LBM

Question 1

f(c, x, t)

Answer 1

represents the probability at time t, of a particle being positioned at position x with velocity c

Question 2

DnQm teminology

Answer 2

• n refers to the number of physical dimensions
• m refers to the number of discretised velocities, more relevant as one considers particle energies
• D2Q9 has 3 speeds {0, 1, sqrt(2)}
• D3Q19 has 4 speeds {0, 1, sqrt{2), sqrt(3)}

Question 3

BGK model

Answer 3

• in the collision operator, current particle distribution is relaxed over time (tau) back towards the equilibrium distribution (f^eq) for a given set of conditions
• equilibrium distribution is computed directly from the discretised Maxwell-boltzmann function
• BGK model limited to isothermal flows at low Reynolds number

Question 4

LBM advantages

Answer 4

• efficient for parallel computation due to local nature of computation and algorithmic simplicity
• faster computation per iteration due to linear nature of LBM equation and absence of pressure term which is expensive in FVM
• solid boundary conditions are very easy to implement

Question 5

Unknown components

Answer 5

• after the streaming step the unknown components are those which point back into the domain since they have streamed from non-existent lattice sites
• therefore f1, f5 and f7 are the unknown lattice components

Question 6

Bounceback boundary condition disadvantages

Answer 6

• only 1st order accurate

- cannot represent curved surfaces

Question 7

The Mesoscale

Answer 7

• based on kinetic theory
• instead of a single particle we consider a distribution function
• distribution function represents a collection of particles

Question 8

LBM disadvantages

Answer 8

• regular grids are a limitation
• subject to instabilities with low viscosity
• limited to low Reynolds and Mach numbers

Question 9

Pressure Eq from probability function

Answer 9

p=(Cs)^2.rho

Question 10

LBM algorithm steps

Answer 10

• Initialise: all population distribution functions set to equilibrium
• Equilibrium: equilibrium recomputed at all points based on updated velocity field
• Stream: particles are convected along all discrete velocity paths in a single step
• Collide: distribution functions recomputed at all sites as a function of the difference between current and equilibrium state
• Macroscopic: macroscopic quantities are recomputed based on summations of the particle distributions
• Boundary conditions: particle states at boundaries are updated
• End: calculation completed and results saved

Question 11

Recovering quantities from distribution function

Answer 11

Density - recovered by summing all components of the population density function

Velocity in x-direction - recovered by summing those components with a component in the x-direction

Pressure - a thermodynamic quantity non recovered directly, can be approximated from density