In this paper, a three-dimensional two-phase flow solver, SedFoam-2.0, is
presented for sediment transport applications. The solver
is extended from twoPhaseEulerFoam available in the 2.1.0 release of the open-source
CFD (computational fluid dynamics) toolbox OpenFOAM. In this approach the sediment phase is modeled as a
continuum, and constitutive laws have to be prescribed for the sediment
stresses. In the proposed solver, two different intergranular stress models
are implemented: the kinetic theory of granular flows and the dense granular
flow rheology

Sediment transport is the main process that drives the morphological evolution of fluvial and coastal environments. Consequently, the ability to predict sediment transport is a major societal issue for the management of natural systems in order to limit and prevent the impacts related to extreme events exacerbated by climate change and human activities such as the construction of hard structures (dams, harbors, dikes, etc.), land reclamation, and dredging. Addressing these issues requires the development of comprehensive models that account for the variety of complex hydro-sedimentary processes such as particle interactions with hydrodynamic and flow turbulence or particle–particle interactions due to collisions or frictions. However, these complex phenomena are only poorly understood at present and they are incompletely integrated into engineering tools to predict the coastal and river morphodynamics. As a result, our prediction performance is limited. Improving these models is urgently needed for land settlement decision-makers for the management of water resources and environmental issues.

The processes at work in sediment transport are numerous. In classical
sediment transport definitions, particles can be transported as
suspended load, i.e., without contact with the sediment bed, or as bed load,
i.e., with permanent or intermittent contact with the sediment bed by rolling,
sliding, or saltation

From the modeling perspective, the classical modeling approach consists of
dividing the physical domain into two sublayers. The upper layer corresponds
to the water column in which depth-integrated or depth-resolving Reynolds-averaged
Navier–Stokes equations are solved and the sediment
concentration is assumed to be dilute, in which the sediment particles are
treated as passive scalar with a settling velocity difference with the fluid
phase. The lower near-bottom bed-load layer is solved by using a
bed shear-stress-based empirical formula for the sediment flux

During the past two decades, an increasing amount of research efforts have
been devoted to develop two-phase flow models for sediment transport (see a
brief summary in Table

Summary of Eulerian two-phase models for sediment transport applications.

The key closures in the two-phase flow sediment transport models are flow
turbulence and granular stress closures. In terms of the turbulence
closure, the first one that has been tested was a mixing length model by

The focus of the present work is to demonstrate the capabilities included in SedFoam-2.0 to model sediment transport. In particular, the comparison of different combinations of granular stress and turbulence models in the same numerical framework is presented here for the first time.

The paper is organized as follows, in Sect.

The mathematical formulation of the Eulerian two-phase flow model is obtained
by averaging local and instantaneous mass and momentum conservation equations
over fluid and dispersed particles. Different averaging operators can be
used, ensemble averaging

The mass conservation equations for the particle phase and fluid phase are
written as the following:

The drag parameter

Because the present model equations are obtained by averaging over
turbulence, the fluid stresses consist of a large-scale component

The Reynolds stress tensor

In SedFoam-2.0, several different viscosity or turbulence closures are
implemented, and these models can be selected according to specific flow
conditions ranging from laminar to turbulent flows, and in particular, the
mixture viscosity can be selected in combination with the granular rheology model
for the granular stresses (see Sect.

The mixture viscosity model mostly depends on the particle-phase volume
concentration. Four different models are available in SedFoam-2.0. In the
pure fluid model, the mixture viscosity is equal to the fluid one:

The Einstein model

The phenomenological model proposed by

The model proposed by

The choice of mixture viscosity model is made in the file

List of the mixture viscosity model that can be selected through the
keyword

A special treatment of the term

As discussed above, the turbulence-averaged formulation requires a closure
for the eddy viscosity. Three turbulence models are available in SedFoam-2.0:
a mixing length model (only valid for 1-D configuration), the

For laminar-flow applications, the turbulence model is turned off by setting

In the mixing length approach, the eddy viscosity is modeled using a simple algebraic equation:

Finally, the balance equation for the rate of turbulent kinetic energy dissipation

As discussed in

Default coefficient values for the

It is well-known that the original

The turbulent eddy viscosity

The fluid TKE equation reads as

Default coefficient values for

The turbulence model can be selected using the

List of the turbulence models that can be selected through the
keyword

Note that Tables

In sediment transport applications, the particle stresses are important
mechanisms to support a particle's immersed weight in concentrated regions of
sediment transport (

In the modern sediment transport modeling framework, two major threads of
modeling approach for shear-induced/collisional particle normal stress and
shear stress are kinetic theory of granular flows and dense granular flow
rheology. They are implemented in this version of SedFoam-2.0 (see
Sect.

In the kinetic theory model, intergranular interactions are assumed to be
dominated by binary collisions for low to moderate sediment concentration,
and the collisional shear stresses are quantified by particle velocity
fluctuations represented by the granular temperature

In the 1980s, dense-phase kinetic theory of gases

Following

Through the kinetic theory, the particle shear viscosity is calculated as a
function of granular temperature and radial distribution function:

Similarly, the bulk viscosity is calculated as

The closure of granular temperature flux is assumed to be analogous to
Fourier's law of conduction:

The dissipation rate due to inelastic collisions is calculated based on that
proposed by

Due to the presence of the carrier fluid phase, carrier-flow turbulence can also
induce particle fluctuations. Following

In order to extend the model capability to resolve the quasi-static or immobile
sediment bed, the shear stress due to frictional contact is modeled as

The total shear stress

All the simulations presented in this paper have been obtained using the
closures summarized in Table

The other alternative for modeling the particle-phase stress proposed in
SedFoam-2.0 consists of the dense granular-flow rheology or the so-called

List of the kinetic theory closures that can be selected through the different keywords listed in the table together with model equation references.

In the viscous regime, the friction coefficient

Concerning the shear-induced contribution to the particle pressure,

In the grain inertia regime, the friction coefficient depends on the inertial
number

Concerning the shear-induced contribution to the particle pressure, it can be
obtained from the dilatancy law

According to

Inverting Eq. (

Similar to those in the viscous regime, the total particle pressure

The rheology has been originally stated for steady uniform granular flows and
this shear-induced pressure term induces a very strong coupling between the
wall normal and the streamwise components of the particle-phase momentum
balance equation. The granular-flow rheology has been used to simulate with
success the transient flows such as the granular column collapse
configuration

The different closure laws implemented in SedFoam for the dense granular-flow
rheology are summarized in Table

List of the dense granular-flow rheology models that can be selected
through the keyword

The numerical implementation of the present Eulerian two-phase flow sediment
transport model is based on an open-source finite volume CFD library called
OpenFOAM. Taking advantage of the numerical discretization schemes and
framework of finite volume solvers in OpenFOAM, the two-phase flow governing
equations are implemented by modifying the solver twoPhaseEulerFoam

To illustrate the numerical discretization, the fluid-phase momentum equation
is taken as an example. Rearranging the fluid-phase momentum equation
(Eq.

The last term in the above equation, the gradient of fluid-phase shear
stress, can be written according to Eq. (

In the above equation, the first two terms on the RHS are treated implicitly
while the last two terms are treated explicitly. By substituting the expanded
shear stress formulation in the momentum equation, the following equation is
obtained:

It is more convenient to rewrite the above equation into a matrix form:

The matrix

The same process can be carried out for the solid-phase momentum
Eq. (

Following

The advantage of separating the RHS of the momentum equations as
the sum of two terms,

The velocity–pressure coupling and the consequent oscillations in the
pressure fields are resolved by using the Rhie and Chow method

First, the intermediate velocities (

Taking the divergence of the volume-averaged mixture velocity given by the
velocity correction Eq. (

The volume-averaged flux is also corrected according to

In order to ensure the mass conservation an iterative procedure of

The numerical solution procedure for the proposed two-phase flow model is
outlined as follows:

solve for sediment concentration

update the volume concentration of fluid:

update the drag parameter

solve for the fluid turbulence closure, update

solve for the particle-phase stress (kinetic theory model or the dense granular rheology);

PISO-loop, solving velocity–pressure coupling for

construct the coefficient matrices

update the other explicit source terms

calculate

construct and solve the pressure Eq. (

correct fluid and particle velocities after solving pressure
and update fluxes Eqs. (

go to (a–e) if the number of loops is smaller than

advance to the next time step.

The time step,

In this section, four benchmarking cases are presented to validate and verify the numerical implementation of the model. The first one concerns the pure sedimentation of a suspension of non-cohesive spherical particles for which experimental data are available. The second one concerns the laminar bed-load problem for which an analytical solution exists. The third case is unidirectional turbulent sheet flow and the fourth one is about the scour at an apron. For each case, an analytical solution, experimental data, or empirical formula is used to validate the model. All the input files for the four cases described in the paper are available in the tutorial folder of the distribution and Python scripts based on an open-source postprocessing toolbox are available as well for facilitating the training on using SedFoam-2.0.

The first test case corresponds to a pure sedimentation of non-cohesive
particles; this test case allows us to validate the implementation of the
pressure–velocity coupling algorithm when the flow is induced by the sediment
phase. The other component that is tested here is the permanent contact
pressure model (Eq.

The mesh is composed of 200 cells in the vertical direction with a uniform
distribution over a height

Numerical schemes for each term in the momentum and mass
conservation equations used in the validation test Sect.

Figure

Comparison of two-phase flow model results with experiments of

The second test case is inspired by

The numerical domain setup is based on the experimental configuration of

Figure

In Fig.

Comparison of two-phase numerical results with experiments of

In this subsection the model results are compared with experimental results
from

The first case is based on the experimental configuration of

In the numerical configuration, the flow is driven by a pressure gradient
(

Physical parameters for the numerical simulations of

The results are presented in Fig.

Comparison of two-phase numerical results with experiments of

In the

After this calibration of the turbulence models, the results obtained with
the different combinations of granular stress and turbulence models are
discussed. In terms of velocity profiles, both the

In order to further assess the model, the same combinations of granular
stress and turbulence models are applied to

The results in term of volume-averaged velocity profiles, defined as

Sketch of the scour downstream of an apron.

Sediment concentration contour at different times during the scour
process using

Different hypotheses can be proposed to explain the discrepancies presented
above. According to

In order to demonstrate the multidimensional capability of SedFoam-2.0, the
fourth test case corresponding to the development of the scour downstream of an
apron was examined. Following the numerical studies of

The sediment bed is made of sand, density

Summary of the boundary conditions implemented in the 2-D scour
downstream of an apron configuration. The following abbreviations are used:
zG is

Numerical results for the temporal evolution of the normalized
maximum scour depth and upstream bed angle. Different lines represent the
best-fit curves for each run for which the parameters are given in
Table

Summary of the numerical results obtained for the scour at an apron using the different combinations of turbulence and granular stress models and comparison with existing two-phase numerical results on this configuration.

The bottom boundary, the lower part of the inlet (forming the step) and of
the outlet are set as wall boundaries. The upper part of the inlet is an
inlet boundary where the velocity profile is imposed according to the rough
wall log law (Eq.

Four combinations of fluid turbulence models (

In Fig.

In conclusion, this test case shows good capability of the proposed two-phase flow model to deal with multidimensional flow configurations. Further work is needed to improve the model validation and the model sensitivity to flow turbulence and rheological parameters. This requires more detailed experimental data that, to the best of our knowledge, are not available at present.

In this paper, a comprehensive two-phase flow model for sediment transport
applications has been presented and the details concerning its implementation
in OpenFOAM have been given. The proposed model provides different options for
the modeling of flow turbulence (mixing length,

As a general conclusion, the aim of this contribution is to provide a comprehensive two-phase flow sediment transport modeling framework to the scientific community. Intense efforts have been made to ensure its reliability and numerical robustness. This numerical tool is suitable to address various physical problems for which the classical sediment transport approach is not working very well or requires more model assumptions. However, the readers are reminded that two-phase flow simulations are still relatively time-consuming and require finer spatial resolution and smaller time steps than classical sediment transport models.

The code is distributed under a GNU General Public License
v2.0 (GNU GPL v2.0) and is available at

Table of notations.

Continued.

List of the main model input parameters together with their default values when relevant.

The authors declare that they have no conflict of interest.

Julien Chauchat, Tim Nagel, and Cyrille Bonamy are supported by the Region Rhones-Alpes (COOPERA project and Explora Pro grant), the French national programme EC2CO-LEFE MODSED. Zhen Cheng and Tian-Jian Hsu are supported by National Science Foundation (OCE-1537231; OCE-1635151) and Office of Naval Research (N00014-16-1-2853) of USA.

Numerical simulations were carried out on MILLS/FARBER at the University of
Delaware, on HPC resources from GENCI-CINES (Grant 2015-x2016017567) and on
the Froggy platform of the CIMENT infrastructure
(

The authors would also like to acknowledge the support from the program on “Fluid-Mediated Particle Transport in Geophysical Flows” at the Kavli Institute for Theoretical Physics, Santa Barbara, USA. The laboratory LEGI is part of the LabEx Tec 21 (Investissements d'Avenir – grant agreement nANR-11-LABX-0030) and Labex OSUG@2020 (ANR10 LABX56).

We are grateful to the developers involved in OpenFOAM who are the foundation of the model presented in this paper. Edited by: James R. Maddison Reviewed by: two anonymous referees