Effect of time-step size with LES

This sub-section provides a study of the cylinder flow simulations performed at three different time-steps. We have done all simulations using Grid B for the LES model. The choice of a smaller time step might be superfluous in certain points of the flow. Also, the numerical grid will most probably not be fine enough to simulate the length scales which correspond to the micro time scales and so the the subgrid scale (SGS) model will play an important role in the simulation for certain points of the flow. Nonetheless, it is interesting to see the effect of the time-step size on the results of the simulation.

Table 9 contains the experimental data, predicted values of the C_d and the Strouhal number with different time-step size of the LES model. Here, we have considered three time-steps(∆t) as10ms,1.0msand 0.5ms. From table, it is fact that the simulation with1.0ms time-step produces better C_d value and it has good agreement with exper-imental. In addition, we can observe that there are small variation between10ms and 1.0mswhere time-step0.5msoverestimates theC_dvalue. There is no any effect on the Strouhal number when reduction in∆t.

Following Figure 25a and b describe the time histories of C_d and C_l with different ∆t using the LES model. A progressive increment in amplitude of both drag and lift oscil-lations when decrease the time-step size.

Table 9: Effect of time-step size on theC_d and Stusing LES.

Re_D = 1000

Time-step Cd St

Exp — 0.98±0.005 0.21

∆t= 10ms 1.1661 0.2142 LES2 ∆t= 1.0ms 1.1499 0.21

∆t= 0.5ms 1.2271 0.21

Figure 25: C_d andC_l histories with different timestep size.

The dimensionless frequency of vortex shedding is the Strouhal numbers found by dif-ferent conditions with the LES model are compared with the experimental value can be seen in Figure 26. Iso-surface of instantaneous x-vorticity for each of the different time-steps using LES are presented in Figure 27. Each figure is shown with the same contours values, and it can be seen that the time step of10ms results in little weaker vorticity in the wake region of the cylinder. From this study, it is clear that∆thas not much significant effect on vortex shedding in the wake.

Figure 26: Strouhal number of the flow past a circular cylinder compared to experimental data.

Figure 27: Iso-surface of instantaneous x-vorticity for LES using different time-step.

5 Flow past in a staggered tube bundle

In the previous chapter, the simulation of flow past a circular cylinder has been dis-cussed and investigated. In this chapter, we will study about flow past in a staggered tube bundle. There are two main types of tube array configurations: in-line and stag-gered. In the in-line array type, one row of tubes is placed exactly behind the next along the streamwise direction, without displacement in the cross-flow direction. In staggered arrays, every second row of tubes is displaced resulting in several configurations: sym-metric arrays, rotated square arrays, normal triangle arrays, parallel triangle arrays, etc.

Cross-flow in tube bundles has wide practical applications in the design of heat exchang-ers, in flow across overhead cables, and in cooling systems for nuclear power plants [13].

For these reasons, numerous measurements of turbulent cross-flow in tube bundles have been made to advance a physical understanding of such flows.

Experimental studies of such flows have proven that there are unsteady phenomena such as vortex shedding and jet flapping which are governed by different mechanisms depending on the tube row and the position within the tube bundle. Other factors which affect the periodicity of the flow are the Reynolds number, the arrangement and spacing of the tube bundle, tube surface roughness etc [24] [25]. In addition to the complexity arising from the flow instabilities in the tube bundle, one must also consider whether the flow is turbulent or laminar. Flows in heat exchanger tube bundles are usually subcritical or critical. In critical flows, transition to turbulence occurs before separation and turbulence is prominent in the rest of the boundary layer and in the flow inside the bundle (Zukauskas, 1989). The combination of the flow instabilities and the transitional phenomena present in the boundary layers makes this type of flow difficult to model numerically.

Zdravistch et al. (1995) also performed calculations of the flow and heat transfer in stag-gered and in-line tube bundles by solving the Reynolds averaged Navier-Stokes equations.

Their results showed good agreement when compared to experimental measurements but the Reynolds numbers that they studied corresponded to either laminar or fully turbu-lent flow so subcritical flow was not considered [26]. Balabani et al. (1994) employed an LDA technique to measure mean velocities and turbulent intensities in a staggered tube bundle [27]. The measured data were compared with the predictions using a standard κ−turbulence model with and without a curvature modification. Bouris and Bergeles (1999) used an LES approach with the Smagorinsky SGS model to predict the exper-iments of Balabani et al. (1994) [28]. In both studies, they obtained good agreement with the experimental results .

5.1 Computational details

In this work, first of all we have tested the suitability and the applicability of the models on the flow past in a staggered tube bundle using various turbulence RANS models such as the κ− model, the κ−ω model and the SST model. In this study, we have tried to compare simulated results with LDA (Laser Doppler Anemometry) measurement of S.Balabani(1994) for the streamwise and transverse, mean velocities and r.m.s. velocities.

Furthermore, grid independence test and time-step effect have been checked in the same application. The brief details of the simulations are as follows:

• Computational domain

Figure 28a shows a sectional side-view of the tube bundle model used. The bundle consists of 6 rows of tubes with outer diameter of 10 mm arranged in staggered array. As shown in the figure, each row has 1 or 2 full tubes. Half tubes are also mounted along the top and bottom walls of the test model alternately to simulate an infinite tube bundle and minimize the wall boundary layer. The transverse and longitudinal pitch-to-diameter ratios,S_T andS_L, are 3.6 and 2.1, respectively. The length-to-diameter ratio of the rods is 7.2. The origin of the coordinate system is defined to be at the center of the middle tube in the first row. Here, the streamwise and transverse directions are denoted by x and y, respectively. The present study are performed at a Reynolds number of 12,858 which is same with Balabani’s experiments. This Reynolds number is based on the gap velocity and the cylinder diameter. Figure 28b shows the details of the region where most results will be presented using spatial coordinates normalized by the tube diameter. The majority of the results will be presented at the following x/d locations: 0.85, 1.25, 2.95, 3.35, 5.05, 5.45, 7.15 and 7.55 in the region0 ≤y/d≤3.6 and 0 ≤x/d≤8.4, as indicated in Figure 28b. The simulation are carried out with water at 20^◦C used as the working fluid and corresponding to the approach velocity,U∞, of 0.93 m/s.

The full number of tubes in the present experimental study is modeled and, due to symmetry, the computational domain is half of the experimental domain.

Figure 28: (a) Cross-sectional view of the tube bundle, and (b) locations at which results are presented.

• Boundary conditions

The boundary conditions for the solution domain shown in Figure 29a are as fol-lows. At the inlet, an average upstream value of the mean velocity equal to the approach velocity,U∞ =0.93m/sobtained from Balabani’s measurement. In ad-dition, at the inlet the relative turbulence intensity is set equal to 6%. The outlet boundary is defined with an average static reference pressure of0P a. On the con-stanty surface at y= 0, a symmetry boundary condition is applied. A stationary wall (no-slip) boundary condition is prescribed on the constantysurface aty =ST

and the surfaces of the tubes. In theκ−model, a wall function approach is used for the near-wall treatment. The low-Reynolds formulation is employed for both theκ−ω model and the SST model.

• Meshing

Structured quadrilateral and non-uniform grids are generated for the solution do-main shown in Figure 29a. For this purpose we have used ANSYS GAMBIT 2.4 package. Near the tube surfaces, fine mesh have been used for to resolve boundary layer separation. For the grid independence tests we have been used four different

grid and all are structured and non-uniform. Figure 29b displays the mesh genera-tion in the computagenera-tional domain. Figure 29c the closer view near the tube surface of the domain.

Figure 29: (a) Boundary conditions, (b) computational grid, and (C) closer view of the tube surface.

• Simulation set-up

As a CFD code, ANSYS Fluent 12.1. package is used for the numerical solution of the system. For all the cases, unsteady and pressure based solver are used. A least square cell based method is used to calculate gradients. In this work, theκ−,κ−ω and SST turbulence models are used. The pressure-velocity coupling is obtained

using the SIMPLE algorithm. A fully implicit upwind differencing scheme is used for the time integration. The brief details of simulation setup are summarized in Table 10.

Table 10: Simulation settings of flow past in a staggered tube bundle case

Settings Choice

Turbulent kinetic energy 2^nd order upwind Turbulent dissipation rate 2^nd order upwind (forκ−model)

Specific dissipation rate 2^nd order upwind (forκ−ω and SST models )

Energy 2^nd order upwind

Convergence criteria 1.0×10⁻⁵ Boundary conditions:

The first objective of the present study is to perform detailed assessment of the ability of the statistical turbulence models associated with Reynolds averaging of the Navier-Stokes equations such as κ −, κ−ω and SST models, to reproduce the mean flow and turbulent quantities in a staggered tube bundle. Here, the two dimensional time dependent simulations are applied using the RANS models. All simulations have been done using Grid C. Grid C contains 31608cells. The detail information about it which is given in the Table 11. For models comparison purpose, the streamwise and transverse,

mean and r.m.s velocities are compared with experimental data of Balabani (1996).

Turbulent quantities such as turbulent kinetic energy and dissipation rate have been calculated using RANS models and compared with each other.