Surprising i Hydrodynamic d Results Discovered by means of Direct Simulation Monte Carlo

Transcription

1 Surprising i Hydrodynamic d Results Discovered by means of Direct Simulation Monte Carlo Alejandro L. Garcia San Jose State University Lawrence Berkeley Nat. Lab. RTO-AVT-VKI Lecture Series, January 24 th 2011, y von Karman Institute for Fluid Dynamics

2 Discoveries made by DSMC This lecture will describe four interesting discoveries made using Direct Simulation Monte Carlo (DSMC). First two occur due to Knudsen number effects: * Anomalous Poiseuille Flow * Anomalous Couette Flow The other two occur due to thermal fluctuations: * Anomalous Temperature Gradient Flow * Anomalous Diffusion Flow

3 Outline Direct Simulation Monte Carlo Anomalous Poiseuille ill Flow Anomalous Couette Flow Anomalous Temperature Gradient Flow Anomalous Diffusion i Flow

4 Molecular Dynamics for Dilute Gases Molecular l dynamics inefficient for simulating the kinetic scale. Collision Relevant time scale is mean free time but MD computational time step limited by time of collision. DSMC time step is large because collisions are evaluated stochastically. Volume of potential collision partners

5 Direct Simulation Monte Carlo Development of DSMC DSMC developed by Graeme Bird (late 60 s) Popular in aerospace engineering (70 s) Variants & improvements (early 80 s) Applications in physics & chemistry (late 80 s) Used for micro/nano-scale flows (early 90 s) Extended to dense gases & liquids (late 90 s) Granular gas simulations (early 00 s) Multi-scale modeling of complex fluids (late 00 s) DSMC i th d i t i l th d f DSMC is the dominant numerical method for molecular simulations of dilute gases

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

7 DSMC Collisions Sort particles into spatial collision cells Loop over collision cells Compute collision frequency in a cell Select random collision partners within cell Process each collision Two collisions during t Probability that a pair collides only depends on their relative velocity.

8 Collisions (cont.) Post-collision velocities (6 variables) )given by: Conservation of V momentum (3 constraints) cm v 2 v 2 Conservation of energy (1 constraint) v r v r Random collision solid angle (2 choices) v 1 v 1 Direction of v r is uniformly distributed in the unit sphere

9 DSMC in Aerospace Engineering i International Space Station experienced an unexpected degree roll about its X-axis during usage of the U.S. Lab vent relief valve. Mean free path ~ Pa Analysis using DSMC provided detailed insight into the anomaly and revealed that the zero thrust T-vent imparts significant torques on the ISS when it is used. NASA DAC (DSMC Analysis Code)

10 Computer Disk Drives Mechanical system resembles phonograph, with the read/write head located on an air bearing slider positioned at the end of an arm. Air flow Platter

11 Multi-scales of Air Bearing Slider Slider (1 mm long) flies about 30 nm above platter. This is like a 747 flying 1.5 millimeters above ground Sensitivity decreases exponentially with height ht

12 DSMC Simulation of Air Slider Flow between platter and read/write head of a computer disk drive Navier-Stokes 1 st order slip re Inflow 30 nm R/W Head Pressure Outflow Pressur DSMC Cercignani/ Boltzmann Spinning platter Position F. Alexander, A. Garcia and B. Alder, Phys. Fluids (1994).

13 Outline Direct Simulation Monte Carlo Anomalous Poiseuille ill Flow Anomalous Couette Flow Anomalous Temperature Gradient Flow Anomalous Diffusion i Flow

14 Acceleration Poiseuille Flow Similar to pipe flow but between pair of flat planes (thermal walls). Push the flow with a body force, such as gravity. Channel widths roughly 10 to 100 mean free paths (Kn 0.1 to 0.01) periodic v a periodic

15 Anomalous Temperature Velocity yprofile in qualitative agreement with Navier- Stokes but temperature has anomalous dip in center. DSMC - Navier-Stokes DSMC - Navier-Stokes M. Malek Mansour, F. Baras and A. Garcia, Physica A (1997).

16 Heat Flux &Temperature Gradient Heat is generate inside the system by shearing and diti is removed at the walls. Heat is flowing from cold to hot near the center. DSMC - Navier-Stokes HEAT M. Malek Mansour, F. Baras and A. Garcia, Physica A (1997).

17 Burnett Theory for Poiseuille Pressure profile also anomalous with a gradient normal to the walls. Agreement with Burnett s hydrodynamic theory. P but no flow Navier-Stokes Burnett Burnett DSMC N-S F. Uribe and A. Garcia, Phys. Rev. E (1999)

18 BGK Theory for Poiseuille Kinetic theory calculations using BGK theory predict the dip. BGK DSMC Dip term Navier-Stokes M. Tij, A. Santos, J. Stat. Phys (1994)

19 Super-Burnett Calculations Burnett theory does not get temperature dip but it is recovered at Super-Burnett level. DSMC S-B Xu, Phys. Fluids (2003)

20 Pressure-driven Poiseuille Flow periodic reservoir High P Compare accelerationdriven and pressure-driven Poiseuille flows v a periodic Acceleration Driven v P reservoir Pressure Driven Low P

21 Poiseuille: Fluid Velocity Velocity profile across the channel (wall-to-wall) NS & DSMC almost identical NS & DSMC almost identical Acceleration Driven Pressure Driven Note: Flow is subsonic & low Reynolds number (Re = 5)

22 Poiseuille: Pressure Pressure profile across the channel (wall-to-wall) DSMC DSMC NS NS Acceleration Driven Pressure Driven Y. Zheng, A. Garcia, and B. Alder, J. Stat. Phys (2002).

23 Poiseuille: Temperature DSMC NS DSMC NS Acceleration Driven Pressure Driven Y. Zheng, A. Garcia, and B. Alder, J. Stat. Phys (2002).

24 Super-Burnett Theory Super-Burnett accurately predicts profiles. Xu, Phys. Fluids (2003)

25 Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow Anomalous Couette Flow Anomalous Temperature Gradient Flow Anomalous Diffusion i Flow

26 Couette Flow Dilute gas between concentric cylinders. Outer cylinder fixed; inner cylinder rotating. Low Reynolds number (Re 1) so flow is laminar; also subsonic. A few mean free paths Dilute Gas

27 Slip Length The velocity of a gas moving over a stationary, thermal wall has a slip length. Slip length Thermal Wall Moving Gas This effect was predicted by Maxwell; confirmed by Knudsen. Physical origin is difference between impinging and reflected velocity distributions of the gas molecules. Slip length for thermal wall is about one mean free path. Slip increases if some particle reflect specularly; define accommodation coefficient,, as fraction of thermalize (non-specular) reflections.

28 Slip in Couette Flow Simple prediction of velocity profile including slip is mostly in qualitative agreement with DSMC data. = 0.4 = 1.0 = 0.1 = 0.7 K. Tibbs, F. Baras, A. Garcia, Phys. Rev. E (1997)

29 Diffusive and Specular Walls Outer wall stationary Dilute Gas Diffusive Walls When walls are completely specular the gas undergoes solid body rotation so v = r Dilute Gas Specular Walls

30 Anomalous Couette Flow At certain values of accommodation, the minimum fluid speed within the fluid. Minimum (simple slip theory) = 0.1 = 0.05 (solid body rotation) = 0.01 K. Tibbs, F. Baras, A. Garcia, Phys. Rev. E (1997)

31 Anomalous Rotating Flow Outer wall stationary Dilute Gas Dilute Gas Diffusive Walls Minimum tangential speed occurs in between the walls Dilute Gas Intermediate Case Specular Walls Effect occurs when walls ~80-90% specular

32 BGK Theory Excellent agreement between DSMC data and BGK calculations; the latter confirm velocity minimum at low accommodation. DSMC BGK K. Aoki, H. Yoshida, T. Nakanishi, A. Garcia, Physical Review E (2003).

33 Critical Accommodation for Velocity Minimum BGK theory allows accurate computation of critical accommodation at which the velocity profile has a minimum i within the fluid. Approximation is r I c 2 r O Kn

34 Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow Anomalous Couette Flow Anomalous Temperature Gradient Flow Anomalous Diffusion i Flow

35 Fluid Velocity H h ld l l fl id l it How should one measure local fluid velocity from particle velocities?

36 Instantaneous Fluid Velocity Center-of-mass velocity in a cell C u J M N ic mv mn Average particle velocity v 1 N N v i ic i v i Note that u v

37 Estimating Mean Fluid Velocity Mean of instantaneous fluid velocity S t N S j ) ( j C i j i j j j t v t N S t u S u 1 1 ) ( ) ( 1 1 ) ( 1 where S is number of samples where S is number of samples Alternative estimate is cumulative average S S j t N C i j i t N t v u j ) ( ) ( ) ( * g j j t N ) (

38 Landau Model for Students Simplified model for university students: Genius Not Genius Itll Intellect t= 3 Intellect = 1

39 Three Semesters of Teaching First semester Second semester Third semester Sixteen students in three semesters Total value is 2x3+14x1 = 20. Average = 3 Average = 1 Average = 2

40 Average Student? How do you estimate the intellect of the average student? Or Average of values for the three semesters: (3+1+2)/3 + = 2 Cumulative average over all students: (2 x x 1 )/16 = 20/16 = 1.25 Significant difference because there is a correlation between class size and quality of students in the class.

41 Relation to Student Example S t N j i j t v t N S u ) ( ) ( ) ( 1 1 S S j t N C i j i t N t v u j ) ( ) ( ) ( * j C i j j t N S 1 ) ( j j t N ) ( u * u 16 * Average = 3 Average = 1 Average = 2

42 DSMC Simulations Measured fluid velocity Temperature profiles using both definitions. Expect no flow in x for T = 4 closed, steady systems T system T = 2 Equilibrium u 10 mean free paths 20 sample cells N = 100 particles per cell x

43 Anomalous Fluid Velocity Mean instantaneous t J u fluid velocity measurement gives an anomalous flow in a closed system at steady state with T. Using the cumulative mean, u *, gives the expected result of zero fluid velocity. x( L x) T M Hot Cold u J 0 * M Hot Cold

44 Anomalous Fluid Velocity Mean instantaneous fluid velocity measurement gives an anomalous flow in the closed system. u T = 4 T = 2 Using the cumulative mean, u *, gives the expected result of zero fluid velocity. Position Equilibrium

45 Properties of Flow Anomaly kt Small effect. In this example u 10 4 m Anomalous velocity ygoes as 1/N where N is number of particles per sample cell (in this example N = 100). Velocity goes as gradient of temperature. Does not go away as number of samples increases. Similar anomaly found in plane Couette flow.

46 Correlations of Fluctuations At equilibrium, i fluctuations ti of conjugate hydrodynamic d quantities are uncorrelated. For example, density is uncorrelated with fluid velocity and temperature, ( x, t) u( x', t) ( x, t) T ( x', t) 0 0 Out of equilibrium, (e.g., gradient of temperature or shear velocity) correlations appear.

47 Density-Velocity Correlation Correlation of density-velocity fluctuations under T ( x) u( x') DSMC Theory is Landau fluctuating hydrodynamics When density is above average, g, fluid velocity is negative COLD u HOT Position x A. Garcia, Phys. Rev. A (1986).

48 Relation between Means of Fluid Velocity From the definitions, 2 N JN u u u u 1 * 2 N 2 * m N From correlation of non-equilibrium fluctuations, ( x) u( x) x( L x) T ( x) u( x' ) This prediction agrees perfectly with observed bias. x = x

49 Comparison with Prediction Perfect agreement between mean Grad T instantaneous fluid velocity and prediction from correlation of fluctuations. u and u Grad u (Couette) M. Tysanner and A. Garcia, J. Comp. Phys (2004). Position

50 Inflow / Outflow Boundaries Unphysical fluctuation correlations also appear boundary conditions are not thermodynamically correctly. For example, number of particles crossing an inflow / outflow boundary are Poisson distributed. Reservoir boundary Surface injection boundary

51 Anomalous Equilibrium Flow Surface injection boundary Mean In nstantane eous Fluid Velocity Constant Poisson Mean instantaneous fluid velocity is non-zero even at equilibrium if boundary conditions not treated correctly with regards to fluctuations. M. Tysanner and A. Garcia, Int. J. Num. Meth. Fluids (2005)

52 Instantaneous Temperature error in rmeasured mean instantaneous temperature for small and large N. (N = 8.2 & 132) Error goes as 1/NN Mean Inst. Te emperatu ure Error Error about 1 Kelvin Predicted error from for N = 8.2 density-temperature correlation in good agreement. Position Hot Cold A. Garcia, Comm. App. Math. Comp. Sci (2006)

53 Non-intensive Temperature A B Mean instantaneous t temperature t has bias that tgoes as 1/N, so it is not an intensive quantity. Temperature of cell A = temperature of cell B yet not equal to temperature of super-cell (A U B)

54 Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow Anomalous Couette Flow Anomalous Temperature Gradient Flow Anomalous Diffusion i Flow

55 Diffusion & Fluctuations As we ve seen, fluctuations are enhanced when a system is out of equilibrium, such as under a gradient imposed by boundaries. Equilibrium concentration gradient (induced by gravity) Steady-state concentration gradient (induced by boundaries)

56 Giant Fluctuations ti in Mixing i Fluctuations grow large during mixing even when the two species are identical (red & blue). Snapshots of the concentration during the diffusive mixing of two fluids (red and blue) at t = 1 (top), t = 4 (middle), and t = 10 (bottom), starting from a flat interface (phase-separated separated system) at t = 0.

57 Experimental Observations Giant fluctuations in diffusive mixing seen in lab experiments. Experimental images (1mm side) of scatterring from the interface between two miscible fluids (from A. Vailati & M. Giglio, Nature 1997)

58 Diffusion & Fluctuations Using Landau-Lifshitz Lifshitz fluctuating hydrodynamics in the isothermal, incompressible approximation we may write, for the fluctuations of concentration and velocity. Solving in Fourier space gives the correlation function,

59 Diffusion & Fluctuations The total mass flux for concentration species is, where there are two contributions, the bare diffusion coefficient and the contribution due to correlation of fluctuations. ti For a slab geometry (L z << L x << L y ) we have, y L y Notice that diffusion enhancement goes as ln L x L x L z

60 DSMC Measurements Can separate the contributions to the concentration flux as, = D eff c = D 0 c + ΔD c In DSMC we can easily measure and c L y and find the bare diffusion coefficient D 0 and the total effective diffusion coefficient D eff L x L z

61 DSMC Results for D eff and D 0 D eff D 0 L y L x L z A. Donev, J. Bell, A. Garcia, J. Goodman, E. Vanden-Eijnden, in preparation

62 Global Enhancement of Diffusion Spectrum of hydrodynamic fluctuations is truncated at wavenumbers given by the size of the physical system. L y L y L y + L L z L z L z x Lx 2Lx The wider system can accommodate long wavelength y g g fluctuations, thus it has enhanced diffusion.

63 References and Spam Reprints, pre-prints and slides available: DSMC tutorial & programs in my textbook. Japanese

64 RGD 2012 in Zaragoza, Spain Summer 2012 Hosted b ZCAM the Spanish node of the E ropean Centers for Hosted by ZCAM, the Spanish node of the European Centers for Atomic and Molecular Calculations (CECAM)

65 DSMC 2011 Workshop Late September 2011 Santa Fe, New Mexico Hosted by Sandia Nat. Lab.

66 Von Neumann Symposium on July 4-8, 2011 Snowmass, Utah Sponsored by American Mathematical Society Multi-scale Algorithms