Oscillations, Waves, and Interactions - GWDG

100 D. Ronneberger et al.
The term dˆη/dy is the oscillation of the wall-normal spacing of the streamlines or
in other words the oscillation of the thickness of the streamtubes. dˆη/dy is effected
by the compressibility (ρc 2 ) −1 of the fluid and by the oscillation of the length of the
elements of the streamtubes (second term on the right-hand side of Eq. (14)). The
latter is described by the responsible accelerating forces. The terms which depend
on the pressure will be omitted from the further consideration since these terms are
already included in the differential equations which describe the dynamics of the fluid
due to the action of the pressure (e. g., the Rayleigh equation (4) with the inclusion
of the compressibility effects). This includes that the velocity field is assumed to be
composed of two non-interacting parts which are exclusively governed by the pressure
and by the shear stress, respectively.
The oscillation of the Stokes layer thickness is computed by the integration of
dˆη/dy through the Stokes layer. Complying with the assumption that the Stokes
thickness δS is much smaller than the lateral extent |ℜ{α}| −1 of the wave field only
the limit |ℜ{α}|δS → 0 will be considered. The shear stress is assumed to decay from
its value ˆτw at the wall to zero at the outer edge of the Stokes layer according to
ˆτxy(y) = ˆτw · ˆ θ(ˇy) with ˇy = y/δS. Then only the term with dˆτxy/dy = (ˆτw/δS)(d ˆ θ/dˇy)
contributes to the integral and since this term is zero outside the Stokes layer the
integration can be extended to infinity without changing the result:
ˆδS := ˆητ (δS) − ˆητ (0) =
�δS
0
dˆητ iαˆτw
dy =
dy ω2 ρ
�∞
0
dˆ θ/dˇy
dˇy . (15)
[1 − αU(ˇy)/ω] 2
Thus, the amplitude ˆ δS of the oscillation of the Stokes layer thickness is proportional
to the amplitude of the wall shear stress ˆτw = −ρUɛˆvw = iωρUɛ ˆ δw, and consequently
is proportional to the amplitude ˆ δw := ˆv/iω|y=−0 of the fictitious deflection of the
wall. The deflection of the streamlines at the border of the Stokes layer is then given
by ˆ δw + ˆ δS. So the ratio | ˆ δS|/| ˆ δw| can be used to depict the influence of the shear
stress on the boundary condition at the wall.
In order to estimate ˆ θ(ˇy), and with it the integral on the right-hand side of Eq. (15),
we follow the common assumption that the relation between the Reynolds stress
tensor and the respective flow field is local and can be described by the so called
eddy viscosity µ t. For the mean flow through a circular pipe the wall of which is
rough and is assumed to be a fair equivalent to the pervious wall [19], one finds
τ ′′ /(du/dy) =: µ t = ρUµ · (y + y0), wherein Uµ = auU and y0 = ayR (here
au = 0.044, ay = 0.017) are fitted to the mean wall shear stress τ w and the average
mean flow velocity U. Unfortunately much less is known about the relation between
�τ ′′ and �u. Hartmann [46] has investigated the propagation of shear waves in a turbulent
channel flow at low Reynolds numbers by means of a direct numerical simulation
which he validated on the basis of comprehensive experimental data [47–51]. In accordance
with theoretical considerations he finds that the turbulence is equivalent to a
viscoelastic fluid with regard to the response of the shear stress to a shearing deformation.
However, the extrapolation of his results to the present Reynolds numbers and
frequencies yields that the time constant that characterizes the memory of the turbulence
is much shorter than the periods of the considered oscillations. So the ratio