Princeton University Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators C....

Princeton University

Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators

C. Theodoropoulos and I.G. Kevrekidisin collaboration with K. Sankaranarayanan and S. Sundaresan

Department of Chemical Engineering,Princeton University, Princeton, NJ 08544


Outline

• Motivation

• Basics of the Lattice Boltzmann method

• Bubble dynamics

• The Recursive Projection Method (RPM)– The basic ideas– Use of RPM for “coarse” bifurcation/stability analysis of LB

simulations of a rising bubble– Mathematical Issues

• Hybrid Simulations– Gap-tooth scheme

– Dynamic simulations of the FitzHugh-Nagumo model

• Conclusions


Motivation• Bubbly flows are frequently encountered in industrial practice

– Study the dynamics of a rising bubble via 2-D LB simulations•Oscillations occur beyond some parameter (density difference) threshold

• Objectives– Obtain stable and unstable steady state solutions with dynamic LB code

– Accelerate convergence of LB simulator to corresponding steady state

– Calculate “coarse” eigenvalues and eigenvectors for control applications • RPM: technique of choice to build around LB simulator

– Identifies the low-dimensional unstable subspace of a few “slow” coarse eigenmodes

– Speeds-up convergence and stabilizes even unstable steady-states.

– Efficient bifurcation analysis by computing only the few eigenvalues of the small subspace.

– Bifurcation analysis although coarse equations (and Jacobians) are not explicitly available (!!!)


Initialization

Boltzmann

NS

Happens in nature

Happens in computations


Lattice-Boltzmann Method• Microscopic timestepping:

• By multi-scale expansion can retrieve macroscopic PDE’s

• Obtain states from the system’s moments: 1

3 24

6 7

5

8 )y,x(fi

states

)]x(f)t,x(f[1

)t,x(f)1t,x(f eqiiiii

Streaming (move particles) Collision

t t+1t t+1

8,...0i

)y,x(uf ii

moments“Distribution functions” i

(x,y)


LBM background

32

4

2

Mo

Eo

Re

g

dg

Ud

•LBM units are lattice units•Correspondence with physical world through dimensionless groups•LBMNS Reynolds number

Eötvös number

Morton number

1

3 24

6 7

5

8

21

Eo

MoRe2

2

dUWe

23.0MoReTa


Dynamic LB Simulations

Ta = 13.61Ta=2.407

Stable UnstableBubble

rise direction

g


Dynamic LB Simulations

Ta = 13.61Ta=2.407

Stable Unstable

Bubblerise direction

g


Bubble column flow regimes

Chen et al., 1994


LBM single bubble rise velocity

Mo = 3.9 x 10-10

Mo = 1.5 x 10-5

Mo = 7.8 x 10-4

32

4

2

Mo

Eo

Re

g

dg

Ud

FT Correlation: Fan & Tsuchiya (1990)


Wake shedding and aspect ratio

0

1

2

3

4

5

6

0 5 10 15

Ta

Ta

Sr

LBM 2-DLBM 3-DCorrelation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10 100

Ta

h/b

V & E 2-D (1970)

LBM 2-D

V & E 3-D (1970)

LBM 3-D

1/2

V&E: Vakhrushev & Efremov (1970)

Sr = 0.4(1-1.8/Ta)2 , Fan & Tsuchiya, (1990) based on data of Kubota et al. (1967), Tsuge and Hibino (1971), Lindt and de Groot (1974) and Miyahara et al. (1988)

Sr =fd/Urise Ta = Re Mo0.23


Recursive Projection Method (RPM)

• Treats timstepping routine, as a “black-box”

– Timestepper evaluates un+1= F(un)• Recursively identifies subspace of

slow eigenmodes, P• Substitutes pure Picard iteration with

– Newton method in P– Picard iteration in Q = I-P

• Reconstructs solution u from the sum of the projectors P and Q onto subspace P and its orthogonal complement Q, respectively:

– u = PN(p,q) + QF

Initial state un

Timestepper

Convergence?

Final state uf

Subspace P of few slow eigenmodes

Subspace Q =I-P

F(un)

YES

NO

Reconstruct solution:u = p+q = PN(p,q)+QF

Pica

rdite

ratio

ns Newtoniterations

Picard iteration


Subspace Construction

• First isolate slow modes for Picard iteration scheme

• Subspace : – maximal invariant subspace of

– Basis:

•Vp obtained using iterative techniques

• Orhtogonal complement Q– Basis:

•Not an invariant subspace of M

• Orthogonal projectors: – P projects onto P, Q projects onto Q,

• Use different numerical techniques in subspaces– Low-dimensional subspace P: Newton with direct solver

– High-dimensional subspace Q: Picard iteration

NRP

x

FM

pN

p RV

)( pNNq RV

TppVVP T

ppVVIQ

P

Q

PQ

un+1

QFPN(p,q)


RPM for “Coarse” Bifurcations


Stabilization with RPM

g

Unstable Stabilized Unstable Steady State

Ta=13.61

Bubble risedirection


Bifurcation DiagramT

otal

mas

s on

cen

terl

ine

Ta

Hopf point

m=2 m=4

m=6


Eigenspectrum Around Hopf Point

Ta = 8.2 Ta = 10.84

Stable Unstable


Eigenvectors near Hopf point

Stable branch Ta=8.85


Density Eigenvectors near Hopf point

Unstable branch Ta=9.25


X-Momentum Eigenvectors

Unstable branch Ta=9.25


Mathematical Issues• Shifting to remove translational invariance

– Need to find appropriate travelling frame for stationary solution

– Idea: Use templates to shift [Rawley&Marsden Physica D (2000)]

– Alternatively: use Fast Fourier Transforms (FFTs) to obtain a continuous shift

• Conservation of Mass & Momentum (linear constraints)– In LB implicit conservation is achieved via consistent initialization

– RPM: initialization with perturbed density and momentum profiles

•Total mass and momentum changes

– RPM calculations can be naturally implemented in Fourier space


The Gap-tooth Scheme


FitzHugh-Nagumo Model

• Reaction-diffusion model in one dimension

• Employed to study issues of pattern formation

in reacting systems – e.g. Beloushov-Zhabotinski

reaction

– u “activator”, v “inhibitor”

– Parameters:

– no-flux boundary conditions

– , time-scale ratio, continuation parameter

• Variation of produces turning points

and Hopf bifurcations

0.2,03.0,0.4δ 10 aa

)(εδ 012

32

avauvv

vuuuu

t

t


FD Intregration


FD-FD and FD-LB Integration

FD-FD FD-LB

t t


Phase Diagram

t


Conclusions

• RPM was efficiently built around a 2-D Lattice Boltzmann simulator • Coupled with RPM, the LB code was able to converge

– even onto unstable steady states

• “Coarse” eigenvalues and eigenvectors were calculated – without right-hand sides of governing equations !!!

• The translational invariance of the LB Scheme was efficiently removed using templates in Fourier space for shifting.

• Conservation of mass and momentum (linear constraints) was achieved by implementing RPM calculations in Fourier space.

• A hybrid simulator, the “gap-tooth” scheme was constructed – and used to calculate accurate “coarse” dynamic profiles– of the FitzHugh-Nagumo reaction-diffusion model.


Acknowledgements

• Financial support:– Sandia National Laboratories, Albuquerque, NM.

– United Technologies Research Center, Hartford, CT.

– Air Force Office for Scientific Research (Dr. M. Jacobs)

Princeton University Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators C....

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