Amano R.S., Sunden B. (Eds.) Computational Fluid Dynamics and Heat Transfer: Emerging Topics

Подождите немного. Документ загружается.

Sunden CH008.tex 10/9/2010 15: 19 Page 286

286 Computational Fluid Dynamics and Heat Transfer

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Enclosure Rib

−0.2

−4

0 0.2 0.4

x/d

Figure 8.7. Proﬁlesoflocalskinfrictioncoefﬁcientsaroundribalongthehorizontal

wall with x =0 at the middle of the enclosure.

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Enclosure Rib

/Nu

−0.2

0 0.2 0.4

x/d

Figure 8.8. Proﬁles of local Nusselt number around rib along the horizontal wall

with x =0 at the middle of the enclosure.

the total drag is deter mined almost entirely by the pressure drag when the ratio is

within the range from 6 to 20.

From Figure 8.7, the local skin friction coefﬁcient becomes minimum in the

enclosure and maximum at the front corner of the rib in Cases 1 and 2. On the

other hand, in Case 3, although the local skin friction coefﬁcient takes the maxi-

mum at the front corner of the rib, both a large overall decrease and a local peak

in the enclosure cannot be seen; but it changes sign in the enclosure.The reattach-

ment point is located at x/δ =0.04; thus the forward ﬂow region is observed over

0.04< x/δ < 0.21 until separation occurs in the enclosure between the ribs.

The local Nusselt numbers in Cases 1 and 2 distribute similarly to the abso-

lute value of the local skin friction coefﬁcient. However, in Case 3, the Nusselt

number increases over the entire region in the enclosure. This situation is also

observed near the reattachment region of the backward-facing step ﬂows [35], and

it is reported that the heat transfer coefﬁcient reaches maximum there. Similarly,

Sunden CH008.tex 10/9/2010 15: 19 Page 287

Recent developments in DNS 287

0.5

0.3

0.1

0.2

−0.1

U/U

0.5

1.5

0.1

0.2

Rough wall Smooth wall

Plane channel

Case 2 H = 0.1δ

Case 1 H = 0.2δ

Case 3 H = 0.05δ

y/δ

= 150

x/δ = 0

Figure 8.9. Proﬁles of mean velocity.

thelocalNusseltnumberincreasesinapproachingthereattachmentpointx/δ =0.04

and thefollowingforwardﬂowregion(0.04< x/δ < 0.21).From theabove results,

itisconﬁrmedthatinCase3,theheattransferispromoted,yetwithonlyarelatively

small increase in the drag.

8.5.3 Statistical characteristics of velocity ﬁeld and turbulent structures

In a velocity ﬁeld of a channel ﬂow with a rib surface, the ﬂow motions in the

enclosurebetweentheribsstronglyaffecttheﬂowﬁeldabovetheribs.Weexamined

thevariouswaysof averaging,e.g., averageoverthe x–z plane, spanwiseaverageat

themiddleoftheenclosure(x/δ =0)andattheribcrest(x/δ =0.4).Thedifferences

were not seen along the streamwise direction except near the roughness element.

Thus, in the following, the statistical quantities of turbulence in the middle of the

enclosure (x/δ =0) are discussed.

Figure8.9showsthemeanvelocitynormalizedbythebulkvelocityU

.Because

of the effects of the rib, the velocity decreases on the rough-wall side. The corre-

spondingdistributions oftheReynoldsshear stressandturbulentkineticenergy are

showninFigures8.10and8.11.Astheribheightincreases,turbulenceispromoted,

and both the Reynolds shear stress and the turbulent kinetic energy increase near

the wall. This affects the region over the center of the channel. However, in the

near-wall region of the opposite wall, there is only a small effect, in comparison

withtheresultsfortheplanechannelﬂow. Despitethenegativevalueofthevelocity

gradient near the wall in the enclosure as seen in Figure 8.9, the Reynolds shear

stress takes a positive value, thus conﬁrming the occurrence of counter gradient

diffusion.BecausetheproductionoftheReynoldsshearstressismainlymaintained

with thepressure–strain correlation andthepressure diffusion, no production from

the mean shear is observed. An experimental study [29] indicated that the place

where the Reynolds shear stress becomes zero and the mean velocity becomes

maximum is different. However, the difference between them is very small; 2.0%,

2.8%, and 0.8% of a channel width for Cases 1, 2, and 3, respectively.

Sunden CH008.tex 10/9/2010 15: 19 Page 288

288 Computational Fluid Dynamics and Heat Transfer

−0.005

y/δ

0.005

un/U

0.01

0.015

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Plane channel

= 150

x /δ = 0

Figure 8.10. Proﬁles of Reynolds shear stress.

0.01

0.02

y/δ

k/U

0.03

0.04

0.05

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Plane channel

= 150

x /δ = 0

Figure 8.11. Proﬁles of turbulent kinetic energy.

Figure8.12 showsthe turbulence intensitiesnormalized by thefriction velocity

on the rough wall. In all cases, the statistical quantities get together well in the

upper region of the ribs. However, from Figure 8.11, the turbulence intensities

increase relative to the bulk mean velocity. The budget of the turbulent kinetic

energy is shown in Figure 8.13. In Case 1, the convection term is enhanced in

comparison with the planechannel ﬂow.This is because the meanvertical velocity

component near the rib is increased, and also there is a variation in the turbulence

in the streamwise direction. Moreover, the contributions from the turbulent and

pressure diffusion terms are very large; in the enclosure between the ribs, there is

no turbulence production from the mean shear, but turbulent transport maintains

the turbulencethere. In Case3, the distributionof thebudget issimilarto theresult

in the plane channel, in comparison with Case 1, but the contributions from the

turbulent transport (turbulent and pressure diffusions) are large.

Sunden CH008.tex 10/9/2010 15: 19 Page 289

Recent developments in DNS 289

Rough wall Smooth wall

Case 1

Case 2

Case 3

Plane channel

= 150

rms

0.5

1.5

0.5

100 200

Figure 8.12. Distributions of turbulence intensities.

Next, to investigate the anisotropy of turbulence, Figure 8.14 shows the

pressure–straincorrelationtermsseeequation(23)whichareimportantinthenear-

wall region. The pressure–strain correlation is affected because the ﬂow impinges

on the upstream side of the rib wall, and the pressure ﬂuctuations increase over a

wide regionon the rough-wall sideincomparison with the smooth-wallside. Ifwe

compareCase1,wheretheribishighest,withtheplanechannelﬂow, therespective

components of the pressure–strain term become maximum in the region between

the ribs, and a very active energy exchange occurs. In Case 1, the sign of 

becomes positive at y

15. In ordinary wall turbulence without external force,

it is observed that in the near-wall region, the splatting phenomenon caused by

gives the energy to w

as seen in the DNS results [16]. However, in the enclosure

between ribs, the energy does not redistribute to the

component. This can be

attributedto thetwo-dimensional vorticesexistingintheenclosure, whichpromote

Sunden CH008.tex 10/9/2010 15: 19 Page 290

290 Computational Fluid Dynamics and Heat Transfer

Plane channel

Production

Dissipation

Turbulent diffusion

Viscous diffusion

Convection

In enclosure

(cavity)

Case 1 H = 0.2δ

Case 3 H = 0.05δ

Pressure diffusion

= 150

0.2

0.1

−0.1

−0.2

0.2

0.1

−0.1

0.3

0.2

0.1

0.1 0.2

y/d

0.3 0.4

−0.1

LOSS GAINLOSS GAIN LOSS GAIN

Figure 8.13. Budgets of turbulent kinetic energy.

the redistributionfrom the

to u

components.Asfor Case 3, theeffects fromthe

two-dimensionalvortexandthereattachmentarecombinedandtheyaffecttheredis-

tribution mechanism. As a result, as shown in Figure 8.12, in spite of the existing

roughwall,theanisotropyofturbulenceshowsbehaviorsimilartothatinthesmooth

wall, and the roughness promotes the heat transfer with a relatively small drag.

Finally, Figure 8.15 shows the streamwise vortices educed with the second

invariant tensors of the velocity gradient II [36, 37]. In each case, streamwise

vortices are produced in the region above the ribs on the rough-wall side, and

the structures spread over the region above the center of the channel. Moreover,

Sunden CH008.tex 10/9/2010 15: 19 Page 291

Recent developments in DNS 291

Case 1 H = 0.2δ

Case 3 H = 0.05δ

Inside of rib

0.1

−0.1

0.1

−0.1

20 40 60 800

Symbols: plane channel

Figure 8.14. Distributions of pressure strain terms in transport equations for

Reynolds stress.

if we compare Case 1 with Case 3, many more streamwise vortices are produced

in Case 1, where the rib effects are strong. In the enclosure between the ribs,

the streamwise vortices are seen in the whole region of the enclosure in Case 3.

However, in Case 1, because the ﬂow stagnates downstream of the rib surface,

not many streamwise vortices are produced. This corresponds to the results of

the distributions of the skin friction coefﬁcient and the Nusselt number as seen in

Figures 8.7 and 8.8.

8.5.4 Statistical characteristics of thermal ﬁeld and related turbulent

structures

For the thermal ﬁeld, the turbulence statistics at the middle of the cross-sectional

enclosure (x/δ =0) are discussed below. Figure 8.16 shows the mean temperature

distribution normalized by the temperature difference T

. The effect of the rib

causes the mean temperature to become asymmetric in all cases as for the mean

velocity proﬁles, and the temperature increases in the major par t of the channel.

On the other hand, because of the rough-wall side effects, the region of the large

temperature gradient extends to the smooth-wall side. Figure 8.17 shows the tur-

bulent heat ﬂux in the wall-normal direction. The heat transfer is activated from

Sunden CH008.tex 10/9/2010 15: 19 Page 292

292 Computational Fluid Dynamics and Heat Transfer

Case 1 H = 2δ

Case 3 H = 0.05δ

Figure 8.15. Vortex structures visualized by second invariant: II

∗

< −0.5.

0.3

0.2

0.1

102

0 0.2

0.2

0.4

0.6

0.8

Plane channel

t 0

= 150

Pr = 0.71

y /δ

Rough wall Smooth wall

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Θ/∆T

Figure 8.16. Proﬁles of mean temperature.

the enclosure with the ribs, but in the thermal ﬁeld also there are counter gradient

diffusions in Case 1.

Next, Figure 8.18 shows the rms intensities of temperature ﬂuctuation, which

decrease on the rough-wall side, where the velocity ﬂuctuations are promoted.

However, the temperature intensities increase on the smooth-wall side, where

the turbulent velocity ﬂuctuation is suppressed. Apparently, the contrar y situa-

tions occur between the velocity and thermal ﬁelds on each side. To investigate

the cause and effect, we calculated the production and turbulent diffusion terms

of the transport equation for

/2 see equation (11) in each case (Figure 8.19).

Sunden CH008.tex 10/9/2010 15: 19 Page 293

Recent developments in DNS 293

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Plane channel

−v θ/U

∆T

0.002

Smooth wall

y/δ

Rough wall

Figure 8.17. Proﬁles of wall-normal turbulent heat ﬂux.

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Plane channel

Smooth wall

y/δ

120

0.05

rms

/∆T

0.1

Rough wall

Figure 8.18. Proﬁles of rms intensities of temperature ﬂuctuation.

The production terms on the smooth-wall side, where the temperature gradient is

large and the wall-normal heat ﬂuxes increase, contributes greatly in the region

around the peak to the center of the channel. This is why the temperature intensi-

ties become large on the smooth-wall side. On the other hand, on the rough-wall

side, because the contribution from the production decreases and the turbulence

is maintained by the turbulent diffusion, the intensities of temperature ﬂuctuations

decrease, especially in Case 1.

In order to examine the relationship between the velocity and thermal ﬁelds,

the distributions of the turbulent Prandtl number, Pr

, are shown in Figure 8.20.

The spanwise averaged Pr

at the middle of the enclosure and of the rib crest

are compared with that in the plane channel ﬂow. Near the region above the rib,

becomes larger than that on the rib crest. However, it becomes smaller and

then becomes larger again in the enclosure. Thus, the turbulent Prandtl number

is not constant and the analogy between heat and momentum transfer cannot be

expected near the rib roughness. On the other hand, on the smooth-wall side, the

distribution well corresponds to that in the plane channel ﬂow.

Sunden CH008.tex 10/9/2010 15: 19 Page 294

294 Computational Fluid Dynamics and Heat Transfer

0.02

Case 1 H = 0.2δ

Case 2 H = 0.1δ

Case 3 H = 0.05δ

Plane channel

0.01

Production

Turbulent diffusion

0.005

−0.005

y/δ

Figure 8.19. Production and turbulent diffusion terms in temperature variance

budget.

y/ δ

Spanwise average at the

middle of enclosure

Spanwise average at the

middle of rib crest

Plane channel

Case 1 H = 0.2δ

Rough wall Smooth wall

Rib

= 150

Figure 8.20. Turbulent Prandtl number.

Finally, to observe the spatial structures that contribute to the transport of the

passive scalar, Figure 8.21 shows the streamlines, which are spatially averaged

along the streamwise direction, and the temperature ﬂuctuations in the y–z cross

section inthe plane channelﬂowand Case 1. Inthe plane channelﬂow, the stream-

lines show the streamwise vortices in the near-wall region; the sharp variations

in the temperature ﬂuctuations associate with the vortices. On the other hand, in

Case1, large-scalevortexstructuresappear, whichextendtothecenter ofthechan-

nel from the enclosure between the ribs. With this vortex structure, the turbulent

mixing becomes larger in the center of the channel, and the turbulence transport is

promoted.This behavior is consistent with the above-mentioned statistical results,

i.e., theincreasein theheattransfer coefﬁcient, thepronounced changeinthe mean

Sunden CH008.tex 10/9/2010 15: 19 Page 295

Recent developments in DNS 295

Plane channel Cooled wall

Heated wall

Smooth wall (cooled)

Rough wall (heated)

Rib

Case 1 H = 0.2d

Figure 8.21. Streamwise-averaged streamlinesandtemperatureﬂuctuationsin y–z

plane: θ/T

=−0.05 (white) 0.05 (black).

temperature distributions, and the increase in the turbulent diffusion terms in the

budget of temperature variance.

8.6 DNS of Turbulent heatTransfer in Channel Flow with

Arbitrary RotatingAxes: Spectral Method

In order to determine improvements in the modeled expression of rotating, wall-

boundedturbulentshearﬂows,DNSsoffullydevelopedchannelﬂowswithstream-

wise, wall-normal, and spanwise rotationare carried out using the spectralmethod

forthe15casesindicatedinTable8.5[18].DNSresultsofcasesforWNR1∼WNR5

and STR1∼STR5 are shown in Figures 8.22 and 8.23, respectively (Case 3 is not

shown here). In these cases, the spanwise mean velocity appears to be caused by a

rotationaleffect, whichincreaseswiththeincreaseinrotationnumberinbothcases.

Inparticular,withtheincreaseinthespanwisemeanvelocityofCase1,streamwise

mean velocity decreases due to the exchange momentum between the streamwise

and spanwise velocitites as in the following equations obtained from equation (5):

0 =−

∂

∂x

∂

∂y



∂

∂y

−



− 2W (17)

0 =

∂

∂y



∂

∂y

−



+ 2U (18)