The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows

Fujimura, Kaoru

doi:10.1098/rspa.1989.0090

Cited by 29 publications

(23 citation statements)

References 21 publications

(19 reference statements)

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“…Later Crouch and Herbert (1993) . The same problem was also considered by Sen and Venkateswarlu (1983) and Fujimura (1989Fujimura ( , 1991Fujimura ( , 1997) whose papers will be discussed below. Zhou (1982) developed an improved version of the classical Stuart-Watson method of 1960, assuming that both the amplitude A(t) and the angular frequency ω 1 (t) of the unstable wave disturbance vary with time.…”

Section: Evaluation Of Coefficients Of Amplitude Equations and Equilimentioning

confidence: 73%

“…Comparison of the values obtained with those given by the method of multiple scales showed that in all three cases the values of λ 2 and λ 3 , computed by this version of the method of center manifold, approach their values given by the method of multiple scales as the truncation level of the eigenfunction expansion increases. Hence the three papers by Fujimura (1989Fujimura ( , 1991Fujimura ( , 1997, taken together, show that Landau's Eq. (4.41b) given by two versions of the center manifold reduction scheme, the method of multiple scales, and the modified Watson amplitude-expansion method are equivalent to each other if the disturbance amplitude is defined in a consistent way.…”

Section: Evaluation Of Coefficients Of Amplitude Equations and Equilimentioning

confidence: 95%

“…More detailed analysis of assumptions utilized in the rigorous derivations of Eq. (4.41) was undertaken in particular by Herbert (1983b) and Fujimura (1989Fujimura ( , 1991Fujimura ( , 1997) whose papers will be discussed later in this subsection. Stuart (1960) and Watson (1960a) investigated only the temporal nonlinear development of a two-dimensional wave disturbance in a steady plane Poiseuille flow.…”

Section: 7a Below) Hence A(t) Remains Small At All Values Of T Ifmentioning

confidence: 99%

“…Note however that Reynolds and Potter's method and the original Watson methods, considered by Sen and Venkateswarlu, are not of high precision, and many researchers even supposed that they are inapplicable at points (k, Re) which are far from the neutral curve. Fujimura (1989) compared two different methods of derivation of the general Landau Eq. (4.41b) for the complex disturbance amplitude A(t) from the Navier-Stokes equations-his own modification of the amplitude-expansion method of Watson and the above-mentioned method of multiple scales (which can be applied to derivation of Eq.…”

Section: Evaluation Of Coefficients Of Amplitude Equations and Equilimentioning

confidence: 99%

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Stability to Finite Disturbances: Energy Method and Landau’s Equation

Yaglom

Frisch

2012

Fluid Mechanics and Its Applications

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The main part of Chap. 2 and the whole of Chap. 3 were devoted to topics of linear stability theory dealing with the evolution of very small flow disturbances satisfying the linearized fluid dynamics equations. In Chap. 2 it was shown that the classical normal-mode method of the linear theory of hydrodynamic stability often leads to results which strongly disagree with experimental data. It was also indicated there that these disagreements are apparently due to nonlinear effects, which make linearization of the equations of motion physically unjustified. In Chap. 3 it was explained that the necessity for consideration of the full nonlinear dynamic equations often follows from the fact that many solutions of the initial-value problems for linearized fluid dynamics equations grow considerably at small and moderate values of the time t even in the cases when the normal-mode analysis shows that these solutions decay asymptotically (i.e. at t → ∞).The nonlinear theory of hydrodynamic stability has achieved a high level of development. Although the theory is still far from being completed, it has elucidated many formerly mysterious properties of fluid flows which are interesting for physicists and important for engineers. There is now an enormous literature on this subject and only a small part of it, dealing with relatively simple flows of incompressible fluids, will be considered in this book. In the present Chapter two topics from the nonlinear stability theory will be discussed: the energy method of stability analysis (short introductory consideration of this method was included in Sect. 3.4 above) and Landau's approach to the weakly nonlinear stability theory which described the initial period of the nonlinear development of flow disturbances.

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Section: Evaluation Of Coefficients Of Amplitude Equations and Equilimentioning

confidence: 73%

Section: Evaluation Of Coefficients Of Amplitude Equations and Equilimentioning

confidence: 95%

Section: 7a Below) Hence A(t) Remains Small At All Values Of T Ifmentioning

confidence: 99%

Section: Evaluation Of Coefficients Of Amplitude Equations and Equilimentioning

confidence: 99%

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Stability to Finite Disturbances: Energy Method and Landau’s Equation

Yaglom

Frisch

2012

Fluid Mechanics and Its Applications

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“…Several equivalent weakly nonlinear formalisms can be found in the litterature (Stuart 1960;Watson 1960;Reynolds & Potter 1967;Herbert 1983;Fujimura 1989). Our formalism is along the lines of Herbert (1983).…”

Section: Appendix a Weakly Nonlinear Theorymentioning

confidence: 99%

On the nonlinear destabilization of stably stratified shear flow

2013

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A weakly nonlinear analysis of the bifurcation of the stratified Ekman boundary-layer flow near a critical bulk Richardson number is conducted and compared to a similar analysis of a continuously stratified parallel shear flow subject to Kelvin–Helmholtz instability. Previous work based on asymptotic expansions and predicting supercritical bifurcation at Prandtl number $Pr\lt 1$ and subcritical bifurcation at $Pr\gt 1$ for the parallel base flow is confirmed numerically and through fully nonlinear temporal simulations. When applied to the non-parallel Ekman flow, weakly nonlinear analysis and fully nonlinear calculations confirm that the nature of the bifurcation is dominantly controlled by $Pr$, although a sharp threshold at $Pr= 1$ is not found. In both flows the underlying physical mechanism is that the mean flow adjusts so as to induce a viscous (respectively diffusive) flux of momentum (respectively buoyancy) that balances the vertical flux induced by the developing instability, leading to a weakening of the mean shear and mean stratification. The competition between the former nonlinear feedback, which tends to be stabilizing, and the latter, which is destabilizing and strongly amplified as $Pr$ increases, determines the supercritical or subcritical character of the bifurcation. That essentially the same competition is at play in both the parallel shear flow and the Ekman flow suggests that the underlying mechanism is valid for complex, non-parallel stratified shear flows.

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On problems in the weakly nonlinear theory of hydrodynamic stability and its improvement

Zhou

You

1993

Acta Mech Sinica

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The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows

Cited by 29 publications

References 21 publications

Stability to Finite Disturbances: Energy Method and Landau’s Equation

Stability to Finite Disturbances: Energy Method and Landau’s Equation

On the nonlinear destabilization of stably stratified shear flow

On problems in the weakly nonlinear theory of hydrodynamic stability and its improvement

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