A CFD study of flow quantities and heat transfer by changing a vertical to diameter ratio and horizontal to diameter ratio in inline tube banks using URANS turbulence models

TL;DR: In this article, the effect of changing the aspect ratio on the heat transfer and flow quantities over in-line tube banks was investigated and two types of inline arrangements, square and non-square configurations, were employed; the models that were examined are a standard k-e model, SST k-ω model, v2-f model and EB-RSM model.

Abstract: This paper reports the effect of changing the aspect ratio on the heat transfer and flow quantities over in-line tube banks. Two types of in-line arrangements were employed; square and non-square configurations. The models that were examined are a standard k-e model, SST k-ω model, v2-f model, EB k-e model and EB-RSM model. The closer results to the experimental data and LES were obtained by the EB k-e and v2-f models. For the square pitch ratios, the solution has faced a gradual change from a strong asymmetric to asymmetric and then to a perfect symmetry. The strong asymmetric solution was found by the very narrow aspect ratio of 1.2. However, the behaviour of cases of 1.5 and 1.6 became less strong than that predicted in the case of 1.2. In the larger aspect ratio of 1.75, the flow behaviour is seen to be absolutely symmetric for all variables under consideration except Nusselt number. For the very large pitch ratio of 5, the flow has recorded maximum distributions for all parameters on the windward side of the central tube with a perfect symmetric solution around the angle of 180° while the vortex shedding frequency has recorded minimum value and the Strouhal number; therefore, has given the smallest value. However, for the non-square pitch ratio of constant transverse distance, the solution is still asymmetric for all parameters with merely one stagnation at the angle of 52° at the case of the 1.5 × 1.75 while by increasing the longitudinal distance to 2 and 5, the solution provided a comprehensive symmetry for all variables with two vortices are fully developed mirrored in shape on the leeward side of the central tube. On the contrary, for the non-square pitch ratio of constant longitudinal distance, the flow of the case of 1.75 × 1.5 provided two stagnation locations at around 52° and 308° with a very similar solution to the case square ratio of 1.75 for all variables whereas by increasing the transverse distance to 2 and 5, the solution recorded was not perfectly symmetric resulting in two different vortices and one stagnation position located at the leading edge of the cylinder provided by the case of 5 × 1.5. In terms of vortex shedding effect, the reduction in the Strouhal number at a constant transverse pitch is less steep than those at a constant longitudinal pitch.

Summary

Introduction

• This paper reports the effect of changing the aspect ratio on the heat transfer and flow quantities over in-line tube banks.
• In addition to that, the new turbulence model EB k-ε would be used in such application which was not used in literature as well.

2.1. Flow Periodicity

• The periodic boundary conditions can be described as an assortment of boundary conditions which can be selected in order to discretize very large systems by employing a small section from the system.
• In general, the simulation of tube bundles can be significantly simplified by knowing the fact that the flow repeats itself after passing the entrance by a certain length which in turn leads to allow the flow to be periodic at the certain cyclical boundaries.
• The periodic boundary condition as commonly known has a specific condition which is the flow goes out from the one domain is forced to return back to be inflow to another one and thereby the directions would be infinite.
• Generally, the periodic boundary condition should be specified by either a constant pressure drop or a constant mass flow rate between the inner and outer domains.

2.2. Computational Mesh Generation

• The present work has been undergone to study the mesh resolution depending on the case of (ST/D=SL/D=1.6) and similarly the other subsequent cases have been conducted by considering the similar grid parameters.
• Four meshes are examined in order to select the more efficient one according to the accuracy of the results and the stability of the solution.
• These meshes are named; coarse, intermediate, fine and very fine meshes.
• Sixteen cases are simulated and their cell densities and mass flow rates are summarized in table 1.

2.3. Tube bundle domain

• Many preceding periodic investigations have been done such as (Beale and Spalding, 1999, Benhamadouche et al., 2005 and Afgan 2007) concluded that the domain of 2X2 has to be enough in terms of capturing both the unsteady flow physics and mean interested mean characteristics.
• Nevertheless, the domain of 4X4 tubes was investigated numerically by (Benhamadouche et al., 2005 and West 2013) to provide the same flow patterns with another important feature which is there is no difference in the mean characteristics.
• They also reported that the minimum spanwise direction must be twice the tube diameter (Lz=2D) in order to sufficiently cover the flow physics which take place through the inline tube bundles.

3.1 Mesh independence study

• The most important step in the CFD simulations is to achieve a mesh that creates an opportunity to give more accurate results and faster convergence.
• Figure 1 illustrates the pressure coefficient distribution around the central tube using four meshes.
• The more accurate result compared to the very fine grid has been provided by the fine mesh and also the solution is nearly stable as well as the grid provided an independent solution.
• In terms of wall treatment, the type of (all-y+ wall treatment) has been employed which is available in the STAR CCM+ solver.

3.2 The turbulence modelling selection

• The square pitch ratio of 1.6 is chosen in order to judge the performance of different RANS models comparing with the measurements of Aiba et al. (1982) at the Reynolds number of 41000 and LES predictions of Afgan (2007) for 2D URANS and 3D URANS calculations.
• The two parameters are selected for the comparisons which are the normalized pressure coefficient distribution around the central tube and the normalized velocity profile at the wake of the central column.
• In addition to that, the flow patterns using velocity streamlines for 2D URANS cases are selected to compare with the LES flow pattern reported by Afgan (2007).

3.2.1 The normalized pressure coefficient distribution

• Figure 2 shows the normalized pressure coefficient, Cp, profile around the central tube for square pitch ratio of 1.6 for 2D URANS , and 3D URANS , calculations.
• It is obvious that the experimental data (just available for half cylinder 0°-180°) and LES results provided high stagnation pressure on the stream-wise direction located around 45° while the low pressure located at 90°.
• The LES predicted another low-pressure location at 215°.
• Fortunately, all the, 2D URANS and 3D URANS calculations seem to provide good agreement with experimental data and LES prediction in some regions like stagnation point and the minimum pressure except the k-ɛ model.
• Some of them could not match the experimental data and LES prediction in other regions, especially at the second half of cylinder (180°-360°).

3.2.2 The normalized velocity profiles at the wake of the central tube

• The k-ɛ model failed to predict the peaks in the same locations shown by the measurements and LES just in 2D URANS case while the EB-RSM shows good agreement above the central tube but was far away from the experimental data below the central tube.
• The k-ɛ model gave a better prediction in the 3D case but still not close to the measurements and LES.
• Nevertheless, the behaviour of the three rest turbulence models (SST k-ω, v2-f and EB k-ɛ) is approximately the same in all cases and in all regions as well.
• In all cases, their results are nearly close to the experimental data and LES above y=0.04m, whereas below y=0.04m, their behaviour is better and their path is more uniform.

3.2.3 Flow patterns

• Figure 4 presents the flow patterns predicted by URANS models compared with the LES prediction of Afgan (2007) for the square pitch ratio of 1.6.
• In spite of the fact that the k-ɛ model is valid for most engineering applications with results reasonably accurate, its performance becomes very poor in flow with large pressure gradients, high streamline curvature and strong separation.
• Eventually, the results in the next sections are presented either by the EB k-ɛ model or v2-f model or both.
• With increasing the pitch ratio for limited value (say <5), the flow is able to show more uniform presentation and nearly symmetric behaviour like the case of 1.75 with two stagnation points at around 50° and another one around 310° as presented in Figure 5d.
• By increasing the transverse distance to 2, the tube walls became slightly far away from each other and thus the restriction against the flow has been gradually reduced.

3.4 Time-averaged velocity profiles

• The time-averaged velocity distributions divided by the mean velocity at the wake of the central column of the tubes for square inline configurations are presented in Figure 7.
• The results are represented for 2D and 3D simulations by two turbulence models; v2-f and EB k-ɛ.
• This is could be due to the narrow space between tubes that does not allow the flow to move easily and; therefore, when the flow hit the tube, the flow is forced to be deviated from the centreline leading to creating maximum deviated behaviour.
• One can easily notice two things in Figure (7b); the deviation of the velocity stream is significantly reduced and the flow tries to be symmetric.
• The second observation is that the thickness of the profile between tubes became bigger due to increasing the pitch ratio which require bigger mass flow rate and; therefore, the maximum ratio of (U/Uo) decreased from 3.8 to 3.2.

3.6 Flow patterns

• The flow pattern of fluid can easily characterize the pressure drop and heat transfer in tube bundles.
• This is due to the mutual interference of the flow which leads to cause a presence the interesting and unexpected phenomena.
• Another feature one can be taken into account which is the fluid acts to flow diagonally among the tubes instead of following in the stream-wise direction.
• Figure 11 shows the time-averaged velocity streamlines for 2D non-square pitch ratios at a constant longitudinal distance using the EB k-ɛ model.
• If one compares the square pitch ratio of 1.5 shown in figure 13b with the case of 1.75X1.5 observed in Figure 11a, the flow is still denominated by the asymmetrical behaviour with something is interesting can be noted which is the bubble sizes became bigger and also filled the recirculation regions.

3.7 The Nusselt number distribution around the second tube

• In general, the calculations of the Nusselt number of the in-line tube bundles are strongly dependent on three variables.
• All of these variables are set to be constant in the present work.
• Therefore, approximately all the maximum values shown in figure 13, are close to each other while the locations of them are different due to the flow deflection through tube banks.
• That gives an idea that the flow tries to be symmetric but needs further relaxation in the pitch ratio.
• By further increasing the transverse distance to 5, the flow has now been able to provide a perfect symmetry at around 180° and the Nu distribution has concentrated just on the windward side of the central tube (360°-0°).

3.8 Vortex shedding

• Figure 14 presents the variation of stream-wise velocity with time at a point just behind the central tube in the case of square pitch ratio of 1.6.
• For this vortex shedding, it is noticed that the period of time is nearly 0.0225 sec and the corresponding frequency is 44.4 Hz.
• Therefore, the non-dimensional Strouhal number can be computed by multiplying the frequency by the tube diameter and the free-stream velocity.
• The smaller the square pitch ratios, the larger the Strouhal number would occur.
• When the square pitch ratio has been relaxed to 1.5, 1.6, 1.75 and 5, the space between tubes has gradually increased and turbulence accordingly decreased.

4. Conclusion

• Computational investigations were performed for square and non-square in-line tube bundles to study the effect of changing the aspect ratios on the pressure coefficient distribution, velocity profile, turbulence intensity, flow patterns, Nusselt number distribution, and vortex shedding.
• Five turbulence models were tested, namely (EB RSM, SST k-ω, standard k-ɛ, v2-f and EB k-ɛ) and after comparing the results of the square ratio of 1.6 with the experimental data of Aiba et al. (1982) and the LES prediction of Afgan (2007), the best turbulence model was selected for presenting the results.
• The best results are achieved by the EB k-ɛ model and the v2-f model also provides good results.

Figures

Figure 12: Mean velocity streamlines for non-square pitch ratios at a constant transvers distance using the EB k-ɛ model.

Figure 13: Mean Nusselt number distributions around the central tube using the EB k-ɛ model.

Figure 15: A comparison between experimental data of Fitz-Hugh (1937) and predicted results by the EB k-ε model for 3D pitch square ratios.

Figure 4: A comparison between 2D URANS and LES of Afgan (2007) using mean velocity streamlines.

Figure 5: The normalized pressure coefficient around the central tube in square in-line tube bundle using URANS.

Figure 6: The normalized pressure coefficient (Cp) around the central tube in non-square in-line tube bundle using 2D unsteady EB k-ɛ model.

Figure 8: Time-averaged velocity profile at the wake of the central tube for 2D non-square configurations at constant longitudinal distance using unsteady EB k-ɛ model.

Figure 7: Time-averaged velocity profile at the wake of the central column for unsteady, 2D and 3D square configurations.

Figure 9: Time-averaged velocity profile at the wake of the central tube for 2D non-square configurations at a constant transverse distance using unsteady EB k-ɛ model.

Table 2: Free-stream velocities for all cases in the present work.

Figure 1: Pressure coefficient distribution around the central tube of the case of 1.6x1.6 for all grids.

Table 3: Strouhal numbers of all cases using the EB k-ε model.

Figure 11: Mean velocity streamlines for non-square pitch ratios at a constant longitudinal distance using the EB k-ɛ model.

Figure 10: Mean velocity streamlines for the 2D square pitch ratios using the EB k-ɛ model.

Figure 3: A comparison between RANS models and experimental data of Aiba et al. (1982) and LES of Afgan (2007) using normalized mean velocity in the wake of central column; (a) 2D URANS, (b) 3D URANS, and (d) a schematic diagram shows the certain location.

Figure 2: The normalized pressure coefficient around the central tube compared with experimental data of Aiba et al. (1982) and LES of Afgan (2007).

Table 1: Number of cells and mass flow rates (kg/s) of the cases used in the present work.