Instability Waves in High-Speed Jets: Near-and Far-Field DNS/LES Data Analysis

Abstract

Solutions from inviscid linear stability theory (LST) and linear parabolized stability equations (PSE) were used to quantify the extent to which instability waves represent actual near-field dynamics in high-speed forced turbulent jets. The mean and unsteady flow data were taken from a well-validated large-eddy simulation (LES) database (Bodony and Lele, 2005). The jets and their operating conditions were heated high subsonic ( M j = 0.97, T j/ T∞ = 2.3, Re D = 84,000) and unheated supersonic ( M j = 1.95, T j/T∞ = 0.56, Re D = 336,000) jets with the same jet exit velocities. The coherent components of the turbulent jet flow fields were extracted using a filter based on proper orthogonal decomposition (POD). To examine the effects of nonlinear interactions and to test the decomposition method, forced laminar jets were also considered; the flow data were obtained from a direct numerical simulation (DNS) database for an unheated M j = 0.90 and Re D = 4,100 jet (Ryu, 2010). Two methods were used to extract instability waves from the nonlinear flow fields: first, bi-orthogonal relations between regular and adjoint instability waves were formulated from linearized Euler equations to determine the amplitude coefficients of the instability waves in nonlinear flows; then, a least-squares fit was applied to solutions from the PSE, POD, and LES data for the same purpose. The noise radiated by the deduced instability wave modes, POD, and LES were computed using the Kirchhoff surface integral method. Frequencies St = fD j/ U j = 0.1 ∼ 0.5 at azimuthal modes m = 0, 1, and 2 were compared. The adjoint decomposition method showed promising results only for the laminar jet. The eigenfunction shape in the radial direction and the growth and decay of energy in the streamwise direction were well captured. However, this method produced errors in analyses of turbulent jets due to sensitivity in regions where the mode had saturated and was decaying. The results of PSE and POD showed reasonably good agreement in the near field for the first helical mode ( m = 1). However, there was only limited agreement in the far field, and among other modes, in contrast to the good agreements described in recent studies for supersonic unforced jets. This is due to the strong nonlinear effects in forced turbulent jets considered in the present study. This finding confirms a limitation of linear stability theories when predicting the wavepacket dynamics of forced turbulent jets. The PSE solution for the subsonic jet showed very weak acoustic radiation, and the far-field noise displayed strong sensitivity to the location of the Kirchhoff surface. This suggests a limitation of direct propagation approaches for the PSE solutions of moderately heated subsonic jets.