Large Eddy Simulation of a Turbulent Jet Impinging on a Flat Plate at Large Stand-off Distance

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1 Large Eddy Simulation of a Turbulent Jet Impinging on a Flat Plate at Large Stand-off Distance M. Shademan 1, R. Balachandar 2 and R.M. Barron 3 1 PhD Student, Department of Mechanical, Automotive & Materials Engineering, University of Windsor, Windsor, N9B 3P4, Ontario, Canada 2 Professor, Department of Civil and Environmental Engineering, University of Windsor, Windsor, N9B 3P4, Ontario Canada 3 Professor, Department of Mathematics & Statistics, University of Windsor, Windsor, N9B 3P4, Ontario Canada shadema@uwindsor.ca ABSTRACT Large Eddy Simulation (LES) has been performed to evaluate the characteristics of a turbulent impinging jet with large nozzle height-to-diameter ratio. Dynamic Smagorinsky model was employed to simulate the subgrid scale stresses resulting from the filtering of the governing equations. The Reynolds number considered is about 28,000 based on the jet exit velocity and nozzle diameter. Results of the mean normalized centreline velocity in both free jet and impingement regions and also pressure distribution over the plate show good agreement with experimental data. According to this analysis, simulation of the impinging jet using LES can improve the accuracy of the results as compared to the previous RANS simulations, however more computational resource is required. The current comparison presents a robust CFD approach for evaluating the flow characteristics of turbulent impinging jets with large stand-off distance. 1. INTRODUCTION Circular jets impinging on flat surfaces have many practical applications. Most of the experiments on impinging jets have been performed for short stand-off distances, i.e., with an impingement height (H) to nozzle diameter (D) ratio of less than six. Cooper et al. [1] carried out experiments on a jet impinging on a large plane surface and measured the mean and turbulence quantities in different regions of the jet. They considered two Reynolds numbers, 23,000 and 70,000, while the H/D ratio varied from two to ten, with particular focus between two and six. For H/D < 6 the core of the jet is still developing when it reaches the surface (Nishino et al. [2], Hadziabdic and Hanjalic [3]). For larger impingement heights (H/D > 8.3), Beltaos and Rajaratnam [4,5] classified the flow into three different regions: the free jet portion (region I), the impingement zone (region II), and the axisymmetric wall jet portion (region III) as illustrated in Fig. 1. Giralt et al. [6] experimentally evaluated axisymmetric turbulent impinging jets with H/D ratios ranging from 3 to 25, and Re varying from 34,000 up to 80,000. Based on their experimental data, they developed a conceptual model for submerged, axisymmetric, turbulent impinging jets which can be used to analyze the effect of increasing the nozzle height from the plate. In addition to the experiments by Beltaos and Rajaratnam [4,5], Rajaratnam et al. [7] recently performed experiments on an impinging jet with a higher H/D ratio of 18.5 at Re = 100,000 (based on the nozzle exit velocity and diameter) and evaluated the turbulence characteristics in the different regions of the jet. Numerical simulation of a round jet impinging on a flat surface using Reynolds Averaged Navier-Stokes (RANS) simulations has been the subject of extensive research, forming part of the 2nd ERCOFTAC-IAHR Workshop on Refined Flow Modelling [8]. Subsequently, Craft et al. [9] published their research using different turbulence models to analyze the heat transfer in the impingement region of the jet, i.e., region II. They observed that the results were not in good agreement with experimental data, and attributed this to the weakness associated with the eddy viscosity stressstrain relationship in the turbulence models used. They also implemented second-moment closure models. Due to the incorrect response of the wall reflection process, the eddy viscosity model (k - ε) and the basic Reynolds Stress Model (RSM) failed to produce reasonable results while an improved Reynolds Stress Model which takes into account the wall reflection effects generated satisfactory results. Clearly, to analyze a complex fluid

2 flow problem such as an impinging jet, direct numerical simulation (DNS) or large eddy simulation (LES) seem to be the more accurate approaches. However, in order to resolve all scales of motion in DNS, the number of grid points should be of the order of Re 9/4 as suggested by Piomelli [10]. This is a limitation which currently makes DNS practical only for low Reynolds number flows with simple geometries. For example, Chung et al. [11] used DNS to simulate an unsteady slot jet with H/D = 10 and Re = 300, 500 and 1,000. Hattori and Nagano [12] simulated the plane impinging jet using DNS at Re = 9,120 for values of H/D = 0.5, 1 and 2. Both plane and round impinging jets with H/D = 10 were investigated by Tsubokura et al. [13] using DNS for Re = 2,000 and LES for Re = 6,000. In LES, only the large and high energy-containing eddies are resolved and the small ones are modeled. This method demands reasonably fine meshes at higher Reynolds numbers. Most LES jet studies have only dealt with simple plane jets at low Reynolds numbers as, for example, the LES studies on plane impinging jets by Voke and Gao [14] at Re = 6,500, and by Beaubert and Viazzo [15] at Re = 3,000 and 7,500. Recently, Hadziabdic and Hanjalic [3] have investigated the circular impinging jet at Re = 20,000 and H/D = 2 using LES. Based on a review of the literature, there is an apparent lack of information regarding the numerical simulation of turbulent impinging jets with large H/D ratios. Moreover, there is an increasing number of practical applications in which the value of H/D is large. Therefore, it is of interest to evaluate the performance of LES method for modelling the impinging jets with high H/D ratios. In this study, LES was carried out for H/D = 20 at Re = 28,000. The experiments performed by Rajaratnam et al. [7] have been used as the benchmark to validate the numerical model. Experimental data from Bradshaw and Love [16] and Shinneeb et al. [17] have also been used to assess the accuracy of the results. 2. GEOMETRY MODELLING In this simulation, the nozzle exit diameter is 10 mm and the distance between the nozzle and the plate is 20 cm resulting in a H/D ratio of 20. The water jet velocity exiting the nozzle is 2.86 m/s which corresponds to a Reynolds number of 28,000. To ensure that the location of the outlet boundary has negligible influence on the pressure and velocity fields, the computational domain is taken to have a radius of 0.2 m along the plate. In order to mimic the experimental setup, the water is allowed to escape to the ambient through the side boundaries of the computational domain. Therefore, these boundaries are set as pressure outlets. The plate is considered to be a no slip boundary. Three different mesh sizes were used to satisfy the mesh requirement for LES, as discussed below. Details of the computational domain and mesh generated for one of the LES cases are shown in Figs. 2, 3, and 4. The full 3D geometry including the boundary conditions are illustrated in Fig. 2. The fully structured mesh system and surface mesh over the plate are shown in Fig. 3 and 4, respectively. The flow over the wall can be either resolved or modeled in LES by using different mesh resolution close to the wall. Since, in the current simulations, the analysis of the impingement zone is of primary interest, more emphasis is considered on the resolution of the mesh close to the wall. Chapman [18] determined that the resolution needed for the outer layer of a boundary layer is proportional to Re 0.4, while for the wall layer the number of grid points required increases by Re 1.8. The wall surface area increases rapidly with increase of the radial coordinate, resulting in a larger domain and consequently more cells. However, in the area of interest (r/d < 4.0), the mesh resolution is close to the generally approved LES criteria for wall-attached flows suggested by Piomelli and Chasnov [19], which requires that r + < 100, (r θ) + < 20 and z + < 2. The mesh resolution quality can also be evaluated by comparing the mesh size = ( r r θ z) 1/3 to the Kolmogorov length scale η = (ν 3 /ε) 1/4. Here ν is the molecular viscosity and ε is the dissipation rate estimated from our previous RANS simulation using the Realizable k ε turbulence model. For isotropic turbulence, Pope [20] has shown that a grid spacing of 12η is required in order to resolve the major contributions to the dissipation. Therefore, in the current study, attempts were made to keep the Δ/η value less than 12 in regions of interest. Further, using the recommendations from previous studies (Hadziabdic and Hanjalic [3], Piomelli [10]), three types of mesh resolution were generated. Mesh #1 includes 300 points in the axial direction, 120 points in the circumferential direction and 150 points in the radial direction. Mesh #2 contains 400 points in the axial direction, with grid points in the circumferential and radial directions kept the same as Mesh #1. Mesh #3 includes 400 grid points in axial direction, 160 points in circumpherential direction and 150 in radial direction. In all three meshes, in the radial direction, the expansion is 1.04 towards the centerline, while in circumferential direction a uniform distribution is used. The total number of cells in the three meshes is 5,400,000, 7,200,000 and 10,600,000 respectively. The simulations on three different highdensity grids require a large computational effort. The averaged time used for each simulation for each mesh on the Shared Hierarchical Academic Research Computing Network (SHARCNET) clusters was about 2 months using 28 (2.2 GHz AMD Opteron) CPUs.

3 3. NUMERICAL METHOD The governing equations used in LES are obtained by filtering the unsteady Navier-Stokes equations in Fourier space. In this method small eddies with scales smaller than the filter width are removed. The filtered equations are as following: ρ + ρu t = 0 i t ρu i + ρu i u j = μ σ ij p τ ij (2) In these equations σ ij is the stress tensor, defined by σ ij μ( u i + u j ) 2 3 μ u i δ ij (3) and τ ij is the subgrid-scale stress given by τ ij = ρu i u j ρu i u j (4) The subgrid-scale stresses resulting from the filtering operation are unknown, and require modelling. Therefore, the dynamic Smagorinsky method presented by Germano et al. [21] is used for the modelling of the subgrid-scale stresses in current simulations. Fluent 6.3 [22] was used to solve the governing equations. The third-order accurate upwind scheme QUICK (Leonard [23,24]) has been used to discretize the convective terms in the momentum equations. Timemarching is performed using a fully-implicit secondorder scheme. Based on the mesh topology presented and to satisfy the Courant number less than 1 condition for these simulations, the time step is set to be 5e-5. The time step was kept constant during the simulation. The SIMPLE algorithm was used for coupling velocity and pressure. Mean variables were determined by averaging the instantaneous results long after the simulation was started and the flow was in a fully established condition. 4. RESULTS AND DISCUSSIONS Validation of the numerical results is carried out by comparing the mean quantities with the experimental data. Time-averaging is performed on a sufficient number of periodic oscillations far after the initial condition. Figure 5 compares the normalized mean centreline velocity from the current simulations with experimental results for 10 < x/d < 25. The results from the present simulations and the experimental data follow the expected trend observed for different H/D values. Up to about x/d = 15, the flow does not get influenced by the impingement wall and essentially follows the behavior of a free jet. The model shows good agreement with the experimental data as the flow approaches the plate (region II). Figure 6 compares the mean static pressure distribution along the plate. The static pressure values are normalized by the static pressure at the stagnation point, Ps. The radial direction is normalized with r ½, which is the radial position where P = 0.5Ps. The numerical prediction obtained from the current simulation is in good agreement with the measurements by Bradshaw and Love [16] and Giralt et al. [6]. Results of these simulations show the performance of the LES method in capturing accurate flow characteristics in impinging jets with large stand-off distance. ACKNOWLEDGEMENTS This study was conducted using the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET).This research was funded by Ontario Ministry of Research and Innovation through the Ontario Research Fund under the Green Auto Power Train project. REFERENCES [1] D.Cooper, D.C.Jackson, B.E.Launder, and G.X.Liao, "Impinging jet studies for turbulence model assessment - I. Flow-field experiments," Int. J. Heat Mass Transfer 36, 1993, pp [2] K.Nishino, M.Samada, K.Kasuya, and K.Torii, "Turbulence statistics in the stagnation region of an axisymmetric impinging jet flow," Int. J. Heat Fluid Flow 17(3), 1996, pp [3] M.Hadziabdic and K.Hanjalic, "Vortical structures and heat transfer in a round impinging jet," J. Fluid Mech. 596, 2008, pp [4] S.Beltaos and N.Rajaratnam, "Impinging circular turbulent jets,"j. Hydraul. Eng. ASCE 100, 1974,pp [5] S.Beltaos and N.Rajaratnam, "Impingement of axisymmetric developing jets,"j. Hydraul. Res.15(4), 1977, pp [6] F.Giralt, C.Chia, and O.Trass, "Characterization of the impingement region in an axisymmetric turbulent jet," Ind. Eng. Chem. Fund. 16(1), 1977, pp [7] N.Rajaratnam, D.Z.Zhu, and S.P.Rai, "Turbulence measurements in the impinging region of a circular jet," Can. J. Civ. Eng. 37(5), 2010, pp [8] Second ERCOFTAC-IAHR Workshop on Refined Modelling, "Round normally impinging turbulent jet and turbulent flow through tube bank subchannel," 16th Meeting of the IAHR Working Group on Refined Flow Modelling, 1993, University of Manchester Inst. of Sci. and Tech., UK. [9] T.Craft, L.Graham, and B.E.Launder, "Impinging jet studies for turbulence model assessment - II.

4 An examination of the performance of four turbulence models," Int. J. Heat Mass Transfer 36, 1993, pp [10].U.Piomelli, "Large-eddy simulation: achievements and challenges," Progress in Aerospace Sciences 35, 1999, pp [11] Y.M.Chung, K.H.Luo, and N.D.Sandham, "Numerical study of momentum and heat transfer in unsteady impinging jets," Int. J. Heat Fluid Flow 23, 2002, pp [12] H.Hattori and Y.Nagano, "Direct numerical simulation of turbulent heat transfer in plane impinging jet," Int. J. Heat Fluid Flow 25, 2004, pp [13] M.Tsubokura, T.Kobayashi, N.Taniguchi, and W.P. Jones, "A numerical study on the eddy structures of impinging jets excited at the inlet," Int. J. Heat Fluid Flow 24, 2003, pp [14] P.R.Voke and S.Gao, "Numerical study of heat transfer from an impinging jet," Int. J. Heat Mass Transfer 41, 1998, pp [15] F.Beaubert and S.Viazzo, "Large eddy simulation of plane turbulent impinging jets at moderate Reynolds numbers,". Int. J. Heat Fluid Flow 24, 2003, pp [16] P.Bradshaw and E.M.Love, "The normal impingement of a circular air jet over a flat surface,"arc R&M,3205, [17] A.M.Shinneeb, J.D.Bugg, and R. Balachandar, "Quantitative investigation of vortical structures in the near-exit region of an axisymmetric turbulent jet," J. Turbul. 9(19), 2008, pp [18] D.R. Chapman, Computational aerodynamics development and outlook, AIAA J. 17(12), 1979, [19] U. Piomelli and J. R.Chasnov, "Large-eddy simulations: theory and applications. In Transition and Turbulence Modelling," (eds. A. Henningson, K. Hallback, L. Alfredsson & M. Johansson). Kluwer, [20] S.B.Pope, "Turbulent Flows," Cambridge University Press, [21] M. Germano, U. Piomelli, P. Moin, W.H. Cabot, "A dynamic subgrid-scale eddy viscosity model,"phys. Fluids A3, 1991, p [22] FLUENT 6.3 User s Guide, FLUENT Inc., Lebanon, New Hampshire, USA. [23] B.P.Leonard, "A stable and accurate convective modelling procedure based on quadratic upstream interpolation," Comput Meth. Appl. Mech. Engng. 19, 1979, pp [24] B.P.Leonard, "Simple high-accuracy resolution program for convective modelling of discontinuities," Intl J. Numer Meth. Fluids 8, 1988, pp D Jet from nozzle Core jet Free jet region I Impingement region II Wall jet region III Fig. 1. Definition sketch of impinging circular jet Fig. 2. Domain dimensions and boundary conditions

5 P/Ps Fig. 3. Fully structured generated mesh Fig. 4. Generated face mesh over the plate Shinneeb et al. (2008), free jet Rajaratnam et al. (2010), H/D=18.5 LES, mesh #3, H/D=20 U C /U j x/d Fig. 5. Normalized mean centreline velocity r/r 1/2 Fig. 6. Normalized mean static pressure along the wall Bradshaw & Love (1961), H/D=21 Giralt et al. (1977), H/D=22 LES, mesh#3, H/D=20