NUMERICAL VALIDATION OF AN ACOUSTIC IMAGING METHOD FOR DUCT SPINNING MODE WITH IN-DUCT CIRCULAR MICROPHONE ARRAY

Transcription

1 European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 22) J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September -4, 22 NUMERICAL VALIDATION OF AN ACOUSTIC IMAGING METHOD FOR DUCT SPINNING MODE WITH IN-DUCT CIRCULAR MICROPHONE ARRAY Xun Huang, 2 and Edward Peers State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University Beijing, China huangxun@pku.edu.cn 2 Department of Aeronautics and Astronautics, College of Engineering, Peking University Beijing, China Keywords: low-noise engine, spinning mode, acoustic imaging, linearised Euler equations. Abstract. An imaging method of acoustic spinning modes propagating within a circular duct is introduced in this paper. Nowadays, the measurements within a duct have to be conducted using in-duct microphone array, which is unable to provide information of complete acoustic solutions across the test section. The proposed method can estimate immeasurable information by forming a so-called observer. The proposed method is developed in a theoretical way. The fundamental idea behind the testing method was originally developed in control theory for ordinary differential equations. Spinning mode propagation, however, is formulated in partial differential equations. A finite difference technique is used to reduce the associated partial differential equations to a classical form in control. The observer method can thereafter be applied straightforwardly. The algorithm is recursive and can be operated in real-time. A numerical simulation for a straight circular duct is conducted to demonstrate the proposed method. The acoustic solutions reconstructed on the cross section compare well with analytical solutions. The good agreement suggests the potential of the proposed method in acoustic experiments.

2 Introduction Aircraft noise has a significant environmental impact on the communities near airports. To meet the increasingly strict regulations, quiet engine design has been regarded as one of the most important research topics in aircraft propulsion and power systems []. The noise from high bypass ratio turbofans installed on most current transport airports is dominated by fan noise and jet noise. Fan noise has spinning modes generated from fan and stator assembly. Accurate measurements of the propagation of spinning mode noise within the duct of an engine nacelle would be quite helpful for low noise design [2]. Spinning mode sound pressure measurements from microphone arrays installed on engine duct wall [2, 3] have been used to investigate the nature of the flow-induced acoustics. In particular, acoustic field of fan noise sources can be visualized using a carefully designed in-duct beamforming technology [4]. In addition to acoustic pressure perturbations, the knowledge of the associated particle velocity fluctuations is also important for low noise engine design. The techniques of direct particle velocity measurements, such as particle image velocimetry [5], are usually not applicable for aggressive conditions of engine tests. In this work, it was demonstrated that acoustic particle velocity field can be approximately reconstructed from partial measurements of sound pressure. The proposed measurement technique constitutes the main contribution of the paper. Potential applications of the measurement method include low noise design [6] and on-site engine health monitoring [7]. An estimation method of particle velocity fluctuations within engine combustion chamber based on sound pressure measurements has been proposed previously [8]. The physical process of fluid and acoustic was approximately treated as one-dimensional. Beamforming was applied to instantaneously generate sound pressure outputs, which were thereafter decomposed to infer the associated velocity fluctuations. For the relatively complicated three-dimensional spinning mode acoustic problems, a different measurement method was developed in this work. It is well known that fan noise propagating within a cylindrical engine duct consists of various spinning modes [9] which can be described by partial differential equations (PDE) in three-dimensional cylindrical coordinates. A so-called state observer [] was constructed to estimate acoustic particle velocity field from partial sound pressure measurements, given the PDE describing dynamics of the complete acoustic system. The fundamental idea behind the proposed method shares a similar theoretical background with previous beamforming works [, 2, ], where the inherent system dynamics was governed by linear ordinary differential equations (ODE). In contrast, spinning modes are governed by relatively complicated linear PDE. Conceptually, internal states (e.g. velocity) of a linear PDE (e.g. wave equation) could be reconstructed from suitable system outputs (e.g. pressure measurements), as long as the system is observable. We should admit that the observability of linear wave equations is a resolved issue. However, the usage of observer in practical sound measurements is yet rare. Extensive and beautiful mathematical manipulations can be found in the literature [3, 4] and are thus omitted here for brevity. In this work, bearing in mind practical needs, we developed an observer of PDE wave equations for aircraft engine applications. The investigation thus represents an interesting endeavour and should have potentially important implications to more generic scientific applications. The paper is organised as follows. Section 2 introduces the preliminary knowledge of spinning modes and observer theory. Section 3 develops the theoretical model of the proposed method. A numerical case was conducted to validate the proposed method in section 4. Finally, section 5 summarizes the present work. 2

3 2 Background 2. Engine fan noise model Acoustic radiation Acoustic radiation 2 Intake Outer nozzle Figure : Diagram of noise propagation in an aero-engine bypass duct, where: is fan rotor and 2 is stator. The rotating fan and stator assembly (shown in Fig. ) is mainly responsible to the generation of the tonal spinning modes in aircraft engine applications [5]. This research focused on the measurements of spinning modes propagation in the nacelle duct with a slowly varying cross-section. With no loss of generality, an ideally straight cylindrical duct was considered for simplicity. The perturbations of acoustic pressure, density and velocities, (p, ρ, u, v, w ) are generally small compared with the background mean flow variables (p, ρ, u, v, w ). Sound wave propagation can thus be approximately modeled by linearised Euler equations (LEE) [6, 7]. The acoustic disturbances can be represented by Fourier series in terms of azimuthal modes m. For example, the acoustic pressure has the form of p = m= p m(x, r, t)e imθ. In addition, the mean flow in the straight cylindrical duct at the operating Reynolds numbers can be assumed stationary and one-dimensional (u,, ). Hence, the three-dimensional LEE governing a single mth spinning mode propagation is ρ m t + u ρ m x + ρ ( u m x + v m r + v m r + w m r θ ) =, () u m t v m t + u u m x + p m ρ x + u v m x + p m ρ r =, (2) =, (3) w m t + u w m x + p m ρ r θ =, (4) where all variables are nondimensionalised with respect to a reference length, a reference speed and a reference density. For the idealized geometry (a straight and semi-infinite unflanged duct), 3

4 Eqs. () (4) have analytical solutions for sound at a tonal frequency k. The solutions are ρ m(x, r, θ, t) = cj m (k r r)e i(kt kxx mθ), (5) x u ck m (x, r, θ, t) = J m (k r r)e i(kt kxx mθ), (6) k k x M j v m(x, c d[j m (k r r)] r, θ, t) = i e i(kt kxx mθ), (7) k k x M j dr w m(x, cm r, θ, t) = r(k k x M j ) J m(k r r)e i(kt kxx mθ), (8) where M j = u /C, C is the speed of sound, c is the amplitude of the acoustic perturbation (the nondimensional value is normally less than 3 ), and J m is the mth-order Bessel function of the first kind. The nth radial wavenumber k r of the mth spinning mode is the nth solution of the following equation determined by the hard-wall boundary conditions of the duct, i.e. d[j m (Rk r )]/dr =, where R is the radius of the outer duct wall. The axial wavenumber k x of the mth spinning mode can be subsequently calculated using k x = ( M j ± k 2r( M 2j )/k2 ) k/( M 2 j ), where the sign of (±) depends on the upstream/downstream direction of the spinning wave. As an example, Fig. 2 shows the analytical solutions of sound pressure at the spinning mode (m = 2, n = ) and the nondimensional tonal frequency k = 2. Figure 2: Sound pressure instantaneously propagating within an ideally straight cylindrical duct, where the circumferential mode m = 2, the radial mode n = and the nondimensional frequency k = 2. 4

5 2.2 State observer theory The aforementioned LEE model has been extensively used in engine noise simulation [8]. Details of the numerical simulation setup and practical applications can be found in the literature [9, 2]. The LEE model is adopted in this work to assist partial pressure measurements taken at different time steps to recover full sound solutions, and in particular, those associated acoustic particle velocities. In control theory, a system is said to be observable if its internal states can be deduced from its external system outputs. In this work, the physical process of spinning mode propagation is the system, which is described by the LEE model. The internal states of the system consist of sound density (ρ ), pressure (p ) and the associated particle velocity (u, v, w ). The external outputs of the system are sound pressure measurements from a circular microphone array installed on the duct wall (Fig. 3). The essence of the proposed method is to instantaneously estimate time varying acoustic parameters (density, velocities, etc.) of a spinning mode with the knowledge of the inherent dynamic model and partial measurements (sound pressure for the case). For convenience of readers, the fundamental idea of observer theory is briefly introduced in the following. More details can be found in any linear control textbook [2]. r Sensor array Mode noise source u x θ Duct wall Figure 3: The illustration of a spinning mode noise (m = 5) and the setup of a downstream in-duct circular array with 6 sensors (denoted by ). To briefly recapitulate, a dynamic process can be generally described as a so-called state space model in classical control, d x(t) = Ax(t) + Bu(t), dt (9) y(t) = Cx(t), () where t is time, x and u represent internal states and inputs of the model, y denotes model outputs, and A, B and C are dynamic, control and output matrices, respectively. In this work, 5

6 A, B and C are time invariant, and y is the circular sensor array measurements, from which the acoustic variables x are hopefully inferred. The so-called observability in linear control theory defines a measure of how well those internal states could be deduced [2]. The observability of the state space model [Eqs. (9)-()] can be investigated by forming an observability matrix O = ( C, CA,..., CA M ) H, () where M is the rank of A and H denotes conjugate transpose. It has been proved that a system is observable if the rank of O equals M. Once a system is observable, it is possible to construct an observer to approximate the internal state X, with the knowledge of system outputs y. The observer has the form of d = Aˆx(t) + Bu(t) + L(y ŷ), dtˆx(t) (2) ŷ(t) = Cˆx(t), (3) where ˆ() denotes approximations. The only design issue left is the chosen of a suitable observer gain L, which can be resolved for most cases in a couple of steps of trial and error. The estimation error e = x ˆx satisfies d e(t) = (A LC)e, (4) dt which can be straightforwardly achieved by subtracting Eq. (9) from Eq. (2). Hence, e converges to zero when t, as long as the real parts of all eigenvalues of the matrix (A LC) are negative. 3 Theoretical Development The propagation of a spinning mode within a duct is of particular interest in this work. Various analytical methods [6] and numerical simulations [9, 22, 23] have been developed to deep our insights of engine noise transmission and radiation. On the experimental side, it could be interesting and useful to visualize acoustic images of spinning modes in a duct cross section in real-time. A sound pressure imaging technique based on beamforming with microphone array is already available [3, 24]. However, the duct wall prevents laser beams from scanning the inside fluid flow. On the other hand, acoustic velocity sensors are too delicate to be used within a violent duct flow. By now it is still difficult, if not impossible, to directly measure the associated particle velocity. The question to be answered in this work is how to reconstruct acoustic particle velocity from pressure measurements. The inherent dynamics of spinning modes propagation has been represented by LEE [Eqs. (5 8)]. For the construction of an observer, the LEE should be firstly descretized to ODE in the same form as Eqs. (9 ). It is worthwhile to note that all formulations presented in this paper are continuous time version, which is presumably more readable for readers. The practical implementation actually adopts a discrete time version, which is omitted for the brevity of the paper. In addition, the modeling error and measurement error are omitted to focus on central theoretical issues. The method proposed in this work can be straightforwardly extended based on Kalman filter [25] to further address modeling and measurement noise issues. It will be of practical importance to obtain acoustic imagines of spinning modes propagating within ducts. The image plane considered in this work is the duct cross section at the x location of sensor arrays. The imagine plane is descretized into an N r N θ grid, where N r and N θ are 6

7 the gridpoint numbers in r and θ coordinates, respectively. Both values should satisfy Nyquist- Shannon sampling theorem, i.e. N r 2n and N θ 2m, where n is radial mode and m circumferential mode. For simplicity, N θ is set to the sensor number of the circular array. For example, Fig. 4 shows the grids for a (m = 6, n = ) spinning mode; the number of gridpoints in the radial direction is 9; and the number of gridpoints in the circumferential direction is 3. It is implicitly assumed that 3 pressure sensors are located on the outer ring to instantaneously capture sound pressure. θ r A Figure 4: The grids in polar coordinates for the duct test section. For brevity, the subscript m in Eqs. () (8) is omitted in the rest of this paper. To construct a state space model for each single spinning mode, the following relations and approximations are adopted for Eqs. () (4) at each gridpoint (a, b), where a N r, and b N θ : () / x is replaced by ik x (using the spectral method); (2) ρ = p in the nondimensional equations; (3) v a,b / r is approximated by a central difference, (v a+,b v a,b )/(2δr); (4) w a,b / θ is approximated by a central difference, (w a,b+ w a,b )/(2δθ); (5) p a,b / r and p a,b / θ are approximated in similar ways, respectively; (6) v,b / r, p,b / r, () a,b+ = () a,, and () a, = () a,b ; where v a,b = v (a, b), v a,b / r = v (a, b)/ r, and so forth. Using the above relations all spatially differential terms in Eqs. () (4) are replaced and thus the PDE are reformulated to the 7

8 ODE. For example, the ODE at one gridpoint (a, b) are ρ a,b ik x u ik x ρ ρ ρ d u r a,b ik x a,b dt v a,b = ρ ik x u u a,b ik x u v w a,b a,b ik x u w a,b }{{}}{{}}{{} dx a,b /dt A a,b x a,b 2ρ δr ρ a+,b + u 2ρ δr ρ a,b a+,b 2ρ v + u a,b δr a+,b w a+,b 2ρ v δr a,b w a,b }{{}}{{}}{{}}{{} + A ra,b ρ 2rδθ 2ρ rδθ } {{ } A φa,b x a+,b ρ a,b+ u a,b+ v a,b+ w a,b+ } {{ } x a,b+ A ra,b ρ 2rδθ + 2ρ rδθ } {{ } A φa,b x a,b ρ a,b u a,b v a,b w a,b, } {{ } x a,b (5) C a,b p a,b = [ ] }{{}}{{} y a,b ρ a,b u a,b v a,b w a,b }{{} x a,b, (6) where the nondimensional ρ equals p. The complete equations of the state space model on the whole mesh (see Fig. 4) are x a,b d x a,b x a,b = A ra,b A φa,b A a,b A φa,b A ra,b dt x a,b, (7) }{{}}{{} x a,b+ dx/dt A x a+,b }{{} x y a,b }{{} y = C a,b } {{ } C x a,b }{{} x. (8) In most practical tests, Eq. (8) only contains the contributions from those pressure sensors that are surface mounted on the hard wall of the duct, where v and p / r equal. It is easy 8

9 to examine that the radial, axial and circumferential velocities within the duct cross section are normally not observable if only the measurements from on-surface pressure sensors are available. To achieve an observable system model, Eq. (5) is used to extend the system outputs, i.e. the sound pressure at r ( r R) can be inferred from the measurements at outer radius R, according to the relation, ρ (r) = ρ (R) J m(k r r) J m (k r R), (9) With this extrapolation, the system model is observable. An observer [Eqs. (2) (3)] can thereafter be constructed with a carefully chosen observer gain. It is worthwhile to point out the above extrapolation should be regarded as an approximate modeling method and only works for slowly varying cross section, which is the case for almost all aircraft engines. In summary, Eqs. () (4) describe a mathematical model that is obtained by applying first principles to the acoustical physics. In contrast, Eqs. (7) (8) are a reduced mathematical model, in which partial differentials are replaced by finite difference terms. The particular finite difference schemes chosen in the reformulation could affect the reduced model and consequently the accuracy of the observer. A detailed investigation of that topic is still ongoing and is omitted here. In the following numerical validations, we stick to the central finite difference scheme. 4 Numerical Validation A primitive numerical simulation was done to validate the proposed method. A noise source of a single spinning mode m = 6, n = is assumed propagating within a straight duct. The sound is assumed tonal and the frequency is 4 Hz. The related wavenumber k is The nondimensional u =.3. The analytical solutions of sound (ρ, u, v, w ) at the nondimensional time t = and t = are calculated using Eqs. (5) (8), where M j is set to.4, to validate the proposed method. In the simulation, it is assumed that fan noise sources is at x = and a circular array of 3 pressure sensors is surface mounted at x =.. All acoustic velocities on the whole cross section (where r ) and sound pressure within the duct (where r < ) are unknown. An observer is developed to estimate unknown acoustic particle velocities from those partial pressure measurements at r =. It is worthwhile to mention that Eq. (9) is adopted to generate an initial guess of p at r < from the measurements of p at r =. The observer-based method is conduced as the following. Step : Prepare the dynamics matrix A and the sensor matrix C according to Eqs. (7) (8). It should be noticed that the discrete form with a sampling step δt has to be considered for practical cases. The discrete counterpart of A in continuous version is e Aδt. Step 2: Solve the observer gain L using the formula, L = (A P)C, which can be straightforwardly derived from Eq. (4). The observer poles (P) are deliberately chosen to stabilize the observer and thus minimize approximation errors. In this simulation, L is empirically set to.5i, where I is the identity matrix with a rank equals the rank of A. Step 3: Measurements of the sound pressure are generated by the circular sensor array installed on the outer wall. The measurements are simulated using Eq. (5) in this work. In addition, the initial guess of the sound pressure at r < are obtained using Eq. (9). The resulted sound pressure solutions across the r θ section constitute system outputs y in Eq. (2). Step 4: The observer [Eqs. (2) (3)] is conducted to extract the approximation of immeasurable particle velocity. It should be noticed that the acoustic system of interest is autonomous 9

10 and thus the control matrix B is null. Step 5: Step 4 is instantaneously conducted for samples taken at each time steps. The observation error (y ŷ) is examined. In this numerical validation the observation starts from t = to t =. The time step is δt =.. Figure 5 shows the final observation results at t =. It can be seen that particle velocities in x r θ coordinates have been successfully reconstructed with correct profiles. In addition, the results suggest the proposed method is numerically stable. The proposed method has a recursive algorithm that can gradually improve approximation errors to a minimal level. Hence, approximation errors do not build up and the complete calculation does not blow after, iterations. (a) (b) Y.5.5 X -.5 Y.5.5 X (c) (d) Y.5.5 X Figure 5: Observer solutions of the cross section at x =., where t =, (m, n) = (6, ), k = 7.39 and u =.3, (a) ˆρ, (b) û, (c) ˆv and (d) ŵ. -.5 Y.5.5 X To demonstrate the effect of the proposed measurement method, the observation history at a point (A as shown in Fig. 4) is compared to analytical solutions in Fig. 6. The particle velocity (u, v, w ) is initially set to zero and iteratively updated with new pressure measurements. It can be seen that the observer effectively reconstructs those velocity perturbation from partial pressure measurements and the knowledge of the associated inherent model. No phase difference remains in the observations for all velocity estimation. A perceivable damping for the

11 u_hat u 2E-5 u -2E-5 Error of u (%) steps steps 4E-5 v_hat v 4 2E-5 v -2E-5 Error of v(%) E steps steps 4E-5 w_hat w 5 2E-5 w -2E-5 Error of w (%) -4E-5-5 Figure 6: Observation history (left column) and observation errors (right column) at the point A (shown in Fig. 4). approximation of w can be found in the observation. Compared to the analytical solution, the estimation error is consistently within 5%. Figure 7 quantitatively shows the convergence history of the collective observer error over the whole cross section. The approximation error of sound density is defined as a A, b B ρ a,b ˆρ a,b, and so forth for other acoustical parameters. The observer errors for ρ, u, v, w shown in Fig. 7 are nondimensionalized. The sample step in Fig. 7 for the case is.s. It can be seen that the errors oscillate but converge in almost 6 steps. The oscillations of errors should be caused by the imaginary parts in the dynamics matrix A. In the meantime, the errors of ˆρ and ŵ converge more rapidly. In addition, it can be seen that stationary errors remain in the observations. The error is less than % for ˆρ, 2% for v, 5% for u, and 9% for w. The steady state error could appear due to two reasons. Firstly, the matrix A in the acoustic observer includes imaginary parts and therefore (A LC) may be not normal anymore. The error convergence does not solely depend on the related eigenvalues. Secondly, the ODE model adopted in the observer construction is a

12 Error(%) 2 5 rho u v w steps Figure 7: The convergence history of the observer error for the case. discretized version of the LEE and the discretization introduces molding errors. In addition to the central scheme used in this work, we have also tried a first-order upwind scheme and the steady state error approaches 22%. The numerical tests suggest that high-order schemes may be more preferable for acoustic modeling, in terms of accuracy [26]. On the other hand, the hard wall boundary condition (ˆv = ) is implicitly enforced in Eq. (7). Different boundary condition should be evaluated to improve the observation outcomes. Further investigations are ongoing to overcome the issue. Although steady errors remain, the proposed observer method satisfactorily reconstructs most acoustic parameters using measurements from a circular microphone array. In addition, the proposed observer is recursive and hence can be conducted in real-time. On the other hand, the above simulations are conducted for a tonal sound of a single spinning mode. Practical systems generally include various spinning modes at broadband frequencies, which can be decomposed to a tonal single spinning mode by Fourier transform and mode detection. After that the proposed method can be applied to each component to reconstruct the related acoustic field. 5 Conclusions An imaging method for acoustic spinning modes propagating within a duct has been introduced in this paper. The new method was developed in a theoretical way. The fundamental idea behind the method was originally presented in control theory for ODE. Spinning mode propagation in circular duct, however, is formulated by PDE. For each single spinning mode at a tonal frequency, a finite difference technique is used to reduce the associated PDE to a classical ODE form of state space in control. A so-called observer can thereafter be constructed and applied straightforwardly. The complete acoustic solutions, including fluctuations of the associated particle velocities, can be inferred by the observer from the pressure measurements of in-duct circular microphone array. A numerical simulation of a single spinning mode at a tonal frequency for a straight circular 2

13 duct has been conducted. The simulation generates pressure measurements of a circular array on the outer wall. The demonstration shows that the whole acoustic solutions on the test section can be estimated with the proposed method. A generally good agreement compared to analytical solutions was achieved. The results suggest the potentially important implications of the proposed new method in acoustic experiments. ACKNOWLEDGMENTS This work is supported by the NSF Grant of China (grants 727 and 534) and SRF for ROCS, SEM. REFERENCES [] Steering Committee for the Decadal Survey of Civil Aeronautics, N. R. C., Decadal Survey of Civil Aeronautics: Foundation for the Future, The National Academies Press, 26. [2] Castres, F. O. and Joseph, P. F., Experimental Investigation of an Inversion Technique for the Determination of Broadband Duct Mode Amplitudes by the Use of Near-field Sensor Arrays, Journal of Acoustical Society of America, Vol. 22, No. 2, 27, pp [3] Sijtsma, P., Using Phased Array Beamforming to Identify Broadband Noise Sources in a Turbofan Engine, International Journal of Aeroacoustics, Vol. 9, No. 3, 2, pp [4] Sijtsma, P., Feasibility of In-Duct Beamforming, 27, AIAA Paper [5] Shinneeb, A.-M., Bugg, J., and Balachandar, R., Variable Threshold Outlier Identification in PIV Data, Measurement Science and Technology, Vol. 5, July 24, pp [6] Huang, X., Chan, S., Zhang, X., and Gabriel, S., Variable Structure Model for Flowinduced Tonal Noise Control with Plasma Actuators, AIAA Journal, Vol. 46, No., 28, pp [7] Angelov, P., Giglio, V., Guardiola, C., Lughofer, E., and Lujan, J. M., An Approach to Model-based Fault Detection in Industrial Measurement Systems with Application to Engine Test Benches, Measurement Science and Technology, Vol. 7, No. 7, 26, pp [8] Pinero, G., Vergara, L., Desantes, J. M., and Broatch, A., Estimation of Velocity Fluctuation in Internal Combustion Engine Exhaust Systems Through Beamforming Techniques, Measurement Science and Technology, Vol., No. 4, 2, pp [9] Rienstra, S. W. and Eversman, W., A Numerical Comparison between the Multiple-scales and Finite-element Solution for Sound Propagation in Lined Flow Ducts, Journal of Fluid Mechanics, Vol. 437, 2, pp [] Huang, X., Real-time Location of Coherent Sound Sources by the Observer-Based Array Algorithm, Measurement Science and Technology, Vol. 22, No. 4, 2, pp. (655) 9. [] Huang, X., Real-time Algorithm for Acoustic Imaging with a Microphone Array, Journal of Acoustical Society of America, Vol. 25, No. 5, 29, pp. EL9 EL95. 3

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