Exercice I Exercice II Exercice III Exercice IV Exercice V. Exercises. Boundary Conditions in lattice Boltzmann method

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1 Exercises Boundary Conditions in lattice Boltzmann method Goncalo Silva Department of Mechanical Engineering Instituto Superior Técnico (IST) Lisbon, Portugal

2 Setting the problem Exercise I: Poiseuille flow with bounceback walls

3 Setting the problem Setting the problem Governing equation: Reynolds number: ν ux y = a x Re = Umax (ytop y bottom) ν Velocity solution: u x (y) = 1 2ν a x(y y bottom )(y y top )

4 Setting the problem Setting the problem BounceBack walls location: { ybottom = 0.5 y top = NY 0.5 Body force (acceleration) magnitude: a x = 8u max ν (y top y bottom ) 2

5 Setting the problem N steps Number of time steps to achieve steady-state NY Number of nodes along the channel height Re Flow Reynolds number ω Relaxation frequency (LBM parameter) U max Maximum velociy (proportional to Mach number)

6 Setting the problem Question 1: Implement Bounceback BC Question 2: Fix Re = 10 and ω = 1 Question 2.1 For NY = 16, what is U max and L 2? Question 2.2 For NY = 32, what is U max and L 2? Question 2.3 Estimate the convergence rate Hint: α = ln ( ) L2 (u M ) L 2 (u N ) /ln( N M )

7 Setting the problem Question 3 Fix Re = 10 and NY = 16 Question 3.1 What is L 2 for ω = 1, ω = 1.7, ω = 0.5, ω = 0.2 and ω = ? Question 3.2 For each of the above ω values, compute the momentum imbalance F between the applied force and the friction force at walls. Comment the results. Hint: F wall + F prop = F F wall = 2 x t c α fα (x, t) x b S α F prop = a x x b V

8 Setting the problem Exercise II: Poiseuille flow with Zou He walls

9 Setting the problem Setting the problem Governing equation: Reynolds number: ν ux y = a x Re = Umax (ytop y bottom) ν Velocity solution: u x (y) = 1 2ν a x(y y bottom )(y y top )

10 Setting the problem Setting the problem Zou He walls location: { ybottom = 1 y top = NY Body force (acceleration) magnitude: a x = 8u max ν (y top y bottom ) 2

11 Setting the problem N steps Number of time steps to achieve steady-state NY Number of nodes along the channel height Re Flow Reynolds number ω Relaxation frequency (LBM parameter) U max Maximum velociy (proportional to Mach number)

12 Setting the problem Question 1: Implement Zou He BC Question 2: Fix Re = 10 and ω = 1 Question 2.1 For NY = 16, what is U max and L 2? Question 2.2 For NY = 32, what is U max and L 2? Question 2.3 Discuss the solutions accuracy Question 3 Fix Re = 10 and NY = 16 Question 3.1 What is L 2 for ω = 1, ω = 1.7, ω = 0.5, ω = 0.2?

13 Exercise III: Developing Poiseuille flow with Zou He walls and inlet/outlet BC

14 Setting the problem Velocity inlet solution: (u x ) in (y) = 1 12ν a x(y 2 bottom 2y bottomy top + y 2 top) where the driving force term is: a x = 8u max ν (y top y bottom ) 2 Velocity outlet solution: (u x ) out y = 0

15 N steps Number of time steps to achieve steady-state NY Number of nodes along the channel height Re Flow Reynolds number ω Relaxation frequency (LBM parameter) U max Maximum velociy (proportional to Mach number)

16 Question 1: Implement Zou He BC at inlet and outlet Question 2: Fix Re = 10 and ω = 0.9 Question 2.1 For NX = 22 and NY = 22, what is L 2? Question 2.2 For NX = 32 and NY = 32, what is L 2? Question 2.3 For NX = 52 and NY = 52, what is L 2? Question 3 Estimate graphically the entrance length L h for the fully developed condition and compare with semi-analytical solution L h W = 0.05 Re Hint: In command window of MATLAB execute: plot(1:nx,ux(:,round(ny/2)))

17 Exercise IV: Backward facing step flow using Zou He wall boundaries wall influx outflux External corner Internal corner wall

18 Setting the problem Remark I: A comment on the external corner

19 Setting the problem Remark I: A comment on the external corner Zou He framework does not apply for external corners, the problem remains overspecified, e.g. only unknowns are ρ and f 6

20 Setting the problem Remark I: A comment on the external corner Zou He framework does not apply for external corners, the problem remains overspecified, e.g. only unknowns are ρ and f 6 Solution: use bounceback instead

21 Setting the problem Remark I: A comment on the external corner Zou He framework does not apply for external corners, the problem remains overspecified, e.g. only unknowns are ρ and f 6 Solution: use bounceback instead Differences? On corners Zou He reduces to bounceback if u = 0

22 Setting the problem Remark II: A comment on problem geometry Geometry may be simplified if a parabolic profile is used at inlet wall influx outflux wall wall influx outflux wall

23 Setting the problem Influx boundary condition: u x = 1 2ν a x(y y bottom )(y y top inlet ) u y = 0 where: y top inlet y bottom = width of the influx boundary a x = 8u max ν (y top inlet y bottom ) 2 Outflux boundary condition: u x x = 0 u y = 0

24 N steps Number of time steps to achieve steady-state Re Flow Reynolds number ω Relaxation frequency (LBM parameter) U max Maximum velociy (proportional to Mach number)

25 Influx width: y top inlet = Reν U max + y bottom Geometry width Influx ratio = y top inlet y bottom y top y bottom Geometry length L x = L y = y top y bottom

26 Question 1: Implement Zou He BC at left boundary, i.e. {influx left wall} Question 2: Implement Zou He BC at right boundary, i.e. outflux Question 3: Implement Zou He BC at bottom left and top right corners Question 4: Set Re = 20, U max = 0.1 and ω = 0.8 Question 4.1 Set three Influx ratio values, e.g. 1 3, 1 2 and 1 1.2? Question 4.2 Discuss the size of recirculation patterns and the location of the outflow boundary?

27 Exercise V: Flow around circular cylinder influx Solid cylinder wall outflux wall