International Journal of Aerospace Engineering

Volume 2015, Article ID 751029, 7 pages

http://dx.doi.org/10.1155/2015/751029

## Frequencies of Transverse and Longitudinal Oscillations in Supersonic Cavity Flows

^{1}Department of Energy and Environmental Engineering, Kyushu University, 6-1 Kasuga-Koen, Kasuga, Fukuoka 816-8580, Japan^{2}Power Systems Plant Engineering Department, Mitsubishi Heavy Industries, 2-1-1 Shinhama Arai-Cho, Takasago, Hyogo 676-8686, Japan^{3}Plant and Machinery Division, Nippon Steel & Sumikin Engineering, 46-59 Nakabaru, Tobata-Ku, Kitakyushu, Fukuoka 804-8505, Japan

Received 29 July 2015; Accepted 4 October 2015

Academic Editor: Keh-Chin Chang

Copyright © 2015 Taro Handa 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

A supersonic flow over a rectangular cavity is known to oscillate at certain predominant frequencies. The present study focuses on the effect of the cavity length-to-depth () ratio on the frequency for a free-stream Mach number of 1.7. The pressure oscillations are measured by changing the ratio from 0.5 to 3.0, and the power spectral density is calculated from the temporal pressure signals for each ratio. The results demonstrate that the spectral peaks for an ratio of less than ~1 and greater than ~2 are accounted for by the feedback mechanisms of the transverse and longitudinal oscillations, respectively. The results also demonstrate that the spectral peaks in the transition (1 <~ <~ 2) are accounted for by either of the two feedback mechanisms of transverse and longitudinal oscillations; that is, the flows under the transition regime oscillate both transversely and longitudinally.

#### 1. Introduction

A supersonic flow over a rectangular cavity is known to oscillate at certain predominant frequencies. Cavity-induced pressure oscillations cause structural fatigue and sound noise. On the other hand, such oscillations are advantageous in the enhancement of supersonic mixing [1, 2]; that is, the injectant is subjected to disturbances such as pressure waves or a pulsating flow generated by a cavity-induced flow oscillation. A linear stability analysis provides the principle in which the growth rate of a supersonic shear-layer depends on the frequency of the disturbance imposed on the shear layer [3]. Hence, it is important for an enhancement of the supersonic mixing to estimate the frequency of a cavity-induced flow oscillation.

According to Rockwell and Naudascher [4], cavities can be classified into two types with respect to the oscillation direction (direction of pressure-wave propagation) inside the cavity. In general, a cavity is called a “shallow cavity” when the flow oscillates predominantly in a longitudinal direction and a “deep cavity” when the flow oscillates predominantly in a transverse direction. According to Rockwell and Naudascher [4], the transition between transverse and longitudinal oscillations occurs at .

The feedback mechanism of longitudinal oscillation can be expressed as Rossiter’s model [5] or modified Rossiter’s model [6]. Using such models, the predominant oscillation frequencies can be estimated rather accurately for shallow cavities, but the estimation fails for deeper cavities because the effect of the cavity depth is not included in these models. Based on a flow visualization, Handa et al. [7] recently developed a model accounting for the feedback mechanism of a transverse oscillation [8]. Their model estimates the dominant frequencies well for deep cavities whose length-to-depth ratios are lower than 1.0. Before the development of their model, it had been believed for a long time that the empirical formula of East [9] could be used to estimate the predominant oscillation frequencies for supersonic deep-cavity flows. However, to the best of our knowledge, East’s formula cannot reproduce any experimental results for supersonic deep-cavity flows.

When the ratio is increased from a deep to shallow cavity, a transition from a transverse oscillation into a longitudinal oscillation should occur. However, it remains unclear how a flow oscillates under the transition regime because the oscillation frequencies have yet to be discussed in detail under the transition regime; this is due to the fact that, until the formula proposed by Handa et al., there were no formulae developed for estimating the oscillation frequencies for transverse oscillations. In the present study, the pressure oscillations are measured by changing the ratio between 0.50 and 3.0. Peaks appearing in the pressure oscillation spectra are assigned to the oscillation modes accounted for by either of the two feedback mechanisms of transverse and longitudinal oscillations; further, how the oscillation frequencies change with the ratio is discussed herein.

#### 2. Experiments

Figure 1 shows the detailed structure of the test duct. Nitrogen gas flows into the test duct through a nozzle at a Mach number of 1.68. Detailed descriptions of the flow issuing from the nozzle are provided elsewhere [7, 8]; on the basis of the boundary layer thickness, the flow issuing from the nozzle results in a boundary layer thickness of 0.73 mm and a Reynolds number of 1.03 × 10^{4}. The test duct has a rectangular cross section with a height and width of 10.5 and 28.0 mm, respectively. With a width of 28.0 mm, the cavity spans the full width of the duct. The cavity length is 14.0 mm, whereas the depth is adjustable. In the present experiments, is varied from 4.7 to 28.0 mm, which implies that the ratio varies from 0.50 to 3.0.