# Experimental studies on the frequency selection in flat plate wakes:
mean flow stability analyses and low dimensional modeling

Dipankar Dutta1, Indra Kanshana1, Shyam Sunder Gopalakrishnan2,3,†, and A. C.
Mandal1,‡ 1 Department of Aerospace Engineering, Indian Institute of
Technology, Kanpur 208016, India
2 Faculté des Sciences, Université libre de Bruxelles (ULB), CP. 231, 1050
Brussels, Belgium
3 Laboratoire de Physique des Lasers, Atomes et Molécules, CNRS UMR 8523,
Université Lille 1 - 59655 Villeneuve d’Ascq Cedex, France  $†$
<EMAIL_ADDRESS>
$‡$<EMAIL_ADDRESS>
###### Abstract

We investigate the global frequency selection of two-dimensional vortex
shedding in the flat plate wake. The analysis is based on the mean flow
velocity profiles obtained from experimental measurements carried out for two
values of Reynolds number, 1850 and 3350, which are based on the plate
thickness and the free-stream velocity. Two different trailing edge geometries
of the flat plate are considered in this study: blunt and circular. By
performing local spatio-temporal analyses on the measured mean flow velocity
profiles, we estimate the global shedding frequency of the flow. This is in
excellent agreement with the shedding frequency measured experimentally. To
complement the study, we carry out a low-dimensional modeling based on the
proper orthogonal decomposition (POD) of the flow fields which is novel for
flat plate wakes. We observe that a model based on only two POD modes produces
an accurate estimate of the global shedding frequency. Our results also
highlight the role of the nonlinear interaction strength between the mean flow
with the higher harmonics thereby experimentally supporting the theoretical
criterion outlined in Sipp and Lebedev (2007).

##  I Introduction

Ever since the seminal work by Strouhal Strouhal (1878) the frequency
selection of periodic vortex shedding in the wakes of flows past obstacles
have been of interest to both researchers and engineers Roshko (1955); Bauer
(1961); Bloor (1964); Huerre and Redekopp (1990); Williamson (1996). With the
advances in experimental measurements, the frequency selection criterion has
been proposed Chomaz et al. (1991); Williamson (1988, 1996) and has been
successfully built in the framework of weakly non-parallel flows Monkewitz
(1988); Huerre and Redekopp (1990); Monkewitz et al. (1993); Chomaz et al.
(1991); Pier and Huerre (2001); Pier (2002). Interestingly, the selection
criterion when applied to the mean flow of bluff-body wakes have shown to
yield particularly good results for the prediction of the frequency of the
unsteadiness. This has been observed in the wake formed by a rectangular
forebody Hammond and Redekopp (1997), and by a cylinder Pier (2002), and has
been confirmed by performing a global stability analysis on the mean flow of a
cylinder wake Barkley (2006); Mittal (2008). The theoretical support for these
observations were advanced by Sipp and Lebedev (2007) where they outlined the
conditions when the stability of a mean flow would approximately match with
the nonlinear frequency of the unsteady wake. Using a weakly global nonlinear
analysis in the vicinity of the critical Reynolds number, they showed that the
frequency selection is governed by the relative interaction strength of the
mean flow and the higher harmonics. Specifically, the global frequency is
well-predicted by a mean flow analysis when the nonlinear interaction of the
zeroth (mean flow) harmonic with the first harmonic is way higher than the
interaction of the second harmonic with the first harmonic. They observed that
the linear stability analysis of the mean flow yields good results for a
cylinder wake, but does not provide a meaningful insight in the case of an
open cavity flow. In addition, it has been noted in Pier (2002); Hwang and
Choi (2006); Gianetti and Luchini (2007) that the global frequency is well
estimated by the local absolute frequency at the stagnation point in the flow.
In this study, we address these concerns for flat plate wakes using
experimentally measured mean flow velocity profiles which is missing in the
literature.

The wake formed by the trailing edge of a flat plate, or a thin aerofoil at
zero incidence, has been extensively studied in the past in the laminar,
transitional, and turbulent regimes Sato and Kuriki (1961); Ko et al. (1970);
Nishioka and Miyagi (1978); Ramaprian et al. (1982); Wygnanski et al. (1986);
Jovic and Ramaprian (1986); Bogucz and Walker (1988); Julien et al. (2003a);
Lasheras and E. Meiburg (1990); Narasimha and Prabhu (1972). Understanding the
frequency selection in flat plate wakes is of particular importance as they
find direct applications in the aerospace industry, where drag reduction and
noise control are desired Jovic and Ramaprian (1986); Carini et al. (2017), in
the design of turbine and compressor blades Bogucz and Walker (1988), and from
an academic point of view. The flat plate wake is influenced by the profile of
the trailing edge. Carefully made measurements by Nishioka and Miyagi (1978)
has showed that the wake behind a flat plate with a sharp trailing edge can be
kept laminar even at a Reynolds number of 3000 (based on the freestream
velocity and the plate length), even though a wake flow is inherently
unstable. The turbulent wake characteristics of flat plates with a sharp
trailing edge have been well investigated A and Walker (1988); Ramaprian et
al. (1982); Jovic and Ramaprian (1986). In contrast, there exists vigorous
vortex shedding for flat plates with a blunt trailing edge Nakamura et al.
(1991); Rai (2016). The vortex shedding from flat plates with square leading
and trailing edges were experimentally studied in Nakamura et al. (1991) where
they showed that the separated shear layer becomes unstable downstream of the
trailing edge corner. As maintaining extremely sharp trailing edges for flat
plates still pose as engineering challenges, it is pertinent to study the
effect of the trailing edge geometry, and is thus considered in the present
study. In addition to the trailing edge geometry Sieverding and Heinemann
(1990); Taylor et al. (2011); Rai (2016), the characteristics of the flat
plate wake depends on the velocities in the upper and lower surfaces of the
plate (Boldman et al., 1976), and various other boundary layer parameters such
as the boundary layer thickness and shape factor Rowe et al. (2001); Durgesh
et al. (2013); Sieverding and Heinemann (1990).

One of the first experimental measurements on the vortex shedding behind thin
flat plates were made by Bauer (1961). The hydrodynamic resonance criteria
advanced by Koch (1985) based on a local stability analysis predicts well the
observed shedding frequency in Bauer (1961), which is governed by the
streamwise location where the instability character changes from an absolute
to a convective nature. Apart from the global frequency selection criterion
proposed in Koch (1985), there are other criteria such as the maximum growth
criterion by Pierrehumbert (1984), and the initial growth criterion by
Monkewitz and Nguyen (1987). In a strictly linear setting, the first rigorous
selection criterion has been established in Chomaz et al. (1991), with the
global frequency given by a saddle-point condition based on an analytic
continuation of absolute instability frequency in the complex
$x=(x_{r},x_{i})$ plane. This criterion is seen to accurately predict the
selected frequency in the wake of a blunt-edged plate Hammond and Redekopp
(1997) using the time-mean flow. Since then the non-linear instability of
slowly divergent flows has been addressed in Pier and Huerre (2001); Pier
(2002) with the bifurcation to self-sustained oscillation in two-dimensional
wake flows being triggered whenever a region of local absolute instability
exists in the flow. The dominant shedding frequency is imposed by the first
absolutely unstable downstream station. As observed in Hammond and Redekopp
(1997), it has been shown in Pier (2002) that the time average of the
oscillating wake behind a cylinder provided the best profile to predict the
vortex shedding frequency.

It is to be noted that the cylinder wake studied in Pier (2002) goes from a
locally convective nature to a finite sized absolutely unstable region, with
the frequency at this transition station fixing the global shedding frequency,
arising from a balance between upstream perturbation growth and downstream
advection Dee and Langer (1983); Pier (2002). However, the wake of the blunt-
edged plate studied in Hammond and Redekopp (1997) is locally absolutely
unstable at the blunt trailing edge. The first absolutely unstable downstream
station in this case would correspond to the streamwise location proposed in
Koch (1985). This observation is detailed in the criterion proposed in
Monkewitz and Nguyen (1987) where the initial growth of the disturbances
within the convectively unstable region downstream of the bluff-body, the
extent of which then subsequently fixes the spatial scale associated with the
global shedding frequency. As mentioned earlier, using a weakly global
analysis Sipp and Lebedev (2007) theoretically outlined the conditions when
the frequency of the time-averaged mean flows matches with the global shedding
frequency of the wake. This is governed by the interaction strength of the
higher harmonics with the mean flow harmonic. These observations have been
extended and verified for the wake of a cylinder using time-averaged mean
flows in Khor et al. (2008). Such an analysis is missing in the literature for
flat plate wakes which forms the objective of the present study. We
investigate the above discussed characteristics for the wake of a flat plate
with a blunt/circular trailing edge by performing a local stability analysis
using the time-averaged flows, and compare them with the global shedding
frequency. In addition, we also aim to support the theoretical criterion
presented in Sipp and Lebedev (2007) using the experimentally obtained flow
fields.

The characteristics of the wake formed by a circular cylinder, and by a flat
plate with a circular trailing edge are indeed different. This aspect is
studied in Ryan et al. (2005) where they carried out a Floquet stability
analysis to investigate the transition scenario in the wake behind a flat
plate with a circular trailing edge, referred to as an elongated cylinder with
an aerodynamic leading edge, and with a blunt trailing edge. Their study
revealed the existence of three different modes, showing similarities and
differences between the two different wakes, concluding that the transition
scenario in the wake of a flat plate with a circular trailing edge may not be
completely generic, as often assumed based on cylinder wakes. The observations
made in Ryan et al. (2005) have been experimentally confirmed in Naghib-
Lahouti et al. (2012, 2014); Doddipatla (2010) wherein they indicate that the
wake topology and their instability at the trailing edge of a flat plate are
different from those observed in the case of a compact body. Furthermore the
wake characteristics of a flat plate with circular and elliptic trailing edges
were numerically studied in Rai (2016), where they concluded that the
streamlining of a trailing edge results in weaker vortex shedding, with a
smaller separated region.

Similarly, various studies have revealed the differences in the secondary
instability mechanisms and the resulting vortical structures downstream of a
flat plate Julien et al. (2003b, 2004) from that of a cylinder wake Leweke and
Williamson (1998); Barkley (2006). The flat plate wake flows were
experimentally studied at low Reynolds numbers in Julien et al. (2003b) which
were formed by the merging of two parallel laminar streams at the trailing
edge. It is observed that the leading modes are characterised by the same
wavelength unlike in a cylinder wake, indicating that they arise from the same
mechanism, which is a combination of hyperbolic and elliptical instability.
These observations have been corroborated numerically in Julien et al. (2004).
The differences in bluff body wakes from that of a flat plate stem from the
fact that the shear layers in bluff-body wakes separate at a certain location
on the surface of the body, thus introducing a second characteristic length in
the problem. Whereas in a flat plate wake, the flow separation occurs right at
the trailing edge. In addition, the base flow varies rapidly behind a cylinder
and is characterised by flow reversal with highly concentrated vorticity
regions. In this region, the secondary instability is absolutely unstable due
to a global secondary bifurcation Barkley and Henderson (1996) whereas for a
flat plate the instability is convective Julien et al. (2003b).

Figure 1: The experimental set-up used in the present study on flat plate
wakes. Two profiles of the flat plate are considered, with a completely blunt
$(a)$ and circular $(b)$, trailing edges respectively. $(c)$ Illustration of
the TR-PIV measurement setup.

In this work, we investigate the global frequency of flat plate wakes by
performing a local stability analysis on the mean flow fields. A blunt
(square/rectangular) and a circular profile for the trailing edge geometry are
considered in this study which are shown in Fig. 1(a,b). The experiments were
conducted in a wind tunnel, a schematic sketch of which is shown in Fig. 1(c).
The two-dimensional velocity fields were obtained by using a Time Resolved
Particle Image Velocimetry (TR-PIV) technique. The study is carried out for
two values of the Reynolds number, 1850 and 3350, based on the freestream
velocity and the plate thickness. Using the obtained flow fields, we perform a
local spatio-temporal stability analysis on the mean velocity profiles at
different streamwise locations of the unsteady wake. The selected global
frequency is then compared with the different characteristic frequencies
discussed earlier. It has been observed that the local absolute frequency at
the end of the absolutely unstable domain of the flow predicts the global
shedding frequency. A observed in other bluff-body flows, the global frequency
of the wake is well predicted by the local absolute frequency at the
stagnation point in the flow. An excellent pedagogical review on the
theoretical tools used in the present study can be found in Huerre and
Monkewitz (2000).

To complement the study, we analyse the experimental data by carrying out a
proper orthogonal decomposition (POD) of the wake velocity fields. POD is a
useful tool, especially in experimental investigations, which is capable of
extracting information using snapshots of the flow fields Aubry et al. (1988);
Noack et al. (2003); Berkooz et al. (1993); Rempfer and Fasel (1994); Siegel
et al. (2008); Galletti et al. (2004); Luchtenburg et al. (2009); Kanshana et
al. (2018). It provides a set of orthogonal and optimal basis functions, onto
which the Navier–Stokes equations can be projected to construct a reduced
order model Rempfer and Fasel (1993); Rajaee et al. (1994). We perform the POD
analysis on the fluctuations of the wake velocity fields by removing the mean
flow. It is observed that over 80$\%$ of the energy of the unsteady wake is
captured by the first ten POD modes and over 70$\%$ of the energy by the first
two POD modes, even for higher Reynolds number considered here, indicating a
low dimensional nature of the flow under consideration. Therefore, a low
dimensional model based on the dominant POD modes is found to yield an
accurate estimate of the selected global frequency. Similar observations have
been reported for a cylinder wake using POD analyses Siegel et al. (2008). In
doing so, we experimentally support the theoretical criterion outlined in Sipp
and Lebedev (2007) which determines whether a mean flow stability analysis is
meaningful and can predict the global shedding frequency, based on the
nonlinear interaction strength of the mean flow with its higher harmonics.

To this end, we organise the paper as follows. The flow configuration, along
with the theoretical, experimental, and numerical tools used in the current
study are briefly outlined in section II. Following this, in section III we
present the mean flow velocity profiles using which the spatio-temporal
stability analyses are performed. The results from POD analysis with a
description of the flow features of a flat plate wake are presented in section
IV. The paper finishes with a discussion on the global shedding frequency of
flat plate wakes along with some concluding remarks in section V.

##  II Flow configuration and methodology

###  II.1 Experimental set-up

The general flow configuration consists of an incompressible fluid stream over
a flat plate with a blunt (circular) trailing edge, a schematic of which is
shown in Fig. 1. Shape of the leading edge of the plate is a super ellipse,
details of which are available in Balamurugan and Mandal (2017). A wake forms
downstream of the flat plate from the fixed separation points at the trailing
edge. Whole field measurements in the wake of the flat plate were carried out
using the time-resolved particle image velocimetry (TR-PIV) technique in a
low-speed wind tunnel. The wind tunnel used in the present study is an open
circuit suction type wind tunnel. The settling chamber of this tunnel houses a
honeycomb and six turbulence reduction screens. However, the settling chamber
is followed by a contraction cone of $16:1$. The $3000$ mm long test section
of the tunnel has a square cross-section of $610$ mm $\times$ $610$ mm, and is
followed by a diffuser. The three bladed fan of this tunnel is powered by a
$14.5$ kW alternating current motor. The tunnel turbulent intensity measured
in the test section is found to be $0.1\%$ of the freestream velocity (see for
details Balamurugan and Mandal (2017)). The flat plate has a thickness of $12$
mm and a length of $700$ mm, and is mounted horizontally in the mid-plane of
the test section of the tunnel. In the following $x$, $y$, and $z$ represent
streamwise, wall normal, and spanwise directions respectively. The free stream
velocity $U_{\infty}$ and the flat plate thickness $D$ were used as the
velocity and length scales, with the Reynolds number given by
$Re=U_{\infty}D/\nu$. In the present study two values of $Re$ are considered:
$1850$ and $3350$, which are carefully chosen so that the boundary layer
remains laminar till it reaches the trailing edge of the flat plate. The
boundary layer profile at a distance of $0.45D$ from the blunt trailing edge
at $Re=3350$ is shown in Fig. 2. The velocity profile closely matches with the
Blasius solution, thereby ensuring that the boundary layer remains laminar for
both values of $Re$ chosen in this investigation.

Figure 2: Boundary layer profile at $.45D$ (i.e. at $x=-0.45D$) from the blunt
trailing edge of the flat plate for $Re=3350$. Open and filled circles denote
the velocity profiles above and below the flat plate, respectively. The
Blasius solution is shown using the solid black line.

The TR-PIV measurements were carried out at the trailing edge of the flat
plate, as schematically shown in Fig. 1. The TR-PIV system consists mainly of
a $4$ mega pixels CMOS camera with 365 Hz repetition rate in double exposure
mode at full resolution (IDT vision, USA), a Nd-YLF dual PIV laser with
$30$mJ/pulse at $1$ kHz repetition rate (Photonics Industries International
Inc, USA), and an IDT synchronizer. Using the inboard memory of the CMOS
camera, $365$ TR-PIV image pairs were captured per second at full resolution.
The flow was seeded using a fog generator. Using appropriate sheet forming
optics, a thin laser sheet of about $1$ mm thickness was produced at the
region of interest, as shown in Fig. 1. Using the ProVISION XS package
procured from IDT, the images of the seeding particles in the streamwise wall-
normal ($x$-$y$) plane were then acquired over a region of interest of
$\approx$ $66$ mm $\times$ $60$ mm. The acquired images were then processed in
ProVISION XS using a correlation window of $32$ pixels $\times$ $32$ pixels.
Mean quantities are estimated based on 800 PIV realizations. The ProVISION XS
is based on mesh free algorithm as detailed in Lourenco and Krothapalli
(2000). This software package and the PIV system were also used in our various
previous works [e.g. Mandal et al. (2010); Mandal and Dey (2011); Phani Kumar
et al. (2015); Balamurugan and Mandal (2017)].

###  II.2 Local stability analysis

The global shedding frequency of bluff-body wakes can be determined from a
local stability analysis based on the concept of absolute instability Chomaz
et al. (1991); Monkewitz (1988); Monkewitz et al. (1993); Chomaz (2005). There
exists a precise location in the complex $x$-plane which acts as a wavemaker
for the entire field, thereby fixing the global frequency. Following Pier and
Huerre (2001); Pier (2002); Gianetti and Luchini (2007), we boldly ignore the
highly non-parallel nature of the flat plate wake, even though the Reynolds
numbers are on the higher side of the laminar range. We derive the local
characteristics at a streamwise location by freezing the $x$\- coordinate, and
by performing a linear stability on the measured mean flow velocity profiles,
$U(y)=U_{b}(x,y)$. Linearizing around this basic flow by adding a small
amplitude perturbation $(u^{\prime}_{x},u^{\prime}_{y})$, gives us the
linearized Navier–Stokes equations

$\displaystyle\partial_{t}u^{\prime}_{x}+U\partial_{x}u^{\prime}_{x}+u^{\prime}_{y}\partial_{y}U$
$\displaystyle=$
$\displaystyle-\partial_{x}p^{\prime}+Re^{-1}\nabla^{2}u^{\prime}_{x},$
$\displaystyle\partial_{t}u^{\prime}_{y}+U\partial_{y}u^{\prime}_{y}$
$\displaystyle=$
$\displaystyle-\partial_{y}p^{\prime}+Re^{-1}\nabla^{2}u^{\prime}_{y},$
$\displaystyle\partial_{x}u^{\prime}_{x}+\partial_{y}u^{\prime}_{y}$
$\displaystyle=$ $\displaystyle 0.$ (1)

We look for travelling wave solutions in the form of normal modes as
$q^{\prime}(x,y,t)=q(y)\exp[\dot{\imath}(kx-\omega t)]$ where
$\omega(\omega_{r}+\dot{\imath}\omega_{i})$ is the complex frequency, and
$k(k_{r}+\dot{\imath}k_{i})$ the complex wavenumber;
$q^{\prime}(x,y,t)=[u^{\prime}_{x},u^{\prime}_{y},p^{\prime}]^{T}$. The linear
stability of these waves are governed by the Orr–Sommerfeld equation White and
Corfield (2006):

$[(-\dot{\imath}\omega+\dot{\imath}kU)(D^{2}-k^{2})-\dot{\imath}k\frac{d^{2}U}{dy^{2}}-\frac{1}{Re}(D^{2}-k^{2})^{2}]u_{y}=0.$
(2)

$(a)$$(b)$

Figure 3: (a) Parallel model wake velocity profile $U(y)$ and (b) the contours
of $\omega_{\dot{\imath}}$ (greyscale) and $\omega_{r}$ (greylines) at
$Re=11.3$, $R=-1$, $N=2$.

The Orr–Sommerfeld equation (2) along with the boundary conditions
$u_{y}=Du_{y}=0$ (on the wall and in the free stream) is then solved using
standard spectral methods Trefethen (2000) which yields the local dispersion
relation $\omega=\Omega^{l}(k,x)$. For doing this, the measured mean flow
velocity profile at a streamwise location is spatially discretized on
$N_{y}=100$ Chebyshev collocation points in the wall normal direction
($x_{j}=\cos\big{(}\frac{j\pi}{N_{y}}\big{)}\epsilon[-1,1],j=0,1,...N_{y}$).
By applying the classical Briggs–Bers criterion Bers (1961); Briggs (1964);
Pier (2002) of zero group velocity condition the absolute frequency
$\omega_{0}(x)$ is then obtained. As part of the current study, the spatio-
temporal analysis of a parallel model wake flow at a low Reynolds number is
carried out. Following Monkewitz (1988); Hwang and Choi (2006), the profile of
the basic flow is given by

$\displaystyle U(y)=1-R+2RF(y),$ (3) $\displaystyle
R=\frac{U^{*}_{c}-U^{*}_{\infty}}{U^{*}_{c}+U^{*}_{\infty}},$ (4)
$\displaystyle F(y)=[1+\sinh^{2N}(y\sinh^{-1}(1))]^{-1},$ (5)

where $N$ is the shape parameter. The superscript $*$ denotes a dimensional
quantity, with $U_{c}$ being the centerline velocity ($y=0$), and $U_{\infty}$
the freestream velocity. Fig. 3(a) shows the parallel model wake profile at a
Reynolds number $11.3$ (based on the average basic flow velocity and the wake
half-width), $R=-1$, and $N=2$. The variation of $\omega_{r}$ and
$\omega_{\dot{\imath}}$ in the complex $k$ plane is shown in panel (b). The
absolute wavenumber $k_{0}=0.8047-0.5569\dot{\imath}$, and the absolute
frequency $\omega_{0}=1.0086+0\dot{\imath}$ is in excellent agreement with
Monkewitz (1988).

###  II.3 POD analysis and low-dimensional modeling

####  II.3.1 POD methodology

The POD analysis of the unsteady flat plate wake is based on the commonly used
“Method of Snapshots” proposed by Sirovich (1987), to find the dominant POD
modes/eigenfunctions. The discrete data of the fluctuations of the velocity
field are obtained from PIV measurements, and are arranged in vectors Mandal
et al. (2010); Pedersen and Meyer (2002) as

$\mathbf{\Psi}_{i}=\mathbf{V}_{i}-\frac{1}{M}\sum_{j=1}^{M}\mathbf{V}_{j},\quad
i=1,2,....,M,$ (6)

where $M$ is the number of ensemble, and $\mathbf{V}_{j}$ is the instantaneous
velocity field corresponding to the $j$th PIV realization. From these vectors
of the velocity fluctuations, as the mean flow fields are removed, the
elements of a covariance matrix are formed as,

$R_{ij}=(\mathbf{\Psi}_{i},\mathbf{\Psi}_{j}).$ (7)

The covariance matrix $R$ is a $M\times M$ symmetric matrix with non-negative
eigenvalues, $\lambda$. These eigenvalues correspond to the energy of the
respective POD mode. The energy fraction of the $k$-th mode is given by
$E_{k}=\frac{\lambda_{k}}{E}$ where $E$ is the total energy of the POD modes.

The eigenfunctions, $\mathbf{\Phi}^{k}$, are constructed using the
eigenvectors of the co-variance matrix as,

$\mathbf{\Phi}^{k}=\sum_{i=1}^{M}\mathbf{\phi}_{i}^{k}\mathbf{\Psi}_{i},\quad
k=1,....,M,$ (8)

where $\phi_{i}^{k}$ is the $i$-th component component of the $k$-th
eigenvector, and the eigenfunctions are normalized such that
$(\mathbf{\Phi}^{k},\mathbf{\Phi}^{l})$ = $\delta_{kl}$, where $\delta_{kl}$
denotes the Kronecker delta Mandal et al. (2010). Using the eigenfunction
system $\mathbf{\Phi}^{k}$, one can expand the fluctuating velocity field,
$\mathbf{v}(\mathbf{x},t)$, as,

$\mathbf{v}(\mathbf{x},t_{n})=\sum_{k}a^{k}(t_{n})\mathbf{\Phi}^{k}(\mathbf{x}),$
(9)

where the time coefficients, $a^{k}(t_{n})$, are obtained by projecting the
instantaneous snapshots on the eigenfunctions, i.e.

$a^{k}(t_{n})=(\mathbf{v}(\mathbf{x},t_{n}),\mathbf{\Phi}^{k}(\mathbf{x})).$
(10)

####  II.3.2 Low-dimensional modeling

Associated with the Galerkin projection, low-dimensional modeling method
allows one to project the Navier–Stokes equations onto the orthonormal POD
modes and get a system of ordinary differential equations for the time
coefficients, $a_{k}(t)$. For Galerkin projection, we follow the formulation
presented in Rajaee et al. (1994); Kanshana et al. . Since POD eigenfunctions
satisfy the continuity equation, we consider the following momentum equation
for the instantaneous velocity, $\mathbf{V}$. Neglecting the body force term
and considering incompressible flow, the momentum equation for the
instantaneous velocity, $\mathbf{V}$, reads as

$\frac{\partial\mathbf{V}}{\partial
t}+\mathbf{V}\cdot\nabla\mathbf{V}=-\frac{\nabla
P}{\rho}+\nu\Delta\mathbf{V},$ (11)

where $P$ is the pressure and $\nu$ is the kinematic viscosity; here,
$\mathbf{\overline{V}}$ and $\mathbf{v}$ are the mean and the fluctuating
parts of the velocity. Substituting
$\mathbf{V}=\overline{\mathbf{V}}+\mathbf{v}$ in equation (11) and subtracting
the time-averaged momentum equation, we obtain

$\frac{\partial\mathbf{v}}{\partial
t}+\mathbf{v}\cdot\mathbf{\nabla\overline{V}}+\mathbf{\overline{V}}\cdot\mathbf{\nabla
v}+\mathbf{v}\cdot\mathbf{\nabla v}-\overline{\mathbf{v}\cdot\mathbf{\nabla
v}}=-\frac{1}{\rho}\mathbf{\nabla}p+\nu\mathbf{\triangle}\mathbf{v},$ (12)

where $p$ is the fluctuating pressure. Using the Eq. 9 in the above, and
projecting the resulting equation on the POD modes, $\Phi_{k}$, we have

$\frac{da^{k}}{dt}=A^{ki}a^{i}+B^{kil}(a^{i}a^{l}-\overline{a^{i}a^{l}})+C^{k},$
(13)

where the coefficients are given by

$A^{ki}=-(\mathbf{\Phi}^{k},\mathbf{\Phi}^{i}\cdot\mathbf{\nabla\overline{V}})-(\mathbf{\Phi}^{k},\mathbf{\overline{V}}\cdot\nabla\mathbf{\Phi}^{i})+\nu(\mathbf{\Phi}^{k},\triangle\mathbf{\Phi}^{i}),$
(14)
$B^{kil}=-(\mathbf{\Phi}^{k},\mathbf{\Phi}^{i}\cdot\nabla\mathbf{\Phi}^{l}),$
(15) $C^{k}=-\frac{1}{\rho}(\mathbf{\Phi}_{k},\nabla p).$ (16)

Note that $\overline{a^{k}a^{l}}$ = $\delta_{kl}\lambda_{k}$ as the
coefficients are uncorrelated. Utilizing the given POD modes and the mean
velocities, one can calculate the coefficients of Eq. 13. Further, one can
find that $C^{k}=0$, considering homogenous boundary conditions and divergence
free nature of the POD modes (see Rajaee et al. (1994) for further details).

##  III Mean flow fields and shedding frequency

${(a)}$$(b)$$(c)$$(d)$

Figure 4: Mean flow fields of the wake of a flat plate with a blunt trailing
edge. The spatial distribution of the time-averaged mean flow fields of
streamwise velocity component for $(a)$ $Re=1850$ and $(b)$ $Re=3350$. The
zero contour represents the extent of the recirculation zone. The wall-normal
mean velocity profile $U(y)=U_{b}(x,y)$ at $(a)$ $x/D=0.86$ for $Re=1850$, and
$(b)$ $x/D=0.56$ for $Re=3350$. The continuous lines represent the best fit
mean flow velocity profiles given by Eq.(17) which are used in the local
spatio-temporal stability analyses, with $(c)$
$c_{1}=-1.165,c_{2}=0.289,c_{3}=0.996$, and $(d)$
$c_{1}=-1.141,c_{2}=0.217,c_{3}=1.026$.

We now present the mean velocity fields of the flat plate wake obtained for
blunt, and circular, trailing edges for two values of the Reynolds number.
Fig.4 shows the time-averaged velocity profiles of the flat plate wake with a
blunt trailing edge. The spatial distribution of the mean flow fields,
$U_{b}(x,y)$, are presented in Fig. 4$(a,b)$. Directly downstream of the
trailing edge, the flow reverses, as can be seen from the negative values of
the centerline velocity. Moving downstream along the flat plate wake, we can
observe that the centerline velocity decreases up to a point where the flow
reversal is maximum, following which it increases. The variation of the mean
flow velocity along the wall-normal direction at two different streamwise
locations are shown in panels $(c,d)$ for $Re=1850$ and $3350$, respectively.
To perform a local stability analysis of these time-averaged velocity fields,
as in other experimental Asai et al. (2002); Balamurugan and Mandal (2017) and
numerical studies Julien et al. (2004), it is convenient to fit the measured
profiles using an analytical expression as

$U(y)=c_{1}\exp(-y^{2}/c_{2})+c_{3}.$ (17)

Here $c_{1}$, $c_{2}$, and $c_{3}$ are constants which were varied so as to
best predict the measured experimental velocity profiles. Indeed, their values
are different at each streamwise location. These are shown by the solid black
lines in Fig.4$(c,d)$. The length of the recirculation zone, represented using
the zero contour of the streamwise velocity field in panels $(a,b)$, increases
with the Reynolds number. This is elucidated in Fig.5 where the variation of
the centerline velocity downstream of the flat plate with a blunt trailing
edge are shown in panels $(a,b)$. The contours of the complex frequency
$\omega$ in the complex $k$ plane at the respective stagnation points for
$Re=1850$ and $Re=3350$ are shown in panels $(c,d)$.

$(a)$$(c)$$(b)$$(d)$

Figure 5: Variation of the centerline velocity as a function of the downstream
distance behind the blunt trailing edge of the flat plate. Parameter settings:
(a) $Re=1850$, (b) $Re=3350$. Contours of $\omega_{\dot{\imath}}$ (greyscale)
and $\omega_{r}$ (greylines) at $Re=1850$ and $Re=3350$ at the corresponding
stagnation points. The thick black line denotes $\omega_{i}=0$.

Figure 6: Temporal amplification rate $\omega_{0i}$ (circles) and
$\omega_{0r}$ (diamonds) as a function of the streamwise distance for the
different flat plate wakes in the present study. The upper panels $(a,b)$
correspond to the blunt trailing edge, while $(c,d)$ for the circular trailing
edge. The straight vertical line represents the stagnation point
$x_{\textrm{st}}$, while the dashed line indicates the end of the AU region.
Parameter settings: $(a,c)$ $Re=1850$ and $(b,d)$ $Re=3350$.

To delineate the absolutely unstable (AU) region from the convectively
unstable (CU), we extend the local analysis to the entire wake region. Fig.6
shows the variation of the real and imaginary parts of the absolute frequency
$\omega_{0}$ as a function of the downstream distance from the trailing edge
of the flat plate. The AU region where $\omega_{0i}>0$ is demarcated using
dotted lines. It can be seen in Fig.6 that the flow is locally AU both for
blunt, and circular trailing edges directly downstream in the flat plate wake.
The extent of the AU region indeed increases with the Reynolds number, beyond
which the flows are CU. It is this boundary of the AU region which is of
particular interest, as they are defined by real absolute frequencies Pier et
al. (1998); Pier and Huerre (2001). It can be seen from Fig.6 that the
absolutely unstable region closely follows the flow reversal region, extending
slightly beyond it. For $Re=1850$ the distance between the stagnation point
and the front of the AU region is $0.0705D$, while for $Re=3350$ is $0.0912D$.
This feature has also been observed in other bluff-body wake flows Pier and
Huerre (2001); Pier (2002); Hwang and Choi (2006); Hammond and Redekopp
(1997). It can also be noted from Fig.6 that the local frequency at a given
streamwise location in the wake is higher at a larger Reynolds number.

$(a)$$(b)$$(c)$$(d)$

Figure 7: The experimentally measured frequency spectra for the different cases considered in the present study. The global shedding frequency increases with the Reynolds number, and remains roughly the same irrespective of the trailing edge geometry.  Trailing edge | $Re$ | $\omega_{\textrm{st}}$ | $\omega_{\textrm{AU/CU}}$ | $\omega_{\textrm{exp}}$ | $\omega_{\textrm{saddle}}$ | $\omega_{\textrm{POD}}$
---|---|---|---|---|---|---
Blunt | $1850$ | $1.0425$ | $1.0549$ | $1.0487$ | $0.9184$ | $1.0301$
Blunt | $3350$ | $1.2331$ | $1.2745$ | $1.2516$ | $1.0329$ | $1.2480$
Circular | $1850$ | $1.1653$ | $1.1886$ | $1.1995$ | $1.0076$ | $1.2214$
Circular | $3350$ | $1.2721$ | $1.3480$ | $1.3744$ | $1.0931$ | $1.3468$
Table 1: Summary of the different characteristic frequencies ($\omega_{0r}$)
from the present study. Here $\omega_{\textrm{st}}$ and
$\omega_{\textrm{AU/CU}}$ are the frequencies obtained from a local spatio-
temporal analysis at the stagnation point and at the end of AU region
respectively. The experimentally measured shedding frequency is denoted by
$\omega_{\textrm{exp}}$. Here, $\omega_{\textrm{saddle}}$ denotes the circular
frequency obtained from the saddle point criterion, and
$\omega_{\textrm{POD}}$ obtained from the POD based low dimensional model, as
described below.

It has to be pointed out that unlike in cylinder wakes, the flow is AU at the
trailing edge of the flat plate. In the former the flow goes from a locally CU
domain to a finite sized region where the flow is locally AU. The frequency at
this transition station, from local convective to absolute instability, fixes
the global shedding frequency Pier (2002). This arises from a balance between
upstream perturbation growth from the AU region and downstream advection from
the CU region resulting in a stationary front Dee and Langer (1983); Pier
(2002). However, in flat plate wakes the flow is already AU at the trailing
edge, and hence the global selection frequency outlined in Pier (2002) is not
directly applicable to the present flows. We shall see that the other
characteristic frequencies discussed in Pier (2002), notably the real absolute
frequency corresponding to the AU/CU transition station Pier and Huerre
(2001), which corresponds to the criterion outlined in Koch (1985), and the
local absolute frequency at the stagnation point, predicts the observed global
shedding frequency. The dimensional absolute frequency at any location can be
calculated as

$\omega_{0r}=2\pi\Big{(}\frac{fD}{U_{\infty}}\Big{)}.$ (18)

For $Re=1850$ the free stream velocity $U_{\infty}$ is $2.42\textrm{m/s}$,
whereas for $Re=3350$, $U_{\infty}$ is $4.38\textrm{m/s}$. The flat plate
thickness $D$ in the current study is $12\textrm{mm}$. The characteristic
frequencies at some streamwise locations of interest, along with the
experimentally measured values are presented in table 1. Fig. 7 summarises
these experimentally measured global shedding frequencies in the present
study. They are in excellent agreement with the local absolute frequency at
the streamwise location where the flow goes from an absolutely unstable to a
convectively unstable region. It may be noted here that the circular
frequency, $\omega_{\textrm{saddle}}$, is obtained based on the analytical
continuation of $\omega_{0}$ in the complex $x$ plane Chomaz (2005); Hammond
and Redekopp (1997) following a second order polynomial fit through
$\omega_{0}$ as described in Rees (2009). To further investigate this
excellent agreement using the time-averaged velocity profiles, we perform the
POD analysis of the fluctuating velocity data; $\omega_{\textrm{POD}}$ in
table 1 is found from this analysis, and is presented in the following
section.

##  IV Low dimensional modeling

Figure 8: The relative contribution of the POD modes to the total energy. It
can be seen that the first two POD modes, which occur in pairs, contribute
over $80\%$ to the total energy in all the four different cases considered in
the present study.

Figure 9: The spatial distribution of the first six POD modes for the four
different cases considered in the present study. It can be seen that the first
two modes have the same spatial structure, albeit with a slight shift in the
streamwise direction. The fact that the POD modes occur in pairs is also
manifestly evident from their spatial distributions.

Figure 10: The project POD coefficients, $a^{k}$, corresponding to the POD
modes shown in Fig. 9.

Figure 11: Spectra of the POD coefficients, $a^{k}$, shown in Fig. 10. P indicates power in arbitrary scales. Table 2: Relative and cumulative energy for the first 10 POD modes for four different cases. Mode | Blunt | Blunt | Blunt | Blunt | Circular | Circular | Circular | Circular
---|---|---|---|---|---|---|---|---
| $Re=1850$ | $Re=1850$ | $Re=3350$ | $Re=3350$ | $Re=1850$ | $Re=1850$ | $Re=3350$ | $Re=3350$
Number | Relative | Cumulative | Relative | Cumulative | Relative | Cumulative | Relative | Cumulative
| Energy (%) | Energy (%) | Energy (%) | Energy (%) | Energy (%) | Energy (%) | Energy (%) | Energy (%)
1 | 42.6062 | 42.6062 | 37.0986 | 37.0986 | 41.4926 | 41.4926 | 38.0795 | 38.0795
2 | 38.6873 | 81.2934 | 34.4181 | 71.5167 | 38.0011 | 79.4937 | 36.1282 | 74.2077
3 | 2.6852 | 83.9786 | 1.8938 | 73.4105 | 2.1983 | 81.692 | 2.2511 | 76.4589
4 | 2.6349 | 86.6135 | 1.6469 | 75.0573 | 2.1657 | 83.8577 | 2.138 | 78.5968
5 | 1.352 | 87.9655 | 1.6044 | 76.6617 | 1.1788 | 85.0365 | 1.4845 | 80.0813
6 | 1.2912 | 89.2567 | 1.0747 | 77.7364 | 1.1075 | 86.1441 | 1.0535 | 81.1348
7 | 0.8246 | 90.0813 | 1.0381 | 78.7745 | 0.6634 | 86.8074 | 1.0034 | 82.1382
8 | 0.5391 | 90.6204 | 0.9213 | 79.6958 | 0.6358 | 87.4433 | 0.7792 | 82.9174
9 | 0.4842 | 91.1046 | 0.9029 | 80.5986 | 0.5541 | 87.9974 | 0.7426 | 83.66
10 | 0.4131 | 91.5177 | 0.7123 | 81.311 | 0.5075 | 88.5048 | 0.5091 | 84.1691

To quantify the interaction between the mean flow harmonic with the higher
harmonics, which is the crux of the theoretical criterion derived in Sipp and
Lebedev (2007), we look at the reduced order dimensionality of the flat plate
wake. To this end, we compute the energy contribution of the different POD
modes. As outlined in the previous section, the eigenvalue corresponding to
each POD mode represents the proportional contribution to the total energy
content. Fig. 8 describes the relative energy distribution of the different
modes, for the different cases investigated in the present study. The
numerical values of the relative and the cumulative energy of the first ten
POD modes are also listed in table 2. One may notice that the first two POD
modes are the dominant ones. To be specific, the first two modes carry about
$81.3\%,71.5\%,79.5\%$ and $74.2\%$ of the total energy, for blunt $Re=1850$,
blunt $Re=3350$, circular $Re=1850$, circular $Re=3350$ cases, respectively.
Table 2 also shows that the first ten POD modes carry more than $80\%$ energy
for all the cases. It also indicates that some eigenvalues come as pairs, for
example, first two eigenvalues for all the cases; the corresponding modes are
known as degenerate eigenmodes, which originate due to some symmetry in the
flow Deane et al. (1991); Aubry et al. (1992); Rempfer and Fasel (1994);
Mandal et al. (2010), as further discussed below. Similar observations have
also been made in the case of cylinder wakes Noack et al. (2003); Siegel et
al. (2008).

The POD modes or the eigenfunctions corresponding to the first six eigenvalues
are shown in Fig. 9. The corresponding time coefficients obtained from direct
projections (equation 10) and their frequency spectra are shown in Fig. 10 and
11, respectively. The first two near degenerate POD modes are nearly the same
with a slight spatial and temporal shift as seen in Fig. 9 and 10,
respectively. This sort of shift associated with a pair of POD modes is
attributed to some sort of symmetry/traveling disturbance in the flow. Here,
this is due to the vortex shedding at the trailing edge of the flat plate, as
the frequency of these disturbances in Fig. 11 (first row) matches with the
shedding frequency in Fig. 7. Similarly, second and third pairs of modes
corresponding to the second and third pairs of eigenvalues, for the blunt
$Re=1850$ and the circular $Re=1850$ cases, show the first and the second
harmonics associated with the vortex shedding. In contrast, one solitary mode
(mode 3 and mode 5) which does not make any pair is seen to appear for the
blunt and circular cases at $Re=3350$, respectively. This mode is similar to
the shift mode as discussed in Noack et al. (2003) for a circular cylinder.
The spatial structure of this mode is shown in Fig. 9 (second column 3rd mode,
and fourth column fifth mode) and the corresponding time series and frequency
of this mode in Figs. 10 and 11, for the blunt and the circular cases at
$Re=3350$. These observations indicate that this mode is associated with a
slowly moving disturbance. These results show that the trailing edge geometry
has little influence on the vortex shedding characteristics, whereas the
Reynolds number does play a significant role. Further, the present POD
analysis reveals that the flat plate wake at this moderate Reynolds number is
low dimensional in nature as the first ten POD modes can capture more than
$80\%$ of the flow energy.

$(a)$$(b)$$(c)$$(d)$

Figure 12: Comparison of the simulated coefficients (solid line), $a_{1}$ and
$a_{2}$, obtained using different mode models (i.e. 2, 4, 6 modes models),
with their experimental counter parts (red filled circles), calculated from
direct projection, for the different cases considered in the present study.

$(a)$$(b)$$(c)$$(d)$

Figure 13: Comparison of the frequency spectra of the time coefficients,
$a_{1}$ and $a_{2}$ obtained from the simulation (solid line), and from the
direct projection (dashed line). They are in excellent agreement in all the
four different cases investigated.

Since the vortex shedding dynamics is low dimensional, it is interesting to
check whether a reduced order model/low dimensional model can describe the
vortex shedding characteristics. Considering just two POD modes, the equation
13 was numerically solved in Matlab. The numerical solutions for $a_{1}$ and
$a_{2}$ are compared with their experimental counter parts obtained using
direct projection, as shown in Fig. 12, for all the cases. It can be seen that
the coefficients of the first two modes closely follow the experimental data.
The simulated frequencies associated with the first two POD modes, and the
experimentally measured global shedding frequency from the instantaneous
velocity fields are shown in Fig. 13. One may notice an excellent agreement
between the numerical solution and the experimental data, at least for one
shedding cycle. In fact, inclusion of more number of modes while solving the
equation 13, does not improve the solution significantly, as seen in Fig. 12.
Although the amplitudes are found to be comparable with the experimental data
for the four and six mode models, there is a significant deviation of phase as
the solution progresses in time, which is better seen for higher $Re$ cases
considered here.

These findings reaffirm the fact that there is a strong interaction of the
mean flow (zeroth harmonic) with the first harmonic of the flow unsteadiness
as compared to the second and third harmonics of the flow unsteadiness. This
is due to the fact the simulated results were obtained from the numerical
solution of Eq. 13 using the mean velocity and the first two POD modes. This
clearly signifies that the interaction strength between the zeroth and the
first harmonic is the dominant one. This is consistent with the theoretical
finding of Sipp and Lebedev (2007), and in turn explains why the mean flow
velocity fields can give an accurate estimate of the global unsteadiness,
thereby exerpimentally supporting Sipp and Lebedev (2007). It is pertinent to
point out that the theoretical criterion outlined in Sipp and Lebedev (2007)
has been experimentally shown only for the case of a circular cylinder wake in
Khor et al. (2008).

##  V Discussion and concluding remarks

The global shedding frequency of the flat plate wake is well-predicted from a
local spatiotemporal stability analysis based on the mean flow velocity
fields. The mean flow velocity fields were obtained using a time resolved-
particle image velocimetry (TR-PIV) technique in a low-speed wind tunnel. Two
values of the Reynolds number, $Re$, based on the freestream velocity
$U_{\infty}$ and the flat plate thickness $D$ (which is fixed throughout the
study), were considered: $1850$ and $3350$. A blunt, and a circular profile is
used for the trailing edge geometry of the flat plate. Though the time-mean
velocity profiles at different streamwise locations can be obtained using the
hotwire anemometry technique (as in Khor et al. (2008)), the present
investigation uses a non-intrusive PIV measurement technique. This measurement
technique is best suited for separating flows like the present one, and it is
widely used for simultaneous whole field measurements as compared to the
hotwire anemometry technique. The local stability analysis assumes that the
flows are nearly parallel in the regions where the stability analysis is
performed. Despite this, as observed in earlier studies, notably the vortex
shedding from interacting boundary layers at a blunt trailing edge of a
rectangular forebody in Hammond and Redekopp (1997), the selected frequency is
surprisingly well estimated with a deviation within $1\%$. In addition, the
frequency at the end of the absolutely unstable region is seen to be selected
by the global wake in the present study, which is the hydrodynamic resonance
criterion advanced by Koch (1985). Similar observations have been detailed for
a cylinder wake based on the mean flow Pier (2002); Khor et al. (2008).

As discussed in Pier (2002), the linear global frequency corresponding to the
streamwise locations where the flow changes the nature of instability from
convective to absolute, cannot account for the real global frequency selected
by the wake. In addition, the saddle-point criterion gave the best estimate
for the shedding frequency of the fully developed cylinder wake when the
stability analysis was applied on the time-averaged mean flow. This has been
theoretically Sipp and Lebedev (2007), numerically Barkley (2006); Mittal
(2008), and experimentally Khor et al. (2008) supported. The same
characteristics have been observed in the present study based on time-averaged
mean flow fields for a flat plate wake, with the frequencies obtained from a
local analysis providing excellent predictions for the global shedding
frequency. Interestingly, as observed in other wake flow studies, the local
absolute frequency at the stagnation point was found to give an excellent
estimate of the global shedding frequency. We would like to emphasize that to
the best of our knowledge, this is the first experimental study to
quantitatively investigate the global shedding frequency of a flat plate wake,
and compare the different selection criteria available in literature using
mean flow velocity fields.

To further support these observations, we have performed a POD analysis using
the velocity fluctuations. It has been shown that the first two POD modes
(which occur in pairs) account for over $70\%$ to the total energy of the
flow. The frequency of the wake predicted using a low dimensional model with
these two modes are in excellent agreement with the experimentally measured
values. The physical mechanism behind the two theoretical conditions presented
in Sipp and Lebedev (2007) is related to the fact the frequency of the
nonlinearly saturated limit cycle of the unsteady wake depends on the
interaction of the mean flow with the higher harmonics. The fact that the
first two POD modes dominate the total energy supports the criterion outlined
in Sipp and Lebedev (2007). The findings from the present study should
motivate further studies and possible control applications where the accurate
estimation of the global shedding frequency of flat plate wakes are highly
desired.

## ACKNOWLEDGEMENTS

D. D., I. K., and A. C. M. thankfully acknowledge the financial support
provided by IIT Kanpur for the TR-PIV system. Dr. Balamurugan and Mr. Eswar
Sunder are also thankfully acknowledged for their assistance with the data
analyses and the TR-PIV measurements. They also thank Mr. Abhinath Kr. Yadav
and Mr. Dorilal for their help with the experimental setup in the wind tunnel.
S. S. G. acknowledges the support from the French National Research Agency
(LABEX CEMPI, Grant No. ANR-11- LABX-0007) as well as the French Ministry of
Higher Education and Research, Hauts de France council and European Regional
Development Fund (ERDF) through the Contrat de Projets Etat-Region (CPER
Photonics for Society P4S).

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# A counterexample to the CFT convexity conjecture

Adar Sharon<EMAIL_ADDRESS>Simons Center for Geometry and
Physics, SUNY, Stony Brook, NY 11794, U.S.A.  Masataka Watanabe
<EMAIL_ADDRESS>Center for Gravitational Physics and
Quantum Information (CGPQI), Yukawa Institute for Theoretical Physics, Kyoto
University, Sakyo-ku, Kyoto 606-8502, Japan

###### Abstract

Motivated by the weak gravity conjecture, [Phys. Rev. D 104, 126005]
conjectured that in any CFT, the minimal operator dimension at fixed charge is
a convex function of the charge. In this letter we construct a counterexample
to this convexity conjecture, which is a clockwork-like model with some
modifications to make it a weakly-coupled CFT. We also discuss further
possible applications of this model and some modified versions of the
conjecture which are not ruled out by the counterexample.

##  I Introduction

The swampland program [1] (see [2, 3] for reviews) aims to find universal
constraints on low-energy quantum field theories which can be uplifted to UV-
complete theories of gravity. The result has been a series of conjectures
which have gone through numerous stringent tests in various models, and to a
deep understanding of relations between these conjectures.

An intriguing direction of study is to assume that swampland conjectures are
also obeyed in any asymptotically AdS background, in which case the AdS/CFT
duality maps them to statements which apply to arbitrary CFTs. This is very
attractive since CFTs are simpler objects than quantum theories of gravity,
and as such are much better understood. As a result, the CFT versions of
swampland conjectures should be much easier to prove (or disprove). However,
such claims require a significant leap of faith, since many swampland
conjectures have mostly been studied in the context of weakly-coupled and
weakly-curved gravitational theories. Indeed, so far only a handful of these
types of ideas appear in the literature – among them are the CFT distance
conjecture [4, 5], the no-global-symmetries conjecture [6, 7] and the
cobordism conjecture [8, 9].

One of the most famous swampland conjectures is the weak gravity conjecture
(WGC) [10], which states that a $D$-dimensional effective low-energy
gravitational theory with a $U(1)$ gauge field should include a particle whose
mass and charge obey

$\frac{m}{q}\leq\alpha_{D}g(M_{p}^{D})^{\frac{D-2}{2}}\;,$ (1)

with $\alpha_{D}$ some constant, $g$ the gauge coupling and $M_{p}^{D}$ the
$D$-dimensional Planck mass. It was originally motivated by demanding that all
black holes are allowed to decay, and has since been put through many
stringent checks.

Several possible CFT duals of the WGC have been studied in the literature. The
first attempt [11] focused on $d=4$ dimensional CFTs and demanded that black
holes decay in $D=d+1=5$ dimensional AdS, which leads to the existence of an
operator whose dimension $\Delta$ and charge $q$ obey

$\frac{\Delta^{2}}{q^{2}}\leq\frac{9}{40}\frac{c_{T}}{c_{V}}\;.$ (2)

Here, $c_{T}$ and $c_{V}$ are the coefficients of the two-point functions of
the energy-momentum tensor and the conserved current respectively.
Interestingly, this inequality has a simple CFT interpretation, as it also
determines whether twists of double-twist operators have a convex or concave
behaviour at large spin [12, 13].111The calculation in [11] can be extended to
any $d>2$ and this match still occurs. Indeed, both questions essentially
correspond to asking whether gravity is the weakest force in AdS. However,
there are simple examples in which this inequality is not obeyed (like a free
complex scalar in $d=4$), and so it is not the universal result one would have
hoped for. Some additional discussions on the WGC and holography appear in
[14, 15, 16, 17].

A more recent attempt at deriving a universal CFT dual for the WGC was
discussed in [18]. The motivation came from the positive binding energy
conjecture [19, 20], which is an alternative formulation of the WGC. In the
simplest version of the conjecture, the authors assumed a unitary theory in
$d>2$ with a single $U(1)$ symmetry, and defined $\Delta_{\text{min}}(q)$ to
be the dimension of the operator of charge $q$ with the smallest dimension.
Then the abelian convex charge conjecture states that $\Delta_{\text{min}}(q)$
is convex, in the sense that there exists a charge $q_{0}$ of order 1 such
that for any $n,m\in\mathbb{N}$,

$\Delta_{\text{min}}((m+n)q_{0})\geq\Delta_{\text{min}}(mq_{0})+\Delta_{\text{min}}(nq_{0})\;.$
(3)

This conjecture has been discussed and checked in many explicit examples in a
series of recent papers [18, 21, 22, 23, 24]. In particular, some possible
generalizations and refinements have appeared [25, 26].

As discussed in [18], at large enough charge the conjecture is immediately
obeyed in theories with a proper thermodynamic limit,222Free scalar theories
or theories with moduli spaces of vacua are examples of theories with a sick
thermodynamic limit [27]. The spectrum becomes marginally convex in this case.
because of the universal scaling of operator dimensions at large global charge
[28, 29],

$\Delta_{\text{min}}(q)\propto q^{\frac{d}{d-1}}+\text{(lower order in
{\hbox{1/q}})}\;.$ (4)

The content of the conjecture is thus in the statement that $q_{0}$, the
charge at which convexity begins, is of order $1$. In order to find a
counterexample, it is enough to find a theory in which $q_{0}$ can be made
arbitrarily large.

In this letter we construct such a counterexample to the conjecture, and to
most of the refinements in dimension $d>2$ which currently appear in the
literature. The counterexample involves a weakly-coupled CFT in 3d, which we
describe in section II. In short, it is a 3d supersymmetric clockwork-like
model [30, 31] with some modifications to make it a weakly-coupled CFT. We
also discuss some generalizations and implications in section III, and mention
some refinements of the conjecture which are not ruled out by this
counterexample.

We emphasize that the CFT convexity conjecture was only motivated by the WGC,
and is not equivalent to it. As a result, there are no immediate implications
for any version of the WGC from this counterexample. In particular the
counterexample does not have a weakly-coupled Einstein gravity dual.

Naively it might seem as though this letter is a step in the wrong direction.
On the contrary, we would like to emphasize the rigor and simplicity of the
CFT versions of the swampland conjectures, which can be disproven (and
hopefully proven) with relative ease, making them worthy of further study.
Specifically, it is clear that many CFTs do obey the convexity conjecture, and
so conceivably there is a small modification of the conjecture which should
still hold. It would be interesting to find this modified version and attempt
to prove it (see [32] for work in this direction).

##  II The counterexample

###  II.1 The model

We start with a 3d $\mathcal{N}=2$ $U(N)$ Chern-Simons (CS) theory at level
$k$ coupled to $M$ matter multiplets $\Phi_{i}$ in the adjoint of the gauge
group. The literature on these theories is vast, but we will require only the
results from [33, 34], which we now review. We take the large-$N$ ’t Hooft
limit with $k\gg N\gg 1$. Then there exists a weakly-coupled fixed point with
no superpotential for the $\Phi_{i}$’s, whose anomalous dimension
$\gamma_{\Phi}$ is of order $-N^{2}/k^{2}$, sign included. Crucially,
$\gamma_{\Phi}$ is both small and negative.

We can now deform this fixed point by a general superpotential of the form

$W=\sum_{i,j,k,l}\alpha_{ijkl}\mathop{\rm
Tr}\nolimits\left(\Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l}\right)\;.$ (5)

Every $\alpha_{ijkl}$ is weakly relevant and so we can perform perturbation
theory in the $\alpha_{ijkl}$’s. The $\beta$ functions at leading order in
$1/k$ only get contributions from the superpotential, and one finds

$\beta_{ijkl}=4\gamma_{\Phi}\alpha_{ijkl}+\frac{1}{16\pi^{2}}B_{(\underline{i}}{}^{r}\alpha_{r\underline{jkl})}{}+\text{(higher
loop)}\;,$ (6)

where $(\underline{ijkl})$ stands for cyclic symmetrization and

$B_{i}^{j}=\frac{1}{2}N^{2}\alpha_{iklm}\bar{\alpha}^{jklm}\;.$ (7)

We now focus on a specific superpotential:

$W=4\sum_{i=1}^{M-1}\alpha_{i}\mathop{\rm
Tr}\nolimits\left(\Phi_{i}^{3}\Phi_{i+1}\right)\;.$ (8)

For generic $\alpha_{i}$’s, the theory now has two continuous global
symmetries. One is the $U(1)_{R}$ R-symmetry, under which $\Phi_{i}$ has
charge $1/2$. The other symmetry is more interesting – We call it $U(1)_{E}$
(for “exponential”), under which $\Phi_{i}$ has charge $q_{i}=(-3)^{i-1}$.
Importantly, $|q_{i}|$ grows exponentially with the index $i$.

Solving the equations above, one finds a fixed point at

$|\alpha_{i}|^{2}=-\frac{8\pi^{2}\gamma_{\Phi}}{N^{2}}\frac{\left(1-(-3)^{i}\right)\left(1-(-3)^{M-i}\right)}{\left(1+(-3)^{M}\right)}>0\;,$
(9)

where $i=1,\,\dots,\,M-1$. It is crucial that the RHS is always positive and
hence the fixed point is unitary, and that there are no additional continuous
symmetries at the fixed point. In addition, all of the $\alpha_{i}$’s are of
order $O(\sqrt{-\gamma_{\Phi}})=O(N/k)$ and so the fixed point is still weakly
coupled as expected.

###  II.2 Lack of convexity

Let us study convexity for the $U(1)_{E}$ symmetry at the fixed point. First,
in order to have a single $U(1)$ symmetry, we weakly gauge the $U(1)_{R}$ by
projecting out operators charged under it.333Note that by $U(1)_{R}$ we mean
the naive R-symmetry, and not the “true” R-symmetry appearing in the
superconformal algebra, which is some combination of $U(1)_{R}$ and other
$U(1)$ symmetries determined by $F$-maximization [35]. This is done by adding
a CS term to the gauge field with a very large level. Weakly gauging this
symmetry explicitly breaks SUSY, but in a controlled way – we are only
removing some components of SUSY multiplets, but not changing the CFT data of
the remaining operators. This step is technical and is required only in order
to evade any issues with generalizations of the conjecture to more than one
U(1) symmetry; equivalently, one could just consider operators which are
charged only under $U(1)_{E}$ and not under $U(1)_{R}$.

The remaining gauge-invariant operators charged under $U(1)_{E}$ are single-
or multi-trace operators such that the total R-charge vanishes. However, since
the $\Phi_{i}$’s are the only fields charged under the $U(1)_{E}$ symmetry,
and since we are only interested in the lowest-dimension operator at each
charge, we can ignore other fields (like $F_{\mu\nu}$). Denoting the
components of $\Phi_{i}$ by $\phi_{i},\psi_{i}$ which have R-charge $1/2,-1/2$
respectively, we are thus only interested in single- or multi-trace operators
made of $\phi_{i},\psi_{i}$ and their conjugates such that the total R-charge
vanishes. Since the model is weakly-coupled, we are assured that these will be
the only operators which are of interest to us. For example, one can take

$\mathop{\rm
Tr}\nolimits\left(\phi_{i_{1}}...\phi_{i_{n}}\bar{\phi}_{j_{1}}...\bar{\phi}_{j_{m}}\psi_{k_{1}}...\psi_{k_{p}}\bar{\psi}_{l_{1}}...\bar{\psi}_{l_{r}}\right)\;,$
(10)

with $n-m-p+r=0$. We must also rescale the $U(1)_{E}$ charge such that the
minimal charge of all gauge-invariants is $1$, so that now $\Phi_{i}$ has
charge $q_{i}=\frac{(-3)^{i-1}}{2}$.

Now we show that there is no convexity for the $U(1)_{E}$ symmetry for all
$q_{0}$ which obey $q_{0}<\frac{3^{M-3}}{2}$ and such that the lowest-
dimension operators at charges $q\leq 9q_{0}$ consist of less than $N$
fields.444In particular, by taking $N,M$ to be large, all $q_{0}$ up to some
arbitrarily large value can be made to obey these assumptions. Assuming such a
$q_{0}$, there exists some gauge-invariant operator which has minimal
dimension at charge $q_{0}$ of the form discussed above. Since the fixed point
is weakly-coupled, its dimension is555Mixing does not affect this result since
the number of operators which can mix does not depend on $N,k$.

$\Delta_{\text{min}}(q_{0})=\frac{\\#\text{scalars}}{2}+\\#\text{fermions}+O(N/k)\;.$
(11)

Next consider operators of charge $q=9q_{0}$. We can construct an operator at
this charge by taking the lowest dimension operator at charge $q_{0}$ and
shifting all of the indices of the fields by $2$ (in the example above this
corresponds to $i_{s}\mapsto i_{s}+2$ for all $s$, etc.). Since we assumed
$q_{0}<\frac{3^{M-3}}{2}$, the lowest-dimension operator with charge $q_{0}$
consists only of fields whose indices are at most $M-2$, and so shifting these
indices by 2 is allowed. Since this new operator is constructed using the same
number of fields as the operator for $q_{0}$, it has the same dimension up to
corrections of order $N/k$. We thus find

$\Delta_{\text{min}}(9q_{0})\leq\Delta_{\text{min}}(q_{0})+O(N/k)\;,$ (12)

while convexity requires

$\Delta_{\text{min}}(9q_{0})\geq 9\Delta_{\text{min}}(q_{0}).$ (13)

Combined with the unitarity bound $\Delta_{\text{min}}\geq\frac{1}{2}$, we
find that the spectrum is not convex for large enough $k$. We can now make
$k,N,M$ arbitrarily large (while keeping $k\gg N\gg M$), thereby
parametrically delaying convexity as much as we would like and providing a
counterexample to the convexity conjecture.

For completeness we present $\Delta_{\text{min}}(q)$ for $q\leq 200$ and large
enough $k,N,M$ in figure 1 to make the lack of convexity apparent.

Figure 1: $\Delta_{\text{min}}(q)$ for small charges in the counterexample.

##  III discussion

Some generalizations of the convexity conjecture have appeared in the
literature, and we start by remarking on their validity. First, [18] mentions
a version of the conjecture which applies only to scalar operators, but this
conjecture also fails due to the same counterexample discussed above. Some
generalizations have also appeared for theories with more than one $U(1)$
symmetry and for nonabelian symmetries [18, 26, 25], but since the case with a
single $U(1)$ symmetry is a special case of these the counterexample rules
them out as well.

Finally, the 2d version of the conjecture was discussed in [26]. This version
is different from the higher-dimensional version, since there it was already
shown that convexity can be parametrically delayed. However, [26] provided an
upper bound on the delay in convexity which is related to the level of the
corresponding WZW theory. This version of the conjecture was proven explicitly
and so it still stands.

An obvious question is whether there is a small refinement of the conjecture
which rules out the counterexample discussed above. One option appears in
[18], where $q_{0}$ is determined by the charge of the lowest-dimension
charged operator. Computing the anomalous dimensions of various single-trace
operators, we find that this sets $q_{0}\sim 3^{M-1}/2$ at large $M$ and so
this version of the convexity conjecture is not ruled out by the
counterexample (however, $q_{0}$ can still be made to be arbitrarily
large).666We thank Ofer Aharony for discussions on this point. Another option
is to allow convexity to be delayed up to some upper bound which depends
exponentially on the number of fields in the theory. Explicitly, one can
conjecture that $q_{0}\leq e^{N_{D}}$, where $N_{D}$ is some measure of the
number of degrees of freedom in the theory (e.g. $c_{T}$, the $c$ anomaly in
2d, the $a$ anomaly in 4d, or the $F$-coefficient in 3d). Another option would
be to apply the conjecture only to theories with a weakly-coupled Einstein
gravity dual. We will not speculate on such refinements further.

We note that it seems that $\mathcal{N}=2$ SUSY is important in order for the
counterexample to remain tractable (even though we have effectively removed
SUSY by weakly gauging the $U(1)_{R}$ symmetry). One could in principle
consider a $3d$ $\mathcal{N}=1$ or non-SUSY version of this counterexample
(see [36, 37] for recent discussions of such CS theories with adjoints).
However, a practical issue is that in addition to a potential which mimics the
behaviour of the superpotential (8), there are many other terms that can
appear under the RG flow. For example, one can study the $3d$ $\mathcal{N}=1$
CS-matter theory with superpotential (8), but now the RG flow also generates
interactions of the form

$\alpha_{ij}\mathop{\rm
Tr}\nolimits\left(|\Phi_{i}|^{2}|\Phi_{j}|^{2}\right)\;,$ (14)

and one must study whether there exist fixed points for all of these couplings
as well, which complicates the analysis considerably.

Next we discuss possible implications to the WGC. As noted in [18], the charge
convexity conjecture in no way implies the WGC (or the other way around), and
so this counterexample cannot be explicitly used to learn much about quantum
gravity. At best it can be used to perhaps motivate an analogous gravitational
construction, but it is not clear how to do this. First, the CFT we
constructed does not have a weakly-coupled Einstein gravity dual. Second, the
main mechanism used here is the same as in the clockwork model, which is not a
new idea in the context of swampland. However, it is possible that the
counterexample suggests a closer look at clockwork-like models in the context
of the WGC as well.

We end by commenting on some other possible applications of the model we
discussed. First, the model may be interesting to study in the context of the
large-charge expansion, which has been shown to be approximately valid down to
surprisingly low charges in simple examples even when taking into account only
the first few leading terms in the expansion [38, 39, 40, 41]. However, in our
model this will almost definitely not occur, since $\Delta_{\text{min}}(q)$ is
an erratic function of $q$ for small enough charges (see figure 1). It would
thus be interesting to better understand when the large-charge expansion can
be safely extrapolated down to small charges. It is possible that one can
bound the validity of the expansion at small charge by using some function of
the number of degrees of freedom in the CFT, as discussed above. The class of
theories discussed above with more general superpotentials (5) may also have
many possible applications as a weakly-coupled conformal manifold. In
particular, one could try to define an SYK-like theory in 3d by averaging over
this conformal manifold (see [42] for another attempt in 3d). Then observables
like the chaos exponent can be extracted using the formalism discussed in [43,
44, 45], which generalizes SYK-like models to those with general CFTs deformed
by disorder.

## Acknowledgements

We would first like to thank Ofer Aharony and Eran Palti for many interesting
discussions, and especially for their patience and for helping to disprove
many previous attempts at counterexamples. We are also grateful to Simeon
Hellerman for helpful discussions which initialised the project. We would also
like to thank Shota Komatsu, Miguel Montero, Domenico Orlando and Sridip Pal
for illuminating discussions. MW is grateful to the conference “Large charge
aux Diablerets” where this work began, and AS is grateful to the Abu Dhabi
meeting on theoretical physics where this work was finished. MW is supported
by Grant-in-Aid for JSPS Fellows (No. 22J00752).

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