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

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 Happens in nature Boltzmann NS Happens in computations

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

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

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

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

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 FT Correlation: Fan & Tsuchiya (1990)

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)

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

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)

RPM for “Coarse” Bifurcations

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

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

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

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

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)