MODELING THE NEUTRAL ATMOSPHERIC BOUNDARY LAYER BASED ON THE STANDARD k-ε TURBULENT MODEL: MODIFIED WALL FUNCTION

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan MODELING THE NEUTRAL ATMOSPHERIC BOUNDARY LAYER BASED ON THE STANDARD k-ε TURBULENT MODEL: MODIFIED WALL FUNCTION Pingzhi Fang 1, Ming Gu 2, Jianguo Tan 1, Bingke Zhao 1 and Demin Shao 1 1 Laboratory of Typhoon Forecast Technique, Shanghai Typhoon Institute of CMA, Shanghai 23, P. R. China, freedomfpz@yahoo.com.cn, jianguot@21cn.com, zhaobk@mail.typhoon.gov.cn, shaodm@mail.typhoon.gov.cn 2 Corresponding author, State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 292, P. R. China, minggu@tongji.edu.cn ABSTRACT By modifying the wall function proposed by Launder et al., modeling the neutral ABL based on the standard k ε turbulent model is studied again. The modified wall function is employed in modeling the wind fields of exposure categories B and D, which were also physically simulated in TJ-2 Wind Tunnel at Tongji University. The neutral ABLs are modeled effectively by comparing the computational results at the outlet and the inlet flow boundary conditions specified at the inlet according to the simulated results in TJ-2 Wind Tunnel. The modified wall function is shown to be necessary for the effectiveness, especially for the wind field with the larger aerodynamic roughness length. The inlet flow boundary conditions that partially compatible with the standard k ε turbulent model are also discussed. KEYWORDS: NEUTRAL ATMOSPHERIC BOUNDARY LAYER (NEUTRAL ABL), WALL FUNCTION, PHYSICAL ROUGHNESS HEIGHT, AERODYNAMIC ROUGHNESS LENGTH 1. Introduction Reproducing the neutral ABL where the structures are immersed is one of the basic requirements in computational wind engineering (CWE) (Richards et al., 199). The inaccuracy in CWE remains due partly to the errors introduced by modeling the wind field. The first question encountered in modeling the neutral ABL is how to model the inlet flow boundary conditions. This means that the inlet flow boundary conditions should be compatible with the turbulent model. Solving the Reynolds-averaged Navier-Stokes (RANS) equations is preferred in most high-reynolds-number flows. Thus, the second question is how to model the bottom of the computational domain. There are two basic methods to model the bottom of the computational domain: the wall function and the near-wall model. The former is popular because it is economical, robust, and reasonably accurate. Based on the wall function proposed by Launder et al. (Launder et al., 1974), however, some inaccuracy exists, especially for the wind field with the high-value aerodynamic roughness length. In order to reproduce the neutral ABL, many works have been done on the inlet flow boundary conditions. Based on the standard k ε turbulent model, Richards et al. provided one set of the inlet flow boundary conditions, including the mean wind speed, the turbulent kinetic energy and the turbulent dissipation rate. Together with the wall boundary condition proposed there, they reproduced the neutral ABL effectively (Richards et al., 1993). However, for the reason of the wall boundary condition, the method was hardly realized in commercial CFD software. Furthermore, the turbulent kinetic energy inlet flow boundary condition is

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan constant and conflicts with the results from the wind tunnel and on-site measurements. Yang et al. gave another set of the inlet flow boundary conditions for the standard k ε turbulent model (Yang et al., 29). Comparing with the inlet flow boundary conditions provided by Richards et al., the turbulent kinetic energy inlet flow boundary condition given by Yang et al. varies with the height and is more realistic. Blocken et al. gave a general description on the different practical methods and posed four requirements in obtaining the neutral ABL (Blocken et al., 27a; Blocken et al., 27b). All the requirements are related fully or partially to how to model the bottom of the computational domain. Some of the requirements may conflict in modeling the neutral ABL with the wall function proposed by Launder et al.. Hargreaves et. al. also pointed out the importance of the wall function in modeling the neutral ABL (Hargreaves et al., 27). By introducing the equivalent physical roughness height and then modifying the wall function proposed by Launder et al., modeling the neutral ABL based on the standard k ε turbulent model is studied again in the paper. Without loss of generality, the wind fields of exposure categories B and D, which correspond to the wind fields with low-value and highvalue aerodynamic roughness length respectively, are chosen to demonstrate the method of modeling the neutral ABL and the effectiveness of the modified wall function. The inlet flow boundary conditions that partially compatible with the standard k ε turbulent model are also discussed. 2. Method of Modeling the Neutral ABL 2.1. Inlet Flow Boundary Conditions As the indispensable parts of modeling the neutral ABL, the inlet flow boundary conditions that partially compatible with the standard k ε turbulent model in the Cartesian coordinate systemo xz are discussed firstly. The inlet flow boundary conditions include the mean wind speed U, the turbulent kinetic energy k and the turbulent dissipation rate ε. For simplicity, the horizontally homogeneous steady incompressible flow is considered. With the assumption of the generation of k should be equal to its dissipation in local region, the inlet flow boundary conditions that partially compatible with the turbulent model in the paper are: U( z) = u ln( z/ z )/ κ (1) * k = D ln( z/ z ) + D (2) 1 2 1/2 ε = Cμ k U / z (3) Where z is the height, z is the aerodynamic roughness length and κ =.42 is von Karman constant, u * is friction velocity, D 1 and D 2 are the constants and have the same dimensions 2 as that of k, C μ is the model constant and varies with the height. The method of calculating the model constant C μ can be referred to the paper, for example, provided by Richards et al.. 2.2. Modified Wall Function The wall function is often preferred in the high-reynolds-number flows. It was proposed by Launder et al. (Launder et al., 1974) and revised by Cebeci et al. (Cebeci et al., 1977) to consider the sand-grain roughness according to the experimental results given by Nikuradse (Nikuradse, 1933). For the rough surface with the uniform sand-grain roughness, the near wall mean wind speed can be obtained (Prandtl et al., 1969): U / u* = ln( z/ Ks ) / κ + C2 (4) Where C 2 = 8.5, and K s is the physical roughness height. For the non-uniform sand-grain roughness formed by the irregularly distributed structures on the land surface, the equivalent physical roughness height K seq is introduced in the paper. Its counterpart is the aerodynamic roughness length z in the mathematical formula as shown in Eq. (1). The relationship

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan between K seq must be obtained in order to model the neutral ABL. The near wall mean wind speed for the neutral ABL can be expressed as: U / u* = ln( z/ Kseq )/ κ + C2 (5) Letting Kseq = α Ks, the near wall mean wind speed can be further expressed as: U / u* = ln( z/ Ks )/ κ + C2 δ B (6) Where δ B = (ln α) / κ. By comparing Eqs. (4) and (6), it can be seen clearly that the additional term δ B should be considered in order to model the neutral ABL. The near wall mean wind speed should observe both Eq. (1) and Eq. (5) simultaneously, which leads to the relationship between K seq in the case of κ =.42 : Kseq 36z (7) The relationship between K s is also obtained as: Ks 36 z/ α = 36z (8) Where z z / α = can be interpreted as the equivalent aerodynamic roughness length. Eqs. (6) and (8) give the possibility to produce the neutral ABL by selecting an appropriate value of α. Eq. (4) appeared again when α = 1., which is the basis of the wall function proposed by Launder et al.. Using the logarithmic law profile for the mean wind speed provided by Richards et al. and Yang et al.: U( z) = u* ln[( z+ z)/ z]/ κ (9) The relationship between K s can also be expressed as: Ks 36 zpz /[ α( zp + z)] (1) Where z P is the distance from the centre point P of the wall-adjacent cell to the wall. With the assumption of z zp, Eq. (8) is again obtained. However, the above assumption will not be always valid when z P are in a comparable value in case of a sufficiently high mesh resolution in the vertical direction close to the bottom of the computational domain exists, which demonstrates the rationality in using Eq. (1) to obtain the relationship between the two parameters as stated in Eq. (8). 3. Application of the Method and the Modified Wall Function 3.1. Physically Modeling the Neutral ABL in TJ-2 Wind Tunnel Four exposure categories of wind fields including A, B, C and D were simulated with the scale of 1:3 in TJ-2 Wind Tunnel at Tongji University (Luo, 1999). The logarithmic law profile for the simulated mean wind speed with κ =.42 is expressed as: U( z)/ UG = u * ln( z/ z)/ κ (11) Where UG = 5 m/ s means the reference wind speed at the gradient height of z G and 4 zg 1.m. For the wind field of exposure category B, u * =.498, z = 2.638 1 mand for 3 the wind field of exposure category D, u * =.835, z = 6.756 1 m. The turbulent intensities can be roughly expressed by reducing the original ESDU profile (ESDU, 1974) down to 1.3 times, denoted by ESDU/1.3 here. The mean wind speed provided by the logarithmic law profiles and the turbulent intensity provided by the ESDU/1.3 profiles will be used as the inlet flow boundary conditions in modeling the neutral ABL. 3.2. Method of Calculating the Turbulent Kinetic Energy The turbulent kinetic energy must be provided in obtaining the model constant C μ. The turbulent kinetic energy k is defined as: 2 2 2 k =.5( σ + σ + σ ) (12) u v w

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan Where σ u, σ v and σ w are the wind speed fluctuations in the x, y directions. The streamwise turbulent intensity I u is the only physical quantity measured in TJ-2 Wind Tunnel. Thus, it is necessary to estimate the turbulent intensities in the lateral and vertical directions using I u. It can be roughly concluded that the lateral turbulent level is about.6 times of the streamwise turbulent level and the vertical level is about.3 times (Richards et al., 1993). Once the turbulent kinetic energy is obtained, C μ and the other model constants can be calculated consequently. 3.3. Scheme of the Computation The boundary conditions named after Fluent 6.3 are shown in Figure 1. Their mathematical implications will be shown later. The computational domain is 12. 1.5m ( x z). Using the nonuniform structural mesh method, there are totally 24 rectangular Figure 1: Boundary Conditions Named after Fluent 6.3 elements. The smallest scale in z direction near the bottom is.1m. The nodes in x direction are equally distributed, and the streamwise scale is.3m. Green-Gauss node based solver is used in order to get more accuracy results. Velocity-pressure coupled method is SIMPLEC, PRESTO and QUICK discretization methods are used for the pressure and momentum equations. The computation is stopped when the residual error curves of all physical quantities move without variation. Default options are used for the rest setups. More detailed information can be found in Fluent s online help. 3.4. Numerical Results of the Wind Field of Exposure Category B 3.4.1. Boundary Conditions and Model Constants The inlet flow boundary conditions prescribed at the inlet of the computational domain and the wall boundary condition, together with their mathematical implications are shown in Table 1. The physical roughness height Ks = 15z =.3957m is used in order to take into account the effects of the irregularly distributed structures expressed by the aerodynamic roughness length. The value meets the requirement of Ks < z P, and leads to z* 8., where 1/4 1/2 the non-dimensional length z* = ρcμ kp zp / μ and k P is the turbulent kinetic energy corresponding to z P. The model constant C μ =.2 is obtained to model the neutral ABL effectively. The other model constants are C1 ε = 1.44, C2 ε = 1.92 and σ k = σ ε /1.3. Figure 2 shows the computational cell values at the outlet and the inlet flow boundary conditions near the bottom of the wall. It can be seen that the computational results at the outlet agree very well with the prescribed values at the inlet for the mean wind speed. For the turbulent kinetic energy, there exists some difference adjacent to the wall. By using symmetry boundary condition at the top of the computational domain, there also exists some difference between the outlet and the inlet boundaries. Richards et al. and Blocken et al. gave some suggestions to obviate the problems (Richards et al., 1993; Blocken et al., 27b), and further discussions of the problem are out of the range of the paper. Table 1: Boundary Conditions of the Standard k ε Turbulent Model Boundary conditions Mathematical implication U,W U =.249 ln( z/ z ) /.42, W = Velocity-inlet k k = (.4 ln( z/ z ) +.354) ^.5 1/2 ε ε = C k U / z μ Wall Standard wall functions with K = 15z s

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan (a) Mean Wind Speed U (b) Turbulent Kinetic Energy k Figure 2: Comparisons of the Near Wall Values at the Outlet and the Iinlet Boundaries 3.4.2. Effects of the Physical Roughness Height on the Near Wall Flow Several physical roughness heights denoted by Ks / z are selected to evaluate their effects on the near wall flow, which equal to, 5., 1., 12.5, 15. and 17.5 respectively. Computational results are displayed in Figure 3. It can be seen that both of the mean wind speed and the turbulent kinetic energy are affected by the physical roughness height. Generally speaking, difference between the computational results at the outlet and the inlet flow boundary conditions decrease with the increasing of the physical roughness height. It can be concluded that the satisfactory results of modeling the neutral ABL when Ks / z 12.5 are obtained, which is a relatively small value comparing the result given by Eq. (8) whenα = 1. For the terrain with the low-value aerodynamic roughness length, Hargreaves and Wright gave the result of Ks / z = 2. ( Hargreaves et al., 27). For the terrain with the high-value aerodynamic roughness length, however, the additional terms δ B = (ln α) / K in Eq. (6) must be included, which will be shown in simulating the wind field of exposure category D. (a) Mean Wind Speed U (b) Turbulent Kinetic Energy k Figure 3: Effects of the Physical Roughness Height on the Near Wall Flow 3.5. Numerical Results of the Wind Field of Exposure Category D The wind field of exposure category D is simulated following the same procedure as described above. Main results are briefly presented below. 3.5.1. Boundary Conditions and Model Constants All the boundary conditions with their corresponding mathematical meanings are shown in the Table 2. At the beginning of the computation, δ B = 8. is selected, which 4 givesα 29. z / α = = 2.33 1 m. The modified wall function with δ B = 8. is accomplished by using User-defined wall function feature provided by Fluent 6.3. With the above selection, Ks = 11.5z.268m, thus K s < z P, * 11. The model constant C μ =.23 is obtained to model the neutral ABL effectively. Figure 4 gives the cell values at the outlet and the inlet boundaries near the bottom of the wall. Clearly, there exists some difference between the computational results at the outlet with the prescribed values at the inlet for the mean wind speed compared with the wind field of exposure category B. In essence, the difference is caused by the turbulent model. The mean

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan value of the wind speed adjacent to the bottom of the wall is determined by the wall function. By using the modified wall function provided in the paper, the computational value at the outlet and the prescribed value at the inlet adjacent to the wall agree very well, as shown in Figure 4(a). Table 2: Boundary Conditions of the Standard k ε Turbulent Model Boundary condition Mathematical implication U,W U =.418 ln( z/ z ) /.42, W = Velocity-inlet k k = (.134 ln( z/ z ) +.749) ^.5 1/2 ε ε = C k U / z μ Wall User-defined wall functions with K = 11.5z andδ B = 8. s (a) Mean Wind Speed U (b) Turbulent Kinetic Energy k Figure 4: Comparisons of the Near Wall Values at the Outlet and the Iinlet Boundaries 3.5.2 Effects of δ B on the Near Wall Flow The additional termδ B in Eq. (6) is important in success of modeling the neutral ABL. The value of δ B = 8. is obtained for the wind field of exposure category D. Different values of δ B, including δ B =., 2., 4. and 8., are selected to demonstrate the effects on the computation results for the mean wind speed and the turbulent kinetic energy, as displayed in Figure 5. The mean wind speed near the wall is accelerated and the turbulent kinetic energy is underestimated for the smaller value ofδ B. With the increasing of δ B and when δ B = 8., computation results of the mean velocity and the turbulent kinetic energy at the outlet agree considerably well with the prescribed values at the inlet. The main reason of the acceleration is that the mean wind speed of the cell adjacent to the wall is overestimated by the smaller value ofδ B. It should be point out that the wall function used in simulating the wind field of exposure category B above appeared again whenδ B =., which leads to α = 1. Clearly, there exists lager errors in simulating the wind field of exposure category D with the highvalue aerodynamic roughness length whenδ B =.. (a) Mean Wind Speed U (b) Turbulent Kinetic Energy k Figure 5: Effects of δ B on the Near Wall Flow 4. Comparisons between the Computational and the Wind Tunnel Results Computational results and wind tunnel results for the mean wind speed and the turbulent intensity of the wind fields of exposure category B and D are compared in Figure 6

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan and Figure 7 respectively. Results from the power law and the logarithmic law profile for the mean wind speed, and results from the ESDU and the ESDU/1.3 profile for the turbulent intensity are also included. Generally speaking, the neutral ABL is modeled effectively by using the modified wall function, especially for the terrain with the high-value aerodynamic roughness length. The computational results agree well with the logarithmic law profile compared to the power law profile for the mean wind speed. This is mostly because the logarithmic law profile is used in simulating the neutral ABL. (a) Mean Wind Speed U (b) Turbulent Intensity I U Figure 6: Comparisons Between Results for the Wind Field of Exposure Category B (a) Mean Wind Speed U (b) Turbulent Intensity I U Figure 7: Comparisons Between Results for the Wind Field of Exposure Category D 5. Concluding Remarks Modeling the neutral ABL based on the standard k ε turbulent model is studied again in the paper. Without loss of generality, the wind fields of exposure categories B and D are chosen to demonstrate the method of modeling the neutral ABL and the effectiveness of the modified wall function. The computational results at the outlet are compared with the prescribed values at the inlet which are obtained from TJ-2 Wind Tunnel. The agreement is quite well for the wind field with the low-value aerodynamic roughness length as shown by the wind field of exposure category B. Although there exists some difference for the wind field with the high-value aerodynamic roughness length as shown by the wind field of exposure category D, it does not weaken the effectiveness of the method and the modified wall function. The main conclusions are as follow: (1) Wall function is important in modeling the neutral ABL. The original wall function proposed by Launder et al. considers only the uniform sand-grain roughness. However, nonuniform sand-grain roughness formed by the irregularly distributed structures on the land surface is encountered in modeling the neutral ABL. By introducing the equivalent physical roughness height and modifying the wall function proposed by Launder et al., non-uniform sand-grain roughness can be considered in modeling the neutral ABL in the paper. (2) The essence of the acceleration of the mean wind speed near the bottom of the wall is that the wall function proposed by Launder et al. can not provided a smaller mean wind speed when the wind field has a larger aerodynamic roughness length.

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 29, Taipei, Taiwan (3) The inlet flow boundary conditions are studied again. By considering the wall function which gives the relationship among parameters of K s, z P, as stated in Eq.(1), the mean wind speed profile provided in this paper is more rational. The inlet flow boundary conditions in the paper are partially compatible with the turbulent model. There exists some difference between the logarithmic law profile and the power law profile for the terrain with the high-value aerodynamic roughness length, as shown in Figure 7. The logarithmic law profile keeps the shape throughout the computational domain, as shown in Figure 7(a). The question is how about the power law profile, which agrees very well with the wind tunnel results. Following the same procedure and the same model constants, the authors have a try on the question. The preliminary results show that more errors exist for the wind field of exposure category D compared with that shown in Figure 7(a). Whether the power law profile can be used as the inlet flow boundary condition and the errors can be reduced substantially by adjusting the model constants will be studied in the future. Acknowledgements Financial supports of this study from the National Natural Science Foundation of China (Grant Nos. 562162, 97154 and 57814), National Key Technology R&D Program of China (26BAJ6B5) and Shanghai Post-Doctoral Foundation (7R214162) are gratefully appreciated. References Richards, P. J. and Younis, B. A.( 199), Comments on Prediction of the wind-generated pressure distribution around buildings by E H Mathews. Journal of Wind Engineering and Industrial Aerodynamics, 34: 17-11. Launder, B. E. and Spalding, D. B.( 1974), The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 3:269-289. Richards, P. J. and Hoxey, R. P.( 1993), Appropriate boundary conditions for computational wind engineering models using the k ε model. Journal of Wind Engineering and Industrial Aerodynamics, 46-47: 145-153. Yang, Y., Gu, M., Chen, S. and Jin X. (29), New inflow boundary conditions for modelling the neutral equilibrium atmospheric boundary layer in computational wind engineering. Journal of Wind Engineering and Industrial Aerodynamics, 97: 88-95. Blocken, B., Carmeliet, J. and Stathopoulos, T.( 27a), CFD evaluation of wind speed conditions in passages between parallel buildings--effect of wall-function roughness modifications for the atmospheric boundary layer flow. Journal of Wind Engineering and Industrial Aerodynamics, 95: 941-962. Blocken, B., Stathopoulos, T. and Carmelie,t J.( 27b), CFD simulation of the atmospheric boundary layer: wall function problems. Atmospheric Environment, 41: 238-252. Hargreaves, D. M. and Wright, N. G.( 27), On the use of the k ε model in commercial CFD software to model the neutral atmospheric boundary layer. Journal of Wind Engineering and Industrial Aerodynamics, 95: 355-369. Cebeci, T. and Bradshaw, P.(1977), Momentum Transfer in Boundary Layers. Hemisphere Publishing Corporation, New York. Nikuradse, J., Stromungsgesetz in rauhren rohren, vdi-forschungsheft, 1933, 361. (English translation: Laws of flow in rough pipes, Technical report, NACA Technical Memo 1292. National Advisory Commission for Aeronautics, Washinton, DC. 195.). Prandtl, L., Oswatitsch K and Wieghardt K, Fuhrer Durch Die Stromungslehre. Braunschweig, 1969. (Chinese translation, 1981) Luo, P., Wind tunnel measurements on standard models. Master thesis (in Chinese), Tongji University, 24. ESDU. Characteristics of atmospheric turbulence near the ground. Engineering Science Data Unit Numbers 743 and 7431, London, 1974.