2d finite volume numerical schemes for transient … · a great deal of work has been devoted to...

XXX° Convegno di Idraulica e Costruzioni Idrauliche - IDRA 2006 Master Class: Modelli numerici di correnti fluviali su fondo fisso e fondo mobile

2D FINITE VOLUME NUMERICAL SCHEMES FOR TRANSIENT FREE SURFACE FLOW, SOLUTE TRANSPORT AND EROSION/SEDIMENTATION

PROCESSES

J. Murillo1, P. Brufau1 and P. García-Navarro1

(1) Fluid Mechanics, University of Zaragoza - Zaragoza (Spain) e-mail: [email protected]

Parole chiave: equazioni delle acque basse, volumi finiti, trasporto solido, erosione e deposito, trasporto di soluti,

SOMMARIO

Prendendo lo spunto dalle ricerche sviluppate presso il Dipartimento di Meccanica dei Fluidi dell’Università di Zaragoza, la relazione inquadra l'argomento della simulazione matematica di correnti bidimensionali a superficie libera sia su fondo fisso fisso che su fondo mobile con modelli matematici risolti numericamente ai volumi finiti e schemi di integrazione di tipo upwind di diverso ordine di accuratezza. Sono analizzati due solutori accurati al I e al II ordine. L’estensione al II ordine, soprattutto in due dimensioni, presenta notevoli difficoltà in dipendenza della griglia di calcolo utilizzata. Griglie adattative triangolari non strutturate permettono di descrivere correttamente il campo di moto infittendo il reticolo nelle zone di interesse ovvero dove si realizzano i maggiori gradienti, evitando nel contempo di introdurre nel calcolo direzioni preferenziali di propagazione. Oltre al classico solutore proposto da Roe, accurato al I ordine, viene considerata l’estensione al II ordine di accuratezza che utilizza opportune funzioni limitatrici della curvatura della soluzione. I solutori descritti sono stati applicati per la simulazione di esperienze su casi test di laboratorio e su modello fisico: nella simulazione è stata introdotta un tecnica per l'estensione del passo di integrazione temporale oltre il limite posto dalla condizione di stabilità di Courant-Friedrichs-Lewy. I solutori proposti sono infine stati utilizzati per simulare alcuni esperimenti numerici di correnti bidimensionali in presenza di soluti e di correnti iperconcentrate su fondo fisso e mobile.

1 INTRODUZIONE

Many engineering and environmental problems involve the study of unsteady water flows with solute transport and erosion/sedimentation processes. River flows, in particular, are mostly unsteady and, as they are characterized by the presence of a vertical scale much smaller than the horizontal ones, they can be described by the shallow water model (Cunge et al., 1980) which forms a set of non linear hyperbolic equations that, in two dimensions, involve the water depth as well as the two depth averaged components of the velocity (Fig.1)

( ) ( ) ( )USUGUFU ,, yxyxt

=∂∂

+∂∂

+∂∂

( )Tyx qqh ,,=U

1

J.Murillo, P.Brufau, P.García-Navarro

Tyxx

x hqqgh

hq

q

+= ,

2,

22F

,

Tyyx

ygh

hq

hqq

q

+=

2,,

22

G

(1)

where and . The variable represents the water depth,uhqx = vhq y = h g is the acceleration of

the gravity and ( are the averaged components of the velocity vector along the

)vu, ux and coordinates respectively. The source terms in the momentum equations are the bed slopes and

the friction losses along the two coordinate directions, y

(2) ( )Tfyyfxx SSghSSgh )(),(,0 00 −−=S

where,

xzS x ∂∂

−=0, y

zS y ∂∂

−=0 (3)

and the friction losses in terms of the Manning’s roughness coefficient, with

3/4222 / hvuunS fx +=

, 3/4222 / hvuvnS fy +=

(4)

Figure.1: Two-dimensional depth-averaged shallow water model

A great deal of work has been devoted to develop 1D and 2D numerical models for unsteady shallow

flows in the last decades and various computational techniques using finite difference, finite element and finite volume methods have been reported (Cunge et al., 1980; Bellos et al., 1991; Alcrudo & García-Navarro, 1993; Sleigh et al., 1998; Bermúdez et al., 1998; García-Navarro & Vázquez-Cendón, 2000). Several numerical difficulties must be adequately treated to obtain an accurate solution without numerical errors.

Zhao et al. (1994) provided a good historic revision and the features required for a two dimensional

2

2D Finite volumes numerical schemes

river flow simulation model: it should be able to handle complex topography, dry bed advancing fronts, wetting-drying moving boundaries, high roughness values, steady or unsteady flow and subcritical or supercritical conditions.

Among the numerical techniques reported, those belonging to the category of conservative methods have gained acceptance in recent years for their important property of providing a proper discrete representation of the physical conservation laws. Essentially imported from gas dynamics, they have been extended to shallow water problems trying to overcome the relevant differences existing between these two applications. Natural topographies is the main challenge. Dominant source terms and open boundaries are two important difficulties to face when using a conservative method since they both can damage the conservative character of the solution.

Upwind methods in particular are becoming increasingly popular in the hydraulics literature and have proved a suitable way to discretize the shallow water equations (Brufau et al., 2002; Burguete &García-Navarro, 2001; Hubbard & García-Navarro, 2000; Toro, 2000). Being a hyperbolic system of conservation laws, they are a good candidate for application of the techniques developed for the Euler equations in gas dynamics.

The discretization of systems of hyperbolic conservation laws that include source terms following steady state equilibrium criteria is a fundamental step in numerical simulation. Bed slope and friction source terms are of special relevance in hydraulic applications based on a shallow flow model. For that reason, a considerable effort has been recently devoted to this topic in a search for the correct source term discretization given a particular numerical scheme with good properties for the homogeneous case. Following this purpose, Leveque (Leveque, 1998) incorporated the modelling of source terms to his wave propagation algorithm, while Roe’s scheme (Roe, 1981) was modified by a number of authors to include source terms, Glaister (1992), Vázquez-Cendón (1999), Bermúdez & Vázquez-Cendón (1994) and Bermúdez et al. (1998). Following Roe’s scheme, the extension from first to second order in the case of systems of equations was explored by Hubbard & García-Navarro (2000) noting that an adequate evaluation of the source terms ensured a correct balance at least in situations of still water by means of an upwind technique based on the Jacobian matrix diagonalization.

Another challenge is the extension of the cited methods to higher order resolution schemes, able to capture shocks and discontinuities more accurately. The extension of numerical methods to second order in two-dimensional problems is not straightforward for different reasons. One of them is the strong influence of the grid used on the properties of the scheme chosen to be moved from 1D to 2D cases and, therefore, on the quality of the expectable numerical results. Triangular unstructured adaptive meshes offer, not only the freedom to fit the computational domain to the shape of the physical region of interest, but also the possibility to eliminate as much as possible preferential directions in the numerical solution. On the other hand, triangular grids are a challenge for high order numerical methods since, in some cases, their methodology relays on the structured enlargement of the stencil and the step to pass from structured to anisotropic unstructured meshes is restrained by the difficulty in constructing analogous extensions to higher-order accuracy and sometimes is not directly movable to the context of unstructured meshes.

Roe’s scheme (Roe, 1986) is chosen as the basic first order explicit scheme to build improved versions able to be used in two-dimensional problems. The alternative is to move from the first order piecewise constant representation to the reconstruction of the solution by means of slope limiter functions that can be directly formulated on triangular cells (Batten et al., 1996). Those methods are said to achieve an order greater than one in regions of smooth solution and to be devoid of oscillations at discontinuities. Among them, the Limited Central Difference (LCD) approach (Batten et al., 1996), the compressive limiter proposed by Durlofsky et al (Durlofsky et al., 1992) and the Maximum Limited Gradient (MLG) (Batten et al., 1996), are well known. A new reconstruction function with excellent properties can be built as a combination of the MLG reconstruction and the limiter function of Wierse (Wierse, 1997; Murillo et al. 2006).

Another numerical problem analysed is the modelling of wet/dry interfaces between internal cells, that have traditionally represented a difficulty for modellers wanting to solve the shallow water equations over a bed of irregular geometry. Flow over dry bed involves a complicated situation that can be analysed as a boundary condition which is dynamically changing in time with the moving front and continuously expanding or reducing the flow domain. The alternative is to include the wet/dry interfaces in the full

3


domain of computation in which there may be wet cells and dry cells at the same time. In this case the numerical scheme chosen for the discretization must be able to cope with them. In general, cells being flooded or dried during the computation tend to introduce numerical instabilities in the solution, resulting for example in negative water depths or unphysical high velocities. Different approaches have been proposed to handle them. These techniques include modified equations in very shallow regions (Meshelle, 1993), shock capturing schemes or the assumption that a cell is dry if water depth is below a small critical value (Kramer, 2001). Beffa & Connel (2001) reported numerical oscillations when cells switch from dry to wet or vice-versa. George & Stripling (1995) represented the local bathymetry in each cell by a sloping facet rather than by a flat bed to eliminate the spurious shocks in their finite volume model. Some authors working with finite elements solve the problem allowing the controlled use of negative depths (Heniche & Secretan, 2000; Kawahara & Umetsu, 1986; Khan, 2000). Bradford & Sanders (2002) used Neumann extrapolation of the velocity in partially wet cells to bypass the incorrectly estimation of pressure and body forces in such cells. In Brufau et al. (2002) and Brufau et al. (2004), driven by the interest of controlling numerical stability and global mass conservation, a two-dimensional model was presented for unsteady flow simulation where the main strategy was based on a local redefinition of the bed slope at specific locations.

The formulation of the upwind finite volume method can be based on waves. This is different to the most common numerical flux finite volume methods and makes the discretization at the boundary conditions easier and more physically based. The numerical flux is a rigid concept and requires the help of the so called ‘ghost’ cells at the boundaries, which must be artificially created in order to eliminate the non-desired information at the boundary cells. The values of the variables inside the ghost cell must be defined in an ad-hoc manner from the values of the interior cell, depending on the kind of imposed boundary condition (Alcrudo & Garcia-Navarro, 1993; Brufau et al., 2000 and 2002). This may become complicated for certain kind of external boundary conditions. When the updating scheme only considers the in-going contributions to a cell, no ghost cells are needed whatever the boundary condition is, leading to the correct solution in a simpler way.

The mathematical properties of the hyperbolic system of equations (1) include the existence of a

Jacobian matrix, , of the flux normal to a given directionnJ ( )nE ⋅ defined as:

( ) ( ) ( )yx nn

UG

UF

UnEJn ∂

∂+

∂∂

=∂⋅∂

= (5)

making possible the generation of an approximate matrix J , whose eigenvalues ∗n

mλ~ an

eigenvectors me~ (Brufau et al., 2002; Hubbard & Garcia-Navarro, 2000), can be used to define the signals, and also that an upwind treatment can be done over the source terms. The finite volume discretization assumes that, in first order of spatial approximation, the variables are piecewise constant (Fig. 2 left) so that there are contributions or waves defined at the cell edges (Fig.2 right):

(6) ∑∑∫==

=+

NE

k

nkkk

NE

k

e

ekk ldlyx

k

k 11

)(),(1

nEnE δδ

with that travel in or out of the cells depending on the value of the eigenvalues of the normal flux Jacobian. Using at the same time that a decomposition in the basis of eigenvectors is possible as follows:

ijk EEE −=δ

(7) ∑=

=−=λαδ

N

m

mkoiojk

1,, )~( eUUU

4


so that:

(8) ∑∑∑= ==

=NE

k

N

mkk

mmmNE

kkkk ll

1 11, )~~(~ λ

αλδ eUJn

the first order formulation of the upwind scheme is as follows (Murillo et al., 2005a):

tAl

i

kNE

k m

nk

mmmmni

ni ∆−−= ∑∑

=

−+

1

1 ))~)~(( eUU βαλ (9)

Figure 2: First order finite volume scheme

2 EXPLICIT FIRST ORDER SCHEME: EXTENSION TO CFL >1

The allowable time step size is restricted in the explicit case by stability reasons to fulfil the Courant-Friedrichs-Lewy (CFL) condition (Courant et al., 1952). It is possible to relax the condition over the time step size when using explicit schemes. A generalization of the first order explicit upwind and Roe’s method (Roe, 1968), modified to allow large time steps, was explored by Leveque (Leveque, 1981; Leveque, 1985) first in the scalar non-linear case and then adapted to systems of equations. In this work an explicit extension of the two-dimensional upwind finite volume scheme to values of CFL greater than one is defined, where conceptual simplicity is the most valuable characteristic as the variables at a future time can be independently evaluated at every single point. For details on this technique see Murillo et al., 2005b.

The next validation test case for the proposed model deals with a two-dimensional dam break flow problem. The experiment was carried out at the CITEEC, Coruña, Spain (Mendez et al., 2001). The setup consists of a closed pool divided in two parts (Figure 3 left) by a removable gate. In this case the bed is plain. The experiment was performed for an initial depth ratio of 0.5/0.1m assuming a Manning roughness parameter n = 0.01. The initial mesh was generated using a Delaunay solver (Triangle) and retriangularizated as indicated above leading to 7875 cells (Figure 3 right). The results are compared with the measures provided by 17 gauge points distributed all over the domain (Figure 3 left).

5


Figure 3. Model geometry with gauge points (left) and detail of the mesh (right).

Figure 4 shows vector velocity maps at time 2 s for both values of CFL. As it can be seen, the differences between them are negligible. The comparison between numerical results and experimental data at some gauging points of the physical model is displayed in Figure 5, where the numerical results for the different values of CFL are coincident in a single line.

Figure 4. Velocity vectors at time 4 s, computed with CFL = 1 (left) and 4 (right).

6


Figure 5. Comparison between experimental data and numerical data for CFL = 1 and 4.

3 SECOND ORDER APPROACH

The correct extension to second order approach requires special attention in the discretizatrion of the

7


source terms. When applying the MUSCL or MUSCL-Hancock method it is necessary to properly discretize the source term. This can be done using a flux correction to ensure hydrostatic balance in still water conditions (Bradford & Sanders, 2002; Audusse & Bristeau; 2005). A novel and elegant procedure to accurately and efficiently model second order approach is derived constructing the numerical fluxes in a compact form where equilibrium is defined in a natural way. The formulation of second order in space representations is performed via the slope of the gradient of the variable in a cell (Figure 6). The step from the scalar case to the system case introduces a new degree of freedom in the choice of the variables to extrapolate. Also the generation of advection velocities and primitive variable values derived from interpolated variables is not a straightforward question in the case of systems of equations. Special attention has to be put in avoiding numerical oscillations arising from incorrect balance of the source terms at the discrete level for steady-state situations including flow in motion.

The piecewise linear reconstruction of a scalar variable u, over an element with centroid at (xo, yo) is expressed as

iuoiiuooii yxuyxyxuyxu ,,, ),(),(),(),( LrLr +=+= (10)

where r is the position vector from the centroid, and L is the cell slope. Different forms to define and to limit the cell slope can be used. Figure 6 shows the position vectors of the middle points of edge k as well as the slope limited extrapolated values of h.

Figure 6. Second order in space slope limiting technique

The first order formulation as expressed in (9) is extended in spatial second order to:

i

NE

k

N

mk

mkIi

i

NE

k

N

mk

mkJI

ni

ni A

tlA

tl ∆−−

∆−−= ∑∑∑∑

= == =

−−+

1 1,

1 1,

1 )~)~(()~)~((λλ

βαλβαλ eeUU (11)

An effort has been made to formulate the cell-updating algorithm as a sum of waves generated at the cell edges by the joint contribution of normal flux differences and normal source terms. In the second order method these waves are of two kinds, those generated by the jump across the edge and those generated by the jump between the edge and the cell center values.

8


4 COUPLED SOLUTE TRANSPORT

The rules for the upwind finite volume scheme when applied to a general system of hyperbolic

equations with source terms can be applied to the coupled transport-shallow water equations. One fundamental topic is the modelling of passive solute transport by shallow water flows. In practice it is very common to solve the depth averaged solute transport equation apart from the shallow water equations, that is, using a decoupled algorithm in which, first, the flow pattern is known and then the transport in that flow field is calculated (E. Playán et al., 2000; Weiming Wu et al., 2004). The reason for this is the physical assumption that, for low concentrations, the solute dynamics does not influence the flow behaviour, justifying then the use of a simpler decoupled resolution algorithm. It is also frequent to express and to use the solute transport equation in a non-conservative form, assuming that the velocities, depth, and the bottom level vary smoothly in time and space. Also, when the diffusion is not dominant, the transport problem can be considered as a linear advection problem in which the advection velocity is the flow velocity. The traditional numerical techniques applied to the non-conservative form of the decoupled solute transport equation are of the semi-Lagrangian type and do not have, in general, the property of being conservative.

The coupling formulation proves beneficial in avoiding numerical instabilities in the solute concentration when applied to complex situations (see Fig. 8 left). On the other hand, the coupling of the equations and the application of an upwind scheme generate numerical source terms both in the water mass and in the solute mass conservation equation. This leads to the necessity of a complete upwind treatment of the bed variation source terms that ensures the best balance in steady state cases. This has already been pointed out in many previous works in the context of shallow water flows (Brufau et al., 2002; Brufau et al., 2004), but is even more obvious when extending the system of conservation laws to include the solute concentration equation.

Following Murillo et al.(2005), the system of partial differential equations of the form (1) will be formulated here in a coupled form as follows:

( ) ( )USUGUFU

=∂

∂+

∂∂

+∂∂

yxt)(

(12)

being

( )Tyx hqqh φ,,,=U

Tyxx

x uhhqqgh

hq

q

+= φ,,

2,

22F

Tyyx

y vhghh

qhqq

q

+= φ,

2,,

22

G (13)

( )( ) Tfyoyfxox hSSghSSgh φ∇∇−−=

rrKT ),(),(,0

the new quantities areφ , the depth-averaged concentration and the term of solute concentration diffusion, where K is an empirical dispersion matrix that should not be confused with the turbulent diffusivity. In general, K incorporates dispersion due to differential advection as well as turbulent diffusion (Cunge et al., 1980).

An upwind technique is applied to solve the flux terms in both the flow and solute equations and the

bed slope source terms is formulated exclusively from the point of view of the in-going contributions to the cells of the domain. A centred discretization is applied to the diffusion and friction terms. Furthermore, the finite volume scheme used is explicit, which implies a restriction on the time step size

9


that can be severe in presence of diffusion. Various techniques are presented and tested, where the time step is governed by a combination of the Peclet (Pe) and the Courant-Friedrichs-Lewy (CFL) numbers. A splitting technique is adopted to solve diffusion implicitly, avoiding small values of time step and allowing high accuracy, without increasing the numerical diffusion. Figure 7 shows a numerical test with an initial concentration level and the comparison of the numerical results after 10 and 15s obtained with a coupled and a decoupled system of equations is shown in Figure 8.

Figure 7. Initial concentration level.

10


Figure 8. Isolines of concentrations at times T = 10, 15 s, with a coupled system (left), and decoupled resolution (right).

5 VARIABLE DOMAIN LIMITS

When source terms are present in the equations, both the flux of information exchanged across the cell edge and the final state at the neighbour cells depend on them. This leads to a redefinition of the stability condition that must be a function not only of the eigenvalues of the problem but also of the source terms themselves. The modification of the stability region of different numerical schemes in a linear approach has previously been stated (Vázquez-Cendón ,1999). It has been proved that non-centered discretization of the source term enlarges the stability region. Furthermore, when not only source terms are important but also dry/wet interfaces appear in the simulation domain, the stability condition becomes even more restrictive. In order to be able to use the largest possible time step, compatible with stability and volume

11


conservation, a modification of the basic scheme, equivalent to the local redefinition of the bed slopes, is presented. The model summarized in Murillo et al.(2005b) is conservative, ensures bounded values of concentration in all situations and avoids negative water depths, for both existing and generated wetting/drying advance fronts when new dry areas appear.

The Toce river test case has been used here to show the performance of the model in presence of

transient flow over initially dry areas. The river is a watercourse of the occidental Alps, in Italy. An scaled physical model was built at the ENEL laboratory in Milan with a horizontal scale factor of 1:100 and approximate dimensions of 50x11m. The model reproduces a reach of the riverbed, the floodplain, a lateral reservoir designed for flood control purposes and buildings (Fig. 9). Moreover the model reproduces details of the geometry. The friction was modelled using n = 0.0162 s m-1/3. On the upstream side of the model, a tank was installed to supply the inflow of water in the form of a discharge hydrograph, characterized by a sharp peak.

Figure 9. Toce river geometry and location of probes.

The bed valley was initially dry and the subsequent flood wave produced the overtopping of the

reservoir. Several water depth probes were situated in different places along the river bed and in the valley, as Figure 9 displays. In this case, the flooding presents various transitions from subcritical to supercritical, and the wetting/drying techniques presented have to cope with high variable bed regions, in special an initially dry reservoir, which is overtopped. Figure 10 shows good agreement between measured and computed data in the probes located at different points along the domain.

12


13


Figure. 10. Comparison between numerical and measured data for different probes.

6 MOVABLE BED LEVEL SITUATIONS

The flow equations are described assuming a structure similar to those of the coupled shallow water-solute transport equations described previously, introducing the presence of sediment concentration and bed erosion source terms. Brufau et al. (Brufau et al., 2000) presented a revision of mathematical models governing the dynamics of solid-liquid mixture under these assumptions and a one-dimensional model for debris flow, solved by means of an explicit finite volume technique based on Roe’s scheme. The concentration of solute load was solved in cascade at each time step after the momentum balance of the

14


mixture was integrated. The source terms where discretized pointwise in space, leading to great difficulties. The physical details regarding erosion/deposition processes are out of the scope of this work..

The set of equations governing this flow model is:

( ) ( )USUGUFU

=∂

∂+

∂∂

+∂∂

yxt)(

(14)

where

( )Tyx hqqh φ,,,=U

Tyxx

x uhhqqgh

hq

q

+= φ,,

2,

22F

Tyyx

y vhghh

qhqq

q

+= φ,

2,,

22

G (15)

( ) Tssfyoyfxoxs vSSghSSghv φ),(),(, −−=S

where h is the water depth, g is the acceleration of the gravity, uhqx = , are the unit

discharge components, with the depth averaged components of the velocity vector u along the

vhq y =

( vu, )x and coordinates respectively, with y φ the depth averaged dimensionless volumetric sediment

concentration in the mixture, sφ is a parameter related to the bed concentration of sediments (Brufau et

al., 2004) and is the erosion/deposition rate. This rate is the responsible for the bed evolution in time that can be modelled via an equation of the form:

sv

,....),,,( 1πzhvtz

s q−=∂∂

(16)

where the erosion/deposition velocity vs depends on the bed characteristics as well as on the direction of the flow, q, and some other parameters according to a suitable model (Cunge et al.,1980, Egashira & Ashida 1987).

The techniques based on explicit finite volume Roe’s scheme derived for the coupled conservative scheme are applied to the full system of equations in the two dimensional case, proving that the upwind discretization of the source terms modelling bottom surface exchanges provides the key to solve the whole system satisfactory.

In the next test case, a movable bed is considered. The domain consists of two reservoirs connected laterally by a narrow gate. In the first reservoir there is a side where an inlet discharge of a known mixture is supplied. In the second reservoir there is an outlet side where a discharge condition characterized by a subcritical regime with Fr = 0.1 for t s is imposed (Figure 11). The rest of the boundaries are vertical solid walls. The domain was divided in 7875 cells by means of a Delaunay solver (Triangle).

0>

The inlet discharge function and inlet sediment concentration are given by

≥≤

=105.01005.0

)/( 3ttt

smQ 1.0)( =tφ (17)

15


Figure 11. Domain size.

The initial bed elevation function is given by

(18)

−−

==<

≥

374.13.0374.1)º20tan(

)0,,(yifyifx

tyxz

with z in meters. The initial flow conditions are d = h + z = 0.5 m and u = v = 0 as well as zero initial volumetric sediment concentration in the internal points. For the sake of clarity and simplicity the initial slopes depend only on the x coordinate, but it must be remarked that the numerical method can handle all initial configurations, as the erosion/deposition process is defined for both x and y coordinate directions. The evolution in time of the bed slopes are illustrated in next figures.

The sediment carried by the inlet water is characterised assuming 2000=sρ Kg/m3, ϕ = 62.72º

and . The interstitial fluid is water with density 1.0== ∗∗ φφd 1000=lρ m3/s. The dimensionless

volumetric sediment concentration in the mixture at the inlet region is assumed constant in time and equal to 1.0=φ .Under this hypothesis, the steady state bed equilibrium angle is º10=eθ . To provide a common reference level for this and the following test cases, the bottom at the inlet boundary is considered unerodible and at a constant level.

Using the pointwise scheme destructive and unrealistic oscillations in the bed elevation appear. On the other hand, when discretizing the source term in an upwind way, a satisfactory steady state given by the bed slope at the equilibrium angle is achieved. Figure 12a) displays the solution for the bed elevation using the pointwise treatment of the erosion/deposition source term and Figure 12b) the solution when using the upwind discretization at time 2 seconds. Figure 13 displays the results for the bed elevation using the upwind discretization at times t = 5s a), 10s (b), 15s (c), 20s (d), 30s (e) and 400s (f) when finally the steady state has been reached. In the cells near the gate corners, and also in other corners, the velocity is very small due to solid wall boundary conditions, so no erosion takes place and no bottom variation is produced. The original bed elevation in these special regions is preserved also in the

16


following test cases. The evolution of the dimensionless volume concentration of sediments in the mixture is represented in Figure 14 at times t = 5s a), 10s b), 15s c), 20s d), 30s e) and 400s f) showing how, as steady state is approached, the concentration becomes more uniform.

a)

b)

Figure 12. Bed elevation with pointwise discretization a) and with upwind discretization b) at time 2 seconds.

17


a) b)

c) d)

e) f)

Figure 13. 3D contour plot of the bottom elevation at times t = 5 a), 10 b), 15 c), 20 d), 30 e) and 400 f).

18


a) b)

c) d)

e) f)

Figure 14. Dimensionless volumetric concentration of the mixture at times t = 5 a), 10 b), 15 c), 20 d), 30 e) and 400

f).

19


The characteristics of the mixture discharge cited before are used again to carry out another test. The initial bed elevation function is in this test case

(71)

==<

≥

374.10374.1)º10tan(

)0,,(yif

yifxtyxz

with z in meters. The inlet region presents therefore an adverse slope. The initial and transient conditions described are still water and uniform water level, d = 0.8 m. When using the pointwise treatment of the erosion/deposition source term, after one second unreal oscillations appear in the flow that blow up the simulation. The upwind source term discretization copes well with that particular situation.

Figure 15 displays the results for the bed elevation using the upwind discretization at times t = 5 a), 10 b), 15 c), 20 d), 30 e) and 400 f) seconds, when the steady state has been reached. In the cells sharing the junction and the corner no bottom variation appears. Figure 16 displays the evolution of the dimensionless volume concentration of sediments in the mixture at times t = 5 a), 10 b), 15 c), 20 d), 30 e) and 400 f) showing how as steady state is approached the concentration becomes more uniform.

a) b)

c) d)

20


e) f)

Figure 15. 3D contour plot of the bottom elevation at times t = 5 a), 10 b), 15 c), 20 d), 30 e) and 400 f).

a) b)

21


c) d)

e) f)

Figure 16. Dimensionless volumetric concentration of the mixture at times t = 5 a), 10 b), 15 c), 20 d), 30 e) and 400

f).

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