On Sound Propagation in a Linear Shear Flow

Campos, L. M. B. C.; Oliveira, J. M. G. S.; Kobayashi, Marcelo H.

doi:10.1006/jsvi.1998.1880

Cited by 41 publications

(45 citation statements)

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“…The linear profile has the great advantage of an exact solution, but of course the price to be paid is the absence of a critical layer singularity (see Campos, Oliveira & Kobayashi 1999), i.e. a singularity of the solution at y = y c , where ω − αU 0 (y c ) = 0.…”

Section: Discussionmentioning

confidence: 99%

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

2011

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The Ingard–Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a two-dimensional incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical hc and convectively unstable or stable otherwise. This critical hc is by nature independent of wavelength or frequency and is a property of liner and mean flow only. An analytical approximation of hc is given, which is complemented by a contour plot covering all parameter values. For an aeronautically relevant example, hc is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard–Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem.

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Section: Discussionmentioning

confidence: 99%

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

2011

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“…The cases without bias flow are different in each figures 7-11 leading to five distinct comparisons with bias flow. Because the solution (3.8b) of the wave equation of the third order (3.5) is singular for M 0 = 0.0, the case without bias flow was calculated from (2.9b) using methods described in the existing literature [11,24]. acoustic pressure away from the wall.…”

Section: (C) Pressure Fields and Scattering Coefficientsmentioning

confidence: 99%

On the acoustics of an impedance liner with shear and cross flow

Campos¹,

Legendre²,

Sambuc³

2014

Proc. R. Soc. A.

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The acoustic-vortical wave equation is derived describing the propagation of sound in (i) a unidirectional shear flow with a linear velocity profile upon which is superimposed (ii) a uniform cross flow; together with an impedance wall boundary condition representing the effect of a locally reacting acoustic liner in the presence of bias and shear flow. This leads to a third-order differential equation in the presence of cross flow, and in its absence simplifies to the Pridmore-Brown equation (second-order); also the singularity of the Pridmore-Brown equation for zero Doppler-shifted frequency is removed by the cross flow. Because the third-order wave equation has no singularities (except at the sonic condition), its general solution is a linear combination of three linearly independent MacLaurin series in powers of the distance from the wall. The acoustic field in the boundary layer is matched through the pressure and horizontal and vertical velocity components to the acoustic field in a uniform free stream consisting of incident and reflected waves. The scattering coefficients are plotted for several values of the five parameters of the problem, namely the angle of incidence, free stream and cross-flow Mach numbers, specific wall impedance and Helmholtz number using the boundary layer thickness.

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“…The most famous and thoroughly studied approach, the spectral/modal approach (spectral expansion of disturbances in time and space, followed by the eigenfunction analysis), was derived by Orr. 1 A large collection of literature exists on the application of this approach to homogeneous incompressible [2][3][4][5][6] and compressible shear flows, [7][8][9] as well as inhomogeneous stratified or magnetized shear flows. [10][11][12][13][14] A comprehensive introduction to this field can be found, for instance, in the book by Schmid and Henningson.…”

Section: Introductionmentioning

confidence: 99%

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

Hau

Oberlack

Chagelishvili

2017

Journal of Mathematical Physics

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to deceptive conclusions (see, e.g., Ref. 16 and references herein), which do not coincide with experimental observations. 16,17 In the 1990s, the limiting nature of the classical modal approach was recognized and the nonnormal nature of non-uniform/shear flows was finally revealed and rigorously proven (see, e.g., Refs. 15,16,[18][19][20] and references herein)-a major breakthrough in the understanding of linear and nonlinear shear flow dynamics. In fact, the operators involved in the modal analysis of plane shear flows are non-normal, resulting in the non-orthogonality of the corresponding eigenfunctions, and hence strong interference phenomena among the eigenmodes. 19 The so-called non-modal approach, shifting the emphasis from the asymptotic to the short-time dynamics, became a well established alternative, taking into account the shortcomings of the modal approach. This resulted in the understanding and precise description of linear transient phenomena. In one branch of non-modal stability theory, the system's response to initial conditions is investigated, which is central to hydrodynamic stability theory. Here, the Kelvin mode approach, 21 stemming from the 1887 paper by Lord Kelvin, with a time-dependent shearwise wavenumber, has been extensively used for constant shear flows since the 1990s (see, e.g., . This mode represents the "simplest element" of constant shear flow physics 29 and has also been referred to as "flowing eigenfunctions" in expanding fluctuations. 30 In particular, there exist various terminologies for the Fourier mode with a time-dependent wave vector in the different areas of fluid mechanics, e.g., "Kelvin mode," "shear wave," "SH wave," "flowing eigenfunctions." Salhi and Cambon 31 gave a comprehensive survey of the method in the different communities (see Secs. I and II C).We make use of the Lie point symmetry analysis, as founded by the Norwegian mathematician Sophus Lie (1842-1899) and applied successfully in the area of fluid mechanics 32-48 to systematically derive invariant solutions in the context of linear stability of a two-dimensional (2-D) linearly sheared unbounded flow. The notion of invariant solutions was introduced by Sophus Lie, 49 and it denotes solutions whose representations are obtained with the help of any combinations of the admitted groups. This method identifies variable transformation by which new solutions can be generated from existing ones through the use of a differential operator, i.e., the generator of the group. The similarity analysis of systems of partial differential equations (PDEs) leading to group-invariant solution is systemically and well described in several textbooks, for instance, by Refs. 50-54. Compared to a simple dimensional analysis, this method is somehow superior. This is due to the fact that there is no need to analyze the equations and boundary/initial conditions at the same time in order to identify the desired similarity variables-non-dimensional combinations of variables. Further, it is possible to uncover types of similarity th...

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On Sound Propagation in a Linear Shear Flow

Cited by 41 publications

References 0 publications

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

On the acoustics of an impedance liner with shear and cross flow

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

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