Spatial coherence of low-frequency unsteadiness associated with a normal shock wave

Abstract

The present data illustrate a unique analysis approach to explore the physics of shock-wave-flow interactions, based upon correlation and spectral analysis of digitized shadowgraph visualization time-sequence data. As such, demonstration is provided of the use of high resolution, flow visualization data for quantitative correlation and spectral analysis. A shadowgraph system is employed to visualize time-varying, shock wave flow features within the test section of one leg of the transonic/supersonic wind tunnel (which is also referred to as the SS/TS/WT or SuperSonic/TranSonic/WindTunnel), located within the Propulsion Research Center of the University of Alabama in Huntsville. Of interest is a flow field with a well-defined normal shock wave, lambda foot, and separated turbulent boundary layer near the entrance of the lower flow passage (which is part of the test section), produced with a test section inlet Mach number of 1.54. Shadowgraph flow visualization images are processed from different pixel locations to compute frequency spectra. Data associated with two separate pixel region locations, each from a different flow region, are employed to determine magnitude squared coherence values, and associated time lag magnitudes. The data associated with lower Strouhal numbers in vicinity of 0.0013 and 0.0039 (which correspond to respective frequencies of 5 Hz and 15 Hz) illustrate important and pronounced interactions between the normal shock wave and the boundary layer separation zone. As this zone breathes and oscillates, additional flow locations are affected in a subsequent and significant manner. With Strouhal numbers in the vicinity of 0.0091, 0.0104, and 0.0260 (which correspond to respective frequencies of 35, 40, and 100 Hz), the most pronounced and significant interactions between the normal shock wave occur with respect to the downstream boundary layer. When ${\mathit{St}}_r$ equals 0.0091, spatially-varying time lag data show events with significant coherence propagate from a range of locations within the downstream boundary layer to the normal shock wave location. As such, events which originate within the downstream boundary layer at these experimental conditions are far more important (compared to events which originate with the upstream boundary layer) in regard to their effects on shock wave unsteadiness and associated flow motions.

Introduction

A significant number of investigations have been undertaken to consider the origins and propagation of shock wave interaction unsteadiness, and the consequent effects on nearby flow fields. For example, Humble et al. [1] and Ganapathisubramani et al. [2], [3], describe significant coherence between the upstream boundary layer and the unsteadiness in the shock wave interaction region. Others, such as Piponniau et al. [4] and Grilli et al. [5], do not emphasize significant correlation between the upstream boundary layer and the interaction region. Touber and Sandham [6] indicate that low-frequency interaction region unsteadiness is not a result of forcing, either from the upstream or downstream boundary layer, but “an intrinsic property of the coupled system.” These different perspectives may be the result of differences in interaction strength, as described by Clemens and Narayanaswamy [7]. These investigators indicate that interaction strength is directly related to the strength or relative size of separation, which is characterized by the magnitude of separated flow length scales. This strength also determines the degree to which an interaction exhibits sensitivity to upstream or downstream fluctuations. Clemens and Narayanaswamy [7] further indicate that both upstream and downstream mechanisms are present within all interactions, such that the degree of influence of the upstream boundary layer diminishes as separation strength and scale increase. This means that the importance and effects of both mechanisms change as separation strength varies. Thus, for some experimental conditions, the upstream boundary layer is an important source of disturbances, and shock wave unsteadiness is driven by fluctuations in the upstream boundary layer. Alternatively, shock wave unsteadiness can be driven by some large-scale instability intrinsic to the separated flow, which is associated with the influences of downstream mechanisms. Overall, Clemens and Narayanaswamy [7] indicate that the downstream mechanism dominates for strongly separated flows, and combined upstream and downstream mechanisms dominate for weakly separated flows.

For flows with shock wave induced separation, researchers, such as Piponniau et al. [4], Grilli et al. [5], Wu and Martin [8], and Pirozzoli et al. [9], indicate that unsteadiness in the interaction region is related to pulsations of the separation region. The causes of intrinsic separation bubble unsteadiness and reattachment point unsteadiness are believed to be either linked to the upstream boundary layer, as proposed by Pirozzoli et al. [9], or are the result of inherent dynamics between the separation bubble and the shock wave, as proposed by Piponniau et al. [4] and Grilli et al. [5]. Of these investigations, Grilli et al. [5] investigate very-low frequency motions near the foot of a shock wave produced by a compression-expansion ramp. These investigators indicate that frequencies associated with shock wave motions are two or three orders of magnitude smaller than frequencies associated with the incoming boundary layer, a conclusion in agreement with Dolling and Murphy [10] as well as with results from other investigators.

The present investigation considers the coherence and time lag of shock wave unsteadiness in relation to unsteady flow events at other locations, including the (i) upstream boundary layer, (ii) downstream boundary layer, (iii) downstream oblique shock wave leg of the lambda foot, (iv) lambda foot flow region, and (v) boundary layer separation zone. These data are obtained for a flow field with a well-defined normal shock wave, lambda foot, and separated turbulent boundary layer near the entrance of the lower flow passage of a test section with an inlet Mach number of 1.54. A shadowgraph system is employed to visualize time-varying, shock wave flow features within a volume (as a line-of-sight integrated image), which covers most the span of the test section. The resulting gray-scale flow visualization time-sequence data are acquired and digitized, and then processed from different pixel locations to compute frequency spectra, and from two separate pixel locations (or two collections of pixels), to determine magnitude squared coherence values, and associated time lag magnitudes. The experimental sequence of steps to stabilize the normal shock wave, which is presently considered, are described by Ligrani and Marko [11]. The present paper is significantly different from Ligrani and Marko [11] because all of the present results are new in regard to spatial coherence and physical characteristics related to shock wave boundary layer interactions.

The present data thus illustrate a unique analysis approach to explore the physics of shock-wave-flow interactions, based upon analysis of digitized shadowgraph visualization time-sequence data. As such, demonstration is provided of the use of high resolution, flow visualization data for quantitative correlation, time lag, and spectral analysis. The resulting data are significant because they show that normal shock wave unsteadiness is initiated at different flow locations, with different magnitudes of coherence, depending upon the unsteadiness frequency and Strouhal number. The complexity and intricate nature of flow and boundary layer interactions with a normal shock wave and the associated flow structures are illustrated. As such, results from the present experimental study are unique, since most other reported results consider local unsteady interactions between different types of oblique shock waves and nearby flow events. The physical insight provided by the present results thus extends understanding of interactions between normal shock waves and adjacent boundary layers, beyond that provided by other recent related investigations. The results are applicable to development of techniques for shock wave control [12], including their influences on boundary layer interactions [13], and associated flow mixing and unsteadiness [14], [15].