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

J. MAUSS, J. COUSTEIX: Global Interactive Boundary Layer (GIBL) for a Channel
✬
✫
Global Interactive Boundary Layer (GIBL) for a Channel
J. Mauss ♦ and J. Cousteix ♣
♦ IMFT and UPS, 118 route de Narbonne, 31062 Toulouse Cedex - France
Phone: 05 61 55 67 94 - mauss@cict.fr
♣ ONERA and SUPAERO, 2 av. É. Belin, 31055 Toulouse Cedex - France
Phone: 05 62 25 25 80 - Jean.Cousteix@onecert.fr
We consider a laminar, steady, two-dimensional flow of an incompressible Newtonian fluid
in a channel at high Reynolds number. When the walls are slightly deformed, adverse pressure
gradients are generated and separation can occur. The analysis of the flow structure has been
done essentially by Smith [4]. Later, a systematic asymptotic analysis has been performed by
Saintlos and Mauss [3]. With the Successive Complementary Expansion Method, SCEM, we
assume a uniformly valid approximation (UVA) based on generalised expansions. This method,
developed by Cousteix and Mauss [1], has been used by Dechaume et al. [2].
Navier-Stokes dimensionless equations can be written
div −→ V =0, (grad −→ V ) · −→ V = − grad Π+ 1
Re △ −→ V . (1)
The basic plane Poiseuille flow is
v (x) = u0 = y − y 2 , v (y) =0, Π=Π0 = − 2x
Re + p0 . (2)
The flow is perturbed, for instance, by indentations of the lower and upper walls such as
yl = εF (x, ε) , yu =1− εG(x, ε) , (3)
where ε is a small parameter. If we seek a solution in the form
v (x) = u0(y)+εu(x, y, ε) , v (y) = εv(x, y, ε) , Π − p0 = − 2x
the Navier-Stokes equations become
+ εp(x, y, ε) ,
Re
(4)
∂u ∂v
+ = 0 , (5a)
∂x ∂y
�
ε u ∂u
�
�
∂u ∂u du0 ∂p 1 ∂2u + v + u0 + v = − +
∂x ∂y ∂x dy ∂x Re ∂x2 + ∂2u ∂y2 �
, (5b)
�
ε u ∂v
�
�
∂v ∂v
1 ∂2v + v + u0 = −∂p +
∂x ∂y ∂x ∂y Re ∂x2 + ∂2v ∂y2 �
. (5c)
It is clear that, for high Reynolds numbers, the reduced equations are of first order leading
to a singular perturbation. In the core flow, we are looking for approximations coming from
asymptotic generalised expansions such as
u = u1(x, y, ε)+··· , v = v1(x, y, ε)+··· , p = p1(x, y, ε)+··· . (6)
Formally, neglecting terms of order O(ε, 1/Re) , for the core flow, we obtain
∂u1
∂x
+ ∂v1
∂y
∂u1 du0
u0 + v1
∂x dy
∂v1
u0
∂x
1
= 0 , (7a)
= −∂p1
∂x
= −∂p1
∂y
, (7b)
. (7c)
Speaker: MAUSS, J. 116 BAIL 2006
✩
✪