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.

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

OverviewOverview

• The modelling process

• Mathematical models of flood

propagation

• Solution of the Model equations

• Validation

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

The modelling processThe modelling process

REALITY

MATHEMATICAL MODEL

COMPUTER MODEL

Analisis

Numerics &Implementation

Computer Simulation& Validation

Conceptualerrors &

uncertainties

Discretization errors

Data uncertainties

3-D

time dependent

incompresible

free surface

fixed bed

(no erosion – deposition)

turbulent (very high Re)

The flow characteristics

• 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

Mathematical models

• 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

Shallow Water Equations (SWE)

• Depth integrated NS

• Mass and momentum conservation in horizontal plane

• Pseudo compressibility

Simplified mathematical models

h

u

v

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

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

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)

1-D SWE models

x

z

uy

Bz

0

Aux

At

A

212 gISSgAIgAu

xAu

t fo

• 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

• 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.)

• Kinematic & diffusive models

• Arónica et al. (1998)

• Horrit and Bates (2001)

• Flat Pond models

• Tous dam break inundation (Estrela 1999)

Simplified models

Flat pond model of Rio Verde area (Estrela 1999)

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

• 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

• 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

• 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

• 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)

Mesh configurations

• 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

• 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

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

• 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 ?

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