Subsonic Mean Flow Research Articles

Variation of Acoustic Energy Across Sudden Area Expansions Sustaining a Subsonic Flow

The acoustic energy balance of a sudden area expansion is known to be altered in the presence of a mean flow. In this work, Ronneberger’s quasi-steady model describing the acoustic response of a sudden area expansion sustaining a subsonic mean flow of arbitrary Mach number is revisited using the acoustic absorption coefficient, shown to be a function of the inlet Mach number , cross-sectional area ratio , and upstream acoustic reflection coefficient . These analytical predictions are tested using a two-step numerical strategy, whereby the mean flow variables are obtained using Reynolds-averaged Navier–Stokes simulations and the fluctuating variables are computed using in-house linearized Navier–Stokes equations solvers. The agreement between the analytical model and the numerical results is found to be excellent for all geometries, mean flows, and acoustic boundary conditions investigated. The generation of acoustic energy by the flow expansion is observed analytically and numerically for high-Mach-number flows undergoing a slight sudden area increase for given acoustic boundary conditions. Conversely, it is found that substantial acoustic energy damping occurs across sudden area expansions characterized by a wide range of parameters . Moreover, entropy and vorticity fluctuations are found to be generated at the sudden area expansion, but entropy fluctuations are shown to have a negligible impact on the acoustic response of the area expansion at low and intermediate Mach numbers. Finally, the analytical model is found to be reasonably accurate up to Helmholtz numbers , corresponding to the frequency range of many industrial applications.

Acoustic energy balances for sound radiated from duct exit with mean flow

An old model problem for the exchange of energy between sound field and mean flow by vortex shedding has been worked out in numerical detail. The analytically exact solution of the problem of reflection, diffraction and radiation of acoustic modes in a semi-infinite annular duct with uniform subsonic mean flow, including shedding of unsteady vorticity from the duct exit, allows a precise formulation of Myers’ energy for perturbations of an inviscid mean flow. The transmitted power [Formula: see text] in the duct and the radiated power [Formula: see text] in the far field differ by the amounts of hydrodynamic far field powers [Formula: see text] inside and [Formula: see text] outside the wake (vortex sheet) emanating from the duct edge, plus the power [Formula: see text] that disappears into the vortex sheet. This last component represents the source term in Myers’ energy equation. This is evidence of the non-conserved character of acoustic energy in mean flow, owing to the coupling of the acoustic field with the mean flow. [Formula: see text], [Formula: see text] and [Formula: see text] are always positive. This is normally the case too for [Formula: see text] and [Formula: see text]. But for not too high frequencies or other circumstances where shed vorticity produces more sound than was necessary for its creation, [Formula: see text] and even [Formula: see text] may also be negative.

Acoustic and hydrodynamic power of wave scattering by an infinite cascade of plates in mean flow

We study the balance of outgoing and incoming power for acoustic wave scattering by a cascade of flat blades in uniform subsonic mean flow. In nonzero mean flow the Kutta condition at the trailing edge of the cascade plates leads to the production of vorticity in the form of unsteady vortex sheets attached at the trailing edges, which results in a hydrodynamic power flux and the exchange of energy between the unsteady vorticity and the acoustic field. Although the scattering problem itself has been subject to much previous research, a comprehensive study of the composition of the outgoing energy flux for cascade scattering has received little previous attention. As an application of our detailed study we show how the time-averaged energy balance can be used to explain certain symmetries of the field in the angle of incidence with respect to the cascade face when there is no mean flow and the effect of zero acoustic reflection at certain angles of incidence with mean flow. We also show that, depending on the angle of incidence of the acoustic wave and the velocity of the mean flow, the vortex sheets can either absorb or generate acoustic energy. This can, for certain parameter values, result in a net increase in outgoing sound power over the incident sound power, as we demonstrate on a number of numerical examples that are computed using the semi-analytical solution described in recent work by Maierhofer and Peake (J. Sound Vib., vol. 481 (2020), 115418).

Computational investigation of plane wave reflections at the open end of subsonic intakes

The reflection of plane waves at the open end of intakes carrying subsonic mean flows are considered in this work. The investigations are conducted with a three-dimensional numerical scheme based on the lattice Boltzmann method. Two different geometry profiles were considered to represent the intake, including a cylindrical bell mouth and a Borda tube. The comparison between numerical and analytical results provided by the low-frequency theory suggested two different traits. For the cylindrical bell mouth, the incoming flow remains potential and the wave reflection is only influenced by the phase velocity shift caused by convection. In such a situation, the available low-frequency theory holds for the entire frequency range of plane wave propagation. Conversely, for sharp geometries, such as the Borda tube, flow separates at the duct's entrance and the wave reflection becomes mostly influenced by the exchange of energy between the flow and the acoustic field. In this case, the numerical results deviate considerably from the theory, even in the low-frequency limit. The mechanisms involving the transfer of rotational kinetic energy from the flow to the acoustic field are discussed and an empirical formula is proposed in terms of reflection coefficient for the entire range of plane wave propagation.

Numerical investigation of normal mode radiation properties of ducts with low Mach number incoming flow

Normal mode radiation properties of ducts issuing a subsonic mean flow have been thoroughly investigated in the past. The behavior of the radiation characteristics, such as the magnitude of the reflection coefficient and the length correction, are well understood. Nevertheless, the behavior of the same features in the presence of an incoming low Mach number flow has not been investigated in detail, particularly in the case of the length correction. This work presents a numerical study of the reflection coefficient and length correction of unflanged pipes and pipes terminated by circular horns of different radii in the presence of an incoming flow. The investigations are conducted with a tridimensional lattice Boltzmann scheme. The results suggest that the detachment and reattachment of the incoming flow inside the duct play a significant role on the transfer of kinetic energy from the flow to the internal acoustic field. Moreover, the results show that the mechanism of flow detachment and reattachment is ...

An integral formulation for wave propagation on weakly non-uniform potential flows

An integral formulation for acoustic radiation in moving flows is presented. It is based on a potential formulation for acoustic radiation on weakly non-uniform subsonic mean flows. This work is motivated by the absence of suitable kernels for wave propagation on non-uniform flow. The integral solution is formulated using a Green׳s function obtained by combining the Taylor and Lorentz transformations. Although most conventional approaches based on either transform solve the Helmholtz problem in a transformed domain, the current Green׳s function and associated integral equation are derived in the physical space. A dimensional error analysis is developed to identify the limitations of the current formulation. Numerical applications are performed to assess the accuracy of the integral solution. It is tested as a means of extrapolating a numerical solution available on the outer boundary of a domain to the far field, and as a means of solving scattering problems by rigid surfaces in non-uniform flows. The results show that the error associated with the physical model deteriorates with increasing frequency and mean flow Mach number. However, the error is generated only in the domain where mean flow non-uniformities are significant and is constant in regions where the flow is uniform.

Multiple line arrays for the characterization of aeroacoustic sources using a time-reversal method

This letter investigates the use of multiple line arrays (LAs) in a Time-Reversal Mirror for localizing and characterizing multipole aeroacoustic sources in a uniform subsonic mean flow using a numerical Time-Reversal (TR) method. Regardless of the original source characteristics, accuracy of predicting the source location can be significantly improved using at least two LAs. Furthermore, it is impossible to determine the source characteristics using a single LA, rather a minimum of two are required to establish either the monopole or dipole source nature, while four LAs (fully surrounding the source) are required for characterizing a lateral quadrupole source.

Lattice Boltzmann simulations of sound directivity of a cylindrical pipe with mean flow

This paper proposes a numerical scheme based on the lattice Boltzmann method to tackle the classical problem of sound radiation directivity of pipes issuing subsonic mean flows. The investigation is focused on normal mode radiation, which allows the use of a two-dimensional lattice with an axisymmetric condition at the pipe’s longitudinal axis. The numerical results are initially verified against an exact analytical solution for the sound radiation directivity of an unflanged pipe in the absence of a mean flow, which shows a very good agreement. Thereafter, the sound directivity results in the presence of a subsonic mean flow are compared with both analytical models and experimental data. The results are in good agreement, particularly for low values of the Helmholtz number ka. Moreover, the phenomenon known as ‘zone of relative silence’ was observed, even for mean flows associated with very low Mach numbers, though discrepancies were also observed in the comparison between the numerical results and the analytical predictions. A thorough discussion on the scheme implementation and numerical results is provided in the paper.

Green's Functions in Computational Aeroacoustics

The theory of Green's functions for the wave and Helmholtz equations is examined with particular attention to their use in aeroacoustics for the extrapolation of acoustic wavefields from numerical flow simulations. In a new synthesis that permits straightforward generalization of previously published results, spatial and temporal windowing functions are employed to provide equivalent-source expressions to account for both initial and boundary conditions. Detailed results describe the transformation of both source terms and Green's functions to take account of uniform subsonic mean flow, and expressions are given for free-field Green's functions, both with and without flow, in time, frequency and wavenumber domains. A worked example illustrates the non-uniqueness of the Green's function for a simple one-dimensional bounded problem.

Multimode radiation from an unflanged, semi-infinite circular duct with uniform flow

Multimode sound radiation from an unflanged, semi-infinite, rigid-walled circular duct with uniform subsonic mean flow everywhere is investigated theoretically. The multimode directivity depends on the amplitude and directivity function of each individual cut-on mode. The amplitude of each mode is expressed as a function of cut-on ratio for a uniform distribution of incoherent monopoles, a uniform distribution of incoherent axial dipoles, and for equal power per mode. The directivity function of each mode is obtained by applying a Lorentz transformation to the zero-flow directivity function, which is given by a Wiener-Hopf solution. This exact numerical result is compared to an analytic solution, valid in the high-frequency limit, for multimode directivity with uniform flow. The high-frequency asymptotic solution is derived assuming total transmission of power at the open end of the duct, and gives the multimode directivity function with flow in the forward arc for a general family of mode amplitude distribution functions. At high frequencies the agreement between the exact and asymptotic solutions is shown to be excellent.

Influence of mean flow profile and geometrical ratios on scattering of sound at a sudden area expansion in a duct

The scattering of sound at a sudden area expansion in a duct with subsonic mean flow has been modelled with a multimodal method. Technological applications are for instance internal combustion engine exhaust silencers and silencers in industrial duct systems. Both 2D rectangular and 2D cylindrical geometries are considered. The influence of the mean flow profile, and the—in this method—associated application of an acoustic Kutta condition at the edge of the area discontinuity, is investigated. The scattering coefficients for the plane waves are found to change smoothly as the flow profile is changed gradually from one, where the acoustic Kutta condition is applied to one where it is not applied. Furthermore, for high Strouhal numbers no difference is observed in the results for the scattering coefficients obtained for different flow profiles. Also, at low Strouhal numbers the magnitudes of the scattering coefficients are the same for different profiles. The influence of the ratio of the heights (in 2D rectangular geometry), respectively, radii (in 2D cylindrical geometry), of the ducts upstream and downstream of the area expansion on the scattering coefficients is examined. Around a certain Strouhal number, a specific feature in the scattering coefficients is observed when the ratio of the duct heights or radii is less than 0.5. This is found to be connected to a strong interaction between the first evanescent acoustic mode and the hydrodynamic instability mode. For non-uniform flow even an apparent jump between the first evanescent acoustic mode and the hydrodynamic unstable mode and a corresponding jump in scattering coefficients is observed, when employing causality analysis according to the Briggs–Bers or Crighton–Leppington procedure. This implies that in fact an absolute instability occurs.

Sound reflection at the open end of axisymmetric ducts issuing a subsonic mean flow: A numerical study

This paper investigates the reflection of sound waves as they propagate toward the open end of ducts issuing a cold subsonic mean flow in a stagnant fluid. The investigations are conducted by using a relatively new one-step numerical technique known as the lattice Boltzmann method. The results obtained in terms of end correction, l, and magnitude of the pressure reflection coefficient, | R | , for an unflanged pipe model are in very good agreement with theoretical predictions and experimental data for the range of Mach numbers M ≤ 0.15 and Helmholtz numbers ka ≤ 1.5 . Simulations carried out for a similar pipe terminated by circular horns of different curvature radii show that | R | is independent of the horn geometry for Strouhal numbers Sr 0 < 4 , Sr 0 = ka / M . For situations in which M > 0 , the amplification of the reflection coefficient magnitude, that is | R | > 1 , is normally ∼ 20 percent higher than that observed for an unflanged pipe issuing an equal Mach number flow. Furthermore, the amplification maxima is found to occur always at Sr 0 = π / 2 , regardless of whether the pipe is unflanged or terminated by different circular horns. Conversely, the results for the end correction are strongly dependent on the horn curvature. For the low Strouhal number limit, the dimensionless end correction measured at the junction between the horn and the pipe with radius a becomes l / a = 0.31 for a horn whose curvature radius is 2 a , and l / a = 0.62 for a horn with curvature radius equal to 4 a .

Numerical methods for noise propagation in moving flows, with application to turbofan engines

This article reviews the development and application of Computational AeroAcoustics (CAA) to acoustic propagation on subsonic mean flows, with a particular focus on methods used to predict acoustic radiation from turbofan aeroengines. The governing equations are presented and particular issues such as the formulation of impedance and far field boundary conditions, the treament of Kelvin-Helmholtz instabilities, resolution requirements and methods for controlling dispersion error, are discussed. The status of current CAA methods is reviewed. Finally, the matter of validation against benchmark problems and measured data is explored.

Active control of multimodal tonal noise propagated in circular duct with axial subsonic mean flow until M=0.3

As new generation of aircraft engine with lower blade passing frequency appeared in the 1990's, the fan tones radiated from the inlets had become one of the dominant source of sound. Efforts have then been made to develop active noise control. Encouraging results have been obtained but the physical limitation of the fan tones reduction have not been clearly determined, owing mainly to the complexity of the experimental rigs. This paper present an experimental investigation of the control of multimodal tonal noise propagated in circular duct in presence of a mean flow (M⩽0.3). A laboratory wind tunnel has been implemented for this purpose. Two limiting factors for the sound reduction are underlined: (i) the degradation of the secondary transfer matrix conditioning as the number of propagating modes increases in the duct and (ii) the degradation of the hypothesis of the time-invariance of the system to control as the flow velocity is increased. The effect of those limiting factors on the control efficiency are evaluated.

Multimodal method for scattering of sound at a sudden area expansion in a duct with subsonic flow

The scattering of sound at a sudden area expansion in a duct with subsonic mean flow has been modelled with a multimodal method. Technological applications are for instance internal combustion engine exhaust silencers and silencers in industrial duct systems. Both two-dimensional (2D) rectangular and 2D cylindrical geometry and uniform mean flow as well as non-uniform mean flow profiles are considered. Model results for the scattering of plane waves in case of uniform flow, in which case an infinitely thin shear layer is formed downstream of the area expansion, are compared to results obtained by other models in literature. Generally good agreement is found. Furthermore, model results for the scattering are compared to experimental data found in literature. Also here fairly good correspondence is observed. When employing a turbulent pipe flow profile in the model, instead of a uniform flow profile, the prediction for the downstream transmission- and upstream reflection coefficient is improved. However, worse agreement is observed for the upstream transmission and downstream reflection coefficient. On the contrary, employing a non-uniform jet flow profile, which represents a typical shear layer flow downstream of the expansion, gives worse agreement for the downstream transmission- and the upstream reflection coefficient, whereas prediction for the upstream transmission and downstream reflection coefficient improves.

On hydrodynamic and acoustic modes in a ducted shear flow with wall lining

The propagation of small-amplitude modes in an inviscid but sheared subsonic mean flow inside a duct is considered. For isentropic flow in a circular duct with zero swirl and constant mean flow density the pressure modes are described in terms of the eigenvalue problem for the Pridmore-Brown equation with Myers' locally reacting impedance boundary conditions.The key purpose of the paper is to extend the results of the numerical study of the spectrum for the case of lined ducts with uniform mean flow in Rienstra (Wave Motion, vol. 37, 2003b, p. 119), in order to examine the effects of the shear and wall lining. In the present paper this far more difficult situation is dealt with analytically. The high-frequency short-wavelength asymptotic solution of the problem based on the WKB method is derived for the acoustic part of the spectrum. Owing to the stiffness of the governing equations, an accurate numerical study of the spectral properties of the problem for mean flows with strong shear proves to be a non-trivial task which deserves separate consideration.The second objective of the paper is to gain theoretical insight into the properties of the hydrodynamic part of the spectrum. An analysis of hydrodynamic modes both in the short-wavelength limit and for the case of the narrow duct is presented. For simplicity, only the hard-wall flow configuration is considered.

Time-domain impedance boundary conditions for surfaces with subsonic mean flows

The time-domain impedance boundary conditions of Tam and Auriault are extended for impedance boundaries with subsonic mean flows. This extension requires an effective impedance. A model for the effective impedance is proposed, which guarantees the continuity of particle displacement over an infinitely thin shear layer connecting the fluid to the impedance boundary as described by Myers. Together with the effective impedance model, the time domain impedance boundary condition renders the requirement of measuring the effective impedance under flow conditions unnecessary. To demonstrate the feasibility of the extended boundary conditions, a numerical validation using the NASA Grazing Incidence Tube measurements of Jones et al. is carried out. The assumed velocity on the infinitely thin shear layer connecting the flow over the surface to the locally reacting wall is shown to have a strong influence on the performance of the liner. Another important parameter identified is the shape of the mean flow profile, which not only causes convective effects but also may excite possible flow instabilities. It is demonstrated that satisfactory results can be obtained using the extended time domain impedance boundary condition based on the correct effective impedances and mean flow profiles.

Propagation of Acoustical Waves in an Axisymmetrical Duct With Swirling Mean Flow

AbstractA 3D axisymmetric Computational Aero Acoustic (CAA) procedure is extended to predict the propagation and radiation of the unsteady disturbances from a duct with swirling mean flow into the far field. Based on axisymmetric mean flow and acoustical boundary conditions, the 3D task is decomposed into a series of 2D problems using Fourier series expansions in the azimuthal direction. The numerical procedure is validated by the analytical solution within a solid body rotational, subsonic mean flow in a straight circular duct and the numerical results show a very good agreement.

Asymptotic approximation for sound propagation in ducts carrying low subsonic mean flow

This article presents an extension to a previous approach used to describe the two-dimensional problem of an incident plane sound wave traveling along a rigid circular duct. Here the effects produced by a low subsonic mean flow on the values of acoustic impedance are studied using the WKB approximation. The asymptotic approximations for low Mach numbers are applied to a semi-infinite exponential horn. The results show that an increase in the flow velocity reduces the sound power transmitted through the horn.

On sound generation by the interaction between turbulence and a cascade of airfoils with non-uniform mean flow

The sound generated by the interaction between a turbulent rotor wake and a stator is modelled by considering the gust response of a cascade of blades in non-uniform, subsonic mean flow. Previous work by Hanson &amp; Horan (1998) that considers a cascade of flat plates at zero incidence is extended to take into account blade geometry and angle of attack. Our approach is based on the work of Peake &amp; Kerschen (1997), who calculate the forward radiation due to the interaction between a single vortical gust and a cascade of flat plates at non-zero angle of attack. The extensions completed in this present paper are two-fold: first we include the effects of small but non-zero camber and thickness; and second we produce uniformly valid approximations which predict the upstream radiation near modal cut-off. The thin-airfoil singularity in the steady flow at each leading edge is crucial in our model of the sound generation. A new analytical expression for the coefficient of this singularity is derived via a sequence of conformal mappings, and it turns out that in our asymptotic limit this is the only quantity which needs to be calculated from the steady flow in order to predict time-averaged noise levels. Once the response to a single gust has been completed, we use Hanson &amp; Horan (1998)'s approach to determine the response to an incident turbulent spectrum, and find that as well as the noise corresponding to the auto-correlation of the gust velocity component normal to the blade, there is also a contribution from the cross-correlation of the normal and tangential velocities. Predictions are made of the effects of blade geometry on the upstream acoustic power level. The blade geometry can have a very significant effect on the noise generated by interaction with a single gust, with changes of up to 10 dB from the flat-plate noise levels. However, once these gust results have been integrated over a full incident turbulence spectrum the effects of the geometry are rather smaller, although still potentially significant, leading to changes of up to about 2 dB from the flat-plate results. The implication of all this is that the blade geometry can have a significant effect on the tonal noise components generated by rotor–stator interaction (i.e. by single harmonic gusts), but that the broadband part of the noise spectrum is relatively unaffected.