Proceedings of HT2007
2007 ASME-JSME Thermal Engineering Summer Heat Transfer Conference
July 8-12, 2007, Vancouver, British Columbia, CANADA

HT2007-32786
NOTCH ARRANGEMENT EFFECTS ON HEAT TRANSFER
IN A CHANNEL WITH CUT-FINS
Kazuya Tatsumi
Dept. of Mechanical Engineering and Science,
Kyoto University, Sakyo-ku, Kyoto, 606-8501,
JAPAN
tatsumi@mbox.kudpc.kyoto-u.ac.jp

Shintaro Matsuzaki
Dept. of Mechanical Engineering and Science,
Kyoto University, Sakyo-ku, Kyoto, 606-8501,
JAPAN
shintaro.matsuzaki@t03.mbox.media.
kyoto-u.ac.jp

Kazuyoshi Nakabe
Dept. of Mechanical Engineering and Science, Kyoto University,
Sakyo-ku, Kyoto, 606-8501, JAPAN
nakabe@me.kyoto-u.ac.jp

ABSTRACTThe effects of the attack-angle of the fin notch arrayagainst the main flow and size of the clearance at the fin-tip on
the heat transfer and pressure loss performances of a channel
with cut-fins (parallel fins with square notches) mounted on thebottom wall were evaluated in the present article. Threedimensional numerical simulations, PIV measurements andheat transfer experiments employing a modified single-blow
method were conducted to discuss these characteristics.
Larger pressure loss reduction was obtained by the cut-finscase compared with the plain-fins case (parallel fins without
notches) under smaller clearance conditions, while smaller
thermal resistance was achieved with larger clearance. A
maximum peak, therefore, appeared in the overall performancein relation with the clearance size. Larger heat transfer
coefficients were obtained with smaller attack-angles of thenotch array in both experimental and numerical results,
particularly under larger Reynolds number conditions. This wasdue to the spanwise flow generated in the area adjacent to thenotch, by which renewal of the thermal boundary layer was
effectively produced at the trailing edge of the notch.
INTRODUCTION
Parallel plate fins are one of the most popular techniquesby means of enhancing the heat transfer area and coefficient. In
the past few decades, therefore, discussion on the flow and heattransfer characteristics of fins with various kinds of shapes
have been keenly discussed in developing an effective fins forcompact heat exchangers and heat sinks [1-4].

Fig. 1: Computational domain (clearance effect case).

Fig. 2: Computational domain (attack-angle effect case).
When a clearance exists between the channel top wall andfin top surface, a portion of the main flow flows along this area.
This affects the heat transfer performance of the fin. Namely, if
the clearance is small, larger heat transfer coefficient is obtainat the fin surfaces while noticeable pressure loss penalty is
incurred; on the other hand, if the clearance is large, large
amount of fluid flows through the clearance that entails a

1

Copyright © 2007 by ASME

reduction of pressure loss together with deterioration of theheat transfer performance. Thus a trade-off matter between theheat transfer and pressure loss exists in relation to the clearance
size [5-7].
To obtain a higher heat transfer performance in suchparallel plate fins having a clearance at the fin tip, rectangularshape notches are applied to the fins (referred to as “cut-fins”)
in the present study. The notch is expected to convect a portionof the flow in the clearance into the area located between thefins, namely, to enhance mass and heat transfer between thefluids located at each area. This results in an increase of theheat transfer coefficient at the fin sidewalls. In addition to this,reduction of the pressure loss penalty and fin total weight arealso expected as the effects of applying the notch to the fins. In
the authors’ previous works, the effects of the notch shape, fin
spanwise pitch and notch arrangement on the flow and heattransfer characteristics of the channel have been studied byboth numerical and experimental analyses [8, 9].
The advantages obtained by introducing the cut-fins to achannel, as described in the previous paragraph, are believed tobe affected by the clearance size. It is important to evaluate towhat extent these effects are available when the clearance size
is increased. In addition to this, in the authors’ previous works[8, 9], higher heat transfer performance was obtained when thenotch array was set inclined to the main flow particularly in
relatively high Reynolds number regime, Re≥1,000. This was
believed to be attributed to the spanwise flow generated by thenotch array, and further detail discussion should be made on theflow and heat transfer characteristics of such oblique notch
array case including the effects of the attack-angle of the notcharrangement against the main flow.
Under these backgrounds, the effects of the clearance size
and attack-angle of the notch array on the flow and heattransfer characteristics are examined in the present article.
These characteristics are evaluated by conducting threedimensional numerical simulations, velocity measurements
using PIV (Particle Image Velocimetry) and heat transfer
experiments applying modified single-blow method.

NOMENCLATUREc:specific heat capacityf:friction factorh : heat transfer coefficient
H : channel height
Hf : fin height
Hn : notch heightj:Colburn j factor
L : unit length of fin
Lc: length of the test section
Ln : notch lengthm : mass
Nu : Nusselt number
q : heat flux
P : pressure

Pr :
Re :
Wf :
Wp :
Wd :

Prandtl number
Reynolds number
fin thickness
fin pitchchannel width

Greek symbols
β : attack-angle of notch array
λ : thermal conductivity
μ : viscosityρ : densityθ : temperature
Subscripts
b : bulk mean valuef:fluid parth:harmonic mean value
inlet, outlet: values at inlet and outlet
m: mean valuep:plain-fins:solid partw : wall

NUMERICAL PROCEDURE AND CONDITIONS
Figures 1 and 2 show the computational domain of thepresent numerical simulation. In Fig. 1, is shown thecomputational domain of the case conducted to evaluate theeffects of the clearance size between the fin tip and channel top
wall. In this case, periodic boundary conditions were applied toboth streamwise and spanwise directions representing periodic
cut-fin arrays in both directions.
The domain shown in Fig. 2 was employed in the casediscussing the effects of the attack-angle against the main flowof the notch array. In the authors’ previous works [8, 9], when
the notch array was inclined against the main flow, spanwise
flow was observed along the notch. In a channel withsidewalls, interaction between this spanwise flow and sidewalls
might generate a longitudinal vortex-like secondary flow that is
expected to enhance the mass and heat transfer of the fluids inthe channel. To evaluate this effect, a channel with sidewalls
and having 5 rows of cut-fins aligned in the spanwise direction,
as shown in Fig. 2, was used as the computational domain. In
this case, periodic condition was applied to the streamwiseboundaries. The fin number in this case is much smaller thanthat of the test section used in the experiment, and relatively
stronger sidewall effect will appear in the computation than in
the experiment. These numerical results, however, will provide
some hints to understand the flow and heat transfer
characteristics of a channel with oblique cut-fins, at least in aqualitative way.
Three-dimensional incompressible and time-dependent formsof continuity and Navier-Stokes equations, as shown in Eqs. (1)and (2), were solved in the present study.

2

Copyright © 2007 by ASME

Table 1: Parameters of cut-fin array (clearance effect case).
H/Hf
1.25
1.5
1.75
2

H125
H150
H175
H200

Hn/Hf
0.5
0.5
0.5
0.5

Ln/L
0.5
0.5
0.5
0.5

L/Hf
4.375
4.375
4.375
4.375

wf/Hf
0.06250.06250.06250.0625

Flowmeter

Regulator
Needle valve

Wp/wf
7.0
7.0
7.0
7.0

Heater170

500

Diffuser

A90
A75
A60
A45
A30

90
75
60
45
30

H/Hf
1.25
1.25
1.25
1.25
1.25

Hn/Hf
0.5
0.5
0.5
0.5
0.5

Ln/L
0.5
0.5
0.5
0.5
0.5

L/Hf
4.375
4.375
4.375
4.375
4.375

wf/Hf
0.06250.06250.06250.06250.0625

Test section
100 100

250

100

300

Flow
10

20

Table 2: Parameters of cut-fin array (attack-angle effect case).
β [°]Compressed air

Volt slider

Wp/wf
7.0
7.0
7.0
7.0
7.0

150

Thermocouples

30 - CH Fast digital thermometer
Computer

Measuring station

Fig. 3: Apparatus of heat transfer measurements.
Tracer particle generator
Compressed air

∂U i
=0
∂xi

(1)

DU i
∂ ⎛⎜ ∂U i ⎞⎟
∂P
ρf
μ
+
=−
Dt
∂xi ∂x j ⎜⎝ ∂x j ⎟⎠(2)Here, ρ f and μ are the density and viscosity of the fluid,
respectively.
In the computation, both temperature fields of the flow andfin were considered by solving the following energy equationsof the fluid and solid parts.
fluid part: ρ fsolid part: ρsDθ∂ ⎛ μ ∂θ ⎞⎜⎟=Dt ∂xi ⎜⎝ Pr ∂xi ⎟⎠(3)dθ∂ ⎛ λs ∂θ ⎞⎟⎜=dt ∂xi ⎜⎝ cs ∂xi ⎟⎠(4)

ρ s , λs and cs are the density, thermal conductivity andspecific heat capacity of the fin. At the fin surfaces, the heat
flux was calculated by applying a harmonic-mean thermal
conductivity defined in Eq. (5) [10].

q w = −λh

∂θ
∂n

(5)

λh was defined as Eq. (6) using the thermal conductivities ofthe fluid, λ f , and solid, λs .

λh =CCD camera
170

2
1 / λ f + 1 / λs

(6)The physical properties of air and stainless steel were applied
to the fluid and solid parts under constant temperature
condition. Pr was set equal to 0.7.
Isothermal wall condition was applied to the channel bottom
wall, and the side and top walls were set as adiabatic. As
mentioned above, periodic condition was applied to thestreamwise boundaries. The linear pressure loss term due to thechannel pressure loss in the streamwise direction was dealt byapplying an additional inertia force to the source term in Eq. (2)as proposed by Patankar et al. [11]. In the thermal fields,

500

100

250

100

300

Flow
150

Test section

20

10

Double pulse laser
Synchronizer Computer

Fig. 4: Apparatus of PIV measurements.
periodic boundary conditions were applied by employing thedefinition that the non-dimensional temperature was equal ineach periodic streamwise location as shown in Eq. (7).

θ inlet − θ w θ oulet − θ w=(7)θbinlet − θ w θboutlet − θ w
Here θ w and θ b are the wall and bulk mean temperatures,
respectively. The superscripts in the equation denote thelocations of each boundary.
The examined parameters of the cut-fin geometry aretabulated in Tables 1 and 2. The variables shown in each tablecorrespond to those shown in Figs. 1 and 2, respectively. In
Table 1 is shown the values of the case evaluating the effects ofthe tip clearance. In this case, the ratio between the clearance
and fin height were changed, and other variables were set
constant. In the case shown in Fig. 2 and Table 2, the attackangle of the notch array against the main flow, β, was changed
from 90° to 30° by sliding each notch location in thestreamwise direction with a distance of ΔL. Other geometricvalues, in this case, were identical with those of case H125
shown in Table 1.
Reynolds number, Re, was calculated on the basis of thehydraulic diameter of the channel in both cases. In the casediscussing the clearance effect, Re was equal to 200. On theother hand, discussion on β effects was conducted under thecondition of Re=200~2,000.

3

Copyright © 2007 by ASME∂θ s*= θ f* − θ s*
*
∂t
∂θ w*
= b1 (θ f* − θ w* )*∂t
∂θ *f ∂θ f*+ * = b2 (θ s* − θ f* ) + b3 (θ w* − θ f* )
*
∂t
∂x

(8)
(9)
(10)

θ * is the dimensionless temperature defined by the initial
temperature, θ 0 , as θ * = (θ * − θ 0 ) / θ 0 . t * and x * are thedimensionless time and distance, respectively defined ast * = ms cs t / hA and x* = hAx / m& f c f Lc . Here m , m& and Lc
are the mass, mass flow rate and length of the test section,
respectively. cf is the specific heat capacity of the fluid atconstant pressure. h and A are the heat transfer coefficient andheat transfer area of the fin surface. The values b1, b2 and b3 aredefined as follows:hw Aw ms cs
×
hA mw cw
mc
b2 = s s
mf cf
b1 =

b3 =

hw A w ms cs
×
hA m f c f

(11)
(12)
(13)

The subscripts s, w and f represent the parts of solid, channel
wall and fluid, respectively.

-110

present
Liang & Yang (1975)
Nu=6.7Nu=7.47

j

EXPERIMENTAL PROCEDURE AND CONDITIONSIn the present study, heat transfer measurement using
modified single-blow method [12] and flow velocity
measurement using PIV (Particle Image Velocimetry) wereconducted. Figure 3 shows the apparatus of the heat transfer
experiments. The air was supplied to the channel by acompressor located in the upstream. The flow rate was adjusted
by a flow meter and valve that were inserted in between thechannel and compressor.
In the single-blow method, the time-mean overall heattransfer coefficient was obtained by analyzing the unsteadycharacteristics of the flow bulk mean temperature at the inlet
and outlet of the test section. When the inlet temperature ischanged, the bulk temperature of the outlet flow also changes.
Time histories of this outlet flow temperature depend on theinlet flow temperature, heat capacities of the fluid and fin, heattransfer area and heat transfer coefficient of the fin and channel
wall. By appropriately modeling the energy equation of the test
section, therefore, the heat transfer coefficients at the fin andchannel walls can be predicted by finding suitable coefficientsthat adjust the calculated outlet temperature obtained in thecomputation to that obtained in the experiment.
Equations (8), (9) and (10) describe the energy equations,which model the energy balance in the test section used in thepresent study.

-2

10

-3

10

3

10

4

10
Re
Fig. 5: Relation between j factor and Re in parallel platecase.

The partial differential equations (8), (9) and (10) werediscretized by using a second order central scheme of finite
difference in space and using fully implicit method in time. The
obtained algebraic equations were then solved by the iteration
process, to which TDMA was applied. In this case, themeasured temperature of the inlet flow was applied to the inlet
boundary. At the outlet boundary, the temperature gradient wasset equal to zero, namely, Neumann condition was employed.
As shown in Fig. 3, an electric heater was inserted in thechannel in the upstream area of the test section. This heater wasconsisted of a wire mesh of 0.3mm diameter weaved between
the channel top and bottom wall in 10 rows. It was charged by
a voltage-slider (Yamabishi V-130-3) and an electric switch
under constant voltage condition, and a step-like temperaturechange in the inlet flow was obtained.
The flow temperatures at the inlet and outlet were measured
by inserting K-type thermocouples at the inlet and outlet of thetest section. Thermocouples of 0.1mm in diameter were set at
each location, having 5 rows equally spaced in the spanwise
direction. In each row, 3 thermocouples were inserted in equal
distance in the height direction, thus, totally 15 thermocouples
were applied to the inlet and outlet. The thermocouples wereconnected to and measured by a multi-channel thermo-meter(Yokogawa WE500, WE7232).
Considerable difference of velocity distribution from a
simple Poiseuille flow pattern at the outlet was expected owing
to the cut-fin arrangement, and would be a major cause of poor
estimation of the bulk mean temperature. To obtain a moreaccurate value of the bulk mean temperature, a preliminary
experiment of PIV measurements was conducted to measure
the inlet and outlet velocity distributions corresponding to thepositions at which the thermocouples were located.
The PIV system used in the present study was composed of adouble pulsed Nd:YAG laser (New Wave Research Solo3:
wave length 532nm) and a high resolution CCD camera (TSIPIVCAM 10-30: 8bit monochrome, 1Mpixels). During the PIV
experiment, a particle generator (TSI model 9306) was set in
the upstream of the channel, and tracing particles of oil mist
(Di-Ethyl-Hexyl-Sebacate, 1.0mm in nominal diameter) were

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Copyright © 2007 by ASME

12

1.0

10

0.8

f

8

f/fp

0.4
4

0.0

0
1.2

1.4

1.8

H/Hf

2.0

0.20

1.4

1.0
0.10
0.8
0.05

R/Rp

2

R [m K/W]

1.2

R/Rp

0.15

0.6

Cut-fin case
Plain-fin case

0.4

0.00
1.2

1.4

1.6

1.8

H/Hf

2.0

Fig. 7: Fin height to channel height ratio effects (clearance
effects) on thermal resistance, R, (Re=200).

1.25

1.2
1.0

1.00

0.8

1/{(f/fp) (R/Rp)} 0.75

1/3

0.6

1/3

1/{(f/fp)(R/Rp)}

1/{(f/fp)(R/Rp)}

1/{(f/fp) (R/Rp)}

1.50
1.4

0.50
0.0

0.0
1.2

(14)

For comparison, the results of Liang and Yang [12] arealso included in the figure. The lines in the figure illustrate thevalues of Nu =6.7 and 7.47 that respectively represent the casesof parallel plates with isothermal and constant heat flux
conditions. The results obtained in the present study agree very
well with those of the reference, particularly under lower Re
conditions. As Re increases, j slightly deviates from the linear
line and a larger value is obtained in the present case comparedwith that obtained in the reference. This is believed to be
attributed to the developing region of the thermal boundary
layer located at the front edge of the test section. In this area,
lager heat transfer coefficient is obtained that becomes large
enough to affect the overall heat transfer coefficient when Re

1.6

Fig. 6: Fin height to channel height ratio effects (clearance
effects) on friction factor, f , (Re=200).

Preliminary Test Case
To validate the present experimental procedure inmeasuring the heat transfer coefficient, a preliminary test was
carried out in a parallel plate case. The test section wascomposed of two copper plates, which were attached to thechannel top and bottom walls. The plate thickness was 0.85mm
and the length of this section was 100mm. The test section was
mounted to the experimental apparatus introduced in Fig. 3.
The heat transfer coefficients were measured following theprocedure described in the previous section.
Figure 5 shows the relation between the Colburn j-factor
obtained from Eq. (14) and Reynolds number, Re. The
characteristic length in calculating Re is the channel double
height, which is the double value of the distance between thetwo copper plates.

Nu
j = St Pr = 1 / 3
Pr Re

0.2

Cut-fin case
Plain-fin case

2

RESULTS AND DISCUSSION

2
3

0.6

6

f/fp

mixed with the air. As shown in Fig. 4, a laser sheet was
beamed through the channel sidewall. The CCD camera was set
above of the channel, by which a pair of scattering images of
the tracing particles was recorded in a certain time-difference.
The velocity vectors were then calculated by using a PIV
analysis software (TSI insight 3.4).
The test section was made of acrylic resin. The channel
length, width and height were 250, 150 and 10mm,
respectively. The fin was made of aluminum and was glued
onto the channel bottom wall. The fin thickness was 0.5mm,
therefore, the number of row in the spanwise direction was 42.
The number of the notches in the streamwise direction was 7.
The geometric parameters of the notch in the experiment wereidentical to those of the numerical simulation shown in Table 2.
The notch attack-angle to the main flow, β, was changed under
the conditions of β=90°, 75°, 60°, 45° in this case. The flow
Reynolds number employed in the experiment, Re, was on thebases of the channel double height and mean velocity. This
mean velocity was calculated from the mass flow rate measured
by the flow meter located upstream of the channel. Re wasvaried from 1,000 to 2,000.

1.4

1.6

1.8

H/Hf

2.0

Fig. 8: Fin height to channel height ratio effects (clearance
effects) on overall performance (Re=200).
increases due to the insufficient streamwise length of the test
section.
Nevertheless, the results show that the procedures of themeasurement and analysis employed in the present study arereasonable enough to discuss the heat transfer performance ofthe fins under the target regime of Re, which is
1,000≤Re≤2,000.
Clearance Effects
Figures 6 and 7 show the numerical results predicting how
the clearance between the fin tip and channel top wall affects

5

Copyright © 2007 by ASME

40case A30,
case A45,
case A60

1.2

35
30

1.0

f/fp

Nu

25
20

0.8

case A45
case A60
case A75
case A90

15
10
5

0.6
0.00
0

0
1000

1200

1400

1600

1800

1000

1500

Re

2000

Fig. 10: Attack-angle, β, and Reynolds number, Re,
effects on the channel friction factor, f.

the friction factor, f, and thermal resistance, R. Reynolds
number, Re, in this case was Re=200. R is defined as Eq. (15):

1.5

(15)

where θ w and qw are the wall temperature and heat flux. θbinlet
is the flow bulk mean temperature at the inlet of thecomputational domain. For comparison, the results of a plainfin case, which is the case of a parallel fin array without
notches, are also depicted in the figure. The values normalized
by those of the plain-fin case, fp and Rp, are also shown in the
figures scaled by the right hand side axes.
As H/Hf increases, which implies an increase of theclearance, f decreases in both cases of cut-fin and plain-fin.
This is due to the fact that the pressure loss of the flow in theclearance area becomes smaller compared with the area locatedbetween the fins. However, since this accompanies a decreaseof the mass flow rate of the flow in the area between the fins,deterioration of heat transfer is incurred at the fin surfaces. Thisis shown in Fig. 7, where R increases as H/Hf increases
indicating a lower heat transfer performance is obtained underlarger clearance conditions.
Comparing the results obtained by the cut-fins and plainfins, f is smaller in the cut-fin array case than in the plain-fincase (f/fp is smaller than unity) under all H /Hf conditions. A
reduction of the pressure loss penalty is, therefore, achieved byapplying notches to parallel fins. The ratio f/fp becomes small as
H /Hf decreases, which implies that a stronger effect of thepressure loss reduction produced by the notch is obtained in thecases of smaller clearance.
On the other hand, R/Rp shown in Fig. 7 is larger than unity
under all examined H/Hf conditions. Furthermore, R/Rp
increases as H /Hf decreases. This indicates that the heattransfer performances in the two cases are identical or is betterin the plain-fin case; and the effect of applying the notch isnoticeable under larger clearance conditions.
To briefly discuss the aforementioned trade-off trends ofthe clearance in the overall performance of the fins, the values

R/Rpθ w − θbinlet
qw

500

2000

Re
Fig. 9: Averaged Nusselt number obtained in the heattransfer experiment (attack-angle effect).

R=case A75
case A90

1.0case A30,
case A45,
case A60

0.5

case A75
case A90

0.00
0

500

1000

1500

Re

2000

Fig. 11: Attack-angle, β, and Reynolds number, Re,
effects on the distributions of thermal resistance, R.
1 /{( f / f p )(R / R p )} and 1 /{( f / f p ) (R / R p )} are shown in Fig. 8.
1/ 3

Although these values do not precisely describe the overall
performance, one can at least discuss the clearance effects in aqualitative way. Except in the case of H/Hf=1.25, both values
are larger than unity and a maximum peak is observed in thearea of 1.4≤H/Hf ≤1.8, which indicates a better performance isobtained in the cut-fin case compared with the plain-fin case.
Notch Arrangement Effects (Attack-angle effects)
Figure 9 shows the relation between the averaged Nusselt
number, Nu , and Reynolds number, Re, in the cut-fin cases
obtained in the heat transfer experiment. In all attack-angle
cases, Nu increases as Re increases, and higher heat transfer
efficiency is obtained in larger Re regimes. The smallest Nu
among the tested cases is obtained in case A90 in the present
range of Re. Higher heat transfer coefficient is, therefore,
attained by inclining the notch array. Comparing Nu ofdifferent attack-angle, β, although not clearly indicated in the6

Copyright © 2007 by ASME

x/H=0.0 x/H=0.1 x/H=0.2 x/H=0.3 x/H=0.4

2 U/Um

0
2.0

y/H

1.0
0.8
0.6
0.4
0.2
0.0

z/H

1.5
1.0

(a) Re=200, z/H=1.0(b) Re=1500, z/H=1.0

1.0
0.8
0.6
0.4
0.2
0.0

0.5
0.0

y/H

0

1

2

x/H 3

0

1

(a) Re=200
0 2 0

U/Um

0

0

0

(c) Re=200, z/H=0.65

0

U/Um

0

0

0

0

(d) Re=1500, z/H=0.65

2

x/H 3

(b) Re=1500

Fig. 14: Contour maps of streamwise velocity, U, at y/H=0.6(case A90).

Fig. 12: Cross-sectional distributions of streamwise
velocity, U, at z/H=0.65 and 1.0 (case A90).

2 U/Um

0
2.0

1.0
0.5

y/H

1.0
0.8
0.6
0.4
0.2
0.0

z/H

1.5
x/H=0.0 x/H=0.1 x/H=0.2 x/H=0.3 x/H=0.4

0.00(a) Re=200, z/H=1.0(b) Re=1500, z/H=1.0

1.0
0.8
0.6
0.4
0.2
0.0

1

2

x/H 3

(a) Re=200

0

1

2

x/H 3

(b) Re=1500

y/H

Fig. 15: Contour maps of streamwise velocity, U, at y/H=0.6(case A45).
0 2 0

U/Um

0

0

0

(c) Re=200, z/H=0.65

0

U/Um

0

0

0

0

(d) Re=1500, z/H=0.65

Fig. 13: Cross-sectional distributions of streamwise
velocity, U, at z/H=0.65 and 1.0 (case A45).
figure, Nu appears to increase as β decreases, and the largest
value is obtained in case A45.
Figures 10 and 11 show the numerical results of β effects
on the friction factor, f, and thermal resistance, R, respectively.
Both values are normalized by the results of the plain-fin case.
In Fig. 10, f/fp is smaller than unity in all cases, which implies
that a reduction of pressure loss is attained in all β cases
examined in the present study. f/fp increases as Re increases. On
the contrary, in Fig. 11, R/Rp decreases as Re increases
implying that higher heat transfer performance is obtained
particularly under the conditions of larger Re and smaller β.
This trend is believed to be related to the secondary flow
generated by inclining the notch array, which will be discussed
shortly.
Figures 12 and 13 show several cross-sectional
distributions of U at z/H=0.65 and 1.0 in the cases of A90 andA45. z/H=0.65 approximately corresponds to the middle of thearea between two fins. z/H=1.0 is the middle of the fin locatedat the center of the channel. Figures 14 and 15 respectively
show the streamwise velocity, U, contours at the cross-sectional
planes of y/H=0.6 in cases A90 and A45. In these figures, theresults obtained under the conditions of Re=200 and 1500 are
shown, and U is normalized by the channel mean velocity, Um.

In case A90 of Re=200 shown in Fig. 12(a) and (c), U/Umof the flow passing between the fins decreases in the areaadjacent to the notch, while U/Um of the flow inside the notch
increases. In this area, upwash flow from the lower region of
the channel and downwash flow from the clearance wereobserved, by which the fluids in each area were brought near
the fin located at the trailing side of the notch. Thisenhancement of the fluid mixing is one of the reasons why anequivalent heat transfer performance with the plain-fin case
was obtained in this case in spite of the decrease of the heattransfer area. On the contrary, in case A90 of Re=1500 shownin Fig. 12(b) and (d), U/Um distributions do not vary much inthe streamwise direction. In this case, the notch effect becomes
relatively small and effective upwash and downwash flows
were not observed. By the notch, however, an interruption of
the flow boundary layer was obtained and renewal of thethermal boundary layer was produced at the fin located inadjacent to the trailing side of the notch.
In case A45 shown in Figs. 13, a similar velocity
distributions as case A90 are obtained under the condition of
Re=200. On the other hand, in case Re=1500, a different flow
pattern appears particularly at the location of z/H=0.65, i.e.,
two regions of large U/Um are observed in the area near thechannel top and bottom walls and a minimum peak appears
near the area of y/H=0.5. This can also be found in comparison
between Figs. 14 and 15. The U/Um distributions in cases A90
and A45 are similar under the condition of Re=200. However,in case Re=1500, a different pattern is obtained. This difference

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Copyright © 2007 by ASME

case A30,
case A45,

case A60,
case A75

obtained under the conditions of larger Re and smaller β, theseeffects become more conspicuous and higher heat transfer
performance is obtained as discussed in Fig. 11.

case A90

Wn /Um0.10

Re=200

0.00
0.10

Wn/Um

Figure 17 shows the averaged Nusselt numbers, Nu .
Nusselt number of each fin was obtained by using its respective
bulk mean temperature, θbn. θbn was calculated by Eq. (16), into
which the temperature and velocity in the area obtained byequally dividing the cross-section into several parts were
substituted.
θbn = ∫ ρc p uθdA / ∫ ρc p udA
(16)

0.05

Re=1500

0.05

An

0.00

z/H=0

1

2

3

4

5

z/H=2.05
fin number
Fig. 16: Averaged spanwise velocity in the notch area, Wn.
50case A30
case A45
case A60
case A75
case A90
plain-fin

40

Nu

30

10
0
500

Here An is the area with the center located in the middle of eachfin (fin number n) and with the width and height of Wp and H.
These definitions of Nusselt number and θbn for each fin were
employed here due to the fact that the heat transfer
characteristics at the fin surface mainly depend on the flow
passing through the fins together with the spanwise flows.
In Fig. 17, Nu of the plain-fin case is nearly constant andis unaffected by Re. This is due to the fact that since thedistance between the two fins is small, the flow in this area islaminar flow. A fully developed thermal boundary layer oflaminar flow is, therefore, obtained in this area and the heattransfer coefficient at the fin surfaces is constant to Re.

20

0

An

1000

1500

2000

Re
Fig. 17: Nusselt number calculated for each fin.
is considered to be owing to the spanwise flow obtained in theoblique notch array case.
Figure 16 shows the spanwise velocity averaged in thenotch area obtained in cases Re=200 and 1500. In both cases,
Wn/Um takes a positive value indicating a spanwise flow
flowing along the notch array is obtained by the inclination of
the notch arrangement. Wn/Um increases as β decreases and amaximum peak in all cases are obtained at fin number 3, whichis the fin located at the channel center. Furthermore, comparing
the values in cases Re=200 and 1500, Wn/Um is larger in case
Re=1500. A stronger spanwise flow is, therefore, obtained
under smaller β and larger Re conditions.
This spanwise flow is believed to produce two significant
effects that results in the enhancement of the heat transfer
performance compared with case A90. One is that owing to thechannel sidewalls, the spanwise flow along the notch turns its
direction and flow in the reverse spanwise direction in theclearance region, resulting in a generation of a longitudinalvortex-like secondary flow. Although not large in strength, thissecondary flow enhances the mixing between the fluids locatedat the clearance and between the fins. The other effect is thatthe flow passing between each fin is bent in the spanwise
direction in the notch area as shown in Fig. 15(b). This
produces a strong flow sweeping effect at the fin edge at thetrailing side of the notch. Since stronger spanwise flow is

On the other hand, a moderate increase of Nu against Re
is obtained in case A90. As described above, a stronger effectof the renewal of the thermal boundary layer at the trailing edgeof the notch is obtained. This effect increases under larger Re
conditions entailing larger heat transfer coefficient. In cases
A30~60, Nu largely increases as Re increases, and Nu
itself is larger compared with those obtained in cases plain-fin
and A90. This is believed to be attributed to the existence of the
spanwise flow as discussed in above.
CONLUSIONS
Effects of the attack-angles of notch array arrangement andthe clearance size on the flow and heat transfer characteristics
of a channel with cut-fins attached to the bottom wall werestudied using numerical simulations, PIV measurements andheat transfer experiments employing a modified single-blow
method. The major conclusions obtained in the present study
are summarized in the following.
1. A maximum peak was found in the overall performance
ratio between the plain-fin and cut-fin cases against theclearance size. Under smaller clearance conditions, the pressure
loss penalty was largely reduced in the cut-fin case comparedwith the plain-fin, while a smaller thermal resistance wasachieved with larger clearance. The trade-off trend between
these two values produced the aforementioned maximum peak.
2. Larger overall Nusselt number was obtained in the
inclined notch array cases compared with the 90° attack-angle
cut-fin case, particularly under larger Re conditions.

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Copyright © 2007 by ASME

3. Among the examined attack-angles, a stronger spanwise
flow and maximum Nusselt number were obtained under theconditions of 30° and 45° degrees.
4. Heat transfer enhancement was obtained due to the
spanwise flow generated in the notch area. The effects of thisspanwise flow were considered to be twofold. One was that alongitudinal vortex like secondary flow was generated in thechannel, by which the heat and mass transfer of the fluids in thearea located at the clearance and between the fins wereenhanced. The other was that the flow passing between the fins
was bent in the spanwise direction in the region adjacent to thenotch. This accompanied a stronger effect of the renewal of thethermal boundary layer at the trailing side of the notch, which
entailed a higher heat transfer coefficient.
5. Averaged Nusselt number on the basis of a bulk
temperature independently obtained using the flow andtemperature fields around each fin, was constant to Re in theplain-fin. On the other hand, Nu increased as Re increased or
attack-angle decreased, which corresponded to the results
obtained by the experiment. This is due to the fact that the
spanwise flow along the notch was increased in relation withRe.

and without Tip Clearance, ASME J. Heat Transfer, 100, 572579.
[8] K. Tatsumi, M. Yamaguchi, Y. Nishio and K. Nakabe,
2006, Proc. of 13th Int. Heat Transfer Conference, CD-ROM(paper No. THE-15), Sydney.
[9] Tatsumi, K., Noda K. and Nakabe, K., 2005, The Effectof Cut-Fin Array Arrangement on Heat Transfer and Friction
Characteristics, The 6th KSME-JSME Thermal & Fluid
Engineering Conference, Korea, paper-No. EJ06.
[10] Patankar, S. V., Liu, C. H. and Sparrow, E. M., 1980,
Numerical Heat Transfer and Fluid Flow, Hemisphere
Publishing Corporation.
[11] Patankar, S. V., Liu, C. H. and Sparrow E. M., 1977,
Fully Developed Flow and Heat Transfer in Ducts Having
Streamwise-Periodic Variations of Cross-Sectional Area,
ASME J. of Heat Transfer, 99, 180-186.
[12] Liang, C. Y. and Yang W-J., 1975, Modified Singleblow Technique for Performance Evaluation on Heat Transfer
Surfaces, ASME J. Heat Transfer, 97, 16-21.

ACKNOWLEDGMENTS
This work was financially supported by The MIKIYA
Science and Technology Foundation. The authors would also
like to acknowledge the Japan Aerospace Exploration Agency(JAXA) and Kyoto University Supercomputing/Large-scalecomputing service for providing their super-computer resources
to the present numerical computations.
REFERENCES[1] Incropera, F. P., 1988, Convection heat transfer inElectronic Equipment Cooling, ASMA J. Heat Transfer, 110,1097-1111.
[2] Knight, R. W., Goodling, J. S. and Hall, D. J., 1991,
Optimal Thermal Design of Forced Convection Heat SinksAnalytical, ASME J. Electronic Packaging, 113, 313-321.
[3] Morega, A. M, Bejan, A. and Lee, S. W., 1995, Free
Stream Cooling of a stack of parallel plates, Int. J. Heat Mass
Transfer, 38, 519-531.
[4] Wirtz, R. A., Chen, W. and Zhou, R, 1994, Effect ofFlow Bypass on the Performance of Longitudinal Fin Heat
Sinks ASME J. Electronic Packaging, 116, 206-211.
[5] Matsubara, K., Nakabe, K. and Suzuki, K., 1996, A
Three-Dimensional Numerical Study on Heat Transfer
Characteristics of Flow in a Channel with Fins Attached to one
Wall, Heat Transfer Science and Technology Higher Education
Press, Beijing, 156-161.
[6] Min, J. Y., Jang, S. P. and Kim S. J., 2004, Effect of tip
clearance on the cooling performance of a microchannel heatsink, Int. J. Heat Mass Transfer, 47, 1099-1103.
[7] Sparrow, E. M., Baliga, B. R. and Patankar, S. V., 1978,
Forced Convection Heat Transfer for Shrouded Fin Array with

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