1. Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators C. Theodoropoulos and I.G. Kevrekidis in collaboration with K. Sankaranarayanan and S. Sundaresan Department of Chemical Engineering, Princeton University, Princeton, NJ 08544

2. 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

3. 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 (!!!)

4. Initialization
Happens in nature
Boltzmann NS
Happens in computations

5. Lattice-Boltzmann Method
Microscopic timestepping:
By multi-scale expansion can retrieve macroscopic PDE’s
Obtain states from the system’s moments:
Streaming (move particles)
Collision t
t+1 t
t+1 1 3 2 4 6 7 5 8
“Distribution functions”
moments
r(x,y)
states

6. LBM background LBM units are lattice units
Correspondence with physical world through
dimensionless groups
LBMNS
Reynolds number
Eötvös number
Morton number 1 3 2 4 6 7 5 8

7. Dynamic LB Simulationsg
Ta=2.407
Ta = 13.61
Bubble
rise direction
Stable Unstable

8. Dynamic LB Simulationsg
Ta=2.407
Ta = 13.61
Bubble
rise direction
Stable Unstable

9. Bubble column flow regimes
Chen et al., 1994

10. LBM single bubble rise velocity
Mo = 3.9 x 10-10
Mo = 1.5 x 10-5
Mo = 7.8 x 10-4
FT Correlation: Fan & Tsuchiya (1990)

11. Wake shedding and aspect ratio
V&E: Vakhrushev & Efremov (1970)
Sr =fd/Urise Ta = Re Mo0.23
1/2
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)

12. 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
Reconstruct solution:
u = p+q = PN(p,q)+QF
Initial state un
iterations
Picard
Newton
iterations
Timestepper
F(un)
Picard
iteration NO
Subspace
Q =I-P
Subspace P of few
slow eigenmodes
Convergence?
YES
Final state uf

13. Subspace Construction
First isolate slow modes for Picard iteration scheme
Subspace :
maximal invariant subspace of
Basis:
Vp obtained using iterative techniques
Orhtogonal complement Q
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
un+1 Q Q P P QF
PN(p,q)

14. RPM for “Coarse” Bifurcations

15. Stabilization with RPMg
Ta=13.61
Unstable Stabilized Unstable Steady State
Bubble rise
direction

16. Stabilization with RPMg
Ta=13.61
Bubble rise
direction
Unstable Stabilized Unstable Steady State

17. Bifurcation Diagram
m=2
m=4
m=6
Total mass on centerline
Hopf point Ta

18. Eigenspectrum Around Hopf Point
Ta = Ta = 10.84
Stable Unstable

19. Eigenvectors near Hopf point
Stable branch
Ta=8.85

20. Density Eigenvectors near Hopf point
Unstable branch
Ta=9.25

21. X-Momentum Eigenvectors
Unstable branch
Ta=9.25

22. 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

23. The Gap-tooth Scheme

24. 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
e, time-scale ratio, continuation parameter
Variation of e produces turning points
and Hopf bifurcations

25. FD Intregration

26. FD-FD and FD-LB Integration
FD-FD FD-LB t t

27. Phase Diagram t

28. 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.

29. Acknowledgements Financial support:
Sandia National Laboratories, Albuquerque, NM.
United Technologies Research Center, Hartford, CT.
Air Force Office for Scientific Research (Dr. M. Jacobs)