A comparison between classical DES and DDES using the in-house computational code

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1 A comparison between classical DES and DDES using the in-house computational code KAREL FRAŇA AND VIT HONZEJK Department of Power Engineering Equipment Technical University of Liberec Studentská 2, Liberec CZECH REPUBLIC Abstract: - The comprehensive three-dimensional turbulent flow study was performed using the in-house computational code. The simulation of the turbulent effect on the incompressible flows was carried out by the turbulent hybrid model. In general, turbulent flow simulation belongs to the time consuming calculations and requires mostly finer grid resolutions at boundaries. To find appropriate computational conditions, the numerical methods, code compilation and code calculations were investigated. Using optimal compilation options and grid construction, the computational performance can be increased and therefore, higher grid resolutions can be achieved and total required computational time can be reduced. Furthermore, the aspect of the two turbulent flow approaches represented by Delayed-Detached Eddy Simulation (DDES) and classical Detached Eddy Simulation was studied on the benchmarks of the turbulent flow past a cylinder at Reynolds number Key-Words: - simulations, turbulent flows, code compilation, numerical methods 1 Introduction The theoretical numerical investigation of the turbulent flows belongs to the important and useful part of the fluid mechanics and thermodynamics. In general, applied numerical methods opened a new possibility to achieve a wide range of the results describing significant as well as negligible flow effects in particular time and space. Simultaneously, the turbulent flow simulation is a complex problem required a huge computational capacity and knowledge of physics and mathematical backgrounds. Fortunately, there are various known approaches nowadays adopted more or less successfully for turbulent flow investigation [4]. To decide, which approach is the most suitable method for particular turbulent flow patterns, the non-dimensional Reynolds number of the flow is applied. At lower or moderate Reynolds number, a direct numerical simulation is preferred in term of accuracy and easier implementation. However, at higher Reynolds number, spatial scales in flows become to be too small to be resolved by the normal grid spacing and more powerful turbulent approaches have to be applied. The one of them is a hybrid URANS/LES model which combines advantages of different approaches to achieve time effective and sufficiently accurately methods for a simulation of a complex mostly threedimensional turbulent flow. The most popular approach of this particular group is Detached Eddy Simulation originally proposed by Spalart et. al. [2] hereafter referred to as DES97 or easily DES. An advantage of DES is the easy of programming and applications for a complexgeometry. The brief summary about DES applications in various flow problems can be found in [2]. However, the classical DES approach suffers from various deficiencies e.g. early separation caused a premature switch between LES and URANS especially in the region of the flow, where the local Reynolds number is sufficiently low. To depress this weakness the new standard version of DES so called Delayed Detached Eddy Simulation (DDES) has been proposed [10] and [11]. In this new concept, the limiter for the switch between LES and RANS depends not only on the grid size but as well on the solution [5]. The paper is organized as follows: In Section 2, the objective of the flow study including the basic equation system represented by Navier-Stokes equation and by turbulent approach is introduced. Details about numerical method and its application in the in-house computational code are presented in Section 3. Section 4 summarized important numerical results and compare with experimental data and other previous numerical results. Finally, in section 5, the significant conclusion is reported. ISSN: X 84 ISBN:

2 2 Problem formulations An incompressible turbulent fluid flow with the constant material properties molecular kinematical viscosity ν and density ρ past a cylinder at Reynolds number 3900 (based on the diameter of the cylinder D) and it was investigated using DES and DDES approach. The sketch of the flow problem is illustrated in Fig. 1. D ν Dt 2 { [( ν + ν ) ν] + c ( ν) } = cb 1S + b2 ( c f ) w1 w 1 ν σ 2 ν d (5) where the right-hand side includes the production term, the diffusion term and the destruction term for the reduction of the stresses in the vicinity near the solid walls. The production term includes further the scalar quantity S which is expressed by a magnitude of vorticity S plus a near-wall correction and it can be modeled as in the original Spalart-Allmaras model [1] leading to the form ν S + κ d = S f 2 2 υ 2 (6) Fig.1. The sketch of the flow problems The computational grid is consisted by 1.7 mil. tetrahedral elements and the smooth grid resolution was applied at the wall. Basically, this turbulent flow is governed by Navier- Stokes and continuity equations taking the form u + uu= p+ [ ( ν t + v) ( u) ] (1) t u = 0 (2) with Dirichlet and Neumann type boundary conditions u=g [ p+ ( + ν ( u) ] h (3) n ( ν t ) = (4) where u is the velocity, t time, p pressure divided by a density and ν t is a turbulent eddy viscosity, respectively. The effect of turbulence on the flow behavior is involved into a turbulent eddy viscosity calculated using the appropriate turbulent model. For instance, considering the DES model, the calculation of the turbulent eddy viscosity is based on a modified eddy viscosity in the Spalart-Allmaras model [1] and is calculated using the transport equation in the form as follows Equation 5 must be closed with the auxiliary relations and constant that can be found e.g. in [1]. The desired turbulent eddy viscosity ν t is calculated by the modified turbulent viscosity using the relation taking the form of ν =ν (7) t f v 1 In case of the DES model, the wall distance is given by a characteristic length scale d proportional to so that d = min( d, CDES ), max( x, y, z) (8) The recommended value for the adjustable parameter C DES is In practise, the criterion is based on the maximal value of the local grid distance and was originally proposed by Nikitin [3] as an appropriate choice for homogenous grids, especially. The weakness of the classical DES formulation is an unphysical behavior in the attached boundary layers relating to so called grey zone. To suppress the negative effect, various modifications of the DES formulation were proposed, for instance, DDES model. Particularly, a new function f d was additionally appended to the definition of the characteristic length scale d so that the dissipation length scale is now in a form d = d fd max ; DES where { 0 d C } (9) ISSN: X 85 ISBN:

3 f 3 = 1 tanh[(8 ) ] (10) d r d and ν + ν t ν rd = = x u x u κ d Sκ d (11) j i i j In practice, the main objective of this modification is to prevent earlier switch from URANS approach occurring in the attached boundary layer to LES model. 3 Code compilations The computational code was implemented on the top of the MG (Multigrid) library [6]. The both parts of the in-house code were written in programming language FORTRAN 95 and compiled using the Intel Compiler 11.1 for the 64 bit architecture. The MG library itself supports a parallel calculation and applied methods of the grid handling and processor communications were optimized in order to reach the best parallel performance effectiveness. To find more about test performance in respect to the parallel calculations, we refer to [6]. The mesh is decomposed into a specific number of grid partitions and this partitioning process is fully provided by the METIS library. The parallel code running is performed using MPICH 2, version which was compiled at the computational station with the same compiler as used for the computational source code. In general, the Intel compiler provides a wide range of the options that can be used for source code compilation to reach the optimized executable file. There are many aspects that can be taken into account for optimized compilation e.g. the size of the final executable file or more important, the speed optimization of the calculation processes. Table 1 summarizes a computational speed test carried out for a compilation option represented by O0,-O1, -O2, - O3 and Os. The option -O0 represents a compilation without any optimization, the option -O1, -O2, -O3 and -Os optimize a code for maximum speed, but it may cause an increase of the code size. The computational test of the various options was performed on the DELL Blade Server equipped by four cores processors Intel Xeon E5410/ 2,33 GHz with 32 GB memory. Furthermore, the calculated example represented a turbulent unsteady flow past a circular cylinder and the grid consists of tetrahedral elements. The computational domain was bounded by the inlet and outlet type boundary condition, periodic conditions and walls. The total solving times required for the calculation of the 1000 time steps are summarized in Table 1. The average value of the total time was calculated as an arithmetical averaging of the five fully identical computational patterns. option cal. 1 cal. 2 cal. 3 cal. 4 -O O O O Os < T > [s/ts] option cal. 5 avg speed -O O O O Os Tab.1. The total solving time in respect to the option of the compilation process < T > [s/ts] The last column indicates a speed of the calculation process. The code compiled with option -O1 had about 11,4 % lower performance then the code with option - O2. However, the compilation with option -O2 or -O3 did not prove a significant improvement of the computational speed and, in practice, a difference in the solving time between these two options is negligible. This particular conclusion is limited to inhouse developed computational code. In generally, the compilation with option Ox has a significant influence on the total size of the executable file, as well. This fact is demonstrated well in Table 2. option size of the file difference [byte] [ - ] -O O O O Os Tab.2. The size of the executable files compiled using different options. Using option -O1, the size of the executable file can be reduced significantly; however, the computational speed decreases simultaneously as demonstrated in the test above. The option -O0 represents a compilation without any code optimization that, in practice, leads to increase of the size of the executable file about 27 % in comparison to the best compilation with option - O1. The other options were leading to the increase of the code size as well, however, sizes vary between ISSN: X 86 ISBN:

4 these two mentioned thresholds, -O1 and -O0, respectively. Taking into account a time consuming computation of the unsteady turbulent flow, the code optimize in respect to the decrease of the solving time is an essential point of any code optimization. Using an appropriate choice of the compile options the total computational time can be effectively reduced and consequently the higher grid resolution can be used for numerical flow simulations Pressure coefficient Figure 2 illustrates the distribution of the pressure coefficient along the cylinder calculated for DDES. The results where compared with experimental results from Norberg [12]. 4 Results 4.1. Mean velocity field Figure 2 shows a comparison of the time-averaged velocity field at various positions in the flow wake calculated using the DES a DDES approaches and results were confronted by an experiments provided by Lourenco and Shih [8] and by other numerical simulations [4]. In the wake close to the wall of the cylinder, no significant differences were observed. However, far from the cylinder, the obvious deviation between both turbulent approaches DES and DDES is clearly detected. In general, the time-averaged velocity profile calculated using the DDES approach corresponds well to the other calculation based on the LES techniques and as well to experiment. The simulation using DES approach overestimated results especially in the wake region e.g. at positions x/d=1.54 or Fig.3: The angle dependence of pressure coefficient 4.3. Reynolds stress tensor and power energy spectra Figure 4 shows a distribution of the resolved part of the normal Reynolds stress tensors. The difference between distributions of the Reynolds tress tensor is obvious in the whole region of the wake of the flow. The maximal intensity of the streamwise normal Reynolds stress tensor is in both calculations at the same level. Fig.4: The resolved part <u x u x > of the Reynolds stress tensor. Fig.2: The time-averaged velocity field at positions: x/d=1.06,1.54 and ISSN: X 87 ISBN:

5 Figure 7 and 8 discuss the power energy spectra calculated from the kinetic energy at the particular positions. The both turbulent approaches revealed the same energy distribution along the frequency and wave number space. Fig. 5: The resolved part of the Reynolds stress tensors at positions: x/d=1.06,1.54 and 2.02 Figure 5 depicts profiles of the resolved part of the normal <u x u x > Reynolds stress tensor at positions x/d=1.06, 1.54 and The both turbulent approach provided qualitatively satisfactory agreement to the experiments and other numerical results, however, the slight difference can be identified at position far form the cylinder. Generally, DDES approach calculated the distribution of the normal Reynolds stress component more accurately in respect to the experiments. It is obvious as well, the both approaches suffered by the same inaccuracy that caused results deviation from the experimental data. In practice, LES calculation could provide better results; however, these results could be lead only by calculation on the relatively very smooth grid resolution leading further definitely to the timeconsuming calculation conditions. Fig. 6: The resolved part of the turbulent kinetic energy. Fig.7: The frequency spectra at the positions P1 and P2. Fig.8: The wave number spectra at the positions P1 and P2. The wave number and frequency spectra were resolved completely and the slope 5/3 was detected in the intermediate region of the time and space scales. Conclusions Because of the time consuming methods that are used for calculations of the turbulent effect on the flow behavior, the optimalization of the code compilation was carried out. This test performed on the particular FORTRAN compiler and computational station revealed that the best option for the source compilation is option with -O2. Using this option for compilation of the executable file, a reduction about 11 % of the solving time can be reached and even 115 % in case where no option is applied for code compilation. ISSN: X 88 ISBN:

6 The turbulent flow past a cylinder at Reynolds number was investigated using Delayed Detached Eddy Simulation and classical Detached Eddy Simulation and both turbulent approaches were compared with experiments and other numerical simulations. This test study was carried out under same conditions (computational grid, time step, initial and boundary conditions etc.) revealed that the improved variation of the DES approach so called DDES can reach results that were in the better agreement to the experiments and other numerical simulations in respect to the velocity field, Reynolds stress tensors etc.. However, form the point of the view of the wave number spectra and frequency spectra, no significant differences was detected. Acknowledgement This paper was financially supported by the Student Research Grand 2823 at the Technical University of Liberec provided by Ministry of Education of the Czech Republic. References: [1] Spalart, P.R, Allmaras, S.R.: A one-equation turbulence model for aerodynamic flows. La Recherche Aerospatiale, 1994; 1, pp [2] Spalart, P.R., Jou, W.H., Strelets, M., and Allmaras, S.R.: Comments on the Feasibility of LES for Wings, and on a Hybrid RANS/LES Approach, First AFOR Int. Conference on DNS/LES, edited by C.Liu and Z. Liu, Greyden, Columbus, OH, 1997 [3] Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D., Spalart, P.R.: An approach to wall modeling in large-eddy simulations, Phys. Fluids 12, 2000 [4] Fureby Ch., Liefvendahl, M., Svennberg U., Persson L. and Persson T.: Incompressible Wall- Bounded Flows, Implicit Large Eddy Simulation, Computing turbulence fluid dynamics, editor: Grinstein F., F., Margolin L.G. and Rider W.J., Implicit Large Eddy Simulation, Cambridge University Press, 2007 [5] Spalart, Ph.R.: Detached-Eddy simulation, Annual Rev. Fluid. Mech., 41: p , 2009 [6] Stiller J., Nagel W.E.: MG A Toolbox for Parallel Grid Adaptation and Implementing Unstructured Multigrid Solvers. In: E.H. D Hollander et al. (Eds.): Parallel Computing. Fundamentals & Applications, Imperial College Press, 2000 [7] Stiller J., Fraňa K., Grundmann R., Fladrich U., Nagel W.E.: A parallel PSPG Finite Element Method for direct Simulation of Incompressible flow, Euro-Par 2004, Parallel Processing (LNCS 3149), edited by M.Danutello, D.Laforenza and M. Vanneschi, Springer-Verlag, p , 2004 [8] Lourenco L.M., Shih C.: Characteristics of the plane turbulent near wake of a circular cylinder, a particle image velocimetry study, (data taken from Reference [9]), 1993 [9] Breuer M.: Large eddy simulation of the subcritical flow past a circular cylinder: Numerical and modeling aspects, Int. J. for Numer. Meth. Fluids, 28, , 1998 [10] Spalart P.R., Deck S., Shur M.L., Squires K.D., Strelets M. Kh., Travin A.: A new version of detached-eddy simulation, resistant to ambiguous grid densities, Theor. Comput. Fluid Dyn., 20, p , 2006 [11] Paik J., Sotiropoulos F., Port-Agel F.: Detached eddy simulation of flow around two wallmounted cubes in tandem, Int. Journal of Heat and Fluid Flow, 30, p , 2009 [12] Norberg C.: Effects of Reynolds number and lowintensity free stream turbulence on the flow around a circularcylinder, Publ. No. 87 :2, Department of Applied Thermoscience and Fluid Mech., Chalmers University of Technology, Gothenburg, Sweden, 1987 ISSN: X 89 ISBN: