APPLICATION OF HIGH ORDER SCHEMES TO
AERODYNAMICS OF HOVERING ROTORS
Ebru Usta
Graduate Research Assistant
School of Aerospace Engineering
Georgia Institute of Technology, Atlanta, GA 30332

0150
gt7885c@prism.gatech.edu
Lakshmi N. Sankar
Regents Profes
sor
School of Aerospace Engineering
Georgia Institute of Technology, Atlanta, GA 30332

0150
lsankar@ae.gatech.edu
ABSTRACT
The aerodynamics and vortex dissipation characteristics of rotors using the eighth order Symmetric
Total
Variation Diminishing (STVD) scheme with both third order MUSCL and fifth order Weighted Essentially
Non

Oscillatory (WENO) scheme are presented in hover.
The finite volume code TURNS (Transonic Unsteady
Rotor Navier

Stokes) is modified to get better aer
odynamics
characteristics of rotors.
The original solver,
TURNS, uses a third order upwind scheme.
To date, the combination of high order schemes, such as STVD for
dispersion term and WENO for numerical dissipation term has not been explored for use in ro
tary wing
applications.
From the earlier work, it was found that lower order scheme MUSCL inherent in the original solver
limits the effectiveness of the eighth order STVD scheme although it has good dispersion characteristics.
Therefore, high order versio
n of the filter term by using fifth order WENO scheme is developed
.
The new
methodology, referred to as TURNS

STVD8

WENO5, is eighth order accurate in space for the dispersion part
and fifth order accurate for the numerical diffusion portion. TURNS

STVD8

MUSCL3 is again eighth order
accurate for dispersion part and third order accurate in space for numerical viscosity term.
Calculations are
presented for the performance of the UH

60A rotor in hover.
Encouraging improvements for the tip vortex and
the pe
rformance of UH

60A rotor in hover have been obtained by the present methodology TURNS

STVD8

WENO5. Additional work is needed to improve the convergence characteristics of the higher order schemes to
reach the steady

state solution in the present methodol
ogy.
Presented at the American Helicopter Society Aerodynamics, Acoustics, and Test and Evaluation Technical Specialists
Meeting S
an Francisco, CA, January 23

25, 2002. Copyright
2002 by the American Helicopter Society International,
Inc. All rights reserved
.
INTRODUCTION
A main focus in the helicopter industry is to
design highly maneuverable, agile helicopters with
good handling qualities. Next generation helicopters
are expected to have low noise and high performance
capabilities so that their usag
e in search, rescue,
executive transport, traffic monitoring, and a wide
variety of military applications can increase. The first
step for achieving these high capabilities of
helicopters is to understand the noise generation and
improve the aerodynamic pe
rformance of the main
rotor.
The vortical wake, in particular, plays a crucial
role in generating helicopter noise, rotor airloads, and
vibrations. The presence of the vortical wake structure
beneath the rotor influences the flow field and makes
it very co
mplex around the rotor. Therefore, the
prediction of the rotor wake is still one of the most
challenging problems in rotorcraft computational fluid
dynamics (CFD).
Over the past two decades, many Navier

Stokes and Euler solvers have been used to model the
near wake of rotors. In these approaches, the near
wake are all captured from first principles. Purcell
1
,
and Srinivasan and Baeder
2
employed the NASA
Ames CFD solver TURNS (Transonic Unsteady Rotor
Navier

Stokes) to investigate noise sources in hover
a
nd forward flight. Some researchers have used the
OVERFLOW code which uses a third order
"MUSCL" scheme to model the rotor HSI and BVI
noise
3
. These numerical methods and computer codes
suffer from numerical dissipation and dispersion
errors. Dissipation
causes a gradual decrease in the
amplitude of acoustic waves and the magnitude of the
tip vortex filaments between
the time these entities
leave the blade surface and the time they reach an
observation location. This can lead to an
underestimate of the no
ise, induced inflow, and BVI
loads. Dispersion causes waves of different
wavelengths originating at the blade surface to
incorrectly propagate at different speeds. As a result,
the waves may distort in a nonphysical manner by the
time they arrive at the ob
server location. Improved
algorithms for modeling rotary wing aerodynamics
with low dissipation and low dispersion errors are
urgently needed
.
Tam
4
and Webb recently developed a new
numerical scheme called the Dispersion

Relation

Preserving (DRP) finite
difference scheme.
This
approach is aimed at reducing the dispersion errors.
Nance
5
et al extended the DRP ideas to curvilinear
grids and developed a
Low Dispersion Finite Volume
(LDFV) scheme
. Wang
6
et.al. implemented
the
LDFV scheme into TURNS and stud
ied shock noise
and hover performance of rotorcraft. A side benefit
of
these schemes
is their reduced dissipation, or
numerical viscosity that is proportional to
5
where
is the grid spacing.
The easiest way to reduce dissipation errors
is by increasin
g the formal accuracy of the upwind
scheme. For example, a first order upwind scheme
will have dissipation errors that are proportional to
1
, where
is the grid spacing. Third order schemes
such as those found in TURNS
7
and OVERFLOW
will generate dissi
pation errors proportional to
3
.
Fifth order and seventh order upwind schemes, with
dissipation errors of order
5
and
7
respectively, have
been explored by Hariharan and Sank
ar
8
.
Wake
9
et.
al. studied the evaluation of a line vortex in space and
time
using a fifth order spatially accurate scheme.
Very recently, Wake
10
et. al. have presented ninth
order results in a fixed wing mode.
Another way to reduce the numerical
diffusion of the tip vortices is to enhance the grid
resolution. Tang
11
et. al.
re
cently have developed a
grid redistribution method that clusters the grid points
near the tip vortices and reduces the numerical
diffusion of vorticity. Ahmad
12
et. al
used overset

grids with high resolution on the rotor blades and a
systematic variation
of grid resolution in the rotor
wake for hovering rotor cases. Hariharan and
Sankar
13
used an adaptive overset grid to resolve the
tip vortex from a fixed wing for over 100 chord
lengths downstream of the wing. Results for a
hovering rotor with tip vortex
tracking grids using
OVERFLOW have been presented by Egolf
14
et. al
.
Strawn
15
et al.
have systematically studied the effects
of grid refinement on the hover performance
prediction characteristics of rotors.
SCOPE OF THE PRESENT WORK
To date, the co
mbination of high order
schemes, such as Symmetric Total Variation
Diminishing (STVD) for dispersion term and
Weighted Essentially Non

Oscillatory (
WENO)
16

17
for numerical dissipation term has not been explored
for use in rotary wing applications.
With th
e hope to
obtain superior non

diffusive results of the rotorcraft
vortex resolution and performance prediction, the
main purpose of this study is the development and
validation of spatially higher order accurate methods
for capturing the tip vortex better
in hover so that
helicopter rotor applications can have ideal
methodologies in modern CFD. This work is an
extension of the Symmetric Total Variation
Diminishing scheme (STVD) originally developed by
Helen Yee
18
. Yee has documented good results for a
num
ber of steady and unsteady problems including a
Direct Numerical Simulation (DNS) of turbulence.
Yee’s idea is that high

order central difference
schemes may be coupled to lower order dissipation
terms to yield accurate results.
In an earlier work, the
pr
esent authors
19
applied Symmetric TVD algorithm
to rotary wing application in TURNS code. But in that
study, the numerical viscosity term was calculated
using third order MUSCL scheme. A
ccording to
Helen Yee
18
, the numerical viscosity may be
calculated usi
ng a (relatively) low third order MUSCL
scheme in shock capturing techniques, even when the
symmetric part of the algorithm is of a higher order.
But i
t
19
was found that due to computed low order
numerical viscosity, the solution of the flowfield was
highl
y diffusive and the peak vorticity values were
diminished as the original discontinuity spreads and
convects through the fluid. Since this
artificial
diffusion limits the effectiveness of these STVD8
scheme, the higher order version of the numerical
visco
sity term such as fifth order WENO scheme is
worthwile to use to get the better tip vortex resolution.
The present p
aper deals with higher order version of
the dissipation term of the eighth order STVD
algorithm
and its application to a modern UH

60A
rotor
.
The UH

60A rotor is chosen for hover
validation here because the experimental data of
Lorber et al
20
for this configuration is generally
accepted as being the most comprehensive hover data,
and includes wake trajectory, blade loading, blade
deformation
s, and performance measurements. The
various versions of the algorithm discussed here are
referred to as TURNS

STVD8

MUSCL3 and
TURNS

STVD8

WENO5, where "8" refers to the
formal spatial accuracy of the “symmetric” part of the
algorithm while "3" and "5" re
fer the formal spatial
accuracy of the “dissipation” part of the algorithm.
Comparisons with both original TURNS code, the
modified version TURNS

STVD8

MUCL3 and the
new version TURNS

STVD8

WENO5 a
re given.
Encouraging improvements for the tip vortex and t
he
performance of UH

60A rotor in hover have been
obtained by the present methodology, TURNS

STVD8

WENO5 over the baseline method. But the
modified method still did not improve the
convergence characteristics of symmetric TVD
scheme. The tip vortex still i
s passing above the blade,
which is not consistent with the experimental behavior
although the new methodology pushed the vortex
down at the blade compared to other schemes.
Additional work is needed to improve the modeling of
the tip vortex structure in
the present methodology.
MATHEMATICAL AND NUMERICAL
FORMULATION
Computational Grid
A hyperbolic
single C

H
grid generator
21
supplied with the TURNS code is used in all the
calculations. The three

dimensional grid is
constructed from a series of two

dim
ensional C

grids
with an H

type condition in the azimuthal direction.
Since the flowfield around the hovering rotor is
periodic, only one blade can be
modeled, which
allows much computer savings.
For the viscous results presented here for
UH

60A rotor, th
e grid used in the present paper has
808,000 points. There are 149 points in the streamwise
direction with 91 points on the body. There are 89
grid points in the spanwise direction with 40 radial
stations on the blade surface. In the normal direction,
61
points are used. The spacing in the normal
direction is taken as 10

5
. The outer boundaries of
the
grids are located at least two radii away in all
directions for the UH

60A simulations.
TURNS Flow Solver
A public domain code called Transonic
U
nsteady Rotor Navier

Stokes (TURNS) solver
2
has
been modified in this study. This code solves the
strong conservation form of three

dimensional Navier

Stokes equations using a finite volume scheme.
TURNS
uses an LU

SGS (Lower

Upper Symmetric

Gauss

Seidel)
implicit time
marching scheme
which
has
good stability and convergence characteristics.
An
option for
using Newton

type sub

iterations at each
time step allows for the reduction of the linearization
and factorization errors, and improves temporal
accuracy
. The inviscid fluxes crossing the cell face are
evaluated in this formulation using an approximate
Roe solver.
The baseline TURNS solver uses a third
order accurate interpolation scheme called the Van
Leer Monotone Upstream

centered Scheme for the
Conserv
ation Laws (MUSCL)
22
approach to
interpolate the flow properties stored at cell centers to
cell faces.
In the vicinity of shocks and other sharp
gradients, the accuracy of the scheme is reduced to
first order through the use of
a Koren differentiable
lim
iter. This makes the scheme third

order accurate in
space in smooth regions of the flow, and first order in
space near
shocks and vortex cores
.
Boundary Conditions
Four types of boundary conditions are used in
the flow solver TURNS. No slip
boundary conditions
are used at the blade surface. At the wake cut and
outboard of the blade tip interpolation of flow
properties from neighboring cells is used. Periodic
boundary condition is used at the front and rear
boundaries of the two

dimensional
C

grids at each
spanwise station. At the far field boundaries, mass
source and sink based boundary conditions are used to
ensure that appropriate amounts of mass enter and
leave through the boundaries
2
. These conditions
23
are
all applied explicitly, and l
ag the interior point values
by one time step. In all calculations, CFL number is
taken as 20.Since the calculations are performed for
hover, one Newton iteration is done.
Symmetric TVD Scheme
To explain the essential ingredients of the
sym
metric TVD scheme used in this study, it is easier
to look at 1

D unsteady Euler equations on a Cartesian
grid:
0
x
F
t
q
(1)
Here, q is the flow properties (state) vector, and F
contains th
e flux terms. A semi

discrete form of this
equation at a typical node 'i' is given by:
0
2
1
2
1
x
F
F
dt
dq
i
i
(2)
Here
F
is some numerical approximation to the
physical flux F. In classical upwin
d schemes,
including the Roe scheme, the quantity
F
is skewed
towards the direction from which the information is
flowing. For example, in a supersonic flow where the
local velocity is directed from the node i towards the
node i+1,
F
at i+1/2 is simply F at i.
In the symmetric TVD scheme,
F
is viewed
as the sum of two parts

the physical flux F that is
always symmetrically computed, and a numerical
viscosity or diffusion term. Helen Yee
18
recommends
the following second order form:
0
)
(


)
(


2
1
2
/
1
1
2
/
1
1
1
x
q
q
A
q
q
A
x
F
F
dt
dq
i
i
i
i
i
i
i
i
(3)
where A =


q
F
, a 5x5 matrix, computed using
“Roe

averages” of q at adjacent points. Notice that
the second term is simply a second order accurate
approximation
to
x
F
, and is symmetric with
respect to i, with no bias toward i+1 or i

1. The third
term may be viewed as a “numerical” viscosity or
diffusion term while the second one may be viewed as
a "dispersion" term.
Note that eq
uation (3) may be written in the
form given by equation (2) if we define:
i
i
i
i
i
q
q
A
q
F
q
F
F
1
1
2
/
1
2
1
)
(
)
(
2
1
(4)
If we want to change the formal accuracy of the
scheme it is only necessary to increase the stencil size.
As the stencil size broadens, it
becomes somewhat
simpler to interpolate the flow property q (as in the
MUSCL scheme) than the flux F directly. Our eighth
order symmetric TVD scheme therefore uses:
L
R
i
i
k
k
k
i
q
q
A
q
a
F
F
2
1
)
(
4
3
2
/
1
(5)
where
k
a
are the coeffic
ients of the related fluxes.
These were obtained using a Lagrangean curve fit
which takes into account the nonlinear grid spacing
between nodes
and given by:
4
,..,
2
,
3
4
3
3
4
3
2
/
1
)
(
)
(
i
i
i
k
x
x
x
x
a
i
k
l
i
l
l
i
i
k
l
i
l
l
i
k
where a typical factor
)
(
2
1
i
i
x
x
should be
interpreted as t
he physical distance between the node
'i

1' and the node 'i

2' shown in the figure.
Near the boundaries, a large stencil may not be
available. Progressively lower order schemes, that use
fewer neighbor points, are used near the boundaries.
The second term
L
R
q
q
A
2
1
occurring
in equation (5) is a "numerical viscosity" term
designed to filter out high frequency nonphysical
oscillations in the solution at every time step. T
his
term is proportional to O(
3
) in both the baseline
TURNS code and the present TURNS

STVD8

MUSCL3 code. It should be noted that when a low

order accurate upwind scheme is used, the result will
have no oscillations but the solution will be highly
diffusive. As a result, the original
discontinuity will
spread as it convects through the fluid, and the peak
vorticity values will be diminished.
This artificial
diffusion limits the effectiveness of this STVD higher

order scheme discussed in this study. Therefore, the
high order version of
this filter term is used. Note that
in new methodology, TURNS

STVD8

WENO5, the
first term F
)
(
4
3
i
i
k
k
k
q
a
occurring in equation (5) is
calculated as same as the one in TURNS

STVD8

MUCL3.
Treatment of the "Numerical Viscosity" Term
a)
MUSCL Scheme
:
The modified version of the solver referred to as
TURNS

STVD8

MUSCL3 uses third order
“Monotone Upwind Scalar Conservation Law”
(MUSCL) scheme
which is
proportional to O (
3
)
to
calculate the numerical viscosity occurred in equation
(5). This
interpolation is simply given as:
i
i
i
i
i
R
i
i
i
i
i
L
q
q
q
q
q
q
q
q
q
q
q
q
1
1
2
1
1
1
3
1
6
1
6
1
3
1
(6)
Note that
R
q
and
L
q
may be thought of q at (i+1/2)
biased to the right (i.e. i+1) or to the left (i.e. i),
respectively as shown in t
he figure.
* * * *
Right
Stencil for q left
Stencil for q right
Left
i

1 i i+1 i+2
Cell face i+1/2
Figure 2: Stencil for Three

Point Scheme
x
i

1

x
i

2
distance along the coordinate
line
i

3
i

2
i+2
i+3
i+4
i

1
i+
1
i
Figure 1: Stencil used for 8
th
order STVD on
non

uniform grid
b)
WENO Scheme:
The modified version of the solver referred to as
TURNS

STVD8

WENO5 uses fifth order
Weighted
Essentially Non

Oscillatory Scheme to calculate the
numerical dissipation term in equation (5).
WENO
17
scheme us
es the idea of adaptive stencils in the
reconstruction procedure based on local smoothness of
the numerical solution to automatically achieve high
order accuracy and non

oscillatory property near
discontinuities. The beauty of WENO scheme lies
beneath usin
g a convex combination of all the
candidate stencils. It is developed such a way that
each stencil is assigned a nonlinear weight which
depends on the local smoothness of the numerical
solution based on that stencil.
For the br
evity, only the steps to calculate
L
q
are
explained detailly.
R
q
is calculated by shifting the all
stencils according to Figure 3. The main idea for the
fifth order WENO scheme is to establish the three
third order
stencils as follows:
)
3
(
2
/
1
3
)
2
(
2
/
1
2
)
1
(
2
/
1
1
ˆ
ˆ
ˆ
i
i
i
L
q
w
q
w
q
w
q
(8)
where
)
(
2
/
1
ˆ
i
i
q
are three third order primitive
variables on three different stencils. These stencils are
given as:
2
1
)
3
(
2
/
1
1
1
)
2
(
2
/
1
1
2
)
1
(
2
/
1
6
1
6
5
3
1
ˆ
3
1
6
5
6
1
ˆ
6
11
6
7
3
1
ˆ
i
i
i
i
i
i
i
i
i
i
i
i
q
q
q
q
q
q
q
q
q
q
q
q
When the abov
e three stencils are combined all
together, overall accuracy of the scheme is 5
th
order.
This is because the accuracy is equal to (2k

1) where k
is the accuracy of the stencil chosen. Therefore, the
fifth order polynomial for
L
q
usuall
y will involve
nodes (i

2), (i

1), (i), (i+1), (i+2). The one for
R
q
will
involve (i

1), (i), (i+1), (i+2), (i+3) as seen in Figure3.
In equation (8),
3
2
1
,
,
w
w
w
are the nonlinear weights
given below:
3
1
~
~
k
k
i
i
w
w
w
where
2
)
(
~
k
k
w
(9)
In equation (9),
k
is the linear weights given by:
5
3
,
5
3
,
10
1
3
2
1
and
k
is the smoothness indicator. They are
calculated as follows:
2
2
1
2
2
1
3
2
1
1
2
1
1
2
2
1
2
2
1
2
1
)
4
3
(
4
1
)
2
(
12
13
)
(
4
1
)
2
(
12
13
)
3
4
(
4
1
)
2
(
12
13
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
The parameter
occurring in equation (9) is used to
avoid the denominator to become 0 and is taken as
6
10
in the computation.
Near the boundaries, since a large
stencil may not
be available, third order WENO scheme which uses
fewer neighbor points is constructed to calculate
L
q
and
R
q
.
RESULTS AND DISCUSSION
An eighth order symmetric TVD scheme
with the use of third
order MUSCL and fifth order
WENO scheme has been developed and applied to
rotor in hover. Calculations have been performed for a
four

bladed, scaled (1:5.73) model of the UH

60A
(Black Hawk) main rotor in hover. The flow field has
been examined with regard
to the rotor wake, blade
surface pressures, and performance. The UH

60A
rotor blade is untapered, and has 20 degrees of
rearward sweep that begins at r/R=0.93. It has an
aspect ratio of 15.3 and a solidity of 0.0825. The
blades use a 9.5% thick SC

1095 s
ection and its
higher

lift variant SC

1095R8 airfoil section.
The results from the original code TURNS,
and the present modified flow solver TURNS

STVD8

MUSCL3 and TURNS

STVD8

WENO5 are
presented and compared with the experimental data
for a UH

60
A Black Hawk rotor in hover. Since the
original experimental data is not available, the
experimental data is scanned from the original paper
20
at Georgia Tech. All the calculations have been
performed on a similar grid for a Reynolds number of
1.25 million
. Calculations have been done for a range
* * * * * *
Right
Stenci
l for q left
Stencil for q right
Left
i

2 i

1 i i+1 i+2 i+3
Figure 3: Stencil for Five

Point Scheme
Cell face i+1/2
of thrust values.
Approximate estimates of the elastic
twist and coning angle were also taken into account
while generating the grid. It was found that the higher
order schemes and the baseline TURNS both
do not
completely converge to steady

state values.
Efforts are
needed to improve the convergence characteristics of
the higher order STVD scheme.
Pressure Distributions:
The TURNS code and the higher order
extensions give detailed pressure distri
butions all over
the rotor blade. Sample pressure distributions
are
shown
at two typical radial station at 92% and 99% in
F
igures 4

5.
The
pressure distributions are compared
with the scanned experimental data at C
T
/
= 0.085.
For the brevity, only TURN
S

STVD8

WENO5
methodology is compared to experimental data since
the other methods including the baseline TURNS, give
reasonable C
P
distributions, although the
recompression near the trailing edge at the 99%
location is not predicted well by any of these m
ethods.
The suction peak is also over

predicted at 99% and by
all these methods. The over prediction of the
predicted suction peaks near the tip may be related to
the inaccurate strength and location of the passing
vortex and an inadequate resolution of
the local tip
vortex formation.
It may be misleading to conclude that a given
scheme is accurate purely based on surface pressure
distributions. While the baseline TURNS code gives
quite acceptable Cp distributions, the integrated loads
are still unacce
ptable, as discussed below.
Performance of the UH

60A Rotor:
A systematic study of the UH

60A rotor
performance in hover has been attempted. This is
done by varying the collective pitch setting, and
computing the
T
C
and
Q
C
values from first principles.
Figure 6 shows the variation of thrust
coefficient with the collective pitch with TURNS,
TURNS

STVD8

MUCL3 and TURNS

STVD8

WENO5 codes. It is clear that the prediction steadily
improves as the al
gorithm changes from 3
th
order ROE
scheme to 8
th
order STVD. Then the use of fifth order
WENO scheme as a diffusion term along with 8
th
order STVD scheme gives more improvement over
STVD8

MUSCL3 combination. The 3
rd
order baseline
scheme overshoots the
T
C
values when compared to
the higher order schemes.
Figure 7 shows the variation of torque
coefficient with the collective pitch angle. Among the
different algorithms studied, TURNS

STVD8

WENO5 version of the modified solver gives the
best
agreement with the experimental data. The predicted
Figure
of Merit for the Black Hawk rotor is shown in
Figure 8.
It is seen that the higher order calculations
give better agreement with the experiments than the
baseline TURNS. In particular, the com
bination of the
eighth order with fifth order WENO gives the best
agreement with experimental data. In general, the
predicted
Figure of Merit is about 1

2 points (0.01

0.02) under the experimental data. Helicopter
manufacturers consider this level of agree
ment to be
acceptable.
Rotor wake:
The tip vortex structure predicted by
TURNS

STVD8

MUSCL3 and TURNS

STVD8

WENO5 for the UH

60A rotor is also studied. The
qualitative comparison with the baseline TURNS,
TURNS

STVD8

MUSCL3and TURNS

STVD
8

WENO5 predictions is given in Figure 9. This
qualitative figure is obtained using the same vorticity
contour numbers and the magnitude for C
T
/
= 0.085
setting. It is seen that higher order scheme especially
TURNS

STVD8

WENO5 is less dissipative than t
he
low order ROE scheme. The magnitude of the core of
the vortex at 41 degree vortex age is 0.5 by TURNS

STVD8

WENO5 while baseline TURNS gives 0.25.
From the previous work, the computed tip vortex was
considerably diffused by the time it reached the
follo
wing blade. It was attributed to high levels of
numerical viscosity used in both TURNS and
TURNS

STVD8

MUSCL3 methodology. This is
because in TURNS

STVD8

MUSCL3 methodology
the physical flux is calculated to a very high order of
accuracy, while the numeric
al viscosity is calculated
using a third order scheme as in the baseline TURNS
code. From Figure 9, it is seen that fifth order WENO
scheme gives improvement over the baseline TURNS
and TURNS

STVD8

MUSCL3. Although, the vortex
is pushed little down the bla
de, still the tip vortex
passes above the following blade which is not
consistent with the experimental behavior.
r/R=0.920
2
1.5
1
0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
Chord
Cp
Experiment
TURNSSTVD8WENO5
Figure 4
:
Pressure Distribution along the surface of
UH

60A at r/R=0.920 radial location,
Grid Size 149x89x61
CONCLUDING RE
MARKS
Eighth order Symmetric Total Variation
Diminishing Schemes has been implemented in the
TURNS code. Since the amount of numerical
dissipation term was high enough to diffuse the tip
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.05
0.1
0.15
CT/
FM
Experiment
TURNS
TURNSSTVD8MUSCL3
TURNSSTVD8WENO5
Figure 8: Figure of Merit versus C
/
T
Grid Size 149x89x61
TURNS
TURNS

STVD8

MUSCL3
TURNS

STVD8

WENO5
Figure 9: Vorticity Contours, Grid Size 149x89x61
Vortex Age 41 degree.
Vortex from previous blade is pushed down.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
2
4
6
8
10
12
Collective Pitch Angle(deg.)
CT/
Experiment
TURNS
TURNSSTVD8MUSCL3
TURNSSTVD8WENO5
FFigure 6: C
/
T
vs. Collective Pitch Angle,
Grid Size149x89x61
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0
0.05
0.1
0.15
CT/
CQ/
Experiment
TURNS
TURNSSTVD8MUSCL3
TURNSSTVD8WENO5
Figure 7:
C
/
Q
vs. C
/
T
, Grid Size
149x89x61
r/R=0.99
2
1.5
1
0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
Chord
Cp
Experiment
TURNSSTVD8WENO5
Figure 5
:
Pressure Distribution along the surface of
UH

60A at r/R=0.99 radial location,
Grid Size 149x89x61
vortex, fifth order WENO scheme is applied to
numerically c
alculate dissipation term The hover
performance of the UH

60A rotor is more accurately
modeled with the higher order schemes compared to
the baseline TURNS code. In particular, the
combination of STVD8 and WENO5 method is found
to give the best Figure of M
erit predictions for the
UH

60A rotor. Figure of Merit values using third
order baseline TURNS are 5

6 points higher than
higher order results.
The TURNS+STVD8+MUSCL3 or
TURNS+STVD8+WENO5 methodology described
here needs additional improvements
to be used in
hover performance studies. Therefore, the following
shortcomings are addressed:
1. The method converges relatively slowly. This is
attributable to the fact that the factorization scheme
has not properly been modified to reflect the use o
f a
larger stencil in the STVD8 scheme. Convergence
may be enhanced by either properly linearizing the
STVD8 terms at each time step prior to their use in the
LU factorization. Low

Mach pre

condinitioning
technique may also help to improve the convergence.
2. The pressure predictions at r/R=0.99 radial location
is very high. It is believed that this is due to an
inadequate resolution of the tip grid.
3. When C

H type of grid is used, the grid lines are
all skewed and distorted because of the periodicit
y
condition in hover calculations. To get better tip
vortex prediction, the efforts are needed to improve
the grid used.
4. Although the new methodology is using high order
algorithms to calculate the inviscid fluxes, the overall
code is not using high o
rder schemes. Viscous fluxes,
metrics, Jacobian are all computed by using second
order central difference scheme. To be more
consistent in the new methodology, high order metrics
and Jacobian calculations are currently under
development for 4
th
order centr
al scheme.
5. In this study, all calculations are obtained by using
Baldwin

Lomax algebraic turbulence model. Since
Baldwin

Lomax uses the near

wall vorticity
distribution to obtain a turbulent length scale, this
vorticity distribution may be affected by
the close
proximity of the preceding blade vortex. The new
study done by Roger
Strawn
15
has showed that the
choice of turbulence model has an important effect on
the amount of vorticity. Therefore, one equation
model, Spalart Allmaras, will be used to see
the
turbulence effect in this new methodology.
ACKNOWLEDGEMENTS
Technical tasks described in this document
include tasks supported with shared funding by the
U.S. rotorcraft industry and government under the
RITA/NASA Cooperative Agreement No. NCCW

0076,
Advanced Rotorcraft Technology, August 15,
1995.
The work was also supported by the Rotorcraft
Center of Excellence. Dr. Yung Yu was
the technical
monitor.
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