...

Catalano2003.pdf

by user

on
Category: Documents
5

views

Report

Comments

Transcript

Catalano2003.pdf
International Journal of Heat and Fluid Flow 24 (2003) 463–469
www.elsevier.com/locate/ijhff
Numerical simulation of the flow around a circular cylinder at
high Reynolds numbers
Pietro Catalano
a,*
, Meng Wang b, Gianluca Iaccarino b, Parviz Moin
b
a
b
CIRA––Italian Aerospace Research Center, 81043 Capua (CE), Italy
Center for Turbulence Research, Stanford University/NASA Ames Research Center, Stanford, CA 94305-3030, USA
Received 30 November 2002; accepted 23 March 2003
Abstract
The viability and accuracy of large-eddy simulation (LES) with wall modeling for high Reynolds number complex turbulent flows
is investigated by considering the flow around a circular cylinder in the supercritical regime. A simple wall stress model is employed
to provide approximate boundary conditions to the LES. The results are compared with those obtained from steady and unsteady
Reynolds-averaged Navier–Stokes (RANS) solutions and the available experimental data. The LES solutions are shown to be
considerably more accurate than the RANS results. They capture correctly the delayed boundary layer separation and reduced drag
coefficients consistent with experimental measurements after the drag crisis. The mean pressure distribution is predicted reasonably
well at ReD ¼ 5 105 and 106 . However, the Reynolds number dependence is not captured, and the solution becomes less accurate
at increased Reynolds numbers.
Ó 2003 Elsevier Science Inc. All rights reserved.
Keywords: Large-eddy simulation; Wall modeling; Unsteady RANS; High Reynolds number flows; Circular cylinder; Navier–Stokes equations
1. Introduction
The severe grid-resolution requirement in the nearwall region has been the major roadblock to the use of
large-eddy simulation (LES) for practical applications.
This arises because of the presence of small but dynamically important eddies in the near-wall region. To resolve these vortical structures, the number of grid points
required scales as the square of the friction Reynolds
number (Baggett et al., 1997), which is nearly the same as
for direct numerical simulation.
As a practical remedy, LES can be combined with a
wall-layer model. In this approach, LES is conducted on
a relatively coarse mesh which scales with the outer flow
scale, thus making the computational cost only weakly
dependent on the Reynolds number. The dynamic effects
of energy-containing eddies in the wall layer (viscous
and buffer regions) are determined from a wall model
*
Corresponding author. Tel.: +39-0823-623244; fax: +39-0823-623028.
E-mail address: [email protected] (P. Catalano).
which provides to the outer LES a set of approximate
boundary conditions, often in the form of wall shear
stresses.
In recent years, wall models based on turbulent
boundary layer (TBL) equations and their simplified
forms have received much attention. These models, used
with a Reynolds-averaged Navier–Stokes (RANS) type
eddy viscosity, have shown good promise. The approach
has been successfully tested by Balaras et al. (1996) in a
plane channel, square duct, and rotating channel, and
by Cabot and Moin (2000) in a plane channel and
backward-facing step. More recently, Wang and Moin
(2002) employed this approach to simulate the flow past
the trailing-edge of an airfoil at chord Reynolds number
of 2.15 106 , and obtained very good agreement with
solutions from the full LES (Wang and Moin, 2000) at a
small fraction of the computational cost.
The main objectives of the present work are to further
assess the viability and accuracy of LES with wall
modeling for high Reynolds number complex turbulent
flows and to compare this approach with RANS models. To this end, the flow around a circular cylinder at
Reynolds numbers (based on the cylinder diameter D)
0142-727X/03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved.
doi:10.1016/S0142-727X(03)00061-4
464
P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469
o
oui 1 op oui
o
þ
ðm þ mt Þ
¼
þ
ui uj ;
ox2
ox2 q oxi ot oxj
i ¼ 1; 3
ð1Þ
where x1 , x2 , and x3 denote the streamwise, wall-normal,
and spanwise directions, respectively, and u1 , u2 and u3
the corresponding velocity components. Eq. (1) has been
implemented in a body-fitted, locally orthogonal coordinate system as shown in Fig. 1. To make a distinction,
the fixed Cartesian coordinates (see also Fig. 1) and
velocities are denoted by x, y, z and u, v, w. In the present
work a simplified version of the wall model is employed,
X1
The numerical method for LES and wall model implementation are the same as in Wang and Moin (2002).
The energy-conservative scheme, written in a staggered
grid system in body-fitted coordinates, is of hybrid finite
difference/spectral type (Mittal and Moin, 1997); it employs second-order central differences in streamwise and
wall-normal directions and Fourier collocation in the
spanwise direction. The fractional step approach, in
combination with the Crank–Nicholson method for viscous terms and third order Runge–Kutta scheme for the
convective terms, is used for time advancement. The
continuity constraint is imposed at each Runge–Kutta
substep by solving a Poisson equation for pressure using
a multigrid iterative procedure. The subgrid scale (SGS)
stress tensor is modeled by the dynamic Smagorinsky
model (Germano et al., 1991) in combination with a leastsquare contraction and spanwise averaging (Lilly, 1992).
Approximate boundary conditions are imposed on
the cylinder surface in terms of wall shear stress components swi (i ¼ 1,3) estimated from a TBL equationbased wall model (Balaras et al., 1996; Cabot and
Moin, 2000; Wang and Moin, 2002), which has the
following general form in Cartesian coordinates:
2
2. Numerical method for LES
y
X
of 0.5 106 , 1 106 and 2 106 is considered, and the
results are compared to those from steady and unsteady
RANS and the available experimental data. The flow
around a circular cylinder, with its complex features,
represents a canonical problem for validating new approaches in computational fluid dynamics. To take the
best advantage of wall modeling, we have concentrated
on the super-critical flow regime in which the boundary
layer becomes turbulent prior to separation. This is, to
the authorsÕ knowledge, the first such attempt using
LES. A related method, known as detached eddy simulation (DES), in which the entire attached boundary
layer is modeled, has been tested for this type of flow by
Travin et al. (1999). Recently an LES study has been
conducted by Breuer (2000) at a high sub-critical Reynolds number of 1.4 105 , and a good comparison with
the experimental data, especially in the near wake, has
been shown.
θ
0
D/2
x
Fig. 1. Coordinate systems used. z and x3 axes are out of the plane.
in which the substantial derivative (last two terms) is
dropped from the right-hand side of Eq. (1). Since the
pressure is assumed x2 -independent in the thin wall layer
and is imposed from the outer flow LES solution, one
can then integrate Eq. (1) to the wall to obtain a closed
form expression for the wall stress components (Wang
and Moin, 2002)
Z
oui q
1 op d dx2
swi ¼ l
¼
u
dx
R
di
2
d dx2
q oxi 0 m þ mt
ox2 x2 ¼0
0 mþmt
ð2Þ
where udi denotes the tangential velocity components
from LES at the first off-wall node, at distance d from
the wall. In attached flows these nodes are generally
placed within the lower edge of the logarithmic layer. In
the present flow, however, dþ (in wall units) is found to
vary from 0 to 130 depending on the local skin friction.
The eddy viscosity mt is obtained from a mixing length
model with a near-wall damping
mt
þ
2
¼ jywþ ð1 eyw =A Þ
ð3Þ
m
where ywþ is the distance to the wall in wall units, j ¼ 0:4
is the von Karman constant, and A ¼ 19.
Simulations are conducted employing a C-mesh in the
planes perpendicular to the span. The computational
domain extends approximately 22D upstream of the
cylinder, 17D downstream of the cylinder, and 24D into
the far-field. In the streamwise direction 401 grid points,
144 of which on the cylinder surface and 2 129 points
in the wake, are employed. In the wall-normal direction
120 points are used. The spanwise domain size is 2D,
over which the flow is assumed periodic, and 48 grid
points are distributed uniformly. Note that the spanwise
domain size is shorter than the typical values used for
lower Reynolds number flows (e.g. pD for ReD ¼ 3900 in
Beaudan and Moin (1994) and Kravchenko and Moin
(2000)). This is justified because of the reduced spanwise
P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469
3300
465
4. Results and discussion
300
3000
250
2700
2400
∆x+1 ∆x+3
1800
150
1500
1200
∆x+2
200
2100
100
900
600
50
300
0
0
30
60
90
θ
120
150
0
180
þ
þ
Fig. 2. Sizes of the wall-adjacent cells: (- - -) Dxþ
1 ; (—) Dx2 ; (––) Dx3 .
correlation length for higher Reynolds number flows.
The same spanwise domain size of 2D was used by
Breuer (2000) at ReD ¼ 1:4 105 . Potential-flow solutions are imposed as boundary conditions in the farfield, and convective boundary conditions are used at
the outflow boundary.
The sizes of the wall-adjacent grid cells for LES are
shown in Fig. 2 in wall units. They are extremely large,
because the use of a wall model in principle bypasses
the need to resolve the inner scales. In the staggered grid
system used, the first off-wall nodes for the tangential
velocities are located at dþ ¼ Dxþ
2 =2, where the LES and
wall-model velocities are required to match. Note that
the wall units in Fig. 2 are affected by the skin friction
errors to be discussed later, and should thus be viewed
as a crude estimate.
To obtain the LES results presented here, the simulations have advanced more than 300 dimensionless time
units. The statistics are collected over the last 200 time
units. Running at a maximum CFL number of 1.5, the
non-dimensional time step DtU1 =D typically varies between 0.0030 and 0.0045. Three Reynolds numbers,
ReD ¼ 0:5 106 , 1 106 , and 2 106 , have been considered. The discussion will be mainly focused on the
case of ReD ¼ 1 106 , with emphasis on important flow
parameters, such as the drag coefficient, the base pressure coefficient, the Strouhal number, and their dependence on the Reynolds number.
The mean pressure distribution on the cylinder surface is compared to two set of experimental data in Fig.
3. A very good agreement is observed between the LES
at ReD ¼ 1 106 and the experiment by Warschauer
and Leene (1971) which was performed at ReD ¼ 1:2 106 . The original data of Warschauer and Leene (1971)
exhibit some spanwise variations (see Zdravkovich,
1997), and for the purpose of comparison the average
values are plotted. The unsteady RANS also provides a
mean pressure coefficient in satisfactory agreement with
both LES and the experimental data, while, as expected,
the steady RANS yields a poor result. Relative to the
measurements of Falchsbart at ReD ¼ 6:7 105 , the
numerical results show lower values in the base region.
It is worth noting that FalchsbartÕs data contain a kink
near h ¼ 110°, indicating the presence of a separation
bubble. This type of separation bubble is characteristic of the critical regime, and is difficult to reproduce
1
RANS simulations are carried out using a commercial CFD code, FLUENT. It is based on a second order
finite volume discretization and the SIMPLE pressure
correction technique for enforcing the divergence-free
condition of the velocity field; the time integration is
three-level fully implicit. The eddy viscosity is obtained
using the standard k–e model (Launder and Spalding,
1972) with wall functions. Although unstructured grids
can be used in FLUENT, for the simulations presented
here we employed the same C-mesh as used in the LES.
Both steady and unsteady RANS calculations have
been performed for comparison. Steady simulations are
performed using only half of the computational domain
ðy P 0Þ. Starting from this solution and its mirror image
in the region y < 0, a disturbance was imposed to break
the symmetry, and a time-accurate simulation was carried out. A non-dimensional time-step DtU1 =D of 0.01
was used and the simulation was run for 300 time units
(D=U1 ).
0.5
0
-0.5
Cp
3. Numerical method for RANS
-1
-1.5
-2
-2.5
-3
0
30
60
90
θ
120
150
180
Fig. 3. Mean pressure distribution on the cylinder: (—) LES at ReD ¼
1 106 ; (- - -) RANS at ReD ¼ 1 106 ; (––) URANS at ReD ¼
1 106 ; () experiment by Warschauer and Leene (1971) at ReD ¼ 1:2 106 (spanwise averaged); (M) experiment by Falchsbart (in Zdravkovich, 1997) at ReD ¼ 6:7 105 .
466
P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469
experimentally or numerically due to sensitivity to disturbances.
The contours of the vorticity magnitude, as computed
by LES and URANS for ReD ¼ 1 106 at a given time
instant and spanwise plane, are plotted in Fig. 4. In the
LES results, some coherent structures are visible in the
wake, but they are not as well organized as in typical
K
arm
an streets at sub-critical and post-critical Reynolds
numbers. The rather thick layers along the cylinder
surface consist mostly of vorticity contours of small
magnitude. These levels are necessary for visualizing
the wake structure, but are not representative of the
boundary layer thickness. The true boundary layer, with
strong vorticity, is extremely thin and mostly laminar in
the attached flow region. The shear layers are more
coherent in the URANS than in the LES. A clear vortex-shedding pattern is exhibited in the URANS results.
The mean streamwise velocity distribution (time and
spanwise averaged) obtained by LES is presented in the
lower half of Fig. 5. Compared to flows at lower Reynolds numbers (Kravchenko and Moin, 2000; Breuer,
2000), the boundary layer separation is much delayed,
and the wake is narrower, resulting in a smaller drag
coefficient. The time-averaged URANS velocity distribution is plotted in the upper half of Fig. 5, which shows
a thicker wake, resulting in a higher drag coefficient. A
LES
URANS
Fig. 4. Instantaneous vorticity magnitude at a given spanwise cut for
flow over a circular cylinder at ReD ¼ 1 106 . 25 contour levels from
xD=U1 ¼ 1 to xD=U1 ¼ 575 (exponential distribution) are plotted.
Fig. 5. Mean streamwise velocity distribution predicted by LES and
URANS. 45 contour levels from U=U1 ¼ 0:2 to U=U1 ¼ 1:7 are
plotted.
quantitative comparison between LES and URANS in
terms of the mean streamwise and vertical velocity
profiles in the cylinder wake is shown in Fig. 6. The
upper and lower parts correspond to two streamwise
locations inside and outside of the mean recirculation
region, respectively. The LES predicts a wide range of
flow scales and hence more mixing of the flow. The velocity deficit from LES is larger at x=D ¼ 0:75, which is
in the middle of the mean recirculation bubble as predicted by LES, but it is soon surpassed by the URANS
prediction further downstream in the wake. Unfortunately, there is a general lack of detailed experimental
data at super-critical Reynolds numbers. In particular,
velocity and Reynolds-stress profile measurements are
non-existent, making an experimental validation of the
wake profiles impossible.
Another comparison between the LES and URANS
is made in Fig. 7 in terms of lift and drag time histories.
It is again clear that the URANS predicts a very well
organized and periodic flow at this Reynolds number,
whereas the LES results have broadband turbulence
characteristics. The overly dissipative nature of the
URANS calculations is also evident by observing the
small amplitudes of the lift and, especially, drag oscillations.
The drag coefficient, base pressure coefficient,
Strouhal number, and mean recirculation length for the
flow at Reynolds number of 1 106 are summarized in
Table 1. The agreement with the measurements of Shih
et al. (1993) is reasonably good. The LES overpredicts
the drag coefficient compared to Shih et al. (1993), but
underpredicts the CD relative to Achenbach (1968) (cf.
Fig. 9). The Strouhal number of 0.22 from Shih et al.
(1993) is for a rough cylinder. It is generally accepted
that periodic vortex shedding does not exist in the supercritical regime for smooth cylinders (Shih et al., 1993;
Zdravkovich, 1997). From our simulations, a distinct
spectral peak is observed at St 0:35, as shown clearly
in the Evv spectra in Fig. 8. This figure depicts the frequency spectra of the streamwise and vertical velocities
at x=D ¼ 0:70 and 1.50, y=D ¼ 0:15. It can be argued
that the discretization of the cylinder surface and the
numerical errors due to the under-resolution may act as
equivalent surface roughness, causing the flow field to
acquire some rough cylinder characteristics. The wide
scatter of CD and St among various experiments in the
literature (Zdravkovich, 1997), listed at the bottom of
Table 1, suggests high sensitivity of the flow to perturbations due to surface roughness and free-stream turbulence in the super-critical regime. Our simulation
results fall easily within the experimental range.
To assess the robustness of the computational method,
LESs at ReD ¼ 5 105 and 2 106 have also been performed. The predicted mean drag coefficients are plotted in Fig. 9 along with the drag curve of Achenbach
(1968). The CD at the two lower Reynolds numbers is
1.2
0.3
0.8
0.15
V/U∞
U/U∞
P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469
0.4
0
467
0
-0.15
-0.4
-1
-0.5
0
0.5
1
-0.3
-2
-1.5
-1
-0.5
y/D
0
0.5
1
1.5
2
0.5
1
1.5
2
y/D
1.2
0.3
1
0.15
V/U∞
U/U∞
0.8
0
0.6
-0.15
0.4
0.2
-1
-0.5
0
0.5
1
-0.3
-2
-1.5
-1
y/D
-0.5
0
y/D
Fig. 6. Mean streamwise and vertical velocities at x=D ¼ 0:75 (upper figures) and x=D ¼ 1:50 (lower figures): (—) LES; (––) URANS.
0.5
CD
0.4
0.3
0.2
150
2
00
2
00
t U∞ / D
250
3
00
0.6
CL
0.3
0
-0.3
-0.6
150
t U∞ / D
250
300
Fig. 7. Time histories of lift and drag coefficients. (—) LES; (- - -) URANS.
predicted rather well, but the discrepancy becomes large
at ReD ¼ 2 106 . More significantly the LES solutions
show relative insensitivity to the Reynolds number, in
contrast to the experimental data that exhibit an increase in CD after the drag crisis. Poor grid resolution,
which becomes increasingly severe as the Reynolds
468
P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469
Table 1
Drag, base pressure coefficient, Strouhal number, and recirculation
length for the flow around a circular cylinder at ReD ¼ 1 106
CD
CPbase
0.31
0.39
0.40
0.24
0.17–0.40
0.32
0.33
0.41
0.33
–
St
0.35
–
0.31
0.22
0.18–0.50
Lr =D
1.6
1.04
1.37
–
–
1.2
CD
LES
RANS
URANS
Exp. (Shih et al., 1993)
Exp. (Others, see
Zdravkovich, 1997)
2
0.8
number increases, is the primary suspect. Similar Reynolds number insensitivity is shown by the URANS
results.
The skin friction coefficients predicted by the wall
model employed in the LES computations are presented
in Fig. 10 together with the experimental data of Achenbach (1968) at ReD ¼ 3:6 106 . The levels are very different on the front half of the cylinder but are in
reasonable agreement on the back half. The boundary
layer separation and the recirculation region are captured rather well, indicating that they are not strongly
affected by the upstream errors. The different Reynolds
numbers between the LES and the experiments can account for only a small fraction of the discrepancy. Note
that the computed Cf values are comparable to those
reported by Travin et al. (1999) using DES at ReD ¼
3 106 . Travin et al. (1999) attribute the overprediction
of the Cf before the separation to the largely laminar
boundary layer that has not been adequately modeled.
This is also the case in the present simulation. Both
experiments and numerical simulations suggest that
even at these super-critical Reynolds numbers, the
boundary layers remain laminar in most of the favorable
pressure gradient region. In our simulations no effort
was made to trigger transition, nor was the wall model
0 4
10
E vv /U∞ D
10
10-4
10-4
10-5
10-6
10-6
f D /U∞
10
7
-3
10-5
0
10
A bold numerical experiment has been performed to
compute the flow around a circular cylinder at super-
-3
10
6
5. Concluding remarks
10-2
-1
10
modified for laminar flow application. Grid resolution is
another potential culprit in the present work. In addition, an overprediction of the skin friction by the wall
model adopted in the present LES computations has
also been observed by Wang and Moin (2002) in the
acceleration region of the trailing-edge flow, suggesting
that this simplified model may have difficulty with
strong favorable pressure gradients.
10-2
10
5
Fig. 9. Drag coefficient as a function of the Reynolds number. (—)
Achenbach (1968); (d) LES; (j) URANS.
10-1
10-7
10
ReD
10-1
10
Euu /U∞ D
0.4
1
10-7
10
-1
10
0
10
1
f D /U∞
Fig. 8. Frequency spectra of streamwise (left figure) and vertical (right figure) velocities at two wake stations: (—) x=D ¼ 0:70, y=D ¼ 0:15;
(- - -) x=D ¼ 1:50, y=D ¼ 0:15.
P. Catalano et al. / Int. J. Heat and Fluid Flow 24 (2003) 463–469
facilities at NASA Ames Research Center and on facilities at the Center for Turbulence Research. We would
like to thank Professor Javier Jimenez for valuable discussions.
0.015
0.01
Cf
0.005
References
0
-0.005
-0.01
-0.015
469
0
60
120
180
θ
240
300
360
Fig. 10. Skin friction distribution on the cylinder: (- - -) LES at ReD ¼
0:5 106 ; (—) LES at ReD ¼ 1 106 ; (––) LES at ReD ¼ 2 106 ; ( )
experiments by Achenbach (1968) at ReD ¼ 3:62 106 .
critical Reynolds number using LES. The simulations
have been made possible by the use of a wall model that
alleviates the near-wall grid resolution requirements.
Preliminary results are promising in the sense that they
correctly predict the delayed boundary layer separation
and reduced drag coefficients consistent with measurements after the drag crisis. The mean pressure distributions and overall drag coefficients are predicted
reasonably well at ReD ¼ 0:5 106 and 1 106 . However the computational solutions are inaccurate at
higher Reynolds numbers, and the Reynolds number
dependence is not captured. It should be noted that the
grid used near the surface, particularly before separation, is quite coarse judged by the need to resolve the
outer boundary layer scales. The effect of the wall model
under coarse grid resolution and in the laminar section
of the boundary layer is not clear. A more systematic
investigation is needed to separate the grid resolution
and the wall-modeling effects, and to fully validate the
numerical methodology for this challenging flow.
Acknowledgements
This work was supported in part by the US Air Force
Office of Scientific Research Grant No. F49620-00-10111. Computations were carried out on the NAS
Achenbach, E., 1968. Distribution of local pressure and skin friction
around a circular cylinder in cross-flow up to Re ¼ 5 106 . J. Fluid
Mech. 34, 625–639.
Baggett, J.S., Jimenez, J., Kravchenko, A.G., 1997. Resolution
requirements in large eddy simulations of shear flows. Annual
Research Briefs––1997. Center for Turbulence Research, Stanford
University/NASA Ames, pp. 51–66.
Balaras, E., Benocci, C., Piomelli, U., 1996. Two-layer approximate
boundary conditions for large eddy simulation. AIAA J. 34, 1111–
1119.
Beaudan, P., Moin, P., 1994. Numerical Experiments on the Flow Past
a Circular Cylinder at Subcritical Reynolds Number. Report no.
TF-62, Department of Mech. Engr., Stanford University.
Breuer, M., 2000. A challenging test case for large eddy simulation:
high Reynolds number circular cylinder flow. Int. J. Heat Fluid
Flow 21, 648–654.
Cabot, W., Moin, P., 2000. Approximate wall boundary conditions in
the large eddy simulation of high Reynolds number flow. Flow
Turb. Combust. 63, 269–291.
Germano, M., Piomelli, U., Moin, P., Cabot, W.H., 1991. A dynamic
subgrid-scale eddy viscosity model. Phys. fluids A 3, 1760–1765.
Kravchenko, A.G., Moin, P., 2000. Numerical studies of flow over
a circular cylinder at ReD ¼ 3900. Phys. Fluids 12, 403–417.
Launder, B.E., Spalding, D.B., 1972. Mathematical Models of
Turbulence. Academic Press, London.
Lilly, D.K., 1992. A proposed modification of the Germano subgrid
scale closure method. Phys. Fluids A 4, 633–635.
Mittal, R., Moin, P., 1997. Suitability of upwind-biased finite difference
schemes for large eddy simulation of turbulent flows. AIAA J. 35,
1415–1417.
Shih, W.C.L., Wang, C., Coles, D., Roshko, A., 1993. Experiments on
flow past rough circular cylinders at large Reynolds numbers.
J. Wind Eng. Indust. Aerodyn. 49, 351–368.
Travin, A., Shur, M., Strelets, M., Spalart, P., 1999. Detached eddy
simulations past a circular cylinder. Flow Turb. Combust. 63, 269–
291.
Wang, M., Moin, P., 2000. Computation of trailing-edge flow
and noise using large-eddy simulation. AIAA J. 38 (12), 2201–
2209.
Wang, M., Moin, P., 2002. Dynamic wall modeling for large-eddy
simulation of complex turbulent flows. Phys. Fluids 14 (7), 2043–
2051.
Warschauer, K.A., Leene, J.A., 1971. Experiments on mean and
fluctuating pressures of circular cylinders at cross flow at very high
Reynolds numbers. In: Proc. Int. Conf. on Wind Effects on
Buildings and Structures, Tokyo, Japan (see also Zdravkovich
1997), pp. 305–315.
Zdravkovich, M.M., 1997. Flow Around Circular Cylinders. Fundamentals, vol. 1. Oxford University Press, 1997 (Chapter 6).
Fly UP