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)