11
2D Mathematical Modeling of Discontinuous Shallow Sediment-laden Flows Ricardo Canelas Under supervision of Rui Miguel Lage Ferreira Dep. Civil Eng. IST, Lisbon, Portugal October 11, 2010 Abstract This work presents a 2DH mathematical model suited for highly unsteady discontinuous geomorphic flows over complex geometries. In order to numerically simulate the flow a finite-volume method based on the Roe Riemann solver is used, where the weak solutions are generated with a reviewed version of the solver. For simplified scenarios and fixed bed, it is shown that the 2DH model remains conservative with the careful introduction of source terms and reevaluation of the stability domain. The model is also able to find non-oscillatory solutions, is well-balanced and the correct evaluation of the discrete source terms is well made, ensuring energy dissipating solutions when necessary. Keywords: 2DH simulation, Discontinuous flows, Finite-Volume Methods 1 Introduction Open-channels with mobile boundaries are subjected to entrainment, erosion, transport and deposition of sediments. During intense floods, such as those orig- inated by dam or slope failure, hyperconcentrated flows can occur, where the concentration of sediments is such that it can be considered constant and uniform in the fluid column. Frequently these flows exhibit a high formative potential, generating important amounts of deposits and large incision volumes. These flows will herein be designated geomorphic flows. Since the phys- ical system comprises a mixture of fluid and granular matter, conservation equations must take into account such presence along with the influence of sediment on inertia and pressure terms in the momentum balance [6]. The purpose of this work is to present a 2DH mathe- matical model suited for potentially discontinuous geo- morphic flows over complex geometries. The model is based on the clear-water conservation equations consid- ering the presence of granular material, together with their own set of conservation equations. The numerical model is developed within the Finite-Volume framework and based on the model presented by Murillo & Garc´ ıa- Navarro (2010), allowing for a fully conservative simu- lation. 2 Conservation Equations Considering ρ = ρ (w) (1 + C(s - 1)) the density of the mixture, understood as a continuum [1], where ρ (w) is the water density, C is the depth-averaged sediment concentration and s is the specific gravity of sediment grains, also the interaction with the bed sediment layer, promoting a bed normal flux between the fluid column and the bottom, the application of the Reynolds Trans- port Theorem (RTT) to the scalar quantity mass, yields 1

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Page 1: 2D Mathematical Modeling of Discontinuous Shallow Sediment-laden Flows · Open-channels with mobile boundaries are subjected to entrainment, erosion, transport and deposition of sediments

2D Mathematical Modeling of Discontinuous Shallow Sediment-laden

Flows

Ricardo Canelas

Under supervision of Rui Miguel Lage Ferreira

Dep. Civil Eng. IST, Lisbon, Portugal

October 11, 2010

Abstract

This work presents a 2DH mathematical model suited for highly unsteady discontinuous geomorphic flows over

complex geometries. In order to numerically simulate the flow a finite-volume method based on the Roe Riemann

solver is used, where the weak solutions are generated with a reviewed version of the solver. For simplified

scenarios and fixed bed, it is shown that the 2DH model remains conservative with the careful introduction of

source terms and reevaluation of the stability domain. The model is also able to find non-oscillatory solutions,

is well-balanced and the correct evaluation of the discrete source terms is well made, ensuring energy dissipating

solutions when necessary.

Keywords: 2DH simulation, Discontinuous flows, Finite-Volume Methods

1 Introduction

Open-channels with mobile boundaries are subjectedto entrainment, erosion, transport and deposition ofsediments. During intense floods, such as those orig-inated by dam or slope failure, hyperconcentrated flowscan occur, where the concentration of sediments is suchthat it can be considered constant and uniform in thefluid column. Frequently these flows exhibit a highformative potential, generating important amounts ofdeposits and large incision volumes. These flows willherein be designated geomorphic flows. Since the phys-ical system comprises a mixture of fluid and granularmatter, conservation equations must take into accountsuch presence along with the influence of sediment oninertia and pressure terms in the momentum balance[6].

The purpose of this work is to present a 2DH mathe-matical model suited for potentially discontinuous geo-morphic flows over complex geometries. The model is

based on the clear-water conservation equations consid-ering the presence of granular material, together withtheir own set of conservation equations. The numericalmodel is developed within the Finite-Volume frameworkand based on the model presented by Murillo & Garcıa-Navarro (2010), allowing for a fully conservative simu-lation.

2 Conservation Equations

Considering ρ = ρ(w)(1 + C(s − 1)) the density of themixture, understood as a continuum [1], where ρ(w) isthe water density, C is the depth-averaged sedimentconcentration and s is the specific gravity of sedimentgrains, also the interaction with the bed sediment layer,promoting a bed normal flux between the fluid columnand the bottom, the application of the Reynolds Trans-port Theorem (RTT) to the scalar quantity mass, yields

1

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the mass conservation equation

∂t (h) + ∂x (hu) + ∂y (hv) = −∂t (Zb) (1)

where h is the fluid height above the bed elevation, uand v are the x and y depth averaged velocities, respec-tively, Zb is the bed elevation and ∂t (Zb) is the result ofthe vertical mass fluxes between the bed and the fluid.This term should be regarded as a source term as thebed is not considered part of the control volume.Applying the same methodology to the solid fraction ofthe total mass, Ch, having into consideration that thevertical mass fluxes are now exclusively sediment massfluxes, given by

φfs = Cφf,b, φbs = (1− p)φb,f (2)

where C is the concentration in the fluid, p is the bedporosity, (1 − p) is the sediment concentration on thebed and φb,f and φf,b are the velocities associated withthe vertical mass flux from the bottom to the fluid andfrom the fluid to the bottom, respectively, the sedimentmass conservation equation can be written as

∂t (Ch) + ∂x (Chu) + ∂y (Chv) = −(1− p)∂t (Zb) (3)

For hyperconcentrated flows, the assumption that C isconstant and closely related to the bed concentration isvalid, effectively collapsing equation (3) into equation(1). According to Takahashi (2007), C should take avalue close to 0.9 (1− p), since this is the threshold forsediment movement in these types of flows. The con-servation equation for the bed mass is formulated, fromcinematic considerations,

(1− p)∂t (Zb) = D − E (4)

where D stands for deposition of material in the bedand E for erosion, and both should be regarded as timeintegrated mass fluxes, expressed as a volume over anarea, making equation (4) dimensionally homogeneous.

In order to derive the conservation equations for the mo-mentum in x and y, application of the RTT is invokedagain, leading to

∂t (uh) + ∂x

(u2h+

12gh2

)+ ∂y (uvh)

=1ρ

(∂x(hT11)− ∂y(hT12)− ρgh∂x(Zb)− τb,x)(5)

and

∂t (vh) + ∂y

(v2h− 1

2gh2

)+ ∂x (ρuvh)

=1ρ

(∂y(hT32)− ∂x(hT31)− ρgh∂y(Zb)− τb,y)(6)

where Tij is the depth averaged turbulent stress tensor,∂xi

(Zb) represents the bottom slope contribution for themomentum and τb represents the friction exerted by thebed on the fluid. Of special importance is the note that,although the contribution of the bed slope and the tur-bulent stresses are expressed as a flux, they are not aphysical flux and should regarded as numerical sourcein a fluxes form only.

3 Closure Models

3.1 Rheology and Bottom Friction

Five different types of shear stresses are of importancefor the phenomena, yield stress, viscous stresses, tur-bulent, dispersive and frictional stresses [7]. The totalshear stress can be written as the sum of these fourstress components.

τ = τy + τv + τt + τd + τf (7)

where τy represents the yield stress, τv the viscousstress, τt the turbulent stresses τd the dispersive stressesand τf the frictional stresses. The yield stress can beregarded as the result of the cohesive nature of thefine sediment particles and the Mohr-Coulomb shear.The viscous stress, τv, take in to consideration thefluid-particle viscous interaction [10] and the dispersivestresses τd account for the inter-particle collisions [6].The turbulent stresses are addressed in two separatefronts, since the contributions from the bottom inter-action and the turbulent structures developed in thedepth of the flow can be considered in different ways.τt stands for the first contribution only. The frictionalstresses, τf are typical of geotechnical considerations

ρ(w)ghC tan(ϕb)(s− 1) (8)

where ϕb is the internal friction angle of the material.Expressing these stresses in terms of shear rates, thenτy and τy are independent from the velocity gradient,τv varies linearly with the velocity gradient, represent-ing the classical Newtonian relation, and τd and τt are

2

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associated to the second power of the velocity gradi-ent, since they are related to the kinetic energy of themedium [10]. In the bed, these considerations result in

|τb| = τy + τf + µ|u|h

+ ρd2sCf

(|u|h

)2

(9)

where µ is the dynamic viscosity of the mixture and Cfis an inertial friction coefficient, that represents the ef-fects of τd and τt. This quadratic rheological equationshows the non-Newtonian nature of the hyperconcen-trated sediment flow, typical of a Bingham and dilatantfluid. Cf can be regarded as the sum of the dispersiveand turbulent effects near the bed. For the dispersivepart, Takahashi (2007) proposes the following expres-sion

Cdf =45

√1

15π1 + e√1− e

C2G (10)

where e is the reconstitution coefficient of the sedimentmaterial and G is the radial distribution function, givenby

G =C

22− C

(1− C)3(11)

The turbulent parcel of Cf , Ctf , is assumed constant inthe flow ([4], cap. 5.2).

3.2 Bottom Variation, Erosion and De-

position

Applying the Rankine-Hugoniot shock conditions to themomentum transfer across an assumed frictional layerat the interface between bed and fluid, disregarding thepressure variation, yields

−∂t (Zb) (ρu1 − ρu0) = τ1 − τ0 (12)

where −∂t (Zb) is the shock speed of an erosion event,u1 and τ1 are the flow velocity and the shear stress atZf elevation from the bottom, in the top of the fric-tional layer, respectively, and u0 and τ0 are the flowvelocity and the shear stress at the bed, respectively.τ1 depends mostly on dispersive, turbulent and viscousstresses that caracterize the fluid, while τ0 is governedby the yield stress and τf . The net time integrated massflux in equation (12) is now given by

(D−E) =τy + ρ(w)ghC tan(ϕb)(s− 1)− (µuh + ρd2

sCfu2

h2 )ρu1

(13)

since u0 = 0. u1 can be approximated by 0.4|u|, ac-cording to results by Ferreira, (2006) from a series ofnumerical simulations of granular systems.

4 Discretization Scheme

The hyperbolic, non-homogeneous, first order, quasi-linear, system (equations (1), (5) and (6) ) in compactnotation can be written as

∂t (U(V )) + ∂x (F (U)) + ∂y (G(U)) = H(U) (14)

where V is the primitive variables vector, U the inde-pendent variables vector, F and G are the flux vectors,in x and y direction and H is the source terms vector.To obtain the FVM discretization, the system (14) isintegrated in a cell, i

∂t

∫Ωi

U(V )dS +∫Ωi

∇ ·E(U)dS =∫Ωi

H(U)dS (15)

Applying Gauss theorem to the divergence term in (15),assuming that if the cell area is Ai, the representationis piecewise, and performing the boundary integral onthe ni edges of cell i, equation (15) becomes

Ai∆ 〈U i〉

∆t+

ni∑k=1

Lk∆ik 〈E · n〉 = Ai 〈Hi〉 (16)

where E ·n = Fnx+Gny and n = (nx, ny)T is the out-ward unit normal vector to cell, the 〈 〉 represents thespatial average in the cell, and Lk is the k edge length.The fluxes trough the k edge of cell i represent, in thesimplest interpretation, the differences between the val-ues of the independent variables on the adjacent cells jand i, separated by edge k. The flux variations can beexpressed as a function of the independent conservativevariables using a Jacobian matrix, Jn,ik, orthogonal tothe edge in question

∆ik 〈E · n〉 = Jn,ik∆ik 〈U〉 with

Jn,ik =∂E · nik∂U

=∂F

∂Unx +

∂G

∂Uny

(17)

Even if the system is hyperbolic it is not usually possibleto construct a diagonal matrix for x and y simultane-ously for the Jacobian of the flux vector since the fluxvectors are not homogeneous functions of the depen-dent variables, for the shallow water type system. This

3

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makes it impossible to use a natural Jacobian. Assum-ing a local linearization, an approximate Jacobian ma-

trix,vJn,ik can be used for the homogeneous part of the

flux [[?]]. This approximate Jacobian is used trough thedefinition of approximate variables

vuik =

ui√hi + uj

√hj√

hi +√hj

(18)

vv ik =

vi√hi + vj

√hj√

hi +√hj

(19)

vc ik =

√ghi + hj

2(20)

The explicit construction of the matrix is avoided bythis method, but the eigenvalues and eigenvectors arecomputable and given by (21) and (22), respectively.These represent the structure of the homogeneous solu-tion of the system

(1)

ik =(

vu ·n− v

c)ik

(2)

ik =(

vu ·n

)ik

(3)

ik =(

vu ·n+

vc)ik

(21)

ve

(1)

ik =

1vu − v

c nxvv − v

c ny

ik

ve

(2)

ik =

0− vc ny

vc nx

ik

ve

(3)

ik =

1vu +

vc nx

vv +

vc ny

ik

(22)

The independent variables variation is projected on anew base, formed by the eigenvectors of the system

∆ik 〈U〉 =3∑

n=1

α(n)ik

ve

(n)

ik (23)

where the α(n)ik coefficients are the wave strengths taken

directly by solving algebraically equation (23). Thisrenders the fluxes also susceptible to be projected onthe the homogeneous system eigenvectors base,

∆ik 〈E · n〉 =3∑

n=1

(n)

ik α(n)ik

ve

(n)

ik (24)

Following the work of Burguete (2001), taking advan-tage of the way the bottom pressure terms are ex-pressed, in a numerical flux form, the scheme can beredesigned to handle the flux in a unified way, i.e.

Ai∆U i

∆t+

3∑k=1

Lk(∆E − T )ik · nik = 0 (25)

where T is a suitable numerical source matrix [2], repre-senting the contribution of the source terms to the flux.These numerical fluxes are also discretized in the sameconservative way as the advective flux terms, i.e., theyare also projected in the homogeneous system Jacobianeigenvectors

AiHi =3∑k=1

3∑n=1

β(n)ik

ve

(n)

ik Lk (26)

where β(n)ik are the coefficients that express the magni-

tude of the flux derived from the source terms. Equation(25) can now be expressed as

Ai∆U i

∆t+

3∑k=1

Lk

3∑n=1

(vλ

(n)

α(n) − β(n)

)−ik

ve

(n)

ik = 0

(27)Following Godunov’s method [8], the desired flux-basedfinite volume scheme can be written as

Un+1i = Un

i −∆tAi

3∑k=1

Lk

3∑n=1

(vλ

(n)

α(n) − β(n)

)−ik

ve

(n)

ik

(28)

where only the negative part of the eigenvalues,vλ

(n)

ik

and the β(n)ik coefficients are used, with λ

(n)±ik =

12 (λ± |λ|)(n)

ik and β(n)±ik = 1

2 (β ± |β|)(n)ik , so that only

in-coming fluxes are use in the update of the conservedvariables.When cell averaging the solution the time step is chosensmall enough to guarantee the that no interaction be-tween two adjacent RP waves is observed. The stabilityregion considering the homogeneous part of the systemis

∆t 6 CFL ∆tvλ ∆t

vλ =

min (χi, χj)

max |vλ

(n)

|n=1,2,3

(29)

where the CFL value is taken close to 1 for triangularunstructured meshes and

χi =Ai

max (Lk)i,k=1,2,3

(30)

4

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In order to take into account the influence of the sourceterms in the stability region [9], new intermediate statesgenerated by the source terms in the solution of the RPare used to define

∆t∗∗∗ = χj

2vλ

(3)

ik

hnj

hnj −h∗∗∗j

if h∗∗∗j < 0 and hnj 6= 0

∆t∗ = χi

2vλ

(1)

ik

hni

hni −h∗i

if h∗i < 0 and hni 6= 0(31)

where

h∗i = hi +(α− β

)(1)

ik

h∗∗∗j = hj −(α− β

)(3)

ik

(32)

define the fluid heights of these intermediate steady so-lutions, resulting in

∆t ≤

min(

∆t∗∗∗, ∆t∗, ∆tvλ)

if(

(1)vλ

(3))ik

≤ 0

∆tvλ otherwise

(33)

4.1 Treatment of the Source Terms

The complete differential form of the source term vectorof the system is given by

H(U) =(D − E1− p

,pb,xρ− τb,x

ρ,pb,yρ− τb,y

ρ

)T(34)

The bottom source terms are treated by expressingthem as fluxes and integrating them in the upwind al-gorithm. The β coefficients in equation (28) are givenby

β(1)ik = − 1

2vc ik

(pbρw

); β

(2)ik = 0; β

(3)ik = −β(1)

ik (35)

where

(pbρ

)k

=

max

((pb

ρ

)a,(pb

ρ

)b)k

if δdikδzik > 0and (

vuik nik)δzik > 0(

pb

ρ

)bk

otherwise

(36)in order to ensure energy dissipation solutions whenneeded, resulting in an optimization of the solution. ziis the bottom elevation of cell i and di = (hi + zi) and(

pb

ρ

)ak

= −g(vh δz

)ik(

pb

ρ

)bk

= −g(hr − |δzik|

2

)δz′ik

(37)

with

r =

i if δzik > 0j otherwise

(38)

δz′ik =

hi if δzik > 0 and di < zj

hj if δzik < 0 and dj < zi

δzik otherwise(39)

This linearization of the source terms, together withthe requirements on the properties of the numerical so-lution when these are involved can lead to much smallervalues of the time step than using the classic CFL con-dition, since ∆t∗∗∗ or ∆t∗ can be several orders of mag-nitude smaller than ∆t

vλ . This can however be avoided

by means of a proper reconstruction of the approximate

solutionvU (x′, t), [9]. Forcing positivity on equations

(32), leads to

β(1)ik ≥ β

(1)min, β

(1)min = −

(hni + α

(1)ik

)|

(1)

ik |

β(3)ik ≥ β

(3)min, β

(3)min = −

(hnj − α

(3)ik

) vλ

(1)

ik

(40)

This reconstruction is to be applied only to subcriticalwet/wet RP cases, as in dry/wet RP the occurrence ofnegative water depths in the approximate solution al-lows for a correct tracking of the flooding advance, andin supercritical cases positivity is naturally assured.Due to the simplicity and inability to harm the conser-vative properties of the scheme, the mass and frictionsource terms are treated in a point-wise, semi-implicitmanner, rendering equation (28) as

Un+1i = Un

i −∆tAi

3∑k=1

Lk

3∑n=1

(vλ

(n)

α(n) − β(n)

)−ik

ve

(n)

ik

+∆t(Rn+1i

)(41)

where the extra term is R =[D−E1−p −

τb,x

ρ − τb,y

ρ

]T,

and the expressions are (9) and (13).

4.2 Entropy Correction

Unphysical results derived from the construction of theconservative method can occur, since there is no guaran-tee that the entropy condition1 is satisfied by the com-puted weak solution. In order to force a physically cor-rect solution to the problem, an entropy correction must

1Detailed in [8].

5

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take place. A version of the Harten-Hyman entropy fixpresented in [8], is implemented, were the objective isto forcefully promote the propagation of information inany occurring expansion wave by diffusing the shock.To do so, [9] decomposes the initial jump associated tovλ

(1)

ik in two to jumps

−λ

(1)

ik = λ(1)ik

(1)jk −

(1)

ik

)(λ

(1)jk −λ

(1)ik

) ∧λ

(1)

ik = λ(1)jk

(vλ

(1)

ik −λ(1)ik

)(λ

(1)jk −λ

(1)ik

)

−λ

(1)

ik +∧λ

(1)

ik =vλ

(1)

ik

(42)

andvλ

(3)

ik .

−λ

(3)

ik = λ(3)jk

(vλ

(3)

ik −λ(3)ik

)(λ

(3)jk −λ

(3)ik

) ∧λ

(3)

ik = λ(3)ik

(3)jk −

(3)

ik

)(λ

(3)jk −λ

(3)ik

)

−λ

(3)

ik +∧λ

(3)

ik =vλ

(3)

ik

(43)

The same procedure is applied to β(n)ik , but in a way

that a conservative splitting is verified.

4.3 Wetting-Drying Algorithm

It is still possible to find regions of the solution wherenegative values of water depth is expected, namely inwet/dry interfaces with discontinuous bed level, whereaccording to (31), the time step in those conditions be-comes nil. In order to ensure positivity and conservationin the solution for all cases, the fluxes for the update ofthe conserved variables are computed as

If hnj = 0 and h∗∗∗j < 0 then(∆E − T )−ik = (∆E − T )k and (∆E − T )−jk = 0

If hni = 0 and h∗i < 0 then(∆E − T )−jk = (∆E − T )k and (∆E − T )−ik = 0

(44)These restrictions efficiently prevent the appearance ofnegative water depths by the redefinition of the fluxsplitting at the referred interfaces, avoiding the exis-tence of flux for the cell to which an intermediate statepredicts a negative height due to the update, also al-lowing for a simple control over numerical dissipationon the wave front and drying region.

5 Results

For the dam-break test case, the classical Stoker solu-tion, as well as solutions over discontinuous bed levels,where a energy dissipation and stationary shocks areexpected, are presented and compared with numericalresults. It is important to define the fundamental non-dimensional parameters that describe the initial condi-tions [5],

α ≡ hR + |min(0, Ybl)|hL +max(0, Ybl)

; δ ≡ YblhL +max(0, Ybl)

(45)were hR and hL are the water heights of the initial rightand left states, respectively, and Ybl is the bed elevationon the left side considering the right side the referencehorizontal plane. For a case where hL = 10.0 m, hR = 1m, α = 0.1, δ = 0, CFL = 0.8, no friction is regardedand triangular cells have an average side of 0.65 m, Fig-ure 5 shows the profile of the analytical Stoker solutionagainst the computed weak solution. The plot shows aleft moving expansion wave and a right moving shockwave, with correct amplitudes and positions accordingto the exact solution. The effects of the first orderscheme are clear in the interface between the expansionwave and the undisturbed state upstream, since numer-ical dispersion results in the smooth region observed,instead of the clear distinction between the states sub-jected or not to information of the downstream condi-tions. For α = 0.1 the flow upstream is subcritical anddownstream supercritical [4], forcing the entropy cor-rections to take place. The effects of the corrections arenot visible, and no numerical oscillations in the solutionare detected.For a case where hL = 10.0 m, α = 0.0, δ = 0 andCFL = 0.8, Figure 5 shows the profile of the ana-lytical solution against the computed solution. Twoweak solutions are plotted, corresponding to two dif-ferent mesh densities. The weak solutions contain theexpected left moving expansion wave and a right mov-ing wetting advancing front. The water height profileshows a good match to the exact solution, again present-ing a smoothed discontinuity in the interface to the leftof the expansion wave. The velocity profile shows moresevere effects of the first order scheme, since numeri-cal diffusion renders the maximum speed bellow the ex-pected for very small water heights. Also, the presented

6

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−20 −15 −10 −5 0 5 10 15 20 25 300

2

4

6

8

10

x(m)

h(m

)

u

(ms−1

)

ES − h ES − u WS − h WS − u

Figure 1: Dam-break problem, Stoker solution. Exact Solution - ES; Weak Solution - WS.

−20 −15 −10 −5 0 5 10 15 20 25 300

5

10

15

20

x(m)

h(m

)

u(m

s−1)

ES − h ES − u WS(1) − h WS(1) − u WS(2) − h WS(2) − u

Figure 2: Dam-break problem, Ritter solution. Exact Solution - ES; Weak Solution - WS. Cell sides with an average length

of 0.65 m for WS(1) and 0.15 m for WS(2).

7

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wetting-drying algorithm reinforces this behaviour bylimiting the flux in the wetting front in order to ensurepositivity of the solution. A small mass accumulationtakes place immediately upstream of the interface dueto the flux redistribution presented in equations (44).

For a case where hL = 10.0 m, α = 0.2, δ = −0.1 andCFL = 0.8, Figure 5 shows the profile of the analyticaland weak solutions for t = 1 s. In the solution it is visi-ble the left undisturbed state and in the downstream anexpansion wave intersected by a constant state, a sta-tionary shock in the step section, followed by anotherconstant state and the second shock upstream of theundisturbed state of the right side. The weak solutioncorrectly follows the analytical solution, indicating thata correct energy dissipation induced by the considera-tion of the bottom source terms took place, and that theshock speeds are correctly described by the discretiza-tion. The effects of the numerical dissipation are onceagain visible in the interfaces between states.

For a case where hL = 10.0 m, α = 0.1, δ = −0.1and CFL = 0.8, Figure 5 shows the profile of the an-alytical and weak solutions for t = 1 s. Again, theweak solution correctly follows the analytical solution,but expectedly the velocity at the wave front is under-estimated, for which the numerical dissipation and thewetting-drying algorithm are responsible. Of great im-portance is the notion that the distortion of the profileat the wave front for these ideal solutions is not a fac-tor that should greatly compromise the quality of thesolution on a real case simulation, since the inclusion offriction should to some extent mask these effects.

The initial conditions for a 2D dam-break problem arehL = 10.0 m, α = 0.0, δ = 0.0 and CFL = 0.8. Theseare presented in Figure 5, demonstrating a square ob-stacle with 3.0 m height, in order to promote over top-ping. For t=1.0 s, impact on the structure has alreadytaken place, and the fluid free surface and the velocityfield are represented in Figure 5. It is possible to seethe accumulation of mass upstream of the obstacle askinetic energy is dissipated, and the beginning on theover top of the structure. For t=2.35 s the structure isalready completely submerged, and the hydraulic jumpupstream is fully formed. The distortion on the veloc-ity fields are quite clearly represented in Figure 5. Fort=4.70 s, major recirculation zones are present, due to

the reflection of the mass in the end of the flume andthe influence of the submerged structure. Figure 5 rep-resents the computed weak solution.

6 Conclusions

In this dissertation theoretical and computational workwas done in order to develop a general shallow watermodel based on the Finite Volume Method, employingan extended and reviewed version of Roe’s solver. Basedon work by Murillo (2010), a state of the art model waspresented and implemented.Considerable effort was placed into the implementationof modeling capabilities for simple non-Newtonian rhe-ologies, believed to best describe homogeneous hyper-concentrated flows. These rely on closure models thatadds complexity to conceptual model and extra difficul-ties to the discretization technique.Overall, it can be claimed that the model is highly ro-bust, since no situation was detected were the modelresponded in a clearly inadequate way, especially in sim-ulations where typically other schemes of the same na-ture have acute problems. The generation of the solu-tion always came about with an acceptable CPU time,i.e. the stability region and consequent time step werealways optimised, without compromising the solution.Having in mind the previous conclusions, the model isconsidered capable of providing meaningful solutions forapplications as

• studies for emergency plans regarding floods ona local and regional scale, with possible repercus-sions on general regional urban planning;

• the study of the impact of dam-break flows ondownstream regions;

• study of flooding in heavily urbanized areas.

References

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−20 −15 −10 −5 0 5 10 15 20 25 300

2

4

6

8

10

x(m)

h(m

)

u(m

s−1)

WS − h WS − u Bottom Elevation ES − h ES − u

Figure 3: Dam-break problem with discontinuous bottom step and wet bottom, ∆Zb = 1.0 m, Exact Solution - ES; Weak

Solution - WS. t = 1 s

−20 −15 −10 −5 0 5 10 15 20 25 300

5

10

15

20

x(m)

h(m

)

u(

m−1

)

WS − h WS − u Bottom Elevation ES − h ES − u

Figure 4: Dam-break problem with discontinuous bottom step and dry bottom, ∆Zb = 1.0 m, Exact Solution - ES; Weak

Solution - WS. t = 1 s

Figure 5: Initial conditions for 2D Dam-break with low obstacle.

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Figure 6: 2D Dam-break with square obstacle. 3D view and velocity vectors. t=1.0 s.

Figure 7: 2D Dam-break with square obstacle. 3D view and velocity vectors. t=2.35 s.

Figure 8: 2D Dam-break with square obstacle. 3D view and velocity vectors. t=2.35 s.

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