Laminar Flow Patterns Around Three Side-By-Side Arranged Circular Cylinders Using Semi-Implicit Three-Step Taylor-Characteristic-Based-Split (3-TCBS) Algorithm

Abstract

One limitation of the classical characteristic-based-split (CBS) algorithm is that its computational time step is relatively small because it is an explicit scheme with conditional stability. We present in this paper a semi-implicit form of a three-step Taylor-characteristic-based-split (3-TCBS) Galerkin finite element (FE) method in the framework of incremental projection method to numerically solve incompressible fluid flow problems. First, the velocities are semi-implicitly estimated by a three-step process. The computational code is then verified by using the benchmark problems of lid-driven cavity and of flow around a fixed circular cylinder at Reynolds numbers in the laminar regime. Comparisons between the current method and the classical CBS algorithm show that the present 3-TCBS scheme can provide a larger time step with more accurate results. Then, this method is employed to investigate the problem of laminar flow over three cylinders which are arranged side-by-side. The Reynolds numbers range from 40 to 160 and the spacing ratios were set as 1.2, 1.4, 1.6, 1.8, 2.2, 2.5, 3.2, and 4.0. Eight different wake patterns were systematically categorized and their relationships with the Reynolds number and spacing ratio are presented.

A cost-effective FE method for 2D Navier–Stokes equations

“…In Figure 13 are plotted the profiles of the u and v velocity components along the vertical and horizontal centerlines. The comparison of the graphics shown in Figures 12 and 13 and those in some of the above papers (Han et al, 2013;Kumar et al, 2010;Pai et al, 2013) point out some differences; in particular the velocity profiles appear to be affected by viscous effects. All that should not be surprising because the graphics are built by the interpolation of the nodal values calculated by means of a quite coarse mesh (it has only 3009 nodes).…”

“…In fact, the lid-driven cavity problem is considered the prototypical recirculation flow because the streamlines pattern exhibits a central primary vortex and for high Reynolds number (Re ≥ 1000) the possible formation of counter-rotating secondary vortices at the bottom corners of the cavity. In order to check the accuracy and the computational efficiency of the proposed method, we compare our results of problem 5.3 (with Re = 1000) with those found in four recent papers (Han et al, 2013;Kumar et al, 2010;Kumar et al, 2009;Pai et al, 2013). In these papers the problem 5.3 (and other problems) was solved by using different numerical techniques, with different meshes or grids.…”

“…Through them we can deduce that at time t = 5 two recirculation zones start to be present, at t = 10 two secondary vortices arise at the bottom corners, at t = 40 the two secondary vortices are almost completely shaped. In the third paper (Han et al, 2013), the authors solved the 2D Navier-Stokes equations in primitive variables by means of a FE approach in which the elements are nine-node quadrilaterals both for velocities and pressure. The convective term was approximated by a semi-implicit form of three steps Taylor-characteristic-based-split scheme, while the diffusive term by the usual FE technique (namely by the stiffness matrix).…”

“…The new method can be considered optimal in the sense that its formulation attempts to obtain a good compromise between numerical properties (accuracy and stability) and computational efficiency. The main features of the method are: using a triangular (curved edges) FE spatial approximation with polynomials of degree two for velocity and pressure (and temperature); time-advancing by a fractional step approach in which the nonlinear convective term is approximated by characteristics and the pressure correction is calculated by solving a Poisson equation (Codina & Badia, 2006;Gervasio & Saleri, 2006;Han, Zhou, & Tu, 2013;Shirokoff & Rosales, 2011); addition of a suitable stabilization technique in order to overcome the instabilities inherent in the equalorder choice. A well-known technique of stabilization is the one based on the definition of subscale velocities formulated by Codina and Soto (2004).…”

Flow characteristics and dynamic responses of a rear circular cylinder behind the square cylinder with different side lengths

“…The computation procedure forms into a semi-implicit CBS scheme, which is expected to be more stable and robust than the classical explicit one [21,26,27]. Details of the CBS algorithm have been shown in our previous publication [21].…”