Abstract:In this paper, we have developed a new approach based on a combination of the Arnoldi and frontal methods, which is suitable for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigen-solution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computati… Show more
“…These points agree well with the neutral stability curve of Hao et al. (2016), and the slight differences are attributed to the difference between the two constitutive laws (the critical Reynolds number is displaced by less than 1 % for all points tested). In general, the lower static branch becomes unstable as the Reynolds number increases (similar to Heil 2004; Stewart 2017; Herrada et al.…”
Section: Resultssupporting
confidence: 90%
“…(2008) (see also Hao et al. 2016) showed that this neutral stability curve is non-monotonic and for large the system restabilises again as the Reynolds number becomes sufficiently large, although we did not investigate this regime. Figure 5.An overview of the parameter space spanned by the Reynolds number and extensional stiffness, summarising the steady and unsteady solutions of the model.…”
Section: Resultsmentioning
confidence: 99%
“…These predictions were updated slightly by Hao et al. (2016) using a global stability eigensolver, and their neutral stability curve is shown as a solid black line in figure 5. The corresponding neutral stability points from the lower static branch computed using our numerical method are shown as filled black circles.…”
Section: Resultsmentioning
confidence: 99%
“…At each timestep, a frontal method is used to assemble the global matrix equation and a Newton–Raphson iteration scheme is applied to solve the global matrix equation for (Luo & Pedley 1996; Hao et al. 2016). The numerical method enables us to evaluate the terms in the fully nonlinear energy budget in (2.36) by post-processing.…”
Section: The Modelmentioning
confidence: 99%
“…2016) or prescribed upstream pressure (Liu, Luo & Cai 2012; Hao et al. 2016). In this study we focus on the flow driven system, where it has been shown that the system admits multiple families of self-excited oscillations in a novel cascade structure: the system has isolated regions of stability between unstable regions which each correspond to a different number of extrema in the wall profile (Luo et al.…”
We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.
“…These points agree well with the neutral stability curve of Hao et al. (2016), and the slight differences are attributed to the difference between the two constitutive laws (the critical Reynolds number is displaced by less than 1 % for all points tested). In general, the lower static branch becomes unstable as the Reynolds number increases (similar to Heil 2004; Stewart 2017; Herrada et al.…”
Section: Resultssupporting
confidence: 90%
“…(2008) (see also Hao et al. 2016) showed that this neutral stability curve is non-monotonic and for large the system restabilises again as the Reynolds number becomes sufficiently large, although we did not investigate this regime. Figure 5.An overview of the parameter space spanned by the Reynolds number and extensional stiffness, summarising the steady and unsteady solutions of the model.…”
Section: Resultsmentioning
confidence: 99%
“…These predictions were updated slightly by Hao et al. (2016) using a global stability eigensolver, and their neutral stability curve is shown as a solid black line in figure 5. The corresponding neutral stability points from the lower static branch computed using our numerical method are shown as filled black circles.…”
Section: Resultsmentioning
confidence: 99%
“…At each timestep, a frontal method is used to assemble the global matrix equation and a Newton–Raphson iteration scheme is applied to solve the global matrix equation for (Luo & Pedley 1996; Hao et al. 2016). The numerical method enables us to evaluate the terms in the fully nonlinear energy budget in (2.36) by post-processing.…”
Section: The Modelmentioning
confidence: 99%
“…2016) or prescribed upstream pressure (Liu, Luo & Cai 2012; Hao et al. 2016). In this study we focus on the flow driven system, where it has been shown that the system admits multiple families of self-excited oscillations in a novel cascade structure: the system has isolated regions of stability between unstable regions which each correspond to a different number of extrema in the wall profile (Luo et al.…”
We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.
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