Oscillations, Waves, and Interactions - GWDG

92 D. Ronneberger et al.
As usual the oscillating part of the flow field is decomposed into independent modes
each of which is characterized by the cross-sectional distribution of the oscillating
quantities {û, ˆv, ˆp}n and by the dispersion relation ω = Ω n(α); one such example has
already been discussed in connection with the Rayleigh equation (4). In the case of
uniform mean flow (‘flat’ velocity profile) the whole set of the modes is a complete
base for the description of the unsteady part of the field in the duct. A further reason
why a flat velocity profile is assumed in many theoretical studies is the advantage
that both, the mean and the oscillatory part of the flow have a potential in this case,
and that the distributions of the velocity and the pressure {û, ˆv, ˆp}n can be described
by well-known analytic functions in simple cross-sectional geometries like the 2D and
the circular channel. The eigenvalue problem by which Ω n(α) and {û, ˆv, ˆp}n are
determined is then reduced to the solution of a transcendental equation.
4.1 Disregarding of shear stress and uniform mean flow
4.1.1 Boundary condition at the wall
The flat velocity profile exhibits a discontinuity at the wall, i. e., the boundary layer
is replaced by an infinitesimal vortex layer. So it is not trivial which of the unsteady
quantities remain continuous when passing through the vortex layer and how, consequently,
the boundary condition, i. e. the acoustical admittance of the wall, is to
be translated to the field in the interior of the channel. In any case the pressure has
to be continuous since otherwise the lateral pressure gradient and the lateral acceleration
of the fluid elements would tend to infinity. In addition the lateral deflection
of the fluid elements on both sides of the vortex layer are assumed to be equal. This
assumption however is justified only in the absence of shear stress as will be shown
in Sect. 4.2.1.
The unsteady deflection of the wall and of the vortex layer, which sticks to the
wall, so to speak, can be clearly defined. However the extension of this term to
the flow field is problematic even if an infinitesimal oscillatory flow is superimposed
on the mean flow. The vortex layer can be described as a set of streamlines which
are determined by the integration of the field of flow directions. Thus the unsteady
deflection of a streamline generally depends on the arbitrary starting point of the integration.
To avoid this problem some local information about the temporal and the
spatial development of the oscillating streamline is assumed to exist such that the relation
between the transverse component v ′ of the flow velocity and the hypothetical
deflection η ′ of the streamline can be used to define the deflection
v ′ (x, t) = ∂η′ (x, t)
∂ t
+ U(x) ∂η′ (x, t)
∂ x
=: Dη′ (x, t)
Dt
. (10)
Herein x = (x, y, z) t is the 3D position vector, and the x-direction is aligned with
the mean-flow direction (u(x) = [U(y, z), 0, 0] t ). Then v ′ and η ′ are 2D vectors in
the (y, z)-plane. The abbreviation D/Dt := ∂/∂ t + U∂/∂ x stands for the temporal
derivative in a frame of reference that is moving at velocity U. Thus, from Eq. (10)
and from the assumption that the deflection η ′ of the streamlines is solenoidal, i. e.
that the compressibility of the fluid can be ignored on the infinitesimal scale of the