## Publications

121 results found

Schmidt OT, Schmid PJ, 2019, A conditional space-time POD formalism for intermittent and rare events: example of acoustic bursts in turbulent jets, *JOURNAL OF FLUID MECHANICS*, Vol: 867, ISSN: 0022-1120

Yang D, Sogaro FM, Morgans AS,
et al., 2019, Optimising the acoustic damping of multiple Helmholtz resonators attached to a thin annular duct, *Journal of Sound and Vibration*, Vol: 444, Pages: 69-84, ISSN: 0022-460X

Helmholtz resonators (HRs) are widely used to damp acoustic oscillations, including in the combustors of aero-engines and power gas turbines where they damp thermoacoustic oscillations. The geometries of such combustors are often annular in shape, which means that low frequency acoustic modes exhibit both longitudinal and circumferential modeshapes, the latter across different circumferential wave numbers. For linear acoustic disturbances downstream of the flame, the presence of HRs leads to modal coupling and mode shape shifts, which makes design and placement of multiple HRs very complicated. A procedure which ensures that the design and placement of the HRs can be optimised for good acoustic damping performance would be very valuable, and such a procedure is presented in this work. A simplified linear, low-dimensional model for the acoustic behaviour in a hot annular duct sustaining a mean flow is extended to account for the attachment of multiple HRs. The HRs are assumed to sustain a cooling mean bias flow through them, towards the combustor, such that they can be modelled using linear, lumped element Rayleigh conductivity models. An optimisation method based on the gradient derived from adjoint sensitivity analysis is then applied to the low order network acoustic modelling framework for hot annular ducts incorporating HR models, for the first time. It is used to optimise over multiple HR geometry and placement parameters, to obtain optimum acoustic damping over all acoustic modes in a given frequency range. These optimisation procedures are validated via multi-dimensional parameter sweep results. Thus a novel and efficient tool for HR optimisation for thin annular ducts is presented.

Edstrand AM, Sun Y, Schmid PJ,
et al., 2018, Active attenuation of a trailing vortex inspired by a parabolized stability analysis, *JOURNAL OF FLUID MECHANICS*, Vol: 855, ISSN: 0022-1120

Skene CS, Schmid PJ, 2018, Adjoint-based parametric sensitivity analysis for swirling M-flames, *Journal of Fluid Mechanics*, Vol: 859, Pages: 516-542, ISSN: 0022-1120

A linear numerical study is conducted to quantify the effect of swirl on the response behaviour of premixed lean flames to general harmonic excitation in the inlet, upstream of combustion. This study considers axisymmetric M-flames and is based on the linearised compressible Navier–Stokes equations augmented by a simple one-step irreversible chemical reaction. Optimal frequency response gains for both axisymmetric and non-axisymmetric perturbations are computed via a direct–adjoint methodology and singular value decompositions. The high-dimensional parameter space, containing perturbation and base-flow parameters, is explored by taking advantage of generic sensitivity information gained from the adjoint solutions. This information is then tailored to specific parametric sensitivities by first-order perturbation expansions of the singular triplets about the respective parameters. Valuable flow information, at a negligible computational cost, is gained by simple weighted scalar products between direct and adjoint solutions. We find that for non-swirling flows, a mode with azimuthal wavenumber is the most efficiently driven structure. The structural mechanism underlying the optimal gains is shown to be the Orr mechanism for and a blend of Orr and other mechanisms, such as lift-up, for other azimuthal wavenumbers. Further to this, velocity and pressure perturbations are shown to make up the optimal input and output showing that the thermoacoustic mechanism is crucial in large energy amplifications. For these velocity perturbations are mainly longitudinal, but for higher wavenumbers azimuthal velocity fluctuations become prominent, especially in the non-swirling case. Sensitivity analyses are carried out with respect to the Mach number, Reynolds number and swirl number, and the accuracy of parametric gradients of the frequency response curve is assessed. The sensitivity analysis reveals that increases in Reynolds and Mach numbers yield higher gains, through a

Eggl MF, Schmid PJ, 2018, A gradient-based framework for maximizing mixing in binary fluids, *JOURNAL OF COMPUTATIONAL PHYSICS*, Vol: 368, Pages: 131-153, ISSN: 0021-9991

Schmid PJ, Garcia-Gutierrez A, Jimenez J, 2018, Description and detection of burst events in turbulent flows, Third Madrid Summer School on Turbulence, Publisher: Institute of Physics (IoP), ISSN: 1742-6588

A mathematical and computational framework is developed for the detection and identification of coherent structures in turbulent wall-bounded shear flows. In a first step, this data-based technique will use an embedding methodology to formulate the fluid motion as a phase-space trajectory, from which state-transition probabilities can be computed. Within this formalism, a second step then applies repeated clustering and graph-community techniques to determine a hierarchy of coherent structures ranked by their persistencies. This latter information will be used to detect highly transitory states that act as precursors to violent and intermittent events in turbulent fluid motion (e.g., bursts). Used as an analysis tool, this technique allows the objective identification of intermittent (but important) events in turbulent fluid motion; however, it also lays the foundation for advanced control strategies for their manipulation. The techniques are applied to low-dimensional model equations for turbulent transport, such as the self-sustaining process (SSP), for varying levels of complexity.

Edstrand AM, Schmid PJ, Taira K,
et al., 2018, A parallel stability analysis of a trailing vortex wake, *Journal of Fluid Mechanics*, Vol: 837, Pages: 858-895, ISSN: 0022-1120

Trailing vortices are generated in aeronautical and maritime applications and produce a variety of adverse effects that remain difficult to control. A stability analysis can direct flow control designers towards pertinent frequencies, wavelengths and locations that may lead to the excitation of instabilities, resulting in the eventual breakup of the vortex. Most models for trailing vortices, however, are far-field models, making implementation of the findings from stability analyses challenging. As such, we perform a stability analysis in the formative region where the numerically computed base flow contains both a two-dimensional wake and a tip vortex generated from a NACA0012 at a angle of attack and a chord-based Reynolds number of . The parallel temporal and spatial analyses show that at three chord lengths downstream of the trailing edge, seven unstable modes are present: three stemming from the temporal analysis and four arising in the spatial analysis. The three temporal instabilities are analogues to three unstable modes in the spatial analysis, with the wake instability dominating in both analyses. The helical mode localized to the vortex co-rotates with the base flow, which is converse with the counter-rotating instabilities of a Batchelor vortex model, which may be a result of the formative nature of the base-flow vortex. The fourth spatial mode is localized to the tip vortex region. The continuous part of the spectrum contains oscillatory and wavepacket solutions prompting the utilization of a wavepacket analysis to analyse the flow field and group velocity. The structure and details of the full bi-global spectrum will help navigate the design space of effective control strategies to hasten decay of persistent wingtip vortices.

Sogaro F, Schmid P, Morgans AS, 2018, Sensitivity analysis of thermoacoustic instabilities, Pages: 2063-2070

Copyright © (2018) by International Institute of Acoustics & Vibration.All rights reserved. Thermoacoustic instability is a phenomenon that occurs in numerous combustion systems, from rockets to land-based gas turbines. The thermoacoustic oscillations of these systems are of significant importance as they can result in severe vibrations, thrust oscillations, thermal stresses and mechanical loads that lead to fatigue or even failure. In this work we use a low-order network model representation of a combustor where linear acoustics are solved together with appropriate boundary conditions and flame jump conditions. Special emphasis is directed towards the interaction between instabilities associated with acoustic modes and flame-intrinsic modes. Adjoint methods are used to perform a receptivity and sensitivity analysis of the spectral properties of the system to changes in the parameters involved. To better analyse the extreme sensitivities that arise in the neighbourhood of a modal interaction, we compare the results with a time domain model that allows us to perform a pseudospectra analysis. The results provide key insights into the interplay between mode types.

Karami S, Stegeman PC, Theofilis V, et al., 2018, Linearised dynamics and non-modal instability analysis of an impinging under-expanded supersonic jet, Third Madrid Summer School on Turbulence, Publisher: Institute of Physics (IoP), ISSN: 1742-6588

Non-modal instability analysis of the shear layer near the nozzle of a supersonic under-expanded impinging jet is studied. The shear layer instability is considered to be one of the main components of the feedback loop in supersonic jets. The feedback loop is observed in instantaneous visualisations of the density field where it is noted that acoustic waves scattered by the nozzle lip internalise as shear layer instabilities.A modal analysis describes the asymptotic limit of the instability disturbances and fails to capture short-time responses. Therefore, a non-modal analysis which allows the quantitative description of the short-time amplification or decay of a disturbance is performed by means of a local far-field pressure pulse. An impulse response analysis is performed which allows a wide range of frequencies to be excited. The temporal and spatial growths of the disturbances in the shear layer near the nozzle are studied by decomposing the response using dynamic mode decomposition and Hilbert transform analysis.The short-time response shows that disturbances with non-dimensionalised temporal frequencies in the range of 1 to 4 have positive growth rates in the shear layer. The Hilbert transform analysis shows that high non-dimensionalised temporal frequencies (>4) are dampened immediately, whereas low non-dimensionalised temporal frequencies (<1) are neutral. Both dynamic mode decomposition and Hilbert transform analysis show that spatial frequencies between 1 and 3 have positive spatial growth rates. Finally, the envelope of the streamwise velocity disturbances reveals the presence of a convective instability.

Schmid PJ, Fosas de Pando M, Peake N, 2017, Stability analysis for n-periodic arrays of fluid systems, *Physical Review Fluids*, Vol: 2, ISSN: 2469-990X

A computational framework is proposed for the linear modal and nonmodal analysis of fluid systems consisting of a periodic array of n identical units. A formulation in either time or frequency domain is sought and the resulting block-circulant global system matrix is analyzed using roots-of-unity techniques, which reduce the computational effort to only one unit while still accounting for the coupling to linked components. Modal characteristics as well as nonmodal features are treated within the same framework, as are initial-value problems and direct-adjoint looping. The simple and efficient formalism is demonstrated on selected applications, ranging from a Ginzburg-Landau equation with an n-periodic growth function to interacting wakes to incompressible flow through a linear cascade consisting of 54 blades. The techniques showcased here are readily applicable to large-scale flow configurations consisting of n-periodic arrays of identical and coupled fluid components, as can be found, for example, in turbomachinery, ring flame holders, or nozzle exit corrugations. Only minor corrections to existing solvers have to be implemented to allow this present type of analysis.

Horn S, Schmid P, 2017, Prograde, retrograde and oscillatory modes in rotating Rayleigh-Benard convection

Horn S, Schmid PJ, 2017, Prograde, retrograde, and oscillatory modes in rotating Rayleigh-Benard convection, *JOURNAL OF FLUID MECHANICS*, Vol: 831, Pages: 182-211, ISSN: 0022-1120

Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative set-ups: first, a fluid with Prandtl number , corresponding to water, in a cylinder with a diameter-to-height aspect ratio of ; second, a fluid with , corresponding to or air, confined in a slender cylinder with ; and third, the main focus of this paper, a fluid with , corresponding to a liquid metal, in a cylinder with . The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number , the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For , DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low- fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of

Sogaro F, Schmid P, Morgans AS, 2017, Sensitivity analysis of thermoacoustic instabilities, 24th International Congress on Sound and Vibration 2017 (ICSV 24)

Thermoacoustic instability is a phenomenon that occurs in numerous combustion systems, from rockets to land based gas turbines. The resulting acoustic oscillations can result in severe vibrations, thrust oscillations, thermal stresses and mechanical loads that lead to fatigue or even failure. This propensity to instability has been found to occur much more frequently in lean premixed combustion, one of the recent methods used in the gas turbine industry of aeroengines and power gas turbines to reduce NOx emissions. In this work we consider a simplified combustion system, and analyse the sensitivity of its thermoacoustic modes to small changes in the flame and combustor geometry parameters. Such a sensitivity analysis offers insights on how best to change the combustion system so as to "design-out" instability. The simplified combustor is modelled using a low order network representation: linear plane acoustic waves are combined with the appropriate acoustic boundary and flame jump conditions and a linear n-tau flame model. A sensitivity analysis is then performed using adjoint methods, with special focus on the sensitivity of the modes to parameters, such as reflection coefficients and flame model gain and time delay. The gradient information obtained reveals how the thermoacoustic modes of the system respond to changes to the various parameters. The results offer key insights into the behaviour and coupling of different types of modes - for example acoustic modes and so-called "intrinsic" modes associated with the flame model. They also provide insights into the optimal configuration for the design of such combustors.

Symon S, Dovetta N, McKeon BJ,
et al., 2017, Data assimilation of mean velocity from 2D PIV measurements of flow over an idealized airfoil, *EXPERIMENTS IN FLUIDS*, Vol: 58, ISSN: 0723-4864

Qadri UA, Schmid PJ, 2017, Effect of nonlinearities on the frequency response of a round jet, *Physical Review Fluids*, Vol: 2, ISSN: 2469-990X

We investigate the effect of nonlinearities on the frequency response of a round, incompressible jet. Experiments show that axisymmetric structures dominate the response of forced and unforced jets. In contrast, linear stability and frequency response analyses predict the asymmetric mode (m=1) to be locally more unstable and globally more amplified than the axisymmetric mode (m=0). We perform a weakly nonlinear expansion of the response of the flow to harmonic forcing and derive an asymptotic expression for the sum of this divergent series beyond its limit of validity. This expression compares reasonably well with the nonlinear gain up to forcing amplitudes an order of magnitude greater than the limit of validity of the weakly nonlinear expansion. For equal forcing amplitudes, the asymmetric mode dominates over the axisymmetric mode. This suggests that the projection of environmental forcing onto the individual azimuthal modes plays an important role in the preferred dynamics of round jets.

Schmid PJ, Sayadi T, Low-dimensional representation of near-wall dynamics in shear flows, with implications to wall-models, *Royal Society of London. Philosophical Transactions A. Mathematical, Physical and Engineering Sciences*, ISSN: 1364-503X

Pier B, Schmid PJ, 2017, Linear and nonlinear dynamics of pulsatile channel flow, *Journal of Fluid Mechanics*, Vol: 815, Pages: 435-480, ISSN: 0022-1120

The dynamics of small-amplitude perturbations, as well as the regime of fullydeveloped nonlinear propagating waves, is investigated for pulsatile channel flows.The time-periodic base flows are known analytically and completely determined bythe Reynolds number Re (based on the mean flow rate), the Womersley number Wo(a dimensionless expression of the frequency) and the flow-rate waveform. This paperconsiders pulsatile flows with a single oscillating component and hence only threenon-dimensional control parameters are present. Linear stability characteristics areobtained both by Floquet analyses and by linearized direct numerical simulations.In particular, the long-term growth or decay rates and the intracyclic modulationamplitudes are systematically computed. At large frequencies (mainly Wo > 14),increasing the amplitude of the oscillating component is found to have a stabilizingeffect, while it is destabilizing at lower frequencies; strongest destabilization is foundfor Wo ' 7. Whether stable or unstable, perturbations may undergo large-amplitudeintracyclic modulations; these intracyclic modulation amplitudes reach huge valuesat low pulsation frequencies. For linearly unstable configurations, the resultingsaturated fully developed finite-amplitude solutions are computed by direct numericalsimulations of the complete Navier–Stokes equations. Essentially two types ofnonlinear dynamics have been identified: ‘cruising’ regimes for which nonlinearitiesare sustained throughout the entire pulsation cycle and which may be interpreted asmodulated Tollmien–Schlichting waves, and ‘ballistic’ regimes that are propelled intoa nonlinear phase before subsiding again to small amplitudes within every pulsationcycle. Cruising regimes are found to prevail for weak base-flow pulsation amplitudes,while ballistic regimes are selected at larger pulsation amplitudes; at larger pulsationfrequencies, however, the ballistic regime may be bypassed due to

Fosas de Pando M, Schmid PJ, 2017, Optimal frequency-response sensitivity of compressible flow over roughness elements, *Journal of Turbulence*, Vol: 18, Pages: 338-351, ISSN: 1468-5248

Compressible flow over a flat plate with two localised and well-separated roughness elements is analysed by global frequency-response analysis. This analysis reveals a sustained feedback loop consisting of a convectively unstable shear-layer instability, triggered at the upstream roughness, and an upstream-propagating acoustic wave, originating at the downstream roughness and regenerating the shear-layer instability at the upstream protrusion. A typical multi-peaked frequency response is recovered from the numerical simulations. In addition, the optimal forcing and response clearly extract the components of this feedback loop and isolate flow regions of pronounced sensitivity and amplification. An efficient parametric-sensitivity framework is introduced and applied to the reference case which shows that first-order increases in Reynolds number and roughness height act destabilising on the flow, while changes in Mach number or roughness separation cause corresponding shifts in the peak frequencies. This information is gained with negligible effort beyond the reference case and can easily be applied to more complex flows.

Qadri UA, Schmid PJ, 2017, Frequency selection mechanisms in the flow of a laminar boundary layer over a shallow cavity, *Physical Review Fluids*, Vol: 2, ISSN: 2469-990X

We investigate the flow over shallow cavities as a representative configuration for modelling smallsurface irregularities in wall-bounded shear flows. Due to the globally stable nature of the flow, weperform a frequency response analysis, which shows a significant potential for the amplification ofdisturbance kinetic energy by harmonic forcing within a certain frequency band. Shorter and moreshallow cavities exhibit less amplified responses, while energy from the base flow can be extractedpredominantly from forcing that impacts the cavity head-on. A structural sensitivity analysis,combined with a componentwise decomposition of the sensitivity tensor, reveals the regions ofthe flow that act most effectively as amplifiers. We find that the flow inside the cavity plays anegligible role, whereas boundary layer modifications immediately upstream and downstream ofthe cavity edges contribute significantly to the frequency response. The same regions constitutepreferred locations for implementing active or passive control strategies to manipulate the frequencyresponse of the flow.

Fosas de Pando M, Schmid PJ, Sipp D, 2017, On the receptivity of aerofoil tonal noise: an adjoint analysis, *Journal of Fluid Mechanics*, Vol: 812, Pages: 771-791, ISSN: 1469-7645

For moderate-to-high Reynolds numbers, aerofoils are known to produce substantial levels of acoustic radiation, known as tonal noise, which arises from a complex interplay between laminar boundary-layer instabilities, trailing-edge acoustic scattering and upstream receptivity of the boundary layers on both aerofoil surfaces. The resulting acoustic spectrum is commonly characterised by distinct equally spaced peaks encompassing the frequency range of convectively amplified instability waves in the pressure-surface boundary layer. In this work, we assess the receptivity and sensitivity of the flow by means of global stability theory and adjoint methods which are discussed in light of the spatial structure of the adjoint global modes, as well as the wavemaker region. It is found that for the frequency range corresponding to acoustic tones the direct global modes capture the growth of instability waves on the suction surface and the near wake together with acoustic radiation into the far field. Conversely, it is shown that the corresponding adjoint global modes, which capture the most receptive region in the flow to external perturbations, have compact spatial support in the pressure surface boundary layer, upstream of the separated flow region. Furthermore, we find that the relative spatial amplitude of the adjoint modes is higher for those modes whose real frequencies correspond to the acoustic peaks. Finally, analysis of the wavemaker region points at the pressure surface near 30 % of the chord as the preferred zone for the placement of actuators for flow control of tonal noise.

Schmid PJ, Sayadi T, Low-dimensional representation of near-wall dynamics in shear flows, with implications to wall-models, *Journal: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences*, ISSN: 1471-2962

Noack BR, Stankiewicz W, MorzyĆski M,
et al., 2016, Recursive dynamic mode decomposition of transient and post-transient wake flows, *Journal of Fluid Mechanics*, Vol: 809, Pages: 843-872, ISSN: 1469-7645

A novel data-driven modal decomposition of fluid flow is proposed, comprising key features of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The first mode is the normalized real or imaginary part of the DMD mode that minimizes the time-averaged residual. The NNth mode is defined recursively in an analogous manner based on the residual of an expansion using the first N−1N−1 modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. Recursive DMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al., J. Nonlinear Sci., vol. 22, 2012, pp. 887–915) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have purer frequency content while retaining a residual of comparable order to POD. In contrast to DMD, with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit-cycle resolution from the same snapshots. Robustness of these observations is demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. Recursive DMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, in particular for nonlinear transient dynamics.

Noack B, Stankiewicz W, Morzynski M, et al., 2016, Recursive dynamic mode decomposition of a transient wake flow

Schmid PJ, Sipp D, 2016, Linear control of oscillator and amplifier flows, *Physical Review Fluids*, Vol: 1, ISSN: 2469-990X

Linear control applied to fluid systems near an equilibrium point has important applications for many flows of industrial or fundamental interest. In this article we give an exposition of tools and approaches for the design of control strategies for globally stable or unstable flows. For unstable oscillator flows a feedback configuration and a model-based approach is proposed, while for stable noise-amplifier flows a feedforward setup and an approach based on system identification is advocated. Model reduction and robustness issues are addressed for the oscillator case; statistical learning techniques are emphasized for the amplifier case. Effective suppression of global and convective instabilities could be demonstrated for either case, even though the system-identification approach results in a superior robustness to off-design conditions.

Fosas de Pando M, Schmid PJ, Sipp D, 2016, Nonlinear model-order reduction for compressible flow solvers using the Discrete Empirical Interpolation Method, *Journal of Computational Physics*, Vol: 324, Pages: 194-209, ISSN: 0021-9991

Nonlinear model reduction for large-scale flows is an essential component in many fluid applications such as flow control, optimization, parameter space exploration and statistical analysis. In this article, we generalize the POD–DEIM method, introduced by Chaturantabut & Sorensen [1], to address nonlocal nonlinearities in the equations without loss of performance or efficiency. The nonlinear terms are represented by nested DEIM-approximations using multiple expansion bases based on the Proper Orthogonal Decomposition. These extensions are imperative, for example, for applications of the POD–DEIM method to large-scale compressible flows. The efficient implementation of the presented model-reduction technique follows our earlier work [2] on linearized and adjoint analyses and takes advantage of the modular structure of our compressible flow solver. The efficacy of the nonlinear model-reduction technique is demonstrated to the flow around an airfoil and its acoustic footprint. We could obtain an accurate and robust low-dimensional model that captures the main features of the full flow.

Edstrand AM, Davis TB, Schmid PJ,
et al., 2016, On the mechanism of trailing vortex wandering, *Journal of Fluid Mechanics*, Vol: 801, ISSN: 0022-1120

The mechanism of trailing vortex wandering has long been debated and is often attributed to either wind-tunnel effects or an instability. Using particle image velocimetry data obtained in the wake of a NACA0012 airfoil, we remove the effect of wandering from the measured velocity field and, through a triple decomposition, recover the coherent wandering motion. Based on this wandering motion, the most energetic structures are computed using the proper orthogonal decomposition (POD) and exhibit a helical mode with an azimuthal wavenumber of |m|=1 whose kinetic energy grows monotonically in the downstream direction. To investigate the nature of the vortex wandering, we perform a spatial stability analysis of a matched Batchelor vortex. The primary stability mode is found to be marginally stable and nearly identical in both size and structure to the leading POD mode. The strikingly similar structure, coupled with the measured energy growth, supports the proposition that the vortex wandering is the result of an instability. We conclude that the cause of the wandering is the non-zero radial velocity of the |m|=1 mode on the vortex centreline, which acts to transversely displace the trailing vortex, as observed in experiments. However, the marginal nature of the stability mode prevents a definitive conclusion regarding the specific type of instability.

Edstrand A, Davis TB, Schmid P, et al., 2016, On the mechanism of trailing vortex wandering

Dunne R, Schmid PJ, McKeon BJ, 2016, Analysis of flow timescales on a periodically pitching/surging airfoil, *AIAA Journal*, Vol: 54, Pages: 3421-3433, ISSN: 0001-1452

Time-resolved velocity fields around a pitching and surging NACA 0018 airfoil were analyzed to investigate the influence of three independent timescales associated with the unsteady flowfield. The first of these timescales, the period of the pitch/surge motion, is directly linked to the development of dynamic stall. A simplified model of the flow using only a time constant mode and the first two harmonics of the pitch surge frequency has been shown to accurately model the flow. Full stall and leading-edge flow separation, however, were found to take place before the maximum angle of attack, indicating that a different timescale was associated with leading-edge vortex formation. This second, leading-edge vortex, timescale was found to depend on the airfoil convection time and compare well with the universal vortex formation time. Finally, instantaneous non-phase-averaged measurements were investigated to identify behavior not directly coupled to the airfoil motion. From this analysis, a third timescale associated with quasi-periodic Strouhal vortex shedding was found before flow separation. The interplay between these three timescales is discussed in detail, particularly as they relate to the periodic velocity and angle-of-attack change apparent to the blades of a vertical axis wind turbine.

Sujith R, Juniper M, Schmid P, 2016, Non-normality and nonlinearity in thermoacoustic instabilities

Sujith RI, Juniper MP, Schmid PJ, 2016, Non-normality and nonlinearity in thermoacoustic instabilities, *International Journal of Spray and Combustion Dynamics*, Vol: 8, Pages: 119-146, ISSN: 1756-8285

Analysis of thermoacoustic instabilities were dominated by modal (eigenvalue) analysis for many decades. Recent progress in nonmodal stability analysis allows us to study the problem from a different perspective, by quantitatively describing the short-term behavior of disturbances. The short-term evolution has a bearing on subcritical transition to instability, known popularly as triggering instability in thermoacoustic parlance. We provide a review of the recent developments in the context of triggering instability. A tutorial for nonmodal stability analysis is provided. The applicability of the tools from nonmodal stability analysis are demonstrated with the help of a simple model of a Rjike tube. The article closes with a brief description of how to characterize bifurcations in thermoacoustic systems.

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