Oliver T. Schmidt

Modal Analysis of Fluid Flows: An Overview

Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades. For a non-expert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (Balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet.

Viscous and inviscid instabilities of supersonic flow in a streamwise corner

The linear stability of supersonic ow in a streamwise corner is examined. A steady base ow at a free-stream Mach number of Ma = 1.5 is calculated as a solution to the parabolized Navier-Stokes equations. Temporal growth rates for incoming and outgoing waves are compared. A new inviscid odd-symmetric corner mode not present in the incompressible spectrum is identi ed. Additionally, fast and slow traveling acoustic modes are calculated in an enlarged computational domain and categorized with respect to bisector-symmetry and wall-boundedness.

Amplitude scaling of turbulent-jet wavepackets

Wavepackets modelling large-scale coherent structures are related to the peak noise radiation by subsonic jets. Such wavepacket models are well developed in the literature, and are often based on a linearization of the Navier-Stokes system; solutions of the resulting linear problem have a free amplitude, which can be obtained by comparison with experiments or simulations. In this work we determine amplitudes of turbulent-jet wavepackets by comparing large-eddy simulation (LES) data from Br`es et al. of a Mach 0.9 jet and fluctuation fields using the parabolized stability equations (PSE) model (Sasaki et al.). Projection of the leading mode from spectral proper orthogonal decomposition (SPOD), applied to the LES data, onto the PSE model solutions is a way to determine the free amplitude, and by analyzing such amplitudes for different Strouhal numbers and azimuthal modes of the turbulent jet, it is possible to notice a clear pattern of the scaling factor with varying St. Azimuthal wavenumbers m = 0, 1 and 2 show an exponential dependence of wavepacket amplitude with Strouhal number. This sheds light on how wavepackets amplitudes behave and how they are excited upstream.

Large-eddy simulations of co-annular turbulent jet using a Voronoi-based mesh generation framework

Large eddy simulations are performed for a cold ideally-expanded dual-stream jet issued from cylindrical co-axial nozzles, with supersonic primary stream (Mach number M1 = 1.55) and subsonic secondary stream (M2 = 0.9). The geometry includes the internal screw holes used to fasten the two nozzles together and to the plenum chamber. These slanted cylindrical holes over which the secondary stream flows were not covered in the experiment and were seamlessly captured in the computational mesh thanks to a novel grid generation paradigm based on the computation of Voronoi diagrams. A simulation with the screw holes covered is also performed and the preliminary results tends to indicate that these features have minimal impact on the flow and acoustic fields for the present operating conditions. As expected, the present dual-stream configuration with subsonic annular stream surrounding the primary supersonic stream features a reduced shear-layer growth, a longer potential core and a lack of strong Mach wave radiation. A long LES database is currently being collected for analysis and modeling of wavepackets and noise sources in such complex turbulent jets.

Resolvent analysis for jet noise source identification

We use resolvent analysis and spectral proper orthogonal decomposition (SPOD) to deduce the acoustic sources for an isothermal Mach 1.5 round jet. Both physics-based resolvent analysis and data-driven SPOD (using a high-fidelity, experimentally-verified, large-eddy simulation (LES) database) provide a basis for predicting the perturbation field. Singular value decomposition of the resolvent operator based upon the LES baseflow provides optimal volumetric forcing modes, or sources, and their associated linear responses. To identify physically relevant resolvent modes, comparisons are made between the highest gain responses and the highest energy SPOD modes computed directly from LES realizations. The prevalence of the associated resolvent forcing modes in the data are then assessed by projecting them onto the full LES nonlinear terms. The resulting distributions are presented and a jet noise model leveraging these forcing statistics is discussed.

Spectral analysis of jet turbulence

Informed by 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 the 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 sub-dominant 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.

One Way Navier-Stokes and resolvent analysis for modeling coherent structures in a supersonic turbulent jet

A linear analysis of the mean flow of an isothermal ideally-expanded Mach 1.5 turbulent jet is conducted. Optimal response modes describing the fluctuating hydrodynamic and acoustic fields are obtained in a computationally efficient way by spatially marching the linearized One-Way Navier-Stokes equations. For this purpose, an adjoint-based optimization framework is proposed and demonstrated for calculating optimal boundary conditions and optimal volumetric forcing. The optimal modes are validated against modes obtained in terms of global resolvent analysis. Two scenarios are considered in the present analysis. In the first case, no restrictions are applied to the spatial forcing distribution. In the second scenario, the forcing is restricted to the nozzle plane. The resulting optimal and suboptimal modes are compared to spectral proper orthogonal modes obtained from a high-fidelity large eddy simulation. The implications of these observations are discussed in detail.

Wavepacket intermittency and its role in turbulent jet noise

The intermittent behavior of large-scale coherent structures in turbulent jets is studied. These structures are the primary source of jet noise, and their emitted sound is in turn characterized by rapid amplitude modulations of the pressure field. These high-energy bursts are well portrayed in the frequency-time domain by means of time-local analysis techniques. Scaleograms obtained from wavelet transforms of a single-point pressure signals, for example, enable the identification of such loud events at specific locations. Our interest, however, is in the time-local behavior of the coherent structures as a whole to gain a physical understanding of jet noise generation. For that purpose, a time series of large-eddy simulation snapshots is projected onto two sets of modal basis functions that describe the large-scale structures in the frequency-domain. The first modal basis consists of frequency-domain, or spectral, POD modes that are empirically deduced from the data. The second basis is comprised of resolvent response modes that are obtained from a linear frequency-response analysis of the mean flow. The proposed method allows us to visualize the intermittent behavior of the modal solutions in the frequency-time domain in terms of magnitude contours of the projection coefficient. The results can then be interpreted in an analogous way to wavelet scaleograms. The limitations, benefits and the potential of the method to yield a low-order representation of the flow in the time domain are discussed.

Leading-Edge Receptivity to Free-Stream Vorticity of Streamwise Corner-Flow

The receptivity of a streamwise corner-flow configuration with super-elliptical leading edges is investigated by means of direct numerical simulations. For this purpose, vortical perturbations are induced into the computational domain upstream of the leading edge to study the influence of frequency and wave obliqueness on perturbation amplification and pattern formation. Additionally, we investigate the flow response to artificial homogeneous and isotropic free-stream turbulence. It is found that the near-corner region is particularly sensitive to vortical disturbances aligned with the free-stream, and prone to the formation of anti-symmetric disturbance patterns that lead to high local amplification levels.