Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models
This study presents, for the first time, a detailed linear stability analysis (LSA) of bedform evolution under low-flow conditions using a one-dimensional vertically averaged and moment (1D-VAM) approach. The analysis focuses exclusively on bedload transport. The classical Saint-Venant shallow water...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/12/1997 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849432287441059840 |
|---|---|
| author | Mohamed Hassan Elgamal Mohd Aamir Mumtaz |
| author_facet | Mohamed Hassan Elgamal Mohd Aamir Mumtaz |
| author_sort | Mohamed Hassan Elgamal |
| collection | DOAJ |
| description | This study presents, for the first time, a detailed linear stability analysis (LSA) of bedform evolution under low-flow conditions using a one-dimensional vertically averaged and moment (1D-VAM) approach. The analysis focuses exclusively on bedload transport. The classical Saint-Venant shallow water equations are extended to incorporate non-hydrostatic pressure terms and a modified moment-based Chézy resistance formulation is adopted that links bed shear stress to both the depth-averaged velocity and its first moment (near-bed velocity). Applying a small-amplitude perturbation analysis to an initially flat bed, while neglecting suspended load and bed slope effects, reveals two distinct modes of morphological instability under low-Froude-number conditions. The first mode, associated with ripple formation, features short wavelengths independent of flow depth, following the relation <i>F<sup>2</sup> = 1</i>/<i>(kh</i>), and varies systematically with both the Froude and Shields numbers. The second mode corresponds to dune formation, emerging within a dimensionless wavenumber range of 0.17 to 0.9 as roughness increases and the dimensionless Chézy coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></semantics></math></inline-formula> decreases from 20 to 10. The resulting predictions of the dominant wavenumbers agree well with recent experimental observations. Critically, the model naturally produces a phase lag between sediment transport and bedform geometry without empirical lag terms. The 1D-VAM framework with Exner equation offers a physically consistent and computationally efficient tool for predicting bedform instabilities in erodible channels. This study advances the capability of conventional depth-averaged models to simulate complex bedform evolution processes. |
| format | Article |
| id | doaj-art-409ed76fc00a4f28ad7f3afc5995b8c0 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-409ed76fc00a4f28ad7f3afc5995b8c02025-08-20T03:27:24ZengMDPI AGMathematics2227-73902025-06-011312199710.3390/math13121997Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) ModelsMohamed Hassan Elgamal0Mohd Aamir Mumtaz1Civil Engineering Department, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaCivil Engineering Department, College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaThis study presents, for the first time, a detailed linear stability analysis (LSA) of bedform evolution under low-flow conditions using a one-dimensional vertically averaged and moment (1D-VAM) approach. The analysis focuses exclusively on bedload transport. The classical Saint-Venant shallow water equations are extended to incorporate non-hydrostatic pressure terms and a modified moment-based Chézy resistance formulation is adopted that links bed shear stress to both the depth-averaged velocity and its first moment (near-bed velocity). Applying a small-amplitude perturbation analysis to an initially flat bed, while neglecting suspended load and bed slope effects, reveals two distinct modes of morphological instability under low-Froude-number conditions. The first mode, associated with ripple formation, features short wavelengths independent of flow depth, following the relation <i>F<sup>2</sup> = 1</i>/<i>(kh</i>), and varies systematically with both the Froude and Shields numbers. The second mode corresponds to dune formation, emerging within a dimensionless wavenumber range of 0.17 to 0.9 as roughness increases and the dimensionless Chézy coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></semantics></math></inline-formula> decreases from 20 to 10. The resulting predictions of the dominant wavenumbers agree well with recent experimental observations. Critically, the model naturally produces a phase lag between sediment transport and bedform geometry without empirical lag terms. The 1D-VAM framework with Exner equation offers a physically consistent and computationally efficient tool for predicting bedform instabilities in erodible channels. This study advances the capability of conventional depth-averaged models to simulate complex bedform evolution processes.https://www.mdpi.com/2227-7390/13/12/1997stability analysis diagramevolution of bedformslow-regime bedformsvertically averaged and moment modelsvelocity over varied topographymoment-based Chezy formula |
| spellingShingle | Mohamed Hassan Elgamal Mohd Aamir Mumtaz Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models Mathematics stability analysis diagram evolution of bedforms low-regime bedforms vertically averaged and moment models velocity over varied topography moment-based Chezy formula |
| title | Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models |
| title_full | Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models |
| title_fullStr | Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models |
| title_full_unstemmed | Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models |
| title_short | Stability Diagrams of Bed Evolution for Vertically Averaged and Moment (VAM) Models |
| title_sort | stability diagrams of bed evolution for vertically averaged and moment vam models |
| topic | stability analysis diagram evolution of bedforms low-regime bedforms vertically averaged and moment models velocity over varied topography moment-based Chezy formula |
| url | https://www.mdpi.com/2227-7390/13/12/1997 |
| work_keys_str_mv | AT mohamedhassanelgamal stabilitydiagramsofbedevolutionforverticallyaveragedandmomentvammodels AT mohdaamirmumtaz stabilitydiagramsofbedevolutionforverticallyaveragedandmomentvammodels |