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

Abstract

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.

A comparison of impedance boundary conditions for flow acoustics

“…The boundary condition proposed by Rienstra and Darau [12,13] assumes a boundary layer with a small thickness d, a linear velocity profile and a uniform mean density. This two-dimensional boundary condition was derived in the incompressible limit and was devised to provide a good approximation of the hydrodynamic oscillations of the boundary layer (see also [15]).…”

“…This defines a family of boundary conditions characterized by the parameter s. Originally, s was set to zero [12], but subsequently it was suggested to use s ¼ 1=3 to remove the second-order derivative in v [13]. The case s ¼ 1 is also considered here since it is more consistent with the special cases discussed in Section 3.…”

“…(7) to three dimensions follows that in [13], and the derivation is outlined in Appendix A. However, it was found that the last step of the derivation differs from [13], and the following version of the boundary condition is proposed here:…”

“…While this assumption is acceptable when modeling the hydrodynamic oscillations of the boundary layer (as shown in Ref. [13,15]), it is an issue if this boundary condition is used to model sound absorption by a lined surface.…”

A wave expansion method for acoustic propagation in lined flow ducts

“…These two issues are problematic for time-domain simulations, where errors generate perturbations at every frequency, and may result in difficulties with numerical convergence. For frequencydomain computations, however, they are of minor practical importance [23]. It has also been shown that the Myers model fails to result in a single impedance value for wave propagation downstream and upstream in a lined duct when the acoustic boundary layer is not much smaller that the mean flow boundary layer thickness, i.e.…”