Landslides and Engineered Slopes: Protecting Society through Improved Understanding – Eberhardt et al. (eds) © 2012 Taylor & Francis Group, London, ISBN 978-0-415-62123-6
A two-phase model for dry density-varying granular flows
over general topography
Y.C. Tai
Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Taiwan
C.Y. Kuo
Research Center for Applied Science, Academia Sinica, Taiwan
L.T. Shen & S.S. Hsiau
Department of Mechanical Engineering, National Central University, Taiwan
ABSTRACT: Landslides, debris flows or granular avalanches are comprised of mixtures of grain-particles (sand, gravel and rocks) and an interstitial fluid (water or air). This work is concerned with an application of two-phase model for avalanches, debris flows or landslides down arbitrary natural topographic terrains of shallow curvatures, in which the moving mass is treated with air as the interstitial fluid. At the flow surface, air entrainment/extrusion is considered. The rate of entrainment is assumed to depend on the divergent or convergent behavior of the solid constituent. A Non-Oscillatory Central (NOC) scheme with high order cell reconstruction is implemented for numerical investigation. The key features and the capability of the proposed model are illustrated by numerical simulations of landslides over non-trivial topography.
system for two-dimensional basal surface, and its extension for three dimensional topography is available by Tai et al. (2011). However, in the afore-mentioned models the variation of the density is assumed to be minor during the motion of the material, and is thus often omitted in the dynami-cal theories.
It is of no doubt that the interstitial fluid may alter the behavior of the flows. Iverson (1999) and Iverson & Denlinger (2001) have emphasized the necessity of including the effect of interstitial fluid in the constitutive relations. Based on the assump-tion of small relative velocity with respective to the solid constituent, they derived a simplified equa-tion system, in which the effect of the pore fluid is taken into account but the force of interaction between the solid constituent and the interstitial fluid is neglected. With the account of the inter-action between the constituents, Wang & Hutter (1999) treated the fluid-saturated flows in a thermo-dynamically consistent manner. This work is also extended later by Hutter & Schneider (2010 a, b). By means of a phenomenological approach to the phase interactions, Pitman & Le (2005) proposed a two-fluid model. In the above theories the intersti-tial fluid is generally assumed to be water.
Shen et al. (2011) adopted the concept of the two-phase mixture model by Chen & Tai (2008), 1 INTRODUCTION
In natural environments, rock-falls, landslides, debris flows or snow-slab avalanches are examples of granular avalanches that take place in geophysi-cal contexts. The flows observed in silos, hoppers or rotating drums are examples in industrial appli-cations. Such flows are in general driven by gravity and consist of solid particles and pore fluids (water or air).
Many models have been proposed for describing the complex behavior of granular flows/avalanches. Most of them are either based on the concept of continuum mechanics or on the discrete element methods (DEM). One of the pioneering works based on the concept of continuum is proposed by Savage & Hutter (1989) (SH), in which the flowing layer is assumed to be of an incompressible mate-rial exhibiting the Mohr-Coulomb behavior. Based on the core concept of SH theory, extensions for complex topography were made by Hutter et al. (1993), Gray et al. (1999), Pudasaini & Hutter (2003), Patra et al. (2005). For flows over general topographic surfaces, Bouchut & Westdickenberg (2004) (BW) presented an arbitrary coordinate system and modelled the gravity-driven flows on the proposed coordinate system. Tai & Kuo (2008) proposed model with a deformable coordinate
that the dry granular material is regarded as a binary mixture of a solid and an interstitial fluid (air). That is, the volume fraction of the air is asso-ciated with the variation of density of the flows. By the advantage of excessively large difference of density between the two constituents, the force of the interaction become dominant in the momen-tum equation for the fluid phase, whilst it is neg-ligible for the momentum of the solid constituent. In addition, Shen et al. (2011) also introduce a mechanism of entrainment/extrusion of air, so that the total volume may vary according to the stretch-ing/ contracting behavior of the flowing body.
The present study is an extension of Shen et al. (2011) for flows of variable density over a non-trivial basal surface. It is implemented to a terrain-fitted coordinate system, so that the evaluation the physi-cal variables are computed in a mesh system fitted to the topographic surface. The non-oscillatory central scheme proposed by Jiang & Tadmor (1997) is adopted for numerical calculation. Numerical tests are performed in investigating the behavior of a finite mass of granular material flowing down an inclined plane onto a horizontal zone.
2 MODEL EQUATIONS
2.1 Topography-fitted coordinate system
In a vertical-horizontal Cartesian coordinate system (x, y, z) the topographic surface is given by z = b (x, y, z), then the unit normal vector is given by
(1)
with c=⎡⎣
( )
∂xxb 2+ ∂( )
yb 2+ ⎤⎦⎤⎤−1 2 1/
(see Bouchut & West-dickenberg 2004). Let rrr be the position vector of a b point at the topographic surface,
ττττ ττττ ττττ ττττ τττ τ τττ τττ ξ ξ ττττζ ξ ττττζ ξ ≡∂ ∂ ∂ ∂ r ττττ ∂rrrb rr ττττ ∂ , τττττττττ ≡η a ∂ ττττ nd η ξ≡ =n (2)
represent the natural basis of the tangent space to the topographic surface, and be the unit nor-mal vector, respectively. For a point at a distance ζ above the topography, its position vector r can be decomposed as
(3)
On the topographic surface one can define
an arbitrary coordinate system Oεηζ as proposed
by Bouchut & Westdickenberg (2004). For a deforming coordinate system, one can rely on the UC method (e.g., Hui 2007) to relate the Cartesian coordinates and an arbitrary coordinates by.
dr d d d d ΩΩΩ ξΩ (4) or in index notation dr dr dr r r r r r r x rr y rr z rr x rr rrx y rr rry z rr rrz ⎛ ⎝ ⎜ ⎛⎛ ⎜ ⎜⎜ ⎜⎝⎝ ⎜⎜ ⎞ ⎠ ⎟ ⎞⎞ ⎟ ⎟⎟ ⎟⎠⎠ ⎟⎟ = ∂ r ∂ ∂ ∂ r ∂ ∂ ∂ r ∂ ∂ ⎛ ⎝ ξrrrxx η ζ ξrrrxx η ζ ξrrrxx η ζ ⎜⎜ ⎛⎛⎛⎛ ⎜ ⎜⎜⎜⎜ ⎜⎝⎝ ⎜⎜ ⎞ ⎠ ⎟ ⎞⎞ ⎟ ⎟⎟ ⎟⎠⎠ ⎟⎟ ⎛ ⎝ ⎜ ⎛⎛ ⎜ ⎝⎝ ⎜⎜ ⎞ ⎠ ⎟ ⎞⎞ ⎟ ⎠⎠ ⎟⎟ d d d ξ η ζ .
As given in Bouchut & Westdickenberg (2004), the Jacobian matrix in (4) is given by.
ΩΩ Ω Ω ≡Grad = I s ξ s 1 , d T r c c ⎛ ⎝⎜ ⎛⎛ ⎝⎝ ⎞ ⎠⎟ ⎞⎞ ⎠⎠
(
d− ∂ ∂)
⎛ ⎝⎜ ⎛⎛ ⎝⎝ ⎞ ⎠⎟ ⎞⎞ ⎠⎠ ∂∂)
∂ξx 0 0 (5) where ∂ ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎛⎛ ⎝⎝ ⎞ ⎠ ⎟ ⎞⎞ ⎠⎠ ∂ ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎛⎛ ⎝⎝ ⎞ ⎠ ⎟ ⎞⎞ ⎠⎠ xs = = x x y x x y y y s ∂y s ∂y x ∂ y ∂ and x ξ ∂ x∂η ξy ∂ηyy2.2 Equations of mass and momentum conservation
The dry granular material is regarded as a binary mixture composed of a solid and an interstitial fluid (air), where the sum of their volume fractions adds up to unity. Since the flow speed considered here is much less than the speed of sound, the interstitial air is supposed to be a quasi-ideal fluid of constant density. Let ρs and ρf be the partial density of
the solid and the interstitial fluid, respectively. With the assumption of constant true densities,
and and the volume fraction of the solid phase
being denoted by ϕ, one obtains the partial
densi-ties by and In case of no
mass exchange between the two constituents, the balance equations for mass and momentum read.
∂tρψ + ∇+ ∇ ⋅+ ∇ ⋅ ρρρρψψ ψψ)= ,0 (6) ∂ + ∇ ⊗ = ∇ ⋅ , ⎛ ⎝⎝⎝ ⎞⎠ ⎛⎝⎛⎛ ⎞⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞ t ρ ⎠⎠⎠ + ∇+ ∇ ⎝⎝⎝⎛⎛⎛ρρ ρ ψ ψ ρψ ψ ψ ψ ψ ψ v ρ + ∇ ⋅⎛ ⎝ ⎛⎛⎛ ⎝⎝ ⎛⎛⎛⎛ ψρψ v Tψ + ψ +ρψg int (7)
where the superscript ψ ∈{s, f} denotes the solid and fluid phases, respectively. Here, vi represents
the velocities of the corresponding constituents,
Ti denotes the solid/fluid stress tensors, fi
int indi-cates the interaction forces and g is the gravita-tional acceleration. Since the sum of the balance
equations of the constituents in a mixture should behave as a single constituent material (Truesdell 1984), we have fs
int+ f
f
int= 0.
Since the interstitial air is supposed to be a quasi-ideal fluid, we have Tf = −pI with p being
the fluid pressure and I the identity tensor. The pressure p can be decomposed into two parts, p = (1 − ϕ) p + ϕ p, where the second term on the right hand side (RHS) corresponds to the buoyancy force of the solid species. Following
Iverson (1997), if Π repre—sents the resultant
non-buoyant interaction force per unit volume, we obtain fs
int= Π−∇(ϕp) = − ffint. The momentum Equations (7) can be then recast to.
(8)
(9)
In Pitman & Le (2005), an empirical proposal has been suggested the non-buoyant interaction force Π to be proportional to the relative velocity with a phenomenological constant αint, i.e.
ΠΠ Π
Π =αint .
f s
( ff − s) (10)
Considering the one-dimensional steady sedi-mentation and neglecting the stress of the solid phase, one obtains the well-known Richardson-Zaki relation,
(11)
where the momentum balance Equations (8) and (9) have been used (cf. Richardson & Zaki 1954). In (11), vT represents a terminal velocity of a typical representative solid particle falling in a fluid under gravity and M is a coefficient tending to be a constant value of 2.39 when the Reynolds number is greater than 500. For the dry granular material, the interstitial fluid is the air, so that the Reynolds number is greater than 500 and the value of M is taken to be 2.39 in present study.
2.3 Boundary conditions and material equations The balance Equations, (6) and (8)–(9), are subject to the kinematic and dynamic boundary condi-tions for the solid and fluid phases at the upper and basal surfaces. As the flow surface is defined by the solid phase, the kinematic boundary condi-tion of the solid species reads
∂ ∂ + ⋅ ∇ = , F t h F F h s h v 0 (12)
where Fh = 0 is the function of the flow surface and vs
h denotes the solid phase velocity at the flow
surface. The stretching/contracting behavior of the flow body allows the air to flow across the flow surface, so that the flow surface is a non-material surface with respect to the fluid phase. We define Vsurf = (vf
h − v s
h) ⋅ nh to be the volume flux of the
fluid phase across the flow surface with nh being the unit normal vector of the flow surface. The kinematic boundary condition of the fluid species can be the given by
∂ ∂ + ⋅ ∇ = ∇ = ∇ . F t ⋅ ∇ h F F h f h h f h s h= ∇ h surf v ⋅ ∇ =h ( h − ) f h s v v (13)
With the assignment of the direction of nh
pointing outward from the flowing body, a positive value of Vsurf indicates movement of the interstitial fluid out from the flowing body whilst a negative value means the ambient fluid (air) is moving into the flowing body. From the viewpoint of physical intuition, when the flow body is contracting, the particles become closer to one another, so that the interstitial fluid flows into the ambient environ-ment. In contrast, the ambient air penetrate into the flow body once the particles become more diluted (e.g., stretching). Shen et al. (2011) sug-gested an entrainment rate to be a function of the divergence of the local solid velocity, i.e.
Vsurf f
V
V = f⎛⎝⎝⎝∇⋅vs⎞⎠⎞⎞⎞⎠⎠⎞⎞⎞⎞. (14)
The dynamic boundary conditions of both of the solid and fluid phases are assumed to be trac-tion free, viz.
T nf
h=0 a d T nT nss hhh =0. (15)
We consider a rigid topographic surface, so that neither the solid phase nor the fluid phase can flow through the basal surface. Hence, the kinematic boundary conditions of both phases read.
vbs b b f b F ⋅ ∇ vb ∇ =b 0, f vbf vbf ∇ = b Fb 0 a d (16)
respectively, where the subscript b indicates the evaluation at the basal surface. The solid species are assumed to satisfy a Coulomb dry-friction slid-ing law, T n n n T n v v b T Ts b nb b TTbs b s s Nbb ⋅ nb − nn = || || , ⎛ ⎝ ⎛⎛ ⎝⎝ ⎞⎠⎞⎞⎞⎞⎠⎠⎞⎞⎞⎞ tanδ (17)
with nb denoting the unit normal vector of the base, Nb the normal pressure at the basal surface, tanδ the basal friction coefficient. Due to the low speed of the interstitial air, its basal stress is assumed to be negligible.
For the behavior of dry granular, a common constitutive relation is the Mohr-Coulomb law, in which the normal stresses of the solid species are connected with the normal solid stress at the basal surface by the earth pressure coefficient, Ka/p, which is a function of the angles of internal fric-tion and the fricfric-tion between the particles and the chute surface (c.f. Savage & Hutter 1989). In the present study, the value of Ka/p is taken to be 1, to isolate the complex effect due to the earth pressure coefficient.
2.4 UC formulation and model equations
Since there are two coordinate systems, the Carte-sian and the UC systems, every vectorial quantity can be expressed using either system. We shall use i∈{x, y, z} and m, n∈{ξ, η, ζ} for the index opera-tion, unless otherwise specified. Following Tai & Kuo (2008) or Tai et al. (2011), the equation sys-tem, (6) and (8) – (9), can be rewritten in the UC formulation, ∂t
( ))
( )
)
))
)
+ ∂+ ∂+ ∂∂∂mmm((
(
(
)
= ,0 (18) ∂t⎡⎣⎡⎣⎡⎡(
(
)
)
⎤⎦ ∂⎦ m⎡⎣(
)
f ⎤⎦⎤⎤ = , m q)
⎤ ∂ ⎡J(
)
− ⎤⎦⎤⎤ + ∂ ⎡)
)
))
)
⎦⎤⎤ + ∂⎦⎤⎤ + ∂ ⎡⎣⎡⎡⎦⎤⎤ ∂ J⎦⎤⎤ + ∂mmm⎣⎣J(
(
−))
0 (19) (20) (21)for i∈{x, y, z} and m∈{ξ, η, ζ}, where J = det Ω. In (20) and (21), the solid stress and the pressure are of the relations
T p
( )
Σims =ΩΩΩΩ( )
( )
Tsss( )
( )
Tsssijij ΩΩΩΩ−T(
( )
( )
iiiimim =pΩΩΩΩ, (22)respectively. Equations, (18)–(21), are essentially the conservation system of the Cartesian com-ponents, but in a mesh system fitted to the topo-graphic surface.
With the assumption of small curvature and shallow flow depth, the term ζ∂xs in (5) is negligi-ble, so that the Jacobian matrix reduces to
ΩΩ Ω Ω
Ω ΩΩΩΩΩΩΩΩΩΩΩΩΩΩb d d ΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩ d ΩΩΩΩΩΩΩΩΩΩΩΩΩΩ =ΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩbbb JJbbbb. (23)
With the help of the boundary consitions, (12)–(14) and (16)–(17), and the Leibniz rule, the depth-integrated equations read
∂t bb s)+ ∂ξ( bb s sqξ + ∂+ ∂η bbb s sqη)= ,0 (24) ∂
( )
+ ∂(
)
∂(
)
= − t(
b f b f f b f f h h surf J hb J hb qf)
+ ∂(
Jbh qf Jh Vs ξ ξ η(
(
Jbbh qη ( −− hhhhh) (25)for the mass balance, and
(26)
(27) for the momentum conservation. In (27), the last term on the RHS
(28) is the interaction force, which is negligible for the solid constituent due to the excessive density differ-ence, see Shen et al. (2011). The depth-integrated interaction in (28) term is phenomenological and highly non-linear. In the current study, it is approximated by
as suggested in Pitman & Le (2005). However, fur-ther study on its approximation is still needed.
Following the scaling analysis by Tai & Kuo (2008) or Tai et al. (2011), we adopt the assump-tions of small curvature and shallow flow thick-ness, yielding that the normal stresses of the con-stituents can be approximated by
where the second terms in the square bracket are the centrifugal acceleration due to the local
curvature. In case of an almost uniform distribution of the volume fraction along the flow depth, one obtains the solid/fluid normal stress at the basal surface
(29)
and the depth-averaged values
(30)
see Tai & Kuo (2008) or Tai et al. (2011) for details. With the help of (29), the fluid basal stress reads
Pb p N n
P
Piζ pbbΩΩΩΩ= fni, (31)
and the Coulomb friction sliding law (17) turns to be Σs b i bs i bs si s N nbs Nbs v , n vsi s, ζ ≈ − − δ v (32)
where δ is the angle of basal friction. The depth-averaged solid stress becomes
Σ Σ Σ Σ Σ Σ s x s s y s s z s T s c Ts ξ Σxη ξ Σyη ξ Σzη ⎛ ⎝ ⎜ ⎛⎛ ⎜ ⎜⎜ ⎜ ⎜⎜ ⎜ ⎜⎜ ⎜⎝⎝ ⎜⎜ ⎞ ⎠ ⎟ ⎞⎞ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟ ⎟⎟ ⎟⎠⎠ ⎟⎟ ≈ ∂ ⎛ ⎝ ⎜ ⎛⎛ ⎜ ⎜⎜ ⎜⎝⎝ ⎜⎜ ⎞ ⎠ ⎟ ⎞⎞ ⎟ ⎟⎟ ⎟⎠⎠ ⎟⎟ ξx s x ξξ ξξ ηη 0 0 TTTs ⎛ ⎝ ⎜ ⎛⎛ ⎜ ⎝⎝ ⎜⎜ ⎞ ⎠ ⎟ ⎞⎞ ⎟ ⎠⎠ ⎟⎟. (33)
In addition to the uniform distribution of the volume fraction, a further assumption is employed for the distribution of the velocity through the ava-lanche depth, viz. all sliding with little differential shear. Thus, the depth-averaged value of the prod-ucts in (18)–(22) can be factorized (e.g., Savage & Hutter 1989 or Gray et al., 1999) by
2 2
vi ≈vvii , ϕvvii ϕvvvii for iii
{ }
{ }
{ }
s f,,ff . (34) In order to simplify the model notation, all the overbars are dropped in the equations henceforth. The resultant governing equations then read∂t b + ∂ + ∂ = , s) s s + ∂ s s b ξ( bb qξ η bbb qη) 0 (35) ∂
( )
+ ∂(
)
∂(
)
= − t(
surf)
+ ∂(
Vs J ξ η((
ϕ ( −ϕ) (36)for the mass balance, and
(37)
(38) for the momentum balance equation.
3 NUMERICAL INVESTIGATION
The resultant model Equations (35)–(38) comprise a nonlinear system, for which a high resolution non-oscillatory central (NOC) scheme is applied to solve the conservative variables, Jbhf, J
bh s, J bh s-vx s, Jbh fvx f, Jbh svy s and Jbh fvy f. In the computation,
the time step Δt is determined by the
Courant-Friedrichs-Lewy (CFL) condition.
We consider a simulation of a finite mass of mixture sliding down an inclined flat chute (incli-nation angle 40°) onto a horizontal plane. The transition zone lies between ξ = 1.75 m and 2.45 m. The initial shape of the mass is a parabolic cap of base radius 0.24 m. Its maximum height is 0.5 m. The center of the cap locates at (0.46, 0.0) on the ξ-η-surface, the tangential components of the ini-tial velocity are given by
v h h v x y 0 0 = 40 0 0.0 h 0
( )
ξ η,, ⎨⎨⎧⎧⎨⎧⎧⎧⎧⎧(
1.2 ((ξξ −0 46)/0 24)
≠ ⎩ ⎨⎨ ⎩⎩ ⎨⎨⎨⎨ 46)/0 cos for fo 0 0( )
ξ,ξ,,ξξξξ η .The angle of basal fruction is 23° and the initial solid volume fraction is set to be 0.6. Following Shen et al. (2011), the entrainment rate Vsurf is sug-gested to be connected with the velocity gradient of the solid phase, i.e.
Vsurf h
V
V αEEh
(
∂ qqqqqqqqs +∂ qqqs)
(39)with an emperical coefficient αE= 0.1. For the inter-action force (28), the terminal speed vT in the is set to be 100 m/s, the density of air is 1.225 kg/m3 and
2800 kg/m3 for the sand.In the computation, the
Minmod slope limiter is applied; the CFL number is 0.4; the mesh size is Δξ × Δη = 0.02 m × 0.02 m. For a physically meaningful value, the solid volume fraction is limited to a maximum value of 0.7. Once the solid volume fraction is zero, it is equivalent to
the fact that the interstitial fluid flows out from the flow body, so that the total depth vanishes.
In the result figures (Figs. 1 to 3), the transition zone lies between the two dash-dotted lines. Figure 1 illustrates the thickness contour plots, in which the levels of the contour lines are 0.001, 0.01 and from 0.04 to 0.46 at increments of 0.03. The evolution of the solid volume fraction with the flow body is shown in the Figure 2. The flowing
mass accelerates on the chute of the inclined sec-tion and deposits on the horizontal plane. While sliding on the inclined section of the chute, the front part extended rapidly by the advantage of the negative value of the depth gradient. At this stage, the front part exhibits the lower volume fraction of solid constituent as well as the small density. The sequential cross-sections along ξ at η = 0.0 are shown in Figure 3. The black dashed lines repre-sent the equivalent depth of solid phase (hϕ), the dotted lines indicate the equivalent depth of the air within the flow body, and the solid lines mark the free surface of the flow.
4 CONCLUSIONS
A two-phase depth-averaging thin layer model for dry granular flows of variable density flowing over a non-trivial basal surface of small curvature is introduced. The interaction force between the two constituents is suggested to follow the Richardson-Zaki relation in the extreme case of high Reynolds number. The evolution of the volume fraction results in the variation of the density. The entrain-ment rate is suggested to be function of the veloc-ity gradient of the solid phase.
The features of the proposed model are illus-trated by numerical simulations, in which a finite mass of dry granular material dlides down an inclined flat chute to be deposited on a horizon-tal plane. The evolution of the volume fraction is illustraed. The almost uniform distribution of
arclength ξ (m) η t = 0.202 s −0.5 0 0.5 arclength ξ (m) η t = 0.505 s −0.5 0 0.5 arclength ξ (m) η t = 0.808 s −0.5 0 0.5 arclength ξ (m) η t = 1.111 s −0.5 0 0.5 arclength ξ (m) η t = 1.414 s −0.5 0 0.5 arc length ξ (m) η t = 1.717 s 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −0.5 0 0.5
Figure 1. Contour plots of the flowing mass.
arc length ξ (m) η t = 0.202 s Volume fraction −0.5 0 0.5 0 0.2 0.4 0.6 0.8 arc length ξ (m) η t = 0.505 s −0.5 0 0.5 0 0.2 0.4 0.6 0.8 arc length ξ (m) η t = 0.808 s −0.5 0 0.5 0 0.2 0.4 0.6 0.8 arc length ξ (m) η t = 1.111 s −0.5 0 0.5 0 0.2 0.4 0.6 0.8 arc length ξ (m) η t = 1.414 s −0.5 0 0.5 0 0.2 0.4 0.6 0.8 arc length ξ (m) η t = 1.717 s 0 0.5 1 1.5 2 2.5 3 3.5 4 −0.5 0 0.5 0 0.2 0.4 0.6 0.8
Figure 2. Evolution of the solid volume fraction.
0 0.01 0.02 Arclength ξ (along η =0 .0) η t = 0.202 s 0 0.01 0.02 Arclength ξ (along η =0 .0) η t = 0.505 s 0 0.01 0.02 Arclength ξ (along η =0 .0) η t = 0.808 s 0 0.01 0.02 Arclength ξ (along η =0 .0) η t = 1.111 s 0 0.01 0.02 Arclength ξ (along η =0 .0) η t = 1.414 s 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.01 0.02
Arc length ξ (along η = 0.0)
η t = 1.717 s
the velocity and volume fraction is nevertheless a strong assumption to real granular flows. In addi-tion, the suggested entrainment rate is based on a simple intuitive approach. With these simplica-tions, the present work only shets a light on this topic. Since the present formulation is based on the arbitrary coordinate sytem, it may have a wide range of potential application in geopogh and geo-technical engineering.
ACKNOWLEDGEMENTS
The authors are grateful to the financial support by National Science Council, Taiwan (Project No.: NSC 100-2628-E-006-207- and 100-2116-M-001-008-).
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