Current affiliation: ]Stanford University, Stanford, CA, USA.

# Contribution of spanwise and cross-span vortices to the lift generation of
low-aspect-ratio wings: Insights from force partitioning

Karthik Menon<EMAIL_ADDRESS>[  Rajat Mittal<EMAIL_ADDRESS>Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD,
USA

###### Abstract

This study reports on the vortex-induced lift production mechanisms in low
Reynolds number flows over a low aspect-ratio rectangular wing. We use a
rigorous force partitioning method which allows for the estimation of the
pressure-induced aerodynamic loads due to distinct flow features or vortex
structures in the flow around the wing. The specific focus of this work is on
distinguishing the effect of spanwise and cross-span oriented vortex
structures on pressure-induced lift production. We quantify the lift induced
on the wing by these different vortices, and also estimate their influence
within different regions of the flow-field around the wing and in the wake. We
show that for this low aspect-ratio wing, the spanwise vorticity contributes
only about 26% of the total lift whereas 60% of the lift is generated by
cross-span oriented vortices. Furthermore, the spanwise vorticity in the near
wake is capable of producing net negative lift on the wing and this is
explained by separating and quantifying the influence of vortex cores and
regions of strain in the wake. The results demonstrate the utility of the
force partitioning method for dissecting the flow physics of vortex dominated
flows.

††preprint: APS/123-QED

##  I Introduction

Coherent vortex structures are known to play a key role in a large variety of
problems in fluid dynamics. In many fundamental problems such as aerodynamic
force generation, separation bubbles and laminar to turbulent transition, past
work has highlighted the importance of vortex structures which exhibit
specific shapes and orientations that have a large influence on the dynamics
of the flow. Examples include vortex rings [1], spanwise rollers, streamwise
streaks, etc. Particularly in the domain of problems relating to aerodynamics
and hydrodynamics, the distinct roles played by different kinds of coherent
vortex structures, such as streamwise-oriented tip vortices and predominantly
spanwise-oriented leading-edge vortices, has been a focus of significant
research and has been shown to be of fundamental importance [2, 3].

Efforts to analyze the dynamics of distinct vortex structures and their
interactions have been mainly motivated by the need to understand their
influence on aerodynamic load generation. For example, the importance of
streamwise-oriented tip-vortices, which have a significant impact on the lift
produced by finite-span wings, has been recognized in the earliest models of
inviscid flow [4]. Similarly, the leading-edge vortex (LEV) has been shown to
be a major lift-enhancement mechanism in a variety of aerodynamics
applications [5, 3]. The motion and shedding of LEVs is implicated in
phenomena crucial to the aerodynamic loads generated on wings, such as in
dynamic stall [6, 7]. This has spurred significant research into the stability
and timing of LEV shedding, and how it relates to force production in various
applications from insect and bird flight [8, 9] to swimming locomotion [10,
11].

Several previous studies have also developed methods to quantify the
aerodynamic loads induced by individual vortices from flow-field data. For
instance, [12] and [13] analyzed the aerodynamic loads induced by the leading
and trailing-edge vortices over airfoils undergoing various maneuvers. The
common features of these types of approaches are: (a) the assumption of two-
dimensionality; (b) use of inviscid models with point vortices in the field
combined with conformal mapping; (c) parameterization of vortex circulation
from data; and (d) calculation of induced aerodynamic load via the Blasius
formula [14]. The predictions from such models can be quite good and lead to
excellent insights into vortex-induced aerodynamic loads. However, the
extension of these inviscid-model based methods to three-dimensional flow
fields is non-trivial, and would be particularly difficult for low-aspect-
ratio wings, where three-dimensional effects are substantial.

For finite-span wings such as the one that is the focus of the current study,
three-dimensional vortex structures have been shown to exhibit a highly
complex evolution due to their mutual interactions in the vicinity of the wing
as well as in the wake [15, 16]. Due to this, the dynamics of these dominant
vortex structures and their influence on aerodynamic loads has been shown to
depend on a variety of factors. For instance, while the tip-vortex is
generally considered detrimental to lift on static wings, it can be beneficial
in flapping flight [17, 18]. Similarly, interactions between the LEV and tip-
vortices in flows over finite wings effect their influence on the production
of aerodynamic forces on such wings [15, 7]. These complexities have motivated
numerous studies of the effect of wing shape, kinematics and other factors on
the three-dimensional vortex dynamics over such wings as well as their
associated aerodynamic load generation [19, 20, 21, 22, 23]

While these past studies have gone a long way in highlighting the important
vortex structures and their influence on aerodynamic load generation in flows
over finite wings, they have mostly resorted to _qualitatively_ correlating
the evolution of particular vortices in the flow to the observed aerodynamic
loads on the wing. However, a more _quantitative_ analysis that directly
relates these vortices and their spatio-temporal dynamics to aerodynamic loads
would be useful for the aerodynamic design of control surfaces as well as the
the development of effective flow control methods that can more precisely
target the most important flow features. Furthermore, in recent work [24] we
showed that although the majority of past work has focused on the role of
vortices, i.e. _rotation-dominated_ regions of the flow, in aerodynamic force
generation, these vortices are in fact surrounded by regions of _strain-
dominated_ flow, which can have a significant impact and whose contributions
are often overlooked.

The focus of this work is therefore to address the aforementioned issues by
providing new insights into the fundamental physics of lift generation on
finite wings and highlight the important vortex dynamics underlying the
aerodynamics of such problems. To that end, we demonstrate the use of a force
partitioning method (FPM) to _quantitatively_ evaluate the distinct
contributions of important types of vortex structures on aerodynamic load
generation on finite wings. In addition, this method allows us to separate the
influence of rotation-dominated regions from strain-dominated regions of the
flow in the generation of aerodynamic loads.

The force partitioning method used here is based on the work of Quartappelle
and Napolitano [25], Wu [26] and Howe [27] with later extensions by Chang
[28], Zhang et al. [29], and Menon and Mittal [30, 31]. This method allows for
the precise estimation of the loads induced on bodies immersed in a fluid flow
by different physical mechanisms, such as unsteady and viscous effects, as
well as distinct flow structures. In previous work by the present authors
[31], we combined this force partitioning method with a data-driven framework
to track individual vortices in complex two-dimensional flow-fields and
quantify the loads induced by these vortices on immersed bodies. This force
partitioning method (and closely related variations of it) has been used to
uncover fundamental aspects of aerodynamic force generation in insect flight
[29], flow-induced oscillations [30], flow over delta wings [32] as well as
other problems in unsteady aerodynamics [33, 24].

While the task of quantifying the forces induced on finite wings by individual
vortices is significantly more complicated in three-dimensional flow-fields,
this paper builds off our previous work and demonstrates an application of the
force partitioning method in quantitatively estimating the distinct roles that
spanwise and cross-span oriented vorticity play in the generation of lift on
finite wings. The focus on distinguishing spanwise from cross-span vorticity
contributions is motivated by the importance of the spanwise-oriented LEV in
lift production. Furthermore, we also analyze the influence of the rotation-
dominant and strain-dominant regions associated with these flow structures
over the wing as well as in its wake. As we will show, the analyses highlight
interesting aspects of the spatio-temporal evolution of these flow structures
and their influence on lift generation over finite wings.

##  II Force partitioning method and vortex-induced loads

A key aspect in this work is the ability to estimate vortex-induced
aerodynamic forces due to pressure loads on bodies within fluid flows. This is
accomplished by using the force partitioning method described in detail in
previous papers by Menon & Mittal [30, 31]. Here we provide a concise
description of this method, with reference to the schematic in figure 1. We
are interested in quantifying the different mechanisms that contribute to the
pressure-induced aerodynamic forces on a body with surface $B$, which is
immersed in a flow domain given by $V_{f}$. The domain is bounded externally
by the surface $\Sigma$, and the unit normal pointing outward from the fluid
domain into the internal and external boundaries, $B$ and $\Sigma$, is given
by $\hat{n}$.

Figure 1: (a) Schematic of the setup for the force partitioning method (FPM),
along with relevant symbols. (b) Schematic showing a projection of the cone
separating spanwise and cross-spanwise vorticity in the $X$-$Z$ plane.

For the partitioning of pressure-induced loads in the $i$-direction on the
body, the first step in this method is the calculation of an auxiliary
potential field, $\phi_{i}$, which satisfies the following equation:

${\nabla}^{2}\phi_{i}=0,\ \ \mathrm{in}\ \ V_{f}\ \ \mathrm{with}\ \
\bm{\hat{n}}\cdot\bm{\nabla}\phi_{i}=\begin{cases}n_{i}\;,\;\mathrm{on}\;B\\\
0\;\;,\;\mathrm{on}\;\Sigma\end{cases}\ \ \ \mathrm{for}\ i=1,2,3.$ (1)

Here $n_{i}$ is the component of the unit vector $\hat{n}$ in the
$i$-direction. In the current problem, $i=2$ corresponds to the direction of
lift forces on the wing and the auxiliary potential field of interest is
therefore $\phi_{2}$. We note that this auxiliary potential is a function only
of the instantaneous position and shape of the immersed body and the outer
domain boundary.

The next step is to project the Navier-Stokes momentum equation onto the
gradient of the auxiliary potential and integrate over the volume of the fluid
domain as follows,

$-\int_{V_{f}}\bm{\nabla}p\cdot\bm{\nabla}\phi_{i}\
dV=\int_{V_{f}}\rho\left[\frac{\partial\bm{u}}{\partial
t}+\bm{u}\cdot\bm{\nabla}\bm{u}-\nu\nabla^{2}\bm{u}\right]\cdot\bm{\nabla}\phi_{i}\
dV\ \mathrm{for}\ i=1,2,3$ (2)

By the definition of $\phi_{i}$ in equation 1, as well as the
incompressibility constraint and the divergence theorem, the above equation
can be manipulated to express the pressure-induced force on the body as
follows,

$\displaystyle F_{i}=\int_{B}p\ n_{i}\ dS=$
$\displaystyle-\rho\int_{B}\bm{\hat{n}}\cdot\left(\frac{d\bm{U}_{B}}{dt}\phi_{i}\right)\
dS$ (3)
$\displaystyle+\rho\int_{V_{f}}\bm{\nabla}\cdot\left(\bm{u}\cdot\bm{\nabla}\bm{u}\right)\phi_{i}\
dV$
$\displaystyle-\mu\int_{V_{f}}\left(\bm{\nabla}^{2}\bm{u}\right)\cdot\bm{\nabla}\phi_{i}\
dV$
$\displaystyle-\rho\int_{\Sigma}\bm{\hat{n}}\cdot\left(\frac{d\bm{u}}{dt}\phi_{i}\right)\
dS\ \ \mathrm{for}\ i=1,2,3$

where $\bm{U}_{B}$ is the local velocity of the immersed surface.

The four terms on the right-hand-side of equation 3 represent distinct
components of the force induced by the surface pressure distribution on the
body. The first term contains the well-known linear acceleration reaction
force (or added mass in potential flow) [30] as well as the centripetal
acceleration reaction, which was shown to be an important lift generation
mechanism for insect wings [29]. In the present case, this unsteady term is
zero due to the fact that we are studying force production over static wings.
The third and fourth terms represent the effects of viscous diffusion and flow
acceleration at the outer boundary of the domain. The latter of the two is
generally negligible for large domains [34]. Further details of these terms
and their contributions to aerodynamic forces as well as moments can be found
in the work of Zhang et al. [29, 34] and Menon & Mittal [30, 31, 24].

The second term on the right-hand-side of equation 3, which is the main focus
of this study, can be expressed as

$F_{i}^{\omega}=\rho\int_{V_{f}}\bm{\nabla}\cdot\left(\bm{u}\cdot\bm{\nabla}\bm{u}\right)\phi_{i}\
dV\equiv-\rho\int_{V_{f}}2\ Q\ \phi_{i}\ dV\ \ \ \mathrm{for}\ i=1,2,3.$ (4)

Here the quantity $Q$ is defined as
$Q=\frac{1}{2}\left(||\bm{\Omega}||^{2}-||\bm{S}||^{2}\right)$, where
$\bm{\Omega}$ and $\bm{S}$ are the anti-symmetric and symmetric parts of the
velocity-gradient tensor ($\bm{\nabla u}$) respectively. This scalar quantity
signifies the relative strength of local rotation versus strain in the flow-
field; positive values of $Q$ correspond to regions where rotation dominates
over strain and vice versa. This rotation-dominance condition, $Q>0$, is
commonly is used to detect vortices in a flow [35].

Given the centrality of $Q$ (and therefore vortices) in this force component,
we identify this term as the “vortex-induced force”. This vortex-induced force
is found to be a dominant component in a variety of flows including those over
flapping wings [29], bluff bodies [30], pitching foils [31, 24] and delta
wings [32]. As we will show, this is also true for the flow configuration that
is a focus of the current paper. It is also useful to note that the volume-
integral form of the vortex-induced force term in equation 4 allows us to
compute the local force contribution from any localized region in the flow-
field (or flow feature) of interest. This simply involves isolating the
volume(s) in the flow-field corresponding to the feature(s) or region(s) of
interest, and computing the integral given in equation 4 over the appropriate
volume(s). This partitioning of the influence of different spatial regions in
the flow has been used to effectively dissect the physics of force generation
in previous studies [29, 30, 31, 24]. Finally, we note that equation 4 also
indicates that rotation-dominant regions ($Q>0$) and strain dominant regions
($Q<0$) induce pressure forces of opposite signs on the immersed control
surface. This simple observation will be useful in dissecting the effect of
various vortices on the surface loads.

A specific focus in this work is on isolating the distinct roles that spanwise
and cross-spanwise oriented vortices play in pressure-induced lift production
over three-dimensional wings. Our interest in the analysis of spanwise
vortices in particular stems from the well-established influence of the
leading-edge vortex in a wide variety of practical applications. While there
are many methods to identify the volumes in the flow-field occupied by
vortices aligned in particular directions, we use a simple method based on the
direction of the vorticity unit vector at every point in the flow-field. In
our coordinate system, $X$ and $Z$ are the streamwise and spanwise directions
respectively, and the three components of the vorticity vector, $\bm{\omega}$,
are denoted as $\omega_{x}$, $\omega_{y}$ and $\omega_{z}$. The component of
the vorticity unit vector in the spanwise direction is therefore given by,

$\eta_{z}=\frac{\omega_{z}}{\sqrt{\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}}}.$
(5)

The magnitude of $\eta_{z}$ in equation 5 represents the cosine of the angle
between the spanwise direction ($Z$-axis) and the local vorticity vector at
any point in the flow-field. Here we identify “spanwise oriented” vorticity
very simply as regions where this angle is less than $45^{\circ}$, which
implies $|\eta_{z}|>\mathrm{cos}(\pi/4)$. Geometrically, this condition is
satisfied at any point in space if the local vorticity vector lies within
either of two symmetric cones whose axes are aligned with the positive and
negative $Z$-axes (spanwise direction), with each cone having a half-angle of
$45^{\circ}$ and its apex at the point of interest. A schematic with the
geometry of these cones is presented in figure 1(b). Vorticity vectors that
lie within this cone are oriented more in the spanwise direction than in any
other direction. Conversely, the remaining vorticity vectors are designated as
cross-spanwise oriented. For the cases analyzed here, we will show using flow
visualizations in subsequent sections that this cross-spanwise component
primarily consists of streamwise oriented vortex structures. This simple
method for separating spanwise from cross-spanwise vorticity that is used here
is sufficient for the current purposes but other complex flows would likely
require more sophisticated methods, including modal decomposition [36], to
separate the effect of different vortex structures on aerodynamic loads.

We perform this analysis of spanwise and cross-spanwise vorticity within a
subdomain of the full computational grid. The subdomain has dimensions
$6C\times 3C\times 3C$ in the $X$, $Y$ and $Z$ directions. It is positioned
symmetrically with the wing in the $Y$ and $Z$ directions, and includes $4.55$
chord-lengths in the wake of the wing in the streamwise direction. We have
verified that this subdomain captures the bulk of the force (about 95%)
induced on the wing by the surrounding flow. In this study, we are
specifically interested in the analysis of the lift production over three-
dimensional wings. We report our results in terms of the lift coefficient,
$C_{L}=F_{L}/(\frac{1}{2}\rho U^{2}_{\infty}A)$, where $F_{L}$ is the
dimensional form of the lift and $A$ is the area of the wing. We express the
lift coefficient associated with the vortex-induced force as $C^{\omega}_{L}$,
and the spanwise and cross-spanwise vorticity components of this lift
coefficient are $C^{\omega z}_{L}$ and $C^{\omega xy}_{L}$.

##  III Computational method

###  III.1 Flow solver

The incompressible flow simulations in this study have been performed using
the sharp-interface immersed boundary method of [37] and [38]. This method is
particularly well-suited to flows around complex geometries as it allows us
the use of a simple non-conformal Cartesian grid to simulate a variety of
different shapes and motions of the immersed body. Further, the ability to
preserve the sharp-interface around the immersed boundary ensures very
accurate computations of surface quantities. The Navier-Stokes equations are
solved using a fractional-step method. Spatial derivatives are discretized
using second-order central differences in space, and time-stepping is achieved
using the second-order Adams-Bashforth method. The pressure Poisson equation
is solved using a geometric multigrid method. This code has been extensively
validated in previous studies for a variety of three-dimensional stationary
and moving boundary problems [39, 37, 38], where its ability to maintain local
(near the immersed body) as well as global second-order accuracy has been
demonstrated. Further, the accuracy of surface measurements has been
established for a wide variety of stationary as well as moving boundary
problems in these studies.

###  III.2 Problem setup

Figure 2: (a) Geometry of the rectangular wing along with spanwise and
streamwise slices showing contours of $\phi_{2}$. (c) Computational domain and
the Cartesian computational grid.

In this paper, we analyze the three-dimensional incompressible flow over flat-
plate wings with rectangular planforms. The rectangular wing studied has a
$2:1$ span-to-chord aspect-ratio. The cross section of the wing is a flat
plate with a thickness of $15\%$ of the chord. Figure 2(a) show a schematic of
the rectangular wing.

We denote the chord-length of the wing as $C$, and the wing is placed in a
uniform incoming freestream flow which has velocity $U_{\infty}$. The chord-
based Reynolds number of the flow is $Re=U_{\infty}C/\nu=1000$. The flow over
both wings is simulated in a large computational domain that has dimensions
$18C\times 20C\times 20C$ in the streamwise, spanwise and “wall-normal”
direction. The domain is discretized using a Cartesian grid which is isotropic
around the wing and is expanded away from the wing. The grid consists of
$192\times 128\times 256$ grid cells in the $X$, $Y$ and $Z$ directions. This
corresponds to a resolution of approximately 72 points across the chord and
144 points across the span, which we have demonstrated in previous work [40]
results in well-resolved flow features in the low Reynolds number flows
studied here. A schematic of the computational domain and the Cartesian grid
is shown in figure 2(b). A Dirichlet velocity boundary condition is specified
at the upstream (inlet) boundary of the domain, and homogeneous Neumann
boundary conditions are specified at all other boundaries.

##  IV Results

Figure 3: Vortex-induced lift on the rectangular wing. (a)-(b) Isometric and
top-view, respectively, of iso-surfaces of $Q$ at levels [-2.0,2.0], coloured
by $-2Q\phi_{2}$. (c) Comparison of total lift coefficient ($C_{L}$) with
vortex-induced lift ($C^{\omega}_{L}$).

We begin our analysis with a discussion of the vortex-induced lift on the
rectangular wing described in section III.2 above. Figures 3(a)-(b) show
instantaneous snapshots of the flow, visualized using iso-surfaces of $Q$ (at
levels [-2.0,2.0]). The rotation-dominant and strain-dominant structures shown
are colored by the integrand of the vortex-induced force, $-2Q\phi_{2}$, which
indicates their local contributions to lift production. We see that this flow
consists of a variety of vortex structures, both in the vicinity of the wing
as well as in the wake. In particular, we highlight the generation and
shedding of a vortex caused by the roll-up of the leading-edge shear layer
over the suction surface of the wing as well as prominent tip-vortices. We
refer to the former as the leading-edge vortex (LEV). We see that as the LEV
sheds into the near-wake, its structure gets slightly deformed from its
initial span-wise orientation. Farther downstream, the effect of vortex
tilting and stretching transform it into a horseshoe-type structure. We will
show in the subsequent discussion that these effects have important
consequences on lift production. Figure 3(c) compares the total lift
coefficient on the wing ($C_{L}$) to the vortex-induced contribution
($C^{\omega}_{L}$). It is clear that the pressure load due to vortex-induced
effects accounts for the bulk of the lift production on the wing (about 90%),
as is expected. The small difference between the total lift and vortex-induced
lift is primarily due to the pressure force induced by viscous momentum
diffusion (third term in equation 3), which accounts for 10%. Viscous shear
produces a very small and negative contribution. We do not focus on these
viscous effects in this work.

###  IV.1 Vortex-induced lift: Contribution of spanwise and cross-spanwise
vorticity

We now examine the roles that different vortex structures (and their
associated strain-fields) play in the production of lift on the wing. As
mentioned in section II, of particular focus in this work is the separate
contributions of spanwise and cross-spanwise oriented vorticity in the flow.
These contributions are isolated in a very simple manner using the angle of
the vorticity unit vector, defined as in equation 5, computed at every point
in the domain. In figures 4(a) and 4(b), we visualize the flow features that
result from this segmentation by plotting $Q$ iso-surfaces (levels [-2.0,2.0])
colored by $-2Q\phi_{2}$ as in figure 3, and masking the regions that
correspond to $\eta_{y}<0.7$ and $\eta_{y}>0.7$ respectively.

Figure 4(a) shows that the spanwise-oriented vorticity, $\eta_{y}>0.7$, mainly
isolates the LEV and shear-layer over the surface of the wing, along with the
shedding of the LEV in the near-wake. Farther downstream, there are smaller
regions of spanwise vorticity that result from vortex interactions, breakdown,
etc. On the other hand, the regions of $\eta_{y}<0.7$ shown in figure 4(b)
correspond to structures that are largely streamwise oriented, consisting of
the tip vortices (although figure 4(a) shows some spanwise vorticity near the
tips, this has a small effect) and a dense region of vortex structures in the
mid-span portion of the near-wake. These streamwise structures are a result of
vortex-tilting and stretching due to shear in the flow, and include the “legs”
of the horseshoe vortex that originate from the LEV. Overall, this simple
segmentation of rotation-dominated and strain-dominated structures based on
the orientation of the vorticity unit vector does a reasonable job of
isolating important features of the flow – the shear-layer, LEV, tip vortices,
as well as features that result from the shedding, deformation and
tilting/stretching of these structures.

Figure 4: Spanwise and cross-spanwise vorticity for the rectangular wing.
(a)-(b) Top-view of $Q$ iso-surfaces (levels [-2.0,2.0]) colored by
$-2Q\phi_{2}$. Spanwise vorticity structures are shown in (a), by masking
$|\eta_{z}|\leq\mathrm{cos}(\pi/4)$ regions; cross-spanwise vorticity
structures are shown in (b), by masking $|\eta_{z}|>\mathrm{cos}(\pi/4)$
regions. (c) Comparison of total vortex-induced lift coefficient
($C^{\omega}_{L}$) with contributions from spanwise ($C^{\omega z}_{L}$) and
cross-spanwise ($C^{\omega xy}_{L}$) vorticity.

We can now use equation 4, along with the segmentation described above, to
quantify the contributions of the spanwise and cross-spanwise vorticity-
containing regions to the total vortex-induced lift. Figure 4(c) shows this
comparison as a function of time, from which several interesting observations
can be made. We see that early in the simulation, as the flow develops over
the wing, the total vortex-induced lift ($C^{\omega}_{L}$) goes through a
large-magnitude transient phase. This large lift peak is caused by the
formation and eventual shedding of the spanwise oriented dynamic stall vortex.
Soon after this transient, we observe a large drop in the spanwise lift
contribution, along with a growth in the cross-spanwise contribution. In fact,
the spanwise vorticity eventually has a lower overall contribution to the
total lift than the cross-spanwise vorticity. After the initial transient, the
contribution of spanwise vorticity to the total lift and vortex-induced lift
are approximately 26% and 29% respectively, while the cross-span vorticity
contribute approximately 60% of the total lift and 66% of the vortex-induced
lift (the remaining 5% of vortex-induced lift is accounted for by the flow
outside the analysis sub-domain described in section II). This is a surprising
and counter-intuitive result given our conventional understanding that the
lift over a wing would be dominated by spanwise oriented vortex structures.
Another interesting aspect of this comparison is that the growth and
subsequent fluctuations of the lift induced by cross-spanwise vorticity
mirrors the corresponding fluctuations in spanwise vorticity. This hints at
the role of vortex tilting in the creation of cross-spanwise vorticity from
spanwise vorticity, which is initially the dominant component according to
figure 4(c). In the rest of this paper, we further dissect these lift
components into spatial regions as well as rotation-dominated and strain-
dominated contributions to uncover additional insights and explain the
counter-intuitive dominance of cross-spanwise vorticity in lift production.

###  IV.2 Lift induced by spanwise and cross-spanwise vorticity in the wake

In order to better understand the relative contributions of cross-spanwise and
spanwise vorticity to the total vortex-induced lift, we now analyze their
local contributions within different spatial regions over the wing as well as
in the wake of the wing. Our interest in analyzing contributions in the wing’s
wake is motivated by the observation in section IV.1 and figure 4 that the
growth of cross-spanwise vorticity-induced lift is driven by the shedding,
tilting, and stretching of spanwise vorticity in the wake. This is evident
from the emergence of significant cross-spanwise vorticity in the wake of the
wing, as shown in figure 4(b), as well as the correspondence between the lift
induced by cross-spanwise and spanwise vorticity which follows the initial
dominance of the spanwise component in figure 4(c). Therefore in this section,
we analyze this behavior by quantifying the local lift induced by these
structures at their inception over the wing and also at increasing downstream
distances from the wing as they evolve with the flow in the wake.

Figure 5: (a) Schematic of distinct cuboidal integration volumes at different
downstream distances from the wing. Iso-surfaces and colored contours are $Q$
and $-2Q\phi_{2}$ respectively (see figure 3). Note that schematic does not
show full extent of integration region in the wake. (b) Time-averaged vortex-
induced lift coefficient on the rectangular wing as a function of downstream
distance from the mid-chord due to all vortex structures ($C^{\omega}_{L}$),
spanwise vorticity containing structures ($C^{\omega z}_{L}$) and cross-
spanwise vorticity containing structures ($C^{\omega xy}_{L}$).

The lift induced by structures corresponding to spanwise and cross-spanwise
vorticity within these local regions is estimated by dividing the analysis
domain into cuboids that enclose the different regions of interest. The
streamwise extent of each of these cuboids is shown in the schematic in figure
5(a), where the vertical lines are planes normal to the streamwise direction.
These streamwise-normal planes serve as a simple way to demarcate regions of
the flow at increasing downstream distances from the wing. In this analysis,
each region has a length of 1 chord-length in the streamwise direction, and
they span the size of the analysis subdomain (see last paragraph of section
II) in the other two directions. The region of interest over the wing is
placed so that it is centered at mid-chord, i.e. its upstream and downstream
ends are close to the leading and trailing edges respectively. The lift
induced by spanwise and cross-spanwise vorticity within each of these regions
is then quantified by performing the volume integral in equation 4 within each
of these regions separately.

In figure 5 we plot time-averages of the total vortex-induced lift coefficient
($C^{\omega}_{L}$), as well as spanwise ($C^{\omega z}_{L}$) and cross-
spanwise ($C^{\omega xy}_{L}$) vorticity contributions, within each of these
regions as a function of downstream distance from the wing. We first point out
that the largest portion of lift is indeed induced within the region over the
wing (downstream distance equals 0), and this lift production is dominated by
spanwise oriented vorticity. As shown earlier in figure 4(a), the spanwise
oriented vorticity over the wing largely corresponds to the LEV and leading-
edge shear-layer, which we expect to have a significant contribution to lift.
Figure 5 also shows that the cross-spanwise vorticity in this region, which we
saw earlier in figure 4(b) mainly consists of the tip vortices, has a negative
effect on lift. This is expected, and is in line our understanding of
classical (inviscid) aerodynamics and the downwash-induced effect of tip
vortices on finite wings. Therefore on the whole, the lift production
mechanisms in the region over the wing align with expectations from theory and
prior work.

As we move into the wake, we see that the total vortex-induced lift displays
an unexpected non-monotonic trend with downstream distance. It first begins to
reduce in magnitude within the near-wake compared to the region over the wing.
Farther downstream at about 2 chord-lengths from mid-chord (or 1.5 chord-
lengths from the trailing-edge) the vortex-induced lift in fact has a negative
value with magnitude approximately 27% of the peak value over the wing. This
is rather surprising as we expect the effect of vortices far away from the
wing to have a monotonically diminishing effect. It is clear that this
negative value of vortex-induced lift in the wake is driven by a precipitous
drop in the lift induced by spanwise oriented vorticity, which also turns
negative within this region. In fact, the magnitude of the negative lift
induced by spanwise vorticity approximately two chord-lengths downstream is
roughly 53% of its peak value over the wing, which is quite significant.
Simultaneously, we see a growth in the lift induced by cross-spanwise
vorticity, although its magnitude is not large enough to offset the negative
lift induced by spanwise structures. All three curves peak at approximately 2
chord-lengths downstream, and subsequently approach zero as we move farther
downstream due to their increasing distance from the wing. This non-monotonic
trend in vortex-induced lift as we move into the wake of the wing, and
particularly the fact that spanwise vorticity in the wake induces negative
lift on the wing while cross-spanwise wake-vorticity induces positive lift, is
an intriguing result that is subjected to further analysis.

###  IV.3 The role of rotation-dominated and strain-dominated regions in lift
generation

It has been shown in previous work that although much of the focus in the
literature concerning aerodynamics has been on the role that vortices play in
force production, vortices are also associated with regions of strain that in
fact, have a significant dynamical effect [24]. In the context of this study,
we hypothesize that the drop in lift due to spanwise structures and the
subsequent negative lift might be a result of the development and growth of
vortex-induced strain associated with these vortices after they shed into the
wake of the wing. The strain-dominated regions, which correspond to $Q<0$,
would induce negative lift according to equation 4. Furthermore, it can be
shown using simple vortex models that vortex cores, i.e. regions of $Q>0$, are
generally associated with higher vorticity magnitudes whereas the regions of
strain around vortices are larger diffused region with lower magnitudes of
vorticity. As a result, these different regions would be affected to different
degrees by vortex stretching and tilting due to the non-linear nature of these
mechanisms, and can therefore have different effects on the lift production in
the wake.

To investigate the effect of rotation and strain further, we decompose the
vortex-induced lift associated with spanwise and cross-spanwise vorticity into
their corresponding rotation-dominant and strain-dominant regions. This is
done by first identifying spanwise and cross-spanwise oriented structures
using the vorticity unit vector as before, and then further decomposing each
of these into regions of $Q>0$, corresponding to vortex cores, and $Q<0$,
corresponding to strain-dominated flow. These 4 separate components of the
flow-field are then used to estimate their contributions to lift within each
of the integration regions described earlier using equation 4.

Figure 6: Time-averaged vortex-induced lift coefficient on the rectangular
wing as a function of downstream distance from the mid-chord due to vortex
cores ($Q>0$) and strain regions ($Q<0$) corresponding to: (a) Spanwise
vorticity containing structures ($C^{\omega z}_{L}$); (b) cross-spanwise
vorticity containing structures ($C^{\omega xy}_{L}$).

In figure 6(a) we show the time-averaged vortex-induced lift due to the
rotation-dominated and strain-dominated regions associated with spanwise
oriented vorticity. We first note that within the region around the wing
(integration volume index $0$ in figure 6), the positive lift due to rotation-
dominated regions of spanwise vorticity, shown in red in figure 6(a), are the
dominant lift producing mechanism. As discussed before, this corresponds to
the effect of the LEV and the leading edge shear layer. Conversely, from
figure 6(b), which shows the corresponding rotation and strain-dominated
contributions from cross-spanwise oriented vorticity, we see that the cross-
spanwise vorticity in the vicinity of the wing (i.e. primarily the tip
vortices) produces negative total lift, as highlighted earlier. By decomposing
the effect of cross-spanwise rotation and strain, we see that the core of the
tip vortices ($Q>0$) in fact produces positive lift, but the total effect of
the tip vortices is negative due to the strain regions ($Q<0$). We point out
that this is in fact consistent with classical finite-wing-theory where the
induced velocity due to tip vortices is implicated in lift reduction. In this
case, the regions of strain around the tip vortex core correspond to the
induced velocity.

As we move downstream into the wake, figure 6(a) indicates a large drop in the
magnitude of positive lift induced by spanwise vortex cores ($Q>0$),
accompanied by a smaller change in the magnitude of negative lift due to
spanwise strain-dominated ($Q<0$) regions. The combined effect of this
disparity between these strain and rotation-dominated regions is the
production of net negative lift in the wake due to spanwise oriented
vorticity. On the other hand, figure 6(b) shows the opposite behaviour for
flow structures containing cross-span oriented vorticity in the wake. There is
a large increase in the magnitude of positive lift induced by cross-spanwise
vortex cores ($Q>0$) two chord-lengths downstream of the wing, and a smaller
increase in the magnitude of negative lift due to cross-spanwise straining
($Q<0$) regions. The net effect of this is positive lift induced in the wake
by cross spanwise oriented vorticity. Therefore we see that the vortex cores
and strain-dominated regions associated with spanwise and cross-spanwise
oriented vorticity behave very differently as we move into the wake –
resulting in negative lift induced by spanwise vorticity and positive lift
induced by cross-span vorticity in the wake.

It is important to note here that the trends in lift induced by the rotation
and strain-dominated regions of spanwise and cross-span vorticity shown in
figure 6 can in fact be used as a proxy to analyze the relative strength of
the four vorticity components (i.e. rotation and strain-dominated regions for
spanwise and cross-span vorticity). This is clear by noting that
$C^{\omega}_{F_{i}}\sim\int_{V_{f}}Q\,\phi_{i}\,dV$ and $\phi_{i}$ is the same
for spanwise and cross-span vorticity components within each integration
volume. Therefore, the different rates of increase/decrease in the lift
induced by vortex cores and their associated strain-dominated regions
highlights some important flow physics that affects the balance of these
vorticity components in the wake.

For one, as pointed out earlier, the larger magnitude of vorticity within
vortex cores than their associated strain regions makes vortex cores more
susceptible to vortex tilting and stretching due to the non-linearity of these
mechanisms. Therefore, we expect the spanwise vortex cores to experience more
vortex tilting than their associated straining regions, resulting in more
spanwise vortex cores being converted to cross-span vorticity. This is
reflected in figure 7(a) in the more substantial decrease of lift induced by
spanwise vortex cores as compared with their associated strain regions as we
move into the wake. Furthermore, the growth of vortex-induced strain after the
shedding of the LEV and other spanwise oriented vortices in the wake is
another factor that augments strain-induced (negative) lift associated with
spanwise vorticity in the wake. This strengthening of strain-dominated flow
(in addition to less tilting) leads to a much smaller change in the magnitude
of negative strain-induced lift as we move into the wake of the wing. To
illustrate this latter effect, figure 7 shows snapshots of the time-averaged
$Q$ field in the wake of the rectangular wing. Figure 7(a) shows a slice of
the field in a $X$-$Y$ plane at the mid-span of the wing and figure 7(b) shows
a slice of the field at a location between mid-span and the wing-tip. At both
locations, the time-averaged distribution of $Q$ shows the emergence of strong
region of strain ($Q<0$ in blue) in the wake of the wing. This region would
therefore enhance the negative lift induced in the wake, as per equation 4,
and contributes to a net negative lift induced by spanwise vorticity in the
wake.

Figure 7: Contours showing the time-averaged $Q$ field around the rectangular
wing. (a) $X$-$Y$ slice of the domain at mid-span. (b) Slice between mid-span
and wing-tip.

In the case of the cross-spanwise vorticity in the wake of the wing, the
larger increase in the strength of vortex cores compared to strain-dominated
regions seen in figure 7(b) can be explained by the preferential tilting of
spanwise oriented vortex cores, as discussed above, to create cross-spanwise
vortex cores (streamwise and horse-shoe vortices), as well as the preferential
stretching of the streamwise vortex cores. This therefore strengthens the
positive lift induced by vortex cores to a larger degree than than the
magnitude of negative lift induced by strain-dominated regions – thus leading
to net positive lift induced by cross-span vorticity in the wake. Hence we see
that the relative strengths of the the rotation and strain-dominated flow
corresponding to spanwise and cross-span oriented vorticity, which is mediated
by non-linear vortex stretching and tilting mechanisms, has interesting
consequences for the overall lift induced on the wing.

##  V Summary and concluding remarks

We have demonstrated the use of a rigorous force partitioning method to
dissect the physics of lift production in the vortex-dominated flows over a
low aspect-ratio wing. We specifically focus on partitioning the vortex-
induced force induced on a 2:1 aspect-ratio rectangular wing by flow
structures containing spanwise and cross-spanwise oriented vorticity. The
force partitioning method allows us to quantify the pressure-induced force
contributions due to these distinct vorticity components, as well as to
analyze their influence within different spatial regions of the flow-field.

We showed that the lift production on this low-aspect ratio wing is dominated
by the overall contribution of cross-spanwise vorticity. Flow structures
containing cross-spanwise vorticity contribute approximately 60% of the total
lift and 66% of the vortex-induced lift, which is significantly higher than
the approximately 26% and 29% contributed by spanwise vorticity to the total
lift and vortex-induced lift respectively. This is counter-intuitive as we
expected the spanwise vorticity within the LEV to dominate lift production. To
understand this better, we further decomposed these lift-producing components
into their contributions within regions of the flow over the wing and in the
wake of the wing. Surprisingly, we found that the spanwise component of
vorticity in the near-wake region induces a negative lift on the wing.
However, the cross-spanwise vorticity induces positive lift in this region. We
explained this behaviour by separately quantifying the influence of vortex
cores and their associated regions of strain in the wake for the spanwise and
cross-spanwise vorticity. We showed that while the positive lift production
due to spanwise vortex cores drops rapidly in the wake due to the tilting of
spanwise vortex cores, the negative lift induced by the strain-dominated
regions of spanwise vorticity does not change as much because of the growth of
vortex-induced strain associated with the shed vortices in the wake. This
difference in strengths of the spanwise oriented vortex cores and their
associated strain regions in the wake of the wing therefore results in a net
negative lift induced by spanwise oriented vorticity in the wake. On the other
hand, positive lift induced by cross-spanwise oriented vortex cores in the
wake is enhanced by the tilting of spanwise vortex cores to form cross-span
structures and the preferential stretching of these vortex cores. Since their
associated strain-dominated regions are affected less by these mechanisms,
cross-spanwise vorticity in the wake produces net positive lift on the wing.
Hence we showed that the interplay between rotation and strain-dominated
regions of spanwise and cross-spanwise oriented vorticity, which is dictated
by vortex stretching and tilting, encodes interesting physics that plays a
significant role in lift generation.

We expect that a better understanding of the physics underlying vortex-induced
lift production will aid the development of physics-based flow control
techniques as well as the design of novel wing designs/features for lift
enhancement. For instance, several efforts have been made to modify the wing
tip of aircrafts to reduce the deleterious effects of the tip-vortices [41,
42], and the analysis described here could help in designing and assessing
such modifications. While this study has demonstrated the utility of the force
partitioning method in uncovering fundamental aspects of the force production
in such flows, there are several questions that remain to be answered. For
one, we have not analyzed how the behavior discussed in this work changes for
wings of different aspect-ratios and Reynolds numbers. The work of Taira and
Colonius [15] has shown that aspect-ratio has a significant effect, while the
effect of Reynolds number (at low $Re$) is somewhat smaller. Such analyses are
therefore future directions for this work.

###### Acknowledgements.

RM acknowledges support from the Army Research Office (Cooperative Agreement
W911NF2120087) for this work. This work also benefited from the computational
resources at the Advanced Research Computing at Hopkins (ARCH) core facility
(rockfish.jhu.edu), which is supported by the AFOSR DURIP grant
FA9550-21-1-0303, and the Extreme Science and Engineering Discovery
Environment (XSEDE), which is supported by National Science Foundation grant
number ACI-1548562, through allocation number TG-CTS100002.

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