Transcript 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
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
The modelling process
REALITY
Computer
Simulation
& Validation
COMPUTER
MODEL
Analisis
Data
uncertainties
Conceptual
errors &
uncertainties
Discretization
errors
Numerics &
Implementation
MATHEMATICAL
MODEL
The flow characteristics
Z
 3-D
Y
 time dependent
 incompresible
ZB
 free surface
 fixed bed
(no erosion – deposition)
 turbulent (very high Re)

X
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
Mathematical models
• Tracking of the free surface
• VOF method (Hirt & Nichols 1981)
• MAC method (Welch et al. 1966)
• Moving mesh methods
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...
Simplified mathematical models
Shallow Water Equations
(SWE)
• Depth integrated NS
Z
• Mass and momentum
conservation in horizontal
plane
Y
• Pseudo compressibility
h
v
ZB

u
X
• Inertial & Pressure fluxes
• Convective Momentum
transport
• Hydrostatic pressure
distribution
U 

 F  Fd   G  G d   H  I
t  x
y
hu
hv
h




 
 2



2
U   hu  F   hu  gh 2  G  
huv

 hv 


 hv 2  gh 2 2 
huv
 




• Diffusive fluxes
• Fluid viscosity
• Turbulence
• Velocity dispersion
(non-uniformity)








 0 
 0 


u 
u 
Fd    h  G d    h 
x 
y 


  h  v 
  h  v 





x

y




ε   fluid   turb  D
û
Benqué et al. (1982)
 f ree surf ace 2
 f ree surf ace
û dz 
û v̂ dz  D  div (grad u)


bottom
bottom
x
y
• Sources
• Bed slope
• Bed friction (empirical)
• Infiltration / Aportation (Singh et al. 1998
Fiedler et al. 2000)


0


H   gh(S ox  S fx ) 


 gh(S oy  S fy ) 


 ir




I    1 / 2  u  ir 
  1/ 2  v  i 
r

1-D SWE models


A   Au  0
t
x

 Au    Au 2  g  I1  gAS o  S f   gI 2
t
x


z
y
zB

A
u
x
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)
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)
Simplified models
• Kinematic & diffusive models
• Arónica et al. (1998)
• Horrit and Bates (2001)
• Flat Pond models
• Tous dam break inundation (Estrela 1999)
Flat pond model of Rio Verde area (Estrela 1999)
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
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)
• Finite volumes
• Both structured & unstructured grids
• Cell-centered or cell-vertex
• Extremely flexible & intuitive
• Many practical applications (CADAM 19981999, Brufau et al. 2000, Soares et al. 1999,
Zoppou 1999)
• Most popular
• 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
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
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)
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)
• 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)
• 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 ...
• 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)
• 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
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
• 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)
• 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 ?
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