Global Galerkin Method for Stability Studies in Incompressible CFD and Other Possible Applications

Abstract

In this paper the author reviews a version of the global Galerkin that was developed and applied in a series of earlier publications. The method is based on divergence-free basis functions satisfying all the linear and homogeneous boundary conditions. The functions are defined as linear superpositions of the Chebyshev polynomials of the first and second klinds that are combined into divergence free vectors. The description and explanations of treatment of boundary conditions inhomogeneities and singularities are given. Possible implementation for steady state solvers, path-continuation, stability solvers and straight-forward integration in time are discussed. The most important results obtained using the approach are briefly reviewed and possible future applications are discussed.

An extended and improved particle‐spectral method for analysis of unsteady inviscid incompressible flows through a channel of finite length

“…With initial data serving as the stable steady-state regime (K = 20), there is essentially no change in error during the whole calculation, see Table 2. Similar behavior is exhibited by other characteristic ( 37)- (39). Figure 3 displays graphs illustrating their time dependence.…”

“…These purposes require knowledge of the velocity field at any point of the flow domain. The spectral methods provide this property when the velocity field is approximated by a finite Fourier series 38,39 . Thus, the most suitable method for studying flows can be a method that keeps the advantages of both listed numerical approaches.…”

“…The spectral methods provide this property when the velocity field is approximated by a finite Fourier series. 38,39 Thus, the most suitable method for studying flows can be a method that keeps the advantages of both listed numerical approaches. A variant of such a method was proposed some time ago 40,41 by the author of the present article, and here we extend and develop it in two directions.…”

Multiple solutions and hysteresis in the flows driven by surface with antisymmetric velocity profile*

“…For such a complex flow problem, it is difficult to obtain the unstable steady solutions and quantitative bifurcation processes only by the direct numerical simulation. [15,16] The numerical modelling and analysis of bifurcation problems in fluid mechanics has been extensively discussed in literature, [15,[17][18][19][20] where the numerical analysis of bifurcation problems in the incompressible fluid mechanics was discussed and the convergence theory for several important bifur-cations was described for the projection-type, finite difference and mixed finite element methods. Generally, there are three sequential parts in the computational approach.…”

“…The third part of the computational work is to access the linear stability of the solution state. Following this idea, Tuckerman et al [15,20,21] have proposed a global instability analyzing methodology, by which an explicit-implicit time integration code could be easily transformed to carry out for steady-state solving, bifurcation points, continuation, linear stability analysis, and nonlinear transient growth analysis in fluid mechanics. Here, we use this methodology [16,[22][23][24][25][26] of nonlinear global stability and bifurcation theory to solve the multiple solutions and hysteresis in the symmetric driven square cavity flow.…”