(Received 11 February 2014; revised 17 June 2014; accepted 19 August 2014) The axisymmetric wave propagation in a viscous fluid with the presence of a uniform flow confined by a circular pipeline is investigated. Particular considerations are imposed on the features of the acoustic wave propagating in the liquid where the thermal conduction is neglected. The boundary constraints at the wall are reasonably discussed for both lined-walled and rigid-walled pipelines. Numerical comparisons of the phase velocity and wave attenuation among three different boundary configurations (rigid wall, steel-composed wall, and aluminum-composed wall) are presented. Meanwhile, the effects of the fluid viscosity and acoustic impedance are coherently analyzed. In the end, parametric analysis of the influence of the acoustic impedance is given in the case of a steelC 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4894801] composed pipeline. V PACS number(s): 43.55.Rg, 43.35.Bf, 43.20.Mv, 43.20.Hq [JDM] I. INTRODUCTION

Wave propagation in a moving fluid confined by a circular pipeline is a common configuration existed in many industrial applications such as ultrasonic flow measurement,1,2 noise attenuation,3,4 and so forth. Present paper takes into consideration the effects of the fluid viscosity3,5–8 and acoustic impedance3,4,9–11 of the wall as the two mechanisms bring about energy dissipation. Accounting the effects of the fluid viscosity and thermal conductivity in a stationary gas, Kirchhoff12 first proposed a complex transcendental acoustic equation in the case of a lossless rigid wall. Tijdeman5 gave a numerical solution to the Kirchhoff formulation and summarized consecutive work. In the case of a uniform pipeline flow, Dokumaci3,6 investigated the fundamental acoustic mode based on the Zwikker and Kosten approximation. Numerical study showed that the assumption of a uniform flow could closely predict the features of an acoustic wave propagating in the shear flow. In the case of a stationary liquid, Elvira-Segura13 assumed the acoustic wave to be isentropic, neglecting the process of thermal conduction. Chen et al.7 expanded the problem in the case of a uniform flow profile. If the wall is not rigid, its influence may alter the propagation speed and attenuation as well. Roughly speaking, two different methodologies exist in the literature to analyze the influence of the wall on wave propagation. By expressing the displacement and stress of the wall and describing the boundary condition at the fluid-wall interface, the features of wave propagation in the stationary fluid can be numerically analyzed. Such a conception was adopted by Grosso,14 Greenspon and Singer,15 Lafleur and Shields,16 Elvira-Segura,13 Sinha et al.,17 Plona et al.,18 and Leighton’s group,10,11 to name a few.

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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Pages: 1692–1701

On the other hand, many researchers prefer to establish the boundary condition at the fluid-wall interface through the acoustic impedance of the wall. In the framework of an inviscid fluid, the theory of the Ingard–Myers boundary condition9,19–21 was widely used in the literature. However, such a method neglects the energy loss due to the fluid viscosity.22 By analyzing the features of wave propagation in the viscous boundary layer, different types of modified boundary condition were proposed by Brambley et al.,19 Rienstra and Darau,9 Auregan and co-workers,22–24 and so on. Although the viscous dissipation was taken into consideration at the viscous boundary layer, the governing equation in the fluid was yet based on the inviscid assumption. The present paper coherently analyzes the effects of the fluid viscosity and acoustic impedance on the acoustic wave propagating in the uniform pipeline flow. Although the uniform flow is controversial in reality, such an approximation can give a reasonable prediction of wave propagation in the shear flow as revealed by Dokumaci.3,6 As the present paper pays particular attention to the acoustic wave in the liquid flow, an isentropic acoustic assumption with the influence of thermal conduction omitted7,13 is reasonable. II. MATHEMATICAL FORMULATION

In this section, the comprehensive mathematical formulation of the isentropic wave propagation is deduced from the conservation of mass and momentum. A viscous fluid is assumed to move uniformly along a circular pipeline while the

FIG. 1. Geometric configuration of the problem in the circular cylindrical coordinate system. r, h, and z denote the radial, circumferential, and axial directions, respectively. y is the vertical axis.

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thermal conduction is neglected. The isentropic acoustic wave is considered to be linear and axisymmetric. Figure 1 presents the configuration of the problem in the cylindrical coordinate system. Specifically, R and Z denote the inner radius of the pipeline and the acoustic impedance of the wall, respectively. q, p, and v represent the fluid density, pressure, and velocity. The uniform flow profile is expressed by U0 ¼ const. A. Governing equation

The basic equations of the problem are the conservation of mass and momentum, expressed by @q þ r ðqvÞ ¼ 0; @t

(1)

@v rp g 2 1 g þ ð v rÞ v ¼ þ r vþ f þ rðr vÞ; @t q q q 3 (2) where g and f are the coefficients of the shear and bulk viscosity which are assumed to be constant.3,10,13 When the viscous fluid experiences a small-amplitude acoustic disturbance (q0 , p0 , and v0 ), its ambient physical variables change to q ¼ q0 þ q0 ; p ¼ p0 þ p0 , and v ¼ v0 þ v0 , where the variables with the subscript 0 denote the steady mean flow. As the steady density and velocity satisfy the conditions of q0 ¼ const and v0 ¼ ½0; 0; U0 (expressed in the cylindrical coordinate system as shown in Fig. 1), one obtains rp0 g 2 þ r v0 q0 q0 1 g þ f þ rðr v0 Þ ¼ 0: q0 3

ðv0 rÞv0 ¼

(3)

If the acoustic wave is considered to be linear, Taylor expansion of Eqs. (1) and (2) can be simplified to @q0 þ ðv0 rÞq0 þ q0 r v0 ¼ 0; @t q0

p0 ¼ c20 q0 ;

(6)

where c0 represents the adiabatic sound speed which is assumed to be constant in this paper. Then Eq. (4) can be simplified to @p0 þ ðv0 rÞp0 þ q0 c20 r v0 ¼ 0: @t

(7)

If a harmonic axisymmetric wave is presumed, the acoustic variables can be expressed as exp ½iðxt k0 KzÞ with xð¼ 2pf Þ, K, and k0 ¼ x=c0 being the angular frequency, the dimensionless axial wavenumber, and the inviscid wavenumber, respectively. Then Eqs. (7) and (5) can be reduced to7 ixð1 KMÞp0 þ q0 c20 r v0 ¼ 0 ) p0 ¼

q0 c20 r v0 ; ixð1 KMÞ (8)

ixq0 ð1 KMÞv0 ¼ rp0 þ gr2 v0 g þ f þ rðr v0 Þ; 3

(9)

where M ¼ U0 =c0 represents the flow Mach number. Insertion of Eq. (8) into Eq. (9) yields " # 2 q c g 0 0 þ fþ ixq0 ð1 KMÞv0 ¼ gr2 v0 þ 3 ixð1 KMÞ rðr v0 Þ:

(10)

(4)

@v0 þ ðv0 rÞv0 þ ðv0 rÞv0 @t g ¼ rp0 þ gr2 v0 þ f þ rðr v0 Þ: 3

Due to the assumption of the isentropic acoustic wave, the acoustic pressure can be expressed by

(5)

By expanding Eq. (10) in the cylindrical coordinate system and non-dimensionalizing the radial coordinate by x ¼ r=R with x 2 ½0; 1, the governing function of the acoustic velocity v0 ¼ ½v0 r ; v0 z (due to the axisymmetric acoustic assumption, the circumferential component of the acoustic velocity is omitted) can be deduced into

" # g @ @v0 r 1 0 c20 1 g @ 1@ 2 2 2 0 0 0 ð xv r Þ ik0 RKv z ; x þ fþ ixR 1 KMÞv r ¼ 2 v r k0 R K v r þ @x q0 x@x x 3 @x x @x ixð1 KMÞ q0 2ð

0

(11) " # g @ @v0 z c20 1 g 1@ 2 2 2 0 ð xv0 r Þ ik0 RKv0 z : x þ fþ ixR 1 KMÞv z ¼ k0 R K v z ik0 RK @x q0 x@x 3 x @x ixð1 KMÞ q0 2ð

0

(12) Using the separation-of-variables principle, the components of the acoustic velocity can be expressed by J. Acoust. Soc. Am., Vol. 136, No. 4, October 2014

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v0 r ¼ ur ðxÞ exp ½iðxt k0 KzÞ;

v0 z ¼ uz ðxÞ exp ½iðxt k0 KzÞ;

(13)

where the functions ur ðxÞ and uz ðxÞ are stepwise and regular in the interval x 2 ½0; 1. Insertion of Eq. (13) into Eqs. (11) and (12), respectively, gives " # 2 g d du u c 1 g d d r r 2 2 2 0 2 R k0 K u r þ ixR2 ð1 KMÞur ¼ x xu fþ Ku þ iRk ð rÞ 0 z ; dx x q0 xdx 3 dx xdx ixð1 KMÞ q0 (14) " # g d duz c20 1 g d 2 2 2 2 ixR 1 KMÞuz ¼ R k0 K uz ik0 K þ x R xu fþ k Ku iR ð rÞ 0 z : dx q0 xdx 3 xdx ixð1 KMÞ q0 2ð

(15)

Obviously, the axisymmetric acoustic wave propagating in the uniform flow can be governed by a set of two secondorder differential equations with the unknown functions ur ðxÞ and uz ðxÞ plus the dimensionless axial wavenumber K.

v0 r ¼ ð1 KMÞ

In the rigid-walled pipeline, the non-invasive condition at the wall leads to the vanishment of the radial acoustic velocity3,7 v0 r ðxÞ ¼ 0 ) ur ðxÞ ¼ 0 at x ¼ 1:

(16)

Furthermore, the fluid viscosity promises the non-slip condition3,6,13 with v0 z ðxÞ ¼ 0 ) uz ðxÞ ¼ 0 at x ¼ 1:

(17)

As a result, Eqs. (16) and (17) constitute the boundary condition in the case of a rigid-walled pipeline. It should be noticed that the non-slip constraint on the steady flow is relaxed, which prevails in the literature.3,6 If the wall’s effect is taken into consideration, the noninvasive condition collapses but the non-slip condition [Eq. (17)] holds. According to the work of Auregan et al.,22–24 the acoustic pressure and radial velocity in the viscous fluid satisfy @ @ p0 @v0 r ¼ ; (18) þ ð1 b Þc0 M @t @t @z Z where b represents the transfer of momentum into the lined wall induced by the fluid viscosity and Z is the acoustic impedance of the wall. If the acoustic frequency is large enough, b vanishes22–24 and the Ingard–Myers boundary condition9,19 recovers @v0 r @ @ p0 ¼ : (19) þ c0 M @t @t @z Z Under the assumption that the acoustic impedance is independent of the axial coordinate, one obtains 1694

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(20)

Substituting Eq. (8) into this equation results in v0 r þ

B. Boundary conditions

p0 : Z

q0 c20 r v0 ¼ 0 ) ur ð xÞ ixZ q c2 d þ 0 0 xu Ku iRk ð rÞ 0 z ¼ 0: ixZ xdx (21)

Using the non-slip condition [Eq. (17)] yields q0 c20 dur ð xÞ q0 c20 u ð xÞ ¼ 0; at x ¼ 1: þ 1þ ixRZ dx ixRZ r

(22)

As a result, Eqs. (17) and (22) constitute the boundary condition of wave propagation in the lined-walled pipeline. Furthermore, the axisymmetric wave promises the vanishment of the radial acoustic velocity at the pipeline center with ur ðxÞ ¼ 0; at x ¼ 0. Meanwhile, the axial acoustic velocity remains finite.

III. SOLUTION BASED ON FOURIER–BESSEL THEORY

According to the Fourier–Bessel theory,25 the bounded functions ur ðxÞ and uz ðxÞ may be expressed by ur ðxÞ ¼

1 X

Crn J1 ðkrn xÞ;

(23)

Czn J0 ðkzn xÞ:

(24)

n¼1

uz ðxÞ ¼

1 X n¼1

The functions J0 ðkzn xÞ and J1 ðkrn xÞ are Bessel functions of the zeroth and first orders, respectively. In the rigidwalled pipeline, krn and kzn are determined by Eqs. (16) and (17), Chen et al.: Wave propagation in lined pipeline

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J1 ðkrn Þ ¼ J0 ðkzn Þ ¼ 0:

(25)

In the lined-walled pipeline [Eq. (22)], the constraint equations are changed to r r q0 c20 r q0 c20 J0 kn J2 kn kn þ 1 þ J1 krn ¼ 0; i2xZR ixZR

(26)

J0 ðkzn Þ ¼ 0:

(27)

According to the orthogonal property of Bessel function, the rigid-walled configuration leads to ð1 r r J22 krm dmn ; J1 kn x J1 km x xdx ¼ 2 0 ð1 z z J12 kzm dmn ; J0 kn x J0 km x xdx ¼ 2 0

(28)

(29)

where the symbol dmn denotes the Kronecker delta function. Given specific acoustic velocity components [ur ðxÞ and uz ðxÞ], the corresponding coefficients in Eq. (23) can be calculated by ð1 2 r u ðxÞJ1 krn xdx; Cn ¼ 2 r (30) J2 kn 0 r Czn

2 ¼ 2 z J1 kn

ð1 0

uz ð xÞJ0 krn xdx;

(31)

which shows that these coefficients are independent of the radial coordinate. In the lined-walled pipeline, the orthogonal property of Bessel function can be expressed by " # ! ð1 r r r 2 1 1 r 1 2 r J0 km J2 km 1 r 2 J1 km dmn ; J1 kn x J1 km x xdx ¼ þ 8 2 0 km ð1 0

J0 kzn x

z J12 kzm dmn : J0 km x xdx ¼ 2

(32)

(33)

As in the case of the rigid-walled pipeline, the coefficients (Crn and Czn ) are independent of the radial coordinate. If the Fourier–Bessel sequences [Eqs. (23) and (24)] are substituted into Eqs. (14) and (15), respectively, one obtains 1 X

1 X n¼1

g r 2 kn þ R2 k02 K 2 Crn J1 krn x q 0 n¼1 " ) # z r 2 r r c20 1 g z z ikn Rk0 KCn J1 kn x kn Cn J1 kn x þ þ fþ ; 3 ixð1 KMÞ q0

ixR2 ð1 KMÞCrn J1 krn x ¼

n¼1

2ð

ixR 1

KMÞCzn J0

kzn x

( 1 X

( 1 X

g z 2 kn þ R2 k02 K 2 Czn J0 kzn x q0 n¼1 " ) # c20 1 g r r r þ Rkn Cn J0 kn x iR2 k0 KCzn J0 kzn x fþ : ik0 K 3 ixð1 KMÞ q0

¼

(34)

Some rearrangements yield " r 2 # 1 2 r 2 X c k k g 4g 0 n þ n ixR2 ð1 KMÞ þ R2 k02 K 2 þ fþ Crn J1 krn x q0 q0 3 ixð1 KMÞ n¼1 " # 1 z z X c0 k RK ik Rk0 K g n þ n fþ ¼ Czn J1 kzn x ; ð1 KMÞ q0 3 n¼1 J. Acoust. Soc. Am., Vol. 136, No. 4, October 2014

Chen et al.: Wave propagation in lined pipeline

(35)

(36)

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# z 2 2 2 g k ixR K 1 4g n R2 k02 K 2 Czn J0 kzn x þ ixR2 ð1 KMÞ þ fþ ð Þ q q 3 1 KM 0 0 n¼1 " # 1 r r X c0 kn RK ik k0 RK g Crn J0 krn x : fþ ¼ þ n ð1 KMÞ q0 3 n¼1

1 X

"

Multiplying Eq. (36) by J1 ðkrm xÞx and Eq. (37) by J0 ðkzm xÞx, respectively, and then integrating over the interval x 2 ½0; 1 lead to ðRRÞm Crm þ

1 X ðRZ Þnm

# c0 kzn RK ikzn Rk0 K g fþ þ Czn ¼ 0; ð1 KMÞ q0 3 ðZZ Þm Czm þ

1 X

(38)

# c0 krn RK ikrn Rk0 K g Crn ¼ 0; fþ þ ð1 KMÞ q0 3

(39)

where g 2 2 2 R k0 K q0 r 2 1 c20 4g r 2 km ; þ þ fþ k q0 3 ixð1 KMÞ m (40)

ðRRÞm ¼ ixR2 ð1 KMÞ þ

ixR2 K 2 g z 2 ðZZÞm ¼ ixR2 ð1 KMÞ k þ ð1 KMÞ q0 m 1 4g 2 2 2 k R K ; þ fþ q0 3 0 ð1 ðRZÞnm ¼ ðHRÞm J1 ðkzn xÞJ1 ðkrm xÞxdx;

(41)

(42)

0

ðZRÞnm ¼

2 z 2 J1 km

ð1 0

J0 krn x J0 kzm x xdx;

(43)

with ðHRÞm ¼ 8ðkrm Þ2 =½ðJ0 ðkrm Þ J2 ðkrm ÞÞ2 ðkrm Þ2 þ 4ððkrm Þ2 1ÞJ12 ðkrm Þ in the lined-walled pipeline and ðHRÞm ¼ 2=J22 ðkrm Þ in the rigid-walled pipeline. If the number of the Bessel functions in Eqs. (23) and (24) is N, Eqs. (38) and (39) can be expressed by GðKÞX ¼ 0;

(44)

where X ¼ ½Cr1 ; Cr2 ; …; CrN ; Cz1 ; Cz2 ; …; CzN T is the coefficient-composed vector. GðKÞ is a matrix of 2N 2N whose element is a function of the dimensionless axial wavenumber K if M, R, and xð¼ 2pf Þ are specified. According to Eqs. (23) and (24), it can be learned that the coefficients of the Fourier–Bessel series do not vanish simultaneously due to the non-zeros of the acoustic velocity, thus one may 1696

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(45)

As a result, the dimensionless axial wavenumber K can be numerically solved.7,8,26 IV. SPECIAL CASE: INVISCID FLUID

ðZRÞnm

n¼1

"

obtain the constraint of X 6¼ 0. Physically speaking, the condition of X ¼ 0 reveals that the acoustic velocity disappears. Consequently, the corresponding determinant of Eq. (44) vanishes, detðGðKÞÞ ¼ 0:

n¼1

"

(37)

In the framework of an inviscid fluid, the convected wave equation can be represented as a function of the acoustic pressure p0 ¼ up ðxÞ exp ½iðxt k0 KzÞ.1,19 If the mean flow is uniform, an analytical solution exists with qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (46) up ðxÞ ¼ J0 ðk0 R ð1 KMÞ2 K 2 xÞ: On the neglect of the fluid viscosity, substituting Eq. (46) into Eq. (9) gives the expression of the radial acoustic velocity, ur ðxÞ ¼

dup ð xÞ 1 : ixq0 Rð1 KMÞ dx

(47)

In the lined-walled pipeline, insertion of Eqs. (46) and (47) into Eq. (19) yields the constraint function of the acoustic pressure, 2 dup q xRð1 KMÞ þi 0 up ¼ 0 at x ¼ 1: dx Z

(48)

In the rigid-walled pipeline with Z ¼ 1, Eq. (48) can be simplified to dup ð xÞ ¼ 0 at x ¼ 1: dx

(49)

V. NUMERICAL STUDY

In what follows, wave propagation in water is considered. The constant parameters27 are q0 ¼ 1000 kg=m3 , c ¼ 1500 m=s, g ¼ 1 103 kg=ðs mÞ, f ¼ 2:4g, R ¼ 4 mm, and f ¼ 1 MHz. If the lined wall is composed of Helmholtz resonators,9 the acoustic impedance can be expressed by xD ~ iq0 c0 cot Z ðxÞ ¼ Z0 þ ixm ; (50) c0 ~ where Z0 is the specific acoustic resistance of the wall, mð¼ 0:02q0 þ ð1=3Þq0 DÞ is the damping inertance, and D is the Chen et al.: Wave propagation in lined pipeline

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TABLE I. The acoustic impedance of the two configurations. Material

Z0 : Pa s=m

Absolute value of Z: Pa s=m

Phase of Z: deg.

Steel Aluminum

4:48 107 1:73 107

1:38 108 1:32 108

71:07 82:47

liner depth. Equation (23) reveals that the number N of the Bessel functions should be large enough to make sure that the numerical calculation of the axial wavenumber is converged. According to the previous research,7 the selection of N ¼ 50 can give an acceptable numerical result. In the numerical calculation, particular considerations are placed on the phase velocity (cp ¼ c0 =KR ;) and attenuation coefficient (A ¼ j8:686k0 KI j : dB=m), where the subscripts “R” and “I” denote the real and imaginary components, respectively. To get a normalized expression of the phase velocity, the relative phase velocity is defined by cp =c0 ¼ 1=KR .

configurations (rigid wall, steel-composed wall, and aluminum-composed wall11) are given. As an example, the liner depth is assumed to be D ¼ 2 mm. From Eq. (50), the acoustic impedance can be calculated as shown in Table I. Special concentrations are given to the features of the first two modes while the discussions of other modes are omitted. Comprehensive analysis of higher order modes propagating in the uniform flow confined by the rigid wall can be found in Chen et al.7 1. Phase velocity

In this subsection, comparisons of the relative phase velocity and attenuation coefficient among three different

Figure 2 demonstrates the relative phase velocity of the first mode as a function of the Mach number propagating in the downstream (a) and upstream (b) directions. Meanwhile, Fig. 3 illustrates the corresponding relative phase velocity of the second mode. Obviously, the relative phase velocity of each mode increases along with the Mach number in the downstream propagation but decreases against the Mach number in the upstream propagation. Physically speaking, as the downstream propagation is along the flow direction, the effect of the steady flow accelerates the propagation speed. On the other hand, as the

FIG. 2. The relative phase velocity of the first mode confined by the three different walls in the downstream (a) and upstream (b) propagation.

FIG. 3. The relative phase velocity of the second mode confined by the three different walls in the downstream (a) and upstream (b) propagation.

A. Rigid wall and lined wall

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upstream propagation is against the flow direction, the flow convection decelerates the propagation speed. If the flow Mach number is higher, the effect of flow convection on the propagation velocity becomes more obvious. A careful comparison of the relative phase velocity among the rigid, steel-composed and aluminum-composed walls shows that the elastic vibration of the wall speeds up the propagation velocity. With the increase of the specific acoustic resistance (Z0 ), the relative phase velocity of each mode slows down in the downstream and upstream directions. However, the difference of the relative phase velocity is minor between the steel-composed and aluminumcomposed walls as the absolute values of acoustic impedance are nearly the same. Physically speaking, if the specific acoustic resistance (Z0 ) is higher, the absolute value of the corresponding acoustic impedance (Z) becomes larger (see Table I). Then the rigid property of the wall shows more obvious (the absolute value of the acoustic impedance of the rigid wall can be assumed infinite). From Fig. 2, it can be learned that the impact of the three different walls on the relative phase velocity of the first

While Fig. 4 displays the attenuation coefficient of the first mode in the downstream (a) and upstream (b) propagation, Fig. 5 illustrates the scenarios of the second mode. With the increase of the Mach number, the attenuation coefficient of each mode decreases in the downstream propagation but increases in the upstream propagation depending on the configuration of the wall. Especially, the energy dissipation due to the fluid viscosity and acoustic impedance becomes slight in the downstream propagation. Physically speaking, the effect of steady flow accelerates the acoustic propagation, the processes of viscous dissipation in the fluid and wave absorption at the wall become less obvious

FIG. 4. Attenuation coefficient of the first mode confined by the three different walls in the downstream (a) and upstream (b) propagation.

FIG. 5. Attenuation coefficient of the second mode confined by the three different walls in the downstream (a) and upstream (b) propagation.

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mode is very small, which applies to the second mode as shown in Fig. 3. Furthermore, comparison between Figs. 2 and 3 shows that the relative phase velocity of the second mode is larger than that of the first mode in the downstream and upstream propagation. Numerical comparisons of 1=KR among different modes can be found in Chen et al.7 2. Wave attenuation

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compared with the case of the stationary fluid. With the increase of the uniform flow profile, the attenuation coefficient caused by the two mechanisms becomes smaller and smaller. On the other hand, the energy dissipation in the upstream propagation is strengthened as the processes of the viscous dissipation and wave absorption are reinforced by the decelerated propagation speed. If acoustic impedance is considered, careful investigation reveals that as the flow Mach number goes up, the increment ratio of the attenuation coefficient in the upstream propagation is more rapid than the decrement ratio in the downstream propagation. Such a phenomenon indicates that the influences of the flow convection on the acoustic wave between the downstream and upstream propagation are asymmetric with respect to the case of the stationary fluid. Among the three configurations, the attenuation coefficient in the steel-composed wall is the largest while the attenuation coefficient in the rigid wall is the smallest. In the rigid-walled pipeline, the source of wave attenuation is only from the viscous loss. In the lined-walled pipeline, the

FIG. 6. Attenuation coefficient of the first mode due to the effects of the viscosity and acoustic impedance in the downstream (a) and upstream (b) propagation. J. Acoust. Soc. Am., Vol. 136, No. 4, October 2014

energy dissipation from the wall impedance is added, which leads to a greater attenuation coefficient. An interesting phenomenon is that the attenuation coefficient in the steel-composed wall is bigger than that in the aluminumcomposed wall, even though the absolute value of the acoustic impedance in the steel-composed wall is nearly identical to that in the aluminum-composed wall (see Table I). It can be seen from Table I that the phase of the acoustic impedance in the steel-composed wall is 71:07o while the phase in the aluminum-composed wall is 82:47o . This may be a possible interpretation of the distinct difference between the steelcomposed and aluminum-composed walls. It has been demonstrated that the effects of the fluid viscosity and wall impedance lead to the energy dissipation in wave propagation. An attractive question may be that whether the two mechanisms of energy dissipation take effect independently. Figure 6 gives a numerical analysis in the case of the steel-composed wall. Specifically, Fig. 6 displays the attenuation coefficient of the first mode in the downstream [Fig. 6(a)] and upstream [Fig. 6(b)] propagation. Clearly, the attenuation coefficient in the presence of the fluid viscosity and acoustic impedance (“vis þ impedance”)

FIG. 7. The absolute values of the amplitude (a) and phase (b) of the acoustic impedance as functions of the liner depth. Chen et al.: Wave propagation in lined pipeline

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is not the sum of the coefficients contributed independently by the fluid viscosity (“vis þ rigid”) and the acoustic impedance (“inv þ impedance”). The inequality is more obvious in the upstream propagation when a larger Mach number is present. These phenomena illustrate that one should consider both the effects of fluid viscosity and wall impedance to get a precise prediction of wave attenuation in the lined-walled pipeline. B. The effect of the liner depth

In this subsection, the effect of liner depth (D) on wave propagation is discussed. Figure 7 exhibits the amplitude [Fig. 7(a)] and phase [Fig. 7(b)] of the acoustic impedance [Eq. (50)] as functions of the liner depth. Numerical calculation is proceeded for the first acoustic mode with M ¼ 0:1 confined by the steel-composed wall. 1. Phase velocity

Figure 8 presents the effect of liner depth on the relative phase velocity in the downstream [Fig. 8(a)] and upstream

FIG. 8. The effect of liner depth on the relative phase velocity in the downstream (a) and upstream (b) propagation. 1700

J. Acoust. Soc. Am., Vol. 136, No. 4, October 2014

[Fig. 8(b)] propagation. At the same time, the difference between the inviscid (“inv”) and viscous (“vis”) assumptions is clarified. Specifically, the relative phase velocity in the inviscid fluid is larger than that in the viscous fluid in the downstream and upstream propagation. It then can be revealed that the existence of fluid viscosity decelerates the propagation speed. Although the variation trend of the relative phase velocity with respect to the liner depth is complex, the relationship between the relative phase velocity and the amplitude of the acoustic impedance [Fig. 7(a)] may be simple. Generally speaking, a larger amplitude value of the acoustic impedance corresponds to a smaller propagation velocity in the viscous and inviscid assumptions. Although the amplitude range of the corresponding acoustic impedance is large, the change interval of the relative phase velocity remains short. Such a phenomenon can also be found in Figs. 2 and 3. Furthermore, the influence of acoustic impedance on the relative phase velocity is more obvious in the viscous fluid than that in the inviscid fluid.

FIG. 9. The effect of liner depth on the attenuation coefficient in the downstream (a) and upstream (b) propagation. Chen et al.: Wave propagation in lined pipeline

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2. Wave attenuation

Figure 9 illustrates the effect of liner depth on the attenuation coefficient in the downstream [Fig. 9(a)] and upstream [Fig. 9(b)] propagation. Comparison between the viscous and inviscid assumptions is simultaneously shown. The variation of attenuation coefficient as a function of the liner depth is sharper compared with the case of the relative phase velocity. If the amplitude of the acoustic impedance is high [Fig. 7(a)], the absolute value of the corresponding phase may reach the maximum point of 90o [Fig. 7(b)]. The attenuation coefficient then goes down to the case of the rigid wall as shown in Fig. 9. Comparison between Figs. 7 and 9 reveals that the tendency of the attenuation coefficient with respect to the absolute value of the phase [Fig. 7(b)] may be simple while the relationship between the attenuation coefficient and liner depth is complicated. As the absolute value of the phase increases and finally goes to the maximum point of 90o , the attenuation coefficient decreases and eventually simplifies to the case of the rigid-walled configuration. Similar results can be found in Figs. 4 and 5. It should be noted that the difference between the viscous and inviscid assumptions is more apparent under the condition of a smaller phase. As a result, to get a comprehensive description of wave propagation with high quality, the effects of fluid viscosity and wall impedance should be taken into consideration synchronously. VI. CONCLUSIONS

Present paper investigates the axisymmetric wave propagation in the viscous fluid with uniform flow confined by a circular pipeline. As particular considerations are given to the phase velocity and wave attenuation in the liquid, the effect of thermal conduction can be neglected. The effects of acoustic impedance at the wall and fluid viscosity on phase velocity and attenuation are analyzed synchronously. Numerical calculations reveal the following results. (1) The phase velocity of each mode seems dominantly determined by the amplitude of the acoustic impedance. As the amplitude of the acoustic impedance goes up, the phase velocity decreases and finally goes down to the rigid-walled configuration (see Figs. 2 and 3, and 8). Furthermore, the phase plays a more important role on the wave attenuation of each mode compared with the amplitude. As the absolute value of the phase goes up to 90o , the attenuation coefficient goes down to the case of the rigid wall (see Figs. 4 and 5, and 9). (2) The energy dissipation due to the fluid viscosity and acoustic impedance should be considered synchronously to get a comprehensive description of wave propagation. The two processes coherently impose influences on the phase velocity and wave attenuation. With the increase of the acoustic impedance of the wall, its effect becomes small and the wall finally behaves rigid. ACKNOWLEDGMENTS

The work described in this paper is funded by the National Natural Science Foundation of China (Grants Nos. J. Acoust. Soc. Am., Vol. 136, No. 4, October 2014

11404405, 91216201, 51205403, and 11302253). The authors gratefully acknowledge the funding. 1

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