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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 analysis is a powerful tool for modelling and analysing transitional and turbulent flows and, in particular, for approximating coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this paper we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction, i.e. one in which the mean flow changes gradually. First, we derive a well-posed and convergent one-way equation describing the downstream-travelling waves supported by the linearized Navier–Stokes equations. The method is based on a projection operator that isolates downstream-travelling waves. Integrating these one-way Navier–Stokes (OWNS) equations in the slowly varying direction, which requires significantly less CPU and memory resources than a direct solution of the linearized Navier–Stokes equations, approximates the action of the resolvent operator on a forcing vector. Second, this capability is leveraged to compute approximate resolvent modes using an adjoint-based optimization framework in which the forward and adjoint OWNS equations are marched in the downstream and upstream directions, respectively. This avoids the need to solve direct and adjoint globally discretized equations, therefore bypassing the main computational bottleneck of a typical global resolvent calculation. The method is demonstrated using the examples of a simple acoustics problem, a Mach 1.5 turbulent jet and a Mach 4.5 transitional zero-pressure-gradient flat-plate boundary layer. The optimal OWNS results are validated against corresponding global calculations, and the close agreement demonstrates the near-parabolic nature of these flows.

Resolvent analysis has demonstrated encouraging results for modeling coherent structures in jets when compared against their data-educed counterparts from high-fidelity large-eddy simulations (LES). We formulate resolvent analysis as an acoustic analogy that relates the near-field forcing to the near-field pressure field and the far-field acoustics. We use an LES database of round, isothermal, Mach 0.9 and 1.5 jets to produce an ensemble of realizations for the acoustic field that we project onto a limited set of resolvent modes. In the nearfield, we perform projections on a restricted acoustic output domain, r/D = [5, 6], while the far-field projections are performed on a Kirchhoff surface comprising a 100-diameter arc centered at the nozzle. This allows the LES realizations to be expressed in the resolvent basis via a data-deduced, low-rank, cross-spectral density matrix. We observe substantial improvements to the acoustic field reconstructions with the addition of a RANS-derived eddyviscosity model to the resolvent operator and find that a single resolvent mode reconstructs the most energetic regions of the acoustic field across Strouhal numbers, St = [0 − 1], and azimuthal wavenumbers, m = [0, 2]. Finally, we present a simple function that results in a rank-1 resolvent model agreeing within 2dB of the peak noise for both jets.