Phys527 Final Presentation Tong Zhu

8/12/2019 Phys527 Final Presentation Tong Zhu

1/16

A Simplified Two-dimensional

DSMC ModelPhys.527 Final Project

Tong Zhu

A Sim lified Two-dimensional DSMC Model . 1/16


2/16

rect s mu at on onte ar oof Gas Flows

Monte Carlo method is a generic numericalmethod for a variety of mathematical problemsbased on computer generation of random

numbers. Direct simulation Monte Carlo (DSMC) method

is the Monte Carlo method for simulation of

dilute gas flows on molecular level, i.e. on thelevel of individual molecules.



3/16

Dilute Gas Dilute gas is a gas where the density parameter

(volume fraction) is small

=nd3

1(1)nis the numerical concentration of gas molecule,dis the diameter of gas molecules.

In dilute gas, only binary collisions between gasmolecules are important.

In Earth atmosphere, air can be considered as adilute gas at any altitude, e.g. on the Earth

surface,= 1.4 103.



4/16

pp cat on o mu a-tions.

Figure 1: Application of DSMC Simulations.A Sim lified Two-dimensional DSMC Model . 4/16

t


5/16

ecoup ng o movementand collision

This model is developed from the MolecularDynamics model in Chapter 8, Sec.6 of our textbook.There are several major differences between DSMC

and MD methods. In this simplified model, thefollowing major features are implemented:

The movement and collision of particles are

decoupled by carefully selecting the time stepdt(Instead of a real-time" interaction via theLennard-Jones potential.) All the particles move

with constant velocity for the duration oft t+dt and at the end of the time step, weselect collision pairs and perform collisions

which update their velocities using Monte Carlomethod.



6/16

wo- mens ona o e(2) Collision models

There are various collision models, Hardsphere(HS), Variable Hard Sphere(VHS) andVariable Soft Sphere(VSS) models are among the

most popular ones. HS model is used here inwhich particles are considered to be solid balls offixed radius.

In a binary collision of theith andjth particle,the collision cross-section can be obtained as:T=

4(ri+rj)

2in whichri andrj are the radii

of the two particles.



7/16

wo- mens ona o e(3) Selection of collision pairs

The collision pairs are selected using theAcceptance-rejection method. We calculate this

value:(Tcr)for each pair of particles and accept

the pair for collision when

RND < Tcr

(T

cr)max

(2)

is satisfied.cr the relative velocity (magnitude) of a pair of

particles;RN Dis a computer-generated random number.



8/16

o ce o compu a onaparameters

There are several parameters that are essential to thevalidity of DSMC calculation:

dt The selection of a time-step that is less thanthe mean collision time (The mean time betweencollisions for a particle) is of primary importance.

Mean free path The mean distance traveledby a particle before it collides with another

particle. Not all N(N1)

2 pairs are tested for

collision, but only those within the mean freepath as we are considering collisions within themean collision time.(This is important for a

Direct Simulation"!)



9/16

re m nary mu a on esu s (1)

A comparison between the DSMC and MD methodsis performed on the system of 8 8 particles in thebox of size 20 15. The system is regarded as dilute

gas with the density parameter= 0.30. (Parametersof the DSMC method remains to be optimized.)

First, Fig.2shows the visualization of theparticle motions of the DSMC and MD methodsand we can see that the DSMC methods does nothave a non-zero minimum distance between two

particles, which is the result of decoupledmovement and collision of particles.If we are to follow the trajectory of a particle, we

can see the randomness of its movement due tothe collisions selected by Monte Carlo methods.

A Sim lified Two-dimensional DSMC Model . /16

re m nar m a on es s


10/16


Figure 2: Comparison between the DSMC and MD

methods.


re m nary mu a on esu s


11/16


Second, Fig.3shows the comparison ofVxhistograms between the DSMC and MD methodsand we can see that they essentially agree with

each other for this simulation condition, whichsuggests that the current DSMC methodthermalizes the system properly.


re m nary mu a on esu s


12/16


Vx

P

robability

-2 0 2 40

0.002

0.004

0.006

0.008

0.01

0.012

DSMC

MD

Figure 3: Comparison ofVxhistograms.



13/16

Discussion

Here we are studying the effect ofNcoll number ofcollisions to perform within each time stepdt. Fig.4shows the comparison amongNcoll =1,Ncoll =10 and

Ncoll =100 for the DSMC method.We can see that there is some difference in theprobability distribution ofVxand we need to be

careful in the selection ofNcoll.



14/16

Discussion

Figure 4: The effect of Ncoll number of collisions

within each time stepdt.



15/16

Future work

In the bookMolecular Gas Dynamics and the DirectSimulation of Gas Flows(by G.A.Bird), moreadvanced discussions of DSMC methods are

presented and can be used to accurately simulate thesystem of dilute gas.Future work will include a standard way to divide the

computation domain into cells of approximately thesize of mean free pathwhich has the followingmerits:

Enabling more efficient selection of collisionpairs.

Enabling a better estimation of number of

collisionsNcoll to perform based on the cell sizeand particle number density.



16/16

Thank you!


Documents

Phys527 Final Presentation Tong Zhu