Investigation of Flow Dynamics of a Subsonic Circular to Rectangular Jet

Transcription

1 Investigation of Flow Dynamics of a Subsonic Circular to Rectangular Jet Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Soumyo Sengupta Graduate Program in Mechanical Engineering The Ohio State University 2016 Thesis Committee: Datta V. Gaitonde, Advisor Seung Hyun Kim

2 Copyright by Soumyo Sengupta 2016

3 Abstract Large Eddy Simulation based investigation is performed on a circular to rectangular Mach jet to understand the effect of the non-axisymmetric geometry on the jet flow dynamics. The nozzle has a circular cross-section at the inlet and a 2:1 aspect ratio rectangular cross-section at the exit. The Reynolds number of the flow is 73,242 and the nozzle has an equivalent diameter of meters. Comparison with experiments, indicate a good match of the normalized mean and rms velocities along the jet centerline. Q-criterion iso-levels confirm the existence of large hairpin like vortices, which initially are dominant along the major axis but further downstream appear along the minor axis as well. To highlight the influence of different frequency ranges, the velocity field at the core collapse location is first decomposed using Empirical Mode Decomposition (EMD). The spectrum in these different ranges is then correlated with observations along representative lip-line at the major and minor axes and the corner. The presence of the large scale structures corresponding to low frequency ranges is predominant along the corner compared to major and minor axes. The correlations provide information on the ii

4 effect of specific scales at the end of potential core along the streamwise direction. The major components of low frequency structures in the propagated signal occur between St = 0.05 and St = 0.3. The correlation analysis shows that structures formed along the corners and minor axis dominate the large scale dynamics of the flow. The Joint Probability Density Function (JPDF) analysis is used to study flow along major and minor axis of the rectangular jet as well as an equivalent circular nozzle. Variations in entrainment and ejection-like patterns along the minor and major axes of the rectangular jet are quantified. Results with an equivalent circular nozzle are compared which indicate enhanced ejection and entrainment like motions for the rectangular nozzle compared to the equivalent circular nozzle. iii

5 Dedication This thesis is dedicated to my family and friends for their constant love, support and advice through the years. iv

6 Acknowledgments I would like to thank my graduate advisor Dr. Datta V. Gaitonde, for his guidance and support and for the chance to work with him in his research group. His ideas and inspirations have been extremely fruitful and helped me to gain so much both academically and personally. I would like to thank all the faculty members within the department for their assistance along these last two years. I would also like to thank Dr. Seung Hyun Kim for being a member on my thesis committee. Lastly I want to acknowledge all my fellow lab mates who have taught me so much and made this effort possible. I would like to give special thanks to Unnikrishnan Sasidharan Nair, Kalyan Goparaju and Swagata Bhaumik for their friendship and advice which has been extremely helpful along the way. v

7 Vita Delhi Public School Kalyanpur, Kanpur, India B.E. Mechatronics Engineering, Manipal Institute of Technology, Karnataka, India Publications Conference Publications Sengupta, S., Agostini, L., Unnikrishnan, S., Gaitonde D.V., Investigation of Rectangular Jet Issuing From a Varying Cross-Section Nozzle. 54th AIAA Aerospace Sciences Meeting. January Sengupta, S., Agostini, L., Unnikrishnan, S., Gaitonde D.V., Effect of Asymmetric Nozzle Configuration on Jet Flow Characteristics. AIAA Aviation and Aeronautics Forum and Exposition. June vi

8 Fields of Study Major Field: Mechanical Engineering vii

9 Table of Contents Abstract ii Dedication Acknowledgments Vita iv v vi List of Tables x List of Figures xi Chapter 1 Introduction and Objectives Overview of Flow Physics Previous Works Present Work Experimental and Computational Details Experimental Setup Nozzle Configuration and Computational Setup Results Experimental Validation Features of the Flowfield viii

10 3.3 Scale specific correlation study Comparison with an Equivalent Circular Nozzle Comparison of entrainment and ejection events features by Joint (u u r) probability density function Conclusions Bibliography Appendix A Post Processing Tools A.1 Smoothing Techniques A.1.1 Smooth function of MATLAB A.1.2 P-Welch Method A.1.3 EMD based FFT Smoothing ix

11 List of Tables 3.1 Comparison of entrainment and ejection events for rectangular and equivalent circular nozzle x

12 List of Figures 1.1 Interacting hairpin vortices for a square, aspect ratio =1 jet [3] Velocity contours along an elliptic jet demonstrating axis switching [7] Experimental setup at Glenn Research Center, NASA[13] Schematic of the Computational Nozzle Geometry Entire Computational grid with boundary conditions Schematic of the rectangular nozzle and important locations Schematic of array of points selected along the minor axis center Schematic of array of points selected along the major axis center Comparison of normalized centerline mean velocity on the two grids with experimental data for the 2 : 1 aspect ratio rectangular nozzle Comparison of normalized centerline rms velocity on the two grids with experimental data for the 2 : 1 aspect ratio rectangular nozzle Isometric view of the u-velocity contours at indicated locations Front View of the u-velocity contours at above mentioned four locations Time-averaged iso-levels of Q for the 2 : 1 aspect ratio rectangular nozzle xi

13 3.9 Auto-correlation used to calculate local timescales along the three mentioned locations Local timescales along three different locations in the flow Turbulent Kinetic Energy along four locations in the flow Pressure fluctuations along four locations in the flow Steps involved in sifting process [31] IMFs of EMD applied to source signal (potential core collapse point) Comparison of PSD of each IMF of source signal with PSD of the total signal Schematic showing key points used in EMD analysis Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF1 of the source signal (x/d = 4) from the jet exit Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF3 of the source signal (x/d = 4) from the jet exit Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF4 of the source signal (x/d = 4) from the jet exit Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF5 of the source signal (x/d = 4) from the jet exit Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF6 of the source signal (x/d = 4) from the jet exit Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF7 of the source signal (x/d = 4) from the jet exit xii

14 3.23 IMFs of EMD applied to source signal (potential core collapse point) separating signal into high frequency and low frequency scales Two point correlation of u-velocity perturbation along a series of points along the minor axis with IMFs corresponding to high and low frequency scales Schematic of the computational equivalent circular nozzle geometry Comparison of normalized mean velocity for the rectangular and circular nozzles Comparison of normalized rms velocity for the rectangular and circular nozzles Isometric view of the u-velocity contours at indicated locations for the circular nozzle Front view of the u-velocity contours at above mentioned four locations for the circular nozzle Conceptual representation of the ejections and entrainment events inside a shear layer Joint PDF of u -w along the centerline of the major axis Joint PDF of u -v along the centerline of the minor axis Joint PDF of u -v along the lip line of the equivalent circular nozzle. 51 A.1 Raw PSD plots for IMFs at the potential core collapse point A.2 Smoothed PSD plots for IMFs at the potential core collapse point 62 A.3 Comparison of Smoothed PSD plots for the two above mentioned methods A.4 Raw PSD plot to highlight EMD based smoothing A.5 Smoothed PSD plots using EMD based smoothing xiii

15 Chapter 1 Introduction and Objectives Non-circular jets have been a topic of extensive research for many years. The popularity of non-circular jets is based on the fact that they provide an efficient and cost effective technique for passive flow control. Significant improvements are observed in practical systems at relatively low cost as non-circular jets rely on changes in the geometry of the nozzle. Non-circular jets further provide ease of integration with the air-frame and thus become an attractive candidate in applications where enhanced thrust and entrainment are desired. Non-circular jets have vast applications which include improved mixing in low and high speed flows, enhanced combustor performance by reducing combustion instabilities and unwanted emissions. Some other applications include heat transfer, noise suppression and thrust vector control. 1

16 1.1 Overview of Flow Physics Entrainment of ambient air by a jet is important in mixing enhancement and noise mitigation applications. The process of mixing occurs in two stages: first stage consisting of an initial phenomena of large-scale stirring, and a second stage associated with small-scale velocity fluctuations [1]. Brown and Roshko [2] showed that jet entrainment, i.e., the rate of shear layer growth through entrainment is controlled by the development and evolution of large-scale coherent vortical structures. They observed that these coherent structures convect at nearly constant speed, and increase their size and spacing discontinuously by interacting with neighboring structures as shown in figure 1.1. Therefore it is vital to understand the dynamics of the coherent structures and how jet properties are affected by the formation, interaction, merging and breakdown of coherent structures. A natural method to attain desired spreading and mixing characteristics is thus to modify the evolution of the coherent structures in the mixing layer. The formation of circular, azimuthally coherent vortex rings and their subsequent merging dominate the shear layer growth[4] in the case of a non-axisymmetric jet. This is followed by dominant three-dimensional effects, which are dependent on initial conditions determined by the nozzle geometry and upstream conditions[5]. In non-circular jets, deformation dynamics of asymmetric vortices play a major role in jet evolution. They are observed to be inherently more unstable than ax- 2

17 Figure 1.1: Interacting hairpin vortices for a square, aspect ratio =1 jet [3] isymmetric jets because of the amplification of high-order instability modes which are dominant in the nearfield region[6]. Experimental observations show that noncircular jets consistently exhibit higher entrainment rates compared to circular jets for all Mach number regimes [7]. The main underlying cause for the enhanced entrainment properties of non-circular jets, relative to comparable circular jets, is self-induced Biot-Savart deformation of vortex rings with non-uniform azimuthal curvature and interaction between azimuthal and streamwise vorticity [7]. Due to Biot-Savart self-induction, portions of the vortex with small radius of curvature, such as the major axis section will move downstream faster than the rest, leading to their deformation. As the vortex convects downstream, the deformations yield a complex topology, which results in redistribution of energy between azimuthal 3

18 and streamwise vortices. The subsequent interaction increases the small-scale content of the jet. 1.2 Previous Works Various efforts have been devoted to investigate properties of jet flows from noncircular jets. Experimental studies using elliptical nozzles [7 9] and nozzles with corners (rectangular, triangular) [10] have observed that as the jet spreads, its crosssection evolves with the same shape as that at the jet exit. Under certain conditions axis switching occurs due to faster growth rate of the jet s shear layer in the minor axis plane compared to that in the major axis plane as shown in figure 1.2. This difference in rate of growth rates results in a crossover location at certain distance downstream from the nozzle exit. The ability to obtain enhanced mixing between jet flow and surroundings played a key role in interest of studying non-circular jets. Elliptic and rectangular jets have been shown to exhibit significantly larger entrainment rates compared to circular or two-dimensional jets due to vortex self-induction effects [7]. For both subsonic and supersonic jets, enhanced mixing was observed in the case of non-circular jets. Aspect ratio and the corner effects of the rectangular nozzle affect the development of the jet. In addition, it has been shown that the spreading rate is usually higher along the minor axis [11]. Different nozzle geometries were compared to study the effect on jet noise in 4

19 Figure 1.2: Velocity contours along an elliptic jet demonstrating axis switching [7] 5

20 high-subsonic and supersonic jets [12]. They showed that the circular jet produces highest levels of noise. Noise produced by a rectangular jet was lower, but a higher level was emitted at wider sides. The addition of tabs to the rectangular jet was ineffective and instead showed higher broadband noise. Flow field surveys for three rectangular nozzles were documented for low subsonic conditions [13]. Three nozzles of aspect ratios of 2 : 1, 4 : 1 and 8 : 1 were studied. Mean velocity, turbulent normal and shear stresses up to a streamwise distance of sixteen equivalent diameters from the nozzle exit were tabulated. This provided us with a comprehensive database to validate our computational results and to make qualitative observations for entrainment, ejections and mixing. 1.3 Present Work Several of these unique features of non-axisymmetric jets can be analyzed by studying the formation, interaction and breakdown of coherent structures in the core. The current analysis will thus focus upon the evolution of coherent features with emphasis on their asymmetry along the major and minor axis. The difference in entrainment levels of a rectangular jet with its axisymmetric counterpart has been a topic of extensive research. Majority of findings in the case of jets explain mixing, ejection and entrainment motions in qualitative or visual terms[1, 14]. The main purpose of this study is to show the effect of geometry by comparing entrainment and ejection like motions for a rectangular jet and an equivalent circular nozzle 6

21 and to investigate the influence of non-axisymmetric geometry on the dynamics of a jet plume. For comparison of ejection and entrainment like motions, an equivalent circular nozzle was simulated and compared with the rectangular nozzle. We focus on the flow along the liplines of the major and minor axes and the corners to highlight the major differences in a rectangular nozzle compared to a circular nozzle. The simulation considers a Mach jet with circular cross-section at the inlet and a 2:1 aspect ratio rectangular cross-section at the exit. The inlet conditions for carrying out the Large Eddy Simulations (LES) were chosen to facilitate validation with the experimental results provided by Zaman [13]. In chapter 2, a description of the experimental setup, governing equations, structure of the mesh and boundary conditions is provided. Chapter 3 contains all of the analysis on the rectangular jet as well as on an equivalent circular jet. Section 3.1 compares mean axial velocity and axial rms velocity with experimental data [13] to validate our computational results along with a brief discussion on grid resolution study. The effect of the non-circular jet on the flow field is studied through local timescales and turbulent kinetic energy in Section 3.2. A visual representation of flow structures is provided by the instantaneous Q-criterion iso-levels to highlight differences in presence of hairpin like vortices along the major and minor axes. The intermittency of the signal in the nearfield is analyzed in Section 3.3 by EMD analysis and spatio-temporal correlations. The design of an equivalent circular nozzle and the comparison of its streamwise mean and rms velocity with the rectangular nozzle is analyzed. Differences in ejection and entrainment like 7

22 motions is analyzed along the major and minor axes of the 2 : 1 aspect ratio rectangular jet and for the equivalent circular nozzle in Section 3.5 using Joint Probability Density Function. 8

23 Chapter 2 Experimental and Computational Details 2.1 Experimental Setup Figure 2.1 depicts the experimental setup at Glenn Research Center, NASA. Various experiments were carried out on different aspect ratio rectangular nozzles[13] with and without chevrons. Out of these, we select a 2 : 1 aspect ratio rectangular nozzle for our analysis. The nozzle has an equivalent diameter of D = 0.054m at the exit. Data was collected primarily by hot-wire anemometry for the chosen configuration in the experimental setup. 2.2 Nozzle Configuration and Computational Setup The computational nozzle geometry, depicted in Fig. 2.2, is qualitatively identical to the nozzle used in the experiment. A CAD model of the experimental nozzle was used to generate the computational model. The nozzle has circular geometry 9

24 Figure 2.1: Experimental setup at Glenn Research Center, NASA[13] at the inlet and a rectangular geometry at the exit. The Mach number and inlet conditions are chosen to match those of the experimental conditions. The computational domain is represented in Fig A structured grid is used where ζ, η, ξ are the coordinates in the streamwise, radial and azimuthal directions, respectively. A fine grid is used near the nozzle exit and the grid is gradually stretched in the far-field. The domain extends to 20D in both radial and streamwise directions, where D is the equivalent nozzle diameter. The grid was clustered both inside and outside the nozzle in the radial direction to accurately capture the boundary layer and the shear layer. Earlier simulations were performed on a coarser mesh of approximately 39 million points with in the ζ, η, ξ directions respectively. The nominal computational grid consists of approximately 74 million points with mesh points as in the ζ, η, ξ directions 10

25 Figure 2.2: Schematic of the Computational Nozzle Geometry respectively. The comparison of results on the two grids and comparison with experimental data is shown later in Section 3.1. Non-reflecting characteristic boundary conditions[15] are applied in the downstream, far-field and upstream boundaries outside the nozzle. In the azimuthal direction, a branch cut is introduced and a five-point overlap is used. Periodic condition is enforced by equating points on the boundary edges with the corresponding points in the interior of the domain on the other side of the branch cut. The centerline corresponds to a singularity which is treated as a boundary condition by enforcing solution continuity. The reference conditions at the jet exit, U ref =78.02 m/s, T ref =296.6 K and ρ ref = 1.225kg/m 3, match the experimental data. The jet exit Mach number is M =

26 Figure 2.3: Entire Computational grid with boundary conditions 12

27 which corresponds to a stagnation pressure of p stagnation = kP a. The strong conservation form of the full, three dimensional, compressible, unsteady Navier-Stokes equations are solved in curvilinear coordinates[16, 17]. These equations can be written as: τ ( ) U J + ˆF [ ξ + Ĝ η + Ĥ ζ = 1 ˆF ] v Re ξ + Ĝv η + Ĥv ζ (2.1) where U = {ρ, ρu, ρv, ρw, ρe} is the solution vector, J = (ξ, η, ζ, τ) / (x, y, z, t) is the transformation Jacobian, and ˆF, Ĝ and Ĥ are the inviscid fluxes given by: ˆF = ρû ρuû + ˆξ x p ρvû + ˆξ y p ρwû + ˆξ z p (ρe + p) Û ˆξ t p Ĝ = ρ ˆV ρu ˆV + ˆη x p ρv ˆV + ˆη y p ρw ˆV + ˆη z p (ρe + p) ˆV ˆη t p Ĥ = ρŵ ρuŵ + ˆζ x p ρvŵ + ˆζ y p ρwŵ + ˆζ z p (ρe + p) Ŵ ˆζ t p (2.2) where Û = ˆξ t + ˆξ x u + ˆξ y v + ˆξ z w (2.3) ˆV = ˆη t + ˆη x u + ˆη y v + ˆη z w (2.4) Ŵ = ˆζ t + ˆζ x u + ˆζ y v + ˆζ z w (2.5) 13

28 E = T (γ 1)M (u2 + v 2 + w 2 ). (2.6) Here, ˆξ x = J 1 ξ/ x, with similar definitions for the other metric quantities. Anderson[18] may be consulted for the form of the viscous fluxes ˆF v, Ĝv and Ĥv. u, v, w, p, ρ, T used in the above equations are the Cartesian velocity components, pressure, density and temperature respectively. Reference values at jet exit are used to non-dimensionalize the flow variables except pressure which has been normalized by ρ u 2. The perfect gas relationship p = ρt/γm 2 is also assumed. The Roe scheme[19] is used to discretize the inviscid terms as follows. G j+ 1 2 = 1 2 [ G ( U L ) + G ( U R)] 1 2 ˆQ ˆΛ ˆQ 1 ( U R U L) (2.7) where QΛQ 1 = G/ U and ( ˆ ) indicates evaluation at the Roe averaged state between U L and U R. Detailed expressions for each term are presented in Morrison[20]. The MUSCL approach of van Leer[21], is used to obtain a thirdorder upwind-biased method, subject to limiting described below, in which U L and U R are derived from values at the nodes. U R j+ 1 2 U L j+ 1 2 = U j+1 1 [ ] (1 η) 4 j+ 3 + (1 + η) 2 j+ 1 2 = U j + 1 [ ] (1 η) 4 j 1 + (1 + η) 2 j+ 1 2 (2.8) (2.9) where, in terms of a limiter, L, j 1 2 ( = L j+ 1 2, j 1 2 ) for example, and η is 14

29 specified to be 1/3. The Van Leer Harmonic limiter is employed: L ( 1, 2 ) = ɛ (2.10) where ɛ is a small number to ensure that the denominator is not zero in regions of small gradients. A second order central difference scheme is used to solve the viscous terms. A second-order diagonalized[22] approximately factored Beam- Warming method[23] is employed for time integration. This is employed within a sub-iteration strategy to minimize errors due to linearization, factorization and explicit boundary condition implementation. 15

30 Chapter 3 Results Section 3.1 presents a validation study of the LES model for the case presented by Zaman [13]. The normalized mean and rms axial velocities are compared with the experimental results to ensure that the computational results correlate well with the experimental values. In the following Section 3.2, we shift our focus to understand the flow dynamics of the rectangular jet. A visual scope of the flow structures dominating the near and far field is observed using iso-levels of Q-Criterion. The effect of the non-circular jet on the flow field is studied through local timescales and turbulent kinetic energy. The intermittency of the signal in the nearfield is analyzed in Section 3.3 by Empirical Mode Decomposition (EMD) analysis and spatio-temporal correlations to observe the effect of specific scales at end of potential core along space in the streamwise direction. An equivalent circular nozzle is designed and its streamwise mean and rms velocities are compared with the rectangular nozzle. Differences in ejection and entrainment like motions is analyzed along the major and minor axes of the 2 : 1 aspect ratio rectangular jet and for the 16

31 Figure 3.1: Schematic of the rectangular nozzle and important locations equivalent circular nozzle in Section 3.5 using Joint Probability Density Function. Figure 3.1 is used to explain the nomenclature of some key points where data has been highlighted in this study. These points help to highlight the effects of the geometry of the non-axisymmetric nozzle. The center of the longer edge represents the major axis center represented as point 1 and the center of the smaller edge represents the minor axis center depicted as point 3. The top right corner is labeled corner 4 (point 4). Point 2 is the jet center, whose streamwise location is x/d = The end of potential core represented by point 5 (velocity is 95% of jet exit velocity[24]), is at x/d = 6.2 or 4 jet diameter downstream from the jet exit. Most of the significant flow features can be summarized from the results obtained by analyzing axial arrays of points along the major axis, minor axis, jet centerline and 17

32 Figure 3.2: Schematic of array of points selected along the minor axis center a corner. Figure 3.2 and figure 3.3 demonstrate the array of points along which data has been stored along the minor axis center and major axis center in the streamwise direction respectively. Data is stored along the points on the red line spaced at half jet diameters (x/d) from each other. The points are chosen from the jet exit till twelve jet diameters downstream. Similarly points are stored along the corners and the jet centerline. The contours shown in these figures is the time averaged u-velocity along the Y=0 plane and the Z=0 plane respectively. 3.1 Experimental Validation A grid resolution study has been performed on two separate grids. The first grid has 39 million points with 614, 180 and 358 points in the ζ, η, ξ directions respectively and the second grid has 74 million points with 650, 290 and 405 points in 18

33 Figure 3.3: Schematic of array of points selected along the major axis center the ζ, η, ξ directions respectively. A comparison of streamwise mean velocity ( u U ) normalized by the jet exit velocity for the two grids is shown in Fig. 3.4 and streamwise rms velocity fluctuation ( urms U = u 2 ) along the jet centerline is presented in Figs The normalized mean axial velocity and axial rms velocity fluctuation are compared in along with the experimental measurements of Zaman[13]. The computational results on the fine mesh captures the trend of the mean jet centerline velocity observed in the experiments fairly accurately. The computational data has been shifted axially by x/d = 1 to match the end of the potential core: this accounts for the unknown experimental boundary layer thickness[25]. Subsequent to this however, the rate of decay of axial velocity is similar for computations and experiment. The computations on the finer mesh reveal a slightly faster decay rate beyond the potential core collapse. Our computations are able to capture the vena contracta effect where the mean velocity exiting from a subsonic jet exceed the value of one. 19

34 The rms velocity shown in Fig. 3.5 indicate experimental core initial fluctuations of about 3% at the nozzle exit. These values are not enforced in the simulation. This delays the growth of fluctuations on the centerline at the nozzle exit for the coarser mesh. We are able to capture the peak fluctuation on the coarse mesh but initially the growth of fluctuations on the centerline near the nozzle exit is fairly low. The finer mesh shows some core turbulence due to a transitional boundary layer prior to the nozzle exit. Results on this mesh are accurate as the peak values of u rms and the decay rate are reproduced accurately by the simulation. This instills confidence in predicting turbulent phenomena inside the core of the jet. 3.2 Features of the Flowfield In this section, we provide an overview of the flow. The results discussed in this section are on the 39 million grid. The u-velocity contours at the four axial locations are selected as shown in Fig They are plotted at four locations with values of x/d = 2.25, x/d = 3.50, x/d = 5.0 and x/d = 7.0 as shown in Fig Near the nozzle exit at the first location, the axial velocity contours are rectangular but further downstream at locations 2 and 3, the cross-section evolves into an ellipselike shape. At about x/d = 7, the effect of the rectangular nozzle shape at the exit is no longer evident and the contour is generally circular, as depicted in Fig This is consistent with the observations of Zaman[26], who attributed this to induced 20

35 Figure 3.4: Comparison of normalized centerline mean velocity on the two grids with experimental data for the 2 : 1 aspect ratio rectangular nozzle 21

36 Figure 3.5: Comparison of normalized centerline rms velocity on the two grids with experimental data for the 2 : 1 aspect ratio rectangular nozzle 22

37 Figure 3.6: Isometric view of the u-velocity contours at indicated locations velocities of streamwise vortex pairs causing the jet to spread and eventually take a circular shape as in Fig To highlight the instantaneous flow structure, Q-criterion iso-levels are plotted in Fig. 3.8 at a typical statistically stationary state. The contour is colored with u- velocity values. Hairpin vortices are generated near the nozzle exit, and they grow in size as they propagate downstream. These vortices are initiated along the corners. The hairpin vortices have heads in the outer, lower speed region while their legs are in the inner higher speed region. Therefore, these vortices elongate along the streamwise direction. Upstream of x/d = 6, the hairpin structures are predominantly present along the major axis. Further downstream from about x/d = 6 to x/d = 8 the structures are present in both major and minor axes. Further down- 23

38 Figure 3.7: Front View of the u-velocity contours at above mentioned four locations Figure 3.8: Time-averaged iso-levels of Q for the 2 : 1 aspect ratio rectangular nozzle 24

39 Figure 3.9: Auto-correlation used to calculate local timescales along the three mentioned locations. stream beyond x/d = 8, the hairpin vortices become stronger along the minor axis compared to the major axis. To quantitatively analyze the scales in the flow, we use local integral timescales by considering the auto-correlation of the pressure signal. An example of such a pressure signal, at x/d = 4 along the major axis, is depicted in Fig The vertical axis represents the correlation coefficient. The horizontal axis represents non dimensional lags. The auto-correlation of a signal at zero lag has been normalized to unity. For discrete signal correlation, lag represents displacement of the destination signal with respect to the source signal. A unit increment in lag is thus equal 25

40 Figure 3.10: Local timescales along three different locations in the flow lags represents 1000 time steps, where time step is one millisecond. 26

41 to the time interval between two successive samplings of the signal. The difference of two successive sampling is equal to the time step of our simulations. The local timescale is defined as the lag at which the auto-correlation function decays to zero for the first time. For a signal oscillating around the zero correlation coefficient as seen in the figure it is not required to integrate the entire signal to get the local timescales. As the positive and negative oscillations about the zero line cancel each other on integration it is sufficient to take the first zero crossing for our local time scale calculation. In the following analysis, local timescales are calculated for array of points along the major axis, minor axis and the corners in the streamwise direction as shown in Fig The timescale calculation in the streamwise direction at the three locations enables us to classify the flow into three regions of interest. The first region is between the nozzle exit up to the collapse of the potential core (x/d = 6). In this region the minor axis has the smallest timescale in Fig. 3.10, which results in prominent fine-scale fluctuations. The major axis has the largest timescale corresponding to relative large scale fluctuations. The second region extends from x/d 6 to x/d 8. Here the timescales along the major and minor axis are comparable. This can be explained by the azimuthal coherence of the hairpin structures in the flow field. In this region, hairpin like vortices in Fig. 3.8 are present along both axes. The footprint of these structures are present along the major, minor axes and the corner in this streamwise region of the flow leading to comparable local time scales. 27

42 The third region is downstream of the nozzle, beyond x/d 10. From Fig. 3.10, here the corners have the largest timescale suggesting that the the corner have significant low frequency content. On the other hand, the major axis has the smallest timescale and should therefore, have negligible presence of these low frequency components. This has been further explained using scale specific correlation in Section 3.3. To understand the turbulence intensity along the major, minor axes and the corner in the streamwise direction, Turbulence Kinetic Energy (TKE), defined as T KE = u 2 + v 2 + w 2 is calculated and shown in Fig Along the minor axis the TKE increases and reaches a maximum value at about x/d = 5, which corroborates with the observation in Fig The minor axis had the smallest timescales or prominent fine-scale fluctuations in the first region of interest which contribute to the highest peak in TKE. Conversely, minor axis has the smallest peak due to largest local timescales or least fluctuations which results in the lowest peak as depicted in Fig The TKE plot along the jet centerline shows a peak at the potential core collapse point as depicted by the black line in Fig Fig shows the pressure fluctuations along the major and minor axes, jet centerline and the corner along the streamwise location. The pressure fluctuations show a peak along the major axis center before the potential core collapse point near the nozzle exit. Further downstream the pressure fluctuations become similar for both major and minor axes. This trend is observed due to formation of hairpin like structures initially along the major axis near the nozzle exit. Further 28

43 Figure 3.11: Turbulent Kinetic Energy along four locations in the flow 29

44 Figure 3.12: Pressure fluctuations along four locations in the flow downstream the hairpin structures are present along both major and minor axes. 3.3 Scale specific correlation study Nearfield of jets has been found to exhibit prominent intermittent phenomena that contribute to noise. Due to these intermittency phenomena, and to highlight the associated scales, the influence of component scales should be studied separately. For circular jets, previous efforts have shown from correlations that the region near 30

45 the end of potential core is the location of the dominant noise source[27, 28]. A similar correlation analysis is performed on the non-axisymmetric jet where the potential core collapse point is selected as the source point. The velocity signal at the core collapse point (x/d = 4) is an fluctuating signal in time and shows a broad spectrum as shown in the top left frame of Fig We enhance the correlation study by splitting the signal at this point into a range of frequencies and correlating each range of frequencies with destination points, thus providing a detailed understanding of the signature of the source point. The technique to perform scale specific correlation is explained next. Empirical Mode Decomposition method, proposed by Huang et al.[29, 30], is designed for data that is non-stationary and nonlinear. The process begins by creating an upper and lower envelope that defines the local maxima and minima respectively around the original signal. The envelope is formed by taking the cubic interpolation on the maxima and minima points. The mean of the upper and lower envelope is calculated. The mean of the signal is then subtracted from the original signal to obtain the sifted signal. If the signal is not converged enough the above steps are repeated. This entire process is known as sifting as shown in Fig This procedure is repeated till the sifting process results in a symmetric output with no negative local maxima and positive local minima. Once these conditions along with IMF conditions are met the first IMF is obtained. A mode is called an IMF if it satisfies the following two criteria: 31

46 1. The number of extrema and number of zero-crossings should be equal to or differ by atmost one. 2. The mean value of the envelope should be zero at any point in the mode. The first IMF is then subtracted from the original signal and the sifting process is performed to the new signal till another IMF is obtained. Each subsequent IMF is found by taking the original signal and subtracting the previous IMF from it. After a sufficient number of IMFs are found, the remaining curve is monotonic function and is defined as the residue as seen in Fig Each IMF accounts for a range of frequencies and intra-wave frequency variations characteristics of non linear system. Each of these IMFs progressively extracts the smallest local oscillatory scales existing in the original signal. As a result, higher IMFs indicate larger scales of oscillation and energy of each IMF is limited within a narrow band of frequencies. The power spectral density (PSD) of each IMF is compared with the total signal to quantify its spectral contribution and to highlight its narrow-band character in Fig The red curve indicates the PSD of the total signal while the black curve represents the PSD of each IMF. The dashed vertical lines indicate values of St = 0.05, 0.1, 0.3, 0.5, 0.7, 1.0, 2.0. It is observed from the Fig that IMF 1 shows the highest range of frequencies in the signal above St = 2. The PSD of IMFs 3 and 4 correspond to St = 0.1 to St = 0.3, while IMFs 5 through 8 correspond to relatively larger scale fluctuations below St =

47 Figure 3.13: Steps involved in sifting process [31] 33

48 Figure 3.14: IMFs of EMD applied to source signal (potential core collapse point) Figure 3.15: Comparison of PSD of each IMF of source signal with PSD of the total signal 34

49 Figure 3.16: Schematic showing key points used in EMD analysis EMD has been used successfully to study intermittency of coherent structures in turbulent flows[32], and to highlight the directivity of the small and larger scales from the source point (potential core collapse point). Scale specific correlation analysis is performed to show the directivity of the structures by correlating u-velocity perturbation signals along the major, minor axes and corner with the individual IMFs of the source signal. Fig highlights some of the key points used in scale specific correlation analysis. The solid white line represents the jet centerline. The white dot represents the end of potential core. The u-velocity at this point is broken into its IMFs. The three dashed lines represent array of points along the major axis center, minor axis center and one of the corners. The IMFs at the potential core collapse point are 35

50 then correlated array of points along the above three mentioned location to see the effect of specific scales at the end of potential core along space in the streamwise direction. The correlation contours to compare each individual IMF of the source signal with the distribution of points along the centerline of the major axis, minor axis and corners are shown in figures below. In each of the below figures the left most frame is the major axis center, the middle frame is minor axis center and the right most frame is the corner. The horizontal axes in the subplots represent the nondimensional lag values in time and the vertical axes denote non-dimensional axial distance. The plotted contours represent the correlation coefficient. The horizontal dotted line depicts the end of the potential core which is the source point. The part above the black line represents downstream locations of the potential core collapse point. Regions below represent upstream locations. Fig compares the correlation of IMF 1 along the major axis, minor axis and the corner. We see negligible correlation both upstream and downstream of the potential core collapse point. IMF 1 shows structures corresponding to St 2.0. The presence of negligible correlation streaks shows absence of these high frequency structures both upstream and downstream of the potential core collapse point. IMF 2 has not been shown in the figures below since they also show negligible correlation and is similar to IMF1. Fig shows the correlation of IMF3 with the three locations. Slightly stronger correlation streaks are observed for the case of IMF3. IMF 3 shows structures corresponding to St 0.5. The correlation patterns 36

51 Figure 3.17: Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF1 of the source signal (x/d = 4) from the jet exit. are present only upstream of the potential core collapse point. Fig shows the correlation of IMF 4 with array of points along the three locations. IMF 4 shows structures corresponding to St 0.3. Correlation patterns are stronger than the previous IMFs. The correlation streaks are present predominantly just above and below the potential core collapse point. This indicates that these scales peak directly above the source location. Fig shows the correlation of IMF 5 with the array of points along the three locations. IMF 5 corresponds to structures of St 0.1. For the major axis correlation streaks are observed primarily downstream of the potential core collapse point and are weaker than the streaks obtained for the minor axis and the corner. The minor axis center shows strong correlation streak predominantly downstream of the potential core collapse point. The corner however shows strong correlation streaks both upstream and downstream of collapse point. this indicates greater influence of source point at minor 37

52 Figure 3.18: Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF3 of the source signal (x/d = 4) from the jet exit. axis compared to major axis for smaller frequencies corresponding to IMF 5. This was also observed during local time scale discussion and see in Fig Fig corresponds to correlation of IMF 6 with array of points along the three locations. This IMF corresponds to structures of St The major axis shows small correlation patterns downstream of the potential core collapse point. For the minor axis correlation streaks are present only downstream while upstream of the core collapse point negligible correlation is observed. However the corner shows strong correlation patterns downstream of the potential core collapse point. Fig depicts the correlation of IMF 7 with array of points along the three locations. IMF 7 corresponds to structures of St Strong correlation patterns are only observed in the case of the corner compared to the major axis and minor axis centers. These streaks are present in the upstream and downstream locations indicating major influence of source point along the corners for the given frequency 38

53 Figure 3.19: Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF4 of the source signal (x/d = 4) from the jet exit. Figure 3.20: Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF5 of the source signal (x/d = 4) from the jet exit. 39

54 Figure 3.21: Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF6 of the source signal (x/d = 4) from the jet exit. range corresponding to IMF 5,6 and 7 (St = 0.01 to St = 0.1). For these large-scale structures, prominent alternate positive negative correlation streaks are observed, indicating emission of these large scale structures downstream. Comparison of the EMD analysis at the three locations suggests that the array of points along the corner are important and show strong presence of frequencies corresponding to St = 0.05 to St = 0.3 as compared to minor and major axes. This corroborates with the discussion in the section of timescales which showed that for x/d > 8 the corner has the maximum timescales corresponding to largescale structures (low Strouhal Number). The signature of the core collapse point is significantly larger in the case of the corner confirming that the corners play an important role in large scale dynamics of a non-axisymmetric jet flow. Since EMD provides a meaningful scale separation, it is used for selectively distributing the signal into only smaller scales and larger scales. Small-scale struc- 40

55 Figure 3.22: Comparison of two-point correlation of u-velocity perturbation along a series of points on the three locations with IMF7 of the source signal (x/d = 4) from the jet exit. tures, calculated by adding the first four IMFs of the signal at the core collapse point is depicted in Fig The sum of the IMFs will be devoid of larger structures which are present in the residue. The sum of IMFs is removed from the signal to form the residue, containing only the larger-scale structures Figure 3.24 shows the result of two point spatio-temporal correlation analysis with the signal containing only small-scale components and signal containing only larger-scale structures with array of points along the minor axis in the streamwise direction. For the finer scale structures the correlation streaks are present only upstream of the potential core. In the case of larger-scale structures, the presence of inclined alternate positive and negative streaks indicate crest and trough of intermittent wave packets propagating in the downstream direction. The influence of the small-scales propagating out is masked, as the source signal no longer contains any trace of it. This highlights the downstream directivity 41

56 Figure 3.23: IMFs of EMD applied to source signal (potential core collapse point) separating signal into high frequency and low frequency scales. of the larger scale structures. The small scale structures are present only upstream of the potential core collapse point. 3.4 Comparison with an Equivalent Circular Nozzle To analyze the difference in flow evolution and to study the ejection and entrainment patterns as a function of streamwise location, an equivalent circular nozzle was designed from the 2 : 1 aspect ratio rectangular nozzle. A circular nozzle was generated by using the areas of the inlet and exit of the circular to rectangular nozzle and using the distance between the inlet and exit of the rectangular nozzle as the distance between the inlet and exit of the circular nozzle as shown in Fig The same initial conditions used for the rectangular nozzle are used for 42

57 (a) IMFs corresponding to high frequency (b) IMFs corresponding to low frequency scales Scales Figure 3.24: Two point correlation of u-velocity perturbation along a series of points along the minor axis with IMFs corresponding to high and low frequency scales. the equivalent circular nozzle for simulation. The normalized mean velocity along the centerline of the two nozzles and the rms velocity are plotted along the lipline of the major axis center for the rectangular jet and for the circular jet. The figures show a good match for the two cases and show the same trend further downstream from the nozzle. We capture the venacontracta effect for both cases as expected for such low Mach number flow. The equivalence facilitates the difference between the dynamics of the rectangular and circular configurations. Fig shows the comparison of the normalized mean centerline velocity for the rectangular nozzle and the equivalent circular nozzle along the centerline. The plots show a similar trend for both the nozzles and validated the accuracy of our equivalent circular nozzle. Fig compares the normalized rms velocity along the lipline for the rect- 43

58 Figure 3.25: Schematic of the computational equivalent circular nozzle geometry Figure 3.26: Comparison of normalized mean velocity for the rectangular and circular nozzles 44

59 Figure 3.27: Comparison of normalized rms velocity for the rectangular and circular nozzles angular nozzle and the equivalent circular nozzle. The plots show a similar trend with the circular nozzle showing a slightly higher peak value. After comparing mean and rms velocity for both nozzles the accuracy of the equivalent circular nozzle was validated and used for further analysis. The mean u-velocity contours at the four axial locations as shown in Fig are selected to show evolution of the shape of the contours in the streamwise direction. The four locations are at x/d = 1.2, x/d = 3.0, x/d = 5.0 and x/d = 7.5. At all 45

60 Figure 3.28: Isometric view of the u-velocity contours at indicated locations for the circular nozzle locations near the nozzle and further downstream from the nozzle exit the velocity contours are circular in shape as shown in Fig, In the case of the rectangular nozzle, the contours near the nozzle exit were dependent on the nozzle exit shape. The shape evolved from rectangular to ellipse like shape to finally a circular shape in the streamwise direction. 3.5 Comparison of entrainment and ejection events features by Joint (u u r) probability density function The Joint Probability Density Function (JPDF) at locations on the major and minor axis has been examined to understand the trend in the formation of coherent structures along the two axes and compared with the equivalent circular nozzle. The 46

61 Figure 3.29: Front view of the u-velocity contours at above mentioned four locations for the circular nozzle Figure 3.30: Conceptual representation of the ejections and entrainment events inside a shear layer. 47

62 formation of ejection or entrainment like motions has been studied to examine the flow along the two unequal edges which play a major role in rectangular nozzle flow dynamics and to differentiate between flow evolution of a rectangular and an equivalent axisymmetric nozzle. A representative velocity profile is depicted in Fig to discuss the ejection and entrainment motions. The Joint Probability Density Function (JPDF) at point B is considered as shown in Fig When a particle moves outwards along the radial direction it has positive radial velocity perturbation (u r > 0) and vice versa. If a particle is moving from a point A which has lower u velocity to a point B which has higher u velocity then u < 0 for the particle at point B. A lower momentum particle at point A is entraining to a higher momentum particle at point B. Similarly, a particle moving from point C to B (higher to lower u velocity) then u > 0. Ejections are thus characterized by u > 0 and u r > 0. Entrainment on the other hand reflect u < 0 and u r < 0 The joint (u u r) JPDFs along the major axis and minor axis center are conveyed by Figs.3.31 and 3.32 respectively, where (u r) is the fluctuating radial component of velocity at that point. These points were chosen to study the difference in ejection and entrainment of structures along the major and minor axes as noticed in Fig The JPDF at a point is taken to correlate the two perpendicular components of velocity at that point. We take u and w at major axis center and u and v at minor axis center. In Figs. 3.31, 3.32 and 3.33, the x- axis represents the u-velocity perturbations. The y-axis represents the other perpendicular component of veloc- 48

63 Figure 3.31: Joint PDF of u -w along the centerline of the major axis ity perturbation at that point (v or w ). The contour plots indicate the number of times a event is occurring at that specific point (ie. number of times an ejection or entrainment like motion is occurring). The angle defined between u and u r brings to light the correlation between the two orthogonal components[33]. A joint PDF performed on a signal with itself will be a straight line with an angle of 45, while a joint PDF performed on random noise with Gaussian distribution will be a circle. First analysis is performed on the rectangular jet. Figure 3.31 shows the joint PDF at points along the major axis center. Near the jet exit at x/d =2, 3,4, strong correlation between u and w in the third quadrant is observed where both u and w are negative, indicating regions of entrainment. As seen during the discussion of the q-criteria plots, the hairpin like structures initially are present along the ma- 49

64 Figure 3.32: Joint PDF of u -v along the centerline of the minor axis. jor axis center which corresponds to these higher entrainmemt like motions seen in JPDF contour plots. Further downstream from the jet exit, the shape and size of the contours in the third quadrant become narrower representing reduction in entrainment. The high values of contours occur in the first quadrant where u and w are both positive indicating regions of ejection. Further downstream beyond the end of potential core more symmetrical contours with small values of correlation indicative of negligible entrainment and ejections are observed. At about x/d = 10, no correlation is observed between u and w indicating negligible ejection or entrainment like motions. Figure 3.32 shows the joint PDF at points along the minor axis center. An immediate observation is that near the nozzle exit at x/d = 2,3 there is lower cor- 50

65 Figure 3.33: Joint PDF of u -v along the lip line of the equivalent circular nozzle. relation compared to the corresponding points along the major axis center. At x/d = 4, the contours show correlation between u and w in the third quadrant again indicative of entrainment like motions. This corroborates with the q-criterion plots which show that at about x/d =4, the formation of the hairpin like structures along the minor axis center becomes apparent. In comparison with contours along the major axis at x/d = 10, higher correlation is observed in both the first and third quadrant. This indicates presence of entrainment and ejection which is negligible in the case of equivalent points along the major axis center. Coherent structures are present along the minor axis center further downstream compared to the major axis center indicating that entrainment is stronger along the minor axis at these downstream locations. 51