Recent extreme hurricanes caused a huge loss in damages to critical civil infrastructure. To estimate hurricane wind loading on structures, spectral methods are widely used to generate neutral atmosphere boundary layer winds, which however are limited to describe extreme wind elds that are non-stationary and more turbulent. To overcome this limitation, a high- delity high-resolution computational model is developed to simulate hurricane wind eld with detailed physics. A large eddy simulation (LES) solver is developed using a sub-grid-scale model based on open source program OpenFOAM. The simulated wind eld is validated through comparison with observations. The generated wind eld is applied to analyze structural response of a power transmission system. The proposed hurricane boundary layer (HBL) model and a neutral atmosphere boundary layer (ABL) model are compared in tropical storm and category-3 hurricane scenarios. Compared with the HBL model, the ABL model doesn't consider the mesoscale terms and overestimates the crosswind velocity and the turbulent kinetic energy (TKE) near the ground. As a result, the ABL model overestimates the dynamic responses of the wires and towers. The developed HBL model captures the main characteristics of hurricane wind and is applicable for modeling civil infrastructure exposed to hurricanes at a large scale. Keywords: Hurricane boundary layer; Large-eddy simulation; Spectral density; High voltage tower line system 1. Introduction Hurricane induced multiple hazards(winds, storm surge/waves and rains) cause huge economic loss and damages to critical civil infrastructure, which severely weakens the community security and resilience. To protect and enhance the resilience of infrastructure exposed to hurricanes, an accurate representation of hurricane boundary layer (HBL) wind elds is essential to understand the wind loading e ect on structures and the complex structural behavior. Hurricane wind eld composes of a horizontal primary circulation and a vertical-radial secondary con- vection [1]. To characterize the hurricane structures, various models have been developed to simulate the radial and vertical wind structures. The models for radial hurricane wind structure include parametric, two dimensional (2D) slab, and physical models. Parametric models (Holland-type, Georgiou parametric models and improved asymmetric models) predict gradient wind speed at a distance rfrom hurricane center [2, 3, 4, 5, 6]. 2D slab models were developed using vertically averaged equations of horizontal motion driven by a prescribed pressure eld [7, 8]. The 2D slab model approach represents hurricane features better than parametric models because it considers air-surface temperature di erence and sea-land friction e ect on wind eld asymmetries. The above models are based on prescribed pressure gradient and key hurricane pa- rameters, including translation speed, moving direction, central pressure di erence and radius of maximum winds (RMW). In addition to these parametric models (empirical) and 2D slab models (semi-empirical), Corresponding author Email address: csun@lsu.edu (Chao Sun) Preprint submitted to Journal Name July 10, 2020 physical models predict tropical cyclone radial structure based on conservation of energy, entropy, and an- gular moment [9, 10, 11]. During hurricane landfall, the roughness sea-land transition will change the storm size, intensity and wind eld structure. Filling rate models have been developed to describe the hurricane decay since its landfall [4, 12, 13]. Guo introduced a water-land-based wind speed conversion process to model the staggered pattern of the landfalling hurricane wind eld [14]. With the gradient hurricane wind eld estimated, the hurricane wind vertical pro le can be determined using a log- or power-law [4, 14, 15]. However, unlike the neutrally stable ABL with log-law or power-law vertical wind pro les, the HBL has super-gradient wind regions, which are generated by the strong inward advection of angular momentum [16]. The vertical wind pro les of HBL over ocean and land were investigated and modeled based on GPS dropsondes observations [17, 18, 19]. However, measurement over land is scarce and insucient to capture the comprehensive HBL characteristics for di erent topographies. With the horizontal wind structure and vertical wind pro les modeled, the averaged wind eld can be obtained. Although averaged hurricane wind eld can be well modeled, the high intensity turbulences of extreme hurricane wind elds are not clearly understood. In HBL, high intensity and spatially coherent turbulences are dominant [20, 21]. While most existing literature has been focused on non-hurricane wind turbulence, hurricane turbulence modeling has received little research e ort. To characterize the high intensity turbu- lence, Yu et al. [22] estimated the hurricane turbulence spectra, cospectra and integral turbulence scales through analysis of measured wind data. The authors found that the turbulent energy of hurricane winds at lower frequencies is higher than that of non-hurricane winds. On the contrary, Li et al. [23] found that HBL has a higher level of energy at higher frequencies, which is contradictory to the conclusions of [22]. With reference to [22, 23], the spectral methods are limited to describe hurricane wind, which has di erent turbu- lence structures from non-hurricane wind, and hence are inadequate for engineering application. Currently, there are few consistent spectra models for hurricane turbulence. In comparison, a three dimensional model of a tropical cyclone can better represent the HBL characteristics, including the super-gradient wind pro le, the surface roughness variations, and roll vortexes generation. The simulation of a tropical cyclone, which spans several hundreds of kilometers in the horizontal direction, is much time consuming. To reduce the computational cost, several models for local HBL within a smaller computational domain were developed [20, 24, 25]. In Refs. [20, 24, 25], large-scale inertial and pressure-gradient acceleration terms are included in the governing equations via mesoscale tendency terms, which allows domain sizes to be with the order of magnitude of kilometers in horizontal extent, making the simulation more ecient. The large-eddy simu- lation (LES) was applied to study the ABL which can predict the statistical properties of turbulent ows. Church eld et al. [26] applied OpenFOAM to perform LES of ABL wind, where the ABL is simulated with the original Smagorinsky model. Although a 3D hurricane model can capture more detailed characteristics, there is little published work on applying a 3D hurricane model to engineering structures. The widely used methods for structural design, resilience analysis and risk assessment under hurricane wind impact are empirical models, 2D slab models and physical models for average wind eld [14, 27]. Only the averaged hurricane wind velocity is considered for structural assessment [28]. However, spatially and temporally variant hurricane wind elds will induce substantial uctuating loads, which will cause complex structural responses and potential severe fatigue issues. The HBL high intensity and spatial coherent turbulences will cause large structural response, especially in long-span and slender structures, such as long- span bridges and power transmission systems [29]. To ensure structural safety, assessment of infrastructure performance in extreme wind conditions should consider the strong turbulences. Currently, in engineering community, the hurricane wind velocity is generated using statistical and spectral methods. Zhang et al. [30] estimated the fatigue life of a bridge under non-stationary hurricane wind, with the averaged wind speed generated using a 2D slab model and the turbulence generated using Kaimal spectrum. It was concluded that the non-stationarty hurricane wind can induce large stress ranges. An et al. [29] analyzed the failure mechanism of transmission line system under typhoon, which was modeled with an improved slab model [31] and a spectrum model. Through comparing with averaged wind static analysis, the authors found that static analysis is inadequate as the uctuating typhoon winds ampli ed the tower axial force and caused the tower risk factor sharply increased. Amirinia [32] compared the e ects of di erent hurricane turbulence spectrum models on the responses of wind turbine. It was concluded that the hurricane spectrum model with high 2 level of energy in higher frequencies caused larger base moments and displacements. Fu et al. [33] found that the uctuating wind speeds generated using di erent spectra has a great in uence on the fragility curve. Hence, application of spectrum methods in real engineering might cause confusions for dicision-makers. Also, large span and sag of power transmission line necessitates consideration of the wind spatial coherence which can hardly be accurately modeled using the spectrum method. In comparison, the LES results of turbulent velocity elds can caputure the three dimensional coherent structures of hurricane wind elds. Doubrawa [34] compared the response of wind turbines under turbulent ABL ow generated using stochastic turbulence models and the LES method respectively. It was found that turbulent wind elds generated by stochastic models failed to accurately represent the spatial coherence and wind loads. To further analyze the e ect of HBL turbulence on the performance and reliability of infrastructure, the present study develops an LES-based solver to generate the HBL for a xed target eld. The developed HBL solver is validated using observational data. Then the simulated wind elds are applied to analyze a power transmission tower-line system, which span several hundreds of meters, to investigate HBL wind induced structural response. The remaining of the present paper is structured as follows. An LES model for HBL wind eld is developed and simulated using OpenFOAM in Section 2. Simulated hurricane wind elds are veri ed through comparing average wind structure and turbulence statistics with observations in Section 3. Section 4 presents the results of a power transmission tower-line system in the generated HBL wind eld. Conclusions are summarized in Section 5. 2. Model Description An LES model of HBL is derived in this section. The moist processes are neglected. As shown in Fig. 1, this study focuses on simulation of the turbulent ow in a relatively small domain(the simulation domain size is within [100101] kilometers), compared with the whole tropical cyclone. The intensity of the hurricane is regarded as a known parameter. The model is applicable for hurricanes over ocean and land. The di erences of surface roughness and air temperature between ocean and land will cause circular wind elds over land to be less symmetric. In the present study, for hurricane landfall, wind structure and speed at the top of the boundary layer is assumed to remain unchanged [19]. uru uzuvw Simulation domain Figure 1: Conceptual schematic of the modelling framework 2.1. Hurricane Boundary Layer Model Governing equations of an incompressible Newtonian ow (density and viscosity are constant) at a speci c location with a distance R from the hurricane eye are expressed in the following. For the speci c location, the interested domain is smaller than the scale of an entire tropical cyclone.To produce the same 3 mean wind pro les with that of an entire tropical cyclone, the mesoscale centrifugal force term, advection term and large scale pressure gradient accelerations are introduced to the governing equations (Appendix A presents detailed derivations). The continuity equation is: @u @x+@v @y+@w @z= 0 (1) The moment equations in the Cartesian coordinates are: @u @t+u@u @x+v@u @y+w@u @z=vhui R+v@hui @R@p @x+r2u+v 3w 2 @v @t+u@v @x+v@v @y+w@v @z=vhvi Rhuiu R+U2 g R+ 3Ug@p @y+r2vu 3+w 1 (2) @w @t+u@w @x+v@w @y+w@w @z=@p @z+r2w(gz)r 0 The equation of the virtual potential temperature is: @# @t+u@# @x+v@# @y+w@# @z =r2# (3) The density is represented as:  01##0 #0(4) wherehirepresents a horizontal average of variables at a certain height; u,v,ware the velocities in x,y andzdirections; Ugis the gradient wind speed and 1, 2, 3are the rotation velocities in x,yandz directions;is the thermal di usivity; 0is the reference density; #is the virtual potential temperature. The mesoscale terms derived from an entire tropical cyclone are included in the governing equations for this HBL model in a much smaller eld. vhui=Randhuiu=R are the centrifugal acceleration terms. v@hui=@R andvhvi=Rare the radial advection terms. U2 g=R+ 3Ugis the pressure-gradient acceleration. The mesoscale terms in Eqn. (2) vary with height, which is expected to simulate the wind eld near the surface more accurately than the model presented in Ref. [25]. It is noted that the models developed by Nakanishi and Green [24] didn't consider the e ect of mesoscale advection, which however should be taken into consideration when the radial gradient of tangential velocity cannot be neglected. In the present model, the simulation domain can be seen as a grid of the hurricane basic-state ow which is assumed to be in a gradient wind balance. 2.2. LES Model The LES equations are obtained by ltering the variables in Eqns. (1 3). The equations for the ltered variables are: @u @x+@v @y+@w @z= 0 (5) @u @t+@uui @xi= vhui R+v@hui @R@p0 @x+r2u+ v 3w 2 @v @t+@vui @xi=vhvi Rhuiu R+U2 g R+ 3Ug@p @y+r2vu 3+ w 1 (6) @w @t+@wui @xi=@p0 @z+r2w(gz)r 0 4 @# @t+@#ui @xi=r2# (7) where() represents ltered variable. The unknown terms uiuj,#uiare represented as uiuj= uiuj+ r ij #ui=#ui+ r j (8) where r ijis the sub-grid-scale (SGS) stress tensor. r jis the SGS potential temperature ux vector. r ij=2tSij+1 3ij r kk r j=t Pr@# @xj(9) where Sijis the rate-of-strain tensor, ijis the Kronecker's delta, Pr is the Prandtl number, tis the turbulent eddy viscosity, which is modeled using the Smagorinsky-Lilly SGS model, t= (Cs)2(2SijSij)1=2(10) where  is the grid scale and Csis the model constant, which is set as 0.168 [35]. 2.3. Boundary Conditions A periodic boundary condition is applied on the horizontal boundaries. On the upper boundary, the vertical gradient of potential temperature is xed; a slip boundary condition is applied to the velocity; the pressure gradient is obtained from the moment equation to ensure that the ux is zero. At the bottom surface, the temperature ux is xed; a wall shear stress model is applied to model the ground roughness. The total stress at the bottom surface includes the viscous and SGS stress, which is expressed as: =2 40 0 tot 13 0 0 tot 23 tot 13 tot 23 03 5 (11) The model proposed in [36] is applied to compute the total wall surface shear stress vector. total 13 =h i ucentl 1 ucentl 1  jhucentlij total 23 =h i ucentl 2 ucentl 2  jhucentlij(12) where super script \centl" indicates values at the center of cells adjacent to the lower boundary. h iis the horizontal mean surface shear stress, which is de ned as: h i= tot 13 2+ tot 23 21=2(13) Near the lower boundary, the near surface velocity can be approximated using rough wall log law:  ucentl 1 2+ ucentl 2 21=2 h i1=2=1  lnz z0  m(L) (14) 5 whereis Von Karman constant, with a value of 0.41. m(L) represents the atmospheric stability-related function[37, 38]. m(L) =8 >>< >>:0 neutral  z=L stable 2 ln1+x0 2 + ln 1+x2 0 2 2 tan1(x0) + 2unstable(15) wherex0= (1 z=L )1=4; and are empirical constant values; L is the Obukhov length. z0is roughness height, which is dependent on locations, e.g., the ocean surface or above the ground. 2.4. Numerical Method The simulations are performed using OpenFOAM (Open-source Field Operations And Manipulations) which is an open-source library. A new solver for HBL has been developed based on a validated LES solver for computing atmospheric boundary layer ows [26]. The mesoscale terms are included in moment equations and are treated explicitly. The PIMPLE (combination of PISO and SIMPLE) algorithm is used to solve the momentum and pressure equations. 3. Model Veri cation To verify the developed LES model, simulated results of hurricane wind eld over open ocean and land regions are compared with observations. The simulation eld is selected at a distance R= 40 km from the hurricane center. The simulation domain has a size of 3km 3km3km. Horizontal resolution is set as 15 m. Vertical resolution is 6 m below the height of 1km and 15 m above the 1 km height. Boundary layer meshes are added at the bottom surface to more accurately simulate wind eld near the surface. The geostropic velocity at the top boundary of the simulation domain is 37 m/s. Based on [39], the potential temperature pro le in the HBL usually contains a stable layer (d dz>0), a well-mixed layer (d dz0), and a near surface unstable layer (d dz<0). Above the HBL, there is a capping inversion layer and an undisturbed stable layer[24]. The lapse rate in the stable layer is 5 K/km. Parameters for hurricanes over ocean and at landfall are slightly di erent. The roughness height and heat ux, which have large in uence on wind eld, are signi cantly di erent for over ocean and overland hurricanes. The uxes of enthalpy and momentum are key factors for the development and maintenance of tropical cyclones. The enthalpy ux is the sum of sensible heat ux, which will change air temperature, and latent heat ux corresponding to heat change during phase change. In this study, the moisturizing process is neglected while the sensible heat ux is considered. The surface latent ux from the ocean contributes the primary energy for the development of tropical cyclone and directly in uence the hurricane intensity. In the present study, the intensity of a hurricane is assumed as a known factor. The uxes from the ocean surface and land are di erent. Land temperature is lower than water temperature, which leads to a lower sensible heat ux. Roughness over land is higher than ocean surface, which enhances the surface friction and moment ux. For the ocean surface, the roughness height is set as 0.0028 m [25]. The ocean surface sensible heat ux is determined using a standard bulk formula, FH=cpCHU10(#0#10) (16) wherecpis the heat capacity of air; CHis the surface exchange coecient,which is set as 1 :2103[40]; U10is the averaged wind speed at 10 m height; #10and#0are the potential temperature at 10 m height and at surface. The potential temperature ux at the surface is approximated as f#=CHU10(#0#10). The di erence of potential temperature at 10 m and surface is set as 1 K. The potential temperature ux is approximated as 0.03 K-m/s. For the uxes over land, the roughness height is selected as 0.1 m in the present study. The surface potential temperature ux is not considered overland. With the boundary conditions and associated parameters set, the simulation of hurricane wind eld is conducted by adding small random perturbations to the basic-state wind with an automatic time step of 6 around 0.2 s. The simulation is performed until the statistical characteristics are stationary[41]. The large scale hurricane boundary layer led is simulated parallelly with the entire domain divided into 512 subdo- mains. The averaged value of mean velocity and velocity variance at various vertical levels are calculated. The wind velocity contour plot for a case over ocean is shown in Fig. 2, where there are high intensity turbulences at the bottom of the simulation domain, signaling that the hurricane winds will probably cause signi cant dynamic e ect on structrues. Figure 2: Contour of simulated hurricane wind eld. (The domain size is 3km 3km3km) 3.1. Vertical Pro le Simulated results of vertical wind pro les for HBL are veri ed with GPS dropsone observations obtained from the National Oceanographic and Atmospheric Administration (NOAA). The vertical wind pro les of horizontal mean wind velocities from simulation and observations are compared in Fig. 3, where the observed data is from [25]. Fig. 3 indicates that the vertical wind pro les of mean velocities from the HBL solver agree well with the dropsonde observations. In addition, simulation of hurricane wind eld at landfall is conducted and compared with the measured composite pro les in Fig. 4. The composite pro les were generated based on measurements of landfalling tropical cyclones [18, 42]. Fig. 4 shows that the simulated wind pro les over land are consistent with observed data. In addition to the mean wind pro le, simulated turbulence is compared with measurements. The total turbulence stress , which is generated by shear stresses and buoyant forces, can be calculated as: j j =h hu0w0i2+hv0w0i2i1=2 (17) where ()0represents turbulent velocity, which equals the di erence between the instantaneous and the hor- izontal averaged wind velocity ( u0=u< u > ). The vertical pro le of turbulence stress is shown Fig. 5. The observed data of turbulence stress over ocean was determined by [43]. Fig. 5 shows good agreement 7 (a) (b) Figure 3: Vertical wind pro les of horizontal mean velocities in x direction (a) and y direction (b) over oceans. The red lines represent the simulated results using the developed HBL solver in the present study. The black circles denote the dropsonde observations. (a) (b) Figure 4: Vertical wind pro les of horizontal mean velocities in x direction (a) and y direction (b) at the landfall. The red lines represent the simulated results in the present study. The black circles denote the dropsonde observations between simulated results from the HBL model and observations in the over ocean case. Fig. 5 also indi- cates that due to the increased roughness of land, the overland turbulence stress near ground of the HBL is signi cantly larger than that over oceans. 3.2. Velocity Spectrum In addition to wind pro les, the time series of wind velocities at a selected point (1500, 1500, 200) m in x, y, z directions are shown in Fig. 6. The autocovariance and power spectrum of given wind velocity time series are: Rii(X; ) =D u0 i(X;t1)u0 i(X;t1+ )E (18) 8 Figure 5: Vertical wind pro le of total turbulence stress . The red line denotes one-hour averaged total stress over ocean simulated using the HBL model. The dash-dotted line denotes one-hour averaged total stress overland using the HBL model. The black dots denote observed values of turbulent stress. Shaded areas in each case represent the corresponding range within one hour. Sii(!) =1 2Z1 1Rii( ) exp(i! )d (19) Figure 6: Time series of wind velocity in x,y,zdirections at a selected point (1500, 1500, 200) m The power spectra density of auto-covariance Ruuof wind velocity u in x direction is calculated and compared with the results presented in [25] in Fig. 7(a). In the present study, the mesh size in horizontal direction is set as 15 m. Fig. 7(a) indicates that the power spectrum density obtained in the present study agrees well with the results for dx=15.6 m in [25]. The spectrum of the wind velocity in zdirection is shown in Fig. 7(b), where remarkably good agreement can be observed between the simulated result and result from [25]. It is reported in [25] that the results are better using ner meshes. The power spectral densities decrease rapidly when the frequency is larger than a critical frequency ( fc< u > = (6x)). In high frequency range, the power spectra density decays quickly where small-scale turbulence is not resolved due to mesh resolution limitations. From Fig. 7(a) and Fig. 7(b), the power spectra density agrees well with the observation data within the inertial subrange. Sections 2 and 3 present the theoretical model and computational results of the hurricane boundary layer model. Comparison between simulated results and observational data indicates that the developed HBL model/solver is valid to simulate hurricane wind eld at given locations. The following section will apply 9 (a) (b) Figure 7: Power Spectra densities of wind velocities in xdirection (a) and zdirection (b) this HBL model to analyze the complex dynamic responses of a power transmission system. 4. Application The developed high- delity model for HBL can capture the hurricane wind turbulence details. Simulated hurricane wind elds can be applied to study complex dynamic response of critical civil infrastructure exposed to extreme hurricane winds with high wind velocities and strong turbulences. Until now, in computational wind engineering, the CFD methods for generating turbulent wind elds mainly include turbulence in ow models using inlet boundary conditions and precursor simulation based on ABL models. The turbulence in ow models which are set with prescribed mean velocity pro les, turbulence spectra, and spatial coherence are limited for modeling hurricane winds because there are few consistent spectra models and coherence models for hurricane turbulence. In the neutral ABL model, the ow only has translational velocity with large-scale geotrophic pressure gradient and can not obtain the super-gradient wind pro le. To show the improvement brought by the developed HBL model, cases based on neutral ABL are simulated and compared with the HBL. For engineering application, the mean wind velocity at a height of 10 m is taken as a reference wind velocity. To compare the model developed for tropical cyclone wind and that for neutral atmospheric conditions, large eddy simulations based on HBL and neutral ABL models over land are performed with the same averaged wind velocity at 10 m height. Simulated turbulent winds are applied to a power transmission system to evaluate wind loading e ects using the two models. 4.1. Comparison of HBL and ABL To compare HBL and ABL models, two wind speed scenarios are simulated: mean wind velocity U10= 21m=s andU10= 42m=s. The 1-minute maximum sustained wind speeds for the two scenarios are calculated to be 29 m/s (tropical storm) and 55 m/s (category 3 hurricane). To produce the target wind velocity U10= 21m=s andU10= 42m=s in the two scenarios, the gradient wind velocities in the HBL model are 37 m/s and 74 m/s. However, in the ABL model, to obtain the same wind speeds at the reference height of 10 m, the gradient wind velocities are 66 m/s and 124 m/s. This is because solver for neutral ABL doesn't consider the mesoscale terms and needs a much higher gradient wind speed to balance the frictional drag forces. It is noted that the required much higher gradient wind velocity in ABL models is unrealistic and computationally inecient. The vertical pro les of the horizontal averaged wind velocities are shown in Fig. 8(a) and 8(b), where one can nd that the ABL model slightly underestimates the averaged wind velocity in x direction and remarkably overestimates the averaged velocity in y direction. The di erence of mean 10 velocity pro les between the HBL model and the ABL model is larger in the case of U10= 42m=s than that inU10= 21m=s. Vertical pro les of turbulence kinematic energy (TKE) for HBL and ABL are shown in Fig. 8(c). Compared with the results from the HBL model, the solver of ABL predicts higher TKE near the ground surface as a result of a large gradient velocity at the top of the ABL. Noticeable di erence of TKE can be observed between the ABL and HBL in the case of U10= 42m=s. The higher velocity at the upper region of ABL generates larger shear production of turbulence, which is transported to the near ground region. U10=21 m/s HBL U10=21 m/s ABL U10=42 m/s HBL U10=42 m/s ABL (a) (b) (c) Figure 8: Vertical wind pro les of horizontal mean velocities in x direction (a) and y direction (b) and TKE (c). The red lines denote the 10 minutes averaged simulated results using the HBL model and the black lines denote the simulated results using the ABL model. The solid and dashed lines denote the U10= 21m=s andU10= 42m=s wind speed scenarios. Shaded areas represent the corresponding range in 10 minutes. In addition to the ow pro les, coherent structures of the turbulence are important features when rep- resenting ow elds. The coherence is de ned as a magnitude-squared cross-spectrum Cijbetween input signals at two locations normalized by the power spectrum of each signals: 2 ij(f) =jCij(f)j2 Sii(f)Sjj(f)(20) The cross-covariance and cross-spectrum of given wind velocity time series at two locations are de ned as: Rij(; ) =D u0 i(X;t1)u0 j(X+;t1+ )E (21) Cij(f) =Z1 1Rij( ) exp(i2f )d (22) whereis the distance between the two locations. Spatial coherence of the turbulent ow, which is a function of distance, is the coherence of velocity at speci ed locations. Coherence of wind velocity component is treated separately. In the present study, the coherence of wind velocity component uis calculated. The horizontal coherence of HBL and ABL at a height of 40 m are calculated and compared with IEC exponential coherence model [44], which is only used for atmospheric in ow under neutral strati cation. Fig. 9 shows the spatial coherence for horizontal separation distances of 45m and 75m. Compared with the exponential coherence 11 model, HBL and ABL models yield larger coherent values for the separation of 45 m. The di erence of coherence predicted by exponential model and LES is caused by the low grid resolution of the numerical method, which can only capture eddies whose sizes are larger than 2 x. Compared with turbulence generated from ABL, the HBL turbulence is more coherent. The high coherence in HBL is caused by the boundary roll structure of the HBL [24, 45]. When estimating the coherence in HBL, the exponential function model underestimates the coherent values, which is consistent with the conclusion in Ref. [46]. Figure 9: Comparison of coherence of wind velocity u 4.2. Analysis of Power Transmission Systems Exposed to Hurricane Wind As a key component of critical civil infrastructure, power transmission towers and wires are vulnerable to strong winds which pose severe threats to the power system security under extreme windstorms. In the present study, the windstorm induced dynamic response of a power transmission system is studied. Wind induced vibration of a tower-line system is simulated through coupling the CSD (Computational structural dynamics) and CFD (Computational Fluid Dynamics) models. The nite element method (FEM) solver, CalculiX[47], is used to simulate the nonlinear tower-line dynamic response. The developed OpenFOAM based HBL solver is used to simulate unsteady hurricane wind eld. In a power transmission system, large displacements of wires are expected due to high wind velocities and strong turbulences. Therefore, geometric nonlinearity of the wire is considered in the FEM models. A three dimensional nite element model of the HVTLS (High voltage transmission tower-line coupling system) with 3 towers and 4 spans is developed, as 12 shown in Fig. 10. In the FEM model, the translation and rotation of the tower bottom nodes and end nodes of conductors and ground lines are xed. The cross section of the tower members consists of Q345-type angle steel with a yielding stress of 310 Mpa. LGJ630/45 type conductors are adopted and their parameters are listed in Table 1. The length of the insulator string is 5 m with a mass of 370.29 kg. XZ Y Figure 10: A three-tower four-span HVTLS Table 1: Parameters of the conductor and ground line Conductor Ground line Span length (m) 380 380 Mass(kg/m) 2.06 0.639 Sag(m) 10.32 7.92 Section area(mm2) 666.55 152.81 Elastic modulus (Mpa) 63000 91100 Poisson ratio 0.3 0.3 4.2.1. Eigenanalysis of the Tower-line System CalculiX is used to build the FE model of the transmission tower-line system. The conductor, tower members and insulator are modeled with B31 element in CalculiX. The 4-bundled conductor is simpli ed as one conductor. With the gravity force acting on the system, the tower-line model is in an initial stress condition. The axial tension force of the conductor is around 35.32 kN. Through performing eigenvalue analysis with respect to the FE model, the natural frequencies and mode shapes are obtained. Natural frequencies of the rst six modes (three in-plane and three out-of-plane) of the conductor and the rst three modes of the tower are presented in Table 2 and the mode shapes are shown in Fig. 11 and Fig. 12. 4.2.2. Structural response of transmission line system under ABL and HBL winds To compare the HBL solver developed for tropical cyclone wind and that for neutral atmospheric condi- tions, section 4.1 compares the simulated wind elds based on HBL and neutral ABL models. This section will further evaluate the dynamic response of the tower-line model subjected to the simulated wind eld. Time histories of the along-wind displacement at the wire mid-span under the two simulated wind scenarios 13 Table 2: Natural frequencies of conductor and tower mode Conductor (Hz) mode Tower (Hz) out-of-plane 1st0.174 1 1.65 in-plane 1st0.246 2 1.94 in-plane 2nd0.347 3 3.20 out-of-plane 2nd0.349 in-plane 3rd0.490 out-of-plane 3rd0.525 Out-of-plane 1stmode Out-of-plane 2ndmode XZ YOut-of-plane 3rdmode In-plane 1stmode In-plane 2ndorder In-plane 3rdmode Figure 11: The rst six mode shapes of the wire XZ YyZ xyZ x 1stmode 2ndmode 3rdmode Figure 12: The rst three mode shapes of the tower 14 are shown in Fig. 13(a) and Fig. 13(b), where one can nd that the wire experiences a larger displace- ment along wind direction under wind simulated using the ABL solver. It is noted that the larger response amplitude of wire under ABL wind is caused by overestimation of wind turbulence near ground using the ABL model, as shown in Fig. 8(c). Fig. 14 shows the power spectrum density of the along-wind acceler- ation at the wire mid-span under the two wind scenarios. It can be found that the rst-order out-of-plane mode dominates the wire dynamic response in the case of U10= 21m=s while the second-order out-of-plane mode dominates in the case of U10= 42m=s. Hence, higher wind speed and stronger turbulence will induce higher-order modes of the wire. Comparison between Fig. 14(a) and Fig. 14(b) shows that the rst three out-of-plane mode frequencies increases from 0.17 Hz, 0.35 Hz, and 0.53 Hz to 0.29 Hz, 0.55 Hz and 0.88 Hz, respectively, indicating that wire natural frequency increases with the increasing wind speed which causes larger wire tension force, displacement, and more signi cant hardening geometric nonlinearity. In addtion, Fig. 14(b) includes a natural frequency of f= 1:46Hzwhich corresponds to the tower out-of-plane vibration. This indicates that high level category hurricane wind can induce coupled vibration of the tower and the conductor. Time histories of cross-wind displacement at the wire mid-span and the response spectrum of the corresponding acceleration are shown in Fig. 15 and Fig. 16, where similar conclusions as obtained from Fig. 13 and Fig. 14 can be drawn. Under ABL wind, the wire experiences a larger displacement in the cross-wind direction. The di erence is more pronounced in the higher wind-speed scenario. Fig. 16 shows that the cross-wind structural response is mainly contributed by the rst three in-plane modes in the two scenarios. (a)U10=21 m/s (b)U10=42 m/s Figure 13: Displacements in along-wing direction at wire mid-span under generated wind elds using HBL model and ABL model: (a) U10=21 m/s, (b) U10=42 m/s Fig. 17 illustrates the dynamic axial force of the wire. It is found that the mean axial force in the case of hurricane category 3 is 90 kN, which is roughly 2 times larger than that under tropical storm. The maximum wire axial force is around 250 kN, which is 4 times larger than that under tropical storm, and exceeds the conductor ultimate strength of 141.3 kN. The dynamic tension force along wire will cause the wire to break under high intensity hurricane turbulent wind. The amplitude of axial force in ABL is larger than that in HBL in the two scenarios. Fig. 18 shows the response spectra of axial forces, revealing the substantial 15 (a)U10=21 m/s (b)U10=42 m/s Figure 14: Response spectrum of acceleration in along-wind direction at wire mid-span: (a) U10=21 m/s, (b) U10=42 m/s (a)U10=21 m/s (b)U10=42 m/s Figure 15: Displacements in cross-wind direction at wire mid-span under generated wind elds using HBL model and ABL model: (a) U10=21 m/s, (b) U10=42 m/s di erence of PSD peak values between HBL and ABL. It is noted that large axial force in high wind speeds sti ens the wire and increases the natural frequencies of wire. Comparison between Fig. 18 and 14 indicates that the axial force is mainly caused by the second-order mode of the wire, whose value is f=0.37 Hz in tropical storm and f=0.53 Hz in category 3 hurricane wind. Fig. 19 illustrates the time history of tower tip displacement in the along-wind direction. The di erence of along-wind structural response under HBL and ABL winds is mainly caused by the di erent dynamic axial forces from the wires. In hurricane category 3, the ABL model overestimates the wire dynamic axial force and the displacement of tower in the along-wind direction. Additionally, the ABL model also overestimates the tower tip displacement in the cross-wind direction, as shown in Fig. 20, which is more pronounced in the 16 (a)U10=21 m/s (b)U10=42 m/s Figure 16: Response spectrum of acceleration in cross-wind direction at wire mid-span: (a) U10=21 m/s, (b) U10=42 m/s (a)U10=21 m/s (b)U10=42 m/s Figure 17: Time history of wire axial forces under generated wind elds using HBL model and ABL model: (a) U10=21 m/s, (b)U10=42 m/s higher wind-speed scenario. In the tropical storm scenario, the estimated mean displacement based on the ABL model is 30% larger than that based on the HBL model. And the former is around two times larger than the later in the case of hurricane category 3. The overestimation of mean displacement in cross-wind direction is caused by the overestimation of wind velocity in y direction, as shown in Fig. 8(b). However, the cross-wind structural response has a larger deviation in the HBL model than that in the ABL model. The tower structural response ranges from -25 cm to 42 cm in category 3 hurricane winds. The cross-wind (out-of-plane) structural response of the tower is mainly caused by the unbalanced axial forces from the wires of neighbouring spans, as shown in Fig. 21. The amplitude of the unbalanced axial force in the HBL model is 2.9 kN larger that in the ABL model for tropical storm wind scenario, and is 6.6 kN larger for hurricane 17 (a)U10=21 m/s (b)U10=42 m/s Figure 18: Response spectrum of wire axial forces under generated wind elds using HBL model and ABL model: (a) U10=21 m/s, (b) U10=42 m/s category 3 wind scenario. The unbalanced axial force is essentially caused by the spatial variation of the wind eld. The higher spatial coherence of the turbulence in the HBL model, with roll vortices generated in the boundary layer, could be the reason for the stronger spatial variation in HBL. (a) (b) Figure 19: Along-wind displacement time history at tower tip under generated wind elds using HBL and ABL models: (a) U10=21 m/s, (b) U10=42 m/s The presented results in this section indicate that the ABL model overestimates the response of the wire and tower especially for high wind-speed scenarios. The tower cross-wind displacement is mainly caused by unbalanced wire axial forces of neighbouring spans. High category hurricane winds will excite higher modes of wires, and cause large dynamic axial forces. In addition, coupled vibration of tower and wire will be excited under extreme hurricane wind. The large axial dynamic forces and coupled vibration of tower-line 18 (a) (b) Figure 20: Cross-wind displacement time history at tower tip under generated wind elds using HBL and ABL models: (a) U10=21 m/s, (b) U10=42 m/s (a) (b) Figure 21: Unbalanced axial force from the neighbouring spans: (a) U10=21 m/s, (b) U10=42 m/s 19 system will cause the potential failure of wires and towers. 5. Conclusion Hurricane winds have signi cantly di erent physical structures and characteristics from the neutral ABL winds and hence are more challenging to be accurately modeled. The present study develops an LES-based solver for hurricane boundary layer at a constrained domain using the open source program OpenFOAM considering the centrifugal force, radial advection and pressure gradient. The ground boundary condition with shear stress and heat ux is introduced to model the surface condition. The high fedility high resolution LES-based solver for hurricane boundary layer models the hurricane wind eld well and is veri ed with measured data. The generated wind eld is applied to analyze the dynamic response of a power transmission tower line system. Based on the presented results, the following ve key conclusions can be drawn: 1. Compared with neutral ABL, the introduced mesoscale terms of HBL have a great e ect on generated wind elds. Unlike the vertical wind pro le (described using a log- or power-law) in ABL, a super- gradient layer is shown in HBL, which is caused by the introduction of mesoscale terms in the HBL model. Through comparison, with the same reference velocity at a 10 m height, the ABL model overestimates the wind velocity in y direction and the TKE near the ground especially for high wind- speed scenarios. Without considering the mesoscale terms, the ABL model needs a much higher gradient wind speed to obtain identical near-surface wind speed. 2. The turbulence is more coherent in HBL winds than that in ABL winds. The exponential coherence model underestimates the coherence for HBL, which might be inapproprite to generate hurricane turbulence elds. 3. The overestimation of wind velocity in y direction and TKE using the ABL model will cause overesti- mation of the structural response (displacement and inner forces) of the wire and tower. 4. The spatial variance of the HBL ow eld causes large unbalanced wire axial forces acting on the tower. This will cause signi cant out-of-plane dynamic response of the tower. 5. Extreme wind conditions (high wind speeds and strong turbulences) will excite higher-order modes of wires in addition to the fundamental mode and will cause coupled vibration of the wire and tower. As a summary, the proposed HBL model is validated and applied to analyze the dynamic response of a power transmission system. In the future, experimental and eld data from instrumented power transmission systems will be collected to further verify the results. It is worthy to mention that this proposed HBL model is applicable for risk analysis and vulnerability assessment of a wide range of civil infrastructure exposed to hurricanes at a large scale. Acknowledgement This work was supported by the Louisiana State University Economic Development Assistantship and the research was conducted using high performance computing resources provided by Louisiana State University. The authors are grateful for all the support. 20 Appendix A. Derivation of Hurricane Boundary Layer Model Adopting Boussinesq approximation method to represent density, which ignores density di erences in the inertia term but conserves the variation in the gravity term, the governing equation of an incompressible Newtonian ow in cylindrical coordinate ( r;;z ) are expressed in the following. The incompressible continuity equation is: 1 r@(rur) @r+1 r@(u) @+@uz @z= 0 (A.1) The moment equation is: @ur @t+ur@ur @r+u r@ur @+uz@ur @zu2  r=@prgh @r+ r2urur r22 r2@u @ +fr @u @t+ur@u @r+u r@u @+uz@u @z+uru r=1 r@prgh @+ r2uu r2+2 r2@ur @ +f (A.2) @uz @t+ur@uz @r+u r@uz @+uz@uz @z=@prgh @z+r2uz+fz(gz)r 0 The equation of the virtual potential temperature is: @# @t+ur@# @r+u r@# @+uz@# @z =#r2# (A.3) where0is the reference density and density is calculated as 01(##0) #0.#is the virtual potential temperature, ur,u,uzare the velocities in the r,,zcylindrical coordinate directions, prghis represented asprgh= (pgr)=0,fr,f,fzare the Coriolis components, which can be calculated f=u , #is the thermal di usivity. The above governing equations are for the entire tropical cyclones. The LES simulation domain is much smaller than the scale of the entire tropical cyclones. The keys in deriving the governing equations for a small domain (similarly equal the grid size of mesoscale model) are to downscale the terms in the moment equations for that in scale larger the domain size. These terms are needed to produce mean wind pro les that are similar to observed pro les in tropical cyclones. As shown in Fig. 1, a small simulated domain is located at r=R,which is long away from the hurricane center. The domain can be analyzed as a horizontally homogeneous boundary layer that has characteristics of the HBL with the distanceR. The horizontal mean of velocity huiiover the domain can be treated as a reference velocity which satis es the governing equations for axisymmetric basic-state hurricane BL mean ow. Introducing a local Cartesian coordinate to represent the simulation domain, the axial xis in the direction of r,yis in the direction of andzis in the direction of z, as shown in the Fig. 1. Replacing r!R,dr!dx, rd!dy, the small terms O R2 and the terms O R1 which multiply perturbation terms are neglected, while the terms O R1 which multiply a mean ow term are kept. The pressure-gradient term consists the large-scale pressure gradient and gradient in this small domain,@p @r=@^P @R+@p @x. The large-scale pressure gradient@^P @Rcan be determined from a gradient wind relation. The tangential velocity above the boundary layer balance the pressure gradient via Coriolis and centrifugal terms, which satis es U2 g r 3Ug+@^P @r= 0 (A.4) The centrifugal force terms uru=randu2 =rcan be approximated as 21 u r(u;ur)hui R(u;ur) (A.5) The angle brackets represent a horizontal average at a certain height. The advection terms consist the advection on mesoscale scales (the scales larger than the domain size) and microcale advections. The gradient of tangential velocity along the hurricane vortex radius was taken into consideration. Then the radial advection term ur@ur @r,ur@u @r,ur@uz @rare approximated in the following. ur@ur @rur@huri @R+ur@ur @x ur@u @rur@hui @R+ur@u @x(A.6) ur@uz @rur@huzi @R+ur@uz @x The radial gradient of tangential velocity@hui @Rcan be obtained from the pro le of hurricane wind along radius. In [25], it is concluded that the wind eld is sensitive to the parameter of@hui @R. For actual cyclone cases, the value of@hui @Ris uncertain and can be roughly determined by the decay rate n, as@hui @R=nhui R. Typical values for n for tropical cyclones of various intensity were discussed and summarized by Mallen et al.[48].@huri @Rcan be get from the static state continuity equation,@huri @R=huri R@huzi @z. The vertical gradient of vertical velocity is negligible for axisymmetric simulation of outside of the eyewall Uz0[25]. With the large scale pressure gradient term, mesoscale centrifugal force term and advection term are included, the moment equations for this small domain in the Cartesian are: @u @t+u@u @x+v@u @y+w@u @zuhui Rhviv R=U2 g R 3Ug@p @x+r2u+v 3w 2 @v @t+u@v @x+v@v @y+w@v @z+uhvi R+u@hvi @R=@p @y+r2vu 3+w 1 (A.7) @w @t+u@w @x+v@w @y+w@w @z=@p @z+r2w(gz)r 0 Withu,v,ware the velocities in the x,y,z(x1,x2,x3) directions, Ugis the gradient wind speed and 1, 2, 3are the rotation velocities in the x,y,zdirections. Rotate the simulation domain around zaxial with 90the governing moment equations in the new coordinates are: @u @t+u@u @x+v@u @y+w@u @z=vhui R+v@hui @R@p @x+r2u+v 3w 2 @v @t+u@v @x+v@v @y+w@v @z=vhvi Rhuiu R+U2 g R+ 3Ug@p @y+r2vu 3+w 1(A.8) @w @t+u@w @x+v@w @y+w@w @z=@p @z+r2w(gz)r 0 22