Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 3, pp. 1051-1062, Warsaw 2016
DOI: 10.15632/jtam-pl.54.3.1051

ON THE TURBULENT BOUNDARY LAYER OF A DRY GRANULAR

AVALANCHE DOWN AN INCLINE. I. THERMODYNAMIC ANALYSIS

Chung Fang

National Cheng Kung University, Department of Civil Engineering, Tainan City, Taiwan

e-mail: cfang@mail.ncku.edu.tw

Characteristics of the turbulent boundary and passive layers of an isothermal dry granu-
lar avalanche with incompressible grains are studied by the proposed zero-order turbulence
closure model. The first and second laws of thermodynamics are applied to derive the equ-
ilibrium closure relations satisfying turbulence realizability conditions, with the dynamic
responses postulated within a quasi-static theory. The established closure model is applied
to analyses of a gravity-driven stationary avalanche down an incline to illustrate the distri-
butions of the mean solid content, mean velocity, turbulent kinetic energy and dissipation
across the flow layer, and to show the influence of turbulent fluctuation on the mean flow
features comparedwith laminar flow solutions. In this paper, detailed thermodynamic ana-
lysis and equilibrium closure relations are summarized, with the dynamic responses, the
complete closuremodel and numerical simulations reported in the second part.

Keywords: closure model, dry granular avalanche, thermodynamics, turbulence

1. Introduction

Drygranular systemsare collections of a large amountofdispersive solidparticleswith interstices
filled with a gas. When in motion, the interactions among the solid grains result from short-
-term instantaneous elastic and inelastic collisions and long-term enduring frictional contact
and sliding. They play a structural role in the macroscopic behavior, and are characterized
as the microstructural effect (Aranson and Tsimring, 2009; Ausloos et al., 2005; GDR MiDi,
2004;Mehta, 2007; Pöschel andBrilliantiv, 2013; Rao andNott, 2008). Depending on dominant
microstructural grain-grain interactions, the quasi-static, dense and collisional flow states are
defined, by which the avalanche is classified as a flow with rapid speed (flow in collisional
state) (Pudasaini and Hutter, 2007; Pöschel and Brilliantiv, 2013; Rao and Nott, 2008). The
microstructural effect significantly depends on flow speed, leading to a dry granular avalanche
exhibiting distinct rheological characteristics (Oswald, 2009; Pudasaini and Hutter, 2007).
A dry granular flow experiences fluctuations on its macroscopic quantities, a phenomenon

similar to turbulentmotion of Newtonian fluids in three perspectives: (i) it results from two-fold
grain-grain interactions, in contrast to that of Newtonian fluids induced by incoming flow in-
stability, instability in the transition region or flow geometry (Batchelor, 1993; Tsinober, 2009);
(ii) it emerges equally at slow speed in contrast to that of Newtonian fluids, which is dependent
significantly on flow velocity, characterized by the critical Reynolds number; and (iii) while tur-
bulentfluctuation inducesmost energyproductionwithanisotropic eddies andenergydissipation
with fairly isotropic eddies at the scales similar to the integral andKolmogorov length scales in
Newtonian fluids, respectively. Granular eddies at the inertia sub-range or Taylor microeddies
are barely recognized. These imply that a dry granular flow can be considered a rheological fluid
continuum with significant kinetic energy dissipation. Turbulent fluctuation induces energy ca-
scades from the stress power at themean scale toward the thermal dissipation at the subsequent
length and time scales (Pudasaini and Hutter, 2007; Rao and Nott, 2008).



1052 C. Fang

Field observations suggest that turbulent intensity is responsible for the entrainment of
geophysical mass transportation, and a dry granular avalanche is conjectured to consist of two
distinct layers: a very thin turbulent boundary layer immediately above the base, and a relatively
thick passive layer above the former (Pudasaini and Hutter, 2007; Wang and Hutter, 2001). In
the turbulentboundary layer, thegrains collide intensively onewith another, resulting in reduced
base friction so that the avalanche can travel unexpected long distance. On the contrary, the
dominant grain-grain interaction in the passive layer is a long-term one, causing the grains to
behave as a lump solid.

To take into account the effect of turbulent fluctuation onmean flow features, a conventional
Reynolds-filter process is applied to decompose the variables into the mean and fluctuating
parts to obtain the balance equations of the mean fields with ergodic terms. They need be
prescribed as functions of the mean fields, known as the turbulent closure relations, to arrive at
a mathematically likely well-posed problem. By different prescriptions of the closure relations,
turbulence closure models of different orders can be established (Batchelor, 1993).

Studies on turbulent characteristics of dry granular systems are so far yet complete. Various
models for slow creeping and dense laminar flows (e.g. Faccanoni and Mangeney, 2013; Fang,
2009a, 2010; Jop, 2008; Jop et al., 2006; Kirchner, 2002) and for rapid laminar flows (e.g.
Campell, 2005; Dainel et al., 2007; Fang, 2008a, 2008b; Savage, 1993; Wang and Hutter, 1999)
haven beendeveloped.The turbulentmodels byAhmadi (1985), Ahmadi andShahinpoor (1983)
andMa andAhmadi (1985) used Prandtl’s mixing length to account for the turbulent viscosity,
with applications to simple shear flows. Although the influence of velocity fluctuation on the
linear momentum balance was accounted for by Reynolds’ stress, the fluctuating kinetic energy
was not taken into account. Effort has been made to account for the fluctuating kinetic energy
by the granular temperature (e.g. Goldhrisch, 2008; Luca et al., 2004; Vescoci et al., 2013;Wang
and Hutter, 2001). Only the equilibrium closure relations were obtained; numerical simulations
of Benchmark problems compared with experimental outcomes were insufficient; the conjecture
of a two-layered avalanche remains unverified systematically.

Recently, the granular coldness, a similar concept to the granular temperature, has been
extended to account for the influence of weak turbulent fluctuation induced by two-fold grain-
-grain interactions (Fang, 2016a; Fang andWu, 2014a). Akinematic equationwas used to descri-
be the time evolution of the turbulent kinetic energy, with the turbulent dissipation considered a
closure relation or an independent field resulting respectively in zero- and first-order turbulence
closure models.While the mean porosity and velocity coincided to the experimental outcomes,
the turbulent dissipation was shown to be similar to that of conventional Newtonian fluids in
turbulent boundary layer flows. Although the first-order model was able to account for the in-
fluence of turbulent eddy evolution, the zero-ordermodel was sufficient to capture the turbulent
kinetic energy and dissipation distributions (Fang andWu, 2014b).

Thus, the goal of the study is to establish a zero-order closure model for isothermal dry
granular avalanches with incompressible grains. The first and second laws of thermodynamics,
specifically theMüller-Liu entropy principle, are used to derive the equilibrium closure relations
with the dynamic responses postulated within a quasi-static theory. The established model is
applied to analyses of a gravity-driven stationary avalanche down an incline, compared with
laminar flow solutions to illustrate the distributions of turbulent kinetic energy and dissipation
with their influence on themean flow features, and to verify the characteristics of the turbulent
boundary and passive layers with their similarities to those of Newtonian fluids. The study is
divided into two parts. Analysis of the Müller-Liu entropy principle and derived equilibrium
closure relations are summarized in this paper, with the complete closure model and numerical
simulations reported in the second part (Fang, 2016b). The mean balance equations and tur-
bulent state space are given in Section 2, followed by the thermodynamic analysis in Section 3.



On the turbulent boundary layer of a dry granular avalanche... 1053

The derived equilibrium closure relations are summarized in Section 4, with the work concluded
in Section 5.

2. Mean balance equations and turbulent state space

2.1. Mean balance equations

Following thebalance equations for laminarmotion and theReynolds-filter process, themean
balance equations for a turbulent flow are given by Batchelor (1993) and Fang (2009a, 2014a)1

0= ˙̄γν̄ + γ̄ ˙̄ν + γ̄ν̄∇· v̄ (2.1)

bf0= γ̄ν̄ ˙̄v− div(̄t+R)− γ̄ν̄b̄ (2.2)

0= t̄− t̄T 0=R−RT 0= Z̄− Z̄T (2.3)

0= γ̄ν̄ ˙̄e− t̄ ·D̄+∇· (q̄+Q)− γ̄ν̄ε− γ̄ν̄r̄− ℓh̄ ·∇ ˙̄ν + γ̄ν̄f̄ℓ ˙̄ν − γ̄ν̄H (2.4)

0= γ̄ν̄ ˙̄η+∇· (φ̄+φ′)− γ̄ν̄σ̄− π̄ (2.5)

0= γ̄ν̄ℓ¨̄ν −∇· (h̄+H)− γ̄ν̄f̄ (2.6)

0=˚̄Z− Φ̄ (̊Z̄≡ ˙̄Z− [Ω̄,Z̄]) (2.7)

0= γ̄ν̄k̇−R · D̄−∇·K+ γ̄ν̄ε (2.8)

0= γ̄ν̄ṡ− ℓH ·∇ ˙̄ν −∇·L+ γ̄ν̄H (2.9)

with the ergodic terms,

0= Rij + γ̄ν̄v
′

iv
′

j 0= Hj − ℓRij
∂ν̄

∂xi
0= φ

′

j − γ̄ν̄η
′v
′

j

0= Qj − γ̄ν̄e
′v
′

j 0= γ̄ν̄ε− t
′

ij

∂v
′

i

∂xj
0=Mij − ℓh

′

iv
′

j

0= γ̄ν̄k−
1

2
γ̄ν̄v

′

iv
′

j 0= Kj − t
′

ijv
′

i−
1

2
Riij

0= γ̄ν̄s+
1

2
ℓ2Rij

∂ν̄

∂xi

∂ν̄

∂xj
0= γ̄ν̄H −Mij

∂2ν̄

∂xixj
− γ̄ν̄d

0= Rijk+ γ̄ν̄v
′

iv
′

jv
′

k
0= γ̄ν̄d− ℓ

(

h
′

i

∂v
′

j

∂xi
− γ̄ν̄f ′v

′

j

)

∂ν̄

∂xj

0= Lj −Mji
∂ν̄

∂xi
−
1

2
ℓ2Rijk

∂ν̄

∂xi

∂ν̄

∂xk

(2.10)

in which ν̄ is the mean volume fraction defined as the total mean solid content divided by
the volume of a representative volume element (RVE), and A ·B ≡ tr(ABT) = tr(ATB),
[A,B]≡AB−BA for two arbitrary second-rank tensors A and B. Variables and parameters
arising in (2.1)-(2.10) are defined in Table 1, with KE and div abbreviations of kinetic energy
and divergence, respectively. Quantities in (2.10), with the others shown later, are classified as
closure relations.

Equations (2.1)-(2.5) are respectively the conventional mean balances of mass, linear mo-
mentum, angular momentum, internal energy and entropy for a fluid continuum in turbulent
motionwith the symmetry of themeanCauchy stress demanded and themean density ρ̄ decom-
posed into ρ̄ = γ̄ν̄. This introduces ν̄, considered an internal variable, with its time evolution

1It is equally possible to conduct the study by using the Naghdi-Green approach with original Rey-
nolds’ treatment, see e.g. Bilicki and Badur (2003).



1054 C. Fang

Table 1.Variables and parameters in the mean balance equations

b̄ mean specific body force D̄ symmetric part of v̄⊗∇
ē mean specific internal energy f̄ mean production associated with ℓ ˙̄ν
h̄ mean flux associated with ℓ ˙̄ν H specific turbulent dissipation of γ̄ν̄s
H turbulent flux associated with ℓ ˙̄ν k specific turbulent KE
K flux associated with γ̄ν̄k ℓ constant internal length
L flux associated with γ̄ν̄s q̄ mean heat flux
Q turbulent heat flux r̄ mean specific energy supply
R Reynolds’ stress s specific turbulent configurational KE
T transpose t̄ mean Cauchy stress
v̄ mean velocity Z̄ mean internal friction
α arbitrary quantity ᾱ time-averaged value of α
α′ fluctuating value of α α̇ material derivative of α with respect to v̄
γ̄ mean truemass density of solid grains ε specific turbulent dissipation of γ̄ν̄k
η̄ mean specific entropy ν̄ mean volume fraction
π̄ mean entropy production σ̄ mean specific entropy supply
φ̄ mean entropy flux φ′ turbulent entropy flux
Ω̄ mean orthogonal rotation of RVE ∇ Nabla operator

described by the revised Goodman-Cowin model (2.6) for rapid flows (Fang, 2009a). It descri-
bes a self-equilibrated stress system, with constant internal length ℓ denoting the characteristic
length of the grains (Fang, 2008a, 2009a;Wang andHutter, 1999). Since the grain arrangements
are assumed to be independent of motion of the granular body, variation in ℓ ˙̄ν provides extra
energies to the internal energy balance, as indicated by (−ℓh̄ ·∇ ˙̄ν + γ̄ν̄f̄ℓ ˙̄ν) in (2.4).

To account for the rate-independent characteristics, an Euclidean frame-indifferent, symme-
tric second-rank tensor Z̄ is introduced as an internal variable for the mean internal frictional
andother non-conservative forces inside a granularmicrocontinuum(SvendsenandHutter, 1999;

Svendsen et al., 1999), with ˚̄Z denoting the Zaremba-Jaumann corotational derivative of Z̄. It
is a phenomenological generalization of the Mohr-Coulomb model for a granular material at
low energy and high-grain volume fraction (see e.g. Fang, 2009b), with its time evolution de-
scribed kinematically by (2.7). The mean internal friction Z̄ with its time evolution (2.7) are
widely used for laminar and turbulent formulations (e.g. Fang, 2008a, 2009a, 2010; Fang and
Wu, 2014a; Kirchner, 2002) and provide the foundation of a dry granular heap at an equilibrium
state, when the grains are incompressible (Fang, 2009a).

Equation (2.8) is the evolution of the turbulent kinetic energy, γ̄ν̄k, derived by taking the
inner product of the velocity with the balance of linear momentum followed by the Reynolds-
-filter process. It is considered to account for the influence of the energy cascade in turbulent
flows (Batchelor, 1993; Pudasaini andHutter, 2007; Rao andNott, 2008). The turbulent kinetic
energy is generated by Reynolds’ stress through mean shearing at the mean scale, transferred
subsequently via the fluxK at various length and time scales, and eventually dissipated at the
smallest scale (theKolmogorov scale) by the turbulent dissipation, γ̄ν̄ε. Similarly, the fluctuation
of the characteristic velocity ℓν̇ in (2.6) also contributes an additional turbulent kinetic energy,
the turbulent configurational kinetic energy, γ̄ν̄s, with its time evolution kinematically described
by (2.9)with the fluxL and turbulent configurational dissipation, γ̄ν̄H (Luca et al., 2004). Thus,
two turbulent kinetic energies appear: γ̄ν̄k for the turbulent fluctuation on the bulk material
velocity, and γ̄ν̄s for that on the characteristic velocity of the grain configuration. Two-fold
turbulent dissipations γ̄ν̄ε and γ̄ν̄H are considered closure relations, with their eventual thermal
effects indicated in (2.4). Since the time evolutions of Z̄,k and s aredescribedbyusingkinematic
equations, their variations do not provide additional energy contributions in the internal energy
balance equation.



On the turbulent boundary layer of a dry granular avalanche... 1055

2.2. Turbulent state space

With these, the quantities

P = {γ̄, ν̄, v̄,Z̄,ϑM,ϑT ,ϑG}

C = {t̄,R, ē, q̄,Q, η̄,φT , h̄,H, f̄,Φ̄,k,s,K,L,ε,H}
(2.11)

are introduced respectively as the primitivemean fields and closure relations, bywhich C should
be constructed based on the turbulent state space given by

Q= {ν0, ν̄, ˙̄ν,g1, γ̄,g2,ϑ
M,g3,ϑ

T ,g4,ϑ
G,g5,D̄,Z̄} C= Ĉ(Q) (2.12)

with g1 ≡ ∇ν̄, g2 ≡ ∇γ̄, g3 ≡ ∇ϑ
M, g4 ≡ ∇ϑ

T , g5 ≡ ∇ϑ
G and φT ≡ φ̄+φ′. In (2.11)1,

ϑM is the material coldness, which can be shown to be the inverse of an empirical material
temperature θM for simplematerials (Hutter and Jöhnk, 2004). The turbulent kinetic energy is
expressed conventionally by the granular temperature θT (Goldhirsch, 2008; Vescovi et al., 2013;
Wang andHutter, 2001) or alternatively by the granular coldness ϑT (Fang andWu, 2014a; Luca
et al., 2004).2 In a similar manner, the turbulent configurational kinetic energy is expressed by
the granular configurational temperature θG, or alternatively by the granular configurational
coldness ϑG (Luca et al., 2004). Two-fold granular coldnesses are used as implicit indices of the
variations in γ̄ν̄k, γ̄ν̄s and their dissipations.

State space (2.12) is proposed based on Truesdell’s equi-presence principle and principle
of frame-indifference, with ν0 the value of ν̄ in the reference configuration, included due to
its influence on the behavior of flowing granular matter (Luca et al., 2004). The quantities
ν0, ν̄, ˙̄ν, g1, γ̄ and g2 are for the elastic effect, corresponding to ρ̄, ˙̄ρ and ∇ρ̄ for complex
rheological fluids; ϑM and g3 represent temperature-dependency of physical properties; ϑ

T and
g4 stand for influence of turbulent kinetic energy and dissipation; ϑ

G and g5 denote the effect
of turbulent configurational kinetic energy and dissipation; while D̄ and Z̄ are for viscous and
rate-independent effects, respectively. Since {t̄, q̄, h̄,Φ̄} are objective, and v′ (the fluctuating
velocity) is also objective, the quantities {R,Q,H,K,L} are equally objective, withwhich (2.3)
is fulfilled.

3. Thermodynamic analysis

3.1. Entropy inequality

Turbulence realizability conditions require that during a physically admissible process, the
second law of thermodynamics with a local form of non-negative entropy production, and all
balance equations should be satisfied simultaneously (Rung et al., 1999; Sadiki and Hutter,
2000). This can be achieved by considering the mean balance equations as the constraints of
inequality (2.5) via the method of Lagrange multipliers, viz.3

2The simple relation between θM and ϑM does not hold for θT and ϑT in dry granular systems. Only the
relation of ϑT = ϑ̂T(θT , θ̇T) is understood, which holds equally for ϑG and θG. In addition, the coldness idea is
more physicallymeaningful whenmaterials exhibit nomemory effect. Since a dry granularmatter in rapidmotion
exhibits a relatively insignificant memory effect, the coldness is used as the first approximation.
3We follow the Müller-Liu approach, from our own experience it can deliver more possible findings than

the classical Coleman-Noll approach for granular media thermodynamics. However, the classical Coleman-Noll
approach is widely used to conduct thermodynamic analyses, see e.g. Cieszko (1996), Kubik (1986), Schrefler et
al. (2009), Sobieski (2009), Wilmański (1996).



1056 C. Fang

π̄ = γ̄ν̄ ˙̄η+∇·φT − γ̄ν̄σ̄−λγ̄( ˙̄γν̄ + γ̄ ˙̄ν + γ̄ν̄∇· v̄)−λv̄ ·
(

γ̄ν̄ ˙̄v− div(̄t+R)− γ̄ν̄b̄
)

−λē
(

γ̄ν̄ ˙̄e− t̄ ·D̄+∇· (q̄+Q)− γ̄ν̄ε− γ̄ν̄r̄− ℓh̄ ·∇ ˙̄ν + γ̄ν̄f̄ℓ ˙̄ν − γ̄ν̄H
)

−λν̄
(

γ̄ν̄ℓ¨̄ν −∇· (h̄+H)− γ̄ν̄f̄
)

−λZ̄ · ( ˙̄Z− [Ω,Z̄]− Φ̄)

−λk(γ̄ν̄k̇−R ·D̄−∇·K+ γ̄ν̄ε)−λs(γ̄ν̄ṡ− ℓH ·∇ ˙̄ν −∇·L+ γ̄ν̄Ht)­ 0

(3.1)

with λγ̄, λv̄, λē, λν̄, λZ̄, λk and λs being the corresponding Lagrange multipliers. Since
{ϑ̇M, ϑ̇T , ϑ̇G} are not considered in (2.11)1, it is assumed that (Hutter and Jöhnk, 2004)

λē = ϑM λk = ϑT λs = ϑG

ϑMψT ≡ ϑM ē+ϑTk+ϑGs− η̄
(3.2)

with ψT the specific turbulent Helmholtz free energy. Since material behavior is required to
be independent of external supplies, it follows (−γ̄ν̄σ̄ + γ̄ν̄λv̄ · b̄+ϑMγ̄ν̄r̄) = 0, an equation
determining the mean entropy supply. Substituting (2.11)-(2.12) and (3.2) into (3.1) yields

π̄ =a ·X + b ­ 0 (3.3)

with X = { ˙̄v, ¨̄ν, ˙̄γ, ϑ̇M, ϑ̇T , ϑ̇G, ġ2, ġ3, ġ4, ġ5,
˙̄D, ˙̄Z, Ω̄,∇ν0,∇ ˙̄ν,∇g1,∇g2,∇g3,∇g4,∇g5,

∇D̄, ∇Z̄}, and ġ1 = ∇ ˙̄ν − g1∇v̄ has been used. Since the vector X is the set of time- and
space-independent variations of Q, and the vector a and scalar b are only functions of (2.12),
(3.3) is linear inX . SinceX can take any values, it would be possible to violate (3.3) unless

a=0 and b ­ 0 (3.4)

Equation (3.4)1 leads to

ψT,α =0 α ∈{ ˙̄ν,g2,g3,g4,g5,D̄} (3.5)

for the restrictions on ψT ; and

λv̄ =0 λγ̄ =−γ̄ϑMψT,γ̄ λ
Z̄ =−γ̄ν̄ϑMψT

,Z̄

λZ̄Z̄= Z̄λZ̄ λν̄ =−ϑMℓ−1ψT
, ˙̄ν

(3.6)

for the Lagrange multipliers, determined with the prescribed ψT ; and

ē = ψT +ϑMψT
,ϑM

k = ϑMψT
,ϑT

s = ϑMψT
,ϑG

(3.7)

for three specific energies; and

0=φT,ν0 −ϑ
M(q̄+Q),ν0 +ϑ

T
K,ν0 +ϑ

G
L,ν̄0

0=φT
, ˙̄ν
−ϑM(q̄+Q), ˙̄ν +ϑ

T
K, ˙̄ν +ϑ

G
L, ˙̄ν + ℓ(ϑ

M
h̄+ϑGH)− γ̄ν̄ϑMψT,g1

0=φT,A−ϑ
M(q̄+Q),A+ϑ

T
K,A+ϑ

G
L,A

0=sym
(

φT,g−ϑ
M(q̄+Q),g+ϑ

T
K,g+ϑ

G
L,g

)

(3.8)

with A ∈ {D̄,Z̄} and g ∈ {g1,g2,g3,g4,g5}. In obtaining (3.8), ψ
T 6= ψ̂T(·, ˙̄ν) has been

assumed, with which λν̄ vanishes. This is motivated by that ψT is influenced significantly by
the variations in ν̄, but not its time rate of change (Fang, 2009a; Kirchner, 2002; Luca et al.,



On the turbulent boundary layer of a dry granular avalanche... 1057

2004; Pudasaini and Hutter, 2007; Wang and Hutter, 1999) and justified for most dry granular
systems in the collisional state.
Equation (3.4)2 yields the residual entropy inequality, viz.

π̄ =
(

− γ̄ν̄ϑMψT,ν̄ + γ̄
2ϑMψT,γ̄ − γ̄ν̄ϑ

Mf̄ℓ
)

˙̄ν +
∑

g

(

φT,g−ϑ
M(q̄+Q),g+ϑ

T
K,g

+ϑGL,g
)

·∇g+
(

ϑM t̄+ϑTR+ γ̄ν̄ϑMψT,g1 ⊗g1+ γ̄
2ν̄ϑMψT,γ̄I

)

· D̄

+ γ̄ν̄
(

ε(ϑM −ϑT)+H(ϑM −ϑG)−ϑMψT
,Z̄
· Φ̄
)

­ 0

(3.9)

with ⊗ denoting the dyadic product; I the second-rank identity tensor; and g ∈ {ν̄, γ̄,
ϑM,ϑT ,ϑG}.

3.2. Extra entropy flux

Define the extra entropy flux ξ, viz.

ξ≡φT −ϑM(q̄+Q)+ϑTK+ϑGL (3.10)

Substituting (3.10) into (3.8)1 and (3.8)2,3, results respectively in

0= ξ,ν0 0= ξ,A 0= sym(ξ,g) (3.11)

It follows from (2.12) and (3.11)1−2 that ξ= ξ̂(ν̄, ˙̄ν, γ̄,ϑ
M,ϑT ,ϑG,g1−5). Integrating (3.11)3 one

by one, yields (Wang, 1970, 1971; Wang and Liu, 1980)

ξ=A1 ·g1+B1 ·g2+C1 · (g1⊗g2)+d1 =A2 ·g2+B2 ·g3+C2 · (g2⊗g3)+d2

=A3 ·g3+B3 ·g4+C3 · (g3⊗g4)+d3

=A4 ·g4+B4 ·g5+C4 · (g4⊗g5)+d4 =A5 ·g1+B5 ·g5+C5 · (g1⊗g5)+d5

(3.12)

with {A1−5,B1−5} and C1−5 being respectively the second- and third-rank skew-symmetric
tensors, andd1−5 the isotropic vectors. Since ξ is an isotropic vector, it follows that0=A1−5 =
B1−5, and 0 = C1−5, because there are no isotropic second- and third-rank skew-symmetric
tensors.With these, equation (3.12) reduces to

ξ= ξ̂(ν̄, ˙̄ν, γ̄,ϑM,ϑT ,ϑG)=0 (3.13)

since there exist no isotropic vectors with only scalar arguments.
Equation (3.13) indicates that for rapidflows, the granular systemas awhole and solid grains

aremoving in a way that combined contributions of two-fold turbulent kinetic energy fluxes are
collinear with the entropy flux, a distinct result from that of dense flows (Fang andWu, 2014a).
For in rapid flows, the short-term grain-grain interaction is dominant, causing the grains to
evolve in amore dispersivemanner from the granular system. In dense flows, the long-term one
dominates, resulting inmore orientated grain arrangement from the granular body.With ξ=0,
equation (3.8)2 reduces to

ℓ(ϑMh̄+ϑGH)= γ̄ν̄ϑMψT,g1 H= ℓRg1 (3.14)

With (3.5)-(3.7) and (3.13)-(3.14), equation (3.4)1 has been fully explored.
Residual entropy inequality (3.9) is recast, viz.

π̄ =
(

ϑM ν̄p̄I+ϑM t̄+ϑTR+ γ̄ν̄ϑMψT,g1 ⊗g1
)

·D̄+(q̄+Q) ·g3

−K ·g4−L ·g5+ϑ
M(p̄− β̄ − γ̄ν̄f̄ℓ) ˙̄ν + π̄int ­ 0

(3.15)



1058 C. Fang

with the internal dissipation π̄int

π̄int = γ̄ν̄ε(ϑ
M −ϑT)+ γ̄ν̄H(ϑM −ϑG)− γ̄ν̄ϑMψT

,Z̄
· Φ̄ (3.16)

and the abbreviations

p̄ ≡ γ̄2ψT,γ̄ β̄ ≡ γ̄ν̄ψ
T
,ν̄ (3.17)

known respectively as the turbulent thermodynamic pressure and turbulent configurational pres-
sure. They are extended versions of their counterparts in laminar flows (Fang, 2009a; Kirchner,
2002; Wang and Hutter, 1999).

4. Equilibrium closure relations

Thermodynamic equilibrium is defined to be a time-independent processwith uniformvanishing
mean entropy production, viz. (Hutter andWang, 2003)

π̄
∣

∣

∣

E
=0 (4.1)

with the subscript E indicating that the indexed quantity is evaluated at thermodynamic equ-
ilibrium. In view of (2.12) and (3.15)-(3.16), equation (4.1) motivates the following equilibrium
and dynamic state spaces, viz.

Q
∣

∣

∣

E
≡ (ν0, ν̄,0,g1, γ̄,g2,ϑ

M,0,ϑT ,0,ϑG,0,0,Z̄)

QD ≡ ( ˙̄ν,g3,g4,g5,D̄)
(4.2)

with the superscript D denoting the dynamic state space. The dynamic variables ( ˙̄ν, g3, g4,

g5, D̄) should vanish at the equilibrium state, with Q
D
∣

∣

∣

E
= 0. In addition, under sufficient

smoothness, π̄ has to satisfy

π̄,a

∣

∣

∣

E
=0 a ∈QD (4.3)

and the Hessian matrix of π̄ with respect to QD at the thermodynamic equilibrium should be
positive semi-definite. Since the latter condition constrains the sign of the material parameters
in the closure relations, only equations (4.1) and (4.3) will be investigated.

First, applying (4.1) and (4.2)1 to (3.15) and (3.16), yields

0= (ϑM −ϑT)γ̄ν̄ε
∣

∣

∣

E
+(ϑM −ϑG)γ̄ν̄H

∣

∣

∣

E
− γ̄ν̄ϑMψT

,Z̄
· Φ̄
∣

∣

∣

E
(4.4)

indicating that two-fold turbulent dissipations γ̄ν̄ε and γ̄ν̄H at the equilibrium state result from
the internal friction, a justified result. It can be fulfilled with the assumptions

0= ε
∣

∣

∣

E
0= H

∣

∣

∣

E
0= Φ̄

∣

∣

∣

E
(4.5)

for all productions cease at the thermodynamic equilibrium state.



On the turbulent boundary layer of a dry granular avalanche... 1059

Secondly, incorporating (4.2)2 and (4.3) into (3.15) and (3.16), results respectively in

0= ϑM(p̄− β̄ − γ̄ν̄f̄
∣

∣

∣

E
ℓ)+(ϑM −ϑT)γ̄ν̄ε, ˙̄ν

∣

∣

∣

E
+(ϑM −ϑG)γ̄ν̄H, ˙̄ν

∣

∣

∣

E

− γ̄ν̄ϑMψT
,Z̄
· Φ̄, ˙̄ν

∣

∣

∣

E

0=(q̄+Q)
∣

∣

∣

E
+(ϑM −ϑT)γ̄ν̄ε,g3

∣

∣

∣

E
+(ϑM −ϑG)γ̄ν̄H,g3

∣

∣

∣

E
− γ̄ν̄ϑMψT

,Z̄
· Φ̄,g3

∣

∣

∣

E

0=−K
∣

∣

∣

E
+(ϑM −ϑT)γ̄ν̄ε,g4

∣

∣

∣

E
+(ϑM −ϑG)γ̄ν̄H,g4

∣

∣

∣

E
− γ̄ν̄ϑMψT

,Z̄
· Φ̄,g4

∣

∣

∣

E

0=−L
∣

∣

∣

E
+(ϑM −ϑT)γ̄ν̄ε,g5

∣

∣

∣

E
+(ϑM −ϑG)γ̄ν̄H,g5

∣

∣

∣

E
− γ̄ν̄ϑMψT

,Z̄
· Φ̄,g5

∣

∣

∣

E

0= ϑM t̄
∣

∣

∣

E
+ϑTR

∣

∣

∣

E
+ ν̄ϑMp̄I+ γ̄ν̄ϑMψT,g1 ⊗g1+(ϑ

M −ϑT)γ̄ν̄ε,D̄

∣

∣

∣

E

+(ϑM −ϑG)γ̄ν̄H,D̄

∣

∣

∣

E
− γ̄ν̄ϑMψT

,Z̄
· Φ̄,D̄

∣

∣

∣

E

(4.6)

While equation (4.6)1 indicates that the internal friction and two-fold turbulent dissipations
affect the evolution of ν̄ via two-fold pressures, justified in view of the microstructural grain-
grain interactions, it also yields an equilibrium expression for f̄. Equation (4.6)2 delivers that
the equilibriummean and turbulent heat fluxes are related to the internal friction and two-fold
turbulent dissipations, according to observations. It reduces to vanishing mean and turbulent
heat fluxes for isothermal flows. Equation (4.6)3 delivers an equilibrium expression for K in
terms of the internal friction and two-fold turbulent dissipations. Since turbulent dissipation is
(negative) productionof γ̄ν̄k, (4.6)3 and (4.4) implyan energy cascade fromthe turbulentkinetic
energy flux toward turbulent dissipation through the effect of internal friction in the presence
of a non-uniform granular coldness gradient, a phenomenon similar to that of Newtonian fluids
in turbulent shear flows (Batchelor, 1993; Tsinober, 2009). A similar situation is also found
between L and γ̄ν̄H, as indicated by (4.6)4. Equations (4.6)3 and (4.6)4 also imply that K
and L are mutually influenced, a result already indicated by (3.13). Lastly, equation (4.6)5
yields an expression that should be satisfied by t̄|E and R|E in terms of the internal friction,
two-fold turbulent dissipations and the mean volume fraction gradient.

In the above, γ̄ν̄k and γ̄ν̄s are expressed respectively as functions of ϑT and ϑG; they are
determined once ψT is prescribed.Thus, equations (3.5), (3.7)2−3, (3.17), and (4.4)-(4.6) deliver
implicitly the equilibrium closure relations for h̄, H, f̄, t̄,R, K,L, ε, H and Φ̄ in the context
of the zero-order turbulence closure model.

Remarks

1. The derived equilibrium closure relations apply for both compressible and incompressible
grains. For the incompressible grains, p̄ is no longer determined by (3.17)1, and should be
computed from the momentum equation.

2. Equation (4.5) ismadebasedonobservations ofNewtonianfluids in turbulentmotion,with
which equations (2.8) and (2.9) are automatically fulfilled, whenK andL are assumed to
depend explicitly on g4 and g5, respectively.

3. In the second part, a hypoplastic model is used for Φ̄ with Φ̄ = ˆ̄Φ(ν̄,D̄,Z̄), with which
equation (4.5)3 is fulfilled, and equations (2.8) and (2.9) reduce to

0=∇·
(

(ϑM −ϑT)γ̄ν̄ε,g
∣

∣

∣

E
+(ϑM −ϑG)γ̄ν̄H,g

∣

∣

∣

E

)

g= {g4,g5} (4.7)

additionally restrictions that should be fulfilled by ε|E and H|E.



1060 C. Fang

4. Equation (4.6)5, by using Truesdell’s equi-presence principle, is decomposed into

ϑM t̄
∣

∣

∣

E
=−ν̄ϑMp̄I− γ̄ν̄ϑMψT,g1 ⊗g1+ γ̄ν̄ϑ

MψT
,Z̄
· Φ̄,D̄

∣

∣

∣

E

ϑTR
∣

∣

∣

E
=−(ϑM −ϑT)γ̄ν̄ε,D̄

∣

∣

∣

E
− (ϑM −ϑG)γ̄ν̄H,D̄

∣

∣

∣

E

(4.8)

in which t̄
∣

∣

∣

E
is generated through the mean fields, with R|E mainly induced via the qu-

antities related to turbulent fluctuation (e.g. two-fold turbulent dissipations), a procedure
widely used for Newtonian fluids in turbulent flows. For laminar flows, equation (4.8)2
yields a vanishing R|E, while equation (4.8)1 delivers that a dry granular heap at the
equilibrium state can be accomplished either by the internal friction or a non-uniform
gradient of ν̄, coinciding with the previousworks by Fang (2009a), Kirchner (2002),Wang
and Hutter (1999).

5. Concluding remarks

The Reynolds-filter process is used to decompose the variables into the mean and fluctuating
parts to obtain the balance equations of the mean fields. Two-fold granular coldnesses are in-
troduced as primitive fields to index the variations in the turbulent kinetic and turbulent confi-
gurational kinetic energies, with their dissipations considered closure relations. TheMüller-Liu
entropy principle is investigated to derive the equilibrium closure relations.

Equations (3.13) and (4.6)3,4 demonstrate that the flux of the turbulent kinetic energy and
that of the turbulent configurational kinetic energyaremutually influenced.This implies that the
turbulent fluctuation of the granular system as a whole tends to drive the grains to arrange and
fluctuate in away that their combined contributions provide the onlydeviation between the heat
and entropy fluxes. This is justified, since laboratory experiments and field observations suggest
that there exists a turbulent boundary layer immediately above the base, in which the grains
collide one with another vigorously, resulting in a dominant short-term grain-grain interaction.
On the other hand, this conclusion does not hold for dense flows with weak turbulent intensity,
for only the collinearity between the fluxes of the turbulent kinetic energy and evolution of the
mean volume fraction can be deduced.

The implementation of the closuremodel, and numerical simulation of an isothermal, isocho-
ric, gravity-driven stationary dry granular avalanchewith incompressible grains downan incline,
compared with the laminar flow solutions are reported in the second part.

Acknowledgements

The author is indebted to theMinistry of Science and Technology, Taiwan, for the financial support

through the projectMOST103-2221-E-006-116-.The author also thanks the editor andProfessor Janusz

Badur for detailed comments and suggestions which led to improvements.

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Manuscript received January 29, 2015; accepted for print February 23, 2016