HYBRID STOCHASTIC SIMULATION OF PRESSURE FLUCTUATIONS ON TALL BUILDINGS

C.M. Chan1, M.F. Huang2, I.W.H Lau3 and K.C.S. Kwok4

1Associate Professor, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. E-mail: cecmchan@ust.hk

2Postdoctoral Researcher, Institute of Structural Engineering, Zhejiang University, Hangzhou 310027, P.R.China. E-mail: hmfust@gmail.com

3PhD candidate, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. E-mail: ilau@ust.hk

4Emeritus Professor, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, E-mail: kkwok@ust.hk

ABSTRACT

With the great advances in of computer power, numerical simulations utilizing Computational Fluid Dynamics (CFD) are gaining much attention in estimating the wind effects on tall buildings under extreme wind conditions. Both Reynolds-Averaged Navier-Strokes (RANS) simulation and Large Eddy Stimulation (LES) have been developed to solve the complex flows around the building-type bluff bodies. While the steady state RANS simulation has been widely used to predict mean characteristics of turbulent flow, the LES technique is aimed at capturing the fluctuating nature of turbulent flow field. However, the current LES technique is still unable to solve the practical problems of building aerodynamics for numerically predicting wind-induced dynamic loads with reasonable computation cost. In this work, by combining the techniques of RANS and Kinematic Simulation (KS), a computationally efficient method hereafter referred to as the RANS-KS method is introduced. Based on the mean kinetic energy and energy dissipation rate obtained from RANS, an artificial fluctuating velocity field can be generated by KS. The Poisson equation is then solved to determine the pressure fluctuations corresponding to the fluctuating velocity field. The effectiveness of RANS-KS is demonstrated through a benchmark full-scale tall building, and the results of pressure fluctuations show good agreements with that of the wind tunnel experiment as well as LES.

KEYWORDS: PRESSURE FLUCTUATIONS; REYNOLDS AVERAGED NAVIER-STOKES (RANS) SIMULATION; KINEMATIC SIMULATION (KS); TALL BUILDINGS

Introduction

In the past several decades, wind-tunnel testing has been the best practice to measure wind-induced load effects on tall buildings [Cermak (2003)]. With the advances in computer technology, numerical simulations using computational fluid dynamics (CFD) have been gradually gaining increased attention as an alternative approach for assessing wind effects on buildings [Senthooran et al. (2004), Huang et al. (2007)].

The Reynolds-Averaged Navier-Stokes (RANS) simulation has been one of the most popular CFD approaches due to its efficiency and robust performance, especially with the development of various K-ε turbulence models [Murakami (1998)]. In RANS simulation, turbulence is only modeled by time-averaged statistics. In fact, the major accomplishment of RANS simulation is its ability to predict the mean velocity and pressure fields of turbulent

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

flows around a bluff body with reasonable accuracy and time for engineering design purpose. The major drawback of RANS is lack of ability on predicting transient behavior of fluctuating flow field. The fluctuating pressure of flow field is particularly important in dynamic wind load estimation of tall buildings.

More recently, time dependent large eddy simulation (LES) has become a powerful CFD tool for turbulent flow analysis as it resolves the large-scale unsteady motions and accounts for smaller eddies using a subgrid-scale turbulence model [Tamura (2008)]. In contrast to RANS, LES employs spatial averaging to filter smaller subgrid scale eddies, but to retain the stochastic nature of turbulent solutions to the full Navier-Stokes equation with the aid of homogeneous turbulence theory [Iaccarino et al. (2003)]. While LES gives more accurate results of fluctuating flow field than RANS, the computational cost to estimate the wind effects on full scale buildings, which has Reynolds number in order of 107 or higher, is still unaffordable due to a prohibitively large amount of meshes required [AIJ (2005), Tamura (2008)], especially in the vicinity of the near-wall layers of the flow. Although LES can be performed in a model scale similar to that of a wind tunnel test, the model scale simulation can only be adequate to solve the overall mean flow quantities. The fluctuating quantities of unsteady flow are highly dependent on Reynolds number [Lim et al. (2007)]. Thus, there is still a demand to develop an efficient CFD method to predict wind-induced pressure fluctuations on full scale tall buildings associated with a realistic high Reynolds number.

In this paper, a hybrid RANS-KS method is presented to predict the pressure fluctuations and wind load effects on tall buildings by combining RANS and a spectral method, known as kinematic simulation (KS) first proposed by Kraichnan (1970) and later rigorously developed by Fung et al. (1992). The detailed techniques of RANS and KS are discussed. The effectiveness of the hybrid RANS-KS technique is demonstrated with an example of the CAARC (Commonwealth Advisory Aernautical Research Council) building.

RANS simulation of mean flow field

Air flows around bluff bodies in wind engineering problems are usually considered as incompressible viscous fluids. The time-average continuity and momentum equations for the mean flow field can be given respectively as follows:

0i

i

Ux

∂=

∂ (1)

1i ij i j

j j j i

U U PU v u ux x x xρ

⎛ ⎞∂ ∂∂ ∂= − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(2)

where Ui is the mean velocity component in Cartesian coordinates (x1,x2,x3); p is the mean pressure; ρ is the wind density and ν is the kinematic viscosity. The over-bar denotes Reynolds-averaged quantities and tensor notation is used. In most popular RANS turbulence models such as K-ε, K-ω, the Reynolds stress model (RSM) or the term i ju u in Eq. (2) has the form based on the assumption of a Newtonian fluid as

23

jii j t ij

j i

UUu u Kx x

ν δ⎛ ⎞∂∂

− = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ (3)

where ijδ is the Cartesian Kronecker symbol; the eddy viscosity tν is given as

2

tKCμνε

= (4)

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

Although the two-equations K-ε or K-ω model, can be used in the RANS-KS method, the RSM is incorporated into the RANS-KS method in order to better capture anisotropic effects of the boundary layer flow.

Kinematic simulation of fluctuating velocity field

Anisotropic Atmospheric Turbulence near the ground Accurate kinematic simulation of Atmospheric Boundary Layer flow in the

computational domain is imperative to obtain reasonable and reliable prediction of wind load effects on tall buildings. In this study, the turbulent wind velocity fields were generated based on the energy spectra derived from full-scale measurements in the Earth’s surface boundary layer. Two modified spectral equations, one for longitudinal (u) gustiness and another one for the lateral (v) or vertical (w) gustiness, are given to characterize the anisotropic atmospheric turbulence near the ground [ESDU (2001)]. The original frequency spectra given in ESDU 85020 were rewritten into the forms of wave number spectra as shown:

( )( )

2

5/ 62 Kol

2 1 exp 21 /

uu u

ee

K kE k Ak k

k k

⎡ ⎤⎛ ⎞⎢ ⎥= − ⎜ ⎟⎢ ⎥⎝ ⎠⎡ ⎤+ ⎣ ⎦⎢ ⎥⎣ ⎦

(5)

( ) ( )

( )( )

22

11/ 62 Kol

/2 exp 2 or 1 /

ess s

ee

k kK kE k A s v wk k

k k

⎡ ⎤⎛ ⎞⎢ ⎥= − =⎜ ⎟⎢ ⎥⎝ ⎠⎡ ⎤+ ⎣ ⎦⎢ ⎥⎣ ⎦

(6)

where Au, Av and Aw are the respective spectral constants; ke is the effective wave number corresponding to the peak vale of spectral energy density. The Kolmogorov wave number can be defined as

1/ 4

Kol 3k εν⎛ ⎞= ⎜ ⎟⎝ ⎠

(7)

Given the total kinetic energy K derived from the RANS simulation, the gust components of kinetic energy Ku, Kv and Kw can be distributed by the ratio of fluctuating velocity components, which are given in ESDU 85020 as

41 0.22cos2

vv

u

zh

σ πβσ

⎛ ⎞= = − ⎜ ⎟⎝ ⎠

(8)

41 0.45cos2

ww

u

zh

σ πβσ

⎛ ⎞= = − ⎜ ⎟⎝ ⎠

(9)

in which z = the height of interest; h = the boundary layer height, which can be taken as

( )* 4/ 6 10u × where *u = the ABL friction velocity; uσ , vσ and wσ represent the RMS values

of the fluctuating velocity components.

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.01 0.1 1 10 100 1000 10000wave number k (m-1)

Ei(k

)(m

3 s-2)

Longitudinal gustinessLateral gustinessVertical turbulenceIsotropic turbulence

Figure 1: The energy density functions of gustiness components at the separation point A

Kinematic simulation Using kinematic simulation approach, the synthetic wind velocity field can be

generated by the sum of N random Fourier modes as [Nicolleau and Elmaihy (2004) ]

( ) ( ) ( ) ( ) ( ) ( )1 1

ˆ ˆ, , cos sinN N

n n n n n n n n nn n

t t t tω ω= =

⎡ ⎤= = × + + × +⎣ ⎦∑ ∑u x u x a k k x b k k xi i (10)

where the wave vector nk with its unit vector ˆnk is characterized by its spherical orientation

as

sin cos

ˆ sin sincos

n n

n n n n n n

n

θ φθ φ

θ

⎛ ⎞⎜ ⎟= = ⎜ ⎟⎜ ⎟⎝ ⎠

k k k k (11)

The orientation angles nθ and nφ of wave vector nk can be picked randomly with the probability ( ) ( )Pr sin / 2n nθ θ= and ( ) ( )Pr 1/ 2nφ π= in each mode and realization. The prescribed probability distributions ensure statistical isotropy of turbulence with respect to

nk . The coefficients na and nb are random and uncorrelated vectors, and can be chosen from

a 3-D Gaussian distribution with a zero mean vector and covariance matrix 2n ijδu , in

which denotes an ensemble average operation. It should be noted from the synthetic velocity field of Eq. (10) that the coefficients ( na and nb ) of the nth Fourier mode are chosen to be normal to nk , so the continuity condition of the flow field can be ensured [Fung et al. (2008)]. The characteristic angular frequency nω in Eq. (10) also takes into account the unsteadiness associated with the nth wavemode and can be chosen to be proportional to the eddy-turnover time of the nth wavemode [Nicolleau and Elmaihy (2004)) as

( )3n n nk E kω λ= (12)

whereλ is the unsteadiness parameter and may be expected to be of the order of 1; n nk = k denotes the discretized wave number within the wave number ranges.

In order to properly capture the energy contribution of larger eddies with lower wave numbers, the wave number range is better discretized using a logarithmic distribution of the N wave numbers as

1log log1

Nn

k kkN−

Δ =−

(13)

Wind

A

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

( )1exp log 1n nk k n k= + − Δ⎡ ⎤⎣ ⎦ (14) where k1 and kN are the wave number corresponding to the Kolmogorov wave number defined by Eq. (7) and the largest eddy defined by

1 3 / 22kKεπ= (16)

The elements of covariance matrix 2n ijδu can be associated with the energy density function

evaluated at each wave number point nk as

( ) ( )2, 2 , ,n s s nu E k s u v w= = (15)

At a given position in the computational domain, the energy density functions (i.e. , ,u v wE E E ) must be properly evaluated with the spectral constants (i.e. Au, Av and Aw) and the effective wave number ke explicitly defined. When determining the energy density function Ev for the lateral v-component wind gusts on the side faces of a tall building, the effective wave number ke representing the wave number corresponding to the peak value of spectral energy density should also agree with Strouhal number.

Pressure fluctuations associated with simulated velocity fields For an incompressible wind flow, the Poisson equation for pressure fluctuations p can

be derived from the divergence of the momentum equation Eq. (2) as

( ) ( )2 2

2 , 1,2,3jii j i j

i i j i i j

uUp u u u u i jx x x x x x

ρ ρ∂∂∂ ∂

= − − − =∂ ∂ ∂ ∂ ∂ ∂

(16)

where Ui denotes the mean velocity component along the Cartesian coordinate axis xi obtained from the RANS simulation and ui denotes the fluctuating velocity component along the Cartesian coordinate axis xi generated by the KS method. The first and second source terms at the right-hand side of Eq. (16) represent the mean-shear-turbulence interaction and the turbulence-turbulence interaction respectively. The Poisson equation can be solved at each time step consistent with the KS method to obtain the time series of pressure fluctuations.

At the inlet of the computational domain, the pressure fluctuation field can be specified by the following empirical formula [Senthooran et al. (2004)] associated with the longitudinal u-component gustiness as

2

2up ρ

= (17)

Zero gradient boundary conditions are applied to the ground, building surfaces as well as symmetric side planes

0i

px∂

=∂

(18)

The spatial discretisation of the Poisson equation using finite volume method requires a subdivision of the whole computational domain into control volumes. The systems of Poisson equation imposing on all control volumes are then treated in a segregated way, meaning that pressures are defined at the nodes of an ordinary control volume while velocity components on staggered grids are centered on the cell faces. The staggering of the velocity avoids the unrealistic behavior of the discretized equation for spatially oscillating pressure, and velocities are generated at exactly the locations where they are required for the scalar transport. The linear algebraic equations arising from finite volume discretization for the Poisson equation are then solved by a parallel version of the Stone’s strongly implicit iterative procedure [Reeve et al. (2001)] based on central and upwind difference schemes.

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

Illustrated Example: The CAARC building

Wind tunnel based pressure measurements The CAARC building has long been used for calibration purposes by different

boundary layer wind tunnels [Melbourne (1980)]. The prototype building has an overall height of 180m and a rectangular floor plan dimension of 30 m by 45 m. For comparison purpose, a wind tunnel test was conducted using 1:400 scale rigid model measured by a synchronous multi-pressure sensing system (SMPSS) at the CLP Power Wind/Wave Tunnel Facility of the Hong Kong University of Science and Technology (HKUST).

The boundary layer wind model corresponding to a countryside open terrain (Category 2) in the wind code (AS/NZS 1170.2: 2002) was simulated at the wind tunnel. The power law exponents of the mean wind speed profile and the turbulence intensity profile are approximately 0.15 and -0.18, respectively. A mean wind velocity of 32 m/s corresponding to a 5-year return period of typhoon wind in Hong Kong was used in the prototype scale calculation.

Numerical simulation All calculation of RANS, KS and Poisson solver were performed using an open-

source CFD software OpenFOAM (Open Field Operation And Manipulation). The computational domain of the simulation is presented in Figure 2. The volumes near the solid wall are sub-divided to account for boundary layer effect. A polyhedral hex mesh with 8.5 million cells has been adopted in this study. In term of numerical discretization scheme, upwind and QUICK schemes are used for diffusion terms and convection terms. Semi-Implicit Method for Pressure Linked Equations (SIMPLE) is used for pressure calculation in the RANS simulation with a multi-grid solver. Wind direction normal to the wider face of the CAARC building was simulated in this study. Symmetry planes were applied on both two lateral sides and top boundary. Zero gradient boundary condition was used at the outlet because free stream flow was expected.

Results and discussion Based on the values of turbulent kinetic energy and dissipation rate obtained from

RANS, the energy density functions for three components of gustiness at each volume of the computational domain were determined. The fluctuating velocity field in the whole computational domain was then simulated by KS. Since the inlet wind flows along the stream wise direction, larger eddies in the stream wise direction and smaller eddies in the lateral and vertical directions are observed. Due to anisotropic effect of atmospheric boundary layer, different standard deviations of fluctuating velocity are also expected in three orthogonal directions.

Figure 3 presents the simulated RMS pressure coefficients at both 1/3H and 2/3H. Both wind tunnel result and LES result [Huang et al. (2007)] were plotted together for comparison. The overall results of the RANS-KS method capture the similar tend and match well with the wind tunnel test results. For point to point comparison, the result difference between RANS-KS and experiment is approximately 5-15%, but this discrepancy is within the acceptable range for wind engineering application for buildings. The same order of discrepancy with 5-15% for both mean and fluctuating pressures obtained from different wind tunnel studies are often observed in practice.

In term of computational efficiency, the RANS-KS method can achieve significant saving of computational cost. Unlike LES, the fluctuation of velocity field is directly generated rather than by iteratively solving the Navier-Stokes equations. It took LES 133 CPU hours to simulate the CAARC building in a 1:250 scale for a 4-second transient duration on a supercomputer at the Shanghai Supercomputer Center [Huang et al. (2007)]. In contrast,

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

the RANS-KS method took only about 50 hours to obtain the encouraging pressure fluctuation results in full scale based on an 8-core workstation.

Figure 2: Geometry and overall mesh configuration of the computational domain

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Distance (m)

RM

S pr

essu

re c

oeff

icie

nt

HKUST @112mHybrid simulation @120mLES (Huang et al., 2007)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Distance (m)

RM

S pr

essu

re c

oeff

icie

nt

HKUST @67m

Hybrid simulation @67m

Figure 3: Result comparison among RANS-KS, wind tunnel and LES

Conclusion

The hybrid RANS-KS technique presented in this paper can be summarized as the follows:

1) Instead of running simulation in model scale, a full-scale simulation can be performed by the RANS-KS technique.

2) The hybrid RAN-KS technique is an explicit time-marching method without solving the Navier-Stokes equations, and it gives a significant saving of computational cost.

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

3) Anisotropic effect is very important for estimating wind load on tall buildings and can be partially captured by splitting kinetic energy into three orthogonal velocity components.

4) The periodic vortex shedding effects due to cross-wind excitation on a tall building can be taken into accounted by adjusting the spectral peak position at the Strouhal number for the lateral component gustiness.

Acknowledgements The work presented in this paper was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region (project no. 611006) and the National Natural Science Foundation of China (project no. 90815023).

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