– Abstract –

The work done in this dissertation was situated in the field of Computational AeroAcoustics. The exponential growth in computational power over the last decade haslead to a growing domain of research where the use of computational fluid dynamicstechniques is applied in the field of aero acoustics. Present state-of-the-art methods use ahybrid approach where the calculation of the sound generating process is separated fromthe calculation of the transport of the sound to the far field. In this work this hybridapproach was followed and the necessary tools were researched and developed. The mainidea was to extend the existing compressible flow solver Euranus towards Large EddySimulations for near field noise computation and to develop a code which could computethe sound, based on the LES computation, for an observer far away from the sound source.

In this work high order upwind type schemes with low diffusion and dispersion errorswere developed and discussed. A new way for optimizing time integration schemes tominimize the total diffusion and dispersion errors was proposed. Two different types ofboundary conditions were implemented and tested on their performance in CAA com-putations. For the far field noise computations, two codes were developed, tested andcompared to eachother. Finally, all those tools were brought together to perform a LEScalculation of a circular cylinder in cross flow, on which an acoustic post processing wasdone with the developed far field solver.

i

– Acknowledgements –

”I did it!”, was the first thought I had when writing the last few words of this disser-tation. However, soon after I realized that this was maybe not a total honest thought.Many persons around me helped me, supported me or gave me the chances which lead tothe end result of this PhD. Through these few words I would like to thank them all foreverything they did for me the last few years.

Firstly, I would like to thank my promotor Chris. He gave me the chance to enter theworld of research and experience six years of learning opportunities and discovery. It washard sometimes but it has allowed me to develop myself as a scientist and a person, in away which wouldn’t have been possible elsewhere.

Secondly, I would like to thank my colleagues of who I sometimes jokingly said that Ihad spend more time with them than with my wife. Sergey, it was fun to share six yearsthe office with you. Thanks for all the discussions we had, whether it was about science,philosophy, magic or martial arts, they were always entertaining. Tim, without you thisthesis wouldn’t have been finished now. Thanks for your constant support and inspiringdrive you always had. Don’t loose it! Mark, thanks for making me laugh with your jokesduring those years. Also a word of thanks to Alain, who was always prepared to act asour last hope when we couldn’t solve our technical problems, to Jenny who took care thateverything ran smoothly in the department and to Ghader with who I had a pleasantcooperation the last 2 years. A special thanks goes to professor Hirsch for who, althoughshortly, I also had the pleasure to work. It was challenging and interesting.

Thirdly, the person to who I owe the biggest word of thanks is my wife Jenni. It isdifficult to put down in words how much I appreciate what you have given up for me tobe able to finish this work. Thanks for trying to understand me and being there for me.I learned tremendously a lot from you and although that it was sometimes difficult herein Belgium, we always had fun together. Let’s continue! They might not be able to readthis for a while to come but I still would like to address my son Onni and daughter Siiriwith a word of thanks for their source of joy, smiles, hugs and kisses when things were abit hectic.

Another special thanks goes to the rest of my family. My parents, for having alwayssupported my choices and decisions and for being an example of a happy, loving and wisecouple of parents to me. To my brothers Tom and Tim: I didn’t see you guys as much asI wanted during those years, but know that you guys are just the most crazy and funniestbrothers I could imagine. Keep up ”The Clan”! Raija and Kari, thanks for giving mea second home in Finland and for the wise life lessons I sometimes got. Those holidaysduring Christmas and in the summer hold a special place in my heart. That many stillmay come.

ii

Lastly, I would like to thank all my friends who make life a joy: Yvi, Lalena, Simme,JP, Cathy, Vinne, Barbara, Smisse, Katrien, Selle, Els, Jallu, Jussi, Sofia, Kukke, Ji-iPee, Jari, Kaisa, Anna, Hissu, Sanni, Francois, Fernando, Peter, Shawn, Tampopo, Eva,Torbjorn, Matti.... and all the others I forgot in this list.

Gambatte!

Brussels, Jan RamboerAugust 2005

iii

Contents

1 Introduction 11.1 Sound in our society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Computational Aero Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Goal and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Basic concepts of Aero Acoustics 62.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Compact Upwind Spatial discretization for use in CAA 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Finite Volume Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Compact Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 ICEU-schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 IUEC-schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.3 IUEU-schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 2D Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5.1 1D Advection Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5.2 Rotating 2D Gaussian Pulse . . . . . . . . . . . . . . . . . . . . . . 233.5.3 Non Linear Acoustic Pulse . . . . . . . . . . . . . . . . . . . . . . . 253.5.4 Subsonic Vortical Flow . . . . . . . . . . . . . . . . . . . . . . . . . 273.5.5 LES channel results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Optimized Time Integration Schemes 334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Dispersion and dissipation errors . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1 Dispersion and dissipation errors of the spatial scheme . . . . . . . 354.2.2 Dispersion and dissipation errors of temporal scheme . . . . . . . . 374.2.3 Dispersion and dissipation errors of total scheme . . . . . . . . . . . 38

4.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.1 Optimizations for Central Spatial discretizations . . . . . . . . . . . 434.3.2 Optimizations for Upwind Spatial discretizations . . . . . . . . . . . 48

iv

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.1 1D Convection Equation . . . . . . . . . . . . . . . . . . . . . . . . 524.4.2 Linearized Euler Equations . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Boundary Conditions 685.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3 Non-reflecting Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 74

5.3.1 Far Field Boundary condition . . . . . . . . . . . . . . . . . . . . . 745.3.2 Characteristic boundary conditions . . . . . . . . . . . . . . . . . . 75

5.4 Boundary Condition Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Far Field Noise Solvers 826.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Ffowcs-Williams-Hawkings Formulation . . . . . . . . . . . . . . . . . . . . 836.3 Linearized Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4 Test case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.4.1 Solution with LEE . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.4.2 FWH equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 LES 1027.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.2.2 Subgrid Scale Models . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.3 External flow past a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 1077.3.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 1097.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Conclusions and Future Challenges 123

v

Chapter 1

Introduction

1.1 Sound in our society

Sound can be defined in general as the sensation perceived by the sense of hearing. In ourdaily life sound is constantly present, and the absence of it would make the organizationof society difficult. The main transfer of knowledge between people relies on speech andthus on the hearing sense. Sound in the form of music can uplift our mood and bringjoy and entertainment. Our orientation ability is, besides input obtained by our vision,based on our auditory sense. Physical reactions to sound are ingrained in our instincts.A parent’s alertness will instinctively raise when hearing the crying of his own child.

However, besides the positive effects which sound brings to us, it can also have anegative side. Usually when talking about sound in a negative context it is referred to asnoise. Noise will influence our general performance in daily life due to its effect on ourhealth. Many people living near airports have shorter periods of sleep and wake up morefrequently during the night, making them tired during daytime. Chronical exposure tonoise can also have an impact on long term memory and recognition. Besides the effectsof noise on human performance, it has also a direct health affect on blood pressure, heartrate, breathing and hearing sense. Loud noise is known to cause direct injuries to theear with loss of hearing as a result. Table 1.1 gives an overview of the different levels ofnoise intensity with examples of sounds for each level. At a Sound Pressure Level of 110dB hearing becomes painfull, at 120 dB it becomes deafening and short term exposureto Sound Pressure Levels above 140 dB causes instant pain and hearing loss may resultif the exposure is prolonged. At levels of 180 dB humans become tired, experience facialpain and the skin may become burnt.

According to the World Health Organisation these health effects, in turn, can lead tosocial handicap, reduced productivity, decreased performance in learning, absenteeism atwork and school, increased drug use and accidents.

It is common to encounter sound at levels that can cause adverse health effects. Infact, it is difficult today to escape noise pollution completely. Therefore noise is becomingmore and more internationally accepted as a form of environmental pollution. This forcesgovernments to adopt laws, regulations and guidelines for noise producing machines andvehicles. Well known examples are the prohibition for airplanes to land or take off duringnight time if their noise emission is too high, especially when urban areas are close to the

1

Decibel Level Example of Sound30 Soft whisper35 Noise may prevent the listener from falling asleep40 Quiet office noise level50 Quiet conversation60 Average television volume, sewing machine, lively conversation70 Busy traffic, noisy restaurant80 Heavy city traffic, factory noise, alarm clock90 Cocktail party, lawn mower100 Pneumatic drill120 Sandblasting, thunder140 Jet Airplane180 Rocket Launching Pad

Table 1.1: Decibel ratings and hazardous levels of noise

airport. With the prediction of a continuous growth in the near future of the transportindustry, noise reduction becomes increasingly important in the design process. Industrialnoise pollution is caused by several physical mechanisms, such as friction , vibration, com-bustion, shocks and aerodynamic perturbations. This last one is a major contributor tovehicle noise since it becomes dominant at travelling speeds over 100 km/h. Aerodynamicnoise produced by aircrafts is generated by the flaps, air brakes, landing gear, wings andthe free jet flow from the engine. Internal noise is also present due to the transmissionof the externally produced aerodynamic noise through the structure to the inner com-partment. With the development of faster trains, driving at speeds over 200 km/h, alsohere the aerodynamic noise will become more dominant than the mechanical noise gener-ated by the vehicle. Besides aerodynamic noise produced by vehicles, other areas whereaerodynamic noise is becoming increasingly important are wind turbines and cooling andclimate systems. For large wind turbines with low rotational speed, the main contribu-tion to aerodynamic noise is the narrow-band noise caused by blade-tower interaction.Smaller and fast rotating wind turbines experience noise radiation caused by vortex-bladeinteracting. Power units of cooling systems often emit tonal noise at a constant frequencywhich is transported through the duct system.

The need to take noise production into account in the design process is becomingclearly increasingly important and as a result the demand for fast and robust computa-tional aero acoustic methods is constantly growing.

1.2 Computational Aero Acoustics

Aero acoustics is the field of research where the discipline of fluid mechanics and classicalacoustics meet. It studies the phenomena of aerodynamic noise which can be defined assound generated by aerodynamic forces induced by, whether or not turbulent, flows. Al-though, acoustics is a very old discipline, reaching back to the beginning of our calendar,aero acoustics is a relatively young discipline. In the early days, middle of the previous

2

century, the research was limited to experimental work deriving mainly empirical rela-tions in order to estimate noise production of certain fluid flow. Gutin’s paper [51], firstpublished in 1936 in Russia, dealing with the problem of propeller noise is generally con-sidered as the beginning of modern aero acoustics. Probably the most famous scientist inAero acoustics is Sir James Lighthill [81], who introduced the idea of representing soundas the difference of the actual flow and a reference flow. Later on he derived his presentlywell known Acoustic Analogy which was developed in the frame work of jet noise andwhich got later extended by Ffowcs Williams and Hawkings [38] for moving surfaces em-bed in a flow. At the mean time, as a result of the development of the computer, the firststeps in computational fluid dynamics (CFD) were taken. This discipline matured overthe past decades separately from the field of aero acoustics. Although CFD is nowadaysat the level where many industrially relevant problems can be solved, it was not until afew years ago that computers became powerful enough to make a numerical approach foraero acoustics attractive.

Since computational aero acoustics (CAA) is a fairly recent domain of research, a clearway of solving industrial problems does not exist. Up until now many different approacheshave been investigated usually limited to the area for which they were developed. Thedeceptively most logic step to take is to apply the methods developed in the world of CFDdirectly to the field of CAA. This would require to solve the complete compressible Navier-Stokes equations for the flow as well as the acoustic field, from the area of aerodynamicsound generation down to the far field observer. These direct methods do not includeany modelling of the sound, since the noise producing pressure fluctuations are inherentlypresent in the Navier-Stokes equations. Since aero acoustics is, especially at low Machnumber, a multi scale problem, this would due to time and spatial resolution requirementsto resolve the small acoustic fluctuations and the large distance over which the noiseis transported to the observer, lead to ridiculous high amount of computational powernecessary. Another aspect to consider is that the schemes designed for applications inCFD were tuned for different properties than those necessary for CAA. Because CFD isusually only interested in the near field of the problem at hand, schemes used for thesepurposes are tuned to suppress acoustic waves which is logically not compatible with theidea of CAA. Up until now, no scheme is known to be compatible with the requirementsof both CFD and CAA.

Because the aerodynamic noise generating process is highly non linear, while propaga-tion of sound to the far field observer is quasi linear, the most successful methods currentlydeveloped are of the hybrid type. In this approach the area of sound generation is decou-pled from the problem of sound propagation, which allows to use techniques specificallysuited for both different problems. This leads to a combination of a sound generationmethod together with a transport method. For the sound generation method one has thechoice between two options. The first is to use a CFD method which is directly coupled toa transport method by providing sound data in the coupling region. This requires a highcontrol of the dispersion and diffusion errors of the schemes used, however they are lessdemanding than for the direct method because the the accurate transport of the soundwaves has to be only sustained over a relative short distance.

The other option for sound generation methods is the use of semi-empirical data fromCFD computation. By using information on the turbulent length and time scales coming

3

from a steady state classic CFD calculation, empirical methods can be used to constructsound source spectra which are then coupled to one of the transport methods at hand.The success of this depends highly on accuracy of the empirical model and the validationof the data, however for specific problems this can give fast and reliable results.

Also the transport methods can be divided in two different methodologies. The com-putational transport methods solve a set of partial differential equations in the entirefield up to the observer. They are similar to the CFD methods, but focus only on solvingfor the acoustic field instead of the whole aerodynamic flow. This leads to simplificationswhich allow schemes to be highly tuned for efficient and accurate solutions. A downside ofthese methods is that they solve for all the space in between the observer and the sourceswhich can lead to high computational cost when the observer is far away. The advantageof these type of field calculations is that they allow the visualization of the whole acousticfield which can lead to better understanding of the problem solved for. When only aspectrum of the sound is needed for a low number of positions in the far field the Ana-lytical Transport methods are the choice of preference. Several different methods exist,such as the Kirchhoff method [69] and Ffwocs Williams-Hawkings method [38], which areall extensions of the Acoustic Analogy of Lighthill [81]. They use an integrated form ofthe governing equation which has the advantage of averaging out the numerical errors.However in the case of bodies moving at transonic speeds these integrals become highlysingular due to the Doppler effect, which leads to problems of instability.

As mentioned before, the choice of the methods being used is highly depending on theproblem which is being considered. Aerodynamic noise occurs because of two basicallydifferent phenomena. Impulsive noise is a result of moving surfaces which cause aerody-namic loads on the body surface generating pressure fluctuations which are radiated assound. This kind of noise is relatively easy to extract from CFD simulation since this iskind of noise is deterministic. Unsteady Reynolds Averaged Navier-Stokes calculationsare here accurate enough to obtain acceptable results since the need for spatial and tem-poral resolution is less stringent. The other noise mechanism is the result of turbulencewhich is present in almost every practical application. The noise generated by turbulencehas a broad frequency spectrum which is a direct result of the stochastic nature of tur-bulent flow. Since turbulence is highly chaotic, containing flow structures in all lengthand time scales, direct numerical simulation (DNS) is a candidate for solving the nearfield turbulent noise generation. As mentioned before, solving for all the length scales inthe flow will lead to unfeasibly high calculation times. Since it is known that mainly thelarger flow structures contribute to the noise production [127], an interesting alternative isLarge Eddy Simulation (LES). Large Eddy Simulation calculates directly the larger flowstructures while it filters out the small scale flow features and models their effect on theflow using a so called sub grid scale model. Currently the most general methodology forthe prediction of far-field noise in turbulent flow is to compute the near field using LESin conjunction with an Acoustic Analogy.

4

1.3 Goal and challenges

The main goal of this work was to build up the necessary know-how to perform CAAcalculations based on the gained experience in CFD and LES of the research group ofFluid Dynamics. It was chosen to follow the most promising approach which was ahybrid method where the existing LES implementation of the compressible Navier-Stokessolver Euranus [74, 79, 80, 113] would be used for the near-field noise calculation andan Analytic Transport method based on the Ffwocs Williams Hawkings equation forthe propagation of the sound to the far field. In order to do this the research focussedon the main problems encountered in sound computations. The first challenge whichposes itself is that of the numerical errors introduced by the discretization process ofthe partial differential equations governing the problem. The main problem lies in thewide range of energy, length and time scales present in CAA problems. The ratio ofnoise energy to mechanical energy is on the order of 10−9 for low Mach number flows to10−5 for transonic flows. This poses severe restrictions on the allowable level of errorscoming from the discretization process. To be able to capture the physical values of thesound radiating pressure fluctuations the level of errors has to be at least 5 to 9 ordersof magnitude smaller than the error level which is allowed for classical CFD problems.Therefore higher order methods for the spatial and temporal discretization needed to bedeveloped. These schemes should be tuned to minimize diffusion and dispersion errors toavoid the numerical errors overwhelming the physical acoustic waves. In computationalmethods the area of simulation is usually not extended to infinity and boundary conditionsat the border of the region of interest need to be imposed. Since sound waves in physicalproblems are ever expanding, sooner or later they will leave the computational domain.The boundary conditions in CAA should therefore be as transparent as possible to reducethe reflection of acoustic energy in the domain.

The outline of this dissertation is as follow: in chapter 2 an introduction to the basicconcepts and definitions of Aero Acoustics is given. Next, in chapter 3 a discussion ofdeveloped schemes for the spatial discretization in CAA is given. chapter 4 continues withan detailed analysis of the the numerical errors introduced by the discretization processand a new way of minimizing these is proposed. A comparison of two different types ofboundary conditions to be used for practical applications in CAA is given in chapter 5.The far field transport methods are discussed in chapter 6, where a comparison is donebetween a computational transport method based on the Linearized Euler Equationsand an analytical transport method based on the Ffwocs Williams Hawkings equation.The results of the simulation of a near field calculation with Large Eddy Simulation arereported in chapter 7 where also initial results of the coupled far field noise calculation aregiven. Finally the main conclusions and future challenges are given in chapter 8. Everychapter is introduced with a literature survey.

5

Chapter 2

Basic concepts of Aero Acoustics

2.1 The Wave Equation

It is hard to give an exact definition of what sound actually is. From a physical pointof view, sound can be defined as the mechanical radiant energy that is transmitted bypressure waves in a material medium (as air) and which is perceived by the sense ofhearing. However, this leaves us with the question of what waves are. It is easy to give anexample of wave motion but it is difficult to give a definition without exceptions. In thebook on acoustics by Blackstock [12] a few observations about wave motions are given:

• A wave is a disturbance or deviation from a pre-existing condition. The motionof the disturbance constitutes a transfer of information from one point in spaceto another. Frequently the pre-existing condition is complete quiet, such as staticequilibrium.

• Time plays a vital role. Static displacement of a string or rubber membrane isindeed a disturbance, but it is not a wave. Moreover, a wave travels with finitespeed. If one end of a perfectly rigid rod is hit by a hammer, all points along therod feel the blow instantaneously and the rod moves as a unit. No wave motion hasoccurred, only rigid body translation.

• All mechanical waves travel in a material medium. Sound, for example, needs afluid or a solid. It cannot travel through a vacuum.

• Gross movement of the medium is not a necessary part of the process of wave motion.A ripple on the surface of a stream may travel upstream or downstream, but theflow of the stream is not a requisite for the propagation even though it may modifythe apparent propagation speed.

• The disturbance need not be oscillatory in order to be a wave. Frequently wavesare thought of having sinusoidal shape, but spikes, rectangular pulses and noise areall perfectly valid waveforms for waves.

6

An idealization of many types of wave motion is embodied mathematically in what iscalled the wave equation:

∂2u

∂t2− c24u = 0 (2.1)

where 4 is the laplacian, u is a physical property associated with the disturbance of thesignal and c is a constant representing the speed at which the wave travels. It can be saidthat all solutions of Equation 2.1 represent waves, but not all waves obey this equation.More complicated forms of the wave equations exist which also describe wave motion.Equation 2.1 can be written generally in spherical coordinates:

∂2u

∂t2− c2

(∂2u

∂r2− a

r

∂u

∂r

)= 0 (2.2)

where a is a constant which depending on its value describes three types of waves. a = 0, 1and 2 correspond respectively to plane, cylindrical and spherical waves. General solutionsto wave equation can be found for plane and spherical waves. For the plane wave, it is clearthat any function of the form f(r − ct) satisfies Equation 2.2 and corresponds to a wavesteadily translated along the r direction. Similar it can also be proven that any functionof the form g(r + ct) satisfies the plane wave equation. The latter solution corresponds toa backward travelling wave. Because the wave equation is linear, superposition may beused. The most general solution to the plane wave equation is therefore:

u = f(x − ct) + g(x + ct) (2.3)

Along the same lines it can be shown that the most general solution for spherical wavesis given by:

u =f(r − ct)

r+

g(r + ct)

r(2.4)

The first term in Equation 2.4 represents an outgoing, spherically diverging wave of whichthe amplitude diminishes as 1

r. The second term corresponds to an incoming spherically

converging wave of which the wave amplitude increases. A general solution for cylindricalwaves does not exist and a further discussion of these types of waves can be found in [12].

2.2 Sound Waves

In the field of aero acoustics one is interested in sound waves occurring in fluida (suchas air). The most general partial differential equations describing the flow of fluids arethe Navier-Stokes equations, which describe the conservation of mass, momentum andenergy:

∂ρ

∂t+

∂

∂xi

(ρui) = 0 (2.5)

∂

∂t(ρui) +

∂

∂xj

(ρuiuj − pδij + τij) = ρfi (2.6)

7

∂

∂t(ρE) +

∂

∂xj

(ρujE) =∂

∂xj

(k

∂T

∂xj

)+

∂

∂xj

(τijui) + ρfiui + qH (2.7)

where ui are the velocity components of the fluidum, ρ the density, p the pressure, τ arethe internal forces of the fluidum due to viscosity, fi are external forces acting on thefluidum (e.g. gravity), T the temperature, E the total energy and qH the heat source.

To simplify the discussion only the one dimensional conservation equations for loss-lessnon dissipative fluids will be further considered.

∂ρ

∂t+

∂

∂x(ρu) = 0 (2.8)

∂

∂t(ρu) +

∂

∂x

(ρu2 − p

)= 0 (2.9)

Since it is assumed that no losses occur no separate equation for energy is neededand only an equation of state, giving the relation between the thermodynamic variablesis considered. For a perfect gas the equation of state is:

p = ρRT (2.10)

where R is the universal gas constant (for air R = 287J/kgK). The speed of sound isdefined as:

c2 ≡ ∂p

∂ρ

∣∣∣∣s

(2.11)

where derivative is taken at constant entropy s. In the absence of sound the fluid isassumed to be quiet and this state of reference is given by the values p = p0, ρ =ρ0 and u = u0 for pressure, density and velocity. This reference state is assumed tobe independent on spatial coordinates. Because sound pressure waves are disturbancesof the reference state which are several orders of magnitude smaller in amplitude thanthe reference pressure, the non linear equations 2.8 and 2.9 can be linearized using thefollowing definitions for the disturbances of the flow variables:

ρ′ = ρ − ρ0

u′ = u − u0

p′ = p − p0

(2.12)

Introducing these variable in the conservation of mass equation 2.8 and dropping thenonlinear terms, which is justified by the assumption that the disturbances are small, thefollowing linear form of the mass equation is obtained:

∂ρ′

∂t+ ρ0

∂u′

∂x= 0 (2.13)

The momentum equation can be linearized accordingly:

ρ0∂u′

∂t+

∂p′

∂x= 0 (2.14)

8

and the isentropic equation of state becomes:

p′ = c20ρ

′ (2.15)

In the next step the density disturbance ρ′ is eliminated between equations 2.8 and 2.15:

∂p′

∂t+ ρ0c

20

∂u′

∂x= 0 (2.16)

and finally by taking the time derivative of equation 2.16 and subtracting this from thelinearized momentum equation 2.14 the linear wave equation for sound waves is obtained:

∂2p′

∂t2− c2

0

∂2p′

∂x2(2.17)

This manipulation of the mass and momentum equations to obtain the linear wave equa-tion, shows that the process of sound propagation is inherently present in the Navier-Stokes equations. Although several assumptions were made to simplify things, the samecan be done without simplifications which will lead to the the famous Acoustic Analogyof Lighthill [81].

2.3 Definitions

Some common quantities used in acoustics are described in this section. The human eardoes not experience loudness in a linear way. If the amplitude of the pressure fluctuationsare doubled, we will not experience it as a doubling of loudness. Roughly our hearingexperiences loudness in a logarithmic scale. This is why most of the noise levels are ex-pressed in units based on logarithmic scales. The Sound Pressure Level (SPL) is measuredin decibels and defined by:

SPL = 20 log10

(p′rms

pref

)(2.18)

where pref = 2 10−5 Pa, which is the threshold of human hearing at 1 kHz and p′rms theroot mean squared pressure fluctuations:

p′2rms =1

T

∫ T

0

p2 dt (2.19)

where T is the time period over which the pressure is averaged. The time averaged energyflux of the acoustic wave is called the sound intensity I (W/m2). The sound intensitylevel (IL) is also measured in decibels and is defined by:

IL = 10 log10

(I

Iref

)(2.20)

where Iref is the reference intensity corresponding to the reference pressure pref and equals10−12Wm−2.

9

The time averaged power P (W ) generated in a volume of space, is the flux integral ofthe intensity I over the surface enclosing the considered volume. The Sound Power Level(PWL) is defined as:

PWL = 10 log10

(P

Pref

)(2.21)

where Pref = 10−12W .

10

Chapter 3

Compact Upwind Spatialdiscretization for use in CAA

3.1 Introduction

Over the past decade certain fields in computational fluid dynamics like Direct Numer-ical Simulation (DNS), Large Eddy Simulation (LES) or Computational Aero Acoustics(CAA), became more feasible thanks to the exponential increase of the computer tech-nology. All these fields require tremendous amounts of computational power because oftheir need for high spatial resolution to resolve the fine scales of the flow; whether it is tocapture the turbulence or the small sound producing fluctuations of the flow. Thereforemost of the applications in these fields resort to schemes of higher order accuracy forthe spatial discretization. Classical explicit methods require large stencils to meet thisrequirement. As an attractive alternative to very accurate spectral methods there hasbeen an increasing interest in high order finite difference and finite volume schemes. Foracademic applications, which are usually flows in simple domains and with simple bound-ary conditions, spectral methods are a very natural and most appealing approach. Strongdependence upon the grid geometry and type of boundary conditions, however, makesthem less attractive for problems of practical interest where one is usually confrontedwith complex geometries. In the light of all this, compact (or Pade type) high orderschemes have recently become a subject of intensive research [42, 67, 77]. These schemes,as was shown in [77], combine a high order of approximation with a compact stencil.More importantly, the resolution of the shorter length scales of the solution is better thanfor classical finite difference methods, which brings them closer to spectral methods andmakes them especially attractive for above mentioned DNS, LES and CAA applications.A finite volume(FV) formulation for these type of schemes has been developed in a seriesof papers [75, 122, 124].

However as these schemes are purely central, the use of some form of artificial dissipa-tion [127] or some filtering procedure [139] will be required for higher Reynolds numberflows to guarantee stability. The inclusion of artificial selective damping has been inves-tigated by Smirnov et al. [121] and Broeckhoven et al. [21]. An alternative, which isdescribed in the present chapter, is to use compact upwind schemes [110, 111]. They can

11

be incorporated in a FV formulation in the same way as the compact central schemes,using an implicit interpolation of the cell-face averaged variables. Other approaches usingupwind compact schemes can be found in [30, 33, 40, 84, 95, 112, 117].

3.2 Finite Volume Formulation

It is important that the compact schemes can be used in multi-dimensions and that theiraccuracy on general, i.e. non-cartesian and non-uniform grids, remains adequate. In afinite difference context [42, 46, 138, 141] this is usually dealt with by formulating thecompact schemes in computational space. However the use of the Jacobian transformationcan lead to an important reduction of the accuracy of the scheme in case of non-smoothlyvarying mesh spacing because of the numerical errors in the determination of the deriva-tives of the transformation appearing in the Jacobians. Although in the framework ofthe finite difference approach the compact schemes are relatively easy to construct onirregular grids, special attention must be paid to the conservation of the scheme as it isnot automatically guaranteed. The finite volume method on the other hand is inherentlyconservative but since one works immediately in physical space the formulation of com-pact schemes is less straightforward. A formulation of a compact scheme within the FiniteVolume context was proposed in [41, 70] but both papers only deal with linear equationsand uniform cartesian grids. In an attempt to formulate a higher order accurate FVformulation on arbitrary grids another route was followed by Smirnov et al. [122]. Asimilar approach was independently developed by Kobayashi and co-workers [101] but theproposed procedure is less general than the one developed by Smirnov et al. [122] as itis restricted to cartesian grids. The approach of Smirnov et al. [122] was here extendedtowards Compact Upwind Schemes [110, 111].

A brief overview of the FV compact schemes developed by Smirnov et al. [122] is givenhere before going further with the description of the FV compact upwind schemes.

Let us consider a finite volume approximation of a 1D advection equation

∂u

∂t+

∂f(u)

∂x= 0 (3.1)

In the FV approach equation (3.1) is integrated over the mesh cells and for a cell i, withcell faces i − 1/2 and i + 1/2, one obtains:

∂

∂t

∫ xi+1/2

xi−1/2

udx + fi+1/2 − fi−1/2 = 0 (3.2)

This is still an exact relation and to discretize it, both the integral containing the timederivative and the fluxes on the interfaces x = xi+1/2 and x = xi−1/2 must be approxi-mated. Before doing so, one has to choose whether point-wise (defined, for example, inthe center of the cell) or cell-averaged values of u

ui =1

∆x

∫ xi+1/2

xi−1/2

u(x)dx (3.3)

12

are used. In the ”point-wise” approach one has to use a complex high order quadratureformula for the calculation of the integral containing the time derivative in order tomaintain the high order accuracy for unsteady calculations. If one is interested only in asteady state solution, the formula for calculating the integral in question does not haveto be high order accurate, as the time derivative vanishes at the steady state. But forunsteady problems the ”cell-averaged” approach is preferable, as it allows to calculate theintegral exactly. Equation (3.2) becomes

∂ui

∂t∆x + fi+1/2 − fi−1/2 = 0 (3.4)

As our major concern is unsteady calculations, the cell-averaged approach will be assumedin the remainder of this chapter.

To evaluate the fluxes f(xi+1/2) and f(xi−1/2) a so-called variable averaging approachis used. One first evaluates the values of u on the interfaces (ui−1/2, ui+1/2) by means ofan interpolation formula and then calculate the fluxes as

fi+1/2 = f(ui+1/2

)fi−1/2 = f

(ui−1/2

)(3.5)

The general form of the interpolation formula for the cell face values is

αui−3/2 + βui−1/2 + ui+1/2 + γui+3/2 + δui+5/2 (3.6)

= aui−2 + bui−1 + cui + dui+1 + eui+2 + fui+3

3.3 Compact Upwind Schemes

In the general case three different types of upwind-biased interpolations can be distin-guished: the implicit (left-hand side) part purely central and the explicit (right-handside) part upwind-biased (ICEU); implicit upwind-biased and explicit central (IUEC);both implicit and explicit part upwind-biased (IUEU). Only the properties of the interiorstencil are discussed in this chapter. One has to be careful with the boundary and near-boundary stencils because they can make the total scheme unstable. For the test cases inthis chapter only periodic boundaries were used. In more practical cases where e.g. inletor outlet boundaries are used, special upwind stencils near the boundaries will be needed.A detailed analysis of the stability of the boundary stencils, is given in chapter 5.

3.3.1 ICEU-schemes

The ICEU schemes are the first class of schemes considered. The left-hand-side (lhs)of the interpolation is kept purely central, while in the right-hand-side (rhs) an upwindformulation is used. The general form of these schemes is:

βui−3/2 +αui−1/2 +ui+1/2 +αui+3/2 +βui+5/2 = aui−3 + bui−2 + cui−1 + dui + eui+1 (3.7)

13

Shown below in Table 3.1, these type of schemes are indicated by ICxEUyz. Here x is thenumber of points in the lhs and y the number of points in the rhs. z indicates the positionof ui. The coefficients are determined to obtain the highest possible order of accuracy.

Table 3.1 lists the coefficients and the order of the different investigated schemes.Other possible schemes, like IC3EU21 and IC3EU32, are not listed in the table becausethey reduce to purely central schemes. Figure 3.1 shows the real and imaginary part of themodified wave number for the schemes listed in Table 3.1. For fully upwind schemes, i.e.with points only left of the cell face to which u is being interpolated, the modified wavenumber overshoots the exact wave number. Only the scheme IC3EU22 is an exceptionto this. The upwind-biased scheme IC3EU43 shows to have better dispersion propertiesthan the the fully upwind scheme IC3EU44. At the same time the upwind-biased schemeis less dissipative. Another thing to notice is that switching to a pentadiagonal system(IC5EU33) does not improve the properties of the scheme. In fact, comparing IC3EU33to IC5EU33 shows that the dispersion properties in the mid wave number range of the5th order IC5EU33 scheme are worse than those of the 4th order IC3EU33 scheme. Asfor the diffusion accuracy, there is almost no improvement in the low wave number range.Considering the amount of extra computational cost a pentadiagonal system requires, theIC5EU33 scheme will not be considered any further. By far the most interesting scheme inthis class of schemes is the upwind-biased scheme IC3EU43. It combines good dispersiveaccuracy with some damping in the higher wave number range.

Scheme order β α a b c d e

IC3EU22 3 0 −15

0 0 − 310

910

0

IC3EU33 4 0 −18

0 18

−58

54

0

IC3EU43 5 0 − 314

0 184

− 584

6784

1928

IC3EU44 5 0 − 314

− 19268

319804

−797804

1195804

0

IC5EU33 5 19354

− 53177

0 559

−2559

5059

0

Table 3.1: Overview of the different ICEU-schemes

3.3.2 IUEC-schemes

The IUEC-schemes use an upwind formulation for the implicit part of the interpolationand a central formulation for the explicit part. They are given by the following formula:

αui−7/2 + βui−5/2 + γui−3/2 + δui−1/2 + ui+1/2 + εui+3/2 = (3.8)

a (ui + ui+1) + b (ui−1 + ui+2) + c (ui−2 + ui+3)

and are labeled as IUxyECz. Here x is the number of points in the lhs and z the numberof points in the rhs. y indicates the position of ui+1/2. Only schemes with 3 and 5 points

14

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Phi

Re(

Phi

appr

)

ExactIC3EU22IC3EU33IC3EU43IC3EU44IC5EU33

,0 0.5 1 1.5 2 2.5 3

−5

−4

−3

−2

−1

0

1

Phi

Im(P

hiap

pr)

ExactIC3EU22IC3EU33IC3EU43IC3EU44IC5EU33

Figure 3.1: Dispersion and dissipation errors of ICEU schemes

in the lhs are considered because schemes with 2 points in the lhs result into explicitschemes and schemes with 4 points result in central schemes.

Other possible schemes, like IU54EC4 and IU54EC6, are not listed in the table becausethey reduce to purely central schemes. Figure 3.2 shows the real and imaginary part ofthe modified wave number for the schemes of Table 3.2. It is interesting to note thatthe IUEC schemes have, in contrast to ICEU (section 3.3.1) and IUEU (section 3.3.3)schemes, a decreasing dissipation for the higher wave numbers, with even no disspationat the Nyquist limit. The 1st order upwind scheme combined with a Runge-Kutta timeintegration, formulated by Sengupta [118] in the framework of flux-vector splitting, showsa similar behavior. Although such schemes are perfectly stable (the vanishing dissipationcorresponds to a dispersion error that completely filters the signal) they are not consideredhere, as the dispersive behavior of the IUEC schemes, even for pentadiagonal systems,is inferior to that of the ICEU and IUEU schemes. Moreover the need to resort toa pentadiagonal formulation increases the cost of these schemes considerably.For thesereasons none of the schemes above will be considered for further calculations.

Scheme order α β γ δ ε a b c

IU33EC2 3 0 0 17

−27

0 37

0 0

IU54EC2 5 0 − 1124

131

− 631

− 29124

4562

0 0

IU55EC2 5 29269

−146269

294269

−266269

0 90269

0 0

IU55EC4 6 145

− 215

1445

−1445

0 67135

− 7135

0

IU55EC6 7 3533

− 18533

42533

− 42533

0 7211230

− 9017995

19915990

Table 3.2: Overview of the different IUEC-schemes

15

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Phi

Re(

Phi

appr

)

ExactIU33EC2IU54EC2IU55EC2IU55EC4IU55EC6

,0 0.5 1 1.5 2 2.5 3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Phi

Im(P

hiap

pr)

ExactIU33EC2IU54EC2IU55EC2IU55EC4IU55EC6

Figure 3.2: Dispersion and dissipation errors of IUEC schemes

3.3.3 IUEU-schemes

αui−3/2 + βui−1/2 + ui+1/2 + γui+3/2 = aui−2 + bui−1 + cui + dui+1 (3.9)

These type of schemes are indicated by IUxyEUzw. Here x is the number of points inthe lhs and z the number of points in the rhs. y indicates the position of ui+1/2 and windicates the position of ui . Other possible combinations of schemes are not listed inthe table because they reduce to purely central schemes. Figure 3.3 shows the real andimaginary part of the modified wave number for the schemes of Table 3.3. The 3rd orderupwind biased scheme IU22EU21 has good dispersion properties although it is only 3rdorder accurate. The diffusion is quite high but nonetheless the scheme is interesting dueto its bidiagonal character in the implicit part which requires less computational time tosolve. An interesting fact to notice is that the 5th order scheme IE32EU32 has a betterdispersion accuracy than the 6th order compact central scheme IC3EC4. On the otherhand, the combination of an upwind biased implicit part and a fully upwind explicit part(like scheme IU32EU33) leads to schemes with positive imaginary parts of the modifiedwave number. Such schemes amplify the waves instead of damping them. They aretherefore unstable and will not be considered below.

Scheme order α β γ a b c d

IU22EU21 3 0 12

0 0 0 54

14

IU32EU32 5 0 12

16

0 118

1918

59

IU32EU33 5 0 11162

− 362

− 593

107186

413186

0

IU43EU21 5 − 157

819

1057

0 0 1 1119

Table 3.3: Overview of the different IUEU-schemes

16

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Phi

Re(

Phi

appr

)

ExactIU22EU21IU32EU32IU32EU33IU43EU21IC3EC4

,0 0.5 1 1.5 2 2.5 3

−4

−3

−2

−1

0

1

2

3

4

5

Phi

Im(P

hiap

pr)

ExactIU22EU21IU32EU32IU32EU33IU43EU21IC3EC4

Figure 3.3: Dispersion and dissipation errors of IUEU schemes

3.4 2D Euler equations

Now let us consider the 2D Euler equations written in the form

∂U

∂t+

∂Fx

∂x+

∂Fy

∂y= 0 (3.10)

where U is the set of conservative variables and Fx, Fy are components of the advectiveflux vector:

U =

ρρuρvρE

(3.11)

Fx =

ρuρu2 + pρuv(ρE + p)u

Fy =

ρvρuvρv2 + p(ρE + p)v

(3.12)

with ρ, u, v, p, E respectively density, x and y components of velocity, pressure, totalenergy per unit mass.

Consider the structured mesh of Figure 3.4. Integrating equation (3.10) over the celli,j and making use of the Gauss theorem, one can write

∫ ∫

Si,j

∂U

∂tdxdy +

∮

ABB′A′A

(−Fydx + Fxdy) = 0 (3.13)

As was described above, before discretizing this equation, one has to choose whetherpoint-wise or cell-averaged (V i,j ≡ 1

Si,j

∫ ∫Si,j

V (x, y)dxdy) values of primitive variables

V =

ρuvp

(3.14)

17

A’’

i,j

i,j+1

i+1,j+1

i,j-1

i+1,j-1

O

B

A

x

y

normal

i+1,j

ß

B’

A’

B’’

Figure 3.4:

will be used. In this chapter the cell-averaged approach is chosen, as it is preferable forunsteady calculations, as mentioned in section 3.2. Let us now consider the discretizationof the line integrals for the convective fluxes in (3.13) . As all the integrals are approxi-

mated in the same manner, we will only consider∫ B

AFxdy. Instead of passing via cell face

values of the primitive variables, the line integrals itself are directly calculated [101, 122].Although the fluxes are now non-linear functions of the unknowns, the first step is similaras for the linear case.

For the IU22EU21 scheme the interpolation formula for the integral of the variablesalong the cell face is

βi−1/2Vi−1/2,j + Vi+1/2,j =1∑

n=0

1∑

m=−1

an,mV i+n,j+m (3.15)

where

Vi−1/2,j =1

yB′ − yA′

∫ B′

A′

V dy (3.16)

Vi+1/2,j =1

yB − yA

∫ B

A

V dy

Vi+3/2,j =1

yB′′ − yA′′

∫ B′′

A′′

V dy

The second step of the method consists in evaluating the integrals of the fluxes which,in contrast to the scalar convective case, contain non-linear terms. This can be done

18

using an approximation method. However, this approximation introduces an error ofsecond order and therefore a reconstruction needs to be done to maintain its formalorder of accuracy. More details on this can be found in the work of Smirnov et al.[21, 124, 122]. The discretization of the line integrals for the convective fluxes is based ona Flux Difference Splitting (FDS) scheme. Hereby the flux is calculated according to:

Fx =1

2(FL

x + FRx ) − 1

2|A|(UR − UL) (3.17)

where L and R indicate respectively an upwind/downwind quantity and A is the Roe-averaged Jacobian based on the upwind and downwind quantities. By using implicitupwind-biased interpolations as discussed above, upwind compact schemes are obtained.A way to control the amount of dissipation in a Roe FDS scheme was suggested in [22, 83].Equation (3.17) can be interpreted as the sum of a central scheme, 1

2(FL

x + FRx )), and a

dissipation term, −12|A|(UR − UL). With this interpretation the amount of dissipation

can be controlled using a parameter in front of the Roe dissipation term which rangesbetween 0 and 1. This leads to the formulation:

Fx =1

2(FL

x + FRx ) − ε

2|A|(UR − UL) (3.18)

When ε is put to zero, the scheme reduces to a central scheme (e.g. IU32EU32 reducesto the 6th order IC3EC4 scheme). For ε = 1, one has the fully upwind scheme.

3.5 Numerical Results

3.5.1 1D Advection Test

Uniform mesh

A series of numerical experiments has been carried out for the 1D linear advection problemin a domain −0.5 < x < 0.5 given by the equation:

∂u

∂t+ a

∂u

∂x= 0 (3.19)

A 1D Gaussian wave and a block wave were used as initial solutions and periodic boundaryconditions were imposed. For each test case the wave was propagated during 1 secondwith a speed of a = 1m/s. In all simulations a six stage low storage RK scheme was usedto advance in time. The RK coefficients were taken as 1/6, 1/5, 1/4, 1/3, 1/2, 1. Thisscheme is 6th order accurate in time for linear problems. Since only the spatial accuracywas to be examined, the time step was taken very small to make sure that the errorsoccurring due to time integration are negligible as compared to those arising from spatialdiscretization. Figures 3.5 and 3.6 show the result for the 2 upwind schemes as well asthe 4th order compact central scheme on a uniform mesh of 30 points. It clearly shows

19

that the dissipation for the IU22EU21 scheme is quite high, while this is not the case forthe IU32EU32 scheme. Comparison of the upwind schemes to the central scheme, showsthat for the central scheme the high wave number waves are not damped and lead tooscillations all over the domain, while for the upwind schemes the waves with the biggestdispersive error are damped out, leading to a more smooth solution.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.2

0

0.2

0.4

0.6

0.8

1

1.2

X

Am

plitu

de o

f sig

nal

original signalIC3EC3 IU22EU21 IU32EU32

Figure 3.5: 1D Gaussian wave with central and upwind compact schemes

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X

Am

plitu

de o

f sig

nal

original signalIC3EC3 IU22EU21 IU32EU32

Figure 3.6: 1D block wave with central and upwind compact schemes

As can be seen from Figure 3.6 both central and upwind schemes are unable to handlethe discontinuity. To resolve this problem, limiters are needed [54, 30, 112, 95], but thisis beyond the scope of the present thesis.

An investigation of the order of accuracy is provided in Figure 3.7. It shows the log ofthe error as a function of the log of the grid spacing. The measured order of accuracy is

20

indicated above each line. It is seen that each of the schemes reaches its theoretical orderof accuracy (being resp. 3,4, and 5 for IU22EU21, IC3EC2, IU32EU32).

−2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4−7

−6

−5

−4

−3

−2

−1

Log(dx)

Log(

erro

r)

IC3EC2 IU32EU32IU22EU21

O(2.9868)

O( 4.0137)

O(5.0151)

Figure 3.7: Order of accuracy

Non Uniform mesh

The influence of non uniformity on the performance of the schemes is investigated ina first step by the same 1D advection test but on a non uniform mesh. Only the 3rdorder compact upwind scheme IU22EU21 was adapted to take the metrics of the meshinto account. The same adaptation for the 5th order compact scheme IU32EU32 is alsopossible, but more cumbersome and has not been implemented so far.

The x coordinates of the domain were altered with a random function:

xi = xi + εφ∆xi (3.20)

where ε is a parameter controlling the amount of distortion and φ a random numberbetween 0 and 1.

The schemes were tested for 2 different values of ε. The first test case on a slightlydistorted mesh used a value of ε = 0.1 and the second test case used a heavily distortedmesh with ε = 0.4. An example of the distortion for ε = 0.4 is shown in Figure 3.8 whichshows ∆x for all the cells.

The influence of taking the metrics of the mesh into account is shown in Figure 3.9.Here the solution is shown for the 3rd order compact upwind scheme with constant andvariable coefficients on a mesh with 40 percent distortion (ε = 0.4).

Figure 3.10 show the error convergence plot on a mesh with 10 percent distortion(ε = 0.1) for the 3rd order compact upwind scheme IU22EU21 with constant coefficientsand with variable coefficients, the 4th order compact central scheme IC3EC2 with variablecoefficients (which is only 2nd order accurate on a non uniform mesh [75, 124, 122]) andthe 5th order compact upwind scheme IU32EU32 with constant coefficients.

21

0 50 100 150 200 250 300 350 400 450 5001.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3x 10

−3

i

dx

Figure 3.8: ∆x for a mesh with ε = 0.4

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

0

0.05

0.1

0.15

0.2

0.25

x

Sig

nal

InitialIU22EU21 con.coef.IU22EU21 var.coef.

Figure 3.9: 1D Gaussian wave;non uniform mesh ε = 0.4

The same calculations were done on a heavily distorted mesh with ε = 0.4. Figure3.11 shows the results.

Table 3.4 below shows the numerical order of accuracy for the different schemes.

It is clear from figures 3.10 and 3.11 and Table 3.4 that the schemes with constantcoefficients can not maintain their order of accuracy on neither of the non uniform meshes.The 3rd order compact scheme with variable coefficients has no problems to maintain its3rd order of accuracy and is very robust with respect to the amount of distortion of themesh. On the other hand the compact central scheme with variable coefficients, whichshould maintain 2nd order of accuracy [75, 122, 124] on non uniform meshes, seems tofail when the mesh gets heavily distorted (ε = 0.4).

22

−3.5 −3 −2.5 −2 −1.5 −1−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

Log(dx)

Log(

erro

r)

Random distorted mesh (10 percent)

IU22EU21 var coefIUU22EU21 const coefIC3EC2 var coefIU32EU32 const coef

Figure 3.10: Error 1D Gaussian wave; non uniform mesh ε = 0.1

−3.5 −3 −2.5 −2 −1.5 −1−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

Log(dx)

Log(

erro

r)

Random distorted mesh (40 percent)

IU22EU21 var coefIUU22EU21 const coefIC3EC2 var coefIU32EU32 const coef

Figure 3.11: Error 1D Gaussian wave; non uniform mesh ε = 0.4

3.5.2 Rotating 2D Gaussian Pulse

A Gaussian wave rotating around the origin x = 0, y = 0 is described by equation

ut + (au)x + (bu)y = 0 (3.21)

with a = −2πy, b = 2πx (varying advection coefficients). This problem has been solvedwith the different compact schemes with and without flux correction. The grid used inall calculations was an ”O” type grid. The isolines of the solution after one rotation ofthe signal are presented in Figure 3.12 for the 3rd order compact scheme with variablecoefficients and flux correction.

23

Scheme ε = 0.1 ε = 0.4

IU22EU21 var. coef. 2.99 2.98

IU22EU21 con. coef. 1.14 0.90

IC3EC2 var. coef. 2.16 1.56

IU32EU32 con. coef. 0.90 1.01

Table 3.4: Order of accuracy

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Solution

Figure 3.12: Solution after one rotation

Figure 3.13 shows the same solution along the midline of the Gaussian Pulse.

When looking at the error convergence in Figure 3.14 it is noticed that there is nodifference between the 3rd order compact scheme with constant coefficients and variablecoefficients. Also the 4th order compact central scheme, which reduces to 2nd order onnon uniform meshes, shows to have a better accuracy than the 3rd order compact schemewith variable coefficients. This can be explained by the fact that when refining the mesh,the non uniformity becomes smaller and smaller and one ends up locally with an almostuniform mesh. The influence of the flux reconstruction is shown in Figure 3.15. Again,this does not make much difference and can also be explained by the fact that the meshbecomes uniform when refining.

24

0 0.5 1 1.5 2 2.5 3−0.05

0

0.05

0.1

0.15

0.2

0.25

initial solutioncalculated solution

Figure 3.13: Solution along the midline after one rotation

−2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

Log(dx)

Log(

erro

r)

Rotating Gauss pulse

IU22EU21 var coefIUU22EU21 const coefIC3EC2 var coefIU32EU32 const coef

Figure 3.14: Error convergence 2D Gaussian wave on a curvilinear mesh

3.5.3 Non Linear Acoustic Pulse

A typical test in CAA is the non-linear acoustic pulse. Here a pressure pulse is superposedon a uniform flow and results into two pressure waves travelling to the right and to the left.The domain used in this calculation stretches from −15 m < x < 15 m and the uniformvelocity is 34.74042m/s. The Euler equations were solved on a 1D Cartesian mesh of120 cells for the standard second order central scheme, the standard second order upwindscheme with Van Leer limiters, the 4th order compact central scheme IC3EC2 and the5th order compact upwind scheme IU32EU32 with ε = 0.1, 0.5, 1 . In all simulations afour stage low storage RK scheme was used to advance in time. The time step was chosensmall enough to eliminate time integration errors. The reference solution was created on

25

−2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3−4.2

−4

−3.8

−3.6

−3.4

−3.2

−3

−2.8

−2.6

−2.4

−2.2

Log(dx)

Log(

erro

r)

Rotating Gauss pulse flux correction

IU22EU21IUU22EU21 flux cor.IU22EU21 const.coef. flux cor.IU22EU21 const coef

Figure 3.15: Influence of flux reconstruction

a fine mesh of 480 cells with the 6th order compact scheme IC3EC4. Figure 3.16 showsthe right travelling pressure wave after 1.7482 10−2 s.

0 5 10 151.986

1.988

1.99

1.992

1.994

1.996

1.998

2

2.002

2.004

2.006x 10

5

x (m)

P (

Pa)

Reference solution2nd O cen.2nd O upwIC3EC2IU32EU32 eps=1

Figure 3.16: Non-linear acoustic pulse for central and upwind schemes

The figure shows clearly the superior spatial accuracy of the compact schemes com-pared to the standard schemes used in most CFD codes. The standard second orderscheme shows a significant dispersion error which is not damped out and results in oscil-lations. On the other hand, the second order upwind scheme shows no oscillations butthe pressure wave is far too much damped. The two compact schemes have a high enoughspatial resolution to adequately capture the pressure wave. Hereby the compact upwindscheme efficiently damps out the oscillations resulting from the dispersion errors in thehigh frequency range.

26

Figure 3.17 shows the influence of ε on the dissipation of the scheme. As noted beforereducing ε limits the amount of damping. For lower values of ε the pressure wave is lessdissipated.

0 5 10 151.986

1.988

1.99

1.992

1.994

1.996

1.998

2

2.002x 10

5

x (m)

P (

Pa)

Reference solutionIU32EU32 eps=0.1IU32EU32 eps=0.5IU32EU32 eps=1

Figure 3.17: Non-linear acoustic pulse for different ε

3.5.4 Subsonic Vortical Flow

The capabilty of the numerical schemes to accurately advect vortical structures is animportant issue in DNS and LES of turbulence. Therefore, as a problem chosen todemonstrate the accuracy of the proposed high-order compact method, a subsonic in-viscid vortical flow was chosen. In the present testcase a vortex is convected in otherwiseuniform flow with freestream velocity U∞ = 10m/s. The initial solution is imposed byprescribing a vortical structure centered around x = 0, y = 0 as follows

u = U∞ − Cy

R2exp(−r2/2) (3.22)

v =Cx

R2exp(−r2/2)

p = P∞ − ρC2

2R2exp(−r2), r =

√x2 + y2

R2

The vortex is propagated in x-direction with the advection speed U∞, the shape of thevortex being preserved. As a result, an analytical solution, with which the results ofnumerical simulations can be compared, is available. In all the calculations presentedhere, the values for C and R were chosen as follows

C = 2, R = 0.1 (3.23)

27

Periodic boundary conditions were imposed in streamwise direction. Figure 3.18 showsthe magnitude of vorticity along a centerline through the vortex after it has traveled withthe uniform flow for a distance of 1.5 meters. The used mesh is very coarse (30x30 cells).

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

16

18

20

X

|vor

ticity

|Initial solutionIC3EC2 IU32EU32 IU22EU21

Figure 3.18: 2D Euler vortex for central and upwind compact schemes

In the figure all Compact Upwind schemes are used with the parameter ε put to 1.It shows that the 4th order compact central scheme solution gets spoiled by the waveswith high dispersion errors because there is no damping. For both upwind schemes, thewaves with the highest dispersion errors are damped out. The 5th order upwind scheme,IU32EU32, performs well with only a small damping of the magnitude of the vorticity.The 3rd order upwind scheme IU22EU21 shows to have significant dissipation.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80

2

4

6

8

10

12

14

16

18

20

x

|vor

ticity

|

Initial IU32EU32 eps=0 IU32EU32 eps=0.1IU32EU32 eps=0.5IU32EU32 eps=1

Figure 3.19: 2D Euler vortex for central and upwind compact schemes

Figure 3.19 shows the same vortex problem for ε = 0, 0.1, 0.5 and ε = 1. It is clear

28

from the figure that for ε close to zero, the magnitude of the vortex is dissipated less thanfor values of ε close or equal to 1. With ε = 0 the scheme is reduced to the 6th ordercentral scheme which performs well but does not dissipate the high frequency dispersionerrors.

3.5.5 LES channel results

In a first step towards using compact upwind schemes in Large Eddy Simulation (seechapter 7), both upwind schemes, IU32EU32 and IU22EU21, were used in a standardLES calculation, namely the fully developed channel flow at Reynolds 3300. The aim ofthis calculation was to see the effect of the dissipation inherent to these schemes on thequality of LES predictions. For all the calculations ε was put to 1.

The dimensions of the channel are taken as 4πδ and 2πδ in streamwise and spanwisedirection, with δ the channel half-width as shown in Figure 3.20. The Reynolds number

y

z x

y

δx

zL = 2

L =4

L = 2

π δ

δ

π

Figure 3.20: Dimensions of channel flow test case

based on the wall shear velocity and the channel half-width is 180 :

Reτ =Uτδ

ν(3.24)

with the wall shear velocity defined as

Uτ =

√τw

ρ(3.25)

29

In the streamwise and spanwise directions of the channel, periodic boundary conditionsare imposed and a no-slip condition is imposed on the (adiabatic) walls. The flow is drivenby a constant streamwise pressure gradient introduced as a source term in the momentumand energy equations. Three different cases are considered:

Case Reτ Gridsize Scheme C

1 180 33 × 65 × 33 IU32EU32 0.012

2 180 33 × 65 × 33 IU32EU32 0.0

3 180 33 × 65 × 33 IU22EU21 0.0

4 180 33 × 65 × 33 IC3EC2 0.012

C is the constant used in the Smagorinsky Subgrid Scale Model (SGS) which is de-scribed in chapter 7. Figure 3.21 shows the non dimensionalized streamwise space averagedvelocity U+ = <U>

Uτand the turbulent intensities v′v′ (u′u′ is not shown, but shows similar

tendencies as v′v′). The reference data are DNS data of [66]. Comparing the differentturbulent intensity profiles of the upwind schemes to the central scheme, it is noticed thatboth compact upwind schemes are too dissipative. The IU32EU32 scheme damps lessthan the IU22EU21 scheme, as expected. The profile of the streamwise velocity is over-predicted by the upwind schemes; because they damp more the turbulence, the velocityprofile tends towards a laminar profile. This was also reported before in [43, 90]. Sincein the channel flow the small scale fluctuations play a crucial role, a too strong dampingof the upwind scheme in the high wave number range seems to explain the poor results,cf. [90] where good results could be obtained for flows dominated by low wave numbereffects (e.g. vortex shedding).

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

5

10

15

20

25

log(Y+)

U+

reference data MoinIU32EU32 with SGS IU32EU32 no SGS IU22EU21 no SGS IC3EC2 with SGS

,−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Y

<vp

vp>

reference data MoinIU32EU32 with SGS IU32EU32 no SGS IU22EU21 no SGS IC3EC2 with SGS

Figure 3.21: LES channel flow

In cases 2,3 the subgrid scale model was switched off to have less dissipation. Inthis case it is assumed that the dissipation of the scheme dissipates the energy in the

30

SGS, as in the MILES [43] approach. It is expected that by avoiding the dissipation ofthe Smagorinsky model, the results would improve. However, as can be observed on thefigures, there is no significant effect: the results with the IU32EU32 scheme with andwithout SGS model are very similar. The numerical dissipation of the compact upwindschemes overwhelms that of the Smagorinsky model.

In a further assessment of the upwind schemes the same channel flow was calculatedwith the scheme IU32EU32 and with a value of ε = 0.1 as suggested by [22, 83]. Figure3.22 shows the streamwise space averaged velocity and the turbulent intensities v′v′ (u′u′

is not shown, but shows similar tendencies as v′v′). The reference data are DNS data ofKim et al. [66].

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

5

10

15

20

25

log(Y+)

U+

reference data MoinIU32EU32 with eps=0.1IU32EU32 with eps=1IC3EC2

,−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Y

<vp

vp>

reference data MoinIU32EU32 with eps=0.1IU32EU32 with eps=1IC3EC2

Figure 3.22: LES channel flow, IU32EU32 with epsilon=0.1

It is clear from the figures that by lowering the value of ε the turbulence in the flowis less damped. The velocity profile for the IU32EU32 scheme with ε = 0.1 approachesthe DNS data of Kim et al. [66], although it is still a bit worse than that of the IC3EC2scheme. This is to be expected since there is still some damping present. When lookingat the turbulent intensities in Figure 3.22 (only space averaged results are shown, hencethe asymmetry) it is noticed that the level of turbulence has increased by lowering thevalue of ε.

This second test has shown that it might be possible to do LES calculations withcompact upwind schemes if the dissipation is sufficiently controlled. Also note that thepresent test case was especially sensitive to the dissipation because of its low Reynoldsnumber. It is expected that this effect will be less for higher Reynolds number cases.

3.6 Conclusions

A Finite Volume Formulation for Compact Upwind Scheme on arbitrary meshes withcontrollable upwind dissipation was thoroughly analyzed and implemented in the full3D Navier-Stokes Solver Euranus [74, 113]. From the different possible stencils of thecompact upwind schemes, 2 schemes were selected for in depth testing, namely the 3rd

31

order compact upwind scheme IU22EU21 for its low computational cost and robustnesson arbitrary meshes and the 5th order compact scheme IU32EU32 for its high accuracyon uniform meshes. The 1D advection equation on uniform meshes showed the goodperformance of the 5th order compact upwind scheme in damping out the high frequencyoscillations. The same test on non uniform meshes indicated the need to take the metricsof the mesh into account. The 3rd order compact scheme is very robust to distortionsof the mesh compared to the 4th order compact central scheme. The influence of fluxcorrection was studied with a rotating 2D gaussian pulse and revealed that on slight nonuniform meshes the error of the flux calculation can be neglected for compact upwindschemes. The compact upwind schemes in a FDS formulation for the Euler equations,implemented in the Navier-Stokes code Euranus [74, 113], were tested on a vortex in auniform flow. It showed again the good accuracy of the IU32EU32 scheme while theIU22EU21 scheme seems to be too dissipative when the full upwind dissipation is used.The non-linear acoustic pulse was calculated and showed the superior spatial accuracy ofcompact schemes. It also showed how the dissipation can be controlled by varying theparameter ε. LES results of a fully developed channel flow at Reτ = 180 were shown.Both of the upwind schemes seem too dissipative to get the correct turbulent intensitieswhen used with the full upwind part in the Roe formulation. By reducing the dissipationterm with a scaling factor, there was a significant improvement of the channel flow results.Although in the LES channel flow test case the central compact scheme performs best,Upwind Compact schemes might be preferable for LES of higher Reynolds number flowsor for flows with shocks. This however is still to be confirmed in further research.

32

Chapter 4

Optimized Time Integration Schemes

4.1 Introduction

In the field of Computational Aero Acoustics (CAA) one is interested in the predictionof sound far from its source. Several characteristics of the acoustic waves require thatspecial attention is paid to the spatial and temporal discretization schemes. First of all,acoustic waves are non-dispersive and non-dissipative in their propagation and thereforecan travel over long distances in all directions. Secondly, the amplitudes of acoustic wavesare several orders of magnitude smaller than normal aerodynamic perturbations. Thesecharacteristics put more stringent requirements on the numerical schemes used in CAA.

The last few years, much attention has been paid to the use of spatial discretizationsin CAA. The accuracy of the spatial discretization schemes needed in CAA is higher thanthose of classical CFD codes, where usually a second order discretization is sufficient.However, taking into account the acoustic properties of the sound waves in CAA, theactual order or the scheme is of less importance than its dispersive and dissipative behav-ior. This information, which is related to the resolution of the scheme (as opposed to theaccuracy) can be obtained from a Fourier mode analysis [5]. The resolution is then theability of the scheme to accurately represent Fourier modes of increasing wave number.The dispersion errors of the scheme cause an error in the phase of the Fourier waves,as compared to the exact solution from the differential equation, whereas the dissipationerrors cause an error in the amplitude of the Fourier waves. In CAA compact or Padetype schemes are often used in a Finite Difference formulation [77, 67]. Recently FiniteVolume formulations were developed, e.g. [76, 21], which are directly applicable on ar-bitrary structured meshes and [100] where a Jacobian transformation to Cartesian gridshas to be used.

Similar arguments apply to the temporal discretization. The acoustic waves have tobe tracked accurately in time. Time integration schemes can be optimized for differentproperties, such as linear and non linear stability, accuracy efficiency, error control reli-ability and dissipation and dispersion accuracy [65]. When one is especially concerned

33

about the dissipative and dispersive behavior of discretized equations, optimization ofdissipation and dispersion properties of the time integration scheme is of paramount im-portance. By far the most popular schemes in CAA are the Runge-Kutta schemes. Theyare explicit and can be formulated up to an arbitrary order of accuracy. Many authorsuse the classical Runge-Kutta schemes, whether or not in a low storage form to reducememory usage. Until now not many authors have considered optimizing time integrationschemes. A few articles mention optimized Runge-Kutta schemes with better dissipativeand dispersive behavior.

A commonly used optimized time integration scheme is the one by Hu et al. [62].Several Runge-Kutta schemes are optimized for their dispersive and dissipative behaviorand are baptized as low-dissipation and low-dispersion Runge-Kutta schemes (LDDRK).Hu et al. give the coefficients in [62] for a four, five and a six stage optimized Runge-Kutta scheme. All of these are kept second order accurate in time while minimizing thedispersion and dissipation errors. A second method used by Hu et al. [62] to optimizetime integration schemes is to use different coefficients for the Runge-Kutta schemes in twoalternating time steps. This leads to a further reduction of the dispersion and dissipationerrors while maintaining a higher order of accuracy. Two fourth order optimized schemesare described in [62], a 4-6 alternating scheme and a 5-6 alternating scheme.

In his extensive study on different numerical methods used in CAA Goodrich [49] con-cludes that the six stage optimized low dissipation and dispersion Runge-Kutta scheme byHu [62] in combination with at least a sixth order spatial differencing can provide betweenone and two orders of magnitude decrease in error at a given grid density. However thefourth order central and the DRP spatial differencing by Tam et al. [133] do not obtain anadvantage of the optimized scheme. Ewert et al. [36] use the fourth order alternating twostep low dissipation and dispersion Runge-Kutta scheme (LDDRK 5-6) formulated by Hu[62] in combination with the fourth order spatial DRP scheme of Tam et al. [133] in theirtwo step CFD/CAA approach. In the ALESIA code used by Bogey et al. [18, 19] the fourstage optimized scheme, described in [62], is applied. Also Morris et al. [93] use the fourstage optimized Runge-Kutta scheme to integrate the Non Linear Perturbation equationsin time. Recently, Bogey et al. present an optimization of Runge-Kutta schemes basedon similar ideas as in Hu et al. The integrals to be optimized are defined somewhat differ-ently leading to different schemes [17]. Another recent proposed optimized scheme is theone by Calvo et al. [23]. The scheme is a 2N-storage six-stage Runge-Kutta scheme. Itwas optimized both for its dissipative and dispersive properties, as well as for its stability.Tam et al. [133] are one of the few authors who optimized a four level time integrationscheme. The procedure is comparable to the one followed by Hu et al. [62].

In this chapter a new way of optimizing the temporal integration schemes is presented.The optimization processes found in the literature so far, took only the dissipation anddispersion errors introduced by the temporal discretization into account. However the per-formance of a code will depend on the total dispersion and dissipation errors introducedby spatial and temporal discretization. By taking into account the spatial discretizationduring the optimization process of the temporal scheme, the total dissipation and disper-sion error can be minimized instead of only the temporal error. Because the optimizationof the time integration scheme is coupled to the spatial discretization, it can also be ap-plied to upwind type schemes, as opposed to optimization procedures along the imaginary

34

axis.The outline of the chapter is as follows. Section 4.2 gives a short introduction in the

Fourier analysis of the errors introduced by the spatial and temporal discretization. Next,in section 4.3 the common optimization process and the newly proposed optimizationprocess are explained. Further on, in section 4.4, the results are shown for a series of testproblems. The main conclusions are given in the final section.

4.2 Dispersion and dissipation errors

Consider the scalar convection equation

∂u

∂t+ a

∂u

∂x= 0 (4.1)

and a Fourier waveu = F (t)eIkx (4.2)

Substitution of (4.2) in (4.1) gives the following equation for F:

dF

F= −Iakdt (4.3)

Solution of this equation givesF = e−Iakt (4.4)

and the Fourier wave of type (4.2) satisfying equation (4.1) exactly is given by

uex = eIk(x−at) (4.5)

4.2.1 Dispersion and dissipation errors of the spatial scheme

We consider again equation (4.1), but semi-discretized in the sense that a spatial dis-cretization scheme is used for the spatial derivative. Equation (4.1), written in point ibecomes:

∂ui

∂t= R(ui, ui−1, ui+1, ui−2, ui+2, ...) (4.6)

with R the discretization stencil of −a∂ui

∂x. The letter R is used to indicate that this is the

so-called residual. Depending on the scheme used, the value of u in point i, ui, as well asin surrounding cells (ui−1, ui+1, ui−2, ui+2, ...) may be used.

Substitution of the Fourier wave (4.2) leads to the following equation for F:

dF

F= Rdt (4.7)

where R is the so-called Fourier footprint of the residual, which can be decomposed in areal part, Rr, and an imaginary part, Ri:

R = Rr + IRi (4.8)

35

The solution of (4.7) is

F = eRrteIRit (4.9)

Combining equation (4.2) and equation (4.9), the solution of the semi-discretized equationis therefore

u = eRrteIk(x+Rik

t) (4.10)

Comparing equation (4.5) with (4.10) it can be observed that because of the discretiza-tion, two errors arise:

|rspace|e−Iδspace =eRrteIk(x+

Rik

t)

eIk(x−at)(4.11)

• an error on the amplitude of the wave: the exact amplitude is 1 whereas, with spatialdiscretization, an amplitude |rspace| = eRrt is obtained. This is the dissipation error.

• an error on the propagation speed of the wave: the exact speed is ’a’ (for all wavenumbers ’k’) whereas, with spatial discretization, the wave speed depends on thewave number and is given by δspace = −Ri

k. This is the dispersion error.

Some authors define a numerical wave number k∗ as

R ≡ −Iak∗ (4.12)

The solution of the semi-discretized equation (4.9) becomes:

u = eIk(x− k∗

kat) (4.13)

The difference between numerical and actual wave speed describes the error. The realpart of k∗ corresponds to dispersion errors, whereas the imaginary part corresponds to adissipation error.

As an example, consider the second order central scheme:

∂ui

∂t+ a

ui+1 − ui−1

2∆x= 0 (4.14)

The Fourier footprint is given by:

R = −Iasin(k∆x)

∆x(4.15)

The Fourier footprint is purely imaginary and hence the central scheme does not introducea dissipation error. The dispersion error, Φdisp,defined as the ratio of numerical wave speedand exact wave speed, is given by:

Φdisp =sin(k∆x)

k∆x(4.16)

36

4.2.2 Dispersion and dissipation errors of temporal scheme

We consider the semi-discretized equation (4.6) and introduce a temporal discretization.To fix thoughts, consider the Euler explicit scheme:

un+1i − un

i

∆t= R(un

i , uni−1, u

ni+1, u

ni−2, u

ni+2, ...) (4.17)

where the superscript indicates the time level of the solution. Substitution of the Fourierwave (4.2) gives

F n+1 − F n

∆t= RF n (4.18)

andF n+1

F n= 1 + ∆tR (4.19)

To study the stability of the temporal discretization it suffices to check that∣∣∣F n+1

F n

∣∣∣ ≤ 1.

For a study of dispersion and dissipation properties, the correct evolution of F in timehas to be considered. This is given by equation (4.9) and hence

(F n+1

F n

)

ex

= eR∆t (4.20)

Comparison of equations (4.19) and (4.20) shows that, for a given spatial discretizationscheme, the temporal discretization in general introduces both an error in the module ofF (the dissipation error) and in the phase of F (the dispersion error).

This analysis is easily extendable to other discretization schemes. E.g. for ExplicitRunge-Kutta schemes equation (4.19) becomes:

F n+1

F n= G(R∆t) (4.21)

where G is a polynomial of degree q, with q the number of Runge-Kutta stages. Thispolynomial is an approximation of the exponential function eR∆t

The numerical error introduced by the temporal discretization is given by comparingG(R∆t) with eR∆t:

G(R∆t)

eR∆t= |r|e−Iδ (4.22)

where |r| is the dissipation error introduced by the temporal discretization and δ thedispersion error.

Figure 4.1 shows the temporal dissipation and dispersion error for a standard 4 stageRunge-Kutta scheme (RK4) with a fourth order Finite Volume Compact Scheme for thespatial discretization [21] at a CFL number of 1.5. The figure shows the residual footprint(R) of the spatial discretization, the stability limit of the discretization (|G(R∆t)| = 1)and the contour lines of the temporal dissipation and dispersion error.

By plotting the temporal error, given in Figure 4.1, along the imaginary axis, thedissipation and dispersion error are found in function of the wave number (for centralschemes only). These one dimensional plots are often used in other articles such as that

37

−4 −3 −2 −1 0 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Real

Imag

inar

y

|rtime

|

0.4

0.4

0.6

0.6

0.8

0.8

0.8

11

1

11

1

1

1

1.2

1.2

1.2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Residual footprintStability zone (|G|=1)

,−4 −3 −2 −1 0 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Real

Imag

inar

y

δtime

−2−2

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1

−1

−1

−1

−1

−0.6

−0.6

−0.6

−0.6

−0.6−0.6

−0.6

−0.6

−0.2 −0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

0

0

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Residual footprintStability zone (|G|=1)

Figure 4.1: |r| (left) and δ (right) for the temporal discretization with a RK4 scheme,

CFL=1.5

of Hu et al. [62]. The plots of Figure 4.1 however also give some insight in the errorbehavior of the upwind schemes. A time integration scheme with low dissipation anddispersion errors is found when the Fourier footprint of the spatial discretization is in theregions of isolines |r| = 1 in the left plot and in the region of δ = 0 in the right plot.

4.2.3 Dispersion and dissipation errors of total scheme

The total dissipation and dispersion error introduced by discretizing equation (4.1) intime and space, is obtained by comparing G(R∆t) with e−Iak∆t. From equation (4.23) itis easily seen that the total dissipation error is the product of the spatial dissipation errorwith the temporal dissipation error, while the total dispersion error is the sum of both.

G(R∆t)

e−Iak∆t=

G(R∆t)

eR∆t

eR∆t

e−Iak∆t= (|rspace||rtime|)e−I(δspace+δtime) = |rtotal|e−Iδtotal (4.23)

Figure 4.2 shows the total dissipation and dispersion errors plotted as a function of k∆xfor a standard 4 stage Runge-Kutta scheme (RK4) with a fourth order Finite VolumeCentral Compact Scheme for the spatial discretization [21] at a CFL number of 1.5.

Since the spatial discretization is done with a central scheme, only the time discretiza-tion is responsible for the dissipation error.

38

0 0.5 1 1.5 2 2.5 30.4

0.5

0.6

0.7

0.8

0.9

1

1.1Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−2

−1

0

1

2

3

4

5Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.2: Dissipation and Dispersion Errors, CFL=1.5

4.3 Optimization

In the present chapter the optimization of time advancing schemes, in particular theRunge-Kutta schemes, is considered. The following initial value problem is considered:

dUd t

= F (t,U (t)) U (t = 0) = U0 (4.24)

with U the vector of unknowns. The most general q-stage explicit Runge-Kutta scheme,[65], to integrate from time level tn to time level tn+1, can be written as:

U(i) = U(n) + ∆ti−1∑

j=1

aijF(j) (4.25)

U(n+1) = U(n) + ∆t

q∑

j=1

bjF(j) (4.26)

where:

F(i) = F(t(i),U(i)

)(4.27)

t(i) = t(n) + ci∆t (4.28)

ci =s∑

j=1

aij (4.29)

and the stage number i runs from 1 to q. The above formulation requires a lot of memorybecause at stage i all F(j) on the previous stages need to be known. The Runge-Kuttaschemes can be implemented in a form which requires less storage, given by:

U(i) = U(n) + ∆tαiF(i−1) (4.30)

U(n+1) = U(n) + ∆t

q∑

i=1

βiF(i) (4.31)

39

A classical Runge-Kutta scheme used in CFD applications is:

U(i) = U(n) + ∆tαiF(i−1) (4.32)

U(n+1) = U(n) + ∆tF(q) (4.33)

If q = 4 this scheme is fourth order accurate in time for linear advection problems withconstant speed. In case of non linearity this scheme has a maximum accuracy in timeof second order. The amplification factor of the Runge-Kutta scheme given by equation(4.32) is given by

G = 1 +

q∑

j=1

aj(R∆t)j (4.34)

Here the coefficients aj relate to αi (α1 = 0) by

a2 = αq

a3 = αqαq−1

...

aq = αqαq−1...α2 (4.35)

In CAA applications, where one is especially concerned about the dissipative anddispersive behavior of discretized equations, optimization of dissipation and dispersionproperties of the time integration scheme is of paramount importance.

A very commonly used optimized time integration scheme is the one by Hu et al., [62].Several Runge-Kutta schemes are optimized for their dispersion and dissipation behaviorand are baptized as low-dissipation and low-dispersion Runge-Kutta schemes (LDDRK).The optimization is achieved by minimizing the difference between the numerical ampli-fication factor, G(R∆t), and the exact temporal amplification factor, eR∆t, as explainedin section 4.2.2.

The following integral is therefore minimized:

∫ Γ

0

|G(−Iσ) − e−Iσ|2dσ (4.36)

with σ = IR∆t and Γ specifies the range of σ in the optimization. However Hu etal. do not give an indication which value Γ is given during the optimization. It can beshown that the integral minimizes the sum of the dissipation and dispersion errors [62].The coefficients are given in [62] for a four, five and a six stage optimized Runge-Kuttascheme (LDDRK4, LDDRK5, LDDRK6). All of these are kept second order accurate intime while minimizing the dispersion and dissipation errors.

This optimization implicitly supposes that one deals with a central spatial discretiza-tion because one integrates over σ, thus assuming σ is a real number. Looking in thecomplex plane, the optimization will only be valid on the imaginary axis. Upwind typespatial discretizations have however complex modified wave numbers k∗.

Also, Hu et al. minimize only the temporal errors by minimizing the difference betweennumerical amplification factor, G(R∆t), and the exact temporal amplification factor,eR∆t. The maximum allowable CFL number is then calculated by imposing an accuracy

40

0 0.5 1 1.5 2 2.5 30.999

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.3: Dissipation and Dispersion Errors standard RK6, CFL=0.837

limit. For the fourth order Finite Volume Compact Central scheme [21] the maximumallowable CFL number according to Hu is 0.837. Figure 4.3 shows the standard six stageRunge-Kutta scheme used with a fourth order Finite Volume Compact Central scheme[21]. Looking at the total dissipation error, it is immediately seen that the scheme isunstable because the error is bigger than one for a certain wave number range. Theoptimized six stage Runge-Kutta scheme by Hu is shown in Figure 4.4. It is noticedthat the scheme is stable and that the dissipation error is minimized. Compared to thetotal dispersion error of the standard six stage Runge-Kutta scheme, little improvementis found as the largest contribution to the total dispersion error comes from the spatialdispersion errors, which are not optimized.

0 0.5 1 1.5 2 2.5 30.999

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.4: Dissipation and Dispersion Errors Hu optimized RK6, CFL=0.837

In the recent paper of Calvo et al. [23], a 2N-storage six-stage Runge-Kutta scheme isdeveloped to maximize the stability as well as to optimize its dissipative and dispersive

41

properties. The optimization is done by maximizing the continuous function 12S + 1

2L,

where S is the border of the stability zone:

S = maxδ ≥ 0; |G(Iν)| ≤ 1,∀ν ∈ [0, δ] (4.37)

and L is a measure associated to both dispersion φ(ν) = ν − arg(G) and dissipationd(ν) = 1 − |G|:

L = maxλ ≥ 0; |d(ν)| ≤ |φ(ν)| ≤ 1.25 × 10−3,∀ν ∈ [0, λ] (4.38)

with ν = −IR∆t. From equation (4.38) it is noted that also for this optimization onlythe temporal errors and not the total errors are minimized. This is illustrated in Figure(4.5) which shows the dissipation and dispersion errors of the scheme proposed by Calvoet al [23]. Since ν is assumed to be a real number, also here the optimization is donealong the imaginary axis and is thus only valid for central schemes. (4.5)

0 0.5 1 1.5 2 2.5 30.999

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.5: Dissipation and Dispersion Errors Calvo optimized RK6, CFL=0.837

In this chapter it is proposed to minimize the total dissipation and the total disper-sion error for an arbitrary scheme (central or upwind) by plugging the residual footprintin the integral and minimizing the difference between numerical amplification factor,G(R(k∆x)∆t), and the exact amplification factor, e−Iak∆t.

The following integral is therefore minimized:

∫ Γ

0

|G(R(k∆x)∆t) − e−Iak∆t|2d(k∆x) (4.39)

with the constraint for stability

|G(R(k∆x)∆t)| < 1 ∀k∆x ∈ [0, π] (4.40)

and Γ specifies the range of exact wave number k∆x in the optimization. The differencewith the optimization done by other authors is that the time discretization will be opti-mized locally in the residual footprint and not just for an arbitrary part of the imaginary

42

axis. This allows to optimize the time integration also for upwind type schemes. Theintegration limit Γ is also more physical because it is the exact wave number range forwhich the scheme will be optimized.

Since in the optimization process, it is not possible to impose the constraint in equation(4.40) for all k ∈ [0, π], it is only imposed for a discrete amount of wave numbers.

|G(R(k1)∆t)| < 1 − ε1, k1∆x ∈ [0, π]

|G(R(k2)∆t)| < 1 − ε2, k2∆x ∈ [0, π] (4.41)...

A minimum number of constraints is taken to force |G(−Iak∗∆t)| < 1. εi is a smallnumber which allows to control the maximum dissipation for wave number ki. This isnecessary sometimes to prevent overshoots of the dissipation error.

4.3.1 Optimizations for Central Spatial discretizations

Optimizations for the classical six stage Runge-Kutta scheme given by equation (4.32)have been performed for a fourth order Finite Volume Compact Central (FVCC) spatialdiscretization [21] at different CFL numbers. The first four coefficients were kept on theirdefault value (an = 1

n!) to keep the scheme fourth order accurate for linear problems

(second order accurate for non linear problems). In this optimization process only oneconstraint was used. Using more constraints with only 2 degrees of freedom (the 5th and6th coefficient of the Runge-Kutta scheme) leads to imaginary solutions for the coefficients.

In the first optimization, the temporal error was minimized and not the total error.This was done to check whether the optimization process yields similar results as otherauthors. The following integral was minimized:

∫ Γ

0

|G(R(k∆x)∆t) − eR(k∆x)∆t|2d(k∆x) (4.42)

Notice the difference with the integral of Hu et al. in equation (4.36); here the Fourierfootprint of the spatial discretization is still plugged in, but the integral minimizes thetemporal errors. The constraint was imposed for the wave number k1∆x = 2π

3which

corresponds with the maximum numerical wave number knum∆x =√

3 of the spatialdiscretization scheme (see Figure 4.6) . This wave number corresponds with the highestpoint of the residual footprint on the imaginary axis and therefore the highest totaldissipation error. ε1 was tuned until the total dissipation error was smaller than one andΓ was chosen to be equal to 2π

3.

Figure 4.7 shows the temporal dissipation and dispersion errors for the optimizationfor the temporal errors together with those of the schemes proposed by Hu et al. [62]and Calvo et al. [23]. The figure shows that the optimization process gives similarresults as the scheme proposed by Hu et al., with a better behavior for the dispersionerrors. Comparing the new scheme with the scheme proposed by Calvo et al. [23], it isnoted that the latter shows a better behavior for the temporal dissipation errors, but aconsiderably worse behavior for the dispersion errors.

43

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

k ∆ x

k num

∆ x

ExactFVCC O(4)

(2 π /3 , sqrt(3) )

Figure 4.6: Resolution of the fourth order FVCC

0 0.5 1 1.5 2 2.5 30.999

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003Temporal Dissipation Error

k ∆ x

|r|

OptimizedHuStandardCalvo

,0 0.5 1 1.5 2 2.5 3

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3 Temporal Dispersion Error

k ∆ x

δ

OptimizedHuStandardCalvo

Figure 4.7: Temporal dissipation and dispersion Errors

However when looking at the total errors instead of the temporal errors of the opti-mized scheme, see Figure 4.8, the optimization has an unnoticeable influence on the totaldispersion error. This is due to the fact that the dispersion error introduced by the spatialdiscretization is several magnitudes bigger than that of the temporal discretization error,hence the influence of the optimization can be neglected. For the dissipation error thetemporal error equals the total error because the spatial discretization does not introduceany dissipation error. This shows the importance of optimizing for the total error insteadof the temporal error.

Thus an optimization was done where the total errors of the six stage Runge-Kuttatime integration scheme with fourth order FVCC scheme [21], were minimized with theintegral given in equation (4.39). The constraint was imposed for the wave number k1∆x =2π3

. ε1 was tuned until the total dissipation error was smaller than one and Γ was chosento be equal to 2π

3. Table 4.1 gives the optimized coefficients a5 and a6 at different CFL

numbers.

44

0 0.5 1 1.5 2 2.5 30.999

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−1

0

1

2

3

4

5x 10

−3 Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.8: Total dissipation and dispersion Errors

Beyond a CFL number of 2.1 no more stable schemes could be found. A curve fittingwas done on the coefficients. This allows the scheme to be implemented without havingto do each time an optimization. Figure 4.9 shows the coefficients as a function of theCFL number.

0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

CFL

a 5

a5

Curve fit

,0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

CFL

a 6

a6

Curve fit

Figure 4.9: a5 and a6 for the optimized RK6 scheme with Compact Central spatial dis-

cretization

Figure 4.10 and Figure 4.11 shows the temporal resp. total errors for the RK6 schemeoptimized for a fourth order FVCC spatial discretization [21] at a CFL of 0.837. Theoptimization stabilized the scheme which is obvious from the total dissipation error (Fig-ure 4.11). When comparing the dispersion error of this optimized scheme (Figure 4.12shows a close-up) to the one of Hu in Figure 4.4 or to the one of Calvo et al. in Figure4.5, it is clearly seen that the total dispersion error is better. This is due to the factthat Hu and Calvo only optimized for the temporal error, and did not take into accountthe spatial error. However, the dissipation error of this optimization is worse than the

45

CFL a5 a6

0.5 0.156062 0.228422

0.6 0.0611325 0.104581

0.7 0.0255565 0.0528761

0.8 0.0113002 0.0286611

0.9 0.00559034 0.01638

1 0.00352957 0.00979459

1.1 0.0030558 0.00612248

1.2 0.00324681 0.00401757

1.3 0.00367836 0.00278719

1.4 0.00415627 0.00205826

1.5 0.00459567 0.00162286

1.6 0.0049638 0.00136202

1.7 0.00525207 0.00120613

1.8 0.00546249 0.00111362

1.9 0.00560131 0.00105914

2.0 0.00567624 0.00102689

2.1 0.00569541 0.00100682

Table 4.1: Optimized coefficients for six stage RK

optimized schemes of Hu and Calvo. The reason is that the optimization tries to com-pensate the large spatial dispersion errors with the temporal dispersion errors to get goodtotal dispersive properties. However, in doing so, the total dissipation errors deteriorate.Still the total error integral given by equation (4.39) is lower for the optimized six stageRunge-Kutta discussed above, than for that of Hu et al. and Calvo et al.

The optimization process will search for the most optimal scheme for a specific CFLnumber. Therefore stable schemes for higher CFL numbers can be found. In realisticCAA applications however, waves will travel at different speeds and therefore locally theCFL number can be lower then the maximum CFL number for which the scheme was

46

−4 −3 −2 −1 0 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Real

Imag

inar

y

|rtime

|

0.4

0.4

0.6

0.6

0.8

0.8

1

1

1

1

11

1

1

1.2

1.2

1.2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Residual footprintStability zone (|G|=1)

,−4 −3 −2 −1 0 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Real

Imag

inar

y

δtime

−2

−2

−2

−2

−2

−2

−2−2

−2

−2

−1−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−0.6 −0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.2 −0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

0 0 0

0

0

0

0

0

0

0

0

00

0

0

0

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

1

1

1

1

1

1

11

1

2

2

2

2

2

2

2

2

2

2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Residual footprintStability zone (|G|=1)

Figure 4.10: Temporal dissipation and dispersion error for optimized RK6 with fourth

order Compact Central scheme, CFL=0.837

0 0.5 1 1.5 2 2.5 30.988

0.99

0.992

0.994

0.996

0.998

1

1.002Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.11: Total dissipation and dispersion error for optimized RK6 with fourth order

Compact Central scheme, CFL=0.837

optimized. Figure 4.13 shows the six stage Runge-Kutta scheme optimized for a CFLnumber of 1.2. On the left figure the total dissipation errors are shown and it is notedthat for lower CFL numbers the dissipation errors converge monotonically towards one.The right figure shows the total and the spatial errors for the optimized Runge-Kuttascheme. For smaller CFL numbers the total dispersion error converge towards the spatialdispersion error. This is due to the fact that for smaller CFL numbers the temporalerrors converges faster to zero then the spatial errors and thus the spatial errors will bethe dominant term in the total dispersion errors. This will be true for any Runge-Kuttascheme, whether optimized or not.

47

0 0.5 1 1.5 2 2.5 3−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3 Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.12: Zoom of the Total dispersion error for optimized RK6 with fourth order

Compact Central scheme, CFL=0.837

0 0.5 1 1.5 2 2.5 30.94

0.95

0.96

0.97

0.98

0.99

1

1.01Total Dissipation Error

k ∆ x

|r|

CFL=1.2CFL=0.9CFL=0.6

,0 0.5 1 1.5 2

0

0.05

0.1

0.15

0.2

0.25

Total and Spatial Dispersion Error

k ∆ x

δ

Total CFL=1.2space CFL=1.2Total CFL=0.9space CFL=0.9Total CFL=0.6space CFL=0.6

Figure 4.13: Dissipation and dispersion error for RK6 optimized for CFL=1.2 with fourth

order Compact Central scheme

4.3.2 Optimizations for Upwind Spatial discretizations

Using the same strategy as for the central spatial discretizations, optimizations for a sixstage Runge-Kutta scheme, given by equation (4.35), were performed for a 3rd orderFinite Volume Upwind scheme (4.43).

ui+ 12

= −ui−1

6+

5ui

6+

ui+1

3(4.43)

In a first attempt, the same approach as for the central schemes was used. The firstfour coefficients were kept on their default value, while the last two were left free for theoptimization process. One constraint was used to keep the total dissipation error lower

48

than one. The wave number where the constraint was imposed, was chosen arbitrarilybecause for upwind schemes it is not clear at which wave number the maximum dissipationerror occurs (cf. central schemes). Table 4.2 gives the optimized coefficients a5 and a6 forthe optimized six stage Runge-Kutta time integration at different CFL numbers.

CFL a5 a6

0.5 0.233527741907065 −1.811438359723931

1 0.060972922581029 −0.029022426455235

1.5 0.021010664194455 0.001785523026951

2 0.010501303982894 0.002144008620393

Table 4.2: Coefficients for RK6,with 3rd order upwind spatial discretization and 1 con-

straint

Figure 4.14 and Figure 4.15 show the temporal resp. total errors for the RK6 schemeoptimized for a 3rd order Finite Volume Upwind spatial discretization at a CFL of 1.

−2 −1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Real

Imag

inar

y

|rtime

|

0.4

0.4

0.4

0.6

0.6

0.60.6

0.8

0.8

0.8

0.8

0.8

1

1

1

1

1

1

1

1

1

1

1.2

1.2

1.2

1.2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Residual footprintStability zone (|G|=1)

,−2 −1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Real

Imag

inar

y

δtime

−2

−2

−2

−2

−2

−2

−1

−1

−1

−1−1

−1

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.2−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

0

00

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.2

0.2

0.20.2

0.2

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Residual footprintStability zone (|G|=1)

Figure 4.14: Temporal dissipation and dispersion error for optimized RK6 with a 3rd

order upwind scheme, CFL=1

Figure 4.15 shows that the total dispersion error is negative for the lower wave number.This means the waves in that wave number range will run ahead and thus one will haveboth trailing and leading error waves.

To prevent this, a second optimization was done. In this optimization a third co-efficient of the six stage Runge-Kutta time integration was left free so that an extra

49

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−3

−2

−1

0

1

2

3

4

5Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.15: Total dissipation and dispersion error for optimized RK6 with third order

upwind scheme, CFL=1

constraint, keeping the total dispersion error positive, could be imposed. This was doneusing equation (4.44):

|G(R(k1∆x)∆t)| < 1 − ε1 k1∆x ∈ [0, π]

δ(R(k2∆x)∆t) > ε2 k2∆x ∈ [0, π] (4.44)

Here ε2 is an arbitrary small positive number, tuned so that the total dispersion erroris positive. For the optimization the parameters where chosen as follow: Γ = 0.7 andε2 = 0.001. The other parameters (k1, ε1, k2) were tuned to get a stable scheme. Theoptimization was done for several CFL numbers given in Table 4.3.

Figure 4.16 and Figure 4.17 show the temporal resp. total errors for the RK6 schemeoptimized with 2 constraints, one to keep the dissipation error lower than 1 and the otherto keep the dispersion error positive, at a CFL of 1.

Figure 4.17 shows that the total dispersion error is positive for all the wave numberswith this optimization process.

50

CFL a4 a5 a6

0.5 0.6103585735342473 0.7275827775574703 1.011484539315444

0.6 0.3752938322936234 0.3819408030144663 0.3900640950322403

0.7 0.2447703272913577 0.2237926707106628 0.1293917897285167

0.8 0.1805088055415769 0.1443231536384547 0.07799879077996832

0.9 0.1298698302104733 0.09592895796572629 0.03311940330994816

1 0.1067896454477023 0.06931071638970879 0.02214886830986389

1.1 0.09153849890351894 0.05218741811771738 0.01652759824747786

1.2 0.081584 0.0411344 0.0120665

1.3 0.07213944773744652 0.03262501366485934 0.008502824530748406

1.4 0.06792744758580115 0.02723553471697291 0.008424947675876551

1.5 0.06243260779861243 0.0225065918724392 0.006160913210227391

Table 4.3: Coefficients for RK6,with 3rd order upwind spatial discretization and 2 con-

straints

−2 −1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Real

Imag

inar

y

|rtime

|

0.4

0.40.6

0.6

0.6

0.8

0.8

0.8

0.8

1

1

1

1

1

1

1

1

1

1.2

1.2

1.2

1.2

1.2

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Residual footprintStability zone (|G|=1)

,−2 −1.5 −1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Real

Imag

inar

y

δtime

−2

−2−2

−2

−2

−1

−1

−1

−1

−1

−1

−0.6

−0.6

−0.6

−0.6

−0.6

−0.6

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

−0.2

0 00

0

0

0

0

0

0

0

0

0

0.2

0.2

0.2

0.20.2

0.2

0.6

0.6

0.6

0.6

0.6

1

1

1

1

1

2

2

22

2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Residual footprintStability zone (|G|=1)

Figure 4.16: Temporal dissipation and dispersion error for optimized RK6 with 2 con-

straints, CFL=1

51

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

Dissipation Error

k ∆ x

|r|

|rspace

||r

time|

|rtotal

|

,0 0.5 1 1.5 2 2.5 3

−3

−2

−1

0

1

2

3

4

5Dispersion Error

k ∆ x

δ

δspace

δtime

δtotal

Figure 4.17: Total dissipation and dispersion error for optimized RK6 with 2 constraints,

CFL=1

4.4 Results

4.4.1 1D Convection Equation

Central Spatial Discretizations

To confirm the results of the previous sections and check the behavior of dissipation anddispersion error, equation (4.1) is solved for a one-dimensional disturbance propagatedover a long distance. The initial disturbance at t = 0 is given by:

u(x) = sin

(2πx

a∆x

)exp

(−ln(2)

( x

b∆x

)2)

(4.45)

where a = 16, b = 3, ∆x = 1 and the propagation speed is 1. The disturbance waspropagated over a distance of 300∆x. Figure 4.18 shows the result for a CFL number of0.837, which corresponds to the accuracy limit of the scheme of Hu. The left figure showsthe results with the optimized Runge-Kutta scheme proposed by Hu, while on the rightfigure the results are shown for the present, total error optimized Runge-Kutta scheme.Looking at the trailing oscillations it is clear that the dispersive properties of the latterscheme are better, as predicted in previous section. Figure 4.19 shows the error given by|u−uinitial| of both calculations. It is clearly seen that the optimization for the total erroryields better results.

The test case above was also calculated at a CFL number 1.2 to compare the optimizedRK6 scheme with the one of Calvo et al [23]. Figure 4.20 shows the signal after it waspropagated over a distance 300∆x. The left figure shows the results for the scheme ofCalvo, while the right figure shows the result for the total error optimized scheme. Alsohere the latter scheme shows less trailing oscillations. Figure 4.21 shows the error given by

52

270 280 290 300 310 320

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

,270 280 290 300 310 320

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

Figure 4.18: optimization of Hu (left) Own optimization (right); CFL=0.837 (central)

270 280 290 300 310 3200

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

,270 280 290 300 310 3200

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

Figure 4.19: optimization of Hu (left) Own optimization (right); CFL=0.837 (central)

270 280 290 300 310 320 330

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

,270 280 290 300 310 320 330

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

Figure 4.20: optimization of Calvo (left) Own optimization (right); CFL=1.2 (central)

53

|u−uinitial| of both calculations. Comparing the amplitudes of the error it is noticed thatthe scheme of Calvo shows less dissipation than the scheme by Hu et al.(even though thiscalculation was run at a higher CFL). The total error optimized scheme shows a similaramount of dissipation than the scheme by Calvo. However the former scheme has a lessspread out error, indicating a better dispersive behavior.

270 280 290 300 310 320 3300

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

,270 280 290 300 310 320 3300

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

Figure 4.21: optimization of Calvo (left) Own optimization (right); CFL=1.2 (central)

Upwind Spatial Discretizations

To test the optimized Runge-Kutta schemes for an Upwind spatial discretization the sametest case as in section 4.4.1 was done. Figure 4.22 shows the result for a CFL numberof 0.97358, this is the CFL number given by the accuracy limit of Hu et al (to be ableto compare with his scheme). The left figure shows the standard six stage Runge-Kuttascheme while the right figure shows the optimized scheme.

70 80 90 100 110 120 130

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

,70 80 90 100 110 120 130

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

Figure 4.22: Standard RK6 (left); Optimized RK6 (right); CFL=0.97358 (upwind)

Figure 4.23 shows the error given by |u − uinitial| of both calculations. The error ismore or less reduced by 50 percent in amplitude.

54

70 80 90 100 110 120 1300

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

,70 80 90 100 110 120 1300

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

Figure 4.23: Standard RK6 (left); Optimized RK6 (right); CFL=0.97358 (upwind)

For comparison the test case was rerun with the optimized scheme of Hu et al. atthe same CFL, Figure 4.24. It is observed that almost no improvement is obtained ascompared to the standard six stage Runge-Kutta scheme. This is not surprising as theHu scheme only optimizes along the imaginary axis without accounting for the spatialscheme, which, in this case, has its footprint in the full complex plane.

70 80 90 100 110 120 130

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X/∆ X

U

U calculatedU exact

,70 80 90 100 110 120 1300

0.05

0.1

0.15

0.2

0.25

0.3

X/∆ X

|u−

u init|

Figure 4.24: Optimized Hu RK6 (left); Optimized Hu RK6 error (right); CFL=0.97358

(upwind)

4.4.2 Linearized Euler Equations

To test the performance of the optimizations for CAA applications the one dimensionalLinearized Euler Equations (LEE) given by equation (4.46) were solved.

∂U

∂t+ A

∂U

∂x= 0 (4.46)

55

where U, the vector of primitive variables, and A are given by:

U =

ρup

, A =

u0 ρ0 00 u0

1ρ0

0 γp0 u0

(4.47)

Equation (4.46) can be written under diagonal form:

∂W

∂t+ B

∂W

∂x= 0 (4.48)

where W, the vector of characteristic variables, and B are given by:

W =

12

(p

ρ0c− u

)

ρ − pc2

12

(p

ρ0c− u

)

, B =

u0 − c 0 00 u0 00 0 u0 + c

(4.49)

and c =√

γ p0

ρ0is the speed of sound. In case of a subsonic regime, 0 < u0 < c, the acoustic

wave w1 will travel upstream and w3 downstream with speeds resp. equal to u0 − c andu0 + c. The entropy wave w2 travels downstream with a speed equal to u0.

The calculation was done using the fourth order FVCC schemes described in [21] to-gether with the optimized six stage Runge-Kutta schemes. On the boundaries periodicitywas imposed. The domain with length L = 100 was divided into cells of width ∆x = 0.025.The CFL number was chosen to be CFL = 0.83732. The time step was calculated fromthe CFL number:

∆t = CFL∆x

u0 + c(4.50)

resulting in ∆t = 0.01608. The speed was chosen to be u0 = c10

. The other parameterswere set to: γ = 1.4, p0 = 1 and ρ0 = 1. The initial conditions were:

ρ(x, 0) = e−250(x−50)2

u(x, 0) = 2 e−250(x−50)2

p(x, 0) = 0

(4.51)

This initial flow field is a superposition of two gaussian acoustic waves (left and righttravelling) and a gaussian entropy wave. All results shown below are obtained after 1000time steps or t = 16.08.

Figure 4.25 shows the density, while Figure 4.26 shows the error on the density. Im-portant oscillations can be observed behind the waves. This is due to the rather bigdispersion error of the standard RK6 scheme (given in Figure 4.3). It is clear that theerrors are the lowest for the entropy wave and the highest for the right travelling acousticwave. This is due to the fact that the entropy wave travels at a speed u0 while the righttravelling acoustic wave travels at a speed of u0 + c. This means that locally the CFLnumber for the entropy wave is a lot smaller resulting in smaller errors.

The results of the optimized schemes are shown in Figure 4.27. One notices immedi-ately the lower trailing oscillations behind the acoustic waves. The trailing waves behindthe entropy wave are of the same order as those of the standard RK6 scheme.

56

35 40 45 50 55 60 65 70

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Std RK6exact

,50 50.5 51 51.5 52 52.5 53 53.5 54

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Std RK6exact

32 32.5 33 33.5 34 34.5 35 35.5 36

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Std RK6exact

,68 68.5 69 69.5 70 70.5 71 71.5 72

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Std RK6exact

Figure 4.25: Density plot for Standard RK6: whole field (top left); entropy wave (top

right); left acoustic wave (bottom left); right acoustic wave (bottom right)

The errors of the acoustic waves for the optimized RK6 scheme, shown in Figure 4.28,are smaller in amplitude and less spread out in space, compared to those of the standardRK6 scheme (shown in Figure 4.26). The errors for the entropy wave are of the sameorder of magnitude as those of the standard RK6 scheme. As mentioned above, the leftand right travelling acoustic waves are travelling at speeds of resp. u0 − c and u0 + c,while the entropy wave travels at a speed of u0. Since the optimizations are done for acertain CFL number, based on the speed u0 + c, the optimizations will be most effectivefor waves travelling at speeds around u0 + c. For low speed flows where u0 << c, and thus|u0+ c| ≈ |u0− c| ≈ c the optimizations will show significant improvement for all acousticwaves. The optimization will be less efficient on the entropy waves. However since herethe CFL number is so small, and thus so are the errors, the need for minimizing the errorsis less stringent.

Figure 4.29 and Figure 4.30 show resp. the density and the error on the density forthe optimized RK6 scheme by Hu et al. [62]. The scheme still shows lots of trailing wavesdue to its total dispersion error (see Figure 4.4). There is only little difference with thestandard RK6 scheme because although the temporal dispersion error was minimized, theoptimization had little effect on the total dispersion error (see Figure 4.4).

The same calculation was done at a higher CFL number of 1.2. The results shownbelow are after 800 time steps or at time t = 18.43973. They are compared with theoptimized scheme by Calvo et al. [23]. Figure 4.31 and Figure 4.32 show resp. the

57

35 40 45 50 55 60 65 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Std RK6

,50 50.5 51 51.5 52 52.5 53 53.5 540

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Std RK6

32 32.5 33 33.5 34 34.5 35 35.5 360

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Std RK6

,68 68.5 69 69.5 70 70.5 71 71.5 720

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Std RK6

Figure 4.26: Density Error plot for Standard RK6: whole field (top left); entropy wave

(top right); left acoustic wave (bottom left); right acoustic wave (bottom right)

density and the error on the density for the total error optimized RK6. Comparing theseresults with the ones obtained with Calvo’s optimized scheme in Figures 4.33 and 4.34,same conclusions as before can be drawn: the total error optimized scheme shows a betterdispersive behavior then the scheme proposed by Calvo. The error is less spread in spaceand the amplitudes are of similar order, showing again the efficiency of optimizing for thetotal errors instead of the temporal errors.

58

35 40 45 50 55 60 65 70

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimisedexact

,50 50.5 51 51.5 52 52.5 53 53.5 54

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimisedexact

32 32.5 33 33.5 34 34.5 35 35.5 36

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimisedexact

,68 68.5 69 69.5 70 70.5 71 71.5 72

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimisedexact

Figure 4.27: Density plot for Optimized RK6: whole field (top left); entropy wave (top

right); left acoustic wave (bottom left); right acoustic wave (bottom right)

59

35 40 45 50 55 60 65 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimised

,50 50.5 51 51.5 52 52.5 53 53.5 540

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimised

32 32.5 33 33.5 34 34.5 35 35.5 360

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimised

,68 68.5 69 69.5 70 70.5 71 71.5 720

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimised

Figure 4.28: Density Error plot for Optimized RK6: whole field (top left); entropy wave

(top right); left acoustic wave (bottom left); right acoustic wave (bottom right)

60

35 40 45 50 55 60 65 70

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Hu RK6exact

,50 50.5 51 51.5 52 52.5 53 53.5 54

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Hu RK6exact

32 32.5 33 33.5 34 34.5 35 35.5 36

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Hu RK6exact

,68 68.5 69 69.5 70 70.5 71 71.5 72

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Hu RK6exact

Figure 4.29: Density plot for RK6 of Hu et al.: whole field (top left); entropy wave (top

right); left acoustic wave (bottom left); right acoustic wave (bottom right)

61

35 40 45 50 55 60 65 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Hu RK6

,50 50.5 51 51.5 52 52.5 53 53.5 540

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Hu RK6

32 32.5 33 33.5 34 34.5 35 35.5 360

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Hu RK6

,68 68.5 69 69.5 70 70.5 71 71.5 720

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Hu RK6

Figure 4.30: Density error plot for RK6 of Hu et al.: whole field (top left); entropy wave

(top right); left acoustic wave (bottom left); right acoustic wave (bottom right)

62

30 35 40 45 50 55 60 65 70 75

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimized RK6exact

,50 50.5 51 51.5 52 52.5 53 53.5 54

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimized RK6exact

29 30 31 32 33 34

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimized RK6exact

,70 71 72 73 74 75

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Optimized RK6exact

Figure 4.31: Density plot for Optimized RK6 CFL=1.2: whole field (top left); entropy

wave (top right); left acoustic wave (bottom left); right acoustic wave (bottom right)

63

30 35 40 45 50 55 60 65 70 750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimized RK6

,50 50.5 51 51.5 52 52.5 53 53.5 540

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimized RK6

29 30 31 32 33 340

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimized RK6

,70 71 72 73 74 750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Optimized RK6

Figure 4.32: Density Error plot for Optimized RK6 CFL=1.2: whole field (top left);

entropy wave (top right); left acoustic wave (bottom left); right acoustic wave (bottom

right)

64

30 35 40 45 50 55 60 65 70 75

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Calvo RK6exact

,50 50.5 51 51.5 52 52.5 53 53.5 54

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Calvo RK6exact

29 30 31 32 33 34

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Calvo RK6exact

,70 71 72 73 74 75

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

ρ

Calvo RK6exact

Figure 4.33: Density plot for RK6 of Calvo et al. CFL=1.2: whole field (top left); entropy

wave (top right); left acoustic wave (bottom left); right acoustic wave (bottom right)

65

30 35 40 45 50 55 60 65 70 750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Calvo RK6

,50 50.5 51 51.5 52 52.5 53 53.5 540

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Calvo RK6

29 30 31 32 33 340

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Calvo RK6

,70 71 72 73 74 750

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

|ρ−

ρ ex|

Calvo RK6

Figure 4.34: Density Error plot for RK6 of Calvo et al. CFL=1.2: whole field (top left);

entropy wave (top right); left acoustic wave (bottom left); right acoustic wave (bottom

right)

66

4.5 Conclusions

In this article the errors arising from spatial and temporal discretization were discussed.A new way of optimizing time integration schemes has been formulated. The proposedapproach allows to optimize time integration schemes, taking also the spatial discretizationinto account. This leads to a minimization of the total dissipation and dispersion errors.Optimizations were done on a six stage explicit Runge-Kutta scheme in combinationwith a fourth order Finite Volume Central Compact scheme and a third order FiniteVolume Upwind scheme. The optimizations were compared with the optimizations of Huet al. [62] and Calvo et al. [23] and found to show better properties. Calculations weredone for a 1D convection of a wave at a low and a higher CFL number. The resultsconfirmed the improvements of the optimizations and also showed that one should becareful when using classical optimized temporal schemes in combination with dissipativespatial discretizations. The improved accuracy using the present methodology was alsoconfirmed in a Linearized Euler test case.

67

Chapter 5

Boundary Conditions

5.1 Introduction

To numerically simulate the flow of a fluidum a set of differential equations, e.g. theNavier-Stokes equations, have to be solved on a computational domain. However it isimpossible to extend the space domain to infinity and therefore the computational domainhas to be restricted to the regions of interest of the flows. While the governing equationswill be solved in the interior of the computational domain, at the edges of the domain,boundary conditions -which model the free field conditions- need to be imposed.

Most of the classical boundary conditions are concerned only with obtaining the correctmean-flow properties. Usually those boundary conditions do not allow vortices or acousticwaves to leave the domain of computation without generating spurious waves which arereflected into the domain. In CAA on the other hand, it is crucial that no spurious wavesare reflected on the boundaries, because they can mask the physical acoustic field radiatedby the flow in the region of interest.

In recent years different strategies have been followed to design boundary conditionswhich reflect a minimal amount of spurious wave. Four main approaches are found in theliterature:

• the absorbing zone techniques.

• the perfectly matched layer techniques.

• the characteristic non-reflecting boundary conditions.

• the radiation boundary conditions.

The absorbing zone techniques make use of zones of extra grid points which are addedat the borders of the physical domain. In these zones several numerical techniques areused to attenuate the out-going disturbances and thus reducing the reflections when theyhit the boundaries. Various techniques for attenuating the out-going disturbances have

68

been reported. The governing equations can be modified in the absorbing zone to mimica physical dissipation mechanism. For the Euler and Navier-Stokes equations this canbe done by increasing slowly the artificial dissipation in the absorbing zone [39, 71].Attenuation of the solution can also be achieved by modifying the properties of the meshat the boundaries. Rai et al. [108] and Colonius et al. [31] did this by stretching andcoarsening the grid in the absorbing zones. Disturbances reaching this zone can not beresolved anymore and therefore are dissipated. Special attention has to be paid to theway how the grid is stretched [61, 137]. Visbal et al. enhanced the attenuation of thenumerical scheme by applying low-pass numerical filters in these zones [140]. Anotherway of preventing disturbances to be reflected is to modify gradually the mean flow in theabsorbing zone to make flow locally supersonic and thus preventing any waves to travelupstream [126].

A variation on the absorbing zone techniques is the method called Perfectly MatchedLayer (PML). Here the PML equations are constructed in such a way that the interfacebetween the PML zone and the physical domain is reflectionless for all frequencies andangles. Therefore PML zones are usually a lot shorter than the absorbing zones mentionedin the previous paragraph. The first PML formulation was developed by Berenger [11] forthe Maxwell equations. Stable PML formulations for the Linearized Euler Equations hasbeen given by Hu [58, 59]. The PML technique appears to be very accurate, however thePML equations are not yet available for every practical situation. Recent developmentsare given in [3, 52, 60].

The characteristic boundary conditions are based on the Euler equations written incharacteristic form. Depending on whether the characteristic is leaving or entering thedomain, the values are extrapolated or imposed. Several different approaches are describedin the literature. A robust and well tested boundary condition of this type is the onedescribed by Hirsch [54]. Thompson developed characteristic boundary conditions for theEuler equations, assuming that the flow is locally normal to the boundary [135]. Thisleads to reflection for incoming waves at other angles. However this limitation can becircumvented by several available techniques [6, 53]. Poinsot et al. [105] extended themethod for the Navier-Stokes equations.

The last category of non reflecting boundary condition, the radiation boundary cond-tion, is based on an asymptotic solution in the far field [8]. The differential equationsare then derived in such a way that they satisfy the asymptotic expansion of the solutionup to a certain order. This set of differential equations have only outward going char-acteristics and are called the radiation boundary conditions. Tam et al. [133] developedtwo-dimensional non reflective boundary conditions using far field asymptotic expressionsof the Linearized Euler Equations in uniform mean flow. Later these were extended fora non uniform mean flow [131]. Hixon et al. [56] compared the boundary conditions byThompson [135] with these of Tam for a uniform mean flow and found the reflected soundto be smaller for the latter. A later paper by Bogey et al. [16] describes an extension ofthe radiation boundary by Tam for three dimensional problems.

Another aspect of boundary conditions for CAA which needs to be taken into account isthe stability and accuracy of the numerical schemes. To reduce the amount of dissipationand dispersion errors, high-order schemes, which can be tuned, are used in the inner

69

domain (ID). In most cases, the stencil of those schemes does not fit on the boundariesand some sort of upwind type schemes are to be used. However the use of these upwindtype schemes on the boundary may affect the overall accuracy and stability of the solutionin the inner domain. Several authors [26, 105] suggested that in a finite difference context,if the formal order of the boundary schemes is of one order lower than that of the schemeused in the interior domain the overall accuracy is preserved. However Kobayashi [70]concluded that implicit schemes are more sensitive to the boundary closure and that theoverall order of accuracy is determined by the order of accuracy of the boundary scheme.In the same paper it is shown that, in contrary to the Finite Difference approach, in aFinite Volume context it is possible to find stable boundary schemes with the same orderof accuracy as the schemes used in the inner domain and thus preserving the overall orderof accuracy.

This chapter starts with a stability analysis of the boundary schemes which are usedin combination with the Compact Schemes described in chapter 3. Next, a description oftwo different types of boundary conditions for CAA applications, the far field boundarycondition falling in the characteristic approach and the radiation boundary conditiondeveloped by Tam [133], which were implemented in the Navier-Stokes Solver Euranus[74, 113] is given in section 5.3. These boundary conditions were tested on a test casetaken from the 2nd Workshop on Benchmark Problems [1]. The results are given in section5.4 and the chapter is concluded in section 5.5.

5.2 Stability

The one dimensional Euler Equations [54] can be written in characteristic form:

∂W

∂t= −Λ

∂W

∂x(5.1)

where

Λ =

u 0 00 u + c 00 0 u − c

(5.2)

and

∂W =

∂ρ − 1c2

∂p∂u + 1

ρc∂p

∂u − 1ρc

∂p

(5.3)

Equation 5.1 is a system of hyperbolic equations and the problem is well posed if thesolution W (x, t) depends smoothly on the initial and boundary data [26]. When boundaryconditions for the system of equations are imposed in characteristic form, it suffices tocheck the stability of the numerical scheme on one of the scalar equations [50]. Thereforethe stability analysis was done on the one dimensional convection equation:

∂w

∂t= −λ

∂w

∂x(5.4)

70

where λ is a positive real constant. Equation 5.4 can be discretized in space using a FiniteVolume approach. The continuous domain [0, 1] is divided in N cells with constant width∆x. When integrating Equation 5.4 over a cell j, a system of ODE’s is obtained which iswritten in the form:

d

dt

...wj...

= C

...wj...

(5.5)

where C is an N×N matrix containing information of the numerical discretization schemein the inner domain (ID) and the numerical boundary scheme (NBS). In a FV contextthe matrix C has the form:

Ci,j = − λ

∆x(Fi+1,j − Fi,j) (5.6)

where F is a (N +1)×N transfer matrix describing the calculation of the fluxes based onthe values of the variables in the cell centers. When the fluxes are calculated implicitly(see chapter 3), the transfer matrix F is given by:

M1F = M2 (5.7)

where M1 describes the implicit part of the scheme, and M2 the explicit part and thusF = M−1

1 M2.A solution of equation of 5.5 can be found by writing it in diagonal form:

dE

dt= SE (5.8)

where S is the diagonal matrix containing the eigenvalues of matrix C. The solution tothis equation is:

Ej (t) = eSjtEj (0) (5.9)

In this form the solution depends exponentially on the eigenvalues Sj and will grow intime if the eigenvalue has a positive real part. For the solution to remain bounded in time,ensuring stability, all eigenvalues of matrix C have to be in the left half of the complexplane.

Stable numerical boundary schemes were developed for the three Finite Volume Com-pact schemes described in section 3.3.3 and section 6.3.2.

Firstly the 4th order Central Compact scheme (IC3EC2) in the inner domain wascombined with a 4th order upwind NBS:

wN+1/2 =25

12wN − 23

12wN−1 +

13

12wN−2 −

3

12wN−3 (5.10)

This gives for matrix M1

M1 =

1 0 0 . . . 014

1 14

0 . . . 0. . .

0 . . . 0 14

1 14

0 . . . 0 1

(5.11)

71

and for matrix M2:

M2 =

0 . . . 034

34

0 . . . 0. . .

0 . . . 34

34

0 . . . − 312

1312

−2312

2512

(5.12)

The matrices M1 and M2 assume that the physical boundary condition is on the left ofthe domain where a zero flux is imposed and that the numerical boundary condition ison the right side of the domain. Figure 5.1 shows the numerically calculated eigenvaluesfor this configuration. Clearly all the eigenvalues are in the left half of the complex plane

−12 −10 −8 −6 −4 −2 0−150

−100

−50

0

50

100

150

REAL

IMA

G

Figure 5.1: Eigenvalues IC3EC2 in the ID and 4th order upwind NBS

and therefore the NBS is stable.

For the 3rd order Compact Upwind scheme (IU22EU21) described in paragraph 3.3.3,a third order explicit upwind scheme was used at the boundaries:

wN+1/2 =11

6wN − 7

6wN−1 +

2

6wN−2 (5.13)

Here M1 equals:

M1 =

1 0 0 . . . 012

1 0 . . . 0. . .

0 . . . 0 12

1 00 . . . 0 1

(5.14)

72

and M2:

M2 =

0 . . . 054

14

0 . . . 0. . .

0 . . . 0 54

14

0 . . . 0 26

−76

116

(5.15)

Figure 5.2 shows the numerically calculated eigenvalues for this configuration and provesthat the schemes are stable.

−160 −140 −120 −100 −80 −60 −40 −20 0−150

−100

−50

0

50

100

150

REAL

IMA

G

Figure 5.2: Eigenvalues IU22EU21 in the ID and 3rd order upwind NBS

For the 5th order Compact Upwind scheme IU32EU32 described in paragraph 3.3.3,special care needs to be taken at the physical boundary condition because the largerstencil on the right hand side does not fit for the second flux. Therefore the flux on thesecond face is calculated with the downwind version of the inner scheme:

1

6w1/2 + w3/2 +

1

2w5/2 =

5

9w1 +

19

18w2 +

1

18w3 (5.16)

For the NBS on the right hand side a 5th order explicit upwind scheme is used, given by:

wN+ 12

=137

60wN − 163

60wN−1 +

137

60wN−3 −

21

20wN−3 +

1

5wN−4 (5.17)

This leads to the following forms for matrices M1 and M2:

M1 =

1 0 0 . . . 016

1 12

0 . . . 012

1 16

0 . . . 0. . .

0 . . . 0 12

1 16

0 . . . 0 1

(5.18)

73

M2 =

0 . . . 059

1918

118

0 . . . 0118

1918

59

0 . . . 0. . .

0 . . . 0 118

1918

59

. . . 15

−2120

13760

−16360

13760

(5.19)

The stability of the schemes is demonstrated in Figure 5.3. The schemes were used for

−60 −50 −40 −30 −20 −10 0−150

−100

−50

0

50

100

150

REAL

IMA

G

Figure 5.3: Eigenvalues IU32EU32 in the ID and 5th order upwind NBS

several test cases described in chapters 3 and 7.

5.3 Non-reflecting Boundary Conditions

5.3.1 Far Field Boundary condition

The first set of boundary conditions was already present in the Finite Volume solverEuranus [74, 113]. The NBS used in these boundary conditions was changed to the higherorder schemes described in the previous paragraph in order to be used with the 4th orderCompact Central scheme (IC3EC2).

The Far Field boundary condition (FFBC), [54, 134], in the Euranus solver is based onthe characteristic variables or Riemann invariants which are for isentropic flows definedas:

R±n =

−→V .−→n ± 2c

γ − 1(5.20)

74

where −→n is the normal of the boundary and c the local speed of sound. For subsonicinflow, R+

n corresponds to a incoming characteristic. R+n is then obtained from the free

stream values:

R+n =

−→V ∞.−→n +

2c∞γ − 1

(5.21)

where−→V ∞ is the free stream velocity and c∞ the free stream speed of sound. On the

other hand, R−n corresponds to an outgoing characteristic and has to be estimated from

inside the computational domain by an appropriate extrapolation:

R−n =

−→V ext.

−→n − 2cext

γ − 1(5.22)

the subscript ext refers to values extrapolated from the interior with the numerical schemesmentioned in the previous paragraph. The normal velocity and the local speed of soundcan now be obtained on the boundary by adding and subtracting both Riemann variables:

−→V boundary.

−→n =R+

n + R−n

2(5.23)

and

cboundary =(R+

n − R−n

) γ − 1

4(5.24)

The entropy on the boundary is set to the free stream entropy. All the flow variables canthen be obtained. Corresponding formulas hold for a subsonic outflow with the differencethat R+

n and the boundary entropy are obtained from interior values.

5.3.2 Characteristic boundary conditions

The three dimensional non reflective boundary conditions which were implemented in theEuranus code were developed by Bogey et al. [16] and are an extension of the two dimen-sional non reflective boundary conditions of Tam et al. [130]. The boundary conditionsare based on an asymptotic solution of the Linearized Euler Equations. Two differentboundary conditions were given: radiation conditions for boundaries reached only byacoustical waves, and outflow boundary conditions for boundaries where besides acousticwaves, also entropy and vortical waves are leaving the domain. The boundary conditionsare written in spherical coordinates r, θ, φ. In case only acoustical waves are leaving theboundary, the radiative boundary condition can be written as:

∂

∂t

ρ − ρ∞−→u −−→u ∞

p − p∞

+ vg

(∂

∂r+

1

r

)

ρ − ρ∞−→u −−→u ∞

p − p∞

= O

(r−5/2

)(5.25)

where vg = (−→u ∞ + c∞).−→1r . These equations are solved in the dummy cells of the domain

with the same time integration as in the inner domain and a fitted upwind scheme forthe spatial discretization. Due to the error term depending on r−5/2 the order accuracyof the spatial discretization is not maintained. However this error can be controlled byincreasing the size of the domain and adding artificial damping near the boundaries.

75

The outflow boundary conditions by Bogey et al. [16] are given by the followingformulation:

∂(ρ−ρ∞)∂t

+ (−→u −−→u ∞)∇(ρ − ρ∞) = 1c2

(∂(p−p∞)

∂t+ (−→u −−→u ∞)∇(p − p∞)

)

∂(−→u −−→u )∂t

+ (−→u −−→u ∞)∇(−→u −−→u ∞) = − 1ρ−ρ∞

∇(p − p∞)∂(p−p∞)

∂t+ vg

(∂∂r

+ 1r

)(p − p∞) = 0

(5.26)These differential equations are to be used in case that besides the acoustical waves, alsovorticity and entropy waves are crossing the boundary. They were implemented in asimilar way as the acoustic radiation boundary condition.

5.4 Boundary Condition Test

To investigate the effectiveness of the boundary conditions mentioned in previous para-graph for the full non linear Euler equations, a test case was selected from the 2ndWorkshop on Benchmark Problems in CAA [1]. The test case was specifically designed totest the effectiveness of radiation, inflow and outflow boundary conditions for CAA. It isan initial value problem (IVP) with a superposition of an acoustic wave, a vorticity waveand an entropy wave embedded in a uniform mean flow parallel to the domain (Figure5.4). It was solved using the Euler equations in the Euranus code [113, 74]. The problem

Figure 5.4: Test case 1

was solved on a computational domain with −100 ≤ x ≤ 100 and −100 ≤ y ≤ 100. Theoriginal test case was given for a uniform flow at a mach number Mx = 0.5. Howeversince the implemented radiation boundary conditions [16] were developed for the LEE,

76

the mach number was lowered to Mx = 0.05 to avoid non linear effects. The initial valuesof the flow variables were:

p = p∞ + p∞ e

(−ln(2)x2+y2

9

)

ρ = ρ∞ + ρ∞ e

(−ln(2)x2+y2

9

)

+ 0.1 ρ∞ e

(−ln(2)

(x−67)2+y2

25

)

u = u∞ + 0.04 u∞ y e

(−ln(2)

(x−67)2+y2

25

)

v = v∞ − 0.04 v∞ (x − 67) e

(−ln(2)

(x−67)2+y2

25

)

(5.27)

The domain was discretized using 100 by 100 cells. At the inner domain the fourth ordercentral compact schemes (IC3EC2) were used combined with the NBS discussed in theprevious section. The calculations were done using the Far Field boundary conditions [54]and the radiation and outflow boundary condition by Bogey [16]. For the time integrationa standard 4-stage Runge Kutta scheme was used with a CFL number of 0.25. Artificialdissipation was used to control the high frequent reflections. To estimate the magnitudeof acoustic waves reflected into the computational domain after the waves have left thephysical domain, the time evolution of the residual fluctuating pressure Lp and streamwisevelocity Lu was recorded:

Lp =

√√√√ 1

N2

N∑

i,j

(p − p∞)2 (5.28)

Lu =

√√√√ 1

N2

N∑

i,j

(u − u∞)2 (5.29)

The initial acoustic pulse will reach the right boundary first, afterwards the entropyand vorticity wave hit the boundary. Figure 5.5 and Figure 5.6 show resp. the pressureand vorticity at time t = 0.3949 s when the acoustic wave has already hit the boundary,at time t = 1.8309 s when the vorticity and entropy wave have already hit the boundaryand t = 3.59 s when all waves have left the physical domain. Looking at the pressure inFigure 5.5 at time t = 0.3949 s (corresponding with the top two figures), it is noticed thatthe reflections of the acoustic wave on the boundaries are of similar order of magnitude.A drawback of the Far Field boundary conditions is that although they do not reflect thelow frequent waves, they generate high frequent spurious noise. This is made more clearin Figure 5.7 where the pressure at time t = 0.3939 s in the streamwise direction at themidline of the domain is shown. At this moment of time the acoustic wave has just leftthe physical domain. The pressure wave on the right side is a result of the non linearityof the Euler equations and corresponds to the position of the vorticity and entropy wave.This high frequency noise can lead to instabilities and must be controlled by artificialdissipation. The reflection ratio of the acoustic wave, defined as the ratio between theamplitude of the initial wave and the amplitude of the reflected wave, is for the Radiationboundary conditions 1.7% and for the Far Field boundary conditions 0.8%. The Far Fieldboundary conditions generate less high frequent acoustic reflections. The reflection ratiofor the acoustic wave for both boundary conditions is within acceptable range for practical

77

Figure 5.5: Pressure, Far Field BC (left) and Radiation BC (right)

applications. For the vorticity wave, given in Figure 5.6, the reflection ratio for the FarField and Radiation boundary condition is respectively 1.3% and 0.06%. The Radiationboundary conditions are more transparent for vorticity waves than the Far Field boundaryconditions.

78

Figure 5.6: Vorticity, Far Field BC (left) and Radiation BC (right)

79

−100 −50 0 50 100

1

1.005

1.01

1.015x 10

5

X

P (

Pa)

−100 −50 0 50 100

1

1.005

1.01

1.015x 10

5

X

P (

Pa)

Figure 5.7: Pressure for Y=50, Far Field (left) and Radiation (right)

Figure 5.8 shows the residual fluctuating pressure and streamwise vorticity as a func-tion of time. The two crosses indicated on the time axis show the moment when theacoustic wave hits the right boundary and the moment when the vorticity wave hits theright boundary. Up until t = 1.5 s both types of boundary conditions show similar be-

0 0.5 1 1.5 2 2.5 3 3.5 410

−4

10−3

10−2

10−1

100

t (s)

Lp/L

p(0)

Far Field BCRadiation BC

0 0.5 1 1.5 2 2.5 3 3.5 410

−2

10−1

100

t (s)

Lu/L

u(0)

Far Field BCRadiation BC

Figure 5.8: Lp and Lu for IVP

havior: when the acoustic wave hits the boundary the residual fluctuations drop, which isexpected. However, on the second time point, when the vorticity wave hits the boundary,Lp and Lu drop both for the Radiation boundary conditions but for the Far Field bound-ary conditions a clear rise is observed. Only at time t ≈ 3.5 s Lp and Lu drop again.This corresponds to the reflection of the vorticity wave which is then damped out by theartificial dissipation.

80

5.5 Conclusions

In this chapter the stability of numerical boundary schemes was analyzed. The 4th orderCompact Central scheme and the 3rd and 5th order Compact Upwind scheme, werecombined with upwind type schemes on the boundaries and proven to be stable. Two typesof non reflecting boundary conditions were implemented in the Finite Volume Navier-Stokes solver Euranus [113, 74] and were tested on a typical boundary problem in CAA[1]. The results showed that both methods performed equally well for the acoustic wavesbut that the Radiation boundary condition generated less reflection for the vorticity waves.However, a draw back of the latter boundary condition is that it generates high frequencynoise. Although this can be controlled with artificial damping, it can lead to instabilities.For practical problems it was concluded that the Far Field boundary conditions are to bepreferred.

81

Chapter 6

Far Field Noise Solvers

6.1 Introduction

Generally the problem of sound generated in flows is governed by the well known Navier-Stokes equations. They describe all the physical phenomena occurring in a flow includingthe acoustic perturbations of the pressure. Since CAA is a multi-scale domain the mostexact way to compute the sound would be to solve the Navier-Stokes equations directly forthe whole domain of interest. However this looks misleadingly attractive. In a practicalproblem the distance between the source of sound and the observer is large, while on theother hand to be able to resolve all the small pressure fluctuations a high resolution isrequired. Thus meshes with a huge amounts of cells are needed. This leads to requirementson the computational power which is not available to most researchers. Therefore the mostsuccessful aeroacoustic tools nowadays make use of an hybrid approach, where the directmethods are only used in the domain where the sound is produced, whereas transportmethods are used to solve for the transport of the generated sound to the observer.

The transport methods can be divided in two main approaches: the analytical trans-port methods and the computational transport methods.

The analytical transport method was developed during the early years of the historyof aero acoustics. One of the first aero acoustic formulations was the Gutin’s theory forpropeller noise in 1937 [51]. In 1952 Lighthill [81] introduced his Acoustic Analogy. Thisanalogy was restricted to deal with problems of jet noise and turbulent flow embedded inan infinite homogeneous fluid in the absence of solid boundaries. The method is based onan integrated form of an acoustic propagation equation, which is obtained by rearrangingthe continuity and momentum equations. The pressure at the observer point in space andtime is obtained by an integration over a source term along a surface or a compact volumecontaining the sources. The advantage of the integral formulation is that random errors inthe sound source are averaged out. Later on, several authors extended Lighthill’s AcousticAnalogy to include the effects of solid boundaries [32, 106]. In 1969 Ffowcs Williams andHawkings generalized the Lighthill Acoustic Analogy to include the effects of generalsurfaces in arbitrary motion [38]. Today, the Ffowcs-Williams-Hawkings (FWH) equationis a dominant tool in predicting the sound produced by motion of complex solid bodies.Paragraph 6.2 describes the method in the special case of rotating bodies.

82

In the computational transport approach a set of partial differential equations is solvedin the entire field up to the observer. Because the method only has to deal with transportof acoustic waves several assumptions can be made which allow to simplify the governingNS equations without compromising the accuracy of the results. One of these methods isthe Linearized Euler Equations (LEE) avoiding due to their linearity many problems whichare encountered with the full set of Navier-Stokes equations. Another advantage is thattheir simplicity allows the discretization schemes to be highly tuned to reduce dissipativeand dispersive errors. This has been done extensively in the past decade by many authorsfor the spatial discretization, [42, 67, 70, 75, 77, 84, 123, 122, 121, 133] and for the temporaldiscretization, [17, 23, 62]. Special attention has to be paid to the boundary conditions forthe computational transport methods which need to be acoustical non-reflective. Manydifferent approaches to avoid reflections at the boundaries of the computational domainhave been developed, such as the characteristic non-reflection boundary condition, [45,105, 135], the radiation boundary condition [16, 132] or perfectly matched layers, [58, 59].The LEE method is further discussed in paragraph 6.3.

Both, the LEE and the FWH method, have their place in CAA. Since the computationof flow produced sound is still in its childhood, there is not one preset way of solving noiseproblems. Which method to use will highly depend on the problem to be solved. Theanalytical transport methods allow for a quick calculation in specific observer points.However a draw back of them is that they need the development of Green functions whichcan become quite cumbersome when complex solid bodies are present. Since they assumethat the observer is located in an unbounded domain, this methods are generally notapplied for acoustics of confined flows. The LEE offer the flexibility that many differenttypes of problems can be solved. The problem of boundaries is only limited to a problemof avoiding reflections at the boundary of the computational domain. When the wholeacoustical field between source and observer needs to be studied they allow an efficientway of obtaining this field. However, computational effort limits the distance over whichthe method can be applied.

In this work a code was developed based on the FWH equation. Another one, basedon the LEE equations, was developed in cooperation with Van den Abeele [4]. As avalidation of both codes the fan noise problem proposed by Tam et al [129] was solved.This test case was part of the 3rd CAA Workshop on benchmark problems [2], which hasbeen organized since 1994 as one of the main references for CAA validation test cases.

This chapter first gives an overview of the formulation used in both codes, the onebased on the FWH equation and the one based on the LEE. Then the benchmark problemfor both cases are described followed by the results for both test cases and a comparisonbetween the two methods. In the final section the conclusions are given.

6.2 Ffowcs-Williams-Hawkings Formulation

Generally the problem of sound generated in flows is governed by the well known Navier-Stokes equations. By rearranging the continuity equation with momentum equationsLighthill described his acoustic analogy [81]. However his analogy was made in the frame-

83

work of jetnoise and did not allow a solution if solid boundaries were present. The FWHequation is a generalisation of the Lighthill acoustic analogy which takes into account theeffect of moving surfaces.

Considering the continuity equation and the momentum equations:

∂ρ

∂t+

∂

∂xi

(ρvi) = q (6.1)

∂

∂t(ρvi) +

∂

∂xj

(ρvivj − pδij + τij) = fi (6.2)

where q is a mass source term related to local mass injection or extraction and fi is anexternal force, which can be a volume or surface force such as surface tension or pressure.

Taking the time derivative of equation 6.1 and combining this with the divergence ofequation 6.2, the following inhomogeneous wave equation is obtained:

∂2ρ

∂t2− c2

04ρ =∂2

∂xi∂xj

((p − c2

0ρ)δij − τij + ρvivj

)− ∂fi

∂xi

+∂q

∂t(6.3)

Defining the acoustic flow variables as the difference between the density and pressurewith their constant surrounding field seen by the observer (ρ0, p0):

p′ = p − p0

ρ′ = ρ − ρ0(6.4)

and introducing this into equation 6.3 with ∂ρ0

∂t= 0, the Lighthill equation is found:

∂2ρ′

∂t2− c2

04ρ′ =∂2Tij

∂xi∂xj

− ∂fi

∂xi

+∂q

∂t(6.5)

where Tij is the Lighthill tensor defined as:

Tij =(p′ − c2

0ρ′)δij − τij + ρvivj (6.6)

This equation is valid when the background field is constant and when the observer isfar away from the sound production processes. The FWH equation is a generalization ofequation 6.5 to include the effect of moving bodies. When a moving body is present itwill generate forces and mass sources in the relative system. Figure 6.1 shows a surface

Σ with normal −→n and which moves at speed−→U in the reference system of the observer.

In case the surface is permeable there is a flow through the surface equal to −→v . Therewill be a contribution to the mass sources because of the mass flux through the surface,which equals:

q = ρ(−→v −−→

U)

.−→n δ (Σ) (6.7)

The forces related to the momentum flux through the surface are:

−→f =

(−→v(ρ

(−→v −−→U

).−→n

)+ p.−→n − τ .−→n

)δ (Σ) (6.8)

84

Figure 6.1: Moving Surface Σ with speed U and normal n

Since ρ0 is the density in the reference system of the observer, the material derivativewith respect to time and along the surface Σ should be zero.

∂ρ0

∂t+−→U .

−→∇ρ0 = 0 (6.9)

Taking the time derivative of equation 6.9 and taking into account that the surface Σ isalso a constant ρ0 surface, it follows that

∂2ρ0

∂t2= −

∂(ρ0

−→U .−→n

)δ (Σ)

∂t(6.10)

Introducing these mass source and forces into equation 6.3 and taking into account equa-tion 6.10, leads to the FWH equation:

∂2ρ′

∂t2− c2

04ρ′ (6.11)

=∂2Tij

∂xi∂xj

−−→∇(−→v

(ρ

(−→v −−→U

).−→n

)+ p.−→n − τ .−→n

)δ (Σ)

+∂

∂t

[(ρ

(−→v −−→U

)+ ρ0

−→U

).−→n δ (Σ)

]

If the surface is a solid surface, i.e.−→v = 0, the FWH equation simplifies to:

∂2ρ′

∂t2− c2

04ρ′ (6.12)

=∂2Tij

∂xi∂xj

−−→∇(p.−→n − τ .−→n

)δ (Σ) +

∂(ρ0

−→U .−→n

)δ (Σ)

∂t

85

In most cases, when thermal processes are neglectable, one can use the isentropicrelation ∂p′ = c2

0∂ρ′ to get the FWH equation for the fluctuation of pressure:

1

c20

∂2p′

∂t2− c2

04p′ (6.13)

= q (−→x , t) =∂2Tij

∂xi∂xj

−−→∇(p.−→n − τ .−→n

)δ (Σ) +

∂

∂t

[(ρ0

−→U .−→n

)δ (Σ)

]

Equation 6.13 is written so that the left hand side resembles the homogenous wave equa-tion while the right hand side is the sum of the three types of sound sources created bydifferent physical phenomena. The first term on the rhs of equation 6.13 is the quadrupolesource and accounts for the flow non-linearities. The second term is the loading noise ordipole source and is related to the force applied by the solid body on the fluid. Thelast term is the monopole source or also called thickness noise and is related to the dis-placement of the fluid by the solid body. Thus when the sound sources are known, thepropagation of the sound to the observer can be calculated. This can be done by integrat-ing equation 6.13 directly or by constructing an analytical solution. In case of an externalflow in an unbounded domain, the latter is done by making use of Green functions. Thissolution will be an integral equation which describes the effect of the sources, propagation,boundary conditions and initial conditions.

A free space Green function G (−→x , t|−→y , τ) is the response of the wave equation on apulse released at source point −→y at time τ in absence of solid boundaries:

1

c20

∂2G

∂t2−4G = δ (−→x −−→y ) δ

(−→t −−→τ

)(6.14)

The response on the pulse is observed at location −→x and at time t.As described in Goldstein [47] the solution of the inhomogeneous wave equation is

given by the convolution of the source term with the pulse response G (−→x , t|−→y , τ). Inmore general cases when the domain is bounded by certain boundaries, the Green functionwill have to be defined further by proper boundary conditions. However, the tailoring ofGreen functions is a complex matter and there is no mathematical procedure to find asolution for all situations. A thorough analysis is given in Goldstein [47].

The free field Green function, or pulse response, for the three dimensional wave equa-tion is:

G (−→x , t|−→y , τ) =1

4π Rδ

(t − τ − R

c0

)(6.15)

where R = |−→x −−→y | is the distance between the observer and the source. A solution forthe pressure at observer position −→x and at time t is found by convoluting the free fieldGreen function of equation 6.15 with the source q in equation 6.13. This yields:

p′ (−→x , t) =1

4π

∫ t

−∞

∫

V

q (−→y , τ)δ(t − τ − R

c0

)

Rd−→y dτ (6.16)

With the retarded time te, implicitly defined as te = t − R(te)c0

, and Mr =−→U .

−→1R

c0the Mach

number of the source velocity−→U projected on the R direction, the integral of equation

6.16 can further be reduced to:

86

p′ (−→x , t) =1

4π

∫

V

[q (−→y , t)

R (1 − Mr)

]

e

d−→y (6.17)

The square brackets indicate that all the values have to be taken at the retarded time.Doing a similar reasoning for the separate sources, the contributions of the monopole,dipole and quadrupole are obtained. The expression for the monopole is:

p′M (−→x , t) =1

4π

∂

∂t

∫

Σe

Q2 (−→y , te)

Re (1 − Mr)e

dΣe (6.18)

where Q2 is the displacement term and equals Q2 = ρ0

−→U .−→n . The contribution of the

dipole is given by:

p′D (−→x , t) =1

4π

−→∇ .

∫

Σe

−→Q1 (−→y , te)

Re (1 − Mr)e

dΣe (6.19)

where−→Q1 is the force term and equals

−→Q1 = p.−→n − τ .−→n . Because the displacement and

force terms are defined on the surface Σ, the volume integral is reduced to a surfaceintegral.

With the aim of using the code for rotating bodies these general formulas can befurther worked out and the formulation 1A of Farrasat is obtained [37]:

p′M (−→x , t) =1

4π

∫

Σe

Q′2e

Re (1 − Mr)2e

dΣe +1

4π

∫

Σe

Q2eM ′

e.−→Re − c0M

2e + c0Mr

R2e (1 − Mr)

3e

(6.20)

p′D (−→x , t) =1

4π

∫

Σe

−→Q ′

1e.−→R e − c0

−→Me.

−→Q 1e

c0R2e (1 − Mr)

2e

dΣe +1

4π

∫

Σe

(−→Q 1e.

−→1 R

) M ′e.−→Re + c0 (1 − M2

e )

c0R2e (1 − Mr)

3e(6.21)

where−→Me =

−→Ue

c0and the primes denote derivatives with respect to retarded time.

6.3 Linearized Euler Equations

6.3.1 Governing Equations

In cases where an analytical solution of the FWH equation can not be developed, forexample if complex solid boundaries are present in the domain, one will have to resortto direct integration to obtain the acoustic field at the observers position. A commonapproach for this is the use of the Linearized Euler Equations (LEE), which are derivedfrom the Navier-Stokes equations under certain conditions. In case the viscous effects canbe neglected the Navier-Stokes equations will reduce to the Euler equations, which aregiven by 6.22:

∂U

∂t+

∂f

∂x+

∂g

∂y+

∂h

∂z= Q (6.22)

87

where U is the vector of the conservative variables:

U =

ρρuρvρwρE

(6.23)

and f ,g, and h are the inviscid flux vectors given by:

f =

ρuρu2 + p

ρuvρuwρuH

, g =

ρvρuv

ρv2 + pρvwρvH

, f =

ρwρuwρvw

ρw2 + pρwH

, (6.24)

where E is the total energy, H the total enthalpy and Q the source terms.However, when the domain of interest is far away from the noise sources the acoustic

phenomena are dominated by linear propagation of sound waves with very little backcoupling to the aerodynamic field. In this case the Euler equations can be linearized forthe flow perturbations around the stationary mean flow.

Considering the primitive variables as the sum of their mean value and the perturbationfrom the mean value:

ρ = ρ0 + ρ′

u = u0 + u′

v = v0 + v′

w = w0 + w′

p = p0 + p′

(6.25)

Substituting these relations into the Euler equations 6.22 and neglecting non linear terms,the LEE are obtained. These are given by:

∂U ′

∂t+ A

∂U ′

∂x+ B

∂U ′

∂y+ C

∂U ′

∂z= Q (6.26)

with:

U ′ =

ρ′

u′

v′

w′

p′

A =

u0 ρ0 0 0 00 u0 0 0 1

ρ0

0 0 u0 0 00 0 0 u0 00 γp0 0 0 u0

B =

v0 0 ρ0 0 00 v0 0 0 00 0 v0 0 1

ρ0

0 0 0 v0 00 0 γp0 0 v0

C =

w0 0 ρ0 0 00 w0 0 0 00 0 w0 0 00 0 0 w0

1ρ0

0 0 0 γp0 w0

(6.27)

The LEE (equation 6.26) describe the generation and propagation of small perturba-tions of the flow quantities around their stationary mean values.

88

6.3.2 Numerical Scheme

The discretization of the LEE, given by equation 6.26, was done in a Finite Volumecontext. For the discretization the 4th order compact central scheme (IC3EC2), describedin [21], was implemented for non-uniform cartesian meshes. This scheme has a tridiagonalsystem with a right hand side involving only cells in the immediate neighborhood of thecell face where the flux is calculated:

α1ui−1/2 + ui+1/2 + α2ui+3/2 = a1ui+1 + a2ui (6.28)

where u indicates the cell averaged value of the primitive variable and u the cell faceaveraged value. This scheme can be made 4th order accurate on non-uniform meshesusing the following expressions for α1, α2, a1 and a2:

h2i+1

(hi + hi+1)2ui−1/2 + ui+1/2 +

h2i

(hi + hi+1)2ui+3/2 (6.29)

=2 (h3

i + 2h2i hi+1)

(hi + hi+1)3 ui+1 +

2(h3

i+1 + 2h2i+1hi

)

(hi + hi+1)3 ui

where hi is the width of cell i. On a uniform mesh where hi = hi+1 this equationsimplifies to:

1

4ui−1/2 + ui+1/2 +

1

4ui+3/2 =

3

4(ui+1 + ui) (6.30)

The dispersion error of the scheme is given in Figure 4.6.Stability problems or spurious numerical oscillations may arise because of the lack of

built-in numerical dissipation of the 4th order IC3EC2 scheme. To overcome this problem,a dissipative term has to be added. In recent years, a so-called artificial selective damping(ASD) [68, 128] has been applied to a number of CAA applications. The effect of thebackground smoothing term in ASD makes sure that damping occurs only in the narrowband of high wave numbers leaving the rest of the spectrum intact. In regions of sharpgradients (shock waves, contact discontinuities) the second order dissipation is switchedon to ensure the monotonicity of the solution. A dissipative term is introduced:

Di = − ε

hi

3∑

j=−3

cjui+j (6.31)

which can be written in a conservative form [68] as a difference of numerical dissipationfluxes:

Di =di+1/2 − di−1/2

hi

(6.32)

where

di+1/2 = −ε

3∑

j=−2

bjui+j (6.33)

The coefficients cj and bj are related through the following relation:

bj =3∑

n=j

cn (6.34)

89

c0 = 0.483583

c1 = −0.366267

c2 = 0.150908

c3 = −0.0264322

Table 6.1: ASD coefficients by Smirnov [121]

By a minimization process, [21], the coefficients cj, given in Table 6.1, are obtained. Figure6.2 compares the Fourier transform of the damping of the ASD formulation developed bySmirnov [121], with formulations from literature.

Figure 6.2: Comparison of ASD formulation [68, 121, 128]

The time integration is performed by a four-stage low storage Runge-Kutta methodbecause of its good stability and memory usage properties.

6.3.3 Boundary Conditions

For the boundary conditions special care needs to be taken so that the waves leaving thecomputational domain do not reflect on the boundaries. A thorough analysis was doneby Van den Abeele [4] on the use of non-reflective boundary conditions for LEE. Theboundary conditions by Tam and Webb [133], which are based on a asymptotical solutionof the LEE in the far field, were compared against the characteristic boundary conditions,clearly indication that the non-reflective boundary conditions by Tam and Webb weresuperior [4] for the LEE. In case oly acoustical waves are leaving the boundary, the

90

radiation boundary condition can be written in cylindrical coordinates as:

∂

∂t

ρ′

−→u ′

p′

+ vg

(∂

∂r+

1

r

)

ρ′

−→u ′

p′

= O

(r−5/2

)(6.35)

where vg = (−→u ∞ + c∞).−→1r . These equations are solved in the dummy cells of the domain

with the same time integration as in the inner domain and a stable numerical boundaryscheme for the spatial derivatives.

6.4 Test case

In the 3rd Computational Aero Acoustics Workshop on Benchmark Problems [2], one ofthe test cases was the calculation of the noise generated by an unducted fan. The rotor,as shown in Figure 6.3, was defined by a specific blade force and Tam [129] developed ananalytical solution of the LEE to this problem. In this test case the rotor is defined as

Figure 6.3: Unducted fan

having m = 8 blades and rotating with an angular velocity Ω. The blade forces are givenin cylindrical coordinates (x,r,θ):

Fx (x, r, θ, t)Fr (x, r, θ, t)Fθ (x, r, θ, t)

= Re

Fx (x, r)0

Fθ (x, r)

eim(θ−Ωt)

(6.36)

where Re is the real part of. The frequency of the radiated sound will be mΩ. Thebody force distributions in r and x were chosen so that a simple analytical solution couldbe developed. They are given by

Fθ (x, r) =

F (x) rJm (rλmN) r ≤ 10 r > 1

(6.37)

Fx (x, r) =

F (x) Jm (rλmN) r ≤ 10 r > 1

(6.38)

91

F (x) = e−(ln2)(10x)2 (6.39)

where Jm() is the mth order Bessel function and λmN is the Nth root of derivative of theBessel function J ′

m. For this problem N = 1 and thus λ8,1 = 9.64742.

6.4.1 Solution with LEE

The three dimensional LEE in cylindrical coordinates can be reduced to a two dimensionalproblem for this test case by factoring out the azimuthal dependence. This leads to thefollowing set of equations:

∂v′

∂t+ ∂p′

∂r= 0

∂w′

∂t+ im

rp′ = Fθ (r, x) e−imΩt

∂u′

∂t+ ∂p′

∂x= Fx (r, x) e−imΩt

∂p′

∂t+ ∂v′

∂r+ v′

r+ imw′

r+ ∂u′

∂x= 0

(6.40)

All the variables above have been non dimensionalized with respect to the followingscales:

scale symbol meaning

length b length of blade

velocity c∞ ambient sound speed

time bc∞

density ρ∞ ambient gas density

pressure ρ∞c2∞

body force ρ∞c2∞b

The mesh was designed in such a way that it could capture accurately the noisesources as well as the acoustic field. Since the source length scale is determined bythe geometry and loading of the blades a high resolution of the mesh near the blade isneeded. On the other hand the length scales of the acoustic field are determined by therotational frequency and the number of the blades of the rotor and thus less resolutionof the mesh is needed in the far field. The calculation was done on a domain going from(x; r) = (−17.05; 0) to (r; x) = (17.05; 17.01). A total number of 285 cells was used in theradial direction with the smallest cell in the vicinity of the blade equalling ∆r = 0.02 andthe biggest cell in the acoustic field ∆r = 0.08. The total number of cells in the x directionwas 496 with the smallest cell width being ∆x = 0.02 and the biggest ∆x = 0.08. Thecell width of 0.08 corresponds to a resolution of 11.5 points per wavelength for the 4thorder IC3EC2 scheme. A time step of 0.002 was used, corresponding to a maximum CFLnumber of 0.1. Near the boundaries the ASD was gradually increased following a gaussian

92

function to a value 10 times that of the background damping (which is ε = 0.4) in orderto damp out spurious waves. Near the rotor tip the ASD was gradually increased to 30times the background damping because of the discontinuity of the body forces over there.The calculation was performed by Van den Abeele [4] for a subsonic tip speed rotation ofΩ = 0.85.

6.4.2 FWH equation

To test the code based on the FWH equation Tam’s benchmark problem was adaptedby Hirsch et al [55]. The body force was redefined by removing the exponential of them-harmonic from the blade force definition, which is for a body force defined as:

Fx (r, θ, x, t)Fr (r, θ, x, t)Fθ (r, θ, x, t)

= Re

Fx (r, x)0

Fθ (r, x)

(6.41)

This will lead to a solution with multiple harmonics of which the first harmonic shouldequal the solution developed by Tam [129]. The blade geometry is derived from the forcesin equation 6.41 for a domain of −0.8 < x < 0.8 and 0 < r < 1. The calculation used 127time steps over one blade period ( T

m).

6.4.3 Results

LEE calculation

Figure 6.4 shows the acoustic field for the fan test case described in paragraph 6.4.1 fortime t = 160. The radiation pattern is inclined slightly to the right due to the fact thatthe axial force (equation 6.38 is positive along the whole rotational axis.

The waves which are visible in the lower right and left corner of each figure are spuriouswaves coming from reflections on the boundaries. The physical acoustic radiation in thosedirections is very low, therefore the spurious waves overwhelm the solution. The samephenomena can be observed in Figures 6.5 and 6.6 in which the pressure in the timedomain and the Sound Pressure Level (SPL) over one blade passing period are given.Note that the pressure and SPL are non-dimensionalized and given for several observerpositions at a radius R ≈ 10.

The results for the pressure in the time domain show good agreement for observerangles β between 45o and 120o. Outside this range the results are overpredicted. Alsonote that the exact solution has only one harmonic given by the blade passing frequencyBPF = 2π

m Ω(shown in the SPL figures by the square symbol). Although there are higher

harmonics appearing in the LEE calculation, due to numerical discretization errors, thedecay with increasing frequency is very steep (figures have logarithmic scales).

The same numerical phenomena can be observed in the directivity pattern shown inFigure 6.7 where the directivity is defined as:

D (θ) = limR→∞

R2p′2 (6.42)

93

Figure 6.4: Acoustic field rotor test case

where p′2 is the time average of the squared pressure fluctuation. A very good overallagreement with the analytical results is obtained . Figure 6.8 shows on the left side azoom on the directivity pattern for angles between β = 30o and β = 60o. For these anglesoscillations in the pattern are observed. These reconfirm the observations made beforethat the spurious waves overwhelm the results close to the rotational axis. On the rightside a zoom on the top of the directivity pattern is shown. The maximum amplitudeof the analytical solution is 5.97 10−6 at an observer angle β = 82.6o. The maximumamplitude for the LEE calculation is observed at an angle β = 82.4o and has a value of5.94 10−6. This corresponds to a relative error of 0.5% for the directivity and is withinacceptable range.

FWH calculation

The data obtained with the FWH code was obtained in cooperation with Mr. G. Ghor-baniasl and a brief overview of the results will be given here. More results can be foundin [55].

Figures 6.9 and 6.10 give the pressure in the time domain and the Sound PressureLevel (SPL) over one blade passing period at a radius of R = 10 and for different observerangles β. Also here the pressure and SPL are shown non-dimensionalized. Only for theobserver angle β = 30o a small deviation from the analytical solution can be seen. All theother angles show an almost perfect match. The only source for errors in this analyticalformulation is the calculation of the retarded time and integration over the blade surface.The overall errors remain small, since the integration averages them out.

Figure 6.11 and Figure 6.12 show the directivity pattern for the calculation with theFWH code. The maximum amplitude of the directivity is 5.98 10−6 at an observer angleof 82.6o, which corresponds to a relative error of 0.1%.

94

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

t

p’

LEEAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

LEEAnalytic

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

t

p’

LEEAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

LEEAnalytic

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4x 10

−4

t

p’

LEEAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

LEEAnalytic

Figure 6.5: Pressure fluctuation and SPL for β = 30o,β = 45o and β = 60o

Comparison

Table 6.2 and Figure 6.13 show the absolute error on the SPL for the first harmonic. Forobserver angles between β = 450 and β = 1200 both methods have errors of a similarorder of magnitude. While for the FWH code the errors are of the same order for allobserver angles, the LEE results show a big deterioration for observer positions near therotational axis.

While the LEE code gives the flexibility to immediately calculate the whole acousticfield, it still has to deal with discretization errors, which increase with the size of thecomputational domain, and with spurious reflected waves. The FWH code allows accurateresults for this test case, with errors independent from the observer position.

95

0 0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

2

4

6

8x 10

−4

t

p’

LEEAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

LEEAnalytic

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−4

t

p’

LEEAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

LEEAnalytic

0 0.2 0.4 0.6 0.8 1−4

−3

−2

−1

0

1

2

3

4x 10

−5

t

p’

LEEAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

LEEAnalytic

Figure 6.6: Pressure fluctuation and SPL for β = 90o,β = 120o and β = 135o

β LEE FWH

30 3.827 0.364

45 0.224 0.250

60 0.386 0.138

90 0.209 0.034

120 0.409 0.088

135 1.494 0.029

Table 6.2: Error on SPL for first harmonic (in dB)

96

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7x 10

−6

β (degrees)

Dire

ctiv

ityLEEAnalytic

Figure 6.7: Directivity LEE

35 40 45 50 55 600

0.2

0.4

0.6

0.8

1

1.2x 10

−6

LEEAnalytic

70 75 80 85 90 955.4

5.5

5.6

5.7

5.8

5.9

6

6.1

6.2x 10

−6

LEEAnalytic

Figure 6.8: Zoom Directivity LEE

97

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

−5

t

p’

FWHAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

FWHAnalytic

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

t

p’

FWHAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

FWHAnalytic

0 0.2 0.4 0.6 0.8 1−4

−2

0

2

4x 10

−4

t

p’

FWHAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

FWHAnalytic

Figure 6.9: Pressure fluctuation and SPL for β = 30o,β = 45o and β = 60o

98

0 0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

2

4

6

8x 10

−4

t

p’

FWHAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

FWHAnalytic

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−4

t

p’

FWHAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

FWHAnalytic

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3x 10

−5

t

p’

FWHAnalytic

100

101

−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

Frequency

SP

L

FWHAnalytic

Figure 6.10: Pressure fluctuation and SPL for β = 90o,β = 120o and β = 135o

0 20 40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7x 10

−6

β (degrees)

Dire

ctiv

ity

AnalyticalFWH

Figure 6.11: Directivity FWH

99

35 40 45 50 55 600

0.2

0.4

0.6

0.8

1

1.2x 10

−6

AnalyticalFWH

70 75 80 85 90 955.4

5.5

5.6

5.7

5.8

5.9

6

6.1

6.2x 10

−6

AnalyticalFWH

Figure 6.12: Zoom Directivity FWH

20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

3.5

4

β (degrees)

Err

or(d

B)

LEEFWH

Figure 6.13: Error on SPL for first Harmonic

100

6.5 Conclusions

Two numerical codes have been developed: one based on the FWH equation for thecalculation of the acoustic field generated by a moving body and another one based onthe LEE equations. As a validation for the FWH code a modified form of the test caseproposed by Tam was performed. The pressure signal in the time domain was comparedwith results obtained from the LEE code. The results of the LEE calculation were withinacceptable accuracy, though suffered from errors depending on the observer position dueto numerical discretization and boundary reflections. The results obtained with the FWHcode showed an almost perfect match with the analytical solution, where the errors arenot depending on the observer position. The FWH formulation is therefore to be preferredfor noise radiation in unbounded domains by moving bodies. The LEE however allow toget an immediately overview of the whole acoustic field for all the flow variables and canbe used for problems where the FWH formulation is not valid, such as confined flows.

101

Chapter 7

LES

7.1 Introduction

Over the past decades a huge progress has been made in CFD and at the same time theamount of computational power has grown exponentially. This lead to more and moreadvanced applications of CFD to different fields of interests, of which one is aero acous-tics. As a logic step, classical CFD techniques, such as Reynolds Averaged Navier-Stokes(RANS) simulations, were first used to calculate the generation and transport of soundin turbulent flows. However, although unsteady RANS calculation can be performed,RANS is usually assumed to be time independent and thus more sophisticated ways forthe prediction of noise generation in turbulent flows, which is a highly complex unsteadyphenomenon, is needed. The most straight forward approach would be to simulate theentire flow field down to the turbulent Kolmogorov length scales. This technique calleddirect numerical simulation (DNS) has been used in the past by Mitchell et al. [90] andMankbadi et al. [87] who focussed on the noise generation by supersonic jets. To beable to resolve all the turbulent scales existing in a flow, extremely fine grids are needed.A dimensional analysis can be done to estimate the amount of grid points needed for aDNS calculation. The Kolmogorov length scale is defined as η = (ν3/ε)1/4 where ν isthe kinematic viscosity (m2/s) and ε(m2/s3) the dissipation per unit mass which can beapproximated as ε ∼ u3/L where u is a characteristic velocity of the flow. Therefore ifthe domain of the calculation is characterized by a length scale L the number of pointsfor the DNS is of the order:

N ∼ (L

η)3 ∼ (

uL

ν)9/4 = Re9/4

This shows that for practical flows, which have high Reynolds numbers, the DNS methodis not applicable for turbulent acoustic calculations. A more promising approach, at themoment applied by many authors, is the hybrid approach where a separate computationis done to obtain the near-field velocities and pressure fluctuations, is coupled to a secondcalculation for the far field acoustics.

102

It is known that the large scale structures are contributing the most to the noisegenerated in many types of flows. Therefore the Large Eddy Simulation (LES) techniqueseems to be very suitable to calculate the near-field acoustic sources. In LES spatiallength scales are separated by a filtering procedure. Hereby the large scale structuresof the flow (bigger than the filter width) are directly calculated while the effect of thestructures smaller than the filter width on the larger structures is modelled by a subgridscale model (Figure 7.1). This method is justified by the fact that the larger structures

Figure 7.1: Large scale and subgrid scale structures

contain most of the energy and depend on the nature of the problem, while the small scalestructures are more universal in nature and thus easier to model. Since the smallest scalesare being modelled in LES, the grid does not need to resolve the turbulent fluctuationsup till the Kolmogorov length scales anymore, which makes the computational cost morefeasible.

The use of LES for noise calculation is a hot topic in current reseach. In the field ofjet noise initial work has been done by Mankbadi et al. [86] and Morris et al. [94], whoboth performed LES calculations on supersonic jet noise using the standard Smagorinskymodel. A DNS for the near-field and far-field sound radiated by subsonic and supersonic,two dimensional axisymmetric jets was done by Mitchel et al. [89]. The predicted acousticfar field was found to agree with Lighthill’s acoustic analogy predictions. More recent jetnoise calculations using LES for the near field were done by Boersma et al. [15] and Bogeyet al. [19]. Another area of interest which is being studied presently is the noise generatedby flows around airfoils and cylinders. Wang et al. [142] performed LES computationsof the flow past an asymmetric trailing edge of a flat strut which was used for the farfield acoustic calculation based on the Ffowcs Williams Hawkings equation (see chapter6). A similar approach was done by Manoha et al. [88] on a blunt trailing edge of a thickplate. Another hybrid method was done by Schroder et al. [115] who performed an LEScalculation for the near-field and solved the linearized Euler equations to transport thesound waves to the far field. Noise propagation from a cylinder in cross flow was studiedby Montavon et al. [92]. They performed LES of cylinder flow at two different Reynoldsnumber and transformed the fluctuating dipole and quadrupole sources to the frequencydomain which were then used with a direct boundary element method to calculate theradiated far field noise. Prot et al. [102] computed both, the noise generated by a cylinderand a wall-mounted half-cylinder in the framework of studying noise generation by sidemirrors of cars. Also here LES was used for the near field and the FWH formulation tocalculate the far field noise radiation.

103

This chapter focuses on the use of Large Eddy Simulation with the goal of using theobtained data to perform far field noise calculations. In the first paragraph a short intro-duction to the formulation of LES is given. The second paragraph discusses extensivelythe calculation of the cross flow around a cylinder. In the last paragraph conclusions aregiven.

7.2 Formulation

The Flow of a compressible Newtonian fluid is governed by the Navier-Stokes equationswhich can be expressed as:

∂ρ

∂t+

∂ρui

∂xi

= 0 (7.1)

∂ρui

∂t+

∂ρuiuj

∂xj

= − ∂p

∂xi

+∂σij

∂xj

(7.2)

∂ρE

∂t+

∂(ρEuj + puj)

∂xj

=∂σijui

∂xj

− ∂qj

∂xj

(7.3)

where ρ, ρui and E are the density, the momentum components and the total energy,respectively. The viscous stress tensor and the heat flux are referred to as σ and q. Inthis framework it is assumed that the fluid is a perfect gas and the system can thus beclosed with the state equation p = ρRT , where R is the perfect gas constant (for airR = 287J/(kgK))and T the absolute temperature (in K).

7.2.1 Filtering

The basic idea in LES is to filter out the small scale turbulent fluctuations and onlycompute the three dimensional large scale structures of the flow. The effect of the smallscale turbulence on the larger scales of the flow is than modelled by a so called subgridscale model (SGS).

The filtering of the Navier-Stokes equations is done by applying the convolution op-erator given in Equation 7.4:

u(x, t) =

∫

D

G(∆, x − x′)u(x′, t)dx′ (7.4)

where ∆ is the arbitrary cut-off length. The variable u can then be decomposed in itsresolved component u and the small scale part u′ defined as:

u′(x, t) = u(x, t) − u(x, t) (7.5)

In the filtering procedure of the Navier-Stokes equations it is assumed that the filter usedis linear and commutes with the derivatives.

Table 7.1 gives an overview of the most common used filters in LES. The Gaussian andthe Top Hat filter are smooth filters because they do not leat to a sharp scale separation

104

Name G(∆, x − x′) G

Gaussian√

6/φ∆2exp(

−6|x−x′|2

∆2

)exp

(−k2 ∆2

24

)

Sharp Cut-off sin(|x−x′|kc)(|x−x′|kc)

1 ifk < kc

0 otherwise

Top Hat

1∆

if |x − x′| ≤ ∆2

0 otherwise

sin(k ∆2 )

(k ∆2 )

Table 7.1: Filters

between resolved and subgrid scales in Fourier space. In a Finite Volume approach thediscretization process acts as an implicit top hat filter. In case of compressible flow amass weighted averaging (Favre averaging) is needed to filter the Navier-Stokes equationsdefined by:

u =ρu

ρ(7.6)

Applying the filtering procedure to the Navier-Stokes equations leads to:

∂ρ

∂t+

∂ρui

∂xi

= 0 (7.7)

∂ρui

∂t+

∂ρuiuj

∂xj

= − ∂p

∂xi

+∂σij

∂xj

(7.8)

∂ρE

∂t+

∂(ρEuj + puj)

∂xj

=∂σijui

∂xj

−∂qj

∂xj

(7.9)

The momentum equation can be further worked out as:

∂ρui

∂t+

∂ρuiuj

∂xj

= − ∂p

∂xi

+∂σij

∂xj

+∂(σij − σij)

∂xj︸ ︷︷ ︸A

− ∂τij

∂xj︸︷︷︸B

Term A describes the non linear interaction between the viscous stresses and is usuallyneglected. In other words it is assumed that σij given in Equation 7.10 equals σij inEquation 7.11

σij = µ(∂ui

∂xj

+∂uj

∂xi

− 2

3

∂uk

∂xk

δij) (7.10)

σij = µ(∂ui

∂xj

+∂uj

∂xi

− 2

3

∂uk

∂xk

δij) (7.11)

Term B is the divergence of the SGS stresses, given in equation 7.12, and is the partthat needs to be modelled.

τij = ρ(uiuj − uiuj) (7.12)

105

7.2.2 Subgrid Scale Models

In order to obtain a closed set of equations the subgrid tensor τij given in Equation7.12, needs to be expressed as a function of the resolved quantities. The basis for themodelling of the SGS stresses is based on the dynamics of isotropic tubulence. It hasbeen shown in the past that the effect of the subgrid scales is to drain energy from theresolved scales [27, 72, 78], which is direct consequence of the Energy Cascade. The inverseprocess, energy injected in the larger scales by the subgrid scales, is called backscatter butthis process is neglected in most of the LES simulations since it gives rise to numericalinstabilities and has so far not produced satisfactory results. The SGS model shouldaccount for this drain of energy and this can be done by describing it as an additionaldissipation. Therefore, in the same lines as RANS modelling, an eddy-viscosity µt isdefined:

τij −1

3τkkδij = −2µt(Sij −

1

3Smmδij) (7.13)

Where Sij = 12(ui,j + uj,i) is the magnitude of large-scale stress-rate tensor. The trace of

the SGS stresses, τii, can be either modelled [91], or simply be neglected [35]. In the LESimplementations of the Euranus code [74, 113] this was neglected.

To close the problem an expression for µt needs to be given. The most popular andoldest model for µt is the SGS model proposed by Smagorinsky [120]. It was obtained byperforming a simple dimensional analysis and assuming that there exist a balance betweenthe flux of energy to the subgrid scales and the viscous dissipation. More models have beendescribed in the literature. Metais et al. [98] derived a structure function model which isvery similar to the Smagorinsky model. Bardina et al. [7] assumed that the most activesubgrid scales are close to the cut of frequency and interact with the resolved scales justabove this frequency. Based on this assumption he developed the scale-similarity model.A different category of models are the one-equation models, in which an extra transportequation is solved for the subgrid-scale kinetic energy. One-equation models have beenused by Schumann [116], Horiuti [57] and Carati et al. [25], among others. However,as stated by Piomelli [104], based on the results obtained using one-equation models,the expense involved in solving an additional equation does not seem to be justified byimprovements in the accuracy. A good overview is given by Sagaut [114].

In the Euranus code [74, 113], used for the LES simulations in this work, the Smagorin-sky model was implemented. The eddy viscosity in this model is defined as:

µt = ρC∆2|S| (7.14)

with ∆ = (∆x∆y∆z)1/3 the filter width, defined implicitly by the grid, and |S| =

(2SijSij)1/2. Although the trace of the SGS stresses, τii, can be modelled [91], in this

code it was neglected as suggested by Erlebacher et al. [35]. To account for the pres-ence of walls, the eddy viscosity should be reduced to zero in the neighborhood of solidboundaries. Therefore a Van Driest damping [34] is introduced in the model which finallybecomes:

µt = ρC[∆(1 − e−y+25 )]2|S| (7.15)

where y+ = uτ yν

is the non dimensionalized wall distance. The coefficient C is a constantwhich is usually set to 0.012.

106

Germano et al. [44] originally extended the Smagorinsky model where the constantC was dynamically calculated. Later Lilly et al. [82] reformulated the model using aleast squares technique to minimize the difference between the closure assumption andthe resolved stresses. Although this model is present in the Euranus code [74, 80, 79, 113]it was not used for the calculations performed in this work (see next paragraph).

7.3 External flow past a Cylinder

Cylinder flows form an important part of turbulent flows because of the complex inter-action between large scale vortical structures of the size of the cylinder and small scaleturbulence. Many different physical phenomena occur in this flow, such as unsteady sep-aration, reattachment and vortex shedding, which makes it an suitable study case forLarge Eddy Simulation. Besides the theoretical aspect of this type of flows, also manyapplications in the real world of the cylinder flow can be found. One of these applicationsare the aeolian tones. Most people have heard the wind singing through telephone wiresor leafless trees or the rigging of ships. The aeolian tones were first studied in 1878 byStrouhal, who was primarily concerned with their frequency. The nature of the flow abouta cylinder moving through a fluid at velocity U is determined by the Reynolds numberRe = UD

νbased on the cylinder diameter D. Different flow regimes can be observed for

varying Reynolds numbers:

• Re < 0.1 This flow is also known as creeping flow. Here the inertia forces are negligi-ble and the flow remains attached over the entire surface resulting in a symmetricalflow field.

• Re = 1 − 50 In this range separation occurs near the rear stagnation point. Twosymmetrical vortices are formed which remain attached to the cylinder. The flow iscomplectly laminar and shown in Figure 7.2.

Figure 7.2: Flow around cylinder with 1 < Re < 50

• Re = 50 − 1000 Increasing the Reynolds number further causes the vortices toelongate in the streamwise direction. At a Reynolds number of 100 the vorticeswill detach themselves alternately from either side of the cylinder. The flow fieldcomprises a laminar boundary layer over the surface of the cylinder followed byvortex shedding and a wake which becomes turbulent. This stream of vorticesis known as the Karman vortex street of which the frequency of the shedding is

107

expressed by the Strouhal number (St = fDU

). The flow regime is shown in Figure7.3.

Figure 7.3: Flow around cylinder with 50 < Re < 1000

• Re = 1000− 200000 In this range the flow field remains unchanged and the bound-ary layer separates just forward of the point of maximum thickness. Transition toturbulence occurs in the shear layers. The flow regime is shown in Figure 7.4.

Figure 7.4: Flow around cylinder with 1000 < Re < 200000

• Re > 200000 In the supercritical regime the boundary layer becomes turbulenton the surface of the cylinder and therefore is more resistant to separation. Theseparation point will move behind the point of maximum thickness. The flow regimeis shown in Figure 7.5.

Figure 7.5: Flow around cylinder with Re > 200000

108

These different types of flows have been for decades extensively studied [9, 10, 96]. Arecent and comprehensive review is given by Williamson [143].

The test case considered in this chapter is the cylinder flow at Re = 3900. Thissubcritical flow is a severe test case for LES because it requires the calculation of thetransition process to turbulence. It is known from the literature that in this flow regimethe separating shear layers are very sensitive to free-stream turbulence level [13], acousticnoise [26], cylinder vibrations [29], boundary conditions [107] and cylinder aspect ratio[97]. Beaudan et al. [9] were the first to attempt a comprehensive large eddy simulationof a flow over a circular cylinder at Re = 3900. This Reynolds number was chosen be-cause of the existence of the particle image velocimetry experimental data for this testcase [85]. High-order upwind-biased schemes were used in their simulations and they ob-tained reasonable agreement with the experimental data for the mean velocity and theReynolds stresses. Differences in the inside of the recirculation zone were attributed toexperimental errors as manifested in the large asymmetry of the experimental data [9].This was later confirmed by Kravchenko et al. [73]. New experimental data was providedby Ong et al. [99] who provided data in the near wake of the cylinder downstream of therecirculation zone. Simulations with different subgrid-scale models were performed andit was concluded that the effect of the model did not appear to be significant. Mittal etal. [90] performed the same calculation using second-order conservative central-differenceschemes. However there were still noticeable discrepancies between numerical and exper-imental results at several downstream locations. This was attributed to the employed loworder scheme. Breuer [20] did an extensive investigation of the influence of discretizationschemes, subgrid scale models and spanwise resolution for the large eddy simulation of thecylinder at Re = 3900. In his study he emphasized the importance of using low-diffusiveschemes and the sufficiently high spanwise resolution.

The cylinder flow at Re = 3900 was found to be a good test case to perform a LargeEddy Simulation in the framework of CAA. The Reynolds number is low enough to bewithin the capacities of available computational power, as well does it offer source datawhich can be used to perform far field noise calculations (see chapter 6). First a coarsemesh calculation was performed to test the LES implementations in the Euranus code[74, 79, 80, 109, 113], which was then followed by a second calculation on a fine mesh.

7.3.1 Numerical Simulations

Grid

Two LES simulations have been performed using the Finite Volume Navier-Stokes SolverEuranus [74, 113] which was extended for LES [79, 80, 109]. The first simulation had acoarse mesh, and was run on one single processor (DS10L with EV67 600MHz processor).The second one used a finer mesh consisting out of eight blocks and was run on a parallelmachine (Transtec 800L Itenium2 1.5Ghz) on 4 processors. The cylinder had for bothcalculations a diameter of D = 1m and the spanwise length of it was taken to be πD. Thislength is crucial for capturing the larger spanwise structures which have wavelengths ofthe order of λz

D= 1 according to Chyu and Rockwell [28]. An O-type grid was used which

109

extended to 15 times the diameter in radial direction. For the coarse grid 48 cells wereused in the circumferential direction, 80 in the radial direction and 32 in the spanwisedirection. This mesh was stretched towards the wall to resolve the viscous sub-layer andtowards the wake zone. A uniform spacing of the mesh was used in the spanwise direction.The minimal value of y+ = uτ y

νin the first layer of cells was equal to 1. The properties

of this grid allowed only a highly under-resolved LES. The fine mesh is shown in Figure7.3.1. The same dimensions of the domain as the coarse grid were used for the fine grid.

Figure 7.6: Grid Fine Mesh

The grid consisted out of 8 blocks, each containing 128 cells in the circumferential andradial direction and 6 cells in the spanwise direction. This gave the same resolution as inthe simulations of Kravchenko et al. [73], Beaudan et al. [9] and Mittal et al. [90]. Alsohere the mesh was stretched towards the wall to resolve the viscous sub-layer and towardsthe wake zone. Uniform spacing of the mesh was used in the spanwise direction. Table7.3.1 gives an overview of the mesh properties for both cases.

Case Blocks Ncirc Nrad Nspan Ntotal Rdom × Lz

coarse 1 48 80 32 122880 15D × πD

fine 8 128 128 6 786432 15D × πD

Spatial discretization and SGS model

For the coarse grid simulation the classical second order spatial discretization was used.The fine grid simulation used the central compact scheme IC3EC2 (see section 6.3.2) inthe radial and circumferential directions. In the spanwise direction a fourth order explicit

110

central scheme was used. This was necessary in order to allow the simulation to be run inparallel. For both simulations a small amount of Jameson type artificial dissipation wasadded to stabilize the calculations.

To reduce computational time the standard Smagorinsky model was selected over thedynamic model. This was supported by reports in the literature [9, 73] on the smallinfluence of the SGS model for this test case. The Smagorinsky constant was set to itsstandard value of C = 0.012.

Time integration

The coarse grid simulation was run with a second order dual time stepping method [63, 79,80, 109] where for the inner cycles a 4-stage Runge-Kutta approach was used. Preconditionand a multigrid strategy were used to speed up the convergence in the inner cycles. Thisapproach allowed to take a time step which is of 2 orders magnitude larger than for anexplicit Runge-Kutta scheme. The time step was ∆t = 10−3s.

For the fine grid simulation an explicit 4-stage Runge-Kutta time integration hadto be used. The use of the central compact schemes did not allow for multigrid to beused and rendered the dual time stepping method inefficient. Moreover the physical timeintegration in dual time stepping implemented in the code was of second order accuracywhich could spoil the spatial accuracy gained with the central compact schemes. Thetime step used here was ∆t = 1.98 10−6s.

Boundary conditions

For both test cases the Far Field boundary conditions discussed in chapter 5 were applied.In the spanwise directions periodicity was imposed. For the coarse grid simulation astreamwise velocity of 10m

swas applied on the boundary while for the fine grid simulation

this was increased to 50ms. This allowed to avoid using preconditioning, which is necessary

for low mach number flows in a compressible solver and which would destroy the timeaccuracy. Stable numerical boundary schemes were used in both test cases.

7.3.2 Results

Both calculations were run until statistical convergence was obtained. Figure 7.7 shows onthe left the typical time histories of the computed lift coefficient CL and drag coefficient CD

as a function of the non dimensionalized time tU∞

Dfor the fine mesh calculation. Besides

the oscillations coming from the vortex shedding and the high frequency oscillations dueto the turbulence, also a low frequency modulation is observed which is typical for cylinderflows in this regime. This was also reported by Strelets [125] who observed a modulation inthe amplitudes of the lift coefficient with a period of ten times that of the vortex sheddingperiod. A frequency analysis was done on the time history of the lift coefficient which isshown on the right in Figure 7.7. The Strouhal number, which is the non dimensionalizedfrequency of the vortex shedding St = fD

U∞, obtained from this analysis equals St =

111

0.20987. This corresponds well with the experimental value of St = 0.215±0.005 reportedby Ong et al. [99].

0 10 20 30 40 50 60 70 80−1

−0.5

0

0.5

1

1.5

Time

CL a

nd C

D

CL

CD

10−2

100

102

104

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Frequency (f*D/Uinf

)

Pow

er

Strouhall number = 0.20987

Figure 7.7: Time history CL and CD

Instantaneous velocity field

Instantaneous velocity fields for the streamwise, crosswise and spanwise velocity are com-pared for the coarse and the fine grid calculation in Figures 7.8, 7.9 and 7.10 respectively.The black line in the figures corresponds to a value of zero. Light shades correspond tohigh values and dard shades correspond to low values of the velocity. Figure 7.8 showsclearly the unsteady three dimensional recirculation zone. It is observed that the flowstructures increase in size along the wake. While for the fine mesh calculation both, smalland large scale structures, are present in the wake, the coarse mesh calculation seems toonly have small scale structures very close to the cylinder. This is a result of the under-resolved grid resolution. Figure 7.9 shows the crosswise component of the velocity. Again,small scale structures are found near the cylinder which grow in size downstream in thewake. For the fine mesh calculation the Karman Vortex street is clearly visible severaldiameters downstream in the wake which is not the case for the coarse mesh calculation.Figure 7.10, which shows the spanwise velocity component, proves again that the coarsemesh can not capture the correct phenomena further downstream in the wake. Clearlyhere all the small scale structures have dissipated after a relatively short distance.

112

Figure 7.8: Instaneous streamwise velocity; fine mesh (top), coarse mesh (bottom)

Figure 7.9: Instaneous crosswise velocity; fine mesh (top), coarse mesh (bottom)

Figure 7.10: Instaneous spanwise velocity; fine mesh (top), coarse mesh (bottom)

113

The magnitude of the vorticity is shown in Figure 7.11. The two shear layers seperatingfrom the cylinder and the Karman vortex street are clearly seen for the fine mesh calcula-tion. The vortices arising from the unstable shear layer mix in with the primary Karmanvortices before propagating downstream, this is consistent with other observations in theliterature [29, 73]. To show the transition of the laminar shear layer to turbulent statean iso-surface of the vorticity is shown in Figure 7.12. The laminar structure of the shearlayer close to the cylinder is observed, evolving to a chaotic state further downstream.Again, it is noticed that the coarse mesh calculation dissipates all the small structures ina short distance downstream in the wake.

Figure 7.11: Contour lines of instantaneous vorticity 0.5 < |ω|D/U∞ < 10; Fine mesh

(top), Coarse mesh (bottom)

114

Figure 7.12: Iso-surface of instantaneous vorticity |ω|D/U∞ = 2.5; Fine mesh (top),

Coarse mesh (bottom)

Mean Flow and Turbulent Intensities

Statistics were compiled by averaging in time and in the spanwise direction. The statisticsfor the coarse calculation were compiled over a period of 6500 time steps (T = 6.5s) whichcorresponds approximately to 13 vortex shedding cycles. The statistics for the fine calcu-lation were compiled over a period of 700.000 time steps (T = 1.386s) which correspondsapproximately to 14 vortex shedding cycles. This averaging time is approximatly doublethe time as that of Beaudan et al. [9] and Kravchenko et al. [73] who both averagedover 7 vortex shedding cycles. The pressure coefficient for both calculations is comparedon the left in Figure 7.3.2 with experimental data of Norberg [96]. On the right the skinfriction coefficient Cf is compared to the LES calculation by Breuer [20]. Both calcula-tions correspond well to the reference data. Only a small improvement for the fine gridcalculation is observed due to the fact that the difference in radial resolution near the

115

cylinder surface is rather small.

−0.4 −0.2 0 0.2 0.4 0.6−1.5

−1

−0.5

0

0.5

1

1.5

X

Cp

Exp. datacoarsefine

−0.4 −0.2 0 0.2 0.4 0.6−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

XC

f

finecoarseLES Breuer

Figure 7.13: Pressure Coefficient (left) Exp. data [96]; Skin Friction coefficient (right)

LES data [20]

The averaged streamlines are similar for both calculations. They are shown in Figure7.14 where the double recirculation zones can be observed. The length of the recirculationzone for the fine mesh calculation is Lr = 1.054 which is clearly shorter than the reportedexperimental value of Lr = 1.33 ± 0.2 in [26, 96]. For the coarse mesh calculation arecirculation zone with length Lr = 0.947 was obtained. It is expected that the lowfrequent modulation of the Karman vortex street, reported by Strelets [125] and seen onthe time evolution of the drag and lift coefficient in Figure 7.7, is responsible for this. Sincethis modulation has a time length scale which is ten times that of the vortex shedding timelength scale, very long averaging is needed to get accurate results for the recirculationlength. The mean streamwise velocity along the wake of the cylinder is shown in Figure

Figure 7.14: Averaged streamlines

7.15 and compared with experimental data of Lourenco et al. [85]. Although the correct

116

recirculation length is not captured for both calculations, the correct behavior and valuesare obtained.

0 1 2 3 4 5 6 7 8−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

X

<U

>/U

inf

fineCoarseExp.

Figure 7.15: Averaged Streamwise velocity along the wake

Mean streamwise and crosswise velocity inside the recirculation zone are shown at astreamwise distance X = 1.06 and X = 1.54 in Figure 7.16 (here X is rescaled withthe recirculation length of the calculation). Both calculations are compared to DNS datafrom Tremblay et al. [136]. A good match of the streamwise velocity with the DNS datafor the fine mesh and coarse mesh calculation is found. Only outside the wake the finermesh calculation seem to recover better to the far field values. The crosswise componentmatches the DNS data less good near the cylinder (X = 1.06). However the the overallshape is very similar which was not the case in the simulations of Kravchenko et al. [73].The turbulent intensities within the recirculation zones are shown at the same downstreampositions from the cylinder in Figure 7.17. A good overall agreement, both in amplitudeand shape, compared to the DNS data of Tremblay et al. [136] is obtained for the finemesh calculation. As expected the coarse mesh calculation does not capture the correctamplitudes for the turbulent intensities, although similar shapes then in the fine meshcalculation are observed.

As already known from the instantaneous velocity fields, the coarse mesh calculationcan not capture the small scale structures of the flow further down the wake of thecylinder due to lack of resolution. This is also demonstrated in Figures 7.18 and 7.19where the streamwise velocity and the Reynolds stress are compared with the hot-wiremeasurements of Ong et al. [99] outside the recirculation zone. Figure 7.18 shows that atthese positions (X = 6 and X = 7) the fine mesh calculation still shows good agreementwith the experimental data. The coarse mesh calculation, in contrast to the positionsinside the recirculation zone, is not able anymore to capture the correct behavior of thevelocity. The same conclusions can be drawn for the turbulent intensities shown in Figure7.19.

117

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

0

0.5

1

1.5

y/D

U/U

inf

finecoarseDNS

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

y/D

V/U

inf

finecoarseDNS

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

y/D

U/U

inf

finecoarseDNS

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

y/D

V/U

inf

finecoarseDNS

Figure 7.16: Averaged Streamwise and Crosswise velocity at X = 1.06 (top) and X = 1.54

(bottom), DNS from [136]

118

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

y/D

u’u’

/U2 in

f

finecoarseDNS

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

y/D

v’v’

/U2 in

f

finecoarseDNS

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

y/D

u’u’

/U2 in

f

finecoarseDNS

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

y/D

v’v’

/U2 in

f

finecoarseDNS

Figure 7.17: Averaged u′u′ and v′v′ at X = 1.06 (top) and X = 1.54 (bottom), DNS from

[136]

−3 −2 −1 0 1 2 3

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

y/D

U/U

inf

finecoarseExp

−3 −2 −1 0 1 2 3

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

y/D

U/U

inf

finecoarseExp

Figure 7.18: Averaged Streamwise velocity at X = 6 (left) and X = 7 (right), DNS from

[136]

119

−3 −2 −1 0 1 2 3−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

y/D

u’u’

/U2 in

f

finecoarseExp

−3 −2 −1 0 1 2 3−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

y/D

u’u’

/U2 in

f

finecoarseExp

−3 −2 −1 0 1 2 3−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

y/D

u’v’

/U2 in

f

finecoarseExp

−3 −2 −1 0 1 2 3−0.03

−0.02

−0.01

0

0.01

0.02

0.03

y/D

u’v’

/U2 in

f

finecoarseExp

Figure 7.19: Averaged u′u′ and u′v′ at X = 6 (left) and X = 7 (right), DNS from [136]

120

Acoustic Results

As an initial step towards using LES data of the near field, for far field noise calcula-tions using the FWH code described in chapter 6, the obtained results for the fine meshcalculation were used as input to the FWH code. The code was extended by Mr. G.Ghorbaniasl [55] to calculate dipole noise produced by unsteady flows. As an input forthis code, the pressure on the surface of the cylinder was sampled over a time period of 3vortex shedding cycles. The solution was sampled every 150 time steps, corresponding toa sampling frequency of 3367 Hz. since the code requires to perform a Fourier transfor-mation the pressure signal had to be multiplied with a Hanning window function to makethe signal periodic. The calculation was done for two observer positions at a distanceof 100D from the cylinder at observer angles of β = 0o -inside the wake- and β = 90o.The results are shown in figure 7.20. For the observer in the wake the frequency pattern

0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t*Uinf

/D

p’ (

Pa)

10−1

100

101

−50

0

50

100

Frequency (Hz)

SP

L

0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t*Uinf

/D

p’ (

Pa)

10−1

100

101

−50

0

50

100

Frequency (Hz)

SP

L

Figure 7.20: Top: Pressure in time and frequency domain for β = 0o; Bottom: Pressure

in time and frequency domain for β = 90o

looks similar then results obtained by the acoustic calculation of Prot et al. [102]. For theobserver standing at an observer angle of 90o the frequency pattern picks up the vortex

121

shedding frequency, which was to be expected. As an estimation of the Sound PressureLevel the well known Philips correlation formula [103] for the root mean squared pressurefluctuations, defined in Equation 7.16, coming from cylinder flow noise given in equation7.17, was used.

p′2 =1

T

∫ T

0

p′2 dt (7.16)

p′2 = 0.041ρ2LDSt2U6

∞

c2∞R2

(7.17)

This formula predicts for the settings of the LES calculation a SPL of 84.3 dB. The SPLlevel based on the computed rms pressure in the far field equals 81.5 dB, which is an underprediction of 2.8 dB. The cause for this under prediction is the use of the Hanning windowfunction. The Hanning window function tends to spread out and diminish the peaks ina fourier analysis. Since only 3 vortex shedding cycles were sampled, the frequency binwas on the low side to be able to capture the vortex shedding frequency. On top of thatthe Hanning window function spread out the amplitude even more. During the processof sampling it was noted that with longer sampling the amplitude on this frequencyincreased. To further improve the results longer sampling is needed.

7.4 Conclusions

With the goal of calculating the near-field of a cylinder in cross flow for later use for farfield noise predictions, two Large Eddy Simulations were performed on a coarse and afine mesh. The results were validated and the fine mesh calculation showed very goodagreement with reference data. Where for the fine mesh calculation all the flow phenomenawere accurately captured, the coarse mesh calculation was too diffusive and showed onlyacceptable results near the surface of the cylinder. As a initial step towards far field noisecalculation using LES for the near field, a calculation was done using the FHW codedescribed in chapter 6. The results looked qualitatively promising but it is too early tomake quantitative conclusions.

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Chapter 8

Conclusions and Future Challenges

The goal of this PhD research work was to establish the necessary know-how for Compu-tational Aero Acoustic simulations. It focussed on a hybrid approach where the existingNavier-Stokes solver Euranus was extended for near field noise computations in combina-tion with an analytical transport method to propagate the noise to the far field.

Chapter 1 gives a short introduction to computational aero acoustics and chapter 2introduced the basic concepts and definition of this research area. The research resultson the spatial and temporal discretization were presented and discussed in chapter 3 andchapter 4. Chapter 5 gives a comparison of two types of boundary conditions and finallychapter 7 shows the results of a near field Large Eddy Simulation calculation.

Spatial discretization for CAA applications was discussed in chapter 3. Based onprevious developments of implicit higher order schemes in a Finite Volume context, anextension towards Finite Volume Compact Upwind schemes was performed and a thor-ough analysis of the different possible schemes was given. Two promising schemes, athird order and a fifth order scheme, were selected and extended towards controllable up-wind dissipation for computations on arbitrary meshes. These were further investigatedon several typical CAA test cases which showed that the third order compact upwindscheme is very robust to distortion of the mesh. Both schemes were implemented in theNavier-Stokes solver Euranus [74, 113] and tests showed the need for controllable upwinddissipation. As a last part the use of compact upwind schemes for LES calculation wasinvestigated but was found too dissipative for turbulent flows at low Reynolds number.

In chapter 4 a thorough overview of the combined errors arising from spatial and tem-poral discretization were given. A study was done on the optimization process of timeintegration schemes and a new method for optimization of time integration schemes wasdeveloped. The new approach is based on an optimization which combines the temporaland spatial errors to minimize the total dissipation and dispersion errors. This allows todevelop time integration schemes which are optimized for upwind type spatial discretiza-tions. The optimizations were compared to other schemes proposed in the literature andfound to show better properties. Results on CAA type problems confirmed the improve-ments of the optimizations.

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Chapter 5 discussed the stability analysis which was performed in order to obtain stablenumerical boundary schemes for the developed spatial discretizations discussed in chapter3. Secondly non reflecting boundary conditions were investigated and implemented in theEuranus code. Results showed the better ability of the radiation boundary condition toconvect voriticity waves out of the domain, but were found less robust than the Far Fieldboundary conditions for Euler calculations.

Two codes for far field noise calculations, one based on the Linearized Euler Equationsand another on the Ffwocs Williams Hawkings equation, were developed and discussedin Chapter 6. Both of the codes were tested on a fan noise problem and found to yieldaccurate results. However it was found that a drawback of the LEE code is that close tothe boundary of the computational domain the errors can be significant. The FWH farfield noise solver on the other hand showed very little influence of the observer positionon the level of errors. It was concluded that for rotational problems in an unboundeddomain the FWH solver is to be preferred.

Finally, near field calculations of a circular cylinder in cross flow at Re = 3900 usingLarge Eddy Simulations were performed and reported in Chapter 7. The results werevalidated with reference data found in the literature and showed very good agreement.The calculation was able to resolve all the physical phenomena present in a flow overa circular cylinder. The near field data was sampled in time and used as input forthe developed far field solver based on the Ffwocs William Hawkings equation. Correctqualitative results were obtained, however it was concluded that the data sampling wasnot long enough to make quantitative conclusions.

With this work the initial know-how for the numerical calculation of aerodynamicinduced noise was established. However with the current available tools it is still to earlyto tackle more advanced industrial problems. More research and development needs tobe done in order to make CAA a valuable tool for the industry. A personal vision andoverview is given below.

Although Large Eddy Simulation is by now a well researched method, specific studiesof the effect of the subgrid scale models on the radiated sound by the smallest turbulentfluctuations are few. Bodony et al. [14] were the first to suggest an acoustic subridscale model for jet noise and the only other study so far on the contribution of subgridscale models to noise that could be found in the literature was the work of Seror et al.[119]. Clearly this area is still an open question and more investigations can be donehere. With the aim of moving towards more practical and industrial CAA problems theuse of unstructured grids is unavoidable. So far the majority of the CAA research wasconducted on structured grids of which a big part were cartesian. The CFD world ismoving more and more towards use of unstructured codes with grid adaptation and it ismore than logic that the world of CAA should follow in this direction. In order to do thisthe main challenges are to extend the topics researched in this work towards unstructuredgrids. So far most unstructured codes are maximum second order accurate for the spatialdiscretization and although a few authors developed higher order schemes [48, 64], theywere not specifically developed with the problems encountered in CAA, such as diffusionand dispersion errors, in mind. Extension of the Compact Schemes towards unstructuredgrids is thus of paramount importance. The question of boundary conditions posses itselfagain for unstructured methods. Little references in the literature could be found on

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the performance of the non reflective boundary conditions on unstructured grids. Forthe calculations in the near field using Large Eddy Simulations questions on the subgridscale modelling raises. It is not clear whether the SGS models developed for structuredgrids perform well on unstructured grids. Especially the question of the filter width tobe used is unanswered. Research on this topic is up until now limited [24]. A last topicof future research coming to mind is the aspect of adaptivity of unstructured codes. Theunstructured codes presently available have local adaption models to refine the grid inzones where important flow phenomena are present. Since resolution is an importantaspect in CAA it would be interesting to look at criteria based on detection of noisesources to obtain grid refinement in important zones of noise production.

Aside from this unstructured grid approach, also more work can be done on the toolsdeveloped in this work. A problem still remaining is that of the use of compact schemesfor the spatial discretization in parallel codes. Since Large Eddy Simulation requirescalculations on large clusters of processors it is important to have a fully parallelizedcode. The solving of a tridiagonal system, necessary for the compact schemes, is notstraightforward parallelized. A second cumbersome problem is that of data management.Sampling data from the Large Eddy Simulations for the far field noise solver, produceshuge amounts of data. Research need to be done to identify new ways which allow aminimum amount of data to be stored. Another approach would be to couple the far fieldnoise solver directly to the near field solver and run them simultaneously. This howeverdecreases the flexibility for researching the near field noise generation. . Lastly it wouldbe interesting to consider higher Reynolds number flows which will require more work.Limiters for the compact upwind schemes need to be developed in case shocks occur inthe flow. The effect of the subgrid scale model becomes more important with increasingReynolds number and this also needs further investigation.

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