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CFD simulations are performed using the Large Eddy Simulation (LES) sub grid scale model originally proposed by Smagorinsky (1963) and modified by Geomano et al. (1991) to tune the model constant based on the flow dynamics. The commercial software package Fluent (2006) is utilized to solve the governing flow expressed by Equations 4.25. Conductor velocity, Vc,, is introduced into the governing equations to account for the conductor movement. Solving the FSI for an entire span of a conductor placed in the wind using LES can be very time consuming. That is because LES requires grid meshing in the order of the conductor cross section which is very smaller than the length of an entire span, making the entire process computationally expensive. Therefore, CFD simulations are performed on a conductor segment similar to what was used by Keyhan et al. (2013). The segment length is taken equal to 4 times the conductor diameter, D, similar to that used by Ochoa and Fueyo (2004). Murakami and Mochida (1995) indicated the importance of using a three dimensional domain to accurately predict the drag coefficient using LES. Summary of the proposed technique that performs the FSI is given by the flowchart shown in Figure 4.10. From the flowchart, a User Defined

Function (UDF) is developed as part this study and is integrated with the software FLUENT to solve for the conductor motion, which is treated as a single degree of freedom system considering the first out of plane vibration mode. Treating the conductor as a single degree of freedom system requires the modal force, F(t)modal, to be known.

Such a modal force can be calculated by integrating the applied forces along the entire span as indicated by Equation 4.26. By assuming a full correlation among those forces, a scaling is introduced to the force obtained from the computational segment, F(t)seg, in

order to obtain the required modal force, F(t)modal, as indicated by Equation 4.27.

Dimensions of the computational domain and the boundary conditions employed are shown in Figure 4.11. The domain length and width are chosen equal to 41D and 21D, respectively, to eliminate the effect of the boundary conditions on the flow near the conductor. The wall unit, y+, defined by Equation 4.28, is maintained less than 1.0 for all simulations. A number of 10 grids are used along the sides of the conductor to reasonably resolve the flow near the conductor. In order to resolve for the vortex shedding, dense grids are introduced near and behind the conductor with a grid size equal to D/10 as shown in Figure 4.11. Discretization schemes for the flow quantities and parameters of the utilized solver are summarized in Table 4.3.

Employing the technique summarized in Figure 4.10 provides time history of the conductor responses. For the cases of a conductor subjected instantaneously to a steady wind, the displacement response shows damped oscillatory movement towards the static displacement. By fitting the peaks of such decaying response with the logarithmic decrement of damping, conductor aerodynamic damping can be calculated. For the cases where the wind is non-stationary, as in downburst, there is no single value for the aerodynamic damping. Therefore instead of comparing the aerodynamic damping, the resulting displacement response can be directly compared with that obtained by solving the equation of motion employing the aerodynamic damping that is suggested in the current study. If compatible responses are found, that means it validates the suggested expression for estimating the aerodynamic damping.

Before the technique summarized in Figure 4.11 can be used to assess the accuracy of the suggested damping expression, the technique itself is examined by estimating the

aerodynamic damping of a conductor placed in a uniform steady wind and comparing the result with those obtained through well-established techniques as will be shown in section 4.3.2. Once accuracy of the technique is confirmed, it is used to assess the accuracy of the suggested damping expression, as will be shown in section 4.3.3.

(u V ) i ci ₀ x i u u _{1 P} i _{(u u )} i _{( 2 S )} j gj ij ij t x x x j i j u u u u ij i j i j u u 1 _i j S ij _{2 x x} j i 1 2 S ij ₃ ij kk e ij 2 2 (C . ) 2S S e s ij ij       _{ }                 _       _{ }                 _{ }   where

i=1,2,3 correspond to the directions x, y and z, respectively

The over bar represents the filtered quantities

ui, ugi ,p, t,τij and ν: fluid velocity, grid velocity, pressure, time, the

SGS Reynolds stress and molecular viscosity coefficient, respectively. Sij,_e, , C_s,: strain rate tensor, eddy viscosity, grid size, Smagorinsky

constant which is determined instantaneously based by the Geomano identity in the dynamic model (Geomano et al., 1991).

Equations 4.25

modal x 1 F(t) =L . f (n). (n).dn

_

where

f(n): wind force applied on a conductor segment dn

1(n)

 : value of the first mode shape at n

x

modal seg 1 seg

seg seg

L 2L

F(t) =F(t) . . (n).dn F(t) .

L



where:

L, Lseg: the conductor actual length and the conductor segment length in the CFD model

Equation 4.27 p p y .V y   where

yp: Distance to the conductor from the first grid Vp: Velocity at the first grid point

ν: Kinematic viscosity

Equation 4.4.28

Table 4.3 Discretization schemes and solution technique for the CFD simulations

Parameter Type

Time discretization Second order implicit

Momentum discretization Bounded central difference Pressure discretization Second order

Pressure-velocity Coupling Pressure-implicit with splitting operators (PISO) Under Relaxation Factors 0.7 for the Momentum and 0.3 for the Pressure

Figure

Figure 4.10 Schematic of the utilized technique to perform the FSI

Figure

Schematic of the utilized technique to perform the FSI

Figure 4.11 the CFD Domain and its Meshing

Schematic of the utilized technique to perform the FSI

the CFD Domain and its Meshing

Schematic of the utilized technique to perform the FSI

the CFD Domain and its Meshing

Schematic of the utilized technique to perform the FSI

the CFD Domain and its Meshing

4.3.2

Simulating the Conductor under a Steady Uniform Winds