Acoustic Duct Modes Research Articles

Uniform Asymptotic Approximations for Duct Acoustic Modes in a Thin Boundary-Layer Flow

Analytical approximations for the acoustic modes in a duct carrying a uniform core flow with a thin shear layer at the walls are developed using the method of matched asymptotic expansions. Both two-dimensional and cylindrical duct propagation are considered. Numerical results for eigenvalues calculated using the theory are presented for the two-dimensional problem and compared with results from earlier analyses. It is found that the new approximations yield a significant increase in accuracy. HE problem of sound transmission through a duct carrying a uniform core flow with a thin boundary layer at the walls has been of interest for some time. Such a flow has practical applications in modeling a fully developed real flow profile. Perhaps the strongest motivation for studying the problem, however, is the possibility of obtaining analytical results using asymptotic analyses which can yield physical understanding of shear flow propagation phenomena in more realistic flows which must be treated numerically. Analytical studies based on the assumption of small boundary-layer thickness have been carried out by several researchers during the past decade. Eversman and Beckemeyer1 developed an inner expansion for acoustic pressure in the layer from which they were able to derive an approximate equivalent boundary condition applicable at the outer edge of the layer. Using this condition they showed that the formulation of the shear layer problem reduces to that of the associated uniform flow problem with the well-known particle displacement continuity boundary condition as the layer thickness vanishes. In a later paper, Eversman2 employed the equivalent boundary condition to formulate the eigenvalue problem for acoustic modes in an annular duct flow. Using the known uniform flow solutions in the core flow, an approximate eigenvalue equation is developed. Numerical solution of this single equation yields the acoustic wave numbers and circumvents the task of numerically integrating the differential equation and iterating for the shear flow eigenfunctions. The procedure is equally applicable to rectangular duct geometries, for which it leads to an eigenvalue equation equivalent to one later derived less formally by Tester. 3 A slightly different analysis by Swinbanks,4 also for rectangular ducts, yields similar approximate results. Other studies less directly related to the present work are discussed in Sec. IV.B ofRef. 5. The work reported in the present paper arose in the course of some research on shear flow propagation in which it was desirable to obtain analytical representations of the acoustic

Investigation of the Coupling of Unsteady Lift to Lower Order Acoustic Duct Modes in an Axial Flow Fan

Abstract : By using flow distortion screens, the plane wave (0,0) and first higher order (-1,0) and (1,0) spinning modes were generated in a low-speed axial flow fan. The perpendicular unsteady lift force on a segment of a nine-bladed rotor and the acoustic pressure along the duct were measured simultaneously by a strain gage sensor and a flush-mounted microphone, respectively. The total pressure field downstream of the screens was measured by circumferentially traversing a Kiel probe. The typical signal-to-noise ratio was sufficiently high for the desired signal isolation with a spectrum analyzer. A phase-locked ensemble averaging technique was used to obtain amplitude and phase measurements of the periodic unsteady signals. Standing waves set up in the duct between the rotor and duct inlet seemed to cause a back-reaction effect which resulted in an increase and reduction of the unsteady lift force above and below expected levels. If these effects are taken into account, theoretical predictions coupling unsteady lift to acoustic far field pressure levels conform to measured data. Therefore, it is concluded that there is a definite coupling between unsteady lift and acoustic pressure in the duct, yet determination of an unambiguous coupling factor would require more extensive data acquisition.

Influence of Exit Impedance on Finite Difference Solutions of Transient Acoustic Mode Propagation in Ducts

The time-dependent governing acoustic-difference equations and boundary conditions are developed and solved for sound propagation in an axisymmetric (cylindrical) hard-wall duct without flow and with spinning acoustic modes. The analysis begins with a harmonic sound source radiating into a quiescent duct. This explicit iteration method then calculates stepwise in real time to obtain the steady solutions of the acoustic field. The transient method did not converge to the steady-state solution for cutoff acoustic duct modes. This has implications as to its use in a variable-area duct, where modes may become cutoff in the smal-area portion of the duct. For single cutoff mode propagation the steady-state impedance boundary condition produced acoustic reflections during the initial transient that caused finite instabilities in the numerical calculations. The stability problem is resolved by reformulating the exit boundary condition. Example calculations show good agreement with exact analytical and numerical results for forcing frequencies above, below, and nearly at the cutoff frequency.

A modal separation measurement technique for broadband noise propagating inside circular ducts

A measurement technique which separates broadband noise propagating inside circular ducts into the acoustic duct modes is developed. The technique is also applicable to discrete frequency noise. The acoustic modes are produced by weighted combinations of the instantaneous outputs of microphones spaced around the duct circumference. The technique is compared with the cross spectral density approach presently available and found to have certain advantages, and disadvantages. Considerable simplification of both the new technique and the cross spectral density approach occurs when no correlation exists between different circumferential mode orders. The properties leading to uncorrelated modes and experimental tests which verify this condition are discussed. The modal measurement technique-is applied to the case of broadband noise generated by flow through a coaxial obstruction (nozzle or orifice) in a pipe. Different circumferential mode orders are shown to be uncorrelated for this type of noise source.

Mode selective transfer of energy from sound propagating inside circular pipes to pipe wall vibration

Experimental results are presented which show a mode selective transfer of energy from sound propagating inside a circular pipe to pipe wall vibration. The experiments utilize broadband noise generated by flow through a restriction in the plastic (PVC) pipe. For each higher acoustic duct mode, the energy transfer occurs in a narrow frequency band located slightly above the higher-mode cut-on frequency. A match in axial phase velocity between the higher acoustic duct mode and a compatible pipe wall vibrational mode is proposed as the mechanism for the energy transfer. Theoretical predictions for the frequency at which the axial phase velocity match occurs show good agreement with the experimental results.

Radiation of cylindrical duct acoustic modes with flow mismatch

An analysis is presented for the radiation of spinning acoustic modes from a round duct. The model, based on a plug flow exhaust jet, allows for a flow temperature and velocity mismatch. Solutions to the convected wave equations within and outside the jet are devised, the continuity of acoustic pressure and particle displacement match at the jet interface being used. Formal solution is obtained by applying Fourier transforms, inversion being accomplished by using the Wiener-Hopf technique coupled with the Carrier-Koiter approximations. The analysis also is extended formally to include the inlet problem, but with a plug type inlet flow. The calculations suggest that the influence of the Mach number discontinuity is to offer a conflict between the effects of convection and refraction, the former becoming increasingly more significant at high subsonic Mach numbers by the presence of a strong lobe in the shadow zone. Effects of flight conditions are to weaken such conflicts so that in the limit of uniform Mach number, one recovers Carrier's well-known solution. The effect of a temperature mismatch is rather more dramatic, resulting in a sharp beaming of sound off to the side. This is consistent with the kinematic picture of a plane wave impinging on a temperature discontinuity.

A note on the radiative directivity patterns of duct acoustic modes

The classical solution for the external radiative pattern of acoustic modes in an unflanged circular duct has been presented in a form more suitable for practical computations. This expression predicts the same locations for the principal lobe and zeroes as would be obtained from the baffled model, and has the added advantage of being able to predict sideline and aft radiation. Simple rules for locating the principal lobe and zeroes, involving only hand calculations, are presented and shown to correlate well with sample computer calculations of the full expression for the large azimuthal mode numbers typical of modern turbofans.

Sound waves in two-dimensional ducts with sinusoidal walls

The method of multiple scales is used to analyze the wave propagation in two-dimensional hard-walled ducts with sinusoidal walls. For traveling waves, resonance occurs whenever the wall wavenumber is equal to the difference of the wavenumbers of any two duct acoustic modes. The results show that neither of these resonating modes could occur without strongly generating the other.

Space-Time Structure of Acoustic Propagation in Ducts in the Presence of Shear Flow

The effects of shear flow on the modal phase velocity, the group velocity, and the cutoff frequencies of acoustic modes propagating in ducts are studied analytically. The convection effect of the shear flow is found to be equivalent to the convection effect of a uniform mean flow with some effective Mach number. Using the effective Mach number, the uniform mean flow dispersion relation can be generally used to express the relation between the space-time parameters for any shear flow Mach number distribution. Acoustic duct modes which may be expected to propagate under the assumption of uniform mean flow will decay because of shear flow effects. The calculated propagation constant of different modes propagating upstream or downstream in annular, circular, or two-dimensional ducts are compared with all available data calculated by time-consuming sophisticated techniques and published by previous investigators. The comparison shows an excellent agreement. By using the effective Mach number to obtain the propagation constant, the solution of the sound field in the duct—and thus the refraction effects of the shear flow—becomes a simple matter of integrating an ordinary differential equation. [Research supported by National Research Council of Canada.]

A note on the radiation efficiency of acoustic duct modes

Graphs are presented for the radiation efficiency of higher-order acoustic modes in a baffled annular opening. The variation of radiation efficiency with frequency is generalized in terms of mode cut-off.