Stability analysis of a class of unsteady nonparallel incompressible flows via separation of variables

Burde, Georgy I.; Nasibullayev, I.Sh.; Zhalij, Alexander

doi:10.1063/1.2814296

Cited by 5 publications

(4 citation statements)

References 36 publications

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“…Analytical solutions were only possible for some special cases (see also Zhalij et al 2006). The numerical solution reveals unstable modes in the latter three-dimensional flows (Burde et al 2007). Nold and Oberlack (2013) presented a unifying theory for both the normal mode and the Kelvin mode ansatz in plane shear flow.…”

Section: Introductionmentioning

confidence: 97%

“…Their method has been applied to differential equations in mathematical physics Zhalij 1999a, 1999b). Burde et al (2007) applied this method to stability problems of unsteady base flows and nonparallel flows. Using eigenfunctions with a combination of algebraic and exponential terms, the stability of a flow between two concentric cylinders was investigated.…”

Section: Introductionmentioning

confidence: 99%

“…Using eigenfunctions with a combination of algebraic and exponential terms, the stability of a flow between two concentric cylinders was investigated. Due to a time-dependent spatial coordinate, in the one case permeable cylinder walls, in the other case cylinders with time-dependent radii are examined in Burde et al (2007). Analytical solutions were only possible for some special cases (see also Zhalij et al 2006).…”

Section: Introductionmentioning

confidence: 99%

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Algebraic stability modes in rotational shear flow

Gebler¹,

Plümacher²,

Kahle³

et al. 2021

Fluid Dyn. Res.

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We investigate the two-dimensional (2D) stability of rotational shear flows in an unbounded domain. The eigenvalue problem is formulated by using a novel algebraic mode decomposition distinct from the normal modes with temporal evolution $\exp(\omega t)$. Based on the work of \citeasnoun{NoldOberlack2013}, we show how these new modes can be constructed from the symmetries of the linearized stability equation. For the azimuthal base flow velocity $V(r)=r^{-1}$ an additional symmetry exists, such that a mode with algebraic temporal evolution $t^s$ is found. $s$ refers to an eigenvalue for the algebraic growth or decay of the kinetic energy of the perturbations. An eigenvalue problem for the viscous and inviscid stability using algebraic modes is formulated on an infinite domain with $r \to \infty$. An asymptotic analysis of the eigenfunctions shows that the flow is linearly stable under 2D perturbations. We find stable modes with the algebraic mode ansatz, which can not be obtained by a normal mode analysis. The stability results are in line with Rayleigh's inflection point theorem.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic stability modes in rotational shear flow

Gebler¹,

Plümacher²,

Kahle³

et al. 2021

Fluid Dyn. Res.

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“…However numerical methods are seeing increased attention, albeit with significant restrictions on the nature of the basic state or the forcing. For example, Burde et al [32] develop techniques using separation of variables to study the linearized equations in simple unsteady incompressible flows. More recently, Towne et al [33] use LNS with "jittering", to more accurately predict the acoustic radiation of jets with the wave packet formulation.…”

Section: Introductionmentioning

confidence: 99%

A high-fidelity method to analyze perturbation evolution in turbulent flows

Nair

Gaitonde

2016

Journal of Computational Physics

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Small perturbation propagation in fluid flows is usually examined by linearizing the governing equations about a steady basic state. It is often useful, however, to study perturbation evolution in the unsteady evolving turbulent environment. Such analyses can elucidate the role of perturbations in the generation of coherent structures or the production of noise from jet turbulence. The appropriate equations are still the linearized Navier-Stokes equations, except that the linearization must be performed about the instantaneous evolving turbulent state, which forms the coefficients of the linearized equations. This is a far more difficult problem since in addition to the turbulent state, its rate of change and the perturbation field are all required at each instant. In this paper, we develop and use a novel technique for this problem by using a pair (denoted "baseline" and "twin") of simultaneous synchronized Large-Eddy Simulations (LES). At each time-step, small disturbances whose propagation characteristics are to be studied, are introduced into the twin through a forcing term. At subsequent time steps, the difference between the two simulations is shown to be equivalent to solving the forced Navier-Stokes equations, linearized about the instantaneous turbulent state. The technique does not put constraints on the forcing, which could be arbitrary, e.g., white noise or other stochastic variants. We consider, however, "native" forcing having properties of disturbances that exist naturally in the turbulent environment. The method then isolates the effect of turbulence in a particular region on the rest of the field, which is useful in the study of noise source localization. The synchronized technique is relatively simple to implement into existing codes. In addition to minimizing the storage and retrieval of large time-varying datasets, it avoids the need to explicitly linearize the governing equations, which can be a very complicated task for viscous terms or turbulence closures. The method is illustrated by application to a well-validated Mach 1.3 jet. Specifically, the effects of turbulence on the jet lipline and core collapse regions on the near-acoustic fields are isolated. The properties of the method, including linearity and effect of initial transients, are discussed. The results provide insight into how turbulence from different parts of the jet contribute to the observed dominance of low and high frequency content at shallow and sideline angles, respectively.

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