| m | \lambda = 0.00 \pm 0.00i | \lambda = 0.25 \pm 0.00i | \lambda = 0.50 \pm 0.00i |
| 250 | -2.499\ 530 \times 10^{-3} \pm 0.716\ 816 | -2.362\ 255 \times 10^{-6} \pm 0.730\ 913 | -2.666\ 849 \times 10^{-6} \pm 0.730\ 912 |
| 500 | +3.029\ 965 \times 10^{-3} \pm 0.726\ 480 | -2.668\ 530 \times 10^{-6} \pm 0.730\ 912 | -2.668\ 473 \times 10^{-6} \pm 0.730\ 912 |
| 1000 | +2.230\ 634 \times 10^{-4} \pm 0.729\ 683 | -2.668\ 474 \times 10^{-6} \pm 0.730\ 912 | -2.668\ 473 \times 10^{-6} \pm 0.730\ 912 |
+1309
+09 May 2026 02:44:09
+Phys. Fluids, Vol. 16, No. 5, May 2004
+M. Sahin and R. G. Owens
+
+
+
+FIG. 2. Reciprocal Ritz values for unbounded flow around a circular cylinder at Re546.76 computed on mesh M3 with Krylov space dimension m 5250 and shift parameter l50.5060.00i (b50.01).
+
+
+
+
+
+FIG. 3. Comparison of Strouhal number versus Reynolds number for unbounded flow around a circular cylinder with other results in the literature: $( - ) .$ experimental work of Williamson ~Ref. 23!; ~¯!, numerical results of Henderson ~Ref. 25!; ~s!, numerical results of Posdziech and Grundmann ~Ref. 24!; ~h!, present ( b50.01, mesh M2!.
+
+
+
+
+FIG. 4. Comparison of drag coefficient versus Reynolds number for unbounded flow around a circular cylinder with other results in the literature: $( - ) ,$ , numerical results of Henderson ~Ref. 25!; ~s!, numerical results of Posdziech and Grundmann ~Ref. 24!; ~h!, present (b50.01, mesh M2!.
+
+
+# IV. FLOW PAST A CONFINED CIRCULAR CYLINDER „0.1ËbË0.9…
+Flow around a confined circular cylinder ~as opposed to the unbounded case! is an attractive benchmark problem in numerical simulation since it does not suffer from any of the difficulties associated with far-field boundary conditions ~particularly at very low Reynolds numbers! and permits the use of grid points more efficiently in smaller computational domains. Somewhat surprising, therefore, is that the only numerical linear stability analysis of Newtonian flow past a confined cylinder available in the literature would seem to be that of Chen $e t a l . ^ { 4 }$ These authors went no further than identifying the curve of neutral stability for the supercritical Hopf bifurcation at blockage ratios up to $\beta { = } 0 . 7$ . This is regrettable, because as we shall see in the paragraphs to follow, the linear stability properties of the flow become rich and therefore interesting at higher blockage ratios and Reynolds numbers than those considered by Chen et al. In the present study we consider two-dimensional flow at Reynolds numbers up to 280 and for blockage ratios in the range 0.1–0.9.
+# A. Linear stability analysis
+The curves of neutral stability computed from the GEVP with a Krylov subspace dimension $m = 2 5 0$ on mesh M2 for $\beta \in \left[ 0 . 1 , 0 . 9 \right]$ and $R e < 2 8 0$ are presented in Fig.5. Our discussion of these curves will focus on the five distinct curve sections labeled AB, BC, CD, CE, and $F G$ in the same
+1310
+09 May 2026 02:44:09
+Phys. Fluids, Vol. 16, No. 5, May 2004
+A numerical investigation of wall effects
+https://cdn-mineru.openxlab.org.cn/result/2026-05-11/e29ea199-8ede-444f-8a3a-062a2f820b92/7143dec957189df04b69e5cf9e774f69213002910deed08e6e9428081f7c6fee.jpg
+FIG. 5. Change of critical Reynolds number corresponding to both Hopf and pitchfork bifurcations with blockage ratio $\beta ,$ computed on $M 2 . A C \mathrm { : }$ Curve of neutral stability for Hopf bifurcations about symmetric solution; CD: Transition curve from asymmetric vortex shedding ~smaller $\beta )$ to a steady asymmetric solution ~larger $\beta ) ; C E \colon$ : Neutral stability curve for pitchfork bifurcation of steady symmetric solution ~smaller $\beta )$ to a steady asymmetric state ~larger $\beta ) ; F G ;$ Hopf bifurcation of an asymmetric solution ~smaller $\beta )$ to asymmetric vortex shedding ~larger $\beta ) . ~ C$ is a co-dimension 2 point where Hopf and pitchfork bifurcations occur simultaneously.
+
+TABLE IV. Convergence of critical Reynolds number for different blockage ratios with $\lambda = 0 . 0 0 \pm 0 . 0 0 i$ .
+
+