Zhang Yue 1, Guo Wei 2, Wang Xin 3, Li Jiawu 4 1 School of High way, Chang an University, Xi an, Shanxi, China,

The Eighth Asia-Pacific Conference on Wind Engineering, December 10 14, 2013, Chennai, India Analysis of Two Kinds of Boundary Conditions for Simulating Horizontally Homogenous Atmosphere Boundary Layer Based on the Standard k- Turbulence Model Zhang Yue 1, Guo Wei 2, Wang Xin 3, Li Jiawu 4 1 School of High way, Chang an University, Xi an, Shanxi, China, zhangyue1990bridge@163.com 2 School of High way, Chang an University, Xi an, Shanxi, China, guowei-14@163.com 3 School of High way, Chang an University, Xi an, Shanxi, China, 511817065@qq.com 4 Professor of Bridge Engineering, Chang an University, Xi an, Shanxi, China, 34086100@qq.com Abstract Constructing horizontally homogenous atmospheric boundary layer (HHABL) in computational domain is a key factor affecting reliability and accuracy of numerical simulation. This paper made analysis on two boundary conditions for achieving HHABL, then these two methods were applied to set boundary conditions and model constants to simulate class A wind field, which is suggested by the Chinese Wind-resistant Design Specification for Highway Bridges, in full scale with Fluent. Via comparing the results, differences between the two boundary conditions were studied. Based on these works, a method to improve simulation result via modifying boundary conditions and model constants was proposed in the latter part of the paper. Keywords k- model, the balanced atmospheric boundary, boundary conditions Introduction As wind tunnel tests should reproduce fully developed, which means being unchanging, atmospheric boundary layer in test section, achieving HHABL in computational domain is a key factor to ensure accurate and reliable calculation results in CFD simulations. As for so-called horizontally homogenous, it means the flow variables defined at the maintain unchanged till the, passing through the computational domain without disruptors. However, achieving such a HHABL is not easy, which needs three factors, flow boundary, turbulence model and wall conditions, to be coordinated. Changes of the flow will be reflected in simulation results mixed with disturbed effects caused by the objects in the computational domain, and the incident flow affecting the objects is different from the one. These two effects decrease the reliability and accuracy of the numerical simulation results. Many scholars had conducted researches on the question of how to achieve HHABL in numerical simulation. Blocken et al. (2007) pointed out that the roughness length included in the velocity profile should match the wall function, otherwise, the flow will develop into a new equilibrium atmospheric boundary in computational domain, resulting in the unexpected gradient flow. However, adjusting the wall parameters according the method Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds) Copyright c 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1 doi:10.3850/978-981-07-8012-8 282 658

Blocken proposed, the flow still changes in the computational domain. Hence, it is drawn that HHABL still cannot be achieved only with the wall function matched. So far, a widely used CFD method to achieve HHABL is proposed by Richards and Hoxey (1993). Through a series of assumptions on the atmospheric boundary layer, researchers derive a set of boundary conditions and corresponding model constants from the standard k- turbulence model equations. The method was recommended in the guides of Science and Technology in Europe Cooperation Organization (COST) and the Architectural Association in Japan (AIJ). The model constants setting method based on such boundary conditions is relatively simple, the given turbulent kinetic energy profile, however, can only be a constant, which is inconsistent with the actual situation. Starting with the model control equations, Yang et al. (2005) derived a set of boundary conditions whose turbulent kinetic energy could vary with the height. Model constants needs to be adjusted when using Yang s boundary, while the researchers did not give definite modification methods. This paper firstly compared theory differences of the two boundary conditions proposed by Richards and Yang separately, and then used standard k- turbulence model to simulate full scale class A wind field suggested by the wind-resistant design specification for highway bridges (hereinafter referred to as the specification) by two methods with commercial CFD software Fluent. By comparing the results of two examples, the differences between the two boundary conditions and their respective applicable conditions were analyzed. Finally, based on the simulation results, a method adjusting the model constants and boundary conditions was proposed, which improved the simulation results of HHABL. 1 Theory conditions of HHABL The standard k- turbulence model control equations of unsteady flow are: (1) (2) Turbulent viscosity in the equation is defined as : (3) 1.1 The first boundary condition This boundary condition was derived by Richards and Hoxey (1993). Firstly, The researchers made following assumptions for physical quantities of HHABL: (1) The vertical velocity in the boundary layer is 0, thus for two-dimensional flow field, there is, (2)The horizontal shear stresses distributed along the boundary layer height are equal, namely: is the atmospheric internal shear stress, is the shear velocity. The assumption is approximately fit in partial atmospheric boundary layer area. (4) 659

Secondly, the researchers use a logarithmic rate expression to fit the mean wind velocity profile at the of the computational domain: Where K is the von Karman constant, whose value is usually 0.4 to 0.42 Formula can be reached by equation (1) with conditions that the wind velocity u, the turbulent kinetic energy k and the turbulent dissipation rate have no gradients in the flowing direction, namely: (6) Since is much smaller than, it can be ignored. Solving equations (3) to (5), it is easy to draw the expressions of the turbulent kinetic energy k at and its dissipation rate that Richards et al. had recommended: The boundary condition equations (5), (7) and (8) derived from the k- turbulence model equation (1) certainly satisfy the equation (1), then with formula model equation (2) can be satisfied. 1.2 The second boundary condition The second boundary condition was derived by Yang et al., where the shear stress was not assumed as a constant, substituting equations (3), (5) and (6) into equation (1), we got:, namely is a constant. Since k only changes with the height z, the latter equation can be transformed into differential equation:. After separating the variables, we obtain the equation general solution: (9) (5) (7) (8) Then substitute it into equation (6), reaching: Equations (5), (9) and (10) are boundary conditions that yang proposed, hereinafter refers to as the second boundary conditions. Substituting equations (5), (6) and (8) into equation (2), we get:. Visibly, the boundary condition yang derived does not limit the model constants directly, however, it should be checked whether F 1 distributed along the height is 0 after all the boundary conditions are defined at the. If F 1 is not 0, the model control equation (2) will not be met, hence the flow will change until it generates a new balanced boundary layer in the computational domain. 660

1.3 Contrast of the two boundary conditions It can be seen from the derivation process, the greatest difference between the two boundary conditions is the assumption that whether the atmospheric boundary layer shear stress is equivalent. This assumption holds in limited range of the atmospheric boundary layer approximately, which is slightly different from real situation. Using this assumption and introducing equation (4) makes the turbulent kinetic energy k of first boundary condition remain unchanged in vertical direction, but it does not match the real atmospheric boundary layer. However, the first type own the following advantages: the shear velocity in mean wind velocity profile has a clear physical meaning, the dissipation rate of turbulent kinetic energy expresses a simple form, at the same time, the constants of standard k- model have a fixed proportional relationship. The shear stress is not assumed as a constant in the second boundary condition. From the standard k- model (1), using the same mean wind velocity profile with the first boundary condition, the general solution (9) of model control equation (1) can be reached directly. The turbulent kinetic energy setting at the fits actual situation better when the parameters A and B in the formula are selected correctly to fit the turbulent kinetic energy k which varies with the height. However, such improvement also introduces a new problem: the mean wind velocity profile, turbulent kinetic energy profile and turbulent kinetic energy dissipation rate profile derived from equation (1) can satisfy equation (1) naturally, however, equation (2) needs to be satisfied as well. However, we cannot find definite relationships among the model constants from the right side of the equation (2), in its finished expression. Substituting equation (3), (6) into the shear stress expression, we get. It can be seen from this, with the turbulent kinetic energy k changes along the height, the vertical distribution of horizontal shear stress in the atmospheric boundary layer varies. Thus, the in the mean wind velocity profile expression loses its physical meaning of shear velocity. 2 Calculation methods 2.1 Boundary conditions settings at computational domain The flow s mean wind velocity at the, turbulent kinetic energy and other physical quantities were set on the basis of relevant parameters of Class A wind field recommended by the Chinese norm. First, wind velocity at the reference height of 10m is assumed as 10m/s. The norm stipulates that the boundary layer roughness length of Class A wind field is 1m, thus the velocity profile is defined as:. Because the norm gives only the flow turbulence intensity at different heights of Class A wind field, we need to convert it into the turbulent kinetic energy profile. Under the premise of only concerning about the turbulence intensity in flow direction, the conversion 661

can be conducted by the formula with further assuming that the vertical pulsating velocity is 0m/s. The conversion results are shown in Table 1: Table 1 The turbulent kinetic energy conversion values of Class A wind field z u I 10 10 15% 3.38 20 11.00 14% 3.56 30 11.59 13% 3.41 40 12.01 12% 3.11 50 12.33 12% 3.28 70 12.82 11% 2.98 100 13.33 11% 3.23 150 13.92 10% 2.91 200 14.34 10% 3.08 2.2 Computing model In order to analyze the applicability of the two boundary conditions, two-dimensional models were applied for trials. The size of the computational domain was 400 5000m(y x), without any structures in it. Structured grid was applied to scatter the computational domain, grid node interval in the horizontal direction was 10m, vertical grid height of the first floor was 0.32m, for which growth rate was 1.08, and the center height of first floor grids is, The total number of grid was 60000. SIMPLEC algorithm was applied for pressure-velocity coupling, the convective terms and the central differencing scheme for the diffusion terms in the momentum equation and the turbulence model equation was QUICK. Set monitoring points at 50m off the computational domain when calculating. Criteria for judging calculation convergence was both the mean wind velocity and turbulence intensity in the monitoring point stayed unchanged and the non-dimensional residuals of each physical quantity was less than 10-5. The commercial CFD software Fluent was applied as the computing platform. 2.3 The computational domain flow settings of the first boundary conditions 200 Spec values Fitted values 150 100 50 0 2.5 3.0 3.5 4.0 k (m 2 /s 2 ) Fig. 1 Fitting results of turbulence intensity The Von Karman constant was taken K = 0.42,, thus,. For the turbulent kinetic energy k in the first boundary condition could only 662

be a constant, took k = 3.4 here,, substitute K and into formula, we got. 2.4 The computational domain flow settings of the second boundary conditions The mean velocity profile was the same with the first boundary conditions. A=-0.45, B=14 were Selected to fit turbulence intensity profiles, obtaining. The fitting results are shown in fig. 1. With reference to the first boundary conditions, was 13 and was 3.68, thus we got the kinetic energy dissipation rate. 2.5 Other parameters settings The standard wall function given by Fluent manual is. In the formula, is the mean wind velocity of first floor grid at the center height, is the dynamic viscosity, is the roughness constant. is the shear velocity in the boundary layer, whose value is, is an empirical constant, which is about 9.793. =, is the roughness height of the boundary wall. If, ). Wall shear velocity. In a balanced atmospheric boundary layer, and the wall function expression is. Setting, substitute the mean velocity profile equation (5) into the wall function, we can get =97. It should be paid attention to that the mean wind velocity profile of the second boundary condition is the same as the first boundary conditions, but in the formula does not have the physical meaning of shear velocity. The actual shear velocity at the center height of the first floor grid should be, then substitute it into the wall function, setting, we get, these are wall parameters of the second boundary conditions. In addition, it should be noted that. In accordance with the first boundary conditions,. At the top of the computational domain,, then, which is larger than the default limit value 1 10 5 in Fluent. Hence the default value in Fluent should be revised, otherwise, the turbulent kinetic energy will decrease rapidly near the. Other parameters are,, =1. 663

3 Calculation results 4.0x10 2 The First Boundary Condition 4.0x10 2 The Second Boundary Condition 8 9 10 11 12 13 14 15 16 v (m/s) 8 9 10 11 12 13 14 15 16 v (m/s) (a) Mean wind velocity comparison of the two boundary conditions at and 4.0x10 2 The First Boundary Condition 4.0x10 2 The Second Boundary Condition 0 1 2 3 4 5 6 k (m 2 /s 2 ) 0 1 2 3 4 5 6 k (m 2 /s 2 ) (b) Turbulent kinetic energy comparison of the two boundary conditions at and 4.0x10 2 The First Boundary Condition 4.0x10 2 The Second Boundary Condition 0 5 0.10 e (m 2 /s 3 ) 0 5 0.10 e (m 2 /s 3 ) (c) Turbulent kinetic energy dissipation rate comparison of the two boundary conditions at and 4.0x10 2 The First Boundary Condition 4.5x10 2 The Second Boundary Condition 4.0x10 2 8 0.10 0.12 0.14 0.16 0.18 Turbulence Intensity(%) 8 0.10 0.12 0.14 0.16 0.18 Turbulence Intensity(%) (d) Turbulence intensity comparison of the two boundary conditions at and Fig. 2 Calculation results comparison of the two boundary conditions Comparing the simulation results of the two boundary conditions, we can find that the boundary conditions changed little after through 5000 meter-long computational domain without structures, which basically maintains a level of homogeneity. The results of the two boundary conditions hold the same trend: (1) from Fig. 2(a) we can see, in the area close to the wall the mean wind velocity at the is higher than the one at the, while at high-altitude location it is lower than the one at the. The max error occurred at the top of 664

the computational domain, which is caused by unharmonious boundary condition at the top, is only about 3% and that is acceptable. The change is minor in the second boundary conditions. (2) Fig. 2 (b) and (d) show that the turbulent kinetic energy at the computational domain simulated by the two boundary conditions is less than the value set in the. Similarly, the change of turbulent kinetic energy with the second boundary conditions applied is slightly smaller than the one with the first. The reduced turbulent kinetic energy value at is 10% less than the one at the, and the reduction of the turbulence intensity is less than 1%. Fig. 2 (c) shows that the turbulent kinetic energy dissipation rate values derived from the two boundary conditions are almost fully consistent at the and the. It can be seen from the results, the two boundary conditions both can basically meet the requirements of the numerical simulation. The setting method of the first boundary conditions is more simple and convenient, which is suitable for the situation that turbulent kinetic energy along the height changes little. The second boundary condition owns better horizontal homogeneity and can set the turbulent kinetic energy varies along the height, which is more realistic and suitable for the circumstances requiring high calculation accuracy and the turbulent kinetic energy intensity changing along the height. 4 Modification of the second boundary conditions For the turbulent kinetic energy obtained by the second boundary conditions is slightly small, in order to obtain a more uniform boundary layer, this paper started with meeting the requirements of the control equation (2). Firstly, whether is checked. It can be seen from the formula, the closer to the wall, where z is smaller, the larger F 1 absolute value is. The distribution of F 1 along the height is shown in Fig. 3. 1.0 0.5 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Fig. 3 Distribution of F 1 near the wall along the height It can be seen from Fig. 3, F 1 near the wall is larger than 0. For reducing will reduce F 1, make changes to accordingly. F 1 Durbin (1995) pointed it out,, whose effective ranges are between 1.3 to 1.55. Without changing the boundary conditions and other model constants, this paper makes changes to, and. With different, F 1 is shown in Fig. 4. It can be seen from the calculation results, with the decrease in, there is almost no change in the mean wind velocity profile. But the change of turbulent kinetic energy has its own feature: the turbulent kinetic energy in high-altitude location (> 200m) has tiny difference, while the turbulent kinetic energy in the location lower than 200m increases with decreasing, and the closer to the wall, the more the value increases. 665

position (m) 1.0 0.8 0.6 =1.40 =1.45 =1.50 0.4 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 F 1 Fig. 4 Different F 1 with different The calculation results with different are shown in Fig. 5. 400 300 200 =1.40 =1.45 =1.50 4x10 2 3x10 2 2x10 2 =1.40 =1.45 =1.50 100 1x10 2 0 8 9 10 11 12 13 14 15 16 v (m/s) 0 2.6 2.8 3.0 3.2 3.4 3.6 k (m 2 /s 2 ) (a) Mean velocity profiles with different (b)turbulent kinetic energy profiles with different Fig. 5 The results with different In order to further reduce the attenuation of the turbulent kinetic energy, considering that the turbulent kinetic energy at high altitude decreases to the extent that doesn t change with, this paper attempts to modify other boundary conditions. It can be noted that even with the first boundary conditions, the physical quantities of the flow still change within the computational domain when the two control equations of the standard k- turbulence model fully meet the HHABL conditions. This should be due to the slight difference between the boundary conditions assumed and the actual situation of the flow field in computational domain, for example the vertical velocity was assumed as 0 in computational domain. Such error is difficult to predict, therefore, in view of the simulation results above showed that the turbulent kinetic energy at the is less than the one at, this paper attempts to add a coefficient to the expression of the turbulent kinetic energy dissipation rate profile at the, making. D was taken as 0.9, 0.8 and 0.7, the calculation results are shown in Fig. 6. It can be easily seen from Fig. 6, the change of turbulent kinetic energy dissipation rate set at the of the computational domain also does not affect the mean wind velocity at the, yet which has an impact on the turbulent kinetic energy values of all the height ranges at the. The specific performances are: while the turbulent kinetic energy dissipation rate decreases at the, the turbulent kinetic energy value increases in all height ranges at the, but the increased value is different. The value added of the turbulent kinetic energy at high altitude is larger, but it is smaller when getting closer to the wall. When D was 0.7, the turbulent kinetic energy at high altitude is well consistent with the conditions, but the turbulent kinetic energy value at the position below 200m is higher than the one at. When D was taken 1 and 0.9, the turbulent kinetic energy at is less than the one at. When D was 0.8, the turbulent kinetic energy of computational domain 666

and has a good matching degree. 400 300 200 D=1 D=0.9 D=0.8 D=0.7 y(m) 400 300 200 D=1 D=0.9 D=0.8 D=0.7 100 100 0 0 2.6 2.8 3.0 3.2 3.4 3.6 8 9 10 11 12 13 14 15 16 Velocity Magnitude(m/s) k (m 2 /s 2 ) (a)mean velocity profiles with different D (b)turbulent kinetic energy profiles with different D Fig. 6 calculation results with different D 5 Conclusions (1)The setting method of the first boundary conditions is relatively simple, which is suitable for the boundary layer simulation with little change in turbulent kinetic energy. The second boundary conditions allow the turbulent kinetic energy to be a change profile along the height, which is more consistent with the actual situation and suitable for the boundary layer simulation with high accuracy and drastic changes in turbulent kinetic energy. (2) Changing the model constant can better meet the model equation (2) in the near-wall region. That decreases will increase the turbulent kinetic energy near the wall of the computational domain, but it does not affect the one at high altitude. Simultaneously, almost does not affect the mean velocity profile. (3) Reduce at the, the turbulent kinetic energy value in the computational domain can increase, and the effect is more obvious with height increasing. The change of has little impact on the mean wind velocity profile. References P.J. Richards and R.P. Hoxey. (1993), Appropriate boundary conditions for computational wind engineering models using the k- turbulence model, Journal of Wind Engineering and Industrial, 46-47, 145-153. YANG Wei, Gu Ming, CHEN Su-qin. (2005), A set ofturbulence boundary condition of k- model for CWE, Acta Aerodynamica Sinica, 23(1), 97-102. YANG Wei, JIN Xin-yang, GU Ming, et al. (2007), A study on the self-sustaining equilibrium atmosphere boundary layer in computational wind engineer ing and its application, China Civil Engineering Journal, 40(1), 1-5. Blocken B, Stathopoulos T, Carmeliet J. (2007), CFD simulation of the atmospheric boundary layer: wall function problems, Atmos. Environ., 41(2), 238-252. FANG Ping-zhi, Gu Ming, TAN Jian-guo. (2010), Numerical wind fields based on the k- turbulent models in computational wind engineering, Chinese Journal of Hydroynamics, 25(4), 475-483. Yang Yi, Gu Ming, Chen Su, et al. (2009), New inflow boundary conditions for modelling the neutral equilibrium atmospheric boundary layer in computational wind engineering,. J. Wind Eng. Ind. Aerodyn., 97(2), 88-95. Svetlana Poroseva, Gianluca Iaccarino. (2001), Simulating separated flow using k- model, Annual Research Briefs 2001, Center for Turbulence Research, Standford University, 375-383. Durbin PA. (1995), Separated flow computations with the - - 2 model, AIAA Journal, 33: 659-664. Fluent Inc. (2003), Fluent 6.1 Documentation. JTG-T D60-01-2004, Wind-resistent Design Specification for Highway Bridges. 667