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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

View Table of Contents: http://aip.scitation.org/toc/phf/29/8

Published by the American Institute of Physics

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

one; it was subjected to pressure gradients varying in space and time. Published by AIP Publishing.

[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

brought about by the different radii of the rounded corners.

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 J∂ξm

∂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¯ui−J 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

i−1

3σk

kδij =J∂ξm

∂xjνSGS ¯

Sij

=J∂ξm

∂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˜

¯ui−J

JUk¯ui,Mk

i=4

H

S

H

¯

Sk

i−I

¯

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

i−1

3σk

kδij =Lm

ij +J∂ξm

∂xjνSGS ¯

Sij

=Lm

ij +J∂ξm

∂xj 2C¯

∆2q2¯

Sij ¯

Sij ¯

Sij !, (5)

where Lm

ij =J∂ξ m

∂xj¯ujui−J∂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

i−Hm

iMm

i

2¯

∆2Mm

iMm

i

, (6)

where Lm

i=

I

JUm˜

¯ui−J

JUm¯ui,Hm

i=

K

J∂ξ m

∂xj¯uj˜

¯

¯ui−

K

J∂ξ m

∂xj¯uj¯

¯

ui, and

Mm

i=4

˜

¯

S

˜

¯

Sm

i−I

¯

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

D≈0.01, 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 θ≈150, 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 ∆z∆y 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, σu≈0.15 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 10∆z.

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 10∆z= 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 10∆z.

sampling spatial resolution of time-dependent data, which is

larger than the TBL thickness (δ≈0.02–0.03). 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 10∆z.

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

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