NUMERICAL FLOOD SIMULATION BY DEPTH …...NUMERICAL FLOOD SIMULATION BY DEPTH AVERAGED FREE SURFACE FLOW MODELS A. I. Delis Department of Sciences, Technical University of Crete, Chania,

NUMERICAL FLOOD SIMULATION BY DEPTH AVERAGED FREE SURFACE FLOW MODELS

A. I. Delis Department of Sciences, Technical University of Crete, Chania, Crete, Hellas. Institute of Applied and Computational Mathematics, Foundation for Research and Technology, Heraklion, Crete, Hellas.

N. A. Kampanis Institute of Applied and Computational Mathematics, Foundation for Research and Technology, Heraklion, Crete, Hellas.

Keywords: Floods, Numerical Simulation, Depth averaged models, Shallow water equations, Finite Volume Schemes, Source terms, Wetting and Drying, Benchmarking.

Contents

1. Introduction 2. Mathematical Modeling of Flood Propagation 3. Numerical Modeling for Flood Propagation in Natural Topographies 4. Benchmarking and Validation 5. Concluding Remarks Related Chapters Glossary Bibliography Biographical Sketches

Summary

Numerical flood simulation has matured significantly over the last decades and has come to the point where a quite realistic picture of potential flood threats can be produced at reasonable cost. The quality of a flood simulation model fully depends on its descriptive capabilities of the physical system in terms of topographic and roughness data, the representativeness of the equations, and the numerical method applied. This contribution reviews and explains the current practice and state of the art of numerical flood simulation models. In particular, the most important and widely used depth averaged model is reviewed and discussed along with its numerical treatment. Proceeding with the two-dimensional flow description and its numerical discretization principles, numerical discretization of depth averaged two-dimensional shallow water equations are presented with special attention given to the correct and robust modeling within the finite volume framework. Some popular finite volume models for solving the depth averaged two-dimensional shallow water equations are described, with special attention to the modeling of shock (bore) waves, the treat of natural topographies, and the appearance of wet/dry fronts.

1. Introduction

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The risk and impact of floods in rural as well as in urban areas has been increased in the last few decades as population and urbanization processes rapidly increase and subsequently more and more people and properties are being concentrated in flood-prone coastal zones and river flood-plains. In addition, there is an increasing awareness about climate changes and extreme weather conditions that can lead to the emergence of natural disasters such as, flash floods and failures of flood defense structures, including dams, weirs and flood dykes. Moreover, floods in urban areas can be much more devastating than any other areas and they can pose a significant threat to human life. Worldwide, coastal, riverine and flash floods are responsible for more than 50% of the fatalities and for about 30% of the economic losses caused by all natural disasters. The state of the art of numerical flood simulation has progressed significantly over the last decades and has come to the point where a quite realistic picture of potential flood threats can be produced at reasonable costs. New data collection techniques have emerged which alleviate the traditional problem of lack of data for topographic and terrain modeling. In addition, numerical techniques have matured, providing robustness and efficiency in model simulation.

Modeling and simulation of flood events are necessary to understand the mechanisms of the process and therefore to better protect urban areas and increase public safety; for example, they can be important for developing emergency plans. The information provided from simulations about potential floods must include such data as (i) time of the flood wave arrival at some points in a valley or a city, (ii) extreme water levels in the flooded area, (iii) duration and range of flooding and (iv) water depths and velocities in the flooded zones. As such, flood inundation and propagation modeling can be defined as the art of quantitatively describing the evolution and characteristics of the flow that is set up when a large amount of water moves along the earth surface in an uncontrolled way (Mandych, 2004). The progress of flood propagation models is linked directly to (a) the understanding the flow processes relative to the problem, (b) the formulation of appropriate mathematical laws, (c) the development of effective numerical techniques to solve them and (d) the validation of the model output against benchmark, experimental and real life data. The underlying mathematical models describing the flood process are basically variations of models for free surface water flow. These models are mainly governed by unsteady non-linear Partial Differential Equations (PDE’s) in general three dimensional (3D) domains, with a free surface boundary condition.

An important feature of free surface flows is that they are unbounded in space, the limits of the spatial domain being an unknown of the problem to be solved. Problems in which the limits of the fluid are unknown and unsteady include among others, dam-break induced flows, tidal flow in estuaries and flood propagation in rivers. In these situations it is necessary to compute a non-stationary wet/dry front, which is part of the solution we are looking for. The full flow field can be described by the Navier-Stokes equations. However, qualitative and/or quantitative approximations of the actual solution are given by approaches based on simplified equations. This is done in a systematic effort to overcome the need, usually, of excessively demanding numerical techniques to resolve the Navier-Stokes equations (supplied possibly with appropriate turbulence closure models). A widely used approach is that of the 2D depth averaged models. The 2D character of the free surface flow is usually enforced by a horizontal length scale which is much larger than the



vertical one, and by a velocity field quasi-homogeneous over the water depth. This small ratio between the vertical and horizontal length scales characterizes flood situations as well as many engineering applications, mainly in river and coastal engineering. Despite their shortcomings, depth averaged models are effectively used in engineering practice in order to model environmental flows in rivers and coastal regions, as well as shallow flows in hydraulic structures. Concerning topographies of flooded territories and the complexity of city structures, flow simulations in 2D in the horizontal plane are indispensable. As such many geophysical flows can be modeled by the shallow water (SW) or Saint-Venant (SV) system of equations or their variations. Extensions of these equations are useful to model sedimentary flows, tsunamis, avalanches, river mouths and junctions, marine flows. The choice of methods and algorithms for computing solutions to these equations is very wide. Among them the Finite Volume formulation, is nowadays the most applied modeling strategy for such computations.

Depending on their objective, flood simulation models may differ in their requirements. Criteria for the selection of the appropriate tools are often based on the required speed of computation, completion time for a simulation, level of accuracy in the results, data requirements, numerical robustness, user-friendliness of the software, and possible others, depending of the model. These objectives may be related to flood risk analysis, flood forecasting and control and may be based upon a variety of causes, such as, storms, dam or dike breaks, hurricanes and geologically induced tsunamis. The development of suitable numerical techniques and that of powerful computer equipment has enable to produce reliable simulations for practical applications. In the area of numerical flood simulation some frequently used tools, that are currently available commercially or as freeware and utilize 2D depth averaged models, are the Mike 11 (1D) and Mike 21 (2D) modeling systems of the Danish Hydraulics Institute, the SOBEK modeling system of Delft Hydraulics, the ISIS tool and InfoWorks of Walling-ford Software, the TUFLOW software of BMT WMB Consultants, the HEC-RAS system of the US Army Corps of Engineers, the LISFLOOD-FP flood inundation system of the University of Bristol, the TELEMAC2D modeling system of Electricité de France (EDF), the BASEMENT software of the Swiss Federal Institute of Technology (ETH), the ANUGA Hydrodynamic Modeling of Geosciences Australia, and the CARPA modeling system of the Flumen research group. In addition, several research projects and consortiums have been initiated (e.g. the CADAM, IMPACT and FLOODsite European research projects and the Floodrisk British consortium ) in order to establish cutting edge research to enhance flood risk management practice and to deliver tools and techniques to support improvements in flood modeling and simulation.

This presentation is focused on the mathematical and numerical modeling of 2D free surface flows under the influence of gravity. It reviews and summarizes some previous theoretical, numerical and experimental studies about the simulation of flood propagation using depth averaged models. The derivation of the 2D SW equations is summarized and discussed, in order to understand the limitation of these equations and asses the numerical results obtained from them. In addition, state of the art finite volume numerical schemes and discretization techniques implemented in flood flow simulations and solve the 2D SW equations are described and discussed in more detail.



2. Mathematical Modeling of Flood Propagation

2.1. Navier-Stokes (NS) and Related Models

Flood propagation over the earth’s surface is a 3D time dependent, incompressible, fluid dynamics problem with a free surface. By not considering the erosion and deposition effects, which are a subject of a separate branch of study, the flow can be considered as a single phase flow. The well known Navier-Stokes (NS) equations (Bardos, 2005), in 3D, perfectly describe the dynamics of a portion of fluid. However, the main drawback to a fully 3D approach is its computational cost, especially in environmental problems, where the size of the spatial domain can be very large and there are flow patterns of different length scales involved in the flow. The flow is turbulent and of geographical size, and the cascade of length and time scales present is huge what impairs any attempt to solve the 3D NS equations by any means; only in very simple geometry configurations it is possible to solve directly the NS equations using appropriate numerical methods such as Direct Numerical Simulations (DNS) (Wagner, 2006). In a DNS it is necessary to resolve all the scales of motion appearing in the flow, since they interact with each other; in order to do that, the computational mesh size must be smaller than the smallest significant scale motion, and the simulation time step must be small enough to resolve the highest frequency oscillations appearing in the flow. This constitutes a significant reservation for using DNS.

In order to circumvent the problem of turbulence the NS equations can be averaged in time in order to obtain the so-called Reynolds-Averaged Navier-Stokes equations (RANS) that describe the mean flow. The effects of the turbulent fluctuations on the mean flow are taken care of by means of turbulence models, i.e. formulations whereby the stress due to turbulence are related to the mean flow variables. Currently there are dozens of turbulence models in use, each adapted to a particular fluid dynamics situation. The RANS equations are of wide use in industrial fluid mechanics and aerodynamics (Chabard and Laurence, 2004) but are still too complex to be applied in order to describe flood propagation, mainly due to the resolution that would be needed to make such a procedure meaningful. Furthermore, since turbulence models are developed to be well suited to specific situations, those currently available may not even make sense in a flood propagation scenario.

Further to the problem of turbulence, the NS and RANS based models have the added difficulty of the air-water interface movement. The free surface moves with the velocity of the fluid particles located at the boundary and therefore its position is one of the unknowns that must be solved for during a computational procedure. The problem lies in that, the equations of motion only apply to the space occupied by the fluid which is not known a priori. Several methods have been developed to circumvent these difficulties, mostly relying on iterative procedures.

A rather general classification distinguishes between mesh methods and meshless methods. Meshless methods use a Lagrangian formulation in order to compute the movement of fluid particles applying Newton’s Second law. The most popular meshless method is the Smoothed Particle Hydrodynamics



(SPH) method. It has the advantage of being able to treat complicated free surface deformations, but it has problems with the correct modeling of boundaries.

Mesh methods can be classified in moving grid methods and fixed grid methods. Moving grid methods use a Lagrangian formulation in order to move the grid nodes and boundaries with the fluid. The free boundary is computed with a front tracking technique. The main disadvantage is the computational cost, since the nodes of the mesh move at each time iteration, and thus, the geometric properties of the mesh need to be recomputed. Lagrangian methods are mainly used when the movement of the free surface is small, because otherwise it is necessary to add or remove some nodes from the mesh in order to avoid a large distortion of the elements.

Fixed grid methods are more commonly used. They can use a fully Eulerian formulation (interface capturing) or a combined Eulerian-Lagrangian formulation (interface tracking). Among the Eulerian methods the Volume of Fluid (VOF) method and Marker in Cell (MAC) methods have gained a reputation of accuracy and robustness, but their application to flooding problems has not yet been possible due to the extraordinary computational power needed for their application. Simulations of propagating and breaking waves as well as dam beak flows with these methods have been obtained however, these simulations are limited to idealized, two dimensional (in the vertical plain) cases with no practical interest or to limited size industrial applications. Furthermore, in order to simplify the problem either only laminar flows are considered or the diffusive re-dropped from the NS equations thus solving the inviscid (Euler) equations. Fully 3D simulations are usually limited to steady or slow flows which in not the case in flooding scenarios, or applied only to solve local flow effects.

As it was mention earlier, the main drawback of a fully 3D approach is its computational cost, specially in environmental problems, where the spatial domain is very large and there are flow patterns of very different length scales involved in the flow. For that reason, it is not yet efficient to use the fully 3D approach in most environmental hydraulic flows. In shallow water flows it is possible to simplify the 3D RANS equations assuming a hydrostatic pressure distribution. In such a case the vertical momentum equation is simplified to the hydrostatic pressure equation, and therefore, only the two horizontal momentum equations need to be solved in a 3D computational mesh. The continuity equation is used in order to compute the free surface level, which in turn, defines the hydrostatic pressure distribution. A computational mesh in this case is often built as a 2D horizontal mesh with several layers in the vertical direction and with in this way well oriented simplified mesh generation can be archived for some common environmental flow problems such as stratified flows. This approach is usually called a 3D SW equations computation and it has been used for simple or simplified geometries. Some work has been done using this 3D approach to model free surface flows in complex geometries using a singled value height-function formulation in order to track the free surface.

Given their increased computational demand, it seems that NS or even Euler (inviscid) based models can be outflanked, regarding their practical effectiveness, by reduced simulation models for use in realistic flood propagation modeling.



2.2. Depth Averaged Models (2D Shallow Water Equations)

In order to simplify the above mentioned mathematical models for flood propagation, the usual approach is to derive the depth averaged shallow water equations also known as Saint-Venant (SV) equations or 2D SW equations. The equations are obtained after vertical integration of the 3D SW equations. Alternatively, they can be derived from mass and momentum conservation in the plane of motion (e.g. the earth surface). The depth average procedure eliminates from the start the free surface location problem which is now simply placed as the depth above a topography surface. The depth averaged approach formulation has been successfully applied to different problems, obtaining quite accurate results with a relatively low computational cost when compared with 3D approaches. It has also the advantage of being very robust for computing accurately the water depth, even in unsteady problems with free surface shocks, as it happens to be the case of advancing surges and bores in dam-break simulations. The 2D depth averaged formulation has been extensively used to model the dam-break problem, the propagation and run-up of shallow water long waves, flooding and drying problems, flow in rivers and estuaries and flow in coastal regions. Especially the treatment of unsteady wet/dry fronts, which will typically appear in such simulations, is also much simpler and stable than in 3D approaches.

In attention to their importance in flood propagation modeling the 2D SW equations are presented and discussed here in more detail. The process of the mathematical derivation of the 2D SW equations basically consists in assuming a hydrostatic pressure distribution (hydrostatic equilibrium), integrating the horizontal RANS equations over the water depth, applying Leibnitz’s rule, and using the kinematic free surface and bed (topography) surface conditions. Several approximations and simplifications are done through the mathematical derivation and it is important to have them in mind in order to understand the limitations of the equations and to interpret the results obtained from them correctly. A schematic representation of a 2D domain is shown in Figure 1. The 2D SW system reads as:

(1)

(2)

(3)



Figure 1. The 2D shallow water configuration

where is the velocity field, is the water depth, is the gravitational acceleration, is the bed elevation, , are the two horizontal components of the bed friction (vertical viscous) stress, , are the horizontal surface stress (wind-stress components), is the Coriolis parameter, is the atmospheric pressure at the free surface and is the (constant) water density. Moreover, is a kinematic viscosity (horizontal diffusion) coefficient that accounts for the kinematic viscosity, the turbulent eddy viscosity and the apparent viscosity due to velocity fluctuations about the vertical average.

Coriolis forces effects, wind stress and atmospheric pressure have minimal influence on flood propagation and these terms are included in Eqs. (1)-(3) for completeness. Additional terms can be added to the equations as for example, the infiltration rate into the ground and sinks (e.g. sewers) in urban areas or the inflow rate of the rain or any additional water source.

The bed friction stresses are usually represented by means of empirical quadratic formulas. Traditionally such empirical formulae like the Manning, Chèzy or Srtickler laws, that scale with the square of the depth averaged velocities have been assumed (Lee and Sharp, 2004). The Manning formula is given as

while the Chèzy formula as

where and are the Manning and Chèzy coefficient respectively.

The validity of the friction formulae has been experimentally verified for uniform flow, but not many things can be said for highly unsteady situations. It is well known that the friction law significantly diverges from the uniform flow in unsteady and oscillatory pipe flow and hence a similar effect can be expected in free surface flows. The quantitative effect of this simplification may be not be large enough to make a big difference, and most probably is smaller than other assumptions made that can hide its influence. However, another issue related to friction can have significantly more influence. The friction coefficients are usually almost universally considered constant along the flow path and as such disregard variations in bed roughness and topography. The effect of roughness distribution across a region can have a considerable effect in the flood propagation characteristics, and hence allowance must be made for a variable friction coefficient in realistic flood propagation simulations.



Through the derivation of the above depth averaged SW equations several assumptions have been made. These pretences are:

1. Constant density (incompressible flow): Deriving the 2D SW equations from the incompressible RANS model, density variations with the pressure gradients are neglected, this is a reasonable hypothesis in water flows.

2. Hydrostatic pressure field: Assuming a separation of vertical and horizontal characteristic scales hydrostatic pressure distribution comes as a result. This occurs when both the horizontal length, , and velocity are larger than the vertical ones, this is a typical characteristic of quasi-2D flows. The definition of the horizontal and vertical scales is not trivial and depends on the flow conditions and geometry. In a long shallow wave propagation case the horizontal scale is given by the wave length, while the vertical scale is given by the water depth, since it is over those distances that the velocity and pressure changes occurs. In some cases the vertical length scale, , is given by the variations in the bed and free surface elevation instead of the water depth, and therefore the condition is actually a restriction on the free surface and bed slopes. Therefore, the SW equations may be applied to flows with large water depth if the bed and surface slopes are small. One can make a distinction between shallow water flow and deep water flow depending on the ration between the internal and bed friction forces, although the SW equations may be applied in both situation. It should be noted that, the fact that of assuming separation of length scales does not mean that the vertical velocity is neglected. Finally, in order to assume hydrostatic pressure distribution, these are another two conditions which must be fulfilled: the Reynolds number must be much larger than 1, and the turbulence intensity must be smaller than one. Both of these conditions are usually fulfilled in shallow water flows.

3. Homogeneous behavior in the vertical direction (assumption of uniform horizontal velocity across the water layer). This approximation implies that in the boundary layer, extended from the bed to the free surface, the horizontal velocity field is considered uniform across the depth of the water layer. This fact may also be responsible for important deviations between model predictions and real observations. When deriving the SW equations by vertical integration, over the water depth, of the convective terms, in the horizontal RANS equations, the dispersion effect due to non-uniformity of the velocity profile can be quantified and these are usually approximated as a diffusion of momentum:

where, and are the deviation

(fluctuations) of the velocity components with repeat to the depth averaged values and . Then, is a diffusion coefficient with dimensions of a kinematic viscosity. Its effect is analogous to the viscous and turbulent stresses. Hence, the formal expression for the kinematic (effective) viscosity coefficient on Eqs. (1)-(3) is

where is the water kinetic viscosity and is the turbulent eddy viscosity coefficients.

4. Depth averaged viscous and turbulent stress. Usually the equivalent



kinematic viscosity, µ, may be more than an order of magnitude than the kinematic viscosity of the fluid, , thus competing with the turbulence effects. Despite this fact the dispersion effects due to the non-uniformity of the velocity profile are not usually taken into account or not reflected in the technical literature. Estimation of the diffusion coefficient can be a difficult task but may be worth some more attention in flood propagation models.

The turbulent contribution to the momentum equations in hydraulics has received more attention. In cases which include tidal studies in coast lines and harbors, circulation in lakes as well as atmospheric SW simulations some form of turbulence model is included. However, when it comes to flood propagation turbulence modeling is not considered an important matter. In some cases a constant eddy viscosity coefficient is used which seems to play more the role of a tuning parameter rather than a characterization of the turbulence characteristics. At this level is assumed constant, so that horizontal stress terms become . Altogether it seems that the importance of diffusion (either due to turbulence or to dispersion of the velocity profile) has not been evaluated in SW models of flood propagation.

We note here that, mathematically speaking, the 2D SW equations model reduces the problem from a 2D free surface NS/Euler incompressible (velocity divergence free) problem to a 2D compressible one (having the same appearance as the Euler equations for compressible flows with no state and separate energy equations). This is somehow peculiar since their derivation comes from incompressible flow equations. However, an apparent compressibility is caused by the presence of the free surface where the depth of the water layer plays the role of density.

Leaving apart the terms that don’t affect flood modeling situations, Eqs. (1)-(3) reduce to

(4)

(5)

(6)

The 2D SW Eqs. (4)-(6) are a system of three partial differential equations with three unknowns defined over a 2D spatial domain. This is an important reduction in terms of the computational cost comparing to the original 3D RANS equations, where for equations are defined over a 3D domain with the additional inconvenience of the free surface moving boundary. Despite its shortcomings, most mathematical models of flood propagation currently in use, as listed in the introduction, are based upon the SW equations and it seems that it will continue to be for the years to come.

System (4)-(6) is written in the so-called conservation (or divergent) form which can be rewritten in vectorial form as



(7)

where is the space-time domain over which solutions are sought, and the vector of conserved variables and fluxes are given by

where the source term models the effects of the shape of the bed and friction and diffusion on the flow. The geometrical source term along the two axes is given as where

The source term component is the friction vector given as

and

The flow can be characterized as sub-critical, super-critical or trans-critical depending on the local Froude ( ) number where is the characteristic water wave propagation velocity. We point here that solutions of non-linear hyperbolic systems, such as SW type equations, admit discontinuous solutions. These can be developed by the non-linearity of the equations, even when the initial conditions are smooth. Discontinuities are generally called shocks and reflect physical processes. Then the problem of solving numerically equations that admit shocks has to be faced. In SW problems the terminology bore or surge is also used for a moving shock and hydraulic jump for describing a stationary shock. For example dam-break flows in natural river valleys and urban areas are both rapidly varied and often supercritical. Additionally, flow in a city area has some specific features, such as interaction with buildings and other structures, that lead into significant variations of the flow profiles. Moreover, flow discontinuities, such as hydraulic jumps, can occur due to wave reflections from walls for example. Considering these features it is clear that rapidly varied flows with discontinuities must be modeled efficiently, for both rural and urban areas, in order to provide realistic flood simulations.

2.2.1. Saint-Venant (SV) Equations for Channel Flows

Another simplification of considerable practical importance, especially when the flow is markedly unidirectional and the transversal effects are of little

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importance, are the St. Venant equations (or the 1D SW equations) for open channel flows. They can be derived by writing the mass and momentum along the direction of motion and can be written in vector form as follows

(8)

in which

with = wetted cross sectional area, =the water discharge and = is the lateral discharge pre unit time and length into the channel. The hydrostatic pressure force , and the pressure force due to longitudinal width variation are defined as

where = integration variable indicating distance from the channel bottom; = channel width at distance from the channel bed, expressed as

. The pressure force integrals are calculated in accordance with the geometrical properties of the channel.

Hydraulic modeling techniques in 1D are currently based, nearly exclusively, upon the numerical solution of the SV equations. Other reduced complexity models for predicting floodplain inundation have been developed. In flood modeling, there exist practical examples where flows may be described by combinations of 1D and 2D schematizations and as such the use of hybrid 1D/2D models has emerged.

2.2.2. Conservative or Non-Conservative Form?

An important practical, and theoretical, issue is conservation. In SW and SV equations it is possible to write the system in non-conservative or primitive form in terms of depth and velocity. A critical issue is which formulation is better in order to numerically describe a physical situation. No definite answer can be given but few remarks can be done on this matter. The non-conservative form yields a simpler form of the viscous terms. In view of numerical approximations, in this form the convective terms can be discretized by an upwind technique in a straight forward manner upon the velocity direction.

The conservative form, where the vector of unknowns is the vector of physically conserved variables, enjoys some other advantages. First, for convection dominated flows (such as floods) across the shock fronts the Rankine-Hugoniot conditions are automatically satisfied due to the divergence form of the convective terms. In the non-conservative form momentum equations are not defined when . Moreover, the elevation gradient in the momentum equations and the divergence in the continuity equation are often the most important terms, and many physical situations are sufficiently well described by them alone. The differential operator associated with the



conservative form is non-negative and this fact enables the use of suitable numerical methods exploiting this property. Lastly, at the numerical level, one cannot construct a conservative numerical method if the equations do not have a conservative form since non-conservative methods are known for giving incorrect wave propagation speeds by introducing dispersion errors.

We point here that, it is possible to write a system in conservation form using the depth and velocity as the unknowns but this form is physically incorrect, as the equations then state conservation of mass (correct) and velocity (incorrect). These equations are valid only for smooth flows since in the presence of shocks the corresponding jump conditions give a shock with the wrong speed. 3. Numerical Modeling for Flood

Propagation in Natural Topographies

©UNESCO-EOLSS Encyclopedia of Life Support Systems



NUMERICAL FLOOD SIMULATION BY DEPTH AVERAGED FREE SURFACE FLOW MODELS

A. I. Delis Department of Sciences, Technical University of Crete, Chania, Crete, Hellas. Institute of Applied and Computational Mathematics, Foundation for Research and Technology, Heraklion, Crete, Hellas.

N. A. Kampanis Institute of Applied and Computational Mathematics, Foundation for Research and Technology, Heraklion, Crete, Hellas.

Keywords: Floods, Numerical Simulation, Depth averaged models, Shallow water equations, Finite Volume Schemes, Source terms, Wetting and Drying, Benchmarking.

Contents

1. Introduction 2. Mathematical Modeling of Flood Propagation 3. Numerical Modeling for Flood Propagation in Natural Topographies 4. Benchmarking and Validation 5. Concluding Remarks Related Chapters Glossary Bibliography Biographical Sketches

3. Numerical Modeling for Flood Propagation in Natural Topographies

In the last decades many efforts have been devoted to develop one and two-dimensional numerical models for unsteady shallow flows that the most realistic mathematical framework for flood propagation is the one based on the 2D SW equations. Several numerical difficulties must be adequately treated to obtain an accurate solution without numerical errors. A flood simulation model should be able to handle complex topography, dry bed advancing fronts, wetting-drying moving boundaries, resistance and bed slopes, steady and unsteady flows and subcritical and supercritical conditions. These issues are common to any solution of the SW equations and apply to numerical flood propagation in general.

The majority of the numerical models used today perform a separate spatial-temporal discretization, whereby the spatial derivative terms are firstly discretized and then the resulting ordinary differential equation is integrated in time. First order accurate methods can be easily constructed and can provide satisfactory results in some cases but, formally, second order accurate operators both in space and time can also be obtained with little extra effort.

3.1. Spatial Discretization Strategies for the 2D SW Equations

The various computational techniques currently in use for solving the SW equations fall into one of the following three categories, as regards the space discretization: Finite Difference (FD), Finite Volume (FV) and Finite Element (FE) methods.

The implementation of FD methods is straightforward and well defined. Several

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practical applications demonstrate the feasibility of this approach however, their popularity is being limited due to their inflexibility in handling physical domains with complex geometries. FD methods are primarily designed for structured Cartesian grids and the use of a curvilinear transformation of coordinates is necessary then. However, this often results in very complicated PDEs and boundary conditions in the transformed computational domain. The main utility of the FD formulation resides in its simplicity for developing new numerical schemes (especially in 1D) that then can be generalized to FV and several dimensions.

The FE method relies upon a variational formulation (Saiac, 2004) of the motion equations. Its main advantage stems from its rigorous mathematical foundation that allows a posteriori error estimation. Successful applications of the FE method to flood propagation and dam beak flood propagation, river inflows and tidal flows, wave propagation and advancing fronts, have been reported. However, its conceptual difficulty makes the method less favorable in practice. More recent works deal with discretization of the SW equations by Discontinuous Galerkin (DG) FE methods. In the more general framework of conservation laws, the development of DG has been stimulated by several advantages such as their high order of accuracy and their sharp evaluation of shocks.

The FV formulation, (Dubois, 2003; Abgrall, 2003), is nowadays the most applied modeling strategy for the 2D SW equations approximation. The method can be applied to both structured and unstructured meshes and as such the physical domain under study can be divided into a certain number of finite volumes and the SW equations cast in integral form can be applied individually to each one of them. This procedure guarantees, a priori, the conservation of physical quantities like mass and momentum, is extremely flexible and conceptually simple. FV schemes can be categorized as of the cell-centered or the cell-vertex type. In 1D FV and FD are equivalent and, depending on the type of discretization and mesh used this can be true in higher dimensions as well. Due to its simplicity, conceptual consistency and straightforward enforcement of the conservation properties the FV approximation has become the most popular approach in simulating flood propagation phenomena. Note that some FV methods can be proved equivalent to appropriate FE formulations.

The mathematical property of hyperbolicity of the SW equations has, in the main, been responsible for the transfer of numerical methodology, formerly applied in the field of aerodynamics (Dubois, 2003), to hydraulics and related areas in geophysics. Godunov’s approach holds an exceptional position among numerical methods. Godunov schemes have been applied in field-scale modeling applications with success and comparisons of the performance of several flood inundation models, in urban flood modeling applications, found that the cost of a Godunov method can be similar to other explicit and implicit SW models. The distinctive features of a Godunov-type FV approach are, (a) the use of the conservative form of the equations to produce a relation between integral averages of the conserved variables and inter-cell fluxes and (b) the use of wave propagation information, the so called upwinding, into the discretization scheme to compute the inter-cell fluxes and thus produce a numerical method. Many researchers have adopted triangular, unstructured grid formulations of Godunov-type SW models because triangles allow for localized grid refinement and easily conform to domains with irregular shapes; recent urban flood modeling studies have utilized triangular grids to resolve buildings.

Within a FV approximation the partial differential equations (2.7) must be first cast in integral form, over a fixed volume, as follows



(9)

and application of Gauss’s divergence theorem to the flux integral leads to

(10)

where is the boundary of the volume and is the unit outward normal vector. A discrete approximation to Eq. (10) is applied in every computational cell so that, volume integrals represent integrals over the area of the cell and the surface integrals represent the total flux through the cell boundaries. By denoting the average value of the conserved quantities over the volume at a given time, from Eq. (10) the following conservation equation can be written for every cell

(11)

It is common practice, that, given the uncertainties inherent in quantifying horizontal turbulent momentum transfer in 2D models, the term is neglected and its effect to be subsumed into the friction term. For that reason and for brevity, this presentation will concentrate on the spatial discretization of the inviscid fluxes, the topography and friction terms.

By assuming a fixed mesh in time, the above expression indicates that changes in the conserved variables inside the cell are the results of the flux balance across its boundary plus the contribution of the sources. Separate discretization of the flux and source terms is performed. Dealing first with the flux balance, the contour integral is approached via a mid-point rule, i.e.,

where is the numerical flux tensor, is the length of the -th edge of cell and represents the edge index of the cell and depends on the cell type (e.g. for triangles and for quadrilaterals). Now the key to the above discrete form is to evaluate the numerical flux suitably. Different implementations and characteristics of the scheme arise depending mainly on the numerical flux approximation.

One of the most important issues in flood propagation is the location and propagation of wave fronts. The outstanding majority of researchers rely upon shock capturing methods in order to accurately represent the celerity and intensity of water fronts. In the original formulation of the Godunov method the wave propagation information was furnished via local exact solutions of the conservation laws subject to special initial conditions consisting of two constant states separated by a discontinuity. This particular initial-value problem is called Riemann problem. Utilizing algorithms, formerly applied in the field of aerodynamics, shock fronts can be computed with high accuracy and robustness. These methods are mainly based in Riemann solvers, either exact or approximate. The search for simplicity and efficiency has lead to the use and development of sophisticated approximate (linearized) Riemann solvers. However, these are prone to a number of shortcomings. In the presence of trans-critical, or sonic flow, one must incorporate a so-called entropy fix in order to avoid the computation of un-physical solutions and for very shallow flows linearized Riemann solvers may compute negative water



depths.

Among approximate Riemann solvers, Roe’s technique has led the way to many others and, through different adaptations to the SW equations, is the most widely used approach as a shock capturing operator that leads to satisfactory results. In searching for solvers that avoid the above shortcomings other approximate solvers have been developed namely the HLL and HLLE Riemann solvers have been introduced. An HLL Riemann solver assumes a simplified wave structure of the solution, admitting only two wave families and as such this structure is only correct for a 1D system of equations. A known way of correcting this shortcoming and restore the missing wave in the structure of the solution of the Riemann problem is the HLLC Riemann solver.

3.2. Numerical Flux Functions

At each interface , according to Godunov, the problem can be taken as a local one-dimensional Riemann problem in the direction normal to the cell edge, so the numerical flux could be obtained by an approximate Riemann solver. By naming

and the reconstructions of on the right and left side of respectively, and using different Riemann solvers and different reconstructions various methods can be obtained.

3.2.1. Roe’s Riemann Solver Numerical flux

The numerical flux corresponding to Roe’s Riemann solver for the 2D SW equations can be written, for any cell edge, as

where is the matrix whose eigenvalues are the modulus of the eigenvalues of matrix , which is the Roe approximate matrix of the system flux Jacobian J. The Jacobian matrix, J, of the normal flux is evaluated as

and can be expressed as

(12)

The eigenvalues of are

(13)

with corresponding left eigenvectors

(14)

From its eigenvectors, two matrices and can be constructed with the property that they diagonalize the Jacobian J as



where is the diagonal matrix with the eigenvalues in the main diagonal. Roe’s average values for are given by

In the case of an advancing front over dry bed the average velocities are calculated in the form

3.2.2. HLL-Type Riemann Solver Numerical Flux

In this solver the numerical flux is constructed as follows

where and are the left and right wave speeds, respectively, estimated as

where

If the cell on the right or on the left of the interface is dry, the wave speeds become respectively,

and

So far the schemes presented are of first order spatial accuracy. In order to achieve second or higher order accurate space discretization (i.e. high-resolution schemes) some form of interpolation is needed (either based on the flux or on the variables) and these must maintain monotonicity or preserve the total variation of the solution in order that this solution is free from spurious oscillations. The accuracy of the Godunov scheme can be increased to second order following various strategies. A class of non-linear second-order schemes that have been very successful are the so called Total Variation Diminishing (TVD) schemes. These methods are able to capture large gradients of the solution, or even discontinuities, free of oscillations. These methods are also called high-resolution methods and have been widely applied to solve the SW equations.

Unfortunately, there does not appear to be a consensus in the literature about constructing high-resolution schemes. Other approaches that lead to high order of accuracy are beginning to see their way through to applications to shallow water flows. The Essentially Non-Oscillatory (ENO) approach, allows the construction of numerical schemes of accuracy greater than two and has recently find its way to applications. In addition, central-upwind type Godunov-type scheme and relaxation schemes have emerged for the numerical solution of the 2D SW equations.

3.2.3. Higher Order Schemes with MUSCL Extrapolation



A very popular technique of contracting TVD schemes of second order spatial accuracy is the so called Monotone Upwind-Centered Schemes for Conservation Laws (MUSCL) reconstruction procedure. This reconstruction of UL and UR consists of a linear extrapolation of the corresponding variables at cell interfaces in order to achieve higher order spatial accuracy. Unfortunately, the application of the MUSCL technique without suitable adjustments leads to oscillatory solutions near discontinuities. In order to avoid these oscillations, it is necessary to limit the extrapolated solution slope. This can be obtained introducing a non-linear function, called limiter, of the ratio between adjacent gradients. For the characterization of this function the total variation diminishing (TVD) property is introduced. A solution that satisfies this condition is non-oscillatory, of an order of accuracy grater than one, and preserves monotonicity. In recent years many researchers have developed high resolution MUSCL-TVD FV schemes utilizing the Roe or HLL-type solvers and reported impressive results for rapidly varying inviscid flows.

The MUSCL approach used for the reconstruction of and can be described as follows: Let be a cell which has a common edge with the cell and denote the averaged of the conserved variables stored at the centers of and by and , respectively. Two reference quantities and are introduced, where

is an average stored at the node which is opposite to edge of cell , and is an average stored at the node which is opposite to edge of cell . Then, the reconstructions of and are given by

where , and is the limiter function. Several such limiters are described in the literature. For instance the minmod function

is one of the most robust, with a better ability to avoid non-physical negative depths when working on a dry bed.

3.3. Time Integration

The system of ODEs, one of every computational cell, that comes from the semi-discrete system of equations

where , must be integrated in time. Time integration schemes for time-

dependent advective problems are traditionally divided in two main categories, according to the way the time derivative is discretized, as explicit and implicit. A general time integration scheme can be written as

,

where superscripts and denote previous and next time levels on the numerical grid. When the Euler explicit scheme is obtained while for is the Euler implicit scheme.

Implicit schemes can offer numerical stability (not always unconditional, however) at the extra cost of having to deal with the resolution of an algebraic, and often nonlinear, system. Usually a linearization technique is applied in order to avoid



solving nonlinear systems and apply iterative processes. Traditionally implicit schemes have been the most attractive for steady and gradually unsteady flows. However, in cases of unsteady and highly discontinuous flows the allowable time step is limited and instabilities appear in the resolution of moving front waves due to the poor treatment of the linearization of the implicit flux terms, which often are locally evaluated. The use of implicit in practical flood simulations has not become popular, in part due to their higher programming complexity and also to the difficulties in solving the implicit operator in 2D simulations.

The vast majority of researchers and existing software use explicit methods, mostly two-step or Runge-Kutta, that combines well with formally second (or higher) order spatial discretization. The allowable time step size is nevertheless restricted in the explicit case by stability reasons in order to fulfill the Courant-Freidrichs-Levy (CFL) condition although, recently explicit extensions of the 2D upwind FV scheme to values of CFL grater than one have been developed and more recently Local Time Stepping (LTS) techniques have been applied to explicit Godunov-type scheme in order to improve run-time efficiency.

For a first order accurate simulation, a simple forward Euler scheme in time has been proven simple and efficient

.

To maintain stability and obtain second order accuracy in time, an optimal TVD Runge-Kutta time stepping method can be applied

;

.

By numerical stability requirement, the time step must be restricted by a CFL-like condition. For the Roe-MUSCL and HLL-MUSCL methods, the maximum time step is limited by the following

(15)

In Eq. (15), the minimum is taken over all cells in the computational mesh and the maximum is taken over the adjacent cells of .

One popular approach of extending to second order accuracy in space and time, known as the MUSCL-Hancock method, is to compute a predictor solution at

time level that is then linearly reconstructed prior to computing the fluxes. This efficient approach has been adopted by many researchers in order to solve the 2D SW equations using either the Roe or HLL-type Riemann solvers.

3.4. Bed Topography and Friction Terms Discretization

Amongst the numerical techniques reported in the literature, conservative FV methods have gained acceptance for their important property of providing a proper discrete representation of the physical conservation laws. A particular difficulty in using the flux conservative SW equations within the finite volume framework is the approximation of the terms representing the bed slope and friction, usually referred as source terms. Dominant source terms, non-differential terms that are functions of



the unknowns of the problem such as, the bed topography and friction, are of special relevance in flood simulations based on a SW model and can cause important difficulties when using a conservative method since they both can damage the conservative character of the solution. Natural topographies pose the main challenge. This difficulty arises in the flux conservative form of the SW equations since the pressure gradient term, that drives the flow, has been split between the flux and the source terms. For a number of years it has been recognized that source terms can be as challenging as that of treating differential terms. For that reason, a considerable effort has been recently devoted to this topic in a search for a correct source discretization, given a particular numerical scheme that possess good properties for the homogeneous case. One such property, that is usually deemed desirable between numerical modelers, is that a numerical scheme must preserve the properties of a quiescent flow (flow at rest). In other words, there must be a balance between flux and source discretization so that no un-physical oscillations appear when no physical waves are present.

One of the standard ways addressing the source term problem is the method of time operator splitting. This means including the topography (and friction) source term in an integration step separate from that with the fluxes. In this context, the splitting allows consideration of only the fluxes at one step and then the source terms is treated separately to a desired order of accuracy. However, this approach has some disadvantages which include complicated strategies for the integration step, and the inability to properly treat the quiescent flow problem since the source term integration step and that dealing with the fluxes use different initial data. Nevertheless, the operator splitting approach remains a useful solution tool for some practical applications where quiescent flow preservation is not a high priority.

Roe advanced the idea that source terms could also be discretized using wave propagation information, or upwinding. However, upwinding by itself is not sufficient to correctly treat source terms and has been recognized that for steady or nearly steady solutions the correct balance between fluxes and source terms is important. For shallow water flows the upwinding approach was put on a firm basis by the work of Bermudez and Vazquez-Cend n, and entails incorporating the source terms into an upwinded decomposition exactly analogous to that used in the fluxes. Numerical schemes that balance the effect of fluxes and sources have been termed well-balanced. Moreover, high-resolution models also require careful discretization of the bed slope term in order to yield accurate solutions. The construction of well-balanced schemes for shallow flows has become a very popular research topic over the last decade. The topography source term upwind strategy is intimately linked to upwind discretization of the flux terms. This approach has the advantage of incorporating the source term within the solution at each time step; this incorporation occurs within the numerical approximation itself and not as an initial mathematical transformation. For every cell edge of cell the discrete bed source term is decomposed into inward and outward contributions

being

(16)

The average value is computed with



(17)

and the bed increments in each direction are computed in the form

where is the distance between the centroids of the (R)and (L)cells that share the same edge.

For every cell the total contribution of the edge bed source term to the cell source term is made of the sum of the parts associated to inward normal velocity at every edge k

The above strategy ensures a conservative discretization of the topography source term. The above up-winding approach can be extended to second order accuracy (for either flux or slope limited schemes).

An alternative strategy to upwinding has been recently proposed named the surface gradient method (SGM) and the modified SGM (MSGM). In this approach, the water surface level is chosen as the basis for data reconstruction. The merits of SGM are that the resulting scheme is no more complicated than the conventional method for the homogeneous terms and the method is generally suitable for both steady and unsteady SW problems.

Bed friction usually has a destabilizing effect when the water layer is thin (near wet-dry areas) and care must be exercised to prevent blow up of the computation. In order to reduce the instability, a simple semi-implicit treatment can be applied. Introducing an appropriate coefficient, , to weight the variables at the current time step and a coefficient to weight the variables at the previous time step , one obtains

.

Introducing the Jacobian matrix and after some algebraic manipulation, the expression of the time increment of becomes

(18)

The parameter controls how much implicit the computation of the friction source term is; corresponds to an Euler implicit scheme and to a totally explicit one. In most practical cases θmust be equal or grater than 1/2 for stability reasons. In cases where the bottom is abrupt, as it happens in natural topographies, must usually be held as one to keep the computation from blowing up.

3.5. Wetting and Drying

In flood flows the wetting and drying process of the affected areas is a natural behavior. In parts of the affected domain the depth of the water is or becomes zero, that is, there can be dry regions, commonly called dry-bed or dry-bottom regions. In dry regions no flow occurs and the governing equations should behave accordingly.



Numerically the problem is the one dealing with the interface between wet and dry regions. Considering the Riemann problem for the SW equations with one wet and one dry state, the exact solution consists of a single rarefaction wave and even if one uses an exact solver for this particular problem there are still a number of potential numerical difficulties. First, the speed of the wet/dry front is much larger than any of the characteristic speeds of the SW system which may result in computation of time steps, based on eigenvalues, that are too large and violate the stability condition of an explicit method. Second, computing the mean flow velocity from the ratio of updated values for the momentum and the water depth, both being very small quantities near the wet/dry front, will result in large errors. Thus, a typical behavior of the numerically computed wet/dry front can be characterized by un-physical oscillations behind the front, and positional errors that grow as a function of time, potentially leading to erroneous predictions for front arrival times. As regards friction instabilities, when the wetting front arrives, very small depths may be computed that produce huge friction slopes and trigger flow reversals; these may change sign and amplified in time leading to negative depths.

Many numerical methods present an unstable behavior in the wet/dry edge where the transition from zero to a finite depth occurs. This problem does not affect some methods for instance Roe’s or HLL-type methods if properly formulated and care is taken for irregular topographies. A widely spread technique is the threshold technique or to wet the bed by adding a small amount of water to the dry cells in the vicinity of the detected dry front or in dry cells every time the water depth falls below an arbitrary threshold value. This technique is only a programming artifact that may destroy mass conservation but has proven very useful in practical computations. Usually mass conservation is not a problem the threshold parameter needed for stabilizing a robust method as the ones described above is extremely small, values of between 10-6 and 10-9 are common in practical computations. When this technique is used, the local Riemann problem has a different structured to that of the exact one and by wetting the bed one therefore changes the wave speeds. If one then adds source terms due to slope of the bed and higher order terms of accuracy, the situations becomes even more complicated. Nevertheless, if logic is built into the equations and numerical implementation, the algorithm can be shield against most instabilities that can be encountered in practical computations.

On the other hand, flow over dry bed involves a complicated situation that can be analyzed as a boundary condition which is dynamically changing in time with a moving front and continuously expanding or reducing the flow domain. The wetting front advanced over a dry bed can be considered as a moving boundary problem in the context of a depth averaged 2D model. As such an optimum way to deal with it is to find the physical law that best defines the dynamics of the advancing front and use it as physical boundary condition, plugged into the general procedure. In advance over adverse dry bed, the water column tends to zero smoothly and the free surface and bed level tend to reduce to one point, where both the free surface and bed boundary conditions apply simultaneously. This approach is by no means easy and does not solve the discrete problem in a simple way.

From a different point of view, wetting fronts over dry bed can be reduced to Riemann problems in which one of the initial depths is zero, and for simplified conditions the problem can analytically studied. The solution when dealing with adverse slopes identifies a subset of conditions incompatible with fluid motion (stopping flow). Therefore, this technique when combined with a FV method that utilizes the above presented Riemann solvers is unable to correctly simulate still water in a domain of irregular shape, generating spurious velocities in the wet/dry interface and often violating mass conservation.

In order to generate a simple and efficient technique for defining a wetting front

Page 10 of 24NUMERICAL MODELING FOR FLOOD PROPAGATION IN NATURAL TOP...


over adverse slope, one can consider the variable water level and compare it between any two cells (L and R). Then two situations can be identified: (i)

, where nothing has to be done (ii) that corresponds to the stopping conditions and hence something has to be done to modify the basic procedure. Some authors proposed a solid wall treatment at that point but this option does not prove optimal since imposing zero velocity does not guarantee that the mass equation is fulfilled, generating inaccurate jumps in the water depth. Driven by the interest of controlling numerical instability and global mass conservation the following technique can be applied that accurately resolves case (ii). Considering the water at rest , it can be seen that the discretization of the mass equation in order to ensure the water at rest steady state at the LR interface leads to the equilibrium condition

(19)

The above requirement can also be written

,

thus predicting the appearance of negative depths at the outside of the wetted domain. Moreover, the equilibrium condition is not always fulfilled due to the piecewise constant representation of the variables in each cell, leading to the appearance of numerical velocities without physical meaning and thus the problem of solving steady flow is converting into an unsteady one.

In order to avoid these numerical errors the technique proposed is to enforce a local redefinition of the bottom level difference at the interface in order to fulfill the equilibrium condition and therefore mass conservation:

.

While with this modification of the discretized topography function one can treat situations of emerging topography for a flow at rest, further modifications have to be made for a flow in motion because otherwise the water could overtop steps of arbitrary size. This is important in the cases considered in the presentation that involves the advance of flood waves on complex topographies. One way to treat these situations is by reducing to zero the velocity components at the interface LR in order to simulate the fact that the water mass-flow across a wet/dry front is zero. However, with this treatment the computed velocities of an advancing wet/dry front are not always correctly simulated. A further improvement can be applied, and extended to high resolution flux limited schemes, based on the remark that this is only natural at the wet/dry front and not in the whole wet cell and therefore it has to be treated like an internal boundary.

3.6. Boundary Conditions

Concerning the number of physical or external boundary conditions required for a given flow domain, the theory of characteristics provides us, depending on the both the value of the normal velocity through the boundary and the local number, with different possibilities. The rest of the required information at the domain boundaries has to come through the so called numerical boundary conditions and the procedure used to obtain them.

When using a FV method, the idea of using a Riemann solver in order to calculate the flux at the interface of a computational cell can be extended to the boundaries. The variables stored, say, at the center of each cell then the boundary conditions are also imposed there. The values of the variables not prescribed, are calculated from



the usual FV formula, and for that the flux across the edged lying on the boundary can be estimated by means of a ghost cell outside the domain. This is a very popular technique and usually, a ghost cell just duplicates the boundary cell. For the case where the boundary is a solid wall, the ghost cell is a mirror cell in which the depth variable has the same value with that of the boundary cell and the velocities are the same with opposite sign. Then the solution of the corresponding Riemann problem gives the correct zero value for the velocity and the water depth value that corresponds to it.

3.7. Topography Modeling

Accurate bathymetric and topographic data is crucial for flood inundation simulations. In addition to numerical methods, new technologies for the collection of data have recently led to complete changes in the selection of numerical models. In particular, the development of global positioning systems (GPS), differential GPS technology and Geographical Information systems (GIS) has led to far cheaper methods of collecting such data (Venkatachalam, 2007). This has led to the gradual replacement of the hydrograph-based hydrologic models by the bathymetric and topographic-based hydrodynamics or hydraulics models. Similarly, the collection of detailed digital terrain data in river and coastal flood plains has led to the replacement of 1D models by 2D models. Accurate terrain data is increasingly available in the form of digital elevation models (DEMs) obtained by photogrammetric and remote sensing methods such as light detection and ranging (LiDAR) (Lloyd, 2004). DEMs based on LiDAR are preferred because of horizontal resolution, vertical accuracy (similar to 0.1 m) and the ability to separate bare-earth from built structures and vegetation. Other technologies are advancing rapidly enabling large amount of data to be collected at relatively low cost. DEMs based on airborne interferometric synthetic aperture radar (IfSAR) have good horizontal resolution but gridded elevations reflect built structures and vegetation and therefore further processing may be required to permit flood modeling. IfSAR and shuttle radar topography mission (SRTM) DEMs suffer from radar speckle, or noise, so flood plains may appear with non-physical relief and predicted flood zones may include non-physical pools. DEMs based on national elevation data (NED) are remarkably smooth in comparison to IfSAR and SRTM but using NED, flood predictions overestimate flood extent in comparison to all other DEMs including LiDAR, the most accurate. Similarly, river and coastal bathymetric information can be obtained, near river and coastal floodplains, utilizing the LiDAR technology.

In addition, complementary analysis of remotely-sensed data has shown potential for providing quantitative roughness measures. At ~1m horizontal resolution, typical of aerial LiDAR surveys, detailed topographic features such as buildings can be resolved and researchers have began to simulate flooding at this scale. However, an inevitable consequence is the increase in the computational cost of flood simulation at scales approaching 1m is high. In a 2D FV mesh required to be representative of some topography, as the number of cells used to create the mesh increases, the discrete representation of the real problem improves, so the accuracy of the results is enhanced, but the computer simulation time grows. A 1m grid corresponds to 106 cells/km2 leading to fine grid resolution and as such, in order to maintain stability and accuracy, may require up to 1012 operations/km2/day. The practical implication is that, the ratio of computational run time to real time may exceed unity on serial computational platforms which limits the utility of such models for engineering practice. Performance improvements can be achieved with faster hardware as well as more efficient algorithms.

Flow simulations can be more sensitive to mesh resolution than topography data sampling. Experience says that, when the topography is smooth, larger cells can be



used to simulate a flooding event (conforming a coarse mesh) but, if the topography is highly irregular, the number of cells must be increased (fine mesh) in order to allow a correct flow representation. One solution is to generate a mesh with local refinement, in order to reduce the calculation effort and, at the same time, be able to achieve the accuracy that a globally fine mesh would provide. This is of particular interest when the flow pattern is complicated, but also when the topography is irregular and highly variable at some places. Abrupt slope changes in space mean abrupt geometries and a higher influence of the terrain on the flow behavior and thus smaller cells are needed to represent both geometry and flooding. A criterion that can be used in such cases, in order to decide the degree of mesh refinement required, is the variation of the slope in distance i.e. the gradient of the bed elevation.

4. Benchmarking and Validation

In order to assess the accuracy, performance and limitations, and hence validate, a numerical model developed for the simulation of flood propagation using the 2D SW equations, comparisons of the model’s output with data obtained either form laboratory experiments and physical scale models or from field observations are in need. Since real case extreme flood events are not frequently recorded, very few cases are well documented. On the other hand, reproducible physical model scale experiments can provide well documented data. In all cases the data used are almost exclusively water depth histories at different (gauging) locations. Once a model has been satisfactorily validated i.e. its accuracy assessed against real data, one is capable of using it to make predictions. Two popular benchmark test cases are reported here. The first is that of a physical scale model and the other is that of a real event.

4.1. Flood Event in the Toce River

This is a benchmark case used in the EU funded CADAM project. This benchmark test problem has been used by many researchers in order to validate their simulation models. A 5km reach of the Toce River (located in the Occidental Alps, Italy) has been reproduced in a 1:100 scale physical model at ENEL HYDRO laboratories. The model, built in concrete, was 50m long and 13m wide. It reproduces the river-bed, flood plain and some details in the real geometry such as a reservoir (polder), two bridges, a dam and several buildings, as seen in Figure 2. The height of buildings was given separately as the decision whether to include them or not in numerical modeling was up to the project’s participants.

The reach has several geometrical irregularities that lead to the coexistence of subcritical and supercritical flow stares. Several gauge points have been placed along the model to measure the time evolution of water depth during experiments, see Figure 3. The model geometry was defined by

Figure 2. General view of the Toce River physical model

Figure 3. Topography of the Toce river valley and stage gauge locations



ENEL-CESI using a DEM. The physical model was covered by a square grid with 5cm intervals and as such it reproduces the details of the real geometry. The suggested Manning friction coefficient was . The upstream boundary was connected with a tank and its inflow was controlled by a computer regulating pump, with a maximum discharge of 0.5 m3/s. When the water level in the tank rises above the bottom of the first section of the model, water flows into the valley. The model is initially dry with the outflow boundary set to be an open one. The rest of the boundaries are considered as walls (solid boundaries). The flow easily submerges buildings and overtops bridges, producing very high vertical flow jets, which demonstrate the intensity of the event.

The average slope in the valley was about 2%, with some local slopes far exceeding this value; hence the capability of shock-capturing is essential for a numerical model to simulate this flow well. On top of that, the valley shows some narrowing, which leads to jump formations. A numerical model should be able to correctly represent such transitions through the critical state (from supercritical to subcritical and vice versa), in order to obtain reliable results. In natural environments such drastic variations in the local bed elevations and geometry are inevitable.

In order to simulate the Toce flood experiment using a FV scheme, the calculation domain can be covered by an unstructured triangular mesh as seen for example in Figure 4. Since the initial topography was given as squares of 5×5cm, resulting in a (very) fine grid (of approximately 2×106 elements), it has to be modified to the coarser triangular one if a small simulation time is required. Node or cell center values can be obtained by interpolation of the provided DEM. Using a local refinement technique, a good representation of the bathymetry can be achieved in those places presenting a more complex geometry. Mesh resolution should not over or underestimate considerably the height of structures. Houses and buildings in the physical model can be represented in the numerical model as higher points in the topography or as regions with an increased friction coefficient. Gauges could be located at any positions in the physical model, whereas numerical predictions have to be taken form discrete points (cell centroids for example). As such, discrepancies between numerical results and the physical measurements may appear. A grid of approximately 13,000 triangles is shown in Figure 4.

Figure 4. Numerical grid for the Toce flood event simulations

The created event showed the same features as a dam-break occurrence. The resulting dynamic was that of a sharp front, which moves forward through the dry valley with high propagation speed. After passing the reservoir section, the flood developed strong 2D features as it was leaving the channel and spread through the flood-plains. Flood showed an abrupt acceleration in the downstream part of the domain, due to the raise of the bottom slope and the rapid reduction of flow area. Some quantitative comparisons between measured and simulated (by FV schemes) water levels at gauges P2, P9 and P19 are presented in Figure 5.

After comparing the numerical and measured water levels, it is of importance to check if the predicted wave speed agrees with the one in the physical model, for that the experimental travel time was measured between gauges P1 and P26 and is of 40s. The front is considered to arrive when the depth is grater that 10-5 cm, which is the gauge precision.

Discrepancies between numerical results and the physical ones must be attributed to



the intrinsic limits of validity of the 2D SW equations. Particular attention should be devoted to that area in the flow field where streamline curvature is important (non-hydrostatic pressure distribution) and/or important recirculation phenomena take place.

4.2. The Malpasset Dam-Break

The Malpasset dam was located in a narrow gorge of the Reyran river valley in France, see Figure 6. It was a 66.5m high arch dam with a crest of 223m and a maximum reservoir capacity of 55×106m3. The dam was breached on December 2, 1959, at 21:13, causing more than 400 casualties. The entire wall collapsed, and only a few blocks remained on the right bank. Even today, there are still pieces of the dam scattered in the area. The breach created a water flood wall 40 meters high and moving at 70 km/h, destroying two small villages -Malpasset and Bozon, a highway construction site, and in

Figure 5. Comparisons between measured vs simulated water levels at gauges P2, P9 and P19

21 minutes reaching Frejus, still standing 3 meters high. Various small roads and railroad tracks were destroyed as well, water flooding western half of Frejus and finally reaching the sea. Field data of the event were made available by Electricité de France (EDF).

The overall dimensions of the study domain are 17.5km×9km. Elevation of the valley bottom ranges from -20m ABS to +100m ABS. Except for inside the reservoir and the sea, the bottom is dry. The computational domain can be discretized by 13,541 points and 26,000 triangles as was proposed by EDF and is shown in Figure 7. The only open boundary condition is the sea with constant water level (0m). Therefore, boundary conditions are free flow. For the initial conditions, the water level is considered equal to 100m inside the reservoir. The dam was not completely destroyed and the concrete remains can be modeled as a weir of which the crest level is 56.8m ABS, but it is not necessary in a first computation.

This dam-break wave propagation is of a super-critical character and we are interested in simulating front wave arrival times and maximum water levels. During the catastrophic event the local police collected detailed data of the flood wave propagation and its maximum elevation on the right and left banks of the valley. The measured data collected by the police are considered to be of limited accuracy. The propagation times of the front wave are known through knowledge of the exact shutdown time of three electric transformers. The arrival times at the village of Frejus and the sea are known, that respectively occurred at 21:35 and 21:50.

In 1964 a 1:400 was contacted at EDF-LNH in order to get a better understanding of the flooding

Figure 6. The Malpasset dam area map



Figure 7. Numerical grid and locations of: ▲Electrical transformers, ●Police survey points and gauges in the scale model

gauges in the scale model process. The maximum water levels at gauges, located at the center of the valley, was measured and were in good agreement with observed high water marks. The front arrival times were also measured at these gauge points. All the above data can be used to validate a numerical model. A global Manning friction coefficient was estimated by EDF between 0.033 and 0.025. As advised by various researchers, a uniform Manning coefficient of 0.033 provides the best overall agreement. Some of these quantitative comparisons in terms of maximum water level and arrival times in the aforementioned locations are presented in Figure 8.

Figure 8. Comparisons of FV numerical results with real data and physical model experiments for the Malpasset test case

It should be noted that the topography used in the computations was digitized from maps dating back to 1931 because of the dramatic changes in the topography after the event. This may explain why the ground elevation at some locations is not in agreement with the maximum level surveyed by the police. As a result, some flooded areas may be predicted dry by the numerical model. These discrepancies can be considered quite acceptable, bearing in mind the limited accuracy of the field survey and the simplifications adopted in the numerical solutions.

It is also advisable to perform a sensitivity analysis on the friction coefficient and on mesh size with the aim of finding possible improvements on the results produced by a numerical model, as well as explaining discrepancies that may be found with experimental data. Friction has a strong influence on wave celerity. The question whether a variable friction coefficient should be used may be also addressed. As regards mesh refinement studies, it is advisable that, given certain conditions, to reach convergence and produce a mesh independent simulation output.

5. Concluding Remarks

Over the past decades significant progress has been made in the numerical solution of the unsteady flow equations both in 1D and 2D. On the basis of bathymetric and topographic information, robust and efficient simulations of flood wave propagation can be produced. Numerical models are now able to model, without problems, flooding and drying of flood-plains as well as modeling sudden transitions between sub-and supercritical flow states. Such simulation models have proven to be of great value for flood forecasting, flood control, dam-break analysis and flood damage assessment.

Simulation models and tools based upon the SW equations can provide a mathematical framework that is complex enough to represent most physics of actual flood flows while still being computationally tractable. Nevertheless, it has to be recognized that such models are not a true description of reality and as such an



acknowledgeable source of error is introduced in flood propagation simulations. More complex mathematical descriptions, though still of limited practical applicability in this specific field, should be further exploited. The numerical complexity of state of the art models is high, but this is the price paid in order to successfully cope with the difficulties encountered when simulating realistic floods. The choice of methods and algorithms available for computing numerical solutions to the SW equations is very wide. However, few have been thoroughly validated against experimental or more importantly real life data. In depth validation should be performed if the confidence and accuracy of flood propagation simulations is to be raised. Application of such models to practical extreme floods is much more difficult since the spatial and time scales are several orders of magnitude larger when compared to scaled models and the available data is usually not accurate or complete, hence, several hypothesis and approximations have to be made (topography, bed resistance, boundary conditions etc.).

A major impediment in the progress of hydrologic modeling, in general, is not only the inability to simulate physical processes mathematically, or to solve the resulted equations, but also the inability to explicitly consider the spatial variation of model parameters. The sources of uncertainty in flood propagation modeling are several and often difficult to control. The recent trend towards spatial hydrologic modeling is due to advances in the field of geographic information systems (GIS). Spatially distributed models take into account of the spatial-temporal variability of the computational domain and allow users to extract parameters for each computational mesh from the available data in GIS database. In the 2D flood simulation model we need to deal with the irregular computational domain, topography, ground details, and the complicated boundary conditions. In FV methods the conservative forms of the 2D SW equations were discretized upon the computational domain that can be represented by diverse computational mesh with geometry from square to arbitrary form. In most cases these mesh are geometrically fairly simple. The three nodes triangle or the four nodes quadrilateral, are commonly used to generate the arbitrary computational mesh. The advantage of arbitrary computational mesh is that it can fit the boundary of computational domain very well.

Flood propagation models are routinely being used by industry for many years now. However, there are still important questions regarding their applicability and range of use such as, validity of model equations in certain conditions, convergence of model output to a unique flooding scenario and long computational times in practical applications that involve large areas or long periods of time. Further research efforts should primarily aim at increasing the speed and efficiency of the present models while preserving all the built in capabilities and technology (i.e. high spatial and temporal accuracy, shock capturing operators, monotonicity, proper source term integration and wetting/drying ability).

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Glossary

Boundary Conditions

: Constraints and values of variables that are assigned at the boundaries of flow domain.

CFL condition

: A necessary condition for the stability of any explicit numerical method for numerical solving partial differential equations.

Dam : A structure that holds back water, up to a certain level. Explicit : The predicted variables are calculated at each grid time step



Bibliography

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Scheme independently by solution of algebraic equations including the grid values of the variables calculated at the previous time steps.

Finite Difference Method

: Numerical solution of differential equations where approximations of the spatial derivatives are based on a spatial grid of nodes and Taylor series.

Finite Element Method

: Numerical solution of differential equations based on the representation of time and space derivatives in terms of integrals of interpolation functions involving variables defined at grid points of an irregular partition, into elementary parts, of a chosen solution do-main.

Finite Volume Method

: Numerical solution of differential equations where the approximation of the spatial derivatives based on a decomposition of the system into elementary volumes, for which elementary balance equations are expressed.

Flash Flood : A flood event of short duration with a rapidly rising flood wave and a rapidly rising water level.

Flood (flooding)

: A short-term or seasonal inundation of Earths dry land areas by water from any source and due to action of different forces and under the influence of various factors.

Free Surface Flow

: The flow of water on the surface of the soil and in the river channel system

Godunov Methods

: Framework and algorithm in finite volume method, which calculates the approximate fluxes of the field.

Hydrostatic approximation

: The assumption that fluid is in static equilibrium; it is a state where a pressure vertical gradient is balanced only by a gravity force.

Implicit Scheme

: The predicted grid variables are calculated at a given time step simultaneously by solution of systems of algebraic equations including the grid values of the variables at the given time step along with values at the previous time steps.

Mesh (grid) Cells

: Partition of the simulation domain into elementary parts.

Numerical Flux Function

: The cell averages of the solution are evaluated by differences of the flux functions on the boundaries of the cells.

Numerical model

: A system of algebraic equations and discrete correlations which describes some physical processes, natural phenomena, etc.

Shallow-water theory

: A water-wave theory based on the depth-averaged conservation of mass and linear momentum with the assumptions of hydrostatic pressure field and uniform horizontal velocity over the depth.

Simulation : The practice of modeling as much the behavior of a system as possible.

Slope Limiter : Condition that insures, for any scheme, the TVD property. Time operator splitting

: Numerical solution procedure which proceeds in two basic steps (a) approximation of spatial derivatives and (b) time integration.

Total Variation Diminishing (TVD)

: Property at least necessary for proving the convergence of the numerical solution towards the solution of the conservative equation.



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Biographical Sketches

Anargiros I. Delis is an Assistant Professor in the Department of Sciences at the Technical University of Crete (TUC), Greece and a collaborative member at FORTH-IACM. He holds a Mathematics degree form the University of Crete, Greece, an MSc degree in Numerical Analysis and Computing from Manchester University and UMIST, U.K., and received his PhD in Computational and Applied Mathematics from the University of West of England, Bristol in 1998. His research interests are focused on the numerical solution of PDE’s, Computational Fluid Dynamics and Scientific Computing with emphasis to environmental and geophysical flows.

Nikolaos A. Kampanis is a Senior Researcher at FORTH-IACM. He received his PhD in Numerical Analysis (1992), from University of Crete. His research focuses on computational fluid dynamics (environmental, physiological and technological incompressible flows) and on computational wave propagation (underwater and atmospheric sound propagation). He participates to the ERCIM Environmental Modelling Group, organized Workshops of the group and other scientific associations. He participates in EU and national research and development projects. He has 25 journal papers, 1 special issue as editor, 2 book chapters, 1 book as editor, and several papers in conferences proceedings.

To cite this chapter A. I. Delis, N. A. Kampanis, (2009), NUMERICAL FLOOD SIMULATION BY DEPTH AVERAGED FREE SURFACE FLOW MODELS, in Environmental Systems, [Ed. Achim Sydow], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford ,UK, [http://www.eolss.net] [Retrieved February 5, 2010]

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NUMERICAL FLOOD SIMULATION BY DEPTH …...NUMERICAL FLOOD SIMULATION BY DEPTH AVERAGED FREE SURFACE FLOW MODELS A. I. Delis Department of Sciences, Technical University of Crete, Chania,