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A state of the art review on mathematical modelling of flood propagation

A state of the art review on mathematical modelling of flood propagation . First IMPACT Workshop Wallingford, UK, 16-17 May 2002. F. Alcrudo University of Zaragoza Spain. Overview. The modelling process Mathematical models of flood propagation Solution of the Model equations Validation.

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A state of the art review on mathematical modelling of flood propagation

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  1. A state of the art review on mathematical modelling offlood propagation First IMPACT Workshop Wallingford, UK, 16-17 May 2002 F. Alcrudo University of Zaragoza Spain

  2. Overview • The modelling process • Mathematical models of flood propagation • Solution of the Model equations • Validation

  3. The modelling process • Understanding of flow characteristics • Formulation of mathematical laws • Numerical methods • Programming • Validation of model by comparison of results against real life data • Prediction: Ability to FOREtell not to PASTtell

  4. The modelling process REALITY Analisis Computer Simulation & Validation Data uncertainties Conceptual errors & uncertainties Discretization errors COMPUTER MODEL MATHEMATICAL MODEL Numerics & Implementation

  5. The flow characteristics • 3-D • time dependent • incompresible • free surface • fixed bed (no erosion – deposition) • turbulent (very high Re)

  6. Mathematical models • 3-D Navier-Stokes (DNS) • Chimerical • 3-D RANS • Turbulence models ? • Still too complex • Euler (inviscid) • Simpler, requires much less resolution • Could be an option soon

  7. Mathematical models • Tracking of the free surface • VOF method (Hirt & Nichols 1981) • MAC method (Welch et al. 1966) • Moving mesh methods

  8. NS, RANS & Euler • 2-D dam break and overturning waves • Zwart et al. 1999 • Mohapatra et al. 1999 • Stansby et al. (Potential) 1998 • Stelling & Busnelli 2001... • River flows • Casulli & Stelling (Q-hydrostatic) 1998 • Sinha et al. 1998, Ye &McCorquodale 1998...

  9. h v u Simplified mathematical models Shallow Water Equations (SWE) • Depth integrated NS • Mass and momentum conservation in horizontal plane • Pseudo compressibility

  10. Inertial & Pressure fluxes • Convective Momentum transport • Hydrostatic pressure distribution

  11. Diffusive fluxes • Fluid viscosity • Turbulence • Velocity dispersion (non-uniformity) Benqué et al. (1982)

  12. Sources • Bed slope • Bed friction (empirical) • Infiltration / Aportation (Singh et al. 1998 Fiedler et al. 2000)

  13. 1-D SWE models

  14. Issues in SWE models • Corrections for non-hydrostatic pressure, non-zero vertical movement • Boussinesq aproximation (Soares 2002) • Stansby and Zhou 1998 (in NS-2D-V) • Flow over vertical steps (Zhou et al. 2001) (Exact solutions Alcrudo & Benkhaldoun 2001) • Corrections for non-uniform horizontal velocity ? (Dispersion effects)

  15. Issues in SWE models (cont.) • Turbulence modelling in 2D-H • Nadaoka & Yagi (1998) river flow • Gutting & Hutter (1998) lake circulation (K-e) • Gelb & Gleeson (2001) atmospheric SWE model • Bottom friction • Non-uniform unsteady friction laws ? • Distributed friction coefficients (Aronica et al. 1998) • Bottom induced horizontal shear generation (Nadaoka & Yagi 1998)

  16. Simplified models • Kinematic & diffusive models • Arónica et al. (1998) • Horrit and Bates (2001) • Flat Pond models • Tous dam break inundation (Estrela 1999)

  17. Flat pond model of Rio Verde area (Estrela 1999)

  18. Solution of the model equations(Restricted to SWE models) • Discretization strategies • Mesh configurations • Numerical schemes • Space-Time discretizations • Front propagation • Source term integration • Wetting and drying

  19. Discretization strategies • Finite differences • Decaying use (less flexible) • Usually structured grids • Scheme development/testing (Liska & Wendroff 1999, Glaister 2000 ...) • Practical appications (Bento-Franco 1996, Heinrich et al. 2000, Aureli et al. 2000)

  20. Finite volumes • Both structured & unstructured grids • Cell-centered or cell-vertex • Extremely flexible & intuitive • Many practical applications (CADAM 1998-1999, Brufau et al. 2000, Soares et al. 1999, Zoppou 1999) • Most popular

  21. Finite elements • Variational formulation • Conceptually more complex • More difficult front capture operator (Ribeiro et al. 2001, Hauke 1998) • Practical applications • Hervouet 2000, Hervouet & Petitjean 1999 • Supercritical / subcritical, tidal flows, Heniche et al. 2000

  22. Mesh configurations • Structured • Cartesian / Boundary fitted (mappings) • Less flexible / Easy interpolation • Unstructured • Flexible but Indexing / Bookkeeping overheads • More elaborated Interpolation (Sleigh 1998, Hubbard 1999) • Easy refining (Sleigh 1998, Soares 1999) and adaptation (Benkhaldoun 1994, Ivanenko et al. 2000) • Quad-Tree

  23. Mesh configurations • Quad-Tree • Cartesian with grid refining/adaptation • Hierarchical structure / Interpolation operators • Needs bookkeeping • Usually specific boundary treatments (Cartesian cut-cell approach Causon et al. 2000, 2001) • Practical applications (Borthwick et al. 2001)

  24. Numerical schemes • Space – Time discretization • Space discretizations + • Time integration of resulting ODE • Time integration • Explicit usu 2-step, Runge-Kutta (Subject to CFL constraints) • Implicit (not frequent)

  25. Front propagation • Shock capturing or through methods • Approximate Riemann solvers (Most popular Roe, WAF second) • Higher order interpolations + limiters (either flux or variables), TVD, ENO • Mostly in FV & FD but progressively incorporated into FE (Sheu & Fhang 2001) • Plenty of methods (or publications)

  26. Multidimensional upwind • Wave recognition schemes (opposed to classical dimensional splitting) • Consistent Higher resolution of wave patterns • Usually in unstructured (cell vertex) grids (mostly triangles) • Considerably more expensive • Hubbard & Baines 1998, Brufau & Garcianavarro 2000 ...

  27. Source term integration (bed slope) • Flow is source term dominated in most practical applications • Flux discretization must be compatible with source term • Source term upwinding (Bermudez & Vazquez 1994) • Pressure – splitting (Nujic 1995) • Flux lateralisation (Capart et al. 1996, Soares 2002) • Surface gradient method (Zhou et al. 2001) • Discontinuous bed topography (Zhou et al. 2002)

  28. Wetting-drying • Intrinsic to flood propagation scenarios • Instabilities due to coupling with friction formulae and to sloping bottom (Soares 2002) • Threshold technique (CADAM 1998), simple, widely used but no more than a trick • Fictitious negative depth (Soares 2002) • Boundary treatment at interface (Bento-Franco 1996, Sleigh 1998), modification of bottom function (Brufau 2000) • Bottom function modification, ALE (Quecedo and Pastor (2002) in Taylor Galerkin FE

  29. Validation • Model accuracy • Differences between model output & real life • Determined with respect to experimental data • Accuracy loss: • Uncertainty Due to lack of knowledge • Errors Recognizable defficiencies

  30. Main losses of accuracy in flood propagation models • Errors in the math description (SWE or worse) • Uncertainties in data (topography, friction levels, initial flood characteristics) • Additional errors • Inaccurate solution of model equations (grid refining)

  31. Much validation work of numerical methods against analytical /other numerical solutions • Chippada et al., Hu et al., Aral et al. 1998 • Holdhal et al., Liska & Wendroff , Zoppou & Roberts etc ... 1999 • Causon et al., Wang et al., Borthwick et al. etc ... 2001 • Validation against data from laboratory experiments • CADAM work, Tseng et al. 2000, Sakarya & Toykay 2000 etc ... • Validation against true real flooding data • CADAM 1999, Hervouet & Petitjean (1999), Hervouet (2000), Horritt (2000), Heinrich et al. (2001), Haider (2001) • Sensitiviy analysis (usually friction) • Urban flooding ?

  32. Conlusions • Present feasible mathematical descriptions of flood propagation are known to be erroneous but ... • Better mathematical models are still far ahead • The level of accuracy of present models has not yet been thoroughly assessed • There are enough methods at hand to solve the mathematical models (most are good enough) • Exhaustive validation programs against real data are needed

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