The of Turbulent Flows

M. E. GOLDSTEIN Research and Technology Directorate NASA Glenn Research Center 21000 Brookpark Rd. Cleveland Ohio 44135 USA [email protected]

Abstract:-Aerodynamic noise prediction has been an important and challenging research area since James Lighthill first introduced his Acoustic Analogy Approach over fifty years ago. This talk attempts to provide a unified framework for the subsequent theoretical developments in this field. It assumes that there is no single approach that is optimal in all situations and uses the framework as a basis for discussing the strengths weaknesses of the various approaches to this topic. But the emphasis here will be on the important problem of predicting the noise from high speed air jets. Specific results will presented for round jets in the 0.5 to 1.4 Mach number range and compared with experimental data taken on the Glenn SHAR rig. It is demonstrated that non- parallel mean flow effects play an important role in predicting the noise at the supersonic Mach numbers. The results explain the failure of previous attempts based on the parallel flow Lilley model (which has served as the foundation for most jet noise analyses during past two decades).

Key-Words: -Aeroacoustics, Jet Noise, , Aerodynamic , Turbulent Flow

1 Introduction and since the governing equations of that Aeroacoustics is concerned with sound motion (i.e., the Navier-Stokes equations) generated by forces and stresses resulting are well established and have been for some from the motion of a rather than the time now (over 150 years) you might think externally applied forces and motions of that the only thing that we need to do here is classical acoustics. Music (i.e., harmonious write down these equations and be done sound) is often generated in this fashion. But with it. But the solution to these equations— this paper is concerned with aerodynamic especially for the turbulent flows that are the noise, (i.e., cacophonous sound), which is of usual source of aerodynamic noise-- has, to engineering interest because of the demand put it mildly, proved to be very elusive. In for quieter transportation vehicles-such as fact it is probably no exaggeration to say aircraft, busses and automobiles. In that fluid turbulence is still ranked among particular the noise from jet engine exhausts the great unsolved problems of classical has been a major concern since the type of physics. engine was introduced over fifty years ago So there is a need to obtain useful and and wind noise from automobiles is now interesting information about the sound field receiving considerable attention since other without actually solving the full Navier- previously dominant noise sources have Stokes equations—which typically involves been reduced to relatively low levels. introducing some sort of reduced order The focus of this paper is on the model for these equations —which, in the fundamental theory. But since aerodynamic present context, is usually referred to as an sound is just a byproduct of fluid motion acoustic analogy (following James Lighthill

1 who was the first to do such analysis)[1]. computational method, which means that it This is, of course, analogous to the can, at best, be expected to provide an situation in turbulence modeling, which adequate representation of the most is almost always based on some form of energetic part of the flow and perhaps (with the filtered (usually Favre [2] filtered) enough computational effort) even the low Navier-Stokes equations such as the frequency components of the sound field-but Reynolds averaged Navier-Stokes (RANS) certainly not all of the spectral components equations[3]. While there is some that are of interest the high Reynolds disagreement about the proper choice of the number flows of practical importance. For turbulence modeling equations, the situation example the noise from full jet engine in Aeroacoustics is considerably more exhausts has a peak frequency in the 200- contentious with significant disagreement 500 Hz range, but the ear is most sensitive to about the appropriate starting equations sound in the 2,000 to 5,000 Hz frequency (perhaps because there is no single set of range. So jet noise prediction methods need equations that is optimal in all to maintain their accuracy over an situations).There does, however, seem to be enormously broad range of frequencies. a consensus about some of the requirements So as a practical matter the base flow for such equations. First of all, they should will have to be determined from some be derivable from the Navier-Stokes reduced order model of the Navier-Stokes equations and secondly; they should, as equation. To my knowledge, these models argued in the well known text by Ann are almost always based on the usual Dowling & Ffowcs Williams, be formally hyperbolic conservation laws that can be linear [4]. written (fairly compactly) as [5,6,7,8]

D ρ = 0, (2) 2 Fundamental Equations o It, therefore, makes sense to begin by dividing the flow variables in the Navier-

Stokes equations, the density ρ, the ∂ pe ∂ Dvoiρ +=eij (3) p, the enthalpy h, the velocity vi , etc, into, ∂∂xxij say, their base flow components , ρ,p ,h ,and vi and ‘residual’ components ⎛⎞γ pp1 ∂ ee2 ρ′′,,p h′and ,v′ , which are essentially Dvo ⎜⎟+−ρ i ⎝⎠γ −∂12 t (4) defined by ∂ ⎡ 1 ⎤ = ⎢ eev+ ⎥ ∂−x ()γ 1 4i ij j ρ =+ρρ′′,,p =+pp i ⎣ ⎦

(1) hhh=+ ′′, v = v + v, where the Latin indices range from 1 to 3, iii the summation convention is being used, with the idea being that base flow is to be found from some relatively inexpensive

2

∂v ∂ Du++ ui p′ oi j e ∂∂xxji ∂f ∂ Dfo ≡+ ( vfj ) (5) (7) ∂∂txj ′ ρ ∂∂ −=τij eij′′ ρ ∂∂xjjx for any function f , we assume that the base flow variables, as well as the original Navier -Stokes variables satisfy an ideal gas law ∂ Dp′++ cu2 , with γ being the specific oe j ∂xj heat ratio, p denotes a pressure-like e ⎛⎞∂∂vu τ variable that can differ from the γ −− 1 p′ jii j ()⎜⎟e thermodynamic pressure, and with ∂∂xjjρ x eλ j ⎝⎠ λ = 1, 2, 3, 4 denotes an, as yet, arbitrary ∂ ∂vi =+−ee4′′ji()γ 1 ′′j, (8) 4x3 dimensional stress . (Greek ∂xjj∂x indices will always range from 1 to 4.) These equations include, among other where things, the Euler equations, the Navier Stokes equations themselves (which p c2 = γ=γRT (9) correspond to setting the first three ρ components of equal to the Newtonian eλ j stress and the fourth component to the heat is the base flow sound speed squared and flux vector). And more generally, the 4x3 dimensional stress tensor can be inputted τ ij≡ δ ijp e − eij (10) through a turbulence model, which also determines the difference between the denotes the total base flow stress tensor. effective pressure pe and the These equations have been put into the thermodynamic pressure p --but in a purely linearized conservation law form by passive fashion. However, this difference is introducing the new dependent variables γ−1 largely irrelevant since these equations will ppp′ ≡− + ρv′2 (11) form a closed system of 5 equations in 5 e e 2 unknowns which can be solved for pe independently of its relation to the thermodynamic pressure p once a particular uvi≡ ρ i′, (12) turbulence model has been introduced. It can now be shown that the remaining as well as the new source strengths residual variables are determined by the five eν′′j , ν = 1234, , , given by the difference formally linear equations [5,6,7,8]

eee′j′′μ ≡ jjμ− μ (13)

∂ Duojρ′ +=0, (6) ∂xj between the generalized Reynolds stress

3 latter non-linearity is dealt with by intruding a specific model for the generalized stress evvνν′′′jj≡−ρ + tensor (14). This is analogous to the γ−1 (14) δρv′2 +σ situation in conventional turbulence 2 ννjj modeling. But the modeling is now (based on the residual velocities and considerably more complex and being able to simplify the requirements by lumping all enthalpy) and the base flow stress eν j , the modeling into a single tensor is an where important consideration. So we might think of requiring that the fundamental acoustic ⎛1 2 ⎞ equations satisfy the following two vh4′′≡γ−()1 o ≡()γ−1 ⎜ h ′ + v′⎟ ⎝⎠2 conditions in addition to the two mentioned (15) in the introduction: (1) The base flow about which the implied linrearization is carried out satisfies the usual hyperbolic conservation laws. And σγ−−4i ≡−()1 (qviijj σ ′ ) (16) (2)All of the modeling requirements can be inputted through a single stress tensor with σij being the viscous stress, qi the heat flux vector and δν j denoting the 4x3 These four requirements taken together dimensional in the usual appear to be just sufficient to restrict notation. the overall form of the fundamental acoustic The true non-linearity of these equations, equations (i.e., the acoustic analogy which can be written more compactly in equations) to trivial variations of the operator form, as linearized hyperbolic conservation laws (6) to(8). Interpretation of the right side of these μ equations as acoustic sources implies that LDμνuν = νj e′′ j for ν (17) that the left sides account for all of the μν=,,,,, 1 2 3 4 5 interaction of this sound with the base flow-- including the propagation of the sound through that flow. The apparent linearity of μ the equations can be exploited to separate where Lμν and Dν j are first order linear out these interaction/propagation effects from the unsteady source fluctuations that operators and uν denotes the five produce the sound by using the 4th dimensional solution vector component of the adjoint vector Green’s

a function g,νμ ( yxτ ,t) , which is related to {uu,p,ν } ≡ρ{ ie′′}, (18) th the 4 component of the ordinary vector is hidden in the non-linear dependent Greens’ function g,t,4ν (x y τ)by the variables (11)and (12) as well as in the non- a linear source strength (14). The former reciprocity relation g,,ν4 ()yxτ t non-linearity is again largely irrelevant here = g,t,x y τ and satisfies the because the variable of principle interest (the 4ν ( ) pressure-like variable, p′ ) reduces to the inhomogeneous linear equation [9] e ordinary pressure fluctuation in the far field where the sound field is to be calculated. And in the acoustic analogy approach the

4 rudimentary being obtained by setting the Lgaa yx,,τ t= base flow stresses ()μνy,τ ν 4 () eν j and the velocity vi to zero. The (19) hyperbolic conservation laws then imply that the remaining base flow dependent δ μ 4δδ()(xy−−t τ) variables ρ, pe will be equal to constants, with Delta-function source term where La say to their values at infinity. This is μν essentially the original Lighthill [1] is the adjoint of the operator Lμν that approach as modified by Lilley [10,11] to appears in equation separate out the isentropic and non- Error! Reference source not found.. And isentropic components of the setting aside solid surface interaction effects, pressure/density term that appears in for simplicity sake, we can use Green’s Lighthill’s source function. This turns out to formula to show that the pressure like be important because it is only the latter variable p′ is given by the tensor product component that can be associated with e viscous and heat conduction effects that are

almost universally neglected at the high

Reynolds numbers of interest in most ∞ Aerodynamic noise problems. The former ′′′ ptejj()xy,,=−∫∫ γμμ()τ xy,,te ()τ ddy τ can be an important noise source that has to V −∞ be modeled in this type of acoustic analogy. (20) But the main criticism of the Lighthill approach is that the residual velocities v′ of the source strength (13) with the i that appear in the residual equation source propagator γτμ j (y,x,t)which is related term have steady flow components that to the adjoint Green’s function introduce steady and linear terms into the g,,a y τ x tby residual stress tensor. But the former cannot 4μ () generate any sound and therefore do not a ∂τg,,t4μ ()yx belong in the source function, while the γτ≡y, x,t − μ j () latter are at least in part associated with ∂y j (21) propagation effects and might, therefore, be best book kept on the left hand side of the ∂vμ a ()γ− 1 g,,t44 ()yxτ , equations. ∂y j This criticism is at least partially overcome by the next most complicated form of the analogy which amounts to again with v4 ≡ 0 ,and, therefore, accounts for all of the base flow interaction effects. setting the stress tensor eν j to zero but now requiring that the base flow dependent Notice that pe′ (x,t)reduces to the ordinary variables v , ρ, p satisfy the resulting pressure fluctuation p′(x,t)when the i e Euler’s equations. However, the interest is observation point x is in the far field. usually in the steady flow solutions to these equations because for many turbulent flows the turbulent stresses act only slowly over 3 Specific Analogies long distances and the Euler equation The present results are, of course, solutions are, therefore, expected to provide general enough to allow for many possible a good local approximation to the actual analogies--each of which corresponds to a mean flow. This is especially true for the different choice of base flow--with the most important class of nearly parallel flows, such

5 as jets wakes and shear layers. In which case evvvv≡−ρ − + the relevant Euler equation solution is the ij( i j i j ) (23) uni-directional transversely sheared flow γ−1 22 δρ v − v 2 ij ( )

e ≡ vU ==δρρ()xx,() 4 j iij TT (22) ⎡ 1 ⎤ ,pe = constant −γ−()1 ρhv − hv + ev + ev ⎢ ()00jj2 iijiji⎥ ⎣ ⎦ where subscript T denotes transverse (24) and with pe related to p by components, the velocity vUiij= δ ( xT ) is in a single direction but, along with the γ −1 22 density ρρ= (x ), can vary in planes pe =+pvρ −v (25) T 2 ( ) perpendicular to that direction. The hyperbolic conservation laws (or in this case where the overbars denote filtered variables, the Euler equations) then imply that the the tildes denote the Favre (ref.2) filtered pressure like variable pe will be constant. variables The linear operator on the left hand side of the residual equations now reduces to • ≡ρ•ρ/ (26) the usual Rayleigh operator of linear ( ) stability theory fame and, as is well known, and the usually unimportant viscous terms the residual equations can be reduced to a have, for simplicity, been omitted. single inhomogeneous equation for the It is possible to choose a filter that only pressure like variable p′ which, in the filters out the components of the motion that e actually radiate to the far field [12]. So the Aeroacoustics context, is usually referred to base flow will be completely silent and all of as a Lilley’s equation [10]. And while it can the residual flow will radiate away as sound. be argued that this equation has served as In which case, it would not be completely the basis of most of the theoretical jet noise unreasonable to interpret the apparent source research for the past two decades, it has a terms on the right side of the residual number of issues which will be discussed equations as the “true sources of sound”. subsequently. But it is first necessary to But no one has as yet succeeded in finish cataloguing the various specific developing a base flow closure model for analogies. this type of filter. While equations (24) and The true reduced order modeling (24) are very suggestive, they do not comes into play when specific turbulence actually provide a means for constructing models are introduced for the base flow such a model—beyond, perhaps, providing stress tensor. As noted in the introduction, some constraints that they have to satisfy. these models are usually based on the filtered Navier Stokes equations, which are obtained by applying a linear filter to these equations. From our point of view a linear 4 Actual Mean Flow Analogy filter is nothing more than a linear There is much more that can be done with transformation of the continuous functions this. But I’m definitely not going to get into into a simpler subspace that commutes with the vast subject of turbulence modelling differentiation. The resulting filtered Navier- here, except to point out that the most well Stokes equations are special cases of the known filter, namely the time average T general base flow equations with the stress 1 •≡lim •()x ,tdt , (27) tensor now given explicitly by T →∞ 2T ∫ −T

6 also produces a completely silent base flow, which is the quantity that is usually but it also filters out much of the non- measured in Aeroacoustic experiments, can radiating (i.e., non-acoustic) components of be expressed as the convolution product the motion. The effective pressure is now given by ∞ 2 pt()xx,;=γνμjl( yη,t +τ )Rνμjl ∫∫∫ γ−1⎛2 2 ⎞ −∞ V p′≡+pv′′⎜ ρ −ρv′⎟ (28) e 2 ⎝⎠ ()y ;η , τ dddy η τ

(31) and more importantly, the base flow stress of a “propagator” γνjlμ , which is related to tensor eν j is now equal to the time average the original propagator γ by the e′ of the generalized residual μ j ν j convolution velocity/enthalpy Reynolds stress. So the γ+τ≡xy;η,t actual stress tensor (acoustic source νμjl( ) strength) that appears in the residual ∞ equations (6) to (8) is just the difference ∫ γ++τγ+νμjl()xy,,tt11()xy η tdt1 −∞ (32) eeeν′′j = νν ′j− ′ j (29)

with the two point time-delayed correlation between this Reynolds stress and its time average--which means that it has zero time 1 T average as would be expected for a true R ()yyy;,ηητ ≡+e′′()(,,τ e ′′ ττ+ )dτ acoustic source. νμjl ∫ νμj0l 0 0 2T −T The base flow equations now turn out (33) to be the usual Reynolds averaged Navier- Stokes equations for which the turbulence ′′ modelling is certainly highly developed [3]. of the source strength eνj (y, τ ), which is This form of the analogy completely directly related to the generalized velocity removes the mean flow propagation effects /enthalpy Reynolds stress autocovariance from the source term, which is expected to tensor significantly improve the medium and lower frequency predictions, but a steady base flow analogy cannot provide the requisite 1 T ⎡⎤′′ ′′ structure for inputting the long range Rνμjl()yy;,η τ ≡ρ−ρ∫ ⎣⎦vvννjj0 vv (),τ turbulent scattering effects, which tend to 2T −T become more important at the higher ⎡⎤ρ−ρ+vv′′ vv ′′ y η, ττ +dτ frequencies ⎣⎦μμll() 00 The Greens’ function solution (20) can (34) now be used to show that the far field pressure autocovariance by the relatively simple linear transform 1 T p2 xxx,,,tptpttdt≡+′′ , ()00∫ ()() Rνμj l = Rνμjl− 2T −T 2 γ−11⎛⎞γ− (30) δ+δ+ννjjRR⎜⎟ δδ R. 22()kkμμν l l jkk μl iikk ⎝⎠ (35)

7 Since this autocovariance tensor is about as ridiculous conclusion that the radiated sound close as you can get to what is actually is infinitely loud. measured in turbulent flows, these results But none of these difficulties would provide an essentially exact relation between occur it the base flow were taken to be the the quantities that are typically measured in actual mean flow field—even in the Aeroacoustic experiments—as would be important case of a nearly parallel shear expected from any good theory. flow. Because, while the linear instabilities can initially grow in such flows, the slow The tensor Rνμjl(y;,η τ )is also the divergence of the flow will eventually cause quantity that ultimately has to be modeled in them to decay—keeping the Green’s the acoustic analogy approach. The hope is function and, therefore, the propagator that most of the non-local “propagation finite. And even more importantly at effects” which would be very difficult to supersonic speeds, these flows do not posses distinguish from the turbulent fluctuations-- critical layers and, therefore, have no critical and, therefore, very difficult to model-- have layer singularities. But the propagator is been removed from these stresses. much more expensive to compute in this But the result (31) to (33) is actually case---in fact, prohibitively so for many more general than this and applies to any applications. steady base flow, including the uni- directional mean flow. This greatly simplifies the computation of the 5Application to Slowly Diverging propagator γ , but, of course, only applies Mean Flows νμjl There is, however, a way to obtain the to nearly parallel shear flows. And even in best of both worlds—which amounts to that, admittedly important, case has, as using a perturbation approach that takes mentioned in the previous section, a number advantage of the small spread rate, say ε , of of serious issues. First of all the source term the mean flow. The base flow velocities contains steady as well mean flow will then expand like [7,8] interaction terms because vi′ still has a mean (steady) flow component. A more serious difficulty is that the Rayleigh operator that vUY = ( ,,yy) +ε UY(1) ()+… (36) now appears on the left hand side of the 1 TT residual equations can support linear instability waves that grow without bound 2 (1) far downstream in the flow. And when the vVyTT= εε(YY,,) + V() y T+…(37) Greens’ function is required to satisfy where causality, which seems like a reasonable thing to do, the corresponding homogeneous Y≡ ε y (38) solutions to the Rayleigh equation contribute 1 to the adjoint Greens’ function and therefore denotes a slow streamwise (source) variable to the propagator γτy,x,t causing μ j () that varies on the streamwise length scale of them to become infinite there as well. An the mean flow , and even more serious difficulty at supersonic speeds is that the Rayleigh operator has a yT = {yy2, 3} (39) singularity at the so called critical layer which produces a much stronger non- integrable singularity in the propagator vT = {vv23, } (40) γτ(y,x,t)when the observation point μ j denote cross flow variables. Of course, the x is in the far field—leading to the other base flow dependent variables will

8 have similar expansions. And it is assumed spectrum is given be the following purely that all lengths have been normalized by algebraic expression[8] some characteristic cross flow dimension and that all velocities have been normalized Iω ( xy) → by some appropriate characteristic streamwise velocity with similar obvious normalizations for the density, pressure and 2 temperature. The lowest order terms in this ⎛⎞2π 2πω ∗ ⎜⎟ sinθΓκλjT()xy ,YY Γ lT() xy, expansion correspond to the unidirectional ⎝⎠xc∞ transversely sheared flow with the slow streamwise variable entering only ⎛⎞ωω Φ *κλjl⎜⎟y; cosθ∇ω− , SM ,() 1c cos θ parametrically. ⎝⎠cc∞∞ th It might be expected that the 4 component of the adjoint Greens’ function as x →∞ (42) would posses a corresponding expansion of the form where S denotes the Eikonal of the

corresponding geometric acoustics solution, aa,0 a ,1 ggνμ=+vv σε g σ +… (41) the turbulence source correlations enter only through the spectral tensor But in so far as this expansion is concerned ∗ the lowest order base flow solution still acts Φωνμjl( y;k1,, kT ) like a uni-directional transversely sheared mean flow. So nothing much has been accomplished here because the lowest order ∞ 1 −ωτiMik()11ξ+kTTiξ term satisfies Rayleigh’s equation and, ≡ ∫∫ede Rνμjl()y,,ξξττd therefore, still posses a critical layer 2π −∞ V singularity. But embedding this solution in (43) this more global context allows us to construct a new “inner solution” in the vicinity of the critical layer that brings in where the asterisk is being used to denote nonparallel mean flow effects to eliminate complex conjugates, Γ xy,Y is the singularity there. Standard singular ν jT() perturbation techniques [13] can then be proportional the far field expansion of the used to combine these two solutions into a M Fourier transform of γνj , Rνμjl()y,,ξ τ is single uniformly valid result that remains the moving frame correlation defined in finite everywhere in the flow and is not that terms of the fixed frame correlation (33) by much more expensive to compute than the [8] original parallel flow result. The resulting

“propagator” still becomes large when the M source point is at the critical layer but, Rνμjl( y,,ξ τ≡) (44) unlike the parallel flow result, now remains ˆ finite there. Rνμjl()y ;ξ + iUc τ , τ This result can now be inserted into the formula (31) for the pressure autocovariance (where U denotes the convection velocity which can then be Fourier transformed to c show, upon assuming only that the of the turbulence) and I have introduced transverse length scale of the turbulence is Iω ( xy| ) ,the acoustic spectrum at the short compared to the transverse length scale observation point x due to a unit volume of of the mean flow, that far field acoustic turbulence at the source point y ,for transparency sake. Obviously, this quantity

9 has to be integrated over the entire noise of the generalized Reynolds stress producing region of the flow to calculate the autocovariance tensor, which can be related actual acoustic spectrum to Φ *ijkl by using the linear transform (35).

But even this quantity has only been Iωω()x= ∫ Id()xyy (45) infrequently measured (see, for example, V Harper-Bourne)[14] -and then only for very And finally I have neglected the contribution low Mach number flows. The only recourse of the linear instability waves primarily is to mode l the Reynolds stress because they do not seem to be all that autocovariance tensor itself and to work important at the relatively low supersonic backward through these formulas to Mach numbers that we will be dealing with calculate the spectrum Φ *ijkl that actually here. appears in the acoustic equation(42).

There are two main requirements for

such models. The first is that they must 6 Application to the Jet Noise reduce the large number of independent Problem spectral components to a manageable level The results of the previous section apply (there are 45 of these in all even when the to any parallel shear flow but, for enthalpy fluctuations are neglected, there are definiteness, the reminder of the paper will 78 when they are not) and the second is that be restricted to the technologically important the models for the remaining components jet noise problem. The propagator must be relatively inexpensive to compute. The first of these is usually met by Γκ jT()xy,Y ultimately depends only on introducing some sort of symmetry and/or the mean flow which, in today’s statistical assumptions such as local isotropy environment, would almost certainly be and quasi-normality. But these assumptions determined from a RANS computation. But do not seem to be all that viable at the high the spectral tensor Mach number end of the range of interest. ∗ Φωνμjl(y;k1,, kT )depends on the We, therefore, assume only that the turbulence anisotropy is Mach number turbulence statistics, which can not be independent and that the turbulence itself is obtained from any steady flow solution. It, axysymmetric [15]—which is consistent therefore, has to be modeled, but the model with the experimental observation that the can be paramatized and the parameters can turbulence statistics in the various cross flow be determined from some reduced order directions are much more similar to one flow computation such as a RANS solution. another than they are to those in the Ideally, we would like to model the streamwise direction. spectral tensor Φ *ijkl itself, but the models The second requirement usually implies must, at least at present, be based on that the integrations that arise in the Φ *ijkl experimental data and the experimentalists calculation can be done in closed form, or are unlikely to measure this quantity any nearly so, because of the relative expense in time in the near future. A fall back position doing numerical integrations over and over might be to develop models for the spectra again. But the models must also be flexib le

enough to provide an accurate representatio n Ψωijkl ()y;k1,, kT of the turbulence structure because the strong streamwise retarded time variations ∞ cause the radiated sound field to be very 1 −ωτiMi()k ξ+k iξT ≡ee11 T R()y,,ξξτ ddτ sensitive to that structure. 2π ∫∫ ijkl −∞ V Appropriate models were introduced in (46) reference [8]. They involve a large number

10 of adjustable parameters (actually infinitely Mach numbers where non-linear many) which, as noted above, have to be propagation effects are believed to be determined from some reduced order flow unimportant. Fortunately, this also computation—which in today’s world would corresponds to the Mach number range of again have to be a RANS computation. But most technological interest. The critical this type of calculation can only provide layer only appears when the observation enough information to determine a small angle (as measured from the downstream number of these parameters. However, axis) is fairly small (<45o) and gradually newer higher fidelity reduced order flow moves inboard from the nozzle lip line with computations, such as URANS, VLES or increasing downstream distance until it even hybrid RANS/LES, are rapidly coming reaches a point beyond the end of the on line and these codes should be able to potential core where it quickly moves onto determine many more of these constants in the jet axis and suddenly disappears. But the the near future. propagator is still very close to being But the results that will be presented in singular, and can consequently be relatively this paper are based on a pure RANS code, large, for a significant distance downstream namely the NPARK WIND code, which is of this point. It was, therefore, necessary to widely used in the United States [16]. This construct an additional inner solution for this code was used to calculate the mean flow region as well—especially since much of the from cold (i.e., unheated) round jets with small angle sound field will actually be generated in this relatively localized portion acoustic Mach numbers M JJ≡ Uc/ ∞ of of the jet when the acoustic Mach number is 0.50, 0.90, and 1.4, where U is the jet exit J close to unity. velocity and c∞ is the speed of sound at Finally, in order to put these results into infinity. The upstream conditions were perspective Fig. 2 shows a comparison with specified in terms of the nozzle temperature the best previous calculation using the Glenn and pressure ratios. The results were then JeNo code with an ad hoc correction to used in equation (42) to calculate their far- eliminate the critical layer singularity [18]. field acoustic spectra on the arc x / D = 100 The results clearly demonstrate that there is J enormous improvement in predictive and compared with jet noise measurements capability with the present code. taken NASA Glenn SHJAR rig [17] with the same upstream conditions and with the atmospheric absorption removed from the References: data in order to make the comparisons on a lossless basis. The results are shown in [1] Lighthill, M.J.,1952, On Sound Fig.1. The lower curves are the acoustic Generated Aerodynamically: I. General spectra at 90o to the downstream axis and Theory, Proceedings R. Society Lond., A the upper curves are the spectra at 30o 211, pp. 564–587 (which is close to the peak radiation direction). [2] Favre, A.,1969, Statistical Equations of The overall agreement appears to be Turbulent Gases”, Problems of quite good but there is a tendency to under Hydrodynamics and Continuum Mechanics, predict the higher frequency components of SIAM, Philadelphia, pp.1-7. o the 90 spectrum in the supersonic case. This is because the flow is not correctly expanded here and the present analysis does not [3] Pope, S. B.,2000, Turbulent flows, account for the resulting shock associated Cambridge University Press noise (shown cross hatched in the figure). Computations were only carried out at the relatively low (<1.5) supersonic (acoustic)

11 [4] Dowling, A.P. & Ffowcs Williams, J. E, on Aerodynamic Noise, Journal of the 1983, Sound and Sources of Sound, Ellis Acoustical Society of America, vol. 54, no. Horwood Limited, Chichester 3, pp. 630-645

[5] Goldstein, M.E., 2002, A unified [16] Nelson, C. C. and Power, G.D., 2001, approach to some recent developments in jet "CHSSI Project CFD-7:The NPARC noise theory International Journal of Alliance Flow Simulation System," AIAA Aeroacoustics, vol. 1,no. 1, pp. 1−16. Paper 2001-0594.

[6] Goldstein, M.E., 2003, A Generalized Acoustic Analogy, Journal of Fluid [17] Bridges, J. and Podboy, G. G., 1999, Mechanics, vol. 488, pp. 315–333. Measurements of Two-Point Velocity Correlations in a Round Jet with Application to Jet Noise , AIAA Paper #1999–1966, [7] Goldstein, M.E. and Leib, S. J., 2005, [18] Khavaran, A Bridges, J. & Georgiadis, The role of instability waves in predicting prediction of turbulence Generated in Hot jet noise Journal of , Jets NASA /TM-2005-213827 Volume 525, pp 37-72

[8] Goldstein, M.E. and Leib, S. J., 2008, The Aeroacoustics of Slowly Diverging Supersonic Jets Journal of Fluid Mechanics. To be published. [9] Morse, P.M. and Feshbach, H.,1953, Methods of Theoretical Physics, McGraw-

Hill,.

[10] Lilley, G.M., 1974, “On the Noise from Jets,” Noise Mechanism, AGARD–CP–131 [11] Lilley, G.M., 1996,The Radiated Noise from Isotropic Turbulence with Applications to the Theory of Jet Noise, Journal of Sound Vibration, Vol. 190, pp. 463–476.

[12] Goldstein, M.E., 2008, On Identifying the Sound Sources in a Turbulent Flow International Journal of Aeroacoustics, To be published.

[13] Van Dyke, M., 1975, Perturbation Methods in Fluid Mechanics, the Parabolic Press, Stanford CA

[14] Harper-Bourne, M., 2003, Jet Noise Turbulence Measurements, AIAA Paper

[15] Goldstein, M.E. and Rosenbaum, B. M., 1973, Effect of Anisotropic Turbulence

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PPart a) M J = 0.5 Part b) M J = 0.9

Part c) M J = 1.4

Fig. 1) Comparison of Jet Noise Predictions with Measurements (Taken from [8])

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Fig. 2) Comparison with JeNo Results. (Taken From [8])

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