The cusp map in the complex-frequency plane for absolute instabilities

Abstract

It is well known that absolute instabilities can be located by prescribed mappings from the complex frequency plane to the wave-number plane through the dispersion relation D(w, k) = 0. However, in many systems of physical interest the dispersion relation is polynominal in w while transcendental in k, and the implementation of this mapping procedure is particularly difficult. If one maps consecutive deformations of the Fourier integral path (originally along the real k-axis) into the w-plane, points having (8D/8k) = 0 are readily detected by the distinctive feature of their local maps. It is shown that a simple topological relationship between these points and the image of the real k-axis determines the stability characteristics of the system, without mapping from the w-plane back into the k-plane.

An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

“…The saddle point therefore satisfies the Briggs criterion for contributing to absolute instability (the ω-cusp [6] at the α-pinchpoint is shown in figure 1b). This result was verified further by a so-called ray analysis (e.g.…”

“…This criterion has been extended to fluid dynamics by a number of authors [3,[5][6][7][8]. The approach is based on a local eigenvalue analysis of the OS equation, which forms the key ingredient in global analyses: (i) a necessary condition for absolute instability is if the imaginary part of the frequency is positive at a saddle point in the complex α-plane: ω i0 := ω i (α 0 ) > 0, where α 0 solves dω/dα = 0; (ii) to obtain a sufficient condition for absolute instability, the saddle point α 0 in the complex α-plane must be the result of the coalescence of spatial branches that originate from opposite half-planes at a larger and positive value of ω i (this coalescence or 'pinching' of the spatial branches is accompanied by the formation of a cusp at ω i0 in the complex ω plane [6]). Typically, the saddle-point and Briggs criteria are checked using a fast numerical eigenvalue solver [9,10].…”

Reduced order modeling of reacting shear flow

“…Therefore, these two equations completely describe the POD reduced order model, and it is possible to simulate the flow field. Figure 13 shows a comparison between the actual data and the POD reconstruction using 4, 6 and 10 modes, and show that the reduced order model in (22) and (23) is reasonably accurate. The model behavior at two different points in the flow is presented in the figure, and corresponds to the anti-node and the node of the first mode.…”

“…Obviously, due to the assumption that the underlying flow contains only one dominant mode, the results in section II yielded only the most dominant Strouhal number. Equation (22) represents the flow field as the combination of the basis functions, i Φ , and equation (23) determines the coefficients of these functions in time. Therefore, these two equations completely describe the POD reduced order model, and it is possible to simulate the flow field.…”

“…The complex frequency of the absolute mode, determined by Eq. (11), is evaluated using the cusp-map method [21,22]. The following velocity profile is used to model a separating flow downstream a backward facing step; (16) where β is the ratio of backflow to the forward flow, δ defines the shear layer thickness, and w δ is the boundary layer thickness at the wall.…”