Lattice Boltzmann Simulation of Flows II: Applications Bodies Dr Giacomo Falcucci, PhD University of Rome Tor Vergata, Via del Politecnico 1, 00100, Italy Outline Bodies 1 2 Reconstruction 3 Problem Statement Model 4 5 - #1 Fixed Obstacles Test Cases 6 Bodies LBM Implementation of Bodies 7 8 9
- #1 Bodies Multi-scale fluid-structure interaction; di erent aspects from a theoretical point of view; di erent technicalities in the implementation. - #1 Bodies Fluid-Structure interaction phenomena are ubiquitous:
Fluid-Structure interaction phenomena are ubiquitous: - #1 Bodies - #1 Bodies Fluid-Structure interaction phenomena are ubiquitous:
Problem Statement Reconstruction - #1 Bodies un-homogeneous pattern in the catalyst surface after having worked; no (apparent) chemical-related reason; possible relation to the reactant distribution?! Fluid dynamic field? Reconstruction Problem Statement Nanoporous gold catalyst samples for the methanol oxidation reaction were prepared by dealloying bulk Ag70 Au30 alloy discs; First: let s discuss about the geometry: how to reconstruct a porous media that derives from a eutectoid?? - #1 Bodies Two possible ways: reconstruct by ourselves; import a SEM micrography as a binary file and process it.
Reconstruction Problem Statement Nanoporous gold catalyst samples for the methanol oxidation reaction were prepared by dealloying bulk Ag70 Au30 alloy discs; First: let s discuss about the geometry: how to reconstruct a porous media that derives from a eutectoid?? - #1 Bodies Two possible ways: reconstruct by ourselves; import a SEM micrography as a binary file and process it. Reconstruction Problem Statement Nanoporous gold catalyst samples for the methanol oxidation reaction were prepared by dealloying bulk Ag70 Au30 alloy discs; First: let s discuss about the geometry: how to reconstruct a porous media that derives from a eutectoid?? - #1 Bodies Two possible ways: reconstruct by ourselves; import a SEM micrography as a binary file and process it.
Reconstruction Reconstruction 2D simulations of the Catalytic Ingot; We chose to recreate the porous media in a way which is similar to the real one; Starting from a solid region in our domain, we dig circular holes inside it, till we reach the desired porosity (" = V void V tot ) [checking permeability and tortuosity, as well.] Bodies Reconstruction Reconstruction 2D simulations of the Catalytic Ingot; We chose to recreate the porous media in a way which is similar to the real one; Starting from a solid region in our domain, we dig circular holes inside it, till we reach the desired porosity (" = V void V tot ) [checking permeability and tortuosity, as well.] Bodies
Reconstruction Reconstruction 2D simulations of the Catalytic Ingot; We chose to recreate the porous media in a way which is similar to the real one; Starting from a solid region in our domain, we dig circular holes inside it, till we reach the desired porosity (" = V void V tot ) [checking permeability and tortuosity, as well.] Bodies Reconstruction Reconstruction 2D simulations of the Catalytic Ingot; We chose to recreate the porous media in a way which is similar to the real one; Starting from a solid region in our domain, we dig circular holes inside it, till we reach the desired porosity (" = V void V tot ) [checking permeability and tortuosity, as well.] Bodies
Acquisition Reconstruction SEM micrographies kindly o ered by Harvard; Post-processing of the images to recover a simple ASCII file; Bodies Acquisition Reconstruction SEM micrographies kindly o ered by Harvard; Post-processing of the images to recover a simple ASCII file; Bodies
Acquisition Problem Statement Model - #1 Bodies Framework Selective methanol oxidation was used as a test reaction: 2 CH3 OH + O2! HCOOCH3 + 2 H2 O ; (1) The reaction was carried out at 150 C in a continuous flow of reactant (6.5%MeOH - 20%O2-73.5% He) at 50 ml/min; 3 species: an inert Carrier (C), a Reactant Species (R) and a Product species (P ); R and P are transported by the carrier. Acquisition Problem Statement Model - #1 Bodies Framework Selective methanol oxidation was used as a test reaction: 2 CH3 OH + O2! HCOOCH3 + 2 H2 O ; (1) The reaction was carried out at 150 C in a continuous flow of reactant (6.5%MeOH - 20%O2-73.5% He) at 50 ml/min; 3 species: an inert Carrier (C), a Reactant Species (R) and a Product species (P ); R and P are transported by the carrier.
Acquisition Problem Statement Model - #1 Bodies Framework Selective methanol oxidation was used as a test reaction: 2 CH3 OH + O2! HCOOCH3 + 2 H2 O ; (1) The reaction was carried out at 150 C in a continuous flow of reactant (6.5%MeOH - 20%O2-73.5% He) at 50 ml/min; 3 species: an inert Carrier (C), a Reactant Species (R) and a Product species (P ); R and P are transported by the carrier. LBM implementation of f i (+ci, t + 1) fi (, t) =![fieq, (, t) fi (, t)], e Species interconversion due to catalytic reactions at the pore surface is accounted for by a local exchange of populations; Sputtering boundary condition: Problem Statement Model - #1 Bodies the populations of reactants and products that enter or exit (superscripts in and out ) a node obey the following equations: R,in fi () = (1 R ps ) b() X R,out Si,j fj ( cj ) j=1 P,in fi () = R (p ps ) b() X j=1 R,out Si,j fj ( cj ) + (1 P ps ) b() X P,out Si,j fj ( cj ) j=1 (2)
LBM implementation of fi (, t) =![fieq, (, t) f i (+ci, t + 1) fi (, t)], e Species interconversion due to catalytic reactions at the pore surface is accounted for by a local exchange of populations; Sputtering boundary condition: Problem Statement Model - #1 Bodies the populations of reactants and products that enter or exit (superscripts in and out ) a node obey the following equations: R,in fi () = R (1 ps ) b() X R,out Si,j fj ( cj ) j=1 P,in fi () = R (p ps ) b() X R,out Si,j fj P ( cj ) + (1 ps ) j=1 b() X P,out Si,j fj ( cj ) j=1 (2) LBM implementation of Problem Statement Model - #1 Bodies R,in fi () = (1 R ps ) b() X R,out Si,j fj ( cj ) j=1 P,in fi () = R (p ps ) b() X j=1 R,out Si,j fj ( cj ) + (1 P ps ) b() X P,out Si,j fj ( cj ), j=1 (3) Si,j is a random sputtering matrix, expressing the probability of a molecule leaving the bulk along direction j, to re-enter along direction i. P Coefficients pr S, ps and p account for the sticking event, thus the sputtering Pb() matrix obeys the conservation rule i=1 Si,j = 1; we needed to calibrate our model on the experimental measurements.
LBM implementation of Problem Statement Model Bodies b() f R,in i () = (1 p R S ) X S i,j f R,out j ( c j ) j=1 b() f P,in i () = (p p R S ) X j=1 b() S i,j f R,out j ( c j )+(1 p P S ) X j=1 S i,j f P,out j ( c j ), S i,j is a random sputtering matrix, expressing the probability of a molecule leaving the bulk along direction j, to re-enter along direction i. Coe cients p R S, pp S and p account for the sticking event, thus the sputtering matrix obeys the conservation rule P b() i=1 S i,j = 1; we needed to calibrate our model on the experimental measurements. (3) LBM implementation of : Model Problem Statement Model Bodies The main dimensionless parameters are: Reynolds (Re): Re = U in h/ ; In our simulations, U in =0.08 lu/ t, h = 200 lu, =7/3 lu 2 / t (lu represents the grid spacing and t is the lattice time step) providing Re LBM 10, which is in good agreement with the experimental one (Re exp = 10); Knudsen (Kn): Kn = c s ( t/2) 1/ ; In the present simulations, 4 t, 3 and the lattice speed of sound c s =1/ p 3, yielding Kn 0.6, marginally in the Knudsen di usion regime; Damköhler (Da): Da = ch /( /v th ); Peclét (Pe): Pe = U in h/d; q where D is the di usion coe cient of the gas and v th = speed. k B T m is the gas thermal The conversion e ciency is defined as = M P y=l M R y=0 0.2 and it is controlled by the ratio between the chemical reaction time scale ch = /p =1/p in lattice units, and the mass transport time scale tr = /v th.
LBM implementation of : Model Problem Statement Model Bodies The main dimensionless parameters are: Reynolds (Re): Re = U in h/ ; In our simulations, U in =0.08 lu/ t, h = 200 lu, =7/3 lu 2 / t (lu represents the grid spacing and t is the lattice time step) providing Re LBM 10, which is in good agreement with the experimental one (Re exp = 10); Knudsen (Kn): Kn = c s ( t/2) 1/ ; In the present simulations, 4 t, 3 and the lattice speed of sound c s =1/ p 3, yielding Kn 0.6, marginally in the Knudsen di usion regime; Damköhler (Da): Da = ch /( /v th ); Peclét (Pe): Pe = U in h/d; q where D is the di usion coe cient of the gas and v th = speed. k B T m is the gas thermal The conversion e ciency is defined as = M P y=l M R y=0 0.2 and it is controlled by the ratio between the chemical reaction time scale ch = /p =1/p in lattice units, and the mass transport time scale tr = /v th. LBM implementation of : Model Problem Statement Model Bodies The main dimensionless parameters are: Reynolds (Re): Re = U in h/ ; In our simulations, U in =0.08 lu/ t, h = 200 lu, =7/3 lu 2 / t (lu represents the grid spacing and t is the lattice time step) providing Re LBM 10, which is in good agreement with the experimental one (Re exp = 10); Knudsen (Kn): Kn = c s ( t/2) 1/ ; In the present simulations, 4 t, 3 and the lattice speed of sound c s =1/ p 3, yielding Kn 0.6, marginally in the Knudsen di usion regime; Damköhler (Da): Da = ch /( /v th ); Peclét (Pe): Pe = U in h/d; q where D is the di usion coe cient of the gas and v th = speed. k B T m is the gas thermal The conversion e ciency is defined as = M P y=l M R y=0 0.2 and it is controlled by the ratio between the chemical reaction time scale ch = /p =1/p in lattice units, and the mass transport time scale tr = /v th.
LBM implementation of : Model Problem Statement Model Bodies The main dimensionless parameters are: Reynolds (Re): Re = U in h/ ; In our simulations, U in =0.08 lu/ t, h = 200 lu, =7/3 lu 2 / t (lu represents the grid spacing and t is the lattice time step) providing Re LBM 10, which is in good agreement with the experimental one (Re exp = 10); Knudsen (Kn): Kn = c s ( t/2) 1/ ; In the present simulations, 4 t, 3 and the lattice speed of sound c s =1/ p 3, yielding Kn 0.6, marginally in the Knudsen di usion regime; Damköhler (Da): Da = ch /( /v th ); Peclét (Pe): Pe = U in h/d; q where D is the di usion coe cient of the gas and v th = speed. k B T m is the gas thermal The conversion e ciency is defined as = M P y=l M R y=0 0.2 and it is controlled by the ratio between the chemical reaction time scale ch = /p =1/p in lattice units, and the mass transport time scale tr = /v th. LBM implementation of : Model Problem Statement Model Bodies To retrieve the desired conversion e reaction probability p = 0.45; ciency of 0.2, we found a value for the The characteristic time of the reaction ch is a function of the time step expressed in physical units, t exp, and the reaction probability p, ch = t exp /p; Taking p as 0.45 and retrieving t exp form the compliance between experimental and numerical viscosities ( t exp = LB x 2 / exp ), we have ch 25 ps, (4) providing Damköhler number Da 0.4 (fast chemistry regime). Finally, we find Pe 8, which is very close to the experimental value of Peclét number, Pe exp = 10.
LBM implementation of : Model Problem Statement Model Bodies To retrieve the desired conversion e reaction probability p = 0.45; ciency of 0.2, we found a value for the The characteristic time of the reaction ch is a function of the time step expressed in physical units, t exp, and the reaction probability p, ch = t exp /p; Taking p as 0.45 and retrieving t exp form the compliance between experimental and numerical viscosities ( t exp = LB x 2 / exp ), we have ch 25 ps, (4) providing Damköhler number Da 0.4 (fast chemistry regime). Finally, we find Pe 8, which is very close to the experimental value of Peclét number, Pe exp = 10. Bodies Lattice units Physical Units L 250 2.5 10 6 m H 500 5 10 6 m h 200 2 10 6 m w 10 10 7 m inlet 1 0.833 ml/s 3 3 10 8 m v 1/ p 3 1000 m/s tr 3 p 3 5 10 11 s ch 2 2.5 10 11 s p R S 0.95 NA p P S 0.00 NA p 0.45 NA Table: Main physical and chemical parameters adopted in our simulations, with the chosen values in lattice non-dimensional units and the corresponding parameters in physical units; the parameters are computed for the reference case, corresponding to w = 10 and p = 0.45.
Bodies (a) R species (b) P species Bodies Most of the chemical conversion takes place very close to the outer surface of the porous catalyst sample that faces the inlet of the computational domain; For all values of p, the flux of products exhibits two trends across the ingot: a first, steep trend, at the ingot surface on the stream-facing side, and a second, slight slope for the flux of products through the ingot; For higher values of p, the depth of conversion remains almost unchanged and that the large part of reactant conversion takes place in the first few layers of the ingot.
Engineering Application: SCR Reconstruction of porous media Bodies Engineering Application: SCR Concentration Fields: No and N 2 Bodies (b) Reactant Species (c) Product Species
- #1 Fixed Obstacles Test Cases Bodies LBM a suitable choice for hetrerogeneous catalysis; reasonable reliability even for non-negligible numbers; necessity to increase! higher-order lattices; necessity to include thermal e ects for re-shaping - #1 Fixed Obstacles Test Cases Bodies LBM a suitable choice for hetrerogeneous catalysis; reasonable reliability even for non-negligible numbers; necessity to increase! higher-order lattices; necessity to include thermal e ects for re-shaping
Other LBM schemes for fluid-structure interaction Fixed Obstacles Test Cases Bodies We have implemented our simulations according to the following: Sinlge-phase problems: - laminæ, wedges and cylinders (rigid and compliant);! second-order accurate o -lattice wall boundary conditions;! immersed boundary; Two-phase problems: VOF & Pseudo-Potential approaches; - wedges (rigid and compliant);! second-order accurate o -lattice wall boundary conditions;! immersed boundary; Bodies Fixed Obstacles: Problem Statement Filippova & Hänel Implementation: Fixed Obstacles Test Cases Bodies in which, where f (x ī b,t)=(1 ) f i (x f,t)+ fi (x b,t)+2w i (x b,t)[c u(x ī w,t)]/c 2 s apple fi (x b,t)=w i (x f,t) 1+ c i u c 2 + [c i u(x f,t)] 2 ku(x f,t)k 2 s 2c 4 s 2c 2 s ( u =[( 1)u(x f,t)+u(x w,t)]/, =[2 1]/ if 1/2 u = u(x f,t), =[2 1]/[ 1] if < 1/2 = x f x w / x f x b
Basic Test Cases Fixed Obstacles Test Cases Bodies 1D Bodies Bodies LBM Implementation of Bodies s are modeled as slender beams with thin rectangular cross sections.! The beam is considered as infinitely thin in the direction of vibration and, for low flexural modes, the flow of the encompassing fluid is assumed to be two dimensional (2D) in the beam cross section plane.
1D Bodies Bodies LBM Implementation of Bodies movement is imposed by an external control unit. 1D Bodies Filippova & Hänel Implementation, but: Bodies LBM Implementation of Bodies movement larger than one lattice site;! need for a refill procedure;
1D Bodies Filippova & Hänel Implementation, but: Bodies LBM Implementation of Bodies movement larger than one lattice site;! need for a refill procedure; 1D Bodies Refill Procedure Bodies LBM Implementation of Bodies the obstacle moves and overcomes a row of nodes; the previously blue nodes turn into red : communication is unphysical!!;
1D Bodies Refill Procedure Bodies LBM Implementation of Bodies the obstacle moves and overcomes a row of nodes; the previously blue nodes turn into red : communication is unphysical!!; Refill Procedure Bodies LBM Implementation of Bodies The Refill Procedure is accomplished in the following steps: the red-turned nodes density is fixed as the same as the starred red nodes below them; macroscopic velocity of red-turned nodes is imposed to be the same as the moving obstacle; f i (x,t) new red nodes = f i (x,t) eq new red nodes ;
Refill Procedure Bodies LBM Implementation of Bodies The Refill Procedure is accomplished in the following steps: the red-turned nodes density is fixed as the same as the starred red nodes below them; macroscopic velocity of red-turned nodes is imposed to be the same as the moving obstacle; f i (x,t) new red nodes = f i (x,t) eq new red nodes ; Refill Procedure Bodies LBM Implementation of Bodies The Refill Procedure is accomplished in the following steps: the red-turned nodes density is fixed as the same as the starred red nodes below them; macroscopic velocity of red-turned nodes is imposed to be the same as the moving obstacle; f i (x,t) new red nodes = f i (x,t) eq new red nodes ;
Refill Procedure Bodies LBM Implementation of Bodies Refill Procedure Caveats: grid size small enough, in order to impose the same density at two rows of nodes; obstacle velocities small enogh in order not to make errors in imposing it at the red-turned nodes; Results Bodies (d) Velocity magnitude contours (e) Density magnitude contours
Results Bodies We measured phase delay between cantlever and fluid acceleration; the force resultant per unit length ˆF exerted by the fluid on the oscillating lamina: ˆF (!) =( /4)! 2 L 2 (", )A in which! and A are, respectively, the oscillation frequency and amplitude and is the hydrodynamic function. Results Bodies (f) Phase delay between accelerations. (g) Real (left) andimaginary(right) partsofthehydrodynamicfunction
Results (a) (b) (c) Bodies (d) (e) (f) Figure: Representative flow fields: velocity magnitude, panels (a) and (d); relative pressure, panels (b) and (e); and out-of-plane vorticity, panels (c) and (f). Bodies LB successfully and conveniently adopted for phenomena of technical interest; Reliable predictions in complex porous media and for moving bodies; Results in line with theoretical predictions and experimental measurements; Interesting possibilities for future applications.
Bodies LB successfully and conveniently adopted for phenomena of technical interest; Reliable predictions in complex porous media and for moving bodies; Results in line with theoretical predictions and experimental measurements; Interesting possibilities for future applications. Bodies LB successfully and conveniently adopted for phenomena of technical interest; Reliable predictions in complex porous media and for moving bodies; Results in line with theoretical predictions and experimental measurements; Interesting possibilities for future applications.
Bodies LB successfully and conveniently adopted for phenomena of technical interest; Reliable predictions in complex porous media and for moving bodies; Results in line with theoretical predictions and experimental measurements; Interesting possibilities for future applications. Bodies