Aeroacoustics of Turbulent Mixing Layers a Dissertation

AEROACOUSTICS OF TURBULENT MIXING LAYERS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Arjun Sharma December 2011

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/hy183xv9184

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Sanjiva Lele, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Brian Cantwell

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Parviz Moin

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii iv Abstract

Jet noise is an important contributor to overall aircraft noise. The flow near the jet nozzle is a spatially developing mixing layer and influences flow dynamics further downstream. The present study focuses on the influence of inflow conditions on mixing layer development and noise generation mechanisms using large-eddy simulations. Large-eddy simulations of spatially developing, turbulent mixing layers with splitter plate included in the computational domain are presented. Different inflow condition cases with initially laminar boundary layers (abbreviated as LBL) and turbulent boundary layers (abbreviated as TBL) are considered. Effect of heating, keeping the velocity ratio fixed, is analyzed for both. For each case, the mean and turbulent intensity profiles collapse when plotted in similarity coordinates. The development distance to achieve self-similarity in the mean velocity profile is found to be shortest for cases with turbulent exit boundary layers. The growth rate of the shear layer and peak self-similar values of the turbulent intensities are found to be in agreement with available experiments. It is observed that with heating, the initial instability is accelerated but the saturation self-similar amplitude of Reynolds stress components do not vary. The saturation amplitudes of density fluctuations were found to increase proportionally to difference in free-stream densities whereas near-field pressure fluctuations were found to decrease with heating. A simple scaling is suggested for the near-field pressure fluctuation amplitude. The observed scaling laws were also confirmed by simulation data from two-dimensional direct numerical simulations. For LBL, sound radiation is observed in downstream direction peaked roughly at 30 degrees. The vortex pairing and breakdown to turbulence contribute significantly

v to the radiated sound. For TBL, the acoustic field near the shear layer is significantly weaker and noise due to passage of boundary layer eddies over the trailing edge is observed. For both the cases, a reduction in overall sound pressure levels in the far-field is observed with heating. Analysis of relative importance of Reynolds stress autocorrelation tensor components is presented to explain the effects of heating.

vi Acknowledgements

Firstly, I would like to thank my advisor Prof. Sanjiva Lele for his constant encour- agement and for being a great teacher. Thanks are due to Prof. Moin and Prof. Cantwell for serving on my reading committee and their helpful suggestions. I also thank Profs. MacCormack and Papanicolaou for serving on my examination committee and Prof. Pitsch for sharing his insights on turbulent flows and Large-Eddy Simulations. The code used for simulations in the present work was provided by Dr. Santhanam Nagarajan and Dr. Bhaskaran Rathakrishnan. I thank them for sharing their codes with me. I thank Prof. Lele and Prof. Moin for involving me in the jet noise project. It was a great opportunity and learning experience to work with Dr. J. Nichols, Dr. S. Mendez, Dr. M. Shoeybi and Dr. F. Ham. I thank Dr. Nichols for sharing his knowledge on stability analysis of shear layers and Dr. Ali Mani for sharing his insights on the use of computer code. I am thankful to the past and present members of the Lele group for their help. I thank Sankaran Ramakrishnan, Shashank and Supreet Bahga for their good company over all the years. I thank my parents, brother and fiancée,Preeti, for their love, support and patience during this long journey. This work was supported by a grant from the Supersonics Element of the Fun- damental Aeronautics Program of National Aeronautics and Space Administration (NASA) under NRA grant NNX07AC94A. Computational support for the simulations was provided by the HPC-DoD system and local Certainty cluster under the following award. MRI-R2: Acquisition of a Hybrid CPU/GPU and Visualization

vii Cluster for Multidisciplinary Studies in Transport Physics with Uncertainty Quan- tiﬁcation.

viii Nomenclature

Lx, Ly, Lz Domain Lengths in streamwise, transverse and spanwise directions

M Mach number

Mc Convective Mach number

Nx, Ny, Nz Number of grid points in streamwise, transverse and spanwise directions

t Ruu Temporal correlation of streamwise velocity ﬂuctuations

Re Reynolds number

t Suu Temporal spectra of streamwise velocity ﬂuctuations

St Strouhal number

T Temperature

U Free stream velocity

Uc Convection velocity

Uj Jet velocity

∆U Diﬀerence in free stream velocities

∆t Time step

∆x, ∆y, ∆z Grid spacing in streamwise, transverse and spanwise directions

ix δθ, δω Momentum and vorticity thickness of the shear layer

γ Ratio of speciﬁc heats

µ Dynamic viscosity

ρ Density

c Speed of sound

p Pressure

p0, ρ0 Pressure and density ﬂuctuations

r Velocity ratio (low speed stream over high speed stream).

s Density ratio (low speed stream over high speed stream).

t Time

u00, v00, w00 Velocity ﬂuctuations in streamwise, transverse and spanwise directions

u, v, w Velocity components in streamwise, transverse and spanwise directions

wp Width of the splitter plate

x, y, z Streamwise, transverse and spanwise coordinates

Operators e· Favre average

· Reynolds average

Subscripts

1 Relative to higher speed stream

2 Relative to lower speed stream

x Contents

Preface v

Abstract vi

Acknowledgements viii

1 Introduction 1 1.1 Shear layer ﬂow development ...... 1 1.2 Previous analytical and computational work ...... 5 1.3 Sound generation in jets and mixing layers ...... 8 1.4 Outline and Accomplishments ...... 14

2 Numerical Framework for LES computations 16 2.1 Governing equations ...... 16 2.2 Spatial Discretization ...... 20 2.3 Time Integration ...... 20 2.4 Overset method ...... 21 2.5 Rescaling-Recycling method ...... 22 2.6 Numerical solution of laminar boundary layer equations ...... 24

3 Acoustic Analogy Computations 27 3.1 Goldstein’s acoustic analogy (GAA) ...... 28 3.2 Connection to Lighthill’s equation ...... 31 3.2.1 Solution using Green’s function ...... 35

xi 3.3 Veriﬁcation tests for acoustic analogy calculation ...... 36 3.3.1 Source placed in uniform subsonic and supersonic ﬂow . . . . 36 3.3.2 Scattering of plane waves by a compressible vortex ...... 40

4 Two-dimensional shear layer dynamics 42 4.1 Problem Description ...... 44 4.2 Numerical Technique ...... 45 4.3 Results ...... 46 4.3.1 Near field flow ...... 46 4.3.2 Impulse response ...... 55 4.3.3 Acoustic near field ...... 55 4.4 Summary ...... 67

5 Turbulent mixing layer computations 69 5.1 Problem Description ...... 69 5.2 Cases with incoming laminar boundary layers ...... 71 5.2.1 Computational parameters ...... 71 5.2.2 Mean ﬂow ...... 74 5.2.3 Impulse response ...... 83 5.2.4 Turbulence statistics ...... 85 5.3 Cases with incoming turbulent boundary layers ...... 99 5.3.1 Computational parameters ...... 99 5.3.2 Mean ﬂow ...... 100 5.3.3 Turbulence statistics ...... 105 5.4 Summary ...... 112

6 Sound generation in turbulent mixing layers 115 6.1 Cases with incoming laminar boundary layers ...... 115 6.1.1 Near-ﬁeld spectra ...... 116 6.1.2 Far-ﬁeld spectra ...... 120 6.1.3 Space time correlations ...... 120 6.2 Cases with incoming turbulent boundary layers ...... 127

xii 6.2.1 Near-ﬁeld spectra ...... 127 6.2.2 Far-ﬁeld spectra ...... 129 6.2.3 Space time correlations ...... 137 6.3 Summary ...... 140

7 Conclusions 147 7.1 Future Work ...... 149

xiii List of Tables

1.1 Summary of some of the previous computational studies of turbulent mixing layers. The study type distinguishes between Direct Numeri- cal Simulations (DNS) and Large-eddy simulations (LES) and temporal mixing layers (TML) from spatially evolving mixing layers (SML). Temporal annular mixing layer (TAML) is indicated for Freund et al. (2000). RANS-based studies or those based on other modelling approaches are not included. The simulations do not contain a splitter plate unless mentioned. The indicated Reynolds number is based on velocity diﬀerence of the two streams and splitter plate thickness if a plate is present, or vorticity thickness. Only one representative case with the highest Reynolds number and grid size is reported for studies

with multiple cases. M1,M2 represent the Mach numbers of the fast and slow streams respectively. The incompressible ones are indicated by “inc.”...... 9

4.1 The parameters for the test case considered. Reynolds number, Reδθ,1 = ∆Uδθ,1 , is about 160, based on free stream conditions on the boundary ν1 layer thickness in the high-speed side, velocity diﬀerence ∆U and kinematic viscosity based on free stream conditions on the high-speed side.

The quantity, U1/c2, corresponds to jet acoustic Mach number and is

kept constant in the three cases. Uc,is is the convection velocity based on the isentropic estimate...... 45

xiv 5.1 The parameters for the test case considered. Reynolds number, Reδθ,1 = ∆Uδθ,1 , is based on free stream conditions on the boundary layer thick- ν1 ness in the high-speed side, velocity diﬀerence ∆U and kinematic viscosity based on free stream conditions on the high-speed side. . . . . 71 5.2 Mesh parameters for three-dimensional calculations. ∆, L and N refer to the mesh spacing, domain length and number of points in each direction respectively. Reference length used to non-dimensionalize

spatial dimensions is the plate thickness, wp...... 73 5.3 Mesh sizes for laminar boundary layer simulation cases...... 73 5.4 Comparison of growth rate and self-similar turbulent intensities for cases A1 through A5 ...... 88 5.5 Mesh parameters for cases A4 and A5. The numbers with superscript + are in plus-coordinates based on skin friction coeﬃcient at the inlet to body-ﬁtted mesh. Otherwise reference length used to non-

dimensionalize spatial dimensions is the plate thickness, wp. ∆, L and N refer to the mesh spacing, domain length and number of points in each direction respectively. For the boundary layer mesh, the x mesh spacing is constant, minimum y-spacing is at the first y grid point and maximum at the last. For the body-fitted mesh, the x and y directions are the wall-tangent and wall-normal directions, respectively. The minimum spacing is at the trailing edge and maximum at the inlet and away from the wall. The minimum spacing location for the background mesh is at its interface to the body fitted mesh...... 100

xv List of Figures

3.1 Contours of real part of density fluctuations obtained from acoustic analogy solution for source place in subsonic uniform flow (at M = 0.5) on the left and supersonic uniform flow (at M = 2.5) on the right. . . 38

3.2 Comparison of real (on left) and imaginary part (on right) of density ﬂuctuations obtained from acoustic analogy solution (solid line) and analytical solution (shown in symbols) for source place in subsonic uniform ﬂow (at M = 0.5)...... 39

3.3 Comparison of real (on left) and imaginary part (on right) of density ﬂuctuations obtained from acoustic analogy solution (shown as solid lines) and analytical solution (shown in symbols) for source place in supersonic uniform ﬂow (at M = 2.5)...... 39

3.4 Contours of real (on left) and imaginary (on right) parts of scattered ﬁeld normalized by amplitude of incident plane waves...... 40

3.5 Comparison of scattered ﬁeld intensity obtained from acoustic analogy solution (solid line) and numerical simulation by Colonius et al. (1994) (shown in symbols). Shown for r/λ = 10 (left) and r/λ = 2 (right). . 41

4.1 Schematic of ﬂow problem setup...... 44

4.2 Contours of vorticity magnitude (normalized by ∆U/wp) plotted in logarithmic increments from 0.01 to 4 shown for cases A (top), B(middle) and C(bottom)...... 47

xvi 4.3 Proﬁles of mean velocity in (a) and density in (b) as a function of normalized transverse coordinate at diﬀerent streamwise locations shown for case B. (c) Comparison of vorticity thickness growth rates for different temperature ratios along the streamwise coordinate. Dash dot line: Temperature ratio = 1; Solid line: 1.4; Dashed line: 1.8...... 48

4.4 Variation of peak root mean square (r.m.s) ﬂuctuations of (a) streamwise velocity (b) transverse velocity (c) density (d) pressure (e) vorticity (f) dilatation in the streamwise direction. Case A: solid line; case B: dashed line; case C: dashed dotted line...... 50

4.5 (a) Variation of peak root mean square (r.m.s) ﬂuctuations of pressure 2 scaled by ρ2(Uc − U2) . Case A: solid line; case B: dashed line; case C: dashed dotted line. (b) The vortex center locations (symbols) based on pressure minima for case A. The straight lines correspond to slope

0.7 which is used as an estimate for convection velocity, Uc...... 51

4.6 Transverse profiles of (a) streamwise velocity fluctuations, (b) transverse velocity fluctuations, (c) density fluctuations and (d) pressure fluctuations for case A shown as a function of transverse coordinate

normalized by local vorticity thickness, δω(x). Streamwise locations:

x/wp = 163: Solid line; x/wp = 223: dashed line; x/wp = 256: dashed

dotted line; x/wp = 275: dotted line...... 52

4.7 Transverse profiles of (a) streamwise velocity fluctuations, (b) transverse velocity fluctuations, (c) density fluctuations and (d) pressure fluctuations for case B shown as a function of transverse coordinate

normalized by local vorticity thickness, δω(x). Streamwise locations:

x/wp = 203: Solid line; x/wp = 224: dashed line; x/wp = 245: dashed

dotted line; x/wp = 266: dotted line...... 53

xvii 4.8 Transverse profiles of (a) streamwise velocity fluctuations, (b) transverse velocity fluctuations , (c) density fluctuations and (d) pressure fluctuations for case C shown as a function of transverse coordinate

normalized by local vorticity thickness, δω(x). Streamwise locations:

x/wp = 118: Solid line; x/wp = 139: dashed line; x/wp = 160: dashed

dotted line; x/wp = 181: dotted line...... 54

4.9 Time history of pressure in the shear layer (y/wp = 0.8) at various

streamwise locations for case A. (a) x/wp = 52, (b) x/wp = 111, (c)

x/wp = 136 and (d) x/wp = 192...... 56

4.10 Time history of pressure in the shear layer (y/wp = 0.8) at various

streamwise locations for case B. (a) x/wp = 32.5, (b) x/wp = 47, (c)

x/wp = 75 and (d) x/wp = 130...... 57

4.11 Time history of pressure in the shear layer (y/wp = 0.8) at various

streamwise locations for case C. (a) x/wp = 18, (b) x/wp = 47, (c)

x/wp = 75 and (d) x/wp = 130...... 58

4.12 Pressure spectra for (a) case A, (b) case B and (c) case C at diﬀerent

streamwise locations. For case A: x/wp = 52: solid line; x/wp = 111:

dashed line; x/wp = 136: dashed-dotted line; x/wp = 192: dotted line.

For case B: x/wp = 32.5: solid line; x/wp = 47: dashed line; x/wp = 75:

dashed-dotted line; x/wp = 130: dotted line. For case C: x/wp = 18:

solid line; x/wp = 47: dashed line; x/wp = 75: dashed-dotted line;

x/wp = 130: dotted line...... 59

4.13 Contours of instantaneous pressure disturbance. (a) Case A (b) Case C...... 60

4.14 Pressure amplitude | A | plotted against the similarity coordinate,

(x − xo)/t, where xo is the initial location of disturbance. (a) Case A (b) Case C...... 60

xviii 4.15 (a) Observer locations located on a circular arc of radius 150 centered at 75 are shown in black symbols. Acoustic ﬁeld at St = 0.045 is shown in the background for case A. (b) Root mean square pressure ﬂuctuations as a function of angle made with respect to the x-axis. Case A: solid line; case B: dashed line; case C: dashed dotted line. . . 61

4.16 Real part of pressure DFT obtained from direct calculations shown for (a) St = 0.015, (b) 0.03, (c) 0.045 and (d) 0.06 for case A (density ratio = 1)...... 63

4.17 Real part of pressure DFT obtained from direct calculations shown for (a) St = 0.015, (b) 0.03, (c) 0.045 and (d) 0.06 for case C (density ratio = 2.7)...... 64

4.18 (a) Observer locations located on a circular arc of radius 150 centered at 75 are shown in black symbols. Pressure spectra for (a) case A, (b) case B and (c) case C...... 65

4.19 Fluctuation amplitude (r.m.s levels) of acoustic sources as deﬁned by Eqs.3.4-3.7 for the time averaged mean ﬂow obtained from the direct

calculations. (a) Su is the source term appearing in GAA equation 0 0 0 0 for variable u1 = ρv1/ρ, (b) Sv for variable u2 = ρv2/ρ and (c) Sp for 0 variable pe. Case A: solid line; case B: dashed line; case C: dashed dotted line...... 66

4.20 Fluctuation amplitude (r.m.s levels) of acoustic sources as deﬁned by Eq. 3.10 for uniform base ﬂow obtained from the direct calculations.

(a) Smom is the ﬁrst term on the right hand side of Eq. 3.10 and (b)

Sent is the second term. For (a) and (b) Case A: solid line; case B: dashed line; case C: dashed dotted line. (c) Comparison of spectra for observer location r = 150, θ = 30 for case C. Direct calculation: solid line; Full source term: dashed line; With only enthalpy ﬂux term: dashed dotted line...... 68

5.1 Schematic of ﬂow problem setup...... 70

xix 5.2 Grid used for the simulations. (a) view of full computational domain for case A1. Here, the shaded region denotes sponge used at domain boundaries. (b) zoom in view of the mesh around the plate and the surrounding background mesh. The spatial extent is shown in terms

of plate width, wp...... 72

5.3 (a) Visualization of contours of transverse velocity for case A1 in the X- Z plane along the lipline. The domain extends from 0 < z < 90 in the spanwise direction. (b)Visualization of density contours (plotted from

0.95ρ1 to 1.05ρ1) for Case A1 in the mid X-Y plane. The portion of the computational domain shown above extends from −20 < x < 340 and transverse extent from −100 < y < 100. Origin is located at the trailing edge of the splitter plate...... 75

5.4 (a) Visualization of contours of transverse velocity for case A3 in the X- Z plane along the lipline. The domain extends from 0 < z < 90 in the spanwise direction. (b)Visualization of density contours (plotted from

5.5 Top: Visualization of contours of streamwise vorticity for case A3 in the X-Z plane along the lipline. The domain extends from 0 < z < 60 in the spanwise direction. Bottom: Instantaneous visualization of density

contours (plotted from 0.5ρ1 to 1.1ρ1) in the mid X-Y plane for Case A3. The portion of the computational domain shown above extends from −20 < x < 500 and −75 < y < 75. Origin is located at the trailing edge of the splitter plate...... 77

5.6 Velocity streamtraces for case A1 close to the plate in (a) and far from the plate in (b)...... 78

xx wp ∂p 5.7 (a) Streamwise pressure gradient ( 2 ) on the surface of splitter ρ2∆U ∂x p02 plate for case A1. (b) Mean square of pressure ﬂuctuations ( 2 )on ρ2∆U the surface of plate for case A1. Solid line: y > 0 and Dashed line: y < 0...... 79 5.8 (a) Growth of mixing layer vorticity thickness as a function of distance from splitter plate tip in the downstream direction for cases case A1 (solid line) and case A2 (dashed line). (b) Variation of normalized

growth rate with convective Mach number, Mc. : present calculation; N: Pantano and Sarkar (2002);: Debisschop and Bonnet (1993); F: Papamoschou and Roshko (1988); I: Samimy and Elliot (1990); −: Langley curve...... 80

5.9 Variation of mean velocity (Umean = u/e ∆U) in the transverse direction (y/δω(x)) for case A1. (a) Close to the plate. −: x/wp = 6; −−:

x/wp = 15; − · −: x/wp = 24; ···: x/wp = 33; −4−: x/wp = 41. (b)

and far from the plate in the self-similar region. −: x/wp = 209; −−:

x/wp = 234; − · −: x/wp = 259; ···: x/wp = 284; −4−: x/wp = 318. 81 5.10 Variation of mean velocity (Umean = u/e ∆U) in the transverse direction (y/δω(x)) for case A3. (a) Close to the plate. Legend: −:

x/wp = 17.5; −−: x/wp = 33; − · −: x/wp = 47; ···: x/wp = 60;

−4−: x/wp = 73. (b) Far from the plate in the self-similar region.

Legend: −: x/wp = 245; −−: x/wp = 293; − · −: x/wp = 342; ···:

x/wp = 390; −4−: x/wp = 438...... 82 5.11 Variation of mean streamwise velocity for case A1 in (a) and A2 in (b).

Legend: −: x/wp = 2; −−: x/wp = 18; − · −: x/wp = 38...... 83 5.12 Variation of wavepacket amplitude with group velocity for cases A1

(a-c) and A2 (d-f). (a,d) x/wp = 2, (b,e) x/wp = 18 and (c,f) x/wp = 38. 84 5.13 Variation of Reynolds stresses along the lipline for (a) case A1 and (b) 2 2 2 A2. −: σu, − · −·: σw, −−: σv , ··· : σuv...... 86 5.14 Comparison of streamwise variation of Reynolds stresses for cases A1 2 2 2 and A2. (a) σu, (b) σv , (c) σw and (d) σuv. Line with symbols: case A2. Solid line: variation along lip line; dashed line: peak stresses. . . 87

xxi 5.15 Comparison of streamwise variation of r.m.s levels of density in (a) and pressure in (b) for cases A1 and A2. Line with symbols: case A2. Solid line: variation along lip line; dashed line: peak stresses...... 88 5.16 Self similar proﬁles of turbulent intensities for case A1 plotted as func- 2 2 2 tion of y/δω(x). (a) σu, (b) σv , (c) σw, (d) σuv, (e) mu and (f) mv as

deﬁned in Eqs. 5.3 and 5.4. −: x/wp = 266; −−: x/wp = 287; − · −:

x/wp = 309; ··· x/wp = 330...... 89 5.17 Self similar proﬁles of turbulent intensities for case A1 plotted as func- 2 2 2 tion of y/δω(x). (a) σu, (b) σv , (c) σw, (d) σuv, (e) mu and (f) mv as

deﬁned in Eqs. 5.3 and 5.4. −: x/wp = 266; −−: x/wp = 287; − · −:

x/wp = 309; ··· x/wp = 330...... 90 5.18 Comparison of peak self-similar Reynolds stresses with experimental

data for case A1. (a) σu, (b) σv and (c) σuv. ◦: Goebel and Dutton (1991); : Samimy and Elliot (1990); .+: Urban and Mungal (2001); O: Gruber et al. (1993); ×: Oster and Wygnanski (1982); N: Present study...... 91 5.19 Comparison of self similar proﬁles of Reynolds stresses for Case A3

plotted as function of y/δω(x) to experimental data of Samimy and Elliot (1990) (shown in symbols 4). (a) uf002/∆U 2 (b) vf002/∆U 2 (c) 002 2 00 00 2 wg/∆U (d) ugv /∆U . Legend: −: x/wp = 342; −−: x/wp = 366;

− · −: x/wp = 390; ··· x/wp = 415...... 92 x 5.20 (a) Two point axial correlation of the streamwise velocity, Ruu, (b) t z Temporal correlation, Ruu, (c) Spanwise correlation, Ruu, for case A1 at various streamwise locations. Legend: −−: x/wp = 355; −: x/wp =

300; −4−: x/wp = 250 ;− · −: x/wp = 190; −−: x/wp = 125; ··· :

x/wp =80...... 94 x 5.21 (a) Two point axial correlation of the streamwise velocity, Ruu, (b) t z Temporal correlation, Ruu, (c) Spanwise correlation, Ruu, for case A2 at various streamwise locations. Legend: same as in Fig. 5.20. . . . . 95 5.22 Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A1 at various streamwise locations...... 96

xxii 5.23 Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A2 at various streamwise locations...... 97

5.24 (a) Spectra obtained from temporal correlations in Fig. 5.20(b) for case A1; (b) Spectra obtained from spanwise correlations in Fig. 5.20(c) for case A1; (c) Spectra obtained from temporal correlations in Fig. 5.21(b) for case A2; (d) Spectra obtained from spanwise correlations in

Fig. 5.21(c) for case A2. Legend: −−: x/wp = 80; −·−·: x/wp = 190;

−: x/wp =355...... 98

5.25 Grid used for the simulations. Shaded region shown is the sponge used at the domain boundaries...... 99

5.26 Visualization of density contours overlayed with vorticity magnitude contours in the mid X-Y plane for Case A4. The density contours are

plotted between ρ2 ±0.05ρ2 and vorticity (normalized by ∆U/wp where

wp is the plate width) contours between 0 and 5...... 101

5.27 Visualization of vorticity magnitude contours in the spanwise plane at y = 0.51 (located just above the splitter plate in the high speed stream) for Case A4. The location of trailing edge is at (0, 0) ...... 102

5.28 Visualization of isosurfaces of second invariant of velocity gradient tensor for case A4. Plotted for invariant value, Q = 0.25...... 103

5.29 (a) Velocity streamtraces for case A4 close to the plate. (b) Shear layer vorticity thickness for case A4 (solid line) and A5 (dashed line). . . . 103

5.30 Variation of mean velocity (Umean = u/e ∆U) in the transverse direc- ∗ tion (y = y /δω(x)) for case A4. (a) Close to the plate (on left) −:

x/wp = 2; −−: x/wp = 3.6; − · −: x/wp = 5.2; ···: x/wp = 7;

−4−: x/wp = 8.75. (b) and far from the plate in the self-similar

region (right) −: x/wp = 16.7; −−: x/wp = 21.2; − · −: x/wp = 26;

···: x/wp = 31.2; −4−: x/wp = 37...... 104

xxiii 5.31 Proﬁles of mean Van-Driest transformed streamwise velocity in (a) and Reynolds stresses in (b) in y+ coordinate for the simulated turbulent boundary layer. (a) Blue line: from present computations, dashed lines: reference lines u+ = y+ and u+ = 2.5 lny+ + 5.5; (b) Legend: −−: u+, − · −·: v+ and −: w+. Symbols: DNS data of Spalart (1988) Lines: from present computations...... 105 5.32 Variation of turbulent intensities along the lipline for (a) case A4 and (b) case A5. Legend: −: uf002/∆U 2, −−: vf002/∆U 2, − · −·: wg002/∆U 2 and ··· : ug00v00/∆U 2...... 106 5.33 Self similar proﬁles of Reynolds stress components for case A4 plotted as function of tranvserse coordinate normalized by local vorticity thick- 002 2 002 2 002 2 00 00 2 ness, y/δω(x). (a) uf /∆U (b) vf /∆U (c) wg/∆U (d) ugv /∆U .

Legend: −: x/wp = 25.3; −−: x/wp = 32; − · −: x/wp = 39.2; ···

x/wp = 47.2...... 107

x z 5.34 (a) Axial correlation, Ruu (b) Spanwise correlation, Ruu, and (b) Tem- t poral correlation of the streamwise velocity, Ruu, for case A4 at various

streamwise locations. Legend: ··· : x/wp = 5.75; −−: x/wp = 12;

− · −·: x/wp = 19; −−: x/wp = 27; −: x/wp = 41; −4−: x/wp = 58. 108 x z 5.35 (a) Axial correlation, Ruu (b) Spanwise correlation, Ruu, and (b) Tem- t poral correlation of the streamwise velocity, Ruu, for case A5 at various streamwise locations. Legend: same as in Fig. 5.34 ...... 109 5.36 Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A4 at various streamwise locations...... 110 5.37 Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A5 at various streamwise locations...... 111

6.1 (a) Two point temporal correlation (b) Two point spanwise correlation of pressure for case A1. (c) Temporal spectra (d) Spanwise spectra of

pressure ﬂuctuations for case A1. −−: x/wp = 355; −: x/wp = 300;

−4−: x/wp = 250 ;−·−: x/wp = 190; −−: x/wp = 125; ··· : x/wp = 80.117

xxiv 6.2 (a) Two point temporal correlation (b) Two point spanwise correlation of pressure for case A2 (c) Temporal spectra (d) Spanwise spectra of pressure ﬂuctuations for case A2. Legend: same as Fig. 6.1...... 118 6.3 Pressure spectra for case A1 in (a) and case A2 in (b) at observer locations shown in (c). The observers are located at a common transverse

coordinate of y/wp = −65. Dashed line: x/wp = 48; dashed dotted

line: x/wp = 84; dotted line: x/wp = 120; Solid line with symbols:

x/wp = 153; Solid line: x/wp = 183 (also the locations of observers 1 to 5 in (c))...... 121 6.4 Comparison of pressure spectra from acoustic analogy predictions and LES computations shown for case A1 and observer locations as shown in Fig. 6.3(c): 1 in (a), 3 in (b) and 5 in (c). −: LES data; −− Volume integral; − · −·: Surface integral...... 122 6.5 Schematic showing the (r, θ) coordinate system used for far-ﬁeld observer locations...... 123 6.6 Noise directivity in terms of OASPL for cases A1 and A2. Solid line: A1; dashed line: A2...... 123

6.7 Sound pressure levels as a function of Strouhal number(St = fwp/∆U)

for case A1 at various observer locations at a distance of R/wp = 6500. 124

6.8 Sound pressure levels as a function of Strouhal number(St = fwp/∆U)

for case A2 at various observer locations at a distance of R/wp = 6500. 125 6.9 Comparison of temporal autocovariance functions for diﬀerent compo-

nents of Rijkl (Eq. 6.3) for cases A1 (a-c) and A2 (d-f) at three loca-

tions: (a) x/wp = 89, (b) x/wp = 200 and (c) x/wp = 284. Figs. (d)-

(f) are at corresponding locations for case A2. Legend: −4− : R1111,

−− : R2222, − · −· : R3333, ··· : R1212, −− : R1122, − B − : R1133,

− ◦ − : R1112 (with a negative sign)...... 128 6.10 Comparison of temporal autocovariance functions for diﬀerent com-

ponents of Rνjµl as deﬁned in Eq. 6.5 for cases A1 (a) and A2 (b)

at x/wp = 89. Legend: −4− : R4141, −− : R4242, − · −· : R4343,

··· : R4111, −− : R4122 and − B − : R4133...... 129

xxv 6.11 Space time correlations of streamwise velocity and estimate for con-

vection velocity for case A1. (a) At x/wp = 89. Legend: −4−: ξ = 0; −: ξ = 2.8; −−: ξ = 5.6; − · −·: ξ = 8.4; ··· : ξ = 11.2; −−:

ξ = 14; − ◦ −: ξ = 16.8. (b) At x/wp = 200. Legend: −4−: ξ = 0; −: ξ = 5.6; −−: ξ = 11.2; − · −·: ξ = 16.8; ··· : ξ = 22.3; −−:

ξ = 27.9; − ◦ −: ξ = 33.5. (c) At x/wp = 284. Legend: Same as (b). (d) ξ − τ plot (streawise separation plotted against the time delay for

peak correlation values) for estimation of convection velocity, Uc. (d)

− ◦ −: At x/wp = 89; −4−: At x/wp = 200; −−: At x/wp = 284. . 130 6.12 Space time correlations of streamwise velocity and estimate for convection velocity for case A2. Legend: same as in Fig. 6.11...... 131

00 00 6.13 Space time correlations of ρu1u1 as deﬁned in Eq. 6.3 and estimate for convection velocity for case A1. Legend: Same as in Fig. 6.11. . . . . 132

00 00 6.14 Space time correlations of ρu1u1 as deﬁned in Eq. 6.3 and estimate for convection velocity for case A2. Legend: Same as in Fig. 6.11. . . . . 133

00 00 00 00 00 00 6.15 Space time correlations of ρu2u2 (in a, d), ρu3u3 (in b, e) and ρu1u2

(in c, f) as deﬁned in Eq. 6.3 shown for x/wp = 200 for cases A1 (a-c) and A2 (d-f). Legend: −4−: ξ = 0; −: ξ = 5.6; −−: ξ = 11.2; − · −·: ξ = 16.8; ··· : ξ = 22.3; −−: ξ = 27.9; − ◦ −: ξ = 33.5...... 134 6.16 (a) Two point temporal correlation (b) Two point spanwise correlation of pressure (c) Temporal spectra for case A4...... 135 6.17 (a) Two point temporal correlation (b) Two point spanwise correlation of pressure (c) Temporal spectra for case A5...... 136 6.18 Noise directivity in terms of OASPL for cases A1, A2, A4 and A5. Solid line: A1; dashed line: A2; dashed dotted line: A4; dotted line: A5.137

6.19 Sound pressure levels as a function of Strouhal number(St = fwp/∆U)

for case A1 at various observer locations at a distance of R/wp = 6500. 138

6.20 Sound pressure levels as a function of Strouhal number(St = fwp/∆U)

for case A2 at various observer locations at a distance of R/wp = 6500. 139

xxvi 6.21 Space time correlations of streamwise velocity and estimate for con-

vection velocity for case A4. (a) At x/wp = 20. Legend: −4−: ξ = 0; −: ξ = 0.75; −−: ξ = 1.5; − · −·: ξ = 2.2; ··· : ξ = 3; −−: ξ = 3.7;

− ◦ −: ξ = 4.4. (b) At x/wp = 35. Legend: −4−: ξ = 0; −: ξ = 0.9; −−: ξ = 1.9; − · −·: ξ = 2.8; ··· : ξ = 3.7; −−: ξ = 4.6; − ◦ −:

ξ = 5.5. (c) At x/wp = 50. Legend: −4−: ξ = 0; −: ξ = 1.1; −−: ξ = 2.2; − · −·: ξ = 3.3; ··· : ξ = 4.4; −−: ξ = 5.5; − ◦ −: ξ = 6.5. (d) ξ − τ plot (streawise separation plotted against the time delay for

peak correlation values) for estimation of convection velocity, Uc. (d)

− ◦ −: At x/wp = 20; −4−: At x/wp = 35; −−: At x/wp = 50. . . 141 6.22 Space time correlations of streamwise velocity and estimate for convection velocity for case A5. Legend: same as in Fig. 6.21...... 142 00 00 6.23 Space time correlations of ρu1u1 as defined in Eq. 6.3 and estimate for convection velocity for case A4. Legend: same as in Fig. 6.21. . . . . 143 00 00 6.24 Space time correlations of ρu1u1 as defined in Eq. 6.3 and estimate for convection velocity for case A5. Legend: same as in Fig. 6.21. . . . . 144 6.25 Comparison of temporal autocovariance functions for different com-

ponents of Rijkl (Eq. 6.3) for cases A4 (a-c) and A5 (d-f) at three

locations: (a) x/wp = 20, (b) x/wp = 35 and (c) x/wp = 50. Figs. (d)-

(f) are at corresponding locations for case A5. Legend: −4− : R1111,

−− : R2222, − · −· : R3333, ··· : R1212, −− : R1122, − B − : R1133,

− ◦ − : R1112 (with a negative sign)...... 145 6.26 Comparison of temporal autocovariance functions for diﬀerent com-

ponents of Rνjµl as deﬁned in Eq. 6.5 for cases A4 (a) and A5 (b)

at x/wp = 35. Legend: −4− : R4141, −− : R4242, − · −· : R4343,

··· : R4111, −− : R4122 and − B − : R4133...... 146

xxvii xxviii Chapter 1

Introduction

The mixing layer that forms between two streams flowing with different velocities is an important canonical flow that has been used to study the dynamics of turbulence (Dimotakis, 2005; Roshko, 1976). Since the processes of sound generation and turbulent mixing are connected, the study of mixing layers has significant implications for applications such as the noise radiation from the near-nozzle region of high speed turbulent jets. This fundamental study of mixing layers has also been motivated by the need for efficient scramjet combustor designs. Here the mixing layer is a simple configuration to study effects associated with compressibility on mixing rate and entrainment. There is a considerable body of literature attributed to the study of mixing layers in various combinations of incompressible, compressible, reacting or non-reacting conditions. However, only a selective review that is pertinent to the present investigation on the problem of jet noise is given.

1.1 Shear layer ﬂow development

An initially laminar spatially developing mixing layer is characterized by the two- dimensional spanwise vortical structures and regions of high strain or braids caused

1 2 CHAPTER 1. INTRODUCTION

due to Kelvin-Helmholtz instability. The mixing layer is unstable to oblique disturbances especially so at higher Mach numbers and this results in increased three- dimensionality. The bending of the spanwise vortex rollers in the streamwise direction leads to generation of streamwise vorticity in the braid regions, observed as arrays of rib vortices extending from bottom of one roller to the top of next (Lin and Corcos, 1984). The Reynolds number based on the local thickness increases downstream as the shear layer grows. The transitioning to a turbulent mixing layer is marked by the appearance of finer scales. Turbulent mixing layers can grow either by a pairing mechanism (Winant and Browand, 1974) or by tearing (Moore and Saffman, 1975) before eventually becoming self-similar. Brown and Roshko (1974) showed the sig- nificance of these large scale coherent structures in the dynamics of turbulent mixing layers. Even at high Reynolds numbers, these structures were found to persist with small scale turbulence superposed over them. However, there has been evidence that the occurence of two dimensional structures might be dependent on the experimental facility and may not necessarily be a universal feature. Hussain (1983) suggests that incoherent turbulence and coherent structures are comparably important in the dynamics of the layer. The evolution of mixing layers is sensitive to its initial and boundary conditions. For example, the velocity and density ratios of the two streams, the free stream turbulence (Chandrasuda et al., 1980), curvature and angle of merging streams (Batt, 1975), Reynolds number (Hussain and Zedan, 1978a,b) or the vibrations of the splitter plate (Pui and Gartshore, 1979) are known to influence the development of mixing layers. One of the main factors influencing the mixing layer evolution is the state (laminar or turbulent) of incoming boundary layer (Bell and Mehta, 1990). However, even though the evolution close to splitter plate may be different, the asymptotic mean velocity and turbulence profiles are found to be comparable (Bell and Mehta, 1990).

Self-similar mixing layers have been a subject of several experimental (Samimy and Elliot, 1990; Papamoschou and Roshko, 1988; Oster and Wygnanski, 1982) and computational (Rogers and Moser, 1992; Pantano and Sarkar, 2002; Kourta and Sauvage, 2002) studies. A plane mixing layer is known to become self-similar after a suﬃcient distance downstream from the splitter plate edge (Rogers and Moser, 1994). The 1.1. SHEAR LAYER FLOW DEVELOPMENT 3

self-similar state is characterized by a linear spreading rate and the independence of shape of mean and turbulence quantities with respect to downstream distance when scaled by mixing layer thickness and the velocity diﬀerence. From a practical point of view, it is important to study both the early evolution as well as the asymptotic self-similar state of the mixing layers.

At high Mach number, the growth of turbulent shear layer is substantially reduced compared to its incompressible counterpart (Brown and Roshko, 1974; Goebel and Dutton, 1991; Clemens and Mungal, 1995; Urban and Mungal, 2001). Parameter called convective Mach number (Mc) has been proposed to quantify compressibility of mixing layer (Bogdanoff, 1983) and has been used frequently to correlate the reduced growth rate of a compressible mixing layer. This means that independent of the choice of the thickness measure, the growth rate normalized by the incompressible growth rate must only be a function of convective Mach number, and independent of velocity ratio (r = U2 ) and density ratio (s = ρ2 ) (subscript 1 denoting upper high-speed U1 ρ1 stream). If the flow is assumed isentropic, in a two-dimensional flow, assuming that the stagnation point occurs between the two eddies, the convective Mach number can be estimated as (Papamoschou and Roshko, 1988),

γ1 γ2 γ − 1 γ1−1 γ − 1 γ2−1 1 + 1 M 2 = 1 + 2 M 2 (1.1) 2 c1 2 c2

Here, c1 and c2 refer to the speed of sound in the two streams whereas γ is the ratio

U1−Uc Uc−U2 of speciﬁc heats. Mc1 = and Mc2 = are the convective Mach numbers for c1 c2 c2U1+c1U2 the two streams. For γ1 = γ2, we have Uc = . Furthermore, the relation can c1+c2 be written as,

∆U U1 (1 − r) Mc = = (1.2) √1 c1 + c2 c1 1 + s

We note that the isentropic assumption is reasonable as long as shocklets are not formed at high convective Mach numbers. More recently, an alternative to the parameter, convective Mach number, has been proposed by Slessor et al. (2000) especially 4 CHAPTER 1. INTRODUCTION

for the case when the two streams have widely diﬀerent densities. This parameter, de-

fined as Πc = maxi[(γi −1)/ci]∆U, has been found to explain experimentally observed mixing layer growth rates well. For compressible flows, the equation for turbulent kinetic energy has additional terms: compressible dissipation and pressure dilatation correlation. The early studies by Zeman (1990), Sarkar et al. (1991) and Durbin and Zeman (1992) focussed on examining the role of these additional terms to explain the reduced growth rates due to compressibility. Whereas the compressible dissipation term acts as a sink of turbulent kinetic energy, the pressure dilatation correlation describes reversible energy exchange between turbulent kinetic energy and mean internal energy (Lele, 1994). However, the reduction of mixing layer growth rates with increasing Mc cannot be explained by contribution from additional dilatational terms. Rather, it is an attribute of reduced turbulent kinetic energy production with increasing Mc. This was confirmed by Vreman et al. (1996), Freund et al. (2000) and Pantano and Sarkar (2002) in their studies of a temporally evolving mixing layer. Sarkar (1995) showed a decrease in turbulent kinetic energy associated with reduced production with increasing gradient

Mach number (defined as Mg = Sl/c, where S is the strain rate, l is the integral length scale and c is the speed of sound) in his simulations of homogeneous shear flow. Vreman et al. (1996) showed a direct relationship between transversely-integrated Reynolds stress and momentum thickness growth rate of mixing layer. A reduction in Reynolds stress associated with the disturbance field of an isolated vortex placed in a shear layer was also reported by Papamoschou and Lele (1993) in their two- dimensional simulations. Compressible dissipation remains negligible even at higher Mach numbers. The reduction of turbulent production occurs with a simultaneous decrease in pressure fluctuations via decrease in the pressure-strain rate correlation term. These observations were confirmed in simulations of temporally evolving annular mixing layer by Freund et al. (2000) and that of temporally evolving plane mixing layer by Pantano and Sarkar (2002). A similar question can be asked regarding the trend of anisotropic part of the 0 0 p 02p 02 Reynolds stress tensor with Mc. The quantity −u v /( u v ), is often taken as 1.2. PREVIOUS ANALYTICAL AND COMPUTATIONAL WORK 5

a measure of anisotropy and compared between experiments and simulations. The numerical values obtained from different experiments and simulations show large scat- ter, but each shows a decrease in value with increasing Mc (Gatski and Bonnet, 2009). 02 02 The trend of (u /v ) |max is less clear. While some studies have shown it does not vary with Mc, others show an increase with Mc (Gatski and Bonnet, 2009). The latter trend is observed due to a constant longitudnal intensity while a decreasing transverse intensity with Mc (for example, Goebel and Dutton (1991)). Direct numerical simulations (DNS) of Pantano and Sarkar (2002) also showed an increase with an increase in Mc consistent with the latter trend. For more details, we refer to texts by Smits and Dussauge (2006) and Gatski and Bonnet (2009). Another aspect associated with compressible turbulent flows is the sound generation process and its propagation to far field, which is described later in this chapter. It is difficult to obtain reliable experimental data for some compressibility terms, such as the pressure-strain, which appears to play a principal role in the dynamics. It is possible to establish budgets of turbulent kinetic energy in supersonic mixing layers by using DNS and can be used to validate closure models and comparing the structure of the Reynolds stress tensor with the one obtained in incompressible flows (Gatski and Bonnet, 2009; Freund et al., 2000).

1.2 Previous analytical and computational work

Most of the early numerical/analytical studies of mixing layer focussed on the linear stability analysis. The growth of small disturbances may be well described by an linearized inviscid Euler equations. With the assumption of a slowly varying mean flow, the disturbance profiles are assumed to be functions of the cross stream coordinate. With disturbances of the form u0 =u ˆ(y)exp[i(αx + βz − ωt)], where tanθ = β/α, the temporal problem involves finding a complex ω with prescribed real α and the spatial problem involves finding a complex α with prescribed real ω for the resulting eigen value problem. Example of application of linear analyses to incompressible mixing layer appear in study by Michalke (1965) and to compressible mixing layers in work by Jackson and Grosch (1989); Grosch and Jackson (1991), Sandham (1989), Day 6 CHAPTER 1. INTRODUCTION

et al. (1998) and Cheung (2007). There have been efforts to test the linear theory predictions with experiments or direct numerical simulations (DNS). It has been emphasized that the spatial stability analysis be only used for convectively unstable flows like mixing layer in order to make comparisons with the experiment (Huerre & Monkewitz (1985)). The spatial analysis based on the mean flow obtained from solution of boundary layer equations shows a

linear relationship between growth rate and λ = (U1 − U2)/(U1 + U2) as also found in experiments. In the context of compressible mixing layers, the trend of growth rate attenuation with increasing convective Mach number (deﬁned above) is captured by the spatial ampliﬁcation rate of the most unstable mode (Day 1998). Sandham and Reynolds (1991) considered the temporal evolution of a three dimensional mixing layer with random initial conditions. The most unstable linear disturbance was found to satisfy Mc cos θ ≈ 0.6. Three regimes can be distinguished: a) A low convective mach number regime 0 < Mc < 0.6, where the two-dimensional (θ = 0) instability is the

dominant one, b) Regime of 0.6 < Mc < 1, where the growth rate of oblique modes is higher than the two dimensional modes and thus the ﬂow is three dimensional and c)

where Mc > 1 and there is a marked reduction in the amplification of two dimensional disturbnaces. Consequently their effect on flow dynamics is minimal. Shocklets have been observed both in experiments and in computational simulations of Kourta and

Sauvage (2002) and Sandham and Reynolds (1991) for Mc > 1. Nonlinear structure which develops from a pair of equal and opposite oblique instability waves resembles a pair of inclined Λ-vortices which are staggered in the streamwise direction. However, the base flow is really not parallel, its thickness increases with downstream distance. The shear layer velocity profile has a wake defect immediately downstream of the splitter plate trailing edge. The influence of wake defect in the meanflow profiles on the predictions of linear analyses for compressible mixing layers has been studied by Zhuang and Dimotakis (1995). Two unstable modes, the shear-layer mode and the wake mode, exist for shear layers with a wake defect in the meanflow. Similar findings have been reported by studies of incompressible layers. The wake mode was found to grow more stable with increasing convective Mach number (Zhuang and Dimotakis, 1995). 1.2. PREVIOUS ANALYTICAL AND COMPUTATIONAL WORK 7

The nonlinear character can be partially addressed by Parabolized stability equations (Herbert, 1997; Day et al., 2001; Cheung and Lele, 2009). In contrast to the parallel flow linear stability theory (LST), parabolized stability equations (PSE) have the capability to model the effects of weakly non-parallel and non-linear effects. The PSE operator differs from the linear stability operator in that the streamwise derivative terms and diffusion terms are included. However, it works strictly for convectively unstable flows. PSE fails to capture the vortex dynamics accurately in the presence of large density gradients. This is mainly due to the approximations made to the streamwise pressure gradient and the elimination of upstream propagating disturbances in the PSE formulation (Cheung and Lele, 2009). Global mode analysis has the ability to incorporate full non-parallel mean flow effects and capture upstream acoustic radiation. Such an analysis has been carried out by Nichols and Lele (2011) for cold supersonic jets. A detailed review of the global analysis can be found in Chomaz (2005).

Attempts have been made to tackle the problem of aeroacoustic prediction from jets and mixing layers using large-scale computations. Some of the numerical studies of mixing layer are detailed in the table below. The full three dimensional direct numerical calculation (DNS) of a spatially evolving jet or mixing layer is computationally costly. The DNS study of jet by Freund (2001) was restricted to Reynolds number of 3600 but required approximately 25 million mesh points. The recent DNS of a mixing layer by Kleinman and Freund (2008) used about 1 billion mesh points to simulate a temporally evolving plane mixing layer at Reynolds number of 414 based on the initial momentum thickness. The spatial mixing layer DNS study with a zero thickness splitter plate by Sandham and Sandberg (2009) required close to 500 million mesh points. Large-Eddy simulation (LES) is a viable tool to simulate turbulent jets and mixing layers at realistic Reynolds numbers and aid in the quantitative prediction of far-field noise. A number of studies have also focussed on using LES to simulate the near-field of compressible turbulent jet (Bodony and Lele, 2005; Shur et al., 2005; Ham et al., 2009; Uzun and Hussaini, 2006; Mendez et al., 2010) or mixing layer flow (Vreman et al., 1996; Foysi and Sarkar, 2009; Balaras et al., 2001). A number of key issues in the current predictive capability of LES of jet noise have been discussed in a 8 CHAPTER 1. INTRODUCTION

recent review by Bodony and Lele (2008a). Inflow forcing or its absence by explicitly including the jet nozzle geometry has been found to impact current LES predictions. Since no explicit inflow forcing is applied in an experimental setting, it has been argued that this situation should be reproduced in numerical simulations. Similarily, in case of mixing layer simulations, including a splitter plate is necessary to reproduce experimental conditions. The effect of boundary layer thickness at the nozzle exit on acoustic radiation of a round jet has been studied numerically by Bogey and Bailly (2010). In their study, the jets originate from a pipe nozzle, with prescribed laminar boundary layer profiles with or without random forcing at the pipe inlet. Their study reveals that inlet forcing results in reduction of noise generated due to vortex pairing in the initial jet shear layer with good comparison to experiments. The effect of inlet forcing on jet noise and on vortex pairing has also been experimentally studied (Za- man, 1985, 1986). However, much thinner boundary layers as well tripped turbulent boundary layers need to be accounted for in the simulations to match the early shear layer development to experiments. Most of the reported computations have focussed on temporally developing mixing layers (Fortunéet al., 2004; Kleinman and Freund, 2008). A relatively smaller number of studies have focussed on spatially evolving mixing layers (Lui and Lele, 2001; Nelson and Menon, 1998). Temporal mixing layers are computationally convenient due to their periodicity in both the streamwise and spanwise directions. However, temporally developing flows only model the experimental mixing layers in the limit when the velocity difference of the two streams is negligible compared to the average velocity (Pope, 2000). They miss the entrainment asymmetry present in a spatially developing flow (Dimotakis, 1984) and because of streamwise periodicity they cannot be expected to match the radiated sound.

1.3 Sound generation in jets and mixing layers

Turbulence associated noise (also called ’mixing noise’) is an important contributor to overall aircraft noise generated by modern airplane jet engines. It is the dominant contributor to the radiated noise in subsonic jets, whereas in supersonic jets, it is 1.3. SOUND GENERATION IN JETS AND MIXING LAYERS 9

Study Type Re M1, M2 Mesh Domain size Laizet et. al. DNS SML 400 inc. 44M (108, 96, 13.5)h (2010) splitter plate (162, 96, 13.5)h thickness h Sandham et. al. DNS SML 2300 0.6, 0.06 462M (314, 226, 17.4)δd (2009) splitter plate 0 thickness Kleinman et. al. DNS TML 414 0.9 1.3B (2000, 2000, 750) (2008) δθ,0 Fortun´e et. al. DNS TML 400 0.8 5.8 M (31, 30, 31)δω,0 (2004) Georgiadis et. al. Hybrid RANS- 1400 1.91, 1.36 2.4 M (1000, 96, 12)h (2003) LES SML splitter plate thickness h Pantano et. al. DNS TML 640 1.1 17 M (345, 172, 86)δθ,0 (2002) Kourta et. al. DNS TML 800 3.2 25 M (30, 30, 12)δω,0 (2002) Lui & Lele DNS SML 350 0.8 32 M (100, 44, 20) δω,0 (2001) Bodony & Lele LES SML 500 1.2, 0 5.4 M (130, 60, 5) δω,0 (2004) Balaras et. al. LES TML 300 inc. 4.2 M (220, 56, 110) δθ,0 (2001) Freund et. al. DNS TAML 3200 3.5 12.5 M (3.5,2π,21)r0 (2000) Vreman et. al. DNS TML 100 2.4 7 M (34,30,34)δω,0 (1996) Rogers & Moser DNS TML 300 inc. 20M (125,∞,31.25)δθ,0 (1994)

Table 1.1: Summary of some of the previous computational studies of turbulent mixing layers. The study type distinguishes between Direct Numerical Simulations (DNS) and Large-eddy simulations (LES) and temporal mixing layers (TML) from spatially evolving mixing layers (SML). Temporal annular mixing layer (TAML) is indicated for Freund et al. (2000). RANS-based studies or those based on other modelling approaches are not included. The simulations do not contain a splitter plate unless mentioned. The indicated Reynolds number is based on velocity diﬀerence of the two streams and splitter plate thickness if a plate is present, or vorticity thickness. Only one representative case with the highest Reynolds number and grid size is reported for studies with multiple cases. M1,M2 represent the Mach numbers of the fast and slow streams respectively. The incompressible ones are indicated by “inc.”. 10 CHAPTER 1. INTRODUCTION

accompanied by two other principal components, namely, broadband shock noise and screech tones (Tam, 1995). We brieﬂy review some of the characteristics of jet mixing noise here, mainly with respect to the parameters, jet velocity or the jet acoustic

Mach number (deﬁned as Ma = Uj/c∞, where Uj is the jet velocity and c∞ is the ambient velocity of sound) and the temperature ratio of the jet to ambient. Most of the noise from jets is generated starting from the region close to nozzle exit and extending upto ten diameters downstream of the end of potential core. Lighthill’s acoustic analogy (Lighthill, 1952) has provided a theoretical basis to much of the work on the jet noise problem. That the noise radiated from a jet scales with the eighth power of jet velocity, is a classical result obtained by Lighthill and underlies the concept of high bypass ratio turbofan engines which rely on increased mixing and lower jet exhaust velocity to achieve noise reduction.

Jet noise has been found to vary with location of observer with respect to the jet axis: it is peaked at low angles (roughly 30◦) in the downstream direction and reduces at larger angles away from the jet axis. The spectrum at low angles is observed to be dominated by low frequency sound and have a broadband peak whereas at larger angles, high frequency sound is more important with comparatively flatter spectral shape. The peak frequency of noise at large angles shows a dependence on jet velocity (shows a Strouhal number scaling) whereas at small angles, it has been found to be independent of Uj (shows Helmholtz scaling) (Lush, 1971). The original scaling with eighth power of jet velocity is valid only for observers located close to 90◦ from the jet axis for low subsonic jets. The Doppler effect due to eddy convection was included in Lighthill’s analysis (Lighthill (1952) and Ffowcs Williams (1963)) to explain the peak noise radiation in jets. It was argued that the Doppler effect would result in a directivity factor proportional to (1 − M cos θ)−5 (Ffowcs Williams, 1963), where M is the eddy-convection speed and θ is the angular position of the observer from the jet axis, and gives rise to prefential radiation at angles close to the jet axis. The result due to Goldstein (1975) on the alteration of quadrupole directivity as 4 02 −9 cos θ|Mj |(1 − M cos θ) at low frequencies accounted for the presence of mean 0 flow (due to presence of gradient of mean Mach number, Mj). It was in better agreement than Lighthill’s correction with experimental data from Lush (1971) at 1.3. SOUND GENERATION IN JETS AND MIXING LAYERS 11

observer angles close to the jet axis. This result was consistent with the picture of direction-independent ’self-noise term’ due to turbulence-turbulence interaction and ’shear-noise term’ due to turbulence-mean shear interaction directed downstream preferentially. Another similar explanation of jet noise directivity stems from work by Tam (1995), Tam and Aurialt (1999) and Tam et al. (2008). Tam et al. (2008) provide evidence of the existence of two distinct jet noise sources: large-scale and fine-scale turbulence. The spectra at the shallow angles and the side-line direction collapse to two different universal spectra regardless of jet exit Mach number and temperature. It is suggested that this different scaling is a result of two different underlying sources. For the intermediate range of angles, a ’combination’ of the two types of spectra was required to fit the measurements.

A wide number of variations to Lighthill’s acoustic analogy are possible depending on the method of rearrangement of Navier-Stokes equations. This gives rise to the difficulty that different interpretations of sound sources and sound propagation effects are possible depending on the choice of acoustic analogy. Generalization of acoustic analogy due to Lilley (1974) focussed on rearrangement of Navier Stokes equations to derive an equation in terms of logarithmic pressure variable. Lilley’s equation has a non-linear left hand side. Usually, solving it requires linearization about some mean flow and bringing the non-linear terms to the right hand side (Goldstein, 2010). If a parallel mean flow approximation is made, the left hand side of this equation simplifies to the Pridmore-Brown operator. Recently, Goldstein (2003) proposed an acoustic analogy in which the equations rearranged into the same form as that of inhomogeneous linearized Euler equations. A feature of this is that it can be shown to reduce to other acoustic analogies depending on the choice of mean flow. For example, choosing a parallel base flow reduces it to Lilley’s equation. Acoustic analogy due to Lighthill or its variants by Ffowcs Williams and Hawkings (1969) have been widely and successfully used in a variety of applications mainly because of the ease and low cost of implementation. For example, in study by Mendez et al. (2010), Ffowcs Williams and Hawkings acoustic analogy has been shown to yield far-field sound predictions in good agreement with experiments. The usual method to solve Lighthill’s analogy is to convolve the free-space Green’s function for the wave operator 12 CHAPTER 1. INTRODUCTION

with the sound source field. Despite its restrictive assumptions, Lighthill’s analogy yields good predictions as long as the entire noise source field is represented accurately and the physical complexities of generation and propagation are handled within the convolution integral. A large number of experimental studies have focussed on the effects of heating on jet noise (Tanna et al. (1975), Hoch et al. (1973), Doty and Mclaughlin (2002), Viswanathan (2004), Harper-Bourne (2007), Bridges (2006) and Bridges and Wernet (2007)). Keeping the jet velocity fixed, increasing the static temperature ratio has been found to increase the far-field noise above a jet acoustic Mach number (defined as the ratio of jet velocity normalized and the speed of sound in the ambient medium) of 0.7 and vice versa for acoustic Mach number below 0.7. High frequency noise is found to decrease with heating whereas the low frequency noise increases with heating at low acoustic Mach numbers and decreases with heating at high acoustic Mach numbers. At low Mach numbers, the differences in noise characteristics of heated jets is usually attributed to the existence of a dipole sound source due to temperature fluctuations in addition to the quadrupole source due to Reynolds stress fluctuations present in isothermal jets. At 90◦ to the jet axis where the convection effects would be absent, the noise intensity due to heat-associated dipole source scales 6 2 with Uj (∆T/Tj) . Analyses in studies by Morfey (1973); Morfey and Szewczyk (1978) using the experimental data of Tanna et al. (1975); Hoch et al. (1973) confirmed these observations. The existence of dipole source and scaling of sound intensity with sixth power of jet velocity has been refuted by Viswanathan (2004, 2009) who raised concerns about the reliability of Tanna’s experimental database. More recently, trends observed in Tanna’s experimental database have been reconfirmed in studies by Bridges and Wernet (2007); Harper-Bourne (2007). In Lighthill’s analogy, a natural way to identify a noise source due to heating is to decompose the Lighthill’s stress tensor into the momentum flux tensor and the isotropic entropy term. However, the entropy term has been found to have a significant contribution even in the isothermal case. The net sound field is therefore a result of mutual cancellation or addition of sound fields due to individual source terms (Freund, 2003). Bodony and Lele (2008b) showed that the Lilley’s form of Lighthill 1.3. SOUND GENERATION IN JETS AND MIXING LAYERS 13

equation may yield a better decomposition of noise sources as hydrodynamic and thermodynamic source terms. More dicussion on acoustic analogies and discussion of heating effects can be found in Chapter 3 and 5 of this report. Numerical predictions of far-field noise from heated or unheated jets require information about the sound sources present in the flow field and sound propagation effects usually in the context of an acoustic analogy. The source description can be input through empirical models (as in NASA’s ANOPP code (Bridges et al., 2008)) or statistical noise source models (as in NASA’s JeNo code (Bridges et al., 2008; Khavaran and Kenzakowski, 2007) or study by Morris and Farassat (2002) and Tam and Aurialt (1999)) usually based on Reynolds Averaged Navier Stokes (RANS) solution or can come from detailed unsteady flow data from LES or DNS computations (Bodony and Lele, 2005), in increasing order of computational time. The RANS based methods usually rely on and are sensitive to assumptions made about the two-point space-time correlations of turbulent sources (Morris and Farassat, 2002; Afsar, 2010). The sound source statistics are hard to measure experimentally and only few such as the DNS study of Freund (2003) have focussed on providing detailed information for low Reynolds number turbulent jet. Attempts to actively or passively control jet noise has also generated significant interest in the dynamics of early shear layers. Wheras passive control of jets and mixing layers is accomplished by geometrical modifications of the nozzle (in jets) or trailing edge of splitter plate in case of mixing layers, active control employs forcing the near field flow via steady or unsteady actuation. The actuation frequency can be close to or lower than the frequency associated with most unstable mode for jets or mixing layers. Passive control methodology underlies introduction of chevrons or tabs into nozzle geometry (Zaman, 1999). The fluidic injection using microjets (Krothapali et al., 1997) or plasma based actuators have been used to actively control jets with or without feedback (Samimy et al., 2007; Sinha et al., 2010). Dimotakis and Brown (1976) observed persistent oscillations in the autocorrelation function even beyond any obviously relevant time scale. Their data suggested a coupling mechanism between longer time period large scale structures downstream and the dynamics of fluctuations close to the splitter plate. This implied that the 14 CHAPTER 1. INTRODUCTION

spectrum computed from data taken anywhere in the ﬂow ﬁeld will contain lowest frequency that corresponds to L/Uc where L is the domain size. Laufer and Monke- witz (1980) argued that such a feedback loop must be responsible for self-sustained instability in the near-nozzle region of jet. In this picture, the vortex interactions downstream would re-initiate instability by perturbing the receptive shear layer near the trailing edge via upstream travelling acoustic waves. This picture suggested a simple relationship where the sum of time taken by instabilities to convect to the vortex pairing location and the time taken by acoustic waves to travel upstream from the vortex pairing location to nozzle tip would be an integer multiple of the vortex pairing time period. The relationship has been tested in the studies by Kibens (1980) and Grinstein et al. (1986) and supports the existence of feedback loop.

1.4 Outline and Accomplishments

The main accomplishments from the present work are summarized below.

• Large Eddy Simulations of spatially developing, compressible, turbulent mixing layers using high order overset method have been conducted. The overset method enabled the inclusion of splitter plate inside the computational domain while retaining good quality of mesh around the turning splitter plate trailing edge. Cases with initially laminar boundary layers (abbreviated as LBL) and turbulent boundary layers (abbreviated as TBL) are considered. Eﬀect of heating keeping the velocity ratio is analyzed for each.

• It is observed that with heating in LBL, the initial instability is accelerated but the saturation self-similar amplitude of Reynolds stress components is not aﬀected.

• For LBL, the saturation amplitudes of density fluctuations were found to increase proportionally to difference in free-stream densities whereas near-field pressure fluctuations were found to decrease with heating. A simple scaling is deduced for the near-field pressure fluctuation amplitude. 1.4. OUTLINE AND ACCOMPLISHMENTS 15

• For LBL, sound radiation is observed in downstream direction peaked roughly at 30 degrees. The vortex pairing and breakdown to turbulence contribute signiﬁcantly to the radiated sound.

• For TBL, the flow adjusted from the wake affected regime to the shear-layer behavior without any major instability resulting in a significantly weaker acoustic field near the shear layer due to passage of boundary layer eddies over the trailing edge.

• For both LBL and TBL, a reduction in overall sound pressure levels in the far-ﬁeld is observed with heating.

• The analysis of Reynolds stress autocorrelation tensor reveals the relative importance of its various components. The autocorrelation amplitudes at zero spatial separation and time delay were found to decrease with heating. This was consistent with the reduction in mean density with heating and would be the most likely cause for reduction in far-field noise levels with heating. The enthalpy flux autocorrelation amplitudes and enthalpy flux momentum flux cross-covariances were also found to have significant amplitudes with heating. But they cancel each other’s effect leading to an overall reduction in far-field sound. Chapter 2

Numerical Framework for LES computations

The governing equations for LES of mixing layer in curvilinear coordinates are summarized below. Spatial discretization and time integration schemes of Nagarajan et al. (2003) are used to solve the equations numerically. Communication between multiple domains is done using overset method of Bhaskaran (2010). The governing equations are written in a tensorially invariant form to preserve the advantage of using staggered variables in general curvilinear coordinates. The high-order staggered method ensures high-order of accuracy and good resolution at high wavenumbers while con- serving mass, momentum and energy in the inviscid limit and in the absence of time integration errors. The overset method extends the method to multiple domains while maintaining high-order of accuracy.

2.1 Governing equations

The governing equations for compressible ﬂuid ﬂow in a curvilinear co-ordinate system (x1, x2, x3) and general non-rotating frame of reference are (Aris, 1989),

16 2.1. GOVERNING EQUATIONS 17

∂ρ + (ρvk) = 0 ∂t ,k ∂ (ρvi) + (ρvivj) = − gijp + τ ij (2.1) ∂t ,j ,j ,j ∂E + (E + p)vj = κgijT + τ ijg vk ∂t ,j ,i ,j ik ,j

where density, temperature and pressure are denoted by ρ, T and p respectively, and the total energy E and viscous stress tensor τ ij are given by

p 1 p 1 E = + ρ |v|2 = + ρg vivj (2.2) γ − 1 2 γ − 1 2 ij 2 τ ij = µ gjkvi + gikvj − gijvk (2.3) ,k ,k 3 ,k

In these equations, the general tensor notation due to Einstein is used (subscripts ij and superscripts denote covariant and contravariant tensors respectively). gij and g are the covariant and contravariant metric tensors. The subscript , j indicates the covariant derivative. √ √ √ √ √ T With the choice of the vector of dependent variables as q = gρ gρv1 gρv2 gρv3 gE , the governing equations can be cast into the following form,

∂ √ ∂ √ ( gρ) + gρvk = 0 ∂t ∂xk ∂ √ ∂ √ √ ∂ √ 2 gρvi + gρvivj + Γi gρvivj = g gij p + µvk + τ ij ∂t ∂xj qj ∂xk 3 ,k √ 2 + Γi g gij p + µvk + τ ij (2.4) qj 3 ,k ∂ √ ∂ √ ∂ √ ∂T ( gE) + g(E + p)vj = g κgij + τ ijg vk ∂t ∂xj ∂xj ∂xi ik

√ 1/2 where J = g = |gij| is the Jacobian and Γ is the Christoffell symbol. To derive the LES equations, define a filter G(x) with compact support in the 18 CHAPTER 2. NUMERICAL FRAMEWORK FOR LES COMPUTATIONS

domain of interest Ω. Then, any tensor Ai1···im may be ﬁltered as j1···jn

ZZZ √ A¯i1···im (x, t) = Ai1···im (x0, t)G (x − x0) gdx0 (2.5) j1···jn j1···jn Ω The ﬁltered continuity equation leads to the curvilinear equivalent of Favre ﬁlter- ing

√ gρAi1···im i ···i j1···jn Ae 1 m = √ (2.6) j1···jn gρ

which leads to a continuity equation with no unclosed terms. The ﬁltered non- dimensional governing equations then become

√ ∂ gρ ∂ √ + gρ v˜k = 0 (2.7) ∂t ∂xk ∂ √ ∂ √ √ ∂ √ √ ( gρ v˜i) + gρ v˜iv˜j + Γi gρ v˜qv˜j = − gij g p − Γi gqj g p ∂t ∂xj qj ∂xj qj ∂ √ √ + gτ ij + Γi g τ qj ∂xj qj ∂ τ ij − sgs − Γi τ qj (2.8) ∂xj qj sgs √ ∂ g E ∂ h√ i ∂ √ ∂ √ + g(E + p)v ˜j = − gq˜j + gτ ij g v˜k ∂t ∂xj ∂xj ∂xj ik ∂ − qj (2.9) ∂xj sgs

where the resolved and sub-grid stress tensor and heat ﬂux vector are

µ˜ 2 τ˜ij = gjkv˜i + gikv˜j − gijv˜k (2.10) < ,k ,k 3 ,k µ˜ ∂Te q˜j = − gjk (2.11) < Pr ∂xk ij √ i j i j sgsτ = gρ v v − v˜ v˜ (2.12)

j √ j j sgsq = gρ T v − Tev˜ (2.13) 2.1. GOVERNING EQUATIONS 19

In addition, the equation of state becomes

√ γ − 1√ gp = gρT˜ (2.14) γ and the expression for pressure in terms of the conserved variables becomes,

√ √ 1√ 1√ gp = (γ − 1) g E − gρ g v˜iv˜j − gρ g vivj − v˜iv˜j (2.15) 2 ij 2 ij

The LES equations are closed by models for the sub-grid scale (SGS) stress tensor ij j sgsτ and heat ﬂux vector sgsq . The sub-grid models in curvilinear co-ordinates are developed following that of the dynamic sub-grid models for LES (Moin et al., 1991; Germano et al., 1991) with the modiﬁcations proposed by Lilly (1992). The sub-grid stress is modelled as,

1 √ 1 τ ij − τ klg gij = −2 gρ C∆2|S˜| S˜ij − gijgmnS˜ (2.16) sgs 3 sgs kl 3 mn where ∆ is a characteristic length scale, |S˜| is the absolute value of the strain rate tensor and C is a dynamically computed as,

0ij kl L gikgjlM C = ij kl (2.17) hM gikgjlM i

√ 1 √ 1 M ij = −2∆ˆ 2|S˜ˆ| gρ S˜ˆij − gijgmnS˜ˆ +2∆2|S˜| gρ S˜ij − gijgmnS˜ (2.18) 3 mn 3 mn

√ 1 √ √ Lij = gρ v˜iv˜j − gρvi gρvj (2.19) √ gρ

Here, h·i represents averaging along a homogeneous direction andL0ij is the trace free 20 CHAPTER 2. NUMERICAL FRAMEWORK FOR LES COMPUTATIONS

ij j part of L . The model term for the sub-grid heat ﬂux sgsq is written as

√ 2 ˜ j gρ C∆ |S| ij sgsq = − g Te,i (2.20) PrT

i j C hgijN K i = − i j (2.21) PrT hgijN N i √ 1 √ √ Kj = gρ T v˜j − gρ T gρ v˜j (2.22) e √ e gρ

ˆ √ ∂Te √ ∂Te N j = gρ ∆ˆ 2|S˜ˆ|gij − gρ∆2|S˜|gij (2.23) ∂xi ∂xi

2.2 Spatial Discretization

The governing equations are discretized using the high-order staggered compact scheme 0 df of Nagarajan et al. (2003). The ﬁrst derivative f = dx at nodes j = 0, ..., N is approximated by

f − f f − f αf 0 + f 0 + αf 0 = b j+3/2 j−3/2 + a j+1/2 j−1/2 (2.24) j−1 j j+1 3∆x ∆x 3 1 Here α = 9/62, a = 8 (3 − 2α) and b = 8 (−1 + 22α). Staggered variable arrange- ment involves mid point interpolation which is carried out using

f + f f + f αf I + f I + αf I = b j+3/2 j−3/2 + a j+1/2 j−1/2 (2.25) j−1 j j+1 2 2 where f I s are the interpolated values. The coeﬃcients α = 3/10, a = 3/2 and b = 1/10 yield a sixth order accurate formula.

2.3 Time Integration

In zones away from walls, the third order compact storage Runge-Kutta scheme is used to advance the equations in time. When applied to integrate from tn to tn+1 a 2.4. OVERSET METHOD 21

general equation of the form

dy = f (y, t) (2.26) dt this scheme may be written as

8 yn+1/3 = yn + ∆tf (yn, tn) 15 1 5 yn+2/3 = yn + ∆tf (yn, tn) + ∆tf yn+1/3, tn+1/3 (2.27) 4 12 1 3 yn+1 = yn + ∆tf (yn, tn) + ∆tf yn+2/3, tn+2/3 4 4 where the intermediate time stations are

8 2 tn+1/3 = tn + ∆t; tn+2/3 = tn + ∆t (2.28) 15 3

The two step approximate factorization scheme of Beam & Warming is used for implicit time marching near the wall. Suppose that the governing equations can be written as,

dU + G(U) = 0 (2.29) dt The implicit method is obtained by

3U n+1 − 4U n + U n−1 = −G(U n+1, tn+1) (2.30) 2∆t

Factorization is performed on the right hand side, followed by diagonalization of the implicit matrix in the x and z directions . The details of the scheme can be found in Nagarajan (2004).

2.4 Overset method

The inter-grid interpolation used is based on the fourth order Hermite interpolation scheme of Delfs (2001). In two dimensions, the Hermite interpolation at an overlap 22 CHAPTER 2. NUMERICAL FRAMEWORK FOR LES COMPUTATIONS

point is constructed using the function values and the ﬁrst derivatives at the four surrounding points in the donor grid which bound the interpolated point in the uniform computational space. A local grid system (ξ, η) is constructed in the computational space, with the origin of the coordinate system at the center of the box formed by the points bounding the interpolated point. The coordinates of the interpolated point in this local coordinate system are found by the inverse mapping (ξ, η) = M −1(x, y). The forward mapping M(ξ, η) is deﬁned at all points using the Hermite interpolation scheme, and the inverse mapping M −1(x, y) is found using a Newton-Raphson procedure. The interpolation formula is given by,

1 " # X ∂f ∂f f(ξ, η) = C0 (ξ, η)f + Cξ (ξ, η) + Cη (ξ, η) +O(∆4) lk i+l,j+k lk ∂ξ lk ∂ξ l,k=0 i+l,j+k i+l,j+k (2.31) 0 ξ η Here, the coeﬃcients of interpolation Clk, Clk and Clk are given by

1 1 C0 (ξ, η) = ( − (−1)lξ)( − (−1)kη)× (2.32a) lk 2 2 1 1 1 − 2(−1)lξ( + (−1)lξ) − 2(−1)kη( + (−1)kη) 2 2 1 1 1 Cξ (ξ, η) = (−1)l( − ξ2)( − (−1)lξ)( + (−1)kη) (2.32b) lk 4 2 2 1 1 1 Cη (ξ, η) = (−1)k( − η2)( − (−1)lξ)( − (−1)kη) (2.32c) lk 4 2 2

Further details of implementation and tests to check the implementation of the overset method can be found in Bhaskaran (2010).

2.5 Rescaling-Recycling method

The direct numerical simulation of a spatially developing boundary layer from the initial laminar state through transition to a fully developed turbulent state requires a huge streawise domain and is therefore computationally very expensive. For example, Wu et al. (1999) used 51 million points, Jacobs and Durbin (2001) used 71 2.5. RESCALING-RECYCLING METHOD 23

million points in their simulations. Recently, Wu and Moin (2009) used around 210 million mesh points for DNS of a zero pressure gradient boundary layer. Thus, much of published work has focussed on generating turbulent inflow boundary condition for simulation of complex spatially developing flows. Spalart (1988) utilized a coordinate transformation to make the streamwise coordinate periodic by following the boundary layer growth in the streamwise direction. Lund et al. (1988) developed a simple technique for simulating incompressible spatially evolving boundary layer in which the disturbances and the mean flow at a downstream location are rescaled and reintroduced from an upstream location. Urbin and Knight (2001) extended rescaling-recycling technique to compressible boundary layers by using the Van Dri- est transformation. This method has been successfully used to provide turbulent boundary layer inflow for various flows, for example, jet in cross flow considered by Kawai and Lele (2010). Various other techniques also focussing on generating turbulent boundary layer exist in the literature, the pros and cons of these have been discussed by Morgan et al. (2011) recently. In this section, we briefly describe the rescaling-recycling technique that we use to generate turbulent boundary layer inflow to simulate spatially developing mixing layer. The details of the method can be found in the study by Urbin and Knight (2001). The flow variables ρ, u, v, w are first decomposed into mean (time averaged) and fluctuating components. Van-Driest transformation of the mean streamwise velocity is taken using,

U∞ −1 U UVD = sin A (2.33) A U∞ where

v u (γ−1) 2 u M∞P rt A = t 2 (2.34) (γ−1) 2 1 + 2 M∞P rt

Here, γ = 1.4 is the gas constant, P rt = 0.89 is the mean turbulent Prandtl number and M∞ is the free-stream Mach number. The mean and ﬂuctuation components are rescaled taking compressibility eﬀects into account as, 24 CHAPTER 2. NUMERICAL FRAMEWORK FOR LES COMPUTATIONS

inner + outer ∞ UV D,inlet = βUV D,recycle(yinlet), UV D,inlet = βUV D,recycle(y/δinlet) + (1 − β)UVD (2.35a) inner + outer Vinlet = βVrecycle(yinlet), Vinlet = Vrecycle(y/δinlet) (2.35b)

inner + outer Tinlet = βTrecycle(yinlet), Tinlet = Trecycle(y/δinlet) (2.35c)

0 0 + 0 0 uinlet,inner = βurecycle(yinlet), uinlet,outer = βurecycle(y/δinlet) (2.35d)

Here, β is the ratio of friction velocity between the inlet station and the recycled station, prescribed by the following empirical relation,

δ 1/10 x − x 5/60 β = recycle = 1 + recycle inlet 0.276/5Re−1/5 (2.36) δinlet δinlet δinlet

Now, the inlet variables are constructed following the procedure of Urbin and Knight (2001)

inner 0 inner 0 uinlet = (Uinlet + uinlet,inner)(1 − f(y/δinlet)) + (Uinlet + uinlet,inner)f(y/δinlet) (2.37a)

1 1 4x − 0.8 f(x) = 1 + tanh (2.37b) 2 tanh(4) 1.2 − 0.4x

2.6 Numerical solution of laminar boundary layer equations

The Thin Layer Navier Stokes equations are solved to provide self similar laminar boundary layers at the inﬂow boundaries. The governing equations for continuity, streamwise momentum, and energy equation are,

∂ρu ∂ρv + = 0 (2.38) ∂x ∂y 2.6. NUMERICAL SOLUTION OF LAMINAR BOUNDARY LAYER EQUATIONS25

∂u ∂v 1 ∂ ∂u ρu + ρv = µ (2.39) ∂x ∂y Re ∂y ∂y

∂T ∂T 1 ∂ ∂T µ ∂u2 ρu + ρv = µ + (2.40) ∂x ∂y Re Pr ∂y ∂y Re ∂y

The mean pressure is assumed to be constant P = 1/γ and ideal gas law p = (γ − 1)ρT/γ is assumed. A similarity solution can be obtained by using the transformation,

s Re Z y ζ = x, η = ρdy0 (2.41) ζ 0

Using u(η) = f(η) and T (η) = g(η), the equations reduce to following system of coupled ordinary diﬀerential equations.

2 µ 0 ff 00 + f 00 = 0 (2.42) γ − 1 g

2 1 µ 0 2 µ g0 + fg0 + f 002 = 0 (2.43) γ − 1 P r g γ − 1 g

The power-law viscosity relation expressed in similarity coordinates used is µ(η) = [(γ − 1)g]n. In case of a boundary layer, the boundary conditions used are,

0 0 f (η → ∞) = M1, f (η = 0) = 0, f(0) = 0. (2.44)

1 g(η → ∞) = , g(η = 0) = T (isothermal) or g0(η = 0) = 0(adiabatic) γ − 1 wall (2.45) A shooting algorithm is used to solve the above equations. The function ode45 was used in matlab for integration in similarity coordinate. The converged values (upto a tolerance level of 10−9) are found for f, f 0, f 00, g and g0 and are plugged into the following relations to ﬁnd relations for the physical variables. 26 CHAPTER 2. NUMERICAL FRAMEWORK FOR LES COMPUTATIONS

1 1 ρ(η) = (2.46) γ − 1 g(η)

u(η) = f 0(η) (2.47)

1 ηf − f 0 Z y ∂ρ v(η) = √ − f 0 dy0 (2.48) ρ 2 Rex 0 ∂x

T (η) = g(η) (2.49)

The variables are then transformed from the similarity coordinates to physical coordinates using the following relations.

r x Z η x = ζ, y = (γ − 1) g(η0)dη0 (2.50) Re 0

The initial starting point for the mixing or the boundary layer is set with a desired value of vorticity thickness, δω. Chapter 3

Acoustic Analogy Computations

A full non-linear calculation for the entire flow-field encompassing the near-field hy- drodynamics and the far-field acoustics is computationally prohibitive. Instead, an approach where the full non-linear flow calculation is required only in the near-field is more viable. Lighthill’s acoustic analogy is an exact rearrangement of the mass and momentum conservation equations into an inhomogeneous wave equation. The linear wave operator governs the propagation of linear pressure disturbances in a uniform ambient medium. The inhomogeneous source terms represent sound sources present in the flow. The coupled approach of solving full non-linear equations in the near flow field and using acoustic analogy for far-field has been used successfully in a variety of applications including airframe noise (Singer et al., 2000) and helicopter noise (Farassat, 1980), jet noise (Freund, 2001). For more complete discussion of various approaches for predicting the sound radiation from unsteady fluid motion, including hybrid methods which use the acoustic analogy approach, we refer to recent review articles (Colonius and Lele (2004) and Wang et al. (2006)). Sound is not intrinsically defined inside a non-uniform compressible flow. Any formal definition requires first the selection of a base flow through which sound propagates. This introduces ambiguity in separating propagation from generation. Lighthill’s acoustic analogy has been generalized to include effects of sound production due to bodies undergoing unsteady motion (Ffowcs Williams and Hawkings, 1969), or effects of interaction of propagating sound with the non-uniform mean flow

27 28 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

(Dowling et al., 1978). Another general approach to include the latter effect within the propagation operator of the acoustic analogy was to consider fluctuations about a parallel or weakly non-parallel mean flow field. The result is a third order wave equation (Lilley, 1974) which includes effects of mean flow-acoustic interaction in a detailed manner. However, the difficulty is that the exact Lilley’s equation has a non- linear left hand side. Usually, solving it will require linearization about some mean flow and bringing some of the non-linear terms to the right hand side and possibly neglecting others. This equation simplifies if a parallel mean flow approximation is made where the left hand side reduces to the Pridmore-Brown operator. The solution to the Lilley’s equation also include the non-trivial instability wave solutions of the homogeneous equation which further require distinction from the acoustic field. Some studies have focussed on suppressing instability wave contribution.The recent generalized acoustic analogy (Goldstein, 2003) has the feature that it can be shown to reduce to other acoustic analogies depending on the choice of mean flow. For example, choosing a parallel base flow reduces it to Lilley’s equation. Another property it has is that the left hand side of this equation is the same as the linearized compressible Euler equations. Therefore, the representation of sound propagation is similar to the linear inviscid fluctuations about a prescribed mean flow. In their study of a weakly non-parallel flow, Goldstein and Leib (2005) found that linear instability waves must also be included in order to develop the appropriate casual solution.

3.1 Goldstein’s acoustic analogy (GAA)

The Generalized Acoustic Analogy (GAA) (Goldstein, 2003; Goldstein and Leib, 2008) is an exact rearrangement of compressible Navier-Stokes equations, governing the fluctuations about an arbitrary base flow. Goldstein (2003) discusses various choices for the base flow. In the present study, we focus on a specific choice of steady base flow, i.e. the Favre-averaged mean flow obtained from the numerical simulations of mixing layers. The flow variables [ρ, vi, p] are decomposed into a time-averaged 0 0 0 and fluctuating part as [ρ + ρ , vei + vi, p + p ]. In the present case, base flow variables satisfy, 3.1. GOLDSTEIN’S ACOUSTIC ANALOGY (GAA) 29

∂ρv ej = 0 (3.1) ∂xj

∂vei ∂τeij ρvej = − (3.2) ∂xj ∂xj

∂ρehovj ∂Heovj ∂Hej e = e + (3.3) ∂xj ∂xj ∂xj

0 0 ρvf2 Here, eho is the base flow stagnation enthalpy, τij = δijp + ρvgivj, Heo = − and e 2 Hej = −ρ hgovj − ehovej . In writing the equations above and the following equations, the viscous and heat conduction terms are neglected. The dependent variables of the GAA are nonlinear functions of the flow variables and are expressed as, Φ0 = 0 0 0 0 0 γ−1 02 02 0 0 [ρ , ui, pe]. Here pe ≡ p + 2 (ρv − ρvf) and ui = ρvi/ρ. The GAA equations for the chosen steady base flow are (Goldstein, 2003),

0 D ρ ∂ 0 ρ + ρuj = 0 (3.4) Dt ρ ∂xj

0 0 0 Dui 0 ∂vei ∂pe ρ ∂τeij ∂ 0 ρ + uj + − = eij − eij (3.5) Dt ∂xj ∂xi ρ ∂xj ∂xj

0 Dpe 0 ∂vej ∂ 0 0 ∂τeij 0 ∂vei ∂ 0 +γpe +γ puj −(γ − 1) ui = (γ − 1) eij − eij + ηj − ηej Dt ∂xj ∂xj ∂xj ∂xj ∂xj (3.6) D ∂ ∂ 0 0 Here ≡ + vj . The terms, e − eij and η − ηj, that appear on the right Dt ∂t e ∂xj ij e j e hand side of the above equations are given by,

0 0 0 γ − 1 0 0 0 0 γ − 1 0 0 e − eij = −ρv v + δijρv v + ρvgv − δijρvgv (3.7a) ij e i j 2 k k i j 2 k k ! 0 0 ^0 0 0 0 0 0 vkvk 0 0 vivkvk η − ηi = −ρv (T + ) + ρ vgT + (3.7b) i e i 2 i 2

The linear diﬀerential operator on the left hand side of the GAA equations is same 30 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

as that of linearized Euler equations. Some of the verification tests we performed to check the solution from our implementation are described in the appendix. The source terms on the right hand side have a zero mean and are the fluctuating stresses forcing the acoustic field. Another feature of it is that it can be shown to reduce to other acoustic analogies depending on the choice of mean flow. For example, choosing a unidirectional transversely sheared base flow, which satisfies the homogeneous inviscid form of base flow equations (3.1) to (3.3), reduces it to Lilley’s equation. When this choice of base flow is made, the equation for the dependent variable reduces to

2 D ∂ ∂e0 ∂u ∂2e0 D ∂η0 ∂u 0 2 ij e 2 ij j 0 e Lpe = ce − 2ce − (γ − 1) 2 + e1j (3.8) Dt ∂xi ∂xj ∂xi ∂x1∂xj Dt ∂xj ∂xj

Here,

2 ! D ∂ ∂ D ∂u ∂ ∂ γp 2 e 2 2 L ≡ ce − 2 − 2 ce and ce = (3.9) Dt ∂xi ∂xi Dt ∂xj ∂x1 ∂xj ρ

For a uniform base ﬂow, the equations for the dependent variables reduce to

2 0 2 0 2 0 2 0 1 ∂ pe ∂ pe ∂ eij γ − 1 ∂ ηj 2 2 − = − + 2 (3.10) c0 ∂t ∂xj∂xj ∂xi∂xj c0 ∂t∂xj The GAA equations can be conveniently expressed in an operator form in time- and frequency-domain as

∂ + L Φ0 = S and (iω + L ) Φˆ 0 = Sˆ (3.11) ∂t G G respectively. The above equations are solved using pseudo time-stepping to steady state. The equations are written in the form,

∂ + iω + L Φˆ 0 = Sˆ (3.12) ∂τ G An explicit ﬁve-stage time marching technique of the Runge-Kutta-Jameson (Jame- son, 1985) type is employed to march the solution to steady state. The solution to 3.2. CONNECTION TO LIGHTHILL’S EQUATION 31

above equation also contains contribution from the shear instabilities which are known to corrupt the acoustic solution (Agarwal et al., 2004). Karabasov and Hynes (2005) use preconditioning and Bogey et al. (2002) use a simpliﬁed linearized Euler’s equations operator to eliminate the instability contribution. In the present study, Eq. 3.12 is modiﬁed as

∂ + Γ (iω + L ) Φˆ 0 = ΓSˆ (3.13) ∂τ G

Here, the preconditioner matrix Γ is a diagonal matrix with elements [τ1, τ1, τ2, τ2]

(τ1 τ2). It is used to suppress instabilities that may be generated due to presence of mean density or velocity gradient.

3.2 Connection to Lighthill’s equation

Lighthill’s acoustic analogy is expressed as,

2 0 2 0 2 ∂ ρ 2 ∂ ρ ∂ Tij 2 − c0 = (3.14a) ∂t ∂xk∂xk ∂xi∂xj 0 2 0 Tij = ρuiuj + p − c0ρ δij − σij (3.14b)

The three terms appearing in the Lighthill source term are labelled momentum, entropy and viscous source terms. Among these, at high Reynolds numbers, the contribution of viscous term to far-ﬁeld noise can be neglected (Lighthill, 1952; Freund, 2003). Goldstein (2010) derived another form of Lighthill’s acoustic analogy by using energy equation to decompose the entropy term into an isentropic and non-isentropic parts. Writing Lighthill’s equation in terms of pressure (Lilley’s form),

2 0 2 0 2 2 1 ∂ p ∂ p ∂ 1 ∂ 0 2 0 2 2 − = (ρuiuj − σij) + 2 2 p − c0ρ (3.15) c0 ∂t ∂xk∂xk ∂xi∂xj c0 ∂t

Writing the energy equation in terms of stagnation enthalpy, hs as,

∂ ∂ρhsui ∂ (ρhs − p) + = − (qi − σijuj) (3.16) ∂t ∂xi ∂xi 32 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

Using the continuity equation to write the above equation in terms of stagnation enthalpy ﬂuctuations, we have

0 ∂ 0 ∂ρhsui ∂ (ρhs − p) + = − (qi − σijuj) (3.17) ∂t ∂xi ∂xi 0 2 where hs = hs − ho. Replacing ho = c0/(γ − 1) and hs = p/(γ − 1) + ρukuk/2, the above equation can be written as,

0 2 0 γ−1 ! 0 ∂ p − c0ρ + 2 ρukuk ∂ hs ∂ qi − σijuj 2 + ρui = − (3.18) ∂t c0 ∂xi ho ∂xi ho

Taking time derivative of the above equation we have,

2 2 0 2 1 ∂ 0 2 0 ∂ ρuihs + qi − σijuj ∂ (γ − 1)ρukuk − 2 2 p − c0ρ = + 2 2 (3.19) c0 ∂t ∂t∂xi ho ∂t 2c0

0 0 Introducing new dependent variable pe ≡ p + (γ − 1)ukuk/2 and substituting expression for entropy term in Eq. 3.19 into Eq. 3.15,

2 0 2 0 2 0 2 0 1 ∂ pe ∂ pe ∂ eij γ − 1 ∂ ηi 2 2 − = − + 2 (3.20) c0 ∂t ∂xk∂xk ∂xi∂xj c0 ∂t∂xi where

γ − 1 e0 = −(ρu u − σ − δ ρu u ) (3.21) ij i j ij 2 ij k k

0 0 ηi = −(ρuihs + qi − σijuj) (3.22)

Ffowcs Williams and Hawkings acoustic analogy (Ffowcs Williams and Hawkings, 1969) extends Lighthill’s work to include sound generated by turbulence and surfaces undergoing arbitrary motion. Suppose the body surface is deﬁned by the level set function f(x, t) = 0. The Ffowcs Williams Hawkings (FW-H) equation can be written as (Ffowcs Williams and Hawkings, 1969; Sharma and Lele, 2008), 3.2. CONNECTION TO LIGHTHILL’S EQUATION 33

2 2 2 ∂ 2 ∂ 0 ∂ ∂ ∂ 2 − c0 [H(f)ρ ] = [Qδ(f)] − [Fiδ(f)] + [TijH(f)] (3.23) ∂t ∂xi∂xi ∂t ∂xi ∂xi∂xj

where the three source terms are the monopole, dipole and quadrupole terms respectively, and are given by

∂f Q = [ρ0vi + ρ(ui − vi)] (3.24) ∂xi

∂f Fi = [Pij + ρuj(ui − vi)] (3.25) ∂xj

2 0 Tij = ρuiuj + Pij − c0 ρ δij (3.26)

Here, ui is the velocity field in an inertial frame, vi is the velocity field defined by the level set f = 0, with f > 0 in the fluid and f < 0 inside the body. H(f) is the Heaviside step function. Extension to the case where the surface described by f = 0 can be taken away from the body surface (permeable surface) is described by Francesantonio (1997). Pij is the compressive fluid stress tensor with ambient pressure subtracted and c0 is the ambient sound speed.

The modiﬁed form of Lighthill’s equation derived here (Eq. 3.20) can now be extended to a form similar to FW-H acoustic analogy (Eq. 3.23). The steps are summarized as follows. First, Eq. 3.23 is written with pressure perturbations as dependent variable. This causes the term involving double time derivative of Lighthill’s entropy term to appear on the right hand side. The energy equation is cast in terms of generalized functions and the expression for double derivative of entropy term is substituted into the FW-H equation written for pressure perturbations.

In terms of pressure perturbations the FW-H equation can be written as, 34 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

2 2 1 ∂ ∂ 0 ∂ ∂ 2 2 − [H(f)p ] = [Qδ(f)] − [Fiδ(f)] + (3.27) c0 ∂t ∂xi∂xi ∂t ∂xi 2 2 ∂ 1 ∂ 0 2 0 [(ρuiuj − σij)H(f)] − 2 2 [(p − c0ρ )H(f)] ∂xi∂xj c0 ∂t

Suppose a volume V is divided into two regions 1 & 2 by surface of discontinuity

f(x, t) = 0 which moves with velocity vi. Then the generalized energy equation valid over the entire volume V can be written as,

∂ 0 ∂ 0 0 (2) ∂f (ρhs − p) + (ρuihs + qi − σijuj) = [ρhs(ui − vi) − pvi + qi − σijuj](1) δ(f) ∂t ∂xi ∂xi (3.28) The right hand side consists of the jump condition to account for the presence of discontinuity. The temporal and spatial derivatives are deﬁned in a generalized sense here and the mathematical identities used can be found in Farassat (1994) and Lighthill (1964). In region 1, the volume enclosed by surface, the ﬂuid is assumed to 0 0 be at rest with p = 0 and ρ = 0. The surface is assumed to be stationary (vi = 0). Replacing the variables by their assigned values in both the regions and introducing the heaviside step function (H(f)) on the left hand side,

∂ 0 ∂ 0 0 ∂f (ρhs − p) H(f) + (ρuihs + qi − σijuj) H(f) = [ρuihs + qi − σijuj] δ(f) ∂t ∂xi ∂xi (3.29) Rearranging and taking the temporal derivative (similar to Eq. 3.19),

2 2 0 1 ∂ 0 2 0 ∂ ρuihs + qi − σijuj − 2 2 p − c0ρ H(f) = H(f) (3.30) c0 ∂t ∂t∂xi ho 2 ∂ (γ − 1)ρukuk + 2 2 H(f) ∂t 2c0 ∂ ρu h0 + q − σ u ∂f − i s i ij j δ(f) ∂t ho ∂xi 3.2. CONNECTION TO LIGHTHILL’S EQUATION 35

0 Using the above equation in Eq. 3.27 and changing the dependent variable to pe,

2 2 ∂ 2 ∂ 0 ∂ ∂ ∂ 2 − c∞ [H(f)pe] = [Qδ(f)] − [Fiδ(f)] + [Eδ(f)] − (3.31) ∂t ∂xi∂xi ∂t ∂xi ∂t 2 2 ∂ 0 γ − 1 ∂ 0 [eijH(f)] + 2 [ηjH(f)] ∂xi∂xj c0 ∂t∂xj

where

0 ρuihs + qi − σijuj ∂f γ − 1 0 ∂f E = = − 2 ηi (3.32) ho ∂xi c0 ∂xi

3.2.1 Solution using Green’s function

For a uniform base ﬂow, the Goldstein acoustic analogy (GAA) equations reduce to

2 0 2 0 2 0 2 0 1 ∂ pe ∂ pe ∂ eij γ − 1 ∂ ηj 2 2 − = − + 2 (3.33) c0 ∂t ∂xj∂xj ∂xi∂xj c0 ∂t∂xj The solution can be written in terms of the free-space Green’s function.

2 ZZ 2 ZZ 0 ∂ 0 δ(g) γ − 1 ∂ 0 δ(g) 4πpe = − eij dydτ + 2 ηj dydτ (3.34) ∂xi∂xj V r c0 ∂t∂xj V r

The three-dimensional, free space Green’s function for a stationary medium is,

1 r δ(g) G(x, t; y, τ) = δ t − τ − = (3.35) 4πr c0 4πr

where r = x − y. The function g ≡ t − τ − r/c0. Using the identities from Francesantonio (1997); Farassat (1980); Farassat and Brentner (1988),

∂ δ(g) 1 ∂ rbiδ(g) rbiδ(g) = − − 2 (3.36) ∂xi r c0 ∂t r r 2 ∂ δ(g) 1 ∂ rbirbjδ(g) 1 ∂ (3rbirbj − δij)δ(g) (3rbirbj − δij)δ(g) = 2 2 + 2 − 3 ∂xixj r c0 ∂t r c0 ∂t r r (3.37) 36 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

The ﬁnal result is

2 Z 2 Z 0 0 1 ∂ Trr γ − 1 ∂ ηr 4πpe(x, t) = 2 2 dV − 3 2 dV (3.38) c0 ∂t V r τ c0 ∂t V r τ Z Z 0 1 ∂ 3Trr − Tii γ − 1 ∂ ηr + 2 dV − 2 2 dV c0 ∂t V r τ c0 ∂t V r τ Z 3Trr − Tii + 3 dV V r τ

0 Here Tij = −eij with Trr = Tijrirj and Tii = T11 + T22 + T33.

3.3 Veriﬁcation tests for acoustic analogy calculation

In this section, we show some verification tests for the solution of generalized acoustic analogy using model problems: sound due to a localized Gaussian source placed in subsonic/supersonic flow and scattering of plane waves due to Taylor vortex. The first problem was also used in the study of Bailly and Juvé(2000) to verify solutions of linearized Euler equation solver and the second problem was considered by Colonius et al. (1994). We use the analytical result for the first problem and results from direct calculations of Colonius et al. (1994) for comparison of our acoustic analogy solution. The acoustic analogy solution is obtained in frequency domain.

3.3.1 Source placed in uniform subsonic and supersonic ﬂow

Both the cases of subsonic (at M = 0.5) and supersonic (at M = 2.5) ﬂow are considered. The resulting overall ﬁeld is shown in contours in Fig. 3.1. The analytical result for this problem can be derived in following manner. For a source (prescribed

function) placed in uniform ﬂow at velocity U∞, density ρ∞ and pressure p∞, the linearized Euler equations reduce to, 3.3. VERIFICATION TESTS FOR ACOUSTIC ANALOGY CALCULATION 37

∂ρ0 ∂ρ0 ∂u0 ∂v0 + U + ρ + = S (3.39a) ∂t ∞ ∂x ∞ ∂x ∂y ∂u0 ∂u0 1 ∂p0 + U∞ + = 0 (3.39b) ∂t ∂x ρ∞ ∂x ∂v0 ∂v0 1 ∂p0 + U∞ + = 0 (3.39c) ∂t ∂x ρ∞ ∂y ∂p0 ∂p0 ∂u0 ∂v0 + U + γp + = c2 S (3.39d) ∂t ∞ ∂x ∞ ∂x ∂y ∞

2 We use scales ρ∞, ρ∞c∞ and c∞ to non-dimensionalize fluctuations in density, pressure and velocity respectively. Using φ0 = φêxp(iωt) for a general fluctuating variable φ0, the above equations can be written in frequency domain (hat variables) as,

∂ρˆ ∂uˆ ∂vˆ ikρˆ + M + + = Sˆ (3.40a) ∂x ∂x ∂y ∂uˆ ∂pˆ ikuˆ + M + = 0 (3.40b) ∂x ∂x ∂vˆ ∂pˆ ikvˆ + M + = 0 (3.40c) ∂x ∂y ∂pˆ ∂uˆ ∂vˆ ikpˆ + M + + = Sˆ (3.40d) ∂x ∂x ∂y

Here k = ω/c∞ and the prescribed frequency is ω = 2π/6. The prescribed source function is Sˆ = 0.1 exp [A(x2 + y2)] where A = −0.8ln2. With ∆ as the Laplacian operator, the equations can be reduced to a single equation in pressure as,

∂pˆ ∂2pˆ ∂Sˆ k2pˆ − 2ikM − M 2 + ∆ˆp = −iωSˆ − M (3.41) ∂x ∂x2 ∂x For M = 0, the left hand side reduces to the standard Helmholtz operator, for i 2 which the Green’s function is G = 4 H0 (kr). Denoting (ξ,η) and (x,y) as the source and observer coordinates respectively, the Green’s function for the above equation is (Lockard, 2000; Bailly and Juv´e,2000), 38 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

Figure 3.1: Contours of real part of density fluctuations obtained from acoustic analogy solution for source place in subsonic uniform flow (at M = 0.5) on the left and supersonic uniform flow (at M = 2.5) on the right.

i iMk(x − ξ) kr G(x, y; ξ, η) = exp H2 , when M < 1 (3.42a) 4α α2 0 α2 1 iMk(x − ξ) kr G(x, y; ξ, η) = exp − J H(x − ξ) (3.42b) 2β β2 0 β2 H[(x − ξ)2 − β2(y − η)2], when M > 1

2 Here, H is the Heaviside step function, H0 is the Hankel function of the second kind and order zero and J is the Bessel function of order zero. For M < 1, r = 0 √ p(x − ξ)2 + α2(y − η)2, α = 1 − M 2 whereas r = p(x − ξ)2 + β2(y − η)2, β = √ M 2 − 1 for M > 1.

The solution can then be written as a convolution of Green’s function with the source function asp ˆ(x, y) = R R Sˆ(ξ, η)G(x, y; ξ, η)dξdη. This integral is numeri- Ωξ Ωη cally computed. The comparison of analytical result and acoustic analogy prediction is shown in Figs.3.2 & 3.3. 3.3. VERIFICATION TESTS FOR ACOUSTIC ANALOGY CALCULATION 39

0.1 0.02

0 0.05 ) ) ˆ ˆ ρ ρ −0.02 ( ( e m R I 0 −0.04

−0.06 −0.05 −20 −10 0 10 20 −20 −10 0 10 20 x x

Figure 3.2: Comparison of real (on left) and imaginary part (on right) of density ﬂuc- tuations obtained from acoustic analogy solution (solid line) and analytical solution (shown in symbols) for source place in subsonic uniform ﬂow (at M = 0.5).

0.06 0.02 0.04 0 ) ) 0.02 ˆ ˆ ρ ρ ( ( e

m −0.02 R 0 I

−0.02 −0.04

−0.04 −0.06 −20 0 20 −20 0 20 x x

Figure 3.3: Comparison of real (on left) and imaginary part (on right) of density ﬂuc- tuations obtained from acoustic analogy solution (shown as solid lines) and analytical solution (shown in symbols) for source place in supersonic uniform ﬂow (at M = 2.5). 40 CHAPTER 3. ACOUSTIC ANALOGY COMPUTATIONS

Figure 3.4: Contours of real (on left) and imaginary (on right) parts of scattered ﬁeld normalized by amplitude of incident plane waves.

3.3.2 Scattering of plane waves by a compressible vortex

The problem of scattering of plane waves by a Taylor vortex was studied by Colonius et al. (1994). The acoustic analogy solution is verified by comparing to the results from direct simulations of Colonius et al. (1994). The flow field due to Taylor vortex is given by

1 − r2 u = u r exp , u = 0 (3.43a) θ θ,max 2 r γ 1 2 γ−1 γ ρ∞c∞ γ − 1 2 2 γp p = 1 − uθ,max exp(1 − r ) , ρ = ρ∞ 2 (3.43b) γ 2 ρ∞c∞

The contours of perturbation pressure due to scattering of incident plane waves using solution of acoustic analogy is shown in Fig. 3.4. The acoustic analogy solution is found to be in good agreement with results from the previous study as seen in Fig. 3.5. 3.3. VERIFICATION TESTS FOR ACOUSTIC ANALOGY CALCULATION 41

0.2 0.2

) 0.15 ) 0.15 0.5 0.5 /r) /r) λ λ ( (

inc 0.1 inc 0.1 /(p /(p rms rms p 0.05 p 0.05

0 0 −100 0 100 −100 0 100 θ θ

Figure 3.5: Comparison of scattered ﬁeld intensity obtained from acoustic analogy solution (solid line) and numerical simulation by Colonius et al. (1994) (shown in symbols). Shown for r/λ = 10 (left) and r/λ = 2 (right). Chapter 4

Two-dimensional shear layer dynamics

In this chapter, sound radiation from mixing layers is studied by numerical solution of two-dimensional unsteady compressible Navier-Stokes equations and acoustic analogy predictions. The splitter plate is included inside the computational domain using high-order overset mesh method. No inflow forcing is provided. Three different cases are considered where the free stream density ratio (or the inverse of temperature ratio) of slower to faster stream is varied from 1 to 2.7, keeping the velocity ratio and Reynolds numbers fixed. Large scale structures or instability waves play a significant role in the generation of noise from shear layers. Supersonically convecting instability waves are known to produce Mach wave radiation. At subsonic Mach numbers, the growth, saturation and decay of instability waves along the mean flow direction leads to a highly di- rectional sound radiation. Pressure fluctuations associated with subsonic instability waves decay exponentially fast away from the hydrodynamic flow region. Far from the near-field hydrodynamic region, the flow solution comprises of propagating acoustic waves which decay algebraically. Solution to the classical compressible hydrodynamic stability does not include acoustic radiation from instability waves. Tam and Mor- ris (1980) showed an analytical procedure to extend the method of multiple scales solution for the linear instability wave to the far-field for low Mach number flows.

42 43

For mixing layers at supersonic convective Mach numbers, Tam and Burton (1984a) further refined their method to predict Mach wave radiation from linear instability waves. Extensions to calculate sound radiation from nonlinear evolution of instability waves have been provided by Wu (2005) for supersonic and Sparks and Wu (2008) for subsonic waves. Crighton and Huerre (1990) showed that the non compactness associated with the sound sources due to convecting instability waves can lead to superdirective sound fields. This was also observed in experiments by Laufer and Yen (1983) of low Mach number forced jets and in direct numerical simulation of two dimensional compressible mixing layer by Colonius et al. (1997). Avital et al. (1998a,b) considered sound radiation from temporally mixing layers. Golanski et al. (2005) studied sound radiation by a non-isothermal mixing layer using a low Mach number approximation coupled with Lighthill’s acoustic analogy. The classical Linear Stability Theory (LST) predicts the evolution of instability waves of small amplitude over a parallel mean flow. The Parabolized Stability Equations (PSE) (Herbert, 1997) provide a generalization to LST for a weakly non- parallel mean flow and including non linear effects in the evolution of instability waves. Cheung and Lele (2009) show that capturing the non-linear effects helps accurately describe the evolution and noise radiation from both subsonic and supersonic mixing layers. The studies by Sandham et al. (2006a) and Sandham et al. (2006b) have also emphasized including non-linear effects to predict noise radiation from subsonic jets. An alternative method is, of course, to capture the full dynamics by solving the compressible Navier Stokes equations. Most of the previously reported computations prescribe shear layer profile forced with disturbances superimposed over the mean profile. We include the splitter plate in the mixing layer calculations to help provide the inflow conditions in an unam- biguous way. In prior computational studies which aim to understand the effect of heating there is some ambiguity due to the inflow forcing used. It is unclear how to specify input disturbance amplitude when Mach number or heating are varied. The results may depend on the assumption regarding which quantity should be held constant when comparing different cases. The study reported here, although uses a two-dimensional problem, is fully consistent since no forcing/inflow disturbance is 44 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

Figure 4.1: Schematic of ﬂow problem setup. required.

4.1 Problem Description

The schematic of the flow problem is shown in Fig. 4.1. Two streams separated by a splitter plate merge downstream of the splitter plate edge, defined as x = 0, to form a mixing layer. The higher speed stream, at Mach number M1, is on the upper side of the plate, whereas, the slower stream, at Mach number M2, is on the lower side of the splitter plate. The upper and lower stream densities are ρ1 and ρ2 respectively. The convective Mach number is denoted by Mc, which is defined here as ∆U Mc = , where c1 and c2 denote the free stream speed of sound in upper and lower c1+c2 streams, respectively. This definition of Mc follows from Papamoschou & Roshko’s (Papamoschou and Roshko, 1988) general definition when the two streams have the same γ (ratio of specific heats), as is the case in the present simulations. We use

∆U = U1 −U2 to represent the velocity-diﬀerence and wp to denote the splitter width. These also provide suitable reference scales for non-dimensionalization. All variables used in the simulations are non-dimensional with length, time, velocity, pressure and 2 temperature scaled by wp, wp/∆U, ∆U, ρ2∆U and (γ − 1)T2. For physical clarity other scales for velocity and pressure are sometimes used in the results shown later. They will be deﬁned when used. 4.2. NUMERICAL TECHNIQUE 45

U2 ρ2 Case r = s = Mc M1, M2 U1/c2 Uc,is/c2 U1 ρ1 A 0.17 1.0 0.5 1.2, 0.2 1.2 0.7 B 0.17 1.8 0.42 0.9, 0.2 1.2 0.63 C 0.17 2.7 0.38 0.73, 0.2 1.2 0.58

Table 4.1: The parameters for the test case considered. Reynolds number, Reδθ,1 = ∆Uδθ,1 , is about 160, based on free stream conditions on the boundary layer thickness ν1 in the high-speed side, velocity diﬀerence ∆U and kinematic viscosity based on free stream conditions on the high-speed side. The quantity, U1/c2, corresponds to jet acoustic Mach number and is kept constant in the three cases. Uc,is is the convection velocity based on the isentropic estimate.

The test cases we simulate as part of our study are described in the table below. The computational domain for the calculations includes both the hydrodynamic region and a substantial acoustic region. The solution from the direct calculations provides source terms for the acoustic analogy. The acoustic prediction from the acoustic analogy is compared to that from the direct calculations on the same mesh.

4.2 Numerical Technique

The computational results are obtained by solving the compressible, viscous Navier Stokes equations in general curvilinear coordinates (Nagarajan, 2004). The governing equations are discretized using a staggered grid 6th order compact finite difference scheme. Time advancement is done using 3rd order Runge-Kutta scheme combined with Beam-Warming implicit scheme close to the solid boundary. In order to include the splitter plate inside the computational domain, two meshes, a background grid and a body-fitted curvilinear mesh are used for the problem. The body-fitted grid wraps around the splitter plate and is overset on the background grid. The details of the overset method are described by Bhaskaran and Lele (2008, 2010). The plate surface is taken to be a no-slip adiabatic surface. For the laminar incoming flow, profiles above and below the plate surface are calculated by solving the steady state, compressible boundary layer equations (Schlichting and Gersten, 2000). The present method was also used to simulate three-dimensional turbulent mixing layers by Sharma et al. 46 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

(2011) but here the focus is on two-dimensional mixing layers. In the present study, the body-ﬁtted curvilinear mesh has 960 points in the direction along the plate and 160 points in the normal direction. The background mesh has 1440 points in the streamwise direction and 2201 points in the transverse direction.

The background mesh extends upto 400wp downstream and 200wp into the slow and fast stream regions while maintaining suﬃcient resolution of the acoustic waves.

4.3 Results

4.3.1 Near ﬁeld ﬂow

Flow development in terms of vorticity magnitude contours is shown in Fig. 4.2 for each of the cases A-C. The near ﬁeld hydrodynamic region consists of an initial laminar shear layer which separates oﬀ the splitter plate. The shear layer formed is susceptible to inviscid Kelvin-Helmholtz instability. The initial vortex rollup is followed by vortex pairing which is the dominant mechanism for growth of shear layer thickness. It can be observed that the vortex rollup starts early in shear layer

of case C (x/wp ∼ 15) as compared to shear layers of cases B and A where the rollup begins at x/wp ∼ 30 amd x/wp ∼ 50 respectively. Visual inspection of Fig. 4.2 also reveals an increasing trend from cases A to C with respect to lateral spreading of shear layer and appearance of small scale vorticity. The thickness of shear layer is quantified in terms of vorticity thickness defined as, δ = ∆U/[max ∂u˜ ] and is shown in Fig. 4.3. It can be observed that the mean ω y ∂y lateral spreading of shear layer increases with increasing density ratio from cases A to C. The mean density and Favre-averaged mean velocity, denoted by ue, plotted as a function of transverse coordinate normalized by local vorticity thickness, δω, is shown in Fig. 4.3 for case C. The collapse of profiles taken at different streamwise locations indicates self-similar development of mean flow. The peak root mean square (r.m.s) levels for the streamwise and transverse velocities are shown as a function of streamwise distance in Fig. 4.4(a) and (b). The peak 4.3. RESULTS 47

Figure 4.2: Contours of vorticity magnitude (normalized by ∆U/wp) plotted in logarithmic increments from 0.01 to 4 shown for cases A (top), B(middle) and C(bottom). 48 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

6 6

4 4

2 2 ω ω δ 0 δ 0 y/ y/

−2 −2

−4 −4

−6 −6 −0.5 0 0.5 1 1.5 −0.5 0 0.5 1 1.5 (u −U )/(U −U ) (ρ −ρ )/(ρ −ρ ) mean 2 1 2 mean 2 1 2

(a) (b)

20 ω

δ 15

0 50 100 150 200 250 x/w p

(c)

Figure 4.3: Profiles of mean velocity in (a) and density in (b) as a function of normalized transverse coordinate at different streamwise locations shown for case B. (c) Comparison of vorticity thickness growth rates for different temperature ratios along the streamwise coordinate. Dash dot line: Temperature ratio = 1; Solid line: 1.4; Dashed line: 1.8. 4.3. RESULTS 49

r.m.s level of velocity fluctuations does not vary as the density ratio is increased. Den- sity fluctuations increase with density ratio (shown in Fig. 4.4(c)). The peak r.m.s level of density fluctuation increases linearly with difference in free stream densities of the two streams. However, the early rate of growth of intensities with streamwise distance is higher for case C followed by case B and then A as observed in Figs. 4.4(a)- (d). The variance of vorticity and dilatation fluctuations is shown in Figs. 4.4(e) and (f). The initial dilatation and vorticity variance is higher for higher density ratio case but further downstream the dilatation variance is insensitive to density ratio. The increase in vorticity variance with density ratio further downstream is most likely due to the increase in baroclinic generation of vorticity with increasing density ratio. In these shear layers as the high-speed stream is heated, the initial instability is accelerated but the saturation level of pressure fluctuations decreases with heating as shown 2 in Fig. 4.4(d). However, the peak pressure r.m.s curves scaled by ρ2(Uc − U2) are found to collapse for different cases as shown in Fig 4.5. In the frame of reference of the ’large scale structures’ or eddies (which move with the convection velocity Uc), the difference between stagnation pressure and the ambient pressure can be written using the isentropic relation as,

γ γ−1 2 2 ∆p p01 − p∞ γ − 1 2 1 ρ1(U1 − Uc) 1 ρ2(Uc − U2) ≡ = 1 + Mc1 − 1 ≈ = p∞ p∞ 2 2 p∞ 2 p∞ (4.1)

Here, we use p01 to denote the stagnation pressure and the isentropic estimate

c1U2+c2U1 for Uc = . The isentropic estimate of convection velocity is found to be c1+c2 consistent with estimate based on convection velocity of vortex centers (identified as pressure minima and shown in Fig. 4.5 for case A at different time instants). The above relation indicates that the r.m.s pressure fluctuations, which should vary as 2 2 ∆p, scale with ρ2(Uc − U2) = ρ1(U1 − Uc) and explains the trends observed in Figs. 4.4(d) and 4.5. The transverse profiles of fluctuation intensities at various streamwise locations 2 are shown in Figs. 4.6, 4.7 and 4.8 for cases A, B and C respectively. Here σu = 002 2 002 2 uf /∆U and σv = vf /∆U . The collapse of profiles when the transverse coordinate 50 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

0.4 0.4

0.3 0.3 U U ∆ ∆ / 0.2 / 0.2 peak rms peak rms v u

0.1 0.1

0 0 50 100 150 200 50 100 150 200 x/w x/w p p

(a) (b)

0.4 0.2

0.3 0.15 2 U 2 ∆ ρ

/ 2 ρ

0.2 / 0.1 peak rms ρ peak rms p 0.1 0.05

0 0 50 100 150 200 50 100 150 200 x/w x/w p p

0.8 0.04

0.6 0.03 ) ) p p U/w U/w ∆ ∆ 0.4 0.02 /( /( peak rms peak rms ω Ω 0.2 0.01

0 0 50 100 150 200 50 100 150 200 x/w x/w p p

(e) (f)

Figure 4.4: Variation of peak root mean square (r.m.s) ﬂuctuations of (a) streamwise velocity (b) transverse velocity (c) density (d) pressure (e) vorticity (f) dilatation in the streamwise direction. Case A: solid line; case B: dashed line; case C: dashed dotted line. 4.3. RESULTS 51

0.8 300

250 2 ) 0.6 2 200 −U c p (U

2 0.4 150 x/w ρ / 100 peak rms

p 0.2 50

0 0 50 100 150 200 1350 1355 1360 1365 1370 x/w t∆ U/w p p

(a) (b)

Figure 4.5: (a) Variation of peak root mean square (r.m.s) fluctuations of pressure 2 scaled by ρ2(Uc − U2) . Case A: solid line; case B: dashed line; case C: dashed dotted line. (b) The vortex center locations (symbols) based on pressure minima for case A. The straight lines correspond to slope 0.7 which is used as an estimate for convection velocity, Uc. is normalized by local vorticity thickness shows the self-similar development of shear layer. It can be observed that the asymmetry of density fluctuation profiles increases as the density ratio is increased. The increase in peak level of density fluctuations and decrease in pressure fluctuations with increasing density ratio is also observable.

The time history of pressure at various streamwise probe locations in the shear layer is shown in Fig. 4.9, 4.10 and Fig. 4.11 for cases A, B and C respectively. Pressure spectra for the corresponding locations are shown in Fig. 4.12 for cases A, B and C. The growth of pressure fluctuations is much delayed in case A compared to cases B and C. The first streamwise location shown in Fig. 4.9 is at x/wp = 52 where substantial fluctuations were noticed. This can be contrasted to location of the

ﬁrst probe at x/wp = 32.5 for case B and x/wp = 18 for case C. The spectrum for the ﬁrst location shows a distinct peak as the density ratio is increased showing the dominance of a particular frequency. The peak Strouhal number St = fwp/∆U ∼ 0.09 and corresponds to Strouhal number based on the initial momemtum thickness 52 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

0.08 0.1

0.08 0.06

0.06 2 u 2 v

σ 0.04 σ 0.04

0.02 0.02

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y/δ y/δ ω ω

(a) (b)

0.03 0.025

0.025 0.02

0.02 4 U 2 2 0.015 ∆ ρ

/ 2 2

0.015 ρ / 2 rms ρ 0.01 2 rms

0.01 p

0.005 0.005

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y/δ y/δ ω ω

Figure 4.6: Transverse profiles of (a) streamwise velocity fluctuations, (b) transverse velocity fluctuations, (c) density fluctuations and (d) pressure fluctuations for case A shown as a function of transverse coordinate normalized by local vorticity thickness, δω(x). Streamwise locations: x/wp = 163: Solid line; x/wp = 223: dashed line; x/wp = 256: dashed dotted line; x/wp = 275: dotted line. 4.3. RESULTS 53

0.08 0.1

0.08 0.06

0.06 2 u 2 v

σ 0.04 σ 0.04

0.02 0.02

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y/δ y/δ ω ω

(a) (b)

0.05 0.02

0.04 0.015 4 U 2 2 0.03 ∆ ρ

/ 2 2

ρ 0.01 / 2 rms ρ 0.02 2 rms p 0.005 0.01

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y/δ y/δ ω ω

Figure 4.7: Transverse profiles of (a) streamwise velocity fluctuations, (b) transverse velocity fluctuations, (c) density fluctuations and (d) pressure fluctuations for case B shown as a function of transverse coordinate normalized by local vorticity thickness, δω(x). Streamwise locations: x/wp = 203: Solid line; x/wp = 224: dashed line; x/wp = 245: dashed dotted line; x/wp = 266: dotted line. 54 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

0.08 0.1

0.08 0.06

0.06 2 u 2 v

σ 0.04 σ 0.04

0.02 0.02

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y/δ y/δ ω ω

(a) (b)

0.08 0.015

0.06

4 0.01 U 2 2 ∆ ρ

/ 2 2

0.04 ρ / 2 rms ρ 2 rms

p 0.005 0.02

0 0 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 y/δ y/δ ω ω

Figure 4.8: Transverse profiles of (a) streamwise velocity fluctuations, (b) transverse velocity fluctuations , (c) density fluctuations and (d) pressure fluctuations for case C shown as a function of transverse coordinate normalized by local vorticity thickness, δω(x). Streamwise locations: x/wp = 118: Solid line; x/wp = 139: dashed line; x/wp = 160: dashed dotted line; x/wp = 181: dotted line. 4.3. RESULTS 55

of Stθ = fδθ/∆U ∼ 0.02. The spectra get broadband further downstream in the ﬂow and are marked by appearance of low frequency content. The amplitude of pressure ﬂuctuations at various frequencies are observed to be higher for the lower density ratio case A compared to higher density ratio case C.

4.3.2 Impulse response

The behavior of the disturbance placed in a prescribed mean flow is analyzed here. The mean flow is taken from a fixed streamwise location from the simulation statistics and replicated in the streamwise direction giving a parallel mean flow. The initial disturbance is prescribed as a divergence-free vortex. Full linearized Euler equations are solved to compute the dynamics of disturbance evolution. Contours of instantaneous pressure disturbance are shown in Fig. 4.13(a) and (b) for cases A and C respectively. The absolute value of pressure is integrated in the transverse direction to obtain the pressure amplitude, | A | and is plotted against the coordinate, (x − xo)/t in Fig. 4.14. It can be seen that case C is close to being absolutely unstable, but not quite (close to being marginally-absolutely unstable). The peak convective instability can be observed to have a group velocity 0.65 for the isothermal case whereas it decreases to 0.45 for the heated case.

4.3.3 Acoustic near ﬁeld

The amplitude of pressure fluctuations in the near field for observer locations on the slower stream side is shown in Fig .4.15. A factor of two reduction in peak pressure amplitude is observed as the heating ratio is changed from 1 to 2.7. At angles away from the shear layer (θ ∼ 120), there is a slight increase in the pressure amplitude as the upstream sound radiation is increased. Overall, the pressure field becomes more evenly distributed across different angles as the heating ratio is increased. The Discrete Fourier Transform (DFT) of the pressure signal is obtained using Eq. 4.2 and its real part is shown in Fig. 4.16 for case A and Fig. 4.16 for case C for different Strouhal numbers (St). The spectra for different observer locations along a radial arc are shown in Fig. 4.18 for cases A through C. For case A, the 56 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

1 1

0.9 0.9

0.8 0.8 2 2

U 0.7 U 0.7 ∆ ∆

2 2

ρ 0.6 ρ 0.6 p/ p/ 0.5 0.5

0.4 0.4

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 t∆ U/w t∆ U/w p p

(a) (b)

1 1

0.9 0.9

0.8 0.8 2 2

U 0.7 U 0.7 ∆ ∆

2 2

ρ 0.6 ρ 0.6 p/ p/ 0.5 0.5

0.4 0.4

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 t∆ U/w t∆ U/w p p

Figure 4.9: Time history of pressure in the shear layer (y/wp = 0.8) at various streamwise locations for case A. (a) x/wp = 52, (b) x/wp = 111, (c) x/wp = 136 and (d) x/wp = 192. 4.3. RESULTS 57

1 1

0.9 0.9

0.8 0.8 2 2

U 0.7 U 0.7 ∆ ∆

2 2

ρ 0.6 ρ 0.6 p/ p/ 0.5 0.5

0.4 0.4

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 t∆ U/w t∆ U/w p p

(a) (b)

1 1

0.9 0.9

0.8 0.8 2 2

U 0.7 U 0.7 ∆ ∆

2 2

ρ 0.6 ρ 0.6 p/ p/ 0.5 0.5

0.4 0.4

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 t∆ U/w t∆ U/w p p

Figure 4.10: Time history of pressure in the shear layer (y/wp = 0.8) at various streamwise locations for case B. (a) x/wp = 32.5, (b) x/wp = 47, (c) x/wp = 75 and (d) x/wp = 130. 58 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

1 1

0.9 0.9

0.8 0.8 2 2

U 0.7 U 0.7 ∆ ∆

2 2

ρ 0.6 ρ 0.6 p/ p/ 0.5 0.5

0.4 0.4

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 t∆ U/w t∆ U/w p p

(a) (b)

1 1

0.9 0.9

0.8 0.8 2 2

U 0.7 U 0.7 ∆ ∆

2 2

ρ 0.6 ρ 0.6 p/ p/ 0.5 0.5

0.4 0.4

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 t∆ U/w t∆ U/w p p

Figure 4.11: Time history of pressure in the shear layer (y/wp = 0.8) at various streamwise locations for case C. (a) x/wp = 18, (b) x/wp = 47, (c) x/wp = 75 and (d) x/wp = 130. 4.3. RESULTS 59

0 0 10 10

−1 −1 10 10

−2 −2 10 10 pp pp S S −3 −3 10 10

−4 −4 10 10

−5 −5 10 −2 −1 10 −2 −1 10 10 10 10 St St

(a) (b)

0 10

−1 10

−2 10 pp S −3 10

−4 10

−5 10 −2 −1 10 10 St

(c)

Figure 4.12: Pressure spectra for (a) case A, (b) case B and (c) case C at diﬀerent streamwise locations. For case A: x/wp = 52: solid line; x/wp = 111: dashed line; x/wp = 136: dashed-dotted line; x/wp = 192: dotted line. For case B: x/wp = 32.5: solid line; x/wp = 47: dashed line; x/wp = 75: dashed-dotted line; x/wp = 130: dotted line. For case C: x/wp = 18: solid line; x/wp = 47: dashed line; x/wp = 75: dashed-dotted line; x/wp = 130: dotted line. 60 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

(a) (b)

Figure 4.13: Contours of instantaneous pressure disturbance. (a) Case A (b) Case C.

−1 −1 10 10 t=50 −2 −2 10 10 t=70 t=90 −3 −3 10 10

−4 −4

|A| 10

−5 −5 10 10

−6 −6 10 10

−7 −7 10 10 −0.5 0 0.5 1 1.5 −0.5 0 0.5 1 1.5 (x−x )/t (x−x )/t 0 0

(a) (b)

Figure 4.14: Pressure amplitude | A | plotted against the similarity coordinate, (x − xo)/t, where xo is the initial location of disturbance. (a) Case A (b) Case C. 4.3. RESULTS 61

0.025

0.02

0.015 rms p

0.01

0.005

20 40 60 80 100 θ

(a) (b)

Figure 4.15: (a) Observer locations located on a circular arc of radius 150 centered at 75 are shown in black symbols. Acoustic ﬁeld at St = 0.045 is shown in the background for case A. (b) Root mean square pressure ﬂuctuations as a function of angle made with respect to the x-axis. Case A: solid line; case B: dashed line; case C: dashed dotted line. 62 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

sound is directed downstream mainly for higher St, whereas for lower St, there is some upstream radiation as well. For case C, the upstream radiation is stronger as compared to case C. This can also be observed in Fig. 4.18 as the spectra for different observer locations move closer to each other as the heating is increased. The contribution to sound field was observed mainly for St < 0.06. Bodony (2010) observed a similar reduction in pressure amplitude in his analysis of linear fluctuations over a prescribed shear layer mean flow.

1 pˆ = ΣN−1p(t ) exp(−2πijn/N) (4.2) n N j=0 j

The acoustic sources (right hand side terms) are defined by Goldstein’s acoustic analogy (GAA) (Goldstein, 2003) in terms of fluctuating momentum and enthalpy fluxes. As noted by Afsar et al. (2011b), the source term associated with heating effect defined in GAA is proportional to stagnation enthalpy fluctuations in the frame of reference moving with local base flow velocity and becomes negligible in cold jets. This is shown to be the case in Fig. 4.19 where the fluctuation amplitude of sources appearing in the GAA are plotted as a function of streamwise coordinate for different heating ratios. It is observed that the terms associated with fluctuating momentum fluxes remain unchanged with increased heating. But, there is an increase in the enthalpy flux term with increased heating. According to Afsar et al. (2011b), it should be the coupling term between the enthalpy and momentum flux terms that should be responsible for reduction of sound in heated jets. A similar procedure is applied to source terms in Eq. 3.10 which is another form 2 0 ∂ eij of Lighthill’s equation. The term Smom = − is identified as the momentum flux ∂xi∂xj 2 0 γ−1 ∂ ηj source term, whereas, Sent = 2 is the term related to enthalpy flux associated c0 ∂t∂xj with heating. The peak fluctuation level of these source terms is shown in Fig. 4.20(a) and (b). Similar to the source terms defined based on the time-averaged mean flow obtained from direct simulations, the Sent term has an increasing trend with heating. However, the momentum flux source term also increases with heating. The spectra shown in Fig. 4.20(c) are obtained for a specific observer location for case C in three ways: from direct calculations, secondly, from acoustic analogy prediction using both 4.3. RESULTS 63

(a) (b)

Figure 4.16: Real part of pressure DFT obtained from direct calculations shown for (a) St = 0.015, (b) 0.03, (c) 0.045 and (d) 0.06 for case A (density ratio = 1). 64 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

(a) (b)

Figure 4.17: Real part of pressure DFT obtained from direct calculations shown for (a) St = 0.015, (b) 0.03, (c) 0.045 and (d) 0.06 for case C (density ratio = 2.7). 4.3. RESULTS 65

−2 10

−3 10

−4 10 pp

S −5 10

−6 10

−7 10 −2 −1 10 10 St

(a) (b)

−2 −2 10 10

−3 −3 10 10

−4 −4 10 10 pp pp

S −5 S −5 10 10

−6 −6 10 10

−7 −7 10 −2 −1 10 −2 −1 10 10 10 10 St St

Figure 4.18: (a) Observer locations located on a circular arc of radius 150 centered at 75 are shown in black symbols. Pressure spectra for (a) case A, (b) case B and (c) case C. 66 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

0.05 0.04

0.04 0.03

0.03

peak v,rms 0.02 peak u,rms S S 0.02

0.01 0.01

0 0 50 100 150 200 50 100 150 200 x/w x/w p p

(a) (b)

0.25

0.2

0.15 peak p,rms S 0.1

0.05

0 50 100 150 200 x/w p

(c)

Figure 4.19: Fluctuation amplitude (r.m.s levels) of acoustic sources as deﬁned by Eqs.3.4-3.7 for the time averaged mean ﬂow obtained from the direct calculations. 0 0 (a) Su is the source term appearing in GAA equation for variable u1 = ρv1/ρ, (b) Sv 0 0 0 for variable u2 = ρv2/ρ and (c) Sp for variable pe. Case A: solid line; case B: dashed line; case C: dashed dotted line. 4.4. SUMMARY 67

the momentum and enthalpy flux source terms and thirdly, with only the enthalpy source term. The spectra obtained with only the enthalpy flux term in Eq. 3.10 is much higher than that obtained from direct calculation and that obtained from the full source term. The above comparisons suggest that even though the source fluctuations may be stronger for heated cases, there is a cancellation between sound fields produced by the individual momentum and enthalpy flux terms leading to a weaker overall sound field.

4.4 Summary

Numerical simulations of heated and unheated two-dimensional mixing layers are presented. The splitter plate which initially separates the two streams is included inside the domain to avoid the need for any inflow forcing. The density ratio was varied keeping the velocity ratio and Reynolds number fixed to study the effect of heating the higher speed stream. The initial instability is accelerated with heating but the saturation amplitude of streamwise velocity fluctuations do not vary. The saturation amplitudes of density fluctuations were found to increase proportionally to difference in free stream densities whereas the near-field pressure fluctuations were found to decrease with heating. Away from the shear layer on the slower stream side, a reduction in peak pressure fluctuations with increased heating is observed. Analysis of sound source terms in the acoustic analogy suggest that although the source term fluctuations become stronger in the near field with heating, the mutually cancelling effects of individual sources lead to a weaker sound field. This result is consistent with the finding of Bodony and Lele (2008b) using LES data for turbulent heated jets that the enthalpy flux source terms cancel partially with the momentum flux source terms in Lighthill’s acoustic analogy. Alternative form of Lighthill’s stress tensor was also derived by Bodony (2009) that decoupled various components of Lighthill’s tensor to study the effect of heating. We infer here that this type of cancellation is operative in two-dimensional heated shear layer when analyzed using Goldstein’s generalized acoustic analogy. 68 CHAPTER 4. TWO-DIMENSIONAL SHEAR LAYER DYNAMICS

0.5 0.4

0.4 0.3 0.3 0.2 peak ent,rms peak mom,rms

0.2 S S 0.1 0.1

0 0 50 100 150 200 50 100 150 200 x/w x/w p p

(a) (b)

−2 10

−3 10

−4 10 pp

S −5 10

−6 10

−7 10 −2 −1 10 10 St

(c)

Figure 4.20: Fluctuation amplitude (r.m.s levels) of acoustic sources as defined by Eq. 3.10 for uniform base flow obtained from the direct calculations. (a) Smom is the first term on the right hand side of Eq. 3.10 and (b) Sent is the second term. For (a) and (b) Case A: solid line; case B: dashed line; case C: dashed dotted line. (c) Comparison of spectra for observer location r = 150, θ = 30 for case C. Direct calculation: solid line; Full source term: dashed line; With only enthalpy flux term: dashed dotted line. Chapter 5

Turbulent mixing layer computations

Large-eddy simulations of spatially developing, turbulent mixing layers are presented in this chapter. The splitter plate is included inside the computational domain using high-order overset mesh method. Five different inflow conditions are considered: (a) laminar streams with velocity ratio of 0.17. Both unheated and heated (with temperature ratio 1.8) higher speed stream cases are considered, (b) laminar streams with velocity ratio 0.36 and density ratio 0.64 and (c) Similar cases as in (a) but with turbulent boundary layer on high speed side and laminar on slower side. For each case, the mean and turbulent intensity profiles collapse when plotted in similarity coordinates. The development distance to achieve self-similarity in the mean velocity profile is found to be shortest for turbulent boundary layer case, followed by the isothermal laminar stream of (a) and then (b). The growth rate of the shear layer and peak self-similar values of the turbulent intensities are found to be in agreement with available experiments.

5.1 Problem Description

The schematic of the ﬂow problem is shown in Fig. 5.1. Two streams which are initially separated by a splitter plate merge downstream of the splitter plate edge,

69 70 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

Figure 5.1: Schematic of ﬂow problem setup.

deﬁned as x = 0, to form a mixing layer. The higher speed stream, at Mach number

M1, is on the upper side of the plate, whereas, the slower stream, at Mach number M2, is on the lower side of the splitter plate. The upper and lower stream densities are ρ1 and ρ2 respectively. The convective Mach number is denoted by Mc, which is defined ∆U here as Mc = , where c1 and c2 denote the free stream speed of sound in upper c1+c2 and lower streams, respectively. This definition of Mc follows from general definition of Papamoschou and Roshko (1988) when the two streams have the same γ (ratio of specific heats), as is the case in the present simulations. We use ∆U = U1 − U2 to represent the velocity-difference and wp to denote the splitter width. These also provide suitable reference scales for non-dimensionalization. All variables used in the simulations are non-dimensional with length, time, velocity, pressure and temperature 2 scaled by L, L/c2, c2, ρ2c2 and (γ − 1)T2, where L is the reference length scale and c2 is the ambient sound speed. For physical clarity other scales for velocity and pressure are sometimes used in the results shown later. They will be defined when used. The table below describes the test cases we simulate as part of our study. The geometry of the splitter plate is prescribed as a flat plate with the trailing edge defined as a super-ellipse (of aspect ratio 6) given as,

x 4 6y 2 + = 1 (5.1) wp wp 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 71

U2 ρ2 δθ1 δθ2 Case r = s = Mc M1, M2 Incoming boundary layer Reδ U1 ρ1 θ,1 wp wp A1 0.17 1 0.5 1.2, 0.2 Laminar 200 0.1 0.05 A2 0.17 1.8 0.4 0.9, 0.2 Laminar 200 0.1 0.05 A3 0.36 0.64 0.51 1.8, 0.51 Laminar 300 0.17 0.05 A4 0.17 1 0.5 1.2, 0.2 Turbulent 800 0.1 0.05 A5 0.17 1.8 0.4 0.9, 0.2 Turbulent 800 0.1 0.05

Table 5.1: The parameters for the test case considered. Reynolds number, Reδθ,1 = ∆Uδθ,1 , is based on free stream conditions on the boundary layer thickness in the ν1 high-speed side, velocity diﬀerence ∆U and kinematic viscosity based on free stream conditions on the high-speed side.

5.2 Cases with incoming laminar boundary layers

5.2.1 Computational parameters

The domain used for simulation case A1 is shown in Fig. 5.2(a). The overall computational grid consists of two overlapping meshes. The first is a body-fitted C-mesh that wraps around the splitter plate. The second mesh is a curvilinear background mesh that communicates with the body-fitted mesh close to the plate and captures downstream evolution of the mixing layer. The overlapping region of the two meshes is shown in Fig. 5.2(b). A rectangular region of the background mesh that overlays the plate is cut and the remaining overlapping points are used for grid communication. A similar mesh topology is used for cases A2 and A3. The motivation for domain expansion in transverse direction is to accommodate shear layer spreading and provide region of unaffected free stream to be entrained into the spreading layer. The incoming laminar boundary layer requires fine transverse resolution on the plate compared the streamwise or spanwise resolution. Close to the plate edge, the boundary layer separates and gradually evolves as a mixing layer downstream. The transverse resolution is chosen to allow for about 7-10 points per momentum thickness when the mixing layer is still laminar. Further downstream, the mixing layer becomes thicker and fully turbulent. About 40 points per momentum thickness are used in the downstream region. The axial and spanwise resolution requirements are much milder when the mixing layer is still laminar. However, within the turbulent mixing layer 72 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

(a) (b)

Figure 5.2: Grid used for the simulations. (a) view of full computational domain for case A1. Here, the shaded region denotes sponge used at domain boundaries. (b) zoom in view of the mesh around the plate and the surrounding background mesh. The spatial extent is shown in terms of plate width, wp. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 73

Mesh min(∆x, ∆y, ∆z) max(∆x, ∆y, ∆z) ∆t Lx,Ly,Lz A1, A2(body-ﬁtted) (0.01, 0.01, 0.47) (0.1, 0.15, 0.47) 0.0035 36, 10, 90 A1, A2(background) (0.3, 0.07, 0.47) (2, 0.8, 0.47) 0.0035 420, (200, 400), 90 A3(body-ﬁtted) (0.01, 0.01, 0.47) (0.8, 0.8, 0.47) 0.0025 36, 10, 60 A3(background) (0.3, 0.07, 0.47) (2, 0.8, 0.47) 0.0025 520, (150, 300), 60

Table 5.2: Mesh parameters for three-dimensional calculations. ∆, L and N refer to the mesh spacing, domain length and number of points in each direction respectively. Reference length used to non-dimensionalize spatial dimensions is the plate thickness, wp.

Mesh Nx,Ny,Nz A1, A2(body-ﬁtted) 480, 60, 192 A1, A2(background) 1440, 513, 192 Total = 147.3 mil A3(body-ﬁtted) 480, 60, 128 A3(background) 1680, 257, 128 Total = 59 mil

Table 5.3: Mesh sizes for laminar boundary layer simulation cases.

region, the mesh is reﬁned in the streamwise direction to capture the small scales. A uniform mesh is used in the spanwise homogeneous direction. The spatial extent and the number of grid points used for the body-ﬁtted and background meshes are summarized in Tables. 5.2 and 5.3 below.

The simulation was first run over a time of the order 800wp/∆U to let the flow settle down before the statistics are collected. The statistics were collected over a time horizon of 250wp/∆U. A numerical sponge was used on the boundary of the computational mesh to damp the disturbances as shown in Fig. 5.2(a). The sponge width was 50wp on the outflow boundary, 30wp on the top and bottom boundaries and 10wp at the inflow boundary. A high-order numerical filter was used to damp the unphysical high-frequency numerical disturbances as described in Nagarajan (2004). The filter coefficient was set to be α = 0.49. 74 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

5.2.2 Mean ﬂow

The visualizations of instantaneous density (in gray scale) and vorticity magnitude (in color) contours for case A1 in Fig. 5.3 and case A2 in Fig. 5.4 give an overall picture of the flow field. The early region of the mixing layer is laminar followed by a transition region (marked by spanwise rollers with small scale fluctuations superposed over them) and then a turbulent mixing layer region. Visually, the outer boundaries of the mixing layer grow laterally in a linear fashion. A similar picture is shown for case A3 in Fig. 5.5. For cases A1, A2 and A3, the mixing layer appears to grow more into the lower speed region. Radiation of sound into the lower speed stream, dominant between 20◦ and 40◦ angle, is also visible in Figs. 5.3(b) and 5.4(b). The apparent source of this radiation is located between streamwise locations of 30 and 200 from the splitter plate edge. There is also a weak radiation in the upstream direction towards the splitter plate. Similar but much weaker radiation pattern is seen for case A3 as well. The mixing layer is observed to transition earlier in heated case A2 than in case A1. Since the high speed stream is supersonic in cases A1 and A3, a shock is present at the trailing edge of the splitter plate. Both the streams are subsonic for case A2. The contours of transverse velocity are portrayed in Figs. 5.3(a) and 5.4(a) for cases A1 and A2 respectively. The early transitioning shear layer is characterized by oblique, cell- shaped structures in the spanwise direction. Further downstream in the turbulent part, larger, randomly organized structures with superimposed smaller scale fluctuations are observed. The mean flow (averaged in time and spanwise homogeneous direction) streamtraces are shown in Fig. 5.6. The part above the splitter plate consists of the higher speed stream. The streamtraces in the lower speed stream are bent towards the center which means the fluid is entrained from the lower speed stream into the mixing layer. Laizet et al. (2010) showed the effect of different splitter plate edge geometries on the flow near the splitter plate edge, for incompressible mixing layers with incoming laminar boundary layers. In their study, the case with a thin trailing edge showed no vortex formation in the mean flow streamtraces, whereas the blunt trailing edges showed a vortex pattern similar to that observed here. The streamwise pressure gradient along the surface of the plate is shown in Fig. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 75

(a)

(b)

Figure 5.3: (a) Visualization of contours of transverse velocity for case A1 in the X-Z plane along the lipline. The domain extends from 0 < z < 90 in the spanwise direction. (b)Visualization of density contours (plotted from 0.95ρ1 to 1.05ρ1) for Case A1 in the mid X-Y plane. The portion of the computational domain shown above extends from −20 < x < 340 and transverse extent from −100 < y < 100. Origin is located at the trailing edge of the splitter plate. 76 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

(a)

(b)

Figure 5.4: (a) Visualization of contours of transverse velocity for case A3 in the X-Z plane along the lipline. The domain extends from 0 < z < 90 in the spanwise direction. (b)Visualization of density contours (plotted from 0.95ρ1 to 1.05ρ1) for Case A1 in the mid X-Y plane. The portion of the computational domain shown above extends from −20 < x < 340 and transverse extent from −100 < y < 100. Origin is located at the trailing edge of the splitter plate. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 77

Figure 5.5: Top: Visualization of contours of streamwise vorticity for case A3 in the X-Z plane along the lipline. The domain extends from 0 < z < 60 in the spanwise direction. Bottom: Instantaneous visualization of density contours (plotted from 0.5ρ1 to 1.1ρ1) in the mid X-Y plane for Case A3. The portion of the computational domain shown above extends from −20 < x < 500 and −75 < y < 75. Origin is located at the trailing edge of the splitter plate. 78 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

(a) (b)

Figure 5.6: Velocity streamtraces for case A1 close to the plate in (a) and far from the plate in (b).

5.7(a) for case A1. The region near the tip is marked by a region of relatively strong adverse pressure gradient due to which the boundary layers above and below separate. The region above the plate also contains a shock emanating from the splitter plate tip which induces an adverse pressure gradient whereas the region behind the shock is an expansion zone which induces a favorable pressure gradient. The pressure gradients on both sides of the plate slowly decay to a zero value away from the tip. The mean-squared pressure fluctuations on the plate surface are shown in Fig. 5.7(b). On the higher speed stream side, the fluctuations are localized in a region close to the plate edge and decay away from it. On the lower speed side, the fluctuations do not decay to zero away from the plate edge. This is due to the acoustic waves travelling upstream in the subsonic portion of the mixing layer. The mixing layer thickness can be quantified in terms of vorticity thickness. The vorticity thickness is defined as,

∆U δω = (5.2) max ∂u˜ y ∂y

The vorticity thickness growth is shown in Fig. 5.8(a) for cases A1 and A2. Vorticity thickness growth rate normalized by growth rate of incompressible shear layer is found to compare well with other experimental studies as shown in Fig. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 79

−5 x 10 1.5

1 2 p

0.5

0 −10 −5 0 x

(a) (b)

wp ∂p Figure 5.7: (a) Streamwise pressure gradient ( 2 ) on the surface of splitter plate ρ2∆U ∂x p02 for case A1. (b) Mean square of pressure ﬂuctuations ( 2 )on the surface of plate ρ2∆U for case A1. Solid line: y > 0 and Dashed line: y < 0.

5.8(b). The early mixing layer is laminar and its thickness grows with square root of streamwise distance. Much downstream, the thickness grows linearly. The mixing layer in the heated case A2 can be seen to depart from the laminar growth behavior earlier than the unheated case A1. The thickness growth rate for case A2 was found to be slightly higher than that of case A1.

The mean velocity profiles are shown downstream of the plate in Figs. 5.9(a) and (b) for case A1. The profiles close to plate (x/wp . 41) contain a non zero wake component. Farther away, the wake component dies out and the velocity profile can be approximated as having a mixing layer hyperbolic tangent profile. It can be observed that farther downstream, the mixing layer growth becomes roughly linear.

After certain distance (x/wp & 200), the meanflow profiles evolve self-similarily as shown in Fig. 5.9(b). The mean velocity profiles for case A3, as shown in Figs. 5.10(a) and (b), have a prolonged wake component in the streamwise direction (upto about x/wp ∼ 73). The mean velocity profiles collapse after x/wp ∼ 245 when scaled with ∆U and transverse coordinate is scaled with local vorticity thickness. 80 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

40 1.2

1 30

0.8 inc ω δ δ

20 / δ 0.6

10 0.4

0 0.2 50 100 150 200 250 300 0 0.5 1 1.5 x/w M p c

(a) (b)

Figure 5.8: (a) Growth of mixing layer vorticity thickness as a function of distance from splitter plate tip in the downstream direction for cases case A1 (solid line) and case A2 (dashed line). (b) Variation of normalized growth rate with convective Mach number, Mc. : present calculation; N: Pantano and Sarkar (2002);: Debisschop and Bonnet (1993); F: Papamoschou and Roshko (1988); I: Samimy and Elliot (1990); −: Langley curve. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 81

20 8 (a) 6 (b) 10 4 2

y 0 y 0 −2 −10 −4 −6 −20 −8 0 0.5 1 1.5 0 0.5 1 1.5 U U mean mean

(a) (b)

Figure 5.9: Variation of mean velocity (Umean = u/e ∆U) in the transverse direction (y/δω(x)) for case A1. (a) Close to the plate. −: x/wp = 6; −−: x/wp = 15; − · −: x/wp = 24; ···: x/wp = 33; −4−: x/wp = 41. (b) and far from the plate in the self-similar region. −: x/wp = 209; −−: x/wp = 234; − · −: x/wp = 259; ···: x/wp = 284; −4−: x/wp = 318. 82 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

30 8 (a) (b) 6 20 4 10 2

y 0 y 0 −2 −10 −4 −20 −6 −30 −8 0.5 1 1.5 2 0.5 1 1.5 2 U U mean mean

(a) (b)

Figure 5.10: Variation of mean velocity (Umean = u/e ∆U) in the transverse direction (y/δω(x)) for case A3. (a) Close to the plate. Legend: −: x/wp = 17.5; −−: x/wp = 33; − · −: x/wp = 47; ···: x/wp = 60; −4−: x/wp = 73. (b) Far from the plate in the self-similar region. Legend: −: x/wp = 245; −−: x/wp = 293; − · −: x/wp = 342; ···: x/wp = 390; −4−: x/wp = 438. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 83

(a) (b)

Figure 5.11: Variation of mean streamwise velocity for case A1 in (a) and A2 in (b). Legend: −: x/wp = 2; −−: x/wp = 18; − · −: x/wp = 38.

5.2.3 Impulse response

The evolution of a two-dimensional disturbance placed in a parallel mean flow is analyzed here. The mean flow profiles are extracted at a fixed streamwise location and extruded in the streamwise periodic direction. The mean streamwise velocity profiles at the selected locations are shown in Fig. 5.11. It can be noted that the first location which is close to the splitter plate has a substantial wake component and a small layer with negative velocity. The initial disturbance is specified as a divergence free vortex. The wavepacket amplitude is computed as the integral of pressure in the transverse direction. The amplitude is plotted against the group velocity, vg = (x − xo)/t, for different mean flow profiles in Fig. 5.12(a)-(c) for case A2. Here xo = 50 is the location where the disturbance is initialized at t = 0. It can be seen by comparing Figs. 5.12 (a) and (d) that the growth rate of disturbance is higher for case A2 for the first streamwise location. The left edge 84 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

−1 −1 −1 10 10 10

−3 −3 −3 10 10 10 A A A

−5 −5 −5 10 10 10

−7 −7 −7 10 10 10 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 (x−x )/t (x−x )/t (x−x )/t o o o

(a) (b) (c)

−1 −1 −1 10 10 10

−3 −3 −3 10 10 10 A A A

−5 −5 −5 10 10 10

−7 −7 −7 10 10 10 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 (x−x )/t (x−x )/t (x−x )/t o o o

(d) (e) (f)

Figure 5.12: Variation of wavepacket amplitude with group velocity for cases A1 (a-c) and A2 (d-f). (a,d) x/wp = 2, (b,e) x/wp = 18 and (c,f) x/wp = 38. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 85

of the wavepacket in Figs. 5.12 (a) and (d) is very close to the group velocity of zero. This means that it is a very small positive group velocity and is not locally absolutely unstable. It is convectively unstable. This means that the wake-defect and the heating of the high-speed stream are not strong enough to make it absolutely unstable. For the second streamwise location the range of group velocities shifts to higher positive values as shown in Figs. 5.12(b) and (e) because of the diminishing wake-defect.

5.2.4 Turbulence statistics

As noticeable from the contour plots in Figs. 5.3, 5.4 & 5.5, the two boundary layers after merging, remain laminar until a certain distance downstream from the plate edge at x = 0, transition and then become turbulent. We now quantify the Favre- averaged (denoted as ˜·) Reynolds stresses normalized by ∆U 2, denoted in general by σ and deﬁned as,

2 002 2 2 002 2 2 002 2 00 00 2 σu = uf /∆U , σv = vf /∆U , σw = wg/∆U , σuv = −ugv /∆U (5.3)

Turbulent mass fluxes in the streamwise and transverse direction are defined in Eq. 5.4. These fluxes also relate the Favre-averaged and Reynolds-averaged (denoted 0 0 0 as ·) velocities as uei = ui + ρ ui/ρ. Here, denotes fluctuations about the Reynolds- averaged mean and 00 denotes fluctuations about the Favre-averaged mean.

0 0 0 0 mu = ρ u /ρ2∆U, mv = ρ v /ρ2∆U (5.4)

The variation of Reynolds stresses along the lipline is shown in Fig. 5.13 for cases A1 and A2. The stresses are observed to ﬁrst grow and then decay to ﬁnally settle down to a constant value roughly half of their peak values. Their shape is the same for both the cases although the stresses start growing earlier for case A2. They become roughly constant after about x/wp ≥ 200 downstream for both cases A1 and A2. Clearly, the observed trend in terms of magnitude of various components of Reynolds 86 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0.06 0.06

0.05 0.05

0.04 0.04 2 2

σ 0.03 σ 0.03

0.02 0.02

0.01 0.01

0 0 50 100 150 200 250 300 50 100 150 200 250 300 x/w x/w p p

(a) (b)

Figure 5.13: Variation of Reynolds stresses along the lipline for (a) case A1 and (b) 2 2 2 A2. −: σu, − · −·: σw, −−: σv , ··· : σuv.

2 2 2 stresses is σu > σw > σv > σuv. The variation of Reynolds stresses along the lipline and the variation of peak values is compared in Fig. 5.14 for cases A1 and A2. It can be observed that the peak Reynolds stress values are higher for the heated case in the initial transitional region. After an initial rise, the stresses finally reach roughly the same constant levels for both A1 and A2. Initially, the lipline values are differ from the peak values but much downstream the lipline and the peak values are roughly the same. Similar to behavior of Reynolds stress profiles, mean square pressure and density fluctuations also show a increase to peak value and then gradual relaxation to a constant peak self-similar value. It can be seen that the density fluctuations in the heated case A2 are higher than those of isothermal case A1 for all streamwise locations. The pressure fluctuations are higher for case A1 than case A2. The transverse profiles of Reynolds stresses taken at different streamwise locations are shown in Fig. 5.16 for case A1, Fig. 5.17 for case A2 and Fig. 5.19 for case A3. The profiles are observed to collapse when the transverse coordinate is normalized by the local mixing layer thickness. Thus, the region where intensities become roughly constant in the streamwise direction is where the mixing layer mean flow evolves self 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 87

0.06 0.05

0.05 0.04 0.04 0.03 2 u 2 v

σ 0.03 σ 0.02 0.02

0.01 0.01

0 0 50 100 150 200 250 300 50 100 150 200 250 300 x/w x/w p p

(a) (b)

0.05 0.025

0.04 0.02

0.03 0.015 2 w uv σ σ 0.02 0.01

0.01 0.005

0 0 50 100 150 200 250 300 50 100 150 200 250 300 x/w x/w p p

Figure 5.14: Comparison of streamwise variation of Reynolds stresses for cases A1 2 2 2 and A2. (a) σu, (b) σv , (c) σw and (d) σuv. Line with symbols: case A2. Solid line: variation along lip line; dashed line: peak stresses. 88 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0.2 0.1

0.08 0.15 2 U 2 0.06 ρ ∆ / 2

0.1 ρ / rms ρ

rms 0.04 p 0.05 0.02

0 0 50 100 150 200 250 300 50 100 150 200 250 300 x/w x/w p p

(a) (b)

Figure 5.15: Comparison of streamwise variation of r.m.s levels of density in (a) and pressure in (b) for cases A1 and A2. Line with symbols: case A2. Solid line: variation along lip line; dashed line: peak stresses.

dδω dδθ 2 2 2 Case dx dx σu σv σw σuv A1 0.1 0.028 0.023 0.012 0.017 0.009 A2 0.11 0.031 0.023 0.014 0.017 0.008 A3 0.06 0.01 0.027 0.012 0.015 0.009 A4 0.07 0.021 0.023 0.012 0.015 0.009 A5 0.09 0.022 0.023 0.012 0.015 0.008

Table 5.4: Comparison of growth rate and self-similar turbulent intensities for cases A1 through A5 similarily. The self-similar intensity profiles for case A3 are observed to agree well with those obtained from experimental study by Samimy and Elliot (1990). The peak intensity values for case A1 agree well with values observed in experiments at a similar convective Mach number as shown in Fig. 5.18. A comparison of growth rate of shear layers and the turbulent intensities for the five cases is shown in Table below. The growth rates reported in the table are the slopes of least-squares fit to the vorticity and momentum thicknesses. The growth rate of case A3 is much lesser than A1. Between the five cases, self-similar turbulent intensities show little variation. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 89

0.03 0.02

0.025 0.015 0.02 2 u 2 v

σ 0.015 σ 0.01

0.01 0.005 0.005

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

(a) (b)

0.02 0.01

0.008 0.015

0.006 2 w uv

σ 0.01 σ 0.004

0.005 0.002

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

−3 −3 x 10 x 10 2 2

1 1 u v

0 m 0 −m

−1 −1

−2 −2 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

(e) (f)

Figure 5.16: Self similar proﬁles of turbulent intensities for case A1 plotted as function 2 2 2 of y/δω(x). (a) σu, (b) σv , (c) σw, (d) σuv, (e) mu and (f) mv as deﬁned in Eqs. 5.3 and 5.4. −: x/wp = 266; −−: x/wp = 287; − · −: x/wp = 309; ··· x/wp = 330. 90 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0.025 0.02

0.02 0.015

0.015 2 u 2 v

σ σ 0.01 0.01

0.005 0.005

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

(a) (b)

0.02 0.01

0.008 0.015

0.006 2 w uv

σ 0.01 σ 0.004

0.005 0.002

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

0.01 0.01

0.008 0.008

0.006 0.006 u v m −m 0.004 0.004

0.002 0.002

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

(e) (f)

Figure 5.17: Self similar proﬁles of turbulent intensities for case A1 plotted as function 2 2 2 of y/δω(x). (a) σu, (b) σv , (c) σw, (d) σuv, (e) mu and (f) mv as deﬁned in Eqs. 5.3 and 5.4. −: x/wp = 266; −−: x/wp = 287; − · −: x/wp = 309; ··· x/wp = 330. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 91

0.3 0.3

0.2 0.2 v u σ σ

0.1 0.1

0 0 0 0.5 1 0 0.5 1 M M c c

(a) (b)

0.02

0.015

uv 0.01 σ

0.005

0 0 0.5 1 M c

(c)

Figure 5.18: Comparison of peak self-similar Reynolds stresses with experimental data for case A1. (a) σu, (b) σv and (c) σuv. ◦: Goebel and Dutton (1991); : Samimy and Elliot (1990); .+: Urban and Mungal (2001); O: Gruber et al. (1993); ×: Oster and Wygnanski (1982); N: Present study. 92 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0.04 0.02

0.03 0.015 2 u 2 v

σ 0.02 σ 0.01

0.01 0.005

0 0 −2 −1 0 1 2 −2 −1 0 1 2 y y

(a) (b)

0.02 0.01

0.008 0.015

0.006 2 w uv

σ 0.01 σ 0.004

0.005 0.002

0 0 −2 −1 0 1 2 −2 −1 0 1 2 y y

Figure 5.19: Comparison of self similar proﬁles of Reynolds stresses for Case A3 plotted as function of y/δω(x) to experimental data of Samimy and Elliot (1990) (shown in symbols 4). (a) uf002/∆U 2 (b) vf002/∆U 2 (c) wg002/∆U 2 (d) ug00v00/∆U 2. Legend: −: x/wp = 342; −−: x/wp = 366; − · −: x/wp = 390; ··· x/wp = 415. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 93

The correlation functions of the streamwise velocity ﬂuctuations calculated in t x z the temporal, streamwise and spanwise directions are denoted by Ruu, Ruu and Ruu respectively and deﬁned as,

t 00 00 x 00 00 Ruu(τ, x) ≡ hu (x, t + τ)u (x, t)i, Ruu(ξ, x) ≡ hu (x + e1ξ, t)u (x, t)i and (5.5a) z 00 00 Ruu(ξ, x) ≡ hu (x + e3ξ, t)u (x, t)i

Here, e1 and e3 refer to the unit vectors in streamwise and spanwise directions.

The corresponding spectra denoted by Suu are obtained by taking Fourier transform x t in ξ or τ. The correlation functions in the streamwise (Ruu), time (Ruu) and spanwise z (Ruu) directions are shown for diﬀerent streamwise locations in Fig. 5.20 for case A1 and Fig. 5.21 for case A2. The corresponding spectra are shown in Figs. 5.22, 5.23 and 5.24. For upstream locations in the early transition region, the spectra shown in

Figs. 5.22(a), 5.24(a, b) for case A1 clearly have a peak corresponding to kxwp ∼ 0.7,

St ∼ 0.8 and kzwp ∼ 0.2. These peaks are related to the dominant mode of the Kelvin-Helmholtz instability of the initial shear layer. Such peaks cannot be seen in spectra corresponding to the other case A2, but that is most likely because the probe locations are farther downstream. The shear layer in A2 transitions earlier than in A1, so that there are already a broad range of frequencies present in the velocity signal downstream at the probe locations. There is a increase in the widths of the correlation functions progressively in the downstream direction. The high- frequency signal in the early shear layer has a smaller integral scale but its spectrum extends to higher frequencies or wavenumbers. The peaky character of the early transtional shear layer is gradually lost with a gradual appearance of the −5/3 slope region characteristic of turbulent ﬂows. This behavior can be clearly observed in Figs. 5.22(a)-(f). A similar trend is present in the temporal and spanwise correlations where the integral scales are gradually observed to increase. The turbulent character of the shear layer can be assessed by the appearance of inertial subrange and a gradual increase in wavenumber/frequency range over which it is present. 94 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

(a) 0.8 0.8

0.6 0.6

x uu 0.4 t uu 0.4 R R

0.2 0.2

0 0

−0.2 −0.2 100 200 300 400 0 50 100 ξ τ

(a) (b)

0.8

0.6

0.4 z uu R 0.2

−0.2

−0.4 0 10 20 30 40 ξ

(c)

x Figure 5.20: (a) Two point axial correlation of the streamwise velocity, Ruu, (b) t z Temporal correlation, Ruu, (c) Spanwise correlation, Ruu, for case A1 at various streamwise locations. Legend: −−: x/wp = 355; −: x/wp = 300; −4−: x/wp = 250 ;− · −: x/wp = 190; −−: x/wp = 125; ··· : x/wp = 80. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 95

0.8 (a) 0.8

0.6 0.6

x uu 0.4 t uu 0.4 R R

0.2 0.2

0 0

−0.2 −0.2 100 200 300 400 0 50 100 150 200 ξ τ

(a) (b)

0.8

0.6

0.4 z uu R 0.2

−0.2

−0.4 0 10 20 30 40 ξ

(c)

x Figure 5.21: (a) Two point axial correlation of the streamwise velocity, Ruu, (b) t z Temporal correlation, Ruu, (c) Spanwise correlation, Ruu, for case A2 at various streamwise locations. Legend: same as in Fig. 5.20. 96 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

−2 −2 10 10

−4 −4 10 10 x uu x uu S S

−6 −6 10 10

−1 0 −1 0 10 10 10 10 kw kw p p

(a) x/wp = 80 (b) x/wp = 125

−2 −2 10 10

−4 −4 10 10 x uu x uu S S

−6 −6 10 10

−1 0 −1 0 10 10 10 10 kw kw p p

−2 −2 10 10

−4 −4 10 10 x uu x uu S S

−6 −6 10 10

−1 0 −1 0 10 10 10 10 kw kw p p

(e) x/wp = 300 (f) x/wp = 355

Figure 5.22: Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A1 at various streamwise locations. 5.2. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 97

−2 −2 10 10

−4 −4 10 10 x uu x uu S S

−6 −6 10 10

−1 0 −1 0 10 10 10 10 kw kw p p

(a) x/wp = 80 (b) x/wp = 125

−2 −2 10 10

−4 −4 10 10 x uu x uu S S

−6 −6 10 10

−1 0 −1 0 10 10 10 10 kw kw p p

−2 −2 10 10

−4 −4 10 10 x uu x uu S S

−6 −6 10 10

−1 0 −1 0 10 10 10 10 kw kw p p

(e) x/wp = 300 (f) x/wp = 355

Figure 5.23: Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A2 at various streamwise locations. 98 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0 0 10 10

−1 10

−2 10 −2 z uu t uu 10 S S

−3 −4 10 10

−4 10 −2 −1 −2 −1 10 10 10 10 kw St p

(a) (b)

0 0 10 10

−1 10 −2 10 −2 z uu t uu 10 S S

−3 −4 10 10

−4 10 −2 −1 −2 −1 10 10 10 10 kw St p

Figure 5.24: (a) Spectra obtained from temporal correlations in Fig. 5.20(b) for case A1; (b) Spectra obtained from spanwise correlations in Fig. 5.20(c) for case A1; (c) Spectra obtained from temporal correlations in Fig. 5.21(b) for case A2; (d) Spectra obtained from spanwise correlations in Fig. 5.21(c) for case A2. Legend: −−: x/wp = 80; − · −·: x/wp = 190; −: x/wp = 355. 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 99

Figure 5.25: Grid used for the simulations. Shaded region shown is the sponge used at the domain boundaries.

5.3 Cases with incoming turbulent boundary layers

5.3.1 Computational parameters

The domain used for simulation cases A4 and A5 is shown in Fig. 5.25. In addition to the two meshes used for the cases A1, A2 and A3, here a third mesh is used for turbulent boundary layer simulation. The instantaneous Y-Z slice of data taken from a section from this simulation is then fed as inﬂow to the body-ﬁtted C-mesh. The turbulent boundary layer imposes strict mesh resolution requirements, par- ticularly, in the spanwise direction. A much shorter domain but higher resolution mesh is used in all the directions compared to the cases when the incoming boundary layer is laminar. A summary of domain sizes, number of points and the resolution in the three directions is given in Table 4.

The simulation was ﬁrst run over a time of the order 200wp/∆U to let the ﬂow 100 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

Mesh min(∆x, ∆y, ∆z) max(∆x, ∆y, ∆z) ∆t Lx,Ly,Lz Nx,Ny,Nz Boundary Layer (23, 0.6, 12)+ (23, 75, 12)+ 0.002 28, 6.5, 10 480, 160, 256 Body-ﬁtted mesh (2, 0.6, 12)+ (15, 75, 12)+ 0.002 12, 6.5, 10 720, 160, 256 Background mesh (0.07, 0.05, 0.08) (0.4, 0.7, 0.08) 0.002 120, 100, 10 720, 513, 128 96.4mil

Table 5.5: Mesh parameters for cases A4 and A5. The numbers with superscript + are in plus-coordinates based on skin friction coefficient at the inlet to body-fitted mesh. Otherwise reference length used to non-dimensionalize spatial dimensions is the plate thickness, wp. ∆, L and N refer to the mesh spacing, domain length and number of points in each direction respectively. For the boundary layer mesh, the x mesh spacing is constant, minimum y-spacing is at the first y grid point and maximum at the last. For the body-fitted mesh, the x and y directions are the wall-tangent and wall-normal directions, respectively. The minimum spacing is at the trailing edge and maximum at the inlet and away from the wall. The minimum spacing location for the background mesh is at its interface to the body fitted mesh. settle down before the statistics are collected. The statistics were collected over a time horizon of 80wp/∆U. A numerical sponge was used on the boundary of the background computational mesh to damp the disturbances as shown in Fig. 5.2(a).

The sponge width was 45wp on the outflow boundary, 5wp on the top and bottom boundaries and 5wp at the inflow boundary. A high-order numerical filter was used to damp the unphysical high-frequency numerical disturbances as described in Na- garajan (2004). The filter coefficient was set to be α = 0.48.

5.3.2 Mean ﬂow

The flow for case A4 and A5 consists of high-speed turbulent boundary layer in the upper stream and a lower speed laminar boundary layer on the lower side of the plate. The instantaneous density contours overlayed with vorticity magnitude contours are depicted in Fig. 5.26 for case A4. There is a visible signature associated with the small adjustment of turbulent boundary layer data to the trailing-edge body-fitted mesh which supports a finer x-mesh. The sound associated with this adjustment appears to be limited to the region downstream of the Mach line emanating from start of this mesh. There is also a small pressure adjustment as the turbulent boundary layer 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 101

Figure 5.26: Visualization of density contours overlayed with vorticity magnitude contours in the mid X-Y plane for Case A4. The density contours are plotted between ρ2 ±0.05ρ2 and vorticity (normalized by ∆U/wp where wp is the plate width) contours between 0 and 5. detaches from the splitter plate and sets up a weak shock wave. Close to the splitter plate trailing edge, sound radiation into the lower speed stream is observable. This radiation is directed, both dowstream and upstream in the low speed stream and appears to emanate from the trailing-edge region. This radiation is associated with the noise generated by the passage of the turbulent boundary layer over the splitter plate tip. The noise radiated by the turbulent mixing associated with the shear layer appears to be much weaker and is not visible at the contour levels plotted here. The linear spreading of the turbulent shear layer can be observed from the vorticity magnitude contours. The streamwise elongated structures present in the incoming turbulent boundary layer flow over the plate reorganize themselves into finer structures as they move past the edge. This can be observed in the spanwise view shown in Fig. 5.27. As shown in Fig. 5.28 where isosurfaces of the second invariant of velocity gradient tensor are plotted, the fine hairpin-type structures present in the early turbulent boundary layer aggregate to form larger structures. There is a substantial decrease in vorticity 102 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

Figure 5.27: Visualization of vorticity magnitude contours in the spanwise plane at y = 0.51 (located just above the splitter plate in the high speed stream) for Case A4. The location of trailing edge is at (0, 0)

magnitude past the splitter plate trailing edge as is notable in Fig. 5.27.

The velocity streamtraces for case A4 are shown in Fig. 5.29(a). The vortex at the splitter plate edge is smaller compared to the laminar boundary layer case (see Fig. 5.6(a)) and shifted upwards towards the centre of the plate edge. The vorticity thickness growth for cases A4 and A5 are shown in Fig. 5.29(b). The shear layer of the heated case A5 has a higher thickness growth rate than the unheated case which was also previously observed for the laminar boundary layer cases, A1 and A2. The normalized values of the growth rates based on both are comparable to values observed in experiments in Fig. 5.8(c). The mean velocity proﬁles close to the plate edge and far downstream in the self similar region are shown in Fig. 5.30. The wake component in the mean velocity vanishes in a much shorter distance from the plate edge, about x/wp ∼ 5. As a result, the self-similarity in mean velocity is attained much earlier compared to cases with laminar boundary layers (A1-A3). 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 103

Figure 5.28: Visualization of isosurfaces of second invariant of velocity gradient tensor for case A4. Plotted for invariant value, Q = 0.25.

6 ω

δ 4

0 10 20 30 40 50 60 70 x/w p

(a) (b)

Figure 5.29: (a) Velocity streamtraces for case A4 close to the plate. (b) Shear layer vorticity thickness for case A4 (solid line) and A5 (dashed line). 104 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

8 8 (a) (b) 6 6 4 4 2 2

y 0 y 0 −2 −2 −4 −4 −6 −6 −8 −8 0 0.5 1 1.5 0 0.5 1 1.5 U U mean mean

(a) (b)

Figure 5.30: Variation of mean velocity (Umean = u/e ∆U) in the transverse direction ∗ (y = y /δω(x)) for case A4. (a) Close to the plate (on left) −: x/wp = 2; −−: x/wp = 3.6; − · −: x/wp = 5.2; ···: x/wp = 7; −4−: x/wp = 8.75. (b) and far from the plate in the self-similar region (right) −: x/wp = 16.7; −−: x/wp = 21.2; − · −: x/wp = 26; ···: x/wp = 31.2; −4−: x/wp = 37. 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 105

30 3

25 2.5

20 2 + , w + VD 15 +

U 1.5 , v + u 10 1

5 0.5 0 0 1 2 3 0 10 10 10 10 0 0.2 0.4 0.6 0.8 1 y+ y/δ

(a) (b)

Figure 5.31: Proﬁles of mean Van-Driest transformed streamwise velocity in (a) and Reynolds stresses in (b) in y+ coordinate for the simulated turbulent boundary layer. (a) Blue line: from present computations, dashed lines: reference lines u+ = y+ and u+ = 2.5 lny+ + 5.5; (b) Legend: −−: u+, − · −·: v+ and −: w+. Symbols: DNS data of Spalart (1988) Lines: from present computations.

5.3.3 Turbulence statistics

The statistics for the incoming turbulent boundary layer are shown in Fig. 5.31. The Van-Driest transformed mean velocity shows a distinct log law which can be expected at a Reynolds number Reδθ = 800. The turbulent intensities are found to be in agreement with the DNS data of Spalart (1988). For the Reynolds stresses to reach self-similarity, their profiles along the lipline which are close to their peak values, should reach a constant value when normalized by ∆U. This is shown to be the case in Fig. 5.32 where the Reynolds stresses gradually rise from their turbulent boundary layer values to the self-similar mixing layer values. The transverse profiles of the turbulent intensities at a few streamwise locations are shown in Fig. 5.33 and show a good collapse indicating self-similarity after the transverse coordinate is normalized by the local vorticity thickness. The correlation functions in the streamwise, spanwise and time directions, as defined in Eq. 5.5, are shown in Figs. 5.34(a)-(c) and 5.35(a)-(c) for a few streamwise 106 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0.03 0.03

0.025 0.025

0.02 0.02 2 2

σ 0.015 σ 0.015

0.01 0.01

0.005 0.005

0 0 10 20 30 40 50 10 20 30 40 50 x/w x/w p p

(a) (b)

Figure 5.32: Variation of turbulent intensities along the lipline for (a) case A4 and (b) case A5. Legend: −: uf002/∆U 2, −−: vf002/∆U 2, − · −·: wg002/∆U 2 and ··· : ug00v00/∆U 2.

locations along the lipline for cases A4 and A5. The corresponding spectra are shown in Figs. 5.36 and 5.37. As the shear layer grows in the downstream direction the correlation length scales also grow. This trend is evident in Fig. 5.34(a). The spectra have a distinct inertial subrange region following a −5/3 power of the wave number. The spectra indicate a frequency shift to lower wavenumber in the downstream direction. This is consistent with the growth of the shear layer. Spanwise and temporal correlations for case A4 are shown in Fig. 5.34(b) & (c) and 5.35(b) & (c) for cases A4 and A5 respectively. The peaky character of spectra associated with the Kelvin- Helmholtz instability of the early shear layer with initially laminar boundary layers, as seen in Figs. 5.22(a) and 5.24(a), cannot be observed in Fig. 5.36. The spanwise correlation length increases significantly with downstream distance. The last station shown (x/wp = 58 in Fig. 5.34(b)) is perhaps beginning to show the effect of domain size. To compute this flow over a larger streamwise domain will also require that the spanwise domain size be increased. 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 107

0.025 0.02

0.02 0.015

0.015 2 u 2 v

σ σ 0.01 0.01

0.005 0.005

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

(a) (b)

0.02 0.01

0.008 0.015

0.006 2 w uv

σ 0.01 σ 0.004

0.005 0.002

0 0 −4 −2 0 2 4 −4 −2 0 2 4 y/δ y/δ ω ω

Figure 5.33: Self similar proﬁles of Reynolds stress components for case A4 plotted as function of tranvserse coordinate normalized by local vorticity thickness, y/δω(x). 002 2 002 2 002 2 00 00 2 (a) uf /∆U (b) vf /∆U (c) wg/∆U (d) ugv /∆U . Legend: −: x/wp = 25.3; −−: x/wp = 32; − · −: x/wp = 39.2; ··· x/wp = 47.2. 108 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

0.8 0.8 0.6 0.6 0.4 x uu 0.4 z uu R R 0.2 0.2 0

0 −0.2

−0.2 −0.4 10 20 30 40 50 60 70 0 1 2 3 4 5 ξ ξ

(a) (b)

0.8

0.6

t uu 0.4 R

0.2

−0.2 0 10 20 30 40 τ

(c)

x z Figure 5.34: (a) Axial correlation, Ruu (b) Spanwise correlation, Ruu, and (b) Tem- t poral correlation of the streamwise velocity, Ruu, for case A4 at various streamwise locations. Legend: ··· : x/wp = 5.75; −−: x/wp = 12; − · −·: x/wp = 19; −−: x/wp = 27; −: x/wp = 41; −4−: x/wp = 58. 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 109

0.8 0.8 0.6 0.6 0.4 x uu 0.4 z uu R R 0.2 0.2 0

0 −0.2

−0.2 −0.4 10 20 30 40 50 60 70 0 1 2 3 4 5 ξ ξ

(a) (b)

0.8

0.6

t uu 0.4 R

0.2

−0.2 0 20 40 60 τ

(c)

x z Figure 5.35: (a) Axial correlation, Ruu (b) Spanwise correlation, Ruu, and (b) Tem- t poral correlation of the streamwise velocity, Ruu, for case A5 at various streamwise locations. Legend: same as in Fig. 5.34 110 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

−2 −2 10 10

−4 −4 x uu 10 x uu 10 S S

−6 −6 10 −1 0 1 10 −1 0 1 10 10 10 10 10 10 kw kw p p

(a) x/wp = 5.75 (b) x/wp = 12

−2 −2 10 10

−4 −4 x uu 10 x uu 10 S S

−6 −6 10 −1 0 1 10 −1 0 1 10 10 10 10 10 10 kw kw p p

−2 −2 10 10

−4 −4 x uu 10 x uu 10 S S

−6 −6 10 −1 0 1 10 −1 0 1 10 10 10 10 10 10 kw kw p p

(e) x/wp = 41 (f) x/wp = 58

Figure 5.36: Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A4 at various streamwise locations. 5.3. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 111

−2 −2 10 10

−4 −4 x uu 10 x uu 10 S S

−6 −6 10 −1 0 1 10 −1 0 1 10 10 10 10 10 10 kw kw p p

(a) x/wp = 5.75 (b) x/wp = 12

−2 −2 10 10

−4 −4 x uu 10 x uu 10 S S

−6 −6 10 −1 0 1 10 −1 0 1 10 10 10 10 10 10 kw kw p p

−2 −2 10 10

−4 −4 x uu 10 x uu 10 S S

−6 −6 10 −1 0 1 10 −1 0 1 10 10 10 10 10 10 kw kw p p

(e) x/wp = 41 (f) x/wp = 58

Figure 5.37: Spectra obtained from axial correlation function of streamwise velocity, x Ruu, for case A5 at various streamwise locations. 112 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

5.4 Summary

The present results demonstrate the viability of simulating spatially developing, compressible mixing layers using large-eddy simulation. The high computational cost for direct numerical simulation (DNS) is overcome by using an overset grid approach. The near-wall flow region is fully resolved whereas and the mesh in region away from the wall is progressively coarsened. In this way, the boundary layer turbulence in the high speed side is physically represented. The simulations require a large streamwise domain to accomodate self-similar flow development. As the correlation lengths in the transverse and spanwise direction also grow going downstream, it automatically means that a large transverse and spanwise domain is also needed with increasing streamwise extent. In the self-similar region, the mean flow, growth rate of shear layer and the turbulent intensities were found to be in good agreement with available experimental data. In cases A1, A2 and A3 with laminar boundary layers, the region of transition from laminar boundary to turbulent mixing layer is marked by peaks in turbulent intensities which gradually relax to self-similar values, roughly half of the peak values. A similar trend is observed in profiles of mean square pressure fluctuations. Sound radiation is observed to originate from the transition region. The development distance to achieve self-similarity in the mean velocity profile is found to be shorter for cases with turbulent boundary layers. Development distance was found to be shorter for cases A1 and A2 than A3. An extended region of wake effect and presence of two-dimensional rollers prolongs the distance for breakdown to turbulence in case A3. Two-point correlations with streamwise, spanwise, and temporal separation were analyzed for the turbulent mixing layers simulated here. These further confirm the self-similar development of the flow and provide reassurance that the spanwise domain was sufficiently large. A Kolmogorov inertial subrange is observed for energy spectra in terms of the streamwise wavenumber, and with temporal frequency. The sound radiation was strongly affected by the state of the boundary layer exiting the splitter tip: for laminar boundary layers the quasi-two-dimensional roll up and pairing and the breakdown to three-dimensional flow dominated the sound generation. 5.4. SUMMARY 113

For turbulent boundary layers the flow adjusted from the wake affected regime to the shear-layer behavior without any major ‘instability’ resulting in a significantly weaker acoustic near-field which resembled the trailing-edge noise radiation. Jet LES predictions have been found to be strongly influenced by inflow conditions. In earlier studies of jets, such as those by Bodony and Lele (2005), the usual way was to prescribe a mean shear layer profile superimposed with disturbances in order to induce an early development of turbulence in the shear layers such that the downstream properties in terms of mean and fluctuating quantities were consistent with the experimental data. In order to avoid ambiguity in specifying inflow disturbances, it is suggested that the nozzle be included inside the computational domain. The boundary layers near the nozzle exit in an experimental setting are thin with a momentum thickness, δθ ∼ 0.001D, where D is the jet diameter. They could be laminar or turbulent. It is computationally expensive to resolve such thin shear layers not to mention the turbulent ones which require increased azimuthal resolution as well. Assuming a hypothetical jet of diameter D around the mixing layers in the present study, the aforementioned relationship for momentum thickness and jet diameter yields a total domain size of ∼ 4.2D for the laminar cases and 1.2D for turbulent cases in the streamwise direction. The number of grid points used in the present study (∼ 147 million for laminar and ∼ 100 million for turbulent cases) reveals how many grid points may be needed to simulate a jet with comparable resolution. A jet would require a much larger domain in all three directions in order to accomodate the large scale structures. The potential core ends at a distance of roughly 5 to 10 diameters from the nozzle and the domain size to include the noise producing region downstream of the potential core end may extend upto 20D in the streamwise direction. If the early thin shear layers are not well resolved, it is incumbent on the SGS model to accurately represent the dynamics of the unresolved scales. The laminar to turbulent transition in early shear layers of jets has been observed to be controlled by choice of the numerical method and the SGS model. The observed potential core lengths in simulations are almost always smaller than those in experiments. Bogey and Bailly (2010) report that thinner shear layers transition early and have a slower 114 CHAPTER 5. TURBULENT MIXING LAYER COMPUTATIONS

growth rate which lead to longer potential core lengths. In the present study, we observe that a mixing layer which forms from an initially turbulent boundary layer has a lower growth rate than the one with an initially laminar boundary layer. This is consistent with experimental observations of Karasso and Mungal (1996). This suggests that jets with initially turbulent boundary layers will lead to still longer potential core lengths and points to the need to match the experimental boundary layer state for favorable comparison with experimental data. The result from the current study that heating increases the growth rate of the shear layer whether the initial boundary layer is laminar or turbulent implies a reduction in potential core lengths with heating. Indeed, this trend has been reproduced in both simulations and experiments. Another eﬀect of boundary layer state, laminar or turbulent, is on the Reynolds stress variation along the lipline. An initial turbulent boundary layer leads to a gradual increase of Reynolds stresses from values that correspond to boundary layer to those of mixing layer. The mixing layer with initially laminar boundary layers, instead, have an overshoot before relaxing to the mixing layer values. This explains observation in the study by Mendez et al. (2010) who report higher ﬂuctuations of streamwise velocity compared to experiments. Chapter 6

Sound generation in turbulent mixing layers

Analysis of sound sources in turbulent mixing layers is presented in this chapter. The LES database for configurations described in the previous chapter are used here. Sound pressure signals obtained from both from LES computations directly and acoustic analogy (outlined in Section 3.2) predictions using source terms obtained from LES are analyzed. Effects of heating the higher speed stream and the state of initial boundary layer on the near-field and far-field sound characteristics are elaborated upon. Connections to the problem of jet noise are made.

6.1 Cases with incoming laminar boundary layers

LES data obtained for cases A1 and A2 (see Tables. 5.1 and 5.2) is analyzed in this section. About 600 snapshots for case A1 and 500 snapshots for case A2 of entire ﬂow

ﬁeld data are saved every 100 time steps. The sampling rate is fswp/∆U = 1/0.35 and the total period over which data is saved is T ∆U/wp = 210 for case A1 and

T ∆U/wp = 175 for case A2.

115 116 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

6.1.1 Near-ﬁeld spectra

The visualizations of instantaneous density (in gray scale) and vorticity magnitude (in color) in Fig. 5.3 (for case A1) of the previous chapter clearly showed sound radiation into the slower stream. The dominant radiation was observed visually to occur between 20◦ and 40◦ angle. The sound waves were found to mainly originate

from the region roughly between 30wp and 200wp downstream from the splitter plate edge. Clearly, this region corresponds to the transitional region of the mixing layer (see for example Fig. 5.13 in the previous chapter) before it eventually behaves self- similarly. The pressure fluctuations in the transitional region are predominantly hydrodynamic fluctuations. Sound is generated as a result of unsteady events in the shear layer. Away from the shear layer, the hydrodynamic fluctuations decay and the pressure fluctuations are associated with sound waves. The spectral content of the pressure fluctuations at some streamwise locations along the lipline can be observed in Fig. 6.1 for case A1 and Fig. 6.2 for case A2. An ideal mixing layer on its own would keep growing unabated. Mixing layer simulations, such as the present one, would be constrained by computational cost and would require that the mixing layer domain be truncated at some point. The finite domain size would thus limit the accuracy to which a true shear layer can be simulated. At some point simulation data would be corrupted at length scales of the order of domain size. In the present case, the spanwise domain size imposed the most severe restriction. It can be observed that the decay rate of spanwise correlation functions of pressure fluctuations (shown in Figs. 6.1(b) and 6.2(b)) does not change much after about x/wp ∼ 250. The spanwise domain size is about four times the spanwise length scale at this location if the point of zero crossing is taken to be a crude measure of the length scale. Given that at least four length scales be contained in a given domain size, we would expect that beyond this point, the shear layer simulation would contain artificial effects of limited domain size and associated imposed numerical boundary conditions. Henceforth, we restrict our attention to those spanwise length scales found upto 6.1. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 117

0.8 0.8

0.6 0.6 0.4 t pp 0.4 z pp R R 0.2 0.2 0 0 −0.2 −0.2 −0.4 0 50 100 0 10 20 30 40 τ ξ

(a) Temporal correlations for case A1 (b) Spanwise correlations for case A1

−1 10

−2 10

−3 z uu t pp 10 S S −4 10

−5 −6 10 10 −2 −1 −2 −1 10 10 10 10 kw St p

Figure 6.1: (a) Two point temporal correlation (b) Two point spanwise correlation of pressure for case A1. (c) Temporal spectra (d) Spanwise spectra of pressure ﬂuc- tuations for case A1. −−: x/wp = 355; −: x/wp = 300; −4−: x/wp = 250 ;− · −: x/wp = 190; −−: x/wp = 125; ··· : x/wp = 80. 118 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

0.8 0.8

0.6 0.6 0.4 t pp 0.4 z pp R R 0.2 0.2 0 0 −0.2 −0.2 −0.4 0 50 100 150 0 10 20 30 40 τ ξ

(a) (b)

−1 10

−2 10

−3 z uu t pp 10 S S −4 10

−5 −6 10 10 −2 −1 −2 −1 10 10 10 10 kw St p

Figure 6.2: (a) Two point temporal correlation (b) Two point spanwise correlation of pressure for case A2 (c) Temporal spectra (d) Spanwise spectra of pressure ﬂuctu- ations for case A2. Legend: same as Fig. 6.1. 6.1. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 119

x/wp ∼ 250. Given that the shear layer vorticity thickness at this point is approximately δω/wp ∼ 20 (or momentum thickness δθ/wp ∼ 5) and the characteristic instability frequency of shear layer is given by Stθ = fδθ/U = 0.032 (Ho and Huerre,

1984), where U is the average velocity of the two streams and δθ is the local momentum thickness, our focus would be on frequencies in the range St = fwp/∆U ≥ 0.005 for far-field acoustic predictions shown in the next section. For quantitative predictions, statistics of pressure field are presented. The important quantity of interest that is commonly used to quantify loudness of sound is the overall sound pressure level (OASPL) expressed in decibel (dB) and is defined as,

p02 OASP L = 10 log10 2 (6.1) pref

Human perception of sound strongly depends on frequency content of sound signal. Sound pressure level (SPL) expressed in decibel (dB) per sampling frequency (Hz) characterizes the spectral content of sound and is deﬁned as,

Spp(ω) SPL(ω) = 10 log10 2 (6.2) pref

t where Spp (same as Spp) is the power spectral density of pressure fluctuations and the reference pressure used is pref = 20µP a. Pressure spectra shown in Fig. 6.3 are for observer locations in the near-field yet away from the shear layer where hydrodynamical fluctuations would have decayed significantly. Their positions relative to the shear layer are marked in Fig. 6.3(c). There is a clear systematic increase in the pressure amplitude at all frequencies going from observer 1 to observer 5 of Fig. 6.3(c). This is because the later observers are located in the range of angles where sound radiation peaks. Further down, the amplification at lower frequencies can be observed which is most likely due to frequency dependent directivity of sound radiation. The increase in pressure amplitudes, from observer 1 location to observer 4 location is more rapid for case A1 than for case A2 as seen in Figs. 6.3(a) and (b). Furthermore, the amplitude of pressure fluctuations for case A2 are lower than those for case A1. This is a clear effect of reduction of 120 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

near-field pressure fluctuations with heating of higher-speed stream. The far-field sound prediction method has been verified by comparing the acoustic analogy predicted spectra with the spectra obtained directly from simulation data at observer locations 1, 2 and 3 (as shown in Fig. 6.3(c)) for case A1. The good agreement between the two provides assurance of reliable prediction of far-field spectra using the acoustic analogy method.

6.1.2 Far-ﬁeld spectra

Noise radiation directivity is shown in Fig. 6.6 in terms of OASPLs for cases A1 and A2. In presenting the OASPL data, the observer locations are ﬁxed at a common

radial distance of R = 6500wp from the splitter plate edge but at diﬀerent angles (θ) to the lipline as shown in Fig. 6.5. The peak radiation is observed to occur at roughly 30 degrees. The peak is slightly broader for the heated case. The noise levels are observed to decrease away from the peak radiation angle. The sound pressure spectra at observer locations at same distance from the splitter plate edge but at diﬀerent angular positions is shown in Fig. 6.7 for case A1 and Fig. 6.8 for case A2. It can be observed that the sound pressure levels from the unheated case dominate over the heated case for low frequencies over all angles. At small angles, θ = 10 or θ = 27, lower frequencies dominate. For higher frequencies, the noise levels around the peak radiation angle is almost similar for the heated and unheated case but at larger angles ( θ ≥ 60), the sound pressure levels for the heated case are slightly higher than the unheated case. Since the contribution of sound at Strouhal numbers, St & 0.1 is small, this trend does not show up in the OASPL levels observed in Fig. 6.6.

6.1.3 Space time correlations

Since the LES or DNS computations are computationally expensive, it would be ad- vantageous if radiated noise could be predicted using acoustic analogy but without recourse to a LES computation. To this end, statistical noise prediction approaches 6.1. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 121

−4 −4 10 10 pp pp

S −6 S −6 10 10

−8 −8 10 −2 −1 10 −2 −1 10 10 10 10 St St

(a) (b)

(c)

Figure 6.3: Pressure spectra for case A1 in (a) and case A2 in (b) at observer locations shown in (c). The observers are located at a common transverse coordinate of y/wp = −65. Dashed line: x/wp = 48; dashed dotted line: x/wp = 84; dotted line: x/wp = 120; Solid line with symbols: x/wp = 153; Solid line: x/wp = 183 (also the locations of observers 1 to 5 in (c)). 122 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

−4 −4 10 10 pp pp

S −6 S −6 10 10

−8 −8 10 −2 −1 10 −2 −1 10 10 10 10 St St

(a) (b)

−4 10 pp

S −6 10

−8 10 −2 −1 10 10 St

(c)

Figure 6.4: Comparison of pressure spectra from acoustic analogy predictions and LES computations shown for case A1 and observer locations as shown in Fig. 6.3(c): 1 in (a), 3 in (b) and 5 in (c). −: LES data; −− Volume integral; − · −·: Surface integral. 6.1. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 123

Figure 6.5: Schematic showing the (r, θ) coordinate system used for far-ﬁeld observer locations.

150

145

140

135

OASPL(in dB) 130

125

120 0 20 40 60 80 100 120 θ

Figure 6.6: Noise directivity in terms of OASPL for cases A1 and A2. Solid line: A1; dashed line: A2. 124 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

140 140

120 120

pp 100 pp 100 S S

80 80

60 −2 −1 60 −2 −1 10 10 10 10 St St

(a) θ = 10◦ (b) θ = 27◦

140 140

120 120

pp 100 pp 100 S S

80 80

60 −2 −1 60 −2 −1 10 10 10 10 St St

Figure 6.7: Sound pressure levels as a function of Strouhal number(St = fwp/∆U) for case A1 at various observer locations at a distance of R/wp = 6500. 6.1. CASES WITH INCOMING LAMINAR BOUNDARY LAYERS 125

140 140

120 120

pp 100 pp 100 S S

80 80

60 −2 −1 60 −2 −1 10 10 10 10 St St

(a) θ = 10◦ (b) θ = 27◦

140 140

120 120

pp 100 pp 100 S S

80 80

60 −2 −1 60 −2 −1 10 10 10 10 St St

Figure 6.8: Sound pressure levels as a function of Strouhal number(St = fwp/∆U) for case A2 at various observer locations at a distance of R/wp = 6500. 126 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

based on Lightill’s acoustic analogy have been developed where the far-field pressure power spectral density formula is expressed in terms of two-point space time correlations of the Lighthill’s stress tensor. This correlation function is a turbulence property and is usually approximated or modelled as a convected gaussian function with local parameters determined from a Reynolds averaged Navier-Stokes solution. Prediction methods based on Lilley’s or Goldstein’s generalized acoustic analogy also rely on space time correlations of turbulent sources but with fluctuations defined over a prescribed mean flow. In unheated flows, the relevant correlation function is the

Reynolds stress autocovariance tensor Rijkl (with i, j, k, l = 1, 2, 3) deﬁned as,

00 00 00 00 00 00 00 00 Rijkl(y, ξ, τ) = [ρui uj − ρui uj ](y, t)[ρukul − ρukul ](y + ξ, t + τ) (6.3)

00 Here, ui = ui − uei for i = 1, 2, 3 denotes the fluctuating velocity components in the three directions about the Favre-averaged mean if used in the sense of Goldstein’s generalized acoustic analogy with a Favre-time-averaged base flow (Karabasov et al., 2010; Karabasov, 2010) or about a uniform base flow is used in the sense of Lighthill’s

acoustic analogy (Morris and Farassat, 2002). The components of Rijkl(y, ξ, τ) are assumed to have a simple gaussian form as in Eq. 6.4. Typically the values of unknown

local parameters, amplitude function Aijkl(y), time scale τs(y) and the length scale

ls(y) are obtained from RANS solution.

2 2 2 ξ1 (ξ1 − uτ˜ ) + ξ2 + ξ3 Rijkl(y, ξ, τ) = Aijkl(y) exp − − ln 2 2 (6.4) uτ˜ s(y) ls (y)

In case of heated ﬂows, the Reynolds stress autocovariance tensor can be generalized (Goldstein, 2010; Afsar et al., 2011a) as,

00 00 00 00 00 00 00 00 Rνjµl(y, ξ, τ) = [ρuνuj − ρuνuj ](y, t)[ρuµul − ρuµul ](y + ξ, t + τ) (6.5)

The allowed values for subscripts are j, l = 1, 2, 3 and ν, µ = 1, 2, 3, 4. Here, 00 uµ = uµ − ufµ for µ = 1, 2, 3 denotes the usual ﬂuctuating velocity components in 00 the three directions about the Favre-averaged mean. The fourth component u4 = 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 127

0 00 00 (γ − 1) [h + ukuk/2] is related to the stagnation enthalpy fluctuations in the frame of reference of the local base flow velocity (Goldstein, 2010). The above tensor can be Fourier transformed and its convolution with the propagator tensor over the entire physical space yields the pressure power spectral density for an observer in the far- field. In general, the above tensor will have 63 components (Afsar et al., 2011b) and evaluating each one would be extremely cumbersome. The direct influence of each component on far-field sound is through its magnitude function Aijkl(y). The degree to which each component is important can be compared by evaluating the temporal autocovariance function (zero separation) of different Reynolds stress components. This is shown in Fig. 6.9 at three different locations along the lipline. Since, the enthalpy fluctuations are non-negligible for heated flows, the additional components as in Eq. 6.5 are relevant and are shown in Fig. 6.10. The space-time correlation functions at different have been directly evaluated and shown in Figs. 6.11 for case A1 and Fig. 6.12 for case A2 based on autocovariance of streamwise velocity. 00 00 Similarily, the space-time correlations of ρu1u1 component of Reynolds stress is shown in Fig. 6.13 for case A1 and Fig. 6.14 for case A2. It can be observed in 00 00 00 00 00 00 Fig. 6.15 that the correlation function shape are similar for ρu2u2, ρu3u3 or ρu1u2 components but with different length and time scales.

6.2 Cases with incoming turbulent boundary layers

6.2.1 Near-ﬁeld spectra

As presented in the analysis of laminar boundary layer cases A1 and A2, we restrict our attention to those spanwise length scales found upto x/wp ∼ 40. Given that the shear layer vorticity thickness at this point is approximately δω/wp ∼ 5 (or momentum thickness δθ/wp ∼ 2) and the characteristic instability frequency of shear layer is given by Stθ = fδθ/U = 0.032 (Ho and Huerre, 1984), focus would be on frequencies in the 128 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

0.8 0.4

0.6 0.3

ijkl 0.4 ijkl 0.2 R R

0.2 0.1

0 0 0 5 10 15 20 0 5 10 15 20 τ τ

(a) (b)

0.4

0.25 0.3 0.2

0.15 ijkl 0.2 ijkl R R 0.1 0.1 0.05

0 0 0 5 10 15 20 0 5 10 15 20 τ τ

0.1 0.1 ijkl ijkl R R 0.05 0.05

0 0

0 5 10 15 20 0 5 10 15 20 τ τ

(e) (f)

Figure 6.9: Comparison of temporal autocovariance functions for diﬀerent components of Rijkl (Eq. 6.3) for cases A1 (a-c) and A2 (d-f) at three locations: (a) x/wp = 89, (b) x/wp = 200 and (c) x/wp = 284. Figs. (d)-(f) are at corresponding locations for case A2. Legend: −4− : R1111, −− : R2222, − · −· : R3333, ··· : R1212, −− : R1122, − B − : R1133, − ◦ − : R1112 (with a negative sign). 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 129

0.02 0.2

0.015 0.15 0.01 l l µ µ

j j 0.1

ν 0.005 ν R R 0 0.05 −0.005 0 −0.01 0 5 10 15 20 0 5 10 15 20 τ τ

Figure 6.10: Comparison of temporal autocovariance functions for diﬀerent components of Rνjµl as deﬁned in Eq. 6.5 for cases A1 (a) and A2 (b) at x/wp = 89. Legend: −4− : R4141, −− : R4242, − · −· : R4343, ··· : R4111, −− : R4122 and − B − : R4133.

range St = fwp/∆U ≥ 0.025 for far-field acoustic predictions. The spectra of pressure fluctuations along the lipline are shown in Fig. 6.16 for case A4 and Fig. 6.17 for case A5. It can be observed that going downstream there is a gradual increase in correlation integral time scales. The temporal spectra can be observed to shift to greater contribution from the lower frequencies. However, the extent to which this trend can be seen depends on the amount of shear layer growth that has been captured within the computational domain. The domain for the laminar boundary layer cases was larger than the turbulent boundary layer cases and consequently the percentage shear layer growth from the initial thickness to the final was larger for the former. The feature of shift in frequencies was more obvious for the laminar cases than here. Moreover, close to the splitter plate, the laminar mixing layer underwent transition to turbulence via Kelvin-Helmholtz instability. Since, the initial boundary layers are turbulent in the present cases, the instability frequency is not observable here.

6.2.2 Far-ﬁeld spectra

The OASPLs computed for laminar and turbulent boundary layer cases (A1, A2, A4 and A5) are shown in Fig. 6.18. The observer locations here are same as for 130 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

1 1

0.8 0.8

0.6 0.6

0.4 0.4 t uu t uu R R 0.2 0.2

0 0

−0.2 −0.2 0 10 20 30 40 0 20 40 60 80 τ τ

(a) (b)

1 40

0.8 30 0.6

0.4 t uu ξ 20 R 0.2

0 10

−0.2 0 0 20 40 60 80 100 120 0 10 20 30 40 50 τ τ

Figure 6.11: Space time correlations of streamwise velocity and estimate for convection velocity for case A1. (a) At x/wp = 89. Legend: −4−: ξ = 0; −: ξ = 2.8; −−: ξ = 5.6; − · −·: ξ = 8.4; ··· : ξ = 11.2; −−: ξ = 14; − ◦ −: ξ = 16.8. (b) At x/wp = 200. Legend: −4−: ξ = 0; −: ξ = 5.6; −−: ξ = 11.2; − · −·: ξ = 16.8; ··· : ξ = 22.3; −−: ξ = 27.9; − ◦ −: ξ = 33.5. (c) At x/wp = 284. Legend: Same as (b). (d) ξ − τ plot (streawise separation plotted against the time delay for peak correlation values) for estimation of convection velocity, Uc. (d) −◦−: At x/wp = 89; −4−: At x/wp = 200; −−: At x/wp = 284. 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 131

1 1

0.8 0.8

0.6 0.6

0.4 0.4 t uu t uu R R 0.2 0.2

0 0

−0.2 −0.2 0 10 20 30 40 0 20 40 60 80 τ τ

(a) (b)

1 40

0.8 30 0.6

0.4 t uu ξ 20 R 0.2

0 10

−0.2 0 0 20 40 60 80 100 120 0 10 20 30 40 50 τ τ

Figure 6.12: Space time correlations of streamwise velocity and estimate for convection velocity for case A2. Legend: same as in Fig. 6.11. 132 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

1 1

0.8 0.8

0.6 0.6

0.4 0.4 1111 1111 R R 0.2 0.2

0 0

−0.2 −0.2 0 10 20 30 40 0 20 40 60 80 τ τ

(a) (b)

1 40

0.8 30 0.6

0.4 ξ

1111 20 R 0.2

0 10

−0.2 0 0 20 40 60 80 0 10 20 30 40 50 τ τ

00 00 Figure 6.13: Space time correlations of ρu1u1 as deﬁned in Eq. 6.3 and estimate for convection velocity for case A1. Legend: Same as in Fig. 6.11. 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 133

1 1

0.8 0.8

0.6 0.6

0.4 0.4 1111 1111 R R 0.2 0.2

0 0

−0.2 −0.2 0 10 20 30 40 0 20 40 60 80 τ τ

(a) (b)

1 40

0.8 30 0.6

0.4 ξ

1111 20 R 0.2

0 10

−0.2 0 0 20 40 60 80 0 10 20 30 40 50 τ τ

00 00 Figure 6.14: Space time correlations of ρu1u1 as deﬁned in Eq. 6.3 and estimate for convection velocity for case A2. Legend: Same as in Fig. 6.11. 134 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

1 1

0.8 0.8

0.6 0.6

0.4 0.4 2222 3333 R R 0.2 0.2

0 0

−0.2 −0.2 0 20 40 60 80 0 20 40 60 80 τ τ

(a) (b)

1 1

0.8 0.8

0.6 0.6

0.4 0.4 1212 2222 R R 0.2 0.2

0 0

−0.2 −0.2 0 20 40 60 80 0 20 40 60 80 τ τ

1 1

0.8 0.8

0.6 0.6

0.4 0.4 3333 1212 R R 0.2 0.2

0 0

−0.2 −0.2 0 20 40 60 80 0 20 40 60 80 τ τ

(e) (f)

00 00 00 00 00 00 Figure 6.15: Space time correlations of ρu2u2 (in a, d), ρu3u3 (in b, e) and ρu1u2 (in c, f) as deﬁned in Eq. 6.3 shown for x/wp = 200 for cases A1 (a-c) and A2 (d-f). Legend: −4−: ξ = 0; −: ξ = 5.6; −−: ξ = 11.2; − · −·: ξ = 16.8; ··· : ξ = 22.3; −−: ξ = 27.9; − ◦ −: ξ = 33.5. 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 135

1 1

0.8 0.8

0.6 0.6

0.4 0.4 t pp z pp R R 0.2 0.2 0 0 −0.2 −0.2 −0.4 0 10 20 30 40 0 1 2 3 4 5 τ ξ

(a) (b)

−2 10 t pp

S −4 10

−6 10 −1 0 10 10 St

(c)

Figure 6.16: (a) Two point temporal correlation (b) Two point spanwise correlation of pressure (c) Temporal spectra for case A4. 136 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

1 1

0.8 0.8

0.6 0.6

0.4 0.4 t pp z pp R R 0.2 0.2 0 0 −0.2 −0.2 −0.4 0 20 40 60 0 1 2 3 4 5 τ ξ

(a) (b)

−2 10 t pp

S −4 10

−6 10 −1 0 10 10 St

(c)

Figure 6.17: (a) Two point temporal correlation (b) Two point spanwise correlation of pressure (c) Temporal spectra for case A5. 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 137

150

140

130

120

OASPL(in dB) 110

100

90 0 20 40 60 80 100 120 θ

Figure 6.18: Noise directivity in terms of OASPL for cases A1, A2, A4 and A5. Solid line: A1; dashed line: A2; dashed dotted line: A4; dotted line: A5.

the laminar boundary cases A1 and A2 (R/wp ∼ 6500). There is a clear reduction in radiated noise as the shear layer is heated. This can be seen both if the initial boundary layer is laminar (compare A1 and A2) or turbulent (A4 and A5). The turbulent boundary layer cases also have noise levels much lower than the laminar ones. However, the SPL levels for laminar boundary layer cases are observed to drop off after St & 0.1 (Figs. 6.7 and 6.8) whereas the SPL for cases A4 and A5 have significant contribution beyond St & 0.1 as shown in Figs. 6.19 and 6.20. The reason for higher OASPL for the cases A1 and A2 is, therefore, mainly the difference in contribution at lower frequencies.

6.2.3 Space time correlations

Space time correlations of streamwise velocity ﬂuctuations are shown in Fig. 6.21 for case A4 and Fig. 6.22 for case A5. The estimated convection velocity based on these correlations were found to be smaller for the heated case. The convection velocities can also be computed using the space-time correlations of Reynolds stress tensors 138 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

140 140

120 120

pp 100 pp 100 S S

80 80

60 −1 0 60 −1 0 10 10 10 10 St St

(a) θ = 10◦ (b) θ = 27◦

140 140

120 120

pp 100 pp 100 S S

80 80

60 −1 0 60 −1 0 10 10 10 10 St St

Figure 6.19: Sound pressure levels as a function of Strouhal number(St = fwp/∆U) for case A1 at various observer locations at a distance of R/wp = 6500. 6.2. CASES WITH INCOMING TURBULENT BOUNDARY LAYERS 139

140 140

120 120

pp 100 pp 100 S S

80 80

60 −1 0 60 −1 0 10 10 10 10 St St

(a) θ = 10◦ (b) θ = 27◦

140 140

120 120

pp 100 pp 100 S S

80 80

60 −1 0 60 −1 0 10 10 10 10 St St

Figure 6.20: Sound pressure levels as a function of Strouhal number(St = fwp/∆U) for case A2 at various observer locations at a distance of R/wp = 6500. 140 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

00 00 and it has been shown for the ρu1u1 component in Fig. 6.23 for case A4 and Fig. 6.24 for case A5. However, the convection velocities based on both the estimates were found to be similar and smaller for the heated case. It can also be seen the the decay of correlations is much faster for the turbulent case than its laminar counter for a similar location. The magnitude of autocorrelations at zero separation and time-delay of various components of Reynolds stress components can be compared from Figs. 6.25 and Figs. 6.26. It can be seen there is a considerable diﬀerence in the autocorrelation amplitudes of diﬀerent components which means that the assumption of isotropy of this tensor is invalidated. There is also a decrease in autocorrelation amplitudes going from unheated to heated case and this explains the relative quietness of the heated shear layer (case A5) with respect to the unheated one (case A4).

6.3 Summary

The temporal spectra of the pressure fluctuations were shown both in the near-field hydrodynamical region and near-field and far-field acoustic region in this chapter. It was observed the heated shear layer is quieter than the corresponding unheated shear layer. The same effect was observed for shear layers with both upstream laminar and turbulent boundary layers. Furthermore, the overall sound pressure levels (OASPL) for the cases with laminar boundary layers was found to be more than the ones with turbulent boundary layers. A reduction in near-field hydrodynamical pressure fluctuations was also observed consistent with reduction in acoustic pressure in the far- field. The simple scaling introduced in Chapter 4 for near-field pressure fluctuations was also found to hold in the present three-dimensional cases as well. The analysis of Reynolds stress autocorrelation tensor reveals the relative importance of its various components. The autocorrelation amplitudes at zero spatial separation and time-delay were found to decrease with heating. This was consistent with the reduction in mean-density with heating and would be the most likely cause for reduction in far-field noise levels with heating. The enthalpy flux autocorrelation amplitudes and enthalpy flux- momentum flux cross covariances were also found to 6.3. SUMMARY 141

1 1

0.8 0.8

0.6 0.6

0.4 0.4 11 11 R R 0.2 0.2

0 0

−0.2 −0.2 0 5 10 15 20 0 5 10 15 20 τ τ

(a) (b)

1 8

0.8 6 0.6

0.4 11 ξ 4 R 0.2

0 2

−0.2 0 0 5 10 15 20 0 2 4 6 8 10 τ τ

Figure 6.21: Space time correlations of streamwise velocity and estimate for convection velocity for case A4. (a) At x/wp = 20. Legend: −4−: ξ = 0; −: ξ = 0.75; −−: ξ = 1.5; −·−·: ξ = 2.2; ··· : ξ = 3; −−: ξ = 3.7; −◦−: ξ = 4.4. (b) At x/wp = 35. Legend: −4−: ξ = 0; −: ξ = 0.9; −−: ξ = 1.9; − · −·: ξ = 2.8; ··· : ξ = 3.7; −−: ξ = 4.6; − ◦ −: ξ = 5.5. (c) At x/wp = 50. Legend: −4−: ξ = 0; −: ξ = 1.1; −−: ξ = 2.2; − · −·: ξ = 3.3; ··· : ξ = 4.4; −−: ξ = 5.5; − ◦ −: ξ = 6.5. (d) ξ − τ plot (streawise separation plotted against the time delay for peak correlation values) for estimation of convection velocity, Uc. (d) −◦−: At x/wp = 20; −4−: At x/wp = 35; −−: At x/wp = 50. 142 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

1 1

0.8 0.8

0.6 0.6

0.4 0.4 11 11 R R 0.2 0.2

0 0

−0.2 −0.2 0 5 10 15 20 0 5 10 15 20 τ τ

(a) (b)

1 8

0.8 6 0.6

0.4 11 ξ 4 R 0.2

0 2

−0.2 0 0 5 10 15 20 0 2 4 6 8 10 τ τ

Figure 6.22: Space time correlations of streamwise velocity and estimate for convection velocity for case A5. Legend: same as in Fig. 6.21. 6.3. SUMMARY 143

1 1

0.8 0.8

0.6 0.6

0.4 0.4 1111 1111 R R 0.2 0.2

0 0

−0.2 −0.2 0 5 10 15 20 0 5 10 15 20 τ τ

(a) (b)

1 8

0.8 6 0.6

0.4 ξ

1111 4 R 0.2

0 2

−0.2 0 0 5 10 15 20 0 2 4 6 8 10 τ τ

00 00 Figure 6.23: Space time correlations of ρu1u1 as deﬁned in Eq. 6.3 and estimate for convection velocity for case A4. Legend: same as in Fig. 6.21. 144 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

1 1

0.8 0.8

0.6 0.6

0.4 0.4 1111 1111 R R 0.2 0.2

0 0

−0.2 −0.2 0 5 10 15 20 0 5 10 15 20 τ τ

(a) (b)

1 8

0.8 6 0.6

0.4 ξ

1111 4 R 0.2

0 2

−0.2 0 0 5 10 15 20 0 2 4 6 8 τ τ

00 00 Figure 6.24: Space time correlations of ρu1u1 as deﬁned in Eq. 6.3 and estimate for convection velocity for case A5. Legend: same as in Fig. 6.21. 6.3. SUMMARY 145

0.2 0.2

0.15 0.15

ijkl 0.1 ijkl 0.1 R R

0.05 0.05

0 0 0 5 10 15 20 0 5 10 15 20 τ τ

(a) (b)

0.2 0.1

0.08 0.15 0.06

ijkl 0.1 ijkl

R R 0.04

0.05 0.02

0 0 0 5 10 15 20 0 5 10 15 20 τ τ

0.1 0.1

0.08 0.08

0.06 0.06 ijkl ijkl

R 0.04 R 0.04

0.02 0.02

0 0

0 5 10 15 20 0 5 10 15 20 τ τ

(e) (f)

Figure 6.25: Comparison of temporal autocovariance functions for diﬀerent components of Rijkl (Eq. 6.3) for cases A4 (a-c) and A5 (d-f) at three locations: (a) x/wp = 20, (b) x/wp = 35 and (c) x/wp = 50. Figs. (d)-(f) are at corresponding locations for case A5. Legend: −4− : R1111, −− : R2222, − · −· : R3333, ··· : R1212, −− : R1122, − B − : R1133, − ◦ − : R1112 (with a negative sign). 146 CHAPTER 6. SOUND GENERATION IN TURBULENT MIXING LAYERS

0.02 0.08

0.015 0.06 0.01 l l

µ µ 0.04 j j

ν 0.005 ν R R 0 0.02

−0.005 0 −0.01 0 2 4 6 8 10 0 2 4 6 8 10 τ τ

(a) (b)

Figure 6.26: Comparison of temporal autocovariance functions for different components of Rνjµl as defined in Eq. 6.5 for cases A4 (a) and A5 (b) at x/wp = 35. Legend: −4− : R4141, −− : R4242, − · −· : R4343, ··· : R4111, −− : R4122 and − B − : R4133. have significant amplitudes with heating. But they can be expected to cancel each other’s effect based on the analysis recently done by Afsar et al. (2011a). Chapter 7

Conclusions

Large Eddy Simulations of spatially developing, compressible, turbulent mixing layers using high order overset method have been conducted. The overset method enabled the inclusion of the splitter plate inside the computational domain while retaining good quality of mesh around the turning splitter plate trailing edge, high order numerical accuracy and reasonable computational cost. It also enabled the study of sensitivity of shear layer flow development and noise on the upstream conditions, the effects of heating and state of incoming boundary layer being laminar or turbulent. Five cases with different inflow conditions are considered to study the influence of increasing the temperature of high speed stream on mixing layer development and associated noise. The effect of heating is studied both when the boundary layer upstream is laminar or turbulent. These cases are chosen to keep the velocity ratio and Reynolds number based on the free stream conditions on the high speed stream side fixed while only varying the temperature. This is done to reproduce observed effects in jets where jet velocity (or jet acoustic Mach number) is kept fixed and the temperature varied. This implies a reduction in jet momentum or thrust and a reduction in jet Mach number as the temperature is increased. Another case is chosen corresponding to the experimental conditions of Samimy and Elliot (1990) where the velocity ratio is higher compared to the previous cases and the high speed stream is colder but the convective Mach number is the same. For each case, the mean and turbulent intensity profiles collapse when plotted

147 148 CHAPTER 7. CONCLUSIONS

in similarity coordinates. The development distance to achieve self-similarity in the mean velocity profile is found to be shortest for turbulent boundary layer cases followed by the laminar boundary layer cases with lower velocity ratio. The heated and the unheated shear layers had roughly the same development distance. The self- similar mixing layer thickness growth rate and distribution of self-similar values of Reynolds stress components are found to be in agreement with available experiments. It is observed that with heating in the cases with laminar boundary layers upstream, the initial instability is accelerated but the saturation amplitude of Reynolds stress components is not affected. The saturation amplitudes of density fluctuations were found to increase proportionally to difference in free-stream densities whereas near-field pressure fluctuations were found to decrease with heating. A simple scaling is deduced for the near-field pressure fluctuation amplitude by taking into account the pressure difference between the ambient and the vortex cores in the frame of reference moving with the eddy convection speed. The reduction in pressure fluctuations is found to be proportional to the dynamic head based on free-stream density and difference of convection velocity and the free-stream velocity of either stream. The observed reduction can be accounted for by the reduction of mean density with heating. Direct numerical simulations of two-dimensional shear layers were also conducted and the scaling law for pressure fluctuations was confirmed. In the present study, we observe that a mixing layer which forms from an initially turbulent boundary layer has a lower growth rate than the one with an initially laminar boundary layer consistent with experimental observations of Karasso and Mungal (1996). This suggests that jets with initially turbulent boundary layers will lead to still longer potential core lengths and points to the need to match the experimental boundary layer state for favorable comparison with experimental data. The result from the current study that heating increases the growth rate of shear layer whether the initial boundary layer is laminar or turbulent implies a reduction in potential core lengths with heating. This trend has been observed in both simulations and experiments. Another effect of boundary layer state, laminar or turbulent, is on the Reynolds stress variation along the lipline. An initial turbulent boundary layer leads to a 7.1. FUTURE WORK 149

gradual increase of Reynolds stresses from values that correspond to boundary layer to those of mixing layer. The mixing layer with initially laminar boundary layers, instead, have an overshoot before relaxing to the mixing layer values. This explains observation in the study by Mendez et al. (2010) who report higher fluctuations of streamwise velocity compared to experiments. For cases with initially laminar boundary layers, the role of the feedback from vortex pairing downstream in sustaining the initial instability via interaction with the splitter plate remains to be established. This would be a part of future research. For laminar exit boundary layers, sound radiation is observed in downstream direction peaked roughly at 30 degrees. The vortex pairing and breakdown to turbulence contribute significantly to the radiated sound. For turbulent exit boundary layers, the acoustic field near the shear layer is significantly weaker and noise due to passage of boundary layer eddies over the trailing edge is observed. In the far-field away from the shear layer on the slower stream side, a reduction in peak pressure fluctuations with increased heating is observed for both mixing layers with incoming laminar or turbulent boundary layer. The analysis of Reynolds stress autocorrelation tensor reveals the relative importance of its various components. The autocorrelation amplitudes at zero spatial separation and time-delay were found to decrease with heating. This was consistent with the reduction in mean-density with heating and would be the most likely cause for reduction in far-field noise levels with heating. The enthalpy flux autocorrelation amplitudes and enthalpy flux- momentum flux cross covariances were also found to have significant amplitudes with heating. But they cancel each other’s effect leading to an overall reduction in far-field sound.

7.1 Future Work

It is apparent that several paths can be taken continuing from the present work to enhance our understanding of turbulence generated noise. A few of them are described below.

• A simple generalization is to study the eﬀect of heating at higher temperature ratios (above 1.8 as in the present study) keeping the velocity ratio same. This 150 CHAPTER 7. CONCLUSIONS

can be used to establish the dependence of far-field noise on the temperature ratio quantitatively as a scaling ratio. The present work can also be extended to study the effect of heating at low Mach numbers. The reduction of far-field noise with heating at high Mach numbers observed for shear layers is similar to jets. However, an increase in noise is observed in jets with heating below a Mach number of 0.7. It would be interesting to investigate if such an effect can be reproduced in the context of shear layers and at what Mach number does the reversal of the heating effect takes place.

• The shear layer data presented here can also serve to investigate the effects due to splitter plate geometry modifications. This would be relevant for studies concerning effects of modifying nozzle geometry on jet flow development and noise.

• Coarser LES calculations for same shear layer cases may be done for detailed comparison with LES data provided here to further investigate the eﬀect of subgrid model on noise prediction. Bibliography

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