Stewart Glegg

Florida Atlantic University, Boca Raton, Florida

Noise Generation by Fans With

Supersonic Tip Speeds

NASA/CR—2003-212098

March 2003
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Stewart Glegg

Florida Atlantic University, Boca Raton, Florida

Noise Generation by Fans With

Supersonic Tip Speeds

NASA/CR—2003-212098

March 2003

National Aeronautics and

Space Administration

Glenn Research Center

Prepared under Grant NAG3–2702
Available from

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7121 Standard Drive

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This report contains preliminary

findings, subject to revision as

analysis proceeds.

The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work.

Available electronically at http://gltrs.grc.nasa.gov
NASA/CR—2003-212098 1 

Noise Generation by Fans With Supersonic Tip Speeds 

Stewart Glegg 

Center for Acoustic and Vibration 

Florida Atlantic University 

Boca Raton, Florida 33431 

Introduction 

Fan noise continues to be a significant issue for commercial aircraft engines and there still exists 

a requirement for improved understanding of the fundamental issues associated with fan noise 

source mechanisms. At the present time, most of the prediction methods identify the dominant 

acoustic sources to be associated with the stator vanes or blade trailing edges which are 

downstream of the fan face (Morin 1999). However recent studies (Hanson 2002) have shown 

that acoustic waves are significantly attenuated as they propagate upstream through a rotor, and 

if the appropriate corrections are applied, sound radiation from the engine inlet is significantly 

underpredicted. The prediction models can only be applied to fans with subsonic tip speeds. In 

contrast, most aircraft engines have fan tip speeds which are transonic and this implies an even 

higher attenuation for upstream propagating acoustic waves. Consequently understanding how 

sound propagates upstream through the fan is an important, and not well understood phenomena. 

The objective of this study is to provide improved insight into the upstream propagation effects 

through a rotor which are relevant to full scale engines. 

The focus of this study is on broadband fan noise generated by boundary layer turbulence 

interacting with the trailing edges of the fan blades. If this source mechanism is important 

upstream of the fan, the sound must propagate upstream through a transonic non uniform flow 

which includes large gradients and non linearities. Developing acoustic propagation models in 

this type of flow is challenging and currently limited to low frequency applications, where the 

frequency is of the same order as the blade passing frequency of the fan. For trailing edge noise, 

much higher frequencies are relevant and so a suitable approach needs to be developed, which is 

not limited by an unacceptably large computational effort. In this study we are in the process of 

developing a computational method which applies for the high frequencies of interest, and allows 

for any type of flow field associated with the fan. In this progress report the approach to be used 

and the basic equations will be presented. Some initial results will be given, but these are 

preliminary and need further verification. 

Approach 

As was described in the proposal for this study, the most suitable formulation for the study of 

trailing edge noise is the wave equation developed by Howe, which uses the stagnation enthalpy 

B as the dependent variable and specifies a dipole source term determined by the local vorticity 

ω , the temperature T and entropy S. The equation is given as 

D

Dt

1

c

2

DB

Dt



 

  

 − 1

ρ

∇(ρ∇B) = ∇(ω×u −T∇S) (1) 
NASA/CR—2003-212098 2 

In this equation the density ρ and speed of sound c vary spatially and the total time derivative is 

linearized about the mean flow so that

D

Dt ≡ ∂

∂t

+ U.∇

In order to develop a general high frequency theory, we will consider the homogeneous form of 

this equation and introduce the source terms at the boundaries or the domain. Hence we define a 

region where a function f>0 in which the right hand side of (1) is zero so that 

D

Dt

1

c

2

DB

Dt



 

  

 − 1

ρ

∇(ρ∇B) = 0 f > 0

where

B = Bq f = 0

 (2) 

We will seek a solution in the high frequency limit where ko=ω/co becomes very large and co is 

the speed of sound in a region of uniform flow far upstream of the region of interest. If we 

consider a solution in the form 

B = Boe

−iωt+ikoq(x,y)

 (3) 

Substituting (3) into (2) we obtain 

ko

2 ∇q.∇q − n

2(1−M.∇q)2 [ ]− iko ∇2

q + M.∇ n

2 ( ) (1−M.∇q) +

∇ρ.∇q

ρ

 

  

 

  

      Boe−iωt+ikoq(x,y)

 = 0 

 (4) 

In the high frequency limit the second term can be ignored so we obtain a single equation for q

which gives the non linear equation 

∇q.∇q − n

2(1−M.∇q)2 = 0 (5) 

However in general both terms in (4) need to be considered giving the equation for q as 

∇q.∇q − n

2(1−M.∇q)2 [ ]− i

ko

∇2q + M.∇ n

2 ( (1− M.∇q))+

∇ρ.∇q

ρ

 

  

 

  = 0 (6) 

Simple Examples 

To understand this equation first consider examples where there is no flow and the mean density 

is constant so (6) reduces to 

∇q.∇q = n

2 +

i

ko

∇2q (7) 
NASA/CR—2003-212098 3 

For cylindrical waves this reduces to 

∂q

∂R

 

 

 

 

2

+

1

R2

∂q

∂θ

 

 

 

 

2

= n

2 +

i

ko

1

R

∂

∂R

R ∂q

∂R

 

 

 

 +

1

R2

∂2q

∂θ2

 

 

  

  



 

  

  (8) 

If the boundary condition is independent of θ and the speed of sound is constant then 

∂q

∂R

= ± 1+

i

ko

1

R

∂

∂R

R

∂q

∂R

 

 

 

 

 

 

  

 

 

  

 

  

1/2

 (9) 

In the high frequency limit •q/•R~1 and so to first order in 1/ko we have 

∂q

∂R

≈ ± 1±

i

ko

1

R

∂

∂R( ) R  

 

 

 

 

  

 

  

1/2

≈ ± 1±

i

2koR

 

  

 

  (10) 

and integration yields 

q ≈ ±R +

i

2ko

loge(koR) (11) 

Hence we find as expected that B=exp(±ikoR)/(koR)1/2 but we note that the effect of cylindrical 

spreading is determined by the Laplacian in equation (6). 

Another example is given by a dipole source which can be modeled by a boundary condition 

which depends on cosθ. Then the full form of (8) must be used, but if n is constant we find

R2 ∂q

∂R

 

 

 

 

2

− i

ko

R

∂

∂R

R

∂q

∂R

 

 

 

 

 

 

  

 − R2 = − ∂q

∂θ

 

 

 

 

2

+

i

ko

∂2q

∂θ2 (12) 

The boundary condition implies that q=f(R)+g(θ) where g=1/(iko) log(cosθ). It follows that

R2 ∂f

∂R

 

 

 

 

2

− i

ko

R ∂

∂R

R ∂f

∂R

 

 

 

 

 

 

  

 − R2 = 1

k o

2 ( ) tanθ 2 − 1

k o

2 ( ) secθ 2 = −1

k o

2 (13) 

and so 

∂f

∂R = 1− 1

(koR)

2 +

i

koR

∂

∂R

R ∂f

∂R

 

 

 

 

 

 

  

 

 

  

 

  

1/2

≈ 1+

i

koR

 

  

 

  

1/2

 (14) 
NASA/CR—2003-212098 4 

giving as before the result that 

q ≈ ±R +

i

2ko

loge(koR) − i

ko

loge(cosθ) (15) 

Cartesian Coordinates 

The next step in the analysis is to consider numerical solutions in cartesian coordinates. In the 

absence of a flow we need to solve 

∂q

∂y = n

2 - ∂q

∂x

 

 

 

 

2

+

i

ko

∂2q

∂x

2 +

∂2q

∂y

2

 

 

  

  

 

  

 

  

1/2

 (16) 

When the flow is primarily in the x direction, we can specify equations suitable for numerical 

integration as 

∂q

∂y = n

2(1− Mx

∂q

∂x

 

 

 

 )

2 - ∂q

∂x

 

 

 

 

2

+

1

iko

(1− n

2Mx

2)

∂2q

∂x

2 +

∂2q

∂y

2

 

 

  

  

 

  

 

  

1/2

 (17) 

Extensions to More Complex Environments 

Three dimensional problems are also readily solved by adding the term (∂q/∂z)2 to equation (5) 

and specifying the boundary condition on the surface y=constant as a function of x and z. For 

problems in cylindrical coordinates (r,θ,z) the natural extension of the above approach is to 

specify the boundary conditions on r=const and consider radial propagation. For ducted flow 

however it may be more suitable to specify the boundary conditions on a cross sectional surface 

z=const and propagate waves up or downstream. This is equivalent to marching the solution in 

the x direction using (6) with boundary conditions specified on x=const. 

Trailing edge noise can be calculated using this approach by specifying the appropriate source 

distribution in the vicinity of the trailing edge of the blade. This source distribution is well 

known and specified in the literature (Howe, 1978) for a spatially harmonic disturbance in 

uniform flow. The most convenient form for the trailing edge noise source distribution is given 

by the Greens function for a source close to a sharp edge in a uniform flow of Mach number M, 

which is given by 

G = −i

8π n=0

∞

∑ ε n(Hn / 2

(1) (kor'/β)Jn / 2(kor

o'/β)H(r'−ro')

 + Hn /2

(1) (koro'/β)Jn / 2(kor'/β)H(ro'−r'))cos( n(θ −π)

2 ))eikoM (x− xo) /β2

 

where r’=(x2

/β

2

+y2)

1/2 and the subscript o refers to the observer location. Note that εn=1, n=0 

and εn=2, n>0. Also β

2 =1-M2

. 
NASA/CR—2003-212098 5 

Preliminary Numerical Results 

Figure 1 shows calculations for the source distribution defined by a Gaussian distribution of 

pressure on the x axis and illustrates the propagation of a Gaussian beam. This shows that the 

procedure can be used successfully to propagate sound waves over many wave lengths. Similar 

examples have also been computed for monopole and dipole source fields. In general the 

procedure works well until the value of the Laplacian in equation (16) or (17) becomes negative. 

The correct choice of the branch cut is crucial to the numerical procedure and work continues to 

fully understand this issue. 

Figure 1: Sound propagation of a Gaussian Beam obtained numerically. 
NASA/CR—2003-212098 6 

Computation of the Flow 

To obtain the flow through which the sound will propagate, we need to develop a computational 

method which can be used specifically for the problem being considered. The computation must 

provide the flow over a fan blade when the free stream Mach number is between M=0.7 and 

M=1.4. To achieve this we will use transonic small disturbance theory which is based on the 

following assumptions: 

1. The airfoil is thin, with a small camber and a small angle of attack 

2. The horizontal and vertical components of the velocity field are not very different from 

their free stream value 

The transonic small disturbance equation which will be solved is: 

1− M∞

2 −( ) γ +1 M∞2 φx

V∞



 

  



 

  φ xx + φyy = 0 

where φ xx denotes the double derivative of the disturbance velocity potential and M∞represents 

the free stream Mach number. 

The solution of this nonlinear equation is subject to boundary conditions at the solid boundary of 

the airfoil, along a cut downstream of the airfoil trailing edge, and at the far field boundaries. On 

the airfoil, the disturbance velocity potential must satisfy the following condition: 

φ y =V∞ 1+ φ x

V∞



 

  



 

  ⋅

dY

dx

For small disturbance, we can neglect the term φx

V∞

 and use the simpler condition: 

φ y =V∞

dY

dx

If the airfoil is producing lift, the circulation around the body will not be zero. Far field boundary 

conditions are also important and for the subsonic free stream case, we use the analytical result: 

φ = Γ

2π

arctan 1− M∞

2 y

x



    

For the supersonic free stream case, the lateral boundaries are computed with the method of 

characteristics, the disturbance velocity potential at the upstream boundary is set to zero 

(no perturbation can radiate upstream), the downstream boundary is computed as any other point 

in the flow. 
NASA/CR—2003-212098 7 

To limit the computation time and still obtain good accuracy, we use a stretched grid to define 

the computational domain. The mesh is fine where the velocity potential gradient is important 

(near the airfoil) and becomes larger as we get further from the body. This allows having 

accurate results with a matrix 100*120 instead of a matrix 1380*400 (this divides the 

computation time by 46). 

This approach has been implemented and an example of the computation of with a free stream of 

Mach number M=0.99 over a double-cosine-shape airfoil is shown in Figure 2. 

Figure 2: Computation of the flow with a free stream of 

Mach number M=0.99 over a double-cosine-shape airfoil. 

Future Work 

The next steps in this study are 

1) Introduce the trailing edge noise source distribution to the acoustic computations, and 

study the effect of flow on the radiated sound. 

2) Introduce the flow computed by transonic small disturbance theory to the acoustic 

computations and study the effect of more realistic shocks on the acoustic computation. 

3) Investigate the effects of shocks on trailing edge noise in two dimensional flow. 

4) Introduce a cascade of blades and repeat the above study. 

5) Extend the work to three dimensional flows in circular ducts. 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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13

Noise Generation by Fans With Supersonic Tip Speeds

Stewart Glegg

Fans; Aerodynamic noise; Supersonic; Broadband

Unclassified -Unlimited

Subject Category: 02 Distribution: Nonstandard

Florida Atlantic University

Center for Acoustic and Vibration

Boca Raton, Florida 33431

Project Manager, Edmane Envia, Structures and Acoustics Division, NASA Glenn Research Center, organization

code 5940, 216–433–8956.

The focus of this study is on broadband fan noise generated by boundary layer turbulence interacting with the trailing

edges of the fan blades. If this source mechanism is important upstream of the fan, the sound must propagate upstream

through a transonic non uniform flow which includes large gradients and non linearities. Developing acoustic propagation

models in this type of flow is challenging and currently limited to low frequency applications, where the frequency is of

the same order as the blade passing frequency of the fan. For trailing edge noise, much higher frequencies are relevant

and so a suitable approach needs to be developed, which is not limited by an unacceptably large computational effort. In

this study we are in the process of developing a computational method which applies for the high frequencies of interest,

and allows for any type of flow field associated with the fan. In this progress report the approach to be used and the basic

equations will be presented. Some initial results will be given, but these are preliminary and need further verification.