Altogether, model selection applied to finite mixture models is a problem with
no clear solution. This has been shown in various simulation studies in other
fields, as well as in the simulations presented in this paper. The specifics
of identifying compositional subpopulations in exoplanet transit survey data —
three-dimensional data, a hierarchical model with observational uncertainties,
non-uniform mixtures, model misspecification — increase the severity of the
challenges. With these challenges in mind, care must be taken when first
formulating mixture models, implementing physical principles and constraints
wherever possible. Model selection techniques should be considered one aspect
of evaluating models, rather than the end-all and be-all, and validated with
simulations where possible. As these simulations have brought to light, data-
driven models can easily lead to overfitting and over-interpretation of
results, even when physical principles are implemented.

###  5.2 Do Water Worlds exist?

Our paper aims to assess the claims of the existence of water worlds in the
sample of Kepler exoplanets, as put forth by Zeng et al. (2019) and others. In
order to do so, we compile a range of models that either incorporated icy
composition planets alongside rocky composition planets, without invoking
photoevaporation (Model Z), or included planets with icy compositions within
models that include photoevaporation (Models Icy3a - Icy6).

While we cannot definitively prove the existence of water worlds, we find that
both models that include water worlds and those that do not are valid fits to
the Kepler sample. Our model selection results, while subject to the caveats
discussed in Sections 4.3 and 5.1, suggest that Model 1 and 2 are disfavored.
The remainder of the models, however, are all within $1\sigma$ of each other
in terms of their expected log predictive density, with no clear best model.
This suggest that models without water worlds (Model 3), models with water
worlds and no photoevaporation (Model Z), and models with water worlds and
photoevaporation (Models Icy3a through Icy6) are all valid fits to the data.
In fact, if you select the best model through Method 2 of He & Fan (2019),
discussed in Section 5.1, where you choose the model with the fewest
parameters within $1\sigma$ of the model with the highest score, Model Z would
be the selected model, as it has the fewest parameters among comparable
models.

With the validity of both models that include and do not include water worlds,
we assessed the constraint on the fraction of icy composition planets in
Section 3.4. We found a wide range across all models, with a general upper
limit of $50\%$, and a lower limit of $0\%$. Including all compositional and
formational subpopulations of planets in Model Icy6 resulted in a broad
posterior distribution of the fraction of icy planets, within the limits of
$0\%$ to $50\%$ and peaking at roughly $25\%$.

The wide uncertainty in the fraction of icy composition planets is reflected
in the large degree of degeneracy between planet subpopulations, as
demonstrated by the ternary diagrams in Figure 6. While there is a high
density of planets that are definitively rocky, there is significant overlap
between planets that could be rocky or gaseous, and an even larger overlap
between planets that could be icy or gaseous. This overlap is demonstrated in
the radius-period and mass-radius plots of Figures 2 and 3 as well.
Intrinsically icy planets and gaseous planets below $4R_{\oplus}$ occupy
similar territory in mass-radius-period space. Distinguishing between these
two subpopulations for planets between $2$ and $4R_{\oplus}$ is difficult.

This difficulty is primarily due to three different facets: the dataset, the
model, and inherent degeneracies in composition. The majority of planets in
the dataset lack mass measurements, and only contain radius and period
information. With the high degree of overlap in radius-period space between
intrinsically icy planets and gaseous planets, these planets are extra
difficult to identify as belonging to one subpopulation over the other. For
planets with mass measurements, the uncertainties on the mass and radius
measurements are high enough that this ambiguity in the identification of an
individual planet’s composition still persists. Even for planets with perfect
mass and radius information, our model parametrization and inherent
degeneracies will make this identification difficult.

In our model parametrization, we have abstracted the complexity inherent to
planet compositions by choosing three broad categories of composition: rocky,
icy, and gaseous, choosing a mean mass-radius relation for each, and adding
intrinsic scatter to simulate compositional diversity within these
subpopulations. The mass-radius relation for the gaseous subpopulation is
retrieved in the model rather than fixed to a theoretical curve as it is for
the rocky and icy subpopulations. As a result, there is significant overlap in
mass-radius space between the icy and gaseous subpopulations. While overlap
between these subpopulations is expected, the retrieved scatter in the gaseous
mass-radius relation is likely higher than physically expected. The envelope
mass fraction and core composition is not directly modeled in the gaseous
subpopulation, and as a result it is possible to have gaseous planets in the
model with higher densities than pure ice planets. Furthermore, each
subpopulation uses the same functional form for the mass and period
distributions. These distributions also tend to be broad, with the consequence
that it is difficult to distinguish between these subpopulations by mass and
period alone.

Regardless of our parametrization, there is also the fact that overlap between
gaseous and icy planets is inherent to some regions of parameter space,
leading to degeneracies in composition when assessing any individual mass-
radius-period point. As a result, even with a perfect model and a perfect
dataset, some planets will necessarily have substantial probabilities of
belonging to either subpopulation when considering mass-radius-period alone.
The question of compositional categorization is a probabilistic one, given the
broad distributions of these subpopulations in mass-radius-period space.

One possible solution to break the degeneracy (as much as we can) is to
increase the amount of physical constraints and information into the model, by
shifting the framework to model composition directly rather than purely
dealing with the abstractions of mass and radius. With core composition, core
mass, and envelope mass fraction as fundamental parameters, we avoid the
situation of gaseous planets with higher density than rocky or icy planets,
although some overlap with icy planets will persist. The three compositional
subpopulations will have greater separation in mass-radius-period space. With
this greater separation, we may be able to better constrain the underlying
mass and period distributions of these compositions. Constraining the core
mass and envelope mass fraction distributions will yield some important
insights on their own as well. Incorporating a more sophisticated mass loss
prescription may help break these compositional degeneracies, as low-density
planets with short mass loss timescales present a possible region of parameter
space to identify water worlds.

###  5.3 Model Caveats

As mentioned in the previous section, model misspecification is a significant
limiting factor in our ability to perform model selection on this collection
of models. Metrics such as k-fold cross-validation have a tendency to over-
select high complexity models, and this problem is exacerbated by model
misspecification. Given the nature of our models as three-dimensional mixture
models with up to six subpopulations of planets, the potential for model
misspecification is high.

For each subpopulation, we assume the mass distribution to be characterized by
a log-normal distribution, and the period distribution to be characterized by
a broken power-law. While these were chosen based on their ability to fit the
data better than several other alternatives, such as a normal distribution for
the mass distribution, and a power-law for the period distribution, there are
many possibilities that we did not explore. With the log-normal only having
two free parameters, and the broken power-law having only three free
parameters, there is the potential for additional complexity in these
distributions that these parameterizations cannot account for. For example,
due to the log-normal distribution having long, symmetric tails in log-space,
the intrinsically rocky subpopulations extend to higher masses than is
physically likely. An exponential cutoff might be a feature of the mass
distribution that could limit the masses of intrinsically rocky planets, as
above a certain mass these planets will have accreted significant hydrogen-
helium gaseous envelopes. Our assumption of a period distribution independent
from mass and radius (aside from photoevaporation) may break down if we
extended our period range beyond 100 days.

Additionally, in our models there is the added assumption that each mixture
has a mass and period distribution that follows the same form. It could be the
case that a lognormal mass distribution is a decent approximation for one
subpopulation, but inadequately describes another subpopulation. The
possibility of different mixture subpopulations being described by distinct
distributions compounds the potential for model misspecification.

Our models assume three distinct compositions for exoplanets: rocky, icy, or
gaseous. Of course, compositions are better described by a smooth spectrum
rather than discrete options. In order to account for this spectrum, we
incorporate a scatter in the mass-radius relation, so that a given mass planet
for a given composition can have a range of radii. This is a convenient
abstraction, but a better way to incorporate the complexities of composition
would be to directly model the composition of exoplanets, rather than the mass
and radius.

The prescription we use to incorporate photoevaporation into our models, given
by Equations (8) and (9), is another potential source of model
misspecification. This photoevaporation timescale depends on several factors,
including the typical age of a star, the XUV flux, and the mass-loss
efficiency, for which we choose nominal values but recognize them to be highly
uncertain. To compensate for this uncertainty, we include a free parameter
$\alpha$ to scale the mass-loss timescale. However, this parameter is strongly
influenced by the prior, and takes disparate values depending on the model,
ranging from $0.3$ to $8.5$. This indicates that this parameter acting more to
scale the number of evaporated planets rather than modeling a specific
physical process.

Core-powered mass loss is an alternative explanation to the Kepler radius gap
that we have not explored in this work. Rogers et al. (2021) have shown that,
similarly to our conclusions with respect to water worlds versus
photoevaporation, the CKS-Gaia dataset is insufficient to distinguish between
core-powered mass loss and photoevaporation. They find that a survey of $5000$
planets with sufficient coverage in host star mass and $5\%$ level
uncertainties would be able to distinguish between the two scenarios by
constraining the slope of the radius gap in radius-host star mass and radius-
incident flux space. The mixture models we presented here could be adapted to
include core-powered mass loss, but model selection difficulties and dataset
limitations would remain a barrier to distinguishing between models.

Finally, as discussed in NR20, the mass-radius-period distribution of
exoplanets is just one 3D projection of the many-dimensional distribution of
exoplanets. Exoplanet occurrence rates can also depend on the multiplicity of
planetary systems, host star spectral type and metallicity, and the galactic
location, among other star-planet system parameters. These additional
dimensions will tend to smear out these mass-radius-period distributions even
more, adding more sources of model misspecification.

While there remain many avenues for improvement, this work has made several
important steps forward compared to the models in NR20. We have explored a
much wider range of models, considering models with up to six subpopulations
of exoplanets. We have included subpopulations with icy-core compositions
alongside subpopulations with rocky-core compositions, opening the door for a
broader range of compositional diversity. Careful tests of model selection
using simulated planet catalogs have shown that overfitting remains a problem
when considering high complexity models. As a result, several of these models
remain viable fits to the Kepler dataset, and higher quality datasets or
improved modeling techniques may be necessary to distinguish between the
competing theories of planet formation processes. Though our results were
inconclusive in ruling a subpopulation of water worlds in or out, our study
presents an in-depth case study on the challenges of model selection applied
to mixture models and the population-level degeneracies that persist when
interpreting a statistical sample of planet mass, radius, and orbital period
measurements.

###  5.4 Future Prospects

Our ability to fit multidimensional distributions including multiple
subpopulations of planets is limited by the quality of the data that we use.
As seen in Section 4.3, the quality and quantity of data is not a complete
solution to the difficulties of model selection with mixture models. However,
improving the dataset (as in our PLATO simulations, compared to Kepler
simulations), may help distinguish between models in the case of high model
complexity and large separation between subpopulations. Improving the dataset
will thus allow us to uncover additional complexities in the mass-radius-
period distribution, just as improving the Kepler sample with higher precision
stellar radius measurements led to the discovery of the Kepler radius gap
(Fulton et al., 2017). These improvements would include more transiting
planets in the sample, more planets with mass measurements, lower mass and
radius uncertainties, and improved sensitivity to smaller planets and longer
period planets.

TESS is an ongoing all-sky transit survey looking at the brightest nearby
stars that has identified 2241 exoplanet candidates to date, with over 100
confirmed exoplanets (Guerrero et al., 2021). TESS also aims to have follow-up
radial velocity measurements of 50 small planets below $4R_{\oplus}$. With its
two-year Prime Mission, its sensitivity is mostly limited to planets with
periods shorter than 100 days. Additionally, it is limited in its ability to
detect smaller planets below $1.5R_{\oplus}$, currently having detected less
than 100 candidates out of 2241. Nonetheless, TESS offers a dataset comparable
in size to Kepler, and will add considerably to the sample of small planets
with radial velocity mass measurements. TESS will also enable more detailed
studies of host star dependence for planet occurrence rates, given the
increased fraction of planets around low-mass stars.

PLATO is a space telescope scheduled for launch in 2026, that is designed to
detect planetary transits around bright stars. The PLATO mission has a goal of
detecting 7000 transiting systems, with radial velocity mass measurements of
400 of those planets to a rough precision of $10\%$, in addition to TTV mass
measurements of an undetermined subset of planets. PLATO also pushes the
bounds in radius and period, with the goal of detecting $1R_{\oplus}$ planets
orbiting in the habitable zone of G-type stars to $3\%$ precision (PLATO Study
Team, 2017). PLATO’s nominal science mission lifetime of 4 years, with a
possible extension to 8 years, will enable it to better study long-period
planets compared to Kepler and TESS. PLATO will represent a significant
upgrade to Kepler by offering a larger dataset with higher precision radii, a
massively expanded RV sample with higher precision masses, and improved
sensitivity to smaller planets on longer orbits.

In the future, we will have several high-quality datasets available to us,
from Kepler, TESS, PLATO, and others. Rather than fitting a mass-radius-period
distribution to each dataset independently, a better way to leverage these
datasets would be to combine them into one single fit. The inhomogeneous
Poisson process we use to model this distribution offers a possible way to
incorporate multiple surveys. Our ability to include multiple compositional
subpopulations in our mass-radius-period models will thus be greatly improved
with the advent of additional Kepler-like surveys.

##  6 Conclusion

We include various subpopulations of planets with icy compositions into our
joint mass-radius-period distribution modelling. We create a suite of ten
mixture models of varying complexity that combine six fundamental
subpopulations of planets - gaseous planets with rocky or icy cores,
evaporated rocky or icy cores, and intrinsically rocky or icy planets - in
different permutations in order to assess the evidence for water worlds in the
Kepler dataset. We find that these icy compositional subpopulations have
significant overlap with existing gaseous and rocky subpopulations, leading to
large degeneracies in identifying any particular planet as belonging to a
specific subpopulation. We apply k-fold cross-validation as a form of model
selection, and find that while the lowest complexity models are disfavored, we
cannot strongly distinguish between the majority of our ten models. Models
that include icy composition planets either with photoevaporation, or without
photoevaporation, or models that include photoevaporation but not icy
composition planets, all have support in the data. We set a rough upper limit
of $50\%$ on the fraction of icy composition planets, although this fraction
varies widely between models. We propose that improved modeling approaches,
such as fundamentally modeling composition, and improved datasets, such as
those from PLATO or TESS, could further probe the existence of water worlds.

##  7 acknowledgements

We thank our anonymous referee for providing valuable feedback and suggestions
that improved the paper. This research has made use of the NASA Exoplanet
Archive, which is operated by the California Institute of Technology, under
contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program. This work was also completed in part with
resources provided by the University of Chicago’s Research Computing Center.
JL thanks the University of Chicago Physics/MRSEC REU Program, and the
National Science Foundation. LAR gratefully acknowledges support from NASA
Habitable Worlds Research Program grant 80NSSC19K0314, from NSF FY2016 AAG
Solicitation 12-589 award number 1615089, and the Research Corporation for
Science Advancement through a Cottrell Scholar Award.

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# End boundary effects on wakes dynamics of inclined circular cylinders

Kai<EMAIL_ADDRESS>Yan Bao1<EMAIL_ADDRESS>Dai Zhou1
Zhaolong Han1  1State Key Laboratory of Ocean Engineering, School of Naval
Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University,
Shanghai 200240, China 2Department of Mechanical and Aerospace Engineering,
Rutgers University, Piscataway, NJ 08854, USA

###### Abstract

We perform direct numerical simulations to characterize the three-dimensional
wake dynamics of long inclined circular cylinders with inhomogeneous end
boundary conditions. Three Reynolds numbers, $\Rey=100$, 200 and 300,
corresponding to the regimes of laminar flow, mode A* wake, and mode B wake,
respectively, are considered to reveal the roles of the intrinsic secondary
instabilities and the extrinsic end boundary effects in shaping the three-
dimensional flows. At $\Rey=100$, the end boundary effects are felt over the
entire cylinder span by inducing oblique vortex shedding, which is associated
with stronger spanwise flow in the wake than a parallel shedding. The Strouhal
number of the oblique shedding is related to that of the parallel shedding of
straight cylinder by the cosine law, considering the combined inclination
angle and oblique angle. At $\Rey=200$, the intrinsic secondary instability
results in large-scale vortex dislocation, precluding the propagation of the
end boundary effects towards further span. Nevertheless, for the inclined
cases, oblique shedding is still observed within limited span from the
upstream end boundary. The oblique vortices feature intact and straight vortex
cores, and are related to the two-dimensional flow at $\Rey=200$ (that are
void of vortex dislocations) from the viewpoint of cosine law. Further along
the span, the oblique vortices destabilizes with the formation of small-scale
vortices, and the flow transitions to the typical mode A* wake. At $\Rey=300$,
the highly three-dimensional flow near the end boundary creates disturbances
that travel along the cylinder span, creating vortex dislocations for cases
with low inclination angles. For high inclination angle, oblique vortex
shedding is again observed over the cylinder span, and is not disrupted by
vortex dislocations of either intrinsic or extrinsic causes. The present study
offers renewed insights into the three-dimensional wake dynamics of inclined
cylinders, and lays the foundation for the designs of long flexible
engineering structures and related flow control techniques.

##  1 Introduction

Flow over a circular cylinder has received sustained attention due to its
fundamental and practical importance. As the key feature of the cylinder wake,
the Kármán vortex street is characterized by periodic shedding of counter-
rotating vortices from the cylinder, and is related with various phenomena
such as the unsteady hydrodynamic forces, flow-induced vibrations, noise, etc.
Over the past few decades, substantial studies have been carried out to
understand various aspects of cylinder wake (Williamson, 1996b; Zdravkovich,
1997a, b; Norberg, 2001; Jiang et al., 2016; Cheng et al., 2017; Garcia et
al., 2019; Fan et al., 2019; Mittal et al., 2021). In particular, the three
dimensionality in the wake of the cylinder has been the focus of extensive
research.

Three-dimensional wake can arise from both “intrinsic” and “extrinsic”
mechanisms. For an infinitely long circular cylinder that is void of extrinsic
boundary effects, the three-dimensional flow emerges as the Reynolds number
increases beyond $\Rey\sim 188$ (Barkley & Henderson, 1996; Henderson &
Barkley, 1996). This intrinsically three-dimensional flow forms via the
secondary instabilities of the periodic vortex street, and manifests itself
successively as the mode A, mode A*, and mode B wakes, with increasing
Reynolds number. The mode A wake is characterized by wavy spanwise vortex
cores interlaced with streamwise vortex pairs, with a wavelength of about 4
diameters. The formation of mode A is shown to be originated from an elliptic
instability of the primary vortex cores, and the streamwise vortex pairs are
formed through Biot-Savart induction (Williamson, 1996a; Leweke & Williamson,
1998; Thompson et al., 2001). The pure mode A wake occurs in the beginning
phase of the three-dimensional transition, and only lasts for a short period
of time before it evolves into the mode A* featuring large-scale vortex
dislocation. Further increasing the Reynolds number to $\sim 230$, the mode B
wake characterized by fine-scale streamwise vortices (wavelength of $\sim$1
diameter), but more stable spanwise vortex cores, becomes the main form of
three dimensionality in the cylinder wake. Unlike the mode A, the streamwise
vortices in mode B are formed in the braid shear layer region due to a
hyperbolic instability (Williamson, 1996a; Leweke & Williamson, 1998; Thompson
et al., 2001). The transitions of these wake modes are associated with
discontinuities in the shedding frequencies of the Kármán vortex streets with
respect to the Reynolds number.

In reality, the wake dynamics of cylinders are inevitably influenced by the
end boundary conditions, which serve as “extrinsic” mechanism for the
emergence of three-dimensional flows. The flow structures near the
inhomogeneous boundary are highly three-dimensional due to the formation of
tip vortices in the case of a free end (Park & Lee, 2000; Roh & Park, 2003;
Krajnović, 2011; Sumner, 2013; Cao et al., 2022), and the interaction with
boundary layer flow in the case of a nonslip wall (Slaouti & Gerrard, 1981;
Pattenden et al., 2005; Krajnović, 2011; Mittal et al., 2021). At low Reynolds
numbers, such complex three-dimensional flows typically occupy 10 – 20
diameters near the end boundary (Williamson, 1989). The oblique vortex
shedding are observed for the rest of the cylinder span, even hundreds of
diameters in length. Depending on the boundary condition, the Reynolds number
and the aspect ratio of the cylinder, the vortex shedding can be split into
different cells with distinct shedding frequencies, separated by vortex
dislocation in between (Behara & Mittal, 2010; Mittal et al., 2021). Taking
advantage of the high sensitivity of the cylinder wake to the end conditions,
Williamson (1989) achieved both oblique shedding and parallel shedding behind
a cylinder by manipulating the end plates. This flow control technique has
also been reported to be effective for higher Reynolds number flows (Prasad &
Williamson, 1997; Luo & Xia, 2005).

The wake of an inclined cylinder presents another type of inherently three-
dimensional flow, due to the strong axial velocity generated behind the
cylinder (Van Atta, 1968; Matsumoto et al., 1990; Najafi et al., 2016; Zhang
et al., 2018). In many cases, the wake is featured by shedding of vortex tubes
that are parallel with the cylinder axis, and the spanwise flow component can
be decoupled from the rest two dimensions. This leads to the famous
independence principle (also known as cosine law), which states that the
aerodynamic characteristics of an inclined cylinder are related to those of a
straight cylinder with a factor of $\cos\beta$, where $\beta$ is the
inclination angle. The validity of independence principle has been
theoretically proven by Sears (1948) for boundary layer flows over infinite
yaw cylinders. For the separated wakes over circular cylinders, the
independence principle is shown to hold true for cases with inclination angle
smaller than $\sim 45^{\circ}$ (Lucor & Karniadakis, 2003; Zhao et al., 2013;
Bourguet et al., 2015). However, most studies on inclined cylinders
(particularly in computational studies) have employed the assumption of
spanwise homogeneity, and have not considered the effects of end boundary
condition on the flow.

The wake dynamics of an inclined cylinder with inhomogeneous end conditions
can exhibit complex features. For short inclined cylinders, the flows are
dominated by vortical structures that develop near the ends. Choi et al.
(2007) experimentally studied the heat transfer around the circular tie-rods
(with length-to-diameter ratio less than 4) that connect the steam turbine
rotors. Their results revealed that the interaction between the horseshoe
vortices near the upstream end wall and the wake shed from the cylinder varies
with the inclination. They also observed a jet-like stream in the cylinder
wake that impinges on the downstream wall, and enhances the heat transfer. Hu
et al. (2015) carried out large eddy simulations to study the flows over cable
bridge pylon in the form of finite inclined square prism with height-to-width
ratio of 18. In the forward inclination case, a dipole wake consisting a pair
of counter-rotating tip vortices is formed, and is related to the strong
downwash in the near wake. In the backward inclination case, the wake is
dominated by base vortex pair, while the tip vortices are suppressed. Zhang et
al. (2020a) and Zhang & Taira (2022) studied the laminar wake dynamics of
finite-aspect-ratio swept wings at high angles of attack. For both forward-
and backward-swept wings, the steady vortices formed near the end boundaries
not only stabilize the flow, but also provide additional vortex lift that
enhances the aerodynamic performance. The strong end effects described in
these researches significantly influence the wake dynamics of low-aspect-ratio
inclined cylinders (and prims, wings), and invalidate the independence
principle.

Relatively less attention has been paid to the end boundary effects on flows
over long inclined cylinders. Interests in such configurations arise in the
flows over trailing antenna, yawed stay cables, deepwater risers, just to name
a few. Among the few who have addressed this issue, Ramberg (1983) showed that
wake can exhibit various flow patterns that are dictated by the end
conditions. With low inclination angle, the wake divides into multiple
shedding modes with different angles of vortex tubes shed at distinct
frequencies. As inclination angle increases, the unsteady part of the wake
moves away from the upstream boundary, and the majority of the span features
steady trailing vortices on alternating sides of the cylinder with opposite
circulation. Such steady flow pattern has also been observed in the wakes of
slender cylindrical bodies at large incidence (Allen & Perkins, 1951;
Sarpkaya, 1966; Thomson & Morrison, 1971; Zhang et al., 2020a). The complex
flow features described above are significantly different from those found in
the wakes of inclined infinite cylinders.

The wake dynamics of inclined cylinders exhibits highly three-dimensional
characteristics originated from various mechanisms such the end effects, the
Reynolds number effects, as well as the inclination angle. While a lot of
fascinating flow physics has been uncovered in the early work of Ramberg
(1983), the roles played by the different mechanisms in the formation the
three-dimensional flows are not clear. The evolution of the vortical
structures along the span is not described in detailed in previous studies. In
addition, the implications of the three-dimensional flow structures on the
forces on the structures are yet to be unveiled. The answers to these
questions provide renewed insights into the complex wake dynamics observed in
the key work of Ramberg (1983), and lay the foundation for the design of long
flexible engineering structures and related flow control devices.

To address the above problems, we carry out direct numerical simulations to
study the flows over circular cylinders with inhomogeneous end boundary
conditions. We cover three representative Reynolds numbers, $\Rey=100$, $200$
and $300$, corresponding to the different flow regimes of two-dimensional
laminar wake, mode A* wake, and mode B wake, respectively. Relatively large
length-to-diameter ratio is considered to allow the end-condition-induced
three dimensionality to fully develop along the cylinder span. These settings
allow for a comprehensive understanding of the interplay between the extrinsic
end boundary effects and intrinsic three dimensionality. In what follows, we
present the computational setup in §2. The numerical results are discussed in
§3, where we analyze the wake vortical structures and the corresponding
hydrodynamic forces at different Reynolds numbers in detail. We conclude this
study by summarizing our findings in §4.

##  2 Computational setup

The flows over the circular cylinders are studied numerically by solving the
three-dimensional Navier-Stokes equations:

$\begin{split}\bm{\nabla}\cdot\bm{u}&=0,\\\ \frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\bm{\nabla}\bm{u}&=-\bm{\nabla}p+\frac{1}{Re}\bm{\nabla}^{2}\bm{u},\end{split}$
(1)

in which $\bm{u}=(u_{x},u_{y},u_{z})$ is the velocity vector, and $p$ is the
pressure. The Reynolds number is defined as $\Rey\equiv U_{\infty}D/\nu$,
where $U_{\infty}$ is the freestream velocity, $D$ is the diameter of the
cylinder, and $\nu$ is the kinematic viscosity of the fluid. This study
considers three representative Reynolds numbers, i.e., $\Rey=100$, 200 and
300, which lie in the regimes of two-dimensional flow, the mode A* wake, and
the mode B wake.

Figure 1: Schematic of the computational setup. The black solid lines and gray
dashed lines represent the setups for flows over straight ($\beta=0^{\circ}$)
and inclined ($\beta\neq 0^{\circ}$) cylinders, respectively. The sketch is
not to scale.

A schematic of the computational setup is shown in figure 1. The center of the
circular cylinder is located at $15D$ from the inlet, and $30D$ from the
outlet. The width of the computational domain is $30D$, which results in a
blockage ratio of 3.3%. The length of the cylinder in the $z$ direction is
fixed at $100D$. The circumference of the cylinder is discretized with 160
grids. The mesh is concentrated in the vicinity of the cylinder to better
resolve the near wake. The height of the first-layer mesh on the cylinder wall
is $0.02D$. The spanwise direction is discretized with 2000 grids, resulting
in a uniform spanwise grid resolution of $\Delta z=0.05D$. The total number of
the grid numbers amounts to $\sim 4.6\times 10^{7}$. The time-step is fixed at
$\Delta t=0.01D/U_{\infty}$, resulting in a maximum CFL number below 1. To
simulate the flows over inclined cylinders, the computational domain is
sheared with the inclination angle $\beta$ in the $x$ direction as shown by
the dashed frame in figure 1, while the direction of the freestream velocity
is kept fixed. Note that in such setup, the incoming flow “sees” the section
of the inclined cylinder still as circular shape. In what follows, all
variables are reported in their dimensionless forms by normalizing spatial
quantities with the cylinder diameter $D$, velocity with $U_{\infty}$, and
time with $D/U_{\infty}$.

|  $\beta=0^{\circ}$ |  $\beta=15^{\circ}$ | $\beta=30^{\circ}$
---|---|---|---
$\Rey=100$ |  slip, nonslip |  cyclic, nonslip, slip |  cyclic, nonslip, slip
$\Rey=200$ |  slip, nonslip |  cyclic, slip |  cyclic, slip
$\Rey=300$ |  slip, nonslip |  cyclic, slip |  cyclic, slip
Table 1: End boundary conditions for $z=0$ plane. Italicized cases are
considered spanwise homogeneous.

The finite-volume-based incompressible solver _pimpleFoam_ of the open-source
CFD toolbox OpenFOAM (Weller et al., 1998) is used to carry out the direct
numerical simulations. Both space and time are discretized with second-order
accurate schemes. Different boundary conditions, including slip ($u_{z}=0$),
nonslip ($\bm{u}=0$) and cyclic ($\bm{u}|_{z=0}=\bm{u}|_{z=100}$) are imposed
on the $z=0$ plane. The other side of the spanwise boundary ($z=100$), which
is positioned downstream in the inclined cases, is generally treated as slip,
except for the cyclic condition used together with that at $z=0$. A summary of
the simulated cases with different boundary conditions are shown in table 1.
These cases allow us to inspect the flow physics in both spanwise homogeneous
and inhomogeneous settings. The rest of the boundary conditions are specified
as follows. The inlet is prescribed with a uniform inflow velocity
$\bm{u}=(U_{\infty},0,0)$. The side walls of the computational domain are
treated as slip boundaries. Nonslip boundary condition is applied to the
cylinder surface. A reference pressure $p_{\infty}=0$ is set at the outlet
domain, with a zero-gradient condition for velocity.

| $\Rey=100$ | $\Rey=200$ | $\Rey=300$
---|---|---|---
Jiang et al. (2016) | — | — | 1.30
Canuto & Taira (2015) | 1.34 | — | —
Qu et al. (2013) | 1.32 | 1.25 | —
Present | 1.35 | 1.27 | 1.31
Refined | 1.35 | 1.26 | 1.31
Table 2: Comparison of the time-averaged drag coefficient $\overline{C_{D}}$
for flow over a two-dimensional circular cylinder at $\Rey=100$, 200 and 300.
The refined case employs increased spatial grid numbers in all three spatial
directions by a factor of 1.4. Figure 2: Comparison of Strouhal number of flow
over a circular cylinder at different Reynolds numbers. The present 2D
simulations are carried out using one cell in the spanwise direction, thus
suppressing the development of the intrinsic three dimensionality. The present
3D simulations refer to the spanwise homogenous cases with developed intrinsic
three dimensionality.

To validate the above computational setup, we present a comparison of the
time-averaged drag coefficient $\overline{C_{D}}$ for flows over spanwise-
homogeneous circular cylinders at different Reynolds numbers in table 2. The
drag and lift coefficients are defined as $C_{D}=F_{D}/(\rho
U_{\infty}^{2}DH/2)$ and $C_{L}=F_{L}/(\rho U_{\infty}^{2}DH/2)$, where
$F_{D}$ and $F_{L}$ are the drag and lift forces on the cylinder, and $H$ is
the length of the cylinder. It is observed that fair agreement between our
numerical results and those from previous studies is achieved, demonstrating
the accuracy of the current numerical method. The results have also reached
mesh independency, as a 1.4-fold refinement in grid resolution results in
negligible changes in the drag coefficients. We further compare the the
Strouhal number (defined as $St=fD/U_{\infty}$, where $f$ is the vortex
shedding frequency) of with Jiang et al. (2016), Barkley & Henderson (1996)
and Williamson (1996b) in figure 2 for the spanwise homogeneous cases. The
different Strouhal numbers in 2D and 3D simulations at $\Rey=200$ and 300 are
accurately predicted.

##  3 Results

To reveal the different roles of the three-dimensional mechanisms play in the
wake dynamics of inclined cylinders, we first present the numerical results on
the extrinsic end boundary effects on the cylinder wakes at $\Rey=100$, where
the intrinsic three dimensionality is absent. The second part of this section
discusses the mode A* ($\Rey=200$) and mode B ($\Rey=300$) wakes of both
straight and inclined cylinders without the end boundary effects. At last, the
flows subjected to both the extrinsic and intrinsic three dimensionality are
presented.

###  3.1 End boundary effects at $\Rey=100$

####  3.1.1 Wake vortical structures

Figure 3: Flows over circular cylinders at $\Rey=100$. ($a$)
$\beta=0^{\circ}$, ($b$) $\beta=15^{\circ}$ and ($c$) $\beta=30^{\circ}$. In
$(b)$ and $(c)$, the results from both the nonslip and symmetry boundaries are
shown. The vortical structures are visualized using isosurfaces of
$QD^{2}/U_{\infty}^{2}=0.1$.

The flows at $\Rey=100$ is generally laminar and offer a “clean” condition to
study the extrinsic end boundary effects. The wake vortical structures
visualized by isosurfaces of $QD^{2}/U_{\infty}^{2}=0.1$ ($Q$ is the second
invariant of velocity gradient tensor) for flows at $\Rey=100$ are shown in
figure 3. Under the nonslip end condition, the wake for the $\beta=0^{\circ}$
case exhibits typical cellular structures that have been described in previous
studies (Behara & Mittal, 2010; Zhang et al., 2020b; Mittal et al., 2021). The
first cell near the nonslip wall occupies $0<z\lesssim 15$, and is featured by
highly three-dimensional flow structures, including the braid-like vortices
that mainly consists of streamwise and crossflow vorticity ($\omega_{x}$ and
$\omega_{y}$), and the curved spanwise vortex tubes. It has been shown that
the braid-like vortices connects the spanwise vortical structures to form
vortex loops (Taneda, 1952; Zhang et al., 2020b; Mittal et al., 2021). The
second cell, which dominates most of the cylinder span, features oblique
vortices with $\theta=12.5^{\circ}$, where $\theta$ is termed the oblique
angle denoting the angle between the vortex tube and cylinder axis. The vortex
shedding frequency in the former cell is smaller than that of the second cell,
thus creating vortex dislocation at $z\approx 15$. Near the $z=100$ plane, the
inclined vortex tubes are bent to be parallel to $z$ axis to respect the
symmetry boundary condition.

For the inclined cases, we assess the effects of the nonslip and slip
boundaries, both of which impose inhomogeneous end conditions to the Kármán
vortex shedding. At $\beta=15^{\circ}$, the wakes with both boundary
conditions are dominated by vortex shedding obliqued at an angle to the
already inclined cylinder axis, as observed in figure 3($b$). For the nonslip
case, the vortex tubes are aligned at an angle of $\theta=23^{\circ}$ to the
cylinder axis, while for the slip case, the oblique angle is
$\theta=15.2^{\circ}$. This comparison suggests the high sensitivity of the
oblique angle on end condition at low inclination angles. Similar observation
has also been reported in (Behara & Mittal, 2010) for the straight cylinders,
in which the shedding angle is shown to be dependent on the boundary layer
thickness of the nonslip wall, and the aspect ratio of the cylinder. In fact,
Mittal & Sidharth (2014) and Hélder Marques Ribeiro et al. (2022) have shown
that for straight cylinder, there exists a vast number of unstable eigenmodes
for the wake, each corresponding to a certain angle of oblique vortices. Aside
from the difference in the oblique angle between the two cases, the nonslip
wall induces a larger region of steady flow near the boundary than the slip
wall. As the inclination angle increases to $\beta=30^{\circ}$, the steady
region expand drastically to $z\approx 35$ – 40. An oblique shedding angle of
$\theta=18^{\circ}$ is observed for both boundaries, suggesting its weak
dependence on the end conditions at large inclination angle.

Figure 4: Instantaneous $\omega_{z}$ fields for the case
$(Re,\beta)=(100,30^{\circ})$ with slip end condition. Slices are taken at
($a$) $z=5$, ($b$) $z=20$, ($c$) $z=30$ and $(d)$ $z=50$. The $x$ coordinates
of these slices are shifted so that the cylinder center resides at
$(x,y)=(0,0)$. Blue and red indicate negative and positive $\omega_{z}$ in the
range of $[-1,1]$.

Let us take a closer look at the transition from the locally steady flow to
the unsteady vortex shedding in the case of $\beta=30^{\circ}$. As shown in
figure 4, at $z=5$, the sectional flow (visualized by the $z$-component of
vorticity vector $\bm{\omega}=\nabla\times\bm{u}$) is featured by a pair of
counter-rotating vortices in the near wake. The vorticity advects to further
wake at $z=20$. At $z=30$, the flow in the far wake starts to oscillate, and
at $z=50$, the flow becomes fully unsteady featuring periodic shedding in the
near wake. The progressive development of the flow along the spanwise
direction as described above is similar to the temporal evolution of the two-
dimensional wake of a symmetric bluff body started impulsively from rest
(Braza et al., 1986; Tamaddon-Jahromi et al., 1994; Laroussi et al., 2014).
This is also reflected in the sectional force coefficients depicted in figure
5. For the case of $\beta=30^{\circ}$, the steady flow region is associated
with lower drag and zero lift, while the unsteady shedding region corresponds
to higher drag and lift forces. The transition between the two states is
similar to the destabilization of the flow from unstable steady state to the
saturated unsteady flow (Noack et al., 2003). Similar impulsive flow analogy
has also been reported for the wakes of slender cylindrical bodies with large
incidence (Allen & Perkins, 1951; Sarpkaya, 1966; Thomson & Morrison, 1971;
Zhang et al., 2020a), although in their cases the flow is steady.

Figure 5: Sectional force coefficients for cases with $\Rey=100$. ($a$) time-
averaged drag coefficient and ($b$) rms lift coefficient. ‘n’ and ‘s’
represent nonslip and slip boundary conditions.

We note that the flow characteristics in the direct vicinity of the boundary
walls can be significantly different depending on the end conditions. This is
reflected by the distinct behaviors of the sectional drag coefficients as
shown in figure 5. With nonslip wall, the sectional drag is low on the $z=0$
plane and then increases up to $z\approx 1$. In the case of slip wall
condition, the $C_{d}$ is high near the end, and decrease monotonically in the
spanwise direction. Despite the difference near the wall, the flows far away
from the end boundaries feature oblique vortex shedding. In view of this, we
only adopt the slip boundary condition in studying the wake dynamics of
inclined cylinders at higher Reynolds numbers, as will be discussed in §3.3.

Figure 6: Time averaged $z$-component vorticity (left), $x$-component
vorticity (middle), and $z$-component velocity (right) fields for parallel and
oblique (with nonslip boundary condition) vortex shedding at $\Rey=100$. $(a)$
$\beta=0^{\circ}$, $(b)$ $\beta=15^{\circ}$ and $(c)$ $\beta=30^{\circ}$.

We further assess the similarity of the flows in parallel and oblique vortex
shedding from the time-averaged fields shown in figure 6. From the
$\overline{\omega_{z}}$ fields viewed on the $x$-$y$ plane, the flows are
characterized by a pair of counter-rotating vortex cores, and not much
difference is discerned in the mean flow between parallel and oblique
shedding. This observation is in accordance with Mittal & Sidharth (2014), in
which the authors reported that the linear stability modes of parallel and
oblique shedding of straight cylinders are similar. However, the oblique
shedding is associated with noticeably stronger streamwise vorticity
$\omega_{x}$, and is also related to intensified spanwise flow, as depicted in
the $\overline{u_{z}}$ fields. The steep gradient of $u_{z}$ in the $y$
direction contributes to stronger streamwise vorticity component, and produces
a slanted vorticity vector (with respect to the $z$-component) leading to
oblique shedding.

####  3.1.2 Spatio-temporal periodicity of the oblique vortex shedding

The oblique vortex shedding is associated with periodic flow-induced forces in
both space and time. We present the instantaneous sectional lift coefficients
along the span in figure 7. For the spanwise regions occupied by the oblique
vortex shedding, the sectional lift exhibits clear spatial periodicity that is
characterized by the wavelength $\lambda$. In general, larger oblique angle is
associated with smaller wavelength, and vice versa. Within each wavelength,
the positive and negative lift coefficients cancel out, resulting in much
weaker fluctuations in the integrated forces on the long cylinder. The spatial
wavelengths are presented in table 3 as a summary.

Figure 7: Instantaneous sectional lift coefficients along the cylinder span. $\lambda$ denotes the spanwise wavelength of the oblique shedding. | b. c. |  $\theta$ |  $\lambda$ |  $St$ |  $St_{\rm{p},0}\cdot\cos(\beta+\theta)$
---|---|---|---|---|---
$\beta=0^{\circ}$ |  slip |  $0^{\circ}$ |  $\infty$ | 0.167 | 0.167
nonslip |  $12.5^{\circ}$ | 24.5 | 0.163 | 0.163
$\beta=15^{\circ}$ |  cyclic |  $0^{\circ}$ |  $\infty$ | 0.160 | 0.161
nonslip |  $23^{\circ}$ | 12.8 | 0.132 | 0.132
slip |  $15.2^{\circ}$ | 18.6 | 0.145 | 0.144
$\beta=30^{\circ}$ |  cyclic |  $0^{\circ}$ |  $\infty$ | 0.143 | 0.145
nonslip |  $18^{\circ}$ | 14.5 | 0.110 | 0.112
slip |  $18^{\circ}$ | 14.5 | 0.110 | 0.112
Table 3: Spatio-temporal periodicity of oblique shedding at $\Rey=100$.
$\lambda$ is the spatial wavelength, and $St$ is the shedding frequency
measured from DNS. $St_{p,0}=0.167$ is the Strouhal number for parallel
shedding at $(Re,\beta)=(100,0^{\circ})$.

Temporally, the oblique shedding is characterized by the Strouhal number
$St=fD/U_{\infty}$, as also presented in table 3. In both cases of parallel
shedding of inclined cylinders (Van Atta, 1968; Lucor & Karniadakis, 2003),
and oblique shedding of straight cylinders (Williamson, 1989; Mittal et al.,
2021), the shedding frequency can be predicted by
$St=St_{p,0}\cdot\cos\gamma$, where $St_{p,0}$ is the Strouhal number for
parallel shedding of straight cylinders, and $\gamma$ represents either the
inclination angle in the former case, or oblique angle in the latter case.
However, it remains elusive how the cosine law can be applied to the case of
oblique shedding in the wake of inclined cylinders. We show in table 3 that
the using the combined angle $\beta+\theta$ with $St_{p,0}$, the predicted
frequencies by cosine law are in good agreement with the Strouhal numbers
measured from DNS. This finding shows that the independence principle can be
generalized to the oblique shedding in the inclined cylinders, and will prove
useful in interpreting results at higher Reynolds numbers in §3.3.

###  3.2 The intrinsic three dimensionality

Figure 8: Flows over circular cylinders with homogeneous end condition at
$\Rey=200$ (left) and 300 (right). ($a$) $\beta=0^{\circ}$, ($b$)
$\beta=15^{\circ}$ and ($c$) $\beta=30^{\circ}$. Shown are isosurfaces of
$QD^{2}/U_{\infty}=0.1$. Figure 9: Spatio-temporal distribution of sectional
lift coefficients $C_{l}$ at $\Rey=200$ for ($a$) $\beta=0^{\circ}$, ($b$)
$\beta=15^{\circ}$ , ($c$) $\beta=30^{\circ}$. Blue and red colors represent
negative and positive $C_{l}$ in the range of $[-1,1]$. ($d$) shows the
averaged spanwise correlation coefficients of lift coefficients.

At $\Rey=200$ and 300, secondary instabilities develop in the cylinder wake,
giving rise to the intrinsically three-dimensional flows. The wake vortical
structures of spanwise homogeneous cylinders at these two Reynolds numbers are
shown in figure 8 for different inclination angles. For $\Rey=200$, the wakes
of straight and inclined cylinders are featured by vortex shedding with large-
scale dislocations. The occurrence of vortex dislocations induces locally bent
vortex tubes, however, the Kármán vortex shedding are mostly parallel to the
cylinder axis regardless of the inclination angle. At $\Rey=300$, the flows
are characterized by coherent Kármán vortex shedding interlaced with small-
scale vortices that are mostly aligned perpendicular to the spanwise vortex
tubes. These features are in accordance with the descriptions of mode A* and
mode B instabilities as described extensively in previous studies (Williamson,
1996a, b; Braza et al., 2001; Jiang et al., 2016).

To further assess the intrinsic three dimensionality in the cylinder wake, we
present the spatio-temporal distributions of the sectional lift coefficients
in figure 9($a-c$) for $\Rey=200$ with different inclination angles. For the
straight cylinder, the spanwise coherence of vortex shedding is disrupted by
the random occurrence of large-scale vortex dislocations, as represented by
the white dots in figure 9($a$). The vortex dislocations are generally compact
in space, but can developed over an extended range of time. For
$\beta=15^{\circ}$ and $30^{\circ}$, the vortex dislocations extend not only
temporally, but also spatially, as evidenced by the stretched white threads in
figure 9($b,c$). This observation indicates that the points of dislocations
travel along the span, due to the spanwise flow in the inclined cases.

For spanwise-homogenous cases, the three dimensionality of the cylinder wake
can be quantified using the spanwise correlation coefficient of the sectional
lift coefficients, which is defined as

$r(z_{i},z_{j})=\frac{\overline{(C_{l,i}-\overline{C_{l,i}})(C_{l,j}-\overline{C_{l,j}})}}{\sqrt{\overline{(C_{l,i}-\overline{C_{l,i}})^{2}}}\sqrt{\overline{(C_{l,j}-\overline{C_{l,j}})^{2}}}}.$
(2)

In this matrix, the elements in each of the diagonals parallel to the leading
diagonal contains the correlation between two spanwise locations separated by
the same distance $s$. By averaging the elements in each diagonal, the
information in the correlation coefficient matrix (2) is condensed to a single
curve as a function of $s$. The averaged spanwise correlation coefficients $r$
for the sectional lift coefficients are shown in figure 9($d$). Due to the
occurrence of large-scale vortex dislocations at $\Rey=200$, the spanwise
vortex shedding loses coherence over small distance. The spanwise correlation
decreases more dramatically with $s$ for the inclined cases, due to the
spanwise velocity component that prolongs the vortex dislocation process along
the cylinder axis. For $\Rey=300$, the spanwise correlation remains much
higher than that at $\Rey=200$, suggesting that the intrinsically developed
large-scale vortex dislocation is not prevalent. For this reason, the spatio-
temporal distributions of the sectional lift at $\Rey=300$ are not presented
here for brevity.

###  3.3 Combined effects of extrinsic and intrinsic three dimensionality

####  3.3.1 Boundary effects at $\Rey=200$

Figure 10: Wake vortical structures for cylinders with inhomogeneous end
conditions at $\Rey=200$. $(a)$ $\beta=0^{\circ}$, ($b$) $\beta=15^{\circ}$
and ($c$) $\beta=30^{\circ}$. Shown are isosurfaces of
$QD^{2}/U_{\infty}=0.1$. $(d)$ and $(e)$ show the wake structures of inclined
cylinders at $\Rey=160$ for $\beta=15^{\circ}$ and $30^{\circ}$, respectively.
Reproduced from Ramberg (1983) with permission from the Cambridge University
Press. Figure 11: Sectional lift coefficients (left) and their amplitude
spectrum for cylinders with inhomogeneous end conditions at $\Rey=200$. $(a)$
$\beta=0^{\circ}$, ($b$) $\beta=15^{\circ}$ and ($c$) $\beta=30^{\circ}$.

In this section, we examine the flows over inclined cylinders at $\Rey=200$
and 300 to evaluate the roles of both extrinsic and intrinsic three
dimensionality in shaping the wake dynamics. The wake vortical structures for
$\Rey=200$ with inhomogeneous end conditions are shown in figure 10. The
corresponding spatio-temporal distributions of sectional lift coefficients and
their spectrum are shown in figure 11. For the straight cylinder, the effects
of nonslip boundary condition are localized near the nonslip boundary, where a
layer of braid-like vortices with a low shedding frequency is observed. The
remaining part of the span features spanwise vortex tubes interlaced with
large-scale vortex dislocation at a primary frequency of $St\approx 0.185$,
which is typical of the mode A* wake as described in §3.2. This differs from
the scenario at $\Rey=100$ where the boundary effects are felt all over the
span by inducing the oblique shedding. This observation suggests that the
occurrence of vortex dislocation in mode A* is able to preclude the
propagation of the boundary effects to further span. Similar observations were
reported by Prasad & Williamson (1997), in which the authors reported loss of
flow control efficacy using end plates on the cylinder wake at $\Rey\approx
190-250$.

The wake dynamics for $\beta=15^{\circ}$ with inhomogeneous end boundary
condition at $\Rey=200$ exhibit rich features along the span. The immediate
vicinity of the upstream boundary is featured by a region of steady flow, and
is followed by the braid-like vortices up to $z\approx 5$. For $5<z\lesssim
35$, organized oblique vortex shedding that is similar to the analogous case
at $\Rey=100$ is observed. These oblique vortices exhibit intact cores with
slight waviness, but no vortex dislocation is developed. The shedding
frequency of these oblique vortices is $St=0.150$ as shown in figure 11($b$).
It is well acknowledged that the Strouhal numbers of the cylinder wake at
$\Rey=200$ are different in 2D and 3D simulations (Williamson, 1988; Jiang et
al., 2016). The 2D simulation suppresses vortex dislocation, and results in a
shedding frequency of $St_{2D}=0.196$, while the 3D simulation yields
$St_{3D}=0.185$, as illustrated in figure 2. With the oblique angle
$\theta=25.2^{\circ}$ at the inclination of $\beta=15^{\circ}$, the
application of the cosine law using the two- and three-dimensional shedding
frequencies gives $St_{2D}\cdot\cos(\theta+\beta)=0.150$ and
$St_{3D}\cdot\cos(\theta+\beta)=0.141$, the former being in close agreement
with the observed shedding frequency in figure 11($b$). This suggests that the
oblique shedding vortices observed here are related to the two-dimensional
vortex shedding without the occurrence of vortex dislocation.

Towards the end of the oblique vortices, the vortex tubes break down with the
formation of small-scale vortices that are mostly perpendicular to the oblique
vortex cores. This is reflected as more noticeable wavy patterns in the
spatio-temporal lift distribution in $30\lesssim z\lesssim 60$, as shown in
figure 11($b$). This type of flow is reminiscent of the mode A wake with
organized long-wavelength streamwise vortices, as described in (Zhao et al.,
2013; Jiang et al., 2016). Such mode-A-like wake develops from purely two-
dimensional Kármán vortex shedding, and exists only for a finite period of
time before it transitions to the mode A* wake with large-scale vortex
dislocation. In the wake of the inclined cylinder, such mode-A-like flow also
occupies limited spanwise region. From $z\approx 60$, the mode-A-like vortices
destabilizes, and the conventional parallel vortex shedding with large-scale
vortex dislocation dominates the wake, indicating the termination of end
effects. The development of the flow features in the spanwise direction, i.e.,
from the oblique vortex shedding with intact vortex cores, to the appearance
of the mode-A-like flows, and finally to the mode A* flow along the cylinder
span, presents another case of temporal-like flow evolution as previous
described with regards to figure 4.

Further increasing the inclination angle to $\beta=30^{\circ}$, the immediate
vicinity of the upstream boundary features steady flow up to $z\approx 20$.
The organized oblique vortices are angled at $\theta=30^{\circ}$ to the
cylinder axis, and shed at a frequency of 0.094, which is close to the
prediction of the cosine law using the 2D frequency
$St_{2D}\cdot\cos(\beta+\theta)=0.098$. These oblique vortices break down at
$z\approx 65$, and the flow transitions to parallel vortex shedding with
large-scale vortex dislocation. In contrast to the scenario in
$\beta=15^{\circ}$ where the mode-A-like flow is observed in between the
organized oblique vortices and parallel vortices, the transition between these
two flows appear more direct. The general flow features revealed in the
current direct numerical simulations are in good agreement with those observed
in the early experiments by Ramberg (1983), as shown in figure 10($d,e$). The
combined effects of extrinsic inhomogeneous end boundary and intrinsic
secondary instability produce highly complex flow physics involving oblique
shedding, parallel shedding, vortex dislocation, cellular shedding, etc., all
in a single wake.

####  3.3.2 Boundary effects at $\Rey=300$

Figure 12: Wake vortical structures for cylinders with inhomogeneous end
conditions at $\Rey=300$. $(a)$ $\beta=0^{\circ}$, ($b$) $\beta=15^{\circ}$
and ($c$) $\beta=30^{\circ}$. Shown are isosurfaces of
$QD^{2}/U_{\infty}=0.1$.

The wake vortical structures of the inclined cylinders at $\Rey=300$ are
presented in figure 12, with the corresponding spatio-temporal lift
distributions and lift spectra shown in figure 13. For $\beta=0^{\circ}$, the
flow generally resembles the mode B wake as described in figure 8, and the
oblique vortex shedding that is observed at lower Reynolds numbers does not
present a persistent feature at $\Rey=300$. Along the span, the coherence of
the spanwise vortex tubes is disrupted by several instances of vortex
dislocations (e.g., $z\approx 25$, 35, 45 at $t=660$). Noticing that the
large-scale vortex dislocation is a rare event in the spanwise homogeneous
case as discussed in §3.2, the dislocations observed in the current case
should be considered as induced by the inhomogeneous boundary condition.
Indeed, from figure 13$(a)$, we observe that the occurrence of dislocations
are mostly originated from the region near the nonslip boundary, and slowly
moves inwards along the span. For the simulated range of time, the propagation
of the vortex dislocation is limited to $z\lesssim 50$, beyond which the
vortex shedding is generally parallel to the cylinder axis.

Figure 13: Sectional lift coefficients for cylinders at $\Rey=300$. $(a)$
$\beta=0^{\circ}$ with nonslip end condition, ($b$) $\beta=15^{\circ}$ with
slip end condition and ($c$) $\beta=30^{\circ}$ with slip end condition. Blue
and red indicate positive and negative values ($C_{l}\in[-1,1]$),
respectively.

Increasing the inclination to $\beta=15^{\circ}$, the vortex shedding near the
inhomogeneous boundary is slightly obliqued to the cylinder axis. This is
reflected by the smaller shedding frequency of $St\approx 0.176$ at
$0<z\lesssim 20$ compared with the rest of the span. Similar to the scenario
at $\beta=0^{\circ}$, the disturbances generated near the boundary generate
vortex dislocations that travel along the span. With the spanwise flow behind
the inclined cylinder, the dislocations propagate at a faster rate compared to
the case of $\beta=0^{\circ}$, and in some cases they can penetrate a larger
region in the spanwise direction. The vortex dislocations also result in
broadband lift spectrum at $20\lesssim z\lesssim 70$. Beyond $z\approx 80$,
parallel vortex shedding associated with a primary frequency of $St\approx
0.192$ becomes dominant.

At $\beta=30^{\circ}$, the direct vicinity of the upstream end boundary is
occupied by steady flow, although this region is much shorter compared to the
analogous cases at $\Rey=100$ and 200. The rest of the span features oblique
vortex shedding with $\theta\approx 10^{\circ}$ and a primary frequency of
$St=0.158$. Towards the downstream end of the span, the lift spectra gradually
becomes broadband. However, no clear occurrence of vortex dislocation is
observed in the flow. We speculate that the lack of vortex dislocation at
$\beta=30^{\circ}$ is due to the steady flow formed near $z=0$ plane. Without
the vibrant vortical activities near the end boundary triggering vortex
dislocations as observed in the cases of $\beta=0^{\circ}$ and $15^{\circ}$,
the boundary effects at $\beta=30^{\circ}$ manifest as oblique shedding
similar to the scenario at $\Rey=100$.

##  4 Conclusions

We have carried out direct numerical simulations to study the wake dynamics of
inclined long circular cylinders with spanwise-inhomogeneous boundary
conditions at $\Rey=100$, 200 and 300. The aim is to clarify the different
roles of the end boundary conditions, the intrinsic secondary instabilities,
and the inclination in the formation of the three-dimensional vortex shedding
behind the cylinders.

At $\Rey=100$ where the intrinsic three dimensionality is absent, the effects
of the inhomongeous end boundary are felt over the entire span, with the
formation of oblique vortex shedding for both straight and inclined cylinders.
Compared with parallel shedding, the oblique shedding is associated with
stronger spanwise velocity, which creates additional streamwise vorticity that
results in oblique angle. The oblique shedding features periodicity in both
space and time. Along the span, the positive and negative sectional lift
coefficients cancel out within one wavelength, resulting in much smaller
fluctuating lift over the entire cylinder. Temporally, we found that the
frequency of the oblique shedding is related to that of the parallel shedding
of straight cylinder by a factor of $\cos(\beta+\theta)$, where $\beta$ is the
inclination angle and $\theta$ is the oblique angle. This observation
indicates that the cosine law can be generalized to the oblique shedding in
inclined cylinders.

At $\Rey=200$, the secondary instability develops in the cylinder wake as
large-scale vortex dislocations, resulting in intrisically three-dimensional
flow. For the straight cylinder with nonslip end condition, the occurrence of
vortex dislocations precludes the propagation of the boundary effects towards
further span. For inclined cylinders, the inhomomgenous end boundary induces
oblique shedding within a finite region along the span. These oblique vortices
are related to the intrinsically two-dimensional flows at $\Rey=200$ without
large-scale vortex dislocations. This is evidenced by the observation that the
Strouhal number of the oblique shedding is in close agreement with that
predicted by the cosine law, if the two-dimensional shedding frequency (with
secondary instabilities suppressed) is used. Further along the span, the
intact oblique vortex cores destabilize with the formation of small-scale
vortices, and the flow transitions to the typical mode A* wake.

At $\Rey=300$, the spanwise-homogeneous cylinder features the mode B wake with
stable Kármán vortex shedding interlaced with fine-scale streamwise vortices.
With inhomogeneous end boundary conditions, the oblique shedding that is
prevalent at lower Reynolds numbers does not present a persistent feature for
cases with small inclination angle. The three-dimensional unsteady flow near
the end boundary creates disturbances that propagate along the span, and later
develops into vortex dislocation. For high inclination angle, the oblique
vortex shedding is again observed throughout the cylinder span, and is not
disrupted by vortex dislocations of either intrinsic or extrinsic cause.

The above analysis have highlighted the complex interactions between the
effects of extrinsic and intrinsic three dimensionality in forming the
cylinder wake. For flows without intrinsic instabilities, the extrinsic end
effects propagate along the cylinder span unchallenged, resulting in oblique
vortex shedding for cylinders even hundreds diameters in length. The
propagation of the end boundary effects can be terminated by the occurrence of
large-scale vortex dislocations that develop intrinsically in the flow, giving
rise to end-effects-free flow. At higher Reynolds numbers, while the intrinsic
three dimensionality in the form of mode B wake rarely creates vortex
dislocations, the highly three-dimensional flow near the inhomogeneous
boundary can be a source of perturbations that later develop into vortex
dislocations. The present study provides new insights into the three-
dimensional wake dynamics of inclined cylinders. The knowledge obtained here
is fundamental to the understanding of higher-Reynolds-number flow and fluid-
structure interactions of flexible bodies. It also lays the foundation for the
designs of large length-to-diameter-ratio engineering structures and related
flow control techniques.

## Appendix I: Cosine law for flow over inclined two-dimensional cylinders

Figure 14: Comparison of the Strouhal number between two-dimensional DNS and
the cosine law.

In the current study, the flow over inclined cylinder is simulated by shearing
the computational domain with fixed incoming velocity. As a result, the
velocity component that is normal to the cylinder axis “sees” the cross-
sectional shape of the cylinder as an ellipse. This is different from some of
the previous studies, which employed a setup where the velocity direction is
altered to change the inclination angle, such that the velocity component that
is normal to the cylinder axis still “sees” a circular shape (Zhao et al.,
2013; Zhang et al., 2018). To assess the validity of the independence
principle in the current setup, we carry out two-dimensional simulations (with
the intrinsic secondary instabilities suppressed) of flow over inclined
cylinder, and compare the results with the cosine law. As shown in figure 14,
the results from the 2D DNS are in good agreement with those predicted by the
independence principle, particularly at low inclination angles. At higher
$\beta$, the cosine law slightly overpredicts and underpredicts the shedding
frequencies for $\Rey=100$ and 300, respectively. We conclude that the
independence principle can be used for estimating the shedding frequencies of
inclined cylinders with the computational setup adopted in this study.

## Acknowledgement

KZ is grateful for the Office of Advanced Research Computing (OARC) at
Rutgers, The State University of New Jersey for providing access to the Amarel
cluster, on which some of the simulations were performed. DZ thanks the
support from Program for Intergovernmental International S&T Cooperation
Projects of Shanghai Municipality, China (grant no. 22160710200). ZLH
acknowledges the financial support from National Science Foundation of China
(grant no. 52122110).

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