Abstract
The initial stage of the laminar–turbulent transition of semi-infinite flows can be characterized as either an absolute or convective instability, naturally associated with localized wave packets. A convective instability is directly linked to an absolute instability in a different reference frame. Therefore, our aim is to determine the absolute stability of a flow in a given but arbitrary reference frame, which can only be directly inferred from the absolute eigenvalue spectrum. If advective processes are present, the associated absolute eigenfunctions grow exponentially in space in the advective direction. The eigenvalue spectrum is usually computed numerically, which requires truncating the domain and prescribing artificial boundary conditions at these truncation boundaries. For separated boundary conditions, the resulting spectrum approaches the absolute spectrum as the domain length tends to infinity. Since advective processes result in spatially exponentially growing eigenfunctions, it becomes increasingly difficult to represent these functions numerically as the domain length increases. Hence, a naive numerical implementation of the eigenvalue problem may result in a computed spectrum that strongly deviates from the (mathematically correct) absolute spectrum due to numerical errors. To overcome these numerical inaccuracies, we employ a weighted method ensuring the convergence to the absolute spectrum. From a physical point of view, this method removes the advection-induced spatial exponential growth from the eigenfunctions. The resulting (absolute) spectrum allows for a direct interpretation of the character of the pertinent perturbations and the eigensolutions can be used to construct and analyse the evolution of localized wave packets in an efficient way.
Highlights
Classifying the growth and the character of unstable perturbations on wall-bounded flows is the central goal of linear stability theory in aerospace applications
Using this definition of absolute instability, it follows that, in the selected fixed reference frame, the flow is: (I) absolutely unstable, if at least one such wave packet grows in time; (II) absolutely stable, if all such wave packets decay in time; (III) convectively unstable, if the flow is absolutely stable in the fixed reference frame, but absolutely unstable in at least one other reference frame; (IV) absolutely and convectively stable, if there is no reference frame in which the flow is absolutely unstable
Sandstede and Scheel [12] prove that the use of any type of separated (Dirichlet, Neumann or Robin type) boundary condition yields a spectrum that dictates the absolute stability as L → ∞
Summary
Classifying the growth and the character of unstable perturbations on wall-bounded flows is the central goal of linear stability theory in aerospace applications. For flows on a semi-infinite domain, the stability is assessed via perturbations with a finite support; wave packets having a front in the up- and downstream direction. Absolute instability is determined by the amplitude evolution of a wave packet whose up- and downstream fronts propagate toward infinity in their respective direction Note that this requires selecting a fixed reference. The bridge to extract the stability characteristics of wave packets from continuum modes is provided by applying Briggs’s method, as described in the detailed treatments by Briggs [7]; Huerre and Monkewitz [8]; Brevdo [9]; Schmid and Henningson [5].
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