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\title{Investigation of anomalous spectra in stratified wake flows}

\author{ \bf{Thomas S. Lund} and {\bf Donald P. Delisi}\\[0.5in]
        Northwest Research Associates \\
        4118 148th Ave NE\\
        Redmond, WA 98052-5164\\[1.5in]
        }

\maketitle

\section{Introduction}

This study was motivated by two separate sets of stably-stratified wake 
laboratory measurements that display an unexpected 
$k^{-4}$ power law in the velocity spectra.  The first data set it due
to Delisi\cite{delisi} who observed wakes behind a variety of bodies in
both stratified and unstratified fluids.  When using a sphere and for 
stratified cases only, Delisi found that the expected $k^{-5/3}$ inertial 
range transitioned to a $k^{-4}$ range instead of entering a non-power 
law viscous dissipation range (see Figure \ref{fig:delisi:data}, do be 
discussed below).  This behavior was not seen close to the sphere, but became 
apparent with increasing downstream distance.  The second data set is due
to Pao\cite{pao70}, who observed wakes from transverse-mounted circular
cylinders in both statified and unstratified fluids.  Like Delisi, Pao
observed a $k^{-4}$ range only for stably-stratified cases.

Before reviewing the data in more detail it is useful to discuss the 
important non-dimensional parameters the characterize stratified wake 
flows.  The relative stable stratification strength is quantified in 
terms of the Froude number, defined as
%
\begin{equation}
Fr = \frac{U_\infty}{ND}
\end{equation}
%
where $U_\infty$ is the free-stream speed, $D$ is the body diameter, and 
$N$ is the buoyancy frequency defined as
%
\begin{equation}
N^2 = \frac{g}{\rho_0}\left(\totderiv{\bar{\rho}}{z}\right)
\end{equation}
%
where $g$ is the gravitational acceleration, $\bar{\rho}(z)$ is the 
undisturbed background density distribution, $\rho_0$ is a reference value 
(taken to be the value of $\bar{\rho}$ on the wake centerline), and $z$
is the vertical coordinate.  The Froude number is an inverse measure
of statification strength, meaning that a value of $\infty$ is unstratified
and low values correspond to strong stable stratification.

For reference it is useful to estimate the Froude number for undersea 
vehicles.  The buoyancy frequency over the bulk of the ocean ranges 
from 0.2-15 cycles/hr, or 1.556e-5 to 4.167e-3 cycles/sec.  A submarine 
with diameter $D=10$ m, moving at 30 km/hr (8.33 m/s, 16.2 kt) is then 
characterized by a Froude number in the range 200-15000.  Since both the 
laboratory measurements and numerical simulations to be discussed in 
this report are for much lower Froude numbers (order 1), it is of 
interest to estimate the likely minimum value for a operational vehicle.  
Based on the definition, a low Froude number can be achieved for 
combinations of low velocity, large body diameter, and strong stable 
stratification.  The largest submarine (Russian Typhoon) has an effective 
diameter of about 18 m.  Although the speed could approach zero, a 
practical lower limit may be 1 km/hr (0.278 m/s, 0.54 kt).
In a thermocline, N may be as large as 0.01 cycles/sec.\cite{padman85}  
Even this combination of extreme values still produces a Froude number of 5.6.

Before dismissing the laboratory measurements and numerical simulations
as being irrelevant, it is important to note that a relevant similarity
parameter for a wake in a stratified fluid is the non-dimensional time,
$Nt$.  Assuming an advection velocity close to $U_\infty$, the non-dimensional
distance a fluid particle travels downstream of the body during the time $t$
is
%
\begin{equation}
\frac{x}{D} = \frac{U_\infty t}{D} = \left(\frac{U_\infty}{ND}\right)Nt = Fr*Nt
\label{eq:wake:age}
\end{equation}
%
Viewed in this way, the Froude number plays the role of a magnification
factor that dictates the non-dimensional streamwise distance required to 
achieve a particular non-dimensional wake age.  Thus a
low Froude number wake goes though its life cycle much closer to the
body as compared to a high Froude number wake.  This provides partial
justification for the low Froude numbers typically found both in
laboratory experiments and in numerical simulations, where it is impractical
to observe wakes at great distances from the body.

The other relevant non-dimensional wake parameter is the Reynolds number,
defined as
%
\begin{equation}
Re = \frac{U_\infty D}{\nu}
\end{equation}
%
where $\nu$ is the kinematic viscosity.  The Reynolds number for a 
full-scale submarine ($D=10$ m) operating at 30 km/hr (8.33 m/s, 16.2 kt) is
about $6.4\times 10^7$.  In contrast, laboratory experiments and numerical
simulations consider Reynolds numbers in the range $10^3$ - $10^5$.  Although
this represents a large mismatch, even a Reynolds number of order $10^3$ is
high enough to produce a turbulent wake with about one decade of inertial
range.  It is generally believed that the gross behavior of turbulence becomes
Reynolds number invariant as this parameter becomes very large.  While the
Reynolds numbers to be considered here probably do not fall within this limit,
the flows may still be reasonably representative of the high Reynolds number
limit.
%
\begin{figure}
\centerline{\includegraphics[width=6.0in]{figures/delisi_data.png}}
\caption{Streamwise velocity spectra at various downstream stations in 
the wake behind a sphere as measured by Delisi.  Fr=11.5, $Re=7.0\times 10^4$.}
\label{fig:delisi:data}
\end{figure}

Returning now to a discussion of the laboratory measurements, A typical plot 
of the presence of a $k^{-4}$ range is displayed in Figure 
\ref{fig:delisi:data}, which shows streamwise velocity spectra taken at 
several different downstream locations in the wake of a sphere in a 
stratified environment.  All the measurements are on the wake centerline 
and stratification was achieved by varying the salt concentration in 
Delisi's tow tank.  The Froude number is Fr=11.5 and the Reynolds number
is $Re=7.0\times 10^4$.  The data highlighted is for $x/D=60$, which
corresponds to a non-dimensional wake age of $Nt=5.2$.  Spectra at the
stations closer to the sphere show a more rounded transition to a viscous
range without a clear power law scaling.  Likewise spectra from an
unstratified run (not shown) show rounded high wavenumber spectra with
no power law behavior.
%
\begin{figure}
\centerline{\includegraphics[width=6.0in]{figures/Pao_fig10_crop_annotate.png}}
\caption{Streamwise velocity spectra for the wake behind a 
transverse-mounted circular cylinder as measured by Pao.  The
Froude number for the stratified case is Fr=3.8 and the Reynolds
number for both cases is Re=4200 and the spectra are taken at
$x/D=20$, $z/D=1$.}
\label{fig:pao:data}
\end{figure}

A second data set showing the presence of a $k^{-4}$ range is due to 
Pao\cite{pao70}, who observed wakes behind transverse-mounted circular 
cylinders in stratified fluids.  Pao only took measurements at one 
downstream location, but considered both unstratified and stratified 
fluids.  He mainly concentrated on a Froude number of Fr=3.8 at a
Reynolds number of $Re$=4200.  As shown in Figure \ref{fig:pao:data}, Pao's 
data show a similar $k^{-4}$ range at high wavenumbers, again only when 
the medium is stably stratified.

Steep power laws such at the $k^{-4}$ range observed in data sets discussed
above have not received much attention in the mainstream turbulence 
literature.  However some interesting references can be found, mainly
in the field of solar physics.\cite{bruno13,bratanov13}  These works consider
additional energy transfer mechanisms such as Alfv\'{e}n waves.  According
to theory and limited measurements, extra energy sink mechanisms can lead
to power law spectral ranges.  Curiously, slopes close to $-4$ are predicted
by one theory\cite{bruno13}.  Since the $k^{-4}$ range in wake turbulence
is only found in stratified flow, it seems  likely by analogy that this 
phenomenon is due to an exchange between kinetic and gravitational 
potential energy.  This hypothesis would also explain the gradual development
of the $k^{-4}$ range with increasing downstream distance since the relative
importance of buoyant effects increases as the wake kinetic energy decreases.

\section{Objectives}

The main objective of this project is to use numerical simulations 
to study the $k^{-4}$ spectral range observed in laboratory measurements 
of stratified wakes.  This broad objective can be broken down into the
following list of sub-objectives:
%
\begin{enumerate}
%
\item Survey existing numerical simulation wake data bases, looking for
the presence of power law ranges close to $k^{-4}$.
%
\item Perform new in-house numerical simulations as necessary in order to 
produce additional data.
%
\item Work with other research groups that are actively simulating 
stratified turbulence wakes to interrogate more data and to help define
future simulations.
%
\item Attempt to identify the physical mechanisms leading to and $k^{-4}$
ranges observed in the data.
%
\end{enumerate}

Each sub-objective was given careful consideration and these form the bases
of the remainder of this report.  To summarize, section \ref{sec:survey}
contains a survey of existing numerical data for time-developing stratified 
turbulent wakes computed by Gourlay {\it et al.}\cite{gourlay01}, 
de Stadler {\it et al.}\cite{destadler10},
Diamessis {\it et al.}\cite{diamessis11}, 
Redford {\it et al.}\cite{redford15}, and 
Watanabe {\it et al.}\cite{watanabe16}.  Spectra from each of these data sets
were reviewed for the presence of a $k^{-4}$ spectral range.  Section
\ref{sec:cylinder} discusses a suite of cylinder wake simulations,
performed by us, designed to replicate the conditions in Pao's experiments.
Section \ref{sec:sphere} describes detailed analysis of a set of very
recent sphere wake simulation data produced by Pal, Sarkar, and others at
UCSD.  Finally Section \ref{sec:conclusion} discusses our current understanding
of the mechanisms responsible for the $k^{-4}$ range. 

\section{Results}

\subsection{Survey of data from existing high-resolution time-evolving 
            stratified wake simulations}
\label{sec:survey}

The first step in our study was to survey existing turbulent wake
numerical simulation databases looking for the presence of a $k^{-4}$
spectral range.  All of the data surveyed were produced by the so-called 
time-evolving simulation method where the mean flow gradients in the 
streamwise direction are ignored.  By a large margin this is the most 
common computational and it has been proven to faithfully reproduce
statistics from equilibrium wakes that are well separated from the
generating body.

To derive the time-evolving approach
one first adopts a coordinate system fixed in a stationary fluid through 
which the wake-creating object moves.  When viewed in a fixed plane 
perpendicular to the wake axis, the wake will evolve only in time in 
this frame.  One then imagines translating this plane upstream and 
downstream short distances to form a rectangular computational box.  
While the wake mean characteristics at any given instant at the upstream 
end of the box are 
different from those at the downstream end, the differences may be small 
enough to be ignored.  This approximation is made, which is equivalent 
to the assumption that the mean flow streamlines are parallel to the 
wake axis.  Under this assumption the wake turbulence must be 
statistically homogeneous in the streamwise direction, making it possible
to use periodic boundary conditions in this direction.  

While the time-evolving method is only appropriate for equilibrium wakes
far downstream from the body, it has tremendous computational advantage.
Foremost among these is that it is not necessary to include the 
wake-generating body in the simulation.  Since the mean component of the
solution only evolves in time, it is possible to cast the simulation as 
an initial value problem, starting from an idealized mean state such as 
a Gaussian profile.  The turbulent component can be initialized via 
random fluctuations which will quickly organize into realistic 
turbulence once the simulation is started.  Periodic boundary conditions 
can be used in all three directions, making it possible to use highly 
accurate and highly efficient spectral methods.

While dozens of time-evolving wake simulations have been reported in the
literature, only a handful of these consider a stratified environment.
We were able to access raw data, or at least plotted spectra, from five
of the existing stratified simulations.  The data sets considered here are
due to Gourlay {\it et al.}\cite{gourlay01},
de Stadler {\it et al.}\cite{destadler10},
Diamessis {\it et al.}\cite{diamessis11},
Redford {\it et al.}\cite{redford15}, and
Watanabe {\it et al.}\cite{watanabe16}.
%
\begin{figure}
\centerline{\includegraphics[width=6.0in]{figures/kops_spec.png}}
\caption{Typical spectra from the time-evolving stratified wake simulation of
Watanabe {\it et al.}}
\label{fig:watanabe}
\end{figure}

Wavenumber spectra are generated directly from the time-evolving simulation 
data by simply taking a Fourier transform in the streamwise direction.
Since both the effective Reynolds and Froude numbers decay with time
in these simulations, it is trivial to compute spectra for different 
combinations of the parameter values during the course of a single 
simulation.  Collectively the data surveyed spans a Reynolds
number range of $10^2-10^4$ and a Froude number of $0.1-12$.  When looking
over this wide parameter space, none of the corresponding spectra displayed
a convincing $k^{-4}$ range.  Instead, a smooth transition between
the inertial and dissipation ranges was invariably observed.  A typical 
result is shown in Figure \ref{fig:watanabe}, which shows data from the 
simulation of Watanabe {\it et al.}\cite{watanabe16}.  This spectrum shows 
a short inertial range followed by a non-power law dissipation range.  

One interesting aspect of the spectrum shown in Figure \ref{fig:watanabe}
is the presence of a $k^{-7/5}$ range in place of the expected $k^{-5/3}$
inertial range.  This is same scaling has been observed in Kelven Helmholtz
simulations performed by our group and we attribute it to the effects of
stable stratification.  

\subsection{Direct numerical simulation of body-inclusive
            stratified cylinder wakes}
\label{sec:cylinder}

A series of circular cylinder wake simulations were undertaken in order
to investigate the influence of stable stratification on the turbulence
energy spectrum.  The simulations target measurements made by Pao in a
tow tank capable of various levels of background stable stratification
via a salinity gradient in the vertical direction.  Pao reported results
for three Froude numbers, Fr=$\infty$, 3.8, and 0.64.  The Reynolds numbers
vary slightly from case to case but are all close to Re=4200.  Pao computed 
frequency spectra from a streamwise velocity time series taken at a position
$x/D=20$ downstream of the cylinder and displaced vertically $z/D=1$ above 
the wake centerline.

\begin{figure}
\centerline{\includegraphics[height=3.0in]{figures/grid_zoom0_skip5.png}}
\centerline{\includegraphics[height=3.0in]{figures/grid_zoom1_skip5.png}}
\centerline{\includegraphics[height=3.0in]{figures/grid_zoom2_skip0.png}}
\caption{Computational mesh for the circular cylinder simulations.
Top - overall domain showing only one out of every five mesh lines, middle -
intermediate zoom showing only one out of every five mesh lines, bottom -
high zoom showing all mesh lines near the cylinder.}
\label{fig:cylmesh}
\end{figure}

The mesh used for the simulations is shown in Figure \ref{fig:cylmesh}.  
The top panel shows the entire domain, but with only one out of every five mesh 
lines drawn.  If the resolution were not reduced in drawing the image, 
it would appear as a solid object (i.e.  the image is not able to 
resolve individual mesh lines).  The middle panel shows a zoom of the mesh 
around the cylinder, again with a skip factor of five.  The bottom panel 
shows an increased zoom level with a skip factor of one (all mesh lines 
are shown).

The mesh itself is a C-mesh topology containing 3000 points in the
streamwise/azimuthal direction, 640 points in the normal direction,
and 320 points in the spanwise direction.  The mesh points are highly
clustered near the cylinder surface in order to resolve the 
thin laminar boundary
layers.  There is also clustering along the wake centerline and along
vertical lines near emanating from the aft end of the cylinder.
These latter two features are not really desired, but are natural
consequences of the orthogonal mesh generation procedure in connection
with the mesh point distribution near the cylinder surface.  The
clustering artifacts do not really cause a problem, except for the fact that
the mesh points involved could be better used elsewhere in the domain.
The domain extends six cylinder diameters in front of, as well as above and
below the cylinder.  It extends 25.5 cylinder diameters downstream and
three diameters in the spanwise direction.

Each simulation is started with low amplitude random velocity fluctuations
but no mean motion. The mean velocity is then ramped up to the value 
$U_\infty$ over a period of $D/U_\infty$ time units.  This mimics the 
laboratory procedure where the cylinder is initially at rest and then 
accelerated up 
to speed $U_\infty$ over a short time interval.  The wake thus evolves in 
both space and time during the initial portion of the simulation.  The wake 
completely fills the streamwise extent of the domain at a time of about 
$U_\infty t/D=30$.  Data collection for time series begins no earlier than a 
time of $U_\infty t/D=38$ and continues for 20 non-dimensional time units.  
Time series of all solution variables are taken on planes located at 
$x/D=5$, 10, 15, 20, and 25, at a sampling interval of 
$U_\infty\Delta t/D=0.0125$.  By averaging over the spanwise direction and
over time, these data are sufficient to form profiles of mean velocity, 
velocity fluctuations, Reynolds stress, and buoyancy flux.  Velocity s
spectra were computed at each point in the data plane and then averaged
in the spanwise direction.  In order to reduce noise in the spectra further,
averages were also taken for equal positive and negative displacements 
from the wake centerline (i.e. the spectra for $z/d=-0.5$ are averaged
with those at $z/D=0.5$).  As shown below the averaging is sufficient
to produce fairly smooth spectra.

\subsubsection{Fr=$\infty$}

The first case to be discussed is unstratified (Fr=$\infty$).  This case is
included as a baseline to aid in identifying the effects of stable 
stratification present in the other two cases.

\begin{figure}
\centerline{\includegraphics[width=6.5in]{figures/{vort_mag.0583}.png}}
\caption{Instantaneous vorticity magnitude from the Fr=$\infty$ circular 
cylinder simulation.}
\label{fig:Foovortmag}
\end{figure}

An image of the vorticity magnitude field near the end of the simulation is 
show in Figure \ref{fig:Foovortmag}.  This figure indicates that, while the
wake is highly turbulent, it is also rather intermittent due to the presence
of coherent vortex shedding structures.  These structures begin to diffuse 
and merge by the downstream end of the simulation but the wake is still 
rather inhomogeneous, especially for the outer sections of the wake 
($z/D > 2$).

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/Fr_inf_mean_u.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr_inf_flct_u.png}}
\centerline{\includegraphics[width=3.0in]{figures/Fr_inf_flct_w.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr_inf_flct_uw.png}}
\caption{Velocity statistics for the Fr=$\infty$ circular cylinder case.
From top to bottom, left to right: mean streamwise velocity, streamwise
velocity fluctuation, vertical velocity fluctuation, and Reynolds stress.}
\label{fig:Foovel}
\end{figure}

Velocity statistics are shown in Figure \ref{fig:Foovel}.  The mean velocity
shows the expected defect within the wake and accelerated flow outside of it.
Convergence of the statistics is only moderate with visible asymmetries
with respect to the wake centerline and residual noise.  Although it would
have been nice to have smoother profiles, the simulations are expensive and
the amount of data recorded is sufficient to compute fairly smooth spectra.

The velocity fluctuations and Reynolds stress display the expected decay 
and broadening with increasing downstream distance.  The decay is modest,
however, with the peak velocity fluctuations and Reynolds stresses reducing 
by roughly factors of three to four between the $x/D=5$ and $x/D=25$ stations.
Thus, as confirmed by the vorticity magnitude image in Figure 
\ref{fig:Foovortmag}, the wake is actively turbulent at all five measuring
stations.

\begin{figure}
\centerline{\includegraphics[width=6.0in]{figures/Froospec_w_pao.png}}
\caption{Comparison of streamwise velocity spectra from the unstratified 
circular cylinder case with the data of Pao.  $x/D=20$, $z/D=1.0$.}
\label{fig:compareFoospec}
\end{figure}

Pao took measurements at $x/D=20$, $z/D=1.0$ and computed streamwise velocity
spectra from the resulting data.  Figure \ref{fig:compareFoospec} shows
a comparison of spectra computed from the simulation with the data from Pao.
The agreement is seen to be quite good over most of the wavenumber range.
The main disagreement is at the high wavenumber end where Pao's data appear to 
show an earlier roll off to a dissipation range.  Both the simulation and
experiment show over one decade of inertial range scaling.


\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/{spec1_1.0}.png}\hfill
            \includegraphics[width=3.0in]{figures/{spec2_1.0}.png}}
\centerline{\includegraphics[width=3.0in]{figures/{spec3_1.0}.png}\hfill
            \includegraphics[width=3.0in]{figures/{spec4_1.0}.png}}
            \includegraphics[width=3.0in]{figures/{spec5_1.0}.png}\hfill \\
\caption{Streamwise velocity spectra for the Fr=$\infty$ circular cylinder 
case at several downstream stations.  From top to bottom, left to right: 
$x/D=5$, 10, 15, 20, and 25.}
\label{fig:Foospecx}
\end{figure}

Streamwise velocity spectra at the five $x/D$ measuring stations are 
displayed in Figure \ref{fig:Foospecx}.  All of these spectra are taken
at $z/D=1.0$, which is near the position of maximum Reynolds stress for
most of the stations.  Spectra at all five $x/D$ stations display at
least one decade of inertial range ($k^{-5/3}$) scaling.  The low-wavenumber
portion of this range is interrupted by a Strouhal peak at a non-dimensional
frequency of about $fD/U=0.25$.  The amplitude of the peak decreases
with increasing downstream distance but is still visible at the $x/D=25$
station.  At the $x/D=5$ and 10 stations, there is a slight tendency for 
the spectrum to pause at a $k^{-4}$ scaling before entering a non-power 
law dissipative range at the high-wavenumber end.  This effect is progressively
diminished and ultimately eliminated at the $x/D=15$, 20, and 25 stations.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/{spec1_0.0}.png}\hfill
            \includegraphics[width=3.0in]{figures/{spec1_0.4}.png}}
\centerline{\includegraphics[width=3.0in]{figures/{spec1_0.8}.png}\hfill
            \includegraphics[width=3.0in]{figures/{spec1_1.2}.png}}
\centerline{\includegraphics[width=3.0in]{figures/{spec1_1.7}.png}\hfill
            \includegraphics[width=3.0in]{figures/{spec1_2.2}.png}}
\caption{Streamwise velocity spectra for the Fr=$\infty$ circular cylinder
case at several vertical stations for $x/D=5$.  From top to bottom,
left to right: $z/D=0.0$, 0.4, 0.8, 1.2, 1.7, 2.2}
\label{fig:Foospec5z}
\end{figure}

The variation in the spectra with vertical position is shown in Figure
\ref{fig:Foospec5z} where data from $x/D=5$ are plotted for several
$z/D$ stations.  The first five of these spectra are for positions
within the turbulent portion of the wake and are all relatively 
similar to the spectra shown in Figure \ref{fig:Foospecx}.  The final
spectrum (at $z/D=2.2$) is outside of the turbulent zone, as measured
by the Reynolds stress shown in Figure \ref{fig:Foovel}.  This spectrum
is remarkably different from the others.  It displays more than one
decade of $k^{-4}$ range, bounded by the Strouhal peak at the low-wavenumber
end and a dissipation range at the high-wavenumber end.  There is no
visible inertial range, which is consistent with the sampling point
lying in a largely non-turbulent zone.

The $k^{-4}$ range for the spectrum taken outside the wake is likely do to 
the presence of ``irrotational fluctuations'' first postulated by O.M Phillips 
in 1955\cite{phillips55}.  According to Phillips, motions within a turbulent 
layer act as sources that drive potential flows in the otherwise quiescent 
region beyond the layer edge.  Later, Bradshaw\cite{bradshaw67} measured 
fluctuations outside of a turbulent boundary layer and showed that they 
produce spectra with steep power laws, close to $k^{-4}$.  We expect that
vigorous turbulence and strong intermittency caused by the coherent vortical
structures act as strong sources for the irrotational fluctuations.  It
stands to reason that we pick up this effect when sampling just outside the
statistical wake edge.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/{spec2_3.0}.png}\hfill
            \includegraphics[width=3.0in]{figures/{spec3_3.7}.png}}
\caption{Streamwise velocity spectra for the Fr=$\infty$ circular cylinder
case taken just beyond the wake edge.  Left figure: $x/D=10$, right $x/D=15$.}
\label{fig:Foospec-4}
\end{figure}

Although not quite as compelling, spectra outside of the wake continue to
show a $k^{-4}$ range at the next few $x/D$ stations.  Figure 
\ref{fig:Foospec-4} shows spectra from the $x/D=10$ and 15 stations, 
taken at vertical positions just beyond the wake edge (as inferred from
the Reynolds stress).  Time series were only stored out to $z/D=4$, which
is not quite outside the wake edge for the $x/D=20$ and 25 stations.
Thus, while we do not see $k^{-4}$ ranges at these stations, presumably they
exist at slightly larger vertical displacements.

\subsubsection{Fr=3.8}

The second circular cylinder case is for Fr=3.8.  This case is of primary
interest since Pao quotes this Froude number for the spectra that displays
a $k^{-4}$ range.

\begin{figure}
\centerline{\includegraphics[width=6.5in]{figures/{vort_mag.0130}.png}}
\caption{Instantaneous vorticity magnitude from the Fr=3.8 circular 
cylinder simulation.}
\label{fig:F38vortmag}
\end{figure}

An instantaneous image of the vorticity magnitude is shown in Figure
\ref{fig:F38vortmag}.  When compared with Figure \ref{fig:Foovortmag}
for the unstratified case, we see that stratification limits the
wake growth and generally reduces the intermittency.  There is also
a visible tendency for the turbulence to weaken more rapidly with
increasing downstream distance.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/Fr3_8_mean_u.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr3_8_flct_u.png}}
\centerline{\includegraphics[width=3.0in]{figures/Fr3_8_flct_w.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr3_8_flct_r.png}}
\centerline{\includegraphics[width=3.0in]{figures/Fr3_8_flct_uw.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr3_8_flct_wr.png}}
\caption{Velocity and density statistics for the Fr=3.8 circular cylinder case.
From top to bottom, left to right: mean streamwise velocity, streamwise
velocity fluctuation, vertical velocity fluctuation density fluctuation, 
Reynolds stress, and buoyancy flux.}
\label{fig:F38vel}
\end{figure}

Velocity and density statistics are shown in Figure \ref{fig:F38vel}.  All 
of the profiles are smoother as compared with the Fr=$\infty$ case 
(Figure \ref{fig:Foovel}) as a direct result of the reduced intermittency.  
The mean velocity defect is initially larger in the Fr=3.8 case but then
steadily decreases to a smaller value than in the Fr=$\infty$ simulation
by the final measurement station. The wake spreading is also reduced
noticeably.  Both the streamwise and vertical velocity fluctuations, as
well as the Reynolds stress are reduced in the Fr=3.8 case.  

\begin{figure}
\centerline{\includegraphics[width=6.0in]{figures/Fr3_8spec_w_pao.png}}
\caption{Comparison of streamwise velocity spectra from the Fr=3.8 
circular cylinder case with the data of Pao.  $x/D=20$, $z/D=1.0$.
The ``shifted'' values have been multiplied by a factor of 10.}
\label{fig:compareF38spec}
\end{figure}

As in the unstratified case, Pao took measurements at $x/D=20$, $z/D=1.0$ 
for his Fr=3.8 case and computed streamwise velocity spectra from the 
resulting data.  Figure \ref{fig:compareF38spec} shows a comparison of 
this data with the results of our numerical simulation.  The open symbols
correspond to the data as read from Pao's plot.  These data clearly
fall way below our our computed results.  When compared with the unstratified
case, our Fr=3.8 case produce spectra that are about a factor of 3 lower.
Pao's data, on the other hand, show about a factor or 30 when his unstratified
and Fr=3.8 cases are compared.  While this drop seems unrealistically large,
Pao comments on this fact saying ``However, the auto-spectral density of the
stratified case is nearly two orders of magnitude lower than that of the
non-stratified case.''  While we are hesitant to contradict Pao, it is
interesting to note that a factor of 10 shift in Pao's values bring the
simulations and measurements in line.  While the shift generally collapses
the two data sets, our computed results do not show an extended $k^{-4}$
range at the high-wavenumber end.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures1/{spec1_1.0}.png}\hfill
            \includegraphics[width=3.0in]{figures1/{spec2_1.0}.png}}
\centerline{\includegraphics[width=3.0in]{figures1/{spec3_1.0}.png}\hfill
            \includegraphics[width=3.0in]{figures1/{spec4_1.0}.png}}
            \includegraphics[width=3.0in]{figures1/{spec5_1.0}.png}\hfill \\
\caption{Streamwise velocity spectra for the Fr=3.8 circular cylinder 
case at several downstream stations.  From top to bottom, left to right: 
$x/D=5$, 10, 15, 20, and 25.}
\label{fig:F38specx}
\end{figure}

Spectra at $z/D=1.0$ for the five $x/D$ locations are shown in Figure
\ref{fig:F38specx}.  These spectra are generally quite similar to those
for the unstratified case shown in Figure \ref{fig:Foospecx}.  The main
differences are that the energy density is uniformly lower in the Fr=3.8
case and as is the magnitude of the Strouhal peak.  As with the unstratified
case, there is only a weak tendency for the spectra to develop a $k^{-4}$
range at the first few $x/D$ stations.

Scans of spectra in the vertical direction show that the shapes shown in
Figure \ref{fig:F38specx} are representative of spectra at all positions
across the wake.  As with the unstratified case, the behavior shifts radically
beyond the wake edge, with an erosion of the inertial range and the
emergence of a $k^{-4}$ range.  Evidence of this change is shown in Figure
\ref{fig:F38spec-4}.  The $k^{-4}$ scaling is not as clear as in the 
unstratified case and a bump at non-dimensional frequency of 2 appears
at the later stations.  This bump is likely due to internal waves that are
not present in the unstratified case.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures1/{spec1_2.3}.png}\hfill
            \includegraphics[width=3.0in]{figures1/{spec2_2.7}.png}}
\centerline{\includegraphics[width=3.0in]{figures1/{spec3_2.6}.png}\hfill
            \includegraphics[width=3.0in]{figures1/{spec4_3.0}.png}}
\caption{Streamwise velocity spectra for the Fr=3.8 circular cylinder
case taken just beyond the wake edge.  From top to bottom, left to right: 
$x/D=5$, 10, 15, 20.}
\label{fig:F38spec-4}
\end{figure}

\subsubsection{Fr=0.64}

The third Froude number considered by Pao is Fr=0.64.  This Froude number
is representative of extreme stable stratification that probably is never
experienced by undersea vehicles.  While we would not normally be interested
in this case, we decided to simulate it on the off chance that Pao incorrectly
mislabeled the plot showing the $k^{-4}$ range and actually plotted data
from his Fr=0.64 case instead of Fr=3.8.  It seems more reasonable
that the spectral energy density could for Fr=0.64 could be a factor of 30 
lower as compared with the unstratified case.

\begin{figure}
\centerline{\includegraphics[width=6.5in]{figures/{xz1_vort_mag.0600}.png}}
\caption{Instantaneous vorticity magnitude from the Fr=0.64 circular 
cylinder simulation.}
\label{fig:F64vortmag}
\end{figure}

An instantaneous image of the vorticity magnitude is shown in Figure
\ref{fig:F64vortmag}.  When compared with Figures \ref{fig:Foovortmag}
and \ref{fig:Foovortmag} for the unstratified and Fr=3.8 cases, we see 
that the current extreme level of stratification almost eliminates 
turbulence by about eight diameters downstream.  The wake remains rather
thin as only sheet-like structures survive near the downstream end of
the simulation.  Although not terribly clear in the vorticity magnitude
image, the flow also contains strong internal waves (these are seen better
in velocity contour plots).  The outer boundary conditions do not handle
these strong waves perfectly, which results in the visible artifacts. 

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/Fr0_64_mean_u.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr0_64_flct_u.png}}
\centerline{\includegraphics[width=3.0in]{figures/Fr0_64_flct_w.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr0_64_flct_r.png}}
\centerline{\includegraphics[width=3.0in]{figures/Fr0_64_flct_uw.png}\hfill
            \includegraphics[width=3.0in]{figures/Fr0_64_flct_wr.png}}
\caption{Velocity and density statistics for the Fr=0.64 circular cylinder case.
From top to bottom, left to right: mean streamwise velocity, streamwise
velocity fluctuation, vertical velocity fluctuation density fluctuation, 
Reynolds stress, and buoyancy flux.}
\label{fig:F64vel}
\end{figure}

As shown in Figure \ref{fig:F64vel}, the presence of strong internal waves 
results in highly anomalous velocity and density statistics.  The normal 
turbulent wake signatures are simply lost in the fluctuations due to the
superimposed wave field.  It is difficult to infer much information about
the wake from these plots and thus we will not discuss them further.  They
are mainly included for completeness.

\begin{figure}
\centerline{\includegraphics[width=6.0in]{figures/Fr0_64spec_w_pao.png}}
\caption{Comparison of streamwise velocity spectra from the Fr=0.64 
circular cylinder case with the data of Pao.  $x/D=20$, $z/D=1.0$.
The ``rescaled'' values have been adjusted such that each division on
Pao's plot is one order of magnitude instead of 0.5 orders.}
\label{fig:compareF64spec}
\end{figure}

As in the the other two cases, Pao took measurements at $x/D=20$, $z/D=1.0$ 
for the Fr=0.64 case and computed streamwise velocity spectra from the 
resulting data.  Figure \ref{fig:compareF64spec} shows a comparison of 
this data with the results of our numerical simulation.  The open symbols
correspond to the data as read from Pao's plot.  Once again there appears
to be something fundamentally wrong with the data as it shows a
rather limited drop from the low to high-wavenumber end.  Pao's plot also
contains a $k^{-5/3}$ reference line, but the slope of this line is not 
consistent with the tic marks on his plot.  Consistency can be restored, and
the data in Figure \ref{fig:compareF64spec} can be made to show the expected 
energy drop if each division on Pao's plot is changed from 0.5 to 1.0 decades,
starting at the position of the first data point on the low-wavenumber
end.  Doing this results in the points labeled ``Measurements, rescaled''.
The rescaled data exhibit a $k^{-5/3}$ range and are of the expected energy
level.

While rescaling Pao's data restores some level of consistency, it looks 
noting like the results from our simulation.  Consistent with the vorticity
magnitude image and velocity statistics, the flow is almost non-turbulent
at the $x/D=20$ station.  Accordingly, our data do not display a $k^{-5/3}$ 
range and are very highly damped at the high wavenumber end.  The most visible
features are the Strouhal peak and its first harmonic.  Pao's data, on the
other hand, shows energy levels nearly identical to his $Fr=3.8$ case,
complete with fairly clear $k^{-5/3}$ and $k^{-4}$ ranges.  It is hard to 
imagine how the spectra could be so little changed by a six fold increase
in stable stratification.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures2/{spec1_0.5}.png}\hfill
            \includegraphics[width=3.0in]{figures2/{spec2_0.5}.png}}
\centerline{\includegraphics[width=3.0in]{figures2/{spec3_0.5}.png}\hfill
            \includegraphics[width=3.0in]{figures2/{spec4_0.5}.png}}
            \includegraphics[width=3.0in]{figures2/{spec5_0.5}.png}\hfill \\
\caption{Streamwise velocity spectra for the Fr=0.64 circular cylinder 
case at several downstream stations.  From top to bottom, left to right: 
$x/D=5$, 10, 15, 20, and 25.}
\label{fig:F64specx}
\end{figure}

Notwithstanding the problems with Pao's data, we continue with the analysis
of the spectra computed from our simulation.  
Spectra at $z/D=1$ for the five $x/D$ locations are shown in Figure
\ref{fig:F64specx}.  All stations display a pronounced Strouhal peak and
first harmonic.  While there is some tendency for the peak to diminish with
downstream distance, it is still prevalent at the $x/D=25$ station.  the 
first two stations show a limited $k^{-5/3}$ range, which is consistent with
the turbulence visible in the vorticity magnitude plot.  As with the prior
two cases, these stations also show a weak tendency to display a short 
$k^{-4}$ range.  Spectra at $x/D=15$, 20 and 25 look progressively less
turbulent with a general erosion of the inertial range and very high
attenuation at high wavenumbers.

Although the wake is quite thin at this Froude number, limited scans of 
spectra in the vertical direction show that the shapes shown in
Figure \ref{fig:F64specx} are representative of spectra at all 
positions across the turbulent wake.  As with the other two cases, 
the behavior shifts radically beyond the wake edge, where a 
$k^{-4}$ range develops.  Figure \ref{fig:F38spec-4} shows this effect
where spectra are plotted in the non-turbulent region adjacent to the
wake.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures2/{spec1_0.8}.png}\hfill
            \includegraphics[width=3.0in]{figures2/{spec2_0.9}.png}}
\centerline{\includegraphics[width=3.0in]{figures2/{spec3_1.0}.png}\hfill
            \includegraphics[width=3.0in]{figures2/{spec4_1.2}.png}}
\caption{Streamwise velocity spectra for the Fr=0.64 circular cylinder
case taken just beyond the wake edge.  From top to bottom, left to right: 
$x/D=5$, 10, 15, 20.}
\label{fig:F64spec-4}
\end{figure}


\subsection{Analysis of high-resolution, body-inclusive stratified
            sphere wake simulations}
\label{sec:sphere}

In this section we discuss the analysis of data from a series of 
high-resolution, direct numerical simulations of sphere-inclusive
wakes in stratified flows.  The simulations were performed by Anikesh
Pal and others, under the direction of Sutanu Sarkar at UCSD\cite{pal17}.
We considered their Fr=1, 3, and $\infty$ cases, all of which
are at Re=3700.  For each case, we were given time series of velocity data
sampled along horizontal and vertical cuts at several downstream stations
within the wake.  More specifically, for each Froude number, data
is sampled at the four downstream locations $x/D$=3.43, 21.43, 40.18, 66.73.
At each of these $x/D$ locations, time series are taken at 10 uniformly
spaced points within the wake along a horizontal line ($y$ axis) connecting the 
wake centerline to the wake edge (as computed from two wake velocity
half-widths).  An additional four uniformly spaced points are included along the
horizontal line beyond the wake edge.  An analogous sampling strategy is
used in the negative $y$ direction, as well as the positive and negative
vertical ($z$) directions. A sampling point is also included at the wake
centerline, giving a total of 4(10+4)+1=57 sampling points at each $x/D$
location.  Although there is some variation from case to case, the time
series contain between 1150-1650 points, sampled at a rate of 
$U\Delta t/D \simeq 0.05$.  The data is sufficient to compute vertical and
horizontal profiles of mean velocity and density, velocity and density
fluctuations, Reynolds stresses and buoyancy flux.  It is also sufficient
to compute frequency spectra.

Although we are most interested in the spectra, profiles of velocity and
density statistics aid in placing the spectra in context in terms of the
relative degree of mean shear and turbulent activity.  Thus in what follows,
for each Froude number, we show several profiles, followed by spectra at
select locations.  A complete collection of all data can be viewed at
https://www.cora.nwra.com/\~lund/sphere/

\subsubsection{Fr=$\infty$}

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/Foo_u_avg_y.png}\hfill
            \includegraphics[width=3.0in]{figures/Foo_u_avg_z.png}}
\caption{Lateral (left) and vertical (right) profiles of mean streamwise
         velocity defect at several downstream locations for the Fr=$\infty$
         case.}
\label{fig:Froomean}
\end{figure}

Mean streamwise velocity defect profiles at several downstream locations
are shown in Figure \ref{fig:Froomean}.  Since this case is unstratified,
the lateral and vertical profiles are nearly identical.  By virtue of 
the maximum defect being less than one, we see that the $x/D=3.4$ measurement
station is downstream of the mean re-circulation zone. The figures also
show a dramatic drop in the maximum mean defect with downstream distance,
reducing by factors of $\simeq$ 12, 21, and 34 from the value at $x/D=3.4$
at the subsequent stations $x/D=21.4$, 40.2, and 66.7.  A similar trend
will be seen in the velocity fluctuations and Reynolds stresses, all of
which indicate that the wake decays quite rapidly at this low Reynolds
number.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/Foo_u_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/Foo_u_rms_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/Foo_w_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/Foo_w_rms_z.png}}
\caption{Lateral (left) and vertical (right) profiles of streamwise (top row)
         and vertical (bottom row) velocity fluctuations at several 
         downstream locations for the Fr=$\infty$ case.}
\label{fig:Frooflct}
\end{figure}

Streamwise and vertical velocity fluctuation profiles are shown in Figure
\ref{fig:Frooflct}.  The streamwise velocity fluctuation displays the expected
double peak structure where the vertical velocity fluctuation forms a single
peak.  Similar to the mean velocity defect, the maximum fluctuation values
decrease rapidly with downstream distance.  Spreading of the turbulent zone
is seen more readily in these profiles as opposed to the mean velocity
defect.  Based on velocity fluctuations, the disturbed zone appears to 
extend from $\pm 3D$ at the $x/D=66.7$ station.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/Foo_uv_y.png}\hfill
            \includegraphics[width=3.0in]{figures/Foo_uw_z.png}}
\caption{Lateral (left) and vertical (right) profiles of Reynolds stresses
         at several downstream locations for the Fr=$\infty$ case.}
\label{fig:Froostrs}
\end{figure}

Turbulent Reynolds stress profiles are shown in Figure \ref{fig:Froostrs}.
The Reynolds stresses display dramatic decay with downstream distance,
dropping by a factor of almost 200 between $x/D=3.4$ and $x/D=66.7$.

Collectively the profiles depict a canonical turbulent wake that is decaying
rapidly under the action of viscosity.  The decay can be quantified in terms
of the instantaneous wake Reynolds number, formed from the maximum 
streamwise velocity deficit and the estimated wake width.  The instantaneous
wake width is given in Pal {\it et al.}\cite{pal17}, whereas the maximum
mean velocity defect is obtained from Figure \ref{fig:Froomean}.  
Such a calculation shows that the instantaneous wake 
Reynolds number decays from 800 at $x/D=3.4$ to 158, 118, and 93 at the
subsequent stations $x/D=21.4$, 40.2, and 66.7.  The Reynolds numbers at
the latter stations are sufficiently small that we can expect only weak
turbulence with limited dynamic range.

Spectra were formed from each of the 57 time series at each $x/D$ location.
In order to reduce noise, each time series was broken into 15 segments, each
overlapping its neighbors by 50\%.  A cosine window was then applied to
each segment, the spectra formed, and the results of the 15 segments averaged
together.  In order to reduce noise further, the spectra were also averaged
across the wake centerline (i.e. spectra at y/D=-0.2 and y/D=0.2
were averaged together).  While some noise is still present, the averaging
procedure is sufficient to detect power laws.

\begin{figure}
\centerline{\includegraphics[width=2.5in]{figures/{Foox03y0.33_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{Foox03z0.33_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{Foox21y0.78_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{Foox21z0.78_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{Foox40y1.01_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{Foox40z1.01_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{Foox67y1.30_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{Foox67z1.30_spec}.png}}
\caption{Spectra at several streamwise positions the Fr=$\infty$ sphere
wake case.  The left 
figure is for a lateral position of $y/\delta_2\simeq 0.40$, whereas the 
right figure is for a vertical position of $z/\delta_3\simeq =0.40$,
where $\delta_2$ and $\delta_3$ are the estimated lateral and vertical
wake half-widths.}
\label{fig:Frooallspec}
\end{figure}

Spectra at several downstream stations are shown in Figure
\ref{fig:Frooallspec}.  The $(y,z)$ locations displayed correspond closely to 
the maximum in the Reynolds stress or, equivalently, the maximum in the 
streamwise velocity fluctuation.  Since the wake is axisymmetric in the mean, 
the spectra displaced in $y$ and $z$  should be statistically equivalent.  
This is indeed seen to be the case with only minor differences when comparing
the left and right graphs in each row.  Starting at $x/D$ = 3.4, we see that
the spectra display about one decade of inertial range $k^{-5/3}$,
followed by about a half decade of $k^{-4}$ scaling.  Although we did not 
expect to see a $k^{-4}$ range for this case since it is unstratified, the
data seem quite compelling.

Spectra at the further downstream locations are not nearly as easy to
interpret as those at $x/D=3.4$.  Although there is still a tendency to display
power laws, these extend over more limited wavenumber ranges.  
Beyond $x/D=3.4$, the transition point between the $k^{-5/3}$ and the $k^{-4}$ 
ranges moves to a much lower wavenumber, thereby reducing the extent of the
inertial range.  While the wavenumber shift provides an opportunity for a more
extended $k^{-4}$ range, a viscous dissipation range erodes the
$k^{-4}$ scaling from the high wavenumber end.  The net result is about
a half decade of $k^{-4}$ scaling for the $x/D=21.4$ station, and even less
for the subsequent stations.

The dissipation range present for the $x/D=21.4$ station and beyond shows the
expected behavior for decaying turbulence.  The onset wavenumber generally
moves to a lower value with increasing downstream distance and the energy
contained in the high frequency modes steadily decreases.  These features
are consistent with the low Reynolds numbers noted above for the latter
several stations.

Returning now to the $x/D=3.4$ station, we found that the presence of the 
$k^{-4}$ range is fairly robust within the turbulent portion of the wake.
Evidence of this is shown in Figure \ref{fig:Froo34zspec}, which shows
spectra at several vertical positions.
Although the turbulent energy drops by more than an order of magnitude
from the wake centerline to the $y/D=0.75$ position, the relative shape
of the spectra remains much the same.  Both $k^{-5/3}$ and $k^{-4}$
continue to exist and the transition point remains reasonably fixed.
Nearly identical behavior is found for a horizontal scan through the wake.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/{Foox03z0.00_spec}.png}\hfill
            \includegraphics[width=3.0in]{figures/{Foox03z0.25_spec}.png}}
\centerline{\includegraphics[width=3.0in]{figures/{Foox03z0.50_spec}.png}\hfill
            \includegraphics[width=3.0in]{figures/{Foox03z0.75_spec}.png}}
\caption{Spectra at $x/D=3.4$, $y/D=0$ for the Fr=$\infty$ sphere wake case.
From left 
to right, top to bottom the spectra are taken at at $z/D=$ 0, 0.25, 0.50, 
and 0.75.}
\label{fig:Froo34zspec}
\end{figure}

Beyond the wake edge the spectra become much more variable and no longer
display a convincing inertial range.  As with the circular cylinder wake
results, certain stations just beyond the wake edge display a well-developed
$k^{-4}$ range.  This behavior is seen in Figure \ref{fig:Froo34latespec} 
which shows spectra at two vertical stations outside of the turbulent zone.  
As in the case with the cylinder wake, we believe that the $k^{-4}$ range
seen in these plots is due to irrotational fluctuations.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/{Foox03z1.03_spec}.png}\hfill
            \includegraphics[width=3.0in]{figures/{Foox03z1.64_spec}.png}}
\caption{Spectra at $x/D=3.4$, $y/D=0$ for the Fr=$\infty$ sphere wake case 
outside of the
turbulent zone.  From left to right the spectra are taken at $z/D=$ 1.03 and 
1.64.}
\label{fig:Froo34latespec}
\end{figure}

\subsubsection{Fr=3}

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/F03_u_avg_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F03_u_avg_z.png}}
\caption{Lateral (left) and vertical (right) profiles of mean streamwise
         velocity defect at several downstream locations for the Fr=3
         sphere wake case.}
\label{fig:Fr3mean}
\end{figure}

The first stratified case to be discussed is for Fr=3.  This Froude number 
represents a significant level of stable stratification and thus we can
expect pronounced differences with the unstratified case discussed in
the previous section.

Mean velocity profiles are displayed in Figure \ref{fig:Fr3mean}.  When 
compared with Figure \ref{fig:Froomean} for the unstratified case, we
see that the mean velocity defect decays more slowly and the wake spreading
rate is reduced under the influence of stable stratification.  There is
also a pronounced asymmetry between the horizontal and vertical wake widths,
with the latter being more than a factor of two smaller than horizontal
width at the $x/D=67$ station.  All of these effects are manifestations
of reduced turbulent momentum transport due to buoyant damping.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/F03_u_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F03_u_rms_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/F03_w_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F03_w_rms_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/F03_r_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F03_r_rms_z.png}}
\caption{Lateral (left) and vertical (right) profiles of rms velocity
         and density fluctuations at several downstream locations for 
         the Fr=3 sphere wake case.  From top to bottom, $u^\prime$, 
         $w^\prime$, $\rho^\prime$.}
\label{fig:Fr3rms}
\end{figure}

Velocity and density fluctuations are shown in Figure \ref{fig:Fr3rms}.  
While the levels of the streamwise velocity fluctuations are similar to
the unstratified case, there is a more rapid decay of the vertical
fluctuations.  The asymmetry of the wake width is also evident in these 
plots.  Reduction in the vertical velocity fluctuations is a direct
effect of buoyant damping.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/F03_uv_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F03_uw_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/F03_vr_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F03_wr_z.png}}
\caption{Lateral (left) and vertical (right) profiles of Reynolds
         stress (top) and buoyancy flux (bottom) at several downstream 
         locations for the Fr=3 sphere wake case.}
\label{fig:Fr3str}
\end{figure}

Profiles of turbulent Reynolds stress and turbulent buoyancy flux are
shown in Figure \ref{fig:Fr3str}.  As expected, both the Reynolds stress
and the buoyancy flux decay rapidly with downstream distance.

The foregoing results indicate that the Fr=3 case is strongly affected
by stable stratification.  Buoyant forces add a second damping mechanism
that works together with viscosity to accelerate the turbulent decay.

\begin{figure}
\centerline{\includegraphics[width=2.5in]{figures/{F03x03y0.44_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F03x03z0.26_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{F03x21y0.60_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F03x21z0.42_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{F03x40y0.71_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F03x40z0.31_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{F03x67y0.79_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F03x67z0.31_spec}.png}}
\caption{Spectra at several streamwise positions the Fr=3 sphere wake case.  
The left 
figure is for a lateral position of $y/\delta_2\simeq 0.40$, whereas the 
right figure is for a vertical position of $z/\delta_3\simeq =0.40$,
where $\delta_2$ and $\delta_3$ are the estimated lateral and vertical
wake half-widths.}
\label{fig:Fr3allspec}
\end{figure}

In spite of the increased damping, spectra from the Fr=3 case are remarkably
similar to those from Fr=$\infty$.  Convincing simultaneous $k^{-5/3}$ and
$k^{-4}$ ranges are visible for most positions withing the turbulent portion
of the wake at $x/D$=3.4.  Less convincing but plausible simultaneous 
$k^{-5/3}$ and $k^{-4}$ ranges continue to be seen downstream, together
with a steeper dissipation range.  All of these features are seen in
Figure \ref{fig:Fr3allspec}, which displays spectra at roughly equivalent
lateral and vertical positions within the wake for the four streamwise
stations.

\subsubsection{Fr=1}

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/F01_u_avg_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F01_u_avg_z.png}}
\caption{Lateral (left) and vertical (right) profiles of mean streamwise
         velocity defect at several downstream locations for the Fr=1
         sphere wake case.}
\label{fig:Fr1mean}
\end{figure}

The second stratified case is for Fr=1.  This Froude number represents an 
extreme level of stable stratification and the results discussed here
may be more of academic interest as opposed to a model for what is observed
in the ocean.  We proceed nonetheless since the the data is readily available
and it may provide additional insight on how stratification affects the
velocity statistics and the spectra.

Mean velocity profiles are displayed in Figure \ref{fig:Fr3mean}.  The 
$x/D=3.4$ displays a complex structure in the vertical direction with
three central peaks and negative lobes beyond $y/D > 1$, that indicate
accelerated flow.  The maximum defect actually increases between the
$x/D=3.4$ and $x/D=21.4$ stations, after which it reduces.  The decrease
is much more modest than the other two cases, however, with a maximum
defect greater than 0.12 at the $x/D=66.7$ station.  The asymmetry between 
the horizontal and vertical wake widths is quite a bit more pronounced
than in the Fr=3 case.  The wake vertical extent is severely limited, 
indicating that vertical momentum transport is reduced to very low levels
at this Froude number.

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/F01_u_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F01_u_rms_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/F01_w_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F01_w_rms_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/F01_r_rms_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F01_r_rms_z.png}}
\caption{Lateral (left) and vertical (right) profiles of rms velocity
         and density fluctuations at several downstream locations for 
         the Fr=1 sphere wake case.  From top to bottom, $u^\prime$, 
         $w^\prime$, $\rho^\prime$.}
\label{fig:Fr1rms}
\end{figure}

Velocity and density fluctuations are shown in Figure \ref{fig:Fr1rms}.  
The velocity fluctuations are systematically lower than in the Fr=3
case (compare with Figure \ref{fig:Fr3rms}).  Decay of the vertical
velocity fluctuation is so severe in this case that $w^\prime/U_\infty$
in the vertical profile is limited to no more than 0.0018 at the 
$x/D=66.7$ station.  This is about a factor of 6 lower than 
$u^\prime/U_\infty$ or $v^\prime/U_\infty$, indicating a high degree of
anisotropy as a result of buoyant damping.

The density fluctuations are larger than the Fr=3 case 
(Figure \ref{fig:Fr3rms}) at the $x/D=3.4$, but then decay more rapidly
with downstream distance.  As with the vertical velocity fluctuation,
the density fluctuation reduces to a very low level by the $x/D=66.7$
station.  

\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/F01_uv_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F01_uw_z.png}}
\centerline{\includegraphics[width=3.0in]{figures/F01_vr_y.png}\hfill
            \includegraphics[width=3.0in]{figures/F01_wr_z.png}}
\caption{Lateral (left) and vertical (right) profiles of Reynolds
         stress (top) and buoyancy flux (bottom) at several downstream 
         locations for the Fr=1 sphere wake case.}
\label{fig:Fr1str}
\end{figure}

Profiles of turbulent Reynolds stress and turbulent buoyancy flux are
shown in Figure \ref{fig:Fr1str}.  Both the Reynolds stress and the 
buoyancy flux are small initially and then decay rapidly with downstream 
distance.  The decay is so severe that values can not be resolved on the 
plots beyond the $x/D=21.4$ station.  At the $x/D=66.7$ station the maximum
values of $u^\prime v^\prime/U_\infty^2$, $u^\prime w^\prime/U_\infty^2$,
and $\rho^\prime w^\prime/U_\infty\Delta\rho$ are $8.5\times 10^{-6}$,
$1.7\times 10^{-6}$, and $4.3\times 10^{-8}$, respectively.  These levels
are small enough that the flow can be characterized as effectively 
non-turbulent.

The foregoing results indicate that the Fr=1 case is supremely affected
by stable stratification.  The flow appears to be driven to a largely
laminar state beyond the $x/D=21.4$ station.  Vorticity images\cite{pal17}
confirm an absence of apparent turbulent structure beyond this point.

\begin{figure}
\centerline{\includegraphics[width=2.5in]{figures/{F01x03y0.70_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F01x03z0.16_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{F01x21y0.67_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F01x21z0.17_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{F01x40y0.66_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F01x40z0.19_spec}.png}}
\centerline{\includegraphics[width=2.5in]{figures/{F01x67y0.67_spec}.png}\hfill
            \includegraphics[width=2.5in]{figures/{F01x67z0.22_spec}.png}}
\caption{Spectra at several streamwise positions the Fr=1 sphere wake case.  
The left 
figure is for a lateral position of $y/\delta_2\simeq 0.40$, whereas the 
right figure is for a vertical position of $z/\delta_3\simeq =0.40$,
where $\delta_2$ and $\delta_3$ are the estimated lateral and vertical
wake half-widths.}
\label{fig:Fr1allspec}
\end{figure}

Spectra at several streamwise positions are shown in Figure 
\ref{fig:Fr1allspec}.  While the data at $x/D=3.4$ continue to exhibit 
simultaneous $k^{-5/3}$ and $k^{-4}$ ranges, the extent of the inertial
range is more limited as compared with the Fr=3 case 
(Figure \ref{fig:Fr3allspec}).  Unlike the Fr=3 case, the spectra 
offset in the vertical direction contains a pronounced Strouhal peak 
together with its first harmonic.  These features disrupt the inertial range,
further limiting its extent.  There may be a hint of an inertial range
for the spectra offset in the horizontal direction at the $x/D=21.4$ station
but, outside of this case, there are no $k^{-5/3}$ ranges visible in any of
the spectra downstream of $x/D=3.4$.  This observation is consistent with
the Reynolds stress statistics that indicate very little turbulent activity
downstream of the first two stations.  Although the spectra for the
stations downstream of $x/D=3.4$ appear to show short segments obeying
the $k^{-4}$ scaling, these are limited to no more than half a decade
in wavenumber.  There is a general tendency for the extent of the segment
to reduce and to move to lower wavenumbers with increasing downstream
distance.

In summary, all three Froude numbers studied show reasonably convincing
simultaneous $k^{-5/3}$ and $k^{-4}$ ranges at the $x/D=3.4$ station.
In all cases this dual scaling behavior is present over most of the turbulent
zone, from the centerline to about 70\% of the wake width.  In addition,
the $k^{-4}$ range extends all the way to the highest frequencies measured
in each case.  While a steeper, viscous dissipation range presumably exists,
the data were not sampled at a high enough rate to reveal this feature.

Downstream of $x/D=3.4$ there is a universal tendency for the $k^{-4}$
region to be terminated by a steeper, apparent viscous range.  This
feature generally reduces the extent of the $k^{-4}$ range.  There is
also a universal tendency for the transition wavenumber separating the
two scaling ranges to move to lower wavenumber with downstream distance.
In all cases there a pronounced drop in the transition wavenumber
when moving from the $x/D=3.4$ to the $x/D=21.4$ station.  Perhaps
surprisingly, stable stratification has only a mild effect on the spectra.
As might be expected, the extent of the inertial range decreases with
increasing downstream distance and with increasing stable stratification.
The extent of the $k^{-4}$ is rather limited downstream of the $x/D=3.4$
station for all three cases.  This is likely due to the rather low Reynolds
number, which allows for significant turbulent decay even by the $x/D=21.4$
station.  

\section{Conclusions}
\label{sec:conclusion}

The major findings of this study are:
%
\begin{enumerate}
%
\item No evidence of a $k^{-4}$ spectral range was found in numerical 
simulation data for time-developing wakes.
%
\item A $k^{-4}$ range was observed in our circular cylinder-inclusive
wake simulations for both stratified and unstratified cases, but only 
beyond the wake edge.
%
\item Simultaneous $k^{-5/3}$ and $k^{-4}$ ranges were observed in sphere-
inclusive wake simulations for both stratified and unstratified cases.
%
\item The $k^{-4}$ range observed beyond the wake edge appears to be due
to non-turbulent irrotational fluctuations.  The $k^{-4}$ range observed
within the turbulent portion of the sphere wake appears to be due to 
non-equilibrium processes rather than stable stratification.
\end{enumerate}
%

With regard to the first point above, data was examined from five
different time-developing stratified wake numerical simulations, each 
computed by a different research group.  Since this numerical approach
assumes that the flow is homogeneous in the streamwise direction,
wavenumber spectra are computed directly via Fourier transforms in
space.  Both the effective Reynolds and Froude numbers decay with time
in these simulations, making it easy to sample a wide range in these
parameters.  However, All of these cases being direct numerical simulations,
the initial Reynolds numbers are modest at best.  With the exception of
the work by Redford {\it et al.}\cite{redford15} the initial Froude numbers 
tend to be low as well.  Collectively the data surveyed spans a Reynolds 
number range of $10^2-10^4$ and a Froude number of $1-12$.  When looking
over this wide parameter space, none of the corresponding spectra displayed
a convincing $k^{-4}$ range.  Instead, a smooth transition between 
the inertial and dissipation ranges was observed.  
%
\begin{figure}
\centerline{\includegraphics[width=3.0in]{figures/pao_raw.png}\hfill
            \includegraphics[width=3.0in]{figures/pao_rescaled.png}}
\caption{Pao's laboratory data compared.  Left: data as read from his plots,
right: data remeasured with Pao's tic marks adjusted}
\label{fig:compare:pao}
\end{figure}

Our direct numerical simulations of a transverse-mounted circular cylinder
were partially successful at reproducing Pao's laboratory measurements.
As shown in Figure \ref{fig:compareFoospec}, spectra from our unstratified
simulation matches Pao's data quite well.  However, Figures 
\ref{fig:compareF38spec} and \ref{fig:compareF64spec} show very poor agreement
with Pao's data as read from his figures.  After careful analysis of Pao's
data, it appears likely that he made mistakes with the tic marks on the
energy axis.  This hypothesis is tested in Figure \ref{fig:compare:pao}.
The left panel shows a plot of the data as read from Pao's plots.  When 
compared, this plot appears to indicate that the Fr=0.64 case contains the
most energy at high wavenumbers, followed by the Fr=$\infty$ case, followed
by the Fr=3.8 case.  It seems extremely unlikely that this is true.  A much 
more plausible situation is achieved if the energy in the Fr=3.8 case is 
shifted up by one decade and the tic marks on Pao's Fr=0.64 plot are changed
from 0.5 decades to 1.0 decades.  The right panel in Figure 
\ref{fig:compare:pao} shows a replotting of the data with these adjustments.
Now the spectra look much more reasonable, with all three displaying an
inertial range and the high wavenumber energy ordering as expected with Froude
number.  One remaining mystery is that the data for the Fr=0.64 case falls
nearly on top of that for the Fr=3.8 case over the entire inertial range.
By contrast, our simulations for the Fr=0.64 case show a substantial 
reduction in energy together with an abbreviated inertial range.  It does
not seem reasonable that Pao's data are almost unaffected when the Froude
number drops by almost a factor of six (from 3.8 to 0.64).  It is probably
unwise to speculate but, given the other obvious problems with Pao's data,
it is certainly possible that Pao's data labeled as Fr=0.64 is actually for
a higher Froude number.

Notwithstanding the problems with the original plotting of Pao's spectra,
it can be said that our simulations for the stratified cases do not display 
simultaneous $k^{-5/3}$ and $k^{-4}$ ranges as seen in Pao's data.  We do see
extended $k^{-4}$ ranges just beyond the wake edge for both stratified and
unstratified cases, but these are most likely due to non-turbulent
irrotational fluctuations.

Due to the absence of simultaneous $k^{-5/3}$ and $k^{-4}$ ranges in our
cylinder wake data, we decided to move onto other data sets.  We were 
fortunate to gain access to very recent sphere-inclusive direct numerical
simulation data produced by Sarkar's group at UCSD.  Special care with 
segmentation of the time series and averaging across the wake centerline 
allowed us to produce spectra with acceptable noise levels.  These spectra 
display simultaneous $k^{-5/3}$ and $k^{-4}$ ranges, very similar to those 
observed by Delisi and Pao in their laboratory data.  However, unlike the 
laboratory data, we see the dual power law ranges across the entire range 
of Froude numbers, including the unstratified case.  We also see the dual
ranges across the entire turbulent portion of the wake.  The extent of 
the $k^{-4}$ range, however, decreases with increasing downstream distance 
in all cases.

Thus, while some similarities exist between the sphere-inclusive numerical
simulation data and the laboratory measurements, the details are different.
Both Delisi and Pao only show $k^{-4}$ ranges for stratified cases.
Furthermore, Delisi only observed the $k^{-4}$ range at considerable 
distance (x/D=60, Nt=5.2) from his sphere.  The numerical simulation
data, on the other hand, mainly shows the simultaneous ranges close to
the sphere, with or without stable stratification.

There are several possible explanations for the discrepancies between
the simulations and laboratory data.  If we just focus on comparing Delisi's 
data with the sphere-inclusive simulations, it becomes apparent that there
is a fairly large mismatch in the governing parameters.  Foremost is the
mismatch in Reynolds number with the lab data being for Re=$7.0\times 10^4$
and the simulations being for Re=$3.7\times 10^3$.  The Froude numbers are
also quite different, with the lab data using Fr=11.5 whereas the simulation
data are for Fr=3 and Fr=1.  Both the relatively low Reynolds and Froude
numbers in the simulations will cause the wake to decay much more rapidly 
than in Delisi's laboratory set up.  Based on Eq. (\ref{eq:wake:age}),
Delisi's $x/D=60$ measuring station corresponds to a non-dimensional wake
age of $Nt = (x/D)/Fr = 60/11.5 = 5.2$.  This same relative wake age is
achieved at $x/D=15.6$ for the Fr=3 numerical simulation and at $x/D=5.2$ 
for the Fr=1 simulation.  While this equilibrium scaling certainly does
not apply so close to the sphere, it at least highlights the fact that
the numerically-simulated stratified wakes are evolving much more rapidly
than the laboratory case.  An analogous development could be undertaken
with the Reynolds number (using a viscous time scale) and the same conclusion
that the simulated wakes evolve more rapidly would be reached.

While the above argument may help explain discrepancies in the downstream
location where the simultaneous power laws are observed, it does not
address the fact that the simulations clearly show the simultaneous
ranges for the unstratified case whereas the laboratory experiments do
not.  The simulation data strongly suggests that the presence of a 
$k^{-4}$ range is not a buoyancy-related effect {\it per se}.  

In an effort to rectify the stratified vs unstratified $k^{-4}$ discrepancy
we have developed a hypothesis that may explain the situation.  Our idea
is that the $k^{-4}$ range is due to streamwise inhomogeneity.  Nearly
all textbook treatments of spectral energy dynamics begin with the 
homogeneous turbulence assumption.  Under this view the spectrum is only
affected by production, conservative transfer, and dissipation.  In reality
the spectrum is also affected by inhomogeneous effects such as advection
by the mean due to systematic gradients in turbulent kinetic energy. This
term is given by
%
\begin{equation}
U_j\parderiv{q^2}{x_j}
\end{equation}
%
where $U_j$ is the mean velocity and where $q^2 = 1/2<u^\prime_i u^\prime_i>$
is the turbulent kinetic energy.  This term vanishes under the homogeneous
turbulence assumption since there are no spatial gradients in $q^2$.  The
term is also small for an equilibrium wake (unstratified far downstream) 
where the streamwise gradients in $q^2$ are small and the mean flows in 
the vertical and lateral directions are weak.  In contrast, the near wake
is characterized by large gradients in $q^2$ and by significant vertical
and lateral mean flows.  The mean advection term is not necessarily small
in this case and could lead to the $k^{-4}$ anomaly.

Similarly, the mean advection term can become significant in a stratified wake
as it transitions from one state to another.  Of particular interest here 
is non-equilibrium (NEQ) regime\cite{spedding96,spedding97}, which represents
a transition between the initially 3D state and the final quasi-2D state.
The NEQ regime typically spans a non-dimensional time of 
$2<Nt<50$\cite{diamessis11}.  This interval occurs in the near wake for the
low Froude numerical simulations considered here, but further downstream
in Delisi's laboratory experiment.  In particular, in Delisi's experiment,
Nt=2 corresponds to
$x/D=23$ and Nt=50 corresponds to $x/D=575$.  Delisi's measurements at
$x/D=60$ correspond to Nt=5.2, which is squarely in the NEQ regime.

Working with out hypothesis further, it is important to note that the mean
advection term is zero by construction in the time-evolving numerical 
simulations.  This is due to the parallel flow assumption necessary to
allow for periodic boundary conditions in the streamwise direction.  The
time-evolving approach was designed to study equilibrium wakes where 
the turbulence is nearly homogeneous in the streamwise direction and 
where the mean flow streamlines are only slightly inclined from the
wake centerline.  Thus it is incapable of simulating the near wake region,
and may be less than perfectly suited for studying stratified wakes
in the non-equilibrium regime.  It may be no coincidence that none of
the time-evolving numerical data sets that we surveyed display a $k^{-4}$
spectral range.  If this phenomenon is truly due to mean advection affects
the time-evolving simulation approach would be incapable of reproducing this
effect.


\section{Recommendations for future work}

While this project has helped to identify the conditions under which
a $k^{-4}$ spectral range occurs, the mechanisms behind this phenomenon
are not yet understood.  We have hypothesized that the $k^{-4}$ range is
due to mean advection of kinetic energy and a natural extension of this
work would be to test this hypothesis using numerical simulations.  Such
an undertaking would also allow us to cover a wider parameter space, to
cover a larger downstream distance, and to take better statistics.

The sphere wake simulations that we utilized were conducted using an immersed
boundary method in conjunction with a structured Cartesian grid.  While
this method is capable of simulating the flow over the sphere as well as
its wake, it is somewhat inefficient.  The structured grid framework
dictates that the fine mesh near the sphere must be carried all the
way to the boundaries in each coordinate direction.  This results is a
large overhead where high resolution is used in regions of low flow
gradients.  The inherent inefficiency in the immersed boundary method
limits the Reynolds number to values less than $10^4$, and also dictates
that the computation run only to $x/D\sim 100$.

In a follow-on project we would propose to switch to an unstructured
mesh method.  This approach gains considerable efficiency with its ability
to coarsen or refine the mesh arbitrarily in space.  This means that it is
possible to have a very fine mesh just for the boundary layers on the sphere
and for the turbulent wake behind.  Once outside of the high-gradient regions,
the mesh can be coarsened in all three directions.  The mesh can also be
coarsened as appropriate with downstream distance within the wake region since
the effective Reynolds number decreases steadily with downstream distance.
These added efficiencies will allow us to match the conditions of Delisi's
laboratory experiment at Re=$7.0\times 10^4$ and Fr=11.5.  We can also
carry the simulation further downstream that even the laboratory measurements
(to $x/D=200$ or more).  Finally, we can accumulate statistics necessary
to test the mean flow advection hypothesis and improve the data sampling
rate to extend the frequency range for spectra by one to two orders of 
magnitude.

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\end{document}

