Chapter 3 Dam-break wave routing - WIT Press · Chapter 3 Dam-break wave routing D. De Wrachien1,...

Chapter 3

Dam-break wave routing

D. De Wrachien1, S. Mambretti2

1State University of Milan, Milan, Italy 2Politecnico di Milano, Milan, Italy

Abstract

Studies of dam-break consider, mainly, situations of clear-water surges. However, under natural conditions a dam-break fl ow can generate extensive debris or encoun-ter fl oating debris in the valley downstream of the dam. Debris fl ows differ from other natural unconfi ned fl ows in the nature of both the fl oating material and the fl ow itself, which is rapid, transient and may signifi cantly infl uence surge height and speed.

Phenomenological analysis defi nes debris fl ows as mixtures of water and clastic material with high coarse particle content, in which collisions between particles and dispersive stresses are dominant mechanisms in energy dissipation. Therefore, their nature mainly changes according to the sediment concentration and characteristics of the sediment size.

The rheological property of a debris fl ow depends on a variety of factors, such as suspended solid concentration, cohesive property particle size distribu-tion, particle shape, grain friction and pore pressure. So, modelling these fl ows requires both a rheological model and constitutive equations for sediment–water mixtures.

To be reliable, a rheological model should possess three major features. It should describe: the dilatancy of the sediment–water mixtures; the soil yield criterion, as proposed by Mohr–Coulomb; and the role of inter-granular fl uid.

Achieving a set of debris-fl ow-constitutive equations is a task which has been given particular attention by the scientifi c community. To properly tackle this problem, relevant theoretical and experimental studies have been carried out during the second half of the last century.

Research work on theoretical studies has traditionally specialized in differ-ent mathematical models. They can be roughly categorized on the basis of three characteristics: the presence of bed evolution equation, the number of phases and the rheological model applied to the fl owing mixture.

Because a debris fl ow essentially constitutes a multi-phase system, any attempt at modelling this phenomenon that assumes, as a simplifi ed hypothesis, homogeneous mass and constant density conceals the interactions between the phases and prevents the possibility of investigating further mechanisms such as the effect of sediment

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www.witpress.com, ISSN 1755-8336 (on-line) WIT Transactions on State of the Art in Science and Engineering, Vol 36, © 2009 WIT Press

doi:10.2495/978-1-84564- 2- /14 9 03

86 DAM-BREAK PROBLEMS, SOLUTIONS AND CASE STUDIES

separation (grading). Modelling the fl uid as a two-phase mixture overcomes most of these limitations and allows for a wider choice of rheological models.

In this context, debris fl ows resulting from a sudden collapse of a dam are often characterized by the formation of shock waves.

It is commonly accepted that a mathematical description of these phenomena can be accomplished by means of 1D or 2D De Saint Venant (SV) or shallow water (SW) equations. These equations yield discontinuous solutions in the form of shocks or bores, which can be diffi cult to represent accurately without the use of appropriate shock-capturing schemes.

In recent years, great effort has been devoted to the construction of effi cient and accurate procedures to solve these problems. Along this line of work, further efforts need to be made to analyse dam-break waves and show the distribution of different size of the solid phase within the mixture profi le in a better way.

With reference to these issues, the chapter aims to provide the state-of-the-art of shock waves and debris-fl ow rheology, modelling and laboratory investiga-tions, along with a glance to the direction that in-depth studies regarding these phenomena are likely to follow in future.

1 Introduction

Debris fl ows and surges, such as those triggered by the sudden collapse of dams, are unique unsteady-fl ow phenomena in which the fl ow changes rapidly and the properties of the moving fl uid mixture of debris and water are very different from those of clear-water fl oods.

Compared with the open-channel fl ow of pure water, where the resistance behaviour is mainly attributed to the boundary turbulent shear stress, the resist-ance behaviour of the debris fl ow depends on the relative importance of the shear stresses arising from different sources. These sources include:

1. the turbulent shear stress due to the channel boundary roughness;2. the solid–liquid mixture viscous stress and yield stress;3. the mixture’s dispersive stress due to sustained frictional contacts;4. the inelastic collisions of solid particles within the fl uid mixture.

Consequently, the characteristics of debris fl ows may differ signifi cantly between events. For these reasons, clear-water surges and debris fl ows triggered by dam-break phenomena will be studied on the basis of the most advanced numerical and experimental investigations nowadays available.

2 Clear–water wave routing

In practical applications, fl ood forecasting involves two steps. First, a fl ood rout-ing model is used to obtain the fl ood wave by routing fl ood events between stream fl ow gauging stations. This fl ood wave must then be put into a hydraulic model based on detailed channel geometry to forecast fl ood levels at key sites. The fi rst

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DAM-BREAK WAVE ROUTING 87

step, fl ood routing, can be defi ned as the process of determining progressively the timing and shape of a fl ood wave at successive points along or throughout a reser-voir. In other words, fl ood routing is a mathematical procedure for predicting the changes in magnitude, speed and shape of a fl ood wave as a function of time at one or more points along a watercourse, waterway or channel (Fread [1]).

Flood routing may be classifi ed as either hydrologic (lumped) or hydraulic (distributed).

Hydrologic routing involves the balancing of infl ow, outfl ow and volume of storage through the use of continuity equation. A second relationship, the storage–discharge relation, is also required between outfl ow rate and storage in the system. This approach has been widely used in the past by hydrologists, because of its simplicity in implementation, calibration and use, but it will not be covered in this book because it has been replaced by more physically based models.

2.1 Governing equations

Hydraulic routing is based on the solution of the continuity and momentum equa-tions for unsteady fl ow in open channel. These two differential equations, known as the SV or SW equations, can be written under the following hypotheses:

1. The fl ow is 1D. This implies that both depth and velocity vary only in the longitudinal direction of the channel.

2. The fl ow is assumed to vary gradually along the longitudinal direction so that the hydrostatic pressure prevails and vertical accelerations can be neglected.

3. The longitudinal axis of the channel can be assumed as a straight line.4. The bottom slope of the channel is small and the channel bed is fi xed.5. Resistance coeffi cients for steady uniform turbulent fl ow are applicable.6. The fl uid is incompressible and therefore of constant density.

These hypotheses do not hold in the case of dam-break phenomena, where the fl ow is multi-dimensional and the vertical accelerations cannot be neglected. To deal with these problems different approaches have been proposed, such as those based on the impulsive wave equations (Favre [2]; Arredi [3]). Anyway, the SV equations have proved to be effective even in cases when not all of the above-mentioned hypotheses hold, but some treatments are required to prevent instability in the integration phase.

The mass-conservation equation can be written as:

∂∂

+∂∂

=Qx

At

0 (1)

and the momentum equation as:

∂∂

+∂

∂+

∂∂

+ − =Qt

Q Ax

gAhx

gA S S( / )

( )2

0 0f (2)

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where A(x, t) is the wetted cross-sectional area; Q(x, t) the fl ow rate; x the spatial co-ordinate; t the temporal co-ordinate; g the acceleration due to gravity; S0 the bed slope; Sf the bed resistance term or friction slope, that can be modelled using different rheological laws (Fread [4]) and h depth of fl ow.

In the past, owing to the mathematical complexity of the SV equations, simplifi -cations were necessary to obtain feasible solutions for important characteristics of a fl ood wave and its movement. This resulted in the development of many simplifi ed fl ow-routing methods such as the so-called diffusive wave, by neglecting accelera-tion terms, which can be written as:

gAhx

gA S S∂∂

+ − =( )f 0 0 (3)

For negligible pressure terms, the equation may be further simplifi ed to yield:

gA S S( )f − =0 0 (4)

known as kinematic wave.While the diffusive wave model neglects the local and convective acceleration

terms but incorporates the pressure term, the kinematic wave’s assumption is that the inertial and pressure effects are unimportant and that the gravity force of fl uid is approximately balanced by the resistive force of bed friction. This means that fl ood waves can be observed as a uniform rise and fall in water surface over a relatively long period of time, and therefore represent the characteristic changes in discharge, velocity and water-surface elevation with time at any location on an overland fl ow plane or along the stream channel. Kinematic waves can be classi-fi ed as uniform and unsteady fl ow.

More powerful expressions of the SV equations are their conservation form with additional terms to account for different characteristics such as lateral fl ows (Stoker [5]), off-channel storage areas (Dronkers [6]) and sinuosity effects (De Long [7]). The extended SV equations consist of the mass conservation equation:

∂∂

+∂ +

∂− =

Qx

s A At

qe ( )0 0 (5)

and the momentum equation:

∂

∂+

∂∂

+∂∂

+ − +⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+ + =

( ) ( / )s Qt

Q Ax

gAhx

S S S L W Bmf ec f

β 2

0 0 (6)

where, q is the lateral infl ow or outfl ow per linear distance along the watercourse (infl ow is positive and outfl ow is negative); A0 is the inactive cross-sectional area (off-channel storage), se and sm are depth-weighted sinuosity coeffi cients that cor-rect for the departure of a sinuous in-bank channel from the x axis of the fl ood-plain; β is the momentum coeffi cient for non-uniform velocity distribution within

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the cross-section (often let equal to 1.0); Sec is the expansion–contraction (large eddy loss) slope; L is the momentum effect of lateral fl ows; WfB is the wind effect on the water surface and B is the wetted top width of the active fl ow portion of the cross-section.

The different terms may be evaluated with a number of formulas.Friction slope Sf is usually evaluated by means of the Manning equation:

Sn Q Q

A Rf =2

2 4 3/ (7)

where n is the Manning roughness coeffi cient and R the hydraulic radius.The term Sec can be computed as follows:

SK Q A

g xecec=

∆∆

( )2

2 (8)

where Kec is the expansion and contraction coeffi cient (negative for expansion, positive for contraction) which varies from −1.0 to 0.4 for an abrupt change in sec-tion. It is obviously equal to 0.0 when the watercourse is straight (Skogerboe et al. [8]; Fouladi Nashta and Garde [9]). The term Δ(Q/A)2 represents the difference in velocity at two adjacent cross-sections separated by a distance Δx.

The lateral fl ow momentum L can be evaluated as:

L qvx= − (9)

where vx is the velocity of lateral infl ow in the x direction of the main channel fl ow.

The wind effect Wf, most often neglected, can be evaluated as:

W V V cf w w w= (10)

where Vw is the velocity of the wind relative to the water along the direction of the main channel, and cw the wind friction coeffi cient.

2.2 Initial and boundary conditions

2.2.1 Initial conditionsValues of water-surface elevation h and discharge Q for each cross-section must be specifi ed initially at time t = 0 to obtain solutions of the SV equations. Initial conditions may be obtained from any of the following:

1. observations at gauging stations, or interpolated values between gauging stations for intermediate cross-sections in large rivers;

2. computed values from a previous unsteady-fl ow solution;

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3. computed values from a steady-fl ow solution, which is the most commonly used method.

In the case of dam break, the initial conditions are sketched in fi g. 1. The prob-lem is due to the presence of the discontinuity in the section of the dam when it is completely and instantaneously removed at time t = 0+.

The fi rst solution was found by Ritter [10], who integrated the equations in the case of sudden collapse of a dam on a horizontal rectangular channel with infi nite extension and no roughness. From the Ritter’s solution, the following value for the water depth in the section where the dam is positioned can be obtained:

h t h( )= =+049 0 (11)

where h0 is the initial water depth at the dam site, which can be used as internal condition for the solution of the numerical problem.

Most authors prefer not to have an internal condition and to adopt more elegant self-contained solutions (Aureli et al. [11]).

To this end, a number of shock-capturing schemes have been developed, as will be described in another paragraph. Moreover, to avoid numerical (non-physical) instabilities artifi cial dissipation terms are sometimes introduced in the equations.

Most codes solve the set of eqns. (1) and (2), neglecting all terms that have been introduced in eqn. (6). Nevertheless, a number of authors (Miller and Chaudry [12]) focused on the effects of curves in channels and proposed numerical models suitable to simulate converging channels, presence of a hydraulic jump, fl ow near a spur-dike and other cases (Dammuller et al. [13]; Molls and Chaudry [14]).

One key assumption of SV equations is that streamline curvature effects are small, so that the pressure distribution is hydrostatic. This feature, however, was taken into account by Basco [15] who extended the SV equations to the Boussinesq system.

X

t � 0

t � 0�

t � 0� DAM

X0

Figure 1: Initial and boundary conditions for dam-break wave.

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Initial stages of the waves have similar problems and should be simulated tak-ing in due account the turbulence terms (Shigematsu et al. [16]).

Other problems are mainly related to the scheme adopted for the numerical integration and to the sediment transport.

2.2.2 Boundary conditionsBoundary conditions can be external and internal.

Values for the unknowns at external boundaries (the upstream and down-stream extremities of the routing reach) of the watercourse must be specifi ed to obtain solutions to the SV equations. In fact, in most unsteady-fl ow applications the unsteady disturbance is introduced at one or both the external boundaries.

Either a specifi ed discharge or water-surface elevation time series (hydrograph) can be used as the upstream boundary condition. The hydrograph should not be affected by downstream fl ow conditions. Boundary conditions have to be assigned upstream and downstream when the fl ow is subcritical, and both of them have to be assigned upstream when the fl ow is supercritical.

Specifi ed water-surface elevation time series, or a tabular relation between discharge and water-surface elevation (single-valued rating curve) can be used as the downstream boundary condition. Another downstream boundary condition can be a loop-rating curve based on the Manning equation.

This condition allows the unsteady wave to pass the downstream boundary with minimal disturbance by the boundary itself, which is desirable when the routing is terminated at an arbitrary location along the watercourse and not at a location of actual fl ow control such as a dam. The downstream boundary condi-tion can also be a critical fl ow section such as the entrance to a waterfall or a steep reach.

When the downstream boundary is a stage–discharge relation (rating curve), the fl ow at the boundary should not be otherwise affected by fl ow conditions farther downstream. Although there are often some minor effects due to the presence of cross-sectional irregularities downstream of the chosen boundary location, these usually can be neglected unless the irregularity is so pronounced as to cause signifi cant backwater or drawdown effects. When these situations are unavoidable, the routing reach should be extended downstream to the dam in the case of the reservoir or to a location downstream of where the major tributary enters. Sometimes the routing reach may be shortened by moving the downstream boundary to a location farther upstream where backwater effects are negligible.

Along a watercourse, there are locations such as a dam, bridge or waterfall (short rapids) where the fl ow is rapidly varied rather than gradually varied in space. In these cases, empirical water elevation–discharge relations, as weir fl ow, are utilized for simulating rapidly varying fl ow.

Some authors (Miller [17]) considered the role of spatially varying bound-ary conditions and fl ow patterns as determinants of geomorphic effectiveness. Slope-area measurements are typically made along straight reaches of uni-form or nearly uniform width, and therefore cannot account for signifi cant

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longitudinal variations in boundary conditions that may affect fl ood hydraulics elsewhere along the valley. Hydraulic reconstructions based on the standard step method can account for longitudinal variability, but cannot predict lateral variations within a cross-section or the spatial pattern of fl ow vectors. To explore these features of the fl ow fi eld it is necessary to use two-dimensional (2D) fl ow models.

2.3 Cross-sections

Much of the characteristics of a specifi c fl ow-routing application is featured in particular cross-sections located at selected points along the watercourse. These sections may be classifi ed as either active or inactive (dead) sections (fi g. 2).

The portion of the channel cross-section in which fl ow occurs is called active.Cross-sections may be of regular or irregular geometrical shape. Usually, a

cross-section is described by tabular values of channel top width and water-surface elevation which constitute a piecewise linear relationship. Areas or widths associ-ated with a particular water-surface elevation are linearly interpolated from the tabular values. Cross-sections at gauging station locations are generally used as computational points. Such points are also specifi ed at locations along the river where signifi cant cross-sectional or fl ow-resistance changes occur or at locations where major tributaries enter.

There can be portions of a cross-section where the fl ow velocity is negligible rela-tive to the velocity in the active portion. The inactive portion is called off-channel (dead) storage: it is represented by the term A0 in eqn. (5). Off-channel storage areas can be used to effectively account for adjacent embankments, ravines or tributaries which connect at some elevation with the fl ow channel but do not convey fl ow in the main direction; they serve only to store some of the passing fl ow. Sometimes,

Dead storageActive

River Dead storage

A�

A

A-A�

Figure 2: Plan view of river with active and dead storage areas, and cross-section view.

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off-channel storage can be used to simulate a heavily wooded fl oodplain that primarily stores some of the fl oodwaters while conveying a very minimal portion of the fl ow.

Generally, dead storage cross-sectional properties are described by width (dead storage) vs. elevation tables.

2.4 2D fl ow models

2D unsteady-fl ow models, which account for momentum conservation across the waterway perpendicular to the longitudinal direction, can be categorized as complete (all terms in the momentum equation appear) or simplifi ed (inertial or acceleration terms in each momentum equation are neglected).

Complete or simplifi ed 2D fl ow-routing models have been used for unsteady fl ows in complex fl oodplains (Cunge [18]). Generally, the additional accuracy gained does not justify their use to predict water-surface elevations and average fl ows in usual unsteady-fl ow applications dealing with fl oodplains.

2.5 Hybrid models

Until recently, hydraulic models were not reckoned a practical alternative for fl ood routing because they were considered not economically viable to obtain cross-section data over the reaches involved in fl ood routing.

Recent investigations (Hicks [19]) have revealed that hydraulic routing can be successfully used to determine discharge hydrographs in reaches where littlechannel geometry data are available, by approximating the model reach by arectangular channel.

It was found that this ‘limited geometry’ modelling approach – based on 1D SV equations – could accurately determine discharge hydrographs, making it an effective and suitable alternative to hydrologic fl ood routing. It was also found that this hybrid model offers the advantage of operationally combining the fl ood routing and the determination of fl ood levels (Blackburn and Hicks [20]). In addi-tion, the use of a hydraulic model opens up the potential for modelling more dynamic fl ood events such as ice jam release surges, which cannot be handled by traditional hydrological modelling approaches.

3 Debris-fl ow routing

Studies of dam-break fl ows consider, mainly, situations of clear-water surges. However, under natural conditions, a dam-break fl ow can generate extensive debris or encounter fl oating debris in the valley downstream of the dam.

This chapter investigates the effects of fl oating debris on dam-break surges.Debris fl ows differ from other natural unconfi ned fl ows (Johnson and Rodine

[21]; Meunier [22]) in two main respects:

the nature of the fl owing material, constituted by a mixture of water, clay and granular materials;

–

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the nature of the fl ow itself, which is rapid, is transient and includes a steep front mainly constituted of boulder.

The rheological property of a debris fl ow depends on a variety of factors, such as suspended solid concentration, cohesive property, particle size distribution, particle shape, grain friction and pore pressure.

Various researchers have developed models of mud and debris-fl ow rheology. These models can be classifi ed as: Newtonian models (Johnson [23]; Hunt [24]), linear and non-linear viscoplastic models (Johnson [23]; O’Brien and Julien [25]; Liu and Mei [26]; Huang and Garcia [27], [28]), dilatant fl uid models (Bagnold [29]; Takahashi [30]; Mainali and Rajaratnam [31]), dispersive or turbulent stress models (Arai and Takahashi [32]; O’Brien and Julien [25]; Hunt [24]), biviscous modifi ed Bingham model (Dent and Lang [33]) and frictional models (Norem et al. [34]; Iverson [35]). Among these, linear (Bingham) or non-linear (Herschel–Bulkey) viscoplastic models are most widely used to describe the rheology of laminar debris/mud fl ows (Liu and Mei [26]; Jang [36]; Huang and Garcia [28]; Jiang and LeBlond [37]).

Because a debris fl ow is, essentially, a multi-phase system, any attempt at modelling this phenomenon that assumes, as a simplifi ed hypothesis, homogene-ous mass and constant density, conceals the interactions between the phases and prevents the possibility of investigating further mechanisms such as the effect of sediment separation (grading).

Modelling the fl uid as a two-phase mixture overcomes most of the limitations mentioned above and allows for a wider choice of rheological models such as Bagnold’s dilatant fl uid hypothesis (Takahashi and Nakagawa [38]; Shieh et al. [39]), Chèzy type equation with constant value of the friction coeffi cient (Hirano et al. [40]; Armanini and Fraccarollo [41]), models with cohesive yield stress (Honda and Egashira [42]) and the generalized viscoplastic fl uid or Chen’s model (Chen and Ling [43]).

3.1 Flow profi le

Debris-fl ow dynamics can be divided into three distinct time phases (Lorenzini and Mazza [44]). The fi rst is a brief initial phase in which the density of the mix-ture differs very slightly from the water, the solid particles do not interact and rapid ‘saturation’ takes place. This is followed by a longer intermediate phase, which corresponds to the developed motion of the debris fl ow, in which density is particularly high and interactions become predominant. In the third and fi nal phase, the process ends, following deceleration, thus resulting in the deposition of the debris and the formation of a rigid skeleton, corresponding to the consolidated solidifi cation, through which the water fi lters out.

A typical debris-fl ow wave can be spatially divided into three main regions, as shown in fi g. 3.

The anterior section of the fl ow (also known as the snout) forms the head; it has a forward protuberation composed mainly of rock fragments due to the

–

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reverse grading phenomenon (Takahashi [45]). The second region (commonly known as the body), characterized by an increasingly tapered fl ow, is composed of a turbulent mixture of water, fi ne particles and sediment transported along to course, whereas the last region, which is slender and much more diluted than the previous two, is characterized by a high mud concentration and continues to fl ow long after the wave has passed.

3.2 Rheological models

Modelling debris fl ow requires a rheological model (or constitutive equations) for sediment–water mixtures. Various models relating stress, strain and time, among other variables, have been developed.

Advanced theoretical models, despite their general validity and applicability, are too complicated to be useful in practice, whereas the use of simpler semi-empirical models is limited to a narrow range of applications for lack of adaptability. There-fore, the selection of a rheological model has constituted one of the major issues in debris-fl ow modelling.

A general model which can realistically describe the rehological properties of debris fl ow should possess three main features. The model should:

describe the dilatancy of sediment–water mixtures;take into account the soil yield criterion, as proposed by Mohr–Coulomb;assess the role of inter-granular or interstitial fl uid.

On the whole, a rheological model of hyper-concentrated fl ow should involve the interactions of several physical processes.

3.2.1 Non-Newtonian fl uid modelsThe non-Newtonian behaviour of the fl uid matrix is ruled, in part, by the cohe-sion between fi ne sediment particles. This cohesion contributes to the yield stress, which must be exceeded by an applied stress to trigger fl uid motion. For large rates of fl uid matrix shear (as might occur on steep alluvial fans), turbulent stresses may be generated. In these cases, an additional shear stress component

–––

Figure 3: The various regions within a debris fl ow wave.

III region

II regionI region

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arises in turbulent fl ow from the collision of solid particles under large rates of deformation.

In general terms, the total shear stress τ in debris and hyper-concentrated fl ows can be assessed from the summation of fi ve shear stress components (O’Brien et al. [46]):

τ τ τ τ τ τ= + + + +c mc v t d (12)

where τc is cohesive yield stress; τmc Mohr–Coulomb shear stress; τv viscous shear stress; τt turbulent shear stress and τd dispersive shear stress.

When written in terms of shear rates du/dy, the total shear stress can be expressed according to the following quadratic model (O’Brien and Julien [47]):

τ τ µ= +⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟y

dudy

cdudy

2

(13)

where:

τ τ τy = +c mc (14)

and:

c l f c d= +ρ ρm m v s2 2( , ) (15)

with µ: dynamic viscosity; u: local velocity; y: co-ordinate axis; c: inertial shear stress coeffi cient; ρm: mass density of the mixture; l: Prandtl mixing length; ds: sediment size; cv: volumetric sediment concentration.

The Mohr–Coulomb stress can be assessed as:

τmc = p tanΦ (16)

p being the inter-granular pressure and Φ the angle of repose of the material.Bagnold [29] defi ned the function f (ρm, cv) as:v

f c acci( , )

* /

ρ ρm v mv

=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

−⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1 3

1 (17)

where ai is an empirical coeffi cient equal to 0.01; c* the maximum static volume concentration for the sediment particles.

The fi rst two terms in eqn. (13) are referred to as the Bingham shear stresses, due to the fact that they represent the internal resistance stresses of a Bingham fl uid. The sum of the yield stress and viscous stress defi nes the shear stress of a cohesive, hyper-concentrated sediment fl uid in a viscous fl ow regime, while the last term represents the sum of the dispersive and turbulent shear stresses, which depend on the square of the vertical velocity gradient (Julien and O’Brien [48]).

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In view of theoretical soundness behind the development of the non-Newtonian rheological models, Bailard [49], Jenkins and Cowin [50], Bailard and Imman [51] and Hanes [52] have questioned the validity of Bagnold’s empirical relations. Moreover, when the fl ow changes from a quasi-static state to a fully dynamic or macro-viscous state, the relation (16) between the total (grain) shear stress and the inter-granular pressure does not hold. To overcome these problems, Chen [53] developed a generalized viscoplastic fl uid (GVF) model, based on two major rheological properties (i.e. the normal stress effect and soil yield criterion) for general use in debris-fl ow modelling.

3.2.2 GVF modelThe analysis that Chen [54] conducted on the various fl ow regimes of a granular mixture identifi ed three regimes: a quasi-static one, which is a condition of incip-ient movement with plastic behaviour; a macro-viscous one at low shear rates, in which viscosity determines the mixture behaviour and fi nally a granular inertial state, typical of rapid fl owing granular mixtures, dominated by inter-granular interactions. Chen developed his model starting from the assumption that a gen-eral solution should be applicable through all three regimes.

Chen observed that an applicable model, able to realistically describe debris fl ows, should consider both an independent term and one dependent on the shear rate in a perpendicular direction to the main fl ow (fi g. 4) as well as the main rheological properties: the effect of normal stress and the Mohr–Coulomb yield criterion.

By introducing the hypothesis according to which the viscous effect should not coexist with turbulent and dispersive effects, Chen developed the following set of theoretical relations:

τ τ φ φ µη

zx xz c pdudz

( ) cos sin= = + +⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟1 (18)

h

z

Flow

u(z)

Bed

(x,z)

Tzz

Tzx

Txz

Txx Txx

Txz

Tzx

Tzz

0 x

Figure 4: Defi nition diagram for a uniform and steady debris fl ow under a bidimensional state of stress in Chen’s GVF model.

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τ τ µη

zz xx pdudz

( )= = − +⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟2 (19)

with x, z the co-ordinates in the longitudinal and normal (positive, upward) direc-tion of fl ow; τzx, τzz the total shear and normal stresses, respectively, at a point (x,z) as shown in fi g. 4; c the cohesion; µ1 and µ2 the consistency and cross-con-sistency indices, respectively and η the fl ow-behaviour index.

The summation of the fi rst two terms on the right-hand side of eqn. (18) is normally referred to as the yield stress:

S c p= +cos sinφ φ (20)

and represents an extended form of the Mohr–Coulomb yield criterion. The value of η may vary from 1 to 2 (or possibly higher) as the fl ow changes from a macro-viscous to a grain-inertial state, covering the entire spectrum of Newtonian models.

If c and ø do not vary with z, eqns. (18) and (19) can be expressed in the follow-ing differential form (Chen [53], [55]):

ddz

dpdz

ddz

dudz

zxτφ µ

η

=⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ +

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

sin 1 (21)

ddz

dpdz

ddz

dudz

zzτµ

η

= − +⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2 (22)

Further generalization of eqns. (18) and (19) or eqns. (21) and (22) have been proposed by Tsubaki et al. [56], by extending the Bagnold’s model of a binary collision of grains to a more general case of the multiple collision of grains.

3.2.3 Multi-layer models

All the models previously reviewed feature monotonic velocity profi le that, gen-erally, do not agree with experimental and fi eld data. In many experimental tests (Takahashi [57]; Shen and Ackermann [58]) ‘S’ reversed shaped trends have been observed, where the maximum shear rate is not achieved near the bed, but rather between the bed and the free surface. The main discrepancy is derived from the assumption of a debris fl ow as a uniform mixture. In fact, the solid concentration distribution is usually non-uniform due to the action of gravity, so that the lower layer could, consequently, have a higher concentration than the upper layer. This is specifi cally the case of ‘immature’ debris fl ow, while, in mature fl ows, due to the inverse grading effect, larger debris are on the surface (and front), and fi ner ones towards the bottom.

Higher concentration means higher cohesion, friction and viscosity in the fl ow. For this reason, the velocity profi le for non-uniform concentration is altered. To clear this hurdle, Wan and Wang [59] proposed a multi-layer model, known as the laminated layer model, that features a stratifi ed debris fl ow into three regions

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from the bed to the surface: a bed layer, in which an additional shear stress is dominant in momentum exchange; an inertial layer, where the dispersive stress of the grains is dominant and an upper viscoplastic layer, which can be represented by the Bingham’s model.

Later on, Takahashi [60] proposed a so-called unifi ed model of inertial debris fl ow, by altering the constitutive equations from the theory of granular mixtures and suspended-load transport. In this model, the author hypothesized the fl ow as a two-layer system: a lower layer dominated by collisions, and an upper one composed of the turbulent suspension (fi g. 5).

The relative extension of the two layers depends on the concentration and diameter of the particles: the range goes from the stony debris fl ows in which only collision layer exists, to muddy debris fl ows in which the entire fl ow is composed of a turbulent layer.

As previously described, the rhelogical behaviour of a mixture depends on the hydrodynamic characteristics of the fl ow and the properties of the sediment, which can vary from one fl ow to another.

The one-layer models are unable to adequately feature the entire thickness of the fl ow and, therefore, it has recently become common to use multi-layers models that combine two or more constitutive relationships to analyse adequately these phenomena. The coeffi cients of the rheological models have wide ranges of variation and, therefore, in evaluating them considerable errors are committed. On the other hand, some empirical equations of velocity are necessary in any debris-fl ow disaster-forecasting measure, although the hydraulics of debris fl ow is not theoretically comparable to that of a traditional water fl ow, due to the high sediment condition and the interaction between the solid particles. Being more diluted than the head, the main body of the fl ow could be described by a tradi-tional law of resistance such as that of Manning or Chézy, and simulated by a set of mass and momentum equations averaged on the fl ow depth.

3.3 Governing equations

Shock waves with fl oating debris are complex two-phase systems, and so model-ling the fl ow as a two-phase mixture is the best way to predict these phenomena.

Figure 5: Inertial debris fl ow in the multi-layer Takahashi’s model.

Turbulent layer

Collisional layer

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The change in debris-fl ow density can be modelled through mass and momen-tum balance of both phases (solid and liquid) and interactions between the two phases could be assessed by means of appropriate additional terms (Wallis [61]; Wang and Hutter [62]), while the erosion/deposition rate can be controlled by the excess of the local instantaneous concentration over the equilibrium con-centration (Egashira and Ashida [63]; Honda and Egashira [42]). Many other models, the so-called quasi-two-phase models, use different forms of motion equations – SW or SV equations – together with erosion/deposition and mass conservation equations for the solid phase and take into account mixtures of varying concentrations.

Recently, Mambretti et al. [64], [65] and De Wrachien and Mambretti [66] proposed a general model suitable to analyse both non-stratifi ed (mature) and stratifi ed (immature) fl ows, i.e. when the solid/liquid mixture is present in the lower layer, while only water is present in the upper one (fi g. 6). The model, based on 1D SV equations, takes into account mass and momentum conservation balance for each phase and layer and energy exchange between layers. The level of maturity of the fl ow is assessed by an empirical, yet experimental based, crite-rion. The model could also be improved to predict and assess the propagation and stoppage processes of debris and hyper-concentrated fl ows, triggered by extreme hydrological events, once validated on the basis of laboratory or fi eld data.

3.3.1 One-phase modelThe 1D approach for unsteady debris fl ow triggered by dam-break is governed by the SV equations. This set of partial differential equations describes a system of hyperbolic conservation laws with source term (S) and can be written in compact vector form as:

∂∂

+∂∂

=V F

St x

(23)

hmx

hcw

Clear water

Mixture debris/water

Figure 6: Scheme of the immature (stratifi ed) debris fl ow.

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where:

V F S=⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

= ( )+

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟=

−( )+AQ

Q

Q A gI gA i S gIi2

1

0/ 22

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

where A(x, t) is the wetted cross-sectional area; Q(x, t) the fl ow rate; x the spatial co-ordinate; t the temporal co-ordinate; g the acceleration due to gravity; i the bed slope; Si the bed resistance term or friction slope, that can be modelled using different rheological laws (Rodriguez et al. [67]).

The pressure force integrals I1 and I2 are calculated in accordance with the geometrical properties of the channel. I1 represents a hydrostatic pressure form term and I2 represents the pressure forces due to the longitudinal width variation, expressed as:

I H x d I Hx

dh h

10

20

= − = −∂∂∫ ∫( ) ( , ) ( )η σ η η η

ση (24)

where H is the water depth; η the integration variable indicating distance from the channel bottom; σ(x, η) the channel width at distance η from the channel bed, expressed as:

σ ηη

η( , )

( , )x

A x=

∂∂

(25)

To take into account erosion/deposition processes along the debris fl ow propa-gation path, which are directly related to both the variation of the mixture density and the temporal evolution of the channel bed, a mass-conservation equation for the solid phase and an erosion/deposition model have been introduced in the SV approach. Defi ning the sediment discharge as:

q x t EB( , ) = (26)

where E is the erosion/deposition rate; B the wetted bed width, the modifi ed vec-tor form of the SV equations can be expressed as follows:

∂∂

+∂∂

=V F

St x

(27)

where:

V F=

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟= +

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟AQ

c A

Q

Q A gIc Qs s

( / )21

⎟⎟⎟⎟⎟⎟⎟⎟⎟

= − +

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟S

q x tgA i S gI

Ec Bi

( , )( )

*2

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where cs is the volumetric solid concentration in the mixture and c* the bed volu-metric solid concentration.

3.3.2 Two-phase mathematical model3.3.2.1 Non-stratifi ed (mature) fl ows The model includes two mass and momentum balance equations for both the liquid and solid phases, respectively. The interaction between phases is simulated according to Wan and Wang hypoth-esis [59]. The system is completed with equations to estimate erosion/deposition rate derived from the Egashira and Ashida [63] relationship and by the assump-tion of the Mohr–Coulomb failure criterion for non-cohesive materials.

Mass and momentum equations for water can be expressed as:

∂

∂+

∂∂

=Q x t

xc A x t

tl l( , ) ( ( , ))

0 (28)

∂∂

+∂∂

⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟= − −

∂∂

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟−

Qt x

Qc A

gc A i JHx

Fl l

llβ

2

(29)

where Q1(x, t) is the fl ow discharge; cl the volumetric concentration of water in the mixture; β the momentum correction coeffi cient (assumed β = 1); J the slope of the energy line according to Chézy’s formula; i the bed slope and F the friction force between the two phases.

According to Wan and Wang [59], the interaction of the phases at single gran-ule level f is given by:

f cd u u

u u=−

−Dl l s

l sπ ρ50

2

4 2( )

(30)

where cD is the drag coeffi cient; ul the velocity of water; us the velocity of the solid phase; d50 the mean diameter of the coarse particle and ρ1 the liquid density.

Assuming grains of spherical shape and defi ning the control volume of the mixture as:

V BH dx BHdxc = ≈cos ϑ (31)

where � is the channel slope angle, which holds for low channel slopes, the whole friction force F between the two phases for the control volume can be written as:

F c u u u uc

dHBdx= − −

34 50

D l l s l ssρ ( ) (32)

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Mass and momentum conservation equations for the solid phase of the mixture can be expressed as:

∂

∂+

∂∂

=( )

*c A

tQx

Ec Bs s (33)

∂∂

+∂∂

⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟= −

−+

∂∂

+

+

Qt x

Qc A

g c iHx

A F

g

s

s

s

s

s ls

s

βρ ρ

ρρ

221( )

−−− +

−ρρ

δρ ρ

ρl

ss

s l

ssc i tg A g c Ai( )2 1 (34)

where Qs(x, t) is the discharge of the solid rate; ρs the solid phase density.According to Ghilardi et al. [68] and Egashira and Ashida [63], the bed volu-

metric solid concentration c* is assumed to be constant and the erosion velocity rate E a function of the mixture velocity u:

E uk tgE= −( )ϑ ϑf e (35)

with kE: coeffi cient equal to 0.1 according to experimental data (Egashira and Ashida [63]; Gregoretti [69], [70]; Ghilardi et al. [68]).

Positive or negative values of E correspond to granular material erosion or deposition, respectively.

�f and �e represent the energy line and the bed equilibrium angles, respec-tively, expressed as (Brufau et al. [71]):

ϑϑf arc=

⎡

⎣⎢⎢

⎤

⎦⎥⎥

tgJ

cos (36)

ϑρ ρ

ρ ρ ρφe

s s

s sarc =

−− +

⎡

⎣⎢⎢

⎤

⎦⎥⎥tg

cc

tg( )

( ) (37)

where the debris-fl ow density is defi ned as:

ρ ρ ρ ρ= − +( )s l s lc (38)

and φ is the static internal friction angle. u is defi ned as follows:

u c u c u= +s s l l (39)

For J the Takahashi [45] equation can be assumed, according to the dilatant fl uid hypothesis developed by Bagnold [29]:

J Su

d H a c c gR= =

+ −i

b s s l ssen

2

5022 5 1 1(( / )( / )) ( / )[ ( )( / )]λ δ ρ ρ

(40)

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where Si is the friction term and R the hydraulic radius given by:

RAP

= (41)

where P is the wetted perimeter.The quantity λ (linear concentration) depends on the granulometry of the solids

in the form:

λ =−

cc c

s

m s

1 3

1 3 1 3

/

/ / (42)

where cm is the maximum packing volume fraction (for perfect spheres cm = 0.74); ab the empirical constant.

For high values of sediment concentration, the resistance is mainly caused by dispersive stress and the roughness of the bed does not inf luence the resist-ance (Scotton and Armanini [72]). For low values of the same characteristic the energy dissipation is mainly due to turbulence in the interstitial f luid and the inf luence of the wall roughness becomes important. In such a case, Takahashi [45] suggests to use the Manning’s equation or similar resistance laws.

3.3.2.2 Stratifi ed (immature) fl ows As shown in fi g. 6, debris fl ows are cat-egorized as stratifi ed or immature whenever the solid/liquid mixture is present in the lower layer, while only water is present in the upper one.

Assuming hmx and hcw as the depths of the mixture and of the clear water respectively, the total depth of the debris fl ow hdf is equal to:

h h hdf mx cw= + (43)

while the maturity degree dm is assessed as the ratio:

dhhmmx

df= (44)

Larcan et al. [73] has suggested – on the basis of laboratory experiments – to distinguish mature and immature debris fl ow by means of a criterion based on mixture velocity and concentration (fi g. 7).

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The fi gure underlines the effectiveness of the above-mentioned criterion and depicts a boundary between mature and immature debris fl ow. The boundary Cs boundary can be expressed by:

Cs boundaryFr when Fr

when Fr= − ⋅ <≥{0 5 0 08 5

0 1 5. .. (45)

where Fr is the Froude number, while the maturity degree dm can be assessed as:

dC

Cms effective

s boundary= (46)

The experimental tests showed that in the fi rst phase the fl ow is stratifi ed; then, usually, it becomes mature, because the velocities and the concentrations are quite high. Finally, the tail of the wave is characterized by low velocities, due to the fact that the solid phase tends to deposit, and thus the fl ow becomes again stratifi ed.

Mass and momentum equations for clear water in the higher layer (cw) can be expressed in conservative form as:

∂

∂+

∂∂

=Q x t

xA x t

tcw cw( , ) ( , )

0 (47)

∂

∂+

∂∂

⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟= − −

∂∂

−Q

t xQA

gA i JH

xJcw cw

cwcw cw

cwtwo layerβ

2

ss⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟ (48)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9 10 11 12

Froude number

Cs

MatureImmature

Figure 7: Characteristics of mature and immature debris fl ows.

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The resistance term Jcw can be assessed on the basis of bank shear stress, while the slope of the energy line, Jtwo layers, due to the lower layer, according to Chézy’s formula, is expressed as:

Jn u u

Rtwo layerscw mx=

−2 2

4 3

( )/

(49)

n being the Manning’s number and umx the velocity of the lower layer. The drag force Ttwo layers, between the higher layer and the lower one, can be expressed as:

T gA Jtwo layers cw two layers= (50)

In the same ways as eqns. (28) and (29), mass and momentum equations for the liquid phase in the lower layer (mx) can be expressed as:

∂

∂+

∂

∂=

Q x t

x

c A x t

tl,mx l,mx mx( , ) ( ( , ))

0 (51)

∂

∂+

∂∂

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟= − −

Q

t x

Q

c Agc A i Jl,mx l,mx

l,mx mxl,mx mx mxβ

2∂∂

∂

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

− +

Hx

F c T

mx

l,mx two layers

(52)

Ttwo layers is opposite in sign with respect to eqn. (50) due to the fact that the higher layer, with greater velocities, exerts a drag force to the mixture.

Likewise as in eqns. (33) and (34), mass and momentum equations for the solid phase in lower layer can be expressed as:

∂

∂+

∂

∂=

( )*

c A

t

Q

xEc Bs,mx mx s,mx

(53)

∂

∂+

∂∂

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟= −

−Q

t x

Q

c Ag cs,mx s,mx

s,mx mx

s l

ss,mxβ

ρ ρρ

2

(11

1

2

2

+

∂∂

+ +−

−

+−

i

Hx

A F g c i tg A

g c A

)

( )mxmx

s l

ss,mx mx

s l

ss,mx

ρ ρρ

δ

ρ ρρ mmx s,mx two layersi c T+

(54)

3.3.3 Numerical modelThe SV equations for 1D two-phase unsteady debris fl ow can be expressed in compact vector form as follows:

∂∂

+∂∂

+∂∂

=V F F

St x

Cx

' '' (55)

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where, for a rectangular section channel and for a completely mixed fl uid:

V F=

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

=

c Ac AQQ

QQ

Q c A

Ql

l

s

l

s

l

s

l

s

'/ ( )2

22 / ( )c As

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

F ''( / ) ( / )

( / ) (( ) / )( )( / )

=

− +

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

00

1 2

1 2 1

2

2 2

g A B

g i A Bρ ρ ρs l s

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

S = − − −−

0

3 4 250

2

Ec B

gc A i J c u u c A d

gc A i

*

( ) ( / ) ( ) (( ) / )

[(

l D l s s

ss l

s

ρ ρρ

−− + +

−

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ 1

3 4 250

) ]

( / ) ( / )( ) (( ) / )

tg i

c u u c A d

δ

ρ ρD l s l s s

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

C = ( )0 0 c cl s

and for a stratifi ed (immature) fl ow:

V =

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟Ac A

c AQ

Q

Q

l

l

cw

mx mx

s mx mx

cw

mx

s,mx

,

,

,

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

=F ' /

/ (

QQ

QQ A

Q c

cw

l,mx

s,mx

cw cw

l,mx l,mx

2

2 AA

Q c A

mx

s,mx s,mx mx

)

/ ( )2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

F '' ( / ) ( / )

( / ) ( / )

( / ) (( ) / )(

=

− +

000

1 2

1 2

1 2 1

2

2

2

g A B

g A B

g i

cw

mx

s l sρ ρ ρ ))( / )A Bmx2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

c03.indd 107c03.indd 107 8/18/2009 10:13:30 AM8/18/2009 10:13:30 AM



S =

− − −

−

⋅ ⋅

00

2 4 3

E c B

gA i J n u u R

gc A i J

*/( ) (( ( )) / )

(cw cw

2cw mx cw

l,mx mx mmx D l,mx s,mx s,mx mx

l,mx mx

) ( / ) ( ) (( ) / )(( (

− −

+

3 4 250

2c u u c A d

c gA n uccw mx cw

s,mx mx s l

−

− − + +

u R

gc A i tg i cs

) / )

(( ) / )[( ) ] ( / )

/2 4 3

2 1 3 4ρ ρ ρ δ DDls l,mx s,mx

s mx mx s mx mx cw m

ρρ ( )

(( ) / ) (( (, ,

u u

c A d c gA n u u

−

× + −

2

502

xx cw) ) / )/2 4 3R

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

C = ( )0 0 0 1 c cl s

This model can be easily extended to 2D as follows

∂∂

+∂∂

+∂∂

=V E F

St x y

(56)

where

E E c EF F c F

= += +

⋅⋅

' ''

' ''1

2

U =

hu hv hc hu c hv c hc hu c hv c

cw

cw cw

cw cw

l mx

mx l mx

mx l mx

s mx

smx s mx

smx shhmx

E' '

( / )

( / )

( / ) (( ) / )( )

=

− +

01 2

001 2001 2 1

2

2

2 2

gh

gh

g i hx

cw

mx

s l s mxρ ρ ρ00

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E' =

u hu hu v hu c hu c hu v hu c h

mx

cw cw

cw cw

cw cw cw

mx l mx

mx l mx

mx mx

smx s

2

2

mmx

l s

u c hu v c h

c c

smx s mx

smx smx s mx

2

1 0 1 0 0 0 0 0

⋅ ⋅⋅ ⋅ ⋅

= [ ]c

F ' =

v hu v hv hv c hu v c hv c hv c

cw cw

cw cw cw

cw cw

mx l mx

mx mx l mx

mx l mx

smx

2

2

ss mx

smx smx s mx

smx s mx

cw

mxh

u v c hv c h

gh

gh

2

2

2

001 2001 200

F ' '

( / )

( / )=

(( / ) (( ) / )( )1 2 1 2 2g i hyρ ρ ρs l s mx− +

c2 0 0 1 0 0 0 0= [ ]c cl s

S =

−[ ]−⎡

⎣⎢⎤⎦⎥

− −

0

03 4

gh i Jgh i J

c c h D u

x

y

D s

cw cwx

cw cwy

mx mx( / ) (( ) / )( uu gh i Jc c h D v v gh i

xsmx mx m

D s mx mx smx mx

) [ ]( / ) (( ) / )( ) [

2

23 4+ −

− − +xx

yy

x y

D s

JE E

c c h D u ugc h

−+

−+

mx

l s mx mx smx

s mx

y ]

( / ) ( / )(( / ))( )((

3 4 2ρ ρρρ ρ ρ φ

ρ ρρ ρ

s l s

l s mxx s mx

D l

− − −+

) / )[ ( ) tan ']( / )

( / ) ( /

i ig J c h

c

x x1

3 4

2

ss s mx mx smx

s mx s l s

)(( ) / )( )(( ) / )[ ( ) tan

c h D v vgc h i iy y

−+ − − −

2

21ρ ρ ρ φφρ ρ

']( / )+g J c hl s mxy s mx

u and v being the velocities along the x and y axes, respectively.Numerical treatments of such equations, generally, require schemes capable

of preserving discontinuities, possibly without any special shift (shock-capturing schemes). Most numerical approaches have been developed in the last two or three

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decades, which include the use of fi nite differences, fi nite elements or discrete/distinct element methods (Asmar et al. [74]; Rodriguez et al. [67]).

Whatsoever be the solver adopted, at each time step the degree of maturity has to be assessed, to choose the appropriate terms to incorporate in the SV equations.

4 Numerical models and solvers for dam-break shock waves

Debris fl ow resulting from a sudden collapse of a dam (dam-break) are often characterized by the formation of shock waves caused by many factors such as valley contractions, irregular bed slope and non-zero tail water depth. It is commonly accepted that a mathematical description of these phenomena can be accomplished by means of 1D SV equations (Bellos and Sakkas [75]; Bechteler et al. [76]; Aureli et al. [11]).

During the last century, much effort has been devoted to the numerical solu-tion of the SV equations, mainly driven by the need for accurate and effi cient solvers for the discontinuities in dam-break problems.

A rather simple form of the dam-failure problem in a dry channel was fi rst solved by Ritter [10] who used the SV equations in the characteristic form, under the hypothesis of instantaneous failure in a horizontal rectangular channel with-out bed resistance.

Later on, Stoker [77] on the basis of the work of Courant and Friedrichs [78] extended the Ritter solution to the case of wet downstream channel. Dressler [79], [80] used a perturbation procedure to obtain a fi rst-order correction for resistance effects to represent submerging waves in a roughing bed.

Lax and Wendroff [81] pioneered the use of numerical methods to calculate the hyperbolic conservation laws. McCormack [82] introduced a simpler version of the Lax–Wendroff scheme, which has been widely used in aerodynamics prob-lems. Van Leer [83] extended the Godunov scheme to second-order accuracy by following the Monotonic Upstream Schemes for Conservation Laws (MUSCL) approach. Chen [84] and Chen and Ambruster [85] applied the method of char-acteristics, including bed resistance effects, to solve dam-break problems for res-ervoir of fi nite length.

Sakkas and Strelkoff [86], [87] provided the extension of the method of the characteristics to a power-law cross-section and applied this method to a dam-break steady right channel, in the case of rectangular and parabolic cross-section shapes. Strelkoff and Falvey [88] presented a critical review of the method of characteristics of power-law cross-sections. Hunt [89] proposed a kinematic wave approximation for dam failure in a dry sloping channel.

Flux-splitting-based schemes, like that of the implicit Beam-Warming [90], were applied to solve open-channel fl ow problems without source terms and, in general, reported good results. However, these schemes are only fi rst-order accurate in space and employ the fl ux splitting in a non-conservative way. When applied to some cases of dam-break problems, these tools gave much slower front celerity and higher front height when compared to experimental tests. Later, Jha et al. [91] proposed a modifi cation for achieving full conservative form of both

c03.indd 110c03.indd 110 8/18/2009 10:13:31 AM8/18/2009 10:13:31 AM



the continuity and momentum equations, employing the use of the Roe average approximate Jacobian (Roe [92]). This produced signifi cant improvement in the accuracy of the results.

Total variation diminishing (TVD) and essentially non-oscillation (ENO) schemes were introduced by Harten [93] and Harten and Osher [94] for effi ciently solving 1D gas dynamic problems. Their main property is that they are second-order accurate and oscillation free across discontinuities.

Recently, several 1D and 2D models using approximate Riemann solvers have been reported in the literature. Such models have been found very successful in solving open-channel fl ow and dam-break problems. Zhao et al. [95] reported implementation of an approximate Riemann solvers with Osher scheme in fi nite volume and later extended that work by including fl ux-vectors splitting and fl ux-difference splitting (Zhao et al. [96]).

In the past ten years, further numerical methods to solve fl ood routing and dam-break problems have been developed, which include the use of fi nite elements or discrete/distinct element methods (Asmar et al. [74]; Rodriguez et al. [67]).

Finite element methods (FEMs) have certain advantages over fi nite differ-ent methods, mainly in relation to the fl exibility of the grid network that can be employed, especially in 2D fl ow problems.

In this context, Hicks and Steffer [97] used the characteristic dissipative Galerk-ing (CDG) FEM to solve 1D dam-break problems for variable width channels.

The McCormack predictor–corrector explicit scheme is widely used for solv-ing dam-break problems, due to the fact that it is a shock-capturing technique, with second-order accuracy both in time and in space, and that the artifi cial dissipation terms, such as TVD correction, can be easily introduced (Garcia and Kahawita [98]).

Garcia-Navarro and Saviròn, [99] used the TVD-McCormack scheme to compute open-channel fl ows, particularly those involving hydraulic jumps and bores. Yang et al. [100] solved numerically 1D and 2D free surface fl ows by using second-order TVD and ENO schemes, while Delis and Skeels [101], [102] made a comparison with several different TVD solvers to predict 1D dam-break fl ows.

The main disadvantage of this solver regards the restriction to the time step size to satisfy Courant–Friedrichs–Lewy (CFL) stability condition. However, this is not a real problem for dam-break debris-fl ow phenomena that require short time step to describe the evolution of the discharge.

To ease the time step restriction, implicit methods could be considered. In this case, the variables are calculated simultaneously at a new time level, through the resolution of a system with as many unknowns as grid points. For non-linear problems, such as the SV equations, the resulting system of equations is also non-linear and either a linearization or an iterative procedure is required. This extra computation time is, usually, compensated by the possibility of achiev-ing unconditional or near unconditional stability for the scheme or allowing the use of very high CFL numbers. To this end, implicit TVD schemes have been proved to be unconditionally stable, even when a linearization technique is applied to solve a non-linear hyperbolic equation (Yee [103], [104]). Attempts along this line of work were presented by Alcrudo et al. [105] introducing in the

c03.indd 111c03.indd 111 8/18/2009 10:13:31 AM8/18/2009 10:13:31 AM



McCormack scheme TVD corrections to reduce spurious oscillations around discontinuities, for both 1D and 2D fl ow problems, and by Delis et al. [106] devel-oping new implicit TVD methods to solve SV equations.

Mambretti et al. [64], [65] and De Wrachien and Mambretti [66] used an improved TVD–McCormack–Jameson scheme to predict the dynamics of both mature (non-stratifi ed) and immature (stratifi ed) debris fl ows in different dam-break conditions.

4.1 Numerical solutions

The SV set of partial differential eqns. (23) describe a system of hyperbolic con-servation laws with source term (S). The equations, as previously described, are based on the assumption of hydrostatic pressure distribution and incompressibility of water.

The homogeneous part of the system is responsible for most of the diffi culties when it is numerically integrated (Delis et al. [106]).

The fl ux vector F is related to the fl ow variable V through the Jacobian J of F with respect to V as:

∂∂

+∂∂

=V

JV

t x0 (57)

with:

JF

= =−

ddV g A u u

0 1

22( / )σ (58)

The hyperbolic nature of eqn. (57) ensures that matrix J has a complete set of independent and real eigenvectors expressed as:

e1 2 1, ,= ±( )u c T (59)

with u = Q/A fl ow velocity and c g A= ( / )σ wave celerity.The eigenvalues of J are expressed as:

a u c1 2, = ±

(60)

and correspond to the two characteristic wave speeds with their signs providing information about the direction of the wave.

4.2 Homogeneous conservation laws

High-order shock-capturing schemes have been recently proposed from the non-linear scalar case to 1D non-linear systems, based on the use of Riemann solvers.

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Consider the initial value problem for a 1D system of hyperbolic conservation laws represented by:

∂∂

+∂

∂=

V Ft

ux( )

0 (61)

and

V V( , ) ( )x x0 0= (62)

where V0(x) is the initial datum with a single discontinuity:

V VV0

00( )x x

x= <>{ L

R (63)

with VL and VR, respectively, the left and right constant states of the initial dis-continuity x = 0.

4.3 Riemann solvers

According to Riemann, for a given set of initial data, a conservation law may have different weak solutions one of which has physical meaning, known as the entropy-satisfying solution or solution of the Riemann problem. This solution is composed of two types of elementary waves, called shock (bores) and rarefac-tions (depressions) (Delis et al. [106]).

Assuming there are n independent conservation laws, then the solution of the Riemann problem will, in general, consist on n centred waves. A convergent explicit different scheme, in conservation form for eqn. (60), may be written as:

V V F Fin

in

in

int

x+

+ −= + −( )11 2 1 2

∆∆

� �( / ) ( / ) (64)

where i and n are the spatial and temporal grid levels, Δx and Δt the spatial and temporal increments and �Fi+( / )1 2 the numerical fl ux function that approximates the time average fl ux across the cell interfaces.

4.3.1 Local Lax–Friedrichs (LLF) Riemann solverThe simplest Riemann solver is the LLF solver (Lax [107], Davis [108]). The form of the numerical fl ux function is given by:

�F F F V Vi i i i ia+ + += + − −( / ) max[ ( )]1 2 1 112

(65)

where:

a a a a ai i i imax max , , ,= ( )+ +11 2

11

12 (66)

with ak the characteristics’ wavespeeds.

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4.3.2 Roe’s approximate Riemann solverIn the approximate Riemann solver developed by Roe [92], [109], a solution is sought for the linearized form of eqn. (57):

∂∂

+∂∂

=V

JV

t s� 0 (67)

where �J is an approximation of the Jacobian J.For each computational call, �J is chosen such that it satisfi es the following

conditions:

1. for any Vi and Vi+1 the following must hold:

∆ ∆ ∆i i i i i i i+ + + + += = ∀( / ) ( / ) ( / ) ( / )( , )1 2 1 1 2 1 2 1 2F J V V V J V� � (68)

where the operator ∆i i i+ +•( ) = •( ) − •( )( / )1 2 1 and F Fi iV= ( );

2. for V V Vi i= =+1 the matrix � �J V V J VFV

,( ) = ( ) =∂∂

;

3. �J has real eigenvalues with linearly independent eigenvectors.

A matrix satisfying the above conditions is called Roe matrix.From condition (3) it follows that the eigenvalues �a1 2, and the eigenvectors �e1 2,

of �J are given by:

� � �

� �

a u c

a

i i i

i i

+ + +

+ +

= ±

=

( / ),

( / ) ( / )

( / ),

( / ),,

1 21 2

1 2 1 2

1 21 2

1 211e 22( )T (69)

and the problem of fi nding �J is now transferred to that of fi nding the average values of �u and �c that meet the conditions (1)–(3).

The following equations for �u and �c can be obtained (Dalis and Skeels [102]):

�uQ A Q A

A Aii i i i

i i+

+ +

+

=( )+( )

+( / )

/ /1 2

1 1

1 (70)

�cg

I IA A

A A

c gA Ai

i i

i ii i

ii i

i

+

+

++

+

+

=

−

−− ≠

=−( / )

, ,

1 22

1 1 1

11

2 1

0when

σ 111

1 1 1

10 0

−− =

−

−<

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪+

+

+σii i

i i

i iA A

I IA A

when or , , (71)

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The diagonalized form of �J , diag �aik+

⎡⎣⎢

⎤⎦⎥( / )1 2 , can be expressed as:

� � � �J R R= ⎡⎣⎢

⎤⎦⎥ =+ + +

−i i

kia k( / ) ( / ) ( / ) ,1 2 1 2 1 2

1 1 2diag (72)

where �Ri+( / )1 2 represents the right eigenvectors matrix.Condition (3) implies that:

∆i ik

ik

ka+ + +

== ∑( / ) ( / ) ( / )1 2 1 2 1 2

1

2V e� (73)

where:

aQ u c A

ii i i i

++ + + +=

+ − ±

±( / ), ( / ) ( / ) ( / ) ( / )[ ( ) ]

1 21 2 1 2 1 2 1 2 1 2

2

∆ ∆� �

�cci+( / )1 2 (74)

The numerical fl ux for a Roe’s Riemann solver can be expressed as:

� � �F F F eik

i i ik

kik

ika a+ + +

=+ += + −

⎡

⎣⎢ ∑( / ) ( / ) ( / ) ( / )1 2 1 1 2

1

2

1 2 1 212 ⎢⎢

⎤

⎦⎥⎥ (75)

This form of the interface numerical fl ux can be interpreted as the average fl ux plus a ‘dissipation’ term.

An important drawback of this procedure is that Roe’s scheme admits unphys-ical stationary jumps, because it cannot properly distinguish a shock from a rarefaction wave. Therefore an entropy correction is needed to ensure entropy-satisfying solution (Delis et al. [106]).

4.3.3 Harten–Lax–van Lier–Einfeldt (HLLE) Riemann solverThe HLLE solver approximates the solution of any Riemann problem (Einfeldt [110]) with the following three constant states:

VVV( )( / ),

( / )

( / )xx b t

b t xi t

in

il

in

il

+

+

+ +=

<

< <1 2

1 2

12

1 2

whenwhen

�

� bb t

b t x

ir

in

ir

+

+ + <

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

( / )

( / )

1 2

1 1 2V when �

(76)

where �x x xi= −+ 1

2 and b coeffi cients are defi ned as:

b u c a

b u c

ir

i i i

il

i i

+ + + +

+ + +

= +⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

= −

12

1 1 12

1

12

1 1

max ,

min ,

�

�aai+⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟1

2

2 (77)

with �ai+( / ),

1 21 2 : eigenvalues of the Roe’s solver; �bi

r l+( / ),

1 2 : numerical approximations for the maximum and the minimum characteristic speeds at the respective locations.

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The average fl ow vector Vin+( / )1 2 is defi ned as:

VV V F

in i

rin

il

in

ir

il

b b

b b++ + +

+ +

=−

−−( / )

( / ) ( / )

( / ) ( / )1 2

1 2 1 1 2

1 2 1 2

ii i

ir

ilb b

+

+ +

−

−1

1 2 1 2

F

( / ) ( / )

(78)

The numerical fl ux at the cell interface can be written as:

� �F F F eiHLLE

i i ik

kik

ikq a+ + +

=+ += + −

⎡∑( / ) ( / ) ( / ) ( / )1 2 1 1 2

1

2

1 2 1 212 ⎣⎣

⎢⎢

⎤

⎦⎥⎥ (79)

where:

qb b

b bai

i i

i ii+

++

+−

++

+− +=

+

−( / ), ( / ) ( / )

( / ) ( / )( /1 2

1 2 1 2 1 2

1 2 1 21 2� ))

, ( / ) ( / )

( / ) ( / )

1 2 1 2 1 2

1 2 1 2

2−−

++

+−

++

+−

b b

b bi i

i i

(80)

b b

b bi i

r

i il

++

+

+−

+

= ( )= ( )

( / ) ( / )

( / ) ( / )

max ,

min ,1 2 1 2

1 2 1 2

0

0 (81)

The HLLE solver does not require an artifi cial entropy correction.

4.4 Inhomogeneous conservation laws

The overwhelming majority of the real problems in dam-break shock waves include source terms quantity that should be considered when using a numerical solver.

The general form for a 1D implicit numerical scheme of the SV equations can be written as:

V F F S

V

in

in

in

in

in

tx

t

t

+++

−+ +

+

+ ⋅ −( )−

== −

11 21

1 21 1

1

∆∆

∆

∆

ϑ ϑ� �( / ) ( / )

∆∆∆

xti

nin

in1 11 2 1 2−( ) −( )+ −( )+ −ϑ ϑ� �F F S( / ) ( / )

(82)

with 0 1≤ ≤ϑ . For ϑ = 0 the scheme is explicit.Two other important cases for ϑ can be identifi ed. For the case where ϑ = 0 5.

(Crank–Nicholson scheme) second-order accuracy in time can be achieved but severe stability restrictions apply (Delis et al. [106]). The case ϑ =1 (fully implicit scheme) appears to give the preferable scheme for solving open-channel fl ow equations.

4.4.1 Linearized conservative implicit (LCI) solverNumerical schemes for Vi

n+1 in eqn. (82) generally involve systems of non-linear algebraic equations that have to be solved iteratively. To avoid this

c03.indd 116c03.indd 116 8/18/2009 10:13:32 AM8/18/2009 10:13:32 AM



procedure, the implicit operator is linearized and solved (Yee [103]; Alcrudo et al. [105]). In this approach, a family of schemes with the numerical fl ux vector in (82) written as:

� �F F F Ri i i i i+ ± ± += + +( / ) ( / ) ( / )[ ]1 2 1 1 2 1 212

Φ (83)

can be derived for different vector functions Φi+( / )1 2 . �R is the matrix of the eigenvectors of �J (eqn. (72)) and the elements of Φ (dissipater vector) can be calculated according to:

B R Ri i ik

i+ + + +−= −⎡⎣⎢

⎤⎦⎥( / ) ( / ) ( / ) ( / )1 2 1 2 1 2 1 2

1� �diag Φ (84)

where Φik+( / )1 2 are the elements of Φi+( / )1 2 .

Dropping the subscript indexes, the matrix B can be defi ned as:

B =−

− −

−( ) −

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

12 1

1 2 2 1 2 1

1 2 2 1 2 2 1 1� �

� �

� � � �a a

a a

a a a a

Φ Φ Φ Φ

Φ Φ Φ Φ⎥⎥⎥⎥⎥ (85)

The numerical fl ux can, then, be expressed as:

�F F F B Vi i i i i+ ± ± += + +( / ) ( / ) ( / )[ ]1 2 1 1 2 1 212

∆ (86)

The implicit terms Fn+1 and Sn+1, using a Taylor expansion, can be rewritten as:

F F J V

S S J Vin

in

in

i

in

in

S i

O t

O tin

+

+

= + + ( )= + + ( )

1 2

1 2

δ

δ

∆

∆ (87)

where:

∂ = −+V V Vi in

in1 (88)

and Js is the Jacobian of the source term S with respect to V.

JSVs =

∂∂

(89)

The resulting implicit operator is strictly diagonally dominant. The LCI scheme preserves the conservative form of the differencing scheme and it is applicable to both steady- and unsteady-fl ow conditions without stability restrictions.

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4.4.2 Symmetric LCI solverThe elements of Φi+( / )1 2 in eqn. (84) are defi ned as (Yee [104]):

Φ Ψik s

ik i

k

ika

L

ak+ +

+

+( ) = − ( ) −

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

( / ) ( / )( / )

( / )1 2 1 2

1 2

1 2

1� ==1 2, (90)

where the function Ψi+( / )1 2 is the entropy correction to the eigenvalues �ai+( / )1 2 that guarantees the physically valid discontinuities in the solution. The entropy correction can either take the form:

Ψ � �a a( ) = ( )max ,δ (91)

or the smooth one (Harten and Hyman [111]):

Ψ ��

� �a

a a

a a( ) =

≥

+( ) <

⎧⎨⎪⎪⎪

⎩⎪⎪⎪

when

when

δ

δ δ δ2 2 2/ (92)

where ϑ is a small positive number, between 0.1 and 1.0.The limit function L can assume one of the following values:

L m a a aik

ik

ik

ik

+ − + += ( )( / ) ( / ) ( / ) ( / ), ,1 2 1 2 1 2 1 2 (93)

L m a a a a aik

ik

ik

ik

ik

i+ − + + −= +( / ) ( / ) ( / ) ( / ) ( / ), , ,1 2 1 2 1 2 3 2 1 22 2 212 ++( )⎡

⎣⎢⎢

⎤

⎦⎥⎥( / )3 2

k (94)

La a a a a a

ik i

kik

ik

ik

ik

+− + + + −

=+ +

( / )( / ) ( / ) ( / ) ( / ) ( / )

1 21 2 1 2 1 2 3 2 1 2 ii

kik

ik

ik

ik

ik

a a

a a a

+ + +

+ − +

+

+ +

( / ) ( / ) ( / )

( / ) ( / ) ( / )

1 2 1 2 3 2

1 2 1 2 1 22 (95)

L a a a aik

ik

ik

ik

i+ + − + += ( )( / ) ( / ) ( / ) ( / ) (max ,min , , , min1 2 1 2 1 2 3 20 2 11 2 1 2 1 22/ ) ( / ) ( / ), ,kik

ika a− +( )⎡

⎣⎢⎤⎦⎥

(96)

where m a b( , ) is the minmod limiter function defi ned as:

m a b a a a b( , ) sgn( ) max ,min ,sgn( )= ( )⎡⎣⎢

⎤⎦⎥0 (97)

4.4.3 Monotonic upstream scheme for conservation laws (MUSCL) LCI solver

The MUSCL LCI solver, introduced by VanLeer [83], provides a one-parameter set of second-order schemes and one third-order scheme.

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This procedure replaces the arguments Vi and Vi+1 of the numerical fl uxes by Vi

L+( / )1 2 and Vi

R+( / )1 2 , calculated as follows:

V ViL

i i im m+ −+

++= + − + −⎡

⎣⎢⎤⎦⎥( / ) ( / ) ( / )( ) ( )1 2 1 2 1 2

14

1 1∆ ∆ (98)

V ViR

i i im m+ + ++

++= − − + −⎡

⎣⎢⎤⎦⎥( / ) ( / ) ( / )( ) ( )1 2 1 3 2 1 2

14

1 1∆ ∆ (99)

where:

∆ ∆ ∆i i im+−

+ −=( / ) ( / ) ( / )( , )1 2 1 2 1 2V Vβ (100)

∆ ∆ ∆i i im++

+ +=( / ) ( / ) ( / )( , )1 2 1 2 3 2V Vβ (101)

where β is the compression parameter whose value is in the range:

131

1≤ ≤−−

≠βmm

m (102)

Using MUSCL procedure linked with the LCI scheme it is possible to achieve third-order accuracy in space.

4.4.4 McCormack–Jameson solverThe McCormack–Jameson scheme incorporates forward (predictor) and back-ward (corrector) steps to solve the SV equations (McCormack [82]; Jameson [112]) that can be written as:

V V F F F Sip

in

in

in

in

int

xt= − −( ) − −( ) −⎡

⎣⎢⎤⎦⎥ ++ −

∆∆

∆1 1 21 1ϑ ϑ ϑ

V V F F F Sic

in

ip

ip

ip

ipt

xt= − + −( ) + −( )⎡

⎣⎢⎤⎦⎥ ++ −

∆∆

∆ϑ ϑ ϑ1 11 2 1 (103)

V V V d din

ip

ic

in

int

x+

+ −= +( )+ −( ) =11 2 1 2

12

0 1∆∆ ( / ) ( / ) ,ϑ

in which the superscripts ‘p’ and ‘c’ indicate the variables at predictor and cor-rector steps, respectively. The order of backward and forward differentiation in the scheme is governed by θ which can also be cyclically changed during the computations (Chaudry [113]).

In eqns. (103) d is an artifi cial dissipation term that must be added to the original McCormack scheme, to avoid spurious oscillations and discontinuities without any physical signifi cance.

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The artifi cial dissipation terms introduced by Jameson [112], according to the classical theory developed in the fi eld of aerodynamics, can be written as:

d V V V V V Vi i i i i i i i i+ + + + + += − − − + −( / ) ( / )( )

( / )( )( ) (1 2 1 2

21 1 2

42 3 3ε ε ii−1) (104)

where:

ε ε ε

ε α

i i

ii i i

i

i

h h h

h

+

( ) + −

+

= ( )=

− +

+

+( / )( ) ( ) ( )

( )

max ,1 22 2 2

2 2 1 1

1

1

2

22

0

1

14 4

1 22

⋅ +

= −( )

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

−

+( ) ( )

+( )

h hi i

i iε α εmax , ( / )⎪⎪⎪

(105)

where h is the local free surface elevation.The McCormack–Jameson predictor–corrector scheme is widely used for

solving dam-break problems, due to the fact that it is a shock-capturing scheme, with second-order accuracy both in time and in space and that the artifi cial dis-sipation terms can be introduced, to avoid non-physical shocks and oscillations around discontinuities (Garcia Navarro and Saviròn [99]).

Recently, the scheme was applied by Mambretti et al. [64], [65] and by De Wrachien and Mambretti [66] to predict the dynamics of both mature (non-stratifi ed) and immature (stratifi ed) debris and hyper-concentrated fl ows in different dam-break conditions.

5 Sediment concentration of the mixtures

Dam-break hydraulics involves not only water fl ow, but also the fl ow-induced sediment transport and morphological changes of the channel, which in turn modify the fl ow. In this context it is recognized that fl ow, sediment transport and morphological evolution are strongly linked to each other. So, the predic-tion of the grain-size-specifi c sediment transport of sand mixtures plays a role of paramount importance in studies of sorting processes in debris fl ow and surges triggered by the sudden collapse of dams. As a matter of fact, debris fl ows can rapidly transport large volumes of sediment, have a strong erosive force and, dur-ing movement, they collect debris or rocks, eroded from the beds or banks of the channel. They continue fl owing downstream, growing in volume and sediment concentration with the addition of water, sand, mud, boulders, trees and other materials (Hui-Pang and Fang-Wu [114]).

A typical debris fl ow is a dense, poorly sorted, solid–fl uid mixture that com-monly contains more than 50% sediment by volume, and consists of particles that range widely in size from clay to boulders of several meter in diameter (Major and Pierson [115]).

Therefore, the nature of debris fl ow mainly changes according to the sediment concentration and characteristic of the sediment size (Egashira et al. [116]). For

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debris fl ow of very high concentration, it is necessary to assess the equilibrium sediment concentration, and its vertical distribution to properly analyze the triggering, movement and deposition processes of dam-break shock waves with fl oating debris.

This paragraph describes current knowledge on sediment properties, trans-port, velocity distribution and suspended sediment concentration, within debris fl ow’s bodies, and the most advanced analytical and mathematical models, nowa-days available, for sediment-laden fl ows.

5.1 Fundamental parameters of sediment particles

The velocity and concentration profi les in sediment-laden fl ows has been the sub-ject of many theoretical, experimental, and numerical studies over the past fi fty years. The main reason for such interest is that accurate prediction of the profi les allows for accurate prediction of the fl ow depth and suspended sediment dis-charge, key parameters for morphological prediction models.

Early on, Vanoni [117] and Einstein and Chen [118] experimentally observed an increase in the velocity gradient in the presence of suspended sediment. It was hypothesized that the effect was due to a ‘damping’ of the turbulence, thus decreasing turbulent mixing (Vanoni [119]). One mechanism identifi ed as the cause for this ‘damping’ is the vertical density stratifi cation induced by the con-centration gradient. It was also observed that the density gradient creates a buoy-ancy force that makes it more diffi cult to fl ux heavier fl uid/sediment upward into lighter fl uid/sediment, and similarly restricts the downward fl ux of lighter fl uid/sediment into heavier fl uid/sediment.

The main characteristics that affect the above-mentioned processes are the size, shape, concentration and fall velocity of sediment particles (Shen and Julien [120]).

5.1.1 SizeThe size of sediment particle is defi ned on three mutually perpendicular axes labelled a, b, c.

The a axis is the direction of the longest dimension of a sediment particle. The other two mutually perpendicular axes are chosen with the c axis close to the shortest dimension and b being the intermediate perpendicular axis.

The most reliable method to measure sediment size distribution is by labora-tory sieve analysis. Other methods feasible to measure the sediment size distribu-tion are (US Inter-Agency Committee on Water Resources [121]):

visual accumulation tube method, for sediment size between 0.062 and 2 mm;pipette method, for sediment particles between 0.002 and 0.062 mm;hydrometer method, which relies on the relationship between the progressive decrease in density that occurs at a given elevation in a well-dispersed sediment suspension and the sediment diameter of the particles.

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5.1.2 Shape of sediment particlesThe shape of a sediment particles has a strong infl uence on its fall velocity. The shape factor can be defi ned as:

kcab

= (106)

where a, b, c are the particles length, width and height, respectively (Garde and Ranga Raju [122]).

5.1.3 Sediment concentrationSediment concentration measures the amount of sediment carried by the fl ow. Commonly, three parameters are used to express sediment concentration:

Sediment concentration by volume:

SS

S S SVW

G W G=

− −( )1 (107)

where SG is the specifi c gravity of the sediment;SW is the sediment concentration by weight (mass):

SS

S SWm

W G m=

+ −ρ ( ( / ))1 1 (108)

where ρW is the density of water andSm is the sediment concentration by a mixture of weight (mass) and volume

SS S

S S SmW W G

G G W=

− −ρ

( )1 (109)

The sediment concentration is usually expressed as parts per million (ppm) by weight. Another sediment concentration measure is by Newton (weight) per cubic meter (volume).

5.1.4 Fall velocityThe fall velocity is defi ned as the terminal fall velocity of a sediment particle. It depends on the size, shape and density of the particle and the effects of fl uid density and turbulence. According to Schultz et al. [123], the fall velocity of a particle varies greatly with the Reynolds number based on particle size.

5.2 Sediment concentration distribution

Sediment concentration, defi ned as the amount of sediment carried by the fl ow, represents a parameter of paramount importance in debris and hyper-concentrated fl ow movement and deposition.

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Einstein [124] was one of the fi rst to propose a method for the calculation of sediment transport on a grain-size-specifi c basis, proposing separate relations for bed and suspended load. Later on, the log-law velocity profi le and Rouse concen-tration profi le (Vanoni [119]) were used, along with the Einstein bed-load equa-tion, for the near-bed concentration. McLean [125] proposed a similar approach with several improvements that account for density stratifi cation effects in both the velocity and concentration profi les.

Another class of relations take a more empirical approach for partitioning the total sediment transport on a grain-size-specifi c basis. A review of these methods can be found in Molinas and Wu [126].

While these methods predict the total sediment transport by grain size, there are other procedures that predict the grain-size-specifi c bed load and/or sus-pended load separately (Samaga et al. [127]). The suspended load is assumed to be related to the bed shear stress.

More advanced analytical and mathematical models for sediment-laden fl ows are based on the governing equations for single-phase fl ows, and are valid for very-low-sediment situation. Usually a model of this type is referred to as a decoupled one. Examples of this kind are the widely accepted Rouse formula for suspended sediment concentration distribution and the well-known logarithmic velocity profi le developed initially for a single-phase fl ow. Recently, more and more attention has been paid to the infl uences of the interphase (water–sediment) and sediment particle–particle interactions, which become predominant factors and cannot be neglected for dam-break shock waves with fl oating debris, with heavy sediment concentrations.

A two-phase fl ow approach appears to be more appropriate for this kind of phenomena. Up to now, much effort has been performed in this direction. Kobayashi and Seo [128] proposed a model for water–sediment interaction in the bed-load layer on the basis of two-phase fl ow equations. Later on, Cao et al. [129] provided a two-phase hydraulic model, based on mass and momentum conservation equations, feasible to predict the distribution of velocity and sus-pended sediment concentration. The fl ow is divided vertically into a suspension fl ow, a near-bed layer and a viscous sublayer.

To determine the velocity and sediment concentration at the interfaces of the three layers, the shear and dispersive stresses, stemming from the interactions between sediment particles, are taken into account.

5.3 Hydraulic-based routing modelling

The model is based on the continuum assumption for both the fl uid phase (water) and the dispersed solid one (sediment).

5.3.1 Governing equationsIn the following, the Cartesian co-ordinate system and time are denoted by x ii =( )1 2 3, , and t, respectively.

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The equations of mass conservation for the fl uid and solid phases are:

∂ −

∂+

∂∂

− =ρ

ρff f

( )( )

11 0

ct x

c cj

j (110)

∂∂

+∂

∂=

ρρs

s sc

t xcu

jj 0 (111)

The equations of motion for the fl uid and solid phases are:

∂ −∂

+∂

∂−

= − − −∂∂

+∂

∂

ρρ

ρ

f ff f f

f

( )( )

( ) ( )

11

1 1

c ut x

c u u

c g cpx x

T

i

jj i

ii j

ffij if−

(112)

∂∂

+∂

∂= −

∂∂

+∂

∂+

ρρ ρs s

s s s s scut x

cu u cg cpx x

T fi

ji j i

i ji j i

(113)

where ρf , ρs denote the constant densities of the fl uid and solid phases, respec-tively; c the volumetric concentration of the solid phase; u jf , u js : x j-components of instantaneous velocities of the fl uid and solid phases, respectively; p the full pres-sure of the fl uid–solid system; gi the xi-component of gravitational acceleration;Tf the viscous stress tensor of incompressible fl uid phase; Ts the stress tensor stem-ming from interaction between dispersed solid particles; fi the xi-component of interphase interaction force per unit volume; and the subscripts ‘f’ and ‘s’ denote, respectively, the fl uid and solid phases. The fractional pressures of the fl uid and solid phases are assumed to be proportional to their volumetric concentrations, respectively (Pai [130]).

The density of the fl uid–solid mixture is defi ned as:

ρ ρ ρm f s= − +( )1 c c (114)

where (and hereafter) the subscript m denotes the mixture.The continuity and motion equation of the mixture can be written as:

∂∂

+∇( ) =ρ

ρmm m,MFt

u 0 (115)

∂∂

+∂

∂

= −∂∂

+∂

∂+ −

∂∂

ρρ

ρ ρ

mm

m f s s s

Ut x

u u

gpx x

T Tx

c i

i

ii j

ii j

ij ijj

j( ) [ ( ss fi ii− )] (116)

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where ui, uj denote the xi, xj-components of the mixture’s velocity vector um,MF

defi ned as:

ρ ρ ρm m,MF f f s su u u= − +( )1 c C (117)

with uf, us: velocity vectors of the fl uid and solid phases, respectively, and isi, ifi the xi-components of the dispersive velocity vectors of the same phases:

i u uf f m,MF= + (118)

i us s m,MF= +u (119)

Under gravitational action, a solid particle is considered to have an effective settling velocity ω (time independent) relative to the mixture’s velocity in the vertical direction, that is:

is = –ωk (120)

where k represents a unit vector in the vertical direction, positive upward.

5.3.2 Sediment concentration profi leOn the basis of the governing equations, the fl ow, as previously mentioned, is divided into three parts vertically (fi g. 8):

suspension fl ow: y a h∈[ ], ;

near-bed layer: y y a∈[ ]b , ;

viscous sublayer: y y∈[ ]0, b .

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�

�

y

a

yb

0

x

V

U

h

Figure 8: Two-dimensional uniform fl ow.

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For the suspension fl ow, it is assumed that the inertia of the sediment particles is negligible and that the sediment velocity differs from the fl uid–sediment mix-ture’s velocity only in the vertical direction by an effective settling velocity.

This allows to establish the following diffusion equation for suspended sedi-ment concentration:

ε ωρ ρ

ρωs

fy

dcdy

c c= − −−s f 2 (121)

where εsy denotes the y-coeffi cient of the turbulent diffusion of sediments, fea-sible to provide sediment concentration profi les according to different models for turbulent diffusion.

Solution of eqn. (121) requires appropriate initial and boundary conditions.Usually the mean sediment concentration is set equal to a reference concentra-

tion at the interface between the suspension fl ow and the near-bed layer. Moreo-ver, a closure method is required to determine εsy . Conventionally, the following relationship is assumed:

ε βs l= v (122)

where v1 is the coeffi cient of turbulent viscosity; b the coeffi cient usually sup-posed equal to 1 (Van Rijn [131]).

The two-phase model was tested extensively using measured data of Coleman [132] and Einstein and Chen [118]. The agreement is satisfactory for low-sediment concentration fl ows, and fairly good for heavy sediment concentration cases.

5.4 Stochastic approach

Much research on sediment concentration of debris fl ow triggered by dam-break events has been conducted using hydraulic-based routing models. However, debris fl ow always has uncertainties in variables and model parameters. The properties of the moving fl uid mixtures of debris and water are very different from those of purely water fl oods (Jin and Fread [133]).

Unlike sediment-laden fl ow, debris fl ow is composed of a dense mixture of water and solid particles rather than of a continuous medium so that the sedi-ment concentration is not continuous as its profi le. Therefore, the traditional approach to debris-fl ow studies using a physically hydraulic-based is limited. Despite advances made with this approach, major problems and barriers still exist in debris-fl ow analysis. In this context, Hui-Pang and Fang-Wu [114] proposed a model, based upon probability and the concept of entropy, feasible to solve sediment concentration problems of debris fl ow. The entropy concept provided a new approach for investigating the stochastic behaviour of certain debris-fl ow

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phenomena, including vertical distribution of suspended sediment concentration in both open-channel and debris fl ows.

5.4.1 Maximum entropy principleAssuming the entropy as a measure of uncertainty, the probability distribution function of a hydraulic variable, like the sediment concentration of debris fl ow, can be assessed by maximizing the entropy of the variable in an appropriate way (Chiu [134]).

From this assumption, the maximum value of the sediment concentration function H c( ) can be written as:

maximize H c p c p c dc( ) ( ) ln ( )= −∫ (123)

where p c( ) is the probability density function and p c dc( ) the probability of the sediment concentration being between c and c + dc.

From experimental studies, different authors (Tsubaki et al. [56]; Takahashi [45], Egashira et al. [135]) proposed monotonic variations of the sediment con-centration along the debris-fl ow profi le, under steady and uniform fl ow (fi g. 9).

The fi gure shows that the sediment concentration, in a steady uniform debris fl ow with depth h0, decreases monotonously in the vertical direction from the maximum value of cm, at the channel bed, to any arbitrary value ch at the water surface. In such fl ows, c represents the sediment concentration at the vertical distance y y h( )0 0≤ ≤ from the channel bed. Then, at any distance greater than y, the sediment concentration is less than c. Under this hypothesis, the density function p c( ) can be written as:

p c hdcdy

( ) = −⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟

−

0

1

(124)

y

h0

Cm

Ch

C(y)

Cm

Figure 9: Sketch of uniform debris fl ow.

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The density function p c( ) must satisfy the following two conditions:

p c dcc

c

h

( )m

∫ =1 (125)

cp c dc cc

c

h

( )m

D∫ = (126)

The fi rst condition (125) is the defi nition of general probability, the second one (126) defi nes the equilibrium (or mean) sediment concentration by volume, cD.

5.4.2 Equilibrium sediment concentrationSolution of the differential eqn. (124), using the boundary condition c = cm at y = 0, provides sediment concentration profi les. This solution can be written as:

c

c Ee e e

yh

E E c c Eh

m

m= + −( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

1

0ln ( / ) (127)

where the dimensionless entropy parameter E is defi ned as:

E c= λ m (128)

where λ is the Lagrangian multiplier.To validate the model, comparisons have been made between its prediction

and experimental results carried out by different authors (Takahashi [30]). The comparisons have shown that the procedure based on the maximum entropy prin-ciple is feasible to predict the sediment concentration distribution profi le of debris fl ow once its channel gradient and bed materials are known or given.

5.5 Gradual earth-dam failure

Gradual earth-dam failure (dam breach) involves not only water fl ow, but also the fl ow-induced sediment transport and morphological changes, which in turn modify the fl ow. Therefore, the strong interaction between fl ow, sediment and morphological evolution has to be taken into account in any experimental and/or computational study of dam-breach hydraulics (Cao et al. [136]).

In the context of computational studies of dam-breach hydraulics, two basic observations are warranted. First, it has to be recognized that the fl ow, sediment transport and morphological evolution are strongly coupled to each other, the rate of bed deformation being considerable compared to that of the fl ow evolution.

Second, it has to be assumed that the most important mechanism of sediment transport due to dam-breach fl ows is the local entrainment of bed sediment, in response to excessively energetic turbulent motions.

In these processes, erosion and sediment transport are triggered by some low or weak points in the crest or in the downstream face of the dam, by pip-ing or by overtopping. The water overtopping the crest shapes a little breach.

c03.indd 128c03.indd 128 8/18/2009 10:13:34 AM8/18/2009 10:13:34 AM



Progressive erosion, then, widens and deepens the breach, increasing outfl ow and erosion rate.

With reference to these issues, dam-breach models over mobile beds provide a unique opportunity to gain enhanced insight into the interaction between fl ow, sediment transport and morphological evolution. In the context of fl uvial mobile bed hydraulics, it has long been known that the fl ow tends to adapt itself (velocity and depth) to reach a potential dynamic equilibrium. Under this equilibrium state, sediment entrainment from, and deposition to, the channel bed is well balanced and the fl ow bears minimum energy dissipation rate (Yan [137]). Bed erodibility can modify the free surface profi le and hydrographs greatly, and has considerable implication for fl ood prediction. The velocity is strongly reduced over a mobile bed, which is accompanied by an increase in fl ow depth. This behaviour can pro-voke very active sediment exchange between the water column and the bed, and also produce a sharp spatial gradient concentration. These features are sensitive to bed sediment particle size. The fi ner the sediment, the greater the effects bed mobility will have. These processes signifi cantly affect the shape of the mobile bed, which can be scoured. Therefore the rate of bed deformation is not negligible compared to that of fl ow change, characterizing the need for coupled modelling of the interacting fl ow–sediment–morphology system.

An attempt to simulate this interacting system, was made using geotechnical prin-ciples and simplifying hypotheses to obtain empirical equations relating the amount of eroded material to the fl ow through the breach (Cristofano [138]). Later on, Brown and Roger [139] developed a model using the Schoklitsh relationship to compute sus-pended sediment. Ponce and Tsivoglou [140] presented a mathematical model of the gradual failure of an earth dam, which can be considered as the fi rst effort to tackle this problem, within the framework of an implicit numerical solution scheme. After-wards, many other authors have proposed their own models. These models can be grouped into the following categories (Leopardi et al. [141]):

comparative models: they rely on data of similar, previously failed dams, which represent input data to predict the breach evolution and discharge values of the simulated structures;statistical models: correlations are established to link unknown variables and typical dam characteristics (McDonald and Langridge-Monopolis [142]);hydraulic models: they require as input the size of the breach, the time expected for the failure to develop and the height of the surface water at which breaching begins (Fread [4], US ARMY [143]);physically based model: the breach development depends on the erosive strength of the stream fl owing through it. Erosion laws are assumed as linear or quadratic functions of the average fl ow velocity (Singh and Scarlatos [144]). The evolu-tion of the dam-breach process can be based on hydrologic, geometric and geo-technical considerations, or on the use of erosion formulas, like those proposed by Einstein [124]. The physically based models can be coupled with sediment routing schemes to simulate the triggering of a debris fl ow due to the water overtopping the crest of a dam (Takahashi and Nakagawa [38]).

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6 General remarks

Studies of dam-break fl ows consider, mainly, situation of clear-water surges. However, under natural conditions a dam-break fl ow can generate extensive debris or encounter fl oating debris in the valley downstream of the dam. Debris fl ows differ from other natural unconfi ned fl ows in the nature of both the fl oating material and the fl ow itself, which is rapid, transient and may signifi cantly infl u-ence surge height and speed.

Because a debris fl ow is, essentially, a multi-phase system, any attempt at modelling this phenomenon that assumes, as a simplifi ed hypothesis, homogene-ous mass and constant density, conceals the interactions between the phases and prevents the possibility of investigating further mechanisms such as the effect of sediment separation (grading). Therefore, modelling the fl uid as a two-phase mixture overcomes most of the limitations mentioned above and allows for a wider choice of rheological models.

The rheological behaviour of a mixture depends on the hydrodynamic charac-teristics of the fl ow and the characteristics of the sediment. To be reliable, a rheo-logical model should possess three major features (Chen [54]). It should describe: the dilatancy of the sediment–water mixtures; the soil yield criterion, as proposed by Mohr–Coulomb; and the role of inter-granular fl uid.

On the whole, the one-layer models are unable to adequately describe the entire thickness of the fl ow, so, recently, multi-layer models suitable to combine two or more constitutive relationships to fi t experimental velocity and concentra-tion profi les have been proposed (Wan and Wang [59]).

The coeffi cients of these models depend, strongly, on factors such as the dimension, concentration and distribution of the sediments, as well as the fl ow regime. The assessment of this dependency constitutes one of the major chal-lenges of debris-fl ow studies.

Different methods have been proposed for the prediction of the fl ow depth, grain-size specifi c near-bed concentration and bed-material suspended sediment transport rate in sand-bed rivers. The salient improvements, related to the need to enhance the existing procedures to encompass the full range of sand-bed streams, and in particular large, low-slope sand-bed watercourses, can be summarized as follows (Wright and Parker [145]):

the inclusion of density stratifi cation effects, which are particularly relevant for large, low-slope, sand-bed rivers;a new predictor for near-bed entrainment rate into suspension;a new predictor for form drag based on the shear stress component.

However, debris fl ow always has uncertainties in variables and model param-eters. The properties of the moving fl uid mixture of debris and water are very different from those of purely water fl oods (Jin and Fread [133]).

The traditional approach to debris-fl ow studies using a physically hydraulic-based model is limited. Despite advances made with this approach major problems

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and barriers exist in debris-fl ow studies. Therefore, efforts to develop new tools for debris-fl ow forecasting are needed. Some investigations have, recently, explored the feasible application of the theory and concept of entropy, based upon prob-ability, to solve the sediment concentration problems of debris fl ow (Hui-Pang and Fang-Wu [114]). The entropy concept provides a new approach suitable to analyse the stochastic nature of certain hydraulic and morphologic mechanisms, including vertical distribution of the velocity, shear stress and the distribution of suspended sediment concentration in both open-channel and dam-break fl ows.

The stochastic approach proved feasible to predict sediment concentration distribution profi les, within debris-fl ow phenomena, once the channel gradient and bed materials are known or given.

In modern hydraulic engineering practice, there is a need for effi cient and accurate mathematical models and solvers that should be able to numerically rep-resent all the physically signifi cant phenomena in a given fl ow scenario.

In this context, debris fl ows resulting from a sudden collapse of a dam (dam-break) are often characterized by the formation of shock waves. It is commonly accepted that a mathematical description of these phenomena can be accomplished by means of 1D SV equations (Bellos and Sakkas [75]). These equations yield discontinuous solu-tions in the form of shocks or bores, which can be diffi cult to represent accurately without the use of appropriate shock-capturing methods.

Several classical fi nite difference or fi nite element schemes are commonly used in practical applications, but they are highly inaccurate in modelling discon-tinuous fl ow and provide unreliable approximations in subcritical and supercritical fl ow conditions.

In recent years, a great effort has been devoted to the construction of effi -cient and accurate procedures for systems of conservation laws. This resulted in schemes known as high-resolution shock-capturing schemes. Much of the devel-opment of recent high-resolution numerical methods written in conservative form is applied to the fl ow equations via the use of Riemann solvers. Some of these solvers have been analysed in this chapter.

TVD schemes were introduced by Harten [93] for effi ciently solving the Euler equations in gasdynamics. Their main property is that they are second-order accurate (except at extremes) and oscillation free across discontinuities. Their main disadvantage lies in the restriction on the time step which is imposed by the CFL condition. However, this is not a real problem for dam-break debris-fl ow phenomena that require short time steps to describe the evolution of the discharge. Attempts along this line of work were presented by Alcrudo et al. [105] introducing in the McCormack scheme TVD corrections to reduce spurious oscillations around discontinuities, for both 1D and 2D fl ow problems.

Recently, Mambretti et al. [64], [65] and De Wrachien and Mambretti [66] used an improved TVD McCormack–Jameson scheme to predict the dynamics of both strati-fi ed and non-stratifi ed debris fl ow in different dam-break conditions that overcome some of the above-mentioned constraints. Along this line of work further efforts need to be made to analyse reverse grading (sorting) and to better feature the distribution of the material of different size of the solid phase within the mixture profi le.

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Chapter 3 Dam-break wave routing - WIT Press · Chapter 3 Dam-break wave routing D. De Wrachien1,...

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