The main aim of this paper is to construct a model of simultaneous 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 nonhomogeneous fluid. Movement of such fluid is described by the NavierStokes equations with variable density and viscosity and the convectiondifusion equations. Special ratios are used to calculate the density and the viscosity of the medium. The results of test calculations for twodimensional problems of the wave generation are presented.
gerous for buildings located on the shore and settled lands. Under natural conditions 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.
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.
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”.
include shallow water theory equations, ideal fluid movement equations, etc [
According to the manner of discretization, mathematical models of
multicomponent fluid movement can be divided into Lagrangian and Eulerian. Lagrangian
methods are based on recording the equations of the medium movements in
Lagrangian 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 [
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 [
of landslide type, relied on nonlinear shallow water wave theory [
The aim of this paper is to apply the model of three-component viscous
incompressible fluid with variable viscosity and density and with the presence of
mass difusion between the components for the problems of occurrence of
surface waves generated by landslide movement. Previously two-component model
has been used in problems of substance difusion in the branched channel [
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)
μ = ,
where D is the difusion coeficient.
We consider the mixture components to possess the incompressibility property and its interfusion to form an incompressible medium, which density depends only on the concentrations. Let ωt be a control moving volume of such medium. Value of ωt remains constant due to the incompressibility: ∂ρ qn = −D ∂n
, ∫
divV = 0, where ddt is total time derivative.
The equations of mass balance for fluid volume ωt considering the presence of mass difusion, take the following form: (2) (3) (4) (5) (6) (7)
From known formula for the time diferentiation of the integral taken over
the moving volume [
where ∆ is Laplace operator.
(5) and (7) together provide the condition of mass balance conservation in the medium for three-component incompressible fluid.
ρ = { C1ρ1 + C2ρ2, C3 = 0,
C2ρ2 + C3ρ3, C1 = 0. ρ =
{ C3 = 0, C1 ̸= 0, C1 = 0, C3 ̸= 0.
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: (8) (9) (10) (11) (12)
= −∇p + divI + ρF , d ( ρV ) dt divV = 0, dC1 = D1∆C 1, dt
dC3 = D3∆C 3, dt C2 = 1 − C1 − C3, μ = μ1μ2μ3 C1μ2μ3 + C2μ1μ3 + C3μ1μ2 ρ = ρ1C1 + ρ2C2 + ρ3C3.
, (13) 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: ∂v1 ∂v1 ∂v1
∂x1 ∂x2 ∂x3 JV = ∂∂xv21 ∂∂xv22 ∂∂xv23 .
∂v3 ∂v3 ∂v3
∂x1 ∂x2 ∂x3
3
diference method on a rectangular uniform grid with staggered arrangement
of nodes [
the adjacent nodes and avoid the appearance of oscillations in solution, which arise when using central diferences on a combined grid. Also, a staggered arrangement of the nodes automatically allows to satisfy the discrete representation of the continuity equation.
bution (and thus density and viscosity values), a time step for the Navier
2. Using the received values of velocity component, a time step for the “soil” convection-difusion equation is made. 3. Using the received values of velocity component, a time step for the “air” convection-difusion equation is made
ter” concentration is calculated according to the formula (1).
on physical factors [
V˜ − V n ∆t = − (V n · ∇ ) V˜ +
1 ( + ρ −V nD∆ρ + μ∆ V˜ + (∇μ · ∇ ) V˜ + (∇μ · J V˜ )) + F ,
At this, despite the fact that the obtained intermediate velocity field V˜ 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 V˜ : ∑ i ∂ ( 1 ∂pn+1 ) ∂xi ρ ∂xi = ∇ V˜ .
∆t
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 [
V n+1 − V˜ ∆t 1 = − ρ ∇pn+1
(16) with regard to ∇V n+1 = 0. (14) (15) (16)
with approximation of convective components against the stream is used [
4
tom of reservoir that generates waves on the surface of fluid. Here one of the components (more stif 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.
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. The following hydrodynamic parameters were chosen: μ1 = 10 P a · s, μ2 = 10−3 P a · s, μ3 = 10−5 P a · s and ρ1 = 3000 mkg3 , ρ2 = 1000 mkg3 , ρ3 = 1 kg 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.
4.2
Soil movement on the inclined bottom
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
hydrodynamic parameters were used the same as in [
Fig. 5 shows the comparison of water surface elevations with results obtained
in [
is a discrepancy in the fields of high gradients, what can be explained by the form of equation (2) for μ. Diferences 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
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 diferent problems without distinguishing characteristics at the phases boundaries.