Acoustics in a Two-Deck Viscothermal Boundary Layer over an Impedance Surface

Abstract

The acoustics of a mean flow boundary layer over an impedance surface or acoustic lining are considered. By considering a thick mean flow boundary layer (possibly due to turbulence), the boundary layer structure is separated asymptotically into two decks, with a thin weakly viscous mean flow boundary layer and an even thinner strongly viscous acoustic sublayer, without requiring a high-frequency. Using this, analytic solutions are found for the acoustic modes in a cylindrical lined duct. The mode shapes in each region compare well with numerical solutions of the linearised compressible Navier-Stokes equations, as does a uniform composite asymptotic solution. A closed-form effective impedance boundary condition is derived which can be applied to acoustics in inviscid slipping flow to account for both shear and viscosity in the boundary layer. The importance of the boundary layer is demonstrated in the frequency domain, and the new boundary condition is found to correctly predict the attenuation of upstream-propagating cuton modes, which are poorly predicted by existing inviscid boundary conditions. Stability is also investigated, and the new boundary condition is found to yield good results away from the critical layer. A time-domain formulation of a simplified version of the new impedance boundary condition is proposed.

Nonlinear Acoustics in a Viscothermal Boundary Layer over an Acoustic Lining

“…However, Renou and Auregan [13] demonstrated that to correlate mathematical and numerical results with the results of experiments, the effect of viscosity within the mean flow boundary layer must be included, and Khamis and Brambley [14,15] demonstrated that the effects of viscosity on the acoustics are of a comparable magnitude to the effects of shear, and thus both should be taken into account. Viscosity within the mean flow boundary layer was investigated by Aurégan, Starobinski, and Pagneux [7] for thin low-velocity mean flow boundary layers, and by Brambley [2] for thin mean flow boundary layers of arbitrary subsonic velocity, while investigations taking into account both shear and viscosity within the mean flow boundary layer have recently been performed by Khamis and Brambley [16,17]. This approach agrees most closely with results from solving the linearised Navier Stokes equations (LNSE) for the entire duct.…”

“…While asymptotic approximate solutions to equations (19,25,37) are possible [see, e.g. 2,17], here these equations are solved numerically using 4th order finite differences. The resulting 3N × 3N banded matrix system of equations is solved using the LAPACK_ZGBSV routine.…”

“…In conclusion, solving (19) subject to the boundary conditionsT 1 = 0 andû 1 = 0 at y = 0 and (21) at y = Y , wherev 1∞ is given by (17) and (18) yields a unique solution. Requiring also the impedance boundary condition p O1 (1) =p 1 (0) = Zv 1 (0) at y = 0 gives a dispersion relation relating allowable values of k and ω.…”

“…While the use of the asymptotics simplifies the governing equations, numerical solutions are still needed. In the linear case, other additional methods have been used [2,16,17] to derive approximate solutions and an effective impedance Z eff that accounts for the behaviour within the mean flow boundary layer without having to numerically solve differential equations, and such techniques may well be applicable here.…”

On the use of a stress–impedance model to describe sound propagation in a lined duct with grazing flow

“…This shear stress is apparently due to the interaction between turbulent flow and the rough wall which is the interface of the acoustic treatment. Further attempts were made to account for shear stress in terms of viscous stress [9,10] or in term of additional force acting on the walls of a cavity [11]. A modified boundary condition was derived that introduces a coefficient β v that characterizes the transfer, by the normal fluctuating displacement, of axial momentum from the steady flow into the lined wall [9].…”