Flow transitions on a cambered airfoil at moderate Reynolds number

Abstract

A combined experimental and numerical study is performed to investigate the flow field and associated aerodynamic forces on a cambered airfoil. The Reynolds number is low enough to ensure importance of viscous dynamics, and high enough so that instability and transition to turbulence can occur. The flow fields are complex and their correct description is essential in understanding the nonlinear curves describing the variation of lift and drag coefficients with angle of attack, α. As α is increased from 0, the flow states go through a number of qualitatively distinct phases. At low to moderate α, the laminar boundary layer separates before the trailing edge, and as the separation point moves forward, instabilities of the detached shear layer form coherent vortices over the upper (suction) surface. At a critical angle, αcrit, instabilities in the shear layer grow fast enough to transition to turbulence, which then leads to reattachment before the trailing edge. In this flow state, lift is increased and drag decreases. Hence, in order to understand the aerodynamics at this scale, we need to understand the viscous dynamics of the boundary layer, as elegantly described and analyzed by Frank White.

Full text

Phys. Fluids 33, 033326 (2021); https://doi.org/10.1063/5.0039787 33, 033326
© 2021 Author(s).
Lagrangian models of particle-laden
flows with stochastic forcing: Monte
Carlo, moment equations, and method of
distributions analyses
Cite as: Phys. Fluids 33, 033326 (2021); https://doi.org/10.1063/5.0039787
Submitted: 06 December 2020 . Accepted: 22 February 2021 . Published Online: 26 March 2021
Daniel Domínguez-Vázquez, Gustaaf B. Jacobs, and Daniel M. Tartakovsky
COLLECTIONS
Paper published as part of the special topic on In Memory of Edward E. (Ted) O’Brien
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Lagrangian models of particle-laden flows with
stochastic forcing: Monte Carlo, moment
equations, and method of distributions analyses
Cite as: Phys. Fluids 33, 033326 (2021); doi: 10.1063/5.0039787
Submitted: 6 December 2020 .Accepted: 22 February 2021 .
Published Online: 26 March 2021
Daniel Dom
ınguez-V
azquez,
1,a)
Gustaaf B. Jacobs,
1,b)
and Daniel M. Tartakovsky
2,c)
AFFILIATIONS
1
Department of Aerospace Engineering, San Diego State University, San Diego, California 92182, USA
2
Department of Energy Resources Engineering, Stanford University, Stanford, California 94305, USA
Note: This paper is part of the special topic, In Memory of Edward E. (Ted) O’Brien.
a)
Electronic mail: ddominguezvazquez@sdsu.edu
b)
Author to whom correspondence should be addressed: gjacobs@sdsu.edu
c)
Electronic mail: tartakovsky@stanford.edu
ABSTRACT
Deterministic Eulerian–Lagrangian models represent the interaction between particles and carrier flow through the drag force. Its analytical
descriptions are only feasible in special physical situations, such as the Stokes drag for low Reynolds number. For high particle Reynolds and
Mach numbers, where the Stokes solution is not valid, the drag must be corrected by empirical, computational, or hybrid (data-driven)
methods. This procedure introduces uncertainty in the resulting model predictions, which can be quantified by treating the drag as a random
variable and by using data to verify the validity of the correction. For a given probability density function of the drag coefficient, we carry out
systematic uncertainty quantification for an isothermal one-way coupled Eulerian–Lagrangian system with stochastic forcing. The first three
moment equations are analyzed with a priori closure using Monte Carlo computations, showing that the stochastic solution is highly non-
Gaussian. For a more complete description, the method of distributions is used to derive a deterministic partial differential equation for the
evolution of the joint PDF of the particle phase and drag coefficient. This equation is solved via Chebyshev spectral collocation method, and
the resulting numerical solution is compared with Monte Carlo computations. Our analysis highlights the importance of a proper approxi-
mation of the Dirac delta function, which represents deterministic (known with certainty) initial conditions. The robustness and accuracy of
our PDF equation were tested on one-dimensional problems in which the Eulerian phase represents either a uniform flow or a stagnation
flow.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0039787
I. INTRODUCTION
Particle- and droplet-laden flows occur in many anthropogenic
and natural environments. For example, the mixing of liquid fuel spray
and/or solid fuel particles with turbulent gas flows determines the effi-
ciency of many propulsion and energy systems. Environmental pollu-
tion is affected by the dispersion of aerosol particles in environmental
carrier air flow.
The Eulerian–Lagrangian (EL) method provides a natural frame-
work for the modeling of particle- and droplet-laden flows. It uses
Eulerian continuum models to describe the dynamics of the ambient
flow and tracks individual particles along their Lagrangian paths.
1
A “first-principle” EL approach accounts for the surface of a
finite-size particle and computes the flow over each particle. The
excessive computational cost involved in these high-resolution (aka
high-fidelity) simulations limits the number of particles that can be
dealt with on modern computational infrastructure to hundreds.
2–4
Reduced-complexity models, such as the force coupling method
5
and
filtered particle method,
6
can reduce the computational cost and
increase the computationally feasible number of particles. The accu-
racy of these methods deteriorates near walls, which limits their range
of applicability.
Process-scale problems involve a number of particles on the
order of millions to billions. To handle such numbers, it is common to
treat each particle as a point of zero volume, e.g., according to the
Particle-Source-In-Cell (PSIC) method.
1
This method models par-
ticles’ interaction with the carrier flow via singular point source terms
Phys. Fluids 33, 033326 (2021); doi: 10.1063/5.0039787 33, 033326-1
Published under license by AIP Publishing
Physics of Fluids ARTICLE scitation.org/journal/phf

that account for the momentum and energy exchange. The momen-
tum and energy exchange between the point particles and the flow is
determined by the (drag) force and heat transfer between the particle
and the carrier phase. Analytical expressions for the force and heat
transfer coefficient are available only for a few physical conditions,
with limited application range.
7,8
Thus, the widely used Stokes drag
law
9
is valid for steady incompressible flows at low Reynolds numbers
and for spherical particles. The Maxey-Riley (MR) relation
10
accounts
for unsteady effects, but it applies only to low Reynolds number and
spherical particles. More general flow conditions with arbitrary
Reynolds and Mach numbers require empirical corrections to the
Stokes drag,
11,12
number density, slip coefficients,
13
or viscosity ratios
for droplets.
14
Treatment of the energy exchange between a point par-
ticle and the ambient flow is analogous. For low Reynolds number
flows over a spherical particle, the heat transfer coefficient is computed
analytically; to account for high Reynolds, Mach, and/or Nusselt num-
bers, one has to resort to a correction factor.
7,15
The empirical corrections depend on plethora of parameters,
such as particle shape and the Reynolds and Mach numbers, that have
nonlinear effects on the flow around a particle. This naturally trans-
lates into a prediction error of a point-particle model as the momen-
tum and energy exchange are only known within certain bounds. The
twin goals of alleviating this modeling inaccuracy and expanding its
applicability range have inspired multiscale and data-driven modifica-
tions to the PSIC method. Thus, the multiscale method
2,3,16–19
connects accurate high-resolution simulations with the reduced point-
particle method through surrogate models. The latter approximates
the interphase force and heat flux with a surrogate model in a wide
parameter space using high-resolution simulations in a data-driven
manner. In regions of the parameter space with a large uncertainty,
additional high-resolution simulations are conducted to improve the
accuracy and/or validity range of the surrogate model. As more high-
resolution simulations become available, the multiscale method is
updated via a Bayesian procedure. This procedure yields the stochastic
forcing with a computable PDF.
The presence of such a stochastic forcing in both the empirical and
data-driven approaches renders solutions to the corresponding PSIC
model random as well. These solutions are given in terms of a joint
probability density function (PDF) of system states or their ensemble
moments. Monte Carlo (MC) simulations are often used to obtain such
solutions. They are easy to implement, “embarrassingly” parallel, and
free of distributional assumptions; their only approximation stems from
the practical need to rely on a finite number of MC realizations, N
s
,to
compute the sample statistics. A drawback of the MC method is its slow
convergence: its sampling error decays as 1=ffiffiffiffiffi
Ns
p.ThiscanmakeMC
simulations prohibitively expensive if each realization takes a long time
to compute. Various modifications of the standard MC, e.g., multilevel
MC, pseudo-MC, or combinations thereof, can significantly accelerate
this convergence rate, but their performance is not guaranteed especially
when the goal is to compute the full PDF rather than its moments.
20,21
Other sampling-based methods, such as stochastic collocation, require
nontrivial modifications
22
in order to handle hyperbolic problems with
discontinuities. When the stochastic dimension and/or the noise
strength become large, such methods might become slower than the
standard MC even for problems with smooth solutions.
23
Direct numerical alternatives to sampling techniques include
methods based on (generalized) polynomial chaos expansions. These
methods represent uncertain inputs and state variables by orthogonal
polynomials of standard random variables and often exhibit spectral
accuracy. Of direct relevance to high-speed compressible flows
described by the Euler equations with shocks is the multi-element gen-
eralized polynomial chaos method,
24–26
which accommodates the
presence of discontinuities in the stochastic space. Its computational
cost might become comparable to sampling methods,
27
specifically
when the stochastic dimension is large. Like their sampling-based
counterparts, the direct simulation methods do not provide physical
insight into the probabilistic behavior of a system, e.g., the spatiotem-
poral nonlocality of the statistical moments
28
and PDFs
29
of the sys-
tem states.
The method of moments (MoM) alleviates some of these disad-
vantages by deriving deterministic equations for the statistical
moments of a system state. Since the MoM is free of polynomial
expansions, it does not suffer from the “curse of dimensionality,” but
it often requires closure approximations to be computable. It has been
used to derive moment equations for high-speed flows interacting
with a particle phase;
17
the closure terms were learned from the MC
simulations. Practical considerations limit the MoM to the derivation
andsolutionofequationsforthefirsttwomoments—meanand
(co)variance—of a system state, which limits its usefulness for highly
non-Gaussian phenomena. Specifically, the MoM cannot capture rare
events occuring in such problems, which are characterized by fat-
tailed PDFs.
The method of distributions (MoD) overcomes this limitation by
deriving a deterministic equation for either the joint PDF or the joint
cumulative density function (CDF) of the system states. Having its ori-
gins in the statistical theory of turbulence,
30
the MoD has since been
used as an efficient uncertainty quantification technique.
31
It retains
all the advantages of the MoM, but it might require closure approxi-
mations. The MoD yields a closed-form PDF/CDF equation for non-
linear flows in the absence of a shock, e.g., those described by the
inviscid Burgers equation
32
and the shallow-water equations.
33
Within
the MoD framework, shocks and discontinuities can be treated either
analytically, as was done for the Buckley–Leverett equation
34
and
water hammer equations,
35
or by adding the PDF/CDF equation a
kinetic defect-like source term that generally has to be learned from
Monte Carlo runs.
36
Numerical solutions of PDF/CDF equations
admit high-order spectral accuracy
32
and can be up to two orders of
magnitude faster than the standard MC.
37
We deploy the MoD to describe isothermal particle-laden flows
driven by stochastic forcings. The underlying flow model relies on the
Lagrangian point-particle formulation with one-way coupling between
fluid flow and particle transport. The drag on a particle is modeled as
a random variable with a prescribed PDF. The MoD yields a closed-
form partial differential equation for the joint PDF of a particle’s posi-
tion and velocity. We consider two canonical flow scenarios, both in
one spatial dimension: a uniform carrier flow and a stagnation carrier
flow. These are important in their own right and can be used as build-
ing blocks of more general and multi-dimensional flows; for example,
the stagnation flow is a central component to the dynamic description
of attractors and repellers in dynamic systems.
38
Our PDF solutions
are validated against high-fidelity MC simulations and compared with
solutions of the moment equations.
17
ThehyperbolicPDFequationis
solved via the Chebyshev collocation method.
32
Discontinuities in its
solution are captured using the filtering techniques.
39,40
A key result of
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 033326 (2021); doi: 10.1063/5.0039787 33, 033326-2
Published under license by AIP Publishing

our analysis is the derivation of analytical expressions for the position
and velocity of a particle moving in deterministic uniform and stagna-
tion flows. These expressions allow us to generate sufficiently large
numbers of MC realizations. In both flow regimes, the PDF solutions
are non-Gaussian and their moments can increase or decrease
depending largely on the time-dependent increase or decrease in the
inter-phase velocity. Moreover, the stochastic solution can develop dis-
continuities at inflection points of the inter-phase velocity.
In Sec. II, we formulate an Eulerian–Lagrangian stochastic model
of particle-laden flows. The formulation is presented in one-
dimension but applies to multiple spatial dimensions as well. Its statis-
tical treatment is presented in Sec. III via the MoD (Sec. III A)andthe
MoM (Sec. III B). Our numerical strategy for solving the resulting
PDF equation is described in Sec. IV. We apply our methodology to
the uniform and stagnation flows, for which the analytical solutions
are derived in Sec. V.InSec.VI, we use these two canonical settings to
analyze the accuracy and robustness of the MoD and MoM solutions.
Section VII contains concluding remarks and future work.
II. LAGRANGIAN PROBLEM FORMULATION
Dynamics of an isothermal collisionless particle phase in a one-
way coupled unidimensional Eulerian–Lagrangian system with the
point-particle approximation is governed by
1,41
dxp
dt¼up;(1a)
dup
dt¼/uup
sp
:(1b)
Here, tis the non-dimensional time, x
p
is the non-dimensional particle
position, and u
p
is the non-dimensional particle velocity. The non-
dimensional particle response time s
p
is a measure of the response of
the particle to a change in the carrier velocity u. It is expressed as
sp¼d2
pRe=ð18eÞ,wheredp¼d
p=Lis the non-dimensional particle
diameter, Re ¼U1L= is the Reynolds number, Lis a characteristic
length, U1is the reference velocity, and e¼q=qpistherelativeden-
sity ratio of the two phases.
The function /is used to correct the Stokes drag force for flow
conditions outside of Stokes assumptions. Such models for the cor-
rected drag coefficient /are empirical and, therefore, can only be
determined within an uncertainty bound.
2,3,42
For the sake of general-
ity, we postulate
32,42
that /depends on the relative velocity uup
/¼agðuupÞ;(2)
without specifying the functional dependence of the function gðÞ.
This function can be expanded in terms of several random modes.
Here, we consider the first of those random modes and take /¼a.
The random coefficient awith a given PDF faðAÞaccounts for the
uncertainty in /stemming from a broad range of sources, such as
uncertainty in the particle size, shape, or inexactness/empicism of
the drag force, and renders the system of ordinary differential equa-
tions (1) stochastic. Its solution is given in terms of the joint PDF
faxpupðA;Xp;Up;tÞ.
Our model formulation ignores inter-particle collisions. This is
justified if the particle phase is dilute, specifically in one spatial
dimension.
43
III. SOLUTION STRATEGIES
A. Method of distributions
When applied to Eq. (1), the MoD yields an exact PDF equation
(see Appendix A)
@faxpup
@tþ@
@Xp
Upfaxpup

þ@
@Up
AgðUUpÞ
spðUUpÞfaxpup
"#
¼0;
(3)
with A,X
p
,andU
p
denoting the deterministic versions of the stochas-
tic variables a,x
p
,andu
p
.Equation(3) describes the evolution of the
joint PDF of the particle phase and drag coefficient,
faxpupðA;Xp;Up;tÞ,inthephasespaceXspanned by coordinates
ðXp;Up;AÞ. This space can be either infinite or bounded,
X¼½X0
p;X1
p½U0
p;U1
p½A0;A1. In the latter case, (3) is subject to
boundary conditions for the independent variables X
p
and U
p
faxpupðA;Xb
p;Up;tÞ¼fXp
axpupðA;Up;tÞ;(4)
faxpupðA;Xp;Ub
p;tÞ¼fUp
axpupðA;Xp;tÞ:(5)
The boundary functions on the right hand side of these expressions
are specified according to the corresponding marginal distributions;
and using the characteristic velocities of (3) defined as in Eqs. (14) and
(15),Xb
p¼X0
por Xb
p¼X1
pfor CXðX0
pÞ>0andCXðX1
pÞ<0, respec-
tively. Similarly, Ub
p¼U0
por Ub
p¼U1
pfor CUðU0
pÞ>0and
CUðU1
pÞ<0, respectively. The PDF equation (3) is also subject to the
initial condition
faxpupðA;Xp;Up;0Þ¼f0
axpupðA;Xp;UpÞ:(6)
The function form of f0
axpupðA;Xp;UpÞis determined by the degree of
certainty in the initial state of the system, ðxp0;up0Þ. If it is known with
certainty, i.e., deterministic, then
faxpupðA;Xp;Up;0Þ¼faðAÞdðXpxp0ÞdðUpup0Þ;(7)
where dðÞ is the Dirac delta function. We will refer to this as a deter-
ministic initial condition (dIC). If the initial condition is not known
with certainty, then we refer to it as stochastic (sIC).
Once faxpupðA;Xp;Up;tÞis computed from (3)–(6),thePDFs
fxpupðXp;Up;tÞ;fxpðXp;tÞ,andfupðUp;tÞare computed as its margin-
als via numerical integration over the respective variables (see
Appendix A).
B. Method of moments
Solutions of the moment equations have been used to elucidate
many salient features of stochastically forced particle-laden flows.
17
We summarize that analysis and extend it to derive third-moment
equations in order to understand the degree of non-Gaussianity. The
derivation starts by using the Reynolds decomposition to represent all
parameters and state variables as the sums of their respective ensemble
means (denoted by the overbar) and zero-mean fluctuations (denoted
by the prime), e.g., xp¼
xpþx0
pwith x0
p¼0. Substituting these
decompositions into (1) and taking the ensemble average, we obtain
equations for the means
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 033326 (2021); doi: 10.1063/5.0039787 33, 033326-3
Published under license by AIP Publishing

d
xp
dt¼
up;(8)
sp
d
up
dt¼
/ð
u
upÞþ/0u0/0u0
p;(9)
for the variances, r2
xp¼x02
pand r2
up¼u02
p,
dr2
xp
dt¼2x0
pu0
p;(10)
sp
2
dr2
up
dt¼
/ðu0u0
pr2
upÞþ/0u0
pð
u
upÞþ/0u0u0
p/0u02
p;(11)
and for the third central moments, sxp¼x03
pand sup¼u03
p,
dsxp
dt¼3x02
pu0p;(12)
sp
3
dsup
dt¼
/ðu0u02
psupÞþ/0u02
pð
u
upÞ
r2
upð/0u0/0u0
pÞþ/0u0u02
p/0u03
p:(13)
As opposed to the exact PDF equation (3), these moment equa-
tions are not closed since they contain unknown mixed, higher-order
moments. To render them computable, one has to introduce closure
approximations such as the aprioriclosure
41,42
used to analyze the
first two statistical moments or a posteriori closure as used in Eulerian
formulations.
43,44
The moment equations (9)–(13) provide insight into the devia-
tion of the stochastic solution from its deterministic counterpart and/
or general dynamics of the moments. For example, the mean dynam-
ics, described by (8) and (9), differs from the solution of the determin-
istic problem (1) with the mean value of the random parameter 
a.The
difference between the deterministic equation and the averaged equa-
tion is the correlation /0u0and /0u0
p. In some special cases, e.g., when
thecarrierflowisconstantandtherelativevelocityiszero(u¼u
p
),
the mean of the solution is the same as the deterministic solution for

a; the velocity deviation decreases since the right hand side of (11)
contains only the damping term.
Finally, we note that the moment equations (9)–(13) are related
to the PDF equation (3) as they represent the evolution of the first
three moments of fxpand fup.
C. Singularities in the stochastic solution
The characteristic velocities which can be directly inferred
from (3),
CX¼dXp
dt¼Up;(14)
CU¼dUp
dt¼AgðUUpÞ
spðUUpÞ;(15)
can be different which can lead to a crossover of the characteristics’
paths at certain values of A. In general, hyperbolic systems when char-
acteristics cross, a discontinuity is expected to appear in the solution.
Depending on the sign of the relative velocity, UUp, we identify
two settings in which the resulting discontinuities appear in the joint
PDF, faxpupand its marginals. First, for a positive (and constant) rela-
tive velocity [Fig. 1(a)], we consider a cloud of N
a
particles with
uniformly spaced different drag coefficients A
i
(i¼1…Na)suchthat
Aiþ1>Ai. The particle with the greatest forcing, ANa(rightmost par-
ticles), moves fastest, whereas the leftmost particle with a slower
response is left behind. As a result the cloud expands. For a nonlinear
relative velocity, the characteristics could steepen and cross in the
expansion, yielding discontinuities (not exhibited in the graph).
For a second setting, the same initial cloud is considered but for a
negative (and constant) relative velocity, which causes the cloud of
particles to compress. At some point, the leftmost particles overtake
the rightmost particles and the cloud concentrates in a reduced region
or even in a singular point. At that instant, the characteristics of the
hyperbolic system (3) cross. If all of them cross in a single point, then
the PDF solution becomes the Dirac delta distribution. After this sin-
gular event, the cloud expands and asymmetry can reemerge, resulting
in steepening of the left side of the PDF fxpand in its discontinuity, as
it did for the positive relative velocity.
Consistent with the formation of discontinuities in the marginal
PDF, fxp, discontinuities also arise in the marginal PDF of the particle
velocity, fup.InSec.V, we illustrate these phenomena by analyzing the
uniform and stagnation carrier flows with stochastically forced particle
dynamics.
IV. NUMERICAL IMPLEMENTATION
The discontinuities and sharp gradients that can appear in the
solution of the PDF equation (3) require special numerical treatment.
We use a low-dispersive/diffusive Chebyshev collocation method to
approximate the derivatives with respect to X
p
and U
p
.Suchspectral
treatment was shown to be effective or even necessary to solve similar
moment equations in Ref. 17. We also deploy the filtering and
FIG. 1. Evolution of the PDF of x
p
of a cloud of particles initially distributed uni-
formly in space and traveling at the same initial velocity (from left to right) with dif-
ferent drag coefficients A
i
such that Aiþ1>Ai. Under the influence of positive
relative velocity in (a) and negative relative velocity in (b).
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 033326 (2021); doi: 10.1063/5.0039787 33, 033326-4
Published under license by AIP Publishing

regularization techniques designed to capture discontinuities and regu-
larize singularities in a spectral solution while preserving
accuracy.
17,40,45,46
A. Chebyshev collocation method and time
integration
The Chebyshev collocation method, extensively described in the
textbooks,
47,48
is briefly summarized below for the sake of complete-
ness. We do so for one spatial dimension; the multi-dimensional for-
mulation builds upon that as it is defined along lines on a tensorial
grid. In the Chebyshev collocation method, a function y(x) is approxi-
matedbyaChebyshevinterpolantas
yNxðxÞ¼X
Nx
j¼0
yðxjÞljðxÞ;ljðxÞ¼ Y
Nx
k¼0;k6¼j
xxk
xjxk
:(16)
Here j¼0;…;Nx;ljðxÞis the Lagrange polynomial of degree N
x
.The
collocation points are chosen as the Gauss-Lobatto quadrature points
ni¼cos ðip=NxÞ;i¼0;…;Nx(17)
such that the L1norm of the interpolant is minimized on the interval
½1;1.
The derivative of the function y(x) at points x
i
is approximated
by
@y
@xðxiÞX
Nx
j¼0
yðxjÞl0
jðxiÞ;(18)
with l0
jthe derivative of the corresponding Lagrange polynomial. This
approximated derivative is recast in the matrix-vector form
y0¼Dy;(19)
where the differentiation matrix Dhas components Di;j¼l0
jðxiÞ.
The multi-dimensional PDF equation (3) is discretized on a ten-
sorial grid that spans X
p
,U
p
,andAin the domain X.Thespectral
approximation of the distribution function ~
f¼~
fNANXpNUpðA;Xp;UpÞ
on this grid is governed by the semi-discrete equation
d~
f
dtþDXpFXpþDUpFUp¼0;(20)
where the entries of the flux arrays are given by
FXp
i;j;k¼Upi;j;k~
fi;j;k;(21)
FUp
i;j;k¼Ai;j;kgðUi;j;kUpi;j;kÞ
spðUi;j;kUpi;j;kÞ~
fi;j;k;(22)
with counters i,j,kalong the tensors. The matrices DXpand DUpare
the scaled versions of the matrix Dwith the following entries:
DXp
ðÞ
m;j¼@n
@Xp
Dm;j
ðÞ
;DUp
ðÞ
m;k¼@n
@Up
D
ðÞ
k;m(23)
with @n=@Xp¼2=ðXmax
pXmin
pÞand @n=@Up¼2=ðUmax
pUmin
pÞfor
the one-dimensional case. The matrix-vector multiplication DXpFXp
and DUpFUpis performed along grid lines with the counters jand k,
respectively, in (21) and (22). The carrier flow velocity Uis specified at
the particle locations. The semi-discrete system is integrated in time
with the total variation diminishing (TVD) Runge-Kutta scheme.
49
To obtain the marginals, fxpand fupare obtained via the numeri-
cal integration of faxpupalong Aand either U
p
or X
p
,respectively.This
is done via Clenshaw-Curtis quadrature in U
p
and X
p
and via the trap-
ezoidal rule in A. Because the distribution equation does not have
terms with derivatives respect to A, the spectral approximation is not
necessary in this direction.
B. Regularization of Dirac delta function
The numerical solution of the PDF equation (3) with the deter-
ministic initial state (7) requires an approximation of the Dirac delta
function dðÞ. We rely on the kernel that regularizes dðÞ with a class
of high-order, compactly supported polynomials,
45
dm;k
eðxÞ¼ e1Pm;kðx=eÞ;x2e;e
½
0;otherwise;
(24)
where e>0 is the support width or scaling parameter. On the com-
pactly supported interval, the regularized delta function integrates to
unity (i.e., the zeroth moment is one). The polynomial Pm;kis designed
to have the first up to the m
th
moment vanished and to have kcontin-
uous derivatives at the endpoints of the compact support. For it to be
possible for the moments to vanish the regularized delta is permitted
to have negative values on its supported interval. The vanishing
moments ensure that the regularized Dirac delta kernel (a so-called
delta sequence) converges to the exact Dirac delta function at a rate of
Oðemþ1Þ.Thismomentpropertyisnecessaryfortheconstructionof
high-order approximations of singular Dirac delta source terms in
spectral approximations of PDEs as was shown in Ref. 45. To preserve
high-order spatial accuracy, it was further shown that the optimal
value for the compact support must be e¼Nk=ðmþkþ2Þ
x.Thecompact
kernel dm;k
eðxÞin (24) has a maximum at its center. To achieve high-
order accuracy, one has to relax positivity of the kernel, leading to the
undershoots in Fig. 2.
For the approximation of the initial Dirac delta distribution func-
tion in (7), the vanishing moments of the regularized delta function
yield an accurate representation of the zero moments of the determin-
istic initial state. Thus, in that case the regularized Dirac delta provides
both spatial accuracy and the correct statistical properties of the distri-
bution function at the initial time.
A naive alternative is to approximate dðxÞvia a Gaussian PDF
dðxÞ 1
ffiffiffiffiffi
2p
prx
exp x2
2r2
x
"#
;(25)
with small variance r2
x. The Gaussian PDF, however, has no vanishing
moments and can thus not yield high-order approximations to the
Dirac delta. If the initial state is random, than the Gaussian distribu-
tion does correctly represent uncertainty in the initial state of the
system.
C. Filtering for capturing discontinuities
Since (3) admits singularities, we have to regularize these singu-
larities in numerical approximations to avoid numerical instabilities.
To this end, we once again resort to the regularized Dirac delta kernel
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(24). This time the kernel serves as a convolution filter kernel as dis-
cussed in Ref. 40 to smoothen a function y(x) as follows:
~
yðxÞ¼ðxþe
xe
yðsÞdm;k
eðxsÞds:(26)
Using a quadrature rule for approximation of the convolution integral,
the interpolant yNxis filtered as
~
yNxðxÞ¼ðxþe
xe
dm;k
eðxsÞX
Nx
i¼0
yðxiÞliðsÞds¼X
Nx
i¼0
yðxiÞSiðxÞ:(27)
ThediscretefilterS
i
is defined as
SiðxÞ¼ðxþe
xe
liðsÞdm;k
eðxsÞds:(28)
In vector notation, (27) and (28) take the form
~
y¼Sy:(29)
The extension to tensorial form is straightforward. This convolution
filter was shown in Ref. 40 to smoothen shock discontinuities while
providing high-order accurate resolution away from shocks. In some
cases, a weak exponential filter
32
is needed to remove high-frequency
numerical noise that appears in regions near the boundaries.
D. Monte Carlo simulations
The PDF and the moments of the PDF can be computed with a
MC approach. In MC, realizations of x
p
and u
p
are computed by solv-
ing (1) with random coefficients adrawn from a given PDF faðAÞ.
Here, we use analytical solutions that will be discussed in Sec. Vand
that allow for a computationally efficient determination of a significant
number of MC realizations, N
s
. In all the tests considered, we found
Ns¼105realizations to be sufficiently accurate yielding a normalized
error of the third moment less than 0.1%. The kernel density estima-
tion, implemented in the Matlab 2019b subroutine kdensity,
determines the PDFs fxpðXp;tÞ;fupðUp;tÞ,andfxpupðXp;Up;tÞ.The
unknown correlation terms in the moment equations (8)–(13) are
closed using MC realizations. The resulting a priori closed moment
equations are integrated in time via the fourth-order Runge-Kutta
(RK4) scheme.
V. TWO CANONICAL PARTICLE-LADEN FLOWS
We consider two one-way coupled particle-laden flows—a uni-
form flow and an inviscid stagnation flow—for which the carrier phase
velocity is described by analytical expressions. These are both impor-
tant in their own right and serve as building blocks for more complex
flows. Both flows admit analytical solutions for the corresponding
particle-laden flow with constant deterministic forcing, /¼constant.
While this particle solution for the uniform carrier flow is well known,
we are not aware of an analytical solution to the particle-laden stagna-
tion flow. Analytical solutions are derived for both flows in Secs. VA
and VB.
A. Uniform flow
By its definition, a uniform carrier flow is characterized by a con-
stant velocity field u. To derive the analytical solution, we cast the par-
ticle transport equations (1) with the constant uand the initial
conditions xpð0Þ¼xp0and upð0Þ¼up0into the following linear sys-
tem of ODEs:
d
dt
xp
up

¼01
0b

xp
up

þ0
bu

;ba
sp
:(30)
The analytical solution of this system is
xpðtÞ¼xp0þut þ1
bðuup0Þebt 1
ðÞ
;(31)
upðtÞ¼uþðup0uÞebt:(32)
Details of the derivation are provided in Appendix B.
The solution is plotted in Fig. 3 and shows that the response of
the particle initially at rest to a fluid velocity is slower with increasing
b,i.e.,withincreasingeffectiveinertia.Hence,foragivens
p
,higher
values of the correction parameter adecrease the particle’s time
response. At long times on the order of Oð1=bÞ,theparticlevelocity
becomes equal to the carrier flow velocity u. When the relative velocity
between the particle and the carrier phase (also called interphase veloc-
ity) becomes zero, the particle position increases linearly at its constant
advection rate u.
B. Stagnation flow
The stagnation carrier velocity field, u¼ðu;vÞ>, is given by the
Hiemenz analytical solution for an inviscid, irrotational flow
50
in the
domain x2½1;0as follows:
u¼kx;v¼ky;
where yis the coordinate perpendicular to the flow direction, and kis
a constant. (The viscous boundary layer solution near a wall at x¼0is
available as well.
51
It predicts the boundary layer thickness of
d¼ffiffiffiffiffiffiffiffi
=k
p, too thin to affect the particle dynamics.)
Along the centerline y¼0, the flow is one-dimensional with a
stagnation point at x¼0andvelocity
FIG. 2. Regularization of the Dirac delta function, dm;k
e, in comparison with a
Gaussian PDF.
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u¼kx:(33)
With the carrier velocity at the particle location x
p
is u¼kxp(1) can
be cast into a linear dynamic system,
d
dt
xp
up

¼01
kb b

xp
up

:(34)
The analytical solution to this system is derived in Appendix B and is
characterized by the eigenvalues of the 22 matrix in (34)
k1¼bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bðb4kÞ
p2;k2¼bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bðb4kÞ
p2:(35)
Their real and imaginary parts are plotted in Fig. 4,forb2½0;8and
k¼1. For 0 <b<4k, the eigenvalues are imaginary with negative
real part. In that case, the solution of the system (34), i.e., the particle
phase solution, is well-known to tend toward an inward spiral in the
phase plane as plotted in Fig. 5(c). Before it reaches the spiral singular-
ity, however, the particle will have crossed the x¼0 line where the
wall is located. This is, of course, not possible and the particle trace has
to terminate at x¼0. Alternatively, we can interpret the solution as a
physical solution of a particle trajectory in an opposed jet carrier flow.
For b>4k, both eigenvalues are real and negative, in which case the
particle moves toward an inward node in the phase space xpvp.A
bifurcation in particle dynamics from a spiral to a node occurs at
b¼4k.Figure 5(c) shows the particle phase when the stagnation point
is an inward node and an inward spiral for two different initial
conditions.
The analytical solutions for the particle’s position and velocity,
xpðtÞand upðtÞ,vstimeareplottedinFigs. 5(a) and 5(b).Theparticle
reaches the stagnation point for any forcing b¼a=sp.Thecollisionof
the particle with the wall for the stangation flow case is indicated by
the red dot in the graphs 5(a),5(b),and5(c).
C. Impact of stochastic forcing
The effect of a stochastic forcing on the particle-laden uniform
and stagnation flows is studied for the cases and parameters collated
in Table I. For both flow regimes, we consider particles initialized at
rest. For the stagnation flow, we also consider the particles initialized
according to the carrier flow velocity.
For each of these cases, we consider three PDFs, faðAÞ,for
the random variable ain the drag correction factor defined in (2)
including a uniform, normal, and beta distribution, all with the same
mean l
a
andstandarddeviationr
a
(Fig. 6). For the stagnation flow,
faðAÞis selected to have a non-zero probability in the interval
0<a=sp<4kto ensure that all particles reach the wall at a finite
time (according to the deterministic solution). Also investigated is the
effect of deterministic vs stochastic initial conditions.
VI. SIMULATION RESULTS AND DISCUSSION
A. Uniform flow: Monte Carlo results
The PDFs fxpðXp;tÞand fupðUp;tÞobtained via MC solution of
(1) for the uniform flow with the uniform forcing distribution faðAÞ
and deterministic initial conditions are depicted in Fig. 7. Starting
from the deterministic Dirac delta initial distribution, both fxpðXp;tÞ
and fupðUp;tÞfirst widen over time, while showing a skewness, i.e., a
FIG. 3. Time dependence (a) and phase space (b) of the particle dynamics in the
constant uniform carrier flow.
FIG. 4. Imaginary and real part of the eigenvalues k
1
and k
2
in (35) for b2½0;8.
The circle corresponds to b¼0, and the diamond and square correspond to b¼8
for k
1
and k
2
, respectively.
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bias, toward the upper range of the X
p
and U
p
values, where more par-
ticles accumulate. This bias reflects the particles’ asymptotic behavior
in the limit of an infinite response time, sp!1in which case all the
particles congregate on a step function in time. After the initial widen-
ing, the velocity distribution narrows with time as the particles’ veloc-
ity settles to the uniform carrier flow velocity. The temporal evolution
of the PDFs has a characteristic time scale on the order Oðsp=
aÞ.At
later times, the velocity distribution returns to the Dirac delta and the
corresponding position distribution is advected at constant velocity u
without changes in time.
The means 
xpðtÞand 
upðtÞ, plotted with their corresponding two
standard deviation bandwidths in Fig. 8, tell a similar story. The mean
particle velocity 
upðtÞincreases from its zero initial state and settles to
the constant carrier velocity at t!1. Associated with the accelera-
tion and settling is an initial increase in the velocity bandwidth that
then returns to zero at later times. Consistent with the velocity band-
width, the position bandwidth grows initially and then remains con-
stant when the particles settle.
Per definition, and as confirmed by Fig. 8, the mean of the solu-
tion must be contained in the interval of deterministic limit trajecto-
ries. Moreover, because xpðaÞand upðaÞare monotonically increasing
with a, it follows that

xp2min
axpðaminÞ;xpðamax Þ

;max
axpðaminÞ;xpðamax Þ

hi
;

up2min
aupðaminÞ;upðamax Þ

;max
aupðaminÞ;upðamax Þ

hi
;
where amin >0andamax denote a minimum and maximum value of a.
This suggests that 
xpxpð
aÞand 
upupð
aÞ,i.e.,themeansolution
is equal to the deterministic solution at the mean stochastic forcing.
The moment equations provide further insight. Because of the
correlation terms /0u0and /0u0
p,thegoverningequationsforthe
FIG. 5. Solutions for particles released at rest (a) and at flow condition (b) in a
stagnation flow with k¼1 for different values of parameter b. The phase space plot
in (c).
TABLE I. Flow regimes and parameter values considered in the simulations.
Test case xp0up0us
p
Uniform flow, particle launched at rest (UF) 0 0 1 0.25
Stagnation flow, particle launched at rest (SFR) –1 0 xp1
Stagnation flow, particle launched at flow
conditions (SFF)
–1 1 xp1
FIG. 6. Uniform, normal, and beta (U;N;B) PDFs selected for the random
parameter a. All three PDFs have the same mean la¼1 and standard deviation
ra¼0:2, i.e., aU½1ffiffiffiffiffi
12
p=2;1þffiffiffiffiffi
12
p=2;aN½1;0:2, and
aB½2;3þ0:6.
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mean position and velocity in (8) and (9), respectively, are different
from the deterministic equations (1) with a¼
a. But the term /0u0is
zero for the uniform flow case because u0¼0. Moreover, the correla-
tion term /0u0
pin (9) is negligible, but not zero. Thus, to a first approx-
imation, 
xpxpð
aÞand 
upupð
aÞ. For a random solution with a
uniform stochastic forcing distribution, the root mean square differ-
ence over the time interval is 0.0073 for the position and 0.0062 for the
velocity.
With a zero carrier phase velocity perturbation, u0¼0, many of
the correlation terms in the second central moment or variance of the
velocity are also zero or negligible. Significant terms that remain are a
damping term 
/r2
upand the source term, /0u0
pð
u
upÞ.Thelatter
is positive because the relative velocity is positive, ð
u
upÞ>0, and
because /0¼a0and u0
phave the same sign since the particle velocity
u
p
is monotonically increasing with respect to the forcing /¼a.The
positive source term is maximum initially and decreases as the particle
velocity settles to the flow conditions. The damping term reduces the
velocity variance to zero in the limit t!1. Correspondingly, the
PDF fuptends to the Dirac delta distribution [Fig. 7(b)]. The combina-
tion of the temporal damping and forcing by the positive source leads
to a maximum variance at times that are on the order of Oðsp=
aÞ.The
particle position variance depicted in Fig. 9(a) shows an initial increase
consistent with the increasing velocity variance and an increased
spreading of random particle trajectories. When the particles settle to
the constant carrier flow condition, all trajectories are advected at con-
stant velocity. After that time, the particle variance no longer changes.
In Fig. 9(a), the variance of the particle velocity and position are
plotted vs time for three different forcing distributions f
a
(uniform,
normal, and beta). The temporal trends for the different stochastic
forcing are very similar because the mean forcing and its variance are
chosen to be the same for the three forcing distributions. The damping
term in the velocity variance equation, which depends on the mean
forcing and velocity variance only, is therefore not affected by the
shape of the forcing distribution. The source correlation term, how-
ever, is directly dependent on the forcing fluctuations, /0,whichleads
to differences in the velocity and position variances for different shapes
of the forcing distribution of up to 15%.
The third central particle position and velocity moments evolve
in a similar way as the variances [Fig. 9(b)]. The third velocity
moment, sup, experiences a negative growth followed by an asymptotic
decay to zero (or the Dirac delta in the PDF sense as observed above).
The third position moment, sxp, first decreases and then asymptotically
evolves to a constant value. Both the minimum in supand the plateau
in sxpoccur at slightly later times as compared to the minimum in rup
and rxp. The difference in the factors 2/=spand 3/=spin Equations
(11) and (13) is assumed to be at least partially responsible for causing
this shift in the maximum. The similarity in variance and skewness
trends would suggest that the third moment dynamics might also be
primarily affected by a positive sourcing and a damping. To verify this,
the correlation terms in (13) are plotted vs time in Fig. 10.Clearly,the
damping term 
/suphas a major influence on the long term response.
However, there is no single dominant source term. While the term
/0u02
pð
u
upÞplays a similar role as the positive source term in the
variance equation, the other correlation terms are not negligible and
contribute also. Surprisingly perhaps is that the term with fourth order
correlations, /0u03
p, is dominant, an indication that the tail behavior
of the solution PDF and tail behavior of the forcing function has a
FIG. 7. PDF of particle position (a) and velocity (b) for the UF test case carried out
with MC with a uniform forcing distribution.
FIG. 8. Two standard deviation interval along the mean for the test case UF with a
uniform forcing distribution. In dashed green, the particle velocity computed with
MoD and in black with MC. In dashed red, the particle position computed with MoD
and in blue with MC. Dark colors indicate dIC, whereas light ones indicate sIC.
Physics of Fluids ARTICLE scitation.org/journal/phf
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Published under license by AIP Publishing

considerable impact on the higher central moment solution of the
solution. This is confirmed by the deviation in the third central
moment evolution of up to 200% for different forcing distributions.
We plan to report further on the tail behavior of the PDF in the
near future.
The solution with stochastic initial condition (sIC) is plotted
in Figs. 8 and 9. It shows that while the trends in the position and
velocity mean and variance are similar to those determined with a
deterministic initial condition (dIC), the sIC solution is offset as
compared to the dIC solution. The offset is according to the initial
position deviation of rx¼0:05. Time integration of (10) from time
zero to a time tconfirms that exactly this term rxðt¼0Þappears at
the initial time, t¼0. The offset in Fig. 9 does not change signifi-
cantly over the time interval ½0;t, which implies that the term x0
pu0
p
in (10) is small. MC simulations confirm this and show that the
term has a maximum value of 0.002 over the time interval. Because
of the damping term, the velocity variance and third central
moment goes to zero in the asymptotic time limit for both deter-
ministic and stochastic initial conditions.
B. Uniform flow: Method of distributions
The solution of the governing equation for the PDF in (3) is grid-
resolved for the uniform flow case using a spectral grid with NXp
NUp¼300 300 collocation points and a uniform grid in Adirec-
tion with N
A
¼200. The CFL condition is set to 0.8. The Dirac delta
distribution function for dIC is regularized according to dk;m
ein (24)
with an optimal scaling e¼0:05, and m¼5 zero vanishing moments
and smoothness k¼2.
Figure 11 shows snapshots of fxpup(contours), fxp(left and bottom
axes), and fup(right and top axes) at three consecutive times. For refer-
ence, the mean of the particle phase solution (black line) is superposed
in the contour plot. At time t¼0, the marginals are initialized accord-
ing to the regularized Dirac delta as shown in Fig. 11(a).Atalater
time, t¼0.54, the joint PDF fxpuphas traveled along the mean in the
XpUpcoordinate system and has widened and deformed [contours
in Fig. 11(b)]. The marginal fxpand fupshow that the particles have a
bias toward the larger values of the position and the velocity. That is
consistent with the observations in the moments discussed previously;
because the particles with smaller response time, sp=A, travel a dis-
tance greater than the slower responding particles, they cluster at large
X
p
. Those fast particles furthermore reach their terminal settling veloc-
ity faster and thus there is a similar clustering in fup.Theconvexityof
the PDFs is an indication that the clustering is more pronounced
toward larger values. The schematic in Fig. 12 underscores this and
shows how the characteristic paths with non-constant advection veloc-
ity for different A
i
leads to a convex probability density.
At time t¼1.6, the velocity PDF has evolved toward a Dirac delta
function represented by a narrowly supported distribution centered at
U
p
¼1. The numerical solution successfully captures this PDF behav-
ior despite showing some minor fluctuations caused by Gibbs oscilla-
tions. The accuracy of the MoD solution at this time relies largely on
the number of vanishing moments mof the regularization of the Dirac
delta function at t¼0. Because the number of vanishing moments is
specified to be greater than five, the first up to the fifth moment is
accurately preserved even at times when the distribution function
FIG. 9. Computations for the UF case with MC in color lines and with the MoM in
black dots of the (a) second and (b) third central moments for the particle position
and velocity and the three PDFs considered for a(see Fig. 6) and dIC. It is included
the case of sIC for the uniform distribution. The legend in (b) is valid for (a) as well.
FIG. 10. Terms in Eq. (13) vs time tfor the UF case with dIC.
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tends to the singular Dirac delta distribution. This accuracy preserva-
tion is confirmed by the results in Fig. 8 that compares the time evolu-
tion of the mean and variance determined with the MoD and MC
approach and that shows no discernible difference between the solu-
tions of the two approaches.
A few remarks on the accuracy of the numerical solution of Eq. (3):
Remark 1: Consistent with the findings in Ref. 17,theuseofhigh
order methods is necessary to compute an accurate solution of the
joint PDF faxpupsuch that the marginals determined according to
(A15)–(A17) are in good comparison with MC results. To underscore
the importance of numerical discretization, we compare the
Chebyshev spectral discretization with a first and a second order
upwind finite difference (FD) schemes. Figure 13(a) shows that the FD
schemes are overly dissipative as compared to the spectral method if
the same number of grid points are used. The root mean square error
(RMSE) between resolved MC results and the spectral solution is
0.064, whereas the RMSE for the first and second order FD method is
significantly larger at 0.280 and 0.180, respectively. To mitigate the dis-
sipation and inaccuracy, FD requires an excessive resolution for the
computation of the PDF after marginalization.
Remark 2: The spectral solution shows dispersion errors in the
form of high-frequency oscillations in the distribution function. These
are induced by the high-order approximation of the steep gradients in
the PDF that in turn are a result of the steep gradient in the uniform
forcing distribution f
a
. These dispersion errors, however, average out
and turn out to have no significant effect on the numerical accuracy of
the first three moments [see Fig. 13(b) for the third central moment].
The second order FD scheme also exhibits dispersion errors, but the
FD’s oscillations do not average out and the moments are not accu-
rately captured using this discretization.
Remark 3: For deterministic initial conditions, the regularization
of the Dirac delta is necessary to accurately compute the moments of
the evolving distribution function. Particularly, the vanishing moment
condition ensures that the evolution of the third moment [Fig. 13(b)]
is not affected by the numerical approximation as compared to Dirac
delta regularization with only two vanishing moment m¼2in(24),or
a Gaussian distribution function.
C. Stagnation flow
In the particle-laden stagnation flow, the relative (interphase)
velocity is not only affected by the evolution of the particle phase as is
FIG. 11. Marginals fxpand fupat t¼½0;0:54;1:6in (a), (b), and (c) respectively,
for the UF test case with dIC and the uniform distribution for f
a
. Contour plots of the
joint PDF fxpupsuperposed with the mean of the particle phase solution.
FIG. 12. Schematic evolution in time of the particle phase PDF fxpupfor the UF test
case.
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Published under license by AIP Publishing

the case in the uniform flow but also by the evolution of the carrier
phase velocity along the particle’s path. The temporal development of
the random particle position and velocity, therefore, displays a consid-
erably more complex behavior as compared to the uniform flow.
Because of the spatial dependence of the carrier flow, the particle solu-
tion is furthermore non-trivially dependent on its initial condition.
We consider two initial conditions described in Sec. I, one with the
particles starting at rest (Case SFR) and another withthe initial particle
velocity specified at the carrier flow’s velocity conditions (Case SFF).
We discuss the MC and MoD solutions for each case below.
1. SFR case: Monte Carlo results
The mean trends with two standard deviation bandwidth deter-
mined with the MC approach for a uniform forcing distribution are plot-
ted in Fig. 14. To discuss the SFR case, three stages of development are
identified. In the first stage (t<0.6) each particle identified with a coun-
ter iaccelerates in positive x-direction at a rate aiðkxpikupiÞ=sp.
Similar to the uniform flow case, the velocity and position variance both
increase in this stage with varying acceleration of the stochastically forced
particles. The second central moment plotted vs time in Fig. 15(a) con-
firms the growth of the particle variance in this stage.
In a second stage (0:6<t<1:6), the particle with the smallest
response time sp=amax—the fastest responding particle with accelera-
tion rate amax ðkxpmax kupmax Þ=sp—has accelerated to the carrier
velocity (t¼0.6). After that, the flow velocity continues to decrease
(stagnates) along this particle’s path. Because of the particle’s inertial
response, the particle’s velocity, however, does not decrease equally
fast along the particle’s path. Effectively, the relative velocity of this
particle therefor becomes negative. In other words, the particle starts
to decelerate. As more particles with larger response time reach the
carrier flow conditions, more particles start to decelerate until all par-
ticles have a negative relative velocity. During this second stage, the
velocity variance of the particle phase decreases to a minimum at t
1:6[Fig. 15(a)].
In a third stage t>1.6, when the relative particle velocity is smaller
than zero (up>uðxpÞ) for all the particles, the cloud decelerates to a
decreasing carrier velocity and the particle velocity variance increases.
The variance increase mechanism is similar to the first stage and the
uniform flow, in which a time varying carrier velocity in combination
with a random forcing leads to a variance increase in the particle veloc-
ity. In the stagnating flow, the random particle cloud compresses with a
decreasing position variance before the wall is reached.
As opposed to the uniform flow case the carrier flow’s velocity
fluctuation, u0, for the stagnation flow is non-zero which affects several
terms in the moment equations. Even with these extra terms, just like
for the uniform flow, the mean stagnation flow solution described by
(8) and (9) can also be approximated by the solution of the determinis-
tic equation for a¼
a. The latter position and velocity solution have a
root mean square deviation of 0.0073 for 
xpand 0.0053 for 
upas com-
pared to the former. Both the terms /0u0and /0u0
pturn out to be neg-
ligible in (9).
The evolution of the velocity variance as governed by its moment
equation (11) is affected by the second term on the right hand side,
FIG. 13. Comparison between the spectral discretization and finite difference
upwind discretization with first and second order for (a) fupat t¼0.24 and (b) sup.
Both figures are for the UF test case with dIC with a uniform forcing distribution.
FIG. 14. Two standard deviation interval along the mean for the test case SFR with
a uniform forcing distribution. In dashed green, the particle velocity computed
with the MoD and in black with MC. In dashed red, the particle position computed
with MoD and in blue with MC. Dark colors indicate dIC, whereas light ones indicate
sIC.
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Published under license by AIP Publishing

i.e., /0u0
pð
u
upÞ.Att1:6, the relative velocity ð
u
upÞin this
term changes sign when the particle phase begins to decelerate after its
initial acceleration. The sign of /0u0
pin this term changes at t1:6
also as follows: in the first stage u
p
is monotonically increasing for all
the forcing values of /¼aaccording to the analytical velocity solu-
tion (32); in stage two, some particles are accelerating and others are
decelerating which yields different signs for dup=d/depending on /.
Upon ensemble averaging, it turns out that the mean of dup=d/is
positive prior to t1:6 and negative after. In the third stage, u
p
is
monotonically decreasing with respect to /. So, the correlation term
/0u0
pchanges sign at t1:6 and thus the term /0u0
pð
u
upÞ>0.
In addition to the damping term that was discussed for the uni-
form flow case, the first term on the right hand side in the velocity var-
iance equations (11) also involves the term 
/u0u0
pfor the stagnation
flow case. Similar to the sign change of /0u0
pat t1:6, the sign of

/u0u0
pis the same as 
uupbecause of a comparable behavior of
dup=du and dup=d/. As a result, 
/u0u0
pis negative before t1:6and
positive thereafter. The values of the term u0u0
pare between –0.002
and 0.002 and are thus of the same order as the velocity variance r2
up
[see Fig. 15(a)]. The term 
/u0u0
ptherefor has a significant effect on the
variance dynamics. At early times, it reduces the growth of the vari-
ance, and at later times it enhances growth as compared to the uni-
form flow where the term is zero.
The third order correlation terms (the third and fourth term in
(11)) are observed to have a negligible contribution to the particle vari-
ance evolution. In comparing the maximum magnitude of each of the
terms in the right hand side of (11) with respect to the left hand side
over the time interval, we find that the terms /0u0u0
pand u0u02
phave at
most a 3.0% and 1.0% contribution, whereas the first and second
terms have a significant 120% and 152% contribution to the “variance
acceleration.”
Figures 15(a) and 15(b) include the variance evolution for several
distribution functions of the forcing f
a
. As in the uniform flow case,
the effect of the shape of f
a
is small on the order of 5% in the velocity
variance and slightly more (order of 10%) in the position variance.
The general trends are not affected by the shape of the forcing PDF.
The third position moment is negative throughout the time inter-
val considered [Fig. 15(b)], indicating a non-symmetric position distri-
bution that is skewed toward larger values of the particle coordinate.
To understand the evolution of the particle velocity’s third moment,
we differentiate between two stages; first, when the mean interphase
velocity 
u
upis positive and the skewness shows a bias toward
higher velocity values similar to the uniform flow case as also illus-
trated in Fig. 1. Second, when 
u
up0atfirstafterwhichit
becomes negative, i.e., 
u
up<0 with a near zero skewness first and
decreasing after showing a bias toward small values of the particle
coordinate when 
u
up<0 right before the particles hit the wall.
This second stage can be also understood through the evolution of the
PDF that consists of the formation of the singular Dirac delta distribu-
tion for which the skewness is zero and its consequent behavior as
illustrated in Fig. 1(b) with a change of the bias in the PDF.
Like in the uniform flow case, the evolution of the third central
moment is affected by many different terms in the velocity skewness
equation (13) as shown in Fig. 16. The fourth order correlation terms
are important in the stagnation flow also, but because the velocity fluc-
tuation is non-zero, u06¼ 0, the evolution of terms that involve u0are
non-trivial and require a separate and more in-depth analysis. We feel
this is outside the scope of the current paper and we plan to report on
the skewness behavior in more detail in future work.
Stochastic initial conditions do not only alter the evolution of the
mean of the stagnation flow solution with dIC by a constant offset as
wasthecasefortheuniformflow(seeFig. 14), but the difference
between the solutions with dIC and sIC changes considerably over the
time interval and specifically at early times. The variance of the particle
position and velocity is initially offset according to its initial values as
shown in Fig. 15(a), but then the difference with respect to the dIC
casedecreasesastimeevolves.Thisreductioncanbeunderstoodby
considering the stagnation flow solution where particles can nonphysi-
cally cross the wall (i.e., an opposed jet flow). For this flow, all particles
move toward the same final state with x
p
¼0andu
p
¼0intheasymp-
totic time limit, t!1, and thus the position and velocity variance
tend to zero.
Between the initial time and the infinite time, the terms x0
pu0
pand
u0u0
pare responsible for the reduced variances. The contribution of
u0u0
pwhich is negative for t<1.6, particularly, causes a greater increase
in the damping term for sIC as compared to dIC at early times. When
FIG. 15. Computations for the SFR case with MC in color lines and with the MoM
in black dots of the (a) second and (b) third central moments for the particle position
and velocity and the three distributions considered for a(see Fig. 6) and dIC. It is
included the case of sIC for the uniform distribution. The legend in (a) is valid for
(b) as well.
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the interphase velocity changes sign at t1:6, this term becomes pos-
itive and it will have the opposite effect. A physical interpretation is as
follows: a more energetic initial state with higher velocity variance is
more resistant to changes induced by stochastic forcing resulting in
greater damping at early times. The term x0
pu0
pis positive in the accel-
eration stage and negative in the deceleration stage and its magnitude
is greater for dIC as compared to sIC consistent with greater values of
u0
pfor sIC.
Another considerable difference between the dIC and sIC is
that the minimum in the velocity variance at t1:6isnon-zero
for the stochastic case, while it is nearly zero for the deterministic
case. As a consequence, the singularity in the distribution function
when the relative velocity changes sign can be expected to be less
significant and the PDF can be expected to have a broader
support.
2. SFR case: Method of distributions
Using the same grid as was used for the uniform flow case, the
PDF solution for the SFR case with a uniform forcing distribution,
f
a
, and dIC are computed and plotted for three instances in Fig. 17
at t¼1.22, t¼1.60, and t¼2.15. At time t¼0, the initial condition
is identical to the uniform flow case plotted in Fig. 11(a), and it is
therefore not repeated in Fig. 17. The MC results are also plotted
in Fig. 17, and they are in excellent agreement with the MoD
results.
During the first stage (t<0.6), the joint PDF fxpupdeforms along
the mean of the particle trajectory (depicted by the black solid line),
showing a non-linear clustering of the particles in the XpUpplane
toward high values. During the second stage (0:6<t<1:6), some
particles accelerate and others decelerate leading to the near singular
Dirac delta distribution at t1:6[Fig. 17(b)]. At later times (t>1.6),
the PDF of the particle velocity increases on the left front [Fig. 17(c)],
confirming a bias toward lower velocities in a deceleration field as dis-
cussed in the MC results.
The position PDF solution has an increasing bias toward the
large value of X
p
which is consistent with the asymptotic infinite time
behavior of the nonphysical solution where particles are permitted to
cross the wall and where both the particle velocity and position distri-
bution evolve to a Dirac delta centered at X
p
¼0andU
p
¼0.
FIG. 16. Terms in Eq. (13) vs time tfor the SFR case with dIC and a uniform forc-
ing distribution.
FIG. 17. Marginals fxpand fupat t¼½1:22;1:60;2:15in (a), (b), and (c), respec-
tively, for the SFR test case with dIC and the uniform distribution for f
a
. Contour plots
are of the joint PDF fxpupsuperposed with the mean of the particle phase solution.
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Published under license by AIP Publishing

3. SFF case: Monte Carlo results
In a final test, the particle velocity is initialized with the carrier
phase velocity at the particle position. The MC results for the mean
with a two standard deviation bandwidth are plotted vs time in
Fig. 18. In the SFF case, the particle phaseonly decelerates which yields
an evolution that is opposite to the uniform flow evolution as plotted
in Fig. 1(a) or an evolution that is very similar to the third
“deceleration” stage of the SFR case for t>1.6. The mean velocity
decreases monotonically when the mean particle position increases
toward the wall. This evolution is accompanied by an increase in the
variances of both x
p
and u
p
.
Because the SFF case is similar to the other two cases, the
moment evolution results do not shed any additional light on the evo-
lution of the stochastically forced particle phase. It is therefore omitted
here. For completeness, we include the plots for evolution of the
moments in Appendix C (Figs. 20 and 21).
4. SFF case: Method of distributions
The results for the uniform distribution forcing, f
a
,fortheSFF
case are also very similar to the deceleration stage of the SFR case.
Rather than reiterating that discussion, we choose a different sto-
chastic forcing according to a beta distribution for f
a
in Fig. 6
which does not have steep gradients in f
a
like the uniform distribu-
tion. For a grid with the same size as described before, the distribu-
tion results for two different times are shown in Figs. 19(a) and
19(b) with a deterministic initial condition. Clearly, the solution
does not show Gibbs oscillations and the MC results and the MoD
are in excellent agreement.
As time evolves, the PDF of the particle position is advected with
a positive characteristic velocity (14) and the particle velocity with a
negative velocity according to (15).ThePDFswidenintimeasthe
response times of random particles is different for different stochastic
forcing leading to variations in the particles velocities and positions.
Both the position and velocity PDF display a non-Gaussian (non-sym-
metric) trend that is more subtle than for the uniform forcing.
VII. CONCLUDING REMARKS
Several techniques and models including a Monte Carlo
approach, a method of moments, and a method of distributions are
developed and compared for analysis of particle dynamics with sto-
chastic forcing in one-way coupled Eulerian–Lagrangian formula-
tions. Random solutions of two canonical flow problems are
discussed including a particle phase accelerated in a uniform car-
rier flow and a particle phase released in a stagnation carrier flow
with two initial conditions, one at rest and one initialized at the
carrier flow velocity.
Starting from the Lagrangian particle equations for position and
velocity with stochastic forcing, a closed PDF formulation is derived.
A single hyperbolic partial differential equation, whose characteristic
advection velocities are non-constant, governs the evolution of the
PDF solution. In a single spatial dimension, the PDF depends on three
FIG. 18. Two standard deviation interval along the mean for the test case SFF with
a uniform forcing distribution. In dashed green, the particle velocity computed with
the MoD and in black with MC. In dashed red, the particle position computed with
MoD and in blue with MC. Dark colors indicate dIC, whereas light ones indicate
sIC.
FIG. 19. Marginals fxpand fupat t¼½0:87;1:17in (a) and (b), respectively, for the
SFR test case with dIC and a uniform forcing distribution. Contours of the joint PDF
fxpupsuperposed with the mean of the particle phase solution.
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variables at a given time, including the position, the velocity, and a
forcing coefficient.
A high-order spectral method with discontinuity regulariza-
tion is necessary for the accurate solution of the hyperbolic partial
differential equation that admits discontinuities. A polynomial reg-
ularization of a Dirac delta function with mvanishing moments is
shown to accurately capture the first mmoments of the PDF solu-
tion in time.
Moment equations are derived for the first three moments of the
particle position and velocity, representing the mean, variance, and
skewness of the PDF. Monte Carlo results are used to determine corre-
lations terms and to close the system of moment equations.
Analytical solutions are derived for the system of two linear
ODEs that govern the dynamics of particles with a deterministic forc-
ing in a one-dimensional uniform flow and stagnation flow. The parti-
cle solution in the stagnation flow has its final state with a zero velocity
at the wall. Depending on the relative forcing, the particle manifolds in
the phase space (position/velocity space) tend to either a node or
spiral.
Themeansolutionwithrandomforcingcanbeapproximated
within 1% using a mere single deterministic solution at the mean forc-
ing for all flow cases considered.
In flows where all randomly forced particles accelerate or deceler-
ate, the velocity variance increases driven by a single correlation source
term. A damping terms counters this source term. When the particle
velocity settles, the velocity variance reduces to zero because of this
damping. Higher-order correlation terms are generally negligible in
the velocity variance equation. The position variance increases in
accelerating flows and decreases in decelerating flows, i.e., the random
cloud expands and compresses, respectively. When the relative velocity
changes sign, the particle variance approaches zero and the PDF has a
very narrow support.
The skewness of the distribution function has a bias toward the
carrier velocity to which the particle accelerates or decelerates. The
bias of the distribution function is non-linear and more significant
toward to the tail ends of the distribution function. The skewness
equation is driven by a sourcing and a damping similar to the variance
equation, but with different response times. High-order correlation
terms are significant, suggesting a complicated tail behavior of the
PDF.
In near future work, we intend to report on the tail behavior of
the PDF solution of the particle phase. We also plan to develop distri-
bution function models for two-way couple Eulerian–Lagrangian
formulations.
ACKNOWLEDGMENTS
This work was supported in part by the Air Force Office of
Scientific Research under Award Nos. FA9550-19-1-0387 and
FA9550-18-1-0474.
APPENDIX A: DERIVATION OF THE PDF EQUATION
USING THE METHOD OF DISTRIBUTIONS
Here, a PDF equation is developed to solve the stochastic sys-
tem defined by (1) with the so-called method of distributions. First,
we define the fine-grained JPDF Pas
PðA;a;Xp;xp;Up;up;tÞ¼dðAaÞdðXpxpðtÞÞdUpupðtÞ

;
(A1)
where A,X
p
, and U
p
are deterministic magnitudes and dðÞ is the
Dirac delta function. Taking the derivative of Pwith respect to
time and using the chain rule and the Dirac delta properties, we
find
@P
@t¼dxp
dt
@P
@xpþdup
dt
@P
@up¼dxp
dt
@P
@Xpdup
dt
@P
@Up
;(A2)
where we can make use of (1) for writing
@P
@tþup
@P
@Xpþag u up
ðÞ
spðuupÞ@P
@Up¼0;(A3)
that in conservative form is
@P
@tþ@
@Xp
upP

þ@
@Up
agðuupÞ
spðuupÞP
"#
¼0:(A4)
Defining the ensemble mean of an integrable function
hða;xp;upÞwith the joint PDF faxpupas
Ehða;xp;upÞ

¼ððð1
1
hðA0;X0
p;UpÞfaxpupðA0;X0
p;U0
p;tÞdA0dX0
pdU0
p;
(A5)
the ensemble of the function Pfor a particular set of the determin-
istic variables is obtained as
EP
½
¼ððð1
1
PðA;A0;Xp;X0
p;Up;U0
p;tÞ
faxpupðA0;X0
p;U0
p;tÞdA0dX0
pdU0
p
¼ððð1
1
dðAA0ÞdðXpX0
pÞdðUpU0
pÞ
faxpupðA0;X0
p;U0
p;tÞdA0dX0
pdU0
p
¼faxpupðA;Xp;Up;tÞ:(A6)
This procedure suggests how to obtain a partial differential equation
for faxpupfrom (A4) taking the ensemble mean each term. To do so,
we need to use the property that allows us to exchange expectation
with derivatives respect to deterministic variables. For example, for
the deterministic variable time, one has
E@hða;xp;upÞ
@t

¼@Ehða;xp;upÞ

@t:(A7)
Using this property, the ensemble mean of the first term in (A4) is
trivial
E@P
@t

¼@faxpup
@t:(A8)
The second is calculated as
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E@upP

@Xp
"#
¼@
@Xp
Eu
pP
 (A9)
¼@
@Xpððð1
1
U0
pPðA;A0;Xp;X0
p;Up;U0
p;tÞ

faxpupðA0;X0
p;U0
p;tÞdA0dX0
pdU0
pi(A10)
¼@
@Xpððð1
1
U0
pdðAA0ÞdðXpX0
pÞdðUpU0
pÞ

faxpupðA0;X0
p;U0
p;tÞdA0dX0
pdUpi(A11)
¼@
@Xp
Upfaxpup

:(A12)
In the same way, the third term is
E@
@Up
ag u up
ðÞ
spðuupÞP
!"#
¼@
@Up
Ag U Up
ðÞ
spðUUpÞfaxpup
!
;(A13)
where for coherence we make use of Uas the deterministic value of
the carrier flow. In this study, the carrier flow is considered deter-
ministic and therefore u¼U.
Finally, the deterministic equation that governs the joint prob-
ability density function of the solution is
@faxpup
@tþ@
@Xp
Upfaxpup

þ@
@Up
AgðUUpÞ
spðUUpÞfaxpup
!
¼0:
(A14)
This equation has to be solved with deterministic or stochastic ini-
tial conditions defined by (7) or (6) and also be marginalized
according to
fxpupðXp;Up;tÞ¼ð1
1
faxpupðA;Xp;Up;tÞdA;(A15)
fxpðXp;tÞ¼ð1
1
fxpupðXp;Up;tÞdUp;(A16)
fupðUp;tÞ¼ð1
1
fxpupðXp;Up;tÞdXp:(A17)
We also define here the n
th
moment about cof a continuum ran-
dom variable xwith PDF fxðxÞas
ln¼ð1
1
xc
ðÞ
nfxðxÞdx;(A18)
where if n¼1 and c¼0, we obtain the mean l
x
, and if cis selected
to be the mean, we find the n
th
central moment of f
x
.
APPENDIX B: DETERMINISTIC ANALYTICAL
SOLUTION OF THE TEST CASES
1. Uniform flow
For the uniform flow where uis constant, the system of Eq. (1)
can be expressed as a first order ODE system with constant coeffi-
cients for xpðtÞand upðtÞ
d
dt
xp
up

¼01
0b

xp
up

þ0
bu

;(B1)
a second order ODE for xpðtÞ
d2xp
dt2þbdxp
dt ¼bu;(B2)
a first order ODE for upðtÞ
dup
dt þbup¼bu;(B3)
or as a differential equation of separable variables for the phase
space upðxpÞas the system (1) becomes autonomous
dup
dxp¼buup
ðÞ
up
:(B4)
The analytical solution of xpðtÞand upðtÞis trivially obtained solv-
ing any of the above options as
xpðtÞ¼xp0þut þ1
bðuup0Þebt 1
ðÞ
;(B5)
upðtÞ¼uþðup0uÞebt:(B6)
The time can be removed combining the last two equations to find
the solution in the particle phase as upðxpÞ.
2. Stagnation flow
For the centerline of the stagnation flow y¼0, the carrier flow
is u¼kx ¼kxpafter interpolating at the particle location
according to Hiemenz solution.
50
The system (1) can be described
as a first ODE system of constant coefficients
d
dt
xp
up

¼01
kb b

xp
up

;(B7)
a second order ODE for xpðtÞ
d2xp
dt2þbdxp
dt þkbxp¼0;(B8)
or an integrable equation for the particle phase upðxpÞ
dup
dxp¼bkx
pþup

up
:(B9)
Writing the system (B7) as z0ðtÞ¼BzðtÞ, with the initial condition
zð0Þ¼z0, where
z0ðtÞ¼ xpðtÞ
upðtÞ

;z0¼xp0
up0

;B¼01
bk b

;(B10)
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the analytical solution is given by
zðtÞ¼eBt z0:(B11)
Using the eigen decomposition B¼SKS1, the exponential matrix
can be obtained as
eBt ¼SeKtS1;(B12)
with
S¼1þffiffiffiffiffiffiffiffiffiffiffiffiffi
b4k
p
2kffiffiffi
b
p1ffiffiffiffiffiffiffiffiffiffiffiffiffi
b4k
p
2kffiffiffi
b
p
11
2
43
5;K¼k10
0k2

;
(B13)
and
k1¼bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bb4k
ðÞ
p2;k2¼bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
bb4k
ðÞ
p2:(B14)
Finally, the exponential matrix is
eKt¼ek1t0
0ek2t

;(B15)
and the analytical solution can be expressed as
xpðtÞ¼ebt
2bxp0þ2up0
csinh tc
2

þxp0cosh tc
2

;(B16)
and
upðtÞ¼1
2cetcþb
ðÞ
2up0cectþ1
ðÞ
be
ct1
ðÞ
ð2kxp0þup0Þ
hi
;
(B17)
where c¼ffiffiffi
b
pffiffiffiffiffiffiffiffiffiffiffiffiffi
b4k
p.
Just for completion, in the case b¼4k, the matrix Sis singular,
and the Jordan decomposition B¼MJM1is required
52
to find the
exponential matrix defined as eBt ¼MeJt M1with
M¼1=ð2kÞ1=ð4k2Þ
10

;J¼2k1
02k

;(B18)
so that one has
eJt ¼e2kt e2ktt
0e2kt

;(B19)
and the analytical solution changes to
xpðtÞ¼e2kt xp0þt2kxp0þup0

;(B20)
upðtÞ¼e2kt up02kt 2kxp0þup0

:(B21)
APPENDIX C: MOMENT RESULTS FOR THE SFF
CASE
FIG. 20. Computations for the SFF case with MC in color lines and with the MoM in
black dots of the (a) second and (b) third central moments for the particle position
and velocity and the three distributions considered for a(see Fig. 6) and dIC. The
case of sIC for the uniform distribution is included. The legend in (b) is valid for (a)
as well.
FIG. 21. Terms in (13) vs time tfor the SFF with dIC.
Physics of Fluids ARTICLE scitation.org/journal/phf
Phys. Fluids 33, 033326 (2021); doi: 10.1063/5.0039787 33, 033326-18
Published under license by AIP Publishing

DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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