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
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
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
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
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: email@example.com
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
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
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
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
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
2, and ReNis
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
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
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
1D. G. Elvery and K. Bremhorst, ‘‘Wall pressure and effective wall shear
stress in heat exchanger tube inlets with application to erosion–
corrosion,’’ J. Fluids Eng. 119, 948 ?1997?.
2D. G. Elvery and K. Bremhorst, ‘‘Erosion–corrosion due to inclined flow
into heat exchanger tubes—investigation of flow field,’’Proceedings of the
ASME Fluids Engineering Division Summer Meeting, San Diego, Califor-
nia, July 7–11, FED ?Am. Soc. Mech. Eng.? 237, 595 ?1996?.
3J. Postlethwaite and S. Nesic, ‘‘Erosion–corrosion in single and multi-
FIG. 4. Skin-friction coefficient in the main recirculating region (1.7?s/h
?4.9) as a function of the wall-layer Reynolds number.
FIG. 5. Instantaneous reattachment points on a typical spanwise (x–y)
FIG. 6. Instantaneous velocity vectors on the r–z plane perpendicular to the
cylinder surface at s/h?3.5.
1546 Phys. Fluids, Vol. 14, No. 4, April 2002Yang et al.
phase flow,’’ in Uhlig Corrosion Handbook, 2nd ed., edited by W. Revie
?Wiley, New York, 2000?.
4F. Durst and C. Tropea, ‘‘Turbulent backward-facing step flows in two-
dimensional ducts and channels,’’ Proceedings of the Third International
Symposium on Turbulent Shear Flows, University of California, Davis,
5B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Scho ¨nung, ‘‘Experimental
and theoretical investigation of backward-facing step,’’J. Fluid Mech. 127,
6E. W. Adams, J. P. Johnston, and J. K. Eaton, ‘‘Experiments on the struc-
ture of turbulent reattaching flow,’’ Rep. MD-43, Thermosciences Divi-
sion, Department of Mechanical Engineering, Stanford University, 1984.
7R. Friedrich and M. Arnal, ‘‘Analysing turbulent backward-facing step
flow with the lowpass-filtered Navier–Stokes equations,’’ J. Wind. Eng.
Ind. Aerodyn. 35, 101 ?1990?.
8L. Kaiktsis, G. E. Karniadakis, and S. A. Orszag, ‘‘Onset of three-
dimensionality, equilibria, and early transition in flow over a backward-
facing step,’’ J. Fluid Mech. 231, 501 ?1991?.
9H. Le, P. Moin, and J. Kim, ‘‘Direct numerical simulation of turbulent flow
over a backward-facing step,’’ J. Fluid Mech. 330, 349 ?1997?.
10S. Jovic and D. M. Driver, ‘‘Backward-facing step measurement at low
Reynolds number, Reh?5000,’’ NASA Tech. Memo. 108807 ?1994?.
11J. Postlethwaite, M. H. Dobbin, and K. Bergevin, ‘‘The role of oxygen
mass transfer in the erosion–corrosion of slurry pipelines,’’ Corrosion
?Houston? 42, 514 ?1986?.
12B. K. Mahato, S. K. Voora, and L. W. Shemilt, ‘‘Steel pipe corrosion under
flow conditions—I. An isothermal correlation for a mass transfer model,’’
Corros. Sci. 8, 173 ?1968?.
13S. Nesic and J. Postlethwaite, ‘‘Relationship between the structure of dis-
turbed flow and erosion–corrosion,’’ Corrosion ?Houston? 46, 874 ?1990?.
14S. Nesic, J. Bienkowsky, K. Bremhorst, and K.-S. Yang, ‘‘Testing for
erosion–corrosion under disturbed flow conditions using a rotating cylin-
der with a stepped surface,’’ Corrosion ?Houston? 56, 1005 ?2000?.
15D. K. Lezius and J. P. Johnston, ‘‘Roll-cell instabilities in rotating laminar
and turbulent channel flows,’’ J. Fluid Mech. 77, 153 ?1976?.
16M. Rosenfeld, D. Kwak, and M. Vinokur, ‘‘A fractional step solution
method for the unsteady incompressible Navier–Stokes equations in gen-
eralized coordinate systems,’’ J. Comput. Phys. 94, 102 ?1994?.
1547 Phys. Fluids, Vol. 14, No. 4, April 2002 Turbulent flow around a rotating stepped cylinder