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Engineering Turbulence Modelling and Experiments - 5 657

W. Rodi andN.Fueyo (Editors)

2002 Elsevier Science Ltd. All rights reserved.

NUMERICAL SIMULATION OF THE FLOW AROUND A CIRCULAR

CYLINDER AT HIGH REYNOLDS NUMBER

P.

Ca talan o\ M. Wang^, G. laccarino^, and

P.

Moin^

^ CIRA - Italian Aerospace Research Center,

81043 Capua (CE), ITALY

^ Center for Turbulence R esearch

Stanford University/NASA Ames Research Center,

Stanford, CA 94305-3030, USA

ABSTRACT

The viability and accuracy of Large Eddy Simulation (LES) with wall modeling for high Reynolds num

ber complex turbulent flows is investigatedbyconsidering theflowaroundacircular cylinder in the super

critical regime. A simple wa ll stress model is employed to provide approximate boundary conditions to

the LES. The results are compared w ith those obtained from steady and unsteady RAN S and the avail

able experimental data. The LES simulations are showntobe considerably m ore accurate than the RANS

results. They capture correctly the delayed boundary layer separation and reduced drag coefficients con

sistent with experimental measurem ents after the drag crisis. The m ean pressure distribution is predicted

correctly. However, the Reynolds num ber dependence is not predicted accurately due to the limited grid

resolution.

KEYWORDS

Large eddy simulation, Wall modeling. Unsteady R AN S, High Reynolds num ber flows. Circular cyUn

der, Navier-Stokes equations

MOTIVATION AND OBJECTIVES

Large eddy simulation of wall boundedflowsbecome s prohibitively expensive at high Reynolds num bers

since the number of grid points required to resolve the v ortical structures (streaks) in the near w all region

scales as the square of the friction Reynolds num ber [2]. This is nearly the same as for direct num erical

simulation.

To circumvent the severe near wall resolution requirements, LES can be combined with a wall layer

model. In this approach, LES is conducted on a relatively coa rse mesh w ith the first off-wall grid point

located in the logarithmic region. The dynam ic effects of energy-containing eddies in the wall layer (vis-

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658

cous and buffer region) are determined from a wall stress model that provides to the outer LES a set of

approximate boundary conditions, often in the form of wall shear stresses.

In recent years, the wall models based on Turbulent Boundary L ayer (TBL) equ ations and their simpli

fied forms [3, 5] have received much attention. These mod els, used with a Reynolds Averaged N avier

Stokes (RA NS) type eddy viscosity, have shown prom ise for complex flow predictions

[15].

Tocompute

the wall stress, the turbulent boundary layer equations are solved on an embedded near-wall mesh. The

tangential velocities obtained by the LES solution are imposed at the outer boundary, and the no-slip con

ditions are applied at the wa ll. The turbulent eddy viscosity is modeled through a mixing length m odel

[14,5].

The main objectives of

this

paper are to further assess the viab ility and accuracy of large eddy simulation

with wall modeling for high Reynolds number complex turbulent flows and to compare this technology

with RAN S m odels. To this end, the flow around a circular cylinder at Reynolds number (based on the

cylinder diameter D ) of 0.5 ,1 and 2 x 10 has been computed using LES w ith a wall stress model, and

the results have been compared to those achieved by steady and unsteady RA NS , and to the available ex

perimental data. The flow around a circular cylinder, with its complex features such as separating shear

layers and vortex formation and shedding, represents a canonical problem to validate 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 authors' knowledge, the first such attempt using LES. A related method, known as detached

eddy simulation (D ES), in which the entire boundary layer is modeled, has been tested for this type of

flow

[13].

Recently an LES study [4] has been conducted at a Reynolds number of 1.4 x 10 , and a good

comparison w ith the experimental data, especially in the near wake, has been show n.

NUMERICAL METHOD

The numerical method for LES employs an energy-conservative scheme of hybrid finite difference/spectral

type written for C meshes[6,10].The fractional step approach, in combination with

the

Crank-Nicholson

method for viscous terms and third order Runge-K utta scheme for the convective term s, is used for the

time advancement. The continuity constraint is imposed at each R unge-Kutta substep by solving a pres

sure Poisson equation using a m ultigrid iterative procedure. The subgrid scale stress (SGS) tensorismod

eled by the dynamic model [7] in combination withaleast-square contraction and spanwise averaging [9].

Approximate boundary conditions are imposed on the cylinder surface in terms of w all shear stress com

ponentsTy i{i= 1,3) estimated through a simplified form of the TBL equa tion model [3, 14, 5] :

C/X2 UXi

where in general

_ 1 ^p dui d

pdxi dt dxj ^

The pressure is assumed to be X2 independent, and equal to the value from the outer flow LES solution.

If the substantial derivative term in Fi is neglected, Eq. (1) can be integrated to obtain a closed form

expression for the wall shear stress component [14]

dui

X2=

Jo u

-

Jo

ly

+

i^t

J

usi - Fi ;dx2 > (3)

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659

where

u si

denotes the outer flow velocity from LES at the first off-wall nodeX2

= S.

The eddy viscosity

vtis obtained from a RA NS type mixing length model with a near wall damping

i^t

: / .y+ l-e^ -/^f 4)

where

y :^

is the distance to the wall in wall units, K ,is the von Karman constant, andA = 19. In the

present work

Fi = -f^.

has been employed. At the inflow boundary the potential flow solution is im

posed, while at the outflow a convective boundary c ondition is used.

RESULTS AND DISCUSSION

Large eddy simulations of the flow around a circular cyhnde r at Reynolds num bers of0 5x 10^, 1 x 10^,

and 2 x 10 have been performed. The com putational domain has a spanwise size of2 D,and theC mesh

extends about 22D upstream,

17D

downstream of the cylinder and 24JDinto the far field. The flow is

assumed to be periodic in the spanwise direction and 401 x 120 x 48 grid points are used. The simula

tion has been advanced for m ore than 300 dimensional time units

AtUoo/D,

and the statistics have been

collected over the last 200 time

units.

For com parison, steady and unsteady RAN S simulations have also

been performed using the same computational grid. A commercial CFD code, using the

k- e

turbulence

model and the classical w all function approach, has been employed. The discussion is focused on the

case of

Ren

= 1 x 10^, with emphasis on important flow parameters, such as the Strouhal number, the

drag coefficient, and the base pressure coefficient, and their dependence on the Reynolds number.

The mean pressure distribution on the cylinder surface is compared to two set of experimental data in

figure

1.

A very good agreement is observed between the LES at

Reo

= 1 x 10 and the experiments by

Warschauer Leene which were performed at

Reo

= 1.2 x 10^ [16]. The unsteady RANS also provides

a mean pressure coefficient in satisfactory agreement w ith both LES and experimental data, wh ile, as ex

pected, the steady RANS yields a poor result. The original data of Warschauer Leene exhibit some

spanwise variations (see [17]), and for purpose of comparison the average v alues are plotted. Relative to

the measurements of Falchsbart at

RCD

= 6.7

X10^,

the numerical results show smaller values in the base

region. It is worth noting that the Falchsbart's data contain a kink near 0 = 110 indicating the presence

of a separation bubble.

The contours of the vorticity magnitude, as computed by LES and URANS, for RCD = 1 x 10^ at a

given time instant and spanwise plane are plotted in figure 2. In the LES results, large coherent struc

tures are visible in the wake, but they are not as well organized as in typical Karman streets at sub-critical

and post-critical Reynolds num bers. T he rather thick layers along the cylinder surface consist mostly of

vorticity contours of small m agnitude. T hese 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 in the attached flow region. T he shear layers are m ore coherent in the URAN S than in the

LES.

A clear vortex shedding is visible in the URA NS results. T he axial velocity distribution (time and

spanwise averaged), obtained by LES, is presented in the lower half of figure 3. Compared to flows at

lower Reynolds numbers[8,4],the boundary layer separation is much delayed, and the wake is narrower

resulting in a smaller drag coefficient. The time-averaged URA NS velocity distribution is also plotted in

the upper half of figure3.;this shows a thicker w ake, resulting in a higher drag coefficient.

The drag coefficient, the base pressure coefficient, and the Strouhal number for theflowat Reynolds num

ber of1X 10^ are summarized in table 1. The agreement with the measurements of Shihet al. [12] is

reasonably good. The LES overpredicts the drag coefficient compared to Shih

et al,

but underpredicts

the CD relative to Achenbach [1] (cf. fig. 4). T he Strouhal number of 0.22 from Shihet al is for a rough

cylinder. It is generally accepted that periodic vortex shedding does not exist in the super critical regime

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660

F

\

F \

F

r

1 1 1

\

\

\

\

\A.

\ A

A

L -

(

1 1 1 1 1

90

e

0.5

0

0.5

-1.5

-2

-2.5

-3

r\

N

F

F

\

\

\

N

\

/ / /

/ / /

/ /o

/ /

o

p

Figure 1: Mean pressure distribution on the cylinder: - LES at

RtD

= 1 x 10^; RAN S at

Rert =

1X 10^ URANS at

Reo

= 1x 10oExperiments by Warschauer Leene [16] at

Reo

= 1.2 x 10^

(spanwise averaged); A Experiments by Falchsbart (in [17]) at

Rep

6.7 x 10^

TABLE 1

DRAG, BASE PRESSURE COEFFICIENT AND STROUHAL NUMBER

FOR THE FLOW AROUNDACIRCULAR CYLINDERATR ev = 1x10^

LES

RANS

URANS

Exp.

(Shihera/. [12])

Exp.

(Others, see [17])

D

0 307

0 385

0.401

0.24

0.17-0.40

- ^ n a s e

0.32

0.33

0.41

0 . 3 3 ^

-

St

0.28

-

0.31

0.22

0.18-0.50

for smooth cylinders [12, 17]. In the present simulations, a broad spectral peak of the unsteady lift at

St ^

0.28 is found. It can be argued that the discretization of the cylinder surface and the numerical

errors due to the under resolution m ay act as equivalent surface roughness, causing the flow field to ac

quire some rough cylinder characteristics. The wide scatter of

Co

and

S t

among various experiments in

the literature [17], listed at the bottom of the table 1, gives evidence of the high sensitivity of the flow

to perturbations due to surface roughness and free-stream turbulence in the super critical regime. The

lack of detailed experimental data in the super critical flow regime m akes a more com plete com parison

impossible.

An additional comparison between the LES and URANS is reported in Fig. 4 in terms of lift and drag

time histories. It is again clear that the UR ANS predicts a very well organized and periodic flow at this

Reynolds num ber, whereas the LES results (especially in terms of drag coefficients) have a distinct tur

bulent character. The over-dissipative nature oftheURAN S calculations is also evident by observing the

limited amplitude of the lift and, especially, drag oscillations.

To assess the robustness of the computational method, large eddy simulations at Rep = 5 x 10^ and

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661

LES

URANS

Figure 2: Instantaneous vorticity magnitude at a given span wise cut for flow over a circular cylinder at

Reo =

1 X 10 . 5 0 contour levels from

uD/Uoo

= 1 to

uD/Uoo =

20 (exponential distribution) are

plotted.

2 X 10^, have also been performed. The predicted m ean drag coefficients are plotted in figure 5 along

with the drag curve of Achenbach [1]. The CD at the two lower Reynolds number is predicted rather

well, and the discrepancy becom es larger atR en = 2 x 10^. More significantly the LES solutions show

relative insensitivity to the Reynolds number, in contrast to the experimental da ta that exhibit an increase

in CD after the drag crisis. Sim ilar Reynolds number insensitivity has been shown also for other quanti

ties such as the base pressure coefficient. Poor grid resolution, which becomes increasingly severe as the

Reynolds num ber increases, is the primary suspect.

The skin friction coefficients predicted by the wall model employed in the LES computations are pre

sented in figure 6 together with the experimental data of Achenbach [1] at

R eo

= 3.6 x 10^. 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 we ll, indicating that they are

not strongly affected by the upstream errors. The different R eynolds numbers between the LES and the

experiments can account for only a small fraction of the discrepancy. The com puted

C f

values are compa

rable to those reported by Travin

e tal

[13] using the detached eddy simulation. Travin

etal

attribute the

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Figure 3: Axial velocity distribution (time and spanwise averaged). 45 contour levels from U/Uoc

-0.2tot/ /[ /oo = 1.7.

Figure 4: Time histories of lift and drag coefficients. - LE S, URANS.

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Figure 5: Drag coefficient as a function of the Reynolds number. A chenbach [1]; L ES; URANS

overprediction oftheC / before the separation to the largely laminar boundary layer that has not been ad

equately modeled in either the simulation. In the present work grid resolution is another potential culprit.

In addition, an overprediction of the skin friction by the wall model adopted in the present LES compu

tations has also been observed by Wang Moin in the acceleration region of the trailing edge flow [15],

suggesting that this simplified model may have difficulty with strong favorable pressure gradients.

CONCLUDING REMARKS

A bold num erical experiment has been performed to compute the flow around a circular cylinder at su

percritical Reynolds number using LES. The simulations have been made possible by the use of a wall

model that alleviates the 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

measurem ents after the drag

crisis.

The mean pressure distributions and ov erall drag coefficients are pre

dicted reasonably well at

Reo =

0.5 x 10^ and 1 x 10^. However the computational solutions are in

accurate at a higher Reynolds number, and the Reynolds num ber dependence is not captured w ell. T he

grid used near the surface, particularly before separation, is quite coarse judged by the need to resolve

the outer boundary layer scales. Furthermore the effect of the wall model under coarse grid resolution

and in the laminar boundary layer is not clear. A m ore systematic investigation is needed to separate the

grid resolution and the w all modeling effects, and to fully validate the numerical m ethodology for this

challenging flow.

References

[1] Achenbach E. (1968) Distribution of local pressure and skin friction around a circular cylinder in

cross-flow up to i?e = 5X10^ J. Fluid M ech., 34, 625-639.

[2] B aggett J. S., Jimenez J. Kravchenko A. G. (1997) Resolution requirements in large eddy simula

tions of shear flows.

AnnualResearchBriefs 1997,

Center for Turbulence Research, Stanford Univer

sity/NASA Ames, 51-66.

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664

o 0

^ i i j I I I

180

0

Figure 6: Skin friction distribution on the cylinder: LES at

Reo =

0.5 x 10^; - LES at

Reo

1 X 10^ LES at

Reo = 2x

10 ^ o Experiments by Achenbach [1] at

Reo =

3.62 x 10^

[3] Balaras E., Benocci C. Piomelli U. (1996) Two-layer approximate boundary conditions for large

eddy simulations. A/A4 7., 34, 1111-1119.

[4] Breuer M . (2000) A challenging test case for large eddy simulation: high R eynolds number circular

cylinder flow.

Int. J. Heat Fluid

Flow,

21,

648-654.

[5] C abot W. Moin P. (20(X)) Approxim ate wall boundary cond itions in the large eddy simulation of

high Reynolds number flows

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Mech.,

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Aerodynamics,

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351-368.

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M.,

Strelets

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Spalart

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[14] Wang M. (1999) LES with wall models for trailing-edge aeroacoustics.

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[15] Wang M. Moin

P.

(2001) Wall modeling in LES of trailing-edgeflow Proceedings of the Second

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