IS S N 1063-7710, Acoustical Physics, 2016, Vol. 62, No. 1, pp. 27—37. © Pleiades Publishing, Ltd., 2016.

______________ CLASSICAL PROBLEMS OF LINEAR ACOUSTICS ______________AND WAVE THEORY

Study of Thermoviscous Dissipation on Axisymmetric Wave Propagating in a Shear Pipeline Flow Confined by Rigid Wall.

Part I. Theoretical Formulation1Yong Chen", Xiaoqian Chen", Yiyong Huang", Yuzhu Bai", Dengpeng Hu*, and Shaoming Feic

a College of Aerospace Science and Engineering, National University of Defense Technology, 410073 Changsha, Chinab Air Force Early Warning Academy, 430019 Wuhan, China

c Electronic Experiment Center, Chengdu University of Information Technology, 610000 Chengdu, China e-mail: literature.chen@gmail.com;yiyong_h@sina.com; chenxiaoqian@nudt.edu.cn; baiyuzhu@gmail.com;

hudengpeng@163.com; fsm@cuit.edu.cn Received: February 25, 2014

Abstract—Axisymmetric acoustic wave propagation in a shear pipeline flow confined by a rigid wall is studied in the two-part paper. The effects of viscous friction and thermal conduction on the acoustic wave propagat­ing in the liquid and perfect gas are respectively analyzed under different configurations of acoustic frequency and shear flow profile. In Part 1 of this paper, mathematical models of non-isentropic and isentropic acoustic waves are formulated based on the conservation of mass, momentum and energy for both liquid and perfect gas. Meanwhile, comprehensive solutions based on the Fourier-Bessel theory are provided, which gives a general methodology of iteratively calculating features of the acoustic wave. Numerical comparisons with previous simplified models verify the validity of the proposed models and solutions.

Keywords: thermoviscous dissipation, duct acoustics, shear flow, Fourier-Bessel theory DOI: 10.1134/S1063771016010061

INTRODUCTIONWave propagation in pipeline flow is of great inter­

est in both theoretical and industrial applications [ 1 ­6 ]. In the aircraft engineering, for example, particular considerations are placed on the prediction and atten­uation of engine noises [7-13]. The prediction of aeroacosutic features is also important in the catalytic converter design of a transport system [14, 15]. In the ultrasonic pipeline flow measurement [16, 17], accu­rate prediction of ultrasonic wave propagation is of great importance on the improvement of measure­ment performance.

In the literature, features of the isentropic acoustic wave propagating in the moving fluid have been widely addressed by many researchers while the thermovis­cous effect has been neglected [18]. As the inclusion of thermoviscous effect brings about acoustic attenuation due to energy dissipation [19], the corresponding investigations are more valuable compared with the inviscid condition. However, consideration of the thermoviscous dissipation complicates mathematical modeling.

In the case ofperfect gas, Kirchhoff[20] firstly pro­posed a complex transcendental acoustic equation when the gas was stationary. Consecutive researchers

1 The article is published in the original.

gave various solutions to the Kirchhoff equations which were discussed by Tijdeman [21]. In the analysis of wave propagation in a capillary tube, Zwikker and Kosten gave a well-known assumption which was used by Dokumaci [22, 23] who analyzed the effect of a uniform flow on the acoustic wave. Furthermore, Peat [14] and other contributors [16] investigated the fea­tures of wave propagation in a shear flow based on the Zwikker and Kosten theory.

In the case of liquid, process of the thermal con­duction was neglected and only the viscous dissipation was considered. Under these simplifications, Elvira- Segura [24] analyzed the isentropic acoustic wave propagation in stationary water. The authors [25] ana­lyzed the influence of a uniform flow on the isentropic acoustic wave using a proposed method based on the Fourier-Bessel theory. Physically speaking, the vis­cous converts the vibration energy to the internal heat friction during the acoustic disturbances, which leads to a temperature change through the thermal conduc­tion. As a result, isentropic acoustic assumption should be discussed in the presence of the thermovis­cous attenuation.

The motivation of the present paper is to analyze the thermoviscous effect on the wave propagating in the presence of a shear mean flow confined by a circu­lar wall. Particular considerations are placed on the

27

28 YO N G C H E N et al.

The Case of LiquidPhysically speaking, the fluid variables satisfy the

conservation of mass, momentum and energy. In a liq­uid one obtains

^P + V-(pv) = 0 , ( 1 )dt

Fig. 1. Geometry of circular cylindrical pipeline. The radial, circumferential and axial directions are denoted by r, 9 and z. p, v, p, and T represent respectively the fluid pressure, velocity, density, and temperature.

comparison of acoustic features between the non- isentropic and isentropic models. This paper is divided into two parts. The first part details the mathematical formulations of the non-isentropic and isentropic acoustic models in the liquid and perfect gas respec­tively while the constraint of Zwikker and Kosten the­ory is relaxed. Furthermore, general solutions based on the Fourier-Bessel theory are comprehensively deduced and a procedure of iteratively calculating the wavenumber is given. Numerical verification of the validity of the proposed model is given in the first part while the features of wave propagation are compre­hensively analyzed in Part 2.

MATHEMATICAL FORMULATION

Geometric configuration of the problem is demon­strated in Fig. 1. Specifically, the pipeline mean flow is assumed to be purely sheared without swirls, which leads to a radial dependent profile of v 0 (r). The pipe­line radius is denoted by R and the steady temperature of the wall is Twall. Further assumptions are that the fluid’s mean density is uniform and that the tempera­ture of the wall keeps constant in the absence of heat­ing or cooling processes.

An acoustic wave propagation through a fluid can add disturbances to the fluid’s pressure, velocity, den­sity and temperature. These perturbations are sup­posed to be so small that linear approximations keep reasonable. Then, the ambient variables can be expressed as the sum of steady mean components (subscripted by zero) and the first-order perturbations (superscripted by a prime). It should be noticed that the acoustic wave can add disturbances to other coef­ficients, such as the shear viscosity^, bulk viscosity Z, thermal conductivity Kth, heat coefficient at constant pressure cp, specific heat ratio у and volume thermal expansion p. However, these disturbances can be neglected compared with the above-mentioned distur­bances. In what follows, governing equations of the axisymmetric wave are formulated in the liquid and perfect gas respectively.

dv_dt

+ (v • V)v = -Vp + nV2v + (z + n)V (V -v), (2 )

П

Pcp

= KthV 2T

3 r

— + (v • V)TL dt .

e T f f ( v V )p (3)

Xdv i dv j 2

л

d x , -- 8 -V- v

dxt 3+ Z (V- v ) 2

i j= 1 s

where S- represents the Kronecker delta-symbol. In the configurations of a shear mean flow, uniform mean density and constant temperature of the wall, the mean flow satisfies

Vpo = pV 2vо, к thV2Ti(4)

+ pp To v о (r) V2v о (r) + n ) = 0 ,

which shows that the steady temperature is a function of the radial coordinate. As a result, the dynamic equations of the acoustic disturbances can be expressed by

■ + (vо -V)p’ + poV- v' = 0 , dt d v '

d t+ (v0 • V)v' + (v' • V)v0

= - Vp + ^ V 2v' + — ( + nW (V • v'),p0 p0 P<A 3/

d T + ( v0 • V) + ( v' •V)d t

v 0 (r ) V2v 0 (r )T'

p0cp

(5)

(6 )

Kth V2T ■p0c p

PTi ^ + (v0 -V ) p + (v' • V)p0 P0cp L dt

(7)

3■X

d v 0 i . 0 j

p 0c p ij = 1 K jd x , d xtdv j , dvj 2

1 + - - 5,-,-V • vdx, dxi 3

According to the work [26], the acoustic density can be expressed in terms of acoustic pressure and temperature:

p ' p ' -P P 0T', (8)c0

where c0 is the adiabatic sound speed in the unbounded fluid which is theoretically a function of the steady pressure and temperature. However, the

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dependence on the fluid pressure is neglected if the change of the steady pressure is not obvious. As a result, the adiabatic sound speed c0 is finally a function of the radial coordinate.

Under the assumption of a harmonic wave, the first-order disturbances can be represented in the form of exp [i (rot - k0Kz)], where ю is the angular wave fre­quency, i is the unit imaginary number, K is the dimen­sionless axial wavenumber, and k0 is the inviscid wave­number. Using the sound speed at the temperature of the wall c0 = c0 (Twall), we get a constant inviscid wave­number expressed by k0 = ю/ C0. As the axisymmetric acoustic wave is presumed, the acoustic velocity can be

expressed by v' with v'r and vZ being the

radial and axial components respectively while the cir­cumferential component is omitted. Through normal­izing the flow profile by M (r) = v 0 (r)/c0 and the radial coordinate by x = r/R, from Eqs. (5) and (8 ) the acoustic pressure can be expressed by

p _ Ppoco t

(9)xv ’r) - ik0RKvZ .

Using the separation-of-variables principle, the acoustic velocity and temperature can be expressed by

p oc oiY roR ( - KM)

Y

1 A.x dx

mR2 (1 - KM) (pz + Rc0 — prdx

= ip R 2k0Kc02

YpT

+ a 1 d \xdpz \ - R X K 2pz p0 Lx dx\ dx

iRk0K +-1 (/yro(1 - KM) p

1 d (xPr) - iRk0K p z Lxdx

( 1 2 )

mR2 (1 - KM) 1 -

Q 2 2rr-TP c 0 T 0

Ycp )ndT0

Pt + R ^ P r dx

K th 1 — \xdpT ] - R 2k X 2pTp0cp L xdx\ dx

вRT°c° ^ d (xPr) - iRk0KPzjcp L x dx

nPcQR 2T0V2MP0cp

2nc0 dM (dp z

p z

p0cp dx v dx- ik0RKp ,

-,2—2n P R c ^ - ^ 2

P0c p

M V M p T .

(13)

v r, v z ,T '( 1 0 )

= [Pr (x), pz (x), Pt (x)] exp [i (cot - kQKz)].Then expanding Eqs. (6 ) and (7) under the con­

straint of acoustic pressure as shown in Eq. (9) gives

mR2 (1 - KM)(pr dc0- PR L 2 d Pt

Y 0 dx_n Г1 A ( A \ _p0 _xdx\ dx )

2 c0 dc{] ,

-2c0dx Pt

Pr -,2 , 2 V2

c2nK dM_iy®(1 - KM) dx iyw(1 - KM)2 dx _

x 1 d (xP r) - iR^K P z L xdx

1 4 + n3fyro(1 - KM) p0

ddx

1 d (xpr) - iRk0Kpz x dx

( 1 1 )

In the case of isentropic acoustic wave, the temper­ature variation in the acoustic disturbance can be omitted, and the governing equations then can be expressed in terms of acoustic velocity:

_ П

mR2 (1 - KM)pr

I A (xdp^) - % - r 2k2K 2p r p0 Lxdx\ dx ! x

+ 2 c0 dc0■ + - clK dM_iyro(1 - KM) dx iym(1 - KM) 2 dx _

x 1 d (xPr) - iR^Kpz L xdx

+ -1 k + n_iyro(1 - KM) p

x d 1 d (xPr) - iRk0Kpzdx Lx dx

(14)

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30 YO N G C H E N et al.

mR2 (1 - KM) pz + Rc0 — ф,dx

= d fx ^ p j - R \ lK 2pzp0 Lxdx\ dx )

iRk0K2

Co, - + -L (z + Пm у (1 - KM) p0\ 3

1 d (xp ,) - iRk)K p г Lxdx

(15)

Compared with the case of the non-isentropic acoustic wave where the governing equations are expressed in terms of the acoustic velocity and temper­ature as shown in Eqs. (11)—(13), the governing equa­tion of the isentropic acoustic wave is extremely sim­plified.

The Case of Perfect Gas

In the perfect gas, the conservation of momentum and energy reduce to

f + (v-V )v

= - Vp + pV 2v + n V(V • v),(16)

Pcp~6T.dt

+ (v -V)T

dp= k thV 2T + ^ + (v -V)pdt

+ n у L v + ^ _ 2 52 , Idx,- dxt 3J=1 4 J ' У

(17)

Compared with the case of liquid as shown in Eq. (4), the steady fluid satisfies

+

Vp0 = pV 2v0, к thV2Ti

П v 0 (r ) v 2 v 0 (r ) + hf — = 0.(18)

Like the case of the liquid, the constant sound speed inthe perfect gas cQ can be defined as ■x/yR0Twall with R0

being the gas constant. Due to the state equation of the perfect gas p = pR0T and the conservation of mass, the acoustic pressure can be represented in terms of acoustic temperature and velocity:

p = P0R0T' p0R0T0 у , v'/ю (1 - KM) ■

(19)

Using the similar deduction procedure as shown in the case of liquid, the governing equations of the non-isen- tropic acoustic wave propagation in the perfect gas are

mR2 (1 - KM) (p, = -R0Rd T + ■ndx p0

1 d (xdPx dx\ dx1 d |xX^ j ) - R"k02K 2p,

xO, 2 T,2

+ R0 dTo RqTqK dM /ю(1 - KM) dx /ю(1 - KM) 2 dx

■ + -

1 d.xdx

(xp ,) - /RkoKpz

RoTo

dx

/ю (1 - KM) 3p0 _

1 d (xp ,) - iRkoKpz xdx

(2 0 )

mR2 (1 - KM) pz + Rc0 — p ,dx

x

= iR0 R2k0 K pT + — Po

1 d {x ^ T V R2ko2K 2pZxdx\ dx J

x

R0T0Rk0K /pRk0 K ю(1 - KM) + 3p0 _

1 d (xp , ) - iRkoKpz , Lxdx _

(2 1 )

mR2 (1 - KM) 1 - Rq

Lp JndT0

Фт + R Ф,dx

_ Kth 1 d (x X P ) - R2klK Vcpp0 Lxdx\ dx

ПCqR2V2M ф - RqTqRCpPo Z cp

1 d ( ф, ) - iRkoK ф zLxdx

r n dM r _ .RkiKф, I.cpp0 dx v dx

(2 2 )

According to the case of liquid, the governing equations of isentropic model can be similarly deduced. Clearly, the governing equations of non- isentropic acoustic wave propagation in either liquid or perfect gas can be governed by coupled second- order differential equations in terms of acoustic veloc­ity and temperature. Meanwhile, the isentropic acous­tic wave can be modeled as coupled second-order dif­ferential equations in terms of the acoustic velocity. To solve these equations, the corresponding boundary conditions should be consequently discussed.

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Boundary Conditions

As the wall is assumed to be rigid and isothermal, the acoustic temperature and radial component of the acoustic velocity disappear at the wall. Furthermore, the effect of the fluid viscosity promises a non-slip constraint which leads to the vanishing of the axial component of the acoustic velocity. As a result, the boundary conditions of the acoustic velocity and tem­perature are

v r = vZ = T = 0 ^ Фг (x ) (23)= фz (x) = фт (x) = 0 at x = 1 .

As the acoustic wave is assumed to be axisymmet- ric, the radial component of the acoustic velocity dis­appear at the origin of the radial coordinate with

v'r = 0 ^ фг (x) = 0 at x = 0. (24)

It should be noticed that the physical existence of the acoustic wave promises that the functions of the acoustic velocity and temperature are bounded in the interval x e [0 , 1 ].

SOLUTION BASED ON FOURIER-BESSEL THEORY

According to the Fourier-Bessel theory [27] and the constraint of the radial component of the acoustic velocity as shown in Eq. (24), the bounded functions of the acoustic velocity and temperature in the interval x = [0,1] can be expressed in terms of Fourier-Bessel sequences:

да

Ф r (x) = X C J i ( > ) ,

n=1

да

Фz (x) = X C J o ( nx) , (25)n=1

да

Фт (x) = X CTnJ о ( Tnx),n=1

where J, (Lrnx ) and J 0 (Lznx) (J0 (XTnx )) are Bessel func­tions of the first and zero-th orders which are orthog­onal and complete in the Lebesgue space !£2x (0,1). The boundary constraints in Eq. (23) shows the coef­ficients Xn, Xzn and XTn are roots of the following Bessel functions:

Ji (Xrn) = J 0 (Xn) = J 0 ( t i ) = 0 . (26)Obviously, the coefficient of the acoustic temperature X n equals to the coefficient of the axial component of the acoustic velocity Xzn. Moreover, the coefficients(Crn, C l, and CT) of Eq. (25) are independent of the radial coordinate.

The Case of LiquidIn the previous section, the governing equations of

the non-isentropic and isentropic acoustic waves in the liquid are mathematically formulated. This sub­section tries to use the proposed method in terms of the Fourier-Bessel sequences to give a semi-analytical solution.

In the case ofnon-isentropic acoustic wave, substitut­ing Eq. (25) into the governing equations in Eq. (11):

X m R1 r v k +-!- ( + 4 ) )2 j , (x :x) c : = X \—n=1 L - 0 - 0

4q x n“ {iY®Ln=1

2(1 - KM)c0 d C 0 + Kc20 dM dx dx _

J 0 (x rnx )

L-0' 2 n R 2k02K 2 2KM - K M 2) (X x)

1 ( + ? ) x )

mR2 (3KM - 3K2M2 + K 3M 3) - (x ( )2 C(l ( 1 ) KM) j , (xrnx)

dx2 (K - K 2M)C° ) + K 2C|2 ,2 dM

0 dx _

i®y

C J 0 (x nx)

ix nRk0

Rkeую L

c02 ( - K 2M ) ( - 2 K2M + K 3M2) ( n

PRY L

x n (C02 - 2 KMC02 + K 2

i®y - 0

M 2C02 ) J 1 (x ( ) - 2 C0 )

i(x z„x)

dC0 (1 - 2 KM + K 2M 2 ) J 0 (x ( CT

(27)

да i- 30

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32 YO N G C H E N et al.

To use the orthogonal properties of the Fourier- Bessel sequences as shown, multiplying Eq. (27) by

xJ 1 (^ rmx j and then integrating over the internal x e [0, 1 ] yield

{HRL)m Cm + X [2 (RRLlYm - 2(R R L 2) K + (RRL3)‘m K ]n=1

C"

+ - K + ^ ( X " ) 2 + П R ^ K 2 [ 2 ( RRL4)nmK - (RRL5)nmK 2] СП_p0 ' 3 I p0 _

+ [/fflR2 ( 3 {RRL4)Km K - 3(R RL5) K 2 + (R R L 6) K 3) - (X",}f RRL7 t - (RRL8) m 4гую

C"

^ [ 2 ((RZL1)"m K - (RZL2)"m K 2) + (RZL3) K 2 ] Cz„ + iX"R^{RZL4)"mK - (RZL5)"mK 2

iyro

(28)

+ (RZL6)"mK - 2 (RZL7)"mK 2 + (RZL8)"mK 3 , 3

P0 l 3С" + ^ [X " ((RZL4)"m - 2 (RZL5)"m K + (RTLA) K2)

2 ((RZL1)m - 2 (RZL2)"m K + (RTL2)"m K 2) Cl l = 0 .

Furthermore, same deductions can be applied to (14) of the isentropic acoustic wave can be trans- Eqs. (12) and (13). Using the similar procedure, Eq. formed to

(HRL) , С + X I [2 (RRL1)m - 2(RRL2)m K + (R R L 3) k ] C"

- К + ) 2 +n R2k20K 2"][2(RRL4) K - (R R L5) K 2] C"LP0 P0

[i®R2 (3(RRL4( K - 3(RRL5)"m K 2 + (RRL6 ) K 3) - — )2 ((R R L 7) - K(RRL8)"mjm y

C" (29)

^ [ 2 (k (RZL1) - к 2 (RZL2)"m) + K 2 (RZL3)"m] C"Z + [ k (R Z L ) - K 2 (R Z L5)] C + Z + +X "Rki iXlRkюу P0

X [ к (RZL6)"m - 2K2 (RZL7)"m + K 3 (RZL8 ) ] C"? [ = 0,

да

да

while manipulation of Eq. (15) is omitted. Further­more, some of the above-mentioned symbols are

(HRL)m = mR2 + ^ R2k^K2 + -1 (z + ^ ( x ^ ) 2 , p 0 p 0 \ 3 '

1

2j 2 ( x m

while other symbols can be found in a similar way. It should be noticed that the integrals near the initial point (x = 0) may be ill-conditioned. In the numerical calculation, the following approximations of Bessel functions are used:

j c i -y - j 0 ( x " x ) j 1 ( x m x ) ) d -x) dx

J 0 ( x ) = 1 - ( ^ ) 2, J 1 (Xx) = *£.

The integral step should be shorter near the initial point than those in other points. Furthermore, the inte­gral step should be short enough to satisfy accuracy.

The Case of Perfect GasAs in the case of a liquid, using the expressions in

terms of Fourier—Bessel sequences shown in Eq. (25), the governing equations (Eqs. (20)—(22)) of the non- isentropic acoustic wave propagation in the perfect gas can be simplified to

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n=1 L^ ю R R V K + (X n )2 J (Kx)cn = ^

Po 3po n=1

X"nRpKTodM + Xr„Rp (1 - KM) d f /ю dx /ю dx J 0 (X„х)СГ

34n (X'O .L3P0 y ' P02 n R 2k2K2 2KM - K 2M2 )J1 ( )c„

raR 2 ( A t2 + v 3M3\- R0(1 - KM) tq Л Л23KM - 3KM Z + K M ) -/ю

(X n) J 1 (X „ л ) c„

W 1 TodM j a (X„х)c „ - RoRko ( - K M c■ J o (Xnx )C zю dx dx

X„RqRk0 ( - K 2m ) Tq ipXz„Rk0 ( - 2K2M + K3M2)ю 3po

J,1 (X „х ) c„

X„RqR (1 - 2KM + K 2M2) ( x „c 'T T I.

(30)

Multiplying Eq. (30) by xJ1 (Xrmx') then integrating these three functions in the interval x e [0,1], give

(HRG) „ C + 2 | X „ R q

n=1 m

(RRG1) K + X„ (RRG8)„m K - (RRG3)„m K

+(RRG2)„m -X „ (RRG7)„mCr

i a (X „ ) + ^- R2k2K2 [ 2(RRG4)„mK - (RRG5)K2] C„ _3p o p 0 J

+ m R 2 _ 3 ( RRG4)„m K - 3 (RRG5)„m K 2 + ( RRG6) K 3 ] Cr„

+ RoRko U RZG3ym K 2 + X „ (RZG4)„m K - (RZG1)„m K 2

- (RZG2)„m K - X „ (RZG5)„m K 2 J C„z

■ inX„Rko _(RZG6)„m K - 2 (RZG7)„m K 2 + (RZG8)„m K 3 J C„ 3 p o

X „ RqR _ ( RZG6) m - 2 (RZG7)„m K + (RZG8) K 2 ] C„T } = 0,

(31)

О

= i®R2 + — r k k 2 +^ (x m )2,Po 3pov >

where

(HRG),

(R R G 1)„ = 2 , . . . 0j 2 ( x m)0J 0 d x

while other symbols can be found in a similar way. Fur­thermore, Eqs. (21) and (22) can be handled by the same procedure and mathematical formulation for the isentropic model is omitted.

■ J t q d-X J o ( X r„ x ) J 1 ( X rmx ) x d x ,

Iterative Calculation of the Axial WavenumberCareful observations of the governing equations of

the non-isentropic acoustic wave in either liquid or perfect gas show that they can be expressed in homog­enous linear equations with

GX = 0, (32)

where X = \c[ ,■■■, CrN, C1z, • • ■, CZN, CT, • • •, CTN ] T (the superscript “ T ’ denotes the transpose operation) rep­resents a vector of the coefficients of the Fourier— Bessel sequences if the number of the elements of these sequences in Eq. (25) is N.

In the numerical calculation, if the acoustic angu­lar frequency ю, pipeline radius R, the flow Mach number M(x) and the properties of the fluid are given, the symbols listed in the appendix can be numerically calculated. As a result, the symbol G represents a 3N x 3N matrix whose element is a function of the axial wavenumber K. Furthermore, Eq. (25) reveals that the coefficients-composed vector X does not equal to 0, which leads to the existence of a non-trivial solution. Thus, one obtains the vanishing of the corresponding determinant with

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Calculated wave features under different numbers N

N__ 30 40 50 60

Non-isentropic 1/Kr 0.9653 0.9668 0.9677 0.9680A 0.5787 0.7818 0.9825 1.0727

Isentropic 1/Kr 0.8414 0.8424 0.8430 0.8432A 0.4668 0.6290 0.8906 0.9458

det (G (K )) = 0. (33)

Obviously, the axial wavenumber K can be solved by various methods, such as iterative Newton-based pro­cedures. Physically speaking, the specific X cannot be obtained, as it is determined by the acoustic amplitude.

f, Hz

Fig. 2. The relative phase velocity and attenuation coeffi­cient of the fundamental isentropic wave propagating in a tube filled with perfect gas. (a) relative phase velocity; (b) attenuation coefficient.

However, if the first coefficient C1r does not equal to 0, which can be obtained by simple mathematical manip­ulation on Eq. (32), the ratio of all coefficients to C[ with

[ i , cn c; ,c zJ c [ ,••■,cn c r ,c t c r , c tn/c[ Jthen can be obtained by solving Eq. (32). Similarly, the proposed procedure can be directly applied to the cal­culation of the isentropic acoustic wave.

CALCULATION VERIFICATION

In the study ofwave propagation in the perfect gas, Peat [14] proposed variational solutions to the non- isentropic and isentropic acoustic waves confined by a capillary tube in the presence of a laminar flow profile. Theoretical models are under the assumptions of Zwikker and Kosten who neglected the possible exist­ence of high-order acoustic modes in the pipeline flow. Specifically, in the case of a non-isentropic acoustic wave, dimensionless axial wavenumber K satisfies

i (1 - MK)

2 , i iMK

a 2 2 PoroR2Kth ’ n

K2 _2 + i - iMK']s2 3 2 J

= 0, (34)

where M denotes the averaged Mach number. Mean­while, the governing equation of the isentropic acous­tic wave is

In the literature, Dokumaci [22, 23] investigated the problem of fundamental acoustic wave propaga­tion in the uniform flow, where mathematical models were simplified based on the assumption of Zwikker and Kosten. Specifically, the governing function [28] of the non-isentropic acoustic wave is

r + Y - 1J 2 M R , K 2 J 2 (PR) = 0 (36)Y J 0 (ар л ) y(1 - KM)2 J 0 (PR) ’

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S T U D Y O F TH ER M O V ISC O U S D IS S IP A T IO N ON A X IS Y M M E T R IC WAVE 35

where the symbol represents p2 = - /ю(1 - KM) p0j n. In the case of an isentropic acoustic wave, the govern­ing function can be simplified to

In the numerical calculation, an acoustic wave is assumed to propagate in the perfect gas without the presence of a moving flow (M = 0) under the configu­rations of y = 1.4, n = 4.15 x 10-5 kg/(s • m), Kth = 0.0786 W/(K • m ), R = 0.01 m,R0 = 287 j/(kg • K ), ^ = 1184 J/(kg • K ),p0 = 1.225 kg m3, and Twal} = 293 K (20°С). In the numerical calculation, particular concentrations are placed on the relative phase velocity (1/KR) and atten­uation coefficient (A = |8.686k0K7| [dB m ]) of the fundamental mode, where the subscripts “R” and “I” denote the real and imaginary components of the dimensionless axial wavenumber K respectively. Table shows the calculated relative phase velocity and atten­uation coefficient under different numbersof selected Bessel functions of Eq. (25) for acoustic wave with fre­quency f = 10 kHz.

Table 1 shows that the calculated relative phase velocity and attenuation coefficient can be converged as the number of selected Bessel functions increases. As a result, the number of Bessel functions can be numerically decided for predefined accuracy while the convergence of this method can be numerically guar­anteed. In the following analysis, the number of Bessel functions is selected to 60.

Figures 2 and 3 compare the numerical results among the three different models ofwave propagation. Specifically, the differences of the isentropic acoustic wave among the three models are demonstrated in Fig. 2, while the differences of the non-isentropic acoustic wave are illustrated in Fig. 3. In the case of the isen- tropic acoustic wave, numerical result from the model of Peat is different from the results from the models of Dokumaci and present work. Under a low acoustic frequency, numerical results from Dokumaci and present work are nearly the same, which verifies the validity of the present model. However, with the increase of the acoustic frequency, numerical result from Dokumaci shows larger than that from the present work. The phenomena reveal that the effect of the radial component of the acoustic wave, which is neglected in the approximation of Zwikker and Kos- ten, is important in the case of a large acoustic fre­quency. Physically speaking, the increment in the acoustic frequency leads to the decrease in the acous­tic wavelength and then increases the ratio of pipeline radius to the acoustic wavelength. Thus, the assump­tion of the capillary pipeline in the theory of Zwikker and Kosten becomes controversial with the increase of the acoustic frequency.

1/KR1.00 Г0.98

0.96

0.94

0.92 |-

0.90

0.88

0.86

0.84

0.82

(a)

Proposition------ Peat- - Dokumaci

2000 4000 6000 8000 10000 f, Hz

A, dB/m (b)

Fig. 3. The relative phase velocity and attenuation coeffi­cient of the fundamental non-isentropic wave propagating in a tube filled with perfect gas. (a) relative phase velocity; (b) attenuation coefficient.

In the case of the non-isentropic acoustic wave, similar distinctions can be found between Peat’s model and the other two models, which is consistent with the case of the isentropic acoustic wave. Further­more, the difference between Dokumaci and present work is more obvious compared with the isentropic acoustic wave, meanwhile the tendency that numeri­cal result from Dokumaci is larger than that from present proposition keeps the same between non-isen­tropic and isentropic acoustic waves. Compared with isentropic acoustic wave, the relative phase velocity and attenuation coefficient are larger in the case of any one of the three models. It can be learned that the effect of thermal conduction is important in the anal­ysis of acoustic wave propagation in the gas.

Figure 4 investigates the influence of the thermal conduction on wave propagating in the stationary

ACOUSTICAL PHYSICS Vol. 62 No. 1 2016

36 YO N G C H E N et al.

1/Kr1.0010 f1.0005

1.0000

0.9995

0.9990

0.9985

0.9980

0.9975

0.9970

-

A, x10-38.5

(a)

------ Non-isentropic------ Isentropic

2000 4000 6000 8000 10000f, Hz

dB/m (b)

2000 4000 6000 8000 10000f , Hz

Fig. 4. Comparison of the relative phase velocity and atten­uation coefficient between the fundamental non-isen- tropic and isentropic acoustic wave propagating in a tube filled with water. (a) relative phase velocity; (b) attenuation coefficient.

water under the configurations of p0 = 1000 kg/m3, Y = 1, Twall = 293 K (20°Q, p = 0.207 x 10-3 1/°C, cp = 4181.3 J/(kg • K ), Kth = 0.5984W/(K • m ),П = 1 x 10-3 kg/(s • m), Z = 2.4^andR = 0.01 m. Under the configurations, the value of pR and apR are large enough that equation (36) can be simplified to

1 - I z l - K = 0. (38)Y Y (1 - KM)

As the fluid is stationary with M = 0, the dimensionless axial wavenumber K of the non-isentropic acoustic wave equals to 1. Similar deduction of equation gives the identical result with K = 1 in the case of isentropic acoustic wave. As a result, the simplified model of

Dokumaci cannot be applied to analyze wave propa­gation in the liquid. As the numerical result of Peat’s model is obviously different from that of Dokumaci’s model, Peat’s model is not applied to the analysis of wave propagation in water.

Roughly speaking, Fig. 4 reveals that the difference between the non-isentropic and isentropic acoustic assumptions in the case of liquid shows less obvious compared with that in the case of perfect gas. While the coefficients of the fluid viscosity and thermal con­ductivity in the water are larger than those in the per­fect gas, the density of the water is extremely larger than the density of the perfect gas, leading to obviously lower ratios of these coefficients to the fluid density in the water than those in the perfect gas. Then the energy dissipation of wave propagating in the water due to the thermoviscous effect is less obvious than that in the perfect gas, which reveals that the isentropic acoustic assumption is more reasonable in the case of liquid than that in the case of perfect gas.

CONCLUSIONSMathematical formulations of axisymmetric

acoustic wave propagation in the thermoviscous fluid with the presence of a shear flow are presented while the circular pipeline wall is assumed rigid. The models of non-isentropic and isentropic acoustic waves in the liquid and perfect gas are deduced and the constraints of the Zwikker and Kosten theory are relaxed. Based on the boundary properties of the acoustic distur­bances, solutions to the non-isentropic and isentropic acoustic models are given based on the Fourier-Bessel theory. As a result, a general procedure of iteratively calculating the wavenumber is given. Numerical com­parisons with some simplified models validate the pro­posed acoustic model. Furthermore, numerical calcu­lations show that the predictions from the simplified models become more and more controversial as the acoustic frequency increases.

The work described in this paper is funded by the National Natural Science Foundation of China (nos.11404405, 91216201, 51205403 and 11302253). The authors gratefully acknowledge the funding.

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