International Journal of Aerospace Engineering

Volume 2017, Article ID 8532507, 7 pages

https://doi.org/10.1155/2017/8532507

## Spatial-Temporal Instability of an Inviscid Shear Layer

School of Astronautics, Beihang University, Beijing 100191, China

Correspondence should be addressed to Li-jun Yang; moc.361@77051362431

Received 23 November 2016; Revised 11 March 2017; Accepted 22 March 2017; Published 9 April 2017

Academic Editor: Corin Segal

Copyright © 2017 Qing-fei Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work, we explore the transition of absolute instability and convective instability in a compressible inviscid shear layer, through a linear spatial-temporal instability analysis. From linearized governing equations of the shear layer and the ideal-gas equation of state, the dispersion relation for the pressure perturbation was obtained. The eigenvalue problem for the evolution of two-dimensional perturbation was solved by means of shooting method. The zero group velocity is obtained by a saddle point method. The absolute/convective instability characteristics of the flow are determined by the temporal growth rate at the saddle point. The absolute/convective nature of the flow instability has strong dependence on the values of the temperature ratio, the velocity ratio, the oblique angle, and number. A parametric study indicates that, for a great value of velocity ratio, the inviscid shear layer can transit to absolute instability. The increase of temperature ratio decreases the absolute growth rate when the temperature ratio is large; the effect of temperature ratio is opposite when the temperature ratio is relatively small. The obliquity of the perturbations would cause the increase of the absolute growth rate. The effect of number is different when the oblique angle is great and small. Besides, the absolute instability boundary is found in the velocity ratio, temperature ratio, and number space.

#### 1. Introduction

A high mixing rate of the fuel and air is desired in scramjet engine for the propulsion of hypersonic aircraft, because the residence time of the fuel and air in the combustion chamber is very short. In the interest of the projected use of the scramjet engine, it is fundamental and also extremely important to understand the stability characteristics of compressible shear/mixing layers. Many experimental studies [1–3] suggested that the mixing rates of shear layers decrease as the Mach number increases from zero. Hence, a major concern in the development of scramjet is the mixing enhancement techniques. Imparting disturbances on the shear layer, which pulsate at some prescribed frequency, is a choice of mixing enhancement techniques. The prescribed frequency can be obtained through linear stability analysis of the compressible shear layer.

There have been numerous literatures on the topic of linear stability analysis of the compressible shear layer, including the earlier studies conducted by Lessen et al. [4, 5] Drazin and Davey [6] performed a temporal stability analysis of a compressible mixing layer, which has a hyperbolic tangent velocity profile and uniform temperature throughout the layer. Jackson and Grosch [7] reported the results of the inviscid spatial stability of a parallel compressible mixing layer. All these studies found multiple stability modes. Zhuang et al. [8] and Ragab [9] both found a strong stabilization effect on the flow when increasing Mach number. Ho and Huerre [10] summarized the studies on linear stability analysis of incompressible shear layer. For the studies on linear stability analysis of compressible shear layer, the reader can refer to the relevant literature in the monograph by Criminale et al. [11].

The studies above are all confined in the scope of temporal or spatial mode. When we study the stability characteristics of shear layers in spatial-temporal mode, which treat both spatial and temporal eigenvalue complex [12, 13], there are two distinct instabilities for spatial-temporal evolving disturbances: convective and absolute instabilities. The concept of absolute and convective instabilities was first put forward by Briggs [14] in the study of plasma instability and then introduced to classical hydrodynamic stability [15]. A flow is convectively unstable if the unsteady response to an impulsive perturbation grows along some rays that pass away from the forcing location but decays at the forcing location itself. A flow is absolutely unstable if the impulse response grows at the forcing location [16].

An absolutely unstable flow is not sensitive to external disturbances and initial conditions; consequently, the “flow management” techniques such as forcing the shear layer at some prescribed frequency could be useless. [17]. Thus, if we want to control the downstream evolution of the flow, it is essential to determine whether the shear/mixing layer is convectively or absolutely unstable. Kulikovskii and Shikina [18–20] studied the asymptotic behavior of localized perturbations on the surface of a shear discontinuity separating two homogeneous steady flows of ideal incompressible fluid in the linear approximation. The effect of surface tension, gravity forces, and viscosity is taken into account. Pavithran and Redekopp [21] investigated the absolute and convective instabilities for a subsonic mixing layer with the hyperbolic-tangent-like profiles in velocity and temperature fields. The study on transition of absolute and convective instability for compressible shear layer is rare. Caillol [22] analyzed the transition of absolute and convective instabilities for an inviscid mixing layer. The Mach number in his work attained high supersonic values. However, he did not consider the density stratification within the mixing layer. Large density stratification is possible for binary mixing layers at high pressures, which is usually encountered in power and propulsion system.

The aim of this paper is to track the transition between the absolute instability and convective instability in a compressible shear layer. The density stratification is taken into account. The effects of flow parameters on the spatial-temporal stability of a compressible shear layer are examined by observing whether an increase of the value of parameter tends to increase or decrease the value of absolute growth rate.

#### 2. Theoretical Framework

The base flow is a shear layer between two streams with its far-stream condition denoted by the subscripts and , as shown in Figure 1. The thickness of shear layer increases with increasing distance downstream, denoting that the base flow is nonparallel. Here, we made a locally parallel flow assumption, implying that the present results yield the instability characteristics of individual profiles to leading order when the shear layer thickness grows slowly with downstream distance [23]. The streamwise direction is defined as and the cross stream is defined as . In the present study, denotes the velocity, the density, and the temperature. Taking the momentum thickness and average mean velocity as the characteristic length and velocity scale, respectively, the flow properties can be normalized. Superscript denotes a nondimensional quantity.