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One-dimensional numerical modelling of dam-break waves over movable beds: application to experimental and field cases

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Abstract

This paper reports a numerical study on dam-break waves over movable beds. A one-dimensional (1-D) model is built upon the Saint-Venant equations for shallow water waves, the Exner equation of sediment mass conservation and a spatial lag equation for non-equilibrium sediment transport. The set of governing equations is solved using an explicit finite difference scheme. The model is tested in various idealized experimental cases, with fairly good agreement between the numerical predictions and measurements. Discrepancies are observed at the earlier stage of the dam-break wave and around the dam location due to no vertical velocity component being taken into account. Sensitivity tests confirm that the friction coefficient is an important parameter for the evaluation of sediment transport processes operating during a dam-break wave. The influence of the non-equilibrium adaptation length (or the lag distance) is negligible on the wavefront celerity and weak on the free surface and bed profiles, which indicates that one may ignore the spatial lag effect in dam-break wave studies. Finally, the simulation of the Lake Ha!Ha! dyke-break flood event shows that the model can provide relevant results if a convenient formula for computing the sediment transport capacity and an appropriate median grain diameter of riverbed material are selected.

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Abbreviations

A :

Wetted area

A b :

Cross-sectional area

B :

Channel width at the free surface

BSS :

Brier Skill Score (Eq. 19)

C :

Conveyance (Eq. 4)

C r :

Courant number

c :

Celerity= \(\sqrt{gA/B}\)

D char :

Lag distance

D 50 :

Particle size for which 50% of the sediment is finer by weight).

F :

Flux vector= \(\left[{\begin{array}{l} Q\\ \frac{Q^{2}}{A}+g I_1 \end{array}}\right]\)

g :

Gravitational acceleration

h :

Flow depth

h0,  hw,u,  hs,u:

Water head, flow depth and initial bottom step in the upstream reach

h w,d :

Flow depth in the downstream reach

I 1 :

Hydrostatic pressure force (Eq. 3a)

I 2 :

Pressure force due to longitudinal width variation (Eq. 3b)

J :

Energy slope

K :

Manning–Strickler coefficient for flow resistance calculations (Eq. 4)

p :

Porosity of bed deposit

Q :

Flow discharge

Q s :

Volumetric sediment discharge

\(Q_s^{cap}\) :

Volumetric sediment transport capacity discharge

\(Q_s^{dep},\, Q_s^{ero},\, Q_s^{I_n},\, Q_s^{O_u} \cdot Q_s^{tra}\) :

Sediment fluxes (Eqs. 8 to 10)

R h :

Hydraulic radius

S :

Source term= \(\left[{\begin{array}{l} 0\\ -gA\frac{\partial z_b}{\partial x}-gA \frac{Q\left| Q \right|}{C^{2}} +g I_2 \end{array}}\right]\)

t :

Time

t 0 :

Characteristic time for dam-break wave propagation = \(\sqrt{h_0/g}\)

U :

Conservative hydraulic variables= \(\left[{\begin{array}{l} A\\Q \end{array}}\right]\)

V b :

Bedload velocity assumed equal to the mean flow velocity

x :

Longitudinal distance

z b :

Bed elevation

ΔA b :

Change in the cross-sectional area due to deposition or scour

Δt :

Time step

Δx :

Space step

\(\delta_i^n\) :

Slope of Q (or A) defined by Eq. 12

\(\delta_{i(1)}^n,\, \delta_{i(2)}^n, \,\delta_{i(3)}^n\) :

Parameters in Eq. 12

κ :

Parameter in Eq. 12

ρ :

Density of water

ρ s :

Density of the sediment

τ :

Bottom shear stress = ρ gRJ

τ 0 :

Shields number= \(\frac{\tau}{g({\rho_s -\rho})D_{50}}.\)

τ 0,cr :

Critical Shields number

χ :

Parameter in Eq. 12

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Authors and Affiliations

  1. Cemagref, Hydrology-Hydraulics Research Unit, 3 bis quai Chauveau CP220, 69336, Lyon, France

    Kamal El Kadi Abderrezzak & André Paquier

  2. Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, 69622, Villeurbanne, France

    Bernard Gay

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  1. Kamal El Kadi Abderrezzak
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Correspondence to Kamal El Kadi Abderrezzak.

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El Kadi Abderrezzak, K., Paquier, A. & Gay, B. One-dimensional numerical modelling of dam-break waves over movable beds: application to experimental and field cases. Environ Fluid Mech 8, 169–198 (2008). https://doi.org/10.1007/s10652-008-9056-9

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