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.

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.

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’.

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).

A theoretical study of viscous effects in peristaltic pumping

1994 ◽  
Vol 279 ◽  
pp. 177-195 ◽  
Author(s):  
Alden M. Provost ◽  
W. H. Schwarz
Keyword(s):  
Wave Propagation ◽  
Pressure Gradient ◽  
Flow Rate ◽  
Mean Flow ◽  
Viscous Effects ◽  
Transverse Waves ◽  
Peristaltic Pumping ◽  
Mean Pressure ◽  
Flow Through ◽  
The Mean

Intuition and previous results suggest that a peristaltic wave tends to drive the mean flow in the direction of wave propagation. New theoretical results indicate that, when the viscosity of the transported fluid is shear-dependent, the direction of mean flow can oppose the direction of wave propagation even in the presence of a zero or favourable mean pressure gradient. The theory is based on an analysis of lubrication-type flow through an infinitely long, axisymmetric tube subjected to a periodic train of transverse waves. Sample calculations for a shear-thinning fluid illustrate that, for a given waveform, the sense of the mean flow can depend on the rheology of the fluid, and that the mean flow rate need not increase monotonically with wave speed and occlusion. We also show that, in the absence of a mean pressure gradient, positive mean flow is assured only for Newtonian fluids; any deviation from Newtonian behaviour allows one to find at least one non-trivial waveform for which the mean flow rate is zero or negative. Introduction of a class of waves dominated by long, straight sections facilitates the proof of this result and provides a simple tool for understanding viscous effects in peristaltic pumping.

The Simulation of Airframe Noise Applying Euler-Perturbation and Acoustic Analogy Approaches

2005 ◽  
Vol 4 (1-2) ◽  
pp. 69-91 ◽  
Author(s):  
R. Ewert ◽  
J.W. Delfs ◽  
M. Lummer
Keyword(s):  
Mach Number ◽  
Scaling Laws ◽  
Perturbation Approach ◽  
Far Field ◽  
Acoustic Analogy ◽  
Mean Flow ◽  
Trailing Edge ◽  
Airframe Noise ◽  
Trailing Edge Noise ◽  
The Mean

The capability of three different perturbation approaches to tackle airframe noise problems is studied. The three approaches represent different levels of complexity and are applied to trailing edge noise problems. In the Euler-perturbation approach the linearized Euler equations without sources are used as governing acoustic equations. The sound generation and propagation is studied for several trailing edge shapes (blunt, sharp, and round trailing edges) by injecting upstream of the trailing edge test vortices into the mean-flow field. The efficiency to generate noise is determined for the trailing edge shapes by comparing the different generated sound intensities due to an initial standard vortex. Mach number scaling laws are determined varying the mean-flow Mach number. In the second simulation approach an extended acoustic analogy based on acoustic perturbation equations (APEs) is applied to simulate trailing edge noise of a flat plate. The acoustic source terms are computed from a synthetic turbulent velocity model. Furthermore, the far field is computed via additional Kirchhoff extrapolation. In the third approach the sources of the extended acoustic analogy are computed from a Large Eddy Simulation (LES) of the compressible flow problem. The directivities due to a modeled and a LES based source, respectively, compare qualitatively well in the near field. In the far field the asymptotic directivities from the Kirchhoff extrapolation agree very well with the analytical solution of Howe. Furthermore, the sound pressure spectra can be shown to have similar shape and magnitude for the last two approaches.

scholarly journals On the spectra of Schrödinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients

2005 ◽  
Author(s):  
◽  
Vladimir Batchenko
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Complex Plane ◽  
Mean Value ◽  
Qualitative Behavior ◽  
Conditional Stability ◽  
One Dimensional ◽  
Jacobi Operators ◽  
The Mean ◽  
Complex Valued

In this thesis we characterize the spectrum of one-dimensional Schrödinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of Schrödinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the Schrödinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.

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