Measuring Fluid Velocity and Temperature in DSMC Alejandro L. Garcia Lawrence Berkeley National Lab. & San Jose State University Collaborators: J. Bell, M. Malek-Mansour, M. Tysanner, W. Wagner Direct Simulation Monte Carlo: Theory, Methods, and Applications

2 Landau Model for Students Simplified model for university students: Genius Intellect = 3 Not Genius Intellect = 1

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

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

5 Fluid Velocity How should one measure local fluid velocity from particle velocities?

6 Instantaneous Fluid Velocity Center-of-mass velocity in a cell Average particle velocity Note that vivi

7 Mean of Instantaneous Fluid Velocity Mean of instantaneous fluid velocity is where S is number of samples or

8 Cumulative Mean Fluid Velocity Alternative estimate is from cumulative measurement Average = 3 Average = 1 Average = 2

Which definition should be used? Are they equivalent? Let’s run some simulations and find out.

10 DSMC Simulations Temperature profiles Measured fluid velocity using both definitions. Expect no flow in x for closed, steady systems  T system  T = 2  T = 4 Equilibrium x Thermal Walls 10 m.f.p. 20 sample cells N = 100 particles per cell

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

12 Properties of Flow Anomaly Small effect. In this example 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 Couette flow.

13 Mechanical & Hydrodynamic Variables Mechanical variables: Mass, M ; Momentum, J ; Kinetic Energy, E Hydrodynamic variables: Fluid velocity, u ; Temperature, T ; Pressure, P Relations: u(M,J) = J / M T(M,J,E), P(M,J,E) more complicated

14 Relation with Mechanical Variables Relation with mass and momentum is so Mean Instantaneous Mean Cumulative

15 Means of Hydrodynamic Variables Mean of instantaneous values Mean of cumulative values (mechanical variables) At equilibrium, Not equivalent out of equilibrium.

16 Correlations of Fluctuations At equilibrium, fluctuations conjugate hydrodynamic quantities are uncorrelated. For example, density is uncorrelated with fluid velocity and temperature, Out of equilibrium, (e.g., gradient of temperature or shear velocity) correlations appear.

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

18 Relation between Means of Fluid Velocity From the definitions, From correlation of non-equilibrium fluctuations, This prediction agrees perfectly with observed bias. x = x ’

19 Comparison with Prediction Perfect agreement between mean instantaneous fluid velocity and prediction from correlation of fluctuations. Position Grad T Grad u (Couette)

20 Reservoir Simulations Equilibrium system has anomalous mean instantaneous fluid velocity when constant number of particles, N, generated in reservoir. Non-equilibrium correlation of density-momentum fluctuations unless Poisson distributed particle number in reservoir. System Reservoir Constant N Mean Instantaneous Velocity Distance from reservoir (mfp) Poisson N

21 Translational Temperature Translational temperature defined as where u is center-of-mass velocity. Even at equilibrium (  u  =0), care needed in evaluating instantaneous mean temperature since

22 Instantaneous Temperature Instantaneous temperature defined as Correct mean at equilibrium but similar bias as with fluid velocity out of equilibrium because density and temperature fluctuations are correlated.

23 DSMC Simulation Results 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. Position Mean Inst. Temperature Error Error about 1 Kelvin for N = 8.2

24 Concluding Remarks Avoid measurement bias by measuring means of mechanical variables and use them to compute the means of hydrodynamic variables. Mean instantaneous values have error that goes as 1/N and as non-equilibrium gradient; typically small error but comparable to “ghost” effects.

25 Concluding Remarks (cont.) Measurement error not limited to DSMC; physical origin so also present in MD. Sometimes one needs the instantaneous value of a hydrodynamic variable (coupling to a CFD calculation in a hybrid; temperature dependent collision rate, etc.). Be careful! Correlations important for radial random walk errors?

26 References "Measurement Bias of Fluid Velocity in Molecular Simulations", M. Tysanner and A. Garcia, Journal of Computational Physics (2004). "Non-equilibrium behavior of equilibrium reservoirs in molecular simulations", M. Tysanner and A. Garcia, International Journal of Numerical Methods in Fluids (2005). "Estimating Hydrodynamic Quantities in the Presence of Microscopic Fluctuations", A. Garcia, submitted to Communications in Applied Mathematics and Computational Science (July 2005). Available at