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

EFFECTS OF INITIAL CONDITIONS ON
MIXING IN WALL-BOUNDED TURBULENT FLOW

Milind Singh
Department of Mechanical Engineering
McGill University
Montréal, Canada
milind.singh2@mail.mcgill.ca

Emmanuel Germaine
Department of Mechanical Engineering
McGill University
Montréal, Canada
emmanuel.germaine@mail.mcgill.ca

Laurent Mydlarski
Department of Mechanical Engineering
McGill University
Montréal, Canada
laurent.mydlarski@mcgill.ca

Luca Cortelezzi
Department of Aerospace Science and Technology
Politecnico di Milano
Milano, Italyluca.cortelezzi@polimi.it

ABSTRACT

where θ̃ represents the instantaneous scalar field, ũ j , j = 1, 2, 3
are the components of the instantaneous velocity vector field,
α is the scalar’s molecular diffusivity within the fluid, and repeated indices imply Einstein’s summation convention.
The present work focusses on the mixing of scalars within
a fully-developed, high-aspect-ratio turbulent channel flow, tomodel the behavior of real flows and the physical processes
therein (as opposed to studying mixing within homogeneousflows). Channel flow is especially appropriate because it is thesimplest realization of an inhomogeneous turbulent flow.
Mixing metrics are often used to quantify the efficacy ofa scalar mixing process. These include i) the unmixedness parameter (Danckwerts, 1952; Dimotakis & Miller, 1990):

Scalar mixing occurs in a wide variety of environmental
and engineering flows. Given that the majority of these flows
are turbulent, a more complete understanding of the mechanisms that underlie the scalar mixing process is required to
predict and control these flows and the evolution of the concentration(s) of the scalar(s) therein. Direct numerical simulations (DNSs) are particularly useful in the study of turbulencesince DNSs resolve the full range of (length and time) scalesin turbulent flows. The present work employs a spectral approach for simulating the hydrodynamic field, and a flux integral method for computing the advection and diffusion of
the passive scalar. Mixing of the latter is studied in a fully
developed turbulent channel flow, and particular attention ispaid to the evolution of the scalar dissipation rate and the differences that arise from different scalar-field initial conditions.
Observed differences in the evolutions for the three initial conditions investigated are explained by an analysis of the different terms that comprise the scalar disspation rate budget. To
promote mixing in a wall-bounded flow, it is recommended
that scalar interfaces be aligned normal to the mean velocity
vector, to enhance the stretching of the interface.

Ξ≡

(2)

where Θ is the average scalar concentration, θ is its fluctuation, and angular brackets denote averages, or ii) a multi-scalenorm, such as the mix-norm (Mathew et al., 2005):
"
#1/2
1
2
µθ = ∑ p|Θ̂k (t)|(3)
1 + 4π 2 ||k2 ||k∈Z 3
where k is the wavevector, and Θ̂k are the Fourier coefficientsof a spectral representation of the scalar field. Note that the two
metrics are very different: the unmixedness quantifies molecular mixing only, whereas the mix-norm quantifies mixing byaccounting for both stirring and molecular mixing. These differences have a notable impact on the results using these metrics. For example, consider the idealized case of mixing in
two dimensions of two hypothetical fluids (one white and one
black, and both having molecular diffusivities equal to zero).
Let one then assume that the fluid could be stirred in such a
way that the end result would be a configuration similar toa chessboard, with white and black squares at the smallest
scales. In this case, the unmixedness would be maximum (because there is no molecular mixing) whereas the mix-norm will
be minimum, because the stirring is complete.
Motivation for the present study stems from observations
by Gubanov & Cortelezzi (2009) and Germaine et al. (2018),

INTRODUCTION
Scalar quantities (e.g. temperature, humidity, chemical
species concentration) are mixed within a wide range of environmental and engineering flows (e.g. the atmosphere, whichtransports temperature and humidity; the oceanic mixed layer,
in which temperature and salinity are mixed; chemical reactors; heat exchangers, combustion processes, etc.). Given thatthe vast majority of these flows are turbulent, an understanding of the turbulent scalar mixing process is required to predict (and, ultimately, control) these flows and the evolution of
the concentration(s) of the scalar(s) therein. Thus, the present
work investigates the turbulent mixing of (passive) scalars,
which are governed by the advection-diffusion equation:∂ θ̃∂ 2 θ̃∂ θ̃+ ũ j=α∂t∂xj∂ x j∂ x j⟨θ 2 ⟩; 0≤Ξ≤1Θ(1 − Θ)(1)1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) ICx(b) ICy(c) ICz

Figure 1: The three scalar-field initial conditions studied herein. Imagery generated using VAPOR (Li et al., 2019).

p
256 × 193 × 192. (Reτ ≡ uτ h/ν, uτ ≡ ν × ∂U/∂ y|y=0 , andν is the kinematic viscosity of the fluid.) The resulting turbulent velocity field is statistically steady and statistically onedimensional, such that ⟨U⟩ = ⟨U⟩(y) only.
The pre-computed velocity field is used as an input to theadvection-diffusion solver, and is interpolated from a spectral
to a finite-volume representation via spectral (exact) interpolation. A very small correction is applied to the streamwise component of the velocity field to ensure the exact divergence-free
condition. The advection-diffusion equation is solved using afully three-dimensional and cost-effective flux integral method
entitled 3DFLUX (Germaine et al., 2013). The fully explicit
and multidimensional nature of the approach ensures that it
is free of splitting errors, and provides a better convergence
rate of the numerical errors when compared to commonly used
one-dimensional methods. A semi-Lagrangian approach is implemented in 3DFLUX by discretizing the spatial domain on
an Eulerian grid and using a Lagrangian frame of reference fortemporal discretization. Solutions for the advection-diffusion
equation are obtained using non-overlapping control volumes
to discretize the computational domain and estimate the flux
exchanged between adjacent cells. DNS of the scalar field isundertaken on the same domain of 2πh × 2h × πh, with a resolution of 514 × 195 × 194 to resolve both the large and small
scales of the scalar field. It is assumed that the magnitudesof the scalar fluctuations are sufficiently small, such that the
dynamics of the velocity field are independent from the scalar
field and the scalar is considered as being passive.
As noted above, the evolution of the mixing with time
will be studied for three different scalar-field initial conditions,
ICx , ICy and ICz , as depicted in figure 1. These fields consist of two equal volumes of uniform concentrations, θ̃ = +1
(black) and −1 (white), separated by two interfaces (due tothe periodic boundary conditions in x and z) of total area
4πh2 (= 2 × 2h × πh) for ICx and 8πh2 (= 2 × 2h × 2πh) for
ICz and a single interface of area 2π 2 h2 (= 2πh × πh) for ICy.
All fields have zero mean concentration. Note that all threescalar fields are subjected to identical turbulent velocity fields,
and thus any differences in the evolutions of the three scalar
fields are the sole result of the orientations of the interfaces
defining the initial scalar fields relative to that of the velocity
field.

who have observed that scalar fields, including the evolution of
the scalar dissipation rate, are strongly affected by the initialconditions of the scalar field, even when these initial conditions are mixed by identical hydrodynamic fields. It is therefore of interest to investigate the nature of the physical processes that lead to such different evolutions of the scalar field.
The objective of the present work is therefore to investigate the effect of scalar-field initial conditions on mixing that
occurs within fully developed, turbulent channel flow. This
will be undertaken by way of an initial comparison between
the evolutions of the two aforementioned mixing metrics andthe scalar dissipation rate in turbulent flows. However, theprimary focus of the present work will be an investigation of
the evolution of the scalar fields for the different initial conditions, by way of an analysis of the various terms in the evolution equation (i.e. budget) of the scalar dissipation rate:(εθ ≡ α ∂∂xθ ∂∂xθ ):j

j∂U j ∂ θ ∂ θ∂ ⟨εθ ⟩∂ ⟨εθ ⟩∂Θ ∂uj ∂θ+U j= − 2α⟨⟩ − 2α⟨⟩∂t∂xj∂ xi ∂ xi ∂ x j∂ x j ∂ xi ∂ xi
∂uj ∂θ ∂θ∂ θ ∂ 2Θ⟩− 2α⟨⟩− 2α⟨u j∂ xi ∂ xi ∂ x j∂ xi ∂ xi ∂ x j∂∂ ⟨εθ ⟩+α− ⟨u j εθ ⟩∂xj∂xj
− 2α 2 ⟨∂ 2θ ∂ 2θ
⟩∂ xi ∂ x j ∂ xi ∂ x j(4)where U j is the mean velocity vector, and u j is the velocity
fluctuation.

SIMULATIONS
The investigations herein employ numerical simulationsof both the turbulent velocity and passive scalar fields, inwhich the latter is advected by the former. The turbulent velocity field is calculated via direct numerical simulation (DNS), which resolves all scales of the flow by solving
the Navier-Stokes equations with constant properties. The
DNS of the velocity field is computed using a code entitled CHANNELFLOW (Gibson et al., 2008; Gibson, 2010;
http://channelflow.org/) which uses spectral discretization in
space (Fourier × Chebyshev × Fourier), and a third-order semiimplicit backwards differentiation scheme in time, with nopenetration / no-slip boundary conditions imposed at the wallsof the channel (y = 0 and y = 2h, where h is the half-height ofthe channel), and periodic boundary conditions in the streamwise (x) and spanwise (z) directions. Simulation of the hydrodynamic field is conducted at Reτ = 190 over a domainof size 2πh × 2h × πh, in the streamwise (x), wall-normal (y)and spanwise (z) directions, respectively, with a resolution of

RESULTS
Inspired by the results of Germaine et al. (2018), whoinvestigated the dependence of local anisotropy of passive
scalars on their initial conditions, we plot in figure 2 the scalar
field distributions that result from the action of the identicalturbulent channel flow (velocity field) on scalar fields subjected to the three aforementioned, distinct initial conditions2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) ICx(b) ICy(c) ICz

Figure 2: Concentration distributions generated at τ = 7 by the action of the same turbulent channel flow on the threescalar-field initial conditions. Imagery generated using VAPOR (Li et al., 2019).

Figure 3: Time-evolution of the unmixedness parameter

Figure 4: Time-evolution of the volume-averaged (i.e.

(solid line) and the mix-norm (dashed line) for the scalar
fields resulting from the three initial conditions.

⟨·⟩ = ⟨·⟩x,y,z ) scalar dissipation rates for the scalar fields
resulting from the three initial conditions.

for a period of τ(≡ tuτ /h) = 7. From this figure, it is readily observed that the scalar field resulting from the ICx initial condition is the best mixed, whereas that resulting fromthe ICy initial condition is the least well mixed. This is already an intriguing result and indicates that the initial conditions can strongly influence the subsequent mixing. Moreover,
it is worth reiterating that all three scalar-field initial conditions were comprised of equal parts “black fluid” (i.e. scalarwith an initial concentration θ̃ = +1) and “white fluid” (i.e.
scalar with an initial concentration θ̃ = −1). Thus the rate at
which mixing will occur in such an inhomogeneous flow (as is
the flow in all real devices) is highly dependent on the initialconfiguration of the two quantities being mixed.
To quantify the mixing, we plot the time-evolutions of
the unmixedness parameter (Ξ) and the mix-norm (µθ ) for thethree scalar-field initial conditions in figure 3. One observes
that the unmixedness starts very close to its maximum value of
1, whereas the initial values for the mix-norm fall between 0.3
and 0.4 (because the latter depend on the initial geometry of
the scalar field). However, shortly after the velocity field starts
mixing the scalar field, the time-evolutions for both mixing
metrics follow a similar pattern; i.e. the evolutions of both Ξand µθ decay exponentially with time, with the most efficientstirring and mixing (i.e. fastest rates of decay of Ξ and µθ )occurring for the ICx case and the slowest for the ICy case.
Given the objective of attempting to tie the evolutions of
the mixing metrics to the underlying physics of the turbulence,
we first plot the evolutions of the (volume-averaged) scalar dis-

sipation rates (⟨εθ ⟩) corresponding to the fields with the three
initial conditions as a function of time in figure 4. Althoughsome differences exist at early times (τ < 1), the long term
evolution of ⟨εθ ⟩ is qualitatively similar to the evolutions ofboth Ξ and µθ – the dissipation rates decrease at rates dependent upon the initial conditions of the scalar field, with thethe fastest decrease in ⟨εθ ⟩ occurring for ICx , corresponding
to the most efficient mixing, and the slowest corresponding
to ICy . At shorter times, ⟨εθ ⟩ increases, presumably due to thestretching and stirring of the interface between the two (i.e. the“black” and “white”) scalar fields, which results in a production of ⟨εθ ⟩. This is in contrast with the short-time evolutionsof both Ξ and µθ , which are both monotonically decreasing
functions of time for all values of τ. Note that the values of⟨εθ ⟩ for the ICx and ICz cases peak around τ ≈ 1/3, whereas
the peak in ⟨εθ ⟩ for the ICy case occurs later, around τ ≈ 2/3.
Before proceeding, we are obliged to discuss the issue of
averaging when presenting the results for the individual terms
in eq. (4). In particular, the homogeneity (or lack thereof) of
both the velocity and scalar fields in this work must be considered in the analysis of the results. Whereas the velocity
field is statistically homogeneous in the x- and z-directions, aswell as being statistically stationary, the natures of the scalar
fields resulting from the action of the velocity field on the threedifferent initial conditions are all different. The scalar field resulting from the ICx initial condition is statistically homogeneous in the z-direction (only). That resulting from the ICy
initial condition is statistically homogeneous in the x- and zdirections. Lastly, the scalar field resulting from the ICz ini-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) ICx(b) ICy(c) ICz

Figure 5: Evolution in time of the space- and time-averages of all 9 terms in the scalar dissipation rate budget in which

statistical moments (⟨·⟩) are assessed in planes parallel to the initial location of the interface in the initial condition. (a)R R
1 1 R 2h R τ
1 1 R Lz R τ′′
ICx with L1x τ1 0Lx 0τ Term dτ ′ dx, (b) ICy with 2h
0 Term dτ dy, and (c) ICz with Lz τ 0 0 Term dτ dz.
τ 0Table 1: Legend for the various terms in the budget of the scalar dissipation rate.

Color

Term #

Term

(i)

∂
∂t ⟨εθ ⟩⟨U j ⟩ ∂∂x j ⟨εθ ⟩∂ ⟨U ⟩
2α ∂ xij ⟨ ∂∂ xθi ∂∂xθj ⟩∂uj ∂θ
2α ∂∂⟨Θ⟩x j ⟨ ∂ xi ∂ xi ⟩2∂θ2α ∂∂xi⟨Θ⟩∂ x j ⟨u j ∂ xi ⟩∂u2α⟨ ∂ xij ∂∂ xθi ∂∂xθj ⟩

(ii)
(iii)
(iv)
(v)
(vi)

Physical interpretation
time rate of change of ⟨εθ ⟩mean flow advection of ⟨εθ ⟩production of ⟨εθ ⟩ by mean velocity gradients
production of ⟨εθ ⟩ by mean scalar gradientsmixed production of ⟨εθ ⟩production of ⟨εθ ⟩ by turbulent vortex stretching

(vii)

2
−α ∂ x∂j ∂ x j ⟨εθ ⟩molecular transport of ⟨εθ ⟩(viii)∂∂ x j ⟨u j εθ ⟩turbulent advection of ⟨εθ ⟩∂ 2θmolecular dissipation of ⟨εθ ⟩

(ix)

∂ 2θ

2α 2 ⟨ ∂ xi ∂ x j ∂ xi ∂ x j ⟩tial condition is statistically homogeneous in the x-direction(only). Moreover, all scalar-field statistics studied in this work
are non-stationary, due to the evolution in time of the scalar
fields from the unmixed to the mixed states. Thus the underlying symmetries and ensuing statistical homogeneities associated with the different initial conditions, as well as the statistically non-stationary nature of the scalar field, must be taken
into account when analyzing the results. A specific consequence is that one can no longer volume-average the terms ineq. (4) if one wants to examine the evolution of all its nineconstituent terms, because all but three of them (the first, sixthand ninth), will be zero when volume-averaged. Secondly,
there is no “universal” approach to spatial averaging that canbe employed in this analysis that will not eliminate some of
the nine terms in eq. (4) in some of the scalar fields. For
example, were one to average scalar-field statistics in the zdirection, certain terms in the ⟨εθ ⟩ budget (such as production
by the mean scalar gradient) will become zero for the ICz case,only, because ∂ ⟨·⟩z /∂ z = 0 (by definition). Lastly, given thatvolume-averaging is no longer feasible and we will therefore
employ averages taken over (spatial) planes (at specific locations within the channel) to investigate the terms in eq. (4), thestatistical convergence of the data is reduced, given that averages will be taken over less data. To address this, local time
averages will be employed (i.e. averaging the value of a term in
the budget of ⟨εθ ⟩ from τ ′ = 0 to τ ′ = τ), which will improve
convergence of the data, while retaining its time-dependence.
Moreover, figure 5 will also calculate local spatial averages

after having performed the time-average, which, when calculated in this way, does not eliminate terms and further aids
convergence.
Having discussed the issue of averaging, we plot in figure 5 the contributions of the terms in the budget of ⟨εθ ⟩(eq. (4)) to further investigate the evolutions of ⟨εθ ⟩ depicted
in figure 4. (A legend corresponding to the 9 terms of eq.
(4) is given in table 1.) Multiple observations can be madefrom the plots therein to provide insight into the evolutionsof ⟨εθ ⟩. For τ > 1, ⟨εθ ⟩ is dominated by a balance between
the production of ⟨εθ ⟩ arising from turbulent vortex stretching, and dissipation of ⟨εθ ⟩ by molecular processes, as firstpredicted by Corrsin (1953). However, further details bearnoting. Firstly, when considering the relative magnitudes ofthe various terms for the different initial conditions, it is clearthat the magnitudes of the terms for the ICx condition are thelargest, followed by those corresponding to the ICz initial condition. Those corresponding to the ICy initial condition are thesmallest. Using the peak value of the molecular dissipation of⟨εθ ⟩ (i.e. term (ix), light green line) as a reference, its value forICx is 4 × 10−5 , for ICz is 2 × 10−5 , and for ICy is 1 × 10−5 .
Thus the larger values of all terms in the scalar field arisingfrom the ICx initial condition lead to the largest peaks of ⟨εθ ⟩,as well as the fastest decay rates of ⟨εθ ⟩, consistent with figure
4. Moreover, the converse holds for ICy , which experiences
the lowest peak value of ⟨εθ ⟩ and slowest decay rates. Secondly, one can observe in figure 5 that the two dominant terms(production of ⟨εθ ⟩ by turbulent vortex stretching and molec-

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) ICx(b) ICy(c) ICz

Figure 6: Evolution in time of the time-averages of all 9 terms in the scalar dissipation rate budget in which statistical
1 Rτ
′

moments (⟨·⟩) are assessed on the y-z plane at x/Lx = 0.50, i.e. τby ⟨·⟩ = ⟨·⟩y,z .
ular dissipation of ⟨εθ ⟩) peak at later times for ICy (at τ > 1)
than they do for the ICx and ICz cases (which peak at τ < 1).
This behaviour is also consistent with the evolutions of ⟨εθ ⟩ infigure 4, noted above.
We further investigate the budget of the scalar dissipation
rate for the three initial conditions by plotting the time evolution of terms when spatially averaged over a single y-z plane infigure 6. Note that the plane over which the spatial averaging
is performed in figure 6 is parallel to the interface between the“black” and “white” fluids in the ICx initial condition, and located at the mid-plane of the channel in the x-direction (i.e.
x/Lx = 0.5), i.e. the location of one of the two interfaces
between the “black” and “white” fluids at τ = 0 for ICx . In
figure 6, it is first worth noting the qualitative similarities ofthe evolutions of the terms in the ⟨εθ ⟩ budget for the ICy and
ICz initial conditions. These two subfigures exhibit similar behaviours due to the absence of terms associated with spatial
gradients of statistical moments in the directions of the averaging in the respective figures. We also note that analogous
similarities are observed when reproducing the equivalent of
figure 6, but spatial averaging in the other two (x-z and x-y)
mid-planes (not shown). However, in contrast to figures 6(b)and 6(c), figure 6(a) is dominated at early times by mean-flow
advection of ⟨εθ ⟩ and turbulent advection of ⟨εθ ⟩. This will befurther discussed with respect to figure 7, which follows.
To investigate the evolution of the terms in the ⟨εθ ⟩ budget that are not subjected to such an artefact, we plot in figure 7
their evolution in time assessed on the mid-plane corresponding to the initial location of the interface for the three initialconditions. It reveals that figure 7(a), which corresponds to theICx initial condition, is significantly different from those corresponding to ICy (7(b)) and ICz (7(c)), which are quite similar. Whereas the dominant terms in figures 7(b) and 7(c) are
the molecular dissipation of ⟨εθ ⟩, the production of ⟨εθ ⟩ byturbulent vortex stretching, production of ⟨εθ ⟩ by mean scalar
gradients, the rate of change of ⟨εθ ⟩ with time, and turbulent
advection of ⟨εθ ⟩, figure 7(a) is distinctly different. Of particular note is the importance played by mean-flow advection of⟨εθ ⟩ at early times, which is dominant for τ < 0.2, and balanced by turbulent advection of ⟨εθ ⟩. Given that fully developed turbulent channel flow must be unidirectional, the absence of the mean-flow advection term in the other directions
is to be expected. However, the early dominance of this term
is noteworthy. In this respect, consideration of the evolution
in time of the interface(s) between the “black” and “white”
fluids is beneficial. For the ICy case, the ⟨Θ⟩ = 0 plane will
not change in time and will be advected along the y/h = 1
plane as the flow evolves, but remaining at the same location

0 Term dτ , where all averages (⟨·⟩) in Term are given

as the initial interface. However, the evolution of the scalar
field in time is quite different for ICx . The interface remains“anchored” to the walls due to the no-slip condition, and isstrongly stretched in the x-direction by the mean flow, while
being stirred by the turbulence. Thus as the interface is advected downstream, it can “blow by” a given downstream location multiple times. This effect can be observed by the oscillations in the unsteady term (dark blue line), and to a lesser
degree the mean flow advection terms (red line) in figure 7(a).
It is worth noting that the period of the observed oscillation
is indeed equal to half of the channel’s “flow-through” time(i.e. 12 τFT ≡ 12 (Lx /⟨U⟩y/h=1 )uτ /h = 0.17). With respect toICz , the ⟨Θ⟩ = 0 plane will be advected downstream the midplane of the channel (z = 0.5Lz ), although its top and bottom
edges will remain anchored to the walls of the channel, thus
emulating aspects of both ICx and ICy .
Given that the results in figure 7 are all averages taken at
the three midplanes of the channel, they are somewhat anomalous. For example, when averaging in the x- and z-directions
at y/h = 1 (as is the case in figure 7(b)), there can be no contribution to ⟨εθ ⟩ arising from production by the mean velocity
gradient because ∂ ⟨U⟩/∂ y = 0 at that one location (and noother). Thus it bears investigating the budget of the scalar dissipation rate for the three initial conditions at non-mid-planelocations. To this end, the evolutions of the terms in the budgetof the scalar dissipation rate are plotted for ICx spatially averaged over an y-z plane located at x/Lx = 0.25 in figure 8(a), for
ICy spatially averaged over an x-z plane located at y/h = 0.5
in figure 8(b), and for ICz spatially averaged over an x-y planelocated at z/Lz = 0.25 in figure 8(c). The aforementioned periodic passing of the interface over the measurement plane for
the ICx case is even more prominent in figure 8(a). However,
figures 8(b) and 8(c) also reveal another phenomenon that explains the slower evolutions of ⟨εθ ⟩ for the ICy and ICz cases.
In these two non-mid-plane cases, one observes an initial period of “inactivity.” The initial period in which the terms in thebudget of ⟨εθ ⟩ are effectively zero results from the interface
having not yet “reached” the measurement location. In thesetwo cases, the interface can only travel laterally by the actionof the turbulence, given that there is no mean velocity in they- and z-directions. This phenomenon i) is not observed forthe ICx case, because the interface can be advected by the nonzero mean flow, and ii) further explains the slower evolutionsof ⟨εθ ⟩ for the ICy and ICz cases, given that regions locatedincreasingly far away from the interface experience a delay
before they begin to mix. This latter effect therefore serves
to retard the total mixing, because the interface surface is notnormal to the mean velocity vector.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a) ICx(b) ICy(c) ICz

Figure 7: Evolution in time of the time-averages of all 9 terms in the scalar dissipation rate budget in which statistical

moments(⟨·⟩) are assessed on the mid-plane corresponding to the initial location of the interface in the initial condition:
1 Rτ
Term
dτ ′ . (a) ICx with statistical moments evaluated over the y-z plane at x/Lx = 0.50, (b) ICy with statistical0τ
moments evaluated over the x-z plane at y/h = 1, and (c) ICz with statistical moments evaluated over the x-y plane atz/Lz = 0.50.

(a) ICx(b) ICy(c) ICz

Figure 8: Evolution in time of the time-averages of all 9 terms in the scalar dissipation rate budget in which statistical

moments (⟨·⟩) areR assessed in planes parallel to the initial location of the interface in the initial condition, but away fromthe mid-plane: τ1 0τ Term dτ ′ . (a) ICx with statistical moments evaluated over the y-z plane at x/Lx = 0.25, (b) ICy withstatistical moments evaluated over the x-z plane at y/h = 0.5, and (c) ICz with statistical moments evaluated over the x-yplane at z/Lz = 0.25.
CONCLUSIONS AND FUTURE WORK

Dimotakis, P. E. & Miller, P.L. 1990 Some consequences of theboundedness of scalar fluctuations. Phys. Fluids A 2, 1919.
Germaine, E., Mydlarski, L. & Cortelezzi, L. 2013 3DFLUX:
A high-order fully three-dimensional flux integral solver forthe scalar transport equation. J. Compute. Phys. 240, 121.
Germaine, E., Mydlarski, L. & Cortelezzi, L. 2018 Persistenceof local anisotropy of passive scalars in wall-bounded flows.
Phys. Rev. Fluids 3(1), 014606.
Gibson, J. F. 2010 Channelflow: A spectral Navier-Stokes
simulator in C++, licensed under the GNU GPL, http://channelflow.org/. Tech. Rep.. University of New
Hampshire.
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing
the geometry of state space in plane Couette flow. J. Fluid
Mech. 611, 107–130.
Gubanov, O. & Cortelezzi, L. 2009 Sensitivity of mixing optimization to the geometry of the initial scalar field. In Analysis and Control of Mixing with an Application to Micro andMacro Flow Processes (ed. L. Cortelezzi & I. Mezić), pp.
369–405. Vienna: Springer.
Li, S., Jaroszynski, S., Pearse, S., Orf, L. & Clyne, J. 2019
VAPOR: A visualization package tailored to analyze simulation data in earth system science. Atmosphere 10 (9).
Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211, 23.

In an attempt to better understand mixing within wallbounded flows, with the ultimate goal of optimizing mixing
processes, the present work has investigated the mixing of ascalar within a fully developed turbulent channel flow whensubjected to three different initial conditions. A study of thetemporal evolution of the terms in the scalar dissipation rate
budget for the different initial conditions investigated why theICx case mixed most rapidly, and the ICy case most slowly. It
was found that the magnitude of the terms in the ⟨εθ ⟩ budget
were consistent with both large values of ⟨εθ ⟩ and rapid decay rates of ⟨εθ ⟩. The stretching of the interface by the mean
velocity field was also hypothesized to play an important role.
To promote mixing in a wall-bounded flow, it is recommended
that scalar interfaces be aligned normal to the mean velocity
vector, to enhance the stretching of the interface. Whereas thegeneral trends in the unmixedness and mix-norm were similar
to those of ⟨εθ ⟩ for τ > 1, future work should investigate theirrelationship during the initial phases of the mixing process.

REFERENCES
Corrsin, S. 1953 Remarks on turbulent heat transfer. Proc.
Iowa Thermodynamics Symposium, State University of
Iowa, Iowa City, pp. 5–30.
Danckwerts, P. V. 1952 The definition and measurement ofsome characteristics of mixtures. Appl. Sci. Res. A3, 279.

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