Vital Role of Shock-cell and Mach disk in Noise Production of Jet Engines

Contents

Abstract

Acknowledgments

List of Figures

List of Abbreviations

1 Introduction
1.1 Motivation
1.2 Objectives
1.3 Contributions
1.4 Organization of Thesis

2 Background
2.1 Supersonic Nozzle
2.2 Supersonic Jet Flow
2.2.1 Shock-cell Structure
2.2.2 Shock Associated Noise
2.2.3 Mach diamonds
2.2.4 Shear Layer
2.2.5 Broadband Shock-Associated Noise

3 Experimental Facility and Setup
3.1 The Heated Jet Noise Facility
3.2 Air Supply System 3.3 Convergent-Divergent Nozzle 3.4 Test Matrix 3.5 High Speed Shadowgraph

4 Circular C-D Nozzles
4.1 Experimental approach
4.2 de Laval Nozzle
4.2.1 The Baseline design & MOC Nozzle
4.2.2 Circular Nozzle (Large Nozzle)
4.2.3 Baseline design nozzle with Chevrons

5 Rectangular Nozzles
5.1 Nozzle Geometry
5.1.1 Flow Visualization
5.1.2 Results
5.1.3 Rectangular vs Circular Nozzle

6 Circular and Rectangular Nozzles with Flat plate
6.1 Baseline design & MOC nozzle with plate
6.2 Rectangular Nozzle with plate

7 Conclusion and Future work
7.1 Conclusions
7.2 Future Works

8 Appendix

An Abstract of

Vital Role of Shock-cell and Mach disk in Noise Production of Jet Engines

by

Gibin Mannathikulathil Raju

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the
Master of Science Degree in Aerospace Engineering

University of Cincinnati
August 2018

The aim of this thesis is to discuss the vital role of shock-cell and Mach disk in noise pro­duction of jet engines. This study focuses on calculation of shock-cell spacing and Mach disk from various configuration of C-D Nozzles using Shadowgraph technique and detailing on how it varies with the test conditions. In the past three decades, the air traffic has nearly increased by about 40% and have always called for alternative propulsion techniques to replace or support the current traditional propulsion methodology in order to travel longer distances in shorter frame of time. In the light of current demand, the thesis draws motivation from increase in noise levels due to technological advancements of meeting higher specific thrust thus, achiev­ing higher efficiency. The thesis emphasizes on the study of shock-cell noise by developing a complete view of the shock-cell structure and associated parameters with an imperfectly ex­panded supersonic jet on CD Nozzles over range of under-expanded, design and over-expanded operating conditions .

In the entire thesis study, the heated jets from the different C-D Nozzle are analyzed for proper visualization for the shock-cell structure. Shadowgraph is used in current study as it offers very high flexibility and acts as an excellent technique for visualizing the shock diamond exhaust structure and turbulent eddies. A systematic analysis has been performed to quantify the shadowgraph data through semantic video analysis and measured the shock-cell spacing over different temperature ratios (TR) and nozzle pressure ratios (NPR). Circular, rectangular and method of characteristics (MOC) convergent-divergent nozzles of 1.5 design Mach num­ber with and without plates were tested to interpret the results. The data recorded during the experiment was processed to obtain an averaged image. Further, the averaged images were utilized to compute the shock-cell spacing of each nozzle. The computed shock-cell spacing is compared over the theoretical shock-cell length using empirical formula to validate the results. A good match has been established between theoretical and experimental results for the tested nozzles expect for over-expanded operating conditions.

To my loving family and friends

Acknowledgments

I will instruct you and teach you in the way you should go; I will counsel you with my loving eye on you

Psalm chapter from the bible has this verse to sing the glory of Teacher. He was asked, If both, Teacher and God were to appear at his door step, whose feet will he worship first? He answers, It has to be the Teacher' s feet, because without the Teacher how could he recognize God?

First, I would like to express my sincere gratitude and respect to my Teacher and advisor, Dr. Ephraim Gutmark. From the first day, I started working at the laboratory; he has always provided me continuous support, encouragement and motivation during my graduate studies at the University. He is a great source of inspiration for me always to excel on my research . I am thankful to him for his time and expertise which he has passed on to me .

I would also like to sincerely thank my other committee members Professor Mark Turner, and Dr Kailas Kailasanathan for taking the time from their busy schedules to give me the much needed advise and direction to complete this work. They have always motivated me towards my research and bring out the maximum outcome.

I owe special thanks to my lab senior Prof Pablo Mora and Florian Baier, for helping me for all the intellectual discussions and with whom I worked extensively over the entirety of my graduate research.

I owe much to Dr. Rodrigo Villalva, with whom I worked from the day one I joined the lab. He has always guided me in the right path with constant feedback and has been have enabled me to pursue new research avenues and, in many ways, define my own research questions during my time at UC.

I am extremely thankful to my friends Rumit, Nalini, Emlin, Catherine, Yashmin, Kuldeep,

Karthic and Anjali for supporting me during hard times. They have provided me much needed mental and emotional support to stay focused and progress towards my final goal.

Finally, I want to dedicate my graduate research thesis to my father, mother, my brother and my pet Julie. I want to tell them that I am blessed to have them in my life and I will keep working hard to make them proud.

Gibin Raju

List of Figures

1-1 Aircraft noise sources

2-1 Quasi 1-D Flow Model

2-2 Mach number and A/A* ratio

2-3 Flow properties variations in an isentropic C-D nozzle [1]

2-4 Mach number and A/A* ratio

2-5 Supersonic jet flow [19]

2-6 Shock-cells in Circular C-D Nozzle

2-7 Jet shear layer

2-8 Broadband shock-associated noise. Frequency spectrum from a (a) perfectly expanded jet, (b) under-expanded jet and (c) over-expanded jet

3-1 Baseline design. (All dimensions are in

3-2 High-speed Shadowgraph, 2-Mirror Setup

4-1 Algorithm for the calculation of shock-cell spacing and mach disk location of the jet

4-2 Baseline design nozzle of exit diameter 0.813 in at NPR=2.5 and TR=1

4-3 Circular Nozzles at NPR = 2.5 and TR =1

4-4 Sectional front view of the baseline design in Solidworks. All dimensions mentioned in the figure are in inches

4-5 Refractive index vs x/D plot of the baseline nozzle at NPR 2.5 and TR 1

4-6 Ls/Dn vs NPR plot for baseline and MOC nozzle at NPR= 2.5 and TR= 1.0 for the I first shock

4-7 Averaged image of MOC CD Nozzle at under-expanded condition on the left side and over-expanded condition on the right

4-8 Shock-cell spacing measurements of baseline nozzle and MOC nozzle for TR = 1.0 &TR = 3

4-9 Measurements on location of Mach-disk of baseline nozzle and MOC nozzle for TR = 1.0 & TR = 3

4-10 Averaged Images of C-D Nozzle at NPR= 2.5 and 4.5 to visualize the width of Mach-disk

4-11 Shock-cell spacing measurements on baseline nozzle ,MOC nozzle and medium nozzle for TR = 1.0]

4-12 Measurements on location of Mach-disk of baseline nozzle ,MOC nozzle and medium nozzle for TR = 1.0]

4-13 Chevron nozzle CD Nozzle i) Baseline; ii) Chevron. De = 0.813, Md =1

4-14 Measurements of baseline, MOC and chevron nozzle at TR = 1

4-15 Measurements of baseline, MOC and chevron nozzle at TR = 1.0 & TR = 3.0 for second and third shock

4-16 Plot showing second and third shock-cell spacing for chevron nozzle and baseline design for TR 1 and

4-17 Far-field acoustic spectra, ^ = 140[0], Baseline and Chevron Nozzles [19]

5-1 Schematic of the rectangular C-D conical nozzle of 2:1 aspect ratio, De = 0.813 in and Md = 1

5-2 Instantaneous shadowgraph images of the jet plane (minor axis field of view, with- out plate) for NPR = 3

5-3 Rectangular Nozzles at NPR = 4.0 and TR =1

5-4 Shock-cell spacing of Rectangular nozzle without plate for TR =1.0 & TR =3.0 .

5-5 Mach-disk location of Rectangular nozzle without plate for TR =1.0 & TR =3.0 .

5-6 Comparison of Circular vs Rectangular Nozzle at TR = 1

5-7 Mach-disk location of rectangular nozzle vs circular nozzle for TR =1.0 & TR =3.0

6-1 Coordinate system for observation angles of rectangular nozzle

6-2 Comparison of first shock-cell spacing for TR 1 for all circular nozzles with exitdiameter of 0.813-in and the same nozzle with a plate mounted at an offset h/De [=Z3

6-3 Mach disk location for circular CD nozzles with and without plates at TR = 3.0

6-4 Shock-cell spacing measurements of rectangular Nozzles with and without plates for TR = 1

6-5 Shock-cell spacing measurements of rectangular Nozzles with plates in sideline I direction for TR = 1.0 & TR = 3

6-6 Time averaged shadowgraph images of the jet plane (major axis) for TR = 1.0 and NPR 4.0; a) nozzle without the plate and b) with the plate at h/De = 0

6-7 Shock-cell spacing measurements of rectangular Nozzles with plates in reflected direction for TR = 1.0 & TR = 3

6-8 Shock-cell spacing measurements of rectangular Nozzles with plates in reflected direction for TR = 1.0 & TR = 3

6-9 Shock-cell spacing measurements of rectangular Nozzles with plates in reflected I direction for TR = 1.0 & TR = 3

6-10 Far-field SPL spectra for the circular and rectangular nozzles (minor and major axes), at NPR = 3.67 and TR = 3.0

6-11 Far-field acoustics of the rectangular nozzle with and without the plate at different h/De positions and at the sideline direction (p = 90°), for TR = 3.0 and NPR = 3.67 at SPL spectra p = 136[0]

List of Abbreviations

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Chapter 1

Introduction

1.1 Motivation

Aviation industry has shown fast paced developments over the past two decades. Air travel has gained momentum and has resulted in various new designs of aircraft’s with varying pay­loads for civilian and military applications. Propulsion system being heart of the aircraft, has undergone tremendous improvements to cater the needs of current demand. Irrespective of en­gines being used in different applications for the aviation industry, the highest all time need was to travel faster in shorter time. From the past decade of technological development in air­craft, propulsion system design has met requirements of higher specific thrust achieving high efficiency. Meeting higher specific thrust limits is always associated with higher jet speed and thus, there will be significant increase in the levels of noise associated. This creates a perilous environment to living beings.

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Figure 1-1: Aircraft noise sources

The supersonic jet exhausted by engine nozzle is the strongest noise source components of the overall noise generated by aircraft [5]. Figure 1-1 shows the aircraft noise sources in a com­mercial aircraft, including nose and main landing gear, air-frame surfaces such as slats, flaps etc; the engine and the interaction between the last two, is the place where the engine radiates the maximum level of noise [10]. The radiated noise from exhaust through the nozzle becomes the dominant component of overall noise and thus become the subject of this research. In high speed aircraft technologies, smaller geometric profiles are required to reduce drag, and conse­quently propulsion systems are designed to contain smaller bypass ratios on the engine.These propulsion systems are also designed to achieve higher specific thrust and become more ef­ficient. Both these scenarios are linked to heated jets with higher exhaust speeds and as a consequence a significant increase in noise levels.

Due to increased levels of noise possessing threat to workers and crew members during its maintenance work. This situation arises the need for a solution to reduce in noise levels of high-speed heated jets, and engines with afterburners. Previous efforts have made to understand the physics behind the jet noise generation process, designing noise mitigation techniques, and advancing jet noise prediction tools. Aerodynamic design parameters always play a vital role in the level of noise produced which cannot be altered. Noise control techniques have to be found effective at different nozzle area ratios/operating conditions, and by maintaining small geometries and also chevrons attached to each flap of variable area nozzles. These methods have shown to enhance the mixing between cores high-speed flow and ambient, affecting jet noise sources and reducing intense levels of low frequency. Jet noise sources are also external to the engine, with dominant sources located near the breakdown of potential core [19].

1.2 Objectives

The research primarily focuses on the visualization of heated jets of different C-D Nozzles using shadowgraph technique. This thesis will address the following aspects in this research:

- Study the jet flows on supersonic nozzles and noise associated with it
- Analysis of shock-cell structure using Shadowgraph
- Calculation of shock-cell spacing and Mach disk from the recorded data
- Understanding the impact of nozzle pressure ratio (NPR) and temperature ratio (TR) on shock-cell structure for different nozzles
- Validation by comparing the experimental results to the results obtained from theoretical formulation
- Study variation of shock-cell structure in different nozzle diameter of same C-D Nozzles as well in shapes of same diameter of C-D Nozzles
- Impact of internal contours geometry on circular nozzles.
- Effect of ground and near-by-jet surfaces on shock-cell structure within the installation of plates to circular and rectangular C-D Nozzle.

1.3 Contributions

The current work will serve as pipeline into further research in the field of jet noise reduc­tion and its applications. It is also one step closer towards establishing a better understanding of shock-cell and its associated noise. The main contributions from this thesis are:

- Calculation of shock-cell spacing for rectangular and circular nozzles.
- Comparing the structure of the jet from a conical nozzle and a smooth contoured nozzle designed by the method of characteristics.
- Understanding the differences between rectangular and circular nozzles.
- Study the Mach disk formation on different nozzle configurations.
- The effect of NPR and TR on the formation of Mach disk formation for all the tested nozzles are explained.
- The effect of installation of plates at the exit of circular vs rectangular nozzles.
- The list of publications resulting from the present work is provided in the appendix.

1.4 Organization of Thesis

This thesis consists of seven chapters. The first chapter is the introduction. Chapter 2 is a literature review of theory of supersonic nozzles and its flow. An overview of supersonic jet noise and the theory associated to noise focusing on shock-cell structure are described. Chapter 3 deals with experimental setup and procedures. Chapter 4 presents the entire analysis on different diameters circular nozzle for shock-cell spacing and Mach disk formation. It includes circular C-D nozzles designed from MOC method and baseline C-D nozzle fitted with chevrons. Chapter 5 deals with analysis and the results from rectangular nozzle, Chapter 6 presents the effect of introduction of plate near the exit of circular and rectangular C-D nozzle in major and minor axis. Chapter 7 presents the conclusive remarks, future works and summarizes this thesis by discussing the contributions and objectives achieved through this research work. The list of publications resulting from this work is provided in the appendix.

Chapter 2

Background

The aircraft propulsion system, incorporating the engine is responsible for the generation of required mechanical power to sustain the flight [37]. The aircraft propulsion has been evolving over the past decades, starting from simple bent bow propulsion system in 1978 to todays ultra­complex aircraft engines producing supersonic thrust. The majority of the engine applications are limited to sub-sonic transport aircraft, cargo aircraft and the engines specially designed for military applications. The main function of the subsonic passenger and cargo aircraft engine is to provide sufficient thrust to balance the drag of the aircraft during cruise and provide exces­sive thrust to accelerate the aircraft during other maneuvers. In case of fighter aircraft and other supersonic aircraft, thrust required will be higher. Excessive thrust is required to overcome high drag produced as a results of increases speed. Engines for such high speed applications are designed with afterburner or high bypass fans.

In the time lapse of past 40 years, various new engine technologies and operating mech­anisms have been experimented and a handful of them have sustained in the market. There are various areas of engine operation which still prove to be challenging. The researchers are actively involved in reducing the noise and emission with every new variant of the engines released. The aim is to go greener and softer. All the progress made in recent technologi­cal development aimed at reducing the noise of the jet noise are more focused at the study of nozzles and its modification. NASA has been always conducting and sponsoring research, aimed at continuing this trend, especially since air traffic grows and the impact of noise on the community will increase if the noise levels are not reduced.

2.1 Supersonic Nozzle

In order to meet the thrust requirement levels of supersonic aircraft, the exhaust of the nozzles have to be accelerated in supersonic speeds when they exit through C-D Nozzles. Gov­erning equations are derived from a quasi-one-dimensional flow model to better understand the physics of the gases passing through the C-D Nozzles as described by Anderson [1] and John and Keith [12]. In this 1-D Flow model approximation, pressure (P), density (p),temperature (T), velocity (U) and the cross-sectional area (A) vary only with axial distance (x),as shown in Figure (2-1). For the quasi-one-dimensional flow, flow properties vary axially due to changes in area, as compared to the one-dimensional flow model, where properties vary due to a normal shock wave, heat addition and/or friction.

Abbildung in dieser Leseprobe nicht enthalten

Figure 2-1: Quasi 1-D Flow Model

The area-velocity relation, in Equation 2.1, was derived from the integral and differential forms of the conservation equations in a controlled volume, and by assuming steady flow, no internal forces, and adiabatic flow. In the derivation process, the conservation equations were also combined with the isentropic flow relations, since there are no dissipative mechanisms in adiabatic inviscid flows. In equation 2.1, M is the Mach number.

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This equation dictates that if M<1, velocity can be increased by reduction in area (con­vergent section) and when M>1, the velocity can be increased by increase in area (divergent section) and when M = 1 corresponds to minimum area, A*. Assuming calorically perfect gas, area-Mach number relation can be obtained as

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From the above equation 2.2, it is clear that there is a well established relationship where M is function of area ratio (A/A*) and specific heat ratio. A graph as shown in Figure 2-2 is plotted between M and A/A* and it is basically used for designing the supersonic nozzle exit area by knowing the required Mach number at the nozzle exit. In this study, all nozzles have 1.5 Mach number with A/A* = 1.176 [12].

The entire study in this thesis is mainly focused on the supersonic test conditions. Fig 2-3 demonstrates the variation of flow properties through an isentropic C-D nozzle. So the flow at the nozzle expands to supersonic speeds, starting from a very large inlet (Ai/At* ^ to) with almost stagnant air, M= 0, P= Po, T=To. As the flow moves through the nozzle, Mach number increases and the static pressure and temperature decrease isentropically, as described by Equations 2.3 and 2.4. Both equations are a function of Mach number only, they depend only on A/At* and y. [1]

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Figure 2-3: Flow properties variations in an isentropic C-D nozzle [1]

The difference in pressure levels drives the flow across nozzle. In an isentropic C-D nozzle, as the one shown in Figure 2-4 a, the back pressure (Pb) equals the exit pressure (P1). This is also referred as ideal condition, or operating at the nozzle design Mach number, and is shown in the schematic Figure 2-4 b.

Underexpanded Nozzle

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Perfect Nozzle

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Overexpanded Nozzle

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Figure 2-4: Mach number and A/A* ratio

If Pb starts increasing (P1 < Pb), an oblique shock is formed at the nozzle exit due to the mismatch of pressure, as shown in Figure 2-4 a. This is called the over-expanded condition.

If the back pressure is increased further, a normal shock develops at the nozzle exit, and then it moves upstream, until the back pressure is high enough to generate subsonic flow across the entire C-D nozzle. And if Pb starts decreasing (P1 > Pb), expansion waves are formed at the exit of the nozzle, as shown in Figure 2-4 c, expansion wave are generated. This is called under-expanded condition.

Assuming fully isentropic expansion; Mach number and temperature of the jet is calculated from equation 2.3 and 2.4.

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2.2 Supersonic Jet Flow

Jets exhausted from supersonic CD nozzles into ambient atmosphere contains the following characteristics as shown in Figure 2-5: the shear layer, potential core, and shock cells. Viscous forces generates a velocity gradient from the exit of the nozzle, between the supersonic jet flow and the ambient flow, which generates the mixing/shear layer. The region of supersonic flow, shown in the center in Figure 2-5, contains irrational flow is called as the potential core of the jet. From the exit of the nozzle, the shear layer begin with a very thin and grows with axial dis­tance until it collapses and encloses the potential core region. This axial distance is measured as the potential core length. The potential core length is always dependent on operating condi­tions. It is important to understand that the potential core boundary is not constant, but varies with the function of Reynolds number. The transition zone contains turbulent flow, and the velocity profiles are transitioning until reaching the fully developed region, where the velocity profiles become similar.

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Figure 2-5: Supersonic jet flow [19]

The supersonic region of the jet contains shock and expansion waves that start at the nozzle exit. These waves reflect of the sonic line and generate a series of shock and expansion waves, shown in Figure 2.6, known as the shock-cells/diamond structures. The shock cells extend outside the potential core, into the shear layer. In Figure 2-6, a set of shock cells is also generated starting at the throat in conical nozzles [8]- [21]. Shock-cell spacing and shock strength strongly influence the amplitude and frequency of the noise generated by the shocks cells and turbulent structures interaction [30].

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Figure 2-6: Shock-cells in Circular C-D Nozzle

2.2.1 Shock-cell Structure

In an imperfectly expanded supersonic jet, the structure of a shock-cell is formed by oblique or normal shocks and expansion fans. Dissimilarity of static pressures at the nozzle lip causes formation of these shocks and expansion fans. In an under-expanded jet, an expansion fan is the consequence of incomplete expansion of fluid and allows the static pressure to decrease grad­ ually to that of local atmospheric condition. In an over-expanded jet, an oblique shock forms, as passing through the shock, the static pressure increases abruptly to match with outside jet. Across the jet flow, the expansion fan or shock moves until it impinges on mixing layer. The impinged expansion fan or shock is reflected into the jet plume. This process of reflection by the mixing layer upswings the quasi-periodic shock cells and it continues until it is depraved by turbulence. In other words, the jet flow is considered as a wave guide for the disturbances that form the shock-cell.

Prandtl [[14],[34]] analyzed the shock-cell of the supersonic jet using a linear vortex sheet jet model and developed a partial solution to estimate the length of the shock-cell. Pack [24] completed the analysis using the same method of linear vortex sheet jet model. The solution framed by Pack had cylindrical co-ordinates (r, , x). If ps is the pressure associated with the noise,

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where J0 and J1 are the Bessel functions of 0 and 1. 8p is the static pressure difference between inside and outside of the jet. The above equation is obtained as the solution of the Euler equations, linearized around a perfectly expanded supersonic jet, if pe - pm remains small. The complete expression for the amplitude, Ai, the radial distribution of each mode (pi and its wave number ki can be found in Tams paper [32]. At a fully expanded condition, the diameter of the jet from a supersonic jet nozzle is calculated with a relation to nozzle exit diameter by

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Waveguide mode of the jet flow is basically calculated from this equation 2.7. Solving all these equations 2.7 to 2.8 [32],[34] , the shock-cell spacing is given by

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where Dj is the fully expanded jet diameter at Mj and the parameter is defined as,

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For imperfectly expanded supersonic jets of weak nature,

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where D is the referred as the nozzle diameter.

By the analysis, Tam performed with the measurements of shock cell from Norum and Seiner [34] research. He discovered that for a moderately imperfectly expanded supersonic jet, the first shock was very strong and after that the shock-cell structure weakens depicting that of a weak imperfectly expanded jet. This is observed in the results which would be discussed later.

2.2.2 Shock Associated Noise

Broadband shock-associated noise and screech tones are generated only when a quasi- periodic shock cell structure is present in the jet plume. It will be shown later that the quasi­periodicity of the shock cells plays a crucial role in defining the characteristics of both the broadband and discrete-frequency shock noise. A number of investigations have been reported on sound generated by the passage of turbulence and entropy spots through a plane normal or oblique shock. It must be emphasized that the noise generated by this plane shock-turbulence / entropy spots interaction has no relationship to the observed broadband shock-associated noise under consideration. The importance of quasi-periodicity of the shock cells to broadband shock associated noise was first recognized by Harper-Bourne & Fisher (1974) in their pio­neering work on this subject. They made use of the quasi-periodicity to explain some of the prominent characteristics of this noise component.

For simplicity, we use a one-dimensional model of large turbulence structures-shock cell interaction to explain the noise generation process. A complete three dimensional analysis can be found in the work of Tam (1987). In terms of the shock cell spacing Li = 2/k turbulence con­vection velocity uc, and frequency f = 2nm equation of the spectral frequency may be rewritten as

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As determined by Tam and Tanna (1982)[32], Equation 2.11 gives the relationship between the spectral peak frequencies and the direction of radiation of broadband shock-associated noise. This characteristic property is explained in the Tams paper. Since the shock cells are consid­ered as composed of a number of wave-guide modes, Equation 2.12 implies that the total noise spectrum is made up of a superposition of many spectral peaks. The spectral peak frequen­cies can be calculated from Equation 2.12. The above 1-D model illustrates the crucial noise generation processes of broadband shock-associated noise. Basically, this noise component is generated by the constructive scattering of the large turbulence structures of the jet flow by the stationary quasi-periodic shock cells in the jet plume. Since the shock cells can be conceived of as made up of a superposition of wave-guide modes of different wavelengths, each mode scatters of sound in a preferred direction. This results in a multi-peak noise spectrum and the observed directional dependence of the spectral peak frequency.

2.2.3 Mach diamonds

Mach disks are an evolution of stationary wave patterns that are visible in the supersonic exhaust plumes of jet engines when they are operated in the atmospheric conditions. Generally, these disks (Shock diamonds) are formed from a complex flow field that appears due to abrupt changes in local density and pressure caused by standing shock waves. Mach disks forms su­personic exhaust from a propelling nozzle that is slightly over-expanded, i.e. static pressure of gases exiting the nozzle is less than ambient air pressure. This increase in pressure in the ex­haust and reduction in velocity cause the temperature to increase. The ambient air compresses the flow by oblique shock waves inclined at an angle to the flow when they exist the nozzle. The compressed flow is then expanded by Prandtl-Meyer expansion fans and each diamond will be formed pairing with an oblique shock with an expansion fan. When the compressed flow comes parallel to the centre line of the nozzle, normal shock wave forms perpendicular to the flow and this would be the location of first Mach disk. The distance from the nozzle [39] can be approximated by

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As exhaust passes through this normal shock wave, its temperature increases causing the excess fuel to burn which makes the disks more visible. Eventually, flow expands so that its pressure drops below ambient from where the reflection of expansion fan originates and these are called compression waves. These waves are strong that their compression causes another oblique wave to form creating a second shock diamond. These processes continue till turbulent shear reduces with distance. Similarly, when a nozzle is under-expanded i.e. the exit pressure is higher than the ambient. In this case, the expansion fan is first to form, followed by the oblique shock.

2.2.4 Shear Layer

At the exit of the nozzle, jet flow interact with the ambient flow creating a velocity gra­dient between them causing the generation of shear layer around the jet. Initially, jets with high Reynolds numbers were believed to contain only random disorderly turbulent structures. Studies showed that fully turbulent jets (Re ~ 10[5]' showed that the shear layers also contained orderly coherent structures and jet with high Reynolds number to contain highly disordered turbulent structures [7] - [38].

[...]