Resolvent Modes Research Articles

Acoustic resolvent analysis of turbulent jets

Abstract We perform a resolvent analysis of a compressible turbulent jet, where the optimisation domain of the response modes is located in the acoustic field, excluding the hydrodynamic region, in order to promote acoustically efficient modes. We examine the properties of the acoustic resolvent and assess its potential for jet-noise modelling, focusing on the subsonic regime. Resolvent forcing modes, consistent with previous studies, are found to contain supersonic waves associated with Mach wave radiation in the response modes. This differs from the standard resolvent in which hydrodynamic instabilities dominate. We compare resolvent modes with SPOD modes educed from LES data. Acoustic resolvent response modes generally have better alignment with acoustic SPOD modes than standard resolvent response modes. For the optimal mode, the angle of the acoustic beam is close to that found in SPOD modes for moderate frequencies. However, there is no significant separation between the singular values of the leading and sub-optimal modes. Some suboptimal modes are furthermore shown to contain irrelevant structure for jet noise. Thus, even though it contains essential acoustic features absent from the standard resolvent approach, the SVD of the acoustic resolvent alone is insufficient to educe a low-rank model for jet noise. But because it identifies the prevailing mechanisms of jet noise, it provides valuable guidelines in the search of a forcing model (Karban et al. in J Fluid Mech 965:18, 2023). Graphical abstract

Transient growth of wavelet-based resolvent modes in the buffer layer of wall-bounded turbulence

In this work, we study the transient growth of the principal resolvent modes in the minimal flow unit using a reformulation of resolvent analysis in a time-localized wavelet basis. We target the most energetic spatial wavenumbers for the minimal flow unit and obtain modes that are constant in the streamwise direction and once-periodic in the spanwise direction. The forcing modes are in the shape of streamwise rolls, though pulse-like in time, and the response modes are in the form of transiently growing streaks. We inject the principal transient forcing mode at different intensities into a simulation of the minimal flow unit and compare the resulting nonlinear response to the linear one. The peak energy amplification scales quadratically with the intensity of the injected mode, and this peak occurs roughly at the same time for all forcing intensities. However, the larger energy amplification intensifies the magnitude of the nonlinear terms, which play an important role in damping the energy growth and accelerating energy decay of the principal resolvent mode. We also observe that the damping effect of the nonlinearities is less prominent close to the wall. Finally, we find that the principal resolvent forcing mode is more effective than other structures at amplifying the streak energy in the turbulent minimal-flow unit. In addition to lending support to the claim that linear mechanisms are important to near-wall turbulence, this work identifies time scales for the nonlinear breakdown of linearly-generated streaks.

Prediction of turbulent energy based on low-rank resolvent modes and machine learning

A modelling framework based on the resolvent analysis and machine learning is proposed to predict the turbulent energy in incompressible channel flows. In the framework, the optimal resolvent response modes are selected as the basis functions modelling the low-rank behaviour of high-dimensional nonlinear turbulent flow-fields, and the corresponding weight functions are determined by data-driven neural networks. Turbulent-energy distribution in space and scales, at the friction Reynolds number 1000, is predicted and compared to the data of direct numerical simulation. Close agreement is observed, suggesting the feasibility and reliability of the proposed framework for turbulence prediction.

Computational Workflow for Refining AlphaFold Models in Drug Design Using Kinetic and Thermodynamic Binding Calculations: A Case Study for the Unresolved Inactive Human Adenosine A3 Receptor.

A structure-based drug design pipeline that considers both thermodynamic and kinetic binding data of ligands against a receptor will enable the computational design of improved drug molecules. For unresolved GPCR-ligand complexes, a workflow that can apply both thermodynamic and kinetic binding data in combination with alpha-fold (AF)-derived or other homology models and experimentally resolved binding modes of relevant ligands in GPCR-homologs needs to be tested. Here, as test case, we studied a congeneric set of ligands that bind to a structurally unresolved G protein-coupled receptor (GPCR), the inactive human adenosine A3 receptor (hA3R). We tested three available homology models from which two have been generated from experimental structures of hA1R or hA2AR and one model was a multistate alphafold 2 (AF2)-derived model. We applied alchemical calculations with thermodynamic integration coupled with molecular dynamics (TI/MD) simulations to calculate the experimental relative binding free energies and residence time (τ)-random accelerated MD (τ-RAMD) simulations to calculate the relative residence times (RTs) for antagonists. While the TI/MD calculations produced, for the three homology models, good Pearson correlation coefficients, correspondingly, r = 0.74, 0.62, and 0.67 and mean unsigned error (mue) values of 0.94, 1.31, and 0.81 kcal mol-1, the τ-RAMD method showed r = 0.92 and 0.52 for the first two models but failed to produce accurate results for the multistate AF2-derived model. With subsequent optimization of the AF2-derived model by reorientation of the side chain of R1735.34 located in the extracellular loop 2 (EL2) that blocked ligand's unbinding, the computational model showed r = 0.84 for kinetic data and improved performance for thermodynamic data (r = 0.81, mue = 0.56 kcal mol-1). Overall, after refining the multistate AF2 model with physics-based tools, we were able to show a strong correlation between predicted and experimental ligand relative residence times and affinities, achieving a level of accuracy comparable to an experimental structure. The computational workflow used can be applied to other receptors, helping to rank candidate drugs in a congeneric series and enabling the prioritization of leads with stronger binding affinities and longer residence times.

Interpolatory input and output projections for flow control

Eigenvectors of the observability and controllability Gramians represent responsive and receptive flow structures that enjoy a well-established connection to resolvent forcing and response modes. However, whereas resolvent modes have demonstrated great potential to guide sensor and actuator placement, observability and controllability modes have been leveraged exclusively in the context of model reduction via input and output projections. In this work, we introduce interpolatory, rather than orthogonal, input and output projections, that can be leveraged for sensor and actuator placement and open-loop control design. An interpolatory projector is an oblique projector with the property of preserving certain entries in the vector being projected. We review the connection between the resolvent operator and the Gramians, and present several numerical examples where we perform both orthogonal and interpolatory input and output projections onto the dominant forcing and response subspaces. Input projections are used to identify dynamically relevant disturbances, place sensors to measure disturbances, and place actuators for feedforward control in the linearized Ginzburg–Landau equation. Output projections are used to identify coherent structures and place sensors aiming at state reconstruction in the turbulent flow in a minimal channel at$Re_{\tau }=185$. The framework does not require data snapshots and relies only on knowledge of the steady or mean flow.

Investigation of Oxide Thickness on Technical Aluminium Alloys—A Comparison of Characterization Methods

In this study the oxide layer of technical 6xxx aluminium surfaces, pickled as well as passivated, were comparatively investigated by means of transmission electron microscopy (TEM), Auger electron and X-ray photoelectron spectroscopy (AES, XPS), the latter in two different operating modes, standard and angle resolved mode. In addition, confocal microscopy and focused ion beam cutting were used for structural studies of the surfaces and for specimen preparation. The results illustrate in detail the strengths and weaknesses of each measurement technique. TEM offers a direct way to reliably quantify the thickness of the oxide layer, which is in the range of 5 nm, however, on a laterally restricted area of the surface. In comparison, for AES, the destructiveness of the electron beam did not allow to achieve comparable results for the thickness determination. XPS was proven to be the most reliable method to reproducibly quantify the average oxide thickness. By evaluating the angle resolved XPS data, additional information on the average depth distribution of the individual elements on the surface could be obtained. The findings obtained in this study were then successfully used for the investigation of the increase in the aluminium oxide thickness on technical samples during an aging test of 12 months under standard storage conditions.

Resolvent and dynamic mode analysis of flow past a square cylinder at subcritical Reynolds numbers

Flow-induced vibration (FIV) of bluff bodies can occur at subcritical Reynolds numbers (i.e., below the Re of the vortex shedding from fixed bodies). To analyze the mechanism of this subcritical FIV phenomenon, resolvent and dynamic mode analyses are introduced in this work. For laminar flow past a square cylinder, both resolvent and dynamic modes are extracted and investigated. The results indicate that the dominant dynamic mode decomposition mode and the leading response mode are similar. Both modes lead to vortex shedding at supercritical Reynolds numbers, and they vanish below Re = 19 along with the dominant forcing mode. In addition, the first and second resolvent gains separate near the characteristic flow frequency and overlap at Re = 19, indicating the disappearance of the first-order resolvent mode. The disappearance of these critical modes indicates the lowest Reynolds number of FIV instability for flow past a square cylinder.

A data-driven statistical-stochastic surrogate modeling strategy for complex nonlinear non-stationary dynamics

We propose a statistical-stochastic surrogate modeling approach to predict the response of the mean and variance statistics under various initial conditions and external forcing perturbations. The proposed modeling framework extends the purely statistical modeling approach that is practically limited to the homogeneous statistical regime for high-dimensional state variables. The new closure system allows one to overcome several practical issues that emerge in the non-homogeneous statistical regimes. First, the proposed ensemble modeling that couples the mean statistics and stochastic fluctuations naturally produces positive-definite covariance matrix estimation, which is a challenging issue that hampers the purely statistical modeling approaches. Second, the proposed closure model, which embeds a non-Markovian neural-network model for the unresolved fluxes such that the variance of the dynamics is consistent, overcomes the inherent instability of the stochastic fluctuation dynamics. Effectively, the proposed framework extends the classical stochastic parametric modeling paradigm for the unresolved dynamics to a semi-parametric parameterization with a residual Long-Short-Term-Memory neural network architecture. Third, based on empirical information metric, we provide an efficient and effective training procedure by fitting a loss function that measures the differences between response statistics. Supporting numerical examples are provided with the Lorenz-96 model, a system of ODEs that admits the characteristic of chaotic dynamics with both homogeneous and inhomogeneous statistical regimes. In the latter case, we will see the effectiveness of the statistical prediction even though the resolved Fourier modes corresponding to the leading mean energy and variance spectra do not coincide.

On the origin of quadrupole sound from a two-dimensional aerofoil trailing edge

Tonal noise emitted from the trailing edge of an airfoil is considered using modal analysis techniques to investigate secondary quadrupole tones. We examine the origin of quadrupole sound generated from two-dimensional unsteady laminar flow over a NACA0012 airfoil. In this paper, we consider two flow configurations at Mach numbers of $M_\infty = 0.1$ and 0.05 that lead to different acoustic characteristics: the former has a significant high-frequency quadrupole noise source, whereas the latter does not. We use vortex sound theory, dynamic mode decomposition (DMD), and resolvent analysis to analyze the sound source. First, we employ DMD modes to reveal that the quadrupole sound is only observed in the higher Mach number case. Next, the vortex dynamics in the vicinity of the trailing edge are studied to identify the origin of quadrupole sound. It is found that the quadrupole sound is caused by vortex shedding in the vicinity of the trailing edge. The complex vortex interaction between both sides of the airfoil strengthens the quadrupole source in the higher Mach number case, while it is negligible in the lower one. Furthermore, we perform resolvent analysis to examine the vortex generation over the airfoil. The resolvent mode indicates that the interaction between the vortices on both sides of the airfoil causes a multi-scale vortex structure on the suction-side wall.

Parabolic resolvent modes for streaky structures in transitional and turbulent boundary layers

Resolvent analysis is a powerful tool in fluid mechanics, both for laminar and turbulent flows. Depending on application, computational cost is still a limiting factor. We develop a parabolic method for the calculation of low-frequency resolvent modes, associated with streaky structures, which can enable more than an order of magnitude speed up compared to usual global calculations. The method is applied to both laminar and turbulent boundary layers, unveiling physical mechanisms such as those associated with streak growth due to free-stream turbulence, in the laminar case, and the double peak in the frequency spectrum of velocity fluctuations, in a turbulent boundary layer.

Improved convergence of the spectral proper orthogonal decomposition through time shifting

Spectral proper orthogonal decomposition (SPOD) is an increasingly popular modal analysis method in the field of fluid dynamics due to its specific properties: a linear system forced with white noise should have SPOD modes identical to response modes from resolvent analysis. The SPOD, coupled with the Welch method for spectral estimation, may require long time-resolved datasets. In this work, a linearised Ginzburg–Landau model is considered in order to study the method's convergence. Spectral proper orthogonal decomposition modes of the white-noise forced equation are computed and compared with corresponding response resolvent modes. The quantified error is shown to be related to the time length of Welch blocks (spectral window size) normalised by a convective time. Subsequently, an algorithm based on a temporal data shift is devised to further improve SPOD convergence and is applied to the Ginzburg–Landau system. Next, its efficacy is demonstrated in a numerical database of a boundary layer subject to bypass transition. The proposed approach achieves substantial improvement in mode convergence with smaller spectral window sizes with respect to the standard method. Furthermore, SPOD modes display growing wall-normal and spanwise velocity components along the streamwise direction, a feature which had not yet been observed and is also predicted by a global resolvent calculation. The shifting algorithm for the SPOD opens the possibility for using the method on datasets with time series of moderate duration, often produced by large simulations.

Efficient global resolvent analysis via the one-way Navier–Stokes equations

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.

Global stability and resolvent analyses of laminar boundary-layer flow interacting with viscoelastic patches

The attenuation of two-dimensional boundary-layer instabilities by a finite-length, viscoelastic patch is investigated by means of global linear stability theory. First, the modal stability properties of the coupled problem are assessed, revealing unstable fluid-elastic travelling-wave flutter modes. Second, the Tollmien–Schlichting instabilities over a rigid wall are characterised via the analysis of the fluid resolvent operator in order to determine a baseline for the fluid-structural analysis. To investigate the effect of the elastic patch on the growth of these flow instabilities, we first consider the linear frequency response of the coupled fluid-elastic system to the dominant rigid-wall forcing modes. In the frequency range of Tollmien–Schlichting waves, the energetic flow amplification is clearly reduced. However, an amplification is observed for higher frequencies, associated with travelling-wave flutter. This increased complexity requires the analysis of the coupled fluid-structural resolvent operator; the optimal, coupled, resolvent modes confirm the attenuation of the Tollmien–Schlichting instabilities, while also being able to capture the amplification at the higher frequencies. Finally, a decomposition of the fluid-structural response is proposed to reveal the wave cancellation mechanism responsible for the attenuation of the Tollmien–Schlichting waves. The viscoelastic patch, excited by the incoming rigid-wall wave, provokes a fluid-elastic wave that is out-of-phase with the former, thus reducing its amplitude.

Subgrid-scale parametrization of unresolved scales in forced Burgers equation using generative adversarial networks (GAN)

Stochastic subgrid-scale parametrizations aim to incorporate effects of unresolved processes in an effective model by sampling from a distribution usually described in terms of resolved modes. This is an active research area in climate, weather and ocean science where processes evolved in a wide range of spatial and temporal scales. In this study, we evaluate the performance of conditional generative adversarial network (GAN) in parametrizing subgrid-scale effects in a finite-difference discretization of stochastically forced Burgers equation. We define resolved modes as local spatial averages and deviations from these averages are the unresolved degrees of freedom. We train a Wasserstein GAN (WGAN) conditioned on the resolved variables to learn the distribution of subgrid flux tendencies for resolved modes and, thus, represent the effect of unresolved scales. The resulting WGAN is then used in an effective model to reproduce the statistical features of resolved modes. We demonstrate that various stationary statistical quantities such as spectrum, moments, autocorrelation, etc. are well approximated by this effective model.

Resolvent-based modeling of turbulent jet noise.

Resolvent analysis has demonstrated encouraging results for modeling coherent structures in jets when compared against their data-educed counterparts from high-fidelity large-eddy simulations (LES). We formulate resolvent analysis as an acoustic analogy that relates the near-field resolvent forcing to the near- and far-field pressure. We use an LES database of round, isothermal, Mach 0.9 and 1.5 jets to produce an ensemble of realizations for the acoustic field that we project onto a limited set of resolvent modes. In the near-field, we perform projections on a restricted acoustic output domain, r/D=[5,6], while the far-field projections are performed on a Kirchhoff surface comprising a 100-diameter arc centered at the nozzle. This allows the LES realizations to be expressed in the resolvent basis via a data-deduced, low-rank, cross-spectral density matrix. We find that a single resolvent mode reconstructs the most energetic regions of the acoustic field across Strouhal numbers, St=[0-1], and azimuthal wavenumbers, m=[0,2]. Finally, we present a simple function that results in a rank-1 resolvent model agreeing within 2 dB of the peak noise for both jets.

The Signature of Geomagnetic Field External Drivers in Virtual Observatory 30‐day Means Derived From Swarm Data

AbstractThe separation of contributions from different sources in the magnetic field signal measured at satellite altitude is an open challenge. An approach to this problem, using Principal Component Analysis, is here applied to geomagnetic external field series at Virtual Observatories (VOs). These series are computed from an enlarged data set of Swarm data covering all local times and all geomagnetic activity levels between January 2014 and December 2019. For each 30‐day time window, the Equivalent Source Dipole technique is used to reduce all measurements inside a cylinder to one single “observation” at its axis and 500 km altitude. Our results reveal a first principal mode with dipolar geometry and time variation following very closely the RC‐index of geomagnetic activity. They display a resolved second principal mode with annual periodicity and of approximately zonal quadrupolar radial pattern, reminiscent of results in a previous study using VO series from a filtered satellite data set and with lower time resolution. We resort to the recent comprehensive model CM6 to identify a possible source for this second mode. We propose that the dipolar mode is the expression of the magnetospheric ring current dynamics, at 30‐day time resolution, and the quadrupolar mode is the expression of the annual asymmetry between local summer and winter Sq current vortices. Two fainter modes could be related to the equinoctial amplification of Sq vortices and the ionospheric dynamo modulation by nonmigrating tides. We show that a more uniform local time sampling could contribute to better resolve ionospheric structures.

Numerical dispersion effects on the energy cascade in large-eddy simulation

Implicitly filtered large-eddy simulation (LES) is by nature numerically under-resolved. With the sole exception of Fourier-spectral methods, discrete numerical derivative operators cannot accurately represent the dynamics of all of the represented scales. Since the resolution scale in an LES usually lies in the inertial range, these poorly represented scales are dynamically significant and errors in their dynamics can affect all resolved scales. This Letter is focused on characterizing the effects of numerical dispersion error by studying the energy cascade in LES of convecting homogeneous isotropic turbulence. Numerical energy and transfer spectra reveal that energy is not transferred at the appropriate rate to wavemodes where significant dispersion error is present. This leads to a deficiency of energy in highly dispersive modes and an accompanying pile up of energy in the well resolved modes, since dissipation by the subgrid model is diminished. An asymptotic analysis indicates that dispersion error causes a phase decoherence between triad interacting wavemodes, leading to a reduction in the mean energy transfer rate for these scales. These findings are relevant to a wide range of LES, since turbulence commonly convects through the grid in practical simulations. Further, these results indicate that the resolved scales should be defined to not include the dispersive modes.

An azimuthal Fourier domain formulation of the Ffowcs Williams and Hawkings equation.

The acoustic field of high-speed turbulent jets is dominated by a small number of low-wavenumber azimuthal Fourier modes. Accordingly, it is of interest to directly obtain individual azimuthal modes of the acoustic field from simulation data or models of the jet near-field. To this end, we manipulate the Ffowcs Williams and Hawkings (FW-H) equation to obtain a new formulation that lives entirely in the azimuthal Fourier domain-it delivers individual azimuthal modes of the acoustic field as a function of the same azimuthal modes of the near-field FW-H source term or, upon linearization of the source terms, as a function of the same azimuthal modes of the near-field flow variables. As an added benefit, all surface integrals are converted into line integrals in the streamwise-radial plane. After verifying and validating our formulation using a monopole problem with an exact solution and large-eddy simulation data, respectively, we show how our method can be used to efficiently and naturally compute the acoustic field associated with resolvent modes of a Mach 1.5 jet, thus avoiding the need to compute the modes on a large computational domain to capture their acoustic radiation.

Efficient computation of global resolvent modes

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Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets

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