COMPARISON OF SUBGRID-SCALE MODELS IN LARGE EDDY SIMULATION OF FLOW OVER A RECTANGULAR OBSTACLE

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1 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan COMPARION OF UBGRID-CALE MODEL IN LARGE EDDY IMULATION OF FLOW OVER A RECTANGULAR OBTACLE Qi Li and Qingshan Yang 2 chool of Civil Engineering, Being Jiaotong University, Being, 00044, @btueducn ABTRACT: This paper focus on influence of ubgrid scale (G) models on the accuracy of Large eddy simulations (LE) In the application of LE, performance of the G model is essential Contemporary research of G models in building wind engineering is reviewed here Popular models are listed and classified and detailed mathematics descriptions of them are given respectively Dynamic modify methods are summarized and detailed mathematics descriptions of some corrected models are presented also Furthermore, LE will be performed with the constant-coefficient magorinsky model (M), the mix scale model (MM), the dynamic model (DM) and the dynamic model mixed with similarity scale theory (DMM) on a problem of turbulent flows over a twodimensional rectangular obstacle model which represents typical large span buildings By comparing these computational results to the result comes from direct numerical simulation (DN), the contribution of the subgrid scale component computed by using each model will be obtained KEYWORD: LARGE EDDY IMULATION, UBGRID-CALE MODEL, NUMERICAL COMPUTATION, RECTANGULAR OBTACLE Introduction Large eddy simulation (LE) is a powerful tool in researching turbulent flow In LE, the large scales of motion are computed explicitly while the small-scale motions are modeled by ubgrid-scale (G) model It is well suited to wind engineering where interest centers on quantities such as forces, moments and their fluctuations which are primarily due to largescale motions In general, the higher the grid resolution (especially near the building surface) the more precise the results are if the numerical error is not considered Recently, in the context of the rapid growth of computational resources, extremely high-resolution LE computations have been attempted using massive parallel computing systems, however, for practical problems, in which numerous case studies are usually required, it seems to be more pragmatic to apply relatively low- resolution LE from the viewpoint of the computational cost In the application of low-resolution LE, the contribution of the unresolved part, that is to say, the subgrid-scale component becomes larger, and the performance of the G model for estimating it becomes essential[] Therefore, the investigation of the relative performance of various G models is very important for practical applications of low-resolution LE Research on G models could be divided into two categories, fundamental research and application research In general, the fundamental research is to introduce new G model or to amend existing G models, and check up its validity, while the application research used to compare the performance of several G models on one idiographic flow problems This paper focused on the influence of G models on the accuracy of large eddy simulation about wind flow over rectangular obstacle, so it falls into the application research By this time, the application research on G model already hold in many academic fields such as indoor air flow, fire test, pipe flow and so on

2 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan This paper is arranged as follows In section, basic models in common use are listed and classified, and their mathematics descriptions are given in detail respectively After pointing out the inherent flaw of basic model, section 2 and 3 summarized two modify approach and give some examples of corrected model Four models are chosen to implement large eddy simulation in the problem described in section 4, and their performances are compared in section 5 and 6 with two different ways At last section, a brief summary is given Basic G models Most of G model in use nowadays belongs to eddy-viscous family, it is assumed a linear relationship between the deviatoric part of the subgrid scale stress tensor τ and the resolved strain tensor, like equation () shows τ τkkδ = 2νG () 3 The second term on the left side of equation () is a diagonal tensor, ust like the pressure term of N equation, so they used to add together and form into one equivalent pressure term: Pe = P + τ kkδ 3 o, it can ignored when establish G model, and equation () become u u τ = 2νG, In which = + (2) 2 x x i The superbar means grid filter, and the proportional coefficient v G is usually called G eddy viscosity coefficient The G stress tensor could express as a function of resolved velocity after the mathematical formula of v G is determined Therefore, a closed and solvable governing equation for large scale motion is obtained The most popular basic models in common use are the constant-coefficient magorinsky model (M), the mixed scale model (MM) by agaut and Loc and so on magorinsky model (M) is the first G model proposed by magorinsky in 963 It can express as equation (3) ν (, ) ( ) 2 G Δ u = C Δ (3) In which = 2, is the filter truncation scale, and it could be calculated using Δ= ( Δ ) 3 xδyδ z, C is a constant coefficient between 0 and 02 Mixed scale model (MM) is proposed by agaut and Loc, who thought the G eddy viscosity is dependent on the resolved vorticity ω = u, the kinetic energy of the highest resolved frequencies and the filter truncation scale, ν G ( Δ, u) = C3 ω qc Δ (4) Where qc = uu i i 2 denotes the kinetic energy of the test filter u = u u, the denotation ~ represents the second filter operation, usually been called as test filtering, its truncation length scale should be larger than the grid filtering, that is Δ >Δ The value of C 3 is usually adopting 0 The coefficient C from all three models list above are constant and equal in the whole flow field, so this three models are suitable to simulating full developed and isotropic turbulent flow, they may bring dramatic errors into the numerical results when used in the underdeveloped and inhomogeneous flow near wall From the view of precisely predict the wind pressure on building surface, the flow field around building is more important than the fully developed flow far away from the building, therefore, the inherent flaw of these basic G model makes them unacceptable in building wind engineering, the revise to model is

3 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan necessary cholars made great effort in this problem, and lots of revised models are brought out The dynamic modification approaches is described in detail below 2 Dynamic models Eddy viscous model assumed a linear relation between G stress tensor and the resolved strain tensor, that is to say, the coefficient C is a constant and equal in the whole flow field As discussed before, in underdeveloped turbulence flow, it is not right o in 99, Germano proposed a dynamic modification method to fix this problem He allowed the coefficient C to change with time and space, the basic idea of his dynamic model is: using the Germano identical equation to relate the G tensor and the equivalent tensor obtained at another filtering level, thereby change the constant coefficient in eddy viscous model to a variable of time and space It is based on the assumption of similarity between different scales in the inertial range of the energy spectrum 2 The original dynamic model Germano modified the magorinsky model by using dynamic technique, the revised model is usually called dynamic magorinsky model (DM), it is also called original dynamic model because it is the first one The detailed description is given below Besides grid filtering operator, a second filtering operator with width larger than the width of grid filter is introduced, Grid filter operator and Test filter operator are denoted by - and ~ respectively herein after The filtered N equation for the two filtering levels is written below, u ( uu ) i p u u τ + = + ν + (5a) t x ρ xi x x x i x u ( uu ) i p u u T + = + ν + (5b) t x ρ xi x x x i x In whichτ = uu uu, T = uu uu are G stress tensor and subtest stress tensor T and τ are related by the Germano identity [3] L = T τ (6) When connected with the definition of T andτ, the resolved turbulent stress tensor L could rewritten as follows, L = uu uu (7) Model the T andτ in the same way using constant coefficient magrinsky model, τ = 2 C Δ, T = 2( C Δ ) 2 (8) Combine equation (6)~(8), we get ( ) ( ) L = T τ = 2 C Δ + 2Δ C (9) Assuming the coefficient C could taking out of the filtering operator, that is to say, 2 2 C C, then the equation (9) could rewritten as 2 L = T τ = 2C M (0) 2 2 Where M = Δ +Δ There is only one unknown quantity C in the equation (0) Both L and M are tensors, so (0) is an overdetermined equation system Applying a least square technique to this equation, a least error value for the coefficient is obtained

4 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan LM C = () 2M klm kl This model could produce a correct decreased turbulence near the wall But if the model coefficient C becomes a negative value somewhere in the flow field, numerical instability occurs In order to reduce the instability, the coefficient could be averaged in the homogeneous directions, if no homogeneous direction existed, averaged in time 22 Dynamic Mixed Models The original dynamic model assumes the coefficient C is remain unchanged after test filtering, that is to say, C 2 2 C, but this is unreal in an actual flow In order to overcome this flaw, Bin Zhou (2006) modified the dynamic model based on scale similarity theory, in this way a dynamic mixed model (DMM) is formed This model is used in large eddy simulation of lid-driven cavity flow with Re=0000, numerical results showed that it is well improved the statistical estimate of flow The basic idea of dynamic mixed model is: the subtest stress tensor is still modeled by basic eddy viscous model, but the subgrid stress tensor will be modeled by scale similarity model, in this way, the coefficient will not appears under the filtering operator symbol, so the assumption to its spatial distribution is avoid Using the definition, ubgrid stress tensor could be written as follows, τ = uu uu = ( ui + u i)( u + u ) ui + u i u + u (2) = uu + ( uu uu ) + ( uu uu ) + ( uu uu ) uu The scale similarity assumption is proposed by Bardina He thought the subgrid stress tensor is mainly caused by the interaction between the largest unsolved scale and the least resolved scale, so these equations uu i uu i, uu i uu i, uu i uu holds, therefore the subgrid stress tensor could be written asτ = uu uu ubtest stress tensor is still modeled using basic eddy viscous model M, T = 2 C Δ ( ) 2 Using Germano identity and the expression of T andτ, we can obtain 2 L = T τ = 2 CΔ uu + uu (3) ( ) This is also an overdetermined equation system with only one unknown quantity τ = 2 C Δ to After the C is obtained by least square technique, it is substitute into ( ) 2 get the value of subgrid stress tensor instead ofτ = uu uu 3 Numerical example and the code 3 example description The problem studied here is numerical simulation of wind flow over 2D rectangle obstacle Figure shows the sketch of computational filed and the model size When the simulation started, the initial value of pressure is set as zero in the whole field, and velocity is set as velocity profile of atmosphere boundary layer in exposure category B The far-field boundary condition is set as slip wall, the boundary condition of ground and the obstacle surface is set as non-slip wall, and the outlet boundary condition is laminar approximation condition et one zero pressure point on the ground as a reference point Re= 0 3 We use non-uniform grid system, and the grid resolution is about in LE

5 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan U h 0h 8h 5h 2h Figure ketch of wind flow over rectangular obstacle (2D) 32 Programme description This paper chooses common software to generate mesh and to do post-processing The core code for applying boundary conditions and for solving the N equation is selfprogrammed Like other CFD solvers, there are some essentials need to be determined, such as discretization method for equations spatial discrete scheme, time marching scheme, the uncoupling method for pressure and velocity, the iteration mode for solving discrete equation system and so on In the self-code, the stagger mesh under Cartesian coordinate is used, Finite Difference Method with central discrete scheme of second-order accuracy is used for dominate equations And Adams-Bashfort method is used for time advancing The time advancing form of N equation is shown in equation (4) It is obviously that there are two unknowns in it, u n+ and p n+ n n n ( u n+ ) 2( ) n n n i p ν ν e t + + ui = ui +Δt u + (4) x ρ xi x For uncoupling them, the Fractional step method is used It consists of three steps: Predict intermediate velocity: ignore the unknown pressure term, and obtain an approximate velocity field, it usually dissatisfy the continuum equation, it will fixed in the third step n n n ( u ) 2( ν + ν ) * n n i t ui = ui +Δt u + (5) x x Predict pressure filed: the pressure pn+ is obtained as a solution of the pressure- Poisson equation by means of successive over relaxation (OR) iterative technology, Pressure Poisson Equation (7) is deduce by using (5), (6) and continuum equation 2 n+ * pe ρ ui = (6) 2 xi Δt xi Revise velocity: n+ n+ * Δt pe ui = ui (7) ρ xi 33 elf-code verification The code self-programmed is verified by comparing results of the same flow problem with commerce software called ADINA The extremum of pressure obtain from both programme is given in table Table the extremum of pressure From Form Pressure ADINA the Code Maximum Minimum

6 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan It is shown that the absolute values of pressure obtained from the code are bigger This discrepancy could be cause by the different initial velocity Initial velocity field in ADINA is smaller, and equal in the whole flow field, but in the self-programmed code, the velocity profile of atmosphere boundary layer Isoline of instantaneous pressure distribution around obstacle from the software and the code is given in figure 2 a) Results from ADINA b) Result from the code Figure 2 instant pressure distributions around obstacle From figure 2, approximate distribution of instant pressure around the rectangle obstacle could be observed, that means the results come from self programmed code agree with the reference data from ADINA 4 A priori test The procedure of a priori test (Piomelli, 988) is as follows, Exact filtered quantities, such as subgrid scale stress, is obtained from the data of direct numerical simulation (DN) which produced by a self-programmed code, DN denoted by τ model-name 2 Predict the subgrid scale stress by chosen models, denoted by τ model-name DN 3 Compare τ with τ by calculating correlation coefficient 4 DN data As we all know, the least grid scale in DN should be smaller than the Kolmogorov dissipative scale η In order to make sure our grid is fine enough, DN with two different grid resolutions (70 280vs ) are carried out, and their results are compared The comparison of wind pressure on the top surface of the obstacle derived from two DN is shown in figure 3 The abscissa represents the position on the top surface of obstacle, and the vertical axis represents the value of wind pressure a) Time-averaged pressure b) Instantaneous pressure Figure 3 Comparison of wind pressure on top surface From figure 3, we can observe that the results from two DN with different grid resolution are almost the same o, the grid resolution is reasonable 42 Comparison of G model ince the shear G stresses directly affect the large scale field, so we consider the correlations between the modeled and computed components of the G stress tensor The

7 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan modeled G stress tensors are compared with the exact one by their correlation coefficients which defined as follows: (, ) r X Y X Y ( X X)( Y Y) XY n 2 2 ( ) ( ) ( ) ( ) = = X X Y Y X X n X X n In which X and Y are two generic quantities The correlation coefficient of each modeled G stress and exact one are listed in table 2 Table 2 Correlation coefficient of G stress, modeled by models and from DN data r G stress τ τ 2 ( τ 2 ) τ 22 MvsDN DMvsDN MMvsDN DMMvsDN The contribution of each G stress component to the final results of numerical simulation is the same The cask theory tells that the cubage of a cask is dependent on the shortest wood plate In this case, that means the correlation coefficient of τ 2 dominates Judged by this point, the D model have the largest correlation coefficient of τ 2, so it should be the best model to this flow problem we focused on among the four chosen models The viewpoint nowadays says that although a priori test could soundly expose the G transport mechanism, it is not completely determine the performance of G models, and it can only be an assistant method for studying performance of G models 5 A posteriori analysis Large eddy simulations with four models denoted as M, DM, MM and DMM are carried out on the problem of flow over 2D rectangle obstacle The performance of each G model is checked by comparing the numerical results to DN data Figure 4 shows the timeaveraged velocity field with streamlines which are computed by LE with four models and by DN The time averaging mode to wind velocity is 3 seconds averaging a) DN b) M

8 The eventh Asia-Pacific Conference on Wind Engineering, November 8-2, 2009, Taipei, Taiwan c) DM d) MM 6 summary e) DMM Figure 4 Time-averaged velocity field with streamlines The position of reattached point is compared in table 3 Table 3 comparison of reattachment position DN M DM MM DMM The emphasis of this paper is G models influence on the accuracy of the simulations The performance of four G models, namely the standard magorinsky model, Dynamic model, Mixed scale model and dynamic Mixing model, are compared using a priori test and a posteriori analysis References agaut, P and E Montreuil, et al (999) "Assessment of some self-adaptive G models for wall bounded flows" Aerospace cience and Technology 3 (6): 335~344 Hongrui Gong, hiyi Cheng, et al (2000) "A second-order dynamic subgrid-scale stress model" Applied mathematics and mechanics 2 (2): 47~53 ABBA, A and C Cercignani, et al (2003) "Analysis of subgrid scale models" International Journal of Computers and mathematics: 52~535 Bing zhou, Guixiang Cui, et al (2006) "Dynamic procedure based on the scale-similarity hypotheses for largeeddy simulations" Journal of Tsinghua University(cience and Technology) 46 (8): 438~44,446 Iizuka, HKondo (2006) Large-eddy simulations of turbulent flow over complex terrain using modified static eddy viscosity models Atmospheric Environment 40: 925~935 Lund T, Novikov E A Parameterization of subfrid-scale stress by velocity gradient tensor[j] Ann Res Briefs, CTR,992,27