The fluid motion is governed by the incompressible, filtered Navier–Stokes equations: 1∂u∂t+∇.(u⊗u)=-1ρf∇p+g+∇.(2νS+τf),∇.u=0, where the operator ⊗ is the dyadic product, u the fluid velocity, p the fluid pressure, g the gravitational acceleration, ρf the fluid density, ν the fluid kinematic viscosity, and S=12[∇u+(∇u)T] is the strain rate tensor. τf is a tensor which definition depends on the type of filter used. It can either be the opposite of the specific Reynolds stress tensor to run Reynolds Averaged Simulations (RAS), a subgrid scale stress tensor when performing Large Eddies Simulations (LES), or the null tensor for laminar simulations or Direct Numerical Simulations (DNS). The model takes full advantage of the wide range of options provided by OpenFOAM, and the type of filtering applied to Eq. () can be freely selected by the user. In this study, however, the numerical simulations are limited to either laminar cases (with no filtering of the Navier–Stokes equations) or unsteady RAS simulations.

At this stage of model development, no feedback of the suspended load on the hydrodynamics is considered, i.e. the fluid density ρf is assumed to be constant, an assumption appropriate for dilute suspensions where density effects and particle drag are negligible.

As stated previously, in the case of RAS filtering, the tensor τf in Eq. () is equal to the opposite of the specific Reynolds stress tensor τf=-u′⊗u′, with the Reynolds operator and u′ the fluctuating velocity field. A total of 6 additional unknowns (the velocity fluctuation correlations) are introduced in the system of equations by the use of the Reynolds stress tensor. The system as such is undetermined and the classical Boussinesq assumption is used as a closure. It expresses the Reynolds stress tensor as a function of the eddy viscosity νt and k=12u′.u′, the turbulent kinetic energy (TKE): 2τf=2νtS-23kI3, where I3 is the identity matrix. Then, a turbulence model is used to compute νt. OpenFOAM offers multiple turbulence models to users, many of which are two-equation models based on k and either ϵ or ω, the rate of dissipation of TKE, and the specific rate of dissipation of TKE, respectively. A transport equation is then solved for each variable.

Of the various turbulence models available for the RAS approach in OpenFOAM (k-ϵ, k-ω, RNGk-ϵ …), only the well-known k-ω Shear Stress Transport (SST) model introduced by is employed in this study, although the other turbulence models available in OpenFOAM are also compatible with the solver. The choice of this model was motivated by its capability to simulate both free shear flows and boundary layers, as well as its accuracy in capturing flow separation caused by adverse pressure gradients. The k-ω SST consists of a combination of two other classical turbulence models, the k-ϵ and the k-ω models. The aim is to take the best out of those two models. Indeed, the k-ϵ model is known to perform well for free shear flows but exhibits poor accuracy in the presence of adverse pressure gradients, rendering it unsuitable for flows involving boundary layer separation. Conversely, the k-ω model is better suited for capturing flows with adverse pressure gradients and boundary layers, but it is less efficient than the k-ϵ for simulating free shear flows in regions outside the range of influence of the solid boundaries (e.g., rigid walls, sediment bed). The k-ω SST model transitions between the two models using blending functions that take the distance to the nearest wall as input. In the version implemented in OpenFOAM, the eddy viscosity νt is expressed as follows: 3νt=a1kmax⁡(a1ω,b1F23‖S‖), where F23=F2F3 is the product of two blending functions (F2 and F3) and ‖S‖=2S:S is a scalar measure of the strain rate tensor, with : the double inner product defined as S:S=tr(SST), where tr is the trace operator. The temporal evolutions of k and ω are described by two transport equations: 4∂k∂t+u.∇k=P̃-β*kω+∇.((ν+αkνt)∇k),5∂ω∂t+u.∇ω=γνtP-βω2+∇((ν+αωνt)∇ω)+2(1-F1)αω21ω∇k.∇ω where F1 is another blending function. The production term P is defined as P=νt∇u:[∇u+(∇u)T]. In the equation for k (Eq. ), a limiter is applied on the production rate: 6P̃=min⁡P,c1β*kω. The different blending functions and constants of the model are detailed in Appendix .

In sedExnerFoam, the suspended load is described by the suspended sediment volume fraction cs=Vs/(Vs+Vf) where Vs and Vf stand for the volume of sediment and the volume of fluid, respectively. The evolution of cs in space and time is governed by an advection-diffusion equation: 7∂cs∂t+∇.[(u+ws)cs]=∇.(ϵs∇cs), where ws is the sediment settling velocity and ϵs is the turbulent diffusivity for the suspended sediment. It is expressed as the ratio of the turbulent eddy viscosity and the Schmidt number σc as ϵs=νt/σc. The possibility of using an additional diffusivity ϵw in near-bed areas is discussed in Sect. . The suspended sediment concentration is supposed to behave as a passive scalar being transported with the flow and settling due to the effect of gravity. The settling velocity is computed as follows: 8ws=ws0Fh(cs)eg, where ws0 is the terminal sediment settling velocity of a single particle in a quiescent fluid, eg=g/|g| is a unit vector oriented with gravity, and Fh is a hindrance function that takes values between 0 and 1 and is a decreasing function of cs. It represents the effect of particles hindering each other as they fall, leading to a drop in their settling velocity when cs increases . The terminal settling velocity is estimated using empirical formulations that either explicitly predict the settling velocity or implicitly determine it by computing the drag coefficient CD and solving the force balance between drag, buoyancy, and weight. In the present model, all available formulations take the dimensionless diameter D* as an input, defined as follows: 9D*=d((s-1)gν2)1/3, where d is the grain size, s=ρs/ρf the density ratio, and ρs and ρf are the density of the sediment and the fluid, respectively.

The values adopted for the Schmidt σc number have remained a topic of debate to this day, with no consensus yet reached. proposed a formula to estimate σc from the sediment settling velocity and the friction velocity u*: 10σc=11+2wsu*2,for0.1<wsu*<1. This yields a Schmidt number smaller than one, which corresponds to suspended sediment being dispersed more effectively than momentum is mixed by turbulence. This can be explained by the fact that turbulent diffusion is not the only mechanism responsible for sediment dispersion; additional processes, such as particle collisions, particle inertia, and lift forces, can also enhance sediment diffusivity. Because these mechanisms are not accounted for in this classical approach, the Schmidt number is generally treated as a tuning parameter. In this model, the Schmidt number is treated as constant, with the preceding equation (Eq. ) serving as a useful guideline for choosing its value.

The final key aspect of this approach is the enforcement of the bed boundary condition, i.e., the exchange of mass between the sediment bed and the suspended load. This topic is discussed in Sect. , which addresses erosion and deposition rates.

Sediment transport modeling seeks to quantify how bedforms and channel morphology evolve under fluid flow. The section begins with the Exner equation description, which links bed elevation changes to sediment-flux divergence, followed by the motion threshold and bedload transport formulations that describe the onset and rate of particle motion. Slope-driven avalanching and bedload saturation further constrain near-bed dynamics, while erosion and deposition terms associated with suspended load exchange complete the framework for capturing morphodynamic evolution.

The Navier-Stokes equations (Eq. ) and the transport equation for suspended load (Eq. ) are both solved using the finite volume method. The computational domain is discretized into a multitude of polyhedral control volumes over which the partial differential equations are integrated. The Exner equation, however, is solved over a surface (the sediment bed) using the finite area method (FAM). FAM is an adaptation of the finite volume methods on a surface curved in 3D space. It was initially developed by for the numerical study of the transport of a surfactant at the interface between two fluids and has since then been successfully applied to other problems, such as dense-flow avalanches . In the present model, the finite area mesh is coupled with the volumetric mesh patch representing the sediment bed, such that the finite area mesh coincides with the bed boundary of the finite volume mesh. This approach ensures seamless interaction between flow and sediment transport without the need for multiple meshes. The discretization of the partial differential equations using the finite area method was originally developed to account for surface curvature; however, curvature effects are not considered in the present treatment of bed morphology evolution. Accordingly, the Exner equation is solved on a plane projected normal to the gravity vector g.

The sequence of operations performed during a time iteration and their sequence are represented in Fig. . After solving the mesh deformation, the differential equations for the velocity field u, the pressure p (Eq. ) as well as transport equations for fields related to turbulence modeling (Eqs. , ) are first solved through the PIMPLE algorithm for transient solution which is detailed in . It consists of a mix of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) from and the PISO (Pressure Implicit with Splitting of Operators) from . An additional corrector loop called the PIMPLE loop is added above the PISO loop. During each time step, the velocity flux through the mesh faces is updated at every PIMPLE loop iteration, preserving the simulation stability at a higher Courant number (Co>1). The PISO algorithm behavior is restored by disabling the PIMPLE loop. Once the hydrodynamics have been solved, the shear stress exerted on the bed is computed as well as the associated bedload and erosion flux. The transport equation for suspended sediment transport is solved, and the deposition flux is deduced from it. Lastly, the bed boundary motion is computed by explicitly solving the Exner equation. At the beginning of the next time iteration, the mesh is updated to match the new bed position.

Let us integrate the Exner equation (Eq. ) over the projection of a face f: 21∂zb∂t|f=-1Sfp∑e(qb,ep.nep)lep+(D-E)f, where Sfp=Sf(nf.eg) is the projected area of face f with Sf face f area, and nf is the face normal unit vector oriented outward of the computational domain. lep is the length of the projected edge, qb,ep the projected bedload flux at edge e, and nep the projected normal edge vector, oriented outward with respect to face f. The users have the choice to use either an explicit Euler scheme or a second-order Adams-Bashforth scheme for temporal integration. The discretization scheme for the divergence flux term can be either centered (i.e. linear), upwind first order or upwind second order (i.e. linear upwind).

From Eq. (), at each time step, an increment of bed elevation δzb is computed for each face center. In OpenFOAM, the mesh geometry is defined by the vertices' coordinates. Thus, to impose a mesh motion, the vertical displacements computed at face centers need to be interpolated on the vertices. Particular caution must be given to the interpolation scheme to ensure mass conservation. A straightforward approach would be to linearly interpolate zb from face centers to vertices. Although mass-conservative for 1D bathymetries on structured meshes, the method fails to preserve mass for 2D bathymetries on unstructured meshes. provided a detailed review of the various methods available for solving the Exner equation and analyzed their respective advantages and limitations. He proposed a mass-conservative interpolation scheme, which is the one implemented in the present model.

A dual mesh is constructed as depicted in Fig. . Each vertex v of the primary mesh serves as the center of a face in the dual mesh. The vertices defining this dual face consist of the neighboring primary faces fi that share vertex v, as well as the centers of the edges for which v is an endpoint. The mass increment of the sediments contained under a face f is mf=ρsSfpδzb|f, where δzb|f is the face elevation increment. To ensure mass conservation during the interpolation process, the sum of the mass contained under every face must be the same when computing this sum for both the initial mesh and the dual mesh. The vertical displacement of each vertex δzb|v is then a linear combination of the displacements of the faces sharing this vertex. The weight associated with each face is proportional to the area of the quadrilateral defined by the face center, the vertex, and the centers of the two edges belonging to the face and sharing the vertex (see Fig. ). Let us note the area of this quadrilateral Sdf, the value of the elevation increment δzb|v associated to a vertex v is computed: 22δzb|v=1Sv∑fSdfδzb|f, where Sv is the area of the dual face whose center is the vertex v. It is equal to the sum of the area of each face associated quadrilateral: Sv=∑fSdf. Thus, the sum of the interpolation weights is equal to 1. This interpolation method is mass-conservative and also acts as a filter, as the bed increment is interpolated back and forth between the face centers, where the Exner equation is solved, and the vertices, where the mesh motion is imposed, which in turn updates the positions of the face centers. The final bed displacement at each face is computed in two steps: first, interpolation from faces to vertices, and then from vertices back to faces, with the face centers defined as the center of mass of the vertices composing each face. This filtering effect contributes to maintaining the numerical stability of the Exner equation solution.

At each time step, solving the Exner equation gives a displacement for the bed boundary. For the finite volume mesh to adapt to the bed boundary motion and to preserve the mesh quality throughout the simulation, a mesh motion solver based on a Laplacian equation for cell center displacements is used: 23∇.(Γc∇ΔXc)=0, where Γc is the mesh diffusivity and ΔXc is the displacement of the cell centers. Solving Eq. (), new positions of the mesh cell centers are obtained. The mesh vertices' new coordinates are then interpolated from ΔXc. The motion solver is defined in the file constant/dynamicMeshDict, and this study utilizes the displacementLaplacian solver.

Using a spatially non-uniform mesh diffusivity (Γc) makes it possible to prioritize mesh quality and control cell sizes in specific regions. Areas with lower Γc are more prone to mesh distortion, whereas regions with higher values help prevent excessive cell shrinking or expansion, provided that bed movement remains moderate. Several approaches are available for prescribing Γc, giving the user flexibility in defining its spatial distribution. As a general guideline, it is recommended to assign a high mesh diffusivity near the sediment bed interface to preserve mesh quality in this critical region. The following options, which can be selected in the configuration file constant/dynamicMeshDict, ensure this behavior:

inverseDistance: Γc=1/lsb

One drawback of using the finite-volume method to compute mesh motion is the need to interpolate vertex displacements from the cell-centered displacements obtained from Eq. (). This interpolation step can degrade mesh quality in regions where bed motion is highly non-uniform. In the worst-case scenario, severe distortion may cause some cells to collapse, ultimately leading to simulation failure. Among all the simulations conducted, one problematic case has been identified: the migration of a steep bedform. When the crest is sharp, the vertices located just above the crest, but not belonging to the bed boundary, may be displaced below the bed surface during the interpolation step. Reducing the aspect ratio of the near-bed cells has proven to be an effective way to mitigate this issue.

The numerical setup is defined through a set of input files that specify the computational domain, physical parameters, boundary and initial conditions, and numerical options. Each file must be properly configured by the user before running the simulation. The following sections describe the purpose and required content of each file. The basic directory structure – i.e., all files required to run a sedExnerFoam simulation – is shown in Fig. . It is organized into three main folders, further explained below.

A classical test is the suspension of sediment in a straight flume under equilibrium conditions, which has been extensively studied . A fully developed flow in a channel is considered. The channel is supposed to be long enough so that the vertical profiles of velocity and turbulent eddy viscosity are stationary. Under equilibrium conditions, the vertical profile of suspended sediment concentration is the result of a balance between the gravity, which makes the particles settle at a velocity ws, and the mixing induced by turbulence. Let ws denote the magnitude of ws, the transport equation for the suspended load (Eq.  ) reduces to: 24ddzwscs+ϵsdcsdz=0.

Depending on the shear stress exerted on the bed, granular material is eroded and suspended in the water column. Then, the turbulent diffusion uplifts the particles until an equilibrium is reached. Assuming a parabolic turbulent viscosity profile, νt(z)=u*κz(H-z), where κ=0.41 is the von Kármán constant, the solution of Eq. () between the reference level δzb*, where the concentration is the reference concentration cb*, and the top of the water column H is the so-called Rouse profile: 25cs(z)=cb*(H-zzδzb*H-δzb*)Ro, where Ro=σcws/κu* is the Rouse number.

To validate the suspended load component of the model, numerical results are compared with experimental data from . The experiment was conducted in a 13 m-long and 26.7 cm-wide flume with a bottom covered by a layer of sand. Flow and suspended sediments concentration measurements were made approximately 9 m downstream of the channel entrance. The experiment parameters are summarized in Table .

For the four tests, measurements of the velocity field, the velocity correlation, and the suspended sediment concentration profiles are available.

The four equilibrium bed experiments are reproduced numerically. A 1D mesh is employed, consisting of a column of 120 cells oriented along the vertical z-direction. Cyclic boundary conditions are applied in the stream-wise x-direction, and the mesh is refined near the bed. Only the x-component u of the velocity field is non-zero. For all four simulations, the k-ω SST turbulence model is employed, and the mesh resolution near the bed is maintained at z+≈1 to ensure adequate resolution and an accurate estimation of the bed shear stress. Here, z+=u*z1/ν denotes the distance of the first cell center from the bed boundary in wall units, where z1 is the distance to the sediment bed boundary.

The free surface is not considered, and a rigid lid is applied at the top, with zero gradient condition for the turbulent kinetic energy k, a Dirichlet condition for ω, and a slip boundary condition for the velocity u. To take into account the bed roughness effect on the hydrodynamics, the boundary condition for ω proposed by is used: 26ω=u*2νSR, where SR is defined as a function of the roughness Reynolds number ks+=u*ks/ν as follows: 27SR=200ks+2for ks+≤5,28SR=100ks++200ks+-100ks+e5-ks+for ks+>5.

The transient problem is solved, and the simulations are run until a steady state has been reached, taking a few seconds on a single CPU core. The numerical results are plotted alongside the experimental data from in Fig. .

Overall, the numerical results show good agreement with the experimental data. For the suspended sediment profiles, it is observed that in cases 2565 and 1957, the suspended sediment concentration is slightly overestimated, particularly in the upper part of the water column. To quantitatively assess the agreement between the numerical and experimental profiles, the symmetric mean absolute percentage error (SMAPE) of the logarithm of the sediment volume fraction is computed as follows: 29SMAPE=2N∑|log⁡10(csnum)-log⁡10(csexp)||log⁡10(csnum)|+|log⁡10(csexp)|, where N is the number of measurements available in the experimental test considered, csexp is the experimental sediment volume fraction, and csnum is the sediment volume fraction obtained from the numerical simulation and linearly interpolated to the elevations corresponding to the experimental data. The resulting errors are 4.34 % for case 1565, 8.29 % for case 1965, 6.64 % for case 2565, and 2.46 % for case 1957. These results were obtained without any calibration of the model coefficients, and an improved fit could likely be achieved by adjusting parameters such as the turbulent Schmidt number σc, the near-bed diffusivity coefficients ϵw0 and ξw (Eq. ), or the equivalent sand roughness height ks.

Another test for the suspended load is the development of suspension in a channel, when the flow encounters an abrupt transition from a non-erodible bed to an erodible bed. Initially, the flow is clear, and it becomes loaded with sediments until an equilibrium is reached. For this problem, the results are compared with a pseudo-analytical solution derived by . In order to obtain this solution, some assumptions are made.

The sediment is uniformly advected at the mean flow velocity u‾.

Based on those assumptions, simplified the transport equation for the suspended load and derived an analytical solution. They performed a separation of variables and used the Sturm-Liouville theory to obtain a solution, which only depends on the Rouse number. A first numerical simulation is performed for which all assumptions apart from the fourth one are respected. A water depth of H=0.1m is considered, the mean velocity is u‾=0.9ms-1, and the Rouse number is equal to 0.5, which corresponds to a regime in which suspended load is the dominant sediment transport mode. The results are presented in Fig. .

In this case, the numerical and pseudo-analytical solutions are almost identical, suggesting that the stream-wise turbulent diffusivity (assumption 4) is indeed negligible. However, some of the assumptions from are normally not verified. The vertical velocity profile is not uniform, the concentration at the reference level may vary in space and reach an equilibrium after some distance from the inlet, and lastly, the turbulent eddy-viscosity profile is not exactly parabolic.

The particle diameter was set to d=0.12mm, corresponding to a settling velocity of ws=0.773cms-1 and a Rouse number of Ro=0.5. Although tests were conducted with different Rouse numbers, only the case Ro=0.5 is presented in this work. The same mean flow velocity u‾=0.9ms-1 is taken, and the resulting shear stress exerted on the bed corresponds to the bed friction velocity u*=3.77cms-1. The k-ω SST turbulence model and the rough wall boundary from (Eq. ) condition is used for ω with a roughness height ks=2.5 d.

A first 1D simulation is performed without sediment to obtain vertical profiles for u, k, and ω corresponding to a fully developed channel flow. The fields u, k and ω are extracted from this first simulation and used as the inlet boundary condition for the second simulation, for which suspension is activated. The flow entering the domain being already fully developed, only cs varies with the x-position. The mesh consists of a 2D structured mesh, more refined close to the bed to ensure the condition z+≈1 (nx=2000, nz=100). The results are extracted after 50 s of simulation, requiring 10 h of wall-clock time on 5 CPU cores. Isolines of cs/cb* values from the model and the pseudo-analytical solution are presented in Fig. . To compute the pseudo-analytical solution, the reference level was chosen equal to δzb*=0.05H as in the work of , and the reference concentration is taken equal to cb*=0.025 and applied as a boundary condition at the elevation z=δzb*.

Compared with the situation where the assumptions on the flow is enforced (see Fig. ), the model results do not match the pseudo-analytical solution, but the global behavior remains the same. Starting from no suspension, the suspended sediment quantity gradually increases with the distance to the inlet until reaching an equilibrium situation where the settling and the turbulent diffusion cancel each other out. Figure  shows the vertical suspended sediment volume fraction cs profiles at different positions along the channel and shows the convergence toward an equilibrium solution close to a Rouse profile.

As stated previously, the discrepancies with the pseudo-analytical solution arise from the unrealistic assumptions made in its derivation. These include the assumption that suspended sediments are advected by the mean flow, the use of a parabolic eddy viscosity profile, and the assumption of local equilibrium at the reference level. A better agreement could yet be found, for instance, by adjusting the boundary conditions for ω at the top and bottom boundaries, which would affect the shape of νt profile. Another calibration parameter is the reference concentration at the reference level, which is the bottom boundary condition of the pseudo-analytical solution.

As a first benchmark for the Exner equation (Eq. ), an idealized 1D dune migration model for which an analytical solution exists is presented. In this case, a highly simplified flow is considered in order to focus on the behavior of the Exner equation without the added complexity of the hydrodynamics (see Fig. ). The fluid is topped by a rigid lid placed at an elevation H from the bottom. The flow is assumed vertically uniform with a constant discharge per unit width Q. The depth-averaged velocity is obtained by conservation of mass, U=Q/(H-zb).

In this simplified case, only bedload transport is considered. To be able to derive an analytical solution of the Exner equation, the bedload qb must be expressed as a function of the bed elevation zb. This is done by assuming that the bedload is a power law of the depth-averaged velocity U, qb=αdUβd, where αd and βd are two positive constants. The Exner equation (Eq. ) simplifies to: 30∂zb∂t+c(zb)∂zb∂x=0,31c(zb)=∂qb∂zb=αdβdQβd(H-zb)βd+1, where c(zb), is the celerity of the bedform. Starting with a given initial bedform zb(x,t=0)=F0(x), the solution to Eq. () is obtained with the method of characteristics leading to zb(x,t)=F0(x-ct). Depending on F0, shocks may develop as the bedform migrates. A shock occurs if, over at least one interval I∈R, the function G:x→c(F0(x)) is decreasing. The dune celerity c being an increasing function of zb, shocks will arise if the initial bedform F0 exhibits at least one negative slope. In this idealized dune migration case, the initial dune profile is Gaussian: 32F0(x,t)=hde-x-xd0σd2, with hd the height of the dune, xd0 the initial position of the dune's crest and σd a parameter linked to the dune width such that F0(xd±ln⁡(2)σd)=0.5hd.

With this initial dune profile, a shock wave will form where the bed slope becomes vertical on the lee side of the dune. To know the position and time of the shock, it is necessary to find the position x0* defined as follows: G′(x0*)=min⁡x∈RG(x). It corresponds to the initial position of the point belonging to the characteristic line on which the first shock occurs. The breaking time is then obtained as t*=-1/G′(x0*) and the shock position x* as well by advection of x0* along its characteristic line, x*=x0*+G(x0*)t*.

The following parameters are chosen:

flow properties, H=1m and Q=1m2s-1

For this configuration, the breaking time is t*=24.67s and the shock position x*=4.51m. A solution to Eq. () is sought for the time interval between t=0 and the breaking time t*. A second-order Adams-Bashforth scheme is used for time integration and a linear-upwind scheme for the advective term discretization. A comparison between the model results and the analytical solution is presented in Fig. .

Overall, the model fits well with the analytical solution except when the time gets close to the breaking time, where a small instability starts to develop at the dune's crest. Figure  illustrates how the chosen numerical schemes affect the solution stability and precision.

At the dune front, the gradient of zb becomes significant, and, consequently, the gradient of zb also increases. Depending on the numerical schemes used, this can trigger oscillations. Using the Euler explicit scheme for time integration, the use of a second-order scheme for advection leads to instabilities appearing on the crest of the dune. On the other hand, the low-order upwind scheme introduces numerical diffusion, resulting in reduced predictive accuracy, but ensures numerical stability. A better match between the numerical results and the analytical solution is achieved using a second-order scheme for the temporal term (Adams-Bashforth 2) as the numerical solution no longer oscillates.

As stated in Sect. , the interpolation from faces to vertices needed to enforce mesh motion acts as a filter. However, depending on the case, it may not be sufficient to suppress the appearance of numerical instabilities, in particular in regions presenting steep slopes. The use of an avalanche model (Eq. ) brings more stability by limiting the maximum bed slopes. However, it is not used in this example as no analytical solution can be derived for this problem if the avalanche mechanism is taken into account.

From the results presented in Fig. , the combination of a second-order Adams-Bashforth scheme for the time integration and a linear or linear-upwind scheme for the bedload flux discretization, both being second-order schemes, seems to offer the best compromise between stability and accuracy. Choosing an Euler explicit time scheme tends to trigger instabilities, while the use of an order 1 upwind scheme leads to more stability at the cost of accuracy. To further illustrate the different behaviors of the possible scheme combinations and for different mesh refinements, multiple simulations are performed by varying the grid size and the numerical schemes, and the results are compared with the analytical solution.

The stability of the numerical solution is related to the mesh refinement through the maximum Courant number, whose evaluation is straightforward as the celerity of bedforms c is known (Eq. ). The maximum Courant number is then max⁡(Co)=c(zb=hd)δt/δx where δt is the time step value and δx the width of the mesh faces in the x-direction, the mesh being uniform. The accuracy of the numerical solution is evaluated using the Root Mean Square Error (RMSE): 33RMSE=1NF∑f(zbn|f-zbs|f)2, where the index f stands for the finite area mesh faces, NF the number of faces of the mesh, zbn|f is the elevation of face f center (numerical solution) and zbs|f is the analytical bed elevation at face f center. Each simulation is represented by a point in Fig. .

The simulations were performed using a constant time step. For low Courant numbers, associated with poor mesh quality, all simulations remain stable and exhibit similar errors. As the mesh quality increases, the RMSE decreases but at a faster rate for second-order schemes for bedload advection until the solution becomes unstable when the maximum Courant number gets close to 1. The sudden rise of RMSE values for high mesh resolution is a sign of those instabilities. The use of an upwind scheme allows for the use of a higher Courant number without the simulation failing. This is due to the numerical diffusion that this first-order scheme involves. When using an Euler explicit time scheme along with one of the second-order schemes for advection, it is observed that the RMSE no longer depends on the mesh resolution for values of maximum Courant number of 0.1 and higher until the appearance of instabilities. It shall be recalled here that a filtering process is applied on the numerical solution at each time step as the bed elevation increment is interpolated from faces to vertices (discussed in Sect. ). The results presented support the use of a combination of a second-order Adams-Bashforth scheme along with a linear or linear-upwind scheme to ensure both stability and accuracy.

These simulations complete in a few seconds on a single CPU core.

A still basin of depth H=1m is initially uniformly loaded with a volume fraction cs0=0.05 of suspended sediment corresponding to a mass concentration of 132kgm-3. As the suspended sediment deposit, the bed level rises up and reaches a final elevation zbed=cs01-λsH, where λs is the porosity of the deposited granular material. The settling velocity being set to ws=3.59cms-1, the time at which the last sediment deposit on the bed is t=H-zbedws. The variation over time of the sediment mass distribution between suspension and deposited sediments is represented in Fig. .

During each time iteration, the equation for the concentration of suspended sediments is solved, and the erosion/deposition flux is computed, resulting in an updated sediment bed elevation via the Exner equation (see Fig. ). Since the mesh motion is resolved at the beginning of the time iteration, the bed level increment computed at a given step only affects the mesh geometry in the subsequent time step. Consequently, there is a one-time-step delay in the morphological response of the bed, which introduces a temporary error in the total sediment mass. However, this error vanishes once all suspended sediments have settled.

Another settling case is presented, this time a 2D case with non-uniform settling. The computational domain is conic-shaped, wider at the bottom (5 cm) and narrower at the top (1 cm).

Initially, only water is present in the domain. The repose angle is taken equal to βr=32°. The particles diameter is d=0.29mm and their density ρs=2600kgm-3. The sediment settling velocity ws0=3.5cms-1 is computed using the formula from (see Table ) and is considered uniform as the hindrance effect is not taken into account. A constant flux of sediment is injected during 7 s by imposing a constant suspended sediment volume fraction at the top boundary cs=0.05. The sediment settle under the action of gravity and deposit on the bed. Because the settling is not spatially uniform, pronounced slopes form at the margins of the deposition mound, where avalanching occurs, producing a conical shape similar to that seen in an hourglass. The simulation is then run for 3 more seconds so that all sediment have settled by the end of the simulation (see Fig. ).

The mass repartition of sediments between suspension and deposition is represented in Fig. . At the beginning of the simulation, all the sediments are suspended, and their quantity increases linearly over time until t=1.14s where the sediments start depositing on the bed. As the domain bottom boundary rises, the space occupied by suspended sediments shrinks, leading to a diminution of the mass of suspended sediments. At 7 s, sediment injection into the domain ceases, and approximately one second later, all sediment has settled, with the simulation completing in a few seconds on a single CPU core. Once again, the mass error is evaluated by comparing the mass of sediment, which has been injected into the domain, to the sum of the suspended mass and the bed mass. Just as in the 1D case, the one-time step delay in the bed morphology response induces an error on the total mass in the domain (see Fig. ). The error subsequently disappears as the sediments settle, thereby verifying mass conservation.

Different regimes of a lone dune migration over a starved bed have been investigated by . The author identified the existence of two regimes: the stationary regime and the mass loss scenario. These regimes depend on the flow conditions and on the initial dune mass. In the present work, the focus is on the stationary regime, which exhibits two stages. During the initial phase, the dune morphology evolves rapidly from an initial conical shape formed by sediment deposition in still water. After this transient phase, the dune reaches a stationary state, migrating at a constant speed in the flow direction with minimal deformation.

The experimental facility consists of a hydraulic tunnel working in closed circuit with an experimental section made of a straight channel of length Lc=900mm, of height Hc=90mm, and of width Wc=6.03mm. The flume is closed on the top by a rigid lid. The granular material consists of highly spherical glass beads with a diameter of d=0.4mm and a density ρs=2500kgm-3. The particle's terminal fall velocity in water is ws0=7.67cms-1, which is noticeably higher than values obtained with the models presented in Table (ws0≈5cms-1). The friction velocity upstream of the dune is u*=2.78cms-1 which corresponds to a Rouse number Ro=6.73. This value indicates that bedload is the main transport mechanism for this problem. The critical Shields number (θc0=0.079) obtained experimentally is large compared to what is expected from the formulas in Table . This could be due to the confinement of the particles in the flume, the ratio of the channel width to the particle diameter being only equal to 16.

In the following, a stationary regime configuration is reproduced numerically. Experimentally, 10g of particles is introduced through a hole drilled in the channel cover. They deposit under the influence of gravity, forming a conical-shaped mound with slope angles equal to the angle of repose of the granular material. Once the initial pile has formed, a motor is activated to create a left-to-right flow in the test section with a bulk velocity u‾=0.43ms-1. Initially, the particles are loosely packed with a volume fraction of cs≈0.54 in the particle's bed. As the dune is migrating downstream, the particle bed is compacting, and the sediment volume fraction in the bed increases, resulting in the apparent dune volume decreasing over time until the volume fraction reaches a constant value of about cs≈0.6.

Section presents the two-dimensional numerical simulations used to reproduce the stationary dune migration regime. It describes the numerical setup, modeling choices, and boundary conditions, and assesses the model’s ability to match the experimental observations. A sensitivity analysis then highlights the key parameters controlling the simulated dune dynamics.

Section  presents three-dimensional numerical simulations of the dune migration problem to assess the model’s performance. The 3D setup is derived from the 2D case and applied under the same physical and numerical assumptions. The results are compared with the 2D simulations to evaluate consistency, robustness, and computational performance.

The grid of the 2D case presented in the previous section is just extruded in the y-direction to match the experimental channel width (Wc=6 mm) using 10 cells. This resolution allows us to maintain an aspect ratio close to unity between the streamwise and spanwise grid sizes (Δx≈0.72 mm, Δy≈0.6 mm). A grid resolution of nx=1100, ny=10, and nz=80 is employed, corresponding to 880 000 cells. The numerical schemes and empirical parameters used in the sediment transport model for the 3D configuration are identical to those employed in the 2D case. It should be noted that the lateral wall boundary conditions are set as symmetry planes, while the source term accounting for lateral wall friction remains active in the present simulations.

The results of the 3D simulations with suspension are shown in Fig.  at three different times during the migration process: t=0, 5 and 10s. In this figure, the flow streamlines are colored according to their velocity magnitude. Also shown is a color plot of the suspended concentration, cs, in a vertical plane at y=0.006 m on a log scale. The color plot on the bed patch represents the qsat, and the bed grid is illustrated by the white lines. As shown in the figure, the flow recirculation cell downstream of the dune is clearly visible. The size of the cell reduces as the dune migrates and its height decreases. The suspended load region encompasses the entire recirculation cell, with a solid volume fraction ranging from 10-3 to 10-2. This seems to be an overestimation compared to the experimental observations made by . Not much oscillation is observed at the bed, except perhaps at the downstream toe of the dune, where suspended particles accumulate due to the recirculation cell. This confirms the robustness of the numerical implementation for two-dimensional bathymetry. One final observation is the repartition of the qsat, which shows a clear maximum near the crest of the dune. It may be difficult to discern from this figure, but the maximum actually occurs at the crest at t=5 s and t=10 s in the stationary migration regime. This is consistent with the physical arguments presented, for example, in , who argue that saturation of the bedload flux constitutes the stabilizing mechanism for bedform growth. A bedform's maximum amplitude is reached when the maximum occurs at the crest.

Figure shows a comparison of the bed profiles at different times during the migration process (t=0, 4, 8, and 12s). The results are compared with the 2D case shown in the Fig. . The differences between the two configurations are very small, demonstrating the reliability of the 3D implementation of the numerical model. The 3D simulation results without suspension exhibit small oscillations of the bed elevation on the stoss side of the migrating dune at t=12s. These oscillations do not degenerate into numerical instabilities. This is likely due to the use of an upwind scheme for the divergence of the bedload flux in the solution of the Exner equation. The model also has a centered scheme for this term; it runs smoothly in the 2D case but crashes in the 3D case, as illustrated in Fig.  in Appendix . Further work is needed to improve the accuracy and the stability of the coupling between hydrodynamics, sediment transport (including bedload flux saturation), and the morphological evolution. This includes developing a filter for the bed evolution increment e.g., developing high-order, non-oscillating schemes, such as WENO or NOCS e.g., and/or using a predictor-corrector algorithm for the Exner equation solution.

The hydrodynamic part of the model benefits from the native parallelization available in OpenFOAM. However, the Exner solver is currently only available sequentially. One workaround for running parallel simulations is to decompose the numerical domain while ensuring that all the bed cells remain on a single CPU core. While not optimal, this method enables 3D simulations to be performed in a reasonable amount of time. To demonstrate the performance of sedExnerFoam, the 3D simulation without suspension presented in this subsection took 1 h and 53 min on 32 CPU cores to simulate 5 s of physical time. This performance is acceptable compared to the 2D case run on a single CPU core. Nevertheless, parallelizing the Exner solver will be necessary to take full advantage of the OpenFOAM parallelization capabilities.