PartI Comparative Study and Improvement in Shallow Water

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

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 2

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 3

Governing Equations and projection method Shallow Water Equations: Continuity Equation Momentam Equations Dr. Rajendra K. Ray 16. 09. 2014 4

Governing Equations and projection method … Projection Method: Convection-diffusion step Wave propagation step Dr. Rajendra K. Ray 16. 09. 2014 5

Governing Equations and projection method … Velocity correction step Equations (4)-(8) have been integrated by a technique based on Green’s theorem and then discretised by an Unstructured Finite-Volume Method (UFVM). The convection terms are handled by a 2 nd 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 6

Steady wetting/drying fronts over adverse steep slopes in real and discrete representations Dr. Rajendra K. Ray 16. 09. 2014 7

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 8

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 threshold value as the minimum water depth (h) If the cell will be considered as dry and the water depth for that cell set to be fixed as for that time step Dr. Rajendra K. Ray 16. 09. 2014 9

Conservative Property Definition: If a numerical scheme can produce the exact solution to the still water case: 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). Dr. Rajendra K. Ray 16. 09. 2014 10

Numerical Validation Parabolic Bowl : To test the capacity of the present model in describing the wetting and drying transition The bed topography of the domain is defined by positive constant and The water depth , where is a is non-zero for The analytical solution is periodic in time with a period The analytical solution is given within the range as Dr. Rajendra K. Ray 16. 09. 2014 11

Numerical Validation … Parabolic Bowl … For computation purpose, , and respectively The computational domain ( The threshold value Dr. Rajendra K. Ray and are fixed as , ) is considered as a square region with the origin at the domain centre is set as 16. 09. 2014 12

Numerical Validation … Parabolic Bowl … Dr. Rajendra K. Ray 16. 09. 2014 13

Numerical Validation … Parabolic Bowl … Dr. Rajendra K. Ray 16. 09. 2014 14

Numerical Validation … Parabolic Bowl … Mesh size [13 X 13] Rate 0. 006361 Rate 0. 002829 1. 478 [25 X 25] 0. 003004 0. 001530 0. 001506 Mesh size [13 X 13] Rate 0. 000884 Dr. Rajendra K. Ray 0. 000712 0. 000491 0. 001739 1. 384 1. 181 1. 182 0. 000509 1. 346 0. 000271 1. 403 [200 X 200] 0. 001328 0. 000685 0. 003458 Rate 1. 378 1. 410 [100 X 100] 0. 000211 0. 001268 0. 006943 1. 397 Rate 1. 416 [50 X 50] 0. 000412 0. 000211 0. 008975 1. 425 1. 413 1. 143 [25 X 25] 0. 000837 0. 000421 0. 000385 1. 363 1. 407 1. 403 [200 X 200] 0. 001554 0. 000834 0. 000758 1. 410 1. 354 1. 409 [100 X 100] 0. 003004 1. 377 1. 412 [50 X 50] Rate 1. 365 0. 000273 1. 396 0. 000139 1. 401 0. 000139 16. 09. 2014 15

Numerical Validation … Parabolic Bowl … Average Rate of convergence 1. 33 0. 84 1. 4 0. 5 1. 4 Bunya et. al. (2009) Ern et. al. (2008) Present Relative error in global mass conservation is less than 0. 003% Dr. Rajendra K. Ray 16. 09. 2014 16

Application to the Dam-Break of Malpasset Back Grounds The Malpasset Dam was located at a narrow gorge of the Reyran River valley (French Riviera) with water storage of 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 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 17

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 d’Hydraulique (LNH) of EDF in 1964 Dr. Rajendra K. Ray 16. 09. 2014 18

Application to the Dam-Break of Malpasset … Results and Discussions Water depth and velocity field at t =1000 s Dr. Rajendra K. Ray Water depth at t =2400 s, wave front reaching sea 16. 09. 2014 19

Application to the Dam-Break of Malpasset … Results and Discussions … Table 5. Shutdown time of electric transformers (in seconds). Electric Transformers A B C Field data 100 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 1240 1420 16. 09. 2014 20

Application to the Dam-Break of Malpasset … Results and Discussions Arrival time of the wave front Dr. Rajendra K. Ray Profile of maximum water levels at surveyed points located on the right bank 16. 09. 2014 21

Application to the Dam-Break of Malpasset … Results and Discussions maximum water levels at surveyed points located on the left bank Dr. Rajendra K. Ray Maximum water level 16. 09. 2014 22

Dr. Rajendra K. Ray 16. 09. 2014 23

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 doesn’t affect maximum water levels Dr. Rajendra K. Ray 16. 09. 2014 24

References Bermudez A. , Vázquez M. E. , 1994. Upwind Methods for Hyperbolic Conservation Laws with Source Terms. Comput. Fluids, 23, p. 1049– 1071. 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. 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. 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. 2 D Shallow-Water Model Using Unstructured Finite-Volumes Methods. J. Hydr Engrg. , ASCE, 132(3), p. 258– 269. 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. Technical Report HE-43/97/016 A, 1997. Electricité de France, Département Laboratoire National d’Hydraulique, 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), 460– 472. Dr. Rajendra K. Ray 16. 09. 2014 25

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

–Physical oceanography of the Gironde estuary - - - Confluence of the GARONNE and DORDOGNE: 70 km to the mouth Width: 2 km - 14 km Average depth : 7 -10 m 2 main channels : NAVIGATION & SAINTONGE Partially mixed and macro-tidal estuary Amplitude : 1, 5 -5 m Averaged river discharge (1961 -1970) : 760 m 3/s Solid discharge (19591965): 2, 17 million tons/year 27

Body fitted mesh for Dordogne river River Discharge Free Water Surface Imposed 28

Body fitted mesh for Garonne river River Discharge Free Water Surface Imposed 29

Body fitted mesh for Gironde Estuiry Free Water Surface Imposed (tidal) Free Water Surface Imposed (node) 30

PALM coupling for Gironde Estuary 31

Results and Discussions 32

Results and Discussions 33

Thank you Dr. Rajendra K. Ray 16. 09. 2014 34