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The Aeroacoustics of Turbulent Flows

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The Aeroacoustics of Turbulent Flows The Aeroacoustics 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, Turbulence, Aerodynamic Sound, 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 fluid 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 pressure ∂ 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 ∂vi ∂ Duoi++ u j pe′ ∂∂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 equation of state, 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 tensor. (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 Kronecker delta 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).
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