Three Surprising Hydrodynamic Results Discovered by Direct Simulation Monte Carlo Alejandro L. Garcia San Jose State University & Lawrence Berkeley Nat. Lab. Aerospace Engineering, UT Austin, March 30th, 2006 Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow Anomalous Couette Flow Anomalous Fourier Heat Transfer Molecular Simulations of Gases Exact molecular dynamics inefficient for simulating a dilute gas.
P ree ath Computational time step limited by time of collision. F an Me Interesting time scale is mean free time. Collision Direct Simulation Monte Carlo
Development of DSMC DSMC developed by Graeme Bird (late 60s) Popular in aerospace engineering (70s) Variants & improvements (early 80s) Applications in physics & chemistry (late 80s) Used for micro/nano-scale flows (early 90s) Extended to dense gases & liquids (late 90s) Used for granular gas simulations (early 00s) DSMC is the dominant numerical method for
molecular simulations of dilute gases 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 Select and execute random collisions Sample statistical values Example: Flow past a sphere 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. Collisions (cont.) Post-collision velocities (6 variables) given by: Conservation of
momentum (3 constraints) Conservation of energy (1 constraint) Random collision solid angle (2 choices) v1 v1 Vcm vr v2 vr Direction of vr is uniformly distributed in the unit sphere
v2 DSMC in Aerospace Engineering International Space Station experienced an unexpected 20-25 degree roll about its X-axis during usage of the U.S. Lab vent relief valve. 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) Mean free path ~ 1 m @ 101 Pa
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 DSMC Simulation of Air Slider Flow between platter and read/write head of a computer disk drive
30 nm Pressure Navier-Stokes Pressure R/W Head Inflow 1st order slip Cercignani/ Boltzmann Outflow DSMC
Spinning platter Position F. Alexander, A. Garcia and B. Alder, Phys. Fluids 6 3854 (1994). Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow Anomalous Couette Flow Anomalous Fourier Heat Transfer 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 Anomalous Temperature Velocity profile in qualitative agreement with NavierStokes but temperature has anomalous dip in center. DSMC DSMC
- Navier-Stokes - Navier-Stokes M. Malek Mansour, F. Baras and A. Garcia, Physica A 240 255 (1997). 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. 76 1399 (1994) Burnett Theory for Poiseuille
Pressure profile also anomalous with a gradient normal to the walls. Agreement with Burnetts hydrodynamic theory. Navier-Stokes Burnett Burnett DSMC N-S F. Uribe and A. Garcia, Phys. Rev. E 60 4063 (1999) Super-Burnett Calculations Burnett theory does not get temperature dip but it is recovered at Super-Burnett level. DSMC S-B Xu, Phys. Fluids 15 2077 (2003)
Pressure-driven Poiseuille Flow Compare accelerationdriven and pressure-driven Poiseuille flows periodic reservoir v v High P P
a periodic reservoir Acceleration Driven Pressure Driven Low P 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) Poiseuille: Pressure Pressure profile across the channel (wall-to-wall) DSMC NS
Acceleration Driven DSMC NS Pressure Driven Y. Zheng, A. Garcia, and B. Alder, J. Stat. Phys. 109 495-505 (2002). Poiseuille: Temperature DSMC DSMC NS
Acceleration Driven NS Pressure Driven Y. Zheng, A. Garcia, and B. Alder, J. Stat. Phys. 109 495-505 (2002). Super-Burnett Theory Super-Burnett accurately predicts profiles. Xu, Phys. Fluids 15 2077 (2003) Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow
Anomalous Couette Flow Anomalous Fourier Heat Transfer 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 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. Slip in Couette Flow Simple prediction of velocity profile including slip is mostly in qualitative agreement with DSMC data. = 1.0 = 0.7 = 0.4 = 0.1 K. Tibbs, F. Baras, A. Garcia, Phys. Rev. E 56 2282 (1997)
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 Anomalous Couette Flow At certain values of accommodation, the minimum fluid speed within the fluid. (simple slip theory) Minimum = 0.1 = 0.05 = 0.01 (solid body rotation) K. Tibbs, F. Baras, A. Garcia, Phys. Rev. E 56 2282 (1997) Anomalous Rotating Flow
Outer wall stationary Dilute Gas Dilute Gas Diffusive Walls Specular Walls Dilute Gas Minimum tangential speed occurs in between the walls Intermediate Case Effect occurs when
walls ~80-90% specular BGK Theory Excellent agreement between DSMC data and BGK calculations; the latter confirm velocity minimum at low accommodation. DSMC K. Aoki, H. Yoshida, T. Nakanishi, A. Garcia, Physical Review E 68 016302 (2003). BGK Critical Accommodation for Velocity Minimum BGK theory allows accurate computation of critical accommodation at which the velocity profile has a minimum within the fluid. Approximation is
rI c Kn 2 rO Outline Direct Simulation Monte Carlo Anomalous Poiseuille Flow Anomalous Couette Flow Anomalous Fourier Heat Transfer Fluid Velocity How should one measure local fluid velocity from particle velocities?
Instantaneous Fluid Velocity Center-of-mass velocity in a cell C N mv J u iC M mN i Average particle velocity 1 v N
N v i iC Note that u v vi Estimating Mean Fluid Velocity Mean of instantaneous fluid velocity N (t j )
1 1 1 u u (t j ) vi (t j ) S j 1 S j 1 N (t j ) iC S S where S is number of samples Alternative estimate is cumulative average S u
v (t ) N (t ) j * N (t j ) i iC S j
j j Landau Model for Students Simplified model for university students: Genius Intellect = 3 Not Genius Intellect = 1 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
Average Student? How do you estimate the intellect of the average student? Average of values for the three semesters: ( 3 + 1 + 2 )/3 = 2 Or Cumulative average over all students: (2 x 3 + 14 x 1 )/16 = 20/16 = 1.25 Significant difference because there is a correlation between class size and quality of students in the class. Relation to Student Example N (t j ) 1 S 1 u vi (t j )
S j 1 N (t j ) iC u u* S u v (t ) N (t )
j * N (t j ) i iC S j j j 3 1 2 2
3 2 3 14 1 1.25 16 Average = 3 Average = 1 Average = 2 DSMC Simulations Measured fluid velocity using both definitions. Expect no flow in x for closed, steady systems
Temperature profiles T = 4 T = 2 T system Equilibrium u 10 mean free paths 20 sample cells N = 100 particles per cell x Anomalous Fluid Velocity Mean instantaneous u
fluid velocity measurement gives an anomalous flow in the closed system. Using the cumulative mean, u* , gives the expected result of zero fluid velocity. T = 4 T = 2 Equilibrium Position Properties of Flow Anomaly
4 u 10 Small effect. In this example kT m Anomalous velocity goes 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. Correlations of Fluctuations
At equilibrium, fluctuations of conjugate hydrodynamic quantities are uncorrelated. For example, density is uncorrelated with fluid velocity and temperature, ( x, t )u ( x' , t ) 0 ( x, t ) T ( x' , t ) 0 Out of equilibrium, (e.g., gradient of temperature or shear velocity) correlations appear. Density-Velocity Correlation Correlation of density-velocity fluctuations under T Theory is Landau fluctuating ( x)u ( x' ) hydrodynamics DSMC
When density is above average, fluid velocity is negative COLD u HOT Position x A. Garcia, Phys. Rev. A 34 1454 (1986). Relation between Means of Fluid Velocity From the definitions,
N 2 JN u u u * 1 u * 2 2 N m N From correlation of non-equilibrium fluctuations,
( x)u ( x) x( L x)T This prediction agrees perfectly with observed bias. ( x)u ( x' ) x = x Comparison with Prediction Perfect agreement between mean instantaneous fluid velocity and prediction from correlation of fluctuations. M. Tysanner and A. Garcia,
J. Comp. Phys. 196 173-83 (2004). Grad T Grad u (Couette) u u and Position Measured error in mean instantaneous temperature for small and large N. (N = 8.2 & 132)
Error goes as 1/N Predicted error from density-temperature correlation in good agreement. Mean Inst. Temperature Error Instantaneous Temperature Error about 1 Kelvin for N = 8.2 Position A. Garcia, Comm. App. Math. Comp. Sci. 1 53-78 (2006) Non-intensive Temperature A
B Mean instantaneous temperature has bias that goes 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) References and Spam Reprints, pre-prints and slides available: www.algarcia.org DSMC tutorial & programs in my textbook. Japanese
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