An Analytic Green's Function for a Lined Circular Duct Containing Uniform Mean Flow

2005 ◽  
Author(s):  
Sjoerd Rienstra ◽  
Brian Tester
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Mean Flow ◽  
Circular Duct

Effect of non-parallel mean flow on the Green’s function for predicting the low-frequency sound from turbulent air jets

2012 ◽  
Vol 695 ◽  
pp. 199-234 ◽  
Author(s):  
M. E. Goldstein ◽  
Adrian Sescu ◽  
M. Z. Afsar
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Low Frequency ◽  
Source Location ◽  
Parallel Flow ◽  
Numerical Calculations ◽  
Mean Flow ◽  
Frequency Sound ◽  
The Mean ◽  
Radiated Sound

AbstractIt is now well-known that there is an exact formula relating the far-field jet noise spectrum to the convolution product of a propagator (that accounts for the mean flow interactions) and a generalized Reynolds stress autocovariance tensor (that accounts for the turbulence fluctuations). The propagator depends only on the mean flow and an adjoint vector Green’s function for a particular form of the linearized Euler equations. Recent numerical calculations of Karabasov, Bogey & Hynes (AIAA Paper 2011-2929) for a Mach 0.9 jet show use of the true non-parallel flow Green’s function rather than the more conventional locally parallel flow result leads to a significant increase in the predicted low-frequency sound radiation at observation angles close to the downstream jet axis. But the non-parallel flow appears to have little effect on the sound radiated at $9{0}^{\ensuremath{\circ} } $ to the downstream axis. The present paper is concerned with the effects of non-parallel mean flows on the adjoint vector Green’s function. We obtain a low-frequency asymptotic solution for that function by solving a very simple second-order hyperbolic equation for a composite dependent variable (which is directly proportional to a pressure-like component of this Green’s function and roughly corresponds to the strength of a monopole source within the jet). Our numerical calculations show that this quantity remains fairly close to the corresponding parallel flow result at low Mach numbers and that, as expected, it converges to that result when an appropriately scaled frequency parameter is increased. But the convergence occurs at progressively higher frequencies as the Mach number increases and the supersonic solution never actually converges to the parallel flow result in the vicinity of a critical- layer singularity that occurs in that solution. The dominant contribution to the propagator comes from the radial derivative of a certain component of the adjoint vector Green’s function. The non-parallel flow has a large effect on this quantity, causing it (and, therefore, the radiated sound) to increase at subsonic speeds and decrease at supersonic speeds. The effects of acoustic source location can be visualized by plotting the magnitude of this quantity, as function of position. These ‘altitude plots’ (which represent the intensity of the radiated sound as a function of source location) show that while the parallel flow solutions exhibit a single peak at subsonic speeds (when the source point is centred on the initial shear layer), the non-parallel solutions exhibit a double peak structure, with the second peak occurring about two potential core lengths downstream of the nozzle. These results are qualitatively consistent with the numerical calculations reported in Karabasov et al. (2011).

A Green’s Function Solution for Acoustic Attenuation by a Cylindrical Chamber With Concentric Perforated Liners

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
D. Veerababu ◽  
B. Venkatesham
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Velocity Potential ◽  
Transmission Loss ◽  
Analytical Models ◽  
Mean Flow ◽  
Cylindrical Chamber ◽  
Function Solution ◽  
Grazing Flow ◽  
The Mean

Abstract In this study, a Green’s function-based semi-analytical method is presented to predict the transmission loss (TL) of a circular chamber having concentric perforated screens. Initially, the Green’s function is developed for a single-screen configuration as the summation of eigenfunctions of the inner pipe in the absence of the mean flow. The inlet and the outlet ports are modeled as oscillating piston sources. A transfer matrix is formulated from the velocity potential generated by the piston sources. The results obtained from the proposed method are validated with the numerical and analytical models and with the experimental results available in the literature. Later, the method has been extended to the double-screen configuration. The effect of the additional perforated screen on the TL is studied in terms of the surface impedance of the chamber. Along with grazing flow considerations, guidelines are provided to incorporate more concentric perforated screens into the formulation.

The acoustic analogy in an annular duct with swirling mean flow

2013 ◽  
Vol 726 ◽  
pp. 439-475 ◽  
Author(s):  
H. Posson ◽  
N. Peake
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Stokes Equations ◽  
Swirling Flow ◽  
Base Flow ◽  
Order Differential Operator ◽  
Source Terms ◽  
Acoustic Analogy ◽  
Mean Flow ◽  
Annular Duct

AbstractThis paper is concerned with modelling the effects of swirling flow on turbomachinery noise. We develop an acoustic analogy to predict sound generation in a swirling and sheared base flow in an annular duct, including the presence of moving solid surfaces to account for blade rows. In so doing we have extended a number of classical earlier results, including Ffowcs Williams & Hawkings’ equation in a medium at rest with moving surfaces, and Lilley’s equation for a sheared but non-swirling jet. By rearranging the Navier–Stokes equations we find a single equation, in the form of a sixth-order differential operator acting on the fluctuating pressure field on the left-hand side and a series of volume and surface source terms on the right-hand side; the form of these source terms depends strongly on the presence of swirl and radial shear. The integral form of this equation is then derived, using the Green’s function tailored to the base flow in the (rigid) duct. As is often the case in duct acoustics, it is then convenient to move into temporal, axial and azimuthal Fourier space, where the Green’s function is computed numerically. This formulation can then be applied to a number of turbomachinery noise sources. For definiteness here we consider the noise produced downstream when a steady distortion flow is incident on the fan from upstream, and compare our results with those obtained using a simplistic but commonly used Doppler correction method. We show that in all but the simplest case the full inclusion of swirl within an acoustic analogy, as described in this paper, is required.

Mean flow refraction effects on sound radiated from localized sources in a jet

1998 ◽  
Vol 370 ◽  
pp. 149-174 ◽  
Author(s):  
CHRISTOPHER K. W. TAM ◽  
LAURENT AURIAULT
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Acoustic Waves ◽  
Flow Direction ◽  
Jet Flow ◽  
Parallel Flow ◽  
Mean Flow ◽  
Flow Approximation ◽  
Radiated Sound ◽  
Localized Sources

It is well-known that sound generated by localized sources embedded in a jet undergoes refraction as the acoustic waves propagate through the jet mean flow. For isothermal or hot jets, the effect of refraction causes the deflection of the radiated sound waves away from the jet flow direction. This gives rise to a cone of silence around the jet axis where there is a significant reduction in the radiated sound intensity. In this work, the mean flow refraction problem is investigated through the use of the reciprocity principle. Instead of the direct source Green's function, the adjoint Green's function with the source and observation points interchanged is used to quantify the effect of mean flow on sound radiation. One advantage of the adjoint Green's function is that the Green's functions for all the source locations in the jet radiating to a given direction in the far field can be obtained in a single calculation. This provides great savings in computational effort. Another advantage of the adjoint Green's function is that there is no singularity in the jet flow so that the problem can be solved numerically with axial as well as radial mean flow gradients included in a fairly straightforward manner. Extensive numerical computations have been carried out for realistic jet flow profiles with and without exercising the locally parallel flow approximation. It is concluded that the locally parallel flow approximation is valid as long as the direction of radiation is outside the cone of silence.

scholarly journals Modelling and prediction of the peak-radiated sound in subsonic axisymmetric air jets using acoustic analogy-based asymptotic analysis

2019 ◽  
Vol 377 (2159) ◽  
pp. 20190073 ◽  
Author(s):  
Mohammed Z. Afsar ◽  
Adrian Sescu ◽  
Stewart J. Leib
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Analysis ◽  
Green’S Function ◽  
Green's Function ◽  
Exact Formula ◽  
Convolution Product ◽  
Acoustic Analogy ◽  
Mean Flow ◽  
Partial Differential

This paper uses asymptotic analysis within the generalized acoustic analogy formulation (Goldstein 2003 JFM 488 , 315–333. ( doi:10.1017/S0022112003004890 )) to develop a noise prediction model for the peak sound of axisymmetric round jets at subsonic acoustic Mach numbers (Ma). The analogy shows that the exact formula for the acoustic pressure is given by a convolution product of a propagator tensor (determined by the vector Green's function of the adjoint linearized Euler equations for a given jet mean flow) and a generalized source term representing the jet turbulence field. Using a low-frequency/small spread rate asymptotic expansion of the propagator, mean flow non-parallelism enters the lowest order Green's function solution via the streamwise component of the mean flow advection vector in a hyperbolic partial differential equation. We then address the predictive capability of the solution to this partial differential equation when used in the analogy through first-of-its-kind numerical calculations when an experimentally verified model of the turbulence source structure is used together with Reynolds-averaged Navier–Stokes solutions for the jet mean flow. Our noise predictions show a reasonable level of accuracy in the peak noise direction at Ma = 0.9, for Strouhal numbers up to about 0.6, and at Ma = 0.5 using modified source coefficients. Possible reasons for this are discussed. Moreover, the prediction range can be extended beyond unity Strouhal number by using an approximate composite asymptotic formula for the vector Green's function that reduces to the locally parallel flow limit at high frequencies. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.

Solution of the Parallel sheaR Layer Green's Function Using Conservation Equations

2009 ◽  
Vol 8 (6) ◽  
pp. 585-602 ◽  
Author(s):  
M. Z. Afsar
Keyword(s):  
Wave Propagation ◽  
Green’S Function ◽  
Green's Function ◽  
Acoustic Analogy ◽  
Mean Flow ◽  
Function Variable ◽  
Convective Derivative ◽  
The Mean ◽  
Parallel Shear Flow ◽  
Set Of Points

A parallel shear flow representation of a jet is a standard way to solve for the wave propagation terms in jet noise modeling using the acoustic analogy. In this paper we show by introducing a new primary Green's function variable, proportional to the convective derivative of the pressure-like Green's function, the wave propagation equations reduce to an exact conservation form that does not include any derivatives of the mean flow. We analyze this Green's function variable numerically and show its utility when the mean flow is defined by a CFD solution and known only at a discrete set of points.

Effects of jet flow on jet noise via an extension to the Lighthill model

1996 ◽  
Vol 321 ◽  
pp. 1-24 ◽  
Author(s):  
Herbert S. Ribner
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Green's Function ◽  
Source Term ◽  
Jet Noise ◽  
Convective Motion ◽  
Jet Flow ◽  
Shielding Effect ◽  
Mean Flow ◽  
The Mean

The Lighthill formalism for jet noise prediction is extended to accommodate wave transport by the mean jet flow. The extended theory combines the simplicity of the Lighthill approach with the generality of the more complex Lilley approach. There is full allowance for ‘flow-acoustic’ effects: shielding, as well as the refractive ‘cone of (relative) silence’. A source term expansion yielda a convected wave equation that retains the basic Lighthill source term. This leads to a general formula for power spectral density emitted from unit volume as the Lighthill-based value multiplied by a squared ‘normalized’ Green's function. The Green's function, referred to a stationary point source, delineates the refraction dominated ‘cone of silence’. The convective motion of the sources, with its powerful amplifying effect, also directional, is accounted for in the Lighthill factor. Source convection and wave convection are thereby decoupled, in contrast with the Lilley approach: this makes the physics more transparent. Moreover, the normalized Green's function appears to be near unity outside the ‘cone of silence’. This greatly reduces the labour of calculation: the relatively simple Lighthill-based prediction may be used beyond the cone, with extension inside via the Green's function. The function is obtained either experimentally (injected ‘point’ source) or numerically (computational aeroacoustics). Approximation by unity seems adequate except near the cone and except when there are coaxial or shrouding jets: in that case the difference from unity will quantify the shielding effect. Further extension yields dipole and monopole source terms (cf. Morfey, Mani, and others) when the mean flow possesses density gradients (e.g. hot jets).

scholarly journals FUNGSI GREEN UNTUK PERSAMAAN DIFUSI-ADVEKSI DENGAN SYARAT BATAS DIRICHLET

2020 ◽  
Vol 14 (2) ◽  
pp. 211-222
Author(s):  
Josua Josua ◽  
Evi Noviani ◽  
Fransiskus Fran
Keyword(s):  
Green’S Function ◽  
Regular Solution ◽  
Green's Function ◽  
Fluid Transport ◽  
Transport Processes ◽  
Advection Equation ◽  
Function Concept ◽  
Mean Flow ◽  
Principal Solution ◽  
The Fourier Transform

Diffusion-advection is the process of transportation of matter from one part of a system to another as a result of random molecular motions involving fluid transport processes in the form of mean flow or currents which are driven by gravity or pressure forces and in the form of horizontal motions. Mathematically, diffusion-advection equation can be written as ut +Vux = Duxx, where u  is concentration of material in the fluid, V stands for the advection velocity, and  D for diffusion coefficient. In this research, a solution u(x,t) is sought by using the Green’s function concept. The general solution for Green’s function that can be solved in two parts, namely, the principal solution and the regular solution. The principal solution is obtained by applying the Fourier transform to the variable ξ which is denoted by Ȗ and then calculate the inverse of its transform. A regular solution is obtained based on an inspection approach that is designed on a negative heat source.

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