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A Coupled Hydro-Morphodynamic Model for Sediment Transport using the Moment Approach

Afroja Parvin, Giovanni Samaey, Julian Koellermeier

Abstract

Sediment transport is crucial in the hydro-morphodynamic evolution of free surface flows in shallow water environments, which is typically modeled under the shallow water assumption. In classical shallow water modeling for sediment transport, the vertical structure of the flow is collapsed into a depth-averaged and near-bed velocity, usually reconstructed empirically, e.g., using a parameterized logarithmic profile. In practice, large variations from such empirical profiles can occur. It is therefore essential to resolve the vertical structure of the velocity profile within the shallow water framework to better approximate near-bed velocity. This study introduces a model and simulations that incorporate vertical velocity variations and bottom erosion-deposition effects in sediment transport, providing a computationally efficient framework for predicting sediment dynamics in shallow water environments. We employ the so-called moment model approach for the velocity variation, which considers a polynomial expansion of the horizontal velocity in the scaled vertical direction. This allows the use of a complex velocity profile with an extended set of variables determined by the polynomial basis coefficients, resolving the vertical structure as part of the solution. The extended model comprises four components: (1) the standard shallow water equations; (2) moment equations governing evolution of the basis coefficients; (3) an evolution equation for sediment concentration; and (4) a transport equation for the bed. This enables a coupled model for bedload and suspended load transport. We use a hyperbolic regularization technique to ensure model stability and realistic eigenvalues. Several numerical tests, including dam-break cases with and without wet/dry fronts, validate our results against laboratory data.

A Coupled Hydro-Morphodynamic Model for Sediment Transport using the Moment Approach

Abstract

Sediment transport is crucial in the hydro-morphodynamic evolution of free surface flows in shallow water environments, which is typically modeled under the shallow water assumption. In classical shallow water modeling for sediment transport, the vertical structure of the flow is collapsed into a depth-averaged and near-bed velocity, usually reconstructed empirically, e.g., using a parameterized logarithmic profile. In practice, large variations from such empirical profiles can occur. It is therefore essential to resolve the vertical structure of the velocity profile within the shallow water framework to better approximate near-bed velocity. This study introduces a model and simulations that incorporate vertical velocity variations and bottom erosion-deposition effects in sediment transport, providing a computationally efficient framework for predicting sediment dynamics in shallow water environments. We employ the so-called moment model approach for the velocity variation, which considers a polynomial expansion of the horizontal velocity in the scaled vertical direction. This allows the use of a complex velocity profile with an extended set of variables determined by the polynomial basis coefficients, resolving the vertical structure as part of the solution. The extended model comprises four components: (1) the standard shallow water equations; (2) moment equations governing evolution of the basis coefficients; (3) an evolution equation for sediment concentration; and (4) a transport equation for the bed. This enables a coupled model for bedload and suspended load transport. We use a hyperbolic regularization technique to ensure model stability and realistic eigenvalues. Several numerical tests, including dam-break cases with and without wet/dry fronts, validate our results against laboratory data.

Paper Structure

This paper contains 33 sections, 2 theorems, 116 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Shallow water moment model for sediment transport
  3. Hydrodynamic equations
  4. Mapped reference system and moment expansion
  5. Depth-averaging the mass balance
  6. Depth-averaging the momentum equation
  7. Higher-order moments of the momentum equation
  8. Morphodynamic equations
  9. Depth-averaged volumetric sediment concentration
  10. Bedload mass balance equation
  11. Morphological conditions
  12. Coupled hydro-morphodynamic model
  13. Lower-order moments of SWEMED
  14. SWEMED0
  15. SWEMED1
  16. ...and 18 more sections

Key Result

Theorem 3.1

The transport matrix $\tilde{A}_{ED}(W) \in \mathbb{R}^{(N+4)\times(N+4)}$ of HSWEMED has the following characteristic polynomial with $\chi_{A_{2}}(\lambda-u_m)= det \left(A_{2}- \left(\lambda-u_m \right)I \right)$ where $I \in \mathbb{R}^{N\times N}$ is a unit matrix and $A_{2} \in \mathbb{R}^{N \times N}$ is defined as follows with values $c_i = \frac{i+1}{2i+1}\alpha_1$ and $a_i = \frac{i-1}

Figures (7)

  • Figure 1: Sketch of sediment transport dynamics, illustrating bedload movement along the bed, suspended load within the water column, and zones of erosion and deposition controlled by variations in flow velocity and shear stress.
  • Figure 2: The mapping from physical $z-$space to transformed $\zeta-$space CiCP-25-669.
  • Figure 4: \ref{['academic dam-break']}: (Left) Free surface $h+h_b$ & sediment bottom evolution $h_b$, and vertical profiles of velocity $u$ (black lines) at model points $x=-2,\,-1,\,0,\,1,\,2,\,3$ and (right) volumetric sediment concentration $c$ in the suspension at $t=1,$ computed with the HSWEMED (solid), the SWEED (dash-dotted), and the HSWEM (dotted).
  • Figure 6: \ref{['exp_1']}: Free surface & bottom evolution at time $t=1$ computed with the HSWEMED (solid), the SWEED (dashed), and the HSWEM (dotted). Results are compared with experimental data (dash-dotted) for the bed materials (left) PVC pellets and (right) uniform coarse sand. (bottom) Volumetric sediment concentration in the suspension at $t=1,$ computed with HSWEMED (solid) and the SWEED (dashed) for PVC and sand bed.
  • Figure 8: \ref{['exp_2']}: Free surface & bottom evolution at time $t=1$ computed with the HSWEMED (solid), the SWEED (dashed), and the HSWEM (dotted). Results are compared with experimental data (dash-dotted) for the bed materials (left) PVC pellets and (right) uniform coarse sand. (bottom) Volumetric sediment concentration in the suspension at $t=1,$ computed with the HSWEMED (solid) and the SWEED (dashed) for PVC and sand bed.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 3.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem C.1