Research Article Numerical Simulation on the Effect of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 170317 8 pageshttpdxdoiorg1011552013170317

Research ArticleNumerical Simulation on the Effect of Turbulence Models onImpingement Cooling of Double Chamber Model

Zhenglei Yu Tao Xu Junlou Li Hang Xiu and Yun Li

College of Mechanical Science amp Engineering Jilin University Changchun 130025 China

Correspondence should be addressed to Yun Li yunjlusinacom

Received 5 August 2013 Revised 17 October 2013 Accepted 17 October 2013

Academic Editor Waqar Khan

Copyright copy 2013 Zhenglei Yu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Investigation of the effects of impingement cooling for the different turbulence models and study of the aerodynamic behaviorof a simplified transition piece model (TP) are the two themes of this paper A model (double chamber model) of a one-fourthcylinder is designed which could simulate the transition piece structure and performance The relative strengths and drawbacksof renormalization group theory 119896 minus 120576 (RNG) the realizable 119896 minus 120576 (RKE) the V2 minus 119891 the shear stress transport 119896 minus 120596 (SST) andlarge-eddy simulation (LES) models are used to solve the closure problem The prediction of the inner wall temperature coolingeffectiveness and velocitymagnitude contours in various conditions are compared in five different turbulencemodels Surprisinglythe V2 minus 119891 and SST models can produce even better predictions of fluid properties in impinging jet flows It is recommended as thebest compromise between solution speed and accuracy

1 Introduction

Given the large number of sustained operational hoursrequired for industrial turbines two important demandsplaced on such engines are component life and overall engineperformance These demands are somewhat conflictingbecause high temperatures are required at turbine combustorin order to achieve high performance however increasingcombustor outlet temperature in turn reduced componentlife high repair costs and downtime costs Impingementcooling is an enhanced heat transfer method capable ofcooling a transition piece (TP) without injecting cool airdirectly into the gas chamber Cooling the transition piecefrom the gas inlet enables engineers to dissipate the heatload andmaintainsmore uniform temperatures in the turbineregion needed for efficient turbine [1]

In the example of turbine cooling applications [2]impinging jet flows may be used to cool several differentsections of the engine such as the combustor case (combustorcan walls) turbine caseliner and the critical high tempera-ture turbine blades General applications and performance ofimpinging jets had been discussed in a number of reviews[3ndash6] The jet impingement angle has an effect on heattransfer and was studied frequently [3 4] Goppert et al [5]

investigated the effects of an unstable precessing jet over afixed target plate Hwang et al [6] altered the flow pattern inthe initial shearing layer by using coaxial jets

Numerical modeling of impinging jet flows and heattransfer is employed widely for prediction sensitivity anal-ysis and device design Finite element finite difference andfinite volume computational fluid dynamics (CFD)models ofimpinging jets have succeeded in making rough predictionsof heat transfer coefficients and velocity fields Turbulentimpinging jet CFDemploys practically all available numericalmethods that will be critically reviewed in the followingsections

An earlier critical review of this topic was conducted byPolat et al [7] in 1989 Since that date the variety of numericalmodels have been established and computational research inpredicting the physical behavior of impinging jets Tzeng et al[8] compared seven low Re modifications of the 119896 minus 120576 modelusing a confined turbulent slot jet array problem with threeadjacent jets at 119867119861 = 1 Heck et al [9] showed the RNGmodel that provided a closematch ofNu in thewall-jet regionbut an error up to 10 in the stagnation region Modelingby Cziesla et al [10] demonstrated the ability of LES topredict local Nu under a slot jet within 10 of experimentalmeasurements Silieti et al [11] investigated the numerical

2 Mathematical Problems in Engineering

prediction of cooling effectiveness in 2D and 3D gas turbineend wallsshrouds for the cases of conjugate and adiabaticheat transfer models They considered different cooling holegeometries that is cooling slots cylindrical and fan-shapedcooling holes at different blowing ratios They incorporatedthe effects of different turbulence models in predicting thesurface temperature and hence the cooling effectiveness

There is a great interest in the application of impingementcooling to protect the transition piece from high temperaturegas streams The model created in the paper looks like aquarter torus with the curved double chambers simulatingthe structure of the transition piece The relative strengthsand drawbacks of renormalization group theory 119896 minus 120576 model(RNG) the realizable 119896 minus 120576 model (RKE) the V2 minus 119891 modelthe shear stress transport 119896 minus 120596 model (SST) and large-eddysimulationmodel (LES) for impinging jet flow and heat trans-fer are compared As well the velocity and temperature fieldsin addition to centerline and two-dimensional impingementcooling effectiveness will be presented

2 Turbulence Models

21 The Renormalization Group Theory 119896 minus 120576 Model (RNG)The RNG 119896 minus 120576 turbulence model is derived from theinstantaneousNavier-Stokes equations using amathematicaltechnique called renormalization group (RNG) methodsborrowed from quantum mechanics The analytical deriva-tion results in a model with constants different from those inthe standard 119896minus120576model and it results in additional terms andfunctions in the transport equations for the turbulent kineticenergy (119896) and for dissipation rate (120576) Amore comprehensivedescription of RNG theory and its application to turbulencecan be found in [12 13] The governing equations for thismodel are as follows

119896 equation

120588119863119896

119863119905=

120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 1205831199051198782

minus 120588120576 minus 119892119894

120583119905

120588Pr119896

120597120588

120597119909119894

minus 2120588120576119896

120574119877119879

(1)

120576 equation

120588119863120576

119863119905=

120597

120597119909119895

[(120572120591120583eff)

120597120576

120597119909119895

] +120576

119896(1198621120591

1205831205911198782

minus 120588120576119862lowast

2120591) (2)

where119863119863119905 is the convective (or substantial time) derivative119863

119863119905=

120597

120597119905+ sdot nabla (3)

and 119878 is related to themean strain tensor 119878119894119895

= (12)(120597119906119894119909119895

+

120597119906119895119909119894) as 119878 = radic2119878

119894119895119878119895119894

1198621120591

= 142 1198622120591

= 168 119862120583

=

00845 120578 = 119878(119896120576) 120583119905

= 120588119862120583(1198962120576) and 119862

lowast

2120591= 1198622120591

+

((1198621205831205881205783(1 minus (120578

11205780)))(1 + 120573120578

3))

22 The Realizable 119896 minus 120576 Model (RKE) The term realizablemeans that the model satisfies certain mathematical con-straints on the normal stresses consistent with the physics

of turbulent flows In this model the 119896 equation is the sameas in RNG model however 119862

120583is not a constant and varies

as a function of mean velocity field and turbulence (009 inlog-layer 119878(119896120576) = 33 005 in shear layer of 119878(119896120576) = 6)The equation is based on a transport equation for the mean-square vorticity fluctuation [14] as

120588119863120576

119863119905=

120597

120597119909119895

[(120583 +120583119905

120590)

120597120576

120597119909119895

] + 1198621119878120588120576 minus 119862

2

1205881205762

119896 + radicV119890 (4)

where 1198621

= max[043 120578(120578 + 1)] and 1198622

= 10 This model isdesigned to avoid unphysical solutions in the flow field

23 The V2 minus 119891 Model Durbinrsquos V2 minus 119891 model also knownas the ldquonormal velocity relaxation modelrdquo has shown someof the best predictions to date with calculated Nu valuesfalling within the spread of experimental data [15] The V2 minus

119891 model uses an eddy viscosity to increase stability withtwo additional differential equations beyond those of the119896 minus 120576 model forming a four-equation model The additionalequations are defined as

119863119896

119863119905=

120597

120597119909119895

[(V + V1015840)120597119896

120597119909119895

] + 2V1015840119878119894119895

119878119894119895

minus 120576

V1015840 = 119862120583V2119879scale

119863120576

119863119905=

1198881015840

12057612V1015840119878119894119895

119878119894119895

minus 1198881205762

120576

119879scale+

120597

120597119909119895

[(V +V1015840

120590120576

)120597120576

120597119909119895

]

119863V2

119863119905= 119896119891wall minus V2

120576

119896+

120597

120597119909119895

[(V + V1015840)120597V2

120597119909119895

]

119891wall minus 1198712

scale120597

120597119909119895

120597119891

120597119909119895

=(1198621

minus 1) ((23) minus (V2119896))

119879scale+

11986222V1015840119878119894119895

119878119894119895

119896

119871 scale = 119862119871max(min(

11989632

120576

1

radic3

11989632

V2119862120583radic2119878119894119895

119878119894119895

)

119862120583(V3

120576)

14

)

119879scale = min(max(119896

120576 6radic

V120576

) 120572

radic3

119896

V2119862120583radic2119878119894119895

119878119894119895

)

1198881015840

1205761= 144 (1 + 0045radic

119896

V2)

(5)

where 119862120583

= 019 1198881205762

= 19 120590120576

= 13 1198621

= 14 1198622

= 03119862120578

= 700 119862119871

= 03 and 120572 = 06 Similar equations exist forpredicting the transport of a scalar (eg thermal energy) witha Pr1015840 as a funtion of V and V1015840

Mathematical Problems in Engineering 3

Coolant flow

Mainstream

Figure 1 Impingement-cooling concave model

24 The Shear Stress Transport 119896 minus 120596 Model (SST) There aretwo major ways in which the SST model differs from thestandard 119896minus120596 modelThe first is the gradual change from thestandard 119896minus120596model in the inner region of the boundary layerto a high Reynolds-number version of the 119896 minus 120576 model in theouter part of the boundary layer The second is the modifiedturbulent viscosity formulation to account for the transporteffects of the principal turbulent shear stress The SST (shearstress transport) model consists of the zonal (blended) 119896 minus

120596119896 minus 120576 equations and clips of turbulent viscosity so thatturbulent stress stays within what is dictated by structuralsimilarity constantThe 119896 equation is the same as the standard119896 minus 120596 model whereas the resulting blended equation for 120596 is[16]

120588119863120596

119863119905=

120574

V119905

120591119894119895

120597119880119894

120597119909119895

minus 1205881205731205962

+120597

120597119909119895

[(120583 +120583120591

120590120596

)120597120596

120597119909119895

]

+ 2120588 (1 minus 1198651) 1205901205962

1

120596

120597119896

120597119909119895

120597120596

120597119909119895

(6)

where1198651

= 1 in the inner layer and1198651

rarr 0 in the outer layerand 120590

1205962= 1168

25 Large-Eddy Simulation (LES) Large-eddy simulation(LES) is a technique intermediate between the direct simu-lation of turbulent flows and the solution of the Reynolds-averaged equations In LES the contribution of the largeenergy-carrying structures to momentum and energy trans-fer is computed exactly well only the effect of the smallestscales of turbulence is modeled Since the small scales tend tobe more homogeneous and universal and less affected by theboundary conditions than the large ones there is hope thattheir models can be simpler and require fewer adjustmentswhen applied to different flows than similar models for theRANS equations [17 18]

3 Computational ModelBoundary Conditions and Grid

31 Model Description Transition piece develops heat trans-formation on both internal and external walls to elimi-nate resonant frequency concerns As well the transitionpiece conducts gas flow directly from the correspondingcombustion liners toward the first stage of the gas turbine

Mainstream inlet

Uniform

Coolant inlet

Coolant outlet

Mainstream outlet

Closed

Inner surface

Outer surface

6 times 1206011026

525

68

1050

38162Y

X

Z

(a)

Closed

Mainstream

Coolant flow

Mainstream outlet

Mainstream inlet

Coolant outlet

Coolant flow and droplet inlet Outer

wall

Inner wall

YX

Z

(b)

Figure 2 Computational domain showing boundary conditions

(stator) The discrete coolant jets forming a protective filmchamber on the side of transition piece are drawn from theupstream compressor in an operational gas turbine engineFrom the supply plenum the coolant is ejected through theseveral rows of discrete holes over the external boundarylayer against the local high thermal conduction on the otherside of the transition pieces [19] The model as a one-fourth cylinder could simulate the transition piecersquos structureand performance to elaborate the turbulence effect on theimpingement-cooling performance over a concave surface(see Figure 1) [4]

A schematic diagram of the flow domain along withboundary conditions and dimensions is given in Figure 2(a)As shown in the figure the one-fourth cylinder model hastwo layers of chambers with a length of 1050mm and anouter radius and an inner radius of 200mm and 162mmrespectively In the diagram one side of the outer chambercalled the coolant chamber is closed contrarily both sides ofthe mainstream chamber as the inner chamber are unfoldingin which gas could flow through from one side to the otherThere are 18 holes distributed uniformly in three rows on thesurface of the outer wall The distance between the two rowsis 68mm and the diameter of all the holes is about 1026mmIn this scenario the velocity of the coolant flow is set at 6msthe temperatures of the coolant flow and themainstream floware set as 300K and 1300K respectively


Table 1 Boundary conditions

Component Boundary conditions Magnitude

Mainstream inlet

Mass flux rate 3146 (kgs)Gas temperature 1300 (K)Turbulent intensity 5 ()Hydraulic diameter 0324 (m)

Mainstream outlet

Pressure 1512 (MPa)Turbulent intensity 5 ()Hydraulic diameter 0324 (m)Convection coefficient 10 (Wm2K)

Coolant chamber

Air temperature 300 (K)Pressure 14552 (MPa)Pressure recovery coefficient 095Turbulent intensity 5 ()Hydraulic diameter 001026 (m)

32 Boundary Conditions Boundary conditions are appliedto specific faces within the domain to specify the flow andthermal variables that dictate conditions within the modelFigure 2(a) discloses the boundary conditions used in themodel in which the cooling air and gas are moving alongrespectively in the two layers of chambers in opposite direc-tions In the cooling chamber the simulation is performed byusing air as the cooling flow while velocity and temperatureare set on the jet holes with pressure on the exit In anotherchamber it is assumed that the mainstream is a mixture ofO2 H2O CO

2 N2 and some rare gasesThemodel created is

considered as boundary condition (Table 1) [4] Assumptionof the solid wall of the quarter torus is formed with ahypothesis of negligible thermal resistance by conduction thethermal properties of the material were considered byNimonic 263

33 Meshing and Simulation Procedures The computationaldomain incorporates the model the HEXAmesh in the soft-ware ICEMCFD used to generate the structuredmultiblockand the body-fitted grid system In this study the grid systemassociated with the parts of the mainstream and the coolantsupply plenum is H-type Figure 3 shows the grids of thecomputational domain and the total number of the cells forthe 3D domain is 776828 Local grid refinement is used nearthe hole regions

The wall 119884-plus (7) is a nondimensional number similarto local Reynolds number determining whether the influ-ences in the wall-adjacent cells are laminar or turbulenthence indicating the part of the turbulent boundary layer thatthey resolve

119910+

=119906120591air119910

Vair (7)

The subdivisions of the near-wall region in a turbulentboundary layer can be summarized as follows [20]

(a) 119910+

lt 5 in the viscous sublayer region (velocity pro-files are assumed to be laminar and viscous stressdominates the wall shear)

Z X

Y

Figure 3 Meshes

020406080

100120140160180200220240260280300

0 200 400 600 800 1000Z (mm)

Y-p

lus

RNG2 minus f

RKE

SSTLES

Figure 4 119884-plus distribution on the inner wall surface

(b) 5 lt 119910+

lt 30 buffer region (both viscous and turbu-lent shear dominate)

(c) 30 lt 119910+

lt 300 fully turbulent portion or log-lawregion (corresponds to the region where turbulentshear predominates)

As it can be seen in Figure 4 for all cases all nodes on theinner wall surface have the 119884-plus value smaller than 300

This study is using a commercial CFD code based on thecontrol-volume method ANSYS-FLUENT 12016 which inorder to predict temperature impingement-cooling effective-ness and velocity fields All runs were made on a PC clusterwith sixteen Pentium-4 30GHz personal computers Theconvergence criteria of the steady-state solution are judged bythe reduction in the mass residual by a factor of 6 typicallyin 2000 iterations


RNG

RKE

SST

LES

2 minus f

1300e + 003

1280e + 003

1260e + 003

1240e + 003

1220e + 003

1200e + 003

1180e + 003

1160e + 003

1140e + 003

1120e + 003

1100e + 003

1080e + 003

1060e + 003

1040e + 003

1020e + 003

1000e + 003(K)

Temperature contour 1

Y

X

Z

(a)

10401060108011001120114011601180120012201240126012801300

Tem

pera

ture

(K)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)

Figure 5 Comparative analysis of inner wall temperature using different turbulence models (a) temperature distribution of inner wall (b)temperature along the center line

4 Results and Discussion

41 Inner Wall Temperature Figure 5 shows the temperaturecontours predicted by five turbulence models The figureillustrates that the temperature at the starting point of thewall is high and then starts to go down The starting pointis cooled by the coolant holes at the curve 119885 = 430mmon the outer wall while the same temperature is maintainedthroughout the coolant holes After the coolant strikes theinner wall there are vortexes formed The jet impingementand the vortex formed out of the coolant flow cool the surfaceof the inner wall The delicate color region (orange yellowand green region) is approximately the region where thecoolant strikes the inner wall after being reflected by thecoolant holes in which the inner wall is cooled purely byimpingement cooling These figures confirm that the V2 minus 119891

model has well simulation of heat transfer in the doublechamber model The calculated temperature at the centerline of the inner wall is presented in Figure 5(b) there isa difference among the predictions of SST LES and RKEmodels Both the temperature contours and the color lineshave the same trend in the five turbulence models whereasthe calculated temperatures between 119885 = 400 and 600mmare diverse for all turbulence models All turbulence modelscontinue to predict similar wall temperatures while the RNGmodel underpredicts the temperature by up to 5ndash8 in thesame location

42 Cooling Effectiveness To define cooling effectiveness thesurface temperature downstream of the cooling hole has tobe measured The adiabatic cooling effectiveness (120578) is usedto examine the performance of cooling The definition of 120578 is

120578 =119879119898

minus 119879aw119879119898

minus 119879119888

(8)

where 119879119898is the mainstream hot gas inlet temperature which

is a fixed value for calculation of the adiabatic coolingeffectiveness of any location and 119879

119888is the temperature of the

coolant which is assigned as a constant of 300K in this issue119879aw is the adiabatic wall temperature [21]

Figures 6(a) and 6(b) exhibit the comparison of coolingeffectiveness on distribution and averaged cooling effective-ness along the 119911-axis respectively It could be concludedthat the effectiveness is high at the beginning of the filmwhile decreasing gradually in downstream The distributionof the cooling effectiveness of the V2 minus 119891 and SST modelcases is significantly different The centerline effectiveness byusing the five turbulencemodel cases is shown in Figure 6(b)Overall particularly the V2 minus 119891 model gave better resultscompared to other models Values of cooling effectivenesscompare with the RNGmodel case there is an approximately30 difference in V2 minus 119891 model case


Y

X

Z

2800e minus 001

2613e minus 001

2427e minus 001

2240e minus 001

2053e minus 001

1867e minus 001

1680e minus 001

1493e minus 001

1307e minus 001

1120e minus 001

9333e minus 002

7467e minus 002

5600e minus 002

3733e minus 002

1867e minus 002

0000e minus 000

RNG

RKE

SST

LES

2 minus f

(a)

000

005

010

015

020

025

030

Coo

ling

effec

tiven

ess

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)

Figure 6 Comparative analysis of cooling effectiveness in five turbulence models (a) cooling effectiveness distribution of inner wall (b)averaged cooling effectiveness

43 Characteristics of the Flow Field Since the thermal fieldof a jet-in-crossflow interaction is dictated by the hydrody-namics the flowfield resultswere predicted by five turbulencemodels Figure 7(a) shows the velocity contours at the sectionplane of coolant chamber As it can be seen all modelspredicted the low momentum region along the downstreamedge and the corresponding high momentum or jettingregion along the upstream edge within the cooling holesThepredicted reattachment region by SST and V2 minus 119891 model casesis approximately at the inlet of the cooling chamber whereasLES RKE and RNG predict a larger separation bubble andthe flow separates behind the cooling holes The computednear inner wall velocity contours (ms) along the centerlineplane are shown in Figure 7(b) where the turbulence closurewas simulated using the five different turbulence models Allpredictions are extremely close to each other with a skewedupstream velocity profile

The actual computational cost will of course vary withmodel complexity and computing power With the parallelcomputing resources of a desktop computer available at thetime of writing six Pentium-4 typically 3GHz processorsfor a high-resolution two-dimensional problem the steadytime-averaged eddy viscosity models (RNG RKE) will havecomputation times of a few hours (05ndash15) In comparisonthe more complex SST and V2 minus 119891 could take 2ndash4 hours

depending on how smoothly the model converges Based onrecent work unsteady LESmodels have computation times atleast two orders of magnitude higher a well-resolved three-dimensional LES impinging jet model could take a day toprovide a solution

5 Conclusion

Anumerical simulation has been performed to study the flowandheat transfer of impinging cooling on the double chambermodel and a comparative study indicating the ability of fiveturbulence models is presented The research of turbulencemodel tasks is important to improve the design and resultingperformance of impinging jets

During this investigation numerical simulation isimpacted with five turbulence models which has somepractical value for real processing and guiding significance fortheory To date the SST and V2minus119891models offer the best resultsfor the least amount of computation time It is very importantto consider the effect of heat conduction within the metalon the predictions of an accurate surface temperature andhence impingement-cooling effectiveness The validationof the present study has confirmed angle cases and willbe employed in future studies of impingement-coolingparameters optimization


RNG

RKE

SST

LES

2 minus f

XY

Z

Velocityvelocity contour

(m sminus1)

2450e + 0022287e + 0022123e + 0021960e + 0021797e + 0021633e + 0021470e + 0021307e + 0021143e + 0029800e + 0018167e + 0016533e + 0014900e + 0013267e + 0011633e + 0010000e + 000

(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)

Figure 7 Coolant velocity contours predicted by various turbulencemodels (a)119883-119884 cross-section velocity distribution (b) averaged coolantvelocity

Nomenclature

119863119886 Diameter of coolant chamber

119863119892 Diameter of mainstream chamber

119871 Length of the model119879 Absolute static temperature119883 119884 119885 Nondimensional coordinates in diameter

spanwise and mainstream directions

Greek Symbols

120578 Cooling effectiveness

Suffixes

119898 Mainstream flow119888 Coolant flowaw Adiabatic wall

Conflict of Interests

The authors declare that there is no conflict of interests

Acknowledgment

This research is supported by the Technology Developmentof Jilin Province (no 20126001)

References

[1] J C Han S Dutta and S V Ekkad Gas Turbine Heat Transferand Cooling Technology Taylor and Francis New York NYUSA 2000

[2] N Zuckerman and N Lior ldquoImpingement heat transfer corre-lations and numerical modelingrdquo Journal of Heat Transfer vol127 no 5 pp 544ndash552 2005

[3] A A Tawfek ldquoHeat transfer studies of the oblique impingementof round jets upon a curved surfacerdquo Heat and Mass Transfervol 38 no 6 pp 467ndash475 2002

[4] Z L Yu T Xu J L Li L Ma and T S Xu ldquoComparisonof a series of double chamber model with various hole anglesfor enhancing cooling effectivenessrdquo International Communica-tions in Heat and Mass Transfer vol 44 pp 38ndash44 2013

[5] S Goppert T Gurtler HMocikat andH Herwig ldquoHeat trans-fer under a precessing jet effects of unsteady jet impingementrdquoInternational Journal of Heat and Mass Transfer vol 47 no 12-13 pp 2795ndash2806 2004

[6] S D Hwang C H Lee and H H Cho ldquoHeat transfer and flowstructures in axisymmetric impinging jet controlled by vortexpairingrdquo International Journal of Heat and Fluid Flow vol 22no 3 pp 293ndash300 2001

[7] S Polat B Huang A S Mujumdar and W J M DouglasldquoNumerical flow and heat transfer under impinging jets areviewrdquo Annual Review of Numerical Fluid Mechanics and HeatTransfer vol 2 pp 157ndash197 1989

[8] P Y Tzeng C Y Soong and C D Hsieh ldquoNumerical investi-gation of heat transfer under confined impinging turbulent slotjetsrdquoNumericalHeat TransferA vol 35 no 8 pp 903ndash924 1999


[9] U Heck U Fritsching and K Bauckhage ldquoFluid flow and heattransfer in gas jet quenching of a cylinderrdquo International Journalof Numerical Methods for Heat and Fluid Flow vol 11 no 1 pp36ndash49 2001

[10] T Cziesla G Biswas H Chattopadhyay and N K MitraldquoLarge-eddy simulation of flow and heat transfer in an imping-ing slot jetrdquo International Journal of Heat and Fluid Flow vol22 no 5 pp 500ndash508 2001

[11] M Silieti E Divo and A J Kassab ldquoFilm cooling effectivenessfrom a single scaled-up fan-shaped hole a CFD simulation ofadiabatic and conjugate heat transfer modelsrdquo in Proceedings oftheASMETurbo Expo Power for Land Sea andAir pp 431ndash441June 2005 paper no GT2005-68431

[12] S H Park S W Park S H Rhee S B Lee J-E Choi and S HKang ldquoInvestigation on the wall function implementation forthe prediction of ship resistancerdquo International Journal of NavalArchitecture andOcean Engineering vol 5 no 1 pp 33ndash46 2013

[13] S-E Kim and S H Rhee ldquoEfficient engineering predictionof turbulentwing tip vortex flowsrdquo Computer Modeling inEngineering and Sciences vol 62 no 3 pp 291ndash309 2010

[14] Z L Yu T Xu J L Li T S Xu and Y Tatsuo ldquoComputationalanalysis of dropletmass and size effect onmistair impingementcooling performancerdquoAdvances inMechanical Engineering vol2013 Article ID 181856 8 pages 2013

[15] M Behnia S Parneix Y Shabany and P A Durbin ldquoNumericalstudy of turbulent heat transfer in confined and unconfinedimpinging jetsrdquo International Journal of Heat and Fluid Flowvol 20 no 1 pp 1ndash9 1999

[16] T H Park H G Choi J Y Yoo and S J Kim ldquoStreamlineupwind numerical simulation of two-dimensional confinedimpinging slot jetsrdquo International Journal of Heat and MassTransfer vol 46 no 2 pp 251ndash262 2003

[17] Y Huang S Wang and V Yang ldquoSystematic analysis of lean-premixed swirl-stabilized combustionrdquo AIAA Journal vol 44no 4 pp 724ndash740 2006

[18] V Moureau P Domingo and L Vervisch ldquoFrom large-eddysimulation to direct numerical simulation of a lean premixedswirl flame filtered laminar flame-PDFmodelingrdquo Combustionand Flame vol 158 no 7 pp 1340ndash1357 2011

[19] J Benoit C Johnston and M Zingg ldquoEnhancing gas turbinepower plant profitability chronic transition piece and turbinepart failures in some 501F gas turbines led to a replacement partredesignrdquo Power Engineering vol 111 no 11 pp 140ndash144 2007

[20] MMoshfeghi Y J Song andYHXie ldquoEffects of near-wall gridspacing on SST-K-120596 model using NREL Phase VI horizontalaxis wind turbinerdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 94ndash105 2012

[21] H Wu J Wang and Z Tao ldquoEvaluation of predicted heattransfer on a transonic vane using V2 minus 119891 turbulence modelsrdquoJournal ofThermal Science and Technology vol 6 no 3 pp 424ndash435 2011

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


prediction of cooling effectiveness in 2D and 3D gas turbineend wallsshrouds for the cases of conjugate and adiabaticheat transfer models They considered different cooling holegeometries that is cooling slots cylindrical and fan-shapedcooling holes at different blowing ratios They incorporatedthe effects of different turbulence models in predicting thesurface temperature and hence the cooling effectiveness

There is a great interest in the application of impingementcooling to protect the transition piece from high temperaturegas streams The model created in the paper looks like aquarter torus with the curved double chambers simulatingthe structure of the transition piece The relative strengthsand drawbacks of renormalization group theory 119896 minus 120576 model(RNG) the realizable 119896 minus 120576 model (RKE) the V2 minus 119891 modelthe shear stress transport 119896 minus 120596 model (SST) and large-eddysimulationmodel (LES) for impinging jet flow and heat trans-fer are compared As well the velocity and temperature fieldsin addition to centerline and two-dimensional impingementcooling effectiveness will be presented

2 Turbulence Models

21 The Renormalization Group Theory 119896 minus 120576 Model (RNG)The RNG 119896 minus 120576 turbulence model is derived from theinstantaneousNavier-Stokes equations using amathematicaltechnique called renormalization group (RNG) methodsborrowed from quantum mechanics The analytical deriva-tion results in a model with constants different from those inthe standard 119896minus120576model and it results in additional terms andfunctions in the transport equations for the turbulent kineticenergy (119896) and for dissipation rate (120576) Amore comprehensivedescription of RNG theory and its application to turbulencecan be found in [12 13] The governing equations for thismodel are as follows

119896 equation

120588119863119896

119863119905=

120597

120597119909119895

[(120583 +120583119905

120590119896

)120597119896

120597119909119895

] + 1205831199051198782

minus 120588120576 minus 119892119894

120583119905

120588Pr119896

120597120588

120597119909119894

minus 2120588120576119896

120574119877119879

(1)

120576 equation

120588119863120576

119863119905=

120597

120597119909119895

[(120572120591120583eff)

120597120576

120597119909119895

] +120576

119896(1198621120591

1205831205911198782

minus 120588120576119862lowast

2120591) (2)

where119863119863119905 is the convective (or substantial time) derivative119863

119863119905=

120597

120597119905+ sdot nabla (3)

and 119878 is related to themean strain tensor 119878119894119895

= (12)(120597119906119894119909119895

+

120597119906119895119909119894) as 119878 = radic2119878

119894119895119878119895119894

1198621120591

= 142 1198622120591

= 168 119862120583

=

00845 120578 = 119878(119896120576) 120583119905

= 120588119862120583(1198962120576) and 119862

lowast

2120591= 1198622120591

+

((1198621205831205881205783(1 minus (120578

11205780)))(1 + 120573120578

3))

22 The Realizable 119896 minus 120576 Model (RKE) The term realizablemeans that the model satisfies certain mathematical con-straints on the normal stresses consistent with the physics

of turbulent flows In this model the 119896 equation is the sameas in RNG model however 119862

120583is not a constant and varies

as a function of mean velocity field and turbulence (009 inlog-layer 119878(119896120576) = 33 005 in shear layer of 119878(119896120576) = 6)The equation is based on a transport equation for the mean-square vorticity fluctuation [14] as

120588119863120576

119863119905=

120597

120597119909119895

[(120583 +120583119905

120590)

120597120576

120597119909119895

] + 1198621119878120588120576 minus 119862

2

1205881205762

119896 + radicV119890 (4)

where 1198621

= max[043 120578(120578 + 1)] and 1198622

= 10 This model isdesigned to avoid unphysical solutions in the flow field

23 The V2 minus 119891 Model Durbinrsquos V2 minus 119891 model also knownas the ldquonormal velocity relaxation modelrdquo has shown someof the best predictions to date with calculated Nu valuesfalling within the spread of experimental data [15] The V2 minus

119891 model uses an eddy viscosity to increase stability withtwo additional differential equations beyond those of the119896 minus 120576 model forming a four-equation model The additionalequations are defined as

119863119896

119863119905=

120597

120597119909119895

[(V + V1015840)120597119896

120597119909119895

] + 2V1015840119878119894119895

119878119894119895

minus 120576

V1015840 = 119862120583V2119879scale

119863120576

119863119905=

1198881015840

12057612V1015840119878119894119895

119878119894119895

minus 1198881205762

120576

119879scale+

120597

120597119909119895

[(V +V1015840

120590120576

)120597120576

120597119909119895

]

119863V2

119863119905= 119896119891wall minus V2

120576

119896+

120597

120597119909119895

[(V + V1015840)120597V2

120597119909119895

]

119891wall minus 1198712

scale120597

120597119909119895

120597119891

120597119909119895

=(1198621

minus 1) ((23) minus (V2119896))

119879scale+

11986222V1015840119878119894119895

119878119894119895

119896

119871 scale = 119862119871max(min(

11989632

120576

1

radic3

11989632

V2119862120583radic2119878119894119895

119878119894119895

)

119862120583(V3

120576)

14

)

119879scale = min(max(119896

120576 6radic

V120576

) 120572

radic3

119896

V2119862120583radic2119878119894119895

119878119894119895

)

1198881015840

1205761= 144 (1 + 0045radic

119896

V2)

(5)

where 119862120583

= 019 1198881205762

= 19 120590120576

= 13 1198621

= 14 1198622

= 03119862120578

= 700 119862119871

= 03 and 120572 = 06 Similar equations exist forpredicting the transport of a scalar (eg thermal energy) witha Pr1015840 as a funtion of V and V1015840


Coolant flow

Mainstream




120588119863120596

119863119905=

120574

V119905

120591119894119895

120597119880119894

120597119909119895

minus 1205881205731205962

+120597

120597119909119895

[(120583 +120583120591

120590120596

)120597120596

120597119909119895

]

+ 2120588 (1 minus 1198651) 1205901205962

1

120596

120597119896

120597119909119895

120597120596

120597119909119895

(6)

where1198651



1205962= 1168




Mainstream inlet

Uniform

Coolant inlet

Coolant outlet

Mainstream outlet

Closed

Inner surface

Outer surface

6 times 1206011026

525

68

1050

38162Y

X

Z

(a)

Closed

Mainstream

Coolant flow

Mainstream outlet

Mainstream inlet

Coolant outlet


wall

Inner wall

YX

Z

(b)







Mainstream inlet


Mainstream outlet


Coolant chamber







119910+

=119906120591air119910

Vair (7)


(a) 119910+


Z X

Y

Figure 3 Meshes

020406080

100120140160180200220240260280300

0 200 400 600 800 1000Z (mm)

Y-p

lus

RNG2 minus f

RKE

SSTLES


(b) 5 lt 119910+


(c) 30 lt 119910+





RNG

RKE

SST

LES

2 minus f

1300e + 003

1280e + 003

1260e + 003

1240e + 003

1220e + 003

1200e + 003

1180e + 003

1160e + 003

1140e + 003

1120e + 003

1100e + 003

1080e + 003

1060e + 003

1040e + 003

1020e + 003

1000e + 003(K)


Y

X

Z

(a)

10401060108011001120114011601180120012201240126012801300

Tem

pera

ture

(K)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)






120578 =119879119898

minus 119879aw119879119898

minus 119879119888

(8)







Y

X

Z

2800e minus 001

2613e minus 001

2427e minus 001

2240e minus 001

2053e minus 001

1867e minus 001

1680e minus 001

1493e minus 001

1307e minus 001

1120e minus 001

9333e minus 002

7467e minus 002

5600e minus 002

3733e minus 002

1867e minus 002

0000e minus 000

RNG

RKE

SST

LES

2 minus f

(a)

000

005

010

015

020

025

030

Coo

ling

effec

tiven

ess

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)





5 Conclusion




RNG

RKE

SST

LES

2 minus f

XY

Z


(m sminus1)


(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)


Nomenclature





Greek Symbols


Suffixes




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Coolant flow

Mainstream




120588119863120596

119863119905=

120574

V119905

120591119894119895

120597119880119894

120597119909119895

minus 1205881205731205962

+120597

120597119909119895

[(120583 +120583120591

120590120596

)120597120596

120597119909119895

]

+ 2120588 (1 minus 1198651) 1205901205962

1

120596

120597119896

120597119909119895

120597120596

120597119909119895

(6)

where1198651



1205962= 1168




Mainstream inlet

Uniform

Coolant inlet

Coolant outlet

Mainstream outlet

Closed

Inner surface

Outer surface

6 times 1206011026

525

68

1050

38162Y

X

Z

(a)

Closed

Mainstream

Coolant flow

Mainstream outlet

Mainstream inlet

Coolant outlet


wall

Inner wall

YX

Z

(b)







Mainstream inlet


Mainstream outlet


Coolant chamber







119910+

=119906120591air119910

Vair (7)


(a) 119910+


Z X

Y

Figure 3 Meshes

020406080

100120140160180200220240260280300

0 200 400 600 800 1000Z (mm)

Y-p

lus

RNG2 minus f

RKE

SSTLES


(b) 5 lt 119910+


(c) 30 lt 119910+





RNG

RKE

SST

LES

2 minus f

1300e + 003

1280e + 003

1260e + 003

1240e + 003

1220e + 003

1200e + 003

1180e + 003

1160e + 003

1140e + 003

1120e + 003

1100e + 003

1080e + 003

1060e + 003

1040e + 003

1020e + 003

1000e + 003(K)


Y

X

Z

(a)

10401060108011001120114011601180120012201240126012801300

Tem

pera

ture

(K)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)






120578 =119879119898

minus 119879aw119879119898

minus 119879119888

(8)







Y

X

Z

2800e minus 001

2613e minus 001

2427e minus 001

2240e minus 001

2053e minus 001

1867e minus 001

1680e minus 001

1493e minus 001

1307e minus 001

1120e minus 001

9333e minus 002

7467e minus 002

5600e minus 002

3733e minus 002

1867e minus 002

0000e minus 000

RNG

RKE

SST

LES

2 minus f

(a)

000

005

010

015

020

025

030

Coo

ling

effec

tiven

ess

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)





5 Conclusion




RNG

RKE

SST

LES

2 minus f

XY

Z


(m sminus1)


(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)


Nomenclature





Greek Symbols


Suffixes




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Mainstream inlet


Mainstream outlet


Coolant chamber







119910+

=119906120591air119910

Vair (7)


(a) 119910+


Z X

Y

Figure 3 Meshes

020406080

100120140160180200220240260280300

0 200 400 600 800 1000Z (mm)

Y-p

lus

RNG2 minus f

RKE

SSTLES


(b) 5 lt 119910+


(c) 30 lt 119910+





RNG

RKE

SST

LES

2 minus f

1300e + 003

1280e + 003

1260e + 003

1240e + 003

1220e + 003

1200e + 003

1180e + 003

1160e + 003

1140e + 003

1120e + 003

1100e + 003

1080e + 003

1060e + 003

1040e + 003

1020e + 003

1000e + 003(K)


Y

X

Z

(a)

10401060108011001120114011601180120012201240126012801300

Tem

pera

ture

(K)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)






120578 =119879119898

minus 119879aw119879119898

minus 119879119888

(8)







Y

X

Z

2800e minus 001

2613e minus 001

2427e minus 001

2240e minus 001

2053e minus 001

1867e minus 001

1680e minus 001

1493e minus 001

1307e minus 001

1120e minus 001

9333e minus 002

7467e minus 002

5600e minus 002

3733e minus 002

1867e minus 002

0000e minus 000

RNG

RKE

SST

LES

2 minus f

(a)

000

005

010

015

020

025

030

Coo

ling

effec

tiven

ess

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)





5 Conclusion




RNG

RKE

SST

LES

2 minus f

XY

Z


(m sminus1)


(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)


Nomenclature





Greek Symbols


Suffixes




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










RNG

RKE

SST

LES

2 minus f

1300e + 003

1280e + 003

1260e + 003

1240e + 003

1220e + 003

1200e + 003

1180e + 003

1160e + 003

1140e + 003

1120e + 003

1100e + 003

1080e + 003

1060e + 003

1040e + 003

1020e + 003

1000e + 003(K)


Y

X

Z

(a)

10401060108011001120114011601180120012201240126012801300

Tem

pera

ture

(K)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)






120578 =119879119898

minus 119879aw119879119898

minus 119879119888

(8)







Y

X

Z

2800e minus 001

2613e minus 001

2427e minus 001

2240e minus 001

2053e minus 001

1867e minus 001

1680e minus 001

1493e minus 001

1307e minus 001

1120e minus 001

9333e minus 002

7467e minus 002

5600e minus 002

3733e minus 002

1867e minus 002

0000e minus 000

RNG

RKE

SST

LES

2 minus f

(a)

000

005

010

015

020

025

030

Coo

ling

effec

tiven

ess

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)





5 Conclusion




RNG

RKE

SST

LES

2 minus f

XY

Z


(m sminus1)


(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)


Nomenclature





Greek Symbols


Suffixes




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Y

X

Z

2800e minus 001

2613e minus 001

2427e minus 001

2240e minus 001

2053e minus 001

1867e minus 001

1680e minus 001

1493e minus 001

1307e minus 001

1120e minus 001

9333e minus 002

7467e minus 002

5600e minus 002

3733e minus 002

1867e minus 002

0000e minus 000

RNG

RKE

SST

LES

2 minus f

(a)

000

005

010

015

020

025

030

Coo

ling

effec

tiven

ess

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)





5 Conclusion




RNG

RKE

SST

LES

2 minus f

XY

Z


(m sminus1)


(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)


Nomenclature





Greek Symbols


Suffixes




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










RNG

RKE

SST

LES

2 minus f

XY

Z


(m sminus1)


(a)

0

25

50

75

100

125

150

Velo

city

(ms

)

0 200 400 600 800 1000Z (mm)

RNG

RKE

SSTLES2 minus f

(b)


Nomenclature





Greek Symbols


Suffixes




Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Numerical Simulation on the Effect of