Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 101
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Re{wavenumber}
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Im{wavenumber}
Figure 15. The amplitude ratio | ˆ δS|/| ˆ δw| of the thickness of the Stokes layer and the
deflection of the wall with contour lines at | ˆ δS|/| ˆ δw| = 1 and | ˆ δS|/| ˆ δw| = 5 is plotted as a
function of the complex wavenumber of the modes that propagate in the lined duct. The
wavenumber is normalized to the duct radius. Parameters: f = 1100 Hz and U/c = 0.25.
˜µ t = ˆτ ′′ /(dû/dy) should be real and should assume the value ˜µ t ≈ 2µ t according to
the somewhat uncertain extrapolation of Hartmann’s results. In a WKBJ approximation
the wavenumber γ = (−iωρ/˜µ t) 1/2 of shear waves in homogeneous fluids is transferred
to the actual inhomogeneous case, yielding ûτ (y) ≈ ûτ (0) exp( � y
0 γ(y′ )dy ′ ). So
Eq. (15) can be evaluated with ˆ θ(y) ≈ (˜µ tγ ûτ )(y)/(˜µ tγ ûτ )w.
The ratio | ˆ δS|/| ˆ δw| for a typical combination of parameters (1100 Hz, U/c = 0.25)
is depicted in Fig. 15 as a function of the wavenumber α. For wavenumbers that have
been observed in the experiments (see Fig. 13), | ˆ δS|/| ˆ δw| assumes values which are
typically much greater than unity. So obviously the oscillation of the Stokes layer
thickness turns out to be the governing factor in the boundary condition at the wall.
4.2.2 Influence of the oscillation of the Stokes layer thickness on the dispersion
relations and open questions
First computational results reveal that the dispersion relations are indeed strongly
affected by the oscillation of the Stokes layer thickness. Similar to the experimental
observations one of these dispersion relations (not shown here) exhibits maxima of
the spatial growth rate (−ℑ{α}), and at the same frequencies, a positive slope of the
phase constant. However, this occurs at frequencies below the resonance frequency,
and the phase constant is much too small and mostly even negative. Furthermore
the dependency on the flow velocity is still contrary to the experiment, and the
bifurcations have not disappeared from the dispersion relations.
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