## CALCULATIONS OF TURBULENT FLOW THROUGH A STAGGERED TUBE BANK
## 摘要The primary aim of this paper is to numerically investigate the crossflow in a staggered tube bank by using a variable-resolution method. Experimental data of Simonin and Barcoude (1988) is available in the ERCOFTAC database. There are also few ERCOFTAC workshops, e.g. 1993, 19999, which were considering this test case primarily for checking the performance of the Reynolds-Averaged Navier-Stokes (RANS) models. Therefore, there are number of results with very different models which can be found in the literature. The work presented here aims to add one more set of results but this time with recently advanced variable resolution method, namely the Partially-Averaged Navier-Stokes (PANS). This method (Girimaji, 2006) belongs to so called bridging or seamless methods. The PANS approach adjusts seamlessly from the Reynolds-Averaged Navier-Stokes (RANS) to the Direct Numerical Solution (DNS) of the Navier-Stokes equation. The results are largely improved by using the PANS as for example shown in Basara (2015). This turbulence bridging method is derived from the RANS model equations. It inevitably improves results when compared with its corresponding RANS model if more scales of motions are resolved. This is done by varying the unresolved-to-total ratios of kinetic energy and dissipation. In the practice, the parameter which determines the unresolved-to-total kinetic energy ratio is defined by using the grid spacing and calculated integral length scale of turbulence. When the grid size is smaller, then more of the turbulent kinetic energy can be resolved. Usually, the integral scale of turbulence is obtained by summing up resolved turbulence, calculated as difference between instantaneous filtered velocity and the averaged velocity field, and unresolved turbulence obtained from its own equation. The turbulence model adopted in the present PANS variant is the four-equation ζ − f formulation (Hanjalic et al., 2004) which is the variant of more known v2-f model based on the elliptic relaxation concept. ). As this model represents a practical and accurate RANS choice for a wide range of industrial applications, especially when used in conjunction with the universal wall approach (Popovac and Hanjalic, 2007, Basara, 2006), its PANS variant therefore guarantees that the proper near-wall model is used when Simulations are performed at a Reynolds number of 18000 based on the average velocity stated to be 1.06 m/s and a tube diameter of 21.7 mm. Two computational meshes were used, the coarse with 3 million cells and the fine with 6 million cells. Except walls of pipes, all other boundaries employed periodic type of boundary conditions. The computational domain and the mesh is shown in Fig. 1a. A typical values of the instantaneous resolution parameter f _{k} (note that this is the ratio of unresolved to total kinetic energy) is shown in Fig. 1b. For f_{k} equal unity, PANS will produce results equivalent to those produced by the RANS ζ-f model. If this parameter is below 0.2, one could expect result more similar to those obtained by Large Eddy Simulation. For values between 0.2 and 1, one could expect that PANS produces more accurate results not only compared to RANS but also to LES as more equations are solved to simulate un-resolved turbulence as in the case of LES. Note also that a dynamic parameter f_{k} changes at each point at the end of every time step, and then it is used as a fixed value at the same location during the next time step. Numerical meshes are usually coarse near the wall and it is also difficult to achieve so called a wall-resolved LES. Clearly this issue is present in PANS calculations as well: one could expect that f_{k} is equal or close to unity near the wall which means that the RANS model is used there. Therefore, in the case of PANS, we expect that the accuracy is improved also by the employment of the near-wall RANS modelling including also the heat transfer modelling, see Basara (2015). Calculations by using the PANS method should be performed as with the LES, see an instantaneous velocity magnitude shown in Fig. 2a. As f_{k} is the high near the wall, it should be expected that also there, the un-resolved kinetic energy is high, which is confirmed in Fig. 2b. The averaged mean velocity profiles as predicted on coarse and fine meshes at x = 0 mm, 11 mm, and 16.5 mm, are shown in Figure 3 and 4a. All profiles are well captured and also the coarse mesh provided correct behavior. It is also important to note that if PANS calculations produce results which contain a small portion of unresolved kinetic energy than the effects of convection schemes could significantly contaminate the final calculation results. Therefore, due attention should be given to differencing scheme especially when compared to LES results. Here, we used the central differencing scheme. Also the rate of change is discretized by using three time level implicit scheme of second order accuracy. |

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