Mesh‐free model for Hopf's bifurcation points in incompressible fluid flows problems

Abstract

In this article, a new numerical model is proposed to detect numerically the Hopf bifurcation point of incompressible fluid flows problems. The concept of this model consists to propose a Hopf bifurcation indicator which is solved by a high order mesh free algorithm (HO‐MFA) using the moving least squares (MLS) approximation. This numerical model is based on a strong formulation of Navier–Stokes equations. This procedure allowed us to recover the results of the literature in 3D$$ 3D $$ using only a 2D$$ 2D $$ modelization for low Reynolds number. This new model is tested on the classical flow around a cylindrical obstacle and the backward‐facing step flow to show the advantage and ability of the proposed model to detect the bifurcation points and to determinate the flow periodicity.

Efficient resolution of incompressible Navier–Stokes equations using a robust high‐order pseudo‐spectral approach

“…The third part concerns a flow around a square obstacle. The literature survey of the experimental and numerical works carried out on flow around obstacle indicates that limited studies are reported on cylindrical objects 10,13,28,29 . Motivated by this fact, the objective of this part consists to investigate the streamlines for the flow around a square cylindrical obstacle at several values of Reynolds number.…”

“…Then the proposed mesh‐free approach was also coupled with the homopic technique to study the compressible flows using a dimensional analysis 12 . In the context of dynamic flow, these approaches have been improved by a Hopf bifurcation study for an incompressible flow in Reference 13. Trias et al 36 investigated a turbulent flow around a square cylinder for a Reynolds number of 22,000.…”

“…The use of such a numerical approach could be limited or motivated by testing this factor. Among the best‐known meshless methods in the literature, we find Smooth Particle Hydrodynamics (SPH), 6‐8 Moving Least Squares (MLS) approximation, 2,9‐13 Reproducing Kernel Particle Method (RKPM), 14 Element‐Free Galerkin Methods (EFGM), 15 Taylor meshless method, 16 and Method of Fundamental Solutions (MFS), 17,18 Radial Basis Function 19‐21 …”

“…The first studies conducted on fluid flow consist on two‐dimensional lid‐driven cavity and rectangular cavities. These studies are devoted to the analysis of velocities and pressure fields and flow instabilities by investigating static 11,23‐25 and Hopf bifurcations 13,26,27 . The investigation of the instability is performed also on the symmetric sudden expansion flow experimentally and numerically.…”

“…The use of such a numerical approach could be limited or motivated by testing this factor. Among the best-known meshless methods in the literature, we find Smooth Particle Hydrodynamics (SPH), [6][7][8] Moving Least Squares (MLS) approximation, 2,[9][10][11][12][13] Reproducing Kernel Particle Method (RKPM), 14 Element-Free Galerkin Methods (EFGM), 15 Taylor meshless method, 16 and Method of Fundamental Solutions (MFS), 17,18 Radial Basis Function. [19][20][21] Regarding incompressible fluid flows, the need for careful software to treat the coupling between pressure and velocity fields in (N-S)-equations leads to develop mathematical models and numerical simulations as prediction means.…”