On hydrodynamic and acoustic modes in a ducted shear flow with wall lining

Abstract

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.

Compressible turbulent channel flow with impedance boundary conditions

“…Such hydrodynamic instability is, however, driven by the wall-normal wave propagation resulting from resonant excitation of the tuned IBCs (as discussed in section V A), therefore leading to the designation of hydro-acoustic instability. This type of instability was theoretically predicted by Rienstra and co-workers [8][9][10] and experimentally identified in few previous works 11 in very similar flow configurations.…”

“…Numerous theoretical investigations by Rienstra and co-workers 2,[8][9][10] , together with some companion experimental efforts 11 , have looked at the stability properties of boundary layers over homogeneous IBCs. In particular, the presence of hydro-acoustic instabilities was predicted under specific conditions, which were deemed to be rarely found in aeronautical practice.…”

Spatial Instability of Boundary Layer Along Impedance Wall

“…In contrast to the much more general theory of [18,19], the configuration considered here is taken as simple as possible in order obtain canonical and generic results. In particular, the mean flow variables are uniform everywhere except for a ∼ tanh-profile of the axial velocity (see Figure 1) The resulting system of equations (note slight notation differences with [18,19])…”

“…In particular, the mean flow variables are uniform everywhere except for a ∼ tanh-profile of the axial velocity (see Figure 1) The resulting system of equations (note slight notation differences with [18,19])…”

“…Using the Pridmore-Brown model for duct modes of given frequency in shear flow along a lined wall, developed earlier [18,19], we were able to analyse the spatial modes in more detail, in particular as a function of complex frequency. When the axial wave number k i (ω) ∈ C can be traced to the other complex half plane as we let ω ∈ C vary far enough into the complex plane, arguments of causality applied to the time-Fourier transformed field tell us that k i really is an instability [20][21][22][23].…”