Abstract
This paper presents a new landslide-generated wave (LGW) model based on incompressible Euler equations with Savage-Hutter assumptions. A two-layer model is developed including a layer of granular-type flow beneath a layer of an inviscid fluid. Landslide is modeled as a two-phase Coulomb mixture. A well-balanced second-order finite volume formulation is applied to solve the model equations. Wet/dry transitions are treated properly using a modified non-linear method. The numerical model is validated using two sets of experimental data on subaerial and submarine LGWs. Impulsive wave characteristics and landslide deformations are estimated with a computational error less than 5 %. Then, the model is applied to investigate the effects of landslide deformations on water surface fluctuations in comparison with a simpler model considering a rigid landslide. The model results confirm the importance of both rheological behavior and two-phase nature of landslide in proper estimation of generated wave properties and formation patterns. Rigid slide modeling often overestimates the characteristics of induced waves. With a proper rheological model for landslide, the numerical prediction of LGWs gets more than 30 % closer to experimental measurements. Single-phase landslide results in relative errors up to about 30 % for maximum positive and about 70 % for maximum negative wave amplitudes. Two-phase constitutive structure of landslide has also strong effects on landslide deformations, velocities, elongations, and traveling distances. The complex behaviors of landslide and LGW of the experimental data are analyzed and described with the aid of the robust and accurate finite volume model. This can provide benchmark data for testing other numerical methods and models.
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Abbreviations
- A :
-
Coefficient matrix
- a p :
-
Wave positive amplitude
- a n :
-
Wave Negative amplitude
- B :
-
Coupling term matrix
- b :
-
Bottom level
- c :
-
Characteristic wave velocity
- D :
-
Diagonal matrix of eigenvalues
- df :
-
Generalized Roe flux difference matrix
- Err :
-
Relative error
- Err c :
-
Computational error
- F :
-
Numerical flux matrix
- G 1 :
-
Source term matrix of bed level
- G 2 :
-
Source term matrix of bed curvature/coupling term
- \( \overrightarrow{g} \) :
-
Gravitational acceleration vector
- g :
-
Gravitational acceleration
- H :
-
Flow thickness normal to the bed
- H′ :
-
Characteristic depth
- h :
-
Flow thickness h′/cos2 θ
- h 0 :
-
Still water depth
- h 0C :
-
Landslide initial distance from still water surface
- h′ :
-
Flow depth
- I :
-
Computational cell
- J :
-
Jacobean of transformation matrix
- K :
-
Earth pressure coefficient
- κ :
-
Local eigenvector matrix
- L :
-
Characteristic length
- m :
-
Number of computational cells
- n :
-
Number of time steps
- n s :
-
Unit normal vector of flow surface
- n b :
-
Unit normal vector of bottom
- n i :
-
Unit normal vector of interface
- P :
-
Pressure tensor
- P XX :
-
Normal pressure along X
- P ZZ :
-
Normal pressure along Z
- P ZX :
-
Longitudinal stress along X
- P XZ :
-
Longitudinal stress along Z
- P xx :
-
Normal pressure along x
- P zz :
-
Normal pressure along z
- P zx :
-
Longitudinal stress along x
- P xz :
-
Longitudinal stress along z
- P 1 :
-
Roe correction matrix κ|D|κ − 1
- P 2 :
-
Projection matrixes κ sgn(D)κ − 1
- P s :
-
Pressure tensor of solid phase
- P f :
-
Pressure tensor of fluid phase
- q :
-
Flow discharge hu
- r :
-
Density ratio ρ 1/ρ 2
- r c :
-
Computational speed Δt/Δx
- S :
-
Water surface level normal to the bed
- S 1 :
-
Source term matrix related to bed level
- S 2 :
-
Source term matrix of bed curvature/coupling term
- S 3 :
-
First numerical θ related part of the flux term matrix
- S 4 :
-
Second numerical θ related part of the flux term matrix
- T :
-
Coulomb friction matrix
- T s :
-
Wave period
- t :
-
Time
- U :
-
Flow velocity parallel to the bottom
- U b :
-
Sliding velocity along bottom
- u :
-
Horizontal flow velocity
- ū :
-
Roe-averaged velocity
- V :
-
Flow velocity perpendicular to the bottom
- V′ :
-
Flow velocity vector (u,v)
- v :
-
Vertical flow velocity
- W :
-
Unknown matrix
- W* :
-
Predicted state values in the first step
- X :
-
Local coordinate component along non-erodible bed
- \( \overrightarrow{X} \) :
-
Cartesian coordinate vector (x,z)
- \( {\overrightarrow{X}}^{\prime } \) :
-
Local coordinate vector (X,Z)
- x :
-
Horizontal component of Cartesian coordinate system
- Z :
-
Local coordinate component perpendicular to the bed
- z :
-
Vertical component of Cartesian coordinate system
- ρ 1 :
-
Water density
- ρ 2 :
-
Coulomb mixture density
- ρ a :
-
Air density
- ρ s :
-
Solid grain density
- ψ 0 :
-
Porosity
- θ :
-
Local bed slope angle
- δ :
-
Basal friction angle
- δ 0 :
-
Angle of repose
- δ mod :
-
Modified basal friction angle
- φ :
-
Internal friction angle of granular material
- ε :
-
Small parameter of dimensional analysis H′/L
- ℑ :
-
Coulomb friction term
- σ c :
-
Critical friction resistance of the bottom
- Λ 1 :
-
Numerical coefficient
- Λ 2 :
-
Numerical coefficient
- λ :
-
Local eigenvalue
- λ 1 :
-
Pressure distributing factor at interface
- λ 2 :
-
Pressure distributing factor along the second layer
- λ′ :
-
An empirical coefficient
- μ 1 :
-
Factor = 1
- μ 2 :
-
Factor = tanδ 0
- Λ f :
-
An empirical factor
- γ :
-
A small parameter ∈ (0, 1)
- τ crit :
-
Critical longitudinal stress of the bottom
- η :
-
Water surface wave height
- Δx :
-
Computational cell size
- Δt :
-
Computational time step
- ∇:
-
Gradient vector (∂/∂x, ∂/∂z)
- \( \tilde{.} \) :
-
Dimensionless parameters
- \( \overset{\smile }{.} \) :
-
Depth-averaged parameters
- . *:
-
Predicted values at the first step
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Acknowledgments
The authors wish to thank the Editor-in-Chief Professor Sassa for his thoughtful comments and also two anonymous reviewers for their valuable comments which helped to improve the final manuscript. The first author is grateful for the support of Civil and Environmental Engineering Department of University of California, Irvine, especially for valuable knowledge of Professor B.F. Sanders during her research visit. Authors appreciate Sharif University of Technology, Dr. A. Najafi-Jilani, and Mr. A. Nik-Khah contributions during experimental works.
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Yavari-Ramshe, S., Ataie-Ashtiani, B. A rigorous finite volume model to simulate subaerial and submarine landslide-generated waves. Landslides 14, 203–221 (2017). https://doi.org/10.1007/s10346-015-0662-6
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DOI: https://doi.org/10.1007/s10346-015-0662-6