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

Abstract

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.

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

“…Using the penalty method, 13,15,29 we reformulate the second equation of (1) by eliminating the pressure term using a large penalty parameter 𝜉 as follows:…”

“…where || • || represents the Euclidean norm. In Equation (15), 𝛿 init 0 is the solution of the following system.…”

“…Using the penalty method, 13,15,29 we reformulate the second equation of (1) by eliminating the pressure term using a large penalty parameter $$$ \xi $$$ as follows: $$$$ P=-\xi \nabla .V. $$$$ In order to apply the proposed approach and taken into account (2), the previous nonlinear Equations (1) are rewritten in the following quadratic form representing by the velocity field only: $$$$ M\left(\dot{V}\right)+L(V)+Q\left(V,V\right)=\lambda F, $$$$ where $$$ M(.)$…”

“…Numerically, this means that the indicator is computed at a fixed Reynolds number using the algorithm HO-MFA. [13][14][15] The Hopf bifurcation point is determined according to the angular frequency where a zero value of the indicator is reached for a critical Reynolds value.…”

“…In this article, the computation of Hopf bifurcation points is carried out by associating a bifurcation indicator with the high order mesh‐free approach (HO‐MFA) proposed in References 13‐15). Numerically, this means that the indicator is computed at a fixed Reynolds number using the algorithm HO‐MFA 13‐15 .…”