APPLICATION OF HIGH ORDER SCHEMES TO AERODYNAMICS OF HOVERING ROTORS

usefultenchMechanics

Feb 22, 2014 (3 years and 8 months ago)

102 views

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
TURNS-STVD8-WENO5
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
TURNS-STVD8-MUSCL3
TURNS-STVD8-WENO5
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
TURNS-STVD8-MUSCL3
TURNS-STVD8-WENO5
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
TURNS-STVD8-MUSCL3
TURNS-STVD8-WENO5

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
TURNS-STVD8-WENO5
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.

REFERENCES


1.

Purcell, T. W., “A Prediction of High
-
Speed
Rotor Noise,” AIAA Paper 89
-
1130, presented

at
the AIAA 12
th

Aeroacoustics Conference, San
Antonio, TX, Apr. 10
-
12, 1989.

2.

Srinivasan, G. R., Baeder, J. D., Obayashi, S., and
McCroskey, W. J., “Flow
-
field of a Lifting Rotor
in Hover: A Navier
-
Stokes Simulation,”
AIAA
Journal
, Vol. 30, No. 10, Oct. 1
992.

3.

Strawn, R. C., Ahmad, J., Duque, E. P.
N,“Rotorcraft Aeroacoustics Computations with
Overset Grid CFD Methods,” 54
th

AHS Annual
Forum, Washington DC, May 22
-
24, 1996.

4.

Tam, C. K. W., and Webb, J. C., “Dispersion
-
Relation
-
Preserving Schemes for Computa
tional
Aeroacoustics,”
Journal of Computational
Physics
, Vol. 107, 1993, pp. 262
-
281.

5.

Nance, D. V., Viswanathan, K., and Sankar, L.
N., “Low
-
Dispersion Finite Volume Scheme for
Aeroacoustic Applications,”
AIAA Journal
, Vol.
35, No. 2, 1997, pp. 255
-
262.

6.

Wang, Gang, Sankar, Lakshmi N., and Tadghighi,
Hormaz, “Prediction of Rotorcraft Noise with a
Low
-
Dispersion Finite Volume Scheme,”
AIAA
Journal
, Vol.38, No.3, March 2000, pp.395
-
401.

7.

Wake, B.E. and Baeder, J.D., “Evolution of a
Navier
-
Stokes Analysis for
Hover Performance
Prediction,”
Journal of the American Helicopter
Society
, Vol.41, No.1, Jan. 1996, pp.7
-
17.

8.

Hariharan, N., “High Order Simulation of
Unsteady Compressible Flows Over Interacting
Bodies With Overset Grids,” Ph.D. Thesis,
Georgia Institute o
f Technology, Georgia, USA,
August 1995.

9.

Wake, B.E. and D. Choi, “Investigation of
Higher
-
Order Upwinded Differencing for Vortex
Convection
,” AIAA Journal
, Vol.34, pp.332
-
337,
1995.

10.

Wake, B.E., Egolf, T.A. and D. Choi, “Resolution
and Convection of Tip Vo
rtex, Under preparation.

11.

Tang, L., Baeder, J.D., “Improved Euler
Simulation of Hovering Rotor ip Vortices with
Validation,” presented at AHS 55
th

Annual
Forum, Montreal, Canada, May 25
-
27,1999.

12.

Calculations with an Overset
-
Grid Navier
-
Stokes
Solver,” prese
nted at AHS 55
th

Annual Forum,
Montreal, Canada, May 25
-
27,1999.

13.

Hariharan, N.S., Sankar, L.N., “First Principles
Based High order Methodologies For Rotorcraft


Flowfield Studies,” presented at AHS 55
th

Annual
Forum, Montreal, Canada, May 25
-
27,1999.

14.

Egolf,

T.A., Wake, B.E. and Berezin, C.,

Recent
Rotor Wake Simulation and Modeling Studies at
United Technologies Corporation,” Invited Paper,
AIAA
-
2000
-
0115, 38
th
AeroSciences Meeting and
Exhibit, Jan.10
-
13, Reno, 2000.

15.

Strawn, R.C., Ames Research Center and M
. J.
Dijomehri, Computer Sciences Corp.,
“Computational Modeling of Hovering Rotor and
wake Aerodynamics,” Proceedings of the 57
th

Annual American Helicopter Society Forum,
Washington, D.C., May 9
-
11, 2001.

16.

Shu, W., “Essentially Non
-
Oscillatory and
Weighte
d Essentially Non
-
Oscillatory Schemes
for Hyperbolic Conservation Laws”, NASA CR
206253, November 1997.

17.

Shu, W., “High Order Finite Difference and
Finite Volume WENO Schemes and
Discontinuous Galerkin Methods for CFD”,
NASA CR 210865, May 2001.

18.

Yee, H.C.,
Sandham
, N.D., Djomehri, M.J., “Low
Dissipative High Order Shock
-
Capturing
Methods Using Characteristic
-
Based Filters,”
Journal of Computational Physics
, November
1998.

19.

Usta, E., Wake,.B.E, Egolf, T.A. and Sankar,
L.N, "Application of a Symmetric Total V
ariation
Diminishing Scheme to Aerodynamics and
Aeroacoustics of Rotor", 57
th

AHS Forum,
Washington D.C, May 9
-
11 2001.

20.

Lorber, P.F., Stauter, R.C., and Landgrebe, A.J.,
“A Comprehensive Hover Test of the Airloads
and Airflow of an Extensively Instrumented

Model Helicopter Rotor," Proceedings of the 45
th

Annual American Helicopter Society Forum,
Boston, MA, May 1989.

21.

Steger,J.L., Chaussee, D.S, “Generation of Body
-
fitted coordinates using Hyperbolic Partial
Differential Equations”, SIAM
Journal of
Scientifi
c and Statistical Computing,
Vol.1,
December 1980.

22.

Van Leer, B., “Towards the Ultimate
Conservative Difference Scheme, V: A Second
Order Sequel to Godunov’s Method,”
Journal of
Computational Physics
, Vol.32, 1979, pp.101
-
136.

23.

Vinokur, M.,“Analysis of Fin
ite
-
Diffrence and
Finite
-
Volume Formulations of Conservation
Laws”,
Journal of Computational Physics,
Vol.81, March 1989.