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.

/pdf/spectral-proper-orthogonal-decomposition-and-its-422122q6iv.pdf

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

With the advent of low noise high bypass ratio turbofan engines airframe noise gained significant importance with respect to the overall aircraft noise impact around airports. Already around 1970 airframe noise, originating from flow around the landing gears and high-lift devices, was recognized as a potential “lower aircraft noise barrier” at approach and landing. Since then, the outcome of extensive acoustic flight tests and aeroacoustic wind tunnel experiments enabled a detailed description and ranking of the major airframe noise sources and the development of noise reduction means. In the last decade advances in numerical and experimental tools led to a better understanding of complex noise source mechanisms. Efficient noise reduction technologies were developed for landing gears while the benefits of high-lift noise reduction means were often compensated by a simultaneous degradation in aerodynamic performance. The focus of this paper is not on the historical sequence of airframe noise research but rather aims to provide a concise survey of the achievements in airframe noise source description and reduction over the last 40 years worldwide. Due to the vast amount of work focused on a variety of airframe noise problems, this review can only provide examples but does not claim to be complete.

Almost 40 Years of Airframe Noise Research: What Did We Achieve?

We present applications of modal analysis techniques to study, model, and control canonical aerodynamic flows. To illustrate how modal analysis techniques can provide physical insights in a complementary manner, we selected four fundamental examples of cylinder wakes, wall-bounded flows, airfoil wakes, and cavity flows. We also offer brief discussions on the outlook for modal analysis techniques, in light of rapid developments in data science.

https://arc.aiaa.org/doi/pdf/10.2514/1.J058462

Modal Analysis of Fluid Flows: Applications and Outlook

https://hal.archives-ouvertes.fr/hal-00518478/document

A high-order finite volume method for systems of conservation laws-Multi-dimensional Optimal Order Detection (MOOD)

Part I. The Surface Vorticity Boundary Integral Method of Fluid Flow Analysis: 1. The basis of surface singularity modelling 2. Lifting bodies, two-dimensional aerofoils and cascades 3. Mixed-flow and radial cascades 4. Bodies of revolution, ducts and annuli 5. Ducted propellers and fans 6. Three-dimensional and meridional flows in turbo-machines 7. Free vorticity shear layers and inverse methods Part II. Vortex Dynamics and Vortex Cloud Analysis: 8. Vortex dynamics in inviscid flows 9. Simulation of viscous diffusion in discrete vortex modelling 10. Vortex cloud modelling by the boundary integral method 11. Further development and applications of vortex cloud modelling to lifting bodies and cascades 12. Use of grid systems in vortex dynamics and meridional flows Appendix.

Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems: Vortex cloud modelling by the boundary integral method

The present investigation of aerofoil self-noise generation and propagation concerns the effects of mean flow and quadrupole sources on the broadband noise that arises from the interaction of turbulent boundary layers with the aerofoil trailing edge and the tonal noise that arises from vortex shedding generated by laminar boundary layers and trailing-edge bluntness. Compressible large-eddy simulations (LES) are conducted for a NACA0012 aerofoil with rounded trailing edge for four flow configurations with different angles of incidence, boundary layer tripping configurations and free-stream Mach numbers. The Reynolds number based on the aerofoil chord is fixed at . The acoustic predictions are performed by the Ffowcs Williams & Hawkings (FWH) acoustic analogy formulation and incorporate convective effects. Surface and volume integrations of dipole and quadrupole source terms appearing in the FWH equation are performed using a three-dimensional wideband multi-level adaptive fast multipole method (FMM) in order to accelerate the calculations of aeroacoustic integrals. In order to validate the numerical solutions, flow simulation and acoustic prediction results are compared to experimental data available in the literature and good agreement is observed in terms of both aerodynamic and aeroacoustic results. For low-Mach-number flows, quadrupole sources can be neglected in the FWH equation and mean flow effects appear only for high frequencies. However, for higher speeds, convection effects are relevant for all frequencies and quadrupole sources have a more pronounced effect for medium and high frequencies. The convective effects are most readily observed in the upstream direction.

Convective effects and the role of quadrupole sources for aerofoil aeroacoustics

We present a numerical methodology for construction of reduced-order models (ROMs) of fluid flows through the combination of flow modal decomposition and regression analysis. Spectral proper orthogonal decomposition is applied to reduce the dimensionality of the model and, at the same time, filter the proper orthogonal decomposition temporal modes. The regression step is performed by a deep feedforward neural network (DNN), and the current framework is implemented in a context similar to the sparse identification of nonlinear dynamics algorithm. A discussion on the optimization of the DNN hyperparameters is provided for obtaining the best ROMs and an assessment of these models is presented for a canonical nonlinear oscillator and the compressible flow past a cylinder. Then the method is tested on the reconstruction of a turbulent flow computed by a large eddy simulation of a plunging airfoil under dynamic stall. The reduced-order model is able to capture the dynamics of the leading edge stall vortex and the subsequent trailing edge vortex. For the cases analysed, the numerical framework allows the prediction of the flow field beyond the training window using larger time increments than those employed by the full-order model. We also demonstrate the robustness of the current ROMs constructed via DNNs through a comparison with sparse regression. The DNN approach is able to learn transient features of the flow and presents more accurate and stable long-term predictions compared to sparse regression.

Construction of reduced-order models for fluid flows using deep feedforward neural networks

Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet

This work describes the implementation and analysis of high-order accurate schemes applied to high-speed flows on unstructured grids. The class of essentially non-oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third-and fourth-order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2-D Euler equations in a cell centred finite volume context. High-order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge-Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high-order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high-speed flow simulations are presented with the objective of assessing the implemented capability.

High‐order ENO and WENO schemes for unstructured grids

Installed jet noise is studied by means of a simplified configuration comprising a flat plate in the vicinity of a round jet. The effects of Mach number, jet-plate radial distance, and trailing-edge sweep angle are explored. Acoustic measurements are performed using a traversable 18-microphone azimuthal array, providing pressure data at 360 points on a cylindrical surface surrounding the jet-plate system. Key observations include a decrease, with increasing Mach number, of the relative level of the scattered field in comparison to the uninstalled jet; an exponential dependence of the scattered sound pressure level on the radial jet-plate separation; and considerable sideline noise reductions with increasing sweep angle, with which there is an overall reduction in acoustic efficiency. The measurements are compared with results obtained using a kinematic wavepacket source model, whose radiation is computed in two ways. A TGF for a semi-infinite flat plate is used to provide a low-order approximation of the scattering effect. Use of a more computationally intensive boundary element method provides additional precision. Good agreement between model predictions and experiment, encouraging from the perspective of low-cost prediction strategies, demonstrates that the models comprise the essential sound generation mechanisms.

William Wolf

Papers

Convective effects and the role of quadrupole sources for aerofoil aeroacoustics

Construction of reduced-order models for fluid flows using deep feedforward neural networks

Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet

High‐order ENO and WENO schemes for unstructured grids

Scattering of turbulent-jet wavepackets by a swept trailing edge