12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

RESOLVENT ANALYSIS OF TURBULENT PIPE FLOW LADEN WITH LOW
INERTIA PARTICLES

Rasmus Korslund Schlander
Department of Aeronautics
Imperial College London
London SW7 2AZ, United Kingdom
r.schlander19@imperial.ac.uk

George Papadakis
Department of Aeronautics
Imperial College London
London SW7 2AZ, United Kingdom
g.papadakis@imperial.ac.uk

Stelios Rigopoulos
Department of Mechanical Engineering
Imperial College London
London SW7 2AZ, United Kingdom
s.rigopoulos@imperial.ac.uk

ABSTRACT

tension of the resolvent analysis framework to particle transport which is modelled using an Eulerian description that is
valid for particles of low inertia. This allows us to analyse theinteraction between coherent structures and inertial particles,
and can potentially allow us to derive low rank descriptions ofthese complex interactions.

We extend the resolvent framework to turbulent flows
laden with low inertia particles. The particle velocities aremodelled using the equilibrium Eulerian model, which is valid
for Stokes numbers up to 1. We analyse a vertical turbulent
pipe flow with a Reynolds number of 5300 based on diameter and bulk velocity, for Froude numbers Fr = 0.4, −0.4 andStokes numbers St + = 0 − 1. A direct numerical simulation(DNS) for a pipe with a length of 7.5 diameters (D) is performed with the particles released uniformly at the pipe inlet.
The resolvent formulation can predict some of the physicalphenomena observed in inertial particle flows such as localized high concentration due to the vortical centrifuge effect,turbophoresis and gravitational effects. It is shown that the upward flow increases particle concentration in the log layer ofthe pipe. The downward flow increases concentration near thecentre of the pipe: both features have been observed in previous Lagrangian simulations as well as experiments. The main
effect of Stokes number is the amplification of smaller streamwise wavelengths, therefore, increasing local scale clusteringof particles. The effect of the direction of gravity was also
reproduced using a simplified resolvent model which did not
require a mean concentration profile as input and simplifies theanalysis since no prior simulation or experiment is required forthe model to work.

METHODOLOGY
We consider the turbulent flow inside a circular pipe witha particles injected at the inlet. The flow is assumed to be incompressible and the continuity, momentum and particle transport equations are written in Cartesian tensor notation as:

INTRODUCTION
The interaction between inertial particles and coherent
flow structures plays an important role in particle deposition inwall-bounded turbulent flows. Understanding this interaction
can lead to better control over mixing and deposition in pipes
and channels. One of the methods used for analysing coherent
structures in flows is resolvent analysis. This is an equationdriven approach, where the non-linear terms are treated as random forcing to the linear system. The resolvent operator thenmaps these inputs (forcings) to the outputs (responses). Recently, this method has been extended to scalars (Dawson et al.
(2018)) and geophysical flows using the Boussinesq approximation (Ahmed et al. (2021)). This paper presents another ex-

∂ ui
= 0,
∂ xi

(1)

∂ ui ∂ ui u j∂p
1 ∂ 2 ui
+
=−
+
,
∂t
∂xj
∂ xi Re ∂ x j ∂ x j

(2)

∂c
∂c
1
∂ 2c
+vj
=
,
∂t
∂xj
ReSc ∂ x j ∂ x j(3)

where ui , vi are the instantaneous fluid and particle velocities
in the i-th direction respectively, p is the pressure, t is the time,and c is the particle concentration. The notation (x1 , x2 , x3 ) for
the spatial coordinates is used interchangeably with the notation (x, y, z), where z = x3 is the axial, i.e. streamwise, direction. Due to rotational symmetry, it is also convenient to use
polar coordinates (r, θ , z), where (x, y) = (r cos θ , r sin θ ). Allvariables are non-dimensionalised with the reference quantities being the bulk velocity UB , the pipe diameter D, and theinlet concentration cI . The Reynolds number is defined as
Re = UνB D , where ν is the kinematic viscosity, and the Schmidt

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
number as Sc = αν , where α is the particle diffusivity. The
mean (i.e. time-average) and fluctuating quantities are denoted
using an overbar and a prime, for example c and c′ , respectively.
The particle velocity, vi , is estimated by assuming that thematerial derivative of the particle is approximately equal to thematerial derivative of the fluid and is known as the Eulerian
method, Ferry & Balachandar (2001):
vi = ui − τ

Dui
− gi
Dt

A database of mean velocity and concentration profiles are needed for the Resolvent analysis and 20 different flow configurations were therefore simulated for St + =0.1, 0.2, 0.5, 1.0 and Fr = −4, −0.4, 0, 4, −4 as well as the special case where gravity is omitted (Fr → ∞) for a pipe flow
with Reτ = 180. The mean and RMS plots are, once again,
omitted here for brevity.
An in-depth introduction to the resolvent framework can
be found among others in McKeon & Sharma (2010); McKeon(2017). In this paper, we extend the resolvent analysis further
to understand the behaviour of low inertia particles in turbulent pipe flows. We analyse a fully developed turbulent pipe
flow that transports low-inertia particles. The flow is assumed
homogeneous in the streamwise direction and stationary intime, which means the instantaneous velocity u = [uz , ur , uθ ],pressure and particle concentration fields, defined together as
q = [u, p, c], can be expressed as a superposition of Fouriermodes with streamwise wavenumber kz , azimuthal wavenumber kθ (constrained to be an integer), and frequency ω,(4)where τ is the particle relaxation time,ρp + 12 ρf D2p
τ=
18µf

(5)

and ρ p is the particle density, ρ f is the fluid density, D p is theparticle diameter, µ f is the dynamic viscocity of the fluid andD
Dt is the material derivative. By inserting this into equation 3q′ (r, θ , z,t) = ∑

Z ∞ Z ∞  û (r; kz , kθ , ω)  p̂ (r; kz , k , ω) ei(kz z+kθ θ −ωt) dkz dωθĉ (r; kz , kθ , ω)(7)
ˆ denotes a Fourier transformed variable. This dewhere ()composition leads to each wavenumber-frequency combination k = (kz , kθ , cw = ω/kz ) and represents a helical wave moving downstream with wave speed, cw , for k = (kz , kθ , ω) ̸=(0, 0, 0). Furthermore, wavelengths are defined as λz = Lz π/kz
and λθ = Dπ/kθ . Finally, the critical layer is defined as where
the wave speed, cw , is equal to the mean streamwise velocity, i.e., cw = u(yc ). Using this Fourier representation, at eachwavenumber-frequency combination, k, the convection diffusion equation can be expressed in dimensionless form as:

U2

B
B
and St = τU
and non-dimensionalizing using Fri = gi D
D , weobtain:

kθ




∂uj
∂uj
∂c
1
∂c
1
∂ 2c
+ u j + St
−
− uk
=
,
∂t
Fri
∂t
∂ xk
∂ x j ReSc ∂ x j ∂ x j(6)This model for particle flow assumes that the particle concentration is small, which means there will be no collisions, andthat all particles within a cell move in the same direction, i.e.
no particle crossings.
We here note that 3 different Stokes numbers are usedthroughout this work; St = τUB /D based on bulk values,
St + = τu2τ /ν based on wall values, where uτ is the friction
 2Dvelocity and Stk = τ/τk = (2ρ p + 1)/36 ηp is based on the
Kolmogorov time and length scale, τk and η, respectively. In
the present simulation, St and St + are found to be maximum
0.04 and 1.0, respectively. Stk depends on the Kolmogorov
length scale, which changes with the radial direction in thepipe and it is found to be maximum 1.12.
We validate the velocity field obtained using the equilibrium Eulerian method, to the Lagrangian simulations ofa turbulent channel flow with Reτ = 180 from Ferry et al.
(2003) and a pipe flow simulation with Reτ = 323 comparedto the experiment of Oliveira et al. (2017) and the DNS of
Veenman (2004). The channel flow is simulated using Nc =9.3 × 106 cells, with (Nx , Ny , Nz ) = (256, 128, 256) and dimensions, (Lx , Ly, Lz ) = (4π, 2, 4/3π). We can observe in figure 1athat for St + = 1 the mean radial particle velocity matches the
Lagrangian simulation well, and reasonably well for St + = 3.
However, for the difference in average streamwise fluid andparticle velocity in figure 1b, we start to observe some differences near the peak for St + = 1 as well as a significant deviation in most of the pipe for St + = 3. This means that themodel is deemed accurate for Stokes numbers up to St + = 1.
The pipe flow simulation has Nc = 2.6 × 107 cells on a gridthat near the centre is H-type and transitions to O-type closer+
to the wall to fit the cylindrical boundary, with ∆rwall= 0.1,+++∆θ rmax = 4.84, ∆rcenter,max = 2.63 and ∆zmax = 8.2. The meanof the mean particle field, plotted in figure 1c, show an excellent agreement for St + = 1.47. Mean and RMS values of thefluid velocity field have also been validated for both simulations, but are omitted here for brevity.

−∞ −∞

′
′ ∂c
(−iω
 + ikz uz )c + ur ∂ r +
St iωu′r ∂∂ cr − 2ikz u′r ∂∂urz c − ikz u′r uz ∂∂ cr +

(8)

St
′
−1 −1 2 ′Fr ikz c − Re Sc ∇ c − Θs = f cwhere a source term, Θs , is added to ensure streamwise homogeneity. Equation (8) and the corresponding equations for theflow are re-arranged to obtain a forcing–response relationship:q̃ = Hk fk(9)where fk = [ fu , 0, fc ] is the velocity and scalar forcing and
Hk is the resolvent operator. The resolvent operator is discretized in the radial direction using a Chebyshev pseudospectral method.
We chose the following energy norm, also previously previously been analysed in Dawson et al. (2018),
E=

1
2

Z 


αu2 + β c2 dV

(10)

Ω

where α is the weight of the velocity field, β is the weightof the particles in the norm. For β = 0, we obtain the kinetic
energy norm. Here it must be noted, that by setting α = 0, we
simplify the resolvent operator to the equation:∆St
c ikz uz − iω −+ ikz
− Θs,c = fc
ReSc
Fr
′

2

(11)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
two wave speeds, cw = uz (y+ = 15) figure 3 (top row), andcw = uz (y+ = 150) figure 3 (bottom row). Figure 3a,d show
the case of no gravity, figure 3b,e show the case of a downward flow and figure 3c,e show an upward flow. Here we observe that by increasing St/Fr, We increase the low-rank region for upward flow in the near wall region, and reduce it forthe downward flow as seen in the top row. Similarly, we increase the low-rank region in further from the wall for a downward flow and reduce it for the upward flow as shown in bottom row. Similar observations for the mean and rms particle
concentrations were also made in the Lagrangian simulationsof Nilsen et al. (2013) and Capecelatro & Desjardins (2015).

which is independent of c. By performing the SVD on the
resolvent operator, Hs , we obtain the following decomposition∞∗

Hs (k) = ∑ σ j (k)ψ̂ j (r; k)φ̂ j (r; k)

(12)

j=1

where φ is the forcing mode, ψ is the response mode and σ isthe gain related to each response mode.

RESULTSThe flow structures are now analysed using the resolvent
framework as shown in figures 4 for the full system and 5 forthe simplified equation modes. The most energetic scales forthe various Stokes and Froude numbers under consideration
for the different wall-normal heights still coincide somewhat
with the passive scalar case, even though the inertial and gravitational effects changes which exact wavelengths causes thesystem to become low rank. However, to simplify the analysis, it is necessary to limit the analysis to only a few modes.
We therefore analyse the mode kz = π/2, kθ = 4 since it is low
rank at the different wavespeeds, cw (y+ = 15), cw = uz (y+ =30), cw = uz (y+ = 120), as well as the special case of the locally clustered mode, which is shown at kz = 10π, kθ = 4,
cw = uz (y+ = 150) which was shown to increase in gain withincreasing St + . We plot σ1 |φ1,c | to show both the shape of themode, but also its relative importance.

The resolvent gain, σ , provides information about how
the system reacts to a forcing. The larger the gain, the moresensitive the system is to the corresponding forcing (or input).
The system is said to be low rank if a significant part of thetotal gain is captured by the first mode. Individual weights
for the modes are needed for describing the full forcing andresponse system, however, past studies such as McKeon &Sharma (2010), Sharma & McKeon (2013) have shown thatbroadband forcing can be used to describe important featuresin the flow, which can also be observed in experiments. We are
therefore checking if the operator is low rank i.e.,∗Hs (k) ≈ σ1 ψ̂ 1 φ̂ 1 .

(13)

Low-rank behaviour typically indicates that there is a dynamically significant spatio-temporal structure defined by k. We
study how long rank behaviour changes with St + by plotting
the gain of the first resolvent mode relative to the gain fromall modes in the model for streamwise and azimuthal wavelengths, and this is shown in figure 2. It can be observed thatthe leading response mode contains more than 80 % of the total
response over a large range of wavelengths, and for λz > 1 andλθ > 0.1 what indicates that the operator is low rank. First,
we investigate the effect of the choice of norm, with α = 0or α = 1 and β = 0 or β = 1, on the relative gain of the firstmode for various streamwise and azimuthal wavenumbers witha phase speed, cw = uz (y+ = 15) for St + = 0.1 − 1 and gi = 0
in figure 2. When α = 1, β = 0, shown in figure 2 (left column), the gain plots become independent of St and Fr which
is obvious since only the Navier-Stokes equations are considered. The gain plot then becomes identical to similar plots infor instance McKeon & Sharma (2010) or Luhar et al. (2014b).
Similarly, when α = 0, β = 1, shown in figure 2 (middle col1
= 0, which showsumn) the gain plot is also constant when Fr
that equation 11 in this case is independent of Stokes number as well. It is important to note that in equation 11 only
the concentration fluctuations are taken into consideration, so
there are no interactions with the fluid velocity. Finally, by
setting α = 1, β = 1 (right column), the two previous gainplots combine. Here, it seems like the low rank behaviour isswitched from the λθ = 2π (kθ = 1) to λθ = 2π/3 (kθ = 3)
for the largest streamwise modes (λz > 200, kz < 1). It seems
the main effect of increasing the Stokes number is to increase
the low order nature of the smallest streamwise wavenumbers.
This could indicate the existence of local clustering, observedin among others Marchioli et al. (2007). This could also help
explain the existence of the long narrow clusters of particles
observed near the wall, which are often much longer and narrower than velocity streaks, (Sardina et al. (2012); Brandt &Coletti (2022)).
Next, the effect of Fr is tested, by looking at the gainfrom equation 11 which is the simplified system in figure 3 at

In figure 4, the most energetic mode for at the different
wave speeds is plotted for no-gravity (top row), downward(middle row) and upward flow (bottow row). The main effect of increasing the Stokes number is shown in figure 4a-d
and seems to be to amplify the locally clustered modes, further away from the wall. This is primarily done for smaller
more narrow shapes. This is caused by particles moving to regions of lower turbulence intensity, which has also been observed in for instance Sardina et al. (2012). The effect of
downward flows, shown in figure 4e-h, show that the concentration peak of the modes moves somewhat closer to the wall
for the smaller modes. Furthermore, the concentration fluctuations seems to increase significantly in the bulk region of theflow. This is also observed in several Lagrangian simulations,of Nilsen et al. (2013) and Capecelatro & Desjardins (2015).
In Nilsen et al. (2013), the increase in the bulk region was explained with downward gravity increasing the average particle
velocity and therefore aligning the particles more with fastermoving bulk flow structures, whereas the opposite is the casefor upward flow. The effect of the upward flow, figure 4i-l,
seems to move the peak somewhat away from the wall, but toincrease them in the log region their amplitude when comparedto the downward or no-gravity flow. This was also shown inNilsen et al. (2013), where the concentration in the logarithmic
region increased for upward flows. In both the upward anddownward flow, we see that the gravitational effects seem todominate the inertial effects, since figure 4h,l no longer show
the effect of inertia. This would change if the Stokes number
was increased, or the gravitational effect was reduced.
Finally, the flow is analysed using the simplified resolvent
operator described in equation 11 in figure 5 for the same values as in 4. As in figure 5, σ |φ1,c | is plotted in order to show
both the shape of the mode, but also the relative magnitude.
In figure 5 we can see that inertial effects no longer plays arole, since the no gravity case of figure 5a-d no longer changes
with Stokes number. However, gravitational effects still seemto be visible. For a downward flow, figure 5e-h, it can still3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
REFERENCES

be observed that the modes are mainly amplified closer to thecentre, whereas, for the upward flow, figure 5i-l, the modes areamplified in the log layer of the wall.

Ahmed, M. A., Bae, H. J., Thompson, A. F. & McKeon, B. J.
2021 Resolvent analysis of stratification effects on wallbounded shear flows. Physical Review Fluids 6 (8), 1–25.
Brandt, Luca & Coletti, Filippo 2022 Particle-Laden Turbulence: Progress and Perspectives. Annual Review of Fluid
Mechanics 54 (1), 159–189.
Capecelatro, Jesse & Desjardins, Olivier 2015 Mass Loading
Effects on Turbulence Modulation by Particle Clustering in
Dilute and Moderately Dilute Channel Flows. Journal of
Fluids Engineering, Transactions of the ASME 137 (11).
Dawson, Scott T.M., McKeon, Beverley J. & Saxton-Fox,
Theresa 2018 Modeling passive scalar dynamics in wallbounded turbulence using resolvent analysis. 2018 AIAA
Fluid Dynamics Conference p. 4042.
Ferry, J. & Balachandar, S. 2001 A fast Eulerian method fordisperse two-phase flow. International Journal of Multiphase Flow 27 (7), 1199–1226.
Ferry, Jim, Rani, Sarma L. & Balachandar, S. 2003 A locally
implicit improvement of the equilibrium Eulerian method.
International Journal of Multiphase Flow 29 (6), 869–891.
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014a On the structure and origin of pressure fluctuations in wall turbulence:
Predictions based on the resolvent analysis. Journal of Fluid
Mechanics 751, 38–70.
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014b Opposition
control within the resolvent analysis framework. Journal of
Fluid Mechanics 749, 597–626.
Marchioli, Cristian, Picciotto, Maurizio & Soldati, Alfredo
2007 Influence of gravity and lift on particle velocity statistics and transfer rates in turbulent vertical channel flow. International Journal of Multiphase Flow 33 (3), 227–251.
McKeon, B. J. 2017 The engine behind (wall) turbulence: Perspectives on scale interactions. Journal of Fluid Mechanics
817, 1–86.
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. Journal of Fluid Mechanics
658, 336–382.
Nilsen, Christopher, Andersson, Helge I. & Zhao, Lihao 2013
A Voronoı̈ analysis of preferential concentration in a vertical channel flow. Physics of Fluids 25 (11).
Oliveira, J. L.G., Van Der Geld, C. W.M. & Kuerten, J. G.M.
2017 Concentration and velocity statistics of inertial particles in upward and downward pipe flow. Journal of Fluid
Mechanics 822, 640–663.
Sardina, G., Schlatter, P., Brandt, L., Picano, F. & Casciola, C. M. 2012 Wall accumulation and spatial localizationin particle-laden wall flows. Journal of Fluid Mechanics
699 (May 2014), 50–78.
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure inwall turbulence. Journal of Fluid Mechanics 728, 196–238.
Veenman, MPB 2004 Statistical analysis of turbulent pipe
flow: A numerical approach. PhD thesis, Eindhoven University of Technology.

Despite using only a few modes for this representation,
the overall features of the flow concentration intensities arestill captured, with an increase in the bulk region of the flow
for the downward flow, and an increase in the near-wall regionof the flow for the upward flow, and finally, a sharp increase for
locally clustered modes when the Stokes number is increased.
The gravitational effects can be found using the simplified system, which means that the mean concentration profile is notneeded. The gravitational effect of particles can therefore beanalysed for a wide range of Reynolds numbers, provided only
the velocity profile is obtained first.

CONCLUSION AND FUTURE WORKIn this paper, we investigated the transport and wall transfer in a turbulent pipe flow at Re = 5300 (based on bulk velocity) using the equilibrium Eulerian model for particle velocity and adapted the model to the resolvent framework. The
equilibrium Eulerian model was validated against other experiments and simulations and was found to correctly capture thetrends observed in no gravity, upward and downward verticalturbulent pipe flows. The resolvent methodology presented has
allowed us to analyse which modes become amplified depending on their Stokes and Froude number. It was found that theresolvent model was able to capture the characteristics of inertial particles in wall-bounded effects. The model predicted theoccurrence of smaller regions with a high concentration of particles, that the bulk region of the flow gets amplified for downward flows and that the near-wall region gets amplified for upward flows. The trends solely related to the direction of gravity
could be observed using a simplified resolvent model, that did
not require concentration profiles. This means that key insightcan be provided into these types of flows, in a few seconds,
using a standard laptop. Finally, the model also showed how
gravity affects particle distribution. It was shown that modesin the log region become amplified in upward flow and that
downward flow amplifies modes in the centre, but also moves
some modes closer to the wall.
Future work will explore the potential of the methodology for gaining insight into more complex settings that may
include chemical reactions and different flow configurations.
The current configuration of the resolvent model was closed
using mean velocity and concentration profiles. The model
could become more sophisticated by modelling the forcing
term of the resolvent operator which would make it possible
to obtain accurate flow statistics such as rms values. Other
future work includes using the resolvent operator for flow control to modify deposition of inertial particles, as was done fora velocity only analysis of a turbulent pipe flow by Luhar et al.
(2014a).

ACKNOWLEDGEMENTS
We would like to thank the UK Turbulence Consortiumand the UK Materials and Molecular Modelling Hub for computational time at ARCHER and THOMAS via EPSRC Grant
No. EP/R029326/1 and (EP/P020194/1), and the Leverhulme
Trust (RPG-2018-101) for financial support.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
1.4

0.02
St + =1 Present DNS

0.15

0.01

St + =1 Ferry et al. 2003
St + =3 Present DNS00.1

St + =3 Ferry et al. 2003

1.2
1
0.8

-0.01
0.05

-0.02

0.6

-0.03

0.4

+
St =1.56

0-0.04

St + =0.1
Experiments of Oliveira et al. (2017)

0.2

-0.05

-0.05
0

50

100

150

0

50

(a)

100

150

0
0

0.2

(b)

0.4

0.6

0.8

1

(c)

Figure 1: Wall-normal particle velocity (a) and difference between streamwise average particle velocity and fluid velocity(b) of a channel flow simulation with Reτ = 180 compared to the Lagrangian simulation from Ferry et al. (2003), where
solid lines are the present simulations and (o) are the Lagrangian simulation, blue line color is St + = 1 and red line color is
St + = 3. Velocity profiles for particles with St + = 0.1 and St + = 1.56 for a downward flow with ReB = 10300, comparedwith an experiment by Oliveira et al. (2017) (c).
1

100
0.5

10-1
0
1

100
0.5
-1

10

100

102

100

102

100

102

0

Figure 2: Contour plots of the energy contained in the leading response mode relative to all modes, σ12 / ∑n=1,Nr σn2 for

cw = u(y+ = 15), where the resolvent is found using a norm with kinetic energy, α = 1, β = 0, (left column) the simplified
resolvent operator from equation 11, α = 0, β = 1, (middle column), and kinetic energy and particles, α = 1, β = 1, (right1
column), where (top row) is for St + = 0.1 and (bottom row) is for St + = 1.0 and Fr=0.

Figure 3: Contour plots of the energy contained in the leading response mode relative to all modes, σ12 / ∑n=1,Nr σn2 for
cw = uz (y+ = 15) (top row) and cw = uz (y+ = 150) (bottom row), for the simplified equation 11, i.e α = 0, β = 1. (leftcolumn) 1/Fr = 0 (i.e. zero gravity), St + = 1, (middle column) Fr = 0.4, (right column) Fr = −0.4.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

400

1000

St=0
St=0.2
St=0.5
St=0.75
St=1.0

200

500
20

500

0

0

400

1000

200

500

0

0

500

1000

10
0

0
20

500
10
0

0

500

20
10

500
0

0

0.5

1

0

0

0.5

0

1

0

0.5

1

0

0

0.5

1

Figure 4: Mode shapes for concentration fluctuations for no gravity (a-d), downward (e-h) and upward flow (i-l), where(a,e,i) is for cw = u(y+ = 15), (b,f,j) cw = u(y+ = 30), (c,g,k) cw = u(y+ = 150) and (d,h,l) is for local clustering. The
black vertical line indicate the location of the critical layer of the velocity field, and the dashed-dotted vertical line is thelocation of the critical layer for the concentration profile.

100

100

20

50

50

10

0

0

0

0

50

100

50

St=0
St=0.2
St=0.5
St=0.75
St=1.0

0

50

0

50

100

10

0

0

0

200

100

100

0

0.5

1

20

200

0

20

50

0

0.5

0

1

10
0

0.5

1

0

0

0.5

1

Figure 5: Mode shapes for concentration fluctuations of the simplified resolvent mode described in equation 11 for no

gravity (a-d), downward (e-h) and upward flow (i-l), where (a,e,i) is for cw = u(y+ = 15), (b,f,j) cw = u(y+ = 30), (c,g,k)
cw = u(y+ = 150) and (d,h,l) is for local clustering. The black vertical line indicate the location of the critical layer of thevelocity field, and the dashed-dotted vertical line is the location of the critical layer for the concentration profile.

6