Mesoscopic numerical methods for reactive flows

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

Candidate: Antonio F. Di Rienzo

Advisors: Prof. P. Asinari, Dr. E. Chiavazzo

External Collaborators: Dr. J. Mantzaras, Dr. N. Prasianakis (Paul
Scherrer Institute)

Energy Department, Politecnico di Torino, Torino, Italy

PhD Defence, Torino 20 April, 2012
Antonio F. Di Rienzo (Politecnico di Torino) PhD Defence Torino, April 20th 2012 1 / 46

Outline of The Talk

1 Summary of the talk

2 Introduction

3 The Lattice Boltzmann Method (LBM)

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

5 Radiative Transfer Equation

6 Combustion

7 Conclusions & Acknowledgements


Summary of the talk

Outline Compass


2 Introduction




6 Combustion



Summary of the talk

Topics

Link-wise Artificial Compressibility Method 1 : CFD by kinetic
mock-up models.

The Radiative transfer equation (RTE) is solved by means of
lattice Boltzmann method (LBM) formalism 2 : intensity is updated
according to lattice velocities.

A consistent lattice Boltzmann model for reactive flows 3 is
presented. It addresses the lack of accurate combustion models in
LBM literature (in collaboration with the Paul Scherrer Institute,
Switzerland).
1
Asinari P., Ohwada T., Chiavazzo E., Di Rienzo A.F., Link-wise Artificial Compressiniblity Method, Journ.
Comp. Phys., 2012 (preprint), (Impact Factor: 2.345)
2
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, Int. Journ. Num. Meth.
Heat Fluid Flow, 21 (5), 640-662, 2011, (Impact Factor: 0.53)
3
Di Rienzo A. F., Asinari P., Chiavazzo E., Prasianakis N., Mantzaras J., A Lattice Boltzmann model for reactive
flows simulation, EuroPhys. Lett., 2012 (accepted), (Impact Factor: 2.752)

Introduction

Outline Compass


2 Introduction




6 Combustion



Introduction

Simulating Reactive Flows

Reactive flows can be found in several energy systems: internal
combustion engines, industrial burners, gas turbine combustors...
Numerical modeling of reactive flows is a challenging task per se:
simultaneous processes must be taken into account (turbulent
mixing, multi-phase fluid-dynamics, radiative heat transfer).
Furthermore, in some applications, the numerical model must be
able to deal with different length scales.
The demand of energy systems with high energy efficiency and
low environmental impact requires the development of new energy
systems to be used in industrial and domestic applications.
Porous media combustion (PMC) proved to be a feasible option.
Porous media (PM) burners offer some advantages compared to
classical burners. The internal process of heat recovery makes
the combustion process more efficient, which comes with a lower
emission of polluntants, such as NOx and CO.

Introduction

Porous Media Burner

Abdul Mujeebu at al., Application of porous media combustion technology - A review,
Applied Energy, 2009

Introduction

Fluid Flow in Porous Media

If studies are carried out at the level of the pore-scale, detailed
local informations of the ﬂow can be obtained.
In Lattice Boltzmann Method (LBM) the link between the
pore-scale and the macro-scale is readily available.
LBM is often referred to as mesoscopic approach.

The Lattice Boltzmann Method (LBM)

Outline Compass


2 Introduction




6 Combustion




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 (satisfying some target PDEs) can be derived.
The operative formula consists of the (a) relaxation process and
the (b) advection process:

fi (x + vi ∆t, t + ∆t) − fi (x, t) = ω [fieq (x, t) − fi (x, t)].
ˆ (1)

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



Dealing with Complex Boundaries



State of Art


Link-wise Artiﬁcial Compressibility Method (LW-ACM)

Outline Compass


2 Introduction




6 Combustion




Limitation of Lattice Boltzmann Method
LBM works with larger set of unknowns, including the higher-order
moments beyond hydrodynamics:
ntimes mtimes
Πxx···x yy···y (xx · · · x, yy · · · y) = ˆn ˆm
vi,α vi,β fi .
i

They are unessential as long as the continuum limit is the main
concern and can lead to numerical instabilities 4 .
Ohwada & Asinari 5,6 revived the Artificial Compressibility Method
(ACM) as an high-order accurate numerical method for the
incompressible Navier-Stokes equations (INSE).
The asymptotic analysis of the LBM updating rule delivers the
same governing equations of ACM.
4
Dellar P., Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations,
PRE, 2002, 65:036309
5
Ohwada T., Asinari P., Artificial compressibility method revisited: asymptotic numerical method for
incompressible Navier-Stokes equations, JCP, 2010, 229:1698-1723
6
Ohwada T., Asinari P., Yabusaki D., Artificial compressibility method and lattice Boltzmann method: similarities
and differences, Comp. Math. Appl., 2011, 61:3461-3474


ACM vs LBM

Unlike LBM, ACM deals with macroscopic variables only. It offers
the opportunity of exploiting all existing Finite-Difference (FD)
technologies.

ACM relies on standard meshing techniques for dealing with
complex boundaries. Unstructured body-ﬁtted meshes are used in
order to adapt the computational grid to the real object. This
requires advanced algorithms, which imply an additional
computational overhead.

A possible improvements would be to have the LBM updating rule
of the distribution function depend only on the local equilibrium,
but retain the link-wise formulation.



In the attempt of making ACM more similar to LBM, the following
formula is proposed 7 :
(e)
fi (x, t + ∆t) = fi (x − vi ∆t, t)
ˆ
ω−1 (e,o) (e,o)
(2)
+2 fi (x, t) − fi (x − vi ∆t, t) ,
ˆ
ω
Similarly to LBM, the updating rule of Eq.(2) is link-wise.
(e) (e,o)
fi and fi are local functions of ρ = i fi and ρu = i vi fi ,
ˆ
but they do not depend on the higher-order moments.
Eq. (2) recovers the incompressible Navier-Stokes equations with
a kinematic viscosity deﬁned as:
1 1 1
ν= − .
3 ω 2
7
Asinari P., Ohwada T., Chiavazzo E., Di Rienzo A.F., Link-wise Artiﬁcial Compressibility Method, Journ. Comp.
Phys., 2012 (preprint)


Simple Boundaries

L2 [¯ ]
u L2 [¯ ]
u
∆x M a ∝ ∆t/∆x 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

∆x M a ∝ ∆t/∆x 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: Streamlines

Comparison between LW-ACM and BGK at Re = 5000.



2D Lid Driven Cavity: Pressure Contours

Comparison between LW-ACM and BGK at Re = 5000.



LW-ACM vs other INSE solvers

2D lid driven cavity ﬂow at Re = 5000: comparison between the LW-
ACM and alternative solvers for INSE from literature.
Scheme Grid (xp , yp ) (xlr , ylr ) Energy Enstrophy
Present 128 × 128 (0.51652, 0.53754) (0.81081, 0.079079) 0.039845 29.247
ACM, 2010 128 × 128 (0.52052, 0.53954) (0.82883, 0.071071) 0.027430 41.249
Opt. ACM, 2010 128 × 128 (0.51652, 0.53854) (0.80981, 0.072072) 0.038371 37.704
BGK-LBM 128 × 128 unstable unstable unstable unstable
MRT-LBM 128 × 128 (0.51652, 0.53554) (0.80881, 0.075075) 0.043600 37.404
Bruneau, 2006 128 × 128 (0.51562, 0.53906) (0.80469, 0.070313) 0.043566 30.861
Present 256 × 256 (0.51552, 0.53554) (0.80581, 0.074074) 0.044391 34.821
ACM, 2010 256 × 256 (0.51652, 0.53654) (0.80881, 0.072072) 0.040896 43.198
Opt. ACM, 2010 256 × 256 (0.51451, 0.53654) (0.80380, 0.072072) 0.048114 42.290
BGK-LBM 250 × 250 (0.51752, 0.54054) (0.80781, 0.074074) 0.041614 40.455
MRT-LBM 256 × 256 (0.51552, 0.53554) (0.80681, 0.074074) 0.045222 40.833
Bruneau, 2006 256 × 256 (0.51562, 0.53516) (0.80859, 0.074219) 0.046204 34.368
Bruneau, 2006 2048 × 2048 (0.51465, 0.53516) (0.80566, 0.073242) 0.047290 40.261


Radiative Transfer Equation

Outline Compass


2 Introduction




6 Combustion




Radiative Lattice Boltzmann Model

Combustion generally incorporates also radiation process, which
plays a signiﬁcant role if high temperature and a participating
medium are involved. Treatment of thermal radiation is of key
importance to develop mathematical model of combustion system.

Due to its kinetic nature, the Radiative Transfer Equation (RTE)
can be formulated according to the LBM formalism: the intensity is
already a particle (photon) distribution function.

Recently a LBM model 8 has been proposed to solve RTE in a
participating medium. However, further improvements are
required: the polar angle is not discretized, spoiling the accuracy
of the method with respect to standard Finite Volume Method
(FVM).

8
Asinari P., Mishra S.C., Borchiellini R., A lattice Boltzmann formulation to the analysis of radiative heat transfer
problems in a participating medium, NHT B, 2010, 57:1-21


For an absorbing, emitting and scattering participating medium
and under the assumption of isotropic scattering, radiative LBM
reads:
Iij (x + v∆t, t + ∆t) = Iij (x, t) + Vi,j (κa + σs ) [Ib (x, t) − Iij (x, t)].
(3)
The link-wise formulation makes unnecessary to march from each
single corner (e.g. Finite Volumes).
The data structures are the same as the ﬂuid ﬂow.



Azimuthal Angle Discretization

The azimuthal angle δ is discretized by introducing a ﬁnite number
of discrete velocities vi = (vx,i , vy,i ) lying on the lattice.
Even if not discretizing the polar angle saves a lot of computational
time, this approximation spoils the accuracy of the method.


Azimuthal Angle Discretization

Dimensionless heat
ﬂux as function for
different extinction co-
efﬁcients β = κa + σs .
The polar angle is not
discretized (Asinari P.
et al., NHT B, 2010).

The error between
LBM and FVM is 2%.



Polar Angle Discretization
An additional velocity vz,j is introduced along the z-axis (off the
lattice) in order to discretize the polar angle 9 :
π
vz,j = tan − γj vi (4)
2
The projection on the lattice of the total velocity Vij must overlap
vi .

9
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, Int. Jour. Num. Meth. Heat
Fluid Flow, 2011, 21:640-662


2D Square Enclosure

Dimensionless heat ﬂux for
β = 2 and β = 5. Radiative
equilibrium condition is as-
sumed, i.e. · qrad = 0.

The error between LBM
and FVM is reduced to 0.5%.

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

Combustion

Outline Compass


2 Introduction




6 Combustion



Combustion

Simulating Reactive Flows: What’s Required?

Despite the last decade has witnessed a significant improvement
in describing various problems (flow in porous media, thermal
flows, etc.), application of LBM to combustion is still limited.
"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
satisfactorily fulfill this requirement.
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.


Combustion

Governing Equations for Reactive Flows
The system of macroscopic governing equations reads:
∂t ρ + · (ρu) = 0, (5a)

∂t (ρu) + · (ρu ⊗ u + pI) = · Π, (5b)
N
dp
∂t (ρhs ) + · (ρuhs − κ T ) = +Π: u− h0 ωk Wk −
k˙ · qrad ,
dt
k=1
Qh
(5c)
∂t (ρYk ) + · (ρuYk ) = · (ρDk Yk ) + ωk Wk .
˙ (6)
The corresponding kinetic equations reads:
2∆t 2∆tτ
gt+∆t = gt + (f eq − gt ) + [Ψt + Φt + Qh ], (7)
∆t + 2τ t ∆t + 2τ
(∗) eq(∗)
ξt+∆t = ξt + ωk ξt − ξ t + ωk W k .
˙ (8)

Combustion

Navier-Stokes-Fourier in the Compressible Limit

2∆t 2∆tτ
gt+∆t = gt + (fteq − ft ) + [Ψt + Φt + Qh ] (9)
∆t + 2τ ∆t + 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.10,11
This partially bridges over the lacks of the existing models: also
species transport equation must reckon with compressibility
effects.
10
Prasianakis N. I., Karlin I. V., Lattice Boltmann method for simulation of thermal ﬂows on standard lattice, PRE,
2007, 76:016702
11
Prasianakis N. I., Karlin I. V., Lattice Boltzmann method for simulation of compressible ﬂows on standard
lattices, PRE, 2008, 78:016704

Combustion

Species Equation: Basic Model

Standard LBM12 emulates the species transport equation by
means of the following equation:
eq
ξi,k (x + vi ∆t, t + ∆t) = ξi,k (x, t)+ωk ξi,k (x, t) , −ξi,k (x, t) +ωk Wk ,
ˆ ˙
(10)
where
eq
ξi,k = wi ρYk [1 + 3 (ˆ i · u)] ,
v (11)
and
1 1 1
Dk = − . (12)
3 ωk 2
Eq. (10) recovers the following equation in the continuum limit:

∂t (ρYk )+ ·(ρuYk ) = ·(ρDk Yk )+ · (Dk Yk ρ)+ ωk Wk . (13)
˙
12
Yamamoto K., He X., Doolen G.D., Simulation of combustion ﬁeld with lattice Boltzmann method, JSP, 2002,
107:367-383

Combustion

Species Equation in the Compressible Limit

Standard LBM solves the species transport equation with the
following deviation term:

· (Dk Yk ρ). (14)

Deviation in the species equation is activated in case of signiﬁcant
compressibility effects (i.e. large ρ).
Due to the simpler nature of the equation, the deviation can be
removed without adding any corrections term. A strategy fully
relying upon the LBM formulation can be followed, instead.
The equilibrium distribution function is modiﬁed as follows:

eq(∗) 2 eq(∗) 1
ξ0,k = ρYk 1 − ϕ , ξ1,...,4,k = ρYk [ϕ + 3 (ˆ i · u)],
v (15)
3 6

with ϕ = ρmin /ρ.

Combustion

Species Equation in the Compressible Limit

The relaxation frequency is redefined as follows:

(∗) 1
ωk = . (16)
1 1 1 1
+ −
2 ϕ ωk 2

Through the suggested procedure13 , the advection-
diffusion-reaction equation in the compressible limit is recovered.
The particular cases of weak-compressible and incompressible
flows are readily provided, by setting ϕ = 1.

Since compressibility effects are accounted for both in Navier-
Stokes-Fourier and species equations, the proposed model is
actually suitable for simulating reactive flows.
13
Di Rienzo A.F., Asinari P., Chiavazzo E., Prasianakis N.I., Mantzaras J., A lattice Boltzmann model for reactive
flows simulation, EPL, 2012 (accepted)

Combustion

Reactor of Compact Gas-Turbines

Karagiannidis et al.,Experimental and numerical investigation of a propane-fueled, cat-
alytic mesoscale combustor, Catalysis Today, 2010

Combustion

Combustion in a Narrow Channel
We consider combustion of premixed stoichiometric H2 -air
reactive mixture. Radiation is neglected.


Combustion

Reactive LBM at Work: Uin = 0.85 m/s

Density, x-velocity and
temperature along the
horizontal symmetry plane
(top).
Species mass fractions
(bottom) along the channel
walls predicted by stan-
dard LBM (dashed-line)
are compared to those
recovered by the pro-
posed model (solid-lines).
Symbols are the FLUENT
solution: triangles H2 ,
squares O2 , circles H2 O.


Combustion

Reactive LBM at Work: Uin = 0.48 m/s

Density, x-velocity, tem-
perature (top) and species
mass fractions (bottom)
along the horizontal sym-
metry plane.
The maximun temperature
ratio is 5.5, twice greater
than previous models.

Maximum relative errors
are: 2% for the density,
4.2% for the velocity, 5.7%
for the temperature, 3% for
the species.


Conclusions & Acknowledgements

Outline Compass


2 Introduction




6 Combustion




Conclusions

1 The Link-wise Artificial Compressiblity Method (LW-ACM) has
been presented. The updating rule of the distribution function
depends only on the hydrodynamic variables. This fully alleviates
the problems of additional higher-order moments (e.g. at
boundaries) in pseudo-kinetic schemes.
2 With the LW-ACM, the ability of LBM to easily deal with complex
boundaries is preserved (no body-fitting is required) and all
existing techniques for classical computational fluid dynamic
(CFD) readily available.
3 The LW-ACM represents an alternative to both classical CFD and
LBM, as long as incompressible and weak-compressible flows are
investigated. It is a first step towards the development of models
for reactive flows. Compressibility effects are not taken into
account and its application to combustion simulation will be
considered in a near future.


Conclusions

4 A general lattice Boltzmann scheme for simulating reactive flows
at the low Mach number limit has been developed, so as to
compensate for the limitations of the models available in literature.
5 Accounting for compressibility effects in the Navier-Stokes-Fourier
and species equations, significant density (temperature) variations
can be handled. This extension allows to apply lattice Boltzmann
method (LBM) to a wide range of phenomena, which were not
properly addressed so far.
6 The accuracy of the previous radiative LBM model has been
improved by discretizing also the polar angle. The possibility to
use the same data structures makes the coupling with the reactive
flow solver straightforward.



Journal Publications
1 Di Rienzo A. F., Asinari P., Chiavazzo E., Prasianakis N. I., Mantzaras J., A
lattice Boltzmann model for reactive ﬂows simulations, EuroPhysics Letters,
2012 (accepted), (Impact Factor: 2.752)
2 Asinari P., Ohwada T., Chiavazzo E., Di Rienzo A. F., Link-wise Artiﬁcial
Compressibility Method, Journal of Computational Physics, 2012 (preprint),
(Impact Factor: 2.345)
3 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, Int. Jour. Num. Meth. for Heat
and Fluid Flow, 2011, 21:640-662, (Impact Factor: 0.53)

Proceedings
1 Di Rienzo A. F., Asinari P., Chiavazzo E., Link-wise equilibrium-based lattice
Boltzmann method, Proceedings of the XXIX UIT Heat Transfer Conference,
2011



Conference Presentations
1 Di Rienzo A. F., Asinari P., Chiavazzo E., Prasianakis N. I., Mantzaras J.,
Coupling lattice Boltzmann model with reduced chemical kinetics for combustion
simulations, 8th International Conference for Mesoscopic Methods in
Engineering and Science (ICMMES), July 4-8 2011, Lyon, France
2 Di Rienzo A. F., Asinari P., Chiavazzo E., Link-wise equilibrium based lattice
Boltzmann method, 29th UIT Heat Transfer Conference, June 20-22, 2011, Turin,
Italy
3 Di Rienzo A. F., Chiavazzo E., Asinari P., Radiative lattice Boltzmann method
applied to combustion simulation and reduced chemical kinetics, 19th
International Conference on the Discrete Simulation of Fluid Dynamics (DSFD),
July 5-9, Rome, Italy
4 Di Rienzo A. F., Izquierdo S., Asinari P., Mishra S. C., Borchiellini R., The lattice
Boltzmann method in solving radiative heat transfer in a participating medium,
First International Conference on Computational Methods for Thermal Problems
(ThermaComp), September, 8-10, 2009, Naples, Italy
Award
1 Best presentation at the Energetics PhD Day, Politecnico di Torino, December
20, 2011



Acknowledgements

Dr. John Mantzaras, Dr. Nikolaos Prasianakis



Thank you for your attention!


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