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3D S IMULATION OF PARTICLE MOTION IN LID - DRIVEN CAVITY FLOW BY MRT LBM A RMAN S AFDARI
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L UDWIG E DUARD B OLTZMANN Born in Vienna 1844 University of Vienna 1863 Ph.D. at 22 University of Graz 1869 Died September 5, 1906
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L ATTICE B OLTZMANN AIM The primary goal of LB approach is to build a bridge between the microscopic and macroscopic dynamics rather than to dealt with macroscopic dynamics directly.
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LBM L ITERATURE
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LBM U SAGE IN V ARIOUS F IELDS LBM is new & has been mostly confined to physics literature, until recently. 1
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No combine Fluids/Diffusion (No Interaction) No combine Fluids Single Component Multiphase Single Phase (No Interaction) Number of Components Interaction Strength Streamlines Phase Separation Diffusion Oil & water LBM C APABILITIES
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T HE B OLTZMANN E QUATION Equation describes the evolution of groups of molecules Advection termsCollision terms f : particle distribution function c : velocity of distribution function
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BGK (Bhatnagar-Gross-Krook) model most often used to solve the incompressible Navier-Stokes equations a quasi-compressible come, in which the fluid is manufactured into adopting a slightly compressible behavior to solve the pressure equation can also be used to simulate compressible flows at low Mach- number It perform easily as well as its reliability
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D ISCRETE VELOCITY MODEL The direction of distribution function is limited to seven or nine directions 9 velocity model 7 velocity model
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3D Lattice 27 components, and 26 neighbors 19 components, and 18 neighbors 15 components, and 14 neighbors 2 11 7 1 8 4 96 0 10 5 3 14 18 19 17 13 12 15 16 22 25 21 20 23 24
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B HATNAGAR -G ROSS -K ROOK (BGK) C OLLISION MODEL
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BGK B OLTZMANN EQUATION Equilibrium distribution function
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C OLLISION AND S TREAMING Collision w a are 4/9 for the rest particles (a = 0), 1/9 for a = 1, 2, 3, 4, and 1/36 for a = 5, 6, 7, 8. relaxation time c maximum speed on lattice (1 lu/ts) Streaming
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o MRT (Multiple-Relaxation-Time) model The BGK collision operator acts on the off-equilibrium part multiplying all of them with the same relaxation. But MRT can be viewed as a Multiple-Relaxation-Time model o Regularized model better accuracy and stability are obtained by eliminating higher order, non-hydrodynamic terms from the particle populations This model is based on the observation that the hydrodynamic limit only on the value of the first three moments (density, velocity and stress tensor) Entropic model The entropic lattice Boltzmann (ELB) model is similar to the BGK and the main differences are the evaluation of the equilibrium distribution function and a local modification of the relaxation time.
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M RT L ATTICE B OLTZMANN M ETHOD D2Q9
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M RT L ATTICE B OLTZMANN M ETHOD D3Q15
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So the matrix M is then given by :
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B OUNDARY C ONDITION Bounce back is used to model solid stationary or moving boundary condition, non-slip condition, or flow-over obstacles. 1-B OUNCE B ACK T YPE O F B OUNCE B ACK BC 1 2 3
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2-E QULIBRIUM AND N ON -E QULIBRIUM D ISTRIBUTION F UNCTION The distribution function can be split in to two parts, equilibrium and non-equilibrium. 3- O PEN B OUNDARY C ONDITION The extrapolation method is used to find the unknown distribution functions. Second order polynomial can be used, as :
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3- P ERIODIC B OUNDARY C ONDITION Periodic boundary condition become necessary to apply to isolate a repeating flow conditions. For instance flow over bank of tubes. 4- S YMMETRY C ONDITION Symmetry condition need to be applied along the symmetry line.
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B OUNDARY CONDITION (Z OU AND H E MODEL )
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P ARTICLE EQUATION
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CMWR 2004 Convection by LBM This represents the mixing that would occur when saltwater is sitting on top of freshwater.
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CMWR 2004 Convection by LBM This is a fun simulation of heat rising from below causing convection currents.
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A DVANTAGES OF L ATTICE B OLTZMANN M ETHOD Macroscopic continuum equation, Navier Stoke, the LBM is based on microscopic model. LBM does not need to consider explicitly the distribution of pressure on interfaces of refined grids since the implicitly is included in the computational scheme. The lattice Boltzmann method is particularly suited to simulating complex fluid flow Represent both laminar and turbulent flow and handle complex and changing boundary conditions and geometries due to its simple algorithm. 3D can be implemented with some modification It is not difficult to calculate and shape of particle
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S IMULATION ALGORITHM
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T HANK YOU I hope, this research can contribute to human development.
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