PRED:CT:ON OF T:ME‐ MEAN CHARACTERiST:CS
AND PERIOD:CAL FLuCTUAT10N OF VELOC:TY AND THERIVIAL F:ELDS
OF A BACKWARD― FACINC STEP

K.TATSUM:,H。 :WAl,E.C.NEO,K.:NAOKA and KoSuZUK:
Depattment of Mechanical Engineeri?g,Kyoto University
YoShida Honrnachi,Sakyo‐^.\BSTRACT

The present study presents the results of a threecomponent decomposition approach for the two- andthree.dimensional computations of a turbulent flow
over a backward-facing step. By this approach, effect of periodical fluctuation on the studied flow andtemperature field incurred by the discrete counterrotating vortices appearing in the free shear layer
is discussed. ConsequentlS significance of the influence of this periodical fluctuation to the instantaneous fluid and heat transfer characteristics andeventually to the time mean momentum and heattransfer is demonstrated.

INTROOUCTION
Significant achievemeut has been obtained in thepast decades for the turbulence modeling and numerical simulation of a variety of turbulent flows
employing the proposed models. However, it is still
difficult to fully predict the detailed structure of turbuleht flows. It is also difficult to develop a generalz{urbulence model applicable both for wall-boundednd free shear turbulent flows. Most of the turbulent
flows include not only the turbulent fluctuation but
also the large time scale fluctuation of quantities.
One good sx"-ple is the fluctuations of velocity andtemperature fields accompanying the Kdrm6n vortex or large-eddy structure in the shear layer. [tis important to discriminate apparent shear stressand heat flux produced by these fluctuations fromthe one of turbulent fluctuations in order to havea correct analysis of the velocity and temperaturefields. This is demonstrated in the present study
through numerical computation of a turbulent flow
over a backward-facing step.
Inaoka et al. (1998) recently showed that thedissimilarity between momentum transfer and heattransfer observed in a turbulent boundary layer disturbed with an inserted square rod can be numerically predicted with a three-component decomposition
With this approach, it was demonthe apparent shear stress and heat flux
stratedku, lく yoto 606‑8501・ 」
apan

produced by the periodical fluctuation of velocity
and temperature can significantly affect the timemean momentum and heat transfer in the disturbed
boundary layer. The same approach can be applied
to other types of similar flows in which periodicallychanging velocity and temperature fiel& can play
an important role in momentum and heat transfer.
Thus, in this study, an effort has been made to apply
this three-component decomposition approach to a
turbulent backward-facing step flow.
FIow separation and subsequent reattachment of
the separated layer are encountered in many wall
flows of practical interest and significantly influence

the local heat transfer characteristics. Therefore,
comprehensive knowledge of detailed structure andprediction of such separated flow and accompanied
heat tr^.sfer are of practical importance. One of
the simplest geometry of such flows is a flow passing a backward-facing step. For this reason, large
amount of experimental works and numerical calculations has been carried out in the past few decades.
However, physical structure ofthe flow and the heattransfer processies are not yet fully understood.
In the separated shear layer, counter rotating
vortices are generated. Especially io low Reynolds
number range, periodical components of velocity
and temperature fluctuations relating to the vorticalstructure contribute significantly to the momentum
and heat transfer there, For example, significanceof the periodical components can be seen in the location of flow reattachment point, which fluctuatesback and forth in the streamwise direction. However, their effects have not been discussed in detail.
For an accurate prediction of the reattaching flow,such discussion is indispensable. Therefore, in thisstudy, time mean and periodically changing parts ofvelocity and temperature fields are calculated witha three-component decomposition approach for theflow over a backward-facing step. Both of 2-D and
3-D unsteady computations have been carried out

in this study.

1167

TABLE 1

NOMENCLATURE.
WD = width of the duct
AR = W, aspect ratio

HD = height of duct upstream of the stepH = height of the step
ER = bilZ,expansion ratio
iVu = I11fff6ll, mean Nusselt number
ru = wall shear stress?- = wall temperaturep = density of air
I = fluid thermal conductivity
Pr1 = turbulent Prandtl number

fu ='#,Reynolds number based on step height
6nffi' mean skin friction coefficient

C1 =g- = wall heat flux
To = inlet Temperature
cp = specific heat capacity at constant pressure
U,cf = inlet freestream mean velocity

COMPUTATION METHOD AND+C4μ t(:)2(sた Ω:,+Sた 」Ω

̀)S贖PROCEDURE
Three-component decompositiou approach isbased on the following manipulation of the governing equations (Reynolds and Hussain, 1972). Considering the unsteadiness of flows as a superpositionof slow periodical fluctuation and turbulent irregular fluctuation, all variables are decomposed into
three components; namely time mean value, periodical fluctuation and turbulent fluctuation respectively. This decomposition is mathematically expressed by the following equations respectively forvelocity compotretrt ui in ci direction, pressure p andtemperature 0:
■

u-i
一
一

Ｐ

* rii * ul =( ui ) *tl(1)θ

F*i*p'=1 p)*p'
0+6+d =<c>*dOl(s)where .A denotes the time mean value of a quan-tity, i the periodical part of the fluctuation of thequantity and A'the turbulent stochastic componentof the fluctuation of the quantity. () denotes the'phase averag€.
The governing equation for ui may be written as

十C5ル (言 )2tΩ ‖ῺmS",+SIΩ―ing equation for < ut >.

Ou;.+<"i>Td< u; )__0<p> - L 02<u> - A

oo; ' R"ffi

arr'

sij

=

ptVi = 2.
prsri
ikpflri -0la"rr,! (",.s*, - |sosoe,i)

▲

(指

+讐

),Ω

̲ ̲
s- 舌v島
島/2,Ω
た

Pt=

ρ
%ル

可
=(指 ―
讐)
■

′………―――………

Ω
:J/2
=舌 νΩ
●

■2

=

Sirnilarly,energy equatiOn in terlms of<θbe expressed as follows:
∂θ

> can

∂く θ>

房+<均 > 瓦下=扁 葛戸 ―再 <■ び> (8)
=「

!o1e>
-<vlo,> = Prt
0r;

(9)

Turbulent Prandtl number was assumed to be constant in this study and was taken to be Pr6 : Q.g.
The computational domain in the present computations is illustrated in Fig. I with the coordinate
system to be used. The origin ofthe coordinate system is located at the center of the bottom corner of1 ヽ´マ

the step.

uiuj x6)

The third term of the right hand side of the above
equation is modeled with a non-linear turbulencemodel proposed by the UMIST group (Suga, 1995).

ttS山 島十 :い )＼:Ω 餞
噛)
+C2μ

ISu

+C7ル (:)2Ω た ΩたJΩ

(5i

and operating phase average, one obtains the follow-

Sた

i,)
̀δ

where

(4)

where .Re is the flow Reynolds number. Substituting Eqs.(l)-(3) into Eq.(s) taking account ofEq.(a)

Ω
m・・

C6μ 。
十
(4)2s

follows:

y =o
0't
&t;
?lu;
Ap L 02u;
d*"i6 = -;L,"+6ffi:SInΩ

Im ΩmJ)

C3μby Papadopoulos et al. was at the Reynol& number
26,500. This was to decrease the grid aumber so as
to complete calculation within a reasonable computation time. For boundary conditions at walls, allvariables except temperature were set equal to zero.
For the thermal︱ ¶︱︱︱仁1168

The 3-D simulations performed in the present
work were carried out under conditions clos€ tothose studied by Papadopoulos et al. (1995). The
expansion ratio was set to be BR = 2 for &D
computation and ER = 1.2 for 2-D computation.
The aspect ratio in 3-D computation was set to be
AR = 4. The boundary layer thickness at the inlet
was 6 / H
= 0.16. 5, 200 was chosen for the Reynolds
number in the present study while the flow studied

Flow

2‐

Ph優,elWD

HcatedPhase 2Phase 3Phase 4Mean

Dl ER=1.2
5.90ff
7.47″
8.41″
5,94rr
7.60″

3‐

D ER=2)8.12″
8.50fr
8.13″
8.53″
8.54″

Figure 1 Computational domain.
changes its sign from negative to positive. The loca-

30

tions vary by large amount moving back and forth

Phase 225

Phasc 3centering the mean value.
Figure 4 illustrates the spatial distributions bothof an apparent wall-normal heat flux produced by
the periodical fluctuations of velocity and temperatve i0 and another apparent heat flux producedby the stochastic fluctuations of velocity and tem-

20

ミ15
10
5

0

0

5

ソlπ

perature < o'0' > at four different phases. nf iscalculated automatically as a part o[ the convection term of Eq.(8) while ( u'e' .> is calculated
through the model of Eq.(9) in the present computatiou. In this figure, the shaded parts correspondto the regions where the plotted quantities take positive yalue while the unshaded parts to those where
they take uegative value. Deeper gray color corre.
spoo& to larger heat flux. Compa^ring the rralues ofboth quantitie, < a'0' ) is larger than 06 on thewhole. However, the positions of the positive peaks
of apparent heat flux due to the periodical fluctuatioas match the positions of the maximum peaksof the Nr.selt number at every phase. This reveals
that periodical fluctuation contributes to the over-20

Figure 2 1nstantaneOus Nusselt nuniber

at agerent ph¨ es.(2‑D)⌒

Phase lむPhase 4Phase 3
2
5

15ノ

10

冨

20

Figure 3 Instantaneous skin friction coefrcient
at difrerent phases. (2-D)

boundary conditions, the bottom wall downstreamof the step was set to be heated at a constant heat

flux and other walls were treated as adiabatic.
RESULTS AND DISCUSSIONFirst the results of the 2-D computation will be
discussed. Instantaneous Nusselt number and skincoefficient distributions at difterent phases^friction
are shown in Fig. 2 and Fig.3 together with their- time mean rralues. Both values noticeably vary fromone phase to another. This is due to the unsteadiness of the flow related to the periodic formation ofvortex in the separated shear layer. Comparing thetwo figures, at some positions, peak instantaneous
Nusselt number is found to appear around the dent
in the distribution of skiu friction coefficient. This
indicates the generation of the instantaneous dissimilarity between the heat transfer and momentum
transfer as in the case of the turbulent boundary
layer disturbed by an iaserted cylinder (Inaoka etal., 1998, 1999). The locations of the reattachment
point for each time phases are t\pulated in Table 2.
The reattachment point is definedys the fust pointfrom the step at which the skin fric\ion coefficient

all heat transfer significantly.
Figure 5 shows the spatial distributions both of
an apparent wall-normal Reynolds shear stress produced by the periodical velocity fluctuations, -ii,
and aaother apparent Reyaolds shear stress produced by the stochastic velocity fluctuations, - (u'u' ), in the same manner as seen in Fig.4. In
this figure, again the shaded parts correspond to
the regions where the plotted quantities take positive value and unshaded parts to those of negativevalue. As in the case of the apparent heat flux justdiscussed, stochastic part of apparent shear stresstakes larger yalue thal the periodical fluctuation
counterpart. However, the position of the minimum
peaks of -ii matches the location of the minimum
values of instantaneous wall skin friction coefficient.
Combining all the results discussed in the above,
one can say that the intensi$cation of the apparent
heat flux contributed from ilO and the suppression of
the apparent shear stress caused by the contributiou
from -ti produces the instantaneous dissimilarity
between heat transfer and momentum transfer.
Next, the results obtained from a 3-D computation will be discussed. Iustantaneous Nusselt number and skin friction coefficient obtained along thecenterline at different phases are shown in Fig. 6 andFig.7 together with their mean values. First, as in
the 2-D computqtion, instantaneous and time meandissimilarity appears in some regions where

\
I

I

1169

「

―r

‐

"―

1

T

―

T‐

つ

7

―――r― 一■――
―

●

4

●Ｊ

/ ヽ

し/

凛ヽ

５ ４ ３ ２ １ ０凛ヽ

５ ４ ３ ２ １ ０

4

6
5
10
〔
め

15,x47a)5

4

0

5

'5

0う

10
Vb'>
く

15ノ 冨

つ

2

う０

工ヽ

５ ４ ３ ２ １ ０

４︐識ヽ

５ ４ ３ ２ １ ０

5

10

15 .ノ

7

(→ ■

7r 20

‐
5

4
0

0)‐

5

10

くu'V'>

Figure 5 1nstantaneous maps of apparent shear stress contours,′(a)̲こぅamd(b)̲く ごυ >(shad‐ ed parts:POsitive value).(2‐ D)

1170

l

′ヽ

Figure 4 1nstantaneous lnaps of apparent heat aux Contours,′
ed p劉 由s:pOsitive tlue).(2‐ D)(a)Oθ alld(b)く υダ >(sh

0

1

‐

15

″, 20
」
νH

30
Phase

20

e15

Mean

't0
5

15

Ａυ
一

︐

10

5

r/H

Figure 6 Instantaneous Nusselt number
at different phases. (&D, z - 0)

Phase

5
Mean

0

s

10

15ilH2O

Figure 7 Instantaneous skin friction coefficient
at different phases. (3-D, z = Q)
heat transfer increases while skin friction coefficientdecreases. Comparing the results of &D computation with the 2-D counterparts discussed above,
the plotted fluctuations in these figures are smaller.
However, still a large variation lrom one phase toanother is observed especially downstream the reattachment point. The reattachment location is tabulated in Table 2. Here also, the location shifts as
the phase changes. However, comparing with the
2-D counterparts, the variations of the movementsof the location are smaller.
In Fig.8 are plotted the streamwise distributions along the centerline of wall-normal apparent
heat fluxes obtained at several different phases. td' and < a'd > arc produced by the periodical andstochastic fluctuations of velocity and temperaturefields respectively. Comparing the two quantities
a'|t > and ii with each other, ( u'0' > takes a'^< larger value
0d. This is qualitatively the same
' as the resultsthanof the 2-D computation. Just as in
the 2-D computation, the locations of the maximum
peaks of instantaneous Nu number plotted in Fig.6
match well the locations where 0d shows maximum-td shows a minimum peak matches also well to thelocation where the skin friction coefficient shows its
minimum peak. Additionally, increase of dd is foundat the location where the increase of the Nusselt
number is observed and decrease of the apparent-il6 at the location where decrease of skin frictiou
coefficient is observed. Thus, significant effect ofthe
periodical fluctuations on the over-all characteristicsof flow and thermal fields has been confumed also
in the 3-D computation.
CONCLUSION
A three-component decomposition approach wasapplied both in the 2- and 3-D computations for aflow over a backward-facing step in order to discuss
the contribution of the periodical fluctuations andthe stochastic fluctuations to the over-all characteristics of the flow and thermal fields. It is demonstrated that the periodical fluctuation seriously affects the characteristics of iastantaneous flow andthermal fields. Effects of periodical components are
more pronounced both at locations downstream thereattachment point and inside the recirculating flow
region. Stochastic fluctuation term shows a larger
contribution than the periodical term even at the
Reynolds number 5,200. However, the periodical
fluctuation still seriously affects the characteristics
of both flow and temperature fields. Consequently,
apparent shear stress and heat flux produced by theperiodical fluctuation of velocity and temperaturesignificantly affect the time mean characteristics ofmomentum and heat transfer.

REFERENCES
Inaoka, K., Yamamoto, J., and Suzuki, K., uDissimilarity between heat transfer
momentumtransfer in a disturbed turbulent
layerwith insertion of a rod", Proc. Int.
in Tur,14, May
bdeat Heot Transfer 2, Vol.t, pp.
3l-June 5 1998.
Inaoka, K., Yamamoto, J.,
K., "Dissimilarity between heat transfer and momentum

transfer in a disturbed turbulent boundary layer
with iasertion of a rod - modeliag and numericalsimulation -", Iflt. J. Heot ond, Fluid, FIou, 1999.
(in printing)
Papadopoulos, G., Otiigen, M. V., "Separating
and Reattaching Flow Structure in a Suddenly Ex-

Peaks.

panding Rectangular Duct", .Iotrnal ol Ffuids En-

Figure 9 illustrates the spatial distributions along
the centerline of the apparent wall normal Reynolds
shear stresses, -i0 and - 1 u'a' ) calculated
at several different phases. Comparing with theabove.discussed results of 2-D computation illustrated in Fig.5, influence of the periodical fluctuations is somewhat less pronounced in the case of 3-Dcomputation. However, just as discussed in the 2-D

mechanics of an organized wave in turbulent shear
flow, Part 3: Theoretical models and comparisonswith experiments', J. Fhljd Mechonics, 34-2, 263288, 1972.
Suga, K., nDevelopment and application of a nonlinear eddy viscosity model sensitised to stress andresults, the quantity - 1 u'u' ) shows a larger value
comparing to the quantity -r7i. The location where

gineering, VoI.117, pp. 17-23, 1995.

Reynolds, W.C., and Hussain, A.K.M.F., "Thestrain invariants", UMIST, Dept of Mechanical Engineering, Report TFD/95/11, 1995.

11'71

Phase l

Phase I

0,ヽ

7

・ ヽヽ

0 .

F
Pllase 3

01。

′
′

へ ＝ν

Phase 2\\[ !

Phase 4

ミ

0

ゝ

10

15

1

σ

0
‐5

0

5
(→

T

2

凛1
ヽ
0

‐
5

'T

ヤ
や

Phase 4
0

χ′リ

5

0)
d Parts:positi

10

15ノ冒

"

⌒

:

value).(3‐D,2=0)

0

1

Phase 2′>(Sha∵"'0'>

Figure 8 1nstantaneOus nlaPs OF apparent heat llux contours,(a)"and(b)<詢'‐ … q3

￨

20

等5

r

Phase 3ｎＵ

2…:Phase 2

。 ヽ
0

3

Qヽ

0
戸 彙

1

4

3ヽ

0

ｏ

ヽ

普

2
,H

0
5

(a)

15

-ii

ノH

20

5

10

〃冨

(b) -<u'v'>

Figure 9 lnstaataneous maps of appa,rent shear etress contours,.
(") -ii and (b) - < r'o' > (Shaded parts: positive ralue). (3-D, z = 0)1172

15