UPTEC F09 038

Examensarbete 30 hpApril 2009

Acoustic wave propagation and modeling turbulent water flow with acoustics for district heating pipes

Pablo Antonio Vallejos Olivares

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Acoustic wave propagation and modeling turbulentwater flows with acoustics for district heating pipes

Pablo Antonio Vallejos Olivares

The acoustic wave propagation has been studied and simulations on district heatingpipes have been performed for a new leak detection method. The new method is amodified correlation equation which can pinpoint the leak with high accuracyregardless if the sound velocity upstream is not equal to downstream. It turns outthat the dispersive nature of waves inside pipes is extremely complex and dependenton several factors e.g. shell thickness and pipe radius. The propagating modes at lowfrequencies are studied and it turns out that there are two non-dispersive anddetectable modes at low frequency for leak detection. The theory is compared toexperimental data from district heating pipes and the results are discussed. Thedistrict heating pipes are simulated in order to study the temperature and pressureinfluence on the noise. Pipes with high pressure gradient have higher noise levelcompared to low pressure gradient. Low temperature has an impact on the viscosityof the water. The high viscous flows have the ability to generate noise in the lowfrequency range which is clearly visible as a large peak in the 500 Hz region, which forlow viscous flows does not appear. The results indicate that the best frequency regionfor leak detection is between 1 kHz – 2 kHz.

Key words: Cross-correlator, iron-pipe, leak-detection, hot water flow, turbulence,viscosity, acoustic-waves, cylindrical waveguide, circular duct, dispersion, fundamentalnon-torsional modes, attenuation, Large Eddy Simulation, Ffowcs Williams-Hawkings,FLUENT, GAMBIT.

ISSN: 1401-5757, UPTEC F09 038Examinator: Tomas NybergÄmnesgranskare: Per LötstedtHandledare: Eugen Veszelei

1

Acknowledgements

I would like to thank my supervisor, E. Veszelei for guidance and his work in leak detection.

Thanks to Vattenfall which made it possible to do this thesis in the search of leaks in district

heating pipes in Uppsala, Sweden.

Many thanks to ANSYS Sweden AB for providing me the required software (FLUENT and

GAMBIT) for modeling flows and acoustics. I want to thank the support Fredric Carlsson, my

contact in ANSYS Jill Andersson and Eva Tilson.

The magnificent work in acoustic wave propagation inside iron pipes performed by M.

Lowe et al has given huge knowledge in this field, many thanks to M. Lowe et al and the

publisher of their papers Elsevier and the Royal Society for permitting having material from

the papers in this thesis.

2

Sammanfattning

Fjärrvärmenätet står för värme i t.ex. hushåll och industrier i Sverige. Man värmer upp vatten

till ungefär 100oC och skickar det via nedgrävda rör till hushållen. Det finns ett rör för

framledning av hett vatten med högt tryck och ett rör som returnerar vattnet vid lägre tryck

och temperatur. Med åren slits dessa rör och läckor uppstår i fjärrvärmenätet. Läckor i

fjärrvärmenätet är något som drabbar alla kommuner i Sverige och kostnader för reparationer

är höga. Därför har man i Generic i ett projekt för Vattenfall utvecklat en ny

läcksökningsmetod där man mäter ljudhastigheten i röret och lyssnar på läck-ljudet med

sensorer och utför en s.k. kors korrelering av signalerna som ger en tidsförskjutning.

Sensorerna är uppbyggda av piezoelektriskt material och klarar av att mäta frekvenser upp till

5 kHz. Förutom tidsförskjutningen behövs information om avståndet mellan sensorerna och

ljudhastigheten i röret för att peka ut läckan. Metoden är flexibel, effektiv och ekonomisk.

Det visar sig att ljud i rör drabbas av kraftig dispersion. Detta betyder att ljudets hastighet i

röret varierar med frekvens. Ljud i vatten är i normala fall omkring 1500 m/s, i fjärrvärme rör

har man mätt hastigheter i området 500 -1200 m/s beroende på vilken frekvens man mäter i.

Förutom dispersionen kan ljud i rör som innehåller vatten och är omslutna av ett medium

propagera i olika moder d.v.s. vid en given frekvens kan ljudet propagera med flera olika

hastigheter.

Simuleringar i FLUENT av fjärrvärmerör med läckor har utförts i detta arbete. Med

simuleringar är det möjligt att studera detaljer i rören och läckorna som kan vara svårt och

dyra i verkligheten. Man har bl.a. studerat hur flödet och akustiken beter sig för olika tryck

och temperaturer. Resultaten visar att tryck temperatur har en inverkan på ljudet. Högre tryck

i röret leder till ett högre tryck fall vid läckan som resulterar i högre läck ljud. En mätning

med läcksöknings utrustningen tar ungefär 20 sekunder. Fjärrvärme rörens tryck kan variera

varje minut och vilken inverkan de har på läck ljudet är därför viktig. Rören innehåller vatten

vid två olika temperaturer som har en stor inverkan på viskositeten. Viskositet är ett mått på

hur tjockt vattnet är och skiljer sig mycket vid 100 o

C och 40 oC. Viskositeten är en av

faktorerna som bidrar till att ljud bildas vid läckan och därför kan ljud vid olika temperaturer

skilja sig.

Detta gör att ljudets natur i rör är extremt komplicerad eftersom det finns flera moder där

alla drabbas av dispersion dessutom kan tryck och temperatur ha en stor inverkan på läckan.

Moderna och dispersionen är dessutom beroende av rörets dimensioner, vattnet och det media

röret är inbäddad i.

Detta examensarbete kommer att studera fysiken i rör för att förstå hur läck ljud bildas och

för vilka frekvenser, hur det påverkas av tryck och temperatur, vilka moder som existerar för

låga frekvenser och vilka som är detekterbara och bra kandidater för läcksökning.

3

CONTENTS

CHAPTER 1 – LEAK DETECTION AND THE CROSS CORRELATOR ................................................... 5

1.1 CLASSICAL CROSS-CORRELATOR .................................................................................................................. 5 1.2 MODIFIED CROSS-CORRELATOR .................................................................................................................... 7

1.2.1 Derivation of the Modified Cross-Correlation Equation ...................................................................... 7 1.2.2 The Instrument Steelpecker and the Time-Of-Fly .................................................................................. 9 1.2.3 The Software ........................................................................................................................................ 11

1.3 DISTRICT HEATING PIPES IN UPPSALA, SWEDEN ......................................................................................... 12 1.3.1 Difficulties in Locating Leaks .............................................................................................................. 13

1.4 THE GOALS OF THIS THESIS ......................................................................................................................... 14 REFERENCES...................................................................................................................................................... 15

CHAPTER 2 – FLUID MECHANICS AND TURBULENCE THEORY ...................................................... 16

2.1 CHARACTERISTICS OF FLUIDS ...................................................................................................................... 16 2.2 FUNDAMENTAL CONCEPTS IN FLUID MECHANICS ....................................................................................... 16

2.2.1 Density and Velocity Field .................................................................................................................. 17 2.2.2 Stress and Pressure Field .................................................................................................................... 18 2.2.3 Viscosity .............................................................................................................................................. 20

2.3 EQUATIONS IN FLUID DYNAMICS ................................................................................................................. 21 2.3.1 Continuity Equation ............................................................................................................................ 21 2.3.2 Conservation of Momentum for Viscous Fluids .................................................................................. 21 2.3.3 Equation of State ................................................................................................................................. 22

2.4 TURBULENT FLOWS ..................................................................................................................................... 23 2.4.1 Laminar and Turbulent Flows ............................................................................................................. 23 2.4.2 Eddies in Turbulent Flows................................................................................................................... 23 2.4.3 Reynolds Number ................................................................................................................................ 24 2.4.4 Turbulent Boundary Layer .................................................................................................................. 25 2.4.5 Velocity Profile Near Walls ................................................................................................................. 26

2.5 SUMMARY AND CONCLUSIONS FOR CHAPTER 2 ........................................................................................... 28 REFERENCES...................................................................................................................................................... 29 FIGURES ............................................................................................................................................................ 29

CHAPTER 3 – SOUND PROPAGATION IN WATER-FILLED PIPES ...................................................... 30

3.1 SOUND WAVES ............................................................................................................................................ 30 3.1.1 Speed of Sound in Water ..................................................................................................................... 30

3.2 SIMPLE MATHEMATICAL ANALYSIS OF WAVES INSIDE CYLINDERS WITH HARD WALLS ............................ 31 3.2.1 Radial Boundary Condition ................................................................................................................. 31 3.2.2 The Wave Equation in Circular Waveguides....................................................................................... 31 3.2.3 Axisymmetric Modes ............................................................................................................................ 32 3.2.4 Asymmetric Modes .............................................................................................................................. 34 3.2.5 Conclusions ......................................................................................................................................... 35

3.3 GENERAL SOLUTION TO WAVE PROPAGATION IN HOLLOW CYLINDERS ...................................................... 35 3.3.1 The Phase Dispersion Curves for a Vacuum-Pipe-Vacuum System .................................................... 36 3.3.2 The Non-Dispersive-Leak-Noise Velocity............................................................................................ 37 3.3.3 Modes and Displacements in Water-Pipe-Vacuum Systems ................................................................ 37 3.3.4 Pipes Surrounded by a Medium and Attenuation Characteristics....................................................... 40 3.3.5 Modes and Displacements in Pipes where csurround < VNDLN ................................................................. 41 3.3.6 Attenuation Characteristics ................................................................................................................. 43

3.4 ACOUSTIC WAVE PROPAGATION INSIDE PIPES CONTAINING SHEAR FLOW ................................................. 46 3.4.1 The Convected Wave Equation in Circular Waveguides ..................................................................... 46 3.4.2 Effects of Shear Flows in Pipes ........................................................................................................... 48

3.5 MODE CANDIDATES FOR LEAK DETECTION ................................................................................................. 49 3.5.1 Excluded modes ................................................................................................................................... 49 3.5.2 The α and α3 modes ............................................................................................................................. 50

4

3.6 EXPERIMENTS ON DISTRICT HEATING PIPES ................................................................................................ 51 REFERENCES...................................................................................................................................................... 55 FIGURES ............................................................................................................................................................ 56

CHAPTER 4 – MODELING TURBULENCE AND ACOUSTICS WITH LES AND FW-H ..................... 58

4.1 TURBULENCE MODELS IN FLUENT ............................................................................................................ 58 4.2 LARGE EDDY SIMULATION .......................................................................................................................... 59

4.2.1 Near-Wall Treatment for Turbulent Models ........................................................................................ 60 4.2.2 Meshing Turbulent Flows Near Walls ................................................................................................. 60 4.2.3 LES Grid Size ...................................................................................................................................... 61 4.2.4 LES Time-step and Courant Number................................................................................................... 62

4.3 MODELING ACOUSTICS ................................................................................................................................ 62 4.3.1 FW-H Method ...................................................................................................................................... 63

REFERENCES...................................................................................................................................................... 64 FIGURES ............................................................................................................................................................ 64

CHAPTER 5 – MODELING DISTRICT HEATING PIPES IN FLUENT ................................................... 65

5.1 DISTRICT HEATING PIPES PROBLEM DESCRIPTION ...................................................................................... 65 5.1.1 Considerations before Modeling in GAMBIT and FLUENT ............................................................... 66

5.2 INCOMPRESSIBLE TWO DIMENSIONAL PIPE WITH LES AND FW-H AT 1000C AND 10 ATM .......................... 67

5.2.1 The Solid Model and Mesh .................................................................................................................. 68 5.2.2 Stationary Flow with κ-ω Turbulence Model ...................................................................................... 70 5.2.3 Unsteady Flow with LES ..................................................................................................................... 71 5.2.4 Calculating the Acoustic Sound at the Walls ....................................................................................... 72

5.3 INCOMPRESSIBLE TWO DIMENSIONAL PIPE WITH LES AND FW-H AT 400C AND 3 ATM .............................. 73

5.3.1 Unsteady Flow with LES ..................................................................................................................... 74 5.3.2 Calculating the Acoustic Sound at the Walls ....................................................................................... 75

5.4 RESULTS FOR AN INCOMPRESSIBLE TWO DIMENSIONAL PIPE WITH LES AND FW-H AT 1000C AND 3 ATM 75

5.5 RESULTS FOR AN INCOMPRESSIBLE TWO DIMENSIONAL PIPE WITH LES AND FW-H AT 400C AND 10 ATM 77

5.6 CONCLUSIONS FOR TWO DIMENSIONAL SIMULATIONS ................................................................................ 78 5.7 COMPRESSIBLE THREE-DIMENSIONAL PIPE WITH LES AND FW-H AT 100

0C AND 10 ATM ......................... 79

5.7.1 The Solid Model and mesh .................................................................................................................. 80 5.7.2 Stationary solution with κ-ω turbulence model ................................................................................... 83 5.7.3 Unsteady compressible flow with LES................................................................................................. 83 5.7.4 Calculating the Acoustic Sound from the Leak Interior ...................................................................... 85

5.8 COMPRESSIBLE THREE-DIMENSIONAL PIPE WITH LES AND FW-H AT 400C AND 3 ATM ............................. 86

5.8.1 Unsteady compressible flow with LES................................................................................................. 87 5.8.2 Calculating the Acoustic Sound from the Leak Interior ...................................................................... 88

5.9 CONCLUSIONS FOR THREE DIMENSIONAL SIMULATIONS ............................................................................. 90

CHAPTER 6 – SUMMARY AND CONCLUSIONS ....................................................................................... 91

6.1 SUMMARY AND CONCLUSIONS .................................................................................................................... 91 6.2 FURTHER WORK ........................................................................................................................................... 92

APPENDIX 3A .................................................................................................................................................... 93

APPENDIX 5A .................................................................................................................................................... 96

APPENDIX 5B .................................................................................................................................................... 98

APPENDIX 5C .................................................................................................................................................. 100

APPENDIX 5D .................................................................................................................................................. 102

APPENDIX 5E .................................................................................................................................................. 104

APPENDIX 5F .................................................................................................................................................. 105

APPENDIX 5G .................................................................................................................................................. 107

APPENDIX 5H .................................................................................................................................................. 109

5

CHAPTER 1

Leak detection and the Cross Correlator

Leakage from pipes is a major issue for water distribution companies since it is a waste both

resources and money. Detecting noise from the buried pipes is the most efficient way to locate

leaks. Since buried pipes operate without any supervision, the leakage volume is high when

noticed. Several methods have been developed to detect leaks in pipes. Acoustic leak

detection equipment is normally used to locate the leaks e.g. cross-correlators, aquaphones

and listening rods which can be placed directly on the pipe. The methods work fine for

metallic pipes and have a high accuracy in finding leaks; however, there are complications

when measuring on district heating pipes. This causes problems for many water distribution

companies all around the world due to high costs in repairing.

1.1 Classical Cross-Correlator

The cross-correlator is a computer based method which measures the vibrations or sound

generated by the leak. Two signals are cross-correlated in order to obtain the time-lag between

them. The leak can be pin pointed by knowing the time-lag, the distance between the sensors

and the speed of sound inside the leak with simple algebraic equations. An illustrative picture

of the cross correlator can be found in figure 1.1. This has been for many years the most

useful method to detect and pinpoint leaks in buried pipes. Compared to other leak finding

systems, the cross-correlator is a low-cost system, flexible and has high accuracy in finding

leaks. The process with the cross-correlator can be subdivided into three steps; data

acquisition, signal conditioning and signal processing.

The correlator consists of sensors for detecting acoustic waves, wire-less transmitters and

receivers and an electronic processing unit. Acoustic waves can travel several hundred meters

inside pipes without attenuating too much. The leak noise is studied on both sides of the leak

with piezoelectric sensors, usually in valves or fire hydrasants. The sensors are normally

accelerometers and are attached directly on the pipe. The sensors communicate wireless in

order to avoid cables which can be disturbed by cars or other objects. The sensors will

measure the sound generated by the leak and transmit it to the electronic processing unit,

normally a computer where the two signals are processed and studied.

It is assumed that the noise propagates at a constant velocity, however, it turns out that the

nature of leak noise is highly dispersive and must be taken into account when using the cross-

correlator since the leak location becomes highly unreliable e.g. when measuring on pipes

with large cross diameter.

The raw data from the data acquisition step consist of frequencies up to a certain frequency

determined by the sampling frequency. However, not all frequencies are of interest e.g. there

can be frequencies from the electrical components, resonance response of the pipe etc. In

order to avoid unwanted frequencies and distortion, the signal will be filtered with a bandpass

filter.

6

Fig 1.1; The cross correlator and sensors. D is the distance between the sensors, L1 is the distance from the leak

to the first sensor and L2 is the distance from the leak to the second sensor. The sensors are made of

piezoelectric materials.

There are three methods in the signal processing mode that give valuable information about

the signal; auto-correlation, coherence function and cross-correlation.

The auto-correlation gives information about the frequency content of the signal (in this

case the leak). The coherence function is a measure of how related the two signals are with

each other. The cross-correlation gives information of the time-lag of the two signals. Denote

two continuous signals as S1(t) and S2(t) and the cross correlation as Rcorr(t), the cross

correlation can be obtained from

dtSStRcorr )()()( 21 (1.1)

The cross-spectral density can then be obtained from

dsesR si

corr )()( (1.2)

which will give valuable information of the frequency content of the signals.

7

The cross-correlation function has a peak that corresponds to a time-lag which is the time

difference between the arrivals of the signals at the sensors. The time-lag can be obtain from

the following equation

c

LLlag

2121 (1.3)

where η1 and η2 is the time it takes for the noise to reach the sensors from the leak, c is the

speed of sound, L1 and L2 are the distance from the sensors to the leak (see figure 1.1).

Let D be the sum of the distances L1 and L2 which is the distance between the sensors. Then

equation 1.3 can be rewritten as

2

21

1 lag

lag

cDL

c

DL (1.4)

which is the classical correlation equation where L1 is the only unknown. The distance D is

measured at site, the ηlag is calculated from equation 1.2 and the speed of sound c is assumed

to be constant in both directions of the pipe.

1.2 Modified Cross-Correlator

The correlation equation 1.4 is very limited since the speed of sound is not measured at site

and is assumed a value based on other calculations. The speed of sound is not necessarily the

same in both directions in the pipe and there are several modes in which the noise can travel

with different speed. For these reasons, a new correlation technique has been developed at

Generic Solutions Sweden in Uppsala by E. Veszelei [4].

1.2.1 Derivation of the Modified Cross-Correlation Equation

Let the distances L1 and L2 in figure 1.2 be

111 cL (1.5a)

222 cL (1.5b)

where c1 is the speed of sound from the leak to point 1 and c2 is the speed from the leak to

point 2, the time-lag can then be obtained from

2

2

1

121

c

L

c

Llag (1.6)

8

Fig 1.2; The modified cross correlator technique assumes that the velocity upstream in a pipe is not necessarily

equal to downstream. The method takes into account two velocities, one upstream and one downstream.

Denote the distance between the sensors D. The speed of sound from sensor 1 to 2 and vice

versa can be obtained from

12

12

Dc (1.7a)

21

21

Dc (1.7b)

At this point, there are four velocities where c12 and c21 is from sensor 1 to 2 and 2 to 1

respectively and c1 and c2. Assume that one mode is the strongest mode in direction 1 to 2 and

has a certain velocity c12, then the same mode must be strongest at the leak propagating in the

same direction with the same velocity from the leak to sensor 2, i.e. c12 = c2, and vice versa

from 2 to 1, the velocity c21 = c1. With these assumptions and by replacing L2 = D – L1,

equation 1.6 can be rewritten as

1221

12

1121211 ))((1 lag

lag DLLDLD

(1.8)

which is the new correlation equation. Assuming that the velocities are equal in upstream and

downstream i.e. η12 = η21, equation 1.8 falls into the classical correlation equation 1.4, as

expected.

9

1.2.2 The Instrument Steelpecker and the Time-Of-Fly

The times η12 and η21 will be referred as the time-of-flies. In order to obtain the time-of-flies

and the velocities c12 and c21, the new equipment will need a new instrument (see figure 1.3

and 1.4). The new devices are attached to the pipe surface at both ends near the sensors. The

devices, called Steelpecker will knock the pipe shell once and send a defined impulse that

propagate through the pipe, measured at the other end with respect to time and thereby

measuring the time-of-flies η12 and η21. The speeds c12 and c21 are calculated with equations

1.7a-b.

Fig 1.3; The modified correlator. Two knocking devices are needed in order to estimate the speed of sound in

both directions

Consider two modes, M1 and M2 that can exist inside the pipe with velocities c1=1 km/s and

c2 = 0.5 km/s. Assume that the Steelpeckers produce both modes which then propagate within

the pipe and that the leak also produces both modes (see figure 1.5). The mode with highest

intensity will correspond to the measure time η12 and η21 in sensor 1 and 2 and be calculated in

equation 1.8. This consideration will result in four cases;

M1 dominates at sensor 1 and 2

M2 dominates at sensor 1 and 2

M1 dominates at sensor 1 and M2 dominates at sensor 2

M2 dominates at sensor 1 and M1 dominates at sensor 2

10

Fig 1.4; The piezoelectric sensor and knock device Steelpecker.

Fig 1.5; A schematic picture over the produced sound from the leak and the knocking devices. The leak is

located in the middle yielding L1 = L2

Example 1: Consider the case where M1 is the strongest mode at sensor 1, sensor 2 and leak

which means that c1 is the dominating velocity. Let L1 =50 m, L2=100 m and D = 150 m.

Since the modes propagate with the same velocity in both directions, equation 1.7a (or 1.7b)

yields η12 = η21 = 0.15 s and equation 1.6 yields time lag ηlag = -0.05 s . Equation 1.8 and 1.4

yields L1 = 50 m i.e. both the classical and modified correlation equation give the correct

leak location.

Example 2 : Consider the case where M1 dominates in direction 1 to 2 and M2 in 2 to 1 such

that η12 = 0.15 s, η21 = 0.3 s and the measured time lag ηlag = 0. Let L1 =50 m, L2=100 m and

D = 150 m. By using equation 1.8 yields L1 = 50 m, however, the classical correlation

equation 1.4 cannot be used for this case, hence only the modified correlation equation give

the correct leak location.

11

These examples show that the new correlation equation 1.8 is correctly. However, it is

worth to mention that if M1 dominates in direction 1 to 2, M2 dominates in direction 2 to 1,

but if the leak noise is dominated by M2 in both directions away from the leak the new

correlation function will not work. The correlation in direction 2 to 1 is still correct, but in

direction 1 to 2 is incorrect because the dominating velocity c12 ≠ c2 due to different

dominating modes in that direction i.e. the difficulties is to identify the time-of-flies τ12 and

τ21 with the corresponding time-lag τlag. However, in reality it can be really difficult to pick

the time-of-flies and time-lag since there are several modes that can propagate inside the pipe

which results in several peaks.

1.2.3 The Software

The software of the equipment is implemented in LABVIEW, The software will plot the

correlation of the measured knocks from the Steelpeckers. Figure 1.6 shows a measure of a

real leak with the equipment. The upper and middle plots show the time it takes for the sound

to reach one sensor to the other where the sound is generated by the Steelpeckers. Several

peaks arise which correspond different velocities i.e. the noise in pipes does not travel with

one constant velocity c = 1500 m/s which is the speed of sound in water, but instead travel at

velocities in the range 600 – 1200 m/s. The noise is indeed influenced by the pipe. It turns out

that the phase velocity of the traveling modes is also frequency dependent. The lower plot

shows the time-lag where several peaks arise.

Fig 1.6; A picture of the program implemented in LABVIEW. The figure show the three plots; the upper show the

knock from sensor 2 to 1, the middle show the knock from sensor 1 to 2 and the lower plot the cross correlation.

The time-lag is not so identifiable but could be one of the two peaks at η ~= -10 ms.

12

The program will identify the time-of-fly of the mode with highest intensity, in figure 1.5 at

the lower left corner, the time-of-fly in one direction is η21 = 94.5 ms and in the opposite

direction η12 = 64.3 ms, which clearly shows that the velocities are not equal for the strongest

propagating mode in different directions. The program will also automatically identify the

strongest time-lag ηlag = -6.5 ms (see figure 1.5 at the lower left part). The values are

measured within a time period of 10 s.

1.3 District Heating Pipes in Uppsala, Sweden

The site at which the equipment was tested and developed is Uppsala, Sweden on district

heating pipes. The district heating pipes covers the whole town and the network consists of

two types of pipe, one hot water distributing pipe at high pressure and one water returning

pipe with cooler temperature at lower pressure. The daily leakage could be around 100-1000

m3 a day and the costs in repairing are tremendous. Therefore, it is of great importance to

pinpoint the leak with high accuracy in order to reduce the costs. A summarize of the district

heating pipes can be found in table 1.1. Note that the values given in table 1.1 can vary a lot

(denoted with ~), but this values agrees with many of the pipes in the field and will be used in

this thesis.

The pipe is embedded with insulation. The pipes are made of iron and contain water. A

cross-section of the pipes can be found in figure 1.6.

Table 1.1; Summarize of district heating pipes in Uppsala, Sweden, values that vary a lot are denoted with ~.

Common pipe diameter ~300 mm

Common pipe shell thickness ~5.6 mm

Pipe material Iron

Material Young modulus 196 GPa

Water bulk modulus 2.2 GPa

Average flow velocity ~2 m/s

Average leak diameter ~1mm

Pressure ~3 and ~10 atm

Temperature ~40 oC and ~100

oC

Total pipe length in town 450 km

Total daily leakage 100-1000 m3 (usually below 500 m

3)

Repairing cost/leak 20 000-200 000 Swedish kr

13

Fig 1.7; A cross section of the pipe.

1.3.1 Difficulties in Locating Leaks

The distance D is easy to measure, the speed of sound and time-lag can get more complicated

as described in section 1.2. The equipment has shown superior performance for metallic pipes

containing cold water; however, the equipment works fairly well for district heating pipes.

The difficulties are as follow

The speed of sound is difficult to determine even if it is measured.

The several time-lags appear and seem to be frequency dependent. (See figure 1.8)

The phase velocity of the sound is frequency dependent due to several unknown

reasons.

For one frequency, the leak sound can travel at several velocities (i.e. there are several

modes in which the sound can propagate).

The pipe pressure is time-dependent and unknown and can influence the sound

characteristics every minute.

Main differences between hot and cold water. Can the characteristics of the leak

sound change with respect to temperature?

Leak noise that propagates inside water-filled pipes cannot be assumed to travel at one

constant velocity for all pipes because it is dependent on geometry, shell thickness and

material. If the dispersive nature of leak noise is not taken into account, the correlation and

sound velocities can be difficult to determine. The time-lag is hard to identify and sometimes

it cannot be identified at all since the dispersion of waves blur the correlation. However, due

to varying pressure and temperature, the water properties like density and viscosity tends to

vary a bit. This in turn can influence noise generated by the turbulence at the leak which

affects the measured time-lag i.e. the shape of the correlation can vary with time.

14

Fig 1.8; The time-lag at a) 2 kHz b) 1 kHz

1.4 The Goals of this Thesis

The purpose of this thesis is to investigate and simulate pipe with leaks and an attempt to find

out why these difficulties (e.g. speed of sound and time-lag) occur on district heating pipes.

There are several things to investigate and some of the questions can be found below:

What water- and flow properties play a significant role and what are the main

differences between hot and cold water flows? Water and flows have several

properties e.g. pressure, compressibility and temperature, but it is not certain that all of

them influences the noise generated at the leak. A study of the water properties need to

be done in order to exclude all properties with no influence on the noise.

Since properties changes with temperature, hot and cold water must have some

differences e.g. the turbulence, boundary layer of a flow, viscosity, density etc. Those

differences must in turn have some relation to the generated frequencies at the leak

e.g. can leak in hot water pipes generate higher frequencies than cold water pipes do.

Chapter 2 will indentify some differences between hot and cold water and present a

study of the turbulent flows.

Mathematical analysis of waves inside cylindrical waveguides. By solving the wave

equation inside a cylinder the cut-off frequencies related to the Bessel functions are

obtained. This will in turn show that there can only exist discrete modes inside pipes

characterized by e.g. velocity, indices and mode. Lower frequencies needs to be

studied in order to understand the dispersive nature of low frequencies modes inside

pipes. Chapter 3 will discuss the dispersive nature of propagating modes and how the

outer medium and flows influences the propagating noise.

What mechanism generates the noise and what frequency content does it have? Does

the noise come from the pipe walls, from the turbulent flow or core etc.? Chapter 4

will present different tools in FLUENT in which the noise in a pipe can be studied.

There are several things that have to be considered before setting up a model e.g. what

15

kind of models to choose in between and why they satisfy this problem. Simulating

both cold and hot water pipes at different pressures and studying the frequencies can

yield important information about the system e.g. how pressure gradient with constant

viscosity affects the flow and noise. Chapter 5 will present a simulation of district

heating pipes in two and three dimensions where properties like pressure, viscosity,

water properties, noise and velocity field are discussed.

The thesis is divided into three parts, a pre-study of fluid dynamics inside pipes, a study of

how the noise propagates inside pipes and a study of what happens at the leak and how the

frequency content looks like. The thesis will improve the understandings of leak correlation

by studying a complex model of acoustics inside pipes and how temperature and pressure

influence the leak noise.

References

[1] Yumei Wen, Ping Li, Jin Yang and Zhangmin Zhou, Information Processing in Buried

Pipeline Leak Detection System, University of Chongqing 2004

[2] Hunaidi, O.; Wang, A., LeakFinder - New pipeline leak detection, System, National

Research Council Canada, Institute for Research in Construction, Ottawa, Canada

[3] R. Long, P. Cawley, M. Lowe, Acoustic wave propagation in buried iron water pipes,

Proc. R. Soc. Lond. A (2003) 459, 2749-2770

[4] E. Veszelei, Eugen.veszelei@generic.se, Generic Solutions Sweden, Uppsala

16

CHAPTER 2

Fluid Mechanics and Turbulence Theory

Fluid mechanics is the physics of liquid and gases. Fluid mechanics can be divided into two

fields, fluid at rest (fluid statics) and fluid in motion (fluid dynamics). The mathematics of

fluid mechanics can be very complex and numerical tools are often needed.

The main objective of this chapter is to describe the fluid dynamics of flowing water inside

a pipe. This chapter will describe briefly the study of fundamental concepts and equations,

followed by the theory of turbulent flows and turbulent boundary layers. The last section will

summarize the chapter and discuss conclusions.

2.1 Characteristics of Fluids

Fluids can be divided into gases and liquids. The most important difference between liquids

and gases is that liquids are incompressible fluids, which mean that they are difficult to

compress and gases are compressible fluids since they are much easier to compress.

Compressibility is a measure of a volume change in response to a pressure change. All

matter is compressible more or less. Consider a gas element. Increasing the pressure

surrounding the element will cause the element to shrink in size. When the gas element

shrinks in size, the density and/or temperature will vary in response to this. Gases in general

are highly compressible and are dependent on the Mach number. Liquids on the other hand

are difficult to compress and are usually classified as incompressible fluids and the density is

considered constant with respect to pressure. A pressure drop for water from 210 atmospheres

to 1 atmosphere causes the density to change with 1% [2].

2.2 Fundamental Concepts in Fluid Mechanics

Fluids are composed of molecules in motion separated with empty space between, but in most

applications, only the macroscopic effects are of interest, and not concern about individual

molecules. A complete analysis would have to take into account the action from each

molecule, but since this is not possible, only the average conditions of velocity, pressure,

temperature and density will be of interest. The fluid is assumed to be a continuum that is a

matter with no empty space. This assumption is valid for common application, but could fail

for certain conditions e.g. a gas flow in extremely low pressure. However, in this thesis, the

fluid inside pipes will be treated as a continuum.

As a result, the properties will then be represented as continuous functions such as pressure,

density, velocity and temperature with respect to the position in the fluid with no discontinues

jumps in time.

17

2.2.1 Density and Velocity Field

At any given point in time, there will be density in the fluid given by

V

m (2.1)

where m is the mass and V is the volume. However, it is of interest to know the density at a

point and not only the average on a volume. The density at a point is

V

m

VV 1

lim (2.2)

where δV1 is the lower limit in defining a density at a point. The density will depend on the

space coordinates in an instance of time, but since the density could vary in time due to for

instance temperature variations, a complete description will then be as follow

),,,( tzyx (2.3)

It is known that water is heaviest around 4oC. The density will decrease with 4% as it

increases from its heaviest form to 100oC boiling water. The density of a material can in

general vary when changing the pressure or temperature. Increasing the pressure results in

higher density and increasing the temperature will decrease the density. As a result, the

density must be pressure and temperature dependent. However, water is an incompressible

fluid; the density changes with respect to pressure are very small and can be neglected for

general applications. Figure 2.1 shows the density with respect to pressure and temperature.

Note the small difference in density with respect to the pressure.

The velocity vector v in a fluid is dependent on the coordinates x, y and z. The velocity

might vary from one point to another, and from one instant to another, therefore a complete

description of the velocity would be

),,,( tzyxvv

(2.4)

The velocity could be described by scalar components denoted by u, v and w, then

v = ui + vj + wk (2.5)

18

The components will in general be functions of x, y and z. If a flow at any point does not

change with time, the flow is said to be steady, if the flow vary in time, it is said to be

unsteady.

Fig 2.1; Density with respect to pressure and temperature for water. For atmospheric pressure 1 bar, there is a

peak at 4oC, as expected. The figure also shows that density variations with respect to pressure are very small.

2.2.2 Stress and Pressure Field

The stress is defined as average amount of force exerted per unit area. It can be determined

from the equation

A

F (2.6)

Force acting with its normal component is called normal stress, and a force acting with its

tangential component is called shear stress. If a fluid is at rest, then there are no shear forces

acting on it. Shear forces cause displacements on fluids, and are only possible if relative

movement between layers can take place. No matter how viscous a fluid is, the slightest net

shear force will cause the fluid to flow.

The normal stress εn and the shear stress ζn are then defined as

19

n

n

An

A

F

0lim (2.7)

n

t

An

A

F

0lim (2.8)

where δFn is the normal force acting on an area δAn and δFt is the tangential force acting on

δAn..

Since δF is a vector quantity, it is convenient to use components in an orthogonal

coordinate system. Assume that the normal to the plane δAn is in the x direction, thus giving

δAx, equation 2.7 and 2.8 can be written as

x

x

Axx

A

F

x 0lim

x

y

Axy

A

F

x 0lim (2.9)

x

z

Axz

A

F

x 0lim

In equation 2.9, tensor notations have been used. The first subscript indicates the plane in

which the stress acts, and the second in which direction the stress acts. If the normal to the

plane A turned out to be in the y or z direction, the definitions would be similar.

A tensor of these stresses could provide a complete description by specifying the stresses

acting on three mutually perpendicular planes through a point C. The stress tensor has nine

components and could be written as

zz

yz

xz

zyzx

yyyx

xyxx

ijT (2.10)

where ε is used to denote the normal stress, and ζ the shear stress.

A fluid has always pressure, which is the result of the collisions of molecules. A fluid

element will have a force exerted on it by the surrounding fluid or solid. When all shear

stresses are zero, the fluid is said to be at rest, this means that all non-diagonal elements in the

stress tensor are zero. The ones that are left are the diagonal components (normal stresses),

caused by the pressure, which is defined as the normal force exerted by a fluid per unit area

[1]. Pressure is a quantity that has the same magnitude in all directions, in other words, it is a

scalar quantity. This is also applicable for shear forces (fluids in motion).

20

2.2.3 Viscosity

Relative motions between fluids will cause a shear force (frictional force). Consider a relative

motion between two layers in a fluid, the acting force must be parallel to the surfaces, and the

resisting force that comes up is in the exact opposite direction (Newton’s third law), also

parallel to the surface. This resistance between fluids is called viscosity. Viscosity is

dependent not only on external conditions but the fluid itself. A more quantitative definition

of viscosity is given as follows; consider a fluid element in motion on a plate that is fixed. All

particles are moving in the same direction, but the different layers moving with different

velocity. It can be assumed that the particles are moving in a straight line along the x

direction. The rectangular element PQRS deformed to P’Q’R’S’ as it moved in figure 2.2.

The deformation rate is equal to dα/dt. Assume that the velocity xyvv ˆ)( , it can be shown

that the deformation rate is equal to the velocity gradient du/dy for small angles, as shown in

equation 2.11 [1].

dy

dv

dt

d (2.11)

Fig 2.2; a) A moving fluid element on a plate. The fluid is moving in the x direction b) A moving fluid element on

a plate at some instance later. The deformation was caused by the different velocities on the layers

The equation shows that a fluid under deformation is equivalent to the velocity gradient. For

most fluids, the rate of deformation is directly proportional to shear stress and on account to

equation 2.11, it can be written as

dy

dv

A

Fyx

(2.12)

Fluids in which shear stress is directly proportional to the rate of deformation are Newtonian

fluids [1]. Fluids like water, air and gasoline are Newtonian.

Consider two different and separated fluids equally in size under influence of a shear stress.

The deformation rate will be different for the two fluids due to that one of the fluid is more

viscous than the other one. Thus, equation 2.12 can be written as

21

dy

dvyx

(2.13)

where μ is a the constant of proportionality and called the dynamic viscosity. The viscosity

turns out to be temperature dependent for liquids and gases i.e. μ= μ(T).

2.3 Equations in Fluid Dynamics

Since the mathematics in fluid mechanic is complex, it is difficult to achieve analytical

solutions. It is only possible to achieve analytical solutions by doing simplifying

assumptions. For a full description of a moving fluid, there are five quantities to determine,

three velocity components, pressure and density.

2.3.1 Continuity Equation

The continuity equation is one of the fundamental equations in fluid motion. It is a

mathematical statement of the principle of conservation of mass. Consider a region where

mass can flow through and can be accumulated. Since matter is neither created nor destroyed,

the rate of mass that enters the region must be equal to the sum of the rate of mass that leaves

the region and the accumulated mass in the region. The differential continuity equation can be

written as

0)( vt

(2.14)

where ρ is the density, and v is the velocity vector.

2.3.2 Conservation of Momentum for Viscous Fluids

The momentum equations in fluid dynamics are called the Navier-Stokes equations. The

Navier-Stokes equation is the most general equation in describing the motion of both

compressible and incompressible viscous fluids. Navier-Stoker equation can be derived from

Newton’s second law with the assumption that the fluid stress is the sum of a diffusing

viscous term and a pressure term. The Navier-Stokes equation is written as

)()()( 2 vbavpvvt

v (2.15)

where ρ is the density, v is the velocity, p is the pressure, μ is the viscosity and b is a constant

independent of velocity or its gradients. This equation is unsteady, non-linear, second-order

and partial differential equation and solutions are only obtained for very simple flows.

Equation 2.15 can be simplified for incompressible fluids and has the form

22

vpvvt

v 2)( (2.16)

There are four unknowns in the equation, three vector components and the pressure, and only

three equations to solve. Therefore there must be a fourth equation in order to be able to solve

the problem, the continuity equation.

2.3.3 Equation of State

The continuity equation together with the Navier-Stokes equation includes four scalar

quantities. However, there is a fifth quantity that needs to be solved, namely the density. A

fifth equation is needed to complete the system of equations. The equation of state must be

added to the system of equations. The equation of state is a thermodynamic equation that

describes the state of matter of a system under some physical conditions. The equation

involves fluid quantities like pressure, temperature, volume and internal energy. The simplest

case of equation of state is reduced to

p(ρ)=c2ρ (2.17)

The equation of state for ideal gases is known as the ideal gas law [2].

RTV

mRTp (2.18)

where m is the mass, R is the gas constant, T is the temperature and V is the gas volume. The

ideal gas law is inaccurate for problems involving high pressure or lower temperature.

The case of incompressible fluids, the density is assumed constant for any value of the

pressure. For compressible liquids, the pressure and density will weakly fluctuate in space

and time; denote the pressure and density fluctuations as

ref

refppp

where p and ρ are the absolute pressure and density and pref and ρref be the reference values of

the pressure and density. Let the speed of sound be written as in equation 3.1, then the

equation of state can then be written as

23

B

pcp

ref

1

2 (2.19)

which is the equation of state for compressible liquids where B is Young´s modulus for

water and [4]

ref

B

B

pBc 12 (2.20)

2.4 Turbulent Flows

Flows inside pipes are normally turbulent, which is why flows inside pipes are complicated.

Turbulent flows are irregular and its velocity, pressure and other parameters vary as well. For

a full understanding of turbulent flow inside pipes, a boundary layer theory is also required.

2.4.1 Laminar and Turbulent Flows

Fluid flows can be categorized into two types of flow, laminar and turbulent flow. The

laminar flow is smooth and the fluid particles flow in streamlines and there is no mixing

between the adjacent fluid layers. The turbulent flow is chaotic where the fluid particles have

randomly motions in all three dimensions [1].

If a dye is dropped into a turbulent flow, the drop will disperse quickly into the fluid and

will be colored, compared to laminar flows where the color will follow a streamline. This

behavior occurs because there is macroscopic mixing between the adjacent layers. Turbulent

flows are very complex, mainly because the randomly motions occur in all directions for both

small and large particles. Laminar flow can be described mathematically without additional

information from experiments, but not for turbulent flow. Statistical techniques and

experimentally determined models are required to describe the motion of the particles but

have brought little benefit in practice. Turbulent flows cannot be described as steady, but

taking the taking the average velocity and pressure over a longer time, the term steady flow

could be used for turbulent flows [1].

2.4.2 Eddies in Turbulent Flows

In a turbulent flow, the fluid quantities like mass, momentum and scalar species fluctuate with

time and space. They arise when flows come in contact with walls or when two layers of

different velocity come in contact.

Turbulence contain eddies of varying size interacting with each others. These eddies are

identified as swirling patterns and reversed current in flows. Eddies can for instance appear in

flows behinds obstacles where the flow will create a swirl followed by a reversed current

towards the obstacle. This phenomenon is clearly visible in rivers with rocks.

24

Fig 2.3; Eddy energy cascade, the larger eddies gain their energy from the mean flow, while smaller eddies gain

their energy from the larger eddies. This process continues until molecular diffusion becomes important. At this

stage the eddies will dissipate and the energy will convert into thermal energy.

Larger eddies will carry smaller eddies. The larger eddies gain energy from the mean flow

and transfer it to the smaller ones by vortex stretching. It is an energy cascade that occurs

from the larger eddies to the smaller one. The energy from the smallest eddies convert from

kinetic to thermal energy through molecular viscosity. At which rate the smallest eddies

dissipates is determined by Kolmogorov length scale. Eddies are characterized by their

length scale L, their velocity scale u and the timescale L/u. The smallest eddies are universal

according to Kolmogorov [1]. From experimental results it has been concluded that if a flow

have sufficiently large Reynolds number, small scale fluctuations must be homogeneous and

isotropic [1].

The existence of eddies implies that there is vorticity. Vorticity is a mathematical concept

in fluid mechanics and is related to a local angular rate of rotation. This can in general not be

supported in two dimensions which is why turbulence is a physically three dimensional

phenomena. Turbulence has been stated as very intense study there are still many unsolved

problems involving turbulence.

2.4.3 Reynolds Number

Turbulent flows where first notices by Hangen 1839 when he studies flows inside pipes,

followed by systematic investigations by Reynolds in 1843 [1]. It was clear that the transition

occurred when the velocity of the fluid increased, and at some point, it changed from laminar

to turbulent. However, Reynolds showed that the velocity was not the essential for the

transition, but a dimensionless quantity named after him, Reynolds number. Reynolds

number is not only dependent on the velocity, but the viscosity, density and the characteristic

dimension of the flow.

udRe (2.21)

25

For all flows, there exists a critical value of Reynolds number Recritical,, when a transition

occurs. Experiments shows that flow are laminar if Re ≤ 2000. There is no complete theory

that explains the transition of laminar-turbulence.

2.4.4 Turbulent Boundary Layer

Fluids are usually in contact with some solid boundary. These areas are most affected by

viscosity. The molecules at the boundary will have a zero velocity and stick to it. This

condition is called No-Slip condition and is a characteristic of fluids that they do not slip at

boundaries [1].

.

Fig 2.4; The velocity profile in the boundary layer. The velocity is zero at the boundaries and rapidly increases

until the region where viscosity effects can be neglected, the inviscid region.

Since the velocity at a boundary of any fluid is zero, there must be a zone there the velocity

rapidly increases to the value of the mean velocity of the flow. This region is very thin and is

called the boundary layer. An illustration of the boundary layer can be found in figure 2.4.

The velocity will increase until it reaches the main stream where the viscosity can be

neglected; this region is called the inviscid region. The layer closest to the boundary slows

the adjacent layer due to stresses. It is the viscous force that causes the no-slip condition. The

viscosity is of great importance near boundaries like for instance walls. The boundary layer

can be laminar or turbulent, and can be determined by Reynolds number in particular regions.

Laminar flows move in streamlines, therefore it is obvious that fluid elements move in

streamlines also in the boundary layer with decreasing velocity as shown in figure 2.4, but

considering turbulent flows in the boundary, the fluid elements along the boundary cannot

move since there is no-slip at the boundaries, however, the random movements in turbulent

flows occur in all three dimensions. The question is then; can particles at the wall randomly

move in a direction perpendicular to the surface? The answer is no, the random movement in

turbulent flows perpendicular to the surface must die out when approaching the boundary i.e.

turbulence cannot exist in the region which has immediately contact with a solid boundary.

As a result, even if the main flow has a very high Reynolds number and has a turbulent

boundary layer, there is a thin layer adjacent to the boundary in which the flow is laminar,

26

known as the laminar sub-layer [2]. An illustration of the laminar sub-layer can be found in

figure 2.5. Note that a laminar boundary layer is not the same as a turbulent one. The region

in between the laminar sub-layer and the turbulent core is called the buffer region.

Fig 2.5; The layer adjacent to the boundary is laminar, i.e. there are no eddies or turbulence in that region. This

layer is extremely thin but important.

2.4.5 Velocity Profile Near Walls

The mean velocity profile can be described by the law-of-the-wall, which is a semi-empirical

formula and can be written as

)(yFU (2.22)

where U+ and y

+ are dimensionless velocities and distance from the wall given by

yuy

u

UU

where U is the local velocity, u* is the friction velocity at the wall, ρ is the density, y is the

distance from the wall and μ is the viscosity. The velocity for very small y+ reduces to [4,5]

yU (2.23)

For larger y+ > 30, the velocity can be written as [4,5]

27

ByU )ln(1

(2.24)

where κ=0.419 is the von Karman constant, B=5.1 is a constant. The region is also called the

logarithmic region. The buffer region will be influences both by the linear and logarithmic

part. The composite velocity profile can be found in figure 2.6. This model is used in

numerical calculations where the turbulent boundary layer cannot be resolved due to lack of

computer resources. The first grid point must fall somewhere between 30 < y+ < 300 [4].

In order to get calculate u*, the centre line flow can be calculated by

BU

U core

0

0

Reln

1 (2.25)

where U0+ is the core velocity and Recore is Reynolds number in the core. The equation must

be solver numerically.

Fig 2.6; The sub-divisions of the near-wall regions. The graphs are the approximation of the turbulent boundary

layer. For small y+ the velocity is linearly increasing, for high y

+ the velocity is logarithmic and the buffer layer

if the overlap between the log part and the linear part.

28

2.5 Summary and Conclusions for Chapter 2

From this section, a few conclusions can be made regarding water and its properties. In

section 2.2.6 it is stated that water is an incompressible fluids i.e. the density can be

approximated as constant with respect to pressure variations. However, some applications in

computational fluid dynamics require compressible solutions for liquids e.g. direct numerical

calculations for acoustics. This will in turn require an equation of state that describes the

pressure variations with respect to the density and for this case, equation 2.19-2.20 can be

used.

The temperature, on the other hand can change the density with at most 4%, this could in

turn have an impact on the speed of sound of a traveling wave, since the speed of sound can

be obtained from the equation of state which in turn is both density and pressure dependent.

However, the temperature will be seen as constant throughout the pipe and therefore the

equations will not be temperature dependent.

The temperature has a larger impact on the viscosity. A table of the water viscosity can be

found in table 2.1. The viscosity at 100oC is only 28% of the viscosity at cold 20

oC water,

which is a significant difference. Since cold water is more viscous than hot water, it results in

a thicker boundary layer at the walls which also could have a big impact on the frequency

content in the leak. This will be simulated in chapter 5.

Table 2.1; The water viscosity with respect to temperature.

Temperature [oC] Water viscosity [Pa∙s]

10 1.308 × 10−3

20 1.003 × 10−3

30 7.978 × 10−4

40 (simulated) 6.531 × 10−4

50 5.471 × 10−4

60 4.668 × 10−4

70 4.044 × 10−4

80 3.550 × 10−4

90 3.150 × 10−4

100 (simulated) 2.822 × 10−4

The type of flow can be examined with Reynolds number. Let the velocity be 2 m/s, density

998 kg/m3, the characteristic length 300 mm as stated in table 1.1, and viscosity as indicated

in table 2.1 for water at 100oC. Using equation 2.21 gives

6

4101.2

108.2

30.02998Rehot

which indicates that the flow is highly turbulent. For cold water at 40 o

C, the Reynolds

number is

29

6

41092.0

105.6

30.02998Recold

which indicates that the flow is turbulent, but less turbulent than for hot water. Flows in

general are turbulent inside pipes, in order to obtain a laminar flow, the velocity and the

characteristic length must decrease several orders e.g. for a laminar flow in a pipe with 30 cm

in diameter, the velocity must be of the order of 1 mm/s.

It is not necessary that Reynolds number is the same inside the leak as for the pipe. Assume

that the average velocity is 30 m/s inside the leak. The characteristic length of the leak is 2

mm, giving a Reynolds number of

6

4102.0

108.2

002.030998Re leak

which is turbulent. The average velocity inside the leak is unknown, but it is obvious that it

should be at least one order in magnitude faster. Even if the velocity would be the same as in

the pipe, the flow would be turbulent, but this is not the case in this problem. Flows inside

pipes are rarely laminar.

References

[1] Vladimir Pavlenko, Lisa Rosenqvist, Fluid mechanics Compendium version 2,

Department of Astronomy and Space Physics, Uppsala University 2007

[2] B.S. Massey, Van Nostrand Reinhold 1975, Mechanics of fluids 3rd

edition, 1975

[3] Julian L. Davis, Mathematics of Wave propagation, Princeton University Press 2000

[4] FLUENT user guide, Sep 2006

[5] N. K. Agarwal, M. K. Bull, Acoustic wave propagation in a pipe with fully developed

turbulent flow, Journal of Sound and Vibration 1989, 132(2) 275-298

Figures

Fig 2.1,

http://content.answers.com/main/content/img/McGrawHill/Encyclopedia/images/CE092500F

G0010.gif

Fig 2.4,

http://hydrogen.kaist.ac.kr/DigitalClass/Kaist/kaist/images/kaist_lecture02_02_02.gif

Fig 2.5, http://hydrogen.kaist.ac.kr/DigitalClass/Kaist/kaist/images/kaist_lecture02_02_02.gif

30

CHAPTER 3

Sound Propagation in Water-Filled Pipes

Pipes can be seen as cylindrical waveguides which will influence the acoustics inside the

pipe. Due to the circular geometry, the waves inside a cylinder are highly complex three

dimensional problems. The attenuation mechanism leads to various distortion phenomena of

the leak noise. Therefore, a study of cylindrical waveguides will be presented in this chapter.

The study will show that there are different cut-off frequencies that appear due to the wave

guide. It turns out that the propagating modes are extremely dispersive and dependent on

various factors.

At low frequency it is well known that the fundamental L(0,1) axisymmetrical mode exist.

Another mode appears when considering a water-filled pipe which at low frequencies is

characterized as predominantly axial water-borne displacements, the α mode. The different

propagating modes that appear due to the containing fluid and the embedding medium

characteristics will be discussed. The aero acoustical effects will also be discussed.

A measurement on a real leak will be discussed and compared to the theory in this chapter

and chapter 1. The last section will summarize the chapter and discuss the results.

3.1 Sound Waves

Sound waves are longitudinal waves which arise from an alternate compression and

expansions in fluids rapid enough to be an adiabatic process. Compared to electromagnetic

waves, sound waves can only travel in medium which contains matter and not vacuum. Sound

waves can also travel between several mediums.

3.1.1 Speed of Sound in Water

The speed of traveling waves can be obtained from

),(

)(

Tp

pBcliquid (3.1)

where ρ is the density and B is the bulk modulus. Note that the density is both temperature

and pressure dependent (as stated in section 2.2.1), and the bulk modulus is pressure

dependent i.e. the speed of the waves is not necessarily constant. The density can vary with at

most 4% from its heaviest form to its lightest. The bulk modulus for water is 2.15 ∙109

Pa, but

can be higher for higher pressures. Table 3.1 summarizes the speed of sound in an infinite

medium of water at different temperatures.

31

Table 3.1; The speed of sound in water with respect to temperature. Note that the highest speed is around 70-

80oC.

Temperature [oC] Speed of sound in water [m/s]

0 1403

10 1447

20 1481

30 1507

40 (simulated) 1526

50 1541

60 1552

70 1555

80 1555

90 1550

100 (simulated) 1543

3.2 Simple Mathematical Analysis of Waves inside Cylinders with

Hard Walls

District heating pipes are typically long hollow tubes with circular cross-section. Waves can

travel several hundred meters inside steel pipes in air since the attenuation is controlled by the

material attenuation of steel which is low and the leakage to the surrounding air is minimal.

The acoustic waves propagate inside the pipes and will be distorted and affected by the

boundaries of the pipe. However, acoustic waves can travel between different media, which

will not be taken into account in this section. For the pipe case, there are three medium in

which the waves can propagate; water, pipe shell and outer medium (see figure 1.2). In order

to do a simple mathematical analysis on the pipe, a few assumptions will be made.

3.2.1 Radial Boundary Condition

The pipe is much stiffer than the water and will be considered infinitely rigid in this section.

This assumption can be made since the bulk modulus of water is 2.15 GPa and iron around

118 GPa, the water stiffness is therefore much less than the stiffness of iron, thus the

displacements at the boundary are very small and the iron will undergo little deformation

compared to the water. Therefore the radial displacement or velocity must vanish at the

boundary.

3.2.2 The Wave Equation in Circular Waveguides

The wave equation will be solved for waves inside pipes with cylindrical coordinates. The

wave equation is written as

32

2

2

2

2 1

t

u

cu (3.4)

Let the propagation constant be kz, the solution for the wave equation in a cylinder with

cylindrical coordinates in positive z-direction is

)(

21 )())sin()cos((),,,(tzki

cnzerkJnCnCtzru (3.5)

where n is an integer and, C1 and C2 are constants and

222

zc kkk (3.6a)

kc (3.6b)

where k is the wave number. The derivation to this result is found in Appendix C. The

boundary conditions in radial direction are left to solve. The radial boundary condition at the

walls according to section 3.2.1 must be in this case

0

Rrr

u (3.7)

where R is the radius of the pipe, however, the boundary condition at r = 0 must be

0

0rr

u (3.8a)

00r

u (3.8b)

The boundary conditions 3.8a-b will give two types of waves that can propagate along pipes.

3.2.3 Axisymmetric Modes

Applying boundary condition 3.8a and 3.7 yields

0)0(nJ (3.9a)

33

0)( RkJ cn (3.9b)

Equation 3.9a can only be fulfilled for the Bessel function corresponding to n=0, since it is

the only Bessel function that has a zero at r = 0, equation 3.5 can be reduced to

)(

01 )(),,,(tzki

czerkJCtzru (3.10)

which will be axially symmetric in z-direction and therefore are called axisymmetric waves.

The solution to equation 3.9b is the roots of the derivative of the Bessel function with n = 0

i.e.

R

pkpRk m

cmc0

0 (3.11)

where mp0 is the m:th root of the derivative of the Bessel function with n=0, which can be

found in mathematical tables. Hence, by using equation 3.6a and 3.11 yields

2

02

R

pkk m

z (3.12)

The propagation constant is real only if k2≥

2

0

R

p m which means that a wave with wave

number k must satisfy the condition and be real in order to propagate. If a wave has a wave

number k such that k<

2

0

R

p m , then the propagating constant kz will be imaginary. Denote the

propagating constant kz = iα, where α is real and called the attenuation constant, then

tiztiziktzkieeeee zz )(

which means that the exponential zikze will no longer oscillate but will instead decay as

ze i.e. all waves where k<

2

0

R

p m will decay exponentially. By letting kz = 0, equation 3.12

can be written as

34

R

pcf

R

pk m

cutmm

cutm0

,00

,02

(3.13)

Table 3.1; Cut-off frequencies for axisymmetric modes in the audible range.

Mode m Root p´0m Cut-off frequency

f0m,cut [kHz]

1 0 0

2 3.83 6.1

3 7.02 11.17

4 10.17 16.19

5 13.32 21.2

6 16.47 26.21 (out of range)

where f0m,cut is the cut-off frequency for the mode m in a water-filled pipe with radius R. Let

the speed of sound be c = 1500 m/s, the cut-off frequencies for the first Bessel function can be

found in table 3.1. The cut-off frequency defines a frequency range in which a mode with

indices (0,m), can propagate, e.g. consider a mode with indices m = 3 and n = 0, then

according to table 3.1, the mode can propagate in the frequencies range that is larger than

11.17 kHz.

3.2.4 Asymmetric Modes

Applying boundary condition 3.8b and 3.7 yields

0)0(nJ (3.14a)

0)( RkJ cn (3.14b)

Equation 3.16a can only be satisfy if n>0 because all Bessel functions are equal to zero at r =

0 except n = 0, equation 3.5 can be rewritten as

)(

21 )())sin()cos((),,,(tzki

cnzerkJnCnCtzru (3.15)

where n>0. The equation is asymmetric around the z-axis and the modes are therefore called

asymmetric modes. Equation 3.14b yields similar results as in the previous section.

However, the cut-off frequencies for asymmetric waves are

35

Table 3.2; Cut-off frequencies for asymmetric waves in the audible range.

Mode m Root p´1m Cut-off frequency

fm,cut [kHz]

1 1.84 2.93

2 5.33 8.48

3 8.57 13.64

4 11.71 18.64

5 14.86 23.65 (out of range)

R

pcf nm

cutnm2

, (3.16)

where n>1. The cut-off frequencies for n = 1 can be found in table 3.2.

3.2.5 Conclusions

According to table 3.1 and 3.2, only one mode with indices m = 1 and n = 0 (the fundamental

L(0,1) mode) can exist at low frequencies below 5.71 kHz for district heating pipes. By

knowing how many modes that can propagate at lower frequencies and knowing its properties

e.g. the phase velocity curves and the attenuation curves with respect to frequency can be very

useful in leak detection, however, the simple model in this section does not provide any

information about the phase velocity and attenuation. The following sections will describe the

dispersive nature of waves inside pipes and how the pipe shells, inner and outer media can

influence the propagating modes in a pipe, resulting in even more modes propagating at lower

frequencies.

3.3 General Solution to Wave Propagation in Hollow Cylinders

In order to achieve a more general solution to the harmonic waves inside hollow circular

cylinders, the linear theory of elasticity and shell theory must be taken into account. In 1958,

Gazis [10] derived an intractable equation which had to be solved numerically. From the

solution of Gazis equation, three type of waves were recognized. Longitudinal waves

correspond to waves with displacement in the radial and axial direction. Torsional waves

have its displacement in θ-direction. The modes will be denoted as L(n,m) and T(n,m).

However, longitudinal and torsional waves must have equivalents in the flat plate, and

therefore n = 0 for these waves (axisymmetric waves). Waves which have n=1,2,3… are non

axially symmetric and called flexural waves (asymmetric waves). The waves are designated

as follow;

1. Longitudinal axially symmetric modes

L(0,m) m=1,2,3…

2. Torsional axially symmetric modes

36

T(0,m) m=1,2,3…

3. Flexural non axially symmetric modes

F(n,m) n=1,2,3… m=1,2,3…

Silk and Bainton [7] denoted modes with the indices m and n. This family of waves is the

ones that can propagate along the axial direction in a pipe. The waves are characterized by

their cut-off frequency , phase-velocity and attenuation.

3.3.1 The Phase Dispersion Curves for a Vacuum-Pipe-Vacuum System

The equipment described in chapter 1 are usually low frequency measurements. The

equipment can only measure frequencies up to 5 kHz, therefore, modes at higher frequency

than 5 kHz will not be discussed into detail. The only modes that exist at low frequencies are

L(0,1), T(0,1) and F(1,1) for vacuum-pipe-vacuum systems, but only non-torsional waves will

be taken into account. Torsional waves are excluded because the sensors are not sensitive to

these waves. L(0,2) exist only above its cut-off frequency which does not exist below 5 kHz,

however, it is worth to mention that F(1,2) has it cut-off frequency at 2.75 kHz, but will not

be taken into account either.

Fig 3.1; Phase velocity dispersion curves for vacuum-pipe-vacuum. The bore is 152 mm with wall thickness of 8

mm. The plot is obtained from the software DISPERSE.

Figure 3.1 shows a phase velocity dispersion curve for a vacuum-filled pipe with a bore of

152 mm and wall thickness of 8 mm plotted with the software DISPERSE. These are the

fundamental modes that exist for all pipes. There exist only two modes in the low frequency

range, L(0,1) and F(1,1). The figure has the x-axis as frequency-radius product, if the ratio of

the pipe bore and the wall thickness was constant, then pipes of any bore would scale on the

x-axis [9]. The ratio between the pipe in figure 3.1 is 152/8 = 18.75 and for the district heating

37

pipes 300/5.6 = 53.6, unfortunately, the plot in figure 3.1 cannot be used for district heating

pipes, but can give an idea of how it could look like. The wall thickness has minimal effect on

the phase velocity, but do instead affect the attenuation due to leakage. The next sections will

discuss how the wall thickness and surrounding medium influences the shape of the curves in

figure 3.1.

3.3.2 The Non-Dispersive-Leak-Noise Velocity

It has been found that four waves are responsible for most of the energy transfer for low

frequencies [6]. Three of those waves correspond to n = 0 (axisymmetric waves), and one to n

= 1 (flexural, non-axially symmetric waves). By solving the equilibrium forces for

axisymmetric waves, useful relationships of the wave number can be obtained for different

modes. The non-dispersive-leak-noise velocity is the phase velocity of a dominating

axisymmetric mode in the low frequency range that only exist when the pipe is filled with

liquid (the mode is not shown in figure 3.1) and can be written as

2

1

21

Eh

BacV fNDLN (3.17)

where cf is the velocity of sound in the contained liquid (water), B is the bulk modulus of the

medium, E is the Young´s modulus of the pipe material, a is the internal pipe radius, h is the

pipe thickness. The approximation is valid for low frequencies up to 1 kHz. The non-

dispersive-leak-noise is assumed to be a leak location technique derived by Pinnington et al

[6].

3.3.3 Modes and Displacements in Water-Pipe-Vacuum Systems

The discussions are based on the work done by R. Long et al for a 10 inch bore cast iron pipe

with a wall thickness of 16 mm [5]. The phase velocity dispersion curve for a water-pipe-

vacuum system is similar to the one shown in figure 3.1, but there is one important difference,

an extra mode appears for water-filled pipes called α shown in figure 3.2 (note that α does

not refer to the attenuation constant in section 3.2.3). This is the wave mentioned earlier in the

previous section 3.3.2. The existence for the α mode is validated in the experiments

performed by Aristérgui et al [8].

Figure 3.2 show the phase velocity dispersion for the α mode and the approximation derived

by Pinnington et al [6]. The approximation agrees for very low frequencies but deviates for

higher frequencies [5]. However, the non-dispersive-leak-noise velocity will serve as an

asymptote for the phase velocity of the α mode as shown in figure 3.3.

Figure 3.4 shows a plot for axial (blue lines) and radial (red lines) displacement for the

different modes for near zero frequencies. The amplitude (y-axis) can be arbitrary and the

scales shown in figure 3.4 correspond to normalized power that is carried by the modes in the

pipe. The F(1,1) mode has insignificant axial displacement and constant radial displacement

both inside the pipe and shell. The L(0,1) and α mode have dominant axial displacement

which are concentrated at the wall for the L(0,1) mode and inside the pipe for α mode. These

38

modes have insignificant radial displacement at low frequencies. The α mode tends to be

constant and the dominant mode for low frequencies [9]. Note that piezoelectric sensors can

only detect modes with radial displacements in the shell, which is why it is important to study

the displacements.

When going up in frequency, all modes seem to converge to one velocity (including higher

modes, see figure 3.3). However, this happens at very high frequencies, much higher than the

frequencies of interest. According to R. Long et al [5], frequencies up to 5 kHz can be

detected for signals near the leaks; therefore it is consistent to study higher frequencies up to 5

kHz. Figure 3.5 and 3.6 shows the displacements at 2.5 kHz and 10 kHz for the L(0,1) and α

mode. The results are not exact, but show how the axial and radial displacements vary with

respect to frequency.

Fig 3.2; Phase velocity dispersion for the L(0,1), α and the approximation derived from Pinnington et al.

Fig 3.3; Phase velocity dispersion curves for a 10 in bore pipe in a water-pipe-vacuum system.An extra mode α

appears for water-filled systems. The non-dispersive leak noise VNDLN is the low frequency asymptote for the α

mode.

39

Fig 3.4; The radial (red) and axial (blue) displacements at near zero frequency for a) F(0,1) mode b) L(0,1)

mode and c) α mode

L(0,1) is predominantly in axial displacement in the walls at 2.5 kHz, but at 10 kHz the

modes axial displacement has decreased in the wall and instead increased inside the pipe (see

figure 3.5). Comparing the displacements with low frequencies, the mode seems to be

dominant at the walls for low frequencies up to 2.5 kHz where the axial displacement in the

middle slightly increases, the transition occurs somewhere between 2.5 – 10 kHz and at 10

kHz and above the mode is dominant in axial direction in the pipe bore. The radial

displacement however, varies slightly for all frequencies and can be assumed to be constant

and low.

Fig 3.5; The radial (red) and axial (blue) displacements for the L(0,1) mode. a) shows the displacements at 2.5

kHz, the axial displacement is dominantly in the walls. b) The axial displacement in the wall decreases and

increases in the center.

40

Fig 3.6; The radial (red) and axial (blue) displacements for the α mode. a) shows the displacements at 2.5 kHz

and b) at 10 kHz

The α mode on the other hand is predominant in axial displacement in the bore at 2.5 kHz,

however, the axial displacement decreases in the bore at 10 kHz and is comparable with the

radial displacement. Note that the mode is predominant in radial displacement inside the

walls for all frequencies. If the mode can propagate for long distances without attenuating too

much, the mode can be detectable due to the radial displacements in the wall.

The F(1,1) mode displacements have similar properties as the α mode since they are

interlaced at higher frequencies, but the mode is more attenuative and therefore will not be

shown.

3.3.4 Pipes Surrounded by a Medium and Attenuation Characteristics

The discussed modes thus far have been surrounded by vacuum in which acoustic waves

cannot propagate. It turns out that the surrounding medium is a key parameter for the

propagating modes. In the case of infinite medium, let the waves inside the pipe have the bulk

velocity cf, the velocity of the waves in the surrounding medium to csurround. There will be

three general cases;

1. csurround < VNDLN

2. csurround > cf

3. VNDLN < csurround < cf

Note that cf is always greater than VNDLN which was mentioned earlier. The case of district

heating pipes is the one with csurround < VNDLN.

The attenuation of the waves inside the pipes depends on what the pipe is surrounded with.

When the surrounding media is a fluid, leakage is possible if the phase velocity of the wave

exceeds csurround i.e. if a mode has higher velocity than the waves in the surrounding medium,

the pipe will radiate energy in form of waves. If the pipe is embedded by a solid, both

longitudinal and shear waves will radiate and attenuate fast. Pipes embedded in vacuum will

41

have no attenuation if the pipe is elastic, and the attenuation will only depend on the

properties of the material and geometry. The propagating mode will be exited if the phase

velocity of the propagating mode is higher than the bulk velocity csurround of the embedding

medium. Exited waves will carry energy outside the pipe at a angle given by Snell’s Law. But

if the phase velocity of a mode is slower than any of the bulk velocities, the leakage becomes

imaginary and no energy will leak from the pipe [9]. Displacements still exist outside the pipe

but will decay exponentially with distance.

Once the pipe is embedded with some medium e.g. soil or insulation, then the nature of the

modes displacements that occur in the outer medium becomes important. The radial

displacements of the modes are coupled to longitudinal bulk waves and axial displacements

are coupled to bulk shear waves. The leakage into the surrounding medium is dependent on

both material properties of the pipe and the properties of the soil which in turn depend on the

frequency-radius product.

3.3.5 Modes and Displacements in Pipes where csurround < VNDLN

The results from this section are from the work of R. Long et al [5]. The phase velocity

dispersion curves for a case where csurround < VNDLN (e.g. air or insulation) can be found in

figure 3.7. Since there is a medium outside the pipe the waves can exist outside the pipe walls.

csurround is lower than the phase velocity of the α mode and therefore the mode is leaky, figure

3.8 shows that the modes radiates energy instead of decaying exponentially, however, the

mode is still propagating due to water born displacements. Besides that, a second and a third

additional mode α2 and α3 appears.

Fig 3.7; The phase velocity dispersion curves for a 10 inch bore pipe. A third mode α3 appears. The dashed lines

are the water-pipe-vacuum paths

The α2 mode follows the old path of L(0,1) at higher frequencies. The previous section

42

stated that the attenuation was dependent of the bulk velocity of the embedded medium and

the velocity of the propagating mode, however, the α2 mode is very attenuative at low

frequencies even if the phase velocity is much less than csurround. The mode can be considered

non-propagating at low frequencies and therefore the mode is not appropriate for leak finding

[5].

The asymptote of the α3 mode is csurround and is always non-leaky because the velocity is

always below csurround. The mode decays exponentially in the embedding medium but can still

be a good candidate for leak finding due to its radial displacements in the wall, see figure 3.9.

The displacement for the L(0,1) mode is similar to the ones shown in figure 3.5 except that

its leaky and attenuative for all frequencies. The mode can be detected at higher frequencies

due to that the mode has more water borne displacements and radial displacements in the

walls (see figure 3.5). The F(1,1) mode is highly attenuative at low frequencies and is

therefore not appropriate for leak finding. For a more detailed study on modes inside pipes

see [5].

Fig 3.8; The radial (red) and axial (blue) displacement for the α mode in a 10 inch bore iron pipe. a) the

displacements at 2.5 kHz b) the displacements at 10 kHz

Fig 3.9; The radial (red) and axial (blue) displacement for the α3 mode in a 10 inch bore iron pipe. a) the

displacements at 2.5 kHz b) the displacements at 10 kHz

43

3.3.6 Attenuation Characteristics

The results from this section are from the work of R. Long et al [9]. The material properties of

this section are found in table 3.3. Figure 3.10 shows the attenuation dispersion curves for the

modes L(0,1), F(1,1) and α mode for a pipe containing water surrounded by saturated soil

where the bulk shear velocity Cs vary between 25 to 100 m/s, the longitudinal velocity CL =

1500 m/s. The longitudinal velocity inside the pipe is cf =1500 m/s. The y-axis is plotted as

the product of wall thickness (mm) and attenuation (dB/m) and the x-axis as the frequency-

radius (MHz mm). The peaks occur due to resonance phenomena.

The L(0,1) modes attenuation at low frequencies is dominated by the shear velocity Cs in

the soil. Since the mode is predominant in axial displacements in the wall, bulk shear waves

will be excited into the surrounding medium since Cs < CL(0,1) where CL(0,1) is the phase

velocity of L(0,1) i.e. the phase velocity exceeds the bulk velocity in the soil.

The F(1,1) and α modes have their phase velocity below the bulk longitudinal velocity CL

for all frequencies and will not leak longitudinal waves into the soil, but will instead leak

shear waves. Figure 3.10 shows that the α mode has less attenuation at lower frequencies due

to water-borne displacements, low displacements in the pipe wall and is the dominant mode

over long distances which is promising for leak detection. At higher frequency-radius product

the F(1,1) and α mode seem to have similar attenuation characteristics, however, only

frequencies below 5 kHz are of interest which correspond to a frequency-radius product of

approximately 0.38 and less. F(1,1) is the mode which suffer most attenuation at lower

frequencies (compare with α2 properties, modes which phase velocity goes to zero as the

frequency goes to zero are attenuative). Note, district heating pipes have almost double the

bore size of the pipes described in figure 3.10 and the resonance peak could fall lower in the

frequency-radius range.

Table 3.3; Material properties of the water-pipe-soil system

Material Density [kg/m3] Bulk longitudinal

velocity CL [m/s]

(radial dependent)

Bulk shear velocity CS

[m/s]

(axial dependent)

Water 1000 1480 0

Iron pipe 7100 5500 3050

Saturated soil 1000 1500 25-100

Unsaturated soil 1900 250-1250 100

Figure 3.11 shows another case of water-pipe-unsaturated soil system where the shear

velocity of in the soil is Cs=100 m/s and the longitudinal velocity CL varies between 250 to

1250 m/s. The attenuation dispersion curves differ from the ones shown in figure 3.10.

The L(0,1) mode does not differ with the different values on the longitudinal velocity CL,

which means that it is governed by Cs. The F(1,1) and α mode on the other hand attenuate

much more than the case with saturated soil. The reason is that they leak more since the phase

velocity exceeds the longitudinal velocity CL and therefore is governed by it. When the

velocity CL is greater than the phase velocity if F(1,1), the dispersion curve is very similar to

figure 3.10b and the mode is leaking shear waves only (the case water-pipe saturated soil).

44

When CL is below the phase velocity of F(1,1) the attenuation increases rapidly due to

longitudinal and shear wave leaking.

Fig 3.10; The attenuation dispersion curves for a water-pipe-saturated soil system with 152 mm bore and 8 mm

thick wall cast iron pipe where CL=1500 and Cs varies from 25-100 m/s. a) L(0,1), b) F(1,1) c) α

45

Fig 3.11; The attenuation dispersion curves for a water-pipe-unsaturated soil system with 152 mm bore and 8

mm thick wall cast iron pipe where, Cs=100 m/s and CL varies from 250-1500 m/s. a) L(0,1), b) F(1,1) c) α

The α mode have still low attenuation at low frequencies due to water borne characteristics.

The attenuation however, will increase much more rapidly at higher frequencies than the

previous case. The reasons are the same as for F(1,1) mode, they leak both longitudinal and

shear waves. The α mode is in this case to the dominating mode for long distances.

Experiments performed by R. Long et al [9] verified that α mode was the dominating mode

for the case water-pipe-saturated soil at 1 kHz which correspond to a frequency-radius

product of 0.08, unfortunately, both L(1,0) and F(1,1) where lost in noise since they suffered

most attenuation as expected.

The α2 mode is considered non-propagating at low frequencies and will therefore be not

discussed.

46

Fig 3.12; The α3 mode properties for a 10 inch iron pipe with wall thickness of 16 mm for a varying soil bulk

longitudinal velocity CL from 700 to 1200 m/s a) phase velocity b) attenuation

Figure 3.12 shows the phase velocity and attenuation for the α3 mode. Note the similarities

between the α and α3 mode. The mode have low attenuation at low frequencies but increases

rapidly at higher frequencies (faster than α) and remain very high even after the resonance

peak. The mode is clearly detectable at lower frequencies for longer distances according to

figure 3.12.

3.4 Acoustic Wave Propagation inside Pipes Containing Shear Flow

The work thus far has described acoustics mode inside a water-filled pipe surrounded with

vacuum and a medium. In the real case, the pipes contain a turbulent water flow which can be

described by Navier-stokes equation (equation 2.16 for incompressible fluids). The

phenomena can be described by aeroacoustics, which is the science of motion of fluids

interacting with objects.

3.4.1 The Convected Wave Equation in Circular Waveguides

The convected wave equation derived by Mungur and Plumblee [11, 12] can be written as

0221 2

02

22

2

2

2

2p

z

v

r

Mcp

z

pM

tz

p

c

M

t

p

c (3.21)

where M is the Mach number, p is the pressure, ρ is the density and c is the velocity of sound

the equation can be derived from the Navier-Stokes, continuity equation and equation of state

described in section 2.3. The effects of viscosity and thermal conductivity are neglected and

the system is considered isentropic with constant temperature through the system which

means that the velocity c is constant. Equation 3.21 falls into equation 3.4 if the Mach number

M = 0 which means that the water is not flowing. In the case of M = constant, equation 3.21

47

can get rid of the fourth term, however, the real case is that M = M(r).

The solution to equation 3.21 is [11]

)(

21 )())sin()cos((),,,(tzki zerPnCnCtzrp (3.22)

where n is an integer that appears due to continuity boundary condition in circumferential

direction similar to equation 3.5, C1 and C2 are constants. P(r) corresponds to the radial

distribution function and has the form [11]

)()()1(1

212

22

2

2

2

rPaR

cnakaMk

c

a

dR

dP

dR

dM

aMk

ak

RdR

Pdzz

z

z (3.23)

where R = r/a is a non-dimensional variable, a is the pipe radius, kz is the propagation

constant in axial direction. If M = 0, equation 3.23 becomes the normal Bessel functions as

expected for a non-moving medium. The boundary condition are the same as the regular wave

equation and can be found in equations 3.7 – 3.8a-b, and yields both axisymmetric (n = 0) and

asymmetric modes (n > 0).

Considering the case M = 0, the solutions to the equations falls to equation 3.10 and 3-15

that has already been discussed. The case where M > 0 and dM/dR=0 (uniform flow, the no-

slip condition is not taken into account), the solution reduces to [11]

22 )()()( akaMkkaa

rJrP zzn (3.24)

where k is the wave number. Applying the boundary condition 3.7 yields

2

2222 )()(0))()((a

pkMkkakaMkkaJ nm

zzzzn (3.25)

where p´nm correspond to the roots to the derivative of the Bessel functions. For axisymmetric

waves n = 0, by letting M = 0 equation 3.25 can be rewritten as equation 3.12, as expected. It

is by now clear that the flow has influence on the modes cut-off frequency and is governed by

the flow Mach number, however, the Mach number for district heating pipes is very low

(approximately 0.00133), and the flow compared to the acoustic waves can be seen as frozen.

The effects of flow are minimal and can be neglected. Cut-off frequencies will be unaffected

by the flow (see [11] for detail studies of the cut-off frequencies).

48

3.4.2 Effects of Shear Flows in Pipes

The results of this sections are based on the work of N.K. Agarwal and M.K. Bull [11]. They

solved equation 3.23 numerically for shear flows.

For the dispersion curves, it turns out that the curve for shear flow lies in between for those

of no flow and uniform flow (see figure 3.13). The flow will translate the dispersion curves

towards lower frequencies, and lower the values of the propagating constant kz. The shapes of

the three curves are similar. Note where the dispersion curves for M = 0.2 and M = 0, the

dispersion curves are very near each other, a flow with very low Mach number should have

minimal effects on the dispersion and can be neglected.

The special case of Kx = 0 is the only point where the dispersion curve for no flow, uniform

and shear flow coincide. This is the case where kz = 0 which according to equation 3.24

correspond to the cut-off frequencies to the no flow case. This can be derived by letting kz=0

in equation 3.23.

The modes properties are also more affected by the flow at higher frequencies than low

frequencies [11]. Near its cut-off frequency or near zero frequency, modes are less unaffected

by the effects of flow which is another argument to neglect effects of flow. For more details

of effects of shear flow on acoustic waves, see [11].

Fig 3.13; The dispersion curves for the modes L(0,1), F(2.4) and F(5.5) for Mach number M=0, 0.2, 0,6. --- no

flow, – - – uniform flow and ——— fully developed turbulent flow. For negative values of Kx, the wave is

propagating in opposite direction in the pipe.

49

3.5 Mode Candidates for Leak Detection

The L(0,1) and F(1,1) modes are fundamental and exist for all pipes. The α mode exists for

any water-filled pipe with outer medium and is therefore governed by the water. The α2

appears whenever there is any outer medium where waves can propagate and the α3 mode

appears for the case csurround < VNDLN (e.g. air). There are in total five modes which can

propagate in district heating pipes.

The leak correlator described in chapter 1 has sensors and Steelpeckers placed directly on

the pipe. The sensors, which are usually based on piezoelectric materials, are mainly sensitive

to modes that have radial displacement in the pipe shell. The district heating pipes are

surrounded by insulating material which can be approximated as air, therefore the dispersion

curves in figure 3.10 is the closest model to the real case in this chapter. However, there are a

lot of parameters and characteristics to consider and there is still a lot do to before a good

model can be presented. The modes which can yields good measurements results with the

leak correlator should have the following properties:

Radial displacements in the pipe shell

Not purely water-borne (cannot be detected in the outer pipe surface)

Velocity less than the speed of sound in water (<1500 m/s) due to attenuation, the

higher velocity the higher attenuation.

Not too attenuative or leak too much

Not too many modes in the frequency range, which can yield many peaks in the

measurements which can get confusing.

3.5.1 Excluded modes

The L(0,1) is not appropriate for leak detection due to several reasons. The mode phase

velocity is very high for low frequency and is therefore leaky and highly attenuative. The

mode is also axially dominant at near zero frequencies in the shell which the piezoelectric

sensors cannot detect, however, even if the mode is not good for leak detection, the mode

becomes more water-borne axially dominant at higher frequencies and have radial

displacements in the shell which means that it is detectable. The mode is also sensitive to pipe

joints [9].

The F(1,1) is also excluded due to high attenuation at low frequencies. The mode phase

velocity is very low at low frequencies. The mode is predominant in radial displacements but

experiments verify that the mode is lost in noise. Not that at higher frequencies the mode

seems to have similar properties as the α mode but is still too attenuative to detect.

The α2 mode for the case csurround < VNDLN, the mode has special behavior at low frequencies

and is considered non-propagating and therefore excluded. At higher frequencies the mode

has similar behavior as the L(0,1) mode. For both cases, the mode is hard to detect and have

not been experimentally verified [5].

50

Table 3.8; Summarize of propagating modes at low frequencies in district heating pipes, results are based on

both empirical studies and the work of R.Long et al [5,9].

Mode Detectable Frequency Non-dispersive Velocity

L(0,1) Hard to detect due to dominant in axial motion

and high phase velocity but is detectable

>1kHz ~2.5 km/s

F(1,1) No, too attenuative None Dispersive

α2 Never been detected None Dispersive

3.5.2 The α and α3 modes

The α mode show promising results and can be detected with a phase velocity of

approximately 1.185 km/s according to the non-dispersive-leak-noise equation (see section

3.6). The mode is unlikely detected at very low frequencies (below ~300 Hz) since the mode

is purely water borne with negligible radial displacement. At frequencies above 2 kHz the

mode can be hard to detect for long distances since it is leaky and attenuate. This mode

suffers less from pipe joints, attenuation and can be detected at a distance of roughly 175 m

according to experiments performed by R. Long et al [5]. The α mode is the dominating mode

in pipes and is appropriate for leak finding in the frequency range ~0.3 – 2.5 kHz, depending

on pipe geometry, soil and wall thickness. The VNDLN can be a good start point.

The α3 mode is always non-leaky since the velocity is always below csurround and has

considerable displacement in the outer medium and is also detectable due to radial

displacements in the pipe shell. The mode has a large amount of its energy in the outer

medium and the attenuation is governed by the embedding medium and can therefore be hard

to predict since it will require attenuation knowledge of the embedding medium at site. The

mode can be detected to up to 175 m according to experiments performed by R. Long et al

[5]. The asymptote for the α3 mode is unknown for district heating pipes, but empirical studies

shows that the phase velocity is around ~630 m/s, which could be used as a start point for this

mode.

Figure 3.14 shows where α and α3 mode are dispersive and non-dispersive. At low

frequencies the modes can be assume to propagate at constant velocity. The dispersive region

starts around 1.7 – 2 kHz and the modes are highly dispersive above 2.5 kHz. It is expected to

find two time-lags on the measurements which correspond to α and α3. It is important to know

which peak corresponds to which time-of-fly (η12 and η21) in order to have a good correlation,

if not the correlation will yield false leak locations. The challenge is to identify the time-of-fly

and a good start point is to look at figure 3.14 or calculate the VNDLN, for the pipe in order to

know what to expect.

Table 3.9; Summarize of propagating modes at low frequencies in district heating pipes, results are based on

both empirical studies and the work of R.Long et al [5,9].

Mode Detectable Frequency Non-Dispersive Velocity

α Yes, dominating mode over long distances 0.3 – 2.5 kHz ~1.2 km/s

α3 Yes, mode is non-leaky, can be detectable at

higher frequency

0.3 – 4 kHz ~630 m/s

51

Fig 3.14; The α and α3 mode phase velocities. The figure shows where the modes are dispersive and non-

dispersive. The model is based on empirical studies. The asymptote for α mode is calculated from equation 3.17

and for α3 on empirical studies.

3.6 Experiments on District Heating Pipes

The non-dispersive-leak-noise for the pipes described in this chapter can be a key equation for

α modes since it describes phase velocity of the mode at low frequencies. Using equation 3.17

with the values of table 1.1 gives a non-dispersive-leak-noise of

smmmGPa

cmGPasmVNDLN /1185

6.5196

152.221/1500

2

1

The approximation of α mode phase velocity in equation 3.17 is only valid for low

frequencies. In order to have a good approximation, the frequency must stay below or around

1 kHz. At higher frequency than 1 kHz, the velocity slightly starts to decrease.

The α3 mode is also expected under its asymptote which is unknown, but assuming that the

insulating material sound velocity is close but above the velocity of sound in air, it should be

above 340 m/s (probably around 600 m/s due to empirical studies).

This section will compare the theory and an experimental data taken on district heating pipe

in Uppsala, Sweden.

Leak is located at L1 = ~26 m

Sensor distance D = 62 m

α mode expected at ~50 ms

α3 mode expected at ~95-100 ms (assuming that the speed is between 600 - 650 m/s)

52

Fig 3.15; The figure is taken from the software described in chapter 1 for the modified cross correlator at 500

Hz with a sensor distance of 62 m. a) Knocks measured in sensor 1 performed at sensor 2, b) knocks measured

in sensor 2 performed at sensor 1 and c) the cross correlation between the signals.

Figure 3.15a shows a measurement of the time-of-fly η21 which is the time it takes for the

sound to reach sensor 1 from sensor 2 at 500 Hz and figure 3.15b shows the measurement of

η12 which is the time it takes to reach sensor 2 from sensor 1 (use equation 1.7a-b to convert

into velocities). The α mode is not identified in figure 3.15a-b and the α3 mode could be the

peak at 95 ms in fig 3.15a and the broad peak at 92 in figure 3.15b.

The software automatically pick the peaks with highest amplitude which is for this case ηlag

=-14 ms and the time-of-fly η21 = 64 ms and η12 = 82 ms and by using equation 1.8 yields L1

= 28.9 m.

Choosing the peaks corresponding to α3 yields L1,α3 = 25.9 m which is closer. This could

indicate that the time-lag at -14 ms correspond to the α3 mode.

53

Fig 3.16; The figure is taken from the software described in chapter 1 for the modified cross correlator at 1000

Hz with a sensor distance of 62 m. a) Knocks measured in sensor 1 performed at sensor 2, b) knocks measured

in sensor 2 performed at sensor 1 and c) the cross correlation between the signals.

Figure 3.16 show the same as figure 3.15 but at frequency 1 kHz. The plot tends to have

more noise than the previous one and also more peaks. The α mode is identified at 50 ms and

51 ms in figure 3.16-a-b respectively. The α3 mode is not clearly identified in figure 3.16a but

could be the peak at 98 ms. The peak at 95 ms in figure 3.16b is a good candidate for the α3

mode. Both the α and α3 mode are visible at 1 kHz.

The correlation shows a time-lag ηlag = -7 ms and the software will pick the time-of-fly η21 =

65 ms and η12 = 95 ms which yields L1 = 34.1 m, almost 8 m away from the leak.

By applying the correlation equation 1.8 on the α mode peaks at η21 = 50 ms, η12 = 51 ms

and using the same time lag ηlag = -7 ms yields L1,α= 27.0 m which is better since it is only 1

m away from the leak.

Note the peak at ηlag = -14 ms which in the previous example was considered to correspond

to α3, by applying this time-lag to the α3 peaks at η21 = 98 ms and η12 = 95ms yields L1,α3

=26.0 m.

For this measurement, the leak could be located with two different modes. Note also that the

intensity of the time-lag is also higher than the previous measurement; a possible reason could

be that the radial displacements are higher at 1 kHz than 500 Hz.

54

Fig 3.17; The figure is taken from the software described in chapter 1 for the modified cross correlator at 2000

Hz with a sensor distance of 62 m. a) Knocks measured in sensor 1 performed at sensor 2, b) knocks measured

in sensor 2 performed at sensor 1 and c) the cross correlation between the signals.

Figure 3.17 shows measurements at 2 kHz. The α3 mode is clearly identified at 98 ms and

95 ms in figure 3.17 a-b respectively. New peaks have arise at -28 ms in figure 3.17a and at

21 μs in figure 3.17b which could be the L(0,1) mode.

The software will pick the time-lag at ηlag = -14 ms according to figure 3.17c, the time-of-

fly η21 = 103 ms and η12 = 98 ms in figures 3.17a-b yielding L1 = 25.9 m.

By taking the peak at 98 ms instead of 103 ms in figure 3.17 a instead will yields L1,α3 =

26.0 m. The calculations are likely based on the α3 mode. Note the high intensity of the time-

lag.

The α mode is not visible for figure 3.17a. In figure 3.17b the α mdoe could be the very

small peak at 52 ms. Figure 3.17c shows that there is a secondary peak at -9 ms, if this peak

correspond to the α mode, the change from -7 ms to -9 ms could only be explained by that the

mode is starting to become slower and dispersive.

55

Fig 3.18; The figure is taken from the software described in chapter 1 for the modified cross correlator at 4000

Hz with a sensor distance of 62 m. a) Knocks measured in sensor 1 performed at sensor 2, b) knocks measured

in sensor 2 performed at sensor 1 and c) the cross correlation between the signals

Figure 3.18 shows the plot at 4 kHz. The dominating peak is located at -70 ms for figure

3.18a and at 118 ms for 3.18b. Note that at 4 kHz, the modes are highly dispersive and the

velocity for both the α and α3 are expected to be lower. There is a clear peak at -70 ms for

both figure 3.18a and 3.18b. This corresponds to a velocity of 890 m/s and could be a

candidate for the α mode. The peaks located at 118 ms in 3.18 a-b are a candidate for the α3

mode with a corresponding velocity of 530 m/s. However, the time-lag is hard to identify and

the leak cannot be pinpointed.

The experimental data in this section show that it is not an easy task to pinpoint the leak, the

α mode did not show up as the strongest peak with the Steelpeckers and did only show up

clearly at 1 kHz with a speed of ~1.2 m/s to both directions as expected. At 2 kHz, the α was

not identifiable but instead the α3 mode showed up as a broad peak around 630 m/s. At higher

frequencies, the time-lag is hard to identify and the leak could not be pinpointed. With these

modes and corresponding time-lags, the results show that the leak is located at L1 ≈ 26 m.

References

[1] Mathematics of Wave propagation, Julian L. Davis, Princeton University Press 2000

[2] Computational Fluid Dynamics, T.J. Chung University of Alabama in Huntsville,

Cambridge University Press 2002

[3] Antonio Kondis, Acoustical wave propagation in buried water filled pipes, Massachusetts

Institute of Technology 2005

56

[4] Richard Haberman, Applied Partial Differential Equations with Fourier Series and

Boundary Value Problems fourth edition, Pearson Education 2004.

[5] R. Long, P. Cawley, M.Lowe, Acoustic wave propagation in buried iron water pipes,

Proceedings of the Royal Society 2003 459, 2749-2770

[6] Pinnington, R. J. & Briscoe, A. R. 1994, Externally applied sensor for axisymmetric

waves in a fluid filled pipe, Journal of Sound and vibration 173, 503-516

[7] Silk, M. G. & Bainton, K. F. 1979, The propagation in metal tubing of ultrasonic modes

equivalent to Lamb waves. Ultrasonics 17, 11-19

[8] C.Aristégui, M.J.S. Lowe, P. Cawley, Guided waves in fluid-filled surrounded by different

fluids: Prediction and measurements, in: D.O. Thompson, D.E. Chimenti (Eds.), review of the

progress in Quantitative ND E, Plenum Press, New York, 1999, vol 18A, 159-166

[9] R. Long, P. Cawley, M.Lowe, Attenuation characteristics of the fundamental modes that

propagate in buried iron water pipes, Ultrasonics 41 (2003) 509-519

[10] C. Gazis, Three-Dimensional Investigation of the Propagation of Waves in Hollow

Circular Cylinders. I. Analytical Foundation, Journal of the Acoustical Society of America

volume 31, 568-578

[11] N. K. Agarwal, M. K. Bull, Acoustic wave propagation in a pipe with fully developed

turbulent flow, Journal of Sound and Vibration 1989, 132(2) 275-298

[12] P. Mungur, H.E. Plumblee, 1969 NASA SP-207, 305-322 Propagation and attenuation of

sound in a soft walled annular duct containing shear flow

Figures

Fig 3.1, R. Long, P. Cawley, M.Lowe, Attenuation characteristics of the fundamental modes

that propagate in buried iron water pipes, Ultrasonics 41 (2003) 509-519, Figure 1,

Fig 3.2, R. Long, P. Cawley, M.Lowe, Acoustic wave propagation in buried iron water pipes,

Proceedings of the Royal Society2003 459, 2749-2770, Figure 3

Fig 3.3, R. Long, P. Cawley, M.Lowe, Acoustic wave propagation in buried iron water pipes,

Proceedings of the Royal Society2003 459, 2749-2770, Figure 3

Fig 3.17, R. Long, P. Cawley, M.Lowe, Acoustic wave propagation in buried iron water

pipes, Proceedings of the Royal Society2003 459, 2749-2770, Figure 9

Fig 3.10, R. Long, P. Cawley, M.Lowe, Attenuation characteristics of the fundamental modes

that propagate in buried iron water pipes, Ultrasonics 41 (2003) 509-519, Figure 6

Fig 3.11, R. Long, P. Cawley, M.Lowe, Attenuation characteristics of the fundamental modes

that propagate in buried iron water pipes, Ultrasonics 41 (2003) 509-519, Figure 7

Fig 3.12, R. Long, P. Cawley, M.Lowe, Acoustic wave propagation in buried iron water

pipes, Proceedings of the Royal Society2003 459, 2749-2770, Figure 16

Fig 3.13, R. Long, P. Cawley, M.Lowe, Acoustic wave propagation in buried iron water

pipes, Proceedings of the Royal Society2003 459, 2749-2770, Figure 17

57

Fig 3.14 N. K. Agarwal, M. K. Bull, Acoustic wave propagation in a pipe with fully

developed turbulent flow, Journal of Sound and Vibration 1989, 132(2) 275-298, Figure

58

CHAPTER 4

Modeling Turbulence and Acoustics with

LES and FW-H

The science of computational fluid dynamics (CFD) is the prediction of fluid flow, heat and

mass transfer, chemical reactions and other related phenomena by putting up a set of

mathematical equations and solving them with numerical tools. This chapter will present a

short description of different turbulence models, a more detailed description of LES,

boundary layer and the FW-H acoustic method. These methods are used in the modeling in

chapter 5.

4.1 Turbulence Models in FLUENT

There are several models for turbulence in FLUENT: one-equation models, two equation

models, second order closure models and two hybrid models. The computational cost in table

4.1 increases from top to bottom where Spalart-Allmaras is the cheapest and most economic

model and Large Eddy Simulation is the most expensive model.

Table 4.1; Available turbulence models in FLUENT.

Models

One equation models

Spalart-Allmaras RANS-based

Two equation models

Standard κ-ε RANS-based

RNG κ-ε RANS-based

Realizable κ-ε RANS-based

Standard κ-ω RANS-based

SST κ-ω RANS-based

Second order closure model

Reynolds-Stress model RANS-based

Hybrid

Detached Eddy Simulation DNS/RANS

Large Eddy Simulation DNS/RANS

59

DNS stands for Direct Numerical Calculations which uses the Navier-Stokes system of

equations. Due to the randomly motions in turbulence, the DNS is not always feasible for

modeling turbulence and a statistical description is needed. The RANS equations stands for

Reynold Average Navier Stokes equation and describes the time-averaged quantities of flows,

where the mean quantities are separated from the fluctuations and therefore there is need of

additional equation to represent the new unknowns. The RANS method models all eddies in

turbulent flows, from the largest to the smallest eddies.

4.2 Large Eddy Simulation

LES is the most suitable model for acoustic applications and is the future of flow simulation.

The model is a compromise between DNS and RANS. In DNS, all scales are numerically

solved, and in RANS, all scales are modeled. LES will filter the transport equations such that

only larger eddies are resolved, and the smaller eddies modeled. There are some difficulties in

doing this since eddies are anisotropic, dependent on history effects and depend on upcoming

flow configuration and boundaries. The smaller eddies are isotropic and more friendly to

model. LES is an efficient method in achieving good results in turbulent flows [2].

Large eddies carries momentum, mass, energy and other fluid quantities. These eddies are

dependent on geometry, mesh and boundary conditions. Small eddies are less dependent on

geometry and tend to be more isotropic and universal. Smaller eddies have higher change in

finding a universal turbulence model than larger ones [2].

Fig 4.1; The figure shows an energy cascade of eddies. A large eddy is created to the left, giving its energy to

smaller and smaller eddies, until the smallest eddies dissipate, and the energy turns from kinetic energy to

thermal energy.

LES requires that only larger eddies are resolve which allows using coarser mesh and larger

time step compared to DNS, but still requires a much finer mesh compared to other turbulent

models. In order to be able to achieve good results, LES has to run for a long flow-time so

statistics can be obtained and the flow modeled. Therefore, computational costs in LES are

60

higher than other steady RANS models in terms of RAM and CPU and high-performance

computing e.g. parallel computing is necessary.

4.2.1 Near-Wall Treatment for Turbulent Models

Walls are a major issue when modeling turbulent flows. The viscous sublayer could either be

resolved completely (normally for lower Reynolds numbers) or not resolved at all. For high

Reynolds number (Re>106) the viscous layer near the wall cannot be resolved since it cost too

much computer resources. Experiments show that there is little gain resolving it and that the

core turbulent is more important [5].

The no-slip condition must be satisfied at the wall. According to section 2.4.2, the turbulent

vanishes when approaching to the wall, i.e. the turbulence is changed by the presence of the

wall. The solution variables have large gradients in the near wall region. In order to predict

successful turbulent flows, accurate representation of the walls is necessary.

Fig 4.2; To the left is an illustration of the wall functions. The laminar sub-layer and buffer layer is resolved in

the near-wall model approach.

The turbulent boundary layer can roughly be divided into three regions, the laminar sub-

layer where the molecular viscosity is important to consider, the outer layer called the

turbulent core, and the region in between called the buffer layer where both turbulence and

viscosity is important [6].

In turbulence modeling, there are two approaches for the near wall region. The first one is to

resolve the laminar sub-layer and the second one is to not resolve it and instead use

experimental formulas called the wall functions, which is the bridge between the turbulent

core and the wall (see figure 4.2).

Depending on the turbulence model, there are four different choices of wall functions;

Standard Wall Functions, Non-Equilibrium Wall Functions, Enhanced Wall Treatment

and User-Defined Wall Functions [3]. A short description of wall functions is described in

section 2.4.2.

4.2.2 Meshing Turbulent Flows Near Walls

In order to obtain successful results in CFD, the meshing requires some consideration.

Turbulent quantities e.g. eddies are usually very small and if high accuracy is needed, the

mesh must be fine enough to resolve all those quantities. Turbulent flows are very grid

61

dependent and it is recommended that the mesh is sufficiently fine in regions where quantities

change rapidly.

FLUENT has powerful tools available in the postprocessing panels to check near-wall

mesh. The values of y+, y

* and Rey can all be displayed and studied. The quantity y

+ describes

the distance from the wall adjacent grid point. The quantity can be calculated from equation

yuy (4.1)

which shows that y+ is unit less. For standard wall functions, the first cell should be located

within the range 30< y+<150, most desirable is y

+≈30 [3]. y

+ is also solution dependent e.g. if

y + has a certain value for some solution, then it is not necessarily the half by halving the wall

distance. As shown in equation 4.1, the viscosity, density and the friction velocity can change

the value of y+. The distance of the first grid point can also be measured with y

* that should

have a value y* < 11.225.

For enhanced wall treatments, at least ten cells should be in the viscosity affected region to

be able to resolve it, and the first cell should be in the order of y+≈1.

4.2.3 LES Grid Size

LES turbulent model requires mesh sufficiently fine to resolve high energy containing eddies.

Therefore, the mesh has significant influence on the results for LES. However, in order to

decide the grid size, the integral length scale has to be considered, which describes the size

of the largest eddies. The integral length scale can be calculated from equation 4.2.

2

3

kLt (4.2)

where k is the turbulent kinetic energy and ε is the turbulent dissipation rate. However,

eddies in a flow inside a pipe cannot be larger than the size of the pipe. For fully developed

flows inside duct flows with characteristic length D, the turbulent length scale is

DLturb 07.0 (4.3)

where Lturb is the largest turbulent length scale inside the pipe. Equation 4.3 states that the

largest eddies of a flow in a duct are 7% of the characteristic length. The characteristic length

for pipes is the diameter. Note that equation 4.3 is not suitable for all applications and is just a

rough approximation of eddies inside pipes [3].

62

Assuming that 80% of the total turbulent kinetic energy must be resolved, eddies of roughly

the half size of the integral length scale must be resolved [5]. In this case, the integral length

scale and the turbulent length scale are of the same order; therefore the turbulent length scale

can be used in order to have a rough estimation of the grid size.

4.2.4 LES Time-step and Courant Number

The Courant number is a dimensionless quantity that compares the time-step to a

characteristic time of transit of a fluid element across a volume. The Courant number can be

calculated from

fluid

cell

vx

tCourant (4.4)

The time-step should be chosen so Courant < 1 [3]. However, if the required time-step yields

a Courant number greater than one, there are two options, the time-step should be decreased

or the mesh should be coarsened. Note that there are other factors that affect the time-step e.g.

acoustic applications require that a small time-step is settled in order to resolve high

frequencies.

4.3 Modeling Acoustics

Aeroacoustics is the study of sound generated by turbulent flows or fluids in motion

interacting with surfaces. When fluids undergo high velocity gradients (e.g. leaks), vertical

waves are generated that produces pressure fluctuations i.e. the sound comes from the eddies

and vorticity in turbulent flows. There is no complete theory of aeroacoustics, and in practice

the aeroacoustic studies relies on acoustic analogy, where a wave equation is derived from

the fluid dynamics equations with pressure as a fluctuating variable and fluid flow variables

as sources.

The challenges in modeling acoustic are that fluids have much higher energy than sound-

waves. The difference is of several orders of magnitude, e.g. the pressure inside the pipe is or

the order of 106 Pa and the sound-waves fluctuate with the order of a few Pascal.

The main elements in aeroacoustic applications are sound source, the propagating medium

and the receiver. These applications are extremely unsteady and time-dependent. For achieve

good results in aeroacoustics, spatial resolution is necessary to capture small eddies. There are

four approaches in FLUENT for computing sound-waves shown in table 4.2

The CAA method is not feasible for this thesis 3D, since it requires extremely fine mesh and

a compressible solution. The Coupled CFD requires an external program like for instance

SYSNOISE. Unfortunately, SYSNOISE is not available for this thesis. The FW-H is the best

choice for this thesis. The disadvantage is that is does not take into account walls and

reflection in the pipe. The pipe geometry will influence traveling waves which suffer from

dispersions and other factors. The broadband give to inaccurate results and will not give

satisfying results, but can be used as an initial simulation to identify sources.

63

Table 4.2; Approaches to Aeroacoustics.

CAA Coupled CFD with

acoustic code

FW-H Broadband noise

method

Computational

effort

Very high High High Low

Reflection and

scattering

Yes Yes No No

Sound

propagation

through shells

No Yes No No

Account effects of

flow on sound

Yes No No No

Solution required Transient Transient Transient Steady state

Compressible

solution

Yes No No No

Accuracy Good Good Good Limited

4.3.1 FW-H Method

In order to use FW-H, a high quality unsteady LES or DES solution is required. The method

will represent certain fluid processes as acoustic sources which generate sound. The acoustics

will be described by the wave equation. This method is the most general acoustic integral

formulation. It can solve problems for mid- to far-field noise.

The acoustic sources are classified as monopole-, dipole- and quadrupole-sources. The

monopole-source is generated from unsteady mass injection i.e. a fluid flowing into a region

(see figure 4.3). The dipole-sources are generates noise due to interaction with flow and

bodies (or walls) and external forces. The quadrupole-sources generates noise due to unsteady

stresses and turbulence, these sources can be negligible when mono- and dipole-sources are

present for low mach numbers. This tells that the dominating noise comes from the mass

injection (injection in the leak) and the walls of the pipe and leak [7]. In order to study the

dipole sources, incompressible solution is sufficient, however, in order to study the monopole

(and quadrupole) a compressible solution is required.

There are several disadvantages with the FW-H method, however, this method has its focus

on the sound sources than the environment. Reflection, scattering dispersion are not

accounted for and inhomogeneities and the effect of flow are also ignored.

64

Fig 4.3; The mono-, di- and quadru-pole sources in turbulent flows. An aeroacoustic source classification.

References

[1] Vladimir Pavlenko, Lisa Rosenqvist, Fluid mechanics Compendium version 2,

Department of Astronomy and Space Physics, Uppsala University 2007

[2] T.J. Chung, Computational Fluid Dynamics, University of Alabama in Huntsville,

Cambridge University Press 2002

[3] FLUENT user guide, Sep 2006

[4] ANSYS FLUENT, Introductory FLUENT Training – Choosing a Solver, FLUENT v6.3

Dec 2006

[5] ANSYS FLUENT, Advanced FLUENT CFD training, Turbulence, Turbulence Modeling

Options April 2005

[6] B.S. Massey, Van Nostrand Reinhold 1975, Mechanics of fluids 3rd

edition, 1975

[7] ANSYS FLUENT, Advanced FLUENT CFD training, Acoustic Analogy Modeling, Jan

2008,

Figures

Fig 4.3; ANSYS FLUENT Advanced FLUENT CFD training, Introduction to Acoustic

modeling, Jan 2008,

65

CHAPTER 5

Modeling District Heating Pipes in

FLUENT

The simulation will consist of three parts, meshing in GAMBIT v2.4, setting up the solver

settings in FLUENT v6.3, and postprocessing in FLUENT v6.3. This section will describe

how the pipe model was created, meshed, explain the chosen models and discussed the

results. The sections 5.2 – 5.6 will describe and discuss results of a simulation of a

incompressible two dimensional pipe with different pressure and viscosity. The sections 5.7 –

5.9 will describe and discuss results of a three dimensional compressible simulation of the

pipes.

5.1 District Heating Pipes Problem Description

The district heating pipe dimensions have a diameter of 30 cm and a wall thickness of 5.6

mm. According to statistic, most leaks have an average diameter of 1 mm and penetrate

through the wall. Unfortunately due to difficulties with meshing the diameter of the leak in

the simulation will be larger than 1 mm and will be approximated as a cylinder. The pipe

model in GAMBIT will be as follow;

Length l=32 cm with velocity inlet and pressure outlet boundary condition

Radius r=16 cm surrounded by walls

Leak radius rleak=1 mm with outlet boundary condition

Leak length lleak=6 mm surrounded by walls

The two dimensional case will be solved with water as an incompressible liquid. For

incompressible solutions, there are four different boundary conditions to determine; inlet,

outlet, the leak outlet and the walls. The inlet boundary condition will be a velocity inlet with

a velocity of 2 m/s over the whole boundary. The outlet however cannot be a velocity outlet.

The most essential outlet boundary condition is pressure-outlet for both the outlet and the leak

outlet. The pressure inside the high temperature pipe is 10 atm and for the lower temperature

3 atm and therefore the outlet boundary condition must be 10 and 3 atm respectively. The leak

outlet releases water into the air, an obvious pressure condition in this case would be 1 atm.

All walls have no-slip condition as stated in chapter 2. The boundary conditions are

summarized in table 1.

66

Table 5.1; The boundary conditions for incompressible solution

Boundary type Value

Inlet Velocity 2 ms-1

Outlet Pressure 10 atm/3 atm

Leak outlet Pressure 1 atm

Wall No-slip none

The three dimensional case will be solved with water as a compressible liquid. The equation

of state requires special attention when considering water as compressible, especially if the

Mach number is low. This will be solved by using a user defined function for the density and

the speed of sound. The boundary conditions for compressible flows can be found in table 5.3.

Note that for compressible flows, velocity inlet boundary conditions are not appropriate and

instead the mass flow inlet is used.

Table 5.2; The boundary conditions for compressible solution

Boundary type Value

Inlet Mass flow inlet ~160 kg/s

Outlet Pressure 10 atm/3 atm

Leak outlet Pressure 1 atm

Wall No-slip none

The essential water properties for this simulation are the density and dynamic viscosity. For

incompressible solution, the density will remain constant. For compressible solution, the

density is determined by a user-defined function. The viscosity at 100oC and 40

oC can be

found in table 2.1.

5.1.1 Considerations before Modeling in GAMBIT and FLUENT

Before modeling in GAMBIT and FLUENT, there are several factors to consider in order to

obtain good results;

1. Near-wall treatment. The boundary cannot be resolved since it cost too much,

therefore the first grid point must be placed at 30< y+

<150 as stated in section 4.4.2

(preferable to the lower limit). The first grid point must be estimated since it is

difficult to calculate and evaluated with FLUENT postprocessing tools.

2. Meshing the turbulent core. The mesh at the turbulent core must at least resolve half of

the size of the largest turbulent scale. This can be as an upper limit of the coarsest

mesh the problem can have.

67

3. The Courant number. This dimensionless quantity must be lower than 1 and is related

to the time-step. In order to decrease the Courant number when it is too high there are

two options; put a coarser mesh or decrease the time-step. Putting a coarser mesh can

lead to problems with resolving eddies as stated above. Choosing a smaller time-step

can lead to time-consuming iterations, by decreasing it with a factor two makes the

simulation time to increase with a factor two.

4. Low frequency resolution with FH-W. Frequencies of 20 Hz have a time period of

0.05 s. In order to obtain good results for low frequencies with FW-H method, the

flow time must be at least 10 times of the time-period corresponding to the lowest

frequency, in this case, having a time-step of 1 μs, as it is the case for pipes with a

pressure of 10 atm, the simulated flow time must be at least 0.25 s long, resulting in

250 000 iterations. Assuming that one iteration with LES takes 2 seconds for two

dimensional pipe and 30 seconds for three dimensional pipes, the total time is 5.8 and

86 days respectively, which in three dimensional is not feasible.

It is also recommended to start with incompressible solutions in two dimensions due to

several reasons. Two dimensional incompressible simulations do not require too much time

and computational effort compared to compressible three dimensional simulations. Therefore,

the system will be studied in two dimensions for incompressible solutions first. The noise

sources will be restricted to the walls only in two dimensions. The acoustic in the interior is

studied in three dimensions only.

5.2 Incompressible Two Dimensional Pipe with LES and FW-H at

1000C and 10 atm

This part can be roughly divided into three steps, the pre-processing step where the solid

model is created and meshed, the solver execution where the model is simulated and the post-

processing part where the results are evaluated and studied. The solution procedure is as

follows:

Create and mesh a two dimensional pipe in GAMBIT

Perform a two dimensional steady simulation in FLUENT with κ-ω model to

evaluate mesh and predict the time-step

Perform a two dimensional unsteady simulation in FLUENT with LES

Activate the FW-H acoustic model in FLUENT

o Select Export Acoustic Source Data

o Select source surfaces (walls) from available FLUENT zones in Acoustic

Sources Panel

o Specify write frequency, the upper limit is the time-step of the solution, in order

to have higher frequencies, the time-step must be reduced, however, the write

frequency can be coarsened in order to save computer resources. The

frequency of interest must be identified before reducing the write frequency.

68

Iterate for at least ten cycles of the lowest frequency. The lowest frequency of interest

must be identified in order to choose the amount of iterations (having a very short

time-step and a very low frequency results in very long simulations).

min

min

110

ft imum

Calculate acoustic signal at the walls (dipole source)

o Specify receivers at locations in Acoustic receiver panel

o Compute/write the acoustic signal

Postprocess the results

5.2.1 The Solid Model and Mesh

The dimensions of the pipe are given in section 5.1.1. The length of the pipe must be

determined. It is of advantage to have a short pipe, but the pipe must also be long enough to

be able to study effects of the flow. The length of the pipe will therefore be the as the

diameter, 32 cm which based on assumptions of the affected area length and how much length

before the flow reaches a reasonable uniformity. The leak is approximated to a small cylinder

of 1 mm in radius and placed 12 cm from the inlet.

The regions of the pipe can be found in figure 5.1. By divide the model into regions, it is

possible to put conditions at the boundaries of the regions when meshing.

Table 5.3; Mesh summary for hot water pipes. In wall cell, the first factor is y+ and the second factor is the

distance of the cells along the wall.

Region Total cells

Leak region 6800

Near leak region 47114

Front leak region 12400

Rear leak region 22320

Middle region 42240

Lower region 13280

Total cells 144154

Smallest dominating

scale

30 μm

69

Fig 5.1; The two dimensional pipe and the regions in GAMBIT

Before meshing, it is important to consider the near-wall treatment as stated in section 4.4.2.

To save computer resources, the model will have finest mesh in the leak region and coarsest

mesh in the lower and middle regions. Since the geometry is pretty simple, a quad mesh will

be used which also gives high quality solutions with fewer cells.

Using equation 4.3 gives a turbulence length scale of 22.4 mm. In order to resolve 80% of

the total kinetic turbulent energy, eddies of approximately half of the size must be resolved,

that means eddies with length scale 11.2 mm. The mesh in the coarsest meshed region

satisfies this roughly condition at the pipe. At the leak, however, the turbulent length scale is

approximately 0.14 mm which means that eddies of the size 0.07 mm must be resolved. The

mesh in this region is also satisfying since the cells are 30 μm, and there are approximately

2.33 grid points in this length scale. Note that this is just a rough approximation of the grid

size in the different regions. The mesh should be evaluated with FLUENT postprocessing

tools before doing further simulations.

70

Fig 5.2; The mesh in the leak. The mesh consists of quad mesh only.

Fig 5.3; Front leak region, near leak region and leak region. The mesh elements in the near leak region

decreases in size approaching the fine meshed leak region.

5.2.2 Stationary Flow with κ-ω Turbulence Model

To evaluate the mesh, a stationary solution is needed. The model will be simulated in

FLUENT with the standard κ-ω turbulence model. The settings can be found in Appendix 5A.

Since this is a RANS-based turbulence model, the turbulent specific energy κ and turbulent

dissipation rate ω must be calculated on each boundary. Instead of calculating the specific

energy and dissipation rate, an alternative way is to determine the turbulent intensity and the

turbulent length scale Lturb. The turbulent intensity is usually around 5-10 % and the length

scale is calculated from equation 4.6. See Appendix 5C for the boundary layer y+ distance

from the walls.

71

The mesh is satisfying and the leak seems to have an average velocity of 20-24 m/s (see

figure 5.4). Note the velocity gradient in the leak inlet. Due to velocity acceleration and

pressure fall, this zone could be a high noise generating region. The cells in the leak are 30 x

30 μm2, which means that it takes 1.25 μs for a fluid element to go through one cell to the

next one, therefore the time step of the unsteady solution should not be higher than 1.25 μs.

However, this is just a stationary solution of the problem and the velocity can differ much

more in an unsteady solution with LES. Assuming that the velocity can vary to up to 70 m/s, a

time-step of 0.43 μs is required in order to fulfill the Courant number criterion. However, the

chosen time-step for this problem is settled to 1 μs in order to save computer resources and

time, which in the worst case would give a Courant number of 2.33 which is acceptable.

Fig 5.4; The velocity profile in the leak. Notice the first cell closest to the walls. The boundary layer is very thin

and not so visible. The average velocity in the leak seems to be around 20-24 m/s.

5.2.3 Unsteady Flow with LES

The settings for LES are based on the FLUENT User guide LES best practice (note that LES

can only be used for unsteady solutions). The settings for LES can be found in Appendix 5B.

For the Courant number, drag- and lift force see Appendix 5C. The time-step is 1 μs and will

iterate for 30000 times which gives a time flow of 0.03 seconds. At this point, the bulk flow

will have crossed the pipe with 6 cm and at the leak with 75 cm, which means that it has

crossed the leak domain about 125 times.

In section 2.4.2, it was stated that there is vorticity in turbulent flows; figure 5.5 can verify

that vorticity appears at the walls of the leak. This vorticity can in turn emit pressure waves

which is the next step in the simulation where the acoustic waves will be simulated with FW-

H method. These vorticity will with time move upwards along the flow. The velocity seems to

72

be highest near the vorticities inside the leak. Vorticity will slow down the flow and increase

it next to it in radial direction of the leak (i.e. to the right or left in figure 5.5) due to that the

mass flux must be constant through the leak. The highest velocities will then be dependent on

the size of the vorticities e.g. a huge vorticity will increase the velocity more than smaller

ones (compare the vorticities in figure 5.5) The highest velocities are shown with red and are

up to 72 m/s. However, inside the vorticity, the speed is around ~8-18 m/s and therefore the

stresses between the fluid layers at the vorticity must be high, i.e. fluids with high viscosity

will suffer from high stresses at the leak. The mass injection into the leak is also important

which can cause noise generation and must be studied with compressible flows with FW-H

method.

Fig 5.5; Pathlines of the fluid elements with color scale of the velocity

5.2.4 Calculating the Acoustic Sound at the Walls

It is not necessary to save acoustic data for each iteration. Since the time-step is 1 μs, the

maximum sampling frequency is 1 MHz. The frequencies of interest are below 5 kHz. The

write frequency can be coarsened with a factor of 20, which give a sampling frequency of 50

kHz. This value is a sampling frequency and according to Nyquist theorem, the highest

resolved frequency will be 25 kHz.

It is recommended to have at least ten cycles for those frequencies of interest; in this case,

the lower limit will set the amount of iterations. With the current time-step and due to time

consuming numerical computations, the maximum flow time for calculating the acoustic

waves will be 0.25 s after obtaining a uniform unsteady LES solution. Since 40 Hz has a time

period of 0.025 s, it will be the lower limit (lower frequencies are not of interest to resolve).

The results are shown in figure 5.6 for a receiver 5 m away from the source and with source

correlation length of 0.1 m up to 10 kHz. The source correlation is necessary in two

73

dimensional solutions because the FW-H method need three dimensional data to compute the

noise i.e. the source correlation length will extrude the two dimensional solution with 0.1 m.

The leak is circular which cannot be taken into account with this method; therefore three

dimensional solutions are required in order to obtain reliable results. At this point, the flow-

time is 0.28 s (0.03 for obtaining an unsteady LES and 0.25 for calculating the acoustic data).

Fig 5.6; The frequency content from the walls in the pipe system to a receiver 5 m away from the leak

Note that this method does not take scattering, dispersion in waveguide or walls into account

in the acoustic calculations. But the results can show what frequencies a leak generates of this

kind. Most of the frequencies are above 4 kHz which is the range with high dispersion

frequencies according to section 3.2. The low frequency noise does not appear for this result

which means that it must come from another sound source than the walls. Note that two

dimensional calculations cannot predict the intensity of the frequencies. In order to do it,

three dimensional simulations are required.

5.3 Incompressible Two Dimensional Pipe with LES and FW-H at

400C and 3 atm

This section is similar to section 5.2 and therefore only the unsteady solution and acoustic

waves will be discussed. The distances y+ from the walls can be found in Appendix 5D. The

same solid model is used as for the hot water pipe model (see section 5.2.1), but the meshing

is a bit different at the boundaries. The viscosity is higher for cold flows which in turn

generates a thicker boundary layer. The viscosity at 400C can be found in table 2.1.

74

Table 5.4; Mesh summary for cold water pipes.

Region Total cells

Leak region 6355

Near leak region 12689

Front leak region 12000

Back leak region 21600

Middle region 42290

Far region 13280

Total cells 108164

Smallest dominating

scale

30 μm

5.3.1 Unsteady Flow with LES

The Courant, drag- and lift force can be found in Appendix 5D. Figure 3.7 shows the path

lines and vorticities. There are more vorticities and they are larger than the previous case

which is expected for a more viscous and slower fluid since the boundary is dependent on

both the viscosity and the velocity. The velocity and stresses are lower due to lower pressure

fall which allows a higher time-step of 2 μs. It is left to find out if the noise is dependent on

the stresses and mass injection in the leak.

Fig 3.7; Pathlines of the fluid elements with color scale of the velocity.

75

5.3.2 Calculating the Acoustic Sound at the Walls

The procedure is similar to section 5.2.5, except that the coarsening factor is 10 which

correspond to maximum frequency of 25 kHz. The flow time for the acoustic calculations will

be 0.25 s, with a lower limit of 40 Hz. The results are shown in figure 5.8 with correlation

length 0.1 m.

Note that the intensity at 5 m away from the leak is approximately one order in magnitude

lower for this case than the case with hot water and pressure of 10 atm. This could depend on

both the pressure, which is much lower for this case, and the viscosity which is much higher

for cooler water. The peaks are narrower than those in figure 5.6 which had one broad peak.

There is also more noise in the region below 4 kHz; two peaks arise at 2 kHz and 3.5 kHz

which did not show up in plot 5.6.

Fig 5.8; The frequency content from the walls in the pipe system to a receiver 5 m away from the leak.

5.4 Results for an Incompressible Two Dimensional Pipe with LES and

FW-H at 1000C and 3 atm

In order to understand how the pressure affects a pipe of 100oC, a pipe with pressure of 4 atm

was simulated in FLUENT. Procedure will not be shown as it is similar to section 5.2 and 5.3.

The results are found in figure 5.9 and figure 5.10.

76

Fig 5.9; Pathlines of the fluid elements with color scale of the velocity.

Fig 5.10; The frequency content from the walls in the pipe system to a receiver 5 m away from the leak.

77

The highest velocities are around 32 m/s, which is similar to the case with 40oC and

pressure 3 atm. For this case, there are less vorticities near the wall compared to the previous

case with cooler water and could be due to that the flow is less viscous for this case i.e. fluids

with high viscosity have more vorticities.

The amplitude of the power spectral density in figure 5.10 is similar to figure 5.8, except at

the frequencies around ~3 kHz, where figure 5.8 have two peaks, and figure 5.10 has one

broader peak i.e. the peaks are merged. One possible reason for the broad peaks is the

viscosity; low viscous flows have broader peaks i.e. the noise is spread out on more

frequencies. The amplitude is dependent on the pressure fall. By having higher pressure, leaks

will generate stronger noise.

5.5 Results for an Incompressible Two Dimensional Pipe with LES and

FW-H at 400C and 10 atm

In this section, the viscosity of the simulated pipe will be compared with the results of 5.2-5.4.

The results are found in figure 5.11 and 5.12.

Fig 5.11; Pathlines of the fluid elements with color scale of the velocity.

Comparing the vorticities in figure 5.11 with the previous flows, there are more vortices for

this case which should be expected for cooler flows i.e. the amount of vorticities are governed

by the viscosity only. The velocity is similar to the case with pressure of 10 atm.

78

The power spectral density can be found in figure 5.12. As expected, since the fluid is more

viscous and due to high pressure gradient, the peaks are narrower and more intense.

Compared to figure 5.6, frequencies in the range around 3-4 kHz appears due to viscosity and

with the high pressure gradient, they are more intense.

Fig 5.12; The frequency content from the walls in the pipe system to a receiver 5 m away from the leak

5.6 Conclusions for Two Dimensional Simulations

It is important to understand that two dimensional simulations cannot predict real amplitudes

of the noise level; it can only predict at which frequency range the noise is strongest.

However, the noise level and shape in the sections 5.3 - 5.6 are clearly dependent on the

viscosity and the pressure in the pipe.

For leak detection, a high pressure fall is preferred, since the noise level will be higher. The

pressure fall will influence the velocity in the leak causing high stresses. Comparing 5.6 and

5.12 with 5.8 and 5.10, the noise level is higher for high pressure pipes.

The viscosity will influence the shape of the peaks. Fluids with high viscosity generate more

vorticity and the noise will be characterized by fewer frequencies due to narrower peaks. The

viscosity have influence on a noise in the range 3 – 4 kHz which appear only of the viscosity

is high.

Three dimensional simulations can predict the noise levels which will be described in

section 5.7. The path lines in figure 5.5, 5.7, 5.9 and 5.11 all show that the mass injection is a

zone which can generate noise. The noise on the frequency range 0 - 3 kHz did not show in

the two dimensional simulations, therefore it is expected that the three dimensional

simulations will show the low frequency noise by simulating the acoustics from the interior of

the pipe. This can only be studied with compressible flows, therefore the three dimensional

79

simulations will be performed with compressible conditions.

5.7 Compressible Three-Dimensional Pipe with LES and FW-H at

1000C and 10 atm

A solution of a compressible liquid with low Mach number is tricky to obtain. An equation of

state for liquids is required since the density is not constant. The equation of state for liquids

is described in section 2.4.3. In FLUENT there is no equation of state for liquids, therefore the

equation of state must be written into a C code and be interpreted in FLUENT. The C code

can be found in Appendix 5E.

The solution procedure is as follows:

Create and mesh the three dimensional model in GAMBIT

o Create interior volumes for the monopole source calculations

Perform a three dimensional steady compressible simulation in FLUENT with e.g. κ-

ω model to evaluate mesh, predict the time-step and use it as start point for LES.

Perform a three dimensional unsteady simulation in FLUENT with LES, starting

with high time-step, and reducing it successive until satisfying Courant number in

order to achieve satisfying solution fast.

Activate the FW-H acoustic model in FLUENT

o Select Export Acoustic Source Data

o Select source surfaces (interior and walls) from available FLUENT zones in

Acoustic Sources Panel, both walls and interior zones

o Specify write frequency, the upper limit is the time-step of the solution, in order

to have higher frequencies, the time-step must be reduced, however, the write

frequency can be coarsened in order to save computer resources. The

frequency of interest must be identified before reducing the write frequency.

Iterate for at least ten cycles of the lowest frequency. The lowest frequency of interest

must be identified in order to choose the amount of iterations (having a very short

time-step and a very low frequency results in very long simulations).

min

min

110

ft imum

Calculate acoustic signal at the walls (mono- and dipole sources)

o Specify receivers at locations in Acoustic receiver panel

o Compute/write the acoustic signal

Postprocess the results

80

5.7.1 The Solid Model and mesh

The solid model will be divided into 7 regions; the leak which is subdivided into two smaller

regions, the near-leak region which is subdivided into four smaller regions, the front- and rear

region, the outer pipe and the core of the pipe (see figure 5.13). Compared to the two

dimensional model, this model is shorter in total length and only the upper half of the pipe is

modeled. The lower part is removed and replaced with a wall in order to save computer

resources. The reason is that it is very expensive to model the whole pipe and the pipe is

mainly influenced by the closets part of the flow.

The model has an interior region in which the acoustic data will be saved and calculated

with the FW-H method (see figure 5.14). This method will not only take the walls into

account, but the whole region.

A summary of the mesh can be found in table 5.5. The meshing was similar as in section

5.2.1 but in three dimensions. Near-wall treatment and turbulence length scale was considered

and is satisfying for the problem. The model was meshed with mostly hexagonal mesh except

the transition region. Due to that the leak has more compact mesh, the near leak wall and

upper region where extruded from the top and it is not necessary to have compact mesh all the

way down to the lower wall, therefore a tetrahedral mesh (see figure 5.16) was used to coarse

the mesh and make if more uniform (compare the mesh element in upper region, transition

region and lower region in the near leak region).

Table 5.5; A summary of the mesh.

Region Total cells

Leak 291 819

-Leak top region 68 354

-Leak lower region 223 795 (acoustic region)

Near leak region 441 189

-Wall region 49 285 (acoustic region)

-Upper region 118 284 (acoustic region)

-Transition region 251 220

-Lower region 22 400

Front leak region 72 000

Rear leak region 72 000

Outer pipe region 604 800

Core 387 360

Total cells 1 869 168

Smallest dominating

scale

30 μm (inside the leak)

81

Fig 5.13; The figure shows the modeled pipe and the different regions. Note that the near leak region and leak is

subdivided into smaller regions

Fig 5.14; The leak and near leak region. The leak is subdivided into a top and lower region and the near leak

region is subdivided into a wall-, upper-, transition- and lower region. The purple filled region is the region

where the acoustic data is calculated.

82

Fig 5.15; The mesh at the leak and the close regions. The mesh at the outer pipe is much coarser than inside the

leak. Note that the mesh can only be seen at the walls.

Fig 5.16; Slice of the pipe with the mesh. The figure shows how the mesh is coarsened in the transition layer.

83

5.7.2 Stationary solution with κ-ω turbulence model

A stationary solution with the κ-ω turbulence model is an excellent start point for LES. The

wall distances y+ can also be studied (see Appendix 5H). A stationary velocity profile can be

found in figure 5.17. The average velocity is roughly 25 m/s and the top velocity 39 m/s. Due

to lack of computer resources, the time step must be 1 μs which in the worst case can yield a

courant number of 2.33 for a velocity of 70 m/s.

Fig 5.17; The velocity profile for two cross-section planes.

5.7.3 Unsteady compressible flow with LES

The solver settings can be found in Appendix 5G. In order to achieve a solution faster, the

iteration will start with a time step of 10 μs for 500 iterations, decrease the time step to 5 μs

for 1500 iterations, decrease it to 2 μs for another 1500 iteration and finally to 1 μs for 2500

iterations. When the solution has reaches a total of 6000 iterations corresponding to a flow-

time of 18 ms and when the drag- and lift force are oscillative the acoustic module can be

defined and start to save acoustic data for the acoustic calculation (the drag- and lift force can

be found in Appendix 5H). The converged flow velocity profile at some instance can be found

in figure 5.18. Note the differences with the stationary flow in figure 5.17. The velocity can

reach up to approximately 63 m/s giving a worse Courant number of approximately 1.8 which

is good enough. A histogram of the Courant number can be found in Appendix 5H. Note how

the velocity near the walls is slower than the core. This part suffers from turbulence or so

called vorticity around the whole leak wall. Figure 5.19 shows a plot of the vorticity in the

leak.

84

Fig 5.18; The velocity profile for a converged LES simulation

Fig 5.19; The vorticity profile for a converged LES simulation

85

Figure 5.19 shows which parts of the leak that suffers from vorticity. Vorticity is a

mathematical concept in fluid mechanics which is related to the spin or “circulation” of a

fluid element. The vorticity is largest at the leak inlet which is a region that contributes to

acoustic noise, however, the region is included in the acoustic region and the sound will be

calculated in the following section.

5.7.4 Calculating the Acoustic Sound from the Leak Interior

It takes 30 seconds to do one iteration with Iterative-Time-Advancement with PISO in

FLUENT. With larger time-step it took up to 1 minute and to achieve a converged flow with

LES it took 60 hours for a time flow of 18 ms. With a time-step of 1 μs and continuing for

35 000 iterations (total of 41 000 iterations), it was possible to reach a time-flow of

approximately 53 ms, where the acoustic data was achieved from a time flow of 35 ms which

correspond to a minimum frequency of 285 Hz (which has oscillated at least 10 cycles). The

total time for the 35 000 iterations is approximately 12 days and in total 15 days of iterations.

Fig 5.20; The power spectral density for the leak noise.

The power spectral density and the sound intensity in dB can be found in figure 5.20-5.21.

The receiver is located 5 m away from the leak. At 1 kHz and lower, the frequency intensity

start to decrease which means that the leak noise is low at those frequencies. The intensity is

highest around 1 - 1.4 kHz and slowly oscillates as going up in frequency. These results show

that it is of advantage to study frequencies higher than 1 kHz. The sound level in dB is

between 65 – 75 dB for frequencies higher than 1kHz.

86

Fig 5.21; The sound pressure level from the leak noise.

5.8 Compressible Three-Dimensional Pipe with LES and FW-H at

400C and 3 atm

The procedure for this simulation is similar to the one described in section 5.7. The C-code

for the density is also used in a similar way (see Appendix 5E). The same model was used.

The mesh is a bit different due to higher viscosity and lower pressure. The differences are

mostly near walls due to thicker boundary layer.

Region Total cells

Leak 338 808

-Leak top region 77 272

-Leak lower region 261 536 (acoustic region)

Near leak region 472 141

-Wall region 44 940 (acoustic region)

-Upper region 112 350(acoustic region)

-Transition region 290 155

-Lower region 24 696

Front leak region 70 560

Rear leak region 70 560

Outer pipe region 573 888

Core 403 576

Total cells 1 929 533

Smallest dominating

scale

30 μm (inside the leak)

87

A summary of the mesh can be found in table 5.6. A stationary solution was obtained first in

order to study the mesh, boundary layer and to settle a reasonable time-step. This will not be

discussed into detailed. This section will only study the converged flow and the acoustic data.

5.8.1 Unsteady compressible flow with LES

To achieve long flow time fast enough, the LES simulation initial time-step was settled to 20

μs for 1500 iterations, then decreased to 10 μs for 1500 iterations finally 2 μs for 3500

iterations reaching a flow time of 52 ms. The iterations took approximately 3 days to achieve.

The reason for choosing a higher time-step for this case is t hat the pressure gradient is

smaller than the one in section 5.7 and therefore a smaller time-step is possible to choose.

A converged velocity profile of the leak can be found in figure 5.22. The top velocity is

around 28 m/s. Note that the left side of the leak has almost the same shape as in figure 5.18,

the right side however, has less irregularities in the velocity than the left side. The right wall

of the leak corresponds to the front wall; the water flow is in the positive x direction i.e. the

flow is from the right to the left. The rear wall is therefore less influenced by turbulence for

this case.

Figure 5.23 show the vorticity. Note the difference between the front and rear leak wall. The

vorticity in the rear leak wall stays near the wall. The front leak wall however spreads into the

middle. For the previous case where the pressure was 10 atm, the rear wall was also

influenced by turbulence (se figure 5.19).

Fig 5.22; The velocity profile for a converged LES simulation

88

Fig 5.23; The vorticity magnitude for a converged LES simulation.

5.8.2 Calculating the Acoustic Sound from the Leak Interior

The time-step is 2 μs, letting the simulation iterate for 20 000 iterations yields a flow time of

40 ms which sets the lower limit at 250 Hz. Each iteration takes approximately 30 seconds

which means that it takes approximately 7 days for 20 000 iterations and in total 10 days with

achieving a converged LES simulation.

The power spectral density and the sound pressure level can be found in figure 5.24 and

5.25 respectively, where the receiver is located 5 m away from the leak. The overall noise

level is lower for this case as expected for a lower pressure gradient in the leak (compare with

figures 5.20 and 5.21). The case with 10 atm reached sound levels of 75 dB, this case has its

overall sound level at 65 dB. A peak has also arise in the range of approximately 400-500 Hz

with intensity of 70 dB as it did for the two dimensional case. This means that this peak is

governed by viscosity.

89

Fig 5.24; The power spectral density for the leak noise.

Fig 5.25; The sound pressure level from the leak noise.

90

5.9 Conclusions for Three Dimensional Simulations

The three dimensional case where the interior of the leak was included to the acoustic

calculations showed that noise also exist in the low frequency region and that the noise is

clearly influenced by temperature and pressure. The power spectral densities give valuable

information about where the largest peaks are located. The leak noise is most intense in the

frequency range of 1 – 2 kHz which is promising for leak detection. Below 1 kHz the noise

level tends to decrease except for the case with 400C where peak exist in the range of 400 –

500 Hz which is not visible for the case of 100oC. A possible explanation for this peak is the

increased viscosity for cold flows. For both and low and high temperature flows the results

show that it is of advantage to study frequencies above 1 kHz. For cold pipes it could be

possible to also study frequencies of 500 Hz. Experiments will be needed in order to verify

this result.

From the sound pressure level figures, the overall sound intensity is lower for low pressure

pipes as expected. This was also clearly visible for two dimensional pipe simulations. It is

therefore of advantage to have high pressure pipes in order to increase the noise from the leak.

Note that in figure 5.25, above 3 kHz the noise level starts to slightly decrease which is

another argument to only study low frequencies for low pressure pipes in leak detection.

Figure 5.26 show a sound pressure level plot of a three dimensional pipe where the noise

sources are the wall. The temperature is 1000C and the pressure 10 atm. The sound level is

extremely low and therefore it can be concluded that the sound does not come from the wall,

but the interior.

Fig 5.26; the Sound pressure level for a three dimensional pipe with 100oC flow and 10 atm.

91

CHAPTER 6

Summary and Conclusions

The results of this thesis will be discussed in this chapter and some recommendations for

future work.

6.1 Summary and Conclusions

The acoustic wave propagation in district heating pipes embedded in insulation has been

discusses and a simulation of district heating pipes in two and three dimension has been

performed.

In chapter 3, it was concluded that there exist several modes in which the noise can

propagate, but after studying phase velocity, displacements and attenuation, it was concluded

that only two modes, α and α3 are detectable with piezoelectric sensors. The reason is that

these modes have low attenuation at low frequencies and have enough radial displacements

inside the shell to be detected.

The α mode which is governed by the water inside the pipe is a leaky mode due to that the

sound velocity in the embedding medium is lower than the phase velocity of the mode. The

word leaky means that the mode will radiate energy into the embedding medium. Leaky mode

tends to attenuate fast if the leak amount is high. The mode phase velocity at low frequencies

can be calculated from equation 3.17 for all pipes and can be considered as non-dispersive at

frequencies up to approximately ~2 kHz. At near zero frequencies, the mode is not detectable

due to negligible radial displacement and dominant axial displacement inside the pipe, the

mode is said to be purely water-borne. At higher frequencies, the energy is mostly inside the

pipe, but radial displacements slowly start to increase with increasing frequency and dominate

in the pipe shell, the mode is therefore detectable.

The α3 however, has the sound velocity in the embedding medium as asymptote and will at

all frequencies propagate below it. The mode can be considered non-dispersive at frequencies

up to ~2 kHz. The mode is governed by the embedding medium but will not be a leaky mode

and can therefore propagate long distances since it is not attenuative. The α3 has a lot of its

energy in the embedding medium but the energy decays exponentially outside which means

that a lot of the mode energy is located at the outer pipe surface. Note that the velocity of the

surrounding medium is unknown in this thesis due to lack of knowledge of the insulation.

The simulations showed the frequency content of the leak noise. The leak noise is created in

the leak interior according to simulations performed in this thesis. The 3D simulation clearly

shows that the leak noise is strong at 1 kHz and above. Below 1 kHz the noise tends to

decrease for both walls- and interior sound sources. For leak detection, this means that leak

noise should be studied at 1 kHz to 2 kHz where the modes starts to become dispersive. At

higher frequencies, the modes become strongly dispersive and more attenuative (see figure

3.10, 3.11, 3.13 and 3.19)

The simulations also showed that high pressure pipes generate higher noise level than low

pressure pipes. For very low pressure pipes the leak noise could fall into the noise of the core

92

flow and therefore be undetectable. The temperatures impact on the viscosity results in a low

frequency peak in the power spectral density plot (see figure 5.25). This means that it could

be possible to study lower frequencies for low temperature flows. The simulations show that

the amount of vortices inside the leak is dependent on the viscosity. Increasing the viscosity

results in more vortices and vice versa.

The study of acoustic wave propagation and the simulations of district heating pipes both

indicate that the leak noise should be studied between 1-2 kHz where the noise is still non-

dispersive, have low attenuation and is most intense.

6.2 Further work

The theory of the propagating modes in chapter 3 has been experimentally verified by R.

Long et al. For district heating pipes the theory has not been experimentally verified. In

chapter 3.5, the theory was compared to one measurement with the new equipment and the

theory seems to agree with those results, however, more experimental data is needed for better

understanding and how to explain the other peaks that does not correspond to the α and α3

modes. The exact phase velocity dispersion and attenuation curves are not obtained for district

heating pipes. These curves can be obtained by the software DISPERSE which is not

available for this thesis. More knowledge of the embedding medium of the pipe is needed in

order to know the phase velocity of the α3 mode.

The simulations of this thesis shows the frequency content of the leak, but does not take into

account walls, flow, pipe, shell and outer medium. In order to take into account other media

and the propagation properties of the sound, software like SYSNOISE is needed or to

simulate with “Direct Numerical Calculations” in FLUENT which will require more CPU

resources which are not available in this thesis.

The leak in the simulations where approximated as a cylinder, in reality this is not the case.

Simulations with other leak dimension should be performed in order to study the leak content

with respect to the leak dimensions.

More detailed studies of the temperature is needed to study all possible sound sources,

unfortunately, the simulations are very time consuming and only had time to perform two

three dimensional simulation where the acoustic interior included almost the whole leak and

part of the near leak region. Further work must sub divide this region into smaller regions

where the noise can be studied in more detailed. The mesh used in the simulation was fair,

more compact mesh is needed in order to resolve smaller eddies in the turbulent flow. This

will require more CPU and RAM resources.

93

Appendix 3A

Solution to the wave equation in a pipe

The wave equation is written as

2

2

2

2 1

t

u

cu (A.1)

Where the Laplacian operator in cylindrical coordinates is

2

2

2

2

22

22 11

zrrrr (A.2)

Applying separation of variables

)(),,(),,,( tTzrtzru (A.3)

Applying the Laplacian operator and separation of variables to A.1 yields

0),,(22 zrk (A.4)

0)(22

2

2

tTckt

T (A..5)

where k is a constant. Applying separation of variable to A.4 with

)(),(),,( zZrzr (A.6)

yields

0),()(11 22

2

2

22

2

rkkrrrr

z (A.7)

94

0)(2

2

2

zZkz

Zz (A.8)

where kz is another constant. Applying separation of variables again to A.7 with

)()(),( RrPr (A.9)

yields

0)())(( 22222

2 rPkrkkr

Pr

r

Pr z

(A.10)

0)(2

2

2

RkR

(A.11)

The wave equation can be described as a system of differential equation as

0)())(( 22222

2 rPkrkkr

Pr

r

Pr z

(A.12a)

0)(2

2

2

RkR

(A.12b)

0)(2

2

2

zZkz

Zz

(A.12c)

0)(22

2

2

tTkct

T (A.12d)

The solutions to the equations are as follow

)()()( 21 rkYArkJArP ckck (A.13a)

)cos()sin()( 21 kBkBR (A.13b)

zikzik zz eCeCzZ 21)( (A.13c)

titi eDeDtT 21)( (A.13d)

95

where kc2=k

2-kz

2, ω=kc and equation A.13a are the Bessel functions. The constant A2 can be

set to zero since Ykθ goes to infinity at zero. Equation A.13b has periodic boundary conditions,

R(θ)=R(θ+2π), and therefore kθ must be equal to an integer N=0,1,2…. The solution to the

wave equation can be written as

))()(())sin()cos((),,,( 2121121

titizikzik

cn eDeDeCeCrkJBnAnAtzru zz (A.14)

where

kn

kc

kkk zc

222

The solution for waves in positive z-direction is

)(

21 )())sin()cos((),,,(tzki

cnzerkJnCnCtzru (A.15)

where C1 and C2 are new constants.

96

Appendix 5A

Setting for the κ-ω turbulent model in

FLUENT

FLUENT Version: 2d, dp, pbns, skw (2d, double precision, pressure-based, standard k-omega) Release: 6.3.26 Models ------------------------------------------------------- Model Settings ----------------------------------------------------- Space 2D Time Steady Viscous k-omega turbulence model Heat Transfer Disabled Solidification and Melting Disabled Species Transport Disabled Coupled Dispersed Phase Disabled Pollutants Disabled Pollutants Disabled Soot Disabled Solver Controls ------------------------------------------------------- Equations Equation Solved ----------------------------------------------- Flow yes Turbulence yes Numerics Numeric Enabled --------------------------------------------------------------------- Absolute Velocity Formulation yes Relaxation Variable Relaxation Factor --------------------------------------------- Pressure 0.30000001 Density 1 Body Forces 1 Momentum 0.69999999 Turbulent Kinetic Energy 0.80000001 Specific Dissipation Rate 0.80000001

97

Turbulent Viscosity 1 Linear Solver Solver Termination Residual Reduction Variable Type Criterion Tolerance ------------------------------------------------------------------------------- Pressure V-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Turbulent Kinetic Energy Flexible 0.1 0.7 Specific Dissipation Rate Flexible 0.1 0.7 Pressure-Velocity Coupling Parameter Value ------------------ Type SIMPLE Discretization Scheme Variable Scheme ----------------------------------------------- Pressure Standard Momentum Second Order Upwind Turbulent Kinetic Energy Second Order Upwind Specific Dissipation Rate Second Order Upwind Solution Limits Quantity Limit ---------------------------------------------------------------- Minimum Absolute Pressure 1 Maximum Absolute Pressure 5e+10 Minimum Temperature 1 Maximum Temperature 5000 Minimum Turb. Kinetic Energy 1e-14 Minimum Spec. Dissipation Rate 1e-20 Maximum Turb. Viscosity Ratio 100000

98

Appendix 5B

Setting for LES turbulent model in

FLUENT

FLUENT Version: 2d, dp, pbns, LES, unsteady (2d, double precision, pressure-based, large eddy simulation, unsteady) Release: 6.3.2 Model ------------------------------------------------------- Model Settings --------------------------------------------------------- Space 2D Time Unsteady, 2nd-Order Implicit Viscous Large Eddy Simulation Sub-Grid Scale Model Smagorinsky-Lilly Heat Transfer Disabled Solidification and Melting Disabled Species Transport Disabled Coupled Dispersed Phase Disabled Pollutants Disabled Pollutants Disabled Soot Disabled Solver Controls ------------------------------------------------------- Equations Equation Solved ----------------- Flow yes Numerics Numeric Enabled --------------------------------------- Absolute Velocity Formulation yes Unsteady Calculation Parameter ---------------------------------------------------------------- Time Step (s) 1e-06 (2e-6 for the case p=3 atm) Max. Iterations Per Time Step 20

99

Non-Iterative Solver Factors Variable Max. Corrections Correction Tolerance Residual Tolerance Relaxation Factor ------------------------------------------------------------------------------------------- Pressure 10 none 0.25 none 9.9999997e-05 1 Momentum 5 none 0.050000001 none 9.9999997e-05 1 Linear Solver Solver Termination Residual Reduction Variable Type Criterion Tolerance -------------------------------------------------------- Pressure F-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Pressure-Velocity Coupling Parameter Value --------------------------------------------------------- Type Fractional Step Discretization Scheme Variable Scheme --------------------------------------- Pressure PRESTO! Momentum Bounded Central Differencing Solution Limits Quantity Limit -------------------------------------- Minimum Absolute Pressure 1 Maximum Absolute Pressure 5e+10 Minimum Temperature 1 Maximum Temperature 5000 Maximum Turb. Viscosity Ratio 100000

100

Appendix 5C

y+, drag- and lift force and Courant

Number for 2D pipe with 100oC and 10

atm

Figure C.1 shows a histogram of the length y

+ for all walls. There are only a few that does not

fulfill the criterion 30 < y+ but are still acceptable.

Fig C.1; a) Almost all grid points ful fill the criterion y+>30. There are a few below 30 but it does not matter if a

few cells are below 30. b) The front wall looks quite good with most cells at y+= 50. c) The rear wall have a few

cell at ~20, most of those cells are very close to the leak in the near leak region. d) The lower wall grid points

have all cells over 30.

101

The near-leak region was a bit tricky to mesh, and the consequences are observed in figure

C.1c. A few cells lays in the near leak region just beyond the leak. However, the results are

still acceptable and should not influence the results.

The drag- and lift force on the walls are shown in figure C.2. When the drag- and lift force

are oscillatory and periodic, then the unsteady solution with LES is ready for setting up an

acoustic model to perform acoustic calculations.

Fig C.2; The drag force to the left and the lift force to the right for the walls.

A histogram of the Courant number for each cell is shown in figure C.3. Most of the cells

have satisfying Courant number, except some at the leak where the velocity can increase to up

to 70 m/s, thus giving a Courant number greater than 1, however, the largest Courant number

is around 2, which is acceptable.

Fig C.3; The cell Courant number for an unsteady flow with LES.

102

Appendix 5D

y+, drag- and lift force and Courant

Number for a 2D Pipe with 40oC and 3

atm

Figure D.1 shows a histogram of the length y

+ for all walls. There are very few cells that does

not fulfill the criterion 30 < y+.

Fig D.1; a) Almost all grid points ful fill the criterion y+>30. There are a few below 30 b) The front wall looks

good with most cells at y+= 50. c) The rear wall have almost all grid points at y

+=40. d) The lower wall grid

points have all cells over 30.

103

The drag- and lift force are oscillative at flow time 0.35 s and the LES solution is ready for

acoustic calculation.

D.2 The drag force to the left and the lift force to the right for the walls.

Figure D.3 show the Courant number, which is satisfying since there are only a few elements

with Courant number higher than 1.

Fig D.3; The cell Courant number.

104

Appendix 5E

Equation of state for liquids in FLUENT

#include "udf.h"

#define BMODULUS 2.2e9

#define rho_ref 1000.0

#define p_ref 101325 //1 atm, 10 and 3 atm are used in the simulations

DEFINE_PROPERTY(superfluid_density, c, t)

{

real rho;

real p, dp;

real p_operating;

p_operating = RP_Get_Real ("operating-pressure");

p = C_P(c,t) + p_operating;

dp = p-p_ref;

rho = rho_ref/(1.0-dp/BMODULUS);

return rho;

}

DEFINE_PROPERTY(sound_speed, c,t)

{

real a;

real p, dp,p_operating;

p_operating = RP_Get_Real ("operating-pressure");

p = C_P(c,t) + p_operating;

dp = p-p_ref;

a = (1.-dp/BMODULUS)*sqrt(BMODULUS/rho_ref);

return a;

}

105

Appendix 5F

Setting for the κ-ω turbulent model in

three dimensions in FLUENT

FLUENT Version: 3d, dp, pbns, skw (3d, double precision, pressure-based, standard k-omega) Release: 6.3.26 Title: Solver Controls --------------- Equations Equation Solved ------------------- Flow yes Turbulence yes Numerics Numeric Enabled --------------------------------------------------------------- Absolute Velocity Formulation yes Relaxation Variable Relaxation Factor --------------------------------------------- Pressure 0.30000001 Density 1 Body Forces 1 Momentum 0.69999999 Turbulent Kinetic Energy 0.80000001 Specific Dissipation Rate 0.80000001 Turbulent Viscosity 1 Linear Solver Solver Termination Residual Reduction Variable Type Criterion Tolerance ----------------------------------------------------------------------- Pressure V-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Z-Momentum Flexible 0.1 0.7 Turbulent Kinetic Energy Flexible 0.1 0.7 Specific Dissipation Rate Flexible 0.1 0.7

106

Pressure-Velocity Coupling Parameter Value ------------------ Type SIMPLE Discretization Scheme Variable Scheme ----------------------------------------------- Pressure PRESTO! Density Second Order Upwind Momentum Second Order Upwind Turbulent Kinetic Energy Second Order Upwind Specific Dissipation Rate Second Order Upwind Solution Limits Quantity Limit --------------------------------------- Minimum Absolute Pressure 1 Maximum Absolute Pressure 5e+10 Minimum Temperature 1 Maximum Temperature 5000 Minimum Turb. Kinetic Energy 1e-14 Minimum Spec. Dissipation Rate 1e-20 Maximum Turb. Viscosity Ratio 100000

107

Appendix 5G

Setting for LES turbulent model in three

dimensions in FLUENT

FLUENT Version: 3d, dp, pbns, LES, unsteady (3d, double precision, pressure-based, large eddy simulation, unsteady) Release: 6.3.26 Title: Solver Controls --------------- Equations Equation Solved ----------------- Flow yes Numerics Numeric Enabled --------------------------------------- Absolute Velocity Formulation yes Unsteady Calculation Parameters ------------------------------------- Time Step (s) 1e-06 Max. Iterations Per Time Step 3 Relaxation Variable Relaxation Factor ------------------------------- Pressure 0.69999999 Density 1 Body Forces 1 Momentum 0.69999999 Linear Solver Solver Termination Residual Reduction Variable Type Criterion Tolerance -------------------------------------------------------- Pressure V-Cycle 0.1 X-Momentum Flexible 0.1 0.7 Y-Momentum Flexible 0.1 0.7 Z-Momentum Flexible 0.1 0.7

108

Pressure-Velocity Coupling Parameter Value ----------------------------------- Type PISO Skewness-Neighbour Coupling yes Skewness Correction 1 Neighbour Correction 1 Discretization Scheme Variable Scheme --------------------------------------- Pressure PRESTO! Density Second Order Upwind Momentum Bounded Central Differencing Solution Limits Quantity Limit -------------------------------------- Minimum Absolute Pressure 1 Maximum Absolute Pressure 5e+10 Minimum Temperature 1 Maximum Temperature 5000 Maximum Turb. Viscosity Ratio 100000

109

Appendix 5H

y+, drag- and lift force and Courant

Number

The wall distances y+ for the leak, the near leak-, front and rear region and the rest of the pipe

can be found in figure H.1. It is important that the leak (fig H.1a) has its wall distances

between 30 < y+ < 150 since it is the region of interest i.e. the acoustic region.

Fig H.1; The wall distances y+ for a) the leak b) near leak-, rear and front regions and c) the rest of the pipe

110

The wall distances are not as satisfying as in two dimensions. Many cells fall below y+ =

30. The main reason is that it is harder to mesh three dimensional pipes with leaks. Figure H.2

shows a plot of the pipe with the wall distances. The range is 20 < y+ < 150. Note the two

small “white holes” near the leak, unfortunately this part falls in the region below y+ = 20. It

is a very difficult task to fulfill the criterion 30 < y+ < 150 everywhere. However, this mesh

should be good enough for calculating noise from the leak.

Fig H.2; The wall distances. The lower limit is 20 (blue) and the upper limit is 150 (red). Note the two “white

holes” in which the criterion is not fulfilled. The flow is in positive x direction.

The drag- and lift force can be found in figure H.3. The forces are oscillative and the

simulation is ready for saving acoustic data.

111

Fig H.3; a) The drag force of the leak walls and b) the lift force of the leak wall.

The Courant number can be found in figure H.4. The Courant number reaches a maximum of

approximately 1.9 which is satisfying. The cell Courant number in the rest of the pipe is

extremely small due to the lower velocity compared to the velocity of the flow inside the leak.

Fig H.4; The cell Courant number for a) the leak and acoustic region and b) the rest of the pipe.