3.4.1 Floquet stability analysis

The time-periodic 2D flow could be unstable to 3D disturbances at a higher Re, consequently becoming 3D. This instability is termed the secondary (or 3D) instability. The onset of the secondary instability leading to 3D flow can be detected by Floquet stability analysis in which an instantaneous velocity field of the flow past a rounded cylinder is decomposed into the 2D base flow with a period $T$ ( $\boldsymbol{U}(x,y,t)=\boldsymbol{U}(x,y,t+T)$ ) and a 3D perturbation-velocity field ( $\boldsymbol{u}^{\prime }(x,y,z,t)$ ) as follows:

Substituting (3.3) into the continuity and Navier–Stokes equations (2.1) and (2.2), and then linearizing them, one can obtain the following governing equations for the perturbation velocity field:

Here, the additional term ( $\boldsymbol{f}^{\prime }$ ) for the immersed boundary method is also included. At the inlet, a Dirichlet boundary condition $(\boldsymbol{u}^{\prime }=0)$ was imposed, while the convective and slip boundary conditions were used at the exit and at the lateral boundaries, respectively.

By defining the operator $L$ so that $L(\boldsymbol{u}^{\prime })$ is the right-hand side of the linearized equation, (3.5) can be written symbolically as $\partial \boldsymbol{u}^{\prime }/\partial t=L(\boldsymbol{u}^{\prime })$ . The general solution of this equation can be expressed as a sum of Floquet modes, $\tilde{\boldsymbol{u}}(x,y,z,t)\exp ({\it\gamma}t)$ , where ${\it\gamma}~(\equiv {\it\sigma}+\text{i}{\it\omega})$ is the Floquet exponent, and each Floquet eigenfunction $\tilde{\boldsymbol{u}}$ is a time-periodic function. To extract the periodic characteristics of the perturbation, a normalized Floquet mode $(\breve{\boldsymbol{u}})$ is defined as $\breve{\boldsymbol{u}}=\tilde{\boldsymbol{u}}\exp (\text{i}{\it\omega}t)$ . Instability of the base flow $\boldsymbol{U}$ is determined by the Floquet multipliers, ${\it\mu}\equiv \exp ({\it\gamma}T)$ ; $|{\it\mu}|>1$ indicates an exponentially growing perturbation. The Floquet multipliers can be obtained from the eigenvalues of $L$ ;  $\tilde{\boldsymbol{u}}$ represents the corresponding eigenfunctions. Since velocity and pressure fluctuations are assumed to be homogeneous in the spanwise direction, they can be expressed by an inverse Fourier transform in $z$ as follows:

where ${\it\beta}=2{\rm\pi}/{\it\lambda}$ represents the spanwise wavenumber and ${\it\lambda}$ is the corresponding spanwise wavelength of a disturbance, non-dimensionalized by $d$ . Since (3.4) and (3.5) are linear, modes with different $|{\it\beta}|$ can be decoupled. The governing equations for each disturbance wave are similar to (3.4) and (3.5) except for the replacement of the gradient operator $\boldsymbol{{\rm\nabla}}$ with $\boldsymbol{{\rm\nabla}}_{{\it\beta}}=(\partial /\partial x,\partial /\partial y,\text{i}{\it\beta})$ . Recently, a 1D power-type method was applied by Sheard et al. (Reference Sheard, Thompson and Hourigan2003) to estimate the maximum magnitude of the Floquet multipliers by computing the following ratio:

where $N(t)$ is the $L_{2}$ -norm of the perturbation-velocity field at an instant of time. This method was verified by Blackburn & Lopez (Reference Blackburn and Lopez2003). In this study we used the power-type method in conjunction with an immersed boundary method (Yoon et al. Reference Yoon, Yang and Choi2010) to calculate the Floquet instability of the periodic wake past a rounded cylinder. For convenience, the term ‘Floquet multiplier’ implies the one that has the maximum magnitude among the Floquet multipliers from now on, and the subscript max is dropped. The governing equations for each ${\it\beta}$ were temporally and spatially discretized in the same way as for the base flow.

The 2D time-periodic base flows were first computed with $800\times 448$ grid cells in the $x$ and $y$ directions, respectively. In particular, $128\times 128$ grid cells were allocated for the rounded cylinder to capture the small-scale structures in the vicinity of the cylinder. Thirty-two instantaneous snapshots were saved over one period ( $T$ ) of the base flow. They were fed to the linearized perturbation equations for each ${\it\beta}$ , being Fourier-interpolated at each time step. For the Floquet stability analysis, the numerical resolution of $608\times 288$ in the $x$ and $y$ directions, respectively, was used. To accurately find the critical values of Re and ${\it\beta}$ , computations were carried out with the small increments of 5 and 0.05 for Re and ${\it\beta}$ , respectively. Random noise of very small amplitude (less than $10^{-5}$  % of the inlet velocity of the base flow) was used as the initial perturbation-velocity field.

The critical Reynolds number ( $Re_{c2}$ ) of a 3D instability mode is determined by the Re value at which $|{\it\mu}|=1$ . The unstable modes manifest themselves in the order of A, B and QP with increasing Re. In the cases of circular ( $r/d=0.5$ ) or square ( $r/d=0$ ) cylinders, the characteristics of A, B and QP are well known from previous studies in the literature (Williamson Reference Williamson1988, Reference Williamson1996; Zhang et al. Reference Zhang, Fey, Noack, Konig and Eckelmann1995; Barkley & Henderson Reference Barkley and Henderson1996; Robichaux et al. Reference Robichaux, Balachandar and Vanka1999; Blackburn & Lopez Reference Blackburn and Lopez2003; Sheard et al. Reference Sheard, Fitzgerald and Ryan2009; Blackburn & Sheard Reference Blackburn and Sheard2010; Yoon et al. Reference Yoon, Yang and Choi2012; Choi & Yang Reference Choi and Yang2014). Mode A reveals the odd reflection–translation (RT) symmetry, and mode B exhibits the even RT symmetry, mathematically expressed as

Here, $T$ and $\breve{{\it\omega}}_{x}$ represent the time period of the base flow and streamwise vorticity of the normalized Floquet mode, respectively. Mode A persists even far downstream, while that is not the case for mode B. It should also be noted that the spanwise wavelength of mode A is approximately five times larger than that of mode B (Barkley & Henderson Reference Barkley and Henderson1996; Williamson Reference Williamson1996; Robichaux et al. Reference Robichaux, Balachandar and Vanka1999).

Marques, Lopez & Blackburn (Reference Marques, Lopez and Blackburn2004) reported that flow with spatio-temporal $Z_{2}\times O(2)$ symmetry, like the flow past a single square cylinder, exhibits three unstable 3D modes, namely two synchronous modes ( ${\it\omega}=0$ ) and one quasi-periodic mode (QP) (Barkley & Henderson Reference Barkley and Henderson1996; Blackburn & Lopez Reference Blackburn and Lopez2003; Blackburn & Sheard Reference Blackburn and Sheard2010; Yoon et al. Reference Yoon, Yang and Choi2010; Sheard Reference Sheard2011). In mode QP the streamwise vorticity of the normalized Floquet mode ( $\breve{{\it\omega}}_{x}$ ) is repeated in magnitude with an approximate time period of $T$ , whereas its sign is alternating. This temporal feature of QP seems to be similar to that of mode C, in that mode C has the time period of $2T$ . However, a close examination reveals that the periodicity of the normalized Floquet QP mode is not exactly identical to twice the time period of the base flow, implying a quasi-periodic feature of QP. Floquet multipliers of QP obtained from the eigenvalues of the operator L turn out to be complex numbers (Blackburn & Lopez Reference Blackburn and Lopez2003; Blackburn & Sheard Reference Blackburn and Sheard2010; Sheard Reference Sheard2011), while those of A, B and C are pure real. One can refer to Blackburn & Lopez (Reference Blackburn and Lopez2003) and Sheard (Reference Sheard2011) to see the behaviour of ${\it\mu}$ on a complex plane with increasing Re, and to find a detailed description on the relation between QP and C. The critical spanwise wavelength of mode QP is approximately an average of those of A and B (Blackburn & Lopez Reference Blackburn and Lopez2003; Blackburn, Marques & Lopez Reference Blackburn, Marques and Lopez2005; Sheard et al. Reference Sheard, Fitzgerald and Ryan2009; Sheard Reference Sheard2011).

Figure 7.

Streamwise-vorticity contours for a rounded cylinder, $r/D=0.25$ : (a,b) mode A, $Re=200$ , ${\it\beta}=1.4$ ; (c,d) mode B, $Re=262$ , ${\it\beta}=7.0$ ; (e,f) mode QP, $Re=395$ , ${\it\beta}=3.1$ ; (a,c,e) instantaneous contours of the Floquet mode; (b,d,f) temporal accumulation on the line $x=2.5$ . The red and blue regions represent negative and positive streamwise vorticity, respectively. Solid and dashed lines represent positive and negative values of spanwise vorticity of the base flow, respectively (lines, ${\it\omega}_{z}=\pm 0.1$ ).

In figure 7 streamwise-vorticity contours of the three distinct instability modes in the flow past a rounded cylinder with $r/d=0.25$ are presented near the criticality. The red and blue regions represent negative and positive streamwise vorticity, respectively. In figure 7(a,c,e), the solid and dashed lines denote positive and negative spanwise vorticity of the corresponding base flow, respectively, whereas temporal accumulation of the streamwise vorticity contours on the line $x=2.5$ is shown in figure 7(b,d,f). It is seen in figure 7 that the topological change of the base flow due to rounding the edges does not affect the main characteristics of A, B and QP, including the RT symmetry of each mode, the fast downstream decay of mode B, the quasi-periodicity of mode QP (Barkley & Henderson Reference Barkley and Henderson1996; Robichaux et al. Reference Robichaux, Balachandar and Vanka1999; Sheard et al. Reference Sheard, Fitzgerald and Ryan2009) and the strong localization of the core of modes A and B in the near wake (Giannetti, Camarri & Luchini Reference Giannetti, Camarri and Luchini2010). The present Floquet stability analysis does not reveal any other instability mode, confirming the affirmation of Marques et al. (Reference Marques, Lopez and Blackburn2004).

Figure 8.

Variation of critical Reynolds number with $r/d$ for flow past a rounded cylinder.

3.4.2 Criticalities of 3D instability modes

Figure 8 presents the critical Reynolds number ( $Re_{c2}$ ) of each 3D instability mode depending upon $r/d$ , and the corresponding critical spanwise wavelength ( ${\it\lambda}_{c}$ ) is shown in figure 9. The present Floquet results on $Re_{c2}$ and ${\it\lambda}_{c}$ are compared with other authors’ data for the circular and square cylinders (Barkley & Henderson Reference Barkley and Henderson1996; Williamson Reference Williamson1996; Blackburn et al. Reference Blackburn, Marques and Lopez2005; Luo, Tong & Khoo Reference Luo, Tong and Khoo2007; Sheard et al. Reference Sheard, Fitzgerald and Ryan2009), being in good agreement with them. It is seen in figure 8 that the emergence of mode A is most delayed (i.e. $Re_{c2,A}$ is maximum) when $r/d=0.25$ . Nevertheless, $Re_{c2,A}~(=198.3)$ at $r/d=0.25$ is slightly lower than $Re_{c2,B}$ ( $=201.0$ ) at $r/d=0$ . Recalling that $Re_{c1}$ is also maximum at the same value of $r/d$ (figure 2), and that mode A is the 3D instability mode that first emerges with increasing Re, one can say that a moderate rounding ( $r/d=0.25$ ) most efficiently stabilizes the flow. This is quite unexpected and surprising because full rounding ( $r/d=0.5$ , circular cylinder) is not the most efficient one. Unlike mode A, the critical Reynolds number of the next emerging 3D mode B ( $Re_{c2,B}$ ) initially increases with increasing $r/d$ , but decreases very slightly between $r/d=0.3125$ and $r/d=0.4375$ , and then increases again to reach the maximum at $r/d=0.5$ (figure 8). This trend is more noticeable for mode QP that emerges at a higher Re than the other two modes (figure 8). A moderate rounding ( $r/d=0.25$ ) is also most effective for mode QP. It should also be noted that even a slight rounding ( $r/d=0.125$ ) of a square cylinder effectively increases $Re_{c2}$ of the three 3D instability modes (figure 8), delaying their emergence. Unlike $Re_{c2}$ , the critical spanwise wavelength monotonically decreases with increasing $r/d$ for all the three modes (figure 9), connecting the two extreme cases (square and circular) almost linearly.

Figure 9.

Variation of critical spanwise wavelength with $r/d$ for flow past a rounded cylinder.

3.4.3 Neutral stability curves

The neutral stability curves of each mode are presented for some selected values of $r/d$ in figures 10–12. It is seen in figure 10 that rounding increases $Re_{c2,A}$ , and significantly reduces the unstable range of spanwise wavelength of mode A. This reveals the effectiveness of rounding the cylinder edges in stabilizing the flow past a square cylinder because mode A that emerges first with ascending Re is suppressed to a large extent by rounding. The lower branches of the neutral stability curves are almost flat, and hardly affected by rounding, meaning that the shortest unstable wavelength for mode A is weakly dependent upon Re and $r/d$ . The critical Reynolds number ( $Re_{c2,A}$ ) reaches the maximum value at $r/d=0.25$ (figure 10 b) as also seen in figure 8. In the case of mode B (figure 11), the neutral stability curve shifts to the right with increasing $r/d$ up to $r/d=0.25$ . Beyond that value, $r/d$ does not seem to significantly affect the criticality of mode B. Mode QP emerges at a rather higher Re than the other two modes, and is most sensitive to rounding. In figure 12(a) one can notice a rapid increase of $Re_{c2,QP}$ again up to $r/d=0.25$ as $r/d$ increases. Further rounding ( $r/d=0.375$ , 0.4375) destabilizes the flow (i.e $Re_{c2,QP}$ decreases, figure 12 b), and then stabilizes the flow again at $r/d=0.5$ , as seen in figures 8 and 12(b). From figures 10–12, it is concluded that small rounding ( $0\leqslant r/d\leqslant 0.25$ ) has more impact on the criticality of the 3D modes (A, B, QP) than large rounding ( $0.25<r/d\leqslant 0.5$ ), which can be explained by the topological change induced by the small rounding as mentioned in § 3.3. The small rounding suppresses the flow separation originally at the upstream edges of the square cylinder (figure 5 a), and delays separation towards the downstream rounded edges (figure 5 b,c). Further rounding, however, does not yield a drastic change in flow topology, resulting in less influence on the criticality of the 3D modes. Evolution of the dominant spanwise wavelength ( ${\it\lambda}_{p}$ ) corresponding to the largest $|{\it\mu}|$ is plotted against Re for each mode and for each $r/d$ in figure 13. It is seen that ${\it\lambda}_{p}$ decreases with increasing $r/d$ in all cases. In particular, a significant decrease of ${\it\lambda}_{p}$ is noticed between $r/d=0$ and $r/d=0.0625$ , implying that a very small rounding definitely affects the dominant spanwise wavelength of each instability mode.

Figure 10.

Neutral stability curves of mode A: (a) entire range of Re investigated, (b) magnified view.

Figure 11.

Neutral stability curves of mode B.

Figure 12.

Neutral stability curves of mode QP: (a) entire range of $Re$ investigated, (b) magnified view.

Figure 13.

Evolution of the dominant spanwise wavelength ( ${\it\lambda}_{p}$ ) corresponding to the largest $|{\it\mu}|$ : (a) mode A, (b) mode B, (c) mode QP.