Assessment of the far-field sound radiation of ducts using the lattice Boltzmann method and a two-dimensional Ffwocs Williams and Hawkings formulation

Transcription

1 Numerical Techniques (others): Paper ICA Assessment of the far-field sound radiation of ducts using the lattice Boltzmann method and a two-dimensional Ffwocs Williams and Hawkings formulation Danilo Braga (a), José P. de Santana Neto (a), André Spillere (a), Andrey R. da Silva (a) and Julio Cordioli (a) (a) Federal University of Santa Catarina - Vibration and Acoustic Laboratory, Brazil, danilo.braga@lva.ufsc.br, 1jpsneto@gmail.com, andre.spillere@lva.ufsc.br, andrey.rs@ufsc.br and julio.cordioli@ufsc.br Abstract The characteristics of the acoustic far-field radiated by a duct highly depends on the geometric aspects found at its open end. This work proposes a simple numerical technique in order to investigate the parameters associated with normal mode radiation of ducts issuing a subsonic mean flow. The technique is based on a hybrid approach involving the lattice Boltzmann method and the Ffowcs Williams and Hawkings formulations for a porous surface. The results are presented in terms of reflection coefficient and sound directivity and compared with the exact analytical solutions provided in the literature for low subsonic mean flows. The good agreement between numerical and analytical solution suggests that this approach can be used to investigate the duct s open end geometry on the parameters associated with sound radiation. Keywords: Lattice Boltzmann Method, Sound Radiation, Ffwocs-Williams and Hawkings sufarce, Far field directivity and Unflanged Duct.

2 Assessment of the far-field sound radiation of ducts using the Lattice Boltzmann method and a two-dimensional Ffwocs Williams and Hawkings formulation 1 Introduction The noise radiation from duct systems is a classical problem in acoustics, which is still addressed due to the new technological advents involving applications such as jet engines, exhaust systems, industrial pipeline openings and burner stacks. Therefore the characteristics of the acoustic far-field radiated by a duct highly depends on the geometric aspects found at its open end. In this context, in the classical theoretical work by Levine and Schwinger [1], the problem of sound reflection and radiation from an open unflanged duct termination has been solved for the absence of a mean flow using the Wiener Hopf technique. In a similar work, Munt [2] developed an accurate model for the directivity of the sound radiated by pipelines carrying subsonic flow. Munt [3] also proposed a model for the reflection coefficient in pipelines for flow velocities up to M = 0.4 assuming uniform flow at duct (plug form) [4], and solving flow mismatch problem between the issuing jet and the ambient fluid using approximate theories [5, 6]. Analytical models by Munt [2, 3] were validated experimentally by Allam and Åbom [7], based on the maximum flow velocity for the Munt s theory and average flow velocity for the experimental data. Most recently, these works were validated numerically by Da Silva et al. [8] and Yong et al. [9] using a non-equilibrium technique known as the lattice Boltzmann method (LBM) with axisymmetric condition proposed by Halliday et al [10]. Using the same numerical method, the analytical model of Levine and Schwinger [1] was validated by Da Silva and Scavone [11]. The objective of this paper is to develop a numerical scheme based on the lattice Boltzmann method for investigating the radiation properties of ducts issuing a subsonic mean flow. The main challenge involved in this task is to define a technique in order to extract the far-field sound using near-field properties. This is achieved by using the Ffwocs Williams and Hawkings (FWH) formulations proposed by [12] for two-dimensional domains. This work is structured as follows: Section 2 presentes the theoretical background involving the numerical method (LBM scheme D2Q9 model [13]) and the two-dimensional formulation of the FWH used in the study. In Section 3, the pipe model is described by an axisymmetric cylinder structure immersed in a fluid domain surrounded by open boundaries. In Section 4, presents the comparison between the numerical results and those provided in the literature. Moreover, the contribution of monopole and dipole effect on far field radiation is discussed. Finally, Section 5 provides a discussion of the results and suggestions for future research. 2

3 2 Theoretical Background 2.1 The Lattice Boltzmann Method The LBM is used to calculate the unsteady flow and its induced noise. Lattice-based methods are by nature explicit, transient and compressible, and are an alternative to traditional CFD methods based on the discretization of the Navier-Stokes equations and derived variations. The LBM is based on the velocity space discretization of the Boltzmann equation to predict macroscopic fluid dynamics. The lattice Boltzmann equation has given by f i (x + c i t,t + t) = C i (x,t), (1) with f i the particle distribution function moving in the i-th direction, according to a finite set of discrete velocity vectors (c i = 0,...,n), c i t and t are respectively space and time increments, which c i is the speed of sound in the i direction. For convenience, we choose the convention t = 1 in the following discussions. The collision term on the right hand side of Eq. (1) adopts the form known as Lattice Bhatnagar Gross Krook (LBGK): C i (x,t) = 1 τ ( fi f eq ) i, (2) with τ the relaxation time parameter, and f eq i the local equilibrium distribution function, which depends on local hydrodynamic properties. Qian et al. [13] propose a LBGK scheme called D2Q9 model, which provides a solution to equilibrium distribution function, given by ( fi M = ρε i 1 + u.c i c 2 + u.c2 i c2 s u 2 ) s 2c 4, (3) s where c s, ε, ρ and u are lattice speed of sound, velocity weights, fluid density and velocity, respectively. For D2Q9 model, the discrete velocity c i connecting each site to its neighbor lattices with scheme shown in Figure 1. The velocity weights to the scheme are ε 0 = 4/9, ε 1 = ε 2 = ε 3 = ε 4 = 1/9 and ε 5 = ε 6 = ε 7 = ε 8 = 1/36 and lattice sound speed is c s = 1/ 3. Source: (Own authorship, 2016) Figure 1: Cell lattice scheme based in D2Q9 model However, to simulate the 3D axisymmetric flow, we built the two-dimensional axisymmetric scheme incompressible LBGK D2Q9 model proposed by Halliday et al [10]. In this condition, Eqs. (1) and (2) can be rewritten in terms of f i (x + c ix,r + c ir,t + 1) = 1 τ ( fi f eq ) (1) i + h i + h (2) i, (4) 3

4 where the discrete velocity c i is describe at position x (axial direction) and r (radial direction), and h (1) i and h (2) i are the source terms given by h (1) i h (2) i = ε iρu r, r (5a) 3ν [ =ε i r p + ρ x u x u r + ρ ] r u r u r + ρ( r u x + x u r )c ix, r (5b) where the gradient terms in Eq. (5b) were evaluated using discrete difference approximations on lattice. The source term h (2) i should be considered to obtain second order accurate in results. Thus, using the discrete velocity, ditribution fuction and relaxation time, we obtain the basic hydrodynamic quantities, in lattice units, such as fluid density ρ = f i, velocity ρu = f i c i, pressure p = ρc 2 s and kinematic viscosity ν = c 2 ( s τ 1 2). For all simulations the relaxation time was set to τ = , which corresponds to a kinematic viscosity of ν = Ffwocs Willians and Hawkings two-dimensional surface The FW-H equation formulates aerodynamic noise in terms of sources that are distributed on a control surface and throughout the volume external to that surface [15]. The FW-H equation can be written in differential form as ( 2 2 t 2 c2 0 x i x i )(H( f s )ρ ) = 2 x i x i (T i j H( f s )) x i (F i δ( f s )) + t (Q iδ( f s )), (6) where, the first term on right side of Eq. (6) is quadripole term, which depends Lighthill stress tensor T i j = ρu i u j + P i j c 2 0 ρ δ i j, the second one is dipole term that involves the contribution of vector F i = (P i j + ρu i (u j v i )) f s x j and last term is monopole term that contains the vector Q = (ρ 0 v i + ρ(u i v i )) f s x i. f s is a function that defines a surface from which the near-field variables are to be extracted. u i and v i are the particle velocities in the horizontal and vertical direction, respectively. δ i j is the Kronecker delta, H( f s ) is Heaviside function, which is 1 for f s > 0 and zero for f s < 0), c 0 is speed sound ambient and ρ 0 is density ambient. The frequency domain solution of the FW-H equation (Eq. 6) can be written in the form [12] H( f s )c 2 0ρ (y,ω) = iωq n (ξ,ω)g(y,ξ )ds F i (ξ,ω) G(y,ξ ) 2 G(y,ξ ) ds T i j dξ, f s =0 f s =0 y i f s >0 ξ i ξ j (7) or in bi-dimensional solution neglecting the quadrupole term: H( f s )c 2 0ρ (y,ω) = iωq n (ξ,ω)g(y,ξ )dl F i (ξ,ω) G(y,ξ ) dl (8) f s =0 f s =0 y i where ξ and y describe the source and observer coordinates, respectively and ω is the frequency. Note Eq. (7) is valid in two or three dimensions depending on the form of the free-field 4

5 Green s function G(y,ξ ). If we set the surface around a turbulent region where T i j is predominant, the third term of Eq. (7). disappears resulting in Eq. (8). Furthermore, it is desirable to avoid the volume quadrupole calculation due to it is computational expense. However, the quadrupole effect can be captured by the remaining sources term Q n and F i given by Q n = (ρ(u i +U i ) ρ 0 U i )) ˆn i, (9) F i = (pδ i j + ρ(u i U i )(u j +U j ) + ρu i U j ) ˆn j, (10) where U i is the cartesian fluid velocity component in the direction i and u i is the cartesian surface velocity components and ˆn j is the outward-directed unity normal vector. The 2-D Green s function for the flow in the y 1 direction is given by [15] G(y,ξ ) = i ( 4πβ e(imk(y 1 ξ 1 )) H (2) k 0 β 2 (y 1 ξ 1 ) 2 + β 2 (y 2 ξ 2 ) ), 2 (11) where M = U i /c s is the Mach number, β = 1 M 2, H (2) 0 is a Hankel function of the second kind and zero order and k = ω/c 0. 3 Numerical Scheme The numerical model based on LBM is represented by a closed-open cylinder inserted in a fluid domain surrounded by anechoic boundaries as illustrated in Figure 2. All fluid domain was built with a square grid is defined with L x = 1000 lattices or cells along horizontal axis and L y = 500 lattices along vertical axis. The duct wall has the length equal to 630 cells and thickness equal to 1 cell. The no-slip condition was asserted to the wall, using a bounce-back scheme that provides second-order acuracy, as provided by Bouzidi et al. [16]. Source: (Own authorship, 2016) Figure 2: Scheme of the axisymmetric domain. 5

6 A absorbing boundary condition reported by da Silva et al. [8] was used around the fluid domain. The thickness of the absorbing boundary was 30 cell, which provides a frequencyaveraged magnitude of the reflection coefficient smaller than 0.001, as reported in [8] and [9], for both perpendicular and oblique incidence. An acoustic source buffer (L source = 60 cells) was implemented using a linear chirp running from ka = 0 to 1.5, where ka = ω c s a is Helmholtz number. TIt is based on the ABC condition, but prescribed by a non-zero target velocity and density given by u source (t) = Mc s + ρ 0 c s [ cs cos ρ 0 ρ source (t) = ρ 0 + ρ 0cos ] ) T t T i ] a (ka (t T i ) t min + ka max [ cs a (ka (t T i ) t min + ka max ) T t T i H(t), H(t), (12a) (12b) where ρ 0 is input density amplitude, t is the increment time, T i initial time iteration, t is the timestep, T t is total number iterations and H(t) is the Heaviside step function that adopt the following condition: H(t) = { 0,t < T i 1,t T i (13) The Heaviside step function is necessary mainly in mean flow cases, where the fluid in the whole domain has to accelerate from stagnation to a steady state. Thus, there should be enough time to stabilize and apply the acoustic source superimposed to the mean flow. Initialization of the acoustic source will occur when the transient time of the flow is less than T i, i.e., T i T i0 + L x /(Mc s ) [9]. The acceleration time T i0 for the source buffer with thickness equivalent to 60 cells is approximately 4000 timesteps. Pressure and acoustic particle velocity was measured at point 1 (see Fig. 2), using averaged values at the section. The discrete Fourier transform was used on the signals, and the impedance at point 1 is calculated as Z( f ) = P( f,1) u( f,1) (14) And the radiation impedance can be obtained by the following expression [17] ( ( ) ) Z Z r = Z 0 i tan arctan + k L, (15) Z 0 i where L is the distance from measuring point to the pipe end and Z 0 = ρ 0 c s is the characteristic acoustic impedance. Dalmont et al. [17] defines the radiation reflection coefficient at the pipe termination and the inertial effect due to the fluid loading at the open end, known end correction as 6

7 R r = Z r Z 0 and (16a) Z r + Z 0 l = 1 ( ) k arctan Zr (16b) Z 0 i In order to calculate the sound directivity in the far field, the FW-H sufarce was placed around the pipe s open end, as depicted in Fig. 2. Thus, the time histories of the fluid density and velocity for each measuring point around the FW-H surface were stored for the entire simulation time. This data was then used in Eq. (12) in order to obtain the far-field acoustic pressure. In this case, the calculations of Eq. (12) were performed for 38 points evenly distributed around the pipe s open end. For this calculation the observer distance used is five times 250 cell (distance used by Yong et al [9]) from the outlet. The normalized pressure directivity was obtained from the far-field data using the following equation: G(θ, f ) = P(θ, f ) 38 n=1 P2 (θ, f ) N, (17) where P(θ, f ) is the acoustic pressure as a function of the source frequency and azimuthal angle. 4 Results 4.1 Sound radiation at the open end Initially, the numerical model proposed in this study was validated with analytical model results proposed by Munt [3] in the presence of mean flow condition using the duct parameters such as reflection coefficient and end correction for this validation. In order to compare the numerical approach, the numerical results from a similar LBM model proposed by Da Silva et al. [8] (with pipe radius and axisymmetric condition different) were used to different Mach numbers. The numerical simulations using M = 0.05 to 0.15 were implemented following the conditions described on in section 3 and the results obtained for the magnitude of the reflection coefficient are shown in Figure (3a) to Figure (3c). The general trend found on the numerical result of the magnitude of the reflection function, shows that in the mid frequencies the reflected energy at the open end is usually bigger than the one predicted by the Munt s theory, especially at range from ka = 0.08 to 0.8. However, the results at the low and high-frequency limits are in good agreement with the analytical solution, whereas Da Silva et al [8] results diverges significantly, particularly when the Mach number increases. Other significant deviation can be see at the low frequency band on the results related to the end correction, as shown int the Figures (3d) to (3f). The deviations can be explained by the following reasons: First, the anechoic boundary is not perfectly anechoic. As a consequence, 7

8 (a) M = 0.05 (b) M = 0.1 (c) M = 0.15 Figure 3: Comparison between analytical [3] (solid line) and numerical results [8] (empty circle) and present work (dash line) to Magnitude of R r (left side) and End correction l normalized by radius a (right side). the acoustic energy reflected at the fluid boundaries return to the measuring points. Moreover, in comparing with Da Silva et al [8] results, the formulation used to calculate end correction using 2 point inside of the pipe can explain the deviations on low frequencies. Nevertheless, the maximum deviation between analytical and numerical results was 7% in terms of the reflection coefficient and approximately 5% in terms of end correction (to ka up 0.2). 4.2 Far field directivity Following the approach proposed in this work, in order to evaluate the directivity in far field of the model, the analytical results by Gabard and Astley [14] and numerical data by Yong et al [9] are used to compare results obtained for M = and 0.15 and the Helmholtz number of ka = 0.74 to Using the 2D FWH formulation (Eq. 8), the directivity results for these mach numbers and frequencies are calculated with the conditions proposed in the previous section and shown in Figure (4). In general, the results are in good agreement for low and high observer angles (until 35 o for M = and 41 o for M = 0.15) with analytical model, which could represent the zone of relative 8

9 (a) ka = 0.74 and M = (b) ka = 1.48 and M = (c) ka = 0.74 and M = 0.15 (d) ka = 1.48 and M = 0.15 Figure 4: Comparison between analytical [14] (solid line) and numerical results [9] (dashline) and present work (empty circle) for directivity, in db silence observed in some experiments [2][6]. The greater discrepancy between the numerical and analytical results takes place for ka = 1.48 and ka = 0.74 to M = 0.15 corresponding to a deviation of db and db at approximately θ = 84 o, respectively. The numerical results by Yong et al [9] demonstrate a better convergence on the low angle range and a greater deviation for higher angles, which can be explained for measures made directly on points at 250 cells from end pipe (probably a region between near field and far field). To understand the tendency of the numerical results it becomes necessary to evaluate the dominance of the source term on directivity. In order to analyze the influence of each term (monopole and dipole) on directivity the Figure 5 presented this comparing to M = 0.15 and for same frequency. As we have expected, the two terms affect the results at all angles, where the contribution of monopole term is bigger on almost all angles and dipole term has more contribution on lower angles. 5 Conclusions This paper investigated an axisymmetric based on the LBM scheme for a pipe model immersed in a fluid domain surrounded by open boundaries. The Numerical model was considered partially validated with exact analytical results [3] to end pipe parameters with maximum deviation of 7% for the reflection coefficient. For the far field prediction, using approximated analytical solution [14], the maximum deviation observed corresponds to db and db for M =

10 (a) M = (b) M = 0.15 Figure 5: Contribution of dipole (empty circle), monopole source (dash line) and both (solid line) to directivity to ka = The deviations can be explained by reflections at the absorbing boundary, which is not perfectly anechoic, moreover, the axisymmetric model may not be able to capture tri-dimensional phenomena involved and approximations on the FWH formulation on implementation. Test cases have demonstrated that several terms are needed (changing FWH sufarce size) as well as better positioning at the end of the pipe to for the Mach numbers involved in this study. Apart from that, the current approach it is efficient enough to be used to perform quick estimates of the far field noise radiating from tailpipes at low Mach numbers. References [1] Levine, H.; Schwinger, J. On the radiation of sound from an unflanged circular pipe. Physical Review, 73 (4), 1948, [2] Munt, R.M. The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe, Journal of Fluid Mechanics, 83, 1977, [3] Munt, R.M. Acoustic transmission properties of a jet pipe with subsonic jet flow: I. The cold jet reflection coefficient, Journal of Sound and Vibration, 142 (3), 1990, [4] Carrier, G. F. Sound transmission from a tube with flow. Quarterly of Applied Mathematics, 13, 1956, [5] Many R. Refraction of acoustic duct waveguide modes by exhaust jets. Quarterly of Applied Mathematics, 30, 1973, [6] S.D. Savkar, Radiation of cylindrical duct acoustic modes with flow mismatch, Journal of Sound and Vibration, 42, 1975, [7] Allam, S. and Abom, M. Investigation of damping and radiation using full plane wave decomposition in ducts. Journal of Sound and Vibration, 292, 2006, [8] Da Silva, A. R.; Scavone, G. P.; Lefebvre, A. Sound reflection at the open end of axisymmetric ducts issuing a subsonic mean flow: a numerical study, J. Sound Vib., 327, 2009,

11 [9] Yong, S.; Da Silva, A. R.; Scavone, G. P. Lattice Boltzmann simulations of sound directivity of a cylindrical pipe with mean flow. Phys. A: Math. Theor., 46, 2013, (13pp). [10] Halliday, I.; Hammond, L. A.; Care, C. M.; Good, K.; Stevens, A. Lattice Boltzmann equation hydrodynamics. Phys. Rev., 64, 2001, [11] Da Silva, A. R. and Scavone, G. P. Lattice Boltzmann simulations of the acoustic radiation from waveguides. J. Phys. A: Math. Theor., 40, 2007, [12] Lockard, D. P. An Efficient, Two-Dimensional Implementation of the Ffowcs Williams and Hawkings Equation. Journal of Sound and Vibration, 229(4), 1999, [13] Qian, Y. H.; D Humieres, D.; and P. Lallemand. Lattice BGK Models for Navier-Stokes Equation. Europhys. Lett., 17 (6), 1992, [14] Gabard, G. and Astley, R. J. Theoretical model for sound radiation from annular jet pipes: far- and near-field solutions. J. Fluid Mech., 549, 2006, [15] Lockard, D. P and Casper, J. H. Permeable Surface Corrections for Ffowcs Williams and Hawkings Integrals. 11th AIAA/CEAS Aeroacoustics Conference. Monterey, California, [16] Bouzidi, M.; Firdaouss, M.; Lallemand, P. Momentum transfer of a Boltzmann-lattice fluid with boundaries, Physics of Fluids 13(11), 2001, pp [17] Dalmont, J. P. ; Nederveen, C.J. and Joly, N. Radiation impedance of tubes with different flanges: numerical and experimental investigations. J. Sound Vib., 244(3), ,