1. An Investigation of Multi-Velocity Treatments for
Multi-Phase Flows Inspired by Gaskinetic Theory
Zakaria Ben Dhia1 and James McDonald2
1 Institute Polytechnique Privé, Université Libre de Tunis,
30 Avenue Kheireddine Pacha, 1002, Tunis, Tunisia
2 Department of Mechanical Engineering, University of Ottawa,
161 Louis Pasteur, Ottawa, Ontario, Canada, K1N 6N5
Email: Zakaria Ben Dhia, zbend027@uottawa.ca
ABSTRACT
This research entails the investigation of novel models
for an inert, dilute, disperse, particle flow to be cou-
pled to a gas. Such flows are important in many en-
gineering situations. Particle phases are often difficult
to model, as Lagrangian methods can be too costly and
many Eulerian methods suffer from model deficiencies
and mathematical artifacts. Often, Eulerian formula-
tions assume that all particles at a location and time
have the same velocity [12]. This assumption leads
to nonphysical results, including an inability to pre-
dict particle paths crossing and a limited number of
boundary conditions that can be applied [12, 13]. Re-
cently, multi-velocity treatments of multi-phase flows
have been proposed that are based on the field of gask-
inetic theory [3, 4, 5].
This work examines and compares multi-velocity for-
mulations for the prediction of gas-particle flows. De-
tails regarding derivation, the mathematical structure,
and physical behaviour of the resulting models are ex-
plored. Finally, a numerical implementation is pre-
sented and results for several flow problems that are
designed to demonstrate the fundamental behaviour of
the models are presented.
1 INTRODUCTION
The behaviours of multi-phase gas-particle flows are
important in many practical engineering situations. As
one example, in internal-combustion engines, fuel is
often injected as a spray of tiny droplets and, dur-
ing combustion, a cloud of tiny soot particles can be
formed. The accurate prediction of the evolution of the
particulate phases is essential in the analysis of these
situations. The design of clean and efficient combus-
tion technologies rests on an ability to make accurate
predictions and analyzes of these situations.
Mathematical models of particulate flows fall into two
categories: Lagrangian and Eulerian. In Lagrangian
treatments, the evolution of individual particles are
tracked. This is often prohibitively computationally
expensive, as, even at dilute concentrations, an im-
mensely large number of particles are often present. In
Eulerian formulations, the particle phase is treated as
a continuum and partial differential equations (PDEs)
are defined which govern the evolution of field vari-
ables with position and time as independent variables.
The most basic Eulerian formulation restricts all parti-
cle velocities at a given position and time to be equal
to that of the fluid. A slightly more flexible model can
be developed by allowing the particle velocity to devi-
ate from that of the gas, but restricting all particles at
a location and time to have the same velocity as each
other [12]. In many cases, this is an acceptable model,
especially in situations when drag forces between the
particles and the gas are dominant and particles tend to
all have velocities near that of the gas. In other situa-
tions, however, this is not true, and forcing all particles
to have the same velocity leads to artifacts in the model
and causes it to produce completely nonphysical solu-
tions [12, 13].
Multi-velocity formulations in which particles are
grouped into “families”, each of which are treated with
extensions of single-velocity treatments, have been
proposed. However, these treatments often group par-
ticles based on a predefined separation of velocity
space [11]. The result is that some of the deficien-
cies of a single-velocity formulation may be removed
for some situations, however all of the deficiencies
of a single-velocity formulation can still be observed

2. for any case that includes multiple particle velocities
within a region of velocity space that has been assigned
to a single family. Also, the resulting model equations
are not Galilean invariant as the segregation of velocity
space is done in a fixed reference frame.
In a quest for new multi-velocity treatments for
particle-gas flows that eliminate some or all of the ar-
tifacts of previous treatments, while maintaining desir-
able mathematical structures (such as Galilean invari-
ance), the field of gaskinetic theory [6] can be used as
a guide. The physical situation of a huge number of in-
dependent, practically indistinguishable solid particles
is very similar to the situation of a gas that is comprised
of an enormous number of indistinguishable atoms or
molecules. It is therefore reasonable to expect that so-
phisticated techniques from gaskinetic theory can be
adopted. Recently, models based on such an idea have
begun to be explored [3, 4, 5]. These models repre-
sent the distribution of particle velocities at a location
and time by a finite collection of delta or Gaussian
distributions. Moment equations are then formulated
which describe the evolution of statistical properties of
the particle-velocity distributions. The result is an ex-
panded system of first-order hyperbolic balance laws
that are amenable to solution using standard numerical
techniques. This is a special case of the technique of
moment closures which can be used to describe gen-
eral non-equilibrium gas flows [7, 9, 10]. Moment clo-
sures provide an extended set of hyperbolic partial dif-
ferential equations describing the transport of macro-
scopic fluid properties. In general, the solution of these
PDEs require considerably less effort than obtaining
solutions using a direct particle simulation method.
This paper explains the construction of multi-velocity
models for particle-laden gas flows based on kinetic
theory of gases. The physical behaviour and limita-
tions of the models are discussed and demonstrated.
Numerical solutions for several flow problems de-
signed to clearly demonstrate the limitations of tradi-
tional models are presented.
2 MOMENT METHODS
Moment closures arise from the field of gaskinetic
theory. This theoretical approach takes into account
the particle nature of gases by defining a probability
density function, F (xi,vi,t), in six-dimensional phase
space which specifies the probability of finding parti-
cles at a given location, xi, and time, t, having a par-
ticular velocity, vi. This treatment can obviously be
immediately applied to solid-particle flow by simply
replacing the gas particles with the particulate. Macro-
scopic “observable” properties of the particle are then
obtained by taking appropriate velocity moments of F .
This is done by integrating the product of the distri-
bution function and an appropriate velocity-dependent
weight, W(vi), over all velocity space,
W(vi)F =
∞
−∞
∞
−∞
∞
−∞
W(vi)F (xi,vi,t)d3
v. (1)
For example, σ = m F , σui = m viF , and Pij =
m cicjF . Here m is the mass of a particle, σ is the
particle density, ui is average (bulk) particle velocity,
ci = vi −ui is the particle random velocity, and Pij is an
anisotropic tensor that is related to the standard deriva-
tion of particles velocities at a location. The symbol P
is used for consistency with kinetic theory of gases,
where it is related to pressure. In this work, the fol-
lowing examples demonstrate the notation that will be
used to describe moments of arbitrary order:
Mx = m vxF , Mxxy = m vxvxvyF , (2)
where the number of times a coordinate appears in
the subscript of the symbol M denotes the number of
times that velocity component appears in the generat-
ing weight. Third- and fourth-order moments of the
random velocity are given the symbols
Qijk = m cicjckF and Rijkl = m cicjckclF . (3)
The Einstein summation convention is used for gen-
eral indices (i, j, k, etc.), but not for specific Cartesian
directions (x, y, z).
The evolution of the velocity distribution function is
described by the Boltzmann equation [2, 1, 6]. This is
a high-dimensional integro-differential equation for F
having the form
∂F
∂t
+vi
∂F
∂xi
+ai
∂F
∂vi
=
δF
δt
. (4)
Here ai is the acceleration due to external forces, drag
between the fluid and particulate is an example of such
a force. The term on the right hand side of the equa-
tion, δF
δt , is the collision integral and represents the
time rate of change of the distribution function pro-
duced by inter-particle collisions.
Transport equations governing the time evolution of
macroscopic quantities can be derived by evaluating
velocity moments of the Boltzmann equation given
above. This leads to Maxwell’s equation of change
which describes the evolution of the moment WF
by
∂
∂t
WF +
∂
∂xi
viWF = ∆ WF . (5)
Here the acceleration field is taken to be zero, as will
be the case throughout the present work. ∆ WF =

3. W δF
δt is the effect of collisions on the moment
quantity, and is also neglected herein. Neglecting
these two terms corresponds to the case of the flow
of non-interacting particles in a vacuum. W is the cho-
sen velocity-dependent weight that corresponds to the
macroscopic quantity of interest.
It is at this point that the problem of closure becomes
apparent. The time evolution of a given moment,
WF , is clearly dependent on the spatial divergence
of viWF , a moment of one higher order in terms
of the velocity, vi. This pattern is repeated, with the
time evolution of every moment being dependent on
a moment of one higher order in vi. In general, an
infinite number of moment equations is required to
fully describe the evolution of a macroscopic quantity,
and solving this infinite system is equivalent to solving
Eq. (4).
One technique used to obtain moment closure is to re-
strict the distribution function to an assumed form [7].
Restricting the form of the distribution function has the
effect of restricting the value of all higher-order mo-
ments to be functions of lower-order known moments,
thus furnishing a closing relationship in the moment
equations.
3 GOVERNING EQUATIONS
The classical, single-velocity model for particle flows
can be derived in a moment closure framework by re-
stricting the distribution function describing particle
velocities to be of the form
F1 = ω(xi,t) δ(vi − ˆvi(xi,t)), (6)
where ω(xi,t) and ˆvi(xi,t) are closure coefficients that
must be chosen to ensure consistency with Eq. (1), and
δ is the Dirac delta function. By substituting Eq. (6)
into Eq. (5), and choosing as velocity weights W1 = m,
and W2 = mvi, the following conservation model for
particle flow can be found:
∂U
∂t
+
∂Fk
∂xk
= 0 (7)
with
U =
σ
σui
and Fk =
σuk
σuiuk
. (8)
This system comprises one scalar and one vector equa-
tion for the conservation of mass and momentum of
particles. This model is very standard and has been ex-
tensively utilized [12, 13]. However, restricting all par-
ticles at one location to have the same velocity obvi-
ously renders this model inappropriate when the situa-
tion comprises multiple particle velocities at any point
in the flow. The practical deficiencies that this can
cause are demonstrated in Section 5.
3.1 Two-Velocity Models
Once particle flows are seen through the lens of gask-
inetic theory, an obvious extension to the model de-
scribed in Eq. (8) is to allow for multiple particle fam-
ilies with arbitrary velocities at a location. For exam-
ple, a two-velocity description is given by restricting
the distribution function to be
F2 = ω(1)
(xi,t) δ(vi − ˆv
(1)
i (xi,t))
+ω(2)
(xi,t) δ(vi − ˆv
(2)
i (xi,t)), (9)
where the superscript (1) and (2) refer to particle fam-
ilies one and two respectively. Once this distribution
function is chosen, all that remains in to chose an
appropriate set of velocity weights, W, and moment
equations can be constructed.
Such a two-velocity distribution function was consid-
ered previously by Desjardins et al. [3]. In this study,
they chose as generating weights (in two dimensions),
m, mvx, mvy, mv2
x, mv2
y, and a strange third-order
weight given by mv3
x + mv3
y. The result is a system of
six conservative moment equations of the form given
in Eq. (7) with solution vector and flux diad given as
U =








σ
Mx
My
Mxx
Myy
Mxxx +Myyy








and Fk =








Mk
Mxk
Myk
Mxxk
Myyk
Mxxxk +Myyyk








.
One peculiarity of this model is that it is not invari-
ant under rotation. This is because the model pro-
vides a treatment for two entries of the second-order
moment, Mxx and Myy, but not the “cross” term, Mxy.
These three second-order moments make up a tensor
and knowledge of all are required for rotation. Also,
the last entry in the solution vector, Mxxx + Myyy, (the
sum of two entries in a third-order tensor) is a some-
what strange choice. This is similar to choosing to
have the sum of the components of a vector as a so-
lution variable. This is obviously not a value that is
invariant under rotation.
Another issue with this chosen set of generating
weights is that, for some moment values, the locations
of the two delta functions in velocity space is ambigu-
ous (i.e. the closure coefficients in Eq. (9) cannot be
uniquely determined from the given state). For exam-
ple, for a state with a non-dimensionalized density of

4. σ = 1, bulk velocity of ui = 0, non-dimensionalized
Mxx = Myy = 1, and third-moment Mxxx +Myyy = 0, at
least two distribution functions can be found. For both
cases, ω(1) = ω(2) = 1
2 , however, the velocities of the
two particle families can be located either at
ˆv(1)
= ˆi+ ˆj and ˆv(2)
= −ˆi− ˆj,
or
ˆv(1)
= ˆi− ˆj and ˆv(2)
= −ˆi+ ˆj,
where ˆi and ˆj are the unit vectors in the x and y di-
rections respectively. In these ambiguous situations,
the closing fluxes cannot be uniquely determined as a
function of the solution vector and multiple consistent
values of Fk are possible in Eq. (7).
In the present study, the distribution function defined
in Eq. (9) is also adopted. However, the generating
weights are chosen such that the resulting system in
both Galilean invariant, and has a closure that is never
ambiguous. The generating weights chosen here are m,
mvi, mvivj, and mvivjvj. This results in the following
solution and flux vectors (for two dimensions):
U =












σ
Mx
My
Mxx
Mxy
Myy
Mxii
Myii












and Fk =












Mk
Mxk
Myk
Mxxk
Mxyk
Myyk
Mxkii
Mykii












. (10)
This is a system of eight first-order PDEs that includes
conservation of mass, momentum, all components of
the second-order tensorial moment, and the contracted
third-order moment. It may seem odd to use a system
of eight equations when there are only six free param-
eters, or closure coefficients, in the presumed distri-
bution function. However, it is found that the extra
information is necessary to maintain rotational invari-
ance and a flux definition that is unambiguous in all
possible cases.
The moments presented in Eq. (10) can be expressed
as functions of the “random” moments, defined in Sec-
tion 2, through the relations [6]:
Mx = σux ,
My = σuy ,
Mxx = Pxx +σu2
x ,
Mxy = Pxy +σuxuy ,
Myy = Pyy +σu2
y ,
Mxxx = Qxxx +3uxPxx +σu3
x ,
Mxxy = Qxxy +uyPxx +2uxPxy +σu2
xuy ,
Mxyy = Qxyy +uxPyy +2uyPxy +σuxu2
y ,
Myyy = Qyyy +3uyPyy +σu3
y ,
Mxxii = Rxxii +2uyQxxy +u2
yPxx
+ux(2Qxxx +2Qxii)+4uxuyPxy
+u2
x(6Pxx +Pyy)
+σu2
x(u2
x +u2
y),
Mxyii = Rxyii +uy(Qxii +2Qxyy)
+ux(Qyii +2Qxxy)
+3Pxy(u2
x +u2
y)+3uxuyPii
+σuxuy(u2
x +u2
y),
Myyii = Ryyii +2uxQxyy +u2
xPyy
+uy(2Qyyy +2Qyii)+4uxuyPxy
+u2
y(6Pyy +Pxx)
+σu2
y(u2
x +u2
y).
The random moments, σ, ux, uy, Pxx, Pxy, Pyy, Qxii, and
Qyii can all be determined from the solution vector. To
close the system, the moments Qxxx, Qxxy, Qxyy, Qyyy,
Rxxii, Rxyii, and Ryyii must be related to those in the
solution vector.
Once the distribution function given in Eq. (9) is
adopted and the moment coefficients are determined
from Eq. (1), the expressions for the needed random
moments can be integrated and are found to be
Qxxx = Qxii
Pxx
Pii
,
Qxxy = Qyii
Pxx
Pii
,
Qxyy = Qxii
Pyy
Pii
,
Qyyy = Qyii
Pyy
Pii
,
Rxxii =
Q2
xii
Pii
+
PxxPii
σ
,
Rxyii =
QxiiQyii
Pii
+
PxyPii
σ
,
Ryyii =
Q2
yii
Pii
+
PyyPii
σ
.
The system of equations defining the model is thus
closed.
In any implementation, special care must be taken to
avoid dividing by zero in these expressions. This can
simply be accomplished by adding a small tolerance

5. to all the denominators. One does not need to worry
about the sign, as the moments that appear in the de-
nominators are all non-negative.
3.2 Multi-Velocity Models
It should be noted that extensions to models with more
than two distinct velocities have been developed and
considered [4, 5]. In these cases, however, it is not
possible to write the fluxes as closed-form functions
of the solution vector. This is because it is not pos-
sible to analytically determine the weights and loca-
tions of the delta functions in the assumed distribu-
tion. Rather, a numerical inversion algorithm must be
used to determine the moment relation each time a flux
evaluation is needed. Though these models hold the
promise of improved accuracy in situations when par-
ticles at a location can have one of many velocities, the
added computational expense is significant. It is hoped
that the two-velocity model developed above brings a
good balance between physical accuracy and numeri-
cal cost.
4 NUMERICAL METHODS
In order to evaluate the predictive capabilities of the
developed model, a two-dimensional flow solver is
constructed. An upwind Godunov-type finite-volume
scheme with piece-wise limited-linear reconstruction
using the Venkatakrishnan slope limiter [14] is used.
Inter-cellular fluxes are determined through the ap-
proximate solution of a Riemann problem using a
flux function developed by Saurel et al. [12] for the
single-velocity formulation, or the HLL flux func-
tion [8] for the two-velocity model. Second-order
accurate predictor-corrector time marching is used.
Wave speeds of the system are approximated through
the calculation of the locations of the two delta func-
tions given in Eq. (9).
As stated above, the solution vector contains eight en-
tries while the distribution function has only six de-
grees of freedom. This means that, after a time step, it
is possible to have a solution vector that does not corre-
spond to any distribution of the form in Eq. (9). Thus,
after each time step, the solution vector is adjusted to
ensure consistency with the distribution function. To
be consistent with a two-velocity distribution, it can be
shown that PxxPyy = P2
xy. Thus, after each update, the
sign of Pxy is maintained, but its magnitude is changed
to be equal to PxxPyy. The angle that the line that con-
nects the two deltas in the distribution function makes
with the x axis is given by the relation
cos2
θ =
Pxx
Pii
,
and it can be shown that the third-order vector, Qij j
must point along this line. Therefore, after an update,
the magnitude of Qij j is maintained, but its direction
is corrected.
5 RESULTS
Two situations are chosen that demonstrate the advan-
tages of a two-velocity description. First, two cross-
ing beams of non-interacting particles is considered.
Single-velocity descriptions are incapable of describ-
ing such crossing of particle trajectories. Second, a
situation of one group of fast-moving particles which
overtakes and passes through a group of slower par-
ticles is studied. Solutions to both situations are ob-
tained for the single-velocity model, Eq. (8), and the
two-velocity model, Eq. (10).
In both of these cases, much of the domain contains
no particles. Numerical and round-off errors can result
in some areas developing slightly non-physical states
(σ slightly less than zero, for example). It is therefore
necessary to ensure the state in each cell is realizable
after each time step. This is simply done by setting
any negative values of σ to zero and to set Pxx = Pxy =
Pyy = 0 if the determinant of this tensor is found to be
negative (which it cannot physically be).
In addition to issues related to non-realizability, it is
also found that, when all moments in the solution vec-
tor are on the order of the truncation error and are es-
sentially random, the state can correspond to a distri-
bution function where the two velocities are also ran-
dom and can be arbitrarily high. Such high veloci-
ties are a serious concern, not only because they can
seriously restrict the time-step size, but because they
can actually lead to large errors in the computation of
the fourth-order moments contained in the flux vec-
tor. This is found to lead to large random oscillations
in the solution. In order to handle this issue, maxi-
mum bounds on the size of particle-family velocities
are employed.
For all cases considered, a mesh of 300 by 300 cells is
used. This yields meshes of 90,000 equally sized cells.
Non-dimensional variables are used in all cases.
5.1 Crossing Beams
The first case considered is that of two crossing jets
of particles. The domain is −0.35 < x < −0.35 and

6. −0.35 < y < −0.35. The boundary conditions are
transmissive everywhere except on the left boundary,
where two jets of particles enter. If −0.3 < y < −0.2
on this wall, a jet of particles, all with σ = 1, ux = 1,
and uy = 1, enters. If 0.2 < y < 0.3 another jet of parti-
cles with σ = 1, ux = 1, and uy = −1 enters. Solutions
were advanced in time until steady-state was reliably
achieved.
As the particles are assumed not to interact with each
other, the exact solution is that the two beams should
continue in straight lines and exit the domain. Com-
puted solution values for particle density σ and the
y component of momentum, My, are shown in Fig-
ure 1. As can be seen in Figures 1a and 1c, when a
single-velocity model is used, the beams cannot cross
and the particles simply “pile up” along the symme-
try line. The two-velocity model solutions, shown in
Figures 1b and 1d, correctly allows the beams to cross.
It is interesting to see that a region of zero y-direction
momentum correctly develops as the beams intersect.
Non-zero momenta re-emerge from this region as the
particle-flow continues.
5.2 Superimposed Families of Particles
The second case considered is that of one family of
non-interacting particles overtaking another. In this
case, at time zero, one family of particles with σ = 1
and ux = 1 spans the region −0.6 < x < −0.4 and
−0.1 < y < 0.1. A second family with σ = 1 and ux =
0.1 spans the region −0.25 < x < 0.15 and −0.2 < y <
0.2. Results at t = 0, t = 0.5, and t = 1 are shown in
Figure 2.
Again, the exact solution should show the faster fam-
ily of particles simply passing through the slower one,
completely unaffected. It can again be seen that the
single-velocity model is wholly unable to predict this
effect as particles again collect in a concentrated re-
gion. The results for the two-velocity formulation are
much better, though not perfect. It can be seen that
the faster particles do pass through the slower group,
however, there is a noticeable interaction. It is thought
at this time that this is largely due to numerical errors
incurred when the two families initially come in con-
tact with each other. Further study of this process is
needed.
6 CONCLUSIONS
A Galilean-invariant, unambiguous, two-velocity
model for particle transport has been shown. This
model has the promise of being more physically ac-
curate in situations when particles at a given location
and time do not all have the same (or nearly the same)
velocity. It is hoped that this model provides a good
balance of improved physical accuracy over the stan-
dard, single-velocity treatment and the more expensive
multi-velocity treatments where the distribution of par-
ticle velocities must be computed using a costly algo-
rithm for every flux evaluation.
It was shown that the two-velocity model does in-
deed produce superior results for canonical particle-
flow problems. The increase in computational cost is
modest when moving from a one-velocity model, but
the possible increase in physical accuracy is large.
Future work will involve the coupling of the particle
model to a fluid model for a background gas flow. The
field of gaskinetic theory provides a framework for this
coupling through the acceleration term in Eq. (4). This
extension should therefore be fairly straight forward to
develop and implement. Collision operators to model
the interaction of particles with each other is also the
subject of possible study.
ACKNOWLEDGEMENTS
This first author’s time in Ottawa to conduct this re-
search was supported by Mr. Achraf Ben Dhia. Both
authors are very grateful for this support.
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7. 0.00
2.00
1.00
(a) Single-velocity model, particle density
0.00
2.00
1.00
(b) Two-velocity model, particle density
-1
1
-0.8
-0.4
0
0.4
0.8
My
(c) Single-velocity model, y momentum
-1
1
-0.8
-0.4
0
0.4
0.8
My
(d) Two-velocity model, y momentum
Figure 1: Solutions for case of two crossing beams of non-interacting particles, computed using one- and two-
velocity formulations.
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8. 0.00
2.00
1.00
sigma
(a) Single-velocity model, t = 0.0
0.00
2.00
1.00
sigma
(b) Two-velocity model, t = 0.0
0.00
2.00
1.00
sigma
(c) Single-velocity model, t = 0.5
0.00
2.00
1.00
sigma
(d) Two-velocity model, t = 0.5
0.00
2.00
1.00
sigma
(e) Single-velocity model, t = 1.0
0.00
2.00
1.00
sigma
(f) Two-velocity model, t = 1.0
Figure 2: Solutions for case of one family of non-interacting particles overtaking another, computed using one-
and two-velocity formulations.