Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

This paper investigates the effects of finite flat porous extensions to semi-infinite impermeable flat plates in an attempt to control trailing-edge noise through bio-inspired adaptations. Specifically the problem of sound generated by a gust convecting in uniform mean steady flow scattering off the trailing edge and the permeable–impermeable junction is considered. This set-up supposes that any realistic trailing-edge adaptation to a blade would be sufficiently small so that the turbulent boundary layer encapsulates both the porous edge and the permeable–impermeable junction, and therefore the interaction of acoustics generated at these two discontinuous boundaries is important. The acoustic problem is tackled analytically through use of the Wiener–Hopf method. A two-dimensional matrix Wiener–Hopf problem arises due to the two interaction points (the trailing edge and the permeable–impermeable junction). This paper discusses a new iterative method for solving this matrix Wiener–Hopf equation which extends to further two-dimensional problems, in particular those involving analytic terms that exponentially grow in the upper or lower half-planes. This method is an extension of the commonly used ‘pole removal’ technique and avoids the need for full matrix factorisation. Convergence of this iterative method to an exact solution is shown to be particularly fast when terms neglected in the second step are formally smaller than all other terms retained. The new method is validated by comparing the iterative solutions for acoustic scattering by a finite impermeable plate against a known solution (obtained in terms of Mathieu functions). The final acoustic solution highlights the effects of the permeable–impermeable junction on the generated noise, in particular how this junction affects the far-field noise generated by high-frequency gusts by creating an interference to typical trailing-edge scattering. This effect results in partially porous plates predicting a lower noise reduction than fully porous plates when compared to fully impermeable plates.

The scattering of sound by a finite rigid plate with a finite poroelastic extension interacting with an unsteady acoustic source is investigated to determine the effects of porosity, elasticity and the length of the extension when compared to a purely rigid plate. The problem is solved using the Wiener–Hopf technique, and an approximate Wiener–Hopf factorisation process is implemented to yield reliable far-field results quickly. Importantly, finite chord-length effects are taken into account, principally the interaction of a rigid leading-edge acoustic field with a poroelastic trailing-edge acoustic field. The model presented discusses how the poroelastic trailing-edge property of owls’ wings could inspire quieter aeroacoustic designs in bladed systems such as wind turbines, and provides a framework for analysing the potential noise reduction of these designs.

This paper presents an analytic solution for aerodynamic noise generated by an unsteady wall pressure gust interacting with a spanwise-variable trailing edge in a background steady uniform flow. Viscous and nonlinear effects are neglected. The Wiener–Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to permit analytical progress. The solution is obtained in terms of a tailored modal expansion in the spanwise coordinate; however, only finitely many modes are cut-on, therefore the far-field noise can be quickly evaluated. The solution gives insight into the potential mechanisms behind the reduction of noise for plates with serrated trailing edges compared to those with straight edges. The two mechanisms behind the noise reduction are an increased destructive interference in the far field, and a redistribution of acoustic energy from low cut-on modes to higher cut-off modes. Five different test-case trailing-edge geometries are considered. The analytic solution identifies which geometries are most effective in different frequency ranges: geometries which promote destructive interference are best at low frequencies, whilst geometries which promote a redistribution of energy are better at high frequencies.

This paper presents an analytic solution for gust–aerofoil interaction noise for flat plates with spanwise-varying periodic leading edges in uniform mean flow. The solution is obtained by solving the linear inviscid equations via separation of variables and the Wiener–Hopf technique, and is suitable for calculating the far-field noise generated by any leading edge with a single-valued piecewise linear periodic spanwise geometry. Acoustic results for homogeneous isotropic turbulent flow are calculated by integrating the single-gust solution over a wavenumber spectrum. The far-sound pressure level is calculated for five test-case geometries; sawtooth serration, slitted $v$ -root, slitted $u$ -root, chopped peak and square wave, and compared to experimental measurements. Good agreement is seen over a range of frequencies and tip-to-root ratios (varying the sharpness of the serration). The analytic solution is then used to calculate the propagating pressure along the leading edge of the serration for fixed spanwise wavenumbers, i.e. only the contribution to the surface pressure which propagates to the far field. Using these results, two primary mechanisms for noise reduction are discussed; tip and root interference, and a redistribution of energy from cuton modes to cutoff modes. A secondary noise-reduction mechanism due to nonlinear features is also discussed and seen to be particularly important for leading edges with very narrow slits.