A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean

Abstract

The flows around a NACA 0018 airfoil at a chord-based Reynolds number of Re = 10250 and angles of attack of α = 0 • and α = 10 • are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity profiles are data-assimilated so that they are solutions of the incompressible Reynolds-averaged Navier-Stokes equations forced by Reynolds stress terms which are derived from experimental data. Spectral proper orthogonal decompositions of the velocity fluctuations and nonlinear forcing suggest different modelling approaches should be taken based on the angle of attack under consideration. For the α = 0 • case, the cross-spectral density tensors of both the velocity fluctuations and nonlinear forcing are low-rank at the shedding frequency and its higher harmonics. In the α = 10 • case, low-rank behaviour is observed for the velocity fluctuations in two bands of frequencies. Resolvent analysis of the data-assimilated means identifies lowrank behaviour only in the vicinity of the shedding frequency for α = 0 • and none of its harmonics. The resolvent operator for the α = 10 • case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by spectral proper orthogonal decomposition. It is also shown that the second linear mechanism, corresponding to the Kelvin-Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as a mean flow for resolvent analysis. This is due to the fact that experimental data are missing near the leading edge of the airfoil. The α = 0 • case is classified as an oscillator where the flow is organized around an intrinsic instability mechanism while the α = 10 • case behaves like an amplifier whose forcing is unstructured. For both cases, resolvent modes resemble those from spectral proper orthogonal decomposition when the operator is low-rank. To model the higher harmonics where this is not the case, we add parasitic resolvent modes, as opposed to classical resolvent modes which are the most amplified, by approximating the nonlinear forcing from limited triadic interactions of known resolvent modes. The amplifier case is modelled without parasitic modes at frequencies where the resolvent is low-rank. The two cases suggest that resolvent-based modelling can be achieved for more complex flows with limited experimental measurements and the nonlinear forcing need not be approximated unless the flow behaves like an oscillator.

Broadband reconstruction of inhomogeneous turbulence using spectral proper orthogonal decomposition and Gabor modes

“…For example, Towne et al (2015) and Schmidt et al (2018) found that the leading SPOD mode at each frequency captures only approximately 10 %-20 % of the flow energy in a turbulent jet, and that more than fifty modes are required at each frequency to capture 95 % of the energy. Similar observations have been made for other turbulent flows, including wakes, especially at high frequencies (Sanjose et al 2019;Symon, Sipp & McKeon 2019) In this paper, we propose to use Gabor modes to enrich the low-order SPOD flow reconstruction in order to recover truncated portions of the flow. Gabor modes are compact support wavepackets, with each mode carrying its own real valued wavevector, k, a real valued spatial location, x, and complex valued velocity vector,û.…”

The colour of forcing statistics in resolvent analyses of turbulent channel flows

“…However, the authors focused on the variance of the solenoidal forcing, so the analysis did not distinguish between the contribution of different frequencies. Symon, Sipp & McKeon (2019) isolated the contribution of each frequency by employing the spectral proper orthogonal decomposition (SPOD) (Lumley 1970;Picard & Delville 2000). However, they used data from particle image velocimetry (PIV) of the flow around a NACA0018 airfoil at a chord-based Reynolds number Re = 10 250, and the forcing is not decomposed in its solenoidal and irrotational parts.…”

Resolvent-based estimation of space–time flow statistics

“…The resolvent operator is derived from the Navier-Stokes equations linearized about the turbulent mean flow and constitutes a transfer function in the frequency domain between terms that are nonlinear and linear with respect to fluctuations to the mean. Resolvent analysis has proven to be a useful tool for understanding and modelling a wide range of flows, including wall-bounded flows (Sharma & McKeon 2013;Morra et al 2019), free-shear flows (Jeun, Nichols & Jovanović 2016;Schmidt et al 2018) and aerodynamic wakes (Thomareis & Papadakis 2018;Symon, Sipp & McKeon 2019;Yeh & Taira 2019). Whereas Beneddine et al (2016) constructed their model using only the first singular mode of the resolvent operator (obtained via singular value decomposition), our model relaxes this a priori assumption and allows the known data to self-select the relevant portion of the resolvent operator.…”