a state of the art review on mathematical modelling of flood propagation first impact workshop...
TRANSCRIPT
![Page 1: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/1.jpg)
A state of the art review on A state of the art review on mathematical modelling ofmathematical modelling offlood propagation flood propagation
First IMPACT Workshop
Wallingford, UK,
16-17 May 2002
F. Alcrudo
University of Zaragoza
Spain
![Page 2: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/2.jpg)
OverviewOverview
• The modelling process
• Mathematical models of flood
propagation
• Solution of the Model equations
• Validation
![Page 3: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/3.jpg)
The modelling processThe 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
![Page 4: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/4.jpg)
The modelling processThe modelling process
REALITY
MATHEMATICAL MODEL
COMPUTER MODEL
Analisis
Numerics &Implementation
Computer Simulation& Validation
Conceptualerrors &
uncertainties
Discretization errors
Data uncertainties
![Page 5: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/5.jpg)
3-D
time dependent
incompresible
free surface
fixed bed
(no erosion – deposition)
turbulent (very high Re)
The flow characteristics
![Page 6: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/6.jpg)
• 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
Mathematical models
![Page 7: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/7.jpg)
• Tracking of the free surface
• VOF method (Hirt & Nichols 1981)
• MAC method (Welch et al. 1966)
• Moving mesh methods
Mathematical models
![Page 8: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/8.jpg)
• 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...
NS, RANS & Euler
![Page 9: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/9.jpg)
Shallow Water Equations (SWE)
• Depth integrated NS
• Mass and momentum conservation in horizontal plane
• Pseudo compressibility
Simplified mathematical models
h
u
v
![Page 10: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/10.jpg)
2
222
22
ghhv
huv
hv
huv
ghhu
hu
hv
hu
h
GFU
• Inertial & Pressure fluxes
• Convective Momentum transport
• Hydrostatic pressure distribution
IHGGFFU
dd
yxt
![Page 11: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/11.jpg)
y
vh
y
uh
x
vh
x
uh
00
dd GF
• Diffusive fluxes
• Fluid viscosity
• Turbulence
• Velocity dispersion (non-uniformity)
)u(Ddzvuy
dzux
surfacefree
bottom
surfacefree
bottom graddiv2
uBenqué et al. (1982)
Dε turbfluid
![Page 12: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/12.jpg)
r
r
r
fyoy
fxox
iv/
iu/
i
)SS(gh
)SS(gh
21
21
0
IH
• Sources
• Bed slope
• Bed friction (empirical)
• Infiltration / Aportation (Singh et al. 1998 Fiedler et al. 2000)
![Page 13: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/13.jpg)
1-D SWE models
x
z
uy
Bz
0
Aux
At
A
212 gISSgAIgAu
xAu
t fo
![Page 14: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/14.jpg)
• 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)
Issues in SWE models
![Page 15: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/15.jpg)
• 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)
Issues in SWE models (cont.)
![Page 16: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/16.jpg)
• Kinematic & diffusive models
• Arónica et al. (1998)
• Horrit and Bates (2001)
• Flat Pond models
• Tous dam break inundation (Estrela 1999)
Simplified models
![Page 17: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/17.jpg)
Flat pond model of Rio Verde area (Estrela 1999)
![Page 18: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/18.jpg)
Solution of the model equationsSolution of the model equations(Restricted to SWE models)(Restricted to SWE models)
• Discretization strategies
• Mesh configurations
• Numerical schemes
• Space-Time discretizations
• Front propagation
• Source term integration
• Wetting and drying
![Page 19: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/19.jpg)
• 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)
Discretization strategies
![Page 20: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/20.jpg)
• 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
![Page 21: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/21.jpg)
• 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
![Page 22: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/22.jpg)
• 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
Mesh configurations
![Page 23: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/23.jpg)
• 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)
Mesh configurations
![Page 24: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/24.jpg)
• 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)
Numerical schemes
![Page 25: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/25.jpg)
• 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)
![Page 26: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/26.jpg)
• 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 ...
![Page 27: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/27.jpg)
• 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)
![Page 28: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/28.jpg)
• 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
![Page 29: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/29.jpg)
ValidationValidation
• 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
![Page 30: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/30.jpg)
• 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)
![Page 31: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/31.jpg)
• 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 ?
![Page 32: 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](https://reader035.vdocuments.net/reader035/viewer/2022070411/56649f585503460f94c7d172/html5/thumbnails/32.jpg)
ConlusionsConlusions
• 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