Strouhal numbers are the primary factors

Transcription

1 The Eighth Asia-Pacific Conference on Wind Engineering, December 10 14, 013, Chennai, India NUMERICAL INVESTIGATION OF WIND FORCES AND STROUHAL FREQUENCIES OF SECONDARY HYPERBOLOID REFLECTOR Eswaran M 1, R K Verma, G.R. Reddy, R. K. Singh and K.K. Vaze Reactor safety division, Bhabha Atomic Research Centre, Mumbai, Corresponding author eswaran@barc.gov.in ABSTRACT In this work, flow around the secondary hyperboloid reflector is studied by numerical solutions of the unsteady Navier-Stokes equations. The computed wind load is applied on the central tower to find its deflection. For validation, a simple two tandem cylinders under the wind load is investigated for a certain range of values of spacing ratio (L/D). These results are compared with few previously published results and good agreement is found. After the validation of present numerical procedure, the forces (drag and lift) and the Strouhal frequencies of solar secondary hyperboloid reflector are determined for operational (40 km/hr) and survival (160 km/hr) wind speeds. Analysis has also been done to find out the deflections due to wind load. The computed wind loads are applied on the Secondary hyperboloid and central tower. Keywords: Secondary reflector; Tandem cylinders; Strouhal frequency; Vortex shedding; Lift and drag force; Wind loads. Introduction Solar energy has attracted more attention during the recent years, is a form of sustainable energy. Today the great verity of solar technologies for electricity generation is available and among many, the application of reflector in large sizes is employed in many systems. The amount of solar radiation entering the aperture of a collector depends on the local solar energy potential. Application of reflectors for solar heating and solar power plant improved in the recent years. Most of the solar power plants installed with reflectors are on flat terrain and they may be subjected to some environmental problems. One of the problems for such a large reflector is their stability to track the sun very accurately (Naeeni and Yoghoubi, 007). Solar thermal power plants are a gifted alternative to cover significant parts of growing energy demand. The Fig. 1 shows the schematic diagram of typical molten salt type solar power plant. In this concept, the central tower has a secondary hyperboloid mirror surrounded by heliostats. The sunrays from all the heliostats are reflected downwards by hyperboloid to a ground based receiver which absorbs the solar radiation and transfers to a molten salt steam generating system. The steam is then used to run a turbine generator system as in a conventional power plant. Wind forces play a significant role in design and operation of large reflectors and need for satisfactory estimates of these forces are becoming increasingly evident (Cohen et al., 006). Studies of wind loads on structure and boundary layer over different bodies such as buildings (Huang and Chen, 007), hills, towers (Armitt, 1980), arch roof, automobiles, heliostat (Pfahl and Uhlemann, 011), parachute, and dish are extensive both experimentally and theoretically. Conversely, wind flow around parabolic and hyperbolic shapes is rare. In engineering, fluid forces and Strouhal numbers are the primary factors considered in the design of structures subjected to cross flow, e.g., chimney stacks, tube bundles in heat Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds) Copyright c 013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: doi: / p8 691

2 exchangers, overhead power cables, bridge piers, stays, chemical reaction towers, power plant towers, offshore platforms and adjacent skyscrapers. IS 875 Part 3 provides force coefficient for most structural shapes. This coefficient when multiplied by design wind pressure and effective area of structure provides wind load on structures. Since IS 875 Part 3 does not provide force coefficient for hyperboloid surface, CFD analysis has been performed to find out the forces due to wind loads on hyperboloid surface. Shedding frequencies has also been obtained by CFD analysis. Hyperbolic reflector is a little more than quarter segment of full hyperboloid. In this work, flow around the secondary hyperboloid reflector is studied and computed wind load is applied on the central tower to find its deflection. After the validation of simple tandem cylinder case, the forces (drag and lift) and the Strouhal frequencies of solar secondary hyperboloid reflector are determined for operational (40 km/hr) and survival (160 km/hr) wind speeds. Analysis has also been done to find out the deflections due to wind load. The computed wind loads are applied on the secondary hyperboloid and central tower. y x Fig. 1. Solar power plant Fig. Secondary hyperboloid reflector z Numerical Methodology Governing equations Based on the Navier-stokes time averaged equations and using Bousssinesq approximation for Reynolds stresses, differential equations governing viscous turbulent flow field can be written as ρ + div( ρx ) = 0 (1) t ( ρu) p + div( ρx u) = div( μ gradu) () t ( ρv) + div( ρ v) = div( μ t eff eff x p gradv) y X (3) where ρ is the fluid density, μ eff the effective viscosity, X the mean flow velocity field, p the pressure and u, v are the mean components of flow field in the x and y directions, respectively. In the present CFD model, the RNG k turbulence scheme presented by Yakhot et al. (199) is used. This scheme differs from the standard k turbulence scheme in that it includes an additional sink term in the turbulence dissipation equation to account for non-equilibrium strain rates and employs different values for the model coefficients. The RNG turbulence model is more responsive to the effects of rapid strain and streamline curvature, flow separation, reattachment and recirculation than the standard k ε model (Jeong et al., 00). Thus, the turbulence of flow field is expressed in turbulence kinetic energy (k) and dissipation rate (ε ), using following equations, ( ρk) + div( ρuk) = div( Γ gradk + G ρε (4) t k ) 69

3 ( 1 η / η ) c ( ρε ) ε μη 0 ε ε + div( ρuε ) = div( Γε gradε ) + c1 G cρ t k βη k k k with G = μt ( uij + u ji ) uij ; μt = ρcμ ; μeff = μ + μt (6) ε where Γk = μ + ( μt / σ k ) and Γε = μ + ( μt / σ ε ) are diffusion coefficients for k and ε, respectively. Here, μ, μ t, σ k and σ ε are the molecular viscosity, turbulent viscosity, turbulent Prandtl number and turbulent Schmidt number respectively. The primary coefficients of the RNG model are provided by Yakot et al. (1986). The coefficients used in the turbulent models c μ, c 1, c and η 0 values are 0.085, 1.4, 1.68 and 4.38 respectively. The 4 th and 5 th terms in Eq. 5 represent the shear generation and viscous dissipation of ε. The extra term in Eq.5 employs the parameter η, which represents the ratio of characteristics time scales of turbulence and the mean flow fields, defined by η = Sk / ε. It can be shown that η is a function of generation of dissipation of k and can be written as: η = cμ ( G / ρε) (7) The standard k ε model along with Boussinesq equation, performs well for the broad range of engineering problems, however in the problems which include unbalanced effects, etc., finally this model reaches to responses which are over diffused, i.e., the μ t values predicted by this model will be large. Computational domain and boundary conditions Numerical simulation of wind flow around the secondary hyperboloid reflector (Fig. ) is studied. The fluid is assumed incompressible. As there is no free surface, the body forces can be ignored. Secondary hyperboloid reflector is placed in a 10 r x 64 r rectangular domain, in which the bottom corner of reflector is located 36 r from the inlet boundary, where r denotes the maximum distance from x axis to reflector in y direction as shown in Fig. 3. In the present CFD model, the RNG k turbulence scheme presented by Yakhot et al. (199) is used. This scheme differs from the standard k turbulence scheme in that it includes an additional sink term in the turbulence dissipation equation to account for non-equilibrium strain rates and employs different values for the model coefficients. The RNG turbulence model is more responsive to the effects of rapid strain and streamline curvature, flow separation, reattachment and recirculation than the standard k ε model (Jeong et al., 00). 3 (5) Fig.3 Computational domain. At the inlet, the Dirchlet boundary conditions are applied (u=u, v=0, k = ( I * U ref ) and ε = cμ ( k ) l, where U ref and I is the free stream velocity and turbulent intensity, and l =0.007L, where L is the diameter of the computational domain of flow).the pressure is prescribed at a point at the inlet. At the outlet, the Neumann-type conditions are employed ( 693

4 u x = 0, v x = 0, k x = 0 and ε x = 0 ). No slip boundary condition is applied at the boundary of a fixed cylinder, i.e., velocity components on the boundaries zero ( u = 0, v = 0 ). The top and bottom boundaries of the computational domain are considered as symmetry boundary to negate the propagation wall effects inside the domain over the time period. Grid generation and discretization In this study, non uniform staggered Cartesian grid is used and the enclosed area of reflector is refined with respect to other areas of the flow field. Boundary layers have also been created around the solid boundary. In such a grid, the velocity components (u and v) are calculated for the points located on the main control volume surfaces, i.e., the staggered points, while pressure, kinetic energy and dissipation rate of turbulent energy (p, k and ε ) are calculated for the points located on the main grid. The flow close to a solid wall is for a turbulent flow, is very different compared to the free stream. This means that the assumptions used to derive the k ε model are not valid close to walls. So that wall functions are used to describe the flow at the walls. This corresponds to the distance from the wall and structure where the logarithmic layer meets the viscous sub-layer. The normal distance from the + structure boundary to the wall boundary, y w is automatically computed from y w = ρμ t yw / μ, where μ t = Cμ k is the friction velocity. The y w value is kept within recommended range of , to satisfy the log law. Spatial discretization is performed using second order upwind scheme and temporal discretization is performed based on second order implicit method which causes much less damping and is thereby more accurate. A Finite volume based commercial solver is used for solving RANS and turbulent equations. The well known Semi- Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) numerical algorithm is employed for the velocity pressure coupling. Fig.4 Grid arrangement Fig.5 Boundary layers around the cylinder. Results and discussion The dimensionless flow parameters are defined as follows, C FD D =, C FL L = [1] ρu L ρu L where F L and F D denote lift and drag forces on the reflector per unit length, respectively. The ρ and U denote the fluid density in kilogram per unit volume and free stream flow velocity in meter per second. Dimensionless flow parameters are also given by ρul fv L tu Re =, St =, T = [] μ U L where Re is the Reynolds number, St is the Strouhal number, T is the dimensionless time, μ is the fluid viscosity, L is the characteristic length (i.e., r for flow over hyperboloid and cylinder diameter for flow over tandem cylinders), t is the time in second at each time step and f v is the frequency of vertex shedding which can be calculated from the oscillating frequency of lift force. Flow over tandem cylinders Validation 694

5 Flow around two tandem cylinders provides a good model to understand the physics of flow around multiple cylindrical structures. In order to better understand the flow characteristics and wake interference, initially, simple tandem cylinder arrangement cases were analysed and compared with Slaouti and stansby (199) and Meneghini et al. (001) results and for high Reynolds number, compared with Moriya et al. (00), Dehkordi et al. (011) results. The present results are very closely matching with their experimental results and numerical results. Fig. 6 shows the notations for staggered configuration. The investigation is performed in the range of L/D ratio with zero staggered angle ( ), where is the angle between the free-stream flow and the line connecting the centers of the cylinders, L is the gap width between the cylinders, and D is the diameter of a cylinder. Fig.6 Notation for staggered configuration. Fig. 7 Streamline plot for tandem cylinder =0 and L/D =3.0 Flow around two tandem cylinders of identical diameters is in general classified into three main regimes (Zdravkovich, 1987; Alam and Zhou, 007) (i) the extended-body regime (L/D < 0.7), (ii) the reattachment regime (L/D = 0.7 to 3.5), (iii) the co-shedding regime (L/D > 3.5). Fig. 7 shows the streamline plot for tandem cylinder =0 and L/D =3.0, where the shear layers separated from the upstream cylinder reattach on the downstream cylinder and the flow in the gap is still insignificant. If increase the L/D ratio further then the shear layers roll up alternately in the gap between the cylinders and thus the flow in the gap is significant. Table 1. Comparisons of flow parameter for two tandem cylinders at =0. Mean Drag Coefficient Strouhal Frequency Parametric Cylinder Slaouti and Slaouti and Meneghini Results Meneghini Present Stansby Present Stansby et al. et al. (001) (199) (199) (001) Re =00 and UC T/D = DC Re=00 and UC NA 0.15 T/D =3 DC NA 0.15 Moriya et Dehkordi et Moriya et al, Dehkordi al, 00* al, et al, 011 Re= UC and T/D = DC UC= Upstream cylinder; DC = Downstream cylinder; *Values are taken from Ghadiri et al, 011. Strouhal number is a significant feature of fluid which has a strong dependence on both Reynolds number and spacing. This non-dimensional number, specify how cylinders response to hydrodynamic forces and when their oscillation frequency reaches to the point near natural frequency whichh can lead to damage of the structure. Thereby, comparisons of the Strouhal numbers and mean drag coefficients in the present work with other data are presented for Re = 00 in Table 1. Table 1 shows comparisons of flow two tandem cylinders at Re =00 and =0, Re =10000 and =0 and Re = and =0. It can be seen from Table 1 that the negative drag has completely eliminated in the co-shedding regime while increasing the L/D ratio. The reattachment of upstream shear layer onto the second cylinder is 695

6 observed. Due to this, negative drag is found on the downstream cylinder which results from the pressure difference between front and back sides of this cylinder. Frequency of the vertex shedding is calculated from the oscillation frequency of lift force. Flow over the secondary hyperboloid reflector In this work, the calculations are made for three wind directions and the two velocities say operational wind speed (40 km/hr) and survival wind speed (160 km/hr). The wind directions are taken as follows, wind flow in positive x direction (case 1), wind flow in negative x direction (case ), wind flow in positive z direction (case 3). The x, y and z directions are correspondence to Fig.. Since case 1 and computational domains are almost similar except the wind direction for case 3 the shape of the object is changed accordingly. Vortex shedding is an oscillating flow that takes place when a fluid flows past a bluff body at certain velocities, and it is depending to the size and shape of the body and Reynolds number of fluid flow. If the structure is not mounted rigidly and vortex shedding frequency matches the resonance frequency of the structure, the structure can start to resonate, vibrating with harmonic oscillations driven by the energy of the flow. Fig.6 Velocity contour for case 1 at operational wind speed at tu L =50. Fig.7 Pressure contour for case 1 at operational wind speed at tu L =50. Secondary vortex Primary vortex Fig. 8 Velocity contour at tu L =50, case1 with operational wind speed Fig. 9 Streamline diagram at tu L =50, case1 with operational wind speed Wind flow in positive x direction (case 1) In this case, operational and survival wind velocities are taken to find the forces and shedding frequency. Velocity and pressure contours and streamline diagram at operational wind speed are depicted in Figures 8-1. The velocity and pressure fluctuations in the wake region create an oscillating flow in rear of the structure. Drag is generated by the difference in velocity between the solid object and the fluid. Fig 8 shows the velocity magnitude contour. Fig. 9 shows the streamline diagram. It is found that the one primary and one secondary vortex are formed behind the structure for all the cases. The shedding frequency has been calculated from lift force fluctuations. The velocity at x and y components are shown in Fig For case 1, the region above the structure has a positive pressure, while the just behind the structure holds the negative u velocity. Alternate convective shedding rolls have been observed from u and v contours. The velocity values are depicted in the respective place in the picture itself. And coefficient of pressure is showed in pressure contour as shown in Fig

7 Fig.10 Velocity at x component (u) contour for case 1 at operational wind speed at tu L =50. Fig.11 Velocity at y component (v) contour for case 1 at operational wind speed at tu L =50 Fig.1 Pressure contour for case 1 at operational wind speed at tu L =50 Fig. 13 shows the drag and lift coefficients fluctuations over non-dimensional time. From these fluctuations the average coefficients are calculated. Wind speeds are varied between operational and survival wind speeds, and strouhal number have been calculated and illustrated as dimensional form in Fig. 14. Drag and lift forces are increasing while increasing the wind velocity. However, the lift force is increasing sharply compared to drag force. The structural first mode frequency has been calculated as 1.0 Hz, as shown in Fig. 14 the wind velocity corresponds to structure frequency is 100 KM/HR. So that, the plant can be operated upto 81 KM/HR after including safety margin of 5% with structure frequency. If the wind shedding frequency matches with tower structural frequency (i.e., 1.0 Hz), the structure can start to resonate, vibrating with harmonic oscillations driven by the energy of the flow and subsequently, deflection will be increased in the tower. C L c D Shedding Frequency in Hz % Resonance region of higher mode Resonance region of first mode - 5 % Fundamental Frequency of Structure (1.0 Hz) Safe region for plant operation Non-dimentional time(tu/l) Fig.13 Drag and lift force coefficients at operational wind speed for case Wind Velocity in km/hr Fig.14 Shedding frequency with wind velocity for case 1 Wind flow in negative x direction (case ) In this analysis, the wind direction is considered in negative x direction as shown in Fig.. The streamline plots for operational and survival wind speeds are shown at T =50 in Figs. 15 and 16. These streamline plots show the primary and secondary vortices. Different recirculation regions can be found on the leeward side of the hyperboloid reflector. The shape of the primary vortex is slightly more in the survival wind speed. Compare to 697

8 operational speed, the secondary vortex size is also more fluctuating with respect to time at survival wind speed. The force coefficients and Strouhal number are shown in Table 1. Fig. 15 Streamline plot at operational wind speed and tu L =50 Fig. 16 Streamline plot at survival wind speed at tu L =50 Wind flow in positive z direction (case 3) While compare to the previous cases, the case 3 computational domain and shape of the object is dissimilar. Here the D cut section is taken from the y z coordinate of hyperboloid. Since wind directions are similar in both directions, i.e., positive and negative z direction. The drag, lift coefficients and Strouhal number for this case are shown in Table 7.. The Strouhal number is calculated from the first mode frequency. Since the shape of the reflector guides the wind flow around structure, it holds low pressure coefficient around the reflector. So that, case 3 shedding frequency is relatively low. The wind force on the reflector increases sharply while the wind speed increases. When the flow direction is in positive x direction (Case 1), then the lift force is negative. That means the lift force on structure is acting towards ground. Flow direction is in negative x direction (case ), now situation is just opposite. i.e., the lift force is acting opposite to gravity. Table 7.: Force coefficients and Strouhal numbers Sl. No Operational (40 km/hr) Survival (160 km/hr) C D C L St C D C L St CASE CASE CASE Using the above coefficient, total wind load on secondary reflector has been calculated for three cases. Structural analysis Finite element analysis has been performed to find out the structural frequencies. Analysis has also been done to find out the deflections due to wind load. Table shows the frequencies and mass participation factors in dominant modes. Hyperboloid has been designed as a truss structure as per IS 800. Structural tubes of different sizes have been used as truss member to reduce deflections under dead weight and wind loads. Depth of truss has been optimized and is 600mm. Truss member has been optimized to reduce weight of hyperboloid structure. This hyperboloid is supported on three towers. Height of the tower is 45m. Base of the tower is 7 7 m and top is m. Tower has been designed to ensure fundamental frequency beyond 1 Hz. Pre-stressed cables have been used to limit the deflections in hyperboloid reflecting surface. A secondary hyperboloid and central tower has been designed as a truss structure as per IS Table 3 shows deflections in secondary hyperboloid due to wind loads acting on structure. Figures 17 and 18 show deformed shape due to survival wind load for case

9 Table : Frequencies and Participation Factors Frequency X-direction Y-direction Z-direction S. No. (Hz) *MPF **%MP MPF %MP MPF %MP Note: MPF- *Mass Participation Factors, **%MP-Percentage Mass Participation Table 3: Maximum Deflections in Secondary Hyperboloid Wind speed Maximum Deflections Bending stress Case 1 Case Case 3 Case 1 Case Case 3 Operational Survival Fig. 17 Deformed Shape of central tower due to survival wind load for case1. (Dead Loads + Imposed Loads + Wind Loads) Fig. 18 Deformed shape of support frames of reflector due to wind load for case1. (Dead Loads + Imposed Loads + Wind Loads). Conclusions The solar power plant secondary reflector has been taken for analysis against wind load. Drag and lift forces are evaluated under operational (40 km/hr) and survival (60 km/hr) wind speeds. Since IS 875 Part 3 does not provide force coefficient for hyperboloid surface, CFD analysis has been performed to find out the forces due to wind loads on hyperboloid surface. Shedding frequencies has also been obtained by CFD analysis. The wind directions are also varied to find these parameters. Here, the forces and vortex shedding frequency is increasing while increasing the wind velocity. From above work, the following conclusions are drawn. 1. Fundamental frequency of the structure is 1.0 Hz, which is quite away from shedding frequency estimated by CFD analysis. Also IS 800 recommends frequency more than 1.0 Hz to avoid wind oscillation. If the wind shedding frequency matches with tower structural frequency (i.e., 1.0 Hz), the structure can start to resonate, vibrating with harmonic oscillations driven by the energy of the flow and subsequently, deflection will be increased in the tower.. Since IS 875 Part 3 does not provide force coefficient for hyperboloid surface, CFD analysis has been performed to find out the forces due to wind loads on hyperboloid surface. 699

10 3. From Fig. 7.16, it can be observed that the wind velocity corresponds to structural fundamental frequency 1.0 Hz is 105 km/hr, considering resonance frequency band of ±5% the wind velocity up to which plant can be operated is ~80 km/hr. 4. Drag and lift forces are increasing while increasing the wind velocity. However, the lift force is increasing sharply compared to drag force (Figure 7.15). When the flow direction is in positive x direction (case 1), then the lift force is negative. That means the lift force on structure is acting towards ground. When flow direction is in negative x direction (case ), the lift force is acting opposite to gravity (Table 8.1). 5. The shape of the reflector guides the wind flow around structure while wind flowing from positive z direction, it holds low pressure coefficient around the reflector resulting relatively low shedding frequency (Table 7.). The calculated stresses are found within acceptable limit for both operational and survival wind speeds. This study can be extended by finding the suitable drag minimization techniques to avoid the resonance matching. References Alam M.M. and Zhou Y., (007), Dependence of Strouhal number, drag and lift on the ratio of cylinder diameters in a two-tandem cylinder wake, 16th Australasian Fluid Mechanics Conference, Australia. Armitt J., (1980), Wind loading on cooling towers, Journal of the Structural Division, 106(3), Cohen E., Vellozzi J. and Suh S.S., (006), Calculation of wind forces and pressures on antennas, Annals of the New York Academy of Sciences, 116(1), Dehkordi B.G., Moghaddam H.S., Jafari, H.H., (011), Numerical simulation of flow over two circular cylinders in tandem arrangement, Journal of Hydrodynamics, Ser. B, 3(1), Huang G. and Chen X., (007), Wind load effects and equivalent static wind loads of tall buildings based on synchronous pressure measurement, Engineering Structures 9, Jeong U.Y., Koh H.M., and Lee, H.S., (00), Finite element formulation for the analysis of turbulent wind flow passing bluff structures using the RNG k- ε model, Journal of Wind Engineering and Industrial Aerodynamics, 90, Meneghini J.R., Satara F., Siqueira C.L.R. and Ferrari Jr. J.A., (001), Numerical simulation of flow interference between two circular cylinders in tandem and side by side arrangements, Journal of fluids and structures, 15(), Moriya M., Alam M. and Takai K, (00), Fluctuating fluid forces of two circular cylinders in tandem circular cylinders in tandem arrangement at close spacing, Transactions of the Japan Society of Mechanical Engineers, 68(669), Naeeni N. and Yoghoubi M., (007), Analysis of wind flow around a parabolic collector (1) fluid flow, Renewable Energy, 3, Pfahl A. and Uhlemann H., (011), Wind loads on heliostats and photovoltaic trackers at various Reynolds numbers, Journal of Wind Engineering and Industrial Aerodynamics, 99, Slaouti A., and Stansby P.K. (199), Flow around two circular cylinders by the random vertex method, Journal of fluids and structures, 6(6), Yakhot V and Orszag S A, (1986), Renormalization group analysis of turbulence, Journal of Scientific Computing, 1 (1986), 3 5. Yakhot, V., Orszag, S.A., Thangam, S., Gatski and T.B., Speziale, C.G., (199), Development of turbulence models for shear flows by a double expansion Technique, Phys. Fluids A 4(7), Zdravkovich, M. M., (1987), The Effects of interference between circular cylinders in cross flow, J. Fluids and Structures, 1,