The object of this study is a steady two-dimensional incompressible flow
past a rapidly rotating cylinder with suction. The rotation velocity is
assumed to be large enough as compared with the cross-flow velocity at
infinity to ensure that there is no separation. High Reynolds number
asymptotic analysis of incompressible Navier-Stokes equations is performed.
Classical Prandtl's approach of subdividing the flow field into two regions,
the outer inviscid region and the boundary layer, was used earlier by
Glauert (1957) for analysis of similar flow without suction. Glauert found
that the periodicity of the boundary layer allows to find uniquely the
velocity circulation around the cylinder. In the present study it is shown
that the periodicity condition does not give a unique solution for suction
velocity much greater than 1/Re. It is found that in fact these non-unique
solutions correspond to different exponentially small upstream vorticity,
which cannot be distinguished from zero considering only a few power terms in
large Reynolds number asymptotic expansion. Unique solutions are constructed
for the flow regimes with suction velocity $V_w$ being an order one quantity,
for $V_w = O(1/Re)$ as well as for $V_w = O(1/\sqrt{Re}). In the latter case
an explicit analysis of the distribution of exponentially small vorticity
outside the boundary layer was carried out.
A new `semi-direct' method to solve viscous-inviscid interaction problems for
high Reynolds number separated flows is developed. Both supersonic and
subsonic flow separation may be studied using this technique. The method is
based upon the vorticity and stream function formulation. It is fully implicit
with respect to the vorticity equation and `interaction law' which describes
the mutual interdependence of the viscous layer near the body surface and the
rest of the flow. The main idea of this approach consists in taking advantage
of the particular structure of the governing equations which allows the entire
flow field to be solved simultaneously by using the Thomas Matrix technique.
The method had better numerical stability characteristics than most of the
traditional techniques and was also faster than many other techniques developed
before.
In this paper the method is used for solving the classical problem of the
boundary-layer separation in compression ramp flow. Supersonic and subsonic
versions of the problem have been studied. In both cases the semi-direct method
allows calculation of flow regimes with extended separation regions
corresponding to large ramp angles which could not be analysed using other
methods.
The separation of the laminar boundary layer from a convex corner on a rigid
body contour in transonic flow is studied based on the asymptotic analysis of
the Navier-Stokes equations at large values of the Reynolds number. It is shown
that the flow in a small vicinity of the separation point is governed, as
usual, by strong interaction between the boundary layer and inviscid part of
the flow. Outside the interaction region the Karman-Guderley equation
describing transonic inviscid flow admits a self-similar solution with the
pressure on the body surface being proportional to the cubic root of the
distance from the separation point. Analysis of the boundary layer driven by
this pressure shows that as the interaction region is approached the boundary
layer splits into two parts, the near-wall viscous sublayer and the main body
of the boundary layer where the flow is locally inviscid. It is interesting
that contrary to what happens in subsonic and supersonic flows, the
displacement effect of the boundary layer is primarily due to the inviscid
part. The contribution of the viscous sublayer proves to be negligible to the
leading order. Consequently, the flow in the interaction region is governed by
the inviscid-inviscid interaction. To describe this flow one needs to
solve the Karman-Guderley equation for the potential flow region outside
the boundary layer; the solution in the main part of the boundary layer was
found in an analytical form, thanks to which the interaction between the
boundary layer and external flow might be expressed via the corresponding
boundary condition for the Karman-Guderley equation. Formulation of the
interaction problem involves one similarity parameter which in essence is the
Karman-Guderley parameter suitably modified for the flow at hand. The
solution of the interaction problem has been constructed numerically.
Laminar boundary-layer separation in the supersonic flow past a corner point
on a rigid body contour, also termed the compression ramp, is considered based
on the viscous-inviscid interaction concept. The `triple-deck model' is used to
describe the interaction process. The governing equations of the interaction
may be formally derived from the Navier-Stokes equations if the ramp angle
$\theta $ is represented as $\theta = \theta _0 Re^{-1/4}$, where $\theta _0$
is an order one quantity and $Re$ is the Reynolds assumed large. To solve the
interaction problem two numerical methods have been used. The first method
employs a finite-difference approximation of the governing equations both with
respect to the streamwise and wall normal coordinates. The resulting algebraic
equations are linearized using Newton-Raphson strategy and then solved with the
Thomas-matrix technique. The second method uses finite-differences in the
streamwise direction in combination with Chebychev collocation in the normal
direction and Newton-Raphson linearisation.
Our main concern is with the flow behaviour at large values of $\theta _0$.
The calculations show that as the ramp angle $\theta _0$ increases additional
eddies form near the corner point inside the separation region. The behaviour
of the solution does not give any indication that there exists a critical value
$\theta _0^{\ast }$ of the ramp angle $\theta _0$, as suggested by
Smith & Khorrami (1991) who claimed that as $\theta _0$ approaches
$\theta _0^{\ast }$ a singularity develops near the reattachment point
preventing the continuation of the solution beyond $\theta _0^{\ast }$.
Instead we find that the solution is in good agreement with Neiland's (1970)
theory of reattachment; the latter does not involve
any restriction upon the ramp angle.
Interaction between the boundary layer on a smooth body surface and the outer
inviscid compressible flow in the vicinity of a sonic point is considered.
First, a local self-similar solution of the inviscid transonic Karman-Guderley
equation in the vicinity of the sonic point on the smooth body surface is
constructed. It is found that pressure has a singularity near this point and
the pressure gradient behaves like $dp_w/dx \sim (-x)^{-1/3}$ as $x \rightarrow
0^-$ here. Acting on the boundary layer occurring on a rigid body surface,
such a singular pressure gradient leads to the interaction between the boundary
layer and the inviscid flow. It is shown that current type of pressure
distribution is associated with logarithmic upstream skin friction behaviour
different from other interaction problems known nowadays. The corresponding
interaction problem is formulated and solved numerically for the case when a
separation zone is localized inside the viscous sublayer and the inviscid part
of the flow is supersonic. Solution is proved to be non-unique. It has three
branches and shows hysteresis-like dependence on the problem governing
parameter.
A semi-direct method for calculating
flows with viscous-inviscid interaction.
Abstract
On laminar separation at a corner point in transonic flow.
Abstract
Once again on the supersonic flow separation near a corner.
Abstract
On the transonic viscous-inviscid interaction.
Abstract