Abstracts - KTH Mechanics

Oscillating growth in convection dominated systems
G. Coppola ∗ , L. de Luca ∗
In the context of the linear stability analysis of non parallel open flows, such as
wakes, jets, boundary layers and separation bubbles, the concept of global modes has
been employed in order to describe the dynamics of the flow disturbances. By considering
the model problem of the linearized Ginzburg Landau equation with spatially
varying coefficients, Cossu and Chomaz 1 firstly demonstrated that the simultaneous
excitation of different global modes in a non normal system can be advocated in order
to describe wave packets convectively propagating in open flow systems. In recent
papers these concepts have been applied to real flows problems such as the stability
of a falling liquid curtain 2 , and the two dimensional temporal analysis of boundary
layer flows along a flat plate 3 .
The case illustrated by the model of the falling curtain presents a situation in which
the non normal character of the operator involved in the global analysis produces
a transient evolution that is characterized by a periodic oscillating pattern, which
appears to be as important for the physical description of the system as the occurrence
of the growth itself. However, while the relations between non normality of a linear
operator and the transient growth of its solution operator norm has been a wide
theme of research in hydrodynamic stability in the past years, the theme of oscillating
transient growth and of its relations with non normality of the operator has received
less attention.
In this work, the study of oscillating norm behavior in linear evolutionary systems
is addressed. A link between the occurrence of oscillating patterns in the energy evolution
of the solutions and the nonorthogonality of linear global modes is illustrated;
moreover, a distinction between transient and asymptotic oscillations is made. The
analysis carried out shows that special frequency signatures associated to transient
oscillations can be explained by means of suitable “synchronizations” between global
modes, and some frequency selection rules are also proposed. The physical importance
of such oscillating behaviors is stressed by reconsidering the model for the linear
stability of the falling liquid curtain studied by Schmid and Henningson and by considering
different physical models arising in the context of the linearized formulation
of some convection dominated systems over finite length domains.
∗ DETEC, Univ. di Napoli ‘Federico II’, Italy.
1 Cossu and Chomaz Phys. Rev. Lett. 78, 4387 (1997).
2 Schmid and Henningson J. Fluid Mech. 463, 163 (2002).
3 Ehrenstein and Gallaire J. Fluid Mech. 536, 209 (2005).
3