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Subcritical Flow Regime of Tandem Interfering Cylinders

The subcritical flow regime of Reynolds number Re = 1.0 x 104 is the subject of the paper in the context of aerodynamic interference of tandem circle cylinders. The two cylinders are in distance of L/D equal to 2.0. The purpose of the paper is to determine aerodynamic drag and lift coefficients and the frequencies for upstream and downstream cylinders and also the distribution of the velocity around the cylinders and in their wakes. In numerical analysis the RNG k – ϵ method was used.


Introduction
Wind flow around civil structures and their parts are the subject of detailed analysis due to practical applications in design and operations of constructions [1][2]. Wind load action on structures cause their vibrations. In addition, behind the structure the vortex street appears which may play significant role in the interaction with elements located in it. Cables of the suspended and stayed bridges and conductors of overhead transmission lines are often in the system of several elements so they may interact. The researchers presented in [3][4][5][6][7] show a high level of complexity of the problem of flow around cylinders with the changing distance between them. Due to the flow around the elements, the pressure differences appear. Followed by separation, the boundary layer is located in the upstream and downstream cylinders. The boundary layer as a disturbance has the different level depending on inertia, viscosity of the media, surface roughness and distance between the bodies. The study includes a numerical analysis of the mechanism of wind flow around two cylinders of circular cross-section, one behind the other, for a given Reynolds number Re = 1.0 x 10 4 . The analysis allows to determine the drag and lift forces, frequencies and velocity distribution.

Mathematical formulation
In the paper Navier-Stokes equations of motion for description of incompressible and viscous air is used. In the analyzed state, the inertial force, pressure and viscosity forces are balanced. The state of medium can be described by three parameters: temperature, pressure, density or volume. These three variables are independent and are connected by equation of state [8][9][10][11][12][13][14]: For analysed case of the air the relationship between T, p, ρ is came from theoretical formulas based on the experimental molecular -kinetic theory. For ideal gas the equation of state has the form: This formula is valid at any volume V, so the integrand must vanish: and in the scalar components: where , , are component of velocity vector.
In the fluid mechanics the constitutive equations are the quantitative expression of the relations found experimentally that occur between the stress field and the velocity field .

Analysed model, discussion and results
Analyzed model with the mesh of flow domain and general assumption of inflow direction and section of cable is shown in figure 1 and figure 2. The analysis is performed for Re = 1.0 x 10 4 for tandem cylinders of circle sections. The assumption that air is incompressible is made. Upstream cylinder is located in the distance from the flow inlet of 19D and 21D from the flow outlet, downstream cylinder is located 21D from the inlet and 19D from the outlet. The initial conditions are: uniform flow inlet with velocities v x = , v y =0. No slip condition is applied on the cylinder surface and v y =v y =0. For improving accuracy of results, 27820 elements were applied. In the analysis the grid was required to be fine to capture the flow distribution near the cylinders and in their wakes.     5 and figure 6).
For comparison the plots for single circle cylinder is shown. The typical velocity contours around the tandem circle cylinders and in their wakes are shown in the figure 7. Figure 8 shows the distribution of pressure coefficient C p along the surface of the upstream and downstream cylinders. For upstream cylinder the magnitude of pressure is zero at ϕ = 37 0 and is minimum C p =-1 at ϕ = 71 0 . For downstream cylinder the magnitude of pressure is maximum C p =-0.25 at ϕ = 60 0 and is minimum at ϕ = 0 0 .

Conclusions
In the paper the wind flow past tandem cylinders with circle cross-section was discussed. Drag and lift coefficients for Reynolds number Re = 1.0 x 10 4 was analyzed. These coefficients were analyzed in the time domain and also power spectra analysis was performed. The average drag coefficient for the upstream cylinder is bigger than for the downstream cylinder. The amplitude of the lift coefficient for upstream is smaller than for downstream cylinder. The frequency of drag force is approximately twice that of lift force. In the paper the distribution of pressure coefficients along the surface of upstream and downstream cylinders is also presented. The received results are the good background for future study on flow past cables with control of the vortex shedding what is especially important for design and minimization of vibrations of civil engineering structures.