In the absence of the wake survey, the equivalent planar motion is a good approximation to a rotating blade in a vertical-axis wind turbine. 30 3.3 The L2 norm of the error of the velocity field in the streamwise direction in a single-.
Motivation
LES is a good choice to simulate three-dimensional and separated flow at such a high Reynolds number. 2007) simulated dynamic stall in a section of a VAWT using LES and detached-eddy simulation (DES) atRe = 50,000 and validated the results by comparing the vortex in the rotor area with particle image velocity (PIV) data. Since the flow experienced by the blades of a VAWT is inherently unsteady, we are interested in taking advantage of this unsteady flow and maximizing the power output of a VAWT through the pitching motion of the blades.
Aerodynamics of VAWT
Only the Coriolis force depends on the fluid velocity and can locally influence the dynamic stall. Therefore, in this thesis we focus on investigating the effect of the Coriolis force on the dynamic stall.
Self-starting of a VAWT
Beri and Yao (2011a,b) numerically investigated the self-launch of VAWT with cambered airfoils and modified trailing edge airfoils. Chen and Kuo (2013) studied the effects of pitch angle and blade twist on self-starting.
Flow control on vertical axis wind turbine
However, the performance of the turbine during start-up, which is important for the study of self-starting problems, was not accurately captured due to the lack of near-wall and transient turbulence models. They showed that VAWT with cam airfoils has the potential to self-start with reduced power coefficient and VAWT with modified trailing-edge airfoils has better self-starting performance at lower tip-to-speed ratios. with normal power factor.
Outline of this thesis and the resulting publications
With the relationship of the time differentiation in the Newtonian frame and the rotating frame. The alternative form of the incompressible Navier-Stokes equations in the non-inertial frame, equation (2.9), with the submerged limiting force.
The evolution of the flow coupled with the motion of the body
Derivation of adjoint equations and control gradients
Note that equation (2.44) is the adjoint of the Navier-Stokes equations and equation (2.47) is the corresponding initial condition. Equation (2.46) is the adjoint of the equation of motion and equation (2.48) is the corresponding initial condition. In equation (2.61), ∆F is the difference of the desired forces to its target values, Fref, and B(φ) is the rotation matrix, which for 2D is given by.
Procedures of optimal control
In particular, the Coriolis force depends on the fluid velocity and is known to locally affect dynamic stagnation. In this chapter, we focus on investigating the effect of the Coriolis force on dynamic stall by comparing a rotating airfoil with one undergoing equivalent plane motion. The relationship of the Coriolis force to the input velocity and angle of incidence variations in VAWT and equivalent planar motions is also examined.
Simulation setup
The Rossby number is the ratio of the inertial force to the Coriolis force, which is used to measure the Coriolis force. Furthermore, inspired by Wang et al. 2010), who studied dynamic stall in a VAWT by investigating an airfoil subjected to a simplified sinusoidal pitching motion, we introduce sinusoidal approximations of the VAWT angle of attack and inlet velocity. The symbol ◦ indicates that the motion is a sinusoidal approximation of the VAWT motion described in equations (3.12) and (3.13).
Qualitative flow features in a VAWT
The generated vortices will propagate downstream into the wake of the VAWT or interact with the wings in the downwind half of a cycle. This vortex pair interacts with the airfoil in the downwind half of a cycle (Figure 3.4(j-l)), which was also observed by Ferreira et al. When the blade rotates in the downwind half of a cycle, the angle of attack becomes negative.
Comparison of VAWT and EPM
Therefore, in this study we will focus on the lift in the upwind half of a cycle. Furthermore, when this vortex pair moves downstream, it interacts with the airfoil in the downwind half of a cycle. Therefore, the Coriolis force acting on the fluid around eddies becomes relatively important in the downstroke phase.
Comparison with an airfoil undergoing a sinusoidal motion
We can see from the second and third columns in Figure 3.5 that the Coriolis force deflects the flow around the rotating airfoil in a clockwise direction. In the stroke phase, the behaviors of CL, EPM, CL, SPM and CL, SSPM are close to that of CL,VAWT due to the low angle of attack. However, it overestimates the lift coefficients in the shock phase due to its inability to predict the wake capture phenomenon.
Effect of tip-speed ratio, Rossby number, and Reynolds number
Nevertheless, the differences between the four lift coefficients are relatively small as the angle of attack increases and after the vortex shedding starts. As the tip-velocity ratio increases, the amplitudes of the angle of attack variation and the corresponding lift decrease. Therefore, EPM motion is a good approximation of VAWT at larger tip-velocity ratios due to the low angle of attack.
Comparisons with experiments
Comparison with Ferreira et al. (2007)
As the VAWT rotates faster, the wake-trapping effect is enhanced on one side due to the intensifying Coriolis force, which corresponds to decreasing Rossby number; on the other hand, it is attenuated due to the decreasing amplitude of the angle of attack variation due to the increasing tip-speed ratio. Therefore, the growth of the discrepancy depends subtly on the increase in the rotating speed of the VAWT. The location of the vortex pair composed of the phase-averaged LEV and TEV agrees with the current simulation, especially at θ = 158◦ (figures 3.9(e) and 3.9(h)).
Comparison with Dunne and McKeon (2014, 2015a,b)
The qualitative agreement in the opposite half of a cycle suggests that wake capture may also occur in the experiment of Ferreira et al. At maximum α= 30◦ the flow is completely separated in both experiment and calculation with a shear layer at the leading edge visible in the experiment, and continuous-scale vortices shed from the leading and trailing edges in the calculation (figure 3.10(d) ). Similar to Figure 3.10, the eddies are more coherent in the calculation due to the lower Reynolds number.
Decoupling the effect of surging, pitching, and rotation
Airfoil undergoing only surging motion
The effect of the experimental frame and the turbine reference frame is clearly visible in the trajectories of the vortices. The vortices in the calculation convect from left to right with the relative free flow of the turbine, while the vortices in the experiment convect behind the airfoil with the relative free flow of the blade.
Airfoil undergoing only pitching motion
Open-loop control on the wake-capturing phenomenon
Summary
The motion of the VAWT is associated with the aerodynamic forces exerted on the blades. The radius of the VAWT is fixed at R = 1.5 c in the present study in order to compare with the experiments conducted by Araya and Dabiri (2015). Again, VAWTs are investigated numerically at low Reynolds number, Re∼O(103), in order to gain more insight into the initiation of a VAWT in a relatively short computational time.
Starting of a three-bladed VAWT at various Reynolds numbers and density ratios
Therefore, VAWTs with low density ratios,φρ∼O(1), have been investigated to explore the effect of density ratio on the initiation of a low Reynolds number VAWT. From figure 4.3(b), we observe a critical density ratio between 3 and 4 that VAWTs with density ratios higher than the critical value cannot start at Re= 2000. When VAWTs are able to start, those with a ratio lower densities achieve a faster stationary rotation, but with a larger fluctuation in the tip-velocity ratio due to the lower moment of inertia.
Flow-driven VAWT and the comparison with motor-driven VAWT
There is a qualitative match in the measured torque trend for similar flow-driven and motor-driven cases. The flux-driven and motor-driven torque coefficients are in good agreement with each other in the calculation and behave similarly to the torque for the flow-driven and motor-driven VAWT measured in the experiments. These show that our load model is sufficient to represent the current-driven VAWT load and that the motor-driven VAWT can reproduce the current-driven VAWT physics at Re= 2000 when λ ranges from 0 .4 to 2.0.
Analysis on the starting torque of a three-bladed VAWT
Shown in Figure 4.8, we investigate the torque contributions from each blade to the initial torque for a three-blade VAWT. These blade torque distributions are compared to the initial torque of a single blade VAWT. Due to the similarity of the torque distribution from each blade, the initial torque of a three-blade VAWT can be well reconstructed from the single-blade torque distribution.
The starting torque distribution of a multi-bladed VAWT
Similarly, we also investigate the torque contributions from each blade to the initial torque for the multi-blade VAWT, as discussed in Section 4.4. These blade torque distributions are compared to the initial torque of a single blade VAWT in Figure 4.11. The average value of the torques during a cycle marked in figure 4.10 increases with the increase in the number of blades.
Effect of the static pitch angle on the single-bladed torque distribution
Torque distributions with different static pitch angles (blue, black and red curves) and the optimal torque distribution (green curve) at att∗= 1 and Re= 2000. The optimal initial configuration for a multi-blade VAWT can be determined by pitching the blades with their own optimal static pitch angles as shown in Figure 4.14(a). The optimal three-blade torque distribution can be reconstructed by linearly summing the torques at these azimuthal angles with their own optimal static pitch angles from Figure 4.13(a).
Summary
The torque distributions of the single clamp VAWT with various static pitch angles were then investigated. Since the VAWT in this case is motor driven at a constant rate of rotation, the input power is equal to the negative of the power generated by the forces exerted by the fluid on the blade, i.e. (P ower)input =−Ω τf b. In this thesis, optimal control aims to minimize the integral of the input. power over a horizon of the lengthTH,.
Optimization results
The improved results of implementations 1 to 5 using JP are achieved by increasing the length of the control horizon. Each horizon in the receding horizon predictive control is chosen such that THi = 1.5T and the corresponding overlap between two horizons is To=23TH= 0.5T. In series 6 to 9, JP is minimized using two to five horizons of length 1.5T, corresponding to a total horizon of length 2.5T to 5.5T.
Influence of the cost function
By the end of the longest control horizon examined, time-invariant pitch angles of about 5◦ and. Comparison of uncontrolled and controlled flows showed that as the angle of attack increases/decreases in the upwind/sand half cycle, the negative/positive blade pitch angle reduces the power input by decreasing the effective angle of attack. Time-invariant positive pitch control successfully reduces input power by eliminating wake-capturing vortex pair TEV.
In the impact phase, the EPM overestimates the mean lift coefficient due to the absence of the aforementioned wake-capturing. By assuming that the inertia of the blade is much greater than that of the fluid, the VAWT can be considered stationary in the flow.
