Advanced methods to compute the rotor thickness noise which is predominant in the case of high speed rotor were developed. These methods were deduced from a previous method by transforming the integral coordinate, commuting the order of integration and differential, and/or performing chordwise integration analytically with some adequate assumption. The necessary computational times and waveforms obtained by the previous and three advanced methods were compared. It was then concluded that the advanced methods could save the computational time compared with the previous method with the same accuracy

Azuma, A.

Nakamura, Y.

English

NASA Technical Reports Server

16 
IMPROVED METHODS FOR CALCULATING 
THE THICKNESS NOISE 
Yoshiya Nakamura and Akira Azuma 
University of Tokyo 
SUMMARY 
Three advanced methods to compute the rotor thickness noise which is pre- 
dominant in the case of high speed rotor have been developed. These methods are 
deduced from a previous method by transforming the integral coordinate, commut- 
ing the order of integration and differentiation, and/or performing chordwise 
integration analytically with some adequate assumptions. The necessary compu- 
tational times and waveforms obtained by the previous and three advanced 
methods were compared. It was then concluded that the advanced methods could 
save the computational time very much compared with the previous method in 
keeping the same accuracy. 
INTRODUCTION 
Farassat has proposed a method to calculate the thickness noise which is 
produced by moving bodies of finite volume or thickness normal to the moving 
direction and applied it to the rotor noise of helicopters (ref.1). 
Based on his work, the authors made clear the cause and the characteris- 
tics of the rotor rotational noise. Through these studies (ref.2 and 3) with 
the concept of the "influential surface", which is an integral region at 
retarded time, and of the distribution of source strength, many acoustic char- 
acteristics of helicopter rotor noise have been clarified by the analytical pre- 
dictions and numerical calculations, and verified by the experimental tests. 
It has been shown that among the rotor rotational noise components the rotor 
thickness effect executes a dominant roll rather than the loading effects such 
as thrust noise and drag noise in the case of high blade-tip speed. 
In the computational process of the thickness noise, however, the calcula- 
tion has the worst converging characteristics and thus needs a lot of computer 
time in getting solutions within enough accuracy. It has, therefore, been 
expected to develop an improved method of calculation for obtaining an advanced 
form of solution which is more convenient to get the result with less computer 
time and enough accuracy. 
323 
._..I ..I.. _.I_. -..-.._-, -m.- --..-._-__-. -..---. ..-.. .__._. - .__. - _..._...__.__ - _-___ -.~.. . 
https://ntrs.nasa.gov/search.jsp?R=19780024889 2020-03-22T02:06:25+00:00Z
1111 II I II llllllllllllllllllllll 
B 
SYMBOLS 
number of blades 
CO 
'h 
f=O 
g=o 
h 
K 
Mr 
P 
R 
RO 
r 
t 
V 
vll 
V n 
X 
Y 
sound speed, m/set 
, 
chord length 
equation of body surface 
n&q, 112) at the upper surface 
= 
-ng-h(nl, ~2) at the lower surface 
equation of acoustic sphere = T-t+[rl 
ret /CO 
blade thickness, m 
integrand of a modified solution of thickness noise, see equation 
(10) 
relative Mach number 
acoustic pressure, kg/m2 
rotor radius, m 
blade cut-off radius, m 
distance between source and observer, =1X-y/, m 
distance between hub center and observer, m 
unit vector in direction of propagation in blade fixed coordinate 
rlsin$-r2cos$ 
observer time 
rotor hub velocity, =(Vl ,V2 ,V3 ) 
T 
, m/set 
blade element velocity normal to blade surface, m/set 
T observer position vector, =(x1,x2,x3) , m 
T source position vector, =(yl ,y2,Y3) , m 
324 
a angle between radiating direction and rotor plane, rad or deg 
r curve of intersection of body with acoustic sphere 
position vector in blade fixed Cartesian coordinate system 
angle between radiating direction and normal direction to blade 
surface, rad or deg 
density of undisturbed fluid, kg*sec/m4 
source time, set 
blade azimuth angle, rad or deg 
rotor rotational speed, rad/sec 
directional parameter in modified method, see equation (8) 
directional parameter in previous method, see equation (3) 
Superscript 
( IT transposed of ( ) 
Subscript 
[ 1 ret value at a retarded time 
Operator 
V gradient operator in fluid fixed coordinate, =a/ay,, l/m 
PREVIOUS ANALYSIS - METHOD (A) 
According to Farassat (ref.l), the rotor thickness noise, p, for a given 
observer position, X, and time, t, is given by ‘\ 
dI'dT (1) 
where h(nl) is the blade thickness distribution as a function of chordwise 
coordinate, nl. Here the relative speed of a blade element with respect to the 
fluid and the directional parameter, A, are respectively given by 
V = Vlsin$-V2cos$+n2R, (2) 
n1 
325 
A = IVf)sinCl = I(-ah/aql, -ah/aq2, flITlsin8 
= I’-“;, l/2 +(ah/a~l)2(l-Sp , 
(3) 
where r is the distance between the observer and the source, and Vf means the 
vector outward normal to the blade surface. Other symbols related to the above 
equations are listed in SYMBOLS. 
The integrations appearing in equation (1) may be understood as follows: 
In order to calculate the noise field of a moving source, the concept of re- 
tarded time designated by [ Iret must be considered, because the pressure 
change generated at different points and times might be received by an observer 
at a given time simultaneously. It is, thus, useful to know all regions of 
noise sources that have influence on the observer at the given time. These 
regions will be formed by the loci or trajectories of closed curves called 
"I?-curves" for the given observer time and position with the change of the 
source time, -m<-cLt. 
An external surface of each region has been named "influential surface". 
All sources distributed on these surfaces must be integrated to give the in- 
stantaneous pressure of the given observer time and position. By considering 
the shape of the respective influential surface, many typical effects of 
various rotor operating parameters on each noise component have come to be 
predictable analytically (ref.2 and 3). 
T2 
The integration j= / 
1 r 
drcodT should be performed on the influential sur- 
faces which are the loci of intersections between an "acoustic sphere," 
g=t-T-[rlret/cO=O, and blade surfaces, f(y,T)=O, for a given observer time, t. 
The times specified by -cl and ~~ are source times at which the acoustic sphere 
enters and leaves the blade respectively. With the lapse of the source time, 
the acoustic sphere contracts toward its center, just where the observer 
locates, with the speed of sound, co, while the blade rotates around the rotor 
axis. Fig.1 shows the geometric arrangements of an influential point. The in- 
fluential point is given as the intersection between an acoustic line and a 
specified point, (rll,nz), on the i-th blade and is determined by the following 
equation: 
F'- cosa*(n2cos$-R)R/co+$ 
(4) 
-(.Qt+2x(i-l)/B-n,/n,) = 0, 
where 
‘zrl l$h 
R 0 9121 R. 
(5) 
The angular velocity of the influential point, a$/%, which plays an im- 
portant role in the noise calculation as will be stated later, can be derived 
by differentiating equation (4) as 
a+/at = i-2/(1-M,), 
326 
(6) 
where M, is relative Mach number of the specified blade position with respect 
to an observer. Since Mr has positive peak at $=90°, a$/at takes maximum value 
at this azimuthal position. 
In the numerical calculation by the previous method, factors which decide 
the precision level of calculation are: i) mesh dimension of the Simpson sum, 
ii) dimension of the observer time increment, iii) precision rank adopted in 
the computation. After some trial computations by using a computer, FACOM 
230/75, it was concluded that the double precision was necessary and sufficient 
for the present calculation and that the number of integrating point .for ob- 
taining reliable results was about 104, which corresponded to the following 
dimension for one element: The tangential partition was (chord length)/20; the 
radial partition was (rotor radius)/300; and b~O.16. This mesh dimension re- 
quired about twenty seconds to compute the total pressure value of a given 
observer time and position. Then it was decided that the number of discrete 
observer times in the one blade passing period was thirty six in order to give 
a total computational time within twenty minutes. 
MODIFIED ANALYSES - METHODS (B) AND (C) 
Undesirable defects on the accuracy in the previous computation were caused 
by the numerical differentiation, a/at, and the numerical double integration on 
the influential surfaces, //drdT. If the observer time differentiation, a/at, 
can be put inside the integration, and be performed analytically, then the 
numerical differentiation will disappear and very much computational time can 
be saved. 
In the previous analysis, the integration on the influential surface was 
performed along a contracting acoustic sphere, drd-r. Here, in the present 
analyses, by using a polar coordinate system associated with the rotor disk, an 
elemental area of the integration, n2d$dn2, can be transformed as follows: 
codrdr rlzWdrl2 
.h/lofT= A 
where 
A = (l+M;-2M,cos@ l/2 
v = n -#fl = +l/lvfl. 
Then, by using the above relation, the solution of wave equation of the rotor 
thickness noise can be rewritten in the new coordinate system as follows: 
327 
where 
K= glVnl/r IVflA- 
(9) 
(10) 
In the ranges of integrations, $1 and $2 are the azimuth angles of the leading 
edge and trailing edge of the influential surface respectively, which are 
functions of the span position, n,, the observer time, t, and position, x; and 
R and Rg are the rotor radius and the blade cut-off radius respectively. The 
integrand, K, is considered to be a function of independent variable, nl or $, 
1123 t, and X. Thus, the double integration, JJn2dQdn2, gives an influential- 
surface integration of a single blade. 
Fig.2 shows two different arrangements of integrating points on the same 
influential surface. It can be seen that the present methods have a well- 
fitting coordinate converging to the integral region. The method (B) is derived 
from equation (9) by performing numerically the differentiation and the 
integration. 
By commuting the 
(9) b ecomes 
p(X,t) = 5 
where aK($)/at should be considered as follows: 
order of the differentiation and the integration, equation 
BR 
c 
I il 
$lct)aK(Q) w2 Wl 
(11) 
i Rg 
Q2w 
at dJ, I dJ,n2 
-K($2kjj7 +Kb!~&g- r12dr12, I 
ret 
+3J 2Jn,,aK ar 
avnlk 
+!yl$L 
ar aq, a vf 
+ aK aA 221 
$ I an aq at 
772,tJ ’ 
(12) 
The method (C) is derived from equation (11) by performing the numerical 
integration. 
328 
Consider the following case for simplicity of analysis: 
(i) hovering state, V 
‘I1 
= 112Q 
and (13) 
(ii) far field, R<<r that is r=rg=const., A=l, and (Vfl=l. 
Then the kernel given by equation (10) and the acoustic pressure can be 
written respectively as follows: 
K = (ah/anl)n2n/r0 (14) 
and 
ANALYTIC METHOD _ METHOD (D) 
=s- 0 
=pR 
27rr0 
$1 
iR I\( 1122 ZL+ atan1 
RO $2 
ah 
all, 
UJ 
ah 
-I an1 Q 
a@2 
at drl2 1 
2 ret 
ae1 ah 
lat- an, 
(15) 
dn,. 
In a double parabolic airfoil, for example, the thickness change 
given by 
ah 2hmax -=-- 
an1 'h 
(1411/Ch) 3 
which is, at the leading edge, n,=O ($=el), and the trailing edge, 
n l=Ch(JI=$2 1, given by 
J 
ret 
can be 
(16) 
ah ah 2hmax 
Gl$l=-anl$2=- 
'h 
. (17) 
329 
Combining equations (16) and (4) yields 
2n2 - yj-- Wt+ F(i-1) 
h 
(18) 
-JI- n2cos$-R)cosa11. 
By substituting the above two relations into equation (15), the acoustic 
pressure can be obtained in a decomposed form as follows: 
where 
PI :)h2 
p(v) = Pl(X,t>+P2(X,t)+P3(X,t) 
R 
.(X&J = ,z,,, 
I 
211252 
$I- c uJ142 
h Ro h 
P2bw = ;zy= 
h ! 
R 
‘$‘hdw 
RO 
R 
n$it$h2. 
Ro 
(19) 
As written above, the chordwise integration and the time differentiation 
have been performed analitically. Three components of equation (19) are cor- 
responding to the first, second, and third terms of equation (11) exactly. An 
example of computed waveform is shown in Fig.3, in which solid lines show 
components of the respective term and a dotted line shows the total acoustic 
waveform obtained by summing up these three components. 
As shown in Fig.4, the influential surfaces of multi-bladed rotor can be 
made from those of one blade at appropriate observer time. Then, once the sound 
pressure of one blade is computed; the acoustic pressure of the multi-bladed 
rotor can be easily obtained by summing up those of each blade one after 
another. Fig.5 shows an example of this process for three bladed rotor. 
COMPUTATIONAL RFSULTS 
The dimensions of an exemplified rotor are given in Table I. The computa- 
tional procedure is shown in Fig.6. Shown in Fig.7 is an example of the 
330 
computed waveforms by four different methods, (A) through (D). It can be seen 
that these four methods give the very good coincidence in the waveform and the 
negative peak value, but a slight difference in the positive peak value. The 
difference between the waveforms of (B) and (C) is too small to be distinguished. 
It is interesting to note that the area made by the positive acoustic pressure 
and zero line in one period is equal to that made by the negative one. 
Fig.8 shows the change of computed pressure amplitude in methods (B) and 
(C) in relation to the spanwise mesh size for a given chordwise mesh size by 
circles, and to the chordwise mesh size for a given spanwise mesh size by tri- 
angles. The difference of sensitivity of the accuracy for the number of span- 
wise and chordwise partitions are obvious. The insensitive tendency of the 
chordwise partition may result mostly from the blade contour of symmetric profile 
without singularity. 
Fig.9 shows the computed pressure amplitudes or peak values of pressure 
versus the number of spanwise partitions in four methods. As the number of 
partitions increases the pressure converges to an expected true value whereas 
the computation time increases predominantly as shown in Fig.10. It can be 
seen that choice of proper coordinate, analytic chordwise integration, and the 
consequently simplified program used in the present methods saved the computa- 
tional time significantly. This guarantees that the present methods (C) and 
(D) need respectively only one tenth and one hundredth of the computational 
time of the method (A) to get the acoustic pressure in the same accuracy. 
CONCLUSION 
Four different methods to compute the rotor thickness noise have been 
compared. They are (A) the previous method based on equation (1); (B) the 
one modified method based on equation (9) in which the surface integration is 
performed in the rotor-fixed-coordinate system; (C) the other modified method 
based on equation (11) in which the numerical differentiation is discarded; 
(D) the still other modified method based on equation (lo) in which the numeri- 
cal chordwise integration is further discarded. All methods have shown the 
good coincidence in both the waveform and the peak amplitude. 
The introduction of the rotor-fixed-polar-coordinate system has brought 
better characteristics in convergence and accuracy-of the numerical integration 
than the previous method (A) in which the observer-fixed-polar-coordinate sys- 
tem was adopted. Specifically, the method (D), in which the chordwise integra- 
tion was performed analytically by assuming the hovering state of rotor and the 
far field location of observer, has saved further computational time. The 
computational time in the method (D) was only one hundredth of method (A) and 
one tenth of methods (B) and (C) in keeping the same accuracy. It can, thus, 
be concluded that the computing method of the rotor thickness noise was 
improved very much by introducing the advanced methods (.C) and (D) in the 
accuracy and speed. 
331 
REFERENCES 
1. Farassat, F.: Theory of Noise Generation From Moving Bodies With an 
Application to Helicopter Rotors, NASA TR R-451, Dec. 1975. 
2. Nakamura, Y.: Helicopter Noise. PhD Thesis, University of Tokyo, March, 
1977. 
3. Nakamura, Y.: The Analysis of Helicopter Rotor Noise, ISAS Report No. 549 
(Vol. 42, No. 4), Institute of Space and Aeronautical Science, University 
of Tokyo, Sept. 1977. 
TABLE I 
Dimensions: 
Operating conditions: 
Observer: 
Observerposition;rg,m w~~=~~~~~*~*~~*~****= 50 
Observer elevation angle;aC,deg l l l l l l l l l l l l l l l l l 0 
332 
I 
._.- ..-_ -- . 
1 I g(t,O)=O- 
ACOUSTIC LINE 
BLADE SURFACE 
/I 'r-LINE l- f( t ,AT)=O J 
LINFLUENTIAL POINT 
Figure l.- Geometric arrangements of an influential point. 
INTEGRATING POIN 
/LEADING EDGE' 
USTIC 
:NES 
LINES OF INSIGHT 
FROM OBSERVER 
(a) Present methods. (b) Previous method. 
Figure 2.- A comparison of integrating mesh between two coordinate system= 
333 
- 
ACOUSTIC 
PRESSURE 
kg/m2 
(l/2) PERIOD 
-60 
-80 
Figure 3.- Computation of waveform. 
,lst BLADE 
(a) Influential lines of leading (b) Influential surfaces of 
edge of one blade at each three bladed rotor at 
observer time. given observer time. 
Figure 4.- An example to obtain influential surfaces of multi-bladed rotor 
from influential lines. 
334 
(l/3) PERIOD 
e 
P I. PII PIu 
c ONE PERIOD OF, 
A SINGLE BLADE 
I 
WAVEFORM OF 
A SINGLE BLADE 
P 
THREE BLADED ROTOR 
WAVEFORM OF THREE BLADED ROTOR 
Figure 5.- An example to obtain waveform of multi-bladed rotor from 
a waveform of single blade. 
READ DATA 
4 
SET BLADE NUBBER 
CAiCULATE BOUNDARY VALUE 
AT LEADING EDGE (Jll ,+, ,K,) 
A; TRAILING EDGE (+2,+2,K2) 
CHORD POSITION ( 71) 
-------- ~ 
I 
I 
L-----l .----- - -_--- ~----- -__-- -1 
SPANWISE INTEGRATION RETHOD (B)&(C) 
Sun’UP THE WHOLE BLADE 
I 
OUT PUT THE RESULTS 
23 END 
Figure 6.- Flowchart of modified mehtods to compute the rotor thickness noise. 
335 
ACOUSTIC 
PRESSURE 
kg/m2 
2 
-4 
4 
p 
- 
c ---- METHOD (A) 
--- METHOD (B)&(C) 
-6 - METHOD (D) 
-8 t 
Figure 7.- Waveforms given by four methods. 
30 
COMPUTED ' 
ACOUSTIC 
PRESSURE 20 - 
AMPLITUDE 
kg/m2 
10 
I 
- 
SPANWISE PARTITION 
LCHORDWISE PARTITION 
1 I I I I I I 
0 5 10 20 40 80 160 
NUMBER OF PARTITIONS 
336 
Figure 8.- Converging tendency with the number of partitions. 
COMPUTED 
ACOUSTIC 
PRESSURE 
AMPLITUDE 
kg/m2 
0 METHOD (A) 
(D>- 
I I I I I I I 
5 10 20 40 80 160 320 
NUMBER OF SPANWISE PARTITIONS 
Figure 9.- Converging tendency of the four methods. 
tl : 
t2 : 
t3 : 
TIME SAVED FOR A GIVEN MESH SIZE 
u IN KEEPING SAME ACCURACY BY METHOD (B>OR(C) 
It II (D) 
CPU TIME 
set 
I I I -J-m_ PlL---J I I 
0 5 10 20 40 80 160 320 
NUMBER OF SPANWISE PARTITIONS 
Figure lo.- A comparison of computational time between four methods. 
337 


Improved methods for calculating the thickness noise

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