Abstracts - KTH Mechanics

Application of the ray-tracing theory to the stability analysis
of three-dimensional compressible boundary layers
R. S. Donelli ∗ ,G.Mazzotti ∗ , P. Luchini †
The traditional approach to transition prediction is based on the linear stability
analysis of viscous flows, generally treated as thin shear layers, consisting in determining
the evolution, in space or in time, of small perturbations superimposed to a basic
flow field 1 , 2 . The transition location is individuated by using the e N method 3 .The
disturbance growth rate is computed in the hypothesis of constant spanwise wavenumber
and frequency, then it is, traditionally, integrated along an assigned path, generally
coincident with the chordwise flow direction, allowing the computation of the N
factor curves.
The analysis is repeated for a wide range of spanwise wavenumbers (or wavelengths,
depending on the approach) and frequencies to select the most amplified
disturbances. Laminar turbulent transition is assumed to take place where the most
unstable disturbances are amplified by a factor e N ,withN determined by correlation
with experiments.
In the present work, a different approach has been introduced to individuate the
transition location. The linear instability of steady compressible laminar boundary
layers, developing on 3D tapered swept wings, has been approached in the framework
of the theory of ray tracing in non-homogeneous anisotropic dispersive wave
systems 4 , 5 . The stability analysis permits, with this approach, to take into account
three-dimensional effects because the N factor curves are, here, the result of the
integration of the disturbance growth rate along the ray of propagation of the disturbances.
The stability analysis is performed starting from points located on the
neutral curve, changing initial spanwise wavenumbers and frequencies. The aim is to
search the directions that show the maximum amplification of the disturbances. This
investigation allows to build iso-N factor curves on the wing surface. As result of this
analysis a map of the transition locations on the wing will be achieved. A second
map built starting from a given N factor at transition will show the upstream regions
that strongly affect the transition. The knowledge of this type of maps allows to
individuate regions of the wing particularly critical for the transition of the boundary
layer from laminar to turbulent because particularly sensitive to disturbances. This
information is extremely important for the design and construction of a laminar wing
since these critical regions require particular care if laminar flow is desired.
∗ CIRA Centro Italiano Ricerche Aerospaziali, 81043 Capua, Italy
† Dipartimento di Ingegneria Meccanica, Universita di Salerno, 84040, Fisciano.
1 Mack, AGARD 709, 51 (1984).
2 Schlichting, McGraw-hill Book 7 th ed., (1979).
3 Van Ingen et all, Douglas Aircraft Co. rept. Es 26388 , (1956)
4 Donelli et all, AIDAA proceeding vol. 2, 1, (1995)
5 Grea et all, Phis. Rev. E. submitted (2005)
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