\section{Introduction}

The properties of white dwarf stars are relatively simple to measure.
Ideally, a complete spectral energy distribution, from the optical to the
ultraviolet, helps establish the white dwarf atmospheric properties, i.e.,
their effective temperature, surface gravity, and chemical composition,
from which we may deduce their age and mass using theoretical mass-radius
relations \citep[e.g.,][]{woo1995}. These measurements lay the foundations for
a study of the white dwarf cooling history (luminosity function).

The 2dF QSO Redshift Survey (2QZ) survey is a deep spectroscopic survey of blue
Galactic and extra-Galactic sources which was conducted at the Anglo-Australian
Telescope (AAT). Over 2000 white dwarfs were discovered during this survey
\citep{cro2004}. \citet{ven2002} obtained atmospheric parameters for many of
these objects. Another survey is the Sloan Digital Sky Survey (SDSS),
which aims to observe 1/4 of the sky in {\it ugriz} photometric bands and
obtain spectroscopy for many of these objects. A catalog of 9316
spectroscopically confirmed white dwarfs from the SDSS 4th data release
was published by \citet{eis2006}.
The Galaxy Evolution Explorer (GALEX) is an orbiting observatory that aims to
observe the sky in the ultraviolet. GALEX provides photometry in two bands,
{\it FUV} (1344-1786 \AA) and {\it NUV} (1771-2831 \AA), and will conduct
several surveys during its mission, including the
All-Sky Imaging Survey (AIS) with a limiting magnitude of 21.5 mag and the
Medium-Imaging Survey (MIS) with a limiting magnitude of 23 mag.

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{fig_1.eps}
\caption{The overlap between 2QZ and GALEX (GR2) surveys.\label{fig_coord_comp}}
\end{figure}

We cross-correlated the 2QZ DA white dwarf catalog with the SDSS 5th
data release to obtain {\it ugriz} photometry. Only the north Galactic cap
(NGP) is covered by the SDSS and therefore we were only able to get {\it ugriz}
photometry for 795 objects. We also cross-correlated the 2QZ DA white dwarf
catalog with the GALEX 2nd data release (GR2) to obtain ultraviolet photometry
for approximately 810 objects, however some of these stars have only been
detected in either the FUV or NUV band. Figure~\ref{fig_coord_comp} shows
the overlap between the 2QZ white dwarf catalog and GR2. Finally we
cross-correlated our SDSS and GALEX samples to obtain 252 stars for which we
have both optical {\it ugriz} and ultraviolet photometry.

\section{Determining the parameters}

\begin{figure}
\includegraphics[width=0.5\textwidth]{fig_2a.eps}
\includegraphics[width=0.5\textwidth]{fig_2b.eps}
\caption{({\it left}) Optical and ultraviolet colors ($g-FUV$ vs. $FUV-NUV$) of 2QZ
white dwarfs compared to synthetic DA white dwarf colors. The full-lined grid
includes the effect of Ly$\alpha$ satellites, and the dotted-lined grid excludes
them. ({\it right}) Optical ($u-g$/$g-r$) colors of 2QZ white dwarfs compared
to synthetic DA white dwarf colors. The effective temperature is indicated in
units of 1000 K and the $\log{g} = 6.0$, $7.0$, $8.0$, and $9.0$ ({\it from
bottom to top}). The main-sequence colors
are also shown.\label{fig_gmf_fmn}}
\end{figure}

We obtained effective temperatures for the 2QZ DA white dwarfs by comparing
the ultraviolet/optical colors ($g-FUV$/$FUV-NUV$) and optical colors
($u-g$/$g-r$) to synthetic DA colors. Figure~\ref{fig_gmf_fmn} shows the
ultraviolet and optical colors of the 2QZ white dwarfs compared to synthetic
DA and main-sequence colors. The optical and ultraviolet photometry of the 2QZ
white dwarfs were corrected for interstellar
extinction using the dust maps of \citet{sch1998} and the extinction
law of \citet{car1989}, where we have assumed $R_V= 3.1$. Note that $R_V$
can vary anywhere between $\sim 2.5$ to $\sim 5.5$ which can change the
extinction in the ultraviolet by a very significant amount \citep{car1989}.
In addition, the true extinction toward a star is possibly only a fraction
of the total extinction in the line of sight.
The synthetic DA colors were calculated using a grid of pure hydrogen LTE
models \citep{kaw2006}. The grid of models extend from $T_{\rm eff} = 7000$ to
84\,000 K at $\log{g} =6.0$ to $9.5$.
The spectra include the effect of Ly$\alpha$ satellites \citep{all1992},
however we also calculated spectra which exclude the effect.
Figure~\ref{fig_gmf_fmn} (left) shows the effect of Lyman satellites on UV
colors, which is most prominent in the cooler white dwarfs
($T_{\rm eff} \lesssim 12000$ K). The color diagrams also show the
main-sequence, which was calculated using Kurucz synthetic spectra
\citep{kur1993}. Figure~\ref{fig_gmf_fmn} (right) shows some stars that have
colors corresponding to the main-sequence rather than the DA white dwarf
sequence, therefore some contamination by main-sequence stars is still
present in the 2QZ DA catalog. We found that the combination of both diagrams
helps distinguish between white dwarfs and main-sequence stars.

Figure~\ref{fig_temp_lysat} shows a comparison between the
temperatures obtained using the $g-FUV$/$FUV-NUV$ colors and the $u-g$/$g-r$
colors. A relatively good agreement between the temperatures is observed
with a few points which appear to give very low optical temperatures compared
to their UV temperatures. A check of the 2QZ spectra for these objects
reveal them to be DA white dwarfs with cool companions and therefore the UV
temperatures should be adopted as the white dwarf temperatures.
Figure~\ref{fig_temp_lysat} also shows a comparison of the UV temperatures
determined with grids which include or exclude the effect of Ly$\alpha$
satellites. Temperatures determined using models which exclude the effect of
Ly$\alpha$ satellites are significantly lower than those which do not for
$T_{\rm eff} \lesssim 12000$ K.

\begin{figure}
\includegraphics[viewport=0 285 570 570, width=\textwidth]{fig_3.eps}
\caption{In the {\it left} panel, the temperatures obtained from the
optical-ultraviolet and optical diagrams are compared.
And in the {\it right} panel, the temperatures determined using the
optical-UV colors determined if Ly$\alpha$ satellites are included compared
to the temperatures determined using the same grid but which excludes the
effect of the Ly$\alpha$ satellites.\label{fig_temp_lysat}}
\end{figure}

\section{Luminosity Function}

We constructed the DA luminosity function using the
2QZ/SDSS5/GR2 and the 2QZ/SDSS5 samples by using
the accessible-volume method \citep[see][and references therein]{boy1989}
and assuming a scale-height of 250 pc. Using the temperatures obtained from
the $g-FUV$/$FUV-NUV$ and $u-g/g-r$ diagrams, we calculated the absolute
magnitudes ($M_V$), assuming $\log{g} =8.0$ and using the mass-radius
relations of \citet{ben1999}. Figure~\ref{fig_lum_GR2} shows the 2QZ/SDSS5
and 2QZ/SDSS5/GR2 luminosity functions compared to the luminosity
functions determined in the PG Survey \citep{fle1986}, the AAT-UVX survey
\citep{boy1989} and a theoretical luminosity function based in the cooling
sequence of \citet{woo1995} and a constant DA birthrate of
$0.5\times 10^{-12}$ pc$^{-3}$ yr$^{-1}$. Both the 2QZ/SDSS5/GR2 and
2QZ/SDSS5 luminosity functions are incomplete at the cool end
($M_V \ge 12.0$ mag). The two main reasons for this is that fewer objects
were selected due to the limiting colors at the cool end and the cooler
objects are very faint in the ultraviolet and hence would not be detected
by the GALEX survey. Also there appears to be an excess of stars in the
$M_V = 11.5$ bin, in particular in the 2QZ/SDSS5/GR2 sample. Interstellar
reddening could be a contributing toward this excess since our extinction
correction did not consider the distance toward the star, and therefore
cooler objects which would be closer would be over corrected resulting in
higher temperatures.

\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{fig_4.eps}
\caption{The DA luminosity function measured using the 2QZ/SDSS/GR2
survey compared to the luminosity function measured using AAT-UVX and
Palomar-Green (PG) survey. The theoretical luminosity function assuming
a DA white dwarf birthrate of $0.5\times 10^{-12}$ pc$^{-3}$ yr$^{-1}$.\label{fig_lum_GR2}}
\end{figure}

\section{Summary and Future Work}

We have presented our initial analysis of DA white dwarfs from the 2QZ survey
using SDSS5 and GR2 photometry. Using these photometric data, we obtained
temperature and absolute magnitudes from which we built a luminosity function.
We will extend this analysis to the He-rich sequence of white dwarfs (DO/DB).
We will investigate the effect of interstellar extinction on the
temperature distribution and hence the luminosity function in more detail.
Also, we will examine the effect of heavy elements on the UV/optical
temperature scales.
Many hot white dwarfs show heavy element lines, and the abundance of
these elements can vary many orders of magnitude.

{\it Acknowledgements:}

This research is supported by NASA/GALEX grant NNG05GL42G.
A.K. is supported by grant GA \v{C}R 205/05/P186.


\section{Introduction}

Amongst the lattice gauge community it has recently become quite
popular to study the distributions of eigenvalues of the Dirac
operator in the presence of the background gauge fields generated in
simulations.  There are a variety of motivations for this.  First, in
a classic work, Banks and Casher\cite{Banks:1979yr} related the
density of small Dirac eigenvalues to spontaneous chiral symmetry
breaking.  Second, lattice discretizations of the Dirac operator based
the Ginsparg-Wilson relation\cite{Ginsparg:1981bj} have the
corresponding eigenvalues on circles in the complex plane.  The
validity of various approximations to such an operator can be
qualitatively assessed by looking at the eigenvalues.  Third, using
the overlap method\cite{Neuberger:1997fp} to construct a Dirac
operator with good chiral symmetry has difficulties if the starting
Wilson fermion operator has small eigenvalues.  This can influence the
selection of simulation parameters, such as the gauge
action.\cite{Aoki:2002vt} Finally, since low eigenvalues impede
conjugate gradient methods, separating out these eigenvalues
explicitly can potentially be useful in developing dynamical
simulation algorithms.\cite{Duncan:1998gq}

Despite this interest in the eigenvalue distributions, there are some
dangers inherent in interpreting the observations.  Physical results
come from the full path integral over both the bosonic and fermionic
fields.  Doing these integrals one at a time is fine, but trying to
interpret the intermediate results is inherently dangerous.  While the
Dirac eigenvalues depend on the given gauge field, it is important to
remember that in a dynamical simulation the gauge field distribution
itself depends on the eigenvalues.  This circular behavior gives a
highly non-linear system, and such systems are notoriously hard to
interpret.

Given that this is a joyous occasion, I will present some of this
issues in terms of an amusing set of puzzles arising from naive
interpretations of Dirac eigenvalues on the lattice.  The discussion
is meant to be a mixture of thought provoking and confusing.  It is
not necessarily particularly deep or new.

\section{The framework}

To get started, I need to establish the context of the discussion.  I
consider a generic path integral for a gauge theory
\begin{equation}
Z=\int (dA)(d\psi)(d\overline\psi)\ e^{-S_G(A)+\overline\psi D(A) \psi}.
\end{equation}
Here $A$ and $\psi$ represent the gauge and quark fields,
respectively, $S_G(A)$ is the pure gauge part of the action, and
$D(A)$ represents the Dirac operator in use for the quarks.  As the
action is quadratic in the fermion fields, a formal integration gives
\begin{equation}
Z=\int (dA)\ |D(A)|\ e^{-S_G(A)}.
\label{path}
\end{equation}
Working on a finite, lattice $D(A)$ is a finite dimensional matrix,
and for a given gauge field I can formally consider its eigenvectors
and eigenvalues
\begin{equation}
D(A)\psi_i=\lambda_i\psi_i.
\end{equation}
The determinant appearing in Eq.~(\ref{path}) is the product of these
eigenvalues; so, the path integral takes the form
\begin{equation}
Z=\int (dA)\ e^{-S_G(A)}\ \prod_i \lambda_i.
\end{equation}
Averaging over gauge fields defines the eigenvalue density
\begin{equation}
\rho(x+iy)={1\over {N}Z}\int (dA)\ |D(A)|\ e^{-S_G(A)}
\sum_{i}\delta(x-{\rm Re}\lambda_i(A))\delta(y-{\rm Im}\lambda_i(A)).
\end{equation}
Here $N$ is the dimension of the Dirac operator, including volume,
gauge, spin, and flavor indices.

In situations where the fermion determinant is not positive, $\rho$
can be negative or complex.  Nevertheless, I still refer to it as a
density.  I will assume that $\rho$ is real; situations where this is
not true, such as with a finite chemical
potential,\cite{Osborn:2005ss} are beyond the scope of this
discussion.

\begin{figure*}
\centering
\includegraphics[width=3in]{eigen0.eps}
\caption{ In the naive continuum picture, all eigenvalues of the Dirac
operator lie along a line parallel to the imaginary axis.  In a finite
volume these eigenvalues become discrete.  The real eigenvalues divide
into distinct chiralities and define a topological invariant.
\label{continuum}
}
\end{figure*}

At zero chemical potential, all actions used in practice satisfy
``$\gamma_5$ hermiticity''
\begin{equation}
\gamma_5 D \gamma_5=D^\dagger.
\label{hermiticity}
\end{equation}
With this condition all non-real eigenvalues occur in complex
conjugate pairs, implying for the density
\begin{equation}
\rho(z)=\rho(z^*).
\end{equation}
This property will be shared by all the operators considered in the
following discussion.

The quest is to find general statements relating the behavior of the
eigenvalue density to physical properties of the theory.  I repeat the
earlier warning; $\rho$ depends on the distribution of gauge fields
$A$ which in turn is weighted by $\rho$ which depends on the
distribution of $A$ \ldots.

\subsection{The continuum}

Of course the continuum theory is only really defined as the limit of
the lattice theory.  Nevertheless, it is perhaps useful to recall the
standard picture, where the Dirac operator
$$
D=\gamma_\mu (\partial_\mu+igA_\mu)+m
$$
is the sum of an anti-hermitian piece and the quark mass $m$.  All
eigenvalues have the same real part $m$
$$
\rho(x+iy)=\delta(x-m) \tilde\rho(y).
$$ The eigenvalues lie along a line parallel to the imaginary axis,
while the hermiticity condition of Eq.~(\ref{hermiticity}) implies
they occur in complex conjugate pairs.

Restricted to the subspace of real eigenvalues, $\gamma_5$ commutes
with $D$ and thus these eigenvectors can be separated by chirality.
The difference between the number of positive and negative eigenvalues
of $\gamma_5$ in this subspace defines an index related to the
topological structure of the gauge fields.\cite{index} The basic
structure is sketched in Fig.~(\ref{continuum}).

The Banks and Casher argument relates a non-vanishing $\tilde\rho(0)$
to the chiral condensate occurring when the mass goes to zero.  I will
say more on this later in the lattice context.

Note that the naive picture suggests a symmetry between positive and
negative mass.  Due to anomalies, this is spurious.  With an odd
number of flavors, the theory obtained by flipping the signs of all
fermion masses is physically inequivalent to the initial theory.

\begin{figure*}
\centering
\includegraphics[width=2.5in]{eigen1.eps}
\caption{ Free Wilson fermions display an eigenvalue spectrum with a
momentum dependent real part.  This removes doublers by giving them a
large effective mass.}
\label{fig2}
\end{figure*}

\subsection{Wilson fermions}

The lattice reveals that the true situation is considerably more
intricate due to the chiral anomaly.  Without ultraviolet infinities,
all naive symmetries of the lattice action are true symmetries.  Naive
fermions cannot have anomalies, which are cancelled by extra states
referred to as doublers.  Wilson fermions\cite{Wilson:1975id} avoid
the this issue by giving a large real part to those eigenvalues
corresponding to the doublers.  For free Wilson fermions the
eigenvalue structure displays a simple pattern as shown in
Fig.~(\ref{fig2}).

As the gauge fields are turned on, this pattern will fuzz out.  An
additional complication is that the operator $D$ is no longer normal,
i.e. $[D,D^\dagger]\ne 0$ and the eigenvectors need not be orthogonal.
The complex eigenvalues are still paired, although, as the gauge
fields vary, complex pairs of eigenvalues can collide and separate
along the real axis.  In general, the real eigenvalues will form a
continuous distribution.

As in the continuum, an index can be defined from the spectrum of the
Wilson-Dirac operator.  Again, $\gamma_5$ hermiticity allows real
eigenvalues to be sorted by chirality.  To remove the contribution of
the doubler eigenvalues, select a point inside the leftmost open
circle of Fig.~(\ref{fig2}).  Then define the index of the gauge field
to be the net chirality of all real eigenvalues below that point.  For
smooth gauge fields this agrees with the topological winding number
obtained from their interpolation to the continuum.  It also
corresponds to the winding number discussed below for the overlap
operator.

\subsection{The overlap}

Wilson fermions have a rather complicated behavior under chiral
transformations.  The overlap formalism\cite{Neuberger:1997fp}
simplifies this by first projecting the Wilson matrix $D_W$ onto a
unitary operator
\begin{equation}
V=(D_W D_W^\dagger)^{-1/2} D_W.
\end{equation}
This is to be understood in terms of going to a basis that
diagonalizes $D_W D_W^\dagger$, doing the inversion, and then
returning to the initial basis.  In terms of this unitary quantity,
the overlap matrix is
\begin{equation}
D=1+V.
\end{equation}
The projection process is sketched in Fig.~(\ref{fig3}).  The mass
used in the starting Wilson operator is taken to a negative value so
selected that the low momentum states are projected to low
eigenvalues, while the doubler states are driven towards $\lambda\sim
2$.

\begin{figure*}
\centering
\includegraphics[width=5in]{eigen3.eps}
\caption{The overlap operator is constructed by projecting the Wilson
  Dirac operator onto a unitary operator.}
\label{fig3}
\end{figure*}

The overlap operator has several nice properties.  First, it satisfies
the Ginsparg-Wilson relation,\cite{Ginsparg:1981bj} most succinctly
written as the unitarity of $V$ coupled with its $\gamma_5$
hermiticity
\begin{equation}
\gamma_5 V\gamma_5 V=1.
\end{equation}
As it is constructed from a unitary operator, normality of $D$ is
guaranteed.  But, most important, it exhibits a lattice version of an
exact chiral symmetry.\cite{Luscher:1998pq} The fermionic action
$\overline\psi D\psi$ is invariant under the transformation
\begin{eqnarray}
&\psi\rightarrow e^{i\theta\gamma_5}\psi\cr
&\overline\psi\rightarrow \overline\psi e^{i\theta\hat\gamma_5}
\label{symmetry}
\end{eqnarray}
where
\begin{equation}
\hat\gamma_5 =V\gamma_5.
\end{equation}
As with $\gamma_5$, this quantity is Hermitean and its square is unity.
Thus its eigenvalues are all plus or minus unity.  The trace
defines an index
\begin{equation}
\nu={1\over 2}{\rm Tr}\hat\gamma_5
\end{equation}
which plays exactly the role of the index in the continuum.

It is important to note that the overlap operator is not unique.  Its
precise form depends on the particular initial operator chosen to
project onto the unitary form.  Using the Wilson-Dirac operator for
this purpose, the result still depends on the input mass used.  From
its historical origins in the domain wall formalism, this quantity is
sometimes called the ``domain wall height.''

Because the overlap is not unique, an ambiguity can remain in
determining the winding number of a given gauge configuration.  Issues
arise when $D_W D_W^\dagger$ is not invertible, and for a given gauge
field this can occur at specific values of the projection point.
This problem can be avoided for ``smooth'' gauge fields.  Indeed, an
``admissibility condition,''~\cite{Luscher:1981zq,Hernandez:1998et}
requiring all plaquette values to remain sufficiently close to the
identity, removes the ambiguity.  Unfortunately this condition is
incompatible with reflection positivity.\cite{Creutz:2004ir} Because
of these issues, it is not known if the topological susceptibility is
in fact a well defined physical observable.  On the other hand, as it
is not clear how to measure the susceptibility in a scattering
experiment, there seems to be little reason to care if it is an
observable or not.

\begin{figure*}
\centering
\includegraphics[width=3in]{circles.eps}
\caption{Inverting a complex circle generates another circle.}
\label{circles}
\end{figure*}

\section{A Cheshire chiral condensate}

Now that I have reviewed the basic framework, it is time for a little
fun.  I will calculate the chiral condensate in the overlap formalism.
I should warn you that, in the interest of amusing you, I start the
argument in an intentionally deceptive manner.

\subsection{He's here}

I begin with the standard massless overlap theory.  I want to
calculate the quantity $\langle\overline\psi\psi\rangle$.  Remarkably,
this can be done exactly.  I start with
\begin{equation}
\langle\overline\psi\psi\rangle=\langle {\rm Tr} D^{-1}\rangle
=\left\langle \sum_i {1\over \lambda_i}\right\rangle
=\left\langle \sum {\rm Re} {1\over \lambda_i}\right\rangle
\end{equation}
where I have used the complex pairing of eigenvalues to cancel the
imaginary parts.  At the end, the average is to be taken over
appropriately weighted gauge configurations.

Now the crucial feature of the overlap operator is that its
eigenvalues all lie on a circle in the complex plane.  An interesting
property of a general complex circle is that the inverses of all its
points generates another circle, as sketched in Fig.~\ref{circles}.

This process is, however, somewhat singular for the overlap operator
itself since the corresponding circle touches the origin.  In this
case the inverted circle has infinite radius, i.e. it degenerates into
a line.  For the circle of the overlap operator, with center at $z=1$
and radius 1, the inverse circle is a line with real part 1/2 and
parallel to the imaginary axis.  This is sketched in
Fig.~\ref{circles1}.

\begin{figure*}
\centering
\includegraphics[width=3in]{circles1.eps}
\caption{Inverting the overlap operator generates a line with real
  part 1/2.}
\label{circles1}
\end{figure*}

This placement of eigenvalues enables an immediate calculation of the
condensate
\begin{equation}
\langle\overline\psi\psi\rangle=
\sum {\rm Re} {1\over \lambda_i}=\sum {1\over 2}= {N\over 2}.
\end{equation}
Here $N$ is the dimension of the matrix, and includes the expected
volume factor.

So the condensate, supposedly a signal for spontaneous chiral symmetry
breaking, does not vanish!  But something is fishy, I didn't use any
dynamics.  The result also is independent of gauge configuration.

\subsection{He's gone}

So lets get more sophisticated.  On the lattice, the chiral symmetry
is more complicated than in the continuum, involving both $\gamma_5$
and $\hat\gamma_5$ in a rather intricate way.  In particular, the
operator $\overline\psi\psi$ does not transform in any simple manner
under chiral rotations.  A possibly nicer combination is
$\overline\psi(1-D/2)\psi$.  If I consider the rotation in
Eq.~(\ref{symmetry}) with $\theta=\pi/2$, this quantity becomes its
negative.  But it is also easy to calculate the expectation of this as
well.  The second term involves
\begin{equation}
\langle\overline\psi D \psi\rangle={\rm Tr} D^{-1} D={\rm Tr}I=N.
\end{equation}
Putting the two pieces together
\begin{equation}
\langle\overline\psi(1-D/2)\psi\rangle=N/2-N/2=0.
\end{equation}
So, I've lost the chiral condensate that I so easily showed didn't
vanish just a moment ago.  Where did it go?

\begin{figure*}
\centering
\includegraphics[width=3in]{circles2.eps}
\caption{As the mass changes sign a pole moves between inside and outside
the overlap circle.  This generates a jump in the condensate.}
\label{circles2}
\end{figure*}

\subsection{He's back}

The issue lies in a careless treatment of limits.  In finite volume,
$\langle\overline\psi(1-D/2)\psi\rangle$ must vanish just from the
exact lattice chiral symmetry.  This vanishing occurs for all gauge
configurations.  To proceed, introduce a small mass and take the
volume to infinity first and then the mass to zero.  Toward this end,
consider the quantity
\begin{equation}
\langle\overline\psi\psi\rangle=\sum_i {1\over\lambda_i+m}.
\end{equation}
The signal for chiral symmetry breaking is a jump in this quantity as
the mass passes through zero.

As the volume goes to infinity, replace the above sum with a contour
integral around the overlap circle using $z=1+e^{i\theta}$.  Up to the
trivial volume factor, I should evaluate
\begin{equation}
i\int_0^{2\pi} d\theta {\rho(\theta)\over 1+e^{i\theta}+m}.
\end{equation}
As the mass passes through zero, the pole at $z=-m$ passes between
lying outside and inside the circle, as sketched in
Fig.~(\ref{circles2}).  As it passes through the circle, the residue
of the pole is $\rho(0) = \lim_{\theta\rightarrow 0}\rho(\theta)$.
Thus the integral jumps by $2\pi\rho(0)$.  This is the overlap version
of the Banks-Casher relation;\cite{Banks:1979yr} a non-trivial jump in
the condensate is correlated with a non-vanishing $\rho(0)$.

Note that the exact zero modes related to topology are supressed by
the mass and do not contribute to this jump.  For one flavor, however,
the zero modes do give rise to a non-vanishing but smooth contribution
to the condensate.\cite{Damgaard:1999ij} More on this point later.

\section{Another puzzle}

For two flavors of light quarks one expects spontaneous symmetry
breaking.  This is the explanation for the light mass of the pion,
which is an approximate Goldstone boson.  In the above picture, the
two flavor theory should have a non-vanishing $\rho(0)$.

Now consider the one flavor theory.  In this case there should be no
chiral symmetry.  The famous $U(1)$ anomaly breaks the naive symmetry.
No massless physical particles are expected when the quark mass
vanishes.  Furthermore, simple chiral Lagrangian
arguments\cite{DiVecchia:1980ve,Creutz:2003xu} for multiple flavor
theories indicate that no singularities are expected when just one of
the quarks passes through zero mass.  From the above discussion, this
leads to the conclusion that for the one flavor theory $\rho(0)$ must
vanish.

But now consider the original path integral after the fermions are
integrated out.  Changing the number of flavors $N_f$ manifests itself
in the power of the determinant
\begin{equation}
\int dA\  |D|^{N_f}\  e^{-S_g(A)}.
\end{equation}
Naively this suggests that as you increase the number of flavors, the
density of low eigenvalues should decrease.  But I have just argued
that with two flavors $\rho(0)\ne 0$ but with one flavor $\rho(0)= 0$.
How can it be that increasing the number of flavors actually increases
the density of small eigenvalues?

This is a clear example of how the non-linear nature of the problem
can produce non-intuitive results.  The eigenvalue density depends on
the gauge field distribution, but the gauge field distribution depends
on the eigenvalue density.  It is not just the low eigenvalues that
are relevant to the issue.  Fermionic fields tend to smooth out gauge
fields, and this process involves all scales.  Smoother gauge fields
in turn can give more low eigenvalues.  Thus high eigenvalues
influence the low ones, and this effect evidently can overcome the
naive suppression from more powers of the determinant.

\section{{\AE}thereal instantons}

Through the index theorem, the topological structure of the gauge
field manifests itself in zero modes of the massless Dirac operator.
Let me again insert a small mass and consider the path integral with
the fermions integrated out
\begin{equation}
Z=\int dA\
e^{-S_g}\
\prod_i (\lambda_i+m).
\end{equation}
If I take the mass to zero, any configurations which contain a zero
eigenmode will have zero weight in the path integral.  This suggests
that for the massless theory, I can ignore any instanton effects since
those configurations don't contribute to the path integral.

What is wrong with this argument?  The issue is not whether the zero
modes contribute to the path integral, but whether they can contribute
to physical correlation functions.  To see how this goes, add some sources
to the path integral
\begin{equation}
Z(\eta,\overline\eta)=\int dA\ d\psi\ d\overline\psi\
e^{-S_g+\overline\psi (D+m) \psi +\overline\psi \eta+ \overline\eta\psi}.
\end{equation}
Differentiation (in the Grassmann sense) with respect to $\eta$ and
$\overline \eta$ gives the fermionic correlation functions.
Now integrate out the fermions
\begin{equation}
Z=\int dA\
e^{-S_g-\overline\eta(D+m)^{-1}\eta}\
\prod_i (\lambda_i+m).
\end{equation}
If I consider a source that overlaps with one of the zero mode
eigenvectors, i.e.
\begin{equation}
(\psi_0,\eta)\ne 0,
\end{equation}
the source contribution introduces a $1/m$ factor.  This cancels the
$m$ from the determinant, leaving a finite contribution as $m$ goes to
zero.

With multiple flavors, the determinant will have a mass factor from
each.  When several masses are taken to zero together, one will need a
similar factor from the sources for each.  This product of source
terms is the famous ```t Hooft vertex.'' \cite{'tHooft:1976up} While it
is correct that instantons do drop out of $Z$, they survive in
correlation functions.

While these issues are well understood theoretically, they can raise
potential difficulties for numerical simulations.  The usual numerical
procedure generates gauge configurations weighted as in the partition
function.  For a small quark mass, topologically non-trivial
configurations will be suppressed.  But in these configurations, large
correlations can appear due to instanton effects.  This combination of
small weights with large correlations can give rise to large
statistical errors, thus complicating small mass extrapolations.  The
problem will be particularly severe for quantities dominated by
anomaly effects, such as the $\eta^\prime$ mass.  A possible strategy
to alleviate this effect is to generate configurations with a modified
weight, perhaps along the lines of multicanonical
algorithms.\cite{Berg:1992qu}

Note that when only one quark mass goes to zero, the 't Hooft vertex
is a quadratic form in the fermion sources.  This will give a finite
but smooth contribution to the condensate
$\langle\overline\psi\psi\rangle$.  Indeed, this represents a
non-perturbative additive shift to the quark mass.  The size of this
shift generally depends on scale and regulator details.  Even with the
Ginsparg-Wilson condition, the lattice Dirac operator is not unique,
and there is no proof that two different forms have to give the same
continuum limit for vanishing quark mass.  Because of this, the
concept of a single massless quark is not
physical,\cite{Creutz:2004fi} invalidating one popular proposed
solution to the strong CP problem.  This ambiguity has been noted for
heavy quarks in a more perturbative context\cite{Bigi:1994em} and is
often referred to as the ``renormalon'' problem.  The issue is closely
tied to the problems mentioned earlier in defining the topological
susceptibility.

\section{Summary}

In short, thinking about the eigenvalues of the Dirac operator in the
presence of gauge fields can give some insight, for example the
elegant Banks-Casher picture for chiral symmetry breaking.
Nevertheless, care is necessary because the problem is highly
non-linear.  This manifests itself in the non-intuitive example of how
adding flavors enhances rather than suppresses low eigenvalues.

Issues involving zero mode suppression represent one facet of a set of
connected unresolved issues.  Are there non-perturbative ambiguities
in quantities such as the topological susceptibility?  How essential
are rough gauge fields, i.e. gauge fields on which the winding number
is ambiguous?  How do these issues interplay with the quark masses?  I
hope the puzzles presented here will stimulate more thought along
these lines.

\section*{Acknowledgments}
This manuscript has been authored under contract number
DE-AC02-98CH10886 with the U.S.~Department of Energy.  Accordingly,
the U.S. Government retains a non-exclusive, royalty-free license to
publish or reproduce the published form of this contribution, or allow
others to do so, for U.S.~Government purposes.

\section*{References}


\section{How to set up a proper comparison?}
\label{sec:analysis}
Measuring (small) changes in aerodynamic drag is not trivial, especially in the turbulent regime, regardless of the numerical or experimental nature of the analysis. Studies employ a variety of approaches, where simulations and experiments presents different approaches and different challenges.

Nowadays, whenever we need to compare the drag of a reference flat surface with that of a rough surface, we are aware of the subtlety of the measurement, of the importance to carefully define and control the Reynolds number of the experiment, to discriminate between internal and external flows, and in general to correctly define the equivalent "flat wall" flow to compare with. In this final Section, we will discuss some of these topics, trying to call the reader's attention to the logical steps that should be followed when designing a meaningful experimental or numerical campaign.

\subsection{Measurement of the drag (difference)}
All the available studies measure the drag difference $\Delta Drag$ by separately measuring the drag forces $Drag_{smooth}$ and $Drag_{dimples}$. As recently discussed in ref.\cite{vannesselrooij-etal-2022} in the context of the description of their novel experimental setup devoted to such measurements, various approaches are available. The simplest among them measure the local friction, and as such are unable to yield satisfactory results for the drag, because the friction contribution to the drag force over a dimpled surface depends on the position, and the same holds for the pressure component. Hence, in an experiment one has to either resort to measuring the drag force with a balance, a challenge by itself owing to the small forces involved, or to deduce the force from the pressure drop across two sections, as done for example by Gatti et al. in \cite{gatti-etal-2015}. With dimples, both approaches have been used. Information about the shear stress was extracted from boundary layer momentum loss in ref. \cite{lienhart-breuer-koksoy-2008}. Direct measurement of the drag through a force sensor was employed in refs.\cite{veldhuis-vervoort-2009, vannesselrooij-etal-2016, vancampenhout-etal-2018}. This type of measurement may be affected by uncertainty and accuracy problems: forces are small, and blurred by spurious contributions, and the experimental setup must be designed and run with extreme care.

In the case of numerical experiments, only the DNS approach provides the required accuracy that is not embedded e.g. in RANS models, constructed and tuned for canonical flows and hence incapable to deal with drag reduction in a quantitatively accurate way. Once DNS is used, two equivalent options are available to compute the drag in internal flows. One possibility is the calculation of the (time-averaged value of) the friction drag and the pressure drag separately, employing their definition as surface integrals of the relevant force component.
In alternative, the (time-averaged value of) the pressure drop between inlet and outlet informs of the total dissipated power, and thus leads to the total drag. This is feasible both in simulations and experiments. Tay and colleagues \cite{tay-2011, tay-khoo-chew-2015, tay-etal-2016, tay-khoo-chew-2017, tay-lim-2017, tay-lim-2018} in fact compared the mean streamwise pressure gradients of both the two flat sections upstream and downstream of the dimpled test section with the mean streamwise pressure gradient within the test section, employing static pressure taps.

Experience accumulated in riblets research, however, tells us that the riblets community obtained its first fully reliable dataset when D.Bechert in Berlin developed on purpose a test rig, the Berlin oil channel \cite{bechert-etal-1992}, where the measured quantity was directly the drag difference: targeting the quantity of interest, i.e. the drag difference under identical flow conditions,  instead of relying on the difference between two separately measured drag forces was key to improve accuracy and reliability.

\subsection{The Reynolds number}

Dynamic similarity is a well known concept in fluid mechanics, and enables meaningful comparative tests provided the value of the Reynolds number is the same. The true question is to understand {\em which} Reynolds number should be kept the same. The Reynolds number is defined as the product of a velocity scale $U$ and a length scale $L$, divided by the kinematic viscosity $\nu$ of the fluid. While e.g. in an experiment the precise measurement of $\nu$ might be difficult, its meaning is unequivocal. Choosing $U$ and $L$, instead, presents more than one option.

For the velocity scale $U$, dimples do not lead to specific issues. While for a zero-pressure-gradient boundary layer over a flat plate the use of the external velocity $U_\infty$ sounds reasonable, for internal flows like the plane channel flow one has to choose among the bulk velocity $U_b$, the centerline velocity $U_c$ and the friction velocity $u_\tau$. The choice of reference velocity has been already discussed in the context of skin-friction drag reduction \cite{hasegawa-quadrio-frohnapfel-2014}: provided drag reduction is not too large, and the flow is far enough from laminarity, choosing $U$ is not critical and should not be regarded as a major obstacle.

For the length scale $L$, instead, the situation is different, as dimples themselves contain one or more length scales that could be used in the definition of $Re$. For example, to avoid the ambiguity implied by the definition of the origin for the wall-normal coordinate, Van Nesselrooij et al. \cite{vannesselrooij-etal-2016} and Van Campenhout et al. \cite{vancampenhout-etal-2018} for their boundary layer experiments decided to define a Reynolds number based on the diameter of their circular dimple. Naturally, achieving the same $Re$ based on flow velocity and dimple diameter is not enough to guarantee dynamic similarity in two different flows.

\begin{figure}
\includegraphics[width=0.9\textwidth]{re}
\caption{Drag change versus bulk Reynolds number $Re_b$.}
\label{fig:Re}
\end{figure}

By isolating all the data sets for which a value for the bulk Reynolds number $Re_b$ is given (either explicitly or deduced from equivalent information), and putting together the reported drag changes, one obtains the picture reported in figure \ref{fig:Re}. Besides showing both drag reduction and drag increase, drag changes exhibit every possible trend with $Re_b$: increasing, decreasing, constant or nearly constant, and non-monotonic with either a maximum or a minimum at intermediate $Re_b$. Without excluding additional possible causes, this can be attributed to the host of parameters that are not kept identical across the dataset, besides the Reynolds number, and stresses once more the importance of experiments where only one parameter is changed at a time.

\begin{figure}
\centering
\includegraphics[width=1\textwidth]{results}
\caption{Present simulations, circular dimples at various sizes and Reynolds numbers with $2690 \le Re_b \le 10450$. Left: drag changes vs dimple depth in inner units. Right: drag changes vs dimple depth in outer units.}
\label{fig:deep-dimples}
\end{figure}

In a turbulent wall flow, the Reynolds number is an essential ingredient to define the proper scaling of important quantities, say the total drag change. If for example only the dimple depth $d$ is varied, its value can be set in wall units ($d^+$) or in outer units ($d/h$), and, if the Reynolds number is also changed, various combinations for $d^+$ and $d/h$ become possible. It is the flow physics which dictates what scaling works best at collapsing results.
We have performed two sets of DNS simulations (see \S\ref{sec:appendix} for details) to understand the scaling of drag changes induced by circular dimples when only their dimensions are changed but its shape is preserved. We have fixed the values of $d/D$ and $r/R$, the value of the depth $d$ (either in inner $d/h$ or outer $d^+$ units) has been varied, and all the other parameters did vary accordingly, as prescribed by equation \eqref{eq:geom_rel}.

Figure \ref{fig:deep-dimples} plots the results and shows that drag changes (in this specific case, drag increases) appear to follow an outer scaling: all the data points collapse onto a single curve when drag changes are plotted against $d/h$. This is an expected result, as these dimples are rather deep, and thus somehow akin to a large-scale $d$-roughness \cite{jimenez-2004}, where the large cavities basically destroy the near-wall layer, i.e the only region where inner scaling would make sense.

\subsection{The equivalent flat wall}

The comparison between flat and dimpled wall can be set up for internal or external flows. The latter, which may be less convenient in numerical simulations owing to their non-parallel nature, present a sensible advantage in this context, since drag and its related changes have simply to be computed for the same plate immersed in the same external velocity, and a reduced drag force is unequivocally advantageous. For internal flows, however, the non-planar dimpled wall brings up the problem of properly defining the location of the equivalent flat wall and, in general, of setting up the comparison properly.

\begin{figure}
\includegraphics[width=\textwidth]{equiv-channel}
\caption{A dimpled wall and two different, equivalent flat channels. The red/blue lines indicate the dimple profile. Left: the channel height $2h$ goes from the top wall to the dimple tip; right: the channel height $2h$ goes from the top wall to the dimple lowest point.}
\label{fig:equiv-channel}
\end{figure}

As shown schematically in fig.\ref{fig:equiv-channel}, for a channel flow, for example, a certain definition of the reference flat wall impacts the reference length $h$ and, eventually, changes the value of the Reynolds number of the flow to compare with. The reference wall might be placed on the flat surface among dimples, on the position of lowest elevation in the cavity, on the average height of the dimpled surface, etc., leading to different flow volumes.

To properly account for this effect, let us start from the usual definition of the bulk Reynolds number $Re_b = U_b h / \nu$, where $h$ is a reference length (e.g. half the witdh of the flat channel) and $\nu$ is the kinematic viscosity. Once the cross-sectional area $A(x)$ of the dimpled channel changes along the streamwise direction, the bulk velocity $U_b$, defined as an average velocity across the section, becomes itself a streamwise-dependent function:
\begin{equation}
  U_b(x)= \frac{1}{A(x)}\int_{A(x)} u(\mathbf{x}) \text{d}A .
\end{equation}
We thus replace this definition with a volume average, and define a new bulk velocity $\mathsf{U_b}$ as an average over the volume to obtain a streamwise-independent quantity:
\begin{equation}
  \mathsf{U_b} = \frac{1}{V} \int_V u(\mathbf{x}) \text{d}V .
\end{equation}
Note that the two quantities $U_b$ and $\mathsf{U_b}$ coincide for a flat wall. A comparison at same flow rate requires that the volumetric flow rate
\begin{equation}
  Q = \int_{A(x)} u(\mathbf{x}) \text{d}A = \frac{1}{L_x} \int_0^{L_x}\int_{A(x)} u(\mathbf{x})\text{d}A\text{d}x = \frac{1}{L_x} \int_V u(\mathbf{x}) d V = \frac{V}{L_x} \mathsf{U_b}
\end{equation}
is the same for the flat and dimpled channels, provided the streamwise length $L_x$ of the channel is the same. This implies that $V_f \mathsf{U_{b,f}} = V_d \mathsf{U_{b,d}}$, where the subscripts $\cdot_f$ and $\cdot_d$ refer to quantities measured in the flat and dimpled channel respectively. In the end, the bulk velocity in the dimpled channel (and the bulk Reynolds number) need to be changed by multiplication of a factor given by the volume ratio:
\begin{equation}
  \mathsf{U_{b,d}} =\frac{V_f}{V_d} \mathsf{U_{b,f}};
  \qquad
  Re_{b,d} = \frac{V_f}{V_d} Re_{b,f}.
\end{equation}

The numerical value of $Re_b$ is thus affected by the choice of the equivalent flat channel. For example, the equivalent flat channel might go from the top wall to the lowest point of the dimple, and $Re_{b,d}>Re_{b,f}$. In contrast, if the equivalent channel goes from the top wall to the tip of the dimple, $Re_{b,d}<Re_{b,f}$. The two bulk Reynolds numbers end up being the same only when the volume is preserved in the reference and dimpled channels (i.e. the equivalent flat channel is located at the average dimple height).

If the comparison is carried out by DNS, one conveniently measures the time-averaged value $\overline{f}$ of the spatially uniform volume force $f$ required to maintain a constant flow rate at each time step. This volume force is interpreted as $f = \Delta P / L_x$, where $\Delta P$ is the pressure drop along the channel. The proper measure of the drag change is:
\begin{equation}
  \Delta Drag = \frac{ V_d \overline{f_d} - V_f \overline{f_f} }{ V_f \overline{f_f} } = \frac{ V_d/V_f \overline{f_d} - \overline{f_f}}{\overline{f_f}};
\end{equation}

Therefore, the change of the fluid volume has to been considered also when measuring the drag change in the controlled case.

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{DC_difference}
\caption{Drag changes, measured by DNS, for circular dimples with $d/h=0.25$ at $Re_b \approx 2800$. Red/blue bars express drag changes when the equivalent channel defines $2h$ as the distance between the top wall and the top/bottom of the dimple (color code is the same of figure \ref{fig:equiv-channel}). Case A: comparison at the same $Re_b$, $\Delta Drag$ computed without accounting for the volume ratio. Case B: as case A, but $\Delta Drag$ is corrected with the volume ratio. Cases C and D are like cases A and B, but the comparison is made at the same flow rate.}
\label{fig:origin}
\end{figure}

Figure \ref{fig:origin} exemplifies the consequences of neglecting these considerations. These are certainly exaggerated by the choice of working with a dimple configuration that causes a large change of drag. However, the relative differences are major; neglecting such considerations would most certainly hinder the true ability of dimples to alter skin-friction drag.

\subsection{The drag reduction metrics}

In closing, we mention a final methodological issue, that affects drag reduction measurements for dimples, riblets, and roughness at large: the proper metrics to express it. It is customary to express drag reduction as (percentage) changes in the skin-friction coefficient at a given $Re$; unfortunately, the coefficient itself contains a dependence on the Reynolds number already for the flat wall case, thus making it impossible to rely on percentage changes for a robust assessment of the drag change properties of a given surface. The complete information would be the $(\Delta Drag, Re)$ pair. In alternative, the proper metric for expressing drag reduction is the vertical shift of the logarithmic portion of the mean streamwise velocity profile expressed in viscous units.

This is a known concept for roughness \cite{jimenez-2004} as well as riblets \cite{luchini-1996, spalart-mclean-2011}, and also extends to some active flow control strategies \cite{gatti-quadrio-2016}. As long as the direct effect of the roughness remains confined within the buffer layer of the turbulent flow, it can be translated into an upward shift $\Delta U^+$ of the logarithmic velocity proﬁle in the law of the wall: a positive $\Delta U^+$ corresponds to drag reduction, and a negative $\Delta U^+$ implies drag increase, as for the conventional $k$-type roughness. Part of the trends seen in figure \ref{fig:Re} for drag reduction data are due to Reynolds effects; properly removing them via analytical relations is possible, as done in ref.\cite{gatti-quadrio-2016} for active spanwise forcing, and would contribute to clarifying the situation, by exposing some remaining "puzzling" trends with $Re$ (to cite words used in ref.\cite{spalart-etal-2019}).

\section[
\section{Conclusions}
\label{sec:conclusions}

In this review paper we have provided a brief and up-to-date description of what we know and what we don't about the potential of dimples for turbulent skin-friction drag reduction.
While we can't obviously offer an answer to the still-standing question whether or not dimples are a suitable technique to reduce turbulent skin-friction drag, it is our hope that this comprehensive overview will at least help the newcomer to frame the problem, quickly identify the key references, and get a glimpse at the complexity of the topic.

While reviewing the state of the art, we have also mentioned some methodological issues that bear a critical importance when attempting to measure drag changes by dimples. Leveraging concepts and procedures (and perhaps facilities altogether) developed over the years for riblets might yield data that are reliable enough to begin understanding the physics behind dimple drag reduction, a necessary and preliminary step to improve their performance.

\section{Introduction}

Reducing the drag generated by a fluid set in relative motion to a solid body is at the same time a fundamental attempt to learn how to favorably interact with turbulence, and a technological challenge with immense potential in so many application fields. The interest for turbulent flow control is steadily increasing, owing to massive economic and environmental concerns.

Skin-friction drag is perhaps the most essential manifestation of the dissipative nature of turbulence, and accounts for the total drag in the case of planar walls (as in a channel flow or a zero-incidence flat plate boundary layer). Several techniques are available to reduce friction drag below the level typical of a smooth solid wall; they can be categorized into active (requiring extra energy) and passive ones.
The former typically provide larger savings, but imply extra complexity and cost, so that the ideal technique for friction reduction remains a passive one, often embodied in surface patterns performing better than the planar flat geometry.

The most prominent example of such patterns is riblets \cite{garcia-jimenez-2011-a}. Introduced by NASA in the '80 of the past century, and intensely studied over the subsequent years for their potential in aeronautical applications, riblets consist of streamwise-aligned microgrooves, and have the proved ability to reduce friction drag. The riblets cross-section can be of several shapes, the triangular one being perhaps the most popular, but an essential feature is a very sharp tip. Although new developments \cite{quadrio-etal-2022,cacciatori-etal-2022} hint at a bright future for riblets in aeronautics and suggest lower cost/benefit ratios, riblets are currently still not deployed in commercial transport aircraft, owing to their limited savings \cite{mclean-georgefalvy-sullivan-1987, kurita-etal-2020} and to critical production and maintainance issues, descending from the microscopic size of riblets and from the requirement of preserving the sharpness of the tip.

A possible alternative to riblets is emerging recently, easier to manufacture and lacking any sharp detail. The pattern to impress on the surface consists of small dimples. Dimples, i.e. small concavities imprinted on a surface, have been extensively studied in the past for their ability to enhance the heat transfer of a surface (see e.g. ref.\cite{leontiev-etal-2017} and references therein). The use of dimples on the surface of bluff bodies (e.g. a golf ball) is well known, and their ability to influence the turbulent boundary layer and the separation on the body is rather well understood \cite{choi-jeon-choi-2006}; the same concept is also being considered in sport-car racing \cite{allarton-etal-2020}. In this paper we concern ourselves with dimples applied to an otherwise flat surface: the goal is to reduce the turbulent skin-friction drag. We limit our review to passive dimples, although also active control by dimples has been proposed \cite{ge-fang-liu-2017}.

The ability of dimples to reduce drag is way less intuitive than that to increase heat exchange, and was considered first at the Kurchatov Institute of Athomic Energy \cite{kiknadze-krasnov-chushkin-1984} in Russia, where hemispherical dimples were placed on the surface of a heat exchanger and found to reduce the flow resistance as well.
In subsequent studies by the same group, a drag reduction of about 15--20\% was mentioned \cite{alekseev-etal-1998}, the highest performance reported so far.
In the last two decades, a handful of research groups devoted their efforts to understanding the drag reduction problem by dimples, attempting to come up with a recipe for the best shape and size.
Unfortunately, to date no consensus has been reached on the effectiveness of dimples in reducing friction drag, and on their working mechanism: some authors confirmed drag reduction, others did not.

Measuring -- in the lab, or with a numerical simulation -- the (very small, if any) drag reduction induced by dimples is by no means a trivial task.
A reduction of friction drag would be unavoidably accompanied by an increase of pressure drag, with a net benefit possible only if the former overwhelms the latter.
There are just so many design variables to be tested, as the geometry of the dimple itself, the size, the spatial layout and relative arrangement of the dimples on the surface need to be carefully considered. This is a daunting task as long as no theory, hypothesis of working mechanism or scaling argument is available to guide the search in such a vast parameter space.
However, it is undeniable that dimples, once proved to work, would provide substantial advantages over riblets, thanks to their simplicity, ease of production, lack of sharp corners and easier maintenance.

The goal of the present contribution is to provide the first comprehensive review of the published information available on dimples for skin-friction drag reduction.
Since the very fact that dimples can actually work is still subject to debate, we will complement the review with a discussion of important procedural aspects that in our view are essential, should one embark in a (numerical or laboratory) experiment to assess the potential for drag reduction.
Such procedures (or, more precisely, their absence) are at the root of the large uncertainty and scatter of the available data, and have hindered so far the answer to such a simple question as: Do dimples actually work to reduce turbulent drag?

The present contribution is structured as follows. Section \S\ref{sec:overview} provides an overview of the experimental and numerical studies on the drag reduction properties of dimples. Section \S\ref{sec:parameters} describes the geometrical parameters defining the dimples, whereas \S\ref{sec:physics} reports the two main physical explanations for the working mechanism of drag-reducing dimples. In \S\ref{sec:analysis} we highlight the problem of properly measuring drag reduction, and guidelines and recommendations on how to properly compare results among different studies are provided. This review paper is closed by brief concluding remarks in \S\ref{sec:conclusions}. Appendix \ref{sec:appendix} contains details of the DNS simulations that have been carried out for the present study.

In the next Subsection, the concept of dimples is introduced first, together with the notation that will be used later to indicate their geometrical parameters.

\subsection{Characterization of a dimpled surface}

\begin{figure}
\includegraphics[width=1\textwidth]{section}
\caption{Cross-section of the parametric dimple geometry introduced by ref.\cite{chen-chew-khoo-2012} (left) and streamwise shift of the deepest point (right).}
\label{fig:spherical_geom}
\end{figure}

A solid wall covered with dimples is described by several geometric parameters: the dimple shape, the relative spatial arrangement of the dimples and the coverage ratio (ratio between non-planar and total surface). Originally, dimples were conceived as spherical recesses, hence with a circular footprint on the wall. One particular class of circular dimples, introduced by Chen et al. \cite{chen-chew-khoo-2012}, has become quite popular thanks to its parametric nature and represents the starting point of our description. This design is the union of a spherical indentation and a torus, meeting tangentially in a regular way that avoids sharp edges. A cross-section of this dimple, which possesses axial symmetry, is drawn in figure \ref{fig:spherical_geom}.

Four parameters define the geometry of this dimple: the diameter $D$ of the circular section at the wall, the depth $d$ of the spherical cap, the curvature radius $r$ at the edge and the curvature radius $R$ of the spherical cap. These four parameters are not independent, but linked by one analytical relation, so that only three degrees of freedom exist. In fact, geometry dictates that:
\begin{equation}
\frac{D}{2} = \sqrt{ d \left( 2R + 2 r -d \right)}.
\label{eq:geom_rel}
\end{equation}

Moreover, a handful of studies extended this baseline circular geometry, by introducing the additional parameter $s$, which describes the shift along the streamwise direction (either downstream for $s>$ or upstream for $s<0$) of the point of maximum depth, which in the baseline geometry lies exactly at the center of the dimple cavity.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{shapes}
\caption{Popular variants of the dimple shape.}
\label{fig:diff_geom}
\end{figure}

It is difficult to overemphasize the importance of a well-defined parametric geometry in the quest for the optimally performing dimple. Although the circular shape is by far the most popular, over the last years a number of alternative dimple shapes have been studied; sketches for the various shapes are drawn in figure \ref{fig:diff_geom}. Some of them derive from a deformation of the circular shape: e.g. the elliptical dimple is the result of a symmetrical stretch of the circular dimple along the streamwise direction. The teardrop dimple has two segments tangent to the circle, preserves the spanwise symmetry and exists in two variants depending on whether the triangle points upstream or downstream. The diamond dimple is the union of the two variants of teardrop and possesses two vertices. Only the triangular dimple differs substantially from the circular shape and --- as for the teardrop dimple --- can have the streamwise-aligned vertex pointing upstream or downstream.

When a single dimple is identically replicated to fully cover the planar surface, the relative spatial arrangement of the dimples is important in determining the overall influence on the flow.
A regular spatial layout of a dimpled surface depends on the distance between two adjacent dimples in both the streamwise and spanwise directions.
Another parameter that is related to the spatial arrangement of dimples is the coverage ratio, that can be defined as the percentage of recessed surface compared to the total surface of the wall.
(The reader will notice an ambiguity, as at the denominator of the coverage ratio one could put either the surface area of the equivalent flat wall, or the wetted area of the dimpled surface. This ambiguity is often ignored, but it is commented upon e.g. in refs.\cite{prass-etal-2019,tay-khoo-chew-2017,ng-etal-2020}.) It is doubtful whether coverage, which is affected by so many parameters, is by itself a useful quantity to describe dimples performance.

\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{layout}
\caption{Most popular dimples layout: staggered (left) and flow aligned (right).}
\label{fig:stag-alig}
\end{figure}

Moreover, dimples can be arranged either irregularly or regularly following a certain pattern. The two most widespread patterns are the staggered and the flow-aligned arrangements. Their definition is not unique. Often, the layout is referred to as staggered when the streamwise projection of one dimple overlaps with the following one, while it is called flow-aligned otherwise (see figure \ref{fig:stag-alig}). However, this definition, that corresponds to the most used arrangement, is not universally accepted. Prass et al. \cite{prass-etal-2019}, indeed, define the staggered arrangement as having the distance in spanwise direction from the centres of two adjacent dimples equal to half the distance between the centres of two non-adjacent dimples. Several additional patterns have been tested, e.g. the hexagonal one.

\section{Do dimples work?}
\label{sec:overview}

Although in the last two decades a number of dimples-related contributions have appeared, many works claim that drag reduction is possible for certain geometries and flow conditions \cite{veldhuis-vervoort-2009, tay-2011, tay-khoo-chew-2015, vannesselrooij-etal-2016, tay-etal-2016, tay-lim-2017, tay-lim-2018, spalart-etal-2019}, whereas others only report drag increase \cite{tay-khoo-chew-2017, vancampenhout-etal-2018, prass-etal-2019, ng-etal-2020}. Notably, one work \cite{spalart-etal-2019} set out to specifically reproduce the experimental drag reduction results described in ref.\cite{vannesselrooij-etal-2016} with a state-of-the-art combined numerical/experimental study, and failed.

Such uncertain situation can be traced back to the lack of a generally accepted standard to measure drag and to compare different geometries among themselves and with the reference flat wall, since there are unavoidable differences when measuring drag in experiments and simulations, and in external (e.g. boundary layer) and internal (plane channel) flows.
An additional reason explaining the scatter of available results consists in the still limited understanding of how dimples modify the surrounding flow field.
Knowing the physics involved in drag reduction by dimples would be extremely useful in the optimization of all the several parameters involved. A description of the effect of the many geometrical parameters involved, and on the conjectures on the working mechanism of dimples are reported later in \S\ref{sec:parameters} and \S\ref{sec:physics} respectively.

We start by presenting an overview of the main results available in the literature, by focusing on the raw drag reduction information.
They are reported in Table \ref{tab:comparison} that contains entries for the best drag reduction figure that could be extracted from each paper; when multiple dimple shapes are present, they are all considered. Drag change is simply defined here as $\Delta Drag = Drag_{dimples} - Drag_{smooth}$, where $Drag_{smooth}$ and $Drag_{dimples}$ are the (measured or computed) friction drag of the reference flat plate and the total drag of the dimpled plate, respectively. Negative values of $\Delta Drag$ thus correspond to drag reduction. Across the several studies, various definitions of the Reynolds number are used, particularly for internal flows. These have been all converted, whenever possible, to value of the bulk Reynolds number $Re_b$, by using the empirical Dean's correlation \cite{dean-1978}. Several other entries are also available in the Table, and will be defined and discussed throughout the text.

\subsection{Experimental studies}
\label{sec:experiments}

The majority of the experimental studies carry out their tests in a wind tunnel and compare the drag measured on a flat plate with the drag measured on a dimpled plate. The flat/dimpled plate lies either on the upper or bottom wall, whereas the other wall of the wind tunnel is smooth. The plate is installed at a certain distance from the entrance section for the flow to become fully developed by the time it reaches the test section. A major difference among the various studies consists in the internal/external character of the flow.

The largest drag reduction, as observed in Table \ref{tab:comparison}, is a whopping 14\% found in the boundary layer experiment by Veldhuis \& Vervoort \cite{veldhuis-vervoort-2009} at the Technical University of Delft. The free-stream velocity was $7.5\ m/s$ and dimples were of circular shape. They found the staggered configuration to be more efficient in reducing drag than the flow-aligned one. Other boundary layer experiments carried out later by the same group at TU Delft reported a significantly smaller but still extremely interesting maximum drag reduction of 4\% \cite{vannesselrooij-etal-2016}, obtained at a Reynolds number based on the free-stream velocity $U_\infty$ and the dimple diameter $D$ of $Re_D \approx 40000$, which corresponds to a Reynolds based on the boundary layer thickness $\delta$ of $Re_\delta = 1500$. In this case, dimples are relatively large (in physical dimensions) shallow circular dimples, with a 50\% smaller coverage ratio than ref.\cite{veldhuis-vervoort-2009}. In a later study  \cite{vancampenhout-etal-2018}, they also measured a drag increase of 1\% for shallow dimples with different layouts at $Re_D \approx 63100$. Van Nesselrooij et al. \cite{vannesselrooij-etal-2016} presented what is described in ref.\cite{spalart-etal-2019} as a "very convincing experimental paper", studying different dimples configuration and finding that the best one consistently involves sparse (low coverage) and staggered dimples for the entire range of considered Reynolds numbers. They also focused on the importance of the depth of the dimples. When made dimensionless with the dimple diameter, shallower dimples are found to work better for each flow condition; however, when depth is compared to the boundary layer thickness, shallow dimples work better at low $Re$ but deep dimples are better at higher $Re$.

Another group that provided significant contribution to the dimples research thread is from the National University of Singapore, with Tay and colleagues. They performed experimental studies on a channel flow and reported up to 7.5\% drag reduction in ref.\cite{tay-lim-khoo-2019} for diamond-shaped dimples at a Reynolds number based on the bulk velocity $U_b$ and the channel semi-height $h$ of $Re_b \approx 30000$ and a layout with full coverage. Large drag reductions are measured also with other non-conventional dimple shapes \cite{tay-lim-2018}, such as the upstream-pointing teardrop at 6\%, or the downstream-pointing teardrop at 5\%, in a flow with $Re_b \approx 30140$ and $Re_b \approx 22270$, respectively. Conversely, the triangular shape was proved to always lead to drag increase \cite{tay-etal-2016}. Circular dimples were found to be less effective than diamond and teardrop shapes. A reduction of drag up to 2\% \cite{tay-2011} and 2.8\% \cite{tay-khoo-chew-2015} are found at $Re_b \approx 17500$ and $Re_b \approx 32100$ for different physical geometrical parameters of the dimple but with an identical layout and coverage ratio of 90\%. At $Re_b \approx 42850$ a drag reduction of 3.5\% is measured in ref.\cite{tay-lim-2018}. In ref.\cite{tay-khoo-chew-2015} they compare the same physical dimples and flow geometry by varying the coverage ratio and find that a dense layout with 90\% coverage performs better than a sparse one with 40\% coverage. They also compare two different dimple depths, measuring slightly higher drag reduction for deeper dimples. Finally, Tay \& Lim in ref.\cite{tay-lim-2017} experiment with shifting the point of maximum depth within the dimple along the streamwise direction, and measure the best performance of 3.7\% when the shift is $s=0.1D$ in the downstream direction.

\subsection{Numerical simulations}
\label{sec:simulations}

For drag reduction studies, numerical simulations need to resort to high-fidelity approaches, like Direct Numerical Simulation (DNS) and highly resolved LES (Large Eddy Simulation). Obviously, such simulations are not very practical for large-scale parametric studies, especially when the Reynolds number becomes large, since their unit computational cost rapidly increases with $Re$. The need for high-fidelity methods, the computational cost and the requirement to handle non-planar geometries are among the reasons why numerical studies for dimples are fewer than experiments. However, simulations (and DNS in particular, which avoids the need of turbulence modeling) are perfectly suited for such fundamental studies and provide us with the full information required to understand the working mechanism of dimples, by e.g. breaking up the drag changes into friction drag and pressure drag changes and by yielding a detailed and complete statistical characterization of the turbulent flow.

Circular dimples in a turbulent channel flow were studied with DNS for the first time in 2008 by Lienhart et al. \cite{lienhart-breuer-koksoy-2008}, who reported a drag increase of 1.99\% at $Re_b \approx 11000$. The same work contains also an experimental study of the same configuration, for which no drag changes were observed.

Ng et al. \cite{ng-etal-2020} at NUS performed one of the most interesting DNS studies, considering a turbulent channel flow at $Re_b=2800$ and examining different dimple geometries. They found that the classic circular dimple increases drag by 6.4\%, an amount that decreases to 4.6\% when the point of maximum depth is shifted downstream by $s=0.1D$. They also studied non-circular dimple shapes, obtaining this time a large drag reduction of 7.4\% for the diamond dimple, 4.9\% for the elliptical dimple and 3.1\% for the upstream-pointing teardrop dimple; the downstream-pointing teardrop dimple, instead, gave 0.1\% drag increase.

Another recent numerical channel flow study is that by Tay and coworkers in \cite{tay-khoo-chew-2017}: they run a Detached Eddy Simulation (in which a baseline LES is complemented with a RANS model for the near-wall region) to replicate their own experimental study described in ref.\cite{tay-khoo-chew-2015}.  The DES yielded 1\% drag increase at $Re_b= 2830$, which does not confirm the experimental study and found drag increase for every case tested, whereas the experiments found smaller drag increase and even a slight drag reduction for a particular geometry. The suitability of DES for such drag reduction studies, however, remains dubious.

Prass et al. \cite{prass-etal-2019} published the only work in which an open channel is considered: with a LES they report a drag reduction of 3.6 \% at $Re_b \approx 6121$. They also considered two different configurations, finding that flow-aligned dimples perform better than staggered dimples.

There is only one DNS study for the boundary layer, i.e. the already mentioned work by Spalart \& at. \cite{spalart-etal-2019}, in which circular dimples at $Re_\delta = 30000$ were considered as the baseline geometry. They additionally studied the effect of the edge radius $r$, and found that with proper smoothing of this edge a drag reduction of -1.1\% is obtained, which descends from the combination of a -1.7\% reduction of friction drag, counterbalanced by a 0.6\% increase of pressure drag.

\newcommand{\specialcell}[2][c]{%
  \begin{tabular}[#1]{@{}c@{}}#2\end{tabular}}

\begin{landscape}
\begin{table}
\caption{Summary of the main parameters and results of the literature. Columns report, in sequential order: 1. the reference and its acronym; 2. the numerical (DNS: direct numerical simulation, LES: large eddy simulation, DES: detached eddy simulation; FVM: finite volume method, SEM: spectral element method) or experimental approach; 3. the flow type: channel flow (CF), half-channel flow (HCF) or boundary layer (BL); 4. the value of the Reynolds number: $Re_b$ for CF and HCF, $Re_\delta$ for BL (other $Re$ definitions for CF are converted to $Re_b$ using the Dean's law); 5. the dimple shape: circular (Circ), triangular (Triang), diamond (Diam), elliptical (Ell), teardrop (Tear); upstream-pointing (Up), Downstream-pointing (Do); 6. spanwise width $D_z$ and streamwise length $D_x$, expressed as a fraction of the reference length $L$ (the channel half-height $h$ for CF and the boundary layer thickness $\delta$ for BL); for a circular dimple $D_z=D_x$, thus only one value is reported; 7. the dimple depth $d$; 8. the edge curvature radius $r$; 9. the curvature radius $R$ of the spherical cap; 10. the shift $s$ of the point of maximum depth; 11. the coverage ratio; 12. the type of layout: S:staggered, A: aligned, H:hexagonal; 13. the percentage drag change. "-" is used when some information required to compute the value is lacking.
\label{tab:comparison}}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\begin{tabularx}{1\linewidth}{CCCCCCCCCCCCC}
\toprule
\hline
\textbf{Article}   & \specialcell[t]{\textbf{Num/} \\ \textbf{Exp}}  &\textbf{Flow}   &\textbf{Re}    &\textbf{Shape} & \specialcell[t]{$\mathbf{D_z/L}$ \\ $\mathbf{(D_x/L)}$}  &$\mathbf{d/D_z \%}$   &$\mathbf{r/D_z}$   &$\mathbf{R/D_z}$    &$\mathbf{s/D_x}$   &\textbf{Cov$\%$} &\textbf{Layout} &$\mathbf{\Delta Drag \%}$  \\
\toprule
\multirow{2}*{\makecell{LBK-08 \\ \cite{lienhart-breuer-koksoy-2008}}} &Exp  &CF  &$\approx 10000$   &Circ  &$0.6$  &$5$ &- &- &$0$  &$22.5$ & A &$0$  \\
                                                  &DNS (FVM)   &CF  &$10935$           &Circ  &$0.6$  &$5$ &- &- &$0$  &$22.5$ & A &$+1.99$  \\
\midrule
\multirow{1}*{\makecell{VV-09 \\ \cite{veldhuis-vervoort-2009}}}      &Exp   &BL       &-                &Circ  &-      &$5$ &- &- &$0$  &$60$ &S &$-14$  \\
\\
\midrule
\multirow{1}*{T-11  \cite{tay-2011}}                    &Exp   &CF  &$\approx 17500$  &Circ  &$5$    &$5$ &$0.84$   &$1.68$ &$0$  &$90$  &S &$-2$  \\
\midrule
\multirow{1}*{\makecell{TKC-15  \\ \cite{tay-khoo-chew-2015}}}          &Exp   &CF  &$\approx 32100$  &Circ  &$5$    &$5$  &$4.2$ &$8.45$ &$0$  &$90$   &S  &$-2.8$  \\
\midrule
\multirow{1}*{\makecell{VVVS-16 \\ \cite{vannesselrooij-etal-2016}}}   &Exp   &BL       &$\approx 1500$   &Circ  &$26.67$ &$2.5$ &$0.5$ &$4.51$ &$0$  &$33.3$  &S &$-4$  \\
\midrule
\multirow{3}*{\makecell{TLKJ-16 \\ \cite{tay-etal-2016}}}             &Exp   &CF  &$\approx 5625$  &Triang Up  &$- (4.67)$ &$5$   &- &- &$-0.5$  &-   &S  &$+4.8$  \\
                                                  &Exp   &CF  &$\approx 5625$  &Triang Do  &$- (4.67)$ &$5$   &- &- &$+0.5$  &-   &S  &$+2.8$  \\
                                                  &Exp   &CF  &$\approx 50350$ &Circ      &$5$    &$5$   &- &- &$+0.1$ &$90$   &S  &$-3.6$  \\
\midrule
\multirow{1}*{\makecell{TKC-17 \\ \cite{tay-khoo-chew-2017}}}         &DES (FVM)    &CF  &$\approx 2830 $  &Circ     &$5$    &$1.5$  &$0.84$ &$1.68$ &$0$  &$90$   &S  &$+1$  \\
\bottomrule
\end{tabularx}
\end{table}
\unskip
\end{landscape}

\begin{landscape}
\begin{table}
\newcolumntype{C}{>{\centering\arraybackslash}X}
\begin{tabularx}{1\linewidth}{CCCCCCCCCCCCC}
\toprule
\hline
\textbf{Article}   & \specialcell[t]{\textbf{Num/} \\ \textbf{Exp}}  &\textbf{Flow}   &\textbf{Re}    &\textbf{Shape} & \specialcell[t]{$\mathbf{D_z/L}$ \\ $\mathbf{(D_x/L)}$}  &$\mathbf{d/D_z \%}$   &$\mathbf{r/D_z}$   &$\mathbf{R/D_z}$    &$\mathbf{s/D_x}$   &\textbf{Cov$\%$} &\textbf{Layout} &$\mathbf{\Delta Drag \%}$  \\
\toprule
\multirow{1}*{\makecell{TL-17 \\  \cite{tay-lim-2017}}}                &Exp   &CF  &$\approx 28600 $  &Circ    &$5$   &$1.5$ &- &- &$+0.1$  &$90$   &S  &$+1$  \\
\midrule
\multirow{3}*{\makecell{TL-18 \\ \cite{tay-lim-2018}}}                &Exp   &CF  &$\approx 42850$   &Circ    &$5$    &$5$   &- &- &$0$  &-   &S  &$-3.5$  \\
                                                  &Exp   &CF  &$\approx 30140$   &Tear Up  &$5 (7.5)$    &$5$   &- &- &$+0.17$  &$84$   &S  &$-6$  \\
                                                  &Exp   &CF  &$\approx 22270$   &Tear Dp  &$5 (7.5)$    &$5$   &- &- &$-0.17$  &$84$   &S  &$-5$  \\
\midrule
\multirow{1}*{\makecell{VVVVS  \\-18 \cite{vancampenhout-etal-2018}}}     &Exp   &BL       &-                 &Circ    &-      &$2.5$     &$0.5$ &- &$0$  &-   &S/A/H  &$+1$  \\
\\
\midrule
\multirow{1}*{\makecell{SSSTPW \\ -19 \cite{spalart-etal-2019}}}           &DNS (FVM)    &BL       &$30000$           &Circ    &$1.33$ &-  &$2.03$  &-  &$0$  &- &S &$-1.1$  \\
\midrule
\multirow{1}*{\makecell{TLK-19 \\ \cite{tay-lim-khoo-2019}}}           &Exp   &CF  &$\approx 30000$   &Diam     &$5(10)$    &$5$         &- &- &$0$  &$99$   &S  &$-7.5$  \\
\midrule
\multirow{1}*{\makecell{PWFB \\ -19\cite{prass-etal-2019}}}            &LES (FVM)    &HCF    &$\approx 6121$  &Circ  &$5.7$ &$2.5$  &$1.5$ &$4.51$ &$0$  &- &S  &$-3.6$  \\
\midrule
\multirow{5}*{\makecell{NJLTK \\-20 \cite{ng-etal-2020}}}               &DNS (SEM)    &CF    &$2800$  &Circ       &$5$  &$5$   &$0.84$ &$1.68$ &$+0.1$  &$90.7$ &S  &$+4.6$  \\
                                                  &DNS (SEM)    &CF    &$2800$  &Ell        &$5 (7.5)$  &$5$   &$0.84$ &$1.68$ &$0$  &$90.7$  &S  &$-4.9$  \\
                                                  &DNS (SEM)    &CF    &$2800$  &Tear Up        &$5 (8.75)$  &$5$   &$0.84$ &$1.68$ &$+0.21$  &$84.4$  &S  &$-3.1$  \\
                                                  &DNS (SEM)    &CF    &$2800$  &Tear Dp        &$5 (8.75)$  &$5$   &$0.84$ &$1.68$ &$-0.21$  &$84.4$  &S  &$+0.1$ \\
                                                  &DNS (SEM)    &CF    &$2800$  &Diam          &$5 (10)$  &$5$   &$0.84$ &$1.68$ &$0$  &$99.5$  &S  &$-7.4$  \\
\bottomrule
\end{tabularx}
\end{table}
\unskip
\end{landscape}

\section{How to design dimples?}
\label{sec:parameters}

Systematic studies which address the influence of each geometric parameter are lacking, so that the design of the optimal configuration to achieve the maximum drag reduction has not been identified yet. This Section describes the little we know, first in terms of the geometrical characteristics of the dimples and then in terms of their arrangement.

\subsection{The shape of the dimple}

\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{depth} \\
\includegraphics[width=0.49\textwidth]{edgeradius-circular}
\includegraphics[width=0.49\textwidth]{radius-circular}
\caption{Drag change obtained with circular dimples versus depth (top), edge curvature radius (bottom left) and radius of the spherical cap (bottom right). Dashed lines connect points for which only the parameter on the abscissa is changed.}
\label{fig:geom_par_dep}
\end{figure}
Figure \ref{fig:geom_par_dep} plots the drag change data measured by several works which adopted the baseline circular geometry. The percentage of drag change is shown as a function of the three independent geometrical parameters $d/D$, $r/D$ and $R/D$, after extracting from each publication the largest drag reduction (or the smallest drag increase). It should be noted that, in general, the various points correspond to simulations or experiments that differ for other, sometimes very important, design parameters. Dashed lines, instead, connect points for which only the parameter on the abscissa is changed.

The influence of $d/D$ on the drag change has been studied by several authors: previous research from heat exchange enhancement suggests the very reasonable idea that this is one of the key factors affecting drag. However, while the optimal $d/D$ is in the range $0.1-0.5$ for best heat exchange \cite{kovalenko-terekhov-khalatov-2010, tay-etal-2014}, several authors report that shallower dimples with $d/D < 0.1$ should be considered for reducing the overall drag, to avoid excess penalty from the ensuing pressure drag.
Data are extremely scattered and clearly indicate that the drag change over a dimpled surface does not depend on the $d/D$ ratio alone. For example for $d/D=0.05$ Veldhuis and Vervoort \cite{veldhuis-vervoort-2009} report a drag reduction of almost 15\%, while Tay et al. \cite{tay-khoo-chew-2017} report a drag increase of approximately 6\%. The experimental measurements from ref.\cite{veldhuis-vervoort-2009} are for a turbulent boundary layer over a dimpled surface with coverage ratio of 60\% at a free-stream velocity in the range $0-29\ m/s$; the results from ref.\cite{tay-khoo-chew-2017} are from a Detached Eddy Simulation of a turbulent channel flow at $Re_b \approx 3000$ with a coverage ratio of 90\%.

It is reassuring, though, to see --- at least in some of the datasets where data points are connected by dashed lines --- a local optimum for intermediate depths, since it is reasonable to expect zero drag changes for $d \to 0$ and an increase of drag as for standard $k$-type roughness for $d \to \infty$.
With the other parameters unchanged, Tay et al. \cite{tay-khoo-chew-2017} and Van Nesselrooij et al. \cite{vannesselrooij-etal-2016} agree on observing a decrease of performance with increasing $d/D$ (in the range $0.015 < d/D < 0.05$), although at a different rate; within the same $d/D$ range, Tay et al. \cite{tay-khoo-chew-2015} and Veldhuis and Vervoort \cite{veldhuis-vervoort-2009} measured a slight increase of drag reduction performance with increasing $d/D$.
The latter study was extended up to $d/D =0.12$, finding that for $d/D>0.05$ the overall drag increases with $d/D$.

The curvature radius $r$ at the edge of the dimple is meant to mitigate the negative effects of pressure drag, by preventing or decreasing flow separation. The second panel of figure \ref{fig:geom_par_dep} shows that also in this case data are highly scattered: for $0.5 \lessapprox r/D \lessapprox 1.5$ the achieved drag change ranges between -4\% \cite{vannesselrooij-etal-2016} and 4.8\% \cite{ng-etal-2020}. The experiments of Tay et al. \cite{tay-khoo-chew-2015} at $Re_b \approx 32100$ show that after a certain value, i.e. $r/D > 4$, the influence of the edge curvature on the drag change is minimal. Spalart et al. \cite{spalart-etal-2019} performed a DNS of a turbulent boundary layer, with a Reynolds number (based on the boundary layer thickness) of $Re_\delta = 7.5 \times 10^3 $ and $ Re_\delta=3 \times 10^4$ and considered $r/D=0.5$. Their data points are not plotted in figure \ref{fig:geom_par_dep}, since their paper does not contain enough information to quantify $r$. However, they confirmed that smoothing the dimple rim is beneficial.

A scattered picture is also obtained when data are plotted against the $R/D$ ratio, as shown in the third panel of fig.\ref{fig:geom_par_dep}, confirming again that for this configuration a single geometrical parameter is unable to fully characterize the influence of the dimples on the flow.

The experiments of Tay \& Lim \cite{tay-lim-2017} and the numerical simulations of Ng et al. \cite{ng-jaiman-lim-2018, ng-etal-2020} agree on the indication that the downstream shift $s$ is beneficial, for a wide range of Reynolds numbers, with the best effect observed when $s=0.1D$ in the downstream direction. When instead the shift is in the upstream direction, i.e. $s<0$, drag increases rapidly.
It should be mentioned that the Reynolds number of the simulations ($Re_b=2800$) is somewhat lower than the lowest Reynolds number of the experiments ($Re_b \approx 4300$). Tay \& Lim \cite{tay-lim-2017} claim that a $0.2D$ downstream shift is equivalent to the axisymmetric case at $Re_b=7000$ with a drag increase of 1.5\%, because the lower drag obtained by the reduced flow separation at the shallower upstream wall is compensated by the higher drag of the flow impinging on the steeper downstream wall. Ng et al. \cite{ng-jaiman-lim-2018, ng-etal-2020}, who can take advantage of DNS to break down the total drag into friction and pressure contributions, find that friction drag is almost unaffected by a downstream shift, since it does not affect the reattachment point.

When it comes to alternative shapes, triangular dimples were considered by Tay et al. \cite{tay-etal-2016}. In their experiment they machined dimples with the bottom surface sloping up from the deepest point at the triangular vertex towards the base of the triangular depression to meet the flat channel surface, hence producing the negative of a wedge. A larger drag was obtained for both upstream- and downstream-pointing triangles, for the whole range of tested Reynolds numbers, i.e. $ 5180 \le Re_b \le 28600$. Moreover, for the downstream-pointing triangle the drag increase is nearly constant with $Re$, whereas for the upstream-pointing triangle the drag increase grows with $Re$.
Tay \& Lim \cite{tay-lim-2018} studied the teardrop dimple and measured drag reduction for both the upstream- and downstream-pointing teardrops, for $4500 \le Re_b \le 44000$, with the former yielding up to 6\% drag reduction and the latter up to 5\%. Tay et al. \cite{tay-lim-khoo-2019} studied the diamond dimple and measured drag reduction up to 7.5\%. More recently, Ng et al. \cite{ng-etal-2020} compared in a numerical study the circular, elliptical, teardrop and diamond dimples in a turbulent channel flow, reporting drag reduction of 4.9\% for the elliptical dimple, 3.1\% for the upstream-pointing teardrop, and 7.4\% for the diamond dimple.

\subsection{The arrangement of the dimples}

When it comes to the spatial arrangement of dimples on the surface, once the other parameters are fixed, the staggered configuration leads to lower drag compared to the flow-aligned one \cite{veldhuis-vervoort-2009, vannesselrooij-etal-2016, vancampenhout-etal-2018, spalart-etal-2019}, a fact that explains why the staggered configuration is the most adopted one. Van Nesselrooij et al. \cite{vannesselrooij-etal-2016} found 3\% of drag increase for flow-aligned dimples and up to 4\% drag reduction for staggered dimples with the same geometrical parameters, coverage and Reynolds number. Spalart et al. \cite{spalart-etal-2019} found drag increase for both configurations, but the drag increase of the flow-aligned dimples was almost twice that of the staggered dimples.

Lashkov and Samoilova \cite{lashkov-samoilova-2002} and Van Campenhout et al. \cite{vancampenhout-etal-2018} considered the drag change also for other, non-standard arrangements. The former study found a large drag increase (up to 50\%) when an hexagonal dimple layout is used. The latter study showed a constant drag increase of about 1\% for each of the several considered layouts.

The effect of coverage ratio was considered by Tay et al. \cite{tay-2011, tay-khoo-chew-2015}, who compared in a channel flow circular dimples with 40\% and 90\% coverage, and found that higher coverage enhances the (positive or negative) effects of the dimples.
Van Nesselrooij et al. \cite{vannesselrooij-etal-2016} experimentally studied the effect of coverage in a boundary layer. They found that a 90\% coverage yields drag increase for a wide range of $Re$, whereas 33.3\% coverage always yields drag reduction within the same Reynolds number range. Performance of both layouts are found to improve by increasing $Re$. Spalart at al. \cite{spalart-etal-2019} in their boundary layer DNS compared the two coverage ratios, and observed about 1\% of drag reduction for the lower coverage, and 2\% of drag increase for the higher one.

\section{How do dimples work?}
\label{sec:physics}

The uncertainties on the true effectiveness of dimples in reducing turbulent drag are accompanied, perhaps unsurprisingly, by a limited understanding of the physics involved. Thanks to the several experimental and numerical works carried out so far, some ideas and hypotheses exist, but consensus is lacking. We describe below two prevailing descriptions of how dimples interact with the overlying turbulent flow.

\subsection{Self-organized secondary tornado-like jets}

The first attempt at explaining drag reduction by dimples is due to Kiknadze et al. \cite{kiknadze-gachechiladze-barnaveli-2012}, who based their explanations uniquely of video records and photographs, even though similar observations were already put forward in previous numerical \cite{veldhuis-vervoort-2009} and experimental \cite{kovalenko-terekhov-khalatov-2010} studies. According to ref.\cite{kiknadze-gachechiladze-barnaveli-2012}, whose authors are affiliated with the Research and Production Centre “Tornado-Like Jet Technologies” in Moscow, the action of dimples can be explained by a so-called tornado-like jet self-organization. In plain words, this is how the flow organizes itself and develops over the double-curvature concavity of a dimple. The flow coming from an upstream flat portion accelerates at the leading edge of a circular dimple, and is lifted off from the surface while trying to follow the curved wall, leading to a reduction of skin-friction drag in the fore half of the dimple. After the streamwise midpoint, the flow converges towards the midline to eventually meet the flat wall past the trailing edge, and the skin friction increases again. Although the skin friction reduction in the fore half might outweigh the increase of the aft half, the recessed geometry of the dimple introduces an additional pressure drag component: hence, to achieve drag reduction the net reduction of the skin-friction drag needs to be larger than the increase of pressure drag. It should be observed, though, that this description is not directly addressing the insurgence of drag reduction, but only constitutes an attempt to draft a simplified description of the local flow modifications induced by the dimple.

If dimples are deep enough, their steep walls make the flow prone to separation on the upstream part of the recess, with creation of spanwise vorticity and recirculation.
The flow reversal has a positive effect on the drag, causing negative skin friction in the first portion of the dimple.
When the flow reattaches, a strong impingement of the flow on the rear slope of the dimple produces a locally high skin friction.
Moreover, flow separation obviously causes a large increase of pressure drag which could cancel out the positive effect of the skin friction drag.
To avoid separation and the consequent increase of pressure drag, more efficient shapes than the classical circular one are used.
Shifting downstream the point of maximum depth of the dimple alters the wall slopes, and affects the total drag by changing pressure drag, whereas the friction drag tends to remain unchanged \cite{ng-etal-2020}.
A (moderate) downstream shift minimizes the negative effects of separation, and offers lower drag than the standard circular geometry.
However, the shift does not significantly affect the location of the reattachment point, except for very large shifts, for which flow reversal may be entirely suppressed, but at the cost of an intense impingement onto the steeper rear wall which negatively affects the drag.

Non-circular dimples induce different drag changes \cite{ng-etal-2020}. Flow separation and flow reversal are not observed for elliptical, upstream-pointing and diamond dimples, leading to a lesser drag compared to the smooth wall. This can be attributed to the gentler upstream slope and to the longer, more streamwise-aligned leading edge.
Other studies which do not report flow reversal even for the circular shape are \cite{vannesselrooij-etal-2016, spalart-etal-2019}; they measure a maximum drag reduction of 4\% and 1.1\%, respectively. Tay et al. \cite{tay-khoo-chew-2017} observe flow separation for circular dimples in the whole range of tested flow conditions for $d/D = 0.05$, but not for $d/D=0.015$; however, they measure drag increase in all the tested cases.

\subsection{Spanwise forcing}
A more recent conjecture on the mechanisms by which dimples attain drag reduction has been put forward independently by the two groups at TU Delft \cite{vannesselrooij-etal-2016} and NUS \cite{tay-khoo-chew-2015}.
Flow visualisations indicate that, near the wall, streamlines coming in straight from a flat surface bend towards the dimple centerline in the upstream portion of the recess, then bend away from it in the downstream portion, thus creating a converging-diverging pattern (see for example \cite{tay-etal-2014}).
Such meandering implies a spanwise velocity distribution with changing sign across the dimple length \cite{vannesselrooij-etal-2016, vancampenhout-etal-2018}, and a consequent alternating streamwise vorticity \cite{tay-khoo-chew-2017} since the spanwise velocity remains confined very near to the wall. Ref. \cite{vannesselrooij-etal-2016} reports an average spanwise velocity of about 2--3\% of the free-stream velocity for a boundary layer; ref.\cite{tay-khoo-chew-2017} measured a maximum spanwise velocity in the range 3.5--8\% of the centerline velocity in the channel. Spalart et al. \cite{spalart-etal-2019} also detected in their DNS study a spanwise motion, although weaker in intensity.

\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{spanvel}
\caption{Instantaneous spanwise velocity component $w$ on a wall-parallel plane at $y^+= 1.3$ from the flat part of the wall. Lengths and velocities are made dimensionless with $h$ and $\mathsf{U_b}$. The velocity field is computed by DNS for a circular dimple, which actually yields drag increase.}
\label{fig:spanwise}
\end{figure}

Figure \ref{fig:spanwise} depicts an instantaneous spanwise velocity field over a circular dimple, taken from one of our DNS simulations of a turbulent channel flow over circular dimples (see Appendix \ref{sec:appendix} for computational and discretization details of our simulations). An alternating spanwise velocity pattern is clearly visible, supporting the idea that the dimple creates a velocity component in the spanwise direction and bends the streamlines in a converging-diverging behaviour. The instantaneous values are very large, up to 40\% of the bulk velocity.

The alternate spanwise velocity resembles the spanwise-oscillating wall \cite{jung-mangiavacchi-akhavan-1992}, an active technique for the reduction of turbulent friction drag, where the wall oscillates in time in the spanwise direction. In the oscillating-wall control, the spanwise  velocity component at the wall $w_w$ is prescribed as a function of time as:
\begin{equation}
w_w(t) = A \sin \left( \frac{2 \pi}{T} t  \right),
\end{equation}
where $A$ is the amplitude of the oscillation and $T$ is its period.
The oscillating wall produces very large reductions of friction drag, although at a significant energy cost. Its detailed performance is determined by the control parameters $A$ and $T$; Quadrio \& Ricco \cite{quadrio-ricco-2004}, after a careful DNS study, identified the link between the value of parameters and the obtained drag reduction. They found an optimum value for the oscillating period of $T^+ \approx 100$, whereas drag reduction monotonically improves with the amplitude (albeit the energy cost of the control rises faster as $A^2$).
Dimples could be considered as a passive implementation of the spanwise-oscillating wall. Van Campenhout et al. \cite{vancampenhout-etal-2018} measured the analogous parameters and defined a period $T$ and a maximum spanwise velocity $w_{max}$ of a fluid particle, averaging over a selected region of the domain. In the oscillating wall, it is known \cite{quadrio-sibilla-2000} that the time-averaged mean spanwise velocity profile coincides with the laminar solution of the Stokes second problem. Ref.\cite{vancampenhout-etal-2018} assumes the same to hold for the flow over dimples, thus deriving an analogous value for the amplitude. For their dimples with $d/D=0.025$, they found $T^+=135$ and $A^+=0.74$. Data from ref.\cite{quadrio-ricco-2004} do not contain information for such small amplitudes, but an extrapolation leads to a drag reduction of about 4\% for this combination of parameters: a value that closely resembles the measurement of 3.8\% from ref.\cite{vannesselrooij-etal-2016}.

It should be noted, first, that a closer analogy should be made between this interpretation of the dimples working mechanism and the spatially modulated spanwise forcing introduced by Viotti et al. \cite{viotti-quadrio-luchini-2009}. However, in that paper it is shown how temporal and spatial oscillations can be easily converted one into the other by using a suitable convective velocity scale at the wall. There are, of course, obvious differences between data collected by Quadrio \& Ricco for a turbulent channel flow at $Re_\tau=200$ or $Re_b=3173$ and the dimple experiments described in refs.\cite{vannesselrooij-etal-2016, vancampenhout-etal-2018} for a boundary layer at $Re_\delta=1226$ (the limited information provided in these references precludes computing the value of the friction Reynolds number).

Other important concepts to be aware of when trying to draw such a parallel is that, with the oscillating wall, a minimum spanwise velocity is required for the active technique to produce its effects: this threshold value $A_{th}^+$, that needs to be of the order of the natural fluctuations of spanwise velocity in the near-wall region, is quantified in ref.\cite{quadrio-ricco-2004} as $A_{th}^+=1$, i.e. similar or larger than the dimples-induced spanwise velocity as determined in \cite{vancampenhout-etal-2018}. Finally, and definitely most important, with a flat wall, even in presence of spanwise forcing, one should be only concerned with friction drag, whereas with dimples both viscous and pressure drag come into play.