towards an immersed boundary methods for particle …wim/euromech549/... · 2013-07-25 · sreenath...

Towards an Immersed Boundary Methods for

Particle Sedimentation in Drilling Muds

Sreenath Krishnan, Eric S.G. Shaqfeh and Gianluca Iaccarino

Mechanical Engineering, Stanford University, Stanford CA 94305, USA

June 19, 2013

Sreenath Krishnan, Eric S.G. Shaqfeh and Gianluca Iaccarino (Mechanical Engineering, Stanford University, Stanford CA 94305, USA)Towards an Immersed Boundary Methods for Particle Sedimentation in Drilling MudsJune 19, 2013 1 / 30

Introduction

Outline

1 IntroductionMotivationViscoelastic fluidsImmersed Boundary method

2 Newtonian FlowsAlgorithm - Basic IdeaImplementation of the algorithmInterpolation KernelResults

3 Viscoelastic FluidsFormulationResults

4 Conclusion


Introduction Motivation

MotivationHydraulic fracturing

Suspensions of solids in polymeric solutions(proppant) are pumped to help prop openthe fracture

The dynamics of suspended particles in thepolymeric solutions is largely unknownparticularly in the vicinity of fractures

On a larger scale, two phase flows in whichsolid particles are dispersed in ambient fluidare very common


Introduction Viscoelastic fluids

Viscoelastic FluidsGoverning Equations

Viscoelastic materials exhibit both instantaneous elastic effectsand creep characteristics when undergoing deformationGoverning equations resemble Navier-Stokes equations fornewtonian flows, but with an additional stress τ pij , due to theelasticity of polymers in the flow

∂uj

∂xj= 0

∂ui

∂t+ uj

∂ui

∂xj= − ∂p

∂xi+

β

Re

∂2ui

∂xj∂xj+

1− βRe

1

Wi

∂τ pij∂xj

Weissenberg Number(Wi)

Ratio of characteristic polymer relaxation timescale and characteristicflow timescale


Introduction Viscoelastic fluids

Viscoelastic FluidsConstitutive equations

Finitely Extensible Nonlinear Elastic-Peterlin (FENE-P) modelAvg. polymer conformation tensor cij = 〈QiQj〉,τ pij =

cij1−ckk/L2 − δij

∂cij∂t

+ uk∂cij∂xk− cik

∂uj

∂xk− ckj

∂ui

∂xk=

1

Wiτ pij


Introduction Immersed Boundary method

Immersed Boundary(IB) Method

Generic term for methods that simulate flowswith embedded boundaries on grids that do notconform to the shape of these boundaries.

Domain grids are typically cartesian grids

But cartesian grids cannot efficiently representa complex fracture geometry

Focus is on IB method for viscoelastic flowswith unstructured domain grids


Newtonian Flows

Outline




4 Conclusion


Newtonian Flows Algorithm - Basic Idea

Basic Idea

Extend Navier-Stokes equations over the entire domain inclusiveof the particle regions

Assume particle regions are filled with fluid with density equal toparticle density (ρp)

Both the fluid and particle regions are incompressible

Motion of fluid inside the particle is constrained to be a rigidbody motion by adding a rigidity constraint body force to themomentum equations

ρ = ρf (1−Θ) + ρpΘ

Θ is the indicator function that has a value 1 inside the particleregion and zero outside


Newtonian Flows Algorithm - Basic Idea

Domain and particle meshes

Domain Mesh

Could be unstructured

Particle Mesh

Immersed in the domain mesh

Mostly unstructured


Newtonian Flows Implementation of the algorithm

Algorithm[Ref: Apte et al 2008]

1 Evaluate Θ, ρ from the initial position of the particle

Θ(~x) =∑Np

k=1 VkI(~x − ~xk)

2 Solve Navier-Stokes equation without any rigidity constraint

3 Project the solved velocity onto the particle grid

~uk =∑N

j=1 VjI(~xj − ~xk)~u(~xj)

4 Evaluate linear and angular momentum of the particle

Mp~U =

∑Np

k=1 ρp~ukVk Ip~ω =∑Np

k=1 ρp(~rkx~uk)Vk

5 Evaluate rigid body motion of the particle

~URBMk = ~U + ~ωx~rk



Algorithm

6 Calculate the rigidity constraint force in the particle grid

~fk = ρp

(~URBMk − ~uk

∆t

)7 Project the rigidity constraint force onto the domain grid

~f (~xj) =∑Np

k=1 VkI(~x − ~xk)~fk

8 Correct the velocity in the domain grid

~u = ~u +~f ∆t

ρ

9 Update the particle position

~X n+1 = ~X n + ∆t

(3

2~Un − 1

2~Un−1

)+R~X n



Special Treatment for collision[Ref: Glowinski et al. 2001]

In general the grid resolution is such that lubrication forces arenot resolved

We need to add a repulsive force to prevent unphysical overlaps

’Soft’ collision assumption

Collision force is modeled as a short range repulsive body force


Newtonian Flows Interpolation Kernel

Interpolation Kernel

In each iteration there are three projection steps involved.

Typically a discrete approximation of delta function is used asthe interpolation kernel

It is straightforward to define a suitable interpolation kernel foruniform grids with grid resolution ∆:

I(~x) = δ3∆(~x) =

1

∆3ξ(

x1

∆)ξ(

x2

∆)ξ(

x3

∆)

ξ(r) =

{f (r), if |r | ≤ 1.5

0, if |r | > 1.5

This kernel satisfies∑N

j=1 I(~x − ~xj) = 1 for any point ~x withinthe domain



Projection Template

However, for unstructured grids, it is difficult to define such asimple kernel.

Instead of defining a kernel, we could define a projectiontemplate

Projection Template

Defines how unity should be projected from the particle grid to aninfinite and continuous domain grid. In other words, it predefines theindicator function.

With a projection template, we can constraint what theprojection weights should sum to, during interpolation



Projection Template for a spherical particle

Projection template is only a function of the geometry of theparticle

Define the template as follows (L - interface smearing length):

F (r) =

1 if r <= R − L

0 if r >= R + L

P( r−RL

) otherwise

P(x) is a cubic polynomial that satisfies the following properties:

P(0) = 0.5, P’(± 1) = 0, P(1) = 0, P(-1) = 1

To project q(xk) from material grid to domain grid (Q(x)):

Q(x) = F (r)

∑Nk=1 Wkq(xk)∑N

k=1 Wk



Parallel Implementation

The code was built upon an existing Newtonian/viscoelastic flowsolver (CDP) developed at Stanford’s Centre for TurbulenceResearch

The code is based on an unstructured finite volume formulationand is capable of computing over many processors in parallel


Newtonian Flows Results

Flow over a fixed sphere in uniform streamMesh: 2M/4k elements, Re: 10-100

Particle velocity explicitly set to zero

Drag coefficients and flow patternsagree well.



Sedimentation of a sphereRep = 31.8, Blocking ratio a/R = 0.3125

Freely falling sphere

Sphere of diameter 0.625m sedimenting in a 2m x 2m x 8m box



Sedimentation of a sphereRep = 31.8, Blocking ratio a/R = 0.3125

Terminal Velocity


Viscoelastic Fluids

Outline




4 Conclusion


Viscoelastic Fluids Formulation

Formulation

The transport equation for conformation tensor is solvedthroughout the domain

Polymers are present everywhere except in the particle regions

This can be expressed by multiplying the polymeric stress termwith the complement of indicator function

∂ui

∂t+ uj

∂ui

∂xj= − ∂p

∂xi+

β

Re

∂2ui

∂xj∂xj+

1− βRe

1−Θ

Wi

∂τ pij∂xj


Viscoelastic Fluids Results

Preliminary ResultsRe=22.9, Wi = 4.2

Sphere sedimenting in viscoelastic fluid



Preliminary ResultsDrag on a sphere



Preliminary Results

The drag coefficient prediction for a fixed sphere case isreasonably accurate

The high polymer stress close to the sphere is not resolved

However in case of moving particle, the results don’t agree verywell

This is because the solution of the conformation tensor is notconsistent within the particle regions

Once the particle moves, this inconsistent solution becomesexposed



Ongoing work

In the previous formulation, we do not use the rigid bodyconstraint in the conformation tensor equations

The conformation tensor is δij for rigid body motion, as there isno shear

We can add a constraint force to the scalar transport equationas in the momentum equation in order to enforce this condition

This will result in an additional corrector step for thecomponents of conformation tensor in addition to the velocity


Conclusion

Outline




4 Conclusion


Conclusion

Conclusion

An unstructured formulation for the fully resolved flow of rigidparticles is developed

The idea of projection template is introduced

The formulation is validated for a variety of Newtonian flows

Using a polymer indicator function is not suitable because of theinconsistent solution within particle regions


Conclusion

References

Apte, Sourabh V., and Neelesh A. Patankar. ”A formulation forFully Resolved Simulation (FRS) of particle-turbulenceinteractions in two-phase flows.” Int. J. Numer. Anal. Model 5(2008): 1-16.

Glowinski, R., et al. ”A fictitious domain approach to the directnumerical simulation of incompressible viscous flow past movingrigid bodies: application to particulate flow.” Journal ofComputational Physics 169.2 (2001): 363-426.

David Richter , Gianluca Iaccarino, and Eric S.G Shaqfeh.”Simulations of three-dimensional viscoelastic flows past acircular cylinder at moderate Reynolds numbers.” Journal ofFluid Mechanics 651 (2010): 415.


Conclusion

Acknowledgements

Stanford Graduate Engineering Fellowship

Certainty Cluster through award MRI-R2

Prof. Eric S.G Shaqfeh and Prof. Gianluca Iaccarino


Conclusion

THANK YOU.!


towards an immersed boundary methods for particle …wim/euromech549/... · 2013-07-25 · sreenath...

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