Nonlinear dynamics in a backward-facing step flow

Rani, H. P.; Sheu, Tony W. H.

doi:10.1063/1.2261852

Cited by 26 publications

(32 citation statements)

References 29 publications

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“…The geometry and boundary conditions are taken from [32] with small adaptation, since laminar-turbulent transition is investigated as a dynamic system so we compare not only benchmark results but bifurcation sequences as well.…”

Section: Initial-boundary Value Problem Mesh Adaptation and Benchmarmentioning

confidence: 99%

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FSM Scenarios of Laminar-Turbulent Transition in Incompressible Fluids

Evstigneev¹,

Magnitskii²

2012

Nonlinearity, Bifurcation and Chaos - Theory and Applications

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The problem of turbulence arose more than hundred years ago to explain the nature of chaotic motion of the nonlinear continuous medium and to find ways for its description; so far it remains one of the most attractive and challenging problems of classical physics. Researchers of this problem have met with exclusive difficulties and there was an understanding of that the problem of turbulence always considered difficult, is actually extremely difficult. This problem is named by Clay Mathematics Institute as one of seven millennium mathematical problems [1] and it is also in the list of 18 most significant mathematical problems of XXI century formulated by S.Smale [2].The nature of turbulence -the disordered chaotic motion of a nonlinear continuous medium, the causes and mechanisms of chaos generation remain the main issue in the turbulence problem. Several models trying to explain the mechanisms of turbulence generation in nonlinear solid media were suggested at different time. Among such models the most known are Landau-Hopf and Ruelle-Takens models, explaining generation of turbulence by the infinite cascade of Andronov-Hopf bifurcations and, accordingly, by destruction of three-dimensional torus with generation of strange attractor. However, these models have not been justified by experiments with hydrodynamic turbulence.The universal unified mechanism of transition to dynamical chaos in all nonlinear dissipative systems of differential equations including autonomous and nonautonomous systems of ordinary and partial differential equations and differential equations with delay argument was theoretically and experimentally proven in number of recent papers by the authors [3][4][5][6][7][8][9][10][11][12]. The mechanism is developing by FSM (Feigenbaum-Sharkovskii-Magnitskii) scenario through subharmonic and homoclinic bifurcation cascades of stable cycles or stable two dimensional or many-dimensional tori.In this chapter we are presenting a consistent numerical solution method for 3D evolutionary Navier-Stokes equations with an arbitrary initial-boundary value problem posed. Then we

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Section: Initial-boundary Value Problem Mesh Adaptation and Benchmarmentioning

confidence: 99%

“…Width of the domain W = 3.5 spans in Z axis direction and is the same for both part. The geometry is almost identical to [32] with L 1 = 10.0. The formed step causes the flow to create recirculation zones inside Ω and transit to turbulence with the growth of Reynolds number.…”

Section: Initial-boundary Value Problem Mesh Adaptation and Benchmarmentioning

confidence: 99%

FSM Scenarios of Laminar-Turbulent Transition in Incompressible Fluids

Evstigneev¹,

Magnitskii²

2012

Nonlinearity, Bifurcation and Chaos - Theory and Applications

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“…Some examples involve supercritical pitchfork (symmetry-breaking) bifurcations in laminar plane sudden expansion flows [1,3,6,[12][13][14]22,34,41] and Hopf bifurcations in backward-facing step flows [18,38], in rotating cylindrical flows [32,44], and in lid-driven cavity flows [4,17,37,42]. One classical approach for examining the stability of a stationary solution is to simulate the discrete, time-dependent NavierStokes (NS) equations directly with some perturbations in the stationary solution and then to investigate whether the time-dependent response solution returns to the original solution or not after certain time steps.…”

Section: Introductionmentioning

confidence: 99%

Parallel pseudo-transient Newton–Krylov–Schwarz continuation algorithms for bifurcation analysis of incompressible sudden expansion flows

Huang¹,

Hwang²

2010

Applied Numerical Mathematics

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“…It is well known that in this case for Re > 800, the flow becomes unsteady, with the initiation of a supercritical Hopf bifurcation (see [29]). In this section, we report the steady-state solution obtained for Re = 800 using the proposed MAPS domain decomposition approach.…”

Section: Backward Facing Stepmentioning

confidence: 93%

Schwarz alternating domain decomposition approach for the solution of two‐dimensional Navier–Stokes flow problems by the method of approximate particular solutions

Bustamante

Power

Florez

2014

Numerical Methods Partial

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The method of approximate particular solutions (MAPS) is used to solve the two-dimensional Navier-Stokes equations. This method uses particular solutions of a nonhomogeneous Stokes problem, with the multiquadric radial basis function as a nonhomogeneous term, to approximate the velocity and pressure fields. The continuity equation is not explicitly imposed since the used particular solutions are mass conservative. To improve the computational efficiency of the global MAPS, the domain is split into overlapped subdomains where the Schwarz Alternating Algorithm is employed using velocity or traction values from neighboring subdomains as boundary conditions. When imposing only velocity boundary conditions, an extra step is required to find a reference value for the pressure at each subdomain to guarantee continuity of pressure across subdomains. The Stokes lid-driven cavity flow problem is solved to assess the performance of the Schwarz algorithm in comparison to a finite-difference-type localized MAPS. The Kovasznay flow problem is used to validate the proposed numerical scheme. Despite the use of relative coarse nodal distributions, numerical results show excellent agreement with respect to results reported in literature when solving the lid-driven cavity (up to Re = 10,000) and the backward facing step (at Re = 800) problems.

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Nonlinear dynamics in a backward-facing step flow

Cited by 26 publications

References 29 publications

FSM Scenarios of Laminar-Turbulent Transition in Incompressible Fluids

FSM Scenarios of Laminar-Turbulent Transition in Incompressible Fluids

Parallel pseudo-transient Newton–Krylov–Schwarz continuation algorithms for bifurcation analysis of incompressible sudden expansion flows

Schwarz alternating domain decomposition approach for the solution of two‐dimensional Navier–Stokes flow problems by the method of approximate particular solutions

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