Proceedings of the International Gas Turbine Congress 2003 Tokyo
November 2-7, 2003

IGTC2003Tokyo TS-082

The Eﬀects of Oblique Discrete Rib Arrangement on
Heat Transfer Performance of a Square Duct
Kazuya TATSUMI1 , Hiroshi IWAI1 and Kenjiro SUZUKI1
1

Department of Mechanical Engineering

Kyoto University
Sakyo-ku, Kyoto 666-8501, JAPAN
Phone: +81-75-753-5219, FAX: +81-75-753-5251, E-mail: ktatsumi@mbd.nifty.com

PpressurePr
Prandtl numberRe
Reynolds number Um D/ν
Uivelocity components
Umchannel mean streamwise velocity
VRvelocity ratio between the cross-flow and jetWDduct widthWr
rib width
Greek symbols
βangle of attack of oblique rib and winglet
Γcirculation
θ
jet skew angleµ
viscosityρ
densityφ
jet pitch angleΩmaxmaximum vorticity
subscripts
0value for a smooth circular tube

ABSTRACT
Three-dimensional numerical simulation was conducted bymaking use of a RANS approach for the flow in a duct accompanying longitudinal vortices and discussions are given
for the related thermal fields. In the first half of this paper, applicability of the three turbulence models adopted in
the present study is discussed based on the comparison withpublished experimental results both for the flow around arectangular winglet vortex generator and for the inclined impinging jets. The results obtained with the non-linear eddy
viscosity models agree reasonably well with the experiments.
Main part of the article treats the numerical results forthe flow around a discrete rib attached obliquely to the flow
direction onto the bottom wall of a square duct in whicha fully developed turbulent flow is established at the inlet.
The rib attack angle to the main flow was varied in several
levels from β = 20◦ to 75◦ . The ratio of the rib height to theduct hydraulic diameter and the duct Reynolds number were
set to be e/D = 0.15 and Re = 20, 000, respectively. Dueto the rib inclination and the gaps existing between the rib
ends and the sidewalls, a noticeable heat transfer augmentation is obtained downstream the rib where the separated flowfrom the leading edge of the rib reattaches. As β decreases,
the region of heat transfer deterioration behind the rib isreduced in size. Furthermore, heat transfer enhancement incurred by the downwash flow and the longitudinal vortex
becomes more conspicuous. Heat transfer enhanced region
however becomes narrower as β decreases since the cross section of the generated longitudinal vortex becomes smaller
then. Combination of the two eﬀects produces the most effective heat transfer enhancement in the case of β = 45◦
among the tested cases.

INTRODUCTION
Use of a rib-roughened channel is one of the basic andeﬀective tools for the enhancement of internal flow convective heat transfer. It has been used in various industrial
applications such as corrugated plate heat exchangers or serpentine cooling passages in high temperature gas turbine
blades. Flow within the rib-roughened channels is highly
three-dimensional especially for the cases of small aspect ratio. So the pattern of accompanying flows and the mechanism of related heat transfer strongly depend on the rib geometry. Therefore, numerous investigations have been conducted for many kinds of rib arrays in the past few decades.
A full-span rib mounted perpendicularly to the flow direction is the standard one and its friction and heat transfer characteristics have been studied first(Han et al., 1984,
Acharya et al., 1997). Oblique ribs and V-shaped ribs possessing certain angle of attack have been studied next(Hanet al., 1985, Fann et al., 1994, Ekkad and Han, 1997, Iacovides et al., 2001, Jang et al., 2001). Inclination of the ribs
to the flow direction invokes secondary flow downstream therib. Mixing to be incurred between the duct core region andthe wall vicinity region eﬀectively enhances the wall heattransfer (Kiml et al., 2001, Mohammad et al., 2002). Furthermore, several references demonstrate that introductionof discrete ribs, the ribs interrupted by gaps in their spanwise direction, is eﬀective in the augmentation of heat transfer (Taslim et al., 1996,Tatsumi et al., 2002). Flow passing
through the gap directly produces the enhancement of heattransfer. It also aﬀects the pattern of recirculating flow behind the rib and reduces the size of the low eﬃciency region

NOMENCLATURE
AR
duct aspect ratiodjet nozzle diameterDhydraulic diameter
Cp
pressure loss coeﬃcienterib height, winglet heightHduct heightkturbulent kinetic energy
Nu
Nusselt number based on D and fluid local bulk
mean temperaturec
Copyright °2003
by GTSJ
Manuscript Received on March 31,2003
1

ity model. Three diﬀerent models are tested in the present
study, i.e. the ones proposed by Launder and Sharma(Launder and Sharma, 1974) (hereafter referred to as LS
model), by Abe, Kondoh and Nagano (Abe et al., 1997)(hereafter referred to as NLAKN model) and by Craft, Launder and Suga (Craft et al., 1996) (hereafter referred to as
CLS model). LS model is one of the classic models and has
been tested most frequently so far by many researchers. It issimple in form and is numerically stable. NLAKN model is
one of the typical non-linear eddy viscosity models and has
been demonstrated to work well for various heat transfer
problems. CLS model is another non-linear eddy viscosity
model retaining third order terms ignored in NLAKN modeland is therefore expected to work better for flows containing
rotating velocity components. For the turbulent heat flux in
Eq. (3), the following gradient type assumption was adopted
in the present computation.

of heat transfer (Tatsumi et al., 2002). Therefore, obliqueribs with spanwise gaps are expected to further improve thewall heat transfer in relation with the secondary flow or thelongitudinal vortices to be generated (Hu et al., 1996) inaddition to the shrinkage of the ineﬀective recirculating flow
region. They may produce even a chance to reduce the pressure loss penalty (Cho et al., 2001). However, flow pattern
to be produced should be complicated and must be studied
in detail.
In the present article, discussion will be developed forthe flow pattern to be produced around an oblique discrete
rib attached onto the duct bottom wall in a fully developed turbulent flow region, based on the results of a threedimensional numerical computation. The discussion aims at
giving insight for the flow pattern and related heat transfer
to be produced in a duct and at providing hints to experimentalists for future detailed study. Unfortunately, Direct
Numerical Simulation is yet very diﬃcult even with the mostadvanced super-computer. So, the present numerical computation adopts a RANS approach which introduces ambiguity
to some extent depending on the applicability of the turbulence models to be employed. In this connection, preliminary
computations were carried out for two types of flows similar
to the one to be treated in the present article in a sense that
longitudinal vortices are generated in a channel. One is thevortex generated with a rectangular winglet vortex generator
and another is the one generated with a skewed and pitched
jet injected into a cross-flow in a channel. The discussions inthis part show that the present numerical computation may
work for the purpose mentioned above, at least, in qualitative sense.

−u0j θ0

DρΘ
Dt

=
=
=

0
−

∂P
∂
+
∂xi
∂xj

∂
∂xj

µ

µ

∂Ui
− ρu0i u0j
∂xj

µ ∂Θ
− ρu0j θ 0
P r ∂xj¶¶(4)

PRELIMINARY TEST CASES FOR TURBULENCE MODELS EMPLOYED
Here is shortly discussed some of the results obtained in
the preliminary tests made for the two types of flows. These
are additional cases to the results of Tatsumi et al.(2002)
where the present code was compared with experiments forflow in a duct with 90◦ full-span ribs and discrete ribs attached onto its wall. The tested two cases are of a flow passing a rectangular vortex generator and an inclined impinging
jet, respectively. Since oblique discrete ribs are believed toinduce longitudinal vortices, it is essential to evaluate the applied turbulence model for such flows. Thus, these two cases
are chosen as test cases additional to 90◦ ribs to validate thepresent numerical code.

(1)

µ

νt ∂Θ
P rt ∂xj

where the turbulent Prandtl number, P rt was set to beconstant and equal to 0.9. Fully implicit forms of finitediﬀerence equivalents of the governing equations were solved
numerically along the time axis in the manner adopted in reference (Iwai et al., 2000). For finite-diﬀerencing of the momentum equations, a fourth-order central diﬀerence scheme
was employed for the diﬀusion terms while a fifth-order upwind scheme was employed for the convection terms. In theenergy equation and the two governing equations for k andε, a first-order upwind scheme was adopted for the convection terms and a central-diﬀerence scheme for the diﬀusion
terms. The pressure gradient in the momentum equation
was evaluated with the SIMPLE algorithm which solves thepressure correction so as to satisfy the finite diﬀerence equivalent of Eq. (1). Alternating direction implicit (ADI) method
was employed to accelerate the conversion in iterative procedure needed to solve the elliptic-type partial diﬀerential
equations.
For the winglet vortex generators and oblique ribs, in order to calculate the flow and thermal fields adjacent to thebodies with high accuracy, a composite grid system was employed in the computation. The grid system is thus composed of two diﬀerent grid frames. One is the main grid
system allocating the grids aligned with the duct geometric structure, i.e. the streamwise, spanwise and transverse
directions. The other is a sub-grid system for which grid
points are aligned with another Cartesian frame of reference
rotated to match the geometry of the winglet vortex generator or oblique rib. The values of the solved quantities oneach of the grid system were adjusted so as to share the common values at their overlapping boundaries at every iterative
process at any time.

NUMERICAL PROCEDURE
Due to the high Reynolds number and the complicatedgeometrical structure of the rib treated here, adoption of Direct Numerical Simulation is impractical as a means to tackle
the problem. Therefore, a RANS approach was employedin the present study by applying eddy viscosity models forsolving the momentum equation. As a numerical procedure,time-dependent forms of the governing equations are solved
for the following reason. For a separating and reattaching
flow, such as the backward-facing step flow, separated shear
layer contains large-scale vortices periodically formed andmoving toward the reattachment point. This produces periodical fluctuation of velocity and other quantities and cansignificantly contribute to the heat transfer characteristics
especially at the reattachment point (Tatsumi et al., 1999).
With an approach to solve the time-dependent forms of the
momentum and energy equations, eﬀects of apparent shear
stresses and heat fluxes to be produced by these large-scalevortical motions can automatically be taken into account in
the computation.
Thus the governing equations solved in the present study
are the following three-dimensional, incompressible, timedependent forms of continuity, momentum and energy equations:
∂Ui
∂xi
DρUi
Dt

=

(2)
(3)

The turbulence models employed here to determine theReynolds stresses, i.e. the last term of Eq. (2), are categorized as a low Reynolds number version of eddy viscos-

2

Fig.1: Computational domain of the winglet vortex generator.

Rectangular Winglet Vortex Generator
The first test flow is the one in a channel in which in-line
arrays of the pairs of rectangular winglet vortex generators(hereafter referred to as WVG) are mounted to its bottom
wall. The results are compared with the experiment conducted by Lau et al.(1999). Geometrical configuration of
the WVG treated in the present study is shown in Figure
1 together with the adopted coordinate system. A pair ofrectangular winglets, having an angle of attack and facing toeach other, is attached to the duct bottom wall in a symmetric pattern. The arrays are arranged in a spatially periodicalmanner in both streamwise and spanwise directions. Considering these conditions for the arrangement of the WVG, periodic and symmetric boundary conditions are employed at thestreamwise and spanwise boundaries, respectively. The coordinate origin is located at the center of the winglet, which
is located at a distance of 4e from the upstream boundary.
β is set to be 30◦ . The geometric condition of the WVG is
set to be identical with the experiment of Lau et al.(1999).
Reynolds number, which is based on the winglet height, istaken to be Re = 10, 000. The turbulence models applied
here in the computation are the LS model and CLS model.
Figures 2(a) and (b) illustrate the cross-sectional distributions of the streamwise velocity, U/Um , and Reynolds stress,2
u0 v 0 /Um, respectively. Here Um denotes the cross sectionalmean velocity in the channel. Each includes three figuresand shows the values obtained at three spanwise locationsof z/WD = −0.125, 0.125 and 0.375 at the streamwise location of x = −2.92e, namely a position of 5.35e downstream
the trailing edge of the WVG. The three spanwise locations
roughly correspond to the downwash flow, the vortex centercore and the upwash flow regions. The results of hot wiremeasurement presented by Lau et al.(1999) are depicted together in the figures. They used two types of hot wire probes
in their study, i.e. a quadruple-wire probe (referred to asprobe-A) and an X-probe (referred to as probe-B). Thus, the2results for u0 v 0 /Um
obtained with both probes are shown in
Figure 2(b). Comparison of the numerical results with theexperimental data plotted in the figure must be discussedwith reservation giving consideration to noticeable diﬀerence
existing between the two measurements.
As for the distribution of U/Um , both models show underprediction in the center region at z/WD = 0.125, a position
close to the vortex core. The LS model shows a better prediction at z/WD = −0.125 whereas it largely diﬀers withthe experimental data at z/WD = 0.375. CLS model overpredicts the mean streamwise velocity at z/WD = −0.125.
However good agreement is observed at z/WD = 0.375. On
the whole, the present results obtained with CLS model show

2
Fig.2: Transverse distributions of U/Um , u0 v 0 /Umand2
u0 w 0 /Um.

Fig.3: Computational domain of the inclined impinging
jet.

a reasonable agreement with the experimental data. As forthe turbulence quantities, it is noticed that CLS model shows
2
values than thea better performance in predicting u0 v 0 /Um
LS model. Additionally to be marked here, the present re2
sults of u0 v 0 /Umshow better accordance with the data ofprobe-B, the probe of simpler geometry. Higher accuracy expected in the measurement with the probe-A may not havebeen achieved because of its positioning diﬃculty related toits complicated geometry.

Inclined Impinging Jets
The second test flow is a skewed and pitched jet injected
into a cross flow in a channel. Such inclined jet is known
to produce longitudinal vortex at the downstream of thejet which induces heat transfer augmentation at the channel walls. This is similar to the flow to be studied as themain target of the present study so that it is taken to be
one of the test flows to study the applicability of the turbulence models. The present results are compared with theexperiment carried out by Nakabe et al.(1998).
The computational domain of this case is schematically3

played in the figure for comparison. Although the peak Nu
is underpredicted and the calculated peak position does notprecisely coincide with the experimental counterpart, both
numerical results capture the essential features of the experimentally obtained distribution of Nu. Especially, thecalculated result with the CLS model agrees fairly well withthe experimental counterpart. The discrepancy between thecalculated and experimental peak positions of N u is to someextent produced by the spanwise flow induced more strongly
by the periodic side boundary condition adopted in the computation. Lower numerical results of Nusselt number comepartially from the numerical negligence of weak developmentof thermal field observed in the experiments at x/d = 0.0. It
is also related to the ambiguity in the inlet flow temperatureand to the possible diﬀerence between the experiments andthe present calculation in the jet velocity profiles.
In Figure 4(b) is plotted the peak Nusselt number at eachstreamwise position, N umax , against x/d. While the numerical result with the NLAKN model is almost flat, theone with CLS model agrees fairly well with the experimental data. The first peak is caused by the jet impingement
and the second peak is due to the eﬀect of the generated
longitudinal vortex.
Considering all of the points raised in the above, CLS
model is judged to work reasonably well in the predictionof the thermal field aﬀected by the generated longitudinalvortices.

Fig.4: (a) Spanwise distribution of N u at x/d = 20
and (b) streamwise distribution of peak Nusselt number,
N ux at each streamwise position.

shown in Figure 3 together with the geometrical parameters and coordinate systems to be used here. The jet isinclined in a way characterized by two angles: the skew angle θ = 90◦ and the pitch angle φ = 45◦ . The origin of thecoordinate system is located at the center of the outlet portof the jet nozzle having elliptic shape. The inlet and outlet
boundaries of the computational domain are located respectively at x/d = −10 and x/d = 60, where d is the diameterof the jet nozzle. Approaching flow to the channel inlet ishydro-dynamically taken to be a fully developed one in achannel. Periodic boundary conditions are applied at bothside boundaries of the computational domain for simplicity, while in the experiments a single nozzle was mounted
in a channel having the aspect ratio of 14.4. The channelheight to jet diameter ratio is H/d = 5 and the channel
domain width to jet diameter ratio is WD /d = 30, respectively. The channel flow Reynolds number defined with thecross-sectional mean velocity and the channel hydraulic diameter is Re = 5, 000. The jet to channel flow velocity ratio
is set to be V R = 3. Although the jet Reynolds number
based on the jet mean velocity and jet nozzle diameter is
ReJ = 1, 500, injected flow from the nozzle is treated in
the computation to be turbulent at the nozzle outlet andthe time mean velocity and other quantities are determinedfrom the corresponding values for a fully developed turbulent
flow in a circular tube. The working fluid is water but noinformation about the inlet flow temperature was suppliedin the reference (Nakabe et al., 1998). In this computation,
the Prandtl number is set to be P r = 6.7. Top wall of thechannel was heated in the experiments and the Nusselt number measured on the top wall inner surface is discussed here.
Thermal field of the channel at the nozzle location was reported in the experiments to be still weakly developing but,in the computation, it is treated to be fully developed at constant heat flux condition for simplicity. For the turbulencemodels, NLAKN model and CLS model are employed in thepresent computation.
Figure 4(a) shows the spanwise distribution of Nusselt
number, N u, calculated at the streamwise location x/d = 20.
The experimental results of Nakabe et al.(1998) are also dis-

RESULTS AND DISCUSSION
Discussion is now given to the flow pattern and its related heat transfer characteristics around a single rib attached with an angle of attack onto one of the walls of aduct having small aspect ratio. The adopted computational
domain is illustrated in Figure 5 together with the geometricparameters and the main coordinate system where its origin
is located at the duct centerline. A fully developed turbulent flow is assumed to approach the inlet boundary, which
is located 10e upstream the rib, and the outlet boundary is
located at a position 35e downstream the rib, where e is therib height. The aspect ratio of the duct, AR, is set to beunity. Reynolds number based on the hydraulic diameter ofthe duct is fixed and is set equal to Re = 20, 000. Rib heightto duct height ratio and rib width to duct width ratio are
set equal at e/H = 0.15 and Wr /WD = 0.6, respectively,
while the rib possesses a square shape cross section. The
angle of attack of the rib to the main flow is varied in several steps, i.e. β = 20◦ , 30◦ , 45◦ , 60◦ and 75◦ . Regarding
the previous discussions concerning the rectangular WVG
and inclined impinging jet, the CLS model is chosen here
for the turbulence model applied to the computation. For
wall boundaries, constant heat flux condition is applied tothe duct walls and to the rib surface.
The grid number is 227 × 57 × 77 for the main grid systemand 36 × 57 × 52 for the sub-grid system. A test for gridindependency of the computation was carried out by varying the grid numbers in the three directions. Since the wallheat transfer value is most sensitive to the grid density inthe normal direction of the wall, the discussion of grid dependency only in the y direction will be shown here. Threekinds of grid node variations were tested, i.e. 227 × 52 × 77,
227 × 57 × 77 and 227 × 75 × 77 for the main grid. The
values of y+ for each grid were approximately 5, 2 and lessthan 1. Only one case of the angle of attack was examined,
i.e. β = 45◦ . The overall heat transfer coeﬃcient decreasedby 3.5 percent between the first two cases and there was nosevere diﬀerence between the latter two cases. A grid inde-4Fig.6: Spatially averaged Nusselt number, N uavg /N u0 .

Fig.5: Computational domain of the oblique discrete rib.

pendent study was also carried out by Suga(1998) and hehas reported that the CLS model produces accurate results
for y + ≤ 4. From these discussions, the aforementioned grid
numbers are chosen in the present study.
Mean Nusselt number, N uavg , spatially averaged over theregion from x/e = −5 to x/e = 20 of the rib-attached ductwall including the rib surfaces is shown in Figure 6. The
values are normalized by Nu0 , which represents the Nusselt
number for a fully developed turbulent flow in a circular tube
correlated by McAdams(1942) as follows:
N u0 = 0.023Re0.8 P r0.4

(5)

N uavg /Nu0 increases as the angle of attack decreases andtakes a maximum peak at β = 45◦ , and then starts to decrease with a further decrease of β. Thus, the largest heattransfer augmentation is obtained at β = 45◦ .
Streamwise local but spanwise-averaged Nusselt number,
N ux /Nu0 , is plotted against x/e in Figure 7(a). Rib surface
area is counted as a part of the total averaging area. In
all the cases, the value of Nux /Nu0 is larger than unity
over the whole span of the streamwise locations so that theattachment of the rib enhances the duct wall heat transfer
over long streamwise distance irrespective of the value of theangle of attack. Most eﬀective enhancement is obtained in
the region 5 < x/e < 10 in every case of diﬀerent value ofβ. In the case of β = 45◦ , N ux /N u0 shows largest value
among the results for diﬀerent value of β at any streamwise
position.
Figure 7(b) shows the pressure loss coeﬃcient, Cp , whichis calculated in the following form.
Cp =

2∆P
2
ρUm

Fig.7: Streamwise distribution of (a) the spanwiseaveraged Nusselt number, N ux /N u0 and (b) pressure
loss coeﬃcient, Cp /Cp0 .

closer to each other, however, the order does not change significantly: Cp /Cp0 is maximum for β = 60◦ and minimumfor β = 20◦ . Therefore, the angle of attack which produces
the largest Nuavg does not correspond to that of the maximum Cp /Cp0 . These results suggest that β = 45◦ may be
the optimum value for oblique discrete ribs in this sense.
Figure 8 shows the velocity vector maps in an x-y planeat three spanwise locations z/WD = −0.15, 0.0 and 0.15 forthe case of β = 45◦ . The contour maps of the spanwise
velocity, W , are superimposed on the graph, in which theshaded parts correspond to the region where W is positive.
Velocity vector maps of the secondary flow in a cross section
calculated at three streamwise locations x/e = 2.0, 5.0 and
10.0 are illustrated in Figure 9 for the case of β = 45◦ .
Figure 10 shows the spatial distribution of the local Nusselt
number, Nu/Nu0 , on the rib-attached wall for various valuesof β. Figure 11 shows the cross-sectional distribution of the2
turbulent kinetic energy, k/Um
, calculated for various values(6)

where ∆P is the diﬀerence of the cross sectional mean pressure between the calculated location and x/e = −5.0. Cp isnormalized by the value Cp0 which represents the pressure
loss coeﬃcient of a circular tube calculated from the friction
factor, f0 , correlated by Petukhov (Webb, 1994).
f0 = (1.58lnRe − 3.28)−2(7)

Cp /Cp0 largely increases where the rib exists and pressure
recovery is observed downstream of the rib for all angles ofattack. In the region of 0 ≤ x/e ≤ 5, Cp /Cp0 takes its largest
value for β = 75◦ and 60◦ and then decreases as β decreases.
Although the present results are for discrete ribs, this orderof Cp /Cp0 is the same as that for oblique ribs without gaps
as reported by Han et al.(1985). Further downstream at
x/e = 20 the values of Cp /Cp0 for the diﬀerent cases become5Fig.8: Velocity vectors and W contour maps at x-ycross-sectional plane (β = 45◦ ).

Fig.10: N u/N u0 distribution on duct bottom wall.

wall at other spanwise positions as found in Figures 8(b) and(c), and is actually bifurcated into two streamlines directed
upstream and downstream at a position detached from thewall. Because of this, velocity of near wall flow becomeslarger with the decrease of the rib angle of attack. In thecase of β = 90◦ , recirculating flow region appears behind therib near the duct centerline and Nusselt number takes low
value there because of small near wall flow velocity. Thisshould be improved in the case of using the oblique discrete
rib having smaller angle of attack due to the appearance oflarger near wall flow velocity as discussed above.
There occurs another eﬀect with the decrease of the rib
angle of attack, which causes the change of the spatial distribution pattern of local Nusselt number. It is related to
the generation of the longitudinal vortex. As is seen in Figure 9, the separated flow streamlines starting from one endof the rib located on negative side of z accompanies flowrotation as well as the above-discussed conspicuous downwash flow at x/e = 2.0. The rotating flow develops into alongitudinal vortex downstream slightly accompanying thedeflection toward the positive z direction as is easily foundin the figures for the downstream positions of x/e = 5.0 and
x/e = 10.0. In the same manner as the previously discussed
inclined jet case, this longitudinal vortex produces eﬀective
heat transfer augmentation at the bottom wall over large
span of streamwise distance as is observed in Figure 7(a).
It is reported in the reference (Tiggelbeck et al., 1991) thatfor delta winglet type vortex generators, a longitudinal vortex is not clearly formed for large angles of attack. Namely

Fig.9: Velocity vectors at the y-z cross-sectional plane(β = 45◦ ).

of β at the streamwise location x/e = 10 on which velocity
vector maps are superimposed. In Figure 8(a), streamlineof the separated flow from the rib leading edge is clearlyobserved to reattach the duct bottom wall. However, thisis not true for Figures 8(b) and (c). As is suggested by theappearance both of the shaded and un-shaded parts near
the wall in Figure 8, near wall flow has spanwise velocity
or the velocity component along the rib axis in noticeable
magnitude. Because of the existence of the spanwise flownear the wall, the separated flow streamline starting fromthe edge of the rib does not reattach at the duct bottom

6

Table 1. Peak vorticity and circulation at x/e = 10βΩmax /Um (1/m)
Γ/Um × 102 (m)

75◦
-

60◦
-34.1
-1.18

45◦
-31.3
-1.49

30◦
-45.3
-1.18

20◦
-58.1
-0.821the critical angle is reported to be βc ' 70◦ and no vortex
was appreciable above that angle. In the present results, although not shown here, a longitudinal vortex of large scale
was recognizable at β ≤ 60◦ resulting in larger heat transfer
augmentation at the bottom wall. This gives another reason why better performance of heat transfer is obtained at
β = 45◦ than at β = 75◦ . However, on the other hand, thegenerated longitudinal vortex becomes slender in its crosssectional size with the decrease of β as is estimated fromthe shrinkage of the cross-sectional area of high turbulent
kinetic energy region shown in Figure 11. Table 1 shows thepeak value of the streamwise vorticity and the circulation
calculated at the location of x/e = 10.0 for the four cases ofdiﬀerent value of β. The circulation is obtained by spatiallyintegrating the streamwise vorticity in the cross section. In
the integration, value smaller than 20% of the peak vorticity
was not taken into account. The values are not shown herefor β = 75◦ since a clear vortex formation was not found forthis case. From the decrease of the maximum vorticity withthe decrease of the angle of attack, it is confirmed that generation of the longitudinal vortex becomes more conspicuous
at smaller value of β. On the contrary, however, the largest
value of the circulation is obtained at β = 45◦ . This suggeststhat the generated longitudinal vortex becomes slender withthe decrease of β. Therefore, with the decrease of β, thelocal Nusselt number becomes larger because of the generated longitudinal vortex but the width of the heat transfer
enhanced-region is reduced. These are the background why
N uavg takes largest value at β = 45◦ . One more point to
be noted here is that the level of turbulent kinetic energy
is larger at larger value of β as is found in Figure 11. This
indicates that a rib of larger angle of attack yields salient interruption of the main flow and therefore generates stronger
turbulence. This further suggests that turbulent kinetic energy takes a minor role on heat transfer enhancement andthat, at smaller value of β, more eﬀective heat transfer isachieved with causing smaller disturbance to the flow in aduct.
Figure 12 shows the contour of the local Nusselt number,
N u/N u0 , at the duct sidewall at z/WD = 0.5 for the case of
β = 45◦ . For a duct of small aspect ratio, as is the present
case, a substantial eﬀect to the duct sidewall heat transfer
occurs. A noticeable peak of Nu/N u0 appearing at x/e ' 5
is caused by a vortical flow induced at the corner, which
is clearly discernible in Figure 9 at x/e = 2.0 and x/e =5.0. This vortical flow sweeps the wall surface and keeps
the thermal boundary layer thinner, thereby heat transfer isenhanced there.

Fig.11: y-z cross-sectional view of turbulent kinetic energy contours and velocity vectors at x/e = 10.

Fig.12: N u/N u0 distribution on the duct sidewall (β =45◦ : z/WD = 0.5).

three-dimensional flows. Namely calculation was made foran array of a pair of rectangular winglets attached onto achannel wall and for a skewed and pitched impinging jet injected to a cross-flow and the obtained results were comparedwith experimental results. The major conclusions drawn inthe discussions are as follows.
(1) Among the treated turbulence models, CLS modelshows the best performance. The predicted streamwise velocity and Reynolds stresses of the rectangular winglet caseand the Nusselt number of the inclined jet are in reasonableagreement with the experimental data.
(2) Oblique rib attached to a heat transfer surface produces augmentation of wall heat transfer at positions downstream the rib having the largest overall Nusselt number at
the angle of attack of β = 45◦ . Due to the rib inclinationand to the provided gaps, the flow separated from the leading edge of the rib turns its direction so as to better sweep

CONCLUDING REMARKS
Three-dimensional unsteady numerical simulations wereconducted using a RANS approach for the flow in a square
duct attached with an oblique discrete rib to its wall. The
rib was mounted with an angle of attack to the approachingflow and side-ends were detached from the sidewalls. The rib
angle of attack has been varied in several steps from β = 20◦
to 70◦ in which the flow and heat transfer characteristics
were investigated. Additional to this main discussion, othertwo test cases were also conducted as a means of evaluating
applicability of the present numerical method to such highly

7

Iacovides, H., Jackson, D. C., Kelemenis, G., Launder,
B. E., Yuan, Y.-M., 2001, “Flow and Heat Transfer in a
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2.

the heat transfer surface. This causes heat transfer augmentation at wall and also contributes to remedying the low heattransfer in the region behind the rib by reducing its size. A
longitudinal vortex is generated by the rib. At rib downstream position, thinning of the thermal boundary layer is
produced by this vortex on the duct bottom. This resultsin heat transfer augmentation there. Although larger maximum vorticity is observed for smaller value of β, the vortex
becomes slender and the width of heat transfer enhancedregion is decreased. As a result of a trade-oﬀ matter of thesefeatures, the aforementioned β = 45◦ shows the highest heattransfer performance among the tested cases.
(3) The inclination of the rib also aﬀects the heat transfer at the duct sidewalls. A corner vortex is generated at
the trailing side-end of the rib. By the combination of this
vortex and the main longitudinal vortex, a thinner thermalboundary layer is produced at one side of the duct sidewalls
located at z/WD = 0.5. Thus enhancement of the wall heattransfer is obtained in this area.
ACKNOWLEDGEMENTS
This article is written based on the results of the research
work of the CREST project ”Micro Gas Turbine and Solid
Oxide Fuel Cell Hybrid Cycle for Distributed Energy System” supported by the Japan Science and Technology Corporation (JST). The article is also supported by the Japan
Society for the Promotion of Science (JSPS) and the firstauthor gratefully acknowledge the scholarship provided by
them.
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