Acoustic modes in a ducted shear flow

The propagation of small-amplitude modes in an inviscid but sheared 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. A numerical method similar to the procedure used by Tam & Auriault is proposed for the solution of the modal equation. Since for sufficiently high Helmholtz and wavenumbers, which are of great interest for the applications, the field equation is inherently stiff, special care is taken to insure the stability of the numerical algorithm designed to tackle this problem. The accuracy of the method is checked against the well-known analytical solution for the uniform flow. The numerical method is shown to be consistent with the analytical predictions at least for the Helmholtz numbers up to 100 and the circumferential wavenumber as large as 50, typical Mach numbers being up to 0.65. In order to gain further insight into the possible structure of the modal solutions and to get an independent verification of the robustness of the numerical scheme, the asymptotic solution of the problem based on the WKB method is derived. The comparisons of theWKB solution against the exact potential flow solution show remarkably good agreement between the two. This permits us to use the asymptotic solution as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available. Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that Ingard-Myers condition is recovered for vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases

Numerical study of acoustic modes in ducted shear flow

The propagation of small-amplitude modes in an inviscid but sheared mean flow inside a duct is studied numerically. 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. Since for sufficiently high Helmholtz and wavenumbers, which are of great interest for applications, the field equation is inherently stiff, special care is taken to insure the stability of the numerical algorithm designed to tackle this problem. The accuracy of the method is checked against the well-known analytical solution for uniform flow. The numerical method is shown to be consistent with the analytical predictions at least for Helmholtz numbers up to 100 and circumferential wavenumbers as large as 50, typical Mach numbers being up to 0.65. In order to gain further insight into the possible structure of the modal solutions and to obtain an independent verification of the robustness of the numerical scheme, comparison to the asymptotic solution of the problem based on the WKB method is performed. The asymptotic solution is also used as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available. The bulk of the analysis concentrates on the influence of the wall lining. The proposed numerical procedure is adapted in order to include Ingard–Myers boundary conditions. In parallel with this, the WKB solution is used to check the numerical predictions of the typical behaviour of the axial wavenumber in the complex plane, when the wall impedance varies in the complex plane. Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that the use of Ingard–Myers condition in combination with an appropriate slip-stream approximation instead of the actual no-slip mean flow profile gives valid results in the limit of vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases

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

Acoustic modes in a ducted shear flow

The propagation of small-amplitude modes in an inviscid but sheared 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. A numerical method similar to the procedure used by Tam & Auriault is proposed for the solution of the modal equation. Since for sufficiently high Helmholtz and wavenumbers, which are of great interest for the applications, the field equation is inherently stiff, special care is taken to insure the stability of the numerical algorithm designed to tackle this problem. The accuracy of the method is checked against the well-known analytical solution for the uniform flow. The numerical method is shown to be consistent with the analytical predictions at least for the Helmholtz numbers up to 100 and the circumferential wavenumber as large as 50, typical Mach numbers being up to 0.65. In order to gain further insight into the possible structure of the modal solutions and to get an independent verification of the robustness of the numerical scheme, the asymptotic solution of the problem based on the WKB method is derived. The comparisons of theWKB solution against the exact potential flow solution show remarkably good agreement between the two. This permits us to use the asymptotic solution as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available. Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that Ingard-Myers condition is recovered for vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases

Spatial instability of boundary layer along impedance wall

A numerical analysis is made of the hydro-acoustical spatial instability, apparently occurring in a mean flow with thin boundary layer along a locally reacting lined duct wall. This problem is of particular interest because unstable behaviour of liner and mean flow has been observed only very rarely. It is found that this instability quickly disappears for increasing boundary layer thickness. Specifically, for boundary-layer-thickness based Helmholtz numbers !??/c0 of the order of 0.1 the growth rate vanishes and the instability disappears. This corresponds to very thin boundary layers for practical values of frequencies that occur in aero-engine applications, which is in turn in good agreement with the fact that in industrial practice no instabilities are observed. For low duct-radius based Helmholtz numbers (?? 1), the instability exists for rather large values of ?? as an almost neutrally stable wave. This is qualitatively in good agreement with the experimental observations of Ronneberger and Auregan. It is shown by a Rayleigh-type stability criterion that impedance related hydrodynamic instabilities of temporal type do not occur for mean flows with strictly negative 2nd derivative (the usual situation)

Spatial instability of boundary layer along impedance wall

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this article we study left-invariant diffusion on the 3D Euclidean motion group SE(3) and its application to crossing-preserving smoothing of high angular resolution diffusion imaging (HARDI), which is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space R3 o S2 of positions and orientations embedded in SE(3) and can be solved by R3 o S2-convolution with the corresponding Green’s functions. We provide analytic approximation formulae and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (non-linear) convection-diffusions on HARDI-data we put emphasis on the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan-connection. This Cartan-connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI-data containing crossing-fibers