I. J. Trans. Phenomena, Vol. 4, pp. 257-274
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© 2002 Old City Publishing, Inc.
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a member of the Old City Publishing Group.
The Influence of Inner Cylinder Rotation on
Laminar Axial Flows in Eccentric Annuli
of Drilling Bore Wells
P. FANG1 AND R. M. MANGLIK2
Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical, Industrial and Nuclear Engineering,
University of Cincinnati, Cincinnati, OH 45221-0072, USA
(Received September 15, 2000; In final form June 30, 2001)
An extended theoretical study of the effects of annulus radius ratio r* and eccentricity ε*,
and inner cylinder rotation of drilling bore wells on the axial flow structure and frictional
losses is presented. Computational results for a wide range of eccentric annuli geometry
(0.1 ≤ r* ≤ 0.9, 0 ≤ ε* ≤ 0.95) and inner pipe rotational Reynolds number (0 ≤ Rer ≤ 150) in
the sub-critical Taylor number regime (Ta ≤ 10,000) are obtained. The inner-core rotation
produces a counter-rotating kidney-shaped vortex in the widest gap of the annulus, which
grows with increasing eccentricity and decreasing radius ratio. This lowers the local peak
axial velocity but increases the wall gradients, which results in higher friction losses. The
combined effects of r*, ε*, and Rer on the friction factor, however, are quite complex and
their parametric influence is delineated.
Keywords: Axial vortices; laminar flow; sub-critical Taylor flow; rotational couette flow; annular duct;
drill wells
INTRODUCTION
tained axially downstream, and the annulus eccentricity and the inner pipe rotation have a significant influence on the drilling hydraulics and performance.
For more than seven decades, since Taylor (1923)
reported his linear stability analysis for flow between
concentric rotating cylinders, this problem has
attracted much attention in the literature (Fang and
Manglik, 1998). Based on flow visualization experiments, Taylor found that beyond a critical rotational
speed, the laminar azimuthal-Couette flow becomes
unstable and it transitions to one characterized by
toroidal vortices. This recirculation, referred to as
The modeling and prediction of axial fluid flows in
the annular passage between the drill pipe and borewell wall is critical for the effective design and operation of oil and gas drilling, and geothermal wells.
The annular duct is usually eccentric, and the rotating
inner drill pipe promotes a swirling fluid motion that
is superimposed on the pressure-driven axial flow in
the bore well. Also, because of the very viscous
nature of the fluid, the flows are primarily in the laminar regime. These developed swirl flows are main1
Present address: LuK Inc., 3401 Old Airport Road, Wooster, Ohio 44691.
Author to whom all correspondence should be addressed: Telephone (513) 556-5704; Fax (513) 556-3390;
and e-mail: [email protected]
2
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FANG and MANGLIK
Taylor vortices, involves fluid motion in the axialradial spatial directions. Extended reviews of the general Taylor-Couette flow problem, the associated
instability and evaluation of critical rotational speeds
have been given by DiPrima and Swinney (1985),
Yamada and Imao (1986), and others. The flow
behavior in annular geometries confined by eccentric
rotating cylinders is quite different from that in concentric annulus (DiPrima and Stuart, 1972; Ballal and
Rivlin, 1976; San Andres and Szeri 1984). For stable,
sub-critical Taylor number (Ta < Tac) flows with a
rotating inner cylinder, the streamline distribution
displays the formation of a recirculating cell with
increasing rotational speed and/or eccentricity in the
wider gap of the annulus. The concomitant issue of
flow stability has been investigated theoretically by
Zhang (1982), and Lockett (1992), and experimentally by Escudier and Gouldson (1997), among others,
who found that stable flows are maintained at larger
rotational speeds, or higher Taylor number, with
increasing annulus eccentricity.
Yamada (1962), and Escudier and Gouldson
(1995) have investigated pressure-driven laminar
axial flow and frictional characteristics in concentric
annuli with inner cylinder rotation. While Yamada
has considered various narrow gap (1 < r* ≤ 0.897)
annuli, Escudier and Gouldson have reported experimental results for flow in an annular channel of r* =
0.506. In both cases, higher f Re values have been
reported for increasing rotational speeds. The onset of
Taylor vortices is found to be delayed by the presence
of axial flow, and stable flows are sustained with relatively higher speeds for inner cylinder rotation. That
the rotational flow tends to be more stable and the
critical Taylor number increases with axial flow Re
has also been established in the theoretical and linear
stability analyses of DiPrima (1960), and Chung and
Astill (1977).
The effects of annulus eccentricity and inner cylinder rotation on axial viscous fluid flows, even though
they have significant design implications for many
practical applications, have received rather scant
attention in the literature (Fang and Manglik, 1998).
Some of the recent efforts include those of Mori et al.
(1987a,b), Nouri and Whitelaw (1997), Escudier and
Gouldson (1997), and Siginer and Bakhtiyarov
(1998). Mori et al. (1987a,b), and Siginer and Bakhtiyarov (1998) have reported theoretical and experimental results for non-Newtonian fluids in narrow
gap annuli (0.8 ≤ r* ≤ 0.96). The experimental work
of Nouri and Whitelaw (1997) deals with turbulent
flows. Using a LDA system, Escudier and Gouldson
(1997) have reported experimental data for the velocity distribution and friction factor in a fixed radius
ratio (r* = 0.506) annulus, but with eccentricities in
the range 0 ≤ ε* ≤ 0.8. For a rotational speed of Rer =
56 in the sub-critical Taylor number regime (Ta ≅
2850), their results indicate that center-body rotation
produces a counter-rotating kidney-shaped vortex in
the radial-tangential flow field in the widest gap of
annulus when the eccentricity is large (ε* ≥ 0.5).
While this flow structure is unaffected by the axial
flow, the peak axial velocity is lowered and a higher
friction factor is obtained, particularly with ε* = 0.5.
In a more recent paper, Escudier et al. (2000) have
reported a computational treatment of this problem.
This paper presents extended and significantly
more accurate numerical simulations for fully developed laminar axial flows in eccentric annuli with a
rotating inner surface. Finite-difference methods are
used to solve the governing flow equations in bipolar
coordinates. The results presented cover a very wide
range of annulus radius ratio (0.1 ≤ r* ≤ 0.9), inner
core eccentricity (0 ≤ ε* ≤ 0.95), and its speed of
rotation (represented by the rotational Reynolds number, 0 ≤ Rer ≤ 150). These rotational speeds for the
various annuli geometry, it may be noted, are restricted to the stable flow, sub-critical Taylor number
regime. The flow physics of the rotation-induced
swirling flows is advanced by delineating the salient
characteristics of the swirling flow behavior and its
influence on the axial flow frictional loss, along with
a parametric assessment of the impact of annuli
geometry and inner-core rotation.
MATHEMATICAL FORMULATION
The eccentric annular flow cross section shown in
Fig.1 can be mapped by a bipolar coordinate system,
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
which is related to the Cartesian coordinates by the
transformation
(1)
where the constant c > 0, 0 ≤ ζ ≤ 2π and –∞ < η < ∞.
The inner and outer cylinder radii, ri and ro, respectively, that bound the annulus geometry, correspond
to η = ηi and η = ηo. Also, from geometrical considerations it follows that
(2)
259
Here the dimensionless stream function ψ, vorticity
ω, and axial velocity w are given by
(8a)
and the rotational Reynolds number Rer is defined as
(8b)
where Ωi is the angular velocity of the rotating inner
cylinder. The no slip and inner core rotation boundary
conditions that constraint Eqs. (5)-(7) can be
expressed as
(3)
where the radius ratio r* and dimensionless eccentricity ε* are given by
(4)
(9a)
For the steady state, fully developed, constant
property, incompressible, laminar flows considered in
this study, by employing the stream function and vorticity formulation the corresponding governing equations can be stated in dimensionless form as
(9b)
and the vorticity boundary conditions can be derived
from Eq. (5) to yield
(5)
(9c)
(6)
(9d)
(7)
Finally, given the velocity distribution, the friction
factor can be evaluated by applying a force balance
across the annular flow cross section. The consequent
hydraulic-diameter based Fanning friction factor is
given by
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FANG and MANGLIK
(10c)
Extended details of the mathematical development
and formulation described above can be found in
Fang and Manglik (1998).
NUMERICAL METHODOLOGY
FIGURE 1
Annular channel cross section in bi-polar coordinates.
(10a)
In order to solve the governing non-linear partial
differential equations that describe the flow field,
finite-difference techniques are employed. Their discretized form is obtained by the control-volumebased procedure outlined by Patankar (1980) for
convection-diffusion problems. The power-law
scheme was used for the axial momentum equation,
and the hybrid scheme for the stream function and
vorticity equation. The power-law scheme was also
used in a few test cases for ψ and ω, and because
there were no differences between the results of this
scheme and the hybrid scheme the later was used in
all cases because of its faster convergence. The firstorder derivatives (∂ψ/∂η) and (∂ψ/∂ξ) were represented by central differencing about the half-node
plane, and complete discretization details are given in
Fang and Manglik (1998).
For the application of the boundary conditions
given in Eq. (9), the stream function ψo(ηo) at the
outer wall of the annulus is arbitrarily set to zero
without any loss of generality. However, at the inner
rotating surface, ψi(ηi) cannot be pre-assigned
because the other constraints on ψ,
where, by their definition
(10b)
(11)
and τw is the average wall shear stress, and Vz , m is the
mean axial velocity. The dimensionless mean axial
velocity can be calculated from the known velocity
distribution as
are not exactly satisfied; using (∂ψ/∂η) leads to a
solution where (∂ψ/∂ξ) ≠ 0, implying fluid penetration at the inner wall. This difficulty can be circumvented by determining ψi, such that it satisfies the
condition that pressure is single-valued at the wall
(Yoo, 1998; Lee, 1992). The application of this crite-
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
261
FIGURE 2
Typical grid refinement and accuracy of numerical solutions.
rion requires that
(12)
where (∂p/∂ξ) is obtained from the azimuthal component of the momentum conservation equation (Fang
and Manglik, 1998). Furthermore, in order to apply
the vorticity boundary conditions of Eqs. (9c) and
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FANG and MANGLIK
(9d), the second-order derivative of ψ was evaluated
by the Taylor-series expansion about the boundary
node and into the flow field to yield the following:
(∆η/∆ξ) = 3 and 4.5, and both sets produced the same
result. The accuracy of the computations is further
verified by the good agreement with the available
experimental data for a few cases, as presented in the
next section.
Sub-critical Taylor Number Flows
(13a)
(13b)
where N is the node point at the inner boundary, and
1 is the node point at the outer boundary of the annulus.
The finite-difference solutions were obtained by
using a uniform grid in ξ and η to describe the computational domain. Depending upon the radius ratio
r* and eccentricity ε* of the annulus, the grid size
varied from Nξ × Nη = 21 × 61 to 81 × 181, where Nξ
and Nη are the number of nodes in the ξ- and η-directions, respectively. The finer mesh was needed for
annuli with small r* and large ε*. A typical grid
structure in the computational flow domain is presented in Fig. 1(b). For all cases the convergence of
solution was ensured by the criterion
where φ i, j and φ i, j are the dependent variables at two
successive iterations in the entire flow field. Also,
grid insensitivity of the numerical solutions was
checked by successively refining the mesh such that
the change in calculated values of f Re was less than
1%. The relative accuracy of computed results is
illustrated in Fig. 2 for a typical case of r* = 0.5, ε* =
0.6, and Ta = 10,000. Numerical values for f Re
obtained by successive grid refinement as well as a
two-dimensional Richardson extrapolation are
graphed. The latter requires results for three different
mesh sizes, where (∆η/∆ξ) is constant for each grid.
Two different grid-size ratios were employed,
n
n −1
For the results reported in this study, the rotational
speeds of the inner cylinder were restricted to the
sub-critical Taylor number regime. Based on his theoretical work, Taylor (1923) has given the following
equation to estimate the critical speed for inner cylinder rotation in a concentric annulus, prior to the onset
of flow instabilities that are characterized by toroidal
vortices:
,
The validity of this correlation, however, is restricted
to large r* or small annular-gap geometries (r* → 1).
For smaller r* annuli, Chung and Astill (1977), and
DiPrima and Swinney (1985) have reported critical
rotational speed results, and these are presented in
Fig. 3(a) along with those of Taylor. It is evident that
small r* annuli sustain stable circular Couette flows
at higher inner-cylinder rotational speeds. Also, as
shown by Chung and Astill (1977), the flow stability
increases in the presence of superimposed axial
flows.
The flow stability is found to further increase in
eccentric annuli, as shown in the theoretical analyses
of Zhang (1982) and Lockett (1992), and experimental measurements of Escudier and Gouldson (1997).
Their results are graphed in Fig. 3(b), and each of
them indicate higher values for the critical Taylor
number Tac with increasing eccentricity ε*. While
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
263
a)
b)
FIGURE 3
Critical Taylor number for onset of flow instabilities in annuli with inner-core rotation: (a) effect of r* and axial flow Re, and (b) effect
of eccentricity ε*.
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FANG and MANGLIK
for rotational speeds in the range 0 ≤ Rer ≤ 150, and
Ta ≤ 10,000, depending upon r*, ε*, and Re, are justifiably for the stable sub-critical Taylor number
regime (see Fang and Manglik (1998) for more
details).
RESULTS AND DISCUSSION
Computational solutions for the velocity distributions (ψ and w) and axial flow friction factors (f Re)
for fully developed laminar flows in different annuli
geometries (0 < r* < 1 and 0 ≤ ε* ≤ 0.95) have been
considered. These results reflect the influences of the
duct geometry as well as the rotational speed (subcritical Taylor number, stable flow regime) of the
inner core, and they are discussed in the ensuing sections.
Fluid Flow Field
FIGURE 4
Axial velocity distribution in an eccentric annulus (r* = 0.506)
with inner-cylinder rotation (Rer = 56) in laminar flow (Re = 110)
[solid lines – present solutions, symbols – experimental data of
Escudier and Gouldson (1997), and dashed line – flow without
rotation].
Lockett finds the ε* dependence of Tac is not influenced by r*, Zhang suggests otherwise and gives relatively larger values for Tac in small r* annuli.
Nevertheless, the results presented in the next section
The influence of the inner core eccentricity ε* on
the flow field in a typical annular channel of radius
ratio r* = 0.506 with a rotating inner cylinder (Rer =
56) is depicted in Figs 4 and 5. These figures also
give a comparison between the present numerical
solutions and the experimental measurements of
Escudier and Gouldson (1997) for the axial (w/wm)
and tangential (Vξ/Ωiri) velocity distributions, respectively. The good agreement between the computational and experimental results that is evident in the two
figures attests to the accuracy of the former. That
eccentricity has a very pronounced effect on the flow
behavior is also seen from these results. Even a small
eccentricity (ε* = 0.2) in the annular channel produces a significant deviation in the tangential velocity
profile (Fig 5) from that for circular Couette flow (ε*
= 0), where the latter is given by
(14)
With increasing eccentricity, the tangential velocity
increases in the narrowest gap (profile C) and
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
FIGURE 5
Angular velocity distribution in an eccentric annulus (r* = 0.506)
with inner-cylinder rotation (Rer = 56) in laminar flow (Re = 110)
[solid lines – present solutions, symbols – experimental data of
Escudier and Gouldson (1997), and dashed line – flow without
rotation].
decreases in the wide gaps (A, B and D). The negative values for (Vξ/Ωiri) in the wide gap for ε* = 0.5
265
and 0.8 are indicative of the flow recirculation in that
region. Furthermore, as seen in Fig 4, the axial velocity distribution in an annulus with a rotating inner
core is altered considerably from that without rotation. When the eccentricity is small (ε* = 0.2), innercore rotation produces a near axisymmetric
axial-velocity profile in the annulus, thereby nullifying, to some degree, the influence of eccentricity.
With higher eccentricities (ε* = 0.5 and 0.8), however, the axial flow distribution tends to revert to that
without rotation, though the magnitude and location
of the peak velocities at different sections are still
quite different.
A more complete picture is presented in Fig. 6,
where contour maps of stream function (ψ) and axial
velocity (w/wm) distributions in the entire cross section of eccentric annuli with r* = 0.5 and ε* = 0.2,
0.5 and 0.8 are presented, for inner cylinder rotation
with Rer = 56. For increasing ε*, the onset and
growth of a kidney-shaped vortex in the wider gap of
the annulus is clearly evident. With the flow separation and reattachment points (end points of the noslip contour at the interface of the re-circulating and
main flows) at the outer surface of the annulus, the
center of this counter-rotating eddy moves in the
direction of rotation (anti-clockwise) with increasing
ε*. Also, as the eddy grows, the separation point
moves upstream (clockwise) and the reattachment
point moves further downstream (anti-clockwise).
This behavior with increasing ε* is in conformity
with the results of Ballal and Rivlin (1976), and San
Andres and Szeri (1984). It may be noted that in the
present problem the radial (η) and tangential (ξ)
momentum equations are decoupled from the axial
momentum equation, but not vice-versa. This implies
that while the presence of axial flow has no influence
on the radial-tangential velocity in the annular cross
section, the axial velocity distribution is altered considerably. A fact also observed by Escudier and
Gouldson (1997) in their experiments. This should
not be misconstrued to imply that axial flow has
absolutely no effect on the flow behavior; as noted in
the previous section, the stability of the flow is
enhanced considerably. More notably, rotation of the
inner cylinder increases the axial fluid throughput in
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FANG and MANGLIK
FIGURE 6
Effect of ε* on the stream function (ψ) and axial velocity (w/wm) distributions for flow in an annulus with r* = 0.506 and Rer = 56.
the narrow regions of the eccentric annulus, where
otherwise the flow tends to stagnate.
In stationary, non-rotating eccentric annular channels, the restriction in the narrowest part of the duct
cross section causes the axial flow to become effectively immobile and “squeeze” higher peak velocity
flow in the wider section. With a rotating inner cylinder, however, the angular distribution of axial flow
tends to become more uniform. This is shown in Fig.
7, where the azimuthal variations in the normalized
peak axial velocity (wmax / wm) are graphed and
which can be considered as an effective measure of
the local fluid mobility around the annulus. Results
for flows with and without rotation (Rer = 0, 50, and
150) of the inner core in an annulus with r* = 0.5 are
presented for two different eccentricities (ε* = 0.2
and 0.6). It is evident that with increasing Rer the
peak velocity decreases in the wider gap and increases in the narrow gap of the annulus, thereby lending
to a relatively increased axisymmetry in the flow distribution. The more pronounced effect of rotation is
seen with a smaller eccentricity (ε* = 0.2) as compared to that with ε* = 0.6. It may be noted that in a
concentric (ε* = 0) annulus of r* = 0.5, (wmax / wm) =
1.507 and it is azimuthally constant.
Figure 8 illustrates the effect of the radius ratio r*
of an eccentric annulus on the stream function (ψ)
and axial velocity (w/wm) distributions. With an inner
core eccentricity of ε* = 0.5 and a rotational speed of
Rer = 40, results for three different radius ratios (r* =
0.25, 0.5, and 0.75) are presented. The separation
point, demarking the location of a kidney shaped
vortex at the outer surface of the annulus, is seen to
move downstream in the direction of rotation as r*
increases; the reattachment point remains relatively
unaffected. The re-circulating eddy is much larger in
a wide gap (r* = 0.25) annulus, and has a more dramatic effect on the axial flow as compared to that in
a narrow annulus (r* = 0.75). Once again, in the former case, the inner core rotation tends to promote
greater momentum convection in the narrower sections of the eccentric annulus, and reduce the flow
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
267
FIGURE 7
Influence of eccentricity ε* and inner-cylinder rotation Rer on the azimuthal distribution of the peak axial velocity.
stagnation considerably. In a narrow annulus, on the
other hand, the axial velocity distribution remains
largely unchanged from that when Ωi = 0.
Increasing the speed of inner cylinder rotation in
an eccentric annulus significantly reduces the
azimuthal variation in the axial velocity distribution.
This is illustrated in Fig. 9, where variations in both
stream function and axial velocity contours in the
flow cross section of an annulus with r* = 0.5 and ε*
= 0.6 are presented. Results for three different subcritical Taylor number rotational speeds (Rer = 10,
50, and 150) are graphed, and the increased axial
flow throughput in the narrow sections of the annulus
at higher rotational speeds is evident. From the
stream function distribution it is seen that the center
of the kidney-shaped vortex near the outer cylinder in
the wider sections of the annulus moves in the direction of rotation (anti-clockwise). Also, the flow separation point moves upstream in the rotating flow and
the reattachment point moves slightly downstream.
That is, the re-circulating flow region tends to engulf
a larger portion of the annulus cross section, thereby
promoting increased axial flow in the otherwise stagnant narrow section.
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FANG and MANGLIK
FIGURE 8
Effect of r* on the stream function (ψ) and axial velocity (w/wm) distributions for flow in an annulus with ε* = 0.5 and Rer = 40.
FIGURE 9
Effect of Rer on the stream function (ψ) and axial velocity (w/wm) distributions for flow in an annulus with r* = 0.5 and ε* = 0.6.
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
269
FIGURE 10
Influence of ε* on f Re in fully developed laminar flows in an eccentric annulus with inner cylinder rotation.
Axial Flow Friction Losses
The influence of the flow cross-section eccentricity in an annular duct with inner core rotation on the
frictional pressure loss is shown in Fig. 10, where f
Re results for an annulus with r* = 0.506, Rer = 56,
and 0 ≤ ε* ≤ 0.95 are depicted. Included are f Re
results for flows without rotation (Rer = 0), and the
experimental data of Escudier and Gouldson (1997).
The generally good agreement between the theoretical and experimental values is evident. More importantly, with constant inner cylinder rotation, the
difference between f Re with and without rotation is
seen to first increase with increasing ε*, reach a maximum and then decrease again to attain a minimum
value when ε* ≈ 0.8; f Re in the former case being
higher. The maximum increase in f Re with rotation is
about 12.2% over that without rotation when ε* = 0.5
for r* = 0.506, and the minimum asymptotic value is
f Re = 12.41. This behavior is also seen in the computational results of Ooms and Kampman-Reinhartz
(1996) graphed in Fig. 10 for flows in an annulus
with r* = 0.6 and inner core rotation with Rer = 90.
The effects of the radius ratio r* of an eccentric
annulus on f Re are depicted in Fig. 11, where results
for ε* = 0.5 and Rer = 40 are graphed. Compared
with the frictional loss in stationary annuli (Rer = 0),
inner-cylinder rotation produces a large increase in
the axial-flow friction factor, particularly in wide gap
annuli (r* < 0.5). For example, in an annulus with r*
= 0.1, f Re (Rer = 40) = 23.983 and f Re (Rer = 0) =
18.44. At the same rotational Reynolds number, however, this increase in pressure-drop reduces with
increasing r*, and completely vanishes in narrow gap
annuli (r* > 0.75). Also shown in Fig. 11 is a compar-
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FANG and MANGLIK
FIGURE 11
Influence of r* of eccentric annuli on laminar fully developed f Re.
ison between the present solutions and the computed
results reported by Ooms and Kampman-Reinhartz
(1996) for the effects of r*, where the annuli eccentricity is kept constant at ε* = 0.5, but the rotational
Reynolds number is different for each r*. This is
given by Rer = 60r*/(1 – r*), from which it follows
that 3.16 ≤ Rer ≤ 540 for 0.05 ≤ r* ≤ 0.9. There is
good agreement between the present computations
and those of Ooms and Kampman-Reinhartz; the
slightly large deviation when r* < 0.2 may be due to
differences in discretization, grid size, and numerical
techniques. In any event, the decreasing influence of
rotation in a small-gap annuli (r* → 1) is very clearly
evident. A similar behavior is seen in the f Re results
with fixed Taylor number in Fig. 11. Here Ta =
10,000 corresponds to rotational speed in the range of
3.7 ≤ Rer ≤ 2,700 for 0.1 ≤ r* ≤ 0.9. The friction loss
increases with r*, then plateaus to a peak value when
0.45 ≤ r* ≤ 0.75, before decreasing again as r* → 1.
The variation in f Re with the rotational Reynolds
number Rer is illustrated in Fig. 12. For an eccentric
annulus of radius ratio r* = 0.5, f Re results for different eccentricities (0 ≤ ε* ≤ 0.8) are presented for a
range of rotational Reynolds number (0 ≤ Rer ≤ 150)
in the sub-critical Taylor number regime. Included
are the results for r* = 0.5, ε* = 0.5, and 0 ≤ Rer ≤
120 reported by Ooms and Kampman-Reinhartz
(1996), and the very good agreement between the two
sets of numerical solutions is obvious. Also, except
for intermediate values of ε* and high rotational
Reynolds number, there is fair agreement with the
more recent computations reported by Escudier et al.
(2000). Figure 12 essentially displays that f Re
increases with Rer in eccentric annuli. The increase in
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
FIGURE 12
Influence of rotational Reynolds number Rer on f Re in eccentric
annuli.
271
axial frictional loss with increasing inner-core rotation is more pronounced with mid-range eccentricities (ε* ≅ 0.5 – 0.6) as compared to that when ε* =
0.8 or ε* ≤ 0.2. These results further verify the strong
influence of inertial forces on f Re, that come into
play due to the inner-core rotation and its eccentricity
in the flow cross-section. The consequent departure
of the flow behavior from rotational-Couette flow
causes a significant increase in the axial flow frictional loss.
Finally, Fig. 13 shows the combined influence of
the annuli geometry (ε* and r*) and inner-core rotation (Ta = 10,000) on the friction loss relative to that
without rotation (f/f0). Reaffirming the previously
stated observations, rotation of the inner surface produces higher friction factors in the axial flow irrespective of the annuli geometry; though the trends are
somewhat complex. For any radius-ratio annulus (r*
= 0.2, 0.5, and 0.8, for example) and fixed Taylor
number, (f/f0) first increases with eccentricity up to a
peak value when ε* ≈ 0.55, it then decreases to a
local minimum when ε* ≈ 0.8, and finally increases
again quite sharply as ε* → 1. The latter, also noted
FIGURE 13
Variation in laminar axial flow with ε* in eccentric annuli with r* = 0.2, 0.5, and 0.8 for fixed Taylor number.
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FANG and MANGLIK
by Escudier et al. (2000)ª, is different from the observation in Fig. 8, where, for r* = 0.506 and Rer = 56
(Ta = 2918), an asymptotic value for f Re is seen to
be obtained. The increase in (f / f0) as ε* → 1 in Fig.
13 can be attributed to the higher rotational speed due
to Ta = 10,000, which corresponds to Rer = 100 for r*
= 0.5. A clear indication of this is found in the results
for r* = 0.5 in Fig. 12. Here the f Re curve for ε* =
0.95 is first seen to lie below that for ε* = 0.90, but
with increasing Rer it crosses over when Rer ≈ 55 and
then lies above. Thus, for r* = 0.5 and as ε* → 1, f
Re displays an asymptotic behavior when Rer ~ 55, a
decreasing trend when Rer < 50, and an increasing
trend (as seen in Fig. 13) when Rer > 60. A qualitatively similar frictional loss performance is obtained
for laminar axial flows in annuli of other radius ratios
as well.
CONCLUSIONS
The effects of inner core rotation and its eccentricity on the axial flow behavior in annular channels
have been numerically investigated. The annulus
radius ratio r*, inner cylinder eccentricity ε*, and its
rotational speed (represented by Rer or Ta) are seen to
have a complex influence on the flow field. With
increasing ε* and/or Rer, a kidney-shaped vortex is
found to grow in the wide-gap flow cross section near
the outer wall of the eccentric annulus. Its size and
flow coverage increases with decreasing r*. While
this lowers the local peak axial velocity magnitude,
the wall gradients are much sharper and they produce
higher friction factors compared to those in stationary
annuli. The increase in f Re due to inner core rotation
is significantly more pronounced in annuli with midrange values of ε*; this effect diminishes with higher
ε*, and f Re again increases very sharply as ε* → 1.
In fact, three regimes can be identified on the f Re –
ε* map: for ε* < 0.5, the rotating inner cylinder drags
the fluid around into the narrow gap region and the
axial flow tends to have a more azimuthally uniform
distribution though significantly higher wall gradients; with 0.5 < ε* < 0.8 the kidney-shaped vortex
grows to engulf much of the wider cross section and
the peak axial velocity reverts to this location, which
causes a relative reduction in f Re (though it is still
greater than that without rotation; finally, with ε* >
0.8 the re-circulating eddy grows to engulf and dominate much of the flow field, the location of the maximum axial velocity reverts to the wider gap region of
the annulus, and f Re again increases rather sharply
as ε* → 1.
ACKNOWLEDGEMENTS
This study was supported in part by the National
Science Foundation (CTS – 9502128), and the Thermal-Fluids & Thermal Processing Laboratory. Professor M. P. Escudier, The University of Liverpool, UK,
very kindly provided the tabulated results from his
computational study. Also, the many constructive
suggestions of Professor M. A. Jog, University of
Cincinnati, and the computational assistance of J. Pillutla are gratefully acknowledged.
NOMENCLATURE
f
Q
ri, ro
r*
Rer
Re
Fanning friction factor, Eq. (10)
volumetric flow rate, m3/s
inner and outer radius of annulus, Fig. 1, m
radius ratio, (ri / ro), Eq. (4)
rotational Reynolds number, (ρΩiri2/µ),
Eq. (8b)
axial flow Reynolds number,
[V 2(r − r / v] , Eq. (10b)
z, m
o
i
ª It may be noted that there is considerable difference between the numerical results of this study and those reported by Escudier et al. (2000) in Fig.
13 for r* = 0.5 and 0.8, and 0.4 ≤ ε* ≤ 0.7. This may perhaps be due to relatively higher errors/uncertainties in the latter’s computations. For example, Escudier et al. (2000) have reported f Re = 18.603 for fully developed laminar flow in an annulus with r* = 0.5, ε* = 0.6, and Ta = 10,000, which
is about 4% smaller than the more accurate value of f Re = 19.325 obtained in the present study (see Fig. 2).
THE INFLUENCE OF INNER CYLINDER ROTATION ON LAMINAR AXIAL FLOWS
Ta
Tac
Vz,
Vz , m
w
wm
wmax
x, y, z
Taylor number, ri(ro – ri)3 (ρΩi/µ)2 =
Rer[(1–r*)/r*]3
critical Taylor number for onset of flow
instabilities
dimensional axial velocity, m/s
dimensional mean axial velocity, m/s
dimensionless axial velocity, Eq. (8a)
dimensionless mean axial velocity, Eq.
(10c)
dimensionless peak axial velocity
cartesian coordinates, Fig. 1
Greek Symbols
ε*
η, ξ
µ
ρ
τw
ω, ω
ψ, ψ
Ωi
dimensionless eccentricity, ε/(ro–ri) , Eq. (4)
bipolar coordinates, Fig. 1
dynamic viscosity of fluid, N⋅s/m2
fluid density, kg/m3
average wall shear stress, N/m2
dimensional and dimensionless vorticity,
Eq. (8a)
dimensional and dimensionless stream
function, Eq. (8a)
rotational speed of inner pipe, rad/s
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