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Multiphase simulations in DualSPHysics. New developments, capabilities
and practical examples.
G. Fourtakas, B. D. Rogers, E.H. Zubeldia and M. M. FariasSchool of Mechanical, Aerospace and Civil Engineering,
University of Manchester, UK
Department of Civil and Environmental Engineering,
University of Brasilia, Brasilia, Brazil
4th DualSPHysics Users Workshop, IST, University of Lisbon 22nd-24th October 2018
Outline of the presentation
o Motivation
o SPH methodology
o Multi-phase model
o Liquid - sediment model o Yield criteria
o Constitutive equations
o Capabilities and XML example
o New developmentso New validation cases and applications
o Conclusions
Courtesy of the National Nuclear Laboratory, UK
Courtesy of the National Nuclear Laboratory, UK
Motivation
o Real life engineering problemso Underwater sand bed trenchingo Local scour around structureso Suspension of hazardous materialso Non-Newtonian flows
o UK Nuclear industry applicationo Industrial tanko Hazardous materialo Sediment agitationo Submerged jets
The SPH method
o Using the total derivative
the Lagrangian form of the Navier-Stokes equations is:
o Momentum equation (conservation of momentum)
o Continuity equation (conservation of mass)
o Tait’s equation of State (weakly compressible SPH (WCSPH))
o Plus other closure models
Navier Stokes equations in Lagrangian form and SPH formalism
d
dt=¶
¶t+ui ×Ñ
0d u
dt x
1dug
dt x
2
0 0
0
1
p
ii
cp
1
Niji
j ijj
Wdm u
dt x
1
Ni j iji
j
j i j
Wdum g
dt x
Multi-phase model
Liquid – sediment model
Multi-phase modelBased on Fourtakas and Rogers (2016)
o Liquid phase(5) Newtonian flow
o Sediment phase
(1) Yield criteria
o Surface yielding, sediment skeleton pressure
(2) Non-Newtonian flow
o Sediment shear layer at the interface, seepage forces
(3) Bed load
o Sediment erosion at the interface using Shield’s criterion
(4) Sediment resuspension
o Entrainment of soil grains by the liquid
Liquid – sediment model
o Weakly compressible SPH (WCSPH)
o Tait’s equation of state to relate pressure to density
o δ-SPH – Density diffusion term
o Turbulence is modelled through a SPS model
o GPU implementation to DualSPHysics
Liquid phase
Multi-phase model
o Multi-phase implementation
with
since
from
Liquid phase
o Newtonian constitutive equation
),,,( mnf yii
i
ijN
j ji
ji
ji
x
Wm
x
1
i
ijN
j
ij
j
j
ix
Wu
m
x
u
i
i
i
i
i
ii
x
u
x
u
x
u
3
1
2
1
1dug
dt x
p
Sediment phase
o Treated as a semi-solid non-Newtonian fluid
o Non-Newtonian flowo Herschel-Bulkley-Papanastasiou (HBP) constitutive model
o Entrained suspended sedimento Concentration based apparent viscosity based on a Newtonian formulation, Vand model
Multi-phase model
o Approximation of seepage forces on the surfaceo Darcy law
o Yield criterion Drucker-Pragero Below a critical level of sediment deformation sediment particles remain stillo Above a critical level of sediment deformation follow the governing equations
o Suspension of sedimento Shield’s criterion
Validation at Fourtakas and Rogers (2016)
Sediment phase
o Saturated sediment rheological characteristics
Multi-phase model
o Mohr-Coulomb stress ( τmc )o Cohesive yield strength ( τc )o Viscous stress ( τv )
o Turbulent stress ( τt )o Dispersive stress (due to collision) ( τd )
Validation at Fourtakas and Rogers (2016)
total mc c v t d
Sediment phase
o Surface Yielding o Drucker-Prager (DP) yield criterion
o For an isotropic material
o Apply the yield criterion
o Yielding occurs when
skeletonPJ2
02 yJ
1y
2tan129
tan
2tan129
3
c
Constants
where c is the cohesion and φ angle of repose
Drucker-Prager (DP) yield surface in principal stress
space
Sediment phase
o Surface Yielding o Drucker-Prager (DP) yield criterion
o Yielding occurs when
1y
Yield strength has a constant (fixed) value defined by the user (τy ≠ 0)
or
Yield strength is being calculated based on the internal friction angle and sediment parameters (τy = 0)
Sediment phase
o Sediment constitutive equationo Simple Bingham
o Herschel-Bulkley-Papanastasiou (HBP)
o Viscous – Plastic (m exponential growth)
o Shear thinning or thickening (n power law)
d
D
y
Bingh
mpap =t y
IID1- e
-m IIDéë
ùû+KD(n-1)/2
Dpapi 2
x
u
x
uD
2
1
Sediment phaseo Sediment constitutive equation
o Non-Newtonian model
mpap =t y
IID1- e
-m IIDéë
ùû+KD(n-1)/2 Dpapi 2
Viscous – Plastic (m exponential growth)
Shear thinning or thickening (n power law)
o Sediment/non-Newtonian model - xml file parameters
The xml file
Define number of phase i.e. 0, 1 etc
Define the density of the phase (saturated)
Define the numerical speed of sound of the phase
Define the polytrophic index of the phase for the EOS
Define the viscosity of the phase
Define the m exponential growth of the phase
Define the shear thinning or thickening (n power law) of the phase
The xml fileo Sediment/non-Newtonian model - xml file parameters
Define the cohesion of the (sediment) phase
Define the internal friction angle of the (sediment) phase
Define the yield strength of the (fluid/sediment) phase
Define if the phase is Newtonian (0) or non-Newtonian- Note if Newtonian is selected viscous stresses linearly proportional to
the local strain rate and the HBP model will be reduced automatically to a Newtonian model
The following combinations are possible:
1. Phase 0 = Newtonian Phase 1 = Newtonian
2. Phase 0 = Non-Newtonian Phase 1 = Non-Newtonian
3. Phase 0 = Sediment Phase 1 = Sediment
4. ANY combination of the above
The xml fileo Final xml parameters
Example from Zilong Li and Xiaofeng Liu, Penn State University, USA
The xml fileo Final xml parameters
ExampleNew: Interaction with floating objects
Zilong Li and Xiaofeng Liu, Penn State University, USA
New developmentsBased on Zubeldia, Fourtakas, Rogers, Farias (2018)
o Shield’s criterion – sediment suspensiono The critical shear stress over a horizontal bed of uniform
sediment is then defined as
τbcr,0 is the critical bed shear stress and d is particle diameter median diameter
o Critical Shields parameter as a function of the sediment Reynolds number (Re*) according to van Rijn (1993)
with and
,0
( )
bcr
cr
s gd
* *
*
*
0.1104760.010595ln(Re ) 0.0027197 for Re 500
Re
0.068 for Re 500
cr
cr
**Re
u d
*
bu
Sediment phaseo Shield’s criterion
o The velocity profile at the interface is defined as
as the combination of a linear function and a logarithmic velocity function for the turbulent layer.
o zo is the bottom roughness length-scale parameter
ks is the equivalent grain roughness
2
*( )
( )
*
for
1ln for >
z
z
o
uu z z
v
u zz
u z
*
11.6v
u
with the thickness, δ, of this layer as
*
*
*
*
*
0.11 5 (smooth flow)
0.033 70 (rough flow)
0.11 0,033 5< 70
s
so s
ss
v k u
u v
k uz k
v
v k uk
u v
Sediment phase
o Shield’s criterionGravitational influence for non-horizontal beds is considered by (upsloping and downsloping) van Rijn (1993)
where φ is the internal friction angle of the sediment, β and γ are the longitudinal and transverse slope, respectively.
The critical bed stress is calculated as
sediment particle is considered to be yielded if
sin( )
sink
0.52
2
tancos 1
tank
,0bcr bcrk k
b bcr
Sediment phase flow chart
Particle located at region (1) or (2)
Calculate
Eq. (15)
Calculate
Eq. (31)y
Particle at the surface?
(condition (i) and (ii) Section
2.5.1)
Calculate
Eq. (18) and Eq. (24)
and b bcr
( )HBP yf
Eq. (25)
b bcr
Particle located at region (3)
Calculate
Eq. (15)( )HBP bcrf
YES
NO
YES
NO
2-D flushing experiment
Sketch of the experimental 2-D flushing test
(Falappi et al., 2007)
Flushing experiment of Falappi et al. (2007) represents the scour of a sediment bed
due to constant water discharge from a tank.
2-D flushing experiment
Ideal test case to assess the effectiveness of the Shields criterion in applications
where erosion occurs at the bed surface only in the absence of sediment yielding
Slope profile at t = 48 s
2-D constant discharge flume experiment
Lutheran University of Brazil (ULBRA), campus Palmas, TO
Zubeldia et al., 2018
Flume experiment
2-D constant discharge flume experiment
Snapshots of the simulations at t = 8s
2-D constant discharge flume experiment
Position of the wave front at t = 8 s for different particle resolution and
experimental result
Case I. Experimental (dotted line) and numerical (continuous line) profiles obtained for the upward bed step case with a bed of PVC (left) and sand (right). a) t = 1.00 s, b) t = 1.50 s
a)
b)
2-D dam-break (regular) over erodible bed
Conclusions
o Weakly compressible SPH (WCSPH)o Newtonian and non-Newtonian multi-phase flowso Sediment transport and scouring using yield criteria
o Currently the solvers is o Closed sourceo Based on DualSPHysics v3.2o Limited features enabled
o Starting from DualSPHysics v4.4 the code will be open source (based on Fourtakas and Rogers (2016) formulation)
o Initially solver will be ported to v4.4o (DSPH) features will be enabled (i.e. floating bodies, periodics etc.)o GUI for pre-processing is still under discussion
o At later releases, Shields criterion will also be included (based on Zubeldia et al., (2018) formulation)