Abbas Khavaran

Science Applications International Corporation, Cleveland, Ohio

James Bridges

Glenn Research Center, Cleveland, Ohio

An Empirical Temperature Variance Source Model 

in Heated Jets 

NASA/TM—2012-217743

October 2012
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Abbas Khavaran

Science Applications International Corporation, Cleveland, Ohio

James Bridges

Glenn Research Center, Cleveland, Ohio

An Empirical Temperature Variance Source Model 

in Heated Jets 

NASA/TM—2012-217743

October 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

This work was sponsored by the NASA Fundamental Aeronautics Program. The authors are grateful to the Acoustics Branch, 

NASA Glenn Research Center for supporting this research, and to Drs. Stewart Leib and James Debonis for their constructive 

comments and suggestions. The authors also acknowledge Mr. V.A. Bhagwandin, currently with the U.S. Army Research Laboratory 

in Aberdeen Maryland, for providing the Wind-US solution to the jet confi gurations listed in Table 2.

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Available electronically at http://www.sti.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. 

This report is a formal draft or working 

paper, intended to solicit comments and 

ideas from a technical peer group.

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
 

ABSTRACT

 An acoustic analogy approach is implemented that models the sources of jet noise in heated jets. The equivalent 

sources of turbulent mixing noise are recognized as the differences between the fluctuating and Favre-averaged 

Reynolds stresses and enthalpy fluxes. While in a conventional acoustic analogy only Reynolds stress components 

are scrutinized for their noise generation properties, it is now accepted that a comprehensive source model should 

include the additional entropy source term. Following Goldstein’s generalized acoustic analogy, the set of Euler 

equations are divided into two sets of equations that govern a non-radiating base flow plus its residual components.

When the base flow is considered as a locally parallel mean flow, the residual equations may be rearranged to form 

an inhomogeneous third-order wave equation. A general solution is written subsequently using a Green’s function 

method while all non-linear terms are treated as the equivalent sources of aerodynamic sound and are modeled

accordingly. 

 In a previous study, a specialized Reynolds-averaged Navier-Stokes (RANS) solver was implemented to 

compute the variance of thermal fluctuations that determine the enthalpy flux source strength. The main objective 

here is to present an empirical model capable of providing a reasonable estimate of the stagnation temperature 

variance in a jet. Such a model is parameterized as a function of the mean stagnation temperature gradient in the jet, 

and is evaluated using commonly available RANS solvers. The ensuing thermal source distribution is compared 

with measurements as well as computational result from a dedicated RANS solver that employs an enthalpy 

variance and dissipation rate model. Turbulent mixing noise predictions are presented for a wide range of jet 

temperature ratios from 1.0 to 3.20.

Nomenclature 

c Sound speed

C , Cτ Empirical constants

DJ Jet diameter

h Enthalpy

ht Stagnation enthalpy ( h + v2

/ 2 ) 

k Wave number 

k Wave number magnitude (ω/c∞ )

κ Turbulent kinetic energy

ε Turbulent dissipation rate

 Turbulence length-scale

Ma Acoutsic Mach number (U/ c∞ )

M j Aerodynamic Mach number at jet exit

An Empirical Temperature Variance Source Model 

in Heated Jets

Abbas Khavaran 

Science Applications International Corporation 

Cleveland, Ohio 44135 

James Bridges 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

NASA/TM—2012-217743 1
 

NPR Nozzle plenum pressure ratio

NTR Nozzle plenum temperature ratio

ω Radian frequency (2π f )

mj Momentum variable (ρv′

j)

π′ Normalized pressure fluctuation 

ρ Density

p Pressure

Pref Reference acoustic pressure (0.0002 μbar)

R Arc distance from jet exit

r Radial distance from jet centerline

R Gas constant 

ξ Spatial separation vector

t Time

τ Time delay

τ o Turbulence time-scale

U Mean axial velocity

vi Velocity components

v′

i Turbulent velocity component

θ Polar angle from downstream axis 

x Observer location 

y Source location

δ ij Kronecker delta

Subscripts

c Convection variable

∞ Ambient condition

t Stagnation value

j Jet exit conditions

Superscripts

– Time average

 Favre average ( q = ρq / ρ ) 

〈 〉 Variance

′ Fluctuating quantity

s Source location

1.0 Introduction

 The Fundamental Aeronautics program at NASA is tasked with achieving technological capabilities necessary to 

overcome air transportation challenges that include reduced noise, emissions, and fuel consumption. The U.S. 

commercial aviation traffic is projected to increase by 70% in the period 2010 – 2030; subsequently more people 

will be exposed to air traffic noise in the communities surrounding airports. Recent noise metrics require a 

cumulative (cutback, sideline and approach) noise reduction in the subsonic transport of 32dB by 2015 and 71dB by 

2025 (referenced to 737-800 aircraft with CFM56 engines). Drastic changes in both propulsion system and engine 

placement are required to achieve these objectives. Computational tools are under research and development that 

address component noise from a host of structural components such as wing flaps, slats, landing gears, as well as 

propulsion system elements such as engine inlet, combustion and exhaust noise. Today’s high bypass ratio engines 

NASA/TM—2012-217743 2
 

have mitigated jet noise substantially such that jet noise during takeoff is now slightly less than fan exhaust-noise, 

and is nearly on par with fan noise along the sideline. In this article, we primarily focus on the turbulence-generated 

noise that is a product of the mixing of the jet exhaust flow with the surrounding air. Since jet mixing-noise is 

broadband with significant levels of noise within a three-octave frequency band, a successful predictive capability 

needs to incorporate sufficient physics to produce a reasonably accurate spectrum over a broad range of conditions.

As part of an ongoing research effort at the NASA Glenn Research Center, a RANS-based jet noise prediction 

methodology (also referred to as physics-based) is being developed to provide the state-of-the-art in source 

modeling and propagation. The goal is to achieve practical levels of accuracy in noise prediction from a generalized 

modeling methodology to help designers with concept evaluation and down selection within a reasonable time 

frame. Physics-based prediction methods have an advantage over the empirical models when design concepts fall 

outside the envelope of the latter method. Empirical models, on the other hand, are computationally efficient and 

may be more advantageous when noise is one component in a more general system study. 

 There have been numerous efforts in the past half-century to experimentally measure the actual sources of 

aerodynamic sound. Measurement techniques such as the phased array measure the relative source strength and its 

frequency content at various regions in a jet. Particle Image Velocimetry (PIV) methods have been instrumental in 

turbulence and correlation measurements. When combined, the two techniques may establish a link between the 

cause and effect, i.e. turbulence and its ensuing far-field sound. However, as pointed out by Ffowcs Williams [Ref. 

1] “there is no uniqueness theorem to guarantee that the source measured by any one of the several existing source 

location schemes is actually the origin and cause of the observed sound field.” Kirchhoff’s theorem shows that the 

same sound field could be generated by sources of different origin. In jets, our physical intuition leads us to focus 

on the transition region in order to link the flow unsteadiness (which is at its peak in this region) to the origin of jet 

noise. 

In his pioneering work, Lighthill [Ref. 2] defined the equivalent sources of aerodynamic sound as the double 

divergence of the stress tensorT

ij =ρvi

v

j

+ ( p − c∞

2

ρ)δ

ij

−τ

ij , where ρ denotes the density, 

v = (ν1

,ν 2,ν 3) is the 

fluid velocity, p is the pressure, is the ambient sound speed, and τ

i j is the viscous stress tensor. In this 

formalism, the actual noise generating fluid is replaced with a distribution of quardrupole sources ∂2

Tij /∂xi∂xj in a 

quiescent medium at a constant sound speed c∞ . For all practical purposes, viscous stresses are considered as 

relatively unimportant in noise generation. Additionally when jets are isothermal, the pressure/density 

difference p−c∞

2

ρ is ignored and subsequently the momentum flux ∂2

(ρviv

j

) /∂xi∂x

j is regarded as the primary 

source of sound. The U8 power law, which is the most quoted scaling law in aeroacoustics, is derived from 

Lighthill’s acoustic analogy when the source is represented as the momentum flux term. It concludes that the 

acoustic power emitted over the surface of a large sphere surrounding a jet is proportional to the eight-power of the 

jet exit velocity. Lighthill’s definition of the equivalent sources of aerodynamic sound has been a subject of wide 

spread scrutiny and interpretations – and other scaling laws have been proposed that relate the far-field sound to 

flow parameters such as velocity and temperature in a variety of forms. 

Experimental observations by Tanna [Ref. 3] and more recent measurements by Bridges et al. [Ref. 4] and 

Viswanathan [Ref. 5] show that the effect of heat addition on jet noise depends on the jet speed. At low speeds, i.e. 

acoustic Mach number of 0.50 and below, heat amplifies the low to mid frequency jet noise with minimal effect at 

the high frequency. At high speeds (i.e. above acoustic Mach numbers of 0.90), it reduces jet noise at all 

frequencies. In between, e.g. at acoustic Mach number of 0.70, the heated spectrum crosses over the unheated 

spectrum due to enhancement of the low-frequency noise and slight weakening of the high-frequency amplitude.

These observations have compelled many researchers in the field to examine the entropy source term ( p −c∞

2

ρ) as a 

potential contributing source. For example, Morfey et al. [Ref. 6] offered a new two-scaling hypothesis, in favor of 

and power laws, and with some dependence on the mean temperature gradient. Methodical examination of 

jet noise data [Ref. 7, 8] reveal a more general power law AU n

, where the amplitude A and exponent n depend on 

NASA/TM—2012-217743 3
 

observer angle as well as jet temperature. A recent development in the utilization of scaling laws for jet mixing 

noise and broadband shock-associated noise in axisymmetric jets is detailed by Khavaran and Bridges [Ref. 9]

Lilley [Ref. 10] proposed a second order wave equation for the pressure fluctuations, similar to Lighthill’s, but 

replaced ∂( p −c∞

2

ρ) / ∂t with a different expression that explicitly displays its isentropic (dipole) and nonisentropic (monopole) components. His contribution to the total acoustic power consisted of a dipole term

∂(ρviv4 ) / ∂xi , which depends on the enthalpy gradient through v4 ≡(γ − 1)(h∞ − ht ) where subscript t denotes a 

stagnation value and γ is the specific heat ratio. In Lilley’s formulation this term exhibits a U6 jet noise scaling, 

and the monopole term is neglected.

 The governing acoustic equations presented in the following section are written for axisymmetric jets, and 

assume a unidirectional, transversely sheared mean flow. Under such idealized conditions, the propagation 

equation simplifies to the second-order compressible Rayleigh operator. The resulting Green’s function has a 

regular singular point in the vicinity of the critical layer at r where 1−U(r)cosθ / c∞ = 0 . Goldstein [Ref. 11] 

describes a weakly non-parallel mean flow analysis to eliminate this singularity. Additional simplification in the 

source model is achieved when we consider the mean static pressure as a constant. In the final analysis two distinct 

space-time correlations of turbulent fluctuations in the jet need to be examined for source modeling:

 Source type-1: a fourth-rank autocovariance tensor with velocity/velocity fluctuations (v′

i

v′

j)A (v′k v′l)B

 Source type-2: a second-rank tensor with velocity/stagnation-enthalpy fluctuations (v′

i

h′

t )A (v′j

h′

t )B . 

Subscripts A and B denote two points separated in space and time, and over-bar is a time-averaged value. Source 

type-1 is known as a momentum flux source term; while source type-2 is an enthalpy flux term. A complementary 

type-3 source is also present due to coupling between momentum flux and enthalpy flux terms, which is also 

referred to as momentum flux–enthalpy flux term. This source is a third-rank tensor (v′

j

h′

t )A (v′k v′l)B , and is 

generally small compared to type-2 source. An estimate of the relevance of type-3 source along the jet lip line as 

provided by Afsar et al. [Ref. 12] suggests that this source might become relatively significant compared to type-2 

source at high Mach numbers. While source type-2 always makes a positive contribution to the spectrum, source 

type-3 is projected to make a positive contribution at high subsonic Mach numbers and a negative contribution at 

supersonic Mach numbers when the observer angle is close to the downstream jet axis. This latter source is 

tentatively ignored here and is not pursued any further. 

In a RANS-based jet noise prediction approach, such as JeNo code that is used in the present computations 

[Ref. 13], the non-radiating background flow is usually taken to be the jet mean flow as calculated with commonly 

available computational fluid dynamic codes that use a Reynolds-averaged Navier-Stokes (RANS) solution with 

some form of a κ-ε turbulence model. The turbulent kinetic energy κ = v′

i

v′

i /2 and its dissipation rate ε are 

sufficient to evaluate the source strength as well as its length- and time-scales in unheated jets, which are dominated 

by source type-1. The variance in stagnation enthalpy is an additional factor that enters the picture as a source of jet 

noise in heated jets (source type-2). A specialized RANS solver was discussed in a previous study [Ref. 14, 15] 

that predicted the variance in stagnation temperature (or enthalpy) in addition to the standard turbulence-related 

parameters. The approach makes use of a baseline κ-ε turbulence model with an additional two-equation scalar 

variance model. Determination of the scalar variance variable is achieved by selecting the appropriate mean flow 

quantity gradient for its production term. In jet noise the model traces the variance in stagnation temperature by 

choosing the gradient in total enthalpy as its production term. The Total Enthalpy Variance model is available as an 

option within the CARFT code [Ref. 16] and will be referred to as CRAFT-TEV. In order to achieve some level of 

confidence in the computational algorithm, the predicted stagnation temperature variance was compared [Ref. 17]

with Rayleigh scattering measurements at a limited number of set points. 

Here an Empirical Temperature Variance (ETV) model is proposed in connection with the type-2 source strength. It

employs a standard RANS solvers, such as CRAFT with a standard κ-ε turbulence model [Ref. 16] or Wind-US

[18, 19] to evaluate the variance in stagnation temperature. Iterations are carried out across a range of jet conditions 

NASA/TM—2012-217743 4
 

to fine tune empirical parameters present in the model. Subsequently the thermal source distribution as evaluated 

with the ETV model is compared with the computational values obtained directly from a dedicated RANS solver

(CRAFT-TEV), as well as with measurements.

The remainder of this paper is organized as follows. The governing acoustic equations in the specialized 

case of a unidirectional parallel mean flow are discussed in section 2. In section 3 we discuss the empirical ETV

model and use that to predict the intensity of thermal sources. Computational fluid dynamics solutions are presented 

for a host of heated subsonic jets using CRAFT-TEV solver, and the variance in total temperature is assessed 

against the ETV model as well as Rayleigh scattering measurements. 

Section 4 presents jet noise predictions using three methods for evaluating the thermal source strength:

 

 (i) Flow input from CRAFT code with a total enthalpy variance model (CRAFT-TEV) .

 (ii) Flow input from CRAFT with a standard κ-ε model (CRAFT-κε) , using ETV source model.

(iii) Flow input with Wind-US and a standard κ-ε model (Wind-κε), using ETV source model . 

The latter RANS solver, i.e., Wind-US [Ref. 18, 19] is additionally used as the flow solver of choice when noise 

predictions are presented at a wider range of jet conditions shown in Table 2. Jet noise spectra are compared with the 

far-field noise measurements gathered at the Small Hot Jet Acoustic Rig (SHJAR) facility at the NASA Glenn 

Research Center [Ref. 20]. A summary is provided in section 5.

2.0 Acoustics Equations 

The far-field sound spectral density at observer location 

x , per unit volume of turbulence at source location 

y , is 

(see Ref. 13, Appendix C)

p2(



x,



y,ω) = G*(



x,



y −



ξ/2,ω)G(



x,



y +



ξ/2,ω)q(



y,



ξ,τ )e

iωτ dτ

−∞

+∞

∫ 

ξ

∫ d



ξ , (1) 

where G is the Green’s function (GF), and q is the two-point space-time correlation between sources of turbulent 

mixing noise at two points A and B separated by space 



ξ and time τ . The product of the GF and its conjugate 

(denoted by *) may be evaluated at the center of the correlation times a phase factor exp(−i



k.



ξ ), where wave 

number 

k is directed as (



x − y) and has a magnitude k = ω/c∞ . 

p2(



x,



y,ω) = G(



x,



y,ω)

2

q(



y,



ξ,τ )e

iωτ e−i



k .



ξ dτ

−∞

+∞

∫ 

ξ

∫ d



ξ (2) 

The above expression is written in a fixed frame of reference. Since the sources of jet noise are convecting

downstream with a convection velocity Uc , the Green’s function G should also conform to that of a convecting 

singularity. As the fixed- and moving-frame source correlation functions are related q(



y,



ξ,τ ) = qm (



y,



ξ m ,τ ), 

where 

ξ m = ξ − ˆiUcτ , it is readily shown that 

q(



y,



ξ,τ )e

iωτ e−i



k .



ξ dτ

−∞

+∞

∫ 

ξ

∫ d



ξ = qm (



y,



ξ m ,τ )e

iωsτ

e

−i



k .



ξm dτ

−∞

+∞

∫ 

ξm

∫ d



ξ m , (3)

where source and observer frequencies are related through the Doppler factor as ω = ωs

/(1 − Mc cosθ) . Polar angle 

θ is with respect to downstream axis, and Mc = Uc /c∞ is the convection Mach number for a noise-generating 

turbulent eddy. The convection velocity Uc (



y) is set equal to an appropriate weighted average of the jet exit 

NASA/TM—2012-217743 5
 

velocity Uj and local mean velocity U(



y). It is noted that transformation (3) is simply an identity, and the true 

effect of source convection on the radiated sound is included in the GF. 

The governing acoustic equation in a locally parallel mean flow (with a constant mean static pressure p ) is the socalled variable density inhomogeneous Pridmore-Brown equation 

Lπ′ = Γ, π′ ≡ p′

γ p

, (4)

where L is a third-order wave operator

L ≡ D(D2− ∂

∂x j

c

2 ∂

∂x j

) + 2c

2 ∂U

∂x j

∂2

∂x1∂x j

 , (5)

D is a differential operator D ≡ ∂ / ∂t + U(x2 , x3 )∂ / ∂x1 , and the sound speed is evaluated from the Favreaveraged local enthalpy c

2 = (γ − 1)

h = γ RT . It is noted that in deriving the acoustic equation we require that 

both fluctuating variables and sources have a zero mean; consequently the true equivalent sources of aerodynamic 

noise are the differences between the fluctuating and Favre-averaged Reynolds stresses and enthalpy fluxes [Ref. 

11]. An expanded form of source term Γ was given in [Ref. 17], where various terms were compared for their 

relative significance to the noise generation. Here we adopt the following approximation

 

Γ ≅ D

∂2

v′

i

v′

j

∂xi∂x

j

− D2 ∂

∂x

j

(v′

j

h′

o



h

) , (6)

where h′

o ≡ h′ + v′i

v′

i / 2 , and it is recognized that the divergences of both terms ρv′

i

v′

j and ρv′j

h′

o are zero in the 

locally parallel mean flow approximation (or are of higher order in a slowly varying mean flow). We also replace 

h′

o with h′t , where h′

t = h′o − v′i

v′

i

 / 2 + viv′

i is the true fluctuation in stagnation enthalpy with a zero mean, h′t = 0 . 

When the mean flow is axisymmetric, the GF to equation (4) corresponding to a convecting singularity

L(Ge−iωt) = D c∞

2 exp(−iωs

t)δ(x1 − Uct)δ(r − r

s ⎡ ) ⎣ ⎤

⎦ (7)

is given as

G(



x,



y,ω) = −i

4π R

(1− M scosθ)

(1− Mc cosθ)

e

i kR f (m)

(rs,k,θ)cosm(φ − φs) m=0

∞

∑ , (8)

where superscript s denotes a source location, and f (m)

is solved numerically as a solution to the self-adjoint 

second-order compressible Rayleigh equation

NASA/TM—2012-217743 6
 

d

dr

r

Φ2

dg m

dr

⎡

⎣

⎢ ⎤

⎦

⎥ +

r

Φ2 k 2

(Φ2 − cos2θ) − m2

r 2

⎡

⎣

⎢ ⎤

⎦

⎥ g(m) = 0, m = 0,1,2,

Φ2 ≡ ρ

ρ ∞

(1− U cosθ / c∞ )

2

, f (m) ≡ g(m)(r,k,θ) / (1− U cosθ / c∞ )

3

.

 (9)

where k = ω/c∞ , R is the far-field arc distance from jet exit, φ is the azimuthal angle in cylindrical coordinates, and 

Mc= Uc /c∞ . The number of azimuthal modes m required for a converged summation in equation (8) increases with 

frequency. The numerical solution to equation (9) starts with f

(m) = amrm as r → 0 [Ref. 21], and continues to the 

jet boundary where the matching conditions are applied to determine the coefficients am . 

Using the quasi-normal approximation for the joint probability distribution of turbulence [Ref. 22], the fourth-order 

autocovariance tensor (v′

i

v′

j)A (v′k v′l)B is written as a product of second-order tensors. The second-order correlations 

may be approximated using an isotropic turbulence with separable space- and time factors. Both spatial and 

temporal decay factors of the correlation are well represented by exponential functions 

f (ξ) = exp(−πξ / ), h(τ ) = exp(− τ / τ o ) . (10)

Here, the two-point (second-order) velocity-correlations are modeled as

(v′

i

)A(v′

j

)B = R

ij(



ξ − ˆiUcτ )h(τ ) (11) 

where the spatial decay is described according to Batchelor’s isotropic turbulence model [Ref . 23] 

R

ij(



ξ ) = v′

1

2

[( f +

1

2

ξ f ′)δ

ij − 1

2

ξiξ j

ξ f ′], f ′= ∂f /∂ξ, ξ

2

= ξ1

2

+ ξ 2

2

+ ξ 3

2

. (12) 

Following the quasi-normal assumption, the spectral density for the axial component of tensor 

(v′

1

v′

1

)A(v′

1

v′

1

)B = 2 v′

1Av′ ( ) 1B

2

becomes

 I

1111(



y,ω) = 4

3

5π2 v′

i

v′ ( )i

2

H(ω)N1(k); H(ω)= h2(τ )exp(iωsτ ) −∞

+∞

∫ dτ = τ o

1+ (ωsτ o/2)2 (13)

Turbulence length- and time-scales are calculated in the usual way from the turbulent kinetic energy and its 

dissipation rate as  = C

κ1.5/ ε andτ o

= Cτκ /ε , and N1 (k) represents the source non-compactness factor that is 

evaluated by considering separation vector as aligned with the direction of the wave number 

k [Ref. 17, Appendix 

B]. Constants C and Cτ were determined previously by calibration with unheated jet noise data when momentum 

flux was the only source term. These constants remain unchanged here and are used for both type-1 and type-2 

sources. More elaborate source models are available in the literature that consider the turbulence as non-isotropic, 

and/or with multiple length-scales, or frequency-dependent length-scales. For example Leib and Goldstein [Ref. 24] 

proposed a non-separable axisymmetric tensor to model source type-1 space-time correlations – and used different 

length-scales in the stream-wise and transverse directions. Our goal at the present time is to keep the model simple 

NASA/TM—2012-217743 7
 

and with minimum empiricism with regard to its directionality and scales. Several assumptions were proposed in 

modeling the enthalpy flux correlation tensor [Ref. 17]. 

First, we apply the quasi-normal approximation to the autocovariance tensor for type-2 source, and express the 

fourth-rank tensor as a superposition of second-rank tensors. Next we assume that the coherence spectra 

γ 2 ≡ RvT ( f )

2

/ Rvv ( f )RTT ( f ) is small compared to unity, where RvT ( f ) is a Fourier transform of the two-point 

correlation function RvT (τ ) ≡ v′

i(t)T ′t(t + τ ) . Subsequently we arrive at the following 

(v′

i

h′

t

) A (v′

j

h′

t

)B ≅ (v′

i

) A (v′

j

)B (h′

t

) A (h′

t

)B . (14)

The enthalpy correlation function is modeled using spatial and temporal functions f (ξ) and h(τ ) defined in 

equation (10)

(h′

t

) A (h′

t

)B = (h′

t

)

2 f (ξ)h(τ ). (15)

Following the steps outlined in [Ref. 17], the spectral density for the two source terms in equation (6) may be 

expressed as 

p2(



x,



y,ω) = ρ ∞

2

I

1111 (



y,ω) k 4(1−Ucosθ / c∞ )

2

(cos2θ + Qsinθ)

2

(4π R)

2

(1−Mccosθ)

2 ×

 

A(cos2

θ + Qsinθ)

2

+B(1− Ucosθ / c∞ )

2 15

16

c∞

2

κ

h′

t

2



h2

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(1+ δ m0 )

m=0

∞

∑ f (m)

f *(m)

 (16)

where Q2

≡ (ρ/ρ ∞)(1 − Ucosθ / c∞ )

2

− cos2θ . Constants A and B are determined by calibrating with jet noise data. 

3.0 Subsonic Flowfield Simulations

A set of eight subsonic jets within the Tanna matrix [Ref. 25] is listed in Table 1. A 2-inch (5.08cm) 

diameter convergent nozzle was considered for flow simulation and jet noise prediction. This set was also examined 

in a previous study [Ref. 17] for flow simulation with the CRAFT-TEV model. Here we present some of the earlier 

computations using CRAFT-TEV solver with an emphasis on the distribution of the temperature variance in the jet 

and comparison with data obtained with a Rayleigh scattering technique [Ref. 26, 27]. We also examine an 

empirical temperature variance (ETV) model that evaluates the total temperature variance using a function of mean 

stagnation temperature in the flow. A more comprehensive set of subsonic conditions will be considered later on at 

the conditions of Table 2 as we implement the ETV model in conjunction with the Wind-κε RANS solver [Ref. 18]. 

Each set point (sp) in Table 1 identifies a jet at a nozzle plenum pressure ratio (NPR), stagnation temperature ratio 

Tt / T∞ , exit static temperature ratio Tj /T∞ , aerodynamic Mach number M j = Uj / cj , and acoustic Mach number 

Ma = Uj / c∞ . These conditions have been studied extensively in the past using the CRAFT-κε RANS solver [Ref. 

16] to predict turbulence in the transition region where most of the jet noise is produced. Comparisons were made 

previously with available PIV data collected at SHJAR facility [Ref. 4, 20]. Simulations with the CRAFT code use 

a 316x171 computational grid, which includes the internal nozzle region well upstream of the exit plane. The 

computational domain extends 50 jet diameters in the streamwise direction from the nozzle exit, and 12.5 diameters

NASA/TM—2012-217743 8
 

in the radial direction, sufficiently distant to minimize boundary condition placement impact on the shear layer 

entrainment path-lines. Nozzle flow boundary conditions were prescribed uniformly as inflow stagnation conditions 

for pressure and temperature imposed well upstream of the nozzle exit. The predicted centerline decay of mean 

axial velocity and turbulent kinetic energy were presented in [Ref. 17] as a function of Witze’s similarity parameter 

and will not be repeated here. 

3.1 An Empirical Temperature Variance (ETV) Source Model

The thermal source strength, see second term in equation (16), is proportional to h′

t

2

/ h2

. We intend for 

the ETV source strength ST to resemble h′

t / 

h in its distribution, and to approach zero under unheated jet 

conditions Tt / T∞→ 1 . 

We define

ST = dTt

dr

Dj

T∞

⎛

⎝

⎜

⎞

⎠

⎟

α

(1 − 1 / NTR)

δ

6 β , 

where

β = βo + (1 − βo ) x1 / Lc , x1 / Lc ≤ 1

1 , x1 / Lc > 1

α = 0.20, βo = 0.70, δ = 1 + 1 / (3 NPR)

 (17)

Axial distance x1 is measured from the jet exit, and Lc is the length of the potential core in the jet. For example, 

Witze’s axial similarity parameter xw in subsonic jets may be used to evaluate Lc [Ref.28]

 

xW = 0.08 ρ ∞/ρ j − 0.16U j

/c ( ∞ )(ρ ∞/ρ j)

−0.22 x1

. (18)

Parameters NTR and NPR in equation (17) denote the jet plenum temperature ratio and pressure ratio, respectively.

Sample RANS predictions with CRAFT-TEV and comparison with the Rayleigh scattering measurements are shown 

in figures 1 to 4. The non-intrusive molecular Rayleigh scattering technique [Ref. 26, 27] is used to measure jet 

velocity, static temperature and density simultaneously at a sampling rate of 10 kHz, and the root-mean-square (rms) 

in total temperature fluctuations < T ′

t > , is obtained as the variance in (T + v1

2

/ 2cp ). Measurements and 

predictions are presented separately, as parts (a) and (b), in order to avoid overcrowded figures. Figures 5 to 8 show 

the axial turbulent velocity component ( u′/Uj ) at indicated streamwise locations as measured with the Rayleigh 

scattering methods. Part (b) in each figure shows a comparison with CRAFT predictions assuming u′

2

= 2κ / 3 . 

Although the centerline values near the jet exit appear less agreeable in several cases, nonetheless the experimental 

data agree qualitatively with the computational results in their peak values as well as radial profiles. Thermal 

fluctuations, in general, display a distribution similar to that of the turbulent kinetic energy. 

 

Figures 9 to 15 show computational results for the square root of the normalized turbulent kinetic energy κ0.50 /Uj

and enthalpy source strength < T ′

t >/T (or equivalently h′

t / 

h ) at conditions of Table 1. A third contour plot 

displays the ETV source strength ST using input CRAFT-κε. The empirical temperature variance model shows a 

source distribution similar to CRAFT-TEV, although small differences in intensity level are noticeable at several 

NASA/TM—2012-217743 9
 

conditions. The source strength is practically insignificant at unheated conditions of sp03 (Figure 15) and sp07 (not 

shown here). We use both ETV model and CRAFT-TEV in our noise predictions later on, and show that the two 

results agree to within the experimental uncertainty in noise measurements. 

Table 1. Tanna Matrix Set Point Conditions

sp NPR Tt T∞ Ma

03 1.197 1.000 0.950 0.50 0.51

23 1.103 1.810 1.760 0.50 0.37

42 1.066 2.750 2.700 0.50 0.30

07 1.861 1.000 0.840 0.90 0.98

27 1.361 1.922 1.760 0.90 0.68

46 1.225 2.861 2.700 0.90 0.54

29 1.890 2.114 1.760 1.33 1.00

49 1.692 3.138 2.700 1.48 0.90

4.0 Acoustic Results 

Noise predictions are carried out according to equation (16) with the correlation coefficient I1111 defined in 

equation (13). A RANS flow solver provides the mean flow as well as the turbulent kinetic energy κ and 

dissipation rate ε , which are necessary in evaluating source type-1. Source type-2 is evaluated using either a 

dedicated RANS solver (CRAFT-TEV), or a standard RANS solver combined with the ETV model described in 

equation (17). Jet noise spectra are evaluated as a constructive contribution from the two source types, and are 

presented as lossless at arc distance R/Dj =100 . Acoustic results are displayed as the sound spectral density per 

Strouhal number (SPD-dB) vs. Strouhal frequency (St) where 

SPD ≡ 10Log p2Uj / Pref

2

( ) Dj , St = f Dj

/Uj . (19)

4.1 Enthalpy Source Noise 

Spectral predictions are carried out at the subsonic jet conditions listed in Table 1 using two different flow 

solvers and three methods for computing thermal source strength as described in section 1. Jet noise predictions are 

compared with narrow-band spectral measurements gathered at the SHJAR facility at the NASA Glenn Research 

Center [Ref. 4, 20].

 

Intensities for type-1 and type-2 sources were examined earlier in figures 9 through 15. The predicted sound 

spectral density is shown at three inlet angles of 90o

, 120o, and 150o in figures 16, 17 and 18. Computations were 

carried out with RANS input from CRAFT code, and with the type-2 source evaluated according to; (i) empirical 

model (ETV); (ii) CRAFT-TEV model. Figure 16 corresponds to two unheated jets at set points sp03 and sp07. 

Figure 17 shows three jets at a common exit static temperature ratio of 1.76 and at acoustic Mach numbers of 0.50, 

0.90, and 1.33. Similarly, figure 18 corresponds to jets at a common static temperature ratio of 2.70 and at acoustic 

Mach numbers of 0.50, 0.90, and 1.48. In general the two predictions are in close agreement. This indicates that the 

empirical model is a viable substitute for type-2 source strength; at least at jet conditions examined here, and could 

replace the enthalpy variance model available in CRAFT-TEV code as an alternative means of evaluating h′

t / 

h

source strength. The two spectral predictions coincide at unheated conditions due to insignificant contributions from 

NASA/TM—2012-217743 10
 

type-2 source (Fig. 16). Overall, the computations are in close agreement with measurements. The major 

discrepancy arises at high speeds, i.e. at set points sp29 and sp49 (figures 17c and 18c), where the predicted high 

frequency noise attenuates faster than measurements and lacks a broadening at the peak that is commonly measured 

at aft angles in heated jets. This feature is particularly visible at set point sp49 at 150o

. The parallel flow 

approximation is known to exaggerate the attenuation of the high frequency noise, and to generate a fairly larger 

cone of silence compared to a spreading jet. Scattering of the high frequency jet noise from the nozzle lip could also 

be another mechanism contributing to the near-axis jet noise. The extension/broadening of the spectrum-peak and 

the appearance of a second peak at a relatively higher frequency is a recurrent feature observed in supersonic jet 

noise measurements.

Next we examine the impact of the flow solver on the predicted noise with a third set of computations that use 

Wind-US [Ref. 18] RANS solver. Since Wind code is not currently outfitted with the enthalpy variance and 

dissipation rate model, the type-2 source strength was evaluated with the ETV model, while type-1 source strength 

uses a κ-ε turbulence model. Figures 19 to 24 present the source distributions for each source type at heated 

conditions considered above, and should be compared with similar results using CRAFT solver presented in figures 

9 to 14. Although the two flow solvers do not provide identical results, the overall source distributions and intensity 

levels remain in close agreement. Subsequent jet noise predictions that use the empirical ETV model with both 

Wind-κε and CRAFT-κε input are also shown in figures 25, 26 and 27 while calibration constants in evaluating the 

time- and length-scales remained unchanged. The two sets of spectra are practically similar and slight differences in 

the flow solvers have inconsequential impact on the predicted jet noise. 

4.2 Model Source Validation – A Comprehensive Set of Subsonic Conditions 

Here we set out to examine the source model proposed in section 3 under a wider range of subsonic jet 

conditions listed in Table 2. Twenty set points are divided into four distinct categories – the first three groups are at 

acoustic Mach numbers of 0.50, 0.70, and 0.90 respectively, with five jets listed within each group in the order of 

increasing temperature. Jets in the fourth group are at Mach number 1.0, i.e. nozzle pressure ratio of 1.89, and with 

a plenum temperature ratio in the order from 1.0 to 3.20. The nozzle of interest in is the 2-inch diameter convergent 

nozzle smc000 studied extensively for aerodynamic performance and jet noise at the SHJAR facility [Ref. 29, 30]. 

Acoustic data repeatability has been determined to be within 0.50 dB in one-third-octave band at all frequencies. 

Mean flow and turbulence calculations were performed with the Wind-US code using a κ-ε turbulence model. 

Subsequent noise predictions were carried out using type-1 and type-2 sources, with the enthalpy-related source 

evaluated according to the empirical ETV model discussed in section 3. 

As before, spectral predictions are presented as lossless and are compared with the narrow-band SHJAR noise 

measurements at two inlet angles of 90o

 and 150o at an arc distance of100Dj . These results are shown in 20 

separate figures, from 28 through 47, where each jet is identified through a Reading No. as listed in column-2,Table 

2. Reasonably accurate jet noise spectra were predicted at all subsonic exhaust-speeds as seen in figures 28 through 

43. Shortfalls develop at aft angles and at supersonic exhaust speeds at the latter four conditions in Table 2. These

 

NASA/TM—2012-217743 11
 

 

 Table 2. Subsonic Set Point Conditions 

results, presented in figures 44 to 47, show that both the spectral peak and its amplitude at the aft angle of 150o

 fall 

short of measurements under supersonic conditions. This discrepancy, noted to increase at higher exhaust 

temperature (velocity), was also discussed earlier in the context of supersonic conditions in Table 1. The broadening 

of the spectrum peak at 150o

 with increasing jet temperature is quite noticeable in the data – and in fact a second 

spectral peak at the Strouhal frequency of ~ 0.30 can clearly be distinguished at acoustic Mach numbers of 1.50 and 

1.63 in figures 46 and 47. As pointed out earlier, we attribute this shortcoming to the overemphasized refraction 

phenomena in a parallel mean flow approximation. As for our primary goal, the empirical ETV source model (i.e. 

type-2 source) appears to have worked successfully as a replacement for the variance of thermal fluctuations as 

calculated with a dedicated flow solver. This model, described in equation (17), depends entirely on the mean flow 

parameters and may readily be implemented with input from a standard flow solver. 

5.0 Summary 

In this study we presented a model for the enthalpy flux source term (i.e. type-2 source) within the 

framework of the generalized Acoustic Analogy. The main parameter of interest was the variance in the jet 

stagnation enthalpy. The goal was to conceive a more practical computational approach in connection with type-2 

source strength that would be independent of a dedicated RANS solver. Following the representation outlined in 

equation (17), the strength of the enthalpy-related source was evaluated from readily available mean flow 

parameters supplied by the standard RANS solvers. The model was subsequently used to evaluate an approximate 

form of the velocity/stagnation-enthalpy autocovariance tensor in the context of a physics-based jet noise modeling 

approach. The spectral computations were carried out with the help of jet noise prediction code “JeNo” according to 

equation (16), and with the propagation Green’s function evaluated following equation (9). The major 

NASA/TM—2012-217743 12
 

simplification in the computations is associated with the parallel flow approximation. We argued that this 

assumption leads to a larger than measured cone of silence near the downstream jet axis at supersonic exhaust 

speeds. We also chose to tentatively ignore the type-3 source, i.e. momentum-flux enthalpy-flux coupling term, due 

to its small contribution to jet noise at subsonic Mach numbers. Predicted spectra were in reasonably good 

agreement with measurements at all subsonic speeds considered in Tables 1 and 2. At supersonic exhaust speeds, 

however, the predicted spectra missed some commonly observed features at aft angles such as spectral-peak 

broadening, and/or the appearance of a second peak at a relatively higher frequency. We attributed this shortcoming 

to the overemphasized refraction phenomena in a parallel mean flow approximation. Computation of the Green’s 

function in a spreading jet may be carried out by solving a vector Green’s function in a set of five linear equations 

that govern the non-radiating base flow in the acoustic equations [Ref. 11]. These computations become very 

intensive and should preferably be carried out with the adjoint form of the governing equations. 

References 

[1] Ffowcs Williams, J. E., “Noise mechanisms, technical evaluation report,” AGARD-CP-131, (1974), pp. vii-xx. 

[2] Lighthill, M. J., “On sound generated aerodynamically: I. general theory,” Proceedings R. Soc. London, A211, (1952), pp. 

564-587. 

[3] Tanna, H.K., “The influence of temperature on shock-free supersonic jet noise,” J. Sound Vib., 39(4), (1975), pp. 429-460.

 

[4] Bridges, J., and Wernet, M. P., “Measurements of the aeroacoustic sound source in hot jets,” AIAA-2003-3130, May 2003. 

[5] Viswanathan, K., “Aeroacoustics of hot jets,” J. Fluid Mechanics, 519, (2004), pp. 39-82. 

[6] Morfey, C. L., and Szewczyk, V. M., “New scaling laws for hot and cold jet mixing noise based on a geometric acoustic 

model,” J. Sound Vib., 61(2), (1978), pp. 255-292.

[7] Tam, C. K.W., “Dimensional analysis of jet noise data,” AIAA Journal, 44(3), (2006), pp. 512-522. 

[8] Viswanathan, K., “Scaling laws and a method for identifying components of jet noise,” AIAA Journal, 44(10), (2006), pp. 

2272-2285. 

[9] Khavaran, A., and Bridges, J., “Development of jet noise power spectral laws using SHJAR data,” AIAA-2009-3378, May 

2009.

[10] Lilley, G. M., “The radiated noise from isotropic turbulence with application to the theory of jet noise,” J. Sound Vib., 

190(3), (1996), pp. 463-476.

[11] Goldstein, M. E., and Leib, S. J., “The aeroacoustics of slowly diverging supersonic jets,” J. Fluid Mechanics, 600, (2008), 

pp. 291-337.

[12] Afsar, M. Z., Goldstein, M. E., and Fagan, A. "The influence of enthalpy flux – momentum flux coupling on the acoustic 

spectrum of heated jets," AIAA-2011-1033, January 2011. 

[13] Khavaran, A., Bridges, J., and Georgiadis, N., “Prediction of turbulence-generated noise in unheated jets,” Part 1: JeNo 

Technical Manual, NASATM-2005-213827, 2005.

[14] Nagano, Y. and Kim. C., “A Two-Equation Model for Heat Transport in Wall Turbulent Shear Flows,” J. Heat Transfer

, 

(110), (1988), pp. 583-589.

[15] Kenzakowski, D.C., “RANS modeling improvements for jets using scalar variance equations,” AIAA-2006-0491, 2006.

[16] Dash, S.M., Kenzakowski, D.C., Seiner, J.M., and Bhat, T.R.S., "Recent advances in jet flowfield simulation: Part I - Steady

flows," AIAA 93-4390, 15th AIAA Aeroacoustics Conf., Oct. 25-27, 1993.

[17] Khavaran, A., Kenzakowski, D. C., and Mielke-Fagan, A. F., “Hot jets and sources of jet noise,” International J. 

Aeroacoustics, 9(4,5), 2010, pp. 491-532.

[18] Nelson, C.C., “An overview of the NPARC alliance's Wind-US flow solver,” AIAA-2010-27, Jan. 2010.

[19] Georgiadis, N.J., Yoder, D.A., and Engblom, W.A., “Evaluation of modified two-equation turbulence models for jet flow 

predictions,” AIAA Journal, 44(12), (2006), pp. 3107-3114.

[20] Bridges, J., “Effect of heat on space-time correlation in jets,” AIAA-2006-2534, May 2006. 

[21] Tam, C. K. W., and Auriault, L. “Mean flow refraction effects on sound radiated from a localized source in jets,” J. Fluid 

Mechanics, 370, (1999), pp. 149-174,

[22] Monin, A. S., and Yaglom, A. M., Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2, MIT Press, 1975.

[23] Batchelor, G. K., The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, 1960.

[24] Leib, S. J., and Goldstein, M. E., “Hybrid source model for predicting high-speed jet noise,” AIAA Journal, 49(7), (2011), 

pp. 1324-1335.

[25] Tanna, H. K., “An experimental study of jet noise, part I: turbulent mixing noise,” J. Sound Vib., 50(3), (1977), pp. 405-428. 

[26] Mielke, A. F., Elam, K. A., and Sung, C. J., “Molecular Rayleigh scattering diagnostic for dynamic temperature, velocity, 

and density measurements,” AIAA-2006-2969, 2006.

[27] Mielke, A. F., Elam, K. A., and Sung, C. J., “Time-Resolved Rayleigh scattering measurements in hot gas flows,” AIAA2008-262, 2008.

[28] Witze, P. O., “Centerline velocity decay of compressible free jets,” AIAA Journal, 12(4), (1974), pp. 417-418.

[29] Brown, C. A., and Bridges, J., “Small Hot Jet Acoustic Rig validation,” NASA TM – 2006-214234, 2006.

[30] Bridges, J., and Wernet, M. P., “Effect of temperature on jet velocity spectra,” AIAA-2007-3628, 2007.

NASA/TM—2012-217743 13
 

 

Figure 1. Variance in total temperature at set point sp23, Table 1: (a) Rayleigh scattering 

measurements; (b) predictions with CRAFT-TEV flow solver. 

Figure 2. Variance in total temperature at set point sp27, Table 1: (a) Rayleigh scattering 

measurements; (b) predictions with CRAFT-TEV flow solver. 

NASA/TM—2012-217743 14
 

Figure 3. Variance in total temperature at set point sp42, Table 1: (a) Rayleigh scattering 

measurements; (b) predictions with CRAFT-TEV flow solver. 

Figure 4. Variance in total temperature at set point sp46, Table 1: (a) Rayleigh scattering 

measurements; (b) predictions with CRAFT-TEV flow solver. 

NASA/TM—2012-217743 15
 

Figure 5. Streamwise turbulent velocity component (rms) at indicated axial locations at set point sp23 from Table 1:

 (a) Rayleigh scattering data; (b) CRAFT RANS predictions.

Figure 6. Streamwise turbulent velocity component (rms) at indicated axial locations at set point sp27 from Table 

 (a) Rayleigh scattering data; (b) CRAFT RANS predictions.

NASA/TM—2012-217743 16

1:
 

Figure 7. Streamwise turbulent velocity component (rms) at indicated axial locations at set point sp42 from Table 1:

 (a) Rayleigh scattering data; (b) CRAFT RANS predictions.

Figure 8. Streamwise turbulent velocity component (rms) at indicated axial locations at set point sp46 from Table 1:

 (a) Rayleigh scattering data; (b) CRAFT RANS predictions.

NASA/TM—2012-217743 17
 

Figure 9. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp23 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

 

Figure 10. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp27 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

NASA/TM—2012-217743 18
 

 

Figure 11. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp29 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

 

Figure 12. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp42 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

NASA/TM—2012-217743 19
 

 

Figure 13. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp46 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

 

Figure 14. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp49 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

NASA/TM—2012-217743 20
 

Figure 15. CRAFT RANS predictions for turbulent kinetic energy (top) and variance in stagnation 

temperature (middle) in the jet at set point sp03 shown in Table 1. The ETV source 

strength with CRAFT input is shown in the bottom.

Figure 16. Spectral predictions using CRAFT RANS input and empirical temperature variance (ETV) 

source model (blue); enthalpy variance source model (red); and measurements (dark) –

Table 1, set points sp03 (bottom curves), sp07 (top curves) at inlet angles of: (a) 90o

; (b) 

120o

; (c) 150o. 

 

NASA/TM—2012-217743 21
 

Figure 17. Spectral predictions using CRAFT RANS input and empirical temperature variance (ETV) 

source model (blue); enthalpy variance source model (red); and measurements (dark) –

Table 1, set points sp23 (bottom curves), sp27 (middle curves), sp29 (top curves) at inlet 

angles of: (a) 90o

; (b) 120o; (c) 150o. 

Figure 18. Spectral predictions using CRAFT RANS input and empirical temperature variance (ETV) 

source model (blue); enthalpy variance source model (red); and measurements (dark) –

Table 1, set points sp42 (bottom curves), sp46 (middle curves), sp49 (top curves) at inlet 

angles of: (a) 90o

; (b) 120o; (c) 150o. 

 

NASA/TM—2012-217743 22
 

 

Figure 19. Wind-US predictions for turbulent kinetic energy (top) and the ETV source strength 

(bottom) at set point sp23. 

Figure 20. Wind-US predictions for turbulent kinetic energy (top) and the ETV source strength 

(bottom) at set point sp237

NASA/TM—2012-217743 23
 

Figure 21. Wind-US predictions for turbulent kinetic energy (top) and the ETV source strength 

(bottom) at set point sp29.

Figure 22. Wind-US predictions for turbulent kinetic energy (top) and the ETV source strength 

(bottom) at set point sp42.

NASA/TM—2012-217743 24
 

Figure 23. Wind-US predictions for turbulent kinetic energy (top) and the ETV source strength 

(bottom) at set point sp23.

Figure 24. Wind-US predictions for turbulent kinetic energy (top) and the ETV source strength 

(bottom) at set point sp49.

NASA/TM—2012-217743 25
 

 

Figure 25. Spectral predictions using Wind-US RANS input and ETV source model (red); CRAFT 

RANS input and ETV source model (blue); and measurements (dark) –Table 1, set points 

sp03 (bottom curves), sp07 (top curves) at inlet angles of: (a) 90o

; (b) 120o; (c) 150o. 

Figure 26. Spectral predictions using Wind-US RANS input and ETV source model (red); CRAFT 

RANS input and ETV source model (blue); and measurements (dark) –Table 1, set points 

sp23 (bottom curves), sp27 (middle curves), sp29 (top curves) at inlet angles of: 

(a) 90o

; (b) 120o; (c) 150o. 

NASA/TM—2012-217743 26
 

Figure 27. Spectral predictions using Wind-US RANS input and ETV source model (red); CRAFT 

RANS input and ETV source model (blue); and measurements (dark) –Table 1, set points 

sp42 (bottom curves), sp46 (middle curves), sp49 (top curves) at inlet angles of: 

(a) 90o

; (b) 120o; (c) 150o. 

NASA/TM—2012-217743 27
 

Figure 28. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1513, Table 2 (NPR =1.188, NTR = 1.04), 

(measurements are shown as jagged lines).

Figure 29. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1521 from Table 2 (NPR =1.188, NTR = 1.047).

NASA/TM—2012-217743 28
 

Figure 30. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1525 from Table 2 (NPR =1.168, NTR = 1.154).

 

Figure 31. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1528 from Table 2 (NPR =1.153, NTR = 1.251).

NASA/TM—2012-217743 29
 

Figure 32. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1531 from Table 2 (NPR =1.128, NTR = 1.479).

Figure 33. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1514 from Table 2 (NPR =1.418, NTR = 1.025).

NASA/TM—2012-217743 30
 

Figure 34. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1523 from Table 2 (NPR =1.389, NTR = 1.10).

Figure 35. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1526 from Table 2 (NPR =1.345, NTR = 1.20).

NASA/TM—2012-217743 31
 

Figure 36. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1529 from Table 2 (NPR =1.318, NTR = 1.30).

Figure 37. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1532 from Table 2 (NPR =1.26, NTR = 1.53).

NASA/TM—2012-217743 32
 

Figure 38. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1515 from Table 2 (NPR =1.834, NTR = 1.017).

Figure 39. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input

and ETV source model at Reading No. 1524 from Table 2 (NPR =1.694, NTR = 1.164).

NASA/TM—2012-217743 33
 

Figure 40. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1527 from Table 2 (NPR =1.616, NTR = 1.26).

Figure 41. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1530 from Table 2 (NPR =1.563, NTR = 1.359).

NASA/TM—2012-217743 34
 

Figure 42. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1533 from Table 2 (NPR =1.452, NTR = 1.592).

Figure 43. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1614 from Table 2 (NPR =1.893, NTR = 1.0).

NASA/TM—2012-217743 35
 

Figure 44. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1584 from Table 2 (NPR =1.893, NTR = 1.80).

Figure 45. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1572 from Table 2 (NPR =1.893, NTR = 2.20).

NASA/TM—2012-217743 36
 

Figure 46. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1565 from Table 2 (NPR =1.893, NTR = 2.70).

Figure 47. Jet noise spectral predictions at inlet angles of 90o

 and 150o using Wind-US RANS input 

and ETV source model at Reading No. 1554 from Table 2 (NPR =1.893, NTR = 3.20).

NASA/TM—2012-217743 37
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4. TITLE AND SUBTITLE

An Empirical Temperature Variance Source Model in Heated Jets 

5a. CONTRACT NUMBER

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6. AUTHOR(S)

Khavaran, Abbas; Bridges, James 

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WBS 561581.02.08.03.45.02.05 

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John H. Glenn Research Center at Lewis Field 

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14. ABSTRACT

An acoustic analogy approach is implemented that models the sources of jet noise in heated jets. The equivalent sources of turbulent mixing 

noise are recognized as the differences between the fluctuating and Favre-averaged Reynolds stresses and enthalpy fluxes. While in a 

conventional acoustic analogy only Reynolds stress components are scrutinized for their noise generation properties, it is now accepted that 

a comprehensive source model should include the additional entropy source term. Following Goldstein’s generalized acoustic analogy, the 

set of Euler equations are divided into two sets of equations that govern a non-radiating base flow plus its residual components. When the 

base flow is considered as a locally parallel mean flow, the residual equations may be rearranged to form an inhomogeneous third-order 

wave equation. A general solution is written subsequently using a Green’s function method while all non-linear terms are treated as the 

equivalent sources of aerodynamic sound and are modeled accordingly. In a previous study, a specialized Reynolds-averaged Navier-Stokes 

(RANS) solver was implemented to compute the variance of thermal fluctuations that determine the enthalpy flux source strength. The main 

objective here is to present an empirical model capable of providing a reasonable estimate of the stagnation temperature variance in a jet. 

Such a model is parameterized as a function of the mean stagnation temperature gradient in the jet, and is evaluated using commonly 

available RANS solvers. The ensuing thermal source distribution is compared with measurements as well as computational result from a 

dedicated RANS solver that employs an enthalpy variance and dissipation rate model. Turbulent mixing noise predictions are presented for a 

wide range of jet temperature ratios from 1.0 to 3.20. 

15. SUBJECT TERMS

Noise; Jet noise; Propulsion noise; Acoustics 

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