Linear Stability Analysis of Chevron Jet Profiles

We investigate the linear stability characteristics of the mean velocity profiles produced by chevron nozzles. We show that chevron instability waves can be decomposed into azimuthal modes analogously to those of round jets. This facilitates a direct comparison of growth rates and mode structure between different nozzles. We find that the three nozzles used in this study share a set of modes, referred to as primary modes. In addition, we find that there exist modes unique to the chevrons nozzles, termed secondary modes. While chevron jets possess a much larger number of unstable modes, the modes with lowest azimuthal structure show strong suppression of growth rates in two different chevron jets. Some preliminary implications on sound generation are discussed. INTRODUCTION With turbofan bypass ratios approaching practical limits, different approaches to noise reduction have been pursued, including the machining of serrations, or chevrons, into the nozzle lip. Examples of such nozzles, commonly referred to as chevron nozzles, are shown in Fig. 1. The chevrons generate streamwise vortices that enhance mixing and shorten the potential core. They also reduce noise at low frequencies and aft angles, but increase noise at high frequencies. The high frequency penalty is explained by the increased levels of small scale turbulence, while the noise reduction at lower frequencies, associated with the large scale structures of the flow, is not well understood. At present, design and deployment of chevron nozzles is based on exhaustive and costly laboratory, and full-scale testing. Due to the complexity of the near-nozzle flow field, there are currently no physics-based prediction methods for noise reduction; properly resolving the large-scale flow structures is not yet possible, even with modeling tools such as Large Eddy Simulation. As a surrogate, we propose to compute the linear instability modes of the time-averaged chevron flow field, and to investigate the extent to which the chevrons modify the stability characteristics of an equivalent-thrust round jet. In a strict sense, the concept of a linear instability mode is not applicable to a spatially spreading flow such as a jet (see cartoon in Fig. 2, left). However, such flows can be assumed locally parallel if their spreading rate is negligible over fluctuation length-scales. Under these circumstances, cross sections of the jet, as shown in Fig. 2, right, are analyzed as if each one was a sample from an unbounded parallel mean flow. This approach was taken by [1], where the hydrodynamic pressure field of a round jet was shown to be consistent with that of the instability modes of the spreading mean flow1. The appropriateness of the quasi-parallel assumption for the chevron jet is analyzed as follows. The potential core length for the most aggressive chevron we consider is shorter than the round jet by about 2 diameters [2]. This corresponds to an increase in (azimuthally averaged) spread rate from about 0.18 for the straight nozzle to about 0.25 for the chevron nozzle. The PIV data we used in the present study also shows that the azimuthally averaged transverse mean velocity in the chevron jet has a maximum value of about 10% of the streamwise velocity at x/D = 2, whereas for the straight nozzle it is 8%. Thus the assumption of locally parallel mean flow is somewhat 1We note that [1] used the same dataset as this study, for nozzle SMC000 (see Fig. 1). 1 Copyright c © 2006 by ASME Figure 1. From left: straight nozzle, chevron nozzles with moderate/aggressive penetration. less appropriate for the chevron jet, but still reasonable as a first step. Standard quasi-parallel stability analysis of jet flows considers, at each streamwise position, a one-dimensional (Ordinary Differential) eigenvalue problem in the cross flow direction and is amenable to direct matrix solution or shooting algorithms. The chevron jet, on the other hand, is inhomogeneous in both coordinate directions normal to the stream and must be solved using a two-dimensional (Partial Differential) eigenvalue problem. As such it is much more computationally intensive than the onedimensional problem. To solve the chevron stability problem, we have developed an approach that couples a high-order accurate compressible flow solver with the iterative eigensystem solver ARPACK, described in the next sections. CALCULATION OF INSTABILITY WAVES Using far-field sound speed a∞, density ρ∞, and nozzle diameter D for non-dimensionalization, we linearize the flow field about a locally parallel mean flow having axial velocity U(y,z), and density ρ(y,z), where coordinate configuration is illustrated in Fig. 2. For this purpose, we assume that flow field variables can be decomposed into mean, and fluctuating components, as u = U +u, v = v, w = w, ρ = ρ+ρ, and p = 1/γ+ p, where primed quantities represent small fluctuations, and γ is the ratio between specific heats. Substituting these expressions into the Navier-Stokes equations, and only retaining terms first order in fluctuations yields ∂q ∂t +L q = 1 Re V q, (1) where q = [(ρu)′,(ρv)′,(ρw)′,ρ′,e′]T , Re is the Reynolds number, based on the jet diameter, kinematic viscosity and centerline velocity. L and V represent linear differential operators, used here for brevity; the full equations are shown in the appendix. Since we have assumed the mean flow to be locally parallel, the operators L and V are homogeneous in the flow direction, so that the solutions q can be decomposed as q(x,y,z, t) = q̂(y,z)e e−iωt , (2) PSfrag replacements Potential core Mixing layer x z y r θ PSfrag replacements Potential core Mixing layer x z