**3.2 Air flow interaction with three-dimensional objects**

A four-ray star prism's interaction with air flow was simulated. A diagram for pressure distribution around this prism at a supersonic velocity of 1.8 Mach (equivalent to 612.5 m/s) is presented in **Figure 18**.

As it is seen from the diagram presented (**Figure 18**), even at high supersonic speeds, the pressure in the boundary layer of the suction zone is visually constant. This confirms the opportunity to apply the above considered analytical formulas for the analysis of flow-prism interaction at supersonic velocities.

Air flow interaction with full and perforated flat plates was studied in order to find out the physical nature of air medium in the suction zone. The distributions of streamlines in the suction zone for full and perforated plates are shown in **Figures 19** and **20**.

As it is seen from the diagram for the full flat plate (**Figure 19**), the vortices and bubbles are formed behind the plate in the suction zone. But for the perforated plate (**Figure 20**), the nature of the air flow in the suction zone is changed fundamentally. There is no vortices and bubbles downstream of the plate. This property should be

#### **Figure 16.**

*Pressure distribution around the rhombic prism 5x5.*


**Figure 17.** *Pressure distribution around the triangle prism 60x0.*

#### **Figure 18.**

*Pressure distribution around the star prism at supersonic velocity.*

#### **Figure 19.**

*Streamlines distribution around the full flat plate.*

**Figure 20.** *Streamlines distribution around the perforated flat plate.* taken into account in analytical calculations by the reducing air flow interaction constant *C* in the suction zone.

The results of numerical modeling confirm that air flow and rigid body interaction phenomena can be analyzed within two completely different zones: the pressure zone and the suction zone. It was shown that pressure in the suction zone along the entire boundary layer is constant. In the pressure zone, the interaction has an analytical relationship, but in the suction zone, it is possible to supplement the formula with a constant parameter *C*. It is found that for the velocity of 10 m/s, the constant parameter for two-dimensional modeling is *C* = 0.5, but for three-dimensional modeling, it is reduced to about *C* = 0.25.

### **4. Experimental investigations in wind tunnel**

Experiments were carried out in the Armstrong Subsonic wind tunnel, available at Riga Technical University. The main specifications of the wind tunnel can be found in [9].

#### **4.1 Experiments with full flat plate**

The object of study is a square flat plate with dimensions 0.159 x 0.159 m, which is about two times less than the dimension of the tunnel working section (0.304 m). The drag force is measured using the concept of balanced weights. The schematic diagram of experimental installation and process parameters is shown in **Figure 21**.

*V*<sup>0</sup> is air flow velocity; *VN* is a normal component of air flow velocity; β is an angle of plate's normal position against the flow direction; *Fx* and *Fy* are horizontal and vertical components of the air interaction force; *L*<sup>1</sup> is a length of the square plate's edge; *L*<sup>2</sup> = 0.005 m is a thickness of the plate.

The main purpose of the experiment was to test the applicability of analytical formulas for calculation of drag force *Fx* (horizontal component of air interaction force). Experimental interconnection between drag force *Fx* and angle of attack (90° + β) is graphically presented in **Figure 22** (results are obtained for the constant air flow velocity *V*<sup>0</sup> = 10 m/s).

**Figure 21.** *Schematic diagram of experiment and process parameters.*

**Figure 22.** *Experimental results for drag force* Fx *for the full flat plate.*

Analytically drag force *Fx* for a flat plate interacting with air flow can be determined by the formula [8]

$$\mathbf{F\_x} = \frac{\mathbf{H(\boldsymbol{\beta})} \bullet \mathbf{B} \boldsymbol{\rho}}{2} \bullet \left[ \mathbf{C} + \frac{(\cos \boldsymbol{\beta})^3 + (\sin \boldsymbol{\beta})^3}{\cos \boldsymbol{\beta} + \sin \boldsymbol{\beta}} \right],\tag{29}$$

where *H*ð Þ¼ *β L*<sup>1</sup> cos *β* þ *L*<sup>2</sup> sin *β*.

Analytical curves *Fx* ¼ *f*ð Þ *β* , constructed by Eq. (29) at the three different values of constant *C* (0.125; 0.25; 0.50), are presented in **Figure 23**. For comparison, experimentally measured values of forces *Fx* are shown on this diagram, too.

**Figure 23.** *Comparison of analytical and experimental results for drag force* Fx*.*

Theoretical curves *Fx* ¼ *f*ð Þ *β* agree qualitatively well with the experimental data (**Figure 23**), i.e., the curves have the same shape. But the quantitative difference is satisfactory and lies within the range from 12–25%. Such difference could be explained by the limited cross-sectional dimensions of the wind tunnel (0.304 x 0.304 m) in comparison with plate dimensions (0.159 x 0.159 m), as well by the operation principle of the tunnel (not pressing, but suction principle). Therefore, it has been experimentally proved that obtained analytical formulas can be used in air flow interaction calculations (tasks of analysis, optimization, and synthesis).

#### **4.2 Experiments with perforated flat plate**

Experiments were held with a perforated flat plate shown in **Figure 24**. During experiments, different orientations of perforated grooves were used: horizontal (as in **Figure 24**) and vertical. The velocity of air flow was constant and equal to 10 m/s.

Experimental interconnection between drag force *Fx* and angle of attack (90° + β) is graphically presented in **Figure 25**. Curves *Fx* ¼ *f*ð Þ 90° þ *β* are constructed for the plates with horizontal (H) and vertical (V) orientations of perforated grooves. Additionally, results of analytical calculations of drag force *Fx* by formula (19) are shown (for the perforated plate with vertical grooves, assuming *C* = 0.5).

On the analysis of experimental results (**Figure 25**), it can be concluded that drag force *Fx* is always higher if grooves are oriented horizontally. This could be explained by the fact that there is an additional air flow interaction with the edges of perforated horizontal grooves. But in the horizontal position of plates (under the β = 90°), drag forces are the same both in the vertical and horizontal grooves orientation (and equal to the drag force for the full plate *Fx* = 0.2 N, see **Figure 22**). This is well understood because the perforation in both plates is covered, if β = 90°.

**Figure 24.** *The geometry of the perforated flat plate (all dimensions are in mm).*

**Figure 25.** *Experimental values of drag force* Fx *for the perforated flat plate.*

Results of analytical calculations of drag force *Fx* by formula (19) agree well with experimental data (see **Figure 25**). Therefore, the mathematical model obtained for perforated plate can be successfully used in air flow interaction calculations.

### **5. Models of wind energy conversion devices**

The above results of the theoretical and experimental analysis are used in the designing of new devices for energy extraction from air flow. Models of wind energy conversion devices equipped with vibrating plates (disks) are developed.

#### **5.1 Wind energy conversion device equipped with rotating perforated disk**

A new model of wind energy conversion device equipped with working head made from two concentric circular flat plates (disks) with alternate flow sectors is synthesized (**Figure 26**). Disks are connected to each other at the center. Besides, the disk whose front area is subjected to the action of air flow has an ability to rotate freely over the other circular non-rotating disk. Both disks have the same surface area and identical sector perforations (holes). During the rotation of one disk, perforations are cyclically opened and closed, and due to this equivalent surface area of the working head is periodically changed in accordance with the given control action [10].

*V* is air flow velocity; *x* is a displacement of the disk in its translation motion; -*bVx* is a force of linear generator; �*cx* is the elastic force of a spring; *ω*<sup>0</sup> is an angular velocity of rotating disk.

Control action for the variation of perforated disk's surface area *A* can be given in the following form:

$$A = A\_0 \left\{ 1 + \frac{\cos^{-1}[\cos(\alpha\_0 t)]}{\pi} \right\},\tag{30}$$

where *A*<sup>0</sup> is a medium surface area of the disk per its one cycle; ω<sup>0</sup> is an angular frequency of harmonic control action. Area *A* variation function (30) graphically is shown in **Figure 27**.

*Methods and Devices for Wind Energy Conversion DOI: http://dx.doi.org/10.5772/intechopen.103120*

**Figure 26.**

*Model of wind energy conversion device with rotating perforated disk.*

**Figure 27.** *Control action by the variation of area* A *of perforated plate.*

Translation motion of the perforated plate in the direction of *x*-axis (**Figure 26**) under the control action (30) is described by the following differential equation:

$$\begin{split} m\ddot{\mathbf{x}} &= -c\mathbf{x} - \left[ F\_0 \text{sign}(\dot{\mathbf{x}}) - b\dot{\mathbf{x}} \right] + (\mathbf{1} + \mathbf{C}) \frac{\mathbf{A}}{\pi} \cdot \left[ \cos^{-1}(\cos \omega\_0 t) + \pi \right] \\ &\cdot \rho (-V\_0 - \dot{\mathbf{x}})^2 \cdot \text{sign}(-V\_0 - \dot{\mathbf{x}}), \end{split} \tag{31}$$

where *m* is a mass of perforated plate; *c* is stiffness coefficient of spring; *F*<sup>0</sup> and *b* are constants of linear damping generator; *C* is an interaction coefficient between air flow and plate; *V*<sup>0</sup> is an air flow velocity; ω<sup>0</sup> is an angular frequency of harmonic control action; *A* is a constant surface area of the plate; ρ is air density.

By the simulation with program Mathcad of disk motion under the Eq. (31), the optimization task was solved. It is shown that maximal power *P* through disk interaction with air flow is generated under the resonant condition *<sup>ω</sup>*<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi *c=m* p . The graph of generated power *P* versus time *t* for the *V*<sup>0</sup> = 10 m/s is shown in **Figure 28**.

As it is seen from the analysis of the graph presented (**Figure 28**), a stationary oscillatory regime with maximal generated power *P* can be achieved after some cycles of a transient process.

**Figure 28.** *Power generated due to air flow interaction with a perforated disk.*

#### **5.2 Air flow generator on the base of a closed track conveyor**

The principal model of the wind energy conversion generator synthesized on the base of a closed track conveyor is shown in **Figure 29**. Closed track conveyor 1 forms a Central part of the generator, besides the track has an ability to move parallel to coordinate plane *x*0*y*. The conveyor is driven by an air flow with velocity *V*0, acting on blades 2 in parallel to the 0*z* axis.

Power is obtained from a generator connected with rotor 3 at the one end (left or right) of conveyor 1 (**Figure 29**). Flat blades 2 is attached tightly to conveyor 1 with a rigid fastening element 4 (welded hinge). Besides, blades 2 are fixed at the angle α toward the *x*-axis. The model of generator has several flat blades 2. Due to the action of air flow *V*0, the translation motion of blades 2 along conveyor's straight and circular sections (in final turns) is excited.

The three-dimensional design of air flow generator made with the program Solid Works is shown in **Figure 30**.

The generation of useful power in the proposed device (**Figures 29** and **30**) is due to the translation movement of the flat blades. Therefore, the wind flow load is uniformly distributed over the lateral surface of the flat blades. This provides a simple way to increase the operational efficiency of the device, which can be achieved by increasing the area *A* of the blade's lateral surface.

#### **5.3 Air flow generator on the base of vibrating flat blade and crank mechanism**

The model of the developed wind energy conversion device is shown in **Figure 31**. Flat blade 1 is a main element of the device, and it is attached to the rotating axle 2 by a cylindrical axial hinge. And symmetry axis *z*<sup>1</sup> of blade 1 simultaneously is a longitudinal axis of axle 2. Besides, rotating axle 2 is rigidly attached to slider 3, which has the ability of translation motion along the *x*-axis. Additionally, the translation motion of slider 3 is limited by elastic springs 4 and shock absorbers 5, but turning of the blade 1 around axis *z*<sup>1</sup> is restricted by a torsional spring 6 and a rotary shock absorber 7. The crank 8 is rigidly attached to the flat blade 1 perpendicular to its side surface. Additionally, there is a connecting rod 9, which opposite ends are hinged to the crank 8 and slider 10 of an electric generator. And slider 10 has the ability to move inside the electric coil 11 along the *x*<sup>1</sup> axis.

*Methods and Devices for Wind Energy Conversion DOI: http://dx.doi.org/10.5772/intechopen.103120*

#### **Figure 29.**

*Principle model of air flow generator on the base of track conveyor: 1 – closed track conveyor; 2 – flat blade; 3 – rotor; 4 – rigid fastening element.*

**Figure 30.** *Three-dimensional design of the air flow generator.*

The operation of the wind energy conversion device starts from the position shown in **Figure 31**. It is assumed that wind flow has a speed of *V*<sup>0</sup> and is directed perpendicular to the *x*-axis. Due to the effect of wind flow on the side surface of the flat blade 1, a force *N* is formed in the direction of normal *n* (**Figures 31** and **32**). The action of the force *N* causes slider 3 to move to the right along *x*-axis. As a result, compressive force *Fk* is formed in the connecting rod 9.

The force *Fk* of the connecting rod 9 acts on the linear generator, consisting of a slider 10 and a built-in electric coil 11. This force holds the rotating flat blade 1 at the left rotary shock absorber 7 (**Figures 31** and **32**). At this device position, the turning angle α reaches the maximum value (α<sup>0</sup> clockwise). Then spring 4 is stretched, and the right shock absorber 5 is deformed until slider 3 stops in the right extreme position.

Then, under the action of the elastic forces, slider 3 moves back to the left. At the beginning of this translational movement, the connecting rod 9 is tensioned, and,

#### **Figure 31.**

*Principle model of the wind energy conversion device: 1 – flat blade; 2 – rotating axle; 3 – slider; 4 – spring; 5 – shock absorber; 6 – torsional spring; 7 – rotary shock absorber; 8 – crank; 9 – connecting rod; 10 – slider of the linear generator; 11 – electric coil.*

#### **Figure 32.**

*Two extreme stopping positions of the blade during operation of the device: 1 – flat blade; 4 – spring; 9 – connecting rod; 10 – slider of linear generator.*

consequently, the force *Fk* acts in the opposite direction. The flat plate 1 rotates counterclockwise about the symmetry axis *z*<sup>1</sup> and reaches the right rotary shock absorber 7. At this position, the angle of rotation α reaches a maximum value (α<sup>0</sup> counterclockwise). As a result, the normal *n* to the flat blade changes its position against the wind flow *V*0. Therefore, a new force *Fk* pushes the flat blade 1 in the direction opposite to the *x*-axis. As a result, slider 3 moves to the left, deforms spring 4 and also the left shock absorber 5 until slider 3 stops in the left extreme position. The cycle then repeats as the compressive force *Fk* again begins to interact with the connecting rod 9.

During the generated cyclic movement, the generator's slider 10 moves backward inside the electric coil 11 along the *x*<sup>1</sup> axis. As a result, electrical energy is produced in the generator (alternating current is generated in the electric coil 11). This dynamic operational principle of a linear generator has been described in the literature [11].
