New Sediment Transport Models for Dam Break Problems

Acta Mech
https://doi.org/10.1007/s00707-025-04316-7
ORIGINAL PAPER
Arno Roland Ndengna Ngatcha ·Daniel Bandji ·Imad Kissami ·
Abdou Njifenjou
A new class of sediment transport models for dam break
problems
Received: 1 October 2024 / Revised: 10 March 2025 / Accepted: 12 March 2025
© The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2025
Abstract During sudden dam breaks in shallow water environments, the water fluctuations with greater lengths
of correlations are more intense. As consequence, the water flow becomes rapid and mobilizes sediments, while
simultaneously modifying the bottom interface. The existing literature does not provide an averaged sediment
transport model (STM) that accounts for both mean and fluctuating motions over an abrupt mobile sediment bot-
tom. Averaged shallow water based models are frequently employed to describe sediment transport in shallow
water environments. However, they are not still applicable when the flow becomes turbulent as observed during
sudden dam breaks. The aforementioned classical models consider solely the mean motion of water and such
consideration provides an inaccurate description of the wavefront profiles. To account the fluctuating motion of
water in a STM, we will assume that |u−u|k≤τkh,k>0, where his the water depth such that |h|=O(ε).
Additionally, we assume a weakly concentrated approximation to derive the model. The new derived model in
this study builds upon previous work by several authors in 1980–2007, 2012, and 2018, while also improving
upon more recent models developed in 2022 and 2023. The hyperbolic STM is subject to various complexities
arising from turbulence and several nonlinear coupled terms. Furthermore, solving this problem still poses a
computational challenge in terms of accuracy, convergence, robustness, and efficiency. To address this chal-
lenge, we propose a new first-order path-conservative method based on HLL Riemann solver named HLL∗∗.
The second-order scheme is achieved by using a modified Averaging Essentially Non-Oscillatory (AENO)
nonlinear reconstruction. This method is noteworthy because it generalizes several HLL-based schemes devel-
oped in recent years. Several dam break tests have been performed to assess the developed modeling. It was
observed that all the admissible weak solutions of the model are numerically well represented. Furthermore,
it was determined that the model improves the description of the movements of the two successive slow/fast
interfaces driven by a shock. The proposed numerical modeling improves also the description of the water
A. R. N Ngatcha (B )·D. Bandji ·A. Njifenjou
National Advanced School of Maritime and Oceanic Sciences Techniques (NASMOST), University of Ebolowa, P.O. BOX 118,
Kribi, Cameroon
E-mail: [email protected]
D. Bandji
E-mail: [email protected]
I. Kissami
College of Computing, University Mohammed VI Polytechnic University, Lot 660, 43150 Benguerir, Morocco
E-mail: [email protected]
A. Njifenjou
Department of Mathematical Engineering, National Advanced School of Engineering, University of Yaounde I, P.O. BOX 8390,
Yaounde, Cameroon
E-mail: [email protected]

A. R. N. Ngatcha et al.
depth profile and the propagation of wavefronts during a dam break. The scheme captures all the admissible
erosional shocks that improve the description the behavior of the bottom interface.
Mathematics Subject Classification 65J08 ·65J18 ·76F10 ·76M12
1 Introduction
Geophysical free surface currents are frequently linked to sediment dynamics. Sediment transport processes
in geophysical flows are divided into three main scales: particle scale, sediment mass scale, and sediment
motion scale. Different processes occurring on each scale are often computed by different numerical methods
[1–3]. At the scale of sediment motion, sediment transport may be influenced by fluctuating motion, which
is caused by turbulence in shallow water contexts. This factor has not been extensively discussed in modern
literature dealing sediment transport. There are various approaches to model sediment transport: the long-
wave approximation approach (also known as the shallow water framework of [4]) and the two-fluid approach
introduced by [5]. The two-fluid equations are based on a continuum scale approach where the fluid phase
(water) and the dispersed sediment particle phase are each considered as a continuum. Each phase has its own
continuity and momentum equations. Sediment transport modeling in a two-fluid approach, based on unsteady
Stokes equations and morphodynamics, has been developed in several works [2,6]. Two-fluid approach is also
used to describe the sediment transport in steady or unsteady turbulent flows [7–13]. By considering two-phase
approach, the motion of water can be described using systems of equations of Navier–Stokes for water phase
and solid mechanic for sediment phase as in [2,14]. There are several drawbacks to these approaches. The
assumption of a continuum for the dispersed sediment phase is questionable. Secondly, solid pressure modeling
relies on empirical or semi-empirical methods that are still poorly understood. Formulations are not always
satisfactory, and the two-phase equations may require significant computing resources that may not always be
readily available. However, models like these are often unsuitable for dam break problems. Sediment transport
models based on classical shallow water equations are often preferred over two-phase-based models due to
their simple formulations and ease of implementation. These models are based on a concept introduced by
[4], also known as first-order long wave approximation models, provide a simpler approximation than the
full Euler or Navier–Stokes equations with a free surface or two-phase models. The first-order approximation
of long wave theory describes water motions with a free boundary. Sediment transport models in shallow
water equations in such situations have been studied numerically by [15–22,24] where their works although
introducing new numerical solutions by well-balanced schemes neglect the turbulence. Such modelings have
also been studied experimentally in famous works such us [23,25–27] and analytically in [28]. Nonetheless, as
evidenced by the aforementioned works, the bedload transport models frequently yield erroneous results when
the flow transitions to a turbulent state. Eddy-resolving techniques can also be used to enhance the prediction
of bedload transport models [29]. The technique allows for a space- and time-dependent description of the
bottom shear stress distribution. The accuracy of commonly used bedload transport models when applied in
combination with high-resolution Navier–Stokes solvers is proved by [29].
During dam breaks, the water flow becomes rapid (due to velocity variation) and arises coherent turbulent
structures, mobilizing fine and coarse sediments and rapidly altering the bottom’s geometry [30]. The variation
of velocity in the time generates shearing effect and this leads to intense exchange mechanical energy (see
Caltagirone et al., [31]). This effect is neglected in the classical sediment transport modeling. Even with
unsteadyflowsusedin[32] or rapid unsteady flow [20], in shallow water context, it cannot accurately describe
sediment transport during dam break because it does not consider the effect of these coherent structures that
modify the wave structure of the flow. In addition, any unsteady flow is not necessarily turbulent. As a result,
sediment transport models based on unsteady SWE are not applicable when the flow becomes turbulent or when
the water flow arises accelerated internal waves. Classical STM does not capture these internal accelerated
waves and this is a relevant problem for dam break applications. The water flow acceleration increases the
bottom scission that affects the Shields parameter, which leads to an increase in the bedload rate. Additionally, it
can impact the rheological behavior of the flow of sediments and water. On larger scales, constitutive equations
for the macroscopic rheological behavior of sediment beds that may decouple the relevant scales for fluid and
sediment motion are exposed in [1]. The widely used shallow water Exner models are limited in their ability to
describe the additional wave arising that occur during a dam break [33,34]. This is because the models do not
account for the propagation of this additional wave into the wave structure. According to [35](seealso[30]),
during dam breaks, flow structures admit waves of small disturbances that can propagate with a finite velocity
depending on component of mean velocity and distortion velocity. Distortion waves occur during the dam

A new class of sediment transport models for dam break problems
Fig. 1 Averaged sediment transport models derived from nonhomogeneous free boundaries Navier–Stokes equations. The classical
SWE have been introduced by [4]. Classical sediment transport models are based on SWE and are widely used in the literature.
The DSWE have developed by [53], multi-velocity DSWE have been recently exposed in [38], and sediment transport in multi-
velocity DSWE is presented in this work. The multilayer models based on DSWE or/and multi-velocity DSWE remain open
problems
break and this is well observed numerically (see Sect. 7). Even if we use a kinetic energy equation as in [36,37]
or the turbulence equations as in [18], it do not make possible to capture fast waves near the sediment bed.
Additionally, the wave structure used here is the same as the one obtained in shallow water models. Turbulence
modifies fluid flow behavior when the horizontal profile velocity is distorted in the vertical direction. The
water distortion effects arise from the coherent structures in the turbulent zones of the water near the bottom
[39]. The developed model contains additional contact and shock waves associated with distortion velocity. In
this work, we show how the characteristic velocity of the distortion plays an important role in the controlling
sediment deposition and re-suspension (see numerical solutions in Sect. 8). Furthermore, it generates a distinct
wave structure that differs from the one observed in shallow water models.
Furthermore, the equation for evolution of distortion velocity is derived to simplify the averaged model.
It significantly reduces computational efforts and facilitates the identification of eigenstructures and Riemann
invariants. The method also helps to determine the characteristic distortion velocity. In order to appear in
an averaged STM, the modification of fluid motion requires the inclusion of two velocities in the modeling:
mean flow velocity and distortion flow velocity. These velocities drive, respectively, the mean motion and
fluctuation motion of the flow. In classical modeling based on shallow water equations, only the mean motion
is considered [16,28,34,37,40,41]. These models are derived by vertical averaging the Navier–Stokes equations
for nonhomogeneous flows following [4] and are used to study turbulent flows during the dam breaks. However,
this approach can result in a non-physical description of sediment bed morphodynamics during a dam break,
where the internal waves of the fluid become accelerated. The new class of sediment transport models developed
here enhance the understanding of sediment transport during dam breaks, avalanches, tsunami and flooding (see
Fig. 1). A comprehensive understanding of sediment transport processes can assist in the efficient management
of water in coastal cities and hydroelectric dams.
The sediment transport models in long-wave approximation that exist in the literature are revisited and
improved following a new methodology. This methodology inspired of [38] involves designing a two-velocity
averaged momentum equation and reformulating the kinetic energy equation derived from the two-velocity
momentum equation with appropriate boundary conditions. The method differs from the one based on

A. R. N. Ngatcha et al.
Fig. 2 Physical modeling: Free surface (h(x,t) + Zb(x,t) + b(x)), water depth (h(x,t)), mobile bed (Zb(x,t)) and non-erodible
(b(x)), where xis the spatial position and tis the time. Both layer of clear water and suspended particles layer constituted the
water depth
Tsekunov’s approximation and used in [30,39]. The aim of this work is to propose a new mathematical
model for sediment transport resulting from interactions between turbulent mixing flows and morphodynam-
ics. The article discusses the hydro-morphodynamics of the riverbed using a single-phase approach. The layer
of clear water, suspended particles, and bedload are treated as a single layer as illustrated in Fig. 2. The gov-
erning equations for each layer are written and then coupled into a system of equations representing coupled
processes, as in [16–18,42,43].
The layer of suspended particles consists of particles that are much smaller than the distance between
them, which asymptotically coincides with the mean distance of the surrounding fluid. The layer of bedload
involves particle transport with a phase lag effect. The sediments move at different velocities than the mean
fluid velocity, and they do so slowly. The model considers the interactions between the suspension layer and
the bedload layer, including erosion, deposition, and transport.
The multi-fluid equations were integrated vertically following Saint-Venant’s assumption [4], accounting
for turbulence (or distortion motion). This is a new contribution to the literature and is more easy to implement
than the LES modeling as the one developed by [29]. In the derivation, special attention was given to the velocity
induced by the turbulence (or water fluctuations). The mathematical model resulting from distortion velocity
consists of a system of nonlinear hyperbolic partial differential equations. These equations are approximated
using a stable and robust finite volume scheme. Various methods can be employed to calculate the flux at
the interfaces between cells. We cite, for example, the well-known HLL (Harten–Lax–Leer) solver of [45]
and its variants (widely used in some sediment transport models based on shallow water equations) [34], Roe
solver of [47] and its variants, Lax–Wendroff flux, Lax flux, CU (central-upwind), finite volume characteristics
[46], path-conservative method. The last method is widely used in the literature for non-conservative systems
[17,19,48,49]. Assuming a linear path between two states, the presence of a non-conservative term (which
is the product of the unknowns and their first-order derivative) indicates a non-unique derivation of jump
conditions. Jump conditions are only valid for weak discontinuities [30]. Formulating jump conditions for
weak discontinuities alone can be limiting and may prevent convergence to an exact solution, if one exists.
This problem fails some representation test cases in recent works of [30,48,49]. We will design a shock-
capturing path-conservative scheme including one intermediate wave between two states. It differ from other
HLL-based methods that include several intermediate waves. However, it was observed in a recent work (see
[30]) that path-conservative schemes including a great number of intermediate waves number does not have
any influence on the convergence of the scheme and on the stability of a solution. This is due to the nature of the
path itself and the validity range of Rankine–Hugoniot relations. The RH relations are applicable only for weak
discontinuities. The discrete path does not always converge really to the analytical path. It is to remark also
that the solutions produced by multi-wave HLL-based Riemann solvers are better than several Godunov-type
methods well known in the literature see [48]and[49]. The intermediate states at the interfaces of the control
volumes can also be constructed using the method of characteristics [50].
The HLL Riemann-based solvers with intermediate waves in a path-conservative framework can be suitable
to solve the proposed model. Since the developed model contains an additional contact and shock waves

A new class of sediment transport models for dam break problems
associated with distortion velocity, it is therefore important to develop a well-balanced scheme able to capture
all the internal waves (shocks, rarefactions, contacts) of the proposed system. Here, we introduce for the first
time a 3-wave approximation HLL Riemann solver in path-conservative framework combined to an AENO
reconstruction without ad hoc assumptions (HLL∗∗-AENO method for short). The proposed shock-capturing
HLL-based scheme is well balanced and positive and captures the steady states near the mobile bottom. Some
important results of this scheme are proposed and discussed. The proposed HLL∗∗-AENO method improves
many HLL-type schemes available in the literature and can be applied to several shallow water-based systems
or other hyperbolic systems [34]. The proposed method extends the HLL-based scheme developed by [51]
for a classical two-velocity shallow water model (see also [52]) since their schemes do not include the non-
conservative terms (due to potential energy loss). The computational challenges offered by the Multi-velocity
Sediment Transport Model (Mv-STM) are very interesting and will help to address several hydraulic problems.
The primary innovations are as follows: (i) The integration of a novel long-wave sediment transport model
(formally derived) that incorporates turbulence in terms of velocity, aligning with the theory presented in [38].
This theory diverges from the one introduced by [53] and utilized in [30]. (ii) The initiation of the inaugural
mathematical analysis of this model. This analysis yielded the establishment of a global weak solution and its
demonstration of significance in preserving the stability of solutions. The third contribution is the development
of a new stable, robust, positive and high-order path-conservative method based on multi-wave approximation
Riemann solver to address the newly developed model. The fourth contribution is the proposal of a numerical
study and numerical validation of the model, which has not yet been published in the existing literature. A
particular focus of our study is the convergence of the computed solution of the model.
The rest of the paper is organized as follows. In Sect. 2, we give some assumptions and basis equations
used in the modeling. Sect. 3proposes a formal derivation of the multi-velocity sediment transport model in
distorted shallow water mixing flows equations. Some properties and new results associated with the model
are given in Sect. 4. We developed a new approximate Riemann solver is designed in Sect. 5. In Sect. 6,the
positive scheme is extended to the second order by using a special nonlinear reconstruction. Several numerical
tests are performed to show the performance of our model and our method in Sect. 7.
2 Assumption, equations of basis and boundary conditions
2.1 Definitions and notations
Let I=[Zb,η]be the vertical integration. We introduce here the averaged part and the fluctuating part of a
function ψ(x,z,t)by:
ψ(x,t)=1
hI
ψ(x,z,t)dz and ψ=ψ−ψ, with 1
hI
ψdz =0(1)
where x,zare the spatial coordinates (horizontal and vertical) of the fluid domain. Here, his the water depth,
Zbis the bed level (or erodible bed) and η=h+Zbis the water free surface.
We introduce also the Leibniz relations:
∂hψ
∂a=∂ ∂aI
ψdz =I
∂ψ
∂adz −ψ(η)∂η ∂a+ψ(Zb)∂Zb ∂a,a=x,t(2)
Above Leibniz’s formula is applied to invert the differential operators and integration.
2.2 General assumptions
The following assumptions are considered:
(i) A small aspect ratio ε=H
L1, where Hand Lare two characteristic scale lengths along the z-axis
and x-axis of the channel introduced in depth-averaged equation. (ii) The fluid is incompressible, no heat
transfer. (The horizontal gradient temperature is zero.) (iii) We consider a dilute suspension zone according
to the Boussinesq assumption [30,39]. Such assumption will permit to consider that the sediment and the
fluid velocity are the same. In addition, this assumption means that the mean distance between neighboring
sediments is not small (i.e., the suspension flow is not dense or hyperconcentrated). (iv) The sediment diameters