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The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. Paper: 405 LARGE EDDY SIMULATION OF FLOW ACROSS IN-LINE TUBE BUNDLES Sofiane, Benhamadouche1, Dominique, Laurence, Electricité de France R&D, MFTT, 6 quai Watier, 78400 Chatou, France. [email protected] Nicolas, Jarrin, Imran, Afgan, Charles, Moulinec. Manchester University, MAME Dept, George Begg Bldg, Sackville St. PO box 88, Manchester M601QD, UK ABSTRACT The cross-flow in a 3D square in-line tube bundle is computed using Large Eddy Simulation (LES) with periodic boundary conditions. Both the in-house EDF code Saturne and CD-Adapco code STAR CCM are tested in the present work. The numerical method in both codes is based on a finite volume approach on unstructured grids with a collocated arrangement for all the unknowns. Two aspect ratios are tested, P/D=1.44 and P/D=1.75. The influence of two parameters is explored herein; the accuracy of the pressure gradient calculation and the spanwise extrusion length (Lz). Drag and Lift coefficients and their rms values are calculated and compared to available experimental data. It is shown that a high accuracy is needed for pressure gradient calculation to obtain a physical solution. In addition, one needs at least Lz = 2D in the homogeneous direction, where D is the diameter of the tubes, to accommodate the use of periodic boundary conditions. Drag and lift coefficients have reasonable values, but their rms values are very sensitive to the extrusion length. Although, rms values are satisfactory for P/D=1.44, they are much larger than experimental data for P/D=1.75. All the cases presented are computed with Saturne, and Star CCM is used on some cases to confirm asymmetric results and sensitivity for the narrow gap case. KEYWORDS Large Eddy Simulation, In-line Tube Bundle, Drag and Lift Coefficients 1. INTRODUCTION Heat exchangers are vital components of power generation systems, which present many issues that are not understood in detail and others that need to be optimized. These include heat transfer enhancement vs. head loss, hot spots, vibrations, fluid structure coupling, in line / staggered / oblique flow direction, two phase flows and dust deposition … A major parametric CFD study of tube bundles would seem timely and worthwhile in view of the billion dollars per year potential savings worldwide, for coal power plants alone (Bouris et al., 2001). However, Reynolds averaged Navier Stokes (RANS) simulations have so far failed to produce reliable predictions of flows across tube bundles. This provocative statement is based on numerous reported RANS simulations on staggered tube bundles: the ERCOFTAC workshops on refined flow modeling of 1993 and 1994, (Rollet-Miet et al., 1999), (Sebag et al., 1991), (Meyer, 1994), and (Bouris and Bergeles, 1999). The general conclusion is that even advanced RANS models such as non-linear, realizable and RNG types of k-epsilon models severely underestimate the high turbulent kinetic energy levels observed in densely packed tube bundles. Paradoxically the standard k-epsilon model returns reasonable predictions of mean velocities 1 Corresponding author 2/12 The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. and global level of turbulent kinetic energy, but this is purely by chance thanks to the erroneous overproduction of kinetic energy on the impinging side of the tubes. This artificially raises the overall turbulence intensity but the locations of the maximum and minimum are erroneous. These discrepancies have led (Hassan and Ibrahim 1997) and (Bouris et al. 1999 and 2001), to resort to «two dimensional LES» with some success including the prediction of turbulence levels. However debatable this approach may be, it yielded better results for the mean velocities and turbulent kinetic energy than RANS models, thus allowing some analysis of stress loading, heat transfer and deposition rates which are highly dependent on the turbulence intensity. (Rollet-Miet et al. 1999) and later (Benhamadouche and Laurence 2003) applied respectively an industrial finite element code, N3S, and an industrial finite volume code, Saturne, to 3D LES of the flow in a staggered tube bundle and obtained good results and observed that the subgrid scale model (whether Smagorinsky or dynamic) had very little effect. The present simulations use the LES implemented in Code_Saturne to compute another configuration of tube bundle, the square in-line one. The commercial code, STAR CCM is then used to confirm the present results. 2. NUMERICAL METHOD A finite volume code for complex geometries, Code_Saturne (Archambeau et al. 2004), is used to solve 3D Navier-Stokes equations on unstructured grids. The flow is assumed incompressible and Newtonian and the density ρ is constant. Let u be the filtered velocity when using LES. The filtered Navier-Stokes equations can be written for LES: ∂ τ ij ⎧∂ u i ∂ u i u j ∂ 2 ui 1 ∂ p ν , i ∈ {1,2,3} + = − + − ⎪ ∂ xj ρ ∂ xi ∂ xi ∂ x i ∂ x j ⎪∂ t (1) ⎨ ⎪∂ u i ⎪∂ x = 0 i ⎩ As (Rollet-Miet et al. 1999) have shown that the subgrid scale model is not crucial in the tube bundle case, only the standard Smagorinsky model is used herein: 1 3 τ ij − τ kk δ ij = −2ν t S ij = −2(C S ∆) 2 S S ij (2) S ij is the filtered strain rate tensor ( S = 2 S ij S ij ), νt the turbulent viscosity and ∆ the filter length. As the cells used in the present work are hexahedral (even if Saturne or STAR CCM accept cells of 1 3 any shape), one can take ∆ = 2Ω , where Ω is the volume of a cell. The Smagorinsky constant, CS, is set to 0.065 . In the collocated finite volume approach used here, all variables are located at the centers of gravity of the cells (which can be of any shape). The momentum equations are solved by considering an explicit mass flux (the three components of the velocity are thus uncoupled). Velocity and pressure coupling is insured by a prediction/correction method with a SIMPLEC algorithm (Ferziger & Peric, 1999). The Poisson equation is solved with a conjugate gradient method. The collocated discretization requires a (Rhie and Chow 1982) interpolation in the correction step to avoid oscillatory solutions. In Saturne, a second order centered scheme (in space and time) is used. It is Crank-Nicolson in time with a linearized convection, and the second order Adams-Bashforth method is used for the part of the diffusion involving the transposed gradient operator, that couples the velocity components. The nonorthogonalities are taken into account with a reconstruction technique explained in (Archambeau et al. 2004) and (Ferziger and Peric 1999). When a non-orthogonal grid is used, the matrix contains the orthogonal contribution only and the non-orthogonal part is included in the right hand side of the equation. This is known as the deferred correction (the non-orthogonal part is explicit), however one can iterate on the system to make it implicit. Typically 5 iterations are performed in Saturne and the CPU cost is double compared to the deferred correction. The effect of these two different methods will be shown later. The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. 3. 3/12 CASE DESCRIPTION The present test-cases require only periodic and wall boundary conditions. All the meshes used in the present work do not require wall functions, thus no-slip boundary conditions are used. In-line configurations of tube bundles are computed in the present work. The bundle is assumed infinite in all directions, thus one needs to choose a periodic cell (as shown in figure 1) to compute the whole bundle. Although several tests have been performed with several cells, one will focus on the cell with two tubes to minimize the computational costs. As periodic boundary conditions are used in the three directions of space, the mass flow is imposed by the mean pressure gradient. This is done in Saturne by adding a source term to the momentum equations as shown in equation (3). dp k +1 = dp + α k 2(U k − U g ) − (U k −1 − U g ) 2 ∆t (3) dp k is the source term (homogeneous to a pressure gradient) at the k th time iteration, U k the bulk velocity calculated in a chosen plane of the computational domain (the first periodic plane in the streamwise direction for example), ∆t the time step and α a relaxation coefficient as this formulation may be unstable at the beginning of the calculation. In STAR CCM, a constant pressure gradient that gives the right bulk velocity is added to the right hand side of the momentum equations. Boundary conditions are shown in figure 2. No particular initial field is used; the initial value of the velocity is zero. The relaxation coefficient has been taken to 10% to stabilize the imposed flux at the beginning of the calculation. Drag and Lift coefficients are calculated for the central tube at each time step. When frequencies of oscillations are given, this is done by using Power Density Spectra (PDS) and localizing the peak values (the most energetic frequency). All the experimental data used in the present paper are taken from (Chen, 1987). P and D stand respectively for the horizontal or vertical distance between two centers of tubes and the diameter of the tubes. The given Reynolds numbers (Re) are based on the diameter of the tubes, the kinematic viscosity of the fluid and the gap velocity. Two configurations are studied with Saturne: P P = 1.44 , Re = 70 000) and ( = 1.75 , Re= 20 000). STAR CCM calculations are done for D D P ( = 1.5 , Re = 45 000). This aspect ratio is close to 1.44 and the two configurations can be compared D ( as they are both at high Reynolds numbers. The same mesh is used for the two configurations with Saturne (figure 3a); whereas mesh for STAR CCM is shown in figure 3b. Table 1 summarizes the different parameters of the main calculations. The number of cells in 2D indicates the number of 2D cells obtained by cutting the computational domain with a plane orthogonal to the spanwise direction (this 2D mesh is then extruded in the z direction to obtain the 3D mesh). Case 1 Case 2 Case 3 Code Saturne Saturne STAR CCM P/D 1.44 1.75 1.50 Re 68 420 20 000 45 000 Table 1: Test-case configurations Number of cells in 2D 11 000 11 000 12 800 The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. 4/12 4. Figure 1: Possible computational cells Figure 2: Boundary conditions Figure 3a: Mesh used with code Saturne Figure 3b: Mesh used with code STAR CCM RESULTS 4.1 Pressure accuracy, P = 1.44 D The need of high accuracy for pressure is shown in this section. As the mesh is non-orthogonal, a reconstruction method of gradients has to be used to take into account the non-orthogonalities that occur particularly with the present meshes. One can either use a deferred non-orthogonality correction (deferred n-o.c.) where the reconstruction is put explicitly on the right hand side of the Poisson equation or iterate on the reconstruction to make it implicit (implicit n-o.c.). The two methods have been tested with Saturne for P = 1.44 . The extrusion length in the spanwise direction is Lz = D . D Figures 4 and 5 show an instantaneous pressure field for the two methods. It appears that the pressure with an explicit n-o.c. exhibits a staggered mode as the peak of the pressure is alternatively on the top or on the bottom of the tubes. According to (Chen, 1987) this pattern occurs for wider crossstream spacing. When implicit n-o.c. is used, the pressure field has the same behavior on all the tubes and shows a high pressure region on the bottom of the tubes, which is believed to be the correct solution according to the experimental lift coefficient. Figure 6 shows instantaneous evolution of the lift coefficient The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. 5/12 computed for the two cases. With deferred n-o.c., the lift coefficient has a non-zero mean value. The mean value of the lift is zero when implicit n-o.c. is used although a non-symmetrical solution is obtained for the pressure. There is no numerical reason to obtain a high-pressure region on the bottom or on the top as the mesh is perfectly symmetrical. The solution is stable and the phenomenon is similar to the Coanda effect obtained with diaphragms. It is mentioned in (Chen, 1987, p. 322) that Heinecke and Mohr observe experimentally a non-symmetrical solution for the pressure field. It is observed that the high-pressure region is compensated with a low-pressure one just downstream (see figure 5), what gives a zero mean lift coefficient. This case is thus sensitive to the pressure resolution. Initially it was believed that this was due to insufficient representative cells so a test case with four tubes (figure 1) was run. Figure 7 exhibits the mean value of pressure for this case. The alternating mode of the stagnation pressure is again obtained. These observations have been confirmed by the code STAR CCM which uses an explicit deferred correction on the pressure. Figure 8 shows the pressure field obtained for Figure 5: instantaneous pressure field Figure 4: instantaneous pressure field ( P = 1.5 with STAR CCM. D P = 1.44 , Lz = D , deferred n-o.c) D ( P = 1.44 , Lz = D , implicit n-o.c) D 0.5 0 −0.5 0 0.5 1 temps (s) 1.5 Figure 6: instantaneous lift coefficient (black: low accuracy, red: high accuracy) 2 Figure 7: mean pressure field ( P = 1.44 , Lz = D , D deferred n-o.c, four tubes) 6/12 The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. Figure 8: mean pressure field ( 4.2 Length extrusion L z , P = 1.5 , Lz = D , deferred n-o.c, STAR CCM code) D P = 1.44 D All the following cases with Saturne will be carried out with an implict n-o.c. in pressure gradient calculations. The same 2D mesh has been used by taking L z = 2 D . The results are qualitatively similar the ones with L z = D . Figure 9a shows the mean pressure field obtained by Saturne. A non-symmetrical solution is observed, as was the case with L z = D . A calculation was also done with STAR CCM code with an explicit deferred correction, P = 1.5 and L z = 2 D , shown in Figure 9b. The same behavior D for the pressure field is noticed. This is strange as the case was sensitive to the pressure accuracy with STAR CCM while using L z = D . This is a first indication that one has to take at least L z = 2 D to compute this case to be insensitive to the periodic boundary conditions in the spanwise direction. Figures 10 and 11 show the mean velocity vector field for Saturne and STAR CCM codes. Two recirculation bubbles coexist, a big one downstream the tube in the bottom due to the acceleration of the fluid and a small one on the top. The shear stress in the bottom of the tube is then higher than in the top. To illustrate this, figure 12 shows the fluctuations in the streamwise direction. Figure 13 gives the evolution of normalized pressure coefficient along the curvilinear coordinate around the central tube. Both codes agree very well with each other, though somewhat shorter time-sampling was collected with Star-CCM. As one can see, the pressure peak is clearly offset from the axis, in correspondence with the velocity plot. Moreover the Cp curve show that the stagnation point pressure is compensated by a low-pressure region (negative values of Cp) on the same side, which gives an overall zero mean lift coefficient. The mean values of the drag and lift coefficients for the pressure forces and their rms values are summarized in table 2. It has been observed that viscous coefficients are small compared to pressure ones (not more that 10%). Drag and lift coefficients are computed using the gap velocity as a reference velocity and the reference surface is S = L z D . The drag and lift coefficients remain unchanged and are close to usual experimental values. The rms values are more sensitive to the extrusion length particularly for the rms value of the drag coefficient which is two times bigger with L z = D . To validate our approach, the experimental values found in (Chen, 1987) for P T = 1.42 , = 1.42 and D D Re = 1360 are used. T is the vertical spacing between the tubes. The coefficients should not be strongly affected by variations of the turbulent Reynolds numbers (this is observed in Chen’s book for The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. 7/12 P = 1.75 ). The rms of the lift coefficient is equal to 0.078 which is close to the numerical value D found in the present work with Saturne and L z = 2 D . It would have been interesting to test a wider P = 1.75 . At this stage one can argue that it is extrusion. This is done in the next section with D mandatory to take an extrusion length at least equal to 2D. It is shown next that this extrusion length is enough for P = 1.75 . D An analytical formula for the Strouhal number (4) is proposed in Chen’s book. It depends only on the tube spacing but not on the Reynolds number which is not really realistic but gives a good order of the Strouhal number. St = For 1 (4) P 2( − 1) D P = 1.44 , one finds S t = 1.13 . By applying PDS to the lift’s signal, two main peaks are obtained D around the frequencies 35 Hz and 50 Hz (see figure 14). These two values correspond respectively to the Strouhal numbers 0.8 and 1.25 which straddles the analytical model. This means that the vortex shedding detected in the shear regions with LES is realistic. The Strouhal number obtained with STAR CCM and based on the gap velocity is around 0.9 which is also close to Saturne results. It is concluded that the length of the extrusion has to be at least 2D to obtain a physical solution and realistic drag and lift coefficients. The previous configuration corresponds to an industrial one. Unfortunately, additional experimental data is only available for the P = 1.75 case. This is presented D in the next section. case Saturne, L z = D CL 0.001 CD 0.339 C’L 0.107 C’D 0.091 Saturne, L z = 2 D 0.013 0.350 0.074 0.049 STAR CCM, L z = 2 D -0.004 0.351 0.032 0.026 Table 2: Drag and Lift coefficients Figure 9a: mean pressure field (Saturne, Figure 9b: mean pressure field (STAR CCM, P = 1.44 , L z = 2 D , implicit n-o.c) D P = 1.5 , L z = 2 D , deferred n-o.c) D The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. 8/12 Figure 10: mean velocity field (Saturne, Figure 11: mean velocity field (STAR CCM, P = 1.44 , L z = 2 D , implicit n-o.c) D P = 1.5 , L z = 2 D , deferred n-o.c) D 1.00 SATURNE STAR CCM 0.75 Normalized Pressure 0.50 0.25 0.00 -0.25 -0.50 -0.75 Azimuthal Angle -1.00 0 Figure 12: u 'u ' (Saturne, P = 1.44 , L z = 2 D , D 45 90 135 180 225 3e−04 2e−04 1e−04 0e+00 0 315 Figure 13: Cp around the central tube (Saturne and STAR CCM, L z = 2 D ) implicit n-o.c) −1e−04 270 50 100 150 frequence (Hz) Figure 14: DSP of the lift coefficient (Saturne, P = 1.44 , L z = 2 D , implicit n-o.c) D 360 The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. 5. ASPECT RATIO 9/12 P = 1.75 D In this section all the calculations are done with Saturne and with implicit deferred n-o.c. The Reynolds number is moderate and equal to 20 000. The parameter which varies in the present section is the extrusion length ( L z = D , L z = 2 D , L z = 2.5 D and L z = 5 D ). All the meshes have the same 2D mesh and exactly the same numerical options. Figures 15 and 16 show respectively the mean pressure field and the mean velocity vector for the case with L z = 2 D . A perfectly symmetrical solution is observed with two high-pressure regions upstream the tubes and two symmetrical recirculation bubbles downstream. This entails naturally a zero mean lift coefficient. Table 3 gives the drag and lift coefficients and their rms values. The last line indicates the range of the rms values in an experiment (Chen, 1987) with exactly the same parameters (as the number of tubes in an experiment is limited, only the middle tubes which are supposed to be independent from the inlet and outlet conditions have been considered). The mean drag and lift coefficients are not available for this set of data. These results are also available in (Zdravkovich, 2003, p. 1137). As it was the case with the previous tube spacing, drag and lift coefficients are not sensitive to the extrusion length. The lift coefficient has almost a zero mean value and the drag coefficient has a value around 0.34. The rms values are very sensitive to L z particularly when this parameter is small. Both the rms values of the drag and the lift have an asymptotic behavior after L z = 2 D . The rms of the drag is around 0.07 whereas that of lift is around 0.3. Table 3 shows that the value L z = 2 D is enough to simulate the present case. Although, the rms value of the drag coefficient is close to the experiment, the rms lift coefficient is still three times higher than the experimental value. The use of the dynamic model seems mandatory and this will be done in a future study. The analytical formula (4) indicates that the Strouhal number should be of the order of 0.7. Table 4 gives the Strouhal numbers for the different cases. All the cases exhibit two main frequencies respectively around 0.7 and 1.2. The first frequency is very close to the analytical result. The case Lz=D is particular as it exhibits the two frequencies but with more energy at the second one (see figure 17). In the three other cases, the behavior is different as the two main frequencies have a comparable energy. It is then highly recommended to take at least the value Lz=2D to simulate the present case. Lz = D Lz = 2D L z = 2 .5 D L z = 5D Chen exp. CL CD C' L C' D -0.011 0.358 0.430 0.138 -0.017 0.331 0.291 0.074 -0.010 0.331 0.313 0.076 -0.015 0.339 0.307 0.070 - - 0.08-0.10 0.04-0.08 Table 3: Drag and lift coefficients and their rms values ( Lz = D Lz = 2D L z = 2 .5 D L z = 5D Analytical formula St1 St2 0.73 1.22 0.74 1.18 0.73 1.19 0.72 1.18 0.7 - P Table 4: Strouhal numbers ( = 1.75 ) D P = 1.75 ) D 10/12 The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. Figure 15: mean pressure ( P = 1.75 , L z = 2 D ) D Figure 16: mean velocity ( Figure 16: DSP of the lift coefficient ( P = 1.75 , L z = 2 D ) D P = 1.75 ) D 6. CONCLUSIONS The in-line configuration of infinite tube bundle is investigated in the present paper. Two tube spacing have been computed with collocated finite volumes using Large Eddy Simulation (LES): and P = 1.44 D P = 1.75 . D For the narrow gap, a strongly asymmetric solution is obtained, with one large and one small vortex in the mean velocity field, and a stagnation point well off the symmetry plane. The larger vortex is on the same side as the stagnation point, which deflects the main street flow towards the same side for the next tube. Hence all tubes in the same line have the same asymmetry. In earlier simulations, a depth of only 1D had been used in the homogeneous direction. In this case a lower time accuracy scheme for the pressure yielded a spatially alternating pattern for the stagnation point, whereas only a higher accuracy pressure scheme seemed to produce the correct results. However, when L z = 2 D , both schemes yielded the same correct pattern. This dependency on the extrusion The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. length has been confirmed with 11/12 P = 1.75 in which the drag and lift values have an asymptotic D behavior after L z = 2 D . This is due to the periodic boundary conditions which require a spanwise length twice larger than the bigger 3D turbulent structures. One assumes that in the case L z = D , two dimensional structures may be created that entail a non-physical solution and wrong rms values for the drag and lift coefficient otherwise. The higher accuracy pressure scheme yielded correct results with L z = D but is not explained. Perhaps it is coincidental, or perhaps higher pressure resolution allows a breakdown of the 2D solution but smaller 3D perturbations. With P = 1.44 and L z = 2 D , the results D are very satisfactory and this promising for future calculations which aim to estimate the derivative of the lift coefficient for a series of fixed displacements of the central tube in the vertical direction. These results can be exploited to compute a realistic fluid-structure with Quasi Linear Models for example (QMI) (Price and Païdoussis, 1997). Unfortunately, the conclusions are not totally optimistic for P = 1.75 . In this case, one shows D that a symmetrical solution is obtained, which is physical because of the tube spacing, but the rms values of the lift coefficient are seriously overestimated. The next step in this work is to test the dynamic Smagorinsky model in which the Smagorinsky constant varies in space and is more representative of the reality in the boundary layers, which will probably affect considerably the lift coefficient. REFERENCES Archambeau, F. Méchitoua, N. and Sakiz, M. 2004. 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Turbulence prediction in two-dimensional tube bundle flows using large eddy simulation, Nuclear Technology 119, 11-28. Meyer K. E. 1994. Experimental and Numerical Investigation of Turbulent Flow and Heat Transfer in Staggered Tube Bundles. Ph. D. Thesis, AFM 94-03, Tech. Univ. of Denmark. Rhie C.M. and Chow W.L. 1982. A numerical study of the turbulent flow past an isolated airfoil with training edge separation. AIAA paper, 82-0998. Perić, M. 2004. Flow Simulation Using Control Volumes of Arbitrary Polyhedral Shape, ERCOFTAC bulletin No. 62, Page 25-29. 12/12 The 11th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics (NURETH-11) Popes’ Palace Conference Center, Avignon, France, October 2-6, 2005. Price S. J. and Païdoussis M. P. 1997. An improved mathematical model for the stability of cylinder flows subject to cross-flow, J. of Sound and Vibration, 97(4), 615-640. Rollet-Miet P., Laurence D., Ferziger J. 1999. LES and RANS of Turbulent Flow in Tube Bundles, I. J. Heat & Fluid Flow, 20, 241-254. Sebag S., Maupu V., Laurence D. 1991. Non-orthogonal calculation procedures using second moment closures, TSF 8. Zdravkovich, M. M. 2003. Flow around circular cylinders, vol 2: Applications. Oxford University Press.