Imperial College London

ProfessorPeterSchmid

Faculty of Natural SciencesDepartment of Mathematics

Chair in Applied Mathematics
 
 
 
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peter.schmid

 
 
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753Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

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173 results found

Margaritis AT, Sayadi T, Marxen O, Schmid PJet al., 2022, Sensitivity of Reacting Hypersonic Boundary Layers to n-periodic Surface Roughness, IUTAM Bookseries, Pages: 599-612

Successful design of aerospace missions requires accurate modelling of the physical phenomena in a hypersonic boundary layer. The behaviour of hypersonic boundary layers is strongly influenced by finite-rate thermochemical effects. These effects can be captured by including finite-rate thermochemistry models in computational tools. Such models are typically highly parametrised, introducing non-equilibrium features into the flow that generate significant uncertainty, which reflects on the output quantities of interest. The way such phenomena interact with solid boundaries and roughness is fundamentally unknown and the additional uncertainty is unquantifiable. In the present work, we propose the investigation of the linear response of n-periodic roughness arrays using an efficient mathematical framework. We extend an existing computational tool to investigate the effect of roughness, including real-gas and finite-rate chemistry effects by coupling with the Mutation++ library. The proposed framework allows to study efficiently and in parallel the linear flow response and wake synchronisation without the restricting idealised periodicity constraint. The results are extracted from reduced-order geometries using automatic linearisation tools. This framework can be potentially combined with sensitivity analysis tools to identify critical roughness configurations. In this paper, we provide the necessary background and present preliminary results for canonical flat plate hypersonic boundary layers.

Book chapter

Pier B, Schmid PJ, 2021, Optimal energy growth in pulsatile channel and pipe flows, Journal of Fluid Mechanics, Vol: 926, ISSN: 0022-1120

<jats:p>Pulsatile channel and pipe flows constitute a fundamental flow configuration with significant bearing on many applications in the engineering and medical sciences. Rotating machinery, hydraulic pumps or cardiovascular systems are dominated by time-periodic flows, and their stability characteristics play an important role in their efficient and proper operation. While previous work has mainly concentrated on the modal, harmonic response to an oscillatory or pulsatile base flow, this study employs a direct–adjoint optimisation technique to assess short-term instabilities, identify transient energy-amplification mechanisms and determine their prevalence within a wide parameter space. At low pulsation amplitudes, the transient dynamics is found to be similar to that resulting from the equivalent steady parabolic flow profile, and the oscillating flow component appears to have only a weak effect. After a critical pulsation amplitude is surpassed, linear transient growth is shown to increase exponentially with the pulsation amplitude and to occur mainly during the slow part of the pulsation cycle. In this latter regime, a detailed analysis of the energy transfer mechanisms demonstrates that the huge linear transient growth factors are the result of an optimal combination of Orr mechanism and intracyclic normal-mode growth during half a pulsation cycle. Two-dimensional sinuous perturbations are favoured in channel flow, while pipe flow is dominated by helical perturbations. An extensive parameter study is presented that tracks these flow features across variations in the pulsation amplitude, Reynolds and Womersley numbers, perturbation wavenumbers and imposed time horizon.</jats:p>

Journal article

Sayadi T, Schmid PJ, 2021, Frequency response analysis of a (non-)reactive jet in crossflow, JOURNAL OF FLUID MECHANICS, Vol: 922, ISSN: 0022-1120

Journal article

Skene CS, Eggl MF, Schmid PJ, 2021, A parallel-in-time approach for accelerating direct-adjoint studies, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 429, ISSN: 0021-9991

Journal article

Margaritis AT, Sayadi T, Marxen O, Schmid PJet al., 2021, Linear analysis of n-periodic fluid systems: An application to wake synchronization, Pages: 1-11

In this manuscript we outline a practical application of the framework presented by Schmid et al. (Phys. Rev. Fluids, 2 (11) 2017) for the linear analysis of fluid systems consisting of periodic arrays of n identical units (i.e. n-periodic systems). We generalize and apply the methodology in the case of large-scale fluid flow solvers, implementing an operator-free approach. Using the roots-of-unity and the corresponding complex transformation, we show that we can decouple the original linear system and massively parallelize its solution. We exploit an operatorfree approach to extract information about any arbitrary number of units from our reduced simulations. In this work, we apply this methodology to a simple flow configuration: a boundary layer over a flat plate with n-periodic synthetic wakes, approximating the flow downstream of a roughness array. The proposed mathematical framework enables the study of wake synchronization downstream of such arrays with a reduced-cost system of independent simulations that can run in parallel. From a limited number of simulations, it is possible to extract the linear response of the full geometry for arbitrary number of units n, thus relaxing the assumption of single-unit periodicity that is commonly imposed in such simulations. We use an in-house high-fidelity direct numerical simulation code to solve the nonlinear and the linearized systems of the Navier-Stokes equations. Potential coupling with the corresponding adjoint solver would enable efficient sensitivity analysis and optimization of the full geometry, with reduced simulation cost compared to the full simulations. This manuscript presents the necessary background and some preliminary results for the proposed methodology. Development of supplementary analysis tools and extension of the applications are part of ongoing work.

Conference paper

Qadri UA, Magri L, Ihme M, Schmid PJet al., 2021, Using adjoint-based optimization to enhance ignition in non-premixed jets., Proc Math Phys Eng Sci, Vol: 477, Pages: 20200472-20200472, ISSN: 1364-5021

Gradient-based optimization is used to reliably and optimally induce ignition in three examples of laminar non-premixed mixture configurations. Using time-integrated heat release as a cost functional, the non-convex optimization problem identified optimal energy source locations that coincide with the stoichiometric local mixture fraction surface for short optimization horizons, while for longer horizons, the hydrodynamics plays an increasingly important role and a balance between flow and chemistry features determines non-trivial optimal ignition locations. Rather than identifying a single optimal ignition location, the results of this study show that there may be several equally good ignition locations in a given flow configuration.

Journal article

Arun R, Dawson STM, Schmid PJ, Laskari A, McKeon BJet al., 2020, Control of instability by injection rate oscillations in a radial Hele-Shaw cell, PHYSICAL REVIEW FLUIDS, Vol: 5, ISSN: 2469-990X

Journal article

Eggl MF, Schmid PJ, 2020, Mixing enhancement in binary fluids using optimised stirring strategies, JOURNAL OF FLUID MECHANICS, Vol: 899, ISSN: 0022-1120

Journal article

Eggl MF, Schmid PJ, 2020, Shape optimization of stirring rods for mixing binary fluids, IMA Journal of Applied Mathematics, Vol: 85, Pages: 762-789, ISSN: 0272-4960

Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the nonlinear Navier–Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman penalization technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilize a Fourier-based approach for parameterizing and optimizing the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimization space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.

Journal article

Sipp D, Fosas de Pando M, Schmid PJ, 2020, Nonlinear model reduction: A comparison between POD-Galerkin and POD-DEIM methods, Computers and Fluids, Vol: 208, Pages: 1-21, ISSN: 0045-7930

Several nonlinear model reduction techniques are compared for the three cases of the non-parallel version of the Kuramoto-Sivashinsky equation, the transient regime of flow past a cylinder at Re=100 and fully developed flow past a cylinder at the same Reynolds number. The linear terms of the governing equations are reduced by Galerkin projection onto a POD basis of the flow state, while the reduced nonlinear convection terms are obtained either by a Galerkin projection onto the same state basis, by a Galerkin projection onto a POD basis representing the nonlinearities or by applying the Discrete Empirical Interpolation Method (DEIM) to a POD basis of the nonlinearities. The quality of the reduced order models is assessed as to their stability, accuracy and robustness, and appropriate quantitative measures are introduced and compared. In particular, the properties of the reduced linear terms are compared to those of the full-scale terms, and the structure of the nonlinear quadratic terms is analyzed as to the conservation of kinetic energy. It is shown that all three reduction techniques provide excellent and similar results for the cases of the Kuramoto-Sivashinsky equation and the limit-cycle cylinder flow. For the case of the transient regime of flow past a cylinder, only the pure Galerkin techniques are successful, while the DEIM technique produces reduced-order models that diverge in finite time.

Journal article

Skene CS, Qadri UA, Schmid PJ, 2020, Open-loop control of a global instability in a swirling jet by harmonic forcing: A weakly nonlinear analysis, PHYSICAL REVIEW FLUIDS, Vol: 5, ISSN: 2469-990X

Journal article

Symon S, Sipp D, Schmid PJ, McKeon BJet al., 2020, Mean and unsteady flow reconstruction using data-assimilation and resolvent Analysis, AIAA Journal: devoted to aerospace research and development, Vol: 58, Pages: 575-588, ISSN: 0001-1452

A methodology is presented that exploits both data-assimilation techniques and resolvent analysis for reconstructing turbulent flows, containing organized structures, with an efficient set of measurements. The mean (time-averaged) flow is obtained using variational data-assimilation that minimizes the discrepancy between a limited set of flow measurements, generally from an experiment, and a numerical simulation of the Navier–Stokes equations. The fluctuations are educed from resolvent analysis and time-resolved data at a single point in the flow. Resolvent analysis also guides where measurements of the mean and fluctuating quantities are needed for efficient reconstruction of a simple example case study: flow around a circular cylinder at a Reynolds number of Re=100. For this flow, resolvent analysis reveals that the leading singular value, most amplified modes, and the mean profile for 47<Re<320 scale with the shedding frequency and length of the recirculation bubble. A relationship between these two parameters reinforces the notion that a wave maker, for which the length scales with the recirculation bubble, determines the frequency and region where an instability mechanism is active. The procedure offers a way to choose sensor locations that capture the main coherent structures of a flow and a method for computing mean pressure by using correctly weighted resolvent modes.

Journal article

Murthy SR, Sayadi T, Le Chenadec V, Schmid PJ, Bodony DJet al., 2019, Analysis of degenerate mechanisms triggering finite-amplitude thermo-acoustic oscillations in annular combustors, JOURNAL OF FLUID MECHANICS, Vol: 881, Pages: 384-419, ISSN: 0022-1120

Journal article

Sogaro FM, Schmid PJ, Morgans AS, 2019, Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities, Journal of Fluid Mechanics, Vol: 878, Pages: 190-220, ISSN: 0022-1120

This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

Journal article

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

Journal article

Yang D, Sogaro FM, Morgans AS, Schmid PJet 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.

Journal article

Nidhan S, Tarin JLO, Chongsiripinyo K, Sarkar S, Schmid PJet al., 2019, Dynamic mode decomposition of stratified sphere wakes, Pages: 1-15

Dynamic mode decomposition (DMD) has been applied to data from simulations of stably stratified flow past a sphere. The simulation data is obtained using direct numerical simulation (DNS) at Reynolds number, Re = 500, and large-eddy simulation (LES) at a higher Re =10, 000. At Re = 500, the wake changes qualitatively from being three-dimensional and unsteady (although not turbulent) with shedding of three-dimensional vortices in the unstratified case at Fr = ∞ to being unsteady (not turbulent) with shedding of quasi two-dimensional vortices at Fr = 0.125. Stratification also leads to the formation of body-generated steady lee waves which progressively dominate the wake as the strength of stratification increases in the Fr = O(1) regime. The Re = 10, 000 case has fully-developed turbulence in the examined cases of Fr = ∞, 3, 1 with significant differences between Fr = 3 and 1. The objective of the present work is to assess the ability of DMD to capture the varied effects of buoyancy on a bluff body wake. The sparsity-promoting variant of DMD that is employed here helps identify the dynamically important modes once the conventional DMD spectra are obtained. Investigation shows that DMD successfully captures vortex shedding, lee waves, and other large-scale features of stratified flow with a few modes. With increasing number of modes, the reconstruction error decreases systematically in the case of Re = 500 while at Re = 10, 000, the error tends to plateau. Thus, although large-scale flow features and their variation with buoyancy can be captured using a relatively smaller number of modes, it may be necessary to supplement DMD with other reconstruction tools for representation of the turbulent wake at high Re.

Conference paper

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

Journal article

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

Journal article

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

Journal article

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.

Conference paper

Edstrand AM, Schmid PJ, Taira K, Cattafesta LNet 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.

Journal article

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

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.

Conference paper

Karami S, Stegeman PC, Theofilis V, Schmid PJ, Soria Jet 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.

Conference paper

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.

Journal article

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

Working paper

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

Journal article

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.

Conference paper

Symon S, Dovetta N, McKeon BJ, Sipp D, Schmid PJet 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

Journal article

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.

Journal article

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