AIAA-00-0114
A REVIEW OF COMPUTATIONAL TECHNIQUES FOR ROTOR WAKE MODELING
Nathan Hariharan*
CFD Research Corporation, Huntsville, AL 35805
Lakshmi N. Sankar †
School of Aerospace Engineering
Georgia Tech, Atlanta, GA 30332-0150
ABSTRACT
This paper discusses existing techniques for modeling
the rotor wake in hover and in forward flight. It also
briefly review extensions of these methods for
modeling complete rotorcraft. The review is followed
by results from promising high order overset methods
for wake capturing, and efficient hybrid near field
Navier-Stokes wake capturing/far-field potential flow
wake convecting methodologies. The paper concludes
with some speculations on the likely advances in
rotorcraft flowfield modeling in the forthcoming years.
INTRODUCTION
The flow field around a rotor, whether in forward flight
or hover, is difficult to model due to the presence of
strong vorticity. The flow phenomena for a rotor differ
from that for a wing in forward flight, because of the
differing influence of their respective wakes. For a
wing in forward flight, the generated tip vortex and the
vortex sheet are quickly convected away from the wing,
and the wake influence on the flow field in the vicinity
of the wing is small. For an adequate numerical
simulation of a wing in forward flight, it is sufficient to
capture the generated tip vortex in the vicinity of the
wing.
In contrast, in the flow field around a rotor, the strong
vortex wake system lingers in the vicinity of the rotor.
In hover, the strong tip vortex coils beneath the rotor,
and significantly alters the effective angle of attack seen
by the rotor [1].
Accurate numerical prediction of aerodynamic
parameters such as thrust coefficient and induced
torque coefficient requires an accurate modeling of the
tip vortex. In forward flight the entire vortex system is
* Senior Engineer, AIAA member
†
Regents Professor, AIAA Associate Fellow
Copyright© 2000 by CFD Research Corporation.
Published by the American Institute of Aeronautics
and Astronautics, Inc., with permission.
swept back leading to strong interaction between bladetip vortices with successive blades, a phenomenon
known as the blade-vortex interaction (BVI). These
blade vortex interactions results in rapid changes in
local flow conditions and are a major source of
aerodynamic noise and structural vibration.
In addition, in high-speed forward flight, the advancing
blade experiences transonic flow conditions leading to
the formation of shocks and shock-boundary layer
interactions. The retreating blade experiences a rapid
variation in the effective pitch angle that causes a threedimensional dynamic stall. "First-principles" based
analyses are needed to understand and predict the highspeed phenomenon. Methodologies that solve for the
flow field from the basic conservation laws without
using additional information (information from
analyses such as other numerical formulations,
analytical formulations, or experimental observations)
are generally referred to as "First-principles" based
methods.
In modern helicopter designs, the rotor disk is placed
very close to the fuselage for compact configurations,
and to reduce drag associated with the mast. In such
configurations the vortex system generated by the rotorblades interacts with the body surface in a complex
unsteady manner. Rotors and airframes designed
without taking into effect the rotor-airframe
aerodynamic-interaction often tend to perform much
below their design capability [2].
The underlying issue in modeling these flows is the
necessity to properly account for the complex vortex
system generated by the rotor. Researchers in the past
two decades have adopted a broad class of
methodologies with various levels of complexity to
model the vortex system.
This paper will review the existing methodologies for
modeling rotors in hover, forward flight, and rotor and
airframe interactions. Particular attention will be given
to Navier-Stokes based methods for modeling the rotor
wake. The next section presents a brief summary of this
paper.
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SCOPE OF THE PRESENT WORK
Several excellent survey articles [McCroskey[46],
Srinivasan and Sankar[71], Landgrebe[79] are available
that summarize rotorcraft flow simulation efforts. The
present work is not intended to be an all-inclusive
survey of rotorcraft wake modeling. In this effort we
briefly review recent efforts at modeling rotor-wake,
with the emphasis on Navier-Stokes based “firstprinciples” wake capturing. This is followed by some of
our cent results in high order NS methods and hybrid
NS/Potential free wake methods that hold good promise
in the forthcoming years. We conclude the paper with
some speculation on the future makeup of NavierStokes rotorcraft wake-capturing in terms of research
and design utility.
REVIEW OF ROTORCRAFT FLOWFIELD
SIMULATION METHODOLOGIES
Simulation of Rotor in Hover and Forward Flight
The earliest methods for modeling rotors were based on
an extension of Prandtl's lifting line theory for wings. In
these techniques, the individual blades were modeled as
"lifting line" vortices, and the wake was modeled as a
deformed
helix.
During
1970's,
Gray[1],
Landgrebe[2,3] and others developed a prescribed wake
model to define this helical geometry. Their models
were based on experimental observations, such as
smoke visualization of the wake. Kocurek and
Tangler[4] and Shenoy[5] refined the lifting line
method by using a lifting surface representation of the
rotor with an improved prescribed tip vortex and inner
vortex geometry. The prescribed wake technique is
simple and effective but its applicability is limited to
rotors having geometry similar to those used in
experiments. The lifting line methods require a table
look up of airfoil load data, and modeled the unsteady
blade wake interactions in a quasi-steady manner.
Free wake analysis methods were subsequently
developed by Scully[6], Summa[7,8], Bliss and
Miller[9] and others. In these methods the tip vortex
was modeled as a series of line vortex filaments and
was tracked using a Lagrangian technique. These free
wake methods accounted for self induced distortions of
a rotor wake in forward flight such as roll up. The
velocity field was computed using the Biot-Savart law
and numerical integration techniques. Johnson [10]
used the free wake technique as an option in his
computer program CAMRAD (Comprehensive
Analytical Model of Rotorcraft Aerodynamics and
Dynamics). This program is widely used even to this
date to compute inflow velocities, airloads, and
rotorcraft performance, stability and control. Caradonna
and Isom[11] extended the transonic small disturbance
theory which became available during the early 1970's
to transonic flow over rotors. This methodology
allowed a first principles based study of forward flight
conditions and tip effects. In this approach, the effect of
the vortex wake was modeled as a table of angle of
attack changes, computed from a comprehensive
analysis case such as CAMRAD.
During early 1980's, Chang[12] modified the full
potential flow solver FLO22 for isolated wings to
model rotors. Egolf and Sparks[13] modified Chang's
work by embedding the vortex element associated with
the tip vortex with the potential flow field. Their
approaches solved either the steady or quasi-steady
form of the potential flow equation. Sankar and
Prichard[14], Sankar et al.[15], Strawn[16], Bridgeman
et al.[17], Strawn and Caradonna[18]
developed
unsteady full potential flow based rotor solvers. In all
these applications the rotor wake effects ere computed
either using a prescribed wake model or a free wake
model such as CAMRAD. The full potential solvers
could analyze transonic flow with mild shocks, though
in an isentropic manner. Viscous effects were
accounted only through a boundary layer correction.
Ramachandran et al.[19,20] solved the full potential
equation and included the rotor wake effects using a
Lagrangian based approach for tracking the vortex
filaments.
During late 1980's, Euler methods matured to a point
where calculation of the rotor flowfield in hover and
forward flight was feasible. These methods solved the
mass, momentum and energy conservation equations in
a time dependent fashion using finite-difference or
finite-volume methods. These solvers did not include
viscous effects but could analyze the transonic flow
with non-isentropic shocks. Sankar et al.[21], Agarwal
and Deese[22], and Hassan et al. [23] developed Euler
solvers for isolated rotors. Again, the wake effects were
modeled as an inflow angle of attack table supplied
from a separate comprehensive analysis.
Wake and Sankar[24] were one of the first few
researchers to develop a Navier-Stokes code to analyze
rotor flow fields in hover and forward flight[25]. They
used a C-H computational grid and a hybrid ADI
implicit time marching algorithm. These terminologies
will be explained in the full paper. The rotor wake
effects were accounted for using the inflow table. These
solvers were computationally expensive, but had all the
features required for modeling the advancing blade
transonic flow and the retreating blade dynamic stall
phenomena. The only drawback of these codes was that
they were dependent on external wake models for
calculation of inflow velocities. Smith and Sankar and
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Tsung et al.[26,27] extended this Navier-Stokes code to
include the aeroelastic effects and improved the spatial
accuracy from second to fourth order using a compact
operator implicit scheme. Narramore et al.[28] used
Navier-Stokes methods to investigate viscous
phenomena over rotors-blades such as flow separation
and dynamic stall. Chen, McCroskey and Obayashi[29]
used an implicit L-U factorization scheme and an
external free wake model to solve forward flight rotor
flows.
Mello[30], Berezin[31] and Moulton et al.[32] have
developed hybrid Navier-Stokes/Full Potential Equation
based solvers in an effort to reduce the computation
time. Here, the computational domain is divided in to
an inner domain comprising of the body and the near
wake and an outer domain far from the body. NavierStokes equations are solved in the inner domain and the
full potential equation is solved in the outer domain
where viscous effects are negligible. They were
successful in reducing the computation time by 50%
when compared to the full Navier-Stokes computation
times. Mello[30] also developed a non-reflective
interface boundary condition procedure to eliminate the
false acoustic reflections at the Navier-Stokes/Full
potential interface boundary. More recently, Moulton et
al.[72], Berkman and Sankar et al.[78] have used
overset grids for the Navier-Stokes and Potential free
wake solutions.
During the early 1990's, taking advantage of the
enormous improvement in computing capabilities
during the 1980s, a new class of Euler/Navier-Stokes
codes were developed in an effort to capture the rotor
wake from first principles without any need for external
wake models. Removing
the need for external
information that depends on rotor geometry is a big step
in the true simulation of the rotor flowfield. These firstprinciples based solvers are particularly useful in
analyzing new or complex rotor blades where no
experimental data available. Strawn and Barth[33],
Srinivasan and McCroskey[34], Srinivasan et al.[35],
Srinivasan and Baeder[36], Srinivasan et al.[37],
Duque[38], and Duque and Srinivasan[39] solved the
hovering rotor flowfields by capturing the rotor wake in
an Eulerian fashion, and from first principles. Hariharan
and Sankar[40] used high order methods to solve the
flow field of rotor in hover from first principles.
Bangalore[41] used first-principles based methods to
investigate high-lift rotor systems in hover and forward
flight. Ahmad and Duque[42] analyzed the AH-1G two
bladed rotor in forward flight mode using structured
embedded grids. The tip vortices in most of the
analyses were captured up to one revolution, beyond
which the vortex diffused due to numerical dissipation.
In an effort to reduce the numerical dissipation,
Steinhoff et al.[43], Wang et al.[44] used vorticity
confinement techniques to prevent the tip vortex
diffusion. Their idea was to add artificial convective
velocities that drive the vorticity field towards the
centroid of the vortices. A structured/unstructured grid
approach was tried by Duque[45] to solve hovering
rotors. A structured grid was used near the blade
surface to resolve the boundary layers and unstructured
grids were used in the wake. Unstructured grids[46]
have a better potential for grid adaptation. It is also
easier to add/delete grid cells as required, compared to
structured grid methods. However grid adaptation/fine
control are difficult to handle in context of unsteady
vortex-wake, and hence these methods have not been
well developed in the context of rotor-blade wake. In
the recent past overset grid structured methods have
(Ahmed, Strawn[72], Hariharan[75]) have moved far
ahead of unstructured methods for wake capturing. The
relative ease of construction of high order schemes with
structured grids and the simplicity of using Cartesian
background overset wake grids have thrust high order
overset methods of various flavors[74-77] as the most
likely candidate to capture the entire wake of a rotorblade without resorting to Lagrangean free-wake
modeling.
Simulation of Rotor-Airframe Interaction
Because of the complexity of the rotor-airframe
interaction, much of the initial research was based on
experiments. Sheridan and Smith[47] reviewed the
interactional problem, and classified the categories of
interactions.
Smith and Betzina[48] studied the
interaction between a helicopter rotor and a model
fuselage at low speeds. Brand, Komerath, and
McMahon[49], Liou et al.[50], Brand[51], Liou[52],
Kim[53], and Liou et al.[54] at Georgia Tech have
extensively documented the interactional effects of a
rotor with a model airframe through a systematic
series of experimental studies. Crouse, Leishman, and
Bi[55] and his coworkers at the University of Maryland
performed similar experiments on an airframe with a
simulated tail boom. Researchers at NASA Langley
have studied a full scale main rotor and body-ofrevolution airframe model[56].
Egolf and Landgrebe[57] used an uncoupled prescribed
wake/source panel analysis to study the interactional
problem. Clark and Maskew[58] developed a fully
coupled analytical method using a vortex sheet of the
rotor wake and the rotor, and doublet panels for the
airframe. Mavris[59], Lorber and Egolf[60], and
Quackenbush and Bliss[61] used variations of
analytical representation of the rotor wake, in their
attempt to simulate the rotor-airframe interaction. These
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methods could accurately model the first order effects
of the rotor on the airframe and vice versa, but could
not model secondary, viscous effects such as vortexsurface interactions. Eulerian approaches to the
interaction problem soon followed. Chaffin and
Berry[62], and Zori, Mathur and Rajagopalan[63] used
embedded body forces to represent the rotor while
solving the viscous flow over the airframe. This
approach modeled the rotor as an actuator disk, and was
quasi-steady. Affes and Conlisk[64], used a
combination of potential flow methods and three
dimensional unsteady boundary layer analyses to
resolve the central features of strong vortices
interacting with curved surfaces.
Modeling both the rotor which rotates in space, and the
stationary fuselage requires two or more grids that are
overset on top of each other and are in relative motion.
Following a suggestion from McCroskey[65],
Duque[66] analyzed the flow field over the Comanche
rotorcraft using overset grids. With this approach,
Meakin[68] used overset grids to simulate the flowfield
around the Osprey rotorcraft. In these approaches, the
formulas that transfer information from one grid to the
next, known as the domain connectivity data, were precomputed and stored as tables of influence coefficients.
Srinivasan and Ahmad[69] used overset grids to study
rotor-center body interaction. The single most
determinant factor in rotor performance is the behavior
of the shed wake which dominates the rotor acoustics
and vibratory loads. Hariharan and Sankar[67] used
high order unsteady overset mechanism to study the
vortex impingement for an experimental rotor-body
setup tested at Georgia Tech.
In the next section, we discuss some of our recent
results from first-principles high order overset methods
and hybrid Navier-Stokes/Potential methodologies for
rotor-blade tip-vortex wake capturing.
HIGH-ORDER METHODS FOR ACCURATE
PREDICTION OF THE WAKE
temporal accuracy. The viscous fluxes are computed
using central differences. The inviscid fluxes are
updated using an approximate Riemann solver, i.e., the
numerical flux on the cell faces is given by,
F q   F1 q R 
F 1 L
 A q R  q L 
2
Baseline Fifth Order ENO Scheme
The fifth order formulation in the solver has been
developed along the lines of
Essentially NonOscillatory (ENO) methods [Hariharan, Ref. 40,67].
The higher order reconstruction comes in the projection
stage of the conservative variable, i.e., qL and qR. For a
smoothly varying function, these projections are based
on the support stencils as shown in the Figure 1a
below.
Figure 1a. Fifth Order Stencils for Computing Left and
Right Primitive Variables
The details of the economic implementation of this
scheme discussed in [Hariharan, Ref. 67]. The increase
in the cost of computation was found to be less than
10% more than the time required for computations of
similar flowfield solutions with the third order spatially
accurate MUSCL projection. In case a discontinuity is
present in the sampling region, i.e., as illustrated in
Figure 1b, the sampling stencils are automatically
shifted to avoid sampling across the discontinuity, as
shown in the Figure 1c below.
The discretized form of 3D, unsteady finite-volume
version of the Navier-Stokes equation is solved.



6 


q J    VF  VG q  F  S 
t
i 1
6
(2)
(1)
 Fv  S
i 1
The above formulation allows for arbitrary motion of
the grids. The temporal discretization is done using a
three point stencil, and the solution update process uses
a Newton iterative solver to achieve third order
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The stencil shifting near the boundaries and
discontinuities
is
similar
to
the
baseline
implementation.
Study of Tip Vortex of Rotor in Hover
Figure 1b. Illustration of a Distribution with a
Discontinuity
This study focuses upon the vortex dissipation
characteristics using the baseline fifth order ENO
scheme for the Caradonna-Tung[11] rotor in hover.
This has been simulated widely, including firstprinciples based Navier-Stokes simulations by
Srinivasan[36], Hariharan and Sankar[48,75]. In this
simulation we look into the velocity distribution across
the tip vortex and assess what is needed to resolve a
potential BVI in forward flight.
A 120*40*60 (streamwise, spanwise, normal) H-H grid
system as shown in Figure 3 was used for this
simulation. Periodic boundary conditions were used to
simulate the second blade. The tip Mach number was
0.44 and the collective pitch was eight degrees. Figure
4a shows iso-vorticity contour (top and side views)
showing the captured tip vortex. The tip vortex
contraction and descent are very well captured as
shown in Figure 4b over the first 180 degrees. The tip
vortex is seen to acquire a downward kick as it
approaches the following blade.
Figure 1c. Adaptive Stencil for Uniformly High Order
Solution
A third order Newton iterative scheme is used to
integrate in time. The time stepping is done in an
implicit manner using directional factorization
[Hariharan, Ref. 40,67].
Seventh Order Scheme
The seventh order extension is considered along similar
lines as above. A wider stencil as shown below in
Figure 2 is used for the left and right projection.
Figure 3. Boundary Conditions for the Solution of
Rotor in Hover
Figure 2. Seventh Order Stencil for Smooth Flow
Conditions
Figures 5a-f show the evolution of the vortex over the
blade (Figures 5a-c — view from behind the rotor) and
its transport underneath the next blade (Figures 5e-f).
The vortex strengths are compared by ascertaining the
tangential velocity variation across the vortex at zero
degree azimuth (right off the blade) and at around 180
degree azimuth (as it passes underneath the next blade).
The comparison is shown in Figures 5g-h. It is clearly
seen that even though the fifth order scheme (with the
current grid structure) has maintained the vortex
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structure well, the strength itself is well reduced over
180 degrees of vortex convection. A much higher
fidelity of the vortex is required if interactional effects
such as BVI are to computed with any accuracy. One of
the main reasons for the tangential velocity dissipation
is that the tip vortex in its natural inboard/downward
motion moves out of “high density” grid region near the
Top View
blade and out into “lower density” grid region in the HH grid system. In the next study, we avoid this by
using overset refinement to a wing-tip vortex and make
sure that enough points are always provided across the
face of the vortex.
Side View
Figure 4a. Vorticity Iso-Surfaces Showing the Tip Vortex
Figure 4b. Comparison of Tip Vortex Position
Figure 5. Vortex Evolution and Strength Comparison for Fifth Order Computation with a Single H.
Grid (120 x 40 x 60
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Evaluation of a Baseline Fifth Order ENO Scheme:
Wing tip vortex capture for 18 chord lengths
A wing grid-vortex grid overset system, as shown in
Figure 6a is considered here. The vortex grid extends
up to 18 chord length behind the wing trailing edge. In
this computation a NACA0015 wing at  = 15°, M =
0.18 (tested by McAllister et al. 1991) was used. The
vortex formation using high order methods have been
validated in an earlier study (Hariharan, Ref. 70). The
existing fifth order spatial ENO/third order temporal
implicit scheme was used for this study. The purpose of
this study is to demonstrate the feasibility of using rotor
grids (C or H-H) with self-deforming vortex grid
systems to fully capture BVI effects, i.e., capture the tip
vortex over 180 degrees of revolution (for a two bladed
rotor) with less than 5-10% dissipation of the peak to
peak variation of the tip vortex. For a typical rotor
blade of aspect ratio (AR) of 6, a 180 degree
convection would entail ~18 (pi*AR) chord lengths of
vortex transport, which is the same length as the one
considered here. The vortex is generated by the wing
grid and transferred to the vortex grid. The vortex grid
used for this study had 100 streamwise points and
30*30 points at every streamwise station.
Movement of a
Streamwise
Plane
Grid-1
Grid-2
Tip Vortex
Figure 6b. Schematic of Unsteady Vortex Grid System
Figures 7a and 7b show the top and rear view of the self
adapted vortex grid and the starting vortex grid. Figure
8 shows the axial momentum component on a crosssection of the vortex. The captured vortex has a positive
axial momentum (jetlike) till around the halfway mark
of the 18 chords. Then it switches over to a negative
axial momentum exhibiting wake like behavior. A real
physical vortex exhibits a similar behavior due to
viscosity. However, the current simulation is inviscid
and such a transition is incorrectly triggered by the
dissipation in the numerics.
The axial velocity
component in a realistic vortex (over a wing or a rotor)
plays a important role in the determination of the vortex
structure A closer look at the variation of the axial and
tangential velocity components reveal the need for
accurately capturing the axial velocity. Figures 9a-g
shows the axial (black) and tangential (red) velocity
components across the vortex at several streamwise
stations.
Top View
Figure 6a. Wing Grid-Vortex Grid Overset System
Figure 6b shows a schematic of the self-adaptive vortex
tracking grid. The vortex grid adapts itself in the
following manner.
Initial
As the computation progresses, each streamwise crosssection of the vortex grids sense the maximum vorticity
and correspondingly shift their position within the
streamwise plane of that station. Vortex Grid-2 (as
shown in Figure 3.1-2) will derive its position from the
position of the streamwise station of Grid-1 that lies in
the periodic plane. As the vortex evolves, the vortex
grids will ensure that the required 8-10 points across the
face of the vortex required by the fifth order
computation is made available, thus preventing
diffusion.
Adapted
Figure 7a. Adapted Vortex Grid
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Back View
Adapted
Initial
Figure 7b. Initial and Adapted Vortex Grid
Figure 8. Streamwise Momentum Contours Across
Spanwise Section of the Vortex Grid
a. x/c = 1.0
b. x/c = 4.0
e. x/c = 9.0
c. x/c = 6.0
f. x/c = 11.0
d. x/c = 8.0
g. x/c = 15.0
Figure 9. Comparison of Axial (red) and Tangential (black) Momentum Variation Across the Vortex
Structure of a 3D Vortex and Its Numerical
Resolution
The following can be ascertained from picture
sequences Figures 9a-g. The vortex forming over a
wing (or a rotor) is a three dimensional entity as shown
in Figure 10. It has a distinct axial velocity component
across the vortex, apart from the peak-to-peak
tangential velocity variation. The fifth order Euler
simulation capturing the vortex (Figures 9a-g) exhibits
the following behavior. If the axial velocity is correctly
represented, i.e., jetlike, then the peak-to-peak
tangential velocity variation remains at a certain value.
If the axial velocity dissipates (due to numerical
dissipation) below zero, it switches to a wakelike
structure changing the peak-to-peak tangential velocity
variation to a different lower value. In the current study
the axial velocity component switches mode around the
40% mark of the required length of the vortex grid.
Thus, to capture the right variation of the peak-to-peak
tangential variation, it is necessary to capture the
correct axial mode (jetlike versus wakelike). Even if
the axial velocity component does not play a big role in
a certain physical interaction (say BVI under some
conditions), for the purpose of correctly capturing the
tangential peak-to-peak variation, the axial velocity
component has to be captured correctly. The axial
component has much steeper gradients and hence it
dissipates faster. The vortex grid used in the current
study had 30*30 points in the cutting plane across the
vortex. This has to be enhanced twice or thrice the size
to capture the vortex without any dissipation using the
baseline fifth order ENO scheme. Alternatively we can
increase the spatial accuracy further and capture the
vortex with the current or even smaller grids. We
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analyze the performance of a seventh order scheme in
the next study.
Table 1. Comparison of Run Time Dec Alpha 500
Workstation
SCHEME
CPU TIME/ITERATION
Fifth ENO
13.5 seconds
Seventh ENO
17.2 seconds
Overset System:
Grid-1 - 90x40x30
Grid-2 - 100x30x30
1 iteration - 3 sub-iterations
The increased computational expense for the cost of
enhancing the fifth to the seventh order scheme is 27%
more compared to the fifth order scheme.
Figure 10. Schematic of Structure of a 3D Tip-Vortex
of a Wing or Rotor
Seventh Order ENO Scheme: Wing Tip Vortex
Capture for 18 Chord Lengths
The analysis of wing vortex convection using fifth
order ENO in the previous sections gave definite
pointers towards the endgame of this complex puzzle of
which method is most practical (at present) towards
capturing rotor-tip vortices accurately. With around 810 points across the vortex core the fifth order scheme
captures the tangential velocity variation, but the axial
velocity tends to get dissipated when the convection
distances are more than 10 chord lengths. The question
arises if increasing the order of accuracy further will be
more effective. To this end the seventh order ENO
scheme as outlined earlier was implemented. Of direct
interest are the three queries, (i) stability, (ii) increased
cost of computation per iteration, and (iii) the pay-offs
in terms of its ability to resolve the vortex.
The same grid system as in the study for fifth order
scheme was considered. To keep the comparisons fair
and simple the vortex itself is generated using the
baseline fifth order scheme and transferred to the vortex
grid. The seventh order scheme is applied to the vortex
grid alone. The computation inside the vortex grid is
uniformly seventh order accurate, using one-sided
stencil shift near the boundaries.
The stability was not affected in any way, given that the
vortex grid is fairly simple. The same time stepping that
was applied using the fifth order computation was
retained (the time stepping translates to a movement of
one chord length for 50/[Mach number] iterations).
The timing comparison between the two schemes is
shown in Table 1 below.
Finally, we consider the pay-offs. The vortex grid is
still 100*30*30. Figure 11 shows the axial momentum
contour across a spanwise plane cutting the vortex. The
axial momentum variation is captured with very little
dissipation over the entire ~18 chord lengths. The axial
momentum contour appear diminished in patches
towards the 70% mark and the 90% mark. This is
because the vortex “wiggles” sideways and the
maximum does not always stay in the same plane.
Figure 12a-c compare the axial (black) and tangential
(red) velocity variation at the 5,50, and 100 percent
mark of the length of the vortex grid respectively. The
axial component has very little dissipation and the
tangential component has no noticeable dissipation. The
objective of “realistically” capturing the tip vortex over
180 degrees of revolution has been achieved and
bettered. The seventh order scheme requires around 3-5
points across the vortex face to capture both the axial
and tangential velocity accurately.
Figure 11. u (axial) Momentum Contours Across a
Spanwise Section of the Vortex Grid
Comparison with the fifth order solution is rather
redundant. The seventh order solution vastly
outperforms the fifth order solution for the extra effort
involved. With the given grid, the fifth order convects
up to a distance of ~7 chord lengths before axial
momentum is totally mis-captured. The seventh order
solution has captured up to ~18 chord lengths with
little dissipation and could potentially capture ~40-50
chord lengths (or in terms of the average rotor ~3-4
half-revolutions) with the given grid.
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Figure 13 shows the axial momentum contours at
several streamwise stations. Even the vortex sheet rollup is captured and can be seen eighteen chord length
away.
Seventh Order ENO Scheme: Wing tip vortex
capture for 50 chord lengths
As a final demonstration of high order vortex capturing,
the simulation in the last section was repeated with the
vortex grid extended to 50 chord lengths. The
streamwise number of points was increased to 300 to
a. x/c = 1.0
keep the streamwise point density approximately the
same. The spanwise distribution was retained at 30*30.
All the other flow conditions were identical to the
simulation in section 4 above. The length of the vortex
grid roughly represents 3-4 half-revolutions of rotor-tip
vortex, for a rotor of aspect ratio 6.
Figure 14 shows the axial momentum iso-surface
contour inside the vortex, over the fifty chord lengths of
simulation. The axial and tangential momentum
variation across the vortex at x/c = 2, 25, and 50 are
also shown in the figure.
b. x/c = 9.0
c. x/c = 18.0
Figure 12. u (axial), w (tangential) Momentum Variation Across the Vortex at Various Streamwise Sections
Figure 13. U-Momentum Contours at Several Streamwise Stations
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a. x/c =2
b. x/c = 25
c. x/c = 50
Figure 14. Axial-Momentum Iso-Surface Showing the Vortex. Graphs Show Axial (black) and Tangential (red)
Momentum Distribution across Vortex
The seventh order scheme captures the vortex in its
entirety with 3-5 points across the vortex. This
simulation has clearly demonstrated the ability of firstprinciples based high order scheme in conjunction with
overset refinement (either vortex grid adaptation or
placing new overset grid wherever the vortex happens
to be) to resolve the vorticity laden flowfields of
rotorcrafts. It has been achieved with grid sizes
reasonable enough for most workstations and even PC’s
in the near future.
Even if such high order methods are used capturing 1520 revolutions of the tip vortex is still a daunting task.
The helical structure
a. is highly unsteady and interacts
with itself. Tracking vortices to such lengths using
vortex-grids becomes very complex and is fraught with
geometrical difficulties and algorithm robustness issues.
Vortex-tracking overset grids demonstrated in this
paper are best suited for the first few revolutions when
it is relatively easy to track the tip vortex. For capturing
15-20 revolutions it is probably best if grid point
density is increased uniformly all over the wake – e.g.,
uniform Cartesian wake grids in a rotor in hover
simulation by Ahmad and Strawn[72]. Even with the
use of high order methods, this will still entail millions
of grid points and routine design computations may be
too costly, at present and in the near future. In the next
section, we describe a hybrid methodology which
tackles wake capturing using a near-field Navier-Stokes
and an efficient far-field potential free wake hybrid
methodology to take the best out of both the methods.
NS/POTENTIAL FREE WAKE HYBRID
METHODOLOGY
The existing finite difference methodologies for
rotorcraft aerodynamics can be divided into two
categories depending on their treatment of wake effects.
First generation finite difference codes [11,14,18]
solved full potential, Euler or Navier-Stokes equations
coupled with an external free or rigid wake model. In
the second generation Navier-Stokes codes, an attempt
was made to capture the shed vorticity and the tip
vortex entirely from first principles [33-36]. This
approach requires significant computer resources even
when high order spatial accuracy schemes are used, due
to the high levels of numerical viscosity present in these
schemes, particularly on coarse grids far away from the
rotor disk. Although some attempts in the past have
been made to reduce the computer time through the use
of a hybrid Navier-Stokes/full potential method [31] in
such methods the rotor wake is captured by Eulerian
approaches and is not adequately resolved away from
the rotor. The only exception to these methods is a
hybrid Navier-Stokes/full potential method of Moulton
and Caradonna [73], where a Lagrangean approach was
used to track the vortex filaments once they leave the
Navier-Stokes zone and enter the potential flow zone.
In this approach, a hybrid method is developed to solve
for the flow field over multi-blade rotors in hover or
forward flight. In the present study, only the hover
calculations are considered.
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The flow is divided into three regimes: (a) a small
viscous region surrounding individual rotor blades, (b)
a potential flow region which carries the acoustic and
pressure waves generated by the rotor to far field, and
(c) a Lagrangean scheme for capturing the vorticity that
leaves the viscous region and convecting it away to the
far field. Figure 15 shows the Navier-Stokes and
potential flow zones, and a typical vortex trajectory.
In the viscous region, the unsteady compressible
Navier-Stokes equations are solved in a finite volume
form. In the inviscid region, the isentropic potential
flow equations are solved. The effect of vorticity
embedded in the potential flow region is computed
using Biot-Savart law. Mathematical details of all the
equations involved is described in Reference 18.
Although there are many similarities between the
present work and earlier works of Berezin [31] and of
Moulton [73], there are several notable differences. For
example in Berezin’s work only the near wake was
captured. The effects of far wake had to be
implemented as a user-supplied induced angle of attack
table. In Moulton’s work, the Navier-Stokes zone was
modeled using a C-grid which extended from one blade
periodic boundary to next, and was large. The present
work uses a very small H-grid enclosing the blade
which can potentially grow or shrink with time. Finally,
the present formulation is cast as an unsteady problem,
so that steady and unsteady forward flight calculations
may be done with a single solver. Of course, time
saving approaches such as use of local time stepping
may be used in our work in hover applications to
accelerate convergence.
UH-60A Rotor in Hover
The hover solver is also validated for a typical current
generation rotor, the four bladed UH-60A rotor. The
blade has an aspect ratio of 15.3 and a maximum twist
of 13. The blade has a rearward sweep of 20 starting
from a rotor radius of 93%. The blade is made up of
two airfoil sections, SC1095 as the main airfoil and
SC1095R8 section in the mid-span [80]. The H-O grid
is shown in Figure 15 has a size of 90*43*80.
Approximately 37% of the nodes lie in the NavierStokes zone.
In these calculations, the tip vortex strength was chosen
initially to be the bound circulation at 99% radius. After
several hundred iterations, the peak bound circulation
was used as the tip vortex strength. The calculations
were started with a rigid, non-contracting wake. At
every 10 blade rotation wake geometry and the wake
induced velocity were updated.
Figure 16 shows the surface pressures at 4 radial
stations. The rotor is at a collective pitch angle of 9
and M tip  0.628 . The results are in reasonable
agreement with the experimental measurements, except
that the suction peak is underestimated at the tip
section. No trimming was used to match the thrust
coefficient values.
Figure 15. H-H Grid Computation for Hybrid NS/Potential Free-Wake Methodology
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Figure 16. Surface Pressure Coefficients for UH-60A Rotor at Various Radial Stations
FUTURE OUTLOOK
Rotorcraft wake modeling efforts are at an interesting
juncture. In the coming years one can expect a
multitude of methodologies — high order overset,
vorticity confinement, compact schemes, hybrid wake
capturing/Lagrangian convection, etc. — being
assembled together to tackle the issue of wake
resolution at different levels.
In this paper the advantages of ENO based high order
methods in capturing wake vorticity were outlined. In
the recent years various other approaches to building
high order schemes for rotor wake capturing have been
proposed. Schemes such as DRP schemes [Hall, Long
Ref. 76] and projected MUSCL [Tang and Baeder, Ref.
77] have been tried out rotorcraft wakes with varying
degree of success. Compact high order schemes –
schemes that use information only from any given cell
and its neighbours – such as Discontinuous Galerkin[
Hariharan, Ref. 81] have been only moderately
successful for 3D unsteady vortex capturing so far.
More research is needed to refine high order methods
further to arrive at an optimal scheme applicable
uniformly in overset grid settings. An alternative to
using high order methods without resorting entirely to
Lagrangean markers is the use of vorticity confinement
[Steinhoff, Ref. 74]. This method modifies momentum
equations to prevent vortex diffusion. It should be
further explored to see how such methods perform for
full-fledged rotor computations. Eventually the gains of
the higher order methods, confinement methods needs
to be incorporated into full-fledged first-principles
computations such as the hover simulation by Ahmed
and Strawn[72]. In this simulation they used
approximately 20 million overset grid cells for solving
rotor flow in hover, and still the wake was far from
adequately resolved. At present, structured overset grid
framework is possibly best equipped to capture the
rotor wake adequately enough and enable correlation
study correlations between rotor-blade shape and tip
vortex strength/trajectory.
It is becoming quite clear that any forthcoming
increases in memory/CPU will be in form of distributed
systems rather than monolithic systems. Therefore the
wake capturing grid points will have to be broken up
with each processor convecting a part of the wake.
Hence any non-dissipative high order scheme - within
the context of structured grids – will have to be accurate
all the way up to the block boundaries.
Even if all the current advances in high order methods
are properly built in a multi-processor, overset
framework and accurate NS computations capturing 1520 revolutions of the tip vortex is accomplished, it may
be impractical and unnecessary for design applications
in the near future. Such computations are necessary for
establishing benchmark computational results and
fundamental confidence that first-principles based
schemes are indeed capable of resolving the highly
unsteady wake for complex rotor-blade shapes. Fullfledged wake capturing methodologies is also essential
for interactional problems. Once such an option is
13
American Institute of Aeronautics and Astronautics
AIAA-00-0114
available – it is still going to be a costly option in the
near future – one can step back and use more
approximate methods for specific design problems.
Blade-vortex interactional characteristic studies, for
instance, require an accurate computation of the first
couple of revolutions. However, to accurately predict
the position even for the first couple of revolutions it is
essential to correctly represent the far-wake with
Lagrangean techniques apart from resolving the near
wake from first-principles.
Hybrid inner-zone (or zones fully capturing the first
couple of revolutions) NS and outer-zone Lagrangean
solutions such as Berkman and Sankar[78], Moulton
and Caradonna[73] with high order accurate NS zones –
Hariharan[75], Tang and Baeder[77] - promises to be a
useful tool in the near future. For industrial purposes
further grid point reduction by using tip vortex tracking
overset grids can be employed. As faster CPU becomes
affordable, more of the wake can be directly computed
and less of it convected using Lagrangean techniques.
ACKNOWLEDGMENTS
Parts of results presented in this paper were done under
NASA SBIR contract NAS2-14226. Dr. Hariharan
would also like to thank Dr Z.J. Wang, and Dr A.J.
Przekwas of CFD Research for their support, and Dr.
Roger Strawn of NASA Ames Research Center for the
various discussions, during this effort. The second
author was supported by the National Rotorcraft
Technology Center (NRTC).
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