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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Figure 1: Discrete global spectrum from a convectively
instable boundary layer. The real
part of the eigenvalue is represented on the Xaxis
and the imaginary part on the Y-axis.
Figure 2: Real part of the horizontal velocity
eigenfunction of a discrete eigenvalue from the
spectrum ( Fig. 1).
with U(x, y) =[U, V ] t and û(x, y) =[û, ˆv] t . The boundary conditions on velocity disturbances
and constraints on pressure (T. N. Phillips & G. W. Roberts (1993), C. Canuto et al. (1988))
complete the eigenvalue value problem. The partial differential stability equations (2) are discretized
using an algorithm based on the collocation method based on Chebyshev Gauss-Lobatto
grid. The algebraic eigenvalue problem (A − ΩB) X = 0 is solved by the QZ algorithm.
4. Two-Dimensional Temporal Modes, results and perspectives
The figure 1 represents a global spectrum of the boundary layer. The stable discrete eigenvalues
appear in concordance with the fact a boundary layer is globally stable. These discrete values
represent spatio-temporal convective modes as it can been shown fig. 2 (structure similar to a
spatial exponential growth). At the conference, this analysis will be also applied on a separated
incompressible boundary layer which is not absolutely unstable but only convectively unstable.
References
[1] M. Gaster , “The development of a two-dimensional wavepacket in a growing boundary
layer ”, Proc. R. Soc. London, 3, 317–332 (1982).
[2] Lin R.S. and Malik M.R , “On the stability of attachment-line boundary layers. ”, J. Fluid
Mech., 3,239–255 (1995).
[3] Theofilis V. Hein S and Dallmann U , “On the origins of unsteadiness and three dimensionality
in a laminar separation bubble ”, Proc. R. Soc. London, 3,3229–3246 (2000).
[4] U. Ehrenstein and F. Gallaire , “On two dimensional temporal modes in spatially evolving
open flows: the flat-plate boundary layer ”, J. Fluid Mech., (2005).
[5] Briley WR., “A numerical study of laminar separation bubbles using the Navier-Stokes
equations.”, J. Fluid Mech., 3,713–736 (1971).
[6] Timothy N. Phillips and Gareth W.Roberts, “The Treatment of Spurious Pressure Modes in
Spectral Incompressible Flow Calculations.”, Journal of Computational Physics, 3,150–164
(1993).
[7] C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang, “Spectral Methods in
Fluids Dynamics”, Springer, (1988).
Speaker: ALIZARD, F. 68 BAIL 2006
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