the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Aug 2025
sedExnerFoam 2412: A 3D Exner-based sediment transport and morphodynamics model
Abstract. The development of an open source numerical model for sediment transport and morphological evolution is presented. It relies on the Arbitrary Lagrangian Eulerian (ALE) method to track the bed interface position over time. The sediment bed acts as a moving boundary whose motion depends on sediment fluxes and a dynamic mesh is employed to adapt the computational domain to the dynamic boundary. The implementation of the different components of the model (bedload, suspended transport, avalanche, etc.) is validated using a series of academic benchmarks. Finally, in order to highlight the model capability, an application to the study of a lone dune migrating under the influence of a steady flow is presented.
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Status: open (until 29 Sep 2025)
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RC1: 'Comment on egusphere-2025-2375', Gabriel Barajas, 12 Aug 2025
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The manuscript entitled “sedExnerFoam 2412: A 3D Exner-based sediment transport and morphodynamics model”, presents a numerical model for sediment transport and morphological evolution, with validations and an application.
From the reviewer point of view, this is a very interesting work, as the topic is quite interesting and relevant. However, before recommending this paper for publication I think that the paper needs to be substantially improved in several specific parts. The reviewer adds a detailed review in the hope that the authors will find it useful.
Major issues:
1.- The abstract is very poor.
2.- All sections need a small introduction.
3.- Section 2 needs to be checked.
There are wrong namings, variables cited before definition, etc.
3.- There is not any comment about the computational cost of the different simulations. I would like to know about the computational times and computational resources used for each of the simulations, Y+ values, etc.
4.- I would like to see a section highlighting the strengths and limitations of the numerical approach.
5.- I expect to see the units of all variables listed in the manuscript.
6.- There are too many variables defined in the manuscript, a table containing all of them should be added, with the units and the naming.
7.- Numerical wave tanks are not described by any means, neither the BC’s applied or sketches of the meshes. There is no grid analysis in the validation cases.
8.- Section 3.2 and 3.3 should be defined in the order of appearance of Figure 5.
Notes about the figures:
Figure 2
Is not referred to in the text.
Figure 3 should be next to Table 2a
Figure 4 should be next to Table 2b
Figure 5 should state clearly which variables are in the input and output for each part of the flow chart.
Figure 7a is very small, I can not see anything. Figure 7b should be wider, as there is space for doing it in the text.
Figure 8, legend is missing
Why figure 10 is referred to in the text before figure 9. Please, correct it.
Figure 12:
What are the black dots? they are not referred to in the legend.
Figure 15.
Black crosses are not referred to in the legend.
Figure 16:
What are the blue and orange zones?
Figure 17:
Legend is missing.
Figure 18.
Legend is missing.
Notes about the Tables:
Table 2.
Is Phib the bedload flux? In line 128 was defined as qb?
Notes about the manuscript:
Line 17:
“The twentieth century saw the development of analytical models (Hjelmfelt and Lenau, 1970) and one-dimensional numerical models (Verwey, 1980; Goutal and Maurel, 2002).”
Please, rephrase.
Line 17:
one dimensional (1D)
two-dimensional (2D)
Add it, as well as three-dimensional (3D).
Line 33:
“the deformation of the associated volume mesh (ALE)..”
What does ALE stand for?
Line 38:
“and was developed to meet the needs of hydraulic engineering.”
Please, rephrase.
Line 39:
“However, the model’s scope extends beyond this to include a wide range of morphodynamics problems”
Please,define which problems.
Line 46:
Please, capital f for Figure 1. Check it through the entire manuscript.
Line 49:
Please, summarize correctly all the different sections of the manuscript.
Line 54:
“The hydrodynamics is described by the incompressible filtered Navier-Stokes equations.”
Please, rephrase.
Line 58:
“It can either be 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 in case of laminar simulation or Direct Numerical Simulation (DNS). The model makes use of the vast panel of possibilities offered by OpenFOAM and let the user choose freely the kind of filtering to be applied to equation 1. This is done by changing entries in a file turbulenceProperties. In this work however, the numerical simulations presented are either laminar cases (no filtering of the Navier-Stokes equations) or unsteady RAS simulations”
Please, rephrase.
Line 64:
“which is a valid assumption only in the case of dilute suspended sediments”
Please, explain correctly. this statement.
Line 66: Section 2.2
Why a RANS turbulence model and not a LES?
Can it capture correctly all the processes of figure 1?
There is not a clear description of the choice of the turbulent model.
Line 99:
“ It is expressed as the ratio of the turbulent eddy viscosity and the Schmidt number σc”.
Write the relation as an equation,
Line 108:
Please, capital t for Table 1. Check it through the entire manuscript.
Line 110:
Add “sediment” to the definition of the settling velocity.
Line 112:
“This yields a Schmidt number smaller than one which corresponds to sediment diffusion being more intense than the turbulent diffusion for the fluid”
Please, justify this statement correctly.
Line 117:
“This topic is covered at the end of the next section in relation to bedload and morphological evolution.”
Please, add the number of the section.
Line 23.
“ an Austrian meteorologist and geologist and geophysicist.
Please remove it.
Line 124:
“ In their article, Paola and Voller mention that Felix Exner initially”“
Who is Paola? Please, correct.
Line 139:
“The bedload flux qb is the specific flux of sediment transported along the bed per unit width. It is computed from the bed shear stress using an empirical formula”.
Table 2 does not refer to qb. Please, correct this section.
Line 142:
The user can choose between one of those models or manually set a value for θc.”
How? Which value is appropriate? Please, specify.
Line 152:
“The slope correction can be activated/deactivated in the file bedloadProperties using the entry slopeCorrection.”
This section is the mathematical model, not the numerical model.
Please, add it correctly in its correct section.
Line 154.
What is Phib?
Correct it.
Line 155:
“A lot of this formulas..”
How many formulas are a lot?
Line 160:
“Another phenomenon that needs…”
Please, cite explicitly which phenomenon has been already addressed, so readers can easily follow the mathematical models.
Line 167:
Is the equation correctly defined?
Line 173:
Add “and” between the two cited works.
Line 177:
“From the file bedloadProperties, the user can activate the saturation and provide values for Tsat and Lsat. The saturated flux qsat is then computed from the bed shear stress using one of the formula from table 2 and the bedload flux qb is the solution of equation 14”
Again, this part is the numerical implementation of the mathematical model. Add it to the correct section.
Line 181:
“As said previously, the modeling of the erosion and deposition fluxes is a hard point of classical sediment transport models”
Please, rephrase it.
Line 189:
“Since then, .”
“.. is not appropriate for a lot of situation..”
“The idea is that the …”
Please, rephrase it.
Line 197:
“One difficulty is then how to impose this reference concentration at the reference level which is above the bed boundary.
Line 203:
“In sedExnerFoam it was chosen to avoid the use of two different meshes”
Please, specify the solution of one single mesh: pros and cons.
Which physical process is chosen to define the cell size, etc.
Line 219:
This flux is imposed as a boundary condition for the suspended-load transport (equation 6). With this method the same mesh can be used for both the suspended load and the hydrodynamics with a fine resolution near the sediment bed. It is to be remembered that the various formulas for c ∗ b existing in the literature are all empirical and are based on measurements made in straight channel flow experiments. Their validity out of this configuration, let alone in the vicinity of an obstacle disturbing the flow, is subject to caution
Rephrase this paragraph.
Line 224:
“The possibility to use an additional diffusivity near the bed for suspended sediments as been introduced in the model after experimenting difficulties to suspend material from the bed to the water column in the case of fine grid resolution and low to medium dimensionless roughness”
Rephrase it.
Line 232:
“The coefficients ϵ 0 w and ξw are both set to 5 by default but can be modified in the file bedloadProperties.”
Again, this part is the numerical implementation of the mathematical model. Add it to the correct section.
Line 244:
“ In the present model, the finite area mesh is mixed with the patch of the volumic mesh corresponding”
What does mixed mean?
Line 271:
Rephrase “a naive approach…”
Line 288:
Please, explain why it is acting as a filter.
Line 299:
Explain just the mesh motion solver used in the work, not the options.
Line 305.
Explain just the mesh diffusivity used in the work, not the options.
Line 326:
There is no description for sediment bed fields or suspension, please add it in this section.
Line 331:
Write down a paragraph describing the purpose of each validation and to which part of the flow chart and equations is related. Specify clearly if they are 1D, 2D or 3D validations.
Line 338:
Why isolating the physical process at a time is a correct approach in this work?
Please, justify it.
Line 335:
Please rephrase “The situation is the following”
Line 350:
Is it four experiments of four tests in the experiment?
If so, update also the naming in Table 3.
Line 370:
Quantify quantitatively the errors from Figure 7. Do the same for all validations, just a qualitative comparison for the different validations is not correct.
Line 372.
Why another test? Justify it properly.
Line 382:
Why isn't the fourth hypothesis respected? How does it affect the simulation?
Line 390:
if just one case is presented in figure 10, then do not write that a series of particle diameter values were selected.
Line 409:
Write down again the unrealistic hypothesis assumed for the simulations.
Line 413:
Please, change it from one-dimensional to 1D.
Check it for 2D and 3D in the entire manuscript.
Line 414:
Please rephrase, “a very simple flow”
Line 424:
Give a reference for the method of characteristics.
Line 431:
Please, rephrase the sentence: “a shock wave will appear somewhere”
Line 436:
Use the SI for all magnitudes, cm is not correct when referring after to m.
Line 453:
Is there any reference for the assumption regarding the avalanche model?
Line 460:
Specify the number of simulations, multiple is not correct.
Line 470:
I do not see the relation between low courant number and poor mesh quality.
This full paragraph is quite questionable, as there are multiple numerical variables that might affect the stability of the simulations.
Please, justify it properly.’
Line 480.
Justify the purpose of this validation.
Is the first validation 1d?
Line 485.
This paragraph news to be rewritten.
Where can I see the error?
Line 492.
Remove black space line.
Line 493:
What does this 2d case differ from the 1d case? justify this validation case.
Line 508.
How is “mass error” named in figure 17?
Line 515:
Justify the selection of the application case.
Validations were done for 1D and 2D cases.
Line 526:
“…which differs slightly from values obtained with the models presented in table 2…”
How does it affect the numerical results? Justify it properly.
Line 529:
“...large compare to what is expected from the formulas in table 2”.
Again, justify how these values affect the numerical results.
Line 542:
Please, specify how Figure 18 is obtained. There is no clear information on the text.
Line 548:
Change two-dimensional to 2D.
Line 558:
Quantify correctly “after some time”. This type of statement is not acceptable in a paper.
Line 560:
“...it was verified that the domain upstream “
How was it verified? Please, specify it.
Line 572:
Which numerical solvers are used for this “new initialization” of the case?
Please, specify them.
Line 577:
Please rephrase the paragraph (“...multiple attempts…”)
Line 578:
Please, add in an appendix a graph that illustrates the importance of a grid analysis.
Line 588:
Please, add in an appendix a graph that illustrates the importance of the choice of the numerical schemes.
Line 599:
Please, add in an appendix a graph that illustrates the importance of the choice of the critical Shields number.
Line 616:
I can not follow how the velocity is better observed in Figure 19 than in Figure 21.
Please, rephrase this paragraph for a better comprehension.
Line 619.
Please, rephrase the first two sentences correctly.
Line 622:
“ but it could also be a consequence of the method used to estimate the dune length”
Please, specify why and if it could be solved with a different method.
Line 625:
Conclusions are very poor. They should resume the structure of the paper, explain different sections and clearly state the main findings and objectives of the manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-2375-RC1 -
RC2: 'Comment on egusphere-2025-2375', Jennifer Keenahan, 05 Sep 2025
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General comments: Overall, this is a high‑quality paper that presents a coherent, open‑source ALE Exner framework in OpenFOAM with clear mathematical formulation, careful numerical implementation, and well‑chosen validations that collectively demonstrate capability for complex morphodynamics around structures and steep bedforms. The narrative is logically structured from model description to targeted benchmarks and a realistic dune application, with figures and tables that effectively support the arguments and trace key design choices such as turbulence modelling, bedload closures, saturation, and avalanche treatment. Reproducibility is bolstered by public code, tutorials, and explicit parameterisation for most tests, although a few tuned constants and boundary‑condition specifics in the dune case could be pinned more precisely in the text for one‑to‑one replication without consulting the repository. The discussion is appropriately balanced - linking observed behaviours to numerical choices and acknowledging limitations like the absence of a free surface, constant porosity, dilute suspension assumption, and RAS grid‑sensitivity - while outlining credible extensions that would broaden applicability. Minor editorial issues (e.g., symbol definitions and occasional notation/ordering inconsistencies) do not detract materially from the science or conclusions, which are well supported by the evidence provided. In sum, the paper is rigorous, useful, and timely, meriting a positive evaluation with only small clarifications needed to reach an excellent standard.
Specific comments:
Slope‐corrected critical Shields number formula (eq. 11) has ambiguous powers; the intended factors are likely sin2(αs)tan2(βs)/μs2sin2(αs)tan2(βs)/μs2, but it is written as “sin(αs)2 tan(βs)2/µ2s”; use unambiguous exponent notation to avoid misinterpretation.
k–ω SST transport equations: in eq. (5) the production term of ω is written “αS,” which is not the standard SST form (normally involves production from Pk/νtPk/νt); if this is a deliberate simplification, state it; otherwise, correct the ω equation to the standard SST form or cite the exact implemented variant.
Lateral wall source term derivation needs clarity on specific vs absolute quantities and dimensions: τwall is introduced without density, and the width‑based source term Fwalls=−f∣u∣u/(4Wc)Fwalls=−f∣u∣u/(4Wc) would benefit from a short derivation to show consistency with the specific (per unit mass) form of the momentum equation used elsewhere (kinematic viscosity, specific pressure gradient)
Directionality of bedload flux: the scalar ϕb and vector qb are related via direction along shear or slope; explicitly define how qb direction is set (e.g., aligned with bed shear stress, corrected for slope via αs) to avoid ambiguity in eqs. (12) and the Exner divergence.
Clarify whether τf denotes the negative Reynolds stress (−u′u′) or the total additional stress; eq. (2) matches the Boussinesq model for −u′u′, but the narrative could mislead; a single precise definition early in Section 2.2 would prevent confusion.
Overstatement: the difference between measured w0w0 (7.67 cm s−1) and model estimates (~5 cm s−1) is described as “slightly,” but the ~50% gap is not slight; rephrase to “noticeably higher than” or quantify the percentage difference.
Suspended‑load additional diffusivity (eq. 18): give default values and guidance on when to disable (e.g., rough‐wall regimes, k+≫1) and how sensitive results are to ϵ0w and ξw, since Section 4.1 notes sensitivity but defaults are only briefly stated.
Bedload saturation (eq. 14): since Section 5 shows saturation is essential to prevent non‑physical crest arrest, specify default Tsat and Lsat and provide a short note on how qb direction is treated in the 2D finite‑area discretisation when aligning with shear.
Provide a short derivation or appendix note for converting Darcy–Weisbach wall shear into the 2D source term used, including assumptions (uniformity across width, use of hydraulic diameter Dh, and whether τwall is per unit mass), to pre‑empt reviewer questions on dimensional consistency and calibration of f and kw.
Technical corrections:
Convective term notation in the Navier–Stokes equation appears incorrect: “∇.(uT u)” should be the divergence of the dyadic product ∇⋅(u u)∇⋅(uu), not a vector–vector inner product with a transpose; as written it is dimensionally and notationally inconsistent for convection in vector form.
Adams–Bashforth order is misstated: the text says “a first order Adams‑Bashforth scheme, which is a second order time scheme,” which is contradictory; AB1 is first‑order and AB2 is second‑order, so either the method or its order needs correction in Section 3.2 and figure captions/discussion where “Adams Bashforth 1” is presented as second order.
Undefined symbol ϖ (varpi) in bedload formulas and avalanche term: ϖ is used multiplicatively in Table 2, eq. (12), and eq. (13) but is never defined (e.g., Heaviside step function, direction/sign, or scaling factor); define it explicitly or remove if redundant.
Inconsistency for Brownlie: Figure 3 legend shows “Brownlie (1981)” while the references list “Brownlie (1983)”; standardise to the correct year (the widely cited Caltech thesis is 1983) and update the figure legend accordingly.
Naming consistency in Table 2 and text: “MeyerPeter” vs “Meyer‑Peter & Müller (1948)” and “vanRijn” vs “Van Rijn (1984)” are inconsistent; standardise hyphenation, spacing, and capitalisation across tables, figures and prose.
Duplicate heading: “Author contributions. Author contributions.” appears twice; remove duplication.
Spelling: “librairies” → “libraries” in Code and data availability.
“pimpleFoam” appears only in Conclusions; consider introducing that inheritance explicitly in Section 3.1 for immediate context.
Citation: https://doi.org/10.5194/egusphere-2025-2375-RC2 -
RC3: 'Comment on egusphere-2025-2375', Sem Geerts, 15 Sep 2025
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Overall, it is a promising paper that is worth publishing as it can be a useful tool for many civil engineers. The manuscript does well in explaining some chosen test cases to explain the model performance. However, I think there are some points worth of attention and major revisions are necessary before final publication.
Please refer to the pdf file for proper formatting of the comments.
General comments
- The model shows no three-dimensional results and performance while this is computationally different. Especially the introduction talks about processes such as scour (lines 41 – 49 and figure 1), which is then not covered as a complete case for the remainder of the manuscript. I think these three-dimensional cases are worth showing to truly trust the model.
- The style of writing of the paper concerns me. To me, it reads like a textbook for students or like an informal review paper describing the history of development of morphodynamic modelling. Overall, the style should be changed and the quality of English should be improved (examples of which are provided in the Technical comments below). Specifically, I recommend to either (1) add the historical framing to the aim of the overall paper, or (2) remove the historical context as it does not contribute scientifically. Some examples:
- Lines 76-83 mention all kinds of different turbulence closures, but in the end only one is implemented.
- Line 122: “…an Austrian meteorologist and geophysicist.” Not sure why this is relevant.
- Line 135: “In the 1930’s, Albert Frank Shields made measurements of the motion threshold already highlighted by Du Boys in 1879.” Vague and misses citations
- Lines 161-165. Unclear why alternatives to modelling avalanches are mentioned when a different implementation is used.
- Section 2.4.3 provides a lot of context and alternatives, while it seems that only one implementation is done in the end (even mentioning “In sedExnerFoam it was chosen to avoid the use of two different meshes” in line 203). Focus on this formulation and avoid confusion.
- Model performance, convergence and computational costs are missing to the manuscript and how this compares to other morphodynamic models. Where should this newly developed model be framed, given the morphodynamic models discussed in the introduction?
- To me, a separate discussion of the weak points of the developed model is missing. I do notice these kinds of points returning in the main text and the conclusion, and I understand that the migrating-dune case is considered the benchmark, but a separate discussion would be appropriate, such that the conclusion can focus on a fundamental summary.
Specific comments
- The abstract should be rephrased, it currently has no conclusion.
- Line 41: Scour is explicitly covered in the introduction, but later, no reference is made to it, and it's not a use case. Furthermore, it’s a 3D process, and it is unclear whether the newly developed model is able to achieve this.
- The introduction misses a clear aim.
- The end of the introduction should clearly state the structure of the paper with corresponding sections.
- Lines 109 – 114. Not sure why all of this is mentioned while a constant value is assumed in the end.
- Line 130: D and E are not fluxes, but a gradient of fluxes (just like q_b is a flux).
- Line 141: Equation 10 has no reference to it and no context or explanation.
- Line 156: Equation 12, and in general, unclear what omega is.
- Line 156: Equation 12 could be removed, as it is just one of the many sediment transport equations available to the user, listed in Table 2.
- Line 167, Figure 4: Not sure whether it is worth it to show this figure, as they are not your results.
- Line 175: Equation 14 is one-dimensional, while bedload q_b is in the text considered a 2D flux.
- Line 354 & 399: unclear what z^+ is.
- Line 365: “For all 4 simulations, the turbulent Schmidt number was set to values slightly above 1, σ^c ∈ [1.1,1.2]” Why is this done? Is this part of a calibration step that is not described?
- Line 369 talks about “adjusting”; is calibration meant by this?
- Lines 384-385: “Rouse number is equal to 0.5 which corresponds to a highly suspended regime.” Is this still valid concerning the assumption that suspended load does not influence hydrodynamics, as mentioned in line 64?
- Lines 456-461: I don’t understand which numerical schemes are implemented here. I understand the Euler explicit time scheme, but I’m only familiar to the notion of an upwind scheme when dealing with spatial discretisation of advective terms. In conclusion, I do not understand how a temporal scheme and a spatial scheme are compared here. Could you explain further?
- In addition to the point made above; the evaluation of the different numerical schemes is not appropriate within Section 4: Validation.
- Line 482: I am not an expert on suspended sediments, but 132 kg/m3 seems like a very high concentration and seems to violate the assumption of dilute sediments only (line 64). Could you comment on this?
- Section 4.4: The presented cases are all purely deposition cases. The time-step delay in the morphological response (lines 489-491) indicates that mass is not conserved. This is not a problem in the example given, but could you comment on how this influences more realistic cases when hydrodynamic conditions are more complex?
- Section 5.1: A lot of calibration seems to be necessary to get representative results. Especially, the critical shields parameter and the bedload transport formula need to be greatly adapted to get proper model validation. How do you reflect on the validity of the results? Are the results shown essentially a “model overfit” for the specific experiment examined, or can the same parameters also be used to model different experiments in the same flume from Sandoungout (2019)?
- Section 5.1: The results would become a lot stronger if a validation of the hydrodynamics over the dune is done as well.
Technical comments
- General technical 1: please use more comments to improve readability. Examples:
- 408: As stated previously the difference with the pseudo-analytical … -> As stated previously, the difference with the pseudo-analytical …
- 442: Overall, the model fits well with the analytical solution except when the …-> Overall, the model fits well with the analytical solution, except when the
- 592: In addition the critical Shields number …-> In addition, the critical Shields number
- General technical 2: The citation format used (especially the use of first names) is inconsistent and not in line with the journal's expectations.
- Line 32: ALE is mentioned but not what it means
- Line 89: Equation 5 has weird formatting
- Line 109: “… hot topic today …”, informal -> Rephrase
- Line 131 – 131: “2 dimensional” -> “two-dimensional”
- Line 144: “Accurately measuring the threshold of motion is a difficult task mainly because of the absence of a universal definition of the motion threshold.” Vague and unclear -> rephrase
- Line 167: adimensioned -> dimensionless
- Line 172: “eolian”, unclear what this is.
- Line 181: “hard point”, informal and vague -> rephrase
- Line 190: “for a lot of situation”, vague and informal -> rephrase
- Line 192: “to handle out of”, unclear -> rephrase
- Line 238: “split” -> “discretised”
- Line 265-267: “The users have the choice to use either an explicit Euler scheme or a first order Adams-Bashforth scheme, which is a second order time scheme, for temporal discretization.” Confusing -> rephrase
- Line 273-275: “Jacobsen (2015) made a detailed review of the different possible methods to solve the Exner equation and analysed their benefits and shortcomings. He proposed a mass conservative interpolation scheme which is the one implemented in the current model.” Informal, rephrase
- Lines 314-325: Rephrase, informal use of English and many redundant words.
- Line 375 & 381: “hypothesis” -> hypotheses
- Line 398: vary -> varies
- Line 440: “looked for”, rephrase
- Line 442: get -> gets
- Line 464: deltax -> \delta x
- Line 470: “all simulation are stables” -> “all simulations are stable”
- Line 485: “ the sediment bed slopes get important on the extremity of the deposition mound” -> rephrase
Model code and software
sedExnerFoam Matthias Renaud, Cyrille Bonamy, Julien Chauchat https://github.com/Renaud-Matthias/sedExnerFoam
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