Implementation of a Rotor Simulation with a Coupled Panel and Vortex Particle Method in State-Space Form

Abstract

This paper describes the implementation of a coupled panel and vortex particle method in state-variable form for rotary wings. More specifically, the coupled panel and vortex particle dynamics are formulated as a nonlinear system of time-periodic, ordinary differential equations in first-order form with the specific objective for it to be self-contained and inherently linearizable. The proposed simulation model is a mesh-free aerodynamic method based on panel and vortex particle methods. Panels are used to model the wing surface as well as the near wake, whereas vortex particles are used to model the far wake. The code is implemented in MATLAB® and demonstrated through an example involving a UH-60-like rotor at hover.

Full text

Implementation of a Rotor Simulation with a Coupled Panel and Vortex
Particle Method in State-Space Form
Hussien Hafez
Graduate Research Assistant
Alessandro Cocco
Postdoctoral Researcher
Umberto Saetti
Assistant Professor
Department of Aerospace Engineering
University of Maryland
College Park, MD 20740
ABSTRACT
This paper describes the implementation of a coupled panel and vortex particle method in state-variable form for rotary
wings. More specifically, the coupled panel and vortex particle dynamics are formulated as a nonlinear system of time-
periodic, ordinary differential equations in first-order form with the specific objective for it to be self-contained and
inherently linearizable. The proposed simulation model is a mesh-free aerodynamic method based on panel and vortex
particle methods. Panels are used to model the wing surface as well as the near wake, whereas vortex particles are
used to model the far wake. The code is implemented in MATLAB®and demonstrated through an example involving
a UH-60-like rotor at hover.
INTRODUCTION
High-fidelity aeromechanics models are increasingly
making their way into rotorcraft flight dynamics simu-
lations for generalized maneuvering flight, flight control
design, and even real-time piloted simulations. Real-
time time-accurate free wake modeling was implemented
in the General Helicopter (GenHel) (Ref. 1) simula-
tion using the CHARM Wake Module (Ref. 2). The
model was used to compute the main rotor’s induced in-
flow, and real-time simulation was demonstrated. The
CHARM Wake Module was subsequently implemented
in Navy training simulators coupled with ship airwake
effects (Ref. 3). Recently, the CHARM Rotor Module
was incorporated into simulations of eVTOL configu-
rations (Ref. 4). The commercial simulation software
FLIGHTLAB can use Viscous Vortex Particle Models
(VVPM) (Ref. 5) to represent rotor inflow and wake in-
teractions between rotors and lifting surfaces. It has been
used for both conventional helicopters and electric verti-
cal take-off and landing (eVTOL) configurations.
Future Vertical Lift (FVL) and Urban Air Mobility
(UAM) configurations feature multiple rotors, high lev-
els of aerodynamic interactions, and, in the case of UAM,
high revolutions-per-minute (RPM) / variable speed ro-
tors. These features drive the need for advanced aerome-
Presented at the 6th Decennial VFS Aeromechanics Special-
ists’ Conference, Santa Clara, CA, Feb 6–8, 2024. Copy-
right © 2024 by the Vertical Flight Society. All rights reserved.
Distribution Statement A. Approved for public release; distri-
bution is unlimited.
chanics models while simultaneously making real-time
speeds (or even sufficiently fast execution speeds for
routine design) significantly more difficult. Higher ro-
tor RPM requires smaller time steps, modeling of aero-
dynamic interactions needs more significant amounts of
wake to be computed, and rigid rotor systems require
more costly structural dynamics models of the blades.
Thus, while more advanced aeromechanics models are
feasible, it is likely that, in many cases, execution speeds
required for real-time simulations or routine design ap-
plications will remain elusive. Therefore, high-fidelity
aeromechanics models must be formulated to be readily
linearized and simplified to extract more tractable and
less expensive models. Linearized state-space models
are particularly attractive from the control designer’s per-
spective. Not only are linearized models used in most
practical control design methodologies, but linear model
analysis provides many physical insights into system dy-
namics. To this end, a highly desirable feature for fu-
ture advanced simulations is a first-order state-variable
implementation of the aeromechanics (or state-space im-
plementation) that can be efficiently linear.
In Ref. 6, a state-space free-wake model was introduced
by Celi. The basis of the model is the University of
Maryland Free Wake (MFW) (Refs. 7, 8), which is ex-
pressed in partial differential equation (PDE) form. Celi
converted this model into first-order state-variable form
using the method of lines. The formulation of the MFW
model is well-suited for state-space implementation due
to the introduction of the wake age parameter. A sim-
ple kinematic equation for a vortex node subjected to the
1

Biot-Savart law is in a first-order state-space form. How-
ever, in the kinematic implementation, the nodes propa-
gate downstream and must be discarded when they reach
the end of each filament. Thus, the equations are not in
proper state-variable form, and there is no tractable state
equilibrium to linearize about (Ref. 9). In the state-space
formulation, a particular wake state represents the posi-
tion of a wake node of a given age.
In Celi’s state-space implementation, the underlying
PDEs are spatially discretized in the wake age using a
fourth-order 5-point biased upwind (5PBU4) scheme fol-
lowing the approach of Ref. 9. The system then be-
comes a set of first-order ordinary differential equations
(ODEs) in time (i.e., a state-space system), which can
be accurately integrated forward in time. More critically,
while the system states do not have a constant equilibrium
due to the time-periodic dynamics associated with a ro-
tor wake, they have a periodic equilibrium. It is worth
noting that if the same approach is applied to a fixed
wing, the wake dynamics will be time-invariant and have
an equilibrium consisting of a fixed point in the region
of attraction. Recently, a state-space free-vortex model
was implemented in the rotor model of the PSUHeloSim
flight simulation code (Ref. 10) for a rotor representative
of the UH-60. The model was linearized using the time-
periodic systems methods described above (Refs. 11–13).
To improve the fidelity of this simulation, a near-wake
model was implemented to model the distribution of cir-
culation along the blade, and shed vortices were used to
model unsteady aerodynamics. A vortex lattice model of
the near wake, as described in Ref. 14, has been imple-
mented in the state-space wake model of Ref. 15. How-
ever, all of these models have three significant limitations
in that they are: (i) formulated in such a way to be suit-
able for rotating wings only (singularities arise if the ro-
tor angular speed is set to zero), (ii) limited to blade load
spanwise predictions as they do not incorporate a chord-
wise discretization of the wing, (iii) limited in modeling
rotor-on-rotor and rotor-on-wing interactions due to in-
herent instabilities in these instances.
These limitations are generally relaxed by mid-fidelity
aerodynamic solvers developed in recent years that make
use of the vortex particle method (VPM) (Refs. 16–19).
This numerical methodology showed a reasonably ac-
curate representation of the aerodynamic interactions
among several bodies typical of complex rotorcraft con-
figurations and an enhanced computational time with
respect to unsteady Reynolds-averaged Navier-Stokes
(URANS) computational fluid dynamics (CFD) simula-
tions and, in general, with respect to grid-based meth-
ods. To cite a few examples, Lu et al. (Ref. 20) de-
veloped an optimization methodology for a helicopter
design based on a viscous VPM model combined with
an unsteady panel hybrid method. Alvarez and Ning
developed a VPM-based code (Ref. 21) to investigate
multirotor configurations. Tan et al. (Ref. 22) used a
vortex-based approach coupled with a viscous boundary
model to study rotor-rotor interactional problems during
shipboard operations. Recently, Politecnico di Milano
developed a novel, flexible mid-fidelity computational
tool called DUST (Refs. 23–26) 1for the simulation of
the aerodynamics of complex rotorcraft configurations.
However, all said codes are not developed using a state-
variable formulation and, as such, cannot be linearized to
extract simplified models for real-time simulation, stabil-
ity analysis, and controller design. Moreover, the time-
marching nature of these codes does not allow for the
solution of unstable equilibria. The solution of unstable
equilibria becomes particularly important when the ro-
tor wake is coupled with the rotorcraft flight dynamics,
which are unstable at hover and across most of the flight
envelope. Moreover, the solution of unstable equilibria is
also relevant to aeroelastic problems (Refs. 19,27).
Outside of the rotorcraft community, Palacios and Mu-
rua (Ref. 28) developed a discrete state-space formula-
tion of the unsteady vortex lattice method coupled with a
non-linear beam model, and a similar approach was used
in (Ref. 29) validating it against the Pazy wing aeroelastic
testbed 2. Other efforts also focused on state-pace, lin-
earized, and reduced-order formulations (Refs. 30–33).
These examples are, however, limited by the modeling of
a single lifting component. By the presence of a panel
wake that is inherently prone to numerical instabilities
when used to model the interaction between aerodynamic
bodies, such as rotor-wing or rotor-rotor interaction.
Most recently, a coupled panel and vortex particle method
in state-space form was proposed by the authors for
fixed wings and relax all aforementioned limitations
of aeromechanics models (Ref. 34). Along with this
method, an analytical linearization algorithm was de-
veloped to abate the cost of linearization over perturba-
tion methods by O(n2), where nis the total number of
states of the system. This reduction in linearization cost
has the potential to enable the linearization of high-order
aeromechanics models of rotary-wings which, in compar-
ison to fixed-wings, require being linearized at incremen-
tal time steps over one rotor revolution to yield a linear
time-periodic (LTP) representation of the rotor wake dy-
namics.
As such, the objective of this paper is to extend the state-
space, coupled panel and vortex particle method pro-
posed in Ref. (Ref. 34) from fixed to rotary wings. As
part of this objective, the coupled vortex and particle dy-
namics will be demonstrated for a UH-60-like main rotor
blade. This requires formulating the coupled vortex and
particle dynamics as a nonlinear time-periodic (NLTP)
system in first-order form for it to be suitable to lineariza-
tion and periodic systems lifting methods. These are nec-
essary steps to transform the NLTP dynamics to the de-
sired linear time-invariant (LTI) form.
The paper begins with a mathematical description of the
coupled panel and vortex particle dynamics in PDE form.
This is followed by a discussion on how these partial dif-
ferential equations are discretized to obtain an equivalent
1https://public.gitlab.polimi.it/DAER/dust
2https://nescacademy.nasa.gov/workshops/
AePW3/public/
2

system of nonlinear time-periodic (NLTP) ordinary dif-
ferential equations in first-order form. Next, the coupled
panel and vortex particle dynamics is implemented and
demonstrated for a UH-60-like main rotor blade. Final
remarks summarize the study’s overall findings and iden-
tify future developments.
COUPLED PANEL AND VORTEX
PARTICLE METHOD
Overview
The coupled panel and vortex particle method is an adap-
tation to that proposed in Ref. 34. This simulation model
is a mesh-free aerodynamic method based on panel and
vortex particle methods. Panels are used to model the
wing surface as well as the near wake, whereas vortex
particles are used to model the far wake. This setup is
shown in Fig. 1. This section provides an overview of
the mathematical background behind this method.
wing panels
µb,σb
wake panels
µw,rw
wake particles
rp,αp
Fig. 1: Simulation model schematic.
PDE Formulation
The mathematical formulation of the present method re-
lies on a vorticity-velocity formulation of the aerody-
namic problem based on Helmholtz’s decomposition of
the velocity field and a Lagrangian description of the vor-
ticity field. The Helmholtz decomposition of the velocity
field states that the velocity field u
u
u(r
r
r,t)can be written as
the sum of an irrotational field characterized by the po-
tential velocity u
u
uϕ=∇
∇
∇ϕ(i.e., the velocity induced by
lifting bodies such as the wing, and by near-wake panels)
and a solenoidal field described by the rotational velocity
u
u
uψ
ψ
ψ=∇
∇
∇×ψ
ψ
ψ(i.e., the velocity induced by vortex parti-
cles):
u
u
u(r
r
r,t) = u
u
uϕ(r
r
r,t) + u
u
uψ
ψ
ψ(r
r
r,t)(1)
Since the rotational velocity is a solenoidal vector field,
the incompressibility constraint reduces to the Laplace
equation for the kinetic potential ϕ
ϕ
ϕ(r
r
r,t)in the fluid do-
main Ω:
0=∇
∇
∇·u
u
u=∇
∇
∇·u
u
uϕ+∇
∇
∇·u
u
uψ
ψ
ψ
|{z}
∇
∇
∇·(∇
∇
∇×ψ
ψ
ψ)≡0
→∆ϕ
ϕ
ϕ=0 (2)
The Laplace equation is solved using Morino’s method
(Ref. 35) to retrieve the pressure distribution on the lifting
bodies.
Because the potential velocity is an irrotational velocity
field, the vorticity field ω
ω
ω(r
r
r,t)acts as a volume forcing
of the Poisson equation for the vector potential ψ
ψ
ψ(r
r
r,t):
ω
ω
ω=∇
∇
∇×u
u
uφ
|{z }
∇
∇
∇×∇
∇
∇ϕ≡0
+∇
∇
∇×u
u
uψ
ψ
ψ→ −∆ψ
ψ
ψ=ω
ω
ω(3)
given the vector identity ∆ψ
ψ
ψ=∇
∇
∇(∇
∇
∇·ψ
ψ
ψ)−∇
∇
∇×∇
∇
∇×ψ
ψ
ψand
the gauge condition ∇
∇
∇·ψ
ψ
ψ=0. Then, the Lagrangian de-
scription of the dynamical equation governing the vortic-
ity field ω
ω
ω(r
r
r,t)of an incompressible flow is:
Dω
ω
ω
Dt = (ω
ω
ω·∇
∇
∇)u
u
u+ν∆ω
ω
ω(4)
where νis the flow kinematic viscosity and Dω
ω
ω/Dt rep-
resents the material derivative of the vorticity associated
with a material particle.
Differential problems defined in the fluid domain Ωare
manipulated using Green’s function method to obtain a
grid-free volume formulation of the aerodynamic prob-
lem. Note that, in the current formulation, viscosity is
omitted.
Wing and Wake Panel Intensity With regards to the
potential portion of Eq. 1, Morino’s method (Ref. 35)
is adopted to solve the Laplace equation using Green’s
functions. Morino’s method discretizes the surface of the
solid bodies ∂Ωbinto panels of uniform source (σ
σ
σb) and
doublet intensities (µ
µ
µb). The sources term is defined as
σ
σ
σb=ˆ
n
n
n·(u
u
ub−U
U
U∞−u
u
uψ
ψ
ψ), where u
u
ubis the body velocity,
U
U
U∞is the free-stream velocity, and u
u
uψ
ψ
ψis the velocity in-
duced by the vortex particles. The potential part of the
wake ∂Ωwis discretized into panels of uniform doublet
intensity µ
µ
µw. The rotational part is discretized into a se-
ries of particles of intensities α
α
αp. A discrete formulation
of the problem is given by:
K
K
K1µ
µ
µb+K
K
K2µ
µ
µTE
w+K
K
K3µ
µ
µw+K
K
K4α
α
αp=B
B
Bw
w
w(5)
where K
K
K1∈RNb×Nb,B
B
B∈RNb×Nb, and Nbis the total num-
ber of wing panels. Additionally, K
K
K2∈RNb×Nsis a rect-
angular matrix where Nsis the number of panels in span-
wise direction. Moreover, K
K
K3∈RNb×Nwis a rectangular
matrix where Nwis the total number of near-wake pan-
els excluding the trailing edge (TE) wake panels. Finally,
K
K
K4∈RNb×Npis a rectangular matrix where Npis the total
number of vortex particles.
Equation (5) is enhanced with three additional algebraic
equations: (i) Kutta’s condition, which provides a link
between the intensity of the trailing edge wake panel and
the first implicit wake panel, (ii) an equation to link the
intensity and direction of the first line of wake particles
with the last row of potential wake panels, as shown in
Ref. 34, and (iii) the transfer of a panel wake intensity
3

from one step to the other. These three equations are pro-
vided below:
K
K
K5µ
µ
µb=µ
µ
µTE
w(6a)
K
K
K6µ
µ
µw=α
α
αp(6b)
K
K
K7µ
µ
µw+K
K
K8µ
µ
µTE
w=µ
µ
µt+1
w(6c)
For more detail on this derivation and on the definition
of the various terms, the interested reader is invited to
consult Ref. 34.
Vortex Particle Intensity The vorticity field can be dis-
cretized with Npvortex particle (Ref. 16):
ω
ω
ω(r
r
r,t) =
Np
∑
i=1
α
α
αpi(t)δr
r
r−r
r
rpi(t)(7)
where r
r
ris the evaluation point position, r
r
rpiis the position
of the ith particle, and α
α
αpiis the intensity of the ith par-
ticle. By applying Green’s method to the Poisson equa-
tion 3, the rotational portion of this velocity field is found:
u
u
uψ(r
r
r,t) = −1
4π
Np
∑
i=1
r
r
r−r
r
rpi
|r
r
r−r
r
rpi|3×α
α
αpi(t)(8)
=K
K
Kψ(r
r
r,r
r
rp)×α
α
αp(t)(9)
In this work, similarly to Ref. 23, the matrix K
K
Kψ(i.e.,
the Rosenhead kernel) is implemented as the regularised
version of the Biot-Savart kernel:
K
K
Kψ(x
x
x,y
y
y) = −1
4π
x
x
x−y
y
y
(|x
x
x−y
y
y|2+δ2)3/2(10)
In this equation, the regularization parameter δis propor-
tional to the radius of the vortex particles (vortex radius).
The matrix K
K
Kψis assembled by taking advantage of the
skew-symmetric matrix that defines the cross product:
K
K
Kψ×=

0
0
0−K
K
KzK
K
Ky
K
K
Kz0
0
0−K
K
Kx
−K
K
KyK
K
Kx0
0
0

3Nb×3Np
(11)
where K
K
Kx,K
K
Ky, and K
K
Kzare the matrices containing the
x,y,zcomponents of the Rosenhead kernel evaluated
for each particle on each surface panel control point.
Each block matrix has dimension Nb×Np. The non-
penetration condition is enforced on the panels:
u
u
up
b=N
N
N·K
K
Kψ×α
α
αp=K
K
K4α
α
αp(12)
where N
N
Nis the matrix that contains the unit vectors nor-
mal to the panel surface. Again, for more detail on this
derivation, the interested reader is invited to consult Ref.
34.
State-Space Formulation
It is desirable to convert the coupled panel and vortex
particle method from PDE to ODE form. Because the
coupled panel and wake dynamics for a rotating blade
are periodic, consider a nonlinear system of time-periodic
ODEs in first-order form:
˙
x
x
x=f
f
f(x
x
x,u
u
u,t)(13a)
y
y
y=g
g
g(x
x
x,u
u
u,t)(13b)
where x
x
x∈Rnis the state vector, u
u
u∈Rmis the control
input vector, y
y
y∈Rlis the output vector, and tis the di-
mensional time in seconds. Additionally, f
f
fare the non-
linear, coupled panel and vortex particle dynamics and g
g
g
are the output equations. The nonlinear functions f
f
fand
g
g
gare T-periodic in time such that:
f
f
f(x
x
x,u
u
u,t) = f
f
f(x
x
x,u
u
u,t+T)(14a)
g
g
g(x
x
x,u
u
u,t) = g
g
g(x
x
x,u
u
u,t+T)(14b)
The fundamental period of the system is T=2π
ωseconds,
where ωis either the angular speed of the main rotor in
rad/s. Note that, assuming a solution x
x
x∗(t)and u
u
u∗(t)ex-
ists, then this solution for NLTP systems will be periodic
such that x
x
x∗(t) = x
x
x∗(t+T)and u
u
u∗(t) = u
u
u∗(t+T). In the
proposed framework, the state vector of the coupled panel
and vortex particle method is:
x
x
xT=r
r
rT
wr
r
rT
pµ
µ
µT
wα
α
αT
pµ
µ
µT
b(15)
where:
r
r
rware the wake panel positions,
r
r
rpare the vortex particle positions,
α
α
αpare the vortex particle intensities,
µ
µ
µware the wake panel intensities, and
µ
µ
µbthe surface panel intensities.
For a 3-dimensional aerodynamic model, the total num-
ber of states is n=3Nn
w+3Np+Nw+3Np+Nb
where Nn
wis the total number of wake nodes excluding
the trailing-edge wake nodes. The control input vector of
the system is defined as the swashplate inputs:
u
u
uT=θ0θ1cθ1s(16)
where θ0is the collective angle, θ1cis the lateral cyclic,
and θ1sis the longitudinal cyclic. The outputs of the sys-
tem are defined as the wing surface pressure distribution
and the total wing aerodynamic loads:
y
y
yT=p
p
pTF
F
FT(17)
Wake Panel Intensity The dynamics of the wake panel
intensity are obtained by approximating the first-order
derivative with a forward difference scheme:
˙
µ
µ
µw=µ
µ
µt+1
w−µ
µ
µw
dt =K
K
K7µ
µ
µw−µ
µ
µw
dt +K
K
K8µ
µ
µTE
w
dt =
=(K
K
K7−I
I
I)
dt µ
µ
µw+K
K
K8
dt µ
µ
µTE
w=W
W
W1µ
µ
µw+W
W
W2µ
µ
µTE
w
(18)
where K
K
K7and K
K
K8are the shifting matrices. Then, the
wake panel dynamics become:
˙
µ
µ
µw=W
W
W1µ
µ
µw+W
W
W2µ
µ
µTE
w
=W
W
W1µ
µ
µw+W
W
W2K
K
K5µ
µ
µb
=(W
W
W1−W
W
W2K
K
K5(K
K
K1+K
K
K2K
K
K5)−1K
K
K3)µ
µ
µw
−W
W
W2K
K
K5(K
K
K1+K
K
K2K
K
K5)−1K
K
K4α
α
αp
+W
W
W2K
K
K5(K
K
K1+K
K
K2K
K
K5)−1B
B
B′w
w
w
(19)
4

The various terms in this equation are defined in Ref. 34.
Wing Panel Intensity Substituting Eq. (6a) into Eq. (5)
yields an explicit form for the wing panel intensity:
µ
µ
µb=−(K
K
K1+K
K
K2K
K
K5)−1K
K
K3µ
µ
µw+K
K
K4α
α
αp−B
B
B′w
w
w(20)
The dynamics of the wing doublet intensities are com-
puted by taking the time derivative of the equation as fol-
lows:
˙
µ
˙
µ
˙
µb=−(K
K
K1+K
K
K2K
K
K5)−1˙
K
˙
K
˙
K3µ
µ
µw+K
K
K3˙
µ
˙
µ
˙
µw
+˙
K
˙
K
˙
K4α
α
αp+K
K
K4˙
α
˙
α
˙
αp−B
B
B′˙w
˙w
˙w(21)
where the evolution ofK
K
K3and K
K
K4is governed by the wake
transport equation. The evolution of µ
µ
µwfollows Eq. (19),
whereas the evolution of αpis governed by Eq. (22).
Vortex Particle Intensity A constant wake age descrip-
tion 9 of the rotational velocity and the vorticity fields
is provided by the kinematic and dynamical equations of
the trajectory of the material particles and the vorticity as
follows (i.e., the transpose method):
Dα
α
α
Dt =(α
α
αp·∇
∇
∇T)u
u
u
=(α
α
αp·∇
∇
∇T
p)(U
U
U∞+u
u
uψ+u
u
uϕ) + ˙
K
K
K6µ
µ
µw+K
K
K6˙
µ
µ
µw
(22)
where u
u
uψ=v
v
vp,u
u
uϕ=v
v
vµ
b+v
v
vσ
b+v
v
vw, and ∇
∇
∇T
pU
U
U∞=0.
The vortex particle vorticity equation, i.e., Eq. (22), is
spatially discretized using the method of lines (Ref. 6):
∂α
α
αp
∂t+∂α
α
αp
∂ ζ
∂ ζ
∂t=(α
α
αp·∇
∇
∇T
p)(U
U
U∞+u
u
uψ+u
u
uϕ)
+˙
K
K
K6µ
µ
µw+K
K
K6˙
µ
µ
µw
(23)
where ζis the wake age parameter at each spanwise lo-
cation:
ζi=Ωri∆t=Ui∆t(24)
In this equation, riand Uiare the radial location and tan-
gential velocity of the ith wake node. The wake age is
a distance measured along the vortex starting from the
vortex release point on the wing trailing edge such that a
particular wake state represents the position of a wake
node/particle of a given age. The partial derivative of
the wake node positions with respect to the wake age
is discretized using a fourth-order 5-point biased upwind
(5PBU4) scheme. This discretization is key to making
the system of ODE time-periodic in that it associates each
state (i.e., wake panel node and particle position) to a par-
ticular wake age ζ. Thus, each state will always have
the same wake age such that, at equilibrium, the position
states will be periodic and oscillating about a fixed mean.
If one were to solve Eq. (22) directly (i.e., kinematic ap-
proach), then the system would be time varying because
each state would flow down the stream with a higher
wake age at each time step. Not only this would make
the system time varying but, to preserve a fixed number of
states, it would also require to: (i) destroy the last raw of
vortex particles at each time step and (ii) to create a new
row of wake panels at each time step (Ref. 9). Both of
these are impractical. The same applies to the remaining
states of the system discussed below – wake panel, vor-
tex particle, and wing panel intensities. It is critical for
the wake dynamics to be a time-periodic system to make
it suitable to harmonic decomposition and thus approx-
imable with a higher-order linear time-invariant system.
This way, the large pool of time-invariant system anal-
ysis tools can be applied for dynamics stability analysis
and control design. The matrix A
A
Aζiis the finite difference
matrix corresponding to the 5PBU4 scheme 6 defined for
each blade radial station i:
A
A
Aζi=−Vi
12




−25 48 −36 16 −3
−3−10 18 −6 1
1−8 0 8 −1
−1 6 −18 10 3



(25)
Then, the vortex particle vorticity equation becomes:
˙
α
α
αp=−A
A
Aζα
α
αp+ (α
α
αp·∇
∇
∇T
p)(U
U
U∞+u
u
uψ+u
u
uϕ)
+˙
K
K
K6µ
µ
µw+K
K
K6˙
µ
µ
µw
(26)
Wake Panel and Vortex Particle Position The position
of the wake panels and particles is found by using the
vortex transport equation:
Dr
r
r
Dt =v
v
vind (27)
where the induced velocity is calculated using Biot-
Savart law (Refs. 6, 36). The induced velocity at each
wake node is given by:
v
v
vind =v
v
vσ
b+v
v
vµ
b+v
v
vw+v
v
vp(28)
In Eq. (28), v
v
vσ
bis the induced velocity contribution due to
the body sources, v
v
vµ
bis the induced velocity contribution
due to the body doublets, and v
v
vwis the induced velocity
contribution due to the wake panel doublets. Equations
for these induced velocities can be found in (Ref. 36).
Additionally, v
v
vpis the contribution due to the wake par-
ticles. The transport equation, i.e., Eq. (27), is also
transformed into a system of ODEs via the method of
lines (Ref. 6), leading to:
∂
∂tr
r
r+∂r
r
r
∂ ζ
∂ ζ
∂t=v
v
vind (29)
By adopting the same 5-point spatial discretization as
above, a time-periodic system of ODEs for the wake pan-
els and particles position is obtained:
˙
r
r
r=−A
A
Aζr
r
r+v
v
vind (30)
Summary The following equations summarize the cou-
pled panel and vortex particle dynamics in state-space
5

form, i.e., the nonlinear function f
f
f(x
x
x,u
u
u,t):
˙
r
r
rw=−A
A
Aζr
r
rw+v
v
vind(r
r
rw,r
r
rp,α
α
αp,µ
µ
µw,µ
µ
µb)(31a)
˙
r
r
rp=−A
A
Aζr
r
rp+v
v
vind(r
r
rw,r
r
rp,α
α
αp,µ
µ
µw,µ
µ
µb)(31b)
˙
µ
µ
µw=(W
W
W1−W
W
W2K
K
K5(K
K
K1+K
K
K2K
K
K5)−1K
K
K3)µ
µ
µw(31c)
−W
W
W2K
K
K5(K
K
K1+K
K
K2K
K
K5)−1K
K
K4α
α
αp
+W
W
W2K
K
K5(K
K
K1+K
K
K2K
K
K5)−1B
B
B′w
w
w
˙
α
α
αp=−A
A
Aζα
α
αp(31d)
−1
4π1
R3α
α
αj
p×α
α
αp−3
R5∆r
r
rj(∆r
r
rj·α
α
αj
p×α
α
αp)
(31e)
+α
α
αp·(∇
∇
∇T
pv
v
vµ
b+∇
∇
∇T
pv
v
vσ
b) + α
α
αp·∇
∇
∇T
pv
v
vw+˙
K
K
K6µ
µ
µw
+K
K
K6˙
µ
µ
µw
˙
µ
µ
µb=−(K
K
K1+K
K
K2K
K
K5)−1˙
K
˙
K
˙
K3µ
µ
µw+K
K
K3˙
µ
˙
µ
˙
µw+˙
K
˙
K
˙
K4α
α
αp(31f)
+K
K
K4˙
α
˙
α
˙
αp−B
B
B′˙w
˙w
˙w
The output equations, i.e., the nonlinear function
g
g
g(x
x
x,u
u
u,t), are:
C
C
Cp=1−1
U2
∞∥u
u
usurf∥2+2˙
µ
µ
µb(32a)
F
F
F=ˆ
A
A
Ap
p
p(32b)
For the complete derivation of these equations the inter-
ested reader is referred to Ref. 34.
RESULTS
The coupled panel and wake method is implemented in
MATLAB®for a UH-60-like rotor blade. The rotor blade
characteristics are reported in Table 1. The time step of
the simulation is chosen as ∆t=2π/(Ω∆ψ) = 0.006464
seconds, corresponding to a rotor angular speed of Ω=
27 rad/s and a non-dimensional time step of ∆ψ=10
deg. To find an equilibrium condition, the coupled panel
and vortex wake dynamics are time marched for a swash-
plate setting corresponding to θ0=19.7 deg, θ1c=0, and
θ1s=0. The collective angle corresponds to the hover
trim condition for a UH-60 with a weight of W=17,000
lb. This trim condition was found with the PSUHeloSim
flight dynamics code (Ref. 10). The simulation features
10 spanwise and 10 chordwise blade panels, 10 spanwise
and 5 chordwise near-wake panels, and 10 spanwise and
177 chordwise vortex particles corresponding to approx-
imately five wake revolutions. Figure 2 shows a three-
dimensional representation of the converged wake after
approximately 8 rotor revolutions. In this figure, the wake
is shown to contract below the rotor disk, according to
momentum theory (Ref. 37). Additionally, the strength of
the particle is maximum at the tip of the blade, forming a
helicoidal vortex structure. Based on these observations,
the coupled panel and vortex particle method appears to
capture the well-known physics of a rotor wake.
Next, the response of the coupled panel and vortex wake
dynamics following a 2 deg doublet in the collective an-
gle θ0is simulated. Results are presented for: (i) a repre-
sentative wake panel at the outermost spanwise location
Table 1: Characteristics of the UH-60-like main rotor
blade (Ref. 1).
Parameter Value Units
Radius, R26.8 ft
Blade chord, c1.73 ft
Blade twist, θtw −13 deg
Flapping hinge offset 1.25 ft
Spar length 2.25 ft
Angular speed, Ω27 rad/s
and with an equivalent wake age of 45 deg, and (ii) a rep-
resentative vortex particle also located at the outermost
spanwise location and with an equivalent wake age of 45
deg. This setup is shown in Fig. 3.
Figure 4 shows the position response of the representative
wake panel. While the longitudinal and lateral positions
of the panel change periodically, the vertical position is
shown to decrease/increase with an increase/decrease of
the collective angle, which suggests that the near wake is
pushed down/up when the inflow is increased/decreased
as a result of an increase/decrease of the collective angle.
Similar observations can be made for the representatie
vortex particle position response, shown in Fig. 5.
CONCLUSIONS
A coupled panel and vortex particle method in state-
variable form was extended from fixed to rotary wings.
The coupled panel and vortex particle dynamics were
formulated as a nonlinear system of time-periodic or-
dinary differential equations in first-order form to be
self-contained and inherently linearizable. The proposed
simulation model consisted of a mesh-free aerodynamic
method based on panel and vortex particle methods. Pan-
els were used to model the wing surface as well as the
near wake, whereas vortex particles were used to model
the far wake. The code was implemented in MATLAB®
and demonstrated through an example involving a rotor
at hover. Based on this work, it is concluded that state-
space modeling of grid-free aerodynamic methods, such
as panel and vortex particle methods for helicopter ro-
tors, is possible. This formulation offers a unique oppor-
tunity to obtain linearized models that can be analyzed
with linear systems tools. These tools include stability
and response analysis, model order reduction, and con-
trol design. Moreover, the coupled panel and vortex par-
ticle dynamics in state-space form can be used to aug-
ment the flight dynamics to provide high-fidelity, time-
accurate simulations of the coupled rigid-body, rotor, and
rotor wake dynamics.
Future work will focus on additional validation and on
the linearization of the state-space coupled panel and vor-
tex particle method using both numerical and analytical
methods (Ref. 34). Linearized models of the coupled
panel and vortex particle dynamics have applications in
the flight dynamics of rotary-wing vehicles, where these
6

-30
-20
-10
20
0
0
30
20
-20 10
0
-10
-20
-30
1600
1800
2000
2200
Fig. 2: Three-dimensional representation of the coupled panel and vortex particle wake at hover for θ0=19.7 deg.
Fig. 3: Schematic indicating the states whose responses
are plotted.
0 0.5 1 1.5 2
-20
0
20
0 0.5 1 1.5 2
-20
0
20
0 0.5 1 1.5 2
-0.4
-0.3
-0.2
Fig. 4: Position response of a representative wake panel
node to a 2 deg doublet in the collective input.
dynamics can be used to augment the rigid-body dynam-
ics. The coupled rigid-body and wake dynamics can be
0 0.5 1 1.5 2
-20
0
20
0 0.5 1 1.5 2
-20
0
20
0 0.5 1 1.5 2
-0.4
-0.2
Fig. 5: Position response of a representative vortex
particle to a 2 deg doublet in the collective input.
leveraged to assess the stability, response characteristics,
and handling qualities of vehicles experiencing aerody-
namic interactions between rotors, wings, and/or obsta-
cles. Moreover, unlike time-marching approaches, state-
space representations of the wake dynamics are suitable
for determinations of unstable equilibria via trim solu-
tion methods. This is particularly important for rotary-
wing aircraft, given their unstable nature across typically
a large portion of their flight envelope.
7

ACKNOWLEDGMENTS
This research was partially funded by the U.S. Govern-
ment under agreement no. N000142312067. The views
and conclusions contained in this document are those of
the authors and should not be interpreted as representing
the official policies, either expressed or implied, of the
U.S. Government.
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