Outline

1. Coarse Bifurcation Studies of Alternative Microscopic/Hybrid SimulatorsC. Theodoropoulos and I.G. Kevrekidisin collaboration with K. Sankaranarayanan and S. SundaresanDepartment 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. 4 3 2 5 1 6 8 7 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+1 t+1 t t moments “Distribution functions” r(x,y) states

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

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

8. Dynamic LB Simulations g 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 Newton iterations Picard iterations Timestepper F(un) Picard iteration NO Subspace Q=I-P Subspace Pof 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 • 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 un+1 Q Q P P QF PN(p,q)

14. RPM for “Coarse” Bifurcations

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

16. Stabilization with RPM g 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 = 8.2 Ta = 10.84 Stable Unstable

19. Eigenvectors near Hopf point Stable branch Ta=8.85

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

21. X-Momentum Eigenvectors Ta=9.25 Unstable branch

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)