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A Model for Jet-Surface Interaction Noise Using Physically 
Realizable Upstream Turbulence Conditions 

Mohammed Z. Afsar 1 

Imperial College London, 180 Queen's Gate, London, SW7, UK 

S .J. Leib 2 

Ohio Aerospace Institute, 22800 Cedar Point Road, Cleveland, Ohio 44142, USA 


Richard F. Bozak 3 

National Aeornautics and Space Administration, Glenn Research Center, Cleveland, Ohio 44135, USA 

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 lOdB 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 de-correlation region. Numerical predictions, based on 
three-dimensional RANS solutions 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 de- 
correlation region 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. 

1 Research Fellow, Member AIAA. 

2 Senior Scientist, Senior Member AIAA. 

3 Researcher, Member AIAA. 


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C oo 








M n 




U c 


v i 













= sound speed 
= ambient sound speed 

= nozzle diameter 
= Green ’s function 
= acoustic spectrum 

= turbulent kinetic energy 

= streamwise wavenumber 
= characteristic length scale 

= acoustic Mach number 

= jet acoustic Mach number 
= pressure 
= time 

= averaging time 

= convection velocity 
= source volume 

= velocity vector 
= observer location 

= source location 
= turbulence intensity 
= angle function 

= specific heat ratio 
= Kronecker delta 

= turbulence dissipation rate 
= separation vector 
= density 

= polar angle measured from jet axis 
= time delay 
= radian frequency 

= gradient operator 
= absolute value 


ijfif = tensor indices =1,2,3 
-L = transverse component 


a = adjoint 

= average 


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= fluctuating quantity 
= Favre average 
= complex conjugate 

I. Introduction 

Jet flows of technological interest are almost always in close proximity to, or confined by, solid boundaries, such 
that the surface defining the boundary plays a direct role in the generation of sound and 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 further develop a prediction method 
for the noise generated by the interaction of a 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 non-homogeneous Rapid-distortion 
theory (RDT) introduced recently in [1] (hereafter referred to as GAL). 

Experiments conducted by Olsen & Boldman [2] and Wang [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. Recent 
experiments at the NASA Glenn Research Center ([4], [5], [6], [7], [8]) have considered 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 [5] have generally confirmed the trends of [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 & 8 in [5]. This low-frequency amplification effect has been called “jet-surface 
interaction” noise Brown [9], owing to its observed dipole characteristic and structural difference to high-frequency 
noise shielding and reflection. In [9] Brown, developed of an empirical model for jet-surface interaction noise, 
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 [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 over which non-linear interaction and viscous dissipation take place Hence 
the basic equation of the problem is 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 [12], 
but used here for the (second-order) transverse velocity auto covariance. In particular, we show that a region of 
negative correlation (or ‘de-correlation’) in this quantity directly impacts the low-frequency algebraic decay (often 
referred to as the ‘roll-off) of the edge-generated 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 ([13]). This result has implications for trailing-edge noise control. 

The prediction capabilities of the method are further developed by using results from Reynolds-averaged 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 
([10] [11]), which provides relatively fast solutions for the geometries of interest. As a check on the quality of these 
results, we include comparisons with measure flow data [6] for selected cases. 

In the next section we summarize the main features of non-homogeneous 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 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 (TKE) near the trailing edge are 
presented and compared with experimental data taken at NASA Glenn. 


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In section II we briefly review the relevant parts of the GAL analysis used in this work. In section III, we 
introduce the new model for the transverse velocity correlations and present the corresponding formula for the 
acoustic spectrum. In section IV 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 “roll-off’ of the spectrum and comparions 
with the model of GAL are shown. In section V, 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 [6] for the mean flow and turbulent kinetic energy distributions near the edge of the 
plate. In section VI 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 VII. 

II. Review of the GAL Formulation 
A. Euler Equations for Small-Amplitude Motion 

Let all lengths and velocities be non-dimensionalized by Dj and U d , respectively; time by Uj / Dj and 

pressure by pJJ ] , where pj and U 3 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 numbers is 
assumed to be large, ie. R = UjDj/v » O(l) and the turbulence Reynolds number is fixed at order 1, i.e. 

R t =aR = 0(l); where v is the kinematic viscosity and a = \v'\jUj «cO(l) 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 non-heat conducting and assume an ideal gas so that the entropy is 
proportional to In [p / p 7 j and the squared sound speed is yp / p , where p denotes the pressure, p the density and 
Y the specific heat ratio. Since the upstream entropy fluctuations are negligible, the inviscid pressure p' = p — p 0 
and momentum flux perturbations, u. = pv', (where V. denotes the velocity perturbation) on a transversely sheared 
mean flow with pressure p Q = constant , velocity and mean sound speed squared c 2 , are governed by the 

linearized momentum and energy equations 



0 + < 5 . u . 
Dt u j 

dU dp ' 

+ — — 

dyj 3v. 

= o 


D oP' , 3c A 

Dt dy. 

= 0 , 

( 2 ) 

where, y = {y l ,y 1 ,y,} = {y v y T } ,y T = {r 2 , y 3 1 and Dq /Dx = 9/5x + [/5/5y 1 denotes the convective 

7. Integration of the Euler equations (1) - (2) 


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Goldstein [14] showed that equation (1) will be satisfied for any function </>(y, t) and any purely convected 
quantity, ) , when p' and u t are determined by 

p '(y’ T )=-^(y’ T )’ 

Dt n\ 

.(j,x)= 5.. — — 

^ ’ lJ Dx n dy. A ’ 

V ' } ) 

1 du 5 

" ijk c^y^y k T'uM* 

where 5 y denotes the Kronecker delta, Ey k the alternating tensor and 

^ ^ | 2 
j dy. Dx dy j dy { 

a kind of generalized particle displacement ([14], eqs. 2.9 & 2.10). 

The energy equation, (2), is then satisfied when the scalar j,t) is determined by: 

A i_ c 4 |_^ 

Dr rJv, I 5v, Dt dy i dy\ I Dr 

which can be integrated to obtain, 


L ^^-d) c (x-y l / u (y T ),yT), 

Dl d J d D n ?U d 

Dt dy. { dy. Dt dy. dy ] 

and (D c (t - yJu(y T ^, y T j is another arbitrary, purely convected, quantity. GAL point out the operator L a is 
the adjoint to the Rayleigh operator, 

Ls Bn A C ;A__3L - 2 ^A c 2 _i_, 

Drldy, dy. Dr 1 ) dy, Sr, dy , ' 

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

Lp =0. nm 

The solution to (f)(y, r) 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) 


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( 11 ) 

Zg(j,T|x,f) = <5(j>-x)<5(T-f) , 

subject to appropriate boundary conditions, and the result inserted into (4) to obtain u. ( j,t) . 

B. Integral Solutions to the Euler Equations for Arbitrary Transversely Sheared Mean Flows 

GAL showed that the formal complete solution to the non-homogeneous Rapid Distortion Theory problem for 
the pressure perturbation, p , is given by 

P'{ X A=\ 

( 12 ) 

-T V 

(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 t^x-j^ j r ) . 

The corresponding solution for the transverse momentum fluctuation is 


pv^(x,f) = w.(*,f)— /\VU\ 



dU / dx. 



J ^ gi{yMx,t)iii c [x- yJu[y T ),y T ) dydx 

-T V 


gi (y,T\x,t) = ^ 

d D n+2 dU d 

dx ( Dt dx i dx { 



This solution involves only the purely convected quantity dfix- yJu(y T ^,y T ^ , that can be specified as an 
input condition within a general boundary value problem involving inhomogeneous boundary conditions. The 
Green’s function y,T \ x,t^j is found by solving (11) with incoming wave behavior as | j| — > o° and appropriate 
boundary conditions (on the bounding surfaces S that generate volume V) for (12) and (13) to hold. 

C. 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, 

g(y,x | x,t) = ( y,x | *,*) + I *,*) (15) 

where g" ^ (jVT | xp is now defined on all space and satisfies 


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( 16 ) 

n. 3[£> 0 3 g (0) (y,t | x,t)/ Dt 3 jdy. = 0 for y T e S 

where S is a doubly infinite surface and w(j) = {w.}(j) is the unit outward-drawn normal to S . 





Flat plate 


J > Streamwise direction 

Schematic of an example RDT problem: flow 
over a flat plate trailing edge 



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

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




J Jg ; ( 0 ) (j,x|x,?)cb c 

-T V 




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 d) c (^T - yjl/^y^, y T j is not a physically measurable quantitity, 
equation (17) provides an (in general) integral relation between CQ c [t - y l /u(y T 'j, y T j and the physical variable, 
pv'^xd) that can, in principle, be inverted (using identity (1.97) in Goldstein ([15]) to solve) for the Fourier 
transform of & c (t - yju( y T ), y T ) in terms of the Fourier transform of pv^ . 

D. Relation Between the d) 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 (Q c [t - y l /u(y T }, y T j and 
in the actual flow is the same as it would in an idealized mathematical representation of a transversely sheared mean 


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flow where R r = aR = 0(l) 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 y 1 ). 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 pv^(x,/) and relating it exactly to the 
Fourier transform of (b c [r - y l ju{ y y T ^, y T ) , but that this relation can only be specified at N discrete transverse 
space points, say , where x[^ is given by solutions to discrete equation, 

u[x^ j = f/(x 2 ), for fi — 1,2, ...TV , leading to a matrix problem to determine the auto-covariance of the transform 
of (b c (x-yJu(y T ),y T ) in terms of the auto-covariance of the transform of pv^ (x,/) . For jet flows, such as those 

considered in the present paper, there are two such points and, following GAL, we take |x^,x^ j = y d , where y d 

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 de-correlation 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 pv^(x,/) and (b c [r - y^/ui^y^, y T j to obtain a relatively simple working 

formula relating the spectrum of the convected quantity d) c f-J'iM-G’J’r) > > 

oo oo 

s(y 2 ,y 2 ;k v (o) = — J J + + 

to the experimentally measurable transverse velocity spectrum 

f( * 2 Ak’A®^ 3 ) = 7 — J e ,[<in ~ k ^f L (x 2 ,i- 2 |®/t/(.y 2 ),® / G v 2 ),!7 3 ,T)t/VT 

U'(y 2 )u'(y 2 ) 

1+ — ~ b o 



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( 22 ) 

E {y 2 ,k v u) ^ y— 

c l y d ) 





[co7[/ 2 (j 2 ) + ^ 2 ] inU^y^ 
7co 2 / 1/ 2 (^ 2 ) + ^ 3 2 - A: 2 c 2 JJ"y d 

Z? 0 is a constant relating the even and odd symmetry components of the Fourier transform of d) c (see GAL) and 
u"{y d ) is the second derivative of the mean velocity profile at y d . We also use this formula for the calculations in 
this paper. 

E. Scattered Solution 

The scattered solution, | 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 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 ^ | . 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 v k^ = ^ O(l) for y 2 = <9(l) . 

This low-frequency approximation is also adopted in this paper. 

F. 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 jet- surface 
interaction noise is given by remarkably simple asymptotic result, 

1 oo f , y coco 

7 ®( x ) = — \e im p s (x,t)p s (x,t + xyiT~ \\ d(o,\i/ ,M(y 2 ))s[v 2 ,yyk^\(o\dy 2 dy, 

ZK -oo \^ X ) 00 V 7 


in which the integrand can be interpreted as the product of the source function s[^ 2 ,y 2 \k^\(Q^ with a non- 

uniform directivity factor, d(#,i/a,m(j/ 2 )) , that encapulates all propagation and surface interaction effects when 

K 0(\) and |x| — » 00 . c m denotes the local acoustic Mach number at the position y 2 and the 

spanwise wavenumber k\ ’ = sin# cos i/a is found by applying the method of stationary phase for the inverse 

Fourier integral in k 3 (see GAL, p. 553). Here, # is the polar observation angle measured relative to the jet center 
line and l/A is the azimuthal angle in the cross-stream plane. 

The directivity factor, given by 

D(e, V ,M(y 2 )) 

_ 3/2 

_M(y 2 )M(y 1 )\ (fi-cosfl) 


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0 2 

and reduces to Goldstein’s ([14], eq. 3.23) result sin 2 — j |^1- M(j; rf )cos0j at the point where the velocity 

profile is maximum y 2 = y 2 = y d since /? = (l- sin 2 # cos 2 y/^j ,is unity at y/ = +n , in the plane perpendicular to 
the plate. In Figure 2 we show that D^6,y/,Ma^ peaks at 90° for subsonic acoustic Mach numbers. The azimuthal 
variation of D(6,\if,Ma} possesses dipole-like structure at any point y 2 = y 2 (see figure 4.1b). 


0 (degrees °) 

ip (degrees °) 

Figure 2 - Directivity factor: (a) 0 -directivity, (b) \j/ -directivity 

III. Physically Realizable Upstream Turbulence Conditions 
A. 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 

I 1 °° 

( v l(*vK ( x i’V 3 + ri v t + r)) = lim— j J v[(x t t)v[(x l ,x 2 ,x 3 + n 3 ,t + 'u)dtdx 3 

-T - °° 


that enters the integrand of (20) is both experimentally determinable and has a well established database ( e.g . 
experiments reported in [16]). 

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 de-correlation region, which we show has a direct impact on the jet surface interaction noise and, in principle, 
could provide a means to reduce it. 


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We use the function ([12]): 

pv' ± (x r y J ,x v t)pv' 1 (x r y J ,x 3 +r] v t + t)) = 

a ° + a ^ + a ^ + - 

where the decay function is, 

( 26 ) 

JT (77, , r7 3 , t) = ^( 77, //, ) 2 +(/?,- t/ c T ) 2 // 0 2 + (773 /Z 3 ) 2 


and the streamwise and spanwise separations are Tj x =x 1 — x x and r/ 3 , respectively. l x and / 3 are turbulence length 
scales in streamwise and spanwise directions, respectively; l Q , on the other hand, measures the scale of turbulence 
as it convects over separation distance 7] 1 with convection velocity, U c . 

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

subsequent Fourier transform integrals, i.e. v F(xJ = x F 0 £ where CL is a small positive number, 0<a<l 

which depends on the symmetric location x x = (x 1 + xJ/2 , since we this quantity is expected to be independent of 
the streamwise coordinate for the nearly parallel flow being considered. *F 0 is expected to scale with the transverse 

component of the mean square turbulence momentum flux (pv') , L { ,L 3 are geometric spatial scales: L 3 being a 
measure of the transverse extent of the turbulence and the L x , is taken to be large in order to insure that the 
correlation (26) is relatively independent of x x to remain consistent with our representation of the upstream 
boundary condition described in section II. 

As shown in Figure 3, allowing the coefficient a x > 0 in equation (26) gives a negative (de-correlation) region 

for the auto-correlation (xmr | 3 = 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. 

Figure 3 - Turbulence model (26) with (27) for T = fj 3 = 0 . (a) GAL model a x =a 2 = 0 ; 
(b) Present Model model a x > 0 . 


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B. Acoustic spectrum formula with a finite de-correlation region 

Substituting the source model function (26) in (18)-(20) for Syy 2 ,y 2 \k\ , Q)J and inserting this result in (23), we 
obtain the final formula for the acoustic spectrum used in this paper 


4 n\x\ l n ) U [ c 

1 \ 

— -jp- (j8-cos d)l 0 (Ma,d,yf) 

where (3 is defined below (24) and 


) {Ma,e,\f/) = 4 J 


tl[(olu(y 2 ),(oy 

0 [l-M(j 2 )cos0] E[y 2 $\a)} [l -pM(y 2 )~ 

dM(y 2 ) 

where Ma = u{y d }l c m denotes the maximum acoustic Mach number and spectral functions Eyy 2 \k^\(D^ is given 
by (22) and u(a>lu(y 2 ),(o,k^\ is defined by: 

H/V.T + Z 

7 >,) + 

(KMr+4 'y u - 


+ (K -<°I u c)K 

c J 

X-x(k v k 3 )~ (kj 3 ) +(k l -co/U c ) l 2 + 1 

The derivation of this formula is summarized in appendix A of Afsar et al. [17]. 

IV. Low-Frequency Roll-Off 

The predictions in GAL were compared against the jet-surface interaction experiments performed at NASA 
Glenn ([5], [6], [7], [8], [9]). The relevant geometric parameters are shown in Figure 4 


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Figure 4 - Nozzle/plate configuration. Figure courtesy Dr. James E. Bridges, NASA Glenn 

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. a { = a 2 = 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 I 0 ^Ma,6,\j/^j defined by (29), for very small 
frequencies. As CO — > 0 , the dominant contribution to integral comes in the vicinity of critical level y 2 = . In 

the appendix, we estimate terms in (29) under this limit by consdering y/ = ±90° (which is the azimuthal location of 

the data for which the GAL predictions were made); = 1 and = 0 here. Substituting equation asymptotic 
properties (38) into acoustic spectrum formula (28) shows that the latter possesses asymptotic properties, 

O(l), for a x - a 2 - 0 

O(co 2 ), 




when CO — > 0 that is now directly dependent on the whether a de-correlation region in (26) exists or not. 

In the absence of a de-correlation region in (26) (i.e. a { = a 2 = 0) the acoustic spectrum does not possess a low 
frequency roll-off as such and tends to 0(l)(i.e. a constant, see equation (6.52) in GAL). A finite de-correlation 
region, however, increases the low frequency ‘roll-off in the prediction of (28) to exhibit an asympotic order of 
o(co 2 ) 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 ([8]) and for a mean velocity profile (eq. 6.55) used in GAL. 


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(a). Parametric increase in a { with (<2 0 = l;a 2 = 0) . (b). Equation (28) using model (26). 

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 /i Pa ) as a function of Strouhal number, for Ma =0.9. 

Plate trailing edge at y d /Dj = 1.2, x d /Dj =5.7 , D d = 2.12", i/a = ±90° and 0 = 90°. Source model constants 
for GAL theory are the same as their figure 4. Source model constants for current predictions are 

'P 0 =0.04(p^) 2 ;(/ 0 ,/ 1 ,/ 3 )/ J D J =(0.53,0.01,0.0l);(L 2 ,Z 3 )/ J D / =(0.5,20), U c = 0.6W d \b a = 0.52 and 

{a Q ,a v a^ = (0.82, 0.88, 0.05) . 

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 S^y 2 , y 2 ;k[ s \co ^ , 

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 


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Figure 6 - \|/ -directivity of RDT prediction (28) at peak Strouhal number (St) and upper limit as shown in 
Figure 5 for M(y 2 ) = Ma = (0. 5, 0.7, 0.9) (colour coding same as Figure 2). Source model constants for RDT 
prediction is same as Figure 5. 

(a). Ma = 0.9 and 0 = 90° . (b). Ma = 0.9 and 6 = 75° 


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RDT prediction 

► Experiment: AR2 

• Experiment: AR4 

■ Experiment: AR8 

_l i i i 

0.1 0.2 0.3 0.4 0.5 

Strouhal number 

(c). Ma = 0.7 and 0 = 90° . (d). Ma = 0.7 and 6 = 75° 

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. 

V. 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 (TKE) 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, TKE and rate of energy dissipation (e) to define the mean 
velocity profile, length scales and amplitude of the source function, S , near the trailing edge. 

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 ([10], [11]) 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. [6]. The results are compared for the same aspect ratio (8) and surface length (12-inches), but, owing to 
facility constraints, at different flow conditions and very slightly different surface offsets. The RANS results have a 
surface offset of y 2 / D s = 1.05 , whereas the offset in the experiment is y 2 / D s = 1.0 . In comparing the RANS and 
hot-wire data, we have normalized the mean axial velocity and turbulent kinetic energy using the ideally-expanded 
jet exit velocity as, Uj and 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. 


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(a). Mean velocity, U/Uj (b). Turbulent Kinetic Energy (TKE), k/u j 

Figure 8- Comparison of normalized mean flow and TKE obtained by RANS SolidWorks® calculation at the 
trailing edge location, y d /D J = 1.2, x d /D J = 5.7 , £^ = 2.12" , with low Mach number & high aspect ratio 
experiment reported in Zaman [6], where jet Mach number is Ma = 0.22 and AR = 8 . 

The results in Figure 8 show that, while the shapes of the axial velocity and turbulent kinetic energy distributions 
are well predicted, the peak TKE is over-predicted by about 25% 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 [19]. 
Cross flow distributions of mean axial velocity and turbulent kinetic energy, shown in Figure 9, near the surface 
trailing edge ( x d /Dj = 5.7 in Figure 4), compare favorably with the experimental results. 

y 3 /o 

(a). U/Uj : Zaman et al. [6]. 

y 3 /D 

(b). U/Uj : RANS calculation. 


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y 3 /o 


















(c). TKE, k/Uj : Zaman [6], (d). TKE, k/u ) : RANS calculaton. 

Figure 9 - Comparison of Normalized mean velocity and TKE obtained by RANS SolidWorks® calculation 
and experiments reported in Zaman [6] for same trailing edge location as the caption in . 

Figure 10 shows values of the normalized RANS-based TKE and length scale, L RANS = k m / £ , 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 TKE 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. 

(a). TKE, k/u j (b). RANS length scale, L S4VS /D ; 

Figure 10 - . Variation of normalized TKE and length scale obtained by RANS SolidWorks® 
calculation with Nozzle Aspect Ratio (AR) and Acoustic Mach number. 


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VI. RANS-Based Jet-Surface Interaction Noise Predictions 

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

y / 

l n - c n k /2 / £ , n = 0,1 and 3 and the amplitude = c^pk 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, 6 = 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 1 1 and Figure 12 we show the RANS-based 6 = 90 spectra for Ma = 0.7 and Ma = 0.9 , respectively 
for aspect ratios AR4 and AR 8 . The low frequency roll-off is predicted well for almost all cases shown in Figures 
11 and 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 (c o ,c i 5 C 3 ) = (l. 4, 0.021, 0.022 ) , U c = 0.60U^; b Q = 0.6 

Cy = 1.0 and(a 0 ,a 15 a 2 ) = ^0.82,0.88,0.05) in all cases, with exception of the gspanwise length L^/Dj , where 

LjDj = 10 for AR=4 case and LjDj = 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 c 0 ,c v c 3 and c ^ ,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 

We note that the model does not predict the oscillations that are present in the data at very low frequency. These 
are believed to be due to scattering of the jet noise by the edge of the plate, rather than being directly produced by 
the edge-noise source treated in this work. 

(a) . AR = 4 (b). AR = 8 

Figure 11 - 6 = 90° spectrum: RDT prediction for Ma = 0.9 compared to noise data . 


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(a). AR = 4 (b). AR = 8 

Figure 12 - 6 = 90° spectrum: RDT prediction for Ma = 0.7 compared to noise data. Parameters same as in 

VII. Conclusions and Future Work 

In this paper we have extended the jet-surface interaction model developed in [1], where it was shown that the 
jet-surface interaction noise spectrum, (28), at the observation point x, is given by the integral of a 

directivity factor (24) and a source function, S^y 2 , y 2 ;k[ s \co ^ , 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 de-correlation 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 de-correlation region (i.e. taking a { > 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|cd 2 j for the acoustic field to be consistent with the dipole-like 

measured ‘roll-off in the experiments reported in Bridges [8]. A finite de-correlation is required for the roll-off to 

be o|co 2 j at very low frequencies where the maximum sound amplification occurs. In constrast, the GAL model 

(eq. 6.50 in that paper) did not include the de-correlation effect and thereby produced a spectrum that tends to O(l) 
at very low frequencies, which is at worst 10 dB greater than experiment (see figure 5b). 

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 addtion 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 empiriscim introduced by tuning the scales from RANS calculation could 
be eliminated by using experimental or LES data on turbulence (as, for example in [18]). 


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Professor J. T. Stuart 4 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 non-homogeneous 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 pv^ ( x,t ) (transverse momentum fluctuation), which Tennekes & Lumley [13] 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 de-correlation region (increasing coefficient, a } ) varies the predictions 

so that they lie between GAL model {a x = 0) to the present results (figure 4.1b) which possesses the correct o|co 2 j 

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

frequency amplification associated with jet- surface interaction. Physically, this condition would indicate the 
presence of larger, more-organized, structures within the turbulence that are more likely to de-correlate than finer- 
scale turbulence. 


Equation (22) shows that E^y 2 \k^\ coj expands as 

£( j2 ;0, W ).4([uU)-f/(j 2 )]+icas)+O(7 2 -y,) 



u (yi) =u (yd) + 0 (y2-yd ) and cl {y2) =c l + °{y2-yd) when = .where s(y d ) is g i 


s(ydh ^JLA 

T T” 

u v 



Integral (29) then expands as 

I 0 (Ma,6,±n/2) = 

^ c^Ma^fl^cd/U d ,co,ti) 

(l- Ma cos df (l- Ma) ^ |is(j; 2 ;0,©)| 


- [ for>' 2 = ofJ 

I J J7l u -n /-,»)! 


where Ma = and U d = U (y d ) . Inserting (33) result into this latter integral shows: 

4 Private communication. 


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I„(Ma,9,±n/2) = 


U A 


(l-Vfacosfl) (l -Ma)l \u(y d )-U(yf\ +(©5) 

;dU(y 2 ), 


,, cjjff^ff^cof) n 
( 1 - Ma cos Of ( 1 - Ma ) 2(0 S ’ 

for y 2 = 0(y d ) 

The integral in (36) is now 

0(l/ co) where S = O(l) . Given that equation (31) shows 1 as 

(0 — > 0 , the function, flyco/U d ,<J),0) (using (30)) expands as, 

Il{(olU d ,(o,0)~\ 

1, for a { = a 2 = 0 

co 2 , for a 0 ~ a { 


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

I 0 (Ma, d+n/2) 

l/ co, for a x = a 2 = 0 

co, for a Q ~ a x 


when CO 0 . 


MZA 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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Flow,” AIAA Paper 2013-2184, 2013. 


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[7] Zaman, K. B. M. Q., Fagan, A., Clem, M and Brown, C. A., “Resonant Interaction of a Rectangular Jet with a Flat-plate,” 
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[11] “Enhanced Turbulence Modelling in SolidWorks Flow Simulation” Dassault Systemes SolidWorks Corporation Technical 

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pp. 1324-1335,2011 

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[14] Goldstein, M.E., “Scattering and Distortion of the Unsteady Motion on Transversely Sheared Mean Flows, J. Fluid Mech. 
Vol. 91, pp. 601-632,1979. 

[15] Goldstein, M. E., Aeroacoustics , McGraw Hill, New York, 1976. 

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[18] Karabasov, S.A., Afsar, M.Z., Hynes, T.P., Dowlong, A.P., McMullan, W.A., Pokora, C.D., Page, G.J. and McGuirk, J.J, 
“Jet Noise: Acoustic Analogy Informed by Large Eddy Simulation,” AIAA Journal , Vol. 48, No. 7, pp. 1312-1325, 2010. 

[19] Dippold, V., “ Acoustic Reference Nozzle with Mach 0.97, Unheated Jet Flow,” NPARC Alliance Validation Archive, NASA 
Glenn Research Center, 


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