Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol. 40, No. 9, p. 655–663 (September 2003)
ORIGINAL PAPER
Numerical Prediction of Secondary Flows in Complex Areas Using Concept of Local Turbulent Reynolds Number Vladimir KRIVENTSEV1 , Hiroyuki OHSHIMA1 , Akira YAMAGUCHI1,* and Hisashi NINOKATA2 1
Japan Nuclear Cycle Development Institute, 4002 Narita, O-arai-machi, Higashi-Ibaraki-gun, Ibaraki 311-1393 2 Tokyo Institute of Technology, 2-21-1 O-okayama, Meguro-ku, Tokyo 152-9550 (Received March 17, 2003 and accepted in revised form June 20, 2003)
A new model of turbulence is proposed for the estimation of Reynolds stresses in turbulent fully-developed flow in a wall-bounded straight channel of an arbitrary shape. The main idea of a Multi-Scale Viscosity (MSV) model can be expressed in the following phenomenological rule: A local deformation of axial velocity can generate the turbulence with the intensity that keeps the value of the local turbulent Reynolds number below some critical one. Therefore, in MSV, the only empirical parameter is the critical Reynolds number. Multi-scale viscosity has been verified on the pipe flow and applied to simulation of turbulence-driven secondary flow in elementary cell of the infinitive hexagonal rod array. Since MSV can predict turbulent viscosity anisotropy in directions normal and parallel to the wall, it is capable to calculate secondary flows in the cross-section of the rod bundle. Calculations have shown that maximal intensity of secondary flow is about 1% of the mean axial velocity for the low-Re flows (Re=8,170), while for higher Reynolds number (Re=160,100) the intensity of secondary flow is as negligible as 0.2%. KEYWORDS: turbulence, secondary flows, hexagonal rod bundle
I. Introduction The present state of turbulence nature understanding is that Navier-Stokes equations can properly describe the turbulent flow. However, direct numerical solution of Navier-Stokes equations is not only too costly but also still impossible in most important cases for not-very-low Reynolds numbers even in the simplest geometries. Instead, averaged NavierStokes, or Reynolds equations, are considered to be sufficient and practical enough to describe turbulent flow in most engineering applications. In this case, a turbulence model should include expressions or/and equations for eddy diffusivity or (in other terms) turbulent viscosity. 1. Mixing-Length Model of Turbulence One of the first and very effective turbulent viscosity models was a “mixing-length” model proposed by Prandtl.1) Prandtl’s hypothesis is that a fluid particle, or eddy, keeps its axial momentum when moving athwart the flow for some finite distance called “mixing length.” According to the mixing length model, turbulent shear stress in the athwart direction can be estimated as 2 du du , (1) τturb = −ρu v = ρl dy dy where l is the “mixing length”, ρ is the density, u=u+u and v=v+v are the axial and cross velocities, y is the distance to the wall, superscript index ( ) is devoted for pulsation component of the value, and values in brackets are ensemble averaged. Using the idea of the Boussinesq’s eddy viscosity model, one can write the following expression for ∗
Corresponding author, Tel. +81-29-267-4141, Fax. +81-29-2663675, E-mail:
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eddy diffusivity in the athwart direction: du 2 du . =l νt = −u v dy dy
(2)
Here, in Eq. (2), the mixing length l should be modeled and Prandtl proposed the simple relation for pipe flow: l=χ y, where χ is the von Karman constant. Since only simple formulas are used here, the Prandtl’s mixing length model can be classified as a “zero-equation” model in contrast to more complicated modeling that uses additional equations in partial-derivatives derived for other turbulent parameters such as kinetic energy of turbulent fluctuations, dissipation and so on. Zero-equation models were popular because they are relatively simple and intuitive. However, these models are very limited because they can be applied to particular problems with similar geometry and are inappropriate for predicting other types of flows in more complicated geometries. Buleev2) (1962) proposed an integral model of turbulence based on consideration of Prandtl’s original hypothesis of momentum transported by stochastic eddy movements. In spite the fact that Buleev’s model could give quite accurate prediction of the velocity distribution in very complicated threedimensional areas, it has never been accepted broadly. It poses some additional calculation complications and, after all, strongly depends on some empirical constants. Here, we recall Buleev’s model because the Multi-Scale Viscosity (MSV) model proposed in this work can be also categorized as an integral zero-equation model. In fact, MSV is a further development and generalization of Prandtl’s and Buleev’s mixing length hypothesis. Now, let us make some physical considerations related mainly to wall-bounded channel flows.
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2. Reynolds Number and Transition to Turbulence The first systematic investigation of a pipe flow was conducted by Reynolds who discovered the law of similarity based on a dimensionless parameter which is named after him: Re = ud/ν,
(3)
where u is the flow velocity, d is the pipe diameter, and ν is the kinematic viscosity of the fluid. In a more general form applicable to any flow, u and d are considered as some “typical” flow velocity and flow scales. One of the most fundamental facts is that there exists some critical Reynolds number Recr where the flow becomes unstable and turbulent for all the higher Reynolds numbers. Reynolds himself found that Recr is between 11,800 and 14,300. This overestimate may have resulted from the use of very smooth pipe inlet in Reynolds experiments. Numerous present experiments indicate that Recr ≈2,000. Nevertheless, it is important to note here that in some other works, in particular the experiment of Pfeninger,3) conducted on very smooth pipes, the laminar flow was observed for Reynolds numbers up to 100,000. Suggesting that the experiments of Pfeninger3) are correct, one should accept that the laminar flow region cannot be limited by Reynolds numbers up to 2,000. In other words, channel flow is stable to infinitively small disturbances but unstable to finite-size, yet small, perturbations. Those perturbations can originate from either wall roughness or inlet conditions. One point is that once flow became unstable, or turbulent, for any reason, it never returns to laminar if the Reynolds number is high enough (>2,300 for a pipe). Whatever is true, the observations show that for the most practical problems, turbulent flow in relatively smooth channels of the same shape definitely behaves very similar (statistically) developing a universal velocity profile.
II. MSV: Multi-Scale Velocity Medel of Turbulence 1. Energy-Balanced Turbulent Reynolds Number First of all, let us suggest possible physical explanations for Reynolds number Re=ul/ν. Generally, the Reynolds number indicates the intensity of turbulence: the higher Reynolds number—the higher flow turbulence. Vice versa, for the low Reynolds numbers, the turbulence does not occur or, at least, is very unlikely. Schlichting4) suggested that physically, the Reynolds number is a ratio of inertia forces to friction forces. This theory is supported by good reasoning for external boundary layers around solid obstacles where flow changes its direction. However, inertia forces exist neither in straight channel flows nor in plane boundary layers. Another possible treatment of the Reynolds number as the ratio of turbulent energy production ρu 3 /l to the corresponding energy dissipation µu 2 /l 2 (where µ is the molecular viscosity) was suggested by Kirillov.5) From analysis of dimensions, some other physical explanations of Reynolds number are possible. Let us suggest a Reynolds number that represents a balance of energies. For example, one can define the “energy” Reynolds number Ree as a ratio of the flow kinetic energy to the work of friction forces within the same volume of fluid:
Kinetic energy K = . (4) Work of friction W Let us consider now what kind of kinetic energy and work of friction forces can be used. To minimize the loss of energy in turbulent flow, two controversial trends, can be considered: The velocity profile tends to be flat. There always exist turbulent vortexes that transport energy from the lower speed region to the higher speed area and vice versa. For the flatter velocity profile, less energy is spent on dissipation. On the other hand, the smaller wall velocity gradient results in lower skin friction and less energy loss. We guess these contradictory factors shape the resulting velocity profile for the turbulent flow. In the case of laminar flow, there are no vortexes and no energy transport by crossflow. Therefore, the laminar flow profile is shaped by only wall velocity gradient for the given pressure drop in axial direction. Keeping the above in mind, let us suggest a universal criterion for the turbulent flow conditions. We define the kinetic energy of velocity profile mixing, or an extra energy that is released when, for the given flow rate, the flow is completely mixed, or becomes flat: ρ u¯ 2 ρV 2 ρu 2 dV − V = (u − u¯ 2 ), (5) K = 2 2 2 Ree =
V
where u¯ means the space-averaged value of u. The work of friction is defined as a total energy loss because of friction including Reynolds shear stress: ∂ u¯ n τi j d V = (6) µ n − u i u j d V, W = ∂n V
V
where τi j is a total shear stress component, that includes both molecular and turbulent friction, and the velocity components are redefined as u 1 =u, u 2 =v, and u 3 =w, and n is a vector normal to the surface of integration volume V . Using Eqs. (4), (5) and (6), the energy-based Reynolds number was calculated for the pipe flow. An experimental Reichardt’s6) formula for the velocity profile is used. Figure 1 shows the numerically calculated Ret =K /W , which we call a turbulent Reynolds number, for the different flow areas in a pipe. Here, R0 =r/d=0.5 is the relative pipe radius and outer limit of integration area, and Ri refers to the inner radius of the integration areas. Accordingly, bigger Ri corresponds to the smaller integration areas near the wall. One can see that Ret grows linearly within laminar region, reaches its maximum near the critical point Re≈2,000, and then goes to some constant value that ranges from about 8 to 11 depending on the integration area. For most integration areas numbers, Ret does not exceed some maximal value. Therefore, we suggest the following be true for every fluid area: There exists a turbulent Reynolds number Ret that never exceeds some critical value which is fixed and universal. Ret is defined as balance of kinetic energy given by Eq. (5) and work of friction forces Eq. (6). Knowing the value of this turbulent Reynolds number, one can calculate the turbulent viscosity distribution for a given velocity profile, and then, calculate the final velocity and turbulent distributions that satJOURNAL OF NUCLEAR SCIENCE AND TECHNOLOGY
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Fig. 1 Turbulent Reynolds number vs. regular Reynolds number for different integration areas Ri and R0 are the inner and outer radii of integration area.
isfy the non-exceeding Ret condition at every point of the fully-developed channel flow. The details will be discussed in the following section. Here, we would like to mention that the critical turbulent Reynolds number is the only experimental value in the MSV model. Its value can be estimated from the following experimental fact. For the all turbulent flows, there exists a viscous sublayer near the wall where u + =y + that is velocity u + =u/u τ , limited by y + 105 , but not immediately after the transition region. (3) The discrepancy between different integration areas is big enough as in the value (8≤Ret ≤10), as well as in the slope. That is natural for laminar flow but in disagreement with critical turbulent Reynolds number theory. To answer, we may suggest that: First, the experimental Reichardt’s formula6) used in these calculations is not perfectly correct. Even smooth and wellfitted experimental velocity profiles do not guarantee the same smoothness in turbulent viscosity and Reynolds shear stress distributions derived by differentiating of the measured data or experimentally obtained formulas. Second, the integration area is not necessary radial as shown in Fig. 1. It may be flat, distance-weighted, etc. For the moment, it is still unclear which type of integration to use. Definitely, it should reflect the way in which vortexes are developed in the flow. In other words, the energy balance should be applied to an area where eddies can move, exchange the momentum and dissipate the energy. The last, but not least, issue is that the actual work of friction forces must include Reynolds shear stresses from the other two directions u w and v w of stress tensor to satisfy the full energy balance. These shear stresses shall be taken into account too. Thus, the calculation procedure should include simultaneous numerical solution of Reynolds equation and an additional integral equation defined as K < Recr , (8) Ret = W where K and W are defined by Eqs. (5) and (6) correspondingly. While the numerical calculation of Eq. (5) for K is just a very simple volume integration of a given velocity distribution, one should find an appropriate distribution of u i u j , that satisfies the integral equation (8). It is not evident how to solve the system. Moreover, there may even be multiply solutions. That is why we also attempted to find a more simple method that is discussed in the following section.
difference between the values of the axial velocity taken at the ends of the corresponding scale l. An extra turbulent viscosity is generated at the middle of l. Again, if we assume that, there exists a viscous sublayer near the wall where u + =y + that is limited by y + Recr , turbulent viscosity is recalculated. (4) Using new turbulent viscosity distribution, the velocity profile is recalculated. (5) Step 2 is repeated until convergence is reached. Typical results of turbulent viscosity calculations in the pipe are shown in Fig. 3. The plots show the relative
l2 l (scale)
Fig. 2 Velocity profiles in a channel
Note, that unlike energy-balanced MSV definition (Eq. (7)), the value of y0+ =5, because the minimal scale l is exactly the double size of viscous sublayer. There is no turbulent viscosity to the middle of l. Also note, that value of Recr in Eq. (10) is very sensitive to the exact size of viscous sublayer, i.e. Eq. (10) gives very approximate value of Recr . Then, in actual calculations, we use the value of critical turbulent Reynolds number Recr =130 because it fits better experimental data for the friction factor. In the following, we shall refer to the MSV model that uses Reynolds number defined by Eq. (9), as simple MSV.
III. Application of MSV to Basic Channel Flows
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(a)
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Fig. 3 Distribution of relative turbulent viscosity within the pipe
IV. Simulation of Turbulence-Driven Secondary Flows Symmetry considerations preclude the existence of secondary flow in uniform geometry, such as a circular tube and an infinitive plane channel considered in the previous section.
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Fig. 4 Calculated and experimental velocity profiles
64/Re (1.82*log10(Re) - 1.64)-2 0.316 / Re0.25 SuperPipe Experiment
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turbulent viscosity distributions versus dimensionless radius r/d. Here, the wall location corresponds to the r/d=0.5 on the right. Turbulent viscosity profiles corresponding to the Reichardt’s6) and three-layers velocity profiles are also shown for comparison. Brief analysis shows that the calculated viscosity profiles are in relatively good agreement with empirical formulas within the wide range of Reynolds numbers. Unfortunately, detailed experimental data are not available for the turbulent viscosity distribution. Figure 4 shows the comparison between velocity profiles obtained with MSV with those measured in “Superpipe” experiment,7) as well as with empirical Reichardt’s and threelayers profiles. Again, one can see that MSV simulations are in a good agreement with experimental data. Figure 5 shows the dependence of friction factor vs. Reynolds number. Famous formulas and Superpipe experiment7) data are given for comparison. It should be noted that the friction factor is very important because it is a non-direct, implicit and integral result of the velocity/viscosity profiles combination. Friction depends on the pressure gradient that is not involved in calculations directly (via boundary conditions) but is the result of a simulation where the mean velocity is given as a boundary condition instead. As one can see, MSV is in a good agreement with experimental data. And even simplified MSV fits the friction curve well (while it cannot predict the physically-correct shape of the eddy diffusivity distribution). Similar calculations were performed for infinitive plane channels and annulus. For the latter, the position of the maximal velocity line was found to be in a good agreement with experiments. The shape of eddy diffusivity distribution also correlates well with physical expectations (in contrast to k- model calculations).
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Secondary flow that may occur in straight non-circular channels was first observed by Nikuradse.8) Nikuradse measured the turbulent velocity distribution in several of these channels and found the constant velocity lines to be distorted in a manner indicating the presence of secondary flows. The lateral variation of wall shear stress is considered as a
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primary cause of secondary flows. From the other hand, that shear stress distortion, in its own turn, may be a result of secondary flow as well. In addition, it was shown that secondary flows do not exist for fully-developed laminar flows. One explanation is that the secondary flow is a result of turbulent velocity fluctuations along the constant-velocity lines causing a net flow normal to these in a region of curvature. Another generally accepted reasoning is that secondary flow has its origin in the inequality of the normal Reynolds stresses combined with a presence of lateral wall gradient. Application of the turbulent energy-balance equation results in the rule that secondary flow transports turbulence-rich fluid away from regions where turbulence production exceeds dissipation. Every explanation above presumes that the turbulent shear stresses vary in different directions. Therefore, secondary flows in straight channels, called turbulent-driven secondary motions, cannot be predicted by isotropic models of turbulence such as k- model. In MSV, the maximal possible scale of turbulence l is equal either to distance to the wall or distance to the symmetry line. Therefore, the turbulent viscosity also depends on the direction and thus anisotropy can be modeled. In this work we apply simplified MSV where the turbulent Reynolds number is defined via velocity deformation and it’s scale by Eq. (9). The maximal distance for each direction in a cross-section of the channel is assumed as distance to the nearest wall along a given direction, or, as a maximal period in the case of periodical coordinate system.
Fig. 7 Velocity vectors and steam lines of secondary flows in a square duct
Table 1 Intensity of turbulence-driven secondary flows in square duct Reynolds number
U¯ scnd (%)
max Uscnd (%)
6,534 58,200
0.25 0.38
1.21 1.93
1. Fully-Developed Turbulent Flow in Square Duct To check the ability of simplified MSV to predict turbulence-driven secondary flow, it has been first applied to fully-developed turbulent flow in a square duct. Figure 6 shows schematic view of secondary flows that occur in such a channel. Velocity vectors and secondary flow streamlines calculated with the simplified MSV model for a symmetric quarter of square duct are shown in Fig. 7 for Re=58,200. Here, the Reynolds number is defined as follows: W¯ B , (11) Re = ν where W¯ is the mean axial velocity, B is the side size of the square duct and ν is the molecular viscosity. The calculated intensity of secondary motions in a square duct is summarized in Table 1. max Here, U¯ scnd and Uscnd are cross-section average and maximal intensity of the secondary motion with regard to the mean Fig. 8 Axial velocity profiles W/W¯ without (right-bottom) and with (left-top) secondary flows in a square duct
Fig. 6 Fully-developed turbulent flow in a square duct
longitudinal velocity. Figure 8 shows axial velocity profiles calculated with and without taking into account secondary motions. The numerical results for square duct calculated with simplified MSV are in agreement with those obtained by Reece9) who compared the Algebraic-Stress Model (ASM) and the
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Reynolds-Stress Model (RSM) with experimental data.
tary domain of an infinitive array of pins. Such an elementary cell can be found by implementing of symmetry lines on the hexagonal rod bundle as shown in Fig. 9. The pitch-todiameter ratio is given as S/D=1.17 which corresponds to the standard parameters of rod bundles of the Soviet BN600 reactor. Again, as in the case of duct flow, numerical calculations were performed for low and high Reynolds numbers, Re=8,170 and Re=160,100. The latter is related to the normal operational conditions of the BN600 sodium-cooled fast breeder reactor.
2. Turbulence-Driven Secondary Flows in Hexagonal Array of Rod Bundles The prediction of turbulence-driven secondary flow in a rod bundle of a nuclear reactor is the one of the most challenging problems reactor thermal-hydraulics ever faced. Taking into account that those secondary motions have never been measured yet in an experiment, only indirect data and side effects such as an unexpected deformation of wall shear stresses indicates an existence of secondary flow in a rod array. From the general point of view, since the geometry of the rod bundle is definitely non-uniform, such a flow is likely to occur. However, as it has been already mentioned, there are no direct experimental measurements available for comparison. It also should be noted that in work of Lemos and Assato,10) calculated secondary flows are compared with “experiments” of Carajilescov and Todreas11) despite the latter describes experimental measurements of axial velocity profiles and calculated results for turbulent-driven secondary flows. In this work, we use MSV model to simulate secondary flow in a rod bundle with numerical calculations of fullydeveloped turbulent flow in the minimal-symmetry elemen-
(a)
Secondary flow vectors
(b)
Stream lines
Fig. 11 Secondary flows in the elementary cell of infinite hexagonal rod array; Re=8,170 Fig. 9 Minimal-symmetry elementary cell in a hexagonal rod bundle
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Fig. 12 Secondary flows in the elementary cell of infinite hexagonal rod array; Re=160,100
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(a) Trupp and Azad12)
(c) Lee and Jang13)
(b) Carajilescov and Todreas11)
(d) Lemos and Assato10)
Fig. 13 Structure of secondary motions in a rod array calculated by different authors
The results of numerical calculations of axial velocity is shown in Fig. 10 for a Reynolds number Re=8,170. For a low Reynolds number, secondary flow affects longitudinal velocity slightly, however there is no visible effect in the case of a bigger Reynolds number Re=160,100. Figures 11 and 12 show velocity vectors and streamlines of secondary flows for the low and high Reynolds numbers. In numbers, for the for-
mer, the maximal cross-velocity in a plane is an about 1% of the mean axial velocity, while for the latter, the maximal intensity of the secondary flow is a negligible 0.2% of the longitudinal velocity. For comparison, Fig. 13 shows secondary flows calculated by different authors. Figure 13(a) (Trupp and Azad12) ) shows hypothetical prediction of secondary flows that based
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on the experimental measurements of axial velocity profiles. Figure 13(b) (Carajilescov and Todreas11) ) shows calculated results. For relatively low Reynolds number (Re=2.7 × 104 ) and similar pitch-to-diamer ratio, calculations11) predict two loops of secondary motions, where the loop in a narrow part of the cell is quite big. For more open channels and higher Reynolds numbers (Re=1.49 × 105 ), this loop becomes smaller and moves close to the wall. In present work, the small second loop is predicted only for the very small Reynolds number (see Fig. 11), while for higher Reynolds numbers, there is only one loop of secondary flow calculated with MSV as shown in Fig. 12. Figures 13(c) (Lee and Jang13) ) and 13(d) (Lemos and Assato10) ) show other calculations. One can see that some authors predict two loops of secondary flows while others only one. In present work, the small second loop of secondary flows has been predicted only for small Reynolds numbers (Fig. 11). It should be noted that intencity of mean secondary motions is very small, even much smaller that turbulent pulsations in the same direction. Thus, it is very difficult not only to measure but also calculate those secondary flows. Calculations are very sensitive to the model of turbulence in use and there is no standard, wide-accepted model. Most of models were constructed specially for the cases of interests and contain many constants that should be adjusted from case to case. However, the MSV model uses only one constant, that is a critical turbulent Reynolds number that is supposed to be universal for any geometry. Thus, our calculations show that secondary motions are negligible small for operational values of the Reynolds numbers. As an explanation of this phenomenon, we propose the following. Unlike the square duct and other rectangular duct flows, the geometry of the elementary rod bundle channel is more “open”. There is no strongly-defined region of “laminarized” flow as in the duct corners. It allows turbulence to develop more “freely” in the wall region, especially for the large Reynolds numbers. In this case, the main drop in velocity is developed in the very narrow boundary layer. From this point, flow development is more “uniform” than those in the duct are. It may be expected that for very close rod arrays, where the pitch-to-diameter ratio is approaching unity, or for deformed rod bundles, the effect of secondary flow would be more significant.
V. Conclusions In this work, numerical results of the Multi-Scale Viscosity (MSV) model of turbulence have been reported. MSV has been applied to fully-developed wall-bounded channels flows including basic shapes such as a circular tube, a plane channel, and annulus, and to an elementary cell of an infinitive hexagonal rod bundle array. The main idea of MSV is to define an appropriate physical criterion, which we call a turbulent Reynolds number, that reflects a level of turbulence within a given area of the channel. In doing so, we assume that the turbulent Reynolds number never exceeds a critical value that is the only experimental parameter used in the MSV model. Two different definitions of the turbulent Reynolds
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number have been proposed and investigated as: (1) A product of the value of axial velocity deformation for a given scale and generic length of this scale divided by accumulated value of the laminar and turbulent viscosity of lower scales (Simplified MSV, Eq. (9)); (2) A ratio of the difference between the total kinetic energy and the “mixed-profile” kinetic energy to the work of the friction forces (Eqs. (4), (5), (6)). The former definition is very simple and logical, though, intuitive. The latter has a clear physical meaning as a balance between two energies: (i) energy necessary to mix the velocity profile to the flat one, minimizing the energy loss due to momentum transport and (ii) work of friction forces or dissipation. Both definitions of the turbulent Reynolds number above have been examined on the basic shape flows: a circular pipe, a plane channel, and an annular channel. The results of calculation show that the MSV model can predict accurately velocity profiles and the friction factor. Even the simplified MSV model predicts correctly the friction factor in circular tube. However, the simplified model fails to calculate the right physical shape of the turbulent viscosity profile. The simplified MSV model has been applied to more complicated geometries such as a square duct and an elementary cell of an infinitive hexagonal rod bundle. Simplified MSV has predicted a reasonable numerical solution for turbulencedriven secondary flows in these geometries. References 1) L. Prandtl, “Uber die ausgebildete turbulenz,” Z. Angew. Math. Mech., 5, 136–139 (1925). 2) N. I. Buleev, “Theoretical model of turbulent exchange in fluid flow and heat-transfer problems,” Heat-Transfer, USSR Academy of Sci., p. 64–98 (1962), [in Russian]. 3) W. Pfeninger, “Transition experiments in the inlet length of tubes at high Reynolds numbers,” Boundary Layer and Flow Control, ed. G. V. Lachman, Pergamon, p. 970–980 (1961). 4) H. Schlichting, Grenzschicht-Theorie, G. Braun, (1968). 5) P. L. Kirillov, Private communications, (2000). 6) H. Reichardt, “Vollstandige Darstellung der turbulenten beschwindigkeitsverteilung in glatten Zeitungen,” Z. Angew. Math. Mech., 31[7], 208–219 (1951). 7) M. V. Zagarola, A. J. Smits, “Mean-flow scaling of turbulent pipe flow,” J. Fluid Mech., 373, 33–79 (1998). 8) J. Nikuradse, Ing.-Arch., 1, 306–333 (1930). 9) G. J. Reece, A Generalized Reynolds-stress Model of Turbulence, Ph.D. diss., Univ. of London, England, (1977). 10) M. J. S. de Lemos, M. Assato, “Simulation of axial flow in a bare rod bundle using a non-linear turbulence model with high and low Reynolds approximations,” Proc. ICONE-10, Arlington, VA, Apr. 14–18, 2002, paper 22300, (2002). 11) P. Carajilescov, N. E. Todreas, “Experimental and analytical study of axial turbulent flows in an interior subchannel of a bare rod bundle,” J. Heat Transfer, 98, 262–268 (1976). 12) A. C. Trupp, R. S. Azad, “The structure of turbulent flow in triangular array rod bundles,” Nucl. Eng. Des., 32, 47–84 (1975). 13) K. B. Lee, H. C. Jang, “A numerical prediction of the turbulent flow in closely spaced bare rod arrays by a non-linear k- model,” Nucl. Eng. Des., 172, 351–357 (1997).
