Grid refinement in LBM based on continuous distribution functions

Transcription

1 Grid refement LBM based on contuous distribution functions Denis Ricot 1 Simon Marié 1,2 Pierre Sagaut 2 1 Renault - Research, Material and Advanced Engeerg Department 2 d Alembert Institute, Université Paris VI 5th ICMMES june 2008

2 Grid refement problem For simulatg complex engeerg flows, mesh refement technique is necessary LB distribution functions g α are not conserved through a refement terface [Filippova & H -anel, 1998] Spatial and time terpolation methods can not be used directly on the distribution functions to evaluate the unknown data

3 Grid refement schemes Vertex centered approach with ([Yu et al., 2002][Dupuis & Chopard, 2003]) our without rescalg ([L & Lai, 2000]) rescalg and terpolations applied on the post-collision distribution functions ([Yu et al., 2002][Filippova & H -anel, 1998]) spatial terpolation performed before time terpolation [Yu et al., 2002][Dupuis & Chopard, 2003] Volumetric approach (cell centered) [Rohde, 2004][Chen et al., 2005 (PowerFLOW)] Explode (c f ) and coalesce (f c) distribution functions : mass conservative scheme No rescalg of distribution functions

4 Presentation outle Contuous distribution functions? Grid refement problem DVBE and LBE Numerical verification Rescalg procedure Overlappg nodes Algorithm description Grid parameters Pulse propagation uniform flow Convected vortex

5 Grid refement problem DVBE and LBE Numerical verification Grid refement problem After one standard LB timestep, some distribution functions are missg Distribution functions are known to be discontuous through an terface between meshes with different grid size : gα c (,, t) gα f (,, t)

6 Grid refement problem DVBE and LBE Numerical verification Base ideas of Filippova & H -anel (1998) Macroscopic properties of the fluid must be conserved (ν, c s) δx c /δt c = δx f /δt f τg c = 1 τ f 2 g + 1 (the relaxation time is not contuous) 4 Macroscopic variables of the flow must be conserved (ρ, u) g eq,c α = g eq,f α The shear stress must be contuous g neq,c α = m τ g c g neq,f τg f α (m = δxc = 2) δx f

7 Grid refement problem DVBE and LBE Numerical verification Why the distribution functions of LBM are not contuous? Boltzmann equation f t + c f i x i = f f eq ɛλ (BE) Quadrature formulae to calculate moments of distribution functions usg a discrete velocity set f α t + c α,i f α x i = 1 τ `fα fα eq (DVBE) By defition the distribution functions of DVBE f α are contuous space and time (and also ν = τθ 0, c s = θ 0, ρ = P f α, ρu = P c αf α)

8 Grid refement problem DVBE and LBE Numerical verification Integrate DVBE along the characteristic c α for a time terval δt The tegral of the BGK collision operator is approximated by the trapezium rule : f α (x + c αδt, t + δt) f α (x, t) = δt {fα (x + cαδt, t + δt) f eq α 2τ (x + c αδt, t + δt)+f α (x, t) fα eq Change of variable ([He et al., 1998][Dellar, 2001]) g α (x, t) =f α (x, t)+ δt 2τ `fα (x, t) fα eq (x, t) (x, t) + O δt 3 Fal Lattice Boltzmann Equation g α(x+c αδt, t +δt)= 1 δt «g α(x, t)+ δt fα eq (x, t) (2) τ g τ g (1) with τ g = τ + δt/2.

9 Grid refement problem DVBE and LBE Numerical verification Numerical verification of the lk between f α and g α LBE and DVBE computations have been performed on the same uniform grid, for the same flow problems and fluid parameters LB model : standard D2Q9, BGK collision operator, equilibrium function truncated at second order DVBE model : D2Q9 velocity model, BGK collision operator, equilibrium function truncated at second order 6th-order central fite difference scheme for space derivatives 5th-order Runge-Kutta scheme for time tegration Fluid is air, τ/δt = 0.93, same δt and δx for LBE and DVBE For two simple flows, comparison of f α, g α and d α = f α + δt eq (fα f 2τ α )

10 Grid refement problem DVBE and LBE Numerical verification Reference cases Acoustic pressure pulse an uniform flow 8 h >< ρ = 1 + a P exp `x1 ln 2 x 0 2 b P 1 + `x2 x 2 i 2 0 u 1 = U 0 >: u 2 = 0 Convected vortex 8 >< >: with a P = 0.03 and b P = 5 ρ = 1 h u 1 = U 0 + a T U 0 `x2 x 2 0 exp `x1 ln 2 b T x `x2 x 2 i 2 0 h u 2 = a T U 0 `x1 x 0 1 exp `x1 ln 2 b T x `x2 x 2 i 0 2 with a T = 0.5 etb T = 25

11 Grid refement problem DVBE and LBE Numerical verification Comparison of the macroscopic variables Density profile of the pressure pulse at time t = 50 along the le x 2 = x Velocity profile u 2 of the convected vortex 0.01 at time t = 100 along the le x 2 = x x/δx LBM; DVBE. x/δx The simulated results are exactly the same term of macroscopic variables gα eq = fα eq

12 Grid refement problem DVBE and LBE Numerical verification Distribution functions at a given pot as a function of time Pressure pulse, α = 6 Convected vortex, α = t/δt ( ):g α ( ):f α ( ):d α = f α + δt(f α f eq α )/2τ t/δt ( ):gα eq ( ):fα eq This numerical test confirms that g α = f α + δt eq (fα f 2τ α )

13 Rescalg procedure Overlappg nodes Algorithm description Conversion g α f α for a given x and t g α = f α + δt 2τ `fα fα eq <=> fα = 2τ 2τ + δt g α + δt «2τ f α eq By defition f α, τ, ρ = P f α, ρu = P c αf α ( f eq α ) are contuous 8 >< >: fα c = 2τ g c 2τ+δt c α + δtc f eq 2τ α fα f = 2τ g f 2τ+δt f α + δtf f eq 2τ α fα f = fα c ==> τ = τg f δtf = τ c 2 g δtc g c 2 α = 1 2 τ g f m = 2 gα f = 1 f ``2 τ 2 τ g g f gc +1 α + fα eq f ``2 τ g + 1 gα f fα eq Remark: these expressions immediately imply that g neq,c α = 2 τ g c g neq,f τ g f α

14 Rescalg procedure Overlappg nodes Algorithm description First approach (implemented with success our 3D LB code (L-BEAM)) Conversion gα c f α for needed terface pots Spatial and temporal terpolations on f α Conversion f α gα f Second approach (preferred this work) Conversion gα c gα f for needed terface pots Spatial and temporal terpolations on g α f In both cases Conversion step needs the equilibrium functions to be known

15 Rescalg procedure Overlappg nodes Algorithm description Fe grid pots must also be calculated as coarse grid pots Conversion g f α out (, t) g c α out (, t)

16 Rescalg procedure Overlappg nodes Algorithm description Conversion g c α `, t + 2δt f g f α `, t + 2δt f can be done because macroscopic variables can be now calculated at pots

17 Rescalg procedure Overlappg nodes Algorithm description Timestep t : all functions g c α and gf α are known everywhere

18 Rescalg procedure Overlappg nodes Algorithm description Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t)

19 Rescalg procedure Overlappg nodes Algorithm description Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots )

20 Rescalg procedure Overlappg nodes Algorithm description Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f

21 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f

22 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Stage 5 : spatial terpolation of gα f,t +δt f usg g f α 1,t +δt f and gα f + 1,t +δt f Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f

23 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Stage 5 : spatial terpolation of gα f,t +δt f usg g f α 1,t +δt f and gα f + 1,t +δt f Timestep t+δt f : all functions gα f are known the fe grid region Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f

24 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Stage 5 : spatial terpolation of gα f,t +δt f usg g f α 1,t +δt f and gα f + 1,t +δt f Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f Timestep t+δt f : all functions gα f are known the fe grid region Stage 6 : collide and propagate all pots the fe region

25 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Stage 5 : spatial terpolation of gα f,t +δt f usg g f α 1,t +δt f and gα f + 1,t +δt f Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f Timestep t+δt f : all functions gα f are known the fe grid region Stage 6 : collide and propagate all pots the fe region Stage 7 : recover g f α,t +2δt f from g c α,t +2δt f

26 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Stage 5 : spatial terpolation of gα f,t +δt f usg g f α 1,t +δt f and gα f + 1,t +δt f Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f Timestep t+δt f : all functions gα f are known the fe grid region Stage 6 : collide and propagate all pots the fe region Stage 7 : recover g f α,t +2δt f from g c α,t +2δt f Stage 8 : spatial terpolation of gα f,t +2δt f

27 Rescalg procedure Overlappg nodes Algorithm description Injection scheme : gα f,t +δt f = g f α,t +2δt f Lear terpolation usg gα,t f +2δt f and g f α (,t) (that must be stored) Stage 5 : spatial terpolation of gα f,t +δt f usg g f α 1,t +δt f and gα f + 1,t +δt f Timestep t : all functions g c α and gf α are known everywhere Stage 1 : convert g f α out (, t) g c α out (, t) Stage 2 : collide and propagate all pots the fe and coarse regions (cludg pots ) Stage 3 : convert g c α, t +2δt f g f α, t +2δt f Stage 4 : time terpolation of g f α,t +δt f Timestep t+δt f : all functions gα f are known the fe grid region Stage 6 : collide and propagate all pots the fe region Stage 7 : recover g f α,t +2δt f from g c α,t +2δt f Stage 8 : spatial terpolation of gα f,t +2δt f Timestep t +2δt f : all functions g c α and gf α are known everywhere

28 Grid parameters Pulse propagation uniform flow Convected vortex Coarse grid : 120x80, fe grid : Acoustic pulse or vortex is itialized outside or side the fe grid M = 0.2, ν = m 2 /s, c s = 340 m/s, δx f = 10 2 m All computations are also done with a uniform coarse grid (reference) 80 U 0 = y /δ x c x/ δ x c

29 Grid parameters Pulse propagation uniform flow Convected vortex Iso-contours of the density fluctuation 80 Uniform grid Injection (no time terpolation) Lear terpolation time y/ δ x c y /δ x c y /δ x c x/ δ x c x/ δ x c x/ δ x c Small pressure reflexion occurs at the grid terface. The error is reduced when lear time terpolation is used. There is an analytical solution for the propagation of the pulse uniform flow precise error quantification is possible

30 Grid parameters Pulse propagation uniform flow Convected vortex Simulation with the uniform grid 5 x ( ) reference analytical solution ( ) LB simulation ρ / ρ x/ δ x c

31 Grid parameters Pulse propagation uniform flow Convected vortex Two-grid simulation with jection (no time terpolation) 5 x ( ) reference analytical solution ( ) results on coarse pots ρ / ρ ( ) results on fe pots x/ δ x c

32 Grid parameters Pulse propagation uniform flow Convected vortex Two-grid simulation with lear terpolation time 5 x ( ) reference analytical solution ( ) results on coarse pots ρ / ρ ( ) results on fe pots x/ δ x c

33 Grid parameters Pulse propagation uniform flow Convected vortex Convergence rate Parameter b p gives the itial width of the pressure pulse, i.e. the spatial resolution 10 6 ( ) slope ( ) Uniform grid ( ) Two-grid simulation without time terpolation (jection) L ( ) Two-grid simulation with lear terpolation time /2 b p

34 Grid parameters Pulse propagation uniform flow Convected vortex Initial pulse the fe grid region 80 Uniform grid Injection (no time terpolation) Lear terpolation time y/ δ x c y /δ x c y /δ x c x/ δ x c x/ δ x c x/ δ x c

35 Grid parameters Pulse propagation uniform flow Convected vortex Convected vortex (lear terpolation time) Uniform grid Lear terpolation time Lear terpolation time y/ δ x c y/ δ x c y/ δ x c x/ δ x c x/ δ x c x/ δ x c Fe to coarse grid convection : spurious vorticity appears side the fe region due to non-physical reflexion at the grid terface

36 Conclusion New theoretical sight the lk between coarse and fe distribution functions fal rescalg expressions are fully equivalent to the previous approaches this new theoretical approach can be useful for other LB models (other collision operator such as MRT models) The proposed grid refement algorithm mimize the unknown functions that must be approximated with terpolations use of overlappg coarse nodes terpolations are done on the distribution functions time terpolation is done before spatial terpolation Accuracy of the grid refement scheme can be improved usg higher order terpolation schemes