Wake Model

Lee et al.
Advances in Aerodynamics (2022) 4:20

https://doi.org/10.1186/s42774-022-00111-3
Advances in Aerodynamics
REVIEW Open Access
Review of vortex methods for rotor

aerodynamics and wake dynamics
H. Lee*, B. Sengupta, M. S. Araghizadeh and R. S. Myong
*Correspondence:
hlee@gnu.ac.kr Abstract
School of Mechanical Electric vertical take-off and landing (eVTOL) aircraft with multiple lifting rotors or prop-
and Aerospace Engineering,
Gyeongsang National rotors have received significant attention in recent years due to their great potential
University, Jinju, Gyeongnam for next-generation urban air mobility (UAM). Numerical models have been developed
52828, South Korea and validated as predictive tools to analyze rotor aerodynamics and wake dynamics.
Among various numerical approaches, the vortex method is one of the most suit-
able because it can provide accurate solutions with an affordable computational cost
and can represent vorticity fields downstream without numerical dissipation error.
This paper presents a brief review of the progress of vortex methods, along with their
principles, advantages, and shortcomings. Applications of the vortex methods for
modeling the rotor aerodynamics and wake dynamics are also described. However, the
vortex methods suffer from the problem that it cannot deal with the nonlinear aerody-
namic characteristics associated with the viscous effects and the flow behaviors in the
post-stall regime. To overcome the intrinsic drawbacks of the vortex methods, recent
progress in a numerical method proposed by the authors is introduced, and model
validation against experimental data is discussed in detail. The validation works show
that nonlinear vortex lattice method (NVLM) coupled with vortex particle method
(VPM) can predict the unsteady aerodynamic forces and complex evolution of the rotor
wake.
Keywords: Vortex methods, Nonlinear vortex lattice method, Viscous vortex particle
method, Rotor aerodynamics, Wake dynamics
1 Introduction
Recently, environmental concerns and energy consumption have motivated the aviation
industry to develop electric vertical takeoff and landing (eVTOL) aircraft with multiple
lifting rotors or prop-rotors. These can be used not only for military and emergency ser-
vices but also as next-generation urban transport systems in highly crowded megacities.
The versatility, VTOL capability, and maneuverability of rotorcraft make them excellent
candidates for many military missions and urban mobility services. Although turbojet
aircraft have recently become capable of vertical take-off and landing, rotorcraft still
have the lowest disk loading among all VTOL aircraft, making them an attractive and
efficient choice for the aviation industry.
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Lee et al. Advances in Aerodynamics (2022) 4:20 Page 2 of 36
Rotorcraft undergo complex aerodynamic phenomena due to sharp gradients of veloc-

ity and pressure near the blade tips, strong wake vortices, compressible dynamic stall,
large fluctuation amplitude, and unsteady flow reversal. Rotorcraft’s complex aerody-
namics create many challenges for engineers. Progress in the aviation industry has come
from initial experimental rotorcraft development, conducted in wind tunnel facilities,
along with simple mathematical calculations [1–4]. However, the limitation of experi-
mental studies is that they can consider only a small number of aerodynamic rotorcraft
layouts. Wind-tunnel tests also cannot be used to carry out multi-objective, multi-
parameter aerodynamic optimizations that address interactional effects. Nonetheless,
wind-tunnel testing is required as the last step in the manufacturing process to validate
numerical simulations [5–8]. A valuable list of empirical research done to study rotor
flow fields is provided in the references [9–11].
With improvements in computing power and resources, engineers are now able to
use numerical methods to analyze complex rotorcraft configurations and improve their
understanding of areas of rotor aerodynamics that had been difficult to investigate via
wind-tunnel tests [12–14]. A variety of numerical methods, ranging from engineering-
level models to intensive high-fidelity approaches, have been developed and validated as
predictive tools to analyze rotor aerodynamics and wake dynamics. The blade element
momentum (BEM) theory was derived by combining the basic principles of the one-
dimensional momentum theory and blade element theory. The BEM theory has a simple
structure and can rapidly calculate the rotor performance with very low computational
cost. Therefore, the BEM theory is an engineering-level model, and has been widely used
for design purposes. However, owing to its inherent assumptions, it can only describe
quasi-steady solutions and ignores mutual aerodynamic interactions between adjacent
blade elements. Computational Fluid Dynamics (CFD) methods were used to simu-
late rotor flow fields and had an undeniable impact on rotorcraft design developments.
CFD methods are the high-fidelity, expensive approaches used to predict unsteady and
transient phenomena by directly solving entire flow-fields containing rotor blades and
downstream regions. As computing performance and resources have begun to grow dra-
matically, industrial and academic research has been attempting to simulate the isolated
rotor, multirotor, and complete rotorcraft configurations using various CFD methods. In
general, they can be classified into the Reynolds-Averaged Navier–Stokes (RANS), Large
Eddy Simulation (LES), and Detached Eddy Simulation (DES) techniques depending on
turbulence models with the range of length and time scales. However, CFD methods
inherently suffer from grid-induced dissipation errors because of the numerical discre-
tization over the flow field. Excessive dissipation error caused by the numerical discre-
tization may result in rapid decay of the intensity of the rotor tip vortex downstream and
a breakdown of the helical rotor wake structure. Moreover, although many modifications
have been made to make CFD methods more efficient [15–18], high computational costs
and complicated equation setup still make these methods unfeasible for designers aim-
ing for rapid feedback to optimize their models [19, 20].
Accordingly, there has been increasing demand for a fast and reliable method to ana-
lyze new eVTOL aircraft. Vortex methods are becoming useful tools to predict the
unsteady flow physics and to study rotorcraft’s complex wake flow. Although vortex
methods are basically used for irrotational and incompressible flow, they can produce
acceptable and accurate results for complex configuration designs in less time than the
CFD methods [21–24]. For these reasons, most current comprehensive rotorcraft analy-
sis codes have adopted vortex methods, in which the lifting surface (e.g., rotor blade) can
be modeled in various ways, including the lifting line theory (LLT), Weissinger’s LLT
(extended lifting-line theory), the lifting surface method, and the source-doublet panel
method. LLT or Weissinger’s LLT use the simplest representation of the blade model
in terms of vortex singularities, called horseshoe vortex filaments, along the spanwise
direction. However, these models do not capture the three-dimensional effects on the
rotor blade because the lifting surface is represented by only a single chordwise vortex
element. The lifting surface method, referred to as the vortex lattice method (VLM), rep-
resents the curvature of the rotor blade using both the chordwise and the spanwise dis-
tribution of vortex ring elements on the actual camber surface. Therefore, the VLM has
been shown to give much better representation of the three-dimensionality of the flow
on the blade. The source-doublet panel method, in which the airfoil involved in the rotor
blade is divided into upper and lower panels, can consider the thickness effect of the
rotor blade. This is a major advantage compared to VLM. The rotor wake model is also
used to represent the wake structure and to describe vorticity fields in the wake through
the use of straight/curved vortex filaments or vortex particles. A vortex filament is the
concentrated vortices along a segment with a singularity at the center, whereas a vortex
particle is the concentrated vortices within a certain volume. These vortices are gener-
ated from the trailing-edge of each blade and convected downstream. Tables 1 and 2
present a simple overview of the blade and wake models, respectively, for the rotor simu-
lation reported in this paper.
The most challenging problems of rotorcraft can be divided into three fields of engi-
neering: aerodynamics, aeroacoustics, and aeroelastics. Researchers have used different
vortex methods to tackle these challenges. The main topics in rotor aerodynamics are
the influences of interactions among components [41, 45–47] and with nearby infra-
structure (e.g., brownout and shipboard operation) [48, 49], on rotorcraft performance,
stability, control, and safety. There have also been several investigations on the design
of rotorcraft configuration to improve aerodynamic performance [50–52]. The next
Table 1 Overview of blade models for rotor simulation

Methods Singularity Blade grid Computation time References
LLM Horseshoe vortex No grid (lifting line) Very fast Leishman [25]
Johnson [26]
Landgrebe et al. [27,
28]
VLM Vortex ring element Lattice on camber Fast (slower than DeYoung [29]
line LLM) Katz and Maskew [30]
Wachspress and Yu [31]
Panel method Source and doublet Panels on upper and Medium (slower than Rubbert and Saaris [32]
elements lower surface VLM) Hess [33], Crispin [34]
Wachspress et al. [35]
Hybrid method Horseshoe vortex Only volume grid Slow (Faster than only Rajmohan and He [36]
for flow field (CFD CFD simulation) Zhao et al. [37]
domain)
Bae and He [38]
Table 2 Overview of wake models for rotor simulation

Methods Singularity Volume grid References
Free-wake method Straight and curved No grid Wie et al. [22]

filaments Leishman and Bhagwat [39]
Lee and Na [40]
Vortex particle method Particles No grid Jang et al. [41], He and Zhao [42]
Winckelmans and Leonard [43]
Singh and Friedmann [44]
rotorcraft design challenge is to reduce their noise, especially in urban environments.

Blade-vortex interaction (BVI) is identified as the dominant source of noise in rotorcraft.
Vortex methods are powerful tools for researchers to study interactions of blade wake
vortices with each other as well as with other rotorcraft components [23, 53–55]. Rotor-
craft are strongly affected by unsteady aerodynamic loads (e.g., BVI), which significantly
contribute to vibration and structural deformation. Aeroelastic analysis has always been
a crucial step in designing rotorcraft. With the new generation of compound rotorcraft,
which generally have hingeless and bearingless rotors, rotor aeroelastic and response
problems are attracting great attention in the aviation industry. Vortex methods are reli-
able and low-cost aerodynamic approaches for studying aeroelasticity in rotary wing air-
craft [56–58].
The main objectives of this paper are to provide an overall review of vortex methods
including principles, advantages, and drawbacks, as well as development and applica-
tions (specifically in the rotorcraft industry). In addition, the authors’ recent progress in
their numerical method of modeling rotor aerodynamics and wake dynamics is intro-
duced, and its applications are discussed. The paper is organized as follows: Section 2
gives an overview of the principles of vortex methods and their fundamental formula-
tion. Section 3 explains the three most common vortex methods used to predict rotor
aerodynamics and introduces hybrid approaches. Applications of the vortex method
for modeling rotor wake dynamics are presented in Section 4, which reviews the time-
marching free-wake method and viscous vortex particle method in detail. Section 5
introduces our proposed nonlinear vortex lattice methods and compares our numerical
results with experimental data. Finally, concluding remarks are made in Section 6.
2 Vortex theory
The vortex method is based on the assumption of inviscid, incompressible, and irrota-
tional flow over the entire domain surrounding the body surface and wake region. The
velocity field is obtained by solving Laplace’s equation using appropriate velocity bound-
ary conditions on the body surface and far-field. The continuity equation for incom-
pressible potential flow, also represented by Laplace’s equation for the velocity potential,
is given by Eq. (1). Here, φ is the velocity potential; the velocity of an irrotational flow
can be defined as in Eq. (2):
∇ 2φ = 0 (1)
V = ∇φ (2)
Green’s second identity can be used to find the general solution to Laplace’s equa-
tion. The wake surface is supposed to be thin; it can be represented as the sum of only
the doublet distribution on the wake surface Sw, and the sum of source (σ) and doublet
(μ) distribution on the body surfaces SB. The integral form used to generate the solu-
tion is derived as in Eq. (3) and Eq. (4), in which surface integration should be made
over all boundaries containing singularity elements. Here, G is Green’s function; β is a
solid angle, of which the value is 4π when the point x is located at the outside of the
body surface boundary for three-dimensional flow; n is the outward unit normal vec-
tor of the surface. σ and μ are the strength of the source and doublet singular elements,
respectively.

βφ(x) = − [G∇(φ − φi ) · n − (φ − φi )∇G · n]dS − φ ∇G · ndS (3)
SB SW

βφ(x) = − [G∇σ · n − µ∇G · n]dS + µ∇G · ndS (4)
SB SW
The difference between the values of external and internal potential on the solid
boundary defines the strength of the doublet elements in Eq. (5); discontinuity in the
normal derivative of the velocity potential on the solid boundary can be referred to as
the strength of the source element in Eq. (6).
µ = φi − φ (5)
∂φi ∂φ
σ = (φi − φ) · n = − (6)
∂n ∂n
Using the definitions of the source and doublet strength, the general solution is rear-
ranged as in Eq. (7).

1 1 1 1 1

φ(x) = − σ − µ∇ · n dS + µ∇ · ndS (7)
4π SB r r 4π SW r
Equation (8) shows the resulting velocity induced by the source and doublet distribu-
tion on the body surface and wake.

1 1 ∂ 1 1 ∂ 1

∇φ(x) = − σ∇ − µ∇ dS + µ∇ dS
4π SB r ∂n r 4π SW ∂n r
(8)
3 Vortex methods for rotor aerodynamics

3.1 Lifting line method
The lifting line method dates back to the development of Prandtl’s lifting-line theory (LLT)
[59]. It was the first analytical method used to predict lift and induced drag on lifting sur-
faces. In this theory, each spanwise section of a finite wing has a section lift equivalent to
that acting on a similar section of an infinite wing having the same section circulation. The
local circulation is related to the local aerodynamic force of a three-dimensional wing using
the two-dimensional vortex lifting law of Kutta and Joukowski. LLT assumes an incom-
pressible and inviscid fluid for which compressibility and viscous effects are negligible for
application areas of interest. However, for Mach numbers less than 0.6, the effect of low-
speed compressibility can be introduced by the Prandtl-Glauert rule [60].
The classical theory assumes an infinitesimally thin and uncambered rigid flat plate with
zero spanwise twist and zero sweep. Wing twist about a spanwise axis can be included as
an alteration to the wing geometric angle of attack. Lifting surfaces, such as wings or rotor
blades, are modeled as bound vortices with the strength of Ŵ(r) at the aerodynamic center.
The goal of LLT is to determine Ŵ(r) as a function of the wing geometric properties; then,
the Kutta-Joukowski theorem is used to obtain the lift per unit span [25, 61]:
1
ρ(V (r))2 c(r)drClα,2d αg − αi (r) (9)

dL(r) = ρV (r)Ŵ(r)dr =
2
where ρ is air density, V(r) is sectional flow speed along wing length, r is wing radial posi-
tion measured from the wing root, and c is the chord. Clα,2d is the airfoil lift curve slope,
αg and αi are the wing geometric angle and the induced angle of attack, respectively.
Thus, Γ(r) can be computed as shown in Eq. (10), where w(r) is the induced downwash
velocity distribution along the wing length.
1
(10)

Ŵ(r) = c(r)Clα,2d V (r)αg − w(r)
2
LLT has wide application in evaluating the aerodynamics of wings with prescribed rotary
and flapping motions. Goldstein [62] applied the original LLT of Prandtl for propellers.
Lerbs [63] extended the formulations for moderately loaded propellers of arbitrary cir-
culation distribution using the induction concept proposed by Kawada [64]. Conlisk [65]
discussed the use of LLT for rotary wings in hover, emphasizing the importance of account-
ing for the influence of linear velocity variation along the blade on the bound circulation
distribution. Johnson [26] emphasized the significance of applying modifications to LLT
to handle specific rotary-wing aerodynamic phenomena such as wake periodicity, whereas
Leishman [25] gave a generic formulation of LLT for rotary-wing motions.
LLT can be applied to a rotating frame with constant angular velocity [66–68]. The rotor
encounters a uniform velocity field (V∞) aligned perpendicular to the rotation axis, rotat-
ing with constant angular velocity (Ω). The relative velocity field is calculated as in Eq. (11).
V = V∞ − × r (11)
The Kutta-Joukowski theorem connects the lift force per unit length of the lifting line (L)
to the total velocity field (V), as in Eq. (12). Here, ρ is the fluid density. The flow is assumed
to be nearly two-dimensional at each radial position along the lifting lines.
L = ρV × Ŵ (12)
The lifting line approach accurately predicts the hover performance of a wide range
of conventional and advanced rotor designs. Landgrebe et al. [27, 28] showed that the
lifting line approach was adequate for predicting the aerodynamic performance of rotor
blades in hover and forward flight conditions. Miller [69] explored the aerodynamics
and dynamics of a rotor and the dynamics and control characteristics of a vehicle by
modeling the rotor with LLT. Analytical modeling and design of the Apache helicop-
ter were performed by Jones and Kunz [70, 71] using CAMRAD II, which uses LLT
for blade modeling. Yeo et al. [72, 73] studied tail rotor flutter, exploring a wide range
of design parameters and examining their effects on whirl flutter speed. Jain et al. [74]
studied rotor performance in hover and forward flight and compared their results with
experimental data. Interference of coaxial rotors in hover was studied by Ho et al. [75].
Wachspress et al. [76] also studied coaxial rotor performance with wake geometry and
aeroacoustics by changing key rotor design parameters. Conceptual design of a slowed-
rotor compound helicopter emphasizing aerodynamic efficiency was performed by
Moodie et al. [77].
Although classical LLT does not consider compressibility or viscosity and has limita-
tions in its consideration of low aspect ratio wings and the thickness effect, it is a pow-
erful tool to model lifting devices, offering a simple approach with low computational
cost for the preliminary design of wings and for aerodynamic predictions. Though the
method has some inherent limitations, researchers have proposed variations and adap-
tations over the years. Guermond [78] and Phillips [79] modified the classical LLT to
account for curvature of swept wings. Several researchers also modified LLT to include
unsteady effects [80–82].
3.2 Vortex lattice method

The vortex lattice method (VLM) is a branch of computational fluid dynamics that
mathematically stems from finite-difference concepts [29]. In the VLM, lifting surfaces
are approximated as thin surfaces represented by vortex sheets of unknown circulation.
Because it allows the rotor blade to be discretized into both chordwise and spanwise
directions, VLM can geometrically represent the curvature of a rotor blade surface,
camber, and swept shape, a significant advantage of VLM compared to LLT [83].
Vortex lattices (or vortex rings) with circulation (Γ) are placed on the mean surface
of a wing (neglecting the thickness of the wing) and the wake. The normal vector (n)
is defined at the center of the vortex ring element, and is referred to as the collocation
point (see Fig. 1). The leading edge of the vortex ring element and collocation point are
located at the quarter and the three-quarter chord lines of each vortex panel, respec-
tively. Because there is no flow separation in potential flow, the velocity field around the
surface will be tangential. Moreover, no flow should cross the solid surface, in what is
usually known as the no-penetration boundary condition, shown in Eqs. (13) and (14).
The zero normal flow condition indicates that the sum of normal velocity components
at each collocation point on the camber line induced by the rotor blade, wake, and time-
dependent kinematic velocity due to unsteady motion of the body must be zero.
(∇φ + V) · n = 0 (13)

Vind,bound + Vind,wake + V∞ − Vrel − × r · n = 0 (14)
Here, the V term is the time-dependent kinematic velocity, which is the sum of the
system’s velocity (V∞), the relative velocity of the body (Vrel), and the body’s frame
Fig. 1 Discretization of lifting surface by vortex lattices
rotational velocity (Ω × r) at each collocation point. Vind,bound, and Vind,wake indicate

the velocity components induced by the rotor blade and wake vortices.
Applying the no penetration boundary condition to the vortex sheet reduces the
problem of finding the vortex strengths to a system of linear algebraic equations, as in
Eq. (15). The bound-vortex circulations (Γi), and thus the self-induced velocity by the
rotor blade can be represented by a combination of influence coefficients (aij). These
influence coefficients are defined as the velocity component normal to the surface
induced by the j-th vortex ring element with unit strength to the collocation point of
the i-th vortex ring element. The right-hand side of the equation includes the contri-
butions of wake-induced velocity and body kinematic velocities. Katz and Plotkin’s
book [60] provides a detailed description of the derivation of VLM.
 Ŵ1   RHS1 
   
a11 a12 · · · a1m 

     
 a21 a22 · · · a2m  Ŵ2   RHS2 

   

 .
 . .. . . ..  .
 = ..  (15)
. . . . 
 .. 
 
 . 
   
am1 am2 · · · amm   
   

Ŵm RHSm

RHSi = − V∞ + Vind,wake − × r i · ni (16)

aij = Vind,bound ij · ni (17)
The VLM mathematical model has seen much improvement and its application
has been extended to compressible fluids [84], unsteady flow [30], and many other
complex aerodynamic phenomena [29, 85]. Because the simplicity and low compu-
tational cost of VLMs increase their applicability, they are still being widely used in
many aviation projects instead of CFD methods, which are far more demanding [31].
VLMs have helped researchers investigate the influences of rotorcraft configuration
and flying environment on its aerodynamic performance [52], noise generation [55,
86], flight control [87], and aeroelasticity response [88].
3.3 Source‑doublet panel method

Although VLM is a useful tool for analyzing the aerodynamics of many rotorcraft con-
figurations, it is not suitable for simulating flow-field over structures like fuselage or
empennage. Engineers usually use the source-doublet panel method (also known as
the panel method) to evaluate the aerodynamic performance of complex planforms
[41, 50, 89–91].
Like VLM, the source-doublet panel methods assume irrotational and incompressible
flow. Therefore, they can also be used to solve Laplace’s equations for potential function in
the vortex field. The general solution of Laplace’s equation for each element in an inertial
system can be derived following Green’s second identity. Considering the source (σ) and
doublet (μ) distribution on the lifting surface and its wakes (see Fig. 2), the general solution
of Eq. (1) can be expressed as:

1 1 1 1 1

φi∗ (x) = − σ − µ∇ · n dS + µ∇ · ndS + φ∞
4π SB r r 4π SW r
(18)
The integral equation can be made into a more straightforward form by applying the Dir-
ichlet boundary condition to each of the collocation points in Eq. (19), where the total inner
potential can be set equal to the freestream potential. To obtain a numerical solution of
the integral equation, the body and wake surfaces are divided into a number of rectilin-
ear panels consisting of N body surface panels and NW additional wake panels with con-
stant-strength singularities. For the constant-strength source (σ) and doublet (μ) elements,
the influences of body panel k and wake panel j at point P can be computed using Eqs.
(21,22,23).
φi∗ = (φ + φ∞ )i = φ∞ (19)
Fig. 2 Surface and wake panels arrangements on wing

N NW N
1 1 1 1 1 1

µ∇ · ndS+ µW ∇ · ndS− σ dS = 0
4π SB r 4π SW r 4π SB r
k=1 j=1 k=1
(20)

1 1

Ck ≡ ∇ · ndS (21)
4π SB r k

1 1

Cj ≡ ∇ · ndS (22)
4π SW r j

1 1

Bk ≡ dS (23)
4π SB r k
The integral equation for finding the unknown doublet distribution (μ) on the body sur-
face can be established as a linear algebraic equation, as in Eq. (24). Here, μW is the strength
of wake doublets, which can be expressed in terms of the unknown surface doublets (μk)
by enforcing the Kutta condition as in Eq. (25). The Kutta condition implies that the wake
doublets are related to the difference between the doublet strengths of the upper and lower
panels of the trailing edge (See Fig. 2). In addition, for unsteady rotor simulation, the source
strength (σ) can be determined by applying the zero normal flow condition on the surface,
defined as in Eq. (26) with the local kinematic velocity.
N
NW
N

Ck µk + Cj µW ,j − Bk σk = 0 (24)
k=1 j=1 k=1
µW = µu − µl
(25)
σ = −(V∞ + Vrel + � × r) · n (26)
Note that the strengths of the source and wake doublet are known at each time step.
Moving the source term of Eq. (24) to the right-hand side of the equation results in a linear
system of N equations with the unknown μk, in the following form [60]:
N
N

Ak µk = Bk σk (27)
k=1 k=1
 µ1   RHS1 
   
a11 a12 · · · a1N 

   
 µ2   RHS2 
   
 a21 a22 · · · a2N    
 .
 . .. .. ..  .
 = ..  (28)
. . . . 
 .. 
 
 . 
   
aN 1 aN 2 · · · aNN 
 
 
 

µN RHSN
Once the linear system of equations is solved, the local velocity components of the
surface panel coordinates and the pressure coefficients can be evaluated. In addition, the
aerodynamic loads are then obtained by integrating the pressure over the lifting surface.
The literature refers to the articles of Hess and Rubbert [32, 33] as pioneering imple-
mentations of the panel method in the aviation industry [34, 92]. Because of their spe-
cial ability in simulating realistic, complex aircraft geometries, and their accuracy,
these source-doublet panel methods are well-known in studying aerodynamic research,
including area of optimizing all rotor design parameters [50, 93] and investigating com-
ponent interaction effects [41, 94]. Rotor performance in the vicinity of other rotorcraft
components, such as fuselage and wings (for tilt rotors), is also important to ensure com-
plete rotorcraft analysis. Not only is a rotor affected by other vehicle components, but it
also causes many changes in the components’ structural dynamic and control behavior.
Therefore, many studies have been conducted to better understand interactive effects
between rotor and airframe and to better predict rotorcraft characteristics.
Recent developments in vortex methods have enabled researchers to analyze full rotor-
craft. Because rotor blades usually have low thickness sections, it is advantageous to use
the VLM to simulate various blade shapes. The source-doublet panel method can ade-
quately represent thick components, such as fuselage. One well-known example of using
these methods for comprehensive analysis of full-configuration rotorcraft is CHARM,
which stands for Comprehensive Hierarchical Aeromechanics Rotorcraft Model. Con-
tinuum Dynamics, Inc. [35, 95] developed this comprehensive analysis model by com-
bining several extended versions of their previous models; they later employed CHARM
to analyze various rotorcraft phenomena such as rotor/airframe flow interaction [35],
vibration [96], and noise propagation [97]. Other noteworthy combinations of VLM and
the panel method for comprehensive analysis of rotorcraft can be found in Jang et al. [41]
and Cao et al. [45], which investigate pressure fluctuation on fuselage surfaces and aero-
dynamic interaction among rotorcraft components, respectively. For more on coupled
VLM and panel analysis, we encourage you to read the NASA Ames report by Wayne
Johnson [98] on developing a comprehensive analysis.
3.4 Hybrid vortex methods

Although vortex methods, including LLT, VLM, and the source-doublet panel method,
are useful approaches for analyzing rotor aerodynamics and simulating a complete
rotorcraft configuration, they do not consider viscous and compressibility effects. These
effects become more important than the influence of rotor wake when the rotorcraft
operates in high-speed forward flight. CFD methods allow us to simulate a wide range
of flow regimes and to accurately capture the complex flow physics occurring in near-
fields around rotor blades. However, CFD simulations suffer from excessive numerical
dissipation on coarse grids; hence, wake structure and vorticity tend to dissipate rapidly
after shedding from rotating blades. Therefore, hybrid approaches combining the advan-
tages of the vortex and CFD methods have been suggested to overcome these inherent
shortcomings.
Advanced Rotorcraft Technology, Inc. developed a coupled vortex particle method
(VPM) and CFD analysis model in which VPM and CFD are employed to resolve the
wake vorticity dynamics and to compute the near-body flow solution, respectively [36,
37]. VPM is used to solve the dynamics of wakes generated by rotors, wings, and other
rotorcraft components, convect the wake vorticity under the combined influence of
the freestream, and account for self-induced wake velocities and flow field perturba-
tions caused by bodies like the fuselage. The CFD solver is used to compute the flow
field around the bodies, including rotor wake-induced velocities. Yang et al. [99] devel-
oped another hybrid method, which combines Navier–Stokes equations near the blade
to consider the viscous effects and near wake, and potential flow analysis in the far-
field to model inviscid isentropic flow. Wie et al. [100, 101] suggested an approach that
reduces computational time by minimization of the computational domain: the strength
and motions of the wake are modeled using the time-marching free-wake approach, and
the rotating blades and flow field around the rotor are estimated using CFD. A moving
overset grid approach was used to consider rotor motions during hovering and forward
flight. The time-marching free-wake method provides inflow and outflow conditions in
the CFD domain from an induced flow at each time step. A similar approach can be
observed in the works of Shi et al. [102] and Zhao et al. [103]. In the present paper, a
coupled VPM/CFD methodology is presented briefly, as follows.
Flow field velocity from the CFD solution, termed uCFD, is treated as an interference
velocity in the VPM simulation. CFD flow field solution directly affects the rotor wake
dynamics, represented by the vortex particles, as in Eqs. (29) and (30). The grid points
and flow properties on the grid are derived from the CFD solution.
dx
= utot = u∞ + uVPM +uCFD (29)
dt
dω
= ω · ∇utot + ν∇ 2 ω + γsrc = ω · ∇(uVPM + uCFD ) + ν∇ 2 ω + γsrc (30)
dt
The governing equation of the near-body CFD solver can be changed to account for
the influence of VPM wake induced velocity, as follows [104]:

∂
Q dV + (Fi − Fv + �F(uVPM , uCFD )) · n dS = 0 (31)
∂t
V S
In the equation above, Q is the flow state variable; Fi and Fv are the inviscid and vis-
cous fluxes. V is the considered fluid domain, S is the boundary of the fluid domain,
and n is the normal vector pointing into the fluid domain. A flux correction term, ∆Fi,
is provided to account for the VPM driven velocity. Due to the coupled approach, the
flux correction is dependent on both the VPM-induced velocity field and the CFD solver
velocity field. The flow state variables for compressible flow are shown in Eq. (32).
 
ρ
 ρuCFD 
Q =  ρvCFD  (32)
 
 ρw 
CFD
E
In the above equation, uCFD = (uCFD, vCFD, wCFD) is the flow field; ρ and E are density
and total energy obtained from CFD simulation. In CFD solvers, for both structured and
unstructured frameworks, the Flux Correction Approach can be used to account for the
VPM-induced velocity. The implementation of VPM-induced velocity contributes to the
CFD solver’s correction of the convective flux term. The VPM-induced velocity correc-
tion terms can be formatted as follows:
ρuVPM · n
 
 ρuCFD uVPM · n 
�Fi (uVPM , uCFD ) · n =  ρvCFD uVPM · n (33)
 

 ρw u
CFD VPM ·n 
(E + p)uVPM · n − p uVPM · n
Here, uVPM and p are the VPM-induced velocity field and the pressure, respectively.
The boundary condition on the body surfaces must be changed in the tightly coupled
VPM/CFD solver. Due to the viscosity effect, the flow velocity on the body surfaces will
be the same as the body velocity. The no-slip boundary condition was enforced in VPM/
CFD coupling in such a way that the combined velocity field, uCFD + uVPM, is the same as
the body velocity, ubody, as follows:
ubody = uVPM + uCFD (34)
According to the no-slip boundary condition, for a stationary body, ubody = 0. There-
fore, the boundary condition becomes uCFD = -uVPM in VPM/CFD coupling analysis.
Similarly, the far-field boundary condition must be applied to the outer boundary of
the CFD grid domain. Riemann invariants are commonly used to impose the far-field
boundary condition. The far-field Riemann boundary condition can be modified to
include VPM/CFD interaction and to account for the effect of the VPM rotor wake on
the CFD far-field boundary. When the flow at the local grid possesses outgoing char-
acteristics, the flow state at the boundary can be extrapolated from the inner CFD grid
nodes. When the flow at the local grid has incoming characteristics, the flow state at
the boundary can take the combined values from the free stream and the VPM-induced
velocity data. For an incompressible flow, the freestream state is written as shown in
Eq. (35), whereas for compressible flow it is a function of the local speed of sound.
u′ u∞ + uVPM
   
′ v∞ + vVPM
 v  
Qfreestream = ′ = (35)

w w ∞ + wVPM

′
� 2 2 2 ′2 ′2 ′2
�
p p∞ + 0.5 u∞ + v∞ + w∞ − u − v − w
4 Vortex methods for rotor wake dynamics

The wake field plays an important role in rotorcraft performance with significant impacts
on rotorcraft’s performance, vibration [105], maneuverability [106], and noise [54, 105].
The dominant vortical structures of the rotor wake are the inboard vortex sheet and the
tip vortices [25]. The vortices generated by a rotorcraft form different structures accord-
ing to the flight condition. Compared to other wake components, the concentrated vor-
tices evolving from the blade tips have the highest contributions to the flow field and
blade aerodynamics [25]. Therefore, it is critical for the wake model to simulate tip vor-
tices accurately.
Many experiments have been conducted to investigate disturbances and instabilities

in rotor wake flow fields; resulting data have become the basis for several semi-empirical
rotor wake models [107–111]. Wake models aim to predict the strengths and structures
of rotorcraft wake vortex fields [112]. Choosing a proper wake model is an important
responsibility for designers of rotorcraft. Therefore, researchers are continuously trying
to improve the ability of wake models to accurately predict the aerodynamic loads of
certain configurations. A common problem with the CFD method is that its inherent
numerical dissipation causes the wake vorticity to diffuse too quickly. This causes prob-
lems in applying far-wake boundary conditions, especially for hovering flight, which are
critical in wake modeling [112–114]. These difficulties of CFD, alongside its computa-
tional burden, reduce its application for practical problems. Therefore, vortex methods
are still being implemented to model the rotor wake flow field [115] since they can offer
accurate solutions with affordable computational burden and model the wake geometry
without numerical dissipation error. References [9, 34, 116] provide exhaustive infor-
mation on the history of wake models, as well as their development and application in
rotorcraft studies. In this section, two wake models based on vortex theory, the free-
wake method (FWM) and the viscous vortex particle method (VPM), are introduced.
4.1 Time‑marching free‑wake method

In the free-vortex or free-wake methods, the wake vortex sheet can be divided into trail-
ing and shed vortices, as shown in Fig. 3. The strength of a trailed vortex is the radial
change between the bound vortices, whereas the strength of a shed vortex is related to
the azimuthal change in the bound vortices with time [60, 83]. The time-marching free-
wake method, in contrast to the prescribed wake model, allows the vortex sheet consist-
ing of vortex filaments behind the rotor to move in free motion as the wake propagates
downstream. In addition, the vortex sheet will deform at each time step based on the
induced velocities by other vortex elements during unsteady rotor simulation [117].
According to the Kutta condition, which acts as a bridge between the wake field and the
lifting surface vortices in the vortex methods, the strength of a nascent shed wake vortex
element is equal to the strength of a blade vortex element placed at the trailing edge in
Fig. 3 Rotor wake model using free-wake method (FWM)

the previous time step. Once the wake vortices are shed, their strengths remain constant
according to Helmholtz’s theorem.
Because the vortex surface has no force exerted on it, a number of discrete vortex
elements are allowed to convect freely with the local stream velocity, which is induced
by other vortex elements in the wake region. Biot-Savart’s law in Eq. (36) provides the
induced velocity components of a Lagrangian marker, which is connected to adjacent
markers by straight or curved vortex filaments [60, 118].
1

r × Ŵdl
V=− (36)
4π c |r|3
Here, V is the velocity of the marker induced by other markers, dl is the vortex line
element, r is the distance between the Lagrangian markers, and Γ is the strength of the
vortex filaments.
The positions of the Lagrangian markers are defined as a function of the blade azimuth
angle (ψ) and wake age (ζ). Thus, the initial helical wake structure will change as the
markers convect downstream. The rate of change of the Lagrangian markers’ position
can be expressed in the partial differential equation (PDE) form, which should be trans-
formed into a finite number of finite difference equation (FDE) to obtain the wake solu-
tion by the time-accurate numerical means. Various time-marching approaches, such
as Euler explicit, Adams–Bashforth, and two predictor–corrector schemes, have been
employed for the time integration of the rotor wake equations. However, the free-wake
method is sensitive to numerical instabilities, particularly in hovering and low-speed
forward flight conditions [155] since the rotor wake is inherently unsteady and unstable
[156]. The time-marching methods affect the overall stability of the free-wake analysis
and the numerical errors could induce nonphysical disturbances. Bhagwat and Leish-
man [39] suggested a predictor–corrector central (PCC) and a predictor–corrector with
second-order backward (PC2B) difference schemes. The PC2B utilizes the solutions at
the three previous time steps for approximating the temporal derivative. This algorithm
implicitly introduces additional effective damping terms, making it more stable with
second-order accuracy. Kini and Conlisk [157] used a fourth-order Adams–Moulton
method, which is implicit and computationally expensive for a free-wake analysis. Gupta
and Leishman [158] performed both linear and nonlinear stability analyses of time-
marching methods used in the free-wake analysis. Bagai and Leishman [159] developed
a pseudo-implicit predictor–corrector method with a five-point central differencing
approach for improving the convergence characteristics of the free-wake method. The
numerical results obtained from free-wake method with a relaxation implementation
were compared with the measurements of the tip vortex locations and flow field in hover
and forward flight. The numerical instability of time marching free-wake modeling is
also caused by the impulsive rotation at the beginning of computation, which creates a
strong starting vortex. Initial treatments for this problem used a helicoidal spiral wake at
the start [119] or assumed a uniform axial velocity [30]. Lee and Na [40, 120] suggested
that, for this problem, it is better to increase the rotating speed of the blade slowly from
zero to the desired speed after some revolution. Their model achieved a close agreement
with experiments [114]. Leishman and Bhagwat [39] provided more detailed informa-
tion on this matter. Abedi et al. [160] suggested the VLM with the free-wake method to
predict aerodynamic loads of wind turbine rotor blades operating the unsteady flow field
by employing tabulated airfoil data and the dynamic stall model. Wachspress and Yu [31]
also proposed the lifting surface blade model with free-vortex wake model for compre-
hensive rotorcraft analysis, and Wachspress et al. applied to the problems of rotor/wake/
body interaction [35] and rotor/airframe noise of eVTOL aircraft [86] by incorporat-
ing fast panel methods and aeroacoustic software. Yeo [161] explored the performance
potential of advanced compound helicopter configurations with a wide range of sizes,
gross weight, and rotor systems using an aeromechanics analysis tool based on lifting
line theory with the free-wake method.
4.2 Viscous vortex particle method

The potential flow assumption in the rotor wake limits the free-wake method’s ability to
forecast the rotor wake dynamics. The effects of turbulence on the diffusive characteris-
tics of the vortex and the vortex stretching are considered empirically. The vortex decay
factor or vortex core growth are modeled empirical formulations with the parameters,
which are often derived experimentally [121]. Therefore, the numerical results of the
free-wake analysis strongly depend on empirical formulations and the values of param-
eters. The viscous vortex particle method (VPM) solves the vorticity-velocity form of the
incompressible Navier–Stokes equations with a Lagrangian description for obtaining the
wake vorticity field. Therefore, consideration of viscous effect utilizing the VPM, which
directly simulates the viscous flow and avoids artificial numerical dissipation, is crucial
for solution accuracy. This method, which was developed for 2-D [122, 123] and 3-D vis-
cous flow [123–126], has been used in rotor flow field analysis [42]. A rotor wake gener-
ated from the trailing edge of the rotor blade is modeled by a number of vortex particles,
which influence each other and induce a velocity in the flow-field in the wake evolu-
tion process, as shown in Fig. 4. Compared to the vortex filament method, VPM has
the advantage that the particles do not necessarily need to maintain connectivity with
adjacent particles because particles propagate independently during the time-marching
step. This property is especially useful for investigating wake interaction phenomena
occurring with rotors or rotor-body configuration. In addition, VPM can be used to eas-
ily treat the wake vortex penetration problem.
The Navier–Stokes equation can be represented in vorticity-velocity form with a
Lagrangian description for incompressible flow. The Lagrangian description of the vorti-
city-velocity form is used in the VPM approach. The vorticity dynamics equation can be
expressed as in Eq. (37), where ω is the vorticity and u is the velocity.
Dω
= ω · ∇u + ν∇ 2 ω (37)
Dt
A natural way to solve the vortex transport problem is to use the vorticity-velocity
form with a Lagrangian description. Grid generation is not required for the VPM simu-
lation. Moreover, the convection term in Eq. (37) is not explicitly treated, resulting in a
dissipation-free approach for the vortex particle model. To solve the vorticity-velocity
governing equation, the vorticity field can be represented by a set of S Lagrangian vec-
tor-valued particles, as in Eq. (38).
Fig. 4 Rotor wake modeling using vortex particle method (VPM)
S
S

ω(x, t) = ξσ (x − xi )α i = ξσ (x − xi )ωi Vi (38)
i=1 i=1
where xi and αi are the position and the vector-valued total vorticity inside the vortex
particle with volume Vi, respectively. The three-dimensional regularization function or
smoothing function (ξσ) can be defined as in Eq. (39), where σ is the smoothing parame-
ter [42]. A Gaussian distribution is one of the distribution functions that can be utilized,
as expressed in Eq. (40) [43, 127].

1 |r|
ξσ (r) = ξ (39)
σ3 σ
1 2
ξ (ρ) = e−ρ /2 (40)
(2π )3/2
The velocity component of the i-th vortex particle (xi) induced by other particles can
be computed using Eq. (41), where σij is a symmetrized smoothing parameter used to
conserve the linear and angular vortex impulses.
S
1
u(xi , t) = − K (ρ)(xi − xj ) × αj (41)
σ3
j=1 ij
1 2
σij = √ σi + σj2 (42)
2
Here, ρ is a non-dimensional distance parameter, Κ(ρ) is the regularized Biot-Savart

kernel used for the velocity calculation, and G(ρ) is Green’s function, used for the stream
function evaluation, as follows:
G(ρ) − ξ(ρ)
K (ρ) = (43)
ρ2

1 ρ
G(ρ) = erf √ (44)
4πρ 2
1
ρ=
σj
x − xj (45)
During the time-marching step for unsteady rotor simulation, the location of parti-
cles xi (t) will be updated using the local convection velocity, which is the sum of the
freestream velocity, self-induced velocity, and wake-induced velocity. Then, the convec-
tion equation governs particle positions xi as follows:
d
xi (t) = u(xi , t) (46)
dt
The governing equation for the vorticity field is defined in Eq. (37). The left-hand side
of the equation is the material derivative of the vorticity. The first term on the right-hand
side of the governing equation for the vortex dynamics represents the stretching effect,
which describes vortex stretching and rotation owing to the velocity gradient. The vis-
cous diffusion term is the second term on the right-hand side of Eq. (37); it describes
the vorticity diffusion due to viscous effects. In the direct scheme, the vortex stretching
effect is accounted for by directly multiplying the velocity gradient matrix by the particle
vorticity [42].

dαi
= αi · ∇u(xi , t) (47)
dt ST
The particle strength exchange (PSE) method [122, 124, 128–131] can be utilized to
consider the viscous diffusion effect, for which the second term is on the right-hand
side of the vorticity governing equation in Eq. (37). The fundamental idea of the PSE
algorithm is to approximate the Laplacian operator with an integral operator, avoiding
numerical differentiation, which has lower numerical precision than the integral opera-
tion. The approximated Laplacian can be written as in Eq. (48).
2

∇ 2 ω(xi ) ≈ 2 ησ (xi − y)[ω(y) − ω(xi )]dy (48)
σ
The kernel ησ in the above equation is considered to have a Gaussian distribution func-
tion. The integral in Eq. (48) can be discretized across all particles by using the midpoint
quadrature, resulting in the following equation:
S
2
∇ 2 ω(xi ) ≈

σ 2
ησ (xi − yj ) ω(yj ) − ω(xi ) Vj (49)
j=1
Koumoutsakos et al. [132] implemented the Neumann type vorticity boundary condi-
tions to consider the no-slip state, which is expressed in terms of vorticity flux. Singh
and Friedmann [133] used viscous VPM to simulate coaxial rotors in hover flight, which
yields complex unsteady aerodynamic interaction effects. The flow separation during the
dynamic stall is an important source of vibrations on a rotor at high advance ratios. They
also used VPM simulation to model the shedding of concentrated vorticity from the air-
foil’s leading edge and study the wake evolution of the coaxial rotors [44]. Su et al. [134]
applied the VPM to an electrically controlled swashplateless rotor to investigate aerody-
namic characteristics and wake structure. He et al. [42] investigated the effects of mod-
eling parameters such as wake cut-off distance, time-step size, and other model tuning
parameters for hovering and forward flight. Helicopter rotor loads were predicted using
a source-doublet panel/VPM hybrid approach combined with a computational struc-
tural dynamics (CSD) algorithm. In a hybrid approach, the panel method was employed
to simulate the rotor blade surfaces and close wakes, and VPM was used to model the far
wake [91]. Alvarez and Ning [162] developed a viscous VPM code to study the unsteady
wake dynamics of individual propellers of distributed propulsion electric aircraft, includ-
ing near-far field transition and vortex breakdown. They presented its capacity to model
rotor-on-rotor aerodynamic interactions in a side-by-side configuration [163]. Tan et al.
[164] suggested coupling VPM and a discrete element method to simulate the helicopter
brownout phenomenon and investigated the particle–surface interactions and the flow
field of a helicopter in ground effect. Huberson et al. [165] employed the panel method
with VPM and Farassat’s formulation 1A to predict the vortex–solid interactions, such
as helicopter blade–vortex interaction noise.
While flow is passing over the rotor blade, the wake elements keep emitting from it
and the number of vortex particles increases at every time step. Although the number
of wake elements determines the prediction accuracy of the wake region, any increase in
the number of vortex particles extends the computational time of the numerical model
[114, 120]. Therefore, an efficient summation rule needs to be incorporated for VPM
simulation. In the literature, there are two broad groups of fast summing algorithms for
resolving the N-body problem. The first is the Tree-Code approach [129–131]; the sec-
ond is the fast multipole method (FMM) [135, 136]. The detailed description is out of
the scope of this paper. Berdowski et al. [166] developed the efficient framework coupled
with actuator disc method, VPM, and open-source FMM library for handling the effi-
cient data-parallelism on a CPU, and Willis et al. [167] suggested a pFFT-Fast Multipole
Tree algorithm to accelerate the calculation of particle-induced velocity.
5 Nonlinear vortex lattice method

A comprehensive simulation tool capable of accurately and efficiently predicting rotor
aerodynamics and wake dynamics is needed to design more advanced rotor systems.
Among the various numerical approaches, VLM is one of the most suitable because it
can provide reasonably accurate solutions with affordable computational cost and can
represent the wake vorticity fields without numerical dissipation error. However, VLM
simulation cannot consider nonlinear aerodynamic characteristics, which are mainly
associated with viscosity, flow separation, and low-Reynolds number flow. To overcome
these drawbacks, the authors of this paper have suggested a nonlinear vortex lattice
method (NVLM) that combines VLM with airfoil lookup tables, semi-empirical mod-
els, and vortex strength correction. Moreover, NVLM is tightly coupled with VPM to
simulate unsteady wake dynamics. Details of the numerical strategies used in NVLM/
VPM are elucidated in previous studies [83, 137]. In the following sections, NVLM/VPM
methodology is briefly presented, and its applications are discussed.
5.1 Airfoil look‑up table

VLM simulation yields accurate results with an affordable computational cost for sub-
sonic attached flow. However, it fails to predict the nonlinear aerodynamic behavior of
rotor blades that occurs at high angles of attack (above stalling or critical angle of attack)
or low-Reynolds number flow. For these flow regimes, an abrupt drop in the lift coeffi-
cient and nonlinear variation in lift coefficient with respect to the angle of attack become
important in determining the rotor aerodynamics. The airfoil look-up table is an effi-
cient and practical way of obtaining aerodynamic information about the airfoils of the
rotor blades operating in both attached and stalled flow conditions. The sectional look-
up table containing pre-computed airfoil aerodynamic data for a wide range of Reynolds
numbers and angles of attack is used in the NVLM/VPM simulation to compute the lift
and drag coefficients at each blade section’s control point. The location of the control
point should be explicitly specified to conduct an airfoil look-up table. The mathematical
equation is derived by applying zero normal boundary conditions, and it is numerically
solved using the root-finding approach. The result shows that the most suitable control
point location is at half of the chord for NLVM [138]. A detailed derivation is outside the
scope of this paper.
The aerodynamic coefficients of the individual blade sections rely largely on the local
inflow velocity and effective angle of attack, which are calculated using Eqs. (50) and
(51), based on VLM solutions.
Vinflow = V∞ − × r + Vind,bound + Vind,wake (50)

Vinflow · a3
αeff = θtwist + θpitch − tan−1 (51)
Vinflow · a1
Here, Vinflow is the local inflow velocity. θtwist and θpitch are the local twist angle and
collective pitch angle at each blade section, respectively. a1 and a3 are unit vectors along
directions tangential and normal to the rotating plane. Once the strengths of the bound
vortices on the rotor blade are determined by solving the linear system of equations
with an instantaneous boundary condition at each time step, the self-induced velocity
(Vind,bound) and wake-induced velocity (Vind,wake) can be calculated by Biot-Savart’s law.
Finally, each blade section’s sectional lift and drag forces can be calculated using Eqs.
(52) and (53).
1
dLtable = ρ∞ (Vinflow · a1 )2 + (Vinflow · a3 )2 cl dA (52)
2
1
dDtable = ρ∞ (Vinflow · a1 )2 + (Vinflow · a3 )2 cd dA (53)
2
According to the force conventions for the rotor blade, all other aerodynamic coef-
ficients, including normal (CN), tangential (CA), thrust (CT), and torque (CQ) can be eval-
uated using the lift (CL) and drag (CD) coefficients in conjunction with their reference
angles. The normal and tangential force coefficients are determined relative to the chord
line plane, while the thrust and torque coefficients are determined relative to the rota-
tion plane.
5.2 Semi‑empirical models for airfoil aerodynamics

The centrifugal and Coriolis forces caused by three-dimensional effects and rotational
augmentation affect the stability of the boundary layer of the rotor blades, resulting in
significantly higher lift coefficients than in the two-dimensional or three-dimensional
non-rotating cases. These events are called as stall delay phenomenon, which is espe-
cially noticeable at the inboard section of the rotating blade, where they strongly influ-
ence the onset of flow separation. The boundary layer can be stabilized, minor separation
bubbles can return to attached flow, and flow separation can be postponed above stall
onset angle corresponding to the two-dimensional situation, despite the comparatively
high angle of attack due to the low rotational speed at the inboard of the blade. As a
result, the rotor blade in the inboard region experiences different post-stall airfoil char-
acteristics than in the two-dimensional scenario. Several stall delay models have been
developed to account for the three-dimensional rotating effect by adjusting the two-
dimensional aerodynamic coefficient data. Among several other models, Raj and Selig’s
model [139] can be utilized to account for the influence of stall delay on the aerodynamic
coefficients at the inboard portion of the rotor blade; their model is an improvement of
Du and Selig’s model [140]. The corrected lift and drag coefficients, cl,3D and cd,3D, can be
calculated in the following way:
cl,3D = cl,2D + gcl cl (54)
cd,3D = cd,2D + gcd cd (55)
where cl,2D and cd,2D are two-dimensional lift and drag coefficients. ∆cl and ∆cd are the
differences between aerodynamic coefficients obtained from the potential theory and
the two-dimensional configuration. Incremental factors for the lift and drag coefficients,
gcl and gcd , are determined as follows:
(r/R)nl
 
1  1.6 (c/r) al − (c/r) dl � r�
gcl = nl 1 − (56)
2 0.1267 bl + (c/r) (r/R) R
dl
(r/R)nd
 
1  1.6 (c/r) ad − (c/r) dd � r�
gcd = nd 2 − (57)
2 0.1267 bd + (c/r) (r/R) R
dd
where c/r and λ are the ratio of a local chord to the local radius and a local speed ratio.
a, b, d, and n are empirical constants for the correction formula; their values were deter-
mined from the measurements.
The dynamic stall phenomenon is also most likely to occur in the inboard regions of
the rotor blade because the effective angle of attack is much higher than in the outboard
regions due to the low rotational speed. This creates significantly unstable aerodynamic
behavior in the blade section, resulting in a situation in which the aerodynamic coef-
ficients enter a hysteresis loop. As a result of the presence of shedding vortices from the
leading edge of the airfoil, nonlinear variations in lift, drag, and pitching moment coef-
ficients as a function of angle of attack occur, and their values in the stall and post-stall
regions are completely different from the static aerodynamic coefficients. Dynamic stall
generally causes unsteady and high loads on the rotor blades, which could cause struc-
tural damage to the rotor blade and other components. To forecast the aerodynamic
loads acting on a rotor blade subjected to unsteady inflow circumstances, it is crucial to
include the dynamic stall effect and the nonlinear aerodynamic features of the airfoils in
the stall or post-stall regions. Various dynamic stall models have been proposed, includ-
ing the ONERA model [141], the Leishman and Beddoes (L-B) model [142, 143], and the
Stig Øye model [144]. The L-B model, devised by Leishman and Beddoes, is the most
widely used semi-empirical modeling method for the hysteresis loop of the aerodynamic
coefficients. This model consists of three parts that were formulated to represent flow
behaviors occurring around rotor blades: unsteady attached flow, trailing-edge separated
flow, and dynamic stall flow. The details of the L-B model are omitted as it is rather com-
plicated and lengthy.
5.3 Vortex strength correction

The sectional aerodynamic forces acting on the rotor blades for separated flow or low-
Reynolds number flow can be evaluated through the look-up table. However, the bound
circulation strength of the rotor blades, derived by solving a linear system of equa-
tions may be significantly over-predicted. Vortex elements placed on the trailing edge
of the rotor blades will shed into the wake during time-marching simulation, and their
strength will remain constant due to Helmholtz’s theorem. As a result, the over-pre-
dicted strengths of bound and wake vortices can cause inaccurate estimation of the rotor
wake geometry and evolution process, leading to errors in induced velocity predictions.
Therefore, the influence of nonlinear aerodynamics on the bound circulation strength
should be included to obtain a more accurate solution. In NVLM, sectional lift forces
from the look-up table are employed to correct the bound vortex strength derived from
VLM simulation. By equating the formulas for sectional lift forces based on airfoil the-
ory in Eq. (52) and the Kutta-Joukowski theorem, the representative circulation strength
at the control point, represented as Γtable, can be newly determined. A correction process
with an under-relaxation factor is iteratively conducted until a convergence criterion is
satisfied. If the difference between the current and updated circulation strengths reaches
a value under 0.001%, then the iterative correction is stopped, and the corrected circula-
tion strength is assigned to both the chordwise and spanwise vortex elements depend-
ing on the ratio of strength of each chordwise element to the average strength. A full
description of the circulation strength correction technique can be found in the refer-
ences [138].
After the bound circulation strength on the rotor blade converges, the corrected
bound vortices located at the trailing edge will shed into the wake according to the Kutta
condition. The rotor blade is rotated at each time step, and wake vortex elements are
generated from the rotor blades’ trailing edges. Once the strength of wake vortices is
determined, the wake structure can be modeled through two approaches, vortex fila-
ments and vortex particles, as mentioned above. In NVLM/VPM, the nascent wake that
was recently shed from the full span of blades is represented using curved vortex fila-
ments during approximately 4 ~ 5 discretized time steps. After that, the vortex filaments
are split into a finite number of vortex particles, except for the nascent wake panels,
to avoid wake instability problems. Vortex particles mutually interact with each other,
allowing them to distort and propagate freely downstream with local convection veloc-
ity. A large number of vortex particles are generated during the time-marching step for
wake evolution. The computing time to evaluate each vortex particle’s local convection
velocity increases exponentially as the number of vortex particles increases. To allevi-
ate the computational burden, parallel computing using the Message Passing Interface
(MPI) library on a multi-core processor is applied to evaluate the induced velocities at
each vortex particle.
5.4 Applications
As previously mentioned, NVLM/VPM has been suggested to overcome the intrinsic
drawbacks of the existing VLM, which is impractical for many applications. NVLM/
VPM has been applied to investigate the rotor aerodynamics, wake dynamics, and
acoustics of various types of rotors, such as propellers [55, 145–148] and wind turbine
blades [137, 138, 149, 150]; its predictive capability has been compared with those of
other numerical predictions and experimental results. The validation results show that
NVLM/VPM can consider the nonlinear aerodynamic characteristics, which are mainly
introduced by viscous effects and low Reynolds number flow. In this paper, specific
application examples are introduced to assess the model accuracy of NVLM/VPM.
5.4.1 Caradonna‑Tung rotor
The experiment on the Caradonna-Tung rotor model was conducted in 1981 [151], and
provided extensive measurements of rotor aerodynamics and wake dynamics that have
been widely used in the rotorcraft field to validate the accuracy of numerical methods.
The model rotor is made of two blades installed on a tall column with a drive shaft. The
Caradonna-Tung rotor is a rectangular blade without twist or swept angles, and its sec-
tional shape is that of a NACA 0012 airfoil. The aspect ratio of the rotor blade is 6, the
chord length is 0.1905 m (0.625 ft), and the rotor diameter is 2.286 m (7.5 ft), as listed
in Table 3. A wide range of test parameters were used under ambient conditions, with a
tip Mach number ranging from 0.226 to 0.890; the collective pitch setting varied from
0 to 12 degrees. The tip vortex trajectory was retrieved using a hotwire approach after
pressure distributions were recorded at five cross-sections of the blade. The specific flow
conditions chosen for the validation work are listed in Table 4.
The thrust coefficient is a non-dimensional parameter representing the aerody-
namic load acting on the rotor blades in a direction normal to the rotating plane. Fig-
ure 5 provides a comparison between measurements and numerical predictions for
the thrust coefficient distribution along the radial direction for various blade pitch
angles. The results of VLM (blue dashed line with triangle symbols) and NVLM (red
solid line with square symbols) are compared with experimental data to validate the
prediction capability of the numerical models. Results obtained from both VLM and
NVLM simulations matched well with the measurements, although VLM tended to
slightly over-predict the sectional thrust force, especially in the blade tip region. The
variation in the integrated thrust force coefficient with respect to collective pitch
angle is shown in Fig. 6. Exact agreement between the experiment and NVLM was
observed, whereas slight overprediction was observed in the case of VLM.
The rotor wake is defined as an unsteady fluctuation flow that generates unsteady
rotor blade aerodynamics and complicates the flow field. The wake vortex particles
have different sizes depending on the circulation strength. The stronger the wake vor-
tex strength, the larger the particle size. The color of the vortex particles also varies
with their circulation strength, just as the size does. The evolution of wake geometries
of the Caradonna Tung rotor with respect to revolutions is shown in Fig. 7. The devel-
opment of near and far wakes is easily predicted as rotor revolutions increase; tip vor-
tex descent and wake contraction can be clearly noticed. Figure 8 shows a comparison
of vorticity magnitude contours on a vertical cross-section (x–z plane) through the
center of the rotating axis with an increase in revolutions. It can be clearly seen that
the periodic shedding of wake vortices behind the rotor plane gives rise to a symmet-
ric wake structure. A comparison of the tip vortex trajectory locus between experi-
ment and computation is provided in Fig. 9, which shows an excellent agreement with
the experiment.
5.4.2 NREL Phase VI rotor

Vortex methods also have broad applicability in simulating wind turbines, which
operate at lower Mach numbers than rotorcraft. Here, we again validate the proposed
Table 3 Model description of Caradonna-Tung rotor [151]

Parameter Value
Number of blades, NB [-] 2

Chord length, c [m] 0.1905
Rotor radius, R [m] 1.143
Twist angle, θtwist [deg.] 0
Blade planform Rectangular blade
Blade sectional profile NACA 0012 airfoil
Table 4 Flow conditions for Caradonna-Tung rotor simulation

Parameter Value
Collective pitch angle, θc [deg.] 5, 8, 12

Rotating speed, Ω [rpm] 1250
Tip Mach number, Mtip [-] 0.439
Fig. 5 Comparison of sectional thrust coefficients: (a) θc = 5º, (b) θc = 8º, and (c) θc = 12º
Fig. 6 Comparison of integrated rotor thrust coefficient depending on collective pitch angle
Fig. 7 Rotor wake structures: (a) 10 rev., (b) 15 rev., and (c) 20 rev
NVLM against extensive and high-quality measurements of an NREL Phase VI wind

turbine operating in both axial and yawed flow conditions. A well-known experimen-
tal study was conducted in the NASA-Ames wind tunnel facility under controlled
wind conditions [152]. The NREL Phase VI wind turbine consists of two-bladed rotors
without hub tilt, coning, and prebend angles. The rotor blade is an S809 airfoil with a
tapered-twisted configuration. Details of the NREL Phase VI model and its operating
conditions for the validation work are provided in Tables 5 and 6, respectively.
Because the NREL Phase VI is a stall-regulated wind turbine, the rotor blades are
designed so that flow begins to separate from their upper surfaces at high wind speeds.
The aerodynamic thrust and power output are controlled by stall effects occurring on
the wind turbine blades. Below a wind speed of 10 m/s, the flow remains fully attached
over the rotor blade. However, with increases in wind speed, separated flow starts to
cover the rotor blade from the inboard to outboard regions. Finally, the flow is com-
pletely separated from the rotor blade at a wind speed of 20 m/s, and massive flow
separation occurs at a wind speed of 25 m/s. Once flow separation occurs, the flow
Fig. 8 Rotor vorticity structures: (a) 10 rev., (b) 15 rev., and (c) 20 rev
Fig. 9 Comparison of tip vortex trajectory in terms of radial and vertical positions for θc = 8º
field around the rotor blades becomes highly unsteady and transient. Figures 10 and
11 respectively provide comparisons of the normal and tangential force distribu-
tions along the radial direction, depending on the wind speed. Above a wind speed of
10 m/s, there is a distinct difference between the results of VLM and NVLM simula-
tions; significant improvements associated with flow separation and the stall effect are
evident. NVLM predictions are seen to be quite close to the measurements, even if
there are minor discrepancies, whereas the VLM simulations show significant over-
prediction of the normal and tangential force coefficients due to the neglecting of the
nonlinear aerodynamic behaviors caused by the flow separation. In Fig. 12, the overall
aerodynamic performances of wind turbines in terms of low-speed shaft torque and
Table 5 Model description of NREL Phase VI rotor [152]

Parameter Value
Number of blades, NB [-] 2

Chord length, cr = 0.75R [m] 0.483
Rotor radius, R [m] 5.029
Blade planform Tapered-twisted blade
Blade sectional profile S809 airfoil
Table 6 Flow conditions for NREL Phase VI rotor simulation

Parameter Value
Collective pitch angle, θc [deg.] 4.815

Tip pitch angle, θtip [deg.] 3
Rotating speed, Ω [rpm] 72
Tip Mach number, Mtip [-] 0.11
Wind speed, V∞ [m/s] 5, 7, 10, 15, 20, 25
Yaw angle, β [deg.] 0, 30
thrust force predicted by VLM and NVLM simulations are compared with experimen-
tally measured data. Above a wind speed of 10 m/s, VLM tends to overestimate the
aerodynamic load on the rotor blades, while the NVLM results are in reasonably good
agreement with experimental data and the CFD results, even if flow separation occurs
at high wind speed.
Figures 13 and 14 show NVLM predictions of unsteady aerodynamic loads on a wind
turbine blade subjected to wind speed of 7 m/s and yaw angle of 30°. Under yawed flow
conditions, the wind turbine blades suffer from cycle-to-cycle variation in aerodynamic
load. This is mainly attributed to the advancing and retreating blade effect and the
skewed wake effect. It can be observed that azimuthal variations in the sectional aerody-
namic loads are much more pronounced at the inboard section because, there, the rotor
blade experiences significant variation in angle of attack due to asymmetric inflow dis-
tribution. The time-averaged normal and tangential force coefficients of the rotor blade
are shown in Fig. 15. Comparing Fig. 15 (a) and (b), we can confirm that NVLM also
accurately predicts wind turbine aerodynamic loads with yaw angle.
Figures 16,17,18 compare the results of NVLM and LLM [153] with NREL measure-
ment data obtained under wind speed of 10 m/s and yaw angle of 30°. At higher wind
speeds, advancing and retreating blade effects become more dominant. This can induce
a dynamic stall of rotor blades; the blades experience periodic variation in angle of attack
with large amplitude, particularly at the inboard section. These results indicate that
NVLM can provide more accurate predictions of aerodynamic load from the inboard
to the outboard regions than LLM, which tends to overestimate the aerodynamic loads.
Based on the authors’ experience, it is difficult to use the LLM approach to accurately
predict tangential forces of rotor blades because a single vortex element in the chord-
wise direction cannot sufficiently represent the three-dimensional geometry and various
planform shapes of the rotor blades [31, 138, 154].
Fig. 10 Comparison of normal force coefficients depending on wind speed: (a) 5 m/s, (b) 7 m/s, (c) 10 m/s,
and (d) 15 m/s
Fig. 11 Comparison of tangential force coefficients depending on wind speed: (a) 5 m/s, (b) 7 m/s, (c)
10 m/s, and (d) 15 m/s
Fig. 12 Comparison of rotor aerodynamic performance depending on wind speed: (a) aerodynamic thrust
force and (b) low-speed shaft torque (LSSTQ)
Fig. 13 Azimuthal variation of normal force coefficient acting at specific radial positions for V = 7 m/s and
β = 30º: (a) r/R = 0.47, (b) r/R = 0.63, and (c) r/R = 0.8
Fig. 14 Azimuthal variation of tangential force coefficient acting at specific radial positions for V = 7 m/s and
β = 30º: (a) r/R = 0.47, (b) r/R = 0.63, and (c) r/R = 0.8
Fig. 15 Comparison of time-averaged force coefficients for V = 7 m/s and β = 30º: (a) normal force and (b)
tangential force coefficients
Fig. 16 Azimuthal variation of normal force coefficient acting at specific radial positions for V = 10 m/s and
β = 30º: (a) r/R = 0.47, (b) r/R = 0.63, and (c) r/R = 0.8
6 Conclusion
This paper provides a brief introduction to the most common vortex methods for
analyzing rotor aerodynamics and wake dynamics. The purpose of this paper is not
Fig. 17 Azimuthal variation of tangential force coefficient acting at specific radial positions for V = 10 m/s
and β = 30º: (a) r/R = 0.47, (b) r/R = 0.63, and (c) r/R = 0.8
Fig. 18 Comparison of time-averaged force coefficients for V = 10 m/s and β = 30º: (a) normal force and (b)
tangential force coefficients
to present an exhaustive review but to present state-of-the-art vortex methods and

address well-known uses of these methods for simulating flow over rotorcraft. The
vortex methods are still appealing today due to their negligible numerical dissipation,
conservation of flow invariants, relaxed stability condition at time steps, and abil-
ity to capture high-resolution wake structure. The vortex methods are coupled with
grid-free wake modeling methods, such as the time-marching free-wake method and
viscous vortex particle method, which do not require any grid generation effort and
minimize the dissipation of vorticities over long distance traveled. Moreover, the vor-
tex methods are useful for preliminary designs and parametric studies because they
produce numerical results much more quickly than grid-based CFD simulation. How-
ever, the numerical methods based on vortex theory fail to provide accurate represen-
tations of the viscous boundary layer and lead to underestimation of the drag force
due to the assumption of potential flow. They are also unable to consider nonlinear
aerodynamic characteristics of airfoils involved in the rotor blade, including viscosity,
flow separation, and low-Reynolds number flow. To overcome the intrinsic shortcom-
ings of VLM, the authors have proposed NVLM; its applications to rotor aerodynam-
ics and wake dynamics were discussed. Simulations indicated that NVLM/VPM has
great capability to assess aerodynamic loads acting on rotor blades for a wide range
of operating conditions and to simulate the generation and evolution of rotor wake,
allowing for higher resolution simulation of the helical wake structure.
Recent increases in efficiency of electric propulsion, particularly in areas of motor
and battery technology, have been driving the development of electric vertical take-
off and landing (eVTOL) aircraft, which use electric power to hover, take off, and land
vertically in highly populated urban areas. Among various configurations, eVTOL
aircraft designed with multiple lifting rotors or prop-rotors are popular in the UAM
market because distributed electric propulsion (DEP) systems using multiple rotors
can improve safety and reduce noise. With the increasing number of rotor systems,
vortex methods have emerged as useful tools for comprehensive analysis of eVTOL
aircraft; these methods can provide a comprehensive solution with an affordable com-
putational cost by combining structural, flight dynamic, and acoustic analysis solvers.
Furthermore, in the presence of fuselage or any other such body, a hybrid method
that combines vortex methods with CFD or a generalized treatment of boundary con-
ditions on solid walls can be used, an efficient and accurate way to compute flow fields
around bodies while considering the effects of the rotor wake.
Acknowledgements
This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science,
ICT & Future Planning (NRF-2017-R1A5A1015311 and 2021R1C1C1010198), South Korea.
Availability of supporting data

Not applicable.
Authors’ contributions
This research is the outcome of joint effort. All authors have read and approved the final manuscript.
Funding
The National Research Foundation of Korea (NRF-2017-R1A5A1015311 and 2021R1C1C1010198), South Korea.
Availability of data and materials

All data and materials are available upon request.
Declarations
Competing interests
The authors declare that they have no competing interests.
Received: 9 December 2021 Accepted: 13 March 2022
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Lee et al.

Advances in Aerodynamics (2022) 4:20

REVIEW Open Access

Review of vortex methods for rotor

Rotorcraft undergo complex aerodynamic phenomena due to sharp gradients of veloc-

Table 1 Overview of blade models for rotor simulation

Table 2 Overview of wake models for rotor simulation

Free-wake method Straight and curved No grid Wie et al. [22]

rotorcraft design challenge is to reduce their noise, especially in urban environments.

3 Vortex methods for rotor aerodynamics

3.2 Vortex lattice method

Fig. 1 Discretization of lifting surface by vortex lattices

rotational velocity (Ω × r) at each collocation point. Vind,bound, and Vind,wake indicate

3.3 Source‑doublet panel method

Fig. 2 Surface and wake panels arrangements on wing

3.4 Hybrid vortex methods

ubody = uVPM + uCFD (34)

4 Vortex methods for rotor wake dynamics

Many experiments have been conducted to investigate disturbances and instabilities

4.1 Time‑marching free‑wake method

Fig. 3 Rotor wake model using free-wake method (FWM)

4.2 Viscous vortex particle method

Fig. 4 Rotor wake modeling using vortex particle method (VPM)

Here, ρ is a non-dimensional distance parameter, Κ(ρ) is the regularized Biot-Savart

5 Nonlinear vortex lattice method

5.1 Airfoil look‑up table

Vinflow = V∞ −  × r + Vind,bound + Vind,wake (50)

5.2 Semi‑empirical models for airfoil aerodynamics

cl,3D = cl,2D + gcl cl (54)

cd,3D = cd,2D + gcd cd (55)

5.3 Vortex strength correction

5.4.2 NREL Phase VI rotor

Table 3 Model description of Caradonna-Tung rotor [151]

Number of blades, NB [-] 2

Table 4 Flow conditions for Caradonna-Tung rotor simulation

Collective pitch angle, θc [deg.] 5, 8, 12

NVLM against extensive and high-quality measurements of an NREL Phase VI wind

Table 5 Model description of NREL Phase VI rotor [152]

Number of blades, NB [-] 2

Table 6 Flow conditions for NREL Phase VI rotor simulation

Collective pitch angle, θc [deg.] 4.815

to present an exhaustive review but to present state-of-the-art vortex methods and

Availability of supporting data

Availability of data and materials

Received: 9 December 2021 Accepted: 13 March 2022

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3 Vortex methods for rotor aerodynamics

3.2 Vortex lattice method

3.3 Source‑doublet panel method

3.4 Hybrid vortex methods

4 Vortex methods for rotor wake dynamics

4.1 Time‑marching free‑wake method

4.2 Viscous vortex particle method

5 Nonlinear vortex lattice method

5.1 Airfoil look‑up table

Vinflow = V∞ − × r + Vind,bound + Vind,wake (50)

5.2 Semi‑empirical models for airfoil aerodynamics

cl,3D = cl,2D + gcl cl (54)

cd,3D = cd,2D + gcd cd (55)

5.3 Vortex strength correction

5.4.2 NREL Phase VI rotor