(1),(2)
Rui M. L. Ferreira
Research Assistant, Dpto. de Engenharia Civil e Arquitectura
Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Tel: ++351 21 841 81 55, E-mail: ruif@civil.ist.utl.pt
João G. A. B. Leal
Research Assistant, Dpto. de Engenharia Civil
Universidade da Beira Interior, Rua Marquês D’Ávila e Bolama, 6200 Covilhã, Portugal
Tel: ++351 275 320 836, E-mail: jleal@ciunix.ubi.pt
António H. Cardoso
Associate Professor, Dpto. de
Engenharia Civil e Arquitectura
Instituto Superior Técnico,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Tel: ++351 21 841 81 54, E-mail: ahc@civil.ist.utl.pt
Abstract: The present work is concerned with the structure of the solution of the generalised dam-break problem, defined as the propagation of the dam-break flood wave over a cohesionless erodible bed. The conceptual model is based on a shallow-water approach with bed-load dominated sediment transport. Under this framework, the properties of each of the l(p)-characteristic fields are analysed regarding non-linearity, monotonicity and entropy conditions are also object of analysis. It was found that the solution must be composed of one expansion wave and two shock waves, connected by two constant states if, initially, the sediment bed is discontinuous. If the initial sediment bed is continuous, the solution is composed by two expansion waves. Numerical finite-difference solutions and experimental evidence were used to verify the theoretical assumptions. It was found that the numerical results, in the form of characteristic paths, for each field and each conceptual situation, confirm the proposed model. It was also found that the numerical results agree well with the experimental data.
Keywords: dam-break, sediment transport, riemann problem
The failure of a dam is a complex process in which water and sediment are set in motion to form a highly unsteady two-phase flow. A simple yet sound enough mathematical description of this flow can be accomplished by admitting the Saint-Venant hypothesis and that the channel bed is composed by cohesionless sediments whose collisions produce a negligible transfer of momentum. The governing equations form a system of quasi-linear, first order, partial differential equations, hyperbolic in nature. Further simplifying, the collapse of the dam can be idealised as a sudden process in which the instantaneous removal of a vertical gate sets in contact two constant states, initially upstream and downstream the gate. Under this framework, the dam-break problem is a Riemann problem whose left and right states are those depicted in Fig. 1 and whose mathematical description is:
(1),(2)
In the above equations, U is the 3×1 vector of dependant variables, F(U) is the 3×1 flux vector, function of U, and G(U) is the 3×1 source terms vector, such that:
;
;
As for the scalar variables, t and x are, respectively, the temporal and spatial co-ordinates and Y is the water elevation above some
datum, defined by
where h is the water depth and h is the bed elevation. Q is the total discharge (water and
sediment) per unit width and HS
represents an equivalent water elevation, defined by
where n represents the bed porosity and Cs the depth-averaged
suspended sediment concentration. Qs
is the total volumetric sediment discharge per unit width, Sf is the friction slope and g is the acceleration due to gravity.
Fig.
1 Graphic depiction of the Riemann problem for the generalised
dam-break problem. The subscripts L
and R stand, respectively,
for the left and right states.
If the bed is
immobile and hL = hR, the
solution for the homogeneous problem (obtained by setting G(U) = 0 in
(1)), with initial conditions (2), is well known. Ritter’s analytical solution
for YR = hR and
Stoker’s generalisation for YR
¹ hR (see Stoker (1958), pp. 313-314 and 333-341) have
been used as a starting point for, virtually, all of the dam break studies ever
since.
The present text is aimed at the clarification of the solution of the generalised dam-break problem. The inclusion of the sediment mass conservation equation in the conceptual model prevents the formulation of an analytical solution as intelligible as the Stoker solution. The analysis of the properties of the characteristic fields is, therefore, of crucial importance since it determines the strategic approach for the design of a solution. The following paragraphs are thus dedicated to the analysis of the characteristic fields, proposal of a solution and its numerical and experimental verification.
The initial condition comprises a discontinuity in both the water and in the bed levels, as shown in Figure 1. Even in non-linear systems, this discontinuity is bound to propagate downstream since there are no diffusive fluxes. The solution will, thus, comprise continuos and discontinuous reaches. In the continuos reach, the conservation laws represented by system (1) apply. In compact, non-conservative form, this system becomes:
(3)
with
and aY = [(1-n)-h(Cs)Y]/(1-n-Cs) » 1, aQ = (Q(Cs)Q)/(1-n-Cs) » 0, aS = 1/(1-n-Cs) » 1/(1-n). As stated before, system (3) (or, equivalently, system (1)) has a complete set of eigenvectors and, therefore, is hyperbolic. Because of this hyperbolic nature, three real and distinct eigenvalues, such that l(1) > l(2) > l(3), are obtained as solutions of the characteristic polynomial |B - lI| = 0. In non-dimensional form, the later reads
(4)
where C = l/U
, U is the depth averaged flow velocity,
and
. The solutions of (4), for Qs/Qs0
ranging from 1 to 50, where Qs0
is a reference sediment transport rate, are shown in Fig. 2 as a function of the
Froude number,
.
Fig.2
Characteristics of matrix B, as function of the Froude number
Figure 2 highlights the fact that the
characteristic fields are strongly influenced by the intensity of sediment
discharge. This is true for the first field (represented by the characteristic C(1)) but only for high Froude
numbers (Fr > 1.3) and very strong
transport rates. For smaller Froude numbers it is acceptable to say that the C(1) is non-influenced by the
presence of sediment and is approximately equal to
(cf. Sieben (1999)). The second and third characteristic fields are
strongly influenced by sediment transport rates, even for Froude numbers as low
as 0.75. Critical flow can be defined as the flow for which C(2) @ |C(3)|, a
situation which occurs near Fr = 1, as
seen in Fig. 2. Some authors (cf.
Sieben (1999)) prefer to substitute this concept by the concept of transcritical
flow, a flow for which the transfer of information regarding bed morphology is
evenly upstream and downstream directed (which prevents the formation of stable
bed forms).
Monotonicity is an important property of characteristics since it determines whether or not discontinuities can evolve from continuos initial conditions or, given as initial conditions, can be maintained throughout the computation domain. The monotonicity of the l(p) characteristics is shown in Fig. 3. It is observable that l(1) and l(2) are indeed monotonically increasing in U which means that, in the absence of dissipative fluxes, a flow with increasing velocity leads inevitably to the formation of a shock wave. l(1) is also monotonically increasing in h, which confirms the fact that its properties are quite similar the l+ properties. l(2) is monotonically decreasing in h, which means that a shock is bound to appear as the Froude number increases. Since l(2) is not a pure hydrodynamical or morphological characteristic, the discontinuity is likely to appear on both types of variables, i.e., Y, Q and HS. As for l(3), it is increasing in U to the left-hand side of a crest line (see Figure 3.c)) and decreasing otherwise. For this reason, shock waves are not expected to develop or are not expected to be stable for this characteristic field.
(a)
(b)
(c)
Fig.3 Monotonicity
of the l(p) characteristics with respect to U and h. U* = U/(gh0)1/2, l* = l/(gh0)1/2,
h* = h/h0, h0
= 0.5 m
It is well known that the Riemann solution for the Euler equations (the shock tube problem) comprises one contact discontinuity and two rarefaction waves, shocks or expansion waves. A contact discontinuity cannot occur if the r(p)-characteristic field is genuinely non-linear, i.e., if
(5)
In equation (5), ÑY,Q,Hs is the gradient operator in the space
of dependent variables and r(p) stands for the right eigenvector
for the pth field, computed
from
and defined
by
It is well known that inequality (5)
is verified for the fixed-bed dam-break problem. In order to verify if system
(3) admits a contact discontinuity, a plot of the left-hand side of inequality
(5) is shown in Fig. 4. It is easy to show that, for all characteristic fields,
inequality (5) holds except in a null-measured set of points. A larger scale
observation (see Fig. 4) shows that the values of
for to the 2nd and 3rd
characteristic fields are close to zero for a large set of Froude numbers.
Contact discontinuities are, thus, not permitted but very weak shocks or
expansion waves are expected.
In the continuos regions it is possible to find the value of the three dependent variables by solving a system of three characteristic equations. Double-valued points occur in discontinuities. Across the discontinuities the application of a control volume analysis yields the following relations:
(6)
where DU = UR
- UL,
DF’ = F’(UR) - F’(UL) with
The
parameter
has the dimensions of a velocity. In the case of a shock it is indeed its
celerity. In the case of an expansion wave, it has no physical meaning.
is the average water depth across
the discontinuity. Equations (6) are known as the Rankine–Hugoniot shock
relations. It should be noticed that F’
incorporates the flux of total hydrodynamical impulsion and the internal source
of momentum (h)x. It is thus implied that the discontinuity in h represents a source of momentum across
a shock wave, whose role cannot be neglected under the penalty of disregarding a
strong coupling between morphological and hydrodynamical processes and, thus,
miss-computing the shock celerity.
(a) (b) (c)
Fig.4 Verification
of the non-linearity of the characteristic fields: (a) l(1)-
field; (b) l(2)-
field; (c) l(3)-
field. G(p) =
The computation of the double-valued
dependant variables requires, in this case, three equations for the left-hand
side state, three others for the right-hand side state and one for the shock
celerity. One of these equations must be a characteristic equation for the
right-hand side. The three remaining equations are supplied by (6) provided that
>
>
(shocks) or
<
<
(expansion waves), for monotonically increasing l(p).
Figure 5 shows the right-states and
the associated shock celerities (or the
-proportionality constants, for expansion waves) as a solution of (6) and for a
given set of left-states. The computation of the shock and right-states was
accomplished numerically by the Newton-Raphson method. The inspection of this
figure shows that entropy conditions are satisfied for l(1)-
and l(2)-characteristic
fields. It shows, furthermore, that while l(1)-characteristics
admit only shock waves, l(2)-characteristics
admit both shock and expansion waves, depending on the intensity of sediment
transport.
(a) (b)
Fig.5 Left-and right-hand sides and
entropy conditions:
(a) strong sediment transport; (b) weak sediment transport.
Discontinuity; 
left
state; 
right state
Given the previous considerations, the
expected solution for the Riemann problem that represents the generalised
dam-break problem can be described as follows. The discontinuity in the water
elevation induces a discontinuity in h
and U, a shock wave in the l(1)-characteristic
field. This shock is sustained as result of the monotonicity of l(1).
Whenever hL > hR, the increase in momentum
given by (h)x induces a shock wave in the l(2)-characteristic
field since l(2)
is positive and monotonically increasing in U and h. This 2-shock is much slower then the
1-shock because l(2)
< l(1)
and since the second characteristic field depends on both morphological and
hydrodynamical variables, the discontinuity will be reflected by a sharp
sediment wave and by an hydraulic jump in the same section. The occurrence of an
hydraulic jump during the dam-break flood wave propagation over a mobile
granular bed was first reported by Capart and Young (1998), working with
cohesionless particles of almost neutral buoyancy and under a debris-flow
framework. The authors concluded that the scour hole near the wave centre (the
gate section) was responsible for the bending of a counter current
characteristic (
) into an upstream moving shock. It should be noted that the hydraulic jump
identified in this work is not of the same nature; it moves downstream and not
upstream and is induced by an initial bed discontinuity and not by a developing
scour hole.
If the bed is flat and hR/hL < 0.1384 or if (hR – hL)/hR is large, the flow is critical at some stage near the gate section because of minimum energy requirements. The connection between the initial left constant state and the middle constant state cannot be accomplished by another shock because the left state features subcritical flow and the lack of monotonicity in the l(3)-characteristic field wouldn’t allow for the development of such shock. Therefore, the solution for the l(3)-characteristic field must be an expansion wave. Figure 6(a) shows the configuration of the solution in the case hL > hR. Whenever hL = hR, the solution approaches the fixed bed case since the l(2)-characteristic field will not develop into a shock wave but on an expansion wave. The particular case hL = YL (vacuum right state) was studied by Fracarollo and Armanini (1999) under a debris flow framework. The configuration of the solution is shown in Fig. 6(b). It should be noted that the downstream moving shock disappears for the simple reason that there is no medium for it to propagate.
(a) (b)
Fig.6 Structure of the solution: (a) hL > hR; (b) hL = hR and hL = YL (vacuum right state)
In the limit, when the sediment discharge approaches zero, the 2- and 3-expansion wave fields will coalesce and the solution becomes the Ritter solution (Fracarollo and Armanini (1999)).
Numerical solutions and experimental data were obtained for the Riemann problem defined by (1) with initial conditions (2). Two conceptual situations were verified: (1) hL = 0.15 > hR = 0.05 and YR = 0.10 > hR = 0.05; and (2) hL = hR = 0.05 and YR = hR. Two different models were used to gather the numerical results, an uncoupled model with an explicit first order flux-splitting discretization based on the Beam and Warming scheme with Roe’s Riemann solvers and a TVD MacCormack (details on Ferreira and Leal (1998)). The experiments were performed in the Hydraulics Laboratory, Instituto Superior Técnico, Lisbon. Figure 7 shows the path of the characteristics, computed numerically, for each field and each situation. Figs. 7(a), (b) and (c) show the results for the l(1)- l(2)- and l(3)-characteristic fields, respectively, when hL > hR and YR > hR. The predicted 1- and 2-shocks (the latter much slower than the former) are observable in Figs. 7(a) and (b). Figure 7(c) shows the expansion wave in the l(3)-characteristic field.
(a)
(d)
(b)
(e)
(c)
(f)
Fig.7 Numerically computed characteristic paths: (a) situation 1, l(1)- field; (b) idem, l(1)- field; (c) ibidem, l(3)- field; (d) situation 2, l(1)- field; (e) idem, l(2)- field; (f) ibidem, l(3)- field
Figures 7(d), (e) and (f) show the results for the l(1)- l(2)- and l(3)-fields, respectively, for situation 2. The 1-shock disappears because there is no water downstream and the 2- and 3-expansion waves (Figs. 7(e) and (f)), intercalated by a constant state, are quite visible.
Figure 8 shows the comparison of numerical results and experimental data. A submerged hydraulic jump and a deposition pattern similar to a sharp sediment wave were indeed observed in Situation 1. Situation 2 shows a good agreement between experimental data though the scour is slightly underestimated.
Fig.8
Comparison of numerical and experimental data
The study of the properties of the l(p)-characteristic fields for the generalised dam-break problem under a shallow-water framework and bed-load dominated sediment transport was carried out in this work. As a result, it was found that the solution must be composed of one expansion wave and two shock waves, connected by two constant states, or by two expansion waves and a shock wave depending on the existence or not of an initial bed discontinuity. The l(1)-characteristic field features a downstream moving bore, common to the fixed bed problem. The second shock arises from the l(2)-characteristic field and is a combination of a downstream moving sharp sediment wave and an associated hydraulic jump. The third characteristic field shows an expansion wave. Is the bed is initially flat, the solution is solely composed of two expansion waves.
Numerical finite-difference solutions and experimental evidence were used to verify the theoretical assumptions. It was found that the numerical results, in the form of characteristic paths for each field and each conceptual situation, confirm the proposed model. It was also found that the numerical results agree well with the experimental data.
Acknowledgements
This paper was prepared under the framework of the project “Mathematical modelling of the morphological evolution of alluvial rivers”, financed by the Portuguese Foundation for Science and Technology.
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