Large Eddy Simulation of Shear-Free Interaction of ...Large Eddy Simulation of Shear-Free Interaction of Homogeneous Turbulence with a Flat-Plate Cascade Abdel-Halim S. Salem Said

Large Eddy Simulation of Shear-Free Interaction of

Homogeneous Turbulence with a Flat-Plate Cascade

Abdel-Halim Saber Salem Said

Thesis submitted to the faculty of the

Virginia Polytechnic Institute and State University

in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

in

Engineering Mechanics

Dr. Saad A. Ragab, Chairman

Dr. Muhammad R. Hajj

Dr. William J. Devenport

Dr. Demetri Telionis

Dr. Surot Thangjitham

July 23, 2007

Blacksburg, Virginia

Keywords: Homogeneous Turbulence, Cascade Interaction, Large Eddy Simulation,

Acoustic Radiation, High-Order Finite Difference

c© Copyright 2007, Abdel-Halim S. Salem Said

Large Eddy Simulation of Shear-Free Interaction of Homogeneous

Turbulence with a Flat-Plate Cascade

Abdel-Halim S. Salem Said

(ABSTRACT)

Studying the effects of free stream turbulence on noise, vibration, and heat transfer on

structures is very important in engineering applications. The problem of the interaction

of large scale turbulence with a flat-plate cascade is a model of important problems in

propulsion systems. Addressing the problem of large scale turbulence interacting with a

flat plate cascade requires flow simulation over a large number of plates (6-12 plates) in

order to be able to represent numerically integral length scales on the order of blade-to-

blade spacing. Having such a large number of solid surfaces in the simulation requires very

large computational grid points to resolve the boundary layers on the plates, and that is not

possible with the current computing resources.

In this thesis we develop a computational technique to predict the distortion of homogeneous

isotropic turbulence as it passes through a cascade of thin flat plates. We use Large-Eddy

Simulation (LES) to capture the spatial development of the incident turbulence and its

interaction with the plates which are assumed to be inviscid walls. The LES is conducted

for a linear cascade composed of six plates. Because suppression of the normal component

of velocity is the main mechanism of distortion, we neglect the presence of mean shear in the

boundary layers and wakes, and allow slip velocity on the plate surfaces. We enforce the zero

normal velocity condition on the plates. This boundary condition treatment is motivated by

rapid distortion theory (RDT) in which viscous effects are neglected, however, the present

LES approach accounts for nonlinear and turbulence diffusion effects by a sub-grid scale

model. We refer to this type of turbulence-blade interaction as shear-free interaction.

To validate our calculations, we computed the unsteady loading and radiated acoustic pres-

sure field from flat plates interacting with vortical structures. We consider two fundamental

problems: (1) A linear cascade of flat plates excited by a vortical wave (gust) given by a 2D

Fourier mode, and (2) The parallel interaction of a finite-core vortex with a single plate. We

solve the nonlinear Euler equations by a high-order finite-differece method. We use nonre-

flecting boundary conditions at the inflow and outflow boundaries. For the gust problem,

we found that the cascade response depends sensitively on the frequency of the convected

gust. The unsteady surface pressure distribution and radiated pressure field agree very well

with predictions of the linear theory for the tested range of reduced frequency. We have also

investigated the effects of the incident gust frequency on the undesirable wave reflection at

the inflow and outflow boundaries. For the vortex-plate interaction problem, we investigate

the effects of the internal structure of the vortex on the strength and directivity of radiated

sound.

Then we solved the turbulence cascade interaction problem. The normal Reynolds stresses

and velocity spectra are analyzed ahead, within, and downstream of the cascade. Good

agreement with predictions of rapid distortion theory in the region of its validity is obtained.

Also, the normal Reynolds stress profiles are found to be in qualitative agreement with

available experimental data. As such, this dissertation presents a viable computational

alternative to rapid distortion theory (RDT) for the prediction of noise radiation due to the

interaction of free stream turbulence with structures.

iii

Dedication

I would like to dedicate this dissertation to my parents, my wife, my daughters Noura and

Menna, and my son Mahmoud.

iv

Acknowledgments

First of all, I would like to express my deep gratitude and love to my advisor, Dr. Saad

Ragab, for his support, guidance and patience. I owe him a big debt. I would like to thank

Dr. Muhammad Hajj for his kind support.

I would like also to thank Dr. William Devenport, and Dr. Jon Larssen for sending us their

experimental data, and Dr. Demetri Telionis, and Dr. Surot Thangjitham for their service

on my committee.

My thanks are due to Mr. Mohammad Elyyan and my daughter Noura for their help in the

presentation and Tecplot.

I would like to thank the Department of Engineering Science and Mechanics for supporting

me financially through GTAs.

My gratitude is due to the Egyptian embassy for supporting me financially throughout my

stay in the USA.

v

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Technical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Numerical Simulation of Turbulent Flows . . . . . . . . . . . . . . . . 2

1.3.2 Rapid Distortion Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.3 Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Flat Plates Interacting with Vortical Structures . . . . . . . . . . . . . . . . 6

1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Accomplished Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Mathematical Model for Large Eddy Simulation 10

2.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Navier-Stokes Equations in Nondimensional Form . . . . . . . . . . . . . . . 11

2.3 Equations For Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Subgrid-scale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Numerical Method and Boundary Conditions 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vi

4 Response of a Flat-Plate Cascade to Incident Vortical Waves - 2D Cal-

culations 25

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Glegg’s Linearized Potential Flow Solution . . . . . . . . . . . . . . . 28

4.2.2 Two-Dimensional Euler Simulations . . . . . . . . . . . . . . . . . . . 30

4.3 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Vortex-Plate Interaction 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Vortex-Plate Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Interaction of Homogeneous Turbulence with a Flat-Plate Cascade -

Comparison with Experimental Data 65

6.1 Inflow Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . . 67

6.2.1 Spatially Decaying Isotropic Turbulence . . . . . . . . . . . . . . . . 67

6.2.2 Computational Domain and Inflow Spectra . . . . . . . . . . . . . . . 69

6.2.3 Comparison of LES with Larsen Experimental Data . . . . . . . . . . 70

7 Interaction of Homogeneous Turbulence with a Flat-Plate Cascade -

Comparison with RDT 75

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2 Graham’s RDT Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.3 Comparison of LES with Graham’s RDT . . . . . . . . . . . . . . . . . . . . 77

7.3.1 Six-Plate Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.3.2 Three-Plate Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Conclusions and Recommended Future Work 116

vii

Bibliography 120

Appendices 126

A Graham’s RDT 126

B Glegg’s Linearized Solution 129

viii

List of Figures

4.1 Flat plate cascade and computational domain. . . . . . . . . . . . . . . . . 26

4.2 Unsteady lift response and sound power using Glegg’s linear solution at

Mach number M∞ = 0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Unsteady lift response and sound power using Glegg’s linear solution at

Mach number M∞ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Comparison of unsteady lift response with Glegg’s linear solution, M = 0.3. 31

4.5 Comparison of unsteady lift response with Glegg’s linear solution, M = 0.5. 32

4.6 Test case 1: A snapshot of pressure contours. . . . . . . . . . . . . . . . . 35

4.7 Test case 1: Pressure amplitudes for propagating mode ν = −1 and decay-

ing mode ν = −7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.8 Test case 1: Sensitivity of surface pressure jump to grid step sizes. . . . . . 37

4.9 Test case 1: Sensitivity of surface pressure jump to streamwise domain length. 38

4.10 Test case two: A snapshot of pressure contours. . . . . . . . . . . . . . . . 40

4.11 Test case two: Pressure amplitudes for propagating mode ν = 2 and decay-

ing mode ν = −4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.12 Test case two: Sensitivity of surface pressure jump to grid step sizes. . . . 42

4.13 Test case two: Sensitivity of surface pressure jump to streamwise domain

length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43





ix

4.18 Test case 4: Pressure amplitudes for propagating mode ν = 1 and decaying

mode ν = −3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49



5.1 Parallel vortex-plate interaction. . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Flow properties of Oseen and Taylor vortices. . . . . . . . . . . . . . . . . 57

5.3 Oseen vortex, a snapshot of vorticity field, t = 2. . . . . . . . . . . . . . . 58

5.4 Oseen vortex, a snapshot of pressure filed, t = 2. . . . . . . . . . . . . . . 58


5.6 Oseen vortex, a snapshot of pressure field, t = 3. . . . . . . . . . . . . . . 58





5.11 Taylor vortex, a snapshot of vorticity field, t = 6. . . . . . . . . . . . . . . 62

5.12 Taylor vortex, a snapshot of pressure field, t = 6. . . . . . . . . . . . . . . 62

5.13 Directivity of pressure amplitude on a circle r = 6.2 centered at x = 3.1, z =

0 at time t = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.14 Pressure signature at x = 0.5, z = 3. . . . . . . . . . . . . . . . . . . . . . 63

5.15 Lift coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1 Energy spectrum function, spatial LES, coarse grid. . . . . . . . . . . . . . 68

6.2 Energy spectrum function, spatial LES, fine grid. . . . . . . . . . . . . . . 68

6.3 Streamwise variation of dynamic model coefficient in spatial decaying tur-

bulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4 Flat plate cascade and computational domain . . . . . . . . . . . . . . . . 69

6.5 Target and numerically generated energy spectra at inflow boundary. . . . 71

6.6 Mid-passage distribution of normal Reynolds stresses and q2 = u2 + v2 + w2. 71

x

6.7 Reynolds stress profiles at x = 0.840. . . . . . . . . . . . . . . . . . . . . . 72

6.8 Normalized Reynolds stress profiles at x = 0.840. . . . . . . . . . . . . . . 72

6.9 Reynolds stress profiles at x = 1.948. . . . . . . . . . . . . . . . . . . . . . 73

6.10 Normalized Reynolds stress profiles at x = 1.948. . . . . . . . . . . . . . . 73

6.11 Spanwise vorticity contours for a single plate placed in isotropic turbulence,

no-slip condition is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.12 Reynolds stress profiles at (x − xLE)/c = 0.92 and (x − xLE)/c = 1.53 for

a single plate, no-slip boundary condition is applied. . . . . . . . . . . . . 73

6.13 Reynolds stress contours u2 for a single plate . . . . . . . . . . . . . . . . . 74

6.14 Reynolds stress contours w2 for a single plate. . . . . . . . . . . . . . . . . 74

7.1 Six-plate cascade and computational domain. . . . . . . . . . . . . . . . . 79

7.2 Case A: 3D-energy spectra of the incident turbulence, inflow (x=-4.836),

and upstream of cascade (x=-0.269). . . . . . . . . . . . . . . . . . . . . . 80

7.3 A snapshot of the instantaneous upwash velocity contours(xz-plane). . . . 81

7.4 A snapshot of the instantaneous upwash velocity contours at plane x = 0.17. 81

7.5 A snapshot of the instantaneous streamwise velocity contours (xz-plane). . 82

7.6 A snapshot of the instantaneous spanwise velocity contours (xz-plane). . . 82

7.7 A snapshot of the instantaneous velocity vectors (xz-plane). . . . . . . . . 83

7.8 A snapshot of the instantaneous velocity vectors (yz-plane) x = 0.17. . . . 83

7.9 A snapshot of the instantaneous pressure fluctuation contours (xz-plane). . 84

7.10 A snapshot of the instantaneous pressure fluctuation contours (yz-plane),

x = 0.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.11 A snapshot of the instantaneous density fluctuation contours (xz-plane). . 85

7.12 A snapshot of the instantaneous density fluctuation contours (yz-plane),

x = 0.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.13 Case A: A snapshot of the instantaneous pressure fluctuation contours (no

plates). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xi

7.14 Case A: A snapshot of the instantaneous density fluctuation contours (no

plates). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.15 Contours of the averaged streamwise-Reynolds stress component. . . . . . 86

7.16 Contours of the averaged spanwise-Reynolds stress component. . . . . . . 86

7.17 Contours of the averaged upwash-Reynolds stress component. . . . . . . . 87

7.18 Contours of the averaged square of the pressure fluctuations (pp). . . . . . 87

7.19 Mid-passage distribution of q2. . . . . . . . . . . . . . . . . . . . . . . . . 88

7.20 Normalized q2 profiles for different ratios of plate spacing to integral length

scale s/L111. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.21 Averaged TKE, Reynolds stresses, and pressure fluctuation (read right) . . 89

7.22 Normal Reynolds stress profiles at x = 0.067. . . . . . . . . . . . . . . . . 90







7.29 Case A: Profiles of q2/q20 at plane x = 0.201. . . . . . . . . . . . . . . . . . 94



7.32 One dimensional energy spectra, Eww(k1) at z/s = 0.5. . . . . . . . . . . 95

7.33 One dimensional energy spectra, Eww(k1) at z/s = 0.0417. . . . . . . . . 95

7.34 One dimensional energy spectra, Euu(k1) at z/s = 0.5. . . . . . . . . . . . 96

7.35 One dimensional energy spectra, Euu(k1) at z/s = 0.0417. . . . . . . . . . 96

7.36 One dimensional energy spectra, Evv(k1) at z/s = 0.5. . . . . . . . . . . . 97

7.37 One dimensional energy spectra, Evv(k1) at z/s = 0.0417. . . . . . . . . . 97

7.38 6-plate cascade and computational domain. . . . . . . . . . . . . . . . . . 98

7.39 3D-energy spectra of the incident turbulence . . . . . . . . . . . . . . . . . 99

xii

7.40 Case B: Snapshot of the instantaneous upwash velocity contours (xz-plane). 100

7.41 Case C: Snapshot of the instantaneous upwash velocity contours (xz-plane). 100

7.45 Case D: A snapshot of the instantaneous velocity vectors (xz-plane). . . . 100

7.42 Case D: Snapshot of the instantaneous upwash velocity contours (xz-plane). 101

7.43 Case B: A snapshot of the instantaneous velocity vectors (xz-plane). . . . 101

7.44 Case C: A snapshot of the instantaneous velocity vectors (xz-plane). . . . 101

7.46 Case B: Snapshot of the instantaneous pressure contours (xz-plane). . . . . 102

7.47 Case C: Snapshot of the instantaneous pressure contours (xz-plane). . . . . 102

7.48 Case D: Snapshot of the instantaneous pressure contours (xz-plane). . . . 103

7.49 Case B: Snapshot of the instantaneous density fluctuation contours (xz-plane).103

7.50 Case C: Snapshot of the instantaneous density fluctuation contours (xz-plane).103

7.51 Case D: Snapshot of the instantaneous density fluctuation contours (xz-plane).104

7.52 Case B: Contours of the averaged streamwise Reynolds stress component. . 105

7.53 Case C: Contours of the averaged streamwise Reynolds stress component. . 105

7.54 Case D: Contours of the averaged streamwise Reynolds stress component. 106

7.55 Case B: Contours of the averaged spanwise Reynolds stress component. . . 107

7.56 Case C: Contours of the averaged spanwise Reynolds stress component. . . 107

7.57 Case D: Contours of the averaged spanwise Reynolds stress component. . . 108

7.58 Case B: Contours of the averaged upwash Reynolds stress component. . . 108

7.59 Case C: Contours of the averaged upwash Reynolds stress component. . . 108

7.60 Case D: Contours of the averaged upwash Reynolds stress component. . . 109

7.61 Case B: Contours of the averaged square of the pressure fluctuations. . . . 109

7.62 Case C: Contours of the averaged square of the pressure fluctuations. . . . 109

7.63 Case D: Contours of the averaged square of the pressure fluctuations. . . . 110

7.64 Averaged streamwise decay of the turbulent kinetic energy. . . . . . . . . . 110

7.65 Case B: Normal Reynolds stress profiles at x = 0.201. . . . . . . . . . . . . 111

7.66 Case C: Normal Reynolds stress profiles at x = 0.201. . . . . . . . . . . . . 111

7.67 Case D: Normal Reynolds stress profiles at x = 0.201. . . . . . . . . . . . . 111

xiii







7.74 Case B: Profiles of q2/q20 at plane x = 0.201. . . . . . . . . . . . . . . . . . 113

7.75 Case C: Profiles of q2/q20 at plane x = 0.201. . . . . . . . . . . . . . . . . . 113

7.76 Case D: Profiles of q2/q20 at plane x = 0.201. . . . . . . . . . . . . . . . . . 114







xiv

List of Tables

3.1 The am coefficients of fourth-order Runge-Kutta scheme [54] . . . . . . . . 19

3.2 The bm coefficients of fourth-order Runge-Kutta scheme [54] . . . . . . . . 19

3.3 Boundary-points fifth-order scheme coefficients . . . . . . . . . . . . . . . 20

7.1 Characteristics of the inflow turbulence and computational domain . . . . 78

7.2 Values of q20 at the planes of comparison for different cases . . . . . . . . . 92

xv

Chapter 1

Introduction

1.1 Motivation

Studying the problem of the effects of free stream turbulence on noise, vibration, and heat

transfer on structures is very important in engineering applications. The problem of the

interaction of large scale turbulence with a flat plate cascade is a model of important problems

in propulsion systems. Some linearized solutions such as the rapid distortion theory (RDT)

are used to predict the response to incident turbulence from a flat plate cascade. Large

eddy simulation (LES) could be used as an alternative to RDT which can relax some of the

limitations of RDT. Our goal is to develop an LES code which can be used as an alternative

means for solving the problem of cascade response to incident turbulence.

1.2 Technical Problem

The interaction of free stream incident turbulence with engineering structures is a significant

source of vibration, noise, unsteady loading, and heat transfer. Bushnell [21] gave an exten-

1

sive review of body-turbulence interaction problems. A significant component of the noise

radiated by shrouded propellers and turbofans is due to interactions of ingested turbulence

with rotor blades and guide vanes. The incident turbulence is usually generated in boundary

layers on surfaces upstream of the propeller such as the vessel hull, control surfaces as well as

in the wakes of rotor blades upstream of guide vanes. It may also be present in the incident

free stream due to environmental effects such as atmospheric turbulence or breaking of inter-

nal or surface gravity waves. The incident turbulence usually contains large-scale structures

with integral length scales comparable to the blade-to-blade spacing or even larger. The

interactions of these structures with rotating blades or guide vanes result in pressure fluctu-

ations, unsteady lift, vibrations and hence noise radiation. Simultaneously, the turbulence

length scales, intensities and wavenumber-frequency spectra are significantly modified by the

interaction with the blades. It is highly desirable to have proper understanding of the flow

characteristics through developing computer codes which can be used to predict the flow

fields.

1.3 Approaches

1.3.1 Numerical Simulation of Turbulent Flows

Direct Numerical Simulation (DNS), solution of Reynolds Averaged Navier-Stokes equations

(RANS), and Large Eddy Simulation (LES) are three approaches to numerical simulation of

turbulent flows. To predict a turbulent flow, one may use a numerical method to solve the

time dependent Navier-Stokes equations for the instantaneous flow variables without the use

of a turbulence model. Such a solution is known as a direct numerical simulation (DNS). The

mesh and time advancement must resolve all of the dynamically relevant turbulent scales

from the largest scales down to the smallest scales. This resolution requirement puts an upper

limit on the Reynolds number that can be successfully simulated on a given computer [38].

2

A more wide spread utilization of the DNS is prevented by the fact that the number of grid

points needed for sufficient spatial resolution scales as Re94 and the CPU-time as Re3 [10].

RANS is the most frequently used approach in engineering applications. In this method, all

turbulent fluctuations are averaged over a long period of time, and their statistical effects

on the mean flow are modeled. The turbulence models are usually complex, because they

are required to consider all of the turbulent scales including the large scales which may

not be the same in different flow fields. Because of the averaging procedure, no detailed

information can be obtained about turbulent structures. On the other hand, DNS represents

the other extreme where all of the dynamically significant eddies are computed and none are

modeled [3].

The LES is a compromise between the DNS and the RANS approaches. In this approach,

large-scale eddies are computed whereas small scales are modeled. Since it is assumed that

small-scale eddies have an isotropic and homogeneous structure, simpler and more universal

subgrid scale models than the models required for RANS can be used. As the large-scale

turbulence is to be computed, the resolution requirement for the mesh is much more than in

RANS, but not as demanding as in DNS because the small scales are modeled. LES is well

suited for detailed studies of complex flow physics including massively separated unsteady

flows, large scale mixing, or aerodynamic noise [3].

1.3.2 Rapid Distortion Theory

Because of its simplicity and efficiency, Rapid Distortion Theory (RDT) has been extensively

used for investigating the interactions of turbulence with blades and prediction of radiated

noise from rotors. Kullar and Graham [27] obtained an integral equation for the loading of

a flat-plate linear cascade due to an incident three-dimensional gust composed of upwash

velocity component superimposed on a uniform stream. They examined the effects of Mach

number and three-dimensionality of gust on acoustic resonances between cascade blades.

3

Glegg [13] also obtained an integral equation for the loading (expressed as a jump in the

velocity potential across the blades), and solved that equation by the Wiener-Hopf method.

He obtained analytical expressions for the unsteady loading, acoustic mode amplitude, and

sound power output of the cascade. One of his conclusions is that the primary effect of sweep

on the radiated sound power is to cause the propagating acoustic modes to become cut off.

This effect depends on the Mach number.

Graham [23] used RDT with simplifying assumptions and obtained analytical solutions for

the turbulence spectra downstream of a flat-plate linear cascade. He noted that “... the

turbulence flow field for these convective flows is inhomogeneous in the streamwise direction

over a distance of order ÃL∞ [integral length scale]. This is the region within which there

is a significant pressure field associated with the interaction between the turbulence and

the leading edge.” Therefore, a simplified RDT in which the turbulence is assumed to be

homogeneous in the streamwise direction does not apply in this region. Also, RDT does not

apply for streamwise distances much greater than ÃL∞ from the leading edge.

Boquilion et al. [2] have also used RDT to analyze the interaction of turbulence with a

linear flat-plate cascade. They used Glegg’s theory which considers plates of finite chord. In

their work as well as Graham’s work, the transverse velocity is zero on the wake centerline.

Because traditional RDT neglects inviscid nonlinear and viscous effects, the vorticity field

of the incident turbulence is frozen as it convects with the uniform free-stream velocity. The

vortex sheets on the plates and the wake induce an irrotational velocity field, and because

of limitations, that field does not have an effect on stretching or tilting of the vorticity in

the incident turbulence. Moreover, the infinitesimally thin trailing vortex sheets remain flat

and parallel to the free stream, and hence the upwash velocity continues to be zero in the

wake on the plates planes. However, these sheets may deform by self induction and induce

transverse velocity perturbations.

Majumdar and Peak [33] used RDT to predict the distortion of ingested free stream tur-

4

bulence by the strain field of non-uniform mean flow upstream of an open or ducted fan.

They assumed that the incident turbulence is given by von Karman spectrum. They used a

strip-theory to predict the unsteady forces on rotating fan blades, and determined the radi-

ated sound by solving the convected wave equation with the help of Green’s function. They

found that the distortion of incident turbulence under static (zero forward speed) conditions

produced high tonal noise levels, whereas the radiated sound is generally broadband under

flight (aircraft approach) conditions.

Atassi et al. [1] examined the effect of mean flow swirl on the acoustic and aerodynamic re-

sponse of a set of guide vanes. The swirl is imparted to the incoming flow by a rotor upstream

of the guide vanes which are modeled by an unloaded (zero-mean lift) annular cascade. They

linearized the Euler equations around a non-uniform mean state and assumed time-harmonic

disturbances. Because of the disturbance equations have variable coefficients, Atassi et al.

used a finite-difference method and solved for the flow in a single blade passage assuming

quasi-periodic conditions in the circumferential direction. They showed that the mean swirl

changes the mechanics of the scattering of incident acoustic and vortical disturbances. They

pointed to the importance of the radial phase of the incident disturbance in the scattering

process.

1.3.3 Experimental Work

The current investigation is motivated by the cascade experiments conducted by Larssen and

Devenport [28] (see also Larssen [29]). The experimental setup consists of a six-blade linear

cascade. They adopted a mechanically rotating “active” grid design in order to generate

the large scale turbulence. They compared the experimental blade-blocking data to linear

cascade theory (RDT) by Graham [23] and showed good qualitative agreement. In our

investigation we did our simulation on a configuration that has geometric properties similar

to their experimental setup for the purpose of comparison.

5

1.4 Flat Plates Interacting with Vortical Structures

The Blade-Vortex Interaction problem is a fundamental problem in aeroacoustics. Re-

searchers have formulated mathematical models of varying levels of fidelity, and obtained

both analytical and numerical solutions. Howe [25] has presented a comprehensive analyti-

cal treatment of sound radiated by the interaction of line vortices with a flat plate, among

other vortex sound problems. Glegg et al. [14] gave a recent review of theories for comput-

ing leading edge noise due to the interaction of a line vortex as it convects past an airfoil

of finite thickness. In Computational Aeroacoustics (CAA), the field equations that de-

scribe the mechanisms of sound generation and propagation are solved numerically. Delfs

et al. [15] solved the linearized Euler equations using a high-order finite difference method,

and determined the noise radiated by the interaction of a finite-core vortex with a sharp

edge. Delfs [16] also solved the same equations for the interaction problem and determined

the sound radiated by a 2D airfoil with a rounded leading edge. Grogger et al. [17] also

solved the linearized Euler equations, and determined the noise generated by the interaction

of localized three-dimensional vorticity with the leading edge of an airfoil. They studied the

effects of the airfoil’s thickness ratio on the strength and directivity of radiated noise. A

good review of experimental work on blade-vortex interaction is given by Wilder and Telio-

nis [18]. They used Laser-Doppler velocimetry to experimentally investigate two-dimensional

airfoil-vortex interaction. They used oscillating airfoil to generate the vortex which interacts

with a NACA632A015 airfoil at two different angles of attack; α = 0◦ (unloaded blade) and

α = 10◦ (loaded blade). Vorticity fields were constructed and surface pressure fluctuations

on the airfoil were determined. The flow Reynolds number is Re=19000 and Mach number is

nearly zero. They found that a vortex skimming over a blade at zero incidence does not in-

duce separation, and that the vortex quickly loses its strength because of the viscous effects.

Casper et al [22]. predicted the loading noise from unsteady surface pressure measurements

on a NACA0015 airfoil immersed in grid-generated turbulence. They predicted the far field

noise by using the time-dependent surface pressure as input to formulation A of Farassat,

6

a solution of the Ffowcs Williams-Hawkings equation. Moreau et al [20] performed an ex-

perimental investigation of the turbulence-interaction noise presented on various bodies of

different relative thickness. They found that the turbulence-interaction noise is reduced sig-

nificantly by increasing the airfoil thickness. Polacsek et al. [35] have presented a numerical

method for predicting turbofan noise due to rotor-stator interaction. Their computational

model is composed of three components: (1) a 3D RANS code to estimate spatial distribu-

tion and strength of noise sources, (2) an Euler solver for near field acoustic propagation

and (3) a Kirchhoff integral for far-field radiation. The source amplitude, obtained from

post-processing RANS output, is over-predicted in comparison with experiments, and it has

to be adjusted to match the in-duct measurements. This indicates that RANS codes are not

suitable for predicting noise sources due to rotor-stator interaction and calls for more detailed

modeling such as large-eddy simulations. Nevertheless, with the adjusted source strength,

the predicted sound pressure level and directivity patterns are in fairly good agreement with

experimental data.

1.5 Objectives

The objective of our work is to develop an efficient computational method, based on large

eddy simulation, to be used as an alternative to RDT in predicting the turbulence-cascade

interaction, and to address different numerical issues, like, using high-order finite difference

schemes, application of different wall boundary conditions, and use of non-reflecting bound-

ary conditions at inflow and outflow boundaries.

1.6 Accomplished Work

To validate the implementation of our code; we computed the unsteady loading and radiated

acoustic pressure field from flat plates interacting with vortical structures. We considered

7

two fundamental problems: (1) A linear cascade of flat plates excited by a vortical wave

(gust) given by a 2D Fourier mode, Ragab and Salem-Said ( [45], [46]). and (2) The parallel

interaction of a finite-core vortex with a single plate. We solved the two-dimensional non-

linear Euler equations over a linear cascade composed of six plates for a range of discrete

frequencies of the incident gust. We use Giles’ [12] nonreflecting boundary conditions at

the inflow and outflow boundaries, and study their performance at different frequencies of

the incident gust. These boundary conditions have been also investigated for the cascade

problem by Hixon et al. [26] and used by Sawyer et al. [42] for aeroacoustic prediction of

rotor-stator interaction noise. Giles’ conditions, being based on a Taylor series expansion for

small ratio of tangential wavenumber to frequency, are approximately nonreflecting. Rowley

and Colonius [39] (see also Colonius [7] for a review) have developed numerically nonreflect-

ing conditions. Yaguchi and Sugihara [51] have also proposed new nonreflecting boundary

conditions for multidimensional compressible flow. Prediction of radiated sound by a cas-

cade of blades due to interaction with turbulence can benefit from these new non-reflecting

conditions.

For the three-dimensional case we used large-eddy simulation to investigate the interaction

of homogeneous isotropic turbulence with a cascade of thin flat plates, and determine the

distortion of turbulence as it passes through the cascade, Salem-Said and Ragab [44]. We

consider a case in which the integral length scale of the incident turbulence is comparable

to the cascade pitch, hence we have to solve the flow over many passages simultaneously.

Periodicity of the instantaneous flow in the direction normal to the plates is determined

by the need to represent the large scales of the incident turbulence rather than by the

cascade pitch. The governing equations are based on the full Favre-filtered compressible

Navier-Stokes equations but with special treatment of solid walls. Because of the large

number of plates (six in the present investigation), it is not feasible to resolve the turbulence

within the viscous regions over those surfaces, especially for practical high Reynolds numbers.

Our approach aims at resolving the inviscid nonlinear mechanisms and the decay of the

incident turbulence. Hence, we impose the zero normal velocity condition and relax the

8

no-slip conditions to milder zero-shear or slip-wall conditions. This treatment amounts to

neglecting generation of wall turbulence but it will capture vorticity shedding from sharp

edges particularly the trailing vortex sheets. Therefore, the distortion of turbulence will be

dominated by the suppression of the normal velocity on the plates. In this way we will be

able to simulate incident flow at high turbulence Reynolds number but at the expense of

losing turbulence generation by mean shear in wall boundary layers and wakes. Our results

for the Reynolds stresses and energy spectra downstream of the cascade agree with Graham’s

RDT theory in its region of validity. We refer to this type of turbulence-cascade interaction

as shear-free interaction. Such boundary treatment have been used by Perot and Moin [40].

9

Chapter 2

Mathematical Model for Large Eddy

Simulation

The equations governing the flow field in our simulation are: Compressible Navier-Stokes

equations, the energy equation, and the equation of state. In this chapter, we present the

governing equations of the flow field in dimensional and nondimensional forms, the equations

for large eddy simulation, and finally, we discuss the subgrid scale model for the momentum

and the energy equations.

2.1 Navier-Stokes Equations

Navier-Stokes equations are written as:

Continuity Equation:∂ρ∗

∂t∗+

∂(ρ∗u∗j)

∂x∗j= 0 (2.1)

Momentum Equation:

10

∂(ρ∗u∗i )∂t∗

+∂

∂x∗j(ρ∗u∗i u

∗j + p∗δij) =

∂σ∗ij∂x∗j

(2.2)

Energy Equation:

∂(ρ∗E∗)∂t∗

+∂

∂x∗j

[(ρ∗E∗ + p∗)u∗j

]=

∂

∂x∗j(σ∗ij u∗i − q∗j ) (2.3)

Equation of State:

p∗ = ρ∗R∗T ∗ (2.4)

where ρ∗, u∗i , p∗, T ∗, and R∗ are density, velocity components, pressure, temperature, and gas

constant, respectively.

E∗ =p∗

(γ − 1)ρ∗+

1

2u∗i u

∗i (2.5)

σ∗ij = µ∗(

∂u∗i∂x∗j

+∂u∗j∂x∗i

− 2

3

∂u∗k∂x∗k

δij

)(2.6)

q∗j = −κ∗∂T ∗

∂x∗j(2.7)

where γ, µ∗, and κ∗ are the specific heat ratio, dynamic viscosity, and thermal conductivity

of the fluid particle at position x∗i , respectively.

2.2 Navier-Stokes Equations in Nondimensional Form

Using a length (C∗) and the free stream flow variables as reference values, we define the

following dimensionless variables:

t =t∗ U∗

∞C∗ , xi =

x∗iC∗ , ui =

u∗iU∗∞

, ρ =ρ∗

ρ∗∞, p =

p∗

ρ∗∞U∗2∞,

T =T ∗

T ∗∞, E =

E∗

U∗2∞, µ =

µ∗

µ∗∞, κ =

κ∗

κ∗∞, cp =

c∗pc∗p∞

11

Using the above dimensionless variables, we rewrite the Navier-Stokes equations in nondi-

mensional form as follows:

Continuity Equation:∂ρ

∂t+

∂(ρuj)

∂xj

= 0 (2.8)

Momentum Equations :

∂(ρui)

∂t+

∂(ρuiuj + pδij)

∂xj

=1

Re

∂σij

∂xj

(2.9)

Energy Equation:

∂(ρE)

∂t+

∂ [(ρE + p) uj]

∂xj

=1

Re

∂

∂xj

[σij ui − 1

Pr (γ − 1) M2∞qj

](2.10)

Equation of State:

p = ρR T (2.11)

where

E =p

(γ − 1)ρ+

1

2uiui (2.12)

σij = µ

(∂ui

∂xj

+∂uj

∂xi

− 2

3

∂uk

∂xk

δij

)(2.13)

qj = −κ∂T

∂xj

(2.14)

The Reynolds number Re, the Mach number M∞ and the Prandtl number Pr are given by

Re =ρ∗∞ U∗

∞ C∗

µ∗∞(2.15)

12

M∞ =U∗∞√

γR∗T ∗∞(2.16)

Pr =µ∗∞ c∗p∞

κ∗∞(2.17)

The nondimensional gas constant is given by

R =R∗ T ∗

∞U∗2∞

=1

γM2∞(2.18)

Here, we assume that µ and κ are functions of temperature, but we assume that c∗p to be

constant, hence cp = 1.

2.3 Equations For Large Eddy Simulation

The Favre-filtered governing equations for the resolved scales are (see Erlebacher et al. [32]

and Ragab and Sheen [37]) given as:

Continuity Equation:∂ρ

∂t+

∂(ρuj)

∂xj

= 0 (2.19)

Momentum Equation:

∂(ρui)

∂t+

∂(ρuiuj + pδij)

∂xj

=1

Re

∂σij

∂xj

− ∂τij

∂xj

(2.20)

Energy Equation:

∂(ρE)

∂t+

∂[(ρE + p)uj]

∂xj

=1

Re

∂

∂xj

[σijui − 1

Pr (γ − 1) M2qj

]

(2.21)

− ∂

∂xj

[cpQj]

Equation of State:

13

p = ρ R T (2.22)

where

E =p

(γ − 1)ρ+

1

2uiui (2.23)

σij = µ

(∂ui

∂xj

+∂uj

∂xi

− 2

3

∂uk

∂xk

δij

)(2.24)

qj = −κ∂T

∂xj

(2.25)

τij = ρ (uiuj − uiuj) (2.26)

Qj = ρ(ujT − ujT

)(2.27)

Here τij, and Qj represent the SGS (subgrid scale) stress tensor and heat flux, respectively.

In order to close the system of equations we need to model these terms.

2.4 Subgrid-scale Model

In large-eddy simulation, information from the resolved scales are used to model the stresses

of the unresolved scales by a sub-grid-scale (SGS) model. There is only one unclosed term in

the momentum equation, i.e., the SGS stress τij. The dynamic version of the Smagorinsky’s

eddy-viscosity model [43] introduced by Germano et al. [55] is used to model the SGS stress.

The model is

τij − 1

3τkkδij = −2Cρ∆2 |S|

(Sij − 1

3Skkδij

)(2.28)

14

where

Sij =1

2

(∂ui

∂xj

+∂uj

∂xi

), (2.29)

|S| =(2SijSij

) 12, (2.30)

∆ = (∆1 ∆2 ∆3)13 , (2.31)

∆ = (∆1 ∆2 ∆3)13 . (2.32)

Here C is the Smagorinsky coefficient that is assumed to be independent of filter width. In

the present study, ∆i = 2∆i is used, where () implies test filtering, see Toh [47].

We let

Lij = ρ ui uj −ρ ui

ρ uj

ρ , (2.33)

Mij = −2∆2ρ|S|Sdij + 2∆2 ¯ρ|S|Sd

ij , (2.34)

Sd

ij =Sij − 1

3Skkδij , (2.35)

Sdij = Sij − 1

3Skkδij . (2.36)

Then Smagorinsky coefficient is given by

C =

(Lij − 13Lkkδij

)Mij

MijMij

(2.37)

To avoid the large fluctuations in the values of C that may cause instability of numerical

solutions, we follow Zhang and Chen [52] and determine C as follows:

15

C =Ld

ijMij

Mij Mij

, Ldij = Lij − 1

3Lkkδij (2.38)

where the symbol represents double filtering, i.e., a grid filter (¯) is applied first followed

by a test filter ().

The unclosed term in the total energy equation is the SGS heat flux Qj.

Using the Smagorinsky’s eddy-diffusivity model, Qj is modeled as

Qj =ρνT

PrT

∂T

∂xj

= −C∆2ρ |S|

PrT

∂T

∂xj

(2.39)

where C is the eddy-viscosity coefficient given by Equation (2.38) and PrT is the turbulent

Prandtl number. Here, we assume PrT = 0.7.

We used the dynamic model, only, for the simulations of homogeneous turbulence without

the cascade, and used Smagorinsky model with constant coefficient C for turbulence cascade

interaction where the value of C is an average value based on the simulations without the

cascade.

16

Chapter 3

Numerical Method and Boundary

Conditions

3.1 Introduction

In large-eddy simulation; information from the resolved length scales are used to model

the stresses of the unresolved length scales by a sub-grid-scale (SGS) model. Therefore, it is

important that the numerical error be sufficiently small. Tolstykh [48] developed a fifth-order

non-centered compact scheme in which artificial dissipation is controllable. This scheme is

very attractive for large eddy simulation because it does not require explicit filtering as

with centered compact schemes and numerical dissipation can be minimized. Ragab and El-

Okda [36] have investigated the application of Tolstykh’s scheme to LES of temporal decay

of isotropic homogeneous turbulence and for flow over a surface-mounted prism. They found

that a procedure that combines a compact sixth-order scheme with Tolstykh’s compact

upwind fifth-order scheme (C6CUD5) produced accurate LES results. Compact centered

schemes are non-dissipative, and hence they require some kind of filtering for de-aliasing and

preventing odd-even decoupling in inviscid flow regions. We use, for the spatial discretization,

the compact sixth-order scheme which is nondissipative, and hence a filter is also used to

17

damp dispersed high wavenumbers. Our LES code contains two options; the first is to

combine the sixth-order scheme with a compact upwind fifth-order scheme due to Zhong [53].

The second is to use a tenth-order compact spatial filter as given by Visbal and Gaitonde

[49]. For the near-boundary points, we use successively lower even-order compact filters.

The low-storage five-stage fourth-order Runge-Kutta scheme of Carpenter and Kennedy [54]

is used to advance the solution in time. The details of the numerical model are given below.

3.2 Temporal Discretization

Following Toh [47], consider the model equation

dq

dt= L(q) (3.1)

where q is a flow variable, t is time and L is a spatial operator. The five-stage fourth-order

explicit Runge-Kutta scheme [54] is used to advance the solution in time. To advance the

solution from step n to n + 1, we use the algorithm

q0 = qn (3.2)

H0 = L(q0) (3.3)

Then for m = 1, ..., 4, we use

qm = qm−1 + bm∆tHm−1 (3.4)

Hm ← qm + amHm−1 (3.5)

18

m am

1 -0.41789047

2 -1.19215169

3 -1.69778469

4 -1.51418344

Table 3.1: The am coefficients of fourth-order Runge-Kutta scheme [54]

m bm

1 0.14965902

2 0.37921031

3 0.82295502

4 0.69945045

5 0.15305724

Table 3.2: The bm coefficients of fourth-order Runge-Kutta scheme [54]

and for the final step, we use

q5 = q4 + b5H4 (3.6)

qn+1 = q5 (3.7)

3.3 Spatial Discretization

We use a sixth-order compact finite-difference scheme (Lele [30]) for spatial derivatives. In

the nonperiodic x−direction, the scheme is given by:

19

c0 -3.314028 e0 -0.201984

c1 11.957647 e1 -1.073161

c2 -25.101298 e2 1.980626

c3 35.549424 e3 -0.986281

c4 -32.180018 e4 0.338185

c5 17.986367 e5 -0.064053

c6 -5.671575 e6 0.008399

c7 0.773480 e7 -0.001730

Table 3.3: Boundary-points fifth-order scheme coefficients

αf ′i−1 + f ′i + αf ′i+1 =1

h[a(fi+1 − fi−1) + b(fi+2 − fi−2)] (3.8)

where α = 1/3, a = 7/9, b = 1/36, and h is the grid step size. This scheme is applied for

i = 3, . . . , n− 2. Fifth-order explicit schemes (Carpenter et al. [6]) are used at the boundary

points i = 1, 2, n and n− 1:

i = 1:

f ′i =1

h(c0fi + c1fi+1 + c2fi+2 + c3fi+3 + c4fi+4 + c5fi+5 + c6fi+6 + c7fi+7) (3.9)

f ′i+1 =1

h(e0fi + e1fi+1 + e2fi+2 + e3fi+3 + e4fi+4 + e5fi+5 + e6fi+6 + e7fi+7) (3.10)

i = n:

f ′i = −1

h(c0fi + c1fi−1 + c2fi−2 + c3fi−3 + c4fi−4 + c5fi−5 + c6fi−6 + c7fi−7) (3.11)

f ′i−1 = −1

h(e0fi + e1fi−1 + e2fi−2 + e3fi−3 + e4fi−4 + e5fi−5 + e6fi−6 + e7fi−7) (3.12)

20

The coefficients are given by Carpenter et al. [6] and reproduced in Table(3.3). Because of

the singularities at the leading and trailing edges, we do not apply the sixth-order compact

scheme at points that straddle these edges on z−planes that coincide with the plates. Instead,

we use the explicit fifth-order schemes Eq (3.10) at the pints i = ile + 1 and ite + 1, and

Eq (3.12) at the points i = ile − 1 and i = ite − 1, where ile and ite are the indices of

the leading and trailing edges of the plate, respectively. The viscous and turbulent stresses

terms are evaluated using a fourth-order central difference scheme. Our present LES code

contains two options. In the first option, we follow the same procedure in Ragab and El-

Okda [36] but we replace Tolstykh’s fifth order scheme by a more efficient upwind scheme

due to Zhong [53]. In the second option, we use a tenth-order compact spatial filter as given

by Visbal and Gaitonde [50, 49]. For the near-boundary points, we use successively lower

even-order compact filters with a filter parameter given by αf = 0.5125− 0.01125mf , where

mf is the order of the filter. For example, αf = 0.40 for the tenth-order filter, αf = 0.4225

for the eighth-order filter, and so on. The filter is applied to the conservative variables once

every time step after the fifth stage of the Runge-Kutta scheme.

3.4 Boundary Conditions

On the upper and lower surfaces of each plate, we have two options: a zero-shear wall and

a slip-wall boundary conditions. For the zero-shear wall boundary condition, we simply

assign the flow variables of the grid points near the wall to the wall points. In the slip-wall

boundary condition, the Euler equations are solved on the upper and lower surfaces of each

plate. Poinsot-Lele [34] slip wall boundary condition is used. The flux vector derivative

parallel to the plate (x−operator) is evaluated using the sixth-order compact scheme as

shown above at the points ile + 2 ≤ i ≤ ite − 2. The velocity derivatives normal to the

wall (z−direction) are evaluated by a one-sided explicit third-order scheme and the pressure

derivative is evaluated by a first-order scheme. The flow variables at the leading and trailing

edges are obtained by averaging the solutions at the two nearest field points above and below

21

the edges.

At the inflow and outflow boundaries, we use Giles’ nonreflecting boundary conditions. Here,

we present them for two-dimensional flow in the xz-plane. They have been generalized to 3D

flow by Hagstrom and Goodrich [24] which we use in the simulations presented in chapter (7).

At the inflow boundary, the incoming flow is composed of two contributions: A uniform flow

which is parallel to the plates (U∞,W∞ = 0, ρ∞ and p∞), and either a turbulent fluctuation,

as in chapters (6,7), or a vortical gust with prescribed velocity components, as in chapter (4),

(u, w) but without pressure or density variations. Additional perturbations at the boundary

are due to the interaction of the turbulence (gust) with the cascade (u′, w′, ρ′ and p′). For

the purpose of applying the boundary conditions only, we write

u = U∞ + u + u′ (3.13)

w = W∞ + w + w′ (3.14)

p = p∞ + p′ (3.15)

ρ = ρ∞ + ρ′ (3.16)

In developing nonreflection boundary conditions, Giles [12] linearized the 2D Euler equations

about a uniform basic state. The one-dimensional characteristic variables are related to the

field perturbations (u′, w′, ρ′ and p′) by

c1 = p′ − ρ∞c∞u′ (3.17)

c2 = c2∞ρ′ − p′ (3.18)

c3 = ρ∞c∞w′ (3.19)

(3.20)

c4 = p′ + ρ∞c∞u′ (3.21)

At the inflow boundary, we determine the time derivatives of the incoming characteristic

variables from Giles’ conditions:

22

∂c2

∂t+ W∞

∂c2

∂z= 0 (3.22)

∂c3

∂t+ W∞

∂c3

∂z+

1

2(U∞ + c∞)

∂c4

∂z− 1

2(U∞ − c∞)

∂c1

∂z= 0 (3.23)

∂c4

∂t+ W∞

∂c4

∂z− 1

2(U∞ − c∞)

∂c3

∂z= 0 (3.24)

where the z−derivative is evaluated by a fourth-order central difference formula. We also

determine the time derivative of the outgoing characteristic variable c1 from

∂c1

∂t= −(U∞ − c∞)(

∂p′

∂x− ρ∞c∞

∂u′

∂x) (3.25)

where the x−derivative is evaluated by a fifth-order one-sided explicit scheme [6] using infor-

mation within the computational domain. The values of the perturbations and characteristic

variables used to evaluate the x− and z−derivatives are updated at each of the five stages

of the Ruge-Kutta scheme. As shown by Hixon et al. [26], the time derivative of the char-

acteristic variables can be used to obtain the derivative of the conservative variables on the

boundary. We take the time derivative of equations (3.17)- (3.21), and then use the known

derivatives of the characteristic variables to determine the derivatives of the fluctuations. To

take into account the gust at the inflow, we determine the time derivatives of the conservative

variables by

∂ρ

∂t=

∂ρ′

∂t(3.26)

∂ρu

∂t= u

∂ρ′

∂t+ ρ(

∂u′

∂t+

∂u

∂t) (3.27)

∂ρw

∂t= w

∂ρ′

∂t+ ρ(

∂w′

∂t+

∂w

∂t) (3.28)

∂ρE

∂t=

1

γ − 1

∂p′

∂t+ ρu(

∂u′

∂t+

∂u

∂t) + ρw(

∂w′

∂t+

∂w

∂t) +

1

2(u2 + w2)

∂ρ′

∂t(3.29)

At the outflow boundary, the time derivative of the incoming characteristic are given by

23

∂c1

∂t+ w

∂c1

∂z+ u

∂c3

∂z= 0 (3.30)

whereas the changes in the outgoing characteristics (c2, c3, c4) are obtained from information

within the computational domain.

24

Chapter 4

Response of a Flat-Plate Cascade to

Incident Vortical Waves - 2D

Calculations

4.1 Introduction

In order to verify the validity of our code and precisely examine the boundary conditions, we

compute the unsteady lift and radiated sound from a flat plate cascade due to an incident

vortical wave (gust). We solve the two-dimensional nonlinear Euler equations over a linear

cascade by a high-order finite-different method. We use Giles’ nonreflecting boundary condi-

tions at the inflow and outflow boundaries. We compare our results with Glegg’s linearized

potential flow solution.

In the present simulations, we consider an unstaggered six-blade linear cascade as shown in

Figure (4.1). The plates have zero thickness and are at zero incidence relative to the mean

flow. The blade-to-blade spacing (s) is 0.806c, where c is the blade chord. This geometric

25

parameter is motivated by the cascade experiments conducted by Larssen and Devenport [28]

(see also Larssen [29]).

C

S

Inflow

Outflow

x=0 x=C

k

y=0

y=6s

Periodic

Periodic

Figure 4.1: Flat plate cascade and computational domain.

4.2 Governing Equations

Euler equations

We solve the full nonlinear Euler equations in the conservative form on a uniform Cartesian

grid.

26

∂ρ

∂t+

∂ρuj

∂xj

= 0, (4.1)

∂ρui

∂t+

∂(ρuiuj + pδij)

∂xj

= 0, (4.2)

∂ρE

∂t+

∂(ρE + p)uj

∂xj

= 0 (4.3)

where

E =p

(γ − 1)ρ+

1

2uiui. (4.4)

A perfect gas with specific heat ratio γ = 1.4 is assumed.

p = ρR T (4.5)

The coordinates and flow variables are made non-dimensional by using the plate chord, c, the

free-stream velocity U∞, density ρ∞, and temperature T∞ as reference values. The reference

pressure is ρ∞U2∞. In this chapter we consider two-dimensional vortical waves only, hence

the gust velocity field (u, w) is divergence free; it is given by:

u = uoei(k1x+k2z−ωt) + cc (4.6)

w = woei(k1x+k2z−ωt) + cc (4.7)

where uo = −k2wo/k1 and ω = k1U∞. In the above equations, “cc” stands for the complex

conjugate of the preceding term. We assume that k1 = k2 = 2πn/(snb), where n is an integer

and nb is the number of blades in the computational domain. All the results reported here

are obtained for wo = 0.01U∞.

27

4.2.1 Glegg’s Linearized Potential Flow Solution

Glegg [13], see Appendix (B), has developed a complete solution to the response of a stag-

gered flat plate cascade to incident 3D plane waves. He solved the compressible linearized

potential flow equation, and accounted for the finite chord of the blades. He provided ana-

lytical expressions for the unsteady normal force and the upstream and downstream sound

power; all as functions of the wavenumber and frequency of the incident gust and the geo-

metric properties of the cascade. We have developed a linear-theory code based on Glegg’s

analytical solution, and checked it by reproducing the normal force and sound power spectra

for all of the cases reported in Glegg [13]. We then use the linear code to provide data for

comparison with the nonlinear Euler calculations.

Following Glegg, we define the normal force coefficient CN by CN = L/πwoρ∞U∞c, where L

is the normal force per unit span; obtained as an integral along the chord of the pressure jump

across the plate. The sound power per unit span, W± is also normalized by w2oρ∞U∞snb/2.

(The upper sign is for upstream radiation and lower sign is for downstream radiation.) The

normal force magnitude and sound power for the cut-on modes at free stream Mach number

M∞ = 0.3 as function of the reduced frequency of the incident gust, κ = ωc/2U∞, are shown

in Figure (4.2). The second (m=1) and third (m=2) acoustic modes are cut-on, each over a

range of frequencies, but they do not overlap. There is a small window of frequencies where

both modes are cut-off (κ = 5.70125 to κ = 5.915). The frequency for wavenumber n = 9

is κ = 5.8466 which falls in that window. Upstream sound power (given by dashed lines)

is less than the downstream sound power (given by the solid lines). The normal force and

sound power results at M∞ = 0.5 are depicted in Figure (4.3).

28

Reduced Frequency

Mag

nitu

de

ofLi

ftC

oef

ficie

nt

Nor

mal

ized

Sou

ndP

ow

er

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

0

0.01

0.02

0.03

0.04

0.05

0.06

Magnitude of Lift Coeff.m=1 Upstream Soundm=1 Downstream Soundm=2 Upstream Soundm=2 Downstream Sound

m=1

m=2

Figure 4.2: Unsteady lift response and sound power using Glegg’s linear solution at Mach

number M∞ = 0.3.

29

Reduced Frequency

Mag

nitu

de

ofLi

ftC

oef

ficie

nt

Nor

mal

ized

Sou

ndP

ow

er

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Magnitude of Lift Coeff.m=1 Upstream Soundm=1 Downstream Soundm=2 Upstream Soundm=2 Downstream Sound

m=1

m=2

Figure 4.3: Unsteady lift response and sound power using Glegg’s linear solution at Mach

number M∞ = 0.5.

4.2.2 Two-Dimensional Euler Simulations

We integrated the unsteady two-dimensional nonlinear Euler equations in time on a Cartesian

uniform grid. First, we present the unsteady lift spectrum for a range of frequencies of the

incident gust. We obtained numerical solutions for twelve separate runs corresponding to

n = 1, 2, ..., 12. In each run, the normal force on each of the six plates was computed by

integrating the pressure jump on the plate, and its time history was recorded. Excluding

a transient period, we decomposed the normal force coefficient into Fourier modes in time.

The magnitude of the mode whose frequency is equal to the gust frequency was averaged

over the six plates. The spectrum of the obtained normal force magnitude is depicted in

30

Figures( 4.4) and ( 4.5) at the two Mach numbers M∞ = 0.3 and 0.5, respectively. Overall,

the agreement between the present Euler calculations and Glegg’s linear solution is very

good. The sensitivity of the surface pressure distribution to grid resolution and domain

length will be discussed next.

Reduced Frequency

Mag

nitu

de

ofLi

ftC

oef

ficie

nt

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Linear Theory (Glegg)2D Euler, M=0.3

6

8

12

9

Figure 4.4: Comparison of unsteady lift response with Glegg’s linear solution, M = 0.3.

31

Reduced Frequency

Mag

nitu

de

ofLi

ftC

oef

ficie

nt

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Linear Theory (Glegg)2D Euler, M=0.5

Figure 4.5: Comparison of unsteady lift response with Glegg’s linear solution, M = 0.5.

We investigate in more detail the pressure field and excited acoustic modes for three fre-

quencies corresponding to n = 11, 8 and 9 at Mach number M∞ = 0.3. As we shall see, each

frequency results in a qualitatively different cascade response. We use five grids as shown in

Table (1). Grids A, B, C and D are used to evaluate sensitivity to the step sizes (∆x = ∆z),

whereas grids C, E, and F are used for sensitivity to the streamwise extent of the computa-

tional domain. In this table, ns and nc are the number of points on the cascade pitch s and

on the plate chord c, respectively. Lx is the streamwise extent of the computational domain

and ∆t is the time step.

Test case 1: n=11

A snapshot of pressure contours (p − 1/γM2∞) for incident gust with mode number n = 11

is shown in Figure( 4.6). For this gust, the reduced frequency is κ = 7.146, for which the

32

Table 1 Grid parameters for test cases 1-3

Grid ns nc Lx ∆t

A 25 31 5.0375 0.4198× 10−2

B 49 61 5.0375 0.2099× 10−2

C 97 120 5.0375 0.1049× 10−2

D 193 239 5.0375 0.5247× 10−3

E 97 120 7.0525 0.1049× 10−2

F 97 120 9.0675 0.1049× 10−2

third acoustic mode is cut-on. Pressure fluctuations propagate towards the lower left corner

upstream of the cascade and towards the lower right corner downstream of the cascade. We

decompose the pressure field p(x, z, t) into a double Fourier series in z and t, each mode is

of the general form

p(x, ν, ωl) = plν(x)e−iωlt+iνk2z (4.8)

The cascade response at the forcing frequency ωl = ω includes only one propagating mode

ν = −1 as shown in figure ( 4.2). The amplitude |plν(x)| for this mode is depicted in

Figure( 4.7) upstream of the leading edge and downstream of the trailing edge for the four

grids A, B, C, and D. The upstream radiated pressure agrees very well with predictions

using Glegg’s [13] linear theory. Reflection from the inflow boundary is negligible. As we

refine the grid, downstream radiation converges to the linear theory prediction. However,

a small wave reflection from the outflow boundary is evident by the weak variation in the

wave amplitude. In addition to the propagating mode (ν = −1), there are other modes that

decay exponentially upstream and downstream of the cascade. The dominant exponentially

decaying mode is (ν = −7), which is also depicted in Figure( 4.7). This mode shows very

little sensitivity to grid resolution and is not influenced by reflection from the inflow or

outflow boundaries.

The pressure jump across a plate ∆p(x, t) is decomposed into Fourier series in time; of which

33

a mode is

∆p(x, ωl) = ∆pl(x)e−iωlt (4.9)

At the gust frequency ωl = ω, the real and imaginary parts of ∆pl(x) are compared to

predictions of Glegg’s linear theory in Figure( 4.8) for grids A, B, C and D. Grid convergence

is shown. We note here that the coarse grid A gives 13 points per wave length whereas the

fine grid D gives 105 points. The singularity in ∆p(x, t) at the leading edge is difficult to

resolve, nevertheless the predicted pressure distribution varies smoothly there. Near the

trailing edge we also see a small “glitch” in the pressure. Sensitivity of the surface pressure

distribution to the extent of the computational domain in the streamwise direction is a good

indicator of the reflections from the inflow and outflow boundaries. With the leading edge

at x = 0, the inflow boundary is placed at x = −2,−3, and -4 for the three grids C, E and F,

respectively. The domain length Lx is given in Table (1). The surface pressure distributions

for the three domains are shown in Figure( 4.9) along with the linear theory prediction. For

this frequency the effects of the domain length are negligible. Reflection from the boundaries

is negligible because the wavenumber vector of the excited acoustic mode makes a small angle

with the normal to the boundary, which is the right condition for the application of Giles’

nonreflecting boundary conditions.

34

Figure 4.6: Test case 1: A snapshot of pressure contours.

35

x/c

p

-2 -1 0 1 2 3-0.001

0

0.001

0.002

0.003

0.004Linear TheoryEuler (Grid ns=25)Euler (Grid ns=49)Euler (Grid ns=97)Euler (Grid ns=193)

Plate

UpstreamRadiation

DownstreamRadiation

Figure 4.7: Test case 1: Pressure amplitudes for propagating mode ν = −1 and decaying

mode ν = −7.

36

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02


Imaginary Part

Real Part

Figure 4.8: Test case 1: Sensitivity of surface pressure jump to grid step sizes.

37

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03Linear TheoryEuler (Lx=5.0375)Euler (Lx=7.0525)Euler (Lx=9.0675)

Imaginary Part

Real Part

Figure 4.9: Test case 1: Sensitivity of surface pressure jump to streamwise domain length.

Test case 2: n=8

A snapshot of pressure contours is shown in Figure( 4.10) for incident gust with mode number

n = 8. (Because the domain of six blades includes two wave lengths in the z−direction, only

half of the domain is shown in the figure.) For this gust, the reduced frequency is κ = 5.197,

for which the second acoustic mode is cut-on. Upstream of the cascade, pressure fluctuations

propagate towards the upper left corner, and downstream of the cascade they propagate

towards the upper right corner. Reflection from the outflow boundary is evident and is more

significant than from that at the inflow boundary. This is because the wavenumber vector has

38

larger tangential component at outflow. Reflection from the outflow boundary contaminates

the pressure field causing significant dependence of the surface pressure distribution on the

locations of the inflow and outflow boundaries.

The cascade response at the forcing frequency ωl = ω includes only one propagating mode

ν = 2. The amplitude |plν(x)| of radiated pressure for this mode is depicted in Figure( 4.11)

upstream of the leading edge and downstream of the trailing edge for the three grids A, B,

and C. With grid refinements, the upstream radiated pressure converges to the predictions

using Glegg’s [13] linear theory. The undulations in the pressure amplitude are about 5% of

the mean value. However, stronger undulations are observed for the downstream radiated

wave because of reflection from the outflow boundary. (Because of the significant reflection

from the downstream boundary, we felt that it is not necessary to obtain results for the

finest grid D.) In addition to the propagating mode (ν = 2), there are other modes that

decay exponentially upstream and downstream of the cascade. The dominant exponentially

decaying mode is (nu = −4), which is also depicted in Figure( 4.11). This mode shows

very little sensitivity to grid resolution and is not influenced by reflection from the inflow or

outflow boundaries.

At the gust frequency ωl = ω, the real and imaginary parts of surface pressure jump ∆pl(x)

are compared to predictions of Glegg’s linear theory in Figure( 4.12) for grids A, B, and C.

Comparison with linear theory predictions is poor. And as shown in Figure( 4.13) the surface

pressure distribution is very sensitive to the locations of the inflow and outflow boundaries.

In this case n = 8 reflection from the boundaries is significant because the wavenumber

vector makes a large angle with the normal to the boundary, for which Giles’ nonreflecting

boundary conditions breakdown; especially at the outflow boundary. This is a challenging

case for nonreflecting boundary conditions.

39

Figure 4.10: Test case two: A snapshot of pressure contours.

40

x/c

p

-2 -1 0 1 2 3-0.001

0

0.001

0.002

0.003

0.004Linear TheoryEuler (Grid ns=25)Euler (Grid ns=49)Euler (Grid ns=97)

Plate

UpstreamRadiation

DownstreamRadiation

Figure 4.11: Test case two: Pressure amplitudes for propagating mode ν = 2 and decaying

mode ν = −4.

41

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

0.03

0.04Linear TheoryEuler (Grid ns=25)Euler (Grid ns=49)Euler (Grid ns=97)

Real Part

Imaginary Part

Figure 4.12: Test case two: Sensitivity of surface pressure jump to grid step sizes.

42

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

0.03


Real Part

Imaginary Part

Figure 4.13: Test case two: Sensitivity of surface pressure jump to streamwise domain length.

Test case 3: n=9

Pressure contours for incident gust with mode number n = 9 are shown in Figure( 4.14).

(Because the domain of six blades includes three wave lengths in the z−direction, only

one third of the domain is shown in the figure.) For this gust, the reduced frequency is

κ = 5.8466, which falls in the frequency range where no acoustic mode is cut-on as shown

in figure ( 4.2). Pressure fluctuations are given by standing waves that are dominant in the

near field and decay exponentially upstream and downstream of the cascade. The pressure

field exhibits a node at x = 0.428 from the plate leading edge.

43

At the gust frequency ωl = ω, the real and imaginary parts of surface pressure jump ∆pl(x)

are compared to predictions of Glegg’s linear theory in Figure( 4.15) for grids A, B, C and

D. Grid convergence is shown, and excellent agreement with the linear theory is obtained.

The surface pressure distributions for different domain lengths are shown in Figure( 4.16)

along with the linear theory prediction. It is evident that the effects of the domain length

are negligible.


44

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

0.03


Real Part

Imaginary Part


45

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

0.03


Real Part

Imaginary Part


Test case 4: n=5

Next, we present results for the benchmark problem considered by Hixon et al. [26]. The

cascade is made of four blades with pitch s = 1. The convected vortical gust is given by

Eqs ( 4.6) and ( 4.7) for n = 5, and the Mach number is M∞ = 0.5. Table (2) gives the

parameters for the five grids used to investigate sensitivity to step sizes and domain length.

A snapshot of pressure contours is shown in Figure( 4.17). For this gust, the reduced

frequency is κ = 7.854, for which the second acoustic mode is cut-on. Upstream of the

cascade, pressure fluctuations propagate towards the upper left corner, and downstream of

46

the cascade they propagate towards the upper right corner. The cascade response at the

forcing frequency ωl = ω includes only one propagating mode ν = 1. The amplitude |plν(x)|for this mode is depicted in Figure( 4.18) upstream of the leading edge and downstream of the

trailing edge for the four grids A, B, C and D. The upstream radiated pressure is 5% higher

than that predicted by using Glegg’s [13] linear theory. Reflection from the inflow boundary

is negligible. As we refine the grid, downstream radiation converges to the linear theory

prediction. However, the weak undulations in the wave amplitude indicate a small wave

reflection from the outflow boundary. In addition to the propagating mode (ν = 1), there

are other modes that decay exponentially upstream and downstream of the cascade. The

dominant exponentially decaying mode is (ν = −3), which is also depicted in Figure( 4.18).

This mode shows very little sensitivity to grid resolution and is not influenced by reflection

from the inflow or outflow boundaries. It dominates the the radiated pressure in the near

field.

At the gust frequency ωl = ω, the real and imaginary parts of ∆pl(x) are compared to

predictions of Glegg’s linear theory in Figure( 4.19) for grids A, B, C and D. Grid conver-

gence is shown. Sensitivity of surface pressure distributions to domain length is shown in

Figure( 4.20) for the grids C, E, and F, and as shown the effects of the domain length are

negligible. In this case, reflection from the boundaries is much smaller than in case 2.

The streamwise wavenumbers for the acoustic waves radiated upstream (k+x ) and downstream

(k−x ) are shown in Table 3. Also shown is cosθ±, where θ is the angle that the wavenumber

vector makes with the outward normal to the boundary. The outflow boundary in Case 2

for which (cosθ− = 0.375) suffers the most wave reflection as we have shown, and it calls for

applications of higher order nonreflecting boundary conditions such as those developed by

Hagstrom and Goodrich [24].

47

Table 2 Grid parameters for test case 4

Grid ns nc Lx ∆t

A 25 25 5 0.6173× 10−2

B 49 49 5 0.3086× 10−2

C 97 97 5 0.1543× 10−2

D 193 193 5 0.7716× 10−3

E 97 97 7 0.1543× 10−2

F 97 97 9 0.1543× 10−2


48

x/c

p

-2 -1 0 1 2 3-0.001

0

0.001

0.002

0.003

0.004

0.005


Plate

UpstreamRadiation

DownstreamRadiation

Figure 4.18: Test case 4: Pressure amplitudes for propagating mode ν = 1 and decaying

mode ν = −3.

49

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Linear TheoryEuler (Grid ns=25)Euler (Grid ns=49)Euler (Grid ns=97)Euler (Grid ns=193)

Imaginary Part

Real Part


50

Table 3 Wavenumbers for

upstream and downstream propagating modes

Case n ν kz k+x k−x cosθ+ cosθ−

1 11 -1 -1.299 -5.924 3.097 0.977 0.922

2 8 2 2.599 -3.107 1.051 0.642 0.375

4 5 1 1.571 -7.530 2.294 0.979 0.825

x/c

(Co

mp

lex)

Dp

0 0.2 0.4 0.6 0.8 1-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Linear TheoryEuler (Lx=5)Euler (Lx=7)Euler (Lx=9)

Imaginary Part

Real Part


51

4.3 CONCLUSIONS

We considered the response of a flat plate cascade to two-dimensional vortical waves (gust).

We solved the two-dimensional nonlinear Euler equations over a linear cascade composed of

six plates for a range of frequencies of the incident gust. The cascade is unstaggered and

the pitch to chord ratio is 0.806. We use Giles’ [12] non-reflecting boundary conditions at

the inflow and outflow boundaries. We analyzed the cascade response in terms of unsteady

normal force, surface pressure distribution and radiated acoustic pressure field for three

discrete frequencies of the incident gust.

The lift spectrum agrees very well with Glegg’s [13] solution to the linearized potential flow

equation for the tested range of reduced frequency (0 < ωc/2U∞ < 8 ). Since Giles’ boundary

conditions are approximately nonreflecting, we have investigated undesirable wave reflection

at the inflow and outflow boundaries and its variation with gust frequency. Certain frequen-

cies excite acoustic modes whose wavenumber vectors are nearly normal to the boundary.

In such cases, minor reflection at the boundary is obtained and the results are insensitive to

the location of the computational domain boundaries (cases 2 and 4 of this chapter). Other

frequencies may still excite acoustic modes whose wavenumber vectors deviate considerably

from the normal direction to the boundary resulting in major reflection that contaminates the

pressure field (case 2). In such a case, the numerical solution depends sensitively on the loca-

tions of the inflow/outflow boundaries. If the gust frequency is such that all acoustic modes

are cut off, the pressure field decays exponentially towards the boundaries, and boundary

treatment poses no problem (case 3). These observations are consistent with the basic as-

sumption in Giles’ derivation of the approximately nonreflecting boundary conditions, which

is based on a Taylor series expansion for small ratio of tangential wavenumber to frequency.

Rowley and Colonius [39] (, Colonius [7] for a review) and Hagstrom and Goodrich [24]

have developed more accurate numerically nonreflecting conditions. Prediction of radiated

sound by a cascade of blades due to interaction with turbulence which includes a spectrum

of frequencies can greatly benefit from these new nonreflecting boundary conditions.

52

Chapter 5

Vortex-Plate Interaction

5.1 Introduction

The Blade-Vortex Interaction problem is a fundamental problem in aeroacoustics. Re-

searchers have formulated mathematical models of varying levels of fidelity, and obtained

both analytical and numerical solutions. Howe [25] has presented a comprehensive analyti-

cal treatment of sound radiated by the interaction of line vortices with a flat plate, among

other vortex sound problems. Glegg et al. [14] gave a recent review of theories for comput-

ing leading edge noise due to the interaction of a line vortex as it convects past an airfoil

of finite thickness. In Computational Aeroacoustics (CAA), the field equations that de-

scribe the mechanisms of sound generation and propagation are solved numerically. Delfs

et al. [15] solved the linearized Euler equations using a high-order finite difference method,

and determined the noise radiated by the interaction of a finite-core vortex with a sharp

edge. Delfs [16] also solved the same equations for the interaction problem and determined

the sound radiated by a 2D airfoil with a rounded leading edge. Grogger et al. [17] also

solved the linearized Euler equations, and determined the noise generated by the interaction

of localized three-dimensional vorticity with the leading edge of an airfoil. They studied the

53

effects of the airfoil’s thickness ratio on the strength and directivity of radiated noise.

In this chapter we solve the two-dimensional nonlinear Euler equations to study the effects

of the internal structure of a vortex (i.e. the radial distribution of vorticity) on the sound

generated by the interaction of a finite-core vortex with a flat plate of zero thickness.

5.2 Vortex-Plate Interaction

The Incident Vortex

A 2D vortex flow that is an exact solution to the steady compressible Euler equations can

be constructed in polar coordinates by assuming vr = 0 and

vθ = f(r) (5.1)

ρ = g(r) (5.2)

where f(r) and g(r) are arbitrary functions of r. It can be shown that the above fields satisfy

the continuity and θ-momentum equations. The pressure p(r) is obtained by integrating the

r−momentum equation,dp

dr= ρ

v2θ

r(5.3)

and the temperature is obtained from the equation of state. Because pressure and density

are functions of r only, it follows that any other thermodynamic property, such as entropy

S, is also a function of r only. Hence the energy equation

DS

Dt= 0 (5.4)

is satisfied. Instead of choosing the density g(r) arbitrarily, we choose the entropy to be

uniform in space, S = So, and use the isentropic equation

p

p∞=

(ρ

ρ∞

)γ

(5.5)

54

The final results arep

p∞=

(1− γ − 1

c2∞F (r)

) γγ−1

(5.6)

T

T∞= 1− γ − 1

c2∞F (r) (5.7)

ρ

ρ∞=

(1− γ − 1

c2∞F (r)

) 1γ−1

(5.8)

where

F (r) =

∫ ∞

r

f 2

rdr (5.9)

and c∞ is the speed of sound at the reference conditions (c2∞ = γp∞/ρ∞).

Oseen Vortex

The velocity profile

vθ(r) = f(r) =B

r(1− e−βr2

) (5.10)

is known as the Oseen vortex [4], for which we obtain

F (r) = βB2[(1− e−ζ)2

2ζ+ E1(ζ)− E1(2ζ)] (5.11)

where ζ = βr2, and E1(ζ) is the exponential integral

E1(ζ) =

∫ ∞

1

e−ζt

tdt (5.12)

The parameters β and B are related to the core radius ro and the corresponding maximum

circumferential velocity vo by

β =co

2r2o

(5.13)

where co = 1.256431208, and

B =voro

1− e−co(5.14)

We can convect the vortex by superposing a uniform flow (U∞,W∞) onto the velocity field

of the vortex. If the vortex center is at (xc, zc), the cartesian components of velocity will be

u(x, z) = U∞ − (z − zc)B

r2(1− e−βr2

) (5.15)

55

w(x, z) = W∞ + (x− xc)B

r2(1− e−βr2

) (5.16)

where r2 = (x− xc)2 + (z − zc)

2.

Taylor Vortex

The velocity profile

vθ(r) = f(r) = Are−αr2

(5.17)

is known as the Taylor vortex [4], for which we get

F (r) =A2

4αe−2αr2

(5.18)

The parameters α and A are related to the core radius ro and the corresponding maximum

circumferential velocity vo by

α =1

2r2o

(5.19)

and

A =vo

ro

√e (5.20)

Similarly, the Cartesian components of velocity will be

u(x, z) = U∞ − A(z − zc)e−αr2

(5.21)

w(x, z) = W∞ + A(x− xc)e−αr2

(5.22)

The circumferential velocity vθ(r), circulation Γ(r) and vorticity ωy(r) for the Oseen and

Taylor vortices are shown in Figure (5.2). We note that the Taylor vortex has a core region

of positive vorticity (clockwise) and an outer region of negative vorticity, and that the circu-

lation approaches zero as r →∞. The Oseen vortex has one sign vorticity and its circulation

is finite as r →∞, and hence its far field decays like 1/r whereas that of the Taylor vortex

decays exponentially.

The coordinates and flow variables are made non-dimensional by using the plate chord, c, the

free-stream velocity U∞, density ρ∞, and temperature T∞ as reference values. The reference

56

x

z

U

ro=0.125Cvo=0.01U

C

Figure 5.1: Parallel vortex-plate interaction.

r/ro0 1 2 3 4

0

2

4

6

8

10

Oseen-VelocityOseen-CirculationOseen-VorticityTaylor-VelocityTaylor-CirculationTaylor-Vortcity

5 v/vo

Figure 5.2: Flow properties of Oseen and

Taylor vortices.

pressure is ρ∞U2∞. The vortex radius is 0.125, where the maximum circumferential velocity

is 0.01. The free stream Mach number is M∞ = 0.5.

The two-dimensional nonlinear compressible Euler equations are solved on a Cartesian grid

in the x− z plane. The computational domain is the square (−9 ≤ x ≤ 11, −10 ≤ z ≤ 10),

and a uniform grid is used where the step sizes are ∆x = ∆z = 0.0078125. The time step

is ∆t = 0.00125. The flow field is initialized by the superposition of the vortex field and

a uniform flow, such a field is an exact solution to the Euler equations. Initially, the plate

and vortex axis are contained in one plane with the axis being parallel to and upstream of

the plate leading edge as shown in Figure (5.1). Therefore, the plate may split the vortex

along its axis as it convects past the leading edge. Vortex shedding from the sharp leading

and trailing edges is captured by the numerical solution to the nonlinear Euler equations.

As the vortex convects with the uniform free stream along the x−axis from left to right, the

flat plate is suddenly introduced with its leading edge at x = 0 and trailing edge at x = 1.

At this instant the vortex axis is upstream of the leading edge at x = −3.

57

Figure 5.3: Oseen vortex, a snapshot of

vorticity field, t = 2.


pressure filed, t = 2.




pressure field, t = 3.

58

Snapshots of vorticity and pressure coefficient fields [cp = 2(p − p∞)/ρ∞U2∞] at different

stages of the interaction of the vortex with the plate are analyzed. The results for the Oseen

vortex are shown in Figures (5.3) to (5.10). Vorticity contours at time t = 2 (vortex center is

at x = −1) show a vortex sheet, that is of the same sign as the incident vorticity, emanating

from the trailing edge as shown in Figure (5.3) (actually vorticity in the wake spreads over

a few grid lines, but we will refer to it as a vortex sheet). Because the velocity field of the

Oseen vortex decays slowly, the sudden application of the wall boundary conditions generates

two transient pulses that propagate above and below the plate. These are shown by the two

outermost pressure arcs in Figure (5.4). As the vortex convects towards the leading edge,

it induces a downwash velocity on the plate which generates compression and rarefaction

waves from the upper and lower sides, respectively. At time t = 3, the vortex center is

now at the leading edge (x = 0), and the thin flat plate splits the vortex core as shown in

Figure (5.5). At time t = 4, the split vortex is now at the trailing edge of the plate (x = 1),

a vortex sheet of the same sign vorticity as the core vorticity is visible in the wake as shown

in Figure (5.7). A vortex sheet of opposite vorticity also commences at the trailing edge.

Immediately after the vortex center passes the leading edge it induces an upwash velocity

there, and acoustic pressure waves, opposite in phase to the earlier waves, propagate above

and below the plate as shown in Figure (5.8). At time t = 6, a vortex sheet opposite in

sign to the core vorticity is sandwiched between the two parts of the split vortex as shown

in Figure (5.9). By this time the interaction is complete, and radiated pressure filed is fully

formed as shown in Figure (5.10).

The interaction of a Taylor vortex with the plate shows similar characteristics as the inter-

action with Oseen vortex, but it is much delayed. The interaction remains negligible until

the vortex center passes by x = −0.25, which is one vortex diameter upstream of the leading

edge. Before this time the vorticity and pressure fields are dominated by the near field of the

vortex flow. Vorticity and pressure fields are shown in Figures (5.11) and (5.12), respectively.

In Figure (5.11), it is interesting to note that because the vorticity of the incident vortex

is of mixed sign (positive in the core and negative outside) the vortex sheet separating the

59





two parts of the split vortex is also of mixed sign. The pressure field in Figure (5.12) shows

that an observer in the far field above or below the plate receives two consecutive pulses.

The pulses above the plate are out of phase relative to those below the plate. The phase is

determined by the sense of rotation of the vortex, which is clockwise in this case. Maximum

acoustic pressure is radiated downstream relative to a line normal to plate at the leading

edge. No radiation in the streamwise direction is detected in the plane of the plate.

60





Howe [25] shows that the linear theory of the low Mach number, two-dimensional interaction

of a line vortex with a flat plate predicts a dipole directivity pattern on a circle in the far

field with center at the midpoint of the plate. In the present simulations, the Mach number

is 0.5, and hence the effect of convection on the directivity cannot be neglected. At t = 6,

the first pressure pulse falls on a circle with radius r = 6.2 and center at x = 3.1 and z = 0.

as a result of convection. Directivity of the pressure amplitude on this circle for the two

vortices is shown in Figure (5.13). The dipole character of the directivity pattern is clear.

However, in the case of Oseen vortex the pattern is inclined towards the upstream.

The pressure signature at a point above the plate (x = 0.5, z = 3) is shown in Figure (5.14)

for the two finite core vortices and for a point vortex, the later is a prediction of an approxi-

mate low Mach number linear theory (Equation 8.1.8, [25]) (Note that [t] in the abscissa is

given by [t] = t− r/a∞). For the Oseen vortex, the pressure signature shows an early pulse

which represents the passage of the transient response caused by the sudden introduction of

the plate in the vortex filed. The Taylor vortex shows more compact pressure signature as

61

Figure 5.11: Taylor vortex, a snapshot of


Figure 5.12: Taylor vortex, a snapshot of


p

0

30

60

90

120

150

180

210

240

270

300

330

0 0.0005 0.001 0.0015

OseenTaylor

+p

-p

Figure 5.13: Directivity of pressure am-

plitude on a circle r = 6.2 centered at

x = 3.1, z = 0 at time t = 6.

62

2U[t]/C

Cp

-7 -6 -5 -4 -3 -2 -1 0 1 2-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004Oseen (Euler)Taylor (Euler)Point Vortex (h/C=0)Point Vortex (h/C=0.07)

Plate

Figure 5.14: Pressure signature at x =

0.5, z = 3.

2Ut/C

Lift

Coe

ffic

ien

t-7 -5 -3 -1 1 3 5

-0.03

-0.02

-0.01

0

0.01

0.02

Oseen VortexTaylor Vortex

Plate

Figure 5.15: Lift coefficient.

compared to the Oseen vortex with qualitative differences in the second pulse that is gener-

ated after the vortex center passes the plate leading edge. The results of the linear theory

would be obtained for an Oseen vortex in the limit of zero core radius. The unsteady lift is

shown in Figure (5.15), the effects of the internal structure of the vortex on the lift is evident.

Taken together, Figures (5.13) to (5.15) show that the radiated sound and plate response in

the vortex-plate interaction problem depend on the internal structure of the vortex.

5.3 Conclusions

We have simulated the parallel interaction of a finite-core vortex with a zero-thickness flat

plate. We have investigated the effects of the internal structure of the vortex (radial variation

of vorticity) on the strength and directivity of radiated sound. We considered two vortices:

(1) The Oseen vortex, whose vorticity distribution is monotone and its circulation at infinity

63

is finite, and (2) The Taylor vortex whose vorticity distribution is of mixed sign and its

circulation decays to zero at infinity. The core radius and maximum circumferential velocity

are the same for the vortices. The simulations indicate that there are qualitative differences

in the radiated sound and unsteady lift produced by the two vortices, and hence one needs

to consider the internal structure of the vortex when studying blade-vortex interactions.

64

Chapter 6

Interaction of Homogeneous


- Comparison with Experimental Data

In this chapter, we are interested in the comparison of our numerical simulation with the

available experimental results. The current investigation is motivated by the cascade exper-

iments conducted by Larssen and Devenport [28] (see also Larssen [29]). They adopted a

mechanically rotating “active” grid design in order to generate large scale turbulence. The

experimental setup consists of a six-blade linear cascade. In our simulation, we use the same

configuration as that of the experiment but with thin flat plates. The numerical results of

this chapter are based on using the options of zero-shear wall boundary conditions and the

sixth-order scheme combined with a compact upwind fifth-order scheme due to Zhong [53].

65

6.1 Inflow Turbulence

At the inflow boundary we want to have control over the incident velocity fluctuations.

Therefore, we specify the three velocity components and temperature. The inflow boundary

is a plane of constant x. The velocity components on that plane are prescribed by Fourier

series in time:

u(y, z, t) = U∞ +N∑

n=1

Anu(y, z)cosωnt + Bn

u(y, z)sinωnt (6.1)

v(y, z, t) =N∑

n=1

Anv (y, z)cosωnt + Bn

v (y, z)sinωnt (6.2)

w(y, z, t) =N∑

n=1

Anw(y, z)cosωnt + Bn

w(y, z)sinωnt (6.3)

where the coefficients of the series are functions of y and z. To generate the Fourier coeffi-

cients, we first generate a box of incompressible (divergence free) isotropic random field in

the wavenumber space (kx, ky, kz) using a method given by Durbin and Rief [9] (page 241).

The modified wavenumber of the sixth-order compact scheme [30] is used so that the inflow

velocity field is divergence free if the divergence is evaluated by that scheme. The 3D energy

spectrum function, E(k), is given by von Karman spectrum

E(k) = q2LCvk(kL)4

[1 + (kL)2]p(6.4)

where

Cvk =Γ(p)

Γ(52)Γ(p− 5

2)

(6.5)

We use p = 17/6 and hence Cvk = 0.484254. And q2 is twice the turbulence kinetic energy,

and L is a length scale that is related to the integral length scale, L111 by:

66

L =4(p− 1)(p− 2)

3πCvk

L111 (6.6)

Moet et al. [19] generated a similar box of homogeneous turbulence and used it as initial

condition in the investigation of the ambient turbulence effects on vortex evolution. For

the cascade problem, we use q2 = 0.02 (The reference velocity is the free stream velocity).

The numerical value of the integral length scale is L111 = 280 mm (s/L1

11 = 0.943), which

is suggested by the cascade experiments of Larssen [29]. The next step in specifying the

inflow velocity is to take the inverse Fourier transform in the y and z plane and interpret

the wave number in the x-direction as a frequency in time by invoking Taylor’s hypothesis,

ωn = U∞kxn. Over the duration of a simulation, the box of turbulence is repeatedly fed at

the inflow boundary, and hence the incident turbulence is perfectly periodic. To reduce the

effects of this periodicity as well as the anisotropy of the incident turbulence, we conduct

six independent simulations using six different boxes at the inflow boundary. Then we take

ensemble average of the statistics of the six simulations.

6.2 Comparison with Experimental Data

6.2.1 Spatially Decaying Isotropic Turbulence

Before we present results for the interaction problem, it is important to establish credibility

of the simulations for spatially decaying isotropic turbulence without plates. Here, we com-

pare LES results with experimental data for grid-generated turbulence: Comte-Bellot and

Corrsin [8] experiments (referenced to as CBC in this chapter). Case ‘a’ of CBC with grid

size M = 5.08 cm is considered. The reference length is M = 5.08 cm, and reference velocity

is U = 10 m/s. The Reynolds number is Re = 34000, and the free stream Mach number in

the simulation is assumed to be 0.4. For spatial simulation, the energy spectrum at the in-

flow boundary matches the experimental data at station x/M = 42 in the CBC experiments

67

kM

E/U

^2M

100 101 10210-9

10-8

10-7

10-6

10-5

10-4

CBCM2 (x/M=42)CBCM2 (x/M=98)CBCM2 (x/M=171)Inflow (x/M=42)C6CUD5 (x/M=98)C6CUD5 (x/M=171)

Coarse grid (449,33,33)

Figure 6.1: Energy spectrum function,

spatial LES, coarse grid.

kM

E/U

^2M

100 101 10210-9

10-8

10-7

10-6

10-5

10-4

CBCM2 (x/M=42)CBCM2 (x/M=98)CBCM2 (x/M=171)Inflow (x/M=42)C6CUD5 (x/M=98)C6CUD5 (x/M=171)

Fine grid (897,65,65)

Figure 6.2: Energy spectrum function,

spatial LES, fine grid.

(usually referred to as Ut/M = 42 in temporal simulations), and the outflow boundary is

taken at x/M = 193.2. The cross plane (yz-plane) is a square of side 10.8M . Periodic bound-

ary conditions are applied in the y and z directions. Comparison with CBC experimental

data is done at stations x/M = 98 and x/M = 171. We have tested two grid resolutions. A

coarse grid of (nx, ny, nz) = (449, 33, 33) points and a fine grid of (nx, ny, nz) = (897, 65, 65)

points. The 3D energy spectrum function E(k) is shown in Figures (6.1) and (6.2) for the

coarse and fine grids, respectively. The symbols in these figures are the experimental data

as listed in table 3 of CBC.

About 2/3 of the modes are well resolved but there is dissipation of the high wavenumbers.

In the dynamic SGS model, Smagorinsky’s constant, C, is a function of space and time. In

the present simulation, we average C over the yz-plane, and hence C is a function of x and

t. We depict C in Figure (6.3) as a function of x. On average, the value of C decays slightly

with the decay of turbulence. The rapid decay near the outflow boundary is caused by the

enhanced artificial damping near the boundary. Based on the variation of C with x as shown

68

x/M-42

SG

Sm

odel

,C

0 40 80 120 1600

0.02

0.04

0.06

0.08

0.1

Coarse grid (449,33,33)Fine grid (897,65,65)

Figure 6.3: Streamwise variation of dy-

namic model coefficient in spatial decay-

ing turbulence.

XY

Z

B

A

C

S

B

B

B=4.836 CA=4.911 CS=0.806 C

Inflow

Outflow

Figure 6.4: Flat plate cascade and com-

putational domain

in Figure (6.3), a constant value of 0.03 for C may be used with the classical Smagorinsky

model which is more efficient than the dynamic model. We used the classical model with

C = 0.03 in the cascade simulations presented in the next section and in chapter (7).

6.2.2 Computational Domain and Inflow Spectra

The current investigation is motivated by the cascade experiments conducted by Larssen and

Devenport [28] (see also Larssen [29]). The experimental setup consists of a six-blade linear

cascade. The geometric properties of the plate are: chord (c) = 327.5 mm and thickness =

6.35 mm. The plate has a semi-circular leading edge and sharp trailing edge. The blade-to-

blade spacing (s) is 264 mm. In our simulations, we also consider a six-blade linear cascade

as shown in Figure (6.4). The plate has zero thickness, and the chord (c) is the same as in

the experimental setup. We use the chord c as a reference length. The pitch s is 0.806 and

the span B is 4.836. The mean free stream velocity (U) is used as the reference velocity.

69

The Reynolds number based on c and U is 2.7 × 105 and the Mach number is assumed to

be 0.3. The cascade is unstaggered and the plates are at zero incidence relative to the mean

flow. A uniform Cartesian grid of (nx, ny, nz) = (321, 145, 145) points is used in the present

cascade simulations.

At the inflow boundary we specify the three velocity fluctuations. The target 3D energy

spectrum function E(k) is specified by von Karman spectrum as given by Eq (6.4). The

integral length scale is specified by L111 = 280 mm as given by Larssen [29] based on his

experimental data. The cascade pitch s = 264 mm, and hence s/L111 = 0.943. However, the

low-wavenumber end of the numerically generated fluctuations is limited by the transverse di-

mension of the computational domain (B) which includes only six blades (B = 6s = 5.66L111).

As a result of this limitation, wavenumbers smaller than the wavenumber where E(k) at-

tains its maximum are not activated in the inflow fluctuations. The highest wavenum-

ber of the spectrum of the generated fluctuations is also limited by the grid resolution,

kmax = π/∆z = πnz/B. The target and numerically generated spectra are shown in Fig-

ure (6.5). The roll off of the 1D energy spectra at high wavenumbers is because we zero out

the spherical shells with radius greater than kmax in the 3D wavenumber space.

6.2.3 Comparison of LES with Larsen Experimental Data

We compare LES results with experimental data provided by private communication with

Larssen (see also [29]). The streamwise decay of the normal Reynolds stresses at mid-

passage (z/s = 0.5) is depicted in Figure (6.6). The experimental data show higher levels

of the normal stresses and turbulence kinetic energy than those of LES. This is because

the experimental data contain energy from the full spectrum (unfiltered) whereas the LES

results represent only the resolved scales. Experimental data and LES results show that the

reduction of turbulence kinetic energy (q2/2) at the mid-passage is almost complete a short

distance into the passage (x = 0.65). Thus the reduction happens over a distance on the

order of one integral length scale L111/c = 0.85. Downstream of the station x = 0.65, the

70

k, k1

E,E

11,

E22

100 101 10210-7

10-6

10-5

10-4

10-3

E(k)- von KarmanE11(k1)- von KarmanE22(k1)- von KarmanE(k)-Inflow BoundaryE11(k1)-Inflow BoundaryE22(k1)-Inflow Boundary

Figure 6.5: Target and numerically gener-

ated energy spectra at inflow boundary.

x

uu,v

v,w

w,q

q-2 -1 0 1 2 3

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

uu-LESvv-LESww-LESqq-LESuu-EXPvv-EXPww-EXPqq-EXP

LE TE

z/s = 0.5

Figure 6.6: Mid-passage distribution of

normal Reynolds stresses and q2 = u2 +

v2 + w2.

decay rate of the kinetic energy is nearly the same as the decay rate upstream of the cascade.

The normal Reynolds stress profiles are shown in Figure (6.7) at a station near the trailing

edge (x = 0.840). The large discrepancy between LES results and experimental data in the

streamwise u2 and spanwise v2 components is due to the more energetic incident turbulence in

the experiments as compared to simulations. The close agreement in the upwash component

w2 is fortuitous, but it serves the purpose of showing that the shape is well predicted. A

more meaningful comparison is obtained if we normalize each profile by the respective values

of Reynolds stresses upstream of the cascade. At the point (z/s = 0.5 and x = −0.95),

the experimental data are (u21 = 0.004477, v2

1 = 0.004687, w21 = 0.004416) whereas the LES

results are (u21 = 0.003686, v2

1 = 0.003870, w21 = 0.003842). These values are then used to

normalize the respective Reynolds stress profiles, which are depicted in Figure (6.8). The

tangential profiles (u2 and v2) predicted by LES are in good agreement with the experimental

data over most of the passage, whereas the upwash profile (w2) is over-predicted by LES. We

71

(uu, vv, ww) / U^2

z/s

0 0.001 0.002 0.003 0.004 0.005 0.006 0.0070

0.1

0.2

0.3

0.4

0.5

uu-LESvv-LESww-LESuu-EXPvv-EXPww-EXP

Figure 6.7: Reynolds stress profiles at x =

0.840.

uu/uu1 , vv/vv1, ww/ww1

z/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5


Figure 6.8: Normalized Reynolds stress

profiles at x = 0.840.

note that the discrepancies in u2 and v2 near the wall is caused by the zero-shear boundary

condition which is directly enforced on the velocity near the wall.

The normal Reynolds stress profiles at a streamwise station (x = 1.948) in the cascade wake

are depicted in Figure (6.9), and the normalized profiles are shown in Figure (6.10). The

normalized profiles show the correct trend except near the wake centerline. The spanwise

profile (v2) is closer to the measured one near the wake centerline whereas the u2 profile

is not. The experimental data for the streamwise profile (u2) show very strong maximum

above the centerline (z/s = 0.04), whereas the upwash profile has its maximum on the wake

centerline (z/s = 0). These local maxima are generated by the instability of the mean

shear of the wake profile. Thus they are not a part of the incident turbulence, although the

instability may have been enhanced by the interaction with that turbulence. The mean shear

in the boundary layers and wakes are not resolved by the current LES cascade simulations.

To verify that the maxima in the fluctuations near the wake centerline are induced by the

mean shear instability, we consider interaction of homogeneous turbulence with a single flat

72

(uu, vv, ww) / U^2

z/s

0 0.001 0.002 0.003 0.004 0.005 0.006 0.0070

0.1

0.2

0.3

0.4

0.5


Figure 6.9: Reynolds stress profiles at x =

1.948.

uu/uu1, vv/vv1, ww/ww1

z/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5


Figure 6.10: Normalized Reynolds stress

profiles at x = 1.948.

x

z

0.5 1 1.5 2 2.5 3-1

-0.5

0

0.5

1

Figure 6.11: Spanwise vorticity contours

for a single plate placed in isotropic tur-

bulence, no-slip condition is applied.

uu, ww

z

0 0.005 0.01 0.015 0.020

0.05

0.1

0.15

0.2

<uu> at x/c=0.92<ww> at x/c=0.92<uu> at x/c=1.53<ww> at x/c=1.53

Figure 6.12: Reynolds stress profiles at

(x−xLE)/c = 0.92 and (x−xLE)/c = 1.53

for a single plate, no-slip boundary condi-

tion is applied.

73

x

z

1.8 2 2.2 2.4 2.6 2.8

-0.4

-0.2

0

0.2

0.4

Plate Trailing Edge

High Levels of uuby wake instability

Figure 6.13: Reynolds stress contours u2

for a single plate .

x

z1.8 2 2.2 2.4 2.6 2.8

-0.4

-0.2

0

0.2

0.4

Plate Trailing Edge

High Levels of wwby wake instabilitySuppression of ww

by plate surface

Figure 6.14: Reynolds stress contours w2

for a single plate.

plate. A schematic of the plate and incident flow (from left to right) is shown in Figure (6.11).

In this case, the plate leading edge is at (x = xLE = 1.0) and the plate chord (c = 1.0).

A zero-thickness flat plate is placed in a uniform stream at Mach number M = 0.6 and

Reynolds number based on chord of Re = 2.70 × 105. At the inflow boundary, velocity

fluctuations are superimposed on the uniform flow. The zero-shear condition used in the

cascade simulations is abandoned, and instead we apply the usual no-slip wall boundary

conditions on the plate.

Reynolds stress profiles at two streamwise stations, one near the trailing edge (x−xLE)/c =

0.92 and another in the wake (x − xLE)/c = 1.53 are shown in Figure (6.12). While the

transverse component w2 is totaly suppressed on the plate, it attains a maximum on the

wake centerline. The streamwise component u2 increases slightly near the plate, but it has a

very strong magnitude in the wake below and above the centerline. The local maxima of (u2)

and (w2) are also shown by the contour plots in Figures (6.13), and (6.14), respectively.

74

Chapter 7

Interaction of Homogeneous


- Comparison with RDT

7.1 Introduction

In this chapter, we study the distortion of homogeneous isotropic turbulence as it passes

through unstaggered cascade of thin flat plates. Spatial large eddy simulation (LES) is

conducted for two linear cascades: a six-plate cascade, and a three-plate cascade. Because

suppression of the normal component of velocity is the main mechanism of distortion, we

neglect the presence of mean shear in the boundary layers and wakes, and allow slip velocity

on the plate surfaces. We enforce the zero normal velocity condition on the plate and relax the

no-slip condition to a zero-shear or slip wall condition. This boundary condition treatment is

motivated by rapid distortion theory (RDT) in which viscous effects are neglected, however

the present LES approach accounts for nonlinear and turbulence diffusion effects by a sub-

grid scale model. We tested two different wall boundary conditions; zero-shear boundary and

75

slip-wall boundary. We have presented LES results for zero-shear conditions in chapter (6).

All the LES results in this chapter are based on the option of slip-wall boundary and the use

of tenth-order filter compined with the compact six-order finite-difference scheme. To test

the applicability of Graham’s RDT solution [23], we introduce homogeneous turbulence of

different spectral content and different intensity to the computational domain and compare

with LES results. The normal Reynolds stresses and velocity spectra are analyzed ahead,

within, and downstream of the cascade.

7.2 Graham’s RDT Solution

Using rapid distortion (RDT), Graham [23], see Appendix (A), has developed analytic so-

lutions for the spectra of isotropic turbulence downstream of a linear cascade of thin flat

plates. Because viscous and nonlinear effects are neglected, the disturbance produced by

the cascade is an irrotational velocity field induced by flat vortex sheets that coincide with

the plates and extend to infinity downstream of the trailing edge. Graham [23] assumes the

incident turbulence velocity field to be homogeneous, and represents it by 3D Fourier inte-

grals, Equation (7.1). He also assumes the turbulence after the introduction of the cascade

to remain homogeneous in planes parallel to the plates, and obtains the Fourier coefficients

of the distorted turbulence in terms of those of the incident turbulence.

The incident turbulence velocity field is given by:

u∞(x, t) =

∫ ∫ ∫u∞(k)ei(ωt−kjxj)dk, j = 1, 2, 3 (7.1)

where ω = k1U∞, and k is the wave number vector. Each Fourier component of the total

flow field can be expressed as:

u(x, t;k) = u∞(k)ei(ωt−kjxj) +∇φ(k, x3)ei(ωt−k1x1−k2x2), (7.2)

76

where φ is the velocity potential due to the blocking effect of the cascade. Solution to

Laplace’s equation (subjected to zero normal velocity on the plates and Kutta condition at

the trailing edge) for each Fourier component in the region between any two plates is given

by:

u1 = u1∞ +ik1

τ

cosh[τx3]e−ik3s − cosh[τ(s− x3)]

sinh(τs)u3∞eik3x3 (7.3)

u2 = u2∞ +ik2

τ



u3 = u3∞ − sinh[τx3]e−ik3s + sinh[τ(s− x3)]


where τ = (k12 + k2

2)12 .

Graham’s solution is valid for a streamwise distance on the order of the integral length scale

x = O(L111) downstream of the leading edge. Assumptions of RDT imply that the solution

is not valid for shorter or much greater distance than the integral length scale.

7.3 Comparison of LES with Graham’s RDT

In comparing LES with RDT, we found it necessary to account for the decay of turbulence,

and hence the input spectra to RDT should be the spectra that would exist at the streamwise

location of comparison but in the absence of the cascade. Therefore, to compare LES results

with Graham’s RDT results, we conduct simulations but without the cascade under identical

conditions (grid, inflow spectra, etc.) as those of the six simulations conducted in the presence

of the cascade. The simulations without the cascade provide the Fourier representation of the

incident turbulence that is required by RDT. In other words, the input to RDT is determined

by simulations of spatially decaying homogeneous turbulence. The velocity field in planes

77

Case No.of plates Domain size Grid size u∞U

A 6 10.881× 4.836× 4.836 325× 145× 145 0.0816

B 3 5.4405× 2.418× 2.418 163× 73× 73 0.0816

C 3 5.4405× 2.418× 2.418 163× 73× 73 0.163

D 3 5.4405× 2.418× 2.418 163× 73× 73 0.231

Table 7.1: Characteristics of the inflow turbulence and computational domain

of constant x are stored as functions of time (t), and then Fourier transform is obtained in

yz-plane and time (t). Taylor’s hypothesis is then used to replace frequency by streamwise

wavenumber.

The characteristics of the inflow turbulence and the computational domain of the two con-

sidered cascades are shown in Table (7.1).

7.3.1 Six-Plate Cascade

For case (A), the integral length scale of the incident turbulence is L111 = 0.940s and the

turbulence intensity is u∞/U = 0.0816, see Table (7.1). The cascade geometry and the

computational domain are shown in Figure (7.1). The plate is represented by 30 grid points

and the number of grid point in z-direction in the passage between two plates is 24 points.

The plate leading edge is located at x = 0 and the plate chord c = 1. The time step ∆t =

0.00279861. The 3D-energy spectrum of the incident turbulence is depicted in Figure (7.2).

The turbulence intensities shown in this table should be considered “nominal” values. They

are used in generating boxes of homogeneous turbulence according to Von Karman spectrum.

However, because of filtering and inflow boundary condition treatment, we noted a fast drop

in the turbulence kinetic energy near the inflow boundary. A more reliable measure of the

actual turbulence kinetic energy is given by its streamwise distributions in the absence of

the cascade which is shown in Figure (7.19) for the 6-plate cascade and Figure (7.64) for the

3-plate cascade.

78

0

1

2

3

4

5

Z

-4

-2

0

2

4

6

X

0

2

4

Y

XY

Z

I nf l ow

Out f l ow

Per i odi c

Per i odi cPer io dic

Per i odic

Six-Plate cascade

s

B

Figure 7.1: Six-plate cascade and computational domain.

79

k

E(k

)

10-1 100 101 10210-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

x=-4.836x=-0.269

Case A

Figure 7.2: Case A: 3D-energy spectra of the incident turbulence, inflow (x=-4.836), and

upstream of cascade (x=-0.269).

Instantaneous Flow Field

A snapshot of the instantaneous upwash velocity contours (w − w), (where, w is the local

mean value.), in xz-plane is shown by Figure (7.3). Large scale fluctuations upstream of the

cascade are suppressed as the turbulence passes through the cascade. It is clear that the

instantaneous flow is non-periodic from one passage to the next, and that the computational

domain must include multiple blades in order to correctly capture the interaction of large

scale turbulence with the cascade. Figure (7.4) is a snapshot of the instantaneous upwash

velocity contours in the yz-plane at x = 0.17. The plates break the large scale structures

into smaller scales.

Figures (7.5) and (7.6) show snapshots of the instantaneous streamwise and spanwise velocity

contours, respectively. The spacial structure of these contours does not show significant

80

Figure 7.3: A snapshot of the instanta-

neous upwash velocity contours(xz-plane).


neous upwash velocity contours at plane

x = 0.17.

distortion by passing through the cascade plates.

Figures (7.7) and (7.8) show instantaneous velocity fluctuation vectors in a region around

the plates in xz- and yz-planes, respectively. The highly turbulent flow field structure is clear

in the figures. As we expect, the velocity vectors are tangent to the plate surfaces because

of the zero-normal velocity boundary condition applied on the plates.

A snapshot of the instantaneous pressure fluctuations (p − p), (where p is the local mean

value.), in xz and yz planes are shown in Figures (7.9) and (7.10), respectively. The pressure

contour levels (which could include acoustic pressure waves) are totally different upstream

and downstream the cascade. The plate surfaces are under varying pressure amplitudes in

the spanwise direction. The periodic flow structure in the spanwise direction is clear in

Figure (7.10).

The instantaneous density fluctuation contours (ρ− ρ), (where ρ is the local mean value), are

81


neous streamwise velocity contours (xz-

plane).

Figure 7.6: A snapshot of the instan-

taneous spanwise velocity contours (xz-

plane).

also shown as snapshots in Figures (7.11) and (7.12). The incoming flow field is supposed

to be divergence free, (i.e. no density fluctuations). (Figures (7.13) and (7.14) show the

corresponding pressure and density contours for the case without plates) But, because of

the presence of the cascade plates which interacts with the incoming turbulence it produces

density fluctuations in the flow field. The density fluctuations also show different spatial

structures upstream than downstream of the cascade. The periodic flow structure in the

spanwise direction is clear in Figure (7.12).

Normal Reynolds Stresses

Figures (7.15), (7.16), and (7.17) show the y-averaged normal Reynolds stress contours.

These contours are the average of six independent runs and are also averaged in the z-

direction by taking the average over the halves of the passages above and below each plate.

The figures show the spatial decay of the turbulence as it convects downstream.

82

x

z

-0.5 0 0.5 1 1.5

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Case AVelocity vector


neous velocity vectors (xz-plane).

y

z1.5 2 2.5

2.5

3

Case AVelocity vector

x=0.17


taneous velocity vectors (yz-plane) x =

0.17.

Figure (7.15) shows the distortion of the streamwise component (uu) by the introduction

of the cascade plates. The streamwise component shows higher values in the vicinity of

the leading edge. It also shows high values in the passage near the plate surfaces and

extends downstream the trailing edge. The streamwise component shows lower values in a

thin region in the wake of each plate in planes coincident with the plane of the plates. To

capture the correct structure of the wake of the plates we need to use much finer grid to

represent the vortex sheet emanating from the trailing edge. However, this does not affect

the solution in the passage outside this thin region. The incoming turbulence is homogeneous

upstream of the cascade. As the turbulence passes through the cascade it becomes no longer

homogeneous. However, because of the spacial decay of the turbulence it starts to become

homogeneous again at distances far from the trailing edge.

Figure (7.16) shows the distortion of the spanwise component (vv) by the introduction of

the cascade plates. It shows almost the same configuration as the streamwise component

83


neous pressure fluctuation contours (xz-

plane).


neous pressure fluctuation contours (yz-

plane), x = 0.17.

except that there is no higher values at the leading edges.

Figure (7.17) shows the distortion of the upwash component (ww) by the introduction of

the cascade plates. The turbulence starts homogeneous upstream of the cascade. As the

turbulence passes through the cascade the cascade suppresses the upwash component. The

suppression of the upwash component in the passages starts at the plane of the plates where

the upwash component is zero and continues to affect the entire passage. The upwash

component continues to have almost zero values in the wake of the plates for a distance

about 1.5 chord but then starts to build up which could be because of the viscosity, the roll

up of the vortex sheet, and the nonlinear effects.

The averaged contours of the square value of the pressure fluctuations (pp) are shown in

Figure (7.18). It is interesting to notice that high pressure fluctuations occur only around

the leading edge and extend short distance downstream stream of the leading edge. It shows

no pressure fluctuations from the trailing edge. The leading edge is the main source of the

84


taneous density fluctuation contours (xz-

plane).


taneous density fluctuation contours (yz-

plane), x = 0.17.

sound radiated from the cascade.

The streamwise decay of q2 = u2 + v2 + w2 at mid-passage (z/s = 0.5) is depicted in

Figure (7.19). The decay of the same quantity without the cascade is also shown. The

LES results show that the reduction of turbulence kinetic energy (q2/2) at the mid-passage

starts upstream of the cascade (x = −0.2) and is complete a short distance into the passage

(x = 0.65), afterwards the decay rate is nearly the same as the case without the cascade. The

RDT results is in full agreement with LES downstream of the streamwise location x = 0.65.

The RDT does not apply near the leading edge of the plate or farther downstream of the

trailing edge. The reduction in the turbulence kinetic energy in the passage persists to

the plate surface. In other words, although the turbulence kinetic energy increases towards

the wall, but it is still reduced relative to its free stream value. It is interesting to note

that Graham showed that in the limiting case in which the ratio of cascade pitch to integral

length scale of the incident turbulence (s/L111) approaches zero, the turbulence kinetic energy

85

Figure 7.13: Case A: A snapshot of the in-

stantaneous pressure fluctuation contours

(no plates).

Figure 7.14: Case A: A snapshot of the in-

stantaneous density fluctuation contours

(no plates).

Figure 7.15: Contours of the averaged

streamwise-Reynolds stress component.


spanwise-Reynolds stress component.

86


upwash-Reynolds stress component.


square of the pressure fluctuations (pp).

is reduced by a half once the cascade is introduced into the flow. Turbulence kinetic energy

profiles, normalized by the free stream value (q21) in the passage, as predicted by Graham’s

RDT analytic solution for s/L111 = 11.7, 0.94 and 0.177 are shown in Figure (7.20). The

profile predicted by LES at x = 0.840 is also shown and indicates excellent agreement with

RDT. As the ratio s/L111 approaches zero, the theoretical limit of 50% is obtained.

The streamwise decay of the averaged (over the yz-plane) turbulence kinetic energy (TKE)

and normal Reynolds stresses are shown in Figure (7.21). The reduction of the TKE is

almost complete a short distance downstream the leading edge of about (x = 0.65). The

TKE shows a further reduction at the trailing edge. Such reduction at the trailing edge could

be minimized by using a finer grid to represent more accurately the vortex sheet emanating

from the trailing edge. The reduction of the TKE happens just at the introduction of

the plates and mainly because of the suppression of the normal component (upwash) of

velocity fluctuation. The decay rate of both the TKE and the normal component (ww) has

different values upstream the leading edge and downstream the trailing edge. However the

87

x

q^2

-4 -2 0 2 40

0.004

0.008

0.012

0.016

0.02

no cascadecascade-LEScascade-RDT

LE TE

Figure 7.19: Mid-passage distribution of q2.

q2/q12

z/s

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

s/L11=11.70 -RDTs/L11=0.940 -RDTs/L11=0.940 -LESs/L11=0.117 -RDTs/L11= 0.0 -RDT

Figure 7.20: Normalized q2 profiles for dif-

ferent ratios of plate spacing to integral

length scale s/L111.

streamwise and spanwise components, (uu) and (vv), have the same decay rate upstream and

downstream the cascade. Figure (7.21) shows also the averaged square values of the pressure

fluctuation (pp) which has almost constant values upstream the leading edge and rapid

increase just ahead the leading edge. The high values of the pressure fluctuation continues

from the leading edge until the trailing edge. The drop of the TKE could be converted as an

increase in the pressure fluctuation and radiated as sound waves. However, a more detailed

investigation is needed to understand the mechanism by which the kinetic energy drops.

88

x

Ave

rag

e(u

u,v

v,w

w)/

2,

TK

E

Ave

rag

e(p

p)

-4 -3 -2 -1 0 1 2 3 4 5 60

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

TKEuu/2vv/2ww/2ppTKE-noplates

LE TE

6-pates

Figure 7.21: Averaged TKE, Reynolds stresses, and pressure fluctuation (read right)

Normal Reynolds stress profiles (normalized by the inlet turbulence intensity u2∞.), see ta-

ble(7.1), are compared in Figures (7.22), (7.23), (7.24) and (7.25) for planes in the passage

at x = 0.067, 0.201, 0.638 and 0.840, respectively. There is excellent agreement between RDT

and LES results for the plane x = 0.840, which is in the region of applicability of RDT, but

there is some deviation which increases at planes which is closer to the leading edge where

RDT does not apply. The profiles for three planes downstream of the trailing edge are shown

in Figures (7.26), (7.27) and (7.28) at x = 1.578, 1.948 and 2.787, respectively. The LES

results continue to agree with Graham’s RDT formulation across the passage except near the

wake center line, where the no penetration condition is relaxed in the LES simulation and

hence the turbulence structures are free to cross the wake center line plane. The LES solu-

tion accounts for viscosity and nonlinear effects and hence the generation of upwash velocity

fluctuations there. We recall that in Graham’s RDT formulation, the trailing vortex sheets

remain flat and parallel to the plates, and hence the upwash velocity is zero on the plate

89

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5uu-lesvv-lesww-lesuu-rdtvv-rdtww-rdt

6-platesu∞=0.0816x=0.067

Figure 7.22: Normal Reynolds stress pro-

files at x = 0.067.

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4


6-platesu∞=0.0816x=0.201


files at x = 0.201.

and on the wake vortex sheets. Therefore, the linearized inviscid rapid distortion theory

correctly predicts the behavior of Reynolds stresses for x = O(L111), but it does not capture

changes in these stresses due to nonlinear effects and viscosity.

We note that if the incident turbulence is perfectly isotropic, then Graham’s RDT formu-

lation predicts identical profiles for the two tangential components u2 and v2. The small

difference between the u2 and v2 profiles presented here is caused by a small anisotropy of

the simulated turbulence. This anisotropy is present in the two simulations with and without

the cascade.

For the comparison with RDT to be quantified and to have a reasonable comparison between

different cases, we normalize the quantity q2 within the passage by the corresponding value

q20 in the case without plates and plot the distribution of q2 as function of z/s at different

streamwise locations. Table (7.2) lists the magnitude of q20 in the case without cascade at

planes x = 0.201, 0.840, and 2.787 for different cases.

90

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4


6-platesu∞=0.0816x=0.638


files at x = 0.638.

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4


6-platesu∞=0.0816x=0.840


files at x = 0.840.

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4


6-platesu∞=0.0816x=1.578


files at x = 1.578.

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4


6-platesu∞=0.0816x=1.948


files at x = 1.948.

91

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4


6-platesu∞=0.0816x=2.787


files at x = 2.787.

x-plane Case A Case B Case C Case D

0.201 0.010678 0.006954 0.023102 0.03972

0.840 0.010391 0.006629 0.021073 0.03494

2.787 0.009539 0.005759 0.015949 0.02357

Table 7.2: Values of q20 at the planes of comparison for different cases

92

The profiles of the normalized values q2/q20 at planes x = 0.201, 0.840, and2.787 are depicted

in Figures (7.29), (7.30), and (7.31), respectively. The comparison of these plots indicates

that both RDT and LES results show a reduction of the TKE within the passage that takes

place from the center of the passage to the plate surface. Figure (7.29) shows a significant

discrepancy between the RDT and the LES solutions at plane x = 0.201. The RDT solution

predicts that the reduction to be complete instantaneously, over the entire passage, once the

cascade is introduced. While, LES solution allows for the reduction to develop gradually in

the passage as the turbulence convects downstream. Figure (7.30) shows excellent agreement

between the RDT and the LES solutions. According to the assumptions of the RDT; the

RDT is applicable at distances of the order of the integral length scale measured from the

leading edge. The RDT solution is not valid for shorter or much greater distance than the

integral length scale.

Figure (7.31) shows disagreement between the RDT and the LES solutions, specially, near the

wake center line. RDT solution assumes the vortex sheet emanating from the trailing edge

is extended downstream as if we have semi-infinite plate. Nevertheless, excellent agreement

is obtained between RDT and LES over the rest of the passage. Both RDT and shear-free

LES do not predict the correct physical behavior in the wake of the plate. Reynolds stresses

in the wake of the plate and the associated instabilities are not captured by the two models.

The distortion of the turbulence spectra downstream of the cascade is of interest to hydro/aero-

acoustic predictions of noise radiated by rotors or guide vanes. The one-dimensional energy

spectra of the upwash velocity component Eww(k1) are depicted in Figures (7.32) and (7.33)

on the mid-passage (z/s = 0.5) and near the plate surface at (z/s = 0.0417), respectively,

for different streamwise locations. The major change in the large scales happens between

the station x = −0.269 upstream of the leading edge and the station x = 0.638 down-

stream of the leading edge. The spectra predicted by Graham’s RDT formulation are in

good agreement with the LES spectra for x > 0.638. The spectra near the plate surface at

(z/s = 0.0417) are shown in Figure (7.33). Because of the suppression of the upwash velocity

by the plate surface there is a significant reduction of the energy at low wavenumbers. Most

93

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

6-platesu∞=0.0816x=0.201

Figure 7.29: Case A: Profiles of q2/q20 at

plane x = 0.201.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

6-platesu∞=0.0816x=0.840


plane x = 0.840.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

6-platesu∞=0.0816x=2.787


plane x = 2.787.

94

k1

Ew

w

10 30 50 70 9010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

x=-0.974 -LESx=-0.269 -LESx= 0.840 -LESx= 0.840 -RDTx= 2.787 -LESx= 2.787 -RDT

6-platesu∞=0.0816Z/S=0.5

Figure 7.32: One dimensional energy

spectra, Eww(k1) at z/s = 0.5.

k1

Ew

w10 30 50 70 90

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2


6-platesu∞=0.0816

Z/S=0.0417


spectra, Eww(k1) at z/s = 0.0417.

of the reduction again happens between the stations x = −0.269 and x = 0.638. The spectra

according to RDT agree very well with LES results at x = 0.840. However, downstream of

the trailing edge at station x = 2.787, the LES results indicate a build up of large scales

transverse fluctuations whereas RDT results predict insignificant changes.

The one-dimensional energy spectra of the streamwise and spanwise velocity components

Euu(k1), and Evv(k1) are depicted in Figures (7.34) and (7.36) on the mid-passage (z/s =

0.5), respectively, and Figures (7.35) and (7.37) near the plate surface at (z/s = 0.0417),

respectively, for different streamwise locations. Both the streamwise and spanwise energy

spectra components have almost the same trend. The change in the large scales, on the

mid passage, of the streamwise and spanwise energy spectra components (Figures (7.34)

and (7.36)) is smaller than the corresponding reduction in the upwash energy spectra com-

ponent. The major change in the large scales happens also between the stations x = −0.269

upstream of the leading edge and the station x = 0.638 downstream of the leading edge. The

RDT results are in good agreement with the LES results. Figures (7.34) and (7.35) show

95

k1

Eu

u

10 30 50 70 9010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2


6-platesu∞=0.0816Z/S=0.5


spectra, Euu(k1) at z/s = 0.5.

k1

Eu

u10 30 50 70 90

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2


6-platesu∞=0.0816

Z/S=0.0417


spectra, Euu(k1) at z/s = 0.0417.

significant reduction in the streamwise and spanwise energy spectra components in all the

turbulence scales between the streamwise locations x = −0.974 and 2.787. Good agreement

over all scales near the passage center line. But near the wake center line, the disagreement

between LES and RDT is expected as discussed before.

7.3.2 Three-Plate Cascade

One of the main assumptions of the rapid distortion theory is that (u′/U∞)(x1/L

111) ¿ 1,

where x1 is distance downstream of the leading edge. To fulfill such condition; the turbulence

should be weak and the region of applicability of the RDT is at distances from the leading

edge of the order of the integral length scale of the turbulence. For more investigation of the

applicability of Graham’s RDT we need to contradict this assumption by both increasing

the incoming turbulence intensity and decreasing the integral length scale of the turbulence.

To decrease the integral length scale of the turbulence we choose to simulate the flow field

96

k1

Evv

10 30 50 70 9010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2


6-platesu∞=0.0816Z/S=0.5


spectra, Evv(k1) at z/s = 0.5.

k1

Evv

10 30 50 70 9010-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2


6-platesu∞=0.0816

Z/S=0.0417


spectra, Evv(k1) at z/s = 0.0417.

of a 3-plate cascade of the same pitch as the previous 6-plate cascade. The domain length

is reduced to half of that of the 6-plate cascade. Then we simulated three different levels

of the incoming turbulence intensity. The different characteristics of the turbulence and

the computational domain are listed for all cases in Table (7.1). The geometry and the

computational domain of the 3-plate cascade are shown in Figure (7.38).

97

0

0.5

1

1.5

2

2.5

Z

-2

-1

0

1

2

3

X

0

1

2

Y

XY

Z

Out f l ow

I nf l ow Per i odi c

Per i odi cPerio

di c

Per io dic

Three-Plate cascade

Figure 7.38: 6-plate cascade and computational domain.

The 3d-energy spectra of the incident turbulence for cases A, B, C, and D are depicted in

Figure (7.39).

Instantaneous Flow Field

Snapshots of the instantaneous upwash velocity contours in xz-plane for the cases B, C,

and D are shown by Figures (7.40), (7.41), and (7.42). The integral length scale of the

turbulence in cases B, C, and D is smaller than that in case A. The large scale fluctuation

structures upstream of the cascade of cases B, C, and D are smaller than those of case A,

see figure (7.3). Most of the large scale structures passes through the cascade without break

down because the large scale structures upsteam of the cascade are smaller than those of

case A while we keep the same pitch (B) for cases B, C, and D as that of case A. It is clear

that the magnitude of the upwash velocity contour levels is becoming larger from cases B,

98

C, to D.

k

E(k)

10-1 100 101 10210-8

10-7

10-6

10-5

10-4

10-3

10-2

Case ACase BCase CCase D

Figure 7.39: 3D-energy spectra of the incident turbulence

Figures (7.43), (7.44) and (7.45) show the instantaneous velocity fluctuation vectors in a

region around the plates in the xz-plane for cases B, C, and D, respectively. The highly

turbulent flow field structure is clear in the figures. The magnitude of the velocity vectors

are getting higher from case B, C, to D.

99

Figure 7.40: Case B: Snapshot of the in-

stantaneous upwash velocity contours (xz-

plane).

Figure 7.41: Case C: Snapshot of the in-


plane).

x

z

-0.5 0 0.5 1 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Case Dvelocity vector

Figure 7.45: Case D: A snapshot of the instantaneous velocity vectors (xz-plane).

100

Figure 7.42: Case D: Snapshot of the in-


plane).

x

z

-0.5 0 0.5 1 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Case Bvelocity vector

Figure 7.43: Case B: A snapshot of the

instantaneous velocity vectors (xz-plane).

x

z

-0.5 0 0.5 1 1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Case Cvelocity vector

Figure 7.44: Case C: A snapshot of the

instantaneous velocity vectors (xz-plane).

101


stantaneous pressure contours (xz-plane).



A snapshot of the instantaneous pressure fluctuations in the xz-plane are shown in Fig-

ures (7.46), (7.47) and (7.48) for cases B, C, and D, respectively. The pressure contour

levels (which could contain acoustic pressure waves.) are totally different upstream and

downstream the cascade. The strength of the pressure fluctuation is getting higher from

case B, C, to D.

The instantaneous density fluctuation contours are also shown as snapshots in Figures (7.49),

(7.50) and (7.51) for cases B, C, and D, respectively. The strength and the spatial structures

of the density fluctuations, produced by the introduction of the cascade into the flow field,

are totally different in the three cases. The density fluctuations also show different spatial

structures upstream than downstream of the cascade.

Normal Reynolds Stresses

The distortion of the streamwise component (uu) by the introduction of the cascade plates

for cases B, C, and D is shown by Figures (7.52), (7.53), and (7.54), respectively. The

102





(xz-plane).



(xz-plane).

103



(xz-plane).

figures show unwanted inhomogeneity in the domain, near the inlet boundary, which can

be eliminated by taking the average over larger number of independent runs. The current

results are obtained by taking the average over six independent runs. The figures show

the streamwise decay of the turbulence as it convects downstream. The magnitude of uu-

contours is increasing from case B, C, to D. The streamwise component shows higher values

in the vicinity of the leading edge. It also shows high values in the passage near the plate

surface and extends downstream the trailing edge. It also shows lower values in a thin region

on the wake of each plate.

Figures (7.55), (7.56), and (7.57) show the distortion of the spanwise component (vv) of

cases B, C, and D, respectively. The spanwise component shows almost the same config-

uration as the streamwise component except that there is no higher values at the leading

edges.

Figures (7.58), (7.59), and (7.60) show the distortion of the upwash component (ww) by the

104

Figure 7.52: Case B: Contours of the av-

eraged streamwise Reynolds stress compo-

nent.

Figure 7.53: Case C: Contours of the av-


nent.

introduction of the cascade plates for cases B, C, and D, respectively. The cascade suppresses

the upwash component as the turbulence passes through it. The suppression of the upwash

component in the passages starts at the plane of the plate where the upwash component is

zero and continues to affect the entire passage. The upwash component continues to have

almost zero values in the wake of the plates for a distance about 1.5 chord but then starts

to build up which is because of the viscosity and the nonlinear effects.

The averaged contours of the square value of the pressure fluctuations (pp) for cases B, C,

and D are shown in Figures (7.61), (7.62), and (7.63), respectively. Again, the high pres-

sure fluctuations occur only around the leading edge and extend short distance downstream

stream of the leading edge. The strength of the pressure fluctuation contours is getting

higher from case B, C, to D.

The streamwise decay of the averaged (over the yz-plane) turbulence kinetic energy (TKE)

for cases A, B, C, and D is depicted in Figure (7.64). It is clear from the figure that the

105

Figure 7.54: Case D: Contours of the av-


nent.

streamwise decay rate of the TKE increases as the turbulence intensity increases. Such

behavior is due to non-linear effects and the increase of the eddy viscosity. Again, the

reduction of the TKE is almost complete a short distance downstream the leading edge of

about (x = 0.65). The TKE shows a further reduction at the trailing edge. The figure shows

also the decay of the TKE for each case without the plate.

Figures (7.65) to (7.73) show the normal Reynolds stress profiles (normalized by the inlet

turbulence intensity u2∞.), see table (7.1), for cases B, C, and D at different streamwise

locations (x = 0.201, 0.840, and 2.787). The RDT solution continues to agree with LES

solution. Same conclusions as those of case A can be obtained.

The normalized quantity q2 within the passage by the corresponding value q20 in the case

without plates at different streamwise locations is a good measure for the comparison. Table

(7.2) lists the magnitude of q20 in the case without cascade at planes x = 0.201, 0.840, and

2.787 for different cases.

106


eraged spanwise Reynolds stress compo-

nent.



nent.

The profiles of q2/q20 at planes x = 0.201, 0.840, and2.787 for cases B, C, and D are depicted

in Figures (7.74) to (7.82). Comparison of cases A and B at location x = 0.201 indicates

that the agreement between RDT and LES is better in case B than that in case A, which

implies that the location of validity of RDT is shifted to a shorter distance downstream of

the leading edge. This is because the integral length scale is smaller in case B than that

in case A. In general, the agreement of LES results with RDT results is better in case A

than in cases B, C, and D. However, unexpectedly, the RDT solution is still valid even if the

condition ((u′/U∞)(x1/L

111) ¿ 1) is contradicted.

107



nent.

Figure 7.58: Case B: Contours of the aver-

aged upwash Reynolds stress component.

Figure 7.59: Case C: Contours of the aver-


108

Figure 7.60: Case D: Contours of the aver-



eraged square of the pressure fluctuations.



109



x

TKE

-2 -1 0 1 2 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Case ACase BCase CCase Dno plates

LE TE

Figure 7.64: Averaged streamwise decay

of the turbulent kinetic energy.

110

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

uu-lesvv-lesww-lesuu-rdtvv-rdtww-rdt

3-platesu∞=0.0816x=0.201

Figure 7.65: Case B: Normal Reynolds

stress profiles at x = 0.201.

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.163x=0.201

Figure 7.66: Case C: Normal Reynolds


(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.231x=0.201

Figure 7.67: Case D: Normal Reynolds


(uu,vv,ww)/u ∞2

z/s

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.0816

x=0.84



111

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.163x=0.840



(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.231x=0.840



(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.0816x=2.787



(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.163x=2.787



112

(uu,vv,ww)/u ∞2

Z/S

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5


3-platesu∞=0.231x=2.787



q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.0816x=0.201

Figure 7.74: Case B: Profiles of q2/q20 at

plane x = 0.201.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.163x=0.201

Figure 7.75: Case C: Profiles of q2/q20 at

plane x = 0.201.

113

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.231x=0.201

Figure 7.76: Case D: Profiles of q2/q20 at

plane x = 0.201.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.0816x=0.840


plane x = 0.840.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.163x=0.840


plane x = 0.840.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.231x=0.840


plane x = 0.840.

114

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.0816x=2.787


plane x = 2.787.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.163x=2.787


plane x = 2.787.

q2/q02

z/s

0.7 0.75 0.8 0.85 0.9 0.95 10

0.1

0.2

0.3

0.4

0.5

RDTLES

3-platesu∞=0.231x=2.787


plane x = 2.787.

115

Chapter 8

Conclusions and Recommended

Future Work

In this dissertation we developed a numerical model for simulations of the interaction of

large-scale free stream turbulence with a flat-plate cascade. For computational efficiency,

we assume that the interaction is shear free. We neglect the shear in boundary layers and

wakes except for the vortex sheet emanating from trailing edges, and treat solid walls as

adiabatic slip walls. However, to capture the spatial development of the incident turbulence

and its distortion by the cascade, we use large eddy simulations. Such a model is a good first

step for the prediction of noise radiation due to interaction of free stream turbulence with

immersed bodies. It represents an alternative computational approach to rapid distortion

theory (RDT) which has limitations for practical engineering applications.

In order to verify the validity of our calculations, we computed the unsteady loading and

radiated acoustic pressure field from flat plates interacting with vortical structures. We

considered two fundamental problems: (1) A linear cascade of flat plates excited by a vortical

wave (gust) given by a 2D Fourier mode, and (2) The parallel interaction of a finite-core

vortex with a single plate.

116

For the response of a flat plate cascade to two-dimensional vortical waves (gust), we solved

the two-dimensional nonlinear Euler equations over a linear cascade composed of six plates

for a range of frequencies of the incident gust. The cascade is unstaggered and the pitch

to chord ratio is 0.806. We use Giles’ [12] non-reflecting boundary conditions at the inflow

and outflow boundaries. We analyzed the cascade response in terms of unsteady normal

force, surface pressure distribution and radiated acoustic pressure field for three discrete

frequencies of the incident gust.

The lift spectrum agrees very well with Glegg’s [13] solution to the linearized potential

flow equation for the tested range of reduced frequency (0 < ωc/2U∞ < 8 ). Since Giles’

boundary conditions are approximately nonreflecting, we have investigated undesirable wave

reflection at the inflow and outflow boundaries and its variation with gust frequency. We

found that certain frequencies excite acoustic modes whose wavenumber vectors are nearly

normal to the boundary. In such cases, minor reflection at the boundary is obtained and

the results are insensitive to the location of the computational domain boundaries. Other

frequencies may still excite acoustic modes whose wavenumbers deviate considerably from

the normal direction resulting in major reflection that contaminates the pressure field. In

such a case, the numerical solution depends sensitively on the locations of the inflow/outflow

boundaries. If the frequency is such that all acoustic modes are cut-off, the pressure field

decays exponentially towards the boundaries, and boundary treatment poses no problem.

These observations are consistent with the basic assumption in Giles’ derivation of the ap-

proximately nonreflecting boundary conditions, which is based on a Taylor series expansion

for small ratio of tangential wavenumber to frequency.

We have simulated the parallel interaction of a finite-core vortex with a zero-thickness flat

plate. We have investigated the effects of the internal structure of the vortex (radial variation

of vorticity) on the strength and directivity of radiated sound. The simulations indicate that

there are qualitative differences in the radiated sound and unsteady lift produced by the two

vortices, and hence one needs to consider the internal structure of the vortex when studying

blade-vortex interactions.

117

Then we investigated the effects of a linear thin flat-plate cascade on the evolution of homoge-

neous isotropic turbulence as it passes through the cascade. Three-dimensional spatial large

eddy simulation (LES) is conducted for a linear cascade composed of six plates. Because

suppression of the normal component of velocity is the main mechanism of distortion of the

incident turbulence, we neglect the presence of mean shear in the boundary layer and wake,

and allow slip velocity on the plate surfaces. We enforce the zero normal velocity condition

on the plate and relax the no-slip condition to a zero-shear or slip wall condition. At the

inflow and outflow boundaries we modified Giles (1990) nonreflecting boundary conditions

to account for three dimensionality as given by Hagstrom (2002). The present LES approach

accounts for nonlinear and turbulence diffusion effects by a sub-grid scale model.

The cascade solidity is c/s = 1.24, and the integral length scale of the incident turbulence

is L111 = 0.94 s, where s is the cascade pitch. The incident turbulence is specified by von

Karman spectrum. Reynolds stress profiles within the cascade passage and in the wake

are compared with experimental data provided by Larssen [29], and qualitative agreement is

obtained. The reduction in turbulence kinetic energy happens over a distance on the order of

one integral length scale downstream of the leading edge. Downstream of station x = 0.65c,

the decay rate of the kinetic energy is nearly the same as the decay rate upstream of the

cascade or in the absence of the cascade.

The LES results are also compared with the predictions of rapid distortion theory as for-

mulated by Graham [23]. To check the limitations of the RDT solution we changed the

characteristics of the incoming turbulence and solved two cascade configurations, a 6-plate

cascade and a 3-plate cascade. The Reynolds stress profiles and energy spectra obtained

by LES are in full agreement with RDT in a region approximately equal to one integral

length scale downstream of the leading edge. Graham’s RDT solution continues to provide

accurate Reynolds stresses and energy spectra on the mid-passage for a distance longer than

one integral length scale downstream of the trailing edge. However, on the plate plane, the

LES results indicate a build up of upwash fluctuations in the wake, whereas in the RDT for-

mulation the upwash is set to zero on that plane. These fluctuations are due to the nonlinear

118

effects which are captured by the LES results but not by RDT. The RDT solution contin-

ues to agree with LES in other cases in which the integral length scale of the turbulence is

smaller and the intensity of the turbulence is as large as 11%. However, both RDT and the

present model do not capture the correct Reynolds stresses on the wake centerline because

the mean shear is neglected.

Recommended Future Work

The following improvements are recommended:

1- Use stretched grid near walls for more accurate representation of the wake and the vortex

sheets.

2- Improved differencing and filtering for handling the discontinuities in the tangential com-

ponent across the wake vortex sheet.

3- Model cascades with realistic airfoil geometry (with rounded leading edge).

119

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125

Appendix A

Graham’s RDT

In this appendix, we summurize Graham’s [23] formulation of the rapid-distortion theory

(RDT) for turbulence cascade interaction. In rapid distortion theory (RDT); nonlinear and

viscous effects are neglected and the mean strain field which distorts the turbulence is applied

rapidly. Conditions which must be satisfied for RDT to apply to convected turbulence are:

1- (u′∞/U∞)(x1/L

111) ¿ 1, where x1 is a distance downstream from the leading edge of the

plate and L111 is the turbulence integral length scale.

2- High-turbulence Reynolds number.

Graham [23] considered a linear unstaggered cascade of thin plates 0 < x1 < c, −∞ <

x2 < ∞, x3 = nh, −∞ < n < ∞,. The incident turbulence has corresponding velocity

components (u1∞, u2∞, u3∞) and is convected at the mean speed U∞. The whole vorticity

field ζ of the turbulence is convected with the free-stream speed as a frozen distribution

and the effect of the plate on the turbulence is irrotational outside the wake. The transport

equation for the vorticity is given by:

∂ζ/∂t + U∞∂ζ/∂x1 = 0.

The incident turbulence velocity field is assumed to be homogeneous and can therefore be

126

represented by Fourier integrals.

The incident turbulence velocity field is given by:

u∞(x, t) =

∫ ∫ ∫u∞(k)ei(ωt−kjxj)dk, j = 1, 2, 3 (A.1)

where ω = k1U∞, and k is the wave number vector. Each Fourier component of the total

flow field can be expressed as:

u(x, t;k) = u∞(k)ei(ωt−kjxj) +∇[φ(k, x3)e

i(ωt−k1x1−k2x2)], (A.2)

where φ is the velocity potential due to the blocking effect of the cascade. Since the velocity

field is divergence-free, φ satisfies:

∂2φ/∂x23 − (k2

1 + k22)φ = 0,

with the boundary conditions on the plate:

∂φ/∂x3 + u3∞e(−ik3nh) = 0 on x3 = nh, n=all ± integers.

Solution to Laplace’s equation (subjected to zero normal velocity on the plates and Kutta

condition at the trailing edge) for each Fourier component in the region between any two

plates is given by:

u1 = u1∞ +ik1

τ


sinh(τs)u3∞eik3x3 (A.3)

u2 = u2∞ +ik2

τ



u3 = u3∞ − sinh[τx3]e−ik3s + sinh[τ(s− x3)]


where τ = (k12 + k2

2)12 .

127

Graham’s solution is valid for a streamwise distance on the order of the integral length scale

x = O(L111) downstream of the leading edge. Assumptions of RDT imply that the solution

is not valid for shorter or much greater distance than the integral length scale.

128

Appendix B

Glegg’s Linearized Solution

Glegg [13]obtained an analytical expression for the unsteady loading, acoustic mode ampli-

tude, and sound power output of a three-dimensional rectilinear cascade of blades with finite

chord excited by a three dimensional gust. Here, we summarize the mathematical expres-

sions obtained by Glegg [13]. The cascade consists of a set of infinitely thin flat plates of

chord c which have a stagger angle χ. The mean flow is parallel to the blade surface with

velocity components (U, 0,W ). The flow normal to the blade surfaces will be zero and so

the incident flow perturbation w will induce a scattered field which must satisfy the linear

equations of fluid motion. The blade will shed vorticity into the wake which is convected

downstream by the steady flow. Kutta condition requires no discontinuity in the pressure

on the wake.

The incident gust is a harmonic upwash velocity given by:

w.n = w0ei(−ω

′t+γ0x+αy+νz). (B.1)

Since the blades have infinite span, the scattered field will be harmonic in time and is defined

as:

φ(x, t) = φ(x, y)ei(−ω′t+νz). (B.2)

129

By specifying ω = ω′ − νW , the velocity potential of the scattered field is given by

φ(x, y) =1

2π

∫ ∞

−∞

∫ ∞

−∞

−iµD(γ)

(ω + γU)2/c20 − γ2 − µ2 − ν2

{∑

n

ein(σ+γd+µh)}e−i(γx+µy)dγdµ.

(B.3)

Note how the velocity potential of the scattered field is specified by the function D which

is the Fourier transform of the discontinuity across the blades and the wakes. To obtain D,

equation B.3 must be combined with the boundary conditions to give an integral equation

which can be solved by using the Wiener Hopf method. The Fourier transform of the

discontinuity in velocity potential is given by

D(γ) = { −iw0

(2π)2(γ + γ0)J+(γ)J−(−γ0)} − {

∞∑n=0

(An + Cn)ei(γ−δn)c

i(ω + γU)(γ − δn)[J−(δn)

J−(γ)]}

−{∞∑

m=1

Bm

(γ − εm)[J+(εm)

J+(γ)]} (B.4)

The functions J−(γ) and J+(γ) are defined by equation (A18) in Glegg [13].

To obtain 4φ(x) we evaluate the inverse Fourier transform of equation B.4

4φ(x) = 2πi

∞∑n=0

Ane−iδnx

i(ω + δnU)+

∞∑n=0

∞∑m=1

(An + Cn)eiεm(c−x)−iδnc

i(ω + εmU)(εm − δn)[J−(δn)

J′−(εm)

]

+∞∑

n=0

∞∑m=1

Bme−iδnx

(δn − εm)[J+(εm)

J′−(δn)

], 0 < x < c. (B.5)

The surface pressure jump 4p(x) is obtained as

4p(x) = −ρ0D4φ

Dt(B.6)

The unsteady loading is defined as the integral of the unsteady pressure over the blade

surface. The non-dimensional unsteady pressure is obtained as

Cp =2iω

U0w0c[{ −iw0

4π2γ0J+(0)J−(−γ0)}+ {

∞∑n=0

(An + Cn)e−iδnc

iωδn

[J−(δn)

J−(0)]}

+{∞∑

m=1

Bm

εm

[J+(εm)

J+(0)]}]. (B.7)

130

The sound power is given by

W± =ω′ρ0Bbπ2

2βse

Re∑m

|ζ±mD(λ±m)|2√κ2

e − f 2m

, (B.8)

where se =√

d2 + (hβ)2 and fm = (σ +κMd−2πm)/se, B is the number of blades, and b is

the span. The (+) sign is for the upstream and the (-) is for the downstream sound power.

131

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