**2. New approach to the air flow interaction with a moving rigid body**

The main focus of the present work is to investigate the stationary air flow interaction with rigid body and extend the interaction concept for non-stationary body-air flow interaction without requiring "space–time" programming techniques [5]. In accordance with the concept proposed, a space around rigid body interacting with the air medium is split into several zones (see **Figure 1**).

It has been found in theory and practice that the non-stationary interaction of air flow can be divided into two parts using the principle of superposition. For this purpose, the interaction can be considered within two zones: the frontal pressure zone and the rear intake zone. In addition, it is possible to separate slow movements from fast air particle movements (i.e., from Brownian chaotic particle movements).

Hereinafter, this approach is used to study air flow interaction with flat blades and space prisms that perform translation motion.

#### **2.1 Object interaction with a windless air medium**

The model of moving rigid body interaction with air medium is shown in **Figure 2**.

**Figure 1.** *Concept of zones (pressure and suction zones) for a rigid body immersed in an air flow.*

**Figure 2.** *Interaction of the rectilinearly translationally moving body (prism) with a windless air medium.*

By applying the theorem of momentum change in the differential form [7] to a very small air element in the pressure zone and accordance with the superposition principle, the following system of equations can be received in the projection on the area normal *n*<sup>1</sup> before and after collisions (air–body interaction), taking account of Brownian motion:

$$\mathbf{m}\_{10} \cdot \mathbf{V} \mathbf{B}\_1 - (-\mathbf{m}\_{10} \cdot \mathbf{V} \mathbf{B}\_1) = -\mathbf{N}\_1 \cdot \mathbf{d}t,\\ \mathbf{m}\_{10} = \mathbf{V} \mathbf{B}\_1 \cdot \mathbf{d}t \cdot \mathbf{d} \mathbf{L}\_1 \cdot \mathbf{B} \cdot \rho \tag{1}$$

$$\mathbf{p}\_{10} = \frac{|\mathbf{N}\_1|}{\mathbf{d} \mathbf{L}\_1 \cdot \mathbf{B}},$$

where *m*<sup>10</sup> is Brownian interaction mass; *VB*<sup>1</sup> is an average value of air normal velocity within the pressure zone; *N*<sup>1</sup> is a force directed along a normal to a small element of air medium; d*t* is an infinitely small-time interval; d*L*<sup>1</sup> is a width of a small element; *B* is a prism height in the direction perpendicular to the plane of motion; *ρ* is air density; *p*<sup>10</sup> is atmospheric pressure in the pressure zone.

Considering body and air interaction at the windward side (pressure side), the following system of equations can be formed:

$$m\_1 V \cos \left(\beta\_1\right) - 0 = -\Delta N\_1 \cdot dt,\\ m\_1 = V \cos \left(\beta\_1\right) \cdot dt \cdot dL\_1 \cdot B \cdot \rho,\tag{2}$$

$$\Delta p\_1 = \frac{|\Delta N\_1|}{dL\_1 \cdot B},$$

where *m*<sup>1</sup> is a mass due to prism interaction with air in boundary layer; *V* is a velocity of prism; *β*<sup>1</sup> is an angle between velocity *V* and normal *n*1; *ΔN*<sup>1</sup> is an additional normal force acting on a prism; *Δp*<sup>1</sup> is an increment of pressure in the windward side.

By the solution of the system of Eqs. (1) and (2), six unknown parameters can be found. From the practical point of view, the most required are parameters *p*<sup>10</sup> and *Δp*1, which can be determined by the following calculations:

$$\mathbf{p}\_{10} = \mathbf{2} \cdot \mathbf{V} \mathbf{B}\_1^2 \cdot \boldsymbol{\rho} \cdot \mathbf{dt},\tag{3}$$

$$
\Delta p\_1 = \rho \cdot dt \cdot \mathbf{V}^2 \left[ \cos \left( \beta\_1 \right) \right]^2. \tag{4}
$$

Besides, it is possible to apply a mathematical model similar to Eq. (1)–(4) in the suction zone (leeward side). However, the task is complicated a little due to the increasing number of momentum differentials in the suction zone. Therefore, it is suggested to find the solution using one or the other hypothesis. Hypotheses should be tested experimentally or by the use of numerical computer programs.

The first hypothesis. In the suction zone, pressure reduction *Δp*<sup>21</sup> over the entire surface is considered as constant and proportional to the square of the velocity *V* in accordance with the following equations:

$$
\Delta p\_{21} = -\rho \cdot \mathbf{C}\_1 \cdot \mathbf{V}^2,\tag{5}
$$

$$p\_{20} = 2\text{VB}\_2^{2} \cdot \rho \cdot dt,\tag{6}$$

where *C*<sup>1</sup> is a constant found according to the experimental or numerical simulation; *VB*<sup>2</sup> is an average air normal velocity in the suction zone.

The second hypothesis. It is assumed that in the suction zone, pressure reduction *Δp*<sup>22</sup> over the entire surface is not constant, but is proportional to the square of the velocity *V* and also depends on the normal *n*<sup>2</sup> to the surface area and position angle *β*2. Thus, the following equations can be obtained:

$$
\Delta p\_{22} = -\rho \cdot \mathbf{C}\_2 \cdot \mathbf{V}^2 \cos \left(\beta\_2\right),
\tag{7}
$$

$$p\_{20} = 2\text{VB}\_2^{2} \cdot \rho \cdot d\text{t},\tag{8}$$

The obtained Eqs. (3)–(8) can be used in the engineering analysis and synthesis tasks in the low-velocity range and for bodies that undergo rectilinear translation motion. For practical engineering calculations, it is recommended to adopt *VB*<sup>1</sup> = *VB*<sup>2</sup> for low-velocity ranges *V* < < *VB*<sup>1</sup> and *V* < < *VB*2. Then it is assumed *p*<sup>01</sup> = *p*<sup>02</sup> = *p*0, where *p*<sup>0</sup> is the mean atmospheric pressure around the given prism.

#### **2.2 Stationary rigid body (prism) interaction with air flow**

The model of airflow interaction with a stationary prism is shown in **Figure 3**.

**Figure 3.** *Model of air flow interaction with stationary prism.*

**Figure 4.** *Model of air flow interaction with moving rigid body.*

Airflow interaction with a stationary prism (**Figure 3**) is dependent on the extra velocity and extra kinetic energy of air particles. However, by applying the interaction concept to relative motion, it is possible to use the Eqs. (3)–(8) in the engineering calculations of systems with air flow velocity.

#### **2.3 Moving rigid body (prism) in an air flow**

The model of air flow interaction with moving prism is shown in **Figure 4**. In this case, the relative motion velocity *Vr* vector in the pressure zone must be recalculated by determining the angle *γ* from the elementary parallelograms with normal directions *n*<sup>1</sup> and *n*<sup>2</sup> (**Figure 4**). By projecting the vectors *V* and *V*<sup>0</sup> onto the *x* and *y* axes, the following formulas are obtained:

$$\mathcal{V}r = \sqrt{\left(-V\_0 \cdot \cos a - V\right)^2 + \left(-V\_0 \cdot \sin a\right)^2},\tag{9}$$

$$\cos \chi = \frac{-V\_0 \cdot \cos a - V}{\sqrt{\left(-V\_0 \cdot \cos a - V\right)^2 + \left(-V\_0 \cdot \sin a\right)^2}},\tag{10}$$

where *Vr* is a relative velocity module; γ is an angle indicating the direction of the vector *Vr* of relative velocity; *V*<sup>0</sup> is a velocity of wind air flow; *V* is a velocity of prism in its rectilinear translation motion; α is an angle indicating the direction of the vector *V*<sup>0</sup> of air flow velocity (see **Figure 4**).

By the use of obtained Eqs. (3)–(10), it is possible to solve various technical problems of air flow and rigid body (prism) interaction. For example, it is possible to solve the problems of energy extraction from an air flow. Besides, body's shape optimization problem can be solved in order to obtain the desired effect along with motion control realization.

#### **2.4 Model of air flow interaction with a perforated flat plate**

Pressure distribution for a flat plate element with a rectangular cross-section is shown in **Figure 5**.

In accordance with the theorem of linear pulse change in the differential form [7], the following equations for the plate's pressure side can be written:

$$d m\_1 \cdot V \cos \beta = d N\_1 \cdot dt,\tag{11}$$

$$dm\_2 \cdot V \sin \beta = dN\_2 \cdot dt,\tag{12}$$

$$dm\_1 = V \cos \beta \cdot dt \cdot dL\_1 \cdot B \cdot \rho,\tag{13}$$

$$dm\_2 = V \sin \beta \cdot dt \cdot dL\_2 \cdot B \cdot \rho,\tag{14}$$

where *dm*1, *dm*<sup>2</sup> are masses of elementary air flow particles with relative velocity *V* against inclined surfaces; *dN*1, *dN2* are elementary impulse forces in the directions of normality toward the surfaces of the elemental area; *β* is an angle between elementary pulse d*N*<sup>1</sup> and air flow; *dt* is an elementary time moment; *dL*1, *dL*<sup>2</sup> are elemental lengths of the surface; *B* is a width of the element, which is considered as constant in the case of a two-dimensional task; *ρ* is a density of air medium.

Using Eqs. (11)–(14), the change in pressure on the sides of the perforated plate can be expressed as follows:

$$
\Delta p\_1 = V^2 \rho \cdot \left(\cos \beta\right)^2; \tag{15}
$$

$$
\Delta p\_2 = V^2 \rho \cdot (\sin \beta)^2. \tag{16}
$$

The suction pressure in a small layer directly along the plate's lower edge is considered as constant and can be expressed with the following equation:

$$
\Delta p\_3 = V^2 \cdot \rho \cdot \mathbb{C}, \tag{17}
$$

where *C* is a constant determined experimentally or by computer modeling [5]. For subsonic velocity flow, the *C* value varies within interval 0 < *C* < 1.

The model of air flow interaction with perforated plate is shown in **Figure 6**. For the length *L*<sup>3</sup> of the perforated gap, the following condition is satisfied:

**Figure 5.** *Pressure distribution for a rectangular element of flat plate.*

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

**Figure 6.**

*Pressure distribution in a cross-section of a rectangular flat perforated plate, where* L*1,* L*<sup>2</sup> are lengths of plate's edges,* L*<sup>3</sup> is a length of the perforated gap.*

$$L\_3 \ge L\_2 \cdot \text{tg}(\beta). \tag{18}$$

Using the laws of classical mechanics for a two-dimensional flat plate [7], interaction force *IFx* in the air flow direction (direction of the x-axis) can be determined by the formula:

$$\text{IFx} = -k \cdot B\_1 \cdot H \cdot V^2 \rho \cdot \left[ C + \frac{(\cos \beta)^3 + d \cdot (\sin \beta)^3}{\cos \beta + d \cdot \sin \beta} \right],\tag{19}$$

where *k* is a total number of elements between perforations; *d* = *L*2/*L*<sup>1</sup> is a ratio of plate edges; *H* ¼ ð Þ *L*<sup>1</sup> cos *β* þ *L*<sup>2</sup> sin *β* is a dimension of the plate's element in the direction perpendicular to air flow. Another notation is the same as in Eqs. (11)–(17).

The mathematical model of perforated plate interaction with air flow is validated by computer simulation with the program Mathcad. Simulation is performed in application to translational motion of two-dimensional perforated plate in air flow with velocity *V*. Plate interacts with a linear spring with stiffness coefficient *c* and a linear damper with damping constant *b* (**Figure 7**).

Following the methods of classical mechanics [7], it is possible to determine relative interaction velocity *Vr* by the formula:

$$V\_r = V + \nu,\tag{20}$$

where *V* is an air flow velocity; *v* is a velocity of a flat plate in the direction of *the x*-axis.

For the plate with the very small thickness (*δ* ≈ 0), the differential equation of its motion along the *x*-axis can be written in the following form:

**Figure 7.** *Model of air flow interaction with perforated plate.*

**Figure 8.** *Motion on phase plane "displacement* x *– velocity* v*".*

$$m\ddot{\mathbf{x}} = -c\mathbf{x} - b\dot{\mathbf{x}} - A\_0(\mathbf{1} - \mathbf{a} \cdot \text{sign}(\dot{\mathbf{x}})) \cdot \boldsymbol{\rho} \cdot \left\{ \mathbf{C} + (\cos \boldsymbol{\rho}\_0)^2 \right\} \cdot (\mathbf{V} + \dot{\mathbf{x}})^2 \frac{\mathbf{V} + \dot{\mathbf{x}}}{|\mathbf{V} + \dot{\mathbf{x}}|}, \tag{21}$$

where *A*<sup>0</sup> is an average value of contact surface area of the plate; *ρ* is the air density; *a* is a constant of area variation; *β*<sup>0</sup> is plate angle against air flow; *m* is mass of the plate; *C* is an air flow and plate interaction constant.

Mathematical simulation of Eq. (21) was performed with program MathCad assuming the following values of main system's parameters: *A*<sup>0</sup> = 0.04 m2 ; *V* = 10 m/s; *ρ* = 12,047 kg/m3 ; *m* = 1.56 kg; *c* = 3061 kg�s �2 ; *b* = 5 kg�s �1 ; *a* = 0.5; *C* = 0.065; *β<sup>0</sup>* = π/6. Results of simulation for the perforated plate translation motion are presented in **Figures 8** and **9**.

From the graphs in **Figures 8** and **9**, it can be concluded, that stable oscillatory movement can be initiated in the aerodynamic system by the variation interaction area of the perforated plate. As it is seen from the analysis of the graph for generated power (**Figure 9**), the almost stationary oscillatory regime with maximal power *P* can be achieved after some cycles of a transient process.

#### **2.5 Model of air flow interaction with a quadrangular convex prism**

An analytical model of a quadrangular convex prism interacting with air flow is shown in **Figure 10**.

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

**Figure 9.** *Power* P *=* b‧v*<sup>2</sup> generated for the given time interval.*

**Figure 10.**

*Model of a quadrangular convex prism:* L*1,* L*2,* L*3,* L*<sup>4</sup> are lengths of edges;* β*1,* β*2,* β*3,* β*<sup>4</sup> are prism's frontal angles;* H *is a height of prism; is a symbol of the direction of air flow on the surfaces of the prism (due to flow-prism interaction).*

By applying the theorem of air flow motion quantity change in the differential form [7], pressures *p*1, *p*2, *p*<sup>3</sup> on frontal planes of the prism (in pressure zone) can be expressed in the following form:

$$p\_1 = (V \cos \beta\_1)^2 \cdot \rho; \qquad p\_2 = (V \cos \beta\_2)^2 \cdot \rho; \tag{22}$$

$$p\_3 = V^2 \rho \cdot \left[ (\cos \beta\_3)^2 - \text{C}\_{12} \cdot \text{C}\_{23} \cdot \cos \beta\_2 \cdot \sin \left(\beta\_3 - \beta\_2\right) \right].$$

where *β*1, *β*2, *β*<sup>3</sup> are angles of lateral orientation of prism sides relative to the air flow; *C*12, C23 are constants for changing the flow rate along with the boundary layer at breaking points of the flow. For example, condition *C*<sup>12</sup> = *C*<sup>23</sup> = 1 means that the speed at the breaking points is not changed and is the same as at the beginning of the entire flow.

Accordingly, the pressure *p*<sup>4</sup> in the suction zone between the two broken edges can be determined by the formula

$$p\_4 = \mathbb{C}\_4 \cdot V^2 \cdot \rho,\tag{23}$$

where *C*<sup>4</sup> is an air flow and prism interaction constant [5].

In calculations, it is necessary to take into account that Eqs. (22) and (23) are applicable to a prism that has curved surfaces in the pressure zone. For example, for the prism shown in **Figure 10**, the following angle relationships must be satisfied: 0 < *β*<sup>4</sup> < *π*/2; 0 < (*β*<sup>3</sup> + *β*4) < *π*; (*β*<sup>3</sup> – *β*2) > 0.

Using the Eqs. (22) and (23), the following projections of the interaction forces on the *x* and *y* axes can be obtained:

$$-F\_{\mathbf{x}} = V^2 \mathbf{B} \boldsymbol{\rho} \cdot \left\{ L\_1 \cdot \left( \cos \beta\_1 \right)^3 + L\_2 \cdot \left( \cos \beta\_2 \right)^3 + L \mathbf{3} \right. \tag{24}$$

$$\cdot \left[ \left( \cos \beta\_3 \right)^3 - C\_{12} \cdot C\_{23} \cdot \cos \beta\_2 \cdot \cos \beta\_3 \cdot \sin \left( \beta\_3 - \beta\_2 \right) \right] + L\_4 \cdot C\_4 \cdot \cos \beta\_4 \}; \tag{25}$$

$$-F\_{\mathcal{V}} = V^2 \mathcal{B} \rho \cdot \left\{ L\_1 \cdot \left(\cos \beta\_1\right)^2 \cdot \sin \beta\_1 + L\_2 \cdot \left(\cos \beta\_2\right)^2 \cdot \sin \beta\_2 + L\_3 \right. \tag{26}$$

$$\cdot \left[ \left(\cos \beta\_3\right)^2 \cdot \sin \beta\_3 - C\_{12} \cdot C\_{23} \cdot \cos \beta\_2 \cdot \sin \beta\_3 \cdot \sin \left(\beta\_3 - \beta\_2\right) \right] + L\_4 \cdot C\_4 \cdot \sin \beta\_4 \right\}, \tag{27}$$

where *Fx* is a resistance force (along the direction of air flow); *Fy* is a lifting force (perpendicular to the direction of air flow).

When analyzing or optimizing forces expressed by Eqs. (24) and (25), the following geometric relationships should additionally be observed:

$$L\_4 \cdot \sin \beta\_4 + L\_3 \cdot \sin \beta\_3 + L\_2 \cdot \sin \beta\_2 - L\_1 \cdot \sin \beta\_1 = 0;\tag{26}$$

$$L\_4 \cdot \cos \beta\_4 - L\_3 \cdot \cos \beta\_3 - L\_2 \cdot \cos \beta\_2 - L\_1 \cdot \cos \beta\_1 = 0. \tag{27}$$

#### **2.6 Model of air flow interaction with a quadrangular concave prism**

An analytical model of a quadrangular concave prism interacting with air flow is shown in **Figure 11**.

#### **Figure 11.**

*Model of a quadrangular concave prism:* L*1,* L*2,* L*3,* L*<sup>4</sup> are lengths of edges;* β*1,* β*2,* β*3,* β*<sup>4</sup> are prism's frontal angles;* H *is a height of the prism.*

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

In this case, the air flow impact force *N*<sup>3</sup> acts on the concave edge of the prism. The force *N*<sup>3</sup> is perpendicular to the edge with the length *L*3, as shown in **Figure 11**. According to the boundary air flow motion change, when the direction of flow is varied from edge *L*<sup>2</sup> to edge *L*3, the impact force *N*<sup>3</sup> is as follows:

$$\mathbf{N}\_3 = \mathbf{L}\_2 \mathbf{B} \mathbf{p} \mathbf{V}^2 \cdot \cos \boldsymbol{\upbeta}\_2 \cdot \sin \left(\boldsymbol{\upbeta}\_2 - \boldsymbol{\upbeta}\_3\right) \cdot [\mathbf{0}.\mathbf{5} + \mathbf{0}.\mathbf{5} \cdot \operatorname{sign}(\boldsymbol{\upbeta}\_2 - \boldsymbol{\upbeta}\_3)].\tag{28}$$

In the case of the concave prism, the following criterion additionally must be satisfied: sin *β*<sup>2</sup> � *β*<sup>3</sup> ð Þ≥ 0*:* Eqs. (24) and (25) remain valid, only negative members should be excluded, since this part of the interaction is equivalent to *N*3.

The resulting relationships (24), (25), (28) make it possible to analyze interactions of air flow with various prismatic forms, solving the tasks of optimization and synthesis.

**Figure 12.** *Results of optimization for* C*<sup>4</sup> = 0.5.*

**Figure 13.** *Results of optimization for* C*<sup>4</sup> = 0.25.*

#### **2.7 Example of shape optimization for a quadrangular prism**

Problem of prism shape (**Figure 10**) optimization is solved by computer simulation. The optimization criterion is resistance force *Fx* in accordance with Eq. (24) and taking account of limitations given by Eqs. (26) and (27). It was assumed that sides *L*<sup>2</sup> and *L*<sup>3</sup> are equal to the constant height *H* but for simplifications *β*<sup>1</sup> = 0 and *β*<sup>4</sup> = 0. Parameters *V*, *ρ*, *B* remained constant (were not varied).

Results of optimization for the criterion *<sup>K</sup> <sup>β</sup>*<sup>2</sup> ð Þ¼ *Fx<sup>=</sup> <sup>V</sup>*<sup>2</sup> *BρH* , which is a resistance coefficient in the direction of air flow, are presented in **Figures 12** and **13**.

As it is seen from the diagrams presented (**Figures 12** and **13**), it is possible to maximize or minimize a resistance force *Fx* by the variation of angle β2. Qualitatively the same results are obtained for two different values of flow rate constant *C*4.

### **3. Computer simulation of air flow interaction with simple form prisms**

The interaction theory discussed above has been tested in computer modeling. Two-dimensional and three-dimensional problems are considered [8].

#### **3.1 Air flow interaction with two-dimensional objects**

Air flow interaction with rhombic and triangular prisms of various shapes was studied by numerical modeling. The aim of the study was to find out the reliability of the formulas obtained in the previous section in the description of air flow interaction with objects. The studied two-dimensional objects are shown in **Figures 14** and **15**.

The multiplication numbers under the prism drawings (**Figures 14** and **15**) indicate the position of the prism side (in angular degrees) relative to normal against the flow in both the pressure and suction zones. Software ANSYS Fluent was used to perform the numerical simulations. All the simulations were made assuming a constant air speed of 10 m/s.

**Figure 15.**

*Parameters of computer-studied triangle section prisms.*

Results of numerical simulation are presented in **Figures 16** and **17** in the form of diagrams for pressure distribution around rhombic and triangles prisms.

Pressure and suction zones around prisms are shown in diagrams (**Figures 16** and **17**). Pressure distribution around the prisms is presented using different colors. As it is seen, the color of the suction zone is almost invariable. Therefore, it can be concluded that pressure in the boundary layer practically is almost constant.
