A TURBOLENT FLOW PAST A CYLINDER *Vít HONZEJK, **Karel FRAŇA *Technical University of Liberec Studentská 2, 461 17, Liberec, Czech Republic Phone:+ 420 485 353434 Email: vit.honzejk@seznam.cz **Technical University of Liberec Studentská 2, 461 17, Liberec, Czech Republic Phone:+ 420 485 353436 Email: karel.frana@seznam.cz The paper deals with an incompressible flow past a circular cylinder in slightly turbulent flow regime. Numerical simulations was carried out by using unsteady Reynolds averaged Navier Stokes (URANS) approach with Spalart-Allmaras (SA) and detached-eddy-simulation (DES) turbulence models. Furthermore, experimental flow visualizations on hydrodynamic tank is presented as well. In the frame of this study parameters such as geometry, computational grid resolutions and mathematical models were considered. Results such as time averaged velocity fields and Reynolds shear stress etc. are compared with experimental data and good agreement was found. Keywords: flows past a cylinder; detached-eddy-simulations; Spalart-Allmaras model; flow visualizations 1 INTRODUCTION Benchmark represented by a flow past a cylinder is used commonly for validations of mathematical models and computational software in fluid mechanics. For turbulent flow simulations, unsteady three-dimensional turbulent model based on the URANS approach in two variation: with SA [1] and DES [2] turbulence models was used. To find more about details relating to mathematical implementation of the turbulent models, we refer to [3]. Usual computational grid resolution can not lead to taking into account small turbulent scales appeared in the high Reynolds number flows. Effect of small scales on the turbulent flow behaviour is represented by turbulent modelling in turbulent models. Spalart-Allmaras and Detached Eddy Simulation turbulence models use so-called turbulent viscosity, which substitute effect of smaller scales. SA model provide good results at simulation turbulent flow near walls, while DES is hybrid turbulence model, which use SA in near wall areas and models only small scales in free flow such as large eddy simulation (LES). Flow visualizations provides view on the creation of turbulence in the wake of the flow past a cylinder. Visualization methods are based on photography paths of metallic dusty particles, which are wafted on surface of black-colourised water. 2 PROBLEM FORMULATIONS 2.1 Numerics simulations The computational grid was created by grid generator Gambit and saved in Necton format. The grid is unstructured, consisted of tetrahedral elements, and it has local finer grid resolution in the area close to cylinder wall (Fig. 1). Parameter D/Δ is approximately 83 on cylinders surface and 13 in free flow area, where D is cylinder diameter and Δ is specified size of element. Number of grids elements is 248378 and corresponding number of nodes is about 60904. Parameter h/d=0.15, where h is a grid distance. The size of the computational domain is depicted in Fig. 2. The problem was solved as an incompressible turbulent flow with constant material properties molecular kinematic viscosity ν and density ρ and at Reynolds number 3900 defined as follows Re= D u inlet =3900 where uinlet is velocity at inlet boundary condition. 1 (1)
Figure 1: Sketch of the computational grid Figure 2: Geometry of computational domain The mathematical model is governed by Navier-Stokes and continuity equations in form (2) and (3) and the turbulent model SA (DES) is represented by transport equation takes a form in (4) with basic additional relations (5). u p u u = [ t u ] t (2) u =0 (3) 2 D 1 =c w1 S {[ ] cb2 2 } c w1 f w Dt d S =S 2 2 f 2 d t = f v1 (4) (5) The original SA model was published in [1]. In Equation (4), d represents a distance from the wall boundary is so-called modified turbulent viscosity. condition, S magnitude of vorticity and The turbulent model DES adopts same relations such as SA model, but d parameter is replaced with which is formulated in (6), where is maximal value of the local grid distance. d =min d,c DES max x, y, z d, (6) The computing software is successfully validated by means of solutions laminar flow and flow generated by magnetics field [4, 5]. The computational code is based on finite element approach with explicit scheme providing further second order accuracy in time and space. The data sampling was made in advanced flow. 2.1 Visualizations on hydrodynamic tank The flow visualization was performed using the hydrodynamic tank (Fig. 3) that illuminates path lines of metallic dusty particles, which are wafted on surface of black-colourised water. The water flows on smooth 2
glass board in horizontal position, and therefore, it is quasi-two-dimensional flow visualization. Photos with appropriate exposure time is compared with vectors of the velocity field computed numerically. The cylinder was located into the canal and sketch of the measurement equipment is presented in Fig. 4. Figure 3: Hydrodynamic tank Figure 4: Sketch of the canal A velocity profile in vertical direction must be considered. Supposing the velocity profile is parabolic shape (Fig. 5), so that the maximal and mean velocity ratio is relate to 2/3. The water level was measured approximately 11 mm and kept constant during the measurement. Photos are taken in Reynolds number Re=3880±210. The deviation of the Reynolds number is evoked by not/constant water level, temperature dependent viscosity, flow rate deviations in inlet and geometry accuracy. Effects of the maximal and mean velocity ratio in vertical direction and additional colour dependent viscosity was not to be considered. 3
Figure 5: Velocity profile in the vertical direction 3 RESULTS 3.1 Numerical solution In Figure 6, the mean x-component velocity and Reynolds stress tensor in three line past a cylinder is displayed. Positions of lines are 1.06, 1.54 and 2.02 of x/d distant. Results from DES and SA is compared with experimental [6] and LES [7] results. Figure 6: X-component mean velocity and Reynolds stress tensor profiles Results were in relatively good agree to experimental and LES results. Results provided by SA turbulent model simulation agrees better then DES results. The Strouhal number was identified as a peek of ycomponent velocity frequency spectrum (Fig. 7) in point, which is located at x/d=9.5 in centreline of cylinder. The experimental investigation identified the Strouhal number for same Reynolds number and it was in interval 0.208-0.212 [8]. 4
Figure 7: y-component velocity Frequency spectrum 3.2 Visualization Longer exposures time Shorter exposure time Figure 8: X-component mean velocity and Reynolds stress tensor profiles In figure 8, the exposure time effect to visualisation velocity field is demonstrated. The photo with longer exposure time displays longer path of metallics particles. The visualisation Vector velocity field of DES solution Figure 9: Comparing the visualization with vector velocity field DES solution Figure 9 shows comparison between experiments and numerics, more precisely pictures of vector velocity field of DES simulations and visualizations photos. The generations and dissipations of small eddies exist mainly in the wake of the cylinder. These phenomena were distinctly captured experimentally and as well numerically. 5
4. CONCLUSIONS Results of mean velocity fields and Reynolds stress tensors computed using SA turbulent model agreed well with provided experimental [5] and LES calculation [6]. By contrast, DES simulations provided less accuracy results compared to experimental investigations. The value of the Strouhal number was identified as the same for SA and DES calculations and it was about Sh=0.228. It differed about 8-10% to experimental data [8]. Photos of flow visualizations were used to compare experimental and numerical results in the wake of the turbulent flow past a cylinder. It shows, that results provided numerically by means of Detached-Eddy Simulation contain an information about an existence of small eddies that are generated and dissipated in the wake of the flow and this phenomena were definitely confirmed experimentally as well. Furthermore, the size of the recirculating zone is approximately the same for experimental and numerical results. REFERENCES [1] SPALART, P.R., ALLMARAS, S.R..: A one-equation turbulence model for aerodynamic flows, La Recherche Aerospatiale, 1994; pp.5-21 [2] SPALART, P.R., JOU W.H., STRELEC, M., ALLMARAS, S.R.: Comments Of the Feasibility of LES for Wings, and on Hybrid RANS/LES approach, In Proceedings of First AFOR Int. Conference on DNS/LES. Greyden, Columbus, 1997 [3] FRAŇA, K., Simulations of the Unsteady Turbulent Flows Unsteady the Finite Element Method, In Proceedings of 1st WSEAS International Conference on Finite Differences - Finite Elements Finite Volumes Boundary Elements (F and B 08). Malta, 2008, Published by WSEAS Press, (ISBN 978960-474-004-8, ISSN 1790-2769) [4] STILLER, J., FRAŇA, K.: Transitional and weak turbulent flow in a rotating magnetic field, Physics of Fluids 19, 2006 [5] FRAŇA, K., STILLER, J.: A numerical study of flows driven by a rotating magnetic field in a square container, European Journal of Mechanics / Fluids B, 27, 2008, pp. 491-500 [6] FUREBY, C., LIEFVENDAHL, M., SVENNBERG, U., PERSSSON, L., PERSON, T.: Implicit Large Eddy Simulation, Incompressible Wall-Bounded Flows, Cambridge University Press, 2007 [7] BREUER, M.: Numerical and Modelling Influences on Large Eddy Simulations for the Flow past Circular Cylinder, Heat and Fluid Flow 19, 1998 [8] ROSHKO, A.: On the Development of Turbulent Wakes from Vortex Streets, NACA Report 1191, 1954 [9] HONZEJK, V.:The Work of Master of Degree: Turbulence Modelling of Unsteady Flow Past a Cylinder, Technical University of Liberec, Liberec, 2008 ACKNOWLEDGEMENT Financial support from Research Grant MSM 4674788501 Ministry of Education of the Czech Republic is gratefully acknowledged. 6