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Technische Universit¨ at M¨ unchen CFD Lab Course The Lattice Boltzmann Method Philipp Neumann 20.5.2011 P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 1

Technische Universit¨ at M¨ unchen Review: The story so far So far: Navier–Stokes • Compute pressure and velocity of the fluid • Nonlinear equations • Equation system to be solved in each timestep However: • Only valid in the continuum regime, that is for Kn := l mfp << 1 L c Kn Knudsen number, l mfp mean free path of fluid molecules, L c characteristic length of flow system • Not that trivial to bring Navier–Stokes to supercomputers P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 2

Technische Universit¨ at M¨ unchen Lattice What??? Lattice Boltzmann Method: Derived from Boltzmann equation ( → microscopic description) • Describes fluid by means of statistics → Find probability f ( � x ,� v , t ) that fluid molecules are in a small surrounding of � x at time t with a certain velocity � v • Macroscopic quantities are obtained by integration over the velocity space: ∞ � f ( � x ,� ρ ( x , t ) = v , t ) dv −∞ (1) ∞ � ρ ( x , t ) � u = f ( � x ,� v , t ) � v dv −∞ with fluid density ρ and fluid velocity � u • Assumption “Ideal gas” ⇒ Pressure p = c 2 s ρ , c s speed of sound P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 3

Technische Universit¨ at M¨ unchen Lattice What??? Problems: • Probabilities live in a 2 D + 1-dimensional space → “Curse of dimensionality” • Integration over infinite domains can be quite hard P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 4

Technische Universit¨ at M¨ unchen Lattice What??? Problems: • Probabilities live in a 2 D + 1-dimensional space → “Curse of dimensionality” • Integration over infinite domains can be quite hard How can we solve these issues? P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 4

Technische Universit¨ at M¨ unchen Welcome to the dark ages: Castle invasion Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen Welcome to the dark ages: Castle invasion Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders • Each bowman can shoot arrows only into the direction of one enemy (and vice versa) P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen Welcome to the dark ages: Castle invasion Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders • Each bowman can shoot arrows only into the direction of one enemy (and vice versa) • If a bowman does not have any more arrows, he gets new arrows from the other bowmen P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen Welcome to the dark ages: Castle invasion Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders • Each bowman can shoot arrows only into the direction of one enemy (and vice versa) • If a bowman does not have any more arrows, he gets new arrows from the other bowmen • The arrows coming from the enemies are collected and re-used by the bowmen P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen Dark ages ↔ Lattice Boltzmann Q Bowmen on a tower Probability densities f i ( i = 1 , ..., Q ) in a (cubic) lattice cell Q lattice velocities � Q directions for the arrows c i for fluid molecules Re-distribution of arrows Collision process between populations f i within a cell Incoming arrows from Q attackers Molecules streamed from Q neighboured cells P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 6

Technische Universit¨ at M¨ unchen Collision and Streaming 1 collision + 1 streaming = 1 timestep P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 7

Technische Universit¨ at M¨ unchen Reduce the effort... Problems: • Probabilities live in a 2 D + 1-dimensional space → “Curse of dimensionality” • Integration over infinite domains can be quite hard Our discrete formulation... P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 8

Technische Universit¨ at M¨ unchen Reduce the effort... Problems: • Probabilities live in a 2 D + 1-dimensional space → “Curse of dimensionality” • Integration over infinite domains can be quite hard Our discrete formulation... • ... lives in a D + 1-dimensional space, having Q unknowns per lattice cell P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 8

Technische Universit¨ at M¨ unchen Reduce the effort... Problems: • Probabilities live in a 2 D + 1-dimensional space → “Curse of dimensionality” • Integration over infinite domains can be quite hard Our discrete formulation... • ... lives in a D + 1-dimensional space, having Q unknowns per lattice cell • ... can easily (and locally!) handle the integration procedure: Q = � f i ρ i = 1 Q f i � ρ� u = � c i i = 1 P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 8

Technische Universit¨ at M¨ unchen Collision • Interaction of local populations f i inside each lattice cell • Denote the post-collision state by f ∗ i • Assume, our system is close to an equilibrium state • If f eq represents the equilibrium state of the fluid system, the i molecular collisions should drive our system even closer towards this equilibrium BGK model (Bhatnagar – Gross – Krook): i ( x , t ) = f i ( x , t ) − 1 f i ( x , t ) − f eq � � f ∗ i τ with relaxation time τ ∈ ( 0 . 5 , 2 ) P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 9

Technische Universit¨ at M¨ unchen The equilibrium distribution • Can be derived from the Maxwell-Boltzmann distribution • Equilibrium state only depends on macroscopic quantities ( ρ,� u ) � 2 � � 1 + � � � c i · � c i · � u u − � u · � u f eq ( ρ,� u ) = w i ρ + i c 2 2 c 4 2 c 2 s s s 1 with c s := 3 speed of sound on the lattice √ P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 10

Technische Universit¨ at M¨ unchen The Lattice Boltzmann Algorithm Given: f i ( x , t start ) everywhere in our computational domain for ( t = t start ; t < t end ; t = t + dt ) ; do done P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen The Lattice Boltzmann Algorithm Given: f i ( x , t start ) everywhere in our computational domain for ( t = t start ; t < t end ; t = t + dt ) ; do Compute density and velocity inside a fluid cell: ρ� f i � ρ = � f i , u = � c i i i done P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen The Lattice Boltzmann Algorithm Given: f i ( x , t start ) everywhere in our computational domain for ( t = t start ; t < t end ; t = t + dt ) ; do Compute density and velocity inside a fluid cell: ρ� f i � ρ = � f i , u = � c i i i f i ( x , t ) − f eq i ( x , t ) = f i ( x , t ) − 1 � ( ρ,� � Local collision: f ∗ u ) i τ done P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen The Lattice Boltzmann Algorithm Given: f i ( x , t start ) everywhere in our computational domain for ( t = t start ; t < t end ; t = t + dt ) ; do Compute density and velocity inside a fluid cell: ρ� f i � ρ = � f i , u = � c i i i f i ( x , t ) − f eq i ( x , t ) = f i ( x , t ) − 1 � ( ρ,� � Local collision: f ∗ u ) i τ Streaming to/ from neighbours: f i ( x + � c i dt , t + dt ) = f ∗ i ( x , t ) done P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen The Lattice Boltzmann Algorithm Given: f i ( x , t start ) everywhere in our computational domain for ( t = t start ; t < t end ; t = t + dt ) ; do Compute density and velocity inside a fluid cell: ρ� f i � ρ = � f i , u = � c i i i f i ( x , t ) − f eq i ( x , t ) = f i ( x , t ) − 1 � ( ρ,� � Local collision: f ∗ u ) i τ Streaming to/ from neighbours: f i ( x + � c i dt , t + dt ) = f ∗ i ( x , t ) done f i ( x + c i dt , t + dt ) = f i ( x , t ) − 1 f i − f eq � � i τ P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen Remarks P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 12

Technische Universit¨ at M¨ unchen Remarks • Explicit method , no equation system that needs to be solved! P. Neumann: CFD Lab Course The Lattice Boltzmann Method, 20.5.2011 12