Milo D. Dahl

Glenn Research Center, Cleveland, Ohio

Abbas Khavaran

ASRC Aerospace Corporation, Cleveland, Ohio

The Effect of Nondeterministic Parameters on

Shock-Associated Noise Prediction Modeling

NASA/TM—2010-216801

September 2010

AIAA–2010–3841
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Milo D. Dahl

Glenn Research Center, Cleveland, Ohio

Abbas Khavaran

ASRC Aerospace Corporation, Cleveland, Ohio

The Effect of Nondeterministic Parameters on

Shock-Associated Noise Prediction Modeling

NASA/TM—2010-216801

September 2010

AIAA–2010–3841

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

16th Aeroacoustics Conference

cosponsored by the American Institute of Aeronautics and Astronautics and

the Confederation of European Aerospace Societies

Stockholm, Sweden, June 7–9, 2010
Acknowledgments

The NASA Supersonics Project of the Fundamental Aeronautics Program supported this work.

Available from

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7115 Standard Drive

Hanover, MD 21076–1320

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Available electronically at http://gltrs.grc.nasa.gov

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. 
Engineering applications for aircraft noise prediction contain models for physical phenomenon that enable solutions to be computed quickly. These models contain parameters

that have an uncertainty not accounted for in the solution. To include uncertainty in the

solution, nondeterministic computational methods are applied. Using prediction models for

supersonic jet broadband shock-associated noise, fixed model parameters are replaced by

probability distributions to illustrate one of these methods. The results show the impact of

using nondeterministic parameters both on estimating the model output uncertainty and

on the model spectral level prediction. In addition, a global sensitivity analysis is used

to determine the influence of the model parameters on the output, and to identify the

parameters with the least influence on model output.

Nomenclature

A Area

B Intercept parameter, see equation (3)

D Diameter

E Amplitude power exponent parameter, see equation (6)

J0 Bessel function of the first kind, order zero

J1 Bessel function of the first kind, order one

L Gaussian source half-width function, see equation (7)

M Mach number

N Total number of random samples

P Pressure

R Radial distance

ST i Total effects sensitivity measure, see equation (1)

St Strouhal number

T Temperature

U Velocity

Xi i-th parameter in vector X

Y output quantity from model with parameters X

X Vector of parameters

X∗ X fixed at a set of values

E{ } Expected value of the enclosed quantity

Var( ) Variance of the enclosed quantity

A

2

Shock cell strength amplitude function, see equation (7)

c Amplitude factor, see equation (7)

c Speed of sound

The Effect of Nondeterministic Parameters on

Shock-Associated Noise Prediction Modeling

Milo D. Dahl

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abbas Khavaran

ASRC Aerospace Corporation

Cleveland, Ohio 44135

NASA/TM—2010-216801 1
f Frequency

j0,ν ν-th zero of a zeroth order Bessel function

m Shock power factor

hp

2

i Mean-square acoustic pressure

x Independent regression variable

y Dependent regression variable

z Linear regression residual error, see equation (5)

Subscripts

∞ Ambient conditions

ν, Nν integer index, maximum value Nν

∼i All parameters in vector not including the i-th parameter

c Convective velocity or Mach number

d Design conditions

e Nozzle exit conditions

i, I integer index, maximum value I

j Fully expanded jet conditions

k, K integer index, maximum value K

r Randomly selected variable

sh Shock-associated noise related

t Total conditions

Symbols

β Shock parameter

χ

2

n Chi-squared distribution with n-degrees of freedom

 Nonlinear regression residual error, see equation (6)

γ Ratio of specific heats

ρ1 Density with one subscript

ρ12 Correlation coefficient with two subscripts

σ Standard deviation

θ Inlet directivity angle

N(μ, σ2) Normal distribution with mean μ and variance σ

2

I. Introduction

The development of tools to computationally predict the sound generated by fluid flow or when fluid

interacts with a surface continues to be an important part of the research effort for developing future environmentally friendly aircraft. For aircraft propulsion system related jet noise, a wide range of computer-based

schemes are included in a hierarchy of these noise prediction tools from numerically intensive, extremely long

running, direct simulations based on solving first principle equations to near instantly computed, empirical

methods using experimental databases. The numerically intensive computations may simulate all the proper

physics for jet noise predictions, but running them is so costly that they currently have little practical usage.

At the other extreme, the empirical models are limited in their ability to predict jet noise beyond a relatively

small set of geometries and operating conditions with which they were developed. In between these two extremes are found methods that use some combination of numerical techniques and physics-based models in

an attempt to increase the practical ability to predict jet noise over a wider range of jet operating conditions

and nozzle geometries. Among these are included tools to compute the unsteady fluid flow using various

types of filtering and turbulence modeling schemes to obtain a noise source field that is used to compute the

radiated noise field, and tools that use acoustic analogy source models based on mean fluid turbulent statistical values to compute a solution for the radiated noise field. Even though these latter numerical methods

still include some level of modeling that reduces total computation time, the time required to compute the

unsteady or the steady mean flow can inhibit the number of cases run for a design study.

For engineering applications in aircraft noise prediction at the component and especially at the system

level, extensive modeling of the physical phenomena is required to obtain many solutions for design, development, and parametric studies. Analytical development and intuition are used to develop models that

describe physical phenomena. In the end, these models will contain parameters that must be determined

NASA/TM—2010-216801 2
by optimization and calibration. These parameters are found by using numerical simulation and/or experimental data. Regression is often used to set these parameters. This deterministic approach may provide

reasonable solutions, but usually within limited conditions. However, the determination or the calibration

of these parameters is often inexact especially when using experimental data that contains uncertainty, that

may be insufficient for good statistics, or that may be of poor quality. Thus there is a question if the best-fit

parameters are truly the optimal parameters for the model. It may be more appropriate to include the parameter uncertainty in the model1 and/or to compute the solution using nondeterministic methods.2

In the

present paper, we consider the introduction of nondeterministic parameters into models for the prediction

of broadband, shock-associated noise from supersonic jets. This was done as part of a broader study and

assessment of jet noise prediction models both as a method of determining the quality of the prediction and

as a method of providing better prediction capability. Nondeterministic approaches are shown to impact the

predicted results and can be used to estimate the output uncertainty and to determine the influence of the

model parameters on the output.

A brief background for this paper is given in the next section. A previous assessment study of jet noise

prediction models, including models for broadband shock-associated noise, determined the uncertainty in the

output from the prediction models. The shock-associated noise prediction models were easily modified to

apply the method of computing with nondeterministic parameters. This is followed by a section describing

some of the details of the prediction models. One model relies on a two-parameter regression analysis of

experimental data to provide a scheme to predict noise amplitude as a function of jet operating conditions

and directivity angle. The regression analysis provides the statistical quantities to define the distribution for

the uncertain parameters. The other model has a large number of parameters with unknown distributions.

The results are then presented in three parts. The first discusses the effects of the input operating condition

uncertainty on model output uncertainty. Next, the results for the model using the regression analysis is

presented. The main issue is which type of regression is appropriate. The last part presents results for the

many parameter model. This part focuses on the choice of distribution for the uncertain parameters and the

determination of the influence of the parameters on the output. This helps identify parameters that most

affect the output uncertainty.

II. Background and Methodology

Bridges et al.3 assessed the capability to predict jet noise and showed results from empirical and acoustic

analogy based tools compared to measured data over a wide range of operating conditions. A general

conclusion was that empirical tools could make overall sound pressure level predictions to within ±6 dB

of measured values for a broad range of nozzle shapes and operating conditions when averaged over all

directivity angles. For simple axisymmetric nozzles, the predictions improved to agree within ±1.2 dB. In

comparison, an acoustic analogy tool was only applicable to a limited range of conditions consisting of

axisymmetric, cold subsonic jets. The accuracy for these comparisons were also within ±1.2 dB. That study

showed some of the issues related to how the results from different jet noise prediction models compare

to the same set of observations and how to judge the quality of those predictions. Khavaran & Bridges4

further analyzed the measured jet noise data at supersonic conditions, separating mixing noise and shockassociated noise spectral components from the total jet noise spectrum, to provide better data for jet noise

model predictions and improvements. The results of that component separation allow us to focus this

study on shock-associated noise prediction capabilities for jets from convergent nozzles. To illustrate the

results from different models, example spectra that compare predictions for shock-associated noise with the

shock-associated noise component of measured data are shown in Figure 1. The plots show power spectral

density as a function of the Strouhal number for under-expanded jets with a fully expanded Mach number

of about 1.5. There are two jet temperatures represented at two directivity angles from the inlet of 50 and

90 degrees. Three predictions are shown from the following models: The analytically based shock-associated

noise prediction model from Tam,5the intensity scaling prediction model based on the work of Khavaran &

Bridges6 and further detailed in Appendix A of this paper (labeled sJet), and the Reynolds-average-NavierStokes-based (RANS) model of Morris & Miller.7 Three different models attempting to predict the same

experimental results.

The shock-associated noise models under consideration here were originally developed to make deterministic, one-time calculations using parameters found through least-squares fitting, averaging, point matching,

or other methods to calibrate computed predictions using experimental data. These fixed parameters in

NASA/TM—2010-216801 3
reality have an uncertainty associated with them that is ignored in the basic usage of these prediction models. To incorporate the uncertainty, knowledge about the quantities that describe the uncertainty must be

available and/or the distribution of the uncertain parameter must be known or estimated. This information may be unknown or hard to obtain. In the case of the sJet code, the uncertain parameters are well

described in terms of their probability distributions and the moments associated with those distributions.

Thus, it is readily possible to replace fixed parameters with their nondeterministic distributions. Once this

is done, a method must be applied to perform the probabilistic analysis. One approach is to perform Monte

Carlo sampling using the original prediction model. Depending on the scale of the problem, the number

of computations may become quite large. Latin hypercube sampling helps reduce the number of sampling

computations required to achieve convergence for good statistics.8 Alternatives are to replace the original

computational model with an analytical model that is more computationally efficient. Such methods include

first and second order reliability methods, advanced mean value methods, and the response surface method.9

The analytical model may be faster to compute, but it is approximate and contains inaccuracies. The two

models to be used here, the sJet and Tam models, are already approximate and are easily amenable to

replacing fixed parameters with nondeterministic distributions. The RANS-based model was not available

for this study. Thus, the Monte Carlo approach was chosen. A software tool that easily performs the Monte

Carlo calculations using the external jet noise prediction codes is called DAKOTA.8 This program can be

set up to perform the following:

1. Generate random inputs from predefined distributions.

2. Execute the prediction model code using the random inputs.

3. Perform statistical analysis on the outputs from the model code.

In addition to this Monte Carlo calculation, the DAKOTA code can perform sensitivity analyses of the

parameters to determine their influence on the output, optimization studies to find the best parameters, and

other related computations.

Sensitivity analysis is often used in conjunction with uncertainty analysis. While uncertainty analysis

attempts to quantify the level of model output uncertainty given a set of model parameter uncertainties,

sensitivity analysis is concerned with attempting to determine how much influence a particular parameter

has on the output or output uncertainty and then rank the parameters in degrees of importance. This allows

parameters of lower importance to be fixed and more attention and resources can be focused on parameters of

higher importance. One sensitivity analysis method supported by the DAKOTA code is called variance-based

decomposition. Let the parameters be denoted by the vector X. If a particular parameter Xi was absolutely

non-influential, then the variance of the output Y due to changes in Xi when all the other parameters in

X are fixed, denoted as X∗

∼i

, must be zero. We compute the conditional variance VarXi(Y |X∼i = X∗

∼i

)

where the subscript Xiindicates the one parameter in X that is being randomly sampled while all the other

parameters are held fixed. The conditional variance must be zero for a second set of X∗

∼i

, a third set of

X∗

∼i

, and so forth. Averaging these conditional variances over the set X∼i will not change this result, hence

the average will be zero, also. Comparing the average to the total variance of the output Y , we obtain the

total effect sensitivity measure ST i for the parameter Xi,

ST i =

EX∼i{VarXi(Y |X∼i)}

Var(Y )

, (1)

where ST i includes both a measure of the direct effect of Xi and a measure of the interaction effects of

Xi with all the other X∼i on the output variance. In actual calculation, ST i will most likely be nonzero;

hence, a small value for ST i represents an Xi parameter with less influence on the output than a different

Xi parameter with a larger ST i value. Since the computation of the total effect sensitivity measure involves

varying all the parameters simultaneously, it is considered to be a global measure in contrast to a local

measure that involves varying only one parameter at a time. The details of variance-based decomposition

and its implementation to obtain sensitivity measures are found in Saltelli et al.10, 11 and the references

therein.

III. Models and Data

This section introduces the two jet broadband shock-associated noise prediction models. The development

approach for these two models were entirely different. The sJet code was developed from a study of the data,

NASA/TM—2010-216801 4
particularly far-field spectral densities at various directivity angles. The Tam model was developed from

derivations of the equations of motion. Only after this development were simplifications and adjustments

made when results were compared to measured data. The section concludes with the experimental conditions

for the data used for comparison with predictions in this paper.

III.A. sJet Model

In the jet noise prediction modeling approach proposed by Khavaran & Bridges,6scaling laws are used

to derive relationships between fundamental jet parameters and the total radiated power computed by

integrating over the appropriate power spectral density. To enable this process for imperfectly expanded

supersonic jets, the radiated broadband noise was separated into distinct spectral components associated

with mixing noise and shock-associated noise. Here, we are only concerned with the model for shockassociated noise. The total shock-noise intensity scales with the shock parameter β, a measure of the shock

strength in the flow given by the amount to which the fully expanded Mach number deviates from the nozzle

design Mach number. In general, this may be expressed as

hp

2

ish ∝ β

m where β2 =



M2

j − M2d





. (2)

Based on experimental measurements, the shock power factor m varies with the noise radiation directivity

angle and the jet temperature.

Following Khavaran & Bridges,6 a linear regression is applied in the form

yi = m(θ, T)xi + B(θ, T) (3)

where the dependent variable yi and the independent variable xi are given by

yi = OASPLsh(θ, T) and xi = 10 log βi, i = 1, 2, . . . , I . (4)

The overall sound pressure level (OASPL) is for the shock-associated noise computed from the shockassociated noise spectral component of the total radiated noise over the Strouhal number range 0.1 ≤ St ≤ 10

where St = fDe/Uj . The intercept parameter B is a measure of the shock noise overall level in decibels

when the shock parameter β equals one. The results of the linear regression are the mean values m and B,

the standard deviations σm and σB, and the correlation coefficient ρmB between the m and B parameters.

Once m and B are determined from multiple linear regressions covering a range of inlet directivity angles θ

and total temperatures Tt, the shock-associated noise levels are now set in general and it remains to compute

the spectral shape. See Appendix A for the detailed equations.

The linear regression model given in equation (3) is based on the general model regression equation

yi = m xi + B + zi (5)

where the zi’s are assumed to be independent, normally-distributed, random values with zero mean and

equal variances representing the random or residual error in the linear regression.12 This assumption is most

likely not strictly maintained in the above linear regression. Equation (3) is a logarithmic transformation of

equation (2). While this creates a linear equation from a nonlinear equation, the errors are also transformed

and may violate the assumption for ziin equation (5).13 In the original formulation, hp

2

ish is an overall power

level computed by adding together the power levels in each frequency bin of the shock-associated noise power

spectral density (PSD). Each power level at each frequency in the PSD has a chi-squared distribution χ

2

n

with n-degrees of freedom after spectral averaging.14 A summation over K frequencies in a power spectral

density with each power level having a χ

2

n distribution results in a total power level with a chi-squared

distribution χ

2

K×n with (K × n)-degrees of freedom. A nonlinear regression equation for the total power

level can then be written as

yi = 10E(θ,T)β

m(θ,T)

i + i (6)

where both yi = hp

2

ish(θ, T) and i, the residual error, have in general a χ

2

K×n

-distribution. Using 10E is a

recognition that hp

2

ish has a potential range covering many orders of magnitude. It can be shown that when

weighting is used and the chi-squared distribution has a large number of degrees of freedom, the nonlinear

regression formulation of equation (6) is appropriate for the given data and that the residual errors more

closely have the desired normally-distributed properties. See Appendix B for details.

NASA/TM—2010-216801 5
III.B. Tam Model

The broadband shock-associated noise model of Tam5, 15 is based on the theory that large-scale turbulent

structures generated in the supersonic jet mixing layer interact with the periodic-like shock cell structure

that exists in imperfectly expanded jets. This interaction causes pressure perturbations that radiate as

sound. The shock cell structure is modeled as a wave guide containing a series of modes and the large

structures are modeled as a sum of instability waves each with their own amplitude envelope and phase

speed characteristics. These characteristics affect the amplitude and directivity of the radiated noise. The

model is applicable to both under-expanded and over-expanded jets, but in this paper, it is applied only to

under-expanded jets issuing from convergent nozzles.

Following extensive algebraic manipulations and modeling approximations, Tam5 derived an equation for

the far field mean-square acoustic pressure

hp

2

ik =

ρj

ρ∞



1 +

γ − 1

2

M2

j

−1

c(cb1)L

2

(hs1, xc1, xc2) Aj A

2

(ab1, ab2)ρ

2

∞c

2

∞U

2

j

R2 Stk fk

X

Nν

ν=1

1

j

2

0,ν J

2

1

(j0,ν)

exp [Fν] (7)

and

Fν = −



fν(vc1, vc2, w11, w12, w21, wν1)

fk

− 1

2

L

2

(hs1, xc1, xc2)

2 ln 2 

Uj

Uc(vc1, vc2)

2

(1 + Mc(vc1, vc2) cos θ)

2

where ρj , Uj , Mj , and Aj are the density, velocity, Mach number, and cross-sectional area, respectively, of

the jet at the fully expanded condition, ρ∞ is the ambient density, c∞ is the ambient speed of sound, R

is the radial observer distance, θ is the inlet directivity angle, fk is the k-th frequency in a spectrum, Stk

is the Strouhal number fkDj/Uj , Dj is the diameter corresponding to Aj , and fν is the peak frequency

corresponding to the mode number ν.. The denominator factor Stk in equation (7) is fixed at 0.35 when

Stk < 0.35. In the summation of waveguide modes, J1 is a first order Bessel function and j0,ν is the ν-th

zero of the zeroth order Bessel function J0. The functional arguments with two subscripts represent the

twelve parameters in equation (7) that were derived by fitting equations to experimental or calculated data.

Table 1 defines these parameters and the detailed equations are given in Appendix C.

Parameter Contributes to Model or Mean Value

ab1 amplitude 3 (Mj/Md ≥ 1), 6 (Mj/Md < 1)

ab2 amplitude 3 (Mj/Md ≥ 1), 5 (Mj/Md < 1)

w11 1st waveguide mode wave number 1.596

w12 1st waveguide mode wave number 0.1773

w21 2nd waveguide mode wave number 1.265

wν1 ν-th waveguide mode wave number 1.1

cb1 amplitude 2.65×10−4

hs1 source half-width length scale 3.3

xc1 potential core length 4.3

xc2 potential core length 1.2

vc1 convective velocity 0.7

vc2 convective velocity 0.025

Table 1: Tam model parameter values determined from experimental data or numerical calculations.

III.C. Experimental Data

The results of exploring the nondeterministic parameters in the shock-associated noise prediction models are

compared to measurements consisting of noise data from supersonic jets issuing from a 2-inch exit diameter,

circular, convergent nozzle. This nozzle was used in an extensive test program to measure the noise from jets

at a variety of nozzle pressure and temperature ratios.4 The measured noise spectrum from this convergent

NASA/TM—2010-216801 6
Case Tt/T∞ Pt/P∞ Mj Uj/c∞ β γj T∞(K) P∞(Pa) c∞(m/s)

Cold 1.01 3.636 1.493 1.245 1.109 1.4000 279.2 97790 334.93

Hot 2.16 3.629 1.494 1.828 1.109 1.3942 279.6 97839 335.18

Table 2: Test conditions for a convergent nozzle used in the prediction models.

nozzle at a variety of operating conditions were recently compared to predictions from both jet mixing noise

and jet shock-associated noise models.16 A subset of those conditions, one a cold jet and one a hot jet, are

used in this study and are given in Table 2. Example spectral comparisons between measured and predicted

shock-associated noise results at the operating conditions of Table 2 are shown in Figure 1.

IV. Results

The results of this paper are divided into three parts. The first part discusses the effects of jet operating

condition uncertainty on the predicted results. This is followed by a presentation of results where the

uncertain parameters have known distributions. In the final part, the prediction model has many parameters

with no clearly defined distributions. Estimates are made for these distributions in order to complete the

analysis.

IV.A. Operating Condition Input Parameter Uncertainty

The validation of a prediction code is performed by comparing computed values with the same values obtained

from an experimental measurement. For jet noise prediction, this comparison is done for various noise

level metrics including overall and spectral levels at specific jet operating conditions. Hence, the operating

conditions from the experiments are used as input to the jet noise prediction code. The primary experimental

measurements are pressures, temperatures, and lengths or distances associated with the geometry of the

nozzle and the location of measurement microphones. Any code specific operating condition input parameter,

such as Mach number, can be computed using the measured values. The purpose here is to demonstrate

the uncertainty of predicted noise levels given the known uncertainty in the experimental jet operating

conditions. For this example, we will use the Tam model.

The ability to accurately measure pressures, temperatures, and lengths at the NASA Glenn Research

Center aeroacoustic facilities has been documented in a recent assessment of noise prediction capability.17

Given those results, Table 3 shows an example of the measured operating conditions and the standard

deviation σ for the assumed normally distributed measured quantities. The table shows the measured total

pressure and temperature, Pt and Tt, the measured ambient pressure and temperature, P∞ and T∞, the

measured nozzle exit diameter De, and the radial distance R from the center of the nozzle exit area to the

microphone. The example conditions are for the noise radiated from a jet issuing from a convergent nozzle

with the hot jet operating conditions listed in Table 2.

Parameter Measured Value (mean) σ

Pt 355058 Pa 139.3

Tt 605.3 K 1.89

P∞ 97839 Pa 139.3

T∞ 279.6 K 1.89

De 0.0508 m 0.000051

R 5.08 m 0.0127

Table 3: Example measured jet operating conditions with standard deviations.

Following the Monte Carlo procedure given above, sampling calculations were performed from 100 to

10000 samples. In each case, the means and 95% confidence intervals were computed for the predicted

NASA/TM—2010-216801 7
shock-noise spectra at directivity angles from 50 to 90 degrees from the inlet direction. From these results,

the maximum width and the average width confidence intervals were determined across all computed spectral

frequencies and directivity angles and presented in Table 4. The increasing number of samples N results

in both more clearly defining the distribution of the computed solution and reducing the estimate of the

standard error σ/√

N, thereby reducing the confidence interval. The table clearly indicates that the effects

of input parameter uncertainty on the predicted shock-associated noise, sound-pressure-level spectrum for

the jet operating conditions listed in Table 3 are less than 0.2 dB at 95% confidence at the maximum and on

average, less than 0.04 dB. These levels of uncertainty would not be noticeable on spectral density plots such

as those shown in Figure 1. In general, this shows that the ability to make accurate measurements of input

parameters will lead to small uncertainties in the predicted values. This does not mean the predictions are

accurate as can be seen in Figure 1 where many decibels of error can separate the predicted values from the

measured values. Thus, we cannot rely on determining prediction uncertainty solely from the uncertainty in

the jet operating condition input parameters, but must consider the uncertainty of the parameters within

the model. We turn to these parameters, next.

No. of Samples N Max. 95% CI Width Avg. 95% CI Width

100 0.198 dB 0.036 dB

200 0.146 dB 0.025 dB

400 0.099 dB 0.018 dB

1000 0.063 dB 0.011 dB

10000 0.020 dB 0.004 dB

Table 4: Maximum and average widths of spectral 95% confidence intervals (CI) at upstream angles.

IV.B. Uncertain Parameters with Known Distributions

The sJet code computes the shock-associated noise spectrum by first computing an overall level value and

then combining that value with a spectral function to produce the final results. For the convergent nozzle

used here, the jet operating conditions and directivity angle set the spectral shape. The only uncertain

values are m and B that were determined by regression and only enter the model prediction through the the

overall level in the first two terms on the right hand side of equation (16). Hence, we compare the outputs

using the uncertain parameters m and B in those two terms to the output obtained using the mean values

m and B. The spectrum result that would follow from equation (22) will have exactly the same level of

uncertainty at all frequencies.

The linear regression analysis performed in Khavaran & Bridges6 using equation (3) produced tables

of m and B as a function of directivity angle and jet total temperature. In addition, the results of linear

regression provide estimates for the standard deviations σm and σB and the correlation coefficient ρmB

between m and B. These are also tabulated. The means and the standard deviations are used to define the

normal distribution functions when the fixed parameters are replaced with nondeterministic parameters. If

the correlation coefficient is nonzero, then the distribution function is jointly normal. For a deterministic

calculation, the jet operating conditions are used as input to the sJet code and the code uses interpolation

to select the appropriate m and B values from the tables. Since the jet operating conditions do not change

in the nondeterministic calculation, additional random inputs must be provided to the code. To simplify

the replacement of m and B with random values, normalized random variables are provided as input and a

linear transformation is performed.

m = m + σmmr

B = B + σBBr

where

mr ∼ Nm(0, 1)

Br ∼ NB(0, 1)

⇒

m ∼ Nm(m, σ2

m)

B ∼ NB(B, σ2

B)

(8)

The symbol N(0, 1) represents a normally distributed random variable with a zero mean and a variance equal

to one. The subscripts m and B on N indicate that the random variables mr and Br are selected separately.

The transformations in equation (8) provide the desired normal distributions for m and B where the jet

operating conditions are used to select m, B, σm, and σB by tabular interpolation. Up to this point, m and

NASA/TM—2010-216801 8
B have been considered to be uncorrelated. In the linear regression process that produced the tables for m,

and so forth, the correlation coefficient ρmB was found to change very little and it was set at a constant value

of 0.45 for all operating conditions and directivity angles. A constant ρmB simplifies the introduction of a

correlation between m and B into sJet. The normalized random variables mr and Br are sampled external

to sJet with a fixed correlation and when provided to sJet the transformation maintains that correlation in

the resulting m and B parameters.

Following the Monte Carlo procedure, the random inputs are provided to the sJet code by DAKOTA

using the latin hypercube sampling technique. The outputs from the sJet code are then stored and analyzed

by DAKOTA. The first set of results compare the amplitude level given by equation (3) with fixed parameters

determined by the linear regression to the amplitudes computed with nondeterministic parameters. Figure 2

is a summary of results for the hot jet case at a directivity angle of 50 degrees where m = 5.75 and B = 117.76.

The figure shows the mean levels and the 95% confidence intervals as a function of the number of random

samples. The single point on the right is an example result after 10000 samples. In addition, the effects

of changing σm and σB from half to twice their original tabulated value without and with correlation are

shown. The following results are deduced from examining Figure 2:

1. The mean value of the output distribution, here 120.5 dB, is basically unaffected by the width of the

distributions of m and B, by whether or not there is any correlation between m and B, and by the

number of samples. This is the expected result when the output level (in dB) is simply treated as a

number in a linear calculation.

2. Wider distributions in the input parameters leads to a wider distribution in the output. Again, this is

due to the linear calculation.

3. The width of the confidence intervals decreases as the number of samples N increases, as expected, since

it varies as 1/

√

N. For the given σm and σB and no correlation, the confidence interval is ±0.39 dB

wide at 100 samples decreasing to ±0.12 dB wide at 1000 samples. Since this is a result of a linear

calculation, the confidence interval is symmetric about the mean value.

4. The presence of correlation between m and B increases the width of the output confidence interval.

On a dB scale, this was not significant as the increase was less than 0.1 dB.

The alternative to treating the decibel output as a number in an uncertainty calculation is to recognize

the nonlinear nature of the power level calculation and perform the uncertainty calculation using equation

(6). Using the same Monte Carlo approach with the normalized random input variables as above, the mean

power levels and the 95% confidence intervals are computed before converting the results to decibels. Figure 3

is the equivalent plot to Figure 2 for the nonlinear approach. The mean parameters for the hot jet case at a

directivity angle of 50 degrees are m = 5.60 and E = 11.505. Using these as fixed parameters results in an

output power level of 117.7 dB. The results for the nondeterministic calculations show the following:

1. The mean of the output level depends on the output distribution generated by the distributions of the

random parameters m and E. Both the size of the parameter distributions and whether the parameters

are correlated affect the output. The fact that the mean output level increases with increasing σ follows

directly from the fact that any random change in the parameters that increases the power level of the

output will increase the mean power in a quadratic summation of all sampled power levels.

2. Using latin hypercube sampling, more samples results in a better definition of the output distribution

and convergence in the statistical values computed from that distribution. Hence the mean output

levels reach an asymptote with an increasing number of samples.

3. The 95% confidence intervals are in general not symmetric about the mean value. Given σm, σE, and

no correlation, the confidence interval is +0.34 dB,−0.37 dB at 100 samples decreasing to a confidence

interval of +0.11 dB,−0.12 dB at 1000 samples.

4. Including the effects of a correlation between parameters m and E results in an increase in the mean

output levels and width of the 95% confidence intervals. In terms of decibels, these increases are small

for the average correlation coefficient ρmE = 0.43, up to 0.2 dB in the mean level and up to 0.08 dB in

the confidence interval width at 1000 samples.

NASA/TM—2010-216801 9
IV.C. Many Uncertain Parameters with Unknown Distributions

The previous model had two uncertain parameters with clearly defined distributions determined from regressions that fit model equations to measured data. We now consider the Tam model, equation (7), that

contains 12 parameters, listed in Table 1, that each have some level of uncertainty. To determine the effects

of these uncertainties, we have to define the probability distributions in order to replace the fixed parameters

with distributions. For the Tam model, the detailed information required to make such distributional specifications is not given in the references describing the development of that model. A parameter for which there

is information available to estimate a probability distribution is the vc1 parameter that is used to determine

the convective velocity ratio Uc/Uj . The data given in Harper-Bourne & Fisher18 and Harper-Bourne,19

have been used to determine a fixed value for vc1 of about 0.7. That same data is used here to determine

a mean value of 0.708 and a variance of 0.00937 for both a normal distribution and an equivalent uniform

distribution. The uniform distribution covers a range from 0.54 to 0.876 or about ±24% from the mean

value. Since the data were insufficient to know the exact behavior of the vc1 distribution, both the normal

and the uniform distributions were applied to replace the fixed parameter. The DAKOTA code, using the

latin hypercube sampling technique, performed 100 samples from the vc1 parameter distribution and computed the mean and the 95% confidence interval (CI) using the pressure-squared output distribution from

the Tam model. The remaining parameters were fixed at the values given in Table 1. The results are shown

in Figure 4 using the cold jet conditions at a 50 degree inlet directivity angle. For all but the lowest frequency

results, the spectra generated using the nondeterministic vc1 parameter with either the normal distribution

or the uniform distribution are essentially the same. Only 24 frequencies were used in the nondeterministic

calculations, hence the mean and 95% CI spectra are not as smooth as the fixed parameter spectrum which

was evaluated in narrow band with a 40 Hz bandwidth.

There are 11 other parameters in the Tam model that were identified as being replaceable by distributions.

Without any specific information about these parameters, we estimate a distribution for each parameter

based on the vc1 distribution results in Figure 4. First, the difference between using a normal or a uniform

distribution was small. Second, the uniform distribution had a range based on the mean value of vc1 ± 24%.

Observation of the vc1 data indicates that this range was perhaps too wide. We reduce the range by about

half or ±10%. Considering this to be conservative, we apply this to all the parameters listed in Table 1

and use their listed values to define uniform distributions with a range given by their individual mean value

±10%. A result of a 100-sample calculation is shown in Figure 5a for the hot jet case at a directivity angle

of 50 degrees where all 12 parameters are nondeterministic and varied independently. This result shows that

the predicted spectrum using the mean parameter values (that is the original model fixed values, shown

by the red line) is not the same as the mean spectrum of an ensemble of spectra computed using model

parameters whose mean values of the distributions are the same as the original fixed parameters. The

variances or distributions of the uncertain model parameters have affected the output distribution that was

used to compute the mean spectrum. To emphasize this, Figure 5b shows an expanded view of the main

spectral peak where both the number of random samples and the range of the uniform distribution where

changed. The increased number of samples provides a better definition of the output distribution. This does

not change the output mean value significantly but does reduce the size of the confidence interval by 1/

√

N

as mentioned before. Reducing the variance of the nondeterministic parameter distributions by reducing

the range from ±10% to ±2% does show the points of the mean spectrum approaching the fixed parameter

spectrum.

The results in Figure 5a for the output variance from the Tam model, as expressed in the displayed

confidence interval, when 12 parameters were varied using uniform distributions are similar to the 1 parameter

distribution results shown in Figure 4. The confidence intervals shown in both figures are about 0.5 dB wide

at the peak. The vc1 parameter with the wider distribution has given as much output variance as all 12

parameters give with smaller distributions. By itself, the vc1 parameter with a uniform ±10% distribution

results in a peak frequency confidence interval width of 0.12 dB. The output confidence interval for each of

the other 11 parameters, computed separately, range from 0 to 0.68 dB in width at the peak. This indicates

that the uncertainty in each parameter does not have the same influence on the output uncertainty and that

computing output uncertainty due to individual parameter uncertainty neglects any interaction between

parameters. To determine which parameters do not have significant influence and to include the effects of

interactions between parameters, the variance-based decomposition technique is used to compute the total

effect sensitivity measure given by equation (1). All 12 parameters have uniform distributions with a range

of ±10% from the mean value. A result of this analysis is shown in Figure 6a for the cold jet case at a

NASA/TM—2010-216801 10
directivity angle of 50 degrees. The figure shows the ST i values for each parameter added on top of each

other as a function of the jet Strouhal number. The width of the ST i band is related to the influence of

the parameter on the output variance. Thus, non-influential parameters have small total effect sensitivity

measures.

Figure 6b shows the nondeterministic parameter spectra compared to the fixed parameter spectrum and

the measured data that corresponds to the results in Figure 6a. Comparing these two figures, we see in

the region of the major peak that 9 of the 12 parameters have small total effect measures. The important

parameters are those associated with the convected velocity (vc1), the lowest mode wave number of the shock

cell structure model (w11), and the half-width of the similarity source (hs1). From the results in the region

of the second peak, we include the parameter associated with the second mode wave number of the shock

cell structure model (w21) as being an additional influential parameter. Fixing the other 8 parameters, the

100-sample calculation with the Tam model was repeated. The resulting spectra are also shown in Figure 6b.

The sizes of the output spectral variance are about the same between the 12-parameter and the 4-parameter

distribution models. Thus, the sensitivity analysis has identified 8 relatively non-influential parameters in

the prediction model.

The same set of plots is shown in Figure 7 for the 90 degree inlet directivity angle. The peak frequency of

the spectra have moved to a higher Strouhal number of 0.6, and the total effect sensitivity measures reflect

this shift. The same 4 parameters as found at 50 degrees have the most influence and only the 4 parameter

nondeterministic spectra are shown compared to the fixed parameter spectrum and the measured data. The

50 degree and 90 degree sensitivity results are shown in Figures 8 and 9, respectively, for the hot jet case.

These results do not show any significant changes in the influential parameters especially in the peak regions

of the spectrum. In all these results, only the hs1 parameter has influence on amplitude uncertainty in the

peak frequency region. The other amplitude parameters such as cb1 and ab1 only have their most influence on

the output uncertainty at the higher frequencies. The parameters with consistent little influence are xc1 and

xc2 that are related to the jet potential core length model and vc2 which attempts to include temperature

effects in the model for convective velocity.

Finally, we consider whether using nondeterministic parameters leads to better predictions. An examination of the spectral results in Figures 6 to 9 shows that the mean spectrum computed after sampling 4

nondeterministic parameters with uniform distributions have lowered peak levels and increased valley levels.

At 50 degrees, the primary and secondary peaks have broadened due to the interaction of the changing wave

numbers and the source width. This broadening reduces peak-to-valley levels. In addition, at 90 degrees the

peak frequency of the mean spectrum shifts to a frequency lower than the fixed spectrum peak frequency and

has better agreement with the measured peak frequency. These changes appear relatively small for the added

complication of a nondeterministic calculation compared to the fixed parameter calculation. However, the

total uncertainty of the Tam model output cannot be determined without varying its internal parameters.

Consider the 4 parameters with the most influence on output spectrum uncertainty. Only the source width

related parameter hs1 was determined from a fit to acoustic data. Its influence is both through the mode

summation in equation (7) that determines spectral shape and directly as an amplitude parameter. The

other parameters vc1, w11, and w21 were determined in stability and waveguide models related to the jet

flow. The true level of uncertainty in these 4 parameters is unknown and small changes in these parameters

affect the output spectrum. If we sample these parameters using the ±10% uniform distribution and plot

the possible spectral outputs, then after 100 samples results such as those shown in Figure 10 are obtained.

A widely scattered set of spectra are shown where the fixed parameter spectrum is but one of the realized

spectra. The best alternative to choosing which parameter set is best is to account for the distribution of

the model parameters, compute the distribution of the output, and take the mean of the output distribution

with its known level of uncertainty as your prediction. This result has not necessarily given a better prediction method, compared to any particular set of measured data, but it does provide a quantified level of

uncertainty for the prediction and the most influential parameters were identified that may aid in developing

an improved model.

V. Concluding Remarks

Engineering applications in aircraft noise prediction rely on modeling of complex physical phenomena in

order to obtain solutions rapidly for design, development, and parametric studies. Parameters within these

models are usually fixed in value even though there are uncertainties related to those values. It may be more

NASA/TM—2010-216801 11
appropriate to allow a parameter to cover a range of values by replacing a fixed value with a distribution of

values. A nondeterministic calculation must then be performed. In this paper, a Monte Carlo method was

used. A random sample from one or more parameter distributions was chosen, processed by the prediction

model, and the output value recorded. This process was repeated until a distribution of output values was

obtained. From this output distribution, mean and confidence interval values were computed representing

the predicted solution and its uncertainty.

The above nondeterministic calculation process was applied to two models for the prediction of supersonic

jet broadband shock-associated noise. The first model, called the sJet code, was developed from a study

of far-field spectral data. It contained two nondeterministic parameters with statistical properties derived

from performing both linear and nonlinear regression. These parameters only affect the overall level of the

prediction and not its spectral shape. Using regression, the statistical properties of the parameters were

obtained and used to define the parameter’s normal distribution. Random samples from these distributions

were used to compute the output acoustic power levels. When averaging a large number of acoustic power

levels, random higher values will raise the average level. Only the results using nondeterministic parameters

derived from the nonlinear regression approach exhibited this behavior.

The second model, the Tam model, was developed from a derivation of the equations of motion followed

by using physical models to simplify the calculations. This model was used to demonstrate that the small

uncertainty in high quality measurements of the jet operating conditions used as code input yielded small

uncertainties in the code output. This is insufficient for demonstrating the model’s output uncertainty and

the uncertainty of internal model parameters must be taken into account. This model contained 12 internal

nondeterministic parameters for which estimates had to be made of their distributions in order to complete

the analysis. External reference data were used to estimate the vc1 parameter distribution. It was found that

the output spectrum confidence interval was not changed much between using a normal distribution and a

uniform distribution. Based on that result, all parameters were assigned a uniform distribution but with a

smaller variance. A global sensitivity analysis performed with these 12 nondeterministic parameters identified

8 non-influential parameters. Using the remaining 4 influential nondeterministic parameters, calculations

were performed to judge the benefit of this approach for predictive capability. The results showed that

the fixed parameter spectral result is only one possible solution and that the mean spectral result with its

accompanying confidence interval computed from a distribution of output spectra would be the preferred

result. This is the best way to quantify the uncertainty in the prediction from this model.

Appendix A – sJet Model Spectrum Calculation

The sJet code computes the far field radiated noise spectra from subsonic and supersonic jets. Both the

mixing noise and the broadband shock-associated noise spectra, as appropriate, are computed as a function

of the inlet directivity angle. For supersonic jets, the predicted radiated noise is modified depending on

whether the jet issued from a convergent or a convergent-divergent nozzle. In addition, if screech is present,

there is the ability to include noise amplification effects in the predicted spectra. The details of the modeling

are found in Khavaran & Bridges.4, 6 Here, only the equations for computing the radiated broadband shockassociated noise spectra are given. The formulation is a modification of the model given by Deneuville20 and

Gliebe et al.21

The following parameters are defined:

β – Shock parameter

β =

q

M2

j − M2

d



 (9)

where Mj is the fully expanded jet Mach number and Md is the nozzle design Mach number with

Md = 1 for a convergent nozzle.

βm – Modified shock parameter for convergent-divergent nozzles, βm = β for convergent nozzles.

For Md > 1.01, βm depends on whether the nozzle pressure ratio Pt/P∞ is greater or less than

the critical pressure ratio.

βm =









βM3/2

d

M−3/2

d

,

Pt

P∞

>

Pcrit

P∞

, under-expanded

βM−3/2

d ,

Pt

P∞

<

Pcrit

P∞

, over-expanded

(10)

NASA/TM—2010-216801 12
Lavg – The average spacing length between shock cells in the jet shock cell structure.

Lavg = 1.1De

q

M2

j − 1 (11)

where De is the nozzle exit diameter.

Uc, Mc – Convective velocity and Mach number

Uc = 0.7 (Uj − U∞) (12)

Mc =

Uc

c∞

(13)

Cuo – Sound level correction for convergent-divergent nozzles, Md > 1

Cuo =







−10{1.04 + tanh[4(βm − 1.49)]} , Mj > Md

−15 1.0 + tanh 7.9



β

2

M2

d

− 0.379, Mj < Md

(14)

θc – Critical angle

θc =







π , Mc ≤ 1

π − tan−1

p

M2

c − 1



, Mc > 1

(15)

For each inlet directivity angle θ`, ` = 1, 2, . . . , Nang, compute the shock-associated noise spectrum if

θ` < θc. First set the maximum sound pressure level

SPLMAX` = 10 m` log βm + B` + 42.37 + 10 log







Ns

8

π

4



De

R

2

1



1 −

U∞

c∞

cos θ`

4







(16)

where Ns is the number of shock cells (typically Ns = 8), R is the radial distance from the nozzle to the

microphone, and m` and B` are determined by the linear regression given by equation (3). If the nozzle is

convergent-divergent, then modify SPLMAX` by

SPLMAX` = SPLMAX` + Cuo Mj > Md . (17)

The fundamental frequency of the peak of the shock-associated noise spectrum is given by

fp,` =

Uc

Lavg (1 + Mc cos θ`)



1 −

U∞

c∞

cos θ`

 . (18)

Finally, compute the spectrum for each frequency for k = 1, 2, . . . , K.

ξk` =

fk

fp,`

(19)

S1 = SPLMAX` + 10 log Uj

Defp,`

+

(

10 log (ξk`)

7

, ξk` ≤ 1

10 log (ξk`)

−4

, ξk` > 1

(20)

S2 = SPLMAX` − 6 + 10 log Uj

Defp,`

+

(

10 log (ξk`/2)2

, ξk` ≤ 2

10 log (ξk`/2)−2, ξk` > 2

(21)

SPLk` = 10 log n10S1/10 + 10S2/10o(22)

NASA/TM—2010-216801 13
Appendix B – Weighting in Regression Analysis

The necessity to use weighting in the nonlinear regression formulation given in equation (6) is due to the

wide range of values found for yi = hp

2

ish. The acoustic power levels considered here cover many orders of

magnitude.

The residual error is related to the variation in the data value yi minus the estimated value ybi from the

nonlinear regression. At a particular data point after the fit, yiis a constant and the variation in the residual

error i = yi − ybi will follow the variation in ybi. Using the analysis in Bendat & Piersol,14 we construct an

error band for the data value yi based on the confidence interval for an averaged spectrum. Let the following

probability interval for the spectral density Gpp

Pr "

n

χ

2

n;α/2

Gbpp < Gpp <

n

χ

2

n;1−α/2

Gbpp#= 1 − α (23)

be the definition of a (1 − α)100% confidence interval for an averaged spectral density Gbpp(f) where Gbpp

has a chi-squared distribution with n-degrees of freedom and χ

2

n;α is the 100α percentage point of the χ

2

distribution. The total acoustic power is then computed as the sum of all components in the spectrum

hp

2

i =

X

K

k=1

Gbpp(fk) (24)

which is distributed as χ

2

K×n

. The confidence interval for hp

2

i is also determined by equation (23) with

(K × n)-degrees of freedom. The size of the variation in the data value is given by

∆hp

2

i =

K × n

χ

2

K×n;1−α/2

hp

2

i − K × n

χ

2

K×n;α/2

hp

2

i. (25)

Thus, if the variance of the residual error

Var [i] ∝ ∆hp

2

ii (26)

then by using weighting and applying equation (25)

i

√yi

=

yi − ybi

√yi

⇒ Var 

i

√yi



=

Var [i]

yi

∝

∆hp

2

ii

hp

2ii

= constant. (27)

By using the weighting given by equation (27), the variances of the residual errors are now equal as desired

for performing the nonlinear regression using equation (6). Furthermore, the distribution of the residual

error is nominally χ

2 but it approaches a normal distribution for a large number of degrees of freedom. For

the overall power level computed from a typical power spectral density with 6000 frequency bins averaged

400 times, the number of degrees of freedom is 2 × 400 × 6000 = 4.8 × 106.

Appendix C – Tam Model Detailed Equations

The parametric form of the detailed model equations for the Tam broadband shock-associated noise

prediction model, given in equation (7), are presented in this appendix. These equations follow from those

found in Tam.5

c – An amplitude factor where cb1 was determined by fitting the far field spectrum to measured data with

Stref = 0.35.

c(cb1) = cb1 × Stref (28)

L – The source model includes a representation of growing and decaying instability waves. L is the half-width

of a Gaussian function that replaces the amplitude envelope of the instability wave. The coefficient hs1 was

determined by fitting the far field spectrum to measured data. Mj,ref = 1.67 is a reference Mach number.

L(hs1, xc1, xc2) = hs1

xc(xc1, xc2, Mj )

xc(xc1, xc2, Mj,ref )

(29)

NASA/TM—2010-216801 14
xc – A model for the jet potential core length. The coefficients xc1 and xc2 were determined by fitting a line

to measured mean flow data.

xc(xc1, xc2, Mj ) =









xc1 + xc2



M2

j + 1 −

Tj

T∞

 Dj ,

Tj

T∞

≤ 1



xc1 + xc2M2

j



Dj ,

Tj

T∞

> 1

(30)

Dj – The diameter of the fully expanded jet derived by using mass conservation.

Dj = De

"

1 +

γ − 1

2

M2

j

 1 +

γ − 1

2

M2

d

−1

# γ+1

4(γ−1) 

Md

Mj

1/2

(31)

A

2

– An amplitude quantity that relates to the shock cell strength. The coefficients ab1 and ab2 were

determined to provide a best fit to acoustic data.

A

2

(ab1, ab2) =







A2

1



De

Dj

2

1 + ab1A

ab2

1

,

Mj

Md

≥ 1

A2

1

1 + ab1A

ab2

1

,

Mj

Md

< 1

(32)

where

A1 =



M2

j − M2

d





1 +

γ − 1

2

M2

d

(33)

fν – The peak frequency of the spectrum associated with the ν-th waveguide mode of the shock cell structure.

fν(vc1, vc2, w11, w12, w21, wν1) = Uc(vc1, vc2)kν(w11, w12, w21, wν1)

2π

(1 + Mc(vc1, vc2) cos θ)

−1

(34)

kν – The wave number for the ν-th waveguide mode of the shock cell structure. The coefficients w11, w12,

w21, and wν1 were chosen to provide better agreement with multiple-scales shock cell calculations.

kν(w11, w12, w21, wν1) =







1.1(w11 − w12Mj )k1,vs, ν = 1

w21k2,vs , ν = 2

wν1kν,vs , ν > 2

(35)

where kν,vs is the wave number for the vortex sheet shock cell structure and j0,ν is the ν-th zero of the zeroth

order Bessel function.

kν,vs =

2j0,ν

Dj

q

M2

j − 1

(36)

Uc – The convective velocity of large turbulent structures in the jet. The coefficients vc1 and vc2 were

determined from stability calculations

Uc(vc1, vc2) =







vc1Uj ,

Tt

T∞

≤ 1



vc1 − vc2



Tt

T∞

− 1

Uj ,

Tt

T∞

> 1

(37)

and the convective Mach number follows.

Mc(vc1, vc2) = Uc(vc1, vc2)

c∞

(38)

NASA/TM—2010-216801 15
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12Ratkowski, D. A., Nonlinear Regression Modeling: A Unified Practical Approach, Marcel Dekker, New York, NY, 1983.

13Bates, D. M. and Watts, D. G., Nonlinear Regression Analysis and Its Applications, Wiley, New York, NY, 1988.

14Bendat, J. S. and Piersol, A. G., Random Data Analysis and Measurement Procedures, Wiley-Interscience, New York,

NY, 2nd ed., 1986.

15Tam, C. K. W., “Forward Flight Effects on Broadband Shock Associated Noise of Supersonic Jets,” AIAA Paper No.

89-1088, 1989.

16Dahl, M. D., “Predictions of Supersonic Jet Mixing and Shock-Associated Noise Compared with Measured Far-Field

Data,” NASA TM-2010-216328, 2010.

17NASA, Assessment of NASA’s Aircraft Noise Prediction Capability, TP-20XX-215653, to be published.

18Harper-Bourne, M. and Fisher, M. J., “The Noise from Shock Waves in Supersonic Jets,” Noise Mechanisms, AGARDCP-131, 1974, pp. 11.1–11.13.

19Harper-Bourne, M., “On Modelling the Near-Field Noise of the High-Speed Jet Exhausts of Combat Aircraft,” AIAA

Paper No. 2002-2424, 2002.

20Deneuville, P., “Simplified Prediction Method of Shock Cell Noise of Supersonic Jets from Conical Nozzles,” SNECMA

YKA No. 5982/76, Societe Nationale d’Edute et de Construction de Moteurs d’Avion, France, 1976.

21Gliebe, P. R., Motsinger, R. E., and Sieckman, A., “High Velocity Jet Noise Source Location and Reduction. Task

6. Supplement. Computer Programs: Engineering Correlation (MS) Jet Noise Prediction Method and Unified Aeroacoustic

Prediction Model (M*G*B) for Nozzles of Arbitrary Shape; Final Report,” FAA-RD-76-79-6A, FAA, 1979.

NASA/TM—2010-216801 16
10-1 100 101

Strouhal Number

90

100

110

120

130

SPL (dB) per unit St

(a) Power spectral density at 50 deg., Mj = 1.493,

Pt/P∞ = 3.636, Tt/T∞ = 1.01

10-1 100 101

Strouhal Number

90

100

110

120

130

SPL (dB) per unit St

(b) Power spectral density at 90 deg., Mj = 1.493,

Pt/P∞ = 3.636, Tt/T∞ = 1.01

10-1 100 101

Strouhal Number

90

100

110

120

130

SPL (dB) per unit St

Measured Data

Tam Model

sJet Model

RANS-based Model

(c) Power spectral density at 50 deg., Mj = 1.494,

Pt/P∞ = 3.629, Tt/T∞ = 2.16

10-1 100 101

Strouhal Number

90

100

110

120

130

SPL (dB) per unit St (d) Power spectral density at 90 deg., Mj = 1.494,

Pt/P∞ = 3.629, Tt/T∞ = 2.16

Figure 1: Power spectral densities from 3 shock-associated noise prediction models compared to the shockassociated noise component of measured data, distance 100De.

NASA/TM—2010-216801 17
50 100 200 400 1000

Number of Samples

119

120

121

122

Mean y and 95% Conf. Int. 

y

± CI (dB)

! × 0.5, "mB = 0.00

! × 1.0, "mB = 0.00

! × 2.0, "mB = 0.00

! × 0.0

! × 0.5, "mB = 0.45

! × 1.0, "mB = 0.45

! × 2.0, "mB = 0.45

Figure 2: Mean (thick line) and 95% confidence intervals (CI)(thin line) for sampling equation (3) with

normally distributed m and B parameters. σ range applies to both parameters where σm = 0.756 and

σB = 1.930. Solid lines have uncorrelated parameters. Dashed lines have ρmB = 0.45. Gray area is region

of 95% CI when ρmB = 0. Symbol on right applies to 10000 samples.

50 100 200 400 1000

Number of Samples

117

118

119

120

121

Mean 10 log y and 95% Conf. Int. 10 log(

y

± CI)

! × 0.5, "mE = 0.00

! × 1.0, "mE = 0.00

! × 2.0, "mE = 0.00

! × 0.0

! × 0.5, "mE = 0.43

! × 1.0, "mE = 0.43

! × 2.0, "mE = 0.43

Figure 3: Mean (thick line) and 95% confidence intervals (CI)(thin line) for sampling equation (6) with

normally distributed m and E parameters. σ range applies to both parameters where σm = 0.538 and

σE = 0.177. Solid lines have uncorrelated parameters. Dashed lines have ρmE = 0.43. Gray area is region

of 95% CI when ρmE = 0. Symbol on right applies to 10000 samples.

NASA/TM—2010-216801 18
0.2 0.4 0.6 0.8 1 2

Strouhal Number

110

120

130

SPL (dB) per unit St

Measured Data

Fixed Spectrum

Mean Spectrum, Uniform

Mean Spectrum, Normal

Figure 4: Power spectral density comparisons at 50 degrees, distance 100De, Mj = 1.493, Pt/P∞ = 3.636,

Tt/T∞ = 1.01. Effect of replacing the vc1 parameter in the Tam model with a uniform distribution ±24%

and a normal distribution both with mean 0.708 and variance 0.00937. N = 100. Dashed lines for 95% CI.

0.2 0.4 0.6 0.8 1 2

Strouhal Number

110

120

130

SPL (dB) per unit St

Measured Data

Fixed Spectrum

Mean Spectrum

(a) Uniform ±10%, N = 100

0.3 0.4 0.5

Strouhal Number

120

122

124

126

128

SPL (dB) per unit St

Measured

Fixed

N=100, 10%

N=1000, 10%

N=10000, 10%

N=10000, 2%

(b) Expanded view, vary N and distribution size

Figure 5: Power spectral density comparisons showing spectra from the Tam model with 12 fixed parameters

and spectra from the Tam model with 12 parameters replaced by uniform distributions with ±10% range

with variance 0.00167 or with ±2% range with variance 6.68×10−5. Results at 50 degree inlet angle, distance

100De, Mj = 1.494, Pt/P∞ = 3.629, Tt/T∞ = 2.16.

NASA/TM—2010-216801 19
0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

0.0

0.5

1.0

1.5

2.0

Cumulated Total Effect STi

ab2

ab1

w

!1

w21

w12

w11

cb1

hs1

xc2

xc1

vc2

vc1

(a) Total effect sensitivity measures derived from variance-based decomposition for

the parameters listed in Table 1. Non-influential variables have smaller widths.

N = 10000.

0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

100

110

120

130

SPL (dB) per unit St

Measured Data

Fixed Spectrum

12 Parameters

4 Parameters

(b) Power spectral density at 50 degrees. Dashed lines for 95% CI. N = 100.

Figure 6: Effects of reducing the number of parameters with distributions from 12 to 4 in the Tam model

after determining variables to set constant using a variance-based decomposition sensitivity analysis. All

parameters set with uniform distributions ±10%. Results at 50 degrees, distance 100De, Mj = 1.493,

Pt/P∞ = 3.636, Tt/T∞ = 1.01.

NASA/TM—2010-216801 20
0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

0.0

0.5

1.0

1.5

2.0

2.5

Cumulated Total Effect STi

ab2

ab1

w

!1

w21

w12

w11

cb1

hs1

xc2

xc1

vc2

vc1

(a) Total effect sensitivity measures derived from variance-based decomposition for

the parameters listed in Table 1. Non-influential variables have smaller widths.

N = 10000.

0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

100

110

120

130

SPL (dB) per unit St

Measured Data

Fixed Spectrum

4 Parameters

(b) Power spectral density at 90 degrees. Dashed lines for 95% CI. N = 100.

Figure 7: Variance-based decomposition sensitivity analysis used to reduce the number of parameters with

distributions from 12 to 4 in the Tam model. All parameters set with uniform distributions ±10%. Results

at 90 degrees, distance 100De, Mj = 1.493, Pt/P∞ = 3.636, Tt/T∞ = 1.01.

NASA/TM—2010-216801 21
0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

0.0

0.5

1.0

1.5

2.0

Cumulated Total Effect STi

ab2

ab1

w

!1

w21

w12

w11

cb1

hs1

xc2

xc1

vc2

vc1

(a) Total effect sensitivity measures derived from variance-based decomposition for

the parameters listed in Table 1. Non-influential variables have smaller widths.

N = 10000.

0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

100

110

120

130

SPL (dB) per unit St

Measured Data

Fixed Spectrum

4 Parameter

(b) Power spectral density at 50 degrees. Dashed lines for 95% CI. N = 100.

Figure 8: Variance-based decomposition sensitivity analysis used to reduce the number of parameters with

distributions from 12 to 4 in the Tam model. All parameters set with uniform distributions ±10%. Results

at 50 degrees, distance 100De, Mj = 1.494, Pt/P∞ = 3.629, Tt/T∞ = 2.16.

NASA/TM—2010-216801 22
0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

0.0

0.5

1.0

1.5

2.0

2.5

Cumulated Total Effect STi

ab2

ab1

w

!1

w21

w12

w11

cb1

hs1

xc2

xc1

vc2

vc1

(a) Total effect sensitivity measures derived from variance-based decomposition for

the parameters listed in Table 1. Non-influential variables have smaller widths.

N = 10000.

0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

100

110

120

130

SPL (dB) per unit St

Measured Data

Fixed Spectrum

4 Parameter

(b) Power spectral density at 90 degrees. Dashed lines for 95% CI. N = 100.

Figure 9: Variance-based decomposition sensitivity analysis used to reduce the number of parameters with

distributions from 12 to 4 in the Tam model. All parameters set with uniform distributions ±10%. Results

at 90 degrees, distance 100De, Mj = 1.494, Pt/P∞ = 3.629, Tt/T∞ = 2.16.

NASA/TM—2010-216801 23
0.2 0.4 0.6 0.8 1 2 3 4

Strouhal Number

90

100

110

120

130

SPL (dB) per unit St

Sampled Spectra

Mean Output Spectrum

Tam Fixed Spectrum

Measured Data

Figure 10: Power spectral density comparisons at 50 degrees, distance 100De, Mj = 1.494, Pt/P∞ =

3.629, Tt/T∞ = 2.16. Scatter of output spectra from the Tam model computed using random samples of 4

parameters with uniform distributions ±10% with mean output spectrum shown. N = 100.

NASA/TM—2010-216801 24
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The Effect of Nondeterministic Parameters on Shock-Associated Noise Prediction Modeling 

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Dahl, Milo, D.; Khavaran, Abbas 

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WBS 984754.02.07.03.17.02 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

Engineering applications for aircraft noise prediction contain models for physical phenomenon that enable solutions to be computed quickly. 

These models contain parameters that have an uncertainty not accounted for in the solution. To include uncertainty in the solution, 

nondeterministic computational methods are applied. Using prediction models for supersonic jet broadband shock-associated noise, fixed 

model parameters are replaced by probability distributions to illustrate one of these methods. The results show the impact of using 

nondeterministic parameters both on estimating the model output uncertainty and on the model spectral level prediction. In addition, a global 

sensitivity analysis is used to determine the influence of the model parameters on the output, and to identify the parameters with the least 

influence on model output. 

15. SUBJECT TERMS

Jet noise; Noise prediction; Sensitivity analysis; Statistical analysis; Accuracy 

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