Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 99
velocity increases from zero to a value Uɛ, nearly discontinuously; Uɛ ≈ U/4 has been
estimated on the basis of the sound reflection from and the transmission through the
resonator section measured at frequencies far below the resonance frequency [19].
Thus the wall-normal flow velocity �vw of the instability wave imposes an oscillating
component �τw = −ρUɛ�vw on the wall-normal transport of axial momentum;
�vw = 〈vw · exp(iωt)〉 · exp(−iωt) is the phase average by which the coherent part at
the frequency ω is filtered out from the fluctuating quantity vw(t). The considered
oscillation �τw of the wall shear stress is by orders of magnitude greater than with
an impervious smooth wall even if the considerable increase of �τw by the turbulent
mixing [41] is taken into account. The transfer of axial momentum to the wall by
the wall-normal particle velocity, and the associated high acoustic shear stress have
scarcely been regarded in aero-acoustics so far. This mechanism is observed also with
openings subject to grazing flow [42], and it can be used with deflected mean flow to
arbitrarily increase the quality factor of resonators [43–45]. Pöthke and Ronneberger
[42] determine the amplitude of the shear force at transverse slits in the wall of the
flow duct; these had the same dimensions as openings of the cavities in the present
experiments. At low frequencies (compared to the ratio between flow velocity and
slit width) the same value of the shear force is obtained as in the aforementioned
investigation by Rebel and Ronneberger [19], but the shear force amplitude increases
considerably as a function of the frequency according to Ref. [42]. Yet, a frequencyindependent
value Uɛ = U/4 is assumed in the following estimates.
The oscillation of the wall shear stress in the lined duct section excites a shear wave
which is propagated by the wall-normal gradient of the shear stress �τ = �τ ′′ + �τµ into
the flow region. The viscous part �τµ = µ·∂�u/∂y of the shear stress is small compared
to the turbulent part and therefore will be disregarded. The layer within which the
shear wave decays is called ‘turbulent Stokes layer’ following the denomination in
the laminar case. Though the thickness δS of the Stokes layer is much greater in the
turbulent case than in the laminar case we assume that δS is still small compared to
the lateral extent of the field of the instability wave which decays approximately like
exp(−|ℜ{α}|y). So we attempt to subsume the influence of the shear stress under
the boundary condition at the wall.
In order to study the influence of the shear stress on the deflection of the streamlines,
the relation (10) is introduced into the equations of conservation of mass and
momentum. For the sake of clearness a 2D flow is assumed composed of the mean
flow and the field of a coherent oscillation while the effect of the turbulent fluctuations
are already taken into account by the Reynolds shear stress. Then one obtains
for the deflection of the fictitious streamlines
ρ D2
Dt2 ∂�η 1
+
∂y c2 D2�p ∂
−
Dt2 ∂x
�
∂�p ∂�τxx ∂�τxy
− −
∂x ∂x ∂y
�
= 0 . (13)
The terms �τxx, �τxy = �τ ′′ + �τµ (and �τyy) are the components of the tensor which
comprises the oscillatory parts of the viscous and the Reynolds stresses. A clearer
version of Eq. (13) is obtained when a single mode is considered such that ∂/∂x and
D/Dt can be replaced by iα and −iˇω, respectively:
dˆη ˆp
= −
dy ρc2 − iαiαˆp − iαˆτxx − dˆτxy/dy
ρ ˇω 2 . (14)