Structural optimization of H-type VAWT blade under fluid-structure interaction conditions

3.1. Optimization model

The strength ratio of the small-scale VAWT blade is much less than 1 at the rated wind speed, confirming that the facture failure will not occur. Thus, the optimal objective is to minimize the wind wheel mass:

where $X$ is the vector of optimal variables, and $m$ is the wind wheel mass.

The optimal variables include the thicknesses of uniaxial glass cloth, biaxial glass cloth, triaxial glass cloth, foam and gel coat, and the positions of two webs. Table 1 lists their initial values and ranges:

The maximum stress ${\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, maximum strain ${\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and maximum deformation $d_{\mathrm{m}\mathrm{a}\mathrm{x}}$ of the wind wheel after the structural optimization should be smaller than the initial maximum stress ${\sigma }_0$, initial maximum strain ${\epsilon }_0$ and initial maximum deformation $d_0$, respectively. Thus, the optimal variables are bounded by:

The Design Exploration is a rapid optimization tool of ANSYS Workbench, and its response surface optimization project classified as the goal-driven optimization can accurately describe the relationship between input and output parameters and has the advantages of short calculation time and high efficiency. The blade’s structural optimization process adopting the response surface optimization is shown in Fig. 3.

Fig. 3Flow chart of the blade’s structural optimization

3.2. Optimization example

An H-type VAWT blade from Ref. 22 is chosen as a testbed. As for the basic parameters, the rated wind speed $v$ is 7 m/s, the rated power $P$ is 100 W, the rated speed $n$ is 200 r/min, the blade number $B$ is 3, the tip speed ratio $\lambda$ is 1.777, the rotation diameter of the wind wheel $d$ is 1.1889291 m, the blade length $l$ is 1.307822023 m, and the chord length of NACA0021SC airfoil is 0.313503278 m. The fluid-structure interaction calculation process of the wind wheel is shown in Fig. 4. During the optimization, the type of the response surface is the genetic aggregation, the screening method is adopted, the quantity and starting point number of the initial sample are 100 and 5, and the optimization iteration is 81.

In the DesignModeler module, the airfoil profile formed by the 3D Curve command stretches symmetrically with the size of 0.5$l$ to obtain the solid model of the blade. Two web planes established through offsetting the $YZ$ plane along the $Z$-axis by 0.2$c$ and 0.5$c$ are taken as the boundaries, and the blade is divided into three parts using the Slice command. The solid model of the wind wheel is shown in Fig. 5 by translating and arraying the three parts. A cuboid of size $22d\times 10d\times l$ and two cylinders with the height of $l$ and diameters of $d+3c$ and $d-3c$ are established, and their attributes are Fluid. Through using the Boolean subtraction retain subtracted parts, the computational domain consisting of two static subdomains and one rotating subdomain is obtained, as shown in Fig. 6. The thicknesses of the leading-edge, main beam, trailing-edge and webs are defined to be 0 by the Thin/Surface command. The left and right surfaces of the computational domain constitute the velocity inlet and pressure outlet, respectively. The moving and adiabatic wall boundary conditions are applied on three blades, namely blade 1, blade 2 and blade3. The connection between static and rotating subdomains adopts the interface boundary, and the upper and lower surfaces of the rotating subdomain are represented by rotating-wall1 and rotating-wall2.

Fig. 4Flow chart of the fluid-structure interaction calculation of the wind wheel

Fig. 5Solid model of the wind wheel

Fig. 6Computational domain of the wind wheel

In the Fluid Flow (Fluent) module, the fluid domain is discretized by an unstructured tetrahedral grid. The airfoil profile and the inner and outer circumferences of the computational domain are evenly divided according to the element sizes of 0.002 m, 0.008 m and 0.04 m, respectively. The edges IL and JK are evenly divided into 25 parts, and AB, BC, CD and DA are uniformly divided according to the element size of 0.07 m. The volume grids consist of 147637 nodes and 746667 elements, as shown in Fig. 7. Moreover, Fig. 8 gives the skewness, aspect ratio and Jacobian ratio of the grids through changing the mesh measurement in Mechanical. The skewness mainly ranges from 0 to 0.6, and its maximum value is 0.84 and less than 0.97. The aspect ratio is mainly around 1.6 and 2.6, and the Jacobian ratio is concentrated nearby 1. These indicate that the grid quality is good, avoiding the negative volume that can lead to the failure of the aerodynamic performance calculation.

The dynamic mesh combining the diffusion smoothing with remeshing is more suitable for a rotating wind wheel. The diffusion parameter is 1.5, and the maximum and minimum element sizes in the remeshing are 1.5 times and half those of the maximum and minimum grids generated, respectively. The skewness of the surface mesh is 0.7, and so is 0.9 for the volume mesh. The surfaces of blade1, blade2 and blade3 are defined as the Fluid Solid Interface, and the rotating-wall1 and rotating-wall2 are the Deformation Plane.

The geometric model and computational domain of the wind wheel are imported into the ACP (Pre) module. The blade surfaces are discretized using the quadrilateral grid with the element size of 0.02 m to obtain the surface grids with 7760 nodes and 8050 elements. The ply materials, layer number and reference coordinate system are set according to Ref. [23]. The initial stacking sequences and numbers of the leading-edge, main beam, trailing-edge and webs are FM+TM+3UM+TM, FM+TM+BM+3UM+BM+TM, FM+TM+3UM+GM and 2BM+GM+2BM (with UM, BM, TM, FM and GM for the uniaxial glass cloth, biaxial glass cloth, triaxial glass cloth, foam and gel coat), respectively, and the ply angle is 0°, as shown in Fig. 9. The layer information is transferred to the Transient Structural module, the gravitational acceleration and angular velocity around the $Z$-axis counterclockwise are applied, and the node displacements are restrained for the parts that the blades are in contact with the arms, as shown in Fig. 10. The Weak Springs is turned off, the Large Deflection is turned on, and the displacement, stress and strain are output.

Fig. 7Grids of the computational domain

a) Sectional view of the computational domain

b) Sectional view of the rotating subdomain

Fig. 8Grid quality parameters

a) Skewness

b) Aspect ratio

c) Jacobian ratio

Fig. 9Blades with initial composite material layers

Fig. 10Constrained wind wheel

In the System Coupling module, the analysis time is 1.2 s and the time step is 0.0015 s. It can be seen from Fig. 11 that the maximum displacement and stress fluctuate greatly in the early stage, indicating that the aerodynamic load calculated in the fluid domain also changes obviously, and their fluctuations gradually become smaller with time. The structure tends to be stable near 0.3 s under the influence of the flow field. So the displacement, stress and strain after 0.3 s are extracted, reflecting the structural performance during operation more accurately.

Fig. 11Maximum displacement and stress of the wind wheel at different moments

a) Maximum displacement

b) Maximum stress

Table 2 gives the optimization results of composite material layers and inner structure parameters. The optimized wind wheel mass is 7.0075 kg and decreases by 6.57 % compared with the original structure mass 7.5 kg.

4.1. Modal analysis

The forces acted on the blade for the static, single pre-stress and multiple pre-stresses states are shown in Fig. 12. Table 3 lists only the early six frequencies due to the greater influence of the lower frequencies on the stability of wind turbines. Under the static and single pre-stress states, the first-order frequencies and critical speeds increase and other frequencies decrease after the blade optimization. The first-order critical speeds are greater than 200 r/min that is 20 % of five times the rated speed, confirming that the resonance will not occur for the blade. The dynamic stiffening effect makes the frequencies under the single pre-stress state smaller than those under the static state. Under the multiple pre-stresses state, the frequencies and critical speed are the same as those under the static state, indicating that the effect of the centrifugal force on the frequency counteracts that of the aerodynamic force. Therefore, the vibration frequencies of the blade with the effect of gravity only are calculated to prevent the resonance.

Fig. 12Force diagrams of the blade under the static, single pre-stress and multiple pre-stresses states

a) Static state

b) Single pre-stress state

c) Multiple pre-stresses state

Fig. 13 only gives the mode shapes of the original blade under the static state since the original and optimized blades with and without pre-stress have the same mode shapes. The first-order mode shape is the first-order torsional vibration, the second-order mode shape is the first-order flapping and shimmy, the third- and fifth-order mode shapes are the second-order flapping and first-order shimmy, the fourth- and sixth-order mode shapes are the second-order torsional vibration and first-order shimmy, and the trailing-edge deformation is obvious along the span-wise.

Fig. 13Mode shapes of the original and optimized blades

a) First-order

b) Second-order

c) Third-order

d) Fourth-order

e) Fifth-order

f) Sixth-order

4.2. Transient analysis

Table 4 and Fig. 14 give the maximum values and distributions of the displacement, stress and strain for the original and optimized wind wheels under different wind speeds. In Table 4, the maximum displacement, stress and strain at the rated wind speed decrease by 2.00 %, 5.05 % and 23.77 % after the blade optimization, respectively, and so are 7.92 %, 6.16 % and 12.18 % at the extreme wind speed. The percentage decreases in the displacement and stress at the extreme wind speed are greater than those at the rated wind speed. In Fig. 14, the maximum displacement occurs at the trailing-edge of the blade, and the maximum stress and strain are at the blade-arm connections. These results indicate that the deformation- and damage- resistant capacities improve in comparison with the original wind wheel.

Fig. 14Displacement, stress and strain distributions of the original and optimized wind wheels

a) Original wind wheel at the rated wind speed

b) Optimized wind wheel at the rated wind speed

c) Original wind wheel at the extreme wind speed

d) Optimized wind wheel at the extreme wind speed