Structural bifurcation analysis of vortex shedding from shear flow past circular cylinder

Abstract

In this paper, the unsteady flow separation of two-dimensional (2-D) incompressible shear flow past a circular cylinder is studied using theoretical structural bifurcation analysis based on topological equivalence. The stream function vorticity form of Navier–Stokes (N–S) equations in cylindrical polar coordinates are considered as the governing equations. Numerical simulations are performed, using higher-order compact finite difference scheme (Kalita and Ray J Comput Phys 228:5207–5236 2009), for the Reynolds number (Re) 100, 200 and shear parameter (K) 0.0, 0.05, 0.1 and 0.2. Through this structural bifurcation analysis, the exact location and time of occurrence of bifurcation points (flow separation points) associated with primary and secondary vortices are studied. In this process, the instantaneous vorticity contours and streakline patterns, center-line velocity fluctuation, lift and drag coefficients, phase diagram are also studied to confirm the theoretical results. Unlike the shear flow past a square cylinder, in this case, the structural bifurcation occurs from the downstream surface of the circular cylinder for the same Reynolds number. All the computed results very efficiently and very accurately reproduce the complex flow phenomena. Through this study, many noticeable and interesting results for this problem are reported for the first time.

Appendix 1

The expressions for the finite difference operators appearing in the above equations are as follows:

$$\begin{aligned} \delta _r\phi _{i,j}= & {} \frac{\phi _{i+1,j}-\phi _{i-1,j}}{2\Delta r},\\ \delta _\theta \phi _{i,j}= & {} \frac{\phi _{i,j+1}-\phi _{i,j-1}}{2\Delta \theta },\\ \delta _r^2\phi _{i,j}= & {} \frac{1}{\Delta r}\bigg \lbrace \frac{\phi _{i+1,j}}{r_\mathrm{f}}-\bigg (\frac{1}{r_\mathrm{f}}+\frac{1}{r_\mathrm{b}}\bigg ) \phi _{i,j}+\frac{\phi _{i-1,j}}{r_\mathrm{b}}\bigg \rbrace ,\\ \delta _\theta ^2\phi _{i,j}= & {} \frac{1}{\Delta \theta }\bigg \lbrace \frac{\phi _{i,j+1}}{\theta _\mathrm{f}}-\bigg (\frac{1}{\theta _\mathrm{f}}+\frac{1}{\theta _\mathrm{b}}\bigg )\phi _{i,j}+\frac{\phi _{i,j-1}}{\theta _\mathrm{b}}\bigg \rbrace ,\\ \delta _r^2\delta _\theta \phi _{i,j}= & {} \frac{1}{2\Delta r\Delta \theta }\bigg \lbrace \frac{1}{r_\mathrm{f}}(\phi _{i+1,j+1}-\phi _{i+1,j-1})-\bigg (\frac{1}{r_\mathrm{f}}+\frac{1}{r_\mathrm{b}}\bigg )(\phi _{i,j+1}-\phi _{i,j-1}) \\&+\frac{1}{r_\mathrm{b}}(\phi _{i-1,j+1}-\phi _{i-1,j-1})\bigg \rbrace ,\\ \delta _r\delta _\theta ^2\phi _{i,j}= & {} \frac{1}{2\Delta r\Delta \theta }\bigg \lbrace \frac{1}{\theta _\mathrm{f}}(\phi _{i+1,j+1}-\phi _{i-1,j+1})-\bigg (\frac{1}{\theta _\mathrm{f}}+\frac{1}{\theta _\mathrm{b}}\bigg )(\phi _{i+1,j}-\phi _{i-1,j}) \\&+\frac{1}{\theta _\mathrm{b}}(\phi _{i+1,j-1}-\phi _{i-1,j-1})\bigg \rbrace ,\\ \delta _r^2\delta _\theta ^2\phi _{i,j}= & {} \frac{1}{\Delta r\Delta \theta }\bigg \lbrace \frac{\phi _{i+1,j+1}}{r_\mathrm{f}\theta _\mathrm{f}}+\frac{\phi _{i-1,j+1}}{r_\mathrm{b}\theta _\mathrm{f}}-\bigg (\frac{1}{r_\mathrm{f}\theta _\mathrm{f}}+\frac{1}{r_\mathrm{b}\theta _\mathrm{f}}\bigg )\phi _{i,j+1}-\bigg (\frac{1}{r_\mathrm{f}\theta _\mathrm{f}}+\frac{1}{r_\mathrm{f}\theta _\mathrm{b}}\bigg )\phi _{i+1,j} \\&+ \bigg (\frac{1}{r_\mathrm{f}\theta _\mathrm{f}}+\frac{1}{r_\mathrm{f}\theta _\mathrm{b}}+\frac{1}{r_\mathrm{b}\theta _\mathrm{f}}+\frac{1}{r_\mathrm{b}\theta _\mathrm{b}}\bigg )\phi _{i,j}-\bigg (\frac{1}{r_\mathrm{f}\theta _\mathrm{b}}+\frac{1}{r_\mathrm{b}\theta _\mathrm{b}}\bigg )\phi _{i,j-1}\\&-\bigg (\frac{1}{r_\mathrm{b}\theta _\mathrm{f}}+\frac{1}{r_\mathrm{b}\theta _\mathrm{b}}\bigg )\phi _{i-1,j} \nonumber \\&+ \frac{\phi _{i+1,j-1}}{r_\mathrm{f}\theta _\mathrm{b}}+\frac{\phi _{i-1,j-1}}{r_\mathrm{b}\theta _\mathrm{b}}\bigg \rbrace ,\\ \delta _r\delta _\theta \phi _{i,j}= & {} \frac{1}{4\Delta r\Delta \theta }\lbrace \phi _{i+1,j+1}-\phi _{i+1,j-1}-\phi _{i-1,j+1}+\phi _{i-1,j-1} \rbrace . \end{aligned}$$