return to rest in finite time for a solid body settling in a yield...

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Return to rest in finite time for a solid bodysettling in a yield stress fluid

Anthony Wachsa, Benjamin Herzhaftb, Guillaume Vinaya, Paulo de Souza Mendesc, Ian Frigaardd, Rajendra Chhabrae,

Xavier Chateauf, Philippe Coussotf(a) Fluid Mechanics Department, IFPEN, France

(b) Chemical Engineering Department, IFPEN, France(c) Mechanical Engineering Department, PUC Rio, Brazil

(d) Mechanical Engineering Department, UBC Vancouver, Canada(e) Chemical Engineering Department, IIT Kanpur, India

(f) Laboratoire Navier, IFSTTAR, France

VPF2013, Rueil Malmaison, France2

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Settling of a solid body in a yield stress fluidis an old problem

Steady state velocity is a result of a balance between Newtonian: buoyancy = viscous drag + pressure drag Viscoplastic: buoyancy = viscous drag + pressure drag + yield stress

resistance If buoyancy cannot overcome yield resistance, no motion, so

threshold for motion is a balance betweenbuoyancy = yield stress resistance

2 basic questions: critical yield stress magnitude to prevent motion (stability criterion) magnitude of finite time for a particle being assigned an initial motion

to return to rest


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Many applications in industrial processesand environmental flows

Drilling operations Waste water treatment Mud flows ...


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Previous works

Large body of experimental & computationalknowledge on this problem for spheres & cylinders Among others: Adachi & Yoshioka, Jossic & Magnin,

Chhabra et al., Mitsoulis et al., Roquet & Saramito, Yu & Wachs, Prashant & Derksen, ...

For other body shapes ? Experimental work of Jossic & Magnin, 2001, AiChe

Computational studies: often look at the reverse problem, i.e., the flow past an obstacle


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Question: how to properly compute the stabilityand the return to rest in finite time for any object shape ?

Experimental measurements: still show discrepancies Computational works:

Flow/no flow and return to rest in finite time in viscoplasticfluid flows require the use of a (fully) implicit time algorithm + Lagrange multiplier technique (ILM)

In the literature: Roquet & Saramito, CMAM, 2003: ILM but flow past an obstacle Prahant & Derksen, CaCe, 2011 and Yu & Wachs, JNNFM,

2007: moving particles but weak coupling between particle and fluid momentum equations


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Numerical methodProblem 1: single-phase viscoplastic fluids

In the literature: regularization techniques: Papanastasiou, Bercovier &

EngelmanNeither flow/no flow nor return to rest in finite time, but reasonable approximations though

Augmented Lagrangian (AL): Glowinski et al and later usedby Vinay et al, Roquet & Saramito, Frigaard et al., ...Both flow/no flow and return to rest in finite time whencoupled to an implicit time algorithm (ILM)

Solution: ILM


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Numerical methodProblem 2: moving particles

In the literature: Lattice-Boltzmann : usually weak coupling Immersed Boundary : usually weak coupling Fictitious Domain (FD): usually strong coupling but decoupled

from the viscous, potentially yield stress, inertia and pressure effects (see Yu & Wachs, JNNFM, 2007)

Strong coupling of FD since it leads to a saddle-point problem

Solution: FD


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Numerical methodProblem 3: fully implicit time algorithm

Fully implicit means implicit treatment of the full balancebuoyancy = viscous drag + pressure drag + yield stress resistance

At each time step, modified AL: While not convergence:

Step 1: solve fluid momentum + mass conservation + FD Step 2: update the true strain-rate tensor d Step 3: update the viscoplastic Lagrange multiplier

Convergence: | d-D(u) | < epsilon Step 1: instead of a standard velocity-pressure problem,

we need to solve a velocity-pressure problem coupled to a FD method for the presence of the rigid bodies


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Numerical methodProblem 3: lack of preconditioner for Step 1

Problem: at the matrix level, problem 1 looks like

Does not converge well due to the lack of a good preconditioner (see Yiantsios, IJNMF, 2011), or unlessthe standard FD is modified

Even the fully coupled Newtonian problem is hard to solve

0gfff

p

Uu

MMMB

MAMAMBA

U

u

FDUu

t

tUU

tu

tu


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Numerical methodProblem 3: an easier variant

Slow flow => slow change of pressure field from one time step to the next Idea: relax the divergence free constraint and use an explicit

prediction of the pressure field as in a second order L2 projection algorithm

Our semi-implicit time algorithm: Full implicit solution of yield stress + buoyancy + viscous with an

explicit pressure field L2 projection to satisfy velocity divergence free Reasonably good convergence properties (but AL is always slow

to converge !!)


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Test problem:a 2D circular cylinder settling in a 4:1 channel

Comparison of drag coefficient withMitsoulis, CES, 2004

Simulation of a no-flow situation, comparison with standard operatorsplitting scheme (Yu & Wachs, JNNFM, 2007)

Computations ran witha platform for DNS of multiphaseflows using a FV/MAC scheme

buoyancy

viscouspressureyield


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Dimensionless numbers

p

sf

DU

Re

f

sr

sp

y

UD

Bn

Reynolds number

Density ratio

Bingham number

Gravity-yield number ?4

4/,2

DlorDLDDL

AVl

glY y

G

Jossic & Magnin : YG[0.04:0.095] for a cylinder


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Drag coefficient as a function of Bn, r=2, Re < 0.03

10

100

1000

0,1 1 10 100

Bn

Fn

Mitsoulis

PeliGRIFF


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Return to rest in finite time, r=2, Re < 0.030<t<Tt: y=0 , t<Tt: y/ YG=0.1

Convergence criterion FD = 10-8

Tt = 0.0375

Final state = completely unyielded domain


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Simulation for YG=0.25 > YG,maxComparison with 1st order splitting (Yu & Wachs, 2007)

splitting error O(Δt)for OP order 1Us O(Δt)

semi-implicitalgorithmUs convergencecriterion


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Discussion & Perspectives Reasonably good and accurate numerical strategy Still sensitive to time step in the vicinity of YG,max

Our computations reveal that YG,max~0.06 Computing cost is very high due to the slow convergence

rate of AL in 2D, velocity matrix is Cholesky factorized using MUMPS in 3D, probably not feasible anymore, ILU or Boomer-AMG multigrid all available in fully parallel in with PETSC

One step towards the DNS of dense suspension of particlesin a yield stress fluid

Might be too demanding for low Bingham number flows in which flow/no flow is not an issue

return to rest in finite time for a solid body settling in a yield...

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