Aeroacoustics in High Speed Trains

Universidad Politécnica de Madrid

Doctoral dissertation: AEROACOUSTICS IN HIGH SPEED TRAINS

Félix Sorribes Palmer

Advisors:

Prof. Dr. Ing. Ángel Sanz Andrés Prof. Dr. Ing. Gustavo Alonso Rodrigo

E.T.S.I. Aeronáuticos, Universidad Politécnica de Madrid, December 2014

Resumen

Este trabajo se centra en el estudio de problemas aeroacústicos en los trenes de alta ve- locidad. Se han considerado dos escenarios en los que las ondas de presión generadas son críticos para el confort de los pasajeros. Uno es el debido a las ondas de presión que genera el tren cuando entra y sale de un túnel, que a su vez producen saltos de presión de baja frecuencia en el tren (cuando se cruzan con él) y en los alrededores del túnel cuando alcanzan la salida. Se estudia este fenómeno, y se propone un sistema aeroelás- tico basado en el galope transversal para disminuir la energía de estas ondas, y se analiza la energía extraíble de las ondas utilizando cuerpos con diferentes secciones transversales [Sorribes-Palmer and Sanz-Andres, 2013]. La influencia de la geometría de los portales en la energía radiada hacia el exterior de túnel es analizada experimentalmente, prestando especial atención a las boquillas porosas. Las ondas de presión en el interior del túnel se han analizado mediante el método de las características. Se han realizado ensayos experimentales para estimar la energía reflejada hacia el interior del túnel al alcanzar las ondas de presión el portal de salida del túnel. Se ha estudiado la formación e interacción entre el portal del túnel y la onda de choque generada en los túneles de gran longitud y pequeña fricción. Se propone un método para describir de forma aproximada el ruido radiado al exterior. Por otro lado se ha estudiado el ruido de media y alta frecuencia de origen aerod- inámico. Se ha estudiado la influencia del desprendimiento de la capa límite sobre el tren. Se propone una metodología basada en una sección de tren característica para predecir rápidamente el nivel de presión de sonido dentro y fuera del tren para todo el rango de frecuencias. Se han realizado medidas experimentales en vía de los espectros de presión sobre la superficie del tren, y de la transmisibilidad de las uniones entre estructura y reves- timiento. Los resultados experimentales se han utilizado en los modelos vibroacústicos. El método de la sección del tren característica es especialmente útil a altas frecuencias cuando todo el tren se puede modelar mediante el ensamblaje de diferentes secciones características utilizando el análisis estadístico de la energía. Summary

This work is focused on the study of aeroacoustic problems in high speed trains. We have considered two scenarios in which the pressure waves generated are critical for passengers comfort. The first one is due to the pressure waves generated by a train entering in a tunnel. These waves generate pressure gauges inside the train (when they find each other) and outside of the tunnel portals. This phenomenon has been studied, and an aeroelastic system based on transverse galloping to reduce the energy of these waves is proposed. The maximum extractable energy by using bodies with different cross-section shapes is analyzed. The influence of the portals geometry in the energy radiated outwards the tunnel is analyzed experimentally, with particular attention to the porous exits. The pressure waves inside the tunnel have been analyzed using the method of characteristics. Experimental tests to estimate the energy reflected into the tunnel when the pressure waves reach the tunnel portal have been performed. We have studied the generation and interaction between the tunnel portal and a shock wave generated in long tunnels with small friction. A method to describe in an approximated way the pressure radiated outside the tunnel is proposed. In the second scenario, middle and high frequency noise generated aerodynamically has been studied, including the influence of the detachment of the boundary layer around the train. A method based on a train section to quickly predict the sound pressure level inside and outside the train has been proposed. Experimental test have been performed on board to evaluate the pressure power spectra on the surface of the train, and the transmissibility of the junctions between the structure and trim. These experimental results have been used in the vibroacoustic models. The low frequency pressure waves generated with the train during the tunnel crossing has been identified in the pressure spectrum. The train characteristic section method is especially useful at high frequencies, when the whole train can be modeled by assembling different sections using the statistical energy analysis. The sound pressure level inside the train is evaluated inside and outside the tunnel. Acknowledgments

First of all I would like to acknowledge the director of Instituto Universitario de Micro- gravedad “Ignacio Da Riva”, Prof. Dr. Ing. José Meseguer, for giving me the opportunity to work in the institute. I have been able to work in different areas and learn a lot from many persons in the institute, specially from my advisors Prof. Dr. Ing. Angel Sanz An- drés and Prof. Dr. Ing. Gustavo Alonso Rodrigo, and i want to thank them their patience to listen and explain every time I needed it. I also want to thank the rest of the people of the institute for their support: Álvaro Cuerva, Oscar López, Santiango Pindado, Sebas- tian Franchini, Javier Pérez, Isabel Pérez, Alejandro Martínez, Mohsen Ghaemi, Rafael García, Sergio Ávila, Alejandro Gómez, etc.. I want to thank the late Prof. Dr. Ing. Jesús López Díez for all his help on introducing me to the world of vibroacoustics. I am really grateful to Universidad Politécnica de Madrid for the scholarship to spend a three months visit at TU Berlin, where they let me use their facilities to perform experimental test and also helped me to understand vibroacoustic modeling, this experience was highly useful to focus my work and to push forward this dissertation. I want to thank my friends for cheering my up during the tough moments and for sharing their time with me: Marcos, Diego, Laura, Joseba, Eduardo, Pedro, etc. I render thanks to Dani for reading everything I gave him. Lastly, I would like to thank my family, for their support and encouragement, because having them by my side made everything look possible. And most of all for my loving girlfriend Julija for her patience, unconditional love and faithful support. A life full of possibilities is open to us now, and I am willing to explore it with you “ljubezen”, rtm. Contents

Contents i

List of Figures iii

List of Tables ix

1 Introduction 1 1.1 Noise sources in high speed trains ...... 1 1.2 Objectives and content structure ...... 9

2 Pressure waves in high speed railway tunnels 11 2.1 Introduction ...... 11 2.2 Mathematical model for prediction of pressure waves inside a tunnel . . . 15 2.2.1 Condition for the existence of a plane wave ...... 16 2.2.2 Propagation of the signature ...... 17 2.3 Hydrostatic pressure influence on high speed railway tunnels cross-section sizing ...... 22 2.4 Non-linear propagation and pressure wave steepening inside a tunnel . . 25 2.5 Pressure wave interaction at the tunnel exit ...... 29 2.6 Experimental set-up ...... 31 2.7 Results and discussion ...... 42 2.7.1 Reflected wave at the horns ...... 42 2.7.2 Reflected and transmitted wave at airshafts ...... 43 2.7.3 Reflected and transmitted wave at a perforated section ...... 44 2.7.4 Reflected wave at perforated exits ...... 45 2.8 Conclusions ...... 49

3 Energy extraction from aerodynamic instabilities 51 3.1 Introduction ...... 51 3.2 Mathematical model ...... 53 3.2.1 Extracted power in a general case ...... 56 3.2.2 Comparison between numerical integration method and polynomial expansion ...... 60 3.2.3 Inﬂuence of the number of discretization points ...... 60

i ii CONTENTS

3.2.4 Comparison of the numerical integration method with the polynomial expansion at a point different from the origin...... 62 3.3 Experimental set-up ...... 65 3.4 Results and discussion ...... 67 3.4.1 Biconvex airfoil ...... 67 3.4.2 D-shape body ...... 70 3.4.3 Rhomboidal cross-section bodies ...... 72 3.4.4 Triangle cross-section bodies ...... 75 3.4.5 Square cross-section bodies ...... 76 3.5 Conclusions ...... 80

4 Interior noise prediction in high speed trains 83 4.1 Introduction ...... 83 4.2 Numerical analysis ...... 85 4.2.1 Basic acoustic concepts and definitions ...... 85 4.2.2 Methodology ...... 93 4.2.3 Structural and fluid models ...... 93 4.2.4 Loads ...... 99 4.2.4.1 Structure-borne ...... 99 4.2.4.2 Airborne ...... 101 4.3 Experimental set-up ...... 102 4.4 Tests definition ...... 104 4.4.1 Aerodynamic noise characterization ...... 104 4.4.1.1 SPL in open field ...... 104 4.4.1.2 SPL inside a tunnel ...... 105 4.4.2 Point junction characterization ...... 106 4.5 Interior noise prediction ...... 107 4.5.1 Flow detachment influence on interior noise ...... 108 4.6 Conclusions ...... 108

5 Conclusions and future work 111 5.1 Conclusions ...... 111 5.2 Future work ...... 114

Bibliography 115

A Appendix 125 A.1 Wave separation ...... 125 A.2 Slowly varying cross section ducts ...... 128 A.3 Experimental set-up calibration ...... 131 A.3.1 Influence of microphone insertion ...... 131 A.3.2 Influence of leakage on microphone holes ...... 131 A.3.3 Influence of relative humidity ...... 131 A.3.4 Influence of pipe junctions ...... 132 A.3.5 Validation of the experimental set-up with analytic expression for the reflection coefficient ...... 133 List of Figures

1.1.1 Variation of the sound power levels Lw with train speed Utr of different sound sources (after Lauterbach et al. [2012])...... 2 1.1.2 Main sources of interior noise in a train and their transmission paths into the vehicle (after Thompson [2009])...... 4

2.1.1 Train reference signature deﬁned by the standard EN14067-5:2006:E [2006]...... 12 2.2.1 Pressure variation in the tunnel as a function of time, validation of the TWS implementation, M=0.24 and φ =0.135. Pressure evolution in the tunnel at 550 m from the tunnel entrance. Experimental results from William-Louis [1999]...... 20 2.2.2 Pressure variation at the train as a function of time, validation of the TWS implementation M=0.24 and φ =0.135. Pressure evolution in the train at 72 m from the nose...... 21 2.2.3 Position x of the pressure waves generated by a high speed train inside a tunnel of length Ltun as a function of time t. C, compression: E, expansion wave...... 21 2.3.1 Tunnel cost (per meter) as a function of the tunnel cross-section area, Atun, for different Rock Mass Ratings (RMR)...... 23 2.3.2 Pressure inside (red) and outside (blue) the train during pass through a tunnel of 20 km length with 25 ‰ ascendant slope...... 24 2.3.3 Critical cross-section area of tunnels with 25‰ of ascendant slope and length 20, 25 and 30 km...... 25 2.4.1 Variation with time t of pressure P of the wave inside a tunnel...... 28 2.5.1 Result for the radiated pressure by the non-linear propagation method. Variation with time of: 1) predicted pressure at the tunnel entrance and exit; 2) pressure gradient at the exit; 3) predicted radiated pressure at r=1 m from the tunnel exit in Pa; 4) predicted radiated pressure at r=1 m from the tunnel exit in dB...... 31 2.5.2 Predicted radiated pressure spectrum (sound pressure level, SPL) in dB by using a Gaussian approximation of the pressure gradient (blue line) and measurements from experiments (red line)...... 31 2.6.1 Scheme of the set-up components: speaker (1), microphones (2), signal generator (3) and computer (4)...... 32

iii iv List of Figures

2.6.2 Variation of the pressure gauge P (in arbitrary units) with time t. The- oretical signal (blue line), actual signal reproduced by the high-speaker (red line)...... 32 2.6.3 Scheme exits A1 and A2. Side view...... 33 2.6.4 Mounting of the porous exit A1...... 34 2.6.5 Mounting of the porous exit A2...... 34 2.6.6 Mounting of the PVC porous exit A2...... 34 2.6.7 Tube with ﬂanged open end...... 35 2.6.8 Perforated cylindrical section...... 35 2.6.9 Area section of the horns...... 35 2.6.10 Intermediate adapter and semi-cylindrical exits (B1, B2, B3, B4 and B5). 36 2.6.11 Mounting of the B1 exit...... 36

2.6.12 Mounting of the airshaft Sc/Sh =1...... 37 2.6.13 Adapter for airshaft of different sections...... 37 2.6.14 Pressure gauge on the train and inside the tunnel with an airshaft of area section 12 m2 at 350 m from the entrance in a tunnel of 90 m2 cross- section area, train speed 241 km/h...... 38 2.6.15 Scheme of the ducts exit porosity distribution...... 41 2.6.16 Porous duct exits of porosity 0.18 %, 0.42 % and 1.16 % tested at TU Berlin...... 42 2.6.17 Experimental set-up to implement the TMTC technique using an aluminum exit of porosity σ =1.16%...... 42

2.7.1 a) Variation with the area ratio Sc/St of effective pressure transmitted by the airshaft RITH multiplied by the inverse of the area ratio. Continuous line: results of Baron et al. [2001], circles: measurements. b) Variation with the area ratio St/Sc of the pressure wave reflection coefficient at the junction between the duct and the airshaft coefRT , transmitted through the duct coefTT, and transmitted to the airshaft coefTC...... 44 2.7.2 Variation with time t of pressure P (arbitrary units) of the incident wave OI (blue line) and the reflected wave OR (red line) at the PVC perforated exit...... 45 2.7.3 Reflection coefficient at the perforated exit as a function of the number of holes opened, for flanged and unflanged exits...... 46 2.7.4 Influence of the open hole as a function of the distance to the exit for different frequencies...... 47 2.7.5 Variation with frequency f of the reflection coefficient R(f) for exit samples with porosity σp 0.18, 0.42 and 1.16 % obtained with TMTC method, compared with flanged and unflanged analytical expression, and experiment results obtained with pulse reflectometry...... 48 2.7.6 Sound pressure level, SPL [dB] radiated from the duct end by hoods with different porosity...... 49

3.2.1 Schematic representation of aerodynamic forces and the angle of attack, α, pitch angle of the velocity, γp, pitch angle of the body, θ; z, x is the inertial reference frame and zb, xb is the reference frame attached to the body...... 54 List of Figures v

3.2.2 Variation of the dimensionless extracted power averaged per cycle, pe, ∗ ∗ A A with the relative amplitude of the motion, ∗ . ∗ is the relative U U pemax amplitude that provides maximum extracted power...... 58 A∗ 3.2.3 Variation of b1 with the relative amplitude . a) Possible equilibrium U ∗ solutions: 1) one solution, 2) 3 solutions, 3) limit case. E: stable, I: unstable. b) no inflection points (case 1); one inflection point (case 2). . . 59 3.2.4 Determination of the maximum dimensionless extracted power from experimental results. a) Variation of dimensionless extracted power, pe,as A∗ a function of relative amplitude, , for two values of the pitch angle θ, U ∗ b) Variation of maximum extracted power, pemax, with the pitch angle of the body, θ...... 59 ∗ ∗ 3.2.5 Variation of coefficient, b1, with the relative amplitude, A /U . Com- parison of the analytical solution, b1a, and the numerical integration, b1n, ◦ for different curves of Cz (polynomial functions) around α =0, for the cases k according to Table 3.2.1. Discretization Δα =1◦...... 61 ∗ ∗ 3.2.6 Variation of coefficient, b1, with the relative amplitude, A /U . Influ- ence of the discretization interval, Δα, of the force coefficient, Cz,in the numerical integration of b1. Case k =3of Table 3.2.1...... 61 3.2.7 Variation of the uncertainty in the relative amplitude at the equilibrium, disc, as a function of the discretization interval Δα of the curve Cz (α). Case k =3of Table 3.2.1...... 62

3.2.8 Variation of the transverse force coefﬁcient, Cz, with the pitch angle of the body, θ. Symbols: pitch angles analyzed in Figure 3.2.9. Case k =1, see Table 3.2.1...... 63

3.2.9 Variation of the transverse force coefficient, Czd (α), considering different pitch angles, θ0, and the antisymmetric function, Czda, used in the numerical integration of b1. Case k =1, for θ0 angles shown in Figure 3.2.8...... 63 A∗ 3.2.10 Variation of coefficient b1 with the relative amplitude , obtained by U ∗ numerical integration, b1n, at configurations with pitch angle θ0 =0 , and ◦ b1 analytic, b1a. Discretization interval Δθ =1. Case k =1in Table 3.2.1. For values of θ0 see Figure 3.2.8...... 64 A∗ 3.2.11 Variation of coefficient b1 with the relative amplitude , obtained by U ∗ numerical integration, b1n, at configurations with pitch angle θ0 =0 , and ◦ analytical solution, b1a. Discretization interval Δθ =4...... 64 3.2.12 Variation with the discretization interval Δθ of the standard deviation, σb1, between the analytic solution b1a and the numerical integration b1n due to the discretization of Cz, for different pitch angles θ0. Case k =1 in Table 3.2.1...... 65 3.3.1 Wind tunnel A4C: a) Instrumentation; b) Detail of the ATI six-component strain-gauge balance (model Gamma SI-130-10)...... 66 3.3.2 Wind tunnel A9: a) Instrumentation; b) Detail of the ATI six-component strain-gauge balance (model Delta FT5575)...... 66 3.4.1 Nomenclature in the case of a biconvex airfoil...... 67 vi List of Figures

3.4.2 Variation with the pitch angle of the body, θ, of the coefficients of lift, cl, and drag, cd, and of the Glauert-Den Hartog parameter, H, over the biconvex airfoil of relative thickness Er =0.817 in the A4C wind tunnel. Symbols indicate the pitch angles θ0 considered in the analysis of Figures 3.4.3 and 3.4.4...... 68 3.4.3 Variation of the force coefficient, Cz, with the angle of attack, α, where ◦ α = θ − θ0, for different values of pitch angle θ0, around θ0 =70 = 1.221 rad. Curve Cz interpolated, CzI, and its antisymmetric part, CzIa, for the biconvex airfoil of relative thickness Er =0.817...... 68 A∗ 3.4.4 Variation with the relative amplitude of the coefficient b1 from the U ∗ n numerical integration, and the specific dissipated energy, p˜d, for a biconvex airfoil of relative thickness Er =0.817 oscillating around a pitch ◦ angle of the body θ0 =1.221 rad = 70 ...... 69 ∗ ∗ 3.4.5 Variation of the relative amplitude of equilibrium A /U |eq as a function of the pitch angle, θ, for a certain specific dissipated power, p˜d, obtained from the parameters in Table 3.2.2...... 69 1 3.4.6 Variation of the maximum specific extracted power p˜ max = b1| ,as e 2 max a function of the pitch angle of the body, θ...... 70 3.4.7 Variation of the maximum dimensionless extracted power, pemax,asa function of the pitch angle of the body, θ...... 70 3.4.8 Nomenclature in the case of the D-shape body...... 71 3.4.9 Variation with the pitch angle of the body, θ, of the coefficients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the D-shape body tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.10)...... 71 A∗ 3.4.10 Variation with the relative amplitude of the coefficient b1 obtained U ∗ n from the numerical integration and the dissipation, p˜d, for the D-shape body, oscillating around a pitch angle of the body close to θ =80◦ =1.4 rad...... 72 3.4.11 Nomenclature in the case of the rhomboidal cross-section airfoil. .... 72 3.4.12 Variation with the pitch angle of the body, θ, of the coefficients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the rhomboidal airfoil Er =11/30 tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.13)...... 73 A∗ 3.4.13 Variation with the relative amplitude of the coefficient b1 obtained U ∗ n from the numerical integration, and the dissipation p˜d, for the rhom- ◦ boidal airfoil of Er =11/30 around θ =23 =0.40 rad tested in the A9 wind tunnel...... 73 3.4.14 Variation with the pitch angle of the body, θ, of the coefficients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the rhomboidal airfoil Er =10/30 tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.15)...... 74 A∗ 3.4.15 Variation with the relative amplitude of the coefficient b1 obtained U ∗ n from the numerical integration, and the dissipation p˜d, for the rhom- ◦ boidal airfoil of Er =10/30 around θ =23 =0.40 rad tested in the A9 wind tunnel...... 74 List of Figures vii

3.4.16 Variation with the pitch angle of the body, θ, of the coefficients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the rhomboidal airfoil Er =12/30 tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.17)...... 75 A∗ 3.4.17 Variation with the relative amplitude of the coefficient b1 obtained U ∗ n from the numerical integration, and the dissipation p˜d, for the rhomboidal airfoil of Er =12/30 around θ =23=0.40 rad tested in the A9 wind tunnel...... 75 3.4.18 Variation with the pitch angle, θ, of the maximum dimensionless extracted power, pemax, for different cross-section bodies...... 76 3.4.19 Variation with the pitch angle, θ, of the maximum specific extracted power, p˜emax, for different cross-section bodies...... 76 3.4.20 Variation of the transverse force coefficient, Cz with the angle of attack, α, experimental data from Ng et al. [2005], Parkinson and Smith [1964], Luo and Bearman [1990] and Parkinson and Brooks [1961]. In this Fig- ure, the 9th and 11th order polynomials of Ng et al. [2005] are coincident. 77 A∗ 3.4.21 Variation with the relative amplitude of the coefficient b1 obtained U ∗ by direct numerical integration of experimental data, b1ne; by integration of a 7th order polynomial, b1n7; and the specific dissipated power, p˜d, for a square-cylinder. Polynomial coefficients (a1 =2.69, a3 = −168, a5 = 6270, a7 = 59900 at Re∼ 22300) from Parkinson and Smith [1964]. . . . 77 A∗ 3.4.22 Variation with the relative amplitude of the coefficient b1 obtained U ∗ by direct numerical integration of 7th, 9th and 11th order polynomial, b1n7, b1n9 and b1n11, respectively; experimental data, b1ne; and the specific dissipated power, p˜d, for a square-cylinder. Experimental data and polynomial coefficients extracted from Ng et al. [2005]...... 78 3.4.23 Variation of the transverse force coefficient Cz, with the angle of attack α (from Parkinson and Brooks [1961]) and 7th, 9th and 11th order polynomials that fit the experimental data, obtained with a Matlab function (polyfit), see Table 3.4.1...... 78 A∗ 3.4.24 Variation with the relative amplitude of the coefficient b1 obtained U ∗ by numerical integration of 7th, 9th and 11th order polynomial, b1n7, b1n9 and b1n11, respectively; experimental data, b1ne; and the specific dissipated power, p˜d, for a square-cylinder. The lift cl and drag cd coefficients to obtain Cz were taken from Parkinson and Brooks [1961], and Cz(α) is shown in Figure 3.4.23...... 79 3.4.25 Damper device based on transverse galloping with active magnetic suspension...... 80

4.2.1 DLF of a common trimmed interior cavity of a vehicle...... 89 4.2.2 Two plates connected through a point junction...... 89 4.2.3 Mobility Y = |v/F| of a common structure used as ﬂoor in high speed trains...... 90 4.2.4 Variation with frequency of the TL of a panel ...... 91 4.2.5 Open ﬁeld vibro-acoustic models...... 94 4.2.6 TL of structures parts of the train...... 96 4.2.7 Some of the DLFs used in the SEA structural subsystems...... 97 viii List of Figures

4.2.8 Modal densities of the FEM structural parts prone to be substituted by SEA subsystems...... 97 4.2.9 Analysis of modal density and radiation loss factor of train roof to interior and exterior acoustic cavities...... 98 4.2.10 CLFs to exterior and interior cavities from wagon roof and floor. .... 98 4.2.11 Difference on CLFs depending on panel excitation...... 99 4.2.12 SEA vibro-acoustic models: a) 2 wagons, b) section model...... 99 4.2.13 a) SPL (dB) predicted numerically with the one section model in the interior and exterior acoustic cavity in 23 m from the nose of the train, due to the relocated load of converter at 29 m and TBL of the whole train; b) SPL (dB) in the interior and exterior of the section at 23 m, due to all the loads in this section and the relocated loads of the converter at 29 m and the TBL of the rest of the train...... 100 4.2.14 Continuity of vibro-acoustic models...... 100 4.2.15 Cross-sections of a) extruded structure Z-shape and b) equivalent flat plate.101 4.3.1 Microphones locations and sections instrumented with accelerometers. . 103 4.3.2 Accelerometers placement layout around the window frame...... 103 4.4.1 SPL (dB) in the microphones at 300 km/h in open field: a) at the head; b) at the tail...... 105 4.4.2 SPL (dB) in the microphones a) at the head b) at the tail at 300 km/h inside a tunnel...... 106 4.4.3 Low frequency noise inside the tunnel. SPL spectrum of microphones averaged M6, M5, M4 and accumulated M6a, M5a, M4a ...... 106 4.4.4 Sound pressure level spectra in thirds of octave bands considered in the vibro-acoustic model...... 107 4.4.5 Acceleration spectral density in the junctions at 300 km/h...... 107 4.4.6 Transmissibility obtained from the accelerometers at 300 km/h [Sorribes- Palmer et al., 2014]...... 108 4.5.1 SPL (dB) predicted inside the train at 300 km/h in a section at 25 m from the nose: a) analytic expression for CLF in point junction b) point junction CLF obtained from experimental test...... 108 4.5.2 SPL inside the train for attached and detached flow configuration at different speeds...... 109 4.5.3 SPL inside the train at 11 meter from the nose at speed: a) 300 km/h b) 350 km/h...... 109

A.1.1 Wave separation using the method proposed in Kikuchi et al. [2009]. . . 127 A.1.2 Wave separation using the proposed method...... 128 A.2.1 Geometry of the considered horns. The length l has been chosen to be 2.4 times the height h of the section at the exit...... 129 A.2.2 Variation of the transmission coefficient τ with the Helmholtz number of conical, exponential and catenoidal horns...... 130 A.3.1 Undesired reflection at a junction...... 133 A.3.2 Tube with unflanged open end...... 133 A.3.3 Tube with flanged open end...... 133 A.3.4 Variation of the reflection coefficient at the duct exit with the Helmholtz number (ka), compared with the expression proposed in Silva et al. [2009].134 List of Tables

2.3.1 Train properties...... 24 2.6.1 Area of holes of exit A2...... 33 2.6.2 Horn dimensions...... 36 2.6.3 Temperature corrections in the air properties [Flecher and Rossing, 1991]. . . 39 2.6.4 Scheme of pressure measurements during microphones position exchange. Fe is the scale factor; αa: plane wave attenuation coefﬁcient; Poa and Pob: reference pressure of cases a and b, respectively...... 39

2.6.5 Scale factor, Fe, attenuation coefﬁcient, αa, inside the PVC tube for different distances between the microphones s...... 40 2.6.6 Frequency ranges under analysis in the different experimental set-ups. The distance between the microphones used in the TMCT was s =76.5 mm. . . . 40

2.6.7 Diameter radius Φ, and porosity σp, of the samples tested with the TMTC technique...... 41 2.7.1 Ratio between the reflection coefficient in the horns and the horn B1...... 43 2.7.2 Transmission loss (TL) through a porous section...... 45 2.7.3 Reflection coefficient at the perforated exit R for different number of open holes in the unflanged perforated exit...... 46 2.7.4 Reflection coefficient at the perforated exit, R, as a function of the number of open holes in the flanged perforated exit...... 46

3.2.1 Polynomial coefﬁcients used in the validation of the numerical integration method, extracted from Barrero-Gil et al. [2009] for different cases k. .... 60

3.2.2 Values used to estimate the specific dissipation power, pd, extracted from Alonso et al. [2012]...... 62 3.4.1 Coefficients of 7th, 9th, 11th order polynomials obtained with Matlab function polyfit for the fitting of the experimental data Cz(α) obtained from Parkin- son and Brooks [1961]...... 79

4.2.1 Deﬁnition of lower, center and upper frequencies in 1/n-th of octave frequency bands. n is the desired fraction of octave (n must be an integer greater than or equal to 1), m is the index of the band (m can be a positive, zero or negative integer), and f0 is a ﬁxed reference frequency (set to 1000 Hz). . . . 86 4.2.2 Energy and modal density of SEA subsystems [Wijker, 2009]...... 92

ix x List of Tables

4.2.3 Models material properties. HPL: High Pressure Laminate; SMC: Sheet Molding Compound; EPDM: Ethylene Propylene Diene Monomer; PVB: Polyvinyl Butyral...... 94 4.2.4 Foam properties...... 94 4.2.5 Methods used to model, depending on the frequency range...... 95 4.2.6 Pressure rout mean square for attached and detached ﬂow...... 102 4.2.7 TBL power spectral density parameter...... 102 4.3.1 Accelerometers used in test campaign...... 104 4.4.1 Measured SPLOA (dB) in open ﬁeld at 300 km/h, with x being the microphone location...... 105 4.4.2 SPLOA (dB) inside the tunnel at 300 km/h, with x being the microphone location...... 105

A.2.1The transmission coefficient τ of conical, exponential and catenoidal horns [Morse, 1948]...... 130 A.3.1Influence in the reflection coefficient R of microphone M2 insertion into the duct...... 131 A.3.2Influence of the leakage through the duct holes...... 131 A.3.3Expression to estimate the pressure attenuation taking into account humidity variation...... 132 Nomenclature

Roman Symbols

A Oscillation amplitude [m]

2 Atr Cross-sectional area of the train [m ]

2 Atun Cross-sectional area of the tunnel [m ] b Characteristic dimension of the body in the direction transverse to the ﬂow [m] b1a Analytic determination of b1

2 Bp Bending stiffness of a plate [N · m ]

a bn n-th polynomial coefficient of the sinus series expansion of Cz b1n Numerical determination of b1 cftr Friction coefficient of the train cftun Friction coefficient of the tunnel fc Coincidence frequency [Hz] cp Specific heat at constant pressure [J/(kg · K)] c Free field sound speed propagation in the air [m/s]

Cz Dimensionless coefﬁcient of the aerodynamic force transverse to the ﬂow

a Cz Antisymmetric part of the dimensionless force coefﬁcient Cz

Czd Dimensionless force coefﬁcient, body pitch angle displaced from origin

Czda Antisymmetric part of the dimensionless force coefﬁcient, body pitch angle displaced from origin, Czd

Dtun Tunnel diameter [m]

Qθ Directivity r Distance [m]

xi xii List of Tables

E Young modulus [Pa] f Frequency [Hz] fcenter Central frequency of the band [Hz] fl Lower frequency of the band [Hz] fu Upper frequency of the band [Hz] fv Vortex shedding frequency [Hz] fz Aerodynamic force in the direction transverse to the ﬂow [N/m] cg Group speed [m/s]

H Glauert-Den Hartog parameter

H12 Transfer function between microphones pressure spectra

H2 Transfer function between accelerometer power spectral density

Z Acoustic impedance [Pa · s/m] k Wave number [rad/m]

L Characteristic dimension of the structure in the ﬂow direction [m]

Lx Characteristic length in x direction [m]

Ly Characteristic length in y direction [m]

M Mach number

Imax Maximum sound intensity [Pa · m/s]

Y Mobility [s/kg] n(ω) Modal density [mode/Hz]

M(ω) Modal overlap [mode/Hz]

2 Ixx Momentum of inertia in x axis [kg · m ] pemaxt Absolute maximum dimensionless extracted power, averaged per cycle cph Phase speed [m/s]

Pd Dissipated power, averaged per cycle [W/m] pd Dimensionless dissipated power, averaged per cycle pd Dimensionless dissipated power, averaged per cycle and per maximum relative amplitude (speciﬁc dissipated power) List of Tables xiii

Pe Extracted power, averaged per cycle [W/m] pe Dimensionless extracted power, averaged per cycle pe Dimensionless extracted power, averaged per cycle and per maximum relative amplitude (speciﬁc extracted power)

Pr Prantdl number prms Pressure root mean square [Pa] q Dynamic pressure [Pa] a Inner radius of the cylindrical duct [m]

Re Reynolds number

R Reﬂection coefﬁcient

Rc Room absorption constant

Sc Scruton number

St Strouhal number

S Surface [m2]

T Oscillation period [s] t Time [s] tr Characteristic residence time [s]

TR Reverberation time [s] tv Time scale of the structure oscillations [s]

Utr Train speed [m/s]

U ∗ Dimensionless velocity of the incident ﬂow

U Velocity of the incident ﬂow [m/s]

V Volume [m3] z0 Transverse displacement to the incident ﬂow [m]

Acronyms

BEM Boundary Element Method

CAA Computational Aeroacoustics

CLF Coupling Loss Factor xiv List of Tables

DAF Diffuse Acoustic Field DES Detached Eddy Simulation DG Discontinuos Galerkin

DLF Damping Loss Factor DNS Direct Numerical Simulation EPNL Effective Perceived Noise Level

EPDM Ethylene Propylene Diene Monomer FEM Finite element method FSP Fluctuating Surface Pressure

FWH Ffowcs-Williams and Hawkings HPL High Pressure Laminate HVAC Heating, Ventilation and Air Conditioning

LBM Lattice-Boltzmann Method

LEE Linearized Euler Equations LES Large Eddy Simulation

NCT Noise Control Treatment

PVB Polyvinyl Butyral

PSD Power Spectral Density

RANS Reynold Averaged Navier Stokes equations

RMR Rock Mass Rating

ROR Rain On the Roof

SEA Statistical Energy Analysis

SMC Sheet Moulding Compound

SIF Semi-Inﬁnit Fluid

SIL Speech Interfeence Level SPL Sound Pressure Level TBL Turbulent Boundary Layer

TL Transmission Loss TNS Train Nearﬁeld Signature TWS Train Wave Signature List of Tables xv

URANS Unsteady Reynold Averaged Navier Stokes equations

WBM Wave-Based Method

Greek Symbols

α Angle of attack [rad]

αa Sound absorption coefﬁcient [dB/m]

αav Average absorption coefﬁcient

φ Blockage ratio between the train and the tunnel area sections

η Dynamic viscosity of the air [kg/(m · s)]

ηs Acoustic radiation loss factor

ηb Border loss factor

ηs Structural loss factor

γ Heat capacity ratio of the air

γp Pitch angle of velocity [rad]

κ Thermal conductivity [W/(m · K)]

ν Poisson modulus

ω Angular frequency [rad/s]

ωn Undamped natural frequency [rad/s]

ω0 Oscillation frequency of the structure [rad/s]

Δpfr Pressure gauge due to friction effects caused by the entry of the main part of the train into the tunnel [Pa]

ϕ Phase between oscillations and the aerodynamic force transverse to the ﬂow [rad]

ΔpHP Pressure gauge caused by the passing of the train head at the measurement position in the tunnel [Pa]

ΔpN Pressure gauge caused by the entry of the nose of the train into the tunnel [Pa]

ΔpT Drop in pressure caused by the entry of the tail of the train in the tunnel [Pa]

ρ Air density [kg/m3]

3 ρs Structure density [kg/m ]

σ Radiation efﬁciency

τ Dimensionless time

τ Transmissibility θpemaxt Pitch angle of the body at absolute maximum extracted power [rad] θ Pitch angle of the body [rad]

σ Porosity

ζ Dimensionless structural damping coefﬁcient

Note: Realize that sometimes the same symbol has been used to refer different variables or parameters. It has been done this way to use the traditional nomenclature used in bibliography. Chapter 1

Introduction

1.1 Noise sources in high speed trains

In the last years the effort invested on the study of high speed trains aerodynamics has increased considerably. Among the main objectives of research, there is the improving of the comfort of train passengers and of the citizens of the tunnel portals surrounding infrastructure. Reducing the minimum tunnel cross-section needed for high speed trains passing through without decreasing their speed has been one of the main topics in railway infrastructure research, because of its impact on construction costs. As train speed has been increasing, the aerodynamic noise has become one of the main train noise sources, and at the same time the center of attention of many researchers. A classical review of the main aerodynamic and aeroacoustic problems of high-speed trains and the strategies followed to alleviate their undesirable effects can be found in Raghunathan et al. [2002]. This paper summarizes the studies performed on the comfort and safety of the passengers and the environment of the train, aerodynamic drag, pressure variations inside train, train-induced ﬂows, cross-wind effects, ground effects, pressure waves inside tunnel, impulse micro-pressure waves at the exit of tunnel (MPW), noise and vibration generated by the train, and its energy consumption. It also presents some expressions to estimate: the trains aerodynamic drag, the pressure gauge of the compression wave generated at the entrance during the train entering in a tunnel, and also the impulsive micro-pressure wave which is radiated outside when the pressure wave reaches the tunnel exit. Another good review of the main sources of noise and vibration on railways, and the concepts needed to understand the physics involved is presented in Thompson [2009]. As it can be appreciated in Figure 1.1.1, special attention should be paid to aerodynamic noise. It should be pointed out that the square of the acoustic pressure depends on the 2 2 4 6 6th power of the Mach number, M=Utr/c,asp¯ ∝ Aρ c M [Curle, 1955], which is dominant above 300 km/h. Aerodynamic noise is caused by the ﬂow of air over the train as it travels at high speed. Near the train nose the boundary layer is laminar, but due to viscosity effects it turns into turbulent. This transition depends mainly on the train surface roughness, negative

1 2 CHAPTER 1. INTRODUCTION

pressure gradients, free-stream disturbances and on the Reynolds number Re = ρUtrL/μ. The Reynolds number of the ﬂow around a train of 3.5 m height traveling at a speed of 7 Utr = 300 km/h is about Re ∼ 2 · 10 . Crespi et al. [1994] measured the boundary layer on a high speed train at different speeds using laser Doppler velocimeter mounted on board. The measurements showed that thickness is relatively constant along the train (some 2 m), and that it increases rapidly towards the lower region of the vehicle due to ground friction and blocking effect of the bogie.

Aerodynamic noise ~U6 Wheel/rail interaction noise ~U4 Engine/generator noise ~U1 Total [dB] w L

1 2 10 10 U [km/h] tr

Figure 1.1.1: Variation of the sound power levels Lw with train speed Utr of different sound sources (after Lauterbach et al. [2012]).

Broad-band noise generated by wheel-rail interaction is clearly dominant at middle speeds. The vibration excited by the contact between the wheel and the rail is highly dependent on the surface irregularities. Squeal noise generated by the wheels during the curves of tight radius is high pitched tonal and can be quite annoying, as well as the impact noise that can appear at discontinuities in the track or the wheel (rail joints, points and crossings). Dominant frequencies of ground-borne vibration can be between 4 and 50 Hz for soft soil, and between 30 and 200 Hz inside a tunnel due to the radiation by vibration of their walls, and can increase the overall level of noise up to 20 dB when crossing through a bridge. The cross wind can increase drag and also moves forward the detachment point of the boundary layer over the train surface, which dramatically increases the aerodynamic noise. A study of the flow around high speed trains (around train nose, a long the side, roof, underbody and the wake) with special attention to cross winds is presented in Baker [2010]. The dominant sources of aerodynamic noise in high speed trains are typically dipole- type. Cavity noise which appear between the train inter-coach gap can be an important source of noise, and the tonal frequencies can be predicted by Rossiter’s equation [Rossiter, 1964]. Other important tonal sources can be the wake generated by the pantograph, roof-mounted equipment, ventilation grilles, etc. The Strouhal number St = fL/U can be of help to identify the characteristic length L of the shedding vortices that 1.1. NOISE SOURCES IN HIGH SPEED TRAINS 3 can appear in the pantograph. If the mean flow velocity U and the frequency are known, assuming St ∼ 0.2 (as is the case for a cylinder above Re > 3 · 105) the characteristic length can be estimated. The tonal frequency for a typical pantograph diameter of 3 cm at 300 km/h is close to 500 Hz. Array microphone measurements in free field or in aeroacoustic wind tunnels have been used successfully to identify the main aeroacoustic sources, but quantification of sources strengths remains still difficult due to contamination from other sources, background noise, etc. The optimization of the methods for analysis and post-processing like beam-forming (least mean square), deconvolution (CLEAN-SC based on source coher- ence), diagonal removal, BiClean algorithm, etc.) are in the center of attention of many researchers. Among the assumptions considered in beam-forming for the characterization of the aerodynamic noise in a train are: the sound sources are located in a plane to the side of the train, all sources are incoherent and they are statistically time independent. Vertical disposition of the microphones are used to find the height of the source meanwhile the horizontal array is used to locate their position along the train. Besides the flow-induced noise caused by turbulence and unsteady flows, another mechanism of sound generation is the vibration of solid bodies (structure-borne sound) which can also generate and radiate sound energy. Meanwhile the fluids can only store energy in compression (longitudinal waves), the solids can store energy in compressional, flexural (transverse or bending), shears and torsional waves. But flexural (bending) waves are the only type of structural wave that plays a direct part in sound radiation and transmission [Norton and Karczub, 2003]. The junctions between the structures and the noise control treatments applied in structures (vehicle floor, walls, windows, doors and roof) play an important roll in the path in which the energy flows between them and the environment. Noise from the air-conditioning system can also require consideration in modern trains as there is often very limited space in which to package the air-conditioning unit and ducts [Thompson, 2009]. Also fixing equipment, such as compressors under the floor structure, or a carriage on the roof, can transmit energy to the train interior by both airborne and structure-borne paths, as shown in the scheme of the main sources in Figure 1.1.2. Mounting stiffness must be carefully chosen in accordance to the characteristic load, this combined with a good mounting practice can reduce significantly noise transmission. Avoiding leaks during the mounting process is critical to reduce direct airborne transmission. High speed trains passing by each other or by stationary structures produce also important effects to be taken into account in the design process of the train. All this aerodynamics effects must be studied to optimize the train shape. For example, if a high speed train with two floor levels is wanted to be built, a compromise must be achieved between the cross-section needed to fit the 2 levels, the aerodynamic resistance (which in- fluences critically in the power consumption), the overturning moment due to crosswinds, the maximum speed to enter in the tunnels railways in which it is going to be operated, the surface section prone to flow detachment, etc. 4 CHAPTER 1. INTRODUCTION

Figure 1.1.2: Main sources of interior noise in a train and their transmission paths into the vehicle (after Thompson [2009]).

In Spain, the infrastructure for high speed trains has experienced a considerable increase over the last 10 years, a big part of the resources have been invested in railway tunnels construction. Inside tunnels, the noise level transmission of rolling noise and aerodynamic noise increase, specially the one generated in the inter-coach gap. A decrease of the tunnel cross-section area can save several MC, and this can be achieved studying the critical situations that can occur in the tunnel during the train passing through. The tunnel portals, at both the entrance and exit, play an important role reducing not just the pressure gradient of the waves generated by the train but also the intensity of the pressure gauge during the reflection process. The reflection of pressure waves at tunnel entrance/exit is analyzed in this work both numerically and experimentally. The reflection of step-wavefronts from perforated exits was studied in Vardy [1978], comparing it with the reflection at a flared portal, concluding that flared is less advantageous than a perforated exit. This paper also highlights that unacceptable pressure gradients will remain unless very long extension of portal hoods are used, and even that will not solve the problem due to pressure transients and non-linear effects. This study leads to the conclusion that is important to attenuate the pressure wave before it reaches the tunnel exit. To do this some used mechanisms are: tunnel side branches [William-Louis and Tournier, 1998], increasing tunnel friction (ballast track is not feasible in long tunnels), using Helmholtz resonator (acoustic liners are commonly used in aircraft engine noise reduction), extracting energy from the pressure waves using appropriated devices, etc. An extensive review on noise reduction methods that could be considered to be applied in railway tunnels can be found in Ingard [2008]. A good review of aerodynamic instabilities like galloping, vortex-induced vibration, 1.1. NOISE SOURCES IN HIGH SPEED TRAINS 5 wake-induced instabilities, etc., can be found in Paidoussis et al. [2011]. Transverse galloping is an aerodynamic instability that can appear in slender structures with non-circular sections, with oscillation amplitudes in cross-wind direction, like in electrical lines after ice deposition, traffic signals, bridge decks, etc. The residence time of a fluid particle is small compared to the period of the oscillating body, so the hypothesis of quasi-steadiness can be assumed, and static pressure measurements can be used to characterize the dynamic behavior. The amplitude of oscillations are limited by non-linear aerodynamic effects, which establish a oscillation limit cycle [Barrero-Gil, 2008], this makes easier the fatigue and damage tolerance estimations of the device. This properties and the fact that the magnitude of the flow velocity induced by the pressure waves generated by a train inside a tunnel is of the order of the transverse galloping critical wind speed, make from transverse galloping a phenomenon easy treatable both analytically and experimentally, which could also be implemented in tunnel railways airshaft, portals, etc. The aerodynamic phenomena during the entering of the train into a tunnel were treated theoretically in Hara [1961]. In this paper the velocities and pressures in each portion of the tunnel are calculated, also an expression for the pressure drag is obtained under the assumption of compressible and inviscid air. The compression wave transient and its reflections in the tunnel portal during the train passing through it are studied numerically in Zeng and Gretler [1995]. A theoretical model for the pressure field in a tunnel generated by high speed train, modeled by a pair of moving monopoles into a semi-infinitely long tunnel, is presented in Sugimoto and Ogawa [1997], the model is validated with CFD by solving directly the Euler equations for inviscid and compressible air. An analytical study of the compression wave generated by a train entering a tunnel and the influence of the train nose geometry in the shape of the wave can be found in Howe [1997a]. In this analytical solution, the train is modeled by a uniformly translating continuous distribution of monopole sources whose density is proportional to the local gradient of the cross-section area of the train. The head wave is expressed as the con- volution product of the sources and an acoustic Green’s function, whose determination is simplified at low Mach numbers when the characteristic thickness of the head wave is large compared to the tunnel diameter. Also a second dipole of comparable strength is attributed to “vortex sound” sources in the shear layers of the black-flow out of the tunnel of the air displaced by the train. The intensity of the wavefront of the compression wave generated by the train can be estimated by using the train’s speed Utr, the blockage ratio, φ - which is the ratio of the cross-section area of the train, Atr and the tunnel, Atun - the nose profile of the train, the geometry of the tunnel entrance and the atmospheric conditions (density, ρ and temperature T of the air). The influence of the Mach number (M = Utr/c) is studied in Howe [1997a], where the models are validated with published experimental measurements presented in Maeda et al. [1993]. The effect of vents in the tunnel entrance portal in the rise time of the wavefront to alleviate the impulsive micro-pressure radiation at the tunnel exit is analyzed in Howe [1998] and Howe [1999]. The maximum compression wave rise time that can be obtained with vents is equal to the time of passage of the front of the train through the perforated section. The Rayleigh’s method for the approximate calculation of potential flow from the open end of a semi-infinite flanged cylinder is applied in Howe [1999b] to obtain analytical representations of the Green’s function describing the generation of sound waves within a flanged cylinder by sources located in the neighborhood of the open end. The expression for the pressure rise across the wave front generated by the train en- 6 CHAPTER 1. INTRODUCTION tering in the tunnel is given approximately in Howe et al. [2000]. The amplitude of this pressure is typically 1-3% of the atmospheric pressure (around 150 dB) when the train Mach number exceeds about 0.2 (at 250 km/h). This paper proofs that a “flared” portal entrance hood without windows behaves as an ideal hood producing a linear increase in pressure with distance across the wavefront, with a constant pressure gradient that is as small as practicable, but also implies a great increase in construction costs and space requirements. The aerodynamic phenomena generated by a train in a tunnel is analyzed by one- dimensional numerical simulation in several tunnel configurations in Baron et al. [2001]. A tunnel design criteria for long-range underground high-speed railways is established and also the positive and negative effects of pressure relief ducts and partial air vacuum are discussed. A reduce-scale (1/77) experimental set-up to study the compression wave generation is used in Bellenoue et al. [2002], and validated with 3-D simulation and full-scale test measurements. The generated compression wave is clearly established as a planar wave once it has propagated some four times the tunnel’s diameter inside the tunnel. So, the 3-D model can be replaced by a 2-D (axially symmetric) model, saving computational time. Meanwhile the optimal longitudinal cross-section area distribution of the train nose to increase the rise time of the compression wave in the tunnel can be estimated using axially symmetrical models, drag and aerodynamic stability optimization in open air must be performed by using 3-D models. The resistance coefficients of a train from full scale measurements inside tunnel during routine operation are calculated in Vardy and Reinke [1999]. We use this values in our implemented code to predict the pressure gauge due to friction. The vented hood as a method to reduce micro-pressure wave by increasing the time rise of the pressure was studied analytically in Howe [1998]. A theoretical analysis of the influence of the unvented hood geometry and its optimal size on the wavefront can be found in Howe et al. [2002]. The optimal distribution and size of the windows in a vented hood is studied in Howe et al. [2003]. The infrasound generated when a antisymmetric model of a train enters along the axis of a a duct is studied in Howe [2003c]. This low- frequency pressure pulses generated by the train entering and leaving the tunnel, also known as tunnel continues waves (TCW), are also studied in Kikuchi et al. [2009]. Here an interesting method to separate the incident and reflected waves from pressure data measured inside tunnel is presented. The objective is to study the amount of energy of the incident wave which is radiated from the portal as micro-pressure waves. The influence of the train nose profiles in the generation of the wavefront is studied in Sato and Sassa [2005] numerically, together with an experimental set-up used to validate the results. The thickness of the wavefront can be made many times larger than its typical thickness order of 5 tunnel diameters Dtun, when the wave is reflected at a tunnel exit with a discrete distribution of rectangular widows as shown in Howe and Cox [2005]. The expansion wavefront reflected at the exit, and the expansion wave caused by the inward propagating pulses generated at the windows are shown for two different windows area distributions, one with equal area and another with linearly increasing area. A fast method to evaluate separately the different contributions to the compression wave generated by a high speed train entering a tunnel with vented hood is proposed in Howe et al. [2006]: the interactions of the train nose with the hood portal, the junction 1.1. NOISE SOURCES IN HIGH SPEED TRAINS 7 between the hood and the tunnel, the effect of the windows, and the contribution of the separated flow along the sides of the train. A genetically optimized tunnel-entrance hood (playing with the dimensions and distribution and sizing of the hood windows) in Howe [2007] with the aim to reduce the maximum pressure gradient is presented. The aerodynamic noise generated in the boundary layer over a high speed train has been widely studied with simulations using different numerical methods such as Susuki [2001], Wang et al. [2008], Muld [2012], etc. The methods used in aeroacoustics have experienced a very fast development due to the need of reducing the computational time of the simulations. Some of the numerical methods used on this field are the direct numerical simulation of the Navier Stokes equations (DNS), large eddy simulation (LES), Reynolds Averaged Navier Stokes equations (RANS) and unsteady RANS (URANS). Computational aeroacoustics (CAA) is developing new techniques to simulate the aerodynamic noise combining different methods. URANS could satisfy the need for grids in engineering problems, but it can lead to relatively large numerical error due to its failure in capturing the sources in sub-grid scales. In Sun et al. [2012] a nonlinear acoustics solver is used to reconstruct the unresolved sub-grid scales from a previous RANS simulation that provides an initial statistically-steady model of turbulent flow. Detached eddy simulation (DES) combines RANS formulation to simulate the adhered portion of the boundary layer with LES formulation for detached flow zone. The aeroacoustic noise generated in the bogie cavity is studied in Takaishi et al. [2002], where the three-dimensional unsteady flow around the train is solved by the LES technique and the distribution of dipole sound sources is predicted numerically by coupling the instantaneous flow properties with the compact Green’s functions. In Paradot et al. [2008] a LES simulation of aeroacoustic sources and their near-field propagation based on the Lattice-Boltzmann method (LBM) is computed. The far-field propagation and structural coupling is calculated using Ffowcs-Williams and Hawkings method (FWH), which used the aeroacoustic analogy [Lighthill, 1952], assuming that sound propagation follows the simple wave equation, which can be solved through the integration on the surface surrounding the nonlinear acoustic sources. This methodology that separates generation from propagation of sound is also used in Chen et al. [2011] to calculate the wind excitation in a car. Linearized Euler Equations (LEE) is also quite used when viscous effects can be considered 2nd order source of sound. A literature survey of the techniques used concerning the internal noise prediction is presented in Lalor and Priebsch [2007]. The common methods used to analyze sound transmission are classified in: finite element method, boundary element method, statistical energy analysis, hybrid methods, ray tracing methods and band-averaged transfer function method. This paper also introduces the main aspects to take into account in noise prediction modeling, like the mechanisms of internal noise generation: the direct transmission (“leakage”), the “mass law” and the acoustic radiation by vibration walls; also highlights the high influence of the damping in the spot-welded and bolted joints in the variability of internal noise in deterministic methods. Among the different techniques mentioned in this paper, FEM, BEM, the Wave-Based Method (WBM) and SEA are presented as the more widely used in interior noise prediction. Lightweight, large area structures, like structures used in high speed trains are very sensitive to acoustic loads. A combined FEM and BEM analysis is commonly used to simulate the fluid structure interaction at low frequencies, specially with exterior fluids. High order schemes used in finite difference methods are extremely sensitive to the generation of spurious high-frequency waves. Discontinuous Galerkin’s method, based on 8 CHAPTER 1. INTRODUCTION high-order polynomials basis functions is also used in CAA. In DG, fluxes between internal elements are calculated, solving an approximated Riemann problem [Lorenzoni, 2008]. SEA is suitable for problems that combine many different sources of excitation, whether mechanical or acoustic. In SEA the structural vibrational behavior of elements (subsystems) is described statistically. For high-frequencies a deterministic modal description of the dynamic behavior of structures is not very useful [Wijker, 2008]. The concept of SEA is simple: the structure or fluid is split up into subsystems characterized by its internal energy, its Damping Loss Factor (DLF) and its Coupling Loss Factor (CLF) which determine the efficiency by which energy is transmitted into vibrational power to its neighbours. If power from an external source is applied to a subsystem, part of the energy will be dissipated within the subsystem due to its damping and the rest will be transferred to the neighboring subsystems, a detail description can be found in Lyon and DeJong [1995]. 1.2. OBJECTIVES AND CONTENT STRUCTURE 9 1.2 Objectives and content structure

The objective of this work is to pay attention to and analyze several of the phenomena that takes place concerning the aerodynamic noise in high speed trains (pressure waves generated in the passing of a train through a tunnel, micro-pressure waves radiated from railway tunnel portals, noise generated aerodynamically around the train, etc) and to study some ideas to alleviate those problems as: mechanisms to reduce noise inside the train, the tunnel and the surroundings; methodologies to analyze aeroelastic instabilities; and on track measurements of aerodynamic noise. Among the acoustic problems analyzed in this work there is the generation, propagation of pressure waves inside tunnel and the micro-pressure radiated outside the tunnel. Several studies on the influence of the main parameters (tunnel portals geometry, track configuration, existence of airshafts, train shape, non-linear propagation) are performed and validated with measurements performed on track, data extracted from bibliography or with scale tests. These data are used in a semiempirical method to predict efficiently in the first steps of design the sound pressure level inside a high speed train using different approaches to characterize, analyze and reduce the airborne and structure borne noise. Vibro-acoustic models are used to analyze the influence of the modelization parameters. This dissertation is divided in five chapters, in the first chapter there is an introduction to the main sources of noise in high speed trains and their influence in passenger comfort and tunnel surroundings, together with a summary of the state of the art of the engineering techniques applied to evaluate and design countermeasures and some of the solutions adopted. In chapter 2 the problems associated to the pressure wave generated by the train entering a tunnel are considered. Among them are: the pressure wave propagation, the micro-pressure wave radiated outside the tunnel portals, the experimental setups built to study the pressure waves propagation, the influence of the hydrostatic pressure in the fulfillment of comfort and health standards, and an alternative indicator to evaluate passenger discomfort to be used in the standard is proposed. In chapter 3 a method to study the extraction of energy from the pressure waves that propagate inside the tunnel based on transverse galloping is proposed together with a method to optimize the energy extraction. In chapter 4 a methodology to predict the noise inside the train is described. Finally, some conclusions and the future work are summarized in chapter 5.

Chapter 2

Pressure waves in high speed railway tunnels

2.1 Introduction

A well-known aerodynamic phenomenon in high speed trains is the propagation of the pressure waves generated when a train passes through a tunnel, which can give rise to several problems. The reflections of these waves at the discontinuities of the tunnel can meet the train several times inside the tunnel and can cause discomfort to the train passengers and also to the pedestrians near the tunnel entrance/exit due to the micro-pressure radiated outside from the tunnel portals. When a train approaches a tunnel entrance portal, the air inside the tunnel begins to be compressed due to the open-air local field preceding the train nose. Once the train head reaches the tunnel entrance, the pressure increases faster until the nose is completely inside the tunnel. This interaction between the tunnel entrance hood and the train nose determines the shape of the compression wave generated, but the total pressure gauge DpN remains constant. The pressure continues to increase slowly due to the viscous effects on the tunnel wall and the train surface Dpfr, until the end of the train reaches the tunnel entrance portal, and a rarefaction wave is then generated with a pressure decrease DpT . The pressure decreases suddenly DpHP when the train nose passes the measurement point. These pressure variations are known as the train reference signature, and is defined in the standard EN1 [2006] (see Figure 2.1.1). When this pressure wave reaches the tunnel exit, part of the wave energy is radiated outside as a micro-pressure wave and part is reflected back inside the tunnel. At very low frequencies the exit radiates as a monopole source, and almost all the energy is reflected back. Discontinuities at the tunnel exit, like porosity or variations of the cross-section area, generate reflections which can increase the frequency of the signal and at the same time the capacity to radiate outside the tunnel. The micro-pressure radiated outside is proportional to the pressure gradient of the incident wave. In the literature, the effort to reduce the micro-pressure have been focused mainly on the optimization of the tunnel entrance and the nose shape of the train, reducing the pressure gradient. The efficiency of this option gets substantially reduced in long tunnels (more than 5 km) with small friction losses between the air and the interior walls, like in slab track tunnels, where

11 12 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Figure 2.1.1: Train reference signature defined by the standard EN14067-5:2006:E [2006]. the non-linear effects generate a steepening of the wave which progressively reduces the compression wave thickness, significantly increasing the pressure gradient and also the disturbance radiated from the tunnel portal. It becomes similar in strength to a sonic boom. A ballasted track is an effective porous sound absorbing material in the low- frequency range, but the high maintenance costs in long tunnels make this option not feasible. One interesting goal would be to find a device with an attenuation capability as much as the ballast but with reduced maintenance costs. The design of railway tunnels for high speed train operation requires accurate models to study the wave propagation process and its interaction with the vehicle which implies long calculation times. However, this process can be somehow alleviated by using simplified models. The main advantage of simplified models is that the computational cost is quite low compared with more complex models. The most common simplified approach to deal with the interaction of pressure waves and vehicle is the use of models based on of the one-dimensional equations of gas dynamics. A method based on the propagation of the signature of the pressure waves inside the tunnel, which requires low CPU cost, is presented in William-Louis and Tournier [2005]. This method of characteristics, 1D finite difference, is used as a numerical scheme, only predicts pressure changes and is only valid for simple tunnels (without abrupt cross- section changes, air shafts, etc.), but allows to investigate trains crossings using different time delays in the entry time of the trains. In the zones where the flow is strongly three-dimensional (around tunnel portals, train ends) other methods have to be used. Concerning the interaction of the train head with the tunnel portal, in Iida et al. [2006] an analytical method based on the aeroacoustic theory developed by Howe is applied to a short “acoustically compact” hood having a window on its side wall. The boundary element method is used to calculate the potential flow through a portal of arbitrary geometry. A parametric analysis and an experimental study is performed in order to show the great influence in its performance of the relative length of the train nose to the distance between the window and the hood entrance. Another parametric numerical study of the micro-pressure wave is performed in Kaoua et al. [2006]. The results are validated with experimental data. Among the parameters considered are the train speed, train nose shape, tunnel length, track form, transitions in cross-section area at the portals, shafts distributed along the tunnel, and cross passages to an adjacent tunnel. The flow generated by a train in a tunnel with side branches is studied in William- Louis and Tournier [1998]. The results confirm that the use of shafts provides a very effective method to reduce pressure fluctuations in tunnels. The shafts divide the waves into secondary waves which can be more effectively damped by viscous dissipation or 2.1. INTRODUCTION 13 acoustic diffusion. A method to predict the sound field in long enclosures with side branches is proposed in Liu and Lu [2009]. The model proposed is validated with experiments, and with an acoustic modeling program ODEON. Field measurements and numerical simulation performed on the distortion of the compression wave generated by a train entry, and its propagation through a slab track with inclined side branches (with orifice-shaped aper- tures at the junction) tunnel is done in Fukuda et al. [2006]. In Ricco et al. [2007] is also studied the pressure wave generated in the tunnel during the passing by of a high speed train numerically an experimentally with a 1:87-scaled setup consisted of a launching mechanism, a 6-meter-long tunnel and a damping system to block the vehicle after the tunnel exit. In their one-dimensional code they take into account the influence of the local separation region, which occurs near the train head for high-angled nose. Numerical simulations of the flow field around high speed trains passing by each other inside a tunnel are done in Fujii and Ogawa [1995], Hwang and Lee [2000] and Hwang et al. [2001]. The pressure waves are reflected at tunnel discontinuities such as changes in cross- sectional area, airshafts, side branches and also every time that find the train. This multiple reflections can create localized regions where constructive interferences produces higher pressure that can also affect the passengers and personnel in tunnel surroundings. A study of the sonic boom generated at the tunnel exit when the compression wave reaches the exit and a review of some passive and active alleviation methods is presented in Vardy [2008]. The compression wave and the micro-pressure wave generated by different hoods (linear horn, discontinuous, equal and enlarged cross-section with and without windows and gradient hood), using 3-D numerical simulations and moving-model experiments are presented in Liu et al. [2010]. The optimal shape of the unvented entrance hood to reduce the micro-pressure wave is studied in Murray and Howe [2010]. And another way to reduce micro-pressure waves using side branches of different lengths with half opened end is presented in Sato [2010]. With this method the reflected waves at the half opened end are added in opposition, this can decrease micro-pressure wave in 60%. Two new methods for the prediction of the micro-pressure wave are proposed in Yoon and Lee [2001]. The first one combines acoustic monopole analysis (generation of compression wave) and the method of characteristics (wave propagation inside tunnel) with the Kirchhoff formulation (computation of a micro-pressure wave). The second one replaces the first two stages by 2D-3D Euler/Navier-Stokes solvers. In this paper it is concluded that the combination of acoustic monopole analysis/method of characteristics- Kirchhoff formulation is a very useful tool for preliminary design. Most of the experimental set-ups launch an asymmetric train model through horizontal circular cylindrical pipe using a driving wheel, an air compressed chamber, or an elastic bundle. In Kim et al. [2003] the distortion of a compression wave reflected from a baffled open end of a shock tube is studied both numerically and experimentally, where the incident expansion wave is generated by a sudden rupture of a diaphragm. The analysis of the experimental measurements of the reflection of the pressure waves at the tunnel portals is performed in the time domain using wave separation methods with multiple microphones in Kemp et al. [2010]. Another way to modify the pressure wave reflected back inside the tunnel is acting on 14 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS the absorption or in the radiation efficiency of the tunnel exit. The radiation from a circular unflanged tube without thickness, for a symmetric excitation is discussed in Levine and Schwinger [1948]. In this paper an analytical solution, valid for the frequency range in which acoustic waves inside the tube can be considered planes, is presented. The acoustic radiation at the end of an infinite circular flanged tube is analyzed theoretically in Norris and Sheng [1989] and approximated formulas are proposed for the reflection coefficient for a flanged and unflanged tube. The influence of the shape of the tube end on the pressure waves is analyzed experimental and numerically in Dalmont et al. [2001]. Different exits shapes are analyzed numerically by using the boundary element method (BEM) in Selamet et al. [2001]. The results are in good agreement with analytical expressions and experimental data. The effect of the radius of curvature of the extreme non-linear losses is discussed in Atig et al. [2004] with the two-microphones-three-calibrations method (TMTC). The acoustic pulse reflectometry (APR) and TMTC methods, outlining the advantages and disadvantages of each one, are compared in Lefebvre et al. [2007]. The acoustic pulse reflectometry does not require calibration since only a microphone is needed, although requires very long lines which makes it less portable installation than the installation TMTC, meanwhile with the TMTC method is easier to measure the impedance at high frequencies, but requires multiple microphones and precise calibrations that increase the time of tuning the facility. Other approximated formulas for the reflection coefficient of the flanged and unflanged tube are proposed in Silva et al. [2009]. The porous side walls have been previously studied in Nishimura and Ikeda [2008]. In this work, their efficiency to reduce low frequency plane pressure waves radiated and reflected at the end of the duct are shown. In case of weak shock waves generation in the tunnel, pseudo-perforated walls sections sufficiently long are presented as efficient attenuator in Sasoh et al. [1998]. This chapter is focused on the study of the generation, propagation and radiation of pressure waves in railway tunnels. The reduction of the energy reflected back inside the tunnel, the absorption along the tunnel, and the energy radiated outside the tunnel are also analyzed looking for the optimum size and holes distribution in the tunnel exit hood. Also the optimum location of damper devices based on aeroelastic instabilities are studied taking into account the train speed, and the train and tunnel lengths. If the energy reflected at the tunnel exits is reduced, the trains could pass through the tunnel without reducing the speed, without decreasing the comfort and security inside the train, and even the cross-area section of the tunnel could be smaller for the same train speed. The reflection coefficient of several scale-model exits using pulse reflectometry and TMTC technique following the norm UNE-EN ISO 10534-2 [1998] is studied. The results obtained are compared with Nishimura work [Nishimura and Ikeda, 2008]. Among the conclusion of the optimum porosity for the tunnels exits can be highlighted that holes bigger than the ones Nishimura proposed still have a considerable effect on the reduction of the reflection coefficient. This makes more feasible the construction of optimum porosity in the concrete hoods to reduce reflection coefficient. The optimized porosity distribution could be added to the existing tunnel entrance hoods without increasing the pressure gradient during the generation of the wave. In this chapter a simplified expression for the variation of the pressure along the tunnel and in the train is proposed. 2.2. MATHEMATICAL MODEL FOR PREDICTION OF PRESSURE WAVES INSIDE A TUNNEL 15 2.2 Mathematical model for prediction of pressure waves inside a tunnel

In this section a method based on the classical method of the characteristics is used to predict the pressure waves inside the tunnel and also inside the train. The unsteady aerodynamic effects in railway tunnels, like pressure waves generated during the train passing through the tunnel can be analyzed by Euler equations, which are particularly useful in preliminary design work, where information about pressure alone is desired with low CPU cost [Tannehill et al., 1984]. To gain a good understanding of the concept of plane wave it is ﬁrst deduced the general solution of pressure waves propagation inside a duct. The assumptions to obtain the one-dimensional wave equation are [Kinsler et al., 2000]:

• Very small density fluctuations (linear acoustics). • Inviscid fluid. • Adiabatic (which is a quite reasonable except in the tunnel portal entrance/exit and near the walls). • Fluid is in local thermodynamic equilibrium (Stokes’ hypothesis). This assumption may partially fail at high frequencies resulting in a dissipation related to the volume changes [Rienstra and Hirschberg, 2011]. • Homogeneous fluid. • Stagnant and uniform fluid (quiescent fluid). • Air as an ideal gas. • Irrotational (which is a quite reasonable except in the tunnel portal entrance/exit and in the train wake).

With all the above considerations and in absence of sources, the continuity equation is

∂ρ + ρ0∇·u =0. (2.2.1) ∂t

where ρ0 is the mean density in a stationary inviscid fluid and ρ , u are the density and speed perturbations. The Euler’s equation (momentum conservation law for a frictionless fluid in absence of external forces) is ∂u ρ0 + ∇p =0. (2.2.2) ∂t 2 where p the pressure perturbation. Using the constitutive equation (p = c0ρ ,asflowis isentropic) we obtain the wave equation for p

2 1 ∂ p −∇2 2 2 p =0. (2.2.3) c0 ∂t The general equation that approximates the time-harmonic acoustic waves propagation inside a hard-wall hollow cylindrical duct is 16 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

1 ∂p ∂2p 1 ∂2p ∂2p 1 ∂2p + + + − =0, (2.2.4) r ∂r ∂r2 r2 ∂ϕ2 ∂x2 c2 ∂t2 where the instantaneous pressure inside the tube depends on the distance to the sound source x, the distance to the longitudinal axis of the duct r, the angle of the point position with the horizontal axis f, the speed of sound in the medium c, and the time t. The solution of the homogeneous equation is (by separation of variables with boundary conditions ∂p | impermeability condition at the boundary =0and p x=L =0) ∂r r=a m=+∞ n=+∞ q r p (x, r, ϕ, t)= c J mn ei(kmnx+mϕ−ωt) , (2.2.5) ω mn m r m=−∞ n=0 i 2 2 where ω is the angular frequency, kmn = ± (ω/c) − (qmn/ri) the wave number, ri cylinder radius, Jm are Bessel functions of ﬁrst kind, and qmn the eigenvalues (zeros of Jm). The coefﬁcients cmn are obtained from the boundary condition (p (0,r,ϕ,t0)= −iωt Fw(r, ϕ)e ), applying the orthogonality properties of Bessel functions ˆ ˆ 1 q2 2π a q r c = mn F (r, ϕ)J mn e−imϕrdrdϕ. mn 2 q r ω m πa 2 − 2 2 mn 0 0 ri (qmn m ) Jm ri (2.2.6)

2.2.1 Condition for the existence of a plane wave The propagation mode (m, n) of an acoustic disturbance that propagates axially along the tube depends on the wave number. The first propagation mode of the disturbance is a plane wave k00 = ω/c, in which there are not traversal fluctuations (qmn =0). The condition of propagation of an acoustic mode is that the wave number kmn must be real, otherwise the wave will decay exponentially, this is known as an evanescent wave. The lowest zero of the Bessel function J1 is q11 =1.8412, therefore the cut-off frequency is fc =1.8412c/(2πa). In a railway tunnel (a ∼ 5 m) for high speed trains this frequency is close to 20 Hz. The wave generated at the tunnel entrance by the train can be considered as a compact source, which means that the tunnel cross-section characteristic length (L =2a) of variation of the sources is small compared with the wave length λ, and the flow can locally be approximated as an incompressible potential flow. This can be done when the head of the wave is larger compared with the tunnel diameter. At the same time, the region of emission, in this case the portal, can also be considered compact. If a source region is compact compared to the sound wavelength can be represented by the sum of basic sources (monopole, dipole, quadrupole). For example the train can be described as a compact source, as moving excitation (F (x, t)=δ(x − Utr,t) where δ is a Dirac function), and a Green function could be used to describe the pressure field. The pressure waves generated by a train entering inside a tunnel can be treated as plane waves after several diameters distance away from the tunnel portals (where the flow is strongly three-dimensional) [Rienstra and Hirschberg, 2011]. 2.2. MATHEMATICAL MODEL FOR PREDICTION OF PRESSURE WAVES INSIDE A TUNNEL 17

2.2.2 Propagation of the signature The train wave signature (TWS) method proposed in William-Louis and Tournier [2005] has been used to predict the pressure wave during the train pass inside the tunnel. It is based on one dimensional Green’s function and the results are validated with experimental measurements. The problem is similar to a piston moving in a finite pipe, but with both sides open because the blockage is much smaller than unity, and the waves that find the train experience a small reflection and practically all the energy is transmitted (pressure waves see the train as a small cross-section area reduction). The hypothesis assumed in this model are summarized as follows, in these groups, concerning the train, the tunnel and the fluid. For the Train:

• Constant speed.

• Slender enough for the ﬂow not to be detached behind the nose.

• Leakage through the train walls is modeled through an sealing constant (values vary from 0 (train window open) up to 20 seconds (in highly tight coaches of modern high speed trains)).

For the tunnel:

• Axilsymmetric, circular cylinder duct.

• Hard walls (no transmission of sound through these walls).

• Constant perimeter and cross sections area.

• The altitude can vary with the distance to the entrance.

• Blockage is small (φ = Atr/Atun 1).

For the Fluid:

• Pressure waves are assumed to be plane waves. We are considering a pressure pulse whose wavefronts (surfaces of constant phase) are parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector.

• Air behaves as an ideal gas, with isentropic, quiescent ﬂuid (uo =0). This hypothesis is modiﬁed in non-linear propagation.

The acoustic field induced by the nearfield of the train (TNS) is considered constant and does not generate any reflections with the wave that finds during the pass through the tunnel but also with the tunnel cross-section area changes (or airshafts). Friction effects are located in a thin layer near the wall of the tunnel. The principle of the TWS method is to propagate the pressure profiles (TNS, and TWS and it’s reflections at the tunnel portals). At each point of the tunnel the sum of the effects due to the passage of the pressure profiles is calculated to obtain the resulting pressure at this location. The amplitude of the reflections of the TWS at the tunnel portals are determined from experimental results. 18 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

The one-dimensional d’Alembert’s solution for wave equation in x axis direction is

p = f (x − c0t)+g (x + c0t) , (2.2.7)

1 u = (f (x − c0t) − g (x + c0t)) , (2.2.8) ρ0c0 where f and g are determined by boundary and initial conditions. They can be interpreted as longitudinal waves (fluid velocity is parallel to the direction of propagation) propagating in positive x direction f; and g in opposite direction. The particle velocity induced by a acoustic plane wave in a stagnant medium is then u = p /ρ0c0, where ρ0c0 is the characteristic impedance of the fluid. The expressions used to describe the wave signatures have been extracted from Howe [1998] and EN1. The expression used to estimate ΔpN is ρU 2 Δp = tr φ (1 + φ) , (2.2.9) N 1 − M2 where Utr is the speed of the train. The total aerodynamic drag of a train in a tunnel is usually expressed [Vardy and Reinke, 1999] in terms of a nose loss coefficient, kN , which 1 implies a stagnation pressure loss as Δp | = k ρU 2 , and a tail loss coefficient, s nose 2 N nose 1 k ,which implies a stagnation pressure loss as Δp | = k ρU 2 . Note that the T s tail 2 N tail velocities in the annulus between the train and the tunnel wall will not, in general, be the same at the nose and tail for a long train. An estimation of these coefficients when kN ∼0.1 or less, for a streamlined nose, is given in Vardy and Reinke [1999]. The expression used for the pressure gauge generated during the entry of the train body is EN1 2L Δpfr = γP0MX4 1+ cftun (1 − MX4) − ΔpN , (2.2.10) Dh,tun with 4Ltr 1 3 X1 = cftun − 1 − (1 − φ) cftr φ + cftu − Dh,tun M 1 (2.2.11) ξn 2 − 1 , (1 − φ) 1 4Ltr 1 1 X2 = + cftun − 1 − 2 , (2.2.12) M Dh,tun M (1 − φ) 2 4Ltr 1 1 X3 = + cftun − 1 − , (2.2.13) M Dh,tun M (1 − φ) 2 −X2 + X2 − X1X3 X4 =1+ , (2.2.14) X1

ξh and ξn =1+ 2 . The coefficient ξn is a stagnation pressure loss coefficient at 1 − (1 − φ) the train nose (common values between 0.03 and 0.05), and ξh is the stagnation pressure loss coefficient at the train in the tunnel. Dh,tun is the hydraulic inner diameter of the tunnel, Ptun the tunnel perimeter, Atun the area section of the tunnel, and the friction 2.2. MATHEMATICAL MODEL FOR PREDICTION OF PRESSURE WAVES INSIDE A TUNNEL 19

coefﬁcients of the train cftr (common values for high speed train are between 0.003- 0.005) and of the tunnel cftun (values between 0.005-0.0065). The expression used for the pressure drop due to the entrance of the train rear, according to EN14067-5:2006:E, is ξn ΔpT = −ΔpN 1 − 2 . (2.2.15) 1 − (1 − φ)

Since the TWS is generated during the train entrance, the characteristic time is the same that the TNS, but as the TWS propagates at the speed of sound the relation between the characteristic lengths, LTWS length of the TWS, and LTNS of the train nearﬁeld (coincides with the train length Ltr)is L + L L tr e , (2.2.16) TWS M where Le is the length of the entrance. The entrance can modify the rise time of the pressure, this is a common method used to reduce the pressure gradient, and with it the micro-pressure radiated at the tunnel portals. The attenuation is taken into account by using the relation [Ozawa et al., 1993]

−αac0t Δp =Δp0e , (2.2.17)

−4 where the coefﬁcient αa is estimated experimentally. It is approximately 1.8 · 10 for a tunnel with ballasted track, and around 3 · 10−5 for slab track tunnels. As the pressure at the tunnel exit must remain constant, the boundary condition is

F1(ξ)+G1(η)=0, (2.2.18) where ξ = Ltun − c0t and η = Ltun + c0t. Thus the reﬂected wave is

G1(η)=−F1(ξ)=−F1(2Ltun − η)=−f(2Ltun − (x + ct)) . (2.2.19)

And taking into account the reﬂection coefﬁcient at the exit Rexit from the experimental measurements, we can use the expression

G1(η)=−Rexit · f(2Ltun − (x − ct)) . (2.2.20)

For the reﬂection at the entrance (x =0, ξ = −ct and η = −ξ) we obtain

− − − · − F2 (ξ)= G1(η)= G1( ξ)=Rentrance f(2Ltun +(x ct)) . (2.2.21)

A recurrent expression for the wave generated by the train entrance and the reﬂection at the exits would be (assuming that the exit and the entrance portals are equal, Rexit = −αac0t Rentrance = R, and considering exponential attenuation e ) −αact in−1 F (x, t)= sgf [(kLLtun + sgx − ct) /Lsig]e R , (2.2.22)

in+1 where sg =sign((−1) ), in the wave counter and the wavenumber kL =2ﬂoor(in/2) (as the function ﬂoor of Matlab calculates the nearest integer, the serie of kL is 0, 2, 2, 4, 20 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

4, 6, 6). For a wave generated at the exit, as for example the wave generated when the train leaves the tunnel, the general expression will be c −αac0(t−t1) in−1 G (x, t)= −sg1g kL1Ltun + Ltun + sg1x − ct /Lsig e R , Utr (2.2.23) in where sg1 =sign((−1) ), the wavenumber kL1 = 2ﬂoor((in − 1)/2) + 1 (the serie of kL1 is then 1, 1, 3, 3, 5, 5, 7, 7), and t1 the instant in which the train reaches the exit (t1 = Ltun/Utr). The TWS results from William-Louis and Tournier [2005] with the TWS implementation with the selected expressions for ΔpN , Δpfr, ΔpT are presented in Figure 2.2.1. The pressure inside the train is calculated with the dynamic sealing constant τdyn (which represent the time for the train inside pressure pi, to became a fraction of the pressure outside pe), dpi 1 = (pe − pi) . (2.2.24) dt τdyn

Pressure Gauge inside the tunnel at x=550m

3 Experiments

2 TWS William−Louis et al. 2005 TWS 1

−1 P [kPa] Δ −2

−3

−4 0 5 10 15 20 25 30 Time [s]

Figure 2.2.1: Pressure variation in the tunnel as a function of time, validation of the TWS implementation, M=0.24 and φ =0.135. Pressure evolution in the tunnel at 550 m from the tunnel entrance. Experimental results from William-Louis [1999].

The pressure in the train exterior at 72 m from the nose is shown in Figure 2.2.2 . There are some situations in which the pressure gauge inside the train or in the tunnel reach critical values or variations. Some of this cases are also mentioned in the standard EN1. The characteristic times of the situation of interest, denoted as 1, 2 and 3 in the case of Figure 2.2.3, are:

• tct, when the end of the TWS reaches the train head (1)

Ltr 1 tct = . (2.2.25) Utr 1 − M tci, when the TWS reﬂected at the tunnel exit portal ﬁnds the train nose (2)

2Ltun tci = . (2.2.26) c0 (1+M) 2.2. MATHEMATICAL MODEL FOR PREDICTION OF PRESSURE WAVES INSIDE A TUNNEL 21

Pressure gauge at the train 2 1.5 TWS William−Louis et al. 2005 TWS 1 0.5 0 −0.5 −1

P [kPa] −1.5 Δ −2 −2.5 −3 −3.5 0 2 4 6 8 10 12 14 16 18 20 Time [s]

Figure 2.2.2: Pressure variation at the train as a function of time, validation of the TWS implementation M=0.24 and φ =0.135. Pressure evolution in the train at 72 m from the nose.

Figure 2.2.3: Position x of the pressure waves generated by a high speed train inside a tunnel of length Ltun as a function of time t. C, compression: E, expansion wave.

The TWS generated during the train exit (a compression wave) finds another compression wave (3). This is the situation in which the pressure inside the tunnel can reaches its maximum, this situation is not as important for the train that just left the tunnel but for another train that can enter in the tunnel just after the first one. A study of the time delays between the trains entering the tunnel must be done to design the minimum tunnel cross-section area. The critical situations depend mainly on the time delay between train entrances, train lengths, train speeds, tunnel lengths, reflection coefficients of the pressure waves at the portals, attenuation along the tunnel, existence of side branches, airshaft, etc. The implemented code can be used to quickly and approximately predict the induced particle velocity inside the tunnel and find the optimal locations for the damper devices based on aerodynamics instabilities, as a function of the train-tunnel operation parameters (train and tunnel lengths, train speed, sound speed, reflection coefficients at the portals, etc.). 22 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS 2.3 Hydrostatic pressure influence on high speed railway tunnels cross-section sizing

In this section an analysis of the increase of long tunnel costs due to the existence of a tunnel slope is presented. As the intensity of the waves is limited by regulations, and also by the effects on passengers and infrastructures, the sizing of the tunnel section area is largely influenced by the maximum train speed allowed in the tunnel. In Europe, the maximum gauge pressure permitted in the train is defined by the manda- tory rules contained in the “Technical Specifications for Interoperability relating to the infrastructure subsystem of the trains - European high-speed rail system (2008)”, the so- called TSI. According to these rules, the maximum pressure variation along the train shall not exceed 10 kPa during the time taken for a train to pass through the tunnel. Therefore, the change in hydrostatic pressure due to the atmospheric pressure variations associated with the sea level altitude differences between both ends of the tunnel must be included in the computation of the maximum pressure variation. This contribution is relevant in long tunnels with large differences in altitudes. Note that a 100 m altitude difference means roughly a 1 kPa gauge in hydrostatic pressure. If this contribution is considerable, the maximum speed of the train has to be limited, to reduce the intensity of the pressure waves produced by the train, and to provide margin to accommodate the hydrostatic pressure as established by TSI. When designing a tunnel the health criteria defined in the TSI Regulation must be fulfilled. Hence the hydrostatic pressure variations has to be considered. The critical cross-sectional area of the tunnel for a train entering in at 300 km/h in a tunnel with an ascendant slope of 2.5% and length over 20 km is determined by the health criterion which implies that the maximum pressure variation along the train shall not exceed 10 kPa.A simple analysis can be performed in the present case, based on the particular circumstances. Thus, we are interested in tunnels that show a large difference in altitude at the tunnel ends. This is only possible in the case of very long tunnels, due to the limitation in tunnel slopes. In the case of long tunnels where friction is not small, the reflected TWS has decreased its intensity when interacts with the train. In a first approach we can neglect the pressure difference due to the friction compared to the one from the train nose entrance in the tunnel, which in the limit of small Mach numbers M, and blockage ratios, φ, by the asymptotic expansion of (2.2.9), is given by 2 ΔpN ρφUtr . (2.3.1)

The maximum gauge pressure, associated to DpN , DpNmax is

2 ΔpNmax = kwρφUtr , (2.3.2) where kw is a parameter which depends on the operations in the tunnel, single train passing, trains crossing, time delay, etc. To obtain the total pressure increment, the pressure variation along the tunnel due to the hydrostatic pressure created by the ground level altitude change, or static pressure, ΔpS must be included. In the critical situation, the combination of both should be less or equal to the maximum value deﬁned in TSI as

DpmaxT SI =ΔpNmax +ΔpS . (2.3.3) Therefore, using 2.3.2 2 − kwρφUtr =ΔpmaxT SI ΔpS , (2.3.4) 2.3. HYDROSTATIC PRESSURE INFLUENCE ON HIGH SPEED RAILWAY TUNNELS CROSS-SECTION SIZING 23 and if the train speed is given, the blockage ratio φ is

ΔpmaxT SI − ΔpS φ = 2 . (2.3.5) ρkwUtr As the static pressure increases, the blockage ratio should also decrease, that is, the tunnel cross-section area should increase. If the unitary cost of the tunnel Cu (cost per meter) can be estimated by a linear function of the tunnel cross-section area, in the range of areas considered (47 to 120 m2) then

Atun kc Atr Cu = k0 + kc = k0 + , (2.3.6) Aref φ Aref where Aref is a tunnel reference area (so that kc is dimensionless). Therefore the unitary cost of the tunnel is 2 kc Atr ρkwUtr Cu = k0 + . (2.3.7) φ Aref ΔpmaxT SI − ΔpS

As DpS approaches DpmaxT SI the tunnel cross-section area increases as well as the cost. Tunnel costs for different values of the Rock Mass Rating (RMR), which is an indicator to classify the rock quality, detailed in Peláez-González and Montenegro-Palmero [2009] are shown in Figure 2.3.1, where it can be pointed out that construction costs increase almost linearly with the tunnel cross-section area.

Figure 2.3.1: Tunnel cost (per meter) as a function of the tunnel cross-section area, Atun, for different Rock Mass Ratings (RMR).

In order to obtain this critical cross-section area and the inﬂuence in it of the static pressure, several simulations have been run with the developed code to evaluate pressure inside train and the tunnel. The program evaluated the pressure in different positions inside and outside of the train, passing through a tunnel of ascendant slope of 25‰ at a constant speed of 300 km/h and changing the length (20, 25 and 30 km). The ascendent is more critical that descendent slope because the train carries a negative pressure gauge. The parameters of the train used for calculations are showed in Table 2.3.1. Note that this slope has been chosen because it is representative of the high speed railway tunnels. 2 The perimeter of the tunnel is 28.7 m, area section 52 m and friction coefﬁcient cftun = 0.0065, and the entrance and exit portals are 30 m length and with a bevel edge. In Figure 2.3.3 is presented the critical cross-section area of tunnels with 25‰ of ascendant slope and length 20, 25 and 30 km. 24 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Table 2.3.1: Train properties.

Train S-130 Length, Ltr [m] 200 Cross-section area of coaches [m2] 12 Nose shape factor, kt 1 Atmospheric pressure at the entrance [Pa] 101325 Pressure loss coefﬁcient of the train head, ξh 0.06 Speed, Utr [km/h] 300

Gauge pressure in the train 2 Exterior Interior

−2

−4

P [KPa] −6 Δ

−8

−10

−12 0 50 100 150 200 250 300 350 T [s]

Figure 2.3.2: Pressure inside (red) and outside (blue) the train during pass through a tunnel of 20 km length with 25 ‰ ascendant slope.

The critical cross-section area of the tunnel of 20 km length with a slope of 25 ‰ (47.2 m2) is less than the minimum area required for tunnel installations 52 m2. For tunnels of 35 km length and longer ones; the cross-section area needed is so big (over 500 m2) that nowadays is non-feasible to construct it. It should be mentioned that the gauge pressure due to hydrostatic pressure for a tunnel of 30 km and 25 ‰ of ascendant slope is around 8.8 kPa, therefore it is very difficult to avoid exceeding the 10 kPa limit. The health criterion statement adjusts perfectly to short tunnels in which the time of a train passing through a tunnel is short. Nevertheless it can become a critical design condition for long tunnels (more than 25 km), where the time that the train needs to cross it is longer than 3 minutes. In the project designing phase for long tunnels, the cross- section areas to fulfill this requirement leads to high costs in construction. During a trains crossing inside a tunnel, the maximum gauge pressure appears when the first train is already outside the tunnel, because an expansion wave is then generated, the second train which is still inside the tunnel is in a more critical situation than in a simple train pass, for this reason trains crossing must be considered in the tunnel design process. 2.4. NON-LINEAR PROPAGATION AND PRESSURE WAVE STEEPENING INSIDE A TUNNEL 25

120

100 ] 2

[m 80 tun A 60

40 20 25 30 L [km] tun Figure 2.3.3: Critical cross-section area of tunnels with 25‰ of ascendant slope and length 20, 25 and 30 km.

2.4 Non-linear propagation and pressure wave steepening inside a tunnel

In long tunnels with small friction, the non-linear effects became signiﬁcant, generating a steepening in the rise time of the pressure wave front. Wave steepening involves large gradients so that the heat conduction and friction cannot be ignored. This steepening can even generate a shock wave. The thickness of a shock wave is only a few times the molecular mean free path so that a continuum theory fails [Rienstra and Hirschberg, 2011]. Apart from discontinuous, the solution is dissipative, as there is production of entropy. To study the problem, a model based on a nonlinear partial-differential equation for plane waves in homogeneous media in one-dimensional ﬂow, neglecting viscous and other dissipative terms, [Pierce, 1989] is used

∂p ∂p +(u + c) =0, (2.4.1) ∂t ∂x where u + c is function of p . Setting p = p0 + p (x, t) and p (x, 0) = f(x) yields

p = f(x − (u + c) t) . (2.4.2) ∂c For small-amplitude acoustic waves the relations u ≈ p /ρc and c ≈ c0 + p ∂p 0 can combine into

c + u ≈ c0 + β0u, (2.4.3) where the constant β0 is ∂c β0 =1+ ρc . (2.4.4) ∂p 0 If p is regarded as a function of entropy s and density ρ 2 1 B ∂p 2 ∂ p β0 =1+ ; A = ρ ; B = ρ 2 , (2.4.5) 2 A ∂ρ 0 ∂ρ 0 26 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS where A and B are coefficients in the expansion of p (ρ, s) at fixed s. For an ideal gas γ where p ∝ ρ at a fixed entropy, A = γp0 and B = γ (γ − 1) p0,soB/A = γ − 1 and β0 =(γ +1)/2.Asβ0 > 0 the waveform with higher overpressure move faster than those with lover overpressure; parts of the wave where pressure increases with time become steeper with increasing time and propagation distance. The relaxation process has minor influence on transition region if the overpressure amplitude is p > 200 Pa [Pierce, 1989]. For a pressure wave witha2kPapeak, the sound speed increases by 6 m/s for the positive peak, and the difference of speed with reflected wave expansion is hence 12 m/s, this is less thana4%ofthespeed of sound, but its importance becomes significant in long tunnels. The non-linear wave equation in characteristic form in presence of source terms (external mass injection ρβm and external force per unit volume fx) [Rienstra and Hirschberg, 2011] ˆ ∂ ∂ dp f c ∂ (ρβ ) +(u ± c) u ± = x ± m . (2.4.6) ∂t ∂x ρc ρ ρ ∂t ± The Riemann invariants Γ remain constant along the characteristics ˆ ± ± 2 ∂ (ρβm) Γ − Γ = c0f ± ρc dt, (2.4.7) 0 x 0 ∂t for an ideal´ gas with constant specific heat and isentropic flow (when no dissipation is dp =2c/(γ − 1) considered) ´ρc . The linear approximation in absence of source terms ± dp + ± ± + is Γ = u ρc = u p/(ρ0c0) along the characteristic C (dxc /dt = u0 + c0) − and C (dxc− /dt = u0 − c0), respectively. Therefore, using the Riemman invariants, and assuming no dissipation and waves propagating into a uniform region, the speed of a + perturbation along the characteristic C which passes through xc, distance traveled along the characteristic, can be deduced. Particularizing the Γ− at the tunnel entrance we obtain 2 2 c (τ ) − u (τ )= c0 , (2.4.8) γ − 1 e D e D γ − 1 where ce(τD) and ue(τD) are the sound speed and flow speed at the entrance, respectively. t = τD is the time at which the perturbation is generated at the tunnel entrance (x =0). + The characteristic C in the instant τD at the tunnel entrance propagates at the speed, where using equation (2.4.8) γ +1 c (τ )+u (τ )=c0 + u (τ ) , (2.4.9) e D e D 2 e D As long as the speed is a constant along the C+, the characteristic are straight lines whose slope is given by dxc+ γ +1 = c0 + u (τ ) , (2.4.10) dt 2 e D By using the invariant Γ+ 2 2 c + u = c (τ )+u (τ ) , (2.4.11) γ − 1 γ − 1 e D e D + and the equation (2.4.8), the solution over C is u(x, t)=ue(τD), where ue(τD) is a function of the characteristic that starts in xc =0at t = τD, and γ − 1 c = c0 + u (τ ) . (2.4.12) 2 e D 2.4. NON-LINEAR PROPAGATION AND PRESSURE WAVE STEEPENING INSIDE A TUNNEL 27

Depending on the derivative of the speed at the entrance u˙ e (τ), the slope of the characteristic C+ will decrease or increase, and the possibility of a multivalued solution can appear. If friction is considered (also under the assumption of being small) and all the other hypothesis are preserved, we obtain the following equation [Barrero Ripoll and Pérez-Saborid, 2006] ∂u ∂u dp λ u2 + u + = − D ≡ D, (2.4.13) ∂t ∂x ∂x 8rh where λD is the Darcy friction coefﬁcient which depends on the Reynolds number and the relative roughness of the walls (it has been assumed λD = Cf0 + cftun with an adjust coefﬁcient Cf0 =0.295). The entropy increases with the friction (due to viscous dissipation) and heat exchange. From the equation of entropy and assuming no heat exchange with the walls through conduction and radiation (q0 =0and qr =0)

∂s ∂s λ u3 T + Tu = D . (2.4.14) ∂t ∂x 8rh Comparing the viscous effects with the term of convection entropy in the case of a gas, we obtain the dimensionless parameter

λ L T D M2 , (2.4.15) rh ΔT

2 which is small if M (ΔT/T) (rh/λL). Therefore the friction can be important in the conservation of momentum equation (2.4.13) but it can be neglected concerning of increase of entropy. The equation (2.4.13) can be used to estimate the effect of the dissipation in the variation of the perturbation speed along the characteristic. To do so we have considered the solution of the case without dissipation ue(x, t)=u(x, t) obtained previously (2.4.12), and we calculate a correction function which takes into account the energy lost due to mechanical dissipation

ud(x, t)=ue(x, t)F (εx) , (2.4.16) where ε = λDL (umax/c0) /8rh is the parameter that takes into account the friction losses, being umax the maximum flow speed induced by the pressure wave in the non dissipative case, rh the hydraulic radius. F (xc) defines the way in which the speed is influenced by dissipation. In equation (2.4.13) the term D is linearized as follows

λ u D = − D max F (εx) , 8rh from where we obtain F (x)=exp(−εxc/L). The equation of the characteristic is then determined from (2.4.10) by subtracting from the propagation speed the reduction due to attenuation ue(τD)(1− F (xc))

dx γ +1 c = c + u (τ ) − u (τ )(1− F (x )) . (2.4.17) dt e 2 e D e D c In a general case we have

dx (t, τ ) c D + u (τ )F (x )=C, (2.4.18) dt e D c 28 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS where γ − 1 C = c + u (τ ) . (2.4.19) e 2 e D With the change of variable y = F (x)=exp(−εx/L) equation (2.4.18) can be written as L y − − u y = C, (2.4.20) ε y e

where y =dy/dt. This equation has a primitive ˆ dy − ε u = Cdt, (2.4.21) e y2 + y L C integrating with the boundary conditions X = x/L =0in t = τD ⎛ u ⎞ 1+ e e−εX ⎝ C ⎠ εC − εX +ln u = (t τD)dt. (2.4.22) 1+ e L C

The expression to simulate the rise time compression or expansion due to non-linear effects during the wave propagation inside the tunnel is ⎡ ⎛ u ⎞⎤ 1+ e e−εX ∼ L ⎣ 1 ⎝ C ⎠⎦ t = τD + X + ln u , (2.4.23) C ε 1+ e C where X =1at the receiver point. The wave steepening reduces the rise time of the pressure wave, increasing the pressure gradient and at the same time the sound radiated outside the tunnel. The rise times near the entrance, at the middle of the tunnel and at the tunnel end, of the pressure wave generated by a train entering in a long tunnel with small friction (area section 52 m2 and length 7230 m) at 291 km/h are shown in Figure 2.4.1. It can be appreciated that a shock wave is generated before the wave reaches the tunnel end. The maximum pressure gradient can reach 250 kPa/s; as it will be commented in section 2.5 this can lead to a peak SPL up to140 dB at the tunnel exit.

Pressure wave evolution inside the tunnel 2500

2000

1500 At the tunnel entrance In middle of the tunnel P [Pa] 1000 At the tunnel end

500

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 t [s] Figure 2.4.1: Variation with time t of pressure P of the wave inside a tunnel. 2.5. PRESSURE WAVE INTERACTION AT THE TUNNEL EXIT 29

This method to analyze the pressure gradient can be used until the vertical slope is achieved and the shock wave is generated; to study the evolution beyond this point another method should be used. At a large distance r from the tunnel exit the micro-pressure radiated outside from the tunnel at x = xE can be approximated by the expression [Kikuchi et al., 2009] 2At ∂ptun r Prad =Δpmpw(r, t) ∼ t − . (2.4.24) Ωc0r ∂t c0 The shape of the pressure gradient for the pressure radiated outside the tunnel portal is approximated using a Gaussian function g(t) (unit area)

t2 − 1 2 g(t)= √ e 2τr , (2.4.25) τr 2π

dptun (t) 2Atun = A · psg(t) , (2.4.26) dt c0rΩ where A is a constant, τr the shock rise time, ps the shock wave overpressure at the exit −αaLtun ps = p0e , p0 being the pressure gauge generated by entrance of the train nose into −5 the tunnel, αa =(λD/8rh)(umax/c0) ∼ 4 · 10 , the directivity Ω=π and r the distance to the tunnel portal exit.

2.5 Pressure wave interaction at the tunnel exit

To analyze the energy reflected at the exit, pulse reflectometry tests were performed in the first place. The effective pressure of the incident wave, pie, was compared with the effective pressure of the reflected wave, pre. The reflection coefficient at the duct end is defined by the ratio between the effective reflected and incident pressures R = pre/pie. The reflection coefficient at the duct end with the TMTC method is obtained using the expression given by the standard UNE-EN ISO 10534-2 [1998] −jks H12 − e R(f)=ej2k(l+s) , (2.5.1) ejks − H12

p1(f) where H12 = is the transfer function between the microphones pressure spectrum. p2(f) The results obtained in the experiments are compared with the analytic expression proposed in [Nishimura and Ikeda, 2008]. The expression for the acoustic impedance of a perforated side wall ZP proposed in Guess [1973] is √ 2 − 2 ZP 8νω t (kd) 1 σp = (1 + )+ + (M0 +0.3M) + ρoc σpc d 8σp σp √ (2.5.2) 8νω t k(t + δ) +i (1 + )+ , σpc d σp where σp is the porosity (evaluated as the area of holes divided by the area of exit using the average surface between the interior and exterior surface), d the diameter of the holes, t the plate thickness, ν the kinematic viscosity, ω angular frequency, δ the correction length 30 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS factor of open end, M0 the Mach number based on the particle velocity, and M the Mach number of the grazing flow. The steps to obtain the analytic expression for the reflection coefficient were proposed in Nishimura [2003]. The expression used to for the specific acoustic radiation impedance from rigid circular unflanged duct end (ka 1)is Z 1 ζ = E = (ka)2 +i0.6133ka . (2.5.3) E ρc 4

The speciﬁc acoustic radiation impedance from surface of the circular duct ⎛ ⎞ 2 Z ⎜ E ⎟ ZR l ⎜ ρc ⎟ ζR = =i ⎜ 2 ⎟ . (2.5.4) ρc kS ⎝ Z ⎠ E − 1 ρc

The speciﬁc acoustic impedance ratio inside and normal to the porous wall is

ZW = ZP + ZR . (2.5.5)

The sound propagation constant is

2 − l ρc αa =1 i . (2.5.6) kS Zw The specific acoustic impedance ratio at the end of the solid duct calculated by the transfer matrix method is ZA 1 αaZE +iρc tan (αakl) ζA = = , (2.5.7) ρc αa ρc +iαaZ tan (αakl) and the expression for the reflection coefficient at the duct exit is Z 1 − A ρc R = . (2.5.8) A Z 1+ A ρc

The reflection coefficient RA is close to unity at very low frequencies. The porosity at the exit increases the frequency of the pressure wave due to the multiple reflections, increasing the radiation at the exit and its attenuation. To predict the micro-pressure wave intensity radiated outside of the tunnel exit in case of shock wave generation, the proposed expression (2.4.26) can be used and from (2.4.24), and after Fourier transform the following expression is obtained

2 (τk) − 2At −αaLtun prad (r, k)=A · p0e e 2 (2.5.9) c0rΩ

The wavefront characteristic time of the shock wave reﬂected back inside the tunnel is independent of the train shape and speed. It just depends on the tunnel exit hydraulic radius rh, its length Le and it is approximately tc ∼ 2(Le +0.63rh) /a0 (around 0.08 s for a tunnel portal of length Le =11m and radius rh =5m). 2.6. EXPERIMENTAL SET-UP 31

Figure 2.5.1: Result for the radiated pressure by the non-linear propagation method. Variation with time of: 1) predicted pressure at the tunnel entrance and exit; 2) pressure gradient at the exit; 3) predicted radiated pressure at r=1 m from the tunnel exit in Pa; 4) predicted radiated pressure at r=1 m from the tunnel exit in dB.

Figure 2.5.2: Predicted radiated pressure spectrum (sound pressure level, SPL) in dB by using a Gaussian approximation of the pressure gradient (blue line) and measurements from experiments (red line).

2.6 Experimental set-up

The experimental set-up to study the reflection coefficient with acoustic pulse reflectometry (APR) technique of the pressure waves of different pipe exits consisted in a wave generator (HP 33120A), a speaker (ALTEC Lansing) placed at one of the ends of the PVC cylindrical tube of length 5 m, inner radius 28.4 mm and thickness 3 mm (see Fig- ure 2.6.1). The traveling pressure wave inside the tube is captured by two microphones (Sitecom TC-221). The data was send to the computer (HP Pavilion dv6) were it was post-processed. Temperature and humidity are also registered (sensor RS 212-124) and taken into account in the determination of the absorption coefficient of the air inside the tube. The junctions used are as the one shown in Figure 2.6.6. The validation of the experimental set-up is presented in section A.3. 32 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Figure 2.6.1: Scheme of the set-up components: speaker (1), microphones (2), signal generator (3) and computer (4).

Due to experimental limitations, there is a frequency range in which the study can be performed. If the incident and reﬂected wave are to be seen separately in the microphone time signal, then the lower frequency is limited by the duct length. The higher frequency limit would be the ﬁrst cut-off frequency (for this duct is approximately 3500 Hz), but in our case the ability of the speaker to reproduce the driving signal with an acceptable distortion (see Figure 2.6.2) settles this limitation in 3000 Hz.

Figure 2.6.2: Variation of the pressure gauge P (in arbitrary units) with time t. Theoretical signal (blue line), actual signal reproduced by the high-speaker (red line).

The frequency of the pressure wave that represents the rise up of the wavefront generated by a high speed train entering a tunnel is f ∼ Ut/lc, where Ut is the train speed and lc is the characteristic length of the tunnel entrance and the train nose. The equivalent scale frequency is calculated so that the Helmholtz number (H =StRe = ka) is preserved. Therefore, the relation between the frequency at full scale and test scale should be proportional to ratio between the inner radius of the tunnel and the inner radius of the tube. The frequency of the pressure waves generated in a tunnel by a high speed train is in the range between 2 and 20 Hz. At frequencies over 20 Hz, the PSD of the signal decreases around 20 dB ([Kikuchi et al., 2009]). The background noise in this set-up is under the 2% of the effective pressure of the incident pressure wave. It has been calculated from the measurements of the second microphone when the signal was passing through the first one. Other effects like reflections at the walls near the exits have been neglected. The test to validate this set-up with the analytic expression for unflanged open pipe was performed in an anechoic chamber at TU Berlin with a different pipe in order to assure concordance. The test performed in the setup was at TU have been crucial to validate our hypothesis of using gradual porosity 2.6. EXPERIMENTAL SET-UP 33 easy to implement on tunnel portals. The set-up at the Technische Akustik at TU-Berlin consists of an anechoic chamber of free volume 1070 m3 with a low frequency limit 63 Hz, the reflection coefficient at the duct exits was measured with the TMTC method. The distance between the microphones was s =76.5 mm, and the first microphone was located at 250 mm from the test sample. The equipment used consisted of 2 microphones of ½ inch PCB Piezotronics MI 260, MI 261, a signal generator acquisition card OROS (input sampling 51.3 kSamples/s), the excitation signal is a random noise and was generated by an OROS System OR34 Loudspeaker Visaton FR 10 Art. No 2021- 8 ohms.

Perforated exits The dimensions of the perforated exits tested A1 and A2, are shown in Figure 2.6.3. The exit A1 has holes of equal size, meanwhile in A2 the holes size decrease linearly with the distance to the exit.

Figure 2.6.3: Scheme exits A1 and A2. Side view.

The size of the holes is 20 mm, separated 20 mm, the distance between the holes center is 40 mm. The exit A1 has 6 equal holes equispaced, whose upper sides coincide with the symmetry plane of the conduit, and the nearest hole to the exit is at 20 mm from the intermediate adapter. The exit A2 has been deﬁned according to the following criteria:

1. The total area of the holes must be the same as exit A1 (24 cm2).

2. The area is distributed linearly (in arithmetic progression).

3. The shape is squared, the same as A1.

4. The geometric center of the holes are the same position as A1 (equispaced 40 mm).

The dimensions of the holes are deﬁned according to criteria 1) and 2). The area of the holes deﬁned starting form the nearest to the end of the exit are shown in Table 2.6.1.

Table 2.6.1: Area of holes of exit A2.

Hole 1 2 3 4 5 6 Area [cm2] 7 5.8 4.6 3.4 2.2 1

The exits deﬁned are shown in Figure 2.6.4 and 2.6.5. A porous exit with the same holes distribution as the porous exit A2 was built on PVC to compare the different absorption between PVC and wood, see Figure 2.6.6. 34 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Figure 2.6.4: Mounting of the porous exit A1.

Figure 2.6.5: Mounting of the porous exit A2.

Figure 2.6.6: Mounting of the PVC porous exit A2.

A porous section was built to analyze the transmission of the different porosity when is located between to ducts in which the were located foams at both ends to avoid reﬂections (see Figure 2.6.8). 2.6. EXPERIMENTAL SET-UP 35

Figure 2.6.7: Tube with ﬂanged open end.

Figure 2.6.8: Perforated cylindrical section.

Horns A horn [Pierce, 1989] is an impedance-matching device that increases the acoustic power output of a source and gives a directional preference to the radiated power. The dimensions of the horns are determined by the height of the center of the circumference Hb and its radius Rb, as it is shown in Figure 2.6.9. The ratio of the area at the junction with the pipe (Sa) and the exit (Sb) is 1.06 (in horns B2 and B4) and 1.11 (in B3 and B5). In Figure 2.6.9 the parameters that deﬁne the section of the horns are shown.

Figure 2.6.9: Area section of the horns.

The dimensions of the horns are speciﬁed in Table 2.6.2. The horns adapter and the mounting details can be appreciated in Figure 2.6.10 and Figure 2.6.11 respectively. 36 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Table 2.6.2: Horn dimensions.

Horn Lb [mm] Hb [mm] Rb [mm] B1 700 8.1 36 B2 700 8.9 40 B3 700 9.4 42 B4 1.060 8.9 40 B5 1.060 9.4 42

Figure 2.6.10: Intermediate adapter and semi-cylindrical exits (B1, B2, B3, B4 and B5).

Figure 2.6.11: Mounting of the B1 exit.

Airshaft The experimental setup used to analyze the pressure wave reflection at an airshaft consisted on PVC pipes assembled as shown in Figures 2.6.12 and 2.6.13. The pressure wave that is transmitted through an airshaft of different cross-section area Sh, and the pressure waves reflected to the duct of cross-section area Sc, have also been tested. The length of 2.6. EXPERIMENTAL SET-UP 37 the airshaft has been chosen in order to be able to distinguish the incident and the reflected waves in the time domain signals recorded by the microphones. The pressure wave transmitted through the airshaft has been evaluated by using the ratio (rITH = pefOT H /pefIH ) between the effective pressure wave transmitted through the airshaft, pefOT H and the effective pressure of the incident wave at the junction between tube and the airshaft, pefOI .

Figure 2.6.12: Mounting of the airshaft Sc/Sh =1.

Figure 2.6.13: Adapter for airshaft of different sections.

Transmission coefficients obtained in the tests, multiplied by the cross-section area are compared with Baron et al. [2001] in Figure 2.7.1, where it is shown that the accuracy of the experimental set up is acceptable. In Figure 2.6.14 the evolution of the pressure at a train entering at 241 km/h in a tunnel of 800 m length and area section of 90 m2 with an airshaft at 350 m of 12 m2 of area 38 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS section is shown. As it can be appreciated, just the first reflection at the airshaft has been taken into consideration. This is because the area ratio between the airshaft and the tunnel is commonly under 0.15, and the signal reflected back at the airshaft junction is around 5% of the incident pressure wave. In its propagation to the tunnel portal and back, it losses around 2 %, and around 8% in the reflection at the portal. So, for an pressure gauge of 2 kPa incident in the airshaft, after the reflections at the airshaft and at the tunnel portal would be around 100 Pa. This is a 5% difference with respect to the incident in the airshaft, which is under the error of the method.

Pressure gauge at the train 0.8 Outside the train 0.6 Inside the train 0.4 0.2 0 −0.2

P [kPa] −0.4 Δ −0.6 −0.8 −1 −1.2 0 1 2 3 4 5 6 7 8 9 10 Tiempo [s]

Pressure gauge in the tunnel x=150 m 1 x=200 m x=300 m

0.5

0 P [kPa]

Δ −0.5

−1

−1.5 0 1 2 3 4 5 6 7 8 9 10 Time [s]

Figure 2.6.14: Pressure gauge on the train and inside the tunnel with an airshaft of area section 12 m2 at 350 m from the entrance in a tunnel of 90 m2 cross-section area, train speed 241 km/h.

The pressure plane-waves attenuation coefﬁcient inside the tube, αa, assuming expo- −αax nential attenuation P = Poe , is evaluated using [Flecher and Rossing, 1991]: ω 1 γ − 1 αa ≈ √ + √ , (2.6.1) c0 rv 2 rt 2

where rv is the dimensionless parameter that takes into account the corrections due to the viscous boundary layer and rt is the parameter that takes into account the corrections due to the thermal boundary layer. Some measurements have been performed with the experimental set-up to validate the expression used. The expressions proposed in Flecher and Rossing [1991] are rv ≈ 632.8a f (1 − 0.0029 (T0 − 300)) , (2.6.2) 2.6. EXPERIMENTAL SET-UP 39 1/2 rt = rvPr ≈ 532.2a f (1 − 0.0031 (T0 − 300)) . (2.6.3) Substituting in the above equations the conditions under which measurements have been performed (a =0.0284 m, T0 = 297 K and P0 = 94768 Pa) and taking into account the corrections of the Table 2.6.3 the attenuation constant obtained is αa=0.0331 (or -0.287 dB/m).

Table 2.6.3: Temperature corrections in the air properties [Flecher and Rossing, 1991]. Variable Temperature corrections −4 ρ 1.1759 · 10 (1 − 0.00335(T0 − 300)) γ 1.4017(1 − 0.0002(T0 − 300)) −5 η 1.846 · 10 (1 − 0.0025(T0 − 300)) 1/2 Pr 0.8410(1 − 0.0002(T0 − 300)) c 347.23(1 − 0.00166(T0 − 300))

The scale factor between the microphones at a certain frequency Fe, is deﬁned as the ratio between the effective pressure measured at one microphone pM2e and the effective pressure measured at other microphone pM1e (Fe = pM2e/pM1e). Fe is evaluated experimentally, using three data from measurements performed with the microphones located at different distances from the speaker, see Table 2.6.5. Subsequently the position of microphones was exchanged and process repeated. The same procedure has been used to evaluate experimentally the absorption coefﬁcient αa. The process followed is explained in Table 2.6.4.

Table 2.6.4: Scheme of pressure measurements during microphones position exchange. Fe is the scale factor; αa: plane wave attenuation coefﬁcient; Poa and Pob: reference pressure of cases a and b, respectively. Measurement with M1 in front Measurement with M2 in front (case a) (case b) M1 M2 M2 M1 (−αax) (−αax) Poa PoaFee PobFe Pobe

In both cases the pressure at microphone M2 is divided by the pressure at the microphone M1 to obtain the following expressions PM2 (−αax) =Fee , PM1 a PM2 (αax) =Fee . PM1 b Thus, we obtain the expression to evaluate α a PM2 P 1 M a (−2αax) = e , PM2 PM1 b and multiplying we get the expression for F e PM2 PM2 2 = Fe . PM1 a PM1 b 40 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Table 2.6.5: Scale factor, Fe, attenuation coefﬁcient, αa, inside the PVC tube for different distances between the microphones s.

s [m] Fe αa 1 1.0557 ± 0.0033 −0.0344 ± 0.0013 2 1.0776 ± 0.0014 −0.0300 ± 0.0012 2.5 1.0781 ± 0.0256 −0.0313 ± 0.0019

Finally the values used to correct the distance attenuation and the difference between the microphones sensibility were αa = −0.034 and Fe =1.0776. Note that the attenuation constant is proportional to the square of the frequency, and the error in its calculation can lead to errors in the reflection coefficient of about 2% for a distance of 3 m. To avoid this, the two-microphone-three-calibration (TMTC) method, is used so that the distance from the microphones to the end of the duct can be smaller. This method reduces significantly the dimensions of the experimental set-up, and also the spectrum for all the frequencies (under the plane wave hypothesis) can be calculated at once. As a representative case of a pressure wave generated by a high speed train, a train speed of 270 km/h and a tunnel radius of 5 m have been chosen. The characteristic frequency of the wavefront at the tunnel end is fpw 10-20 Hz, the Helmholtz number is H = ka = πfpwDtun/c 0.2 − 0.4 [Kikuchi et al., 2009]. The Helmholtz number must be preserved to compare scale experiments with the pressure wave in the tunnel, from this condition we obtain the frequency range of study in the scale test

Dtun fs = fpw . (2.6.4) Dduct where Dduct is the inner diameter of the duct. Depending on the method used to study the reflection at the duct ends, this frequency range is limited for and upper fu and a lower frequency f0. The first and the second rows in Table 2.6.6 correspond to the APR tests, where the frequency of study fs is limited by the speaker fidelity fu, and by the laboratory dimension f0. When using the TMTC method (third row of Table 2.6.6) the frequency of study fs is limited by the distance between the microphones s. The expression to determine this limits are specified in the standard UNE-EN ISO 10534-2 [1998], which are in the case of a cylindrical duct, for the lowest frequency limit 0.05c/s < f0 and for the highest fu < 0.58c/d.

Table 2.6.6: Frequency ranges under analysis in the different experimental set-ups. The distance between the microphones used in the TMCT was s =76.5 mm.

Dduct/2 [mm] f0 fu 15 666.6 6666 28.4 352 3520 50 215 2487

A test to analyze the effect of duct wall porosity (size and distance between holes) in the pressure pulse attenuation was performed. The energy reﬂected and transmitted in the porous section were compared. A porous duct section was built and located between two ducts with foam at the ends to avoid reﬂections. The porous section consists of 60 holes with a diameter 12 mm, distributed in 3 rows of 10 (separated 20 mm) on each side of the 2.6. EXPERIMENTAL SET-UP 41 plane of symmetry (see Figure 2.6.8). The position of the microphones were exchanged to get the following correction 1/2 H12 H12 = . (2.6.5) H21 The absorption in the air inside the duct is calculated with the expression extracted in the standard UNE-EN ISO 10534-2 [1998] ω 1.94 · 10−2f 1/2 k0 = − i . (2.6.6) c c · Dduct

The porosity of the samples, σp, under study was selected to compare the results for Helmholtz number H ∼ 0.5 with [Nishimura, 2003]. The samples are cylinders of aluminum 5 mm thick, with the same internal radius as the PVC duct. The porosity of the samples is distributed in 6 rows, in the longitudinal axis, of 10 holes each. They have a plane of symmetry with 3 rows at each part and separated 30º between them, starting the ﬁrst row at 60º from the plane of symmetry. The diameters of the holes Φ, are constant for each sample, the values are shown in Table 2.6.7.

Figure 2.6.15: Scheme of the ducts exit porosity distribution.

Table 2.6.7: Diameter radius Φ, and porosity σp, of the samples tested with the TMTC technique. Φ [mm] 2 3 5 σp [%] 0.18 0.42 1.16

The built porous exits are shown in Figure 2.6.16. The mounting of the porous exits in the duct, with the speaker at the other side used at TU Berlin is shown in Figure 2.6.17. 42 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Figure 2.6.16: Porous duct exits of porosity 0.18 %, 0.42 % and 1.16 % tested at TU Berlin.

Figure 2.6.17: Experimental set-up to implement the TMTC technique using an aluminum exit of porosity σ =1.16%.

2.7 Results and discussion

2.7.1 Reﬂected wave at the horns The setup used to measure the reﬂected waves at the horns has been described in section 2.6, and shown in Figures 2.6.1 to 2.6.11.

The reflection coefficients rIR of the different horns have been referenced to the reflection coefficient of the horn B1, and are summarized in Table 2.7.1. The part of the wave attenuation due to the different absorption between the PVC and the wood is included in these coefficients. From the results, the existence of undesired reflections at the junctions, 2.7. RESULTS AND DISCUSSION 43 between the pipe and the intermediate adapter, and between this one and the duct exits, can be deduced.

Table 2.7.1: Ratio between the reﬂection coefﬁcient in the horns and the horn B1.

Pipe exit L [m] rIR/rIR|B1 B2 7.62 1.046 ± 0.015 B3 7.62 1.077 ± 0.018 B4 7.93 1.181 ± 0.009 B5 7.93 1.243 ± 0.014 A1 7.37 1.179 ± 0.098 A2 7.37 1.096 ± 0.024

In any case, even if the conical horn has more capacity to radiate at very low frequencies, the required lengths are prohibitive, as is explained in section A.2. From the comparison between the reflection coefficient in B4 and B2 we can extract the attenuation coefficient of the wood as αw =ln((rIR|B2/rIR|B4) /2(LB4 − LB2)) = −0.169, which is much higher than in the PVC. So in order to compare the difference in the reflection coefficients due to the possible differences in shape between the A2 in wood and at the A2 in PVC we should considered the effect of the different absorptions during the propagation through the ducts. We obtain that if the effect of the wood is extracted the reflection coefficient is (rIR|A2WOOD/rIR|A2PVC)exp(αw − αa)=1.0707. We also have to take into account the effect of the different absorption in the intermediate adapter (which is only needed in the A2 made from wood), finally we obtain that the difference between both reflection coefficient is less than 1%. This study highlights the efficiency of increasing the interior surface roughness of the duct in the pressure waves attenuation. This lead us to think on introducing devices to increase the roughness and at the same time have the possibility to extract energy from the pressure wave induced speed, like the proposed in the next chapter.

2.7.2 Reflected and transmitted wave at airshafts The setup used to measure the reflected waves at the airshafts has been described in section 2.6, and shown in Figures 2.6.12 to 2.6.13. Airshafts act as a high-pass filter for pressure wave, and using the appropriated length they can reduce steepening. This length would be λc/4, being λc the wavelength of the rise time of the compression/expansion wavefront. This leads to different length as the wave travels through the tunnel as this time is reduced to nonlinear effects. The pressure wave reflected back at the airshaft is coefRT evaluated by the ratio between the rms of the reflected and incident wave at the junction, also the transmitted through the duct and through the airshaft are related to the incident wave in the coefficients coefTT and coefTC respectively. The airshafts considered in these test are long compared with the wave length in order to distinguish in time domain the different wave reflections. The test results are shown in Figure 2.7.1. It can be appreciated that half of the energy is reflected back at the junction for area ration close to unity. The common cross-section area ratio used for tunnel airshafts is Sc/St ∼0.15 (St/Sc ∼0.67). The reflection coefficient at the airshaft junction is RITH ∼ 0.85 shows the efficiency of the airshafts as a pressure wave attenuators. But airshaft are an expensive resource to attenuate waves if they are not implemented in the tunnel portals. The possibility of using airshafts of different lengths coming out of the tunnel 44 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

0.8 t c S S 0.6

ITH 0.4 R 0.2

0 0 1 2 3 4 5 6 7 8 9 10 Sc/St

Figure 2.7.1: a) Variation with the area ratio Sc/St of effective pressure transmitted by the airshaft RITH multiplied by the inverse of the area ratio. Continuous line: results of Baron et al. [2001], circles: measurements. b) Variation with the area ratio St/Sc of the pressure wave reflection coefficient at the junction between the duct and the airshaft coefRT , transmitted through the duct coefTT, and transmitted to the airshaft coefTC. portals as side branches can be an efficient method not only to reduce the reflection at the exits but also the pressure gradient during the train entrance in the tunnel. The length of the airshaft would be around 25 m and even shorter for long tunnel in which non-linear effects are relevant.

2.7.3 Reﬂected and transmitted wave at a perforated section The setup used to measure the reﬂected waves at the perforated section has been described in section 2.6, and shown in Figure 2.6.8.

The influence of the porosity σp, has been analyzed by varying the number of holes σp = 0, 3, 7, 15 % for a 3-cycle sinusoidal burst frequency of 1000 Hz, and the energy reflected at the porous section is compared with the case in which the inside of the duct is completely smooth. The reflection coefficient obtained through the porous tube is defined as p1i TL[dB] = 20 log10 . (2.7.1) p1t

The results shown in Table 2.7.2 indicate that the wave is practically reﬂected back as an open end termination when the porosity is too high. 2.7. RESULTS AND DISCUSSION 45

Table 2.7.2: Transmission loss (TL) through a porous section. σ [%] TL [dB] 0 0 3 21.33 7 23.14 15 23.58

2.7.4 Reflected wave at perforated exits The set-up used to measure the reflected waves at the perforated has been described in section 2.6, and shown in Figures 2.6.6 to 2.6.7. The duration of the data register used to evaluate the prms of the incident and reflected waves was increased in order to consider all the energy reflected back, because the reflections at the holes change the shape of the reflected wave (see Figure 2.7.2). At the same time we can appreciate how the amplitude of the peaks have been significantly reduced, and with them the pressure gradient that would found a train inside a tunnel, but also the radiation at the other tunnel exit. The frequency of the wave has increased, and with it the radiation efficiency of the duct end. Furthermore, the absorption inside the duct increases as the root of frequency.

Figure 2.7.2: Variation with time t of pressure P (arbitrary units) of the incident wave OI (blue line) and the reﬂected wave OR (red line) at the PVC perforated exit.

The influence of the number of holes in the reflection coefficient has been analyzed by closing the holes, starting at the farthest from the exit. As it can be appreciated in the results presented in Table 2.7.3, the reflection coefficient varies between 0.61, when all holes are opened, and 0.8 when all are closed. The closing of the last hole (the closest to the exit and the biggest) slightly reduces the reflection coefficient, probably due to the way the holes are closed. The possibility of having modified the absorption coefficient during the hole closing process was considered, and also a possible acoustic feed back between pressure radiated at the exit and the hole. To analyze these points, a flange was added between the side hole and the exit (see Figure 2.6.7), and the test were repeated. This time the closing process of the holes started from the nearest to the wall to the farthest. In the results presented in Table 2.7.4 a similar behavior in the reflection coefficient can be appreciated, the values are quite similar (a bite smaller in most of the cases), although it should be noted that the 46 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Table 2.7.3: Reflection coefficient at the perforated exit R for different number of open holes in the unflanged perforated exit.

Number of open R holes 6 0.611 ± 0.002 5 0.613 ± 0.001 4 0.635 ± 0.001 3 0.706 ± 0.001 2 0.780 ± 0.001 1 0.810 ± 0.002 0 0.800 ± 0.002 size of the holes left over to be closed were smaller, and at more distance from the exit that in the other test.

Table 2.7.4: Reflection coefficient at the perforated exit, R, as a function of the number of open holes in the flanged perforated exit. Number of open R holes 6 0.592 ± 0.002 5 0.595 ± 0.002 4 0.614 ± 0.002 3 0.680 ± 0.001 2 0.770 ± 0.001 1 0.820 ± 0.001 0 0.801 ±0.001

The reflection coefficients in the perforated exit for the flanged and unflanged cases are shown in Figure 2.7.3. In the case were just the last hole is open, the reflection coefficient is bigger in the second case. The reflection coefficient decreases in both cases, closing the last hole .

Figure 2.7.3: Reflection coefficient at the perforated exit as a function of the number of holes opened, for flanged and unflanged exits.

To analyze the effect of the wave combination interference at the exit, during the re- ﬂection process at the perforated exit (without a ﬂange), a study at different frequencies 2.7. RESULTS AND DISCUSSION 47

(700, 1000 and 1300 Hz) was performed modifying the position of the open hole at 8, 12, 16, 20 and 24 cm from the center of the exit. A minimum in the variation of the reflection coefficient with the position of the hole can be appreciated in Figure 2.7.4. The distance of the hole to the exit that provides minimum reflection increases as frequency decreases. This variation can be explained as follows: if the distance of the hole to the exit is λ/4, and as the wave changes its phase at the openings, the second half of the sine period due to the reflection of the wave at the hole will be added with the first half of the reflection of the wave at the exit, thus minimizing the reflection coefficient.

Figure 2.7.4: Inﬂuence of the open hole as a function of the distance to the exit for different frequencies.

The setup used to measure the reflected waves at the perforated has been described in section 2.6, and shown in Figures 2.6.15 to 2.6.17. The results for the reflection coefficient at the end of a cylindrical duct for the cases of flanged and unflanged terminations which are shown in Figure 2.7.5, were obtained by the two-microphones-three-calibration (TMTC) technique at the TU Berlin. They are compared with the values obtained by using the expressions proposed in Silva et al. [2009] for flanged and unflanged duct open end. The measurements are in accordance with the theoretical approximations. The radiation from a duct end with different porosity (see Figure 2.7.6) has also been measured. A variation in the directivity can be appreciated specially at high frequencies and at high angles. The intensity of the radiated sound is concentrated in the axis of symmetry when using bigger hole diameters. According to these results, the optimum tunnel exit for a train entering at 300 km/h with a blockage of 0.15 will have 6 diameters in length, and 60 holes of 0.25 m of diameter each. This means a porosity σp = 0.0116 . Three frequency ranges can be distinguish in Figure 2.7.5 in which the smaller reflection coefficient is provides by a different porosity exit. From H=0 to 0.35, the optimum exit porosity is σ =0.18% , from H=0.35 to 0.5 where is σ =0.42% , and from H=0.5 to 0.9 where is σ =1.16% . All this curve show a minimum which presented at higher frequencies and it value is smaller as the porosity increases. This is a economic way of reducing the pressure wave reflected back in the tunnel which could be optimized for the case in which appears steepening and a shock wave is reached, because in this case the frequency of the pressure gradient will be independent of the train shape and speed. In cases where this frequency varies, then the porosity of the portal should be modified in relation with this characteristic frequency. An alternative to a frozen porosity configuration at the tunnel portal, could be a variable porosity system, in which depending on the train shape and speed, and tunnel entrance 48 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Figure 2.7.5: Variation with frequency f of the reflection coefficient R(f) for exit samples with porosity σp 0.18, 0.42 and 1.16 % obtained with TMTC method, compared with flanged and unflanged analytical expression, and experiment results obtained with pulse reflectometry. geometry, the porosity could be modified at the tunnel exit. This way we could modify the porosity to be always on the optimum for each incident wave in the tunnel portal. From equation 2.4.2 we can estimate how varies the pressure rise time along the tunnel for each train. With the inverse of this time (frequency) and the radius of the portal obtain the Helmholtz number and from in Figure 2.7.5 extract the optimum porosity that should be chosen for the portal. One possibility could be a group of wind turbines situated at the holes of the tunnel exit, and the porosity could be modified by varying the collective pitch angle of the turbine blades. The energy extracted (for example with and induction engine) could by used for the control of the blades. The micro-pressure radiated at a long slab track tunnel exit can also be reduced by increasing the friction in the tunnel walls. This could be done by introducing mechanisms that extract energy from the pressure waves generated by the train, as is analyzed in chapter 3. 2.8. CONCLUSIONS 49

Radiation from a duct end with 60 holes d=2mm β=0º Radiation from a duct end with 60 holes d=3mm β=0º f = 200 [Hz] f = 200 [Hz]

90 f = 250 [Hz] 90 f = 250 [Hz]

40 f = 315 [Hz] 40 f = 315 [Hz] 120 60 f = 400 [Hz] 120 60 f = 400 [Hz] f = 500 [Hz] f = 500 [Hz]

30 f = 630 [Hz] 30 f = 630 [Hz] f = 800 [Hz] f = 800 [Hz] f = 1000 [Hz] f = 1000 [Hz]

150 30 f = 1250 [Hz] 150 30 f = 1250 [Hz] 20 20 f = 1600 [Hz] f = 1600 [Hz]

10 10

180 0 180 0

Radiation from a duct end with 60 holes d=5mm β=0º Radiation from a duct end with 60 holes d=5mm β=30º f = 200 [Hz] f = 200 [Hz]

90 f = 250 [Hz] 90 f = 250 [Hz]

40 f = 315 [Hz] 40 f = 315 [Hz] 120 60 f = 400 [Hz] 120 60 f = 400 [Hz] f = 500 [Hz] f = 500 [Hz]

30 f = 630 [Hz] 30 f = 630 [Hz] f = 800 [Hz] f = 800 [Hz] f = 1000 [Hz] f = 1000 [Hz]

150 30 f = 1250 [Hz] 150 30 f = 1250 [Hz] 20 20 f = 1600 [Hz] f = 1600 [Hz]

10 10

180 0 180 0

Radiation from a duct end with 60 holes d=5mm β=90º Radiation from a duct end without hood f = 200 [Hz] f = 200 [Hz]

90 f = 250 [Hz] 90 f = 250 [Hz]

40 f = 315 [Hz] 40 f = 315 [Hz] 120 60 f = 400 [Hz] 120 60 f = 400 [Hz] f = 500 [Hz] f = 500 [Hz]

30 f = 630 [Hz] 30 f = 630 [Hz] f = 800 [Hz] f = 800 [Hz] f = 1000 [Hz] f = 1000 [Hz]

150 30 f = 1250 [Hz] 150 30 f = 1250 [Hz] 20 20 f = 1600 [Hz] f = 1600 [Hz]

10 10

180 0 180 0

Figure 2.7.6: Sound pressure level, SPL [dB] radiated from the duct end by hoods with different porosity.

2.8 Conclusions

A model to predict the pressure inside a tunnel and at the train has been developed and implemented, based on the available information in the literature, that can be used in preliminary design of tunnel sections and to evaluate tunnel costs, and also to ﬁnd the optimal location for damper devices depending on the trains and the travel speeds during the pass through the tunnel. 50 CHAPTER 2. PRESSURE WAVES IN HIGH SPEED RAILWAY TUNNELS

Besides, a simple model of non-linear wave propagation has been developed, that allows the pressure gradient at the tunnel exits to be predicted, and also the micro-pressure radiated outside the tunnel to be estimated. In order to study different systems to reduce the intensity of the pressure waves, two experimental set-ups have been developed one at the IDR/UPM and another one at the TU Berlin. The experimental set-up at IDR/UPM has been used both to evaluate the reflection coefficient at a duct end, and the transmission coefficient through a porous section; the results obtained are in accordance with classical mathematical models. The behavior of airshafts as high-pass filters, and of the junction as low pass filters has been confirmed experimentally. Also the wave separation technique has been validated experimentally at IDR/UPM set-up. The technique used at IDR/UPM set-up was the APR, meanwhile at the TU Berlin was used the TMTC. At TU Berlin we have had the possibility to measure radiation from different porous exits. An validate the hypothesis of an optimum porosity depending on the rise time of the wavefront when it reaches the tunnel portal exit. The reflection coefficients of horns have shown that they could be quite efficient if the secondary reflections at the horns are forced: the absorption in the horns could be increased by using Helmholtz resonator, absorptive materials, cross-section area changes, airshafts of different lengths, etc. The porous exit is an efficient system to decrease the reflection coefficient of the pressure waves that reaches the exit of the duct. The porosity can increase by 40 % the efficiency to reduce the intensity of the reflected wave with respect to the flanged exit and generate a more directional radiation spectrum. Porous exits increase the intensity of the sound radiated outside in the direction of the railway and reduce the intensity in the direction perpendicular to the railway, if the holes are located at the top of the hood. Chapter 3

Energy extraction from aerodynamic instabilities

3.1 Introduction

As it has been shown in chapter 2, the pressure waves generated by a train entering a tunnel induces a flow which can be used to extract energy using an appropriated device, and at the same time reduce the intensity of the waves. This will be specially useful in a long tunnel where the use of ballast is almost prohibitive by the high maintenance costs. In short tunnels could also be an alternative to avoid the ballast pick up, due to the overturning moment produced by the gust generated by the passing train that generates a turbulent Couette flow between the ballast in the track and the train under-body. The ballast train-induced-wind erosion can produce damage to the train under-body and the infrastructure surrounding the tracks [Sorribes-Palmer et al., 2012]. The aim is to design and tune a device matched to the excitation, and extract the maximum energy from the device motion, and at the same time to increase the effective area section of the tunnel to generate partial reflections of the wave. Before starting any deep analysis, let us first consider the energy of the pressure wave propagating through the tunnel, and identify the cases and positions in the tunnel of interest to locate a device to extract energy. Once the position is set, the period and the induced speed will be determined (after some transient time the pressure in the tunnel becomes periodic) along with the available energy to be extracted, taking into account the efficiency of the device. For a device based on transverse galloping depends on the body cross-section shape, dimensions, mass, damping, etc. All this properties have their influence in the first eigenfrequency of the device. A large effort has been already done on the study of energy harvesting from aerodynamic instabilities (transverse galloping, vortex induced vibrations) see Abdelkefi et al. [2013b], Abdelkefi et al. [2013a], Jung and Lee [2011], Kluger et al. [2013] among others. Let us make a rough estimation of the amount of available energy in a tunnel when

51 52 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES a train of 200 m length enters at 300 km/h, the characteristic time of the pressure wave generated is 2.4 s, the pressure gauge of the wave generated by a tunnel-train combination is ΔpN ∼ 1 − 3 kPa depending on the blockage and of the altitude of the tunnel entrance. If we idealize the pressure wave as a squared pulse of 2 kPa, the speed induced by the wave will be u = p/ρc ∼ 5 m/s. The total average energy density per unit area of a plane 2 ∼ 2 wave is E =(1/2)ρumax 15 N/m . If we assume that each device has a frontal area of S ∼ 0.2 · 1m2 =0.2m2 and the efficiency of extraction is η ∼ 0.1 the extractable energy by each device would be Πe ∼ 1.5 W. When using the flow speed induced by the wake of the train (speed can rise up to 20 m/s rms depending on train speed and blockage ratio) this extractable energy can rise till Πe ∼ 100 W. If we suppose a logarithmic decrement of δ ∼ 0.01 with an oscillation frequency of 2 Hz, the time to reduce the oscillation to a 10% of the maximum amplitude would be approximately 10 seconds. This time could be used to extract energy before the next reflection reaches the device again. In this moment the speed induced by the wave would be in the opposite direction. If the body shape used in the device had a symmetric behavior, the energy extracted or damped would be of the same order in both directions. The possibility of using solenoid coils and adapting the damping with the number of coils (varying with it the damping of the device) in accordance to the position of the device could be used to optimize the efficiency depending which moment in the oscillating cycle the airfoil is on. In order not to increase too much the drag of the train inside the tunnel, the bodies could be set in minimum frontal cross-section area when the train is passing. Energy could be also extracted from the wake generated by the train. The air speed induced in the tunnel (by the train wake) can reach till 20-30 m/s depending on the blockage and the train speed. The device to be considered is based on the motion generated by an aeroelastic phenomenon, and this chapter is devoted to find a way to optimize the energy extracted by devices based on such phenomenon. Aeroelastic phenomena are still an important issue in the design of engineering structures (aircraft, wind turbines, bridges, tall buildings, power lines, etc.), among them we find transverse galloping, wake galloping, flutter, vortex shedding, torsional divergence and buffeting. This chapter focuses on the transverse galloping, whose theoretical foun- dations can be found in Crawley et al. [1995]. The galloping occurs when the aerodynamic forces associated with the motion of the structure have a destabilizing character [Barrero-Gil, 2008]. The galloping is caused by a coupling between the aerodynamic forces and the across-wind oscillations induced in the structure, which change the angle of attack, which in turn vary the aerodynamic forces modifying the dynamical response of the structure. The structure, which usually has low stiffness and low damping, moves in the direction normal to the average wind speed. This motion is characterized by oscillations of large amplitude and low frequency. The most important parameters which influence galloping (considered here as a one-degree-of-freedom oscillator subjected to aerodynamic forces) are: the geometric shape, the angle of incidence, the speed of flow, the density and viscosity of the fluid, the turbulence intensity of the flow, and the system’s mechanical properties (mass, stiffness and damping). Until now, the tool usually used to analyze the stability in transverse galloping of a body is based on the shape of the curve of the dimensionless coefficient of the aerodynamic force transverse to the incident flow Cz (α), α being the angle of attack, and its expansion in powers of α or tan α [Alonso et al., 2009]. For the study of a particular case this method can be complex and tedious, because it requires a polynomial fitting 3.2. MATHEMATICAL MODEL 53 to the experimental results. This rises concerns about the suitable number of points to consider, and the range of amplitude of motion around the pitch angle of the body to be taken into consideration in the polynomial fitting. Note that if the amplitude of the oscillations exceeds the range of validity of the fitting, the dynamics obtained from analysis will differ from its real behavior. For the analysis of a generic configuration, the approach in Barrero-Gil et al. [2009], which concludes that the number of inflection points in the curve Cz (α) determines the regions where hysteresis in galloping may appear, is restricted to the simple cases that allow a low order polynomial fitting to be used, with limited physical meaning. In Ng et al. [2005], high order polynomials (ninth and eleventh) are used to fit the curve of Cz (α) in order to reveal more flow physics. But no additional positive real roots appear, only extra negative real and/or complex roots are obtained, concluding that a seventh order polynomial is sufficient to capture the hysteresis phenomenon in a squared cylinder. In Abdelkefi et al. [2013b] the polynomial fitting method is also used to approximate the aerodynamic force to model a galloping-based piezoaeroelastic energy harvester. Both a linear and a nonlinear analysis are performed to determine the effects of the electrical load resistance and the cross-section geometry on the onset of galloping, which is due to a Hopf bifurcation. The results show that the maximum levels of harvested power are accompanied with minimum transverse displacements amplitudes. The aim of this chapter is to present a new method to analyze the problem that does not require the power series expansion for the above mentioned fitting, but is based on the use of the curve Cz (α) obtained directly from the experimental data, without requiring any polynomial fitting.

Depending on the shape of the curve Cz (α), an hysteresis phenomenon may appear linked to the separation and re-adherence of the boundary layer on the upper or lower surfaces of the body. In this case, different amplitudes of the limit cycle oscillations can be achieved depending on whether the ﬂow speed is increasing or decreasing. In this chapter an alternative method to the classical polynomial ﬁtting analysis is proposed to study the stability of the phenomenon of transverse galloping. This chapter is organized as follows: the mathematical model that describes the transverse galloping and the method to analyze the stability is presented in Section 3.2 The experimental set-up used in the measurements is described in Section 3.3. In Section 3.4 the results obtained from the application of the method to experimental measurements are shown. Finally, in Section 3.5, conclusions are drawn.

3.2 Mathematical model

Since the dynamical behavior of a structure placed inside a fluid flow can be a very complex problem, it has become customary in the study of transverse galloping to analyze the problem using a simplified model consisting of a linear one-degree-of-freedom oscillator, a structure which is slender enough to consider two-dimensional flow, and neglecting the effect of the incident flow turbulence [Parkinson and Smith, 1964]. The simplified model of the system is shown in Figure 3.2.1. Under these conditions, common in fluid-structure interaction analysis, the equation 54 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

Figure 3.2.1: Schematic representation of aerodynamic forces and the angle of attack, α, pitch angle of the velocity, γp, pitch angle of the body, θ; z, x is the inertial reference frame and zb, xb is the reference frame attached to the body. that describes the dynamic of transverse galloping can be expressed as

2 1 2 m(¨z0 +2ζω z˙0 + ω z0)=f (α)= ρU bC (α) , (3.2.1) n n z 2 z where z0 denotes the displacement of the point O transverse to the incident flow, ωn is the undamped natural frequency, ζ the dimensionless structural damping coefficient, U the velocity of the incident flow, ρ the fluid density, b the characteristic dimension of the body in the direction transverse to the flow, fz(α) the aerodynamic force in the direction transverse to the flow, and Cz(α) the dimensionless coefficient of the aerodynamic force transverse to the flow c (α)+c (α)tanα C (α)=− l d , (3.2.2) z cos α where 1 dz0 tan α = . (3.2.3) U dt This expression refers to the study of the oscillatory motion around a pitch angle of the body θ considered as reference, with θ =0as origin. If the lift cl(θ) and drag cd(θ) coefficients determined in static conditions (θ = α) are defined as a function of the pitch angle of the body θ, and the reference point of study is θ0 =0 , doing the change of variable α = θ − θ0, the functions cl(α) and cd(α) can be generated for each θ0. The hypothesis of quasi-steady aerodynamic behavior can be assumed, since the transverse galloping appears at high enough velocities, such that the characteristic residence time scale of the flow (tr ∼ L/U ) is small compared with the time scale of the change in boundary conditions of the flow, which can be evaluated as the period of the structure oscillations (tv ∼ 1/ω0). In the above expressions L is the characteristic dimension of the structure in the direction of the flow and ω0 the oscillation frequency of the structure. The vortex shedding phenomenon is not taken into account here because at the flow speeds considered the vortex shedding frequency fv is much higher than the oscillation frequency of the body ω0/2π and both problems can be decoupled. Therefore, the transverse galloping phenomenon can be analyzed using static measurements in a wind tunnel at a pitch angle coincident with the instantaneous angle of the incident flow. The solution to the motion equation (3.2.1) by the Krylov-Bogoliubov method is based on assuming that the system is an undamped oscillator slightly perturbed by both a small 3.2. MATHEMATICAL MODEL 55

damping ζ, and an aerodynamic force transverse to the ﬂow fz(α), so that the motion of the body can be considered harmonic

z0 (t)=A cos(ωnt + ϕ) , (3.2.4) where the amplitude, A, and the phase, ϕ, are slowly varying functions of time, t. The dimensionless dissipated power averaged per cycle (due to friction, or another similar mechanism) can be deﬁned as Pd pd = 1 3 , (3.2.5) 2 ρU b where Pd is the dissipated power averaged per cycle,

2 ˆT 2 ˆ π 1 dz0 1 P = 2mζω dt = 2mζω3A2 sin2 τdτ, (3.2.6) d T n dt 2π n 0 0 where τ = ωnt + φ is the dimensionless time, and T =2π/ωn the oscillation period. Dividing the amplitude of the oscillations by the characteristic dimension of the body A ∗ in the direction transverse to the flow b, z0 = bψ, ψ = cos τ = A cos τ, the mass of the b body by the characteristic mass of the surrounding fluid, m∗ = m/(ρb2); and considering ∗ the dimensionless velocity U = U/(ωnb), the expression for the transverse velocity of the body obtained is dz0 = −bA∗ω sin τ. (3.2.7) dt n The dimensionless dissipated power averaged per cycle defined in (3.2.5), using (3.2.6) and (3.2.7) becomes

2π 2 3 ∗2 ˆ ∗ ∗ 2 1 b 2mζωnA 2 2m ζ A pd = 1 3 sin τdτ = ∗ ∗ . (3.2.8) 2 ρU b 2π U U 0

The extracted power averaged per cycle is

T T ˆ 2 ˆ 1 dz0 ρU b dz0 P = f dt = C dt, (3.2.9) e T z dt 2T z dt 0 0 and the dimensionless extracted power averaged per cycle is given by

ˆT Pe 1 dz0 pe = 1 3 = Cz dt. (3.2.10) 2 ρU b UT dt 0

The first step in our study is to prove that the dimensionless extracted power per cycle A∗ depends only on the parameter . To do this, we will consider that the dimensionless U ∗ coefficient of the aerodynamic force transverse to the flow can always be expanded in power series of tan α n Cz(α)= an tan α, (3.2.11) 56 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES so the expression of the dimensionless extracted power averaged per cycle (3.2.10) becomes ˆT ˆ2π ∗ ∗ n 1 dz0 bA ω bA ω sin τ p = a tann α dt = n (−a sin τ) − n dτ = e UT n dt U2π n U 0 0

ˆ2π ∗ n+1 ∗ 2 ∗ 4 1 A n+1 1 A 3 A = a − (sin τ) dτ = a1 + a3 + ... . 2π n U ∗ 2 U ∗ 4 U ∗ 0 (3.2.12) Note that in equation (3.2.12) just the odd terms n =2p − 1 contribute, and it can A∗ also be concluded that p is a function depending only on , making it easier to ﬁnd e U ∗ the maximum of the dimensionless extracted power per cycle, as this only depends on a single parameter 2 2 A∗ A∗ p = f . (3.2.13) e U ∗ U ∗ A∗ A∗ The maximum extracted power, pemax, is obtained for the value of ∗ = ∗ U U pemax that maximizes pe. In equilibrium, the extracted power (3.2.13) and the dissipated power (3.2.8) are equal, pd = pe, from this equation the expression obtained is 2 2m∗ζ A∗ = f , (3.2.14) U ∗ U ∗ or its inverse 2 A∗ 2m∗ζ = g . (3.2.15) U ∗ U ∗ m∗ζ The value determines the combination of parameters that produces the max- U ∗ pemax A∗ imum energy extracted. It can be obtained from (3.2.14) or (3.2.15) once ∗ is U pemax known from (3.2.13). This procedure can also be used even if the coefﬁcients an of the A∗ expansion (3.2.11) are unknown, but the function f can be determined by other U ∗ methods (e.g. experimentally).

3.2.1 Extracted power in a general case

In this section we consider a general case, in which Cz (α) has a complex shape and can not be easily fitted to such a low degree polynomial that a appropriate analytic solution can be obtained. We will show that it is possible to calculate the power extracted directly a from the experimental data. Let us consider just the antisymmetric part, Cz (α), of the dimensionless coefficient of the aerodynamic force transverse to the flow. Expressions (3.2.9) and (3.2.10) can be rewritten as

ˆT ρU 2b2A∗ P = − Ca(α)sinτdτ , (3.2.16) e 2T z 0 3.2. MATHEMATICAL MODEL 57

ˆT A∗ p = − Ca(α)sinτdτ . (3.2.17) e 2πU∗ z 0 From the expression of the transverse speed (3.2.7) it follows ∗ − 1 dz0 − U b 1 dz0 −U sin τ = = = ∗ tan α, (3.2.18) Aωn dt bωn A U dt A

A∗ A∗ Ca(α)=Ca(tan α)=Ca(− sin τ)=−Ca( sin τ) , (3.2.19) z z z U ∗ z U ∗ ∗ A Aωn where = =tanαmaxc is the maximum amplitude of the transverse speed during U ∗ U a period of oscillation divided by the velocity of the incident ﬂow. Hence αmaxc is the maximum pitch angle of velocity (or angle of attack at condition θ0 =0) during one a oscillation period. Considering Cz (τ), deﬁned in (3.2.19), as a sinus series expansion ˆ2π A∗ N A∗ 1 A∗ Ca(τ, )= b sin nτ; b = Ca(τ, )sinnτdτ. (3.2.20) z U ∗ n n U ∗ π z U ∗ n=1 0

a A∗ The integral in (3.2.16) is the term b1 of the series Cz (τ, U ∗ ) in (3.2.20). Equations a (3.2.18) and (3.2.19) show that varying τ, the argument of Cz varies sinusoidally between − tan αmax and tan αmax during the period of oscillation. a n On one hand, using tan α is inconvenient to fit Cz (tan α). Note that tan α is not an orthonormal function and adding terms to the series expansion would not simplify much the study of the convergence. On the other hand, a Fourier series expansion as a A∗ in (3.2.20) fits Cz (τ, U ∗ ) uniformly, because every term is orthonormal to the others, and the contribution of every harmonic term of the series to the total uncertainty has a clear interpretation. In any case, the chapter is focused on the extracted power, which is proportional to the term b1 in (3.2.17), that is 1 A∗ p = b1 . (3.2.21) e 2 U ∗ A∗ Following the procedure that leads to (3.2.14), in order to find the variation of with U ∗ m∗ζ the parameter , expressions (3.2.8) and (3.2.21) are used to impose the condition of U ∗ equilibrium pe = pd 2 1 A∗ 2m∗ζ A∗ b1 = , 2 U ∗ U ∗ U ∗ A∗ A∗ as b1 = b1 , the solution is the value of that satisfies U ∗ U ∗ 4m∗ζ A∗ = b1 . (3.2.22) U ∗ U ∗ A∗ Note the similarity with condition (3.2.14). The maximum of the curve pe ∗ appears U A∗ at ∗ , as is shown in Figure 3.2.2. U pemax 58 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

Figure 3.2.2: Variation of the dimensionless extracted power averaged per cycle, pe, ∗ ∗ A A with the relative amplitude of the motion, ∗ . ∗ is the relative amplitude U U pemax that provides maximum extracted power. m∗ζ A∗ The value of ∗ that leads to ∗ is obtained from expression (3.2.22) U U pemax −1 m∗ζ 1 A∗ A∗ ∗ = b1 ∗ ∗ . (3.2.23) U pemax 4 U pemax U pemax m∗ζ Once the value of ∗ is known, the mechanical design should be oriented to U pemax find the combination of values of mass, damping coefficient, natural frequency and flow speed that matches it. To study the different types of solutions, the extracted and dissipated power per maxi- A∗ mum relative amplitude in a cycle (or specific power) can be used U ∗ ∗ ∗ pe 1 pd 2m ζ A p = ∗ = b1 p = ∗ = , (3.2.24) e A 2 d A U ∗ U ∗ U ∗ U ∗ A∗ so that the shape of the curve b1 determines the possible solutions. The equilib- U ∗ rium points fulfill the condition b1 =2pd.

As shown in Figure 3.2.3a, there are multiple scenarios possible as the slope of pd is increased. The labels in the curves show the different cases: 1) one solution, 2) three solutions (two stable and one unstable) and 3) limit case. Studying the shape of the curves b1 (independently of the dissipation), if no inflection point exists, the solution is unique and stable. If an inflection point appears in the increasing part of the curve, there are two solutions, one stable and another one unsta- A∗ ble (3.2.3b, case 2). If the complexity of the shape of the curve b1 increases, a U ∗ different behavior can appear. The dimensionless dissipated power averaged per cycle and per maximum relative A∗ amplitude, p , defined in (3.2.24), is a linear function of , so that if U ∗ is small d U ∗ 3.2. MATHEMATICAL MODEL 59

A∗ Figure 3.2.3: Variation of b1 with the relative amplitude . a) Possible equilibrium U ∗ solutions: 1) one solution, 2) 3 solutions, 3) limit case. E: stable, I: unstable. b) no inﬂection points (case 1); one inﬂection point (case 2).

∗ or m ζ is big, the slope of pd increases and a limit, in which no stable solution exists, is reached (Figure 3.2.3, case 3). The determination of the maximum dissipated or extracted power from the experi- A∗ A∗ mental results is deﬁned by the maximum of the curve pe ∗ = pe ∗ U U pemax obtained from (3.2.21). The curves of maximum power as a function of the pitch angle of the body, θ, can be obtained as shown in Figure 3.2.4. From the maximum dimensionless extracted power, pemax, for different pitch angles of the body, the curve of maximum efﬁciency can be determined as a function of the pitch angle of the body (Figure 3.2.4).

Figure 3.2.4: Determination of the maximum dimensionless extracted power from experimental results. a) Variation of dimensionless extracted power, pe, as a function A∗ of relative amplitude, , for two values of the pitch angle θ, b) Variation of U ∗ maximum extracted power, pemax, with the pitch angle of the body, θ. A∗ The main advantage of this method is that b1 can be calculated numerically U ∗ using (3.2.20) directly from experimental results without polynomial ﬁtting, which could be of a high order depending on the shape of the curve. In such a case, the possible advantages of an analytical treatment are lost. 60 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

3.2.2 Comparison between numerical integration method and polynomial expansion To validate the numerical integration method and to estimate the inﬂuence of the parameters involved in the uncertainty of the results, the polynomial expressions given in Barrero-Gil et al. [2009] have been used as reference curves of dimensionless coefﬁcient of transverse force Cz over the body as follows

3 5 7 Cz(α)=a1 tan α + a3 tan α + a5 tan α + a7 tan α. (3.2.25)

The coefﬁcients of the polynomials used for validating the method presented here to an- alyst the galloping stability are shown in Table 3.2.1.

Table 3.2.1: Polynomial coefﬁcients used in the validation of the numerical integration method, extracted from Barrero-Gil et al. [2009] for different cases k.

k a1 a3 a5 a7 1 8 -75 0 0 2 1.50 231 -3800 0 3 4.72 -294 10972 -105000 4 2.70 -168 6270 -60000

An analytic expression for b1 can be obtained from (3.2.25) using deﬁnition (3.2.20) or (3.2.12), as follows

3 5 7 1 A∗ 3 A∗ 5 A∗ 35 A∗ b1 = b1 = a1 + a3 + a5 + a7 . (3.2.26) a 2 U ∗ 8 U ∗ 16 U ∗ 128 U ∗

The formula integrated to obtain b1n is: ˆ 1 2π A∗ b1 = b1n = Cz(τ, ∗ ) sin(τ)dτ. (3.2.27) π 0 U

The results from the numerical method b1n are compared in Figure 3.2.5 with the analytic ◦ expression, b1a, (3.2.26). The data Cz (α) are discretized at an interval Δα =1.

The good agreement between the analytical expression b1a and the numerical integration b1n can be appreciated in Figure 3.2.5 for all the cases that were considered.

3.2.3 Influence of the number of discretization points The number of measurement points required in order to properly characterize the dimensionless coefficient curve of the transverse force, Cz, needs to be determined so that integration (3.2.27) can be made with an acceptable error. A study of the influence of the discretization is performed for Δα =0.5, 1, 2, 4◦ (see Figure 3.2.6) in the polynomial curve k =3from Table 3.2.1. In this case, the experimental data cl (θ), cd (θ) are considered, and the simulated experimental force coefficient Cz (α) is taken directly from (3.2.26) at pitch intervals Δα =Δθ.

The speciﬁc dissipation power, pd, used in Figure 3.2.6, has been estimated from the results reported in Alonso et al. [2012], and also from the wind tunnels tests performed at IDR/UPM with similar bodies (see Table 3.2.2). 3.2. MATHEMATICAL MODEL 61

0.6 b1n(k =1) b1n(k =2) b1n(k =3) b1n(k =4) b1a 0.4

0.2

0 0 0.1 0.2 0.3 0.4 A∗/U ∗

∗ ∗ Figure 3.2.5: Variation of coefﬁcient, b1, with the relative amplitude, A /U . Comparison of the analytical solution, b1a, and the numerical integration, b1n, for ◦ different curves of Cz (polynomial functions) around α =0, for the cases k according to Table 3.2.1. Discretization Δα =1◦.

0.6 ◦ b1n, Δα =1 ◦ b1n, Δα =2 ◦ b1n, Δα =3 ◦ b1n, Δα =4 ◦ b1n, Δα =5 b1a 0.4 2˜pd

0.2

0 0 0.1 0.2 0.3 A∗/U ∗

∗ ∗ Figure 3.2.6: Variation of coefficient, b1, with the relative amplitude, A /U . Influence of the discretization interval, Δα, of the force coefficient, Cz,inthe numerical integration of b1. Case k =3of Table 3.2.1.

The maximum of the curve b1 in Figure 3.2.6 corresponds to twice the maximum speciﬁc extracted power, pemax.

In order to estimate the uncertainty due to the discretization of the curve Cz (α), the parameter disc is deﬁned as ∗ ∗ A | − A | ∗ eqa ∗ eqn U U disc = ∗ , (3.2.28) A | U ∗ eqa A∗ where | is the relative amplitude at the equilibrium for a certain dissipation which U ∗ eqa A∗ is obtained analytically from condition (3.2.14) and the analytic expression of b1. | U ∗ eqn is the relative amplitude at the equilibrium which is obtained from condition (3.2.14) and the numerical integration of b1. 62 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

Table 3.2.2: Values used to estimate the speciﬁc dissipation power, pd, extracted from Alonso et al. [2012].

Characteristic transverse dimension of the body, b [m] [0.1-0.3] Structural damping, ζ 0.014 Natural frequency, ωn/2π [Hz] 2.5 Mass, m [kg/m] 2.65 ∗ Dimensionless velocity, U = U/(ωnb) 7.64

As shown in Figure 3.2.7, for a discretization interval Δα =1◦, the uncertainty due to discretization is under 0.3%, and around 10% for a discretization interval Δα =5◦.

0.15

0.1

disc

0.05

0 0 1 2 3 4 5 Δα[◦]

Figure 3.2.7: Variation of the uncertainty in the relative amplitude at the equilibrium, disc, as a function of the discretization interval Δα of the curve Cz (α). Case k =3of Table 3.2.1.

3.2.4 Comparison of the numerical integration method with the polynomial expansion at a point different from the origin.

In the process of validation of the numerical integration of b1 in the study of transverse galloping, the case of a body at a pitch angle different from the origin (θ = θ0 =0 )was considered. A polynomial of third degree has been considered (case k =1, see Figure 3.2.8), where the coefﬁcients are the ﬁrst row in Table 3.2.1. To calculate the value of the analytic expression of b1, b1a, in the offset galloping (α = θ0 + β, θ0 being the new n angle origin), the expression of Cz has been expressed in terms of β , and b1 calculated analytically:

3 2 2 3 Cz(β)=a0 + a1θ0 + a3θ0 +(a1 +3a3θ0)β +3a3θ0β + a3β , (3.2.29) so that the new coefﬁcients to be used in the expression of b1a (3.2.26) are:

a 3 = a3 a 2 =3a3θ0 a 1 = a1 +3a3θ0² a 0 = a0 + a1θ0 + a3θ0³ 3.2. MATHEMATICAL MODEL 63

and from (3.2.26), b1a can be determined ∗ ∗ 3 1 A 3 A b1 = a + a . (3.2.30) a 2 1 U ∗ 8 3 U ∗

From the new displaced curve, Czd, the antisymmetric part Czda is extracted to proceed with the integration.

−1

−2 −0.4 −0.2 0 0.2 0.4 θ [rad]

Figure 3.2.8: Variation of the transverse force coefﬁcient, Cz, with the pitch angle of the body, θ. Symbols: pitch angles analyzed in Figure 3.2.9. Case k =1, see Table 3.2.1.

Figure 3.2.9 shows that by increasing the pitch angle θ0 the shape of the curve Czda changes till the slope at the origin α =0is horizontal (at θ0 ∼ 0.2 rad), and then galloping does not appear.

θ0 =0 θ0 =0.05 θ0 =0.1 2 2 2 Czd Czda

0 0 0

−2 −2 −2 −0.4 −0.2 α0 0.2 0.4 −0.4 −0.2 α0 0.2 0.4 −0.4 −0.2 α0 0.2 0.4 θ0 =0.15 θ0 =0.2 θ0 =0.25 2 2 2

0 0 0

−2 −2 −2 −0.4 −0.2 α0 0.2 0.4 −0.4 −0.2 α0 0.2 0.4 −0.4 −0.2 α0 0.2 0.4

Figure 3.2.9: Variation of the transverse force coefﬁcient, Czd (α), considering different pitch angles, θ0, and the antisymmetric function, Czda, used in the numerical integration of b1. Case k =1, for θ0 angles shown in Figure 3.2.8.

The comparison between the results from numerical integration b1n and the analytic solution b1a (3.2.30) at different values of pitch angle θ0 is shown in Figure 3.2.10, for a discretization interval Δθ =1◦. 64 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

0.6 b1,θ0 =0 b1,θ0 =0.05 b1,θ0 =0.10 b1,θ0 =0.15 0.4 b1,θ0 =0.20 b1,θ0 =0.25 2˜pd b b 1a 1 0.2

−0.2 0 0.1 0.2 0.3 0.4 A∗/U ∗ A∗ Figure 3.2.10: Variation of coefﬁcient b1 with the relative amplitude , obtained U ∗ by numerical integration, b1n, at conﬁgurations with pitch angle θ0 =0 , and b1 ◦ analytic, b1a. Discretization interval Δθ =1. Case k =1in Table 3.2.1. For values of θ0 see Figure 3.2.8.

A good agreement between both solutions can be appreciated, as numerical and analytical results can not be distinguished. However, a small difference between both solutions can be noticed in Figure 3.2.11, which is due to a larger discretization interval Δθ =4◦. The solid lines next to each numerical solution, b1n, are the respective analytical solutions, b1a, for each case.

0.6 b1,θ0 =0 b1,θ0 =0.05 b1,θ0 =0.10 b1,θ0 =0.15 0.4 b1,θ0 =0.20 b1,θ0 =0.25 2˜pd b1a b 1 0.2

−0.2 0 0.1 0.2 0.3 0.4 A∗/U ∗ A∗ Figure 3.2.11: Variation of coefﬁcient b1 with the relative amplitude , obtained U ∗ by numerical integration, b1n, at conﬁgurations with pitch angle θ0 =0 , and ◦ analytical solution, b1a. Discretization interval Δθ =4.

In order to better evaluate the effect of the discretization interval of the wind tunnel measurements, the standard deviation between the analytical solution, b1a, and the numerical one, b1n, is deﬁned as 2 (b1n − b1a) σ 1 = , (3.2.31) b N where N is the number of samples considered N 90 in most of these cases. 3.3. EXPERIMENTAL SET-UP 65

The standard deviation, σb1, is less than 0.5% for discretization intervals less than Δθ =2◦ and around 4% for Δθ =5◦, as can be seen in Figure 3.2.12. The inﬂuence on σb1 of the pitch angle of the body, θ0, is not very relevant.

0.06 θ0 =0 θ0 =0.2 θ0 =0.3

0.04 σ b1

0.02

0 0 1 2 3 4 5 Δθ [◦]

Figure 3.2.12: Variation with the discretization interval Δθ of the standard deviation, σb1, between the analytic solution b1a and the numerical integration b1n due to the discretization of Cz, for different pitch angles θ0. Case k =1in Table 3.2.1.

3.3 Experimental set-up

To explain the application of the method to some practical cases, measurements have been performed in two different wind tunnels, A4C and A9, both property of IDR/UPM. The A4C wind tunnel is an open-circuit, closed-test-section wind tunnel with a two- dimensional contraction. The dimensions of its test chamber are length 1.2 m, width 0.2 m and height 1.8 m. The diffuser behind the test chamber expands and adapts the ﬂow generated by four centrifugal blowers, each of 7.5 kW. The return circuit is the reduced space between the wind tunnel and the walls of the room. The wind speed proﬁle at the model test section was uniform within ±1% and the turbulence intensity was around 7%. The test bodies were positioned using a NEWPORT EPS100 rotating platform, which allows the angle of attack of the airfoil to be set with an uncertainty of ±0.1º. The blockage of the A4C wind tunnel was around 10% for the biconvex cross-section at θ =72º, which is the maximum blockage reached during the test. The aerodynamic loads were measured with an ATI six-component strain-gauge balance, model Gamma SI-130-10 (see Figure 3.3.1). An AirFlow 048 Pitot tube was located at the entrance of the test chamber and was connected to a Druck LPM5480 pressure transducer. The A9 wind tunnel is also an open-circuit, closed-test-section wind tunnel with a two- dimensional contraction. The dimensions of the test chamber are length 2.0 m, width 1.5 m and height 1.8 m. The aerodynamic loads were measured with an ATI six-component strain-gauge balance, model Delta FT5575 (see Figure 3.3.2). The blockage of the A9 wind tunnel was in the range 3-9% for the rhomboidal cross-section at θ =0◦ and θ = 90◦, respectively. 66 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

The wind speed vertical profile at the model test section was uniform within ±1% and the turbulence intensity was around 2.5%. The Pitot tube was connected to a Druck LPM9381 pressure transducer. In both wind tunnels the sampling frequency used was 200 Hz, to comply with the Nyquist criterion, as the maximum frequency in the flow is that of the vortex shedding, f D f , which calculated from the Strouhal number (St = v ) was under 20 Hz. v U In the case of the rhomboidal cross-section, the value of the parameters are D = 0.11 m (characteristic length of the structure transverse to the incident flow), U =14.5 m/s (velocity of the flow in the A9 wind tunnel test chamber) and St ∼ 0.15, and the measuring period was 30 seconds. Note that measurements in both tunnels have not been corrected for blockage. This effect has not been taken into account due to the fact that the objective of this chapter is to show the feasibility of the proposed analysis method.

a) b) Figure 3.3.1: Wind tunnel A4C: a) Instrumentation; b) Detail of the ATI six-component strain-gauge balance (model Gamma SI-130-10).

a) b) Figure 3.3.2: Wind tunnel A9: a) Instrumentation; b) Detail of the ATI six-component strain-gauge balance (model Delta FT5575). 3.4. RESULTS AND DISCUSSION 67 3.4 Results and discussion

In this section the study of the stability for several cylindrical bodies with different cross- section shapes is performed, by comparing the relative position of the curves of extracted and dissipated power. It is based on the integration of the curves Cz obtained from the experimental results measured with a balance in a wind tunnel. The cross-section shapes analyzed were biconvex, D-shape, rhomboidal of different relative thickness, triangles extracted from Alonso et al. [2012] and the square extracted from Ng et al. [2005], Parkinson and Smith [1964]. The pitch angle intervals at which the body is prone to galloping are determined using ∂c (α) the Glauert-Den Hartog criterion H = l + c (α) < 0. The mechanical properties ∂α d can be synthesized by the Scruton number Sc, defined as 2mζ Sc = , ρb2 from the values in Table 3.2.2, it can be deduced that the Scruton number is between 1 and 7 depending on the characteristic transverse dimension of the body. The values of these mechanical properties, along with the stiffness of the spring, have been chosen so that the body motion speed is slow relative to the speed of a fluid particle, and then the hypothesis of quasi-stationary aerodynamic forces can be justified. In all the cases presented hereafter, the discretization interval was Δθ =1◦ =0.017 rad.

3.4.1 Biconvex airfoil After studying some biconvex airfoils of different relative thickness in the A4C wind tunnel (measuring global forces with a balance), it was observed that the airfoil of relative thickness Er = b/c =0.817 (where b is the characteristic transverse dimension and c the chord of the airfoil) showed the greatest galloping intensity [Alonso et al., 2009]. The pitch angles were measured with regard to the plane of symmetry of the airfoil, as is shown in Figure 3.4.1. The Reynolds number during the measurements, based on the chord of the airfoil c =0.2 m, was Re = 2.01 · 105.

Figure 3.4.1: Nomenclature in the case of a biconvex airfoil.

The results from the measurements of cl, cd, and the Glauert-Den Hartog criterion, H, ◦ are shown in Figure 3.4.2, where a region around θ0 ∼ 70 =1.22 rad can be appreciated in which the necessary condition for galloping, H<0, is satisﬁed. 68 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

1.5 cl cd 1 d c

, 0.5 l c 0 −0.5 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 θ [rad] 5

0 H −5

−10 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 θ [rad]

Figure 3.4.2: Variation with the pitch angle of the body, θ, of the coefﬁcients of lift, cl, and drag, cd, and of the Glauert-Den Hartog parameter, H, over the biconvex airfoil of relative thickness Er =0.817 in the A4C wind tunnel. Symbols indicate the pitch angles θ0 considered in the analysis of Figures 3.4.3 and 3.4.4.

Figure 3.4.3 shows the force coefﬁcient, Cz, obtained from the experimental data using (3.2.2) for different values of the pitch angle of the body, θ0. The origin of the pitch angles, ◦ θ0, were displaced around θ =70 =1.22 rad, and the displaced data were interpolated, ◦ CzI, with a discretization interval Δθ =1=0.017 rad. A linear interpolation - a Matlab function - was used.

0.2 θ0 =1.169 θ0 =1.204 0 θ0 =1.239 θ0 =1.274 zI −0.2 θ0 =1.309 C −0.4

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 α [rad]

θ0 =1.169 0.2 θ0 =1.204 θ0 =1.239 θ0 =1.274

zIa 0 θ0 =1.309 C −0.2 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 α [rad]

Figure 3.4.3: Variation of the force coefﬁcient, Cz, with the angle of attack, α, ◦ where α = θ − θ0, for different values of pitch angle θ0, around θ0 =70 =1.221 rad. Curve Cz interpolated, CzI, and its antisymmetric part, CzIa, for the biconvex airfoil of relative thickness Er =0.817.

The possible points of equilibrium for the biconvex airfoil, for different pitch angles of the body θ0, are placed at the crossing points of the b1 curves with the 2˜pd dissipation 3.4. RESULTS AND DISCUSSION 69

◦ line (see Figure 3.4.4). At θ0 ∼ 76 =1.33 rad a limit case is reached and the existence of an equilibrium solution is no longer possible.

0.15 θ0 =1.169 θ0 =1.187 θ0 =1.204 θ0 =1.222 θ0 =1.239 0.1 θ0 =1.257 θ0 =1.274 b1 θ0 =1.292 θ0 =1.309 θ0 =1.326 p 0.05 2˜d

−0.05 0 0.05 0.1 0.15 0.2 0.25 A∗/U ∗ A∗ Figure 3.4.4: Variation with the relative amplitude of the coefﬁcient b1 from U ∗ n the numerical integration, and the speciﬁc dissipated energy, p˜d, for a biconvex airfoil of relative thickness Er =0.817 oscillating around a pitch angle of the body ◦ θ0 =1.221 rad = 70 .

In Figure 3.4.4 it can also be seen that, for the dissipation chosen, a stable solution exists in the range 1.169 ≤ θ0 ≤ 1.326 rad with a relative amplitude of oscillation 0.15 < A∗ < 0.19 ; besides this stable solution, there is one unstable solution in the range U ∗ 1.169 ≤ θ0 ≤ 1.187 rad, and another in the range 1.292 <θ0 < 1.326 rad.

For a given dissipated power, pd, there is a pitch angle, θ, which provides a maximum ∗ ∗ relative amplitude of equilibrium A /U |eq, as shown in Figure 3.4.5.

0.195

0.19 ∗ ∗ A /U |eq 0.185

0.18

0.175

0.17 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 θ [rad]

∗ ∗ Figure 3.4.5: Variation of the relative amplitude of equilibrium A /U |eq as a function of the pitch angle, θ, for a certain speciﬁc dissipated power, p˜d, obtained from the parameters in Table 3.2.2.

Another approach is to use the dissipation as a parameter, so that for every pitch angle, θ, there is a specific dissipated power, pd, (defined in equation (3.2.24)), which maximizes the specific extracted power, pe. The maximum values of pe, pemax as a function of θ are shown in Figure 3.4.6. Note that, despite fact that the maximum of the relative amplitude 70 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES is reached at θ ∼ 71.62◦ =1.25 rad (for a certain dissipated power), the maximum ◦ specific dissipated power, p˜d, (and the extracted, pe) is obtained for θ = 71.05 =1.24 rad, although in the range 1.22 <θ<1.25 rad the dissipated power is close to the maximum value.

0.06

0.05 p˜emax

0.04

0.03

0.02

0.01

0 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 θ [rad] 1 Figure 3.4.6: Variation of the maximum speciﬁc extracted power p˜ max = b1| , e 2 max as a function of the pitch angle of the body, θ.

In Figure 3.4.7 it can be observed that the dimensionless extracted power averaged per cycle, pe, is one order of magnitude smaller than the speciﬁc extracted power due to A∗ the fact that the relative amplitude ∗ that provides the maximum dimensionless U pemax extracted power averaged per cycle pemax is small compared to unity. The maximum of pemax appears at θ0 =1.27 rad and the maximum of pemax appears at θ0 =1.23 rad. This could be useful for designs in which the relative amplitude could be a limiting factor.

−3 4x 10

3.5 pemax 3

2.5

1.5

0.5

0 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 θ [rad]

Figure 3.4.7: Variation of the maximum dimensionless extracted power, pemax,asa function of the pitch angle of the body, θ.

3.4.2 D-shape body The measurement tests of the D-shape cross-section body, which is a semi circular cylinder of angle 190º, were performed in the A9 wind tunnel. The test conﬁguration is shown 3.4. RESULTS AND DISCUSSION 71 in Figure 3.4.8.

Figure 3.4.8: Nomenclature in the case of the D-shape body.

The Reynolds number during the test, based on the radius of the circular part of the D-shape body, b =0.126 m, was Re = 1.05 · 105. In the results for the lift and drag coefﬁcient and the H are shown in Figure 3.4.9. The coefﬁcient b1 is presented in Figure 3.4.10, it can be appreciated that a stable solution ∗ ∼ ◦ A appears around θ0 64 =1.117 rad with a relative amplitude of oscillation ∗ 0.03 ◦ U and it disappears around θ0 ∼ 74 =1.292 rad. At θ0 ∼ 1.204 rad there are three A∗ solutions: two stable solutions with a relative amplitude of oscillation 0.25 and U ∗ A∗ A∗ 0.66 respectively, and one unstable with 0.35. For higher pitch angles only U ∗ U ∗ A∗ one stable solutions exists, of relative amplitude 0.7 < < 0.9. U ∗

2 cl cd d c

, 1 l c

0 0 0.5 1 1.5 2 2.5 θ [rad] 5

0 H −5

−10 0 0.5 1 1.5 2 2.5 θ [rad]

Figure 3.4.9: Variation with the pitch angle of the body, θ, of the coefﬁcients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the D-shape body tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.10). A∗ Note the complex way in which the shape of the curves b1 varies with θ0, show- U ∗ ing the existence of several types of oscillations as the pitch angle changes. 72 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

0.4 θ0 =1.117 θ0 =1.204 θ0 =1.292 0.3 θ0 =1.379 θ0 =1.466 θ0 =1.553 θ0 =1.641 0.2 θ0 =1.728 θ0 =1.815 b 2˜pd 1 0.1

−0.1

−0.2 0 0.2 0.4 0.6 0.8 1 A∗/U ∗ A∗ Figure 3.4.10: Variation with the relative amplitude of the coefﬁcient b1 U ∗ n obtained from the numerical integration and the dissipation, p˜d, for the D-shape body, oscillating around a pitch angle of the body close to θ =80◦ =1.4 rad.

3.4.3 Rhomboidal cross-section bodies The measurement test of the rhomboidal airfoil characteristics were performed in the A9 wind tunnel. The conﬁguration of the body for the tests is shown in Figure 3.4.11. Several rhomboidal airfoils of different relative thickness Er = b/c =10/30, 11/30 and 12/30 were tested. The Reynolds number during the test, based on the chord of the airfoils 5 c =0.3 m, was Re = 1.03·10 . The airfoil of relative thickness Er =11/30 had rounded leading and trailing edges with a ﬁllet radius of ra =0.52 cm, which has an aerodynamic effect.

Figure 3.4.11: Nomenclature in the case of the rhomboidal cross-section airfoil.

Rhomboidal airfoil Er =11/30

Figure 3.4.12 shows the cl(α) and cd(α) coefﬁcients. As shown in Figure 3.4.13 for the chosen dissipation p˜d, in all the conﬁgurations there is only one solution, which is a stable A∗ one, and the range of the relative amplitude at equilibrium is 0.08 < ∼ 0.11 in the U ∗ range of pitch angles 0.37 <θ0 < 0.43 rad.

Rhomboidal airfoil Er =10/30

In this airfoil the intensity of the galloping is higher than for the relative thickness Er = 11/30, as can be appreciated in Figure 3.4.14 (H is more negative than in Figure 3.4.12) 3.4. RESULTS AND DISCUSSION 73

1.5 cl cd

d 1 c , l

c 0.5

0 0 0.5 1 1.5 θ [rad] 4

H 0

−2 0 0.5 1 1.5 θ [rad]

Figure 3.4.12: Variation with the pitch angle of the body, θ, of the coefﬁcients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the rhomboidal airfoil Er =11/30 tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.13).

0.025 θ0 =0.35 θ0 =0.37 θ0 =0.39 0.02 θ0 =0.41 θ0 =0.43 2˜pd 0.015 b 1 0.01

0.005

−0.005 0 0.02 0.04 0.06 0.08 0.1 0.12 A∗/U ∗

A∗ Figure 3.4.13: Variation with the relative amplitude of the coefﬁcient b1 U ∗ n obtained from the numerical integration, and the dissipation p˜d, for the rhomboidal ◦ airfoil of Er =11/30 around θ =23 =0.40 rad tested in the A9 wind tunnel.

and the maximum value of b1 is larger in Figure 3.4.15 than in Figure 3.4.13.

In Figure 3.4.15, it can be seen shown that for the chosen dissipation, p˜d, there is a stable solution in the range 0.34 <θ0 < 0.44 rad, and also one unstable solution in the range 0.43 <θ0 < 0.44 rad. At θ0 =0.34 rad there are three solutions: one stable A∗ A∗ at ∼ 0.02 and a double solution at ∼ 0.045 .Atθ0 =0.44 rad there are two U ∗ U ∗ A∗ A∗ solutions: one stable at ∼ 0.12 and the other one unstable at ∼ 0.06 . U ∗ U ∗ 74 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

1.5 cl cd

d 1 c , l

c 0.5

0 0 0.5 1 1.5 θ [rad] 4

H 0

−2 0 0.5 1 1.5 θ [rad]

Figure 3.4.14: Variation with the pitch angle of the body, θ, of the coefﬁcients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the rhomboidal airfoil Er =10/30 tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.15).

0.025 θ0 =0.34 θ0 =0.36 θ0 =0.38 0.02 θ0 =0.4 θ0 =0.42 θ0 =0.44 2˜pd 0.015 b 1 0.01

0.005

−0.005 0 0.05 0.1 0.15 A∗/U ∗ A∗ Figure 3.4.15: Variation with the relative amplitude of the coefﬁcient b1 U ∗ n obtained from the numerical integration, and the dissipation p˜d, for the rhomboidal ◦ airfoil of Er =10/30 around θ =23 =0.40 rad tested in the A9 wind tunnel.

Rhomboidal airfoil Er =12/30 In this airfoil, the range of pitch angles of galloping is wider than for the other of smaller relative thickness, as can be seen in Figure 3.4.16. In Figure 3.4.17, it can be observed that for the chosen dissipation, p˜d, there is one stable solution in the range 0.32 <θ0 < 0.48 A∗ rad, the maximum of the relative amplitude of equilibrium appears at θ0 =0.46 rad. U ∗ There is also an unstable solution in the range 0.46 <θ0 < 0.48 rad. 3.4. RESULTS AND DISCUSSION 75

1.5 cl cd

d 1 c , l

c 0.5

0 0 0.5 1 1.5 θ [rad] 4

H 0

−2 0 0.5 1 1.5 θ [rad]

Figure 3.4.16: Variation with the pitch angle of the body, θ, of the coefﬁcients of lift, cl, and drag, cd, and Glauert-Den Hartog parameter, H, over the rhomboidal airfoil Er =12/30 tested in the A9 wind tunnel. Symbols indicate the pitch angles θ0 included in the analysis (see Figure 3.4.17).

0.08 θ0 =0.3 θ0 =0.32 θ0 =0.36 θ0 =0.4 0.06 θ0 =0.46 θ0 =0.48 θ0 =0.5 2˜pd 0.04 b1

0.02

−0.02 0 0.05 0.1 0.15 0.2 0.25 A∗/U ∗ A∗ Figure 3.4.17: Variation with the relative amplitude of the coefﬁcient b1 U ∗ n obtained from the numerical integration, and the dissipation p˜d, for the rhomboidal airfoil of Er =12/30 around θ =23º =0.40 rad tested in the A9 wind tunnel.

3.4.4 Triangle cross-section bodies In Figure 3.4.18 the dimensionless extracted power, for the bodies studied above, are presented. In the same Figure the dimensionless power curves calculated from the results reported in Alonso et al. [2012] for triangular cross-section bodies are included. It can be seen that the rhomboidal cross-section of relative thickness Er =11/30 has the minimum capacity of energy extraction from the ﬂow, and this capacity, as in the case with the biconvex airfoil, is negligible compared with the extracted energy of the D-shape and the triangle cross-section bodies. The triangle of β =60◦, where β is the angle of the isosceles triangle (equilateral in this case) seems to be the most efﬁcient cross-section shape. 76 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

Triangle β =30◦ Triangle β =40◦ 0.1 β ◦ pemax Triangle =60 Biconvex Er =0.817 D − shape 0.08 Rhombi Er =11/30

0.06

0.04

0.02

0 0 0.5 1 1.5 2 2.5 3 3.5 θ [rad]

Figure 3.4.18: Variation with the pitch angle, θ, of the maximum dimensionless extracted power, pemax, for different cross-section bodies.

It is worth noting the difference between the results obtained for the maximum specific extracted power, p˜emax (see Figure 3.4.19), and the maximum dimensionless extracted ◦ power, pemax. The triangle of β =60 can extract the highest amount of energy from the fluid flow, and the D-shape cross-section body provides the highest specific extracted A∗ power, p˜ . This is so because the relative amplitude needed to generate such power e U ∗ is smaller than in the case of the triangular cross-section body. This information can be helpful in the design process of a device extracting energy by transverse galloping.

0.2 ◦ Triangle β =30 Triangle β =40◦ p˜emax ◦ Triangle β =60 Biconvex Er =0.817 0.15 D − shape Rhombi Er =11/30

0.1

0.05

0 0 0.5 1 1.5 2 2.5 3 3.5 θ [rad]

Figure 3.4.19: Variation with the pitch angle, θ, of the maximum speciﬁc extracted power, p˜emax, for different cross-section bodies.

3.4.5 Square cross-section bodies The method has also been used to compare the suitability of the 7th and higher order polynomials to ﬁt the experimental data reported by Ng et al. [2005] for square-section cylinders, see Figure 3.4.20.

The results obtained by numerical integration b1n7 of the 7th order polynomial proposed in Parkinson and Smith [1964], and the integration of the experimental results b1ne, are presented in Figure 3.4.21. Around a 5% difference can be appreciated in the maxi- 3.4. RESULTS AND DISCUSSION 77

0.6 7th order polynomial Ng et al. (2005) C 9th order polynomial Ng et al. (2005) z 0.5 11th order polynomial Ng et al. (2005) 7th order polynomial Parkinson and Smith (1964) C data from Parkinson and Smith (1964) 0.4 z C data from Luo and Bearman (1990) z 0.3

0.2

0.1

0 0 5 10 15 α [º]

Figure 3.4.20: Variation of the transverse force coefficient, Cz with the angle of attack, α, experimental data from Ng et al. [2005], Parkinson and Smith [1964], Luo and Bearman [1990] and Parkinson and Brooks [1961]. In this Figure, the 9th and 11th order polynomials of Ng et al. [2005] are coincident. mum value of b1, due to the fact that the fitting of this polynomial is not accurate enough, compare dotted line and circular symbols in Figure 3.4.20. The difference in the position of the inflection point is significant. The position of inflection point is an important characteristic, regarding the dynamical behavior.

0.3 b1n7 b1 2˜pd b 0.25 1ne

0.2

0.15

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 0.3 A∗/U ∗ A∗ Figure 3.4.21: Variation with the relative amplitude of the coefficient b1 U ∗ obtained by direct numerical integration of experimental data, b1ne; by integration of a 7th order polynomial, b1n7; and the specific dissipated power, p˜d, for a square-cylinder. Polynomial coefficients (a1 =2.69, a3 = −168, a5 = 6270, a7 = 59900 at Re∼ 22300) from Parkinson and Smith [1964].

In Figure 3.4.22 a better polynomial ﬁtting of the experimental data (given by Luo and Bearman, 1990) is observed, and the difference between the integration of the exper- A∗ imental data and the polynomials is small in the range of relative amplitude , from 0 U ∗ to 0.25. However, at higher relative amplitudes differences between the results from 7th and 9th order polynomials ﬁttings can be appreciated. It is also shown that 9th and 11th order polynomials give practically the same results. 78 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

0.25 b1n7, b1n9, b1n11, 2˜pd 0.2 b1ne

0.15 b1

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 0.3 A∗/U ∗ A∗ Figure 3.4.22: Variation with the relative amplitude of the coefficient b1 U ∗ obtained by direct numerical integration of 7th, 9th and 11th order polynomial, b1n7, b1n9 and b1n11, respectively; experimental data, b1ne; and the specific dissipated power, p˜d, for a square-cylinder. Experimental data and polynomial coefficients extracted from Ng et al. [2005].

In order to compare the b1 curves for higher amplitudes, the lift cl and drag cd coefficient data for a square cylinder have been obtained from Parkinson and Brooks [1961] because there are defined till angle of attack α =20◦ =0.3491 rad. The experimental values of Cz(α) are obtained from cl and cd data using equation (3.2.2), and have been fitted by 7th, 9th and 11th order polynomials. To do this, a Matlab function has been employed. The results are shown in Figure (3.4.23).

0.6 Data from Parkinson (1961) C 7th order polynomial z 9th order polynomial 0.4 11th order polynomial

0.2

−0.2

−0.4

−0.4 −0.2 0 0.2 0.4 α

Figure 3.4.23: Variation of the transverse force coefficient Cz, with the angle of attack α (from Parkinson and Brooks [1961]) and 7th, 9th and 11th order polynomials that fit the experimental data, obtained with a Matlab function (polyfit), see Table 3.4.1.

For convenience, the coefﬁcients are given referenced to the maximum relative ampli- ∗ −1 A tude (tan ∗ =αm=0.3491) as follows U max n n n α α Cz(α)= anαm = an αm αm 3.4. RESULTS AND DISCUSSION 79 and the results are listed in Table 3.4.1.

Table 3.4.1: Coefficients of 7th, 9th, 11th order polynomials obtained with Matlab function polyfit for the fitting of the experimental data Cz(α) obtained from Parkinson and Brooks [1961].

a1 a3 a5 a7 a9 a11 0.27 3.79 −8.63 4.17 0 0 1.15 −8.72 38.61 −60.90 29.49 0 1.25 −11.01 52.89 −95.62 65.98 −13.79

The 7th order polynomial shows an “out of phase” behavior, as the curvature has variations opposite to the trends of the experimental curve. The behavior is also shown by comparing the signs of coefficients ai in Table 3.4.1. The 9th and 11th order polynomials fit remarkably well the trends of the experimental points, except the last points, where some overshoot is present. In Figure 3.4.24 a good agreement between b1 obtained from integration of both 9th and 11th order polynomials, b1n9 and b1n11 respectively, and the integration of the experimental data, b1ne, can be shown. The integration of the 7th order polynomial, b1n7, does not show a good agreement in this case (see also Figure (3.4.24)). Concerning the square cylinder case, some conclusions can be drawn. Although the results from polynomial fitting and direct numerical integration from experimental data, b1n7 and b1ne, respectively, shown in Figure 3.4.21 does not coincide, the qualitative variation seems to be in agreement. Therefore the analysis of the stability presented in Section 4 would not be compromised, at least qualitatively. However, if the amplitude range is increased (see results in Figures 3.4.23 and 3.4.24), the 7th order polynomial gives clearly trends and values different from both the experimental results b1ne and polynomial fittings b1na and b1n11. Furthermore, both b1n9 and b1n11 results start to fail close to the ends of the amplitude range (see Figure 3.4.23) while the fitting was quite satisfactory in Figure 3.4.20, for smaller amplitude range.

0.25 b1n7, b1n9, b1n11, 2˜pd 0.2 b1ne

0.15 b1

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 A∗/U ∗ A∗ Figure 3.4.24: Variation with the relative amplitude of the coefficient b1 U ∗ obtained by numerical integration of 7th, 9th and 11th order polynomial, b1n7, b1n9 and b1n11, respectively; experimental data, b1ne; and the specific dissipated power, p˜d, for a square-cylinder. The lift cl and drag cd coefficients to obtain Cz were taken from Parkinson and Brooks [1961], and Cz(α) is shown in Figure 3.4.23. 80 CHAPTER 3. ENERGY EXTRACTION FROM AERODYNAMIC INSTABILITIES

Therefore, it can be deduced that no rule can be given for the order of polynomial to be used in a general case, taking into account the amplitude range, and the suitability should be assessed by qualitatively feeling, in a case-by-case basis. This inconvenience is not shown by the method based on the direct numerical integration of the experimental results, presented in this chapter, as an intermediate polynomial fitting procedure is not needed. Once the aerodynamics behavior of the body shape is characterized, the next step would be to control the damping to assure the maximum efficiency to extract energy from the flow, this could be done using active magnetic suspension (see Figure 3.4.25). This device would composed of: a body shape (1), a hollow cylinder beam (2), springs (3) from which the body shape attached to an axle (4) will be hanging, and then the force induced in the coil (5) could be tuned by modifying the current that passes through the A∗ coils in the hollow cylinder (6) depending on the relative amplitude , to assure that it U ∗ is always working on the maximum of the b1.

Figure 3.4.25: Damper device based on transverse galloping with active magnetic suspension.

3.5 Conclusions

In this chapter a new method to analyze the stability of transverse galloping based on stationary measurements performed in a wind tunnel has been presented. This method does not make use of a polynomial fitting of the experimental data. The influence of the discretization interval of the measurements in the uncertainty of equilibrium positions has been analyzed and quantified. The numerical method presented has been used to analyze the stability of cylindrical bodies with different cross-sections (biconvex, rhomboid, D-shape, triangles and squares), and their respective extracting power capabilities have been compared. Most important, the way of finding the maximum effciency in extracting power has been found for a general case. 3.5. CONCLUSIONS 81

By analyzing the extracted power curves of a body in the neighborhood of the points prone to present galloping (which are only dependent of the aerodynamic characteris- A∗ tic of the body), the value of the relative oscillation amplitude that gives rise to the U ∗ maximum extracted power can be determined, and as a consequence the associated me- m∗ζ chanical parameter can be deduced. As it is a combination of the values of wind U ∗ speed, spring stiffness and damping coefficient, the appropriate values of these variables required to maximize energy extraction from the flow can be determined. In this way, the design process is decomposed into two independent phases and thus is considerably simplified. It has been found that same differences appear in both the maximum extracted power and in the relative amplitude at equilibrium, between the results of the direct integration of experimental data and those of the integration of the polynomial fitted to the same experimental data. These differences appear because the polynomial fitting is an approximation to the real curve Cz (α) and thus, if the polynomial fitting is no good enough, differences come out. In this regard, the new method avoids problems related to polynomial fitting, for in- stance the polynomial order suitable for a given application. Besides, for the determination of maximum extracted power, the associated analytical treatment using a high order polynomial is not practical. Furthermore, if the polynomial fitting is not enough accurate the prediction of the dynamical behavior of the analyzed bodies can be wrong.

Chapter 4

Interior noise prediction in high speed trains

4.1 Introduction

Noise is a key issue during high speed trains design process as environmental requirements for railway operations are becoming more and more restrictive. The noise generated by these different sources can be grouped according to two types of phenomena: noise generated by flow over structural elements (vortex shedding: pantograph and equipment, cavity noise: recess of the pantograph, inter-coach spacing, louvers, bogie, ventila- tors) and noise generated by turbulent flow (turbulent boundary layer: surfaces; boundary layer separation: nose of the power car; unsteady wake: rear power car) [Talotte, 2000]. In modern high speed trains the aerodynamic noise can become dominant above 300 km/h, over wheel-rail noise, engine, gearbox, air conditioning, etc. The experimental characterization of the airborne noise generated by the flow around the train becomes an important step to obtain accurate prediction models. Measurements can be used to validate numerical simulations, for near-field propagation and also for the integration of these aeroacoustic sources within algorithms for far-field propagation. A summary of methods to evaluate the acoustic performance of a rail vehicle at the design state is given in Eade and Hardy [1977]. In this paper a group of correlations to estimate the sound pressure level SPL at a certain speed, based on the SPL at a different speed, are proposed. Among the methods used in noise source identification on trains, the microphone array technique is the most common. The pass-by of the train is usually measured with an array of microphones in Talotte et al. [2003], Gori et al. [2001]. Post-processing techniques like beamforming, least squares, deconvolution, diagonal removal methods are commonly used to find the noise source. Some simplifications have to be done in order to be able to manage the problem due to its complexity (Doppler effect shift in frequency, amplitude attenuation in the received signal, turbulent boundary layer around the microphones and

83 84 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS mock-up supports and ground effect) which modifies the propagation, and the directivity of the sources. An example of the use of the microphone array technique in wind tunnels to locate and quantify the aeroacoustic sound sources is presented in Lauterbach et al. [2011]. Experiments in anechoic wind tunnels have been done to get information of the flow behavior around the pantograph and the inter-coach spacing in Kurita et al. [2006], Noger et al. [1999], Fredmion and Vincent [2000]. Measurement of sound pressure level in open field, in viaduct and in a tunnel with ballast and slab track are presented in Choi et al. [2004]. In Remington [1987b,a] a theoretical model of the wheel-rail interaction and its sound radiation is presented and validated. The main parameters taken into account in the wheel/rail model are: wheel and rail radius, roughness, wheel/rail area filter, number of axes on the train, train speed, contact stiffness, rail density, rail cross-section area, radiation efficiency, ground effects, rail loss factor and rail impedance. The noise generated by turbulent flow phenomena, or flow-induced noise can be classified in: vortices, turbulent eddies, vortex shedding, turbulent boundary layers, boundary layer separation and rotating surfaces in a fluid (propellers, fans). In external aeroacoustics the turbulent flow interacts with a rigid body that radiates noise in free space, as is the case of the pantograph, bogie and separated flow regions. In the train pass by, at 25 meters from the rail, the bogie cavity noise and wheel-rail noise are the principal noise sources. Due to the nature of the sound, the low frequency component of their noise spectrum reaches larges distances before getting attenuated. The structural acoustic process can be subdivided into four main stages: generation, transmission, propagation and radiation. The generation comprises the origins of the oscillation, the transmission includes the transfer of energy from the generation mechanism to a (passive) structure. During the propagation the energy is distributed throughout the structural subsystem. Any structural vibrating part in a fluid environment will inject power to that fluid [Cremer et al., 2005]. In Hubbard [1991] an overview of the methods used to predict interior noise in an aircraft, and a classification of the noise sources for interior and flyover noise prediction, are presented. This technical report also mentions the noise regulations and criteria, and recommended practices detailed in FAR 36, which used the effective perceived noise level, EPNL. In the prediction of noise inside the vehicle, to evaluate the transmission through the structure it is necessary to determine accurately the airborne and structure borne load to be able to compare the test with the vibro-acoustic response of the model . A detailed description of the modeling process of the interior sound field of a entire railway vehicle using statistical energy analysis is validated with scale (1:5) measurements in Forssén et al. [2011]. The prediction of the noise inside the train with vibro-acoustic models is used in the design of the noise control treatments (NCT), before the final train configuration is frozen. An example of trim optimization can be found in Sapena and Blanchet [2009]. The vibro- acoustic behavior of the aluminum extruded structures used on railway vehicles is studied in Xie et al. [2003], Xie et al. [2006] and Kohrs and Petersson [2009]. A full-scale test with microphone measurements flush to the train exterior surface and results of the transmissibility through some junctions between structure and the trim is reported in Sorribes-Palmer et al. [2014]. The results are used to validate the turbu- 4.2. NUMERICAL ANALYSIS 85 lent boundary layer model used in the vibro-acoustic models for interior noise prediction [Cockburn and J.E., 1974]. The characteristic train section analysis method used in this paper is justified previously in Sorribes-Palmer et al. [2013b]. In Gardonio [2002] a review of techniques used in aerospace vibro-acoustic control are presented. One of the used isolator is the “fluid mount” which achieves a compromise between the static and dynamic stiffness by providing additional damping produced by the flow of a fluid through an orifice between two chambers. In Campolina [2012] the effect of the porous layer compression is addressed, and re- ductions measurements of the TL up to 5 dB for a 50% compression of the porous thickness in the mid-frequency range (800 Hz) are reported. A classification of the common control methods used in aircraft is presented in this cited thesis. Porous materials and damping treatments are among the passive ones; the active group is represented by noise canceling techniques and smart-foams. This chapter is organized as follows, first a description of the basic definitions and the methodology used to build the vibro-acoustic models to evaluate interior noise is presented in section 2. A description of the experimental set-up to characterize the airborne and structure borne noise sources is presented in section 3. The tests results from the measurements are described in section 4. The results of the test are presented and compared with the numerical simulations in section 5. Finally the conclusions of the work performed are summarized in section 6.

4.2 Numerical analysis

4.2.1 Basic acoustic concepts and definitions First we introduce some concepts and definitions of acoustics before describing the vibro- acoustic models. The variation of the atmospheric pressure is called Sound Pressure Level, (SPL or Lp), its expression is prms SPL[dB] = 20 log10 , (4.2.1) pref where prms is the root mean squared pressure variation ˆ T 1 2 prms = p(t) dt , T 0 −6 and the pressure reference is pref =20· 10 Pa. The expression for the SPL (or Lp) equivalent, Leq, in a period of time T , is defined as 1 Li Leq =10log t 10 . (4.2.2) 10 T i

where i refers to the frequency band of the corresponding Li to be added. Sound Power Level is deﬁned as W Lw[dB]=10log10 , (4.2.3) Wref 86 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

−6 where the standard reference power level is Wref =20· 10 W. The expression to convert Lw to Lp in a room is

− Qθ 4 Lp =Lw 10 log10( 2 + ) , (4.2.4) 4pr Rc where Qθ is the dimensionless directivity factor of the source in the direction of r, and Rc Sαav is the room constant (Rc = , where S is the total area surface of the room, and 1 − αav αav is the average absorption coefficient in the room). The directivity factor is defined 2 as the ratio of the mean square sound pressure pθ at angle θ and distance r from the 2 actual radiating source to the mean square sound pressure ps at the same distance from a nondirectional source radiating the same acoustic power W . Sound Intensity is defined as the continuous law of power carried by a sound wave through an incrementally small area at a point in space. In an environment in which there are no reflecting surfaces, the sound pressure at any point in any type of freely traveling wave is related to the maximum intensity Imax [Vér and Beranek, 2006] through the relationship

2 prms = Imaxρc , (4.2.5) where prms is the root mean square of the acoustic pressure ﬂuctuations, ρ and c are the density of air and the speed of sound in the air respectively. Wavelength, λ, is deﬁned as the distance that a pure-tone wave travels during a full period and is given by c λ = cT = , (4.2.6) f where T is the period and f the frequency. The center, lower and upper band frequencies of the exact proportional 1/n-th octave bands are presented in Table 4.2.1 (see Kinsler et al. [2000]):

Table 4.2.1: Deﬁnition of lower, center and upper frequencies in 1/n-th of octave frequency bands. n is the desired fraction of octave (n must be an integer greater than or equal to 1), m is the index of the band (m can be a positive, zero or negative integer), and f0 is a ﬁxed reference frequency (set to 1000 Hz).

fl fcenter fu −1/2n m/n +1/2n 2 fcenter 2 f0 2 fcenter

The A-weighting is a curve deﬁned in the International standard IEC 61672:2003 which takes in consideration the sensitivity of the human ear to the sound pressure level. The perception of loudness is related to both the sound pressure level and duration of a sound. The result of applying A-weighting to a SPL is abbreviated as dBA. An acoustic indicator used in aircraft industry is the Speech Interference Level (SIL), computed from the average of acoustic pressure levels in the octaves of 1, 2 and 4 kHz; and overall sum of pressure levels between 44 and 11.3 kHz in dBA. Another important concept to understand the behavior of wave propagation is the phase velocity of the wave, cph. An observer traveling in the direction of wave propagation at this speed would see no change of phase [Fahy, 1985]. The wavenumber, k, is the 4.2. NUMERICAL ANALYSIS 87 magnitude of a vector quantity that indicates the direction of propagation as well as the spatial phase variation, wave phase change per unit distance (k = ω/cph =2π/λ, where ω is the angular frequency in rad/s). Since a harmonic wave is an idealization, any wave is really a packet of waves, with frequencies and wavenumber related by a dispersion relation and localized within a beginning and an end. The packet does not necessarily travel with the phase speed, but with the group speed. The group speed of an almost harmonic wave, νg, with a spectrum concentrated near a dω single wave number k0 can be approximated by νg = . If group velocity varies dk k=k0 with frequency then wave is said to be “dispersive”. The propagation speed of the waves will be used to determine the required mesh characteristic length in FEM and BEM. This concept is important for SEA because is related to the speed at which energy is propagated by a wave. The frequency independent phase 2 speed of quasi-longitudinal waves in a plate is cl = E/ρ(1 − ν ), where E is the Young’s Modulus, and ν is the Poisson modulus. The acoustic meshes were re-meshed in order to reduce the time to calculate the radiation loss, using the expression

cph lc = , (4.2.7) Nfmax where N is the number of elements per wavelength, which we have kept over 6, and fmax is the maximum frequency in which an accurate result is required. As the job done has been more focused on SEA model, due to its fast and simple way to evaluate modiﬁcation and perform sensitivity analysis, some SEA concepts are introduced, as follows. Modal density, n(f), is the number of modes in the band N(f), divided by the bandwidth Δf

N(f) n(f)= =2πn(ω) . (4.2.8) Δf The modal density is an important parameter in SEA because the energy exchange between subsystems is proportional to its modal density. Modal overlap factor, M(ω), is related to the modal density and the damping loss factor (DLF), η, which is a measure of the amount of dissipation in the subsystem. The modal overlap factor is the ratio of the width of a resonant peak to the average modal spacing, given by the expression

M(ω)=ωηn(ω) . (4.2.9)

For M<1 deterministic methods can be used (e.g., FEM, BEM), but if M>1 statistical methods are more suitable [Wijker, 2009]. Damping Loss Factor (DLF) is deﬁned so that the half power bandwidth of a resonant mode is ωη, it is a measure of the width of the resonant peak, and is twice the critical damping of a structure, ζ,(η =2ζ). It is composed of three independent damping mechanisms, and it can be written as [Chen et al., 2011]:

η = ηs + ηb + ηr , (4.2.10) 88 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

where ηs is the structural loss factor due to the material inner friction of subsystem ( −4 ηs ∼ 10 for aluminum), ηb is the loss factor attributed to the border connection damping among subsystems (it is extremely small), and ηr is the loss factor formed by acoustic radiation of the subsystem, given by

ρcσ η = , (4.2.11) r ωm where σ is the radiation efficiency, m is the mass per unit area. For a panel of uniform density ρs and thickness h, the expression for the radiation loss is ηr =(ρ/ρs)(1/kh) σ and rarely exceeds 10−3 for engineering structures vibrating in air [Fahy, 1985]. The radiated power is determined by the acoustical coupling between the vibrating structure and the surrounding fluid. In Fahy and Gardonio [2007] the radiation efficiency from 0 to 5000 Hz for a plain, line-stiffened and a corrugated plate are presented.

Coupling Loss Factor (CLF), ηij, governs the power transfer between subsystems i and j. It will depend upon the properties and nature of the subsystem; and on the way they are coupled to other subsystems. For the bending motion of two plates that are connected along a line is given by Lyon and DeJong [1995]

cgili ηij = τij , (4.2.12) ωπSi where cgi is the group speed (speed at which energy is transported by a wave, cg = ∂ω/∂k) for the incident wave type in plate i, Si the area of plate i, li the length of the line connection, and τij is known as the diffuse wave transmission coefﬁcient, which depends on the reﬂection and transmission of the waves at the junction. The CLF from the cavity to the structural subsystem, taking into account the reci- procity theorem (niηij = njηji) can be written as

ns σρc ns ηcs = ηsc = , (4.2.13) nc ωρs nc where ηsc is the CLF from the structure to the cavity, ns is the modal density of the structural subsystem and nc is the modal density of the acoustic cavity. At low frequencies ηsc can differ signiﬁcantly from ηcs because of the amount of modes in the air is higher than in the plate.

In a Diffuse Acoustic Field (DAF) the energy density is the same through all the space, and all directions of propagation are equally probable.

The reverberation time TR is deﬁned as the time in seconds required for the level of sound to drop by 60 dB. The modal reverberation time Tn is that for which the sound pressure decays in that mode by 60 dB.

The average sound absorption αav, can be calculated with the Sabine formula for the TR ofaDAF

24 ln 10 V TR = . (4.2.14) c Sαav where V is the volume of the room. The DLF of an acoustic cavity can then be calculated from the modal reverberation time spectrum with the expression: 4.2. NUMERICAL ANALYSIS 89

ln106 cα S η = = av . (4.2.15) ωTn 4ωV A common spectrum of an interior trimmed cavity of a vehicle is shown in Figure 4.2.1.

Figure 4.2.1: DLF of a common trimmed interior cavity of a vehicle.

The transmissibility can be used to characterize the behavior of the junction in the vibro-acoustic model. A scheme of the SEA model of the junction is presented in Figure 4.2.2.

Figure 4.2.2: Two plates connected through a point junction.

The point force mobility of a inﬁnity plate driven in the center is given by Fahy [1985]:

1 v 1 Y = = = , (4.2.16) Z F 8 Bpρh where Bp is the bending stiffness per unit width, and ρ is the density per unit length 3 2 of the plate. For a flat plate the bending stiffness is Bp = Eh /12(1 − ν ), where h is the thickness. This expressions are used to determine the power input supplied to a subsystem idealized as infinite plates, using the average velocity from the accelerometer measurements located in the subsystem. A common mobility spectrum of a high speed train floor structure is shown in Figure 4.2.3. The expression proposed by Craik and Smith [2000] for a structure-to-structure CLF through a point junction is:

Re {Y2} η12 = 2 , (4.2.17) 2πfm1|Y1 + Y2| 90 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

Figure 4.2.3: Mobility Y = |v/F| of a common structure used as ﬂoor in high speed trains.

where Y1 and Y2 are the mobilities of the subsystems 1 and 2 respectively, m1 is the mass of the subsystem 1, and Re is the real part of the mobility. If the force and velocity are assumed to have a harmonic time dependence the input power can be written as ˆ 1 T 1 v2 Πin = F (t)v(t)dt = . (4.2.18) T 0 2 Re {Y2}

The average power W per frequency band introduced by a point force into a weakly damped ﬂat plate [Cremer et al., 2005] is given by

1 2 π ΔN W = |FΔ| , (4.2.19) 2 2Sm Δω where the mobility is given by

π ΔN Re(Y )= , (4.2.20) 2Sm Δω with m being the mass per unit surface. If the system 2 is highly damped the expression

2 |v2| Re(Y1) W12 c21 2 = , (4.2.21) |v1| Re(Y2) Wd2 m2ωη2 can be considered as a guide for measurements of the transmissibility τ, ratio between the power transmitted and power incident. The transmissibility at the point junction between the structure and the internal acoustic lining is studied using ﬂat panels with equivalent ﬂexural behavior. The expression for transmissibility is given by 2 Re( 1 ) 2 − 2 Πtransmitted V3 Y3 V3 ρ3 E3 I3 (1 ν1 ) τ = = 1 = 2 . (4.2.22) Π V1 Re( ) V1 ρ1 E1 I1 (1 − ν ) incident Y1 3

For an extruded aluminum I-shape of 3 mm thickness connected by a point junction to a 3 mm ﬂat plate of SMC is of order 4.2. NUMERICAL ANALYSIS 91

2 2 V3 1 1 V3 τ = 2 · 7 · · 1 ∼ . (4.2.23) V1 80 3 V1 Damping Loss Factor (DLF) between two subsystems is approximated by the expression

1 η12 ∝ τ12 , (4.2.24) ωn1 where n1 can be approximated by the modal density of the equivalent ﬂat plate to the extruded aluminum I-shape. The expression to evaluate the effective transmission loss between two acoustic cavities (subsystems 1 and 3) separated by a subsystem 2 is A2ω E1 − n1 TL =10log10 2 2 , (4.2.25) 8π n1c1η3 E3 n3 where A is the effective transmission area of the junction, Ei and ni are the energy and the modal density of subsystem energy i, respectively, and η3 is the CLF of subsystem 3. The TL of a panel can be divided in 4 different frequency ranges (see Figure 4.2.4). The stiffness controlled region which is related to very low frequencies. The resonance controlled region, in which the TL shows a very irregular behavior. The mass controlled region, which is the most important because affects audible frequencies in heavy panels. The mass law (R 20 log 10(mf) − 42 dB) implies that doubling the mass or the frequency the TL is increased by 6 dB (using single wall to obtain high TL is expensive). In the fourth region, the speed of propagation of ﬂexural waves in the panel coincides with the propagation speed of the waves in air. The critical frequency (or lowest coincidence frequency) can be calculated by

2 c mp fc = , (4.2.26) 2π Bp where mp is the mass per unit surface in the panel.

Figure 4.2.4: Variation with frequency of the TL of a panel .

In order to increase the TL of the internal acoustic linings, different fibers and foams can be applied. The main parameters used to model the vibro-acoustic performance of 92 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS an elastic-porous material can be classified in fluid physical (fluid density, fluid speed of sound, kinematic viscosity, specific heat ratio, Prandtl number), fluid bulk properties (flow resistivity, porosity, tortuosity, viscous and thermal characteristic lengths), elastic bulk properties (bulk density, bulk Young’s modulus, Poisson’s ratio, loss factor). The fluid physical properties used to describe the vibro-acoustic behavior of a foam can be found in tables of fluid properties in the literature [Allard, 1993]. The Prandtl number, Pr, of a fluid is defined as the ratio of the momentum diffusivity to the thermal diffusivity, Pr = cpμ/κ where κ represents the coefficient of thermal conductivity, and μ, the dynamic viscosity of the fluid. The flow resistivity, σr =Δp/(vΔx), is a measure of the resistance to fluid flow through the porous material, where Δp is the static pressure differential across a layer of thickness Δx, and v is the velocity of flow through the material. Porosity, φ = Vfluid/Vporous, is defined as the fraction of the porous material (Vporous) that is filled of fluid (Vfluid). The tortuosity, α∞ = φrfluid/rfoam, is a dimensionless quantity relating the average fluid path length through the material normalized by the thickness of the material, rfluid is the resistivity of a conduction fluid, and rfoam the resistivity of the material saturated with the conducting fluid. The viscous and thermal characteristic lengths are average macroscopic dimensions of the cells related to viscous and thermal losses, respectively. The first is also related with the average radius of the smallest porous and the second with the biggest ones. Power Spectral Density (PSD), describes how the power of a signal is distributed over frequency ˆ T lim 1 2 PSD(x)= →∞ x (t)dt. (4.2.27) T 2T −T The amplitude is normalized to 1 Hz bands. PSD can be then converted from one bandwidth ΔfB to other ΔfA with the expression [Vér and Beranek, 2006]

2 2 ΔfB prmsB = prmsA . (4.2.28) ΔfA The energies of SEA subsystems are presented in Table 4.2.2, where L is the characteristic length, A the area, V the volume,I is the moment of inertia, andcB the bending √ EI √ EI wave group speed (in a beam c = ω 4 ) and in a plate c = ω 4 ). Bb ρA Bp (1 − ν2)ρA

Table 4.2.2: Energy and modal density of SEA subsystems [Wijker, 2009].

Subsystem Energy, E Modal density, n(ω) L Beam 2 v m πcBb A 12(1 − ν2)ρ Flat plate 2 Et2 p2 ω2V ωA L Acoustic cavity V + + ρc2 2π2c3 8πc2 πc

As it can be appreciated in Table 4.2.2 the modal density of a flat plate is independent of the frequency, meanwhile in a bar is proportional to ω−1/2. The Power Injection Method (PIM) is a straightforward approach to obtain the entries of the frequency band averaged energy influence coefficients (EIC) matrix, and consists in 4.2. NUMERICAL ANALYSIS 93 injecting power into one subsystem at a time (by applying an excitation to the subsystem) and then measuring or calculate the energy responses of the various subsystems. By exciting each subsystem one after the other it is possible to determine the columns of the EIC matrix. There are various ways that the power injection method can be implemented. The differences between the different implementations relate to the way that the input power and energy are estimated for a system [ESI, 2014]. The basic concept of SEA power balance equations, is that the power input from external source into a subsystem is partially dissipated through damping in the subsystem (Pi,diss), and partially transmitted to other subsystems (Pij):

N E E Π =Π +Π = ωη E + ωη n i − j . (4.2.29) i,in i,diss ij i i ij i n n j=i i j The SEA hypothesis of weak-coupling implies that each subsystem should ideally differ substantially in dynamics properties from its immediate neighbours, so that vibrational/acoustic waves are rather strongly reﬂected at its mutual interfaces [Fahy, 2004]. Any wave reﬂected back to the interface are uncorrelated with the incident waves at the interface. This condition is promoted by high modal density, irregular shape, broadband excitation and multiple, uncorrelated input forces.

4.2.2 Methodology A characteristic section of a train wagon with an equivalent structural and acoustic behavior is modeled to analyze the efficiency of the Noise Control Treatment (NCT). In the first steps of design of a vehicle the body in blue, with no NCT, is analyzed and different NCT are proposed and studied before the final configuration is finally frozen. The train section has been modeled using the most common structures in high speed trains, curved stiffened panels, and also extruded (truss-like cores) and ribbed ones, and multilayer glass windows, derived from the examples found in the bibliography [Forssén et al., 2011, Xie et al., 2006, Orrenius et al., 2010, Králíek and Dupal, 2007].

4.2.3 Structural and fluid models The response of an elastic structure to an acoustic load is a coupled problem in which the pressure of the fluid imposes a load to the structure, and at the same time the movement of the structure acts like a load to the fluid. The boundary condition applied in the interface fluid-structure is the conservation of the perpendicular speed to the surface. The models are separated in parts with similar structural properties, a modal analysis was performed to identify the subsystem partitions. The truss-like cores that compose the structural box were separated in external and internal flat aluminum skins and the core, composed by a set of interconnected aluminum panels which form trapezoidal like sound chambers within the structure. This separation of interior and exterior domains makes easier the connection to the exterior and interior air cavities. Many models were built during the process of validation of the characteristic train cross-section analysis method, due to the difficulty of imposing the appropriated boundary conditions, not only in the structure but also in the air acoustic cavity faces. Finally, the most reliable and conservative, from the point of view of predicted interior noise was selected. The structure box is pinned, and the faces in the acoustic cavity that delimitates the cross-section of the 94 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS train have been modeled as periodic (rigid wall), this approach is highly conservative as the pressure inside is over-predicted. The possible leakage through gaps in gangways on windows has been neglected, this analysis has been considered to be more appropriated in later stages of design. The open field vibro-acoustic models for low, middle and high frequencies ranges are shown in Figure 4.2.5. The Sommerfeld radiation condition of non incoming waves from the infinite is implemented by using a semi-infinite fluid, (SIF) condition, which tries to act as a perfectly matched layer with absorbing boundary conditions. The model have been implemented in the trusted commercial software VAOneESI [2014].The properties of the material used in the models are summarized in Table 4.2.3 (these values were extracted from bibliography [MIL, 2003]).

Table 4.2.3: Models material properties. HPL: High Pressure Laminate; SMC: Sheet Molding Compound; EPDM: Ethylene Propylene Diene Monomer; PVB: Polyvinyl Butyral.

3 Material ρ [kg/m ] E [GPa] ν cl [m/s] Aluminum 6061 2710 69 0.33 5355 Plywood 700 6 0.25 3024 Tempered glass 2500 48.5 0.24 4537 Hard rubber 1100 2.3 0.49 1659 HPL 1350 9 0.5 2981 SMC 1950 10.5 0.25 2397 EPDM 930 6 · 10−3 0.5 93 PVB 1070 2.28 0.42 1608

In Table 4.2.4 the properties of the ﬁbers used in the models are presented (these values were extracted from bibliography [Allard, 1993], [VAO, 2014]).

Table 4.2.4: Foam properties.

Material Polyester foam Light glass ﬁber Density [gr/m2] 40 700 Flow resistance [N·s/m4] 4000 17000 Porosity 0.99 0.99 Tortuosity 1.03 1.03 Viscous characteristic length [mm] 0.32 0.2625 Thermal characteristic length [mm] 0.16 0.12

2 Also a blanket of 10 mm thickness, density 2150 gr/m , with a Rw ∼ 33 dB was applied at the ﬂoor also as a NCT.

Figure 4.2.5: Open ﬁeld vibro-acoustic models. 4.2. NUMERICAL ANALYSIS 95

A scheme of the methodology used to model the different parts of the model is shown in Table 4.2.5. The airborne load is introduce as turbulent boundary layer (TBL), or a pressure constrain in a volume if measurements were available.

Table 4.2.5: Methods used to model, depending on the frequency range.

Frequency range Model part Low Middle High Structure FEM FEM/SEA SEA Air FEM/BEM SEA SEA

The train section structure has been subdivided in 6 different parts: roof, ﬂoor, side wall under and over the window frame, the window frame and the window. The damping use in the FE structure was 1% of the critical damping.

In the FEM model the mesh length has been calculated using the phase propagating speed of ﬂexural waves in a thin plate, cB, of the corresponding material, and keeping the number of elements per wavelength between 5 and 10. Meanwhile, in BEM this number has been kept always over 6 elements per wavelength. As the speed of sound in the air is smaller, the length of the acoustic mesh required is coarser, but this is in conﬂict with the required mesh length to resolve the TBL, in which the convective air speed is used to calculate it. Both FEM and BEM analysis were performed in 1/24 of octave frequency scale.

As the frequency increases, some of the train section parts can be substituted by SEA equivalent panels, thus reducing drastically the computing time. The ﬁnal substituted parts were the roof, ﬂoor, windows and the up window frame structure, because the structure under the window frame does not have 3 modes per band till 500 Hz. All the interior NCT were modeled as SEA in the hybrid model.

The modal densities, DLFs, and CLFs of the SEA equivalent plates in the hybrid FE- SEA model were obtained from the analysis of the FE model, and applied in the hybrid area junctions. The number of modes per frequency band in 1/3 of octave in the SEA plates was higher than 3 modes in the frequency range under study. The dimensions of the small systems with less than 3 modes per wavelength, but with a strong coupling to another subsystem, have been corrected in order to keep the distance that the waves have to travel in the original panels. The known transmission loss (TL) of similar subsystems found in the bibliography were applied in the corresponding area junctions between the structure and the interior acoustic cavities. Junction modelization in FE needs a high level of discretization at high frequencies. Part of the energy that reaches the junction is dissipated internally (structural damping, acoustic radiation, system interaction, friction mechanisms) and is deﬁned by the damping loss factor. The power ﬂow between the subsystems is supposed to be proportional to the difference in modal densities. The NCT’s in SEA model are connected to the structural box by point junctions. The transmissibility obtained in the experimental test are introduced in the models in the point junction between the structure and the interior lining. 96 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

Figure 4.2.6: TL of structures parts of the train.

In the tunnel scenario, the ground has an absorption coefficient due to the ballast track that was assumed as 5% constant spectrum. In the hybrid model, the FE structures are substituted by equivalent SEA plates in which the modal density, the DLF and the CLF between the plate and the acoustic cavities have been modified, taken into account the radiation efficiency, which is defined in the literature [Fahy, 1985] as:

W σ = rad , ¯2 (4.2.30) ρcS vn where Wrad is the power radiated by a uniformly vibrating baffled piston, S is the area of ¯2 the plate, vn is the spatially averaged mean-square vibration velocity of the plate. The radiation efficiency is defined as the ratio of the average acoustic power radiated per unit area of a vibration surface to the average acoustic power radiated per unit area of a piston that is vibrating with the same average mean square velocity. As an approximation of the average absorption of all the NCT, the CLF of the acoustic cavity inside the train could be calculated from the TR. It should be highlighted that the redundant areas from the shared area faces of the acoustic cavities should be subtracted from the area of each cavity, in the way that the total area of the subdivided acoustic cavities has to be the same as they were just one. In Figure 4.2.7 the DLFs for commonly used in engineering structures is presented. The modal densities of the subsystems that are planned to be substituted by equivalent SEA plates in the hybrid model are presented in Figure 4.2.8. For the hybrid model, the CLF between the acoustic cavities and the FE structure replaced by a SEA plate equivalent are calculated using the expression of the radiation loss factor, ηr. In Figure 4.2.9 the model used to calculate the CLF between the train roof structure to the interior and exterior acoustic cavities is shown. A rain-on-the-roof (RoR) method is applied to the structure at a 10% of the total nodes, and the exterior and interior faces are connected to a SIF. The panel is assumed to be baffled and pinned at the point of contact with the surrounding structure. Once the CLFs are calculated, a new SEA plate is created with the same surface but with an equivalent Young modulus, E, and density, ρ, in order to preserve the 4.2. NUMERICAL ANALYSIS 97

Figure 4.2.7: Some of the DLFs used in the SEA structural subsystems.

Figure 4.2.8: Modal densities of the FEM structural parts prone to be substituted by SEA subsystems. mass and also the propagation speed of the bending waves in the plate. The CLF obtained with this process, for the roof and the floor with the interior and exterior cavities, is shown in Figure 4.2.10. There is a clear difference on the CLFs when the excitation is done by a DAF or by rain-on-the-roof method, ROR, see Figure 4.2.11: DAF excitation is generating higher CLFs. This should not happen, because in the context of power injection methods, CLFs should not depend on the applied loads. This could be due to the formulation of the DAF at low frequency; in any case, this CLF will be used above 400 Hz where the spectra are already close. Finally, the CLFs obtained from DAF excitation were used. Only bending waves have been considered when introducing loads and overwriting the plates properties, due to the fact that these waves involve substantial displacements in a direction transverse to the direction of propagation, which can effectively disturb an adjacent fluid; and that the transverse impedance of structures carrying bending waves can be of similar magnitude to that of sound waves in the adjacent fluid, thereby facilitating energy exchange between the two media [Fahy, 1985]. 98 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

Figure 4.2.9: Analysis of modal density and radiation loss factor of train roof to interior and exterior acoustic cavities.

Figure 4.2.10: CLFs to exterior and interior cavities from wagon roof and ﬂoor.

To study the influence of the noise sources located in other sections that in the section under analysis in the case of the train inside a one slab track tunnel, two models were created, a model to simulate the first 2 wagons, composed by 23 sections, in which 9 sections are in front of the noted section at 23 meters from the nose and 13 sections at the back of this section (Figure 4.2.12a) and another one of the characteristic section located at 23 meter from the nose of the train (Figure 4.2.12b). In Figure 4.2.13a) the SPL due to the TBL in the sections different from the analyzed and the relocated load of converter at 29 m from the nose of the train is shown. In Figure 4.2.13b) the SPL adding also the TBL in the section under analysis is shown. A difference of more than 20 dB is appreciated in all the frequency bands, this justifies to use only the loads located in the characteristic section under analysis for an analysis in the train preliminary design phase. In Figure 4.2.14 the continuity of the vibro-acoustic model for low, middle and high frequency ranges is presented. In the low frequency range model (FEM-BEM) the prediction in the acoustic cavity 4.2. NUMERICAL ANALYSIS 99

Figure 4.2.11: Difference on CLFs depending on panel excitation.

a) b) Figure 4.2.12: SEA vibro-acoustic models: a) 2 wagons, b) section model. starts to be inaccurate as the acoustic wavelength starts to be of the same order of the longitudinal length of the cavity (under 160 Hz) due to the boundary conditions applied, rigid wall. As this study is more focused on higher frequencies, where SEA can be applied, over 300 Hz, the acoustic wavelength in the longitudinal direction can be considered small compared with the length of the characteristic section.

4.2.4 Loads 4.2.4.1 Structure-borne The structural loads input has been introduced in the model as a power input calculated with the expression of mobility of an infinite flat plate with equivalent properties to the structure (mass, bending stiffness), and the accelerometers measurements to evaluate the mean squared of the speed of the subsystems. Another way considered was to use the FEM model of the structure, and modify the power input at the connecting points (bogie and HVAC with the structure box) till the sensor located at the same point than the accelerometers would give a similar response. At early state of the design and considering that the TBL as the higher source of noise this second approach can be postponed, but it would be convenient especially at low frequency range where the mobility varies significantly from one point to another in the structure. In order to calculate the power input using the expression of the mobility of a infinite flat plate, we should first evaluate the equivalent properties of the structure, which 100 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

a) b) Figure 4.2.13: a) SPL (dB) predicted numerically with the one section model in the interior and exterior acoustic cavity in 23 m from the nose of the train, due to the relocated load of converter at 29 m and TBL of the whole train; b) SPL (dB) in the interior and exterior of the section at 23 m, due to all the loads in this section and the relocated loads of the converter at 29 m and the TBL of the rest of the train.

Figure 4.2.14: Continuity of vibro-acoustic models. should preserve the bending waves propagation speed plate. As the area of the equivalent plate will be the same, and also the mass, the expression to preserve the ﬂexural waves propagation speed can be written as:

E0I 0 EI xx = xx . (4.2.31) ρ0 ρ

The equivalent density is approximately ρ ∼ 3ρ0 due to the fact that the perimeter of the extruded section is approximately lco ∼ 3lc.

The moment of inertia Ixx of the sections shown in Figure 4.2.15 are

2 L t3 L L I = x + x t y , (4.2.32) xx 12 2 2 for the plane plate of thickness t, and length Lx, separated of the coordinate system half the thickness of the truss like core Ly 4.2. NUMERICAL ANALYSIS 101

Figure 4.2.15: Cross-sections of a) extruded structure Z-shape and b) equivalent ﬂat plate.

3 2 2 2 2 / t Ly Lx Ly Ixx0 =2Ixx +2 2 2 + . (4.2.33) 3 Lx + Ly 2 2 for the initial truss like core section. Comparing the extruded plate (Z-shape) with a ﬂat plate, the ratio between the moments of inertia is approximately Ixx0/Ixx ∼ 2, so the equivalent Young modulus will be E ∼ 6Eo.

4.2.4.2 Airborne Among the airborne noise sources in the train traveling at a speed close to 300 km/h, the most relevant are: the engine (Lw ∼115 dB), traction inverter (Lw ∼100 dB), blowers (converter LwA ∼ 96 dB and inverter/chopper LwA ∼ 99 dB), wheel-rail noise, air conditioning (HVAC), inter-coach gap, pantograph and compressors. The measurement of the acoustic field outside the source region is not sufficient to determine the source uniquely [Ffowcs Williams, 1982], this is the main reason of measuring the TBL with microphones in the near field, flush to the surface. The airborne noise generated by the turbulent boundary layer is introduced in the program VA One as a pressure power spectral density (PSD), using the expression proposed by Cockburn and J.E. [1974]. The spacial correlation function used in VA One between the pressure fluctuations at any two points on the loaded surface is [Larko and Hughes, 2007]

2 − ( ) 2( )+ 2( )+ 1 | | cξ ω kξ ω kη ω ( 3δ∗ ) ξ R(x, x, ω)= e cos (kξωξ) · 2 (4.2.34) − ( ) 2( )+ 2( )+ 1 | | cη ω kξ ω kη ω ( 3δ∗ ) η e cos (kηωη) ,

where the fractions of the convective wave number kc = ω/Uc along and across the ∗ flow are kη(ω)=αkc = α (ω/Uc) and kη(ω)=β (ω/Uc) respectively, and δ = δ/8= 1/5 (0.37/8)(X0/Re ) the TBL thickness, X0 the distance from the leading edge of the train to the center of pressure load on the surface of the subsystem. The default parameters for the TBL aligned with the x axis (α =1, β =0) have been used. The dimensionless spatial correlation coefficients of decay along and across the flow cξ(ω)=0.1, cη(ω)=0.72, a convection velocity Uc =0.7 being U0 the free stream flow velocity. The fluctuation pressure spectrum on the surface can be extracted from the direct measured data of rms pressure or calculated by the following expression 102 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

p2 rms Sp(f)= B , (4.2.35) f A f0 1+ f0 where the root mean square pressure, prms, in indicated in the Table 4.2.6.

Table 4.2.6: Pressure rout mean square for attached and detached ﬂow. p 0.006 Attached rms = q 1+0.15M2 p 0.041 Detached rms = min 0.026, q 1+1.606M2

2 where q, is the dynamic pressure ( q =(1/2)ρU ), and f0 is the reduced frequency, f0 = CU/δ. And the parameters A, B and C depend on the ﬂow conﬁguration and are summarized in the Table 4.2.7.

Table 4.2.7: TBL power spectral density parameter.

Attached Detached A 0.9 0.83 B 2.0 2.15 C 0.346 0.170

These airborne sources can be introduced in the models as pressure constrains in a equivalent acoustic cavity as: a modified DAF in a subsystem surface; a TBL (which can also be user defined, extracted from experiments); a Fluctuating Surface Pressure, FSP; with group of plane waves defined from a equivalent DAF; and monopole sources, depending on the nature of the source. The frequency range and the directivity are one of the most important parameters to look at before choosing the wave to introduce the noise source. If we have the SPL of the source, the way in which this has been characterized has to be taken into account. This problem can be avoided by using the sound power level (Lw). In any case, if the source is going to be introduced as a DAF in a subsystem surface, 3 dB should be added to the power spectrum measured by the microphones in a volume. Several sections at different distances from the nose of the train have been analyzed in order to see the influence of the airframe noise. The airborne noise generated by the turbulent boundary layer is introduced in the program VA One using the expression proposed by Cockburn and J.E. [1974]. The spacial correlation function used in VA One between the pressure fluctuations at any two points on the loaded surface is the one proposed in Larko and Hughes [2007]. VAOne uses in the formulation of the TBL load into a FEM subsystem, the convective wavenumber bases of the flow speed around the train. The mesh of the FE faces must be finer enough to have at least 5 elements per wavelength.

4.3 Experimental set-up

Dynamic tests were carried out with the train at different speeds in order to characterize the noise sources and the subsystems behavior, taking into account the standard [ISO3381_2005E]. 4.3. EXPERIMENTAL SET-UP 103

The microphones used to characterize the airborne excitation where located at the front (at 7, 20.5 and 25 m from the nose of the train) and the rear of the train (at 193, 179.5 and 175 m), see Figure 4.3.1. The microphones M1 and M2 are at the lateral side of the train, and microphone M3 is in the bogie cavity, and microphones M4, M5 and M6 correspond to the same positions but with the train traveling the other direction. The microphone M2 is mounted ﬂush to the surface, meanwhile there is a cylinder air cavity between M1 and the surface of the train. This cavity of 120 mm length and 12 mm diameter acts like a Helmholtz resonator. To characterize the behavior of the junctions between the structure and the NCT once they are mounted in the train, the acceleration of different points at both parts of the junctions were measured. The accelerometers were located at two sections, one at 40 m and the other one at 160 m from the nose of the train.

Figure 4.3.1: Microphones locations and sections instrumented with accelerometers.

The microphones used in the test were Roga Instruments MI-17, the acquisition card NI9233 in a high speed usb carrier NI USB-9162, the sampled time was 2 sec at a sampling rate of 25 kHz.

Figure 4.3.2: Accelerometers placement layout around the window frame.

The accelerometers placement layout is presented in Figure 4.3.2 and the models and sensitivities is summarized in Table 4.3.1, the time sampled was 1.7 sec at a sampling rate of 12 kHz. The acquisition card used for the accelerometers was a LSD Dactron Focus II model, and the software used was RT Pro Focus. The odd-numbered accelerometers were attached to the structure and even ones to the coating, at the other side of the junction. The sampled time has been chosen to have enough signal to make averages taking into account that the boundary conditions change very fast when a train is traveling at high speeds, and that is not easy to maintain constant speed to make a proper averaging. The transmissibility through the junction is evaluated with the ratio between the acceleration power spectral densities in the structure and the NCT. The expression used to 104 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

Table 4.3.1: Accelerometers used in test campaign.

Accelerometer Sensitivity [mV/g] Channel KISTLER 8702B25 203.07 1 KS77C.100 103.99 2 KS77C.100 103.99 3 KS77C.100 103.00 4 4513B-001 103.10 5 4513B-001 102.30 6 PCB 353 B01 22.10 7 PCB 353 B01 21.22 8 evaluate the transmissibility through the junctions, H2(f), is the ratio between the acceleration spectral densities in the structure, asds, and in the coating, asdc [Chen et al., 2011], asd (f) H2 (f)= c . (4.3.1) asds(f)

4.4 Tests deﬁnition

The experimental SPL measurement can be used to validate the aerodynamic noise spectra from the TBL model used in vibro-acoustics, and the acceleration in the structure to be used as a point constrain in the model. The transmissibility through the junctions between the structure and the NCT in the model can be modiﬁed by changing the CLF calculated from the experimental test.

4.4.1 Aerodynamic noise characterization In this part the characterization of aerodynamic noise generated by the turbulent boundary layer in the nose of the train, in the gap between the cars, and in the bogie is presented. This characterization is performed by using microphones measurements. No weighting has been applied to the SPL in order to observe the impulsive character of the noise when the train gets inside the tunnel and the pressure wave generated during the entering process, and also when this wave ﬁnds the train after being reﬂected at the tunnel exit.

4.4.1.1 SPL in open ﬁeld In Figure 4.4.1 the SPL in narrow band (frequency bandwidth is 0.5 Hz) of the microphones at 300 km/h are presented. The SPL have been calculated from the PSD obtained with pwelch function from Matlab, using Hanning window, without overlap using 16 spectra to average. In the Table 4.4.1 the measured sound pressure level overall (SPLOA) at 300 km/h is presented. As shown in Figure 4.4.1, the spectra in the bogie cavity (microphone M3) at the tail or at the head does not vary signiﬁcantly above 2000 Hz, from 50 to 1000 Hz the microphone M4 is 5 dB higher than M3. The microphone M2 shows lower SPL than M5 between 50 to 100 Hz, but higher (10 dB at 1000 Hz) from 100 to 1000 Hz. In the microphone M1 4.4. TESTS DEFINITION 105

Table 4.4.1: Measured SPLOA (dB) in open ﬁeld at 300 km/h, with x being the microphone location.

x [m] 7 20.5 25 175 179.5 193 SPLOA (dB) 123.0 127.1 125.2 127.1 124.1 124.3

110 110 M1 M6 M2 M5 100 M3 100 M4

90 90

80 80

SPL [dB] 70 SPL [dB] 70

60 60

50 50

40 1 2 3 4 40 1 2 3 4 10 10 10 10 10 10 10 10 a) f [Hz] b) f [Hz] Figure 4.4.1: SPL (dB) in the microphones at 300 km/h in open ﬁeld: a) at the head; b) at the tail. the SPL is lower than in M6 from 50 to 1000 Hz, and there is a resonant frequency at approximately 1250 Hz. In M1 can be appreciated harmonics of this resonance at 2500 and 5000 Hz, also the SPL in M1 is higher above 5000 Hz. The characteristic size of the source that generates that tonal noise could be estimated by using the Strouhal number (St = fLc/U), a typical value for this number is St ∼ 0.2 therefore the length obtained is 0.13 m. This noise could come from the inter-coach gap.

4.4.1.2 SPL inside a tunnel The noise during the pass through different tunnels was measured, see Table 4.4.2, and in the spectrums shown in Figure 4.4.2 a tonal resonant mode can be appreciate around 10 kHz, probably due to the noise generated in the bogie, which is reﬂected in the tunnel walls.

Table 4.4.2: SPLOA (dB) inside the tunnel at 300 km/h, with x being the microphone location.

x [m] 7 20.5 25 175 179.5 193 SPLOA (dB) 122.8 127.1 125.0 127.0 123.4 123.7

A Fourier transform of the signal during approximately all the time that the train is inside the tunnel (30 seconds) with the averaged signal is shown in Figure 4.4.3. Finally, the loads considered in the vibro-acoustic model are the airborne noise due to the turbulent boundary layer, for this microphone signals are used to substitute the TBL VAOne models in the regions where we had information from measurements (microphone M2); and the airborne noise generated by the interaction between the wheel and the rail in the bogie cavity (microphone M3). As it can be appreciated in Figure 4.4.4 the pressure signal of microphone M2 is between the attached and the separated model of VAOne for the TBL. The SPL of the microphone M2 has been applied to the SEA plates at both sides of the train and for the rest the TBL attached. 106 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

110 110 M1 M6 M2 M5 100 M3 100 M4

90 90

80 80

SPL [dB] 70 SPL [dB] 70

60 60

50 50

40 1 2 3 4 40 1 2 3 4 10 10 10 10 10 10 10 10 a) f [Hz] b) f [Hz] Figure 4.4.2: SPL (dB) in the microphones a) at the head b) at the tail at 300 km/h inside a tunnel.

115

110

105

100

90 SPL [dB] M6 85 M5 M4 80 M6a M5a 75 M4a

70 0 1 2 10 10 10 f [Hz]

Figure 4.4.3: Low frequency noise inside the tunnel. SPL spectrum of microphones averaged M6, M5, M4 and accumulated M6a, M5a, M4a .

4.4.2 Point junction characterization

The asd of all the accelerometers is shown in Figure 4.4.5, the acceleration used as refer- −6 2 ence was a0 =1· 10 m/s . The accelerations at the structure can be considered as the load and applied as acceleration constrain in the vibro-acoustic model. In Figure 4.4.6 is shown the transmissibility used to evaluate the DLF used in the point junctions between the structure and the internal acoustic lining. 4.5. INTERIOR NOISE PREDICTION 107

Figure 4.4.4: Sound pressure level spectra in thirds of octave bands considered in the vibro-acoustic model.

Figure 4.4.5: Acceleration spectral density in the junctions at 300 km/h.

4.5 Interior noise prediction

The SEA model is valid for frequencies larger than 315 Hz, frequency at which all the subsystems have more than 3 modes per frequency band in 1/3 of octave. The SEA model requires much less computational time than the FE model, allowing the sensitivity of the input parameters to be performed faster. Using the characteristic train section in which we have imposed the pressure measurement in the bogie cavity and the acceleration in junction of the structure with the NCT, we have obtained the SPL shown in presented in 1/3 of octave in the Figure 4.5.1. The SPL predicted are in accordance with expectations extracted from Choi et al. [2004]. 108 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS

Figure 4.4.6: Transmissibility obtained from the accelerometers at 300 km/h [Sorribes-Palmer et al., 2014].

a) b) Figure 4.5.1: SPL (dB) predicted inside the train at 300 km/h in a section at 25 m from the nose: a) analytic expression for CLF in point junction b) point junction CLF obtained from experimental test.

4.5.1 Flow detachment influence on interior noise The power spectral densities of the pressure fluctuation in the train surface proposed by Cockburn and J.E. [1974] are applied on the exterior surface of the structural subsystems in the vibro-acoustic models [Sorribes-Palmer et al., 2013a]. The responses of the models at different speeds for the attached and detached TBL are presented in Figure 4.5.2. The influence of the point detachment and the size of the detached region in the interior noise is shown in Figure 4.5.3. It can be appreciated how a detachment region of 2 meters can increase the noise inside the train some 5 dB.

4.6 Conclusions

A methodology to create vibro-acoustic models to do fast design analysis based on SPL has been presented. The models predict the noise in a characteristic train section, in which there are directly applied loads but also relocated loads from other sections. Several numerical models have be used to cover the whole frequency range to pre- 4.6. CONCLUSIONS 109

Figure 4.5.2: SPL inside the train for attached and detached ﬂow conﬁguration at different speeds.

b) Figure 4.5.3: SPL inside the train at 11 meter from the nose at speed: a) 300 km/h b) 350 km/h. serve the computational time below some reasonable limits. The results obtained allow us to deduce that a good continuity between the models exits and an adequate level of agreement with the usual measured noise for the qualitative character of the model. A procedure to relocate loads has been developed and used for airborne loads, although it is directly applicable to structure-borne. The result of this methodology to the case of a train inside a tunnel shows that the influence of the far loads is quite lower than the in situ ones. A study of the influence of the flow detachment has been presented. The importance 110 CHAPTER 4. INTERIOR NOISE PREDICTION IN HIGH SPEED TRAINS of checking the hypothesis of attachment with the parametric analysis of the detachment point has been highlighted. The effect of separation is bigger than the effect of speed. At the speed of 300 km/h with attached flow, SPL increases at least 10 dB when the flow becomes detached, but SPL increases less than 10 dB if the speed increases from 300 to 350 km/s in the current attached flow field, this shows the importance of predicting the separation regions to optimize the design, and that the flow detachment should be carefully considered in NCT design in high speed trains. An experimental set-up for characterization of the airborne noise have been carried out and the results obtained have been compared with the TBL models. Also transmissibility test have been performed to adjust the point junctions behavior in the vibro-acoustic models. The prediction obtained is inside the region of SPL expected levels. The importance of the characterization of the airborne excitation and the junction behavior has been shown. Some of the difficulties presented during on board tests to characterize the noise generated by the turbulent boundary layer have been highlighted. The SPL of the TBL captured by on board microphones has been presented for the train in open-field and also inside a tunnel. Chapter 5

Conclusions and future work

5.1 Conclusions

The work has been developed within a research program aimed at studying aerodynamic effects on high-speed trains. The work done on the characterization of acoustic loads for the study of noise reduction methods as well as the development of models to predict vibro-acoustic noise inside the train is presented. Among the analyzed loads are the pressure waves generated inside the railway tunnels and their surroundings and the noise due to the turbulent boundary layer around the trains and its influence on the train interior; the relevance of the study of transition between laminar and turbulent flow in the boundary layer is highlighted. A propagation model to study the steepening of the pressure wave in long tunnel and the influence of the dissipation is presented. In the present context of the work, we have built two experimental set-ups to study pressure waves reflection at the duct exits, side branches, the transmission coefficient through porous sections, and the results obtained from the tests have been compared with analytical expression found in the bibliography. One set-up is located at IDR/UPM and another at the TU Berlin. At IDR/UPM different horns, airshafts and porous exits have been tested. At TU Berlin different porous exits have been tested in an anechoic chamber. The test at TU Berlin were useful to validate the setup at IDR/UPM, and also the accuracy of the measurement let appreciate the effect of the small difference in the porosity of the exits on the pressure wave reflection but also in the pressure radiated from the exits. The porous exits are an efficient system to mitigate reflection of the incident pressure waves. Porosity can increase by 40 % the efficiency to reduce the intensity of the reflected wave with respect to the flanged exit, and can also generate a more dispersed wave, which also reduces the intensity of the transmitted wave outside. In order to optimize the reduction of the pressure wave intensity, the porosity of the porous section can be controlled such a system should take into account the speed at which a train enters in the tunnel, and modify its geometry at the exit of the tunnel to optimize the porosity according to the predicted rise time of the wavefront. One possible system could be a group of wind turbines situated at the holes of the porous tunnel exit,

111 112 CHAPTER 5. CONCLUSIONS AND FUTURE WORK and the porosity could be modified by varying the angle of attack of the blades. The energy extracted (for example with and induction engine) could be used to control the blades. Aeroelastic devices based on transverse galloping are proposed as an alternative to use ballast track in medium and long tunnel, to reduce the micro-pressure radiated outside. These devices would increase the apparent friction in the tunnel walls, but also extract energy from the pressure waves that propagate through the tunnel. They can be easily implemented in tunnel side branches or in airshafts at the tunnel portals. Although transverse galloping is a well-kown phenomenon, its application to energy harvesting is a recent issue, mainly concerning the optimization of the extraction of energy. In this regard, a method to analyze the stability of transverse galloping based on stationary measurements performed in a wind tunnel has been presented. This new proposed method does not make use of the classical method of polynomial fitting of the experimental data due to its limitations. The influence of the discretization interval of the measurements in the uncertainty of equilibrium positions has been analyzed and quantified. The numerical method presented has been used to analyze the stability of cylindrical bodies with different cross sections (biconvex, rhomboid, D-shape, triangles and squares), and their respective extracting power capabilities have been compared. A new method to determine the configuration with maximum extracting power capabilities is proposed. It is based on the analysis of the extracted power curves of a body in the neighborhood of the points prone to present galloping (which are only dependent of the aerodynamic characteristic of the body). The value of the relative oscillation A∗ amplitude that gives rise to the maximum extracted power can be determined, and U ∗ m∗ζ as a consequence the associated mechanical parameter can be deduced. As it is a U ∗ combination of the values of wind speed, spring stiffness and damping coefficient, the appropriate values of these variables required to maximize energy extraction from the flow can be determined. In this way, the design process is decomposed into two independent phases and thus is simplified considerably, in order to fit the design to the environmental conditions. Concerning the effect of noise a new methodology to create vibro-acoustic models to perform fast design analysis based on interior sound pressure level (SPL) has been presented. The models predict the noise in a characteristic train section, in which there are directly applied loads but also relocated loads from other sections. The relevant frequency range has been spitted in several regions, and suitable numerical models have been adapted to cover the whole frequency range, to preserve the computational time below some reasonable limits. The results obtained allow us to deduce that a good continuity between the models exits, and an adequate level of agreement with the usual measured noise for the qualitative character of the model. A procedure to relocate loads has been developed and used for airborne loads, although it is directly applicable to structure-borne. The result of this methodology to the case of a train inside a tunnel shows that the influence of the far loads is quite lower than the in-situ ones, specially the load due to the boundary layer when this is detached. The importance of checking the hypothesis of attachment with the parametric analysis of the detachment point has been highlighted. At the speed of 300 km/h with attached flow, SPL increases at least 10 dB when the flow becomes detached, but SPL increases less than 10 dB if the speed increases from 300 to 350 km/s in the attached flow condition, this shows the importance of pre- 5.1. CONCLUSIONS 113 dicting the separation regions to optimize the design, and that the flow detachment should be carefully considered in noise control treatment (NCT) design in high speed trains. A characterization of the aerodynamic noise in high speed train has been performed and the results obtained have been compared with the turbulent boundary layer (TBL) models. The structure borne characterization has been used to adjust the point junctions behavior. The prediction obtained is inside the region of SPL expected levels. The importance of the characterization of the airborne excitation and the junction behavior has been shown. Some of the difficulties presented during on board tests to characterize the noise generated by the turbulent boundary layer have been highlighted. The SPL of the TBL captured by on board microphones have been presented for the train in open-field and also inside the tunnel. 114 CHAPTER 5. CONCLUSIONS AND FUTURE WORK 5.2 Future work

As a future work with respect to the pressure waves in tunnels, it is being considered to study the dynamic response of the pressure waves dampers based on transverse galloping to high speed train wakes by numerical simulation. This simulation could be validated with measurement of a damper device based on transverse galloping (or flutter) in a railway tunnel. Measurements of the energy extracted when using variable magnetic dissipation to maximize the extraction could be performed. The validation of the nonlinear propagation model and the analysis of the evolution of different profiles. Validation of the non-linear propagation model for the pressure waves inside a tunnel, and its application to analyze the evolution of the different wavefronts. In aerodynamic noise source characterization, a combination of microphone array method and on board flush mounted microphones could be an interesting test to perform. This will help to better locate the noise sources and correlate the results for the near-field and far-field. In the field of vibro-acoustic a multi-objective optimization of the train structure response could be implemented. The topology of the car body could be studied optimizing the weight and also the noise inside the train. For a better understanding of the aerodynamic phenomena around the train, an aeroacoustic simulation (using OpenFoam/Ansys) of a scale model and the validation of the results with microphone arrays and PIV measurements in an aeroacoustic wind tunnel, is also considered. Bibliography

Metallic materials and elements for aerospace vehicle structures. AMSC, 2003.

[en 14067-5:2006:08] railway applications - aerodynamics - part 5: Requirements and test procedures for aerodynamics in tunnels, 2006.

Va one 2014 foam module. Technical report, ESI Group, 2014.

A Abdelkeﬁ, M R Hajj, and a H Nayfeh. Piezoelectric energy harvesting from transverse galloping of bluff bodies. Smart Materials and Structures, 22(1):015014, January 2013a. ISSN 0964-1726. doi: 10.1088/0964-1726/22/1/015014.

A Abdelkeﬁ, Z Yan, and M. H. Hajj. Modeling and nonlinear analysis of piezoelectric energy harvesting from transverse galloping. Smart Materials and Structures, 22(2): 025016, 2013b.

J. F. Allard. Propagation of Sound in Porous Media. Elsevier Applied Science, 1993.

G. Alonso, E. Valero, and J. Meseguer. An analysis on the dependence on cross section geometry of galloping stability of two-dimensional bodies having either biconvex or rhomboidal cross sections. European Journal of Mechanics - B/Fluids, 28(2):328 – 334, 2009.

G. Alonso, A. Sanz-Lobera, and J. Meseguer. Hysteresis phenomena in transverse galloping of triangular cross-section bodies. Journal of Fluids and Structures, 33:243–251, May 2012.

M. Atig, J. Dalmont, and J. Gilbert. Termination impedance of open-ended cylindrical tubes at high sound pressure level. Comptes Rendus Mecanique, 332:299–304, April 2004. doi: 10.1016/j.crme.2004.02.008.

Chris Baker. The ﬂow around high speed trains. Journal of Wind Engineering and In- dustrial Aerodynamics, 98(6-7):277–298, June 2010. ISSN 01676105. doi: 10.1016/j. jweia.2009.11.002. URL http://linkinghub.elsevier.com/retrieve/ pii/S0167610509001408.

A. Baron, M. Mossi, and S. Sibilla. The alleviation of the aerodynamic drag and wave effects of high-speed trains in very long tunnels. Journal of Wind Engineering and Industrial Aerodynamics, 89(5):365–401, 2001. ISSN 0167-6105.

115 116 BIBLIOGRAPHY

A. Barrero-Gil. Dinámica de osciladores aeroelásticos. PhD thesis, Escuela Técnica Superior de Ingenieros Aeronáuticos / Universidad Politécnica de Madrid, 2008.

A. Barrero-Gil, A. Sanz-Andres, and G. Alonso. Hysteresis in transverse galloping: The role of the inﬂection points. Journal of Fluids and Structures, 25(6):1007 – 1020, 2009.

Antonio Barrero Ripoll and Miguel Pérez-Saborid. Fundamentos y aplicaciones de la mecánica de ﬂuidos. MCGraw-Hill, 2006. ISBN 9788448198909.

H. Bass, L. Sutherland, A. Zuckerwar, D. Blackstock, and D. Hester. Atmospheric absorption of sound: Further developments. The Journal of the Acoustical Society of America, 97(1):680–683, 1995.

M. Bellenoue, V. Moriniére, and T. Kageyama. Experimental 3-d simulation of the compression wave, due to train-tunnel entry. Journal of Fluids and Structures, 16(5):581 – 595, 2002. ISSN 0889-9746.

B. L. Campolina. Vibro-acoustic modelling of aircraft double-walls with structural links using statistical energy analysis. PhD thesis, Université de Sherbrooke/ Universite Pierre et Marie Curie, 2012.

S. M. Chen, D. F. Wang, and J. M. Zan. Interior Noise Prediction of the Automobile Based on Hybrid FE-SEA Method. Mathematical Problems in Engineering, 2011: 1–20, 2011. ISSN 1024-123X. doi: 10.1155/2011/327170.

S. Choi, C.W. Lee, and J. H. Cho. Interior noise of a korean high-speed train in tunnels. ACOUSTICS, 2004.

J.A. Cockburn and Robertson J.E. Vibration response of spacecraft shrouds to in-ﬂight ﬂuctuating pressures. Journal of Sound and Vibration, 1974.

R.J.M. Craik and R.S. Smith. Sound transmission through double leaf lightweight partitions part I: airborne sound. Applied Acoustics, 61(2):223–245, October 2000. ISSN 0003682X. doi: 10.1016/S0003-682X(99)00070-5.

F. E. Crawley, H. C. Curtiss Jr., D. A. Peters, R. H. Scanlan, and F. Sisto. A Modern Course in Aeroelasticity. Kluwer Academic Publishers, 1995.

L. Cremer, M. Heckl, and B.A.T. Petersson. Structure-Borne Sound - Structural Vibra- tions and Sound Radiation at Audio Frequencies. 2005.

P. Crespi, R. Grégoire, and P. Vinson. Laser doppler velocimetry measurements and boundary layer survey on-board the tgv high speed train. In Proceedings of World Congress on Railway Research, 1994.

N. Curle. The inﬂuence of solid boundaries upon aerodynamic sound. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 231(1187): 505–514, 1955. doi: 10.1098/rspa.1955.0191.

J. Dalmont, C. Nederveen, and N. Joly. Radiation impedance of tubes with diferent ﬂanges:numerical and experimental investigations. Journal of Sound and Vibration, 244(3):505–534, 2001. BIBLIOGRAPHY 117

P.W. Eade and A.E.J. Hardy. Railway vehicle internal noise. Journal of Sound and Vibration, 1977.

EN14067-5:2006:E. Railway applications - Aerodynamics - Part 5: Requirements and test procedures for aerodynamics in tunnels, 2006.

VA One User’s Guide. The ESI Group, 2014.

F. Fahy. Sound and Structural Vibration. Radiation, Transmission and Response. Aca- demic Press, 1985.

F.J. Fahy. A note on the subdivision of a volume of air in a vehicle enclosure into sea subsystems. Journal of Sound and Vibration, 271(3-5):1170–1174, April 2004. ISSN 0022460X. doi: 10.1016/S0022-460X(03)00632-1.

Frank Fahy and Paolo Gardonio. Sound and Structural Vibration Radiation, Transmission and Response. Academic Press, Oxford, 2007. ISBN 9780123736338.

J. E. Ffowcs Williams. Sound sources in aerodynamics-fact and ﬁction. American Insti- tute of Aeronautics and Astronautics, 20(3):307–315, 1982.

N. Flecher and T. Rossing. Physics of musical instruments. Springer-Verlag New York, 1991.

J. Forssén, T. Stefan, A. C. Corakci, A. Frid, and W. Kropp. Modelling the interior sound ﬁeld of a railway vehicle using statistical energy analysis. Applied Acoustics, 2011.

N. Fredmion and N. Vincent. Aerodynamic noise radiated by the intercoach spacing and the bogie of a high-speed train. Journal of Sound and Vibration, 2000.

Kozo Fujii and Takanobu Ogawa. Aerodynamics of high speed trains passing by each other. Computers & Fluids, 24(8):897 – 908, 1995.

Takashi Fukuda, Satoru Ozawa, Masanobu Iida, Toru Takasaki, and Yusuke Wakabayashi. Distortion of compression wave propagating through very long tunnel with slab tracks. JSME International Journal Series B Fluids and Thermal Engineering, 49(4):1156– 1164, 2006.

Paolo Gardonio. Review of active techniques for aerospace vibro-acoustic control. Jour- nal of aircraft, 2002.

L. Gori, P. Giovanni, M. Tosi, and A. G. Violi. Acoustic characterisation of high speed train etr 500. In Proceedings of World Congress on Railway Research, Munich, 2001.

A. W. Guess. Calculation of perforated plate liner parameters from speciﬁed acoustic resistance and reactance. The Journal of the Acoustical Society of America, 54(1): 307–307, 1973. doi: 10.1121/1.1978205.

T. Hara. Aerodynamic force acting on a high speed train at tunnel entrance. The Japan Society of Mechanical Enginners, 1961.

M. S. Howe. The compression wave produced by a high-speed train entering a tunnel. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1974):1523–1534, 1997a. 118 BIBLIOGRAPHY

M. S. Howe. The compression wave produced by a high-speed train entering a tunnel. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 454(1974):1523–1534, June 1998. ISSN 1364-5021. doi: 10.1098/rspa.1998.0220. M. S. Howe. Review of the theory of the compression wave generated when a high-speed train enters a tunnel. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 213(2):89–104, January 1999. ISSN 0954-4097. doi: 10.1243/0954409991531056. M. S. Howe. On rayleigh’s computation of the "end correction", with application to the compression wave generated by a train entering a tunnel. Journal of Fluid Mechanics, 385:63–78, April 1999b. M. S. Howe. On the infrasound generated when a train enters a tunnel. Journal of Fluids and Structures, 17(5):629 – 642, 2003c. M. S. Howe. The genetically optimized tunnel-entrance hood. Journal of Fluids and Structures, 23(8):1231–1250, November 2007. ISSN 08899746. doi: 10.1016/j. jfluidstructs.2007.06.005. M. S. Howe, M. Iida, T. Fukuda, and T. Maeda. Theoretical and experimental investigation of the compression wave generated by a train entering a tunnel with a flared portal. Journal of Fluid Mechanics, 2000. M. S. Howe, M. Iida, T. Fukuda, and T. Maeda. Aeroacoustics of a tunnel-entrance hood with a rectangular window. Journal of Fluid Mechanics, 487:211–243, June 2003. ISSN 00221120. doi: 10.1017/S0022112003004725. M.S. Howe and E.A. Cox. Reflection and transmission of a compression wave at a tunnel portal. Journal of Fluids and Structures, 20(8):1043 – 1056, 2005. M.S. Howe, M. Iida, and T. Fukuda. Influence of an unvented tunnel entrance hood on the compression wave generated by a high speed train. 2002. M.S. Howe, M. Iida, T. Maeda, and Y. Sakuma. Rapid calculation of the compression wave generated by a train entering a tunnel with a vented hood. Journal of Sound and Vibration, 297(1-2):267 – 292, 2006. H. H. Hubbard. Aeroacoustics of flight vehicles: Theory and practice. volum 2: Noise control. Technical report, NASA, 1991. J. Hwang and D.H. Lee. Unsteady aerodynamic loads on high speed trains passing by each other. Journal of Mechanical Science and Technology, 2000. Jaeho Hwang, Tae-Seok Yoon, Dong ho Lee, and Soo gab Lee. Numerical study of unsteady flowfield around high speed trains passing by each other. JSME International Journal Series B Fluids and Thermal Engineering, 44(3):451–464, 2001. Masanobu Iida, Katsuhiro Kikuchi, and Takashi Fukuda. Analysis and experiment of compression wave generated by train entering tunnel entrance hood. JSME Interna- tional Journal Series B Fluids and Thermal Engineering, 49(3):761–770, 2006. Uno Ingard. NOISE REDUCTION ANALYSIS. Jones and bartlett publishers, 2008. ISBN 9781934015315. BIBLIOGRAPHY 119

Hyung-Jo Jung and Seung-Woo Lee. The experimental validation of a new energy harvesting system based on the wake galloping phenomenon. Smart Materials and Struc- tures, 20(5):055022, May 2011. ISSN 0964-1726. doi: 10.1088/0964-1726/20/5/ 055022.

J. N. Kaoua, C. W. Pope, D. A. Henson, and Mott MacDonald Ltd, UK. A parametric study into the factors affecting the development and alleviation of micro-pressure waves in railway tunnels. Technical report, 12th International Symposium on Aerody- namics and Ventilation of Vehicle Tunnels, 2006.

Jonathan a Kemp, Maarten van Walstijn, D Murray Campbell, John P Chick, and Richard af Smith. Time domain wave separation using multiple microphones. The Journal of the Acoustical Society of America, 128(1):195–205, July 2010. ISSN 1520- 8524. doi: 10.1121/1.3392441.

K. Kikuchi, S. Ozawa, H. Takami, and M. Iida. Continuous pressure wave generated by a train running in a tunnel. Journal of low frequency noise, vibration and active control, 2009.

H.-D. Kim, D.-H. Lee, and T. Setoguchi. Study of the Impulse Wave Discharged From the Exit of a Right-Angle Pipe Bend. Journal of Sound and Vibration, 259(5):1147–1161, January 2003. ISSN 0022460X. doi: 10.1006/jsvi.2002.5190.

L. Kinsler, A. Frey, A. Coppens, and J. Sanders. Fundamentals of Acoustics. Wiley, 4 edition, 2000. ISBN 0471847895.

J.M. Kluger, F.C. Moon, and R.H. Rand. Shape optimization of a blunt body Vibro-wind galloping oscillator. Journal of Fluids and Structures, 40:185–200, July 2013. ISSN 08899746. doi: 10.1016/j.jﬂuidstructs.2013.03.014.

T. Kohrs and B. A. T. Petersson. Wave beaming and wave propagation in light weight plates with truss-like cores. Journal of Sound and Vibration, 2009.

J. Králíek and J. Dupal. Vibroacoustical analysis of rail vehicle. Applied and Computa- tional Mechanics, 2007.

T. Kurita, M. Hara, and M. Horiuchi. Reduction of pantograph noise. In JR EAST Tech- nical Review-No.8, 2006.

N. Lalor and H.H. Priebsch. The prediction of low- and mid-frequency internal road vehicle noise: a literature survey. 2007.

J.M Larko and W.O. Hughes. Initial assesment of the ares i-x launch vehicle upper stage to vibroacoustic ﬂight environments. In 14th International Congress on Sound and Vibration, 2007.

A. Lauterbach, K. Ehrenfried, S. Loose, and C. Wagner. Scaling laws for the aeroacoustics of high speed trains. American Institute of Aeronautics ans Astronautics, 2011.

A Lauterbach, K Ehrenfried, S Loose, and C Wagner. Aeroacoustics of a High-Speed Train Double Model at High Reynolds Numbers. In Proceedings of the First Interna- tional Conference on Railway Technology: Research, Development and Maintenance, pages 1–15, 2012. 120 BIBLIOGRAPHY

Antoine Lefebvre, Gary P. Scavone, Jonathan Abel, and Alexander Buckiewicz-Smith. A comparison of impedance measurements using one and two microphones. In ISMA, 2007.

H. Levine and J. Schwinger. On the radiation of sound from an unﬂanged circular pipe. Phys. Rev., 73(4):383–406, 1948.

Lighthill. On sound generated aerodinamically. Cambrige Uinversity Press, 1952.

Jacob Chia-chun Liu and Po-Chien Lu. Study on sound propagation in long enclosures with a vertical or inclined branch of different dimensions. Applied Acoustics, 70(3): 522–529, March 2009. ISSN 0003682X. doi: 10.1016/j.apacoust.2008.05.010.

Tang-Hong Liu, Hong qi Tian, and Xi feng Liang. Design and optimization of tunnel hoods. Tunnelling and Underground Space Technology, 25(3):212 – 219, 2010.

Valerio Lorenzoni. Aeroacoustic Investigation of Rod-Airfoil Noise based on Time- Resolved PIV. PhD thesis, TU Delft, 2008.

S.C. Luo and P.W. Bearman. Predictions of ﬂuctuating lift on a transversely oscillating square-section cylinder. Journal of Fluids and Structures, 4(2):219 – 228, 1990. ISSN 0889-9746.

R. H. Lyon and R. G. DeJong. Theory and Application of Statistical Energy Analysis. Butterworth-Heinemann, 1995.

T. Maeda, T. Matsumura, M. Iida, K. Nakatani, and K. Uchida. Effect of shape of train nose on compression wave generated by a train entering tunnel. The Japan Society of Mechanical Enginners, 1993.

P. M. Morse. Vibration and sound. McGraw-Hill book company, Inc., 1948.

Tomas W Muld. Slipstream and Flow Structures in the Near Wake of High-Speed Trains. PhD thesis, 2012.

M. L. Munjal. Acoustics of Ducts and Mufﬂers With Application to Exhaust and Ventila- tion System Desing. Jonh Wiley & sons, 1987.

P.R. Murray and M.S. Howe. Inﬂuence of hood geometry on the compression wave generated by a high-speed train. Journal of Sound and Vibration, 329(14):2915 – 2927, 2010.

Y.T. Ng, S.C. Luo, and Y.T. Chew. On using high-order polynomial curve ﬁts in the quasi-steady theory for square-cylinder galloping. Journal of Fluids and Structures,20 (1):141 – 146, 2005.

K. Nishimura, M.; Ikeda. A new concept for reducing sound reﬂection coefﬁcient at a duct open end. In inter-noise 2003, 2003.

M. Nishimura and K. Ikeda. Reduction of sound reﬂection coefﬁcient at open duct end by using porous side wall. Journal of Environment and Engineering, 3(2):351–362, 2008.

C. Noger, J. C. Patrat, J. Peube, and J. L Peube. Aeroacoustical study of the tgv pantograph recess. Journal of Sound and Vibration, 1999. BIBLIOGRAPHY 121

A. N. Norris and I. C. Sheng. Acoustic radiation from a circular pipe with an infinite flange. Journal of Sound Vibration, 135:85–93, November 1989. doi: 10.1016/ 0022-460X(89)90756-6. M. P. Norton and D. G. Karczub. Fundamentals of Noise and Vibration Analysis for Engineers. Cambridge University Press, Cambridge, 2003. ISBN 9781139163927. doi: 10.1017/CBO9781139163927. Ulf Orrenius, Vincent Cotoni, and Andrew Wareing. Analysis of sound transmission through periodic structures typical of train carbodies and aircraft fuselages Periodic structures : Modeling concept. pages 1–16, 2010. S. Ozawa, T. Maeda, T. Matsumura, and K. Uchida. Effect of ballast on pressure wave propagation through tunnel. In The International Conference on Speedup Technology for Railway and Maglev Vehicles (STECH), 1993. Michael P. Paidoussis, Stuart J. Price, and Emmanuel de Langre. Fluid-Structure Inter- actions. Cambridge University Press, 2011. ISBN 9780521119429. N. Paradot, E. Masson, F. Poisson, R. Gregioire, E. Guilloteau, H. Touil, and P. Sagaut. Aero-acoustic methods for high-speed train noise prediction. Seoul: World Congress on Railway Research, 2008. G. V. Parkinson and N. P. H. Brooks. On the aeroelastic instability of bluff cylinders. Journal of Applied Mechanics, pages 252–258, 1961. G. V. Parkinson and J. D. Smith. The square prism as an aeroelastic non-linear oscillator. The Quarterly Journal of Mechanics and Applied Mathematics, 17(2):225–239, 1964. Mario Peláez-González and Natalia Montenegro-Palmero. Pressure wave measurements inside tunnel of the spanish railway network and inside a high velocity streamlined test train. In ITA-AITES World Tunnel Congress, 2009. Allan D. Pierce. Acoustics: an Introduction to Its Physical Principles and Applications. Acoustical Society of Amer, 1989. R. S. Raghunathan, H. D. Kim, and T. Setoguchi. Aerodynamics of high-speed railway train. Progress in Aerospace Sciences, 38:469–514, 2002. P. J. Remington. Wheel/rail rolling noise, ii: Validation of the theory. Journal of the Acoustical Society of America, 1987a. P.J. Remington. Wheel/rail rolling noise, i: Theorical analysis. Journal of the Acoustical Society of America, 1987b. Pierre Ricco, Arturo Baron, and Paolo Molteni. Nature of pressure waves induced by a high-speed train travelling through a tunnel. Journal of Wind Engineering and Industrial Aerodynamics, 95(8):781–808, August 2007. ISSN 01676105. doi: 10.1016/j.jweia.2007.01.008. S W Rienstra and A Hirschberg. An Introduction to Acoustics. Technology, (April), 2011. J.E. Rossiter. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Technical report, Aeronautical Research Council Reports and Memoranda, 1964. 122 BIBLIOGRAPHY

J. Sapena and D. Blanchet. Interior noise structureborne path prediction in a high speed train using fe/sea hybrid modeling methodologies. Noise and Vibration: Emerging Methods, 2009.

A. Sasoh, K. Matsuoka, K. Nakashio, E. Timofeev, K. Takayama, P. Voinovich, T. Saito, S. Hirano, S. Ono, and Y. Makino. Attenuation of weak shock waves along pseudo- perforated walls. Shock Waves, 8(3):149–159, June 1998. ISSN 0938-1287. doi: 10.1007/s001930050108.

Miyachi T. Iida M. Sato, S. Reduction of micro-pressure wave emitted from portal of side branch of high-speed railway tunnel. Quarterly Report of RTRI, 2010.

T. Sato and T. Sassa. Prediction of the compression pressure wave generated by a high- speed train entering a tunnel. International Journal of Computational Fluid Dynamics, 19:53–59, January 2005.

A. Selamet, Z. L. Ji, and R. A. Kach. Wave reﬂections from duct terminations. Acoustical Society of America Journal, 109:1304–1311, April 2001. doi: 10.1121/1.1348298.

F Silva, J Kergomard, B Mallaroni, and A N Norris. Approximation formulae for the acoustic radiation impedance of a cylindrical pipe. Aerospace, 2(November 2008): 255–263, 2009.

F. Sorribes-Palmer and A. Sanz-Andres. Optimization of energy extraction in transverse galloping. Journal of Fluids and Structures, 43:124–144, November 2013. ISSN 08899746. doi: 10.1016/j.jﬂuidstructs.2013.09.011.

F. Sorribes-Palmer, F. Navarro-Medina, A. Sanz-Andres, I. Perez-Grande, A. Martinez, E. Vega, A. Mascaraque-Sillero, M. Rodriguez-Plaza, J. Iglesias-Díaz, A. Andres- Alguacil, and D. Alonso-Gimeno. Criteria for ballast ﬂight initiation induced by high speed trains. In Proceedings of the First International Conference on Railway Technol- ogy: Research, Development and Maintenance, 2012.

F. Sorribes-Palmer, G. Alonso, and M. Ghaemi-Nasab. Aeroacoustic noise prediction in high speed trains due to attached and detached ﬂow. In International Workshop on Numerical Prediction of Detached Flows, 2013a.

F. Sorribes-Palmer, M. Ghaemi, M. Chimeno, and G. Alonso. Trade-off approaches for the vibroacoustic analysis of trains. In 11th International Conference on Vibration Problems, 2013b.

F. Sorribes-Palmer, G. Alonso, and A. Sanz-Andres. Vibroacoustic problems in high speed trains. In The Second International Conference on Railway Technology: Re- search, Development and Maintenance, 2014.

N. Sugimoto and T. Ogawa. Acoustic analysis of the pressure ﬁeld in a tunnel, generated by entry of a train. Proceedings: Mathematical, Physical and Engineering Sciences, 454(1976):pp. 2083–2112, 1997.

Zhenxu Sun, Jinjing Song, and Yiran An. Numerical Simulation of aerodynamic noise generated by high speed trains. Engineering Applications of Computational Fluid Me- chanics, 30:254–259, 2012. BIBLIOGRAPHY 123

Masahiro Susuki. Unsteady Aerodynamic Force Acting on High Speed Trains in Tunnel. RTRI, 42(2):89–93, 2001.

Takehisa Takaishi, Akio Sagawa, Kiyoshi Nagakura, and Tatsuo Maeda. Numerical analysis of dipole sound source around high speed trains. The Journal of the Acoustical Society of America, 111(6):2601, 2002. ISSN 00014966. doi: 10.1121/1.1480833.

C. Talotte. Aerodinamic noise: a critical survey. Journal of Sound and Vibration, 231: 549–562, 2000.

C Talotte, P.-E Gautier, D.J Thompson, and C Hanson. Identiﬁcation, modelling and reduction potential of railway noise sources: a critical survey. Journal of Sound and Vibration, 267(3):447–468, October 2003. ISSN 0022460X. doi: 10.1016/ S0022-460X(03)00707-7.

J.C. Tannehill, D.A. Anderson, and R.H. Pletcher. Computational ﬂuid mechanics and heat transfer. Taylor&Francis, 1984.

David Thompson. Railway Noise and Vibration. Elsevier Press, 2009.

UNE-EN ISO 10534-2. Determination of sound absorption coefﬁcient and impedance in impedance tubes, 1998.

A. E Vardy. Reﬂection of step-wavefronts from perforated and ﬂared tube extensions. Journal of Sound and Vibration, 1978.

A. E Vardy and P. Reinke. Estimation of train resistance coefﬁcients in tunnels from measurements during routine operation. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 213, 1999.

A.E Vardy. Generation and alleviation of sonic booms from rail tunnels. Proceedings of The Ice - Engineering and Computational Mechanics, 161:107–119, 2008.

I. Vér and L. Beranek. Noise and vibration control engineering. John Wiley & Sons, 2006.

YiWei Wang, Yang Wang, YiRan An, and YaoSong Chen. Aerodynamic simulation of high-speed trains based on the Lattice Boltzmann Method (LBM). Science in China Series E: Technological Sciences, 51(6):773–783, May 2008. ISSN 1006-9321. doi: 10.1007/s11431-008-0063-3.

J. Wijker. Spacecraft structures. Springer-Verlag, 2008.

J. Wijker. Random Vibrations in Spacecraft Structures Design. Springer Dordrecht Hei- delberg London New York, 2009. ISBN 9789048127276.

M. William-Louis and C. Tournier. A wave signature based method for the prediction of pressure transients in railway tunnels. Journal of Wind Engineering and Industrial Aerodynamics, 93:521 – 531, 2005. ISSN 0022-0833.

M.J.P. William-Louis. Détermination des paramétres aérodynamiques nécessaires pour le calcul 1d de la circulation d’un train dans un tunnel. Technical report, Laboratoire de Mécanique et Energétique de l’Université de Valenciennes, 1999. 124 BIBLIOGRAPHY

M.J.P. William-Louis and C. Tournier. Non-homentropic ﬂow generated by trains in tunnels with side branches. International Journal of Numerical Methods for Heat & Fluid Flow, 8(2):183–198, 1998. ISSN 0961-5539. doi: 10.1108/09615539810201730.

G. Xie, D.J. Thompson, and C.J.C. Jones. The radiation efﬁciency of bafﬂed platesand strips. Journal of Sound and Vibration, 280:181–209, 2003.

G. Xie, D.J. Thompson, and C.J.C. Jones. A modelling approach for the vibroacoustic behaviour of aluminium extrusions used in railway vehicles. Journal of Sound and Vibration, 293(3-5):921–932, June 2006. ISSN 0022460X. doi: 10.1016/j.jsv.2005.12. 015.

T. S. Yoon and S. Lee. Efﬁcient prediction methods for the micro-pressure wave from a high-speed train entering a tunnel using the Kirchhoff formulation. The Journal of the Acoustical Society of America, 110(5):2379, 2001. ISSN 00014966. doi: 10.1121/1. 1409374.

Si-min Zeng and Walter Gretler. Pressure wave propagation in a short tunnel caused by passing trains. Forschung im Ingenieurwesen, 61:304–315, 1995. 10.1007/BF02605749. Appendix A

Appendix

A.1 Wave separation

A method to distinguish between the incident and reflected waves at the tunnel portals was proposed in Kikuchi et al. [2009]. The method consisted on analyzing the pressure signal of two microphones close enough one to each other and at a certain distance from the tunnel exit. Assuming that the signal in microphone located in x1 is F (x1,t− t/2), and has two parts f(x1,t− t/2) and g(x1,t− t/2).IfF (x1,t− t/2) only has the part of τ x1 r the wave traveling to the right f, as the argument is t − − = t − τ − m which is 2 c0 c0 the same value as at the midpoint xm at time t − τ. Instead, if F just has wave traveling τ x1 to the left, g, which has the argument t − + = t + xm which is the same value as 2 c0 the wave at the midpoint xm in time t. When studying F (x1,t+ t/2), another value for f(x1,t+ t/2) is obtained, and also for g(x2,t+ t/2) = g(xm,t) . Subtracting the two signals we obtain the expression τ τ ∼ xm F x2,t+ − F x1,t− = 2τf t − . (A.1.1) 2 2 2 Then the pressure of the wave incident to the portal is obtained integrating the equation A.1.1. The result of applying this technique are shown in Figure A.1.1. Another way to separate the waves is to consider that the incident and the reflected wave have only one harmonic, I (incident wave) and R (reflected wave) respectively. Then the signal recorded by a microphone at location x = x2 at time t, can be described as the addition of I and R with their respective time delays τ τ τ F x2,t+ = I sin ω t − + R sin ω t + . (A.1.2) 2 2 2 From this signal we can calculate τ τ x1 τ x1 F1 x1,t− = I sin ω t − − + R sin ω t − + , 2 2 c0 2 c0 and

125 126 APPENDIX A. APPENDIX

τ τ x2 τ x2 F2 x2,t+ = I sin ω t + − + R sin ω t + + . 2 2 c0 2 c0 The difference is xm xm F2 − F1 = I sin ω t − + τ − sin ω t − − τ + ... c0 c0 x x R sin ω t − m − sin ω t + m , c0 c0 thus the reﬂected wave is exactly canceled (except measurement errors), and only the incident wave remains xm F2 − F1 =2I cos ω t − sin ωτ . c0

If F (x2,t) was composed of a set of incident waves of frequency ωn and amplitude In, and Rn for the reﬂected, the calculation can be repeated, and the reﬂected will also be cancel out, and only the incident would remain xm F2 − F1 =2 In cos ωn t − sin ωnτ. c0

If this expression is expanded in cosine series of ωn xm F2 − F1 ≡ F0 + F21n cos ωnt =2 In cos ωn t − sin ωnτ, c0 identifying (with t = t − xm/c0)isF21n =2In sin ωnτ or In = F21n/2sinωnτ which is an exact expression. If a time series of duration Ts is considered, then the angular frequency of the “n” harmonic is ωn = n2π/Ts, as it is supposed to develop the full signal is periodic with period Ts. If the approach sin ωnτ ∼ ωnτ 1 is valid ( fn c0/ (2πLm) where Lm = x1 − x2 is the distance between microphones) then

∼ F21n In = , 2ωnτ as F21n is as small as it comes from F2 − F1, which are separated a small time τ, the error in In (ratio of two small things) can be big, if F2 − F1 is noisy. If ωnτ is big, this simpliﬁcation can not be done. The results of applying this methodology to obtain the pressure waves separated is shown in Figure A.1.2.

It is important to highlight that if sin (ωnτ) → 0 that is ωnτ → nπ , this case must be taken into account when expanding the signal in cosine series. If we want to increase n to have a more detailed expansion of the signal, τ must be decreased, so that ωnτ do not reach π increasing with this the uncertainty of the correspondent harmonic.

To detect the peaks of input/output station (head, tail) the effective length of the head, Leff = Lhead + Lentrance should be considered, the half period of the signal would be Tc/2=Leff /Utr. As the signal propagates at speed of sound c0, the typical frequency would be fhead =1/Tc = Utr/2Leff and the length in the wave Lhead = c0/fhead = 2Leff /M ∼ 270 m. In order to measure it, the condition is 4πLm Lhead , it should be A.1. WAVE SEPARATION 127

Figure A.1.1: Wave separation using the method proposed in Kikuchi et al. [2009]. notice that this distance will be considerably reduced in long tunnels with small friction due to nonlinear effects. An analysis of the noise level magnitude order, acceptable to be able to apply the separation method is done. From A.1.1, the signal S, is of order S ∼ 2πf ,τ = Lm/c0. The variation of pressure in the nose, in the initial phase can be approximated by

1 C p ∼ ρU 2 w sin (ω t) , 2 tr 1 − φ2 head

Cw is the pressure coefﬁcient at the train nose. So then the expression for the pressure gradient is

1 C p˙ ∼ ρU 2 w ω cos (ω t) . 2 tr 1 − φ2 head head Finally an estimation of p˙ is

1 C 2π πρ C U 3 ∼ 2 w w tr p˙ ρUtr 2 = 2 . 2 1 − φ Tc 4 1 − φ Leff And an approximation of the pressure amplitude is

2L πρ C U 3 ∼ m w tr p 2 . c0 4 1 − φ Leff

The microphone noise Nm should be limited by a factor SNR = S/Nm

2L πρ C U 3 ∼ m w tr Nm 2 , SNRc0 4 1 − φ Leff the smaller the Lm smaller is the signal. The common values of this parameters are:

SNR ∼100; Cw ∼1; φ ∼10/60 = 0.16; Utr ∼=80m/s;Leff ∼30 m; Lm ∼1m.So then the noise amplitude in the signal captured by the microphone must be close to Nm ∼ 2Pa. The wave separation method proposed in [Kikuchi et al., 2009] can be applied to distinguish the incident and reﬂected waves at the exit and also to compare the radiated pressure with the predicted. 128 APPENDIX A. APPENDIX

Figure A.1.2: Wave separation using the proposed method.

A.2 Slowly varying cross section ducts

The behavior of the air inside a horn can be approximated by the equations [Munjal, 1987]

Dρ ∂u uρ dS + ρ0 + =0, (A.2.1) Dt ∂x S dx

∂u ∂p ρ0 + =0, (A.2.2) ∂t ∂x ∂p = c2 . (A.2.3) ∂ρ s Combining the equations (A.2.1), (A.2.2) and (A.2.3) the Webster horn equation is obtained

∂2p ∂2p c2 dS(x) ∂p − c2 + =0. (A.2.4) ∂t2 ∂x2 S(x) dx ∂x The analytic solution of (A.2.4) is used to obtain the orders of magnitude of the lengths needed for conical, hyperbolic and catenoidal horns. The area section of the conical horn is described by the expression

2 x S(x)=πr2 1+ , x0 where x0 is the distance between the throat and the intersection of the cone with the symmetry axis. Using this area section into equation A.2.4 we obtain 2 1 ∂ 2 dp 1 ∂ p 2 (x + x0) = 2 2 =0, (x + x0) ∂x dx c ∂t and the pressure of the outgoing waves is

p0 p = eik(x−ct) , (x + x0) A.2. SLOWLY VARYING CROSS SECTION DUCTS 129

0.7 l 0.6

0.5 Conical Exponential 0.4

(1/2) Catenoidal

S(x) 0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x

Figure A.2.1: Geometry of the considered horns. The length l has been chosen to be 2.4 times the height h of the section at the exit. and the induced speed at x is ⎡ c ⎤ 1 u = ⎣1 − iω ⎦ p. ρc x + x0

The speciﬁc acoustic impedance at the throat of the horn is

ρcx0ω |Z| = , 2 1+(λ/2πx0) where λ is the wavelength. The radiated energy from the conical horn is 2 2 1 2 1 ρcu0πr Π= u0R0S0 = 2 , 2 2 1+(λ/2πx0) where u0 is the particle speed at the throat, R0 is the real part and X the imaginary part of specific acoustic impedance Z = R0 +iX. The radiation efficiency can be evaluated with the transmission coefficient

Π τ = , 1 ρcu2πr2 2 0 where the denominator is the limit of energy that a piston would radiate in a conduit of 1 inﬁnite length ρcu2πr2. 2 0 To compare their efﬁciency to we assume the same length in the x axis and the same area section at the exit as it is shown in Figure A.2.1. The expression for exponential and catenoidal horns are

S(x)=πr2e2x/h exponential

S(x)=πr2 cosh2 (x/h) catenoidal The throat impedance of an inﬁnite exponential horn is 2 2 ρ0 − m m ZA = 1 2 +i 2 , St 4k 4k 130 APPENDIX A. APPENDIX

Table A.2.1: The transmission coefﬁcient τ of conical, exponential and catenoidal horns [Morse, 1948]. Horn τ 1 Conical 2 1+(λ/2πx0) 2 λ Exponential 1 − 2πh 1 Catenoidal 2 c/2πh 1+ f

1.5 Conical Exponential Catenoidal

1 τ

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 H=k ⋅ r

Figure A.2.2: Variation of the transmission coefﬁcient τ with the Helmholtz number of conical, exponential and catenoidal horns.

where St is the area section at the throat, m =(dS(x)/dx)/S(x) is the flare-rate. The throat impedance of an infinite catenoidal horn is −1/2 ρ0 − 1 ZA = 1 2 , St μ where the normalized μ = f/fc, being fc the cutoff frequency, below this frequency the horn transmits nothing and its throat impedance is purely reactive. Exponential and catenoidal have slower flare close to the throat than conical, and thus have much better low frequency loading. The throat cut-off frequency is determined by the rate of expansion of the horn. Doubling the horn length for a given mouth and throat size will halve the flare rate and lower the cut-off frequency by an octave. Halving the horn length will double the cut-off frequency, (raise it by an octave) if the throat and mouth sizes are kept constant. The transmission coefficient τ of the above mentioned horns was already studied in Morse [1948] and is presented in Table A.2.1 and is also shown in Figure A.2.2. The comments about the cut-off frequency of the exponential and catenoidal horns can be appreciated in the mathematical expressions. These horns are then not much efficient for very low frequencies, but also conical need large length to be efficient. A.3. EXPERIMENTAL SET-UP CALIBRATION 131 A.3 Experimental set-up calibration

A.3.1 Influence of microphone insertion Different measurements were performed with microphone, M1 and M2 placed at 3000 mm and 3050 mm from the speaker, respectively. The microphone M2 was introduced at different depths h2 =0, 5, 10, 20 mm into the duct perpendicular to the surface (where h=0 mm means that the microphone is flush to the inner duct surface). The values obtained for the reflection coefficient are summarized in TableA.3.1.

Table A.3.1: Influence in the reflection coefficient R of microphone M2 insertion into the duct. h [mm] R [M1] R [M2] 0 0.6886 ± 0.0022 0.6919 ± 0.0018 5 0.6761 ± 0.0040 0.6843 ± 0.0031 10 0.6857 ± 0.0005 0.6888 ± 0.0012 20 0.5424 ± 0.0016 0.6014 ± 0.0045

The results show that the insertion up to 10 mm does not modify significantly the reflection coefficient. It is not till 20 mm insertion when the pressure field is perturbed by the microphone M2.

A.3.2 Inﬂuence of leakage on microphone holes The holes in the duct modify the acoustic impedance. In order to evaluate the inﬂuence of the leakage of the holes sealing, a test with the sealing of 2 holes where performed. The ratio between the effective pressure measured by the microphones at both sides of the section with the holes (12 mm diameter) are shown in Table A.3.2, microphones are separated 3500 mm.

Table A.3.2: Inﬂuence of the leakage through the duct holes. P 2 Closed holes M PM1 0 0.9419 ± 0.0018 1 0.9556 ± 0.0006 2 0.9621 ± 0.0008

It can be appreciated the reduction in energy measured at the second microphone (M2). The pressure is reduced around 1% for each open hole.

A.3.3 Inﬂuence of relative humidity The relative humidity during the test campaign varied between 26-32%. The inﬂuence of humidity variation with respect to the attenuation due to the thermal and viscous effects is studied using the expressions in Table A.3.3 [Bass et al., 1995]. The frequency used in 132 APPENDIX A. APPENDIX

Table A.3.3: Expression to estimate the pressure attenuation taking into account humidity variation. . Variable Correlation ⎛ ⎛ ⎞1.261 ⎞ T01 ⎝−6.8346⎝ ⎠ +4.61510⎠ · T psat pr 10 psat h hr pa 1 n 2 10log10 (e ) pa 4 0.02 + h frO 24+4.04·10 h pr 0.391 + h ⎛ ⎛ ⎛⎛ ⎞ 1 ⎞⎞⎞ − 3 ⎜ ⎜ T ⎟⎟ ⎝−4 17⎝⎝ ⎠ −1⎠⎠ −1 2 ⎜ . ⎟ p / ⎜ T0 ⎟ f a T ⎜9 + 280he ⎟ rN T0 pr ⎝ ⎠ 2 −1 −3352/T f z 0.1068 · e frN + frN −5/2 2 −1 T − 2239.1 f y 0.01275 · e T frO + + z T0 frO 1 2 p T / s 8.686f 2 (1.84 · 10−11) r + y pa T0 −nsx p p0·e the sine burst was f = 1000 Hz, the test were carried out at a temperature of 22oC and at an atmospheric pressure of 95768 Pa.

In Table A.3.3 Pa is the atmospheric pressure during the test, pr = 101325 Pa the reference pressure, psat the saturation vapor pressure, and po the effective pressure of the signal of the propagation wave at x =0. The frequencies frN and frO are the correspondent to the relaxation of the Nitrogen and Oxygen molecular respectively. The percentage molar concentration of water vapor h in the air and hr is the relative humidity. T is the atmospheric temperature during the test, T0 = 293.15 K is the reference temperature and T01 = 273.16 K is the temperature of the triple point . The factor s·x [dB] is the total absorption with distance to the source sound x The attenuation due to the variation of the relative humidity is close to 0.03% meanwhile the losses due to thermal and viscous effect are around 3%, so we can neglect the effect of humidity ﬂuctuations.

A.3.4 Inﬂuence of pipe junctions

If the junctions between the ducts and horns are not carefully adjusted, undesired reflection can be generated and the results of the measurement can be modified (see Figure A.3.1). In the junctions used in the experimental setup this could reach 2% of the effective pressure of the incidence wave, and should be taken into account in the evaluation of the reflection coefficient. A.3. EXPERIMENTAL SET-UP CALIBRATION 133

Figure A.3.1: Undesired reﬂection at a junction.

A.3.5 Validation of the experimental set-up with analytic expression for the reflection coefficient The results of the reflection coefficient for a flanged (see Figure A.3.3) and unflanged (see Figure A.3.2 ) duct end are presented in Figure A.3.4, compared with the expressions proposed in Levine and Schwinger [1948], Silva et al. [2009].

Figure A.3.2: Tube with unﬂanged open end.

Figure A.3.3: Tube with ﬂanged open end. 134 APPENDIX A. APPENDIX

1 Unflanged from Silva et al. 2009 Experiment unflanged Flanged from Silva et al. 2009 0.9 Experiment flanged

0.8 R

0.7

0.6

0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ka

Figure A.3.4: Variation of the reﬂection coefﬁcient at the duct exit with the Helmholtz number (ka), compared with the expression proposed in Silva et al. [2009].