BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... 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 ✩ ✪
J. MAUSS, J. COUSTEIX: Global Interactive Boundary Layer (GIBL) for a Channel ✬ ✫ It is very interesting to observe the singular behaviour of the solution of (7a–7c) as we approach the boundaries. For instance, when y → 0, we have u1 = −2p10 ln y + c10 + ··· , where p10 and c10 are functions of x and ε. The generalised asymptotic expansions for the velocity are given by v (x) = u0(y)+εu(x, y, ε)+··· , v (y) = εv(x, y, ε)+··· . (8) Using the SCEM, the problem consists of solving the continuity equation together with the momentum equation ∂u ∂v + = 0 , (9a) ∂x ∂y � ε u ∂u � ∂u ∂u du0 + v + u0 + v = −∂p1 ∂x ∂y ∂x dy ∂x + ε3 ∂2u . (9b) ∂y2 But, now, we have to solve simultaneously (9a–9b) and the core equations (7a–7c). The same form as Prandtl’s equations is recovered if we let ∂Π U = u0 + εu , V = εv , ∂x = −2ε3 + ε ∂p1 , (10) ∂x leading to U ∂U ∂U + V = −∂Π ∂x ∂y ∂x + ε3 ∂2U . (11) ∂y2 The boundary conditions are now U = V = 0 on the walls. As four conditions must be satisfied, it is clear that the pressure gradient must be adjusted in order to ensure the global mass flow conservation in the channel. Calculations have been performed with a simplified model coming from the triple deck theory. An example is given in Fig. 1 which gives the evolution of the skinfriction coefficient along the lower and upper walls of a channel whose lower wall is deformed. References Cf upper wall 0,0 2 Re lower wall 1.2 1.0 0.8 0.6 0.4 0.2 R =10 3 x/L -5 -4 -3 -2 -1 0 1 2 3 4 5 Figure 1: Application of GIBL in a channel whose lower wall is deformed [1] J. Cousteix and J. Mauss. Approximations of the Navier-Stokes equations for high Reynolds number flows past a solid wall. Jour. Comp. and Appl. Math., 166(1):101–122, 2004. [2] A. Dechaume, J. Cousteix, and J. Mauss. An interactive boundary layer model compared to the triple deck theory. Eur. J. of Mechanics B/Fluids, 24:439–447, 2005. [3] S. Saintlos and J. Mauss. Asymptotic modelling for separating boundary layers in a channel. Int. J. Engng. Sci., 34(2):201–211, 1996. [4] F.T. Smith. On the high Reynolds number theory of laminar flows. IMA J. Appl. Math., 28(3):207–281, 1982. Speaker: MAUSS, J. 117 BAIL 2006 2 ✩ ✪
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J. MAUSS, J. COUSTEIX: Global Interactive Bo<strong>und</strong>ary Layer (GIBL) for a Channel<br />
✬<br />
✫<br />
It is very interesting to observe the singular behaviour <strong>of</strong> the solution <strong>of</strong> (7a–7c) as we<br />
approach the bo<strong>und</strong>aries. For instance, when y → 0, we have<br />
u1 = −2p10 ln y + c10 + ··· ,<br />
where p10 and c10 are functions <strong>of</strong> x and ε.<br />
The generalised asymptotic expansions for the velocity are given by<br />
v (x) = u0(y)+εu(x, y, ε)+··· , v (y) = εv(x, y, ε)+··· . (8)<br />
Using the SCEM, the problem consists <strong>of</strong> solving the continuity equation together with the<br />
momentum equation<br />
∂u ∂v<br />
+ = 0 , (9a)<br />
∂x ∂y<br />
�<br />
ε u ∂u<br />
�<br />
∂u ∂u du0<br />
+ v + u0 + v = −∂p1<br />
∂x ∂y ∂x dy ∂x + ε3 ∂2u . (9b)<br />
∂y2 But, now, we have to solve simultaneously (9a–9b) and the core equations (7a–7c). The same<br />
form as Prandtl’s equations is recovered if we let<br />
∂Π<br />
U = u0 + εu , V = εv ,<br />
∂x = −2ε3 + ε ∂p1<br />
, (10)<br />
∂x<br />
leading to<br />
U ∂U ∂U<br />
+ V = −∂Π<br />
∂x ∂y ∂x + ε3 ∂2U . (11)<br />
∂y2 The bo<strong>und</strong>ary conditions are now U = V = 0 on the walls. As four conditions must be satisfied,<br />
it is clear that the pressure gradient must be adjusted in order to ensure the global mass flow<br />
conservation in the channel. Calculations have been performed with a simplified model coming<br />
from the triple deck theory. An example is given in Fig. 1 which gives the evolution <strong>of</strong> the skinfriction<br />
coefficient along the lower and upper walls <strong>of</strong> a channel whose lower wall is deformed.<br />
References<br />
Cf<br />
upper<br />
wall<br />
0,0<br />
2 Re<br />
lower<br />
wall<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
R =10 3<br />
x/L<br />
-5 -4 -3 -2 -1 0 1 2 3 4 5<br />
Figure 1: Application <strong>of</strong> GIBL in a channel whose lower wall is deformed<br />
[1] J. Cousteix and J. Mauss. Approximations <strong>of</strong> the Navier-Stokes equations for high Reynolds<br />
number flows past a solid wall. Jour. Comp. and Appl. Math., 166(1):101–122, 2004.<br />
[2] A. Dechaume, J. Cousteix, and J. Mauss. An interactive bo<strong>und</strong>ary layer model compared to<br />
the triple deck theory. Eur. J. <strong>of</strong> Mechanics B/Fluids, 24:439–447, 2005.<br />
[3] S. Saintlos and J. Mauss. Asymptotic modelling for separating bo<strong>und</strong>ary layers in a channel.<br />
Int. J. Engng. Sci., 34(2):201–211, 1996.<br />
[4] F.T. Smith. On the high Reynolds number theory <strong>of</strong> laminar flows. IMA J. Appl. Math.,<br />
28(3):207–281, 1982.<br />
Speaker: MAUSS, J. 117 <strong>BAIL</strong> <strong>2006</strong><br />
2<br />
✩<br />
✪