Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
5.4 Derivation of Wall-laws 155
which ends the proof for the a priori estimates. Again very weak estimates give:
RεL2 (Ω0) ≤ RεL2 (Γ1∪Γ0) +ε|kMk|
·
β −β
ε L2 (Ω0)
≤ c
≤ cε 3
2 .
5.4 Derivation of Wall-laws
5.4.1 Averaging the ansatz
e−1 ε + √ ε∇RεL2 (Ωε\Ω0) +ε3
2
We aim to derive a system of equations defined on the smooth domain Ω0, for which
the effect of the roughness is included as a macroscopic boundary condition on Γ0. First,
averaging wrt the fast variable in the horizontal direction, we get:
Ûǫ,k = û0,k +εû1,k := uε,k.
Though, the averaging process cancels the oscillations, the averaged ansatz still contains
a first order macroscopic correction û1,k accounting for averaged first order effects. This
new averaged quantity solves a problem in the smooth limiting domain Ω0:
⎧
⎪⎨
⎪⎩
Lkuε,k = Ĉk
in Ω0,
uε,k = 0 on Γ1,
uε,k = εMkβ on Γ0,
uε,k is x1-periodic on Γin ∪Γout.
(5.4.1)
We compute the L 2 -error estimate between the exact solution ûǫ,k of problem (5.3.1) and
the averaged first order approximation uε,k.
Proposition 3. There exists one positive constant c5 , depending only of the mode Ĉk
such that:
ûǫ,k −uε,k L 2 (Ω0) ≤ c5ε 3/2 . (5.4.2)
Proof. We write a triangular inequality:
ûǫ,k −uε,k L 2 (Ω0) ≤ ûǫ,k − Ûǫ,k L 2 (Ω0) + Ûǫ,k −uε,k L 2 (Ω0).
The second term in the rhs is explicit :
Ûǫ,k −uε,k = εMk
β
x
−β .
ε