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# Supercritical flows past a square cylinder with rounded corners

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Large-eddy simulations were used to investigate the supercritical aerodynamics of a square cylinder with rounded corners in comparison with those in the subcritical regime. First, the numerical methods, especially the dynamic mixed model, were validated on the basis of their prediction of supercritical flows past a circular cylinder. Then, the supercritical flows past a rounded-corner square cylinder were simulated and systematically clarified. Strong Reynolds number (Re) effects existed in the forces and local pressures as Re increased from o(10⁴) to o(10⁶). Changeover of flow patterns occurred as Re increased. At the supercritical Re, the free stream overall flowed along the cross sections of the cylinder, separated from the leeward corners and generated Karman vortices behind the cylinder. This pattern resulted in a much smaller recirculation region behind the cylinder compared with the subcritical flow. At the micro level, the flow experienced laminar separation and flow reattachment near the frontal corners, followed by the spatial development of turbulent boundary layers (TBLs) on the side faces and turbulent separation near the leeward corners. The feedback by large-scale primary vortex shedding and the small-scale turbulent motions in the high-frequency region with a slope of −5/3 were detected in the TBL. Their interaction affected the spanwise correlations of wall pressure fluctuations. The TBL on the side face differed from the zero-pressure-gradient flat-plate one; it was subjected to pressure gradients varying in space and time.
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Supercritical flows past a square cylinder with rounded corners
Yong Cao, and Tetsuro Tamura
Citation: Physics of Fluids 29, 085110 (2017); doi: 10.1063/1.4998739
View online: http://dx.doi.org/10.1063/1.4998739
PHYSICS OF FLUIDS 29, 085110 (2017)
Supercritical ﬂows past a square cylinder with rounded corners
Yong Caoa) and Tetsuro Tamura
Department of Architecture and Building Engineering, Tokyo Institute of Technology, 4259 Nagatsuta-cho,
Midori-ku, Yokohama, Kanagawa 226-8502, Japan
(Received 24 February 2017; accepted 2 August 2017; published online 23 August 2017)
Large-eddy simulations were used to investigate the supercritical aerodynamics of a square cylinder
with rounded corners in comparison with those in the subcritical regime. First, the numerical methods,
especially the dynamic mixed model, were validated on the basis of their prediction of supercritical
ﬂows past a circular cylinder. Then, the supercritical ﬂows past a rounded-corner square cylinder were
simulated and systematically clariﬁed. Strong Reynolds number (Re) effects existed in the forces and
local pressures as Re increased from o(104) to o(106). Changeover of ﬂow patterns occurred as Re
increased. At the supercritical Re, the free stream overall ﬂowed along the cross sections of the
cylinder, separated from the leeward corners and generated Karman vortices behind the cylinder.
This pattern resulted in a much smaller recirculation region behind the cylinder compared with the
subcritical ﬂow. At the micro level, the ﬂow experienced laminar separation and ﬂow reattachment
near the frontal corners, followed by the spatial development of turbulent boundary layers (TBLs)
on the side faces and turbulent separation near the leeward corners. The feedback by large-scale
primary vortex shedding and the small-scale turbulent motions in the high-frequency region with a
slope of 5/3 were detected in the TBL. Their interaction affected the spanwise correlations of wall
pressure ﬂuctuations. The TBL on the side face differed from the zero-pressure-gradient ﬂat-plate
[http://dx.doi.org/10.1063/1.4998739]
I. INTRODUCTION
Flow separations and vortex streets play a dominant role
in determining the aerodynamic characteristics of ﬂows past a
two-dimensional bluff body. On the other hand, the cross sec-
tion of a given body signiﬁcantly inﬂuences the time-averaged
ﬂow topology and the time-dependent behavior of the vortex-
shedding process. The aerodynamics of circular and square
cylinders, two typical shapes, has been attracting a great deal
of research interest because of its immense scientiﬁc and prac-
tical signiﬁcance. The ﬂow around a circular cylinder strongly
depends on the Reynolds number (Re = U0D/ν, where U0is the
free-stream velocity, Dis the diameter or width of the cylinder,
and νmeans the kinematic viscosity) because the ﬂow separa-
tion point varies due to the continuous and rounded shape. In
contrast, the aerodynamic characteristics of a smooth square
prism with sharp corners are hardly dependent on Re [at least
Re = U0D/ν=o(104)–o(106)] when the free-stream ﬂow is
uniform because the ﬂow fully separates only at the frontal
sharp corners. Consequently, the primary vortex (or the Kar-
man vortex) behind a square cylinder behaves more vigorously
and oscillates to a greater degree than that behind a circular
cylinder. In the light of this, corner modiﬁcation has widely
been introduced in engineering applications as a passive ﬂow
control method to reduce the drag force and the ﬂuctuating lift
force due to the vortex shedding in the wake, including but not
limited to rounded corners, chamfered corners, and recessed
corners (Kwok et al.,1988;Kawai,1998; and Tse et al.,2009).
a)Author to whom correspondence should be addressed: cao.y.aa@
m.titech.ac.jp
However, there are many fewer studies on the aerodynamics
of corner-modiﬁed square cylinders than on circular or square
cylinders, and hence, a lack of knowledge about them. An
unknown fundamental issue is the Reynolds number depen-
dence for such cross sections based on the fact that most of Re
values investigated so far are not close to those encountered by
actual structures in reality (e.g., if the width of a tall building
is 50 m and the air velocity is 5 m/s, then Re 107).
Previous studies on the aerodynamics of two-dimensional
rounded-corner square cylinders focused on subcritical or
lower Reynolds numbers, mostly due to the limitations of
the experimental facilities or computational power. Most of
them concentrated on the variations of aerodynamic features
Park and Yang (2016) recently clariﬁed the effect of rounding
the sharp edges of a square cylinder on the ﬂow instabilities
(respectively, the Hopf bifurcation and the onset of three-
dimensional wakes) by using an immersed boundary method.
It was found that rounding the corners of a square cylinder
tends to stabilize the ﬂow, and the maximum critical Re occurs
when r/D= 0.25 (where ris the projected length of the rounded
corners onto the frontal cylinder wall). Hu et al. (2006) experi-
mentally studied the effect of varying the rounded corner radius
on the near wake of a square prism by using particle imag-
ing velocimetry (PIV), laser Doppler anemometry (LDA), and
hotwire measurements at Re = 2600 and 6000. They found
that as r/Dincreases from 0 (square cylinder with sharp cor-
ners) to 0.5 (circular cylinder), the strength of the shed vortices
attenuates, the Strouhal number (St = fvsD/U0, where fvs is
the frequency of primary vortex shedding) increases by about
60%, and the vortex formation length almost doubles in size.
1070-6631/2017/29(8)/085110/17/30.00 29, 085110-1 Published by AIP Publishing. 085110-2 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) They suggested that the leading edge corner radius plays a more important role than the trailing one in inﬂuencing the near wake structure at these Re values. Furthermore, Zhang and Samtaney (2016) performed a direct numerical simulation (DNS) at Re = 1000; they emphasized the development of the separated and transitional ﬂow around the cylinder inﬂuenced by various rounded-corner radii. Their results showed that the time-averaged wake recirculation length does not monotoni- cally vary with the corner radius, the maximum value being at r/D= 0.125. Rounded corners also postponed the transi- tion onset downstream, occurring at the latest at r/D= 0.125. Tamura et al. (1998) performed numerical simulations of the unsteady ﬂow at around Re = o(104) past a square cylinder with rounded and chamfered corners with r/D= 0.167. The results showed that the rounded corners reduced mean drag by about 35% and ﬂuctuating lift by about 30%, together with an increase in St. Following that, Tamura and Miyagi (1999) experimentally studied the effects of inﬂow turbulence at Re = 3 ×104and found that turbulent inﬂow contributes to the ﬂow reattachment to the side faces of the r/D= 0.167 cylinder. Nevertheless, there is little experimental evidence con- cerning the Reynolds number effects beyond the subcritical regime on the rounded-corner square cylinder. An experiment performed by Delany and Sorensen (1953) about sixty years ago gave the ﬁrst insights into Reynolds number effects, where a rapid jump in the mean drag force was observed at Re 7 ×105for r/D= 0.167. Recently, Carassale et al. (2014) experi- mentally studied the aerodynamic behavior of rounded-corner square cylinders with r/D= 0.067 and 0.13 for Re values rang- ing between 1.7 ×104and 2.3 ×105. Interestingly, strong Re effects were observed for the cylinder with r/D= 0.13 under a free-stream turbulence intensity of 5%. This result could be interpreted as indicating that free-stream turbulence has a simi- lar effect as increasing Re. Nevertheless, one should notice that the Reynolds numbers tested by Carassale et al. (2014) are rel- atively low. Considering that the free-stream turbulence cannot completely replace the effect of increasing Re, a question still remains about the aerodynamics of such bluff bodies at very- high Re regardless of the imposition of free-stream turbulence. Moreover, it should be noted that in contrast with the measure- ments of Delany and Sorensen (1953) showing a drag crisis as Re increases, the time-averaged drag force gradually falls from the high value at Re >5×104. In the previous experiments, the time-averaged drag decreased within a small Re region, e.g., from 5 ×104to 1 ×105(Carassale et al.,2014). Like the classiﬁcation of Re regimes in ﬂows past a circular cylinder, we call the Re regions before and after the signiﬁcant reduc- tion in mean drag the subcritical and supercritical regimes, respectively. Despite the signiﬁcant drag decrease observed by Delany and Sorensen (1953) and Carassale et al. (2014), the limited information available is not sufﬁcient to system- atically understand very-high-Re ﬂow past a rounded-corner square cylinder. Especially at supercritical Re, information is lacking on the ﬂow and its interaction with the cylinder. The increasingly popular and reliable tool, large-eddy simulation (LES), can provide deep insights into supercritical ﬂows around a rounded-corner square cylinder. Theoretically speaking, there is no limitation on Re when the numerical grids are ﬁne enough. Moreover,as it is a numerical method, LES can simultaneously provide detailed ﬂow and pressure ﬁelds that would greatly improve the understanding of the aerodynamics of this bluff body. To the best of our knowledge, no numerical simulation, especially with a high-accuracy LES technique, has been conducted with the purpose of predicting supercriti- cal ﬂows around a rounded-corner square cylinder. However, accurate simulation of critical or supercritical ﬂows is not a trivial task, as has been clearly shown in previous numer- ical tests on ﬂows past a circular cylinder. This is because the ﬂow experiences the laminar separation, transition of the separated boundary layer to turbulence, ﬂow reattachment on the wall, and secondary separation of the turbulent bound- ary layer. As for supercritical ﬂows past a circular cylinder, Catalano et al. (2003) used LES with wall modeling to pass the need for resolving the ﬂow near the wall. The estimated global aerodynamic characteristics were generally promising even when using about 2.3 ×106grid points. But obviously, the accuracy of the predicted complex ﬂow near the cylin- der was not satisfactory. Since then, computational resources have become much more powerful, and this means that more accurate predictions can be made. Consequently, using the wall-resolving LES method to simulate this typical ﬂow has attracted much attention [see the studies of Ono and Tamura (2008), Lehmkuhl et al. (2014), Rodr´ ıguez et al. (2015), Lloyd and James (2016), and Yeon et al. (2016)]. Generally satisfac- tory predictions can now be made, including of the ﬂow process near the wall. More importantly, more experimental data are available on very-high-Re ﬂows past circular cylinders than on past rounded-corner square cylinders. In light of this situation, a circular cylinder would be preferable as a simulation bench- mark for evaluating the numerical performance of predicting supercritical ﬂows past a bluff body. To summarize, the previous studies do not provide ade- quate knowledge on supercritical ﬂows around a rounded- corner square cylinder. Moreover, a question remains about the ﬂows affected purely by Re instead of together with free- stream turbulence. The present numerical study therefore aims at elucidating the aerodynamic characteristics of supercritical ﬂows past a square cylinder with rounded corners under uni- form inﬂow and the effect of Re by making comparisons with the case of the subcritical ﬂow. Speciﬁcally, the main goals of this study include the following: (1) identifying whether increasing Re causes a variation in the aerodynamic forces and local pressures for square-section cylinders and describing these characteristics at different Re levels; (2) describing and understanding the effect of varying Re on the changeover of the mean ﬂow pattern and turbulent statistics around the cylinder, in particular recognizing whether the turbulent transition and ﬂow reattachment occur at supercritical Re even for a rounded- corner square cylinder; (3) explaining how the ﬂow patterns at different Re determine the characteristics of aerodynamic forces and local pressures; (4) describing the dynamics of the wake, separated shear layer, and boundary layer and vortical structures of different scales; (5) understanding the mecha- nism of interactions between turbulent structures in the reat- tached boundary layer and primary vortex shedding and how these interactions affect the spanwise correlations of ﬂuctuat- ing pressures; and (6) clarifying the features of the reattached boundary layer at the supercritical Re for this cylinder shape. 085110-3 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) The paper is organized as follows. Section II is a descrip- tion of the numerical methods. Section III validates the meth- ods and examines the grid resolution on the basis of simulations of supercritical ﬂows past a circular cylinder (Re = 6 ×105). Section IV describes the simulations of ﬂows past a rounded- corner square cylinder at Re = 2.2 ×104and 1 ×106and discusses their results, focusing on the aerodynamic features at supercritical Re compared with those in the subcritical region. II. NUMERICAL METHODS The governing equations are solved in the generalized curvilinear coordinate system. The ﬁltered Navier-Stokes and continuity equations for incompressible ﬂows are written in the dimensionless form as (J¯ui) t+ ξmJ U m¯ui= ξm J ξ m xi ¯p! + ξm 1 Re Jξm xj ξn xj ¯ui ξn+σm i!, (1) (JUm) ξm =0, (2) where Um(=∂ξ m xiui) is the contravariant velocity perpendicu- lar to the faces of the grid cells, and J(= xi ∂ξ m ) is the Jacobian of the transformation between the physical space and compu- tational space. The subgrid-scale (SGS) stress is expressed as σm i=JUm¯uiJ Umui. A. SGS modeling In the dynamic Smagorinsky model (denoted by DSM) proposed by Germano et al. (1991) and Lilly (1992), the SGS stress σm ihas to be modeled using the linear eddy-viscosity model to relate the residual stress to the ﬁltered rate of strain, σm i1 3σk kδij =Jξm xjνSGS ¯ Sij =Jξm xj 2C¯ 2q2¯ Sij ¯ Sij ¯ Sij !, (3) where νSGS means the eddy viscosity and ¯ is the size of the grid ﬁlter. ¯ Sij =1 2¯ui ∂ξ m ∂ξ m xj+¯uj ∂ξ n ∂ξ n xiis the rate-of-strain tensor. The unknown Smagorinsky coefﬁcient Cis determined explicitly as C=DLk iMk iE 2¯ 2DMk iMk iE, (4) where Lk i=g JUk˜ ¯uiJ JUk¯ui,Mk i=4 H S H ¯ Sk iI ¯ S ¯ Sk i, and ¯ Sk i =J∂ξ k xj ¯ Sij. This study takes an advanced approach to SGS modeling, i.e., the dynamic mixed model (denoted by DMM) proposed by Zang et al. (1993). The DMM can be regarded as a combination of the dynamic Smagorinsky model and the scale-similarity model (Bardina et al.,1983). The three components of SGS stress are treated differently: the modiﬁed Leonard term (Lm ij ) is calculated explicitly, while the modiﬁed cross term and the modiﬁed SGS Reynolds stress are modeled with the linear eddy-viscosity approximation, σm i1 3σk kδij =Lm ij +Jξm xjνSGS ¯ Sij =Lm ij +Jξm xj 2C¯ 2q2¯ Sij ¯ Sij ¯ Sij !, (5) where Lm ij =J∂ξ m xj¯ujuiJ∂ξ m xj¯uj¯uiis the modiﬁed Leonard term. The unknown coefﬁcient Cis determined by Eq. (6), which differs from the dynamic Smagorinsky model by adding the term Hm iin the numerator, C=Lm iHm iMm i 2¯ 2Mm iMm i , (6) where Lm i= I JUm˜ ¯uiJ JUm¯ui,Hm i= K J∂ξ m xj¯uj˜ ¯ ¯ui K J∂ξ m xj¯uj¯ ¯ ui, and Mm i=4 ˜ ¯ S ˜ ¯ Sm iI ¯ S ¯ Sm i. As analyzed by Zang et al. (1993), the DMM has several advantages over the DSM: less modeling requirements because of the explicit computation of the modiﬁed Leonard term; the potential of energy backscatter to the resolved scales as pro- vided by the modiﬁed Leonard term; no need for aligning the principal axis of the SGS stress tensor and the resolved strain rate tensor. One could suppose that the DMM provides more accurate predictions of supercritical ﬂows; this conjecture will be examined in Sec. III. B. Discretization in time and space The code is based on the ﬁnite difference method that has been successfully applied to ﬂows past bluff bodies at subcrit- ical Re (Ono and Tamura,2002;Tamura and Ono,2003; and Cao and Tamura,2016). A non-staggered grid is used herein. Speciﬁcally, the velocity and pressure are stored at the cell cen- ters, while the contravariant velocity is stored on cell faces. A fractional step method, originally proposed by Kim and Moin (1985), is used to advance the velocity and pressure solutions in time. The time marching of the momentum equation is a hybrid one; the Crank-Nicolson scheme is applied to the vis- cous term and an explicit third-order Runge-Kutta method is used for the convective term. The temporal treatment of the convective term is to allow a relatively larger time step (Le and Moin,1991). The detailed procedure is in the work of Cao and Tamura (2016). Spatial discretization is generally treated as a second- order central difference. However, the convective term is approximated using a fourth-order central difference scheme. To avoid numerical instability, numerical dissipation is added through the convective term; the amount is controlled by a parameter αND. In this study, αND = 0.2 was used, which means that the numerical dissipation is very small in com- parison with αND = 3 of the third-order upwind scheme pro- posed by Kawamura and Kuwahara (1984) and αND = 1 of the UTOPIA (uniformly third-order polynomial interpolation algorithm) scheme. 085110-4 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) III. SUPERCRITICAL FLOWS PAST A CIRCULAR CYLINDER The numerical experiments examined predictions of supercritical ﬂows past a bluff body. The ﬂow past a circu- lar cylinder at Re = 6 ×105was simulated, and the results were compared with experimental data. A. Numerical conditions Four cases, namely, case 1, case 2, case 3, and case 4, were computed in order to examine the effects of different grid resolutions and SGS models. The computational domain was 30 ×24 ×1. Note that all lengths in this study, unless otherwise stated, have been non-dimensionalized by the cylin- der diameter or width D. The upstream length between the inlet and the cylinder center was 10, while the downstream length from the cylinder center to the outlet was 20. The total height between the upper and lower boundaries was 24, result- ing in a blockage ratio (BR) of 4.2%. BR was deﬁned as the ratio of projected area of the cylinder upon the cross-sectional area of the computational domain or test section; speciﬁcally for two-dimensional circular or square cylinder, it was the ratio of the cylinder width or diameter to the height of the computational domain or test section. The spanwise length was Lz= 1 for the simulations of supercritical ﬂows around a circu- lar cylinder, following the studies of Ono and Tamura (2008) (Lz= 1), Lehmkuhl et al. (2014) (Lz= 0.5π), and Rodr´ ıguez et al. (2015) (Lz= 0.5πand 1). Lzwas uniformly discretized into 160 cells in all four cases, resulting in a spanwise resolu- tion of z= 0.006 25, which can be deemed sufﬁcient on the basis of LES tests performed by Ono and Tamura (2008) with z= 0.0067 and Rodr´ ıguez et al. (2015) with z= 0.0078. In terms of the grid dimensions, cases 1 and 2 used the same grid system, i.e., 601 ×420 ×161 in (ξ,η,ζ) coordinates with a total of 40.6 ×106cells. By comparison, cases 3 and 4 had a grid resolution of 801 ×500 ×161, i.e., 64.5 ×106cells in total. The grid system of cases 3 and 4 on the (ξ,η) plane is shown in Fig. 1. The height of the ﬁrst cell nearest to the cylinder wall was y= 5.3 ×10 5<0.05/Re1/2 = 6.5 ×10 5. The maximum non-dimensional wall distance y+was less than 2, and the cylinder-averaged value was around 1. The details are summarized in Table I. The stretching ratio of cell heights near the wall was 1.024 for cases 1 and 2 and 1.021 for cases 3 and 4. The inlet boundary condition was a uniform steady inﬂow without any velocity ﬂuctuations, i.e., a constant velocity FIG. 1. Grid system on the (ξ,η) plane for case 3 and case 4: (a) whole grid; [(b) and (c)] close-ups. TABLE I. Summary of global quantities including a comparison with previous studies. Case Re Grid (Lz)y+max y+avg Ncell SGS ¯ CDC0 D¯ CLC0 LSt Lf Present Case 1 6 ×105601 ×420 ×161 1.91 1.00 40.6M DSM 0.22 0.010 0.0053 0.062 0.441 1.18 Case 2 6 ×105601 ×420 ×161 1.92 1.01 40.6M DMM 0.20 0.010 0.0055 0.062 0.471 1.08 Case 3 6 ×105801 ×500 ×161 1.67 0.88 64.5M DSM 0.24 0.013 0.0077 0.071 0.416 1.16 Case 4 6 ×105801 ×500 ×161 1.68 0.89 64.5M DMM 0.22 0.011 0.0032 0.069 0.444 1.11 Expt. Bearman (1969) 4 ×105... ... ... ... ... 0.23 ... ... ... 0.46 ... Schewe (1983) 6 ×105. . . . . . . . . . . . . . . 0.22 . . . . . . 0.018 0.47 . . . LES Ono and Tamura (2008) 6 ×105. . . . . . . . . 13.4M DMM 0.21 . . . . . . 0.08 0.48 . . . Lehmkuhl et al. (2014) 6.5 ×105... <2 . . . 83.2M WALE 0.232 0.008 0.027 0.076 0.44 1.10 Rodr´ ıguez et al. (2015) 7.2 ×105... <2 . . . 89.4M WALE 0.213 0.009 0.016 0.075 0.45 1.07 Rodr´ ıguez et al. (2015) 8.5 ×105... <2 . . . 105.1M WALE 0.218 0.009 0.007 0.070 0.45 1.10 085110-5 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 2. Comparison of present time- averaged drag and ﬂuctuating lift coef- ﬁcients with published experiments. proﬁle u= (U0, 0, 0) was imposed; the cylinder was a no-slip wall without any wall functions; the spanwise end boundary conditions were periodic; the upper and lower boundaries were free-slip. The outlet was set as a convective condition for all velocity components and pressure. For the other boundaries, a zero-gradient condition was used for the pressure. With respect to the effects of the SGS models, cases 1 and 3 used the DSM, while cases 2 and 4 used the DMM. The dimensionless time step t*(=tU0/D) was 0.0001 or 0.000 05. The time intervals for the averaging operation were about 27 shedding cycles. B. Results and discussion Table Ishows the global quantities together with those of the previous experiments and LES studies conducted at similar Re, including the time-averaged and root mean square (rms) aerodynamic forces based on the whole spanwise length ( ¯ CD,C0 D,¯ CL, and C0 L), St, and the time-averaged formation lengths of the Karman vortex behind the cylin- der (Lf). Moreover, the mean drags and ﬂuctuating lifts in Fig. 2are compared with those from the previous experiments (Wieselsberger,1922;Bursnall and Loftin,1951; Delany and Sorensen,1953;Achenbach,1968;Jones et al., 1969;So and Savkar,1981;Cheung,1983;Schewe,1983; and Blackburn and Melbourne,1996). First, the time-averaged drags in the experiments are obviously dispersed to a large degree [Fig. 2(a)]. As pointed out by Zdravkovich (1997), the inﬂuencing parameters (e.g., ratio of the spanwise length to cylinder diameter Lz/D, BR, and free-stream turbulence intensity Ti) strongly affect the aerodynamic characteristics. However, it seems that the ¯ CDvalues from the experiments were mostly between 0.1 and 0.3. The exception, 0.6, was measured by Achenbach (1968) with unfavorable inﬂuencing parameters, i.e., Lz/D= 3.3 (too small to avoid the interfer- ence effect caused by the junction of the cylinder and the wind-tunnel walls), BR = 16% (much larger than the accept- able value 5% and tended to augment the blockage effect), and Ti = 0.7% (relatively high free-stream turbulence intensity). Although their combined effects are complex, the unfavorable Lz/Dand BR tend to decrease the base pressures and increase the mean drag forces according to West and Apelt (1982). The present LES results are around 0.22, reasonable values for most of the experiments. The table also indicates that the present ¯ CD is consistent with the previous well-validated LES predictions of supercritical ﬂows. Regarding the rms values of the lift, the present results are in a reasonable region that accounts for the scatter in the experiments [see Fig. 2(b)]. Moreover, the present rms lifts (0.07) agree with the previous LES results. In addition, the drag shows a small degree of ﬂuctuation in time with C0 D0.01, which is slightly larger than the LES predic- tions of Lehmkuhl et al. (2014) and Rodr´ ıguez et al. (2015). The very small time-averaged lifts ¯ CLindicate a symmetric time-averaged ﬂow topology on both sides of the cylinder that characterizes the supercritical regime of ﬂow past a circular cylinder. The time-series of the lifts have well-deﬁned peri- odic frequencies, matching the ﬁndings at supercritical Re by Bearman (1969), which indicates organized vortex shed- ding in the wake. Lfis generally consistent with the recircula- tion lengths obtained by Lehmkuhl et al. (2014) and Rodr´ ıguez et al. (2015). To summarize, it seems that all four cases tested give satisfactory prediction accuracy in terms of the global quantities in comparison with the previous experimental and numerical studies. Figure 3(a) shows the time-averaged pressure distribu- tions along the circumferential direction of the cylinder, where θ= 0 and 180 correspond to the frontal and base stagna- tion points, respectively. Overall, the time-averaged pressures agree well with the experimental values. Note that Flachs- bart’s experimental data (1929) are taken from the work of Roshko (1961). The kink in the numerical pressure distribu- tions is correct. In particular for case 4, there is a near pressure plateau beginning at θls = 100 and a sharp increase between θTr = 105 and θr= 110. These points correspond to the sepa- ration of the laminar boundary layer, transition to turbulence beyond laminar separation, and reattachment of the turbulent FIG. 3. Distributions around the cylin- der: (a) time-averaged pressure; (b) time-averaged wall shear stress. θ= 0 corresponds to the frontal stagnation point. θls,θr, and θts are the angu- lar positions of laminar separation, ﬂow reattachment, and turbulent boundary layer separation for case 4. 085110-6 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) boundary layer. They are represented more clearly by the distribution of the time-averaged wall shear stresses (after non- dimensionalization) in Fig. 3(b), where τ > 0 in the clockwise direction. For case 4, the wall shear stress experiences sign changes at θls = 100, θr= 110, and θts = 114 that correspond to laminar boundary layer separation, ﬂow reattachment, and turbulent boundary layer separation, repectively. The angu- lar positions of laminar separation, turbulence transition, ﬂow reattachment, and secondary turbulent separation were quanti- tatively measured by Tani (1964), Achenbach (1968), and Pfeil and Orth (1990). But the skin friction probe was a miniature “fence” protruding above the surface in the work of Achenbach (1968); hence, the separation bubble marking the supercritical regime was not successfully detected. Tani (1964) used dye visualization in water and found angles of θls = 102, θTr = 110, and θr= 117, while Pfeil and Orth (1990) measured the pres- sure distribution within the boundary layer, ﬁnding θls = 100, θTr = 110, and θr= 114. By comparison, the present results (say, case 4) reproduced the positions of laminar separation, in spite of the slight under-prediction of the transition and reattachment positions of the boundary layer. However, regard- ing the SGS models, only the DMM provided the obvious reattachment after the transition to turbulence of the bound- ary layer; this is based on the fact that the wall shear stress changes sign from negative to positive at θr= 110 for cases 2 and 4 in Fig. 3(b). Similar conclusions about SGS model- ing were obtained by Ono and Tamura (2008). In addition, for the high grid resolutions tested presently, it seems that SGS models play a more important role than grid resolution does because the ﬂow reattachment was still not reproduced even though the grid resolution increased in going from case 1 to case 3. Another critical parameter is the separation point of the turbulent boundary layer (θts). We obtained θts = 114 on the basis of the sign change of the wall shear stress. Achenbach (1968) suggested that the turbulent separation of the boundary layer is at θts = 147 on the basis of skin friction measurements. This value is widely used to validate numerical results (Lehmkuhl et al.,2014;Rodr´ ıguez et al.,2015; and Yeon et al., 2016). However, as mentioned before, the accuracy of the measurement by Achenbach (1968) is worth further discussion because of the possible disturbance from the fence probe. Two factors, at least, are notable in Fig. 5 in the work of Achenbach (1968). One is the failure of a separation bubble to form, from the distribution of skin friction at the supercritical Re. The other is the delay of the starting point of the near plateau of the base pressure compared with other experimental and numer- ical studies; that is, the plateau of the base pressure started at θ150, while most of the other experiments obtained θ = 125–140 [e.g., Flachsbart’s measurements, as cited in Roshko (1961), Bursnall and Loftin (1951), Tani (1964), and Shih et al. (1993) among others]. Thus, it is possible that Achenbach (1968) overpredicted the separation location of the turbulent boundary layer. More experimental measurements are needed to make a numerical validation. The time-averaged velocity streamlines of case 4 in Fig. 4(a) shows that the width of the recirculation region behind the cylinder becomes small because of the delay of the ﬂow separation compared with the well-known subcritical ﬂow. Figure 4(b) depicts the vector ﬁeld of the time-averaged velocity near the separation point. We can clearly see the pro- cess of laminar separation, ﬂow reattachment, and turbulent separation. The closed ﬂow region between separation and reattachment is called the separation bubble. It is worth noting that at this Re, two separation bubbles formed symmetrically on both sides of the cylinder. Beyond the separation bubbles, reverse ﬂow occurs very near the cylinder wall. On the whole, the supercritical ﬂow simulations past a circular cylinder validated well the present numerical schemes and time-marching algorithm. Grid convergence was obtained when using the same SGS model because the grid reﬁnement did not signiﬁcantly change the global quantities and time- averaged ﬂow topology. The DMM is recommended over the DSM because of its accurate prediction of ﬂow reattachment on the cylinder wall. IV. SUPERCRITICAL FLOWS PAST A ROUNDED-CORNER SQUARE CYLINDER A. Numerical model and grid system The numerical methods validated in Sec. III were used for predicting supercritical ﬂows past a rounded-corner square FIG. 4. Time-averaged velocity ﬁeld of case 4: (a) streamlines around the cylinder; (b) the velocity vector near the separation bubble. 085110-7 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 5. Grid system on the (ξ,η) plane for case E6L: (a) whole grid; [(b) and (c)] close-ups. cylinder. Note that only the more advanced DMM was used in this section. The radii of the rounded corners were r= 0.167 scaled by the cylinder width, which is similar to the val- ues in the experiments of Delany and Sorensen (1953) and Carassale et al. (2014). The supercritical regime was repre- sented by Re = 1.0 ×106, namely, case E6L, which is higher than Re that causes a dramatic decrease in drag [see the work of Delany and Sorensen (1953)]. Their computational domain and grid system on the (ξ,η) plane are shown in Fig. 5. Case E6L had a spanwise length of Lz= 4 that has been demon- strated to be sufﬁcient for the objectives of this study (see the Appendix). The typical subcritical Re = 2.2 ×104was also taken into consideration, namely, case E4. In similar grid topologies, case E4 differed from the supercritical cases only in terms of the grid parameters in Table II. The total cell num- bers were 21.8 and 384.9 ×106for case E4 and case E6L, respectively. The spanwise resolutions were z= 0.006 25 for case E6L (the same as in the simulations of supercritical ﬂows past a circular cylinder) and z= 0.05 for case E4. The height of the ﬁrst cell (y) nearest to the no-slip cylin- der wall was less than 0.1/Re0.5 for the subcritical case, while it was less than 0.05/Re0.5 for the supercritical case. The maximum y+was less than 3.0, and the space-averaged y+ was around 1.0. t*was 0.0005 for case E4 and 0.0001 for case E6L. The statistics of the aerodynamic characteristics were computed for a time span of over 20 vortex-shedding cycles. B. Aerodynamic force and pressure This simulation explored how the aerodynamic forces and local pressure vary with increasing Re from o(104) to o(106). The statistics of the global quantities are summarized in Table III, including the time-averaged and rms forces and St. Overall, the effect of Re is strong on the aerodynamic forces, as is apparent from the variations in the quantities as Re increases from 2.2 ×104to 1.0 ×106. The mean drag coefﬁcient ¯ CD experiences two distinct quantitative levels in the subcriti- cal and supercritical regimes, i.e., it decreases substantially from 1.39 to 0.55 as Re increases. These results match the experimental measurements on similar cylinders conducted by Tamura et al. (1998), Delany and Sorensen (1953), and Carassale et al. (2014). The rms values of the forces also decrease when Re goes beyond the subcritical regime. For example, the rms lift coefﬁcient C0 Lfalls dramatically by about 60% as Re increases from 2.2 ×104to 1.0 ×106. Moreover, St nearly doubles over the course of the rise in Re. Figures 6(a) and 6(b) show the distributions of the time- averaged and ﬂuctuating pressure coefﬁcients at different Re. Like the total forces, one can see strong Re effects in the local pressures not only in the distribution shapes but also in the quantitative values. hpihas two valleys near the frontal and leeward corners at Re = o(106). The ﬁrst one is located at θmin1 = 50.9 at Re = 1.0 ×106, which moves downstream as Re increases. The second valley is at θmin2 = 126 in the upstream part of leeward rounded corners. Between them, hpi has a plateau on the side wall. In comparison with the obvious extreme negative values, there is no big change in hpiwith θ after the frontal corner at the subcritical Re. In particular, the base pressure of case E6L is much larger than that of case E4. For this cylinder, the decrease in ¯ CDwith increasing Re to the supercritical regime is primarily due to the horizontal com- ponent decomposed from the negative pressure on the frontal rounded corners and the smaller suction pressure on the rear face. In Fig. 6(b), the supercritical case E6L shows two sharp peaks near the frontal and leeward corners, denoted by θmax1 and θmax2. The exact locations of θmax1 and θmax2 are shown in TABLE II. Summary of grid information for ﬂows past a rounded-corner square cylinder. Lzand zare spanwise lengths and span resolutions, respectively; yis the height of the ﬁrst cell nearest to the cylinder wall; y+max and y+avg are the maximum and span-averaged y+;t*is the non-dimensional time step. Case Re Grid Ncell LzSGS zy y+max y+avg t* Case E4 2.2 ×104301 ×300 ×241 21.8 ×10612 DMM 0.05 5.59 ×10 41.13 0.33 0.0005 Case E6L 1.0 ×1061201 ×500 ×641 384.9 ×1064 DMM 0.006 25 4.08 ×10 52.71 1.06 0.0001 085110-8 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) TABLE III. Summary of global quantities for the ﬂows past a rounded-corner square cylinder at subcritical and supercritical Re. Previous experimental data are also included. Case Re ¯ CDC0 D¯ CLC0 LSt LES Case E4 2.2 ×1041.39 0.06 0.0053 0.73 0.144 Case E6L 1.0 ×1060.55 0.04 0.0100 0.30 0.280 Expt. Tamura et al. (1998) 6.0 ×1041.11 . . . . . . 0.39 0.140 Carassale et al. (2014) 3.7 ×1041.45 . . . 0.05 0.52 0.138 Delany and Sorensen (1953) Subcritical 1.24 . . . .. . . . . . . . Delany and Sorensen (1953) Supercritical 0 .56 . . . . . . . . . . . . FIG. 6. Distributions of time-averaged and ﬂuctuating pressures around the cylinder. TABLE IV. Main angular positions characterizing the distributions of time- averaged and ﬂuctuating pressures and wall shear stress. θfand θlare the ranges of angular positions of the frontal and leeward rounded corners, respectively. Case θfθlθmin1 θmin2 θmax1 θmax2 θs1 θrθs2 Case E4 33.7–56.3 123.7–146.3 46 . . . 47 .. . 50 . . . . . . Case E6L 33.7–56.3 123.7–146.3 51 126 56 130 54 58 127.6 Table IV. They are slightly more downstream than θmin1 and θmin2. Nevertheless, the above characteristic angular positions are still within the ranges of the rounded corners. Furthermore, the magnitude of the σppeak at the leeward corner is slightly larger than that at the frontal corner. Apart from these peaks, σpon the side faces of case E6L is much lower than that in case E4 on average, whereas the σpvalues on the rear faces are comparable. C. Statistical ﬂow ﬁeld The signiﬁcant effect of Re on aerodynamic forces and pressures points to a changeover of ﬂow patterns due to the increase in Re from subcritical to supercritical levels. This section will attempt to describe and understand the mean ﬂow and turbulent statistics at supercritical Re. Meanwhile, the statistical ﬂow ﬁeld will be used to explain the characteristics of the integral forces and local pressures acting on the cylinder. The overall ﬂows at the subcritical and supercritical Re are shown in Fig. 7by using the streamlines of span- and time-averaged velocity with the stream function ψranging between 0.2 and 0.2. The ﬂow topology at the supercritical Re differs substantially from that at the subcritical Re. The most notable point is that the ﬂuid separates completely from the frontal corner at subcritical Re, whereas the free stream ﬂows along the side face and ﬁnally separates from the leeward cor- ner at supercritical Re. Such a ﬂow induces a large curvature in the streamlines near the rounded corners and a mild cur- vature near the leeward corners. As a result, the streamlines at supercritical Re get closer to each other around the frontal and leeward corners, which leads to valleys of mean pressure around the corners [see Fig. 6(a)]. Behind the cylinder, the width of the recirculation region in the vertical direction (dw) is much smaller than that at subcritical Re. In particular, dw>D at subcritical Re, but dw<Dat supercritical Re. Meanwhile, the streamwise length of the recirculation region (denoted by Lf) at supercritical Re is much smaller as compared with case E4; speciﬁcally, Lffalls from 1.49 at subcritical Re to 1.07 at supercritical Re. The reduction in the size of the recircu- lation regions from the subcritical to supercritical regime is FIG. 7. Streamlines of span- and time- averaged velocity: (a) case E4 and (b) case E6L. 085110-9 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 8. Close-ups of streamlines of span- and time- averaged velocity: (a) case E4, Re = 2.2 ×104and (b) case E6L, Re = 1.0 ×106. Note that ψmeans the stream function. responsible for the growth in the time-averaged base pressure and St. In order to describe the ﬂow features near the wall, Fig. 8 shows close-ups of the time-averaged velocity in the vicinity of the side face. Moreover, the quantitative distributions of time-averaged wall shear stress are plotted in Fig. 9. At sub- critical Re, a large-scale recirculation region (also called the secondary vortex) exists immediately behind the frontal corner and beneath the separated shear layer. However, its velocity is actually very low, as evidenced by the small positive wall shear stress when θ[60, 90]. At supercritical Re, the free stream separates from the downstream part of the frontal rounded cor- ner and then reattaches to the side face. The separation bubble is formed between the ﬁrst separation point θs1 and the reat- tachment point θr. But it is so small-scale that it is invisible on the scale of the cylinder width. The reattached ﬂow remains throughout almost the whole side face, before separating again from the leeward corner at θs2. The exact angular positions of ﬂow separation θs1, reattachment θr, and secondary separation θs2 for case E4 and case E6L are listed in Table IV. Taking the typical angular positions in the mean and rms pressure dis- tributions into consideration, the following relations can be easily found: θmin1 < θs1 < θ max1 < θrand θmin2 < θs2 < θmax2 . One can interpret the ﬁrst relation simply as that the separation bubble is exactly in the region with a strongly adverse pressure FIG. 9. Distributions of time-averaged wall shear stress around the cylinder; the clockwise direction is positive. gradient, and σpreaches a maximum in the middle of the separation bubble, implying oscillation of the separation bubble in time. The turbulent statistics of the wake, shear layer, and attached boundary layer are explored for both subcritical and supercritical Re in Fig. 10. Figure 10(a) shows the ﬁeld of normal Reynolds stress. At subcritical Re, the velocity ﬂuc- tuation next to the side walls is even larger than that in the near-wake region; this results from the large ﬂapping degree of the separated shear layers. In contrast, at supercritical Re, the ﬂuctuation in velocity is eliminated by the attached bound- ary layers, and the peaks are located in the near-wake because of the periodic alternation of the low-speed recirculation ﬂow and high-speed potential ﬂow entrained by the Karman vor- tex. Regarding the wake, Fig. 10(b) displays the quantitative proﬁles of the rms velocity σuand σvalong the wake center- line, where the peak value of case E6L is signiﬁcantly reduced compared with case E4 and its location is moved upstream. It indicates that the reduction in the agitation intensity of the ﬂow structures in the wake is possibly offset by their shorter distance to the rear face in case E6L. Consequently, the overall effects on the rear faces are similar in case E4 and case E6L, and this leads to a comparable σpon the rear faces [see Fig. 6(b)]. Regarding the shear layer or boundary layer at subcritical and supercritical Re, Fig. 10(c) shows the proﬁles of σualong the vertical direction at the stations of x/D=±0.33, ±0.30, ±0.25, ±0.20, ±0.15, ±0.10, ±0.05, and 0, where the vertical axis is a logarithmic scale in order to display detailed infor- mation adjacent to the wall. Note that x/D= 0.33 is located on the ﬂat part of the side face with θ= 56.6 but very close to the ends of the rounded corners. As expected, the subcritical case E4 has a larger σuand the heights of the peaks gradually increase in the xdirection along the curvature of the separated shear layer. But for case E6L, the peaks of σuare generally very weak and their heights are very low. This indicates that the boundary layer is very thin. Surprisingly, σu0.15 at x= 0.33 is substantial, which possibly indicates oscillation of the separation bubble near the frontal rounded corner. The relatively high σuis consistent with the high σpnear the frontal corners [see Fig. 6(b)]. 085110-10 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 10. (a) Contours of span-averaged ﬂuctuating streamwise velocity for case E4 and case E6L; (b) proﬁles of σuand σvalong the wake centerline; (c) proﬁles of σuversus the vertical direction in the shear-layer region, where the rule for measuring the magnitude of σuis shown at the top-right corner. D. Instantaneous ﬂow ﬁeld In this section, the instantaneous ﬂow ﬁeld is visualized in order to describe the dynamic behavior of the wake, detached shear layer, and reattached boundary layer at subcritical and supercritical Re. Figure 11 shows the instantaneous ﬂow ﬁeld for case E4 and case E6L within the same region of x= [ 1, 10] and y= [ 2, 2], which is represented by the streamwise veloc- ity (U), magnitude of velocity (magU), and spanwise vorticity component (Vort Z). They are chosen at the instants having local maximum values of total lift. It is apparent that periodic FIG. 11. Instantaneous ﬂow ﬁeld represented by the streamwise velocity, magnitude of velocity, and spanwise vorticity (from top to bottom) for case E4 (left) and case E6L (right) at the instants with extreme values of total lift. 085110-11 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 12. Close-ups of instantaneous ﬁeld around the cylinder at the same instants as the counterparts in Fig. 11: (a) case E4 and (b) case E6L; (c) distribution of instantaneous streamwise velocity component Uon the third mesh layer nearest to the cylinder wall for case E6L. vortex streets form in the wake at subcritical and supercritical Re. At this instant, the ﬁrst primary vortex forms on the upper side of the wake recirculation region. However, the distance between any two Karman vortices decreases in going from case E4 to case E6L, which implies that the vortex shedding frequency increases in going from case E4 to case E6L. The width of the vortex streets is reduced by the increase in Re not only in the near wake but also in the far wake. By comparison, the Karman vortex at supercritical Re is composed of eddies on smaller scales than at subcritical Re. Figure 12 shows close-ups at the same instants shown in Fig. 11 to get further insight into the dynamics of the shear layer and boundary layer. Case E4 at Re = o(104) displays a typical subcritical ﬂow; that is, the ﬂow completely sep- arates from the frontal rounded corners, and the free shear layers ﬂap to a large degree on the two sides of the cylin- der and connect to the primary vortex behind the cylinder. Regarding the separated shear layers, one can observe the evolution of the Kelvin-Helmholtz (K-H) instability and tran- sition to turbulence. Two shear layers ﬂap out of phase with each other on both sides of the cylinder. In contrast, a distinct ﬂow image appears at Re = o(106). Overall, the free stream seems to ﬂow along the shape of cross sections of rounded- corner square cylinders and ﬁnally separates from the leeward corners. Here, the laminar free stream seems to transit to turbu- lence near the frontal corner. After that, turbulence boundary layers (TBLs) with very thin thickness gradually develop on the side faces. The TBLs then start to separate from the lee- ward corners. This means that the separated shear layer at supercritical Re is turbulent, composed of various small-scale eddies. The small-scale eddies accumulate to form the Karman vortex behind the cylinder. Furthermore, the detached shear layers at supercritical Re oscillate at the same frequency as the vortex shedding. In addition, Fig. 12(c) shows the instanta- neous existence of a separation bubble near the frontal corners, indicated by the sign change of Ualong the circumferen- tial direction on the third mesh layer nearest to the cylinder wall. It also oscillates periodically with the Karman vortex shedding. E. Vortical structures of diﬀerent scales and their eﬀects This section qualitatively and quantitatively examines the scales of the vortical structures around the cylinder. It elucidates the interactions between the small-scale turbulent structures in the attached boundary layer and the large-scale vortex shedding as well as sheds light on the effects on spanwise correlations of ﬂuctuating velocity and pressure. Figure 13 displays the one-dimensional energy spectra of the ﬂuctuating streamwise velocity for case E4 and case E6L, where a line with a slope of 5/3 is added for com- parison. It should be noted that both probe points are at x = 0, while the vertical positions yin the ﬁgures are near the time-averaged shear layers for case E4 and the attached boundary layer for case E6L. At Re = o(104), the Strouhal frequency ( fvs ) caused by the Karman vortex shedding in the wake and its harmonics are characterized by sharp peaks. The free shear layers springing from the bluff bodies at such a high Re relate to the K-H instability. The transition eddies with a frequency signiﬁcantly higher than fvs can be measured. FIG. 13. One-dimensional energy spectra of the streamwise velocity compo- nent measured above the upper cylinder walls for case E4 and case E6L, where a line with a slope of 5/3 is added for comparison. The subcritical probe point is located near the separated shear layer at x= 0, while the supercritical one corresponds to three vertical positions y= 0.5005, 0.501, and 0.51 at x= 0. 085110-12 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 14. Spanwise correlations of ﬂuctuating stream-wise velocity at Re = 1.0 ×106. Note that the ﬁrst point to the second one is separated by 10z. For the rounded-corner square cylinders, the broadband peak corresponding to the shear-layer instability also occurs at fsl. Speciﬁcally, the frequency ratio of the shear-layer frequency to the primary vortex is fsl/fvs 20. It is in good agreement with the generalized power law approximation proposed by Prasad and Williamson (1997) for circular cylinders ( fsl/fvs = 0.0235 Re0.67 19). In contrast, for supercritical Re, three positions in the TBL (i.e., y= 0.5005, 0.501, and 0.510) are chosen in order to plot the energy spectra; they correspond to y+= 14.7, 29.4, and 294. There is a prominent peak corresponding to primary vortex shedding. That is, the boundary layer is unceasingly subjected to unsteadily harmonic oscillation fed back from the vortex shedding behind the cylinder. Comparing the spectra at y= 0.5005, 0.5010, and 0.5100 reveals that the energy of the peak frequency relative to those of the surrounding frequencies is larger when the probe point is farther from the wall. How- ever, the frequency of the shear-layer instability cannot be seen any more. Instead, the slope of 5/3 is observed neatly in the high-frequency range, which means turbulent motions on a wide range of small scales in the attached turbulent bound- ary ﬂow on the side faces. Thus, the velocity quantities in this TBL could be decomposed into U=hUi+˜ U+u0, where the angle bracket indicates the time-averaged value, the tilde indicates the periodic component of vortex shedding, and the prime indicates the high-frequency turbulent compo- nents. The high-frequency turbulent components can be seen in the visualization of the ﬂuctuating stream-wise velocity in Fig. 19. Now let us investigate the effects of the different-scale components on the TBL on the spanwise correlations at dif- ferent spanwise distances (see Fig. 14). Indeed, the spanwise distance is scaled by the cylinder size D. The small-scale struc- tures shown in Fig. 19 result in rapid decay of the spanwise correlations of ﬂuctuating stream-wise velocity Ru(s) at the small spanwise distance s. It is worth noting that the ﬁrst and second points are separated by 10z= 0.0625 because of the FIG. 16. Spanwise correlations Rp(s) of ﬂuctuating pressure probed in the middle of upper and lower cylinder walls, where only half of the spanwise lengths are shown, considering the symmetry of Rp(s) around the middle of the spanwise lengths. Note that the ﬁrst and second points are separated by 10z. sampling spatial resolution of time-dependent data, which is larger than the TBL thickness (δ0.02–0.03). After the rapid decay, Ru(s) remains non-zero at large salthough the values are very small. The non-zero Ru(s) is determined by the inﬂu- ence of the Karman vortex shedding covering the whole span [see the wake along the span in Fig. 15(a)]. Furthermore, at large s,Ru(s) tends to be smaller at the points closer to the wall. This phenomenon is consistent with the lower energy of the primary vortex shedding at the points closer to the wall. This is as expected because the impact of the vortex shedding becomes smaller for the points closer to the wall due to the increased wall shear effects. Figure 16 shows the spanwise correlation coefﬁcients Rp(s) of ﬂuctuating pressure coefﬁcients in the middle of the upper and lower cylinder walls. Note that only half of the span- wise lengths are shown, considering the symmetry of Rp(s) around the middle of the spanwise lengths under the peri- odic conditions on both spanwise boundaries. There are few differences between the spanwise correlations based on the upper and lower walls. First, at subcritical Re, Rp(s) gradually decreases from unity with s. The time series of the pressure coefﬁcients versus dimensionless time t*(=tU0/D) shown in Fig. 17(a) was measured in the middle of the side faces with a spanwise spacing Ms= 3. Clearly, the variation in pressure amplitude and the phase lag between the pressures gauged separating Ms= 3 contribute to the decrease in Rp(s). Of more interest is the Rp(s) at supercritical Re. It drops rapidly from unity (note that the probe points have an interval larger than δ), followed by a smooth reduction tendency with rising s. Two time series of pressure coefﬁcients with a span- wise distance of 1 are displayed in Fig. 17(b). Like the velocity series on the TBL, the total series of pressure is composed of a time-averaged constant, sine waves, and higher-frequency components. The pressure ﬁeld in the wake and on the cylinder FIG. 15. Instantaneous distributions on the three surfaces, i.e., the cylinder sur- face, the mid-span plane, and the mid- height plane: (a) the primary vortex shedding covering the whole span; (b) the relation between pressure ﬁeld in the wake and on the cylinder. 085110-13 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 17. Time series of pressure coef- ﬁcients measured in the middle of side walls for case E4 (left) and case E6L (right). is visualized in Fig. 15(b) for the low-frequency sine wave. The low-frequency pressure oscillation is over the cylinder span, together with the vortex shedding in the wake. Examination of the power spectra shows that its frequency is consistent with the vortex shedding frequency. Generally speaking, the wall pressure ﬂuctuations in the large scales can originate from the potential ﬁeld and the outer layers of the TBL (Choi and Moin, 1990 and Farabee and Casarella,1991). Thus, the reason that the spanwise correlations of ﬂuctuating pressure did not die out even at large distances is the feedback from the nearly parallel Karman vortex shedding, which spans the whole domain. But at small distances scomparable to the TBL thickness (<<D), a rapid decrease in spanwise correlations away from unity occurs at both Rp(s) and Ru(s); it is produced by the superimposed high-frequency vortical structures in the TBL. Let us brieﬂy summarize the interaction between the Karman vortices and the TBLs on the side faces. The ﬂows attached to the side faces are inﬂuenced by the Karman vor- tices and random turbulent components simultaneously. In other words, the attached ﬂows are the TBLs under the pres- sure gradients that vary periodically with the primary vortex shedding behind the cylinder. However, the quasi-coherent structures inherent in the TBLs are so small-scale compared with the Karman vortices that we could regard the TBLs as two-dimensional thin ﬂow layers. On the scale of the cylin- der width, the thin ﬂow layer (as a whole) oscillates in time at the same pace as the alternate vortex shedding, together with the separation bubbles near the frontal corners. This is the feedback from the Karman vortex shedding into the TBLs on the side faces. Conversely, small-scale turbulent structures developed in the TBLs separate, accumulate, and comprise the large-scale Karman vortices. F. Discussion on TBL on side faces The TBL attached on the side faces might be different from the standard fully developed TBLs on a ﬂat plate. The initial proﬁle in the front of the side face exhibits a convexity unlike that of a standard laminar or turbulent boundary layer on a ﬂat plate because of the high acceleration around the rounded corners. Following that, the boundary layer is sequen- tially inﬂuenced by several unsteady external effects including an adverse pressure gradient (APG), a zero pressure gradient (ZPG), and a favorable pressure gradient (FPG) in the stream- wise direction. There is also an APG of a certain degree along the vertical direction y. Now let us focus on the effect of the mean pressure gra- dients, most notably, in the streamwise direction. Figure 18(a) shows the distributions of time-averaged pressure coefﬁcients and their gradients ( dhpi dx ) along x. The region from upstream to downstream is divided into APG, ZPG, and FPG. The nearly ZPG regime is bordered by x= 0.1 and 0.2. Figure 18(b) shows the successive variation of hUiproﬁles using outer law variables. First, let us discuss the proﬁles in the APG region. The velocity proﬁle at x= 0.33 immediately downstream from the corner has a ﬂow reversal in the vicinity of the wall (small separation bubble) and an obvious convexity when the distance to the wall is about 0.005 due to the greater upstream acceleration in this local vertical region [Fig. 12(b)]. But the convexity progressively becomes smoother as xincreases, which is a characteristic of APG. In the APG region, the thick- ness of the boundary layer increases quickly downstream (see also the ﬂow ﬁeld). Second, the proﬁles in the ZPG region are almost unchanged, which means the TBL becomes fully developed there and independent of x. However, when mea- sured using the inner law variables y+and u+, the proﬁles in the ZPG region appear laminar-like, i.e., higher than the standard log-law proﬁle. One of the reasons is possibly the accelera- tion phase of the superimposed harmonic oscillation cycles, as observed in the studies by Spalart and Baldwin (1987) and Akhavan et al. (1991). Moreover, the vertical APG with increasing ytends to stabilize the turbulent boundary layer in a similar manner to stable thermal stratiﬁcation. The ﬂow FIG. 18. (a) Distributions of time- averaged pressure coefﬁcients and their gradients along x, divided by the regions of APG, ZPG, and FPG. (b) Proﬁles of the time-averaged streamwise velocity using outer law variables. Red solid line: x= 0.1; blue solid line: x= 0.2. 085110-14 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 19. (a) Instantaneous stream-wise velocity ﬁeld, where the locations of the velocity proﬁles are marked. (b) Distribution of U-hUiat y= 0.501, where the mean velocity hUiis selected on the basis of the x= 0.05 proﬁle. acceleration in the FPG region helps the proﬁles to steepen and gradually rebuild the convexity when the distance to the wall is less than 0.005. In the ﬂow ﬁeld of instantaneous spanwise vorticity [Fig. 19(a)], FPG seems to affect the vortical structures in the TBL. Especially near the leeward corner, there are fewer and weaker vorticity structures in the outer region. Figure 19(b) shows the ﬂuctuating component of U-hUion the y= 0.501 plane, where hUiis selected on the basis of the x= 0.05 pro- ﬁle. The turbulent structures in this FPG region are more ordered, elongated, and aligned in the streamwise direction upstream of the leeward corners. Such features in acceler- ating boundary layers were systematically investigated by Piomelli et al. (2000). V. CONCLUSIONS LES was used to investigate the aerodynamics of a square cylinder with rounded corners at supercritical Re compared with those at subcritical Re. First, the numerical methods, especially SGS models, were validated as to their prediction accuracy of supercritical ﬂows past a circular cylinder. The results show that the dynamic mixed model can accurately predict the process of laminar separation, transition to turbu- lence, ﬂow reattachment, and turbulent separation. After that, the ﬂow past a rounded-corner square cylinder was simulated and systematically clariﬁed. The major ﬁndings are as follows: (1) Signiﬁcant variations of aerodynamic forces and pres- sure were found as Re increased from o(104) to o(106). Speciﬁcally, the time-averaged drag and St undergo two distinct quantitative levels in the subcritical and supercritical regimes. Local pressures change not only in the shapes of their distribution but also in their quantitative levels. In particular, extreme maximum or minimum pressures appear at the frontal and rear corners. (2) The changeover of the mean ﬂow patterns brought about by the increase from the subcritical to supercritical Re was clariﬁed. Unlike the complete separation from the frontal corners at subcritical Re, the free stream overall ﬂows along the cross sections of cylinder, sepa- rates from the leeward corners, and generates the twin vortices behind the cylinder at supercritical Re. At a microlevel, the ﬂow experiences laminar separation and ﬂow reattachment near the frontal corner, resulting in a small separation bubble. This type of ﬂow has a much smaller recirculation region behind the cylinder and a low turbulent kinetic energy. The ﬂow patterns explained the characteristics of the forces and local pressures. (3) The dynamics of the wake and separated shear layer and boundary layer were elucidated. Even at super- critical Re, periodic vortex shedding occurs but with a shorter distance between neighboring vortices. The laminar ﬂow starts to transit to turbulence near the frontal corner, followed by the spatial development of small-scale eddies in the TBL on the side face and tur- bulent separation near the leeward corner. The detached TBL oscillates at the same frequency as the vortex shedding, and the small eddies accumulate to form the primary vortex behind the cylinder. (4) Different-scale vortical structures were qualitatively and quantitatively examined. At subcritical Re, typi- cal frequencies were measured in the shear layer: the primary vortex shedding frequency and its harmonics and the K-H instability frequency. At supercritical Re, besides the turbulent motions in the high-frequency region with 5/3 slope, there is feedback of the pri- mary vortex shedding covering the whole span in the TBL. Moreover, there is a large-scale gap between the turbulent structures in the TBL and the Karman vortex. The small-scale structures in the TBL are responsible 085110-15 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) TABLE V. Comparison of global quantities at different spanwise lengths of the computational domain at Re = 1.0 ×106. Case Grid LzNcell z y+max y+avg ¯ CDC0 D¯ CLC0 LSt Lf Case E6 1201 ×500 ×161 1 96.7 ×1060.006 25 2.71 1.07 0.58 0.11 0.0155 0.41 0.253 1.06 Case E6L 1201 ×500 ×641 4 384.9 ×1060.006 25 2.71 1.06 0.55 0.04 0.0100 0.30 0.280 1.07 FIG. 20. Distributions of time-averaged and ﬂuctuating pressures when the spanwise length of the computational domain is Lz= 1 and 4 at Re = 1.0 ×106. for the rapid drop in the spanwise correlations of the ﬂuctuating velocity and wall pressure from unity at very small spanwise distances relative to the cylinder size. The large-scale vortex shedding leads to span- wise correlations being non-zero at large distances. Moreover, the effect of vortex shedding on the TBL changes depending on the normal distance from the wall. (5) The TBL on the side face was found to be different from the standard zero-pressure-gradient ﬂat-plate TBL, which is sequentially inﬂuenced by several unsteady external effects, including APG, ZPG, and FPG in the streamwise direction. The convexity in the initial velocity proﬁle is progressively smoothened in the APG region and as the boundary layer thickness increases. The proﬁles remain unchanged and appear laminar-like in the ZPG region. The FPG immediately upstream the leeward corner rebuilds the convexity of the proﬁles, and it elongates and aligns the TBL vortical structures in the streamwise direction. This paper reported for the ﬁrst time a big difference between the subcritical and supercritical ﬂows around the same geometry, i.e., a square cylinder with rounded corners. Some issues remain to be studied in the future: the variation from sub- critical to supercritical ﬂows around a rounded-corner square cylinder; the similarities and differences in the transition from the subcritical to supercritical regime between the case of the rounded-corner square cylinder and the case of the canonical circular cylinder. ACKNOWLEDGMENTS The computations were conducted on the Earth Simulator at the Japan Agency for Marine-Earth Science and Technol- ogy (JAMSTEC). Authors are also grateful to the support of oversea doctoral study from the China Scholarship Council (CSC). APPENDIX: EXAMINATION OF THE SPANWISE LENGTH OF THE COMPUTATIONAL DOMAIN AT SUPERCRITICAL RE Here, let us examine the effects of the spanwise length of the computational domain for a supercritical-Re ﬂow past a rounded-corner square cylinder. On the basis of the simulation results of supercritical ﬂows past a circular cylinder, this study tests two commonly accepted spanwise lengths, i.e., Lz= 1 and 4. The mesh information is listed in Table V, together with typi- cal physical quantities. The extension of the spanwise length Lz from 1 to 4 (i.e., from case E6 to case E6L) slightly inﬂuences the time-averaged drag and lift coefﬁcients and the formation length within a deviation of about 5%. However, it somewhat affects the ﬂuctuation-related quantities, i.e., the rms forces and St. In Fig. 20, the hpidistributions are very similar for the two spanwise lengths Lz= 1 and 4; this behavior is similar to that of the mean drag. Even though the shorter Lztends to over- estimate the ﬂuctuating quantities on the side and rear faces, the shapes of the rms pressure distributions at supercritical Re are FIG. 21. One-dimensional energy spectra of the stream-wise velocity ﬂuc- tuation in the wake (e.g., x= 1.5, y= 0), near the separated shear layer (e.g., x= 0.71, y= 0.42), and the attached turbulent boundary layer (e.g., x= 0, y= 0.501). The spectra are shifted along the Euu axis for clarity. 085110-16 Y. Cao and T. Tamura Phys. Fluids 29, 085110 (2017) FIG. 22. Two-point correlation of the spanwise velocity ﬂuctuation in the wake (a), the recirculation region behind the cylinder (b), and the attached turbulent boundary layer (c). Note that the ﬁrst and second points are separated by 10z. generally similar when the spanwise lengths Lzequal 1 and 4. Thus, a small quantitative difference in the rms pressure is expected when extending the spanwise length. However, such a small difference should not inﬂuence the ﬂow physics. Moreover, the spanwise resolution zis carefully treated to be much smaller than those adopted widely for subcritical- Re ﬂows. The value of zlearns the experiences from the well-validated LESs of supercritical ﬂows around a circu- lar cylinder (Ono and Tamura,2008;Lehmkuhl et al.,2014; and Rodr´ ıguez et al.,2015) and was selected to be smaller than the above references while the present study follows the numerical methods of Ono and Tamura (2008) and also has a negligible amount of numerical dissipation. It was also shown to be sufﬁcient in Sec. III. Thus, it is reasonable from the computational point of view to choose Lz= 4 and maintain the high spanwise resolution to capture the small-scale ﬂow structures. On the basis of the supercritical ﬂows around a rounded- corner square cylinder, the sufﬁciency of the spanwise res- olution is further examined. Two ways are used to evaluate the grid resolution for LES. First, the one-dimensional energy spectra are computed and plotted in Fig. 21. Three typical positions are selected: ﬁrst one is located in the wake (beyond the mean recirculation zone behind the cylinder); second one is near the separated shear layer; and the third one is in the attached turbulent boundary layer. The peaks in Fig. 21 are reasonably associated with the frequency of the Karman vor- tex shedding ( fvs) and its harmonic. More notably, just before the dissipation ranges, the large-extent inertial sub-ranges fea- tured by a 5/3 slope are predicted accurately for all three positions. That is, the small-scale turbulent motions are sufﬁ- ciently reproduced in different ﬂow regions of interest around the cylinder. Second, the spanwise correlations of ﬂuctuat- ing velocity provide another measure of spanwise resolutions as shown in Fig. 22. When the correlations reach zero, one can judge how many cells cover the largest scale turbulent motion. As recommended by Davidson (2009), at least 8 cells should be sufﬁcient for LES. In Fig. 22, the largest scales are on average covered by about 8, 70, and 100 cells in the attached boundary layer, near the separated shear layer, and in the wake, respectively. To summarize, the present spanwise resolution is believed to be sufﬁcient after the evaluation by the two com- mon approaches based both on energy spectra and two-point correlations. As shown in Fig. 16, the spanwise correlations of the wall pressure ﬂuctuation are not weak even at the large spanwise distances of supercritical Re. The reason may be the lack of a large phase variation in vortex shedding along the span (e.g., oblique shedding), as shown in the previous studies (Szepessy and Bearman,1992;Szepessy,1994; and Cao and Tamura,2015). Nevertheless, out-of-phase vortex shed- ding occurs intermittently and infrequently compared with parallel shedding. Thus, there should be no change in the main ﬂow physics if the domain length is further elongated from Lz= 4. This study focused on the three-dimensional vortical structures in the ﬂow when parallel shedding dominates [see Fig. 15(a)] (e.g., the various-scale eddies comprising the Kar- man vortex in the wake, the longitudinal vortices connecting the Karman spanwise vortices, and the small-scale turbulent structures in the boundary layer). The spanwise correlations of the spanwise velocity components are plotted in Fig. 22, which were measured in the wake, the recirculation region behind the cylinder, and the attached turbulent boundary layer. They fall approximately to zero for the distances shorter than half the total spanwise length. It illustrates that the development of these three-dimensional vortical structures and their interac- tion are independent of the spanwise length. To summarize, case E6L with Lz= 4 is believed to be reasonably sufﬁcient to discuss the supercritical ﬂow around the rounded-corner square cylinders. 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[8] found that even with smallest corner rounding the agreement with experimental data is better since such rounded corners cause the recirculation bubble at the side faces of the cylinder to be larger, thereby, drag coefficient is decreased. ... ... Widely studied configurations derived from the square cylinder are various modifications of its corners, for example, rounding (Cao and Tamura 2017), chamfering (Tamura et al. 1998;Tamura and Miyagi 1999), and corner recesses (He et al. 2014). Other variations such as fins, strakes, and shrouds have also been explored (see, e.g., Naudascher et al. 1981). ... Article Full-text available The flow past a cactus-shaped cylinder with four ribs is investigated numerically using large eddy simulations (LES) at Reynolds number of 20,000 and experimentally using particle image velocimetry (PIV) at Reynolds number of 50,000. In both approaches, the full range of angle of attack is covered. LES results show a good qualitative and quantitative match of the aerodynamic properties to previous experimental data, although the value of the critical angle of attack is slightly lower. The results confirm that there is no Reynolds number dependency within the investigated range allowing a comparison of the flow fields from the present LES and PIV. Significant variations of the flow patterns with the angle of attack are found and quantified using the recirculation length and wake width. Overall, the observed angle of attack dependence resembles the behaviour of the square cylinder. However, the studied cylinder has a narrower wake at all angular orientations. Proper orthogonal decomposition is used to identify large coherent structures in the flow. At all angles of attack the first two modes remain dominant making it possible to reconstruct the periodic vortex shedding process using a low-order model. ... The main methodologies of passive control are changing the roughness of the bluff body surface and modifying the geometry. Typical passive control methods include implementation of dimpled surfaces 42 or bumps, 29 rounded corners, 8 slits or slots, 23,24 perforated pipes, 9 and affiliated secondary components. 20,26,37 Despite the fact that passive flow control methods can indeed attenuate the mean drag and fluctuating lift forces acting on the bluff body, there is a drawback that cannot be ignored: the inefficiencies inherent in passive control. ... Article Full-text available The flow passing a circular cylinder can trigger flow-induced vibrations such as the vortex-induced vibration. In this paper, the authors investigated an active method to control the cylinder wake flow. The control scheme was achieved by active blowing through a structured porous surface that was manufactured by 3D printing precisely. The blowing momentum was changed by various mass flow rates so that it defined different values of a non-dimensional momentum coefficient Cμ. The experimental investigation was conducted in a wind tunnel. A 2D particle image velocimetry system was used to measure global flow fields. The Reynolds number based on D was 10 000 in the subcritical region, where D is the cylinder diameter. The proper orthogonal decomposition (POD) was utilized as a reduced-order model. Experimental results showed that transformations could be found in POD modal characteristics and vortex shedding frequencies. Fluctuations in the global wake were suppressed. Moreover, intensities of turbulence kinetic energy and elements of the Reynolds stress tensor T were decreased in the near wake region. It can be concluded that active blowing jets through the structured porous surface of the circular cylinder can be used to control the surrounding flow with effective Cμ values. ... The wall-mounted hemisphere can be regarded as either rough elements [1][2][3][4][5][6][7] or obstacles. [8][9][10][11][12][13][14][15] The studies on rough elements focused on their influence on the turbulent boundary layer and the associated heat transfer [16][17][18] and drag reduction. When obstacles are used, very complex flow patterns are introduced, such as an upstream horseshoe vortex system and a recirculation zone with trailing vortices in the wake area. ... Article Full-text available Turbulent channel flows around a wall-mounted hemisphere numerically are investigated by large eddy simulation, and the Reynolds number based on the hemisphere’s diameter is 3 × 10 ⁴ . The statistical characteristics and turbulent structure evolution are revealed in the Eulerian frameworks and Lagrangian frameworks. The vortex identification and Dynamic Mode Decomposition (DMD) are used to study the evolution of turbulent structure in the Eulerian frameworks, and the finite-time Lyapunov exponents are applied to identify Lagrangian coherent structures (LCS) in the Lagrangian framework. It is found that the developing angle of the hairpin vortex is ∼7° at two frameworks. What is more, there are some hairpin vortices formed behind the hemisphere and some turbulent structures formed near the wall by DMD method. The correlation analysis is applied to investigate the angle variation and scale variation of turbulent structures, and it is observed that the angle of turbulent structures is negative at Y/ d ≥ 1.2 and the spanwise length scales of turbulent structures increase as it moves downstream. By studying the LCS behind a wall-mounted hemisphere, there is formation of “kink” caused by viscous interaction between some hairpin vortex legs, which is the characteristic of hairpin vortex deformation. The comparisons of statistical characteristics between Eulerian frameworks and Lagrangian frameworks are conducted by the correlation analysis, the spectrum analysis, and the structure functions. ... The drag coefficient attained minima when the angle of incidence was between 5 and 10 degrees. Cao and Tamura [22] simulated subcritical and supercritical flows around square cylinder with a rounded corner ratio r/D = 0.167. Two Reynolds numbers, Re = 2.2 × 10 4 and Re = 1.0 × 10 6 were considered, denoting the subcritical and supercritical regimes, respectively. ... Preprint Full-text available Tall buildings are often subjected to steady and unsteady forces due to external wind flows. Measurement and mitigation of these forces becomes critical to structural design in engineering applications. Over the last few decades, many approaches such as modification of the external geometry of structures have been investigated to mitigate wind-induced load. One such proven geometric modification involved rounding of sharp corners. In this work, we systematically analyze the effects of rounded corner radii on the flow-induced loading for a square cylinder. We perform 3-Dimensional (3D) simulations at a high Reynolds number ofRe = 1\times 10^5$which is more likely to be encountered in practical applications. An Improved Delayed Detached Eddy Simulation (IDDES) formulation is used with the$k-\omega$Shear Stress Transport (SST) model for near-wall modelling. IDDES is capable of capturing flow accurately at high Reynolds numbers and prevents grid induced separation near the boundary layer. The effects of these corner modifications are analyzed in terms of the resulting mean and fluctuating components of the lift and drag forces compared to a sharp corner case. Plots of the angular distribution of the mean and fluctuating pressure coefficient along the square cylinder's surface illustrate the effects of corner modifications on the different parts of the cylinder. The windward corner's separation angle was observed to decrease with an increase in radius, resulting in a narrower and longer recirculation region. Furthermore, with an increase in radius, a reduction in the fluctuating lift, mean drag, and fluctuating drag coefficients has been observed. ... As shown in the previous numerical simulation performed by Tamura et al. (1998), a rounded corner applied to a square cylinder affected the wake separation and reduced the drag and lift coefficients. Subsequently, that the rounded corner of square cylinders can suppress the wake separation and influence the vortex shedding frequency has been repeatedly observed in experiments and simulations (e.g., Hu et al., 2006;Park and Yang, 2016;Zhang and Samtaney, 2016;Cao and Tamura, 2017). Moreover, an application rounding the roof's trailing edge to suppress the wake separation was employed for a hatchback Ahmed body by Thacker et al. (2012). ... Article Full-text available The wake bi-stability behind notchback Ahmed bodies is investigated by performing wind tunnel experiments and large eddy simulations (LESs). The focus of this study is on the suppression of bi-stable wakes achieved by rounding the roof's trailing edge of the body. The suppression effect is found to depend on the Reynolds number (Re). The wake behind a sharp edge remains bi-stable for all tested Re. However, for a rounded edge with small radius, wake bi-stability at Re=0.5×10^5 and wake symmetrization with 0.75×105≤Re≤1.5×10^5 are observed. Increasing Re with Re≥1.75×10^5, the wake returns to the bi-stable state. Particularly, with Re≥2×10^5, a stable asymmetric wake state with no switches is observed for long periods. Performing LES confirms the expected wake asymmetry at Re=0.5×10^5 and symmetry at Re=1×10^5 for the case of rounded edge with a small radius. Besides, another wake symmetry is observed for the rounded edge with a large radius at Re=0.5×10^5. For the two wake symmetries shown in the LES results, the symmetrization is attributed to wake suppression in the notchback region, forcing the flow separation from the rear roof to attach to the slant on both sides of the body. ... For rounded corners, despite the number of studies assessing the large-Reynolds number case (see for e.g. Lamballais, Silvestrini & Laizet 2008Cao & Tamura 2017), few works have considered the dependence of the onset of the instabilities on the corner curvature. Park & Yang (2016) determined, via linear stability analysis, how the primary two-dimensional and the three-dimensional instabilities are affected by rounding the four corners of a rectangular cylinder with AR = 1, exploring the shapes ranging between the square cylinder with sharp edges and the circular cylinder. ... Article The primary instability of the flow past rectangular cylinders is studied to comprehensively describe the influence of the aspect ratio$AR$and of rounding the leading- and/or trailing-edge corners. Aspect ratios ranging between$0.25$and$30$are considered. We show that the critical Reynolds number ($\textit {Re}_c$) corresponding to the primary instability increases with the aspect ratio, starting from$\textit {Re}_c \approx 34.8$for$AR=0.25$to a value of$\textit {Re}_c \approx 140$for$AR = 30$. The unstable mode and its dependence on the aspect ratio are described. We find that positioning a small circular cylinder in the flow modifies the instability in a way strongly depending on the aspect ratio. The rounded corners affect the primary instability in a way that depends on both the aspect ratio and the curvature radius. For small$AR$, rounding the leading-edge corners has always a stabilising effect, whereas rounding the trailing-edge corners is destabilising, although for large curvature radii only. For intermediate$AR$, instead, rounding the leading-edge corners has a stabilising effect limited to small curvature radii only, while for$AR \geqslant 5$it has always a destabilising effect. In contrast, for$AR \geqslant 2$rounding the trailing-edge corners consistently increases$\textit {Re}_c$. Interestingly, when all the corners are rounded, the flow becomes more stable, at all aspect ratios. An explanation for the stabilising and destabilising effect of the rounded corners is provided. Article This paper presents two-dimensional unsteady Reynolds-Averaged-Navier–Stokes simulations of flow past a circular cylinder beneath a free surface at a Reynolds number of 4.96×104. The effects of the free surface on the wake dynamics and hydrodynamics are systematically examined over a parameter space consisting of the Froude number (0.2–0.8) and the gap ratio (0.1–2.0). For high Froude numbers, it is easier to agitate the water to produce violent surface distortion compared to low Froude numbers. Three surface deformation patterns are identified and the underlying mechanisms are proposed. The wake type transition from the wake featured a jet flow, to the one-sided vortex shedding, to the free-surface modulated Kármán vortex shedding, and to the pure Kármán vortex shedding is discussed in detail. The strength hierarchy between the three shear layers present in the wake plays a decisive role in the transition of the wake types. An extra recirculation zone appears near the free surface due to the blocking effect of the front blunt body. The proximity to the free surface exerts a modulation effect on the temporal and spectral characteristics of hydrodynamics. Relatively large mean drag force with substantial fluctuations can be induced when (unilateral or bilateral) vortex shedding process occurs. When a jet dominates the wake, the mean drag force stays at a low level and the hydrodynamic fluctuations are suppressed. The circular cylinder is always subjected to a downward thrust which increases as the cylinder approaches the free surface. The critical range of parameter combinations is provided when the wake dynamics and hydrodynamics are not influenced by the free surface. Article This paper aims to provide a comprehensive review of the literature and to impart a perspective on the current status and future directions for fundamentals of flow around and heat transfer from a square cylinder with modified corners. The focus is on how rounded, chamfered, recessed, and cut corners of a square cylinder impact the wake structure, flow instability, fluid forces, vortex dynamics, wake width, vortex formation length, boundary layer thickness, displacement thickness, transition to turbulence, drag crisis, and heat transfer. A circular cylinder undergoes drag crisis flow while a square cylinder does not. Sufficiently rounded corners can rejuvenate the drag crisis flow. The corner modifications encompassing heat transfer performance from low to high can be sorted out as sharp, cut, rounded, and chamfered, given the same scale of modifications. Based on the review and analysis of data in the literature, the optimal corner modifications are suggested for reduced forces and enhanced heat transfer. The research gap in the literature is identified and put forward for future investigations. Article The flow around a square cylinder is widely studied as a paradigmatic case in bluff body aerodynamics. The effects of several physical parameters of the setup, and the errors induced by turbulence models, numerical schemes and grid density have been emphasized in a huge number of studies during the past two decades. Surprisingly, the effects of the grid quality on such a class of flow has been overlooked. The lack of a shared approach and suggested best practices for high-quality grid generation among scholars and practitioners follows. The present study aims at filling this gap. The cell skewness and non-orthogonality are adopted as metrics of the grid quality. The errors induced by poor quality cells and the possible corrective measures are discussed in a Finite Volume Method framework. The effects of the cell quality on the simulated flow are systematically evaluated by a parametrical study including four different types of grid boundary layer. The obtained results are compared among them and discussed in terms of instantaneous and time averaged flow fields, stress distribution at wall, and aerodynamic coefficients. Both the overall modelling error and the skewness-induced one are evaluated with reference to a huge number of data collected from previous studies. The local error induced by few, moderately skewed, near-wall cells upwind the cylinder propagates windward because of the convection-dominated problem, and globally affects the boundary layer separation and the vortex shedding in the wake. Skewness around the trailing edge only affects the flow to a lower extent. The skewness error on bulk aerodynamic coefficients may largely prevails on the overall modelling error, in spite of the very simple turbulence model deliberately adopted in the study. Hybrid grid boundary layer made of structured cells along the cylinder sides and unstructured ones around its edges provides results analogous to the ones obtained with a fully orthogonal grid, in spite of some clusters of few skewed cells far from the wall. Hybrid grid boundary layer is recommended as a fine balance between accuracy and flexibility in grid generation, when full orthogonal grid boundary layer is not feasible around real-world engineering applications having complicate geometries with multiple obtuse or acute edges. Article The flow past a square cylinder at Re = 2.2 × 104 is analyzed by large-eddy simulation (LES) using the fine grids in order to represent details of near-cylinder flows. The accuracy of LES on structured and unstructured grids is assessed from the engineering viewpoint, compared with previous studies. The finite differencing method code with 4th order central scheme for the convective term is used for structured LES, while the open-source finite volume method code (OpenFOAM 2.3.0) with 1st–2nd order schemes is applied for unstructured LES. Typical schemes in OpenFOAM are tested, i.e., “LUST” (blending of linear and linear upwind schemes), “limitedLinear” (TVD) and “linearUpwind” (linear upwind). In this study, three effects are emphasized: numerical schemes for convective terms, meshing strategies, and spanwise resolution and length. On the whole, OpenFOAM can obtain fairly accurate prediction of the time-averaged and r.m.s quantities no matter which numerical scheme is used. “LUST” and “linearUpwind” are suggested. Meshing refinement in wake can be a solution to improve the far-wake velocity distribution and to overcome the earlier energy decay of turbulent motions in the inertial subrange caused by artificial dissipation. Different degrees of instantaneous flow reattachment near the trailing edges are found in the results obtained by OpenFOAM. Flow reattachment is closely associated with the roll-up of shear layer, and the flow topology between the shear layer and the side wall featured by the frontal and leeward secondary vortex. In this regard, the refinement of hexahedra cells near cylinder gives a best solution among all present cases tested compared with the previous DNS result. Generally the Kelvin–Helmholtz instability can be accurately predicted by OpenFOAM under the present numerical conditions. In addition, the spanwise resolution seems no big effect in predicted results when less than 0.05D. By contrast, the increase in spanwise length to 14D from 4D plays an important role in obtaining reasonable spanwise correlation of pressure and consequently the overall fluctuating lifts. Article Direct numerical simulation is performed for flow past an isolated cylinder at . The corners of the cylinder are rounded at different radii, with the non-dimensional radius of curvature varying from (square cylinder with sharp corners) to 0.500 (circular cylinder), in which R is the corner radius and D is the cylinder diameter. Our objective is to investigate the effect of the rounded corners on the development of the separated and transitional flow past the cylinder in terms of time-averaged statistics, time-dependent behavior, turbulent statistics and three-dimensional flow patterns. Numerical results reveal that the rounding of the corners significantly reduces the time-averaged drag and the force fluctuations. The wake flow downstream of the square cylinder recovers the slowest and has the largest wake width. However, the statistical quantities do not monotonically vary with the corner radius, but exhibit drastic variations between the cases of square cylinder and partially rounded cylinders, and between the latter and the circular cylinder. The free shear layer separated from the cylinder is the most stable in which the first roll up of the wake vortex occurs furthest from the cylinder and results in the largest recirculation bubble, whose size reduces as further increases. The coherent and incoherent Reynolds stresses are most pronounced in the near-wake close to the reattachment point, while also being noticable in the shear layer for the square and cylinders. The wake vortices translate in the streamwise direction with a convection velocity that is almost constant at approximately 80% of the incoming flow velocity. These vortices exhibit nearly the same trajectory for the rounded cylinders and are furthest away from the wake centerline for the square one. The flow past the square cylinder is strongly three-dimensional as indicated by the significant primary and secondary enstrophy, while it is dominated by the primary enstrophy ( ) for the rounded cylinders. Article Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius$r$in each edge of a square cylinder of height$d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number ( Re ) and the edge-radius ratio ($r/d$). For low Re , the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon$r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon$r/d$. Article Large-eddy simulation of turbulent flow past a circular cylinder at sub- to super-critical Reynolds numbers is performed using a high-fidelity orthogonal curvilinear grid solver. Verification studies investigate the effects of grid resolution, aspect ratio and convection scheme. Monotonic convergence is achieved in grid convergence studies. Validation studies use all available experimental benchmark data. Although the grids are relatively large and fine enough for sufficiently resolved turbulence near the cylinder, the grid uncertainties are large indicating the need for even finer grids. Large aspect ratio is required for sub-critical Reynolds number cases, whereas small aspect ratio is sufficient for critical and super-critical Reynolds number cases. All the experimental trends were predicted with reasonable accuracy, in consideration the large facility bias, age of most of the data, and differences between experimental and computational setup in particular free stream turbulence and roughness. The largest errors were for under prediction of turbulence separation. Article Large eddy simulations of the flow around a circular cylinder at high Reynolds numbers are reported. Five Reynolds numbers were chosen, such that the drag crisis was captured. A total of 18 cases were computed to investigate the effect of gridding strategy, domain width, turbulence modelling and numerical schemes on the results. It was found that unstructured grids provide better resolution of key flow features, when a ‘reasonable’ grid size is to be maintained. When using coarse grids for large eddy simuation, the effect of the turbulence models and numerical schemes becomes more pronounced. The dynamic mixed Smagorinsky model was found to be superior to the Smagorinsky model, since the model coefficient is allowed to dynamically adjust based on the local flow and grid size. A blended upwind-central convection scheme was also found to provide the best accuracy, since a fully central scheme exhibits artificial wiggles which pollute the entire solution. Mean drag, fluctuating lift and Strouhal number are compared to experiments and empirical estimates for Reynolds numbers ranging from 6.31 × 104 − 5.06 × 105. In terms of the drag coefficient, the drag crisis is well captured by the present simulations, although the other integral quantities (rms lift and Strouhal number) less so. For the lowest Reynolds number, the drag is seen to be most sensitive to the domain width, while at the higher Reynolds numbers the grid resolution plays a more important role. Article In contrast with a wide range of applications concerning flows around a circular cylinder at upper subcritical Reynolds numbers (Re), there is no systematic understanding about the fundamentals of so-called random flow patterns, and their effects on intermittent modulations in the time history of pressure or force, and the decrease in their spanwise correlations. This paper employed the large-eddy simulation (LES) technique to predict flows past a circular cylinder at Re=1.3×105 and to provide images based on flow visualization that can clarify the physical mechanism responsible for these outcomes. A reasonably sufficient spanwise length was adopted for the numerical model by taking into consideration the effect of aspect ratios (the spanwise length to the diameter). We found that even at such high Res, a three-dimensional pattern of vortical field is present in the wake resulting in total force modulation and weak spanwise correlation, e.g., obvious oblique shedding. The whole development process of the three-dimensional wake is exhibited as a universal. The results revealed that local phase variations in primary vortex shedding are the starting points of three-dimensional wake patterns, which are induced by the "irregular" streamwise vortex. The three-dimensional near wake following local phase variations is associated with a successive evolution composed of certain stages in order. Quantitative analyses based on the time series of sectional lift coefficients show that intermittent increase in primary shedding periods and sectional lift streak divisions are closely related to local phase variations and vortex division in the development process of the three-dimensional pattern. In addition to the phase difference along the span, the three-dimensional pattern also weakens vortex shedding in cross sections perpendicular to the axis of the cylinder, resulting in modulation of the sectional lift coefficient. Article Large-eddy simulations (LES) of the flow past a circular cylinder are used to investigate the flow topology and the vortex shedding process at Reynolds numbers$Re=2.5 \times10^5- 8.5\times10^5\$. This range encompasses both the critical and super-critical regimes. As the flow enters the critical regime, major changes occur which affect the flow configuration. Asymmetries in the flow are found in the critical regime, whereas the wake recovers its symmetry and stabilizes in the super-critical regime. Wake characteristic lengths are measured and compared between the different Reynolds numbers. It is shown that the super-critical regime is characterised by a plateau in the drag coefficient at about C_D=0.22, and a quasi-stable wake which has a non-dimensional width of d_w/D=0.4. The periodic nature of the flow is analysed by means of measurements of the unsteady drag and lift coefficients. Power spectra of the lift fluctuations are computed. Wake vortex shedding is found to occur for both regimes investigated, although a jump in frequencies is observed when the flow enters the super-critical regime. In this regime, non-dimensional vortex-shedding frequency is almost constant and equal to St=f_{vs} D/U_{ref}=0.44. The analysis also shows a steep decrease in the fluctuating lift when entering the super-critical regime. The combined analysis of both wake topology and vortex shedding complements the physical picture of a stable and highly coherent flow in the super-critical regime.