# Experimental study of turbulent thermal diffusion of particles in an
inhomogeneous forced convective turbulence

E. Elmakies  O. Shildkrot  N. Kleeorin  A. Levy  I. Rogachevskii
<EMAIL_ADDRESS>The Pearlstone Center for Aeronautical Engineering Studies,
Department of Mechanical Engineering, Ben-Gurion University of the Negev,
P.O.Box 653, Beer-Sheva 8410530, Israel

###### Abstract

We investigate experimentally phenomenon of turbulent thermal diffusion of
micron-size solid particles in an inhomogeneous convective turbulence forced
by one vertically-oriented oscillating grid in an air flow. This effect causes
formation of large-scale inhomogeneities in particle spatial distributions in
a temperature-stratified turbulence. We perform detailed comparisons of the
experimental results with those obtained in our previous experiments with an
inhomogeneous and anisotropic stably stratified turbulence produced by a one
oscillating grid in the air flow. Since the buoyancy increases the turbulent
kinetic energy for convective turbulence and decreases it for stably
stratified turbulence, the measured turbulent velocities for convective
turbulence are larger than those for stably stratified turbulence. This
tendency is also seen in the measured vertical integral turbulent length
scales. Measurements of temperature and particle number density spatial
distributions show that particles are accumulated in the vicinity of the
minimum of the mean temperature due to phenomenon of turbulent thermal
diffusion. This effect is observed in both, convective and stably stratified
turbulence, where we find the effective turbulent thermal diffusion
coefficient for micron-size particles. The obtained experimental results are
in agreement with theoretical predictions.

##  I Introduction

Turbulent transport of particles has been a subject of many studies due to
numerous applications in geophysics and environmental sciences, astrophysics,
and various industrial flows [1, 2, 3, 4, 5, 6, 7]. Different mechanisms of
large-scale and small-scale clustering of inertial particles have been
proposed. The large-scale clustering occurs in scales which are much larger
than the integral scale of turbulence, while the small-scale clustering is
observed in scales which are much smaller than the integral turbulence scale.

The large-scale clustering of inertial particles in isothermal non-stratified
inhomogeneous turbulence occurs due to turbophoresis [8, 9, 10, 11, 12, 13],
which is a combined effect of particle inertia and inhomogeneity of
turbulence. Turbophoresis results in appearance of the additional non-
diffusive turbulent flux of inertial particles ${\bm{J}}_{\rm
turboph}=\overline{n}\,\,\overline{\bm{V}}_{\rm turboph}$, where the mean
particle velocity caused by turbophoresis can be written as

$\displaystyle\overline{\bm{V}}_{\rm turboph}=-\kappa_{\rm
turb}\,\,{\mbox{\boldmath$\nabla$}\left\langle{\bm{u}}^{2}\right\rangle\over
2}.$ (1)

Here $\overline{n}$ is the mean number density of inertial particles,
${\bm{u}}$ is the turbulent fluid velocity, $\kappa_{\rm turb}$ is the
turbophoretic coefficient which generally depends on the Stokes number ${\rm
St}=\tau_{p}/\tau_{\nu}$ and the fluid Reynolds number ${\rm
Re}=\ell_{0}\,u_{\rm rms}/\nu$, where $\tau_{\nu}=\tau_{0}/{\rm Re}^{1/2}$ is
the Kolmogorov viscous time, $\tau_{0}=\ell_{0}/u_{\rm rms}$ in the
characteristic turbulent time, $\nu$ is the kinematic viscosity, $u_{\rm
rms}\equiv\sqrt{\langle{\bm{u}}^{2}\rangle}$ is the rms velocity in the
integral turbulence scale $\ell_{0}$ and $\tau_{p}$ is the Stokes time for the
small spherical particles. Due to turbophoresis, inertial particles are
accumulated in the vicinity of the minimum of the turbulent intensity. In
particular, direct numerical simulations (DNS) [13] show that inertial
particles in inhomogeneously forced isothermal turbulent flows are accumulated
at the minima of turbulent velocity. Two turbulent transport processes,
turbophoresis and turbulent diffusion determine the spatial distribution of
the particles. Numerical simulations [13] demonstrate that the non-dimensional
product of the turbophoretic coefficient $\kappa_{\rm turb}$ and the rms
velocity $u_{\rm rms}$ increases linearly with the parameter ${\rm St}_{f}$
for ${\rm St}_{f}\ll 1$, reaches a maxima for ${\rm St}_{f}\sim 10$ and
decreases as ${\rm St}_{f}^{-1/3}$ for large ${\rm St}_{f}$, where ${\rm
St}_{f}=\tau_{p}/\tau_{0}$ is the Stokes number defined using the
characteristic flow time scale $\tau_{0}$ based on the forcing scale of
turbulence. The same large-scale clustering phenomenon caused by turbophoresis
has been studied in DNS of turbulent Kolmogorov flows [14]. Although the
authors do not interpret their results as a balance between turbophoretic and
turbulent diffusive fluxes, they do observe that the large-scale clustering
increases for small ${\rm St}_{f}$ but this trend reverses smoothly at higher
values of ${\rm St}_{f}$. The large-scale clustering due to turbophoresis has
been also observed in DNS of turbulent channel flows [15] and in various
experimental studies [16, 17].

Another example of the large-scale clustering of inertial particles is the
phenomenon of turbulent thermal diffusion that is a combined effect of the
temperature stratified turbulence and inertia of small particles [18, 19].
Turbulent thermal diffusion is a purely collective phenomenon occurring in
temperature stratified turbulence and resulting in the appearance of a non-
zero mean effective velocity of particles in the direction opposite to the
mean temperature gradient. This implies that this phenomenon causes a non-
diffusive turbulent flux of particles in the direction of the turbulent heat
flux. A competition between the turbulent thermal diffusion and turbulent
diffusion determines the conditions for the formation of large-scale particle
concentrations in the vicinity of the mean temperature minimum.

Turbulent thermal diffusion has been intensively investigated analytically
[18, 19, 20, 21, 22, 23, 24, 25] using different theoretical approaches. This
effect has been detected in DNS [26, 27]. Turbulent thermal diffusion has been
observed in geophysical turbulence, e.g., in the atmosphere of the Earth [28]
and the atmosphere of Titan [29], and it also has been discussed in
astrophysical turbulence applications [30]. Moreover, the phenomenon of
turbulent thermal diffusion has been detected in laboratory experiments in
nearly isotropic and homogeneous turbulence produced by two oscillating grids
[31, 32, 33, 25] and in a multi-fan produced turbulence [34]. Recently the
phenomenon of turbulent thermal diffusion has been found in an inhomogeneous
and anisotropic stably stratified turbulence produced by one oscillating grid
in the air flow [35]. These experiments have demonstrated formation of
inhomogeneous distributions of micron-size particles in the vicinity of the
mean temperature minimum.

The main goal of the present study is to investigate experimentally the
phenomenon of turbulent thermal diffusion of the micron-size solid particles
in an inhomogeneous convective turbulence forced by one oscillating grid in
the air flow. In the experiments, we measure velocity fields applying Particle
Image Velocimetry (PIV). We measure temperature field with a temperature probe
equipped with 12 E thermocouples. In addition, we determine spatial
distributions of small solid particles by a PIV system using the effect of the
Mie light scattering by particles in the flow. We perform detailed comparisons
of the obtained experimental results with those in the experiments in an
inhomogeneous and anisotropic stably stratified turbulence produced by one
oscillating grid [35] and in a convective turbulence forced by two oscillating
grids in the air flow [36]. This paper is organized as follows. In Sec. II we
elucidate the mechanism of the phenomenon of turbulent thermal diffusion and
determine the turbulent flux of particles using the spectral $\tau$ approach
for fully developed temperature-stratified turbulence. In Sec. III we discuss
our experimental facilities and instrumentation, and in Sec. IV we describe
the obtained experimental results. Finally, in Sec. V we outline conclusions.

##  II Turbulent thermal diffusion

In this section we determine the turbulent flux of particles in a temperature-
stratified turbulence and elucidate the mechanism related to the effect of
turbulent thermal diffusion. We study dynamics of small non-inertial particles
advected by a turbulent fluid flow. An evolution of the particle number
density $n(t,{\bm{x}})$ in a fluid velocity field ${\bm{U}}(t,{\bm{x}})$ is
determined by the convective-diffusion equation:

$\displaystyle{\partial n\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}\left(n\,{\bm{U}}-D\mbox{\boldmath$\nabla$}n\right)=0,$
(2)

where $D=k_{B}\,T/(6\pi\rho\,\nu\,a_{p})$ is the coefficient of the molecular
(Brownian) diffusion of particles having the radius $a_{p}$. Here $T$ and
$\rho$ are the fluid temperature and density, respectively and $k_{B}$ is the
Boltzmann constant. The fluid velocity is a turbulent field produced by, e.g.,
an external steering force. Equation (2) is a conservation law for the total
number of particles that implies that the total number of particles is
conserved in a closed volume. Here we do not consider a coagulation of
particles or chemical reactions as well as condensation or evaporation of
droplets which change the total number of particles or droplets in a closed
volume.

Assuming for simplicity, that the diffusion coefficient is independent of
coordinates, Eq. (2) can be rewritten as

$\displaystyle{\partial n\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}(n\,{\bm{U}})=D\,\Delta n.$ (3)

We use a point-particle approximation that implies that the size of particles
is very small in comparison with all possible scales of fluid motions. When
the fluid velocity is much less than the sound speed (i.e., for low-Mach-
number fluid flows), the continuity equation for the fluid density can be used
in an anelastic approximation,
$\mbox{\boldmath$\nabla$}{\bf\cdot}(\rho\,{\bm{U}})=0$. This equation can be
rewritten as
$\mbox{\boldmath$\nabla$}{\bf\cdot}{\bm{U}}=-({\bm{U}}{\bf\cdot}\mbox{\boldmath$\nabla$})\ln\rho$,
i.e., the anelastic approximation takes into account an inhomogeneous fluid
density.

We study a long-term evolution of the particle number density in spatial
scales $L_{n}$ which are much larger than the integral scale of turbulence
$\ell_{0}$, and during the time scales $t_{n}$ which are much larger than the
turbulent time scales $\tau_{0}$. We use a mean-field approach in which all
quantities are decomposed into the mean and fluctuating parts, where the
fluctuating parts have zero mean values, i.e., we use the Reynolds averaging.
In particular, the particle number density $n=\overline{n}+n^{\prime}$, where
$\overline{n}=\langle n\rangle$ is the mean particle number density, and
$n^{\prime}$ are particle number density fluctuations and $\langle
n^{\prime}\rangle=0$. The angular brackets $\langle...\rangle$ denote an
ensemble averaging. Averaging Eq. (3) over an ensemble of a turbulent velocity
field, we arrive at the mean-field equation for the particle number density:

$\displaystyle{\partial\overline{n}\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}\left(\overline{\bm{U}}\,\overline{n}+\langle{\bm{u}}\,n^{\prime}\rangle\right)=D\,\Delta\overline{n},$
(4)

where ${\bm{F}}\equiv\langle{\bm{u}}\,n^{\prime}\rangle$ is the turbulent flux
of particles. We consider for simplicity the case $\overline{\bm{U}}=0$.

To derive an expression for the turbulent flux of particles, we obtain the
equation for particle number density fluctuations $n^{\prime}$, by subtracting
Eq. (4) from Eq. (3), which yields

$\displaystyle{\partial n^{\prime}\over\partial t}=-\tilde{\cal
Q}-({\bm{u}}{\bf\cdot}\mbox{\boldmath$\nabla$})\overline{n}-\overline{n}(\mbox{\boldmath$\nabla$}{\bf\cdot}{\bm{u}})+D\Delta
n^{\prime},$ (5)

where $\tilde{\cal
Q}=\mbox{\boldmath$\nabla$}{\bf\cdot}\left(n^{\prime}\,{\bm{u}}-\langle
n^{\prime}\,{\bm{u}}\rangle\right)$ is the nonlinear term. The source term for
particle number density fluctuations,
$-({\bm{u}}{\bf\cdot}\mbox{\boldmath$\nabla$})\overline{n}$, results in a
production of particle number density fluctuations by the tangling of the
gradient $\mbox{\boldmath$\nabla$}\,\overline{n}$ of the mean particle number
density by velocity fluctuations. The other source term,
$-\overline{n}(\mbox{\boldmath$\nabla$}{\bf\cdot}{\bm{u}})$ for particle
number density fluctuations can be rewritten as
$-\overline{n}(\mbox{\boldmath$\nabla$}{\bf\cdot}{\bm{u}})=(\overline{n}/\overline{\rho}\,)({\bm{u}}{\bf\cdot}\mbox{\boldmath$\nabla$})\overline{\rho}$,
where we take into account the anelastic approximation,
$\mbox{\boldmath$\nabla$}{\bf\cdot}{\bm{u}}=-(1/\overline{\rho})\,({\bm{u}}{\bf\cdot}\mbox{\boldmath$\nabla$})\overline{\rho}$,
which is also valid for the mean fluid density, $\overline{\rho}$. This
implies that this source term describes a production of particle number
density fluctuations by the tangling of the gradient
$\mbox{\boldmath$\nabla$}\,\overline{\rho}$ of the mean fluid density by
velocity fluctuations. We use the Péclet number ${\rm Pe}=|\tilde{\cal
Q}|/|D\Delta n^{\prime}|$ defined as the dimensionless ratio of the absolute
values of the nonlinear term $|\tilde{\cal Q}|$ to the diffusion term
$|D\Delta n^{\prime}|$. The Péclet number can be estimated as ${\rm
Pe}=\ell_{0}\,u_{\rm rms}/D$. We consider the case of large Péclet and
Reynolds number. Since the nonlinear equation (5) cannot be solved exactly for
arbitrary Péclet numbers, we consider the case of large Péclet and Reynolds
numbers, which corresponds to our laboratory experiments.

We apply the Fourier transform only in a ${\bm{k}}$ space but not in a
$\omega$ space, because in a fully developed Kolmogorov-like turbulence, the
turbulent time is universally related to spatial scales. We take into account
the nonlinear terms in equations for velocity and particle number density
fluctuations and apply the spectral $\tau$ approach [37, 38] (see also Ref.
[7] for detail discussion).

For simplicity, we consider a one-way coupling by taking into account the
effect of turbulence on the particle number density, and neglecting the
feedback effect of the particle number density on the turbulent fluid flow.
The one-way coupling approximation is valid when the spatial density of
particles $n\,m_{p}$ is much smaller than the fluid density $\rho$, where
$m_{p}$ is the particle mass. First, we consider non-inertial particles, which
means that the particles move with the fluid velocity, i.e., the particle
number density is a passive scalar.

We use a multi-scale approach [39], i.e., we consider the one-point second-
order correlation function as:

$\displaystyle\left\langle
u_{i}(t,{\bm{x}})\,n^{\prime}(t,{\bm{x}})\right\rangle\equiv\lim_{{\bm{x}}\to{\bm{y}}}\left\langle
u_{i}(t,{\bm{x}})\,n^{\prime}(t,{\bm{y}})\right\rangle$
$\displaystyle=\lim_{{\bm{r}}\to 0}\int F_{i}(t,{\bm{k}},{\bm{R}})\exp({\rm
i}\,{\bm{k}}{\bf\cdot}{\bm{r}})\,d{\bm{k}}=\int
F_{i}(t,{\bm{k}},{\bm{R}})\,d{\bm{k}},$ (6)

where $F_{i}(t,{\bm{k}},{\bm{R}})=\int F_{i}(t,{\bm{k}},{\bm{K}})\exp({\rm
i}\,{\bm{K}}{\bf\cdot}{\bm{R}})\,d{\bm{K}}$ and
$F_{i}(t,{\bm{k}},{\bm{K}})=\left\langle
u_{i}(t,{\bm{k}}+{\bm{K}}/2)\,n^{\prime}(t,-{\bm{k}}+{\bm{K}}/2)\right\rangle$.
Here the mean fields depend on “slow” variables
${\bm{R}}=({\bm{x}}+{\bm{y}})/2$, while fluctuations depend on “fast”
variables ${\bm{r}}={\bm{x}}-{\bm{y}}$, which correspond to large-scale and
small-scale spatial variables, respectively. In the Fourier space,
${\bm{k}}=({\bm{k}}_{1}-{\bm{k}}_{2})/2$, corresponds to the small scales, and
${\bm{K}}={\bm{k}}_{1}+{\bm{k}}_{2}$ characterizes the large scales, where we
use the Fourier transform, $u_{i}(t,{\bm{x}})=\int
u_{i}(t,{\bm{k}}_{1})\exp({\rm i}\,{\bm{k}}_{1}\cdot{\bm{x}})\,d{\bm{k}}_{1}$.
For homogeneous turbulence, the correlation function,
$F_{i}(t,{\bm{k}},{\bm{R}})$ is independent of the large-scale variable
${\bm{R}}$, i.e., $F_{i}(t,{\bm{k}},{\bm{R}})=F_{i}(t,{\bm{k}})$.

To obtain expression for the particle turbulent flux, we use Eq. (5) written
in a Fourier space. This allows us to derive equation for the correlation
function $F_{j}(t,{\bm{k}})$ in a Fourier space as

$\displaystyle{\partial F_{j}({\bm{k}})\over\partial t}$ $\displaystyle=$
$\displaystyle-\left(\nabla_{i}\overline{n}-{\rm
i}\,k_{i}\,\overline{n}\right)\,f_{ij}(-{\bm{k}})+\hat{\cal
M}F_{j}^{(III)}({\bm{k}}),$

where for the brevity of notation, hereafter we omit argument $t$ in the
correlation functions. Here $\hat{\cal
M}F_{i}^{(III)}({\bm{k}})=\langle[\partial u_{i}(t,{\bm{k}})/\partial
t]\,n^{\prime}(t,-{\bm{k}})\rangle-\langle u_{i}(t,{\bm{k}})\,{\cal
Q}(t,-{\bm{k}})\rangle$ are the third-order moments appearing due to the
nonlinear terms $\tilde{\cal Q}$ in Eq. (5) and the nonlinear Navier-Stokes
equation. Here $f_{ij}({\bm{k}})=\langle
u_{i}(t,{\bm{k}})\,u_{j}(t,-{\bm{k}})\rangle$ and ${\cal Q}=\tilde{\cal
Q}-D\Delta n^{\prime}$.

We use the spectral $\tau$ approximation [37, 38]. This approximation
postulates that the deviations of the third-moment terms, $\hat{\cal
M}F^{(III)}({\bm{k}})$, from the contributions to these terms afforded by the
background turbulence, $\hat{\cal M}F^{(III,0)}({\bm{k}})$, can be expressed
through the similar deviations of the second moments,
$F({\bm{k}})-F^{(0)}({\bm{k}})$:

$\displaystyle\hat{\cal M}F^{(III)}({\bm{k}})-\hat{\cal
M}F^{(III,0)}({\bm{k}})=-{1\over\tau_{r}(k)}\,\Big{[}F({\bm{k}})$
$\displaystyle\quad-F^{(0)}({\bm{k}})\Big{]},$ (8)

where $\tau_{r}(k)$ is the scale-dependent relaxation time, which can be
identified with the correlation time $\tau(k)$ of the turbulent velocity field
for large Reynolds and Péclet numbers. The functions with the superscript
$(0)$ correspond to the background turbulence with a zero turbulent particle
flux and a zero level of particle number density fluctuations. Consequently,
Eq. (8) reduces to $\hat{\cal
M}F_{i}^{(III)}({\bm{k}})=-F_{i}({\bm{k}})/\tau(k)$. Validation of the $\tau$
approximation for different situations has been performed in various numerical
simulations [40, 41, 42, 43, 26, 44, 27] (see also Ref. [7] for detail
discussion of the ranges of applicability of this approach).

We assume that the characteristic time of variation of the second moment
$F_{i}({\bm{k}})$ is substantially larger than the correlation time $\tau(k)$
for all turbulence scales. This allows us to use a steady-staye solution of
Eq. (LABEL:F5). Applying the spectral $\tau$ approximation and using the
steady-state solution of Eq. (LABEL:F5), we obtain the following formula for
the turbulent flux of particles, $F_{j}({\bm{k}})$ as

$\displaystyle F_{j}({\bm{k}})=-\tau(k)\,\left(\nabla_{i}\overline{n}-{\rm
i}\,k_{i}\,\overline{n}\right)\,f_{ij}^{(0)}(-{\bm{k}}),$ (9)

where since we consider a one-way coupling, we replace the function
$f_{ij}({\bm{k}})$ by $f_{ij}^{(0)}({\bm{k}})$ in Eq. (9).

We use the following model for the second moments of turbulent velocity field
$f_{ij}^{(0)}({\bm{k}})\equiv\langle
u_{i}({\bm{k}})\,u_{j}(-{\bm{k}})\rangle^{(0)}$ of an isotropic and
homogeneous background turbulence in anelastic approximation in a Fourier
space:

$\displaystyle
f_{ij}^{(0)}({\bm{k}})={\left\langle{\bm{u}}^{2}\right\rangle\,E(k)\over 8\pi
k^{2}}\biggl{[}\delta_{ij}-k_{ij}+{{\rm i}\over
k^{2}}\,\big{(}\lambda_{i}\,k_{j}-\lambda_{j}\,k_{i}\big{)}\biggr{]},$ (10)

where $k_{ij}=k_{i}\,k_{j}/k^{2}$, $\delta_{ij}$ is the Kronecker unit tensor,
${\bm{\lambda}}=-({\bm{\nabla}}\overline{\rho})/\overline{\rho}$, the spectrum
function of the turbulent kinetic energy density is
$E(k)=(2/3)\,k_{0}^{-1}\,(k/k_{0})^{-5/3}$ [see Ref. [7] for detail derivation
of Eq. (10)]. Here the wavenumber $k$ varies within the interval $k_{0}\leq
k\leq k_{\nu}$ corresponding to the inertial range of scales, the wave number
$k_{0}=1/\ell_{0}$, the length $\ell_{0}$ is the integral scale of turbulence,
the wave number $k_{\nu}=\ell_{\nu}^{-1}$, where $\ell_{\nu}=\ell_{0}{\rm
Re}^{-3/4}$ is the Kolmogorov (viscous) scale and the turbulent correlation
time is given by $\tau(k)=2\,\tau_{0}\,(k/k_{0})^{-2/3}$, where $\tau_{0}$ is
the characteristic turbulent time. The functions $E(k)$ and $\tau(k)$
correspond to fully developed turbulence with the Kolmogorov scalings.

Substituting Eq. (10) into Eq. (6), we determine the turbulent flux of
particles $F_{i}=\langle u_{i}\,n^{\prime}\rangle$:

$\displaystyle F_{i}$ $\displaystyle=$
$\displaystyle-{\left\langle{\bm{u}}^{2}\right\rangle\over
8\pi}\,\int_{k_{0}}^{k_{\nu}}\,\tau(k)\,E(k)\,dk\int_{0}^{2\pi}\,d\varphi\int_{0}^{\pi}\sin\vartheta\,d\vartheta$
(11)
$\displaystyle\times\left[\left(\delta_{ij}-k_{ij}\right)\,\,\nabla_{i}\overline{n}+\left(\lambda_{j}-\lambda_{i}\,k_{ij}\right)\,\overline{n}\right].$

For the integration over ${\bm{k}}$ in Eq. (11), we use the integrals given by
$\int_{0}^{2\pi}\,d\varphi\int_{0}^{\pi}\sin\vartheta\,d\vartheta\,k_{ij}=(4\pi/3)\,\delta_{ij}$
and $\int_{k_{0}}^{k_{\nu}}\tau(k)\,E(k)\,dk=\tau_{0}$. After integration over
${\bm{k}}$, we obtain the particle turbulent flux $\left\langle
n^{\prime}\,{\bm{u}}\right\rangle$ as

$\displaystyle\left\langle n^{\prime}\,{\bm{u}}\right\rangle={\bm{V}}^{\rm
eff}\,\overline{n}-D_{T}\,{\bm{\nabla}}\overline{n},$ (12)

where the turbulent diffusion coefficient is

$\displaystyle D_{T}={1\over
3}\,\tau_{0}\,\left\langle{\bm{u}}^{2}\right\rangle,$ (13)

and the effective pumping velocity is given by

$\displaystyle{\bm{V}}^{\rm
eff}=-D_{T}{\bm{\lambda}}=D_{T}\,{{\bm{\nabla}}\overline{\rho}\over\overline{\rho}}.$
(14)

Equations (12)–(14) are in agreement with those obtained using dimensional
analysis (see Ref. [7] for detail discussions). Note that the phenomenon of
turbulent diffusion of particles has been predicted about 100 years ago in
Ref. [45].

To understand the mechanism related to the effective pumping velocity
${\bm{V}}^{\rm eff}$, we use the equation of state for a perfect gas,

$\displaystyle P={k_{B}\over m_{\mu}}\,\rho\,T\equiv{R\over\mu}\,\rho\,T,$
(15)

where $P$ is the fluid pressure, $k_{B}=R/N_{A}$ is the Boltzmann constant,
$R$ is the gas constant, $N_{A}$ is the Avogadro number, $\mu=m_{\mu}N_{A}$ is
the molar mass, and $m_{\mu}$ is the molecule mass. We rewrite the equation of
state for the mean fields assuming that
$\overline{\rho}\,\overline{T}\gg\langle\rho^{\prime}\,\theta\rangle$, where
$\rho^{\prime}$ and $\theta$ are fluctuations of the fluid density and
temperature, respectively, and $\overline{T}$ is the mean fluid temperature.
Thus, the equation of state for the mean fields reads:

$\displaystyle\overline{P}={k_{B}\over
m_{\mu}}\,\overline{\rho}\,\overline{T},$ (16)

where $\overline{P}$ is the mean pressure.

Using Eq. (16), we express the gradient of the mean fluid density in terms of
the gradients of the mean fluid pressure ${\bm{\nabla}}\overline{P}$ and mean
fluid temperature ${\bm{\nabla}}\overline{T}$ as

$\displaystyle{{\bm{\nabla}}\,\overline{\rho}\over\overline{\rho}}={{\bm{\nabla}}\overline{P}\over\overline{P}}-{{\bm{\nabla}}\overline{T}\over\overline{T}}.$
(17)

Substituting Eq. (17) into Eq. (14), we obtain the final expression for the
effective pumping velocity of non-inertial particles as

$\displaystyle{\bm{V}}^{\rm
eff}=D_{T}\,\left({{\bm{\nabla}}\overline{P}\over\overline{P}}-{{\bm{\nabla}}\overline{T}\over\overline{T}}\right).$
(18)

To understand different terms in Eq. (18), we compare the molecular and
turbulent fluxes of particles (or gaseous admixtures). Equation for the number
density of particles reads

$\displaystyle{\partial n\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}(n\,{\bm{U}})=-{\bm{\nabla}}\cdot{\bm{F}}_{\rm
M},$ (19)

where the molecular flux of particles ${\bm{F}}_{\rm M}$ is given by

$\displaystyle{\bm{F}}_{\rm M}=-D\left({\bm{\nabla}}n+k_{\rm
t}{{\bm{\nabla}}T\over T}+k_{\rm p}{{\bm{\nabla}}P\over P}\right),$ (20)

which comprises three terms: molecular diffusion $(\propto{\bm{\nabla}}n)$,
molecular thermal diffusion for gases or thermophoresis for particles
$(\propto k_{\rm t}{\bm{\nabla}}T)$, and molecular barodiffusion $(\propto
k_{\rm p}{\bm{\nabla}}P$), where $k_{\rm t}$ is the molecular thermal
diffusion ratio and $k_{\rm p}$ is the molecular barodiffusion ratio. Note
that the phenomenon of molecular thermal diffusion in gases has been predicted
long ago in Refs. [46, 47, 48].

In turbulent flows, the turbulent flux of particles can be rewritten as

$\displaystyle{\bm{F}}_{T}\equiv\left\langle
n^{\prime}\,{\bm{u}}\right\rangle=-D_{T}\left({\bm{\nabla}}\overline{n}+\overline{n}{{\bm{\nabla}}\overline{T}\over\overline{T}}-\overline{n}{{\bm{\nabla}}\overline{P}\over\overline{P}}\right),$
(21)

which is obtained by substitution of Eq. (18) to Eq. (12). Comparing the
molecular flux of particles (20) and the turbulent flux of particles (21), we
can interpret the new additional turbulent fluxes as fluxes caused by the
effects of turbulent thermal diffusion $[\propto
k_{T}({\bm{\nabla}}\overline{T})/\overline{T}]$ and turbulent barodiffusion
$[\propto k_{P}({\bf\nabla}\overline{P})/\overline{P}]$, where

$\displaystyle{\bm{F}}_{T}\equiv\left\langle
n^{\prime}\,{\bm{u}}\right\rangle=-D_{T}\left({\bm{\nabla}}\overline{n}+k_{T}\,{{\bm{\nabla}}\overline{T}\over\overline{T}}+k_{P}\,{{\bm{\nabla}}\overline{P}\over\overline{P}}\right),$
(22)

and $k_{T}=\overline{n}$ is the turbulent thermal diffusion ratio and
$k_{P}=-\overline{n}$ is the turbulent barodiffusion ratio. These phenomena
have been predicted in Refs. [18, 19].

For small inertial particles, the expression for the effective pumping
velocity reads [11] (see also Ref. [7] for detail derivation):

$\displaystyle{\bm{V}}^{\rm
eff}=-\alpha\,D_{T}\,{\bm{\nabla}}\ln\overline{T},$ (23)

where

$\displaystyle\alpha=1+2\,{m_{p}\over m_{\mu}}\,\left({\ln{\rm Re}\over{\rm
Pe}}\right)\,{\overline{T}\over\overline{T}_{\ast}}=1+2\,{V_{g}\,L_{P}\,\ln{\rm
Re}\over u_{0}\,\ell_{0}},$ (24)

where
$\tau_{p}V_{T}^{2}=\gamma\,(m_{p}/m_{\mu})D=\gamma\,V_{g}\,L_{P}\,\overline{T}_{\ast}/\overline{T}$.
Here $\gamma=c_{\rm p}/c_{\rm v}$ is the ratio of specific heats,
$V_{T}=(\gamma\,k_{B}\overline{T}_{\ast}/m_{\mu})^{1/2}$ is the thermal
velocity, $\overline{T}_{\ast}$ is the characteristic mean fluid temperature,
and $L_{P}=|\nabla_{z}\overline{P}/\overline{P}|^{\,-1}$ is the pressure
height scale. In derivation of Eq. (24), we take into account that the Stokes
time can be written as $\tau_{p}=\overline{\rho}\,V_{\rm
g}\,L_{P}/\overline{P}$ with ${\bm{V}}_{\rm g}=\tau_{p}\,{\bm{g}}$ being the
terminal fall velocity of particles, where ${\bm{g}}$ is the acceleration
caused by the gravity field. For large Péclet numbers, ${\rm Pe}\gg 1$, the
turbulent thermal diffusion coefficient $\alpha=1$ for non-inertial particles,
while for inertial particles $\alpha$ depends on the particle mass, the
Reynolds and Péclet numbers.

The non-diffusive turbulent flux of particles, $\overline{n}\,{\bm{V}}^{\rm
eff}$, toward the mean temperature minimum is the main reason for the
formation of large-scale inhomogeneous distributions of inertial particles in
temperature-stratified turbulence. The steady-state solution of the equation
for the mean number density of inertial particles,

$\displaystyle{\partial\overline{n}\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}\,\big{[}\overline{n}\,({\bm{V}}_{\rm
g}+{\bm{V}}^{\rm
eff})-(D+D_{T})\,\mbox{\boldmath$\nabla$}\overline{n}\big{]}=0,$ (25)

satisfying the boundary condition with a zero total particle flux at the
boundary, is given by

$\displaystyle{\overline{n}\over\overline{n}_{\rm
b}}=\left({\overline{T}\over\overline{T}_{\rm b}}\right)^{-{\alpha D_{T}\over
D+D_{T}}}\,\exp\left[-\int_{z_{\rm b}}^{z}\,{V_{\rm g}\over
D+D_{T}}\,dz^{\prime}\right],$ (26)

where the subscripts $({\rm b})$ represent the values of the mean temperature
and the mean particle number density at the boundary $z=z_{\rm b}$. Equation
(26) implies that small inertial particles are accumulated below the mean
temperature minimum due to the gravity field.

The mechanism for turbulent thermal diffusion for inertial particles is as
following. Particles inside the turbulent eddies due to its inertia tend to be
drift out to the boundary regions between the eddies due to the centrifugal
inertial force. Indeed, for large Péclet numbers, molecular diffusion of
particles in equation for the number density of inertial particles,

$\displaystyle{\partial n\over\partial
t}+\mbox{\boldmath$\nabla$}{\bf\cdot}\left(n\,{\bm{u}}^{(p)}-D\mbox{\boldmath$\nabla$}n\right)=0,$
(27)

can be neglected, so that

$\displaystyle\mbox{\boldmath$\nabla$}{\bf\cdot}\,{\bm{u}}^{(p)}\approx-n^{-1}\,\left[{\partial
n\over\partial
t}+({\bm{u}}^{(p)}\cdot{\bm{\nabla}})n\right]\equiv-n^{-1}\,{{\rm d}n\over{\rm
d}t},$ (28)

where ${\bm{u}}^{\rm(p)}$ is the particle velocity. On the other hand, for
inertial particles,
$\mbox{\boldmath$\nabla$}{\bf\cdot}\,{\bm{u}}^{(p)}=\mbox{\boldmath$\nabla$}{\bf\cdot}\,{\bm{u}}+(\tau_{p}/\overline{\rho})\,\mbox{\boldmath$\nabla$}^{2}p$.
Indeed, the solution of the equation of motion for inertial particles,

$\displaystyle{{\rm d}{\bm{u}}^{\rm(p)}\over{\rm
d}t}=-{{\bm{u}}^{\rm(p)}-{\bm{u}}\over\tau_{p}}+{\bm{g}},$ (29)

for $\rho_{\rm p}\gg\overline{\rho}$ and small Stokes time, reads [49]:
${\bm{u}}^{\rm(p)}={\bm{u}}-\tau_{p}\,({\rm d}{\bm{u}}/{\rm
d}t)+\tau_{p}{\bm{g}}+{\rm
O}(\tau_{p}^{2})\approx{\bm{u}}+\tau_{p}\,({\bm{\nabla}}p/\overline{\rho})+\tau_{p}{\bm{g}}+{\rm
O}(\tau_{p}^{2})$. Here $\rho_{\rm p}$ is the material density of particles.
This yields the equation for
$\mbox{\boldmath$\nabla$}{\bf\cdot}\,{\bm{u}}^{(p)}$. Therefore, in regions
with maximum fluid pressure fluctuations (where
$\mbox{\boldmath$\nabla$}^{2}p<0)$, there is accumulation of inertial
particles, i.e., ${\rm d}n^{\prime}/{\rm
d}t\propto-\overline{n}\,(\tau_{p}/\overline{\rho})\,\mbox{\boldmath$\nabla$}^{2}p>0$.
These regions obey low vorticity and high strain rate. Similarly, there is an
outflow of inertial particles from regions with minimum fluid pressure.

In a homogeneous and isotropic turbulence with a zero gradient of the mean
temperature, there is no a preferential direction. This implies that in a
homogeneous and isotropic turbulence there is no large-scale effect of
particle accumulation, and the pressure (temperature) of the surrounding fluid
is not correlated with the turbulent velocity field. The only non-zero
correlation is $\langle({\bm{u}}\cdot{\bm{\nabla}})p\rangle$, which
contributes to the flux of the turbulent kinetic energy density.

In a temperature-stratified turbulence, the turbulent heat flux does not
vanish, so that fluctuations of fluid temperature $\theta$ and velocity
${\bm{u}}$ are correlated, i.e., $\langle\theta\,{\bm{u}}\rangle\not=\bm{0}$.
Fluctuations of temperature result in pressure fluctuations, which cause
fluctuations of the particle number density. Increase of the pressure of the
surrounding fluid is accompanied by an accumulation of particles, and the
direction of the turbulent flux of particles coincides with that of the
turbulent heat flux. The turbulent flux of particles is directed toward the
minimum of the mean temperature. This causes the formation of large-scale
inhomogeneous structures in the spatial distribution of inertial particles in
the vicinity of the mean temperature minimum. In the next sections we will
study this phenomenon in the experiments with a forced convective turbulence.

##  III Experimental setup

In this section we describe the experimental set-up and measurement technique.
We investigate turbulent thermal diffusion of small solid particles in
experiments with a convective turbulence forced by one oscillating grid in the
air flow. We conduct experiments in rectangular transparent chamber with
dimensions $L_{x}\times L_{y}\times L_{z}$ with $L_{x}=L_{z}=26$ cm and
$L_{y}=53$ cm, where $Z$ is along the vertical direction and $Y$ is
perpendicular to the grid plain. The oscillating grid with bars arranged in a
square array is parallel to the side walls of the chamber, it is positioned at
a distance of two grid meshes from the left side wall of the chamber (see Fig.
1).

Figure 1:  Experimental setup with the forced convective turbulence: (1)
digital CCD camera; (2) rod driven by the speed-controlled motor; (3)
oscillating grid; (4) laser light sheet; (5) temperature probe equipped with
12 E - thermocouples; (6) heat exchanger at the top cooled wall of the
chamber; (7) heat exchanger at the bottom heated wall of the chamber.

Two aluminium heat exchangers with rectangular pins $3\times 3\times 15$ mm
are attached to the bottom (heated) and top (cooled) walls of the chamber,
which allow one to form a large vertical mean temperature gradient up to 1.8
K/cm in the main fluid flow and about 7 K/cm close to the walls. We measure
the temperature field using a temperature probe equipped with 12 E -
thermocouples. The thermocouples with the diameter of 0.13 mm and the
sensitivity of $\approx 75\,\mu$V/K are attached to a vertical rod with a
diameter 4 mm, and the mean distance between thermocouples is about 21.6 mm
(see for details Ref. [35]). We measure the temperature field in many
locations. The data are recorded using the developed software based on LabView
7.0, and the temperature maps are obtained using Matlab 9.7.0.

We measure the velocity field with a Particle Image Velocimetry (PIV) system
[50, 51, 52], consisting in a Nd-YAG laser (Continuum Surelite $2\times 170$
mJ) and a progressive-scan 12 bit digital CCD camera (with pixel size
$6.45\,\mu$m $\times\,6.45\,\mu$m and $1376\times 1040$ pixels). As a tracer
for the PIV measurements, we use an incense smoke with spherical solid
particles having the mean diameter of $0.7\mu$m and the material density
$\rho_{\rm p}\approx 10^{3}\rho$, The particles are produced by high
temperature sublimation of solid incense grains (see for details Ref. [35]).

For instance, the velocity fields in our experiments have been measured in a
flow domain $209.09\times 155.43$ mm2 with a spatial resolution of $1376\times
1024$ pixels, so that a spatial resolution 151 $\mu$m /pixel have been
achieved. We analyse the velocity field in the probed region with
interrogation windows of $16\times 16$ pixels. Using the velocity
measurements, various turbulence characteristics (e.g., the mean and the root
mean square (r.m.s.) velocities, two-point correlation functions and an
integral scale of turbulence) have been obtained in our experiments. In
particular, we determine the mean and r.m.s. velocities for every point of a
velocity map by averaging over 530 independent maps. We obtain also the
integral length scales of turbulence $\ell_{y}$ and $\ell_{z}$ in the
horizontal $Y$ and the vertical $Z$ directions from the two-point correlation
functions of the velocity field.

Next, we obtain the particle spatial distribution by the PIV system using the
effect of the Mie light scattering by particles [53]. To this end, we
determine the mean intensity of scattered light in $80\times 64$ interrogation
windows with the size $16\times 16$ pixels. This allows us to find the
vertical distribution of the intensity of the scattered light in 80 vertical
strips composed of 64 interrogation windows. In particular, we take into
account that the light radiation energy flux scattered by small particles is
given by $E_{s}\propto E_{0}\Psi(\pi d_{\rm p}/\lambda;a_{0};n)$. Here $\Psi$
is the scattering function, $d_{\rm p}$ is the particle diameter, $\lambda$ is
the wavelength, $a_{0}$ is the index of refraction. The energy flux incident
at the particle is given by $E_{0}\propto\pi d_{\rm p}^{2}/4$. Note that when
$\lambda>\pi d_{\rm p}$, the scattering function $\Psi$ is determined by the
Rayleigh’s law, $\Psi\propto d_{\rm p}^{4}$. In opposite case for small
$\lambda$, the scattering function $\Psi$ is independent of the particle
diameter and the wavelength. In a general case, the scattering function $\Psi$
is determined by the Mie equations [54].

Finally, we take into account that the light radiation energy flux scattered
by small particles is $E_{s}\propto E_{0}\,n\,(\pi d_{\rm p}^{2}/4)$. This
implies that the scattered light energy flux incident on the charge-coupled
device (CCD) camera probe is proportional to the particle number density $n$.
The ratio of the scattered radiation fluxes at two locations in the flow and
at the image measured with the CCD camera is equal to the ratio of the
particle number densities at these two locations. For the normalization of the
scattered light intensity $E^{T}$ obtained in a temperature-stratified
turbulence, we use the distribution of the scattered light intensity E
measured in the isothermal case obtained under the same conditions. Indeed, as
follows from our measurements applying different concentrations of the incense
smoke, the distribution of the scattered light intensity averaged over a
vertical coordinate is independent of the particle number density in the
isothermal flow. Therefore, using this normalization, we can characterize the
spatial distribution of particle number density $n\propto E^{T}/E$ in the non-
isothermal turbulence.

Note that the measurement technique and data processing procedure described in
this section are similar to those used by us in various experiments with
turbulent convection [55, 36, 56] and stably stratified turbulence [57, 35].
In addition, the similar measurement technique and data processing procedure
in the experiments have been performed previously by us to investigate the
phenomenon of turbulent thermal diffusion in a homogeneous turbulence [31, 32,
33, 25] as well as for study of small-scale particle clustering [58].

##  IV Experimental results

Figure 2:  Mean velocity field in the core flow for (a) isothermal turbulence;
(c) stably stratified turbulence; (b) forced convective turbulence. The
velocity is measured in m/s and coordinates $Y$ and $Z$ are normalized by
$L_{z}=26$ cm.

Figure 3:  Horizontal $u^{\rm(rms)}_{y}$ (upper panel) and vertical
$u^{\rm(rms)}_{z}$ (lower panel) components of the turbulent velocity versus
normalized coordinate $Y$ averaged over the vertical coordinate $Z$ for
isothermal turbulence (red); stably stratified turbulence (black) and forced
convective turbulence (blue). The velocity is measured in cm/s and $Y$ is
normalized by $L_{z}=26$ cm.

In this section we discuss the obtained experimental results in a forced
convective turbulence with one oscillating grid in the air flow. There are two
sources of turbulence in a forced convective turbulence with heated bottom
wall of the chamber and cooling upper wall. In particular, the turbulent
kinetic energy is increased by buoyancy and the grid oscillations. In our
experiments, the frequency $f$ of the grid oscillations is $f=10.5$ Hz, which
yields the maximum turbulence intensity in our experimental set-up.

Note that early laboratory experiments [59, 60, 61, 62, 63, 64, 65] which have
been conducted in isothermal turbulence with one oscillating grid in a water
flow have demonstrated that the r.m.s. velocity behaves as $\sqrt{\langle{\bf
u^{\prime}}^{2}\rangle}\propto f\,Y^{-1}$, while the integral turbulence
length scale increases linearly with the distance $Y$ from a grid. Therefore,
the fluid Reynolds numbers as well as the turbulent diffusion coefficient of
particles are nearly independent of the distance $Y$ from the grid. Our
previous [35] and present studies in turbulence with one oscillating grid
confirm these findings.

In the present study we conduct experiments in a forced convective turbulence
with one oscillating grid for the temperature difference $\Delta T=50$ K
between the bottom and top walls of the chamber. Using the PIV system we
measure velocity field in the chamber for an isothermal and a forced
convective turbulence, which allows us to determine various turbulence
characteristics. In particular, we obtain the spatial distributions of the
mean velocity in convective turbulence with large-scale circulations, the
vertical and horizontal profiles of the r.m.s turbulent velocity and the
integral turbulence length scales. Since the oscillating grid is located near
by the left wall of the chamber, and the amplitude of the grid oscillations is
6 cm, we measure velocity field in the horizontal direction starting 20 cm
away the left wall of the chamber. We compare these results with those
obtained in our recent experiments [35] with stably stratified turbulence
produced by one oscillating grid in the air flow.

Figure 4:  Distributions of the turbulent velocity $u^{\rm(rms)}_{\rm
tot}=[\langle u_{y}^{2}\rangle+\langle u_{z}^{2}\rangle]^{1/2}$ for (a)
isothermal turbulence; (b) stably stratified turbulence and (c) forced
convective turbulence. The velocity is measured in cm/s and coordinates are
normalized by $L_{z}=26$ cm.

Figure 2 with the mean velocity patterns $\overline{U}$ in the main fluid flow
for isothermal, stably stratified turbulence and convective turbulence,
demonstrates that the temperature stratification and additional forcing
strongly affect the mean velocity distributions. Contrary to our previous
experiments with a forced convection with two oscillating grids [36], the
large-scale circulations in the convective turbulence with one oscillating
grid are not destroyed at the frequency $10.5$ Hz of the grid oscillations,
but their structure is strongly deformed (see the bottom panel in Fig. 2).

Figure 5:  Distributions of the anisotropy parameter
$u^{\rm(rms)}_{z}/u^{\rm(rms)}_{y}$ of the turbulent velocity for (a)
isothermal turbulence; (b) stably stratified turbulence and (c) forced
convective turbulence. The velocity is measured in cm/s and coordinates are
normalized by $L_{z}=26$ cm.

Similar effects of the temperature stratification and additional forcing are
also seen in the horizontal profiles of velocity fluctuations (see Fig. 3,
where we plot the horizontal $u^{\rm(rms)}_{y}$ and vertical
$u^{\rm(rms)}_{z}$ components of turbulent velocities as the functions of $Y$
averaged over the vertical coordinate $Z$ for isothermal turbulence, stably
stratified turbulence and convective turbulence). The turbulent velocities for
convective turbulence are larger than for isothermal turbulence, while the
turbulent velocities for stably stratified turbulence are smaller than those
for isothermal and convective turbulence. This is because the buoyancy
increases the turbulent kinetic energy for convective turbulence and decreases
it for stably stratified turbulence.

The oscillating grid strongly affects convective turbulence, as can be seen in
Figs. 4 and 5, where we show the distributions of the turbulent velocity
$u^{\rm(rms)}_{\rm tot}=[\langle u_{y}^{2}\rangle+\langle
u_{z}^{2}\rangle]^{1/2}$ and the anisotropy parameter
$u^{\rm(rms)}_{z}/u^{\rm(rms)}_{y}$ for the turbulent velocity components for
isothermal , stably stratified and convective turbulence. Figure 5
demonstrates that the anisotropy for isothermal and stably stratified
turbulence is more stronger than that for convective turbulence. This is not
surprising since the large-scale circulation enhances the mixing in the
convective turbulence, and it results in decrease of the turbulence anisotropy
parameter $u^{\rm(rms)}_{z}/u^{\rm(rms)}_{y}$. The same tendencies are also
seen for the horizontal $\ell_{y}$ and vertical $\ell_{z}$ integral turbulent
length scales shown in Fig. 6, as well as for the distributions of the
anisotropy parameter $\ell_{z}/\ell_{y}$ of the integral turbulent length
scales (see Fig. 7).

Figure 6:  Horizontal $\ell_{y}$ and vertical $\ell_{z}$ integral turbulent
length scales versus normalized coordinate $Y$ averaged over $Z$ for
isothermal turbulence (red); stably stratified turbulence (black) and forced
convective turbulence (blue). The velocity is measured in cm/s and $Y$ is
normalized by $L_{z}=26$ cm.

Figure 7:  Distributions of the anisotropy parameter $\ell_{z}/\ell_{y}$ of
the integral turbulent length scales for (a) isothermal turbulence; (b) stably
stratified turbulence and (c) forced convective turbulence. The velocity is
measured in cm/s and coordinates are normalized by $L_{z}=26$ cm.

To investigate the phenomenon of turbulent thermal diffusion in a forced
convective turbulence, we measure the spatial distributions of the mean
temperature and the mean particle number density. When small solid particles
are injected into the chamber, their initial spatial distributions are nearly
homogeneous. Due to the effective pumping velocity caused by a combined effect
of temperature-stratified turbulence and particle inertia (described in terms
of turbulent thermal diffusion) the final spatial distributions of the mean
particle number density is expected to be strongly inhomogeneous.

Sedimentation of particles can also result in a formation of inhomogeneous
particle distributions near the bottom wall of the chamber. However, this
effect in our experiments is very weak, because the terminal fall velocity for
the micron-size particles is about $10^{-2}$ cm/s, while the turbulent
velocity in the experiments with the forced convective turbulence is much
larger than the particle terminal fall velocity (it is about $18$ cm/s near
the grid and is more than $12$ cm/s far from the grid). On the other hand, our
estimates for the effective pumping velocity due to turbulent thermal
diffusion shows that it is more than $3-5$ cm/s near the grid and is about
$0.5$ cm/s far from the grid. Note also that the Stokes time for for the
micron-size particles is about $1.5\times 10^{-6}$ s, while in the forced
convective turbulence the Kolmogorov time varies from $7\times 10^{-3}$ s near
the grid up to $4\times 10^{-3}$ s far from the grid. Therefore, the turbulent
and effective pumping velocities in our experiments are much larger than the
terminal fall velocity for micron-size particles.

Figure 8:  Vertical profiles of the relative normalized mean temperature
$(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ in the forced convective
turbulence averaged over different horizontal regions: $Y=4-10$ cm (blue,
circles); $Y=11-16$ cm (black, diamond); $Y=17-25$ cm (green, crosses) and
$Y=24.5-28.5$ cm (red, slanting crosses), where $Z$ is normalized by
$L_{z}=26$ cm.  Figure 9:  Vertical profiles of the relative normalized mean
temperature $(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ (black,
crosses) and the normalized mean particle number density
$\overline{n}(Y,Z)/\overline{n}_{0}$ (blue, circles) in the forced convective
turbulence averaged over horizontal region $Y=4-15$ cm (near the grid). The
coordinate $Z$ is normalized by $L_{z}=26$ cm.  Figure 10:  Vertical profiles
of the relative normalized mean temperature
$(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ (black, crosses) and the
normalized mean particle number density $\overline{n}(Y,Z)/\overline{n}_{0}$
(blue, circles) in the forced convective turbulence averaged over horizontal
region $Y=17-24$ cm (far from the grid). The coordinate $Z$ is normalized by
$L_{z}=26$ cm.

Our experiments with a forced convective turbulence with large-scale
circulations show that the mean temperature is strongly nonuniform. In
particular, as follows from Fig. 8 (where we plot vertical profiles of the
relative normalized mean temperature
$(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ averaged over different
horizontal regions), the normalized mean temperature near the grid increases
with the height $Z$, reaches the maximum and decreases nearly linearly with
the height $Z$, where $\overline{T}_{0}$ is the reference mean temperature.
Far from the grid, the behavior of the mean temperature is even more
complicated, e.g., the normalized mean temperature
$(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ decreases with the height
$Z$, reaches the minimum and increases with the height $Z$ reaching the
maximum, and finally it decreases nearly linearly with the height $Z$ (see
Fig. 8).

To demonstrate the phenomenon of turbulent thermal diffusion in the forced
convective turbulence, we show in Figs. 9 and 10 the vertical profiles of the
relative normalized mean temperature
$(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ (black crosses) and the
normalized mean particle number density $\overline{n}(Y,Z)/\overline{n}_{0}$
(blue circles) near the grid (see Figs. 9) and far from the grid (see Figs.
10). Due to the phenomenon of turbulent thermal diffusion, the behaviour of
the normalized mean particle number density is opposite to the normalized mean
temperature, i.e., the mean particle number density increases in the regions
where the mean temperature decreases, and the mean particle number density
reaches the maximum at the minimum of the mean temperature, and vise versa.
Therefore, Figs. 9 and 10 clearly demonstrate that particles are accumulated
in the vicinity of the minimum of the mean temperature even in very
complicated temperature field.

In the stably stratified turbulence, the behaviour of the mean temperature and
the mean particle number density is more simple than for the forced convective
turbulence [35]. In particular, the mean temperature increases linearly with
the height $Z$ in the flow for the stably stratified turbulence, and the mean
particle number density decreases linearly with the height $Z$ due to the
phenomenon of turbulent thermal diffusion.

Figure 11:  The normalized mean particle number density
$\overline{n}/\overline{n}_{0}$ versus the relative normalized mean
temperature $(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$ averaged over
different horizontal regions: $Y=4-15$ cm (blue, circles) with $\alpha=4.86$
and $Y=16-24$ cm (red, crosses) with $\alpha=1.74$ in the forced convective
turbulence.

To determine the effective turbulent thermal diffusion coefficient $\alpha$
for particles in the forced inhomogeneous and anisotropic convective
turbulence, we show in Fig. 11 the normalized mean particle number density
$\overline{n}/\overline{n}_{0}$ as the function of the relative normalized
mean temperature $(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$, where the
slope of this dependence yields the coefficient $\alpha$. In particular, we
use a solution (26) for Eq. (25) for the mean particle number density
$\overline{n}$ written in a steady-state, where we assume that $D_{\rm T}\gg
D$ and neglect small terminal fall velocity. Thus, we arrive at the following
expression
$\overline{n}/\overline{n}_{0}=1-\alpha\,(\overline{T}-\overline{T}_{0})/\overline{T}_{0}$,
which shows that the effective turbulent thermal diffusion coefficient
$\alpha$ for particles in the forced convective turbulence is $\alpha=4.86$
for particles accumulated in the regions $Y=4-15$ cm, and $\alpha=1.74$ for
particles accumulated in the regions $Y=16-24$ cm (see Fig. 11). Now we take
into account that turbulence far from the grid is less stronger than that near
the grid. Therefore, the effective turbulent thermal diffusion coefficient
$\alpha$ near the grid is larger than that far from the grid. Therefore, this
experimental study has demonstrated the effect of turbulent thermal diffusion
in an inhomogeneous and anisotropic forced convective turbulence.

##  V Conclusions

In the present study, the effect of turbulent thermal diffusion of small solid
particles, resulting in formation of large-scale inhomogeneities in particle
spatial distributions in a temperature-stratified turbulence, has been
investigated experimentally for micron-size particles in an inhomogeneous
convective turbulence forced by one oscillating grid in the air flow. The
obtained experimental results have been compared with the results of our
previous experiments [35, 36] conducted in an inhomogeneous and anisotropic
stably stratified turbulence [35] produced by a one oscillating grid and in a
forced convection with two oscillating grids in the air flow [36]. We have
found that contrary to our previous experiments with a forced convection with
two oscillating grids [36], the large-scale circulations in the convective
turbulence with a one oscillating grid are not destroyed at the maximum
frequency $10.5$ Hz of the grid oscillations, but their structure is deformed
(see Fig. 2). The measured vertical turbulent velocities for convective
turbulence are stronger than for both, isothermal turbulence and stably
stratified turbulence produced by a one oscillating grid, since the buoyancy
increases the turbulent kinetic energy for convective turbulence and decreases
it for stably stratified turbulence. These effects are also observed in the
measured vertical integral turbulent length scales obtained from the two-point
correlation functions for velocity fluctuations.

To study phenomenon of turbulent thermal diffusion, we measure spatial
distributions of the mean temperature and mean particle number density in many
locations. We have found that in the convective turbulence near the grid, the
mean temperature increases with the height reaching the maximum and then it
decreases nearly linearly with the increase of the height. On the other hand,
far from the grid the behavior of the mean temperature in the convective
turbulence is more complicated. The mean fluid temperature decreases with the
height reaching the minimum, and for larger heights it increases with the
height reaching the maximum, and finally it decreases nearly linearly with the
height (see Fig. 8).

The behaviour of the mean particle number density is opposite to the mean
temperature. In particular, the mean particle number density increases in the
regions where the mean temperature decreases, reaching the maximum near by the
minimum of the mean temperature (see Figs. 9 and 10). This implies that our
experiments in convective and stably stratified turbulence with micron-size
solid particles have clearly demonstrated the existence of the phenomenon of
turbulent thermal diffusion, which causes particle accumulation in the
vicinity of the minimum of the mean temperature even in a complicated vertical
profile of the mean fluid temperature. We have determined the effective
turbulent thermal diffusion coefficient using the vertical profiles of the
mean temperature and the mean particle number density. We also have
demonstrated that the obtained experimental results are in agreement with the
theoretical predictions.

DATA AVAILABILITY

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

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N.Q. Hieu, N.H. Khang, L.Q. Hoa, N.Q. Dat 11institutetext: School of Applied
Math. and Informatics, Hanoi University of Science and technology, Hanoi,
Vietnam
(a<EMAIL_ADDRESS>
(b<EMAIL_ADDRESS>22institutetext: School of Information and
Communications Technology, Hanoi University of Science and technology, Hanoi,
Vietnam
<EMAIL_ADDRESS>33institutetext: HUS High School for Gifted
Students, Hanoi University of Science, Vietnam National University, Hanoi,
Vietnam
<EMAIL_ADDRESS>
(*) Corresponding author.

# An approach to extract information from academic transcripts of HUST

Nguyen Quang Hieu 1a1a Nguyen Le Quy Duong 22 Le Quang Hoa (*) 1b1b Nguyen
Quang Dat 33

###### Abstract

In many Vietnamese schools, grades are still being inputted into the database
manually, which is not only inefficient but also prone to human error. Thus,
the automation of this process is highly necessary, which can only be achieved
if we can extract information from academic transcripts. In this paper, we
test our improved CRNN model in extracting information from 126 transcripts,
with 1008 vertical lines, 3859 horizontal lines, and 2139 handwritten test
scores. Then, this model is compared to the Baseline model. The results show
that our model significantly outperforms the Baseline model with an accuracy
of 99.6% in recognizing vertical lines, 100% in recognizing horizontal lines,
and 96.11% in recognizing handwritten test scores.

###### keywords:

Academic transcript; Image Processing; CRNN; CTC; Digit string recognition

##  1 Introduction

At Hanoi University of Science & Technology, after every exam, scores are
recorded in academic transcripts and then transferred to the school’s database
by teachers. Until now, this process has been done manually, which is time-
consuming for the teachers and may lead to accidental mistakes such as the
scores inputted being incorrect or the scores being assigned to the wrong
students.

Currently, machine learning methods have been applied to automate these
processes (see [1], [2], [3], [11], [12], [13], [14], [15], [16], [17]). It
also helps to free up manpower. By utilizing Image-Processing Techniques and
Deep Learning, we can automate this procedure with a system that can extract
necessary data.

This paper contains five sections. The introduction is the first section. The
second section consists of prior research conducted by various authors on
image processing research techniques. The third part is the method studied in
this paper, including our proposed method. The fourth part is our results
based on real data. And the last part is conclusion and acknowledgment.

##  2 Related works

Digit recognition is extremely useful. This is especially the case for schools
where the process of inputting grades into database is still being done
manually. In such schools, the assistance of digit recognition can
significantly increase accuracy and reduce the time allotted to this process.

In 2018, Liu et al. [4] proposed a hybrid CNN-LSTM algorithm for identifying
CO$2$ welding defects. The algorithm is evaluated using 500 images of molten
pools from the Robotics and Welding Technology Laboratory at Guilin University
of Aerospace Technology. The results from CNN-LSTM algorithm were considered
to be better than those of other basic algorithms (CNN, LSTM, CNN-3), with an
accuracy of 85%, 88%, and 80% for 32x32, 64x64, and 128x128 images,
respectively.

In 2019, Rehman et al. [5] utilized a hybrid CNN-LSTM model to analyze
opinions in people’s online comments. This model outperforms conventional
machine learning techniques on the IMDB movie reviews dataset and the Amazon
movie reviews dataset, with a precision of 91%, recall of 86%, an F-measure of
88%, and an accuracy of 91%.

In 2020, Yang et al. [6] compared the performance of CNN-LSTM model with
analytical analysis and FEA algorithm in detecting the natural frequency of
six types of beams. The goal was to assess the first, second, and third-order
frequencies of each beam. Author’s model was concluded to be superior in both
the test on robustness, with 96.6%, 93.7%, and 95.1% accuracy, respectively,
and the test on extrapolability, with 95.4%, 92%, and 92.5% accuracy,
respectively.

In 2019, Sivaram et al. [7] proposed a CNN-RNN combination model to detect
facial landmarks. The proposed model outperforms existing methods, such as
CFAN, SDM, TCDN, and RCPR, on the FaceWarehouse database, with 99% precision,
99% recall, 99% F1-Score, 98.65% Accuracy, and 98.65% AUC or ROC.

In 2018, Xiangyang et al. [8] utilized a hybrid CNN-LSTM model to tackle the
problem of text classification. With the Chinese news dataset (proposed by the
Sogou Lab), the model proved superior to traditional KNN, SVM, CNN, and LSTM,
with a precision of 90.13% under the CBOW model and 90.68% under the Skip-Gram
model.

In 2017, Yin et al. [9] used CNN-RNN and C3D hybrid networks to detect
emotions in videos from the AFEW 6.0 database. The objective was to assign
emotion from 7 categories, namely anger, disgust, fear, happiness, sadness,
surprise, and neutral, to each video in the test dataset. It was found that
with 1 CNN-RNN model and 3 C3D models, an accuracy of 59.02% was achieved,
surpassing the baseline accuracy of 40.47% and last year’s highest accuracy of
53.8%.

In 2017, Zhan et el. [10] introduced a new RNN-CTC model to recognize digit
sequences in three datasets, namely CVL HDS, ORAND-CAR (include CAR-A and
CAR-B), and G-Captcha. Even though the proposed model only achieved a
recognition rate of 27.07% for the CVL dataset, the model outperformed other
state-of-the-art methods on the CAR-A, CAR-B, and G-Captcha datasets, with
recognition rates of 89.75%, 91.14%, and 95.15%, respectively, due to the
absence of segmentation.

##  3 Methodology

###  3.1 Convolutional Neural Networks

A convolutional neural network can be successfully applied for most computer
vision problems. Its characteristics make it more effective than other
conventional methods. Since its inception, CNN has witnessed numerous
optimizations.

However, when it comes to deeper networks, a degradation problem arises. To
tackle this, He et al. proposed a deep residual learning framework, ResNet.
The basic structure of this network is using shortcut connections. Shortcut
connections are those that traverse multiple layers. With this type of
connection, we can overcome the problem of vanishing gradients and construct
more extensive networks; in other words, better feature representations can be
acquired. In practice, shortcut connections can be adjusted on a case-by-case
basis, depending on each specific problem.

In our proposed model, we design a 10-layer residual network that doesn’t have
any global pooling layers. To prevent divergence, we avoid employing CNN that
is excessively profound. In addition, we maximize the use of residual learning
to enhance gradient propagation.

###  3.2 Recurrent Neural Networks

A Recurrent neural network is an architecture where multiple connections
between neurons create a recursive cycle. Self-connection brings the advantage
of utilizing contextual data when making a correspondence between sequences of
input and output. Nevertheless, for conventional RNN, the range of accessible
context is restricted in practice. due to the vanishing gradient problem.
Applying a memory structure to RNN, which produces a cell that is known as a
long short-term memory (LSTM), is one method that can be utilized to solve the
problem. It has been demonstrated that the LSTM variant of RNN is capable of
resolving some of the issues that are inherently present in conventional RNN,
in addition to learning how to resolve issues of long-term dependency. At this
time, LSTM has developed into one of the RNNs that is utilized the most
frequently.

Regarding the sequence labeling problem, it is helpful to have access to
context in both the future and the past. However, the normal LSTM solely
considers information from the past and pays no attention to the context of
the future. The creation of a second LSTM that processes input in the other
direction is one alternate solution. This type of LSTM is known as a
bidirectional LSTM, and its abbreviation is Bi-LSTM. Every training sequence
is conveyed both forward and backward to two distinct LSTM layers, both of
which are connected to the same output layer by the Bi-LSTM algorithm. This
structure provides the output layer with full context for all points in the
input sequence, both in the past and in the future. This context is supplied
throughout the entire time period.

###  3.3 Handwritten digit string recognition with CTC

Sequence characters recognition is a common problem of OCR. In this paper, we
proposed an approach to recognize handwritten digit string. After having
features extracted by a convolutional neural network, the main idea is to use
an output connectionist temporary classification layer to get the final
predicted results after using a recurrent neural network to recognize sequence
information in images. This is done after the convolutional neural network has
been used to extract features.

The input image is one-dimensional tensor (after resizing 40x100x1). For
feature extraction, a backbone network is constructed with convolutional, max-
pooling layers, and residual network. After every convolution layer, we
performed batch normalization to prevent internal covariance shift. Output of
feature extraction block are fed as a sequence into labeling block. To advoid
vanishing gradient problem, we use two Bi-LSTM layers instead of traditional
RNN layer. Finally, a fully connected layer is used to reduce the dimension to
13, before passing CTC layer. The CTC layer serves primarily two functions:
the first of these is to decode the output, and the second is to calculate the
loss. The full architecture is shown in figure 1

Figure 1: Our proposed architecture model

###  3.4 Proposed method

Figure 2: Our proposed method flowchart

The first step of our method is image preprocessing. Transcript images are
binarize, removing noises by Gaussian filter. We deskew the images by
Projection profile algorithm. For class ID recognition, we use Template
matching followed by OCR tools. To recognize and calculate coordinates of
lines in transcript, horizontal and vertical masks generated by morphological
operations are fed into Hough transformation. After having full coordinates of
lines, cells of student IDs and test scores are cropped. For student IDs, are
sequence of printed digts, can easily recognized by available OCR tools. In
our method, we use Tesseract-OCR, which is inherently very accurate in
recognizing printed characters. For test scores, we use our Handwritten digits
recognition model with CTC decode. Finally, student IDs and test scores are
combined.

Figure 3: Result of automatic score-inputting system

##  4 Experiment and results

###  4.1 Evaluation of Image-preprocessing

By using a dataset consisting of images of 126 academic transcripts with 1008
vertical lines and 3859 horizontal lines, the results of the Baseline model
and my improved model in detecting lines are shown below:

Figure 4: Results of the two models in detecting vertical lines

The Baseline model achieved an accuracy of 65.57% for vertical lines, whereas
the improved model had an accuracy of 99.6%.

Figure 5: Results of the two models in detecting horizontal lines

The Baseline model achieved an accuracy of 74.65% for horizontal lines,
whereas the improved model had an absolute accuracy of 100%.

###  4.2 Evaluation of models in recognizing handwritten test scores

By using an extracted dataset with 2139 handwritten test scores, the results
of the CNN Baseline model and the my CRNN model are shown below:

Figure 6: Results of the two models in recognizing handwritten test scores

The Baseline model achieved an accuracy of approximately 45.86%, whereas my
improved CRNN model had an accuracy of 96.11%.

###  4.3 Evaluation of automatic score-inputting system

To evaluate the accuracy of the entire score-inputting system, we tested it on
a dataset of 75 scanned academic transcripts with 162 images of size 1653 x
2337.

####  4.3.1 a) Evaluation of Baseline model

Results of the Baseline model:

  * •

The model was able to accurately detect lines in 92 images and recognize class
IDs of 51 images (20 academic transcripts), achieving an accuracy of 31.4%.

  * •

Among 3596 student IDs, the model correctly recognized 2201 IDs, achieving an
accuracy of 61.2% (The majority of images in which the lines were accurately
detected all had their student IDs recognized by the model).

  * •

Among 3596 test scores, the model was able to accurately recognize 1532 test
scores, achieving an accuracy of 42.6%.

####  4.3.2 b) Evaluation of improved model

Results of the improved model:

  * •

In 22 images, the model misidentified 1 vertical line. However, these
misidentified lines didn’t affect the columns of data that needed to be
recognized. Horizontal lines, on the other hand, were all accurately detected.
In all 162 images, the model correctly recognized the class IDs with an
accuracy of 100%.

  * •

Among 3596 student IDs, the model correctly recognized 3481 IDs, achieving an
accuracy of 96.8%.

  * •

Among 3596 test scores, the model was able to accurately recognize 3451 test
scores, achieving an accuracy of 95.9%.

##  5 Conclusion

In this research paper, we have introduced a new approach to the case of
handwritten test scores into the computer. When using additional auxiliary
features on the printout such as horizontal and vertical lines on the A4
paper, we have achieved very good results in clearly separating handwritten
letters and numbers, thereby increasing adds precision to reading handwritten
letters and numbers into numeric data.

In the future, we will put more research on some related issues, in order to
further increase the accuracy:

\- Identify several records of the same person.

\- Identify both letters and numbers at the same time (points are written in
both numbers and words in the one handwritten paper)

##  6 Acknowledgment

We would like to extend our gratitude to the researchers at Hanoi University
of Science and Technology (Hanoi, Vietnam), who were of great assistance to us
in the process of collecting data and testing the application of the model in
real-world scenarios.

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facing a symmetric reserve price across channels. Likewise, because the final
bid remains unchanged the global revenue does not change by using the
symmetric reserve prices.

In what follows, we denote by $r$ such symmetric reserve price.

Let $V(q)$ the bidder’s value of impressions having publisher reserve price
less than $q$. That is,

$\displaystyle V(q)=$ $\displaystyle\max\sum_{i\in I}v_{i}x_{i}$
$\displaystyle\mbox{s.t. }\;\sum_{i\in I}p_{i}x_{i}\leq q,$
$\displaystyle\qquad x_{i}\in[0,1].$

Observe that $V$ is a continuous function, and hence, can be uniformly
approximated by differentiable functions over compact sets [22]. Hence, by a
simple limiting argument, we can assume that $V$ is differentiable.

Given a symmetric reserve price $r$ and a bid multiplier
$\alpha\geq\max\\{r,T\\}$, the value the bidder obtains is $V(\alpha)$ for a
cost of $rV(r)+\int_{r}^{\alpha}qV^{\prime}(q)dq=\alpha
V(\alpha)-\int_{r}^{\alpha}V(q)dq$.

Therefore, the bidder’s best response when channels set a reserve price $r$ is
given by the solution $\alpha(r)$ to equation

$(\alpha-T)V(\alpha)=\int_{r}^{\alpha}V(z)dz.$ (17)

We derive the following observation from $\alpha(r)$.

Observation 1: $\alpha(r)$ is non-increasing as function of $r$. To see this,
notice that the left-hand-side of Equation (17) is independent of $r$ while
the right-hand-side of the equation is decreasing in $r$. Thus, we get that
$\alpha(r)\leq\alpha(r^{\prime})$ for $r^{\prime}<r$.

To conclude the proof, we define

$\underline{r}=\min\\{r\mbox{ s.t. Equation~{}\ref{eq:V} has solution}\\}$
(18)

.111111Notice that $\underline{r}$ is well-defined as the set of $r$ solving
Equation 17 is compact and non-empty (for $r=T,\alpha(T)=T$ solves the
equation). and claim that $\underline{r}$ is the optimal reserve price.
Indeed, for $r>\underline{r}$ we have that $\alpha(r)<\alpha(\underline{r})$
due to Observation 1. Hence, the total revenue with $r$ is $T\cdot
V(\alpha(r))<T\cdot V(\alpha(\underline{r}))$.

For $r<\underline{r}$, notice that Equation (17) does not have solution. This
means that the bidder is buying all impression without making the tCPA
constraint binding. In other words, the bidder is buying the same impressions
but for a cheaper price. We conclude that $\underline{r}$ is the optimal
reserve price. ∎

The following lemma characterizes the bidding equilibrium of the largest
channel in the local model.

######  Lemma 13 (Local model: large channel reserve).

Consider a single tCPA bidder with constraint $T$ and a (pure strategy)
reserve prices equilibrium satisfying that $\sum_{k\in
K}\gamma_{k}r_{k}>\underline{r}$ for $\underline{r}$ defined in Equation (18).
When channels are not symmetric (i.e. $\gamma_{k}\neq\gamma_{k^{\prime}}$ for
some $k,k^{\prime}$), every large channel $\hat{k}$
($\gamma_{\hat{k}}\geq\gamma_{k^{\prime}}$ for $k^{\prime}\neq k$) sets a
reserve price $r_{\hat{k}}=\underline{r}$.

This lemma shows that in the local model, the competition among channels leads
to larger channels setting efficient reserve prices and providing value to the
bidder, improving the efficiency of the allocation. In turn, small channels
raise their reserve price extracting the value provided from larger channels
and creating revenue inefficiencies when aggregating all channels.

###### Proof of Lemma 13.

Fix an equilibrium for the channels $(r_{k})_{k\in K}$ such that $\sum_{k\in
K}\gamma_{k}r_{k}>\underline{v}$.

Consider a local deviation where one channel decreases the reserve price. Let
$s$ the extra-value the bidder obtains when channel $k$ lowers their reserve
price. The bidder will spend this extra-value on more impressions. That is, in
the subgame, the bidder reacts to the decrease in reserve price by increasing
its bid from $\alpha$ to $\alpha_{s}$, satisfying that

$\int_{\alpha}^{\alpha_{s}}(q-T)V^{\prime}(q)dq=s$ (19)

where $V$ is defined in Equation (17) in Lemma 12.

Under this deviation, the revenue gain by channel $k$ is
$\gamma_{k}\int_{\alpha}^{\alpha_{s}}qV^{\prime}(q)dq$ while it exerts a cost
of $s$. From equilibrium condition we have that
$\gamma_{k}\int_{\alpha}^{\alpha_{s}}qV^{\prime}(q)dq\leq s$. Taking $s\to 0$
(i.e. taking the limit of the deviation on $r_{k}$ to zero), we get that
$\gamma_{k}\alpha V^{\prime}(\alpha)d\alpha_{s}/ds-1\leq 0$. From Equation
(19), we also obtain that $(\alpha-T)V^{\prime}(\alpha)d\alpha_{s}/ds=1$.
Plugging these two expressions, and noticing that $V^{\prime}(\alpha)>0$ and
$d\alpha_{s}/ds<0$,121212$\alpha_{s}$ is decreasing on $s$ by the same
argument used for Observation 1 in Lemma 12. we conclude that a channel does
not benefit by locally reducing its reserve price if and only if
$T\geq\alpha(1-\gamma_{k})$.

Conversely, using the same argument we conclude that a channel does not
benefit by increasing its reserve price if and only if
$T\leq\alpha(1-\gamma_{k})$.

To conclude the proof, suppose for the sake of a contradiction that
$r_{\hat{k}}>\underline{v}$ for one of the largest channel $\hat{k}$. Then, it
is feasible for such channel to reduce their reserve price. Thus, in
equilibrium we must have that $T\geq\alpha(1-\gamma_{\hat{k}})$. Because
channels are not symmetric, there is a channel $k^{\prime}$ with
$\gamma_{k^{\prime}}<\gamma_{\hat{k}}$. Then, for channel $k^{\prime}$ we have
that $T>\alpha(1-\gamma_{k^{\prime}})$. This implies that it is channel
$k^{\prime}$ would increase their revenue by increase their reserve price from
$r_{k^{\prime}}$ to $r_{k^{\prime}}+\epsilon$, for some small $\epsilon$. This
contradicts the equilibrium assumption. ∎

After the preliminaries steps we are now in position to proof Theorem 8.

###### Proof of Theorem 8.

From Lemma 11, we can restrict our attention to the case where the bidder is a
tCPA bidder.

Proof that $PoA\geq 1/k$

Consider $(r_{k})_{k\in K}$ an arbitrary pure-strategy reserve price
equilibrium on the local model. There can be the following 3 possibilities.

Case 1. That $\sum_{k\in K}\gamma_{k}r_{k}<\underline{v}$. This case cannot be
an equilibrium: it implies that the bidder is unconstrained. Hence, one
channel can slightly increase its reserve price while keeping the bidder
unconstrained, and hence, keeping the same the bid. Therefore, a channel would
increase its revenue which is a contradiction.

Case 2. If $\sum_{k\in K}\gamma_{k}r_{k}=\underline{v}$, then the bidder faces
the same reserve as in the global optimal solution. Thus, for that case, the
outcome of the local model is the same as of the global model.

Case 3. When $\sum_{k\in K}\gamma_{k}r_{k}>\underline{v}$. If channels are
asymmetric (i.e., $\gamma_{k}\neq\gamma_{k^{\prime}}$ for some
$k,k^{\prime}$), Lemma 13 implies that the largest channel sets a reserve
$r_{\hat{k}}=\underline{v}$. Now, a feasible solution for the tCPA bidder is
to purchase only impressions from one of the largest channel. By doing this
suboptimal bidding strategy, the bidder gets a total value of
$\gamma_{\hat{k}}kV(\alpha(\underline{r}))$. Thus, in an equilibrium, the
bidding parameter $\alpha_{LocalEQ}$ must satisfy
$V(\alpha_{LocalEQ})\geq\gamma_{k}H(\alpha(\underline{r}))$. Since the bidder
is tCPA-constrained, the total revenue across the channels is the target $T$
times the value the bidder gets in an equilibrium. Using the reserve in the
global model is $\underline{r}$ (Lemma 12), we conclude that

$PoA=\inf_{\alpha_{LocalEQ}}\frac{T\cdot V(\alpha_{LocalEQ})}{T\cdot
V(\alpha(\underline{r}))}\geq\frac{\gamma_{k}V(\alpha(\underline{r}))}{V(\alpha(\underline{r}))}=\gamma_{k}\geq\frac{1}{k}.$

It remains to tackle Case 3. when channels are symmetric, i.e.,
$\gamma_{k}=1/k$. Using the same argument for deviations as in Lemma 13, we
have that either one channel sets reserve $r=\underline{r}$ or all channels
are setting the same reserve price in which case $r$ is such that
$T=\alpha_{LocalEQ}(1-1/k)$. If one channel sets $r=\underline{r}$ the same
proof as the asymmetric case holds. If not, since the optimal bid multiplier
also satisfy Equation (17), we get that

$\displaystyle(\alpha_{Global}-T)V(\alpha_{Global})$
$\displaystyle=\int_{\underline{r}}^{\alpha_{Global}}V(z)dz$
$\displaystyle=\int_{\underline{r}}^{r}V(z)dz+\int_{r}^{\alpha_{LocalEQ}}V(z)dz+\int_{\alpha_{LocalEQ}}^{\alpha_{Global}}V(z)dz$
$\displaystyle=\int_{\underline{r}}^{r}V(z)dz+(\alpha_{LocalEQ}-T)V(\alpha_{LocalEQ})+\int_{\alpha_{LocalEQ}}^{\alpha_{Global}}V(z)dz$
$\displaystyle\leq(r-\underline{r})V(\alpha_{LocalEQ})+(\alpha_{LocalEQ}-T)V(\alpha_{LocalEQ})$
$\displaystyle\quad+(\alpha_{Global}-\alpha_{LocalEQ})V(\alpha_{Global}).$

Where the last inequality holds since $V$ is non-decreasing and
$\alpha_{LocalEQ}\leq\alpha_{Global}$.

Rearranging terms and noticing that
$(r-\underline{r})V(\alpha_{LocalEQ})+(\alpha_{LocalEQ}-T)\leq\alpha_{LocalEQ}V(\alpha_{LocalEQ})$
holds since $\underline{r}\leq r\leq T$, we obtain that

$(\alpha_{LocalEQ}-T)V(\alpha_{Global})\leq\alpha_{LocalEQ}V(\alpha_{LocalEQ}).$

We conclude that $PoA\geq 1/k$ by using the equilibrium condition we have that
$T=\alpha_{LocalEQ}(1-1/k)$. This implies that
$V(\alpha_{LocalEQ})/V(\alpha_{Global})\geq 1/k$.

Proof that $PoA\leq 1/k$

To proof the tightness of the $PoA$, we consider an instance with $k$
symmetric channels ($\alpha_{k}=\frac{1}{k})$. There are two types of
impressions the high impressions $H$, and the low impressions $L$ that each
channel owns. The publisher pricing constraint for the high impressions is
$p_{i}=h$ for $i\in H$ and for the low impressions is $p_{i}=l$ for $i\in L$.
We consider that there are $|H|=n_{1}$ high impressions and $|L|=n_{2}$ low
impressions. The tCPA constraint is $T=1$.

We assume that $n_{1}(h-1)=n_{2}$. This implies that $\underline{r}=0$ and
that $Rev_{Global}=T\cdot(h+l)$ (Lemma 12).

We assert that in the local model, there is an equilibrium where all channels
set a reserve price $r_{k}=1$. To see this, suppose a Channel deviates to
$r^{\prime}_{k}$ and assume that $r^{\prime}_{k}$ is an optimal deviation.
Then, we have the following cases to study.

  * •

If $r^{\prime}_{k}>1$, since the tCPA bidder does not have slack the bidder
does not buy from channel $k$. This is not a profitable deviation.

  * •

If $r^{\prime}_{k}<1$ and the tCPA bidder is not able to buy some of the high
impressions $H$, then the deviation is unprofitable: without the deviation,
the channel is selling impressions $L$ at a price $1>r^{\prime}_{k}$.

  * •

If $r^{\prime}_{k}<1$ and the tCPA bidder is able to buy some impressions in
$H$. Then if $h$ is such that $1<h(1-1/k)$, we have that the channel can
further improve its revenue by slightly increasing its reserve price higher
than $r^{\prime}_{k}$ (the proof for the condition on the profitable deviation
is in Lemma 13). This contradicts the optimality of $r^{\prime}_{k}$.

We conclude that so long as $1<h(1-1/k)$, setting reserve price $r_{k}=1$ is
an equilibrium for the local game.

In this equilibrium of the local model, the total revenue across channels is
$n_{2}$

Therefore,

$\displaystyle PoA$
$\displaystyle\leq\frac{n_{2}}{n_{2}+n_{1}}=\frac{n_{1}(h-1)}{n_{1}(h-1)+n_{1}}=1-\frac{1}{h}.$

By taking the limit when $h\to\frac{1}{1-1/k}$, we conclude that $PoA\leq
1/k$. ∎

##  Appendix E Sketch of results for Welfare

Most of our revenue and Price of Anarchy results carry over to
welfare.131313For welfare, we define the Price of Anarchy of the Local model
vs. Global model as $PoA=\inf\frac{Wel(x_{Local})}{Wel(x_{Global})}$. In
particular for the setting without publisher reserves,

  1. 1.

The revenue lower bound in Theorem 2 carries over easily, as revenue is a
lower bound on welfare, and the benchmark we use is the optimal Liquid
Welfare.

  2. 2.

The example for the revenue upper bound in Proposition 3 can be modified to
get a similar upper bound on welfare. This can be done by adding a few extra
bidders – a couple with TCPA $1$ and a couple with budget $1$.

  3. 3.

A similar modification of the example in Theorem 3 gives us an upper bound on
Price of Anarchy for welfare.

  4. 4.

Putting these together, we get a tight bound (up to constants) on Price of
Anarchy for welfare, similar to Theorem 5.

The proofs are deferred to the full paper.