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Research Papers

Control-Oriented Methods for Turbomachinery Noise Simulation

[+] Author and Article Information
Alex Siu Hong Lau

Department of Mechanical and
Aerospace Engineering,
Hong Kong University of Science
and Technology,
Kowloon, Hong Kong, China
e-mail: alexshlau@ust.hk

Siyang Zhong

Department of Mechanical and
Aerospace Engineering,
Hong Kong University of Science
and Technology,
Kowloon, Hong Kong, China
e-mail: szhongad@connect.ust.hk

Xun Huang

Department of Mechanical and
Aerospace Engineering,
The Hong Kong University
of Science and Technology,
Kowloon, Hong Kong, China;
Department of Aeronautics and Astronautics,
College of Engineering,
Peking University,
Beijing 100871, China
e-mails: huangxun@ust.hk;
huangxun@pku.edu.cn

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received May 24, 2017; final manuscript received September 6, 2017; published online October 17, 2017. Assoc. Editor: Rakesh Srivastava.

J. Turbomach 140(1), 011001 (Oct 17, 2017) (15 pages) Paper No: TURBO-17-1067; doi: 10.1115/1.4038022 History: Received May 24, 2017; Revised September 06, 2017

This paper presents an innovative stability analysis and design approach for time-domain impedance boundary conditions to simulate noise propagation and radiation from a lined turbomachinery duct in the presence of a mean flow. A control-oriented model is developed for the stability analysis of the impedance boundary condition by using generalized function at the lining surface. The mean flow effect and sound propagation are considered in the model as well. Then, the numerical stability issue is analyzed by using the Bode plots before stabilized accordingly by employing the phase lead compensator method, which results in a rational transfer function. Finally, the corresponding time-domain implementation is achieved by using the so-called controllable canonical form rather than an inconvenient convolution operation. The performance of the current proposed approach is first validated in an in-duct propagation case by comparing to analytical solutions obtained by employing the Wiener–Hopf method and then demonstrated in a couple of duct acoustic problems with representative turbomachinery setups. The innovative cross-disciplinary nature of the current proposed approach can shed light on impedance problems and is very useful to time-domain acoustic simulations for turbomachinery applications.

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References

Figures

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Fig. 1

The block diagram of a feedback system that describes the LEE model P(s) (from Eq. (7)) at r = 1 (the feedforward loop) with a lined wall satisfying the Ingard boundary condition (the feedback loop). For clarity δ(r) is not explicitly shown in this figure.

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Fig. 2

The Bode plots of P(s) at various working conditions. Details of each setup are given in the title of the associated panel. The distinctive setups that might eventually lead to computational instability are highlighted by arrows. It can be seen that certain working conditions, such as m (a), ω (d), and the real part of the impedance (e) impose little influence, while n (b), mean flow speed u0 (c) and the imaginary part of the impedance (f) affect the computational stability. The following part of this work will try to improve the computational stability by reducing the sensitivity to those working conditions.

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Fig. 3

The Bode plots of the transfer functions: Ig(s)=(s−iu0κmn+)/s; 1/Z(s)=ω0/(ℜω0+Xs) for X>0; and L(s)=P(s)Ig(s)/Z(s), where m=4,n=1,u0=0.1,ω0=10,Z=ℜ+Xi=1+i

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Fig. 4

The entire system with the inclusion of a control block, C(s)

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Fig. 5

The Bode plots of the control blocks, Ca(s) and Cb(s), to resolve unstable issues at low and high frequency ranges, respectively

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Fig. 8

The phase plot of Ca(s), where ωmax = 100 and ω0=10, (×)Ka=1, (◻)Ka=2, (°)Ka=3, (+)Ka=4, (⋄)Ka=5

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Fig. 9

Setups of computational aeroacoustic simulations for duct (not to scale): (a) the validation case and (b) the generic bypass duct case (mean flow velocity u0 is shown here). A spinning modal sound wave propagates from left to right. Buffer zones are used around the computational domain. Axial-symmetrical boundary condition is used along r = 0. The circumferential angle θ is not shown in this two-dimensional presentation.

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Fig. 10

Numerical results from the LEE model are compared to analytical solutions using the Wiener–Hopf method. The case setup is as follows: m = 9, n = 1, ω = 20, u0 = 0.3, Z = 1 + i, where (a) shows instantaneous sound pressure with ten contour levels between ±10−4, where lined walls are represented by white dashed lines and (b) shows acoustic power across the x axis, where (–) is computational solution, and (⋯) is analytical solution.

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Fig. 11

The difference between the numerical and the analytical results, where Imag(Z)=X is varied, while Real(Z)=ℜ=1, (◻) m=4,n=1,ω=20,u0=0.2, (Δ) m=9,n=1,ω=20,u0=0, (⋄) m=9,n=1,ω=20,u0=0.1, (°) m=9,n=1,ω=20,u0=0.2, and (+) m = 13, n = 1, ω = 20, u0 = 0.2

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Fig. 12

Instantaneous sound pressure field, where u0 = 0, ω = 15, ((a) and (b)) (m, n) = (4, 1), ((c) and (d)) (m, n) = (4, 2), ((a) and (c)) hard-wall, ((b) and (d)) Z = 0.5 − 1.5i

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Fig. 13

As Fig. 12 but for an annular geometry with an infinite central hub

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Fig. 14

Far-field directivity with respect to the elevation angle ϕ (with respect to the downstream direction) for the hollow duct radiation case with two different setups. “APE” and “WH” denote far-field predictions obtained by using the APE model with the Ffowcs-Williams and Hawkings method (for u0 ≠ 0), and the analytical solution generated by the Wiener–Hopf method, respectively.

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Fig. 15

Instantaneous sound pressure field is shown with ten contour levels between ±10−5, where m = 13, n = 1, ω = 28, (a) the hard-walled case; ((b) and (c)) the lined walls at two different positions are highlighted with thick lines and indicated by arrows

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Fig. 16

Far-field directivities of the cases in Fig. 15

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