Transfer
%αss

207

Aichi ShuPPan

ｙ

く Ы
ｄ １
︐

尭
ｅ

and Heat Transfer Characteristics onulent Flows in a Duct with Rib Turbulatorsｅ‰

TatSuml

1,H.Iwail,K.Ina)ka2 and Ko Suzukil
―
―
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ι
οδ
θ
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u,κ ν
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C itν ,Sa ο
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Cん α
げ νθ
,

'Cal Engれ

げ ν
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んαttCal αηd Sν StemS

610-0321, JaPanEng'ηπ ng,Dο SttSん α υη

erS切 ,κ νοtambθ,'υMany studies have been carried out for a turbulent flow in a ribbed channel.
ditail of the flow structure and its related heat transfer characteristics havethe,stillpresents the results of three-dimensional compubeen fullY investigated. The present studylowrib
turbulator
mounted in a channel.
over
a
duct
bulenltur
aofthree-dimensionalitY was observed not only near the duct sidewalls but also near thenear the rib rear wall, transverse vortices were observed at positionsof heat transfer was obtainedis applied to the computation andhrrthermore, aare compared with each other. By
temperature field incurred by theperiodicalonthestudiedfluctuationofefiect,in the free shear layer is capt ured. Consequently this periodical fluctuation
to the instantaneous and., eventually, the time mean heat transfer.

Introduction

zationn

1983

t the actual flow″by such fluctuat ion from those by small time scale turbulent fluctuation inand thermal fields. Such discrimination can be made by Inaking use of threedecomposition approach and may give a better picture of the nuid and heatヽノＬ︻time SCale DeriOdiCal fluctuations, it is important to discriminate the components

20g

3d Int. Symp. on Thrbulence, Heat and Mass TransfsTtransfer characteristics[5][61. In the present study three-dimensional numerical simulatjon
for a turbulent duct flowil;;i;d;rt *itt ttre ffrte6 component decomposition approachflow structure and its
mean
The
wall.
iiurirrg, f"l-rpin rib attached oir the duct bottomisinvestigated.
relatei thermil characteristics--Th:titr..-iornponent decomposition-approach decomposes instantaneous quantities
into perioaiiai flritration, stochastic fluctuition and the time-averaged comPonents. The
and teryPerature.fields are calculft'*.il una p.rioaicaliy changing-parts of velocity effectof stocbastic component t6
thedescribe
trdd h*;duainA a turbullnce m-od6l-to
investigated.
isfield
thermaltoeffecteach
andifroru t*o compo"nents

2. Computational Procedure

(1)

︵σ

∂

聯Ｐ

ρ

Ｉ
一

︐
２一
助
一
一
甲丁
い
叫
２一

Three-dimensional, time dependent continuity, momentum and energy equations of
the following form are solved in the numerical computations.

/∂ 硫 ̲ヽlμ τr― ― ρuμ ,1+∂π, ヽ σπ,"1
/ μ ∂θ

∂
︵σ

瓦 71
‐
´
ノ

ヽPr \xi

I

∂π,

(2)

ノ

Reynolds stress term in eq.(2)is modeled With a non―

(3｀

⌒

:

linear eddy viscosity mOdel proposeQby the UMIST group 141.
頭υブ =:ん ρδ ̲μ tst,

(4)

=プ

+Clμ tt rS■ S■ 一 :SHSHδ lハ +C2μ t:(Ω tた S朽 +Ω ,た S晟 )。

し 、+C3μ t:(Ω たΩメ ー

:Ω

ι

ノ

ι
たΩ凸

Ω
+C5μ (:)2(Ω ι)+C4μ̀mSm,+SIΩ

t(:)2(sた Ωt,+Sた ,Ω :)StiδhΩ m :ShΩ … Ω■ι
)+C6μ t(:)2S,S‖ S履 +C7μ ι(:)2s ΩたΩたι

where

s.j
s

=(詣 ―
づ
詣)

+器

),Ω
(詣
, ̲̲̲̲̲̲̲,^′̲̲̲¨ ̲̲:み島/2,Ω =:yΩ ュ

1.2

J/2,μ t=ρ Oμ

∫
μ
廿

′ ‐、

)

In this conputation, tulbulent heat flux is assumed to be ploportional to the temperat
gradient, i.e.

+y-n,o = Pr2 0r;

L」

(5)

Prancltl number, Pr1 was assumed to be constant -and Prl -.0.9.
Turbulent
F;jl" ir"oli.it for.n. oi nriii. aiff...n.u .quivalents of the above equations were solvednur".ii."ffi ,ioni the time axis. For the firiite-difference me-thod of.-t[e governing equadiffusion terms whtleli;;, ; ioith-ord"er central-difference schemg yas employed for thein the case of mo
terms;ir;-Hfth;;a.i"pr"irJ;;ii.;; was employed {o1 the ionvection
eouatignl
transport
turbulence
the
two
in
and
;;-";; .qr"li"-". in1t. .n.rgy equ-afion
and a
terms
the
convectionforwasadoptedschemeupwi-ndof I ana 6, a first-order.."irri-ain'...n.. sct.*e foi the diffusion ternts. Fbr the solution algorithm of the p.resrui. *ri.Ction equatlLir, SiMpLE algorithm was employed. In order tb relax the solutions
at each time step',i tfr.r. .ttipti. differ.niial equationS, iterative procdure was.appliedwas
also
combined.
methodimplicit-(ADI)
directionp;ffidure,
alternatingi; ihi;

一Ｊ Ｅ旧障Ｉ‐‐ぎVll,￨「7:じ1111,11111111

￨

︲ 壺

Ⅷ知

一

一

一
一
一
一
一
一
一

Ｆ﹂ レ
Ｉ一
Ｊ嘔
電 翻 噸哺 蛹．
■¨
彎里勇 硝 Ｉ ｌｌ⁚ Ｊ瑚¨
驚 咽堰 罰
︲︲ ．
ｃ
ｓ
ａ
ｕ
ｔ
品
¨
ｅ
咄・
蜘
呻
ｍ
ｅ
ｎ
ｒ
ｒ
︲
ｅ
ａ
ｔ
・
︐

．

209

グ α′
.

FIow

為 υ

ｔ
ａ
・

Figure 1: Computational domain.

ｒｐ

The conputational domain in the present study is illustrated in Fig.1 together withsystem to be used. Stindard full-span ribs were investigated. T,he origin,;j""oiainlte
Uav v- Iiii,o coordinate svstem is located at the center-of the rear bottom corner of the rib.
h"'i"f"t to the domain was located at about ten rib heights (10H) upstream the front
ii,iriiu.u, where a steady fully developed turbulent flow-was assumed to flow in. Inlet
ii,i* .traiiions were estaLlished by cairying out a preliminary computation for a 3-D
domain was located at about thirty ribii"t no*. The outlet of the coniputational
gradientsof all variables except pressure in the
rib,
where
the
downstream(30H)[Jiehti
were
set
equal
to zero.
totheoutletsurfacenorinalii.i"tiori*(Hin three steps, i.e. H f H4:9.15,
washeightratiochangedduct
toheieht
iiU
iir.
lHa)
0.2b and 0.25I For each rib-height cbndition, three different levels of Reynolds number
Uu"ua on the hydraulic diameter of the duct were used in tlr-e. present calculation, i.e.
h7=tO,OOO, 15,000 and 20,000. The aspect ratio (AR:HalWa) was set equal to unity,
a sqrate shaped duct, for all computations.
i.e.
-'-For-allvariables except temperature were set equal toboundary-conditions at walls,
all the duct and rib wall surfaces were set
boundary
conditions,the[hermalFor
zero.
constant in heat flux through all the computations.

3. Results and Discussion

3.1. Time mean fluid and heat transfer characteristics
Fundamental characteristics of the two-dimensional flow with a rib turbulator are welldiscussed and knowledge was acquired for flow separation and reattachment. However,study has not fully beei made foi the effect of the duct sidewall, which inevitably grys$in tlie coolant paisage of gas turbine airfoil. In this section, time mean results of 3-D
calculation are-reporled to investigate the three-dimensional effect of the sidewalls on
the flow and therriral fields. Here, fhe case of HlHo=g'29 at Re:10,000 is discussed forｅｐdemonstration.
Figure 2 illustrates the spatial distribution of the mean skin friction, C1 and the mean
Nusselt number, Nu on the duct bottom wall. Fol Cy, the shaded parts correspond toits positive value area while the white parts to negative value area. Flow separates atthe'rib flront edse and reattaches downsfream the rib. The reattachment point-is locatbd
at rlH:8.7 if ireasured at centerline. Circulation appears both in front and backsideof the rib where highly three-dimensional effect is observed. Esqecially for the region︲ｅｌｓ ∝ 卿 出ｈ
﹁唱ｎ
曖 ・
；ｏ
︲ ｄ

just dorvnstream the iib, sidewall effect is obvious and the local heat transfer showsa distinctive deterioration near the sidewalls. This lower heat transfer performance iscaused by the tornado like transverse vortices generated at such positions. Figure 3-(a)shows the velocity vectors and temperature contours in a plane of. y,f H:0.03 near thebottom wall surface. These vortices'are produced by the cohbined e-ffects of the reverse
low in bhe central region toward upstreani from the rlattachment-point and the downward
inclination of.the flo"w from the dirct sidewall regions. Velocity fi'eta is not vigorous near
the center of these vortices. Furthermore, kineiic energy of-turbulence is small at thecenter of these vortices. Thus the related heat transfer is not pronounced in this area.
The flow passine the rib also affects heat transfer on the duct sidewalls and top wall.
-Figure4 illuitratei the distribution of Nu on one of the sidewalls and on the top wall of
the duct. The flow passing over the rib edge sweeps the duct sidewall. Large Nu is3d Int. Symp. on Turbulence, Heat and Mass210

■ ■ 目目 目

0.025

鵞朧噸

0.5

ミ0
0・

‐
0.5

5

0

‐5

20

15

ノH

cdon coemcient,c/

0.5

′ 0.0
ヽ

10

15

一

5

0

‐5

︐

●.5

ｏ■ ■ ■ 目 目 ｏｏ︲(a)skin1oノH(b)Nusselt number,NFigure 2: C1 and Nu on bottom wall'(HlHa:0'20, Re:10,000)
0.5

(0・

‐
0.5

10.5

0

1

‐
0.5

0.0
パ袖

2

NH

0.0

︒
ｏぴ
ｏ■ ■■ ■国目 目 ｌ
︲

、
冬,

0

2/￢ L

(b) y-z plane (with temperature contour)(a) x-z plane

Figure 3: Velocity vectors and temperature contortrs. (HlHa:0.20, Ee:10'000)
1.0
麟

ヽ

0.5
0.0
‐5

5

0

10

15

ノH

く0.0
‐
0.5

5

‐5

10

ガH

20

25
ｏ
ｏ■ ■■ 目 目 ｏ︲(→ SideWall0)TOp Wallヽ′⌒

Figtrre 4: Nusselt nuntber, N.u on side- ancl top wall. (HlHr:9.20, Re:10,000)
perobserved along the streamline of such flow. On the other hand, low heat transfer ot
positionsstreaurwise
theatespecially
come_r,
top
cluct
tlie
io,riun.i i. o6r.r,r.a-;;;;'
th. ufr*arcl flow feneraied in this corner region. Figure;2";iH-ais.-tr,ir i.ri,ih
i-(U;'ttro*s the mean velocity vector map of the secondary.flow together.with the mean
directed
temoerature contours in the cross-sections at xf H:S ancl xlH:15. Flow isconvevsflow
point.
Thisi""ffiffi;;'i.#'ii,". arct bottom *arr aiorna--i# i."tiu.ti*.ni
is ob'
transferheat
effeitive*lrur.
cooler fluid from ,uo". io*"ia in. duct-boiiom-*"ll
upper
duct
ngul.th^eappearvortices
Secoldaryfig.2-(b)tained. This is .onn.*.a in;;;;;;. ,t ,ir.a**it"'pl.iii6"J?o*irJt..am the reattachment point' This flow comes toi;*:1x,",t*"i3-[,]'$t'']ttxi?;']i t]"T.?f".::'4.il'"fl1f,#"'"t].lt
ml,ttxlit :r,t,ftlie up-goingby
conveyedwarner'fluid.is(;i:bnth;;ih;ihand,Fig.4-causes
is indicated in

the lorv

H'r;;;il;;ua Uy'ifr.;e'vortices sweeping utong the duct sidewall. This
heat transfer.

￨￨￨￨￨11111111111

￨

1

1

Tatumi el

211

al

1.0
編

ゝ0.5
0.0

5

0

‐
5

10

(り 科 pl

20

15

ノH

1.0

︒
０■■■■日目目 ｏ
ｌぴ
︲

ミ0.5
Rib)
０ 蘭
Ｑ ´

︒
・
山
０

0.0
‐
0.5

0.5
0.5 ‐

0)ン 2 plme

0.5

爵o.o
000)

‐5

5

10

15

ノII

20

︐

‐
0.5

ｏ■ ■■ 国 目 ｏｏ︲Figure 5: Velocity vectors and temperature contours.(Hf Ha:0.20, .Re:10,000)

Figure 6: Instantaneous Nusselt number, Nu on bottom wall.(H lHa:0.20, rBe:10,000)

3.2. Periodical fluctuation effect-A three-component decomposition approachThroush all the computations. unsteadiness was observed in the flow and thermal fields.
Conspicu6us fluctuatioii was especially found for low Rey^nolds-number,_i.e. Re:10,000∝

for ail series of rib heieht to duct height ratio. In this flow, large eddies appear periodically inside the sepirated free sheal layer developing from the rear top.edge of therib. Tfiese edclies flow downstream toward! the reattbchment point. Thus, the reattachment point fluctuates periodically in the streamwise direction. Furthermore, the flow
s*eeping the duct sicteivall produles longitudinal vortices downstream the reattachment
point. fhese vortices are oBserved in Fi[.5 which shows the velocity v_ector map superilosed with temperature contours in the-rliddle streamwise-sections of the duct and in
the cross-sections at two locations of. xf H:5 and xf H:15, respectively for the case of
HlHa:0.20. .Re:10,000. These vortices flow downstream producing the non-uniformityof'the instantaneous local Nusselt number, Nu on the duct walls. Figure 6 shorvs appearance of the oeaks of the instantaneous value of Nu on the duct bottom wall where overalllarge fluctuition is observed.
Most of the turbulent flows accompany not only small time scale turbulent fluctuation
but also large time scale periodical dr quasi-periirdical fluctuation of quantities. Largeeddy structu"re in the sheai layer is one g6od eiample of this. -Dilcriminating^the apparent
sheir stress and heat flux produced by"these larg-e time scale fluctuations from those of
turbulent fluctuations is important to analyze th-e flow and temperature fields correctly.
Inaoka et al. lSl showed tliat the dissimilarity between momehtum transfer and heattransfer observLd in a turbulent boundarv layer-disturbed with an inserted square rod canbe numerically predicted by decomposin[ initantaneous valrte of 1 quantity into its threecomponents; i. 'e. tfre two different'time-scale fluctuations from the time averaged value.

ねitions
le

〕
r ls

ornes

ョ覆 ■
ロロ ー ー ーー ロ ニ ■■日 １ １ ⁚ ︱Ｉ Ｅ 盪■１ １ １ 日 日 １ ■ニSγ′

3il Int. Symp. on Turbulence, Heat and Mass

212

:(

̀多

1握

0.5

Fヽ 0.0
‐
0.5

lo

5

0

15

0>
(a)く ▼

ノH

1柵

￨

0.5
ぎヽ

日
日
日
日
日

0.0

‐
0.5

1.5

10

5

0

15

(b)く V10'>

Figure 7:
InstantaneOuS nlaps
wぷ 1.(″ /″ d=0.20,Re=10,000)

‐
1.5
ノH

mKノ s

11[覇

of apparent heat flux contours near

OЮ

u

Fi

of

ri; * i; * u'; =<-u; ) *ut;

P - F+F+P'=<P>+P'o - A+6+0'=<0>+0'
where .4 denotes the time mean value of a quantity, I the periodical part of theＳ︲・ation of the quantity and A' the turbulent stochastic component of the fluctuation c)f
DhaSe aVerage.
quantity <> denotes the .̀
γ̀Figure 7 shows the contour lnaps ol apparent wall-normal heat fluxes in a

く
ｎ
Ｒ岬嗣酔ｄ
ｕ
い
Ｏ

lti =

ｒ
ａ
Ｗ ∬ Π ｒ ｔ ︑１

3.3. Rib height to duct height ratio effect
〃/
lS

wall

‐

｀
F:「

::!!!!11

213
むルαη
曖,

pTatsumi et al.

0.5
1.0

く 0.0

105

o.5

o0
■

't5 tN

0
(a)Hノ

1・

25

to

s

‐5

'15

20

ノH

25

(a)i̲vHd=o.15

Hd=o.15

0.5

01

く0.0
10

0

0.0

0

10

Л

15

15

ノH

(b)H/Hd=0.20

150

Hd=0.20
(b)Hノ

０
ロロ ０
５ ■ ■■ロ ロ

lo.5

o.5

1.0

ミ0
0・

ヽ
ｔ
︒ 聖
ｂ
・ｂｙ

婉

lo.5

[

0.0

0

ｅ
ｍ ・

]e scaleおo.4

nuctu̲
of thera3 ̀Эart tott nux.
is wellof thelt low
v」 ue

1∞∞

Hl Ho
is im-,
rall to

1"∞

20000

R

朦

[ ̀

"∞

Rg
umb・ ■
0)S● nt。 ■●

St

ト
l

0

1ま Ю0

lα Ю0

l

(a) Prcssure cocffioicot'

CP=

Figure 10:Meanらmd■.

t batOm Wil

Nus lt number On the Cduιi° nS° f the rea・い・ The l°taぬment pol嵐
aS・
/・
E Re
"d
peak
of
t
WardS
the
rお
t°the
̀inCre証 ghtly m° VeS u"trel■
H/・
d7.8r peCti"ly'Or
鵬
̀=0.15,0.20 ma 025
3re″ /H=1■ 2,9.3師,themean Nus"lt number on the
睡 tachment pol■f the duCt・ As./.d hcre"tWOf01とtheCemer°e0(b).‐ Feい On d thE SCIty°f thiS蜃 ル=20,∞und i n飩
減 ・ The Wl°
tp°
h∞
"1°
m｀「a・ alS° mC "eS"f°attaぬ
D duCt bott°
ば d aow"ound theW'・
related"the d°
D Oneお1踊sご:111°
t盤W hetranStr iS° btaned'
1輩 :そ ::騨に eS'Where l°

謡:郡
穏
譜
品
∬
温
:l糧
ユ
E幅 と
V°
f the t° rn:d° like trttS"rSe ・

E Ond,the SiZe° rlざ F″ /島・鬱:l等 ル,電叫RfE‖ rttectti蟄 灘:l彎 ぜE SreduCedお
i:::鉢

躍ぶ

絲
腱
翠
菫 Ч
■

ristics
rlator.

̲● ̲Hnt‐025

¨

1

■

(8)

020
̲0̲:ク 1 ■曖

1

:nt flow(6)

wall'(Re:20'000)

゛,proach.

(7)

Figure 9: Nu on toP

0.0
o.6

rection,15(c)卜 vHd=0.25(Rs:20,000)
Figure 8: Nu On sidewall.

rsfer.

atically

10 ゴH0(c)H/Hd=0'25:ity andlue of afluctu-

o.5

15o

10コ 腱

畷償 騨

e:littL38,品 d諄

盤et騨

撚聯
ゼ醇111:i∬E堪
rthermOFe,

for hger F/″̀.、t° imuence the"pW組

w「 By thぉ ,1.ger average Nu is obtin
edge St"..。
″/Hd inCe"い'the now passing oVer the rおly島 :認 3椰 i蠍
::b虻

‖
寧
礁
｀
■晏

‐

騰試

:Ic器:tttlp庶
:駐
民
鷲
護
ギ
ユ
癬雌
:;暇
tOp w州 d aVeFage StmtiOn nu部
ヽ■mめ served On the ductCi昴

8her avertte N

the reSu"dr° pC°

leli∫

.

'C'田

er,3・
r the

and工 /1=18 if me"uredお曝 ^Fttre 10 11luStrat∞¨r the astmce between■ /H=5
e ofthe dua
for the who10 sur¨
tよenii
Neragewお. %wtt ctulated。
.n'thermore,d∝ rèse Of St
H/1d=025.For S,"
可
■
E.()雪
師d ib wd■ .As 1/1d increases both CP ald St mcreas

lliは

214

3d Int. Symp. on Turbulence, Heat and Mass Transfer

is observed as Re increases while C, remain almost constant. This matches well with theexperimental results [7].

4. Conclusions
Three-dimensional computations were carried out for a turbulent duct flow over a fullspan rib turbulator. The duct sidewall affects the flow and thermal field significantly andnoticeable three-dimensionality of the flow was observed not only near the sidewall region
but also near the duct centerline.
Large unsteadiness was obtained through all the computation conditions especiallylowesistudied Reynolds number, Re:10,000. By applying the three-c-omponent decomposition approach-to present computat-ion, it had been demonstrated that the large timeicale periddical fluctuation seriously affects the characteristics of instantaneous flow andthermal fields. Consequently, appaient heat flux produced by the periodical fluctuationof velocity and temperatur6 si[nificantly affects the time mean characteristics of heattransfer.
Effects of rib height to duct height ratio has been investigated at three different Reynolds
number. The average Stanton number decreases as .Be increases while the pressure drop
coefficient remains almost constant.

Nomenclature

Ha, H : duct and rib height, m
W6 - duct width, rn
AR: ff, aspect ratio
D : duct hydraulic diameter, m
p : density of air, kgl*'
r- : w&ll shear stress, Nl*'{u, : wall heat flux, Wl*'
AP: pressure drop7., To: wall and inlet Temperatu re, Ktilq - heat capacity at constant pressure, Jlkgr,,\ : fluid thermal conductivity, Wf mK(J,"1 : inlet freestream mean velocity, m/s

A" : W, Reynolds number
Nu : m,Nusselt number
Sι

=

Q=

[ρ

I

I

I

I

ごFz=I;孝 可=覇 雨],StantOn nuIPberび訴]寿万 ,Skin friCtiOn COefnCient

Prl : turbulent Prandtl number ら = i万 フ全テReferencesI,preSSure drOp COemCient'万1. S. C. Lau, R. T. Kukreja and R. D. McMillin. Effects of V-shaped Rib Arrays on
Turbulent Heat tansfer and Friction of Fully Developted Flow in a Square Channel.
Int. J. of Heat Transfer, Vol.34, No.7, pp.1605-1616,1991.
2. K. Balatka, S. Mochizuki and A. Murata. Visualization of Three Dimensional Flowaround Ribs in a Duct. The lst Pocif,c Sympo. on Flow Visualization and Image
Processing, 1997.

3. H. Iacovides, D. C. Jackson, G. Kelemenis, B. E. Launder and Y-M Yuan. Rr pt
Progress in the experimental Investigation of Flow and Local Wass Heat T[anflin
Internal Cooling Passages of Gas-T\rrbine Blades. Int. Conf . Turbulent Heat Tra,"sfer2, t999.

4. K. Suga. Development and application of a non-linear eddy viscosity model sensi-

tised to stress and strain invariants. UMIST, Dept of Mechanical Engineering, Report
TFD/95/11, 1995.
5. K. Inaoka, J. Yamamoto and K. Suzuki. Dissimilarity between heat transfer and momentum transfer in a disturbed turbulent boundary layer with insertion of a rod modeling and numerical simulation -. Int. J. of Heat and Fluid Flow, Yol. 20, pp
290-301 1999.

6. K. Tatsumi, H. Iwai, E. C. Neo, K. Inaoka and K. Suzuki. Prediction of Time-

Mean Characteristics and Periodical Fluctuation of Velocity and Thermal Fields ofa Backward-Facing Step. /st Int. Sympo. Turbulent and Shear Flow Phenornena, pp.
7L67-1L72,1999.
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