Parabolized stability equation models in turbulent supersonic jets

Transcription

1 Parabolized stability equation models in turbulent suersonic jets Daniel Rodríguez,, Aniruddha Sinha, Guillaume A. Brès and Tim Colonius California Institute of Technology, Pasadena, CA Universidad Politécnica de Madrid, Sain Cascade Technologies Inc. Palo Alto, CA The eak noise radiation in the aft direction of high-seed, turbulent jets has been linked to the dynamics of the large-scale structures. We use the arabolized stability equations (PSE) to model these structures as waveackets associated with instability of the turbulent mean flow. Our ast work has demonstrated the utility of these models for subsonic jets; in the resent work we extend these methods to suersonic jets. A large eddy simulation database corresonding to an unheated, ideally-exanded Mach. jet with Reynolds number of, is emloyed to extract the necessary inut for the PSE (the mean flow and initial conditions) and also to erform comarisons and validations of the comuted waveackets. By contrast with subsonic jets, when the jet exit velocity is suersonic with resect to the ambient seed of sound, linear stability theory redicts that multile instability modes, related to resonance of ressure waves within the otential core, can be resent in addition to the inflectional instability. The ossible coexistence of different instability mechanisms, the determination of adequate inlet conditions, and their effect on the waveackets comuted are investigated here. We comare the waveackets redicted by PSE with fluctuations extracted from the LES data. When erforming comarisons, filtering techniques need to be emloyed in order to extract the coherent, low frequency structures associated with waveackets. Largescale fluctuations educed using cross-correlation techniques, such as the roer orthogonal decomosition, are shown to comare reasonably well with the PSE waveackets, but by contrast with subsonic jets, it aears that several POD modes are required to reresent the PSE-redicted waveacket. I. Introduction The noise generated by a turbulent jet and its subsequent radiation to the far-field is a technological roblem of great imortance that has received continuous attention for decades. Investigation of the associated flow hysics numerically, using direct numerical simulations or large eddy simulations at Reynolds and Mach numbers of ractical alication remains challenging and comutationally exensive even with state-of-the-art algorithms and suercomuters. An intermediate aroach is born from the observation that the highly-directional eak noise radiated in the aft direction of turbulent jets is associated with the dynamics of the large-scale coherent structures. Relatively coarse descritions of the non-comact ressure fluctuations, or waveackets, associated with the large-scale structures can be combined with acoustic rojection methods. Statistical descritions of the coherent structures observed in forced jets are reminiscent of instability waves, suggesting their modeling as instability waveackets of the mean turbulent flow. The concetual Marie Curie COFUND fellow, California Institute of Technology, CA, USA and Universidad Politécnica de Madrid, Sain. AIAA Member. Postdoctoral Scholar, Deartment of Mechanical Engineering, California Institute of Technology, USA. AIAA Member. Senior Research Scientist, Cascade Technologies Inc. AIAA Member. Professor, Deartment of Mechanical Engineering, California Institute of Technology, USA, AIAA Associate Fellow. of 9

2 searation of turbulent mixing flow into a mean flow with coherent large-scale and uncoherent small-scale fluctuations (referred to as trile decomosition), and the subsequent modeling of the coherent art as flow instabilities of the turbulent mean flow was formalized by Hussain & Reynolds., Exerimental observations of forced turbulent lane mixing-layers 7, showed that quasi-arallel inviscid linear stability theory delivers good redictions of the statistical large-scale structures for a finite satial region downstream of the slitter late, but the agreement is lost as the fluctuation amlitude increases due to strong non-linear interactions between the coherent, uncoherent and mean flow comonents of the flow. The satial extent of the region of linear behavior was strongly deendent on the forcing amlitude. Following similar considerations, Mankbadi & Liu 9 ointed out that the evolution of the low-frequency, large-scale statistical structures in turbulent jets is mainly governed by the interaction between the mean flow and the articular structure, with no remarkable interaction between different frequencies. The absence of nonlinear interactions between coherent structures was justified by the relatively small energy content in the individual coherent structures, comared to the energy in the mean flow and in the uncoherent or random fluctuations. This aroach was emloyed in the study of forced suersonic jets,,9 for which the measured near field fluctuations were found to be in good agreement with the redictions of linear stability theory. In the case of natural jets, this aroach has only recently begun to deliver satisfactory quantitative redictions. Suzuki & Colonius showed that the ressure fluctuations measured just outside of the jet shear layer were consistent with the evolution of linear instability waves comuted using locally arallel flow analysis of the jet mean flow. One of the difficulties associated with the exerimental observation of the coherent structures in natural jets is the lack of a hase reference, that is trivially determined in forced jets. To overcome this difficulty, Suzuki & Colonius considered a series of measurements erformed at the NASA Glenn SHJAR facility, in which a microhone hased-array was laced just outside of the turbulent mixing-layer, in a region where hydrodynamic fluctuations are exected to behave linearly and acoustic contamination to be minimal. The careful location of the array and the high-quality, hased measurements erformed were instrumental in the successful comarison between theoretical waveackets and exerimental coherent structures. In subsequent work, it was found that as the Mach number or the jet temerature were increased, the comarison between the linear instability wave redictions and the measurements became oorer. The magnitude of the fluctuations associated with acoustic radiation strongly deends on these two factors (jet exit temerature and Mach number), resulting into a contamination of the measured ressure signal. Proer orthogonal decomosition (POD) of the exerimental data was then used in order to filter out the acoustic contamination, and led to outstanding comarisons between linear waveackets and exerimental POD-filtered ressure enveloes and hases for subsonic cold jets, and reasonably good comarisons for heated jets. A ste beyond arallel-flow linear stability theory in the modeling of the waveackets is achieved with the introduction of the arabolized stability equations (PSE). 7 In our revious works,, we emloyed arabolized stability equations to take into account the mild variation of the jet mean flow along the axial direction and the nonlinear interactions between the different frequency and azimuthal wavenumbers. In these comutations, the use of the exerimental turbulent mean flow as base flow in the PSE comutations artially accounted for the nonlinear interactions resent in the flow, and in articular for the Reynolds stresses due to the random, small-scale fluctuations on the establishment of a mean flow, as theoretized by Mankbadi & Liu. 9 This aroach has been already emloyed in the literature both in its linear 9 and nonlinear versions with romising results. Alternatively, global instability analysis could also be used to rovide the descrition of the linear waveackets, but at a much higher comutational exense. A fundamental asect in the waveacket comutation using PSE is the determination of adequate inlet conditions. Besides broader-scoed concerns on the inlet conditions stated in the literature,, the PSE integration requires of reresentative fluctuation shaes, amlitudes and hases for the frequencies and azimuthal wavenumbers of interest. Linear instability theory (LST) for arallel flows shows that subsonic jets can only sustain the Kelvin-Helmholtz instability, either in the form of thin shear-layer instability or as jetcolumn instability. It suffices then to imose the resective Kelvin-Helmholtz eigenfunction as inlet condition for the PSE. However, suersonic jets can sustain additional kinds of instabilities related to the resonance of ressure waves within the otential core. The sole use of the Kelvin-Helmholtz eigenfunction may not be justified in this case, and more comlex inlet conditions may be necessary. If exerimental or high-fidelity simulation data is available, the bi-orthogonal system formed by the direct and adjoint eigenfunctions of the LST can be used in order to roject a given fluctuation on the different eigenmodes, thus determining the relative imortance of the different instability mechanisms at the inlet. of 9

3 In the resent aer, the PSE model is alied to an isothermal ideally-exanded suersonic turbulent jet, with jet Mach number of. and Reynolds number based on the nozzle diameter and jet exit velocity of,. A large eddy simulation (LES) database of this configuration, comuted using Cascade Technologies flow solver Charles,,7, was used in order to extract the necessary inut for the comutation of PSE waveacket models (i.e. the mean flow and initial conditions) and to erform a detailed evaluation of the modeling strategy, as well as to study the hysics of the flow that are imlicated in the noise generation and radiation. In this resect, standard ower sectral density methods were used to extract fluctuation modes in Fourier-Fourier sace for time and azimuthal wavenumber. Furthermore, a refined POD technique is also used to extract the large-scale structures more relevant to the noise eak radiation. The remainder of the aer is organized as follows. The flow configuration and LES database are described briefly in section II, while the POD technique emloyed in the waveacket extraction is resented in III. Section IV discusses the PSE formulation used. The determination of adequate inlet conditions for the PSE waveackets is discussed in section V. The PSE waveackets and their relation with the coherent structures extracted from the LES are discussed in section VI. Finally, some conclusions and ossibilities for future directions are exosed in section VII. II. Descrition of the Large Eddy Simulation database Large eddy simulation data corresonding to a cold M j =. jet are used in the resent work with the twofold objective of calibrating the PSE models with the determination of the turbulent mean flow and adequate inlet conditions and validating them by comaring with the waveackets in the simulation. The LES comutations were erformed using Cascade Technologies flow solver Charles.,7, The simulation considered the isothermal (T j = T ) ideally-exanded jet flow emanating from a convergent-divergent roundnozzle at Reynolds number Re =,. In what follows, non-dimensional form of the hysical quantities will be used excet where stated otherwise. The reference scale is the jet diameter D, the reference velocity is the ambient seed of sound c, yielding the reference time t ref = D/c. Pressure is made dimensionless with the hysical dynamic ressure ρ c, and density with the far-field density ρ. Besides this non-dimensionalization, some quantities are defined with variables different from the reference ones: the Reynolds number is based on the jet exit conditions: Re = ρ j U j D/µ j, that is related to the acoustic Reynolds number Re a = ρ c D/µ by Re = Re a M j. The dimensionless frequency used to show results will be the Strouhal number defined as St = fd/u j instead of the Helmholtz number He = fd/c that would be consistent with the dimensionless form emloyed. This dimensionless form is common to the LES database and the formulation of the arabolized equations. Time-averaged quantities such as the mean flow variables or the averaged Reynolds stresses were comuted using the total simulation time (after initial transients) of t tot D/c, considered to be long enough to have statistical convergence of the stationary quantities. The simulation was erformed on an unstructured mesh with control volumes and the total size of the transient LES database is aroximately Tb. In order to calibrate and validate the PSE waveacket models, the time-deendent flow field variables were first undersamled in satial and temoral resolutions in order to deliver a reasonable-sized dataset. Data were interolated to an uniform structured cylindrical mesh with N x N r N θ = 7 oints, resulting in satial sacings x =.D and r =., and a constant time ste t =. D/c. In addition, the data were interolated to a virtual cone akin to the microhone array used in UTRC exeriments. The time series are double Fourier transformed in the frequency domain and in the azimuthal direction. The corresonding amlitudes are comuted using standard samling techniques as ower sectral densities (PSD), and then square-rooted. The accuracy of this rocess deends on the quality of the used data, i.e. resolution in the azimuthal direction N θ and time ste t and total number of time stes N s. Proer orthogonal decomosition will also be used, the quality of which will also deend on the data quality. Some studies were conducted in order to determine the necessary resolution in time and sace for the convergence of the fluctuation PSDs. These studies, not reorted here, showed that using the total simulation time delivers converged statistics. The satial resolution in the axial and radial directions cannot be further reduced, otherwise imortant details in the li-line region, where fluctuations are stronger, would not be adequately described. The resolution on the azimuthal direction can be reduced to N θ = equally saced oints and still delivers converged results for the azimuthal modes of interest. Finally, for the range of frequencies of interest here, the time ste in the temoral Fourier transforms can be increased to t =.D/c, imlying a two-fold reduction in memory and comuting time. of 9

4 III. Extraction of waveackets from LES data using Proer Orthogonal Decomosition The PSE models are intended to cature the coherent waveacket motions at low frequency. In order to validate these models with LES data, which resolves motions over a broad range of length scales, these structures must be educed with aroriate statistical techniques. The ower sectral density (PSD) is artially effective, isolating motions at low frequencies. But, as we show below, techniques based on cross correlations are needed to educe the coherent structure. In this study, the roer orthogonal decomosition (POD) 9 is emloyed to extract the structures with correlation over a significant satial region. Such an aroach has been reviously used to data from low Reynolds number turbulent jets and with microhone array data from subsonic jets. The LES flow field is considered to be statistically stationary in time and homogeneous along the azimuthal direction, and thus Fourier modes in frequency and azimuth are introduced, in the same manner as is done in the derivation of the arabolized stability equations: q k (x, r; m, n) = π/ω π t= θ= π q(x, r, θ, t)ex [i (mθ nωt)] dtdθ. () The index k denotes the time segment used in the temoral Fourier transform. The use of different norms in the comutation of POD modes of a turbulent jet was first done by Freund & Colonius, showing that the resulting decomosition may be highly deendent on the hysical variables of interest. The flow variable of interest here is the fluctuating ressure field, and the POD should be geared toward extracting its dominant features. Thus, the inner roduct is defined as < q i, q k >= ( i ) k rdrdx, () x with denoting comlex conjugate. The frequency-domain variant of Sirovich s snashot POD method is emloyed. A segment-to-segment cross-correlation matrix is defined now by alying the inner roduct to airs of Fourier transformed erturbation fields, corresonding to different segments of data: r S ij =< q i, q j > () POD modes are the otimal basis of orthogonal functions that minimizes the square mean error between the fluctuations q k and its rojection on a basis of orthogonal functions. The l th POD mode to be comuted can be written as a linear combination of the different transformed segments: φ l (x, r; m, n) = k β l k(m, n) q k (x, r; m, n), () where the coefficients β l k are obtained as the eigenvectors of the eigenvalue roblem S β l = λ l β l. () The non-negative eigenvalues λ arranged in descending order indicate the order of otimality of the resective POD eigenfunctions. The first POD mode corresonds to the largest λ, the second corresonds to the second largest, and so on. The Fourier decomosition in the homogeneous azimuthal direction reduces the dimensionality of the POD roblem by one. The POD erformed on the entire three-dimensional domain reduces to a twodimensional POD in x r; this will be termed lanar-pod henceforth. Alternatively, the LES data on a virtual cone simulating the near-field microhone array used in related exerimental research, can be used exclusively. The POD on the virtual cone reduces to a one-dimensional POD along a meridional line; this will be called line-pod. IV. Parabolized stability equations Parabolized stability equations (PSE) 7 reresent a generalization of the arallel-flow linear stability theory (LST) for flows with a mild variation in the streamwise direction, also ermitting the nonlinear of 9

5 interaction of the different modes. The technique was originally devised for studying transition in laminar flows. 7 Here, PSE is roosed as a technique for reduced-order modeling of the waveackets linked to instability modes that are imlicated in the eak noise radiation to aft angles in turbulent jets. The total flow field q is decomosed into a mean (time-averaged) and axisymmetric comonent q and its fluctuations q. The formulation of the equations used for the PSE considers the velocity comonents in a cylindrical coordinate system u x, u r and u θ, the dimensionless ressure and the secific volume ζ = /ρ as the flow variables. Only round-nozzle jets are considered here. The fluctuating art is then written as a sum of Fourier modes in the azimuthal direction and in frequency where m is the azimuthal wavenumber, ω is the fundamental circular frequency, n is the harmonic number, and χ mn is the modal function corresonding to the mode (m, n). In the ractical solution of nonlinear PSE, the azimuthal and frequency modes must be truncated, involving a finite number (M, N) of modes. The unresolved modes are then formally gathered in the term q, and the fluctuation field is written as N M q (x, r, θ, t) = n= N m= M χ mn (x, r) ex(i(mθ nωt)) + q. () The mean flow is a function of the axial and radial directions (x, r), but a slow variation of its roerties along the axial direction is assumed. This assumtion ermits the decomosition of χ mn into a slowly varying shae function (that evolves in the same scale as the mean flow) and a raidly varying wave-like art: ( ) χ mn (x, r) = A mn (x) ˆq mn (x, r) = ex i α mn (ξ) dξ ˆq mn (x, r). (7) x Here α mn (x) is a comlex axial wavenumber, for which a mild variation is also assumed. Introducing the decomosition (-7) into the comressible Navier-Stokes, continuity and energy equations and subtracting the terms corresonding to the mean flow, we arrive at the system of equations ( L inωl t + L x x + Lr r + imlθ r + + ( V + V x Re x + Vr r + imvθ r m Vθθ r + Vxr x r + imvrθ r ) ˆq + Vxx + Vrr x r r + imvxθ r x ) ˆq = ˆF mn + F mn. () A mn A mn Exressions for the linear oerators L and V can be found in Aendix A. For brevity, the subscrit have been droed from the shae function and wavenumber in the revious exression. The left-hand-side in () is a linear satial oerator for the mode (m, n), while the right-hand-side accounts for all the nonlinear contributions to the modal evolution. Linear PSE are obtained setting the right-hand-side equal to zero. The contributions resulting from interactions of the resolved modes are comrised in the function ˆF mn (x, r), while those interactions in which the unresolved modes are involved are gathered in the function F mn. In nonlinear comutations considering laminar or transitional flows all the dynamically relevant modes are resolved, and F mn is neglected. In the case of the natural turbulent jets of interest here, a broad sectrum exists and the truncation of the modes is always arbitrary. Consequently, the function F mn is not necessarily negligible and may need to be modeled. Equation () requires an additional condition to be closed. Following Herbert, 7 the normalization condition ˆq ˆq r dr = (9) x is imosed individually to every mode, removing the exonential deendence on the shae function ˆq. Here the suerscrit denotes comlex conjugation. The system of equations is discretized using fourth-order central finite differences in the radial direction, closing the domain with the characteristic boundary conditions of Thomson. The boundary conditions at the centerline are derived following Mohseni & Colonius. The streamwise derivative is aroximated using an imlicit first-order Euler scheme with a relatively large ste size x, in order to revent elliticityrelated instabilities in the marching rocedure. For each successive axial location, the system () is solved iteratively: the α and ˆq converged in the revious location are used as initial guesses. of 9

6 Figure. Satial LST eigensectrum at St =., m = and =.. V. Inlet conditions for PSE waveackets using LES data The PSE solution requires an initial condition (radial variation of the PSE mode) to be secified at the ustream boundary of the domain, tyically near the nozzle exit. The Kelvin-Helmholtz mode is the only convective instability in subsonic jets, and so it is exected to be the determinant of the PSE mode shae over any significant axial extent. In the absence of further information, Gudmundsson & Colonius initiated the PSE with the Kelvin-Helmholtz mode obtained from the arallel-flow linear stability analysis alied to the turbulent mean flow field at a cross-section near the nozzle exit. The LES database contains full-field time-resolved flow information. This reresents an oortunity to revisit the initialization of the PSE comutations. The biorthogonal rojection aroach was introduced in the field of hydrodynamic instability by Salwen & Grosch and in flow control by Hill. It has been used in the ast to filter the fluctuation fields from exeriments and numerical simulations.,,7 In the resent work, this method is adoted to determine the initial conditions for the PSE comutations that rovide the best ossible descrition of the instability waveackets while filtering out other erturbations resent at the initial axial station. Section V.A briefly revisits the linear stability analysis. Section V.B describes the biorthogonal rojection technique, and considers its use for filtering the LES data. Section V.C comares the results of linear PSE initiated with different sets of eigenmodes obtained with the above method. A. Parallel-flow linear stability theory The PSE linear oerators are used as the dearture oint in the definition of a local linear stability eigenvalue roblem (EVP), thus analyzing the eigensectrum of the PSE aroximation, instead of the usual comressible linearized Navier-Stokes equations for arallel flow. In articular, second derivatives in the axial direction are neglected on account of the high Reynolds number of the jet. As ointed out by Li & Malik, neglecting these terms eliminates a continuous branch of stable ustream roagating modes that are of no significance in the downstream evolution of the waveackets. After linearization and neglect of non-arallel effects, the discretized matrix formulation of the PSE () can be rewritten as the matrix EVP A ˆq = iα B ˆq () for the comlex axial wavenumber α. The eigenvectors are denoted by ˆq. The generalized matrix eigenvalue roblem () is discretized using exactly the same numerics develoed for the PSE. The generalized Schur decomosition (QZ algorithm) imlemented in the LAPACK routine ZGGEV is then used to comute the full eigensectrum along with the right and left eigenvectors. Figure shows a tyical eigensectrum, obtained for St =. and m = at =.. Aart from the Kelvin-Helmholtz instability of the shear layer, the sectrum contains families of eigenmodes corresonding of 9

7 Table. Ratio between eak ressure and axial velocity magnitudes ( ˆ max/ û x max) of the eigenvectors in figure. Instability waves - subsonic instability and Kelvin-Helmholtz k - - ˆ max / û x max (direct) max / ũ x max (adjoint) r / D.... u x Figure. Linear stability eigenfunctions for St =., m = and =.. Axial velocity (a) and ressure (b) comonents of the eigenvectors k = ( dashed-dotted); k = ( dashed); k = (solid); k = (dashed) and k = (dashed-dotted). to ressure, vorticity and entroy waves inside of the otential core and in the outer field. Only the family of discrete eigenmodes denoted by filled circles in figure is of interest here. This branch comrises two out of the three different instability mechanisms in suersonic jets, namely Kelvin-Helmholtz and subsonic instability waves. Suersonic instability waves are not ossible for this Mach number, as exlained by Tam & Hu. Only a finite number of satially amlified modes are resent. These eigenmodes can be enumerated according to the number of eaks k that the eigenfunctions resent inside the otential core. The first one corresonds to the Kelvin-Helmholtz instability it is characterized by a marked eak at the liline ( =.) and has no eaks inside the core (k = ). Modes k =, k = and so on are unstable waves roagating with subsonic hase seed inside the otential core. To the left of mode k = in the eigensectrum, the other branch of subsonic waves is found, referred to as k =,,.... The axial velocity and ressure comonents of the eigenvectors associated with the different modes are shown in figure, and the eak ressure to velocity ratio are resented in table. The relative imortance of the ressure comonent is increased with the number of core eaks k. The axial velocity rofile has a eak in the li-line region for all the different instability modes. The subsonic modes with the same number of eaks, e.g. k = and k =, have eaks and nodes inside the otential core at the same locations, revealing their resonant nature. Given that the centerline eak dominates the ressure comonent, these eigenfunctions are nearly identical inside the core (figure, right). Outside the mixing region, the decay in the radial direction is different for each eigenmode, on account of the differences in the axial wavenumber α r. In this region the base flow is aroximately uniform, and linear theory redicts a decay rate r / ex( r α r ω ), that is visible in the eigenfunctions (figure ). The eigenmode k = with the smallest wavenumber decays slower than the others, and at first, this may imly its relevance to the noise radiated to the far field. However, as a counteroint, this wave is also quickly damed in the axial direction, making such simlistic deductions difficult. 7 of 9

8 LES S S S r / D u x u x Figure. Fluctuation axial velocity comonent at x =. (left) and x =. (right), extracted from LES (thick black solid), and as the rojection on different subsets of linear instability eigenmodes: S (dashed-dotted), S (dashed), S (dotted). Strouhal and azimuthal wavenumbers are St =. and m =. B. Biorthogonal rojection The linear oerators describing the eigenvalue roblem () are non-orthogonal, a roerty inherited from the linearized Navier-Stokes equations, and consequently the associated eigenvectors ˆq are not orthogonal. The adjoint roblem is introduced in order to construct the following bi-orthogonality relation between the direct and adjoint eigenvectors: (α k α l ) q k B ˆq l = q k B ˆq l = C l δ kl, () where q k is the adjoint eigenfunction corresonding to the eigenvalue α k, δ lk is a Kronecker delta function, and C l = q l B ˆq l. A given erturbation field q, reviously Fourier transformed in time and azimuthal direction to (St, m), can be written as the weighted sum of the desired eigenmodes: q = k a kˆq k + q out, a k = ( q k Bq ) / ( ) q k Bˆq k. () The comonent q out reresents the fluctuations irrelevant to the waveacket modeling; they are to be filtered out based on heuristics elaborated below. The radial shae of the adjoint eigenfunctions determines the regions that have the largest contributions to the amlitude weights a k of each eigenmode. Most of the adjoint eigenfunctions (not shown) have a eak in the vicinity of the inflection oint. The imlication is if the arbitrary erturbation field q contains large amlitude fluctuations at the inflection oint, then large values of weights are to be exected. Following this rocedure, initial conditions for the PSE can be constructed as a linear combination of a chosen subset of the LST eigenmodes, with the relative amlitudes obtained from the numerical simulation data. The PSE modal function at the inlet location x = x in corresonding to equation (7) is then written as χ mn (x in, r) = k a mn,k ˆq mn,k (r). () Figure dislays the fluctuation rofiles of axial velocity in the St =., m = mode at x =. and x =.. A comarison is made between the rofiles extracted directly from the LES and their rojection on different subsets of eigenmodes. Three sets of eigenmodes are considered in the rojection. The first subset S corresonds to the Kelvin-Helmholtz instability (k = ) alone. The second subset S comrises S and the nearer subsonic instability eigenmodes (S k = {,, }). Subset S includes the Kelvin-Helmholtz and the first subsonic instability waves (S k = {,,,, }) in addition of other eigenmodes in the eigensectrum: downstream moving ressure waves, ustream moving ressure waves, core vorticity of 9

9 modes and entroy vorticity modes. A finite and rather small number of eigenmodes are considered, even in the subset S, and thus fine structures cannot be catured, esecially inside the otential core and in the outer region, where resectively infinite discrete and continuous sectra exist. Due to the non-normality of the eigenmodes, the rojection on the eigenmode subsets is not necessarily bounded by the LES rofile, and consequently the convergence of the rojection as the number of eigenmodes is increased is not monotonous. The thick black curve in figure (left) corresonding to the LES data at x =. decays inside the otential core since the LES does not imose any erturbations at the nozzle. As the turbulent mixing layer develos in the axial direction, the disturbances sread towards the jet axis, as can be seen in figure (right). Adequate recovery of this behavior is not attained in the rojection due to the relatively small number of core modes retained. On the other hand, in the mixing region surrounding the inflection oint (r =.) the shar eak of the LES rofile is accurately described even by the rojection on the Kelvin-Helmholtz eigenmode alone (S). This indicates that the disturbance field is dominated in this axial region by the inflectional instability. The eak of the axial velocity magnitude is at least one order of magnitude higher than the erturbations at any other radial location, and dominates over the other velocity comonents. However, the behavior of the Kelvin-Helmholtz shae functions diverges from the LES rofile as r. The addition of the subsonic instability waves imroves the agreement in the outer region slightly. In any case, all the instability waves decay exonentially in the radial direction, and cannot exlain the trend observed in the LES data. When the outer flow ressure waves are introduced in the rojection (S), the LES trend is recovered. C. Sensitivity of linear PSE results to the initial conditions The foregoing discussion has revealed that the linear combination of the Kelvin-Helmholtz eigenmode and the first two subsonic instability modes comrise most of the LES fluctuations in the shear-layer region. However, the Kelvin-Helmholtz eigenmode exhibits much stronger satial amlification in the downstream direction, so that the relevance of the other two eigenmodes in the downstream evolution of the PSE modes is unclear. Figure comares the solution of the linear PSE initialized with different eigenmode rojections with the solution obtained by imosing the LES disturbance rofile directly as the inlet condition. The two eigenmode subsets S and S, and the initial axial locations =.,. and are considered in the comarison. The cross-section at which the PSE integration starts is indicated in the figures. Small differences are observed in the vicinity of the initial axial location that decay fast as the integration evolves downstream. Downstream of the = section, the fields corresonding to S and S are visually indistinguishable. The main difference between the PSE solution initialized with the LES rofile and those with biorthogonal rojections concerns the regions far outside the mixing layer, and more recisely in the manner in which the erturbations decay towards the far-field. This disagreement is exected on account of the rojections shown in figure. It was surmised that the inclusion of the k = mode may be imortant for accuracy in the outer region. However, its highly damed character is evidently recluding this effect. It has been discussed earlier that modeling the waveackets in the near field immediately outside the shear layer is the most ertinent for redicting eak aft angle noise radiation. In this regard, the above analysis indicates that the redictions in this critical region are quite insensitive to the various initial conditions considered, as far as the contribution of the Kelvin-Helmholtz eigenmode is included. There are two main reasons for this. First, the Kelvin-Helmholtz instability is concluded to be the leading mechanism in this articular jet configuration (which is not necessarily true for all jets). Second, the PSE integration accounts for a single axial wavenumber α for each (St, m) mode, and converges to the most unstable mode (Kelvin- Helmholtz instability) thus daming the other contributions. In the remainder of this work, only the Kelvin- Helmholtz mode from the arallel-flow LST is used to determine the inlet conditions for the PSE integration. VI. PSE waveacket models comared with POD modes A. Linear PSE comared with lanar-pod This section discusses a roer orthogonal decomosition of the LES data on the full comutational domain, considering the two inhomogeneous satial directions (x and r) and Fourier modes in the homogeneous azimuthal direction and stationary time direction. In the comutation of these POD modes, all the relevant satial locations within the jet flow are correlated at the same time through the inner roduct (), extracting 9 of 9

10 Figure. Disturbance axial velocity comonent obtained as the solution of the linear PSE, at St =., m =. The initial axial location for each case is x =. (a,b,c), x =. (d,e) and x = (f,g), as denoted by the vertical lines in the different subfigures. The LES data is used directly as initial condition in case (a). Cases (b), (d) and (f) corresond to the rojection using the eigenmodes subset S, and cases (c), (e) and (g) corresond to the subset S. flow structures that are coherent over significant distances. One otential benefit of comuting POD modes of the whole domain is the ossibility of erforming direct comarisons of the PSE modes with the extracted structures, without resorting to interolations localized in articular cross-sections or conical surfaces (as in ) that can be chosen adequately in order to deliver otimal comarisons. For brevity, the resentation will consist of results for three different frequencies St = {.,.,.} and m = and azimuthal modes, which cover the range of large-scale structures most significant for the aft eak noise radiation. Figure deicts the POD eigensectra for the low-order Fourier modes. It is readily aarent that the POD is unable to reveal a single dominant coherent mode in any of the Fourier modes, that may have been directly comarable with PSE redictions. The relative flatness of the sectra means that many mutually orthogonal modes are dynamically significant in the ressure field. It is also noticed that the flatness of the sectra increases at higher frequencies and azimuthal numbers. Figures 7-9 show comarisons of the linear PSE results with the first two POD eigenfunctions for the Fourier modes under consideration. The linear PSE modes have arbitrary absolute amlitude and hase, so that only their relative amlitudes and hase are ertinent in the figures. The POD yields an orthogonal basis set, so that the PSE solution can be orthogonally rojected onto the individual POD modes using the inner roduct defined reviously. The magnitude of the rojection coefficient is a measure of the correlation of 9

11 . St =.. St =.. St =.... m =... λ j / sum(λ)... m = m = POD mode j Figure. Eigensectra of lanar-pod for different St and m. of the PSE solution χ(x, r) (equation 7) with the corresonding POD mode φ j (x, r) (equation ): β j = < χ(x, r), φj (x, r) > χ(x, r) φ j (x, r). () Figure deicts the squared rojection coefficients. Note that due to the orthogonality roerty of the POD modes, the sum of the squared β j coefficients must be equal to unity. For most cases, a small number of POD modes is required in order to reconstruct the PSE modes for the lowest frequencies, so that the sum of the first βj coefficients amounts to around % of the total. The agreement deteriorates as the frequency increases over St =., but it is resently unclear if the disagreement is motivated by hysical reasons or due to a oorly resolved quadrature: in order to exedite the ost-rocessing, the LES data was extracted on a coarsened axial grid with uniform sacing of x =.D which is erhas too coarse for the smallest wavelengths associated with the higher frequencies considered herein. The results for St =., m =, on figure indicate that the first two POD modes together reconstruct the majority of the corresonding PSE solution. Figure 7 demonstrates the visual resemblance. The close connection between POD modes and Fourier modes is extensively described in the literature. 9 In articular, considering POD of an arbitrary waveacket, raid variations in the waveacket s amlitude enveloe or hase seed would require more POD modes in the reconstruction. At lower frequencies, the PSE modes here reresent waveackets with more gradual variations. This may exlain the effectiveness of the POD filtering in these cases. The modeling assumtions that go into PSE (secifically, the forced removal of ustream roagating waves) render it a relatively ineffective redictor of the acoustic signature in the near field. This is seen in the dearture of the ressure contours outside the virtual cone from those catured by the POD modes. The foregoing discussions also aly to the St =. modes. The m = case is efficiently reconstructed using the first few POD modes (see figure ), and the same conclusion can be drawn from figure. The m = case is different in that, the POD modes dislay strong acoustic fields resembling Mach wave radiation atterns. The St =., m = mode is indeed an efficient radiator. The PSE mode dislays a much weaker acoustic field for reasons stated above, and thus requires several POD modes for its reconstruction. Clearly, full-field lanar-pod is not the otimal filter for comarison here. The revious discussion of figure vis-a-vis the St =. modes has anticiated the issue seen in figure 9. Namely, the PSE solution reresents a waveacket with wavelength close to the grid resolution of the extracted LES data. The POD modes are dislaying waveackets with much larger wavelengths, most robably due to aliasing. This case will not be discussed further. In summary, PSE aears to be effectively redicting the waveacket signatures that dominate the hydrodynamic near field. The comarison with POD-filtered data is more challenging than in the subsonic jets, as the increased imortance of the acoustic field in the total fluctuations as Mach number increases may require several POD modes to mutually cancel in order to yield the hydrodynamic field obtained in PSE. of 9

12 St=. m= St=. m= St=. m= St=. m= St=. m=..... St=. m= St=. m= St=. m= St=. m= St=. m= β j..... St=. m= St=. m= St=. m= St=. m= St=. m=..... POD mode j Figure. Squared magnitudes of rojection coefficients β j for the resective PSE solutions on to the lanar- POD modes. However, careful analysis of the lanar-pod results resented herein reveals accetable fidelity of the PSE model. B. Linear PSE comared with line-pod on virtual cone The results in the receding section exhibit a remarkable qualitative agreement between the PSE modes and the structures associated with the first lanar-pod modes for a given St and m numbers. However, making quantitative comarisons using the full domain structures results troublesome due, in art, to the coarse resolution in the axial direction of the extracted LES data, but also to the different hysical mechanisms that are recovered together in the same POD modes. These mechanisms aarently include the linear hydrodynamic waveackets to be comared with the PSE modes, but also acoustic radiation observable towards the far-field and the ossibility of imortant non-linear interactions in the turbulent mixing region and inside of the otential core. Our earlier work on validation of PSE models focused on comarisons with exerimental measurements made at NASA SHJAR on a hased microhone conical cage. This array was designed so as to resolve the signature of waveackets in the hydrodynamic ressure field for the energetically-relevant frequency and azimuthal modes, and allows a comact quantitative evaluation of PSE models. Analogous comarison is done here by extracting the ressure data on an similar virtual cone, the exact location of which corresonds to the rotating cage array that was secifically designed at UTRC to erform measurements on suersonic round jet at the same conditions monitored here. The virtual microhones are located on a cone of 7 o half-angle between = and, with the most ustream sensor being at =.. The location of the virtual cone is shown in figures 7-9 by dashed lines. A rough estimate of the wavelength of the undulations is comuted on this cone from the aroximate zero-crossings near their eak. The hase velocity u h (relative to the ambient seed of sound) is then estimated from this wavelength, the indicated Strouhal number, and the known acoustic Mach number. The footrint of the second POD mode on the virtual cone demonstrates two distinct eaks for all three frequency cases. Thus, two searate hase velocities can be comuted for the two eaks. The hase velocities are indicated on to of the resective sub-figures, with both the ustream and downstream values for the second POD mode aearing in that order. Figure comares the PSE modes amlitude and hase to the unfiltered (ensemble-averaged PSD) virtual cone data, as well as the first and second line-pod modes. The most energetic Fourier modes are of 9

13 First POD mode: uh =.9 First POD mode: uh =. Second POD mode: uh =.7,.9 Linear PSE mode: uh =. Linear PSE mode: uh =.7 Second POD mode: uh =.,.9 (a) m = (b) m = Figure 7. Real comonents of ressure in the first and second lanar-pod modes comared with linear PSE mode at St =.. The dashed line reresents the virtual cone. comared here. Note that the energies of the ositive azimuthal modes are not doubled to account for the corresonding negative modes. Thus, m = and m = modes have very similar energies, whereas the m = modes go down by about a factor of four in energy. As mentioned earlier, the linear PSE modes are arbitrary with resect to the amlitude and hase, and are thus rescaled and hase-shifted a osteriori. For the subsonic jets studied by Gudmundsson & Colonius, the first POD mode of the cone ressure data tyically matched the PSE mode very closely. However, for some Fourier modes in heated transonic jets, the PSE mode showed greater similarity with the second POD mode. It was concluded thereof that as the acoustic art of the ressure field increases in dominance over the hydrodynamic, POD becomes less effective in cleanly filtering a unique waveacket from the total fluctuations, in agreement with the acoustic radiation atterns observed in the lanar-pod modes above. In the suersonic jet here, individual POD modes also aear to be an incomlete match for the corresonding PSE modes for most Fourier modes. The excetions aear to be the lower frequencies and higher azimuthal modes viz. the (St, m) mode airs of (., ), (., ), (., ), (., ) and (., ). For the remaining Fourier modes, the PSE modes amlitude enveloes resemble the ustream eaks of the corresonding second POD modes. The orthogonality constraint on the set of POD modes results in the near-null amlitude of the second POD mode where the first one reaches its maximum. As a consequence, the second POD modes tyically dislay two eaks. The hase lots demonstrate that the hase seeds (the recirocal of the sloes) of the POD modes match those of the PSE modes at ustream ositions. However, the downstream eaks in the second POD modes are tyically associated with acoustic radiation (as shown in figures 7-9), which is not relicated by the PSE modes. Higher hase seeds indicate acoustic instead of convective roagation of oscillations, reinforcing the suggested notion of dominance of the ressure fluctuations of acoustic nature over those associated with hydrodynamics downstream of the otential core closure. It should be remarked here that the final objective of the resent methodology is to emloy the PSE modes to model the noise sources for a subsequent acoustic rojection method to redict the noise emitted to the far field. In our revious work, a Kirchhoff surface method overlaed with the near-field microhone array or the virtual cone was used. Therefore, the virtual cone is the location most adequate to erform quantitative comarisons of the PSE waveackets with the POD modes. The rojection coefficients βj comuted in a of 9

14 First POD mode: uh =. First POD mode: uh =.99 Second POD mode: uh =.,. Linear PSE mode: uh =.9 Linear PSE mode: uh =.9 Second POD mode: uh =.,. (a) m = (b) m = Figure. Real comonents of ressure in the first and second lanar-pod modes comared with linear PSE mode at St =.. similar fashion to () but restricting the quadrature to the cone are shown in figure. Comared with the rojection on lanar-pod modes (figure ), very few POD modes are required to reconstruct the PSE modes, and dominance of the first and second modes is clear for most of the frequencies and azimuthal wavenumbers. The roblem associated with coarse extraction of the LES data to comute the lanar-pod results noted above is found here for St =. modes. VII. Conclusions Parabolized stability equations (PSE) have been roosed as a tool for develoing reduced-order models of the waveackets imlicated in eak sound radiation to the far field for turbulent jets. Previously, we have shown that linear PSE is indeed effective in redicting the near-field hydrodynamic ressure signature of waveackets in turbulent subsonic jets. These signatures were educed from the exerimental data measured at the NASA SHJAR facility by filtering with roer orthogonal decomosition (POD). The very good agreement found between the ressure imrint of linear PSE modes at the location of a conical microhone array with the POD-filtered data suggests that the main nonlinearities existing in turbulent jet flow are those between the mean flow and the small-scale random turbulence, as suggested by Mankbadi & Liu,9 and are already accounted for in taking the turbulent mean flow as the base for the models. Present results show that, with some caveats, this is also aears to be the case for suersonic turbulent jets. The resent work focuses on a Mach. cold jet and uses LES data to validate the PSE aroach. In alying these techniques to suersonic flow, there are two significant challenges. The first challenge is the existence of additional unstable waves aart from the Kelvin-Helmholtz mode that is the sole instability in subsonic jets. This new family of eigenmodes corresonds to ressure waves that resonate inside of the otential core: its members roagate downstream and ustream with resect to the jet stream forming a Mach wave system within the otential core. Similar instabilities have been found in the scoe of a global eigenmode analysis. Comuting PSE modes corresonding to this kind of instability was found to be, at least, roblematic. On one hand, the ustream roagating comonents are necessarily damed by the arabolic marching scheme. On the other hand, the dominant Kelvin-Helmholtz instability would overwhelm the unstable, downstream roagating ressure modes and the PSE aroach would converge to the Kelvin- of 9

15 First POD mode: uh =. First POD mode: uh =. Second POD mode: uh =.,. Linear PSE mode: uh =.9 Linear PSE mode: uh =.9 Second POD mode: uh =.,. (a) m = (b) m = Figure 9. Real comonents of ressure in the first and second lanar-pod modes comared with linear PSE mode at St =.. Helmholtz mode alone. A study of the relative contributions of the different instability modes to the total fluctuations at the inlet of the PSE integration domain was erformed using the bi-orthogonal eigenfunction system delivered by linear stability theory. It was found that the fluctuation s amlitude corresonding to the Kelvin-Helmholtz eigenmode was much higher that those associated with other instability waves, so that most of the fluctuation kinetic energy was contained in this mode alone. The conclusion is that for this articular jet configuration, the Kelvin-Helmholtz instability dominates over the core ressure waves, and the use of PSE as model for the large-scale hydrodynamic structures is justified. The other challenge osed by suersonic jets is the increased acoustic content of the near field, that hinders the extraction of the PSE-modeled hydrodynamic art. Comared to earlier exerimental data, the LES rovides greater wealth of information. One novel ossibility is a POD on the full ressure field (instead of restricting attention to the near-field microhone cage used at NASA SHJAR or UTRC), and this is ursued for the validation of the linear PSE model. The POD eigensectra are relatively flat for most Fourier modes, indicating that no single mode can be exected to retrieve the PSE solution from the unfiltered fluctuations. However, it is shown here that relatively few POD modes ( to ) are required to reconstruct the PSE mode at frequencies u to St =., thereby lending confidence to our modeling technique. At higher frequencies, the resolution on the interolation grid (used to reduce the LES data and make it tractable) roved insufficient to avoid aliasing. The POD modes also dislay significant acoustic roagating to the aft direction, that cannot be catured by the PSE for the reasons exlained above. In fact, for the St =., m = mode, which is close to the most efficient radiation condition, the first POD modes were required for reconstructing the PSE solution. The conclusion can be drawn that POD may not be otimal for extracting the hydrodynamic signatures of the waveackets that PSE is attemting to model, esecially as the relative imortance of acoustic radiation increases with jet temerature and Mach number. However, whenever the contributions from hydrodynamic and acoustic can be discerned, good agreement between the linear PSE model and the hydrodynamic comonent is found. This work was suorted by NAVAIR through an STTR grant N--C- to Cascade Technologies, Inc. and by ONR Grant N---7. The authors would like to thank Dr. Joseh W. Nichols and Prof. Sanjiva K. Lele, of Stanford University, for their contributions to the LES efforts and their inuts on global modes. The LES calculations were carried out on CRAY XE machines at DoD suercomuter of 9