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Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

Journal Article
Aaron Towne, Oliver T. Schmidt, Tim Colonius
Submitted -- arXiv:1708.04393

Abstract

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). SPOD is derived from a space-time POD problem for stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time while space-only POD modes do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients, which are usually treated as deterministic quantities described by an amplitude and phase, should be regarded as statistical quantities, described by their cross-spectral density, in order for the resolvent-mode expansion to properly capture the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet.

Jet-edge interaction tones

Journal Article
P. Jordan, V. Jaunet, A. Towne, A.V.G. Cavalieri, T. Colonius, O.T. Schmidt, A. Agarwal
Submitted -- arXiv:1710.07578

Abstract

Motivated by the problem of jet-flap interaction noise, we study the tonal dynamics that occur when a sharp edge is placed in the hydrodynamic nearfield of an isothermal turbulent jet. We perform hydrodynamic and acoustic pressure measurements in order to characterise the tones as a function of Mach number and streamwise edge position. The distribution of spectral peaks observed, as a function of Mach number, cannot be explained using the usual edge-tone scenario, in which resonance is underpinned by coupling between downstream-travelling Kelvin-Helmholtz wavepackets and upstream-travelling sound waves. We show, rather, that the strongest tones are due to coupling between the former and upstream-travelling jet modes recently studied by Towne et al. (2017) and Schmidt et al. (2017). We also study the band-limited nature of the resonance, showing a high-frequency cut-off to be due to the frequency dependence of the upstream-travelling waves. At high Mach number these become evanescent above a certain frequency, whereas at low Mach number they become progressively trapped with increasing frequency, a consequence of which is their not being reflected in the nozzle plane. Additionally, a weaker, low-frequency, forced-resonance regime is identified that involves the same upstream travelling jet modes but that couple, in this instance, with downstream-travelling sound waves. It is suggested that the existence of two resonance regimes may be due to the non-modal nature of wavepacket dynamics at low-frequency.

Active control of noise from hot supersonic jets

Journal Article
Aniruddha Sinha, Aaron Towne, Tim Colonius, Robert H. Schlinker, Ramons Reba, John S. Simonich, and Daniel W. Shannon
AIAA Journal (in press)

Abstract

We present diagnostic experiments aimed at understanding and mitigating supersonic jet noise from coherent wavepacket structures that are the source of peak aft-angle mixing noise. Both isothermal and heated, nearly perfectly-expanded Mach 1.5 jets were forced in the near-nozzle region with air injection generated by a spinning-valve device designed to excite the jet at frequencies approaching those of the dominant turbulent structures. Substantial reductions in the peak aft-angle radiation were achieved with steady blowing at amplitudes corresponding to 2-6% of the mass flow rate of the primary jet. The noise benefit saturates at mass flow rates above 4%, with as much
as 6 dB reduction in OASPL at aft angles. Increasing mass flow rates yield a monotonically increasing high-frequency noise penalty at sideline, where noise levels in the natural jet are already 15 dB lower than the aft-angle peak, so that the penalty due to actuation is minor. Although both steady and periodic unsteady mass injection are produced by the spinning valve when it rotates, it was calibrated to hold the steady mass flow rate constant as the frequency of unsteady blowing was changed. In this way, the effect of steady and unsteady blowing on the acoustic field could be decoupled.  We show that the noise benefit is uniquely associated with the steady component of blowing, whereas the unsteady component resulted in additive tones in the spectra.  This implied linearity is consistent with theory and experiments showing that the wavepacket structures, which give rise to the dominant aft-angle radiation, evolve in the turbulent mean flow field in a nearly linear fashion from their origin in the near nozzle region. The interpretation of noise reduction is that the steady component of blowing spreads the mean flow more rapidly resulting in weaker wavepackets. Periodic unsteady blowing forces coherent wavepackets that are largely uncorrelated from the random natural ones, which then leads to the observed additive tones.

Acoustic resonance in the potential core of subsonic jets

Journal Article
Aaron Towne, André V. G. Cavalieri, Peter Jordan, Tim Colonius, Oliver Schmidt, Vincent Jaunet, Guillaume A. Brès (a4)
Journal of Fluid Mechanics, Vol. 825, 2017, Pages 1113-1152

Abstract

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and relate them to previously observed tones in the near-nozzle region. The waves are detected in data from a large-eddy simulation of a Mach 0.9 isothermal jet and modelled using parallel and weakly non-parallel linear modal analysis of the Euler equations linearized about the turbulent mean flow, as well as simplified models based on a cylindrical vortex sheet and the acoustic modes of a cylindrical soft duct. In addition to the Kelvin–Helmholtz instability waves, three types of waves with negative phase velocities are identified in the potential core: upstream- and downstream-propagating duct-like acoustic modes that experience the shear layer as a pressure-release surface and are therefore radially confined to the potential core, and upstream-propagating acoustic modes that represent a weak coupling between the jet core and the free stream. The slow streamwise contraction of the potential core imposes a frequency-dependent end condition on the waves that is modelled as the turning points of a weakly non-parallel approximation of the waves. These turning points provide a mechanism by which the upstream- and downstream-travelling waves can interact and exchange energy through reflection and transmission processes. Paired with a second end condition provided by the nozzle, this leads to the possibility of resonance in limited frequency bands that are bound by two saddle points in the complex wavenumber plane. The predicted frequencies closely match the observed tones detected outside of the jet. The vortex-sheet model is then used to systematically explore the Mach number and temperature ratio dependence of the phenomenon. For isothermal jets, the model suggests that resonance is likely to occur in a narrow range of Mach number, 0.82<M<1.

Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability

Journal Article
Oliver T. Schmidt, Aaron Towne, Tim Colonius, André V. G. Cavalieri, Peter Jordan, Guillaume A. Brès
Journal of Fluid Mechanics, Vol. 825, 2017, Pages 1153-1181

Abstract

Coherent features of a turbulent Mach 0.9, Reynolds number jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

Direct numerical simulation of laminar boundary layer instability noise

Conference Paper
M. Sanjosé, P. Jaiswal, C. Pérez Arroyo, S. Moreau, A. Towne, S.K. Lele, A. Mann
Proceedings of the 23rd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2017-3190, 2017

Abstract

A direct numerical simulation with a Lattice Boltzmann Method is performed of the flow field around a modern controlled-diffusion airfoil within an anechoic wind-tunnel at 5 incidence and a high Reynolds number of 1.5e5. The simulation compares favorably with experimental measurements of wall-pressure, wake statistics, and far-field sound. The temporal evolution of wall-pressure fluctuations shows significant unsteadiness especially in the aft region of the suction side. A wavelet analysis allows identifying quiet and intense periods within the simulation. In the former periods the boundary flow remains attached on the suction while in the latter an instable recirculation bubble forms around 65-70% of the chord. Kelvin-Helmholtz instabilities in the separated shear layer yield rollers that break down into turbulent vortices whose diffraction at the trailing edge produces a strong dipole acoustic field. A linear stability analysis of the mean flow
field around the airfoil identifies convective instability in the aft portion of the airfoil where this shedding occurs for frequencies covering the broadband hump, and also provides estimates of the tonal frequencies.

A statistical jet-noise model based on the resolvent framework

Conference Paper
A. Towne, G.A. Brès, S.K. Lele
Proceedings of the 23rd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2017-3706, 2017

Abstract

This paper continues the development of a recently proposed resolvent-based model designed to capture the full second-order statistics of turbulent jets, which are required to obtain accurate noise estimates. The model requires an approximation of the cross-spectral density tensor of certain nonlinear forcing terms, and the focus of this paper is to characterize the properties of these statistics in a high-Reynolds-number subsonic jet. We show that the power spectral density of the forcing is independent of frequency over a range of almost two orders-of-magnitude. The coherence of the forcing consists of peaks that are spatially compact compared to the coherence length-scales of the flow variables. The widths of these peaks depend on spatial location but not frequency, while the streamwise and radial wavelengths of the coherence depend on frequency but not spatial location. We propose a simple fit function in frequency space that captures these properties and show that it leads to good approximations of the LES forcing statistics. Some of the parameters in the model are well-approximated by quantities that could be obtained from a Reynolds-averaged Navier-Stokes simulation. Finally, we show that the properties of the forcing statistics are completely different for a low-Reynolds-number jet, which may be indicative of direct nonlinear interactions amongst wavepackets which are absent in the high-Reynolds-number jet.

Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets

Thesis
A. Towne
Ph.D. Thesis, California Institute of Technology, 2016

Abstract

Jet noise reduction is an important goal within both commercial and military aviation. Although large-scale numerical simulations are now able to simultaneously compute turbulent jets and their radiated sound, lost-cost, physically-motivated models are needed to guide noise-reduction efforts. A particularly promising modeling approach centers around certain large-scale coherent structures, called wavepackets, that are observed in jets and their radiated sound. The typical approach to modeling wavepackets is to approximate them as linear modal solutions of the Euler or Navier-Stokes equations linearized about the long-time mean of the turbulent flow field. The near-field wavepackets obtained from these models show compelling agreement with those educed from experimental and simulation data for both subsonic and supersonic jets, but the acoustic radiation is severely under-predicted in the subsonic case. This thesis contributes to two aspects of these models. First, two new solution methods are developed that can be used to efficiently compute wavepackets and their acoustic radiation, reducing the computational cost of the model by more than an order of magnitude. The new techniques are spatial integration methods and constitute a well-posed, convergent alternative to the frequently used parabolized stability equations. Using concepts related to well-posed boundary conditions, the methods are formulated for general hyperbolic equations and thus have potential applications in many fields of physics and engineering. Second, the nonlinear and stochastic forcing of wavepackets is investigated with the goal of identifying and characterizing the missing dynamics responsible for the under-prediction of acoustic radiation by linear wavepacket models for subsonic jets. Specifically, we use ensembles of large-eddy-simulation flow and force data along with two data decomposition techniques to educe the actual nonlinear forcing experienced by wavepackets in a Mach 0.9 turbulent jet. Modes with high energy are extracted using proper orthogonal decomposition, while high gain modes are identified using a novel technique called empirical resolvent-mode decomposition. In contrast to the flow and acoustic fields, the forcing field is characterized by a lack of energetic coherent structures. Furthermore, the structures that do exist are largely uncorrelated with the acoustic field. Instead, the forces that most efficiently excite an acoustic response appear to take the form of random turbulent fluctuations, implying that direct feedback from nonlinear interactions amongst wavepackets is not an essential noise source mechanism. This suggests that the essential ingredients of sound generation in high Reynolds number jets are contained within the linearized Navier-Stokes operator rather than in the nonlinear forcing terms, a conclusion that has important implications for jet noise modeling.

Trapped acoustic waves in the potential core of subsonic jets

Conference Paper
A. Towne, A.V.G. Cavalieri, P. Jordan, T. Colonius, V. Jaunet, O.T. Schmidt, G.A. Brès
Proceedings of the 22nd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2016-2809, 2016

Abstract

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and that have been previously detected as tones in the near-nozzle region. Using three models (the linearized Euler equations, a cylindrical vortex sheet, and a cylindrical duct with pressure release boundary conditions), we show that these waves can be described by linear modes of the jet and correspond to acoustic waves that are trapped within the potential core. At certain frequencies, these trapped waves resonate due to repeated reflection between end conditions provided by the nozzle and the streamwise contraction of the potential core. Our models accurately capture numerous aspects the potential core waves that are extracted from large-eddy-simulation data of a Mach 0.9 isothermal jet. Furthermore, the vortex sheet model indicates that this behavior is possible for only a limited range of Mach numbers that is consistent with previous experimental observations.

Super- and multi-directive acoustic radiation by linear global modes of a turbulent jet

Conference Paper
O. T. Schmidt, A. Towne, T. Colonius, P. Jordan, V. Jaunet, A.V.G. Cavalieri, G.A. Brès
Proceedings of the 22nd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2016-2808, 2016

Abstract

The mean flow stability of a Mach 0.9 turbulent jet is investigated by means of global
linear theory with a focus on acoustic effects. A novel class of resonant acoustic modes that
are trapped within the potential core, and whose eigenvalues appear as discrete branches
in the global stability spectrum, is studied in detail. A dispersion relation is reconstructed
from the global modes, and shown to accurately predict energy bands observed in the PSD
of a high-fidelity LES. Similarly, the acoustic far-field radiation patterns of the trapped
modes are compared to the LES. A favorable agreement between the global mode waveforms
and coherent structures educed from the LES is found for both the trapped acoustic wave
component inside the core and the far-field radiation.

Tonal dynamics and sound in free and installed turbulent jets

Conference Paper
V. Jaunet, P. Jordan, A.V.G. Cavalieri, A. Towne, T. Colonius, O.T. Schmidt, G.A. Brès
Proceedings of the 22nd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2016-3016, 2016

Abstract

Acoustic waves trapped in the potential core of subsonic turbulent jets have recently been observed and explained by Towne et al. We show that these waves also radiate outside the jet, primarily into the upstream arc. We provide an experimental identification of the Mach-number dependence of the phenomenon, which indicates that the modes are active even when evanescent, probably due to turbulent forcing. Finally, we show that for Mach numbers lower than about 0.8, the strong tonal dynamics and sound radiation (up to 170dB) that occur when a sharp edge is placed close to the jet are related to a resonance mechanism involving convective hydrodynamic wavepackets and a `slow’, upstream-propagating, trapped acoustic mode. A Helmholtz scaling of the resonance at higher Mach number suggests involvement of the `fast’ trapped modes in the range 0.8 < M < 1.

Large eddy simulation for jet noise: azimuthal decomposition and intermittency of the radiated sound

Conference Paper
G.A. Brès, V. Jaunet, M. Le Rallic, P. Jordan, A. Towne, O.T. Schmidt, T. Colonius, A.V.G. Cavalieri, S.K. Lele
Proceedings of the 22nd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2016-3050, 2016

Abstract

To improve understanding and modeling of jet-noise source mechanisms, extensive experimental and numerical databases are generated for an isothermal Mach 0.9 turbulent jet at Reynolds number Re = 10e6.  The large eddy simulations (LES) feature localized adaptive mesh refinement, synthetic turbulence and wall modeling inside the nozzle to match the fully turbulent nozzle-exit boundary layers in the experiments. Long LES databases are collected for two grids with different mesh resolutions in the jet plume. Comparisons with the experimental measurements show good agreement for the flow and sound predictions, with the far-field noise spectra matching microphone data to within 0.5 dB for mos trelevant angles and frequencies. Preliminary results on the radiated noise azimuthal decomposition and temporal intermittency are also discussed. The azimuthal analysis shows that the axisymmetric mode is dominant at the peak radiation angles and that the first3 Fourier azimuthal modes of the LES data recover more than 97% of the total acoustic energy at these angles. The temporal analysis highlights the presence of recurring intermittency in the radiated sound for the low-frequency range and main downstream angles.  At these frequencies and angles, temporally-localized bursts of noise can reach levels up to3 or 4 dB higher (or lower) than the long-time average.

Toward a resolvent-based statistical jet-noise model

Technical Report
A. Towne, G.A. Brès, S.K. Lele
Annual Research Briefs, Center for Turbulence Research, Stanford University, 2016

Toward low order models of wall turbulence using resolvent analysis

Technical Report
K. Rosenberg, T. Saxton-Fox, A. Lozano-Durán, A. Towne. B.J. McKeon
Proceedings of the Summer Program, Center for Turbulence Research, Stanford University, 2016

Abstract

Resolvent analysis for wall turbulence has the potential to provide a physical basis for both sub-grid scale and dynamic wall models for large-eddy simulations (LES), and an explicit representation of the interface between resolved and modeled scales. Toward the development of such a wall model, direct numerical simulation results are used to represent the Reynolds stresses, formulated as the nonlinear feedback (forcing) to the linear(ized) Navier-Stokes equations. It is found from direct calculation of the Reynolds stress gradients that the (solenoidal) nonlinear feedback is coherent and consistent with energetic activity that is localized in the wall-normal direction. Further, there exists a spatial organization of this forcing that is correlated with individual (large) scales. A brief outlook for LES modeling is given.

Laminar boundary layer instability noise

Technical Report
M. Sanjose, S. Moreau, A. Towne, S.K. Lele
Proceedings of the Summer Program, Center for Turbulence Research, Stanford University, 2016

Abstract

A direct numerical simulation is performed of the ow field around a modern controlled-di usion airfoil within an anechoic wind-tunnel at 5 incidence and a high Reynolds number of 1.5e5. The simulation compares favorably with experimental measurements of wall pressure, wake statistics, and far- field sound. In particular, the simulation captures experimentally observed high-amplitude acoustic tones that rise above a broadband hump. Both noise components are related to breathing from a recirculation bubble formed around 65-70% of the chord, and to Kelvin-Helmholtz instabilities in the separated shear layer that yield rollers that break down into turbulent vortices whose diffraction at the trailing edge produces a dipole acoustic field. A linear stability analysis of the mean flow field around the airfoil identifies convective instability in the aft portion of the airfoil where this shedding occurs for frequencies covering the broadband hump, and also provides estimates of the tonal frequencies.

Prediction of supersonic jet noise using non-parallel flow asymptotics and LES data within Goldstein’s acoustic analogy

Technical Report
M.Z. Afsar, A. Sescu, V. Sassanis, A. Towne, G.A. Brès, S.K. Lele
Proceedings of the Summer Program, Center for Turbulence Research, Stanford University, 2016

Abstract

In this study we show how accurate jet noise predictions can be achieved within Goldstein’s generalized acoustic analogy formulation for heated and unheated supersonic jets using a previously developed asymptotic theory for the adjoint vector Green’s function. In this approach, mean flow non-parallelism enters the leading order dominant balance producing enhanced amplification at low frequencies, which we believe corresponds to the peak sound at small polar observation angles. We determine all relevant mean flow and turbulence quantities using Large Eddy Simulations of two axi-symmetric round jets at fixed jet Mach number and different nozzle temperature ratios. Certain empirical coefficients that enter the turbulence length scales formula are tuned for good agreement with the far-field noise data. Our results indicate that excellent jet noise predictions are obtained using the asymptotic approach, remarkably, up to a Strouhal number of 0.5.

One-way spatial integration of hyperbolic equations

Journal Article
A. Towne, T. Colonius
Journal of Computational Physics, Vol. 300, 2015, Pages 844-861

Abstract

In this paper, we develop and demonstrate a method for constructing well-posed one-way approximations
of linear hyperbolic systems. We use a semi-discrete approach that allows the method to be applied
to a wider class of problems than existing methods based on analytical factorization of idealized dispersion
relations. After establishing the existence of an exact one-way equation for systems whose coefficients
do not vary along the axis of integration, efficient approximations of the one-way operator are constructed
by generalizing techniques previously used to create nonreflecting boundary conditions. When physically
justified, the method can also be applied to systems with slowly varying coefficients in the direction of
integration. To demonstrate the accuracy and computational efficiency of the approach, the method is applied
to model problems in acoustics and fluid dynamics via the linearized Euler equations; in particular we
consider the scattering of sound waves from a vortex and the evolution of hydrodynamic wavepackets in a
spatially evolving jet. The latter problem shows the potential of the method to offer a systematic, convergent
alternative to ad hoc regularizations such as the parabolized stability equations.

Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet

Conference Paper
A. Towne, T. Colonius, P. Jordan, A. Cavalieri, G. A. Brès
Proceedings of the 21st AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2015-2217, 2015

Abstract

Recent studies have shown that while linear wavepacket models accurately reproduce experimentally observed, low azimuthal-wavenumber pressure fluctuations in the near field of turbulent jets, they significantly under-predict the intensity of the acoustic radiation produced in the subsonic case. In a linear context, “jittering” of the wavepackets, which can arise due to both stochastic and nonlinear interactions that force the wavepackets, has been hypothesized as a mechanism by which the radiation efficiency of wavepackets is greatly increased. We use data from a carefully validated large-eddy-simulation of a Mach 0.9 turbulent jet to explore this hypothesis. We analyze the LES data in frequency space using windowed segments of a set of snapshots spanning two thousand acoustic time units. We apply the linearized Navier-Stokes operator to this data in order to compute the nonlinear forcing field that occurred in the LES simulations, and propose several techniques for educing the relation between the forcing and the observed flow fields. In particular, we employ empirical techniques to identify high energy modes (via proper orthogonal decomposition) in both the flow and acoustic fields, as well as a set of empirical resolvent modes that maximize either the gain between the forcing and flow fields, or the gain between the forcing and acoustic fields. The high gain modes are similar to the high energy modes in both cases, suggesting that the forcing fields are nearly uncorrelated in each realization. Both flow and acoustic fields appear to be driven by largely incoherent forcing corresponding to turbulence in the region of strong shear and, in particular, close to the critical layer. With the caveat that we have thus far only analyzed the axisymmetric mode of the disturbance fields, the results suggest that accurate linear wavepacket models that capture both the coherent flow and acoustic fields can be constructed if appropriate parameterizations of the stochastic forcing can be found, i.e. such forcings will excite the high gain modes to produce the observed coherent structures in both the near and far field.

 

Modeling intermittent wavepackets and their radiated sound in a turbulent jet

Technical Report
P. Jordan, T. Colonius , G.A. Brès, M. Zhang, A. Towne, and S.K. Lele
Proceedings of the 2014 Center for Turbulence Research Summer Program, CRT-S14, 2014, Pages 241-249

Abstract

We use data from a new, carefully validated, Large Eddy Simulation (LES) to investigate and model subsonic, turbulent jet noise. Motivated by the observation that sound-source dynamics are dominated by instability waves (wavepackets), we examine mechanisms by which their intermittency can amplify their noise radiation. Two scenarios, both involving wavepacket evolution on time-dependent base flows, are investigated.  In the first, we consider that the main effect of the changing base flow consists in different wavepacket ensembles seeing different steady mean fields, and having, accordingly, different acoustic efficiencies. In the second, the details of the base-flow time dependence also play a role in wavepacket sound production. Both short-time-averaged and slowly varying base flows are extracted from the LES data and used in conjunction with linearized wavepacket models, namely, the Parabolized Stability Equations (PSE), the One-Way Euler Equations (OWE), and the Linearized Euler Equations (LEE). All results support the hypothesized mechanism: wavepackets on time-varying base flows produce sound radiation that is enhanced by as much as 20dB in comparison to their long-time-averaged counterparts, and ensembles of wavepackets based on short-time-averaged base flows display similar amplification. This is not, however, sufficient to explain the sound levels observed in the LES and experiments. Further work is therefore necessary to incorporate two additional factors in the linear models, body forcing by turbulence and realistic inflow forcing, both of which have been identified as potentially important in producing the observed radiation efficiency.

Continued development of the one-way Euler equations: application to jets

Conference Paper
A. Towne, T. Colonius
Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2014-2903, 2014

Abstract

An efficient method for calculating linearized disturbances to shear flows that accurately captures their acoustic radiation was recently introduced (Towne & Colonius, AIAA Paper 2013-2171, 2013). The linearized Euler equations are modified such that all upstream propagating acoustic modes are removed from the operator. The resulting equations, called one-way Euler equations, can be stably and efficiently solved in the frequency domain as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integration of the equations. In this paper, we continue the development of this method with the aim of using it to model wavepackets and their acoustic radiation in turbulent jets.  Before turning attention to jets, two dimensional mixing layer noise results computed using the one-way Euler equations are shown to be in excellent agreement with a direct solution of the full Euler equations.  The one-way Euler operator is then shown to accurately represent all downstream modes that exist in supersonic and subsonic parallel jets, while properly eliminating the upstream acoustic modes.  Finally, the method is applied to a turbulent Mach 0.5 jet mean flow obtained from experimental measurements.  The near-field one-way Euler results are similar to those obtained using a previous spatial marching technique called the parabolized stability equations.  However, the one-way Euler solutions also include the acoustic fields.  With further development, the results suggest that the one-way Euler equation could be used to obtain improved accuracy over the parabolized stability equations as a low-order jet noise model.

Simulation and modeling of turbulent jet noise

Conference Paper
T. Colonius, A. Sinha, D. Rodríguez, A. Towne, J. Liu, G. A. Brès, D. Appelö, T. Hagstrom
Proceedings of the Direct and Large-Eddy Simulation 9 ERCOFTAC Workshop, 2013, Pages 305-310

Abstract

Jet noise reduction remains an important long-range goal in commercial and military aviation. Compared with their early counterparts, modern, ultrahigh-bypass-ratio turbofans on commercial aircraft are very quiet, but ever more stringent noise regulations dictate further reductions. In addition, hearing loss by personnel and community noise issues are prompting the military to seek noise reduction on future tactical aircraft. Further increase in bypass ratio not being a practical option, military applications in particular call for nuanced approaches to noise reduction including mixing devices like chevrons or even active noise control approaches using unsteady air injection. In this paper, we briefly review some recent developments in theoretical, experimental and computational approaches to understanding the sound radiated by large-scale, coherent structures in jet turbulence that might guide these noise reduction efforts.

Improved parabolization of the Euler equations

Conference Paper
A. Towne, T. Colonius
Proceedings of the 19th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2013-2171, 2013

Abstract

We present a new method for stability and modal analysis of shear flows and their acoustic radiation.  The Euler equations are modified and solved as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integration of the equations.  The modified equations, which we call one-way Euler equations, differ from the usual Euler equations in that they do not support upstream acoustic waves.  It is necessary to remove these modes from the Euler operator because, if retained, they cause instability in the spatial marching procedure.  These modes are removed using a two-step process.  First, the upstream modes are partially decoupled from the downstream modes using a linear similarity transformation.  Second, the error in the first step is eliminated using a convergent recursive filtering technique.  A previous spatial marching method called the parabolized stability equations uses numerical damping to stabilize the march, but this has the unintended consequence of heavily damping the downstream acoustic waves.  Therefore, the one-way Euler equation could be used to obtain improved accuracy over the parabolized stability equations as a low-order model for noise simulation of mixing layers and jets.

Heat transfer of supercritical carbon dioxide in printed circuit heat exchanger geometries

Journal Article
A. Kruizenga, M. Anderson, R. Fatima, M. Corradini, A. Towne, D. Ranjan
Journal of Thermal Science and Engineering Applications Vol. 3, 2011

Abstract

The increasing importance of improving efficiency and reducing capital costs has led to significant work studying advanced Brayton cycles for high temperature energy conversion. Using compact, highly efficient, diffusion-bonded heat exchangers for the recuperators has been a noteworthy improvement in the design of advanced carbon dioxide Brayton cycles. These heat exchangers will operate near the pseudocritical point of carbon dioxide, making use of the drastic variation of the thermophysical properties. This paper focuses on the experimental measurements of heat transfer under cooling conditions, as well as pressure drop characteristics within a prototypic printed circuit heat exchanger. Studies utilize type-316 stainless steel, nine channel, semi-circular test section, and supercritical carbon dioxide serves as the working fluid throughout all experiments. The test section channels have a hydraulic diameter of 1.16 mm and a length of 0.5 m. The mini-channels are fabricated using current chemical etching technology, emulating techniques used in current diffusion-bonded printed circuit heat exchanger manufacturing. Local heat transfer values were determined using measured wall temperatures and heat fluxes over a large set of experimental parameters that varied system pressure, inlet temperature, and mass flux. Experimentally determined heat transfer coefficients and pressure drop data are compared to correlations and earlier data available in literature. Modeling predictions using the computational fluid dynamics (CFD) package FLUENT are included to supplement experimental data. All nine channels were modeled using known inlet conditions and measured wall temperatures as boundary conditions. The CFD results show excellent agreement in total heat removal for the near pseudocritical region, as well as regions where carbon dioxide is a high or low density fluid.