J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 1
RESOLVED SIMULATION OF A GRANULAR-FLUID FLOW
2
WITH A COUPLED SPH-DCDEM MODEL
3
R.B. Canelas1 , J.M. Dom´ınguez2 , A.J.C. Crespo3 , M. G´omez-Gesteira4 and R.M.L. Ferreira5 1
CERIS, Instituto Superior T´ecnico, Universidade de Lisboa, Portugal.
2
EPHYSLAB, Universidade de Vigo, Ourense, Spain,
[email protected].
3
EPHYSLAB, Universidade de Vigo, Ourense, Spain,
[email protected].
4
EPHYSLAB, Universidade de Vigo, Ourense, Spain,
[email protected].
5
CERIS, Instituto Superior T´ecnico, Universidade de Lisboa, Portugal, ruimfer-
[email protected]. Corresponding author: Ricardo Canelas,
[email protected] 4
ABSTRACT
5
Debris flows represent some of the most relevant phenomena in geomorphological events.
6
Due to the potential destructiveness of such flows, they are the target of a vast amount of
7
research. This work addresses the need for a numerical model applicable to granular-fluid
8
mixtures featuring high spatial and temporal resolution, thus capable of resolving the mo-
9
tion of individual particles, including all interparticle contacts. The DualSPHysics meshless
10
numerical implementation based on Smoothed Particle Hydrodynamics (SPH) is expanded
11
with a Distributed Contact Discrete Element Method (DCDEM) in order to explicitly solve
12
the fluid and the solid phase. The specific objective is to test the SPH-DCDEM approach by
13
comparing its results with experimental data. An experimental set-up for stony debris flows
14
in a slit check dam is reproduced numerically, where solid material is introduced through a
15
hopper assuring a constant solid discharge for the considered time interval. With each sedi-
16
ment particle possibly undergoing several simultaneous contacts, thousands of time-evolving
17
interactions are efficiently treated due to the model’s algorithmic structure and the HPC
18
implementation of DualSPHysics.
19
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The results, comprising mainly of retention curves, are in good agreement with the measurements, correctly reproducing the changes in efficiency with slit spacing and density.
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 21
Keywords: Smooth Particle Hydrodynamics, Distributed Contact Discrete Element Model,
22
Debris Flow, Check-dams, Non-spherical Sediments.
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INTRODUCTION
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Debris flows are regarded as one of the most feared flow-related phenomena, due to the
25
large destructive potential of infrastructure and the danger it poses to human lives. A large
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body of work has been produced concerning its genesis, behavior and mitigation measures.
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It is a dense granular-fluid flow, whose rheology is determined by solid-fluid and solid-solid
28
interactions. These are very difficult to determine experimentally during actual flow. Numer-
29
ical work should be able to provide complementary information and increasingly plausible
30
insights into the mechanics of the flow. Currently most of the modeling strategies rely
31
on assuming a given rheology and treating the solid-fluid mixture as a continuous medium
32
(Stancanelli and Foti 2015), (Ferreira et al. 2009). These models rely on the calibration of
33
a rheological model for accurate results, valid for a specific event. No contributions for the
34
understanding of the inner processes at play on a fluid-solid flow are made.
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This work presents a relatively new approach to model these phenomena, using meshless
36
methods. Recent advances in computer architectures, namely the introduction of massively
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parallel resources for consumer machines paved the way to explore these more computation-
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ally intensive methods. DualSPHysics (Crespo et al. 2015) is a high performance meshless
39
numerical model, based on Smoothed Particle Hydrodynamics (SPH). It has been exten-
40
sively validated for fluid flows (Crespo et al. 2011), (Altomare et al. 2014), (Altomare et al.
41
2015), gracefully dealing with complex, time evolving geometries and unsteady free surface
42
conditions. Introducing solid bodies with no kinematic restrictions led to successful buoy-
43
ancy tests (Canelas et al. 2015). Canelas et al. (2013) introduced a fully arbitrary fluid-solid
44
interaction model with a relatively small set of conceptual assumptions by developing a
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Distributed Contact Discrete Element Method (DCDEM). The SPH-DCDEM approach rep-
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resents a resolved model, where no ad-hoc formulations are used concerning the momentum
47
exchanges between the phases, bypassing the need for a simplified conceptual model for
48
specific scenarios, such as debris flow.
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The objective of this paper is to test the SPH-DCDEM approach by comparing its results
50
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 with experimental data. This should provide a good indication on the qualities of the model
51
concerning macro-scale behavior, specifically, in this paper, the interaction of a stony debris
52
flow with a check-dam.
53
The application of the model to a debris flow is done by reproducing an experimental
54
campaign where debris flow mitigation measures are studied (Silva et al. 2016). A flume is
55
fitted with a check-dam, intended to control the transport and deposition processes of the
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sediments carried downstream by debris flows. Since check-dams are considered as one of
57
the simplest and most effective engineering measures against debris flows (Zeng et al. 2008;
58
Remaitre et al. 2008) they were widely applied all over the world as a short-term mitigation
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measure. Check dams are composed of a weir, two wings and a robust foundation, although
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there are a variety of other check dam solutions (Takahashi 2007). Open-type check dams
61
present a very attractive characteristic: they allow small sediment particles to pass through,
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while trapping larger blocks. The implemented open-type check dam is a slit dam, whose
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effectiveness in debris flows mitigation has been documented in the literature.
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DISCRETIZATION OF GOVERNING EQUATIONS
65
Equations of Motion in SPH
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In SPH, the fluid domain is represented by a set of nodal points where physical properties
67
such as mass, velocity, pressure and vorticity are computed. Since the nodes carry the mass
68
of a portion of the medium and other properties, the notion of particle emerges intuitively.
69
These particles move with the fluid in a Lagrangian manner and their properties change
70
with time due to the interactions with neighboring particles. Details of the method can be
71
consulted in Crespo et al. (2015) and G´omez-Gesteira et al. (2010).
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The nature of the classical SPH formulation renders an incompressible system expensive
73
and inconvenient to model (G´omez-Gesteira et al. 2010), and as a result most studies rely
74
on the discretization of the compressible Navier-Stokes system
75
dv ∇p =− +∇·τ +g dt ρ
(1)
dρ = −ρ∇ · v, dt
(2)
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J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 where v is the velocity field, p is the pressure, ρ is the density and τ and g are the stress
77
tensor and the vector of body forces, respectively. The continuity equation is traditionally
78
discretized by employing the notion that ρ∇ · v = ∇ · (ρv) − v · ∇ρ, rendering equation (2) X dρi = −ρi mj (vi − vj ) · ∇W (rij , h), dt j
(3)
79
where W is the kernel function, h is the smoothing length and rij is the distance between the
80
i and j particles. This produces a zero divergence field for uniform flow conditions. Equation
81
(1) can be written as X dvi =− mj dt j
pi pj + 2 2 ρi ρj
4µrij · ∇W (rij , h) vij + ∇W (rij , h) + mj 2) (ρ + ρ )(|r | i j ij j X τi τj + mj + 2 · ∇W (rij , h) + g ρ2i ρj j X
(4)
82
The first term of the right side is a symmetrical, balanced form of the pressure term (Mon-
83
aghan 2005). The second and third terms are associated to viscous stresses (Morris et al.
84
1997) and sub-particle scale stresses (SPS) (Dalrymple and Rogers 2006), respectively. The
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SPS term introduces the effects of turbulent motion at smaller scales than the spatial dis-
86
cretization. Following the eddy viscosity assumption and using Favre-averaging, the SPS
87
stress tensor for a compressible fluid can be written as 1 2 ταβ = 2νt Sαβ − δαβ Sαβ − CI ∆2 δαβ |Sαβ |2 , ρ 3 3
(5)
88
where ταβ is the sub-particle stress tensor, νt = (CS |r|)2 |Sαβ | is the eddy viscosity, CS is
89
the Smagorinsky constant, CI = 6.6 × 10−3 and Sαβ is the local strain rate tensor, with
90
|Sαβ | = (2Sαβ Sαβ )1/2 (Martin et al. 2000).
91
Newton’s equations for rigid body dynamics are discretized as MI
X dVI dvk = mk ; dt dt k∈I
II
X dΩI dvf = mk (rk − RI ) × , dt dt k∈I
(6)
92
where body I possesses a mass MI , velocity VI , inertial tensor II , angular velocity ΩI and
93
center of gravity RI . dvk /dt is the force by unit mass applied to particle k, belonging to
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body I. This force encompasses body forces, fluid resultants as well as the result of any rigid
95
contact that might occur, as further analyzed in Canelas et al. (2015) and Canelas et al.
96
(2016).
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J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 DEM conceptualization of rigid contacts between bodies
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The contact force FTi acting on particle i resulting from collision with particle j is de-
99
composed into Fn and Ft , normal and tangential components respectively. Both of these
100
forces are further decomposed into a repulsion force, Fr and a damping force, Fd , for the
101
dissipation of energy during the deformation.
102
103
104
In the current work, the normal forces are given by a modified Hertzian model, with a non linear spring-damper system (Lemieux et al. 2008) 3/4 1/4 Fn,ij = Frn + Fdn = kn,ij δij eij − γn,ij δij δ˙ij eij ,
(7)
105
where kn,ij is the stiffness constant of pair ij, δij = max(0, (di + dj )/2 − |rij |) is the particle
106
overlap, eij is the unit vector between the two mass centers and γn,ij is the damping constant.
107
The stiffness and damping constants are given by Lemieux et al. (2008) q √ 4 ∗√ ∗ kn,ij = Ey R ; γn,ij = Cn 6M ∗ Ey∗ R∗ , 3
(8)
108
with Cn being an empirical tunning parameter of the order of 10−5 . The other parameters
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are given by 1 − νI2 1 − νJ2 ri rj mI mJ 1 = + ; R∗ = ; M∗ = , ∗ Ey EyI EyJ ri + rj mI + mJ
(9)
110
where Ey is the Young modulus, ν is the Poisson ratio, r is the particle radius and m is the
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mass of the body. The Young modulus and the Poisson ratio are set for the desired material,
112
using the corresponding values.
113
114
Regarding tangential contacts, the complex mechanism of friction is modeled by a linear
115
dash-pot bounded above by the Coulomb friction law. The Coulomb law is modified with a
116
sigmoidal function in order to make it continuous around the origin regarding the tangential
117
velocity (Vetsch 2011). One can write
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Ft,ij = min(µf,IJ Fn,ij tanh(8δ˙ijt )etij ; Frt + Fdt ),
(10)
Frt + Fdt = kt,ij δijt etij − γt,ij δ˙ijt etij
(11)
where
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J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 µf,IJ , the friction coefficient at the contact of I and J, should be expressed as a function of
120
the two friction coefficients of the distinct materials. The stiffness and damping constants
121
are derived to be kt,ij = 2/7kn,ij and γt,ij = 2/7γn,ij (Hoomans 2000), as to insure internal
122
consistency of the time scales required for stability. This mechanism models the static and
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dynamic friction mechanisms by a penalty method. The body does not statically stick at
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the point of contact, but is constrained by the spring-damper system.
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Stability Region
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The stability region must be modified to accommodate another restriction, the DEM compatible time-step. The CFL condition can be written as s ! " h h ; min ∆t = C min min ; min hvij ·rij i i i |fi | c0 + maxj 2 |rij |
3.21 C50
M∗ kn,ij
25
−1 vn,ij5
!# , (12)
128
where C is the CFL constant of the order of 10−1 determined in accordance with the case
129
and c0 is a numerical sound celerity (Monaghan 2005). The first term results from the
130
consideration of force magnitudes, where fi is the total force acting on particle i, the second
131
is a combination of the CFL condition for numerical information celerities and a restriction
132
arising from the viscous terms (G´omez-Gesteira et al. 2010) and the third term takes into
133
account a theoretical solution for the DEM stability constraints (Sh¨afer et al. 1996), that
134
disregards the CFL.
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GEOMETRY OF THE SIMULATIONS
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The simulations are based on a flume described in Silva et al. (2016), where vertical slits
137
with a spacing s, a 20% slope and a recirculating mechanism are the main characteristics.
138
The numerical flume presents fully periodic conditions, allowing for both fluid and solid
139
phase recirculation. Small reservoir-like sections were introduced, ensuring similar inlet and
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outlet conditions to the experiment. Two slit typologies were tested, named P1 and P2, as
141
defined in (Silva et al. 2016). Three spacings were tested, s/d95 ∈ [1.18; 1.36; 1.49], where
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d95 represents the sieve aperture that retains on average more that 95% of the mass of gravel
143
samples. Figures 1 and 2 detail the slits section and the flume apparatus.
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The scales of the sediment grain interactions are orders of magnitude inferior to the d50
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of the granulometric curve. This may represent a problem for the numerical discretization,
146
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 as even smaller distances need to be evaluated for the force computations (Equation 7), and
147
machine precision can start to affect the computations after a large number of iterations. In
148
order to curb such effect, the geometric scale of the numerical experiment was doubled, as λl =
Lm =2 Lp
(13)
149
where λl is the geometrical scale, Lm is the characteristic length of the model and Lp the
150
same length on the physical prototype. Assuming Froude similarity (viscous and pressure
151
gradient forces are assumed independent from high Reynolds numbers), the discharges scale
152
as λQ =
λV λ3l 5/2 = λl = 1/2 λt λl
(14)
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It is considered that liquid and solid discharges introduced upstream are independent,
154
i.e., it is not intended that the solid discharge corresponds to the capacity discharge for the
155
given liquid discharge, considering the inclination, geometry and roughness of the flume.
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Table 1 shows the used model discharges and the corresponding prototype values.
157
To promote a correct inlet of solid material, a hopper is modeled, placed on top of the
158
channel. It was heuristically dimensioned to ensure an average solid discharge compatible
159
with the one presented in Table 1. Solid sediment grain sizes are generated according to
160
a random algorithm that reproduces a log-normal function, effectively approximating the
161
granulometric curve from (Silva et al. 2016). Sediment shape was obtained based on a
162
3D model of a rock used in the experiments, as shown in Figure 3. The proportions are
163
also varied according to the same log-normal random function before final scaling, while
164
bounding the sphericity coefficient to a 15% variation of the original shape. The grains are
165
dispersed in the hopper and are let to achieve their natural equilibrium positions at the
166
start of the simulation. The proposed initial conditions generator allows to generate two
167
entirely distinct solutions based on the same granulometric curves, corresponding to several
168
experimental runs.
169
The proposed parameters for the material properties, used in Equations 7 to 11, are
170
summarized in Table 2, representing typical values for consolidated lime stone or granite
171
found in material tables.
172
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 The resolution of the model was set at Dp = 0.008 m, resulting in over 1.6 × 106 particles
173
and D50 /Dp ≈ 6.
174
RESULTS
175
An example of the flow is presented in Figure 4. The sediment trapping efficiency is
176
accounted by measuring the solid discharges at a position sufficiently upstream and imme-
177
diately downstream of the dam. The experimental procedure applied volumes, but due to
178
the recirculation of solid particles, for the analysis of the numerical solution such approach
179
is impractical, and discharges are computed directly by analyzing the flux of solid particles
180
crossing a given plane.
181
As the sediment particles are generated with a random arrangement as initial conditions,
182
significant instantaneous variations occur if one compares similar runs. On average, at
183
t = 40.0 s the hopper is exhausted and the flow is assumed to reach equilibrium conditions
184
close to 10 s after that, when the last solid particle reaches the backwater of the dam. A
185
15 s interval was used to count solid discharges and derive retention trapping rates. Table 3
186
and Figure 5 show the relationship between sediment trapping efficiency, E, and the relative
187
spacing s/d95 for each tested solution.
188
The numerical results present a good comparison with the experimental data. A notice-
189
able under prediction of the efficiency for small spacings is shown, for both slit geometries.
190
It is hypothesized that smaller grains are more mobile due to being under resolved, not
191
being retained in the deposition zone so frequently. For D50 /Dp ≈ 6 but Dmin /Dp ≈ 2,
192
resulting in possible solid-solid contact loss of geometrical accuracy and kernel deficiencies
193
affecting mainly the viscous terms in Equation 4. For P2 slits the trend is accompanied
194
with increasing spacing, but no zero efficiency is established for s/d95 = 1.49, contrasting
195
with the experimental results. This is due to single sediment particles getting retained for
196
long enough to affect the measurements, a problem not encountered with the experimental
197
measurement procedure.
198
These differences in measurement procedures, discretization shortcomings for the smallest
199
of sediment particles and differences in the initial conditions from the experimental to nu-
200
merical experiments should provide a reasonable framing for the deviations between results,
201
considering the predicted trends are correct and consistent throughout the scenarios.
202
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 The slit density, γs , is given by the ratio between the sum of all functional openings and
203
the dam width, that influences water and solid discharges. Figure 6 shows the sediment
204
trapping rate E versus the slit density.
205
An increase in slit density appears to induce increased efficiency of the solution, enhancing
206
deposition upstream of the slit-dam. This is in line with the findings of (Wenbing 2006). For
207
the same discharge, the average flow velocity decreases and the local stream power decreases
208
too, resulting in the local reduction of sediment transport capacity. This effect seems to be
209
captured in the numerical solution, albeit with the already discussed differences.
210
CONCLUSIONS
211
The work details the results of a SPH-DCDEM implementation, capable of dealing with
212
arbitrarily defined fluid-solid flows, applied to a stony debris flow. The model has the poten-
213
tial to fully resolve the flow at proper scales and deal with intense topological changes of the
214
free-surface, promising valuable insight at the expense of significant computation resources.
215
An experimental campaign is reproduced, with comparable results being drawn, despite sig-
216
nificant differences in the conditions of the tests. These have been traced to under-resolved
217
smaller scales and the impossibility of matching initial conditions to the experimental runs.
218
Numerical results present maximum deviations of 9% to the experimental data, well within
219
the variation of experimental runs within the same test. The model provides relevant data,
220
that can greatly enrich experimental efforts, either by pre-validating flume designs prior to
221
construction and set-up and/or by allowing to probe small time and spacial scales, beyond
222
the current capabilities of experimental data acquisition. Further work on initial condi-
223
tions and performance aspects is on course, hopefully improving the approximation of the
224
experimental runs. This would allow to use the model as a virtual debris flow laboratory,
225
promoting the collection of novel data.
226
ACKNOWLEDGMENTS
227
This research was partially supported by project RECI/ECM-HID/0371/2012, funded by
228
the Portuguese Foundation for Science and Technology (FCT). First author acknowledges
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FCT for his PhD grant, SFRH/BD/75478/2010. It was also partially funded by Xunta de
230
Galicia under project Programa de Consolidacin e Estructuracin de Unidades de Investi-
231
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 gacin Competitivas (Grupos de Referencia Competitiva), financed by European Regional
232
Development Fund (FEDER) and by Ministerio de Economa y Competitividad under de
233
Project BIA2012-38676-C03-03. The experimental activity was part of the R&D project
234
STOPDEBRIS, which was funded by QREN and developed by a multidisciplinary team
235
from AQUALOGUS, Engenharia e Ambiente, in partnership with CEris-IST. Our acknowl-
236
edgments to Miguel Silva from AQUALOGUS for providing the data.
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J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 APPENDIX I. REFERENCES
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J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
TABLE 1. Model and prototype discharges Discharge Prototype Model Ql (m3 s−1 ) 0.018 0.1018 Qs (m3 s−1 ) 0.00033 0.0018
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
TABLE 2. Sediment mechanical characteristics Ey (GN m−2 ) ν(−) µf (−) 45 0.35 0.35
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331 s/d95 E[-] (Exp) E[-] (Run I) E[-] (Run II) E[-] (Run III) 1.180 0.900 0.786 0.802 0.810
TABLE 3. Sediment trapping efficiency results. P1 type slits.
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
FIG. 1. Experimental dam configuration. Slits P1 and P2 specifications. (Silva et al. 2015)
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
FIG. 2. Flume set-up. (Silva et al. 2015)
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
FIG. 3. Three-dimensional model of debris particle.
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
FIG. 4. Flow example. t = 24 s.
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
FIG. 5. Sediment trapping efficiency results. P2 type slits. Experimental (·) Numerical ()
J. Hydraul. Eng., 2017, 143(9): -1–1 DOI: 10.1061/(ASCE)HY.1943-7900.0001331
FIG. 6. Sediment trapping rate as a function of slit density . P1 type slits for s/d95 = 1.18. Experimental (·) Numerical ()
