerical Validation

To test the capacity of the present model in describing the wetting and drying transition

Parabolic Bowl :

The bed topography of the domain is defined by , where is a positive constant and

2b(x) r 222 yxr

The water depth is non-zero for ),( trh)(

)cos(22 YX

tYXr

The analytical solution is periodic in time with a period 2

The analytical solution is given within the range as

,coscos

1),( 2

222

tYX

rXY

tYXtrh

.2,2cos

sin,,

yx

tYX

tYtxvu


12

Numerical Validation …

Parabolic Bowl …

For computation purpose, , and are fixed as , and respectively

X Y 17106.1 m 11 m141884.0 m

The computational domain ( ) is considered as a square region

with the origin at the domain centre

2]4000,4000[]4000,4000[ m

The threshold value is set as 3103


13


Parabolic Bowl …

6/t 6/2t 6/3t

6/4t 6/5t t


14


Parabolic Bowl …6/t 6/2t 6/3t

6/4t 6/5t t


15


Parabolic Bowl …Mesh size Rate Rate Rate

[13X13] 0.006361 0.002829 0.003004

1.478 1.377 1.410

[25X25] 0.003004 0.001530 0.001554

1.412 1.354 1.363

[50X50] 0.001506 0.000834 0.000837

1.409 1.407 1.425

[100X100] 0.000758 0.000421 0.000412

1.403 1.413 1.397

[200X200] 0.000385 0.000211 0.000211

)(2 hL )(2xqL )(2

yqL

Mesh size Rate Rate Rate

[13X13] 0.008975 0.001268 0.001328

1.143 1.378 1.384

[25X25] 0.006943 0.000685 0.000712

1.416 1.181 1.182

[50X50] 0.003458 0.000491 0.000509

1.410 1.346 1.365

[100X100] 0.001739 0.000271 0.000273

1.403 1.396 1.401

[200X200] 0.000884 0.000139 0.000139

)(2 hL )(2xqL )(2

yqL


16


Parabolic Bowl …

Average Rate of convergence

Average Rate of convergence

Bunya et. al. (2009)

1.33 0.84

Ern et. al. (2008)

1.4 0.5

Present

1.4 1.4

)2/( t )( t

Relative error in global mass conservation is less than 0.003%


17

Application to the Dam-Break of Malpasset

Back Grounds

The generated flood wave swept across the downstream part of Reyran valley modifying its morphology and destroying civil works such as bridges and a portion of the highway

The Malpasset Dam was located at a narrow gorge of the Reyran River valley (French Riviera) with water storage of 361055 m

It was explosively broken at 9:14 p.m. on December 2, 1959 following an exceptionally heavy rain

The flood water level rose to a level as high as 20 m above the original bed level

After this accident, a field survey was done by the local police

In addition, a physical model was built to study the dam-break flow in 1964


18

Application to the Dam-Break of Malpasset …

The propagation times of the flood wave are known from the exact shutdown time of three electric transformers

The maximum water levels on both the left and right banks are known from a police survey

The maximum water level and wave arrival time at 9 gauges were measured from a physical model, built by Laboratoire National d’Hydraulique (LNH) of EDF in 1964

Available Data


19


Results and Discussions

Water depth and velocity field at t =1000 s Water depth at t =2400 s, wave front reaching sea


20


Results and Discussions …

Electric Transformers A B C

Field data 100 1240 1420

Valiani et al (2002) 98 -2% 1305 5% 1401 -1%

TELEMAC 111 11% 1287 4% 1436 1%

Present model 85 -15% 1230 -1% 1396 -2%

Table 5. Shutdown time of electric transformers (in seconds).


21



Arrival time of the wave front Profile of maximum water levels at surveyed

points located on the right bank


22



maximum water levels at surveyed points

located on the left bank

Maximum water level


23Dr. Rajendra K. Ray 16.09.2014

24

Conclusions

We extended the unstructured finite volume scheme for the wetting and drying problems

This extended method correctly conserve the total mass and satisfy the C-property

Present scheme very efficiently capture the wetting-drying-wetting transitions of parabolic bowl-problem and shows almost 1.4 order of accuracy for both the wetting and drying stages

The numerical experience shows that friction has a strong influence on wave arrival times but doesn’t affect maximum water levels

Present scheme then applied to the Malpasset dam-break case; satisfactory agreements are obtained through the comparisons with existing exact data, experimental data and other numerical studies


25

References

Nguyen K.D., Shi Y., Wang S.S.Y., Nguyen T.H., 2006. 2D Shallow-Water Model Using Unstructured Finite-Volumes Methods. J. Hydr Engrg., ASCE, 132(3), p. 258–269 .

Bermudez A., Vázquez M.E., 1994. Upwind Methods for Hyperbolic Conservation Laws with Source Terms. Comput. Fluids, 23, p. 1049–1071.

Ern A., Piperno S., Djadel K., 2008. A well-balanced Runge–Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int. J. Numer. Meth. Fluids, 58, p. 1–25.

Valiani A., Caleffi V., Zanni A., 2002. Case study: Malpasset dam-break simulation using a two-dimensional finite volume method. J. Hydraul. Eng., 128(5), 460–472.

Technical Report HE-43/97/016A, 1997. Electricité de France, Département Laboratoire National d’Hydraulique, groupe Hydraulique Fluviale.

Brufau P., Vázquez-Cendón M.E., García-Navarro, P., 2002. A Numerical Model for the Flooding and Drying of Irregular Domains. Int. J. Numer. Meth. Fluids, 39, p. 247–275.

Hervouet J.M., 2007. Hydrodynamics of free surface flows-Modelling with the finite element method, John Willey & sons, ISBN 978-0-470-03558-0, 341 p.

Shi Y., Ray R. K., Nguyen K.D., 2013. A projection method-based model with the exact C-property for shallow-water flows over dry and irregular bottom using unstructured finite-volume technique. Comput. Fluids, 76, p. 178–195.


Part-II …Two-Phase modelling of

sediment transport in the Gironde Estuary (France)

Collaborators: Prof. K. D. Nguyen, Dr. D. Pham Van Bang & Dr. F. Levy

Speaker: Dr. Rajendra K. Ray Date: 16. 09. 2014

Assistant Professor,School of Basic Sciences,

Indian Institute of Technology Mandi,Mandi-175001, H.P., India

Dr. Rajendra K. Ray

27

–Physical oceanography of the Gironde Physical oceanography of the Gironde estuaryestuary

- Confluence of the Confluence of the GARONNE and GARONNE and DORDOGNE: 70km to DORDOGNE: 70km to the mouththe mouth

- Width: 2km - 14kmWidth: 2km - 14km- Average depth : 7-Average depth : 7-

10m10m- 2 main channels : 2 main channels :

NAVIGATION & NAVIGATION & SAINTONGESAINTONGE

- Partially mixed and Partially mixed and macro-tidal estuarymacro-tidal estuary

- Amplitude : 1,5-5mAmplitude : 1,5-5m- Averaged river Averaged river

discharge (1961-1970) discharge (1961-1970) : 760 m3/s: 760 m3/s

- Solid discharge (1959-Solid discharge (1959-1965): 2,17 million 1965): 2,17 million tons/yeartons/year

28

Body fitted mesh for Dordogne river

River Discharge

Free Water Surface Imposed

29

Body fitted mesh for Garonne river

River Discharge

Free Water Surface Imposed

30

Body fitted mesh for Gironde Estuiry

Free Water Surface Imposed (node)

Free Water Surface Imposed (tidal)

31

PALM coupling for Gironde Estuary

32


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34

Thank you


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Part-I … Comparative Study and Improvement in Shallow Water Model

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