# Effects of Centrifugal Buoyancy and Reynolds Number on Turbulent Heat Transfer in a Two-Pass Angled-RIB-Roughened Channel with Sharp180∘ Turns Investigated by Using Large Eddy Simulation

Authors: Akira Murata; Sadanari Mochizuki
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

The effects of the centrifugal buoyancy and the Reynolds number on heat transfer in a rotating two-pass rib-roughened channel with180∘ sharp turns were numerically investigated by using the large eddy simulation. The effect of the Reynolds number was seen in the finer flow structure. The effect of the aiding/opposing buoyancy contributions was seen more vigorously on the pressure surface than that on the suction surface, though the details depended on the Reynolds number, the rotation number, and the existence of the ribs. As the buoyancy increased, the friction factor dominated by the pressure loss of the sharp turn decreased, and the decreasing rate is smaller for the higher rotation speed case. The Colburn's j factor stayed almost constant irrespective of the rotation speed. As a result, the heat transfer efficiency index slightly increased by the buoyancy, and it became smaller for the higher rotation speed and higher Reynolds number cases.


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## Body

## 1. Introduction

The effective cooling of a gas
turbine rotor blade is essential because the higher efficiency of the turbine
requires a higher inlet gas temperature. Generally, this blade cooling is
performed by film cooling at the external surface of the turbine blade and also
by internal forced-convection cooling which uses winding flow passages inside the
turbine blade. In the internal
forced-convection cooling, the real phenomena are very complicated due
to external forces: the
Coriolis force and the buoyancy force in the centrifugal acceleration field. In
addition to these external forces, the disturbances induced by turbulence
promoters
(ribs) and 180° sharp turns further
complicate the phenomena [1].As
for the heat transfer in smooth and/or rib-roughened channels with the 180°
sharp turn, several researchers
investigated the detailed spatial
variation of the local heat transfer in the
stationary condition with various techniques: wall temperature measurement by
using hundreds of thermocouples [2, 3],
naphthalene
sublimation technique to measure the local mass transfer, which was transformed
into heat transfer by using the analogy between heat and mass transfer [4, 5], unsteady wall temperature
measurement by using temperature-sensitive liquid crystal [6, 7], and wall temperature measurement
by using infrared thermography [8].  In
these studies, the characteristic heat transfer variation induced by the ribs
and the 180° sharp turn was captured; the high heat transfer areas observed in
and after the sharp turn and on and between the ribs. As for the flow field in
the stationary condition, Son et al. [9]
applied
the particle image velocimetry technique to the two-pass channel with the 180°
sharp turn, and the detailed two-dimensional flow field was measured.  For
the rotating condition, however, both the flow velocity and wall temperature
measurements become very difficult because of the following two reasons: the
high centrifugal force preventing the data acquisition system from normal operation and
the difficulty in transferring
data from the rotating system to the stationary system. Nevertheless, some
researchers have performed experiments in the rotating condition by measuring
the wall temperature distribution using thermocouples for the smooth [10–12] and
rib-roughened [13–15] wall two-pass
channels with the 180° sharp turn. Liou et al. [16] conducted the flow
velocity measurement of a rotating two-pass smooth channel with the 180° sharp
turn by using the laser Doppler
anemometer in addition to the detailed heat transfer measurement by using the
transient liquid crystal method.
However, it is extremely difficult to perform the experiments in the
rotating condition that can identify both the flow structure and its influence
on the heat transfer at the same time.
Thus, further progress in experimental studies has so far been prevented, and the authors
decided to investigate the phenomena by performing numerical analysis.In previous numerical studies of the serpentine flow passage with the
180° sharp turn, the Reynolds-averaged Navier-Stokes equation with a turbulence
model was adopted: Banhoff et al. [17] used the k-ε two-equation turbulence model or
the Reynolds stress equation model with the wall function, and Lin et al. [18] used the low-Reynolds number
two-equation turbulence model without the wall function. Although this approach
using the Reynolds-averaged turbulence model could reproduce the heat transfer
of blade cooling to a certain extent, even the Reynolds stress equation model
has empirical constants and functions, and therefore the applicability of the
model should carefully be examined. Recent advancement in computers enables us
to numerically
simulate the fluctuating components of the turbulent flow by using the large
eddy simulation (LES) or the direct numerical simulation (DNS) [19]. Because LES and DNS directly
resolve temporal variation of the fluctuating components, the results are more
universal, in other words, more free from the empirical modeling than the
Reynolds-averaged turbulence models. Although LES also has empirical constants
and functions, the modeling of the turbulence is confined to the subgrid-scale
turbulence, and therefore the effect of the empirical modeling on the result is
ideally less than that in the Reynolds stress equation model.  Recently, the authors performed the numerical
analyses by using a dynamic subgrid-scale model for the two-pass smooth [20] and rib-roughened [21] channels  with the 180° sharp turn in the stationary and rotating
conditions. The effect of the buoyancy force was also
examined for the two-pass smooth [22] and rib-roughened [23] channels where the
aiding and opposing contributions of the centrifugal buoyancy were
clarified.  However, the bulk Reynolds
number of the authors’ previous studies for the rib-roughened duct was confined
to the low values among 3000–5000, though for
the smooth duct [22], the bulk Reynolds number was increased up to 9851. The higher
grid resolution needed for higher Reynolds number demands more computational
resource, and it had
prevented us from simulating high-Reynolds-number
cases.In this study, the bulk Reynolds number
was increased to about104 for the rotating two-pass angled-rib-roughened channel with 180° sharp turns.
The extremely heavy computational load was managed to
be within the feasible computational time
by using the latest supercomputer and the parallel computing technique. How
the centrifugal buoyancy
force affects the heat transfer for different Reynolds number cases was examined. The computations were performed varying
the Rayleigh number for an angled 60°
rib arrangement.


## 2. Numerical Analysis

Figure1 shows the computational domain and coordinate system used in this study. The
duct had a square cross-section with a side length of D. The coordinate
system was fixed to a rotating channel that had an angular velocity of ω with respect to the axis of rotation parallel
to the x-axis. The axial direction of the channel straight pass was parallel
to the z direction; the x and y directions were the
parallel and perpendicular directions, respectively, to leading/trailing walls.
Round-type ribs as used in
[17, 18] were installed with the in-line arrangement on the trailing and
leading walls with the rib angle of 60° with
respect to duct axis of the straight pass. The rib arrangement of this study
gave e/D=0.1 and P/e=10, which was chosen because it was within
the previously reported optimal range for straight ducts considering both the
Nusselt number and the friction factor [24–26]. The rib
angle, 60°, was chosen as the angle which gave the
maximum heat transfer for straight ducts in Han et al. [27]. The 60° rib “NP” arrangement was investigated in
this study. Here, “N” and “P” were from “negative” and “positive” considering
relation between angled-rib induced and sharp-turn induced secondary flow
directions. For example, in the “NP” arrangement the angled-rib induced
secondary flow in the first straight pass circulated in the opposite  direction
to the  turn A induced one, and on the other hand the angled-rib induced secondary flow in the second straight pass circulated in the same direction as the turn A induced one.Figure 1
Schematic of a rotating two-pass rib-roughened square channel with sharp 180° turns (60° rib with NP arrangement).The
present procedure of the numerical analysis was the same as our recent studies [20–23].
After applying a filtering operation to the incompressible Navier-Stokes
equation with a filter width equal to the grid spacing, the dimensionless
governing equations scaled by a length scale, ℓ (=0.5D), and a mean
friction velocity, u*, became a set of dimensionless governing equations with respect to grid
resolvable components indicated by overbars as (u¯, v¯, w¯) under the assumption of constant
fluid properties. In order to simulate a fully
developed situation, the pressure and temperature fields were decomposed into
the steady, x-directionally linear component and the remaining component [28]. By this
decomposition, the latter component of the pressure and temperature fields can
be treated using a periodic boundary condition in the  x direction.The
temperature was made dimensionless by using a linearly increasing component of
temperature,Tlinear,
and a friction temperature, Tr, as θ=(T−Tlinear)/Tr.
Accordingly, the dimensionless energy equation was derived for the grid
resolvable component, θ¯.
The governing equations in the Cartesian coordinates (x,y,z) were transformed into generalized curvilinear coordinates
(ξ,η,ζ)
that were aligned to the structured grid coordinate of this study; ζ  was the duct-axial direction. The
governing equations in the curvilinear coordinate system were expressed as
follows [29]:(1)1J∂JU¯j∂ξj=0,(2)∂u¯i∂t=−U¯j∂u¯i∂ξj−∂ξj∂xi∂p¯∂ξj+1Re*∂ξj∂xℓ∂∂ξj(∂ξk∂xℓ∂u¯i∂ξk)+∂ξk∂xj∂τSGSij∂ξk+Fi,(3)∂θ¯∂t=−U¯j∂θ¯∂ξj−Cθu¯u¯m+1Re*Pr⁡∂ξj∂xℓ∂∂ξj(∂ξk∂xℓ∂θ¯∂ξk)+∂ξk∂xj∂αSGSj∂ξk, where, U¯j was a contravariant component of velocity, and
the following expressions were assumed: J=∂(x,y,z)/∂(ξ,η,ζ), (x1,x2,x3)=(x,y,z), and   (ξ1,ξ2,ξ3)=(ξ,η,ζ).An
external force term,Fi, in the momentum equations (2) had the Coriolis force, the
centrifugal buoyancy force with the Boussinesq approximation, and the mean
pressure gradient term with a value of t
wo as shown
in the following equation:(4)Fi=(22Ro*w¯−Gr*Re*2⁡(θ¯−θ¯∞)yRm−2Ro*v¯−Gr*Re*2⁡(θ¯−θ¯∞)yRm(H0+z)). Because
the dimensionless mean-rotation radius, Rm,
is much larger than y and z  in the real gas turbines, the
following approximations
can be used: y/Rm≒0 and   (H0+z)/Rm≒1. In the dimensionless reference
temperature of θ∞=(T∞−Tlinear)/Tr, Tlinear changes linearly
in the x direction, and therefore  θ¯∞   also changes linearly in the x direction. From the energy balance, the following equation holds:(5)θ¯∞=−Cθxu¯m. Consequently
(4) was approximated as follows:(6)Fi=(22Ro*w¯−2Ro*v¯−Gr*Re*2⁡(θ¯−θ¯∞)).In this study, the friction velocity,u*,
was calculated from the force balance between the mean pressure gradient in the x direction and the wall shear stress as follows:(7)ρu*2=ℓ2|dpdx|m,dim. Here, the mean
pressure gradient with the subscript, “dim,” meant the value with dimension, and it drove the flow in
the x direction. By nondimensionalizing (7), the
dimensionless mean pressure gradient value of 2 came out. Because
of the constant mean pressure gradient, the flow rate varied depending on the Rayleigh number;
therefore, the flow rate was not known a priori, and it was calculated from the
resultant computed flow field after the
statistical steadiness was attained.Subgrid-scale components of stress,τSGSij, and energy flux, αSGSj, were expressed as follows: (8)τSGSij=2νSGSS¯ij,αSGSj=νSGSPrSGS∂ξk∂xj∂θ¯∂ξk, where,(9)S¯ij=12(∂ξk∂xj∂u¯i∂ξk+∂ξk∂xi∂u¯j∂ξk),νSGS=CS2(Δ1Δ2Δ3)2/32S¯ijS¯ij.Because
the flow field of this study has no homogeneous
direction, we adopted the Lagrangian dynamic subgrid-scale model of Meneveau et
al.  [30] that averaged the value of Cs along the path-line
for a certain distance. As for the coefficient of the Lagrangian averaging time scale,
the same value of 1.5 as that in [30] was used. The turbulent Prandtl number
for the subgrid-scale component, PrSGS, was set to 0.5 [31]. The width of the test filter was
double the grid spacing.Discretization was performed by a finite
difference method using the collocated grid system [29]. The spatial and temporal
discretization schemes were similar to those of Gavrilakis [19]: the second-order central
differencing method and the Crank-Nicolson method for the viscous term, and the
second-order differencing method satisfying the conservative property [29] and
the second-order Adams-Bashforth method for the convective term. The external
force term was also treated by the second-order Adams-Bashforth method. The
pressure field was treated following the MAC
method [32]. At the wall boundary, no-slip and constant heat flux conditions
were imposed. The boundary conditions of the intermediate velocities and
pressure were set following the procedure of [33, 34].The local Nusselt number,NuL,
was calculated from the wall temperature as follows:(10)NuL=2Re*Prθ¯w−θ¯b. The
averaged Nusselt number was calculated by using the integrally averaged
temperature difference for the area in question.In this study, the values ofRe*(=u*ℓ/ν)
and Ro*(=ωℓ/u*) were varied
in (Re*,Ro*)=(3000,1) and (4000,2), and the Rayleigh number, Ra*, was varied form 0 to 1×105.  For the case of Ra*=1×105,  Re* was reduced from 3000 to 2500 in order
to maintain the bulk Reynolds number to be 104. This was needed because the increase in Ra* reduced the friction
factor and, as a result, increased the bulk Reynolds number.  Hereafter, this case of (Re*,Ro*)=(2500,1) is also referred to as (3000,1) in this study for the
simplicity. In correlating the experimental results, the effect of the buoyancy
is often expressed by using the Grashof number, Grm,q, which is defined with the wall heat flux. The
following relation holds due to the definition of Grm,q: (11)Grm,q=16Re*PrGr*=16Re*Ra*.The
conversion of the dimensionless numbers of this study (Re*,Ro*,Ra*) defined by the mean friction velocity,
the friction temperature, and the length scale of 0.5D  into those of (Rem,Rom,Grm,q)
defined by the bulk mean velocity,
the wall heat flux, and the hydraulic diameter, D,
was summarized in Table 1.
In Table 1, our previous numerical results for the smooth [22] and
rib-roughened [23] two-pass channels are also included for comparison. In the
real aircraft gas turbine engines, the central region of the operating range is
in the order of Rem~104, Rom~10−1, and Grm,ΔT/Rem2~10−1 [11].  From the definition of Grashof
numbers, the following relation holds between Grm,q  and Grm,ΔT:(12)Grm,q=NumGrm,ΔT. Note that the
computational conditions of this study are within the real operating range. It should also
be noted that  Grm,ΔT/Rem2 is identical to the buoyancy parameter often used in experiments.Table 1
Dimensionless number range
of this study and the duct-averaged values off and Num  (the
values in parentheses in the second line are for higher grid resolution of
101×101×1569. The values with asterisks are for quasi-DNS).
Rib arrangementRe*Ro*Ra*RemRomGrm,qGrm,qRem2Grm,ΔTRem2f/fBNum/Nu∞(=j/j∞)60°
NP3000 (2500)10115951.0300012.792.401×104123430.974.8× 1083.150.0411.472.372×104129660.939.6× 1085.710.0710.522.35141890.858.982.294×104(14100)(0.85)1.9×  1099.540.11(9.08)(2.38)15040*0.80*8.11*2.23*1×105131990.764.0× 10922.960.267.082.3840002099853.2100029.552.571×104101243.166.4× 1086.240.0828.842.592×104102133.131.3×10912.270.1628.392.60(10350)(3.09)(27.74)(2.62)1000 [23]1034471.1600011.872.67(3501)(1.14)(11.56)(2.70)5×10339001.038.0× 1075.260.149.572.661×10442370.941.6× 1088.910.228.282.692×10447790.843.2×  10814.010.326.702.68(4736)(0.84)(6.81)(2.72)Smooth
[22]35002094162.9700025.072.531×10495452.935.6× 1086.150.0924.472.483×10498512.841.7× 10917.520.2423.182.55As shown in Figure1, the structured
grid system was generated by using Gridgen version 15
(Pointwise
Inc., Tex, USA). The grids in the
physical domain were contracted to the walls and the corners. On the
rib-roughened surfaces, the grids were aligned to the ribs. The grid number was
mainly 71 × 71 × 1207 in ξ,η,ζ directions,
and this grid configuration, for example, gave a grid spacing of Δ1+=0.59–70.0,  Δ2+=0.58–60.1, and Δ3+=15.7–85.9 (Re*=3000, Ro*=1, and Ra*=1×105). Here, the inner length scale of ν/u*mod was used, because the friction velocity, u*, defined in (7) overestimated the value.
In (7), the streamwise direction was assumed to be only in the x direction, and therefore it cannot
account
for the longer total flow
distance caused
by the change of the
streamwise direction in the two-pass channel. In order to estimate appropriate
inner length and time
scales, the friction velocity, u*mod, was calculated by
using the resultant flow rate and
friction factor for each condition. The effect of the grid spacing on the
computed result was checked by increasing the grid number to 101 × 101 × 1569 for several cases, and no major
difference was observed as shown in Table 1. The time step interval was Δt=1.0×10−4,
which can be expressed as Δt+=0.046  when
made dimensionless by an inner time scale, ν/u*mod2,
for Re*=3000, Ro*=1, and Ra*=1×105.In order to deal with the very high
computational load of this study, the computational domain was decomposed into 32 subdomains in the duct-axial (ζ)
direction, and the parallel computing technique was applied. Each subdomain's computation was performed on a
different CPU on HITACHI SR11000J (Information Technology Center, The University of
Tokyo). When the algebraic equation for each variable was solved by using the
SOR method, the values at the subdomain boundaries were transferred to the
neighboring subdomains by using MPI functions in each iteration step.The computation was started using the
result of the similar condition as an initial condition. At first, the
calculations were carried out till the statistically steady flow condition was
attained. After that, additional60000
steps (t=6
or t+=2760 for  Re*=3000,  Ro*=1, and Ra*=1×105)
were performed for computing the statistical values. This 60000 step computation needed about 12.1  ×  32 CPU hours for the 71 × 71 × 1207 grid configuration.


## 3. Results and Discussion

At first, the verification of the
present numerical procedure is explained. In our previous studies of smooth
[20]
and rib-roughened [21] two-pass channels, the numerical results were compared
with the experimental results of the stationary condition in the local and
transversely averaged Nusselt numbers and also in the channel averaged f and j factors. The agreement between the numerical and experimental
results was good, and the present numerical procedure was able to reproduce the
heat transfer enhancement in and after the sharp turn and the quick development
of the swatooth-profile
heat transfer augmentation due to the periodically installed ribs in the
straight pass [20, 21]. In order to examine the subgrid-scale model
contribution on the flow and heat transfer, the quasi-DNS was performed for one
condition of  Re*=3000, Ro*=1, and Ra*=4×104.
As shown in Table 1, the quasi-DNS gave the lower friction factor and heat
transfer than the LES results, and this tendency is the same as the other
researcher's result [35].  All the main
features of the flow and heat transfer were reproduced by the quasi-DNS, and they confirmed the
reliability of the present results.Figures2(a) and 2(b) show the time-averaged local Nusselt number (see Figure 2(a))
and the streamwise component of the wall shear stress (see Figure 2(b)) on all
four walls by viewing the flow channel from six different directions. The Nusselt number of this study was
normalized using the following empirical correlation for a fully developed pipe
flow [36]:(13)Nu∞=0.022Rem0.8Pr0.5. It should be noted that the lower part of
“outer wall” in the figure shows the inner wall values because it is visible
through the inlet and outlet of the channel. In Figure 2(b), zero shear
stress boundary is indicated by a white line. In the calculation of the streamwise component of the wall
share stress, the streamwise direction was approximated by the ζ
direction. In this study, the trailing and leading walls of the first straight pass correspond to the
pressure and suction surfaces, respectively, and the opposite relation holds in
the second straight pass. The pressure and suction surfaces are defined with
respect to the secondary flow induced
by the Coriolis
force, which impinges onto the pressure surface. As a comparison, the result of the lower
Reynolds number [23] is shown in Figure 3. When Figures 2 and 3 are compared, the Nusselt numbers
of the lower Reynolds number case (Figure 3(a)) are more enhanced than the
higher Reynolds number case (Figure 2(a)) on the rib-roughened pressure
surfaces and in the turn section,
though the approximate profiles are similar to each other. Here, it should be
noted that the Nusselt numbers in the figures are normalized by Nu∞ being the increasing
function of the Reynolds number, and therefore the larger NuL/Nu∞ does not necessarily mean the higher Nusselt number. When the instantaneous
local Nusselt numbers are examined in Figure 4, the Reynolds number effect is
seen in the finer flow structure and, as a result, the finer heat transfer
distribution in Figure 4(a) for Re*=3000 as compared to that in Figure 4(b) for Re*=1000.Time-averaged profiles of Nusselt number and streamwise component of wall shear stress (60° rib NP, Re*=3000, Ro*=1, and Ra*=0).
(a)
Nusselt number(b)
streamwise shear stressTime-averaged profiles of Nusselt number and streamwise component of wall shear stress (60° rib NP, Re*=1000, Ro*=1, and Ra*=0).
(a)
Nusselt number(b)
streamwise shear stressInstantaneous profiles of Nusselt number (60° rib NP, Re*=(a)3000 and (b)1000, Ro*=1, and Ra*=0).
(a)
Re*=3000(b)
Re*=1000As
discussed in [22, 23], because of the secondary flow induced by the Coriolis
force, the fluid temperature on the trailing (pressure) side of the first
straight pass becomes lower than that on the leading (suction) side; therefore,
on the pressure and suction surfaces of the first straight pass (radially
outward flow), the buoyancy works in the aiding and opposing directions to the
main flow, respectively (see Figure 5(a)). On the other hand, in the second
straight pass (radially inward flow), the situation is reversed to the first straight
pass: on the pressure and suction surfaces of the second straight pass, the
buoyancy works in the opposing and aiding directions to the main flow,
respectively (see Figure 5(b)). In this way, when the buoyancy works, the
radial flow direction becomes an important parameter which controls the flow
and heat transfer in an orthogonally rotating channel. In the previous study
for the lower Reynolds number case [23], when the centrifugal buoyancy was
introduced, the effect of the aiding buoyancy contribution was seen in the
larger variation of the streamwise shear stress on the pressure surface of the
first straight pass, and in the reduced area of the reverse flow between the
ribs on the suction surface of the second straight pass. These changes in the
flow field caused the heat transfer on the corresponding surfaces larger. The
effect of the opposing buoyancy contribution was seen in the intensified and
extended reverse flow region on the suction surface of the first straight pass,
but the Nusselt number there showed a slight increase in the downstream part.
As seen in Figures 6 and 7 by being compared with Figures 2 and 3,
respectively, these effects are observed in the higher Reynolds number case (see
Figure 6) but less clear than the lower Reynolds number case (see Figure 7).Explanation of aiding and opposing buoyancy contributions in first (radially outward flow) and second (radially inward flow) straight passes.
(a)
first straight pass (radially outward flow)(b)
second straight pass (radially inward flow)Time-averaged profiles of Nusselt number and streamwise component of wall shear stress (60° rib NP, Re*=3000, Ro*=1, and Ra*=1×105).
(a)
Nusselt number(b)
streamwise shear stressTime-averaged profiles of Nusselt number and streamwise component of wall shear stress (60° rib NP, Re*=1000, Ro*=1, and Ra*=2×104).
(a)
Nusselt number(b)
streamwise shear stressFigure8 shows the higher rotation number case of  Ro*=2 (Re*=4000). When the rotation speed
increases, the heat transfer on the pressure surface is deteriorated especially
at the downstream region of the straight pass. This tendency was also observed
in the low-Reynolds-number case in [23]. It should be noted that this deterioration of the heat transfer is
not characteristic to the fully developed region of an infinitely long straight duct, because the fully developed results of
smooth [37] and rib-roughened [38] straight ducts do not show the deterioration
of the heat transfer even when the rotation speed is higher. The interaction
between the sharp-turn induced and Coriolis-induced flows must be affecting the
deterioration of the heat transfer.  The buoyancy effect on the heat transfer and
the shear stress was minor for this high rotation speed case (figures not shown).Time-averaged profiles of Nusselt number and streamwise component of wall shear stress (60° rib NP, Re*=4000, Ro*=2, and Ra*=0).
(a)
Nusselt number(b)
streamwise shear stressIn order to examine the buoyancy
effect in the most developed region in the present straight passes, the
area-averaged Nusselt number at the downstream part of the straight pass is
shown in Figure9. The averaging area between 5th and 6th ribs from
the straight pass entrance (between 2nd and 3rd ribs from the downstream turn)
was chosen so as for the flow and temperature fields to be most developed and
at the same time to be still without the downstream turn effect. In Figure 9,
the buoyancy parameter, Grm,q/Rem2, is used as the horizontal axis. On the pressure surface (triangle
symbols), the buoyancy effect on Nuarea is larger
than that on the suction surface (square symbols), and Nuarea of the aiding contribution cases (solid lines) is larger than those of the opposing contribution
cases (broken lines). This tendency is supported by the experimental results of
the rib-roughened channel [14], although the entrance condition for the first
straight pass in [14] was not disturbed by the upstream sharp turn. The aiding
contribution on the pressure surface for the increase of  Grm,q/Rem2 differs depending on
rib existence, Re*, and Ro*. As seen in the figure, the smooth case
for Re*=3500 once increases and then decreases
as  Grm,q/Rem2 increases. For the rib-roughened case of Ro*=1 (Re*=3000), Nuarea on the pressure surface stays almost constant for the increase of Grm,q/Rem2.
When Ro* is increased to Ro*=2 (Re*=4000),  Nuarea on the pressure surface becomes lower
due to the above-mentioned deterioration of the heat transfer, and the clear
buoyancy effect is only seen for the highest buoyancy case for the aiding
contribution side. Nuarea on the suction surface shows much
smaller variation due to Grm,q/Rem2,
and a slight increase is observed due to the increase of Grm,q/Rem2.Figure 9
Buoyancy effect on area-averaged Nusselt number at developed region in straight passes  (Averaging area was between 5th and 6th ribs from straight pass entrance. Open, filled, half-filled, and concentric symbols are for (Re*, Ro*, smooth/rib) = (3500, 2, smooth), (1000, 1, rib), (3000, 1, rib), and (4000, 2, rib), resp.  Symbols with solid and broken lines are for aiding and opposing buoyancy contributions, resp.).In
order to view the time-averaged flow structure of the whole two-pass channel,
the isosurface (surface with the same value) of the second invariant,Q, of the deformation tensor, ∂ui/∂xj, [39] is shown in Figure 10. In the
figure, the time-averaged temperature is also shown as the color contour on the
isosurface of Q. The value of  Q is calculated by (14) for
incompressible fluids, and it is often used to identify vortices because the
positive value of  Q means that the
vorticity exceeds the strain,(14)Q=12{[12(∂ui∂xj−∂uj∂xi)]2−[12(∂ui∂xj+∂uj∂xi)]2}.Time-averaged vortex structure visualized as isosurface ofQ=10 with the color contour of fluid temperature (60° rib NP, Re*=3000, and Ro*=1).
(a)
Ra*=0(b)
Ra*=1×105As seen in Figure10(a), in and around the turn the strong
vortices are produced. On the pressure surfaces, the stronger vortices are shed
behind the angled ribs as compared to the suction surfaces. In Figure 10(b), the
effect of the buoyancy is seen in the extended and intensified vortex region
near the trailing surface which gives the aiding contribution to the fluid
motion. The aiding contribution enhanced the turbulent kinetic energy near the trailing surface (figures not shown). The high
rotation speed case (Ro*=2) gave the drastic decay of both vortices and turbulent kinetic energy in
the straight pass section (figures not shown).Figure11 shows the friction factor, f, normalized by
using fB in the following Blasius equation:(15)fB=0.079Rem−0.25.Figure 11
Buoyancy effect on pressure loss (Legend for the symbols are the same as that in Figure9.  Symbols for Kturn  and  Kstraight  with solid lines are for turn A or first straight pass, and those with broken lines are for turn B or second straight pass).The
precise values
off/fB are shown in Table 1. In order to decompose the total pressure
loss into the sharp-turn and straight-pass contributions, the following
pressure loss coefficients, Kturn and Kstraight, are also
plotted in Figure 11 [40]:(16)Kturn=Δpturn(1/2)ρUm2,Kstraight=Δpstraight(1/2)ρUm2. Here,  Δpturn is the pressure loss
associated with the sharp turn, and it is calculated by linearly extrapolating
the area-averaged wall-pressure profile at the central region of the first and
second straight passes in the direction toward the turn inlet and outlet
locations at z=17 for the turn A and
at z=1 for the turn B [40]. The
area-average was taken with the streamwise pitch of D in the straight pass. The straight pass component, Δpstraight, was calculated from the linear
pressure profile at the central region of the straight pass. A decrease of the f factor by the buoyancy force is
observed in Figure 11.  Without the buoyancy, Kturn is larger than Kstraight.
The buoyancy makes Kturn decrease both in the turns A and B with the exception of some high rotation
speed cases (Ro*=2) in which Kturn shows a slight increase. The value of Kstraight in the second
straight pass becomes negative, due to the positive pressure gradient in the
streamwise direction which is caused by the coincidence of the main flow and
buoyancy force directions; the main flow is driven by the buoyancy in the
second straight pass (radially inward flow). The buoyancy increases and
decreases Kstraight in the
first and second straight passes, respectively, and this opposing variation in Kstraight cancels each other. As a
total, the pressure loss behavior of the channel is controlled by Kturn, and the f factor decreases as Grm,q/Rem2 increases. The variation of Kstraight due to the buoyancy is smaller for the higher Reynolds number case. The large f factor for the high rotation speed (Ro*=2) is due to the increased Kturn.Figure12 shows the Colburn's j factor normalized by
using j∞ calculated from Nu∞   in (13). The precise values of j/j∞ are shown in Table 1.  In
order to further examine the heat transfer efficiency taking the pressure loss
into account, the heat transfer efficiency index, ηeff, was calculated by using the
following equation [41]:(17)ηeff=St/Stsmooth,stationary(f/fsmooth,stationary)1/3. As explained in [41],  ηeff is the index of the
heat conductance for equal pumping power and heat transfer surface area. In
this study, the smooth and stationary result of Re*=1000
[22] was used for calculating (17) for all the cases. As shown in Figure 12,
the j factor is insensitive to the
buoyancy and stays almost constant. As a result,  ηeff slightly increases with the increase of the
buoyancy due to the decreased f factor seen in Figure 11. ηeff   is lowered as the Reynolds and rotation
numbers increase, and it becomes almost constant with respect to the buoyancy
parameter increase for the higher rotation number case (Ro*=2).
This is because the friction factor is less sensitive to the buoyancy for the
higher rotation number as seen in Figure 11; in other words, the friction
factor is dominated by the pressure loss in the turn section which is increased by the high rotation speed and
is insensitive to the buoyancy.Figure 12
Buoyancy effect onj factor and heat transfer efficiency index (legend for the symbols are the same as that in Figure 9).


## 4. Conclusions

The
large eddy simulation of the two-pass angled-rib-roughened square channel with
the 180° sharp turns was performed changing the Reynolds, Rayleigh, and rotation numbers for the rib
arrangement of the 60° rib NP. From the
numerical results, the following conclusions were drawn.When the Reynolds number was increased, the flow structure became finer
which resulted in the finer instantaneous distribution of the local Nusselt
number. When the centrifugal buoyancy was introduced, the aiding and opposing
contributions of the centrifugal buoyancy worked in the different way depending
on the radial flow directions (radially outward and inward flows). The pressure
surfaces of the radially outward and inward flows came to the aiding and
opposing buoyancy contribution sides, respectively, and the heat transfer was
larger in the aiding buoyancy contribution, though the detailed profile differs
depending on the Reynolds number, the rotation number, and the rib existence.
On the other hand, the suction surfaces of the radially outward and inward
flows came to the opposing and aiding buoyancy contribution sides,
respectively, and both showed the slightly increased heat transfer.The
friction factor was decreased
as the buoyancy increased which was dominated by the sharp-turn induced
pressure loss because the increased and decreased pressure losses in the first
and second straight passes, respectively, cancelled each other. Especially for
the high rotation number case, the sharp-turn induced pressure loss became
large and it made the friction factor very high. On the other hand,
the channel averaged Nusselt number was insensitive to the buoyancy,
and it was almost
constant. The
heat transfer efficiency index taking the pressure loss into account gave the
lower values for the higher Reynolds and rotation number cases with a
slight increase due to the buoyancy.Here, it is emphasized
that the novel findings of this study are in disclosing the flow field, the two-dimensional
profile of the local heat transfer coefficient and the local shear stress on
the wall, and the friction factor in the heating and rotating conditions. These
findings cannot be supplied by the presently available experimental methods.



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*Source: 764720-2008-12-04.xml*