K. N. Toosi University of TechnologyFaculty of Mechanical Engineering

Direct Simulation Monte Carlo TRANSITIONAL PAGE (DSMC)by:Behzad Mohajer (M.Sc. Student)

Supervisor: Dr. Mehrzad Shams

Introduction of Micro/Nano Scale Flows Knudsen number mean free pathIntroduction Numerical Methods

characteristic length Represents the rarefaction rate and validity of the continuum models Knudsen number limits on the mathematical models

DSMCDSMC Algorithm Boundary Conditions

Inviscid Flow Continuum Regime Slip Regime

Kn 0 (Re ) Kn 10-3 10-3 Kn 10-1

Euler Equations Navier-Stokes Equations (using no-slip boundary conditions) Navier-Stokes Equations (using slip boundary conditions) Boltzmann Equation (Considering Molecular Collisions) Boltzmann Equation (Without Molecular Collisions)

Transition RegimeFree Molecular Regime

10-1 Kn 10Kn 10 ( Re 0 )

Numerical Methods

Macroscopic

Microscopic

Introduction Numerical Methods

Models

Models Flow Considered as a number of moleculesDescribed by the position, velocity, and internal state of every molecule

Flow considered continuum

DSMCDSMC Algorithm

Consists of density, pressure and velocity gradient terms (in time and position)

Navier-Stokes Equations

Boundary Conditions

Boltzmann Equation

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Numerical Methods Boltzmann EquationsDependant Variable Fraction of molecules in a given location and state (Molecule Distribution Function ) Time, Velocity Components, Spatial Positions of Molecules Phase Space (multi-dimensional space formed by the combination of physical space and velocity space)

IntroductionIndependent Variable

Numerical Methods

Solution Domain

DSMCDSMC Algorithm Boundary Conditions

Collisionless Boltzmann Equation

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Numerical Methods

Numerical Methods for Boltzmann EquationIntroduction Numerical Methods

1) Direct Boltzmann CFD Large number of independent variables

Extremely large computational cost for collision term

DSMCDSMC Algorithm Boundary Conditions

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2) Molecular Method (Direct Simulation Monte Carlo)

using a statistical description in terms of probability distributions simulating the gas flow as a group of individual molecules which have their own positions, velocities, internal energy, index, etc4/20

Direct Simulation Monte Carlo (DSMC)

DSMC is a numerical tool to solve the Boltzmann equation based on direct statistical simulation of the molecular processes described by the kinetic theoryIntroduction Numerical Methods

DSMC method was first introduced by G. Bird in 1976 DSMC models the real gas by a large number of simulated molecules in a computer. Each simulated molecule represents a number of real molecules. This number vary from the order of hundreds to millions. The primary principle of DSMC is to decouple the motion and collision of particles during one time step

DSMCDSMC Algorithm Boundary Conditions

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DSMC Algorithm

1) Grid GenerationIntroduction Numerical Methods

The first step in DSMC is breaking down the computational domain into acollection of grid cells The size of each cell should be sufficiently small to result in small changes in thermodynamic properties across each cell. The cells are divided into sub-cells in each direction The subcells are then utilized to facilitate the selection of collision pairs.

DSMCDSMC Algorithm Boundary Conditions

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DSMC Algorithm

2) Time StepIntroductionThe mean collision time

Numerical Methods

The mean residence time Molecules in each cell do not cross more than one cell during one time step %25 of the length of the cell is usually used to calculate t

DSMCDSMC Algorithm Boundary Conditions

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DSMC Algorithm

3) Initial Positions of The Simulated MoleculesIntroduction Numerical Methods The initial positions are set using random numbers

4) Initial Velocity of The Simulated Molecules

DSMCDSMC Algorithm Boundary Conditions

The velocity components are calculated using the velocity distribution function

The most probable molecular velocity

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DSMC Algorithm

5) Molecular Movements Molecules move in cell by constant velocity (u, v, w)

Introduction Numerical Methods

DSMCDSMC Algorithm Boundary Conditions

6) Gas-Surface Interaction

Specular Reflection Diffuse Reflection

The most probable molecular velocity

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DSMC Algorithm

7) The Probability of Molecular Collision in a CellIntroduction Numerical Methods The proportion of the volume swept out by relative velocity between a pair of molecules to the volume of the cell:

DSMCDSMC Algorithm Boundary Conditions

The number of Probable Collisions

Among all probable collisions, the collisions would happen which apply to:

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DSMC Algorithm

8) Velocity Components after CollisionIntroduction Numerical Methods

DSMCDSMC Algorithm Boundary Conditions11/20

Diffuse assumption:

DSMC Algorithm

9) Molecular IndexingIntroduction Numerical Methods The indexing of the simulated molecules must be changed according to their new spatial positions(cell and subcell)

DSMCDSMC Algorithm Boundary Conditions

10) Sampling of The Macroscopic Properties After achieving steady flow condition, sampling of molecular properties within each cell is fulfilled during sufficient time to avoid statistical scattering. All thermodynamic parameters such as temperature, density, and pressure are then determined from this time averaged data.

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Boundary Conditions in DSMC Method

Supersonic FlowIntroduction Numerical MethodsOutlet Boundary ConditionsInlet Boundary Conditions

DSMCVacuum

Inlet Temperature

Free Stream Mach Number

DSMC Algorithm Boundary Conditions

Moleceules are not allowed to return to the domain13/20

Boundary Conditions in DSMC Method

Subsonic FlowIntroduction Numerical Methods

Type 2

Type 1

DSMCDSMC Algorithm Boundary Conditions

Specified Outlet Pressure

Specified Mass Flow Rate

Specified Outlet and Inlet Pressure

Molecules at outlet are allowed to return to the domain Molecules are allowed to return to upstream14/20

Introduction Numerical Methods

DSMCDSMC Algorithm Boundary Conditions

The End

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