# Scaling description of frictionless dense suspensions under inhomogeneous
flow

Bhanu Prasad Bhowmik  Christopher Ness  School of Engineering, University of
Edinburgh, Edinburgh EH9 3JL, United Kingdom

###### Abstract

Predicting the rheology of dense suspensions under inhomogeneous flow is
crucial in many industrial and geophysical applications, yet the conventional
‘$\mu(J)$’ framework is limited to homogeneous conditions in which the shear
rate and solids fraction are spatially invariant. To address this shortcoming,
we use particle-based simulations of frictionless dense suspensions to derive
new constitutive laws that unify the rheological response under both
homogeneous and inhomogeneous conditions. By defining a new dimensionless
number associated with particle velocity fluctuations and combining it with
the viscous number, the macroscopic friction and the solids fraction, we
obtain scaling relations that collapse data from homogeneous and inhomogeneous
simulations. The relations allow prediction of the steady state velocity,
stress and volume fraction fields using only knowledge of the applied driving
force.

## Introduction.

Dense suspensions are an important class of soft matter system comprising
Brownian or non-Brownian particles mixed roughly equally by volume with
viscous fluid Ness  _et al._ (2022). Their rheology attracts sustained
interest from physicists due to the manifold complex phenomena that arise with
apparently simple constituents Stickel and Powell (2005); Jamali  _et al._
(2020). These include non-equilibrium absorbing state transitions Corté  _et
al._ (2008), shear thickening Barnes (1989), thinning de Kruif  _et al._
(1985), and yield stress behaviour Richards  _et al._ (2020). As well as being
of fundamental interest, characterising this complexity is key to the
extensive use of dense suspensions in various formulation and processing
industries.

A useful model with which to build rheological understanding is the non-
Brownian suspension Guazzelli and Pouliquen (2018), an especially appealing
system when one considers the case of inertialess hard spheres. By analogy to
dry granular systems Jop  _et al._ (2006), a recent study successfully
obtained constitutive laws for this system Boyer  _et al._ (2011), confirming
their rate-independence and finding one-to-one relations between the volume
fraction $\phi$ and each of two dimensionless rheological quantities, the
viscous number $J=\eta\dot{\gamma}/P$ and the macroscopic friction coefficient
$\mu=\sigma_{xy}/P$. Here $\eta$ is the suspending liquid viscosity,
$\dot{\gamma}$ is the shear rate, $P$ is a measure of the particle
contribution to the normal stress, and $\sigma_{xy}$ is the shear stress. This
important result, the so-called $\mu(J)$-rheology, forms the basis of
subsequent models that introduce rate-dependence through additional stress
scales Wyart and Cates (2014); Guy  _et al._ (2018).

The applicability of $\mu(J)$ becomes limited when considering inhomogeneous
flows in which $\dot{\gamma}$ varies spatially Hampton  _et al._ (1997); Oh
_et al._ (2015); Gillissen and Ness (2020). In particular, the lower limit of
$\mu$ (which we denote $\mu_{J}$) is non-zero in all homogeneously flowing
systems irrespective of the particle-particle friction coefficient $\mu_{p}$
Da Cruz  _et al._ (2005); Chialvo  _et al._ (2012); Cheal and Ness (2018) but
can by construction vanish when mechanical balance dictates sign changes in
$\sigma_{xy}$ such as along pipe centrelines. In such scenarios regions that
would otherwise be jammed (_i.e._ with $\mu<\mu_{J}$ and $J=0$) can have non-
zero $\dot{\gamma}$ thanks to facilitation by nearby flowing regions Saitoh
and Tighe (2019); Pouliquen and Forterre (2009). This non-local effect has
been extensively studied in amorphous solids Goyon  _et al._ (2008) and dry
granular systems Kamrin and Koval (2012), often by formulating a fluidity
field with diffusive behaviour characterised by an inhomogeneous Helmholtz
equation. Microscopically it is conceptualized that the fluidity originates
from an activated process that diffuses through the system in a cooperative
way controlled by an inherent length scale Goyon  _et al._ (2008); Bocquet
_et al._ (2009); Kamrin and Koval (2012); Saitoh and Tighe (2019); Bouzid  _et
al._ (2013). Recent works in dry granular matter Zhang and Kamrin (2017); Kim
and Kamrin (2020); Gaume  _et al._ (2020) interpret the fluidity in terms of
particle velocity fluctuations $\delta u$ and density $\rho$, defining a
fourth dimensionless quantity $\Theta=\rho\delta u^{2}/P$ and seeking
constitutive relations linking it to $\phi$, $\mu$ and $I$ Jop  _et al._
(2006) (the dry counterpart to $J$). This successfully collapses data from
homogeneous and inhomogeneous simulations onto a master curve, but is limited
in that the $\Theta$ fields required to make predictions thereafter must be
obtained by simulation. Naturally such findings raise the question of whether
similar constitutive equations exist to unify homogeneous and inhomogeneous
dense suspension rheology.

Here we use particle-based simulation Cundall and Strack (1979) to model dense
suspensions under homogeneous and inhomogeneous conditions, achieving the
latter through an imposed Kolmogorov flow following the approach of Saitoh and
Tighe (2019). We seek to unify the rheology under both sets of conditions by
first defining a dimensionless suspension temperature based on particle
velocity fluctuations, as $\Theta=\eta\delta u/aP$, analogous to the granular
temperature Kim and Kamrin (2020), and then obtaining relations among the four
dimensionless numbers $\phi$, $J$, $\mu$ and $\Theta$. Although the $\mu(J)$
framework was devised based on frictional millimetric grains, recent
experiments demonstrate it is nonetheless applicable to frictionless ones
Etcheverry  _et al._ (2023), and we focus here on the latter. Doing so we find
scalings that can collapse homogeneous and inhomogeneous rheology data onto a
set of master curves that can then be used to predict the rheology of other
flow types.

Figure 1:  Inhomogeneous flow of a frictionless dense suspension. Shown are
(a) a typical configuration of the system for $\bar{\phi}=0.60$, with the red
region highlighting a coarse-graining box; and the steady-state profiles in
$y$ of (b) the $x$-components of the externally applied liquid velocity field
$u_{x}^{\infty}$ (green line) and the coarse-grained velocity field of the
particles ${u}_{x}$ (red points). Velocity is presented here in units of
$\kappa$; (c) the expected shear rate for a Newtonian fluid
$\dot{\gamma}^{\infty}=\partial u_{x}^{\infty}/\partial y$ (green line) and
the measured shear rate $\dot{\gamma}$ (red points), both in units of
$\kappa/a$; (d) the velocity fluctuations $\delta u$ in units of $\kappa$; (e)
the local volume fraction $\phi$, noting that the higher values at low
$\dot{\gamma}$ demonstrate particle migration has taken place; (f) the
pressure $P$ expressed in units of $\eta\kappa/a$; (g) the shear stress
$\sigma_{xy}$ computed from the particle interactions (red points) and by
integrating over the left hand side of Eq. 4 (green points), in the same units
as $P$.

## Simulations details.

We simulate a mixture of frictionless, non-Brownian spheres of radius $a$ and
$1.4a$ mixed in equal number in a periodic box of dimensions $L_{x}$, $L_{y}$,
$L_{z}$, using LAMMPS Plimpton (1995); Ness (2023) (see Fig. 1(a)). Particles
are suspended in a density ($\rho$) matched viscous liquid, and we impose
pairwise contact and hydrodynamic forces as described by Ref.Cheal and Ness
(2018). Briefly, the hydrodynamic lubrication force for particles of radius
$a_{i}$ and $a_{j}$, with center-to-center vector $\bm{r}_{i,j}$, is given by
$\bm{F}^{h}_{i,j}\sim(1/h)\mathbf{u}_{i,j}$, where $\mathbf{u}_{i,j}$ is the
relative velocity of the particles and $h=(a_{i}+a_{j})-|\bm{r}_{i,j}|$.
$F^{h}_{i,j}$ is not computed for $h>0.05a$, and it saturates to
$\sim(1/h^{c})\mathbf{u}_{i,j}$ for $h\leq h^{c}$ (with $h^{c}=0.001a$),
allowing particles to come into contact. Contact forces arise only when
$|\bm{r}_{i,j}|<(a_{i}+a_{j})$ and are given by
$\bm{F}^{c}_{{i,j}}=k\left[(a_{i}+a_{j})-|\bm{r}_{i,j}|\right]\mathbf{n}_{ij}$,
where $k$ is a spring constant and
$\mathbf{n}_{i,j}=\bm{r}_{i,j}/|\bm{r}_{i,j}|$. Particles additionally
experience dissipative drag due to motion relative to the fluid, given by
$\bm{F}^{d}_{i}=6\pi\eta a\left(\bm{u}_{i}-\bm{u}^{\infty}(y_{i})\right)$,
with $\bm{u}_{i}$ the velocity of particle $i$ and $\bm{u}^{\infty}(y_{i})$
the liquid streaming velocity at the position of particle $i$.

Flow is generated by specifying $\bm{u}^{\infty}$ to induce particle motion
through drag. We obtain homogeneous rheology data for fixed-volume systems of
$\phi=0.48$ to $0.65$ by generating simple shear _via_
$\bm{u}^{\infty}(y)=\dot{\gamma}y\bm{\delta}_{x}$, with $y$ the direction of
the velocity gradient and $\bm{\delta}_{x}$ the unit vector along $x$. We
chose our parameters such that $\rho\dot{\gamma}a^{2}/\eta\ll 1$ and
$\dot{\gamma}\sqrt{\rho a^{3}/k}\ll 1$, recovering rate-independence Boyer
_et al._ (2011). To obtain inhomogeneous flow we specify a spatially dependent
liquid velocity as $\bm{u}^{\infty}(y)=\kappa\sin\left(2\pi
y/L_{y}\right)\bm{\delta}_{x}$ (see Fig. 1(b), and the gradient
$\dot{\gamma}^{\infty}$ in Fig. 1(c)), and later test the model with
$\bm{u}^{\infty}(y)=\kappa\sin^{3}(2\pi y/L_{y})\bm{\delta}_{x}$. We run
simulations with $L_{y}=50a$, $100a$ and $200a$ (with $L_{x},L_{z}=20a$) and
systems containing $\mathcal{O}(10^{4})$ particles (we verified that larger
systems produce equivalent rheology results). We simulated systems with mean
volume fraction $\bar{\phi}=0.5$ to $0.63$ (achieved by varying the particle
number), and $\kappa$ is a constant with dimensions of velocity, chosen so
that the measured $\rho\dot{\gamma}a^{2}/\eta$ remains $<0.01$ throughout and
particle inertia is negligible. The stress (a tensor) is computed on a per-
particle basis as
$\mathbb{\Sigma}_{i}=\sum_{j}(\bm{F}_{i,j}^{*}\otimes\bm{r}_{i,j})$, counting
both contact and hydrodynamic forces.

We aim to compare the spatially-variant values of $J$, $\mu$, $\phi$ and
$\Theta$ obtained _via_ inhomogeneous flow with the spatially-invariant ones
obtained _via_ homogeneous flow (the latter follow closely our previous
results Cheal and Ness (2018)). Doing so requires computing the variation in
$y$ of the stress and velocity fields under inhomogeneous flow, which we do by
binning particle data in blocks of width $a$ and volume $V_{b}=L_{x}aL_{z}$,
with the per-block value of a quantity being simply the mean of the per-
particle quantities of the particles with centers lying therein. We compute
the velocity fluctuation (necessary for calculating the $\Theta$ field) of
each particle as $\delta u_{i}=|u_{i,x}-{u}^{\dagger}_{i,x}|$ where $u_{i,x}$
is the $x$-component of $\bm{u}_{i}$ and ${u}^{\dagger}_{i,x}$ is the average
$x$ velocity of all particles with centers lying in a narrow window
$\pm\epsilon$ (taking $\epsilon=\mathcal{O}(0.1a)$) of $y$, and we then bin
$\delta u_{i}$ per block. As all three components of the velocity fluctuations
are statistically equivalent we have used only the $x$ values to compute
$\Theta$. In what follows we report steady state data only foo , averaging
across 6 realizations and at least 500 configurations per realization.

Figure 2:  Relations between the dimensionless control parameters. Shown in
the top row are the relations between the dimensionless viscous number $J$ and
(a) the volume fraction $\phi$ for a range of homogeneous $\phi$ (black data)
and inhomogeneous $\bar{\phi}$; (b) the effective friction coefficient $\mu$
and; (c) the suspension temperature $\Theta$. In the bottom row are the
collapses using the scaling Eqns. 1 (d), 2 (e) and 3 (f), for different
$\bar{\phi}$ and $L$. In (d) we show data for $L/a=50$ to highlight its
deviation from the scaling relation. Black triangles represent homogeneous
data (simple shear) and all other points are for inhomogeneous flow at
different $\bar{\phi}$.

## Results.

Shown in Fig. 1(b)-(g) are, respectively, steady-state profiles in $y$ of the
coarse-grained velocity (in $x$) $u_{x}$, shear rate $\dot{\gamma}=\partial
u_{x}/\partial y$, velocity fluctuations $\delta u$, volume fraction $\phi$,
pressure $P$ ($=(1/3)\text{Tr}(\mathbb{\Sigma}$)), and shear stress
$\sigma_{xy}$, for $\bar{\phi}=0.60$, with each plotted point representing a
block. Although at initialisation the particle density is homogeneous (_i.e._
$\phi\neq\phi(y)$), in the steady state $\phi$ exhibits spatial variation set
up by particle migration to balance the normal stress Hampton  _et al._
(1997); Oh  _et al._ (2015); Morris and Boulay (1999). The velocity profile
follows a similar trend to the applied force, as expected, but is flattened at
the regions of largest $\phi$ leading to significant deviations between
$\dot{\gamma}$ and $\dot{\gamma}^{\infty}$. The pressure becomes spatially
uniform, and the shear stress follows the shear rate in sign. Since $P$ is
spatially invariant in the steady state, one can deduce that the variation of
the quantities $\eta\dot{\gamma}/P$, $\sigma_{xy}/P$ and $\eta\delta u/aP$
follow $\dot{\gamma}$, $\sigma_{xy}$ and $\delta u$ respectively.

We analyse inhomogeneous data by computing the dimensionless control
parameters in each block, defining the scalar shear rate and stress components
on the basis of invariants of the respective tensor quantities so that
$J,\mu>0$. This is done for a range of $\bar{\phi}$, with parametric plots of
$J(y)$, $\phi(y)$, $\mu(y)$ and $\Theta(y)$ shown in Figs. 2(a)-(c). Each
plotted point represents a $y$-coordinate, and colors represent different
$\bar{\phi}$. Shown also (in black) are homogeneous data. Reading across the
data points of a single color from right-to-left represents moves from regions
of high-to-low $\dot{\gamma}$ in the inhomogeneous domain.

The homogeneous $\phi(J)$ and $\mu(J)$ relations follow qualitatively the
result of Boyer  _et al._ (2011), though our frictionless particles render
$\phi_{J}$ and $\mu_{J}$ dissimilar. $\Theta(J)$ follows a power-law relation,
as in dry granular matter Kim and Kamrin (2020) though with a different
exponent. In general large-$J$ inhomogeneous data approximately match
homogeneous data, though they deviate with decreasing $J$ demonstrating the
shortcomings of the existing constitutive laws.

With the help of scaling theory, we next attempt to find constitutive laws
that simultaneously describe the rheology under homogeneous and inhomogeneous
flow. We focus first on how the inverse viscosity
$J/\mu=\eta\dot{\gamma}/\sigma_{xy}$ vanishes as $\phi$ approaches the jamming
point $\phi_{J}$. This trend is followed by all the homogeneous and
inhomogeneous simulations, leading to our first scaling relation

$J/\mu=\alpha\left(\phi_{J}-\phi\right)^{2}\text{,}$ (1)

plotted in Fig. 2(d) with $\alpha=4.1$ and $\phi_{J}=0.6555$.

The next scaling relation is motivated by Kim and Kamrin (2020). In
homogeneous flow, within the range of our data we find $\mu^{2.5}\sim J$ (Fig.
2(b)) and $\Theta^{1.44}\sim J$ (Fig. 2(c)). Since for the range of
$\bar{\phi}$ explored here inhomogeneous data follow homogeneous laws at large
$J$, we expect a scaling of the form $\mu^{2.5}\Theta^{1.44}\sim F_{1}(J)$.
Indeed this results in a good collapse as shown in Fig. 2(e), in which data
are described by the relation

$\displaystyle\Theta^{1.44}\mu^{2.5}=\begin{cases}\beta J^{2}&\text{if
$J>10^{-3}$};\\\ \vartheta J^{1.33}&\text{if $J\leq 10^{-3}$};\\\ \end{cases}$
(2)

with $\beta=3$ and $\vartheta=0.06$.

The final scaling relation is motivated by the relation between granular
fluidity and $\phi$ reported for dry granular matter. Zhang and Kamrin (2017)
write a non-dimensional granular fluidity $\tilde{g}=gd/\delta u$, where
$g=\dot{\gamma}/\mu$, and $d$ is the spatial dimension. We define an
equivalent quantity in terms of the previously discussed dimensionless
numbers, namely $J/\mu\Theta$, though we find a better collapse is achieved
through a change to the exponents as

$\frac{J}{\Theta^{0.8}\mu^{1.2}}=F_{2}(\phi)\text{,}$ (3)

with
$F_{2}(\phi)=\epsilon\left[(\phi-\phi_{m})+\sqrt{(\phi-\phi_{m})^{2}+\zeta}\right]+\lambda\phi$
(see Fig. 2(f)) and $\epsilon=-10.98$, $\phi_{m}=0.618$, $\zeta=0.0004$ and
$\lambda=1.533$. We thus have three scaling relations, Eqs. 1, 2 and 3, that
relate $\phi$, $J$, $\mu$ and $\Theta$. The collapse appears poorer for
$\bar{\phi}=0.5$ (Fig. 2(f)) and $L/a=50$ (Fig. 2(d)), indicating limits to
the range of applicability. An issue in the former case may be that our
simplified hydrodynamics, accounting only for lubrication, becomes nonphysical
at lower $\phi$ and that a more highly resolved fluid field is required.

Given a profile of one of the dimensionless numbers, one could therefore fully
characterise the rheology of the system. In our simulations, however, the only
known input is the externally applied force, which we recall is defined
through $\bm{u}^{\infty}$. To use the scaling relations we need to establish
another relation that can provide us one of these dimensionless numbers from
the knowledge of the applied force profile. Considering the inertia-free
momentum balance $\nabla\cdot\mathbb{\Sigma}=-\bm{f}$ per unit volume, we can
write the following equation for the $k^{th}$ block of the simulation cell
(which we verified in Fig. 1(g)):

$N_{k}6\pi\eta
a\left[Au_{x,k}^{\infty}-u_{x,k}\right]=-\left(\frac{\partial\sigma_{xy,k}}{\partial
y}\right)V_{b}\text{.}$ (4)

Figure 3: Predictions of the scaling relations against simulation data not
used for obtaining the scaling exponents, with
$\bm{u}^{\infty}(y)=\kappa\sin^{3}(2\pi y/L_{y})\bm{\delta}_{x}$. Shown are
(a) the volume fraction $\phi$; (b) the viscous number $J$; (c) the effective
friction coefficient $\mu$; and (d) the suspension temperature $\Theta$, with
predictions given by solid lines and simulation data in points, for
$\bar{\phi}=0.55$ (red) and $0.57$ (green).

Here $N_{k}$, $u_{x,k}^{\infty}$, $u_{x,k}$ and $\sigma_{xy,k}$ are the
particle number in the block, the liquid streaming velocity at the centre of
the block, and the particle velocity and stress averaged over the block, which
has volume $V_{b}$. $A$ is an order unity quantity necessary to account for
small variations in $u_{x,k}^{\infty}$ across the block. The first term of Eq.
4 represents the net applied force and the second represents the net viscous
force exerted by the fluid due to drag. The resultant of these is balanced by
the net stress gradient inside the block. Using the definition of our
dimensionless numbers, Eq. 4 can be rewritten for the streaming velocity at
$y$ as

$u_{x}^{\prime\infty}(y)=\left[\int_{0}^{y}\frac{1}{a}J^{*}(y^{\prime})dy^{\prime}-\frac{2a}{9\phi(y)}\left(\frac{\partial\mu^{*}(y)}{\partial
y}\right)\right]\text{,}$ (5)

with $u_{x}^{\prime\infty}(y)=u_{x}^{\infty}(y)\eta A/aP$ and asterisks
representing multiplication by $\mathrm{sgn}(\dot{\gamma}^{\infty}(y))$,
noting that $P$ is uniform at steady state and using $\phi(y)=(4/3)\pi
a^{3}N(y)/V_{b}$, acknowledging our earlier comment about phase separation foo
. Equation 5 thus relates the externally applied liquid flow field to the
profiles of $J$, $\mu$ and $\phi$.

For a known $\bm{u}^{\infty}$ we solve Eqs. 1, 2, 3 and 5 numerically in the
following way. We first guess a $\phi\left(y\right)$ profile by assuming
accumulation at points where the spatial derivative of the imposed force
vanishes, starting with a simple form as
$\phi(y)=\sum_{j=1}^{n_{p}}a_{j}/[(y-y^{0}_{j})^{2}+b^{2}_{j}]+\phi_{0}$, with
mass conserved through
$\bar{\phi}=\frac{1}{L_{y}}\int_{0}^{L_{y}}\phi\left(y\right)dy$. Here
$y^{0}_{j}$ are the coordinates of the point where the first derivative of the
applied force vanishes, $n_{p}$ is the number of such points and $b_{j}$ is
the width of the Lorentzian function peaked at $y^{0}_{j}$. We then compute
directly $J$, $\mu$ and $\Theta$ using Eqs. 1, 2 and 3, before attempting to
balance Eq. 5. The imbalance of Eq. 5 reflects the accuracy of our guess. We
refine $\phi(y)$ by tuning $\phi_{0}$, $a_{j}$ and $b_{j}$ until Eq. 5 is
satisfied (up to some tolerance). Shown in Fig. 3 are predicted results
compared against ‘unseen’ simulation data (_i.e._ data not used to obtain the
scaling exponents) with $\bar{\phi}=0.55$, $0.57$ and
$\bm{u}^{\infty}(y)=\kappa\sin^{3}(2\pi y/L_{y})\bm{\delta}_{x}$,
demonstrating the degree of success of the scaling relations for predicting
$y$-profiles of $\phi$, $J$, $\mu$ and $\Theta$. Considering the highly non-
linear nature of the scaling relations, the quality of the predictions is
reasonably good.

## Conclusions.

Using particle-based simulation we seek universality amongst flows of dense,
frictionless suspensions. Along with canonical suspension rheology control
parameters $\phi$, $J$ and $\mu$, we introduce a fourth quantity $\Theta$
characterising velocity fluctuations, inspired by recent studies in dry
granular physics Kim and Kamrin (2020). We find a trio of scaling relations
among these quantities that collapse data for homogeneous and inhomogeneous
flow. Utilising a momentum balance we show that from knowledge of the
externally applied force, one can use the relations to predict the features of
a general inhomogeneous flow. Our work raises manifold avenues for future
work. In particular, the microscopic origin of the exponents is not
understood, nor is their generalisation to the broader class of suspensions
that includes polydisperse particles (for which colloidal forces may become
relevant Li  _et al._ (2023)), non-spheres and other complexities. Meanwhile
the question of a diverging lengthscale —apparently a staple of non-local
rheology in dry granular matter Bouzid  _et al._ (2013); Kamrin and Koval
(2012); Tang  _et al._ (2018)— remains open. Computing a granular fluidity
field from our data, we find, similar to Saitoh and Tighe (2019), no
divergence in the characteristic lengthscale, which remains $\mathcal{O}(a)$
everywhere. This raises an important open question regarding what are the
minimal conditions required for a diverging lengthscale in inhomogeneous
particulate flows.

B.P.B. acknowledges support from the Leverhulme Trust under Research Project
Grant RPG-2022-095; C.N. acknowledges support from the Royal Academy of
Engineering under the Research Fellowship scheme. We thank Ken Kamrin, Martin
Trulsson, Mehdi Bouzid, Romain Mari and Jeff Morris for useful discussions.

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# NIST SRE CTS Superset: A large-scale dataset for telephony speaker
recognition

Omid Sadjadi

(August 2021)

##  1 Introduction

This document provides a brief description of the National Institute of
Standards and Technology (NIST) speaker recognition evaluation (SRE) [1]
conversational telephone speech (CTS) Superset. The CTS Superset has been
created in an attempt to provide the research community with a large-scale
dataset along with uniform metadata that can be used to effectively train and
develop telephony (narrowband) speaker recognition systems. It contains a
large number of telephony speech segments from more than 6800 speakers with
speech durations distributed uniformly in the [10s, 60s] range. The segments
have been extracted from the source corpora used to compile prior SRE datasets
(SRE1996-2012), including the Greybeard corpus as well as the Switchboard and
Mixer series collected by the Linguistic Data Consortium
(LDC111https://www.ldc.upenn.edu/). In addition to the brief description, we
also report speaker recognition results on the NIST 2020 CTS Speaker
Recognition Challenge, obtained using a system trained with the CTS Superset.
The results will serve as a reference baseline for the challenge.

##  2 CTS Superset (LDC2021E08)

The NIST SRE CTS Superset is the largest most comprehensive dataset available
to date for telephony speaker recognition. It has been extracted from the
source corpora (see Table 1) used to compile prior SRE datasets
(SRE1996-2012). Table 2 summarizes the data statistics for the CTS Superset.
There are a total of 605,760 segments originating from 6867 speakers (2885
male and 3992 female). Each segment contains approximately 10 to 60 seconds of
speech 222As determined using a speech activity detector (SAD)., and each
speaker has at least three sessions/calls (hence at least 3 segments). Note
that some speakers appear in more than one source corpus, therefore the sum of
the speakers in the table is greater than 6867. Although the vast majority of
the segments in the CTS Superset are spoken in English (including both native
and accented English), there are more than 50 languages represented in this
dataset.

The procedure for extracting segments from the original sessions/calls is as
follows; given a session of arbitrary duration (typically 5–12 minutes) and
speech time marks generated using a speech activity detector (SAD), we start
extracting non-overlapping segments by repeatedly sampling a speech duration
from the uniform distribution [10, 60], until we exhaust the duration of that
session.

Table 1: Original source corpora used to create the NIST SRE CTS Superset Source corpus | LDC Catalog ID | corpusid
---|---|---
Switchboard1 release2 | LDC97S62 [2] | swb1r2
Switchboard2 Phase I | LDC98S75 [3] | swb2p1
Switchboard2 Phase II | LDC99S79 [4] | swb2p2
Switchboard2 Phase III | LDC2002S06 [5] | swb2p3
Switchboard Cellular Part 1 | LDC2001S13 [6] | swbcellp1
Switchboard Cellular Part2 | LDC2004S07 [7] | swbcellp2
Mixer3 | LDC2021R03 [8] | mx3
Mixer4–5 | LDC2020S03 [9] | mx45
Mixer6 | LDC 2013S03 [10] | mx6
Greybeard | LDC2013S05 [11] | gb1
Table 2: Data statistics for the NIST SRE CTS Superset corpusid | #segments | #speakers | #sessions
---|---|---|---
swb1r2 | 26,282 | 442 | 4757
swb2p1 | 33,746 | 566 | 7134
swb2p2 | 41,982 | 649 | 8895
swb2p3 | 22,865 | 548 | 5187
swbcellp1 | 13,496 | 216 | 2560
swbcellp2 | 20985 | 378 | 3966
mx3 | 317,950 | 3033 | 37,759
mx45 | 40,313 | 486 | 4997
mx6 | 70,174 | 526 | 8727
gb1 | 17,967 | 167 | 2188

The LDC2021E08 package contains the following items:

  * •

Audio segments from 6867 subjects located in the data/{subjectids}/
directories

  * •

Associated metadata located in the docs/ directory

The metadata file contains information about the audio segments and includes
the following fields:

  * •

filename (segment filename including the relative path)

  * •

segmentid (segment identifier)

  * •

subjectid (LDC speaker id)

  * •

speakerid (zero indexed numerical speaker id)

  * •

speech_duration (segment speech duration)

  * •

sessionid (segment session/call identifier)

  * •

corpusid (corpus identifier as defined in Table 1)

  * •

phoneid (anonymized phone number)

  * •

gender (male or female)

  * •

language (language spoken in the segment)

For future releases of the CTS Superset we plan to extend the source corpora
to include Mixer 1, 2, and 7 as well.

##  3 Speaker Recognition System

In this section, we describe the baseline speaker recognition system setup
including speech and non-speech data used for training the system components
as well as the hyper-parameter configurations used. Figure 1 shows a block
diagram of the x-vector baseline system. The embedding extractor is trained
using Pytorch333https://github.com/pytorch/pytorch, while the NIST SLRE
toolkit is used for front-end processing and back-end scoring.

Figure 1: Block diagram of the baseline system.

###  3.1 Data

The baseline system is developed using the CTS Superset described in the
previous section. In order to increase the diversity of the acoustic
conditions in the training set, two different data augmentation strategies are
adopted. The first strategy uses noise-degraded (using babble, general noise,
and music) versions of the original recordings, while the second strategy uses
spectro-temporal masking applied directly on spectrograms (aka spectrogram
augmentation [12]). The noise samples for the first augmentation approach are
extracted from the MUSAN corpus [13]. For spectrogram augmentation, the mild
and strong policies described in [12] are used.

###  3.2 Configuration

For speech parameterization, we extract 64-dimensional log-mel spectrograms
from 25 ms frames every 10 ms using a 64-channel mel-scale filterbank spanning
the frequency range 80 Hz–3800 Hz. After dropping the non-speech frames using
SAD, a short-time cepstral mean subtraction is applied over a 3-second sliding
window.

For embedding extraction, an extended TDNN [14] with 11 hidden layers and
parametric rectified linear unit (PReLU) non-linearities is trained to
discriminate among the nearly 6800 speakers in the CTS Superset set. A cosine
loss with additive margin [15] is used in the output layer (with $m=0.2$ and
$s=40$). The first 9 hidden layers operate at frame-level, while the last 2
operate at segment-level. There is a 3000-dimensional statistics pooling layer
between the frame-level and segment-level layers that accumulates all frame-
level outputs from the 9th layer and computes the mean and standard deviation
over all frames for an input segment. The model is trained using Pytorch and
the stochastic gradient descent (SGD) optimizer with momentum ($0.9$), an
initial learning rate of $10^{-1}$, and a batch size of $512$. The learning
rate remains constant for the first $5$ epochs, after which it is halved every
other epoch.

To train the network, a speaker-balanced sampling strategy is implemented
where in each batch 512 unique speakers are selected, without replacement,
from the pool of training speakers. Then, for each speaker, a random speech
segment is selected from which a 400-frame (corresponding to 4 seconds) chunk
is extracted for training. This process is repeated until the training samples
are exhausted.

After training, embeddings are extracted from the 512-dimensional affine
component of the 10th layer (i.e., the first segment-level layer). Prior to
dimensionality reduction through linear discriminant analysis (LDA) to 250,
512-dimensional embeddings are centered, whitened, and unit-length normalized.
The centering and whitening statistics are computed using the CTS Superset
data. For backend scoring, a Gaussian probabilistic LDA (PLDA) model with a
full-rank Eigenvoice subspace is trained using the embeddings extracted from
only the original (as opposed to degraded) speech segments in the CTS
Superset. No parameter/domain adaptation or score normalization/calibration is
applied.

##  4 Results

In this section, we present the experimental results on the NIST 2020 CTS
Challenge progress and test sets obtained using the baseline system. Results
are reported in terms of the equal error rate (EER) and the minimum cost
(denoted as min_C) and defined in the CTS Challenge evaluation plan [16].

Table 3: NIST baseline system performance using on the 2020 CTS Challenge progress and test sets. System | Approach | Training Data | Backend | Set | EER [%] | min_C
---|---|---|---|---|---|---
NIST baseline | x-vector | CTS Superset | Cosine | Progress | 4.37 | 0.190
Test | 4.74 | 0.206
PLDA | Progress | 4.62 | 0.221
Test | 4.67 | 0.224

Table 3 summarizes the baseline results on the 2020 CTS Challenge progress and
test sets. Note that no calibration is applied to the baseline system output.
It is also worth emphasizing here that one could potentially exploit publicly
available and/or proprietary data such as the VoxCeleb to further improve the
performance; nevertheless, this is beyond the scope of the baseline system,
and therefore not considered in this report.

##  5 Acknowledgement

Experiments and analyses were performed, in part, on the NIST Enki HPC cluser.

##  6 Disclaimer

The NIST baseline system was developed to support speaker recognition
research. Comparison of systems and results against this system and its
results are not to be construed or represented as endorsements of any
participant’s system or commercial product, or as official findings on the
part of NIST or the U.S. Government. The reader of this report acknowledges
that changes in the data domain and system configurations, or changes in the
amount of data used to build a system, can greatly influence system
performance.

Because of the above reasons, this system should not be used for commercial
product testing and the results should not be used to make conclusions
regarding which commercial products are best for a particular application.

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