MESHING STRATEGY EVALUATION FOR A SQUARE SHAPED BLUFF BODY UNDER HIGH REYNOLDS NUMBER CROSS FLOW

This work consists in the numerical evaluation of meshes employed in a twodimensional, incompressible, and transient flow, with high Reynolds numbers and forced convection that passes through a square shaped bluff body. The objective of the study is to evaluate different strategies of mesh construction to solve this type of problem, seeking to reduce the computational cost and improve the solution. The simulations were performed for Reynolds and Prandtl numbers of ReD = 22,000 and Pr = 0.71. The turbulence behavior is solved with the RANS SST–-model. The evaluation of the solutions is performed analyzing the drag coefficient (CD), Nusselt number (NuD) and Strouhal number (St). The use of an unstructured mesh with fully structured local refinement was able to validate the fluid dynamics of the case with a relative deviation of 14.95% for the velocity field, compared to the literature. The deviations for CD and St were as low as 0.43% and 0.79% respectively. For thermal verification, the deviation of the mean local Nusselt number along the bluff body (NuD) was 5.93%, and the deviation for the global Nusselt number (Nug) was as low as 0.56% compared to the literature.


INTRODUCTION
Problems with fluid and thermal flows are found in several engineering areas. One of the most studied topics is precisely the phenomena that occur with tubes, bluff bodies and finned channels in cross flow (commonly found, for example, in heat exchangers). Currently, important studies have been carried out in the area of geometric evaluation of arrangements of cylinders and bluff bodies, seeking to improve the performance of the system according to multiple objectives. It has sought to find geometries that maximize the heat transfer (thermal function) and minimize the resistance to the flow (fluid dynamic function). The great majority of these studies analyze the behavior of laminar flows and, since the interest is the phenomenological study of processes such as advection and thermal diffusion, these flows tend to present more predictable and stable numerical results with good convergence and relative low computational cost.
However, most engineering problems with real application fall within the field of turbulent flow regimes (WILCOX, 2006). From the electronic component cooling system to automotive cooling systems and industrial thermal equipments, the flow regime employed is often turbulent. This is due to the need for a large thermal exchange in these equipments, also requiring large momentum of the refrigerant fluid.
The turbulent flows have a very complex physical nature and generally require sophisticated measuring instruments to be studied experimentally, which sometimes becomes economically unfeasible. Thus, few works are found in the literature related to the experimental study of turbulent flows on a bluff body. Of particular note are the studies by Igarashi (1986), Durao et. al. (1988) and Lyn et. al. (1995).
According to Wilcox (2006), the intricate mathematical modeling of turbulent flows, associated with the geometric and physical complexity of the problem, makes its analytical solution impossible. On the other hand, numerical methods have been presented in a technically and economically feasible way for researchers in the area due to the vertiginous development of high-speed computers with large storage capacity (MALISKA, 2004). Nevertheless, many Revista Mundi Engenharia, Tecnologia e Gestão. Paranaguá, PR, v.3, n.2, maio de 2018.

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difficulties are encountered to evaluate this type of phenomenon, ranging from: the choice of numerical model; the implementation of appropriate boundary conditions; as well as the creation of a mesh that can correctly capture the velocity, temperature and pressure field gradients in the studied flow.
The simulation of turbulent flows over a single bluff body has been the subject of important recent studies such as that of Perng & Wu (2007) who used the Large Eddy Simulation (LES) approach to study the effects of natural convection and buoyancy on the bluff body. Subsequently, Ranjan & Dewan (2015) used a PANS (Partially-Averaged Navier-Stokes) model associated with the SST- turbulence modeling to compare two meshes, one structured with wall function and one unstructured with local refinement. Chen & Xia (2017) compared the RANS (Reynolds Averaged Navier-Stokes) Spalart-Allmaras and SST- models to solve the same problem. All these studies have great emphasis on mathematical and numerical modeling, but do not systematically analyze the generation of meshes that use diverse strategies in order to find patterns that reduce the computational cost while allowing studying geometries that are more complex. Therefore, the present work intends to address the issue of the computational mesh for the solution of external and turbulent flows over bluff bodies, with a series of distinct meshing strategies that can, besides reducing the computational cost, lead to as close as possible results as those obtained in the literature, for the main flow parameters (C D , N uD and St). Although the phenomenon of turbulence is inherently three-dimensional, domains with twodimensional meshes will be generated here, since the objective is to ascertain the different creation strategies. It is intended to use the recommendations obtained in this work for future studies of geometric evaluation on arrangements of bluff bodies

MATHEMATICAL MODELING
The problem under study is a two-dimensional, incompressible high Reynolds number flow, in a transient regime, with forced convection and constant thermophysical properties. The domain employed is a two-dimensional reconstruction of the domain used by Wiesche (2006), Ranjan & Dewan (2015) and Chen & Xia (2017) where the cartesian coordinate z was suppressed and the dimension in y was enlarged from 10D to 14D. This was done in order to avoid numerical distortions originating from the symmetry boundary condition, since the former rapidly generated vortices that reached the upper and lower The flow has also a prescribed temperature T ∞ that is lower than the surface temperature of the bluff body T S by 30K, so the forced convective heat transfer occurs due to this temperature difference. The other boundary conditions include the imposition of null heat flux and zero gauge pressure at the domains outlet, besides the condition of symmetry on the north and south faces. The modeling chosen for this study is based on the Reynolds Averaged Navier-Stokes (RANS) conservative equations, altogether with the equations of the SST- model to tackle with turbulence closure. It is known that turbulence is an inherently three-dimensional phenomenon, but the two-dimensional study aims at generating recommendations while reducing computational cost. The SST- turbulence model is a two-equation model proposed by Menter (1993) as an alternative to the original  model that has too much sensitivity in the free-current regions. The SST (Shear Stress Transport) solves this problem, since it changes the behavior of the model resembling the formulation  in the free flow. It is worth mentioning that the analysis was performed when the flow reached the steady state regime (when the turbulence structures started to present repetitions in time). However, the simulations are performed in transient regime, since turbulent flows are naturally time dependent.

Conservative equations and turbulence model
The mean time-averaged conservative equations for mass, momentum in x and y and energy respectively are given as seen in Bejan (2004): where: is the density (kg/m³), x is the x-axis cartesian coordinate (m); u is the velocity component in the x-axis direction (m/s), y is the y-axis cartesian coordinate (m); v is the velocity component in the y-axis direction (m/s); P is the pressure (N/m²); T is the temperature (K); is the specific heat at constant pressure (J/kg.K) and ´´´ is the energy source (W/m 3 ), that in this case is zero.
To solve the turbulence problem, the model chosen is the SST- initially proposed by Menter (1993). It is a model of two equations derived from the original  formulation presented by Wilcox (1988). Its major advantage is that it uses a blending function, which keeps the formulation in the inner parts of the boundary layer close to the wall, changing to a behavior as it reaches the free-stream thus reducing the weaknesses of each model. In addition, this model modifies the definition of kinematic eddy viscosity to include the turbulence Shear Stress Transport (SST). Thus, the equations for the turbulent kinetic energy and for the specific dissipation rate are given by: where: k is the turbulent kinetic energy, is the specific dissipation rate, is the kinematic eddy viscosity, the constants , , 1 , 1 , , , 2 , 2 , 2 and 2 are the same as seen in Menter (1994) and 1 is a blending function between variables and constants defined by:

NUMERIC MODELING
Gmsh, an open source meshing software was adopted for the construction of the computational meshes. Always using rectangular elements, the constructive complexity of the mesh is increased gradually for each simulated case. Six different mesh generation strategies were studied according to

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intentions of this work is to reduce the computational effort by using strategies that are more efficient.
The strategies used were as follows: 1. Structured mesh with horizontal refinement; 2. Structured mesh with progressive cross shaped refinement; 3. Structured mesh with fixed cross shaped refinement; 4. Structured mesh with partially progressive cross shaped refinement; 5. Unstructured mesh with totally structured local refinement; 6. Unstructured mesh with partially structured local refinement

RESULTS AND DISCUSSION
The simulations for the six proposed cases were performed and compared with each other and with the literature. Numerous parameters (field of velocities, mean local Nusselt number, (Nu D ), global Nusselt number (Nu g ), drag coefficient (C D ) and Strouhal number (St)) were used to determine the level of refinement and the strategy with the highest potential to be used with more complex geometries and at the same time keep the computational requirements at levels not prohibitive for future researches. It is worth remembering that the purpose of this work is to evaluate the mesh construction strategy and not the simple complexity or number of computational cells. Therefore, we tried to make a fair comparison respecting the different size of meshes, but analyzing the behavior of the solution (efficiency between a better result and the less time needed to reach it with the same computational resources). Figure 4 shows the solution for one of the evaluated parameters, the domains centerline mean velocity. In order to quantify a relative deviation, the results are compared with the experiments of Lyn et. al. (1995).
As can be seen in Fig. 4 (a), the mesh strategies 1 and 2 present a solution quite distant from the others, which could already be expected in view of the much lower number of computational cells in the meshes (51,000 and 48,000 respectively) especially in the wall region. However, case 2 shows a significant improvement (48.81% versus 75.07% relative deviation of case 1), even with the reduction of the mesh size, only by the change in the constructive strategy, that is, the cross-shaped refinement. Although with some changes, cases 3 and 4 also used cross-shaped refinements, but with considerably more refined meshes. These cases obtained, for the parameter in question, deviations of 28.25% and 9.69%. However, as can be seen by the simulation time required in Table 2, there was a big increase, on the computational effort. Finally, cases 5 and 6, through unstructured meshes that are substantially smaller than those of cases 3 and 4, were able to reach deviations of 14.85% and 17.04%, respectively (tolerable values for this kind of flows over bluff bodies). Figure 4 (b) shows the graphical comparison of case 5 with solutions from the literature, since this strategy, besides achieving acceptable deviation, reduced the simulation time by approximately 40% against the best case (strategy 4) and has the capacity to adapt to more complex geometries. It is worth to notice that cross-shaped mesh constructions are practically impossible to employ on the presence of not aligned bluff bodies for example. This is verified through cases 4, 5 and 6, all with two hundred cells in each face of the obstacle. Here the relative deviation quantification is performed comparing to Ranjan & Dewan (2015). The relative deviations of the abovementioned cases were 9.86%, 5.93% and 4.34% respectively.

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considering that the two are in the acceptable range of deviation for problems with high Reynolds numbers.
Finally, Figs. 6 (a) and (b) illustrate for each case, the instantaneous y + profiles as a function of the bluff body position, an important parameter for checking the refinement state of the mesh in the wall region. According to Wilcox (2006), turbulence models  require that this value be of y + ≤ 1. In this way, only the unstructured mesh cases with local refinement presented mean y + less than 1, being 0.90 for case 5 and 0.67 for case 6.   Igarashi (1985). The comparison with Lyn et. al. (1995) is also satisfactory, with deviations of 9.52% for the C D and 0.79% for the St. In comparison with numerical works, case 5 presents an average relative deviation of 5.50% for the C D and 6.72% for the St compared to Bouris & Bergeles (1999), as well as 5.42% for the Nu g compared to Chen & Xia (2017).

CONCLUSIONS
A numerical study was carried out to evaluate a series of different mesh The results showed that the mathematical and numerical modeling used could accurately represent the time-averaged parameters of turbulent flows such as the Strouhal number, drag coefficient and Nusselt number even in twodimensional domains. However, it is necessary to employ well-constructed meshes, sufficiently refined in the wall regions.
Case 5 presented a time-averaged local mean deviation within the tolerable range with 14.85% for the field of velocities in the fluid dynamic validation, compared to Lyn et.al. (1995) and 5.93% for Nu D in the thermal verification, compared to Ranjan & Dewan (2015). Compared with the literature, the time-averaged global mean deviation for Nu g , C D and St were as low as 0.56%, 0.43% and 0.79% respectively. This case is the more indicated (although it is not the one that leads to the smaller deviations in comparison with the literature) due to its greater flexibility to the assembly of the meshes, which allows its application in more complex geometries like arrangement of obstacles. The cross-shaped mesh of case 4, in spite of the excellent results obtained, is very complex for the assembly of misaligned bluff bodies, which makes difficult its future application in geometric evaluation studies. In future works, it is intended to use the strategy of case 5 to study the effect of geometry in complex arrangements of bluff bodies and on fluid dynamics and thermal performance subject to forced convective turbulent flows.