Edmane Envia

Glenn Research Center, Cleveland, Ohio

Open Rotor Aeroacoustic Modelling

NASA/TM—2012-217740

October 2012
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Edmane Envia

Glenn Research Center, Cleveland, Ohio

Open Rotor Aeroacoustic Modelling

NASA/TM—2012-217740

October 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

15th International Conference on Fluid Flow Technologies (CMFF’12)

sponsored by the Budapest University of Technology and Economics (BME)

Budapest, Hungary, September 4–7, 2012
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subject to revision as analysis proceeds.
NASA/TM—2012-217740 1 

Open Rotor Aeroacoustic Modelling 

Edmane Envia

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

Owing to their inherent fuel efficiency, there is renewed 

interest in developing open rotor propulsion systems that are 

both efficient and quiet. The major contributor to the overall 

noise of an open rotor system is the propulsor noise, which is 

produced as a result of the interaction of the airstream with the 

counter-rotating blades. As such, robust aeroacoustic 

prediction methods are an essential ingredient in any approach 

to designing low-noise open rotor systems. To that end, an 

effort has been underway at NASA to assess current open 

rotor noise prediction tools and develop new capabilities. 

Under this effort, high-fidelity aerodynamic simulations of a 

benchmark open rotor blade set were carried out and used to 

make noise predictions via existing NASA open rotor noise 

prediction codes. The results have been compared with the 

aerodynamic and acoustic data that were acquired for this 

benchmark open rotor blade set. The emphasis of this paper is 

on providing a summary of recent results from a NASA Glenn 

effort to validate an in-house open noise prediction code called 

LINPROP which is based on a high-blade-count asymptotic 

approximation to the Ffowcs-Williams Hawkings Equation.

The results suggest that while predicting the absolute levels 

may be difficult, the noise trends are reasonably well predicted 

by this approach. 

Nomenclature

Ai Ai , ′ [-] Airy function and its derivative

B [-] rotor blade count

0 c [m/s] ambient speed of sound

j e [-] unit vector component in the radiation direction

j f [N/m2

] blade force per unit area

G [1/m-s] free-space, moving-medium Green’s 

function

gc [-] convective phase factor

M0R [-] medium Mach number in the radiation direction

Mtip [-] rotor tip rotational Mach number

N [rev/s] rotational speed

nj [-] surface unit normal vector 

p xt ′( ,) [Pa] acoustic pressure

p x ′( ) [Pa] Fourier harmonic component of acoustic 

pressure

p [Pa] blade surface pressure

 [Pa] acoustic source amplitude

R [m] distance between source and observer

S [m2

] rotor blade surface

B S [m2

] reference blade surface

Tjk [N/m] Lighthill stress tensor

t [s] observer time

0 j U [m/s] medium velocity components

V [m3

] volume exterior to the rotor blades

n v [m/s] normal component of blade surface velocity

, j x x [m] observer Cartesian coordinates

, j y y [m] source Cartesian coordinates

δ [-] Dirac delta function

κ [-] convective amplitude factor

ρ0 [kg/m3

] ambient air density

τ [s] source (i.e., retarded) time

Φ [rad] rotor phase function

, φ φs [rad] observer and source azimuthal angle

coordinates 

Ψ [rad] convective phase factor

Ω [rad/s] rotor angular speed

Subscripts: 

1, 2 front and aft rotor indices

j principal coordinate directions

s typical source point

 thickness noise designator

 loading noise designator
NASA/TM—2012-217740 2 

Introduction

Open rotors are inherently more fuel-efficient than fans due 

to their very high propulsive efficiency. With the rising fuel 

cost and the increasing restrictions on carbon emissions (CO2

emissions are directly related to fuel burn) there is renewed 

interest in developing open rotor systems for commercial 

aviation applications.

The fuel burn advantage of open rotor propulsion systems 

was convincingly demonstrated in the past (for example, in the 

late 1980s and early 1990s as part of a NASA effort called the 

Advanced Turboprop Project (Ref. 1)). A model scale open 

rotor developed by NASA and General Electric (GE) under 

that project and the subsequent engine demonstrator developed 

by GE, called the unducted fan (UDF), are shown in Figure 1.

At that time, some of the fuel burn margin of UDF was traded 

off in order to meet the noise regulations of the time. Changes

to the design paradigm (e.g., blade count increase and tip 

speed reduction) and the advent of the high-fidelity threedimensional aerodynamic simulation tools have now made it

possible to design open rotor systems that can meet the current 

noise regulations (which are more stringent than the 1990s 

rules) without giving up any of their fuel burn advantage. 

As a result, both the United States and Europe have been

pursuing the development of the next generation open rotor 

propulsion systems. In particular, in the U.S., a joint effort 

between NASA and GE Aviation was initiated to investigate 

the aerodynamic and acoustic performance of modern blade 

designs. An important element of this effort has been the 

assessment of existing tools for predicting the aerodynamic 

and acoustic performance of open rotors. For that purpose a 

baseline, vintage 1990s, blade design called the historical 

blade set (also known as F31/A31) was selected for evaluating 

various prediction tools. F31/A31 has 12 front rotor blades 

and 10 aft rotor blades.

Comprehensive aerodynamic, flowfield, and acoustic data 

sets were acquired in the NASA wind tunnels for a ~0.65 m 

diameter sub-scale model of this blade set. The test campaign 

included both low speed testing in the NASA 9- by 15-foot 

(i.e., 2.7 by 4.6 m) wind tunnel to investigate the aero/acoustic 

performance at operating conditions in the approach to takeoff range as well as high speed testing in the NASA 8- by 6-

foot (i.e., 2.4 by 1.8 m) wind tunnel to analyse the 

aerodynamic performance in the speed range between climb 

and cruise conditions (Ref. 2). Data was acquired in an uninstalled configuration, as well as at angles of attack and with 

a generic pylon for some conditions. In all, a sizable matrix of 

rotor blade setting angles, tip speed, and tunnel Mach number 

combinations were investigated. In this paper, the focus is on a 

small subset of the un-installed (zero angle of attack) 

configurations that were run in the low speed regime at one set 

of blade angles, and one tunnel Mach number. The testbed 

F31/A31 open rotor is shown installed in the NASA 9- by 15-

foot wind tunnel in Figure 2.

Figure 1.—Top: GE/NASA Unducted Fan (UDF) model installed 

in NASA wind tunnel (1985). Bottom: GE UDF engine demonstrator installed on the MD-80 testbed aircraft (1987). UDF had 

equal blade counts, eight each on the front and aft rotors.

Figure 2.—Model scale F31/A31 historical baseline blade set 

shown installed in the NASA 9- by 15-foot wind tunnel (2010). 

The front rotor of F31/A31 has 12 blades and its aft rotor has

10 blades. Traversing inflow microphone system used for 

sideline acoustic pressure measurements is visible on the left.
NASA/TM—2012-217740 3 

Aeroacoustic Modelling

The methodologies for predicting open rotor noise can be 

categorized into three main groups. The most basic approaches 

are empirical in nature which, as the name suggests, are based 

on curve fits to database(s) of measured noise as a function of 

the geometry and operating parameters of open rotors. Once 

developed these are relatively easy to use and are often 

employed for trend studies in the early phases of the design

process. Their shortcoming, however, is that they cannot be 

reliably used to predict noise for configurations or operating 

conditions that are well outside of the envelope of the 

databases used for developing the models in the first place. 

On the opposite end of the spectrum are the high-fidelity 

numerical approaches that can be used to solve for the 

unsteady flowfield of the open rotor and include, as part of the 

solution, the noise field of interest. These types of methods 

typically require sizable computational resources due to the 

huge differences in the flow scales involved. For example, 

whereas the background flow pressure levels may be at the 

ambient atmospheric level, the acoustic perturbation levels of 

interest are easily three to four orders of magnitude smaller. 

This disparity necessities highly resolved grids nearly 

everywhere in the domain of interest thus driving up the 

computer memory and computational time requirements. 

It is in the middle ground between these two extremes

where most of the current state of practice resides. Typically, 

the issue of scale disparity is circumvented by linearizing the 

equations of motion thus separating the nonlinear aerodynamic 

field from the linear acoustic field. In the linearized methods, 

the nonlinear aerodynamic field is specified, measured, or 

computed a priori and is introduced as source terms in the 

linearized equations governing the acoustics. Computational 

fluid dynamics (CFD) is most often used for that purpose. 

Once this input information is specified, the acoustic equations 

can be solved to estimate the acoustic field. The solution could 

be affected by solving the linearized equations numerically, 

but most often the solution is obtained using the Lighthill’s 

acoustic analogy approach whose extension to the moving 

surfaces is the Ffowcs-Williams Hawkings (FW-H) equation. 

This equation describes the acoustic field as temporal and 

spatial integrals over aerodynamic source regions of interest 

(e.g., rotor blades and the surrounding flowfield). The 

approach used in this paper for predicting open rotor noise is 

based on an asymptotic approximation to the FW-H equation 

with the necessary aerodynamic input obtained via a 

commercial CFD code. 

Acoustic Modelling

The FW-H equation, see Equation (1a), is an exact formal 

solution of the general equations of motion accounting for all 

of the underlying physics. It expresses the sound field p xt ′( ,)

as integrals of the aerodynamic source distributions. These 

source distributions are traditionally called the thickness, 

loading, and quadrupole sources, corresponding to the three 

terms on the right hand side of Equation (1a), respectively. nv

is the normal velocity of the blade, j fdS is the blade force, 

and Tjk (the Lighthill stress tensor) represents the fluctuating 

stresses in the flowfield surrounding the rotor blades. Here 

only the inviscid parts of j f and Tjk are implied, since 

viscosity does not significantly contribute to the open rotor 

noise field. In fact, we shall neglect the contribution of the 

quadruple source entirely since the focus of this paper is on 

the low speed operating conditions for which the quadrupole 

source does not significantly contribute to the noise field.

( ) ( ) ( )

( )

( ) ( )

( )

( )

( )

0

0

2

0

0

,

 

 

j

n

S

j

j S

jk

j k V

j

D G

p x t v dS y d D

G f dS y d

y

G T dyd

y y

D U

D y

∞

−∞ τ

∞

−∞ τ

∞

−∞ τ

′ =− ρ τ τ +

τ

∂

τ τ +

∂

∂

τ τ

∂ ∂

∂ ∂

= +

τ ∂τ ∂

∫ ∫

∫ ∫

∫ ∫ (1a)

In this equation, G is the free-space, moving-medium Green’s 

function given by

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

0

2 0

0

2 2

0 00 0

0 2

0 00

0

1 /

4

1

, 

, 

1 , 

R

R Rj

j

j j

c

c

j

j j

j

G t gR c R

g M

M M Me

x y

e R x y R

U

M M

c

τ = δ −τ−

πκ

τ = κ−

β

κ τ = +β τ =

− τ = τ = − τ

β= − =

(1b)

where c g and κ are factors that embody the effects of medium 

convection on the propagation time 0 / c gR c and spherical 

spreading rate 1/ R . Where applicable, the explicit 

dependence of various variables on the source time ( ) τ is 

indicated. Assuming that the geometry of the rotor blades, 

their loading distributions, and the surrounding flow field are

known, the integrals in Equation (1a) can be explicitly 

integrated to provide an estimate of the sound field generated 

by the open rotor.
NASA/TM—2012-217740 4 

Figure 3.—Sound pressure level (SPL) of the model scale 

F31/A31 blade set 1.524 m away and 90° from the open 

rotor axis. The SPL is plotted versus the multiples of the 

open rotor shaft frequency (i.e., shaft orders). The shaft 

frequency is 107.3 Hz for this case. 

While in principal, Equation (1a) can be computed 

explicitly by carrying out the indicated spatial and temporal 

integrals using appropriate quadrature schemes, e.g., see 

Farassat (Ref. 3) and Brentner (Ref. 4), most often this

equation is tackled in the frequency domain and by assuming 

that the observer is in the farfield. This assumption results in 

considerable simplification of the integrands in Equation (1a) 

leading to semi-analytic formulas. See, for example, the 

models due to Hanson (Ref. 5) and Parry (Ref. 6). The 

strategy used in this paper for carrying out the integrals in 

Equation (1a), while frequency-domain based, does not 

require such simplification and the resulting formulas are valid 

for any observer (in the nearfield or in the farfield).

Specifically, the formulas used in this paper are extensions 

of the single rotation ones developed in Reference 7 to 

account for the counter rotation configuration. The main 

difference between the single and counter rotation versions is 

the treatment of the loading noise. Another feature of the 

extended theory is that, owing to the linearity of the sound 

field, the analysis is carried out for each rotor separately and 

the resulting sound fields superimposed at the end. The 

derivation of the working expressions will closely follow that 

described in Reference 7 and will use essentially the same 

notation. The details of the derivation are included in the 

Appendix A and only final formulas are given here.

Owing to the dominance of the open rotor spectra by tone 

noise (see Fig. 3), in this study the modelling is focused on the 

tonal component of the noise spectrum. Let the blade counts

and rotational speeds of the two rotors be denoted,

respectively, by B1 & B2 and N1 & N2 , where the subscript 

“1” refers to the front rotor. It should be noted that the open 

rotor blade counts or the rotational speeds of the two rotors

need not (and frequently are not) equal. The tonal acoustic 

field of the front rotor is given by the following formula

( ) ( )

( ) ( )

1 1

11 2 2

,

4 ,

 

m

m k

imB t

m

i mB kB t

m k

p xt p x e

p xe

∞ − Ω

=−∞

∞ ∞ − Ω+ Ω

=−∞ =−∞

π = ′ ′ +

′

∑

∑ ∑





(2a)

where m p′

 is the thickness noise harmonic amplitude given by

the expression

( ) ( )

( )

( )

( )

( )

1

1

1 1

1

2

1

1

2/3 2

1

0, 1/3

1

2/3 2

1

1, 2/3

1

tip 0

0

 

1

m

B

mB i

n S

n

n

s s

p x iB e

Ai mB

d

mB

Ai mB

d ds

mB

M M

μ −Ψ

=

′ ×

   γ     +





   ′ γ   







Ψ = χ +φ −φ β

∑ ∫   









(2b)

with 1 1 Ω= π2 N denoting the angular frequency of the front 

rotor and 1 1 tip 1 tip 0 M Rc = Ω / its tip rotational Mach number. 

γ and μ are defined by Equation (A.11) in the Appendix. 

The Airy function Ai and its derivative Ai′ give an accurate 

asymptotic approximation to the integral over τ appearing in 

the Equation (1a). That integral represents the radiation 

efficiency of a source point over one period of its revolution

about the axis of the rotor. Thus, the expression inside the 

curly bracket provides a closed form formula for the radiation 

efficiency of a given source point. Note that, Equation (2a) 

indicates that thickness noise is produced at the harmonics of 

the front rotor blade passing frequency B1 1 Ω . The larger the 

parameter mB1 is, the more accurate this approximation 

becomes. 

Similarly, the loading noise harmonic amplitude m k, p′

 is 

given by
NASA/TM—2012-217740 5 

( ) ( )( )

( )

( )

( )

( )

1 2

1

1 1

1

2

1

1

2/3 2

1 2

0, 1/3

1 2

2/3 2

1 2

1, 2/3

1 2

CRF tip 0

0

1 22 1

CRF

1 2

 

1

/

m

B

mB kB i

n S

n

n

s s

p x iB e

Ai mB kB

d

mB kB

Ai mB kB

d

mB kB

M M

mB kB

mB kB

− μ −Ψ

=

′ ×

   − γ     +

 − 

   ′ − γ   

 − 



Ψ = η χ +φ −φ β

+ ΩΩ

η = −

∑ ∫   









(2c)

where 2 2 Ω=π2 N is the angular frequency and B2 2 Ω the 

blade passing frequency of the aft rotor. As before, γ and 

μ are defined by Equation (A.11) in the Appendix.

Equation (2c) shows that, in general, loading noise is produced 

at the frequency combinations mB kB 11 2 2 Ω+ Ω where m is 

called the acoustic harmonic and k the loading harmonic. 

For k = 0 , Equation (2c) reduces to the isolated rotor loading 

noise formula derived in Reference 7. Note that the loading 

noise harmonic amplitude is a function of the difference 

parameter mB kB 1 2 − . As will be discussed in the results 

section this parameter plays a crucial role in determining the 

magnitude of the various loading noise tones. 

Nearly identical expressions for the aft rotor tone noise field 

can be derived but with 1 1 ( , ) B Ω pair interchanged with 

2 2 ( , ) B Ω pair in Equations (2a) to (2c). Once the 

contributions from both rotors have been computed, the 

resulting sound fields can be superimposed to estimate the 

tonal sound field of the counter-rotating open rotor.

Equations (2a) to (2c), and their counterpart for the aft rotor, 

are the working expressions used for computing the tone noise 

of F31/A31 blade set reported here. These expressions were 

incorporated into an existing NASA Glenn Research Center 

propeller noise code called LINPROP to predict the tone noise 

of open rotors. The original LINPROP code was developed in 

early 1990s based on the single rotation theory described in 

Reference 7.

Aerodynamic Calculations 

The computational tool used in this paper for the 

aerodynamic calculations is the commercial CFD software 

package FINE/Turbo developed by NUMECA International1

 

1 Additional information regarding FINE/Turbo software package 

may be found at http://www.numeca.com.

FINE/Turbo is a turbomachinery CFD simulation software 

package with integrated meshing and post-simulation analysis 

tools. It is a structured, multi-block, unsteady Navier-Stokes

solver which offers several solution algorithm choices along 

with several acceleration strategies. The simulations can be 

run in full unsteady mode though this could be quite resource 

intensive. To reduce the computational time requirements for 

the time-dependant simulations, FINE/Turbo can also be run 

in the nonlinear harmonic (NLH) mode (Ref. 8) which solves 

for a finite number of the blade passing frequency harmonic 

components of the time-dependent solution, but ignores all the 

other unsteady components. This is a targeted approach 

focusing on the relevant frequency content of the underlying 

unsteady flow. The net result is a very significant reduction in 

the computational time requirements. This is the approach 

employed in this paper for the purpose of computing the 

aerodynamic response. In results to be presented later in the 

paper, the blade loading perturbation pressures associated with 

the first three harmonics of the blade-passing frequency for 

each rotor were retained in addition to the blade mean pressure

field. Only three loading harmonics were chosen to keep the 

computational cost reasonable while still ensuring that the 

relevant tones up to 50th shaft order could be modelled (see

Fig. 3). The implications of this choice will become clear 

when the acoustic predictions are discussed later in the paper.

The FINE/Turbo computational domain used in generating 

the results presented here includes one passage each of the two 

blade rows and their associated ancillary domains like the

spinner, hub, farfield, etc. (see Fig. 4). The total mesh size is 

slightly over 27.1 million grid points with the farfield boundary 

set seven tip radii away. The computations were run in parallel 

mode and depending on the particular simulation, the domain

was partitioned either into 39 or 42 blocks to achieve optimum 

computational load balance.

Figure 4.—The computational domain and grid blocks used for 

nonlinear harmonic FINE/Turbo simulations. Block regions

are distinguished by different colors with the “farfield” blocks 

shown in gray.
NASA/TM—2012-217740 6 

TABLE 1.—ROTOR RPMS CONSIDERED IN THIS STUDY

Case Front rotor,

rpm

Aft rotor,

rpm

1 4620 4620

2 5268 5268

3 5551 5551

4 6068 6068

5 6303 6303

6 6436 6436

In total, six tip speeds were considered ranging from the 

approach rpm to the nominal takeoff rpm. Table 1 shows the 

rotor tip speeds modelled in this paper. All cases considered 

had equal rpms on the front and aft rotors. The nominal blade 

setting angles were 40.1° for the front rotor and 40.8° for the 

aft rotor in all cases. The tunnel Mach number (i.e., open rotor 

“forward flight speed”) was also fixed at 0.2. As was 

mentioned in the introduction, these cases represent a small 

subset of a very large matrix of configurations for which 

aerodynamic and acoustic data was acquired in the wind 

tunnel tests.

The starting point for the simulations was the highest tip 

speed condition, which was converged first. It took a total of 

4000 iterations (3000 of them in NLH mode) to converge the 

solution. The maximum residual, thrust, torque, and pressure 

ratio had all settled after 3500 iterations, but the simulation 

was run another 500 iterations to ensure complete 

convergence. The acoustic calculations based on the 3500 and 

4000 iteration solutions were nearly identical also. Once 

convergence had been achieved for this rpm, the next lower tip 

speed condition was run starting with this converged solution 

as the initial solution. Typically, it took 2500 iterations to 

achieve full convergence. The process was repeated for the 

next lower tip speed condition until all cases were computed. 

The simulations typically used between 14 and 16 cores of a 

48-core compute node that is part of a NASA Glenn computer 

cluster each compute node of which has 128 GB of RAM.

Figures 5 and 6 show comparisons of total thrust and torque 

ratio predicted by FINE/Turbo against the wind tunnel data. 

The agreement for thrust is quite good with an average 

discrepancy of 1.6 percent. The torque ratio comparison is 

only fair with an average error of roughly 9 percent. It should 

be noted that no tweaks were performed to adjust the blade 

angle, rotor tip speeds, or the tunnel Mach number to achieve 

a better match with the data. In other words, the simulations 

represent true “predictions”.2

Figure 7 shows the contours of predicted magnitude of the 

mean and first blade passing frequency harmonic of the static 

pressure distribution on the front and aft blade rows as well as 

on the hub. Note the change in scale of the contour levels

between the mean and perturbation contours. A close

examination of the levels on the front and aft row shows that,

 

2 The blade shapes at the max climb condition supplied by GE 

were used at all speeds studied here. Thus, the small variations in the 

hot blade shape due to the changes in centrifugal force as a function 

of rpm difference from the max climb condition were ignored.

Figure 5.—Measured and predicted F31/A31 rotor thrust. 

Combined thrust produced by both rotors is plotted. 

Figure 6.—Measured and predicted F31/A31 rotor torque ratio.

while the mean loading levels on the front and aft rotors are

comparable, there is a significant difference between the 

magnitudes of perturbation pressures on the two blade rows. 

The aft row clearly experiences much higher levels of pressure 

fluctuations owing to the impingement of the wakes of the 

front rotor, whereas the front rotor is only weakly affected by 

the potential field of the aft rotor. Another noteworthy feature 

is that the suction side perturbations are larger than the 

pressure side. The second and third harmonic pressure 

perturbations show similar trends to the first harmonic 

component, but they are smaller than the first harmonic 

component by roughly a factor of two and four, respectively. 
NASA/TM—2012-217740 7 

Figure 7.—Contours of predicted pressure distributions on the 

hub and rotor blades for F31/A31. (a) shows the mean pressure level and (b) the blade passing frequency harmonic level.

By plugging the blade geometries and aerodynamic pressure 

distributions for each rotor into the Equations (2b) to (2c) the 

harmonic amplitudes of the thickness and loading tone noise 

were calculated and combined to estimate the sound field of 

F31/A31. The noise prediction results are discussed in the next 

section.

Acoustic Predictions

The modified LINPROP code was run for all of the tip 

speed conditions considered in this study. The calculations 

were performed for observer (i.e., microphone) positions that 

are located in a horizontal plane containing the open rotor axis

and on a sideline parallel to the axis of the open rotor and

60 in. (i.e., 1.524 m) displaced from it. This layout duplicates 

Figure 8.—Sketch of the F31/A31 open rotor testbed and the 

sideline traversing microphone track as they were installed

in the 9- by 15-foot wind tunnel. Acoustic data was acquired 

at a total of 18 microphone stops between the observer 

angles 18° and 140°. 

exactly the test arrangement in the NASA 9- by 15-foot wind 

tunnel shown in the sketch in Figure 8.

During the test, the sideline acoustic data was acquired at 18 

microphone positions ranging between 18° and 140° from the 

open rotor axis as shown in the figure. The data was acquired 

using a traversing inflow microphone (see Fig. 2). Spectral 

data was computed from 15-sec long time series acquired at 

200 kHz sampling rate with a 214 point FFT resulting in a 

frequency bin width of 12.2 Hz. Consequently, the measured 

tone energy is spread across multiple bins (see Fig. 3). 

Therefore, to compare the measured levels to the predicted 

ones (which have an effective bin width of zero), it was 

necessary to sum the tone energy in the bins over which the

tone is spread for the measured data. 

As for the predictions, the loading component dominates the 

total level in all cases, hence only the predicted total is 

presented here. In any case, the experimental data cannot be 

parsed into constituent thickness and loading components. Also, 

while the data was acquired at 18 observer angles, the 

predictions were computed on a finer angular resolution of 51 

observer angles to ensure that the nuances of the tone 

directivities were well captured. Lastly, given the number of the 

rpm and tone combinations, only highlights are presented here. 

Figure 9 shows a representative comparison of the predicted 

tone sound pressure level (SPL) from the LINPROP code with

the measured level for the highest tip speed (i.e., 6436 rpm) at

the broadside microphone location (i.e., observer angle of 

90°). Predictions for the tones in the range from approximately

1 to 5.3 kHz are shown (shaft frequency for this tip speed is 

107.3 Hz). This is because, the FINE/Turbo simulations 

include the mean and only the first three blade passing 

frequency harmonics of the blade pressure loading for each 

row, which determines how many open rotor tones could be 

calculated based on the relationship mB kB 11 2 2 Ω± Ω . 

Therefore, the noise predictions are restricted to the tones for 

which the loading harmonic k does not exceed 3.

The tones are labelled based on whether their origin is the 

front rotor, aft rotor, or if they are produced as a result of the 

interaction of the two. The frequency for each tone is given by

mB kB 11 2 2 Ω± Ω where m k , 0,1,2,3 = .

3 The predicted levels 

 

3 m=k=0 does not result in an acoustic tone.
NASA/TM—2012-217740 8 

Figure 9.—Comparison of measured and predicted tone SPL 

at the broadside microphone location (i.e., observer angle 

90o

) at 6436 rpm. Tones are labeled according to their origin 

(i.e., front rotor tones, aft rotor tones, or interaction tones). 

The labels are color coded according to whether the tones 

are based on sum (blue) or difference (brown) combinations.

include the complex conjugate contributions corresponding to 

the associated negative frequencies − Ω± Ω mB kB 11 2 2 . Before 

discussing the data-theory comparisons, it is important to note 

the following in order to better understand the comparisons.

As indicated in Figure 9, most tones can be produced by 

either the sum or difference frequency combinations. In fact, 

while there is only one sum combination for a given tone, 

there are infinitely many difference combinations possible. 

However, tones corresponding to virtually all frequency 

difference combinations have extremely low radiation 

efficiencies and do not contribute to the tone SPL in a 

significant way. This is most easily seen by examining the 

terms inside the curly bracket in Equation (2c).

Note that since for a given frequency combination 

mB kB 11 2 2 Ω± Ω , the tone amplitude is a function of the index 

combination mB kB 1 2  , there can be substantial difference in 

the radiation efficiency of the sum and difference tones. To 

elucidate the point, let’s contrast two tones, one with the 

frequency 1 2 12 10 Ω+ Ω and the other with the frequency 

1 2 12 10 Ω− Ω . The amplitudes of these two tones according 

to Equation (2c) will be functions of the index combinations

12 10 + and 12 10 − , respectively. Since, Ai and Ai′ decay 

rapidly as their arguments increase, the amplitude of the sum 

tone will far exceed the amplitude of the difference tone. The 

plot in Figure 10 depicts the point graphically. The plot shows 

the dependence of the terms in the curly bracket in

Equation (2c) on the parameter mB kB 1 2 − for a source on the 

tip of the blade. The terms are each normalized by their value 

for zero argument and plotted on a logarithmic scale. 

Figure 10.—Radiation efficiency of sum and difference tones 

as a function of the index mB kB 1 2 − . 

A number of tones for which the parameter mB kB 1 2 − falls 

in the range 0 to 40 are identified on the abscissa. Tones 

whose frequencies are above 14 kHz are not noted since 

beyond that limit no discernible tones are detectable above the

broadband level in the measured spectra. Note the rapid fall of 

the Ai and Ai′ as their arguments increase. It is easy to see 

from this plot that sum tones have far higher radiation 

efficiencies than do the difference tones. In fact, there are only 

three difference tones in the range plotted. The radiation 

efficiencies of higher order difference tones are effectively 

negligible. 

Another notable aspect of this plot is that it shows that the 

radiation efficiencies of several interaction (sum) tones are 

significantly higher than the radiation efficiencies of the 

individual rotor blade passing tones (i.e., BPF1 and BPF2). As 

a result, even though the magnitude of the blade loading’s 

higher harmonics, which drive the interaction tones, are much 

smaller than the mean loading, which drives the blade passing 

tones, their comparatively much higher radiation efficiencies

compensate for their source amplitude deficit (recall Fig. 7) 

thus making their levels, especially those for the low order 

interaction tones, on par or even higher than the blade passing 

frequency tone levels. Finally, it should also be noted that the 

higher harmonic tones of the individual rotors (i.e., nBPF1 and 

nBPF2) also have comparatively smaller radiation efficiencies 

and thus are expected to fall off rapidly in amplitude as seen in 

this plot.

Returning to Figure 9, note that the low order interaction 

tones 1BPF1+1BPF2, 2BPF1+1BPF2, and 1BPF1+2BPF2, are 

well predicted by the LINPROP code with discrepancies of

1 dB or less, when considering the experimental uncertainty in 

the measured levels which is ±1 dB. The next set of

interaction tones, 3BPF1+1BPF2, 2BPF1+2BPF2, and 

1BPF1+3BPF2, show larger discrepancies when compared 

with the measured levels and the discrepancy grows as the 
NASA/TM—2012-217740 9 

tone order (i.e., its frequency) increases. Since, as mentioned 

earlier, the interaction tones are driven by the harmonics of the 

unsteady loading on the blades, it is likely that to predict 

higher order tones more computational grid resolution may be 

needed to achieve better accuracy of the blade pressure 

loading harmonics.4

As for the individual rotor tones, the theory over-predicts 

the blade passing frequency tones BPF1 and BPF2, but underpredicts their harmonics, i.e., nBPF1 and nBPF2. Theses

discrepancies may be explained by the following observation. 

The experimental data show a preponderance of tones at all 

shaft orders (recall Fig. 3) including both odd and even order 

ones. Yet, the theory only predicts even order tones (whether 

sum or difference ones) since the blade counts are even

numbers. The paradox can be explained by noting that the 

theory (aerodynamic or acoustic) assumes that all of the blades 

in each rotor disc are identical and that they all experience the 

same loading time history over the course of each rotor 

revolution, but displaced spatially and temporally from that of 

the reference blade. This assumption results in acoustic energy 

being distributed amongst a subset of shaft orders. The actual 

blades in a given rotor disc, however, are not exactly the same

as they are hand finished (they are made of composites) and 

they may not experience exactly the same time history due to 

inherent unsteadiness of the flow. Perhaps more importantly, 

the setting angles for the blades in a given rotor disc vary 

within a small range (typically in the ±0.1° range for F31/A31 

(Ref. 9)) in a random fashion around the rotor disc thus 

slightly modifying the loading distribution on each blade. 

Therefore, for the actual rotors the precise phasing assumed by 

the theory does not occur, a consequence of which is that the 

acoustic energy is distributed amongst all shaft orders rather 

than just those singled out by the theory. As a result, some of 

the energy that according to the theory would have amassed in 

BPF1 and BPF2 tones is leaked into their harmonics nBPF1 and 

nBPF2. The same mechanism could explain the measurable

levels for the difference tones (e.g., 2BPF2-2BPF1, 3BPF2-

2BPF1, and 4BPF2-3BPF1), for which the theory predicts very

small radiation efficiencies and, thus, levels that would be too 

small to be visible in Figure 9.

Next, the variations of SPL on the sideline as a function of 

observer angle for the BPF1, BPF2, and BPF1+BPF2 tones are 

examined (see Fig. 11). While it is difficult to draw general 

conclusions about the data-theory comparisons on the absolute 

level basis, the trends in data appear to be reasonably well 

predicted except for the far upstream and far downstream 

angles for the BPF1 and BPF2 tones. It should be noted that the 

predicted rapid falloff of the SPL for BPF1 and BPF2 away 

from the vicinity of the respective planes of rotation (i.e., ~90° 

observer angle) is consistent with measured single rotation 

results. It is not clear then, why the measured blade passing 

 

4 This was partially borne out by comparing the results to those 

from a single case that was run with a mesh resolution of 42 M grid 

points. However, the grid packing in the vicinity of the blade may be 

a more important factor than grid resolution alone.

tone levels for the individual rotors of F31/A31 should level 

off (or even increase) below about 70° and above 

approximately 130°.

Figure 11.—Sideline variations of SPL as a function of 

observer angle for (a) BPF1, (b) BPF2, and (c) BPF1+BPF2

tones at the 6436 rpm condition. 
NASA/TM—2012-217740 10

Figure 12.—Sideline variations of OASPL as a function 

of the observer angle on the sideline. Panes (a) to (f) 

show the variations of OASPL with tip speed. Comparisons for observer angles in the range 70° to 130° 

only are shown.

The data-theory comparisons look similar for the other tip 

speeds, so results corresponding to those presented in

Figures 9 and 11 for lower tip speeds will not be presented 

here. Instead, the comparisons as a function of tip speed will

be shown on the basis of the overall sound pressure level 

(OASPL). The principal reason for choosing the OASPL

metric goes back to discussion earlier regarding the acoustic

energy being distributed amongst all shaft order tones rather 

than just those orders that the theory predicts. It could be 

argued that in that sense the OASPL should be a good metric 

to use regardless of how the acoustic energy is distributed 

amongst the various tones. It should also be noted that since 

the measured levels include both tone and broadband content, 

it is necessary to subtract the broadband levels from the 

measured data before calculating the OASPL. Lastly, based on 

the discussion of the results in Figure 11, the OASPL 

comparisons will be restricted to the observer angles in the 70° 

to 130° range.

Figure 12 shows the dependence of the OASPL on the rotor 

tip speed. Both measured and predicted OASPLs are shown 

from the highest tip speed (6436 rpm) on the top of the figure 

to the lowest tip speed (4620 rpm) on the bottom of the figure. 

Generally speaking, the data-theory agreement is reasonable at 

the higher tip speeds, but it deteriorates as the tip speed is 

reduced. The basic trend of decreasing OASPL with 

decreasing tip speed in the measured data is captured by the 

theory though not uniformly for all observer angles in that the 

decrease in the predicted levels is slower than that in the 

measured data.

Conclusions

The results discussed in the previous section suggest that 

absolute level predictions on a tone-by-tone basis for open 

rotors (or any turbomachinery system, for that matter) may be

out of reach using the existing suite of tools (i.e., CFD + 

acoustic analogy). This may be true even when more 

sophisticated numerical methods that directly compute the 

noise field from the field equations are brought to bear. This is 

a consequence of the fact that any model that assumes 

identical blades in each blade row, experiencing spatially- and 

temporally-shifted time histories, is incompatible with the real 

geometries, for which there always exist small manufacturing 

and installation variations as well as flow non-stationarity, 

which could in measurable ways alter the acoustic response of 

the blades. Nevertheless, the basic trends seem to be 

reasonably well predicted using the combination of CFD and 

acoustic analogy (specifically, in this case, FINE/Turbo and 

LINPROP). This is enough of a capability for analysis and 

optimization ultimately aimed at developing low-noise open 

rotor designs. It may also be possible to modify the existing 

noise models (such as that underlying the LINPROP code, for 

example) to accommodate small, randomized variations in the 

geometry and loading distributions on a blade-to-blade basis 

to mimic the real physics. The basic question here is whether 

the size of required variations to get more realistic tone spectra 

would turn out to be consistent with the expected variations in 

the actual geometries. It is the aim of a future study to 

investigate this approach and its consequences.
NASA/TM—2012-217740 11

Appendix A—Acoustic Formulas

Taking advantage of the tonal nature of the sound field, one 

can write for the front rotor

( ) ( )

( )

( )

1

1

,

2 21

,

 

1 /

k

i t

i t

k

p xt p x e

p xe

kB

∞ − Ω

=−∞

∞ ∞ − Ω′

=−∞ =−∞

′ ′ = +

′

′ = + +Ω Ω

∑

∑ ∑













 



 (A.1a)

where p′

 and ,k p′

 are the thickness and loading harmonic 

amplitudes. Based on Equation (1a), these are given by

( )

( ) 1

1

1

1

2 /

0

0

0

2

ˆ ˆ i t n

S

p x

D G e v ds y d dt Dt

π Ω ∞

Ω

−∞

Ω ′ = × π

 

  ρ τ

 

∫ ∫∫







(A.1b)

( )

( )

,

1

1

1

1

2 /

0

2

ˆ ˆ

k

i t

k j

j S

p x

G e p n ds y d dt x

π Ω ∞

′Ω

−∞

Ω ′ =− × π

  ∂  τ

∂  

∫ ∫∫







(A.1c)

Note that thickness noise involves only the rotational 

frequency of the front rotor since the surface (i.e., source)

velocity ˆn v is not in any way influenced by the aft rotor. In 

contrast, loading noise depends not only on the front rotor 

rotational rate, but it is also affected by the rate at which the 

aft rotor blades pass by a given blade on the front rotor. In 

other words, the most general frequency of noise produced by 

the loading source depends on the combinations 

( ) 1 21 2 Ω + Ω +Ω kB . In the absence of the aft rotor, or when 

considering only the mean loading on the aft rotor, 

(i.e., k = 0 ), the loading noise frequency reduces to the 

harmonics of the front rotor rotational frequency as expected.

In writing Equations (A.1b) and (A.1c), j f in Equation (1a) 

has been replaced with its inviscid part j pn , and the source 

coordinates have been switched from the stationary frame of 

reference to the ones in a rotating frame fixed to the rotor, i.e.,

y y → ˆ . The integration over the time dependant rotor surface

1S ( ) τ has been correspondingly replaced with integration 

over the time-independent rotor surface 1S , which denotes the 

aggregate surface of all the blades in the rotor disc. The 

transformation is length preserving and hence its Jacobian is 

unity. The source amplitudes ˆnv and ˆk p are also expressed in 

the blade fixed coordinate system.

Finally, the derivatives of the Green’s function with respect 

to the source coordinate in Equation (1a) have been replaced

with those with respect to the observer coordinates via

0 0 / /

/ / i i

D G D D G Dt

Gy Gx

τ = −

∂ ∂ = −∂ ∂

(A.2)

Assuming that all the blades in the rotor disc are identical, 

the integration over the rotor surface 1 S can then be reduced 

to one over the surface of the reference blade B1 S , since the 

contribution to the harmonic amplitude p′

 from the th q rotor 

blade is related to that from the reference (i.e., 0th) blade 

through the spatial phase shift 2 / π 1 iq B e . Therefore, 

Equation (A.1b) can be rewritten as

( )

1

1

1

1

1

1

1 2 /

0

2 /

0

0

0

2

ˆ ˆ

B

B

iqB

q

i t n

S

p e

D G

e v ds y d dt Dt

−

π

=

π Ω ∞

Ω

−∞

Ω ′ = × π

 

  ρ τ

 

∑

∫ ∫∫









(A.3)

The summation over the blade index q is non-zero only when 

 is an integer multiple of the blade count B1 . When = mB1 

(m being an arbitrary integer), the summation is equal to B1

times the contribution from the reference blade. Thus, 

Equation (A.3) reduces to

( )

( ) 1

1 1

1

1 1

2 /

0

0

0

2

ˆ ˆ

m

B

imB t n

S

B

p x

D G

e v ds y d dt Dt

π Ω ∞

Ω

−∞

Ω ′ = × π

 

  ρ τ

 

∫ ∫∫



(A.4)

which shows that thickness noise is produced at the harmonics 

of the blade passing frequency B1 1 Ω . 

The blade pressure loading ˆk p has, in general, the form 

( ) ( ) ˆ 21 2 ˆ ikB

k k p pe

− Ω +Ω τ

τ =  (A.5)

since, due to the influence of the adjacent blade row, the blade 

loading is always unsteady even in a rotor-fixed coordinate 

system.

Next, note that the loading distribution on the th

q blade at 

any time is related to the loading distribution on the reference 
NASA/TM—2012-217740 12

blade via the spatial phase shift 1 2 / iq B e π , which accounts for 

the location shift, and the temporal phase shift ( ) 11 2 2 / iq B e

π Ω +Ω , 

which accounts for the time lead. Therefore, Equation (A.1c) 

can be rewritten as an integral over the reference blade surface 

as follows,

( ) ( )

( ) ( )

1 1

2 1 1

,

21 2

1

1 2 /

1 2 /

0 0

2

ˆ ˆ

k

B

B

iq kB B i t

q

ikB

k j

j S

px e e

G

p e n ds y d dt

x

− π Ω

π + ′Ω

=

∞

− Ω +Ω τ

−∞

Ω ′ = − π

  ∂   × τ

∂  

∑ ∫

∫ ∫



 





(A.6)

where, as before, the summation over the blade index q is nonzero only when 2  + kB is an integer multiple of the blade 

count B1 . When 2 1  + = kB mB , the summation is equal to B1

times the contribution from the reference blade. Thus, 

Equation (A.6) simplifies to

( ) ( )

( ) ( )

1

11 2 2

,

21 2

1

2 /

1 1

0

2

ˆ ˆ

m k

B

i mB kB t

ikB

k j

j S

B

px e

G

p e n ds y d dt

x

π Ω

Ω+ Ω

∞

− Ω +Ω τ

−∞

− Ω ′ = × π

  ∂  τ

∂  

∫

∫ ∫ 



(A.7) 

which shows that, in general, loading noise is produced at the 

frequency combinations mB kB 11 2 2 Ω+ Ω . 

Next, the chain rule can be used to show that

( )

( )

0

0

2

0 00 2

1

1 

j

R j

j

j

j jc

j

G G x

e gM

c t

e MM

R

∂ = ∂

 ∂

=− − +  κ ∂ 

 β +  κ 



 (A.8)

which, in turn, can be used to remove the spatial derivatives in 

Equations (A.4) and (A.7). Afterwards, the integrations over t

can be easily carried out since they involve Dirac delta 

functions. The results are

( )

( ) ( )

1

1

11 0

2 / ( ) 2

1 1

2

0 1

/

 8

ˆ

m

m

B

c

n

n

S n

imB g R c

B

p x

R

e ds y d

π Ω

=

Ω τ+

  Ω ′ =   × π    

τ

∫ ∫ ∑ 





(A.9a)

( )

( )( ) ( ) ( )

1

,

,

1

11 2 2 0 2 1 2

2 / ( ) 2

1 1

2

0 1

/

 8

ˆ

 

m k

m k

B

c

n

n

S n

i mB kB g R c ikB

B

p x

R

e e ds y d

π Ω

=

Ω + Ω τ+ − Ω +Ω τ

  Ω   ′ = × π    

τ

∫ ∫ ∑ 





(A.9b)

where (1)

m

 and (2)

m

 are the farfield and nearfield thickness 

source amplitudes, and ,

(1)

m k

 and ,

(2)

m k

 are the farfield and 

nearfield loading source amplitudes. These are given by 

Equation (A.9c).

Note that, the limits of integration over τ in

Equation (A.9a) and (A.9b) have been set to 1 (0,2 / ) π Ω , 

since one period of the observer time maps exactly into one 

period of the source time. In what follows it will be assumed 

that the mean flow is aligned with the open rotor axis, i.e.,

10 ( ,0,0) M . 

( )

( )

( )

( )

( )

,

,

(1)

1 10 2

0 0

(2) 2

0 00 00 3

(1)

11 2 2 2

0

0

(2) 2

0 00 3

1 ˆ

1 1

1 ˆ

1

ˆ 

1 ˆ

m

j j

m j R j

m k

j

m k R j

n

j c

n j

kj j c

kj j

imB v

M e gM

vM e M M

i mB kB

c

pn e gM

pn e M M

=− Ωρ ×

κ

  − −     κ

=− ρ β +

κ

=− Ω + Ω ×

κ

−

= β+

κ





















(A.9c)
NASA/TM—2012-217740 13

Grouping terms together and introducing a change of 

variable from τ to a new variable θ , both the thickness and 

loading terms can be cast in the form given by

( ) ( ) ( )

( ) ( )

( )

( )

1

2

1

2

1

2

0

ˆ ˆ 8 B

i n

n S

n

n n

B

p x e y ds y

e d

R

− λΨ

=

π

λΦ θ

′ = π

θ = θ

θ

∑ ∫

∫



 

(A.10a)

( ) ( )

( )

1

1

1 1 222

0

1 cos

2 1 , ˆ

s

s s

s s s

s s

ia b

rr b x y r r

θ=Ω τ+φ −φ

Φ θ = θ+ − θ

= χ= −

χ+ + β

(A.10b)

where for thickness noise5

1

1

1 1

1

1

1

1

tip 222

0

tip 0

0

1 tip

tip

0

1

s ss

s s

mB

M

a r r

M M

R

M

c

λ =

= χ+ + β

Ψ = χ +φ −φ β

Ω=

(A.10c)

and for the loading noise 

1

1

1 1

1

1 2

CRF tip 222

0

CRF

tip 0

0

1 22 1

CRF

1 2

/

s ss

s s

mB kB

M a r r

M M

mB kB

mB kB

λ= −

η

= χ+ + β

η Ψ = χ +φ −φ β

+ ΩΩ

η = −

(A.10d)

Here ηCRF is a factor denoting the influence of the counterrotation kinematics on the loading noise. Note that for k = 0 , 

i.e., when considering the influence of only the mean loading 

from the aft rotor on the noise of the front rotor, Equations

(A.10c) and (A.10d) have identical forms, which is to say that 

the thickness and loading noise components have the same 

frequency content. This result is in agreement with the isolated 

rotor results developed in Reference 7. In general, however,

loading noise for a given rotor is strongly influenced by its 

neighbor. It should be noted that in circumstances for 

 

5 The definition of the parameter as as given in the Reference 7 is 

incorrect.

which mB1 = kB2 , the loading term parameters have the 

following simpler forms 

( )

1

1

1 1

1

1

CRF tip 222

0

CRF

tip 0

0

2

CRF

1

1 cos

1 

s s

s ss

s

ia b

mB

M

a r r

M M

Φθ= − θ

λ =

η

= χ+ + β

η Ψ = χ β

Ω

η =+ Ω

(A.10e)

As shown in Reference 7, the integral n in

Equation (A.10a) can be accurately estimated using the 

method of steepest descent when the parameter λ is large 

which is the generally the case due to the large blade counts of 

modern open rotors. The essential steps are twofold: (1) 

conformably map the phase function Φ into a suitably 

constructed polynomial which facilitates the analysis 

immensely; (2) extend the domain of analysis to the complex 

domain through the change of variable ν=θ+ σi which 

allows the path of integration to be deformed into appropriate 

steepest descent paths. In the neighbourhoods of the saddle 

points of these paths, the integral n can then be accurately 

estimated using closed form analytical expressions. The 

intermediate steps are described in detail in Reference 7. The 

final result for a generic source term is given by

( )

( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

( )

2/3 2

0, 1/3

2/3 2

1, 2/3

0,

1,

3

( )

2 2

2

 

2

2

1

2

3

4

n n

n

n n n

n n n

n

n n

Ai

ie d

Ai

d

d

d

d

R d

d

d

λμ

+ −

+ −

 λ γ  π + 

 λ 

′ λ γ 



 λ 



Γ γ + Γ −γ =

Γ γ − Γ −γ = γ

μ= Φ ν +Φ ν    

  γ = Φ ν −Φ ν  

    ν ζ ν Γ ζ=   ν ζ ζ  

ν γ −ζ = ζ Φ νζ ′   

 



(A.11)
NASA/TM—2012-217740 14

where ± ν denote the location of the saddle points in the 

complex plane and , , and μγ Γn are parameters that define 

the conformal map. 0, 1, , and n n d d are the source amplitudes at 

the saddle points. Note that the actual values of these 

parameters depend on whether thickness noise or loading 

noise is being considered. 

The Airy function Ai and its derivative Ai′ provide an 

accurate asymptotic approximation to the integral n . The 

approximation is uniformly valid at subsonic, transonic, and 

supersonic tip speeds. It is also uniformly valid whether the 

observer is in the nearfield or farfield. The approximation is

formally accurate for large values of the parameter λ , but it 

turns out to be surprisingly accurate even for values of λ as low 

as 2.

With the integral n computed, the integration over the 

blade surface B1 S in Equation (A.10a) can now be carried out 

to compute the harmonic amplitudes of the thickness and 

loading noise sources. For general blade shapes, this integrals 

is not tractable analytically, but can be easily computed using 

a quadrature scheme by dividing the blade surface into small 

elements over which the integral can be accurately estimated

(see Ref. 7). For the acoustic results presented in this paper, 

the surface grid generated as part of the FINE/Turbo

simulations was used in the surface quadrature calculations.

References

1. Hager, R., D., and Vrable, D., 1988, “Advanced Turboprop 

Project,” NASA SP-495.

2. Van Zante, D., E., Gazzaniga, J., A., Elliott, D., M., and 

Woodward, R., P., 2011, “An Open Rotor Test Case: 

F31/A31 Historical Baseline Blade Set,” 20th ISABE 

Conference, Gothenburg, Sweden, paper no. ISABE-2011-

1310.

3. Farassat, F., Dunn, M., H., Tinetti, A. F., and Nark, D., M., 

2009, “Open Rotor Noise Prediction Methods at NASA 

Langley–A Technology Review,” AIAA Paper 2009-3133, 

15th AIAA/CEAS Aeroacoustics Conference, Miami, Florida. 

4. Brès, G., A., Brentner, K., S., Perez, G., and Jones, H., E., 

2004, “Maneuvering Rotorcraft Noise Prediction,” J. of 

Sound and Vibration, Vol. 275, Issues 3-5, pp. 719-738. 

5. Hanson, D., B., 1985, “Noise of Counter-Rotation 

Propellers,” Journal of Aircraft, Vol. 22, No. 7, pp. 609-617.

6. Parry, A., B., 1988, “Theoretical Prediction of CounterRotating Propeller Noise,” Ph.D. Dissertation, Dept. of 

Applied Mathematical Studies, University of Leeds. 

7. Envia, E., 1994, “Asymptotic Theory of Supersonic 

Propeller Noise,” AIAA Journal, Vol. 32, No. 2, pp. 239-246.

8. He, L. and Ning, W., 1998, “Efficient Approach for Analysis 

of Unsteady Viscous Flows in Turbomachines,” AIAA 

Journal, Vol. 36, No. 11, pp. 2005-2011.

9. Van Zante, D., E., Private communications.

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Open Rotor Aeroacoustic Modelling 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

Owing to their inherent fuel efficiency, there is renewed interest in developing open rotor propulsion systems that are both efficient and 

quiet. The major contributor to the overall noise of an open rotor system is the propulsor noise, which is produced as a result of the 

interaction of the airstream with the counter-rotating blades. As such, robust aeroacoustic prediction methods are an essential ingredient in 

any approach to designing low-noise open rotor systems. To that end, an effort has been underway at NASA to assess current open rotor 

noise prediction tools and develop new capabilities. Under this effort, high-fidelity aerodynamic simulations of a benchmark open rotor 

blade set were carried out and used to make noise predictions via existing NASA open rotor noise prediction codes. The results have been 

compared with the aerodynamic and acoustic data that were acquired for this benchmark open rotor blade set. The emphasis of this paper is 

on providing a summary of recent results from a NASA Glenn effort to validate an in-house open noise prediction code called LINPROP 

which is based on a high-blade-count asymptotic approximation to the Ffowcs-Williams Hawkings Equation. The results suggest that while 

predicting the absolute levels may be difficult, the noise trends are reasonably well predicted by this approach. 

15. SUBJECT TERMS

Aerodynamic noise; Propeller noise; Noise modeling 

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