=Paper=
{{Paper
|id=Vol-1839/MIT2016-p46
|storemode=property
|title= Numerical simulation of surface waves arising from underwater landslide movement
|pdfUrl=https://ceur-ws.org/Vol-1839/MIT2016-p45.pdf
|volume=Vol-1839
|authors=Yury Zakharov,Anton Zimin
}}
== Numerical simulation of surface waves arising from underwater landslide movement==
https://ceur-ws.org/Vol-1839/MIT2016-p45.pdf
Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
Numerical Simulation of Surface Waves Arising
from Underwater Landslide Movement
Yury Zakharov and Anton Zimin
Kemerovo State University,
Krasnaya Street, 6, 650043 Kemerovo, Russia
zaxarovyn@rambler.ru,sliii@mail.ru
Abstract. The main aim of this paper is to construct a model of simul-
taneous movement of the landslide, internal currents and surface waves
that can come ashore. The idea of a multicomponent fluid movement is
used in this paper. We consider soil, liquid and gas as components of non-
homogeneous fluid. Movement of such fluid is described by the Navier-
Stokes equations with variable density and viscosity and the convection-
diffusion equations. Special ratios are used to calculate the density and
the viscosity of the medium. The results of test calculations for two-
dimensional problems of the wave generation are presented.
Keywords: Navier-Stokes equations, surface wave propagation, land-
slide movement, inhomogeneous fluid, multicomponent fluid
1 Introduction
Tsunami waves generated by underwater and above-water landslides can be dan-
gerous for buildings located on the shore and settled lands. Under natural condi-
tions the underwater landslide is a movement of some soil mass along the slope
of the bottom. Large volumes of moving mass generate surface waves that are
close in their characteristics to the waves generated by tsunamigenic earthquake.
The overview of historical landslides and tsunamis that they generated can be
found in [1,2].
Construction of tsunami wave model generated by landslide movement can
be divided into two tasks: construction of a model of wave propagation on a free
surface and a model of landslide movement on the bottom of water basin.
Free surface of fluid means the border between fluid and gas that is above
it. Due to the fact that fluid density is several times greater than gas density,
influence of gas on the movement of fluid is often neglected, and it is considered
that fluid moves independently of gas movement or, in other words, “freely”.
Models of wave fluid dynamics are the examples of such approach. They
include shallow water theory equations, ideal fluid movement equations, etc [3].
In case of complex wave movements, big splashes and active interaction of two
phases, multicomponent non-homogeneous medium is considered, where fluid
and gas are separate components with their own values of density and viscosity
[4].
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
According to the manner of discretization, mathematical models of multicom-
ponent fluid movement can be divided into Lagrangian and Eulerian. Lagrangian
methods are based on recording the equations of the medium movements in La-
grangian coordinates connected with particles of the moving medium. A set of
nodal points moving together with the medium can be used in order to get
the discrete analogues of such equations. It can be grid points (grid Lagrangian
methods [5,6]) or point particles that are not connected with each other by grid
lines (meshless Lagrangian methods [7-9]). In this approach the position of the
interphase boundary is tracked automatically.
For discretization of medium movement equations in Euler approaches a fixed
computational grid is usually used. At this, the interphase boundary moves on
the grid, and special methods are applied in order to determine its position.
These methods include MAC [10,11], VOF [12,13], Level Set [14,15].
In laboratory studies the underwater landslide can be imitated either by
movement along the slope of a fully submerged solid body [16], or by some
granulated soil slipping down the basin [17].
Several approaches are identified for computational modelling of landslide
movement. It can be a model of movement of an absolutely solid body [18] or an
ensemble of such bodies [19]; or a model of fluid flow that has different density,
viscosity etc. [20].
Until recently, the majority of models applied for modelling of the tsunami
of landslide type, relied on nonlinear shallow water wave theory [21,22]. In many
cases the Boussinesq equations are applied [23]. And also the attempts are made
to apply three-dimensional non-homogeneous models based on Navier-Stokes
equations and the concentration transfer equations [24].
The aim of this paper is to apply the model of three-component viscous in-
compressible fluid with variable viscosity and density and with the presence of
mass diffusion between the components for the problems of occurrence of sur-
face waves generated by landslide movement. Previously two-component model
has been used in problems of substance diffusion in the branched channel [25],
cohesive soil erosion [26] and surface wave propagation [27].
2 Mathematical model
The movement of the medium consisting of three incompressible interfusing fluids
with constant density ρ1 , ρ2 , ρ3 and viscosities µ1 , µ2 , µ3 is considered. Let the
dx
particle of mixture be the solution x = x(t) of the Cauchy problem = V (x, t),
dt
x(0) = x0 , where V (x, t) = (v1 , v2 , v3 ) is a velocity vector of the mixture in point
x = (x1 , x2 , x3 ) and t is a time point. Let C1 (x, t), C2 (x, t), C3 (x, t), µ and ρ
be volume concentrations of the components, dynamic viscosity and mixture
density correspondingly. Components concentrations are interconnected in the
following way:
C1 + C2 + C3 = 1, 0 ≤ Ci ≤ 1. (1)
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
In order to find out the values of viscosity and density of the mixture we
have the following dependence on the concentrations of the components:
µ1 µ2 µ3
µ = ,
C1 µ2 µ3 + C2 µ1 µ3 + C3 µ1 µ2 (2)
ρ=C ρ +C ρ +C ρ .
1 1 2 2 3 3
Mass diffusion occurs between the particles of the mixture according to the
law:
∂ρ
qn = −D , (3)
∂n
where D is the diffusion coefficient.
We consider the mixture components to possess the incompressibility prop-
erty and its interfusion to form an incompressible medium, which density de-
pends only on the concentrations. Let ωt be a control moving volume of such
medium. Value of ωt remains constant due to the incompressibility:
∫
dτ = const.
ωt
From known formula for the time differentiation of the integral taken over
the moving volume [28]
∫ ∫ [ ]
d df
f dτ = + f divV dτ (4)
dt dt
ωt ωt
we obtain the condition of incompressibility:
divV = 0, (5)
d
where dt is total time derivative.
The equations of mass balance for fluid volume ωt considering the presence
of mass diffusion, take the following form:
∫ ∫
d
ρ dτ = − qn dσ, (6)
dt
ωt ∂ωt
where qn is defined in (3).
From (6) taking into account (3) and (5) we get convection- diffusion equation
for density:
dρ
= D∆ρ, (7)
dt
where ∆ is Laplace operator.
(5) and (7) together provide the condition of mass balance conservation in
the medium for three-component incompressible fluid.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
For our objectives we consider three-layered fluid, i.e. we suppose that the
first and the third component do not directly interact in the solution:
{
C3 = 0, C1 ̸= 0,
C1 = 0, C3 ̸= 0.
Then the density equation (2) will be the following:
{
C1 ρ1 + C2 ρ2 , C3 = 0,
ρ=
C2 ρ2 + C3 ρ3 , C1 = 0.
Or considering (1)
{
C1 ρ1 + (1 − C1 ) ρ2 , C3 = 0,
ρ= (8)
C3 ρ3 + (1 − C3 ) ρ2 , C1 = 0.
The diffusion coefficient D can be also expressed as:
{
D = D1 , C3 = 0,
(9)
D = D3 , C1 = 0.
Then (7) can be transformed into the following equations for the component
concentrations:
dC
1 = D1 ∆C1 ,
C3 = 0,
dt
dC3 (10)
= D3 ∆C3 , C1 = 0,
dt
C2 = 1 − C1 − C3 .
From the integral momentum equation
∫ ∫ ∫
d
ρV dx = Pn dσ + ρF dx (11)
dt
ωt ∂ωt ωt
considering (4) and (5) we get:
( )
d ρV
= divP + ρF , (12)
dt
where P is the stress tensor in the mixture, F = (f1 , f2 , f3 ) is the vector of mass
forces.
Then, considering variable viscosity, we obtain a system of equations for the
motion of the mixture of three viscous incompressible interfusing fluids:
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
( )
d ρV
= −∇p + divI + ρF ,
dt
divV = 0,
dC1
= D1 ∆C1 ,
dt
dC3 (13)
= D3 ∆C3 ,
dt
C2 = 1 − C1 − C3 ,
µ1 µ2 µ3
µ= ,
C µ µ
1 2 3 + C 2 µ1 µ3 + C 3 µ1 µ2
ρ = ρ1 C1 + ρ2 C2 + ρ3 C3 .
where p is pressure in the mixture, I = µ∆V + (∇µ · ∇) V + (∇µ · JV ) is viscous
part of stress tensor, JV is Jacobian matrix, arranged as follows:
∂v ∂v ∂v
1 1 1
∂x ∂x ∂x
∂v1 ∂v2 ∂v3
JV = 2
∂x1 ∂x2 ∂x3 .
2 2
∂v3 ∂v3 ∂v3
∂x1 ∂x2 ∂x3
Thus, the given model consists of the convection-diffusion equations for con-
centration of the components, correlations for the determination of density and
viscosity, and also hydrodynamic Navier-Stokes equations for incompressible vis-
cous fluid.
We use a no-slip condition on the solid wall and boundary conditions of the
second kind for concentration equations. Some initial distribution for concentra-
tions is also given.
3 Solution scheme
For discretization of the system (13) in the spacial variables is used a finite-
difference method on a rectangular uniform grid with staggered arrangement
of nodes [29]: pressure, velocity divergence and component concentration are
determined in the centers of cells; velocity vector components are determined on
the borders of cells.
Application of the staggered grid allows to link speed and pressure values in
the adjacent nodes and avoid the appearance of oscillations in solution, which
arise when using central differences on a combined grid. Also, a staggered ar-
rangement of the nodes automatically allows to satisfy the discrete representa-
tion of the continuity equation.
Time motion algorithm consists of the following stages:
1. By taking into account the known velocity vectors and concentration distri-
bution (and thus density and viscosity values), a time step for the Navier-
Stokes equations system is made.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
2. Using the received values of velocity component, a time step for the “soil”
convection-diffusion equation is made.
3. Using the received values of velocity component, a time step for the “air”
convection-diffusion equation is made
4. Knowing the distribution for “air” and “soil” concentrations, a value of “wa-
ter” concentration is calculated according to the formula (1).
5. Recalculation of density and viscosity values in the medium is carried out
according to the formulae (2). After that follows a transition to the first step
of the next time iteration.
To solve the Navier-Stokes equations system there is used a splitting scheme
on physical factors [30] with regard to variable density. It consists of three steps.
At the first step a momentum transfer is carried out due to convection and dif-
fusion; intermediate velocity field is calculated according to an implicit scheme:
Ṽ − V n
= − (V n · ∇) Ṽ +
∆t (14)
1( )
+ −V n D∆ρ + µ∆Ṽ + (∇µ · ∇) Ṽ + (∇µ · JṼ ) + F ,
ρ
In order to solve the system (14) a prediction-correction method is used [31].
The obtained system of algebraic equations is solved by sweep method.
At this, despite the fact that the obtained intermediate velocity field Ṽ does
not satisfy continuity equation, it has a physical meaning because it preserves
vortex characteristics in internal points.
At the second step, with regard to (5) and variable density, the pressure field
is calculated according to the obtained intermediate velocity field Ṽ :
∑ ∂ ( 1 ∂pn+1 ) ∇Ṽ
= . (15)
i
∂xi ρ ∂xi ∆t
Solution of the system of equations obtained as a result of discretization
of the equation in order to find pressure in (15) is one of the most important
and dominant aspects of computational procedure from the viewpoint of machine
resources expenses. Operator of this system can have a complex structure, which
complicates the task significantly. To solve this stage of computational process
a gradient iterative method BiCGStab [32] is used.
At the third step the transfer of momentum is carried out only due to pressure
gradient:
V n+1 − Ṽ 1
= − ∇pn+1 (16)
∆t ρ
The equation (15) obtained by taking the divergence of both sides of equation
(16) with regard to ∇V n+1 = 0.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
To solve the convection-diffusion equations (10) a prediction-correction scheme
with approximation of convective components against the stream is used [31].
The obtained system of algebraic equations is solved by sweep method.
The numerical scheme has first-order temporal and spatial approximations.
4 Results
4.1 Collapse of viscous soil
There was considered a problem of collapse of viscous and stiff soil on the bot-
tom of reservoir that generates waves on the surface of fluid. Here one of the
components (more stiff and viscous) models the behavior of bottom soil, another
one – liquid, and the third one – aerial environment. Fig. 1 shows the geometry
of area and the scheme of component arrangement.
Fig. 1. Geometry and initial distribution of components.
At the initial time the half circle of the wet soil is located at the center of
the area. As time passes, it caves under the influence of gravity F = (0, −9.8) sm2
and causes the movement of the entire medium. Fig. 2 shows the results of
the calculation for various time points. Parts of the bottom soil spread in the
opposite directions, then reflected from the side walls and connected again in the
center of the area. The liquid phase surface followed the bottom soil movements.
Fig. 2. Picture of medium motion for various time points t = 0.9 s, 1.9 s, 2.7 s, 5.4 s.
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
The following hydrodynamic parameters were chosen: µ1 = 10 P a · s, µ2 =
kg kg kg
10−3 P a · s, µ3 = 10−5 P a · s and ρ1 = 3000 m 3 , ρ2 = 1000 m3 , ρ3 = 1 m3
for the soil, liquid and gas phases. The following grid parameters and time step
were used: hx = 5 · 10−2 m, hy = 5 · 10−2 m, τ = 10−2 s. All the area walls
are solid except the upper one. The atmosphere pressure Patm = 101325 P a is
indicated at the top. We consider the boundary between components to take
place at C = 0.4.
The calculation demonstrates possibility of the model to simulate direct in-
teraction between the bottom soil and the surface waves without distinguishing
characteristics at the phases boundaries.
4.2 Soil movement on the inclined bottom
A problem of the soil slip movement on the inclined bottom that generates waves
on the surface of fluid was considered. The scheme of area was taken from [33]
(see Fig. 3). The result of calculation was compared with one of the numerical
model presented in [33] and with laboratory experiment carried out in [34].
Fig. 3. Geometry and initial condition.
At the initial time the wet soil is located on the left side of the inclined
bottom. As time passes, it rolls down by gravity F = (0, −9.8) sm2 and causes
the movement of the entire structure, simulating the soil slip movement. The
kg
hydrodynamic parameters were used the same as in [33]: µ1 = 10 m 3 , µ2 =
−3 kg −5 kg kg kg kg
10 m3 , µ3 = 10 m3 and ρ1 = 1950 m3 , ρ2 = 1000 m3 , ρ3 = 1 m3 for the
soil, liquid and gas phases. The following grid parameters and time step were
used: hx = 5 · 10−2 m, hy = 5 · 10−2 m, τ = 10−2 s. All the area walls are solid
except the upper one. The atmosphere pressure Patm = 101325 P a is indicated
at the top. We consider the boundary between components to take place at
C = 0.4. Fig. 4 shows the results of the calculation for various time points.
Fig. 5 shows the comparison of water surface elevations with results obtained
in [33] and [34].
Calculation demonstrates good agreement with MM3 [33]. This approach
produces similar waveforms and slide deformation geometries. However, there
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
Fig. 4. Picture of medium motion for various time points t = 0.0 s, 0.4 s, 0.8 s, 1.0 s.
0.10
0.05
Free surface height (m)
0.00
-0.05
-0.10
Model presented here
MM3 (Smith et al., 2016)
-0.15
Experiment (Assier-
Rzadkiewicz et al., 1997)
-0.20
-0.5 0 0.5 1 1.5 2 2.5
Distance (m)
0.10
0.05
Free surface height (m)
0.00
-0.05
-0.10
Model presented here
MM3 (Smith et al., 2016)
-0.15
Experiment (Assier-
Rzadkiewicz et al., 1997)
-0.20
-0.5 0 0.5 1 1.5 2 2.5
Distance (m)
Fig. 5. A comparison of water surface elevations for various time points t = 0.4 s, 0.8 s.
Given model (solid red), MM3 (dashed blue, [33]) and experiment (dotted green, [34]).
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Mathematical and Information Technologies, MIT-2016 — Mathematical modeling
is a discrepancy in the fields of high gradients, what can be explained by the
form of equation (2) for µ. Differences do not exceed 12% on the whole surface
for t = 0.4 s; 20% for t = 0.8 s. The surface shape in numerical simulation is
qualitatively similar to that observed in the laboratory experiment. Also it has
a slightly better agreement with experiment surface then MM3.
5 Conclusion
Presented model of three-component viscous incompressible fluid was used for
modeling simultaneous movement of the landslide, internal currents and surface
waves. The advantage of this approach is that it allows one to simulate the
complex phenomenon of the interaction of waves and bottom soil using a uniform
numerical algorithm for a number of different problems without distinguishing
characteristics at the phases boundaries.
Test calculations for two-dimensional problems of the wave emergence and
propagation on the free surface were carried out. The results obtained show the
possibility of modeling such a phenomenon. Agreement with other model and
experiment was demonstrated.
Acknowledgments. The work was carried out with support of state task of
Ministry of Science and Education, project number 1.630.2014/K.
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