- Email: [email protected]
Finite amplitude standing wave in closed ducts with cross sectional area change
Wave Motion 42 (2005) 226–237
Finite amplitude standing wave in closed ducts with cross sectional area change M.A. Hossain a,∗ , M. Kawahashi b , T. Fujioka a b
a Development Group, Anest Iwata Corporation, 3176 Shinyoshida, Yokohama, Japan Department of Mechanical Engineering, Saitama University, 255 Shimo-Okubo, Saitama, Japan
Received 20 September 2004; received in revised form 7 February 2005; accepted 11 February 2005 Available online 5 March 2005
Abstract A linear theory was developed to estimate the resonant frequency and the standing wave mode in closed ducts with variable cross sectional area. The finite amplitude standing wave in closed ducts with area contraction was numerically studied. The effect of area contraction ratio and the gas properties on nonlinear standing wave was investigated. One-dimensional numerical simulation was conducted using fundamental fluid dynamics equations taking the effects of viscosity and frictions into consideration. Finite difference MacCormack scheme was used by preserving the computational accuracy within second-order in time and fourthorder in space, respectively. The dependences of the pressure waveform on the area contraction ratio, the limit of area contraction ratio to obtain shockless high amplitude pressure waveform with the variation of gas properties and the amplitude of pressure waveform with the variation of duct geometries were investigated. In addition, the compression ratios at the closed end of the duct with different area contraction ratios for different duct geometries were presented. This study suggests that changes in duct geometry can have a significant impact on overall performance of acoustic compressors. © 2005 Elsevier B.V. All rights reserved. Keywords: Linear acoustics; Finite amplitude wave; Area contraction ratio; Compression ratio; Numerical analysis; Acoustic compressor
1. Introduction The oscillations of gas column in closed duct at resonant frequency induce large amplitude acoustic standing wave. The large amplitude standing wave phenomena and nonlinear effects induced in gas filled ducts have wide engineering applications. The use of large amplitude pressure fluctuation obtained at the closed end of duct is a special application of the effect in developing the acoustic compressor. Furthermore, the nonlinear phenomena, ∗
Corresponding author. Tel.: +81 45 591 1122; fax: +81 45 593 0524. E-mail address: [email protected] (M.A. Hossain).
0165-2125/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2005.02.003
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
227
namely the acoustic streaming and thermoacoustic effect, can be used to constrain the convection current and to develop thermoacoustic engines and refrigerators. The amplitude of the standing wave and nonlinear phenomena largely depends on the shape of the duct. Finite amplitude wave motion in ducts has been theoretically and experimentally investigated extensively in many earlier works. It has been known that shock wave generate and propagate periodically back and forth in a gas-filled closed cylindrical duct. In this context, Betchov [1] made an earlier theoretical attempt to account for this phenomenon. He introduced a simple theoretical analysis of the inviscid gas motion, taking frictional losses to the walls of the duct into consideration. Saenger and Hudson [2] devised a theory of the steady state motion of the gas in the Kundt tube in its fundamental mode that includes the dissipative effects of wall friction and heat conduction to the duct walls. Chester [3] described nonlinear phenomena in a closed straight duct at near resonant state. His solution included the effect of compressive viscosity and also the effect of the boundary layer at the duct wall. Weiner [4] developed a technique to estimate finite amplitude effects for standing acoustic waves. His method equates the nonlinear energy loss from the fundamental mode of oscillation to the work done on this mode by its higher harmonics assuming that the waveform is a sawtooth. Temkin [5] studied large amplitude oscillations of air column in piston driven resonant straight duct experimentally. He derived the rate of energy dissipation per unit area of such a shock front as proportional to the cube of the wave amplitude. Coppens and Sanders [6] formulated and solved one-dimensional, nonlinear, acoustic wave equation with dissipative term describing the viscous and thermal energy losses encountered in a rigid-wall closed duct. Their solutions lead to a steady state distribution of harmonics of the fundamental, the amplitude and phase of each term being strong functions of frequency and the absorptive process. Cruikshank [7] verified Chester’s theory experimentally and concluded that the agreement is better when the duct is excited above resonance than below the resonant frequency. His results also disclosed that the resonant frequency shifts with the increase in piston oscillating displacement amplitude. Lee and Wang [8] numerically analyzed the nonlinear resonance of an air-filled acoustic chamber using the fully nonlinear Lagrangian wave equation, in which the air was assumed to be excited by an external body force. Researchers and scientists have found that the amplitude of standing wave generated by the oscillations of gas column in straight duct was limited due to shock wave generation. This limitation arises due to the distortion of standing wave that happens when harmonic frequency coincides with the modal frequency of standing wave. However, recent investigations revealed that it is possible to obtain large amplitude standing wave without shock formation by changing the shape of the ducts. Gaitan and Atchley [9] studied finite amplitude standing waves in harmonic and anharmonic (duct with variable cross sections) air-filled closed ducts driven by a piston at resonance. They found that the detuned ducts suppress the energy transfer effectively into the higher harmonics and thereby suppress generation of higher harmonics. The results revealed that the waveform is strongly dependent on the shape of the ducts. Elvira-Segura and Riera-Franco de Sarabia [10] developed a finite element algorithm to study nonlinear standing wave. Recently, Lawrenson et al. and Ilinskii et al. [11,12] studied finite amplitude standing wave in resonant ducts with different shapes numerically and experimentally. They found high amplitude shockless acoustic pressure fluctuation at the closed end of area change ducts. Lucas [13] showed that the high amplitude resonant duct could be used in real applications as a compressor or pump. High intensity applications of acoustic energy in industrial processing are based on nonlinear effects produced by finite amplitude pressure variations. The nonlinear effects generated in closed acoustic ducts are steady circulating streaming (acoustic streaming) and thermoacoustic effect. Kawahashi and Arakawa [14] studied the acoustic streaming phenomena in closed cylindrical ducts by numerical simulation. Classic works on practical applications of the thermoacoustic effect for the design and development of thermoacoustic engines [15] and refrigerators [16] are found in literature. Therefore, a detailed knowledge of the finite amplitude standing wave as well as nonlinear pressure distribution inside the duct is essential for the development of practical systems. Fundamental characteristics of wave motion in the air-filled closed area change duct with different shapes were recently discussed by Kawahashi et al. [17]. This paper deals with the theoretical and numerical investigation of the oscillations of the gas column in closed ducts with area contraction. The cross sectional area of the duct was decreased from the piston end towards the closed end. We investigated the limit of area contraction ratio to get shockless pressure waveform, the effect of gases on
228
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
finite amplitude oscillations in closed area change ducts and the effect of duct lengths. A linear acoustic theory was presented for the analysis of standing wave in closed ducts with area changes. The governing equations were derived by taking viscosity and frequency dependent wall friction into consideration. One-dimensional numerical simulation was carried out using the MacCormack scheme keeping accuracy with second-order in time and fourth-order in space.
2. Theoretical formulation 2.1. Linear acoustics theory The axisymmetric duct with cross sectional area contraction from the driving end towards the closed end is used for developing an acoustic compressor. The phenomena of the wave motion in the duct depend on the area contraction ratio, geometry of the duct shape and driving acceleration. The development of the acoustic compressor is based on the compression ratio measured from the pressure fluctuation at the closed end of the duct. In this section, a linear theory is developed to estimate the resonant frequency and standing wave mode of the gas column oscillations in closed area change exponential ducts. The linear acoustics theory for plane wave propagation in closed axisymmetric area change ducts with cross sectional area contraction towards the closed end is expressed as follows: 2 ∂2 u 1 dA ∂u d 1 dA 2 ∂ u (1) =c + +u A dx ∂x dx A dx ∂t 2 ∂x2 where u is the fluctuating velocity, c the sound speed, A the cross sectional area, t the time and x is the coordinate along the length of the duct. Equation (1) is solved considering the oscillations of the gas column in area contraction exponential closed ducts by using the piston as a sound source. One important characteristic of the exponential area change duct is that the rate of change of cross sectional area contraction (1/A)(dA/dx) along the longitudinal direction of the duct is constant. To solve equation (1), the following boundary conditions are employed: x = 0 (closed end) : x = l (piston end) :
u=0 u = u0 cos ωt
(2)
where l is the duct length, u0 the piston oscillating velocity amplitude and ω is the angular frequency of the oscillation. The geometry of the exponential duct is expressed by following equation: A(x) = A0 exp(mx)
(3)
where A0 is the cross sectional area at the closed end, m is a constant which is defined as the rate of change of area contraction and calculated from the cross sectional area at piston end (Ap ) and the area contraction ratio (Ap /A0 ). Taking sinusoidal oscillations of the gas column into consideration in closed ducts, we can write, u = u(x) exp(iωt)
(4)
Modifying Equation (1) by using Equation (4), we can write, d2 u(x) du(x) + k2 u(x) = 0 +m 2 dx dx Here, k2 =
ω2 . c2
(5)
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
229
The fundamental solution of Equation (5) is finally written as follows: u(x) = −Aα exp(αx) − Bβ exp(βx) 2 m m m2 2 Here, α = − 2 − i k − 4 and β = − 2 + i k2 − m4 . Applying the boundary conditions, constants A and B can be written in the form: u0 , A=− α exp(α) − exp(β)
B=
(6)
u0 β exp(α) − exp(β)
At last, we obtain the following equation to calculate the angular frequency of the gas column oscillations in closed exponential ducts with area contraction: ω 2 = c2
nπ 2 l
+
c 2 m2 4
(7)
In linear theory, the fluctuating pressure amplitude in closed duct is infinite at resonant frequency. However, it is possible to estimate the standing wave mode by the derived linear acoustic theory. The standing wave mode of fluctuating velocity amplitude that is derived from the above expressions is Mode (u) : −u0 [exp(αx) − exp(βx)] By using momentum equation with the assumption of small amplitude fluctuation, ∂u 1 ∂p =− ∂t ρ ∂x The following standing wave pressure mode is obtained: 1 1 Mode (p) : −iωρ0 u0 exp(αx) + exp(βx) α β
(9)
(10)
where p is the pressure, ρ the density and ρ0 is the mean density of gas in closed duct. 2.2. Finite amplitude standing wave When the oscillations of the gas column in a closed duct are conducted at resonant frequency, high amplitude pressure waveform is induced in the duct. In the linear theory described in Section 2.1, the amplitude of the pressure waveform becomes infinite at resonance. However, practically, the amplitude of pressure waveform is limited due to the effect of viscosity as well as various nonlinear phenomena. A shock formation in cylindrical duct is one of the typical examples of such nonlinear phenomena. In this section, the basic equations will be derived to analyze the finite amplitude wave motion in closed axisymmetric ducts
cross sectional area changes. Considering the cylindrical duct as standard, the Stokes number √ with s = D/ ν/ω of standing wave in closed ducts is predicted to be very large. Here, D is the diameter of the duct, ν the kinematic viscosity of the gas frequency of the vibration. In addition, since the value of
and ω is√the angular the oscillating Reynolds number Re = 2u/ ν/ω is less than the critical (according to Merkli and Thomann [18], Re = 400) Reynolds number, the boundary layer is√considered as laminar oscillatory boundary layer. The thickness
of the laminar oscillatory boundary layer δ ≈ 5 ν/ω is very small compared to the diameter of the duct. We assume plane wave propagation along the longitudinal direction of the duct taking the above factors into consideration. Therefore, one-dimensional fundamental fluid dynamics equations are used for analysis of finite amplitude standing wave in closed axisymmetric area change ducts.
230
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
The equations are derived in a conservative form. The continuity equation is written as ∂(ρA) ∂(ρuA) + =0 ∂t ∂x
(11)
where t is the time, x the Cartesian coordinate, A the cross sectional area of the duct, ρ the density and µ is the oscillating particle velocity in the duct. The momentum equation in conservative form can be expressed as ∂(ρuA) ∂(ρu2 + p)A ∂A 4 ∂2 u + =p + µ 2 ∂t ∂x ∂x 3 ∂x
(12)
where p is the pressure and µ is the effective viscosity. The conservation of energy can be written in the following form: ∂(EA) ∂(E + p)uA 4 ∂2 u + = µu 2 ∂t ∂x 3 ∂x
(13)
where E is the total energy composed of internal and kinetic energy. The equation for the calculation of total energy E is written as follows: E=
p 1 + ρu2 γ −1 2
(14)
where γ is the ratio of specific heats. The heat flux as well as heat transfer between the elements of gas was ignored. However, the gas friction was considered, which is calculated by taking steady friction f and frequency dependent unsteady friction f into consideration. By including the friction term in the momentum Equation (12), we can write ∂(ρuA) ∂(ρu2 + p)A ∂A 4 ∂2 u + =p + µ 2 + fw ∂t ∂x ∂x 3 ∂x
(15)
The wall friction fw can be written as fw = f + f
(16) 2
Here, steady friction, f = 32ρu ReD . According to Trikha [19] unsteady friction can be calculated as follows: 16µu t ∂u f = (τ)W(t − τ) dτ D2 0 ∂τ where Re is the Reynolds number and W is a weight function as a function of time.
3. Numerical simulation The model used for numerical simulation is an axially symmetric, three-dimensional duct with exponential area contraction towards the closed end. The wave motions in the ducts are numerically analyzed to have an insight of the effect of gases and also to find out the limit of area contraction ratio for which shockless standing wave are induced in the ducts. It is assumed that the ducts are filled with air or HFC134a. The parameters of the basic equations described in Section 2.2, such as u, p, ρ, E and A, were normalized by the velocity of sound, ambient pressure and density, internal energy at ambient temperature and the closed end area,
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
231
respectively. Finally, the non-dimensional basic equations in matrix form can be written as follows: ω
∂QA ∂FA + =H ∂t ∂x
(17)
Here, Q, F and H are expressed as follows:
ρ Q = ρu , E
ρu 2 F = ρu + p , (E + p )u
0 ∂A 4 µ∗ ∂2 u p 0 + µ 2 + fw ∗ ∗ ∗ H = ∂x 3 ρ0 a0 L ∂x ∗ 2 4 µ0 ∂ u µu 3 ρ0∗ a0∗ L∗ ∂x2
The non-dimensional equations are solved numerically. The finite difference method is used for the numerical simulation. Equations are discretized by Turkel’s [20] method, which is based on the explicit MacCormack method with second-order accuracy in time and fourth-order accuracy in space. In this method, the discretization for the predictor and corrector is expressed as follows: Predictor: backward difference ¯ j = Qn − (t (7Fjn − 8F n + F n ) + Hjn , Q j−1 j−2 j 6(x
j=2∼N
¯ 0 = Qn − (t (4F n − 17F n + 28F n − 15F n ) + H n , j = 0 Q 3 2 1 0 0 0 6(x ¯ 1 = Qn − (t (−F n + 4F n + F n − 4F n ) + H n , j=1 Q 3 2 1 0 1 1 6(x Corrector: forward difference 1 (t n ¯ ¯ ¯ ¯ ¯ Qn+1 Q = + Q + (7 F j = 0 ∼ (N–2) − 8 F + F ) + H j j j+1 j+2 j , j j 2 6(x 1 n ¯ N−1 − (t (4F¯ N − F¯ N−1 − 4F¯ N−2 + F¯ N−3 ) + H ¯ = Q Qn+1 + Q N−1 , j = N − 1 N−1 N−1 2 6(x 1 (t n+1 n ¯ ¯ ¯ ¯ ¯ ¯ j=N QN + QN − (15FN − 28FN−1 + 17FN−2 − 4FN−3 ) + HN , QN = 2 6(x The boundary conditions are same as expressed in Equation (2). Results are analyzed after the waveform comes to the steady state. In numerical calculation, it was necessary to calculate thousand cycles of wave motion for getting steady state because the establishment of the waveform starts from the complete rest at fundamental frequency. 4. Results and discussion Results are presented to recognize the fundamental phenomena of finite amplitude standing wave in ducts with exponential area contraction towards the closed end. The first section discusses the effect of area contraction on wave motion and the limit of area contraction ratio to obtain shockless high amplitude pressure waveform in ducts filled with air. Subsequently, the effect of gases on the limit of area contraction ratio to obtain shockless pressure waveform is investigated. The effect of duct geometry on finite amplitude oscillations of air column is verified. Finally, the results of the effect of duct length on finite amplitude oscillations of air column in closed area contraction ducts are presented to estimate the optimum duct length for acoustic compressors.
232
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
Fig. 1. The change of normalized resonant frequency with area contraction ratio for piston acceleration of 200 m/s2 when the duct is filled with air, f0 = 566.7 Hz.
4.1. Effect of area contraction Calculations were carried out considering the room temperature as an initial temperature, which is assumed to be 15 ◦ C and normal atmospheric pressure (101.3 kPa) as an initial pressure. The geometry of the exponential duct, which was expressed in Equation (3), is further written here A(x) = A0 exp(mx)
for
0≤x≤l
From the above expression, we can write m=
ln(Ap /A0 ) l
The term Ap /A0 is denoted as area contraction ratio. Here, Ap is the cross sectional area (7.85 × 10−3 m2 ) at the piston end and A0 is the cross sectional area at the closed end which is estimated from the limit of area contraction ratio. Results are presented for an area contraction ratio of 1–100. The length of the duct is considered to be 0.3 m. The change of normalized resonant frequency with regards to area contraction ratio is shown in Fig. 1. The resonant frequencies are normalized by the resonant frequency obtained for the oscillations of air column in cylindrical duct (Ap /A0 = 1), which is 566.7 Hz. The resonant frequencies obtained by linear theory and numerical simulations are presented here. It should be noted that the use of the relationship of natural frequency, i.e., fres = c/2l, is valid only for cylindrical ducts. This relationship is not valid for the calculation of resonant frequencies of area change ducts. It is already known that the maximum fluctuating pressure amplitude is obtained at resonant condition. According to linear theory, the amplitude of the pressure waveform is infinite at resonant state. Moreover, the shift in resonant frequency with the variation of input acceleration amplitude has been confirmed by several earlier investigations. Here, the resonant frequencies are estimated for a piston acceleration amplitude of 200 m/s2 . Fig. 1 shows the increase of the resonant frequency with the increase of area contraction ratio. The rate of increase of resonant frequency is linear at first, where the rate decreases at higher area contraction ratios. It is noteworthy that the resonant frequency obtained by numerical simulation has very good agreement with the resonant frequency obtained by linear acoustics theory. Fig. 2 shows the standing wave velocity and pressure node obtained by linear theory for the oscillations of air column in closed area contraction exponential ducts for area contraction ratio of 1, 16, 49, 64 and 100. According to linear acoustics theory, the velocity and pressure amplitude is infinite at resonant state. Therefore, the velocity and pressure amplitudes are estimated near the resonant frequency for piston acceleration amplitude of 200 m/s2 . The velocity and pressures are normalized by the velocity and pressure amplitude obtained at the piston end. As
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
233
Fig. 2. The standing wave velocity and pressure distribution in exponential duct with different cross sectional area contraction ratios for piston acceleration amplitude of 200 m/s2 . (a) Velocity distribution and (b) pressure distribution.
shown in Fig. 2a and b, the velocity loop and pressure node are at the center of the cylindrical duct. However, the loop of velocity fluctuation moves towards the closed end and the node of pressure fluctuation moves towards the piston end with the increase of area contraction ratio. The results also show that the fluctuating pressure amplitude at the closed end of the duct is (Ap /A0 )0.5 times the fluctuating pressure amplitude at the piston end. Fig. 3 presents the pressure distributions at different oscillation modes in closed area contraction ducts for two different area contraction ratios. In Fig. 3a, we observed that the pressure distributions in cylindrical ducts at different oscillation modes are symmetric with respect to the center of the duct. As shown in Fig. 3b, the pressure distributions are no longer symmetric in area change ducts. However, they have the same general behavior, which means half wavelength in the fundamental, full wavelength is the second mode and one and a half wavelengths in the third oscillation mode. The decrease in pressure amplitude with the increase in oscillation mode is clearly observed. Next, the effect of area contraction on finite amplitude standing wave in closed area change ducts will be explained on the basis of numerical simulation results. Fig. 4a illustrates the wave diagram obtained in ducts with cross sectional area contraction ratios of 1 (cylindrical duct). It is assumed that the duct is filled with air and the results are presented for piston acceleration amplitude of 50 m/s2 . As shown in Fig. 4a, an asymmetry between rarefaction and compression zone in the time as in the space is observed. This indicates that there is no real node for the pressure, which varies during a cycle. However, the strength of the wave front is changing in the direction of propagation. Therefore, the corresponding node position can be estimated. The fluctuating pressure amplitude at the closed end of the cylindrical duct for different piston acceleration amplitudes show that the pressure waveform is accompanied with shock even for small piston acceleration amplitude. Fig. 4b shows a wave diagram obtained at fundamental resonant state in an air filled exponential duct with a cross sectional area contraction ratio (Ap /A0 ) of 9 and piston acceleration amplitude of 50 m/s2 . This figure indicates that even if we have a standing wave, there is no real node for the pressure. The corresponding node location changes during a cycle. However, shock wave does
Fig. 3. Pressure distributions along the duct axis at fundamental, second and third oscillation modes for piston acceleration amplitude of 200 m/s2 . Here, p0 = 101.3 kPa. (a) Cylindrical duct (Ap /A0 = 1) and (b) area contraction duct (Ap /A0 = 100).
234
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
Fig. 4. Wave diagram in resonant acoustic ducts with variable area change for piston acceleration amplitude of 50 m/s2 and pressure waveform at closed end of each duct for three different piston acceleration amplitudes when the ducts are filled with air. (a) Cylindrical duct (Ap /A0 = 1) and (b) area contraction duct (Ap /A0 = 9).
not appear in the duct. The pressure distribution is symmetric with the cycle. The pressure fluctuation at the closed end of area contraction duct show that the waveform is not accompanied with shock wave even at high acceleration amplitudes. A very few studies have been performed to understand the effect of gas properties on wave motion in area contraction closed ducts. In the above discussions, results are presented for the wave motion in area change ducts filled with air. Most of the studies were concerned on a particular gas, which develops waveform of large amplitude. Lucas and his group briefly studied the effect of gas properties on wave motion in different duct geometries. The limit of area contraction ratio to obtain shockless pressure waveform with the variation of gas properties has not been explained yet. In this study, to investigate the effect of gas on the limit of area contraction ratio for shockless waveform, two gases, namely air and HFC134a, are used. The gases were chosen to maintain a range of densities and sound speeds. HFC134a is a gas, which is relatively dense, has low sound speed and is used for vapor-compression cooling systems in refrigerators, air conditioners and so on. Numerical simulation is conducted with the assumption that the duct is filled with air or HFC134a at normal atmospheric pressure and room temperature. The limit of cross sectional area contraction ratio to obtain a shockless pressure waveform is illustrated in Fig. 5a and b. The pressure fluctuation at the closed end of an air filled exponential duct with area contraction ratio of 1, 2, 4, 6 and 9, and piston acceleration amplitude of 50 m/s2 is shown in Fig. 5a, while Fig. 5b shows the results for HFC134a-filled exponential duct. Fig. 5a shows that the shock wave appears up to the area contraction ratio of 2 and then disappears for the ratio of 4 and more. The shockless resonance with nearly equal amplitude is observed for the area contraction ratio of 4, 6 and 9. In Fig. 5b, in an HFC134a filled exponential duct, a shock wave propagates up to the area contraction ratio of 4. It is seen that the amplitude of the pressure wave increases with the increase of area contraction ratio in HFC134a filled ducts; however, the waveform has a narrower peak and a broader trough. The change of compression ratio with respect to area contraction ratio is shown in Fig. 5c. This figure shows that the compression ratio increases with the increase of area contraction ratio. In the case of air filled ducts, the compression
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
235
Fig. 5. Pressure waveform and compression ratio at closed end of exponential ducts with cross sectional area contraction ratio when the ducts are filled with air and HFC134a. (a) Pressure waveform in air filled ducts, (b) pressure waveform in HFC134a filled ducts and (c) comparison of compression ratio for two gases.
ratio is nearly constant from area contraction ratios of 9–100, while it increases with the increase of area contraction ratio in HFC134a filled ducts for area contraction ratios of up to 16. This result indicates that the acoustic saturation takes place rapidly in the former at an area contraction ratio of nearly 9. Fig. 5c also shows that for a particular area contraction ratio, the compression ratio is relatively higher in the duct filled with HFC134a than that of air. The reason why the compression ratio is higher in ducts filled with HFC134a is probably due to the high acoustic impedance. 4.2. Effects of duct geometries The limit of area contraction ratio to obtain shockless pressure waveform in closed area change exponential ducts has been explained in the last section. It is noteworthy that the limit of area contraction ratio to which shock wave appears will be varied with the variation of duct geometry. In this section, the effect of duct geometries on finite amplitude oscillations of air column in closed area change ducts will be explained. Among various duct geometries, linear area contraction, exponential area contraction and half cosine function cross sectional area contraction ducts are taken into consideration, which cover all possible simple duct geometries. To compare the effect of duct shapes, the length of duct, radius at both ends and inside gas condition is kept the same. The resonant frequency of gas column oscillations in closed area change ducts strongly varies with the variation of duct geometry. To design acoustic compressors, the consideration of resonant frequency of pressure oscillation is very important. This is because the valve response is related to the frequency of oscillation. The change of resonant frequency with the variation of the duct shape is shown in Fig. 6a for piston acceleration amplitude of 200 m/s2 . The resonant frequencies were normalized by the theoretical resonant frequency of the cylindrical duct. The theoretical resonant frequencies for each duct shape are also shown in the figure. It is seen that for a particular area contraction ratio and a constant duct length, the resonant frequency in a half cosine duct is relatively high as compared to the resonant frequency in conical and exponential duct. The evaluation of finite amplitude standing wave for some
236
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
Fig. 6. The change of resonant frequency, compression ratio and pressure waveform with duct geometries when the ducts are filled with air at normal atmospheric pressure and temperature. Here, piston acceleration amplitude is 200 m/s2 . (a) Resonant frequency, (b) compression ratio and (c) pressure waveform in conical, exponential and half cosine duct with area contraction ratio of 100 (Ap /A0 = 100).
practical applications, especially for acoustic compressors, can be made by the compression ratio obtained at the closed end of the duct, which is estimated by the ratio of the highest to the lowest absolute pressure of the waveform. Fig. 6b shows the compression ratio obtained at the closed end corresponding to the area contraction ratio when the ducts are filled with air. The lengths of each duct are kept the same in the numerical simulation. It is seen that the compression ratio increases with the increase of area contraction ratio in all duct geometries. The rate of increase of compression ratio is linear at first, which comes to the saturation state in each duct after a certain limit of area contraction ratio. The limit of area contraction ratio for acoustic saturation further varies with the variation of piston acceleration amplitudes. For a particular piston acceleration (200 m/s2 ) and a fixed area contraction ratio (Ap /A0 = 100), the compression ratio is high in conical ducts; however, the difference is significantly small. 4.3. Effect of duct lengths The effect of duct length on wave motion in closed axisymmetric ducts needs to be realized to design any practical devices like acoustic compressors. Depending upon the applications, the duct can be scaled in length to affect the frequency of operation and capacity. In this report, the results of numerical simulation are presented for different duct lengths. The cross sectional area contraction ratio of the exponential area contraction duct is 100. Fig. 7 shows the change of compression ratio and resonant frequency with regards to duct length when the duct is filled with air at normal atmospheric pressure and temperature. The piston acceleration amplitude is kept at 200 m/s2 . In each case, the frequency is kept at the first mode resonant frequency. It is observed that the length enlargement decreases the resonant frequency while increasing the compression ratio. Higher frequency operation reduces not only the
Fig. 7. Compression ratio at closed end of exponential duct with cross sectional area contraction ratio of 100 for duct lengths from 0.1 to 1.2 m.
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
237
size of the duct but also its efficiency. Therefore, to design any practical devices, optimization of the duct length is essential according to the application requirement.
5. Conclusions The oscillations of gas column in closed axisymmetric duct with cross sectional area change were analyzed based on theoretical analysis and numerical simulation. The characteristics of wave motion were investigated to explore the limit of area contraction ratio for shockless pressure waveform in closed ducts. The main conclusions drawn from the present study are summarized as follows: (1) A linear theory was derived to explore the effect of area contraction ratio on wave motion. The standing wave mode was estimated by the developed theory. (2) The limit of area contraction ratio to obtain shockless pressure waveform in closed acoustic duct was estimated. For a particular piston acceleration (50 m/s2 ), the limit of area contraction ratio to get shockless pressure waveform is two when the duct is filled with air. This limit of area contraction ratio for shockless pressure waveform varies with the variation of piston acceleration and duct geometry. (3) The limit of area contraction ratio for shockless pressure waveform and amplitude of the standing wave depend on the gas used in the duct. Particularly, the acoustic impedance plays a significant role in the oscillations of gas in a closed duct. The higher the acoustic impedance of the gas, the higher the amplitude of the pressure waveform in closed ducts.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
R. Betchov, Nonlinear oscillation of a column of gas, Phys. Fluids 1 (1958) 205–212. R.A. Saenger, G.E. Hudson, Periodic shock waves in resonating gas columns, J. Acoust. Soc. Am. 32 (1960) 961–970. W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech. 18 (1964) 44–64. S. Weiner, Standing sound waves of finite amplitude, J. Acoust. Soc. Am. 40 (1966) 240–243. S. Temkin, Propagating and standing sawtooth waves, J. Acoust. Soc. Am. 45 (1969) 224–227. A.B. Coppens, J.V. Sanders, Finite-amplitude standing waves within real cavities, J. Acoust. Soc. Am. 58 (1975) 1133–1140. D.B. Cruishank, Experimental investigation of finite-amplitude acoustic oscillations in a closed tube, J. Acoust. Soc. Am. 52 (1972) 1024–1036. C.P. Lee, T.G. Wang, Nonlinear resonance and viscous dissipation in an acoustic chamber, J. Acoust. Soc. Am. 92 (1992) 2195–2206. D.F. Gaitan, A.A. Atchley, Finite amplitude standing wave in harmonic and anharmonic tubes, J. Acoust. Soc. Am. 93 (1993) 2489–2495. L. Elvira-Segura, E. Riera-Franco de Sarabia, Numerical and experimental study of finite-amplitude standing waves in tube at high sonic frequencies, J. Acoust. Soc. Am. 104 (1998) 708–714. C.C. Lawrenson, B. Lipkens, T.S. Lucas, D.K. Perkins, T.W.V. Doren, Measurement of macrosonic standing waves in oscillating closed cavities, J. Acoust. Soc. Am. 104 (1998) 623–636. Y.A. Ilinskii, B. Lipkens, T.S. Lucas, T.W.V. Doren, E.A. Zabolotskaya, Nonlinear standing waves in an acoustical resonator, J. Acoust. Soc. Am. 104 (1998) 2664–2674. T. Lucas, Acoustic compressors, in: Proceedings of the Non-Fluorocarbon Refrigeration and Air-Conditioning Technology Workshop, 1993, pp. 43–50. M. Kawahashi, M. Arakawa, Nonlinear phenomena induced by finite-amplitude oscillation of air column in closed duct (analysis of acoustic streaming), JSME Int. J. 39 (1996) 280–286. G.W. Swift, Thermoacoustic engines and refrigerators, Phys. Today 7 (1995) 22–28. C. Herman, M. Wetzel, Design of a thermoacoustic refrigerator—a case study, in: Proceedings of ASME Thermodynamics and Design, Analysis, and Improvement of Energy Systems, AES-35, 1995, pp. 195–203. M. Kawahashi, M.A. Hossain, K. Koduka, H. Hirahara, Finite amplitude standing wave in an axi-symmetric closed duct with cross-sectional area change, Trans. Jpn. Soc. Mech. Eng. 68 (2002) 1352–1359. P. Merkli, H. Thomann, Transition of turbulence in oscillating pipe flow, J. Fluid Mech. 68 (1975) 567–575. A.K. Trikha, An efficient method for simulating frequency-dependent friction in transient liquid flow, J. Fluid Eng. 3 (1975) 97–105. E. Turkel, On the practical use of high-order methods for hyperbolic system, J. Comput. Phys. 35 (1980) 319–340.
Finite amplitude standing wave in closed ducts with cross sectional area change M.A. Hossain a,∗ , M. Kawahashi b , T. Fujioka a b
a Development Group, Anest Iwata Corporation, 3176 Shinyoshida, Yokohama, Japan Department of Mechanical Engineering, Saitama University, 255 Shimo-Okubo, Saitama, Japan
Received 20 September 2004; received in revised form 7 February 2005; accepted 11 February 2005 Available online 5 March 2005
Abstract A linear theory was developed to estimate the resonant frequency and the standing wave mode in closed ducts with variable cross sectional area. The finite amplitude standing wave in closed ducts with area contraction was numerically studied. The effect of area contraction ratio and the gas properties on nonlinear standing wave was investigated. One-dimensional numerical simulation was conducted using fundamental fluid dynamics equations taking the effects of viscosity and frictions into consideration. Finite difference MacCormack scheme was used by preserving the computational accuracy within second-order in time and fourthorder in space, respectively. The dependences of the pressure waveform on the area contraction ratio, the limit of area contraction ratio to obtain shockless high amplitude pressure waveform with the variation of gas properties and the amplitude of pressure waveform with the variation of duct geometries were investigated. In addition, the compression ratios at the closed end of the duct with different area contraction ratios for different duct geometries were presented. This study suggests that changes in duct geometry can have a significant impact on overall performance of acoustic compressors. © 2005 Elsevier B.V. All rights reserved. Keywords: Linear acoustics; Finite amplitude wave; Area contraction ratio; Compression ratio; Numerical analysis; Acoustic compressor
1. Introduction The oscillations of gas column in closed duct at resonant frequency induce large amplitude acoustic standing wave. The large amplitude standing wave phenomena and nonlinear effects induced in gas filled ducts have wide engineering applications. The use of large amplitude pressure fluctuation obtained at the closed end of duct is a special application of the effect in developing the acoustic compressor. Furthermore, the nonlinear phenomena, ∗
Corresponding author. Tel.: +81 45 591 1122; fax: +81 45 593 0524. E-mail address: [email protected] (M.A. Hossain).
0165-2125/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2005.02.003
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
227
namely the acoustic streaming and thermoacoustic effect, can be used to constrain the convection current and to develop thermoacoustic engines and refrigerators. The amplitude of the standing wave and nonlinear phenomena largely depends on the shape of the duct. Finite amplitude wave motion in ducts has been theoretically and experimentally investigated extensively in many earlier works. It has been known that shock wave generate and propagate periodically back and forth in a gas-filled closed cylindrical duct. In this context, Betchov [1] made an earlier theoretical attempt to account for this phenomenon. He introduced a simple theoretical analysis of the inviscid gas motion, taking frictional losses to the walls of the duct into consideration. Saenger and Hudson [2] devised a theory of the steady state motion of the gas in the Kundt tube in its fundamental mode that includes the dissipative effects of wall friction and heat conduction to the duct walls. Chester [3] described nonlinear phenomena in a closed straight duct at near resonant state. His solution included the effect of compressive viscosity and also the effect of the boundary layer at the duct wall. Weiner [4] developed a technique to estimate finite amplitude effects for standing acoustic waves. His method equates the nonlinear energy loss from the fundamental mode of oscillation to the work done on this mode by its higher harmonics assuming that the waveform is a sawtooth. Temkin [5] studied large amplitude oscillations of air column in piston driven resonant straight duct experimentally. He derived the rate of energy dissipation per unit area of such a shock front as proportional to the cube of the wave amplitude. Coppens and Sanders [6] formulated and solved one-dimensional, nonlinear, acoustic wave equation with dissipative term describing the viscous and thermal energy losses encountered in a rigid-wall closed duct. Their solutions lead to a steady state distribution of harmonics of the fundamental, the amplitude and phase of each term being strong functions of frequency and the absorptive process. Cruikshank [7] verified Chester’s theory experimentally and concluded that the agreement is better when the duct is excited above resonance than below the resonant frequency. His results also disclosed that the resonant frequency shifts with the increase in piston oscillating displacement amplitude. Lee and Wang [8] numerically analyzed the nonlinear resonance of an air-filled acoustic chamber using the fully nonlinear Lagrangian wave equation, in which the air was assumed to be excited by an external body force. Researchers and scientists have found that the amplitude of standing wave generated by the oscillations of gas column in straight duct was limited due to shock wave generation. This limitation arises due to the distortion of standing wave that happens when harmonic frequency coincides with the modal frequency of standing wave. However, recent investigations revealed that it is possible to obtain large amplitude standing wave without shock formation by changing the shape of the ducts. Gaitan and Atchley [9] studied finite amplitude standing waves in harmonic and anharmonic (duct with variable cross sections) air-filled closed ducts driven by a piston at resonance. They found that the detuned ducts suppress the energy transfer effectively into the higher harmonics and thereby suppress generation of higher harmonics. The results revealed that the waveform is strongly dependent on the shape of the ducts. Elvira-Segura and Riera-Franco de Sarabia [10] developed a finite element algorithm to study nonlinear standing wave. Recently, Lawrenson et al. and Ilinskii et al. [11,12] studied finite amplitude standing wave in resonant ducts with different shapes numerically and experimentally. They found high amplitude shockless acoustic pressure fluctuation at the closed end of area change ducts. Lucas [13] showed that the high amplitude resonant duct could be used in real applications as a compressor or pump. High intensity applications of acoustic energy in industrial processing are based on nonlinear effects produced by finite amplitude pressure variations. The nonlinear effects generated in closed acoustic ducts are steady circulating streaming (acoustic streaming) and thermoacoustic effect. Kawahashi and Arakawa [14] studied the acoustic streaming phenomena in closed cylindrical ducts by numerical simulation. Classic works on practical applications of the thermoacoustic effect for the design and development of thermoacoustic engines [15] and refrigerators [16] are found in literature. Therefore, a detailed knowledge of the finite amplitude standing wave as well as nonlinear pressure distribution inside the duct is essential for the development of practical systems. Fundamental characteristics of wave motion in the air-filled closed area change duct with different shapes were recently discussed by Kawahashi et al. [17]. This paper deals with the theoretical and numerical investigation of the oscillations of the gas column in closed ducts with area contraction. The cross sectional area of the duct was decreased from the piston end towards the closed end. We investigated the limit of area contraction ratio to get shockless pressure waveform, the effect of gases on
228
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
finite amplitude oscillations in closed area change ducts and the effect of duct lengths. A linear acoustic theory was presented for the analysis of standing wave in closed ducts with area changes. The governing equations were derived by taking viscosity and frequency dependent wall friction into consideration. One-dimensional numerical simulation was carried out using the MacCormack scheme keeping accuracy with second-order in time and fourth-order in space.
2. Theoretical formulation 2.1. Linear acoustics theory The axisymmetric duct with cross sectional area contraction from the driving end towards the closed end is used for developing an acoustic compressor. The phenomena of the wave motion in the duct depend on the area contraction ratio, geometry of the duct shape and driving acceleration. The development of the acoustic compressor is based on the compression ratio measured from the pressure fluctuation at the closed end of the duct. In this section, a linear theory is developed to estimate the resonant frequency and standing wave mode of the gas column oscillations in closed area change exponential ducts. The linear acoustics theory for plane wave propagation in closed axisymmetric area change ducts with cross sectional area contraction towards the closed end is expressed as follows: 2 ∂2 u 1 dA ∂u d 1 dA 2 ∂ u (1) =c + +u A dx ∂x dx A dx ∂t 2 ∂x2 where u is the fluctuating velocity, c the sound speed, A the cross sectional area, t the time and x is the coordinate along the length of the duct. Equation (1) is solved considering the oscillations of the gas column in area contraction exponential closed ducts by using the piston as a sound source. One important characteristic of the exponential area change duct is that the rate of change of cross sectional area contraction (1/A)(dA/dx) along the longitudinal direction of the duct is constant. To solve equation (1), the following boundary conditions are employed: x = 0 (closed end) : x = l (piston end) :
u=0 u = u0 cos ωt
(2)
where l is the duct length, u0 the piston oscillating velocity amplitude and ω is the angular frequency of the oscillation. The geometry of the exponential duct is expressed by following equation: A(x) = A0 exp(mx)
(3)
where A0 is the cross sectional area at the closed end, m is a constant which is defined as the rate of change of area contraction and calculated from the cross sectional area at piston end (Ap ) and the area contraction ratio (Ap /A0 ). Taking sinusoidal oscillations of the gas column into consideration in closed ducts, we can write, u = u(x) exp(iωt)
(4)
Modifying Equation (1) by using Equation (4), we can write, d2 u(x) du(x) + k2 u(x) = 0 +m 2 dx dx Here, k2 =
ω2 . c2
(5)
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
229
The fundamental solution of Equation (5) is finally written as follows: u(x) = −Aα exp(αx) − Bβ exp(βx) 2 m m m2 2 Here, α = − 2 − i k − 4 and β = − 2 + i k2 − m4 . Applying the boundary conditions, constants A and B can be written in the form: u0 , A=− α exp(α) − exp(β)
B=
(6)
u0 β exp(α) − exp(β)
At last, we obtain the following equation to calculate the angular frequency of the gas column oscillations in closed exponential ducts with area contraction: ω 2 = c2
nπ 2 l
+
c 2 m2 4
(7)
In linear theory, the fluctuating pressure amplitude in closed duct is infinite at resonant frequency. However, it is possible to estimate the standing wave mode by the derived linear acoustic theory. The standing wave mode of fluctuating velocity amplitude that is derived from the above expressions is Mode (u) : −u0 [exp(αx) − exp(βx)] By using momentum equation with the assumption of small amplitude fluctuation, ∂u 1 ∂p =− ∂t ρ ∂x The following standing wave pressure mode is obtained: 1 1 Mode (p) : −iωρ0 u0 exp(αx) + exp(βx) α β
(9)
(10)
where p is the pressure, ρ the density and ρ0 is the mean density of gas in closed duct. 2.2. Finite amplitude standing wave When the oscillations of the gas column in a closed duct are conducted at resonant frequency, high amplitude pressure waveform is induced in the duct. In the linear theory described in Section 2.1, the amplitude of the pressure waveform becomes infinite at resonance. However, practically, the amplitude of pressure waveform is limited due to the effect of viscosity as well as various nonlinear phenomena. A shock formation in cylindrical duct is one of the typical examples of such nonlinear phenomena. In this section, the basic equations will be derived to analyze the finite amplitude wave motion in closed axisymmetric ducts
cross sectional area changes. Considering the cylindrical duct as standard, the Stokes number √ with s = D/ ν/ω of standing wave in closed ducts is predicted to be very large. Here, D is the diameter of the duct, ν the kinematic viscosity of the gas frequency of the vibration. In addition, since the value of
and ω is√the angular the oscillating Reynolds number Re = 2u/ ν/ω is less than the critical (according to Merkli and Thomann [18], Re = 400) Reynolds number, the boundary layer is√considered as laminar oscillatory boundary layer. The thickness
of the laminar oscillatory boundary layer δ ≈ 5 ν/ω is very small compared to the diameter of the duct. We assume plane wave propagation along the longitudinal direction of the duct taking the above factors into consideration. Therefore, one-dimensional fundamental fluid dynamics equations are used for analysis of finite amplitude standing wave in closed axisymmetric area change ducts.
230
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
The equations are derived in a conservative form. The continuity equation is written as ∂(ρA) ∂(ρuA) + =0 ∂t ∂x
(11)
where t is the time, x the Cartesian coordinate, A the cross sectional area of the duct, ρ the density and µ is the oscillating particle velocity in the duct. The momentum equation in conservative form can be expressed as ∂(ρuA) ∂(ρu2 + p)A ∂A 4 ∂2 u + =p + µ 2 ∂t ∂x ∂x 3 ∂x
(12)
where p is the pressure and µ is the effective viscosity. The conservation of energy can be written in the following form: ∂(EA) ∂(E + p)uA 4 ∂2 u + = µu 2 ∂t ∂x 3 ∂x
(13)
where E is the total energy composed of internal and kinetic energy. The equation for the calculation of total energy E is written as follows: E=
p 1 + ρu2 γ −1 2
(14)
where γ is the ratio of specific heats. The heat flux as well as heat transfer between the elements of gas was ignored. However, the gas friction was considered, which is calculated by taking steady friction f and frequency dependent unsteady friction f into consideration. By including the friction term in the momentum Equation (12), we can write ∂(ρuA) ∂(ρu2 + p)A ∂A 4 ∂2 u + =p + µ 2 + fw ∂t ∂x ∂x 3 ∂x
(15)
The wall friction fw can be written as fw = f + f
(16) 2
Here, steady friction, f = 32ρu ReD . According to Trikha [19] unsteady friction can be calculated as follows: 16µu t ∂u f = (τ)W(t − τ) dτ D2 0 ∂τ where Re is the Reynolds number and W is a weight function as a function of time.
3. Numerical simulation The model used for numerical simulation is an axially symmetric, three-dimensional duct with exponential area contraction towards the closed end. The wave motions in the ducts are numerically analyzed to have an insight of the effect of gases and also to find out the limit of area contraction ratio for which shockless standing wave are induced in the ducts. It is assumed that the ducts are filled with air or HFC134a. The parameters of the basic equations described in Section 2.2, such as u, p, ρ, E and A, were normalized by the velocity of sound, ambient pressure and density, internal energy at ambient temperature and the closed end area,
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
231
respectively. Finally, the non-dimensional basic equations in matrix form can be written as follows: ω
∂QA ∂FA + =H ∂t ∂x
(17)
Here, Q, F and H are expressed as follows:
ρ Q = ρu , E
ρu 2 F = ρu + p , (E + p )u
0 ∂A 4 µ∗ ∂2 u p 0 + µ 2 + fw ∗ ∗ ∗ H = ∂x 3 ρ0 a0 L ∂x ∗ 2 4 µ0 ∂ u µu 3 ρ0∗ a0∗ L∗ ∂x2
The non-dimensional equations are solved numerically. The finite difference method is used for the numerical simulation. Equations are discretized by Turkel’s [20] method, which is based on the explicit MacCormack method with second-order accuracy in time and fourth-order accuracy in space. In this method, the discretization for the predictor and corrector is expressed as follows: Predictor: backward difference ¯ j = Qn − (t (7Fjn − 8F n + F n ) + Hjn , Q j−1 j−2 j 6(x
j=2∼N
¯ 0 = Qn − (t (4F n − 17F n + 28F n − 15F n ) + H n , j = 0 Q 3 2 1 0 0 0 6(x ¯ 1 = Qn − (t (−F n + 4F n + F n − 4F n ) + H n , j=1 Q 3 2 1 0 1 1 6(x Corrector: forward difference 1 (t n ¯ ¯ ¯ ¯ ¯ Qn+1 Q = + Q + (7 F j = 0 ∼ (N–2) − 8 F + F ) + H j j j+1 j+2 j , j j 2 6(x 1 n ¯ N−1 − (t (4F¯ N − F¯ N−1 − 4F¯ N−2 + F¯ N−3 ) + H ¯ = Q Qn+1 + Q N−1 , j = N − 1 N−1 N−1 2 6(x 1 (t n+1 n ¯ ¯ ¯ ¯ ¯ ¯ j=N QN + QN − (15FN − 28FN−1 + 17FN−2 − 4FN−3 ) + HN , QN = 2 6(x The boundary conditions are same as expressed in Equation (2). Results are analyzed after the waveform comes to the steady state. In numerical calculation, it was necessary to calculate thousand cycles of wave motion for getting steady state because the establishment of the waveform starts from the complete rest at fundamental frequency. 4. Results and discussion Results are presented to recognize the fundamental phenomena of finite amplitude standing wave in ducts with exponential area contraction towards the closed end. The first section discusses the effect of area contraction on wave motion and the limit of area contraction ratio to obtain shockless high amplitude pressure waveform in ducts filled with air. Subsequently, the effect of gases on the limit of area contraction ratio to obtain shockless pressure waveform is investigated. The effect of duct geometry on finite amplitude oscillations of air column is verified. Finally, the results of the effect of duct length on finite amplitude oscillations of air column in closed area contraction ducts are presented to estimate the optimum duct length for acoustic compressors.
232
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
Fig. 1. The change of normalized resonant frequency with area contraction ratio for piston acceleration of 200 m/s2 when the duct is filled with air, f0 = 566.7 Hz.
4.1. Effect of area contraction Calculations were carried out considering the room temperature as an initial temperature, which is assumed to be 15 ◦ C and normal atmospheric pressure (101.3 kPa) as an initial pressure. The geometry of the exponential duct, which was expressed in Equation (3), is further written here A(x) = A0 exp(mx)
for
0≤x≤l
From the above expression, we can write m=
ln(Ap /A0 ) l
The term Ap /A0 is denoted as area contraction ratio. Here, Ap is the cross sectional area (7.85 × 10−3 m2 ) at the piston end and A0 is the cross sectional area at the closed end which is estimated from the limit of area contraction ratio. Results are presented for an area contraction ratio of 1–100. The length of the duct is considered to be 0.3 m. The change of normalized resonant frequency with regards to area contraction ratio is shown in Fig. 1. The resonant frequencies are normalized by the resonant frequency obtained for the oscillations of air column in cylindrical duct (Ap /A0 = 1), which is 566.7 Hz. The resonant frequencies obtained by linear theory and numerical simulations are presented here. It should be noted that the use of the relationship of natural frequency, i.e., fres = c/2l, is valid only for cylindrical ducts. This relationship is not valid for the calculation of resonant frequencies of area change ducts. It is already known that the maximum fluctuating pressure amplitude is obtained at resonant condition. According to linear theory, the amplitude of the pressure waveform is infinite at resonant state. Moreover, the shift in resonant frequency with the variation of input acceleration amplitude has been confirmed by several earlier investigations. Here, the resonant frequencies are estimated for a piston acceleration amplitude of 200 m/s2 . Fig. 1 shows the increase of the resonant frequency with the increase of area contraction ratio. The rate of increase of resonant frequency is linear at first, where the rate decreases at higher area contraction ratios. It is noteworthy that the resonant frequency obtained by numerical simulation has very good agreement with the resonant frequency obtained by linear acoustics theory. Fig. 2 shows the standing wave velocity and pressure node obtained by linear theory for the oscillations of air column in closed area contraction exponential ducts for area contraction ratio of 1, 16, 49, 64 and 100. According to linear acoustics theory, the velocity and pressure amplitude is infinite at resonant state. Therefore, the velocity and pressure amplitudes are estimated near the resonant frequency for piston acceleration amplitude of 200 m/s2 . The velocity and pressures are normalized by the velocity and pressure amplitude obtained at the piston end. As
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
233
Fig. 2. The standing wave velocity and pressure distribution in exponential duct with different cross sectional area contraction ratios for piston acceleration amplitude of 200 m/s2 . (a) Velocity distribution and (b) pressure distribution.
shown in Fig. 2a and b, the velocity loop and pressure node are at the center of the cylindrical duct. However, the loop of velocity fluctuation moves towards the closed end and the node of pressure fluctuation moves towards the piston end with the increase of area contraction ratio. The results also show that the fluctuating pressure amplitude at the closed end of the duct is (Ap /A0 )0.5 times the fluctuating pressure amplitude at the piston end. Fig. 3 presents the pressure distributions at different oscillation modes in closed area contraction ducts for two different area contraction ratios. In Fig. 3a, we observed that the pressure distributions in cylindrical ducts at different oscillation modes are symmetric with respect to the center of the duct. As shown in Fig. 3b, the pressure distributions are no longer symmetric in area change ducts. However, they have the same general behavior, which means half wavelength in the fundamental, full wavelength is the second mode and one and a half wavelengths in the third oscillation mode. The decrease in pressure amplitude with the increase in oscillation mode is clearly observed. Next, the effect of area contraction on finite amplitude standing wave in closed area change ducts will be explained on the basis of numerical simulation results. Fig. 4a illustrates the wave diagram obtained in ducts with cross sectional area contraction ratios of 1 (cylindrical duct). It is assumed that the duct is filled with air and the results are presented for piston acceleration amplitude of 50 m/s2 . As shown in Fig. 4a, an asymmetry between rarefaction and compression zone in the time as in the space is observed. This indicates that there is no real node for the pressure, which varies during a cycle. However, the strength of the wave front is changing in the direction of propagation. Therefore, the corresponding node position can be estimated. The fluctuating pressure amplitude at the closed end of the cylindrical duct for different piston acceleration amplitudes show that the pressure waveform is accompanied with shock even for small piston acceleration amplitude. Fig. 4b shows a wave diagram obtained at fundamental resonant state in an air filled exponential duct with a cross sectional area contraction ratio (Ap /A0 ) of 9 and piston acceleration amplitude of 50 m/s2 . This figure indicates that even if we have a standing wave, there is no real node for the pressure. The corresponding node location changes during a cycle. However, shock wave does
Fig. 3. Pressure distributions along the duct axis at fundamental, second and third oscillation modes for piston acceleration amplitude of 200 m/s2 . Here, p0 = 101.3 kPa. (a) Cylindrical duct (Ap /A0 = 1) and (b) area contraction duct (Ap /A0 = 100).
234
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
Fig. 4. Wave diagram in resonant acoustic ducts with variable area change for piston acceleration amplitude of 50 m/s2 and pressure waveform at closed end of each duct for three different piston acceleration amplitudes when the ducts are filled with air. (a) Cylindrical duct (Ap /A0 = 1) and (b) area contraction duct (Ap /A0 = 9).
not appear in the duct. The pressure distribution is symmetric with the cycle. The pressure fluctuation at the closed end of area contraction duct show that the waveform is not accompanied with shock wave even at high acceleration amplitudes. A very few studies have been performed to understand the effect of gas properties on wave motion in area contraction closed ducts. In the above discussions, results are presented for the wave motion in area change ducts filled with air. Most of the studies were concerned on a particular gas, which develops waveform of large amplitude. Lucas and his group briefly studied the effect of gas properties on wave motion in different duct geometries. The limit of area contraction ratio to obtain shockless pressure waveform with the variation of gas properties has not been explained yet. In this study, to investigate the effect of gas on the limit of area contraction ratio for shockless waveform, two gases, namely air and HFC134a, are used. The gases were chosen to maintain a range of densities and sound speeds. HFC134a is a gas, which is relatively dense, has low sound speed and is used for vapor-compression cooling systems in refrigerators, air conditioners and so on. Numerical simulation is conducted with the assumption that the duct is filled with air or HFC134a at normal atmospheric pressure and room temperature. The limit of cross sectional area contraction ratio to obtain a shockless pressure waveform is illustrated in Fig. 5a and b. The pressure fluctuation at the closed end of an air filled exponential duct with area contraction ratio of 1, 2, 4, 6 and 9, and piston acceleration amplitude of 50 m/s2 is shown in Fig. 5a, while Fig. 5b shows the results for HFC134a-filled exponential duct. Fig. 5a shows that the shock wave appears up to the area contraction ratio of 2 and then disappears for the ratio of 4 and more. The shockless resonance with nearly equal amplitude is observed for the area contraction ratio of 4, 6 and 9. In Fig. 5b, in an HFC134a filled exponential duct, a shock wave propagates up to the area contraction ratio of 4. It is seen that the amplitude of the pressure wave increases with the increase of area contraction ratio in HFC134a filled ducts; however, the waveform has a narrower peak and a broader trough. The change of compression ratio with respect to area contraction ratio is shown in Fig. 5c. This figure shows that the compression ratio increases with the increase of area contraction ratio. In the case of air filled ducts, the compression
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
235
Fig. 5. Pressure waveform and compression ratio at closed end of exponential ducts with cross sectional area contraction ratio when the ducts are filled with air and HFC134a. (a) Pressure waveform in air filled ducts, (b) pressure waveform in HFC134a filled ducts and (c) comparison of compression ratio for two gases.
ratio is nearly constant from area contraction ratios of 9–100, while it increases with the increase of area contraction ratio in HFC134a filled ducts for area contraction ratios of up to 16. This result indicates that the acoustic saturation takes place rapidly in the former at an area contraction ratio of nearly 9. Fig. 5c also shows that for a particular area contraction ratio, the compression ratio is relatively higher in the duct filled with HFC134a than that of air. The reason why the compression ratio is higher in ducts filled with HFC134a is probably due to the high acoustic impedance. 4.2. Effects of duct geometries The limit of area contraction ratio to obtain shockless pressure waveform in closed area change exponential ducts has been explained in the last section. It is noteworthy that the limit of area contraction ratio to which shock wave appears will be varied with the variation of duct geometry. In this section, the effect of duct geometries on finite amplitude oscillations of air column in closed area change ducts will be explained. Among various duct geometries, linear area contraction, exponential area contraction and half cosine function cross sectional area contraction ducts are taken into consideration, which cover all possible simple duct geometries. To compare the effect of duct shapes, the length of duct, radius at both ends and inside gas condition is kept the same. The resonant frequency of gas column oscillations in closed area change ducts strongly varies with the variation of duct geometry. To design acoustic compressors, the consideration of resonant frequency of pressure oscillation is very important. This is because the valve response is related to the frequency of oscillation. The change of resonant frequency with the variation of the duct shape is shown in Fig. 6a for piston acceleration amplitude of 200 m/s2 . The resonant frequencies were normalized by the theoretical resonant frequency of the cylindrical duct. The theoretical resonant frequencies for each duct shape are also shown in the figure. It is seen that for a particular area contraction ratio and a constant duct length, the resonant frequency in a half cosine duct is relatively high as compared to the resonant frequency in conical and exponential duct. The evaluation of finite amplitude standing wave for some
236
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
Fig. 6. The change of resonant frequency, compression ratio and pressure waveform with duct geometries when the ducts are filled with air at normal atmospheric pressure and temperature. Here, piston acceleration amplitude is 200 m/s2 . (a) Resonant frequency, (b) compression ratio and (c) pressure waveform in conical, exponential and half cosine duct with area contraction ratio of 100 (Ap /A0 = 100).
practical applications, especially for acoustic compressors, can be made by the compression ratio obtained at the closed end of the duct, which is estimated by the ratio of the highest to the lowest absolute pressure of the waveform. Fig. 6b shows the compression ratio obtained at the closed end corresponding to the area contraction ratio when the ducts are filled with air. The lengths of each duct are kept the same in the numerical simulation. It is seen that the compression ratio increases with the increase of area contraction ratio in all duct geometries. The rate of increase of compression ratio is linear at first, which comes to the saturation state in each duct after a certain limit of area contraction ratio. The limit of area contraction ratio for acoustic saturation further varies with the variation of piston acceleration amplitudes. For a particular piston acceleration (200 m/s2 ) and a fixed area contraction ratio (Ap /A0 = 100), the compression ratio is high in conical ducts; however, the difference is significantly small. 4.3. Effect of duct lengths The effect of duct length on wave motion in closed axisymmetric ducts needs to be realized to design any practical devices like acoustic compressors. Depending upon the applications, the duct can be scaled in length to affect the frequency of operation and capacity. In this report, the results of numerical simulation are presented for different duct lengths. The cross sectional area contraction ratio of the exponential area contraction duct is 100. Fig. 7 shows the change of compression ratio and resonant frequency with regards to duct length when the duct is filled with air at normal atmospheric pressure and temperature. The piston acceleration amplitude is kept at 200 m/s2 . In each case, the frequency is kept at the first mode resonant frequency. It is observed that the length enlargement decreases the resonant frequency while increasing the compression ratio. Higher frequency operation reduces not only the
Fig. 7. Compression ratio at closed end of exponential duct with cross sectional area contraction ratio of 100 for duct lengths from 0.1 to 1.2 m.
M.A. Hossain et al. / Wave Motion 42 (2005) 226–237
237
size of the duct but also its efficiency. Therefore, to design any practical devices, optimization of the duct length is essential according to the application requirement.
5. Conclusions The oscillations of gas column in closed axisymmetric duct with cross sectional area change were analyzed based on theoretical analysis and numerical simulation. The characteristics of wave motion were investigated to explore the limit of area contraction ratio for shockless pressure waveform in closed ducts. The main conclusions drawn from the present study are summarized as follows: (1) A linear theory was derived to explore the effect of area contraction ratio on wave motion. The standing wave mode was estimated by the developed theory. (2) The limit of area contraction ratio to obtain shockless pressure waveform in closed acoustic duct was estimated. For a particular piston acceleration (50 m/s2 ), the limit of area contraction ratio to get shockless pressure waveform is two when the duct is filled with air. This limit of area contraction ratio for shockless pressure waveform varies with the variation of piston acceleration and duct geometry. (3) The limit of area contraction ratio for shockless pressure waveform and amplitude of the standing wave depend on the gas used in the duct. Particularly, the acoustic impedance plays a significant role in the oscillations of gas in a closed duct. The higher the acoustic impedance of the gas, the higher the amplitude of the pressure waveform in closed ducts.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
R. Betchov, Nonlinear oscillation of a column of gas, Phys. Fluids 1 (1958) 205–212. R.A. Saenger, G.E. Hudson, Periodic shock waves in resonating gas columns, J. Acoust. Soc. Am. 32 (1960) 961–970. W. Chester, Resonant oscillations in closed tubes, J. Fluid Mech. 18 (1964) 44–64. S. Weiner, Standing sound waves of finite amplitude, J. Acoust. Soc. Am. 40 (1966) 240–243. S. Temkin, Propagating and standing sawtooth waves, J. Acoust. Soc. Am. 45 (1969) 224–227. A.B. Coppens, J.V. Sanders, Finite-amplitude standing waves within real cavities, J. Acoust. Soc. Am. 58 (1975) 1133–1140. D.B. Cruishank, Experimental investigation of finite-amplitude acoustic oscillations in a closed tube, J. Acoust. Soc. Am. 52 (1972) 1024–1036. C.P. Lee, T.G. Wang, Nonlinear resonance and viscous dissipation in an acoustic chamber, J. Acoust. Soc. Am. 92 (1992) 2195–2206. D.F. Gaitan, A.A. Atchley, Finite amplitude standing wave in harmonic and anharmonic tubes, J. Acoust. Soc. Am. 93 (1993) 2489–2495. L. Elvira-Segura, E. Riera-Franco de Sarabia, Numerical and experimental study of finite-amplitude standing waves in tube at high sonic frequencies, J. Acoust. Soc. Am. 104 (1998) 708–714. C.C. Lawrenson, B. Lipkens, T.S. Lucas, D.K. Perkins, T.W.V. Doren, Measurement of macrosonic standing waves in oscillating closed cavities, J. Acoust. Soc. Am. 104 (1998) 623–636. Y.A. Ilinskii, B. Lipkens, T.S. Lucas, T.W.V. Doren, E.A. Zabolotskaya, Nonlinear standing waves in an acoustical resonator, J. Acoust. Soc. Am. 104 (1998) 2664–2674. T. Lucas, Acoustic compressors, in: Proceedings of the Non-Fluorocarbon Refrigeration and Air-Conditioning Technology Workshop, 1993, pp. 43–50. M. Kawahashi, M. Arakawa, Nonlinear phenomena induced by finite-amplitude oscillation of air column in closed duct (analysis of acoustic streaming), JSME Int. J. 39 (1996) 280–286. G.W. Swift, Thermoacoustic engines and refrigerators, Phys. Today 7 (1995) 22–28. C. Herman, M. Wetzel, Design of a thermoacoustic refrigerator—a case study, in: Proceedings of ASME Thermodynamics and Design, Analysis, and Improvement of Energy Systems, AES-35, 1995, pp. 195–203. M. Kawahashi, M.A. Hossain, K. Koduka, H. Hirahara, Finite amplitude standing wave in an axi-symmetric closed duct with cross-sectional area change, Trans. Jpn. Soc. Mech. Eng. 68 (2002) 1352–1359. P. Merkli, H. Thomann, Transition of turbulence in oscillating pipe flow, J. Fluid Mech. 68 (1975) 567–575. A.K. Trikha, An efficient method for simulating frequency-dependent friction in transient liquid flow, J. Fluid Eng. 3 (1975) 97–105. E. Turkel, On the practical use of high-order methods for hyperbolic system, J. Comput. Phys. 35 (1980) 319–340.