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’.

On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation

1949 ◽  
Vol 1 (2) ◽  
pp. 191-198 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Arbitrary Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Partial Differential ◽  
Second Order Differential Equation ◽  
Image Position

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

A Green's Function-Based Design for Deformation Control of a Microbeam With In-Domain Actuation

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Amir Badkoubeh ◽  
Guchuan Zhu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Control Design ◽  
Target System ◽  
Bernoulli Equation ◽  
Test Function ◽  
Deformation Control ◽  
Control Scheme

This paper presents a Green's function-based design for deformation control of a microbeam described by an Euler-Bernoulli equation with in-domain pointwise actuation. The Green's function is first used in control design to construct the test function that enables the solvability of a map between the original nonhomogeneous partial differential equation and a target system in standard boundary control form. Then a regularized Green's function is employed in motion planning, leading to a computationally tractable implementation of the control scheme combined by a single feedback stabilizing loop and feedforward controls. The viability and the applicability of the proposed approach are demonstrated through numerical simulations of a representative microbeam.

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.

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.

scholarly journals Droplet dynamics on chemically heterogeneous substrates

2018 ◽  
Vol 859 ◽  
pp. 321-361 ◽  
Author(s):  
Nikos Savva ◽  
Danny Groves ◽  
Serafim Kalliadasis
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Analysis ◽  
Contact Line ◽  
Computational Cost ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Reduced Model ◽  
Partial Differential ◽  
Droplet Dynamics

Slow droplet motion on chemically heterogeneous substrates is considered analytically and numerically. We adopt the long-wave approximation which yields a single partial differential equation for the droplet height in time and space. A matched asymptotic analysis in the limit of nearly circular contact lines and vanishingly small slip lengths yields a reduced model consisting of a set of ordinary differential equations for the evolution of the Fourier harmonics of the contact line. The analytical predictions are found, within the domain of their validity, to be in good agreement with the solutions to the governing partial differential equation. The limitations of the reduced model when the contact line undergoes stronger deformations are partially lifted by proposing a hybrid scheme which couples the results of the asymptotic analysis with the boundary integral method. This approach improves the agreement with the governing partial differential equation, but at a computational cost which is significantly lower compared to that required for the full problem.

ON THE RESOLVENT OF THE LAPLACE–BELTRAMI OPERATOR IN HYPERBOLIC SPACE

2015 ◽  
Vol 99 (2) ◽  
pp. 186-206
Author(s):  
GUSEIN SH. GUSEINOV
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Asymptotic Analysis ◽  
Green’S Function ◽  
Hyperbolic Space ◽  
Green's Function ◽  
Beltrami Operator ◽  
Hypergeometric Differential Equation

In this paper, a detailed description of the resolvent of the Laplace–Beltrami operator in $n$-dimensional hyperbolic space is given. The resolvent is an integral operator with the kernel (Green’s function) being a solution of a hypergeometric differential equation. Asymptotic analysis of the solution of this equation is carried out.

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