Content uploaded by Chuandong Lin
Author content
All content in this area was uploaded by Chuandong Lin on Mar 26, 2020
Content may be subject to copyright.
arXiv:1308.0653v3 [cond-mat.soft] 31 Aug 2014
Polar coordinate lattice Boltzmann kinetic modeling of
detonation phenomena
Chuandong Lin1, Aiguo Xu2,3,4∗, Guangcai Zhang2,4,5, Yingjun Li1†
1State Key Laboratory for GeoMechanics and Deep Underground Engineering,
China University of Mining and Technology, Beijing 100083, P.R.China
2National Key Laboratory of Computational Physics,
Institute of Applied Physics and Computational Mathematics,
P. O. Box 8009-26, Beijing 100088, P.R.China
3Center for Applied Physics and Technology,
MOE Key Center for High Energy Density Physics Simulations,
College of Engineering, Peking University, Beijing 100871, P.R.China
4State Key Laboratory of Explosion Science and Technology,
Beijing Institute of Technology, Beijing 100081, P.R.China
5State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences,Beijing 100190, P.R.China
(Dated: December 11, 2019)
∗Corresponding author. E-mail address: Xu Aiguo@iapcm.ac.cn
†Corresponding author. E-mail address: lyj@aphy.iphy.ac.cn
1
Abstract
A novel polar coordinate lattice Boltzmann kinetic model for detonation phenomena is presented
and applied to investigate typical implosion and explosion processes. In this model, the change
of discrete distribution function due to local chemical reaction is dynamically coupled into in the
modified lattice Boltzmann equation, which could recovery the Navier-Stokes equations, including
contribution of chemical reaction, via the Chapman-Enskog expansion. For the numerical investi-
gations, the main focuses are the nonequilibrium behaviors in these processes. The system at the
disc center is always in its thermodynamic equilibrium. The internal kinetic energies in different
degrees freedoms around the detonation front do not coincide due to the fluid viscosity. They show
the maximum difference at the inflexion point where the pressure has the largest spatial deriva-
tive. The dependence of the reaction rate on the pressure, influences of the shock strength and
reaction rate on the departure amplitude of the system from its local thermodynamic equilibrium
are probed.
PACS numbers: 47.11.-j, 47.40.Rs, 47.70.-n
2
I. INTRODUCTION
The rapid and violent form of combustion called detonation [1] propagates through deto-
nation wave which is a shock wave with chemical reaction. Given a wide range of application
in science and engineering, shock and detonation have always been of great concern in the
field of science and technology, such as the research and analysis on coal and gas outburst
mechanism [2, 3]. The detonation phenomena are widely used in the acceleration of various
projectiles, mining technologies, depositing of coating to a surface or cleaning of equipment,
etc. Early in 1899 and 1905, Chapmann [4] and Jouguet [5] presented CJ theory. This
theory assumes that detonation front is a strong discontinuous plane with chemical reaction
which immediately completes as soon as the detonation wave passes. In 1940s, Zeldovich
[6], Neumann [7] and Doering [8] presented the well-known ZND model. This model gives
an important conclusion that there is von-Neumann-peak at detonation wave front. Reac-
tant is firstly pre-compressed by shock wave, and there is a continuous reaction zone behind
the shock wave. Physical quantities (density, temperature, pressure and velocity) reach
maximum values within the reaction zone.
Although detonation has been studied for more than one century [9], it remains an active
area of research in both theoretical studies and numerical simulations [10] due to its practical
importance [11]. So far, all chemical reaction models are empirical or semi-empirical formulas
[12], such as the Arrhenius kinetics, forest fire burn, two-step model, Cochran’s rate function
[13], Lee-Tarver model [14], etc. Selecting appropriate chemical reaction kinetics is very
important for describing detonation phenomena under consideration. In this paper, we
adopt Cochran’s rate function for chemical reaction, which is one of the most physically
justifiable models satisfying simulation and experimental results [15, 16].
In recent decades Lattice Boltzmann (LB) method has achieved great success in various
fields of fluid dynamics[17–41]. LB modeling for combustion phenomena [42–59] has been
an interesting topic from early days. In 1997, Succi et al. [42] proposed the pioneering
LB model for combustion systems under the assumptions of fast chemistry and cold flames
with weak heat release. In 1998 and 2000, Filippova and H¨
anel [43–45] proposed a kind
of hybrid scheme for low Mach number reactive flows. The flow field is solved by modified
lattice-BGK model and the transport equations for energy and species are solved by a finite
difference scheme. In 2002, Yu et al. [46] simulated scalar mixing in a multi-component
3
flow and a chemical reacting flow using the LB method. In the same year, Yamamoto et al
[47] presented a LB model for simulation of combustion, which includes reaction, diffusion
and convection. In 2006, Lee et al. presented a new two-distribution LB equation algorithm
to solve the laminar diffusion flames within the context of Burke-Schumann flame sheet
model. In 2007, Chen et al [52] developed a novel coupled lattice Boltzmann model for two-
and three-dimensional low Mach number combustion simulations. In this model, the fluid
density can bear sharp changes. In the following year, another LB model was proposed for
simulating combustion in two-dimensional system by Chen and and his cooperators [53].
Within this model the time step and the fluid particle speed can be adjusted dynamically.
Later, based on their improved models, they presented a number of works [54–59].
However, previous studies on LB model were mainly focused on isothermal and incom-
pressible fluid systems. Those models generally can not recover the correct energy equation
or describe enough the compressibility in the hydrodynamic limit, which makes difficult the
modeling of systems with shock and/or detonation. At the same time, most of those LB
models assume that exothermic reaction has no significant effect on fluid field, which also
constrains the practical application of the models to most cases of combustion. In recent
years, the development of LB models for high speed compressible flows [60–67] makes it
possible to simulate systems with shock and detonation. Very recently Yan, Xu, Zhang,
et al. proposed a Lattice Boltzmann Kinetic Model (LBKM) for detonation phenomena
in Cartesian coordinates [68]. For simulating the explosion and implosion behaviors, a po-
lar coordinate LB model is obviously more convenient. And there are many nice papers
about LB formulations for axisymmetric flows in polar coordinates [69–73]. In 2011 Watari
[74] proposed a finite-difference LB methods in polar coordinate system. Recently we [75]
improved the LBKM by using a hybrid scheme so that it works also for supersonic flows.
Within the improved model, the temporal evolution is calculated analytically and the con-
vection term is solved via a Modified Warming-Beam (MWB) scheme. In this work, a new
polar coordinate LBKM which is similar to and simpler than the one in Ref.[75] is used to
study detonation phenomena.
In contrast to traditional methods based on Navier-Stokes description, the LBKM has
some intrinsic superiority in describing kinetic mechanisms in systems where equilibrium and
non-equilibrium behaviors coexist [37, 67, 68, 75]. The mini-review [67] presented a method-
ology to investigate non-equilibrium behaviors of the system by using the LB method. The
4
non-equilibrium behaviors in various complex systems attract great attention [68, 75–77].
In the work [68] by Yan, Xu, Zhang, et al., some non-equilibrium behaviors around the von
Neumann peak are obtained. In a recent work [75] we studied the non-equilibrium char-
acteristics of the system around three kinds of interfaces, the shock wave, the rarefaction
wave and the material interface, for two specific cases. We draw qualitative information on
the actual distribution function. In this work, we further develop the LBKM with chemical
reaction in polar coordinates to model the implosion and explosion phenomena and investi-
gate the macroscopic behaviors due to deviating from local thermodynamic equilibrium in
the detonation procedure.
The rest of the paper is structured as follows. In section II the polar coordinate LBKM
for compressible fluid with chemical reaction is proposed for the first time. This model
can recovery the Navier-Stokes equations with chemical reaction. The treatment of inner
boundary around disc center is presented, and the manifestations of non-equilibrium char-
acteristics are introduced. In section III we give the chemical reaction model and numerical
verification, simulate implosion and explosion phenomena, and study the non-equilibrium
characteristics of each case. The actual distribution functions around detonation wave are
qualitatively illustrated. Section IV gives the conclusion and discussions.
II. LBKM IN POLAR COORDINATES
A. Modified Boltzmann equation with chemical reaction
Here, for the purpose of simulating detonation which includes chemical reaction, we pro-
pose a novel Botlzmann equation, to the right side of which an artificial term Mis added.
This term is called chemical term. The modified Boltzmann equation with the Bhatanger-
Gross-Krook approximation reads
∂f
∂t +v· ∇f=−1
τ(f−feq) + M. (1)
where f(feq) is the (equilibrium) distribution function; τthe relaxation time; vthe discrete
velocity. To give the special form of M, the following assumptions are made:
1. The flow is describe by single function f. The the relaxation time τis constant (not
a function of density or temperature) and independent of time tor space r.
5
2. The flow is symmetric, and there are no external forces. The radiative heat loss is
neglected.
3. The local particle density ρand hydrodynamic velocity uremains unchanged in the
progress of local chemical reaction, i.e.,
dρ
dt |R(λ)= 0,(2)
du
dt |R(λ)= 0.(3)
The temperature increases due to the chemical energy released.
4. It is an irreversible reaction, the process of which is described by an empirical or
semi-empirical formulas, i.e.,
dλ
dt =R(λ),(4)
where λdenotes the progress rate of reaction.
In fact, the chemical term Min Eq.1 refers to the change of distribution function fdue
to the local chemical reaction. Specially,
M=df
dt |R(λ).(5)
If the physical system has little departure from equilibrium state, f≈feq , Eq.5 gives
M=dfeq
dt |R(λ).(6)
With feq =feq(ρ, u, T ) and d
dt =∂
∂t +u· ∇, Eq.6 gives
M=∂f eq
∂ρ
dρ
dt |R(λ)+∂f eq
∂u
du
dt |R(λ)+∂f eq
∂T
dT
dt |R(λ).(7)
Substituting Eqs.2 and 3 into 7 gives
M=∂f eq
∂T
dT
dt |R(λ).(8)
With the equilibrium distribution function
feq =ρ(1
2πkT )D/2Exp[−(v−u)2
2kT ],(9)
for a D-dimensional physical system where the particle mass is m= 1, we get
∂f eq
∂T =−DT + (v−u)2
2T2feq.(10)
6
With the relation T=2E
Dρ between temperature Tand internal energy per volume E, we get
dT
dt |R(λ)=2
Dρ
dE
dt |R(λ)=2Q
D
dλ
dt .(11)
where Qis the amount of heat released by the chemical reactant per unit mass. Equations
4 and 11 give
dT
dt =2Q
DR(λ).(12)
Substituting Eqs.10, 12 into 8, we get
M=feq −DT + (v−u)2
2T2
2Q
DR(λ).(13)
It is clear that Eq.13 satisfies the following relations
ZMdv=dρ
dt |R(λ)= 0 (14)
ZMvdv=dρu
dt |R(λ)= 0 (15)
1
2ZMv2dv=dE
dt |R(λ)=ρQR(λ) (16)
B. Discrete velocity model
In a polar coordinate system, the LB equation corresponding to Eq.1 reads,
∂fki
∂t +vkir
∂fki
∂r +1
rvkiθ
∂fki
∂θ =−1
τ(fki −feq
ki ) + Mki,(17)
Mki =feq
ki
−DT + (vki −u)2
2T2
2Q
DR(λ),(18)
where r(θ) is the radial (azimuthal) coordinate; fki (feq
ki ) is the discrete (equilibrium) distri-
bution function; vkir (vkiθ) is the radial (azimuthal) component of the discrete velocity vki
as below [74, 78],
vki =vkir er+vkir eθ, vkir =vkcos(iπ/4−θ), vkiθ =vksin(iπ/4−θ),(19)
with unit vectors erand eθ. The subscript k(= 0,1,2,3,4) indicates the k-th group of the
particle velocities with speed vk. One speed is v0= 0, and each of the other group has
8 components, i.e. i= 0,1,···,7. In this work we choose v1= 1.5, v2= 3.5, v3= 7.5,
v4= 12.5.
7
In terms of local particle density ρ(= Pki fki), hydrodynamic velocity u(= Pki fkivki/ρ)
and temperature T(= Pki
1
2fki(vki −u)·(vki −u)/ρ), we get
feq
ki =ρFk[(1 −u2
2T+u4
8T2) + vkiε uε
T(1 −u2
2T) + vkiε vkiπ uεuπ
2T2(1 −u2
2T)
+vkiεvkiπvkiϑuεuπuϑ
6T3+vkiεvkiπvkiϑvkiξuεuπuϑuξ
24T4] (20)
with weighting coefficients
Fk=1
v2
k(v2
k−v2
k+1)(v2
k−v2
k+2)(v2
k−v2
k+3)[48T4−6(v2
k+1 +v2
k+2 +v2
k+3)T3
+(v2
k+1v2
k+2 +v2
k+2v2
k+3 +v2
k+3v2
k+1)T2+1
4v2
k+1v2
k+2v2
k+3T],
F0= 1 −8(F1+F2+F3+F4),
where the subscript {k+l}equals to (k+l−4) if (k+l)>4.
Via the Chapman-Enskog expansion, it is easy to find that Eq.17 could recovery the
following Navier-Stokes equations
∂ρ
∂t +∇ · (ρu) = 0, (21)
∂(ρu)
∂t +∇ · (PI+ρuu) + ∇ · [µ(∇ · u)I−µ(∇u)T−µ∇u] = 0, (22)
∂
∂t (ρE +1
2ρu2) + ∇ · [ρu(E+1
2u2+P
ρ)]
−∇ · [κ′∇E+µu·(∇u)−µu(∇ · u) + 1
2µ∇u2] = ρQR(λ), (23)
in the hydrodynamic limit, where µ(=P τ ) and κ′(= 2P τ) are viscosity and heat conductivity,
respectively.
C. LB evolution equation
The evolution equation, with first-order accuracy, used for Eq.17, reads
ft+∆t
ki =feq
ki + (ft
ki −feq
ki ) exp(−∆t/τ) + f∗
ki,r +f∗
ki,θ +Mki ∆t, (24)
and
f∗
ki,r =
Cr[fki(ir, iθ)−fki(ir−1, iθ)] f or Cr≥0,
Cr[fki(ir+ 1, iθ)−fki(ir, iθ)] f or Cr<0,(25)
8
O
Y
0
υ
1
υ
2
υ
3
υ4
υ
i 3=
i 1=
i 4=
i 0=
i 5=
i 7=
i 2=
i 6=
0
υ
1
υ
2
υ
3
υ4
υ
i 7=
i 5=
i 0=
i 4=
i 1=
i 3=
i 6=
i 2=
FIG. 1: Rotation of the distribution functions from the first to the fifth sector of physical domain
in a disc divided into 8 sections.
f∗
ki,θ =
Cθ[fki(ir, iθ)−fki(ir, iθ−1)] f or Cθ≥0,
Cθ[fki(ir, iθ+ 1) −fki(ir, iθ)] f or Cθ<0.(26)
with Courant-numbers Cr(= vkir ∆t
∆r) and Cθ(= 1
rvkiθ ∆t
∆θ).
D. Boundary conditions
The physical domain under consideration is in a sector which is only 1/8 of an annular
or circular area. The azimuthal boundaries are treated with periodic boundary conditions
[75]. For annular area with radii 0 < R1< R2, inflow/outflow conditions are imposed at
radial boundaries [75]. For circular area with radius R, its outer radial boundary is treated
in the same way. Specially, around the center, the inner boundary is treated as,
f(ir, iθ, k, i) = f(1 −ir, iθ, k, mod(i+ 4,8)),(27)
with ir=−1,0 for the nodes added to the computational domain and the function mod(a, b)
means the remainder of adivided by b. Figure 1 shows the relation between the distribution
functions in the first and the fifth sector of physical domain in a periodic circular area by
rotation.
9
E. Non-equilibrium characteristics
LB model naturally inherits the function of Boltzmann equation describing non-
equilibrium system. The departure of the system from local thermodynamic equilibrium
state can be measured by the high-order moments of fki. As given in [68, 75], the central
moments M∗
mare defined as:
M∗
2(fki) = Pki fkiv∗
kiv∗
ki
M∗
3(fki) = Pki fkiv∗
kiv∗
kiv∗
ki
M∗
3,1(fki) = Pki
1
2fkiv∗
ki ·v∗
kiv∗
ki
M∗
4,2(fki) = Pki
1
2fkiv∗
ki ·v∗
kiv∗
kiv∗
ki
(28)
where v∗
ki =vki −u. The manifestations of non-equilibrium are defined as:
∆∗
m=M∗
m(fki)−M∗
m(feq
ki ).(29)
In theory, M∗
3(feq
ki ) = 0, M∗
3(fki) = ∆∗
3,M∗
3,1(feq
ki ) = 0, M∗
3,1(fki) = ∆∗
3,1,M∗
3,1,r(fk i) =
1
2(M∗
3,rrr (fki ) + M∗
3,rθθ (fki )), M∗
3,1,θ(fki) = 1
2(M∗
3,rrθ (fki ) + M∗
3,θθθ (fki )), ∆∗
3,1,r(fk i) =
1
2(∆∗
3,rrr (fki ) + ∆∗
3,rθθ (fki )), ∆∗
3,1,θ(fki) = 1
2(∆∗
3,rrθ (fki ) + ∆∗
3,θθθ (fki )).
III. DETONATION
Detonation is in a complex process with mutual influence between fluid dynamics and
chemical reaction kinetics. The detonation front propagates into unburnt gas at a velocity
higher than the speed of sound in front of the wave [9]. Physical quantities at two sides of
detonation front satisfy Hugoniot relations [1].
A. Chemical reaction
To describe the chemical process of detonation, we choose Cochran’s rate function pre-
sented by Cochran and Chan [13],
R(λ) = ω1Pm(1 −λ) + ω2Pnλ(1 −λ),(30)
where λ(= ρp/ρ) is the mass fraction of reacted reactant, and ρpis the density of reacted
reactant. The right side of Eq.30 is composed of a hot formation term and a growth term.
10
Pmand Pndescribe the dependence on the local pressure and ω1,ω2,mand nare adjustable
parameters. Furthermore, T > Tth is a necessary condition for chemical reaction, with the
ignition temperature Tth. In this work, we choose m=n= 1, Tth = 1.1.
Via introducing the symbol, a=ω1Pm,b=ω2Pn,λ=λir=λiθ=λ(ir, iθ, t), the
evolution of Eq.30 with first-order accuracy reads
λt+∆t=(a+bλ)e(a+b)∆t−a(1 −λ)
(a+bλ)e(a+b)∆t+a(1 −λ)+λ∗
ir+λ∗
iθ,(31)
and
λ∗
ir=
−ur(λir−λir−1)
∆r∆t for ur≥0,
−ur(λir+1−λir)
∆r∆t for ur<0,(32)
λ∗
iθ=
−uθ(λiθ−λiθ−1)
r∆θ∆t for uθ≥0,
−uθ(λiθ+1−λiθ)
r∆θ∆t for uθ<0.(33)
In the evolution of λ, the velocity uand the pressure Pare calculated from fki. In this way,
the chemical reaction has coupled naturally with the flow behaviors.
It is worth pointing out that the Cochran’s rate function used in this work is similar to,
but different from, the Lee-Tarver model used in the work [68]. The parameters aand b
in Eq.31 depend on the pressure in the former work, while they are given fixed values in
the latter work where the pressure plays no role in the chemical process. Consequently, the
extinction phenomenon can be investigated in this work and can not be simulated in the
latter work.
B. Simulation of steady detonation
1. Validation and verification
In this section, a steady detonation is simulated to demonstrate the validity of the new
model. The initial physical quantities are as:
(ρ, T, ur, uθ, λ)i= (1.35826,2.59709,0.81650,0,1)
(ρ, T, ur, uθ, λ)o= (1,1,0,0,0) (34)
which satisfy the Hugoniot relations for detonation wave. Here the suffixes iand oindex two
parts, 1000 ≤r≤1000.01 and 1000.01 < r ≤1000.1, in an annular area, respectively. The
11
FIG. 2: physical quantities of steady detonation wave at time t= 0.025, with radial range 1000.07 ≤
r≤1000.1: (a) ρ, (b) T, (c) P, (d) ur, (e) λ.
inner radius is given large enough, so that the curvature becomes negligible and the polar
coordinates revert locally to Cartesian coordinates. With this condition, the simulation
results can be compared to the analytic solutions of the 1-dimensional steady detonation
wave. Other parameters are τ= 2 ×10−4, ∆t= 2.5×10−7,Nr×Nθ= 20000 ×1, ω1= 10,
ω2= 1000.
Figure 2 gives LB simulation results, CJ results [1, 4, 5] and ZND results [1, 6–8] of phys-
ical quantities (ρ,T,P,ur,λ) at time t= 0.025, with radial range 1000.07 ≤r≤1000.1,
respectively. The solid lines with squares are for LB simulation results, the dashed lines
are for analytic solutions of CJ theory, and the solid lines are for analytic solutions of
ZND theory. The simulation physical quantities after detonation wave are (ρ, T, ur, uθ, λ) =
(1.36163,2.59366,0.819670,0,1). Comparing them with CJ results gives the relative differ-
ences 0.2%, 0.1%, 0.3%, 0% and 0%, respectively. Panels (a)-(e) show that the LB simulation
results have a satisfying agreement with the ZND results in the area behind von Neumann
peak. There are few differences between them. Physically, the analytic solutions of ZND
theory here ignore the viscosity and heat conduction, and the von Neumann peak is simply
treated as a strong discontinuity. Furthermore, the relative difference is 1.8% between the
12
(a)(b)
FIG. 3: The profile of steady detonation wave in an annular area with radii R1= 1000 and
R2= 1000.1 at time t= 0.025: (a) physical quantities, (b) gradients. From left to right, three
vertical lines are shown to guide the eyes for the rarefaction area, the maximum value of pressure,
the pre-shocked area, respectively.
simulation detonation velocity D= 3.152 and the analytic solution D= 3.09557. In sum,
the current LB model works for detonation phenomenon.
2. Nonequilibriums in steady detonation wave
Figure 3 gives the physical quantities and their gradients versus radius at time t= 0.025,
with radial range 1000.08 ≤r≤1000.0925. Three vertical lines are shown, from left to right,
to guide the eyes for the rarefaction area, the von-Neumann peak and the pre-shocked area,
respectively. Panel (a) shows that the maximum values of density, temperature, pressure,
velocity do not coincide. The radial positions of their maximum values are Rρ= 1000.08740,
RT= 1000.08654, RP= 1000.08712, Ru= 1000.08739. Panel (b) shows that the largest
absolute values of ∇ρ,∇T,∇P,∇uare located at the pre-shocked area, their second largest
values are at the rarefaction area, and their vales are close to zero at the von-Neumann peak.
Figure 4 shows the central moments and their non-equilibrium manifestations in the case
corresponding to Fig.3. The simulation results of M∗
2(fki), M∗
3(fki), M∗
3,1(fki), M∗
4,2(fki),
M∗
2(feq
ki ), M∗
3(feq
ki ), M∗
3,1(feq
ki ), M∗
4,2(feq
ki ), ∆∗
2,∆∗
3,∆∗
3,1,∆∗
4,2are shown in Figs.4 (a)-(l),
respectively. The vertical lines in Figs.4 (i)-(l) coincide with the ones in Fig.3. It’s easy to
get from Fig.4 that, the non-equilibrium system is mainly around the von-Neumann peak.
13
FIG. 4: The simulation results of M∗
m(fki), M∗
m(feq
ki ) and ∆∗
min the same case as Fig.3.
The departure of the system from its equilibrium around the rightmost line is opposite
the one around the leftmost line. Physically, the former is under shock effect, whereas the
latter under rarefaction effect. Furthermore, both ∆∗
2,rr and ∆∗
2,θθ are close to zero at the
von-Neumann peak, i.e., the internal kinetic energies in different degrees of the freedom are
equal to each other at the von-Neumann peak. Comparing Fig.3 (b) with Fig.4 (i) gives that
the internal kinetic energies in different degrees of freedom show the maximum difference at
the inflexion point where the pressure has the largest spatial derivative. Figure 4 (a) shows
that the internal energy in the freedom of rand that in the freedom of θdo not coincide due
to the fluid viscosity. The former travels faster than the latter around the detonation front.
Around the leftmost line in Fig.4 (i), ∆∗
2,rr shows a negative peak and ∆∗
2,θθ shows a
positive peak with the same amplitude, which implies that the distribution function f(vr)
is “thinner”and “higher”than the Maxwellian feq , and f(vθ) is “fatter”and “lower”. The
simulation results of ∆∗
3in Fig.4(j) and ∆∗
3,1in Fig.4(k) indicate that f(vθ) is symmetric,
and the f(vr) is asymmetric. The portion of f(vr) for vr>0 is “thinner”than that for
14
a
eq
f
( )
r
f v
( ) f v
θ
eq
f
( )
r
f v
( ) f v
θ
b
FIG. 5: The sketch of the Maxwellian and actual distribution functions versus velocity vrand vθ,
respectively. (a) the distribution functions at the leftmost line, (b) the distribution functions at
the rightmost line. The long-dashed line is for distribution function f(vr), the shot-dashed one is
for distribution function f(vθ), and the solid line is for feq .
vr<0. Figure 5 (a) shows the sketch of the actual distribution functions, f(vr), f(vθ) and
the Maxwellian feq. For the rightmost line in Fig.4, similarly, Fig.5(b) shows the sketch of
the actual distribution functions, f(vr), f(vθ) and the Maxwellian feq . It can be found that
f(vr) is “fatter”and “lower”than the Maxwellian feq, while f(vθ) is “thinner”and “higher”.
f(vθ) is symmetric, while f(vr) is asymmetric. The portion of f(vr) for vr>0 is “fatter”
and the portion of f(vr) for vr<0 is “thinner”.
Moreover, the simulation result ∆∗
2,rθ = 0 in Fig.4(i) indicate that the contours of the
actual distribution function in velocity space (vr,vθ) is symmetric about the vr-axis or/and
vθ-axis. The above analysis suggests that vr-axis is the symmetric axis. Combining Figs.5
and the sketch of contours of the actual distribution function gives the sketch of the actual
distribution function in velocity space (vr,vθ) [75].
C. Simulation of implosion
For the case of implosion, the initial physical quantities are:
(ρ, T, ur, uθ, λ)i= (1,1,0,0,0)
(ρ, T, ur, uθ, λ)o= (1.5,1.55556,−0.666667,0,1),(35)
where the suffixes iand oindex areas 0 ≤r≤0.098 and 0.098 < r ≤0.1, respectively.
Other parameters are τ= 2 ×10−4, ∆t= 2.5×10−6,Nr×Nθ= 2000 ×1, ω1= 1, ω2= 50.
15
b c
d ef
radius
radial velocity
0.025 0.05 0.075 0.1
-1
-0.5
0
0.5 t=0.0000
t=0.0375
t=0.0425
t=0.0500
radius
temperature
0.025 0.05 0.075 0.1
2
4
6
8t=0.0000
t=0.0375
t=0.0425
t=0.0500
radius
pressure
0.025 0.05 0.075 0.1
10
20
30 t=0.0000
t=0.0375
t=0.0425
t=0.0500
radius
0.025 0.05 0.075 0.1
-0.8
-0.4
0
0.4
t=0.0000
t=0.0375
t=0.0425
t=0.0500
radius
λ
0.025 0.05 0.075
0
0.2
0.4
0.6
0.8
1
t=0.0000
t=0.0375
t=0.0425
t=0.0500
radius
density
0.025 0.05 0.075 0.1
1
2
3
4t=0.0000
t=0.0375
t=0.0425
t=0.0500
a
FIG. 6: Physical quantities versus radius in implosion process at times, t= 0.0000, 0.0375, 0.0425
and 0.0500, respectively: (a) ρ; (b) T; (c) P; (d) ur; (e) λ; (f) ∆∗
2,rr.
Figure 6 shows physical quantities along radius in implosion process: (a) density; (b) tem-
perature; (c) pressure; (d) radial velocity; (e) the parameter for chemical reaction process;
(f) ∆∗
2,rr . There are two stages in implosion process. In the former stage, the detonation
travels inwards, the material behind the detonation front moves inwards, and the density,
temperature and pressure behind detonation wave increase continuously due to the disc ge-
ometric effect. When the detonation wave reaches the center, the density, temperature and
pressure get their maximum values. Meanwhile, the velocity reduces to zero gradually and
then point outwards. In the latter stage, the detonation wave travels outwards. As the
chemical reaction completes, the detonation wave becomes a pure shock wave. The hydro-
dynamic velocity in front of the shock wave points inwards and that behind the wave points
outwards. Consequently, the density, temperature and pressure outside the shock wave still
increase continuously, and those inside reduce.
In addition, Fig.6 (f) shows that the departure of the system from equilibrium increases
16
(reduces) when the detonation or shock wave becomes stronger (weaker). Specially, the
value of ∆∗
2,rr shows a crest and a trough from the time t= 0.0000 to t= 0.0425. The
crest results from compression effect ahead of the detonation front, while the trough results
from rarefaction effect behind. From t= 0.0425 to 0.0500, it is also positive at the crest
and negative behind. In fact, ∆∗
2,rr is always positive at the shock wave and negative at the
rarefaction wave, which can be seen as a criterion to distinguish the two waves. Furthermore,
the system at the disc center is always in its thermodynamic equilibrium.
D. Simulation of explosion
For the case of explosion, the initial physical quantities are:
(ρ, T, ur, uθ, λ)i= (1.5,1.55556,0.666667,0,1)
(ρ, T, ur, uθ, λ)o= (1,1,0,0,0) (36)
where the suffixes iand oindex areas 0 ≤r≤R1and R1< r ≤R, respectively. Here
τ= 2 ×10−4, ∆t= 2.5×10−6,ω1= 1, ω2= 50. Figures 7-9 show the evolution of
physical quantities (ρ,T,P,ur,λ, ∆∗
2,rr ) versus radius. Figure 7 corresponds to parameters
R1= 0.015, R= 0.3, Nr×Nθ= 6000 ×1; Fig.8 corresponds to parameters R1= 0.023,
R= 0.75, Nr×Nθ= 15000 ×1; Fig.9 corresponds to parameters R1= 0.050, R= 1.2,
Nr×Nθ= 24000 ×1.
Figure 7 shows the extinction phenomenon. The area of initial reacted reaction is very
small, i.e., the initial energy is not enough to trigger the detonation. The energy propagates
outward in the form of disturbance wave with amplitude decreasing gradually. The wave
dissipates and vanishes finally. It’s easy to get from Fig.7 (e) that λonly has a little change
at the beginning. The reason is that the initial temperature of reacted reaction is higher
than the temperature threshold Tth and there is a little chemical reaction at the start. The
rate of chemical energy released is smaller than the rate of heat dissipated under the disc
geometric effect. Consequently, the energy of disturbance wave reduces gradually, which
leads to extinction. Figure (f) shows that the system has only a small departure from
equilibrium at the start. The departure reduces gradually and vanishes finally.
Figure 8 shows that (i) A disturbance wave travels outwards from t= 0.0000 to 0.1850.
Part reactant reacts around the disturbance wave. The released chemical energy is added
into the disturbance wave, meanwhile the thermal energy within disturbance wave disperses
17
b c
d ef
radius
radial velocity
0 0.1 0.2 0.3
0
0.2
0.4
0.6 t=0.000
t=0.005
t=0.050
t=0.150
radius
temperature
0 0.1 0.2 0.3
0.9
1
1.1
1.2
1.3
1.4
1.5 t=0.000
t=0.005
t=0.050
t=0.150
radius
pressure
0 0.1 0.2 0.3
1
1.5
2
t=0.000
t=0.005
t=0.050
t=0.150
radius
0 0.1 0.2 0.3
0
0.02
0.04
0.06 t=0.000
t=0.005
t=0.050
t=0.150
radius
λ
0 0.1 0.2 0.3
0
0.2
0.4
0.6
0.8
1t=0.000
t=0.005
t=0.050
t=0.150
radius
density
0 0.1 0.2 0.3
0.8
1
1.2
1.4 t=0.000
t=0.005
t=0.050
t=0.150
a
FIG. 7: For the case R1= 0.015, physical quantities versus radius in explosion process at times,
t= 0.000, 0.005, 0.050 and 0.150, respectively: (a) ρ; (b) T; (c) P; (d) ur; (e) λ; (f) ∆∗
2,rr.
to its adjacent area. The disturbance wave becomes wider in both radical and azimuthal
directions under disc geometric effect, and the maximum values of density, temperature,
pressure and velocity reduce. The value of ∆∗
2,rr shows a crest at the disturbance wave
and a trough behind. (ii) The perturbation wave is transformed into detonation wave from
t= 0.1850 to 0.3000. The heat release rate of chemical reactant increases sharply. The
physical quantities (ρ,T,P,ur) increase suddenly. Meanwhile the number of crest (trough)
of ∆∗
2,rr increases from one to two. (iii) The implosion and explosion waves coexist from
t= 0.3000 to 0.3475. With amplitude defined as the distance from the crest to trough
of ∆∗
2,rr , it is found that the amplitudes at the implosion and explosion waves increase
dramatically, especially the former one increases from 2.78 ×10−2to 1.39. For the purpose
of a clear view, the plot of ∆∗
2,rr at the time t= 0.3475 is given specially within panel (f ).
(iv) From t= 0.3475 to 0.4000, the implosion wave passes the center of the disc, then travels
outwards and changes into a shock wave with the completion of chemical reaction. It is then
18
FIG. 8: For the case R1= 0.023, physical quantities versus radius in explosion process at times,
t= 0.0000, 0.1850, 0.3000, 0.3475 and 0.4000, respectively: (a) ρ; (b) T; (c) P; (d) ur; (e) λ; (f)
∆∗
2,rr.
transformed into a perturbation wave, dissipates and vanishes finally. While the explosion
wave propagates outwards and its peak rises further. The amplitude of ∆∗
2,rr at the inner
outgoing wave reduces, and the one at the explosion wave increases.
Figure 9 shows a direct explosion phenomenon with ignition energy large enough. Panel
(a) shows that the density behind the detonation front is lower than the one outside due
to the disc geometric effect. Panels (b)-(c) shows higher temperature and pressure inside.
Panel (d) shows that the velocity inside is close to zero. Panel (e) shows that chemical
reaction steadily proceeds. Panel (f) shows that the amplitude of ∆∗
2,rr increases gradually.
The detonation wave becomes wider and wider in both radical and azimuthal directions.
In fact, there is competition between the chemical reaction, macroscopic transporta-
tion, thermal diffusion and the geometric convergence or divergence in the detonation phe-
nomenon. The chemical reaction increases the temperature while the thermal diffusion
19
FIG. 9: For the case R1= 0.050, physical quantities versus radius in explosion process at times,
t= 0.000, 0.025, 0.250 and 0.500, respectively: (a) ρ; (b) T; (c) P; (d) ur; (e) λ; (f) ∆∗
2,rr.
decreases the temperature around the detonation wave. If there is enough thermal energy
transformed from chemical energy, the detonation proceeds; otherwise, extinguishes. Spe-
cially in explosion case, if the geometric divergence effects dominate, extinction will occur;
with the the combustion front propagating outwards, geometric divergence makes less effect,
the existing part combustion may result in complete combustion, even detonation.
IV. CONCLUSIONS AND DISCUSSIONS
A polar coordinate Lattice Boltzmann Kinetic Model (LBKM) for detonation phenomena
is presented. Within this novel model, the change of discrete distribution function due to
local chemical reaction is given directly in the modified lattice Boltzmann equation, which
could recovery the Navier-Skokes equations including chemical reaction via Chapman-Enskog
expansion. And the chemical reaction is described by Cochran’s rate function. A combined
20
scheme is used to treat with the LB equation and the Cochran’s rate equation. Both the
temporal evolution of the collision effects in the LB equation and the temporal evolution of
chemical reaction in Cochran’s rate equation are calculated analytically. Both the convection
terms in the LB equation and Cochran’s rate equation are treated using the first-order
upwind scheme. From the numerical point of view, compared with the LBKM in Ref. [75],
the present model has the same accuracy but is simpler. From the physical or chemical
point of view, compared with the Lee-Tarver model used in previous work [68], the pressure
effects on the reaction rate is taken into account in the Cochran’s rate function.
Compared with the previous work in Ref. [75], the inner boundary condition for the disc
computational domain is treated more naturally. In Ref. [75] the disc computational domain
is approximated by a annular domain where the inner radius approaches zero. Consequently,
one needs to construct ghost nodes for the inner boundary condition. In this work the center
of the disc computational domain is considered as a inner point of the system. For periodic
system, no ghost node is needed for the inner boundary condition. Other boundaries are
treated with the same method as [75].
The simulation results of physical quantities in the steady detonation process have a sat-
isfying agreement with analytical solutions. Typical implosion and explosion phenomena
are simulated. By changing initial ignition energy, we investigate three cases of explosion,
including a case with extinction phenomenon. It is interesting to find that the geometric
convergence or divergence effect makes the detonation procedure much more complex. The
competition between the chemical reaction, the macroscopic transportation, the thermal dif-
fusion and the geometric convergence or divergence determines the ignition process. If there
is enough thermal energy transformed from chemical energy, the detonation proceeds; other-
wise, extinguishes. Specially in explosion case, if the geometric divergence effects dominate,
extinction will occur; with the combustion front propagating outwards, geometric diver-
gence makes less effect, the existing part combustion may result in complete combustion,
even detonation.
Moreover, the non-equilibrium behaviors in detonation phenomenon are investigated via
the velocity moments of discrete distribution functions. The system at the disc center is
always in its thermodynamic equilibrium. The internal kinetic energies in different degrees
of freedom around the detonation front do not coincide due to the fluid viscosity. They
show the maximum difference at the inflexion point where the pressure has the largest
21
spatial derivative. The influence of shock strength on the reaction rate and the influences of
both the shock strength and the reaction rate on the departure amplitude of the system from
its local thermodynamic equilibrium are probed. The departure from equilibrium in front
of von-Neumann peak results from shock effect, while the one behind the peak results from
rarefaction effect. The departure increases when the shock or rarefaction effect increases.
Specially, the value of ∆∗
2,rr is positive at shock wave and negative at the rarefaction wave,
which can be seen a criterion to distinguish the two waves. What’s more, the main behaviors
of actual distribution functions around the detonation wave are recovered from the numerical
results of high-order moments of the discrete distribution function.
Finally, further discussions include the following points. (i) For the combustion systems
where both the reactant and product have more than one components, a multi-distribution-
function model is more preferred. Such a work is in progress [81]. (ii) The transport
properties are relevant to the relaxation time τ. It should depend on physical quantities,
such as density and temperature. Consequently, it should be a function of the space and
time. This point should be further investigated in the future. (iii) In numerical simulations
the spatial and temporal steps should be small enough so that the spurious transportation
is negligible compared with the physical one.
Acknowledgements
The authors thank Prof. Cheng Wang for many helpful discussions. AX and GZ ac-
knowledge support of the Science Foundations of National Key Laboratory of Computational
Physics, National Natural Science Foundation of China and the opening project of State Key
Laboratory of Explosion Science and Technology (Beijing Institute of Technology) [under
Grant No. KFJJ14-1M]. YL and CL acknowledge support of National Natural Science Foun-
dation of China [under Grant No. 11074300], National Basic Research Program of China
(under Grant No. 2013CBA01504) and National Science and Technology Major Project of
the Ministry of Science and Technology of China (under Grant No.2011ZX05038-001).
[1] W. Fickett and W. C. Davis, Detonation, University of California Press, Berkeley, 1979.
22
[2] W. Y. Cheng, X. Y. Liu, K. J. Wang, et al, Journal of China Coal Society, 29 (2004) 57 (in
Chinese).
[3] X. J. Li, B. Q. Lin, Coal Geology & Exploration, 38 (2010) 7 (in Chinese).
[4] D. L. Chapman, Philos. Mag, 47 (1899) 90.
[5] E. J. Jouguet, Math. Pures Appl, 1 (1905) 347.
[6] Ya. B. Zeldovich, J. Exp. Th. Phys., 10 (1940) 542.
[7] J. Von Neumann, Theory of Detonation Waves, New York: Macmillan, 1942.
[8] W. Doering, Ann. Phys., 43 (1943) 421.
[9] D. Bjerketvedt, J. R. Bakke, K. Van Wingerden, J. Hazard. Mater, 52 (1997) 1.
[10] Y. Lian, K. Xu, J. Comput. Phys., 163 (2000) 349.
[11] C. Wang, X. Zhang, C. W. Shu, and J. Ning, J. Comput. Phys., 231 (2012) 653.
[12] J. Sun and J. Zhu, Theory of Detonation Physics, Beijing: National Defense Industry Press,
1995 (in Chinese).
[13] S. G. Cochran, J. Chan, UCID-18024, (1979).
[14] E. L. Lee and C. M. Tarver, Phys. Fluids, 23 (1980) 2362.
[15] J. Cao, Explosion and Shock Waves, 6 (1986) 137 (in Chinese).
[16] F. Zhao, C. Sun, Y. Wei and J. Chi, Explosion and Shock Waves, 9 (1989) 338 (in Chinese).
[17] S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University
Press, New York, (2001).
[18] S. Succi, I. V. Karlin, H. Chen, Rev. Mod. Phys. 74 (2002) 1203.
[19] H. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, V. Yakhot, Science 301 (2003) 633.
[20] A. Lamura, S. Succi, Phys. Rev. Lett. 95, 224502 (2005)
[21] M. Sbragaglia, R. Benzi, L. Biferale, S. Succi, and F. Toschi, Phys. Rev. Lett. 97 (2006)
204503.
[22] J. Horbach, S. Succi, Phys. Rev. Lett. 96 (2006) 224503.
[23] R. Benzi, S. Chibbaro, S. Succi, Phys. Rev. Lett. 102 (2009) 026002.
[24] G. Gonnella, E. Orlandini, and J. M. Yeomans, 78 (1997) 1695.
[25] C. Denniston and J. M. Yeomans, Phys. Rev. Lett. 87 (2001) 275505.
[26] G.Toth, C. Denniston, and J. M. Yeomans, Phys. Rev. Lett. 88 (2002) 105504.
[27] D. Marenduzzo, E. Orlandini, and J.M. Yeomans, 92 (2004) 188301.
[28] R. Verberg, C.M. Pooley, J.M. Yeomans, and A. C. Balazs, Phys. Rev. Lett. 93 (2004) 184501.
23
[29] C.M. Pooley and J.M. Yeomans, Phys. Rev. Lett. 93 (2004) 118001.
[30] D. Marenduzzo, E. Orlandini, and J. M. Yeomans, Phys. Rev. Lett. 98 (2007) 118102.
[31] A. Sengupta, U. Tkalec, M. Ravnik, J. M. Yeomans, C. Bahr, and S. Herminghaus, Phys.
Rev. Lett. 110 (2013) 048303.
[32] X. W. Shan, H. D. Chen, Phys. Rev. E, 47 (1993) 1815; Phys. Rev. E, 49 (1994) 2941
[33] S. Chen, G. D. Doolen, Annu. Rev. Fluid Mech., 30 (1998) 329.
[34] Q. Kang, D. Zhang, S. Chen, X. He, Phys. Rev. E, 65 (2002) 036318
[35] H. Fang, Z. Wang, Z. Lin, M. Liu, Phys. Rev. E, 65 (2002) 051925
[36] Z. Guo, C. Shu, Lattice Boltzmann Method and Its Applications in Engineering, World Sci-
entific Publishing Company, (2013).
[37] A. Xu, G. Zhang, Y. Li and H. Li, Prog. Phys. 34 (2014) 136 (in Chinese).
[38] S. Ponce Dawson, S. Chen, and G. D. Doolen. J. Chem. Phys., 98 (1993) 1514.
[39] J. R. Weimar, J. P. Boon. Physica A, 224 (1996) 207.
[40] R. Zhang, Y, Xu, B. Wen, et al, Scientific Reports, (2014) 4.
[41] S. Chen, D. Martinez, R. Mei, Phys. Fluids, 8 (1996) 2527.
[42] S. Succi, G. Bella, and F. Papetti, J. Sci. Comput., 12 (1997) 395
[43] O. Filippova, D H¨
anel, Int. J. Mod. Phys. C, 9 (1998) 1439.
[44] O. Filippova, D H¨
anel, J. Comput. Phys, 158 (2000) 139.
[45] O. Filippova, D H¨
anel, Comput. Phys. Commun., 129 (2000) 267
[46] H. Yu, L. S. Luo, S. S. Girimaji, Int. J. Comput. Eng. Sci., 3 (2002) 73.
[47] K. Yamamoto, X. He, G. D. Doolen. J. Stat. Phys., 107 (2002) 367.
[48] K. Yamamoto, Int. J. Mod. Phys. B, 2003, 17(01n02): 197-200.
[49] K. Yamamoto, N. Takada, M. Misawa, P. Combust. Inst., 30 (2005) 1509.
[50] T. Lee, C. Lin, and L. D. Chen, J. Comput. Phys., 215 (2006) 133.
[51] E. Chiavazzo, I. V. Karlin, A. N. Gorban, et al., Combust. Flame, 157 (2010) 1833.
[52] S. Chen, Z. Liu, C. Zhang, et al., Appl. Math. Comput., 193 (2007) 266.
[53] S. Chen, Z. Liu, Z. Tian, et al, Comput. Math. Appl., 55 (2008) 1424.
[54] S. Chen, M. Krafczyk, Int. J. Therm. Sci., 48 (2009) 1978.
[55] S. Chen, Int. J. Hydrogen Energ., 35 (2010) 1401.
[56] S. Chen, J. Li, H. Han, et al., Int. J. Hydrogen Energ., 35 (2010) 3891.
[57] S. Chen, H. Han, Z. Liu, et al., Int. J. Hydrogen Energ., 35 (2010) 4736.
24
[58] S. Chen, C. Zheng, Int. J. Hydrogen Energ., 36 (2011) 15403.
[59] S. Chen, J. Mi, H. Liu, et al., Int. J. of Hydrogen Energ., 37 (2012) 5234.
[60] F. J. Alexander, H. Chen, S. Chen, et al. Phys. Rev. A, 46 (1992) 1967.
[61] F. J. Alexander, S. Chen, J. D. Sterling. Phys. Rev. E, 47 (1993) R2249.
[62] Y. Chen, H. Ohashi, M. Akiyama. Phys. Rev. E, 50 (1994) 2776.
[63] G. R. McNamara, A. L. Garcia, B. J. Alder. J. stat. phys., 87 (1997) 1111.
[64] X. Pan, A. Xu, G. Zhang, and S. Jiang, Int. J. Mod. Phys. C 18 (2007) 1747.
[65] Y. Gan, A. Xu, G. Zhang, and Y. Li, Physica A 387 (2008) 1721; Phys. Rev. E 83 (2011)
056704.
[66] F. Chen, A. Xu, G.Zhang, Y. Li, S. Succi, EuroPhys. Lett. 90 (2010) 54003.
[67] A. Xu, G. Zhang, Y. Gan, F. Chen, and X. Yu, Front. Phys. 7 (2012) 582.
[68] B. Yan, A. Xu, G. Zhang, Y. Ying, and H. Li, Frontiers of Physics, 8 (2013) 94.
[69] I. Halliday, L. A. Hammond, C. M. Care, et al. Phys. Rev. E, 64 (2001) 011208.
[70] X. D. Niu, C. Shu, and Y. T. Chen, Int. J. Mod. Phys. C 14 (2003) 785.
[71] K. N. Premnath, J. Abraham. Phys. Rev. E, 71 (2005) 056706.
[72] T. Reis and T. N. Phillips. Phys. Rev. E, 75 (2007) 056703.
[73] Z. Guo, H. Han, B. Shi, et al. Phys. Rev. E, 79 (2009) 046708.
[74] M. Watari, Commun. Comput. Phys. 9 (2011) 1293.
[75] C. Lin, A. Xu, G. Zhang, Y. Li, S. Succi, Phys. Rev. E, 89 (2014) 013307.
[76] Y. Gan, A. Xu, G. Zhang, Y. Yang, EPL, 103 (2013) 24003.
[77] F. Chen, A. Xu, G. Zhang, Y. Wang, Front. Phys. 9 (2014) 246.
[78] M. Watari and M. Tsutahara, Phys. Rev. E 67 (2003) 036306.
[79] E. F. Toro. Riemann solvers and numerical methods for fluid dynamics: a practical introduc-
tion[M]. Springer, 2009.
[80] R. F. Warming, R. M. Beam. AIAA Journal”14 (1976) 1241.
[81] C. Lin, A. Xu, G. Zhang, Y. Li, e-print arXiv: 1405.5500
25
