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Turbulent flow around a rotating stepped cylinder

Kyung-Soo Yanga)and Jong-Yeon Hwang

Department of Mechanical Engineering, Inha University, Incheon, 402-751, Korea

Klaus Bremhorst and Srdjan Nesicb)

Department of Mechanical Engineering, The University of Queensland, Brisbane, Qld 4072, Australia

?Received 5 December 2001; accepted 9 January 2002; published 7 March 2002?

Direct numerical simulation ?DNS? of turbulent flow around a rotating cylinder with two

backward-facing steps axisymmetrically mounted in the circumferential direction was performed

and compared with DNS of plane backward-facing step flow ?PBSF? of Le et al. ?J. Fluid Mech.

330, 349 ?1997??. The original motivation of this work stemmed from the efforts to design a simple

device which can generate flows of high turbulence intensity at low cost for corrosion researchers.

It turned out that the current flow shows flow structures quite similar to those of PBSF downstream

of the step, even though configurations of the two flows are totally different from one another. The

stepped cylinder appears to be a cost-effective tool in the generation of flow structures similar to

those of PBSF. © 2002 American Institute of Physics. ?DOI: 10.1063/1.1455625?

Erosion–corrosion is often active at fittings, valves, and

weld beads in a pipe system, at heat exchanger tube inlets,1,2

and in turbo-machinery including pumps, turbines, and

propellers,3where flow separation and reattachment yield

high turbulence intensity. As a model configuration for flow

separation and reattachment, PBSF is most frequently stud-

ied due to its geometrical simplicity. In the last two decades,

both experimental4–6and numerical investigations7,8have

been extensively carried out on this subject for wide ranges

of Reynolds number ?Re? and expansion ratio. Recently, Le

et al.9performed a detailed and extensive calculation using

DNS, and a parallel experiment was carried out by Jovic and

Driver.10

The present work originally stemmed from a numerical

study related to erosion–corrosion. It is well known that ac-

celerated metal loss is often encountered downstream of flow

disturbances due to enhanced

corrosion.11–13Turbulence of high intensity is regarded as

one of the main causes for this phenomenon.13In an earlier

work,14we proposed that a rotating cylinder with surface

roughness ?two backward-facing steps axisymmetrically

mounted on a circular cylinder? be an economical and trac-

table tool which can generate extreme flow conditions for

erosion–corrosion study.

An extensive calculation using DNS was carried out in

order to elucidate the flow characteristics associated with this

particular configuration, and provide the detailed flow data

for the researchers in the field of erosion–corrosion; a full

report will be given elsewhere. Unexpectedly, it turned out

that the turbulent flow structures downstream of the steps of

the rotating cylinder resemble those of PBSF very much. The

two flows are completely different in that the current flow

involves geometrical periodicity, geometrical curvature, and

mass-transfer-controlled

Coriolis force which PBSF does not have. Nevertheless, the

two flows have a strong correlation with one another; a de-

tailed comparison is presented here. The strong correlation

may render the current flow valuable as a model of separated

flows because the flow is much easier to generate in labora-

tories and less expensive to simulate numerically than PBSF.

In this study, computations were carried out with respect

to a reference frame rotating with a constant angular velocity

???. The governing incompressible continuity and momen-

tum equations are

“•u?0,

?1?

?u

?t??u•“?u??1

?“P???2u?2??u,

?2?

where u, ?, and ? denote velocity, density, and kinematic

viscosity, respectively. The last term in Eq. ?2? represents the

Coriolis force. Since the centrifugal force is conservative, it

is included in the pressure term, and does not affect the ve-

locity field.15The governing equations were discretized us-

ing a finite-volume method in a generalized coordinate sys-

tem. Spatial discretization is second-order accurate. A hybrid

scheme is used for time advancement; nonlinear terms and

cross diffusion terms are explicitly advanced by a third-order

Runge–Kutta scheme, and the other terms are implicitly ad-

vanced by the Crank–Nicolson method. A fractional step

method16is employed to decouple the continuity and mo-

mentum equations. The Poisson equation resulted from the

second stage of the fractional step method is solved by a

multigrid method. Unless specified otherwise, all the flow

variables are normalized by the step height ?h? and the cir-

cumferential speed of the cylinder surface upstream of the

step (U0) as the length and velocity scales, respectively.

Simulation was performed for the rotation with ?

?2000 rpm in the air; this corresponds to Re?335 where Re

represents Reynolds number based on U0and h. The shape

of the cylinder cross section was taken exactly the same as

a?Electronic mail: ksyang@inha.ac.kr

b?Present address: Department of Chemical Engineering, Ohio University,

181 Stocker Center, Athens, OH 45701.

PHYSICS OF FLUIDS VOLUME 14, NUMBER 4APRIL 2002

1544 1070-6631/2002/14(4)/1544/4/$19.00 © 2002 American Institute of Physics

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that of the central part of the cylinder in the experiment.14

Since turbulent length scales are much shorter than the dis-

tance between the boundaries and the steps are mounted axi-

symmetrically with respect to the axis of rotation, only one-

half of the domain is considered with the periodic boundary

condition in the circumferential direction in order to double

the numerical resolution ?Fig. 1?. In fact, a test simulation

with the entire flow domain did not reveal any instantaneous

subharmonic structure and the averaged velocity field was

axisymmetric with respect to the axis of rotation. The span-

wise ?axial? direction ?z? is assumed as homogeneous.

The outer boundary of the computational domain is lo-

cated approximately 3 diameters ?0.07 m? away from the

center of rotation, and the spanwise size ?W? of the domain is

set at one diameter ?0.024 m?. The step height is 0.002 m.

Figure 1 exhibits the whole computational domain and grid

system at one cross section in the spanwise direction. The

number of grid points was progressively refined using grid-

refinement study up to 224?128?80 in the circumferential,

normal, and spanwise directions. Based on the friction veloc-

ity (u?) at the location 3h upstream of the step, the minimum

and maximum grid spacing in each direction is ?min

?0.0174, ?max

?0.0086, ?max

in the spanwise direction, respectively. The initial flow field

was constructed such that at any location in the flow field,

the velocity was given as ? times the distance between the

point and the center of rotation.

The no-slip boundary condition was applied on the sur-

face of the cylinder and a periodic boundary condition was

employed in the homogeneous spanwise direction. A proper

boundary condition was devised and employed on the outer

boundary. That is,

?

??3.18 in the circumferential direction, ?min

??0.67 in the normal direction, and ?z

?

??4.7

?v?

?n??,

vr?0,

?vz

?n?0,

?3?

where v?, vr, and vzrepresent the circumferential, normal,

and spanwise velocity components, respectively, and n de-

notes the direction locally perpendicular to the outer bound-

ary. This boundary condition allows use of a reasonably

small computational domain without significantly disturbing

the flow field near the cylinder which is the region of pri-

mary interest. Averaging of the flow variables was carried

out in the homogeneous spanwise direction and also in time.

More than 160 instantaneous flow fields were collected over

7 revolutions of the cylinder.

The averaged streamlines in the vicinity of the step are

shown in Fig. 2. The main recirculating region and a second-

ary one near the corner can be clearly identified. Figure 2

confirms that this flow geometry will create a qualitatively

similar flow pattern, as observed near a sudden pipe expan-

sion or a plane backward-facing step, including flow separa-

tion and reattachment. The mean reattachment point is lo-

cated at s/h?4.9 where s is the coordinate axis along the

cylinder surface as indicated in Fig. 2. The mean reattach-

ment length (Xr) is shorter than that of PBSF due to the

curvature effect; Le et al.9reported Xr?6.28h.

The profiles of mean circumferential velocity component

(Us) at selected s locations are presented in Fig. 3 in the wall

units based on local friction velocity. The profile of rigid

body motion, which was employed as the initial condition of

the velocity field, is also shown for comparison. The velocity

profile at any location should asymptotically approach this

profile as y goes to infinity because the region far away from

the cylinder is not affected by the rotation of the cylinder.

One can notice that the profile of Usis quickly re-established

past the step; see the profiles at s/h?10, 11, and 12. Figure

3 reveals that the Usprofile at s/h?10 is quite different

FIG. 1. Grid system at a cylinder cross section.

FIG. 2. Mean flow field near the step; streamlines.

FIG. 3. Profiles of the mean circumferential velocity component (Us) at

selected s locations in the wall units based on local friction velocity.

1545 Phys. Fluids, Vol. 14, No. 4, April 2002Turbulent flow around a rotating stepped cylinder

Page 3

from those of s/h??5.0, ?4.0, and ?3.0. However, the Us

profile at s/h?12 is almost the same as those of s/h

??5.0, ?4.0, and ?3.0. As the flow approaches the step on

the opposite side (s/h??5, ?4, and ?3?, the profile of Us

gets fully established. It is interesting to note that the recov-

ery of the profile in the region below y??100 very much

resembles that of the PBSF; see Fig. 18 of Le et al.9Near the

step (s/h??2.0), however, the profile is considerably af-

fected by the presence of the step.

A skin-friction law of the form Cf,UN?ReN

posed by Adams et al.6Here, UNis the ‘‘maximum’’ mean

negative velocity at the distance N away from the wall, Cf,UN

is the friction coefficient normalized by

the Reynolds number based on UNand N. In our study, we

measured N in the direction locally perpendicular to the cyl-

inder surface, and UNas the ‘‘maximum’’ negative value of

Us. In Fig. 4, symbols represent the current data for 1.7

?s/h?4.9, i.e., for the main vortex in the recirculating re-

gion. The Cf,UNcorrelation obtained from the computational

data using a least square method is

?1was pro-

1

2?UN

2, and ReNis

Cf,UN?4.13ReN

The slope ??0.97? is closer to ?1 than that determined

by Le et al.9??0.92?. Unlike the PBSF,9the correlation does

not hold for locations in the secondary vortex in the corner

near s/h?0. This is due to the fact that the secondary vortex

?0.97.

?4?

in the current problem is much smaller than that of the PBSF.

Compare Fig. 2 in the present work with Fig. 8 of Le et al.9

The instantaneous reattachment points on a typical

(x–y) plane are depicted during three revolutions of the cyl-

inder in Fig. 5; the symbols represent the reattachment points

as identified by zero s-component of the wall-shear stress. It

is seen that the recirculating region periodically stretches out

on the cylinder surface; simultaneously it breaks into pieces

?e.g., 1.0?t/T?1.2, where T is the period of the cylinder

rotation?. The same trend was observed in the PBSF; see Fig.

3 of Le et al.9They pointed out that as the large-scale struc-

ture created by the shear-layer roll-up grows, the reattach-

ment location travels downstream at a constant speed. They

also noted that the sudden decrease of the reattachment

length indicates a detachment of the turbulent large-scale

structure from the step. All these observations also hold in

the current case.

Figure 6 presents instantaneous velocity vectors on the

r–z plane perpendicular to the cylinder surface at s/h

?3.5. Pairs of strong streamwise vortices are clearly identi-

fied, indicating the high degree of three dimensionality of the

flow. Similar streamwise vortices were observed in the

PBSF. See Fig. 6 of Le et al.9

As indicated by the numerical results presented above,

the structures and unsteady characteristics of the main recir-

culating region in the turbulent flow around a rotating

stepped cylinder have many features in common with those

of PBSF. It remains to be further investigated how the geo-

metrical curvature and Coriolis force affect the flow. Never-

theless, the stepped cylinder appears to be a cost-effective

tool in the generation of flow structures similar to those of

PBSF.

ACKNOWLEDGMENTS

This research was financially supported by Korea Sci-

ence and Engineering Foundation ?98-0200-12-01-3?. The

computing cost was partially covered by the University of

Queensland.

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FIG. 5. Instantaneous reattachment points on a typical spanwise (x–y)

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FIG. 6. Instantaneous velocity vectors on the r–z plane perpendicular to the

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1547 Phys. Fluids, Vol. 14, No. 4, April 2002 Turbulent flow around a rotating stepped cylinder