Eddy viscosity for resolvent-based jet noise models

Abstract

Response modes computed via linear resolvent analysis have shown promising results for qualitatively modeling both the hydrodynamic and acoustic fields in jets when compared to data-deduced modes from high-fidelity, large-eddy simulations (LES). For an improved quantitative prediction of the near-and far-field, the role of Reynolds stresses must also be considered. In this study, we propose a methodology to deduce an eddy-viscosity model that optimally captures the nonlinear forcing of resolvent modes. The methodology is based on the maximization of the projection between resolvent analysis and spectral proper orthogonal decomposition (SPOD) modes using a Lagrangian optimization framework. For a Mach 0.4 round, isothermal, turbulent jet, four methods are used to increase the projection coefficients: linear damping, spatially constant eddy-viscosity field, a turbulent kinetic energy derived viscosity field, and an optimized eddy-viscosity field. The resulting projection coefficients for the optimized eddy-viscosity field between SPOD and resolvent can be increased to over 90% for frequencies in the range St = 0.35 − 1 with significant improvements to St < 0.35. We find that the use of a frequency-independent turbulent kinetic energy turbulent viscosity model produces modes closely inline with optimal results, providing a preliminary eddy-viscosity resolvent model for jets.

Resolvent-based estimation of space–time flow statistics

“…These references and others have shown that including an eddy viscosity substantially improves the predictive capabilities of linearized models of the Navier–Stokes equations for wall-bounded flows, and recent work has shown similar benefits for free-shear flows (Pickering et al. 2019). Of particular relevance, Morra et al.…”

Lift-up, Kelvin–Helmholtz and Orr mechanisms in turbulent jets

“…The influence of viscosity in the linear operator, A m , has previously been based upon the molecular viscosity, Re j , however, recent resolvent analyses incorporating an eddyviscosity model, Pickering et al (2019) for turbulent jets and Morra et al (2019) in channel flow, have shown substantial improvement for SPOD-resolvent comparisons. Thus, we proceed by including an eddy-viscosity model based upon the turbulent kinetic energy (TKE) suggested for turbulent jets by Pickering et al (2019). This model was primarily chosen due to its simplicity and availability of the corresponding quantities from the LES database.…”

“…l m is chosen as the width of the shear layer where the turbulent kinetic energy is more than 10% of its maximum value at each streamwise location, and the scaling constant c = 0.0065 is used considering the favorable SPOD-resolvent alignments previously shown over m = 0 − 3 and St = 0.05 − 1. With the inclusion of an eddy-viscosity model the forward operator becomes Pickering et al (2019)). The discretization scheme and boundary condition treatment are the same as those used in Schmidt et al (2018) and refer the reader to that paper for further numerical details.…”

“…The influence of viscosity on the linear operator, A m , was previously (Schmidt et al 2018) based on a spatially uniform Reynolds number using an altered molecular viscosity. However, recent resolvent analyses for turbulent jets (Pickering et al 2019) and channel flows (Morra et al 2019), have shown substantial improvement in SPOD-resolvent correspondence by using a mean-flow eddy-viscosity model on the fluctuations. We proceed here by including an eddy-viscosity model based upon the turbulent kinetic energy (TKE) suggested for turbulent jets by Pickering et al (2019).…”

Ambiguity in mean-flow-based linear analysis

“…In other studies, eddy viscosity models have been included in the linear operator. This can enhance the extent to which the resolvent modes match the observed turbulence structure (Cossu et al 2009;Hwang & Cossu 2010;Morra et al 2019;Pickering et al 2019). However, physical interpretation of the remaining, unmodelled forcing terms then becomes unclear, as the use of an eddy viscosity amounts to a partial modelling of the effects of nonlinear forcing from Reynolds stress fluctuations.…”