a hybrid particle-continuum method coupling a fluctuating...

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A Hybrid Particle-Continuum Method Coupling a Fluctuating Fluid with Suspended Structures Aleksandar Donev 1 Courant Institute, New York University & Alejandro L. Garcia, San Jose State University John B. Bell, Lawrence Berkeley National Laboratory 1 This work performed in part under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. AMS von Neumann Symposium Snowbird, Utah July 6th, 2011 A. Donev (CIMS) Hybrid 7/2011 1 / 40

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  • A Hybrid Particle-Continuum Method Coupling aFluctuating Fluid with Suspended Structures

    Aleksandar Donev1

    Courant Institute, New York University&

    Alejandro L. Garcia, San Jose State UniversityJohn B. Bell, Lawrence Berkeley National Laboratory

    1This work performed in part under the auspices of the U.S. Department of Energy byLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

    AMS von Neumann SymposiumSnowbird, UtahJuly 6th, 2011

    A. Donev (CIMS) Hybrid 7/2011 1 / 40

  • Outline

    1 Introduction

    2 Particle Methods

    3 Fluctuating Hydrodynamics

    4 Hybrid Particle-Continuum Method

    5 The Importance of Thermal FluctuationsBrownian BeadAdiabatic Piston

    6 Fluctuation-Enhanced Diffusion

    7 Conclusions

    A. Donev (CIMS) Hybrid 7/2011 2 / 40

  • Introduction

    Micro- and nano-hydrodynamics

    Flows of fluids (gases and liquids) through micro- (µm) andnano-scale (nm) structures has become technologically important,e.g., micro-fluidics, microelectromechanical systems (MEMS).

    Biologically-relevant flows also occur at micro- and nano- scales.

    An important feature of small-scale flows, not discussed here, issurface/boundary effects (e.g., slip in the contact line problem).

    Essential distinguishing feature from “ordinary” CFD: thermalfluctuations!

    I focus here not on the technical details of hybrid methods, butrather, on using our method to demonstrate the general conclusionthat fluctuations should be taken into account at the continuumlevel.

    A. Donev (CIMS) Hybrid 7/2011 4 / 40

  • Introduction

    Example: DNA Filtering

    Fu et al., NatureNanotechnology 2 (2007) H. Craighead, Nature 442 (2006)

    How to coarse grain the fluid (solvent) and couple it to thesuspended microstructure (e.g., polymer chain)?

    A. Donev (CIMS) Hybrid 7/2011 5 / 40

  • Introduction

    Levels of Coarse-Graining

    Figure: From Pep Español, “Statistical Mechanics of Coarse-Graining”

    A. Donev (CIMS) Hybrid 7/2011 6 / 40

  • Introduction

    This talk: Particle/Continuum Hybrid

    Figure: Hybrid method for a polymer chain.

    A. Donev (CIMS) Hybrid 7/2011 7 / 40

  • Particle Methods

    Particle Methods for Complex Fluids

    The most direct and accurate way to simulate the interaction betweenthe solvent (fluid) and solute (beads, chain) is to use a particlescheme for both: Molecular Dynamics (MD)

    mr̈i =∑

    j

    f ij (rij )

    The stiff repulsion among beads demands small time steps, andchain-chain crossings are a problem.

    Most of the computation is “wasted” on the unimportant solventparticles!

    Over longer times it is hydrodynamics (local momentum and energyconservation) and fluctuations (Brownian motion) that matter.

    We need to coarse grain the fluid model further: Replacedeterministic interactions with stochastic collisions.

    A. Donev (CIMS) Hybrid 7/2011 9 / 40

  • Particle Methods

    Direct Simulation Monte Carlo (DSMC)

    (MNG)

    Tethered polymer chain inshear flow.

    Stochastic conservative collisions ofrandomly chosen nearby solventparticles, as in DSMC (also related toMPCD/SRD and DPD).

    Solute particles still interact with bothsolvent and other solute particles ashard or soft spheres.

    No fluid structure: Viscous ideal gas.

    One can introduce biased collisionmodels to give the fluids consistenstructure and a non-ideal equationof state. [1].

    A. Donev (CIMS) Hybrid 7/2011 10 / 40

    Graphics/TetheredPolymer.DSMC.2D.mng

  • Fluctuating Hydrodynamics

    Continuum Models of Fluid Dynamics

    Formally, we consider the continuum field of conserved quantities

    U(r, t) =

    ρje

    ∼= Ũ(r, t) = ∑i

    mimiυimiυ

    2i /2

    δ [r − ri (t)] ,where the symbol ∼= means that U(r, t) approximates the trueatomistic configuration Ũ(r, t) over long length and time scales.

    Formal coarse-graining of the microscopic dynamics has beenperformed to derive an approximate closure for the macroscopicdynamics [2].

    This leads to SPDEs of Langevin type formed by postulating awhite-noise random flux term in the usual Navier-Stokes-Fourierequations with magnitude determined from thefluctuation-dissipation balance condition, following Landau andLifshitz.

    A. Donev (CIMS) Hybrid 7/2011 12 / 40

  • Fluctuating Hydrodynamics

    Compressible Fluctuating Hydrodynamics

    Dtρ =− ρ∇ · vρ (Dtv) =−∇P + ∇ ·

    (η∇v + Σ

    )ρcp (DtT ) =DtP + ∇ · (µ∇T + Ξ) +

    (η∇v + Σ

    ): ∇v,

    where the variables are the density ρ, velocity v, and temperature Tfields,

    Dt� = ∂t� + v ·∇ (�)∇v = (∇v + ∇vT )− 2 (∇ · v) I/3

    and capital Greek letters denote stochastic fluxes:

    Σ =√

    2ηkBT W .〈Wij (r, t)W?kl (r′, t ′)〉 = (δikδjl + δilδjk − 2δijδkl/3) δ(t − t ′)δ(r − r′).

    A. Donev (CIMS) Hybrid 7/2011 13 / 40

  • Fluctuating Hydrodynamics

    Landau-Lifshitz Navier-Stokes (LLNS) Equations

    The non-linear LLNS equations are ill-behaved stochastic PDEs,and we do not really know how to interpret the nonlinearities precisely.

    Finite-volume discretizations naturally impose a grid-scaleregularization (smoothing) of the stochastic forcing.

    A renormalization of the transport coefficients is also necessary [3].

    We have algorithms and codes to solve the compressible equations(collocated and staggered grid), and recently also the incompressibleones (staggered grid) [4, 5].

    Solving the LLNS equations numerically requires paying attention todiscrete fluctuation-dissipation balance, in addition to the usualdeterministic difficulties [4].

    A. Donev (CIMS) Hybrid 7/2011 14 / 40

  • Fluctuating Hydrodynamics

    Finite-Volume Schemes

    ct = −v ·∇c + χ∇2c + ∇ ·(√

    2χW)

    = ∇ ·[−cv + χ∇c +

    √2χW

    ]Generic finite-volume spatial discretization

    ct = D[(−Vc + Gc) +

    √2χ/ (∆t∆V )W

    ],

    where D : faces→ cells is a conservative discrete divergence,G : cells→ faces is a discrete gradient.Here W is a collection of random normal numbers representing the(face-centered) stochastic fluxes.

    The divergence and gradient should be duals, D? = −G.Advection should be skew-adjoint (non-dissipative) if ∇ · v = 0,

    (DV)? = − (DV) if (DV) 1 = 0.

    A. Donev (CIMS) Hybrid 7/2011 15 / 40

  • Fluctuating Hydrodynamics

    Weak Accuracy

    Figure: Equilibrium discrete spectra (static structure factors) Sρ,ρ(k) ∼ 〈ρ̂ρ̂?〉(should be unity for all discrete wavenumbers) and Sρ,v(k) ∼ 〈ρ̂v̂?x 〉 (should bezero) for our RK3 collocated scheme.

    A. Donev (CIMS) Hybrid 7/2011 16 / 40

  • Hybrid Particle-Continuum Method

    Fluid-Structure Coupling using Particles

    MNG

    Split the domain into a particle and acontinuum (hydro) subdomains,with timesteps ∆tH = K∆tP .

    Hydro solver is a simple explicit(fluctuating) compressible LLNScode and is not aware of particlepatch.

    The method is based on AdaptiveMesh and Algorithm Refinement(AMAR) methodology for conservationlaws and ensures strict conservationof mass, momentum, and energy.

    A. Donev (CIMS) Hybrid 7/2011 18 / 40

  • Hybrid Particle-Continuum Method

    Continuum-Particle Coupling

    Each macro (hydro) cell is either particle or continuum. There isalso a reservoir region surrounding the particle subdomain.

    The coupling is roughly of the state-flux form:

    The continuum solver provides state boundary conditions for theparticle subdomain via reservoir particles.The particle subdomain provides flux boundary conditions for thecontinuum subdomain.

    The fluctuating hydro solver is oblivious to the particle region: Anyconservative explicit finite-volume scheme can trivially be substituted.

    The coupling is greatly simplified because the ideal particle fluid hasno internal structure.

    ”A hybrid particle-continuum method for hydrodynamics of complex fluids”, A.Donev and J. B. Bell and A. L. Garcia and B. J. Alder, SIAM J. MultiscaleModeling and Simulation 8(3):871-911, 2010

    A. Donev (CIMS) Hybrid 7/2011 19 / 40

  • Hybrid Particle-Continuum Method

    Our Hybrid Algorithm

    1 The hydro solution uH is computed everywhere, including the particlepatch, giving an estimated total flux ΦH .

    2 Reservoir particles are inserted at the boundary of the particle patchbased on Chapman-Enskog distribution from kinetic theory,accounting for both collisional and kinetic viscosities.

    3 Reservoir particles are propagated by ∆t and collisions are processed,giving the total particle flux Φp.

    4 The hydro solution is overwritten in the particle patch based on theparticle state up.

    5 The hydro solution is corrected based on the more accurate flux,uH ← uH −ΦH + Φp.

    A. Donev (CIMS) Hybrid 7/2011 20 / 40

  • Hybrid Particle-Continuum Method

    Other Hybrid Algorithms

    For molecular dynamics (non-ideal particle fluids) the insertion ofreservoir particles is greatly complicated by the need to account forthe internal structure of the fluid and requires an overlap region.

    A hybrid method based on a flux-flux coupling between moleculardynamics and isothermal compressible fluctuating hydrodynamics hasbeen developed by Coveney, De Fabritiis, Delgado-Buscalioni andco-workers [6].

    Some comparisons between different forms of coupling (state-state,state-flux, flux-state, flux-flux) has been performed by Ren [7].

    Reaching relevant time scales ultimately requires a stochasticimmersed structure approach coupling immersed structures directlyto a fluctuating solver (work in progresss).

    A. Donev (CIMS) Hybrid 7/2011 21 / 40

  • The Importance of Thermal Fluctuations Brownian Bead

    Brownian Bead

    Themal fluctuations push a sphere of size a and density ρ′ suspendedin a stationary fluid with density ρ and viscosity η (Brownian walker)with initial velocity Vth ≈

    √kT/M, M ≈ ρ′a3.

    The classical picture of Brownian motion indicates threewidely-separated timescales:

    Sound waves are generated from the sudden compression of the fluidand they take away a fraction of the kinetic energy during a sonic timetsonic ≈ a/c, where c is the (adiabatic) sound speed.Viscous dissipation then takes over and slows the particlenon-exponentially over a viscous time tvisc ≈ ρa2/η, where η is theshear viscosity.Thermal fluctuations get similarly dissipated, but their constantpresence pushes the particle diffusively over a diffusion timetdiff ≈ a2/D, where D ∼ kT/(aη).

    A. Donev (CIMS) Hybrid 7/2011 23 / 40

  • The Importance of Thermal Fluctuations Brownian Bead

    Velocity Autocorrelation Function

    We investigate the velocity autocorrelation function (VACF) for aBrownian bead

    C (t) = 2d−1 〈v(t0) · v(t0 + t)〉

    From equipartition theorem C (0) = kBT/M.

    For a neutrally-boyant particle, ρ′ = ρ, incompressible hydrodynamictheory gives C (0) = 2kBT/3M because one third of the kineticenergy decays at the sound time scale.

    Hydrodynamic persistence (conservation) gives a long-timepower-law tail C (t) ∼ (kBT/M)(t/tvisc)−3/2 that can be quantifiedusing fluctuating hydrodynamics.

    The diffusion coefficient is the integral of the VACF and isstrongly-affected by the tail.

    A. Donev (CIMS) Hybrid 7/2011 24 / 40

  • The Importance of Thermal Fluctuations Brownian Bead

    VACF

    0.01 0.1 1

    t / tvisc

    1

    0.1

    0.01

    M C

    (t)

    / k

    BT

    Stoch. hybrid (L=2)

    Det. hybrid (L=2)

    Stoch. hybrid (L=3)

    Det. hybrid (L=3)

    Particle (L=2)

    Theory

    0.01 0.1 1

    t cs / R

    1

    0.75

    0.5

    0.25

    tL=2

    A. Donev (CIMS) Hybrid 7/2011 25 / 40

  • The Importance of Thermal Fluctuations Adiabatic Piston

    The adiabatic piston problem

    MNG

    A. Donev (CIMS) Hybrid 7/2011 26 / 40

  • The Importance of Thermal Fluctuations Adiabatic Piston

    Relaxation Toward Equilibrium

    0 2500 5000 7500 10000t

    6

    6.25

    6.5

    6.75

    7

    7.25

    7.5

    7.75x(

    t)ParticleStoch. hybridDet. hybrid

    0 250 500 750 10006

    6.5

    7

    7.5

    8

    Figure: Massive rigid piston (M/m = 4000) not in mechanical equilibrium: Thedeterministic hybrid gives the wrong answer!

    A. Donev (CIMS) Hybrid 7/2011 27 / 40

  • The Importance of Thermal Fluctuations Adiabatic Piston

    VACF for Piston

    0 1 2 3

    0

    5×10-4

    1×10-3ParticleStoch. wP=2

    Det. wP=2

    Det. wP=4

    Det. wP=8

    0 50 100 150 200 250t

    -5.0×10-4

    -2.5×10-4

    0.0

    2.5×10-4

    5.0×10-4

    7.5×10-4

    1.0×10-3C(t)

    ParticleStoch. hybridDet. (wP=4)

    Det. x10

    Figure: The VACF for a rigid piston of mas M/m = 1000 at thermal equilibrium:Increasing the width of the particle region does not help: One mustinclude the thermal fluctuations in the continuum solver!

    A. Donev (CIMS) Hybrid 7/2011 28 / 40

  • Fluctuation-Enhanced Diffusion

    Nonequilibrium Fluctuations

    When macroscopic gradients are present, steady-state thermalfluctuations become long-range correlated.

    Consider a binary mixture of fluids and consider concentrationfluctuations around a steady state c0(r):

    c(r, t) = c0(r) + δc(r, t)

    The concentration fluctuations are advected by the randomvelocities v(r, t) = δv(r, t), approximately:

    ∂t (δc) + (δv) ·∇c0 = χ∇2 (δc) +√

    2χkBT (∇ ·Wc)

    The velocity fluctuations drive and amplify the concentrationfluctuations leading to so-called giant fluctuations [8].

    A. Donev (CIMS) Hybrid 7/2011 30 / 40

  • Fluctuation-Enhanced Diffusion

    Fractal Fronts in Diffusive Mixing

    Figure: Snapshots of concentration in a miscible mixture showing the developmentof a rough diffusive interface between two miscible fluids in zero gravity [3, 8, 5].

    A. Donev (CIMS) Hybrid 7/2011 31 / 40

  • Fluctuation-Enhanced Diffusion

    Giant Fluctuations in Experiments

    Figure: Experimental results by A. Vailati et al. from a microgravity environment[8] showing the enhancement of concentration fluctuations in space (box scale ismacroscopic: 5mm on the side, 1mm thick).

    A. Donev (CIMS) Hybrid 7/2011 32 / 40

  • Fluctuation-Enhanced Diffusion

    Fluctuation-Enhanced Diffusion Coefficient

    The nonlinear concentration equation includes a contribution to themass flux due to advection by the fluctuating velocities,

    ∂t (δc) + (δv) ·∇c0 = ∇ · [− (δc) (δv) + χ∇ (δc)] + . . .

    Simple (quasi-linear) perturbative theory suggests that concentrationand velocity fluctuations become correlated and

    −〈(δc) (δv)〉 ≈ (∆χ)∇c0.

    The fluctuation-renormalized diffusion coefficient is χ+ ∆χ(think of eddy diffusivity in turbulent transport).

    Because fluctuations are affected by boundaries, ∆χ is system-sizedependent.

    A. Donev (CIMS) Hybrid 7/2011 33 / 40

  • Fluctuation-Enhanced Diffusion

    Fluctuation-Enhanced Diffusion Coefficient

    Consider the effective diffusion coefficient in a system of dimensionsLx × Ly × Lz with a concentration gradient imposed along the y axis.In two dimensions, Lz � Lx � Ly , linearized fluctuatinghydrodynamics predicts a logarithmic divergence

    χ(2D)eff ≈ χ+

    kBT

    4πρ(χ+ ν)Lzln

    LxL0

    In three dimensions, Lx = Lz = L� Ly , χeff converges as L→∞to the macroscopic diffusion coefficient,

    χ(3D)eff ≈ χ+

    α kBT

    ρ(χ+ ν)

    (1

    L0− 1

    L

    )We have verified these predictions using particle (DSMC) simulationsat hydrodynamic scales [3].

    A. Donev (CIMS) Hybrid 7/2011 34 / 40

  • Fluctuation-Enhanced Diffusion

    Particle Simulations

    4 8 16 32 64 128 256 512 1024

    Lx / λ

    3.65

    3.675

    3.7

    3.725

    3.75

    χKinetic theory

    χeff

    (System A)

    χ0 (System A)

    χeff

    (System B)

    χ0 (System B)

    χeff

    (SPDE, A)

    Theory χ0 (A)

    Theory χ0 (B)

    Theory χeff

    (a)

    Figure: Divergence of diffusion coefficient in two dimensions.A. Donev (CIMS) Hybrid 7/2011 35 / 40

  • Fluctuation-Enhanced Diffusion

    Microscopic, Mesoscopic and Macroscopic Fluid Dynamics

    Instead of an ill-defined “molecular” or “bare” diffusivity, one shoulddefine a locally renormalized diffusion coefficient χ0 that dependson the length-scale of observation.

    This coefficient accounts for the arbitrary division between continuumand particle levels inherent to fluctuating hydrodynamics.

    A deterministic continuum limit does not exist in two dimensions, andis not applicable to small-scale finite systems in three dimensions.

    Fluctuating hydrodynamics is applicable at a broad range of scalesif the transport coefficient are renormalized based on the cutoff scalefor the random forcing terms.

    A. Donev (CIMS) Hybrid 7/2011 36 / 40

  • Conclusions

    Conclusions

    Coarse-grained particle methods can be used to acceleratehydrodynamic calculations at small scales.

    Hybrid particle continuum methods closely reproduce purelyparticle simulations at a fraction of the cost.

    It is necessary to include fluctuations in the continuum solver inhybrid methods.

    Thermal fluctuations affect the macroscopic transport in fluids.

    A. Donev (CIMS) Hybrid 7/2011 38 / 40

  • Conclusions

    Future Directions

    Improve and implement stochastic particle methods (parallelize, addchemistry, analyze theoretically).

    Direct fluid-structure coupling between fluctuating hydrodynamicsand microstructure.

    Develop numerical schemes for Low-Mach Number fluctuatinghydrodynamics.

    Ultimately we require an Adaptive Mesh and AlgorithmRefinement (AMAR) framework that couples a particle model(micro), with compressible fluctuating Navier-Stokes (meso), andincompressible or low Mach solver (macro).

    A. Donev (CIMS) Hybrid 7/2011 39 / 40

  • Conclusions

    References

    A. Donev, A. L. Garcia, and B. J. Alder.

    Stochastic Hard-Sphere Dynamics for Hydrodynamics of Non-Ideal Fluids.Phys. Rev. Lett, 101:075902, 2008.

    P. Español.

    Stochastic differential equations for non-linear hydrodynamics.Physica A, 248(1-2):77–96, 1998.

    A. Donev, A. L. Garcia, Anton de la Fuente, and J. B. Bell.

    Diffusive Transport Enhanced by Thermal Velocity Fluctuations.Phys. Rev. Lett., 106(20):204501, 2011.

    A. Donev, E. Vanden-Eijnden, A. L. Garcia, and J. B. Bell.

    On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics.CAMCOS, 5(2):149–197, 2010.

    F. Balboa, J. Bell, R. Delgado-Buscallioni, A. Donev, T. Fai, A. Garcia, B. Griffith, and C. Peskin.

    Staggered Schemes for Incompressible Fluctuating Hydrodynamics.Submitted, 2011.

    G. De Fabritiis, M. Serrano, R. Delgado-Buscalioni, and P. V. Coveney.

    Fluctuating hydrodynamic modeling of fluids at the nanoscale.Phys. Rev. E, 75(2):026307, 2007.

    W. Ren.

    Analytical and numerical study of coupled atomistic-continuum methods for fluids.J. Comp. Phys., 227(2):1353–1371, 2007.

    A. Vailati, R. Cerbino, S. Mazzoni, C. J. Takacs, D. S. Cannell, and M. Giglio.

    Fractal fronts of diffusion in microgravity.Nature Communications, 2:290, 2011.

    A. Donev (CIMS) Hybrid 7/2011 40 / 40

    IntroductionParticle MethodsFluctuating HydrodynamicsHybrid Particle-Continuum MethodThe Importance of Thermal FluctuationsBrownian BeadAdiabatic Piston

    Fluctuation-Enhanced DiffusionConclusions