## Publications

45 results found

Patel Y, Mons V, Marquet O,
et al., 2024, Turbulence model augmented physics-informed neural networks for mean-flow reconstruction, *Physical Review Fluids*, Vol: 9

Experimental measurements and numerical simulations of turbulent flows are characterized by a tradeoff between accuracy and resolution. In this study, we combine accurate sparse pointwise mean velocity measurements with the Reynolds-averaged Navier-Stokes (RANS) equations using data assimilation methods. Importantly, we bridge the gap between data assimilation (DA) using physics-informed neural networks (PINNs) and variational methods based on classical spatial discretization of the flow equations, by comparing both approaches on the same turbulent flow case. First, by constraining the PINN with sparse data and the underdetermined RANS equations without closure, we show that the mean flow is reconstructed to a higher accuracy than a RANS solver using the Spalart-Allmaras (SA) turbulence model. Second, we propose the SA turbulence model augmented PINN (PINN-DA-SA), which outperforms the former approach by up to 73% reduction in mean velocity reconstruction error with coarse measurements. The additional SA physics constraints improve flow reconstructions in regions with high velocity and pressure gradients and separation. Third, we compare the PINN-DA-SA approach to a variational data assimilation using the same sparse velocity measurements and physics constraints. The PINN-DA-SA achieves lower reconstruction error across a range of data resolutions. This is attributed to discretization errors in the variational methodology that are avoided by PINNs. We demonstrate the method using high-fidelity measurements from direct numerical simulation of the turbulent periodic hill at Re=5600.

Xia C, Zhang J, Kerrigan E,
et al., 2024, Active flow control for bluff body drag reduction using reinforcement learning with partial measurements, *Journal of Fluid Mechanics*, Vol: 981, ISSN: 0022-1120

Active flow control for drag reduction with reinforcement learning (RL) is performed in the wake of a two-dimensional square bluff body at laminar regimes with vortex shedding. Controllers parametrised by neural networks are trained to drive two blowing and suction jets that manipulate the unsteady flow. The RL with full observability (sensors in the wake) discovers successfully a control policy that reduces the drag by suppressing the vortex shedding in the wake. However, a non-negligible performance degradation ( ∼ 50 % less drag reduction) is observed when the controller is trained with partial measurements (sensors on the body). To mitigate this effect, we propose an energy-efficient, dynamic, maximum entropy RL control scheme. First, an energy-efficiency-based reward function is proposed to optimise the energy consumption of the controller while maximising drag reduction. Second, the controller is trained with an augmented state consisting of both current and past measurements and actions, which can be formulated as a nonlinear autoregressive exogenous model, to alleviate the partial observability problem. Third, maximum entropy RL algorithms (soft actor critic and truncated quantile critics) that promote exploration and exploitation in a sample-efficient way are used, and discover near-optimal policies in the challenging case of partial measurements. Stabilisation of the vortex shedding is achieved in the near wake using only surface pressure measurements on the rear of the body, resulting in drag reduction similar to that in the case with wake sensors. The proposed approach opens new avenues for dynamic flow control using partial measurements for realistic configurations.

Poulain A, Content C, Rigas G,
et al., 2024, Adjoint-based linear sensitivity of a supersonic boundary layer to steady wall blowing–suction/heating–cooling, *Journal of Fluid Mechanics*, Vol: 978, ISSN: 0022-1120

For a Mach 4.5 flat-plate adiabatic boundary layer, we study the sensitivity of the first, second Mack modes and streaks to steady wall-normal blowing/suction and wall heat flux. The global instabilities are characterised in frequency space with resolvent gains and their gradients with respect to wall-boundary conditions are derived through a Lagrangian-based method. The implementation is performed in the open-source high-order finite-volume code BROADCAST and algorithmic differentiation is used to access the high-order state derivatives of the discretised governing equations. For the second Mack mode, the resolvent optimal gain decreases when suction is applied upstream of Fedorov’s mode S/mode F synchronisation point, leading to stabilisation, and the converse when applied downstream. The largest suction gradient is in the region of branch I of mode S neutral curve. For heat-flux control, strong heating at the leading edge stabilises both the first and second Mack modes, the former being more sensitive to wall-temperature control. Streaks are less sensitive to any boundary control in comparison with the Mack modes. Eventually, we show that an optimal actuator consisting of a single steady heating strip located close to the leading edge manages to damp the linear growth of all three instability mechanisms.

Scherding C, Rigas G, Sipp D,
et al., 2023, Data-driven framework for input/output lookup tables reduction: Application to hypersonic flows in chemical nonequilibrium, *PHYSICAL REVIEW FLUIDS*, Vol: 8, ISSN: 2469-990X

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Poulain A, Content C, Sipp D,
et al., 2023, BROADCAST: A high-order compressible CFD toolbox for stability and sensitivity using Algorithmic Differentiation, *Computer Physics Communications*, Vol: 283, Pages: 1-23, ISSN: 0010-4655

The evolution of any complex dynamical system is described by its state derivative operators. However, the extraction of the exact N-order state derivative operators is often inaccurate and requires approximations. The open-source CFD (Computational Fluid Dynamics) code called BROADCAST discretises the compressible Navier-Stokes equations and then extracts the linearised N-derivative operators through Algorithmic Differentiation (AD) providing a toolbox for laminar flow dynamics. Furthermore, the gradients through adjoint derivation are extracted either by transposition of the linearised operator or through the backward mode of the AD tool. The software includes base-flow computation and linear global stability analysis via eigen-decomposition of the linearised operator or via singular value decomposition of the resolvent operator. Sensitivity tools as well as weakly nonlinear analysis complete the package. The numerical method for the spatial discretisation of the equations consists of a finite-difference high-order shock-capturing scheme applied within a finite volume framework on 2D curvilinear structured grids. The stability and sensitivity tools are demonstrated on two cases: a cylinder flow at low Mach number and a hypersonic boundary layer.

Sliwinski L, Rigas G, 2023, Mean flow reconstruction of unsteady flows using physics-informed neural networks, *Data-Centric Engineering*, Vol: 4, Pages: 1-23, ISSN: 2632-6736

Data assimilation of flow measurements is an essential tool for extracting information in fluid dynamics problems. Recent works have shown that the physics-informed neural networks (PINNs) enable the reconstruction of unsteady fluid flows, governed by the Navier–Stokes equations, if the network is given enough flow measurements that are appropriately distributed in time and space. In many practical applications, however, experimental measurements involve only time-averaged quantities or their higher order statistics which are governed by the under-determined Reynolds-averaged Navier–Stokes (RANS) equations. In this study, we perform PINN-based reconstruction of time-averaged quantities of an unsteady flow from sparse velocity data. The applied technique leverages the time-averaged velocity data to infer unknown closure quantities (curl of unsteady RANS forcing), as well as to interpolate the fields from sparse measurements. Furthermore, the method’s capabilities are extended further to the assimilation of Reynolds stresses where PINNs successfully interpolate the data to complete the velocity as well as the stresses fields and gain insight into the pressure field of the investigated flow.

Kelshaw D, Rigas G, Magri L, 2022, Physics-informed CNNs for super-resolution of sparse observations on dynamical systems, 36th conference on Neural Information Processing Systems (NeurIPS)

In the absence of high-resolution samples, super-resolution of sparseobservations on dynamical systems is a challenging problem with wide-reachingapplications in experimental settings. We showcase the application ofphysics-informed convolutional neural networks for super-resolution of sparseobservations on grids. Results are shown for the chaotic-turbulent Kolmogorovflow, demonstrating the potential of this method for resolving finer scales ofturbulence when compared with classic interpolation methods, and thuseffectively reconstructing missing physics.

Towne A, Rigas G, Kamal O,
et al., 2022, Efficient global resolvent analysis via the one-way Navier–Stokes equations, *Journal of Fluid Mechanics*, Vol: 948, Pages: 1-45, ISSN: 0022-1120

Resolvent analysis is a powerful tool for modelling and analysing transitional and turbulent flows and, in particular, for approximating coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this paper we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction, i.e. one in which the mean flow changes gradually. First, we derive a well-posed and convergent one-way equation describing the downstream-travelling waves supported by the linearized Navier–Stokes equations. The method is based on a projection operator that isolates downstream-travelling waves. Integrating these one-way Navier–Stokes (OWNS) equations in the slowly varying direction, which requires significantly less CPU and memory resources than a direct solution of the linearized Navier–Stokes equations, approximates the action of the resolvent operator on a forcing vector. Second, this capability is leveraged to compute approximate resolvent modes using an adjoint-based optimization framework in which the forward and adjoint OWNS equations are marched in the downstream and upstream directions, respectively. This avoids the need to solve direct and adjoint globally discretized equations, therefore bypassing the main computational bottleneck of a typical global resolvent calculation. The method is demonstrated using the examples of a simple acoustics problem, a Mach 1.5 turbulent jet and a Mach 4.5 transitional zero-pressure-gradient flat-plate boundary layer. The optimal OWNS results are validated against corresponding global calculations, and the close agreement demonstrates the near-parabolic nature of these flows.

He X, Tan J, Rigas G,
et al., 2022, On the explainability of machine-learning-assisted turbulence modeling for transonic flows, *International Journal of Heat and Fluid Flow*, Vol: 97, Pages: 1-16, ISSN: 0142-727X

Machine learning (ML) is a rising and promising tool for Reynolds-Averaged Navier–Stokes (RANS) turbulence model developments, but its application to industrial flows is hindered by the lack of explainability of the ML model. In this paper, two types of methods to improve the explainability are presented, namely the intrinsic methods that reduce the model complexity and the post-hoc methods that explain the correlation between the model inputs and outputs. The investigated ML-assisted turbulence model framework aims to improve the prediction accuracy of the Spalart–Allmaras (SA) turbulence model in transonic bump flows. A random forest model is trained to construct a mapping between the input flow features and the output eddy viscosity difference. Results show that the intrinsic methods, including the hyperparameter study and the input feature selection, can reduce the model complexity at a limited cost of accuracy. The post-hoc Shapley additive explanations (SHAP) method not only provides a ranked list of input flow features based on their global significance, but also unveils the local causal link between the input flow features and the output eddy viscosity difference. Based on the SHAP analysis, the ML model is found to discover: (1) the well-known scaling between eddy viscosity and its source term, which was originally found from dimensional analysis; (2) the well-known rotation and shear effects on the eddy viscosity source term, which was explicitly written in the Reynolds stress transport equations; and (3) the pressure normal stress and normal shear stress effect on the eddy viscosity source term, which has not attracted much attention in previous research. The methods and the knowledge obtained from this work provide useful guidance for data-driven turbulence model developers, and they are transferable to future ML turbulence model developments.

Giannenas A, Laizet S, Rigas G, 2022, Harmonic forcing of a laminar bluff body wake with rear pitching flaps, *Journal of Fluid Mechanics*, Vol: 945, ISSN: 0022-1120

A numerical study on the response of a 2D bluff body wake subjected to harmonicforcing imposed by two rear pitching flaps is performed. The wake is generated by arectangle at a height-based Reynolds number Re = 100, characterised by laminar vortexshedding. Two forcing strategies are examined corresponding to in-phase “snaking” andout-of-phase “clapping”. The effects of the bluff body aspect ratio (AR = 1, 2, 4), flappingfrequency, flapping amplitude, flap length and Reynolds number are investigated. For thesnaking motion, a strong fundamental resonance of the root mean square (RMS) drag isobserved when the wake is forced near the vortex shedding frequency. For the clappingmotion, a weak subharmonic resonance is observed when the forcing is applied neartwice the vortex shedding frequency resulting in an increase of the lift RMS whereasthe drag RMS remains unaffected. Both resonances intensify the vortex shedding and aconcomitant mean drag increase is observed for the snaking motion. Forcing away fromthe resonant regimes, both motions result in considerable drag reduction through wakesymmetrisation and propulsion mechanisms. The formation of two vortex dipoles peroscillation period due to the flapping motion, which weaken the natural vortex shedding,has been identified as the main symmetrisation mechanism. A single scaling parameter isproposed to collapse the mean drag reduction of the forced flow for both motions over awide range of flapping frequencies, amplitudes and flap lengths. Finally, the assessmentof the performance of the forcing strategies has revealed that clapping is more effectivethan snaking.

Callaham JL, Rigas G, Loiseau J-C,
et al., 2022, An empirical mean-field model of symmetry-breaking in a turbulent wake, *Science Advances*, Vol: 8, ISSN: 2375-2548

Improved turbulence modeling remains a major open problem in mathematical physics. Turbulence is notoriously challenging, in part due to its multiscale nature and the fact that large-scale coherent structures cannot be disentangled from small-scale fluctuations. This closure problem is emblematic of a greater challenge in complex systems, where coarse-graining and statistical mechanics descriptions break down. This work demonstrates an alternative data-driven modeling approach to learn nonlinear models of the coherent structures, approximating turbulent fluctuations as state-dependent stochastic forcing. We demonstrate this approach on a high-Reynolds number turbulent wake experiment, showing that our model reproduces empirical power spectra and probability distributions. The model is interpretable, providing insights into the physical mechanisms underlying the symmetry-breaking behavior in the wake. This work suggests a path toward low-dimensional models of globally unstable turbulent flows from experimental measurements, with broad implications for other multiscale systems.

Kamal O, Rigas G, Lakebrink MT, et al., 2022, Input/output analysis of a Mach-6 cooled-wall hypersonic boundary layer using the One-Way Navier-Stokes (OWNS) Equations

The dominant instability observed in adiabatic-wall flat-plate hypersonic boundary layers is the second Mack mode, which manifests itself as trapped acoustic waves between the wall and the relative sonic line. If the wall is highly cooled, not only is this mode destabilized, but an additional mode may emerge – the supersonic mode, which is characterized by an acoustic emission from the boundary layer. To investigate the input/output behavior of these boundary layers, we use the One-Way Navier-Stokes (OWNS) Equations, which efficiently approximate a rigorous parabolization of the equations by filtering out disturbances with upstream group velocity, resulting in memory and CPU savings compared to global methods on large grids. We investigate the mechanistic shift of the second mode in a 2D Mach-6 flat-plate boundary layer by examining how the optimal response varies with frequency and the wall temperature. Specifically, we tackle the global forced receptivity problem with highly-cooled-wall conditions by parametrically analyzing the optimal forcings and corresponding responses. We demonstrate that the optimal response shifts from the first to the second mode with increasing frequency, along with the excitation of the supersonic mode when the wall is sufficiently cooled. Although the aforementioned conclusions can also be ascertained from locally-parallel, linear stability theory (LST), we demonstrate that inter-modal interactions involving the supersonic mode locally affect the mode shapes that LST fails to capture. Furthermore, spatially transient or non-modal responses are observed in cases where LST predicts all modes to be stable.

Zhu T, Rigas G, Morrison J, 2022, Near Wake Coherent Structures of a Turbulent Axisymmetric Wake

The coherent structures of a turbulent axisymmetric bluff body wake are analysed using proper orthogonal decomposition (POD) and conditional POD based on synchronized near-wake velocity and base pressure measurements. The analysis confirms the persistence of the laminar unstable eigenmodes at high-Reynolds numbers (here ReD = 1.88 × 105). These correspond to a steady spatial-symmetry breaking mode and asymmetric unsteady vortex shedding modes in the near wake. Additionally, an unsteady axisymmetric bubble pumping mode representing the streamwise pulsation of the wake is found in the turbulent wake. The asymmetric modes are extracted in the wake of the axisymmetric body by performing conditional POD. The asymmetry of the wake, quantified using the centre of pressure (CoP), is correlated to the base pressure using conditional averaging. At the limits of the highly asymmetric wake or a perfectly symmetric wake, a high average pressure coefficient (low drag) is obtained. For the most probable state of the wake, a low-pressure coefficient (high drag) is obtained. The above results provide insight on the coherent structures for canonical 3D bluff body geometries and guidance for drag reduction strategies.

He X, Fang Z, Rigas G,
et al., 2021, Spectral proper orthogonal decomposition of compressor tip leakage flow, *Physics of Fluids*, Vol: 33, ISSN: 1070-6631

To identify the spatiotemporal coherent structure of compressor tip leakage flow, spectral proper orthogonal decomposition (SPOD) is performed on the near-tip flow field and the blade surface pressure of a low-speed compressor rotor. The data used for the SPOD analysis are obtained by delayed-detached eddy simulation, which is validated against the experimental data. The investigated rotor near-tip flow field is governed by two tip leakage vortices (TLV), and the near-tip compressor passage can be divided into four zones: the formation of main TLV (Zone I), the main TLV breakdown (Zone II), the formation of tip blockage cell (Zone III), and the formation of secondary TLV (Zone IV). Modal analysis from SPOD shows that a major part of total disturbance energy comes from the main TLV oscillating mode in Zone I and the main TLV vortex shedding mode in Zone III, both of which are low-frequency and low-rank; on the contrary, modal components in Zones II and IV are broadband and non-low-rank. Unsteady blade forces are mainly generated by the impingement of the main TLV on the blade pressure surface in Zone III, rather than the detachment of the secondary TLV from the blade suction surface in Zone IV. These identified coherent structures provide valuable knowledge for the aerodynamic/aeroelastic effects, turbulence modeling, and reduced-order modeling of compressor tip leakage flow.

Callaham JL, Loiseau J-C, Rigas G,
et al., 2021, Nonlinear stochastic modelling with Langevin regression, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, Pages: 1-28, ISSN: 1364-5021

Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

Pickering E, Rigas G, Schmidt OT,
et al., 2021, Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets, *Journal of Fluid Mechanics*, Vol: 917, Pages: 1-34, ISSN: 0022-1120

Response modes computed via linear resolvent analysis of a turbulent mean-flow field have been shown to qualitatively capture characteristics of the observed turbulent coherent structures in both wall-bounded and free shear flows. To make such resolvent models predictive, the nonlinear forcing term must be closed. Strategies to do so include imposing self-consistent sets of triadic interactions, proposing various source models or through turbulence modelling. For the latter, several investigators have proposed using the mean-field eddy viscosity acting linearly on the fluctuation field. In this study, a data-driven approach is taken to quantitatively improve linear resolvent models by deducing an optimal eddy-viscosity field that maximizes the projection of the dominant resolvent mode to the energy-optimal coherent structure educed using spectral proper orthogonal decomposition (SPOD) of data from high-fidelity simulations. We use large-eddy simulation databases for round isothermal jets at subsonic, transonic and supersonic conditions and show that the optimal eddy viscosity substantially improves the agreement between resolvent and SPOD modes, reaching over 90 % agreement at those frequencies where the jet exhibits a low-rank response. We then consider a fixed model for the eddy viscosity and show that with the calibration of a single constant, the results are generally close to the optimal one. In particular, the use of a standard Reynolds-averaged Navier–Stokes eddy-viscosity resolvent model, with a single coefficient, provides substantial agreement between SPOD and resolvent modes for three turbulent jets and across the most energetic wavenumbers and frequencies.

Rigas G, Sipp D, Colonius T, 2021, Nonlinear input/output analysis: application to boundary layer transition, *Journal of Fluid Mechanics*, Vol: 911, Pages: 1-42, ISSN: 0022-1120

We extend linear input/output (resolvent) analysis to take into account nonlinear triadic interactions by considering a finite number of harmonics in the frequency domain using the harmonic balance method. Forcing mechanisms that maximise the drag are calculated using a gradient-based ascent algorithm. By including nonlinearity in the analysis, the proposed frequency-domain framework identifies the worst-case disturbances for laminar-turbulent transition. We demonstrate the framework on a flat-plate boundary layer by considering three-dimensional spanwise-periodic perturbations triggered by a few optimal forcing modes of finite amplitude. Two types of volumetric forcing are considered, one corresponding to a single frequency/spanwise wavenumber pair, and a multi-harmonic where a harmonic frequency and wavenumber are also added. Depending on the forcing strategy, we recover a range of transition scenarios associated with K -type and H -type mechanisms, including oblique and planar Tollmien–Schlichting waves, streaks and their breakdown. We show that nonlinearity plays a critical role in optimising growth by combining and redistributing energy between the linear mechanisms and the higher perturbation harmonics. With a very limited range of frequencies and wavenumbers, the calculations appear to reach the early stages of the turbulent regime through the generation and breakdown of hairpin and quasi-streamwise staggered vortices.

Tan J, He X, Rigas G, et al., 2021, TOWARDS EXPLAINABLE MACHINE-LEARNING-ASSISTED TURBULENCE MODELING FOR TRANSONIC FLOWS

A machine-learning-assisted turbulence modeling framework is proposed to improve the prediction accuracy of the Spalart-Allmaras turbulence model. The case studied is the transonic bump flow, which partially resembles the flow physics of a transonic compressor. A random forest model is trained, cross-validated and tested to construct a mapping between the input features and the eddy viscosity discrepancy. These input features concern the physical effects of pressure gradient, strain versus vorticity, flow misalignment, wall proximity and viscosity ratio. Results show that the proposed approach predicts an interpolation and an extrapolation test case with L1-type errors of 11.1% and 16.5%, respectively. The Shapley additive explanations method is employed to investigate the global and local sensitivities of each input feature. The capability of these input features in identifying specific flow features is discussed. The methods and results of this work provide useful guidance for turbulence model developers.

Tan J, He X, Rigas G, et al., 2021, TOWARDS EXPLAINABLE MACHINE-LEARNING-ASSISTED TURBULENCE MODELING FOR TRANSONIC FLOWS, ISSN: 2313-0067

A machine-learning-assisted turbulence modeling framework is proposed to improve the prediction accuracy of the Spalart-Allmaras turbulence model. The case studied is the transonic bump flow, which partially resembles the flow physics of a transonic compressor. A random forest model is trained, cross-validated and tested to construct a mapping between the input features and the eddy viscosity discrepancy. These input features concern the physical effects of pressure gradient, strain versus vorticity, flow misalignment, wall proximity and viscosity ratio. Results show that the proposed approach predicts an interpolation and an extrapolation test case with L1-type errors of 11.1% and 16.5%, respectively. The Shapley additive explanations method is employed to investigate the global and local sensitivities of each input feature. The capability of these input features in identifying specific flow features is discussed. The methods and results of this work provide useful guidance for turbulence model developers.

He X, Fang Z, Rigas G, et al., 2021, SPECTRAL PROPER ORTHOGONAL DECOMPOSITION OF COMPRESSOR TIP LEAKAGE FLOW, ISSN: 2313-0067

Spectral proper orthogonal decomposition (SPOD) is performed on the near-stall tip leakage flow of a low-speed compressor rotor. The data used for the SPOD analysis is obtained by delayed-detached eddy simulation (DDES), which is validated against experimental data. The flow quantities of interest include the near-tip axial velocity and the blade surface pressure. Results show that the near-stall flow field of the investigated rotor is governed by two tip leakage vortices (TLV). The main TLV initiated from the leading edge exerts an unsteady force on the blade pressure surface. Its modal component is dominated by the leading modes at low frequencies. The secondary TLV originated from the mid-chord creates a weaker unsteady force on the blade suction surface, and its modal component has more high-frequency components due to its interaction with the suction surface boundary layer flows. These findings improve the understanding of the effects of tip leakage flow on compressor aerodynamic and aeroelastic stability.

Kamal O, Rigas G, Lakebrink MT, et al., 2021, Input/Output Analysis of Hypersonic Boundary Layers using the One-Way Navier-Stokes (OWNS) Equations

Accurate prediction of linear amplification of disturbances in hypersonic boundary layers is computationally challenging. While direct numerical simulations and global analysis can be used to compute optimal (worst-case) forced responses, their large computational expense render these tools less practical for large design parameter spaces. At the same time, parabolized stability equations can be unreliable for problems involving multi-modal and non-modal interactions. To bridge this gap, we apply an approximate fast marching technique, the One-Way Navier-Stokes (OWNS) Equations, in iterative fashion to solve for optimal disturbances. OWNS approximates a rigorous parabolization of the equations of motion by removing disturbances with upstream group velocity using a higher-order recursive filter. Using OWNS, we aim to characterize disturbances of flat-plate and complex-geometry hypersonic boundary layers over a range of Mach numbers, and find optimal disturbances under different cost functions that define corresponding receptivity problems. The calculation of optimal disturbances reveals multi-modal transition scenarios depending on the spatial support, frequency, and physical nature of the external disturbances.

He X, Fang Z, Rigas G, et al., 2021, SPECTRAL PROPER ORTHOGONAL DECOMPOSITION OF COMPRESSOR TIP LEAKAGE FLOW

Spectral proper orthogonal decomposition (SPOD) is performed on the near-stall tip leakage flow of a low-speed compressor rotor. The data used for the SPOD analysis is obtained by delayed-detached eddy simulation (DDES), which is validated against experimental data. The flow quantities of interest include the near-tip axial velocity and the blade surface pressure. Results show that the near-stall flow field of the investigated rotor is governed by two tip leakage vortices (TLV). The main TLV initiated from the leading edge exerts an unsteady force on the blade pressure surface. Its modal component is dominated by the leading modes at low frequencies. The secondary TLV originated from the mid-chord creates a weaker unsteady force on the blade suction surface, and its modal component has more high-frequency components due to its interaction with the suction surface boundary layer flows. These findings improve the understanding of the effects of tip leakage flow on compressor aerodynamic and aeroelastic stability.

Pickering E, Rigas G, Nogueira PAS,
et al., 2020, Lift-up, Kelvin-Helmholtz and Orr mechanisms in turbulent jets, *Journal of Fluid Mechanics*, Vol: 896, Pages: 1-36, ISSN: 0022-1120

Three amplification mechanisms present in turbulent jets, namely lift-up, Kelvin–Helmholtz and Orr, are characterized via global resolvent analysis and spectral proper orthogonal decomposition (SPOD) over a range of Mach numbers. The lift-up mechanism was recently identified in turbulent jets via local analysis by Nogueira et al. (J. Fluid Mech., vol. 873, 2019, pp. 211–237) at low Strouhal number ( St ) and non-zero azimuthal wavenumbers ( m ). In these limits, a global SPOD analysis of data from high-fidelity simulations reveals streamwise vortices and streaks similar to those found in turbulent wall-bounded flows. These structures are in qualitative agreement with the global resolvent analysis, which shows that they are a response to upstream forcing of streamwise vorticity near the nozzle exit. Analysis of mode shapes, component-wise amplitudes and sensitivity analysis distinguishes the three mechanisms and the regions of frequency–wavenumber space where each dominates, finding lift-up to be dominant as St/m→0 . Finally, SPOD and resolvent analyses of localized regions show that the lift-up mechanism is present throughout the jet, with a dominant azimuthal wavenumber inversely proportional to streamwise distance from the nozzle, with streaks of azimuthal wavenumber exceeding five near the nozzle, and wavenumbers one and two most energetic far downstream of the potential core.

Brouzet D, Haghiri A, Talei M,
et al., 2020, Role of Coherent Structures in Turbulent Premixed Flame Acoustics, *AIAA JOURNAL*, Vol: 58, Pages: 2635-2642, ISSN: 0001-1452

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Kamal O, Rigas G, Lakebrink MT, et al., 2020, Application of the one-way navier-stokes (Owns) equations to hypersonic boundary layers

Prediction of linear amplification of disturbances in hypersonic boundary layers is challenging due to the presence and interactions of discrete modes (e.g. Tollmien-Schlichting and Mack) and continuous modes (entropic, vortical, and acoustic). While DNS and global analysis can be used, the large grids required make the computation of optimal transient and forced responses expensive, particularly when a large parameter space is required. At the same time, parabolized stability equations are non-convergent and unreliable for problems involving multi-modal and non-modal interactions. In this work, we apply the One-Way Navier-Stokes (OWNS) equations to hypersonic boundary layers. OWNS is based on a rigorous, approximate parabolization of the equations of motion that removes disturbances with upstream group velocity using a high-order recursive filter. We extend the original algorithm by considering non-orthogonal body-fitted curvilinear coordinates and incorporate full compressibility with temperature-dependent fluid properties. We validate the results by comparing to DNS data for a flat plate and sharp cone, and to LST results for local disturbances on the centerline of the HIFiRE-5 elliptic cone. OWNS provides DNS-quality results for the former flows at a small fraction of the computational expense.

Towne A, Rigas G, Colonius T, 2019, A critical assessment of the parabolized stability equations, *Theoretical and Computational Fluid Dynamics*, Vol: 33, Pages: 359-382, ISSN: 0935-4964

The parabolized stability equations (PSE) are a ubiquitous tool for studying the stability and evolution of disturbances in weakly nonparallel, convectively unstable flows. The PSE method was introduced as an alternative to asymptotic approaches to these problems. More recently, PSE has been applied with mixed results to a more diverse set of problems, often involving flows with multiple relevant instability modes. This paper investigates the limits of validity of PSE via a spectral analysis of the PSE operator. We show that PSE is capable of accurately capturing only disturbances with a single wavelength at each frequency and that other disturbances are not necessarily damped away or properly evolved, as often assumed. This limitation is the result of regularization techniques that are required to suppress instabilities arising from the ill-posedness of treating a boundary value problem as an initial value problem. These findings are valid for both incompressible and compressible formulations of PSE and are particularly relevant for applications involving multiple modes with different wavelengths and growth rates, such as problems involving multiple instability mechanisms, transient growth, and acoustics. Our theoretical results are illustrated using a generic problem from acoustics and a dual-stream jet, and the PSE solutions are compared to both global solutions of the linearized Navier–Stokes equations and a recently developed alternative parabolization.

Pickering E, Rigas G, Colonius T, et al., 2019, Eddy viscosity for resolvent-based jet noise models

Response modes computed via linear resolvent analysis have shown promising results for qualitatively modeling both the hydrodynamic and acoustic fields in jets when compared to data-deduced modes from high-fidelity, large-eddy simulations (LES). For an improved quantitative prediction of the near-and far-field, the role of Reynolds stresses must also be considered. In this study, we propose a methodology to deduce an eddy-viscosity model that optimally captures the nonlinear forcing of resolvent modes. The methodology is based on the maximization of the projection between resolvent analysis and spectral proper orthogonal decomposition (SPOD) modes using a Lagrangian optimization framework. For a Mach 0.4 round, isothermal, turbulent jet, four methods are used to increase the projection coefficients: linear damping, spatially constant eddy-viscosity field, a turbulent kinetic energy derived viscosity field, and an optimized eddy-viscosity field. The resulting projection coefficients for the optimized eddy-viscosity field between SPOD and resolvent can be increased to over 90% for frequencies in the range St = 0.35 − 1 with significant improvements to St < 0.35. We find that the use of a frequency-independent turbulent kinetic energy turbulent viscosity model produces modes closely inline with optimal results, providing a preliminary eddy-viscosity resolvent model for jets.

Nogueira PAS, Cavalieri AVG, Schmidt OT, et al., 2019, Resolvent-based analysis of streaks in turbulent jets

Large scale, elongated structures, similar those ones widely studied in wall-bounded flows, are also present in turbulent jets. Several characteristics of these streaks can be identified via reduced order models such as resolvent analysis. The present work involves a resolvent-based study of these structures in turbulent jets. We focus on obtaining the optimal forcing that generates these energetic coherent structures. Results are compared with experimental data post-processed using spectral proper orthogonal decomposition, allowing us to draw conclusions about the nature of the non-linear forcing, since the two analyses should provide equivalent results if this term is modelled as spatially white. By identifying streaks in a global framework, we expect to better understand the mechanism by which they are generated.

Rigas G, Pickering E, Schmidt O, et al., 2019, Streaks and coherent structures in jets from round and serrated nozzles

Hydrodynamic instabilities are directly related to large-scale coherent structures that are correlated with jet noise emission. Unravelling and accurately predicting their fundamental dynamics shows a promising direction for designing quieter jet engines. In this study, we analyze high-fidelity large-eddy simulation data of a turbulent Mach 0.4 round jet and a Mach 1.5 chevron jet. Using spectral proper orthogonal decomposition we identify, beyond the well-known1 Kelvin–Helmoholtz and Orr mechanisms, elongated alternating streamwise streaks of high and low-speed fluid that have been associated with a non-modal lift-up effect in wall-bounded shear flows. In the global three-dimensional domain, the most energetic streaks manifest for azimuthal wavenumber m = 1 and frequency St → 0. Furthermore, for the chevron jet, streaks and streamwise vortices appear due to the presence of the serrated nozzle, and they inherit the periodicity of the nozzle geometry. Finally, local (planar) spectral proper orthogonal decomposition is used to analyze the coherent structures of the chevron jet flow. Near the nozzle exit, antisymmetric and symmetric modes appear to be amplified and linked to the presence of the chevrons/streaks. Further downstream, the most energetic modes share similar characteristics to the ones observed in round jets.

Schmidt OT, Towne A, Rigas G,
et al., 2018, Spectral analysis of jet turbulence, *Journal of Fluid Mechanics*, Vol: 855, Pages: 953-982, ISSN: 0022-1120

Informed by large-eddy simulation (LES) data and resolvent analysis of the mean flow, we examine the structure of turbulence in jets in the subsonic, transonic and supersonic regimes. Spectral (frequency-space) proper orthogonal decomposition is used to extract energy spectra and decompose the flow into energy-ranked coherent structures. The educed structures are generally well predicted by the resolvent analysis. Over a range of low frequencies and the first few azimuthal mode numbers, these jets exhibit a low-rank response characterized by Kelvin–Helmholtz (KH) type wavepackets associated with the annular shear layer up to the end of the potential core and that are excited by forcing in the very-near-nozzle shear layer. These modes too have been experimentally observed before and predicted by quasi-parallel stability theory and other approximations – they comprise a considerable portion of the total turbulent energy. At still lower frequencies, particularly for the axisymmetric mode, and again at high frequencies for all azimuthal wavenumbers, the response is not low-rank, but consists of a family of similarly amplified modes. These modes, which are primarily active downstream of the potential core, are associated with the Orr mechanism. They occur also as subdominant modes in the range of frequencies dominated by the KH response. Our global analysis helps tie together previous observations based on local spatial stability theory, and explains why quasi-parallel predictions were successful at some frequencies and azimuthal wavenumbers, but failed at others.

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