Numerical simulations of square arrayed rod bundles

N

ÁA

a

ARRA

1

fltons

db1e(mimoRott

0d

Nuclear Engineering and Design 247 (2012) 168– 182

Contents lists available at SciVerse ScienceDirect

Nuclear Engineering and Design

jo u r n al hom epage : www.elsev ier .com/ locate /nucengdes

umerical simulations of square arrayed rod bundles

kos Horváth ∗, Bernd DresselREVA NP GmbH, Paul-Gossen-Str. 100, 91052 Erlangen, Germany

r t i c l e i n f o

rticle history:eceived 9 November 2011eceived in revised form 21 February 2012ccepted 2 March 2012

a b s t r a c t

Computational fluid dynamics (CFD) simulations were performed with square arrayed rod bundles featur-ing pitch to diameter (P/D) ratio of 1.194 and 1.326 in order to find an optimal mesh and turbulence modelfor simulations with more complex geometries in the future. With the tighter lattice a mesh sensitivityand turbulence model study were accomplished and the post processed turbulence quantities, velocityfield and wall shear stress were compared with experimental data (Hooper, 1980 Developed single phaseturbulent flow through a square-pitch rod cluster. Nuclear Engineering and Design 60, 365–379.). Thecomparisons show that Reynolds-Averaged Navier–Stokes method with the Reynolds stress model ofGibson and Launder in conjunction with an appropriate mesh can provide reasonable agreement with
the experiment for this lattice. For pure bundle simulations the body fitted structured meshes are sug-gested, since slightly better agreement can be captured considering all quantities with the same numberof cells. Based on the drawn conclusions the procession was repeated for P/D = 1.326, where, due to lack ofexperiment, just the correct tendencies of the turbulence quantities and velocity field were established.The results show Reynolds number independency correctly and the increase of P/D issues in more similarflow to axisymmetric pipe flow.
© 2012 Elsevier B.V. All rights reserved.

. Introduction

It has been known for decades that fully developed turbulentows in straight non-circular ducts differ from those of circularubes and parallel plates. Since the coolant flow inside rod bundlesf nuclear reactors is of such kind, a detailed understanding of itsature is essential and of great interest for many engineers andcientists.

From several experiments the presence of secondary motion,riven by the near wall turbulence anisotropy has been predictedoth for triangular and square arrayed rod bundles (Rowe et al.,974; Trupp and Azad, 1975; Hooper, 1980; Neti et al., 1982), how-ver it was measured for triangular arrangement first by Vonka1988). In addition Rowe et al. (1974) observed that a non-gradient

acroscopic flow process, maybe flow pulsation, affects the mix-ng between neighboring subchannels. For a long time these high

ixing rates have been explained by the secondary flow, but thebservation of Rowe et al. (1974) was confirmed by Hooper andehme (1984) and Möller (1991). They demonstrated that the sec-
ndary flows in subchannels do not contribute significantly tohe mixing rates and that the flow pulsation solely depends onhe geometry (upper threshold for P/D around 1.1) and Reynolds
∗ Corresponding author. Tel.: +49 9131 900 95572.E-mail address: [email protected] (Á. Horváth).

029-5493/$ – see front matter © 2012 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2012.03.004

number. A detailed summary of the flows inside rod bundles canbe found in the paper of Rehme (1992).

Following the development and quick growth in computationaltechnology at the end of 80s, the demand for precise numerical sim-ulation of such flows with the use of computational fluid dynamics(CFD) has been increased. However, direct numerical simula-tion (DNS) of high Reynolds number flows is still not an optionnowadays and the large eddy simulations (LES) for detailed geome-tries have many restrictions too. The usage of Reynolds-AveragedNavier–Stokes (RANS) and unsteady RANS (URANS) methods is theonly option for engineering applications. Most of the performedsimulations used RANS method searching for the correct turbu-lence model for various rod bundle geometries (Lee and Jang, 1997;In et al., 2003, 2004; Baglietto and Ninokata, 2005; Tóth and Aszódi,2010), and their common conclusion is that without modeling theanisotropy (eddy viscosity models) of the turbulent intensities,the agreement of the time mean quantities with experiments israther poor, meanwhile non-linear models give better, but not per-fect agreement. Recent studies with URANS (Chang and Tavoularis,2007; Merzari et al., 2008) and LES (Ikeno and Kajishima, 2010)for pure rod bundle flows for tight lattices (e.g. P/D = 1.06) high-lighted the possibility to simulate the flow pulsation between the
subchannels, the time-averaged velocity and turbulence quantitiesare in better agreement with the experimental data compared toRANS (Merzari et al., 2008). However, the necessity of a very smalltime step (order of 10−4 s) with URANS to simulate the flow field
dx.doi.org/10.1016/j.nucengdes.2012.03.004

http://www.sciencedirect.com/science/journal/00295493

http://www.elsevier.com/locate/nucengdes

mailto:[email protected]

dx.doi.org/10.1016/j.nucengdes.2012.03.004

Á. Horváth, B. Dressel / Nuclear Engineering and Design 247 (2012) 168– 182 169

Nomenclature

a1, C� constants in the equations for the turbulent viscos-ity

C1, C∗1, C1w, C2, C2wC1, C2wC2, C2wC3, C3, C∗

3, C4, C5, Cε1, Cε2, CS,Cεε, E constants in the model transport equations

bij anisotropy tensorD rod diameterDij diffusion rate of Reynolds stressF1, F2 blending functions in SST modelk turbulent kinetic energyln normal distance to the wall�n unit vector normal to the wallNi number of cells per plane of the ith meshP rod pitchPij production rate of Reynolds stressPk production rate of turbulent kinetic energyRe Reynolds numberreff effective grid refinement ratioSij mean strain tensoruiuj Reynolds stressesu� friction velocityUbulk bulk velocityUi time mean velocity componentymax wall distance to duct center liney wall distancey+ dimensionless wall distance

Greek letters˛ angle of evaluation line with respect to rod gapˇ, ˇ∗, ˇ1, ˇ2, ˇ∗

1, ˇ∗2, �, �0, �k, �k1

ω, �k2ω, �h, �ω, �ω1

ω, �ω2,�ω2

ω constants in the model transport equationsıij Kronecker deltaε dissipation rate of turbulent kinetic energyεij dissipation tensor� von Karman constant� dynamic viscosity�t dynamic turbulent viscosityt turbulent viscosity density�w(˛) local wall shear stress�aver

w average wall shear stress of 1/4th segment of a rodω specific dissipation rate of turbulent kinetic energy˚ pressure–strain correlation tensor

pp

mapaacwarg

pn(t

Fig. 1. Coordinate system and evaluation line used for comparison ( = 45◦ and 22◦)with experiment; (Hooper reported measured data for = 0◦ , 15◦ , 22◦ , 30◦ and 45◦).

ij

˝ij mean vorticity tensor

recisely makes the method hardly applicable for large and com-lex models.

The goal of this study is to investigate the effect of theesh size and turbulence models on the flow field in a square

rrayed rod bundle keeping in mind the requirements for com-lex models; tolerable mesh size; using the law of the wall;pplication of RANS (or URANS, but with tolerated time step)nd automated meshing (error free mesh generation by anyommercial software) in complex geometries. Thus the presentork is very similar to those aforementioned ones from many

spects, focusing more on the optimal balance between accu-ate results and acceptable computational resources for complexeometries.

In further studies a pitch to diameter ratio of 1.326 will be
referred (P = 12.6 mm, D = 9.5 mm), but experimental results areot available for this P/D ratio. Hence, the experiment of Hooper1980) has been selected from the open literature for verifica-ion of the model and settings, where the P/D ratio is close to
the desired, meanwhile measurements of velocity field, normaland shear Reynolds stresses are available. Section 2 presents themesh and turbulence model study using the experiments men-tioned above, followed by Section 3 using the larger P/D ratio of1.326. Section 4 summarizes the work and the Appendix describesin detail the adopted turbulence models.

2. Subchannel simulations for P/D = 1.194

Simulations were performed on a single, square arrayed sub-channel to investigate the effect of the mesh and turbulencemodels. The results are compared with the experiments of Hooper(1980) described in Section 2.1.

2.1. Experiment description

Hooper (1980) performed experiment for fully developed turbu-lent air flow in a six-rod cluster air rig for P/D ratio of 1.107 and 1.194with Reynolds number 48,400 and 48,000, respectively. The diam-eter of the rods were 14 cm, the pitch was varied with respect tothe aforementioned P/D ratios. In the experiment the mean velocitydistribution was measured by Pitot probes, the wall shear stress byPreston tubes and the turbulence intensities by hot-wire anemom-etry for the −45◦ to 45◦ segment of the air rig with respect tothe center line. For this comparative study the P/D = 1.194 in con-junction with Reynolds number of 48,000 was selected. Hooperpublished measurement data along five lines, namely = 0◦, 15◦,22◦, 30◦ and 45◦, see Fig. 1. All turbulence quantities are presentedin non-dimensional form as a function of the normalized wall dis-tance (y/ymax) (ymax is the wall distance to duct center line foreach angle) from the rod wall, except for the dimensionless axialvelocity (U

+), which is presented as the function of dimensionless

wall distance (y+ = y · u�(˛) · /�). The square root of the normalReynolds stresses are normalized with the local wall friction veloc-ity, while the shear Reynolds stresses and the turbulent kineticenergy are transformed to dimensionless form by the square of the
local wall friction velocity. Hooper also published the wall shearstress distribution at the rod wall surface being normalized by theaverage shear of the −45◦ to 45◦ segment (�aver
w ). The calculation of

1 ineerin

n

u

Tra13r

dtTi

Fts˛f(c

2

sirmnvom

tcdecliwiMSaftmadr(

r

weab

70 Á. Horváth, B. Dressel / Nuclear Eng

ormalized quantities can be seen in Eq. (1).

2i

+=

√u2

i

u�(˛), uv+ = uv

u�(˛)2, k+ = k

u�(˛)2,

U+ = U(r, ˛)u�(˛)

, �w = �w(˛)�aver

w(1)

he general accuracy of the measured Reynolds stresses wereeported to be around 10–15% with v2 being the least accurate,lthough vw could be in error by up to 100% (Hooper and Wood,984). The inaccuracy of the measured axial velocity is at least%, since two different probes (boundary layer and Pitot probe)esulted in this difference (Hooper, 1980).

The evaluation of the results were done along all five lines, butue to the sake of brevity, in this paper only the results for evalua-ion lines = 45◦ and = 22◦ are presented, see the sketch in Fig. 1.hese two lines give a good overview about the numerical resultsn comparison with the experiment.

In all comparative charts hereinafter (Figs. 4–10, Figs. 14–20,igs. 24–30), where results for both evaluation lines are presented,he experimental and numerical data for = 22◦ is plotted along theecondary Y axis (right Y axis). The predicted numerical results for

= 45◦ are presented by lines without markers, meanwhile the onesor = 22◦ by lines with small triangle markers. The same line typecolor and style) is used for the same mesh and turbulence model,oncerning mesh and turbulence model study, respectively.

.2. CFD model description

The CFD model represents a regular, in-bundle segment of aquare arrayed fuel rod bundle, see Fig. 2. The diameter of the rods D = 10 mm, the pitch is P = 11.94 mm (P/D = 1.194). The same P/Datio with the experiment ensures the geometrical similarity. Theodel extends 11.94 mm in axial direction, which has practically

o importance, since the bottom and top side are linked togetheria integrally coupled cyclic interface, thus simulating fully devel-ped turbulent flow and saving significant number of cells by notodeling a long tube.Future simulations with spacer grids (SG) are planned and due

o hardware limitations it is not feasible to simulate near wall pro-esses, thus this study is used to determine an appropriate meshensity. Altogether six different meshes, plotted in Fig. 3, were gen-rated for the flow domain to investigate mesh dependency. Theommonality of them is that none of them resolves the viscous subayer, and the first prism cell layer has the same thickness result-ng in an average dimensionless distance from the wall for the near

all cells around 25 (only the log-law region of the boundary layers resolved). The meshes can be sorted to two groups; M1, M2 and

3 are predominantly hexahedral trimmed meshes generated bytarCCM+ v.5.06; M4, M5 and M6 are body fitted structured hex-hedral meshes generated in StarCD v.4.12 via user coding. In theormer group M1 is the finest, M3 is the coarsest mesh. Amonghe structured meshes M4 is the finest, while M6 is the coarsest

esh. For details of the meshes see Table 1. In the axial directionll meshes have 40 equally distributed cells, which is less importantue to the cyclic bottom-top interface. The effective grid refinementatios are also presented in Table 1 based on the following equationEq. (2)):

eff =(

Ni

Ni+1

)1/2(2)

here Ni refers to the number of cells per plane of the ith mesh and iquals to 1, 2, 4 and 5 (thus determining reff separately for trimmednd structured meshes). The dimension in flow direction has noteen considered, since that has no real effect on the results, thus

g and Design 247 (2012) 168– 182

the exponent in Eq. (2) is just ½. In case of systematic refinementof a structured grid (duplication of cell number in all direction)reff would be 2. The averages of effective grid refinement ratiosfor the trimmed and structured meshes are about 1.77 and 1.62,respectively.

The simulations were performed with StarCD v.4.12 commercialCFD code of CD-Adapco, which applies the finite volume method tosolve the system of Navier–Stokes equations. The RANS methodwas applied with steady state time domain. The boundary con-ditions can be seen in Fig. 2. The surfaces of the fuel rods weremodeled as no slip smooth walls. The flow in axial direction wasenforced by a prescribed mass flux through the bottom (inlet) andtop (outlet) boundaries, which corresponds to a Reynolds numberof 48,000. This Reynolds number equals to the Reynolds numberof the experiment. The sides between the quarter fuel rods werealso linked together in pairs (x- and y direction, separately) viaintegrally coupled cyclic interface. This boundary condition for thesides allows cross flow through the boundary regions; however, itslows down the simulations since there are less restriction by theboundary conditions. The fluid was water at 40 ◦C and 1.5 bar.

For the turbulence model study, the renormalization group(RNG) k–ε model (Yakhot et al., 1992), the shear stress trans-port (SST) k–ω model (Menter, 1993), the Speziale–Sarkar–Gatski(SSG) Reynolds stress model (Speziale et al., 1991) and the Gib-son and Launder (GL) Reynolds stress model (Gibson and Launder,1975) were applied. The former two models belong to two equa-tion eddy viscosity turbulence models, while the latter two aresecond-order closure models in which the transport equationsfor the individual Reynolds stresses are solved. For the diffusionterm of the Reynolds stress models the Generalised Gradient Dif-fusion hypothesis (GGDH) was chosen to account for anisotropicdiffusivity (Daly and Harlow, 1970). In case of the GL Reynoldsstress model, for the rapid part of the wall reflection term themodel of Craft was chosen (Craft and Launder, 1991). A moredetailed description of the adopted turbulence models can be foundin the Appendix. All model constants of all applied turbulencemodels were the default values as implemented in StarCD v.4.12(CD-adapco, 2006). All models were used in conjunction with stan-dard wall functions. For the discretisation of the convective termsthe Monotone Advection and Reconstruction Scheme (MARS) wasapplied (CD-adapco, 2006) with blending factor of 0.5. MARS isa multidimensional second-order accurate differencing scheme inspace. The discretized equations were solved in a segregated man-ner with the Semi-Implicit Method for Pressure-Linked Equations(SIMPLE) algorithm (Patankar and Spalding, 1972). The algorithmis adopted in an iterative mode. As convergence criteria for allnormalized equation a maximum residual tolerance of 10−7 wasdefined, which was found to be enough to reach a converged solu-tion. At the same time the development of mass flux through thecyclic bottom-top interface was also monitored and at convergedstate the imbalance between them was less than 0.01%.

2.3. Mesh sensitivity study

Extensive mesh sensitivity analysis was completed with the spa-tial discretization introduced in the previous section to investigatethe impact of the mesh fineness on the results. The same bound-ary conditions were applied for all simulations in this section. Asturbulence model the GL Reynolds stress model was selected for allmeshes due to its sophisticated description of the turbulence. It hasthe capability to reproduce anisotropic behaviour of the turbulence,since it calculates the individual Reynolds stresses separately. The
results are compared with each other and with the measurementdata of Hooper (1980) in the coordinate system presented in Fig. 1.
In Figs. 4–11 the comparison of the measured and calculateddimensionless turbulence stresses, turbulent kinetic energy, mean


Fig. 2. Pure rod bundle flow domain and boundary regions.

ction

aaasa

TN

Fig. 3. Cross se

xial velocity and wall shear stress are shown. By taking a look onll these, one can observe the results are affected by the fineness
nd structure of the applied mesh. Nearly in all figures mesh M3hows considerably different results than the others, which is not
surprise, since the resolution of that mesh is the coarsest. Meshes

able 1umber of cells and vertices per plane of the meshes.

Mesh M1 M2

Number of vertices per plane 3061 885

Number of cells per plane (Ni) 2964 840

reff (Eq. (2)) 1.8727 1.6845

of the meshes.

M2 and M6 give slightly better results, but in a few figures the dis-crepancies compared to the finest mesh (M4) are still remarkable,
e.g. Figs. 9 and 11. It can be also stated that M1 gives nearly identi-cal results with M4 in most of the comparative figures (except e.g.Fig. 5). Furthermore, it is hard to distinguish the line representing
M3 M4 M5 M6

333 5336 2340 813296 5125 2205 736

1.5245 1.7308

172 Á. Horváth, B. Dressel / Nuclear Engineering and Design 247 (2012) 168– 182

ts

ddatpmcc

Fig. 4. Distribution of u2+

.

he results obtained with mesh M5 from the ones of M1 and M4,uggesting that the spatial resolution of M5 is almost fine enough.

The larger differences between the experimental and numericalata at some figures could arise from the accuracy of the measuredata, which is already mentioned above to be 10–15% (v2 is the leastccurate), or from the inappropriate turbulence modeling. The lat-er one is investigated in the scope of the turbulence model study
resented in Section 2.4. Concerning the accuracy of the experi-ent, the numerical results are mostly in the range of that. In all
ases the wall effect was considered via the wall function. The dis-repancies at the vicinity of the wall are related to that, and for

Fig. 5. Distribution of v2+

.

Fig. 6. Distribution of w2+

.

instance it is well represented in the figure of the dimensionlessaxial velocity (Fig. 10), where measurement data is available evenfor y+ < 20. (It should be noted that there wasn’t any evaluationpoint for line = 22◦ between y+ = 20 and 30 in contrast to = 45◦,where one point was located in the aforementioned range.)

Fig. 12 shows cross velocity vectors at a cross section of the flow
domain (the flow field is identical at all elevations). Secondary flowshave been expected based on the deduction of many numericalstudies of rod bundles with any kind of Reynolds stress model. Inall cases eight, symmetric secondary flow vortices evolved in the
Fig. 7. Distribution of uv+ .


sttsoMM

co

Fig. 8. Distribution of k+.

ubchannel. Already mesh M3 can capture secondary flows, but dueo the very coarse spatial resolution it is inappropriate to capturehe flow field accurately. The coarsest structured mesh M6 and theemi-fine trimmed mesh M2 provide better resolution of the sec-ndary vortices thanks to the finer meshes. The rests, M1, M4 and5 give nearly identical crossflow distribution, especially M1 vs.
4.The maximum cross flow velocity magnitude (peak value at the
enter line of the subchannel) for the finest meshes is about 1.568%f the average bulk velocity (values are represented in Fig. 12). In

Fig. 9. Distribution of U/Ubulk.

Fig. 10. Distribution of U+ .

Fig. 13 this parameter is plotted as function of number of cells perplane. The asymptotic convergence of this parameter is correctlycaptured. As the number of cells increases the values convergetoward approximately 1.57% independently of the structure of themesh. The meshes M2, M3 and M6 show considerably smaller val-ues, hence, from this aspect they are too coarse. The other meshesshow results closer to the limit value and further refinement of themesh is not worthwhile. Concerning the average cross flow mag-nitude, the structured meshes give nearly identical result, about0.54% of the bulk velocity. On the trimmed grids this parametershows increasing tendency from 0.416% to 0.527% for M3 to M1,respectively. This difference is ascribed to the structure of the mesh,since, for instance, M2 and M5 have nearly equal number of cells buthaving a difference of approximately 0.06% of Ubulk, meanwhile themaximum cross velocity is practically identical (M2: 1.222%; M5:1.2161%).

Based on the mesh study presented above, it can be concludedthat the body fitted structured meshes provide in general slightlybetter results considering all quantities than trimmed meshes withthe same number of cells and for pure rod bundle simulations theywould be preferred. M1, M4 and M5 grids give very similar results,meanwhile M2, M3 and M6 are too coarse meshes. Mesh M1 or abit finer mesh than M2, around 2200 cells per plane, can be a rea-sonable choice. Of course the mesh study performed here for purerod bundle does not mean automatically that it satisfies the meshdependency of a geometry containing SG too, hence a separatestudy should be performed for such geometry in the future.

2.4. Turbulence model study

The turbulence model study was completed using mesh M1in conjunction with four different turbulence models described

Fig. 11. Distribution of �w wall shear stress.


differ

pSecb

u

Tfcw

bS

m

mks

F

Reynolds stress models is accurate; both of them are within therange of the accuracy of the experiment. The turbulent kineticenergy in Fig. 18 shows that GL Reynolds stress model is in thebest agreement with the experiment; the SSG Reynolds stress

Fig. 12. Cross velocity field calculated with

reviously, namely: k–ε RNG, k–ω SST, GL Reynolds stress andSG Reynolds stress. Since the former two ones belong to the twoquation eddy viscosity models, thus the Reynolds stresses can bealculated from the eddy viscosity hypothesis (Eq. (3)), introducedy Pope (2000).

iuj = 23

kıij − t

(∂Ui

∂xj+ ∂Uj

∂xi

)(3)

he Reynolds stresses for the Reynolds stress models are obtainedrom the solution of model transport equations. The comparativeharts of the dimensionless quantities are shown in Figs. 14–21ith the experimental data of Hooper (1980).

Concerning the normal Reynolds stresses, the discrepancyetween the two equation and Reynolds stress models is obvious.ince there is a linear relation between the Reynolds stresses and

2+

ean velocity gradients (Eq. (3)), therefore u is underestimated,

eanwhile v2+

and w2+

are overestimated by the k–ε RNG and–ω SST models compared to measurement data. Both Reynoldstress models can capture the anisotropic behaviour of the normal

ig. 13. Maximum cross velocity magnitude as function of number of cells per plane.

ent meshes and GL Reynolds stress model.

stresses thus evolving secondary motions. The SSG Reynolds stressmodel usually underpredicts the normal Reynolds stresses, whilethe kind of GL approximates better the experimental data. Thenumerical prediction of the shear stress of uv+ (Fig. 17) by the


.


mobl

aSwne

for the adopted models; however, the shape of the lines from the


.

odel can capture the correct shape but with an under predictionf that; the k–ε RNG and k–ω SST models predict nearly the sameehaviour, but they show a flatter profile along both evaluation

ines than the Reynolds stress models do.The normalized axial velocities (by local friction velocity, u�)

s function of the normalized wall distance can be seen in Fig. 20.ince the viscous sublayer was not resolved in any case, it is not
orth to really look up for agreement with the experiment in theear wall region. Over y+ ≈ 30 the agreement is relatively good,specially for = 22◦, where all models are within the range of the

.


accuracy of the axial velocity measurement (≈3%). Fig. 19 showsthe ratio of U/Ubulk as function of y/ymax, and Fig. 22 representsthe same dimensionless axial velocity in a contour plot in the sub-channel for all adopted turbulence models in four quarters. As itcan be seen from the chart, the maximum velocity at the cen-ter of the subchannel ( = 45◦ and y/ymax = 1) is nearly identical

wall towards the center is different. GL Reynolds stress model pre-dicts the lowest maximum speed (U/Ubulk = 1.219) and the k–εRNG does the highest (U/Ubulk = 1.254). Along the evaluation line




˛ifeG

Fig. 20. Distribution of U+.

= 22◦ the tendency is just the opposite. In Fig. 22 one can see thatn the rod gap (tightest gap) the GL Reynolds stress model predicts
ar away the highest velocities than the others, especially the twoquation models, do. Based on these results it is obvious that theL Reynolds stress model generates the smoothest axial velocity

Fig. 22. Normalized axial velocity contour (RSM-GL upper-left; RSM-SSG upper-right; k–ω SST lower-left; k–ε RNG lower right).

field in the subchannel, which predicts stronger secondary motionin the subchannel than in case of the SSG Reynolds stress model.This is discussed after the next paragraph.

Fig. 21 shows the predictions of the relative local wall shearstress over = 0–45◦. It is shown that the k–ε RNG and k–ω SSTproduce very similar predictions: the wall shear stress monoton-ically increases toward the center of the subchannel, in contrastto the experimental data which has a maximum around = 32◦.Similar effect had been noted by Trupp and Azad (1975) for trian-gular array. The anisotropic models clearly show the tendency toflatten the wall shear stress distribution; however, both Reynoldsstress models have their own disadvantage. The SSG model under-predicts the wall shear stress for = 0–15◦, but the decrease of thatover ≈ 35◦ is smaller compared to GL Reynolds stress model. GLReynolds stress model predicts quite well the wall shear stress forsmaller angles, but overpredicts the decrease after reaching themaximum value.

Fig. 23 shows the obtained cross velocity field for the Reynoldsstress models. Since the two equation models produced zero crossflow, which was expected in advance, it is pointless to plot them.Indeed, both Reynolds stress models can predict the secondary vor-tices in the subchannel, however, there is significant differencebetween the obtained tangential velocity fields; GL Reynolds stressmodel predicts 0.527% and 1.568% of the mean bulk velocity as aver-age and maximum cross velocity, respectively, while SSG Reynoldsstress model just gives 0.264% and 0.559%, respectively. Manyresearchers had attempts to measure the velocity in secondary vor-tex, but most of them failed or could give just an approximation formaximum average tangential velocity. Hooper and Wood (1984)reported for square arrayed bundle with P/D = 1.107 that the gen-eral level of the cross flow velocity is smaller than 1% of the meanbulk velocity, however, they could not obtain accurate measure-ments. Neti et al. (1982), based on their experiment for a 3-by-3rod bundle with P/D = 1.4, reported as maximum tangential veloc-ity 1% of the mean bulk velocity. (For triangular array bundle withP/D = 1.3 Vonka (1988) gave 0.1% for average tangential velocity.)Based on the values for square arrayed bundle it seems that the SSGReynolds stress model underpredicts the cross flow velocity mag-nitude while GL Reynolds stress model slightly overpredicts that;however, the obtained maximum velocities appear just in a very
small region along the symmetry lines of the subchannel (exactvalues are mentioned above and represented in Fig. 23). Moreover,the flow field calculated by the SSG Reynolds stress model showsnot only one secondary vortex per 1/8th section of the subchannel


s stre

(scrsps

matTscsfctnc

3

Pibte

(

•

•

•

Fig. 23. Cross velocity field calculated with GL Reynold

Fig. 23), but a small additional vortex (represented by red coloredtreamline in Fig. 23/RSM-SSG) appears at the rod gap rotating inounter direction with respect to the main secondary vortex (rep-esented by black colored streamline in Fig. 23/RSM-SSG) in theame segment of the subchannel. Presence of such vortex was notredicted by Hooper (1980) and Hooper and Wood (1984) even formaller P/D ratio.

The presented turbulence model study has shown that all linearodels adopted produce very similar predictions, but since they

ssume isotropic distribution of the Reynolds stresses by defini-ion, they are inadequate to capture the flow of secondary kind.he GL Reynolds stress model together with the RANS method hashown its capability for the investigated geometry (P/D = 1.194) toalculate the turbulence quantities correctly and to reproduce theecondary flow in the subchannel. URANS simulation was also per-ormed with mesh M1 and GL Reynolds stress turbulence model toheck the effect of the transient approach (first order implicit Euleremporal discretization with 10−3 second time step), but no sig-ificant differences could be identified for the configurations beingonsidered here.

. Subchannel simulations for P/D = 1.326

In this section the simulations with a rod bundle featuring/D = 1.326 is presented. The lack of experimental results makest impossible to judge the real accuracy of these simulations, butased on the conclusions of experiments concerning the effect ofhe geometry (different P/D ratio) and Reynolds number, the pres-nce or lack of tendencies can be judged.

Based on the experiments of Hooper (1980), Hooper and Wood1984) and many other authors, the followings can be stated:

The turbulence intensities and kinetic energy differs more andmore from axisymmetric pipe flow with the reduction of P/Despecially at the tightest gap ( = 0◦), where a significant increasecan be observed for tighter lattices (related to flow pulsation).The mean velocity profiles at all azimuthal positions fits reason-ably well the logarithmic law of the form U+ = (1/�) ln y+ + Bover Re ≈ 40,000 (� = 0.4 and B = 5.5, Hooper (1980)). It is alsofairly independent of the P/D ratio. (Hooper (1980) uses thesame B = 5.5 for P/D = 1.194 and P/D = 1.107, meanwhile Kraussand Meyer (1998) for triangular arrangement suggest B = 4.5 for
P/D = 1.12 and B = 5.0 for P/D = 1.06.)The wall shear stress distribution is highly affected by the P/Dratio; reduction of P/D produces a larger, and more monotonicgradient from = 0◦ to = 45◦. The Reynolds number has less
ss model (left) and SSG Reynolds stress model (right).

effect on the wall shear stress (as the Reynolds number increasesthe dependency is decreasing)

Based on this short summary, it is expected in advance for thesimulations with P/D = 1.326 that they show better agreement withaxisymmetric pipe flow and that Reynolds number independencyalso appears. The results will be compared with the axisymmetricpipe flow distribution of Lawn (1971).

3.1. CFD model description

The CFD model, just like in Section 2.2, represents a regular, in-bundle segment of a square arrayed fuel rod bundle, see Fig. 2. Thediameter of the rod according to P/D = 1.326 is D = 9.5 mm, the pitchis P = 12.6 mm. In axial direction it extends over 12.6 mm (integrallycoupled inlet and outlet region is applied, thus the axial extensionhas no importance).

For meshing, the same general mesh density and method wasused as for M1 (P/D = 1.194), only the near wall cell size wasadjusted to get approximately the same y+ values as at those sim-ulations. Since two different Reynolds numbers (Re = 48,000 andRe = 107,000) were tested, therefore the mesh was adjusted forboth, separately. The higher Reynolds number is calculated basedon the same flow properties as in Section 2.2 (40 ◦C and 1.5 bar), butthe bulk velocity is much closer to reactor condition (Ubulk ≈ 6 m/s).

The boundary conditions and solver settings corresponded withdescriptions in Section 2.2, except the inlet and outlet boundaries,where the mass flow rate was adjusted to satisfy Re = 48,000 andRe = 107,000. As turbulence model “primarily” the GL Reynoldsstress model was used based on the conclusions of the previoussection, nevertheless, since the flow is expected to be more similarto pipe flow due to its higher P/D ratio (semi-open duct flow), thek–ε RNG turbulence model was also tested.

3.2. Results of the simulations

In Figs. 24–31 the post processed turbulence quantities, veloc-ity distribution and wall shear stress are shown for evaluation lines

= 22◦ and = 45◦. Based on all of them, the Reynolds number inde-pendency of the flow field is satisfied. Only small discrepancies canbe identified at some figures, which are related to the slightly differ-ent mesh close to the walls. The independency is also satisfied withboth turbulence models; the curves in all figures can be grouped
based on the applied turbulence model along both evaluation lines.
Concerning the results obtained with GL Reynolds stress model,the turbulence intensities and turbulent kinetic energy along line

= 45◦ are very similar to the results of the model featuring smaller


Pf(sobfifilt

duct center line. The values of U/Ubulk at y/ymax = 1.0 are sum-


.

/D ratio (by comparison with Figs. 14–21). Usually, the curves startrom the same quantity at y/ymax = 0, but at the center of the ducty/ymax = 1) all normal stresses and turbulent kinetic energy showmaller value. Furthermore, the agreement with the distributionf the quantities of axisymmetric pipe flow (Lawn, 1971) is muchetter (but usually under predicted along line ˛ = 45◦) which con-rms the experiments: higher P/D ratio results in more similar flow
eld to pipe flow. The data obtained along the other evaluation
ine, = 22◦, shows really that the CFD simulation could capturehe correct tendencies, since the differences between pipe flow


.


.

distribution and the CFD results are much smaller than they arefor P/D = 1.194. The results obtained with k–ε RNG model show thesame tendencies.

The dimensionless axial velocity U/Ubulk along line = 45◦ withGL Reynolds stress model (Fig. 29) shows nearly the same distri-bution for both P/D ratios with difference in maximum value at

¯
marized in Table 2 for both turbulence models and for both P/Dratios. Considering the results with k–ε RNG turbulence model, thevalues slightly decreased for = 45◦ and increased for = 22◦. The




Table 2Dimensionless axial velocity U/Ubulk at y/ymax = 1.0.

Turbulence model GL Reynolds stress k-� RNG

Evaluation line = 45◦ = 22◦

= 45◦ = 22◦

P/D = 1.194 1.219 1.09 1.254 1.028P/D = 1.326 1.19 1.088 1.213 1.059

Fig. 30. Distribution of U+.

difference between the turbulence models for this dimensionlessvelocity along both evaluation lines is much smaller, than it is forP/D = 1.194, especially for = 22◦.

The other dimensionless form of the velocity, U+ (normalized bythe friction velocity) and the logarithmic law, with constants � = 0.4and B = 5.5, are represented in Fig. 30. The log-law is better satis-fied along the line = 22◦ than along the other one, where usuallyslightly higher gradient is obtained outside the near wall region.The same tendency was identified for P/D = 1.194, thus this is notthe effect of the change in P/D ratio or in the Reynolds number.

The wall shear stress distribution can be seen in Fig. 31. The GLReynolds stress model predicts nearly constant shear stress withmaximum value around approximately = 23◦. The decrease fromthis maximum value is probably slightly overpredicted based onthe conclusions derived in Section 2.4. The k–ε RNG turbulencemodel shows the same monotonic tendency of the wall shear stressfrom ˛ = 0◦ to = 45◦ as for the tighter lattice, but with higherminimum (≈0.85) and smaller maximum (≈1.125) values. Inde-pendently of the applied turbulence model all curves show smallergradient or nearly constant value along the rod surface as they dofor P/D = 1.194, thus the correct tendency could be captured.

The cross velocity field (with GL Reynolds stress model) is rep-resented in Fig. 32 for Re = 48,000. The flow field for Re = 107,000shows the same distribution and shape of the secondary vortices,just the absolute cross velocity magnitudes are higher. The aver-age and maximum cross velocity magnitude for Re = 48,000 are0.352% and 0.94% of the mean bulk velocity, respectively, mean-while for Re = 107,000 the same values are 0.324% and 0.91%. Sinceincrease of P/D ratio results in decrease of cross velocity strength,
these results are not in contradiction with it (all values are smallerthan for P/D = 1.194). The small differences in the values suggestReynolds number independency.

180 Á. Horváth, B. Dressel / Nuclear Engineerin

FP

PttflRitso(mbm

4

wosmcsnvabmkra

wwmtmItlfdIb

ε

The coefficients for this turbulence model are summarized inTable A.1.

Table A.1

ig. 32. Cross velocity field calculated with Re = 48,000; GL Reynolds stress model;/D = 1.326.

Based on the simulation results with the rod bundle featuring/D = 1.326, it can be stated that the correct tendencies can be cap-ured for all characteristic properties. Due to lack of experimentshe results could not be precisely assessed. The differences in theow field and in the turbulence quantities obtained with the GLeynolds stress and k–ε RNG model are smaller. The wider gaps

ssues in more similar flow field to axisymmetric pipe flow, wherehe two equation turbulence models do not fail. Probably, withome restrictions the k–ε RNG model is appropriate for simulationsf rod bundle with P/D = 1.326 containing complex SG geometriese.g.: SG featuring mixing devices thus forcing strong convective

ixing between subchannels, where secondary flows are negligi-le); however, for pure rod bundle simulations the Reynolds stressodel is still preferred.

. Conclusions

In the first part of this paper CFD simulations were presentedith square arrayed rod bundle featuring P/D = 1.194 with the aim

f finding an optimal mesh resolution and turbulence model foruch flows. The RANS simulations were carried out with the com-ercial code StarCD v.4.12 and the post processed results were

ompared with the experimental data of Hooper (1980). The meshensitivity study showed that the effect of the resolution is sig-ificant in the precise reproduction of turbulence quantities andelocity field. Moreover, the structure of the mesh is also importantnd slightly better agreement can be reached with the same num-er of cells but well structured, body fitted mesh. With the selectedesh a turbulence model study was performed applying k–ε RNG,

–ω SST, GL Reynolds stress and SSG Reynolds stress models. Theesults calculated with GL Reynolds stress model were in the bestgreement with the experiment.

In the second part of this paper simulations were performedith the same subchannel, but with a P/D ratio of 1.326 andith two Reynolds numbers (48,000 and 107,000). The automaticesher of StarCCM+ was used based on the settings for the finest

rimmed mesh (with adjusted near wall cell size) and as “pri-ary” turbulence model the GL Reynolds stress model was used.

n order to check how the increase of P/D ratio affects the calcula-ion with k–ε RNG model, as an “optional” turbulence model theatter one was tested again. Due to the lack of experimental data
or P/D = 1.326, the results were reviewed concerning tendencieserived from different experiments and axisymmetric pipe flow.
ndependently of the applied turbulence model the Reynolds num-er independency was reached. The tendencies for all variables

g and Design 247 (2012) 168– 182

were correctly captured, they showed that the flow field is moresimilar – but not equal – to an axisymmetric pipe flow. The resultswith k–ε RNG turbulence model were closer to the ones obtainedwith GL Reynolds stress model, than they were with P/D = 1.194.However, for pure bundle simulations the former one is suggested.For subchannel simulations containing SG (generating strong con-vective mixing) the latter turbulence model can be a candidate,keeping in mind its deficiencies.

Additionally, for more complex problems, like fluid–structureinteraction (FSI) (including fuel assembly bow as quasistationaryand vibration as transient phenomena), further questions comeup. First of all, is it possible to achieve good numerical results(agreement with experiment) with RANS/URANS method insteadof using LES? If yes, does the usage of any anisotropic turbu-lence model provide better agreement than the widely used twoequation turbulence models? As it has been mentioned in theintroduction, promising numerical results are already obtained bysome authors with URANS method modeling flow pulsation (Changand Tavoularis, 2007; Merzari et al., 2008) between subchannels;however, FSI problems of FAs are different and the questions areunanswered. The authors continue working on this subject.

Appendix A.

Description of the adopted turbulence models.

A.1. Two equation turbulence models

A.1.1. Renormalization group (RNG) k–ε model (Yakhot et al.,1992; CD-adapco, 2006)

The transport equation for the turbulence kinetic energy is asfollows:

∂(k)∂t

+ ∂(Ujk)∂xj

= ∂

∂xj

[(� + �t

�k

)∂k

∂xj

]+ Pk − ε (A1)

The transport equation for the turbulence dissipation rate is asfollows:

∂(ε)∂t

+ ∂(Ujε)∂xj

= ∂

∂xj

[(� + �t

�ε

)∂ε

∂xj

]+ Cε1

ε

kPk − C∗

ε2ε2

k(A2)

where

C∗ε2 = Cε2 + C��3(1 − �/�0)

1 + ˇ�3(A3)

and

� = Sk

ε(A4)

The turbulent viscosity is linked to k and ε via the equation

�t = C�k2(A5)

Coefficients of the RNG k–ε turbulence model.

C� �k �h Cε1 Cε2 � �0 ˇ

0.085 0.719 0.719 1.42 1.68 0.4 4.38 0.012


Table A.2Coefficients of the SST k–ω turbulence model.

ω ω ω ω ∗ ∗

AC

f

T

w

S

Fto

�

C

F

C

F

˛

TT

A

t

wp

(d

D

Table A.3Coefficients for the GGDH formulation.

Cε1 Cε2(GL) Cε2 (SSG) CS Cεε � E

1.44 1.92 1.83 0.22 0.18 0.42 9

Table A.4Coefficients of the SSG Reynolds stress model.

∗ ∗

�k1 �ω1 ˇ1 �k2 �ω2 ˇ2 ˇ1, ˇ2 � a1

1.176 2.0 0.075 1.0 1.168 0.828 0.09 0.41 0.31

.1.2. Shear stress transport (SST) k–ω model (Menter, 1993;D-adapco, 2006)

The transport equation for the turbulence kinetic energy is asollows:

∂(k)∂t

+ ∂(Ujk)∂xj

= ∂

∂xj

[(� + �t

�k

)∂k

∂xj

]+ Pk − ˇ∗kω (A6)

he transport equation for the specific dissipation rate is as follows:

∂(ω)∂t

+ ∂(Ujω)∂xj

= ∂

∂xj

[(� + �t

�ω

)∂ω

∂xj

]− ˇω2 + ˛

ω

kPk + Sω

(A7)

here

ω = 2(1 − F1)1

�ω2

1ω

∂ω

∂xj

∂k

∂xj(A8)

1 is the blending function used in the model to switch betweenhe Wilcox k–ω (actived close to the wall) and a transformed formf standard k–ε turbulence model (actived in the outer regions).

The turbulent viscosity for this model is

t = a1k

max(a1ω, SF2)(A9)

The auxiliary relations are the followings:

˚ = F1C˚1 + (1 − F1)C˚2 (A10)

1 = tanh

{{min

[max

( √k

0.09ωy,

500�

y2ω

),

4k

�ωω2CDkωy2

]}4}(A11)

Dkω = max

(2

ω�ωω2

∂k

∂xj

∂ω

∂xj, 10−20

)(A12)

2 = tanh

{[max

(2√

k

0.09ωy,

500

y2ω

)]2}

(A13)

1 = ˇ1

ˇ∗1

− 1�ω

ω1

�2√ˇ∗

1

and ˛2 = ˇ2

ˇ∗2

− 1�ω

ω2

�2√ˇ∗

2

(A14)

he coefficients for this turbulence model are summarized inable A.2.

.2. Reynolds stress models

The Reynolds stresses are solved directly by all Reynolds stressurbulence models using an equation of the form,

∂(uiuj)∂t

+ ∂(Ukuiuj)∂xk

= Dij + Pij + ˚ij − εij (A15)

here Dij is the diffusion term, Pij is the production term, ˚ij is theressure–strain term and εij is the dissipation term.

Since the generalised gradient diffusion hypothesis (GGDH)Daly and Harlow, 1970) was applied to account for anisotropic
iffusivity, the diffusion term has the form.
ij = ∂

∂xk

[(�ıkl + Cs

kukul

ε

)∂uiuj

∂xl

](A16)

C1 C1 C2 C3 C3 C4 C5

3.4 1.8 4.2 0.8 1.3 1.25 0.4

The anisotropic form of the turbulence dissipation is

∂(ε)∂t

+ ∂(Ukε)∂xk

= ε

k(Cε1Pk − Cε2ε)

+ ∂

∂xk

[(�ıkl + Cεε

kukul

ε

)∂ε

∂xl

](A17)

The coefficients for the diffusion and dissipation term are summa-rized in Table A.3.

The production term Pij is

Pij = −

(uiuk

∂Uj

∂xk+ ujuk

∂Ui

∂xk

)(A18)

The pressure–strain term ˚ij re-distributes energy among turbu-lence normal stresses and ensures that, as the turbulence is driventowards isotropy, the shear stress declines. The two Reynolds stressmodels applied in the study of this paper differ in the modeling ofthis term;

A.2.1. Speziale–Sarkar–Gatski (SSG) Reynolds stress model(Speziale et al., 1991)

˚ij = −(C1ε + C∗1Pk)bij + C2ε

(bikbkj − 1

3ıijbklbkl

)+ (C3 − C∗

3˘b)kSij + C4k(

bikSjk + bjkSik − 23

ıijbklSkl

)+ C5k(bik˝jk + bjk˝ik) (A19)

bij, the anisotropy tensor is given by

bij = uiuj − 1/3(ıijukuk)ukuk

(A20)

Pk, the production of turbulence kinetic energy is given by

Pk = −uiuj∂Ui

∂xj(A21)

and

˘b =√

bijbij, Sij = 12

(∂Ui

∂xj+ ∂Uj

∂xi

), ˝ij = 1

2

(∂Ui

∂xj− ∂Uj

∂xi

)(A22)

The coefficients for the pressure–strain term in case of SSGReynolds stress model are summarized in Table A.4.

A.2.2. Gibson and Launder (GL) Reynolds stress model (Gibsonand Launder, 1975)

In this model ˚ij is the sum of three terms:

˚ij = ˚ij1 + ˚ij2 + ˚ijw (A23)

where ˚ is usually referred to as the slow term, ˚ is the rapid
ij1 ij2term and ˚ijw are the wall reflection terms.
˚ij1 = −C1ε

k

(uiuj − 1

3ıijukuk

), ˚ij2 = −C2

(Pij − 1

3ıijPkk

)(A24)

182 Á. Horváth, B. Dressel / Nuclear Engineerin

Table A.5Coefficients of the GL Reynolds stress model (Craft model for ˚ijw2).

C1 C2 C1w C2wC1 C2wC2 C2wC3

˚

˚

wns

s

˚

w

a

Tso

R

B

CC

C

1.8 0.6 0.5 −0.044 −0.08 0.6

ijw corresponding to ˚ij1 is as follows:

ijw1 = C1wε

k

(ukumnknmıij − 3

2uiuknjnk − 3

2ujuknink

) (l

2.5ln

)(A25)

here �n = (n1, n2, n3) is the unit vector normal to the wall, ln is theormal distance to the wall and l = k3/2/ε is a turbulence lengthcale.

To model the rapid part of the wall reflection term, ˚ijw2, corre-ponding to ˚ij2 the form of Craft and Launder (1991) was chosen.

ijw2 = C2wC1∂Ul

∂xmulum(nqnqıij − 3ninj)

(l

2.5ln

)

+ C2wC2kalm

(∂Uk

∂xmnlnkıij − 3

2∂Ui

∂xmnlnj − 3

2∂Uj

∂xmnlni

)×

(l

2.5ln

)+ C2wC3k

∂Ul

∂xmnlnm

(ninj − 1

3nqnqıij

) (l

2.5ln

)(A26)

here

ij = (uiuj − 1/3(ıijukuk))k

(A27)

he coefficients for the pressure–strain term in case of GL Reynoldstress model in conjunction with the Craft model for the rapid partf the wall reflection term are summarized in Table A.4.

Table A.5.

eferences

aglietto, E., Ninokata, H., 2005. A turbulence model study for simulatingflow inside tight lattice rod bundles. Nuclear Engineering and Design 235,773–784.

D-adapco, 2006. STAR-CD Version 4.00, Methodology.
hang, D., Tavoularis, S., 2007. Numerical simulation of turbulent flow in a 37-rod
bundle. Nuclear Engineering and Design 237, 575–590.raft, T.J., Launder, B.E.,1991. Computation of impinging flows using second-

moment closures. In: 8th Symph. on Turb. Shear Flows. Tech. Univ. Munich,pp. 8-5-1–8-5-6.

g and Design 247 (2012) 168– 182

Daly, B.J., Harlow, F.H., 1970. Transport equations in turbulence. Physics of Fluids13, 2634–2649.

Gibson, M.M., Launder, B.E., 1975. Ground effects on pressure fluctuations in theatmospheric boundary layer. Journal of Fluid Mechanics 86, 491–511.

Hooper, J.D., 1980. Developed single phase turbulent flow through a square-pitchrod cluster. Nuclear Engineering and Design 60, 365–379.

Hooper, J.D., Rehme, K., 1984. Large-scale structural effects in developed turbulentflow through closely spaced rod arrays. Journal of Fluid Mechanics 145, 305–337.

Hooper, J.D., Wood, D.H., 1984. Fully developed rod bundle flow over a large rangeof Reynolds number. Nuclear Engineering and Design 83, 31–46.

Ikeno, T., Kajishima, T., 2010. Analysis of dynamical flow structure in a square arrayedrod bundle. Nuclear Engineering and Design 240, 305–312.

In, W.K., Oh, D., Chun, T., 2003. Simulation of turbulent flow in rod bundles usingeddy viscosity models and the Reynolds stress model. In: Proceedings of the 10thInternational Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-10), Seoul, Korea.

In, W.K., Shin, C.H., Oh, D.S., Chun, T.H., 2004. Numerical analysis of the turbulentflow and heat transfer in a heated rod bundle. Journal of the Korean NuclearSociety 36, 153–164.

Krauss, T., Meyer, L., 1998. Experimental investigation of turbulent transport ofmomentum and energy in a heated rod bundle. Nuclear Engineering and Design180, 185–206.

Lawn, C.J., 1971. The determination of the rate of dissipation in turbulent pipe flow.Journal of Fluid Mechanics 48, 477–505.

Lee, K.B., Jang, H.G., 1997. A numerical prediction on the turbulent flow in closelyspaced bare rod arrays by nonlinear k–� model. Nuclear Engineering and Design172, 351–357.

Menter, F.R., 1993. Zonal two equation k–� turbulence models for aerodynamicflows. In: Proc. 24th Fluid Dynamics Conf., Orlando, Florida, USA, 6–9 July, PaperNo. AIAA 93-2906.

Merzari, E., Ninokata, H., Baglietto, E., 2008. Numerical simulation of flows in tight-lattice fuel bundles. Nuclear Engineering and Design 238, 1703–1719.

Möller, S.V., 1991. On phenomena of turbulent flow through rod bundles. Experi-mental Thermal and Fluid Science 4, 25–35.

Neti, S., Eichorn, R., Hahn, O.J., 1982. Laser Doppler measurements of flow in a rodbundle. Nuclear Engineering and Design 74, 105–116.

Patankar, S.V., Spalding, D.B., 1972. A calculation procedure for heat, mass andmomentum in three-dimensional parabolic flows. International Journal of Heatand Mass Transfer, 15.

Pope, S.B., 2000. Turbulent Flows. Cambridge University Press, New York, p. 358.Rehme, K., 1992. The structure of turbulence in rod bundles and the implications

on natural mixing between the subchannels. International Journal of Heat andMass Transfer 35, 567–581.

Rowe, B.S., Johnson, B.M., Knudsen, J.G., 1974. Implications concerning rod bundlecrossflow mixing based on measurements of turbulent flow structure. Interna-tional Journal of Heat and Mass Transfer 17, 407–419.

Speziale, C.G., Sarkar, S., Gatski, T.B., 1991. Modelling pressure–strain correlation ofturbulence: an invariant dynamical systems approach. Journal of Fluid Mechan-ics 227, 245–272.

Tóth, S., Aszódi, A., 2010. CFD analysis of flow field in a triangular rod bundle. NuclearEngineering and Design 240, 352–363.

Trupp, A.C., Azad, R.S., 1975. The structure of turbulent flow in triangular arrayedrod bundles. Nuclear Engineering and Design 32 (1), 47–84.

Vonka, V., 1988. Measurement of secondary flow vortices in a rod bundle. NuclearEngineering and Design 106, 191–207.

Yakhot, V., Orszag, S.A., Thangam, S., Gatski, T.B., Speziale, C.G., 1992. Developmentof turbulence models for shear flows by a double expansion technique. Physicsof Fluids A 4 (7), 1510–1520.

Date post:	11-Sep-2016
Category:	Documents
Upload:	akos-horvath
View:	221 times
Download:	3 times

Numerical simulations of square arrayed rod bundles

Documents