Abbas Khavaran

ASRC Aerospace, Cleveland, Ohio

Milo D. Dahl

Glenn Research Center, Cleveland, Ohio

Acoustic Investigation of Jet Mixing Noise 

in Dual Stream Nozzles

NASA/TM—2012-217226

February 2012

AIAA–2011–2701
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Abbas Khavaran

ASRC Aerospace, Cleveland, Ohio

Milo D. Dahl

Glenn Research Center, Cleveland, Ohio

Acoustic Investigation of Jet Mixing Noise 

in Dual Stream Nozzles

NASA/TM—2012-217226

February 2012

AIAA–2011–2701

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

17th Aeroacoustics Conference

cosponsored by the American Institute of Aeronautics and Astronautics and 

the Confederation of European Aerospace Societies

Portland, Oregon, June 5–8, 2011
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This work was sponsored by the Fundamental Aeronautics Program 

at the NASA Glenn Research Center.

Level of Review: This material has been technically reviewed by technical management. 

This report is a formal draft or working 

paper, intended to solicit comments and 

ideas from a technical peer group.
NASA/TM—2012-217226 1

Acoustic Investigation of Jet Mixing Noise in Dual Stream Nozzles 

Abbas Khavaran 

ASRC Aerospace Corporation 

Cleveland, Ohio 44135 

Milo D. Dahl 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

In an earlier study, a prediction model for jet noise in dual stream jets was proposed that is founded 

on velocity scaling laws in single stream jets and similarity features of the mean velocity and turbulent 

kinetic energy in dual stream flows. The model forms a composite spectrum from four component singlestream jets each believed to represent noise-generation from a distinct region in the actual flow. While the 

methodology worked effectively at conditions considered earlier, recent examination of acoustic data at 

some unconventional conditions indicate that further improvements are necessary in order to expand the 

range of applicability of the model. The present work demonstrates how these predictions compare with 

experimental data gathered by NASA and industry for the purpose of examining the aerodynamic and 

acoustic performance of such nozzles for a wide range of core and fan stream conditions. Of particular 

interest are jets with inverted velocity and temperature profiles and the appearance of a second spectral 

peak at small aft angles to the jet under such conditions. It is shown that a four-component spectrum 

succeeds in modeling the second peak when the aft angle refraction effects are properly incorporated into 

the model. A tradeoff of noise emission takes place between two turbulent regions identified as transition 

and fully mixed regions as the fan stream velocity exceeds that of the core stream. The effect of nozzle 

discharge coefficients will also be discussed. 

Nomenclature 

A Jet exit area 

B Intercept parameter 

c Sound speed 

D Jet diameter 

Di Inner diameter in the secondary jet 

De Effective diameter 

Dˆ Modified diameter 

cp Specific heat at constant pressure 

f , f * Frequency, characteristic frequency 

mg Mass flow rate

M Mach number 

n Velocity exponent 

T, T o Static or stagnation temperatures 

U Mean axial velocity 

 Amplitude adjustment factor 

δ Diameter ratio (/ ) Di p D
NASA/TM—2012-217226 2

γ Specific heat ratio 

λA, λV Area and velocity ratio—secondary divided by primary 

λT Static temperature ratio—primary divided by secondary 

θ Polar angle from inlet axis 

ρ Density 

ξ Normalized frequency ( f / f *

) 

Subscripts 

p Primary jet (core) 

s Secondary jet (bypass) 

e Transition jet 

m Fully mixed jet 

 Ambient conditions 

Superscripts 

(p) Primary jet condition 

(s) Secondary jet condition 

(m) Mixed jet condition 

1.0 Introduction 

In an attempt to develop a robust and effective noise prediction methodology for turbofan engines, 

semi-empirical models have been proposed in a number of previous studies that address both jet mixing 

noise and the broadband shock associated noise under static and flight conditions. Such models use some 

form of physical scaling law, and regard the far field jet noise as a superposition of a number of 

uncorrelated spectral components, each with a distinctive spectral peak attributed to a particular source 

region in a jet. As expected, calibration of scaling laws relies heavily on far field acoustic measurements 

at a range of temperatures and pressures of interest for core and bypass streams. Details of the nozzle 

geometry, for example, noise suppression devices and installation effects such as center plugs and pylons, 

present additional parameters that impact noise. A justification for the utility of such predictions tools 

becomes evident in conjunction with multi-disciplinary design, analysis and optimization (MDAO) 

efforts that address aerodynamic performance, fuel efficiency, NOx emissions, and noise in the design 

and development of future commercial aircraft. High fidelity tools, applicable to each aspect of design in 

isolation, may not be computationally effective in an iterative fashion in an MDAO environment. Jet 

noise prediction models that use variants of the acoustic analogy theory are computationally intensive and 

require dedicated flow solvers to provide input for jet noise prediction in heated flows. 

An example of a semi-empirical prediction code in aeroacoustics is the Aircraft Noise Prediction 

Program (ANOPP) (Ref. 1). It is a conglomerate of a number of functional modules, each conducting 

noise prediction on a model calibrated to a particular component of propulsion or airframe noise. Jet 

related noise modules in ANOPP were primarily developed by Stone et al. (Refs. 2 and 3) and have 

evolved with time to respond to new parameters and engine operating conditions. Stone et al. used a 

so-called hybrid experimental/empirical model to construct a three-component jet noise spectrum. These 

three noise components are attributed to the primary (core), secondary (bypass), and the fully mixed flow 

regions. Each component deploys a general 8th power velocity exponent when jets are unheated. A 

density exponent correction factor, as proposed by Ahuja and Bushell (Ref. 4), lowers this velocity 

exponent to 5 and 6 under heated conditions. 
NASA/TM—2012-217226 3

Fisher et al. (Refs. 5 and 6) pioneered a jet noise scaling approach in dual stream flows from a fluid 

mechanics perspective. In the light of velocity and turbulence measurements, they divided a coaxial dual 

stream jet into several noise-generating segments based on similarity arguments. These noise-producing 

regions have some resemblance to those suggested by Stone, although their noise modeling details were 

different. They later enhanced the source model (Ref. 6) using a dipole-equivalent correction term that 

represented the temperature effect in the transition region of the dual stream jet. Tester and Fisher 

extended this model to coaxial jets with internal mixing, and showed that a two-source model, 

representing the transition and fully mixed regions, provided a good prediction in such flows. Using 

appropriate filters that reduced the low frequency amplitude in the mixed region, and enhanced the high 

frequency noise in the transition region, they were able to model the noise benefit/penalty in the forced 

mixers when the primary flow consisted of a lobed nozzle configuration (Refs. 7 to 9). These studies were 

limited in their scope and applicability of the velocity, area, and temperature ratio between the two 

streams, and did not examine unconventional velocity and temperature profiles that impact the peak 

amplitude as well as the high frequency noise at small aft angles that results from the span-wise inversion 

in mean velocity and/or temperature. 

Noise component modeling considered in the current study resembles those suggested by Fisher et al., 

however component spectra are evaluated using a more general rule commonly known as the velocity 

power laws of jet noise. Velocity power laws are defined in terms of two empirical parameters—velocity 

exponent n and intercept factor B. These parameters depend on jet temperature and observer angle and 

make it possible to scale the jet noise spectra in single stream jets from one set of operating conditions to 

the next. In order to determine parameters n and B for single stream jets, an extensive matrix of 

narrowband (5 Hz bandwidth) jet noise data acquired at the Small Hot Jet Acoustic Rig (SHJAR) 

(Refs. 10 and 11) at the NASA Glenn Research Center were examined in an earlier study (Ref. 12). A 

least-squares fit was implemented on the sound pressure level at each angle, and at exit velocity U using 

10 log(U/c∞) as the dependent variable. The slope n(θ,To

) and intercept B(θ,To) were subsequently 

evaluated at angle θ and stagnation temperature To

. The set of jet operating conditions selected at each 

interpolation point were screened carefully to achieve a good fit. The jet conditions, interpolation points, 

and the confidence bands for parameters n and B are discussed in Reference 12. Figure 1 in Reference 13 

shows the velocity scaling law parameters n and B at a range of angles and temperatures of common 

interest. 

The idea of extending single-stream jet noise scaling laws to dual stream coannular jets is based on a 

constructive superposition of four single stream jets (Ref. 13). Each single stream jet, herein referred to as 

a component, is intended to capture noise emission from a distinct turbulent region contributing to the far 

field noise. Jet mixing noise is evaluated when velocity U, temperature To

, and diameter D, for each of the 

four components, are supplied as input to a module that implements the scaling law. Additionally, 

spectral filters are designed to highlight the scales of turbulence associated with each noise component. 

Following the filtering process a superposition is carried out to form a composite spectrum. 

The dual stream jet noise prediction model developed in a previous work (Ref. 13) is briefly reviewed 

in Section 2. Sample calculations are presented using conventional (normal velocity and temperature 

profiles) as well as inverted velocity and temperature profiles. Arguments based on refraction phenomena 

are presented in Section 3 to explain the dual peak character in the spectrum at small aft angles under the 

latter condition. Modifications that help improve the predictions are presented in Section 4, followed by 

examination of the model in Section 5. A summary of the results and direction for future work is provided 

in Section 6. 

2.0 A Dual Stream Jet Noise Model 

Following the arguments presented in Reference 13, a dual stream jet may be divided into 4 zones 

within three regions as sketched in Figure 1. The initial mixing region consists of two zones that reflect 

the mixing between the primary and the secondary streams (shown in pink), and the mixing between the 

secondary and the ambient streams (shown in blue). A transition region where the two mixing layers 
NASA/TM—2012-217226 4

Figure 1.—Schematic of a dual stream jet showing mixing zones primary/bypass, 

bypass/ambient, the transition, and the fully mixed region.

merge, and subsequently a fully mixed region further downstream follow this initial mixing region. Noise 

generation from each of the four zones is addressed as a component (single stream) jet with three 

parameters U, To

 and D. Herein, the primary, secondary, and the fully mixed streams are denoted using 

subscripts p, s and m respectively. The one-dimensional conservation equations plus the equation of state 

(assuming a constant static pressure) are used to evaluate the fully mixed jet conditions as discussed in 

Reference 13. 

Using the ratios 

,, , o

o

ss s p p

VAT T o

p p sp s

U A T T

UAT T

      

 , (1)

the following are derived from the conservation equation 

2

2

2

2 2

2

( )

() ()

1 , 1

1 / , 1

(1 )( ) / ( ),

1 . 2

o

m V AT

p V AT

o

m V AT T

o

p V AT

mm m p

V AT

pp m p

o o

m m mm p p

o o

p m p pp p p

p

p

p m

p

U

U

T

T

AD U

AD U

T T TU U

T T c TT U

c

c

      

       

     





 



   

 

 (2) 

The specific heat ratios γ

(p)

 and γ

(s) in the primary and the secondary are evaluated at their respective 

temperatures, and are used in cp = γR/(γ–1) to evaluate the constant pressure specific heat cp in each 

stream. The mixed jet specific heat ratio is evaluated as ( ) ( ) () ( )/( ) mp s m m mm p s ps        , where m is the 

mass flow rate. The constant pressure specific heat in the mixed jet is evaluated as 
NASA/TM—2012-217226 5

() () () /( 1) mm m

p c R   (3) 

Similarity features of the mean axial velocity and turbulent kinetic energy in dual stream subsonic jets 

where examined by Ko and Kwan (Ref. 14). Using appropriate normalization factors defined by 

Abramovich (Ref. 15) and Eldred et al. (Ref. 16), they were able to identify flow characteristics similar to 

those in single stream jets within three zones. These consist of the primary/secondary, secondary/ambient, 

and the fully mixed zones. Accordingly, jet noise for each zone (prior to filtering) is evaluated when the 

respective velocity U, temperature T, and diameter D for that zone is supplied as a set of input parameters 

to a noise module that implements the scaling laws in single-stream jets. The fourth component, that is, 

the transition jet, lacks an intrinsic similarity, and is treated as a single stream jet at the primary conditions 

of Up and Tp, but with an effective diameter De per equal thrust to that of the dual stream jets, that is 

2

2

2 1 AT V

p

De

D     (4) 

Since the maximum turbulence intensity ratio u'/Up in the transition region is much less than the usual 

15 to 16 percent measured in the self-similar zones ( u' is the rms in velocity fluctuations), an appropriate 

attenuation factor was deployed within the filter to attenuate the noise accordingly. 

2.1 Spectral Filters 

Nozzle efflux diameters and propulsion inlet conditions for both core and fan streams are supplied to 

an empirical jet noise model (Ref. 13) that forms a composite spectrum for turbulent mixing noise in dual 

stream flows 

( /10) ( /10) ( /10) ( /10) /10 primary secondary fully-mixed transition 10 10 10 10 10 SPL SPL SPL SPL SPL

   (5) 

Here, SPL denotes the sound pressure level at a third-Octave center frequency, and the summation is 

carried out after each component noise has been treated with an appropriate high- or low-pass frequency 

filter. Two low-pass filters are defined as 

23 4

1

2 3

2

( ) (1 ) 25 125 250

( ) (1 ) 2 6

F e

F e





      

     

 (6) 

 where 

m m , 2 f

f     (7) 

The characteristic frequency f *

 depends on the velocity to diameter ratio in each component, and f is 

a third-Octave band center frequency. A high-pass filter is obtained when we subtract a low-pass filter 

from unity. Noise filtering for each component is briefly described here. 

 
NASA/TM—2012-217226 6

Figure 2.—Dual stream coaxial jets.

2.1.1 Fully Mixed Region 

This jet is defined as a single stream jet at Um, Tm and Dm as evaluated from the set of equations (2). 

The characteristic frequency f * in the fully mixed region additionally depends on diameter ratio δ = Di /Dp

(see Fig. 2). 

  *

10 1

1

( ), ( ) 10 log ( ) m

m

U f Filter f F D    (8) 

The above low-pass filter indicates that this component primarily contributes to the low- to midfrequency noise. 

2.1.2 Secondary Jet 

This jet is defined through the secondary jet parameters Us,Ts, and Tm, and is expected to contribute to 

the high frequency noise. The required high-pass filter should be compatible with that in the fully mixed 

region to accommodate the limiting case when the primary jet becomes relatively weak. In that case, the 

fully mixed jet and the secondary are identical and the two filters should recover a single jet at the 

secondary conditions. Parameter δ best fits the characteristic frequency in the secondary jet as 

  * 3

10 1 ( ), ( ) 10 log 1 ( ) s

s

U f Filter f F D     (9) 

2.1.3 Transition Jet 

In the absence of an intrinsic similarity within the region, this jet is relatively more ambiguous and 

requires a more delicate treatment. Under normal flow conditions it is best defined as single stream jet at 

Up,Tp and effective diameter De (see Eq. (3)). Due to a reduced turbulence level in the region (compared 

to the self-similar regions) the noise level is also attenuated by 60 percent—raised to some exponent. A 

proper exponent (7/2) is derived from physics-based jet noise modeling described by Khavaran et al. 

(Ref. 17) that scales jet mixing noise versus turbulence intensity. Additionally, a low-pass filter is used at 

an appropriate characteristic frequency 

 

 

3

* 7/2 2

10 10 3

2

/ (3 / ) , ( ) 10 log (0.60) 10 log ˆ / (3 / )

A

A

p

e

U F

f Filter f D F m

 

 



   

 

 (10) 

 

where m m ˆ   Min( , ) . The filter, denoted as the second term on the RHS of Equation (10), works more 

effectively as λA increases. Parameter λA in Equation (10) is replaced with its limits at λA > 3 or λA < 1 such 
NASA/TM—2012-217226 7

that 1.0 ≤ λA ≤ 3.0. When combined with the high frequency noise from the secondary jet, these two noise 

components should be required to match the total high frequency noise in a dual stream nozzle. 

2.1.4 Primary Jet 

This jet is defined through parameters Up, Tp, and Up, and is considered subject to a flight speed of 

U  Us . The required high-pass filter is bound by the limiting requirements at Us 0 . 

  *

10 1 / , ( ) 10 log 1 ( ) psp f Filter f F    UUD  (11) 

When the secondary jet is relatively insignificant, the fully mixed jet and the primary become 

identical. The two jets subject to the proposed filters recover the primary jet, as expected. 

In the next section, sample calculations are shown using the above model, and new modifications are 

presented that should help improve the predictions when the bypass stream is the higher velocity and/or 

temperature jet. 

3.0 Sample Predictions and New Improvements

Three coannular nozzles shown in Figure 3 were examined by Larson et al. (Ref. 18) as part of a 

study program directed toward acquisition of acoustic data and measurement of aerodynamic performance 

of jets with an inverted velocity profile (IVP). The geometries designated as A, B, and C use different 

diameter ratio (Di / Ds ), which is achieved by positioning the secondary throat at different mean radii. 

The area ratio of the two streams is λA = 1.43 when the throat area (we are interested in subsonic 

conditions) is used as the efflux area. Acoustic data is tabulated at a 15 ft arc using a rescaled geometry 

with a fan diameter of 5 in. Secondary outer diameters Ds , secondary inner diameter Di , and the primary 

diameter Dp are: 

Geometry A: 5.0, 3.45, 3.02 in.; Geometry B: 5.0, 3.75, 2.76 in.; Geometry C: 5.0, 4.15, 2.35 in. 

Spectra have also been corrected for atmospheric attenuation and are presented as lossless (i.e., 

atmospheric attenuation is removed from the measured data). 

Figure 3.—Coannular nozzle geometries A, B, and C. 

 
NASA/TM—2012-217226 8

Conditions for the core and fan streams used in the study of Larson et al. are listed in Table 1. The 

nozzle pressure ratio NPR and the stagnation temperature To

 may be used in the usual one-dimensional 

isentropic relations to evaluate the ideal exit velocity U and static temperature T, and subsequently 

velocity ratio λV and static temperature ratio λT. The correct value for the specific heat ratio (a function of 

static temperature) is found by iteration. 

A dual stream jet noise model labeled “dJet” has been developed that implements the velocity 

power laws in single stream jets, and forms a composite spectrum in a dual stream flow according to 

Equation (5). Figures 4, 5, and 6 show sample results for the three conditions of geometry A listed in 

Table 1. Component noise is color-coded as in Figure 1 to indicate noise emission from various source 

regions. The first condition, i.e., 8301, corresponds to normal profiles in both velocity and temperature 

(NVP, NTP). At 8302, the velocity is slightly inverted (IVP, NTP); and finally at 8305 both velocity and 

temperature are inverted (IVP, ITP). The predictions agree reasonably well with data for conditions 8301 

and 8302. However, the aft angle spectrum for condition 8305 does not agree with measurements. The 

data exhibits a dual peak spectrum (Fig. 6) that is not reflected in the predictions. The noise associated 

with the fully mixed region is over predicted and the noise associated with the secondary shear layer is 

under predicted. These observations point to the significance of refraction effects for the inverted profile 

conditions considered here. 

TABLE 1.—COANNULAR NOZZLE TEST CONDITIONS OF LARSON ET AL. (REF. 18) 

Run# Geometry Primary (Core) Secondary (Fan) A V T

NPR To – R U – fps NPR To – R U – fps 

8301 A 1.51 1464 1421 1.30 1271 1062 1.43 0.75 1.11 

8302 A 1.53 1465 1442 1.79 1275 1548 1.43 1.07 1.20 

8305 A 1.51 1454 1415 1.79 1980 1961 1.43 1.39 0.76 

8201 B 1.53 1529 1476 1.29 1325 1071 1.43 0.73 1.11 

8202 B 1.52 1535 1468 1.81 1327 1595 1.43 1.09 1.22 

8205 B 1.51 1510 1445 1.79 2034 1989 1.43 1.38 0.77 

8401 C 1.55 1437 1448 1.29 1246 1037 1.41 0.72 1.10 

8402 C 1.53 1434 1426 1.78 1249 1525 1.41 1.07 1.20 

8405 C 1.53 1390 1403 1.77 1936 1920 1.41 1.37 0.74 

Figure 4.—Prediction and data in a dual stream jet at 90° and 150° inlet angles—geometry A Table 1, 

Run#8301: A = 1.43, v = 0.75, T = 1.11. 
NASA/TM—2012-217226 9

Figure 5.—Prediction and data in a dual stream jet at 90o and 150o inlet angles—geometry A 

Table 1, Run#8302: A = 1.43, v = 1.07, T = 1.20. 

 

Figure 6.—Prediction and data in a dual stream jet at 90o and 150o inlet angles—geometry A 

Table 1, Run#8305: A = 1.43, v = 1.39, T = 0.76. 

To shed light on the effect of flow non-uniformities on the refraction of sound we consider a plane 

vortex sheet separating two uniform flows at conditions 1 and 2, respectively (Fig. 7). A sound wave with 

a wave normal at angle φ1 refracts at the interface to a new angle φ2. Angle φ2 depends on velocity as well 

as temperature (sound speed c) in each medium 

1 2

1 2

1 2 Sin Sin

c c

  V V

  . (12) 

Equation (12) is a kinematics statement that requires the wavelength along the interface be equal in 

both regions. Figure 8 shows the transmission angle φ2 as a function of the incidence angle φ1 when the 

high-velocity high-temperature medium is in region 1. Under these conditions, the transmission angle 

cannot exceed 41.8o

 (or 33o) when the Mach number M1 in region 1 is 0.50 (or 1.0), respectively. 

Subsequently, a zone of silence is formed in region 2 where no sound wave can penetrate. Conversely, 

 
NASA/TM—2012-217226 10

Figure 7.—Refraction of sound at a vortex sheet. 

when the high-velocity high-temperature medium is located in region 2, certain incidence angles φ1 fail to 

penetrate into region 2. For example, under the conditions of Figure 9, when M1 = 0.50 (or 1.0), waves 

with incidence angles exceeding some critical value φ1c = 37o

 (or 31.5o) do not reach into region 2. These 

conditions lead to a total reflection, and sound pressure in region 2 decays exponentially with distance 

from the boundary. 

Using the examples shown here, it is readily concluded that the high-frequency jet noise emitted from 

the initial mixing region (see Fig. 1) has a higher likelihood of reaching small aft angles in dual stream 

jets with IVP/ITP as compared to NVP/NTP. In fact, in the latter case, a cone of relative silence is formed 

near the downstream jet axis. In the fully mixed region, noise emitted at angles φ1 > φ1c under IVP/ITP 

conditions, goes through a total reflection and decays with distance from the jet; hence low frequency 

noise at small aft angles is expected to have a relatively lower amplitude compared to that in jets with 

NVP/NTP conditions. 

Considering these observations, several modifications are introduced in the model proposed earlier 

that adjust the amplitude as well as characteristics frequency of a number of noise components. These 

adjustments are necessary due to the fact that the scaling laws of jet noise in single stream jets were based 

on noise measurements under NVP/NTP conditions. 

Figure 9.—Refraction at an interface with V2/V1=1.50, 

T1/T2=0.50: (solid line) M1 = 0.50; (dashed line) M1 = 1.0. 

Figure 8.—Refraction at an interface with V2/V1=0.50, 

T1/T2=1.50: (solid line) M1 = 0.50; (dashed line) M1 = 1.0. 
NASA/TM—2012-217226 11

4.0 Model Modifications 

The parametric changes proposed here are intended to improve the dual stream jet noise prediction 

model of Section 3 under inverted velocity (λV > 1) and/or inverted temperature conditions (IVP/ITP). 

4.1 Secondary Jet 

The adjustments made in the secondary/ambient mixing region are intended to modify both noise 

amplitude and its characteristics frequency f *

. As stated before, this component noise is defined in terms 

of a single stream jet at the conditions of the secondary jet Us, Ts (here we use Ts as normalized with 

respect to the ambient temperature). The jet diameter Ds is now modified as Dˆ

s

0.5 / (2 ), 125

/ (2 ), 125 135 ˆ

/ (1.5 ), 135 145

( )/ , 145

A s

A s

s s

A s

A s

T

T

D D

T

F T

  

  

 

  

 

 (13) 

The adjustment factor at small aft angles θ > 145° becomes a function of the area ratio λA, shown as 

F(λA) in Figure 10. When λA < 0.50 this parameter is set equal to 0.35. 

The modifications in the filter are as follows: 

(a) The filter functions F1() in the secondary now consists of an additional parameter s T in the exponent and is 

denoted as 1

ˆ (, ) F Ts 

Figure 10.—Diameter adjustment factor for the fully 

mixed jet as defined in Equation (13). 

 
NASA/TM—2012-217226 12

Fˆ

1 (,Ts )  (1    

2

25

 

3

125

 

4

250 )e

 Ts (14)

where *   2 / f f as defined earlier. 

(b) The characteristic frequency is modified as 

*

0.25

3 0.25

3 0.75

, 130

, 130 140

, 140

s

s

s

s

s

U f D

T

T

T 



            

 (15) 

4.2 Fully Mixed Jet 

In jets with an inverted velocity profile (λV > 1), the fully mixed jet is defined by values of Um, Tm 

and Dˆ

m , where ˆ

D D m m   , and Dm is calculated from Equation (2). Additionally, an adjustment factor 

( )   Fmx V is designed to attenuate the amplitude of this noise component through its filter 

  10 1 ( ) 10 log ( ) Filter f F Fmx     (16) 

The attenuation in the fully mixed noise will be compensated by an increase in the amplitude of the 

transition noise through factor αTm (λV) as discussed shortly. The functional forms of these two parameters 

are shown in Figure 11. 

Figure 11.—Amplitude adjustment parameter in the 

fully mixed and transition jets. 
NASA/TM—2012-217226 13

4.3 Transition Jet 

The transition component of jet noise in dual stream jets with an inverted velocity profile is evaluated 

as before through parameters Up, Tp, and De. The new modification consists of an adjustment factor 

+ αTm (λV) in the filter expression Equation (10). Thus, as the velocity ratio is increased, there is a tradeoff 

of noise between these two latter noise components. 

4.4 Primary Jet 

The modeling of this noise component remains unaltered. 

It should be noted that when the bypass stream dominates the core flow in terms of its total mass 

and jet speed (typically λV > 1.70, and assuming that the bypass ratio BPR = λVλAλT is not small) the 

secondary and the fully mixed regions, in combination, dominate a composite spectrum. The spectrum is 

then expected to look like that in a single stream jet—subsequently the dual peak spectral character 

should disappear. 

5.0 Sample Results Using Modified Model 

The dual stream jet noise methodology “dJet” is applied to jet configurations listed in Table 1, and 

predictions are examined with emphasis on unconventional operating conditions of IVP/ITP. As before, 

the one-third Octave band spectra are shown on a 15 ft arc at 90° and 150°. Figure 12 should be compared 

with Figures 5 and 6 shown earlier. The high frequency amplitude (i.e., the secondary stream component) 

is enhanced under the conditions of the inverted velocity profile. At the same time, a shift in the spectral 

peak of this noise component brings about a second peak at the higher velocity ratio of V = 1.39. Similar 

features are observed in geometry B (Fig. 13) and geometry C (Figure 14). At a velocity ratio of 1.09, the 

second peak is gradually forming and a significant increase in the high frequency noise is noticeable 

relative to the lower velocity ratio of 0.76. The low frequency noise is shaped primarily by the fully 

mixed and the transition components of jet noise. A gradual transition of the acoustic energy from the 

former source to the latter takes place concurrently with the development of the second peak, as seen in 

Figures 12, 13, and 14. This indicates that the transition jet could steadily advance into the fully mixed 

region and eventually the two zones become one. Under such conditions, the dual spectral feature at aft 

angles is expected to disappear as the high frequency noise from the combined transition and fully mixed 

zones overwhelm the contribution from the secondary source. 

 
NASA/TM—2012-217226 14

Figure 12.—Data and predictions at 90o and 150o inlet angles in dual stream jet geometry A of Table 1. Top 

row Run#8302: A = 1.43, v = 1.07, T = 1.20;bottom row Run#8305: A = 1.43, v = 1.39, T = 0.76.

 
NASA/TM—2012-217226 15

Figure 13.—Data and predictions at 90° and 150° inlet angles in dual stream jet geometry B of Table 1. Top row: 

Run#8201: A = 1.43, v = 0.73, T = 1.11; middle row: Run#8202: A = 1.43, v = 1.09, T = 1.22; bottom 

row: Run#8205: A = 1.43, v = 1.38, T = 0.77.

 
NASA/TM—2012-217226 16

Figure 14.—Data and predictions at 90o and 150o inlet angles in dual stream jet geometry C of Table 1. Top row: 

Run#8401: A = 1.41, v = 0.72, T = 1.10; middle row: Run#8402: A = 1.41, v = 1.07, T = 1.20; bottom 

row: Run#8405: A = 1.41, v = 1.37, T = 0.74. 

 
NASA/TM—2012-217226 17

To further validate the spectral features discussed earlier, we now examine a second set of dual 

stream jet noise data as reported by Kozlowski et al. (Refs. 19 and 20). A number of shock-free test 

conditions at two area ratios of 0.76 and 1.22, designated as configurations 2 and 14, are listed in Table 2. 

Geometry details for each configuration may be found in Reference 19. The primary nozzle in each 

configuration is a convergent-divergent nozzle with an exit to throat area ratio of 1.10. An extensive test 

matrix that includes both subsonic and supersonic pressure ratios for the two streams is tabulated in 

Reference 20. Since we are only interested in the subsonic conditions, the throat dimension is used as the 

nozzle exit plane. Diameters of the secondary and primary nozzles at their respective efflux planes are 

listed in the order of Ds, Di, and Dp 

 Configuration 2: 5.0, 3.93, 3.54 in., 

 Configuration 14: 5.0, 3.58, 3.16 in. 

The test points shown in Table 2 represent a combination of normal and inverted velocity and/or 

temperature profiles. Lossless acoustic data are reported on an arc at 36 fan diameters (15 ft) for all 

configurations. The dual stream jet noise model “dJet” is implemented as before, and the component 

spectra are presented at two inlet angles of 90° and 150°. 

Figures 15 through 23 show how these predictions compare against data. Figures 15, 16, 17, and 18 

relate to configuration 2, that is, area ratio of 0.76. The velocity ratio is increased incrementally from a 

normal velocity profile at V = 0.59 to an inverted velocity profile at V = 1.60. Simultaneously, the 

temperature ratio is decreased from a normal temperature profile at T = 1.72 to an inverted temperature 

profile T = 0.57. Model predictions show a steady increase in high frequency noise in the secondary. A 

shift in the peak amplitude of this noise component brings about a second spectral peak. Figure 16 shows 

that the second peak is nearly about to form. Low frequency noise from the combined transition and fully 

mixed components develop into to the first peak at aft angles, and a single peak along the jet sideline 

angles. The two sources interchange strength as the velocity ratio is increased. 

Figures 19, 20, 21, 22, and 23 are applicable to configuration 14, that is, area ratio of 1.22. Here the 

velocity ratio is increased incrementally from 0.74 to 1.57 for a selection of temperatures that include 

both NTP and ITP. The trend is similar to the sequence of the previous figures. At the conditions of 

V = 1.07 and 1.08 a second peak is barely visible in the predictions of Figures 20 and 21, which is 

missing from the measurements. As we will see shortly, the true exit velocities in the core and bypass 

streams of a dual stream jet may be slightly less than the ideal values as a result of losses associated with 

the discharge coefficient in each nozzle. This could have an impact on the prediction of the second peak 

at conditions that may be considered as a borderline between NVP and IVP. 

TABLE 2.—COANNULAR NOZZLE TEST CONDITIONS OF KOZLOWSKI ET AL. (REF. 20) 

Run# Config. Primary (core) Secondary (fan) A V T

NPR To – R U – fps NPR To – R U – fps 

PWK006.04 2 1.52 1246 1333 1.31 695 788 0.76 0.59 1.72 

PWK005.02 2 1.52 687 965 1.78 680 1115 0.76 1.15 1.06 

PWK034.04 2 1.55 1255 1347 1.80 1949 1953 0.76 1.45 0.66 

PWK026.03 2 1.52 697 973 1.80 1270 1552 0.76 1.60 0.57 

PWK418.04 14 1.53 1453 1436 1.30 1275 1065 1.22 0.74 1.09 

PWK418.03 14 1.54 1464 1452 1.82 1275 1557 1.22 1.07 1.22 

PWK428.03 14 1.53 701 982 1.30 1264 1060 1.22 1.08 0.53 

PWK417.02 14 1.53 1454 1437 1.82 1948 1969 1.22 1.37 0.78 

PWK428.02 14 1.52 714 984 1.80 1264 1548 1.22 1.57 0.59 

PWK036.03 2 1.52 1450 1404 1.81 1615 1745 0.76 1.24 0.94 
NASA/TM—2012-217226 18

Figure 15.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 2 of Table 2, Run# 

PWK006.04: A = 0.76, v = 0.59, T = 1.72. 

Figure 16.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 2 of Table 2, Run# 

PWK005.02: A = 0.76, v = 1.15, T = 1.06. 

Figure 17.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 2 of Table 2, Run# 

PWK034.04: A = 0.76, v = 1.45, T = 0.66. 

 
NASA/TM—2012-217226 19

Figure 18.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 2 of Table 2, Run# 

PWK026.03: A = 0.76, v = 1.60, T = 0.57. 

Figure 19.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 14 of Table 2, Run# 

PWK418.04: A = 1.22, v = 0.74, T = 1.09.

Figure 20.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 14 of Table 2, Run# 

PWK418.03: A = 1.22, v = 1.07, T = 1.22. 
NASA/TM—2012-217226 20

Figure 21.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 14 of Table 2, Run# 

PWK428.03: A = 1.22, v = 1.08, T = 0.53.

Figure 22.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 14 of Table 2, Run# 

PWK417.02: A = 1.22, v = 1.37, T = 0.78. 

Figure 23.—Data and predictions at 90° and 150° inlet angles in dual stream jet configuration 14 of Table 2, Run# 

PWK428.02: A = 1.22, v = 1.57, T = 0.59. 

 
NASA/TM—2012-217226 21

Figure 24.—The effect of the discharge coefficient on noise at 90° and 150° inlet angles in dual stream jet 

configuration 2 of Table 2, Run# PWK036.03: A = 0.76, v = 1.24, T = 0.94.

As indicated earlier, the ideal efflux velocities for core and fan streams are evaluated based on the 

one-dimensional isentropic flow relations. In practice the true jet velocity is somewhat less than the ideal 

one, and depends on the nozzle discharge coefficient. Appendix A describes the relationship between the 

ideal velocity and the true velocity once the discharge coefficient for a nozzle is specified. To examine 

the effect of the discharge coefficient on noise, we consider the final set point from Table 2 at 

Run#PWK036.03. The ideal velocities for the core and fan streams are evaluated as 1404 and 1745 fps, 

respectively. We choose a set of discharge coefficients of 0.95 for both core and fan nozzles. The true 

velocities are evaluated as 1346 and 1678 fps, respectively. Under such conditions, slightly less noise 

should be expected relative to that at the ideal conditions. Figure 22 shows the effect of the discharge 

coefficient on noise. The new velocity ratio (V = 1.246) still remains close to the value listed in the 

Table, however it is not uncommon for the velocity ratio to change relative to the ideal conditions. 

6.0 Summary and Future Work 

A semi-empirical model has been proposed to predict turbulent mixing noise in dual-stream jets. The 

model applies a four-component composite spectrum for such flows, where each component contribution is 

founded on velocity scaling laws for single stream jets. Examination of acoustic data in jets under 

unconventional conditions of inverted velocity/temperature profiles required a number of modifications in 

the model. Refraction of sound at a plane vortex sheet was used to shed light on some of the observed dual 

spectral features of jet noise at small aft angles. Subsequently, adjustments were proposed that modify a 

number of parameters such as diameter, characteristic frequency, and the spectral filter in the secondary 

mixing region and the fully mixed region. In addition, the transition and fully mixed regions required 

amplitude adjustments as a function of the velocity ratio, which resulted in a tradeoff of acoustic energy 

between the two. Two sets of experimental measurements gathered at different laboratories were used to 

examine the validity of the model for a range of velocity and temperature in the core and fan streams. For 

the most part, the model was successful and provided a continuous transformation of the spectrum as the 

velocity profile changed from a normal to an inverted. The latter situation resulted in an enhancement of the 

high frequency noise from fan-stream sources that also led to a second peak in the spectrum. 

Supersonic jets under imperfectly expanded conditions emit broadband shock associated noise that 

could dominate the high frequency turbulent mixing noise. A semi-empirical model for shock noise in 

single stream jets has been developed in Reference 12. It implements a power law similar to the jet 

mixing noise, but utilizes a so-called shock intensity parameter as the dependent variable. Similar efforts 

may be required in order to develop a semi-empirical model for shock noise in dual stream flows. 

Installation elements such as center plugs and pylons present additional parameters that demand 

consideration in the context of a comprehensive prediction model for noise in turbofan jet engines. 

NASA/TM—2012-217226 23

Appendix A.—Nozzle Discharge Coefficient and True Velocity 

Nozzle discharge coefficient Cd is defined as the ratio of the actual mass flow rate through a nozzle to 

the ideal mass flow rate based on one-dimensional isentropic flow relations. We use subscripts t and id to 

identify the actual (true) and ideal parameters, and superscript o denotes the stagnation values as before. 

( )

( )

t

d

id

t

id

m VA

C

m VA

   





(A1)

In general, coefficient Cd is measured experimentally and is a function of the overall operating 

pressure ratio, convergence angle in the nozzle, the effect of the nozzle lip and boundary layer thickness, 

that is, the actual flow area at the exit. Once the discharge coefficient Cd and area ratio / C AA A t id  are 

known, Equation (A1) determines the velocity coefficient Vt /Vid. The density coefficient / t id   may be 

expressed in terms of Vt and Vid using the isentropic gas relations. 

1/ 1/ 1/ 1/

, , o o oo o o o o

t t id id t id t

o o

t id id t id

p pp p

pp p p

   

 

         

         

         

 (A2) 

From equation of state 

, ,( ) assumes o o o oo o o o

id id o t t t id p RT p RT T T T     (A3) 

Or, 

o o

t t

o o

id id

p

p

  

 (A4) 

Equations (A2) and (A4) show that 

(1 ) / o

id t

o

t id

p

p

 

     

  

 (A5) 

The relation between the nozzle pressure ratio and Mach number may be used to evaluate the right 

hand side of (A5) 

/( 1) /( 1) 2 2

2

2 2

1 1 1 ,1 , 2 2

o o

id id t t o

o

o o

p Vp V a RT

p ap a

   

 

            

(A6)

Here ao denotes the sound speed at the stagnation temperature T°. From (A6) we have 
NASA/TM—2012-217226 24

/( 1)

2

2

1

2

1

2

( 1)(

( 1)(

1 )

1 )

t

o

t o

o

id id

o

V

p a

p V

a

 

 

 

                

 

(A7)

Equations (A1), (A5) and (A7) are now combined as 

2

2

1

2

1

2

( 1)(

( 1)(

1 )

1 )

id

t o

d A

t id

o

V

V a C C

V V

a

 

 

    

  

  

 

 (A8) 

Equation (A8) solves for Vt when other quantities are known. Once the NPR / o

id p p   and 

stagnation temperature T°at the nozzle inlet are specified, the value of γ at the exit static temperature is 

determined by iteration, and is assumed to remain unchanged due to subsequent adjustments in velocity 

from Vid to Vt. 

Equation (A8) may be rearranged as 

xt

2  z xt 1  0; z  1 xid

2

(Cd / CA )xid

(A9)

The relevant solution to the above quadratic equation is 

xt  1  (z / 2)2  (z / 2) (A10) 

where 

x    1

2

V

ao

, Vt  Vid

xt

xid

(A11)

As an example, consider the nozzle inflow conditions NPR = 1.50, T°/T∞ = 2.50, T∞ = 520 °R, and 

discharge to contraction area ratio of Cd/CA = 0.95. The specific heat ratio  and the static temperature T

are determined as γ = 1.3704, T = 1176 °R, and the ideal velocity is evaluated as Vid = 1309 fps. 

Subsequently Equation (A10) solves for the true velocity Vt = 1255 fps, and a density coefficient of 0.99. 
NASA/TM—2012-217226 25

References 

1. Zorumski, W.E., “Aircraft Noise Prediction Program Theoretical Manual,” NASA–TM–83199, Parts 

1 and 2, 1982. 

2. Stone, J.R., Krejsa, E.A., and Clark, B.J., “Jet Noise Modeling for Coannular Nozzles Including the 

Effects of Chevrons,” NASA/CR-2003-212522, September 2003. 

3. Stone, J.R., Krejsa, E.A., Clark, B.J., and Berton, J.J., “Jet Noise Modeling for Suppressed and 

Unsuppressed Aircraft in Simulated Flight,” NASA/TM-2009-215524, March 2009. 

4. Ahuja, K.K., and Bushell, K.W. “An Experimental Study of Subsonic Jet Noise and Comparison with 

Theory,” J. Sound and Vibration, 30(3), pp. 317–341, 1973. 

5. Fisher, M.J., Preston, G.A., and Mead, C.J., “A Modeling of the Noise from Simple Coaxial Jets, Part 

I: With Unheated Primary Flow,” J. Sound and Vibration, 209(3), pp. 385–403, 1998. 

6. Fisher, M.J., Preston, G.A., and Mead, C.J., “A Modeling of the Noise from Simple Coaxial Jets, Part 

II: With Heated Primary Flow,” J. Sound and Vibration, 209(3), pp. 405–417, 1998. 

7. Tester, B.J., and Fisher, M.J., “A Contribution to the Understanding and Prediction of Jet Noise 

Generation in Forced Mixers,” AIAA–2004–2897, 2004. 

8. Tester, B.J., and Fisher, M.J., “A Contribution to the Understanding and Prediction of Jet Noise 

Generation in Forced Mixers: Part II Flight Effects” AIAA–2005–3094, 2005. 

9. Tester, B.J., and Fisher, M.J., “A Contribution to the Understanding and Prediction of Jet Noise 

Generation in Forced Mixers: Part III Applications” AIAA–2006–2542, 2006. 

10. Bridges, J. and Brown, C.A., “Validation of the Small Hot Jet Rig for Jet Noise Research”, AIAA 

Paper 2005–2846, May 2005. 

11. Brown, C.A., and Bridges, J., “Small Hot Jet Acoustic Rig Validation,” NASA/TM-2006–214234, 

2006. 

12. Khavaran, A., and Bridges, J., “Development of Jet Noise Power Spectral Laws Using SHJAR Data,” 

AIAA Paper 2009–3378, 2009. 

13. Khavaran, A., and Bridges, J., “Jet Noise Scaling in Dual Stream Nozzles,” AIAA Paper 2010–3968, 

2010. 

14. Ko, N.W.M., and Kwan, S.H., “The Initial Region of Subsonic Coaxial Jets,” J. Fluid Mechanics, 

73(2), pp. 305–332, 1976. 

15. Abramovich, N., The Theory of Turbulent Jets, M.I.T. Press, 1963. 

16. Eldred, K.M., et al., “Far-field noise Generation by Coaxial Flow Jet Exhaust. 1. Detailed 

Discussion,” Wyle Lab Report FAA–RD–71–101, 1971. 

17. Khavaran A., and Kenzakowski, D.C., and Mielke-Fagan, A.F., “Hot Jets and Sources of Jet Noise,” 

International J. Aeroacoustics, 9(4, 5), pp. 491–532, 2010. 

18. Larson, S.R., Nelson, D.P., and Stevens, B.S. “Aerodynamic and Acoustic Investigation of Inverted 

Velocity Profile Coannular Exhaust Nozzles Models and Development of Aerodynamic and Acoustic 

Prediction Procedures,” NASA-CR-3168, 1979. 

19. Kozlowski, H., and Packman, A.B., “Aerodynamics and Acoustic Tests of Duct-Burning Turbofan 

Exhaust Nozzles,” NASA CR–2628, Dec. 1976. 

20. Kozlowski, H., and Packman, A.B., “Aeroacoustics Tests of Duct-Burning Turbofan Exhaust 

Nozzle—Comprehensive Data Report Vol. 1—Model Scale Acoustic Data,” NASA CR–134910, 

Feb. 1977. 
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Acoustic Investigation of Jet Mixing Noise in Dual Stream Nozzles 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

In an earlier study, a prediction model for jet noise in dual stream jets was proposed that is founded on velocity scaling laws in single stream 

jets and similarity features of the mean velocity and turbulent kinetic energy in dual stream flows. The model forms a composite spectrum 

from four component single-stream jets each believed to represent noise-generation from a distinct region in the actual flow. While the 

methodology worked effectively at conditions considered earlier, recent examination of acoustic data at some unconventional conditions 

indicate that further improvements are necessary in order to expand the range of applicability of the model. The present work demonstrates 

how these predictions compare with experimental data gathered by NASA and industry for the purpose of examining the aerodynamic and 

acoustic performance of such nozzles for a wide range of core and fan stream conditions. Of particular interest are jets with inverted velocity 

and temperature profiles and the appearance of a second spectral peak at small aft angles to the jet under such conditions. It is shown that a 

four-component spectrum succeeds in modeling the second peak when the aft angle refraction effects are properly incorporated into the 

model. A tradeoff of noise emission takes place between two turbulent regions identified as transition and fully mixed regions as the fan 

stream velocity exceeds that of the core stream. The effect of nozzle discharge coefficients will also be discussed. 

15. SUBJECT TERMS

Noise; Aircraft noise; Aerodynamic noise; Acoustics 

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