dsmc direct simulation monte carlo of gas flows

7/29/2019 DSMC Direct Simulation Monte Carlo of Gas Flows

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Direct Simulation Monte Carlo(DSMC) of gas flows

Monte Carlo method: Definitions

Basic concepts of kinetic theory of gases Applications of DSMC Generic algorithm of the DSMC method Summary

Reading: G.A. Bird, Molecular gas dynamics and the direct simulationof gas flows. Clarendon Press, 1994


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Direct Simulation Monte Carlo of gas flows: Definitions

Monte Carlo method is a generic numerical method for a variety ofmathematical problems based on computer generation of randomnumbers.

Direct simulation Monte Carlo (DSMC) method is the Monte Carlomethod for simulation of dilute gas flows on molecular level, i.e. onthe level of individual molecules. To date DSMC is the basicnumerical method in the kinetic theory of gases and rarefied gas

dynamics.

Kinetic theory of gases is a part of statistical physics where the flowof gases are considered on a molecular level and described in terms

of changes of probabilities of various states of gas molecules inspace and in time.


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Dilute gas DSMC is applied for simulations of flows of a dilute gas Dilute gas is a gas where the density parameter (volume fraction) is small

= n d3

only binary collisions between gas molecules are important


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Flow regimes of a dilute gas Collision frequency of a molecule with diameter d is the averaged number of collisions

of this molecule per unit time

Mean free path of a molecule = g (1 / ) = 1 / ( n )

Knudsen number Kn = /L = 1 / ( n L ), L is the flow length scale

Knudsen number is a measure of importance of collisions in a gas flowKn > 1 (Kn > 10)

Continuum flow Transitional flow Free molecular (collisionless) flow

2d

Collision cross-section = d2

g t

g

Relative velocity of molecules, g= |v1-v2|

gn

t

tgn=

=

L

Local equilibrium Non-equilibrium flows


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Applications of DSMC simulations (I)

Aerospace applications: Flows in upper atmosphere and in vacuum

Planetary science and

Satellites and spacecraftson LEO and in deep space

Re-entry vehicles inupper atmosphere

Nozzles and jetsin space environment

Dynamics of upperplanetary atmospheres

Atmospheres of smallbodies (comets, etc)astrophysics

Global atmosphericevolution (Io, Enceladus, etc)


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Applications of DSMC simulations (II)

Fast, non-equilibrium gas flows (laser ablation, evaporation, deposition)

Flows on microscale, microfluidics

Flows in microchannesFlows in electronic devices

and MEMSFlow over microparticlesand clusters

SootclustersSi waferHD


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Basic approach of the DSMC method

Gas is represented by a set of Nsimulated molecules (similar to MD)X(t)=(r1(t),V1(t),,rN(t),VN(t))

Velocities Vi(and coordinates ri) of gas molecules are random variables. Thus, DSMC isa probabilistic approach in contrast to MD which is a deterministic one.

Gas flow is simulated as a change of X(t) in time due to

Free motion of molecules or motion under the effect of external (e.g. gravity) forces Pair interactions (collisions) between gas molecules

Interaction of molecules with surfaces of streamlined bodies, obstacles, channelwalls, etc.

Computational domain

ri vi

Rebound of a molecule from the wall

External force field fe

fei

Pair collision

In typical DSMC simulations(e.g. flow over a vehicle inEarth atmosphere) thecomputational domain is apart of a larger flow. Hence,some boundaries of a

domain are transparent formolecules and number ofsimulated molecules, N, isvaried in time.


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Statistical weight of simulated molecules in DSMC

Number of collisions between molecules is defined by the collision frequency = n g. For the same velocities of gas molecules, the number of collisions depends on nand .

Consider two flows

Thus, in DSMC simulations the number of simulated molecules can not be equal to thenumber of molecules in real flow. This differs DSMC from MD, where every simulatedparticle represents one molecule of the real system.

Every simulated molecule in DSMC represents W molecules of real gas, whereW= n/nsim is the statistical weight of a simulated molecule. In order to make flow ofsimulated molecules the same as compared to the flow of real gas, the cross-section ofsimulated molecules is calculated as follows

W

n

n

sim

sim ==

If n1 1 = n2 2 ,

then the collision frequenciesare the same in both flows.

If other conditions in both flowsare the same, then two flows are

equivalent to each other.

n1, 1 n2, 2


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DSMC algorithm (after G.A. Bird)

Any process (evolving in time or steady-state) is divided into short timeintervals time steps t

X(t)=(r1(t),V1(t),,rN(t),VN(t))

Xn= X(tn), state of simulated molecules at time tn

Xn+1

= X(tn+1

), state of simulated molecules at time tn+1

= tn

+

t At every time step, the change of Xn into Xn+1 (Xn Xn+1) is splitted into a

sequence of three basic stages Stage I. Collisionless motion of molecules (solution of the motion

equations)Xn X*

Stage II. Collision sampling (pair collisions between molecules)

X* X**

Stage III. Implementations of boundary conditions (interactions ofmolecules with surfaces, free inflow/outflow of molecules throughboundaries, etc)

X** Xn+1

Thus, in contrast with MD, where interaction between particles aredescribed by forces in equations of motion, in DSMC, interactionsbetween particles is described by means of a special random algorithm(collision sampling) which is a core of any DSMC computer code.


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DSMC vs. Molecular Dynamics (MD) simulations

MD simulations: Direct solution of the motion equations at a time step

ei

j

iji

idt

dm ff

v+=2

2

eii

idt

dm f

v=2

2

Ni ,...,1=

=j

iji

idt

dm f

v2

2

Interaction force between molecules iandj External force

Both MD and DSMC are particle-based methods

DSMC simulations:

Splitting at a time step

(*)

Special probabalistic approach for sampling of binary collisions instead ofdirect solution of Eq. (*)

Use of statistical weights NNi sim


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DSMC vs. MD and CFD in simulations of gas flows

Three alternative computational methodology

for simulations of gas flows

MD, Molecular dynamic simulations

DSMC, Direct simulation Monte Carlo

CFD, Computational fluid dynamics

LowModerateHighRelativecomputational cost

No limitations,usually, /L < 0.1

No limitations, usually,/L > 0.01

Less then 1micrometer

Typical flow lengthscale L

Continuum near-equilibrium flows

Transitional and freemolecular non-equilibrium flows

Dense gas flows,phase changes,complex molecules

Where applied

Dilute gasDilute gasDilute gas, densegas, clusters, etc.

Gas state

Navier-Stokesequations

Boltzmann kineticequation

Classical equationsof motion forparticles

Theoretical model

CFDDSMCMD

L


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Stage I. Collisionless motion

X

n

= (r1n

,V1n

,,rNn

,VNn

)Xn X*

For every molecule, its equations of motion are solved for a time step

dri/dt= Vi, midVi/dt= fei, i= 1, , N

ri(t

n

)=r

i

n

,V

i(tn

)=V

i

n

,mi is the real mass of a gas molecule

In case of free motion (fei= 0)

X*=(r1*,V1n,,rN*,VNn)ri* = ri

n+ tVin

If external force field is present, the equations of motion are solvednumerically, e.g. by the Runge-Kutta method of the second order

X*=(r1*,V1

*,,rN*,VN

*)

ri= rin+ (t/2) Vi

n, Vi= Vin+ (t/2) fei(ri

n)

ri*= ri

n+ tVi, Vi

* = Vin+ tfei(ri)


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Stage II. Collision sampling (after G.A. Bird)

X* = (r1

*,V1

*,,rN

*,VN

*)X* X**

Computational domain is divided into a mesh of cells.

For every molecule, the index of cell to which the molecule belongs is calculated(indexing of molecules).

At a time step, only collisions between molecules belonging to the same cell are takeninto account

Every collision is considered as a random event occurring with some probability ofcollision

In every cell, pairs of colliding molecules are randomly sampled (collision sampling in acell). For every pair of colliding molecules, pre-collisional velocities are replaced bytheir post-collisional values.

Cell


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Collision sampling in a cell : Calculation of the

collision probability during time step

)(simij

|| ijijg g=

cell

ijsimij

ij V

tg

P

=

)(

ijij vvg =

ijg

tgij

Relative velocity of molecules i and j

Probability of a random collision

between molecules i and j duringtime step

Cell of volume Vcell containing Ncellmolecules

j

i

Molecules are assumed to be

distributed homogeneously withinthe cell


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Collision sampling in a cell : Calculation of particle

velocities after a binary collision of hard sphere

Velocities

beforecollision

Velocities

aftercollision

Conservation laws of momentum,

energy, and angular momentum

Equations for molecule

velocities after collision

For hard sphere (HS) molecules unity vector n is an isotropic random vector:

==

===

2

1

22

cos1sin,21cos

),2sin(sin),2cos(sin,cos zyx nnn

i is a random number distributed with equal probability from 0 to 1.

In a computer code, it can be generated with the help of library functions,which are called random number generators.


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Collision sampling in a cell: The primitive scheme

i = 1

j = i + 1

Pij = tij(sim)gij/ Vcell

< Pij

)2sin(sin

)2cos(sin

cos

cos1sin

21cos

2

2

2

1

=

=

=

=

=

z

y

x

n

n

n

j = j + 1

i = i + 1

j < Ncell

Does collision

between moleculesi and j occur?

no yes

yes

yes

no

noGo to the next cell

Calculation of the

collision probability

Calculation ofvelocities aftercollision

Are there otherpairs of moleculesin the cell?

Disadvantage of

the primitivescheme:

Number ofoperation ~ Ncell

2

In real DSMC

simulations, moreefficient schemes

for collisionsampling are used,e.g. the NTC scheme

by Bird

i < Ncell - 1

P( < Pij) = Pij


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X**

= (r

1

**

,V

1

**

,,r

N

**

,V

N

**

)X** Xn+1

Implementation of boundary conditions depends on the specifics of the flow problemunder consideration. Typically, conditions on flow boundaries is the most specific partof the problem

Examples of boundary conditions

Impermeable boundary (e.g. solid surface):

rebound of molecules from the wall

Permeable boundary between the

computational domain and the reservoir

of molecules (e.g. Earth atmosphere) :

free motion of molecules through boundary

reproducing inflow/outflow fluxes

Stage III. Implementation of boundary conditions

Computational domainReservoir (Earth atmosphere)

Flow over a re-entry vehiclein Earth atmosphere


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Rebound of molecules from an impermeable wall

Boundary condition is based on themodel describing the rebound of anindividual gas molecule from the wall.

The model should defined the velocity ofreflected molecule as a function of the

velocity of the incident molecule,Vr=Vr(Vi,nw,Tw,).

In DSMC simulations, velocity of everymolecule incident to the wall is replacedby the velocity of the reflected molecule.

In simulations, the Maxwell models of

molecule rebound are usually applied.

Maxwell model of specular scattering: Amolecule reflects from the wall like anideal billiard ball, i.e.

Vrx= Vix, Vry= - Viy, Vrz= Viz

Vr= Vi 2 ( Vi nw) nw

Disadvantage: heat flux and shear drag on

the wall are zero (Vr2 = Vi

2). The model is

capable to predict normal stress only.

Maxwell model of diffuse scattering: Velocity

distribution function of reflected molecules isassumed to be Maxwellian:

Random velocity of a reflected molecule can

be generated using random numbers i

Vi

nw

Vr

x

y

z Tw, wall temperature

)2sin(lg)/(2

lg)/(2

)2cos(lg)/(2

2exp

))/(2()(

21

3

21

2

2/3

=

=

=

=

wrz

wry

wrx

w

r

r

rr

TmkV

TmkV

TmkV

kT

m

Tmk

nf

VV


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At every time step of DSMC simulations

1. All molecules moving from the computational domain intoreservoir is excluded from further simulations

2. Reservoir is filled by N=n

Vmolecules, where Vis the

reservoir volume. Random coordinates of every moleculein reservoir are generated homogeneously, randomvelocities are generated from the Maxwelian distribution.

3. All molecules in the reservoir are moved: Their positionsand velocities are changed with accordance to theirequations of motion during a time step.

4. All molecules from the reservoir that entered thecomputational domain during a time step are included tothe set of simulated molecules. All other molecules fromthe reservoir are excluded from further simulations.

Free inflow/outflow of molecules on a permeable boundary

( )

=

kT

m

Tmk

nf

2exp

))/(2()(

2

2/3

UVV

)2sin(lg)/(2

)2cos(lg)/(2)2cos(lg)/(2

65

43

21

+=

+=

+=

TmkUV

TmkUV

TmkUV

zz

yy

xx

Computational domainReservoir

3121

2121

1121

)(

)(

)(

+=

+=

+=

zzzz

yyyy

xxxx

x

y

x1 x2

Random coordinates:

Random velocities:

Maxwellian velocity distributionin the reservoir:

n, concentration

T, temperature,

U, gas velocity


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Summary

DSMC is a numerical method for simulations of free-molecular, transitional and

near-continuum flows of a dilute gas on a level of individual molecules.

It is usually used for flows where the local state of gas molecules is far from the

local equilibrium

As compared to MD, DSMC has the following distinctive features

Every simulated molecule in DSMC represents Wmolecules in real flow, typicallyW>> 1. It makes DSMC capable for simulation of flows with almost arbitrary lengthscale (e.g., planetary atmosphere).

Interactions between molecules are taken into account in the framework of a specialcollision sampling algorithm, where interactions (pair collisions) are considered as

random events and simulated based on generation of random numbers. Typically, an implementation of DSMC in a computer code relays on two types

of models describing

Pair collision between molecules

Rebound of a molecule from an impermeable wall

Though in this lecture we consider only hard sphere molecules, a variety of modelsexists for both inter-molecular and molecule-wall collisions. These models arecapable to account for many features of molecules in real gases (e.g., internaldegrees of freedom, etc.)

Reading: G.A. Bird, Molecular gas dynamics and the direct simulation of gas

flows. Clarendon Press, 1994.

dsmc direct simulation monte carlo of gas flows

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