1. Introduction
The generation of time-varying flow-induced forces on bodies immersed in flows is central to a multitude of flow problems ranging from biological and bioinspired locomotion in fluids (Birch, Dickson & Dickinson Reference Birch, Dickson and Dickinson2004; Lentink & Dickinson Reference Lentink and Dickinson2009; Seo & Mittal Reference Seo and Mittal2022) to lifting surfaces such as wings (Lambert et al. Reference Lambert, Stanek, Gurka and Hackett2019; Jardin, Choi & Colonius Reference Jardin, Choi and Colonius2021) and rotors (Raghav & Komerath Reference Raghav and Komerath2015; Nabawy & Crowther Reference Nabawy and Crowther2017) and thrust (propulsors, propellers, etc.) and power (pumps, turbines, etc.) generating devices. The time variation in these forces can result from intrinsic variations in the flow (such as through vortex shedding, turbulence etc.), from extrinsic effects (such as acceleration/deceleration of the body) or through the combined effect of both. Understanding the mechanisms and/or flow features that give rise to these transient forces is of critical importance since the transients can either be desirable (i.e. transient lift forces are key to high performance of flapping wings/fins, see Birch et al. Reference Birch, Dickson and Dickinson2004; Lentink & Dickinson Reference Lentink and Dickinson2009; Seo & Mittal Reference Seo and Mittal2022) or undesirable (i.e. transient forces can generate flow noise, see Brentner & Farassat Reference Brentner and Farassat2003; Prakhar, Seo & Mittal Reference Prakhar, Seo and Mittal2025) and drive flow-induced vibrations/flutter (Williamson & Govardhan Reference Williamson and Govardhan2004). Methods that can enable an understanding of the causality of these force transients through analysis of simulation or experimental data could be used to inform design changes, operational modification or flow control strategies that accentuate (in the former case) or mitigate (in the latter) these effects.
Determining the source of these force transients is, however, challenging, particularly in three-dimensional configurations at moderate- to high-Reynolds-number flows. These flows contain a wide range of highly dynamic vortex structures which can all influence the pressure loading to different degrees. Consider the flow of interest here: the flow associated with an impulsively started rotor blade at a span-based Reynolds number of 25 500 (figure 1). Figure 2(a) shows the lift force versus time for approximately the first quarter revolution, and we note three distinct early-time extrema in the lift force; an initial maximum at
$t/T$
= 0.048, a minima at
$t/T$
= 0.109 and a second maximum at
$t/T$
= 0.213.
Flow schematic (not to scale) for the revolving blade showing the problem configuration with the origin shown at the centre of revolution.
(a) The pressure lift coefficient normalised based on the tip velocity,
$\rho$
is the fluid density and the blade area (
$({1}/{2})\rho v_t^2 A_{\kern-1pt B}$
) is shown with the vertical red lines shown at
$t/T=$
0.048, 0.109 and 0.213 and the corresponding flow field shown in (b), (c) and (d), respectively, using iso-surfaces of
$Q$
. (e) Flow field shown at a later time,
$t/T=0.29$
. Here,
$T=2\pi /\varOmega _z$
is the revolution period,
$C_Q=L_Q/(({1}/{2})\rho v_t^2 A_{\kern-1pt B})$
is the vortex-induced lift coefficient and
$L_Q$
is the vortex induced lift force defined later in (2.5).
The vortex structures corresponding to these time instances and
$t/T$
= 0.29 are shown in figure 2(b–e). At
$t/T$
= 0.048, we note a very coherent, initial leading-edge vortex (LEV) that increases in strength towards the blade tip. We also observe a small tip vortex that connects to the starting vortex that is released from the trailing edge into the wake. At
$t/T$
= 0.109, the initial LEV is starting to separate from the outer regions of the blade and braid vortices and other instabilities begin to manifest. A smaller LEV is also observed to form at this time. The situation at
$t/T$
= 0.213 is difficult to parse since the flow has already transitioned to turbulence at this point and all the vortex structures are seen to merge in one conglomeration that extends from approximately 24 % span to the tip and into the wake. At the later time of
$t/T$
= 0.29, the vortex structures look similar to those at
$t/T$
= 0.213, but the lift has reduced by approximately 13 %.
While we generally understand that the formation of a LEV enhances lift (Polhamus Reference Polhamus1966; Eldredge & Jones Reference Eldredge and Jones2019) and the separation of a LEV (also sometimes referred to as dynamic stall) diminishes lift (Tsang et al. Reference Tsang, So, Leung and Wang2008; Eldredge & Jones Reference Eldredge and Jones2019), we still lack quantitative ways of verifying these concepts from data coming from simulations or experiments of complex three-dimensional (3-D) flow configurations. One relevant study in this regard is that of Jardin et al. (Reference Jardin, Choi and Colonius2021), who showed that reasonable estimates of the lift for accelerating wings could be obtained using the circulation and position of the LEV estimated from simulation data. However, it is not clear how their model, which they applied to a low Reynolds number (Re = 500 airfoil) 2-D flow, can be extended to much higher-Reynolds-number flows over 3-D bodies such as the rotor blade. Ōtomo et al. (Reference Ōtomo, Henne, Mulleners, Ramesh and Viola2021) presented a model that combined classic Theodorsen and thin-airfoil theories to predict the unsteady lift on a nominally 2-D flapping wing from wing kinematics only. However, they noted that the interaction of the late-forming trailing-edge vortex (TEV) with the LEV generates errors in the model prediction. Furthermore, their vortex-lift correction requires the calculation of vortex circulation, which is not applicable to 3-D configurations. Indeed in situations such as at
$t/T$
= 0.213 and
$t/T$
= 0.290, it is difficult to even clearly identify distinct vortex structures for which circulation can be estimated.
We therefore need a data-enabled framework that applies to complex flow fields emerging from high-Reynolds-number, fully 3-D flows. Motivated by this, we employ the force partitioning method (FPM, see Menon & Mittal Reference Menon and Mittal2021a , Reference Menon and Mittalb , Reference Menon and Mittalc ) to develop a method to address precisely this issue – the connection of the extrema in the lift of this impulsively accelerated blade to the vortical features of the flow.
2. Methodology
2.1. Flow solver and configuration
We use our in-house flow solver called ViCar3D (Mittal et al. Reference Mittal, Dong, Bozkurttas, Najjar, Vargas and von Loebbecke2008; Seo & Mittal Reference Seo and Mittal2011) to solve the following set of incompressible Navier–Stokes equations written in a non-inertial rotating reference frame (Speziale Reference Speziale1989; Cariglino, Ceresola & Arina Reference Cariglino, Ceresola and Arina2014):
\begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol {u} & = 0 \,\,\, , \\[-12pt] \nonumber \end{align}
\begin{align} \frac {\partial \boldsymbol {u}}{\partial t} + \left ({\boldsymbol {u}-{\boldsymbol{u_b}}}\right )\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol {u} & = -\frac {1}{\rho }\boldsymbol{\nabla }P +\nu {\nabla} ^2 \boldsymbol {u} -\boldsymbol{\varOmega } \times \boldsymbol {u}, \\[6pt] \nonumber \end{align}
where
$\boldsymbol {u}$
is the flow velocity in the absolute frame,
$P$
is the pressure,
$\nu$
is the kinematic viscocity,
${\boldsymbol{u_b}}={\boldsymbol{\varOmega }}\times \boldsymbol {r}$
,
$\boldsymbol{\varOmega }=[0,0,\varOmega _z]$
is the reference frame rotation rate vector and
$\boldsymbol{\varOmega } \times \boldsymbol {u}$
is the Coriolis force term. The solution is second-order accurate in both space and time (Mittal et al. Reference Mittal, Dong, Bozkurttas, Najjar, Vargas and von Loebbecke2008; Seo & Mittal Reference Seo and Mittal2011). A variety of cases where this solver has been used can be found in Mittal & Seo (Reference Mittal and Seo2023) and Mittal et al. (Reference Mittal, Seo, Turner, Kumar, Prakhar and Zhou2024).
The simulations employ a rotor blade which is a zero thickness, aspect ratio
$(\textit{AR})=\zeta /c=5$
rectangular flat plate (here
$c$
is the rotor chord). The blade pitch angle is set to
$45^\circ$
and the Reynolds number based on the span (
$\zeta$
) and tip velocity (
$v_t$
) is 25 500. The domain is
$10R \times 10R \times 8R$
, where
$R$
is the outer radius of the blade, and the revolution centre is placed at the middle of the domain. The blade initiates an impulsive revolution at
$t=0$
with an angular velocity of
$\varOmega _z$
which remains constant afterwards. Numerically, this results in an angular acceleration of
$\varOmega _z/\Delta t$
at
$t=0$
, where
$\Delta t=0.0004/\varOmega _z$
is the time step size used in the simulation. A grid with 220 million points is used and the grid independence study is summarised in Appendix A.
2.2. The force partitioning method
The pressure forces dominate the forces on the rotor blade at these Reynolds numbers and the FPM (Zhang, Hedrick & Mittal Reference Zhang, Hedrick, Mittal and Swartz2015; Menon & Mittal Reference Menon and Mittal2021b ), based on the original formulation of Quartapelle & Napolitano (Reference Quartapelle and Napolitano1983), enables us to decompose the surface pressure force into components associated with vortices (vortex-induced force), the forces associated with acceleration reaction and pressure force due to viscous diffusion of momentum. The details of FPM and its implementation for immersed bodies can be found in Menon & Mittal (Reference Menon and Mittal2021a ,Reference Menon and Mittal b , Reference Menon and Mittalc ) and Seo & Mittal (Reference Seo and Mittal2022) with an application to this same rotor blade in Prakhar et al. (Reference Prakhar, Seo and Mittal2025). However, a brief description is provide here for completeness. In the current case, there is an additional component due to the Coriolis force term but it makes negligible contribution to the lift component. We also note that, at the constant rotation rate beyond the first time step, the acceleration reaction can only contribute to the spanwise force, but is zero for the present rotor blade geometry.
The ‘influence field’ (
$\phi$
) in the direction of lift (
$z$
) is obtained by solving the following equation (see figure 1):
\begin{equation} {\nabla} ^2\phi =0 \text{ in } V; \text{ with} \, \, \boldsymbol{\nabla }\phi \boldsymbol{\cdot }\boldsymbol {n}= \begin{cases} n_z & \text{on $B$},\\ 0 & \text{on $\varSigma $}\, , \end{cases} \end{equation}
where
$\varSigma$
is the domain boundary,
$B$
is the immersed body boundary and
$\boldsymbol{n}$
is the unit normal pointing into the immersed body. Figure 4 shows iso-surface of
$\phi$
along with contours at two spanwise planes over the blade. The gradient of this influence potential is then projected onto the momentum equation. This results in the following equation for the pressure (
$P$
) lift force:
\begin{equation} \begin{aligned}\underbrace {\int _B P n_z {\rm d}B}_{L} = &\underbrace {-\int _B\left ( \rho \frac {D\boldsymbol {u}}{Dt} \boldsymbol{\cdot }\boldsymbol {n} \right ) \phi {\rm d}B}_{L_B} - \underbrace {\int _\varSigma \left ( \rho \frac {D\boldsymbol {u}}{Dt} \boldsymbol{\cdot }\boldsymbol {n} \right ) \phi {\rm d}B}_{L_O} \\ &- \underbrace {\int _{V_f} 2 \rho Q \phi {\rm d}V}_{L_Q} +\underbrace {\int _{B+\varSigma } \left ( \mu {\nabla} ^2 \boldsymbol {u} \boldsymbol{\cdot } \boldsymbol {n} \right ) \phi {\rm d}B}_{L_\mu }.\end{aligned}\end{equation}
Here,
$L_B$
denotes the force due to the acceleration reaction of the body,
$L_O$
denotes the force due to free-stream unsteadiness,
$L_Q$
is the force due to the vortices and
$L_\mu$
is the force due to the viscous diffusion of the momentum, and
$\mu$
is the dynamic viscocity.
The quantitative partitioning of these force components is shown in figure 3. The numerical impulsive start with the angular acceleration of
$\varOmega _z/\Delta t$
at
$t=0$
generates added mass lift at the first time step, but this is zero for the subsequent time steps where
$\varOmega _z$
is constant.
FPM application to a rotor – the total pressure lift coefficient (
$C_L$
) and the four force partitions;
$C_\mu$
: viscous diffusion,
$C_Q$
: vortex induced,
$C_O$
: outer boundary,
$C_B$
: acceleration reaction (all normalised based on the tip velocity and blade area) versus time normalised by revolution period.
The viscous momentum diffusion component (
$L_\mu$
) contributes a small but non-negligible (4.5 %) component to the lift, but overall, as in other high-Reynolds-number cases (Seo & Mittal Reference Seo and Mittal2022; Raut, Seo & Mittal Reference Raut, Seo and Mittal2024) the vortex-induced force is the dominant contributor to lift generation (contributing 95.5 % of the lift). Thus, FPM leads to the following expression for the pressure lift (
$L$
) on the rotor blade:
\begin{equation} \begin{aligned} L = \int _B P \, n_z {\rm d}S \approx L_Q = -2\int _{V} \left (\rho Q \phi \right ){\rm d}V = \int _{V} l_Q \, {\rm d}V \end{aligned}, \end{equation}
where
$n_z$
is the vertical component of the surface unit normal pointing into the body and
$B$
and
$V$
are the body surface and fluid volume, respectively. Also,
$L_Q$
is the vortex-induced lift force, and
$l_Q$
is referred to as the vortex-lift density and it is equal to
$ - 2 \rho Q \phi$
, where
$Q = ({1}/{2}) (|| \boldsymbol{\omega }||^2- || \boldsymbol {S}||^2) \equiv - ({1}/{2})\boldsymbol{\nabla } \boldsymbol{\cdot }(\boldsymbol {u}\boldsymbol{\cdot }\boldsymbol{\nabla } \boldsymbol {u})$
(Hunt, Wary & Moin Reference Hunt, Wary and Moin1988), and
$\omega$
and
$S$
are the anti-symmetric and symmetric component of the velocity gradient tensor respectively. Given equation (2.5), we can estimate the pressure lift force induced on the surface from any volume segment in the fluid domain.
The iso-surface of
$\phi /R$
is shown on the left. Right – 2-D contour of
$\phi /R$
on two planes normal to the span – rotor mid-span and near the tip (97 % span) are shown.
3. Results
3.1. Temporal variation of lift
Figure 5(a) shows the spanwise variation of sectional pressure lift across the span at the key time instances that are our focus here. Additionally, the surface distribution of pressure difference across the rotor surface (
$\Delta \kern-1pt P/\rho v_{t}^2$
) is shown in figure 5(b–e), showing high net pressure difference near the leading edge and a much higher pressure difference near the tip at
$t/T=0.048$
due to the presence of a strong initial tip vortex. This pressure difference is reduced at
$t/T=0.109$
due to the shedding of the LEV and at later times
$t/T=$
0.213 and 0.29, a larger portion of the rotor shows higher net pressure difference due to the shedding of smaller vortices shown in figure 2(d,e).
(a) Sectional lift coefficient at key time instances. The rotor is divided into 15 parts along the span to show the contribution of each part to the lift force as the flow evolves. The vertical lines show the region over which spanwise average is computed for plots shown in figure 6(b–d). The normalised surface pressure difference (
$\Delta \kern-1pt P/\rho v_{t}^2$
, where
$\Delta \kern-1pt P$
is the difference between the pressure and the suction side of the rotor) is shown for the key time instances.
From the plot in figure 5(a), we note that a significant proportion of the total difference in lift between the key time instances is associated with the spanwise region
$0.62\lt r/R \lt 0.83$
. Note that
$t/T=0.048$
seems to be an outlier with the large sectional lift near the tip, but that increased sectional lift is compensated partially by the lower sectional lift for
$r/R \lt 0.6$
. Thus, we initially focus on this region of the blade and figure 6(a–c) shows a series of plots of the
$Q$
-field that is spanwise averaged over the region
$r/R$
of 0.62–0.83. We note that, at
$t/T$
= 0.048, where the first peak in the lift is found, the
$Q$
-field is dominated by a large LEV, a strong TEV and a small secondary vortex near the LEV that is induced by the reverse flow of the large LEV. As pointed by Menon & Mittal (Reference Menon and Mittal2021c
), there are significant regions of negative
$Q$
that surround each vortex core and these regions induce a positive pressure force on the body, as opposed to the negative (suction) pressure force induced by the vortex cores with positive
$Q$
. In fact, Menon & Mittal (Reference Menon and Mittal2021c
) noted that
$\int _V{Q}{\rm d}V$
is theoretically zero for flow around an immersed body. This integral might be non-zero but small in both simulations and experiments due to truncation and measurement errors respectively. Notwithstanding this error, we expect an overall balance between positive
$Q$
and negative
$Q$
in the flow domain and the non-zero vortex-induced force emerges from the difference in the distribution of these two quantities, and particularly, their proximity to the surface. Indeed, this proximity is exactly what is encoded in the influence field
$\phi$
and multiplication of
$Q$
with
$\phi$
results in the vortex force density which directly quantifies the force induced by every elemental region of the flow. Figure 6(d–f) shows contours of
$l_Q$
corresponding to this spanwise segment-averaged
$Q$
field and we note that the multiplication with
$\phi$
eliminates the TEV and also diminishes the outer corona of negative
$Q$
around the LEV. This results in a region of positive
$l_Q$
that is larger and more intense than that for negative
$l_Q$
. This observation can be connected with the presence of high lift at this time instance.
However, the above correlation is still observational in nature and this observational correlation become slightly more difficult to extend to
$t/T=0.109$
where the vortex field is significantly more complicated. At this time, the LEV is in the process of shedding from the rotor blade and the negative
$Q$
regions of both the LEV and the TEV combine to dominate the region around the blade. The
$l_Q$
field for this time instance is highly fragmented, making it difficult to draw clear conclusions, although the drop in lift coinciding with LEV shedding conforms to our understanding of these flows. Interestingly, Ōtomo et al. (Reference Ōtomo, Henne, Mulleners, Ramesh and Viola2021) noted the difficulty of predicting the lift precisely for this stage in the vortex shedding process.
The situation for
$t/T=0.213$
is much worse in this regard since the flow has already transitioned to turbulence and it is not possible to make observational deductions from the
$Q$
or the
$l_Q$
field. Thus, while the FPM improves our ability to focus on the observable (
$l_Q$
) that determines the pressure lift, we need additional ways to quantify the strength and proximity of the pressure inducing structures.
A spanwise-averaged slice in the range
$r/R=0.62 - 0.83$
is used to show the contour of
$Q$
at three key time instance – (a)
$t/T=0.048$
, (b)
$t/T=0.109$
and (c)
$t/T=0.213$
. The corresponding vortex-induced lift force density (
$l_Q$
) is also shown at (d)
$t/T=0.048$
, (e)
$t/T=0.109$
and (f)
$t/T=0.213$
.
(a) The weighted average of
$\phi$
using
$Q$
is denoted by
$\hat {\phi }_{\pm }$
and is shown for strain- (
$\hat {\phi }_{-}$
) and vortex-dominated (
$\hat {\phi }_{+}$
) regions. The net force is shown by multiplying
$\Delta \hat {\phi }_{\pm }$
with
$\hat {Q}$
. (b) The temporal variation of two components,
$\hat {Q}$
and
$\Delta \hat {\phi }_{\pm }$
and their product are shown. Both in this figure and in figure 8(a),
$\hat {\phi }_{\pm }$
is normalised by
$c$
,
$\hat {Q}$
is normalised by
$0.5\rho v_t^2 \zeta$
and
$\hat {Q}\Delta \hat {\phi }_{\pm }$
is normalised by the force coefficient (
$0.5\rho v_t^2 A_B$
).
(a) The spatial variation of
$\hat {Q}$
,
$\Delta \hat {\phi }_\pm$
, their product (
$\hat {Q}\Delta \hat {\phi }_\pm$
) and
$C_Q$
are shown along the span of the rotor. The values are temporal averages between
$t/T=$
0 and 0.3. (b) Iso-surfaces of
$Q$
temporally averaged between
$t/T$
= 0.0 and 0.3 and coloured using
$l_Q$
. (c)–(f) Two-dimensional contours of
$l_Q$
(also temporally averaged) are shown at four spanwise locations.
Here, we introduce a simple data-driven method for separating and quantifying the effects of vortex strength from vortex proximity and extracting greater insights from the FPM as applied to these highly transient vortex-dominated flows. We define
$Q$
-weighted influence fields for regions of positive and negative
$Q$
as follows:
\begin{equation} \hat {\phi }_{\pm }= \frac {-2 \int _{V} \rho \phi \, Q_\pm \,{\rm d}V}{\int _{V} \rho Q_\pm {\rm d}V} = \frac {L_{Q_\pm }}{\int _{V} \rho Q_ \pm {\rm d}V} \,\, , \end{equation}
where,
$Q_\pm (\boldsymbol{x})= ({1}/{2}) (Q (\boldsymbol{x})\pm |Q(\boldsymbol{x})| )$
denotes the value of
$Q$
in an elemental volume that contains either positive or a negative
$Q$
. The metrics
$\hat {\phi }_{\pm }$
may be viewed as factors that are proportional to the averaged proximity of the positive and negative regions of the
$Q$
field to the body of interest (
$B$
). Thus, higher (lower) values of these proximity functions increase (decrease) the induced pressure forces from these two partitions of the
$Q$
field. Now a metric for the overall strength of
$Q$
in the domain called the ‘
$Q$
-strength’ can be defined by
$\hat {Q} = \int _{V} \rho Q_+ {\rm d}V= -\int _{V} \rho Q_- {\rm d}V$
, then by definition
\begin{equation} L_Q = L_{Q_+}+L_{Q_-} = \hat {Q} \big(\hat {\phi }_{+} - \hat {\phi }_{-} \big) = \hat {Q} \Delta \hat {\phi }_{\pm } \end{equation}.
The above expression for vortex-induced lift therefore separates the effect of the net proximity from the body of the regions with positive and negative
$Q$
(via the expression in the parenthesis) from the effect of the overall
$Q$
-strength (
$\hat {Q}$
), and the rise and fall of the lift force can now be examined with respect to the distinct variations of these two effects. We note that Jardin et al. (Reference Jardin, Choi and Colonius2021) showed that, for low-Reynolds-number 2-D flows, reasonable estimates of the lift on accelerating wings could be obtained using the circulation and position of the LEV, and this implicates the
$Q$
-strength and location as the key parameters for lift as well.
Figure 7(a) shows the time variation of the two proximity functions and the time variation of the vortex-induced lift force
$L_Q$
. This plot shows that, indeed, the two proximity functions have different time variations although
$\hat {\phi }_{+}$
is always larger than
$\hat {\phi }_{-}$
, indicating (as expected) that the lift is always positive. Figure 7(b) shows the time variations of
$\Delta \hat {\phi }_{\pm }$
and
$\hat {Q}$
and this plot is highly revealing. We note that
$\hat {Q}$
rises rapidly in the very early stages up to
$t/T =0.20$
due to the continuous flux of vorticity from the body in response to the establishment of a pressure gradient along the body. Beyond this, the overall
$Q$
-strength levels off. In contrast, the proximity difference is initially large, indicating that regions of positive
$Q$
are closer to the body than regions of negative
$Q$
, but this quantity decreases rapidly. The combination of these two curves explains the time variation of the lift force. The first peak in lift at
$t/T=0.048$
coincides with the peak in
$\Delta \hat {\phi }_{\pm }$
. Thus, while the overall vorticity strength is relatively low, the high proximity of the positive
$Q$
(associated with the first LEV) at this time generates a large magnitude of lift. Beyond this time, the proximity difference falls rapidly as the LEV starts to shed. However, the
$Q$
-strength is rising at this time and these competing effects result in the lift minimum at
$t/T=0.109$
. Beyond this, the lift rises due to the increasing
$Q$
-strength and around
$t/T=0.2$
, the growth in
$\hat {Q}$
levels off and the continual slow decrease in the proximity difference starts to assert itself, thereby generating the peak at
$t/T=0.213$
. Thus the various extrema in this highly transient lift behaviour for this complex 3-D flow can be explained via the interplay between the
$Q$
-strength and vortex proximity.
3.2. Spatial variation of sectional lift
In the final section, we demonstrate that, in addition to explaining the temporal variation in forces, this approach can also be used to examine the spatial distribution of forces. The black line in figure 8(a) shows the spanwise variation of the sectional vortex-induced lift force on the blade, time averaged between
$t/T$
= 0 and 0.3, and we note that, after an initial decrease, there is a rapid increase in the sectional lift up to
$r/R$
= 0.36. Beyond this, the sectional lift is nearly constant until approximately
$r/R=0.95$
and then increases in the last 5 % section of the rotor. The plot of
$\hat {Q} \Delta \hat {\phi }_{\pm }$
follows the variation of the sectional vortex-induced lift very closely (it does not match exactly since we are not including the entire flow volume in the calculation of
$\hat {Q} \Delta \hat {\phi }_{\pm }$
) and thus, the spanwise variation in vortex-induced lift can indeed be understood by examining the
$Q$
-strength and the proximity functions.
The sectional variation of
$\hat {Q}$
and the proximity difference
$\Delta \hat {\phi }_{\pm }$
are also plotted in the graph. The
$Q$
-strength drops slightly between
$r/R$
values of 0.25 and 0.32 (due to the tip vortex on the internal tip, see figure 8
b) and stays small up to
$r/R$
= 0.54. It then rises more rapidly between
$r/R$
values of 0.54 and 0.7 where the primary LEV is growing and a secondary attached LEV as well as a trailing-edge vortex are also present, and then has a more gradual rise between
$r/R$
values of 0.7 and 0.92. Finally, the presence of the tip vortex results in a rapid increase in
$\hat {Q}$
between an
$r/R$
value of 0.92 and the tip. The contour of
$l_Q$
is shown at four key spanwise locations in figure 8(c–f).
The proximity difference has a variation that is quite different from that of the
$Q$
-strength. This parameter is also initially small but rises rapidly between
$r/R$
values of 0.25 and 0.5. This is connected with the presence of a very coherent and tightly rolled up attached LEV that is observed in many of the earlier vortex plots and in figure 8(b). Beyond an
$r/R$
of 0.5, the proximity difference parameter falls continuously, indicating that the LEV continues to move away from the rotor blade surface and regions of larger negative
$Q$
form near the blade. Note from figure 8(b) that this is the spanwise location where the LEV first starts to exhibit a full breakdown. The variation of the sectional lift can now be explained based on these observations. The initial decrease of sectional lift between
$r/R$
values of 0.25 and 0.32 is due to a drop in the
$Q$
-strength. Beyond this, the slight rise in
$Q$
-strength coupled with the rapid rise in the proximity difference lead to an increase in sectional lift until
$r/R$
= 0.5, where this growth is stopped due to the reduction in the proximity difference. Between
$r/R$
values of 0.5 and 0.95, the opposite trends in the
$Q$
-strength (increasing) and proximity difference (decreasing) results in a plateau in the sectional lift. The final increase in sectional lift beyond
$r/R$
of 0.95 is due to the increase in the
$Q$
-strength in the tip region.
4. Conclusions
Decades of research in fluid dynamics have revealed strong connections between the time-varying forces experienced by bodies immersed in flow and the formation and evolution of vortical structures over these bodies. These links often stem from the principle that vortex cores generate suction pressures, implicating features such as LEVs, tip vortices, dynamic stall, flow separation and vortex wakes. However, quite often, these connections are qualitative and leave significant room for ambiguity. In the current study, we have introduced a novel data-driven approach for establishing quantitative and more definitive connections between vortical features and the unsteady forces induced on an immersed 3-D body in high-Reynolds-number flow. Using the FPM, we derive two metrics, the
$Q$
-strength and the vortex proximity parameter, that enable us to provide quantitative measures of the effect of complex, dynamic, mutually interacting and evolving vortex structures on induced pressure forces. The method is demonstrated here by applying to simulation data for a revolving rotor blade at a relatively high Reynolds number, explaining the temporal and spatial variations in lift.
While the current focus is on lift, the method readily extends to drag and aerodynamic moments (Zhang Reference Zhang2015; Li et al. Reference Li, Wang, Graham and Zhao2020; Menon & Mittal Reference Menon and Mittal2021c
; Prakhar et al. Reference Prakhar, Seo and Mittal2025). It is applicable to both 2-D and 3-D flows, since vortex strength is quantified using the
$Q$
-field rather than circulation. In principle, the method can also be applied to experimental flow measurements, given prior successful implementation of FPM to Particle image velocimetry data (Zhu & Breuer Reference Zhu and Breuer2023; Zhu et al. Reference Zhu, Lee, Kumar, Menon, Mittal and Breuer2023). This approach holds promise for informing design modifications or control strategies aimed at optimising the performance of aerodynamic devices and control surfaces.
Finally, and more broadly, the present work, in our view, reinforces the unique role of
$Q$
in vortex identification. Unlike criteria such as vorticity or
$\lambda _2$
(Jeong & Hussain Reference Jeong and Hussain1995), which are based purely on flow kinematics,
$Q$
captures the action of vortex structures on bodies immersed in flows, thereby providing a clear purpose for the identification of vortices.
Funding
Support from ARO (W911NF2120087) and ONR (N00014-24-12516) is acknowledged.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Grid convergence
We employ a set of three meshes – coarse, medium and fine, with 8, 65 and 220 million grid points, respectively. The coefficient of the lift force is shown in figure 9(a) for 0.3 revolutions and we note that the results are well converged on the fine mesh with the mean value of per cent difference being around 3.9 % between the coarse and medium meshes and 0.31 % between the medium and fine meshes. The lift extrema, which are the focus of the current paper, are also well converged on the fine mesh. The fine mesh is used for all the results presented in this paper and this simulation consumed a total of approximately 350 000 CPU hours.
(a) Grid convergence of lift coefficient shown for coarse, medium and fine meshes containing 8, 65 and 220 million grid points, respectively. The lift is normalised based on the tip velocity and blade area, and the time is normalised by revolution period.
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