Mean flow boundary layer effects of hydrodynamic instability of impedance wall

Abstract

The Ingard-Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishing boundary layer, is known to lead to an ill-posed problem in time-domain. By analysing the stability of a mean flow, uniform except for a linear boundary layer of thickness h, in the incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical h c and convectively unstable or stable otherwise. This critical h c is by nature independent of wave length or frequency and is a property of liner and mean flow only. An analytical approximation of h c is given for a mass-spring-damper liner. For an aeronautically relevant example, h c 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

“…Following the suggestion in [9], if we adjust d in Eqs. (12) and (13) so that the displacement thickness d 1 is the same for all three profiles, then the results are almost identical and the actual boundary layer profile has in fact very little impact on sound absorption. This conclusion was also observed with the other test cases listed in Table 1.…”

“…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.…”

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

“…This paper is an adapted and extended version of the IUTAM 2010 Symposium on Computational Aero-Acoustics contribution (see Rienstra & Darau 2010).…”

The Effect of Boundary Layers on Bulk Reacting Liners at Low Mach Number Flows

“…Extensive research in the field of aircraft liners has been made and is still being carried out in order to better describe the effect of flow on the impedance boundary condition of the lining. Several competing models for the modified boundary condition have been proposed in this field [1][2][3][4][5][6] . The flow is shown to considerably alter the lining performance at high Mach number flows 7 , although the size of the effect differs between the models.…”