NASA Technical Reports Server (NTRS) 20060051735: A Hybrid RANS/LES Approach for Predicting Jet Noise

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by  NASA Technical Reports Server (NTRS)

A Hybrid RANS/LES Approach for Predicting Jet Noise
M.E. Goldstein*
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio 44135
Hybrid acoustic prediction methods have an important advantage over the current Reynolds
averaged Navier-Stokes(RANS) based methods in that they only involvemodeling of the relatively
universal subscale motion and not the configuration dependent larger scale turbulence.
Unfortunately, they are unable to account for the high frequency sound generated by the
turbulence in the initial mixing layers. This paper introducesan alternative approach that directly
calculates the sound from a hybrid RANS/LES flowmodel (which can resolve the steep gradients
in the initial mixing layers near the nozzle lip) and adopts modeling techniques similar to those
used in current RANS based noise prediction methods to determine the unknown sources in the
equations for the remaining unresolved components of the sound field. The resulting prediction
methodwould then be intermediate between the current noiseprediction codes and previously
proposed hybrid noise prediction methods.
I. Introduction
Current noise prediction methods are usually based on acoustic analogy approaches6,17,5 which (as a minimum)
require modeling the entire unsteady flow that actually produces the sound and are therefore unlikely to beuniversal
beyond the data base for which they are calibrated. This difficulty would not occur if the sound field were
determined from a direct Navier-Stokes (DNS)1 solution or even an ordinary Large Eddy Simulation (LES).
Unfortunately, the computer resources needed to resolve all the relevant length scales are enormous—often well
beyond those availableon presentday machines. A reasonable compromise wouldbe touse LES simulationswith
very broad filter widths.Thisapproachwasadopted by Bastin, Lafon and Candell2, Bodony and Lele7,8,9, andothers.
They found that simulations of this type are reasonably accurate at low frequencies, but significantly underpredict
the high frequency component of the spectrum. It is now known that this occurs because these methods do not
account for the sound generated by the unresolved turbulence scales, which can correspond to the entire unsteady
flow in the thin shear layers near thenozzle lip24 as well as to the sub-filter scale motionfurther downstream9. The
present paper is an attempt to resolve these difficulties by directly calculating the low frequency sound from a
hybrid RANS/LES computation and determining the remaining small scale, or high frequency, sound by modeling
the source terms in an appropriate set of linear equations.
The relevant equationsare obtainedby dividing eachof the flow variables in theNavier-Stokesequations into
a component that satisfies universal set of conservation laws, which include, among other things, the Reynolds
averaged Navier-Stokes (RANS) equations as well as the Favre20 filtered Navier-Stokes equations with arbitrary
closure models for the sub-filter scale stresses (i.e., the equations that are actually used in most LES calculations)
and a residualcomponent that satisfies theNavier-Stokesequations with thebase flowequations subtractedout. By
introducing new (ingeneral non-linear)dependentvariables10,11,12 the latter equations canbe rewritten inthe form of
the linearized Euler equations with sources that have the same form as those that would be produced by external
stress and heat flux perturbations. The corresponding source strengths, which depend on the non-linear residual
Reynolds stresses, can, inprinciple,be modeled, and theresulting linearequations canbe solved by using a Greens’
function approach. The result can then be used to obtain an expression for the far field pressure autocovariance in
terms of the residual-scale turbulent motion. The latter would only depend on the correlation (i.e., the lower order
statistics) of this motion if the base flow were taken to be a steady RANS solution. But the result for any time
dependent base flow, such as an LES solution to the filtered Navier-Stokes (or FNS) equations, appears to depend
on thedetailedinstantaneous residual -scalestresses--which are very difficulttomodel.
Current RANS based noise prediction codes, such as the JeNo15 code, determine the unknown source
strengths from experimental data. This cannot bedonewithprediction methods that involve time dependent base
*Senior Scientist, Fellow AIAA
1
American Institute of Aeronautics andAstronautics
flows because, as a practicalmatter, experimentalists are limited to documenting the reproducible (i.e., non-random)
characteristics of these sources, such as their correlations or other lower order statistics. It is therefore important to
derive a formula for theresidual sound field that depends only on non-random quantities the can be measured with
currently available experimental techniques.Thepresentpaper is an attempt toobtain such an equationby exploiting
the statistical independence (i.e., thedecoupling) of any base flow solution (such as a hybrid RANS/LES simulation)
from the detailed sub-scale fluctuations that wouldoccur in an actual experiment. The only couplingthat can occur
in these simulations is through the base flow stresses and heat flux vector—which are assumed to be modeled in
terms of the remainingbaseflowvariables and theirderivatives in hybrid RANS/LES computations. Thebase flow
solutions are therefore calculated from a closed set of equations that involve only base flow variables. The chaotic
subscale fluctuations in any realization of the flow, i.e., in an actual experiment, can, therefore, not be correlated
with the fluctuations in thesesolutions.
The present paper assumes that the model for the residual (i.e., unresolved) stresses, say (cid:86)(cid:99) , is
(cid:80)j
constructed from appropriately reduced experimental data (which can, in principle, be obtained from PIV
measurements). While the actual (i.e., the experimental) resolved and residual motions are certainly correlated, the
experimentally based model for(cid:86)(cid:99) must reflect the fact that the experimental residual-scale motions are
(cid:80)j
statistically independent of the fluctuations in the base flow (i.e., the hybrid RANS/LES) solution and must,
therefore be uncorrelated with the fluctuations in that solution. This doesnot,however,mean that the expectation or
average value of the residual stresses will be statistically independent of the base flow solution, since the latter is
only required to represent a given realization of the flow on a time average basis and can significantly depart from it
at any instant of time.
As already indicated, thepresent paper introduces a hybrid approach inwhichthe base flow correspondsto
a typical hybrid RANS/LES solution that goes from a RANS type solution near the nozzle lip (where the initial
shear layers are too thin to be resolvedby the necessarily coarse LES computation24) to a (fairly coarse)LES solution
further downstream. We adopt the so called universal modeling approach13 which assumes that the unknown stresses
and heat flux in the base flow equations are modeled by a RANS/LES model that goes from a time average (i.e.,
RANS type) model near the nozzle lip to a pure spatial LES-type model further downstream. There is no need to
complicatematters byrequiring that the base flow satisfythe filtered Navier-Stokesequations withinhomogeneous
filtering as was done, for example, by Germano4 because the combined result (i.e., the base flow plus residual
equation solution) is, in principle, exact. But the subscale stress correlations in the residual pressure equation (that
go continuously from the full fluctuating stress near the nozzle lip to the instantaneous sub-filter scale stresses
furtherdownstream) ultimately have tobe modeled.
At high Reynolds numbers the small scale motion is expected to be statistically independent of large
energy bearing scales in any realization of the flow. This does not, however, imply that all of the resolved motion
will be statistically independent of the subscale motion, because the near cutoff scales are expected to be highly
correlated. It is only the hybrid RANS/LES simulation of the resolved scalesthat is expected tobe uncorrelated with
the actual, i.e. the experimentallymeasured, subscale tensor(cid:86)(cid:99) . But the propagationfactor in the Greens’ function
(cid:80)j
solution is likely to be dominated by the large energy -bearing scales, andKolmogorov’s 3hypothesis (which forms
the bases of many of the current subscale turbulence models13) indicates that the small scale motion should be
statistically independent of these scales. It is, therefore, likely that the present decorrelation assumption will be
satisfied --at least on a global basis-- in the downstream region of the actual flow (where the implied filter width
eventually becomes small relative to the transverse lengthscale).
The statistical independence assumption is trivially satisfied in the upstream region where the filtered
equations reduce to the steady RANS equations because the fluctuating component of the base flowand, therefore,
the covariance of the base flow and fluctuating motion, is identically zero. It will not, however, be satisfied in the
intermediate blending region. But to the extent that the propagation factor is non-local and the blending region is
sufficiently small the present decorrelation assumption should bereasonably well satisfied in any actual realization
of the flow.
A major attraction of the current hybrid approach is that the subscale stresses that have to be modeled in
the downstream region should be much more universal than the large scale stresses that have to be modeledwith the
usual RANS based methods. Unfortunately, the present approach still requires full unsteady flow modeling in the
initialmixing layers. On theother hand it is now believedthat the soundproduced by the larger scale motion at the
end of thepotential core is highly coherentwhile the soundproducedbythe small scale motion in theinitialmixing
layers (as well as by the smaller scale motion at the end of the potential core) is much more random23. The latter
should therefore be less sensitive tovariations in retardedtime then theformer, which implies that it should alsobe
2
American Institute of Aeronautics andAstronautics
less sensitive to the details of the source structure and, therefore, that relatively universal source models can be
constructed.
This is consistent with theprevailingview that RANS based codes can adequately predict the larger angle sound
field, which is primarily generated in the initialmixing layers, but tend to under-predict the small angle soundfield,
which is believed to be generated near the end of the potential core15,23. The present approach can be viewed as a
framework for directly calculating the latter while retaining the current RANS based approach to calculate the
former. But, unlike the RANS approach, the present formalism explicitly accounts for the scattering of the small
scale sound by the large scale motion--an effect that was emphasized by Crighton6. To the extent that this
phenomenon is nonlocal, the present statistical independence assumption should be very appropriate to its
computation.
Equation Section 2
II. Sub-Filter Scale Equations in Vector Form
We decompose the pressure p, density (cid:85) and velocityvi,i (cid:32)1,2,3, into base flow components (cid:85)ˆ,ˆp
and v(cid:4) and residual componentsdefined by
i
(cid:85)(cid:99)(cid:32)(cid:85)(cid:16)(cid:85)ˆ, vi(cid:99)(cid:32)vi (cid:16)v(cid:4)i, p(cid:99)(cid:32) p(cid:16) ˆp (2.1)
where thebase flow components satisfy a universal set of conservation laws,which include, among other things, the
RANS equations as well as the Favre20 filtered Navier-Stokes equations (in which case (cid:85)ˆ,ˆp and v(cid:4) (cid:123)(cid:85)(cid:109)v /(cid:85)ˆ
i i
would correspond to filtered and Favre20filtered quantitiesrespectively ). The residual components can be shown to
satisfy the5 formally linear equations11,12
L u (cid:32)s for (cid:80),(cid:81)(cid:32)1,2,3,4,5 (2.2)
(cid:80)(cid:81) (cid:81) (cid:80)
where
(cid:11)(cid:74)(cid:16)1(cid:12)
(cid:94)u (cid:96)(cid:123)(cid:94)(cid:85)v(cid:99),(cid:83)(cid:99),(cid:85)(cid:99)(cid:96)(cid:123)(cid:94)(cid:85)v(cid:99),(cid:85)v(cid:99),(cid:85)v(cid:99),(cid:83)(cid:99),(cid:85)(cid:99)(cid:96), (cid:83)(cid:99)(cid:123) p(cid:99)(cid:16) (cid:86)(cid:99) (2.3)
(cid:81) i 1 2 3 2 jj
denotes the solutionvector and thefirstorder linear operator L is defined by
(cid:80)(cid:81)
L (cid:123)(cid:71) D(cid:4) (cid:14)(cid:71) (cid:119) (cid:14)(cid:119) (cid:11)c(cid:105)2(cid:71) (cid:14)(cid:71) (cid:12)(cid:14)K (2.4)
(cid:80)(cid:81) (cid:80)(cid:81) 0 (cid:81)4 (cid:80) (cid:81) (cid:80)4 (cid:80)5 (cid:80)(cid:81)
with
1 (cid:119)(cid:84)(cid:4) (cid:167)(cid:119)v(cid:4) 1 (cid:119)(cid:84)(cid:4) (cid:183)
K (cid:123)(cid:119) v(cid:4) (cid:16) (cid:80)j (cid:71) (cid:16)(cid:11)(cid:74)(cid:16)1(cid:12)(cid:168) j (cid:71) (cid:16) (cid:81)j (cid:184)(cid:71) i,j (cid:32)1,2,3; (2.5)
(cid:80)(cid:81) (cid:81) (cid:80) (cid:85) (cid:119)x (cid:81)5 (cid:168)(cid:119)x (cid:81)4 (cid:85) (cid:119)x (cid:184) (cid:80)4
j (cid:169) j j (cid:185)
and
D(cid:4) (cid:123) (cid:119) (cid:14) (cid:119) v(cid:4) ; (cid:84)(cid:4) (cid:123)(cid:71) p(cid:16)(cid:86)(cid:4) ;
0 j ij ij ij
(cid:119)t (cid:119)x
j
(2.6)
(cid:119)
(cid:119)(cid:80) (cid:123) (cid:119)x , (cid:80)(cid:32) j(cid:32)1,2,3; v(cid:4)(cid:80),(cid:84)(cid:4)(cid:80)j,(cid:119)(cid:80) (cid:32)0,(cid:80)(cid:32)4,5
j
The source function s is definedby
(cid:80)
3
American Institute of Aeronautics andAstronautics
(cid:74)(cid:16)1
s (cid:123) D(cid:80) (cid:11)(cid:86)(cid:99) (cid:16)(cid:86)(cid:4) (cid:12)(cid:16) D(cid:80)(cid:11)(cid:86)(cid:99) (cid:16)(cid:86)(cid:4) (cid:12) (2.7)
(cid:80) (cid:81)j (cid:81)j (cid:81)j ii jj jj
2
with
(cid:119) (cid:119)v(cid:4)
D(cid:80) (cid:123)(cid:71) (cid:14)(cid:71) (cid:11)(cid:74)(cid:16)1(cid:12) (cid:81) (2.8)
(cid:81)j (cid:81)(cid:80) (cid:119)x 4(cid:80) (cid:119)x
j j
where
(cid:86)(cid:99) (cid:123)(cid:16)(cid:85)v(cid:99)v(cid:99), v(cid:99) (cid:123)(cid:16)(cid:11)(cid:74)(cid:16)1(cid:12)h(cid:99), v(cid:99) (cid:32)0 (2.9)
(cid:80)j (cid:80) j 4 0 5
(cid:86)(cid:4) (cid:32)0 and (cid:86)(cid:4) and(cid:86)(cid:4) are the base flow stresses and heat flux vector, which would be given by
5j ij 4j
(cid:86)(cid:4) (cid:123)(cid:16)(cid:85)ˆ (cid:11)v(cid:107)v (cid:16)v(cid:4)v(cid:4) (cid:12),(cid:86)(cid:4) (cid:123)(cid:16)(cid:11)(cid:74)(cid:16)1(cid:12)(cid:170)(cid:85)ˆ (cid:11)h(cid:107)v (cid:16)h(cid:4) v(cid:4) (cid:12)(cid:14) 1(cid:86)(cid:4) v(cid:4) (cid:14)(cid:86)(cid:4) v(cid:4) (cid:186) if thebase flowwere chosen tobe
ij i j i j 4j (cid:171) 0 j 0 j ii j ij i(cid:187)
(cid:172) 2 (cid:188)
the FNS equations. Finally, h (cid:123)h(cid:14)1v2, h(cid:99) (cid:123)h(cid:99)(cid:14) 1v(cid:99)2,where c(cid:105)2 (cid:32)(cid:74)RT(cid:4) is the squaredbaseflow sound
0 0
2 2
speed, and, as usual, the viscous terms, which are believed to play an insignificant role in the sound generation
process, havebeen neglected. The Latinindicesgo from1to 3, whilethe Greekindices gofrom1to5.
Equations (2.7) to (2.9) show that the source term in equation (2.2) consists of two parts: a part
(cid:86)(cid:99) associated with the residual (or subscale) motion and a base flow component (cid:86)(cid:4) that is the same as the source
ij ij
that appears in the hybrid RANS/LES equations. The subscale motion generated by this latter source component is
completely determined by the base flow solution because it is generated by that solution and is determined by
equation (2.2) whose coefficients dependon that solution,while the subscalemotion generated by (cid:86)(cid:99) is determined
ij
by the base flow solution as well as by the experimentally measured residual motion and is, therefore, correlated
with boththeseentities.
Then since equation (2.2) is linear, itmakes sense to divide its solutionvector intothe two components
u (cid:32)usg (cid:14)uf (2.10)
(cid:81) (cid:81) (cid:81)
where
(cid:167) (cid:74)(cid:16)1 (cid:183)
L(cid:80)(cid:81)u(cid:81)f (cid:32)(cid:16)(cid:168)(cid:169)D(cid:79)(cid:80)j(cid:86)(cid:4)(cid:79)j (cid:16) 2 Di(cid:80)i(cid:86)(cid:4) jj(cid:185)(cid:184) (2.11)
and
(cid:74)(cid:16)1
L usg (cid:32) D(cid:80)(cid:86)(cid:99) (cid:16) D(cid:80)(cid:86)(cid:99) (2.12)
(cid:80)(cid:81) (cid:81) (cid:79)j (cid:79)j ii jj
2
It is clear thatthese resultswill remain unchanged if (cid:86)(cid:99) and(cid:86)(cid:4) are redefinedby
ij ij
(cid:86)(cid:99) (cid:111)(cid:16)(cid:85)v(cid:99)v(cid:99) (cid:16)(cid:86) (cid:11)x(cid:12) (2.13)
(cid:80)j (cid:80) j (cid:80)j
(cid:86)(cid:4) (cid:111)(cid:86)(cid:4) (cid:16)(cid:86) (cid:11)x(cid:12) (2.14)
(cid:80)j (cid:80)j (cid:80)j
where (cid:86) (cid:11)x(cid:12) is an arbitrary time independent quantity. The overbarwill nowbe used more specifically to denote
(cid:80)j
time averages. Then (cid:86)(cid:99) (cid:11)y(cid:12) would equal (cid:86)(cid:4) (cid:11)y(cid:12) in the upstream region where the base flow is steady if(cid:86)(cid:99) were
(cid:80)j (cid:80)j (cid:80)j
definedby the original equation(2.8),.Appendix B shows that this will also be approximately true in the downsteam
region wherethe original(cid:86)(cid:4) corresponds to the sub-grid stresses and heat flux in the FNS equations and are given
(cid:80)j
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American Institute of Aeronautics andAstronautics
in the paragraph following equation (2.9). We, therefore, take (cid:86) (cid:11)x(cid:12) to be the (approximate) common value of
(cid:80)j
these two quantities and thereby insure that the derived stresses (2.13) and (2.14)have zero time average and, more
importantly, that (cid:86)(cid:4) (cid:111)0in the upstream region where the base-flow is steady. The solution toequation (2.10) is,
(cid:80)j
therefore, entirely driven by the initial conditions in this region. Notice that steady sources cannot generate any
sound at subsonic Mach numbers.
Since, the coefficients and sources in equation (2.11) are all resolved quantities, i.e. they involve only
variables that appear in the base follow equations, it can be solved right along with these equations with no
additional modeling and little additional effort. These two sets of equations (i.e., equations (2.11) and the base flow
equations) are completely closed once thebase flow stressmodel has been selected. The pressure components of the
solution vectors, say p(cid:99) andˆp, can then be added together toobtain the componentof the pressure
f
p (cid:123) p(cid:99) (cid:14) ˆp (2.15)
L f
that is completely determined by the base flow ( i.e., the hybrid RANS/LES) solution. It may even be possible to
calculate these solutions on the samemesh, since uf is expected tobe composedof the larger sub-grid scales7. But
(cid:81)
even if this is not the case, the solution to the linear equation (2.11) should still berelatively inexpensive compared
to thebase flow solution.
The solution to equation (2.10) can also be used to iteratively calculate at least a portion of the modeled
stresses in the hybrid RANS/LES equations in order to gradually introduce the required unsteadiness into the
downstream LES solution.This typeof calculation, whichis sometimes known as meanfield theory, shouldonlybe
required in the blending region between the pure RANS and LES regions. It is expected to provide a better
representationof the actual physics than current techniques for inputting unsteadiness into LES computations.
III. VectorGreens’ Function Solution for(cid:83)(cid:99) Equation Section 3
sg
Solving equations (2.12) is potentially much more difficult than solving equations (2.11), because of the
additional source modeling requirement and because its solution can depend upon both the base flow and residual
motions. To separate out these effects, we introduce the vector Greens’ function21 g (cid:11)x,t y,(cid:87)(cid:12) which satisfies
(cid:81)(cid:86)
the inhomogeneous linear equations
(cid:11)L (cid:12) g (cid:11)x,t y,(cid:87)(cid:12)(cid:32)(cid:71) (cid:71)(cid:11)x(cid:16) y(cid:12)(cid:71)(cid:11)t(cid:16)(cid:87)(cid:12) (3.1)
(cid:80)(cid:81) x,t (cid:81)(cid:86) (cid:80)(cid:86)
with delta-functiontype sourcetermtogether with the causality condition
g (cid:11)x,t y,(cid:87)(cid:12)(cid:32)0 for t (cid:31)(cid:87) (3.2)
(cid:81)(cid:80)
to obtain
(cid:102)
(cid:167) (cid:74)(cid:16)1 (cid:183)
usg (cid:11)x,t(cid:12)(cid:32) (cid:179) (cid:179) g (cid:11)x,t y,(cid:87)(cid:12) D(cid:80)(cid:86)(cid:99) (cid:16) D(cid:80)(cid:86)(cid:99) dyd(cid:87)(cid:32)
(cid:81) (cid:81)(cid:80) (cid:168)(cid:169) (cid:79)j (cid:79)j 2 ii jj(cid:184)(cid:185)
V (cid:16)(cid:102)
(3.3)
(cid:102) (cid:170) (cid:11)(cid:74)(cid:16)1(cid:12) (cid:186)
(cid:16)(cid:179) (cid:179) (cid:74)(cid:4) (cid:11)x,t y,(cid:87)(cid:12)(cid:171)(cid:86)(cid:99) (cid:11)y,(cid:87)(cid:12)(cid:16) (cid:71) (cid:86)(cid:99) (cid:11)y,(cid:87)(cid:12)(cid:187)dyd(cid:87)
(cid:81)j(cid:80) (cid:80)j 2 (cid:80)j kk
(cid:172) (cid:188)
V(cid:16)(cid:102)
where
(cid:119)g (cid:11)x,t y,(cid:87)(cid:12) (cid:119)v(cid:4)
(cid:74)(cid:4) (cid:11)x,t y,(cid:87)(cid:12)(cid:123) (cid:81)(cid:80) (cid:16)(cid:11)(cid:74)(cid:16)1(cid:12) (cid:80) g (cid:11)x,t y,(cid:87)(cid:12) (3.4)
(cid:81)j(cid:80) (cid:119)y (cid:119)y (cid:81)4
j j
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American Institute of Aeronautics andAstronautics
From the acoustics perspective the primary interest is in the 4th pressure-like component of (3.3), which
only involves the 4th component vector Greens’ functiong (cid:11)x,t y,(cid:87)(cid:12). The latter quantity can, in principle, be
4(cid:81)
found by solving be solving the system (3.1). But since this consists of 15 first order equations, it turns out to be
simpler to compute the adjoint Greens’ functionga (cid:11)y,(cid:87) x,t(cid:12). Equations(A1) to (A4) show that it is related to
(cid:81)4
g (cid:11)x,t y,(cid:87)(cid:12) by the reciprocity relation
4(cid:81)
ga (cid:11)y,(cid:87) x,t(cid:12)(cid:32) g (cid:11)x,t y,(cid:87)(cid:12) (3.5)
(cid:81)4 4(cid:81)
and satisfies the system
(cid:11)La (cid:12) ga (cid:11)y,(cid:87) x,t(cid:12)(cid:32)(cid:71) (cid:71)(cid:11)x(cid:16) y(cid:12)(cid:71)(cid:11)t(cid:16)(cid:87)(cid:12) (3.6)
(cid:80)(cid:81) y,(cid:87) (cid:81)4 (cid:80)4
of five first order equations, which are given in component form by (A-5a,b,c). These equations suggest that
(cid:6)
g (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:123) ga (cid:11)y,(cid:87) x,t(cid:12) will be a stationary random function of its third argument, (cid:87), which results
(cid:81)4 (cid:81)4
from the time dependence of the coefficients in (cid:11)La (cid:12) that are determined from the base flow ( i.e., the hybrid
(cid:80)(cid:81) y,(cid:87)
RANS/LES) solution and are, therefore, random functions of time. (Notice that this argument would not appear if
the base flowwere steady.)
It therefore follows from equations (2.3), (3.4) and (3.5) that the 4th component of equation (3.3) can be
written (with the obviousnotation)as
(cid:102)
(cid:83)(cid:99) (cid:11)x,t(cid:12)(cid:32)(cid:16)(cid:179) (cid:179) (cid:74) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:86)(cid:99) (cid:11)y,(cid:87)(cid:12)dyd(cid:87) (3.7)
sg (cid:80)j
j(cid:80)
V (cid:16)(cid:102)
(cid:6)
where, like g (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:123) ga (cid:11)y,(cid:87) x,t(cid:12), the propagation factor
(cid:81)4 (cid:81)4
(cid:11)(cid:74)(cid:16)1(cid:12)
(cid:74) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:123)(cid:16)(cid:79) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:14) (cid:71) (cid:79) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12) (3.8)
j(cid:80) j(cid:80) 2 j(cid:80) kk
along with
(cid:173) (cid:79) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12) (cid:80)(cid:32)k (cid:32)1,2,3
jk
(cid:176)
(cid:176)
(cid:79) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:123)(cid:174) (3.9)
j(cid:80) (cid:176) (cid:119)ga (cid:11)y,(cid:87) x,t(cid:12)
44
(cid:80)(cid:32)4
(cid:176)
(cid:119)y
(cid:175) j
shouldbe random functionsof their third arguments(cid:87),since the third argumentsof the six independent components
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(cid:79) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(cid:123)
jk
1(cid:170)(cid:119)ga (cid:11)y,(cid:87) x,t(cid:12) (cid:119)ga (cid:11)y,(cid:87) x,t(cid:12)(cid:186) (cid:11)(cid:74)(cid:16)1(cid:12)(cid:167)(cid:119)v(cid:4) (cid:119)v(cid:4) (cid:183) (3.10)
(cid:171) 4k (cid:14) 4j (cid:187)(cid:16) (cid:168) k (cid:14) j (cid:184)ga (cid:11)y,(cid:87) x,t(cid:12)
2(cid:171)(cid:172) (cid:119)yj (cid:119)yk (cid:187)(cid:188) 2 (cid:168)(cid:169)(cid:119)yj (cid:119)yk (cid:184)(cid:185) 44
(cid:6)
of (cid:79) (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12)(which arise from the third argument of g (cid:11)y x:(cid:87),t(cid:16)(cid:87)(cid:12) as well as the (cid:87)-dependence of
jk (cid:81)4
(cid:119)v(cid:4) /(cid:119)y )also have this property.
k j
Equation Section 4
IV. Subscale Pressure Autocovariance
Since(cid:83)(cid:99) (cid:11)x,t(cid:12)(cid:111) p(cid:99) (cid:11)x,t(cid:12) asx(cid:111)(cid:102),the far fieldpressureautocovariance
sg sg
T
1
(cid:51) (cid:11)x,(cid:87)(cid:12)(cid:123) lim (cid:179) p(cid:99) (cid:11)x,t(cid:12)p(cid:99) (cid:11)x,t(cid:14)(cid:87)(cid:12)dt for large x (4.1)
sg sg sg
T(cid:111)(cid:102)2T
(cid:16)T
becomes
T (cid:102)
1
(cid:51) (cid:32) (cid:179) (cid:179)(cid:179)(cid:179)(cid:42) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:16)t(cid:99)(cid:12) (cid:86)(cid:99) (cid:11)y,t(cid:99)(cid:12)(cid:86)(cid:99) (cid:11)y (cid:14)(cid:536),t(cid:99)(cid:99)(cid:12)dyd(cid:536)dt(cid:99)dt(cid:99)(cid:99)
sg 2T i(cid:86)j(cid:80) (cid:86)i (cid:80)j
(cid:16)T(cid:16)(cid:102)V
T (cid:102)
1
(cid:32) (cid:179) (cid:179)(cid:179)(cid:179)(cid:42) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99)(cid:14)t(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) (cid:86)(cid:99) (cid:11)y,t(cid:99)(cid:12) (cid:86)(cid:99) (cid:11)y (cid:14)(cid:536),t(cid:99)(cid:99)(cid:14)t(cid:99)(cid:12)dyd(cid:536)dt(cid:99)dt(cid:99)(cid:99) (4.2)
2T i(cid:86)j(cid:80) (cid:86)i (cid:80)j
(cid:16)T(cid:16)(cid:102)V
where
(cid:102)
(cid:42) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:16)t(cid:99)(cid:12) (cid:123) (cid:179) (cid:74) (cid:11)y x:t(cid:99),t(cid:14)(cid:87)(cid:16)t(cid:99)(cid:12)(cid:74) (cid:11)y(cid:14)(cid:536) x:t(cid:99)(cid:99),t(cid:16)t(cid:99)(cid:99)(cid:12)dt
j(cid:86)l(cid:80) j(cid:86) l(cid:80)
(cid:16)(cid:102)
(4.3)
(cid:102)
(cid:32) (cid:179) (cid:74) (cid:11)y x:t(cid:99),t(cid:14)(cid:87)(cid:14)t(cid:99)(cid:99)(cid:16)t(cid:99)(cid:12)(cid:74) (cid:11)y(cid:14)(cid:536) x:t(cid:99)(cid:99),t(cid:12)dt
j(cid:86) l(cid:80)
(cid:16)(cid:102)
is a stationary random functionof its fourthand fifth arguments. It follows that
(cid:42) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99)(cid:14)t(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) and
i(cid:86)j(cid:80)
R (cid:11)y;(cid:536),t(cid:99)(cid:99),t(cid:99)(cid:12)(cid:123)(cid:86)(cid:99) (cid:11)y,t(cid:99)(cid:12) (cid:86)(cid:99) (cid:11)y (cid:14)(cid:536),t(cid:99)(cid:99)(cid:14)t(cid:99) (cid:12) (4.4)
(cid:86)i(cid:80)j (cid:86)i (cid:80)j
are bothstationary randomfunctionsof t(cid:99).
It is appropriate to regardthese two quantities as beinguncorrelated in the present approach because the base
flow solution, which is determined from a closed set of equations thatonly involve the base flow variables, should
be statistically independent of (i.e., decoupled from) the chaotic subscale motion in the actual experiment used to
model these quantities. (In fact the entire experimental flowshould eventually become de-correlated from the hybrid
LES/RANS simulation where the randomness comes from the numerical round off errors and/or the random initial
conditions-- which are themselves uncorrelated with the initial conditions in the experiment18,19. It is therefore
reasonable torequire that any experimentally based model used to represent (cid:86)(cid:99) be statistically independent of the
(cid:80)j
baseflow motion (and consequently with the coefficientsofLa ,).
(cid:80)(cid:81)
This means that the covariance
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1 T
lim (cid:179)(cid:42)(cid:99)(cid:99) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99)(cid:14)t(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) R(cid:99)(cid:99) (cid:11)y ;(cid:536),t(cid:99)(cid:99),t(cid:99)(cid:12)dt(cid:99) (4.5)
T(cid:111)(cid:102)2T i(cid:86)j(cid:80) (cid:86)i(cid:80)j
(cid:16)T
where
(cid:42)(cid:99)(cid:99) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99)(cid:14)t(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) (cid:123)
i(cid:86)j(cid:80)
(4.6)
(cid:42) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99)(cid:14)t(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) (cid:16)(cid:42) (cid:11)y,(cid:536) x:t(cid:99)(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12)
i(cid:86)j(cid:80) i(cid:86)j(cid:80)
R(cid:99)(cid:99) (cid:11)y ;(cid:536) ,t(cid:99)(cid:99),t(cid:99)(cid:12)(cid:123) R (cid:11)y ;(cid:536) ,t(cid:99)(cid:99),t(cid:99)(cid:12)(cid:16)R (cid:11)y ;(cid:536) ,t(cid:99)(cid:99)(cid:12) (4.7)
(cid:86)i(cid:80)j (cid:86)i(cid:80)j (cid:86)i(cid:80)j
1 T
(cid:42) (cid:11)y,(cid:536) x:t(cid:99)(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) (cid:123) lim (cid:179)(cid:42) (cid:11)y,(cid:536) x:t(cid:99),t(cid:99)(cid:99)(cid:14)t(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12)dt(cid:99) (4.8)
i(cid:86)j(cid:80) T(cid:111)(cid:102)2T i(cid:86)j(cid:80)
(cid:16)T
and
1 T
R (cid:11)y ;(cid:536),t(cid:99)(cid:99)(cid:12)(cid:123) lim (cid:179) R (cid:11)y ;(cid:536),t(cid:99)(cid:99),t(cid:99)(cid:12)dt(cid:99) (4.9)
(cid:86)i(cid:80)j T(cid:111)(cid:102)2T (cid:86)i(cid:80)j
(cid:16)T
should equal zero. It therefore, follows that (cid:51) (cid:11)x,(cid:87)(cid:12) canbewritten as an integral
sg
(cid:102)
(cid:51) (cid:11)x,(cid:87)(cid:12)(cid:32) (cid:179)(cid:179)(cid:179)(cid:42) (cid:11)y,(cid:536) x:t(cid:99)(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) R (cid:11)y;(cid:536),t(cid:99)(cid:99)(cid:12) dyd(cid:536)dt(cid:99)(cid:99) (4.10)
sg i(cid:86)j(cid:80) (cid:86)i(cid:80)j
(cid:16)(cid:102)V
ofthe product oftwo non-randomfunctions.
The first of these functions, whichcan beexpressed as the integral
(cid:102)
(cid:42) (cid:11)y,(cid:536) x:t(cid:99)(cid:99),(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12) (cid:32) (cid:179) (cid:74) (cid:11)y,(cid:536) x:t(cid:99)(cid:99),t(cid:14)(cid:87)(cid:14)t(cid:99)(cid:99),t(cid:12) dt (4.11)
i(cid:86)j(cid:80) i(cid:86)j(cid:80)
(cid:16)(cid:102)
over time of the correlation
1 T
(cid:74) (cid:11)y,(cid:536) x:t(cid:99)(cid:99),t(cid:14)(cid:87)(cid:14)t(cid:99)(cid:99),t(cid:12) (cid:123) lim (cid:179) (cid:74) (cid:11)y x:t(cid:99),t(cid:14)(cid:87)(cid:14)t(cid:99)(cid:99)(cid:12)(cid:74) (cid:11)y(cid:14)(cid:536) x:t(cid:99)(cid:14)t(cid:99)(cid:99),t(cid:12)dt(cid:99) (4.12)
i(cid:86)j(cid:80) T(cid:111)(cid:102)2T i(cid:86) l(cid:80)
(cid:16)T
of the two random propagationfactors (cid:74) (cid:11)y x:t(cid:99),t(cid:14)(cid:87)(cid:14)t(cid:99)(cid:99)(cid:16)t(cid:99)(cid:12)and (cid:74) (cid:11)y(cid:14)(cid:536) x:t(cid:99)(cid:99),t(cid:12),canbe thought of as an
i(cid:86) l(cid:80)
expected or mean propagation factor. It can be determined as part of the hybrid LES/RANS computation. The
second function, which, can be thought of as a source function, describes the low order statistics of the unknown
component of the residual fluctuations. It has to be modeled,but it should be much easier to do this than to model
the instantaneous values of these fluctuations (which would have to be done if the source and propagation
fluctuations were not de-correlated).Notice that this de-correlation only implies that the fluctuation (cid:42)(cid:99)(cid:99) (and
i(cid:86)j(cid:80)
not(cid:42) itself) is independent of the unresolved scales. Itmay,therefore,be appropriate to paramatize themodel for
i(cid:86)j(cid:80)
this quantity (i.e., the functional form inferred from the experimental data) and determine the parameters from the
hybrid RANS/LES solution.
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We can also think of, ˆp,v(cid:4) and(cid:85)ˆ and p(cid:99),v(cid:99) and(cid:85)(cid:99) as the filtered and unfiltered components of the
i i
pressure, velocity and density in the downstream region of anactual flow. Then the adjoint Greens’ function, and
therefore the propagation factor (cid:42) , should be dominated by the large energy -bearing scales of that flow, while
i(cid:86)j(cid:80)
Kolmogorov’s 3hypothesis (which forms the bases of many of the current subscale turbulence models13) indicates
that the smallscalemotion should be statistically independent of these scales. It is, therefore, likely that the present
decorrelation assumptionwill be satisfied --at least on a globalbasis-- inthe downstream regionof that flow (where
the implied filterwidth eventually becomes small relative to the transverse length scale).
The statistical independence assumption is trivially satisfied in the upstream region where the base flow
equations reduce to the steady RANS equations because the fluctuating component of the base flowand, therefore,
the covariance(4.5) is identically zero. It will not,however,be satisfied in the intermediate blending region. But to
the extent that the propagation factor is non-local and the blending region is sufficiently small this should have
relatively little effecton the predicted soundfield.
V. Conclusions
An exact equationfor the sound generation by the residual turbulence scales in a hybrid RANS/LES based
simulation11,12 is used to obtain a formula for the residual -scale acoustic pressure autocovariance in terms of the
residual -scale turbulence correlation tensor by exploiting the statistical independence of the hybrid RANS/LES
solution and the experimentally determined residual motion. The residual acoustic radiation can therefore be
calculated by introducing appropriate models for this relatively universal correlation while determining the
configuration dependent largescale sounddirectly from thehybrid RANS/LES solution.
Theapproachalsoaccountsforthescattering of the small scale sound bythe large scale motion-an effect
that was emphasized by Crighton6. To the extent that this phenomenon is non-local, the present statistical
independenceassumption shouldbevery appropriate to its computation.
At high Reynolds numbers the small scale motion should be statistically independent of the large energy
bearing scales in any realization of the flow. But this does not imply thatthe resolved scales in the downstream LES
region(i.e., the scales larger than the filter width) are all statistically independent of thesub-scale motion, especially
near the filter cut off. It is only the hybrid RANS/LES simulation of the filtered scales that is expected to be
uncorrelated with the actual, i.e. the experimentally measured, residual-scale tensor(cid:86)(cid:99) . But to the extent that the
(cid:80)j
propagation factor (cid:42) is non-local and dominated by the large energybearingscales,Kolmogorov’s3 hypothesis
i(cid:86)j(cid:80)
that the small scale motion is statistically independent of these scales suggests that the present de-correlation
assumptionwill besatisfied in any actual realization of the flow. Thehope is that theresulting equation(4.10)will
accurately predict the subscale pressure autocovariance in that flow to the same extent as the hybrid RANS/LES
solutionpredicts its large scale statistics.
A major attractionof the currenthybrid approach is that the sub-scale stresses that have tobemodeledin
the downstream region should be much more universal than the large scale Reynolds stresses that have to be
modeled with the usual RANS based methods. Unfortunately, the present approach still requires full unsteady flow
modeling in the initialmixing layers. Thehope is that it will be possibleto obtain analytically based modelsfor this
region that exhibit a fair degree of universality. Other extensions and modificationsof this approach are, of course,
possible. The base flow can, for example, be taken as a hybrid URANS/LES solution in order to make the near
nozzle modeling more universal. It may also be possible to use quasi-linear instability wave models to account for
the soundgenerated in this region14.
Appendix A
The adjoint Greens’ function ga (cid:11)y,(cid:87) x,t(cid:12), which satisfies the adjoint equation21,22,8
(cid:81)(cid:86)
(cid:11)La (cid:12) ga (cid:11)y,(cid:87) x,t(cid:12)(cid:32)(cid:71) (cid:71)(cid:11)x(cid:16) y(cid:12)(cid:71)(cid:11)t(cid:16)(cid:87)(cid:12) (A1)
(cid:80)(cid:81) y,(cid:87) (cid:81)(cid:86) (cid:80)(cid:86)
where
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D(cid:4) (cid:167) c(cid:105)2 (cid:183)
La (cid:123)(cid:16)(cid:71) (cid:16)(cid:11)(cid:74)(cid:16)1(cid:12)(cid:71) (cid:119) (cid:14)(cid:168) (cid:71) (cid:14)(cid:71) (cid:184)(cid:119) (cid:14)K (A2)
(cid:80)(cid:81) (cid:81)(cid:80)D(cid:87) (cid:80)4 (cid:81) (cid:168)(cid:74)(cid:16)1 (cid:81)4 (cid:81)5(cid:184) (cid:80) (cid:81)(cid:80)
(cid:169) (cid:185)
and
D(cid:4) (cid:119) (cid:119)
(cid:123) (cid:14)v(cid:4) (cid:11)y,(cid:87)(cid:12) (A3)
D(cid:87) (cid:119)(cid:87) i (cid:119)y
i
is related to the directGreens’ function g (cid:11)x,t y,(cid:87)(cid:12) by the reciprocity relation
(cid:86)(cid:81)
ga (cid:11)y,(cid:87) x,t(cid:12)(cid:32) g (cid:11)x,t y,(cid:87)(cid:12) (A4)
(cid:81)(cid:86) (cid:86)(cid:81)
Itisdetermined bythe system(3.6)which, when written outin full,becomes8
D(cid:4)ga (cid:119)v(cid:4) c(cid:105)2 (cid:119)ga (cid:74)(cid:16)1(cid:119)(cid:84)(cid:4) (cid:119)ga
(cid:16) i4 (cid:14)ga j (cid:14) 44 (cid:14) ij ga (cid:14) 54 (cid:32)0 (A5a)
D(cid:87) j4 (cid:119)y (cid:74)(cid:16)1 (cid:119)y (cid:85) (cid:119)y 44 (cid:119)y
i i j i
D(cid:4)ga (cid:167)(cid:119)ga (cid:119)v(cid:4) (cid:183)
(cid:16) 44 (cid:16)(cid:11)(cid:74)(cid:16)1(cid:12)(cid:168) i4 (cid:14)ga j (cid:184)(cid:32)(cid:71)(cid:11)x(cid:16) y(cid:12)(cid:71)(cid:11)t(cid:16)(cid:87)(cid:12) (A5b)
D(cid:87) (cid:168) (cid:119)y 44 (cid:119)y (cid:184)
(cid:169) i j (cid:185)
D(cid:4)ga 1 (cid:119)(cid:84)(cid:4)
(cid:16) 54 (cid:16) ij ga (cid:32)0 (A5c)
D(cid:87) (cid:85)(cid:119)y i4
j
Thenwhen x,y(cid:111)(cid:102), c(cid:105)2 (cid:111)c2 (cid:32)constantand
0
(cid:119)ga c2 (cid:119)ga
(cid:16) i4 (cid:14) 0 44 (cid:32)0 (A6a)
(cid:119)(cid:87) (cid:74)(cid:16)1 (cid:119)y
i
(cid:119)ga (cid:119)ga
(cid:16) 44 (cid:16)(cid:11)(cid:74)(cid:16)1(cid:12) i4 (cid:32)(cid:71)(cid:11)x(cid:16) y(cid:12)(cid:71)(cid:11)t(cid:16)(cid:87)(cid:12) (A6b)
(cid:119)(cid:87) (cid:119)y
i
(cid:119)ga
54 (cid:32)0 (A6c)
(cid:119)(cid:87)
which shows that ga satisfies the inhomogeneouswave equation
44
(cid:119)2ga (cid:119)2ga (cid:119)
(cid:16) 44 (cid:14)c2 44 (cid:32)(cid:71)(cid:11)x(cid:16) y(cid:12) (cid:71)(cid:11)t(cid:16)(cid:87)(cid:12) (A7)
(cid:119)(cid:87)2 0 (cid:119)y(cid:119)y (cid:119)(cid:87)
i i
Therelevant solutionwhich satisfies thecausality condition
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