Stefan Stefanov Bulgarian Academy of Science, Bulgaria Ali Amiri-Jaghargh Ehsan Roohi Hamid Niazmand Ferdowsi University of Mashhad, Iran

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1 Stefan Stefanov Bulgarian Academy of Science, Bulgaria Ali Amiri-Jaghargh Ehsan Roohi Hamid Niazmand Ferdowsi University of Mashhad, Iran

2 Outlines: Introduction DSMC Collision Schemes Results Conclusion

3 Introduction: Microfluidic systems: Microchannel Micro-turbines Micro-Air-Vehicles (MAV s) Micro-nozzles

4 Introduction: Flow behavior: Ordinary devices: Continuum Based Models MEMS/NEMS: Continuum Based & Molecular Models Criterion: Knudsen number Kn = λ / D h Continuum Slip Transition Free Molecular Kn

5 Introduction: Classification of Flow Regimes based on Modeling Approach Kn Model 0 (continuum, without molecular diffusion) 10 3 (continuum, with molecular diffusion) 10 3 < Kn 0.1 (continuum transition slip) 0.1 < Kn 10 (transition) Kn > 10 (free molecular) Euler equations Navier-Stokes equations with no-slip wall boundary conditions Navier-Stokes equations with 1 st order slip boundary conditions at wall Burnett equations with higher order slip boundary conditions at wall; Moment equations; DSMC; Lattice Boltzmann Collision-less Boltzmann equations; DSMC; Lattice Boltzmann

6 DSMC: Originally proposed by Bird Decoupling the motion free molecular movement binary intermolecular collision DSMC method converges to Boltzmann equation

7 DSMC: DSMC Algorithm: Initialize system with particles Loop over time steps Create particles at open boundaries Move all the particles Process any interactions of particle & boundaries Sort particles into cells Sample statistical values Select and execute random collisions

8 Collision Schemes: DSMC Collision Schemes: No Time Counter (NTC) Majorant Frequency Scheme (MFS) Simplified Bernoulli Trials (SBT)

9 Collision Schemes: NTC Scheme: N c = 1 2 f numn l < N l > l σg max t/ l Determination of number of collision pair N c pair (i,j) choosing collision pair (i,j) randomly from N l particle in cell l. p ij = σ ijg ij σg max if accepted checking the collision probability movment change the particle velocities

10 Collision Schemes: MFS Scheme: ν m = N N 1 2 / cell F N CVM Define the Majorant frequency for cell τ = ln 1 R f ν m Compute τ time to the next eventual collision using Poisson distribution; set t = t + τ If t>time step stop the loop choosing collision pair randomly checking the collision probability change the particle velocities

11 Collision Schemes: SBT Scheme: p ij = 1 2 kf num tσ ij g ij / l local cross-referencing of particles in cell l to number particles from 1 to N l choosing first particle in sequence from list: i = 1,...,N l -1 choosing the other particle from j=i+1+int((n l -i)*random) checking the collision probability change the particle velocities

12 Collision Schemes: The SBT scheme: Provides reasonable results with much lower number of particles per cell. Avoid the production of at least part of the eventually successively repeated collisions

13 Collision Schemes: To get the best results the SBT scheme should be accompanied with staggered mesh. x 2 y 2

14 SBT-DSMC flowchart: B Start Read Data, Set Constants Initialize Molecules and Boundaries, t ref = t Move Molecule, Compute interaction with boundaries A Compute Collision Reset Molecule Indexing compatible with standard grid t = t ref Reset Molecule Indexing compatible with standard grid t= t/2 Compute Collision Sample Flow Properties B Sample Flow Properties No N >N iteration Yes Print Results STOP Reset Molecule Indexing compatible with shifted cells A

15 Results: B U lid = 100 m/s > C V / U lid ^y x A > D < L = m > X / L Geometrical configuration of micro cavity. Grid independency test.

16 T Results: X / L Kn = NTC Algorithm <N> = 250 <N> = 125 <N> = 20 <N> = 5 <N> = 2 Effect of number of particle per cell in accuracy of the NTC collision scheme.

17 T Results: V / U lid Kn = SBT <N> = 2 NTC <N> = U / U lid Kn = SBT <N> = 2 NTC <N> = 20 NTC <N> = 2 MFS <N> = 20 MFS <N> = X / L, Y / L Y / L Comparison of the SBT and NTC schemes in prediction of flow field. Comparison of the SBT with NTC and MFS schemes in prediction of thermal pattern.

18 Results: Contours of local Mach number calculated with SBT scheme.

19 Results: Relative computational time <N> time Algorithm SBT Algorithm NTC Percentage of CPU time usage in DSMC steps. SBT NTC<N>=20 % relative % relative Movement Indexing Collision Sampling Other

20 Results: Comparison of x-velocity and stream lines for (a) SBT scheme (b) NTC scheme < N > = 20

21 Results: Kn = SBT NTC 8 Kn = SBT NTC u slip / U lid T jump X / L X / L Velocity slip (left) and Temperature jump (right) along the driven lid computed by SBT and NTC schemes.

22 Results: Ability of FCT filtering in removing stochastic noises Flood contours: unfiltered early solution Black line: filtered early solution Red line: final filtered solution

23 Conclusions: In this work, we suggested a combination of simplified Bernoulli trial (SBT) algorithm and dual grid strategy to simulate low-speed/low-knudsen rarefied flows at micro/nano scales efficiently The SBT Scheme provides us with: accurate calculations smaller number of particles per cell (< N > 2) less computation time compared to the standard NTC scheme

24 Thanks for your attention

25 References: 1. Stefanov, S.K., "On DSMC calculation of rarefied gas flow s with small number of particles in cells". SIAM J. Sci. comput., Vol. 33 (2), pp Beskok, A. and Karniadakis, G.E., "A model for flows in channels, pipes and ducts at micro and nano scale".j. MicroscaleThermophys. Eng.,Vol. 3, pp Schaaf, S.A. and Chambre, P.L., 1961."Flow of Rarefied Gases". Princeton University press, USA. 4. Karniadakis, G., Beskok, A., and Aluru, N., "Microflows and Nanoflows: fundamentals and simulation". Springer-Verlag, New York. 5. Barber, R.W. and Emerson, D.R., 2006."Challenges in modeling gas-phase flow in microchannels: from slip to transition". Heat Transfer Engineering, Vol. 27 (4), pp Mizzi, S., Emerson, D.R., and Stefanov, S.K., 2007."Effect of rarefaction on cavity flow in the slip regime". Journal of computational and theoretical nanoscience, Vol. 4 (4), pp Chapman, S. and Cowling, T.G., "The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases". 3rd ed., Cambridge university press, Cambridge, UK. 8. Grad, H., 1949."On the kinetic theory of rarefied gases". Communication on pure applied mathematics, Vol. 2 (4), pp

26 References: 9. Kogan, M.N., 1969."Rarefied gas dynamics". 1st ed., Plenum press, New York. 10. Bird, G.A., 1976."Molecular Gas Dynamics". Clarendon Press, Oxford. 11. Wagner, W., "A convergence proof for Bird s direct simulation Monte Carlo method for the Boltzmann equation". J. Statist. Phys., Vol. 66, pp Bird, G.A., 1963."Approach to translational equilibrium in a rigid sphere gas". Physics of fluids, Vol. 6 (10), pp Celenligil, M.C. and Moss, J.N., 1992."Hypersonic rarefied flow about a delta wing direct simulation and comparison with experiment". AIAA journal, Vol. 30 (8), pp Fan, J. and Shen, C., 2002."Micro-scale gas flows". Adv. Mech., Vol. 32, pp Kaplan, C.R. and Oran, E.S., 2002."Nonlinear filtering for low-velocity gaseous microflows". AIAA Journal, Vol. 40, pp Bird, G.A., 1994."Molecular gas dynamics and the direct simulation of gas flows", 2nd ed., Oxford university press, Oxford. 17. Ivanov, M.S. and Rogasinsky, S.V., "Analysis of the numerical techniques of the direct simulation Monte Carlo method in the rarefied gas dynamics". Soviet J. Numer. Anal. Math. Modelling, Vol. 3(6), pp Ivanov, M.S., Markelov, G.N. and Gimelshein, S.F., AIAA Paper

27 References: 19. Boris, J.P and Book, D.L, 1997."Flux-corrected transport: I. SHASTA, a fluid transport algorithm that works". Journal of computational physics, Vol. 135, pp Alexander, F.J., Garcia, A.L. and Alder, B.J., "Cell size dependence of transport coefficients in stochastic particle algorithms". Physics of Fluids, Vol. 10 (6), pp Lofthouse, A.J., 2008, PhD Thesis, University of Michigan.