A FINITE VOLUME, CARTESIAN GRID METHOD FOR COMPUTATIONAL

AEROACOUSTICS

By

MIHAELA POPESCU

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2005

Copyright 2005

by

Mihaela Popescu

To my daughter Ileana Klein, my parents, my brother and my sister, for their love and support

iv

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my advisor Dr. Wei Shyy for his

subtle yet effective methods of encouraging all of his students. I will always value my

good fortune to have been one of them. Seldom does one encounter individuals with

intellectual caliber, scientific temperament and a spirit of humanity that he embodies. I

am equally grateful for his enduring enthusiasm and boundless patience during the

process of preparing me to be a researcher and contributor to the global scientific

community. Without his enthusiasm and commitment to excellence, my research in this

field/area could not have been accomplished.

I would like also like to express my deepest appreciation for Dr. Mark Sheplak who

assessed my strength and successfully directed my research toward acoustics. I have

benefited substantially from his guidance, experience, knowledge, and philosophy both

professionally and personally.

Similarly, I thank sincerely my committee members Dr. Lou Cattafesta, Dr. Nam

Ho Kim and Dr. Jacob Nan-Chu Chung for their support, encouragement and sharing

freely of their expertise whenever it was needed.

Being part of the Computational Fluid Dynamics (CFD) group was a source of

pride and honor. I have always felt fortune to be part of this group and am grateful

especially for their genuine friendship and support. I benefited from both collaboration

and the pleasant work environment offered by members and visitors of CFD group. They

were a real team as well as family for me. Our advisor and friend, Dr. Wei Shyy, fostered

v

close and sincere relationship between members and visitors which enhanced the

experience for all of us.

I would like to thank my cousin Catalina and Laly Chirita, my friend Rada

Munteanu, my friends from home Florina Carabet, Cosmin Carabet and Lascu Luminita

for finding the most extraordinary and creative ways of being there for me whenever I

needed them.

I would like to thank to my family members: my parents, Aurica and Conatantin

Popescu; my brothers and sisters, Pavel, Teofil, Marinela, Daniel Popescu, and Tabita

Chirita and their family, for their sustaining support and love throughout years. They

were the undisclosed source of energy and strength that I relied on during challenging

times

Finally, I would like to express my deepest love and appreciation to my daughter

Ileana Klein, who demonstrated perseverance and resilience in face of unfavorable

prognosis and the challenging circumstance of her birth.

vi

TABLE OF CONTENTS page

ACKNOWLEDGMENTS ................................................................................................. iv

LIST OF TABLES............................................................................................................. ix

LIST OF FIGURES .............................................................................................................x

ABSTRACT..................................................................................................................... xiii

CHAPTER

1 COMPUTATIONAL ASPECTS IN AEROACOUSTICS ..........................................1

Introduction...................................................................................................................1 Definition of Sound ......................................................................................................3

The Characteristics of Sound.................................................................................3 Viscous Effect in Sound Wave..............................................................................5

Classification of Aeroacoustic Problems......................................................................6 Linear Problems in Aeroacoustics.........................................................................6 Nonlinear Problems in Aeroacoustics ...................................................................9

Lightill’s acoustic analogy ...........................................................................10 Ffowcs Williams – Hawkings equation .......................................................14 The merits of Lighthill and Ffowcs Williams – Hawkings analogies..........16 Nonlinear problem: Shock wave formation .................................................17

Computational Techniques for Aeroacoustics............................................................19 Direct Numerical Simulation (DNS) ...................................................................19 Perturbation Technique .......................................................................................20 Linearized Euler Equation...................................................................................21

Computational Issues..................................................................................................21 The Numerical Approach to Reduce Dissipation and Dispersion.......................22 Complex Geometry .............................................................................................24

Scope...........................................................................................................................27

2 ASSESMENT OF DISPERSION-RELATION-PRESERVING AND SPACE-TIME CE/SE SCHEMES FOR WAVE EQUATIONS..............................................30

Introduction.................................................................................................................30 The Dispersion-Relation Preservation (DRP) Scheme...............................................32

Discretization in Space ........................................................................................32

vii

Time Discretization .............................................................................................37 The Space-Time Conservation Element and Solution Element Method ....................41

a–µ Scheme .........................................................................................................41 a-ε Scheme ..........................................................................................................46

Numerical Assessment of the DRP and Space-Time Schemes ..................................49 Short Wave: b/∆x = 3 ..........................................................................................57 Intermediate Wave: b/∆x = 6..............................................................................58 Long Wave: b/∆x = 20.........................................................................................58

Summary and Conclusions .........................................................................................62

3 FINITE VOLUME TREATMENT OF DISPERSION-RELATION-PRESERVING AND OPTIMIZED PREFACTORED COMPACT SCHEMES FOR WAVE PROPAGAION.....................................................................................63

Numerical Schemes ....................................................................................................65 DRP Scheme........................................................................................................66

Finite volume-based DRP scheme (DRP-fv) ...............................................66 Boundary treatment of the DRP scheme ......................................................69

OPC Scheme........................................................................................................70 Finite-difference-based optimized prefactored compact (OPC-fd) scheme.70 Finite volume-based OPC scheme (OPC-fv) ...............................................71 The boundary treatment of the OPC scheme ...............................................71

Time Discretization – The Low Dispersion and Dissipation Runge-Kutta (LDDRK) Method............................................................................................72

Analytical Assessment of DRP and OPC Schemes....................................................77 Operation Count ..................................................................................................77 Dispersion Characteristics ...................................................................................78 Stability................................................................................................................79

Computational Assessment of the DRP and OPC Schemes.......................................82 Test problem 1: One-Dimensional Linear Wave Equation .................................82 Test problem 2: One-Dimensional Nonlinear Wave Equation............................85 Test problem 3: One-Dimensional Nonlinear Burgers Equation ........................89 Test problem 4: Two-Dimensional Acoustic Scattering Problem.......................91

Summary and Conclusions .........................................................................................94

4 A FINITE VOLUME-BASED HIGH ORDER CARTESIAN CUT-CELL METHOD FOR COMPUTATIONAL AEROACOUSTICS.....................................98

Introduction.................................................................................................................98 Cut-Cell Procedure .....................................................................................................99 Test Cases .................................................................................................................104

Radiation from a Baffled Piston ........................................................................104 General description ....................................................................................104 Directive factor D.......................................................................................107 Pressure on the face of the piston...............................................................109 Low frequency (ka = 2) ..............................................................................110 High frequency (ka = 7.5) ..........................................................................111

viii

Reflection of a Pulse on an Oblique Wall .........................................................112 Wave Generated by a Baffled Piston and Reflected on an Oblique Wall .........115

Conclusion ................................................................................................................117

5 SUMMARY AND FUTURE WORK ......................................................................119

Assessment of DRP and Space-Time CE/SE Scheme..............................................121 Finite-Volume Treatment of Dispersion-Relation-Preserving and Optimized

Prefactored Compact Schemes ............................................................................122 Cartesian Grid, Cut-Cell Approach for Complex Boundary Treatment...................123 Future Work..............................................................................................................124

LIST OF REFERENCES.................................................................................................125

BIOGRAPHICAL SKETCH ...........................................................................................134

ix

LIST OF TABLES

Table page 2-1 The stencil coefficient for N = 3...............................................................................36

3-1 A summary of proposed CAA algorithms................................................................96

3-2 The computational cost for DRP and OPC schemes................................................97

4-1 Published cut-cell approach for different problems ...............................................118

x

LIST OF FIGURES

Figure page 1-1 Sound propagation away from a source .....................................................................3

1-2 Moving surface - Ffowcs Williams and Hawkings equation ...................................15

1-3 Schematic diagram showing the (2n + 1) point stencil on a nonuniform grid ...........................................................................................................................24

1-4 Cartesian boundary treatment of curved wall surface; b) detail around boundary...................................................................................................................27

2-1 xα∆ versus α∆x for the optimized DRP 4th order scheme, 7 point stencil, standard 6th order central scheme and 4th order central scheme ................35

2-2 Dispersive characteristics of DRP scheme...............................................................36

2-3 Scheme of the solution elements (SEs) and conservation elements (CEs) ..............43

2-4 Comparison between analytical and numerical solutions Effect of ε on the accuracy of space-time a-ε scheme ....................................................................52

2-5 The dependence of the error on ε for the space-time a-ε scheme at t = 200: a) b/∆x = 3; b) b/∆x = 6; c) b/∆x = 20.........................................................53

2-6 The dependence of the error as function of ν for short (b/∆x = 3), intermediate (b/∆x = 6) and long (b/∆x = 20) waves................................................55

2-7 Effect of ν on the accuracy of space time a-ε scheme: b/∆x = 3, t = 200 ...............56

2-8 The behavior of the error in function of the wavelength: comparison between DRP and space-time a-ε schemes ..............................................................57

2-9 Accumulation of the error in time for short wave - b/∆x = 3 ...................................59

2-10 Accumulation of the error in time for intermediate wave - b/∆x = 6 .......................60

2-11 Accumulation of the error in time for long wave - b/∆x = 20..................................61

3-1 Grid points cluster for one-dimensional problem ....................................................67

xi

3-2 Grid notation for two-dimensional problem.............................................................69

3-3 Four-six–stage optimized Runge-Kutta of order four scheme: a) dissipation error; b) phase error ...........................................................................74

3-4 Dispersive characteristics of the schemes ................................................................80

3-5 Phase speed error on a logarithmic scale .................................................................80

3-6 Errors with respect to the time step size under a fixed space ∆x, at t = 50 - linear wave equation ..............................................................................................84

3-7 Errors under a fixed CFL = 0.5, at t = 50 - linear wave equation: ...........................85

3-8 Errors with respect to the space step size under a fixed CFL = 0.5, at t = 5; nonlinear wave equation .................................................................................87

3-9 DRP–fd solution - nonlinear wave equation; t = 5; CFL = 0.5 ................................87

3-10 DRP–fv solution - nonlinear wave equation; t = 3; CFL = 0.5 ................................88

3-11 OPC-fd solution - nonlinear wave equation; t = 5; CFL = 0.5................................88

3-12 OPC-fv solution - nonlinear wave equation; t = 5; CFL = 0.5................................88

3-13 Error as a function of Pe - nonlinear Burgers equation............................................90

3-14 Numerical solution obtained by DRP schemes nonlinear Burgers quation......................................................................................................................91

3-15 Numerical solution obtained by OPC schemes nonlinear Burgers equation ....................................................................................................................91

3-16 Instantaneous pressure contours at time t = 7; two-dimensional acoustic scattering problem ....................................................................................................93

3-17 The pressure history at point A, B and C .................................................................93

4-1 Illustration of the interfacial cells and cut-and-absorption procedures ..................100

4-2 Modified cut – cell approach for CAA..................................................................101

4-3 Radiation from a baffled piston (test problem (ii)) ...............................................105

4-4 Radiation from a baffled piston: Beam patterns |D(θ)|; ka = 2 .............................108

4-5 Radiation from a baffled piston: Beam patterns |D(θ)|; ka = 7.5 ..........................108

4-6 Radiation from a baffled piston: Beam patterns |D(θ)|; ka = 12.5 ........................108

xii

4-7 Radiation from a baffled piston: Real of piston radiation impedance....................109

4-8 Radiation from a baffled piston: Numerical solution on axis (ka = 2) ..................110

4-9 Radiation from a baffled piston: Comparison between analytical and computed solutions on axis (ka = 2).......................................................................111

4-10 Radiation from a baffled piston: Contour plot of pressure (ka = 2)......................111

4-11 Radiation from a baffled piston: Comparison between analytical and computed solutions on axis (ka = 7.5) ...................................................................112

4-12 Radiation from a baffled piston: Contour plot of pressure (ka = 7.5) ...................112

4-13 Reflection of the pulse on an oblique wall (test problem (ii)): general description ..............................................................................................................113

4-14 Reflection of the pulse on an oblique wall (test problem (ii)): cell around boundary.....................................................................................................114

4-15 Reflection of the pulse on an oblique wall: history of pressure for different angle of wall ............................................................................................115

4-16 Reflection of the pulse on an oblique wall: α = 630 ...............................................115

4-17 Wave generated by a baffled piston and reflects on an oblique wall: General description of the domain .........................................................................116

4-18 Wave generated by a baffled piston and reflects on an oblique wall: α = 630; a) t = 9; b) t = 14; .....................................................................................116

xiii

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

A FINITE VOLUME, CARTESIAN GRID METHOD FOR COMPUTATIONAL AEROACOUSTICS

By

Mihaela Popescu

August 2005

Chair: Wei Shyy Major Department: Mechanical and Aerospace Engineering

Computational Aeroacoustics (CAA) combines the disciplines from both

aeroacoustics and computational fluid dynamics and deals with the sound generation and

propagation in association with the dynamics of the fluid flow, and its interaction with

the geometry of the surrounding structures. To conduct such computations, it is essential

that the numerical techniques for acoustic problems contain low dissipation and

dispersion error for a wide range of length and time scales, can satisfy the nonlinear

conservation laws, and are capable of dealing with geometric variations.

In this dissertation, we first investigate two promising numerical methods for

treating convective transport: the dispersion-relation-preservation (DRP) scheme,

proposed by Tam and Webb, and the space-time a-ε method, developed by Chang.

Between them, it seems that for long waves, errors grow slower with the space-time a-ε

scheme, while for short waves, often critical for acoustics computations, errors

accumulate slower with the DRP scheme. Based on these findings, two optimized

xiv

numerical schemes, the dispersion-relation-preserving (DRP) scheme and the optimized

prefactored compact (OPC) scheme, originally developed using the finite difference

approach, are recast into the finite volume form so that nonlinear physics can be better

handled. Finally, the Cartesian grid, cut-cell method is combined with the high-order

finite-volume schemes to offer additional capabilities of handling complex geometry. The

resulting approach is assessed against several well identified test problems,

demonstrating that it can offer accurate and effective treatment to some important and

challenging aspects of acoustic problems

1

CHAPTER 1 COMPUTATIONAL ASPECTS IN AEROACOUSTICS

Introduction

Aeroacoustics deals with the sound generated by aerodynamics forces or motions

originating in a flow rather than by externally applied forces or motions of classical

acoustics. Thus, the sounds generated by vibrating violin strings and loudspeakers fall

into the category of classical acoustics, whereas the sound generated by turbulent flow or

the unsteady aerodynamic forces on propellers falls into the domain of aeroacoustics.

A main feature of aeroacoustics is the large difference between the energy levels of

the unsteady flow fluctuation and the sound. This is true even for a very loud noise. For

example, in the near-acoustic field of a supersonic jet (at about 10 jet-diameters away)

the acoustic disturbance amplitudes are about three orders of magnitude smaller than flow

disturbance (Lele, 1997). For supersonic jets one percent of the noise is from the

mechanical power of the jet. In many other cases, the efficiency can be much smaller;

thus the amplitude of disturbance is much less.

Another issue in acoustics is the difference between the scales of the unsteady flow

and sound. This phenomenon is evident in situations in which flow speed is much less

than the sound speed that characterizes the medium. This is because the time scale of the

flow and the sound must match. In low Mach number flow (M « 1) this will give an

acoustic wavelength that is M-1 times longer than the flow length scale. In this case, a

direct numerical simulation of the dynamic flow and generation of sound field will not be

possible.

2

An important challenge in computational aeroacoustics (CAA) lies in the coupling

of time and space that appears in the acoustic wave. An acoustic wave has a wavelength

λ in space, as well as a frequency ω = 2πf in time. These are coupled by the relation

λf = c, where c is the speed of sound in the medium (assumed quiescent). Thus, to

determine an acoustic wave, one must resolve both its wavelength and its frequency.

Numerical spatial and temporal approximation meets the same problem: the dissipation

must be reduced in space, like that in time. However, the CFD community has done much

work to overcome these difficulties, and high accuracy numerical schemes have been

developed. CAA schemes try to minimize dissipation and dispersion error in both space

and time and still maintain a certain order of accuracy.

A further problem lies in the effect of the grid on the solution. In CFD, grids are

often stretched to provide high resolution in regions of high gradients with lesser

resolution where gradients are less steep. However, it is very difficult to propagate an

accurate wave in this kind of grid. The dissipation and dispersion characteristics depend

upon the Courant-Friederich-Lewy (CFL) number (= c∆t/∆x in one dimension), which

can be interpreted as the distance an acoustic wave travels in one time step. If CFL is

changing because the grid has been stretched, strange phenomena can occur.

Vichnevetsky (1987) showed that if a wave is propagated over an expanding grid, the

wave could actually appear to change frequency and be reflected such that it starts

propagating back in the other direction. Anisotropy can appear because of the different

sized grid spacing along different directions. An approach to solving the problem was

suggested by Goodrich (1995), who recommended approximating the solution of the

3

governing partial differential equations rather than approximating the derivatives and

solving the resulting numerical scheme.

Boundary conditions for CAA are also problematic. Often one is interested in

solving the problem in an unbounded space. But for such computation, boundary

conditions have to be imposed; thus the computational domain is finite. The first problem

is the selection of the boundary condition to be imposed. Clearly, the proper boundary

condition depends upon what is outside the computational domain: e.g., source, reflection

boundary and free space. The second problem is in the implementation of these

conditions.

Definition of Sound

The Characteristics of Sound

A wave is the movement of a disturbance or piece of information from one point to

another in a medium (excluding electromagnetic waves, as they do not require a

medium). Sound is a wave that moves parallel to the direction of the propagation

Figure 1-1. Sound propagation away from a source

As a sound wave propagates, it disturbs the fluid from its mean state. These

disturbances are almost always small. We will consider departures from a state in which

the fluid is at rest with a uniform pressure p0 and density ρ0. When this is perturbed by a

sound wave the pressure at position xr and time t changes to p0 + p’ ( xr , t), the density

source

4

changes to ρ0 + ρ’( xr , t), and the fluid particle moves with a velocity ( ),v x tr r . The ratios

0'/p p and 0'/ρ ρ are usually much less than unity.

To illustrate the magnitude of the disturbance in an acoustic wave we will consider

pressure waves in air which can be detected by the human ear. The pressure waves are

referred to as sound. For harmonic pressure fluctuation (a pure tone), the typical range of

frequency that our ear can detected sound is

20 Hz ≤ f ≤ 20 kHz (1.1)

The range described by (1.1) is called the audible range. The maximum sensitivity

of the ear is within the frequency range of 2 kHz to 3 kHz (policeman’s whistle tone) The

acoustic pressure of intense rocket engine noise can be as much as 9 orders of magnitude

greater than the pressure of the weakest sound detectable by human ear. A logarithm

scale was necessary to be able to comprehend this large range: 9 decade in amplitude (18

decades in intensity) (Blackstock, 2000). The logarithm devised is called a level, e.g.,

sound pressure level and intensity level. Although the levels are unitless, they are

expressed in decibels (dB).

SPL = 20 log10(p’/pref) (1.2)

On this scale a fluctuation of one atmosphere in pressure corresponds to 194 dB.

The threshold of pain is between 130 and 140 corresponding to a pressure variation of

only one thousandth of an atmosphere, i.e. p’/p0 = 10-3.

At the threshold of pain, fluid particles in a 1 kHz wave vibrate with a velocity of

about 0.1 m/s, which is only about 1/104 of the speed at which the sound wave is

traveling. The displacement amplitude of the particles is thus between 10-4 and 10-5

5

meters and the wavelength is about one-third of a meter, so the displacement amplitude is

only about 10-4 of the wavelength sound (Dowling and Ffows Williams, 1983).

In the case where sounds can be approximated by small perturbations, several

effects are noteworthy. First, there is no interaction between different acoustic waves; the

sound fields add linearly. Hence, for the study of sound that comes from more than one

source, we can study the sound from each source separately, and then add the final

solutions. Second, the flow variables satisfy the linearized equation of the fluid motion,

so each flow variable is linearly related to every other variable. This leads to a great

simplification in the mathematics. Knowledge of one variable, e.g., pressure, provides a

basis for a simple evaluation of all the other variables such as the density or particle

velocity. Third, if we wish to solve the behavior of the sound numerically, we have to use

a scheme that adds a low level of numerical dissipation.

Viscous Effect in Sound Wave

The effect of Reynolds number on CAA is ambiguous. Although all sound is

ultimately dissipated into heat by viscosity, acoustics is generally considered to be

inviscid fluid phenomena. If the viscous term is considered in the standard linear

analysis, one finds its value to be

0Re 2 /cπ λρ η= (1.3)

where c is the speed of sound, λ is the wavelength, ρ0 is the density, and η is the

coefficient of viscosity. The values in air of these parameters for most practical interests

are (Blackstock, 2000)

6

0

5

331.60.331.293

1.7 10

cλρ

η −

=≅≅

≅ ⋅

(1.4)

Based on these values, Re is approximately 4.8 ×107 at the most audible frequency.

This high value of Reynolds number determines that viscous effects are negligible in a

sound field, because the pressure represents a far greater stress field than that induced by

viscosity. Hence, we will regard sound as being essentially a weak motion of an inviscid

fluid. The effect of viscosity can be taken into consideration after the sound travels about

2πλρ0/η wavelengths (i.e., after ≅ 1.5x108 m). Thus, dissipative loss becomes important

for high frequency sound propagation over long distances.

On the other hand, if one considers the generation of sound by flows rather than

propagation of sound through flows, then a significant Reynolds number effect can be

observed. For example, the most common source of sound in flows is the acceleration of

the vorticity (Powell, 1964), which is only present in viscous flow. Even there, a curious

independence is observed that is apparently due to the fact that the large-scale; hence

efficient sound generation structures in the flow change little with Reynolds.

Classification of Aeroacoustic Problems

Diverse problems of aeroacoustics can be classified on the basis of the physical

phenomena that are expected to play a central role. The main classification is between the

linear and nonlinear problem.

Linear Problems in Aeroacoustics

A major strength offered by computational approaches is the generality to deal with

linear interaction problems of aeroacoustics (where one type of physical disturbance is

7

scattered into another type of disturbance). A linearized solution of the unsteady Euler

equations subject to appropriate boundary condition can provide useful prediction of the

noise generation. The problems of linear interaction are not limited to sound generation

but also contain the “reciprocal” problem of the generation (or receptivity) of vertical

disturbance due to the incident sound (Colonius and Lele, 2004).

This category includes the classical boundary value problems of linear acoustics:

sound propagation in a uniform media in presence of reflecting surfaces, barriers,

absorbing walls, and ducts, and also propagation and scattering of sound in a prescribed

non-uniform medium. Specific examples include as: radiation from a duct opening or

engine-inlet due to a specified source or a specified duct-mode disturbance, radiation

from a specified duct-mode disturbance, and radiation from a specified source across a

finite barrier/sound wall with an absorbing treatment.

Sound propagation in a specified non-uniform time dependent medium including

refraction/scattering in steady and unsteady vortical flows, sound propagation in non-

uniform ducts including the interaction with geometrical changes, linear impedance and

mean-flow variations are considered as linear problems of scattering.

The equation of a linear acoustics wave is deduced for a homogeneous fluid

characterized by ρ = ρ0, P = p0, and 0 'u u u= +r r r . The sound waves minutely disturb the

status of the quiescent fluid

0 0

20 0 0

0

,

,

0 ', '

P p p p c

u u u c

ρ ρ δρ δρ ρ

ρ

= + <<

= + <<

= + <<r r r

(1.5)

8

To obtain the wave equation, we will start with the Euler equation and the equation

of state for an isentropic process

( )0

0t

t

u u

u u u P

ρ ρ ρ

ρ

+ ⋅∇ + ∇ ⋅ =

⎡ ⎤+ ⋅∇ ⋅ + ∇ =⎣ ⎦

r r

ur r r (1.6)

and

0p c δρ= (1.7)

Substituting Eq.(1.5) into (1.6), the following is obtained

( ) ( )0

0 0

0

0

t

t t

u u u

u u u u u u p

δρ δρ ρ δρ

ρ ρ δρ δρ

+ ⋅∇ + ∇ ⋅ + ∇ ⋅ =

+ ⋅∇ + + ⋅∇ + ∇ =

r

ur r r r r (1.8)

The first order terms are small because the perturbation is very small; hence,

second order terms or higher can be neglected. The underlined terms are second order or

higher. The result is

0

020

0

0t

t

u

u p

p c

δρ ρ

ρ

δρ

+ ∇ ⋅ =

+ ∇ =

=

r

ur (1.9)

To reduce the set to a single equation in ur , the equation of state Eq.(1.7) first is

used to eliminate p∇ from the second expression in Eq. (1.9)

( )20 0 0tu cρ δρ+ ∇ =r (1.10)

If the time derivative of this equation is subtracted from the divergence of the first

Eq. (1.9), the result is the classical wave equation

2 20 0ttu c u− ∇ =

s r (1.11)

9

Nonlinear Problems in Aeroacoustics

Nonlinear aeroacoustics is the field of acoustics devoted to very intense sound,

specifically to waves with amplitude high enough that the small-signal assumption is

violated. Retaining the nonlinear term makes the conservation equation much harder to

solve. This includes problems of nonlinear propagation, such as nonlinear steeping and

decay, focusing, viscous effects in intense sound beam (acoustics near field of high-speed

jets), and sonic boom propagation through atmospheric turbulence, sound propagation in

complex fluids and multi-phase systems (such as in lithotripsy), and internal flows of

thermo-acoustic cooling system. Also included in this category are problems of scattering

of nonlinear disturbances into sound, such as airframe noise.

The basic equations that describe the nonlinear behavior are the same that describe

the general motion of a viscous, heat conducting fluid: mass conservation, momentum

conservation, entropy balance, and thermodynamic state.

The mass conservation or continuity equation is

0D uDt

ρ ρ+ ∇ ⋅ =r (1.12)

where ρ is the mass density, ur is the fluid velocity vector, and D/Dt = ∂/∂t + ur ·∇ is the

total, or material, time derivative.

The momentum equation may be written as

( )2 13B

Du P u uDt

ρ µ µ µ⎛ ⎞+ ∇ = ∇ + + ∇ ∇ ⋅⎜ ⎟⎝ ⎠

rr r (1.13)

where P is the pressure, µ is the shear viscosity, and µB is the bulk viscosity. Shear

viscosity accounts for diffusion of momentum between adjacent fluid elements having

different velocity. Bulk viscosity provides an approximation description, valid at low

10

frequency, of nonechilibrum deviation between actual local pressure and thermodynamic

pressure.

Equation of state is given by

( )

( )

,

,

P P TorP P s

ρ

ρ

=

=

(1.14)

where T is the absolute temperature, and s is the specific entropy.

A commonly used equation of state of a perfect gas is

( )00 0

exp / v

P s s cP

γρρ

⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦

⎝ ⎠ (1.15)

where P0, ρ0, and s0 are reference values of the pressure, density, and specific entropy.

γ = cP/cv is the ratio of the specific heats at heat pressure (cp) and constant volume (cv). In

case of arbitrary fluid the equation of state is obtained by expanding of Eq.(1.14) in a

Taylor series about (ρ0, s0).

A more general description take implies relaxation, like vibrational relation of

diatomic molecules (as in air) and chemical relation in seawater (Hamilton and

Blackstock, 1997). The former occurs when the energy associated with molecular

vibration fails to keep in step with molecular translation energy associated with the

fluctuating temperature in gas.

Lightill’s acoustic analogy

The study of flow that generates acoustic waves began with Gutin’s theory (1948)

of propeller noise, which was developed in 1937. However, the theory could not be

considered until 1952, when Lighthill (1952, 1954) introduced his acoustic analogy to

11

deal with the problem of calculating acoustic radiation from a relatively small region of

turbulent flow embedded in an infinite homogeneous fluid.

It is well known that a gas (not monoatomic) allows three distinct fundamental

modes of oscillation: the vortical and entropy modes, both of which are convective, and

the acoustic mode, which is the solution of the wave equation (Chu and Kovasznay,

1958). The three modes may exist independently only if i) the oscillations are small, and

ii) the base flow is the uniform medium at rest or is in uniform motion. If the flow is

inhomogeneous, then they are coupled, and it is not easy to separate them.

To overcome these limitations, an approximation is necessary. In this section, the

acoustic approach introduced by Lighthill (1952, 1954) is presented. This approach is

used to calculate acoustic radiation from relatively small regions of turbulent flow

embedded in an infinite homogeneous fluid where the speed of the sound c0 and density

ρ0 is constant. Density and pressure fluctuations are defined by

0

0

''p p p

ρ ρ ρ= −⎧⎨ = −⎩

(1.16)

where the subscript ‘0’ is used to denote constant reference values that are usually taken

to be corresponding properties at large distance from the flow.

Lighthill’s basic idea was to reformulate the general equations of gas dynamics in

order to derive a wave equation suitable to describe sound propagation.

The continuity and momentum equation are

0jj

ijij i

j i j

ut x

u pu ut x x x

ρ ρ

σρ ρ

∂ ∂+ =

∂ ∂

⎛ ⎞ ∂∂ ∂ ∂+ = − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(1.17)

12

where σij is the component of the viscous stress tensor. For a Stokestian gas it can be

expressed in terms of the velocity gradient

23

ji kij ij

j i k

uu ux x x

σ µ δ⎛ ⎞∂∂ ∂

= + −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ (1.18)

where µ is the viscosity of the fluid. Multiplying the continuity equation by ui, adding the

result to the momentum equation, and combining terms yields

( )i i j ij ijj

u u u pt x

ρ ρ δ σ∂ ∂= − + −

∂ ∂ (1.19)

which upon adding and subtracting the term c02∂ρ/∂xi, becomes

20

iji

i i

Tu ct x x

ρ ρ ∂∂ ∂+ = −

∂ ∂ ∂ (1.20)

where

( ) ( )20 0 0ij i j ij ijT u u p p cρ δ ρ ρ σ⎡ ⎤= + − − − −⎣ ⎦ (1.21)

By differentiating the continuity equation with respect with t, and subtracting the

divergence of Eq.(1.20), Lighthill’s equation is obtained

22

2 202

' ' ij

i j

Tc

t x xρ ρ

∂∂− ∇ =

∂ ∂ ∂ (1.22)

Equation (1.22) has the same form as the wave equation that governs the acoustic

field produced by a quadrupole source ∂2Tij/∂xi∂xj in a nonmoving medium (Goldstein,

1976). Hence, there is an exact analogy between the density fluctuation that occurs in any

real flow and the small amplitude density fluctuations that would result from a

quadrupole source distribution (of strength Tij) in a fictitious (nonmoving) acoustic

medium with sound speed c0. This source will vanish in the region outside certain types

13

of turbulent flows such that Eq. (1.22) reduces to the homogeneous wave equation in

such regions.

In the case where the acoustics generated are not due to a jet with high temperature

and the flow is isentropic, the second term of Lighthill’s turbulence stress in Eq.(1.21)

can be neglected.

For a low Mach number, the third term in Eq.(1.21) can be neglected. It can be

neglected because viscosity and heat conduction cause a slow damping due to conversion

of acoustic energy into heat and have a significant effect only over large distances.

We have therefore shown that Tij is approximately equal to ρuiuj inside the flow

and is equal to zero outside this region. Assuming that the density fluctuation is

negligible within the moving fluid, then

Tij ≅ ρ0uiuj (1.23)

The Lighthill approximation has a great advantage in that it is possible to solve the

equation with a standard Green function. But, this approximation is only for isentropic,

low Mach number flows.

Lighthill’s equation could be used to study the sound generated by unsteady flows

where there are no solid boundaries (or more correctly, by flows where the effect of

boundaries can be neglected). Another limitation of this theory is that the principle of

sound generated aerodynamically as stated by Lighthill is relevant only when there is no

back reaction of the acoustic waves in flow, such as at the trailing edge or in initial shear

layer ( Hirschberg, 1994). In this case, the conversion of mechanical energy into acoustic

energy is only one way, and this is the reason why acoustics can be inferred from an

incompressible description of the flow.

14

Ffowcs Williams – Hawkings equation

Let us consider a finite volume of space containing a disturbed flow and rigid

bodies in an arbitrary motion with the surrounding fluid being at rest. The bodies and the

flow generate a sound. In this case, it is possible to replace both the flow and the surface

by an equivalent acoustic source, assuming that the whole medium is perfectly at rest.

The key assumption is that no flow-acoustics coupling occurs, i.e., the acoustic

field does not affect the flow from which the sound originates. Consequently, this

approach is not valid when some resonant conditions induce an acoustic feedback on the

flow. To represent the real medium with the flow and the obstacle in a convenient way,

Ffowcs Williams and Hawkings (1969), and Ffowcs Williams (1969,1992) defined an

equivalent medium where the rigid bodies are replaced by mathematical surfaces. In

order to preserve the kinematics of the flow and the boundary condition of no cross-flow

on the surface, a discontinuity must be imposed at the surface location by introducing

mass and momentum sources.

The mass and momentum equations are written as

( )

( ) ( ) ( )

0

' '

j sij i

i i j ij ijj j

fu u ft x x

fu u u ft x x

ρ ρ ρ δ

ρ ρ σ σ δ

∂ ∂ ∂+ =

∂ ∂ ∂

⎛ ⎞∂ ∂ ∂+ − = −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

(1.24)

In Equation (1.24), ρ0 is the mean density, usi is the velocity field of a point on the

surface, δ is the Dirac delta function, σ’ij (= σij – (P – P0)δij) is the viscous stress tensor

(P being the static pressure with mean value P0) and f( xr ,t) = 0 defines the kinematics of

the surfaces. If the normal unit vector on the surface is nr , then the boundary condition of

no cross-flow is simply

15

su n u n⋅ = ⋅r r r r (1.25)

Figure 1-2. Moving surface - Ffowcs Williams and Hawkings equation

Following the procedure used to obtain Lighthill’s equation, we can derive the

equation for density variation ρ’ = ρ - ρ0

( ) ( )22 2

20 02 2

' ' 'ijij si

i i j i j i

T f fc f u ft x x x x x t xρ ρ σ δ ρ δ

⎛ ⎞∂ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂− = + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

(1.26)

The values of ρ’ and Tij are zero inside the mathematical surface, because in these

zones the fluid is considered to be at rest.

Roger (1995) showed that in the presence of flow and rigid bodies, fluctuations in

the fluid are exactly the same as those that would exist in an equivalent acoustic medium

at rest, and they are forced by three source distribution:

• a volume distribution ∂2Tij /∂xi∂xj in the outer region of the surface due to the flow

• a surface distribution ∂/∂xi (σ’ij δ(f) ∂f/∂xj ) due to the interaction of the flow with the moving bodies

• a surface distribution ∂/∂t ( ρ0 usiδ(f) ∂f/ ∂xi ) due to the kinematics of the bodies

Like in the Lighthill analogy, the analogy of Ffowcs Williams and Hawkins is

limited to flow where there is no flow acoustic coupling, or where the acoustic field does

not affect the flow from which the sound originates.

Vi(t) Fluid at rest S(t) Surface with

rigid-body motion

Ve Real flow field

sur

nr

16

The merits of Lighthill and Ffowcs Williams – Hawkings analogies

The Lighthill analogy marked an important milestone in the study of flow-

generated acoustics. His theory is able to explain a number of important characteristics of

acoustic radiation from a jet. The solution of the Lighthill analogy could be obtained

analytically using the Green function.

The approach was developed when computation capability was limited. That is

why this analogy is a big step for understanding aeroacoustic propagation. For example,

his theory gave a mathematical representation of sound and pseudosound, an

approximation of the sound emission from subsonic cold-air jet flow (which he called

self-noise, i.e., noise generated by turbulent-turbulent interaction, and shear noise, i.e.,

noise generated by turbulent mean shear), for the first time.

Lighthill’s theory represents the basis for the Ffowcs Williams and Hawkings

analogy, which accounts for sound generation along a boundary, as in a helicopter rotor,

an airplane propeller, an aircraft engine fan, a compressor, or a turbine. The solution of

the second analogy is again solved with the Green’s function. In this, it is difficult to

obtain a general solution, because the analytical solution is available only for certain

boundary shapes.

The theory of Ffowcs Williams and Hawkings appeared when the performance of

the computers increased but was still substantially less than what a computer is capable of

today. Hence, this analogy was very useful for studying sound emission from sources

such as a thin strut in a turbulent flow, propeller noise (Gutin, 1948) and sound

generation near a plate.

The advent of powerful computers has made it easier to study more complex

problems using these analogies since no analytical solutions are required.

17

Nonlinear problem: Shock wave formation

An example where a linear approximation is not suitable is subsonic flow in a pipe,

because waves do not attenuate fast with propagation. In this case a wave can propagate

as much as 102 wavelengths before friction has a significant effect. This implies that even

a nonlinearity of order 10-2 in the wave propagation can have a significant effect

(Hirschberg and Rienstra, 1994). The most spectacular nonlinear effect is the formation

of shock waves as a result of the steepening of the compression side of a wave.

The study of the equation of a shock wave is deduced for a homentropic fluid,

namely, a fluid with constant entropy (Hamilton, 1998). The characteristic form of the

equation is obtained from mass and momentum conservation:

( ) 0dpu c ut x cρ

⎛ ⎞∂ ∂⎡ ⎤+ ± ± =⎜ ⎟⎢ ⎥∂ ∂⎣ ⎦ ⎝ ⎠∫ (1.27)

which implies that

dpJ ucρ

± = ± ∫

(1.28)

is invariant along the characteristic path C± in the (x, t) plane which are described by the

equation

C

dx u cdt ±

⎛ ⎞ = ±⎜ ⎟⎝ ⎠

(1.29)

When a C+ wave propagates into a uniform region, the C- wave emanating from the

uniform region will all carry the same information: J- = J0 is constant. Typically for a

simple wave, the characteristic C+ are simple lines in the (x, t) plane. Eq.(1.28) and

Eq.(1.29) yield

u = 1/2 (J+ + J0) (1.30)

18

and

( )012

dp J Jcρ

+= −∫ (1.31)

In other words u and ∫dp/ρc are constant along C+, the thermodynamic variable. Hence,

speed of sound can be considered constant, so that (u + c) of C+ is constant. For an ideal

gas there is the relation

21

dp ccρ γ

=−∫ (1.32)

A shock wave appears at the point where two characteristics lines of the same

family intersect. The intersection of two C+ occurs after traveling time ts:

( )1

0s

d u ct

dx

−⎛ ⎞+⎡ ⎤

= −⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠ (1.33)

which for an ideal gas with constant γ can be expressed in terms of the space derivative

of pressure at x = 0:

( )

0

0

1s

ptpcx

γ

γ= −

∂⎛ ⎞− ⎜ ⎟∂⎝ ⎠

(1.34)

Because the variation of the speed of sound is negligible, the distance of

appearance of the shock wave is given by

( )

00

0

1s s

px c tpx

γ

γ≅ = −

∂⎛ ⎞− ⎜ ⎟∂⎝ ⎠

(1.35)

For harmonic waves

0max

ˆdp k pdx

⎛ ⎞ =⎜ ⎟⎝ ⎠

(1.36)

19

( )

00 ˆ1s

pk xp

γγ

=+

(1.37)

Hence, for an amplitude 0ˆ /p p = 10-2, the shock is expected appear within 10

acoustic waves (Goldstein 1976).

Computational Techniques for Aeroacoustics

Direct Numerical Simulation (DNS)

Direct simulation methods aim to compute both unsteady flow and the sound

generated by it. These methods must use a domain that includes the noise producing flow

region and at least a part of the near-acoustic-field. The computational mesh needs to be

selected so that both the flow and its sound can be well represented. The first issue in this

approach is that the computational cost of such direct computations is large, hence only

simple flow configurations can be studied using this direct method.

Direct simulation of the acoustic field solves the compressible Navier-Stokes

equations (or Euler equations in those cases where viscosity is not important) without

further approximation. These equations govern the total flow field including the

acoustics, so if one could solve them in a domain reaching out to the far field, then the

one can obtain the acoustic radiation emerging from the fluctuating flow. Gloerfelt at al.

(2003) did a comparison of this method with the acoustic analogy. Both methods gave

solutions that agree with the experimental data.

This approach is limited to flows where the viscosity is not important; it encounters

fundamental difficulties when the Reynolds number is high due to the range of scales

present in the flow field. The characteristic frequency of a source that radiates sound is

given by the Strouhal number, St = fL/U ≅ O(1), where L and U are the characteristic

length and velocity. This implies that the wavelength of sound produced is λ≅ L/M,

20

where M is the Mach number of the flow. On the other hand, the dissipation of turbulent

energy takes place at the Kolmogorov length scale η = L/Re3/4 (Tennekes and Lumley

1972), where Re is the Reynolds number of the flow. Thus, the ratio of the wavelength of

sound to the size of the energy dissipating eddies, λ/η≅Re3/4/M, takes high values at low

Mach numbers, even for moderate Reynolds numbers. Accordingly, the requirement for

spatial and temporal resolutions will simply be beyond computational capabilities in the

near future. Thus, although DNS has been utilized quite successfully in low Reynolds

number (Gloerfelt et al., 2003; Hu et al., 2002) (Re < 1000) , a less direct approach where

separate grids can be designed for the viscous and acoustic phenomena appears to be

preferable for high Reynolds number flows.

Perturbation Technique

A perturbation technique consists of splitting up the flow field calculation into two

parts. First, a time-independent viscous background flow is calculated, and then a

perturbation equation (that describes the sound) about this background flow is derived,

and viscous action on the perturbation is neglected. In this approach, an initial

disturbance is introduced upstream which causes an instability and causes the wave to

grow resulting in the radiation of sound. Tam and Morris (1980) developed this approach

analytically for shear layers, and Tam and Burton (1984) developed it for subsonic jets.

Hardin and Pope (1995) developed a slightly different approach to address

vorticity-dominated flow, an expansion about the time-dependent, viscous,

incompressible, subsonic flow. If the density ρ = ρ0 is constant, then the continuity and

momentum equations become a set of four equations for the three incompressible

velocity components and incompressible pressure. One solves this set of equations by

21

using a grid and numerical scheme designed for the viscous aspects of the flow for the

time dependent viscous incompressible flow. Using the solution from the previous time

step, the compressible flow equation is perturbed about the time dependent

incompressible flow. The differences between the compressible and incompressible flows

are then assumed to be inviscid and isentropic.

Linearized Euler Equation

The linearized Euler equation (LEE) comes from the modified Euler equation. The

principal issue appears when we deal with LEE with specific source term and LEE with

projected source terms (Colonius and Lele, 2004). The numerically difficulty of this

approach stems from the fact the full LEE set admit non-trivial instability wave solution

of the homogeneous equation.

Propagation of linear acoustic waves through a medium with known properties, its

refraction and scattering due to the nonuniform of the medium or the base flow, and

scattering from solid surfaces, and scattering from solid surfaces (with prescribed

boundary conditions) are problem for which computational technique are well suited.

Simulation of such phenomena can be based on linearized field equation subject to the

physical boundary conditions; refraction and scattering effects are automatically

obtained. However in such a direct approach the high frequency limit becomes

computationally demanding.

Computational Issues

Numerical methods for problems of sound generation and propagation must

overcome a host difficulties that arise because acoustic waves are very weak compared to

near-field fluctuation, and because they must propagate with little attenuation over long

distances. In practice this has dictated the use of high-order-accurate numerical methods.

22

The Numerical Approach to Reduce Dissipation and Dispersion

In computational aeroacoustics (CAA), accurate prediction of the generation of

sound is demanding due to the requirement of preserving the shape and frequency of

wave propagation and generation. Furthermore, the numerical schemes need to handle

multiple scales, including long and short waves, and nonlinear governing laws arising

from sources such as turbulence, shocks, interaction between fluid flows and elastic

structures, and complex geometries. It is well recognized (Hardin and Hussaini, 1992),

Tam (Tam and Webb, 1993; Tam et al., 1993) that in order to conduct satisfactory CAA,

numerical schemes should induce minimal dispersion and dissipation errors. In general,

higher-order schemes are more suitable for CAA than lower-order schemes since, overall,

the former are less dissipative. That is why higher-order spatial discretization schemes

have gained considerable interest in computational acoustics (Hixon, 1997; Kim et al.,

1997; Lin and Chin, 1995).

For longer wavelengths, the formal order of accuracy is sufficient to indicate the

performance of a scheme. However, for shorter waves relative to the grid size, it is

known that the leading truncation error terms are not good performance indicators (Shyy,

1985; Shyy, 1997). To handle broad band waves, the idea of optimizing the scheme

coefficients by minimizing the truncation error associated with a particular range of wave

numbers has been used over the years by many researchers, e.g., Hu et al. (1996),

Stanescu and Habashi (1998), Shu (1994), Nance et al. (1997), Wang and Sankar(1999),

Cheong and Lee (2001), Wang and Chen (2001), Ashcroft and Zang (2003), and Tang

and Baeder (1999). A successful approach is the Dispersion-Relation-Preserving (DRP)

finite difference scheme proposed by Tam (Tam and Webb, 1993; Tam et al. 1993). The

basic idea in the DRP scheme is to optimize the coefficients to satisfactorily resolve short

23

waves with respect to the computational grid, namely, waves with wavelengths of ∆x

(defined as 6-8 points per wave or PPW) or shorter. It maximizes the accuracy by

matching the wave number and frequency characteristics between the analytical and

numerical operators in the range of resolvable scales. Recently, Ashcroft and Zhang

(Ashcroft and Zang, 2003) have reported a strategy for developing optimized prefactored

compact (OPC) schemes requiring smaller stencil support than DRP. The prefactorization

strategy splits the central implicit schemes into forward and backward biased operators.

Using Fourier analysis, they have shown that it is possible to select the coefficients of the

biased operators such that their dispersion characteristics match those of the original

central compact scheme. Hixon and Turkel (1998) proved that the prefactored scheme is

equivalent to the initial compact scheme if the real components of the forward and

backward operators are equal to those at the corresponding wave number of the original

compact scheme, and the imaginary components of the forward and backward operators

are equal in magnitude and opposite in sign.

Both DRP and OPC schemes are originally designed based on the finite difference

approach. In order to satisfy the governing laws of the fluid physics, it may be

advantageous to adopt the finite volume approach (Udaykumar et al., 1997; Yang and

Ingram, 1999; Udaykumar, 1999), which ensures that fluxes estimated from different

sides of the same surface are identical, i.e., no spurious source/sink is generated due to

numerical treatment. Furthermore, a finite volume formulation can offer an easier

framework to handle the irregular geometry and moving boundaries. In this work, we

investigate a finite volume formulation (which we call DRP-fv), extending the concept

embodied in the original, finite difference-based DRP scheme (which we call DRP–fd).

24

Similarly, for the OPC scheme, we extend the basic concepts of the original, finite

difference-based OPC (OPC-fd) scheme, in a finite volume formulation, named OPC-fv.

Our ultimate goal is to extend the finite volume version of suitable schemes into a cut-

cell type of Cartesian-grid computational technique that we have developed earlier for

moving and complex boundary computations (Yang and Ingram, 1999; Udaykumar et al.,

1999; Kim and Lee, 1996; Ye et al., 1999; Lahur et al., 2000; Calhoun, 2000).

Complex Geometry

To handle problems of practical interest, a CAA scheme needs to have the

capability of handling irregular and curved geometries. It is challenging to develop

methods exhibiting desirable order of accuracy and controlling dispersion and dissipation

errors while being capable of handling complex geometries. In an attempt to address the

need for flexible grid distributions, Cheong and Lee (2001), proposed a grid-optimized

dispersion-relation-preserving scheme (GODRP). They considered the approximation of

the derivative by

( )1 n

j ij n

u a u x x xx x =−

∂≈ + ∆ ⋅ ∆

∂ ∆ ∑ (1.38)

where

( ) / 2n nx x x n−∆ = − (1.39)

( ) / ( , 1,..., 1, )i ix x x x i n n n n∆ = − ∆ = − − + − (1.40)

x0 x1 x2x-1x-2 xn-1 xnx-n x-n+1

( =x )

x0 x1 x2x-1x-2 xn-1 xnx-n x-n+1

( =x ) Figure 1-3. Schematic diagram showing the (2n + 1) point stencil on a nonuniform grid

25

The wavenumber of the scheme over a nonuniform grid is obtained by using a

Fourier transformation:

i

ni x x

jj n

i a ex

αα ∆ ⋅∆

=−

−=

∆ ∑ (1.41)

The scheme is derived by requiring that i) the real part of the scheme closely match

the analytical solution in the chosen range of wavenumbers, and ii) the imaginary part of

the scheme be as close to zero as possible. These requirements can be achieved by

minimizing the integrated error E, defined as

( )

( ) ( )

2

0

2

0

exp ln 2

r

i

e

real

e

imag

E x x d x

xx Sgn c x d x

α α α

αλ α α ασ

= ∆ − ∆ ∆

⎡ ⎤⎛ ⎞∆+ ∆ + − ⋅ − ∆ ∆⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

∫

∫ (1.42)

The terms er and ei denote the upper limits of the integration intervals of the real

and imaginary parts, respectively. The term λ is the weighting factor, σ is the half-width

of a Gaussian function, and c is the speed of wave propagation in ut + cux = 0.

GODRP schemes of curvilinear grids permit an assessment of the accuracy of the

finite difference method for curvilinear meshes from the wave number point of view.

Through the grid-optimization process, high-order finite difference equations can be

solved with curvilinear grids with a guarantee of local and thus resultant global

dispersion-relation-preserving properties. Hence, the approach can be used with success

for a body-fitted grid to study the generation of sound around a body with complex

geometry.

The coefficients are obtained based on local characteristics of the grid; in other

words, for a 2n+1 – point stencil GODRP spatial discretization, the scheme implies a

26

solution of 2n+1 equations for each point in the grid. This will lead to a considerable

increase in the computational cost.

Tam et al. (2000) proposed to solve the problem around a boundary using ghost

points and an optimized extrapolation scheme around the complex boundary. They

considered an extrapolation scheme that is of desirable accuracy over a wavenumber

range –k ≤ α∆x ≤ k. The goal for the extrapolation is that the scheme works well for

general functions. For this purpose, it is sufficient to consider waves with unit amplitudes

over a desired band of wavenumbers

fα(x) = ei[αx + φ(α)] (1.43)

The total effect on the function f(x) will be weighted by the amplitude A(α).

The general formula for extrapolation is given by

( ) ( )1

0 00

,N

j j jj

f x x S f x x x j xη−

=

+ ∆ = = − ∆∑ (1.44)

where Sj (j = 0, 1, 2, …, N-1) are the stencil coefficients. Their values are obtained by

imposing the following: i) the difference between the left and right sides of Eq.(1.44) is

zero when the single Fourier components of Eq.(1.43) is substituted in the formula; ii)the

error is zero if the approximated function is zero. The constrained optimization problem

is handled by the Lagrange multiplier

2

1 1

0 00

1k N N

i y ijyj j

j jL e S e dy Sη λ

− −−

= =

⎛ ⎞= − + −⎜ ⎟

⎝ ⎠∑ ∑∫ (1.45)

The boundary curve is approximated by line segments joining the intersection

points of the computation mesh and the boundary. For instance, the curved surface

between A and B in Figure 1-4 is replaced by a straight line segment. G2 is a ghost point.

A ghost value is assigned to G2 as a boundary condition. The enforcement point is at E

27

and G2E is perpendicular to AB. The value of the derivatives of the function at points A

and B are found via extrapolation from the points at (1’, 2’, 3’, 4’, 5’, 6’, 7’) and (1, 2, 3,

4, 5, 6, 7), respectively.

The method is accurate only for waves with a wavelength of 8 mesh spacings or

longer. The method induces numerical instability when the wavenumber is high. Hence,

we do not expect that the approach gives accurate results for short waves or high

wavenumber.

G1

G4

G3

G2

G5

A

B

1 2 3 7654

1’

2’

3’

7’

6’

5’

4’

G1

G4

G3

G2

G5

A

B

1 2 3 7654

1’

2’

3’

7’

6’

5’

4’

G2

A

B

E

1’

2’

1 2

G2

A

B

E

1’

2’

1 2

a) b) Figure 1-4. Cartesian boundary treatment of curved wall surface; b) detail around

boundary

Scope

The present thesis has three main contributions. First we investigate two numerical

methods for treating convective transport: the dispersion-relation-preservation (DRP)

scheme, proposed by Tam and coworkers (Tam and Webb, 1993; Tam et al., 1993), and

the space-time a-ε method, developed by Chang (1995). The purpose is to examine the

characteristics of existing schemes capable of handling acoustic problems. The space-

28

time a-ε method directly controls the level of dispersion and dissipation via a free

parameter, ε, while the DRP scheme minimizes the error by matching the characteristics

of the wave. Insight into the dispersive and dissipative aspects in each scheme is gained

from analyzing the truncation error. Even though both methods are explicit in time, the

appropriate ranges of the CFL number,ν, are different between them. Different

performance characteristics are observed between the two schemes with regard to long

and short waves.

Second, the two low dispersion numerical schemes developed via optimization

procedures in the wave number domain, namely, the DRP scheme and the optimized

prefactored compact (OPC) scheme developed by Ashcroft and Zhang (2003), are

extended from their original finite difference framework to the finite volume framework.

The purpose is to extend the capability of these schemes to better handle nonlinearity and

conservation laws of the fluid motion. Linear and nonlinear wave equations, with and

without viscous dissipation, have been adopted as the test problems. By highlighting the

principal characteristics of the schemes and utilizing linear and nonlinear wave equations

with different wavelengths as test cases, the performance of these approaches is studied.

Finally, the Cartesian grid, cut-cell method is extended along with the OPC-based

finite volume scheme so that this high order scheme can treat curved geometry associated

with practical acoustic applications. The approach uses a background Cartesian grid for

the majority of the flow domain with special treatment being applied to cells which are

cut by solid bodies, thus retaining a boundary conforming capability. Surface integrals

around complex geometries are computed using flow properties at the cell surface

interpolated from cell nodes while maintaining desirable accuracy level.

29

The rest of the thesis is structured as follows. Chapter 2 presents the investigation

of the Dispersion-Relation-Preserving (DRP) scheme, and the space-time a-ε method.

The characteristics of these two schemes are emphasized using a simple wave equation

with the initial disturbance in the form of the Gaussian function

Chapter 3 presents principal characteristics and introduces two space discretization

schemes: the DRP scheme and the OPC scheme. A low dispersion and low dissipation

Runge-Kutta proposed by Hu and coworkers (1996) is employed for the time stepping

procedure, and combined with the DRP and OPC schemes. A study of the dispersive

characteristics and stability is presented for these schemes. The boundary treatment is

presented in this chapter. The DRP and OPC schemes are then extended to the finite

volume approach. Four linear and nonlinear test problems are presented to evaluate the

merits of these schemes.

Chapter 4 presents the principal characteristics for the Cartesian grid, cut-cell

approach. Additionally, we present the proposed adjustment for the acoustic approach.

Finally, several test problems are presented to demonstrate the performance of the present

approach.

Finally, in Chapter 5, summary and future perspectives are presented.

30

CHAPTER 2 ASSESMENT OF DISPERSION-RELATION-PRESERVING AND SPACE-TIME

CE/SE SCHEMES FOR WAVE EQUATIONS

Introduction

A number of numerical schemes based on various concepts have been proposed to

treat wave propagation and convective transport, including the concepts of upwinding,

flux splitting, total variation diminishing, monotonicity, non-oscillatory, higher order

differencing, and Riemann solvers. For some representative works, see van Leer (1979),

Roe (1981), Harten (1983, 1989), Osher and Chakravarthy (1983), Shyy (Shyy, 1983;

Shyy et al., 1997), Leonard (1988), Shu and Osher (1988), Hirsch (1990), Liou and

Steffen (1993), Tam (Tam and Webb, 1993; Tam et al., 1993), Chang (Chang, 1995;

Chang et al., 1999), Thakur et al. (Part I, 1996; Part II, 1996), Toro and Billet (1996), Yu

and Chang (1997), Loh et al. (1998, 1999, 2000), Wang and Moin (2000), Oran and

Boris(2001). In this chapter, we focus on two approaches: the dispersion-relation –

preservation (DRP) scheme based on an optimized high-order finite difference concept,

proposed by Tam (Tam and Webb, 1993; Tam et al., 1993), and the space-time concept,

proposed by Chang (Chang, 1995; Chang et al., 1999). These two schemes have derived

based on interesting concepts, and have been applied to compute flow problems requiring

balanced treatment for both dispersion and dissipation.

The well studied wave equation, first order linear hyperbolic equation, is:

0u uct x

∂ ∂+ =

∂ ∂, where a > 0 is a constant (2.1)

31

The exact solution of the wave equation for the initial value problem with initial

data

( ,0) ( )( , ) ( )

u x U xu x t U x ct

== −

(2.2)

A simple equation like this one has served useful purpose to guide the development

in numerical techniques.

The challenge for the numerical solution of the wave equation is to maintain the

correct sharpness of the solution without creating wiggles. In other words, the goal is to

offer satisfactory resolution, especially for the high wave number components, by

balancing numerical dissipation and dispersion. Here we use the term “balancing”

because of the finite resolution possessed by numerical computation.

The two methods optimized higher-order finite difference (DRP) method (Tam and

Webb 1993) and the space-time conservation element and solution element (CE/SE)

(Chang, 1995) address the aforementioned issues from different angles. The philosophy

of the DRP method is to maximize the accuracy by matching the wave number and

frequency characteristics between the analytical and the numerical operators in the range

of resolvable scales. The space-time CE/SE method views the flux calculations in a

joined space-time conservation element, taking into account the unified wave propagation

in a solution element. The present work intends to complement the analyses offered in the

original studies to gain further insight into issues related to dispersion, dissipation, and

resolution. The main features of both approaches will be highlighted via truncation error

analysis and numerical computation to evaluate order of accuracy, stability constraints,

and the implication in spatial and temporal resolutions of each scheme. We contrast the

32

performance of the two schemes by varying the ratio between the dominant wavelength

and numerical mesh spacing from long (~ 20 or higher) to short (~ 3).

In the following, we will first present the concepts and derivation of both schemes,

and then conduct computational evaluations. Some of details presented in Tam and

Webb’s paper (1993) and in Chang’s paper (1995) will be summarized to help discuss the

related analysis and numerical assessment conducted in the present work

The Dispersion-Relation Preservation (DRP) Scheme

In the following, we summarize the method by Tam and Webb (1993) to highlight

the salient features of DRP scheme.

Discretization in Space

As illustrations, x

∂∂

is approximated by the central difference schemes of various

orders considering uniform grids with spacing to ∆x, we give three examples.

• The second order central difference approximation is

21 1 ( )2

l l

l

u uu O xx x

+ −−∂⎛ ⎞ = + ∆⎜ ⎟∂ ∆⎝ ⎠ (2.3)

• Fourth-order approximation is

42 1 1 28 8 ( )12

l l l l

l

u u u uu O xx x

+ + − −− + − +∂⎛ ⎞ = + ∆⎜ ⎟∂ ∆⎝ ⎠ (2.4)

• Sixth-order approximation is

63 2 1 1 2 39 45 45 9 ( )60

l l l l l l

l

u u u u u uu O xx x

+ + + − − −− + − + −∂⎛ ⎞ = + ∆⎜ ⎟∂ ∆⎝ ⎠ (2.5)

The construction of the scheme (Tam and Webb, 1993)

( ) ( )1 N

jj N

u x a u x j xx x =−

∂≅ + ∆

∂ ∆ ∑ (2.6)

33

is based on two goals: (i) the behavior of the numerical solution in the resolvable wave

number range closely matches that of the exact solution, and (ii) the formal order of

accuracy of scheme spanning 2N + 1 nodes is 2(N - 1).

To facilitate the derivation, one can start with the Fourier Transform of a function -

f(x) and its inverse

( ) ( )12

i xf f x e dxααπ

∞−

−∞

= ∫% (2.7)

( ) ( ) i xf x f e dαα α∞

−∞

= ∫ % (2.8)

as well as the derivative and shift theorems

( )

( )

~

i fx

f xα α=

∂

∂% (2.9)

( ) ( )~

if x e fαλλ α+ = % (2.10)

With these tools, the relationship between the differential and discrete operators

(given in Eq.(2.6)) can be approximated as

1 Nij x

jj N

i f a e f i fx

αα α∆

=−

⎡ ⎤≅ =⎢ ⎥∆ ⎣ ⎦

∑% % % (2.11)

where

N

ij xj

j N

i a ex

αα ∆

=−

−=

∆ ∑ (2.12)

We see that α is a periodic function of α∆x with a period of 2π. Obviously the

goal is to ensure that α is as close to α as possible. To accomplish this goal the error is

minimized over a certain wavenumber range, α∆x ∈[−η; η] − the numerical dispersion is

reduced by narrowing the range of optimization (Tam and Webb, 1993)

34

2 ( )E x x d xη

η

α α α−

= ∆ − ∆ ∆∫ (2.13)

It is noted that α is real, and hence the coefficients aj must be anti-symmetric, i.e.,

a0 = 0 and a-j = -aj. (2.14)

On substituting Eq.(2.12) into Eq.(2.13) and taking

Eq.Error! Reference source not found. into account, E can be written as

( )2

12 sin

N

jj

E k a k j dkη

η =−

⎡ ⎤= − ⋅⎢ ⎥

⎣ ⎦∑∫ (2.15)

where k = α∆x

Hence, with the anti-symmetric condition, one can use the method of the least-

squares fitting to minimize E, over a wave number range, namely:

0,l

Ea

∂=

∂ l = 1, 2,…N (2.16)

Furthermore, to ensure that the scheme is accurate to O(∆x2(N-1)), additional

conditions can be made by expanding the right side of Eq.(2.6) as a Taylor series and

then equating terms. In this way we have N -1 independent equations with N unknowns:

(aj)j=1,N. Note that aj = -a-j, j = 1,…,N, and a0 = 0.

Tam (Tam and Webb, 1993) shows the relationship between xα∆ and α∆x for the

4th –order, DRP scheme and, the aforementioned 6th-order, 4th -order and 2th -order central

difference schemes over the interval 0 to π

For α∆x up to αc∆x the individual curves are nearly the same as the straight line

x xα α∆ = ∆ . Thus, the finite difference scheme can provide reasonable approximation for

wave number so α∆x becomes less than αc∆x. If we wish to resolve a short wave with

35

straightforward central difference approximations using a fixed size mesh, we need to use

a scheme with a large stencil. On the other hand, the 7-point DRP scheme has the widest

favorable range of xα∆ . Tam and Webb (1993) demonstrate that DRP can be effective in

improving the performance of a given stencil within certain wave number (see

Figure 2-1). In the following, we will only consider the DRP scheme with a symmetric

stencil and in particular the 7-point formula.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

α∆x

DRP 6th order4th order

Figure 2-1. xα∆ versus α∆x for the optimized DRP 4th – order scheme, 7 point stencil,

standard 6th -order central scheme and 4th -order central scheme

To follow wave propagation, it is important to evaluate the group velocity of a

finite difference scheme. The group velocity is characterized by /d dα α , which should

be almost one in order to reproduce exact result. A way to reduce dispersion is to adjust

the range of the wave number in the optimization process. b shows /d dα α curves of the

DRP scheme, for different ranges of the optimization parameter η. Upon examining the

corresponding /d dα α curves, α∆x = 1.1 gives the best overall fit. In this case based on

the criterion 1.0 .01ddαα

− < , the optimized scheme has a bandwidth 15% wider than the

standard 6th order central difference scheme.

36

For long wave the important range of α∆x is small, and hence any value of η less

than 1.1 is reasonable. For short waves, on the other hand, dispersion can be noticeable

with any choice of η.

Table 2-1. The stencil coefficient for N = 3 η a1

a2 a3

π/2 0.7934 -0.1848 0.0254 1.1 0.7688530 -0.1650824 0.0204372 0.9 0.762145 -0.1597162 0.0190957 0.85 0.7607435 -0.1585945 0.0188153 0.7 0.7571267 -0.1557013 0.0180920 0.6 0.7551720 -0.1541376 0.0177011

0 0.2 0.4 0.6 0.8 10.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01

α∆x

standard-2th orderstandard-4th orderstandard-6th orderDRP scheme-η=1.1

f(α∆x)=1

( )( )

d xd x

αα

∆∆

0 0.2 0.4 0.6 0.8 10.99

0.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01

α∆x

standard-2th orderstandard-4th orderstandard-6th orderDRP scheme-η=1.1

f(α∆x)=1

( )( )

d xd x

αα

∆∆

a)

0 0.2 0.4 0.6 0.8 10.99

0.994

0.998

1.002

1.006

1.01

α∆x

η=0.7 η=0.9 η=1.1 eta=π/2

f(α∆x)=1

( )( )

d xd x

αα

∆∆

0 0.2 0.4 0.6 0.8 10.99

0.994

0.998

1.002

1.006

1.01

α∆x

η=0.7 η=0.9 η=1.1 eta=π/2

f(α∆x)=1

( )( )

d xd x

αα

∆∆

b) Figure 2-2. Dispersive characteristics of DRP scheme: a) ( ) ( )/d x d xα α∆ ∆ versus α∆x for

optimized and standard schemes. .b) ( ) ( )/d x d xα α∆ ∆ versus α∆x for the 4th order optimized scheme for different wave number range of optimization: range is from –η to η

37

Time Discretization

Following Tam and Webb (1993) suppose that u(t) is an unknown vector and the

time axis is divided into a uniform grid with time step ∆t. To advance to the next level,

we use the following 4-level finite difference approximation:

( )3

( 1) ( )

0

n jn n

jj

uu u t bt

−+

=

∂⎛ ⎞≅ + ∆ ⎜ ⎟∂⎝ ⎠∑ (2.17)

Similar to space discretization, the goal is to develop schemes which closely match

the exact solution in the frequency domain, and exhibit desired formal orders of accuracy.

In Eq.(2.17) there are four constants, namely: b0, b1, b2, b3. To determine these

constants and create a scheme of O(∆t3) accuracy, the terms in Eq.(2.17) are expanded in

a Taylor series to match exactly up to order ∆t2. This leaves one free parameter, b0. The

relationship between other coefficients and b0 are (Tam and Webb, 1993; Tam et al.,

1993):

1 0 2 0 3 053 16 233 ; 3 ;12 3 12

b b b b b b= − + = − = − + (2.18)

One can utilize the Laplace transform to determine b0. First, the Laplace transform

and its inverse of a function f(t) are related by

( ) ( ) ( ) ( )0

1 ;2

i t i tf f t e dt f t f e dω ωω ω ωπ

∞−

Γ

= =∫ ∫% % (2.19)

where Γ is a line in the upper half ω-plane parallel to the real- ω -axis above all poles and

singularities. Also, the shifting theorem for Laplace transformation is

( )~

if t e fω− ∆+ ∆ = % (2.20)

To apply the Laplace transform, we need to generalize the equation to one with a

continuous variable, namely,

38

3

0

( )( ) ( ) jj

u t j tu t t u t t bt=

∂ − ∆+ ∆ = + ∆

∂∑ (2.21)

On applying the shifting theorem to four-level scheme presented in Eq.(2.21), we

find that

~

3

0

i t i j tj

j

uue u t b et

ω ω− ∆ ∆

=

⎛ ⎞ ∂= + ∆ ⎜ ⎟ ∂⎝ ⎠

∑% % (2.22)

Hence,

( )

3

0

1i t

ij tj

j

i e

t b e

ω

ωω

− ∆

∆

=

−=

∆ ∑ (2.23)

To optimize the time stepping scheme, the error is optimized

[ ] ( ) ( ) ( )221 Re( ) 1 ImE t t t t d t

ς

ς

σ ω ω σ ω ω ω−

= ∆ − ∆ + − ∆ − ∆ ∆⎡ ⎤⎣ ⎦∫ (2.24)

i.e., 1

0

0.dEdb

=

In Eq.(2.24) σ is the weight and ζ is the non-dimensionalized frequency range

needed for the numerical scheme to match the exact solution. Substituting Eq.(2.18) and

(2.23) into Eq.(2.24) E1 becomes a function of b0 alone,

For σ = 0.36 and ζ = 0.5 the scheme becomes (Tam and Webb, 1993)

0 1 2 32.30256; 2.49100; 1.57434; 0.38589b b b b= = − = = − (2.25)

Eq.(2.23) indicates that the relationship between tω∆ and ω∆t is not one to one.

This means that spurious solutions appear. The stability will be established in function of

the real solution. It can be written in the form:

39

4 3 23 2 1 0 0i ib z b z b z b z

t tω ω⎛ ⎞+ + + + − =⎜ ⎟∆ ∆⎝ ⎠

(2.26)

where z = e iω∆t and the values of bj (j = 0, 1, 2, 3) are given by Eq.(2.26).

To maintain satisfactory temporal resolution while being stable, the imaginary part

of the solution (ω∆t) should be negative but close to zero. The interval 0 0.41tω< ∆ <

satisfies these expectations. Furthermore, Re( ) Re( )t tω ω∆ ≅ ∆ in this range. To ensure

that the damping numerical is minimized, Tam and Webb (1993) suggest the condition

4Im( ) 0.118 10tω −∆ ≤ − ⋅ (i.e., 0 0.19tω< ∆ < is adopted). This condition guarantees

numerical stability and negligible damping.

To compute stability of the scheme we take into consideration Fourier-Laplace

transformation of the wave equation (Eq.(2.1))

% % %12

initiali u ci u uω απ

∗− = − + (2.27)

where α, ω* characterize the PDE . For the long wave we can approximate wavenumber

of the scheme with wavenumber of the PDE

α α≅ (2.28)

which leads to

% % %*initialu c u kuω α= + (2.29)

Hence

* c kkω α= + (2.30)

The condition of the numerical stability is that amplification factor for time

discretization is less than 1, and hence ω*∆t <0.41. It is also noted from Figure 2-1 that

1.8xα∆ ≤ (2.31)

40

hold true. By introducing Eq. (2.31) into (2.30) and upon multiplying by ∆t it is found

that

[ ]1.8 M 1ct kk tx

ω∗∆ ≤ ⋅ + ⋅∆∆

(2.32)

where M is mach number. Therefore to ensure numerical stability it is sufficient by Eq.

(2.32) to restrict ∆t to be less than ∆tmax, where ∆tmax is given by

[ ]max

0.411.8 M 1

xtkk c

∆∆ =

⋅ + (2.33)

Therefore, for ∆t < ∆tmax the schemes are numerically stable. Consequently, the

schemes yield the following criteria for numerical stability:

0.21CFL ≤ (2.34)

Based on Eqs.(2.1), (2.6), and (2.17) one can obtain the final form of the DRP

scheme with 7-point in space and 4-point in time:

3 3

1

0 3

n n n jl l j k l k

j k

tu u c b a ux

+ −+

= =−

∆= −

∆ ∑ ∑ (2.35)

The leading truncation error of the scheme (given in Eq.(2.35)) can be evaluated

using the Taylor series expansion, yielding:

( )

3 33 3

0

4 54 3 3

4 3 5

0 3

16 4

( )32

4! 10 5!

t x j xxxxj

xxxxx

k

j kj k

u c u c j b u x

u x O xcc j j b a k

ν

ν

=

=

= =−

+ ⋅ = − + ∆

− + ∆ ∆

⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎛ ⎞ ⎤+ + +⎢ ⎜ ⎟ ⎥⎦⎢ ⎝ ⎠⎣

∑

∑ ∑ (2.36)

Replacing in Eq.(2.36) by the numerical values of the various coefficients, the

scheme becomes:

( )3 3 4 4 51.6318 1.223 0.012814 ( )t x

xxxx xxxxx

u a u

a u x a a u x O xν ν

+ ⋅ =

− ⋅ ⋅ ⋅∆ − ⋅ + ⋅ ∆ + ∆ (2.37)

41

The following can be summarized in regard to the present DRP scheme:

• the scheme is forth order in space, and third order in time

• ν < 0.21 assures stability and reasonable accuracy

• the first term on the RHS of Eq.(2.37) is dissipative. The accuracy bound, ν < 0.21, indicates that the coefficient of the leading dissipative term is small. This observation suggests that, depending on the relative magnitude of ν and ∆x, dispersive patterns may be dominant in numerical solutions.

The Space-Time Conservation Element and Solution Element Method

The tenet in this method is to treat local and global flux conservation in a unified

space and time domain. To meet this requirement, Chang (1995) introduces solution

elements, which are subdomains in the space-time coordinates. Within each solution

element, any flux vector is then approximated in terms of some simple smooth functions.

In the last step, the computational domain is divided into conservation elements within

which flux conservation is enforced. Note that a solution element generally is not the

same as a conservation element. We summarize in the following the key concepts

adopted in Chang’s approach.

a–µ Scheme

Consider Eq.(2.1), and define F1 = -u, F2 = au, x1 = x, , and x2 = t

Applying Green theorem we obtain:

( )2 11 1 2 2

1 2 S

F F dxdy F dx F dxx xΓ

⎛ ⎞∂ ∂− = +⎜ ⎟∂ ∂⎝ ⎠

∫∫ ∫ (2.38)

where ( )1 1 2 2S

F dx F dx+∫ is contour integral on closed curve S.

Following a vector notation, Eq.(2.38) can be written as

( )1 1 2 2S

F dx F dx+∫ = ( )S V

g ds⋅∫r r (2.39)

42

where 1 2( , )g F F=r and ( )1 2,ds dx dx=

r . (dx1, dx2) is a differential vector associated with a

point (x1, x2) on the closed curve S.

Because Eq.(2.1) is valid anywhere in the definition domain, we obtain

( )

0S V

g ds⋅ =∫r r

(2.40)

where (i) S(V) is the boundary of an arbitrary space-time region V in E2 (a two-

dimensional Euclidean space) with x1 = x and x2 = t; (ii) ( )S V

g ds⋅∫r r is contour integral on

closed curve S(V); (iii) (F2, -F1) is a current density in E2. Note that g ds⋅r r is the space-

time flux of (F2, -F1) leaving the region V through the surface element.

As shown in Figure 2-3a, Ω is a set of mesh points (j, n) that is adapted to discretize

a physical domain, where 2, ij n = ± , with i = 0, 1, 2, 3, … To facilitate the construction

of the present scheme a solution element (SE) associated with (j, n) is illustrated in b.

For any (x, y) ∈SE(j, n), u(x, t), is approximated by u∗(x, t; j, n):

*( , ; , ) ( ) ( ) ( ) ( )n n n nj x j j t ju x t j n u u x x u t t= + − + − (2.41)

with nju , ( )n

x ju , and ( )n

t ju are constants in SE(j,n), ( ), n

jx t are coordinates of the mesh

index (j,n), and

( ) ( )* *

* ( , ; , ) ( , ; , )( , ; , ) , ,n nnj j x tj j

u x t j n u x t j nu x t j n u u ux t

∂ ∂= = =

∂ ∂ (2.42)

43

Figure 2-3. Scheme of the solution elements (SEs) and conservation elements (CEs): a) The index positions of SEs and CEs; b) SE(j,n); c) CE-(j,n); d) CE+(j,n); e) CE+(j-1/2, n+1/2); f) CE_(j+1/2, +1/2);

n+1 n+1/2 n n-1/2 n-1

j-3/2 j-1 j-1/2 j j+1/2 j+1 j+3/2 a)

/ 2x∆ / 2x∆

/ 2t∆

/ 2t∆

(j,n)

b)

c) d)

( -1 2 , -1 2)SE j n⊂

(j,n) A

B(j-1/2, n-1/2)

( , )SE j n⊂

( 1 2, -1 2)SE j n⊂ +

(j,n) A

(j+1/2, n-1/2)

(j,n) (j,n)

f) e)

( , )SE j n

( 1 2 , 1 2)SE j n⊂ + + ( -1 2, 1 2)SE j n⊂ +

44

Eq.(2.41) has the form of the first-order Taylor series expansion. Furthermore, if

one takes into account that ( ) ( )n nt xj j

u c u= − , because u∗(x, t; j, n) satisfies wave equation,

then Eq.(2.41) becomes

*( , ; , ) ( ) [( ) ( )]n n nj x j ju x t j n u u x x c t t= + − − − (2.43)

In each SE(j, n), ( , ) ( ( , ), ( , ))g x t u x t cu x t= −r is approximated by

* * *( , ;, , ) ( ( , ; , ), ( , ; , ))g x t j n u x t j n cu x t j n= −r (2.44)

In order to develop appropriate approximation for the flux, one divides the physical

domain into nonoverlapping rectangular elements, referred to as conservation elements

(CEs). Specifically, CE receive sign “-“ or “+” in function of the slope of the line that

connects the vertex from Ω of a CE. If the slope is negative, CE receives the positive

sign; otherwise the sign is negative

In Figure 2-3c the line that unifies the vertices founded in Ω is positive, and CE

index Ω is (j,n). The final notation is CE-(j, n). The surface of CE belongs to two

different SE, SE(j, n) and SE(j-1/2,n-1/2). To specify that a part of surface is in a certain

SE in Figure 2-3c, that part is around by a certain type of line. Figure 2-3d, Figure 2-3e,

and Figure 2-3f illustrate three other corresponding cases relating the conservation

element to the solution element.

The approximation of Eq.(2.40) is

( )( )( ),

, 0S CE j n

F j n g ds±

± = ⋅ =∫r r (2.45)

for all (j, n)∈Ω. ( )( ),S CE j n

g ds±

⋅∫r r is contour integral over closed path S(CE±(j,n)) and

represents the total flux leaving the boundary of any conservation element is zero. The

45

flux at any interface separating two neighboring CEs is calculated using the information

from a single SE. As an example, the interface AC depicted in Figure 2-3c) and Figure 2-

3d) is a subset of SE(j, n). Thus the flux at this interface is calculated using the

information associated with SE(j, n).

We integrate along the entire boundary S(CE±) of CE± such that CE± is on the left

as we advance in the direction of integration (S(CE±) is traversed counterclockwise).

With the above preliminaries, it follows from (39) that:

( )( ) ( )( ) ( )( ) ( ) ( )1 22 2 1 21

1 222 1 2

2 14 , 1 1n n n nx x j jj j

F j n u u u uxxν

ν ν − −± ±±

⎡ ⎤= ± − + − + −⎣ ⎦ ∆∆

m

(2.46)

where, again, c tx

ν ∆=

∆ is CFL number.

With the aid of Eqs.(2.45) and (2.46), nju and ( )n

x ju can be solved in terms of 1 2

1 2nju −−

and ( ) 1 2

1 2

nx j

u −

±, if (1 - ν2) ≠ 0, i.e.,

( ) ( ) ( ), 1 2, 1 2 1 2, 1 2q j n Q q j n Q q j n+ −= − − + + − (2.47)

where

( )( )

( , )/ 4

nj

nx j

uq j n

x u

⎛ ⎞⎜ ⎟=⎜ ⎟∆⎝ ⎠

for all (j, n) ∈ Ω (2.48)

( )

( )( )

22 1 11 11 1;2 21 1 1 1

Q Qν νν ν

ν ν+ −

⎛ ⎞− − −⎛ ⎞+ −⎛ ⎞ ⎛ ⎞⎜ ⎟= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ + − +⎝ ⎠ ⎝ ⎠ (2.49)

We have employed the Taylor series expansion to evaluate the leading truncation

errors of Eq.(2.46), yielding the following two expressions for F+ and F- respectively:

46

22 4

22 4

( )48 16 16 48

( )48 16 16 48

t x xxx

t x xxx

c c c cu cu u x O x

c c c cu cu u x O x

ν νν

ν νν

⎧ ⎛ ⎞+ = − + + + ⋅∆ + ∆⎪ ⎜ ⎟

⎪ ⎝ ⎠⎨

⎛ ⎞⎪ + = − + − + ⋅∆ + ∆⎜ ⎟⎪ ⎝ ⎠⎩

(2.50)

From our derivation indicated by Eqs.(2.49), and (2.50) we can establish the

following observation for the space-time CE/SE scheme:

• The method is second order in space and time;

• The method is stable if ν2 < 1

• The dispersion strength of the scheme increases when ν (or time step) is reduced because ν appears in the denominator in the leading truncation error term in Eq.(2.50). Hence a small value of ν may not be desirable.

• The third order term O(∆x3) is zero in Eq(2.50), indicating that not only the ∂2u/∂x2 term but also the ∂4u/∂x4 terms are non-present. This is the second reason, in addition to previous observation, why the present scheme can be highly dispersive.

a-ε Scheme

In order to help to reduce the dispersive nature of the present scheme, Chang

(1995) introduces an “artificial viscosity” into original CE/SE scheme. Instead of

F±(j,n) = 0, the following modification is made:

( )( )

( )221

( , )4

nx j

xF j n du

ε ν±

− ∆= ± (2.51)

where ε is an independent parameter of the numerical variables, and

( ) ( ) ( ) ( ) ( )1/ 2 1/ 2 1/ 2 1/ 211/ 2 1/ 22 1/ 2 1/ 2

/n n n n nx x x j jj j j

du u u u u x− − − −+ −+ −

⎡ ⎤= + − − ∆⎣ ⎦ (2.52)

Because the magnitude of the added terms in the scheme is controlled by ε, the

numerical dissipation is controlled by ε. However, because F±(j,n)≠0 if ε ≠ 0, strictly

specking CE+(j,n) and CE-(j,n) are no longer conservation elements in the a-ε scheme. On

the other hand, although the net flux entering the interface separating CE+(j,n) and

47

CE-(j,n) is not zero, but the sum of F+(j,n) and F-(j,n) is zero. Hence the total flux leaving

CE(j,n) vanishes. As a result, CE(j,n) is a conservation element in the a-ε scheme. These

are seemingly subtle but relevant feature of the present scheme.

Using Eqs.(2.47), (2.51), (2.52) one obtains the solution of the a-ε scheme as:

( ) ( ) ( ), 1 2, 1 2 1 2, 1 2q j n M q j n M q j n+ −= − − + + − (2.53)

where

( ) ( ) ( )221 12 2

1 11 1;

1 2 1 1 2 1M M

ν νν νε ε ν ε ε ν

+ −

⎛ ⎞− − −⎛ ⎞+ −= = ⎜ ⎟⎜ ⎟ ⎜ ⎟− − + − − +⎝ ⎠ ⎝ ⎠

(2.54)

To assess the leading truncation errors in the a-ε scheme a modified equation is

derived by substituting Taylor series expansion into Eq.(46) for ( )1212

nx j

u −

±,

1212

nju −±

The resulting equations are:

22

3

22

3

12 4 4 12 3 3 2

6 6 2

12 4 4 12 3 3 2

6 6 2

t x xxx

xxxx

t x xxx

xxxx

c c c c c c xu cu u

c c xu HOT

c c c c c c xu cu u

c c xu HOT

ν ν ε εν ν

νεν

ν ν ε εν ν

νεν

⎧ ⎛ ⎞ ∆⎛ ⎞+ =− + + + − −⎪ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎪

⎪ ∆⎛ ⎞⎛ ⎞⎪ + − +⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎝ ⎠⎨

⎛ ⎞ ∆⎛ ⎞⎪ + = − + − + − +⎜ ⎟⎜ ⎟⎪ ⎝ ⎠⎝ ⎠⎪⎪ ∆⎛ ⎞⎛ ⎞+ − +⎪ ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎩

(2.55)

From Eq.(2.55), we can see that the a-ε scheme introduce dissipative term uxxxx and

reduce the strength of the dispersive uxxx. The net outcome term is that the formal order of

accuracy is maintained while the scheme becomes less dispersive. Again, it is undesirable

to employ a small ν, which is different from the DRP scheme discussed previously. As

48

already mentioned, in the a-ε scheme, both dispersive and dissipative aspects are affected

by ε.

In Refs. (Chang, 1995; Chang et al., 1999), insight is offered in regard to the choice

of ε and ν, based on the von Neumann stability analysis of the a-ε scheme, without

detailed information of the truncation error. The analysis reported in these references

suggests that: (i)ν should be large; (ii) ε shouldn’t be close to 0 or 1; (iii) the spurious

solution can be effectively suppressed for ε = 0.5 in the case of long-wavelength.

Reference (Loh et al., 1998) suggests that for aeroacoustic computation, it is essential to

choose a large CFL number and a small ε.

Further insight can be gained based on the present truncation error analysis. If we

relate ε and ν ( CFL number) by ε = ν + ε1, then, in Eqs. (2.55):

• Coefficient of uxxx in first equation of the system Eq. (2.55) is equal to:

2

11 ( 1)

3 4cν ν ε

ν⎡ ⎤+ −⎛ ⎞ −⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

(2.56)

• Coefficient of uxxx in second equation of system Eq. (2.55) is equal to:

( )2

1

6 113 4

cν νν ε

ν

⎡ ⎤− +−⎛ ⎞ ⎢ ⎥−⎜ ⎟⎝ ⎠ ⎢ ⎥⎣ ⎦

(2.57)

The above equations indicate that if ν is close to 1, and ε and ν are close to each

other, i.e., if ε1 is small, then the leading dispersion error shown in Eq.(2.55) is small. On

the other hand, to maintain adequate numerical dissipation, it is helpful to let ε vary in the

same manner as ν. These observations as well as the analysis in Chang (1995) indicate

that the value of ν and ε should be coordinated. Further evaluation will be made based on

numerical computations, to help establish a more explicit guideline.

49

Numerical Assessment of the DRP and Space-Time Schemes

To assess the individual and relative merits of the DRP and space-time schemes the

following simple test problem is adopted.

0.u ut x

∂ ∂+ =

∂ ∂ (2.58)

The initial condition imposed at time t = 0 are:

2

2

exp ln 2

2ln 2 exp ln 2x

xub

x xub b

⎧ ⎡ ⎤⎛ ⎞= −⎪ ⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎣ ⎦

⎨⎡ ⎤⎪ ⎛ ⎞ ⎛ ⎞= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎩

(2.59)

which is a Gaussian profile.

The exact solution is:

2

2

exp ln 2

2ln 2 exp ln 2x

x tUb

x t x tUb b

⎧ ⎡ ⎤−⎛ ⎞= −⎪ ⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎣ ⎦

⎨⎡ ⎤− −⎪ ⎛ ⎞ ⎛ ⎞= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎩

(2.60)

In this study we evaluate the performance of the schemes in short, intermediate,

and long waves relative to the grid spacing, which is assured by the value b/∆x. In all

cases ∆x =1, and c = 1, so that ∆t = ν.

For the optimized DRP scheme, we adopt the one with 4th level in time and 7-point

in space, and 4th order formal accuracy in space and 3rd order formal accuracy in time:

( )3

1

0

n n n jl l j l

j

u u b Fν+ −

=

= + ∑ (2.61)

where tx

ν ∆=

∆ (because c = 1)

50

( ) ( )3

3

k kl j l j

j

F a u +=−

= −∑ (2.62)

The values of the of bj are given in Eq.(24), and the value aj are as chosen to

minimize difference between ddαα

and 1 (see Table 2-1).

For the space-time a-ε scheme, the following formulas result:

( )

( )

( )

( )

1 122 2

1 12 2

1 122 2

1 12 2

1 12 21 12 2

1 12 21 12 2

1 1 (1 ) ( )2 4

1 (1 ) ( )4

1( ) 1 (2 1 ) ( )2 4

1 (2 1 ) ( )4

n nnj xj j

n n

xj j

n nnx j xj j

n n

xj j

xu u u

xu u

xu u u

xu u

ν ν

ν ν

ε ε ν

ε ε ν

− −

− −

− −

+ +

− −

− −

− −

+ +

⎧ ⎡ ∆= + + −⎪ ⎢

⎣⎪⎪ ⎤∆⎪ + − − − ⎥⎪⎪ ⎦⎨

⎡ ∆⎪ = − + − +⎢⎪⎣⎪

⎪ ⎤∆+ − + − −⎪ ⎥

⎪ ⎦⎩

(2.63)

Before evaluating the DRP and space-time a-ε method, we further address the

influence of the parameters: ε and ν for the space-time a-ε scheme, and ν for the DRP

scheme.

An effort is made in this study to construct a simple guidance appropriate for long

as well as short waves. Evidences based on the test problems, aided by the truncation

error analysis, indicate that in order to reduce numerical dispersion and to maintain

satisfactory resolution, for short wave ( e.g., b/∆x = 3), ν and ε are preferable to be close

to each other (see Figure 2-4 and Figure 2-5). For long and intermediate waves there are

virtually no need to introduce much numerical dissipation, hence ε can be reduced close

to zero. On the other hand, mismatched ε and ν can significantly reduce the accuracy of

the scheme. To demonstrate this fact, Figure 2-4a and Figure 2-4b show two solutions for

51

the intermediate wave, all with ν = 0.5, respectively for ν = 0.9. Changing ε from 0.5 to

0.99 causes significant numerical dissipation. On the other hand, ε = 0 can be a very

acceptable choice in for long wave. Figure 2-5 compiles long and intermediate wave

solution with different value of ν and ε. Taking into consideration the previous

observations, it is decided that ν = ε = 0.99 is a good choice, and is used in the results

presented for the space-time scheme. In more complicated computations involving

coupled system with nonlinear effects, such a choice may not be stable. However, for a

simple wave equation, this coordinated choice is beneficial.

For the present DRP scheme a value of ν less than 0.21 guarantees numerical

stability and negligible numerical damping. If we decrease the value of ν much further,

then, as indicated in Eq.(2.37), the numerical damping, as indicated by the leading

dissipation term uxxxx, may be less than adequate due to the small value of ν. For these

reasons we decided to use ν = 0.1 for longer waves and ν = 0.2 for short wave (b/∆x = 3).

To summarize, the DRP scheme is higher formal order than the space-time a-ε

scheme. The DRP scheme prefers smaller ν (or ∆t), indicating that it is more expensive to

compute than the space-time a-ε scheme.

To evaluate the solution accuracy, we define an error vector as:

[ ]1,...,T

NE E E=r

(2.64)

U(xi) is the exact solution at the point xi, and ui is the numerical solution at the

point xi.

The error norm is adopted as norm one

52

1

N

ii

EE

N==∑

(2.65)

where

( ) , 1i i iE U x u i N= − ≤ ≤ (2.66)

which will be used to help to measure the order of accuracy in actual computations.

x

u(x,

t)

160 180 200 220

-0.2

0

0.2

0.4

0.6

0.8

1numerical solution =0.10E-01analytical solution

ε

x

u(x,

t)

160 180 200 220

-0.2

0

0.2

0.4

0.6

0.8

1numerical solution =0.10E+00analytical solution

ε

x

u(x,

t)

160 180 200 220

-0.2

0

0.2

0.4

0.6

0.8

1numerical solution =0.50E+00analytical solution

ε

x

u(x,

t)

160 180 200 220

-0.2

0

0.2

0.4

0.6

0.8

1

numerical solution =0.70E+00analytical solution

ε

x

u(x,

t)

160 180 200 220

-0.2

0

0.2

0.4

0.6

0.8

1

numerical solution =0.90E+00analytical solution

ε

x

u(x,

t)

160 180 200 220

-0.2

0

0.2

0.4

0.6

0.8

1

numerical solution =0.99E+00analytical solution

ε

a)

x

u(x,

t)

20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

numerical solution =0.50E+00analytical solution

ε

x

u(x,

t)

20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

numerical solution =0.99E+00analytical solution

ε

b) Figure 2-4. Comparison between analytical and numerical solutions – Effect of ε on the

accuracy of space-time a-ε scheme: a) ν = 0.9, b/∆x = 3, t = 200; b) b/∆x = 6, t = 200, ν = 0.5

53

ε0.25 0.5 0.75 1

1.80E-03

2.20E-03

2.60E-03

ν=0.99error

ε0.25 0.5 0.75 1

2.5E-02

3.5E-02

4.5E-02

ν=0.5error

ε0.25 0.5 0.75 1

4.0E-02

6.0E-02

ν=0.1error

a)

ε0.25 0.5 0.75 1

2.0E-02

4.0E-02

6.0E-02

8.0E-02

ν=0.1error

ε0.25 0.5 0.75 1

1.5E-02

2.5E-02

3.5E-02

4.5E-02

5.5E-02 ν=0.5error

ε0.25 0.5 0.75 1

4.65E-04

4.75E-04

4.85E-04

4.95E-04

ν=0.99error

b)

ε0.25 0.5 0.75 1

2.0E-02

4.0E-02

6.0E-02

8.0E-02

1.0E-01

1.2E-01

ν=0.1error

ε0.25 0.5 0.75 1

5.0E-03

1.0E-02

1.5E-02

2.0E-02

2.5E-02ν=0.5error

ε0.25 0.5 0.75 1

4.06E-05

4.07E-05

4.08E-05

4.09E-05

ν=0.99error

e

erro

r

0.25 0.5 0.751.53E-03

1.55E-03

1.57E-03

e

erro

r

0.2 0.4

2.04E-032.06E-032.08E-032.10E-032.12E-03

c) Figure 2-5. The dependence of the error on ε for the space-time a-ε scheme at t = 200:

a) b/∆x = 3; b) b/∆x = 6; c) b/∆x = 20

54

In the following we will present the results based on three categories:

• long wave (b/∆x = 20)

• intermediate wave (b/∆x = 6)

• short wave (b/∆x = 3)

They are defined according to the ratio b/∆x, where b is a parameter that

characterizes the wavelength.

Figure 2-6 offer an overview of the two schemes for long, intermediate and short

wave computation at two different time instants: t/∆t = 200 and 2000. It is noted that

since ∆x is fixed in each plot, varying ν is the same as varying ∆t. First, we analyze errors

for DRP scheme applied to both intermediate and long waves. The following are some of

the main observations: (i) the characteristics are similar for short and long times; (ii) the

error increases with ν: (iii) a slower increase with respect to ν, for ν little.

For short wave, DRP scheme behaves differently as time progresses. As previously

discussed in the context of truncation error analysis, Eq.(30), a small value of ν can cause

insufficient numerical damping. Hence, it is advisable to use a somewhat larger value of

ν, but still within the stability bound of ν ≤ 0.21.

The space–time a-ε scheme has exhibited unusual error variations as a function of ν

(or ∆t): the error changes very little with ν, and then suddenly decreases rapidly for ν

close to 1 (see Figure 2-6 and Figure 2-7). This trend holds for all resolution. This type

of behavior is unusual at first glance, but can be explained from the viewpoint of

truncation error analysis. Specifically, with ν = ε, both uxxx and uxxxx terms are small as ν

approach 1, making the scheme sharply improves the apparent order of accuracy.

Next, we present the error as a function of spatial resolution (b/∆x).

55

ν0.1 0.2

1.3E-03

1.3E-03

1.4E-03

1.4E-03

1.5E-03

1.5E-03

1.6E-03

1.6E-03

DRP scheme

t=200b/ x=3.∆

error

1

2

0.011

ν 0.1 0.2

1.0E-04

2.0E-04

3.0E-04

4.0E-045.0E-04

DRP scheme

t=200b/ x=20.∆

error

1

0.9

0.41

ν0.1 0.2

1.0E-04

2.0E-04

3.0E-04

4.0E-045.0E-04

0.9

1

DRP scheme

t=200b/ x=6.∆

error

0.31

a1)

ν0.1 0.2

7.5E-03

7.6E-03

7.6E-03

7.6E-03

7.7E-03

DRP scheme

t=2000b/ x=3.∆

error

ν0.1 0.2

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-030.4

1

DRP scheme

t=2000b/ x=6.∆

error

0.11

ν 0.1 0.2

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

DRP scheme

t=2000b/ x=20.∆

error

1

0.4

0.11

a2)

ν

erro

r

10-2 10-1 100

5.0E-05

5.0E-04

9.5E-041.4E-031.9E-03

Space - time CE SE scheme

b/ x=20.t=200

∆

ν=ε

ν

erro

r

10-2 10-1 1005.0E-04

5.0E-03

9.5E-031.4E-021.9E-02

Space - time CE SE scheme

b/ x=6.t=200

∆

ν=ε

ν

erro

r

10-2 10-1 100

2.5E-03

1.3E-02

2.3E-02

3.3E-02

b/ x=3.t=200

∆

Space - time CE SE scheme

ν=ε

b1)

ν0.1 0.2

7.5E-03

7.6E-03

7.6E-03

7.6E-03

7.7E-03

DRP scheme

t=2000b/ x=3.∆

error

ν0.1 0.2

6.0E-04

7.0E-04

8.0E-04

9.0E-04

1.0E-030.4

1

DRP scheme

t=2000b/ x=6.∆

error

0.11

ν 0.1 0.2

5.0E-04

6.0E-04

7.0E-04

8.0E-04

9.0E-04

DRP scheme

t=2000b/ x=20.∆

error

1

0.4

0.11

b2)

Figure 2-6. The dependence of the error as function of ν for short (b/∆x = 3), intermediate (b/∆x = 6) and long (b/∆x = 20) waves for: a) DRP scheme; b) Space time a-ε scheme, at two different time instants t = 200 and t = 2000. In all cases ∆t = ν

56

x

u(x,

t)

180 190 200 210 220

0

0.2

0.4

0.6

0.8

1

numerical solution =0.99E+00analytical solution

ν=ε=0.99

ε

x

u(x,

t)

180 190 200 210 220

0

0.2

0.4

0.6

0.8

1numerical solution =0.90E+00analytical solution

ν=ε=0.9

ε

Figure 2-7. Effect of ν on the accuracy of space time a-ε scheme: b/∆x = 3, t = 200

Figure 2-8 depicts the dependence of error for b/∆x between 3 and 300. The graphs

show that the space-time a-ε method is second order accuracy in space. In contrast with

the DRP scheme does not exhibit consistent trends for varying spatial resolutions: (i) for

short and intermediate waves the order accuracy of this scheme is four; (ii) for long wave,

the error tends to become almost independent of space (the slope is close to 0.1).

Comparing the performance of these two schemes, we can deduce that the DRP scheme

gives a better solution for b/∆x less than 10 (for a short and intermediate wave). For long

wave, the space-time CE/SE scheme gives a better order of accuracy; nevertheless both

schemes are satisfactory in practical terms. Finally, we compare these two methods with

respect to their performance with sufficient accumulation of the time steps.

57

b/ x

erro

r

101 10210-5

10-4

10-3

10-2

∆

1

Space - Time schemea −ε

2

ν=ε=0.1t=200

b/ x

erro

r

101 10210-5

10-4

10-3

10-2

∆

DRP scheme

1

4

0.011

ν=0.02t=200

b/ x

erro

r

101 102

10-6

10-5

10-4

10-3

∆

2

1

Space - Time schemea −ε

ν=ε=0.99t=200

b/ x

erro

r

101 102

10-6

10-5

10-4

10-3

∆

DRP scheme

41

0.11

ν=0.1t=200

Figure 2-8. The behavior of the error in function of the wavelength: comparison between

DRP and space-time a-ε schemes

Short Wave: b/∆x = 3

Figure 2-9 summarizes various aspects of error accumulation in time, along with

selected solution profiles for both schemes. It is apparent that the space time a-ε scheme

introduces both dissipation and dispersion errors, while the DRP scheme shows mainly

dispersion errors with less dissipation.

In addition, the two schemes exhibit different levels and growth rates. For the

space-time a-ε scheme, at the beginning, the rate of accumulation is slower. Then, around

t ≅ (103), the dissipation becomes more substantial. Around t ≅(5⋅103), both dissipation

58

and dispersion characterize the solution, but the overall error growth rate decreases

slightly in comparison to the previous state.

Initially, the DRP scheme produces lower error level and grows almost linearly

with time. After t ≅ (103), the error growth rate slows down; the dispersion becomes

substantial, but the solution is not smeared. Around t ≅ (2⋅104), dispersion becomes

highly visible while dissipation is also present at a modest level.

Intermediate Wave: b/∆x = 6

As shown in Figure 2-10, for this case, the DRP scheme has lower error than the

space-time a-ε scheme. The space-time a-ε scheme exhibits similar growth trends in both

cases, intermediate and short wave. Again, both dispersion and dissipation errors are

noticeable.

The DRP scheme behaves differently between short and intermediate waves. The

error growth rate at the beginning is low, and then becomes higher. Throughout the entire

computation, only minor dispersion error appears.

Long Wave: b/∆x = 20

Figure 2-11 presents the solutions for long wave computations. Both schemes

perform well over the interval of time. But the space-time a-ε scheme exhibits noticeably

slower rate of error accumulation. Neither dispersion nor dissipation errors are

significant.

59

x

u(x,

t)

19980 20000 20020-0.2

0

0.2

0.4

0.6

0.8

1numerical solution - t=0.20E+05analytical solution t=0.20E+05

D

x

u(x,

t)

880 890 900 910 9200

0.2

0.4

0.6

0.8

1numerical solution - t=0.90E+03analytical solution t=0.90E+03B

x

u(x,

t)

5260 5280 5300 5320

0

0.2

0.4

0.6

0.8

1

numerical solution - t=0.53E+04analytical solution t=0.53E+04

C

x

u(x,

t)

80 90 100 110 1200

0.2

0.4

0.6

0.8

1

numerical solution - t=0.10E+03analytical solution t=0.10E+03A

t103 104

1.0E-02

1.5E-02

2.0E-02

2.5E-023.0E-02

b/ x=3∆ν=ε=0.99

Space - Time scheme

AB

C

D

1

1

errora -ε

a)

x

u(x,

t)

1475 1500 1525

0

0.2

0.4

0.6

0.8

1

numerical solution t=0.150E+04analytical solution t=0.150E+04B

x

u(x,

t)

80 100 120

0

0.2

0.4

0.6

0.8

1

numerical solution t=0.100E+03analytical solution t=0.100E+03A

x

u(x,

t)

19950 20000-0.2

0

0.2

0.4

0.6

0.8

1numerical solution t=0.200E+05analytical solution t=0.200E+05

C

t102 103 104

2.0E-03

1.0E-02

1.8E-022.6E-02

1

0.8

1

0.65

A

B

CDRP schemeb/ x=3.∆ν=0.2η=0.85

error

b) Figure 2-9. Accumulation of the error in time for short wave - b/∆x = 3; a) space-time

a-ε scheme; b)DRP scheme

60

x

u(x,

t)

20000-0.2

0

0.2

0.4

0.6

0.8

1numerical solution - t=0.20E+05analytical solution t=0.20E+05C

x

u(x,

t)

80 100 120

0

0.2

0.4

0.6

0.8

1numerical solution - t=0.10E+03analytical solution t=0.10E+03A

x

u(x,

t)

3575 3600 36250

0.2

0.4

0.6

0.8

1

numerical solution - t=0.36E+04analytical solution t=0.36E+04B

t103 1045.0E-03

1.0E-02

1.5E-02

2.0E-02

2.5E-02b/ x=6∆ν=ε=0.99

Space - Time scheme

A

B

C

1

1

errora - ε

a)

x

u(x,

t)

80 90 100 110 120

0

0.2

0.4

0.6

0.8

1

numerical solution t=0.100E+03analytical solution t=0.100E+03

A

x

u(x,

t)

19980 20000 20020-0.2

0

0.2

0.4

0.6

0.8

1

numerical solution t=0.200E+05analytical solution t=0.200E+05

B

t102 103 104

1.0E-03

3.0E-03

5.0E-03

b/ x=6.∆

DRP scheme

A

B

ν=0.1 1

1η=0.6

error

b)

Figure 2-10. Accumulation of the error in time for intermediate wave - b/∆x = 6; a) space-time a-ε scheme; b) DRP scheme

61

x

u(x,

t)

50 100 1500

0.2

0.4

0.6

0.8

1numerical solution - t=0.10E+0analytical solution t=0.10E+03A

x

u(x,

t)

19950 20000 200500

0.2

0.4

0.6

0.8

1numerical solution - t=0.20E+05analytical solution t=0.20E+05

B

t103 104

5.0E-03

5.2E-03

5.4E-03

5.6E-03

5.8E-03b/ x=20∆ν=ε=0.99

Space - Time scheme

A

Berror

1

0.12

a - ε

a)

x

u(x,

t)

50 75 100 125 150

0

0.2

0.4

0.6

0.8

1

numerical solution t=0.100E+03analytical solution t=0.100E+03

A

x

u(x,

t)

19950 19980 20010 20040-0.2

0

0.2

0.4

0.6

0.8

1

numerical solution t=0.200E+05analytical solution t=0.200E+05

B

t102 103 104

5.0E-04

1.5E-03

2.5E-03

3.5E-034.5E-03

b/ x=20.∆

DRP scheme

A

B

ν=0.1 10.9

error

η=0.6

b)

Figure 2-11. Accumulation of the error in time for long wave - b/∆x = 20, a) space-time a-ε scheme; b) DRP scheme

62

Summary and Conclusions

It should be noted that he DRP scheme is a multi-step method, which requires more

boundary conditions and initial data, while the space-time a-ε scheme is a one-step

method. Combined with the fact that the DRP scheme performs better with smaller ν, it

can be more expensive to compute than for space-time a-ε scheme. Tam and Webb

(1993) show by means of the Laplace transform that the DRP scheme can be constructed

to use the same number of initial data as the original PDE and the single time step

method. For the boundary treatment, the DRP scheme requires additional points outside

the computational domain. A combination of ghost points and backward difference

operators, based on similar optimization procedures, is employed.

The present study is restricted to the investigation of a simple 1-D linear wave

equation. Obviously, more issues will arise when multi-dimensional geometry,

nonlinearity of the physics, and coupling of the dependent variables need to be

considered. Nonuniform grid also creates varying CFL number even with a constant

convection speed. Nevertheless, it seems useful to consider, in a well-defined framework,

the merits of the two schemes in a well-defined context. In this sense, the present study

can be viewed to establish an upper bound of the performance characteristics of the DRP

and space-time CE/SE methods.

63

CHAPTER 3 FINITE VOLUME TREATMENT OF DISPERSION-RELATION-PRESERVING AND OPTIMIZED PREFACTORED COMPACT SCHEMES FOR WAVE PROPAGAION

In computational aero-acoustics (CAA) accurate prediction of generation of sound

is demanding due the requirement of preserving the shape and frequency of wave

propagation and generation. Furthermore, the numerical schemes need to handle multiple

scales, including long and short waves, and nonlinear governing laws arising from

sources such as turbulence, shocks, interaction between fluid flows and elastic structures,

and complex geometries. It is well recognized (Hardin and Hussaini, 1992; Tam and

Webb, 1993; Tam et al., 1993) that in order to conduct satisfactory CAA, numerical

schemes should induce minimal dispersion and dissipation errors. In general, higher-

order schemes would be more suitable for CAA than the lower-order schemes since,

overall, the former are less dissipative. That is why higher-order spatial discretization

schemes have gained considerable interest in computational acoustics (Hixon, 1997; Kim

et al., 1997; Lin an Chin, 1995). Table 3-1 summarizes several approaches proposed in

the literature.

For longer wavelengths, the formal order of accuracy is sufficient to indicate the

performance of a scheme. However, for shorter waves relative to the grid size, it is

known that the leading truncation error terms are not good indicators (Shyy, 1985; Shyy,

1997). To handle broad band waves, the idea of optimizing the scheme coefficients via

minimizing the truncation error associated with a particular range of wave numbers has

been used over the years by many researchers, e.g., Refs.(Hu et al., 1996; Stanescu and

64

Habashi, 1998; Nance et al., 1997; Wang and Sankar, 1999; Cheong and Lee, 2001;

Wang and Chen, 2001; Ashcroft and Zhang, 2003) . A successful approach is the

Dispersion-Relation-Preserving (DRP) finite difference scheme proposed by Tam and

coworkers (Tam and Webb, 1993; Tam et al., 1993). The basic idea in the DRP scheme is

to optimize coefficients to satisfactorily resolve short waves with respect to the

computational grid, namely, waves with wavelengths of 6-8∆x (defined as 6-8 points per

wave or PPW) or shorter. It maximizes the accuracy by matching the wave number and

frequency characteristics between the analytical and the numerical operators in the range

of resolvable scales. Recently, Ashcroft and Zhang (2003) have reported a strategy for

developing optimized prefactored compact (OPC) schemes, requiring smaller stencil

support than DRP. The prefactorization strategy splits the central implicit schemes into

forward and backward biased operators. Using Fourier analysis, they have shown that it

is possible to select the coefficients of the biased operators such that their dispersion

characteristics match those of the original central compact scheme. Hixon and Turkel

(1998) proved that the “prefactored scheme is equivalent to the initial compact scheme if:

i) the real components of forward and backward operators are equal to those at the

corresponding wavenumber of the original compact scheme; ii) the imaginary

components of the forward and backward operators are equal in magnitude and opposite

in sign”.

Both DRP and OPC schemes are originally designed based on the finite difference

approach. In order to satisfy the governing laws of the fluid physics, it can be

advantageous to adopt the finite volume approach (Udaykumar et al., 1999; Yang et al.,

1999; Udaykumar et al., 1999), which ensures that fluxes estimated from different sides

65

of the same surface are identical, i.e., no spurious source/sink is generated due to

numerical treatment. Such a requirement is particularly important when nonlinearity is

involved, as is typically the case in shock and turbulence aspects of the aeroacoustic

computations. Furthermore, a finite volume formulation can offer an easier framework to

handle the irregular geometry and moving boundaries. In this work, we extend the

concept embodied in the original, finite difference-based DRP scheme (which we call

DRP–fd) to a finite volume formulation (which we call DRP-fv). Similarly, for the OPC-

scheme, we extend the basic concepts of the original, finite difference-based OPC (OPC-

fd) scheme, to a finite volume formulation, called OPC-fv. Our overall goal is to develop

the finite volume version of DRP and OPC schemes into a cut-cell type of Cartesian-grid

computational technique that we have developed earlier for moving and complex

boundary computations (Yang et al., 1999; Udaykumar et al., 1999; Ye et al., 1999) to

treat aero-acoustic problems needed for engineering practices.

We present the finite volume formulation of both DRP and OPC schemes, and

assess both fd and fv versions of the DRP and OPC schemes, using well defined test

problems to facilitate systematic evaluations. Both linear and nonlinear wave equations

with different wavelengths and viscous effects are utilized for direct comparisons. In the

following, we first summarize the essence of the individual schemes, including

derivations, then present assessment of the test cases.

Numerical Schemes

We use the following one-dimensional wave equation to facilitate the development

and presentation of the concept and numerical procedures:

0u uct x

∂ ∂+ =

∂ ∂ (3.1)

66

The equation contains time and space derivative. In our work the space derivative

term is treated with DRP or OPC scheme and the time derivative by a Low –Dissipation

and Low-Dispersion Runge-Kutta (LDDRK) scheme (Hu et al., 1996).

The first scheme used for space derivative is DPR scheme. We present the finite

volume version of DRP, and the boundary treatment of the DRP.

The OPC scheme is the second method considered for the space derivative. The

finite difference procedure of the OPC scheme is offered. This presentation is follow by

the extension of this approach to a finite volume and the specific boundary treatment of

OPC .

LDDRK (Hu et al., 1996) scheme is used to approximate time derivative. The

principal characteristics will be presented

DRP Scheme

Finite volume-based DRP scheme (DRP-fv)

To incorporate the DRP-fd concept into a finite volume framework, let us consider

a one-dimensional linear wave equation:

0ct xφ φ∂ ∂

+ =∂ ∂

(3.2)

To derive the discredized equation, we employ the grid point cluster shown in

Figure 3-1. We focus on the grid point i, which has the grid points i -1, and i+1 as its

neighbors. The dashed lines define the control volume, and letters e and w denote east

and west faces, respectively, of the control volume. For the one-dimensional problem

under consideration, we assume a unit thickness in the y and z directions; thus, we obtain

( ) ( )( ) 0w

e we

dx c A Atφ φ φ∂

+ − =∂∫

(3.3)

67

where (Aφ)e and (Aφ)w are the flux across the east and west face, respectively.

Figure 3-1.Grid points cluster for one-dimensional problem

Hence, the discretized wave equation (3.2) can be written as

( ) ( )( ) 0e w

x c A Atφ φ φ∂

∆ + − =∂

(3.4)

where φ is the averaged value of φ over a control volume.

Taking into account Eq. (3.4) we describe the general form of the approximation of

xφ∂

∂ in 1-D using the control volume concept:

` ( ) ( )( )1e w

A Ax xφ φ φ∂

→ −∂ ∆

(3.5)

The general form of the DRP scheme is:

3

3

1 k

k l kkl

ax xφ φ

=

+=−

∂⎛ ⎞ ≅⎜ ⎟∂ ∆⎝ ⎠∑ (3.6)

where ∆x is the space grid, and coefficients aj are constant.

The DRP scheme has a general form similar to the central difference

approximation. Hence, one can adopt a central difference scheme to express φe in the

neighborhood:

( ) 1 2 2 1 3 4 1 5 2 6 3i i i i i ieφ β φ β φ β φ β φ β φ β φ− − + + += + + + + + (3.7)

( ) 1 3 2 2 3 1 4 5 1 6 2i i i i i iwφ β φ β φ β φ β φ β φ β φ− − − + += + + + + + (3.8)

i e i+1(W) i+2 i+3wi-1(E) i-2

∆x

(δx)w (δx)e

68

Taking into consideration Eqs. (3.5), (3.6), (3.7) and, (3.8) we obtain the values of

the βi, i = 1,…,6 by imposing that the value of φ at the same locations has the same

values as that of the DRP-fd.

3

3

k

k l k e wk

a φ φ φ=

+=−

= −∑ (3.9)

Hence, the values of coefficients β’s are

1 6 3

2 5 2 3

3 4 1 2 3

aa aa a a

β ββ ββ β

= =⎧⎪ = = +⎨⎪ = = + +⎩

(3.10)

To illustrate the above-described concept, we consider the following equation:

1 2 0u u uc ct x y

∂ ∂ ∂+ + =

∂ ∂ ∂ (3.11)

If we integrate Eq. (3.11) on the surface we have (see Figure 3-2):

,1

i jabcd

u Ft S

∂=

∂ (3.12)

where the resulting DRP-fv scheme is

, 1 , , , ,

2 , , , ,

( )

( )

s e n wi j i j s i j e i j n i j w

s e n wi j s i j e i j n i j w

F c u y u y u y u y

c u x u x u x u x

⎡= − ∆ + ∆ + ∆ + ∆ +⎣⎤∆ + ∆ + ∆ + ∆ ⎦

(3.13)

, 1 2, 2 1, 3 , 4 1, 5 2, 6 3,ei j i j i j i j i j i j i ju u u u u u uβ β β β β β− − + + += + + + + + (3.14)

, 1 3 2 2, 3 1, 4 , 5 1, 6 2,wi j i i j i j i j i j i ju u u u u u uβ β β β β β− − − + += + + + + + (3.15)

, 1 , 2 2 , 1 3 , 4 , 1 5 , 2 6 , 3ni j i j i j i j i j i j i ju u u u u u uβ β β β β β− − + + += + + + + + (3.16)

, 1 , 3 2 , 2 3 , 1 4 , 5 , 1 6 , 2si j i j i j i j i j i j i ju u u u u u uβ β β β β β− − − + += + + + + + (3.17)

69

( ) ( )

( )

1 2 1

2

abcd s s e e n n w wV S

s s e e n n w w

u udv c udy c udx S c u y u y u y u yt t

c u x u x u x u x

∂ ∂+ − = + ∆ + ∆ + ∆ + ∆ +

∂ ∂

+ − ∆ − ∆ − ∆ − ∆ +

∫ ∫ (3.18)

d c

e

s

w

nP

ba S

W E

Nd c

e

s

w

nP

ba S

W E

N

Figure 3-2. Grid notation for two-dimensional problem, where (i) P denotes the center of

a cell, (ii) E, W, N, and S denote, respectively, the nodes corresponding to the east, west, north and south neighbors, (iii) e, w, n and s denote, respectively, the center of the east, west, north and south face of the cell, and (iv) a, b, c, and d denote, respectively, the corners of the cell

Boundary treatment of the DRP scheme

The current version of the DRP scheme requires seven grid points in space.

Consequently, it is necessary to impose some supplementary condition for boundary

treatments. In this regard, Tam and Webb (1993) devise ghost points. The minimum

number of ghost points is equal to the number of boundary conditions. For example, for

an inviscid flow the condition of no flux through the wall requires a minimum of one

ghost value per boundary point on the wall. It is desirable to use a minimum number of

ghost points to maintain simplicity in coding and structuring data.

In this work we use only backward difference for grid points near the

computational boundary and a ghost point is used only for wall boundary condition.

70

OPC Scheme

Finite-difference-based optimized prefactored compact (OPC-fd) scheme

To derive the factorized compact scheme Ashcroft and Zhang (2003) define

forward and backward operators FiD and B

iD , such that

( )12

B Fi i

i

u D Dx

∂⎛ ⎞ = +⎜ ⎟∂⎝ ⎠ (3.19)

The generic stencil for the forward and backward derivative operators are then

defined as:

1 2 1 1 21F F F F F F F F F

i i i i i i iD D a u b u c u d u e ux

η β+ + + − −⎡ ⎤⋅ + ⋅ = ⋅ + + ⋅ + ⋅ +⎣ ⎦∆ (3.20)

1 2 1 1 21B B B B B B B B B

i i i i i i iD D a u b u c u d u e ux

β γ − + + − −⎡ ⎤⋅ + ⋅ = ⋅ + + ⋅ + ⋅ +⎣ ⎦∆ (3.21)

The coefficients of the scheme are chosen such that: i) the wavenumber of the

scheme is close to the important wavenumber of the exact solution; ii) the imaginary

components of the forward and backward stencil are equal in magnitude and opposite in

sign, and the real components are equal and identical to original compact scheme; iii) the

scheme preserves a certain order of accuracy. The authors (Ashcroft and Zhang, 2003)

define the integrated error (weighted deviation) as:

( ) ( )0

rE x x W x d x

πα α α α= ∆ − ∆ ∆ ∆∫ (3.22)

where W(α∆x) is a weighting function, and r is a factor to determine the optimization

range (0< r < 1). The integrated error, defined in Eq.(3.22), is different from the one of

Tam and Web (1993) in that it contains the weighting function. The coefficients are

obtained by imposing that, within a given asymptotic order, the error is minimal. In space

71

discretization, one sacrifices formal order of accuracy in favor of wide-band

performance, especially for the short wave components.

Finite volume-based OPC scheme (OPC-fv)

Taking into account Eq.(3.3) that describes the approximation of the first derivative

in the finite volume formulation, equations that describe the OPC scheme, Eq.(3.20) and

Eq.(3.21), and the idea that the general form of approximation of the function for points

at the center of the cell face, namely, e and w assumes similar forms :

0.5( )

0.5( )

Fe Bee

Fw Bww

u u u

u u u

⎧ = +⎪⎨

= +⎪⎩ (3.23)

where uFe, uBe, uFw and uBw are determined from:

1 1Fe Fei i i iu u bu duη β+ ++ = − (3.24)

1 1Fw Fwi i i iu u bu duη β+ −+ = − (3.25)

1 1Be Bei i i iu u bu duβ η − ++ = − (3.26)

1 1Bw Bwi i i iu u bu duβ η − −+ = − (3.27)

where the coefficients are the same as those in the OPC-fd scheme: η = ηF= γB,

β = βF= βB, b = bF= -dB, d= dF. = -bB. These relationships among forward and backward

operators are obtained by Ashcroft and Zhang (2003).

The boundary treatment of the OPC scheme

Boundary Formulation of the OPC scheme employs a biased explicit stencil.

Ashcroft and Zhang (2003) design OPC-fd scheme with the follow boundary stencil:

4

11 3

1 1,N

B Bj j N j j

j j N

D s u D e ux x= = −

= =∆ ∆∑ ∑ (3.28)

and

72

4

1 1 11 3

1 1,N

F FN j j N N j j

j j N

D e u D s ux x+ − + −

= = −

= − = −∆ ∆∑ ∑ (3.29)

where the coefficients sj and ej are determined by matching the Taylor series of the

forward and backward compact interior stencils to third-order accuracy.

The boundary treatment in case of OPC-fv approach is similar to that of OPC–fd,

but the boundary stencil is computed on the face:

Di = (uA)ie –(uA)i

w (3.30)

3 3

11 1

3 3

1 1 11 1

w wi i N i N i

i i

e ei i N i N i

i i

u a u u ru

u a u u ru

−= =

+ − += =

⎧ ⎧= =⎪ ⎪⎪ ⎪

⎨ ⎨⎪ ⎪= =⎪ ⎪⎩ ⎩

∑ ∑

∑ ∑

(3.31)

where the value of the coefficients are:

11 1

2 1 2 2 1

3 1 2 3 3 21

FBN

B FN N

B FN N N

a ea s

a s s a e e

a s s s a e e e−

−−

⎧⎧ == −⎪⎪

= − − = +⎨ ⎨⎪ ⎪= − − − = + −⎩ ⎩

(3.32)

1 1 1

2 2 1 21

3 1 2 33 21

B FN

B FN N

FBN N N

r e r s

r e e r s s

r s s sr e e e−

−−

⎧ ⎧= = −⎪ ⎪

= + = − −⎨ ⎨⎪ ⎪ = − − −= + − ⎩⎩

(3.33)

Time Discretization – The Low Dispersion and Dissipation Runge-Kutta (LDDRK) Method

Hu et al.(1996) consider time integration using the Runge-Kutta algorithm of the

differential equation

( )u F ut

∂=

∂ (3.34)

where the operator F is a function of u. An explicit p-stage algorithm advances the

solution of Equation (3.34) from the nth to the (n + 1)th iteration as

73

( )

( )( )

( )( )( )

( )

0

0(1)

1( )

( )

1

...

1,...,...

n

ii

i n ii

pn

u u

K tF u

K tF u

u u b K i p

u u

−

+

=

= ∆

= ∆

= + =

=

(3.35)

where

• bp = 1,

• u(p), where p indicates the stage in algorithm advances

• un+1, where n indicates the number of iterations for time dependent computation

The vale of the un+1 can be written on short like

1

1 1

j

j nppn n j

l jj l p j

uu u b tt

γ

+

= = − +

=

∂= + ∆

∂∑ ∏14243

(3.36)

The resulting algorithm is obtained by optimizing the dispersion and dissipation

properties. Assuming F(u) is linear and applying temporal Fourier transform to (3.36),

the amplification factor is given by

( )1

*

1

1n p j

jnj

ur i tu

γ ω+

=

⎛ ⎞= = + − ∆⎜ ⎟

⎝ ⎠∑%

% (3.37)

The exact amplification factor is

*i t i

er e eω σ− ∆ −= = (3.38)

The numerical amplification factor r in (3.37) is viewed as an approximation of the

exact factor. The order of the optimized Runge-Kutta scheme is indicated by the leading

74

coefficient in (3.37) that matches the Taylor series expansion of e-iσ. For instance, the

third order algorithm is obtained by setting γj = 1/j! for j = 1, 2 and 3.

To compare the numerical and exact solutions we take into consideration the ratio:

i

e

r r er

δ−= (3.39)

where |r| represents the dissipation rate (obviously, the correct value should be 1), and δ

represents the phase error (or dispersive error) where the correct value should be zero.

Hu et al (1996) obtained coefficients of the low dispersion and dissipation Runge-

Kutta (LDDRK) scheme by imposing that: i) the scheme has certain order of accuracy,

ii) the error of the amplification factor of the scheme is minimized, which means that

both dispersion and dissipation errors are minimized. In other words the following

integral is minimized:

( )2

01

1 minp

j ij

j

c i e dσσ σΓ −

=

+ − − =∑∫ (3.40)

and iii) the amplification factor of the scheme is less than 0ne within the given stability

limit

t

|r|-

1

0 1 2 3

-0.04

-0.02

0

0.02

0.04

LR

LLDDRK = 1.64RLDDDR = 2.52

∆

LDDRK

ω∗

ω ∆

δ

0 1 2 3

-0.04

-0.02

0

0.02

0.04

* t

LDDRK

LLDDRK = 1.85

L

a) b) Figure 3-3. Four-six–stage optimized Runge-Kutta of order four scheme: a) dissipation

error; b) phase error

75

In this work we use a two-step alternating scheme: in odd steps we use four stages

and in the even steps we use six stages. The scheme is a fourth-order accurate scheme in

time for a linear problem and second-order accurate for a nonlinear problem. The

advantage of the alternating schemes is that, when two steps are combined, the dispersion

and the dissipation errors can be reduced and higher order of accuracy can be maintained.

The specific procedure is given bellow.

• four-stage

( ) ( )( ) ( )

( ) ( )

( ) ( )

( )

1

2 1

3 2

4 3

41

141312

n

n

n

n

n n

K tF u

K tF u K

K tF u K

K tF u K

u u K+

= ∆

⎛ ⎞= ∆ +⎜ ⎟⎝ ⎠⎛ ⎞= ∆ +⎜ ⎟⎝ ⎠⎛ ⎞= ∆ +⎜ ⎟⎝ ⎠

= +

(3.41)

• six-stage

( )( )( )( )( )

( )

( )

( )

( )

(1)

1(2)

2(3)

3(4)

4(5)

5(6)

61

0.17667

0.38904

141312

n

n

n

n

n

n

n n

K tF u

K tF u K

K tF u K

K tF u K

K tF u K

K tF u K

u u K+

= ∆

= ∆ +

= ∆ +

⎛ ⎞= ∆ +⎜ ⎟⎝ ⎠⎛ ⎞= ∆ +⎜ ⎟⎝ ⎠⎛ ⎞= ∆ +⎜ ⎟⎝ ⎠

= +

(3.42)

76

In the follow we will give an implementation example of LDDRK scheme when we

use OPC and DRP scheme for space discretization. Base on Eq.(3.1), the value of F in

point l is defined as following:

• DRP – fd:

3

3l i l i

i

cF a ux +

=−

= −∆ ∑ (3.43)

• DRP-fv:

Fl = -c( ul e – ul w )/∆x (3.44) where

1 2 2 1 3 4 1 5 2 6 3el l l l l l lu u u u u u uβ β β β β β− − + + += + + + + + (3.45)

1 3 2 2 3 1 4 5 1 6 2wl l l l l l lu u u u u u uβ β β β β β− − − + += + + + + + (3.46)

In the linear case, the fv and fd schemes are equivalent.

• OPC-fd

( )2B F

l l lcF D D= − + (3.47)

where DlB and Dl

F are obtained from the following system of equations:

( ) ( )1 1 11 , 1,...,F F

i i i i i iD D b u u d u u i Nx

η β+ + −+ = − + − =⎡ ⎤⎣ ⎦∆ (3.48)

( ) ( )1 1 11 , 1,...,B B

i i i i i iD D d u u b u u i Nx

β η − + −+ = − + − =⎡ ⎤⎣ ⎦∆ (3.49)

where N represent the number of grid points in space.

• OPC-fv Fl = - c( ul

e – ul w )/∆x (3.50)

where ul

e = 0.5(ul Be + ul

Fe), and ulw = 0.5(ul

Bw + ulFw) (3.51)

The value of ulBe, ul

Fe, ulBw, ul

Fw are obtained by solving the follow system of equations 1 1 1,...,Fe Fe

i i i iu u bu du i Nη β+ ++ = − = (3.52)

1 1 1,...,Fw Fwi i i iu u bu du i Nη β+ −+ = − = (3.53)

1 1 1,...,Be Bei i i iu u bu du i Nβ η − ++ = − = (3.54)

1 1 1,...,Bw Bwi i i iu u bu du i Nβ η − −+ = − = (3.55)

77

Analytical Assessment of DRP and OPC Schemes

Operation Count

We will compare the cost between the alternative approaches only for the

approximation of the first derivative, because we employ the same time stepping scheme

for both scheme.

The efficient form of general formula for the discretization in space of the DRP–fd

scheme is

( ) ( ) ( )3 3 3 2 2 2 1 1 11

i i i i i i iF a x x a x x a x xx + − + − + −= − + − + −⎡ ⎤⎣ ⎦∆

(3.56)

This scheme requires a total of three multiplications and five additions to evaluate

the first derivatives in a certain point. In case of DRP-fv the most efficient form of the

computations scheme is

( ) ( ) ( )1 3 2 2 2 1 3 11

e i i i i i iu u u u u u ux

β β β+ − + − += + + + + +⎡ ⎤⎣ ⎦∆ (3.57)

( ) ( ) ( )1 3 2 2 2 1 3 11

w i i i i i iu u u u u u ux

β β β− + − + −= + + + + +⎡ ⎤⎣ ⎦∆ (3.58)

DRP–fv requires a greater number of operations than DRP-fd: six multiplications

and eleven additions to compute the first derivatives at a given point.

To see the computational cost of the OPC-fd scheme we adopt the most efficient

form that is

( ) ( )1 1 1

1 12 2 2

F Fi i i i i iD b u u d u u D

xη

β β+ − +⎡ ⎤= − + − −⎣ ⎦∆ (3.59)

( ) ( )1 1 1

1 12 2 2

B Bi i i i i iD b u u d u u D

xη

β β− + −⎡ ⎤= − + − −⎣ ⎦∆

(3.60)

78

where the relation between the coefficients of the forward and backward stencils have

been substituted to highlight the equivalent terms in the two stencils. The operation count

is then four multiplications and five additions per point (Ashcroft and Zhang, 2003).

OPC-fv can be written in the form:

( )1 12 2

F Fe Fwi i iD u u

x= −

∆ (3.61)

( )1 12 2

B Be Bwi i iD u u

x= −

∆ (3.62)

where

1 1

1 1

1

1

Fe Fei i i i

F

Fw Fwi i i i

u bu du u

u bu du u

ηβ

ηβ

+ +

− +

⎧ ⎡ ⎤= − −⎪ ⎣ ⎦⎪⎨⎪ ⎡ ⎤= − −⎣ ⎦⎪⎩

(3.63)

1 1

1 1

1

1

Be Bei i i i

Bw Bei i i i

u bu du u

u bu du u

ηβ

ηβ

+ −

− −

⎧ ⎡ ⎤= − −⎣ ⎦⎪⎪⎨⎪ ⎡ ⎤= − −⎣ ⎦⎪⎩

(3.64)

In this case the operation count is eleven additions and six multiplications per point.

So we can see also, in Table 3-2 the finite volume approach is computationally

more expensive.

Dispersion Characteristics

The characteristics of the OPC and DRP schemes, in the finite difference form over

the interval 0 to π, are shown in

Figure 3-4. One can see that the difference between the effective wave number of

the scheme and the real wave is maintained to be within 2% if α∆x < 1.30 for the DRP

scheme, and α∆x < 1.84 for the OPC scheme. The dispersive characteristics of these

79

schemes can be more clearly seen in Figure 3-5, which shows the phase speed error,

1d xabsd x

αα

∆⎛ ⎞−⎜ ⎟∆⎝ ⎠, as a function of wave number on a log-arithmetic scale. We see that the

DRP scheme has a somewhat larger error than the OPC scheme until around 3π/4. The

error is maintained to be within 2% for α∆x less than 0.85 for the DRP scheme, and less

than 1.53 for the OPC scheme. Overall, the OPC scheme yields slightly less dispersion

error than the DRP scheme.

The dispersive characteristics of LDDRK are obtained by studying the value of |r|

and δ, i.e., dissipation rate and dispersion error (see Eq.(3.39)), respectively. In Figure 3-

3 we can see conditions of stability: |r| < 1 for ω*∆t ≤ 2.52 .To obtain an accurate solution

the dispersive characteristics (|r| and δ) should be close to the exact solution (|r| close to

one and δ close to zero). Hu et al. (1996) considered time accurate criterion ||r| -1| ≤ 0.001

(i.e ω*∆t ≤ 1.64), and δ ≤ 0.001 (i.e., ω*∆t ≤ 1.85). These two conditions are satisfied if

ω*∆t ≤ 1.64.

Stability

The Fourier-Laplace transformation of the wave equation (Eq.(3.1)) is

% % %12

initiali u ci u uω απ

∗− = − + (3.65)

where α, ω* characterize the PDE . For the long wave we can approximate wavenumber

of the scheme with wavenumber of the PDE

α ≅ α (3.66)

which leads to

% % %*initialu c u kuω α= + (3.67)

80

Hence

* c kkω α= + (3.68)

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

IdealDRPOPC

Figure 3-4. Dispersive characteristics of the schemes:. xα∆ versus α∆x

0 0.5 1 1.5 2 2.5 3 3.5

10-4

10-3

10-2

10-1

100

101

α∆x

DRP OPC

Figure 3-5. Phase speed error on a logarithmic scale

The condition of the numerical stability is that amplification factor for time

discretization is less than 1, and hence ω*∆t ≤ 2.52 (see Figure 3-3a). It is also noted from

Figure 3-4 that

1.8 for DRP scheme2.1 for OPC scheme

xx

αα

∆ ≤∆ ≤

(3.69)

Phas

e sp

eed

erro

r

81

hold true. By introducing Eq. (3.69) into (3.68) and upon multiplying by ∆t it is found

that

[ ]

[ ]

1.8 M 1 for DRP scheme

2.1 M 1 for OPC scheme

ct kk tx

ct kk tx

ω

ω

∗

∗

∆ ≤ ⋅ + ⋅∆∆

∆ ≤ ⋅ + ⋅∆∆

(3.70)

where M is mach number. From Figure 3-3a it is clear that the condition of stability is

satisfied if |ω*∆t| is less than 2.52. Therefore to ensure numerical stability it is sufficient

by Eq. (3.70) to restrict ∆t to be less than ∆tmax, where ∆tmax is given by

[ ]

[ ]

max

max

2.52 for DRP scheme1.8 M 1

2.52 for OPC scheme2.1 M 1

xtkk c

xtkk c

∆∆ =

⋅ +

∆∆ =

⋅ +

(3.71)

Therefore, for ∆t < ∆tmax the schemes are numerically stable. Consequently, the

schemes yield the following criteria for numerical stability:

1.4 for DRP scheme1.2 for OPC scheme

CFLCFL

<<

(3.72)

Although it is clear that CFL≤1.4 is the stability condition for DRP scheme, this

limit does not assure accuracy of the solution. In the previous analysis we have

established that the solution is time accurate for 4-6 LDRRK if ||r|-1| ≤ 0.001 and | δ | ≤

0.001. But this limit is not fixed, but depends on the scheme that is used for space

discretization. For example, in the case of the DRP scheme, the solution is considered

time accurate as long as ||r|-1| ≤ 0.02 and | δ | ≤ 0.02, or ω*∆t ≤2.0. Hence, in this case the

condition of being both accurate and stable is CFL ≤ 1.1

82

The OPC scheme is less sensitive to the dispersive characteristics of the LDDRK

scheme; hence CFL<1.2 is a condition of the stability and accuracy for the OPC scheme.

This limit is in concordance with the stability analysis of Ashcroft and Zhang (2003).

Computational Assessment of the DRP and OPC Schemes

To investigate the behavior of the schemes, we will use four test problems. First,

we consider a one-dimensional wave equation with constant speed. The purpose of this

test is to check the accuracy, stability, dissipation and dispersion of the scheme. The

second test problem is one-dimensional nonlinear wave equation with no viscous

dissipation. The purpose of this test case is to i) check the influence of singularities on the

performance of the scheme, and ii) analyze dispersion properties when waves are

coupled. In the third test problem, we consider the one-dimensional viscous Burgers

equation, which contains unsteady, nonlinear convection and viscous terms. In this case

we pay attention to the influence of the viscosity on the solution accuracy. The last test

problem is a 2D acoustic scattering problem from the second CAA Workshop (Tam and

Hardin, 1997). This problem tests the curved wall boundary and the capacity of the

scheme to reproduce different wavelengths.

Test problem 1: One-Dimensional Linear Wave Equation

To assess the behavior of the DRP and OPC schemes the following simple test

problem is studied first.

0.u uct x

∂ ∂+ =

∂ ∂ (3.73)

2

0exp ln 2 x xur

⎡ ⎤−⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

; at t = 0 (3.74)

83

which is a Gaussian profile. This is one of the test problems offered in the second CAA

Workshop (Tam and Hardin 1997)]

The exact solution is

2

0exp ln 2 x x ctUr

⎡ ⎤− −⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(3.75)

In this study we evaluate the performance of the schemes in short, intermediate, and long

waves relative to the grid spacing, which is assured by the value r/∆x.

For time discretization, we previously presented the detailed formulas for the 4-6

LDDRK, see equations (3.41), and (3.42).

Tam (Tam and Webb,1993; Tam et al., 1993) show that xα∆ is related to α∆x, and

in function of α∆x they divided the wave spectrum into two categories; i) the long waves

(waves for which xα∆ , in this case α∆x is less than α∆xc, ii) the short waves (waves for

which α is not close to α ). This difference between long and short waves is totally

dependent upon the grid space. Hence, by inspecting the number of grid points on the

wavelength, we can decide that we have a certain category of wave.

In the following, we will present the results based on three categories

• long wave (r/∆x = 20)

• intermediate wave (r/∆x = 6)

• short wave (r/∆x = 3)

The categories are defined according to the ratio r/∆x, where r is a parameter that

characterizes the wavelength of this problem. This test problem is linear; hence we do not

expect differences between finite difference and finite volume approach.

84

In regard to the time step selection, the CFL number (ν) limit is similar for all

schemes. We can see from Figure 3-6 that the critical CFL number of both schemes is

close to 1.1. From the study of the error in time for linear equations with constant

convection speed it is clear that the DRP-fv and OPC-fv schemes have essentially the

same behavior as the corresponding finite difference approach; hence, we only present

comparison for DRP–fv and OPC-fv schemes.

The error decreases when the grid size in space decreases until a critical value is

reached. For all schemes the errors have slopes consistent with the formal order of

accuracy in space. This conclusion is confirmed in Figure 3-7, where the CFL number is

maintained at 0.5. For the long time scale solution, the accumulation of error for both

DRP and OPC schemes is very close (as seen in Figure 3-7b). Here, we consider:

i) different grid space, so both schemes have almost the same initial error, and ii) the

same CFL number (0.5). This behavior is expected because both schemes present the

same discretization in time.

t

ER

RO

R

0.1 0.20.30.410-4

10-3

10-2

∆

DRP - fdDRP - fvOPC - fdOPC - fv

t

ER

RO

R

0.05 0.1

10-5

10-4

10-3

∆

DRP - fdDRP - fvOPC - fdOPC - fv

a) r/∆x = 3 (short wave); b) r/∆x = 10 – long wave

Figure 3-6. Errors with respect to the time step size under a fixed space ∆x, at t = 50 - linear wave equation

85

x

ER

RO

R

10-2 10-1 100

10-7

10-6

10-5

10-4

10-3

10-2

∆

DRP-fvDRP-fdOPC-fvOPC-fd

4

1

t

ER

RO

R

1000 2000

0.001

0.002

0.0030.004

1

1

DRPOPC

a) b)

Figure 3-7. Errors under a fixed CFL = 0.5, at t = 50 - linear wave equation: a) error with respect to the space size; b) accumulation of the error in time

Test problem 2: One-Dimensional Nonlinear Wave Equation

The finite volume and finite difference schemes are equivalent for a linear

equation. The difference between them appears for the nonlinear convective equation. To

observe the merits and similarities of DRP, and OPC schemes, we restrict ourselves to the

1-D case. In this test, a nonlinear wave equation with a different speed is solved

0u uut x

∂ ∂+ =

∂ ∂ (3.76)

This equation is solved in the conservative form

( )2

0.5 0uu

t x

∂∂+ =

∂ ∂ (3.77)

To better understand the effect of high gradients and discontinuities, we chose the

following initial conditions

0 0

( ,0)1 0

xu x

x≤⎧

= ⎨ >⎩ (3.78)

The solution for this problem can be written

86

0 0

( , ) 0

1

xxu x t x tt

x t

≤⎧⎪⎪= < <⎨⎪

≥⎪⎩

(3.79)

In this case, for both DRP and OPC schemes, the finite difference version behaves

differently from the finite volume version. In the Eqs.(3.41) and (3.42) the function F

takes the form:

• DRP-fd

( )23

30.5i k i k

kF a u +

=−

= − ∑ (3.80)

• DRP-fv

( ) ( )( )2 20.5 e w

i i iF u u= − − (3.81)

where ue and uw are as defined before

• OPC-fd ( )0.25 B F

i i iF D D= − +

(3.82) where Di

B and DiF is backward and forward derivative of u2 in place of u

• OPC-fv

( ) ( )( )220.25

i

e wi iF u u= − − (3.83)

where ue and uw are defined by (3.52) - (3.55)

The similarities and differences for all three categories (short waves [∆x/U = 1.0],

intermediate waves [∆x/U = 0.25], and long waves [∆x/U = 0.06]) are first presented. It

should be noted again that the short, intermediate and long waves are defined based on

the numerical resolution. Here, U is defined as the jump (umax - umin); in our case U = 1,

hence in the following we discuss only the effect of the grid space step (∆x).

87

∆

ER

RO

R

10-2 10-1 100

10-3

10-2

DRP - fvDRP - fdOPC - fvOPC - fd

x

21

Figure 3-8. Errors with respect to the space step size under a fixed CFL = 0.5, at t = 5;

nonlinear wave equation

The evolution of the error as a function of grid spacing (∆x) is similar for both DRP

and OPC schemes; the difference between the finite volume and finite difference versions

are far greater, as shown in Figure 3-8. In the case of finite volume, error decreases with

decreasing grid space (see Figure 3-10 and Figure 3-12). For finite difference, a totally

different behavior is seen. The error not only does not decrease when grid spacing

decreases, but in fact increases, as seen in Figure 3-9 and Figure 3-11).

X

U

-4 -2 0 2 4 6 8 10

-0.2

0

0.2

0.4

0.6

0.8

1

DRP - fdExact

X = 1.∆

X

U

-4 -2 0 2 4 6 8 10

-0.5

-0.25

0

0.25

0.5

0.75

1

DRP - fdExact

X = 0.25∆

X

U

-4 -2 0 2 4 6 8 10-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

DRP - fdExact

X = 0.06∆

a)∆x = 1 b) ∆x = 0.25 c) ∆x = 0.06 Figure 3-9. DRP–fd solution - nonlinear wave equation; t = 5; CFL = 0.5

88

X

U

-4 -2 0 2 4 6 8 100

0.25

0.5

0.75

1

DRP - fvExact

X = 1.∆

X

U

-4 -2 0 2 4 6 8 100

0.25

0.5

0.75

1

DRP - fvExact

X = 0.25∆

X

U

-4 -2 0 2 4 6 8 100

0.25

0.5

0.75

1

DRP - fvExact

X = 0.06∆

a) x = 1 b) -∆x = 0.25 c) ∆x = 0.06 Figure 3-10. DRP–fv solution - nonlinear wave equation; t = 3; CFL = 0.5

For short waves, all solutions show substantial errors, but the finite difference

schemes perform noticeably worse. In the case of intermediate or long waves, the finite

volume schemes exhibit satisfactory or better performance than the finite difference

schemes.

X

U

-4 -2 0 2 4 6 8 10-0.25

0

0.25

0.5

0.75

1

OPC - fdExact

X = 1.∆

X

U

-4 -2 0 2 4 6 8 10-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPC - fdExact

X = 0.25∆

X

U

-4 -2 0 2 4 6 8 10-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

OPC - fdExact

X = 0.06∆

a ) ∆x = 1 b) ∆x = 0.25 c) ∆x = 0.06 Figure 3-11. OPC-fd solution - nonlinear wave equation; t = 5; CFL = 0.5

X

U

-4 -2 0 2 4 6 8 100

0.25

0.5

0.75

1

OPC - fvExact

X = 1.∆

X

U

-4 -2 0 2 4 6 8 100

0.25

0.5

0.75

1

OPC - fvExact

X = 0.25∆

X

U

-4 -2 0 2 4 6 8 100

0.25

0.5

0.75

1

OPC - fvExact

X = 0.06∆

a) ∆x = 1 c) ∆x = 0.25 d) ∆x = 0.06 Figure 3-12. OPC-fv solution - nonlinear wave equation; t = 5; CFL = 0.5

89

Test problem 3: One-Dimensional Nonlinear Burgers Equation

In this test the solution for the one-dimensional nonlinear Burgers equation is

evaluated.

2

2

u u uut x x

µ∂ ∂ ∂+ =

∂ ∂ ∂ (3.84)

The numerical solution will approach equation (3.84) in conservative form;

2 2

20.5u u ut x x

µ∂ ∂ ∂+ =

∂ ∂ ∂ (3.85)

The initial condition is:

( ) 0,0 1 tanh2

x xu xµ

⎛ ⎞−= − ⎜ ⎟

⎝ ⎠ (3.86)

In this case the exact solution is:

( ) 0, 1 tanh2

x x tu x tµ

⎛ ⎞− −= − ⎜ ⎟

⎝ ⎠ (3.87)

The scheme described earlier for inviscid Burgers’ equation can also be applied to

the current equation. This is accomplished by simply adding a second-order central-

difference expression for the viscous term uxx. In other words Fi is replaced by Hi

1 1( 2 ) /i i i i iH F u u u xµ − −= + − + ∆ (3.88)

Because of the viscosity that characterizes the scheme in this case, it is expected

that the solution of both approaches would be stable and similar. Hence this term will

have a large influence over the value of the error.

In our discussion, we will distinguish the following three categories of results:

• short wave (∆x/µ = 10)

• intermediate wave (∆x/µ = 3 )

90

• long wave (∆x/µ = 1)

In this case the numerical performance is affected by two parameters: the CFL

number and the Peclet number (Pe = U∆x /µ).

First we compare the solution of all four schemes as function of the Peclet number

(Pe) under constant CFL number (0.2). The value of CFL number is fixed at 0.2, because

the critical value for all schemes is much lower in the present case than for the linear

case. The behavior of the error is similar among DRP-fv, DRP-fd, OPC-fd and OPC-fv:

the error increases with increasing Peclet number, until a certain value beyond which the

schemes can no longer perform satisfactorily (see Figure 3-13).

Pe

ER

RO

R

100 10110-4

10-3

10-2

10-1 DRP - fvDRP - fdOPC - fvOPC - fd

**

**

***

Figure 3-13. Error as a function of Pe - nonlinear Burgers equation; ∆x = 0.25;

CFL = 0.2; t = 20

For the four schemes (DRP-fv, DRP-fd, and OPC-fv, OPC-fd), the solution and

error are very similar for all categories of wave, as shown in Figure 3-13, Figure 3-14 and

Figure 3-15. For long waves the solution is reproduced with high accuracy with all four

schemes, but the finite volume approach presents a slightly higher accuracy than the

finite difference schemes. The error for the intermediate wave is nearly the same with all

four approaches.

91

X

U

16 18 20 220

0.5

1

1.5

2

2.5

DRP - fvDRP - fdExact

X

U

16 18 20 220

0.5

1

1.5

2DRP - fvDRP - fdExact

X

U

16 18 20 220

0.5

1

1.5

2DRP - fvDRP - fdExact

a) Pe = 10 b) Pe = 3 c) Pe = 1 Figure 3-14. Numerical solution obtained by DRP schemes - nonlinear Burgers quation;

∆x = 0.25; CFL = 0.2; t = 20

X

U

16 18 20 220

0.5

1

1.5

2

2.5

DRP - fvDRP - fdExact

X

U

16 18 20 220

0.5

1

1.5

2OPC - fvOPC - fdExact

X

U

16 18 20 220

0.5

1

1.5

2

OPC - fvOPC - fdExact

a) Pe = 10 b) Pe = 3 c) Pe = 1 Figure 3-15. Numerical solution obtained by OPC schemes - nonlinear Burgers equation;

∆x = 0.25; CFL = 0.2; t = 20

Test problem 4: Two-Dimensional Acoustic Scattering Problem

To check the accuracy of the finite volume schemes in multi-dimensional

situations, we consider a test problem from the Second CAA Workshop (Tam and

Webb,1993; Tam et al., 1993): the two-dimensional acoustic scattering problem. The

physical problem is to find the sound field generated by a propeller scattered off by the

fuselage of an aircraft. The pressure loading on the fuselage is an input to the interior

noise problem. The fuselage is idealized by a circular cylinder and the noise source

(propeller) as a line source so that the computational problem is two-dimensional. The

cylinder has a radius of R = 0.5 and is located at the center of the domain.

The linearized Euler equations in polar coordinates are:

92

0 0 0

1 10 0 00

r

r r

u pu p

t r r ru u up

θ

θ

θ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∂ ∂ ∂⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(3.89)

At time t = 0, the initial conditions are:

ur = uθ = 0 (3.90)

( )2 24( , ,0) exp ln 2

0.04x y

p x y⎡ ⎤⎛ ⎞− +⎢ ⎥= − ⎜ ⎟

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ (3.91)

The test problem asks for the unsteady pressure time history at three points A(r = 5,

θ = 900), B(r =5, θ = 1350) and C(r = 5, θ = 1800), over the interval t = 5 → 10.

The numerical computations were performed over the domain: R∈ [0.5, 10.5] and θ

∈ [0, 2π]. For this problem three kinds of the boundary conditions are needed:

• Wall condition on the wall of the cylinder at R = 0.5

• Periodic condition along both azimuthally boundaries at θ = 0 and θ = 2π

Outfield boundary condition, along of the far field boundary, is the acoustic

radiation of Bayliss and Turkel (1982)].

The wall condition is based on the wall condition of Tam and Dong (1994). This

requires that:

0rdv dpdt dr

= − = (3.92)

This condition is satisfied by imposing the pressure derivatives on the wall to be

zero, and vr = 0 on the wall.

93

X

Y

-10 -5 0 5 10-10

-5

0

5

10

X

Y

-10 -5 0 5 10-10

-5

0

5

10

a) DRP-fv b)OPC-fv Figure 3-16. Instantaneous pressure contours at time t = 7 – two-dimensional acoustic

scattering problem

x

Y

-10 0 10-10

-5

0

5

10

AB

C

time

P

6 8 10

-0.02

0

0.02

0.04

0.06 OPC - fvDRP - fvExact

a) position of the testing points b) A: R = 5 ,θ = 900

time

p

6 8 10

-0.02

0

0.02

0.04

0.06OPC - fvDRP - fvExact

time

P

6 8 10-0.02

-0.01

0

0.01

0.02

0.03

0.04 OPC - fvDRP - fvExact

c) θ = 1350 d) θ = 900 Figure 3-17. The pressure history at point A, B and C – two-dimensional acoustic

scattering problem: finite volume approach

94

For this calculation, a uniformly spaced grid of 101 radial points and 153

azimuthally points was used, with a time step of CFL = 0.5. Figure 3-16 shows an

instantaneous pressure at t = 7. In this figure, the acoustic pulse is reflected by the

cylinder and reaches the outer boundary. We can see that two transients are shown: the

first and larger transient travels directly from the source; the second and smaller transient

is reflected from the cylinder. Both schemes reproduce both transients with acceptable

accuracy. Figure 3-17 compares the solution given by the fourth order schemes: DRP-fv

and OPC-fv. Between the two schemes the OPC-fv scheme performs better.

Summary and Conclusions

The DRP and OPC schemes, originally proposed in the finite difference form, have

been assessed. To better handle nonlinearity and geometric complexities, the finite

volume versions of both schemes have also been developed. Linear and nonlinear wave

equations, with and without viscous dissipation, have been adopted as the test problems.

For the linear wave equation with constant convection speed, the numerical

stability bound posed by the CFL number is comparable between the DRP and OPC

schemes. Both OPC and DRP produce solutions of a comparable order of accuracy, but

the magnitude of the error of the OPC scheme is lower.

For the nonlinear wave equation, the finite volume schemes can produce noticeably

better solutions and can handle the discontinuity or large gradients more satisfactorily.

However, as expected, all schemes have difficulties when there is insufficient mesh

resolution, as reflected in some of the short wave cases.

In conclusion, the finite volume version of both DRP and OPC schemes improve

the capabilities of the original version of the finite difference formulas in regard to

nonlinearity and high gradients. They can enhance performance of the original DRP and

95

OPC schemes for many wave propagation problems encountered in engineering

applications.

96

Table 3-1. A summary of proposed CAA algorithms Scheme The philosophy of the scheme Applications DRP (Tam and Webb, 1993; Tam et al., 1993)

In this scheme a central difference is employed to approximate first derivative. The coefficients are optimized to minimize a particular type of error

Wave propagation

LDDRK (Hu et al., 1996 ; Stanescu et al., 1998)

Traditionally, the coefficients of the Runge-Kutta scheme are optimized to minimize the dissipation and propagation waves.

Wave propagation problem

LDFV (Nance et al, 1997; Wang et al., 1999)

Scheme minimizes the numerical dispersion errors that arise in modeling convection phenomena, while keeping dissipation errors small: special high-order polynomials that interpolate the properties at the cell centers to the left and right sides of cell faces. A low pass filter has been implemented

Shock noise prediction

GODPR (Cheong and Lee, 2001)

It is derived, based on optimization that gives finite difference equations locally the same dispersion relation as the original partial differential equations on the grid points in the nonuniform Cartesian or curvilinear mesh

- Acoustic radiation from an oscillating circular cylinder in a wall - Scattering of acoustic pulse from a cylinder

OWENO (Wang and Chen, 2001)

The idea is to optimize WENO in wave number schemes, following the practice of DRP scheme to achieve high resolution for short wave. But in the same time it retains the advantage of WENO scheme in that discontinuity are captured without extra numerical damping.

Simulation of the shock/broadband acoustic wave

CE/SE (Chang, 1995; Chang et al., 1999)

The method is developed by imposing that: (i) space and time to be unified and treated as a single entity; (ii) local and global flux conservation in space and time to be enforced;

Flow involving shock; acoustic wave

FDo, RKo (Bogey and Bailly, 2004)

Optimized schemes are obtained by similar approach as DRP (space discretization), respectively LDDRK (time discretization). The difference consists that: i) error is minimized taking into account logarithm of the wavenumber; ii) the error is minimized on an interval that starts from ln(π/16). The stability and accuracy increase for these schemes

a)convective wave equation b)subsonic flows past rectangular open cavities; c) circular jet

97

Table 3-2. The computational cost for DRP and OPC schemes Scheme Number of

operation DRP-fd 8 DRP-fv 17 OPC-fd 9 OPC-fv 17

98

CHAPTER 4 A FINITE VOLUME-BASED HIGH ORDER CARTESIAN CUT-CELL METHOD

FOR COMPUTATIONAL AEROACOUSTICS

A finite volume-based high-order scheme with optimized dispersion and dissipation

characteristics in cooperation with the Cartesian cut-cell technique is presented for

aeroacoustic computations involving geometric complexities and nonlinearities. The field

equation is solved based on an optimized prefactored compact finite volume (OPC-fv)

scheme. The cut-cell approach handles the boundary shape by sub-dividing the

computational cells in accordance with the local geometric characteristics and facilitates

the use of numerical procedures with a desirable level of accuracy. The resulting

technique is assessed by several test problems that demonstrate satisfactory performance.

Introduction

In computational aero-acoustics (CAA), one needs to resolve dispersion and

dissipation characteristics in order to preserve form and amplitude of the wave (Hardin

and Hussaini (1992), Tam and coworkers (Tam and Webb, 1993; Tam et al., 1993)).

Furthermore, special care needs to be exercised in treating the boundaries to prevent the

creation of spurious waves while still accounting for wave reflection and/or scattering. In

order to meet these expectations, there are specific issues and challenges associated with

the employment of a structured multiblock grid, an overlapping grid, or an unstructured

grid ((Delfs, 2001), (Shyy et al., 2001), (Henshaw, 2004), (Sherer, 2004), (Dumbser,

2003), (Basel and Grünewald, 2003)), other alternatives should also be pursued. In this

work, we present an approach utilizing the Cartesian cut-cell approach based on the

99

finite-volume - based scheme aimed at optimizing the dispersion and dissipation

treatment. The Cartesian, cut-cell approach has been developed extensively for moving

and complex boundary computations ( see Table 4-1). It uses a background Cartesian grid

for the majority of the flow domain while creating sub-cells in the boundary regions to

meet the geometric requirements. Special algorithms for complex geometries are

constructed using flow properties and appropriate interpolation and extrapolation

procedures.

The low dispersion and dissipation scheme is based on an extension of the finite

difference-based, optimized prefactored compact (OPC) scheme originally developed by

Ashcroft and Zhang (2003) that we call OPC-fd. In the previous chapter we present the

extension of this method using the finite volume concept that we call OPC-fv, creating a

scheme to better satisfy the nonlinearity and conservation laws.

The present technique (which consists of the optimized dispersion and dissipation

characteristics and the cut-cell technique) offers desirable capabilities in both interior and

boundary regions. Several test problems will be presented to highlight the performance

characteristics of the proposed approach.

Cut-Cell Procedure

The cut-cell method rearranges the computational cells in the vicinity of the

interface via sub-division to conform to the specified boundary shape. Depending on the

intersection between the grid line and the interface, the subdivided, or cut, cells can

remain independent or can be merged into a neighboring cell in a given direction, e.g., a

direction approximately normal to the solid face. Accordingly, the interfacial cells are

reorganized along with their neighboring cells to form new cells with triangular,

trapezoidal, or pentagonal shapes.

100

The flux across the cell boundaries can be approximated by

1

m

k kkS

f n f n=

⋅ ≈ ∑∫r (4.1)

where S

f n⋅∫r is contour integral on the closed path S

The flux on the face is computed based on the multi-dimensional interpolation

method (Ye et al., 1999; Ye, 2001). Since the OPC scheme considered here is fourth

order, it is desirable to preserve the same order of accuracy around the boundary.

Figure 4-1. Illustration of the interfacial cells and cut-and-absorption procedures

The Cartesian cut cell mesh approach follows the subsequent steps:

• Locate the intersection of the boundary using a Cartesian mesh.

• Construct the background Cartesian mesh: The cells are flagged as solid cells, flow cells, or boundary cells. The boundary cells are those that either intersect the boundary or have a face in common with the boundary.

• Determine the geometric characteristics of the boundary cells such as cell volume, the direction normal to the boundary, and other information.

• Merge cells as necessary. A minimum acceptable cell area Smin is specified, and when the area of a cut cell is smaller than this value, it is merged into a neighboring cell. Determine the new characteristics of the merged cells.

interface interface

domain 1

domain 2 domain 2

domain 1

101

A

B

C

E

D

2928272625

19 222120 23

31 3332

37

35

34

13 161514 17

7 1098 11

1 432 5 6

12

18

2430

36

A

B

C

E

D

2928272625

19 222120 23

31 3332

37

35

34

13 161514 17

7 1098 11

1 432 5 6

12

18

2430

36

B D

A E

Cfsw

fefw

G

(xi,yj)

B D

A E

Cfsw

fefw

G

(xi,yj)

a) b) Figure 4-2. Modified cut – cell approach for CAA: a) Cartesian cut cell approach;

b) Detail around of the cut cell

For the trapezoid ACDE from Figure 4-2, the finite volume approach can be used

to approximate the wave equation:

0ABCDE

u E F dvt x y

⎛ ⎞∂ ∂ ∂+ + =⎜ ⎟∂ ∂ ∂⎝ ⎠

∫ (4.2)

where dv is a volume element. Appling Stokes’s theorem to Eq.(4.2), we have

∂uc/∂t SABCDE + ( )ABCDE

Edy Fdx−∫ (4.3)

The value of the integral can be approximated by

( ) ( )ABCDE AC CD DE EA

Edy Fdx Edy Edy Fdx Edy Fdx− = + − + −∫ ∫ ∫ ∫ ∫ (4.4)

where ( )ABCDE

Edy Fdx−∫ is contour integral on closed path ABCDE.

The function F and E can be represented generically by the function f. The fluxes

on the faces AC and DE can be approximate by

102

AC AB BC

fdy fdy fdy= +∫ ∫ ∫ (4.5)

An approximation of the value of the flux is given by:

( ) ( )w A B sw B CAC

fdy f y y f y y≈ − + −∫ (4.6)

The value of the flux at point w is given by the specific formula for the boundary

cell. The value of fsw is approximated using a polynomial that is fourth order in the x and

y direction:

4 4

0 0

i jsw ij

i jf b x y

= =

= ∑∑ (4.7)

where the coefficients bij are unknown. This interpolation has fourth-order accuracy in

the evaluation of the flux on the cut cells. In this case, the value of the coefficients is

obtained using the values of f in 25 grid points. An example is given in Figure 4-2 where

the value of the function is approximated using the following 25 points.

To solve bij , we use the following system of equations by expressing the function f

at 25 locations:

4 4 3 4 4 31 1 1 1 1 1 1 11 14 4 3 4 4 3

2 22 2 2 2 2 2 2 2

4 4 3 4 4 325 2525 25 25 25 25 25 25 25

1

1

1

x y x y x y x yf bf bx y x y x y x y

f bx y x y x y x y

⎧ ⎫⎧ ⎫ ⎧ ⎫⎪ ⎪⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪⎪ ⎪=⎨ ⎬ ⎨ ⎬⎨ ⎬⎪ ⎪ ⎪ ⎪⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪⎩ ⎭ ⎩ ⎭⎩ ⎭

L

L

M ML L L L L L

L

(4.8)

The coefficients bij from Eq.(4.7) become the coefficients b1, b2, …, b25 in Eq. (4.8)

. These coefficients can now be expressed in terms of values of f at the twenty five points

by inverting Eq.(4.8), i.e.,

25

1, 1,...25n nj j

jb a f n

=

= =∑ (4.9)

103

where anj are the elements of the inverse of the matrix in Eq.(4.8).

After bn is obtained, the value of f at the center of BC is expressed in the form

4 4 3 4 4 31 2 3 23 24 25sw sw sw sw sw sw sw sw swf b x y b x y b x y b x b y b= + + + + + +L (4.10)

and using Eq.(4.9), the value of fsw can be rewritten as

25

1sw j j

jf fα

=

= ∑ (4.11)

Note that bi, i = 1, 2, …, 25 are coefficients that depend only on the mesh, the

location, and the orientation of the boundary. Therefore, with a fixed geometry, these

coefficients can be computed once at the beginning of the solution procedure.

Now we turn to the calculation of the flux on the immersed face CD of the cell

(i, j). To compute the flux on a solid boundary, we use the reflection boundary condition.

In Figure 4-2b, (xi, yj) is the mass center of the boundary cell. We introduce a point across

the boundary that is symmetrically opposite of the center of mass of face CD, and note it

with G. The variable values in the ghost point G are

( )2G ij

G ij ij

p p

u u u n n

=⎧⎪⎨

= − ⋅⎪⎩r r r r r (4.12)

where nr is the normal vector of the solid boundary, the index ij indicates the value of

variable at point (xi, yj), and the index G indicates values at the ghost point G. The value

of the flux on face CD is approximate by

2

2

G ijCD

G ijCD

p pp

u uu

+⎧=⎪⎪

⎨ +⎪ =⎪⎩

r rr

(4.13)

104

Following simple algebra, we can observe that Eq.(4.13) assures that the velocity is

zero in the normal direction, and the variation of pressure in the normal direction around

the boundary is zero.

Test Cases

Three test cases have been selected to assess the performance of the present

approach: i) radiation from a baffled piston in an open domain; ii) reflection of an

acoustic pulse on an oblique wall; iii) a wave generated by a baffled piston and reflected

on an oblique wall.

The numerical methods used in the test problem are: OPC-fv for discretization in

space, and LDDRK for discretization in time.

Radiation from a Baffled Piston

A piston in a large baffle is a good starting approximation for investigating the

radiation of sound from a boxed loudspeaker. The physical problem is to find the sound

field generated by a piston that is baffled. The problem is solved in a two-dimensional

planar domain. This problem is chosen to evaluate the field equation solver in the

absence of the rigid wall.

General description

We will use a system of coordinate centered in origin of the piston as shown in

Figure 4-3. Dimensionless variables with respect to the following scales are to be used

• Length scale = diameter of piston, 2a

• Velocity scale = speed of sound, c

• Time scale = 2a/c

• Density scale = undisturbed density, ρ0

• Pressure scale = ρ0c2

105

• Frequency = c/2a

The linearized Euler equations are used:

0

0.

p utu pt

∂+ ∇ ⋅ =

∂∂

+ ∇ =∂

r

r (4.14)

The initial conditions are:

( )( )( )

, 0.

, 0.

, 0.

u x y

v x y

p x y

=

=

=

(4.15)

y

2a

y

2a

x

y

Outflow BCRigid bafflePiston

x

y

x

y

Outflow BCRigid bafflePiston

Outflow BCRigid bafflePiston

a) b) Figure 4-3. Radiation from a baffled piston (test problem (ii)): a) general description;

b) boundary condition

For this problem, two boundary conditions are used. The boundary condition on the

wall with piston is

( ) ( ) ( )0 cos ,0,0,

0p

V t x pistonu x t

otherwiseω ∈⎧⎪= ⎨

⎪⎩ (4.16)

where ω is the frequency of the piston, and V0 is the amplitude of the displacement. The

solution is obtained using linear equation; hence the value of V0 influences only the

106

amplitude of the solution and not its behavior. From this reason we take the value of V0

equal to one.

The outflow boundary condition is based on the acoustic radiation condition of

Tam and Web (1994):

0.

cos sin 0.2

u ptp p p pt x y r

θ θ

∂+ ∇ =

∂∂ ∂ ∂

+ + + =∂ ∂ ∂

r

(4.17)

where θ is the angular coordinate of the boundary point, and r is distance from the origin

(center of the piston) to the boundary point.

An analytical solution exists for this problem (Williams, 1999; Morse and Ingard,

1968):

( )

2 2

02 2

',0,

. . '2

p

s

u x tc

p x y t d dx

γ ϕ

ρ ϕπ γ ϕ

∞

−∞

⎛ ⎞+−⎜ ⎟

⎜ ⎟⎝ ⎠=

+∫ ∫

&

(4.18)

where the x′ values are points on the source, γ2 = (x-x’)2 + y2 is distance from a point of

the piston, and pu& is time derivative of the displacement of the piston.

The behavior of the piston is presented in terms of the Helmholtz number, ka,

where k is wavenumber, and a is the radius of the piston

akac

ω= (4.19)

In the following we will present two cases: low frequency (ka = 2) and high

frequency (ka = 7.5).

107

Directive factor D

The directional characteristic of a source is described by the amplitude directivity

factor D, defined as the pressure at any angle to the pressure on the angle of maximum

pressure.

( ) ( )( )

,, / 2

p rD

p rθ

θπ

= (4.20)

where the pressure is computed at any arbitrary time t. It is clear that the radiation is

strongest on the y axis, that is why we took maximum pressure at θ = π/2.

We note that r and θ are spherical coordinate. Because wavenumber kr

is in the

same direction as rr then the same spherical angles describe both of them. Thus in

spherical coordinate we have:

coscossinsin

x

y

k kx rk ky r

θθθθ

=⎧=⎧ ⎪⎨ ⎨ == ⎪⎩ ⎩

(21)

It is clear that the strongest radiation is on the y axis. In case that 1ka , the

directive function has nulls, and between the nulls are secondary radiation maxima, of

monotonically decreasing prominence. The number of nulls and secondary maxima is

determined by the size of ka, where a = radius of the piston, and ka = 2π a/λ . In other

words the number of the lobs increases with the value of ka. In our calculation we

compare the analytical and numerical values of beam pattern for ka = 2 and ka = 7.5. As

shown in Figure 4-2 and Figure 4-3, the numerical solution recovers with high accuracy

the analytical beam pattern, which proves the capacity of the approach to recover the

solution.

108

Figure 4-3 and Figure 4-4 represent, respectively, the beam pattern for ka = 7.5

and ka = 12.5. In these figures we can see that the number of lobs increases in the same

time with Helmholtz number (ka).

Figure 4-4. Radiation from a baffled piston: Beam patterns |D(θ)|; ka = 2

Figure 4-5. Radiation from a baffled piston: Beam patterns |D(θ)|; ka = 7.5

Figure 4-6. Radiation from a baffled piston: Beam patterns |D(θ)|; ka = 12.5

109

Pressure on the face of the piston

In the following the radiance impedance of the piston is calculated. The radiation

impedance of the piston is given as:

avp

av

pZu

= (4.22)

where the average values of the pressure and velocity on the face of the piston are given

by:

/ 2

/ 2/ 2

/ 2

( , )

( , )

a

ava

a

ava

p p x y dx

u u x y dx

−

−

⎧=⎪

⎪⎨⎪ =⎪⎩

∫

∫ (4.23)

The function R1, the resistance is defined by:

( )( )1 0Re /pR Z cρ= (4.24)

To contrast the behavior of resistance is in the domain of “low frequency” (ka « 1)

and “high frequency” (ka » 1), we can see in Figure 4-7 that the resistance is close to zero

at low frequency, and close to one at high frequency. Overall, the numerical solution

closely follows the analytical solution.

Figure 4-7. Radiation from a baffled piston: Real of piston radiation impedance

110

Low frequency (ka = 2)

The computation is done on a uniformly spaced grid ∆x = ∆y = 0.05 and a time step

∆t based on CFL = 0.2, where CFL = tcx

∆∆

.

The challenge of this problem is to solve the discontinuity at time t and at the

points y c t= , when maxc t y≤ , where c is the speed of sound. In this test case, the

variables are non-dimensionalized. Hence, c = 1, and the point of discontinuity will be at

y = t, for t < ymax (see Figure 4-8).

Figure 4-9 compares the solution given by the OPC-fv scheme with the analytical

solution demonstrating that there is good agreement between them.

Figure 4-8. Radiation from a baffled piston: Numerical solution on Oy axis (ka = 2),

t = 7.3 < ymax

Figure 4-10 shows contour plots of pressure at t = 20 and t = 40. The piston

radiation starts as a collimated plane-wave beam that moves from the face to r ≅ radius of

piston, beyond which the beam propagates spherically. This behavior is in accordance

with Blackstock’s study (2000).

111

Figure 4-9. Radiation from a baffled piston: Comparison between analytical and

computed solutions on axis (ka = 2)

a) b) Figure 4-10. Radiation from a baffled piston: Contour plot of pressure (ka = 2) –

numerical solution at: a) t = 20.0; b) t = 40.0

High frequency (ka = 7.5)

The computation is done on a uniformly spaced grid ∆x = ∆y = 0.02 and a time step

∆t based on CFL = 0.2, where CFL = tcx

∆∆

. In this case we increase the resolution to

better capture the wave characteristics. Figure 4-11 compares the solution given by the

OPC-fv scheme with the analytical solution demonstrating that there is good agreement

between them. Figure 4-12 shows contour plots of pressure at t = 20, highlighting the

entire domain as well as the region close to the piston.

112

Figure 4-11. Radiation from a baffled piston: Comparison between analytical and

computed solutions on axis (ka = 7.5)

a) b) Figure 4-12. Radiation from a baffled piston: Contour plot of pressure (ka = 7.5) –

numerical solution at t = 20: a) entire domain; b) reduced domain around piston

Reflection of a Pulse on an Oblique Wall

A simple example that enables us to check the performance of the cut-cell approach

is the reflected sound on an oblique wall. Even though this problem can be solved by

placing the wall parallel to the grid line, we purposely arrange the wall so it is at an

oblique angle to the grid line. This offers a direct evaluation of the cut-cell technique.

113

In this example, the sound hits the wall and reflects. The problem is characterized

by the linearized Euler equations, Eq. (4.14).

The initial condition is

2 2

0 0( , ) exp ln(2) x x y yp x yb b

⎧ ⎫⎡ ⎤− −⎪ ⎪⎛ ⎞ ⎛ ⎞= − +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

(4.25)

where b = 1./6. The values x0 and y0 are chosen so that the distance from the point (x0, y0)

to the oblique solid wall is equal to 1.5.

Figure 4-13. Reflection of the pulse on an oblique wall (test problem (ii)): general

description

The outflow boundary condition is the same as that used in the previous test

problem. The solid wall boundary conditions for pressure and velocity are given by

reflecting boundary condition.

The computation is done on a Cartesian grid that is characterized by ∆x = ∆y = 0.05

and CFL = 0.5. Figure 4-12 illustrates the solid boundary intersecting a Cartesian mesh.

The boundary cell can remain independent, like in ABCK, or it can merge with a

neighboring cell. For example, CID merges with KCIJ. The computation of the parameter

in the cut cell CDIJK can be written for pressure from wave Eq.(4.14) in the form:

114

( )1 0.CD CD JK JK DI DI IJ IJ CK CK CD CDCDIJK

p u dy u dy v dx v dx v dx v dxt S

∂+ + − − − − =

∂ (4.26)

where the values of u and v represent the values of the function in the middle of the

segment. The value of the parameter is approximated using i) the OPC scheme on faces

IJ, JK and KC, ii) fourth order polynomial on faces DI , and iii) reflecting boundary

condition on face DF (boundary).

Figure 4-14. Reflection of the pulse on an oblique wall (test problem (ii)): cell around

boundary

To evaluate the performance of the cut-cell approach, Figure 4-15 shows the

pressure history from time = 0 - 4 and for three wall angles: 900, 810, and 630. As

apparent from the figure, there is a good agreement between these solutions. Figure 4-16

shows the pressure contours at three time instants by placing the solid wall at 630

inclinations.

115

Figure 4-15. Reflection of the pulse on an oblique wall: history of pressure for different

angle of wall

a) b) c)

Figure 4-16. Reflection of the pulse on an oblique wall: α = 630 a) t = 0.8; b) t = 1.6; c) t = 3.2

Wave Generated by a Baffled Piston and Reflected on an Oblique Wall

This problem is based on the combined characteristics of the previous two cases.

The wave generated by a piston is reflected by an oblique wall. The linearized Euler

equations (Eq. (4.14)) is used in this test case, and the initial condition is the same as that

of the baffled piston, Eq. (4.16).

The rectangular domain over which we do the computation is:

( ) [ ] [ ], 6, 6 0,15x y ∈ − × . The bottom of the domain is a piston mounted on a plane rigid

116

baffle; hence, the boundary conditions are given by Eq.(4.16). The velocity and pressure

on the solid and open boundaries are identical to those used in the previous case.

Figure 4-17. Wave generated by a baffled piston and reflects on an oblique wall: General

description of the domain

The piston presents the following characteristics: V0 = 1 and ω = 4 (ka = 2). The

solution is obtained using the uniform grid ∆x = ∆y = 0.05, and with a time step of

CFL = 0.5. Figure 4-18 highlights the pressure contours at different time instants.

a) b)

Figure 4-18. Wave generated by a baffled piston and reflects on an oblique wall: α = 630; a) t = 9; b) t = 14;

The challenges of this problem are to recover:

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• The wave generate by the piston

• The reflection wave

• The combination of the two waves: wave generated by the piston and reflected on the oblique wall.

Figure 4-18 shows the outcome of the computation, exhibiting a series of local

maxims and nulls.

Conclusion

A method based on OPC-fv scheme aimed at optimizing the dispersion and

dissipation properties, and the cut-cell technique aimed at handling geometric variations

is presented. The approach is motivated by the need for handling acoustic problems with

nonlinearities (using finite volume technique) and a complex geometry (using the cut-cell

technique). Selected test cases have been used to demonstrate the performance of the

method. The present approach can offer accurate and versatile treatment to some

important and challenging aspects of acoustic problems.

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Table 4-1. Published cut-cell approach for different problems Authors Test Cases Objective De Zeeuw et al, 1999 ;

Transonic NACA Airfoil “Subsonic” Three-Element Airfoil Supersonic Double Ellipse Supersonic Channel Flow

A method for adaptive refinement of the steady equation

Udaykumar et al, 1997

Deformation of viscous droplets in axisymetric Extensional Stakesian flow Deformation of Droplets in extensional flows with Inertia Effects Deformation of droplets in constricted tubes

The ELAFINT algorithm is developed and applied to compute flows with solid-fluid and fluid-fluid interface

Yang et al, 1999; Causon et al, 1999; Ingram et al, 2003

Open-Ended shock tube Fifteen-degree wedge flow at Mach 2 Muzzle brake flowfields Muzzle Sabot/Projectile flowfields Shallow water flow

The flow is an upwind scheme of the Godunov-type based on MUSCL reconstruction and a suitable approximate Riemann solver

Udaykumar et al, 1999

Inviscid flow around circular cylinder Track solid-liquid boundaries on a fixed underlying grid. The interface is treated as a discontinuity conditions and explicitly tracked

Ye et al, 1999 Two-dimensional stokes flow past a circular cylinder Flow past a circular cylinder immersed in a freestream Flow past a circular cylinder in a channel Application to complex geometries: (i) flow past a random array of cylinders (ii) flow past a cascade of airfoils

A Cartesian grid method has been developed for simulating two-dimensional unsteady, viscous, incompressible flows with complex immersed boundaries

Lahur et al, 2000 Moving piston Moving cylinder

Treat moving body problem

D.Calhoun et al, 2000

Advection and diffusion of a plane wave in a channel Advection and diffusion in an annulus Advection through a field of irregular objects

A fully conservative, high-resolution, finite volume algorithm for advection-diffusion equations in irregular geometries

Verstappen et al., 2000

Flow over a circular cylinder To introduce a novel cut-cell Cartesian grid method that preserve the spectral properties of convection and diffusion

Ye et al, 2001 Bubble dynamics with large liquid-to vapor density ratio and phase change

Treating sharp discontinuity interface for bubble dynamics and phase change

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CHAPTER 5 SUMMARY AND FUTURE WORK

Predicting noise radiation associated with unsteady flows is the central theme in

aeroacoustics. Since the generation of the sound by the flow typically involves complex

physical mechanisms, approximate models are often employed. Colonius and Lele (2004)

have reviewed the modeling as well as the numerical aspects based on recent progress.

Although it is not possible to separate the flow from sound, oftentimes, we can separate

the flow prediction from noise prediction, based on pressure, frequency and length scale

separations.

In these regards, CAA presents a series of challenges. In particular, it is important

to extract small quantities relative to other phenomena in the course of computations. For

example, the pressure fluctuations in a turbulent flow are on order of 10-2 of the mean

pressure (Blake, 1986). However, loud noise can be produced by pressure fluctuation at

the level of 10-5 of the mean pressure (Hardin, 1996). In order to accurately capture the

acoustic field, numerical techniques need to yield low artificial dissipation. In addition,

the numerical dispersion related to frequency and phase constitutes another concern. For

this reason, many tests of “goodness” of CAA solutions have been developed. In light of

these issues, we have investigated schemes aimed at offering refined performance in

numerical dissipation and dispersion, including the dispersion-relation-preservation

(DRP) scheme, proposed by Tam and Webb (1993), the space-time a-ε method,

developed by Chang (1995), and the optimized prefactored compact (OPC) scheme,

proposed by Ascroft and Zhang (2003). The space-time a-ε method directly controls the

120

level of dispersion and dissipation via a free parameter, ε, the DRP scheme minimizes the

error by matching the characteristics of the wave, and the OPC scheme aims at

employing a compact stencil to achieve optimized outcome. Insight into the dispersive

and dissipative aspects in each scheme is gained from analyzing the truncation error.

Another challenge in CAA consists in the reproduction of the nonlinear

propagation, including nonlinear steepening and decay, viscous effects in intense sound

beams (e.g., acoustic near field of high-speed jets), sonic boom propagation, hydrofoil

and trailing-edge noise at low speed (Goldstein, 1976; Wang and Moin, 2000; Manoha

and Herraero, 2002; Takeda et al, 2003; Howe , 1978, 1999, 2000, 2001; Casper and

Farassat, 2004; Roger and Moreau, 2005) and high speed jet noise ( Davies et al, 1963;

Hong and Mingde, 1999; Shur et al, 2003; Fisher et al, 1998; Morris et al, 1997; Bogey et

al, 2003). In this regard, we have developed finite volume treatments of the DRP and

OPC schemes. These schemes, originally based on the finite difference, attempt to

optimize the coefficients for better resolution of short waves with respect to the

computational grid while maintaining pre-determined formal orders of accuracy. In the

present study, finite volume formulations of both schemes are presented to better handle

the nonlinearity and complex geometry encountered in many engineering applications.

Linear and nonlinear wave equations, with and without viscous dissipation, have been

adopted as the test problems. Highlighting the principal characteristics of the schemes

and utilizing linear and nonlinear wave equations with different wavelengths as the test

cases, the performance of these approaches is documented. For nonlinear wave equations,

the finite volume version of both DRP and OPC schemes offer substantially better

solutions in regions of high gradient or discontinuity.

121

Boundary treatment is another critical issue. Instead of employing boundary

conforming grid techniques, we propose a Cartesian grid, cut-cell approach, aimed at

aero-acoustics computations involving geometric complexities and nonlinearities. The

cut-cell approach handles the boundary shape by sub-dividing the computational cells in

accordance with the local geometric characteristics and facilitates the use of numerical

procedures with desirable accuracy. The resulting technique is assessed by several test

problems that demonstrate satisfactory performance.

In the following we summarize the outcome of the present research, as presented in

the previous chapters

Assessment of DRP and Space-Time CE/SE Scheme

First we investigate two numerical methods for treating convective transport: the

dispersion-relation-preserving (DRP) by Tam and Webb (1993), and the unified space-

time a-ε method, developed by Chang (1995). Based on the investigation of the DRP

CE/SE scheme we offer the following summary:

• The two methods exhibit different performance with regard to the CFL number, ν. Both truncation error analysis and numerical testing indicate that better solutions can be obtained for the DRP scheme method if ν is close to 0.2 for short wave, and close to 0.1 for intermediate and long wave. For the space-time a-ε method it is preferable if ν is less than but close to 1.

• Both schemes are dispersive for short waves. The space-time a-ε method directly controls the level of dispersion and dissipation, via the free parameter, ε. The DRP scheme, on the other hand, minimizes the error by adjusting the scheme to match the characteristics of the wave.

• Formally, based on the truncation error analysis, the DRP scheme is fourth order in space and third order in time, while the space-time a-ε scheme is second order.

• Evidence based on the truncation error analysis and the test problems indicate that in order to reduce numerical dispersion and maintain satisfactory resolution, for short wave ( e.g., b/∆x = 3), ν and ε are preferable to be close to each other. For long and intermediate waves there is virtually no need to introduce much numerical

122

dissipation and hence ε can be reduced close to zero. On the other hand, mismatched ε and ν can substantially worsen the performance of the scheme. Hence to achieve the best performance of the CE/SE scheme, one chooses the largest possible ν and then a matching ε.

• The DRP scheme exhibits mainly dispersive errors, while the space-time a-ε scheme exhibits both dispersive and dissipative errors.

• It is advisable to use space-time a-ε scheme for long wave computation because its error grows slower.

• For short and intermediate waves, the DRP scheme yields errors with lower level and slower accumulation rate.

• DRP does not offer an accurate solution for a short wave, α∆x greater than αc∆x (in our case greater than 1.1), because the wavelength is not adequately to condition of the method. Hence the DRP scheme offers a good guidance in regard to temporal and spatial sizes.

It should also be noted that he DRP scheme is a multi-step method, which requires

more boundary conditions and initial data, while the space-time a-ε scheme is a one-step

method. Combined with the fact that the DRP scheme performs better with smaller ν

(ν = CFL), it can be more expensive to compute than for space-time a-ε scheme.

Finite-Volume Treatment of Dispersion-Relation-Preserving and Optimized Prefactored Compact Schemes

The DRP and OPC schemes, originally proposed in the finite difference form, have

been further developed. To better handle nonlinearity and geometric complexities, the

finite volume version of both schemes has also been constructed. Linear and nonlinear

wave equations, with and without viscous dissipation, have been adopted as the test

problems.

For the linear wave equation, the numerical stability bound posed by the CFL

number is comparable between the DRP and OPC schemes. Both OPC and DRP schemes

123

produce solutions of comparable accuracy, but the magnitude of the error of the OPC

scheme is somewhat lower.

For the nonlinear wave equation, the finite volume schemes can produce noticeably

better solutions and can handle the discontinuity or large gradients more satisfactorily.

However, as expected, all schemes have difficulties when there is insufficient mesh

resolution, as reflected in some of the short wave cases.

In conclusion, the finite volume version of both DRP and OPC schemes improve

the capabilities of the original version of the finite difference formulas in regard to

nonlinearity and high gradients. They can enhance performance of the original DRP and

OPC schemes for many wave propagation problems encountered in engineering

applications.

Cartesian Grid, Cut-Cell Approach for Complex Boundary Treatment

A method based on high order, finite-volume schemes aimed at optimizing the

dispersion and dissipation properties, and the Cartesian grid, cut-cell technique aimed at

handling geometric complexities is presented. The approach is motivated by the need for

handling acoustic problems for practical problems involving realistic geometries.

The finite volume-based Optimized Prefactored Compact scheme and the Cartesian

cut-cell approach are combined to offer 4th order accuracy and geometric flexibility. The

computational overhead of the cut-cell approach is modest because the following

information needs to be computed only once, unless, of course, if the geometry is time

dependent:

• Data communication between cells affected by the boundary treatment

• Calculation of area and other geometric information

• Interpolation procedures required for the flux computation in the boundary region.

124

Based on the evaluation of the test cases investigated, we conclude that the present

approach can be effective in treating aeroacoustics problems with irregular geometry.

Future Work

Even though the radiant pressure fluctuation is less than 10-4 of the ambient

pressure, there are situations when the nonlinearity behavior can not be neglected.

Examples include intense sound beam resulting from the near field high-speed after

burner flows, sonic boom, and scattering of high intensity noise caused by the blade and

vortex interaction. For example, the interaction of unsteady disturbances with leading and

trailing edges of fan and compressor blades has motivated numerous studies (Goldstein,

1976; Wang and Moin, 2000; Manoha and Herraero, 2002; Takeda et al., 2003; Howe ,

1978, 1999, 2000, 2001; Casper and Farassat, 2004; Roger and Moreau, 2005). In this

problem, the edge is usually a source of high frequency sound associated with smaller-

scale boundary layer turbulence.

A general description of high frequency, nonlinear acoustics involves

nonequilibrium effects, which arises when, e.g., the acoustic time scale becomes

comparable to that of the vibrational relaxation process of polyatomic gases (Hamilton

and Blackstock, 1997).

Another important example of nonlinear behavior is landing gear noise. This

problem is recognized as one of the major components of airframe noise for commercial

aircraft. The computational investigations should take into consideration the following

aspects: the flow conditions with varying Mach number from 0.18 to 0.24 (Guo, 2005),

the noise radiation involves complex flows around a complex geometry. To be

computationally efficient, adaptive grid refinement techniques are attractive. Recent

efforts reported by Singh et al. (2005) are directly applicable.

125

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BIOGRAPHICAL SKETCH

Mihaela Popescu was born in Romania, in a small village, Berca, known for rare

natural phenomena- mud volcanoes. Mihaela received two bachelor degrees from the

University of Bucharest, Romania. Her first BS was in mathematics, specializing in

mathematics applied in mechanics. Her second BS was in biology, specializing in

ecology. While pursuing her degree in biology, Mihaela worked with modeling of

Chironomid community based on ecosystem characteristics. In adition, for over a year,

Mihaela worked for the Nuclear Energy Reactor Institute in Bucharest, Romania.

Mihaela’s other qualification also entail working as a teacher for two years.

In 1999, she enrolled in the University of Florida to pursue a Doctor of Philosophy

degree in Aerospace Engineering under the guidance of Dr. Wei Shyy.

During her Ph.D. studies she was the recipient of the Alumni Fellowship at the

University of Florida.

Mud volcano cones in the Berca Anticline Depression of the Curvature

Subcarpathians. [Photograph Credit: Dr. Dan Balteanu, Romanian Academy]