Duct spinning mode s particle velocity imaging with in-duct circular microphone array

Transcription

1 AIAA Aviation 16-2 June 214, Atlanta, GA 2th AIAA/CEAS Aeroacoustics Conference AIAA Duct spinning moe s particle velocity imaging with in-uct circular microphone array Qingkai Wei Peking University, Beijing, 1871, People s Republic of China Bao Chen Aviation Inustry Corporation of China, Harbin, 151, People s Republic of China Downloae by PEKING UNIVERSITY on June 25, DOI: / Xun Huang Peking University, Beijing, 1871, People s Republic of China Nowaays, the measurements within a uct have to be conucte using in-uct microphone array, which is unable to provie information of complete acoustic solutions across the test section. In this work, an imaging metho of acoustic spinning moes propagating within a circular uct simply with surface pressure information is introuce. The propose metho is evelope in a theoretical way an is emonstrate by a numerical simulation case. The funamental iea behin the testing metho was originally evelope in control theory for orinary ifferential equations. Spinning moe propagation, however, is formulate in partial ifferential equations. A finite ifference technique is use to reuce the associate partial ifferential equations to a classical form in control theory. The observer metho can thereafter be applie straightforwarly. The algorithm is recursive an thus coul be operate in real-time. The observer error of the metho is analyze then. A numerical simulation for a straight circular uct is conucte. The acoustic solutions on the test section can be reconstructe with a goo agreement to analytical solutions. The results suggest the potential an applications of the propose metho. I. Introuction A new metho to analytically euce the associate particle velocity base on the information of soun pressure was evelope in this work. The particular attention was focuse on the soun propagation an raiation from a straight uct 1 3]. The propose metho can fin applications in low noise turbofan engine tests 4] in particular an shoul be applicable to other uct acoustic applications in general. The rotating fan an stator assembly is mainly responsible for the generation of tonal spinning moes in engines 5, 6]. A theoretical moel to preict soun raiation from annular, straight pipes with a jet flow has been evelope previously, base on the Wiener-Hopf technique 7]. In aition, formulae escribing spinning moe reflecting into a straight, cylinrical uct have been presente 1]. The problem of soun propagation an raiation from a cylinrical uct has also been investigate with a wie range of numerical methos, using finite elements 2, 8], moe-matching strategy 9], base on linearise Euler equations (LEE) 1] an the variants of LEE 11 13]. On experimental sie, the evelopment of suitable testing methos for uct acoustics is of particular interest. Nowaays, a series of experimental techniques have been evelope mainly for pressure sensor apparatuses, incluing in-uct microphone array 14, 15] an outsie-uct microphone array 4]. Multiple moes coul simultaneously propagate within a uct. As a first step, one nees to isolate each spinning moe using the so-calle moe etection technique 16, 17]. Next, soun pressure contours at the region of noise sources can be visualise using beamforming technique 16, 18]. Most existing techniques only prouce soun pressure outputs. In this work, we propose a theoretical metho that coul analytically yiel the associate particle velocity from reaily available soun pressure information. One Ph.D. stuent, Department of Aeronautics an Astronautics. weiqingkai@pku.eu.cn. Senior Engineer, Aeroynamics Research Institute. cariachen@163.com. Professor. Affiliations: (1) Department of Aeronautics an Astronautics; an (2) State Key Laboratory of Turbulence an Complex Systems. AIAA Senior Member. huangxun@pku.eu.cn. 1 of 12 American Institute of Aeronautics an Astronautics

2 Downloae by PEKING UNIVERSITY on June 25, DOI: / may regar this metho as an imaging technique that can simultaneously visualise soun pressure an the associate acoustic particle velocity. The metho was evelope an valiate in this work particularly for uct acoustic applications. It is essentially expecte that the propose metho coul infer unknown states (e.g. the associate acoustic particle velocity) base on some partially available information (e.g. soun pressure), given the mathematical relationship between unknown states an available information. A similar problem has been resolve for linear systems, which are normally escribe by orinary ifferential equations (ODE), using a so-calle observer metho 19]. On the other han, the observability of PDE has long been of interest to mathematicians 2 22]. One of the most recent evelopments is for stochastic PDE by Galerkin approximations 23]. Reaers with interest can refer to above citations an references therein for a more exhaustive literature list. The analysis aopte in this investigation buils upon the previous work 2] that firstly stuie observability of iscrete wave equations. It was followe by theoretical iscussions an pioneering insights to spherical wave systems 22]. In this paper, the funamental iea behin the propose metho shares the same theoretical backgroun of previous works in real-time acoustic beamforming 18, 24] of coherent noise sources. 25] However, the previous beamforming works were conucte in frequency omain an noise sources were explicitly regare as simple monopoles. The noise sources iscusse in this work are spinning moes, having a relatively more complex propagation moel. The moel was formulate in a moern control form an a so-calle observer was esigne to infer acoustic velocities an pressures from the in-uct circular array measurements. Section II briefly introuces the preliminary knowlege of spinning moes an observer, an straightforwarly sets up the theoretical moel of the propose metho. A numerical case was conucte to valiate the propose metho in Sec. III. Finally, Sec. IV summarizes the present work. A. Observer theory II. Metho Introuction A so-calle observer base algorithm has been previously propose 18, 24] for acoustic array beamforming. The basic iea was base on observer theory in classical control. The acoustic imaging metho evelope in this work shares the similar theoretical backgroun. The essence of the propose metho is to recursively estimate time varying acoustic parameters (ensity, velocities, etc.) of a spinning moe with the knowlege of its ynamic moel an measurements (soun pressure for the case). It is also worthwhile to notice the ifferences from the previous observer base beamforming metho. Firstly, the present problem of interest is transmission acoustic fiel imaging instea of noise source imaging. Seconly, the present problem is solve in time omain an the resultant moel is governe by linear partial ifferential equations instea of orinary ifferential equations. The issues ue to those ifferences are to be aresse in the following paper. For convenience of reaers, we firstly introuce the funamental iea of observer theory in this section. More etails can be foun in any linear control textbook 19]. In classical control, a ynamic process can be generally escribe as a state space moel x(t) t = Ax(t) + Bu(t), (1) y(t) = Cx(t), (2) where t is time, x an u represent internal states an inputs of the moel, y enotes moel outputs, an A, B an C are ynamic, control an output matrices, respectively. In this work, A, B an C are time invariant, an y is the circular sensor array measurements, from which the acoustic variables x are hopefully inferre. The so-calle observability in linear control theory efines a measure of how well those internal states coul be euce 19]. The observability of the state space moel Eqs. (1)-(2)] can be investigate by forming an observability matrix O = C CA. CA M 1, (3) where M is the rank of A. It has been prove 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 state X, only with the knowlege of y. The 2 of 12 American Institute of Aeronautics an Astronautics

3 observer has the form of ˆx(t) t = Aˆx(t) + Bu(t) + L(y ŷ), (4) ŷ(t) = Cˆx(t), (5) where ˆ ( ) enotes the approximation an L is the observer gain. It is straightforwarly that the estimation error e = x ˆx satisfies e(t) = (A LC)e, (6) t which is achieve by subtracting Eq. (1) from Eq. (4). It can be seen that e converges to zero when t, as long as the real parts of all eigenvalues of the matrix (A LC) are negative. It is worthwhile to mention that for the corresponing iscrete system, e converges if all eigenvalues of the matrix (A LC) rop in the unit circle. The problem of fining a proper L can be straightforwarly resolve for most cases in a couple of steps of trial an error. B. Analytical solution of spinning moe Downloae by PEKING UNIVERSITY on June 25, DOI: / Viscous issipation an heat conuction are neglecte for the stuie soun propagation within a subsonic mean flow. The compressible Euler equations in cylinrical coorinates are use to moel fluis aroun an axisymmetric uct, written in the conservative form as follows ρ t + (ρu) ρw t ρu t + (ρu2 ) + x ρv t + (ρuv) + x + (ρuw) x ( x + ) ( r + 1 r ( r + 1 r ( + r + 1 r r + 1 r ) (ρv) + 1 (ρw) r θ (ρuw) θ (ρuv) + 1 r ) (ρv 2 ) + 1 (ρvw) + p r θ r + p x ) (ρvw) + 1 (ρw 2 ) + 1 p r θ r θ =, (7) =, (8) =, (9) =, (1) where x is the axial coorinate, r is the raial coorinate, θ is the azimuthal angle, ρ is the ensity, p is the pressure, u is the axial velocity, v is the raial velocity an w is the azimuthal velocity. The perturbations of acoustic pressure, ensity an velocities, (p, ρ, u, v, w ) are generally small compare with the backgroun mean flow variables (p, ρ, u, v, w ). Soun wave propagation can thus be simply moele by linearize Euler equations (LEE). The acoustic isturbances can be represente by Fourier series in terms of azimuthal moes m. For example, the acoustic pressure has the form of p = m= p m(x, r, t)e imθ. For a uniform mean flow (u,, ) in the engine uct, the three-imensional LEE are 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 =, (11) =, (12) =, (13) w m t + u w m x + p m ρ r θ =, (14) where all variables are nonimensionalize using a reference length (R, the raius of the uct), a reference spee (the spee of soun), an a reference ensity (air ensity). For the iealize geometry with a straight an semi-infinite unflange uct, the solutions of Eqs. (11) (14) have analytical forms for mth azimuthal moe at a single frequency k ρ m(x, r, θ, t) = cj m (k r r)e i(kt kxx mθ), (15) u m(x, r, θ, t) = ck x J m (k r r)e i(kt kxx mθ), k k x M j (16) v m(x, r, θ, t) = c J m (k r r)] i e i(kt kxx mθ), k k x M j r (17) w m(x, r, θ, t) = cm r(k k x M j ) J m(k r r)e i(kt kxx mθ), (18) 3 of 12 American Institute of Aeronautics an Astronautics

4 where M j = u /C, C is the spee of soun, c is the amplitue of the acoustic perturbation (the nonimensional value is normally less than 1 2 ), an J m is the mth-orer Bessel function of the first kin. The real parts of the above formulations are acoustic solutions. The nth raial wavenumber k r of the mth spinning moe is the nth solution of the following equation etermine by the har-wall bounary conitions of the uct, J m (Rk r )] r =, (19) where R is the raius of the outer uct wall. The axial wavenumber k x of the mth moe can be subsequently calculate using k k x = M 1 Mj 2 j ± 1 k2 r(1 Mj 2) k 2. (2) The chosen of (±) in Eq. (2) is etermine by the upstream/ownstream irection of the spinning wave. Downloae by PEKING UNIVERSITY on June 25, DOI: / C. State space moel The problem of a spinning moe raiation of a uct is of particular interest in this work. The following work evelops a moel in the form of Eqs. (1 2) for spinning moes propagation. It is worthwhile to notice that all formulations are presente in a continuous time version, which is presumably regare more natural to reaers. In this work, the numerical implementation actually aopte a iscrete time form that is a strictly iscrete analogue of the propose metho. In aition, to simplify the problem, the moeling error an measurement error are omitte here, which enable us to focus on central theoretical issues. It will be of practical importance to obtain acoustic images of spinning moes propagating within ucts. The image plane consiere in this work is the uct cross section at the x location of sensor arrays. The imaging plane is escretize into an A B gri, where A an B are the gripoint numbers in r an θ coorinates, respectively. Both values shoul satisfy Nyquist-Shannon sampling theorem, i.e., A > 2n an B > 2m, where n is raial moe an m circumferential moe. For simplicity, B is set to the sensor number of the circular array. For example, Fig. 1 shows the gris for a (m = 3, n = 1) spinning moe; the number of gripoints in the raial irection is 8; an the number of gripoints in the circumferential irection is 3. It is implicitly assume that 3 pressure sensors are locate on the outer ring to instantaneously capture soun pressure. θ r Figure 1. The gris in polar coorinates for the uct test section. For brevity, the subscript m in Eqs. (11) (18) is omitte in the rest of this paper. To construct a state space moel for each single spinning moe, the following relations an approximations are aopte for Eqs. (11) (14) at each gripoint (a, b), where 1 a A, an 1 b B: (1) / x = ik x, using the spectral metho; 4 of 12 American Institute of Aeronautics an Astronautics

5 Downloae by PEKING UNIVERSITY on June 25, DOI: / (2) ρ = p in the nonimensional equations; (3) v a,b / r is approximate by a backwar ifference, (v a,b v a 1,b )/δr; (4) w a,b / θ is approximate by a central ifference, (w a,b+1 w a,b 1 )/(2δθ); (5) p a,b / r an p a,b / θ are approximate in similar ways, respectively; (6) v 1,b / r, p 1,b / r, () a,b+1 = () a,1, an () a, 1 = () a,b ; where v a,b = v (a, b), v a,b / r = p (a, b)/ r, an so forth. All spatially ifferential terms in Eqs. (11) (14) are replace an the partial ifferential equations are reformulate to the orinary ifferential equations. For example, the governing equations of one gripoint at (a, b) are ρ a,b ik x u ik x ρ ρ r ρ u δr ρ ρ a,b ik a,b x t v a,b = ρ ik x u u δr ρ a 1,b a,b u a 1,b ρ δr ik x u v a,b + ρ δr v a 1,b w a,b {{ x a,b /t + {{ ik x u A a,b ρ 2rδθ 1 2ρ rδθ {{ A ϕa,b ρ a,b+1 u a,b+1 v a,b+1 w a,b+1 {{ x a,b+1 C a,b w a,b {{ x a,b + p a,b = 1 ] {{{{ y a,b {{ A ra,b ρ 2rδθ 1 2ρ rδθ {{ A ϕa,b ρ a,b u a,b v a,b w a,b {{ x a,b w a 1,b {{ x a 1,b ρ a,b 1 u a,b 1 v a,b 1 w a,b 1 {{ x a,b 1, (21), (22) where the nonimensional ρ equals p. The complete equations of the state space moel on the entire mesh (see Fig. 1) are x a 1,b x t a,b = A ra,b A ϕa,b A a,b A ϕa,b x a,b 1, (23) x a,b {{{{ x x/t A a,b+1 {{ x y a,b {{ y = C a,b {{ C x a 1,b x a,b 1 x a,b x a,b+1 {{ x. (24) In most practical tests, Eq. (24) only contains the contributions from those pressure sensors that are surface mounte on the har wall of the uct, where v an p / r equal. It is easy to examine that the raial, axial an circumferential 5 of 12 American Institute of Aeronautics an Astronautics

6 velocities within the uct cross section are normally not observable only with measurements from those on-surface pressure sensors. The analytical solution of Eq. (15) is use to exten the system outputs, i.e. the soun pressure at r ( r R) can be inferre from the measurements at outer raius R, accoring to the relation, ρ (r) = ρ (R) J m(k r r) J m (k r R), (25) where the nonimensional value R is 1. Accoring to control theory, the observability of the system Eqs. (23)-(24)] can be etermine by constructing a matrix using Eqs. (3) an subsequently by examining its rank. If the system is observable, an observer of Eqs. (4) (5) can be constructe with a carefully chosen observer gain. In summary, Eqs. (11) (14) escribe an ieal mathematical moel that is obtaine by applying natural laws to the acoustical physics. In contrast, Eqs. (23) (24) represent a reuce mathematical moel that replaces partial ifferentials by finite ifference terms. The particular finite ifference schemes chosen in the reformulation coul affect the final accuracy an efficiency. It is also worthwhile to note that the present output y only contains pressure information. It is also possible to regar velocity as known information, using high spee particle image velocimetry, an to infer the unknown soun pressure on the uct cross section. The theoretical metho is still the same. Downloae by PEKING UNIVERSITY on June 25, DOI: / D. Observation errors Firstly, we reformulate the previous wave equation an the relate control-oriente moel in a more consistent an succinct form. The acoustic system of interest is autonomous an written as t v c(t, x c ) = A c v c (t, x c ), y c (t, x c ) = C c v c (t, x c ), (26) where v = (ρ, u, v, w ) an y = ρ ; the subscript c enotes that the associate variables are continuous in time; A c represents spatial ifferential operators; an C c = 1 ] T, where T is the transpose. One is force to conuct a spatially iscrete approximation to approximate spatially partial ifferentials in A c. The resultant control-oriente moel on one specifie r θ cross section is t v (t, x ) = A v (t, x ), y (t, x ) = C v (t, x ), (27) where the subscript enotes that the associate variables are in iscrete-time omain. We efine a boune linear operator, Π : C C N that satisfies Πx c = x 2], where N = n r n θ is the number of iscretise gripoints on the spatial omain. Physically, Π projects infinite imensional space to finite imensional space. An observer has the following form, where (t, x ) is omitte for brevity. The estimation error is efine as t ˆv = A ˆv + K(Πy c ŷ ), (28) ŷ = C ˆv, (29) e Πv c ˆv, (3) which represents the evolving ifference between the original infinite-imensional ynamics an the corresponing iscretise finite-imensional ynamics. Proposition 1: The spatial iscretization methos for the iscrete linear wave system solely etermine the stability of the observer. Nevertheless, the convergence rate of the observation error is subject to the original continuous-time system. Proof: By simple algebraic manipulations, one obtains t e = Π(A c v c ) A ˆv K(Πy ŷ ) = (A KC )e + (ΠA c A Π)v c (31) Ae + Bv c, 6 of 12 American Institute of Aeronautics an Astronautics

7 where the relationship ΠC c v c = C Πv c is assume. An B, as δr, δθ, if the aopte finite ifference metho is consistent. Given an initial estimation error e (), the explicit formulation of the estimation error over time is e (t) = e At e () + t Conucting Laplace transform (L) on Eq. (31), one can achieve e A(t τ) Bv c (τ)τ. (32) Downloae by PEKING UNIVERSITY on June 25, DOI: / E (s) = BV c(s) si A, (33) where s is a complex number an E (s) = L {e (t) = e (t)e st t; V c (s) = L {v c (t); an I is an ientity matrix with an appropriate rank. Remark: Firstly, the stability of the observer only epens on A, which epens on the aopte finite ifference schemes an the observer gain. Once A an C are known, one can esign a suitable K to ensure the stability of the observer, i.e., to make sure that the eigenvalues of A is on the left complex plane. Seconly, the approximation error ue to B is proportional to the state (v c ) from the original, continuous-time system. Physically, it means that if soun properties (e.g., pressure an particle velocity) are larger at one position, the observation error will be larger at the same place. To suppress this error, one nees a smaller B = ΠA c A Π, which suggests that a high-orer finite ifference scheme for A is preferable to approximate A c. Thirly, Eq. (33) helps to evelop asymptotic estimations. At high frequency range ( s A ), e (t) t Bv c(τ)τ; while at low frequency range ( s A ), e (t) Bv c (t)/a. One can easily check that the alternative efinition, e v ˆv, hols little physical meaning an is therefore not aopte. E. Observability with real-value measurements One might argue that it is misleaing to irectly examine the observability of Eq. (27), because the time-omain outputs y are complex-value, which are actually impossible in practical tests. All time-omain (pressure, velocity, etc.) reaings in (acoustic) measurements are real-value. Duct acoustics are actually governe by t v v ] = A A {{ A e ] v v {{ v e ], Re(y ) = C C {{ C e ] v v ], (34) where ( ) means element-by-element conjugation; Re(y ) represents the real-value outputs in practical measurements; an the constant coefficient (.5) in C e is omitte. Next, we are going to prove that the extene ynamic system (A e, C e ) is observable, if (A, C ) is observable. Lemma 1: If a linear, time invariant, unforce system (A, C ) is observable, the system (A, C ) is also observable. The proof is straightforwar an omitte for brevity. Proposition 2: If a linear, time invariant system (A, C ) is observable, the extene system (A e, C e ) with real-value outputs is also observable. Proof: If the system (A, C ) is observable, accoring to Lemma 1, (A, C ) is also observable. Then, there shoul exist K T > such that K T v 2 K T v 2 T T Cv (t) 2 t, v V, (35) Cv(t) 2 t, v V, (36) where the relation C = C for the wave system is aopte, because C only contains real-value vectors of (1,,, ). As v, v an v e are on a Hilbert space, it is easy to know that v e 2 v 2 + v 2, an Cv (t) 2 + Cv (t) 2 = Cv (t) + Cv (t) 2 = C e v e (t) 2, because v an v are orthogonal. After applying these two relations an summing Eqs. (35)-(36), one can have K T v e 2 T C e v e (t) 2 t, v e V, (37) 7 of 12 American Institute of Aeronautics an Astronautics

8 Downloae by PEKING UNIVERSITY on June 25, DOI: / which conclues the proof. Remark 1: In contrast, the proof base on algebraic observability theorem is not straightforwar. The erivation is simply escribe below. If (A, C ) is observable, the rank of O = C,, C A M 1 ] T is equal to the system orer M. Accoring to Lemma 1, the rank of O = C,, C A M 1 ] T also equals M. One can have the observability test matrix for the extene system, C. M 1 C O e = A M C A. 2M 1 C A C. C A M 1 C A M. C A 2M 1, (38) where C = C. If the system (A e, C e ) is observable, the rank of O e shoul equal the orer of the extene system, that is, 2M. Note that O e is a matrix having 2MN rows an 2M columns, where N is the number of outputs. We firstly prove that the row rank of O e equals 2M. We notice that the rank of first MN rows of O e is M. If A is nonsingular, the rank of the secon MN rows of O e is also M. To prove the algebraic observability conition, we shoul fin at least M rows from the secon MN rows of O e are linearly inepenent of the first MN rows of O e. Suppose the contrary: that at least one submatrix e.g. the (M + j)th, j < M] is linearly epenent of the first MN rows of O e. Then, we must have C M+j A A M+j ] = a 1 C M 1 A A M 1 ] a 2 C M 2 A A M 2 where a 1,, a M are not all zero. By the Cayley-Hamilton theorem, we have ] ] I a M C, (39) I C A M+j = b 1 C A M 1 b 2 C A M 2 b M C I, (4) where b 1,, b M are not all zero. We can conuct element-by-element conjugation to achieve C A M+j = b 1C A M 1 b 2C A M 2 b MC I, (41) where C = C. Comparing Eq. (39) to Eqs. (4)-(41), we will have a 1 = b 1 = b 1,, a M = b M = b M. In other wors, a 1 M an b 1 M are real-value. However, if j =, then b 1 = ik x u, which is not real for almost all mean flow cases. This is a contraiction. By repeate using of the Cayley-Hamilton theorem, any b 1 can be expresse as the coefficients of the characteristic polynomial of A, et(λi A ). For generic mean flow cases of interest, b 1 for various j shoul not be real. Then, we are force to conclue that M submatrices of the secon MN rows of O e are linearly inepenent of the first MN rows of O e. As a result, the row rank of O e is 2M. In a similar way, we can prove that the column rank of O e is also 2M. This completes the proof. Remark 2: One might prefer to numerically evaluate observability of any system of interest using the algebraic observability theorem, once ynamic an output matrices (A e, C e ) are known. III. Numerical Tests Results The attention of this work is focuse on the theoretical evelopment of a new testing metho. A primitive numerical simulation was one to valiate the propose metho. A noise source of a single spinning moe m = 3, n = 1 is assume propagating within a straight uct. The amplitue of soun source is.1. The soun is assume tonal an the frequency is 4Hz. The analytical solutions of soun (ρ, u, v, w ) at each nonimensional time are calculate using Eqs. (15) (18), where M j is set to.4. The analytical solutions ρ, u, v, w when the errors are stable (1 steps later when t = 1) are shown in the first row of Fig. 2 here, which can be use to valiate the propose metho. In the simulation, it is assume that a circular array of 3 pressure sensors is surface mounte at a ownstream position, x =.5. As a result, only the pressure on the outer uct wall can be measure. All acoustic velocities on the whole cross section (where r 1) an soun pressure within the uct (where r < 1) are unknown. An observer in the form of Eqs. (4) (5) is evelope to reconstruct whole acoustic parameters from those pressure measurements 8 of 12 American Institute of Aeronautics an Astronautics

9 Downloae by PEKING UNIVERSITY on June 25, DOI: / Figure 2. Analytical solutions (first row), observe results (secon row) an error istribution (thir row) of the moal soun parameters ρ, u, v, w at the cross section x =.5 from the spinning moe soun source, where t = 1, (m, n) = (3, 1), k = 4. at r = 1. It is worthwhile to mention that Eq. (25) is aopte to generate an initial guess of p at r < 1 from the measurements of p at r = 1. The observer base metho is conucte as follows: Step 1: Prepare the ynamics matrix A an the sensor matrix C accoring to Eqs. (23) (24). It shoul be notice that the iscrete form with a sampling step δt has to be consiere for practical cases. The iscrete counterpart of A in continuous version is e Aδt. Step 2: Solve the observer gain L using the formula, L = (A P)C 1, which can be straightforwarly erive from Eq. (6). P represents observer poles that are eliberately chosen to stabilize the observer. In aition, suitable values of P can spee up the convergence of estimation errors. In this simulation, L is empirically set to.5i, where I is the ientity matrix with a rank equals the rank of A. Step 3: Measurements of the soun pressure are generate by the circular sensor array installe on the outer wall. The measurements are simulate using Eq. (15) in this work. In aition, the initial guess of the soun pressure at r < 1 are approximate by Eq. (25). The resulte soun pressure solutions across the r θ section constitute y in Eq. (4). Step 4: The observer is conucte to extract the approximation of soun parameters, such as ρ, u etc, using Eqs. (4) (5). It shoul be notice that the acoustic system of interest is autonomous an thus the control matrix B is null. Step 5: Step 4 is instantaneously conucte for each samples. The observation error (y ŷ) is examine. If the error converges in a unacceptable slow spee, or even worse, the error is ivergent, a ifferent observer gain L shoul be consiere. We shoul emphasize that the key point in the above algorithm is the pole assignment of A LC. A suitable pole shoul rop in the unit circle. Otherwise, the approximation of the resultant observer will blow up. When the amplitue of a complex value pole is close to 1, the observer will converge quickly but be more sensitive to moeling an measurement errors. On the other han, the convergence spee will slow but be more robust to external noise. Hence, the value of.5 was empirically chosen here. In aition, for most cases with large orer systems, we have to conuct generalize inversion to achieve C 1. The resultant L after the calculation will iffer from the esire values. 9 of 12 American Institute of Aeronautics an Astronautics

10 A robust an efficient pole assignment for a very large system is still an open problem. The observer results ˆx incluing ˆρ, û, ˆv, ŵ can be plotte on the mesh. Secon row of Fig. 2 show the solutions at t = 1. Compare to the corresponing analytical solutions in the first row, û an ŵ are alreay well reconstructe from array measurements (p ). The thir row of Fig. 2 show the istribution of error between analytical solution an observe results. Downloae by PEKING UNIVERSITY on June 25, DOI: / Error (%) 15 Err ρ Err u Err v Err w steps Figure 3. The convergence history of the observer error for the case. Figure 3 quantitatively shows the convergence history of the observer error for the case. The approximation error of soun ensity is efine as 1 a A,1 b B ρ a,b ˆρ a,b, an so forth for other acoustical parameters. The observer errors shown in Fig. 3 are nonimensionalize by the amplitue, which is.1 for the case. Each sample step is.1s. It can be seen that the errors oscillate but converge in almost 1 steps. The errors of ˆρ an ŵ converge more rapily. The oscillations of errors coul be cause by the imaginary parts in the ynamics matrix A. In aition, it can be seen that stationary errors less than 5% remain. In this work, the observer gain L was carefully chosen to set the eigenvalues (P) of (A LC) to.5. In a iscrete time real value system, those eigenvalues ensure the convergence of observer errors to zero as t. As a result, it seems that those resiues of errors violate the observer theory in classical control. The iscrepancy can be explaine by examining the ynamic matrix A of the acoustic system. The elements of A inclue imaginary parts an (A LC) may not be normal. Hence, it coul be improper to infer the entire ynamics of (A LC) from its eigenvalues. Subtle mathematical manipulations coul be neee, which calls further investigations. The acoustic system is originally escribe by partial ifferential equations Eqs. (7) (1)] an reformulate in orinary ifferential equations Eqs. (23) (24)] by secon-orer finite ifference methos. A higher-orer scheme may be more preferable for acoustic moeling, in terms of accuracy. 26] On the other han, the har wall bounary conition (ˆv = ) is implicitly enforce in Eq. (23). Different bounary conition shoul be evaluate to improve the observation. The above simulations are conucte for a tonal soun of a single spinning moe. Practical systems generally inclue various spinning moes at broaban frequencies, which can be ecompose to a variety of tonal single spinning moes by Fourier transform an moe etection. After that the propose metho can be applie to each component to reconstruct the relate acoustic fiel. Although some issues are still to be aresse, the propose metho satisfactorily reconstructs most acoustic parameters only using measurements from a circular microphone array. In aition, the propose observer coul be performe in real-time in terms of its recursive nature. IV. Summary An imaging metho for acoustic spinning moes propagating within a uct has been introuce in this paper. The new metho is evelope in a theoretical way. The funamental iea behin the metho was originally evelope in control theory for orinary ifferential equations. Spinning moe propagation in circular uct, however, is formulate 1 of 12 American Institute of Aeronautics an Astronautics

11 by partial ifferential equations. For each single spinning moe at a tonal frequency, a finite ifference technique is use to reuce the associate partial ifferential equation to a classical form of state space in control. A so-calle observer can thereafter be constructe an applie straightforwarly. The complete acoustic solutions, incluing fluctuations of velocities an pressure, can be inferre by the observer from the pressure measurements of in-uct circular microphone array. A numerical simulation of a single spinning moe at a tonal frequency for a straight circular uct has been conucte. The simulation generates pressure measurements of a circular array on the outer wall. The emonstration shows that the whole acoustic solutions on the test section can be estimate with the propose metho. A generally goo agreement is achieve, compare with analytical solutions. The results suggest the potential an application of the propose new metho for acoustic imaging of spinning moe soun. For example, it is quite ifficult, if not impossible, to measure the instantaneous fluctuations of acoustic velocities within a uct. In aition, the whole in-uct soun pressure istribution has to be measure using a rotating strut, where microphones are installe in the raial irection. Hence, the propose metho can extensively save those experimental efforts. V. ACKNOWLEDGMENTS Downloae by PEKING UNIVERSITY on June 25, DOI: / This work is supporte by National Science Founation of China (Grants an ) an Aviation Inustry Corporation of China, Commercial Aircraft Engine Co., Lt. 11 of 12 American Institute of Aeronautics an Astronautics

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