BAIL 2006 Book of Abstracts - Institut für Numerische und ...

J. HUSSONG, N. BLEIER, V.V. RAM: The structure of the critical layer of a swirling
annular flow in transition
✬
✫
The structure of the critical layer of a swirling annular
flow in transition
J. Hussong, N. Bleier and V. Vasanta Ram
Ruhr-Universität Bochum, D-44780 Bochum, Germany
Subject of our paper falls under the heading of transition of the swirling flow
in an annulus. Two transition mechanisms with some fundamental differences,
which we call the Taylor and the Tollmien-Schlichting mechanisms, are generally
in competition with each other in this flow. The salient difference referred to
is the presence of a critical layer in the Tollmien-Schlichting mechanism. In
contrast, such a critical layer is absent in the Taylor mechanism. The focus of
attention in our work is the modification swirl causes to the structure of this
critical layer in annular flow.
The characteristic feature of the critical layer is that the propagation velocity
of the disturbance wave is the same as the local velocity in the basic flow.
Mathematically, this results in the equations for the propagation of small amplitude
disturbances exhibiting a turning point, i.e. the coefficient of a crucial
term in the governing differential equations crossing a zero value (see eg. [5],
[6], [2], [3], [7]). For this reason viscous effects gain importance in the critical
layer which may be regarded as an internal layer in the flow. The viscous effects
have to be taken properly into account in the critical layer in order to arrive at
the stability characteristics of the flow of interest undergoing transition.
Our basic flow is the fully developed flow with swirl in the uniform annular
gap between concentric circular cylinders. In the flow in this geometrical configuration,
swirl comes into existence when the axial pressure gradient driving
the flow acts in conjunction with a rotation of a cylinder about its own axis.
We restrict our attention for the present to the case when the outer cylinder is
set in rotation at an angular velocity Ωa and the inner cylinder is pulled axially
at a translational velocity Vwi. Both of these flow boundary conditions tend to
raise the critical Reynolds number of the flow. The geometrical and flow pa-
rameters in our problem are, in a self-explanatory notation: the transverse curvature
parameter ɛR = Ra−Ri
Uref x (Ra−Ri)
, the Reynolds number Re = Ra+Ri 2 ν with
Uref x = (Ra−Ri)2 � �
� dPG
Uref �,
ϕ
8µ dx the swirl parameter Sa = Uref x with Uref ϕ = ΩaRa and
the translational wall velocity parameter Twi = Vwi
Uref x .
We have approached the transition problem in this flow configuration by
conducting a modal analysis of the dynamics of the propagation of disturbances
in the basic flow in question, starting from small amplitude disturbances for
which the governing equations may be linearised. The dispersion relationship
for this linearized problem may formally be written as
F(ɛR, Re, Sa, Twi, λx, nϕ, ω) = 0,
Speaker: RAM, V.V. 37 BAIL 2006
✩
✪