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

L. SAVIĆ, H. STEINRÜCK: Asymptotic Analysis of the mixed convection flow past a
horizontal plate near the trailing edge
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of the plate ∆A interacts not only via the potential flow in the upper deck but also via the
hydrostatic pressure in the main deck with the pressure ∆p in the lower deck. The interaction
law for the pressure difference between the upper and lower side can be written in the form
∆A ′ (x (3) ) + √ 3ash(x (3) �
) x (3)� 1/3
=
� � 0 1 ∆ˆp(ξ) + 2as|ξ|
−
π −∞
1/3
x (3) dξ −
− ξ
1
� ∞ As(ξ) − ash(ξ)|ξ|
π −∞
1/3
x (3) �
dξ
− ξ
Where As(x (3) ) is the displacement thickness obtained for the classical trailing edge problem
[3, 1] and x (3) is the coordinate in the lower, main and upper deck parallel to the plate. Here as
is a constant and h(.) the heayside function. It turns out that the difference pressure ∆p has a
discontinuity at the trailing edge. Thus new sub-layers (in the main and lower deck) to resolve
this discontinuity are introduced.
In the following table we give an overview of the most important layers needed for the
asymptotic analysis. We introduce the following notation according to the stretching factor as
a power of the Reynolds number: x (α) = Re α/8 x and y (β) = Re β/8 y.
α β flow region
0 0 potential flow region
0 4 boundary layer and wake
3 3 upper deck
3 4 main deck
3 5 lower deck
4 4 main deck - trailing edge
5 5 lower deck - trailing edge
Table 1: Scales of the different flow regions
The new results of the analysis will be the local behavior of the difference pressure at the
trailing edge. For both of the newly introduced sub-layers elliptic equations for the local pressure
variation can be derived and will be solved numerically. Thus on triple deck scales there will be
a flow around the trailing edge. Thus the perturbation of the classical triple deck problem by
small buoyancy effects behaves quite differently than by small angles of attack [3].
References
[1] K. Stewartson, On the flow near the trailing edge of flat plate, Mathematika, 16, 106-121
(1969).
[2] A.F. Messiter, Boundary layer flow near the trailing edge of flat plate, SIAM J. Appl. Math.,
18, 241-257, (1970).
[3] R. Chow and R. E. Melnik, Numerical Solutions of Triple-Deck Equations for Laminar
Trailing Edge Stall, Grumman Research Dept., Report RE-526J (1976).
[4] W. Schneider and M. G. Wasel, Breakdown of the boundary-layer approximation for mixed
convection flow above a horizontal plate, Int. J. Heat Transfer, 28, 2307-2313 (1985).
[5] Lj. Savić and H. Steinrück, Mixed convection flow past a horizontal plate, Theoretical and
Applied Mechanics, 32, 1-19 (2005).
Speaker: STEINRÜCK, H. 42 BAIL 2006
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