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

F. ALIZARD, J.-CH. ROBINET: Two-dimensional temporal modes in nonparallel
flows
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Two-dimensional temporal modes in nonparallel flows
F. Alizard and J.-Ch. Robinet
SINUMEF Laboratory, ENSAM-PARIS 151, Boulevard de l’Hôpital, 75013 PARIS, FRANCE
Frederic.Alizard@paris.ensam.fr,
1. Introduction
To describe the evolution of a two-dimensional wavepacket in flow such as growing boundary
layer, the classical stability approach is based on the assumption of locally parallel or weakly non
parallel base flow. These approach was used by Gaster (1982) to characterize the spatio-temporal
dynamic of a perturbation in a boundary layer. However this approach could failed if a wave
length of any perturbation is larger than a characteristic length of the spatial inhomogeneity of
the base flow. Consequently a more general eigenvalue problem was developed by some authors
as Lin & Malik (1995), Theofilis et al. (2000) and Erhenstein & Gallaire (2005) where two spatial
directions are inhomogeneous. In this abstract we will focus on two-dimensional modes for a
convectively unstable attached boundary layer. After the description of the numerical method,
preliminary results on temporal linear stability modes depending on two space directions are
computed for a boundary layer flow along a flat plat.
2. Generalities and Basic Flow
At the conference we will be interested exclusively in computations of convective instabilities
in nonparallel flows. Two types of flows will be studied: a flat plate boundary-layer without
pressure gradient, shown as a preliminary result and a separated boundary-layer. The twodimensional
Navier-Stokes equations for incompressible fluids in the stream function-vorticity
formulation are considered:
∂ω
∂ω 1
+ u∂ω + v =
∂t ∂x ∂y Re
�
∂2ω ∂x2 + ∂2ω ∂y2 �
and ∆ψ = ω, (1)
where ω and ψ are the vorticity and the stream function respectively. System (1) is closed
by classical boundary conditions on ψ and ω (Briley (1971)). A second order finite differences
scheme was used for the vorticity transport equation as well as the poisson equation of stream
function. An A.D.I algorithm has been employed to solve the transport equation and the poisson
equation. Preliminary results, shown on part 4, are realized on an attached boundary layer at
Re=610, with a grid (450x200).
3. Linearized Perturbed Flow and Numerical Procedure
The proposed stability analysis is based on the classical perturbations technique where the
instantaneous flow (q) is the superposition of the basic flow (Q), data of this problem, and
unknown fluctuations (ˆq): q(x, y, t) =Q(x, y)+ˆq(x, y)exp(−iΩt), where Ω is the circular global
frequency of the fluctuation. The two-dimensional generalized eigenvalue problem is obtained
by the linearized Navier Stokes equations:
� �
∆2d
div(û) = 0 and − U.grad û − gradU.û − gradˆp + iΩû =0, (2)
Re
1
Speaker: ALIZARD, F. 67 BAIL 2006
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