Abstracts - KTH Mechanics

Leaky Waves in Supersonic Boundary Layer Flow
J. O. Pralits ∗ , F. Giannetti ∗ and P. Luchini ∗
Linear stability analysis of compressible boundary layer flow is commonly investigated
by solving the Orr-Sommerfeld equations (OSE), either in a temporal or spatial
framework. The mode structure of the OSE is in both cases composed of a finite number
of discrete modes which decay at infinity in the wall-normal direction y, anda
continuous spectrum of propagating modes behaving as exp(±ky) wheny →∞,with
k being real. The number of discrete modes changes with the Reynolds number (Re)
and they further seem to disappear behind the continuous spectrum at certain values
of Re. This behaviour can be visualised by tracing the trajectory of the damped
discrete modes as Re is decreased. In certain problems, such as e.g. leading-edge
receptivity of the boundary layer to external disturbances, it is of importance not
only to capture the evolution of the least stable mode as the Reynolds number is
decreased, but also of the additional damped discrete modes. This is especially important
in supersonic boundary layers where a number of modes become neutral at
nearby locations. In order to enable such computation it is of interest to investigate
if an all-discrete representation of the solution is possible.
This is here done solving the response of the boundary layer forced instantaneously
in space and time. Since the solution of the forced and homogeneous Laplace transformed
problem both depend on the free stream boundary conditions, it is shown
here that an opportune change of variables can remove the branch cuts in the complex
frequency plane. As a result integration of the inversed Laplace transform along
the new path corresponding to the continuous spectrum, equals the summation of
residues corresponding to new discrete eigen values. These new modes are computed
accounting for solutions which grow in the y-direction, and a similar problem is found
in the theory of waveguides, e.g. optical fibers, where so called leaky waves 1 are attenuated
in the direction of the wave-guide, while they grow unbounded perpendicular
to it.
The above analysis was previously performed for incompressible flat plate boundarylayer
flows 23 . There it was shown that a discrete representation of the solution of
the OSE is possible accounting for solutions which grow in the wall-normal direction
to the flat plate, and that an analytical continuation in the complex frequency
plane of the damped discrete modes is obtained. The all-discrete representation obtained
here from the study of supersonic boundary layers can be used to clarify some
mechanisms proposed by past authors 4567 to explain the receptivity of supersonic
boundary layers to acoustic disturbances.
∗ DIMEC, Universitá di Salerno, 84084 Fisciano (SA), Italy
1 Marcuse, Theory of dielectric optical waveguides, Academic press, inc. (1991).
2 Pralits and Luchini, Proc., XVII AIMeTA Congress of Theor. and Appl. Mech., Firenze (2005).
3 Pralits and Luchini, APS, Division of Fluid Dynamics, Chicago, IL (2005).
4 Fedorov and Khokhlov, Fluid Dyn. No. 9, 456 (1991).
5 Fedorov and Khokhlov, ASME Fluid Eng. Conf., Washington FED-Vol. 151, 1 (1993).
6 Fedorov, J. Fluid Mech. 491, 101 (2003).
7 Ma and Zhong, J. Fluid Mech. 488, 31, and 79 (2003).
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