Part-I

1
Part-I Comparative Study and Improvement in
Shallow Water Model
Dr. Rajendra K. Ray
Assistant Professor, School of Basic
Sciences, Indian Institute of Technology
Mandi, Mandi-175001, H.P., India

• Collaborators Prof. Kim Dan Nguyen Dr. Yu-e
  Shi

Speaker Dr. Rajendra K. Ray
Date 16. 09.
2014
2
Outlines

• Introduction

• Governing Equations and projection method

• Wetting and drying treatment

• Numerical Validation
• Parabolic 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

• Many of these flows involve irregular flow
  domains
• with moving boundaries

• 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

Dr. Rajendra K. Ray

16.09.2014
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Governing Equations and projection method

• Shallow Water Equations

• Continuity Equation

• Momentam Equations

Dr. Rajendra K. Ray

16.09.2014
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Governing Equations and projection method

• Projection Method

• Convection-diffusion step

• Wave propagation step

Dr. Rajendra K. Ray

16.09.2014
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Governing Equations and projection method

• Velocity correction step

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

• 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.

Dr. Rajendra K. Ray

16.09.2014
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Steady wetting/drying fronts over adverse steep
slopes in real and discrete representations
Dr. Rajendra K. Ray

16.09.2014
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Modification of the bed slope in steady
wetting/drying fronts over adverse steep slopes
in real and discrete representations
Dr. Rajendra K. Ray

16.09.2014
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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).

Dr. Rajendra K. Ray

16.09.2014
10
Conservative Property
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).
Dr. Rajendra K. Ray

16.09.2014
11
Numerical Validation

• Parabolic Bowl

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

• The analytical solution is given within the range
  as

Dr. Rajendra K. Ray

16.09.2014
12
Numerical Validation

• Parabolic Bowl

Dr. Rajendra K. Ray

16.09.2014
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Numerical Validation

• Parabolic Bowl

Dr. Rajendra K. Ray

16.09.2014
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Numerical Validation

• Parabolic Bowl

Dr. Rajendra K. Ray

16.09.2014
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Numerical Validation

• 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
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
Dr. Rajendra K. Ray

16.09.2014
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Numerical Validation

• 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

• Relative error in global mass conservation is
  less than 0.003

Dr. Rajendra K. Ray

16.09.2014
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Application to the Dam-Break of Malpasset

• Back Grounds

• It was explosively broken at 914 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

• 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

• 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

Dr. Rajendra K. Ray

16.09.2014
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Application to the Dam-Break of Malpasset

• Available Data

• 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 dHydraulique (LNH)
  of EDF in 1964

Dr. Rajendra K. Ray

16.09.2014
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Application to the Dam-Break of Malpasset

• Results and Discussions

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

Dr. Rajendra K. Ray

16.09.2014
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Application to the Dam-Break of Malpasset

• Results and Discussions

Table 5. Shutdown time of electric
transformers (in seconds).
Electric Transformers A A B B C 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
Dr. Rajendra K. Ray

16.09.2014
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Application to the Dam-Break of Malpasset

• Results and Discussions

Profile of maximum water levels at surveyed
points located on the right bank
Arrival time of the wave front
Dr. Rajendra K. Ray

16.09.2014
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Application to the Dam-Break of Malpasset

• Results and Discussions

maximum water levels at surveyed points located
on the left bank
Maximum water level
Dr. Rajendra K. Ray

16.09.2014
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Dr. Rajendra K. Ray

16.09.2014
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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

• 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

• The numerical experience shows that friction has
  a strong influence on wave arrival times but
  doesnt affect maximum water levels

Dr. Rajendra K. Ray

16.09.2014
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References

• Bermudez A., Vázquez M.E., 1994. Upwind Methods
  for Hyperbolic Conservation Laws with Source
  Terms. Comput. Fluids, 23, p. 10491071.

• 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. 247275.

• Ern A., Piperno S., Djadel K., 2008. A
  well-balanced RungeKutta discontinuous Galerkin
  method for the shallow-water equations with
  flooding and drying. Int. J. Numer. Meth. Fluids,
  58, p. 125.

• 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.

• 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. 258269 .

• 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. 178195.

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

• 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), 460472.

Dr. Rajendra K. Ray

16.09.2014
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Part-II Two-Phase modelling of sediment
transport in the Gironde Estuary (France)
Dr. Rajendra K. Ray
Assistant Professor, School of Basic
Sciences, Indian Institute of Technology
Mandi, Mandi-175001, H.P., India

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

Speaker Dr. Rajendra K. Ray
Date 16. 09.
2014
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• Physical oceanography of the Gironde estuary

• Confluence of the GARONNE and DORDOGNE 70km to
  the mouth
• Width 2km - 14km
• Average depth 7-10m
• 2 main channels NAVIGATION SAINTONGE
• Partially mixed and macro-tidal estuary
• Amplitude 1,5-5m
• Averaged river discharge (1961-1970) 760 m3/s
• Solid discharge (1959-1965) 2,17 million
  tons/year

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Body fitted mesh for Dordogne river
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Body fitted mesh for Garonne river
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Body fitted mesh for Gironde Estuiry
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PALM coupling for Gironde Estuary
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Results and Discussions
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Results and Discussions
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Thank you
Dr. Rajendra K. Ray

16.09.2014