Mohammed Z. Afsar

Imperial College London, London, United Kingdom

Stewart J. Leib

Ohio Aerospace Institute, Cleveland, Ohio

Richard F. Bozak

Glenn Research Center, Cleveland, Ohio

A Model for Jet-Surface Interaction Noise 

Using Physically Realizable Upstream 

Turbulence Conditions

NASA/TM—2016-218898

January 2016

AIAA–2015–2529
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Mohammed Z. Afsar

Imperial College London, London, United Kingdom

Stewart J. Leib

Ohio Aerospace Institute, Cleveland, Ohio

Richard F. Bozak

Glenn Research Center, Cleveland, Ohio

A Model for Jet-Surface Interaction Noise 

Using Physically Realizable Upstream 

Turbulence Conditions

NASA/TM—2016-218898

January 2016

AIAA–2015–2529

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for

Aviation 2015

sponsored by the American Institute of Aeronautics and Astronautics

Dallas, Texas, June 22–26, 2015
Acknowledgments

Mohammed Z. Afsar would like to thank financial support from Chapman Fellowship (2013-2014) at Imperial College London, 

Department of Mathematics and would like to thank useful discusions with Professor W. Devenport at Virginia Tech. 

The work was also supported by the NASA Fundamental Aeronautics Program, High Speed Project. 

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either expressed or implied, by the National Aeronautics and 

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at the NASA Glenn Research Center.

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NASA/TM—2016-218898 1

A Model for Jet-Surface Interaction Noise Using Physically 

Realizable Upstream Turbulence Conditions 

Mohammed Z. Afsar

Imperial College London

London, SW7, United Kingdom

Stewart J. Leib

Ohio Aerospace Institute

Brook Park, Ohio 44142

Richard F. Bozak

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

This paper is a continuation of previous work in which a generalized Rapid Distortion Theory (RDT) 

formulation was used to model low-frequency trailing-edge noise. The research was motivated by 

proposed next-generation aircraft configurations where the exhaust system is tightly integrated with the 

airframe. Data from recent experiments at NASA on the interaction between high-Reynolds-number 

subsonic jet flows and an external flat plate showed that the power spectral density (PSD) of the far-field 

pressure underwent considerable amplification at low frequencies. For example, at the 90° observation 

angle, the low-frequency noise could be as much as 10 dB greater than the jet noise itself. In this paper, 

we present predictions of the noise generated by the interaction of a rectangular jet with the trailing edge 

of a semi-infinite flat plate. The calculations are based on a formula for the acoustic spectrum of this 

noise source derived from an exact formal solution of the linearized Euler equations involving (in this 

case) one arbitrary convected scalar quantity and a Rayleigh equation Green’s function. A low-frequency 

asymptotic approximation for the Green’s function based on a two-dimensional mean flow is used in the 

calculations along with a physically realizable upstream turbulence spectrum, which includes a finite 

decorrelation region. Numerical predictions of the sound field, based on three-dimensional RANS 

solutions to determine the mean flow, turbulent kinetic energy and turbulence length and time scales, for a 

range of subsonic acoustic Mach number jets and nozzle aspect ratios are compared with experimental 

data. Comparisons of the RANS results with flow data are also presented for selected cases. We find that 

a finite decorrelation region in the turbulence spectrum increases the low-frequency algebraic decay (the 

low frequency “roll-off”) of the acoustic spectrum with angular frequency thereby producing much closer 

agreement with noise data for Strouhal numbers less than 0.1. Secondly, the large-aspect-ratio theory is 

able to predict the low-frequency amplification due to the jet-edge interaction reasonably well, even for 

moderate aspect ratio nozzles. We show also that the noise predictions for smaller aspect ratio jets can be 

fine-tuned using the appropriate RANS-based mean flow and turbulence properties. 

Nomenclature

c sound speed 

c∞ ambient sound speed

DJ nozzle diameter
NASA/TM—2016-218898 2

g Green’s function

Iω acoustic spectrum

k turbulent kinetic energy

k1 streamwise wavenumber

li characteristic length scale

M acoustic Mach number 

Ma jet acoustic Mach number

p pressure

t time

T averaging time 

Uc convection velocity

V source volume

vi velocity vector

x observer location

y source location

α turbulence intensity 

β angle function

γ specific heat ratio 

δij Kronecker delta 

ε turbulence dissipation rate

η separation vector

ρ density

θ polar angle measured from jet axis

τ time delay

ω radian frequency

∇ gradient operator 

 absolute value

Subscripts

i,j,k,l tensor indices = 1,2,3

⊥ transverse component 

Superscripts

a adjoint

time average ' fluctuating quantity 

~ Favre average

* complex conjugate

1.0 Introduction

Jet flows of technological interest are almost always close enough to, or sufficiently confined by, 

solid boundaries, so that the surface defining the boundary plays a direct role in the generation of sound, 

as well as its propagation. Understanding the basic physics behind this process is of considerable 

importance for present-day and future aircraft that may have complex engine installation geometries. The 

aim of this paper is to further develop a prediction method for the noise generated by the interaction of a 
NASA/TM—2016-218898 3

turbulent jet with the trailing edge of a flat plate. This problem serves as a model of one important aspect 

of engine-installation effects, namely the interaction of the exhaust jet with a wing. The prediction 

method is based on a self-consistent application of the nonhomogeneous Rapid-distortion theory (RDT) 

introduced recently in Reference 1 (hereafter referred to as GAL).

Experiments conducted by Olsen and Boldman (Ref. 2) and Wang (Ref. 3) showed that the presence 

of an external surface enhanced the noise produced by the jet alone for observation points on the same 

side as the jet flow. In Reference 2 it was shown that including mean flow interaction effects significantly 

improves the accuracy of noise predictions for a jet near an external surface compared with approaches 

that neglect this effect (Ref. 4). Recent experiments at the NASA Glenn Research Center (Refs. 5, 6, 7, 8,

and 9) have modelled the jet-wing interaction problem as a jet flow interacting with a trailing edge of an 

external plate. The power spectral density of the far-field pressure (PSD) was measured for unheated, 

high Reynolds number, jet flows across a range of acoustic Mach numbers when the trailing edge was 

positioned above/beneath the flow at various axial/radial locations relative to the nozzle center line.

The findings of Reference 6 have generally confirmed the trends of References 2 and 3. In particular, 

at low frequencies, the PSD is considerably amplified compared to the free jet, and this effect is greatest 

at large polar observation angles to the jet axis (i.e., near 90°); see figures 6 and 8 in Reference 6. This 

low-frequency amplification effect has been called “jet-surface interaction” noise by Brown (Ref. 10), 

owing to its observed dipole characteristic and structural difference to high-frequency noise shielding and 

reflection. In Reference 10, Brown developed an empirical model for jet-surface interaction noise and

extracted the noise due to the jet/trailing edge interaction from measurements of the total noise in various 

jet-plate configurations using the expected dipole characteristics of the edge noise source. 

In this paper we use the general theory developed in Reference 1 to model the jet-surface interactions 

using RDT. Rapid Distortion Theory uses linear analysis to study the interaction of turbulence with, for 

example, solid surfaces. It applies whenever the turbulence intensity is small and the time scale for the 

interaction is short compared to the turn-over time of the turbulent eddies, i.e., the time over which 

nonlinear interactions and viscous dissipation take place. The problem is then linear and inviscid and is 

governed by the compressible Rayleigh equation.

An initial application of the general theory to the jet-surface interaction problem was included in 

GAL. Here, we further develop the prediction method by introducing a more realistic model for the 

statistics of the upstream turbulence whose interaction with the plate edge generates noise. The modelling 

approach is similar to that in Reference 13, but used here for the (second-order) transverse velocity auto 

covariance. In particular, we show that a region of negative correlation (or ‘decorrelation’) in this quantity 

directly impacts the low-frequency algebraic decay (often referred to as the ‘roll-off’) of the edgegenerated noise and provides better agreement with experimental data than our previous results. The 

presence of such negative regions in second-order correlation has been known for some time (Ref. 14).

This result has implications for trailing-edge noise control.

The prediction capabilities of the method are further developed by using results from Reynoldsaveraged Navier-Stokes (RANS) solutions to obtain the mean flow and inform the source model, 

in particular to obtain the turbulent length scales and source amplitude. The use of RANS solutions also 

allows these flow quantities to vary with flow conditions and geometry. The RANS solutions are obtained 

using the SolidWorks Flow Simulation software (Dassault Systèmes SolidWorks Corporation) (Refs. 11

and 12), which provides relatively fast solutions for the geometries of interest. As a check on the quality 

of these results, we include comparisons with measured flow data (Ref. 7) for selected cases.

In the next section we summarize the main features of nonhomogeneous RDT introduced in GAL and 

describe how this theory can be used for the trailing-edge noise problem. As mentioned above, there are 

two novel features of the present paper as compared to the trailing edge model in GAL. Firstly, we use a 
NASA/TM—2016-218898 4

more advanced turbulence model for the two-point correlation of the transverse velocity fluctuations

Secondly, noise predictions using results from RANS solutions to obtain the mean flow profile and 

turbulent kinetic energy near the trailing edge are presented and compared with experimental data taken at 

NASA Glenn. 

In Section 2.0, we briefly review the relevant parts of the GAL analysis used in this work. In Section 

3.0, we introduce the new model for the transverse velocity correlations and present the corresponding 

formula for the acoustic spectrum. In Section 4.0, we illustrate some generic features of the model using 

an analytically prescribed mean flow. The impact of the new source model on the low-frequency “rolloff” of the spectrum and comparisons with the model of GAL are shown. In Section 5.0, the use of 

SolidWorks software to obtain RANS solutions for a rectangular jet near the edge of flat plate is 

described. Results from these solutions are compared with experimental data taken at NASA Glenn 

(Ref. 7) for the mean flow and turbulent kinetic energy distributions near the edge of the plate. In Section 

6.0, we present comparisons of noise predictions using the model developed in this paper, with the 

SolidWorks RANS solutions as input, with data taken at NASA Glenn. Conclusions and discussion of 

potential future work is given in Section 7.0.

2.0 Review of the GAL Formulation

2.1 Euler Equations for Small-Amplitude Motion

Let all lengths and velocities be nondimensionalized by DJ and UJ, respectively; time by DJ/UJ and 

pressure by 2 ρJUJ ,where ρJ and UJ are flow density and velocity, respectively, at nozzle exit and DJ is an 

appropriate reference length scale (such as the nozzle exit equivalent diameter). The flow Reynolds 

number is assumed to be large, i.e., R = UJ DJ /v  O(1) and the turbulence Reynolds number is fixed at 

order 1, i.e., R R1 T =α = O( ) ; where v is the kinematic viscosity and α ≡ v′ U O J  (1) is the 

turbulence intensity of the upstream flow. v′ is the magnitude of the local rms turbulence velocity. 

We suppose that the flow is inviscid and nonheat conducting and assume an ideal gas so that the 

entropy is proportional to ln (p/ργ

)and the squared sound speed is γp/ρ, where p denotes the pressure, ρ

the density and γ the specific heat ratio. The inviscid pressure p′ = p – p0 and momentum flux 

perturbations, i ≡ ρ ′, u v i where ′ vi denotes the velocity perturbation, on a transversely sheared mean flow 

with pressure p0 = constant, velocity U(yT) and mean sound speed squared c2

, are governed by the 

linearized momentum and energy equations 

0

1 0 ∂ ∂ ′ +δ + =

τ ∂∂

i

i j

j i

Du U p u

D yy (1)

and 

2

0 0 ′ ∂

+ =

τ ∂

j

j

D p c u

D y , (2)

where, y y = = {yyy y 123 1 ,, , } { T } , yT = {y y 2 3 , } and D U 0 D y τ τ ≡∂ ∂ + ∂ ∂ 1 denotes the convective 

derivative.
NASA/TM—2016-218898 5

2.1.1 Integration of the Euler Equations (1) and (2)

Goldstein (Ref. 15) showed that Equation (1) will be satisfied for any function φ(y,τ) and any purely 

convected quantity, θ(τ – y1/U(yT), yT), when p′ and ui are determined by 

( ) ( )

3

0

3 , φ ′ τ =− τ

τ

D

p, , D y y (3)

and

( ) ( ) ( )

0 1

1 2

1 , ,   ∂ ∂ ∂   τ = δ −δ λ τ +ε   θ τ−   τ ∂ ∂ ∂     i ij i j ijk T

j j k T

DU U y u , , D y cyy U yy y

y

(4)

where δij denotes the Kronecker delta, εijk the alternating tensor and 

0

1

2 ∂ φ ∂ ∂φ λ≡ +

∂ τ ∂∂ j

j j

D U

yD y y 

(5)

is a kind of generalized particle displacement (Ref. 15, Eqs. 2.9 and 2.10). 

The energy equation, (2), is then satisfied when the scalar φ(y,τ) is determined by:

3

0 0 2 0

3

1

2 0,   ∂ ∂ φ ∂ ∂φ φ       + −=

τ∂ ∂ τ ∂ ∂ τ   ii i  

D DU D

c

D y yD y y D

(6)

which can be integrated to obtain,

L yU a c φ = −ω τ − ( 1 ( y y T T ), ,)  (7)

where

3

0 2 0

3

1

2 . ∂ ∂ ∂∂  ≡− +   τ ∂ ∂ τ ∂∂   a

ii i

D DU L c

D y yD y y (8)

and ω τ− ( 1 ( ), )  c y U y y T T is another arbitrary, purely convected, quantity. GAL point out that the 

operator La is the adjoint to the Rayleigh operator, 

2

0 2 2 0

2

1

2 ,   ∂ ∂ ∂∂ ∂ ≡ −−   τ∂ ∂ τ ∂ ∂ ∂   i i j j

D DU L c c

D y y D yy y (9)

which is obtained by eliminating the momentum flux perturbation between (1) and (2) to derive a single 

equation for the fluctuating pressure 

L p′ = 0. (10)

The solution to φ(y,τ) can then be found by solving (7) as a boundary value problem in terms of the 

Rayleigh equation Green’s function (as shown in GAL) 

Lg t ,, ( y x yx τ =δ − δ τ− | ) ( ) ( t) , (11)
NASA/TM—2016-218898 6

subject to appropriate boundary conditions, and the result inserted into (4) to obtain ui(y,τ). 

2.2 Integral Solutions to the Euler Equations for Arbitrary Transversely Sheared Mean 

Flows

GAL showed that the formal complete solution to the nonhomogeneous Rapid Distortion Theory 

problem for the pressure perturbation, p′, is given by 

( ) ( ) ( ( ) )

3

0

1 3

, , , , |

−

τ ′ = ω τ− τ ∫ ∫ 

T

c T T

T V

Dg t p t y U dd Dty x

x yyy (12)

(where T denotes a very large but finite time interval) when the solid surfaces S (which can be finite, 

semi-infinite or infinite in the streamwise direction) coincide with any level surfaces of the mean velocity 

profile (as in the trailing edge problem considered herein) . The solution for p′ in these cases is 

independent of the convected quantity θ(τ – y1)/U(yT), yT).

The corresponding solution for the transverse momentum fluctuation is also independent of θ and is 

given by 

( ) ( ) ( ) ( 1 ( ) ) /

⊥ , , / ,, | , 

−

∂ ∂∂

ρ ≡ ′ ∇ =− τ ω τ− τ

∂ ∇ ∫ ∫ 

T

i

i i c T T

i

T V

U Ux v t u t U g t y U dd x U

x x y x yy y (13)

where 

( ) ( ) 0 0

1

,|, 2 ,|,   ∂ ∂∂

τ≡ + τ     ∂ ∂∂ i

i i

DDU

g t g t Dt x Dt x x y x y x . (14)

This solution involves only the purely convected quantity ω τ− ( 1 ( ), )  c y U y y T T , that can be 

specified as an input condition within a general boundary value problem involving inhomogeneous 

boundary conditions. The Green’s function g(y,τx,t) is found by solving (11) with incoming wave 

behavior as y→ ∞ and appropriate boundary conditions (on the bounding surfaces S that generate 

volume V) for (12) and(13) to hold. 

2.3 Green’s Function Splitting 

Following GAL, we can think of solution (12) as being the sum of an input disturbance and 

downstream response (Figure 1). The trailing-edge noise (or the output response) is generated by an input 

interacting with streamwise changes in boundary conditions at the plate edge. GAL divided the Rayleigh 

equation Green’s function that appears in (12) into the components,

( ) ( ) ( ) ( ) ( ) 0 ,, ,, ,, τ= τ+ τ ||| s g tg tg t yx yx yx (15)

where ( ) ( ) 0 g t y x , , τ | is now defined on all space and satisfies

( ) ( ) 3 0 3

0 ˆ ∂ τ ∂= ∈   , , | 0 for n D g t Dt y i   y x i Ty S (16)

where S is a doubly infinite surface and ( ) { }( ) ˆ ˆ ny= y ni is the unit outward-drawn normal to S.
NASA/TM—2016-218898 7

Figure 1.—Mathematical model of the jet-surface interaction noise.

Then, by (13) , the corresponding transverse momentum flux is

( )

( ) ( ) ( )

0 (0) 1 / , ,, | , 

T

i

ic T

T

T V

U x y v t g t d d

U U ⊥

−

∂ ∂  

ρ =− ′ τ ω τ−   τ

∇   ∫ ∫ x y x y y

y  (17)

and then represents the input disturbance for the trailing-edge scattering problem and is referred to as the 

gust solution, i.e., a bounded hydrodynamic disturbance on a flow with streamwise homogeneous 

boundary conditions in the absence of any scattering surfaces. Although ω τ− ( 1 ( ), )  c y U y y T T is not a 

physically measurable quantitity, Equation (17) provides an (in general) integral relation between

ω τ− ( 1 ( ), )  c y U y y T T and the physical variable, ( )

( ) 0 v t, ⊥ ρ ′ x that can, in principle, be inverted (using 

identity (1.97) in Goldstein (Ref. 16) to solve) for the Fourier transform of ω τ− ( 1 ( ), )  c y U y y T T in terms 

of the Fourier transform of ( )

( ) 0 v t, ⊥ ρ ′ x .

2.4 Relation Between the ω c Spectrum and Measurable Turbulence Statistics

The correspondence our input gust (17) has to the actual upstream turbulence in a “real” trailing-edge 

noise problem can be reconciled as follows. We are assuming the relation between ω τ− ( 1 ( ), )  c y U y y T T

and ( )

( ) 0 v t, ⊥ ρ ′ x in the actual flow is the same as it would in an idealized mathematical representation of a 

transversely sheared mean flow where RT = αR = O(1) everywhere in the flow and in which bounding 

surfaces present are doubly infinite in the streamwise direction (i.e., where the transverse boundary 

conditions are completely uniform in y1). Hence the upstream boundary condition is determined on this 

streamwise homogeneous flow is assumed to be the same as that in the vicinity of trailing-edge where the 

“real” distortion actually takes place.

GAL showed that the inversion of (17) can be obtained analytically for a two-dimensional mean flow 

by computing the (temporal, streamwise and spanwise) Fourier transform of ( )

( ) 0 v t, ⊥ ρ ′ x and relating it 

exactly to the Fourier transform of ω τ− ( 1 ( ), )  c y U y y T T , but that this relation can only be specified at N
NASA/TM—2016-218898 8

discrete transverse space points, say (1 2 ) ( ) ( )

22 2 , ,..., N

xx x , where ( ) 2

n x is given by solutions to discrete 

equation, ( ) ( 2 ) = ( 2 ), for 1,2,... = n Ux Ux n N , leading to a matrix problem to determine the auto-covariance 

of the transform of ω τ− ( 1 ( ), )  c y U y y T T in terms of the auto-covariance of the transform of ( )

( ) 0 v t, ⊥ ρ ′ x

. For jet flows, such as those considered in the present paper, there are two such points and, following 

GAL, we take ( ) ( ) ( ) 1 2

2 2 xx y , = d , where yd is the location where the velocity profile is maximum. In some 

sense, this gives an upper bound for the upstream boundary condition and allows the role of the 

decorrelation region to be assessed easily.

GAL used a uniformly valid low-frequency asymptotic solution for the gust Green’s function in the 

relation between the Fourier transforms of ( )

( ) 0 v t, ⊥ ρ ′ x and ω τ− ( 1 ( ), )  c y U y y T T to obtain a relatively 

simple working formula relating the spectrum of the convected quantity ω τ− ( 1 ( ), )  c y U y y T T , 

( ) ( 2 2 , ; , 3 ω)  s S k y y ,

( ) ( ) ( ) ( ) 3 3 223 2 3 23 3 3

1

, ; ,, ,, , 2 ,

∞ ∞

ωτ− η

−∞ −∞

ω ≡ ω ω +τ +η τ η

π ∫ ∫     i k S y y k e ty y t y y dd c c (18)

to the experimentally measurable transverse velocity spectrum 

( ) ( )

( ) ( ( ) ( ) ) 3 3 2222 3 22 2 2 3 3 2

1 , , ,, , , / , ,

2

∞ ∞

− ωτ− η ⊥ ⊥

−∞ −∞

ω = ω ω ητ η τ

π ∫ ∫     i k F xxy y k e f xx Uy Uy d d (19)

where 

( ) ( )

( ) ( )

( ) ( )( ) 11 11 0 0

2 211 3 1 2 3 3 1 1 2

1 , ,, , , , , , , 2

∞ ∞

−

⊥ ⊥ ⊥

−∞ −∞

− η τ ≡ ρ ρ ′ ′ +η +τ

π ∫ ∫        i xk xk f x x k k e v t v x x x t dx dx x (20)

as,

( ) ( ) ( )

( ) ( )

( )

( ) ( )

2 2

0 0 22 3

2 2

223

2 2 23 23

1 1 , , ,,

, ;

;, ;,

,

⊥

∗

  − −     + +   ω ′ ′    ω =     ω ω  

    

 d d

d d

d d

b bF yyyy k UyUy S y UyUy Ey k Ey k

yy yy

y y y k (21)

where

( ) [ ( ) ( )]

( )

( )

( )

( ) 2 2 2 2 3 2 0

2 3 0 2 2 2 2

3

2 2

2

2 2

2

; , 1

∞ ∞

− − + π

ω ≡ − + ′′ + −

      ω

    ω

d d

d d d

Uy Uy k iU y b Ey k b

c y k k cU y

y y U y

y U y

(22)

b0 is a constant relating the even and odd symmetry components of the Fourier transform of ω c (see 

GAL) and U′′(yd) is the second derivative of the mean velocity profile at yd. We also use this formula for 

the calculations in this paper. 
NASA/TM—2016-218898 9

2.5 Scattered Solution

The scattered solution, g(s)(y,τx,t) in the split formula (15), on the other hand, represents the 

contribution due to presence of the trailing edge and will satisfy streamwise inhomogeneous boundary 

and jump conditions on the streamwise-discontinuous surface present in the flow.

Imposing appropriate boundary/jump conditions on the plate surface and its downstream extension 

leads to a Wiener-Hopf problem (eqs. 6.6-6.8 in GAL) for the (temporal, streamwise and spanwise) 

Fourier transform of g(s)

(y,τx,t). The general solution to this problem was derived by GAL, but since 

experiments show that the interaction of a turbulent jet and a trailing edge generates noise at relatively 

low frequencies, the analysis and computations were simplified by considering the low-frequency limit 

(k k Ok k O 1 3 , ) = ( ∞ ∞ )and 1  ( ) for y2 = O(1) . This low-frequency approximation is also adopted in this 

paper. 

2.6 Acoustic Spectrum Formula for Jet-Surface Interaction in Planar Flows

For a two-dimensional jet with a planar mean flow, GAL showed that the acoustic spectrum for jetsurface interaction noise is given by the remarkably simple asymptotic result,

( ) ( ) ( ) ( ( )) ( )

2 22 ( 3 ) 2 2

0 0

1 , , ,, , ; , 2 4

∞ 2 ∞ ∞

ωτ ∞ ω

−∞

  ≡ +τ τ≈   θ ψ ω

π π   ∫ ∫∫   s is s k I x xx e p t p t d D M y S y y k dy dy

x , (23)

in which the integrand can be interpreted as the product of the source function ( ) ( 2 2 , ; , 3 ω)  s S k y y with a 

nonuniform directivity factor, D My (θ ψ, , ( 2 )), that encapsulates all propagation and surface interaction 

effects, when k∞  O(1) and x→ ∞. M(y2) = U(y2)/c∞ denotes the local acoustic Mach number at the 

position y2 and the spanwise wavenumber ( )

3 = θψ ∞ sin cos s k k is found by applying the method of 

stationary phase for the inverse Fourier integral in k3 (see GAL, p. 553). Here, θ is the polar observation 

angle measured relative to the jet center line and ψ is the azimuthal angle in the cross-stream plane. 

The directivity factor, which is given by

( ( )) ( ) ( ) ( )

( ) ( ) ( ) ( )

3/2

2 2

2

2 2 22

cos , , ,

1 cos 1 cos 1 1

  β− θ   θψ =

 − θ  − θ  −β   −β        



 

My My D My

My My My My

(24)

reduces to Goldstein’s (Ref. 15, eq. 3.23) result ( ) 2 2 sin 1 cos

2

θ  − θ   M yd at the point where the 

velocity profile is maximum 2 2 = =  yyyd since ( )

1/2 2 2 β≡ − θ ψ 1 sin cos ,is unity at ψ = ±π , in the plane 

perpendicular to the plate. In Figure 2(a) we show that D(θ,ψ,Ma) peaks at θ = 90° for subsonic acoustic 

Mach numbers. The azimuthal variation of D(θ,ψ,Ma) possesses dipole–like structure for any point 

y y 2 2 =  (see Figure 2(b)).
NASA/TM—2016-218898 10

Figure 2.—Directivity factor: (a) θ-directivity, (b) ψ-directivity.

3.0 Physically Realizable Upstream Turbulence Conditions 

3.1 Turbulence Model

The jet-surface interaction noise model (23) was constructed for a two-dimensional jet with planar 

mean flow. Consistent with this approximation, we suppose that the turbulence is spanwise homogeneous. 

The space-time average

( )

( ) ( )( ) ( )

( ) ( )

( ) 0 0 0 0

123 3 123 3 3

1 , , , lim ,, , 2

T

T

T

v t v x x x t v t v x x x t dtdx

T

∞

− −∞

⊥ ⊥ ⊥ ⊥ →∞

′ ′ +η +τ ≡ ′ ′ +η +τ ∫ ∫ x,   x, , (25)

that enters the integrand of (20) is both experimentally determinable and has a well established database

(e.g., experiments reported in Ref. 17). 

In this section, we construct a model for this function that is physically realizable and use it to derive 

a formula for the acoustic spectrum of the trailing edge noise. Different from GAL, however, the present 

model is includes a finite decorrelation region, which we show has a direct impact on the jet surface 

interaction noise and, in principle, could provide a means to reduce it.

We use the function (Ref. 13):

( )

( ) ( )

( )

( ) ( 1 3 )

0 0

1 3 1 33

, , 3 1 0 1 21

1

, ,, , , ,

 ...

d d

X

v xy xt v xy x t

L xaa a e

⊥ ⊥

− τη η

ρ ′ ′ ρ +η +τ =

  ∂ ∂ Ψ +τ +η +     ∂τ ∂η



, (26)

where the decay function is, 

( ) ( ) ( ) ( ) 222 2

1 3 11 1 0 3 3 η η τ= η +η− τ +η , , X c l Ul l (27)
NASA/TM—2016-218898 11

and the streamwise and spanwise separations are η≡ − 111 x x  and η3, respectively. l1 and l3 are turbulence 

length scales in streamwise and spanwise directions, respectively; l0, on the other hand, measures the 

scale of turbulence as it convects over separation distance η1 with convection velocity, Uc. 

Following GAL we allow Ψto decay in the streamwise direction in order to insure convergence of the 

subsequent Fourier transform integrals, i.e., ( ) ( )

2

1 1 1 0 Ψ =Ψ −α x L x e where α is a small positive number,

0 ≤ α < 1 which depends on the symmetric location , since we this quantity is expected to 

be independent of the streamwise coordinate for the nearly parallel flow being considered. Ψ0 is expected 

to scale with the transverse component of the mean square turbulence momentum flux (

)

2

⊥ ρv′ , L1, L3 are 

geometric spatial scales: L3 being a measure of the transverse extent of the turbulence and the L1, is taken 

to be large in order to insure that the correlation (26) is relatively independent of x1 to remain consistent 

with our representation of the upstream boundary condition described in Section 2.0. 

As shown in Figure 3, allowing the coefficient a1 > 0 in Equation (26) gives a negative 

(decorrelation) region for the auto-correlation (τ = η3 = 0) function of (26) which did not appear in the 

model used in GAL (Eq. 6.46). As mentioned in the Introduction, the presence of this negative region is 

an expected characteristic of the second-order transverse velocity correlations. 

 (a) (b) 

Figure 3.—Turbulence model (26) with (27) for τ = η3 = 0. (a) GAL model a1 = a2 = 0; (b) Present model a1 > 0.
NASA/TM—2016-218898 12

3.2 Acoustic Spectrum Formula With a Finite Decorrelation Region

Substituting the source model function (23) in (18)-(20) for ( ) ( 2 2 , ; , 3 ω)  s S k y y and inserting this result 

in (23), we obtain the final formula for the acoustic spectrum used in this paper 

( ) ( ) ( )

2

013 3 0

0 2

1 1

cos , , 4

∞ ω

∞

      Ψ =       β− θ θ ψ

  π π  c  

l ll L k I I Ma

U c

x

x

(28)

where β is defined below (24) and

( ) ( )

( )

( ) ( ) ( )

( ) ( ) ( )

( )

3

2 3 2

0 2 2 2

2 0 2 2 3

, ,

,, 4

1 cos ;, 1

∞ Πω ω

θψ =

 − θ   ω  −β    ∫  s

s

Ma Uy k cM y I Ma dM y

M y Ey k My

(29)

where Ma ≡ U(yd)/c∞ denotes the maximum acoustic Mach number and spectral functions ( ) ( 2 ; , 3 ω) s Ey k is 

given by (22) and  ( ) ( ) Πω ω ( 2 , , 3 ) s Uy k is defined by:

( )

(( ) ) ( ) (( ) ) ( )

2

0 2 012 1 21 1 1 1 2 2 2 0 0

1 3

1 4

, ,

c

c c c c

l

a a a a ak a k U l

l U U U l U

k k =

ω ω

− − + − + −ω

ω +χ ω +χ

Π ω

                    



(30)

where

( ) ( ) ( ) 2 2 2

1 3 33 1 1 χ=χ = + −ω + , 1     k k kl k U l c . (31)

The derivation of this formula is summarized in appendix A of Afsar et al. (Ref. 18). 

4.0 Low-Frequency Roll-Off 

The predictions in GAL were compared against the jet-surface interaction experiments performed at 

NASA Glenn (Refs. 6, 7, 8, 9, and 10). The relevant geometric parameters are shown in Figure 4.

However it is clear from figure 6 in GAL, that for Strouhal numbers less than 0.15 (where the peak 

noise amplification occurs), the positive auto-correlation model (i.e., a1 = a2 = 0 in (26)) they used over 

predicts the the low frequency roll-off for jet-surface interaction noise by as much as 10 dB (see figure 6d 

in GAL).

We can easily prove why this occurs by estimating integral I0(Ma,θ,ψ) defined by (29), for very small 

frequencies. As ω → 0, the dominant contribution to integral comes in the vicinity of critical level 

y2 = O(yd). In the appendix, we estimate terms in (29) under this limit by considering ψ = ±90° (which is 

the azimuthal location of the data for which the GAL predictions were made); β = 1 and ( )

3 = 0 s k here. 

Substituting equation asymptotic properties (38) into acoustic spectrum formula (28) shows that the latter 

possesses asymptotic properties,
NASA/TM—2016-218898 13

( )

( )

( )

1 2

2

0 1

1 , for 0

, for 

O aa

I

O aa

ω

 = =  = 

 ω 

x

 (32)

when ω → 0 that is now directly dependent on the whether a decorrelation region in (26) exists or not.

In the absence of a decorrelation region in (26) (i.e., a1 = a2 = 0) the acoustic spectrum does not 

possess a low frequency roll-off as such and tends to O(1) (i.e., a constant, see equation (6.52) in GAL). 

A finite decorrelation region, however, increases the low frequency ‘roll-off’ in the prediction of (28) to 

exhibit an asymptotic order of O(ω2

) or so, which is more dipole-like and appears to be more consistent 

with the experimental data. This is shown below in Figure 5 by comparing against experimental data 

measured at the shielded location with microphone array below plate (Ref. 9) and for a mean velocity 

profile (Eq. 6.55) used in GAL.

Figure 4.—Nozzle/plate configuration. Figure courtesy Dr. James E. Bridges, 

NASA Glenn.

 (a) (b) 

Figure 5.—Power Spectral Density (PSD) of the far-field pressure fluctuations at 100 equivalent diameters from 

nozzle exit (lossless in dB scale referenced to 20 μPa) as a function of Strouhal number, for Ma =0.9. Plate trailing 

edge at yd/DJ = 1.2, xd/DJ = 5.7, DJ = 2.12′′, ψ ±90° and θ = 90°. Source model constants for GAL theory are the 

same as their figure 4. Source model constants for current predictions are Ψ = 0.04(ρ∞Ud)2; (l0,l1,l3)/DJ = 

(0.53,0.01,0.01); (L2,L3)/DJ = (0.5,20), Uc = 0.68 Ud; b0 = 0.52 and (a0,a1,a2) = (0.82,0.88,0.05). (a). Parametric 

increase in a1 with a0 = 1; a2 = 0. (b). Equation (28) using model (26).
NASA/TM—2016-218898 14

Figure 6.—ψ-directivity of RDT prediction (28) at peak Strouhal number (St) and upper limit as shown in Figure 5 for

M(y2) = Ma = (0.5,0.7,0.9) (colour coding same as Figure 2). Source model constants for RDT prediction is same as 

Figure 5.

In Figure 6 we show that the acoustic spectrum (28) possesses a dipole-like azimuthal structure. Here 

we consider the polar angle of 90° where the jet-surface interaction noise is greatest and a peak Strouhal 

number which looking at Figure 5 occurs at about St=0.12. The turbulence, defined through the source 

term ( ) ( 2 2 , ; , 3 ω)  s S k y y , amplifies the pure propagation effects described through the directivity factor 

shown in Figure 2.

Figure 7 shows that even though the GAL theory is based on a two-dimensional mean flow (i.e., on a 

large aspect ratio rectangular jet nozzle) it provides a reasonable estimate to the lower aspect ratio (AR) 

jet-surface interaction noise.

5.0 RANS Solutions

Although the RDT predictions based on an analytical mean flow shown in Figure 7 are in reasonable 

agreement with data, it is expected that the turbulence length scales and peak turbulent kinetic energy 

levels near the trailing edge will change with nozzle aspect ratio and flow conditions. These variations 

can be accommodated within the RDT model using a RANS-based mean flow, turbulent kinetic energy 

(k) and rate of energy dissipation (ε) to define the mean velocity profile, length scales and amplitude of 

the source function, S, near the trailing edge.
NASA/TM—2016-218898 15

(a) (b) 

(c) (d) 

Figure 7.—RDT predictions compared to noise data from various nozzle aspect ratio (AR) rectangular jets. Source 

model constants and mean flow for RDT prediction is same as Figure 5. (a) Ma = 0.9 and θ = 90°. (b) Ma = 0.9 and 

θ = 75°. (c) Ma = 0.7 and θ = 90°. (d) Ma = 0.7 and θ = 75°.

In this section we show results from three-dimensional SolidWorks Flow Simulations of jet flows 

with acoustic Mach numbers, Ma = (0.5,0.7,0.9) through rectangular jet nozzles with aspect ratios, 

AR = (2,4,8). The SolidWorks Flow Simulation automatic gridding methodology (Refs. 11 and 12) 

provides a way to mesh and solve the flow field around complex geometries and is, therefore, rather 

convenient for the jet surface interaction problem. In Figure 8, mean axial velocity and turbulent kinetic 

energy RANS results are compared with hotwire data from Zaman et al. (Ref. 7). The results are 

compared for the same aspect ratio (8) and surface length (12-in.), but, owing to facility constraints, at 

different flow conditions and very slightly different surface offsets. The RANS results have a surface 

offset of y2/DJ = 1.05, whereas the offset in the experiment is y2/DJ = 1.0. In comparing the RANS and 

hot-wire data, we have normalized the mean axial velocity and turbulent kinetic energy using the ideallyexpanded jet exit velocity as, UJ 
NASA/TM—2016-218898 16

(a) (b) 

Figure 8.—Comparison of normalized mean flow and turbulent kinetic energy obtained by RANS SolidWorks 

calculation at the trailing edge location, yd/DJ = 1.2, xd/DJ = 5.7, DJ = 2.12′′, with low Mach number and high aspect 

ratio experiment reported in Zaman et al. (Ref. 7), where jet Mach number is Ma = 0.22 and AR = 8. (a) Mean 

velocity, U/UJ. (b) Turbulent Kinetic Energy, 2

J k U .

and 2 UJ , respectively. It is expected that these normalized results can be reasonably compared to give 

some idea of the validity of the SolidWorks RANS solutions. Further comparisons are planned once more 

data becomes available. 

The results in Figure 8 show that, while the shapes of the axial velocity and turbulent kinetic energy 

distributions are well predicted, the peak turbulent kinetic energy is over-predicted by about 35 percent

compared to PIV data at the trailing edge location. Nonetheless, these differences are consistent with 

other RANS CFD codes, such as the NASA WIND code (Ref. 20). Cross flow distributions of mean axial 

velocity and turbulent kinetic energy, shown in Figure 9, near the surface trailing edge (xd/DJ = 5.7 in 

Figure 4), compare favorably with the experimental results. 

Figure 10 shows values of the normalized RANS-based turbulent kinetic energy and length scale, 

LRANS = k3/2/ε, at the trailing edge of the plate at the peak mean axial velocity location at the center of the 

span for Ma = 0.5,0.7,0.9 and AR2,AR4,AR8. The results show that the RANS-based turbulent kinetic 

energy and length scale reduce as the nozzle AR reduces for any given acoustic Mach number. On the 

other hand, the variation of these normalized scales is relatively insignificant at a fixed nozzle AR and 

varying acoustic Mach number.
NASA/TM—2016-218898 17

(a) (b) 

(c) (d) 

Figure 9.—Comparison of Normalized mean velocity and turbulent kinetic energy obtained by RANS SolidWorks 

calculation and experiments reported in Zaman et al. (Ref. 7) for same trailing edge location as the caption in 

Figure 8. (a) U/UJ: Zaman et al. (Ref. 7). (b) U/UJ: RANS calculation. (c) 2

J k U : Zaman et al. (Ref. 7). (d) 2J k U : 

RANS calculation.
NASA/TM—2016-218898 18

(a) (b) 

Figure 10.—Variation of normalized turbulent kinetic energy and length scale obtained by RANS SolidWorks 

calculation with Nozzle Aspect Ratio (AR) and Acoustic Mach number. (a) 2

J k U . (b) LRANS/DJ.

6.0 RANS-Based Jet-Surface Interaction Noise Predictions

Using the RANS solutions described in the last section, we define the length scales to be

3 l ck n n n = ε= 2 , 0,1 and 3 and the amplitude Ψ0 = ρ c k Ψ in the jet-surface interaction model, (28).

Moreover, the mean flow U/UJ is obtained directly from the RANS calculation at the trailing edge of the 

plate. The scaling coefficients are then tuned so that the predictions at polar angle, θ = 90 and Ma = 0.9

(where jet-surface interaction is greatest, see Figure 5 and Figure 7) and AR8 for which the theory is 

directly applicable. 

In Figure 11 and Figure 12 we show the RANS-based θ = 90 spectra for Ma = 0.7 and Ma = 0.9, 

respectively for aspect ratios AR4 and AR8. The low frequency roll-off is predicted well for almost all 

cases shown in Figure 11 and Figure 12. There is a some underprediction of Ma = 0.7 and AR = 4. 

They are, however, encouraging given that the model parameters in (28) have been kept fixed at 

(c0,c1,c3) = 1.4,0.021,0.022), Uc = 0.60Ud; b0 = 0.6 cΨ = 1.0 and (a0,a1,a2) = 0.82,0.88,0.05) in all cases,

with exception of the spanwise length L3/DJ, where L3/DJ = 10 for AR = 4 case and L3/DJ = 20 for AR = 8.

The latter reflects the larger spanwise extent of the turbulence in the larger aspect ratio case. 

It is expected that, with further experimentation with c0,c1,c3 and cΨ, the predictions could, 

potentially, be made to agree better with the data, especially if the model parameters in (28) are also tuned 

for each nozzle aspect ratio.

We note that the model does not predict the oscillations that are present in the data at very low 

frequency. These oscillations are believed to be due to interference between direct and edge-generated 

sound. In our case, the purely convected disturbance in a streamwise uniform flow (the gust) does not 

produce direct acoustic radiation at subsonic speeds. However, direct acoustic radiation is produced by 

the nonpurely convected flow disturbances, i.e., the ‘jet noise.’ This noise is also scattered by the edge

and its subsequent interference with the edge-generated noise is most likely responsible for the 

oscillations seen in the data.

The basic assumption of RDT is that the decay time of the turbulent eddies is short compared to their 

nonlinear interaction time, which means that the latter unsteadiness (which produces what we refer to as 

the ‘jet noise’) is being neglected. However, the purely convected gust, which is taken as the input 

disturbance to the RDT (see Eq. (17)), is much larger than the nonlinear sources of jet noise during the 
NASA/TM—2016-218898 19

time over which the interactions take place. As we have shown in this paper, the RDT model predicts the 

main asymptotic behavior of edge-generated noise and also provides predictions that are within 

experimental uncertainty over a range of nozzle aspect ratios.

 (a) (b) 

Figure 11.—θ = 90° spectrum: RDT prediction for Ma = 0.9 compared to noise data. (a) AR = 4. (b) AR = 8.

(a) (b) 

Figure 12.—θ = 90° spectrum: RDT prediction for Ma = 0.7 compared to noise data. (a) AR = 4. (b) AR = 8.
NASA/TM—2016-218898 20

7.0 Conclusions and Future Work 

In this paper we have extended the jet-surface interaction model developed in Reference 1, where it 

was shown that the jet-surface interaction noise spectrum, Iω(x), (28), at the observation point x, is given 

by the integral of a directivity factor (24) and a source function, ( ) ( 2 2 , ; , 3 ω)  s S k y y , that is related to the 

upstream turbulence correlation, (25), by the algebraic correspondence relation (21). In this paper we 

have extended the GAL model to include a finite decorrelation region in the upstream turbulence 

correlation function (25) by introducing the turbulence model (26) which exhibits properties of type 

shown in Figure 3 (the auto-correlation of (25)). We have shown, using simple asymptotic arguments and 

numerical analysis, that the presence of a decorrelation region (i.e., taking a1 > 0 in (26)) directly affects 

the low-frequency algebraic decay of the jet-surface interaction noise spectrum. This decay, often termed 

the low frequency ‘roll-off’, must be O(ω2

) for the acoustic field to be consistent with the dipole-like 

measured ‘roll-off’ in the experiments reported in Bridges (Ref. 9). A finite decorrelation is required in 

our theory for the roll-off to be O(ω2

) at very low frequencies where the maximum sound amplification 

occurs. In contrast, the GAL model (eq. 6.50 in that paper) did not include the decorrelation effect and 

thereby produced a spectrum that tends to O(1) at very low frequencies, which is at worst 10 dB greater 

than experiment (see Figure 5(b)). 

The model we have used in this paper, (29), also gives predictions that are reasonably accurate for 

more three-dimensional flows associated with lower aspect ratio rectangular jets (Figure 7). In addition

we have implemented a RANS-based RDT prediction method that takes into account the reduction in 

length scales and turbulent kinetic energy with nozzle aspect ratio predicted by these flow solutions. This 

approach generally gives predictions within experimental uncertainty. In principle, any empiricism

introduced by tuning the scales from RANS calculation could be eliminated by using experimental or 

LES data on turbulence (as, for example in Ref. 19). 

Professor J.T. Stuart1 recently pointed out that the importance of any noise prediction model is in its 

ability to show how to reduce the far-field noise. The jet-surface interaction model (28) based on the 

nonhomogeneous rapid-distortion theory (RDT) allows the “exact” turbulence conditions to be specified 

as its upstream boundary condition through the algebraic relation (21) that is a function of the two-point 

time-delayed correlation (25) of the stationary random function ρv t ⊥

′ ( x, ) (transverse momentum 

fluctuation), which Tennekes and Lumley (Ref. 14) explained must go negative with increase in spatial 

separation and/or time delay in a time-stationary turbulence field. The parametric study in figure 4.1a 

shows that varying the decorrelation region (increasing coefficient, a1) varies the predictions so that they 

lie between GAL model (a1 = 0) to the present results (figure 4.1b) which possesses the correct O(ω2

) low 

frequency roll-off. In principle, however, further increases in a1, could provide a means to reduce the low 

frequency amplification associated with jet-surface interaction. 

 1 Private communication.
NASA/TM—2016-218898 21

Appendix

Equation (22) shows that ( ) ( 2 3 ; ,ω) s Ey k expands as

( 2 2 ) ([ ( ) ( )] ) ( 2 ) 2

1 ;0,

∞

E y ω≡ − ω Uy Uy S d ++ − d

c

i Oy y (33)

since U(y2) = U(yd) + O(y2 – yd) and ( ) ( ) 2 2 c y c Oy y 2 2 =+ − ∞ d when y2 = O(yd), where S(yd) is given by

( ) 0

2 2

∞

∞

π

′′ − = d

d

d d

cUb

y

Uy c U

S (34)

Integral (29) then expands as

( ) ( )

( ) ( ) ( ) ( ) ( )

3

0 2 2 2 2

2 0

, ,0 1 , , 2 4 , for 

1 cos 1 ;0,

∞ Πω ω

θ ±π = = − θ− ∫ ω

 d

d

Ma

c Ma U

I Ma dM y y O y

Ma Ma E y

(35)

where Ma = M(yd) and Ud = U(yd). Inserting (33) result into this latter integral shows:

( )

( )

( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

0

3

2 2 2 2

2

3

2 2

0

,, 2

, ,0 1 4 , 1 cos 1

, ,0 4 , for 

1 cos 1 2

∞

∞

θ ±π ≈

Πω ω

− θ−  − +ω  

Πω ω π = = − θ− ω

∫ 



d

d d

d

d d

d

U

I Ma

cU U

dU y

Ma Ma Uy Uy S

cU U

y Oy

Ma Ma S

(36)

The integral in (36) is now O(1/ω) where S = O(1). Given that Equation (31) shows χ(ω/Ud,0) → 1 as 

ω → 0, the function, Πω ω ( , ,0)  Ud (using (30)) expands as, 

( )

1 2

2

0 1

1, for 0

, ,0

, for 

d

a a

U

a a

 = =  Πω ω 



 ω

 

 (37)

and, therefore, integral (29) possesses asymptotic properties:

( )

1 2

0

0 1

1 , for 0

,, 2

, for 

a a

I Ma

a a

 ω ==  θ ±π 



 ω



 (38)

when ω → 0.
NASA/TM—2016-218898 22

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