Studies on the $\alpha $- and $\beta $-phenomena, terms coined by Bouard & Coutanceau (J. Fluid Mech., vol. 101, 1980, pp. 583–607) for the flow past an impulsively started circular cylinder, have been confined only to the very early stages of the flow. In this paper, besides making a comprehensive in-depth analysis of these phenomena for a much longer period of time, we report the existence of some tertiary vortex phenomena for the first time, which we term the sub-$\alpha $- and sub-$\beta $-phenomena. The mechanism of unsteady flow separation at high Reynolds numbers for the flow past a circular cylinder developed in the last two decades has been used to understand these flow phenomena. The flow is computed using a recently developed compact finite difference method for the biharmonic form of the two-dimensional Navier–Stokes equations for the range of Reynolds number $500\leq \mathit{Re}\leq 10\hspace{0.167em} 000$. We specifically choose $\mathit{Re}= 5000$ to describe the interplay among the primary, secondary and tertiary vortices leading to these interesting vortex dynamics. We also report a $\beta $-like phenomenon which is very similar to the $\beta $-phenomenon, but slightly differs in details. We offer a new perception of the $\alpha $-phenomenon by defining its existence in a strong and weak sense along with a clearer characterization of the $\beta $-phenomenon. Apart from numerical computation, a detailed theoretical characterization using topological aspects of the boundary layer separation leading to the secondary and tertiary vortex phenomena has also been carried out. We compare our numerical results with established experimental and numerical results wherever available and an excellent match with the experimental results is obtained in all cases.
In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth order spatial discretization. The time discretization has been carried out using using second order Crank-Nicolson. The present scheme shows good dispersion relation preserving property and has been thoroughly investigated for stability. The discrete Fourier analysis shows that the method is unconditionally stable. The fact that the method has been particularly developed for parabolic equations with mixed derivatives makes it suitable for solving incompressible Navier-Stokes (N-S) equations in irregular domains. To verify the proposed method, several problems with exact and benchmark solutions has been investigated. The proposed compact discretization has been extended to tackle flows of varying complexities governed by the twodimensional unsteady N-S equations in domain beyond rectangular. The results show good agreement for all the problems considered.
SUMMARYWe recently proposed an improved (9,5) higher order compact (HOC) scheme for the unsteady twodimensional (2-D) convection-diffusion equations. Because of using only five points at the current time level in the discretization procedure, the scheme was seen to be computationally more efficient than its predecessors. It was also seen to capture very accurately the solution of the unsteady 2-D Navier-Stokes (N-S) equations for incompressible viscous flows in the stream function-vorticity ( − ) formulation.In this paper, we extend the scope of the scheme for solving the unsteady incompressible N-S equations based on primitive variable formulation on a collocated grid. The parabolic momentum equations are solved for the velocity field by a time-marching strategy and the pressure is obtained by discretizing the elliptic pressure Poisson equation by the steady-state form of the (9,5) scheme with the Neumann boundary conditions. In particular, for pressure, we adopt a strategy on the collocated grid in conjunction with ideas borrowed from the staggered grid approach in finite volume. We first apply this extension to a problem having analytical solution and then to the famous lid-driven square cavity problem. We also apply our formulation to the backward-facing step problem to see how the method performs for external flow problems. The results are presented and are compared with established numerical results. This new approach is seen to produce excellent comparison in all the cases.
AbstractIn this paper, a newly developed second order temporally and spatially accurate finite difference scheme for biharmonic semi linear equations has been employed in simulating the time evolution of viscous flows past an impulsively started circular cylinder for Reynolds number(Re)up to 9,500. The robustness of the scheme and the effectiveness of the formulation can be gauged by the fact that it very accurately captures complex flow structures such as the von Kármán vortex street through streakline simulation and the α and β-phenomena in the range 3,000≤Re≤9,500 among others. The main focus here is the application of the technique which enables the use of the discretized version of a single semi linear biharmonic equation in order to efficiently simulate different fluid structures associated with flows around a bluff body. We compare our results, both qualitatively and quantitatively, with established numerical and more so with experimental results. Excellent comparison is obtained in all the cases.
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