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 TeachingFirst 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 + 1 + 2 )/3 = 2

Or

Cumulative average over all students:(2 x 3 + 14 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 VelocityCenter-of-mass velocity in a cell

k

N

kii

k

kk mN

mv

M

Ju

k

Average particle velocity

Note that kk vu

kN

kii

kk v

Nv

1vi

7

Mean of Instantaneous Fluid Velocity

Mean of instantaneous fluid velocity is

S

jjkk tu

Su )(

1

where S is number of samples

S

j

tN

kiji

jkk

jk

tvtNS

u)(

)()(

11

or

8

Cumulative Mean Fluid Velocity

Alternative estimate is from cumulative measurement

S

j jk

S

j

tN

ki ji

ktN

tvu

jk

)(

)()(

*

25.116

11432*

ku

23

213

ku

Average = 3 Average = 1 Average = 2

Which definition should be used?Are they equivalent?

Let’s run some simulations and find out.

10

DSMC SimulationsTemperature profilesMeasured fluid velocity

using both definitions.Expect no flow in x forclosed, steady systems

T systemT = 2

T = 4

Equilibrium

xThermal Walls

10 m.f.p.

20 sample cellsN = 100 particles per cell

11

Anomalous Fluid Velocity

T = 4

T = 2

Equilibrium

Position

kuMean 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.

mkTu 410

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

k

kk M

Ju

k

kk M

Ju

*

k

N

kii

k

kk mN

mv

M

Ju

k

so

Mean Instantaneous Mean Cumulative

15

Means of Hydrodynamic Variables

Mean of instantaneous values

Mean of cumulative values (mechanical variables)

),,( EJMTT ),( JMuu

At equilibrium, *uu

*TT

),(*

JMuu ),,(*

EJMTT

Not equivalent out of equilibrium.

16

Correlations of FluctuationsAt 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.

0),'(),( txutx

0),'(),( txTtx

17

Density-Velocity CorrelationCorrelation of density-velocity fluctuations under T

Position x’

)'()( xux DSMC

A. Garcia, Phys. Rev. A 34 1454 (1986).

COLD HOT

When density is above average, fluid velocity is negative

u

Theory is Landau fluctuating hydrodynamics

18

Relation between Means of Fluid Velocity

From the definitions,

From correlation of non-equilibrium fluctuations,

u

uNm

NJ

N

Nuu

*22

2

*1

TxLxxux )()()(

This prediction agrees perfectly with observed bias.

)'()( xux

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)

u

u and

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

Res

ervo

ir

Constant NM

ean

In

stan

tan

eou

s V

eloc

ity

Distance from reservoir (mfp)

Poisson N

21

Translational TemperatureTranslational temperature defined as

where u is center-of-mass velocity.

M

JE

Mk

muv

kN

mT

N

ii 23

2)(

3

2

2

22222)( uvuvuv

Even at equilibrium (u=0), care needed in evaluating instantaneous mean temperature since

22

Instantaneous Temperature

Instantaneous temperature defined as

N

kiki

kk uv

Nk

mT 2)(

)1(3

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

23

DSMC Simulation ResultsMeasured 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

Mea

n In

st. T

empe

ratu

re E

rror

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 196 173-83 (2004).

"Non-equilibrium behavior of equilibrium reservoirs in molecular simulations", M. Tysanner and A. Garcia, International Journal of Numerical Methods in Fluids 48 1337-1349 (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 www.algarcia.org