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

Подождите немного. Документ загружается.

Sunden CH007.tex 10/9/2010 16: 6 Page 266

266 Computational Fluid Dynamics and Heat Transfer

differences in the experimental conditions, which need further clariﬁcation. In a

previous study [1] by the RANS method, side-wall heating was found to be the

mainreasonfor thediscrepancy. However,theRANS methodﬁltered outtoo many

details, so that the explanation of the reason becomes difﬁcult. In addition, the

uncertainties of the RANS models may become large for complex geometries.

Therefore, a more universal modeling method, large eddy simulation (LES), is

employed to further conﬁrm and explain the ﬁndings from RANS.

LES is a computational method in which the large eddies are computed and

the smallest, subgrid-scale (SGS) eddies are modeled. The underlying premise is

that the largest eddies are directly affected by the boundary conditions, carr ying

most of the anisotropic Reynolds stresses, and must be computed.The small-scale

turbulence is weaker, contributing less to the turbulent Reynolds stresses, is nearly

isotropic, and has nearly-universal characteristics. Therefore, it is more amenable

to model.

InLES,thetransportequationsareﬁlteredfromthe3D,unsteady,Navier–Stokes

equations by a ﬁlter function in space:

∂

∂t

∂(

ρ u

)

∂x

= 0 (1)

∂(

ρ u

)

∂t

∂(

ρ u

)

∂x

=−



∂

∗

∂x

− β



∂

∂x





∂

∂x

∂

∂x



ρτ



(2)

∂(

ρ φ

∗

)

∂t

∂(

ρ u

∗

)

∂x

∂

∂x



∂

∗

∂x

+ ρϕ



− δ

ρ u

γ (3)

where τ

−u

are called the subgrid-scale (SGS) stresses, through which

the small scales inﬂuence the large (resolved) scales. Accordingly, ϕ

=φu

−φu

are called the SGS heat ﬂuxes. They are unknown and must be modeled by a SGS

model.

The implicit model, proposed by Boris et al. [6], is the simplest SGS model

without introducing neither the ﬁltering operation nor the SGS stress tensor. It is

based on the observation that truncation errors in such discretizations of Navier–

Stokes equations introduce numerical dissipation with the implicit effects of the

discretization qualitatively similar to the effects of the explicit SGS models.

However, in the presence of the inviscid unsteadiness, the turbulence becomes

much stronger, such as the separation zone downstream a rib. Consequently, the

numerical diffusion is not enough to dissipate the turbulent energy, especially for

a ﬁne grid. Some additional viscosity is necessary to prevent the buildup of turbu-

lent energy at high frequency regions. Smagorinsky [7] was the ﬁrst to postulate

an explicit model for the SGS stresses. His model assumes the SGS stresses fol-

low a gradient-diffusion process, similar to molecular motion. Consequently, τ

given by:

= 2ν

(4)

Sunden CH007.tex 10/9/2010 16: 6 Page 267

LES of turbulent ﬂow and heat transfer 267

where S



∂u

∂x

∂u

∂x



is called the “resolved strain rate,” and ν

is the

Smagorinsky eddy viscosity given by:

= (C

)

(5)

where  and C

are the ﬁlter width (grid spacing) and the Smagorinsky constant,

respectively, which must be speciﬁed prior to a simulation, varying from ﬂow to

ﬂow.The Smagorinsky constant is set as C

=0.1 in this simulation.

7.2 Numerical Method

The computations are carried out in an in-house parallel multi-block computer

code CALC-MP [8, 9], based on the ﬁnite volume technique. The code uses a

collocatedbody-ﬁtted gridsystem andemploysthe improvedRhieandChowinter-

polation procedure method to calculate the velocities at the control volume faces.

The SIMPLEC algorithm couples the pressure and velocity. An algorithm based

on SIP (Strongly Implicit Procedure) is used for solving the algebraic equations.

Both thediffusionand convection terms arediscretized bythe secondordercentral

difference.The validation of the code has been presented by Jia [10].

7.3 ResultsAnd Discussion

7.3.1 Fully developed pipe ﬂow

Toillustratethenumericalaccuracyofthe presentcode, afully developedpipeﬂow

iscalculatedwithLES,forRe

=460.Implicitmodel isusedformodelingtheSGS

stresses, and a central scheme is used for the convection terms. A butterﬂy-type

multi-blockgrid is generated, and the approximate number of grid pointsisaround

one million.

Although ﬂows such as a turbulent pipe ﬂow are geometrically simple, their

accurate simulation using techniques such as LES is in some way more difﬁcult

than ﬂows in complicated geometries. This is because no inviscid instability is

available, its accurate simulation is an indication of the reliability of the method.

Figure7.1ashows thespectrum aty+=230(i.e., apoint atthecenterline ofthe

pipe), and the cutoff of the high-frequency turbulence energy is clearly illustrated.

Figure 7.1b shows the mean main stream velocity and turbulent Reynolds

stresses by LES, in comparison with the DNS data of Eggels et al. [11]. They

are in very good agreement. At the region close to the wall, the ﬂuctuations in the

main ﬂow direction are much larger than those in the other two directions. From

anyRSTMmodel (Reynoldsstress turbulencemodel), onecan seethat theproduc-

tion term for the ﬂuctuations of the two cross directions is zero. The ﬂuctuations

of these two directions obtain the turbulence energy from pressure strain terms.

The pressure strain terms redistribute energy from the high-ﬂuctuation directions

Sunden CH007.tex 10/9/2010 16: 6 Page 268

268 Computational Fluid Dynamics and Heat Transfer

−1

(a) Turbulent energy spectrum

−5/3

= 230

E = f

−5/3

−5

−6

−4

E(f )

−3

−2

(b) Statistic Values

r/D

U/u

u′v ′/u

u′/u

w ′/u

u ′/u

0.0

−1 −5

0.5

−0.5

LES DNS

(U ′

U′

)

0.5/u

0.5

−0.5

N/0

0 0.5

(d) Instantaneous u′ from DNS

0.5

3.5 4 4.5 5 5.5 6

3 3.5 4 4.5 5 5.5 6

(e) Contours of the instantaneous out-of-plane velocity at a center plane

Figure 7.1. The fully developed pipe ﬂow case Re

=460.

to the low-ﬂuctuation directions to reduce the difference. However, it can not take

awaythedifferencetotally.Therefore, themainstreamdirection ﬂuctuationismuch

larger than those in the other two directions.

Figure7.1cshowsthemain-ﬂowdirectioninstantaneousvelocityﬂuctuationsin

acrosssectionofthepipe.Onecanclearlyobservetheejectioneventsfromthewall

towardthepipe center, and low-speed streaks will form downstream these ejection

Sunden CH007.tex 10/9/2010 16: 6 Page 269

LES of turbulent ﬂow and heat transfer 269

(a) Geometry

(b) Block topology

plane-IV

plane-VI

plane-III

plane-II

plane-I

plane-V

plane-III

Figure 7.2. The computation geometry.

events. These ejection events and streaks produced the majority of the turbulence.

Approximately, there are around six ejection events in the circumferential direc-

tion. Normally the spanwise distance for the ejection events are around s+=100.

Figure 7.1d shows a similar picture from the DNS of Eggels et al. [11].

Figure 7.1e shows the out-of-plane instantaneous velocity in a plane through

the pipe centerline. Superimposed on top of the small-scale turbulence in the core

regionofthe ﬂow, large scalestructure, on the scale ofthe pipe diameter, is clearly

seen. Near the pipe walls, the ejection events described earlier are seen as the dark

and light regionsthatleave the pipe walls at an angle ofapproximately20degrees.

In general, we observed good agreement with the DNS data. This is true for

both mean and rms results.

7.3.2 Transverse ribbed duct ﬂow

ThecomputationalgeometryisschematicallyshowninFigure7.2a,andthecompu-

tational block topology is illustratedinFigure7.2b.Thesquare duct withhydraulic

diameterD

isroughenedbyperiodicallymountedribsofsizee,e/D

=0.1,andthe

pitch is10e.TheV-shape inclineangle may be90-degree, 60-degree, or 45-degree.

When it is 90-degree, transverse ribs prevail. The simulated Reynolds number is

small, around 4,000, turbulent ﬂow is ensured.The selection of a low Re is mainly

becausea relatively smallcell numbercanbe used,and themain featuresofthe tur-

bulenceﬁeldarenotdifferentfromthehigh-Recases. Inaddition,thecontradiction

in the experiments is not Re-dependent.

Flow ﬁeld

Figure 7.3a shows the mean streamlines in the symmetry plane. The streamlines

are characterized by four recirculation zones, accompanying ﬂow separation and

Sunden CH007.tex 10/9/2010 16: 6 Page 270

270 Computational Fluid Dynamics and Heat Transfer

(a) Streamlines

x/e

y/e

012345678910

(b) u′/U

x/e

y/e

0.195

0.293

0.325

0.358

0.390

0.423

0.293

0.260

0.228

0.195

0.455

0.228

0.163

0.260

012345678910

0.228

0.48

0.455

0.42

0.39

0.35

0.325

0.29

0.26

0.22

0.195

0.16

0.13

0.09

0.065

0.03

012345678910

x/e

y/e

0.142

0.160

0.178

0.213

0.249

0.267

0.249

0.231

0.213

0.196

0.178

0.160

0.142

0.178

0.196

0.213

0.231

0.249

0.124

0.160

0.107

0.142

0.178

0.160

0.142

0.124

0.142

0.178

0.213

0.249

0.196

0.213

0.267

0.249

0.231

0.196

0.213

0.178

0.160

0.142

0.124

0.107

0.089

0.071

0.053

0.036

0.018

(d) w′/U

012345678910

x/e

y/e

0.124

0.155

0.124

0.155

0.186

0.217

0.248

0.279

0.248

0.217

0.186

0.155

0.124

0.311

0.373

0.279

0.248

0.279

0.311

0.186

0.217

0.248

0.155

0.279

0.466

0.435

0.404

0.373

0.342

0.311

0.279

0.248

0.217

0.186

0.155

0.124

0.093

0.062

0.031

Figure 7.3. Streamlines and contours of the turbulent Reynolds stresses in the

symmetry plane (plane-I).

Sunden CH007.tex 10/9/2010 16: 6 Page 271

LES of turbulent ﬂow and heat transfer 271

reattachment. The sudden expansion downstream of a rib leads to the generation

of a wide recirculation vortex associated with a strong shear layer. Together with

this main recirculation, a second smaller separation is also identiﬁed, located in

the corner between the downstream face of the rib and the bottom wall. After

reattachment, a new boundary layer grows; the ﬂow is accelerated by the main

stream until it impinges upon the next rib. Finally, a fourth vortex is identiﬁed on

top of the rib.

Figure7.3b showscontours of streamwise turbulence intensity u



.The peak

values of u



are located at the rib top. This is due to acceleration and strong

shear at the rib top. Some local maxima of u



values are found at the top of the

main separation zone behind the upstream rib. They are related to the shear layer

originating from the rib edge and located above the separation region. Contours

with high u



values are closer to the ribbed wall.

Figure 7.3c shows the y-direction velocity ﬂuctuations. High v



values are

foundabovethemainseparationzoneinsidethecavitybetweentwosuccessiveribs.



is small near the bottom wall in the cavity, indicating weakvertical turbulent

motions.

Figure7.3dshowsthat thespanwiseturbulenceintensity, w



, isenhanced by

theribs.Thepeakvalueliesatthelowercornerinfrontoftherib,wherestreamwise

ﬂow is blocked by the rib. A local maximum is also found in the main separation

zone behind the upstream rib. The w



maximum is similar to the u



peak.

High w



values are found upstream the rib, near the reattachment point in the

cavity, and close to the rib front edge.

In general, very strong anisotropy is observed in the distribution of Reynolds

stresses inthreedirections.This makesitvery difﬁcultto predict it by using linear-

eddy-viscosity models.

7.3.3 V-shaped ribbed duct ﬂow

The solver was also employed to simulatethe ductﬂowroughened with 60-degree-

inclined V-shape ribs. The heat transfer coefﬁcients are calculated locally, and

compared with the average value. The deviations are displayed with contours to

help explain the discrepancy between different experimental results.

Figure7.4 shows thedeviation ofthe local Nufrom the averageNu.The reason

forthe experimentalcontradictioncan beexplainedfromthispicture. Ifwe useone

thermocouple (TC) to measure the temperature representing the surface; ideally,

weshouldputtheTCattheplaces,wheretheaveragedvalueislocated, i.e.,the0%-

deviationcontour lines. For the downstream pointingV-shaped ribs, if one puts the

TC at thecenter of the surface, one will obtain areasonablyaccurate average value

(Figure 7.4a), because the value at the center of the surface is very close (around

−5% underprediction) to the average value. For the upstream pointing V-shaped

ribs, however, if one puts the TC at the center of the surface, one will obtain an

over predicted averaged value (Figure 7.4b), because the value at the center of the

surface is far (around 30% over prediction) from the averaged value.

Sunden CH007.tex 10/9/2010 16: 6 Page 272

272 Computational Fluid Dynamics and Heat Transfer

−7

−14

−7

−21

−29

−7

x/e

z/e

x/e

z/e

−4

−2

−4

−2

6 8 10 12

(a) Downstream pointing V-shaped ribs

(b) Upstream pointing V-shaped ribs

4681012

Figure 7.4. Deviation (%) from the average Nu.

With the liquid crystal (LC) method, the numbers of the sampling points for

heat transfer coefﬁcients are doing much better to correctly represent anisotropic

heat transfer behavior.Therefore, correct conclusions are more likely obtained.

In summary, we can conclude that the LC experimental data in [4] and [5] are

more reliable thanthethermocouple datain [2] and[3].The mainreasonis thatthe

precise temperature distribution is important. The downstream pointing V-shaped

ribs are providing better heat transfer efﬁciency.

Sunden CH007.tex 10/9/2010 16: 6 Page 273

LES of turbulent ﬂow and heat transfer 273

7.4 Conclusions

The LES method implemented in the present solver has good accuracy for

predictions of heat transfer and ﬂuid ﬂow.

Based on the simulation for the V-shaped ribbed duct ﬂow, we have found that

the LC experimental data are more reliable than those of TC data, which is the

reason for the discrepancy of the experimental conclusions. It further conﬁrmed

that the V-shaped ribs pointing downstream are performing better in heat transfer

enhancement.

However, for general application of LES in heat transfer simulations, there are

remaining difﬁculties:

1. Treatment of near-wall turbulence: The near-wall turbulence is highly turbu-

lent and small scaled (both in time and space), and plays a too signiﬁcant

role in heat transfer predictions to be neglected. These small scales normally

cannot be captured by the simulations oriented for large scales. Therefore,

subgrid modelsare vitalhere. Consequently, SGS modelswould bevery impor-

tant for heat transfer problems. Some numerical error may play a role here,

but may only have limited success. Some SGS models seem promising, such

as WALE (wall-adapting local eddy-viscosity) and the dynamic Smagorinsky

model (DSM).

2. Computational cost: It takes long time to obtain enough LES statistic samples,

which imposes a heavy burden on the computational speed and capacity. Some

RANS calculations may be ﬁnished in minutes, but it may take days for a

comparable LES calculation.

With the development of even faster and larger computer and quicker commu-

nication hardware, at the same time dipping prices, these computational costs will

become less signiﬁcant, but the software level parallelization is still somewhat left

behind.

In general, we can foresee advancement of computational power and conse-

quently the SGS model will become less important at the end, since mesh can be

ﬁner and time steps can be smaller, and gradually come close to the DNS level,

although the better understanding of near wall turbulence will speedup this proce-

dure. Robustandaccurateschemesandalgorithmswillalwaysbe animportantpart

of research.

Asasummary,ontoday’sparallelcomputerclustersbuiltwithcommoditycom-

ponents, realistic heattransfer problemscan be resolvedwith LES at anacceptable

costanddecentaccuracy, whiletheadvancementofthecomputationaltechnologies

will make the solution faster and more accurate.

Acknowledgment

The Swedish EnergyAgency ﬁnancially supported this research work.

Sunden CH007.tex 10/9/2010 16: 6 Page 274

274 Computational Fluid Dynamics and Heat Transfer

References

[1] Jia, R., Saidi, A., and Sundén, B. Heat transfer enhancement in square ducts with

V-shaped ribs,ASME J. Turbomachinery, 125(4), pp. 788–791, 2003.

[2] Han, J. C., Zhang, Y. M., and Lee, C. P. Augmented heat transfer in square

channels with parallel, crossed, and V-shaped angled ribs, ASME J. Heat Transfer,

113, pp. 590–596, 1991.

[3] Olsson, C. O., and Sundén, B. Experimental study of ﬂow and heat transfer in

rib-roughenedrectangularchannels, Exp.Therm. Fluid Sci.,16(4),pp. 349–365, 1998.

[4] Taslim, M. E., Liu, T., and Kercher, D. M. Experimental heat transfer and friction in

channels roughened with angled, V-shaped, and discrete ribs on two opposite walls,

ASME J.Turbomachinery, 118(1), pp. 20–28, 1996.

[5] Gao, X., and Sundén, B. Heat transfer and pressure drop measurements in rib-

roughened rectangular ducts, Exp.Therm. Fluid Sci., 24(1–2), pp. 25–34, 2001.

[6] Boris, J. P., Grinstein, F. F., Oran, E. S., and Kolbe, R. L. New insights into large eddy

simulation, Fluid Dyn. Res., 10, pp. 199–208, 1992.

[7] Smagorinsky, J. General circulation experiments with the primitive equations. I. The

basic experiment, Mon. Weather Rev., 91, pp. 99–164, 1963.

[8] Jia,R., andSundén, B.A Multi-blockimplementationstrategyfora3Dpressure-based

ﬂow and heat transfer solver, Numer. Heat Transfer B, 44(5), pp. 457–472, 2003.

[9] Jia, R., and Sundén, B. Parallelization of a multi-blocked CFD code via three

strategiesforﬂuidﬂowandheattransferanalysis,Comput.Fluids,33,pp.57–80,2004.

[10] Jia, R., Turbulence modelling and parallel solver development relevant for investiga-

tionofgasturbinecoolingprocesses,Doctoralthesis, DepartmentofEnergySciences,

Lund Institute ofTechnology, 2004.

[11] Eggels, J. G. M., Unger, F., Weiss, M. H., Westerweel, J., Adrian, R. J., Friedrich, R.,

and Nieuwstadt, F. T. M. Fully developed turbulent pipe ﬂow: A comparison between

direct numerical simulation and experiment, J. Fluid Mech., 268, pp. 175–209, 1994.

Sunden CH008.tex 10/9/2010 15: 19 Page 275

8 Recent developments in DNS and modeling

of turbulent ﬂows and heat transfer

Yasutaka Nagano and Hirofumi Hattori

Department of Mechanical Engineering, Nagoya Institute of

Technology, Japan

Abstract

The purpose of this chapter is to review recent studies on direct numerical simu-

lation (DNS) and turbulence models from the standpoint of Computational Fluid

Dynamics (CFD) and Computational Heat Transfer (CHT).Trends in recent DNS

researchanditsroleinturbulencemodelingarediscussedindetail.First,theinstan-

taneous and the Reynolds-averaged governing equations for ﬂow and energy are

writtenin rotationand nonrotationframes. Secondly, asa typicalrecent DNSusing

the high-accuracy ﬁnite-difference method, the detailed DNS results of velocity

and thermal ﬁelds in turbulent channel ﬂow with transverse-rib roughness are pre-

sented. The problem is very important for elucidating the structure, which affects

heat transfer enhancement.Thirdly, our recent DNS of rotating channel ﬂows with

variousrotating axesandheat transfer is described. Rotating ﬂows with heat trans-

fer are encountered in many applications relevant to engineering, such as in gas

turbines. This DNS was conducted using the spectral method. Therefore, two sig-

niﬁcant types of DNSs are presented in this chapter. Finally, on the basis of DNSs,

much emphasis is placed on evaluating the performance of various kinds of turbu-

lencemodels usedin thevelocityand thermalﬁelds.Then, anewturbulence model

is proposedfor the accuratepredictionof rotating turbulentﬂowwith heat transfer.

We should, however, bear in mind that a universal turbulence model can never be

established without further development of DNS.

Keywords:Turbulence, HeatTransfer, DNS,Turbulence Model, Wall Roughness,

Rotation

8.1 Introduction

Various attempts to analyze turbulence phenomena using available computer tech-

niques have been made by many researchers. During the past three decades, direct

numerical simulation (DNS) of three-dimensional time-dependent turbulent ﬂows