∎

11institutetext: J. S. Park 22institutetext: Department of Mechanical and
Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526,
USA 22email<EMAIL_ADDRESS>
# On the underlying drag-reduction mechanisms of flow-control strategies in a
transitional channel flow: temporal approach

Alexander J. Rogge  Jae Sung Park

(Received: date / Accepted: date)

###### Abstract

The underlying mechanisms of three different flow-control strategies on drag
reduction in a channel flow are investigated by direct numerical simulations
at friction Reynolds numbers ranging from 65 to 85. These strategies include
the addition of long-chain polymers, the incorporation of slip surfaces, and
the application of an external body force. While it has been believed that
such methods lead to a skin-friction reduction by controlling near-wall flow
structures, the underlying mechanisms at play are still not as clear. In this
study, a temporal analysis is employed to elucidate underlying drag-reduction
mechanisms among these methods. The analysis is based on the lifetime of
intermittent phases represented by the active and hibernating phases of a
minimal turbulent channel flow (Xi & Graham, Phy. Rev. Lett. 2010). At a
similar amount of drag reduction, the polymer and slip methods show a similar
mechanism, while the body force method is different. The polymers and slip
surfaces cause hibernating phases to happen more frequently, while the
duration of active phases is decreased. However, the body forces cause
hibernating phases to happen less frequently but prolong its duration to
achieve a comparable amount of drag reduction. A possible mechanism behind the
body force method is associated with its unique roller-like vortical
structures formed near the wall. These structures appear to prevent
interactions between inner and outer regions by which hibernating phases are
prolonged. It should motivate adaptive flow-control strategies to exploit the
distinct underlying mechanisms for robust control of turbulent drag at low
Reynolds numbers.

###### Keywords:

Flow control Drag-reduction mechanism Direct Numerical Simulation

##  1 introduction

The presence of coherent structures in wall-bounded turbulent flows plays a
crucial role in turbulent dynamics RobinsonARFM1991 . Near the wall, these
structures are closely related to the self-sustaining process of turbulence as
they are very similar to staggered, counter-rotating quasi-streamwise
vortices, forming low- and high-speed streaks Waleffe1997 . In particular,
these near-wall structures are responsible for the production of turbulent
kinetic energy as they are observed to burst in a very intermittent fashion
Hamilton1995 . Since the bursting process appears to account for over 80% of
the energy in turbulent fluctuations, these near-wall coherent structures are
believed to be the dominant structures associated with the turbulence
production, leading to an increase in skin-friction drag Lumley1998 . Thus,
various control strategies have been exploited to manipulate the near-wall
coherent structures to achieve a significant drag reduction Gad2007 . Although
the gross effects and structural understanding of drag-reduction mechanisms of
the flow-control strategies have been well-documented, a temporal analysis on
elucidating the underlying drag-reduction mechanisms of the control strategies
has yet to be explored until now.

Here, we aim to provide a brief description of the control strategies that
have been explored along with their implications in drag-reduction mechanisms
derived by such strategies. In general, the control strategies are classified
as passive or active, depending on whether it requires actuation or external
energy source Gad2007 . For passive control, one of the most successful
strategies involves using riblets or microgrooves installed on the wall and
aligned in the streamwise direction Choi1993 , Jimenez2004 . Drag reduction
varies depending on the height and alignment of the riblets, resulting in up
to 10% drag reduction. The riblets enforce the streamwise vortices away from
the wall by which the amount of shear stress in the near-wall region is
significantly reduced Choi1993 , Garcia2011 . Another substantially-studied
strategy is via slip by placing hydrophobic surfaces at the walls Luchini1991
, Watanabe1999 , Min2004b . Recently, superhydrophobic surfaces, which are a
combination of surface roughness and surface chemistry at the micro- and/or
nano-scales, have been utilized to produce an effective slip length on the
wall RothsteinARFM2010 , Fukagata2006 . It is viewed that the effective slip
length must be on the same order as the viscous sublayer to alter the
streamwise velocity and subsequently the wall shear stress fluctuations
Park2013d . A reduction in the wall shear stress results in the weakening of
the streamwise vortices and streaks, which subdues the streamwise momentum to
move away from the wall, which is also known as the lift-up mechanism. Since
the lift-up mechanism is reduced by the weakening of the wall shear stress,
the transient growth of perturbations is also reduced Chai2019 . Another most
successful strategy involves the addition of a small amount of long-chain
polymers to a liquid Berman1978 . The polymers are likely to store elastic
energy and release it back into the flow as it travels around the buffer and
log-law layers, whereby the streamwise vortices are suppressed White2008 ,
Graham2014 . The vortices tend to remain closer to the wall but become
elongated in the direction of the flow. A substantial drag reduction up to 80%
can be achieved, yielding a much higher flow rate at a given pressure drop
Graham2014 . For active control, one of the most practical strategies involves
active wall motion, which creates streamwise ridges whose appearance is very
similar to that of riblets Choi2002 , Quadrio2004 , Kang2000 . This wall
motion also attempts to push the high-speed fluid away from the wall to obtain
drag reduction Endo2000 . Another popular strategy is the application of
blowing and suction at the wall, which is equal or opposite to the wall-normal
velocity close to the wall Choi1994 . This control strategy, also known as
opposition control, resulted in a drag reduction of nearly 25%. The opposition
control reduces the spinning of the streamwise vortices and in turn stabilizes
them in space, leading to a reduction of the bursting frequency Coller1994 .
Another method of active control includes the application of external body
forces to the flow, which can be applied to both gases and liquids
Karniadakis2003 , Berger2000 , Wang2013 . The spanwise external body forces
appear to stabilize the low-speed streaks and weaken streamwise vortices,
leading to the weakening of the bursting events. As a summary, almost all of
the various control strategies appear to suggest that their underlying drag-
reduction mechanisms are to manipulate near-wall coherent structures, such as
streamwise vortices or streaks near the wall.

Besides the structural drag-reduction mechanisms, there have been very limited
research to compare different flow-control strategies to elucidate different
or similar drag-reduction mechanisms. Very recently, Chen, Yao, & Hussain have
utilized the energy-box analysis Gatti2018  to compare the three control
methods of the spanwise opposed wall-jet forcing (SOJF), spanwise wall
oscillation (SWO), and opposed wall blowing/suction (OBS) at a friction
Reynolds number of 200 and drag reduction of approximately 20% Chen2021 . They
compared the contributions of the mean, coherent, and random turbulent
dissipations to the overall drag reduction and net power savings. It was found
that for the SOJF method, the coherent dissipation is much smaller than the
other two dissipations, while for the SWO method, the coherent value is on the
same scale as the other two values and much larger than SOJF‘s value. For the
OBS method, the random turbulent dissipation is suppressed without the
appearance of the coherent dissipation since the energy is only introduced
randomly. In addition to the energy-box analysis, it should be noted that
there are other approaches to connect the drag to certain flow quantities,
such as the Fukagata-Iwamoto-Kasagi (FIK) identity Fukagata2002  and the
Renard-Deck (RD) identity Renard2016 . A few examples of exploiting the FIK
identity for different flow control methods are wall deformation Tomiyama2013
, external body forces Mamori2014 , superhydrophobic surfaces Lee2015 , and
blowing/suction Kametani2015 , among others. Although these approaches provide
quantitatively meaningful information about drag-reduction mechanisms of
different flow-control methods, there is still demand to better understand
underlying drag-reduction mechanisms of different flow-control strategies via
distinct approaches.

In addition to turbulent flow control for drag reduction, a wall-bounded
turbulent flow itself exhibits substantial intermittency between high- and
low-drag states. Particularly relevant in this regard is the study by Xi &
Graham Xi2010 , where a direct numerical simulation (DNS) was performed for a
minimal turbulent channel flow at a friction Reynolds number of 85. Turbulent
dynamics are observed to cycle intermittently between “active” intervals with
strong streamwise vortices and “hibernating” intervals with very small
Reynolds shear stress. Similar observations have also been made by Hamilton,
Kim, & Waleffe Hamilton1995 . This temporal intermittency is also found to
reflect the organization of the turbulent dynamics around the exact coherent
states or nonlinear traveling-wave (TW) solutions to the Navier-Stokes
equations Park2015jfm , Park2018 , Graham2021 . Indeed, the hibernating
intervals are approaches to lower-branch TW solutions, while the active
intervals are close to an upper-branch TW solution. Very recently, this low-
and high-drag intermittency is observed and quantified in experiments and in
good agreement with DNS results Rishav2020 . It is worth noting that the
temporal and spatial analyses on the relationship between temporal dynamics in
minimal domains and spatiotemporal dynamics in extended domains also yield
very similar results for the hibernating and active intervals Kushwaha2017  at
friction Reynolds numbers ranging from 70 to 100. While the connection between
the temporal intermittency and drag-reduction mechanisms has been identified
for viscoelastic turbulent flows at low Reynolds numbers Graham2014 , it has
yet to be fully explored for other control strategies and will be investigated
in the present study. Along with the polymer method, the current study
examines the slip surface method (passive control) and external body force
method (active control). The description of such a connection would provide a
basis for a deeper understanding of underlying drag-reduction mechanisms
embedded in different flow-control strategies.

In this paper, we use direct numerical simulations and temporal turbulent
phases to elucidate the underlying mechanisms of drag reduction via three
control strategies, namely the application of polymer additives, slip
surfaces, and external body forces. The problem formulation is reported in
Section 2. The simulation results are presented in Section 3, where the
effects of the control strategies on the temporal intermittency at the
different levels of drag reduction are presented. A summary of main results,
conclusions, and future directions are presented in Section 4.

##  2 Problem Formulation

We consider an incompressible fluid in a turbulent channel flow (plane
Poiseuille) geometry, driven by a constant volumetric flux $Q$. The $x$, $y$,
and $z$ coordinates are aligned with the streamwise, wall-normal, and spanwise
directions, respectively. Periodic boundary conditions are imposed in the $x$
and $z$ directions with fundamental periods $L_{x}$ and $L_{z}$, and solid
walls are placed at $y=\pm h$, where $h$ is the half-channel height. The
laminar centerline velocity for a given volumetric flux is given as
$U_{c}=(3/4)Q/h$. Using the half-height $h$ of the channel and the laminar
centerline velocity $U_{c}$ as the characteristic length and velocity scales,
respectively, the nondimensionalized Navier-Stokes equations for a fluid
velocity $\boldsymbol{u}$ and pressure $p$ are then given as

$\nabla\cdot\boldsymbol{u}=0,\quad\frac{\partial\boldsymbol{u}}{\partial
t}+\boldsymbol{u}\cdot\nabla\boldsymbol{u}=-\nabla
p+\frac{\beta}{Re_{c}}\nabla^{2}\boldsymbol{u}+\boldsymbol{f}_{\textrm{ext}}.$
(1)

Here, we define the Reynolds number for the given laminar centerline velocity
as $Re_{c}=U_{c}h/\nu$, where $\nu$ is the kinematic viscosity of the fluid
and $\beta$ is the ratio of the solvent viscosity and the total viscosity (for
a Newtonian fluid, $\beta=1$), and $\boldsymbol{f}_{\textrm{ext}}$ is the
external force, which can result from a body force or polymer stress in the
present study.

For viscoelastic flows, the momentum equation in Equation (1) includes an
external force from the polymer stress
$\boldsymbol{f}_{\textrm{ext}}=(1-\beta)\nabla\cdot\boldsymbol{\tau}_{p}$,
where the polymer stress tensor $\boldsymbol{\tau}_{p}$ is related to the
polymer conformation tensor $\boldsymbol{\alpha}$. This tensor is then
expressed through the FENE-P constitutive relation based on bead-spring
dumbbells. These polymer conformation and stress tensors are obtained by
solving the following equations:

$\displaystyle\frac{\partial\boldsymbol{\alpha}}{\partial
t}+\boldsymbol{u}\cdot\nabla\boldsymbol{\alpha}-\boldsymbol{\alpha}\cdot\nabla\boldsymbol{u}-(\boldsymbol{\alpha}\cdot\nabla\boldsymbol{u})^{\textrm{T}}=-\frac{1}{Wi}\boldsymbol{\tau}_{p},\quad\boldsymbol{\tau}_{p}=\frac{\boldsymbol{\alpha}}{1-\textrm{tr}(\boldsymbol{\alpha})/b}-\textbf{{I}}.$
(2)

Here, we define the Weissenberg number $Wi=\lambda U_{c}/h$, where $\lambda$
is the polymer relaxation time, and $b$ is the maximum extensibility of the
polymers. For the current study, we fix $\beta=0.97$ and $b=10,000$. Since
$1-\beta$ is proportional to polymer concentration and $b$ to the number of
monomer units, this parameter set corresponds to a dilute solution of a high-
molecular-weight polymer. For slip surfaces, streamwise Navier slip conditions
$u_{s}=L_{s}\dot{\gamma}_{w}$ are applied at both top and bottom walls, where
$L_{s}$ is an effective homogeneous slip length and $\dot{\gamma}_{w}$ is the
shear rate at the wall. For an external body force, the following spanwise
body force is used for the external force term in equation (1):

$\displaystyle
f_{z}=I\textrm{e}^{-y/\Delta}\textrm{sin}\left(\frac{2\pi}{\lambda_{z}}z-\frac{2\pi}{T}t\right),$
(3)

where $I$ is the amplitude of excitation, $\Delta$ is the penetration depth,
and $\lambda_{z}$ and $T$ are the wavelength and period of oscillation,
respectively. For the present simulations, we vary $I$ and $T$, while fixing
$\Delta=0.03$ and $\lambda_{z}=L_{z}/2=\pi/2$.

For inner units, characteristic inner scales are the friction velocity
$u_{\tau}=(\bar{\tau}_{w}/\rho)^{1/2}$ and the near-wall length scale or wall
unit $\delta_{\nu}=\nu/u_{\tau}$, where $\rho$ is the fluid density and
$\bar{\tau}_{w}$ is the time- and area-averaged wall shear stress. As usual,
quantities nondimensionalized by these inner scales are denoted with a
superscript “+”. The friction Reynolds number is then defined as
$Re_{\tau}=u_{\tau}h/\nu=h/\delta_{\nu}$.

Simulations are performed using the open-source code ChannelFlow written and
maintained by Gibson channelflow  from which a modified version was made for
the three different control methods used in the current study. In this study,
we focus on the domains of $L_{x}\times L_{y}\times L_{z}=2\pi\times
2\times\pi$, utilizing the minimal flow unit (MFU) approach Jimenez1991 .
Computational verification of direct numerical simulations for MFUs has been
tested up to $Re_{\tau}=1000$ Davis2021 . The minimum spanwise length scale
used here is about 205$\delta_{\nu}$, which is larger than the length scale of
the near-wall streak spacing of about $100\delta_{\nu}$ Smith1983 . In
particular. a grid convergence and domain dependence have been tested in our
previous studies for no-control cases Kushwaha2017 , Rishav2020  and in the
current study for control cases. A numerical grid system is generated on
$N_{x}\times N_{y}\times N_{z}$ (in $x$, $y$, and $z$) meshes, where a
Fourier-Chebyshev-Fourier spectral spatial discretization is applied to all
variables. A typical resolution used is $(N_{x},N_{y},N_{z})=(48,81,48)$. The
numerical grid spacing in the streamwise and spanwise directions are $\Delta
x_{min}^{+}\approx 11.0$, $\Delta z_{min}^{+}\approx 5.5$. The nonuniform
Chebyshev spacing used in the wall-normal direction results in $\Delta
y_{min}^{+}\approx 0.05$ at the wall and $\Delta y_{max}^{+}\approx 2.5$ at
the channel center. For simulations, $Re_{c}=1800$ is being considered, which
gives $Re_{\tau}$ = 85 for no control and slightly lower $Re_{\tau}$ values
for control methods due to a lower value of $u_{\tau}$. Prior to presenting
simulation results, it should be noted that all simulations presented are
sufficiently far above transition and show the sustained turbulence nature of
the flow, yielding the classical mean flow behavior Kushwaha2017 , Whalley2017
, which will be revisited below.

##  3 Results and discussion

###  3.1 Drag reduction by control strategies

Figure 1: Drag reduction percentages (DR%) due to ($a$) polymer additives,
($b$) slip surfaces, and ($c$) spanwise body forces. The error bars in ($a$)
and ($b$) are computed by the block-averaging method to represent the standard
error of the time-averaged quantity Flyvbjerg1989 .

Figure 1 shows the level of drag reduction due to polymer additives (i.e.
viscoelastic flows), slip surfaces, and spanwise body forces. Drag reduction
percentage is defined as DR% = $(C_{f,un}-C_{f})/C_{f,un}\times 100\%$, where
$C_{f}=2\overline{\tau}_{w}/(\rho U^{2}_{b})$ is the skin-friction coefficient
for the controlled case and $C_{f,un}$ is the friction factor for the
uncontrolled case. Here, $\overline{\tau}_{w}$ is the time-area-averaged wall
shear stress and $U_{b}$ is the bulk fluid velocity, which is kept constant in
simulations. The error bars on the plot are the standard errors of the time-
averaged quantity with the block-averaging method Flyvbjerg1989 . Figure
1($a$) shows DR% for viscoelastic turbulence as a function of $Wi$, which is
in good agreement with the previous studies with regard to drag reduction
amounts and onset $Wi$ for drag reduction Xi2010 , Wang2014 . Figure 1($b$)
shows DR% for slip surfaces as a function of $L_{s}$. At a fixed Reynolds
number, the slip length and drag reduction percentage are almost linearly
correlated, which is also observed with the same streamwise-only slip
condition of Min and Kim Min2004b . It is worth noting that the largest slip
length value is $L_{s}^{+}\approx 5$ for the current study, which ensures that
the homogeneous slip surface employed in the present study would produce
essentially the same outcomes with a heterogeneous microtextured slip surface
or superhydrophobic surface Seo2018 , Picella2019 , Davis2020 , Rowin2019 .
Figure 1($c$) shows DR% for body forces for various values of the amplitude of
excitation ($I$) and the time-period of oscillation ($T$), where we fix
$\Delta=0.03~{}(\Delta^{+}=2.55$) and
$\lambda_{z}=L_{z}/2=\pi/2~{}(\lambda_{z}^{+}=42.5)$. Within parameters
studied, the maximum drag reduction percentage is approximately 25% at
$I=0.55$ and $T=10~{}(T^{+}=40)$ – the values in parentheses are based on no-
control case.

Figure 2: Time series of skin friction coefficients $(C_{f})$ at about ($a$)
7% and ($b$) 25% drag reduction for viscoelastic (green), slip (red), and body
force (blue) methods along with no control (black), its mean value (black
dashed), and laminar (pink dashed).

Before moving forward, it should be again emphasized that the simulations
indeed show characteristics of a sustained turbulence. Figures 2($a$) and
($b$) show a time series of the skin friction coefficients for no control and
three control methods along with laminar value at DR% $\approx$ 7% and 25%,
respectively. The parameters used for the control methods at DR% $\approx$ 7%
include $Wi=20$ for the viscoelastic method, $L_{s}$ = 0.015 for the slip
method, and $I=0.15$ and $T=15$ for body force method. At DR% $\approx$ 25%,
$Wi=31$ for the viscoelastic method, $L_{s}$ = 0.06 for the slip method, and
$I=0.55$ and $T=10$ for the body force method. As shown, the turbulent nature
of the flow is clearly visible, showing the substantial fluctuations, which
are zero in a laminar flow by definition. The skin friction coefficients are
well above the laminar for low drag reduction cases (Fig. 2$a$). Although it
tends to approach the laminar value as drag reduction increases, it is still
above the laminar state and shows noticeable fluctuations even during the
lowest drag periods of DR% $\approx$ 27% at which the maximum DR case is
achieved in the current study with a slip surface. Even in this period, there
are no quasi-laminar regions in the simulation domain that every region always
displays the fluctuations.

###  3.2 Low- and high-degree drag reduction regimes

Prior to proceeding to different drag-reduction regimes, it is worth showing
the mean velocity profiles, which can be used as an indicator for different
levels of drag reduction. Figures 3($a$) and ($b$) show the mean velocity
profiles (uncontrolled and controlled) in inner units at DR% $\approx$ 7% and
25%, respectively. The control parameters are the same as ones for Figure 2.
For comparison, the profiles for the viscous sublayer $U^{+}(y^{+})=y^{+}$ and
the log-law layer $U^{+}=2.5\textrm{ln}(y^{+})+5.5$ are also presented. For
DR% $\approx$ 7%, the polymer and body force methods follow the viscous
sublayer profile well $(y^{+}\leq 5)$, while the slip case starts off with a
greater velocity than the other two methods due to the slip velocity at the
wall. The control profiles begin to slightly deviate from the no-control
profile in the buffer layer at $y^{+}\approx 20$. The no-control profile lies
closer to the log-law profile, but it is placed just above the log-law profile
because of the effects of low Reynolds number Tsukahara2005 . The velocity
profiles of three control methods are elevated beyond the no-control and are
very close to one another, suggesting almost the same lower drag. For DR%
$\approx$ 25%, the mean velocity profiles of the polymer and body force
methods clearly diverge from the log-law slope with a steeper incline. The
slip profile shows similar values with the other two cases in the log-law
layer but with a less steep incline. However, there is a much-increased
velocity at the wall due to higher $L_{s}$.

Figure 3: Time-averaged streamwise mean-velocity profiles for no control
(black), viscoelastic (green), slip (red), and body force (blue) methods at
about ($a$) 7% and ($b$) 25% drag reduction. The black dot-dashed line is the
viscous sublayer, $U^{+}=y^{+}$ and the black dashed line is the log-law
layer, $U^{+}=2.5\textrm{ln}(y^{+})+5.5$.

Regarding the drag reduction percentages, an additional distinction can be
applied to separate a low degree of drag reduction (LDR) and a high degree of
drag reduction (HDR). Warholic et al. analyzed the different effects of
polymers at LDR and HDR regimes through experiments based on the mean velocity
and fluctuation profiles relative to Newtonian turbulent flows Warholic1999 .
This difference has been recently investigated through simulations to show
various ranges for LDR and HDR regimes, where LDR is below 13% drag reduction
and HDR is above 15% drag reduction Zhu2019 . A similar separating point for
the LDR-HDR transition was observed in the range of DR% $\approx$ 20%–30%
Xi2010 . Although some studies reported a higher separating point at DR%
$\approx$ 30%–40% Patasinski2003 , Min2003 , the present drag reduction data
can be divided into the LDR and HDR regimes at DR% $\approx$ 15%–20% based on
the mean profiles, as seen in fig. 3. Thus, the 7% and 25% drag reduction
cases belong indeed to the LDR and HDR regimes, respectively, where the
steeper mean velocity profile slope distinguishes the HDR regime from the LDR
regime.

###  3.3 Temporal analysis on drag-reduction mechanisms

We now aim to describe the underlying drag-reduction mechanisms based on a
temporal characterization of temporal events with varying amounts of drag
relative to the mean. It has been seen that a turbulent flow is observed to
intermittently fluctuate between low and high friction drag. During a low-drag
period, vortical motions are suppressed with less wavy low-speed streaks,
causing low Reynolds shear stress Graham2014 . These low-drag intervals are
termed hibernating turbulence Xi2010 . Periods between the hibernating
intervals are called active turbulence and display high-drag features. The
criteria for hibernating turbulence involve the wall shear stress remaining
below $90\%$ of its mean value for at least three eddy turnover times
$(>\Delta tu_{\tau}/h=3)$ Kushwaha2017 , Rishav2020 . We already addressed the
issue of sensitivity to the chosen values and showed almost identical results
Kushwaha2017 , Park2018 . Note that with the aforementioned criteria for
hibernating turbulence, it can be only detected up to $Re_{\tau}\approx 125$
Whalley2017 .

To quantify the effects of the flow-control methods on the temporal
intermittency of turbulence, the duration and frequency of hibernating and
active turbulence are computed from simulation runs for $t=150000$
$(>80Re_{c})$ for all cases. The average duration of hibernating and active
turbulence and the fraction of time spent in hibernation ($T_{H}$, $T_{A}$,
$F_{H}$, respectively) are calculated as

$\displaystyle
T_{H}=\frac{\sum_{i=1}^{N_{H}}t_{H,{i}}}{N_{H}},\;\;\;T_{A}=\frac{T-\sum_{i=1}^{N_{H}}t_{H,{i}}}{N_{A}},\;\;\;F_{H}=\frac{\sum_{i=1}^{N_{H}}t_{H,{i}}}{T},$
(4)

where $t_{H,i}$ is the duration of the $i$th hibernating interval, and $N_{H}$
and $N_{A}$ are the total number of hibernating and active intervals over the
total duration of the simulation $T$, respectively. Specifically, $F_{H}$ can
be referred to as a temporal intermittency factor for low friction drag
Kushwaha2017 . For temporal investigations, these three quantities $T_{H}$,
$T_{A}$, and $F_{H}$ are considered along with the average number of
hibernations $n_{H}$ over $t=30000$ for the LDR and HDR regimes in comparison
to no-control case.

Figure 4: Temporal data at DR% $\approx$ 7% (LDR): ($a,b$) the average
duration of hibernating and active turbulence, respectively, ($c$) the
fraction of hibernation, and ($d$) the average number of hibernations over
$t=30000$. The results are from viscoelastic ($Wi=20$), slip ($L_{s}$ =
0.015), and body force ($I=0.15,T=15$) cases. The dashed (red) line represents
the no-control case’s values for better visualization of changes that occur.
The error bars represent the standard error.

Figure 4 shows the temporal quantities for a drag reduction of $7\%$ at the
LDR regime. It appears that the polymer and slip methods share similar
characteristics. They show almost the same duration of hibernating intervals
as the no-control case but a decrease in the duration of active intervals.
These changes lead to an increase in the fraction of hibernation and the
number of hibernations compared to the no-control case. These trends are in
good agreement with the previous studies on viscoelastic turbulence at low
$Wi$ Xi2010 , Wang2014 . However, the body force method shows different
behaviours, where the duration of both hibernating and active intervals
increases, while the fraction of hibernation and the average number of
hibernations decrease compared to the no-control values. These trends might
indicate that the body force method could cause a drag increase rather than a
drag reduction of $7\%$ even though $T_{H}$ is larger than the no-control
value. Thus, it is strongly suggested that the body force method is likely to
have a different drag-reduction mechanism compared to the polymer and slip
methods at the LDR regime. Before moving to HDR, it should be noted that given
the almost same drag reduction percentage of 7%, the temporal quantities for
the body force method with $(I=0.15,T=15)$, $(I=0.15,T=20)$, and
$(I=0.2,T=10)$ are almost identical (not shown), indicating the independence
of the temporal analysis the chosen control values for the body force method.

Figure 5: Temporal data at DR% $\approx$ 25% (HDR): ($a,b$) the average
duration of hibernating and active turbulence, respectively, ($c$) the
fraction of hibernation, and ($d$) the average number of hibernations over
$t=30000$. The results are from viscoelastic ($Wi=31$), slip ($L_{s}$ = 0.06),
and body force ($I=0.55,T=10$) cases . The dashed (red) line represents the
no-control case’s values for better visualization of changes that occur. The
error bars represent the standard error.

Figure 5 shows the cases for a drag reduction of $25\%$ at the HDR regime. As
in the LDR regime, the polymer and slip control methods display similar
behaviours. While the trends of $T_{A}$, $F_{H}$, and $n_{H}$ with respect to
the no-control case are similar to the LDR cases, the average duration of
hibernating turbulence does increase and is now larger than the no-control
value. With this increase in $T_{H}$ and the resulting increase in $F_{H}$, a
much higher drag reduction ($\sim$25%) is achieved. For the body force method,
the trend is still similar to its LDR case but shows more noticeable changes
in the quantities. In comparison to the LDR case, $T_{H}$ and $T_{A}$ become
almost doubled, while $F_{H}$ remains almost the same, and $n_{H}$ decreases
slightly. These trends still might indicate that the body force method could
cause a drag increase rather than a drag reduction of $25\%$. Thus, a
different drag-reduction mechanism could be suggested for the body force
method compared to the polymer and slip methods even at the HDR regime.

With the temporal quantities at the LDR and HDR regimes, the underlying drag-
reduction mechanisms can be made for the control methods being investigated.
As seen above, it appears that the mechanism is almost the same for polymer
and slip methods. For LDR, they achieve a drag reduction by decreasing the
duration of active intervals, while the duration of hibernating turbulence
remains almost constant. These trends lead to more frequent hibernation and an
increase in the temporal intermittency factor. For HDR, while they still show
a decrease in the duration of active turbulence, more drag reduction is
achieved by increasing the duration of hibernating turbulence, which results
in an increase in $F_{H}$ compared to the LDR case. Interestingly, the body
force method displays a distinctly different mechanism at both LDR and HDR
regimes. Although it causes less frequent hibernation and even a smaller
$F_{H}$ value than the no-control case, the highly prolonged hibernation
intervals are likely to produce comparable amounts of drag reduction to the
polymer and slip methods.

Figure 6: ($a,b$) Reynolds shear stress profiles in inner units for the flow-
control methods at about 7% and 25% drag reduction, respectively, along with
no-control case.

###  3.4 Mechanisms behind different flow-control strategies

Lastly, we attempt to illuminate the mechanisms behind the different temporal
characteristics for drag reduction between the polymers/slip methods and the
body force method. As alluded by different polymer dynamics during active and
hibernating turbulence in viscoelastic flows Graham2014 , the polymer and slip
methods appear to possess a selective turbulent preventative mechanism. For
LDR, as seen in Figure 4, the polymers and slip surfaces seem to affect only
the duration of the active turbulence phases, while not making any changes to
the duration of the hibernating turbulence phases. It suggests that the
effects of polymers and slip surfaces are amplified with turbulence-induced
shear. In other words, the drag-reducing mechanism actively kicks in when
turbulence is excited, such as during active turbulence. The polymers begin to
stretch and store elastic energy to release it back into the flow. The slip
velocity or slip length becomes larger in inner units when wall shear stress
gets higher. Again, this selective mechanism is noticeable during LDR as the
low-drag states are unchanged and the high drag states are only affected. For
HDR, as seen in Figure 5, it is still observed that the selective mechanism
affects the high-drag states as done for LDR. However, the low-drag states are
now also affected as the hibernating turbulence appears to take over turbulent
dynamics as more drag reduction is achieved Graham2014 , Wang2014 . A detailed
investigation of drag-reduction mechanisms during HDR will be included in
future work.

It has been shown that the effects of flow-control methods on the Reynolds
shear stress and vortical structures may provide a mechanistic basis for drag
reduction in turbulent flows Lumley1998 , Gad2007 . Figures 6($a$) and ($b$)
depict the time-averaged Reynolds shear stress profiles of the different flow-
control methods along with a no-control case for the LDR and HDR regimes,
respectively. For LDR, the three control profiles are slightly lower than the
no-control case. Interestingly, it is observed that the body force profile is
slightly higher than the polymers/slip and no-control profiles in $y^{+}<15$.
For HDR, the three control profiles are fairly reduced compared with the
magnitudes of the no-control profile. The slip profile is still relatively
close to the no-control case compared to the other two cases. More
interestingly, a change in the shape of the body force profile is noticeable,
which is non-monotonic. Mostly, the profile is rather higher than the no-
control case in $y^{+}<10$, with the peak being close to $y^{+}\approx 7$, and
its magnitude falls below the other cases from $y^{+}>18$, with another peak
at almost the same locations as the other profiles. Note that this non-
monotonic shape is kept even for hibernation and active intervals (not shown),
while the magnitudes are different.

Figure 7: ($a,b$) Vortical structures due to the body force method for LDR (DR
= 7%) and HDR (DR = 25%) regimes, respectively. The red tubes are isosurfaces
of 1/2 of maximum swirling strength $\lambda_{ci}$, which corresponds to the
imaginary part of the complex conjugate eigenvalues of the velocity gradient
tensor ZhouJFM1999 .

To elucidate the seemingly distinct characteristics of the body force method
for a drag-reduction mechanism, we plot its vortical structures for which the
swirling strength $\lambda_{ci}$, the imaginary part of the complex conjugate
eigenvalues of the velocity gradient tensor ZhouJFM1999 , are calculated.
Prior to presenting the vortical structures due to the body force, it should
be noted that the main effects of the polymers and slip surfaces on flow
structures are well-documented that vortices are weakened xi2010jfm ,
Davis2020 . Figures 7($a$) and ($b$) show contours of the swirling strength
for half of the channel for the LDR and HDR regimes, respectively, during the
hibernating phases due to the body force method. The contours represent
isosurfaces of 50% of the maximum $\lambda_{ci}$ for each case. Similar to
ones found by Mamori & Fukagata Mamori2014 , the roller-like structures are
clearly observed very close to the wall and formed in the streamwise
direction. However, it should be mentioned that Mamori & Fukagata had the
roller-like structures formed in the spanwise direction due to the Lorentz
force applied in the wall normal direction. As the wavelength of the body
force is set to $L_{z}/2$, four roller-like vortices are formed and separated
by $L_{z}/4$ – a pair of vortices per wavelength. It is observed that the
heights of these vortices are independent of the choice of the other control
parameters such as penetration depth ($\Delta$), force magnitude ($I$), and
period ($T$). The center of the rollers is located at $y^{+}\approx 7$. Given
these streamwise-spanned roller-like vortices, it can be speculated that the
body force is likely to prevent interactions between the inner region and
outer region by which the hibernation intervals tend to be substantially
prolonged. In other words, these streamwise roller-like structures tend to
stay near the wall and block structures from the outer region, which helps
increase the hibernation intervals for the reduction of friction drag. The
observation in vortical structures due to body forces could tie into a shear
sheltering effect, which effectively limits an interplay between the inner and
outer layers hunt1999 . As drag reduction is further increased, the effect of
the body force becomes more noticeable as it causes to form a strong shear
sheltering layer in a buffer layer, which effectively decouples the flow
structures above and below this layer. This blocking mechanism is different
from the selective mechanism of the polymer and slip methods.

The differences in Reynolds shear stress and vortical structures between the
polymer/slip methods and the body force method may provide clear and plausible
mechanisms responsible for distinct underlying drag-reduction mechanisms, for
which further investigation is yet needed. In particular, further
investigation is need to understand how vortical structures separate the flow
regions due to a body force. In addition, as there have been limited studies
to relate a shear sheltering to drag reduction mechanisms ptasinski2003 , a
detailed connection of the body force method to a shear sheltering will be a
subject of interesting future work.

##  4 Conclusion

Through direct numerical simulations in a channel flow (plane Poiseuille)
geometry at low (transitional) Reynolds numbers in a range of
$Re_{\tau}=65-85$, the underlying drag-reduction mechanisms of three flow-
control strategies, namely polymer additives, slip surfaces, and external body
forces, are investigated by utilizing two temporal turbulent phases –
hibernating and active turbulence Xi2010 . Given similar drag-reduction
percentages, the polymers/slip methods demonstrate a similar drag-reduction
mechanism to one another by causing the hibernation phases to occur more
frequently with a decrease in the duration of active phases. The body force
method shows a different mechanism, where the hibernation phases happen less
frequently, however the duration of these hibernation phases is prolonged due
to roller-like vortical structures formed near the wall. These vortical
structures appear to prevent interactions between the inner and outer regions
to make hibernating phases longer. At higher drag-reduction regimes, each
control method seems to involve different mechanisms to manipulate the
Reynolds shear stress for which follow-up work is under investigation. More
importantly, a connection of the temporal analysis to the drag-reduction
mechanisms due to the three flow control methods has been made and suggests
that the polymer and slip methods possess a ‘selective’ turbulent preventative
mechanism, affecting only active turbulence, and the body force method
possesses a ‘blocking’ mechanism, affecting both active and hibernating
turbulence.

These distinct underlying drag-reduction mechanisms between the polymers/slip
methods and body force method could suggest that different adaptive and
optimal flow-control techniques could be used to promote more drag reduction,
which will be a subject of interesting future work. For instance, when a flow
enters a hibernation phase with an aid of an external body force, a special
control strategy could be applied at the locations of the streamwise roller-
like structures to stabilize them to make a hibernation phase much longer for
more drag reduction. In addition, a further study is necessary to investigate
the effect of higher Reynolds numbers on the current temporal approach for a
practical relevance in practice. However, it should be noted that hibernation
phases are barely detected beyond $Re_{\tau}\approx 125$ Whalley2017  using
the current hibernation criteria. Thus, the criteria should be somewhat
relaxed for higher Reynolds number flows to detect hibernating turbulence.
This research avernue is currently under investigation Davis2020dfd , Ryu2021
. Finally, the temporal approach of the current study can be combined with the
quantatiative approaches such as the Fukagata-Iwamoto-Kasagi (FIK) identity
Fukagata2002 , Renard-Deck (RD) identity Renard2016 , and energy-box analysis
Gatti2018  to provide more comprehensive information about the underlying
drag-reduction of different flow-control strategies.

###### Acknowledgements.

The direct numerical simulation code used was developed and distributed by J.
Gibson at the University of New Hampshire. The authors also acknowledge the
computing facilities used at the Holland Computing Center at the University of
Nebraska-Lincoln.

Funding The authors gratefully acknowledge the financial support from the
National Science Foundation through a grant OIA-1832976, the Nebraska EPSCoR
FIRST Award supported by the National Science Foundation through a grant
OIA-1557417, and the Collaboration Initiative at the University of Nebraska.

## Declarations

Conflict of interest The authors declare that they have no conflict of
interest.

Availability of data and material The data that support the findings of this
study are available from the corresponding author upon reasonable request.

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#  Stable maps to Looijenga pairs built from the plane

Michel van Garrel  University of Birmingham, School of Mathematics, B15 2TT,
Birmingham, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Choosing a normal crossings anticanonical divisor of $\mathbb{P}^{2}$ leads to
four log Calabi–Yau surfaces, three of which are Looijenga pairs. In this
survey article, I describe how to count rational curves in these that
intersect each component of the divisor in one point of maximal tangency.

###### Contents

  1. 1 Introduction
  2. 2 Genus 0 log Gromov-Witten invariants of maximal tangency
  3. 3 Tropical correspondence and the invariants $N_{d}(\mathbb{P}^{2},D^{\rm toric})$
  4. 4 Degeneration and the invariants $N_{d}(\mathbb{P}^{2},D_{1}+D_{2})$
  5. 5 Scattering and the invariants $N_{d}(\mathbb{P}^{2},D_{3})$

##  1\. Introduction

A log Calabi–Yau surface $(Y,D)$ is a surface $Y$ with a choice of normal
crossings divisor $D$ such that the complement is Calabi–Yau. $(Y,D)$ is a
Looijenga pair if in addition, $D$ has a dimension 0 stratum. Starting with
$Y=\mathbb{P}^{2}$, $D$ is of degree 3 and there are 4 ways of building a log
Calabi–Yau surface: $D=D^{\rm toric}$, the toric divisor given as a union of 3
lines; $D=D_{1}+D_{2}$, the union of a line $D_{1}$ and a conic $D_{2}$ not
tangent to $D_{1}$; $D=D_{3}$ a nodal cubic; and $D=E$ a smooth cubic. All but
the last are Looijenga pairs.

Figure 1.1. The toric divisor $D^{\rm toric}$ and the divisor $D_{1}+D_{2}$
formed of a line and a conic.

Figure 1.2. The divisors formed by resp. a nodal cubic $D_{3}$ and a smooth
cubic $E$.

Denote by $l\leq 3$ the number of irreducible components of $D$ and let
$\mathbb{P}^{1}[l]$ be the complement of $l$ points in $\mathbb{P}^{1}$. Up to
automorphism, we have that $\mathbb{P}^{1}[1]=\mathbb{A}^{1}$,
$\mathbb{P}^{1}[2]=\mathbb{C}^{*}$ and
$\mathbb{P}^{1}[3]=\mathbb{C}^{*}\setminus\\{1\\}$. A surprisingly intricate
question is to study the geometry of $Y\setminus D$ by counting pseudo-
holomorphic maps

$\mathbb{P}^{1}[l]\longrightarrow Y\setminus D$

of degree $d\in\mathbb{Z}_{>0}$ that pass through $l-1$ fixed points. For
example, in degree 1 and for $D=E$, there are 9 such $\mathbb{A}^{1}$-curves
corresponding to the 9 inflection lines of $E$.

The modern formulation of this count is as the genus 0 maximal tangency log
Gromov–Witten invariant $N_{d}(\mathbb{P}^{2},D)$ [Chen14, AbramChen14, GS13].
These invariants, more generally the punctured invariants of [ACGSpunct], play
a crucial role in the mirror constructions of intrinsic mirror symmetry
[GSintr, GScan]. In this survey, I describe the methods used to compute the
$N_{d}(\mathbb{P}^{2},D)$. For the case of $D=D_{1}+D_{2}$, I provide a full
proof, which may serve as an extensive introduction to the scattering diagram
methods of [BBvG2], see also [B] for an introduction emphasing the links with
physics.

For the toric $(\mathbb{P}^{2},D^{\rm toric})$, we can make use of tropical
correspondence results [Mikh05, NiSi, ManRu], which turn
$N_{d}(\mathbb{P}^{2},D^{\rm toric})$ into a count of weighted tropical
curves. A special case of [BBvG1, Theorem 3.2] then states that

$N_{d}(\mathbb{P}^{2},D^{\rm toric})=d^{2},$ (1.1)

reproven in Proposition 3.3 below.

In the case of a line and a conic, $N_{d}(\mathbb{P}^{2},D_{1}+D_{2})$ is
obtained by degeneration. The result of [BBvG2, Example 3.1], or Proposition
4.3 below, is that

$N_{d}(\mathbb{P}^{2},D_{1}+D_{2})=\binom{2d}{d}.$ (1.2)

The invariants $N_{d}(\mathbb{P}^{2},D_{3})$ are encoded by the wall-crossing
function of the central ray of a local scattering diagram. The terms of this
function are conjectured in [GP10, Equation (1.3)], which is proven in
[Rei11]. Applying [GPS], the invariants are given by the equality of power
series expansions

$\sum_{d=1}^{\infty}\,d\,N_{d}(\mathbb{P}^{2},D_{3})\,x^{d}=3\log\left(\sum_{k=0}^{\infty}\,\frac{1}{3k+1}\,\binom{4k}{k}\,x^{k}\,\right).$
(1.3)

The structure of the invariants of $(\mathbb{P}^{2},E)$ carries similarities
with the structure of invariants of K3 surfaces [CGKT2, Bousseau:2020ckw] and
thus sometimes $(\mathbb{P}^{2},E)$ is called a log K3 surface. The
computation of the $N_{d}(\mathbb{P}^{2},E)$ is outside of the scope of this
survey. It has a long and rich history. It was studied for the first time in
the landmark [MR1844627], who stated and provided evidence for 3 influential
conjectures regarding $(\mathbb{P}^{2},E)$. The first one is the first
appearance of the _log/local correspondence_ , which expresses the
$N_{d}(\mathbb{P}^{2},E)$ as the genus 0 local Gromov–Witten invariants of the
local surface given as the total space ${\rm
Tot}\left(\mathcal{O}_{\mathbb{P}^{2}}(-E)\right)$. This conjecture was first
proven in [MR1962055], subsequently generalised in [vGGR], and then extended
in [BBvG1, BBvG2, BBvG3, Bousseau:2020ckw, tseng2020mirror, BNTY, NR22]. The
second conjecture is a BPS version of the log/local correspondance for
$(\mathbb{P}^{2},E)$, that is, a correspondence between the integer-valued
counts that govern the genus 0 log and local theories. It was proven through
the succession of works [CPS, Gra, Bou19a, Bou19b].

The article [MR1844627] also introduces log mirror mirror symmetry of
$(\mathbb{P}^{2},E)$. The author builds a mirror family and computes the
periods on the mirror family. The third conjecture then states that the second
order period written in canonical coordinates is a generating function of the
$N_{d}(\mathbb{P}^{2},E)$. Building on [CPS, Gra, GScan], a geometric proof of
this prediction is given in all dimensions in the upcoming work [vGRS], see
also [vG] for a self-contained proof for the case of $(\mathbb{P}^{2},E)$.

###  1.1. Methods

The approach I take in this survey is to degenerate to _toric models_ for the
Looijenga pairs. These are other – _toric_ – Loiijenga pairs
$(\overline{\mathbb{P}^{2}},\overline{D})$ such that $(\mathbb{P}^{2},D)$ is a
logarithmic modification of a blow-up of
$(\overline{\mathbb{P}^{2}},\overline{D})$ in smooth points of $\overline{D}$.
Concretely this is a diagram

$\textstyle{(\widetilde{\mathbb{P}^{2}},\widetilde{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\pi}$$\textstyle{(\mathbb{P}^{2},D)}$$\textstyle{(\overline{\mathbb{P}^{2}},\overline{D})}$

such that $\varphi$ is a sequence of blow-ups in singular points of (the total
transform of) $D$ and $\pi$ is a toric model, i.e. a sequence of blow-ups in
smooth points of (the proper transform of) $\overline{D}$. By [GHKlog,
Proposition 1.3] such toric models always exist. By [AW] $\varphi$ preserves
the log Gromov–Witten invariants.

The next step is to perform a sequence of standard degenerations to
$(\widetilde{\mathbb{P}^{2}},\widetilde{D})$. Applying the degeneration
formula [MR1938113, zbMATH07283066, KLR, AF16, Chendeg], the outcome is that
the invariants $N_{d}(\mathbb{P}^{2},D)$ are computed in terms of the log
Gromov–Witten invariants of $(\overline{\mathbb{P}^{2}},\overline{D})$, which
are calculated by tropical correspondence, and the log Gromov–Witten
invariants of some standard pieces, which are known. This very general
algorithm was formalised in terms of scattering diagram algorithms in [GPS].

One goal of this note is to unwrap the constructions of [GPS, GHKlog, GHS16]
for the examples of $(\mathbb{P}^{2},D_{1}+D_{2})$ and
$(\mathbb{P}^{2},D_{3})$. For a full treatment, the reader is invited to
consult [BBvG2].

There are other approaches to computing these invariants, all interesting in
their own right. Working at the level of moduli spaces these include [vGGR,
MR3228298, CGKT1, CGKT2, CGKT3, BN20, NR22] and working with Givental-style
mirror symmetry [BN21, tseng2020mirror, BNTY, Y22].

## Acknowledgements

I thank Dhruv Ranganathan for originally suggesting to study the example of
$D_{1}+D_{2}$ leading to [BBvG2, BBvG3]. I thank the Nottingham Online
Algebraic Geometry Seminar, in particular Al Kasprzyk and Livia Campo, for
giving me the opportunity to present these results. I thank my collaborators
Pierrick Bousseau, Andrea Brini, Jinwon Choi, Navid Nabijou, Helge Ruddat,
Yannik Schüler and Bernd Siebert for various discussions linking to several
aspects of this paper. I thank Fenglong You for discussions about parallel
methods for computing these invariants and Tim Gräfnitz for discussions on
[Gra].

##  2\. Genus 0 log Gromov-Witten invariants of maximal tangency

For each Looijenga pair $(\mathbb{P}^{2},D)$, endow $\mathbb{P}^{2}$ with the
divisorial log structure determined by $D$. The log structure is used as a
combinatorial tool to impose tangency conditions along the components of $D$.
For a map

$f:\mathbb{P}^{1}\longrightarrow\mathbb{P}^{2}$ (2.1)

this is achieved by requiring that the pullback map on log structures at
marked points is given by multiplication by the tangency imposed at that
point. Compactifying the space of maps as in (2.1), [GS13, Chen14,
AbramChen14] construct the moduli space
$\overline{\operatorname{M}}^{\log}_{m}(\mathbb{P}^{2},D,d)$ of genus 0
maximally tangent basic stable log maps with $m$ marked points, and its
virtual fundamental class

$[\overline{\operatorname{M}}^{\log}_{m}(\mathbb{P}^{2},D,d)]^{\rm
vir}\in\operatorname{H}_{2(l+m-1)}(\overline{\operatorname{M}}^{\log}_{m}(\mathbb{P}^{2},D,d)).$

I suppress in the notation the $l$ marked points that carry a tangency and
keep track of the $m$ interior points that do not carry a tangency condition.
The notion of basicness/minimality selects a universal log structure needed
for algebraicity and compactness of the moduli space.

For $j=1,\dots,m$, the moduli space admits evaluation maps at the $j$th marked
points

$\operatorname{ev}_{j}:\overline{\operatorname{M}}^{\log}_{m}(\mathbb{P}^{2},D,d)\longrightarrow\mathbb{P}^{2}.$

Imposing passing through a fixed point in $\mathbb{P}^{2}$ corresponds to
capping the virtual fundamental class with $\operatorname{ev}_{j}^{*}([{\rm
pt}])$. Doing this $l-1$ times leads to the invariant

$N_{d}(\mathbb{P}^{2},D):=\int_{[\overline{\operatorname{M}}^{\log}_{l-1}(\mathbb{P}^{2},D,d)]^{\rm
vir}}\,\prod_{j=1}^{l-1}\operatorname{ev}_{j}^{*}([{\rm pt}]).$

Other insertions may be imposed. For example, for the class toric varieties
for which each toric divisor is nef, [BBvG1] computes all invariants with
point and psi classes.

######  Example 2.1.

Fixing a point $P$, there are two lines in the plane that pass through $P$,
intersect $D_{1}$ in one point of order 1 and intersect $D_{2}$ in one point
of order 2, see Figure 2.1.

Figure 2.1. The two lines passing through a point and meeting a conic, resp. a
line, in a point of order 2, resp. 1.

##  3\. Tropical correspondence and the invariants
$N_{d}(\mathbb{P}^{2},D^{\rm toric})$

The Looijenga pair $(\mathbb{P}^{2},D^{\rm toric})$ is toric, thus is its own
toric model, and we can compute $N_{d}(\mathbb{P}^{2},D^{\rm toric})$ by
enumerating weighted tropical curves in the fan $\Xi$ of
$(\mathbb{P}^{2},D^{\rm toric})$.

######  Definition 3.1.

A degree $d$ genus 0 maximal tangency tropical curve $\Gamma$ in $\Xi$ is a
graph $\Gamma$ that may have unbounded edges such that:

  1. (1)

The vertices $v$ of $\Gamma$ are elements of the support of $\Xi$ separated by
edges $e$, which are line segments of rational slope $\in\mathbb{Z}^{2}$ that
carry weights $w(e)\geq 1$.

  2. (2)

Balancing condition: For a vertex $v$ of $\Gamma$ with adjacent edge $e$,
denote by $u_{(v,e)}$ its primitive outgoing vector. Weighted by $w$, these
sum to 0:

$\sum_{v\in e}w(e)\,u_{(v,e)}=0.$

  3. (3)

Writing $D^{\rm toric}=T_{1}+T_{2}+T_{3}$ for the toric divisors $T_{i}$,
$\Gamma$ has exactly one unbounded edge of weight $d$ parallel to the ray of
$\Xi$ corresponding to $T_{i}$, and no other unbounded edges.

  4. (4)

The first Betti number $b_{1}(\Gamma)=0$.

For our purposes, a tropical curve can be thought of as keeping track of the
tangency behaviour of stable log maps.

######  Definition 3.2.

For a trivalent vertex $v\in\Gamma$ with outgoing edges $e_{1},e_{2},e_{3}$,
the weight of $v$ is

$w(v):=w(e_{i})\,w(e_{j})\,\big{|}\det\left(u_{(v,e_{i})}\,u_{(v,e_{j})}\right)\big{|}$

where $i\neq j$ and $\\{i,j\\}\subset\\{1,2,3\\}$. This is well-defined by the
balancing condition. The weight of $\Gamma$, $w(\Gamma)$, is given by the
products of the weights of each trivalent vertex.

######  Proposition 3.3.

Fixing two general points $P_{1},P_{2}\in\Xi$, there is only one genus 0
degree $d$ maximally tangent tropical curve in $\Xi$ that passes through
$P_{1}$ and $P_{2}$. It carries weight $d^{2}$ and hence

$N_{d}(\mathbb{P}^{2},D^{\rm toric})=d^{2}.$

###### Proof.

The only such tropical curve is given in the following diagram. It has weight
$d^{2}$. The result follows from the tropical correspondence [Mikh05, NiSi,
ManRu].

$T_{2}$$T_{3}$$T_{1}$$\bullet$$\bullet$$P_{1}$$P_{2}$$d$$d$$d$$d^{2}$

∎

##  4\. Degeneration and the invariants $N_{d}(\mathbb{P}^{2},D_{1}+D_{2})$

###  4.1. The toric model

Denote by $p$ one of the two points of intersection of $D_{1}$ and $D_{2}$ and
denote by $L$ the line tangent to $D_{2}$ at $p$. I describe a log
modification that is obtained by successively blowing up points which are
dimension 0 strata of the (total transform) of the divisor $D=D_{1}+D_{2}$. I
choose $D_{1}$ and $L$ to be toric divisors and describe a series of
modifications that turn $D_{2}$ into a toric divisor as well (of a different
surface). Every blow-up and blow-down is toric.

######  Convention 4.1.

I denote by the same letter divisors on different surfaces which are related
by strict transforms under blow-ups and blow-downs. This process alters the
self-intersection numbers.

Blow up $p$ which leads to the exceptional divisor $F_{1}$. Then blow up the
intersection of $F_{1}$ with $D_{2}$ and write $F_{2}$ for the exceptional
divisor. This gives a log Calabi-Yau surface
$(\widetilde{\mathbb{P}^{2}},\widetilde{D})$, where $\widetilde{D}$ is the
total transform of $D_{1}+D_{2}$, see Figure 4.1, and where $H$ denotes the
pullback of the hyperplane class of $\mathbb{P}^{2}$.

$D_{1}$$F_{2}$$F_{1}$$L$$H$

Figure 4.1. The log modification $(\widetilde{\mathbb{P}^{2}},\widetilde{D})$.

######  Proposition 4.2.

The Chow group ${\rm CH}_{1}(\widetilde{\mathbb{P}^{2}})$ is generated by
$[H]$, $[F_{1}]$ and $[F_{2}]$ with relations

$[D_{1}]=[L]+[F_{2}],\quad[H]=[D_{1}]+[F_{1}]+[F_{2}],\quad[D_{2}]=[H]+[L],$
$[H]\cdot[D_{1}]=[H]\cdot[L]=1,\quad[L]^{2}=-1,\quad[H]^{2}=1,\quad[D_{1}]^{2}=0,\quad[D_{2}]^{2}=2,\quad[D_{2}]\cdot[L]=0.$

###### Proof.

Immediate either by construction or by the toric relations determined by the
fan. ∎

The toric model $(\overline{{\mathbb{P}}^{2}},\overline{D})$ then is given by
blowing down the strict transform of $L$, which is a $(-1)$-curve, resulting
in $\overline{{\mathbb{P}}^{2}}=\operatorname{\mathbb{F}_{2}}$. This leads to
the fan on the left-hand side of Figure 4.2. Then
$\overline{D}=D_{1}+F_{1}+F_{2}+D_{2}$ with $F_{1}$ the $(-2)$-curve, $D_{2}$
a 2-curve, and $D_{1},F_{2}$ the toric fibres of the bundle projection
$\operatorname{\mathbb{F}_{2}}\to\mathbb{P}^{1}$. Applying the fan
automorphism

$\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}\in{\rm SL}(2,\mathbb{Z})$

we obtain the fan on the right hand side of Figure 4.2. In addition, I label
the boundary divisors with their self-intersections and I keep track of $L$ by
adding a $\times$ on the ray of $F_{2}$ indicating that we blow up a smooth
point of $F_{2}$ in order to obtain a log Calabi–Yau surface logarithmically
equivalent to $(\mathbb{P}^{2},D_{1}+D_{2})$.

N.B.: Through this process, $D_{2}$ became a _toric_ divisor.

$D_{2}$$D_{1}$$F_{2}$$F_{1}$$+$$\longrightarrow$$\cdot\begin{pmatrix}1&0\\\
1&1\end{pmatrix}$

$D_{2}(2)$$D_{1}(0)$$F_{1}(\text{-}2)$$F_{2}(0)$$\times$

Figure 4.2. The toric model of $(\mathbb{P}^{2},D_{1}+D_{2})$.

######  Remark 4.1.

Note that the fan of a smooth toric surface is uniquely determined by the
self-intersections of its toric divisors. Indeed, if
$\rho_{1},\rho_{2},\rho_{3}$ are primitive generators of successive rays in
the fan corresponding to toric divisors $T_{1},T_{2},T_{3}$, then
$\rho_{1}+(T_{2})^{2}\rho_{2}+\rho_{3}=0$. The collection of these equations
determines the fan up to multiplication by an element of ${\rm
SL}(2,\mathbb{Z})$.

###  4.2. Calculation by degeneration

By invariance of log Gromov-Witten invariants under log modifications, $N^{\rm
log}_{d}(\mathbb{P}^{2},D_{1}+D_{2})$ is the genus 0 degree $d[H]$ log Gromov-
Witten invariant of $(\widetilde{\mathbb{P}^{2}},\widetilde{D})$, passing
through one interior point, meeting $D_{1}$ in one point of tangency $d$ and
meeting $D_{2}$ in one point of tangency $2d$ and with no other conditions.

I describe a suitable degeneration of
$(\widetilde{\mathbb{P}^{2}},\widetilde{D})$ into two components. One
component, $(\overline{\mathbb{P}^{2}},\overline{D})$, is toric and thus
tropical correspondence results apply. $L$ specialises into the other
component, which however will be simple enough so that we can compute its
Gromov–Witten invariants.

Start with the trivial family $\mathbb{F}_{2}\times\mathbb{A}^{1}$. The blow
up (degeneration to the normal cone)

$\mathcal{X}^{\prime}:={\rm
Bl}_{F_{2}\times\\{0\\}}\left(\mathbb{F}_{2}\times\mathbb{A}^{1}\right)$

has general fibre $\mathbb{F}_{2}$ and special fibre
$\mathbb{F}_{2}\cup_{F_{2}}\mathbb{F}_{0}$, where
$\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}=\mathbb{P}(N_{F_{2}/\mathbb{F}_{2}}\oplus\mathcal{O}_{F_{2}})$.
Denote by $H_{1},H_{2}$ the two effective curve classes generating
$\operatorname{H}_{2}(\mathbb{F}_{0},\mathbb{Z})$. $\mathbb{F}_{2}$ is glued
to $\mathbb{F}_{0}$ by identifying $F_{2}$ with one of the toric divisors of
$\mathbb{F}_{0}$ of class $H_{1}$. Denote by $\mathcal{F}$ the strict
transform of $F_{2}\times\mathbb{A}^{1}$. Choose a section $\mathcal{S}$ of
$\mathcal{F}\to\mathbb{A}^{1}$ whose image lies in the smooth locus (interior)
of $F_{2}$. Then

$\mathcal{X}:={\rm Bl}_{\mathcal{S}}\,\mathcal{X}^{\prime}$

has general fibre $(\widetilde{\mathbb{P}^{2}},\widetilde{D})$ and special
fibre $\mathbb{F}_{2}\cup_{F_{2}}Y$, where $Y={\rm Bl}_{pt}(\mathbb{F}_{0})$
is the blow-up of $\mathbb{F}_{0}$ at a smooth point of the other toric
divisor in class $H_{1}$. If $\mathcal{L}$ is the exceptional divisor of the
family $\mathcal{X}$, by abuse of notation I denote by $L$ the fibre-wise
exceptional divisors.

Figure 4.3. The two components of the central fibre.

The degeneration formula is a virtual way of keeping track of counts of curves
in the degeneration. The first step is to understand the behavior of curves as
they specialise from the generic fibre to the special fibre. Start with a
curve $C$ in $\widetilde{\mathbb{P}^{2}}$, or more generally an element in
Chow $\alpha\in{\rm CH}_{*}(\widetilde{\mathbb{P}^{2}})$. Then [BvBvG1,
BvBvG2] construct the prelog Chow ring ${\rm CH}^{\rm
prelog}_{*}(\mathbb{F}_{2}\cup_{F_{2}}Y)$ along with a _specialisation
morphism_

$\sigma:{\rm CH}_{*}(\widetilde{\mathbb{P}^{2}})\to{\rm CH}^{\rm
prelog}_{*}(\mathbb{F}_{2}\cup_{F_{2}}Y).$

The ring ${\rm CH}^{\rm prelog}_{*}(\mathbb{F}_{2}\cup_{F_{2}}Y)$ is defined
as the quotient of tuples $(\alpha_{1},\alpha_{2})\in{\rm
CH}_{*}(\mathbb{F}_{2})\oplus{\rm CH}_{*}(Y)$ that satisfy
$\alpha_{1}|_{F_{2}}=\alpha_{2}|_{F_{2}}$ under the identification of classes
on both components which are equal as elements of ${\rm
CH}_{*}(\mathbb{F}_{2}\cup_{F_{2}}Y)$.

######  Proposition 4.3.

The prelog Chow ring ${\rm CH}^{\rm prelog}_{*}(\mathbb{F}_{2}\cup_{F_{2}}Y)$
is generated by the classes

  * •

$[(\mathbb{F}_{2},Y)]$ in degree 2;

  * •

$[(D_{2},H_{2})]$, $[(F_{2},0)]=[(0,H_{1})]$ and $[(0,L)]$ in degree 1;

  * •

$[({\rm pt},0)]=[(0,{\rm pt})]$ in degree 0,

and with intersection products

$[(D_{2},H_{2})]\cdot[(F_{2},0)]=1,\quad[(D_{2},H_{2})]\cdot[(0,L)]=[(F_{2},0)]\cdot[(0,L)]=0.$

###### Proof.

The degrees 2 and 0 are clear. In the case of curves, ${\rm
CH}_{1}(\mathbb{F}_{2})$ is generated by $D_{2}$ and $F_{2}$. ${\rm
CH}_{1}(Y)$ is generated by $H_{1}$, $H_{2}$ and $L$. Moreover, the gluing
$\mathbb{F}_{2}\cup_{F_{2}}Y$ identifies the classes $F_{2}$ and $H_{1}$. Then
the classes are determined by the tuples that agree on the intersection
between both components. Moreover, the intersections can be computed by
choosing representatives on each of the components, see [BvBvG1, Definition
2.2]. ∎

######  Proposition 4.4.

The specialisation of $[H]\in{\rm CH}_{1}(\widetilde{\mathbb{P}^{2}})$ is

$\sigma([H])=[(D_{2},H_{2}-L)]\in{\rm CH}_{1}^{\rm
prelog}(\mathbb{F}_{2}\cup_{F_{2}}Y).$

###### Proof.

By Proposition 4.2, $[H]=[D_{1}]+[F_{1}]+[F_{2}]$. By construction

$\displaystyle\sigma([D_{1}])$
$\displaystyle=[(F_{2},0)],\quad\sigma([F_{1}])=[(F_{1},H_{2})]=[(D_{2},H_{2})]-2[(F_{2},0)],$
$\displaystyle\sigma([F_{2}])$
$\displaystyle=[(0,[H_{1}]-[L])]=[(F_{2},0)]-[(0,L)].$

Since $\sigma$ is a ring homomorphism, we thus have that

$\sigma([H])=\sigma([D_{1}])+\sigma([F_{1}])+\sigma([F_{2}])=[(F_{2}+D_{2}-2F_{2}+F_{2},H_{2}-L)]=[(D_{2},H_{2}-L)].$

∎

######  Remark 4.2.

One may explicitly check that $\sigma$ is a ring homomorphism, at least with
respect to intersections with the class $[H]$.

Let $d\geq 1$. Assume we have two genus 0 curves $C_{1}$ and $C_{2}$ in
$\mathbb{F}_{2}$ and $Y$ respectively, of classes $[C_{1}]=d[D_{2}]$ and
$[C_{2}]=d([H_{2}]-[L])$. Then $C_{1}\cup C_{2}$ can be smoothed to the
generic fibre if and only if $C_{1}$ and $C_{2}$ have the same intersection
pattern with $F_{2}$, i.e. if and only if $C_{1}$ and $C_{2}$ meet $F_{2}$ at
the same points with the same intersection multiplicities. This is called the
_pre-deformability_ condition.

######  Proposition 4.3.

$N_{d}({\mathbb{P}}^{2},D_{1}+D_{2})=\binom{2d}{d}.$

###### Proof.

The degeneration formula [MR1938113, zbMATH07283066, KLR, AF16, Chendeg]
states that virtually counting genus 0 curves $C$ of class $d[H]\in{\rm
CH}_{1}(\widetilde{\mathbb{P}^{2}})$ satisfying a list of properties is the
same as virtually counting genus 0 curves $C_{1}$ and $C_{2}$ of class
$([C_{1}],[C_{2}])=d[(D_{2},H_{2}-L)]\in{\rm CH}^{\rm
prelog}_{1}(\mathbb{F}_{2}\cup_{F_{2}}Y)$ that satisfy the pre-deformability
condition as well as the specialisation of the list of properties. For us the
list of properties is to meet $D_{1}$, resp. $D_{2}$, in one point of tangency
$d$, resp. $2d$, and to pass through a point in the interior of
$\widetilde{\mathbb{P}^{2}}$. Furthermore, the degeneration formula has an
additional multiplicity factor that keeps track of the fact that several
different curves in the generic fibre may specialise to the same curve in the
special fibre.

As $(-1)$-curves admit no infinitesimal deformations, there is only one curve
in $Y$ of class $H_{2}-L$. It is the strict transform of the fibre in class
$H_{2}$ of $\mathbb{F}_{0}$ that meets the point
$\mathcal{S}(0)\in\mathbb{F}_{0}$. As a consequence, the only contributions
coming from $Y$ are unions of multiple covers over this fibre.

The precise expression of the degeneration formula (see [GPS, Proposition
5.3]) is that

$N_{d}(\mathbb{P}^{2},D_{1}+D_{2})=\sum_{m\vdash
d}\prod_{\ell=1}^{d}\frac{\ell^{m_{\ell}}}{m_{\ell}!}\left(\frac{(-1)^{\ell-1}}{\ell^{2}}\right)^{m_{\ell}}N_{m}(\mathbb{F}_{2}).$

  * •

The sum is over the partitions $m=(m_{1},\dots,m_{d})$ of $d$ so that
$d=\sum_{\ell=1}^{d}\ell m_{\ell}$ with $m_{\ell}\geq 0$ and the convention
that $0!=1$. This keeps track of the intersection profile of $C_{1}$,
equivalently $C_{2}$, along $F_{2}$. In this notation, $C_{1}$, equivalently
$C_{2}$, has $m_{\ell}$ points of intersection multiplicity $\ell$ for
$0\leq\ell\leq d$ ($m_{\ell}$ may be zero).

  * •

Since $H_{2}-L$ is a $(-1)$-curve, $C_{2}$ decomposes as a union of multiple
covers of degree $\ell$ over $H_{2}-L$, each meeting $F_{2}$ in multiplicity
$\ell$ at the unique point of intersection of $F_{2}$ with $H_{2}-L$. In order
to lead to a genus 0 connected curve (more precisely a genus 0 connected
stable map), these are glued to a single _connected_ curve (more precisely a
connected stable map) $C_{1}$ in $\mathbb{F}_{2}$. For each $m$, there are
$m_{1}+\cdots+m_{d}$ multiple covers over $H_{2}-L$ that are glued to $C_{1}$
according to the ramification profile determined by $m$.

  * •

The factors $\ell^{m_{\ell}}$ are the multiplicities coming from the
degeneration formula.

  * •

The $1/m_{\ell}!$ are the symmetry factors corresponding to the permutations
of the $m_{\ell}$ multiple covers of degree $\ell$ over $H_{2}-L$. This
adjusts for the over-counting that resulted from counting each constellation
separately.

  * •

The $(-1)^{\ell-1}/\ell^{2}$ are the $Y$-contributions coming from degree
$\ell$ covers of the fibre of class $H_{2}-L$, see [GPS, Proposition 5.2].

  * •

$N_{m}(\mathbb{F}_{2})$ is the genus 0 log Gromov–Witten invariant of
$\mathbb{F}_{2}$ of class $d[D_{2}]$, counting rational curves that pass
through one fixed point in general position in $\mathbb{F}_{2}$ and have

    * –

one point of contact of order $2d$ with $D_{2}$,

    * –

one point of contact of order $d$ with $D_{1}$,

    * –

for every $1\leq\ell\leq d$, $m_{\ell}$ contact points of contact order $\ell$
with $F_{2}$, at the same fixed position on $F_{2}$ corresponding to the
intersection of $F_{2}$ with the unique fibre of $Y$ of class $H_{2}-L$.

The invariant $N_{m}(\mathbb{F}_{2})$ is computed tropically. See Figure 4.4
for $d=2$ and $m=(2,0)$.

$D_{2}(2)$$D_{1}(0)$$F_{1}(\text{-}2)$$F_{2}(0)$$4$$1$$1$$2$$1$$4$$4$$\bullet$$\bullet$$\bullet$
Figure 4.4. For $d=2$, the only tropical curve with $m=(m_{1},m_{2})=(2,0)$.
The numbers indicate the weights of the edges and vertices. The bullets
indicate that we fix a point in the interior of $\mathbb{F}_{2}$ and twice the
same point on $F_{2}$.

There is only one tropical curve contributing to $N_{m}(\mathbb{F}_{2})$. It
has $m_{1}+\cdots+m_{\ell}+\cdots+m_{d}$ infinite edges coming from _fixed_
$F_{2}$-directions, grouped according to $\ell$. The $n$th such curve, for
$0<m_{1}+\cdots+m_{\ell-1}<n\leq m_{1}+\cdots+m_{\ell}\leq d$ carries weight
$\ell$ corresponding to the intersection multiplicity at that point. The
tropical curve has one infinite edge for each of $D_{1}$ and $D_{2}$, of
weights $d$ and $2d$. Requiring the curve to pass through a point in the fan,
all other edges and multiplicites are determined by the balancing condition.
Then

$N_{m}(\mathbb{F}_{2})=\prod_{\ell=1}^{d}\left(\frac{1}{\ell}(2d\ell)\right)^{m_{\ell}}=\prod_{\ell=1}^{d}(2d)^{m_{\ell}}.$

The factors $2d\ell$ are the multiplicities of the vertices. The factors
$1/\ell$ come from the fact that the positions of the contact points with
$\mathbb{F}_{2}$ are fixed. Finally, we obtain

$N_{d}(\mathbb{P}^{2}(1,4))=\sum_{m\vdash
d}\prod_{\ell=1}^{d}\frac{1}{m_{\ell}!}\left(2d\frac{(-1)^{\ell-1}}{\ell}\right)^{m_{\ell}}\,.$

This is the coefficient of $x^{d}$ in the power series expansion of
$\exp(2d\log(1+x))=(1+x)^{2d}$, hence the result. ∎

##  5\. Scattering and the invariants $N_{d}(\mathbb{P}^{2},D_{3})$

We come to the Looijenga pair formed by the nodal cubic $D_{3}$. Denote by $P$
the point of self-intersection. First we get a toric model. To do so, blow up
$P$ leading to an exceptional divisor $E$, and keep track of the two tangent
lines $L_{1}$ and $L_{2}$. Blow up the intersections of $E$ with the proper
transforms of both $L_{1}$ and $L_{2}$ and denote the two new exceptional
divisors by $F_{1}$ and $F_{2}$.

The proper transforms of $L_{1}$ and $L_{2}$ are now $(-1)$-curves, which we
may both blow down to obtain the Hirzebruch surface $\mathbb{F}_{3}$. This
determines a scattering diagram [GPS], which is obtained from the fan of the
Hirzebruch surface by adding two focus-focus singularities on the two toric
fibres.

$\scriptstyle{x}$$\scriptstyle{y}$$D_{3}(3)$$F_{1}(0)$$E(\text{-}3)$$F_{2}(0)$$\times$$\times$$\scriptstyle{1+tx^{-1}}$$\scriptstyle{1+txy^{3}}$

Figure 5.1. The toric model of $({\mathbb{P}}^{2},D_{3})$.

The two walls that meet are directed by vectors $v_{1}$ and $v_{2}$ that have
$|\det\left(v_{1}\,v_{2}\right)|=3\neq 1$. As a consequence, there is infinite
scattering as described e.g. in [GP10]. Each wall carries a wall-crossing
function, which identifies the ring of functions on each side of the wall.
Scattering is a process of inductively adding rays to the scattering diagram
in order to make it consistent order by order. Consistency means that the
sequence of automorphisms of a loop around the origin gives the identity up to
a given order.

The rays are wall-crossing functions whose coefficients are log Gromov–Witten
invariants as shown in [GPS]. The invariant $N_{d}(\mathbb{P}^{2},D_{3})$ has
one point of contact with $D_{3}$. As a consequence, by the correspondence
results of [GPS] between wall-crossing functions and log Gromov–Witten
invariants, $N_{d}(\mathbb{P}^{2},D_{3})$ is encoded by the wall-crossing
function of the central ray of the scattering diagram leading to (1.3).

## References