Turbulent flows around side-by-side cylinders with regular and multiscale arrangements

Direct numerical simulations are performed to investigate turbulent flows behind two different types of cylinder arrays (i.e., one is a widely used regular array and the other is a less studied multiscale array) with the same blockage ratio. Some important characteristics (e.g., mean velocity, turbulence intensity, and pressure drop) of the two flows are evaluated. It is demonstrated that the overall pressure drops in the two cases are almost the same. In the downstream region, the characteristics of the regular flow resemble those of the homogeneous flow, whereas the flow behind the multiscale array exhibits wakelike behavior. For the regular case, the vortex shedding frequencies due to different cylinders are the same since the cylinders involved are virtually identical and the vortex wakes formed by two neighboring cylinders are always in antiphase synchrony. In contrast, the multiscale array can generate wakes with multiple scales of both space and time. To quantitatively evaluate the difference between the features of the coherent structures in the two flows, we adopt the so-called snapshot proper orthogonal decomposition. It is found that immediately behind the regular array, most of the energy is contained in a few of the first modes. In the multiscale case, however, due to cylinders of various sizes, the increase in the energy cumulation is considerably slower than the regular case. This observation is in accord with the suggestion that by using multiscale structures, turbulence with a wider range of scales can be generated. One perhaps interesting and less reported discovery is that in the downstream region for the two cases the profiles of the energy cumulation are somewhat close to each other, although slight deviation can still be identified when normalized by the dominant vortex shedding frequency. This observation seems to tell us that the multiscale flow may not have a long-lasting memory and can eventually forget its multiscale initial conditions. These findings contribute to the understanding of the recently proposed space-scale unfolding mechanism [Phys. Rev. E 86, 046302 (2012) and J. Fluid Mech. 764, 52 (2015)], which outlines a conceptual model for explaining the high mixing efficiency of the multiscale and fractal structures and objects.

Figure 1

Schematic front view (left) and schematic side view (right) of the computational domain. From top to bottom, the two plots correspond to regular array and multiscale array. The dashed lines indicate simulation boundaries. See Table 1 for more geometric details.

Figure 2

Streamwise evolution of the normalized mesh sizes (i.e., $\mathrm{\Delta }X/T_{\mathrm{bar}}$, $\mathrm{\Delta }Y/T_{\mathrm{bar}}$, and $\mathrm{\Delta }Z/T_{\mathrm{bar}}$).

Figure 3

Streamwise evolution of the normalized resolution ${\left(\mathrm{\Delta }X\mathrm{\Delta }Y\mathrm{\Delta }Z\right)}^{1/3}/\eta$ at the two selected $Y$ locations for (a) the array with regular arrangement and (b) multiscale arrangement. The black lines correspond to centerline of the wake behind cylinder bars, whereas the red lines correspond to the centerline of the jet between adjacent cylinders. The data are averaged over time and $Z$ direction.

Figure 4

Streamwise evolution of the normalized mean streamwise velocity $U_1/U_{\mathrm{in}}$ along the centerline. The results from previous experiments [32, 33, 34, 35] (represented by open symbols) and simulations [26] (represented by filled symbols) are also included for comparison.

Figure 5

Streamwise evolution of all three rms velocities along the centerline. (a) Streamwise component $u_1^{\prime }/U_{\mathrm{in}}$, (b) vertical component $u_2^{\prime }/U_{\mathrm{in}}$, and (c) spanwise component $u_3^{\prime }/U_{\mathrm{in}}$. The symbols are the same as those in Fig. 4.

Figure 6

Time traces of the lift coefficient $C_L$ (solid line) and the drag coefficient $C_D$ (dashed line). The origins of time are arbitrarily selected but the same for $C_L$ and $C_D$.

Figure 7

Mean drag coefficient $C_D$ at different ${\text{Re}}_{\mathrm{in}}$. The filled symbols correspond to numerical results (i.e., Portela et al. [26], Sohankar et al. [36], and Trias et al. [37]), whereas the open symbols correspond to experimental results (i.e., Davis et al. [38], Cheng et al. [39], Norberg et al. [40], and Saha et al. [41]).

Figure 8

Distribution of the mean pressure coefficient $C_P$ over the $A\text{−}B$, $B\text{−}C$, $C\text{−}D$, and $D\text{−}A$ sides of the single square cylinder considered. The filled symbols correspond to numerical results (i.e., Portela et al. [26] and Trias et al. [37]), whereas the open symbols correspond to experimental results (i.e., Chen and Liu [42], Bearman and Obasaju [43], and Mizota and Okajima [44]).

Figure 9

Contour plots of the magnitude of instantaneous vorticity field $\mathrm{\Omega }/(U_{\mathrm{in}}/T_{\mathrm{bar}})$ in the $X\text{−}Y$ plane for (a) the regular case and (b) the multiscale case. The contour levels range from 0.1 and 30. The red color indicates a high vorticity $\mathrm{\Omega }/(U_{\mathrm{in}}/T_{\mathrm{bar}})\ge 30$ and blue indicates a low vorticity $\mathrm{\Omega }/(U_{\mathrm{in}}/T_{\mathrm{bar}})\le 0.1$.

Figure 10

Isosurfaces of the second invariant of the velocity gradient tensor $Q$ in the 3D subdomain (i.e., $2\le X/T_{\mathrm{bar}}\le 10$, $-1/2L_Y\le Y\le 1/2L_Y$, and $-1/5L_Z\le Z\le 1/5L_Z$) for (a) the regular case and (b) the multiscale case. The isosurfaces plotted correspond to a positive value, i.e., $Q{T}_{\mathrm{bar}}^2/U_{\mathrm{in}}^2=30$.

Figure 11

Contour plots of the normalized mean streamwise velocity field, i.e., $U_1/U_{\mathrm{in}}$ in the $X–Y$ plane for (a) the regular case and (b) the multiscale case. The red color corresponds to a high value $U_1/U_{\mathrm{in}}\ge 1.5$ and the blue color corresponds to a negative value $U_1/U_{\mathrm{in}}\le 0$.

Figure 12

Contour plots of the normalized streamwise turbulence intensity field, i.e., $u_1^{\prime }/U_{\mathrm{in}}$ in the $X–Y$ plane for (a) the regular case and (b) the multiscale case. The red color corresponds to a high value $u_1^{\prime }/U_{\mathrm{in}}\ge 0.5$ and the blue color corresponds to a zero value $u_1^{\prime }/U_{\mathrm{in}}=0$.

Figure 13

Vertical distribution of the normalized mean streamwise velocity $U_1/U_{\mathrm{in}}$ at five selected streamwise locations ($X/T_{\mathrm{bar}}=2,5,15,25,$ and 35) for (a) regular array and (b) multiscale array. The mean velocity $U_1/U_{\mathrm{in}}$ is averaged over time and $Z$ direction.

Figure 14

Vertical distribution of the normalized mean vertical velocity $U_2/U_{\mathrm{in}}$ at five selected streamwise locations ($X/T_{\mathrm{bar}}=2,5,15,25,$ and 35) for (a) regular case and (b) multiscale case. The mean velocity $U_2/U_{\mathrm{in}}$ is averaged over time and $Z$ direction.

Figure 15

Vertical distribution of the turbulence intensities at five selected streamwise locations ($X/T_{\mathrm{bar}}=2,5,15,25,$ and 35) for (left) regular array and (right) multiscale array. From top to bottom, the three plots correspond to streamwise component $u_1^{\prime }/U_{\mathrm{in}}$ [see panels (a) and (b)], vertical component $u_2^{\prime }/U_{\mathrm{in}}$ [see panels (c) and (d)], and spanwise component $u_3^{\prime }/U_{\mathrm{in}}$ [see panels (e) and (f)]. The lines present the same locations as those in Figs. 13 and 14.

Figure 16

Streamwise evolution of the averaged normalized rms velocities (a) $u_1^{\prime }/U_{\mathrm{in}}$, (b) $u_2^{\prime }/U_{\mathrm{in}}$, and (c) $u_3^{\prime }/U_{\mathrm{in}}$. The data are averaged over time and $Y\text{−}Z$ plane.

Figure 17

Streamwise evolution of (a) the averaged normalized turbulent kinetic energy $K/U_{\mathrm{in}}^2$ and (b) the averaged normalized energy dissipation $\varepsilon /(U_{\mathrm{in}}^3/T_{\mathrm{bar}})$.

Figure 18

Streamwise evolution of the averaged normalized pressure $P/\rho U_{\mathrm{in}}^2$. The data are averaged over time and $Y\text{−}Z$ plane.

Figure 19

Time traces of (a) the lift coefficient $C_L$ and (b) the drag coefficient $C_D$ behind all four cylinders for the regular array.

Figure 20

Power spectral density of (a) the fluctuating component of the lift coefficient, i.e., $C_L^{\prime }$, and (b) the fluctuating component of the drag coefficient, i.e., $C_D^{\prime }$ for the regular array. The two dashed vertical lines indicate $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.18$ and 0.37, respectively.

Figure 21

Time traces of (a) the lift coefficient $C_L$ and (b) the drag coefficient $C_D$ for the multiscale array.

Figure 22

Spectra of the fluctuating component of the lift coefficient, i.e., $C_L^{\prime }$ for (a) the largest cylinder, that is, bar 4, (b) the intermediate cylinder, that is, bar 2, (c) the smallest cylinders, that is, bar 1 and bar 3 in the multiscale case. The three dashed vertical lines indicate the vortex shedding frequencies, i.e., $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.15$, 0.22, and 0.81, respectively.

Figure 23

Spectra of the fluctuating component of the drag coefficient, i.e., $C_D^{\prime }$ for (a) the largest bar, that is, bar 4, (b) the intermediate cylinder, that is, bar 2, and (c) the smallest cylinders, that is, bar 1 and bar 3 in the multiscale case. The three dashed vertical lines indicate twice the corresponding vortex shedding frequency, i.e., $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.30$, 0.44, and 1.62, respectively.

Figure 24

The fraction of the energy $\zeta \left(r\right)$ contained in the first $r$ modes in the five representative subdomains (i.e., $2.5\le X/T_{\mathrm{bar}}\le 5$, $12.5\le X/T_{\mathrm{bar}}\le 15$, $22.5\le X/T_{\mathrm{bar}}\le 25$, $32.5\le X/T_{\mathrm{bar}}\le 35$, and $35.5\le X/T_{\mathrm{bar}}\le 38$) for (a) the regular array and (b) the multiscale array. To highlight the most energetic modes, the profiles of $\zeta \left(r\right)$ corresponding to the first 100 modes are presented. The two dashed horizontal lines indicate $\zeta \left(r\right)=0.8$ and 1.0, respectively.

Figure 25

The same as Fig. 24 but plotted as a function of $r\left(x\right)/\delta \left(x\right)$.

Figure 26

The fraction of the energy $\zeta \left(r\right)$ contained in the first $r$ modes vs $r\left(x\right)/\delta \left(x\right)$ for both arrays in the further most subdomain, i.e., $35.5\le X/T_{\mathrm{bar}}\le 38$. The two dashed horizontal lines indicate $\zeta \left(r\right)=0.8$ and 1.0, respectively.

Figure 27

The fraction of the energy $\zeta$ as a function of $r/N_t$ for (a) the regular case and (b) the multiscale case in the subdomain with $2.5\le X/T_{\mathrm{bar}}\le 5$. Two different time intervals (i.e., $N_t=10$ and 20) are considered. To highlight the most energetic modes, the profiles of $\zeta \left(r\right)$ corresponding to the first 100 modes are presented. The two dashed horizontal lines indicate $\zeta =0.8$ and 1.0, respectively.

Figure 28

The fraction of the energy $\zeta$ as a function of (a) the first $r$ modes and (b) the normalized first $r$ modes (i.e., $r/\text{St}$) in the subdomain with $2.5\le X/T_{\mathrm{bar}}\le 5$. To highlight the most energetic modes, the profiles of $\zeta$ corresponding to the first 100 modes are presented. The two dashed horizontal lines indicate $\zeta =0.8$ and 1.0, respectively.

Figure 29

The same as Fig. 28 but in the subdomain with $12.5\le X/T_{\mathrm{bar}}\le 15$.

Figure 30

The same as Fig. 28 but in the subdomain with $35.5\le X/T_{\mathrm{bar}}\le 38$.

Figure 31

(a) Schematic front view of the computational domain and (b) the corresponding magnitude of instantaneous vorticity field $\mathrm{\Omega }$ normalized by $U_{\mathrm{in}}$ and $T_{\mathrm{bar}}$ in the $X\text{−}Y$ plane. Magnitudes of $\mathrm{\Omega }/(U_{\mathrm{in}}/T_{\mathrm{bar}})$ lower than 0.1 are in blue and magnitudes higher than 50 are in red. The red solid lines represent boundaries in the corresponding case with a smaller cross-sectional area.

Figure 32

Time traces of (a) the lift coefficient $C_L$ and (b) the drag coefficient $C_D$ behind all eight cylinders.

Figure 33

Instantaneous streamwise fluctuating velocity $u^{\prime }$, i.e., (a) the original and (b) the reconstruction field, together with the vertical fluctuating velocity $v^{\prime }$, i.e., (c) the original and (d) the reconstruction field, in the subdomain with $2.5\le X/T_{\mathrm{bar}}\le 5$ for the regular case. With respect to the reconstruction fields, only the first 20 modes are used.

Figure 34

(a) Time traces of the normalized POD coefficients of the first pair modes (i.e., ${\alpha }_1^{+}$ and ${\alpha }_2^{+}$) for the regular case in the subdomain with $2.5\le X/T_{\mathrm{bar}}\le 5$ and (b) the corresponding spectra. The dashed vertical line in plot (b) indicates the primary frequency of the large-scale vortex shedding, i.e., $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.18$.

Figure 35

The same as Fig. 34 but for the multiscale case. The dashed vertical line in plot (b) indicates the vortex shedding frequency of the largest wake, i.e., $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.15$.

Figure 36

Distribution of the normalized coefficients of the first and second modes for (a) the regular case and (b) the multiscale case in the subdomain with $2.5\le X/T_{\mathrm{bar}}\le 5$. The red dashed curves in plots (a) and (b) denote circles ${\left({\alpha }_1^{+}\right)}^2+{\left({\alpha }_2^{+}\right)}^2=R^2$ with the radius $R=0.73$ for the regular case and $R=0.51$ for the multiscale case.

Figure 37

(a) Time traces of the normalized POD coefficients of the third and fourth modes (i.e., ${\alpha }_3^{+}$ and ${\alpha }_4^{+}$) for the regular case in the subdomain with $2.5\le X/T_{\mathrm{bar}}\le 5$ and (b) the corresponding spectra. The dashed vertical line in plot (b) indicates the primary frequency of the large-scale vortex shedding, i.e., $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.18$.

Figure 38

The same as Fig. 37 but for the multiscale case. The dashed vertical line in plot (b) indicates $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.20$, which is quite close to the shedding frequency of the intermediate wake, i.e., $f{T}_{\mathrm{bar}}/U_{\mathrm{in}}=0.22$.

Figure 39

The same as Fig. 36 but for the third and fourth modes. The radiuses are $R=0.20$ and 0.34 for the regular and multiscale cases, respectively.

Figure 40

Distribution of the normalized coefficients of the first and second modes for (a) the regular case and (b) the multiscale case in the subdomain with $35.5\le X/T_{\mathrm{bar}}\le 38$. The radiuses are $R=0.38$ and 0.48 for the regular and multiscale cases, respectively.