Steepened Mach waves near supersonic jets: study of azimuthal structure and generation process using conditional averages

Abstract

The azimuthal structure and the generation process of steepened acoustic waves are investigated in the near field of temporal round jets at Mach numbers of 2 and 3. Initially, the shear layers of the jets are in a laminar state and display instability waves whose main properties are close to those predicted from linear temporal analysis. Then, they transition to a turbulent state and generate high-intensity Mach waves displaying sharp compressions typical of those recorded for jets producing crackle noise. These waves are first shown to be poorly reproduced when only the axisymmetric mode is considered, but to be well captured with the first five azimuthal modes. Their generation process is investigated by performing conditional averages of the flow and acoustic fields triggered by the detection of intense positive pressure peak close to the jets. No steepened waves are visible in the conditionally averaged pressure profiles when the procedure involves only one azimuthal mode at a time. However, sharp compressions are obtained based on the first five modes taken together. In that case, the steep compressions are correlated over a limited portion of the jet circumference and are steeper as more azimuthal modes are considered. Moreover, a direct link is established between the steepened waves and the supersonic convection of large-scale coherent flow structures located in the supersonic core of the jets. This indicates that these waves constitute an extreme, nonlinear case of Mach wave radiation by these structures. In addition, the capacity of flow structures to generate sharp, steepened waves is related to their shapes. More particularly, flow structures with a large extent in the radial direction are shown to produce stronger and steeper Mach waves than those that are elongated in the flow direction.

Potential-core closing of temporally developing jets at Mach numbers between 0.3 and 2: Scaling and conditional averaging of flow and sound fields

“…The forcing strength is therefore of α exc = 0.04u j , resulting in a turbulence intensity of 4% at t = 0 in that case. The effects of the forcing strength on the early development of the instability waves were examined for a jet at M = 2 and Re D = 12 500 in Pineau and Bogey [42]. Except for the amplitude, very similar results were found for α exc = 0.0025u j , 0.01u j , and 0.04u j at t = 5r 0 /u j .…”

“…The numerical framework is identical to that used in recent simulations of spatially developing [49][50][51] and temporally developing [25,42] round jets. The simulations are carried out using an in-house solver of the three-dimensional filtered compressible Navier-Stokes equations in cylindrical coordinates (r, θ, z) based on low-dissipation and low-dispersion, high-order explicit schemes.…”

“…This can be related to the fact that 46.6%, 47.7%, and 34.6% of the energy are contained in the axisymmetric mode for M = 0.3, 0.6, and 1.3, but only 14.8% for M = 2. This reduction of the relative importance of the axisymmetric mode in the hydrodynamic pressure field with the Mach number likely results from the predominance of higher modes of instability waves at supersonic Mach numbers [42]. In terms of spectral content, the spectra peak around k z r 0 = 0.8 for M 1.3, yielding a wavelength λ 8r 0 .…”

“…It is maximum and equal to r = 0.2r 0 for r 16r 0 for the jets at M 1.3 and to r = 0.05r 0 for r 4r 0 for the jet at M = 2, yielding wave numbers of k z r 0 = 7.8 and 31, respectively, for a wave discretized by four points per wavelength. The mesh is finer for the Mach 2 jet than for the others in order to allow for the propagation of shocked waves in the near field of the former jet [42]. The use of n θ = 256 and 512 points in the azimuthal direction leads to r θ = 0.024r 0 and 0.012r 0 at r = r 0 .…”

“…Moreover, in order to highlight the noise generation mechanism, cross-correlations will be computed between centerline flow quantities and pressure fluctuations in the hydrodynamic and near acoustic fields of the jets. More importantly, conditional averaging will also be performed, as was, for instance, the case in the recent studies of Schmidt and Schmid [41] and Pineau and Bogey [42] to isolate the flow event precursor of Mach-wave radiation in supersonic jets. In practice, the DNS fields will be synchronized with the minimum values of centerline velocity at the time of potential-core closing, when the latter are below a given threshold.…”