20-12-11 PhD Day Turin

Mesoscopic numerical methods for
reactive ﬂows: lattice Boltzmann
method and beyond

PhD Student: Antonio F. Di Rienzo

Advisors: P. Asinari, E. Chiavazzo

Energy Department (DENERG), Politecnico di Torino, Torino, Italy

PhD Day, Torino 20 December, 2011

Antonio F. Di Rienzo (Politecnico di Torino) PhD Day Torino, 20th December 2011 1 / 26

Outline of The Talk

1 Research activity: a ﬂip through

2 The Lattice Boltzmann Method (LBM)

3 Radiative Transfer Equation

4 Reactive Flows

5 Link-wise Artiﬁcial Compressibility Method (LW-ACM)


Research activity: a ﬂip through

Outline Compass




4 Reactive Flows



Research activity: a flip through

Over the past three years...

2009 1
The Radiative transfer equation (RTE) is solved by means of
lattice Boltzmann method (LBM) formalism: intensity is updated
according to lattice velocities.
2010-2011 2
A consistent lattice Boltzmann model for reactive flows has been
proposed, so as to overcome lacks of previous existing models (in
collaboration with the Paul Scherrer Institute, Switzerland).
2011 3
Link-wise Artificial Compressibility Method: CFD by kinetic
mock-up (beyond Lattice Boltzmann Method).
1
Di Rienzo A. F., Asinari P., Borchiellini R., Mishra S. C., Improved angular discretization and error analysis of
the lattice Boltzmann method for solving radiative heat transfer in a participating medium, IJNMH&FF, 21 (5),
640-662, 2011
2
Di Rienzo A. F., Asinari P., Chiavazzo E., Prasianakis N., Mantzaras J., A Lattice Boltzmann model for reactive
flows simulation, submitted
3
Asinari P., Ohwada T., Chiavazzo E., Di Rienzo A.F., Link-wise Artificial Compressiniblity Method, submitted

The Lattice Boltzmann Method (LBM)

Outline Compass




4 Reactive Flows



The Lattice Boltzmann Method (LBM)

What is Lattice Boltzmann Method?

"The lattice Boltzmann method (LBM) is used for the numerical
simulation of physical phenomena and serves as an alternative to
classical solvers of partial differential equation (PDEs)"
[www.lbmethod.org]. The main unknown is the discrete
distribution function, from which all relevant macroscopic
quantities can be derived.
The operative formula consists of the (a) relaxation process and
the (b) advection process:
(e)
fi (t + δt, x + ci δt) − fi (t, x) = ω fi (t, x) − fi (t, x) . (1)

The updating of fi is link-wise, in the sense that only the
informations along the directions identiﬁed by the lattice velocity ci
are required.


Radiative Transfer Equation

Outline Compass




4 Reactive Flows




Radiative Lattice Boltzmann Model
For an absorbing, emitting and scattering participating medium
and under the assumptions of isotropic scattering and radiative
equilibrium condition, radiative LBM reads:
G
Ii (t + δt, x + ci δt) = Ii (t, x) + ci β (t, x) − Ii (t, x) . (2)
4π
The link-wise formulation makes unnecessary to march from each
single corner (e.g. Finite Volumes).
The data structures for radiation are the same of those for the ﬂuid
ﬂow.



2D Square Enclosure

β = 2.0 β = 5.0
Nx Nδ Nγ L2 [Ψ] Nx Nδ Nγ L2 [Ψ]
40 8 4 7.31 × 10−2 100 8 4 5.54 × 10−2
80 16 8 3.16 × 10−2 200 16 8 2.01 × 10−2
160 32 16 2.00 × 10−2 400 32 16 1.30 × 10−2

Reactive Flows

Outline Compass




4 Reactive Flows



Reactive Flows

State of Art for Reactive LBM
1 Filippova O., Hänel D., J. Comp. Phys., 2000
An additional factor is introduced in the kinetic equation in order to
model temporal changes of the density: constant density limitation
is removed.
The model behaves like a weak-compressible solver: not suitable to
accurately simulate thermal flows with large density and
temperature variations.
Energy and species transport equations are solved by means of a
finite differences scheme.

2 Yamamoto K., He X., Doolen G. D., J. Stat. Phys., 2002
Fully LBM based.
The flow field is incompressible and not affected by chemical
reactions: density is (nearly) constant.
Significant deviations are observed when compressibility is taken
into account.

Reactive Flows

Simulating Reactive Flows: What’s Required?

"In principle, once lattice Boltzmann models can properly account
for large temperature variation, extension to reactive flows
essentially involves adding appropriate source terms..."[C. E.
Frouzakis, Fluid Mechanics and Its Applications, 2011]

So far, no reactive LBM models has been proposed that satisfies
this requirement.

Combustion problems exhibit significant temperature changes due
to the heat release in chemical reactions.

The LBM model is requested to accurately recover the Navier-
Stokes-Fourier equations, coupled to a transport equation for each
chemical species, and thus to behave macroscopically like a
compressible solver.


Reactive Flows

Governing Equations for Reactive Flows
The system of macroscopic governing equations reads:

∂t ρ + · (ρu) = 0,
∂t (ρu) + · (ρu ⊗ u + pI) = · Π,
(3)
dp
∂t (ρhs ) + · (ρuhs + q) = +Π: u + ST ,
dt

∂t (ρYk ) + · (ρuYk ) = · (ρDk Yk ) + SYk . (4)

The corresponding kinetic equations reads:
2δt (e) 2δtτ
ft+δt = ft + ξt − ft + [Ψt + Φt + ST ], (5)
δt + 2τ δt + 2τ
(e)
gt+δt = gt + ωk gt − gt + SYk (6)


Reactive Flows

Navier-Stokes-Fourier in the Compressible Limit

2δt (e) 2δtτ
ft+δt = ft + ξt − ft + [Ψt + Φt + ST ] (7)
δt + 2τ δt + 2τ

The correction terms Ψ and Φ are added to the discrete kinetic
equation in order to remove the deviations in the momentum and
energy equations.
By means of these correction terms, it is possible to accurately
recover Navier-Stokes-Fourier equations in the compressible
limit.4,5
This partially bridges over the lacks of the existing models: also
species transport equation must reckon with compressibility
effects.
4
Prasianakis N. I., Karlin I. V., Lattice Boltmann method for simulation of thermal ﬂows on standard lattice, PRE,
2007, 76, 016702
5
Prasianakis N. I., Karlin I. V., Lattice Boltzmann method for simulation of compressible ﬂows on standard
lattices, PRE, 2008, 78, 016704

Reactive Flows

Species Equation in the Compressible Limit
Standard LBM solves the species transport equation with the
following deviation term:

· (Dk Yk ρ). (8)

Deviation in the species equation is activated in case of signiﬁcant
compressibility effects (i.e. large ρ).
It can be removed without adding any correction terms, due to the
simpler nature of the equation.
By modifying the equilibrium distribution function, it is possible to
correct the second order moment, so that
(e) 1
c2 gi
i = Π(e) = ρmin Yk , (9)
3
and
∂t (ρYk ) + · (ρuYk ) = · ρmin Dk Yk + SYk . (10)

Reactive Flows

Species Equation in the Compressible Limit

By relating the relaxation frequency to the density-weighted mass
diffusivity as:
1 1 1 2
ρDk = ρmin − c δt, (11)
3 ωk 2
the species equation in the compressible limit is recovered.

Through the suggested procedure, the most general advection-
diffusion-reaction transport equation can be solved. The particular
cases of weak-compressible and incompressible ﬂows are readily
provided.

Since compressibility effects are accounted for both in Navier-
Stokes-Fourier and species equations, the proposed model is
actually suitable for simulating reactive ﬂows.


Reactive Flows

Reactive LBM at Work

We consider combustion of premixed stoichiometric H2 /Air
reactive mixture in a narrow channel.

Reactive LBM is validated against FLUENT for H2 +1/2O→H2 O.

Figure: Lines and symbols represent the LBM and FLUENT solutions, respectively


Link-wise Artiﬁcial Compressibility Method (LW-ACM)

Outline Compass




4 Reactive Flows




Link-wise Artificial Compressibility Method

Let us consider the following formula 6 :
(e)
fi (x, t + δt) = fi (x − ci δt, t)
ω−1 (12)
(e,o) (e,o)
+2 fi (x, t) − fi (x − ci δt, t) ,
ω

The updating rule of Eq.(12) is link-wise in the sense that for
updating fi only the information along the direction identified by
the lattice velocity ci are required. This is similar to LBM.

The updating of fi is ruled only by hydrodynamic quantities, while
LBM works also with the so-called ghost variables, beyond the
hydrodynamic variables.

6
Asinari P., Ohwada T., Chiavazzo E., Di Rienzo A.F., Link-wise Artificial Compressibility Method, submitted


Simple Boundaries: Stealing from CFD
L2 [u] L2 [u]
δx M a ∝ δt/δx ν∝ Re−1 Test 1 Test 2
1 × 10−1 3 × 10−2 3 × 10−2 1.74 × 10−3 4.59 × 10−4
5 × 10−2 1.5 × 10−2 3 × 10−2 4.49 × 10−4 1.21 × 10−4
2.5 × 10−2 7.5 × 10−2 3 × 10−2 1.20 × 10−4 3.11 × 10−5



Complex Boundaries: Stealing from LBM
δx M a ∝ δt/δx ν ∝ Re−1 L2 [u]
5 × 10−2 3 × 10−1 4 × 10−2 1.84 × 10−3
1.5 × 10−2 1.5 × 10−1 4 × 10−2 3.83 × 10−4
1.25 × 10−2 7.5 × 10−1 4 × 10−2 1.11 × 10−4



2D Lid Driven Cavity Flow

Figure: Comparison between LW-ACM and BGK at Re = 5000: streamlines (top) and
pressure contours (bottom).


3D Diagonally Driven Cavity Flow

Figure: 3D diagonally driven cavity (a); ﬂow at the middle plane MP at Re = 2000 (b);
velocity proﬁle along the line ML (c) and the line RL (d) at Re = 700.


3D Diagonally Driven Cavity with Palabos
Palabos "[...] offers an access to the rich world of lattice
Boltzmann, even to an audience with restricted theoretical
knowledge of this method" [www.lbmethod.org].
Implementation of LW-ACM in Palabos is in progress (in
collaboration with the University of Geneva, Switzerland).

Figure: Flow in the plane perpendicular to the direction of the lid at Re = 700. Results
are obtained with the LW-ACM currently implemented in Palabos.

Conclusions

1 A lattice Boltzmann scheme for simulating reactive flows has been
developed, so as to compensate for the limitations of previous
approaches. Accounting for compressibility effects in the
Navier-Stokes-Fourier and species equations, significant
temperature (density) variation can be handled.

2 Radiative LBM can be directly included (same data structures).

3 The LW-ACM has been proposed as an alternative to both
classical CFD and LBM, as long as incompressible and
weak-compressible flows are investigated. The link-wise
formulation preserves the ability of LBM in dealing with complex
geometries (no body-fitting is required), while the formulation in
terms of hydrodynamic quantities makes all existing CFD
technology readily available.


Thank you for your attention!


20-12-11 PhD Day Turin

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