Research

My research focuses on developing physics-based, reduced-order models that can be used to understand, predict, and control the turbulent flow of fluids in various engineering systems.  My approach revolves around identifying and modeling coherent flow structures, i.e., organized motions within otherwise chaotic flows.  These structures provide building-blocks for an improved theoretical understanding of turbulence and also contribute disproportionately to engineering quantities of interest such as drag, heat transfer, and noise emission.  Consequently, strategically manipulating coherent structures can potentially lead to vast performance improvements in a wide range of engineering applications.  Realizing this goal will require the development of accurate, robust reduced-order models that describe the coherent structures to provide a basis for modern control strategies.  I approach model development using a diverse set of tools and ideas from a wide range of disciplines, including hydrodynamic stability theory, advanced large-scale data mining, statistics, applied and computational mathematics, and optimal control theory; a recurring theme in my work is a symbiotic relationship between basic theory and experimental and numerical data.

Some of my main projects are described below.

  • Aeroacoustics / Jet noise
  • Reduced-order models
  • Hydrodynamic instability
  • One-way equations
  • Data decomposition
  • Flow control
  • Numerical methods

Projects

  • Fast solution methods for linear PDEs via spatial marching

    Fast solution methods for linear PDEs via spatial marching

    Many of the models used to describe coherent flow-structures, including those often used for jet-noise, are formulated in terms of linear hyperbolic or mixed hyperbolic/parabolic partial differential equations.  Standard solution techniques for these equations are computationally intensive, making fast, approximate solution methods critical if the models are to be used for design, optimization, or control.  The parabolized stability equations, a regularized form of the flow equations that are solved via spatial integration in the downstream direction, have been used extensively in this regard and are ubiquitous in the greater fluid mechanics community (Herbert 1997).  Despite over thirty years of heavy use, the errors produced by this method have not been mathematically analyzed.  I performed a general analysis that quantifies the error introduced by the approximation in terms of spectral properties of the underlying equations.  The results explained for the first time a number of observations described in the literature and explicitly revealed limitations of the method.

    Then, I developed a new spatial integration method that overcomes many of these limitations.  Using ideas originally developed for constructing high-order non-reflecting boundary conditions, the flow variables are decomposed into upstream and downstream propagating waves, and an approximate evolution equation is derived for the downstream waves.  The method is formally well-posed (unlike parabolized stability equations) and is typically more than an order-of-magnitude faster than traditional methods while achieving a similar level of accuracy.  The method is formulated for general hyperbolic equations (and can also be applied to mixed hyperbolic/parabolic equations under mild assumptions) and thus has potential applications in many fields of physics and engineering.  When applied to linearized flow equations, the method accurately computes shear-flow instability modes that lead to coherent flow-structures and their associated acoustic radiation at a fraction of the computational-cost of previous methods.

    Publications:

    Towne, A. and Colonius, T.  (2015).  One-way spatial integration of hyperbolic equations.  Journal of Computational Physics, Vol. 300.

    Towne, A.  (2016).  Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets, Chapter 2. Ph.D. Thesis, California Institute of Technology.

    Towne, A., Rigas, G. and Colonius, T.  A critical assessment of the parabolized stability equations.  (In prep.)

  • Model reduction for turbulent flows: spectral POD, DMD, and resolvent analysis

    Model reduction for turbulent flows: spectral POD, DMD, and resolvent analysis

    High fidelity simulations and modern experimental techniques such as stereoscopic particle image velocimetry produce enormous data sets; extracting useful information from these data sets is one of the chief challenges in fluid mechanics and computational science in general.  Modal decomposition techniques attempt to educe a set of modes from the data that compactly describe the essential features of the flow, and in particular coherent structures.  Two workhorse methods, a spatial form of proper orthogonal decomposition (POD, Sirovich 1987) and dynamic mode decomposition (DMD, Schmid 2010), have dominated the field in recent years, but I recently showed that neither of these methods are ideal for identifying physically meaningful coherent structures in turbulent flows; POD modes do not describe space-time coherent structures while DMD modes represent one random realization of coherent structures in turbulent flows.  Instead, I showed that the optimal method for describing coherent structures in turbulent flows is given by a different form of POD, sometimes called spectral proper orthogonal decomposition (SPOD, Lumley 1967), that has been largely ignored since the introduction of the other methods.  In fact, SPOD combines the advantages of both POD and DMD; SPOD modes are optimally averaged DMD modes that best describe the statistical variability of coherent structures in turbulent flows.

    Additionally, I establish a connection between SPOD and resolvent analysis (McKeon & Sharma 2010). The key observation is that the resolvent-mode expansion coefficients, which are usually treated as deterministic quantities described by an amplitude and phase, must 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, I showed that SPOD and resolvent modes are identical.  These theoretical results have already led to an improved understanding of coherent structures in turbulent jets and pave the way for new statistical models that could lead to improved control of a variety of turbulent flows.

    Publications:

    Towne, A., Schmidt, O. T., Colonius, T.  (2017)  Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis.  (SubmittedarXiv:1708.04393).

    Towne, A., Brès, G. A. and Lele, S. K.  (2016).  Toward a resolvent-based statistical jet-noise model.  Annual Research Briefs, Center for Turbulence Research, Stanford University.

    Towne, A., Colonius, T., Jordan, P., Cavalieri, A.V.G., Brès, G. A.  (2015).  Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet.  AIAA Paper 2015-2217.

    Schmidt, O. T., Towne, A.,  Rigas, G., Colonius, T. and Brès, G. A.  On the low-rank behavior in turbulent jets.  (In prep.)

     

     

     

  • Statistical jet-noise models based on stochastic forcing of coherent wavepacket structures

    Statistical jet-noise models based on stochastic forcing of coherent wavepacket structures

    Numerous studies have identified large-scale coherent flow-structures within turbulent jets as an important source of noise (Jordan & Colonius 2013).  The average statistics of these coherent structures have been accurately predicted by purely linear models, but the associated far-field noise has been shown to be sensitive to second-order flow statistics that are not properly reproduced by these models, resulting in significant under-prediction of far-field noise.  To address this, I used data from a high-fidelity numerical simulation of a turbulent jet to elucidate the nature and role of the nonlinear terms that are neglected in linear models.  Using data-decomposition techniques, including a novel method (called empirical resolvent-mode decomposition) that identifies structures that are most easily excited, I demonstrated the key role of random turbulent fluctuations in amplifying the acoustic efficiency of large-scale coherent flow-structures.  This improved physical understanding of the noise-production mechanism paves the way for new models that account for these effects and in particular suggests a stochastic representation of the turbulent forcing.

    Motivated by these findings and the theoretical modal decomposition results described earlier, I formulated a new resolvent-based model designed to capture the full second-order statistics of wavepackets in 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 work was to characterize the properties of these statistics in a high-Reynolds-number subsonic jet. I showed that the power spectral density of the forcing is independent of frequency over a range of almost two orders-of-magnitude and that the coherence of the forcing consists of peaks that are spatially compact compared to the coherence length-scales of the flow variables.  These properties makes the forcing statistics amenable to modeling, and I proposed a simple fit function in frequency space that captures the essential characteristics. Some of the parameters in the model are well-approximated by quantities that could be obtained from a Reynolds-averaged Navier-Stokes simulation, which provides a path forward for making the model predictive.

    Publications:

    Towne, A., Colonius, T., Jordan, P., Cavalieri, A.V.G., Brès, G. A.  (2015).  Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet.  AIAA Paper 2015-2217.

    Towne, A.  (2016).  Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets, Chapter 3. Ph.D. Thesis, California Institute of Technology.

    Towne, A., Brès, G. A. and Lele, S. K.  (2016).  Toward a resolvent-based statistical jet-noise model.  Annual Research Briefs, Center for Turbulence Research, Stanford University.

    Towne, A., Brès, G. A. and Lele, S. K.  (2017).  A statistical jet-noise model based on the resolvent framework.  AIAA Paper 2017-3706.

     

  • Acoustic resonance in subsonic jets

    Acoustic resonance in subsonic jets

    I discovered a new instability mode that describes acoustic waves that are trapped within the core of subsonic jets.  At certain frequencies, these waves resonate between the nozzle-exit and a downstream end-condition associated with the streamwise contraction of the jet core.  This downstream end-condition is mathematically described by two saddle-points in the complex wavenumber plane that represent a reflection-transmission process within a weakly-nonparallel spatiotemporal model of the jet.  Extensive comparisons with simulation and experimental data confirm that this mechanism is active in real turbulent jets.

    Furthermore, when a solid edge (representing some aircraft component, e.g., the trailing edge of the wing) is placed in close proximity to the jet, the trapped waves participate in a second type of resonance that produces deafening tones (up to 170 dB!) that could not be explained prior to the discovery of the new instability mode.  The parametric properties of the instability mode provide guidelines for controlling or altogether avoiding these tones in practice, which is especially relevant for next-generation aircraft designs in which the engine is tightly integrated with the airframe.  For this work, I was awarded the Best Student Paper in Aeroacoustics award by the American Institute of Aeronautics and Astronautics and the Council of European Aerospace Societies.

    Publications:

    Towne, A., Cavalieri, A. V. G., Jordan, P., Colonius, T., Jaunet, V., Schmidt, O.T., and Brès, G. A.  (2017).  Acoustic resonance in the potential core of subsonic jets.  Journal of Fluid Mechanics, Vol. 825, pp. 1113—1152.

    Schmidt, O. T., Towne, A.,  Colonius, T., Cavalieri , A. V. G., Jordan, P. and Brès, G. A.  (2017).  Wavepackets and trapped acoustic modes in a Mach 0.9 turbulent jet: a global stability analysis.  Journal of Fluid Mechanics, Vol. 825, pp. 1153—1181.

    Towne, A., Cavalieri, A. V. G., Jordan, P., Colonius, T., Jaunet, V., Schmidt, O. T., and Brès, G. A.  (2016).  Trapped acoustic waves in the potential core of subsonic jets.  AIAA Paper 2016-2809(Best paper award)

    Jordan, P., Jaunet, V., Towne, A., Cavalieri, A. V. G., Colonius, T.,  Schmidt, O. T., and Agarwal A.  Jet-edge interaction tones. (Submitted)

  • Active noise control of supersonic jets

    Active noise control of supersonic jets

    The high noise levels generated by the supersonic jet exhaust from military tactical aircraft are disruptive in communities adjacent to military bases and dangerous to airstrip and flight deck personnel. Strict design and performance requirements for military tactical aircraft preclude the use of high bypass ratio engines that have significantly reduced noise emission from commercial aircraft require innovative noise reduction strategies that subtly alter the jet.  In a series of experiments performed in collaboration with United Technologies Research Center, I investigated an active, deployable approach based on steady and unsteady near-nozzle forcing that could be switched off during mission-critical portions of the flight to avoid performance penalties.  The control strategy is designed to disrupt noise-generating coherent flow-structures, and noise reductions of up to 4 dB in the peak aft-angle direction were achieved.  Using tailored data analysis techniques to distill the influence of the control, I demonstrated that the noise benefit is uniquely associated with the steady component of blowing, with unsteady blowing yielding additional tones.  These findings provide a compelling demonstration of the essential linearity of the dominant noise-producing structures within the turbulent jet and inform future noise-control strategies.

    Publications:

    Sinha, A., Towne, A., Colonius, T., Schlinker, R. H., Reba, R., Simonich, J. S., Shannon, D. W. and Teerlinck, K.A.  (2017).  Active control of noise from hot supersonic jets.  AIAA Journal. (Accepted)

  • Simulation and modeling of laminar boundary layer instability noise

    Simulation and modeling of laminar boundary layer instability noise

    Separated boundary layers on airfoils at moderate Reynolds numbers lead to both broad-band and tonal noise components.  In collaboration with a team from l’Université de Sherbrooke, I am investigating the mechanisms underlying these two noise components using direct numerical simulation and reduced-order modeling.  In particular, we are using statistical analysis coupled with spatial and spatiotemporal linear stability theory to confirm previously proposed instability and resonance mechanism responsible for the two noise components.

    Publications:

    Sanjose, M., Jaiswal, P., Arroyo, C.P., Moreau, S., Towne, A., Lele, S. K. and Mann, A.  (2017).  Direct numerical simulation of laminar boundary layer instability noise.  AIAA Paper 2017-3190. 

    Sanjose, M., Jaiswal, P., Moreau, S., Towne, A. and Lele, S. K.  (2016).  Laminar boundary layer instability noise.  Proceedings of the Summer Program, Center for Turbulence Research, Stanford University.