Generation of artificial inflow turbulence including scalar fluctuation for LES based on Cholesky decomposition

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1 July 23 th, 25 ICUC9 Generation o artiicial inlow turbulence including scalar luctuation or LES based on Cholesky decomposition Tsubasa OKAZE (Tohoku University, Japan) Akashi MOCHIDA(Tohoku University, Japan)

Issue regarding the coupling o LES with MMM In large-eddy simulation cases, an inlow turbulence which satisies not only the turbulent statistics but also the instantaneous turbulent luctuation should

2 Issue regarding the coupling o LES with MMM In large-eddy simulation cases, an inlow turbulence which satisies not only the turbulent statistics but also the instantaneous turbulent luctuation should be generated. Global scale Meso scale Downscaling Building scale Van der Hoven - Journal o Meteorology, 957 Recently, increased access to computing power has led to several attempts to couple large-eddy simulation (LES) and MMM. 2

3 Approaches to generating inlow turbulence The approaches to generating inlow turbulence can be divided into two types: ) Storing the time history o velocity luctuations obtained rom a preliminary or recycling LES computation Lund et al., 998; Kataoka and Mizuno, 22 2) Artiicially generating inlow turbulence which prescribes turbulent statistics without conducting LES computations Lee et al., 992; Iizuka et al., 999; Klein et al., 23; Xie and Castro, 28; Kondo and Iizuka, 22 Jarrin et al.,26 etc. 3

4 Non-isothermal LES In recent years, non-isothermal LES computations within boundary layers have been carried. When LES is applied to a non-isothermal ield, not only the inlow velocity luctuation but also the temperature luctuation should be reproduced. The temperature was treated as passive scalar in the driver section. Kong, H., Choi, H., & Lee, J. S. (2) Physics o Fluids, 2() Hattori, H., Houra, T., & Nagano, Y. (27). International Journal o Heat and Fluid Flow, 28(6),

5 Non-isothermal LES In building scale LES cases, Tamura et al. (22) conducted a nonisothermal LES in Tokyo area with a generated temperature luctuation. Few study have been conducted. In addition, the generation method or temperature luctuation based on preliminary LES sometimes consumes much computational cost. Tamura, T., Nozu, T., Okuda, Y., Ohashi, M., Umakawa, H.: The USB proceedings o 8th International Conerence on Urban Climates, Aug

6 Obective o this study This paper proposes a new method o artiicially generating turbulent luctuations in wind velocity and scalar quantities such as temperature and contaminant based on the Cholesky decomposition o the time-averaged turbulent lux tensors o momentum and scalar. The method was validated by applying it to LES computations o contaminant dispersion in a hal-channel low. 6

7 . Background and obective 2. New method o generating inlow turbulence including scalar luctuation 3. Outline o LES computations 4. Conclusions 7

8 Deinition o values In this study, we express the values o wind velocity and scalar as i, the time-averaged values o i as < i >, and the deviation rom the time-averaged value as i ': i i i i =, 2, 3: the wind velocity components in the streamwise, lateral, and vertical directions (u, v, w) i = 4: the scalar value. 8

9 Matrix o the turbulent luxes o momentum and scalar 9 A regular matrix o the turbulent luxes o momentum and scalar, R i, is deined as w v u w w w w v w u v w v v v u v u u w u v u u R i i =, 2, 3: the wind velocity components in the streamwise, lateral, and vertical directions (u, v, w) i = 4 indicates the scalar value.

10 Cholesky decomposition o R i R i uu vu wu u uv vv wv v uw vw ww w u v w Cholesky decomposition R ik = a ik a k = a a T = A lower triangular matrix, a i, is obtained. a i R R2 a R3 a R 4 a R R R 22 a a a 2 2 a a a a R 43 R a 33 3 a a a a a 42 a 33 R 44 a 2 4 a 2 42 a 2 43

11 Expression o luctuations using a i With the lower triangular matrix, a i and a variable satisying and i i, the luctuations, i, can be rewritten as i i i i a i The transormation was originally proposed by Lund et al. (998) using Reynolds stress tensor. R i uu vu wu uv vv wv uw vw ww Lund T. S., Wu X., Squires K. D., 998: Generation o turbulent inlow data or spatiallydeveloping boundary layer simulations, Journal o Computational Physics, 4(2),

12 Extension o the transormation We extended the transormation to consider the turbulent luxes o scalar. R i uu vu wu uv vv wv uw vw ww R i uu vu wu u uv vv wv v uw vw ww w u v w a i R R2 a R3 a R 4 a R R R 22 a a a 2 2 a a a a R 43 R a 33 3 a a a a a 42 a 33 R 44 a 2 4 a 2 42 a

13 How to give the value o i i i i a i i i To impose time and space correlations or each component o the luctuations, the two-dimensional digital-ilter method proposed by Xie and Castro (28) and then revised by Kondo and Iizuka (22) was employed. 3

14 Prescribed time and space correlations Time rag or space correlation The prescribed time and space correlations are assumed using exponential unctions with an integral time scale, T, and a length scale, L: t t t t exp T r r r r exp L Time rag or space rag : Time rag : space rag 4

15 The time advances o generated luctuations The time advances o artiicially generated luctuations on a grid point (m, n) are expressed as t t, m, n t, m, nexp t t, m, n N N y z t, m, n bmbn rm m, nn m n t T exp 2t T 2 r: a random number satisying <r > = and <r i r > = i N y, N z : number o grid points included in the generated plane in each direction b k : a digital-ilter coeicient or the integral length scale in the generated plane in each direction 5

16 The time advances o generated luctuations t t, m, n t, m, nexp t t, m, n N N y z t, m, n bmbn rm m, nn m n t T exp 2t T 2 b k : a digital-ilter coeicient or the integral length scale in the generated plane in each direction According to the method proposed by Xie and Castro (28), two-dimensional random data are iltered to generate a set o two-dimensional data with the prescribed spatial correlation. Then, these data are combined with those rom the previous time step by using two weighting actors based on the exponential unctions. 6

17 Procedure o generation o luctuations By substituting as obtained using a new dataset o random numbers N y N z t, m, n bmbn rm m, nn m n into the equation below or each time step, t t, m, n t, m, nexp t t, m, n t T or the next time step is obtained. Then, the luctuations, i, are given by substituting into i i Storing temporarily. i i a i exp 2t T 2 7

18 . Background and obective 2. New method o generating inlow turbulence including scalar luctuation 3. Outline o LES computations 4. Conclusions 8

19 A priori LES computations A priori LES computations or a hal-channel were carried out to validate the reproducibility o the low and dispersion ields by applying the artiicially generated wind and scalar luctuations as an inlow boundary condition. Preliminary LES Periodic condition only or wind Sampling Generation o turbulent luctuations o wind and scalar H H Line source y z o x Main LES 9

20 Outline o preliminary LES computation First, a preliminary LES computation was conducted to obtain the turbulent statistics. A line source was placed on the ground cell immediately behind the inlow boundary and a passive scalar was emitted. The time series o the turbulent luctuations o the wind and scalar values were stored on the y-z plane at x = 5.H. Preliminary LES Periodic condition only or wind Generation o turbulent luctuations o wind and scalar Sampling H H Line source y z o x Main LES 2

21 Artiicially generation o luctuations Then, the luctuations o the wind and scalar values were artiicially generated based on the Cholesky decomposition o the time-averaged turbulent lux tensor o momentum and scalar which were obtained rom the database collected at x = 5.H in the preliminary simulation. Preliminary LES Periodic condition only or wind Sampling R i uu vu wu u uv vv wv v Generation o turbulent luctuations o wind and scalar uw vw ww w u v w H H Line source y z o x Main LES 2

22 Outline o main LES computation Finally, the main LES computation was carried out with the artiicially generated turbulent luctuations as the inlow boundary condition o the main computation. The reproducibility o the low and dispersion ields when applying the artiicially generated wind and scalar luctuations as inlow boundary conditions was validated. Preliminary LES Periodic condition only or wind Sampling Generation o turbulent luctuations o wind and scalar H H Line source y z o x Main LES 22

23 Integral length and time scales used in this study t t t t exp T r r r r exp L The integral length scales or prescribing the space correlations o the turbulent luctuations o wind velocity were assumed to be L =.5H (H: Domain height). The integral time scales or prescribing the time correlations o the turbulent luctuations were given based on the rozen turbulence approximation known as Taylor s hypothesis : T L/ U It is assumed that the integral length and time scales or scalar dispersion are equal to those or the wind velocity. 23

24 Turbulent statistics or generated mean values.8 Target Generated.8 Target Generated u / U..2.3 c U / q The generated mean wind velocity and concentration are in agreement completely with the targeted values. 24

25 Turbulent statistics or generated luctuations.8 Target.8 Target Generated.6 Generated c 2 U 2 / q wc / q The variance o the generated concentration and the turbulent scalar lux in the vertical direction as obtained rom the artiicial generation are also in good agreement with the targeted values. 25

26 Comparison o low ield Periodic Preliminary 5.H Main.H.H 3.H 5.H Preliminary LES Periodic condition only or wind Sampling Generation o turbulent luctuations o wind and scalar H H Line source y z o x Main LES 26

27 Streamwise change o mean wind velocity Target x=h x=3h x=5h / U.5 Periodic Target x=h x=3h x=5h The mean wind velocity changes very little in the downstream region and is in good agreement with the target value obtained rom the preliminary simulation. Main.2 5.H.H.H 3.H 5.H Preliminary 27

28 Streamwise change o TKE in grid scale Main Target x=h x=3h x=5h k / U 2 Periodic 5.H.5.H.H 3.H 5.H.5 The turbulent kinetic energy in the grid scale at x =.H is rapidly damped by 4%, relative to the target value. The causes are suspected to be related to the artiicially generated luctuations not generally being able to satisy the continuity and momentum equations (Xie and Castro, 28; Kondo and Iizuka, 22). Preliminary 28

Comparison o concentration ield 5.H 6.H 8.H.H Preliminary Line source Main.H 3.H 5.H Artiicially generated By comparing the results or the concentration ield at x =.H, 3.H, and 5.

29 Comparison o concentration ield 5.H 6.H 8.H.H Preliminary Line source Main.H 3.H 5.H Artiicially generated By comparing the results or the concentration ield at x =.H, 3.H, and 5.H in the main calculation with that obtained with x = 6.H, 8.H, and.h in the preliminary calculation, the reproducibility o the low and dispersion ields when applying the artiicially generated wind and scalar luctuations as inlow boundary conditions was validated. 29

30 Streamwise change o mean concentration Target_5H Generated wind and conc. Target_3H Target_4H Generated wind and conc. Target_2H 系列 7 系列 8 系列 9 系列 Target Generated wind and conc..8.6 Target Generated wind and conc. x =.H x = 3.H x = 5.H c.4 Target_4H Target Preliminary LES (Target) U / q Generated wind and conc. 系列 7 系列 8 系列 9 系列 Target Generated x =.H x = 3.H Generated Main LES wind and conc. The result o the mean concentration obtained rom the main simulation with artiicially generated luctuations is slightly larger c 3.2 c than the result o the preliminary simulation near the surace.

31 Streamwise Target_4H change o mean concentration Target_5H Generated wind and conc. Target_3H Target_4H Generated wind and conc. Target_2H 系列 7 系列 8 系列 9 系列 Target Generated wind and conc..8.6 Target Generated wind and conc. x =.H x = 3.H x = 5.H c.4 Target Preliminary LES (Target) U / q Generated wind and conc. 系列 7 系列 8 系列 9 系列 Target Generated x =.H x = 3.H Generated Main LES wind and conc. This dierence could be attributed to the underestimation o the turbulent diusion o the passive scalar in the upward direction due to the damping o the turbulent kinetic.3energy. near the inlow.2 boundary..6.2 c c U / q 3

32 Streamwise Target_4H change o w c 系列 7 系列 8 mean wind and conc. Target_5H Generated wind and conc. Target_3H 系列 9 系列 mean wind and conc. Generated wind and conc. mean wind and Target_4H conc. Target Generated wind and conc. Generated Target_2H wind and conc. mean wind and conc. Target Generated wind and conc x =.H x = 3.H x = 5.H w'c' / U2.4 The peak values o w c or both simulations are observed at the same height in each measured line. However, the turbulent lux at x =.H is 系列 7 系列 8 系列 9 系列 Target Generated somewhat smaller. than that.2 due.3 to.3 the underestimation..6.2o the turbulent.9.6 c 32 kinetic energy..2 Target Preliminary LES (Target) x =.H x = 3.H / q Generated Main LES wind and conc. c U / q

33 Streamwise change o c 2 Target_4H 系列 7 系列 8 mean wind and conc. Target_5H Generated wind and conc. Target_3H 系列 9 系列 Generated mean wind wind and and conc. Target_4H Generated wind and conc. Target_2H Target Generated wind and conc. 系列 7 系列 8 系列 9 系列 Target Generated Target Generated wind and conc. x =.H x = 3.H x = 5.H c'2 U 2 / q/ 2 U.4 Target Preliminary LES (Target) x =.H x = 3.H Generated Main LES wind and conc. The result o the main simulation are overestimated or each line although the distribution are similar to the results obtained with the preliminary. simulation c c U / q 33

34 . Background and obective 2. New method o generating inlow turbulence including scalar luctuation 3. Outline o LES computations 4. Conclusions 34

35 Conclusions. A new method or generating the turbulent luctuations in wind velocity and scalar such as temperature and contaminants, based on the Cholesky decomposition o the time-averaged turbulent lux tensors o momentum and scalar, was developed. 2. LES computations or a hal-channel were carried out to validate the reproducibility o the low and dispersion ields by applying the artiicially generated wind and scalar luctuations as inlow boundary conditions. 35

36 Conclusions 3. By employing a 5 5 non-singular matrix as a turbulent lux tensor matrix, the proposed method can generate simultaneously time series o wind velocity, temperature, and concentration o contaminants and so on. 4. This method can be applied to other artiicial generation methods based on the Cholesky decomposition o the Reynolds stress, including the synthetic eddy method (SEM) proposed by Jarrin et al. (26). 5. Further investigations into the eect o the integral time and length scales o scalar on reproduced dispersion ield, as well as the applicability o this method to non-isothermal low ields should be undertaken. However, this method will couple LES with MMM easily. 36

RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD

RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD N. Jarrin 1, A. Revell 1, R. Prosser 1 and D. Laurence 1,2 1 School of MACE, the University of Manchester,