Automatic detection and branch switching methods for steady bifurcation in fluid mechanics

Guevel, Yann; Boutyour, H.; Cadou, Jean-Marc

doi:10.1016/j.jcp.2011.02.004

Cited by 32 publications

(55 citation statements)

References 32 publications

(98 reference statements)

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“…By choosing one of those tangents, it makes it possible to switch from a branch to another. Those nonlinear post‐bifurcated branches are computed with a modified ANM path‐following technique as presented in the work of Guevel et al Hence, the nonlinear post‐bifurcated branches are sought as a power series representation as follows:

X^{b i} false(a false) = X_{c} + true \sum_{j = 1}^{N} a^{j} {boldX}_{j}^{b i}, for i = 1, 2 .

…”

Section: Methodsmentioning

confidence: 99%

“…In the case of the Navier‐Stokes equations, the left mode remains to be computed using either an augmented system, or with iterative techniques such as bordering technique as presented in the work of Cochelin and Medale, or with an inverse power method . Then, using the same consideration for the left bifurcation mode Ψ (see Equation ), the augmented system is written as follows:

([], \begin{matrix} array {normalL}_{t}^{c^{⊤}} & array Φ \\ array {boldΦ}^{⊤} & array 0 \end{matrix}) ({}, \begin{matrix} array Ψ \\ array κ \end{matrix}) = ({}, \begin{matrix} array 0 \\ array 1 \end{matrix}) .

…”

Section: Methodsmentioning

confidence: 99%

“…The following Lyapunov‐Schmidt reduction() is used:

{boldU}_{k}^{b i} = {normalλ}_{k}^{b i} boldW + {normalη}_{k}^{b i} boldΦ + {boldV}_{k}^{b i} .

Injecting Equation in Equation (35), the new vector

{boldV}_{k}^{b i}

is solution of

{normalL}_{t}^{c} ((), V_{k}^{b i}) = - true \sum_{j = 1}^{k - 1} boldQ ((), U_{j}^{b i}, U_{k - j}^{b i})

(⟨⟩, V_{k}^{b i}, boldΦ) = 0 .

…”

Section: Methodsmentioning

confidence: 99%

“…Pseudo-arc-length path-following technique 16 based on the ANM is used to perform the continuation of the steady flow solutions. This has been successfully implemented for 2-dimensional (2D) flows in the case of steady incompressible Navier-Stokes equations in other works 12,[17][18][19] and for non-Newtonian fluids in the work of Jawadi et al 20 Recently, 3D flow study has been proposed in the work of Medale and Cochelin. 14 The ANM relies on a linearization technique where the unknowns of the nonlinear problem are sought as power series as follows:…”

Section: Continuation With Perturbation Methodsmentioning

confidence: 99%

See 3 more Smart Citations

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Guevel

Allain

Girault

et al. 2017

Numerical Methods in Fluids

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Summary This paper deals with bifurcation analysis methods based on the asymptotic‐numerical method. It is used to investigate 3‐dimensional (3D) instabilities in a sudden expansion. To do so, high‐performance computing is implemented in ELMER, ie, an open‐source multiphysical software. In this work, velocity‐pressure mixed vectors are used with asymptotic‐numerical method–based methods, remarks are made for the branch‐switching method in the case of symmetry‐breaking bifurcation, and new 3D instability results are presented for the sudden expansion ratio, ie, E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed, which reveal new trends that complete a previous work. Several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio. One of these relies on a fully 3D effect as it breaks both spanwise and top‐bottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio, ie, Ai=1, which was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all post‐bifurcated branches are followed. This makes it possible to plot a complete bifurcation diagram for this 3D case.

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X^{b i} false(a false) = X_{c} + true \sum_{j = 1}^{N} a^{j} {boldX}_{j}^{b i}, for i = 1, 2 .

…”

Section: Methodsmentioning

confidence: 99%

([], \begin{matrix} array {normalL}_{t}^{c^{⊤}} & array Φ \\ array {boldΦ}^{⊤} & array 0 \end{matrix}) ({}, \begin{matrix} array Ψ \\ array κ \end{matrix}) = ({}, \begin{matrix} array 0 \\ array 1 \end{matrix}) .

…”

Section: Methodsmentioning

confidence: 99%

“…The following Lyapunov‐Schmidt reduction() is used:

{boldU}_{k}^{b i} = {normalλ}_{k}^{b i} boldW + {normalη}_{k}^{b i} boldΦ + {boldV}_{k}^{b i} .

Injecting Equation in Equation (35), the new vector

{boldV}_{k}^{b i}

is solution of

{normalL}_{t}^{c} ((), V_{k}^{b i}) = - true \sum_{j = 1}^{k - 1} boldQ ((), U_{j}^{b i}, U_{k - j}^{b i})

(⟨⟩, V_{k}^{b i}, boldΦ) = 0 .

…”

Section: Methodsmentioning

confidence: 99%

Section: Continuation With Perturbation Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Guevel

Allain

Girault

et al. 2017

Numerical Methods in Fluids

Self Cite

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show abstract

“…The asymptotic numerical method (ANM), based on the perturbation technique, has been successfully used to study many nonlinear problems in solid mechanics [43] and in fluid mechanics [44]. It consists of introducing a power series expansion of the unknowns (q and ω) with respect to a perturbation parameter (path parameter a) and starting from a known and regular solution, (q 0 , ω 0 ), into the nonlinear system equation (28) The power series expansions are defined as…”

Section: Asymptotic Numerical Methodsmentioning

confidence: 99%

Nonlinear free vibrations of centrifugally stiffened uniform beams at high angular velocity

Bekhoucha

Rechak

Duigou

et al. 2016

Journal of Sound and Vibration

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Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

Rammane

Mesmoudi

Tri

et al. 2020

Numerical Methods in Fluids

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In this article, we propose to investigate numerically the steady bifurcation points and bifurcated branches in fluid mechanics by employing high‐order mesh‐free geometric progression algorithms. These algorithms are based on the use of the geometric progression (GP) in a high‐order mesh‐free approach. The first proposed algorithm is applied on a strong formulation using the moving least squares (MLS) approximation coupled with GP (HO‐MLS‐GPM). While the second proposed algorithm is applied on a weak formulation using the element‐free Galerkin (EFG) coupled also with GP (HO‐EFG‐GPM). The incompressibility condition is taken by introducing the penalty technique to transform the stationary Navier–Stokes equations verified by the pressure and velocity into ones verified by only the velocity. The high‐order mesh‐free algorithm permits to transform this nonlinear equations into a succession of linear ones. The GP allows to detect with precision the bifurcation points and the Lyapunov–Schmidt reduction is coupled with HO‐MLS‐GPM and HO‐EFG‐GPM as a continuation procedure to follow the many bifurcated branches. The aim of this resolution strategy concerns the treatment of the bifurcation phenomena for a fluid flow through an expansion in several geometries, where the steady flow becomes unstable after a critical Reynolds value. The obtained results are compared with those presented in literature and with those computed using the high‐order finite element algorithm coupled with GP.

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Automatic detection and branch switching methods for steady bifurcation in fluid mechanics

Cited by 32 publications

References 32 publications

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Nonlinear free vibrations of centrifugally stiffened uniform beams at high angular velocity

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

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