**2.3.4.1 Vortex releasing model**

The variation of the lifting surface position in time induces variation of circulation around the lifting surface as well. According to the Kelvin theorem, this variation in circulation must also induce the variation around the wake. According to the unsteady Kutta condition, this can be achieved by successive releasing of the vortices in the airflow.

Fig. 15. Vortex releasing

Suppose that the lifting surface has been at rest until the moment t, when it started with the relative motion with respect to the undisturbed airflow. The vortex releasing, as a way of circulation balancing, is done continually, and in such a way a vortex surface of intensity

The wake influence at large distances from the blade is negligible, so it is possible to neglect the wake distortion at a sufficient distance from the rotor. The number of revolutions for

 is the advance ratio. After that, the wake shape is "frozen" in achieved state, and it moves by flow field velocity, keeping it for the rest of the time. The "frozen" part of the wake still influences the adjacent area in which it is still being distorted. After some distance, even the influence of the "frozen" part becomes negligible, and then it is eliminated from the model.

The boundary condition of impermeability of the lifting surface should be satisfied at any

0 12 ; ,, , *VV n i n i Ti i*

Points at which this condition must be satisfied are called the control points. One of them is placed on each panel, at the three-quarter chord panel positions. By this, at every moment of time, the number of lifting surface impermeability conditions is equal to the number of unknown values of circulations of bound vortices. The equations of motion

free stream velocity and perturbation velocity: *VV w i i* . The perturbation velocity is induced by lifting surface and wake vortex elements. It is calculated by Biot-Savart law. At every moment, the wake shape and circulations of its vortex lines are known, and so the wake-induced velocity at every flow field point is known as well. On the other hand, the circulations of the bound vortices are unknowns (their positions are defined by the lifting surface shape). The boundary condition for the i-th control point can be written

*i j ji i*

*a tb*

where *<sup>i</sup>*, *<sup>j</sup> a* are the coefficients depending of the blade geometry, and *<sup>i</sup> b* are the coefficients containing the influence of the wake and free-stream flows. This way, by writing equations

Along with this equation set, the Kutta condition must be satisfied. In case of numerical solutions, it is customary to satisfy Kutta condition in vicinity of the trailing edge.

<sup>0</sup> *<sup>n</sup>*

the trailing edge panel cord length. By adding the Kutta conditions to the equation set, an over-determined equation set is obtained. It can be reduced to the determined system by the

*tt t*

1 , *n*

for all control points, the equation set of the unknown bound circulations is obtained.

*i*

According to that, we obtain discretized form of unsteady Kutta condition:

*n*

*l t* 

where *n n l* is intensity of the distributed vorticity at the trailing edge panel *<sup>n</sup>*

*V*

as well. At each flow field point, velocity can be divided to the

, where

of all characteristic points,

, and *<sup>n</sup>*

*l* is

which it is necessary to calculate the wake distortion can be determined as *m* 0 4.

moment of time, *t k <sup>k</sup>* 012 , , in a finite number of points of lifting surface

of the lifting surface are known, as well as the velocities*VT i*

**2.3.4.3 Definition of equation set** 

and their normals *ni*

as:

 *t* is formed. At the next moment *t t* , the flow model will look like in Figure 15. The circulation of the vortex element joined to the trailing edges is equal to the difference in circulations at moments *t t* and *t*.

Fig. 16. Unsteady panel scheme

We will discretize the vortex "tail" by replacing it with the quadrilateral vortex loop, whose one side is at the trailing edge, and the opposite side is at the finite distance from the trailing edge (shed vortex). By this, we can obtain the final model for unsteady case.

### **2.3.4.2 Discretization of wake**

The established vortex-releasing model is appropriate for the wake modeling using the "free wake" approach. During the time, by continuous releasing of the quadrilateral vortex loops, the vortex lattice formed of linear trailed and shed vortices is created. The wake distortion is achieved by altering the positions of node points in time, by application of kinematics relation *rt t rt Vt t i ii* .

The velocities of the collocation points are obtained as sums of the undisturbed flow velocity and velocities induced by other vortex elements of the flow field. Induced velocities are calculated using the Biot-Savart law. In order to avoid the problems of velocity singularities, line vortex elements are modeled with core. The core radius varies with the gradient of the bound circulation at the position where vortex is released.

Fig. 17.Discretized wake

 *t* is formed. At the next moment *t t* , the flow model will look like in Figure 15. The circulation of the vortex element joined to the trailing edges is equal to the difference in

We will discretize the vortex "tail" by replacing it with the quadrilateral vortex loop, whose one side is at the trailing edge, and the opposite side is at the finite distance from the trailing

The established vortex-releasing model is appropriate for the wake modeling using the "free wake" approach. During the time, by continuous releasing of the quadrilateral vortex loops, the vortex lattice formed of linear trailed and shed vortices is created. The wake distortion is achieved by altering the positions of node points in time, by application of kinematics

The velocities of the collocation points are obtained as sums of the undisturbed flow velocity and velocities induced by other vortex elements of the flow field. Induced velocities are calculated using the Biot-Savart law. In order to avoid the problems of velocity singularities, line vortex elements are modeled with core. The core radius varies with the gradient of the

edge (shed vortex). By this, we can obtain the final model for unsteady case.

circulations at moments *t t* and *t*.

Fig. 16. Unsteady panel scheme

**2.3.4.2 Discretization of wake** 

relation *rt t rt Vt t i ii* .

Fig. 17.Discretized wake

bound circulation at the position where vortex is released.

The wake influence at large distances from the blade is negligible, so it is possible to neglect the wake distortion at a sufficient distance from the rotor. The number of revolutions for which it is necessary to calculate the wake distortion can be determined as *m* 0 4. , where is the advance ratio. After that, the wake shape is "frozen" in achieved state, and it moves by flow field velocity, keeping it for the rest of the time. The "frozen" part of the wake still influences the adjacent area in which it is still being distorted. After some distance, even the influence of the "frozen" part becomes negligible, and then it is eliminated from the model.

#### **2.3.4.3 Definition of equation set**

The boundary condition of impermeability of the lifting surface should be satisfied at any moment of time, *t k <sup>k</sup>* 012 , , in a finite number of points of lifting surface

$$\left(\vec{V}\_i - \vec{V}\_{Ti}\right) \cdot \vec{n}\_i = 0; \quad i = 1, 2, \dots, m$$

Points at which this condition must be satisfied are called the control points. One of them is placed on each panel, at the three-quarter chord panel positions. By this, at every moment of time, the number of lifting surface impermeability conditions is equal to the number of unknown values of circulations of bound vortices. The equations of motion of the lifting surface are known, as well as the velocities*VT i* of all characteristic points, and their normals *ni* as well. At each flow field point, velocity can be divided to the free stream velocity and perturbation velocity: *VV w i i* . The perturbation velocity is induced by lifting surface and wake vortex elements. It is calculated by Biot-Savart law. At every moment, the wake shape and circulations of its vortex lines are known, and so the wake-induced velocity at every flow field point is known as well. On the other hand, the circulations of the bound vortices are unknowns (their positions are defined by the lifting surface shape). The boundary condition for the i-th control point can be written as:

$$\sum\_{i=1}^{n} a\_{i,j} \Gamma\_{ji} \left( t \right) = b\_{ji}$$

where *<sup>i</sup>*, *<sup>j</sup> a* are the coefficients depending of the blade geometry, and *<sup>i</sup> b* are the coefficients containing the influence of the wake and free-stream flows. This way, by writing equations for all control points, the equation set of the unknown bound circulations is obtained.

Along with this equation set, the Kutta condition must be satisfied. In case of numerical solutions, it is customary to satisfy Kutta condition in vicinity of the trailing edge. According to that, we obtain discretized form of unsteady Kutta condition:

$$V\_{\Leftrightarrow} \frac{\Gamma\_n}{l\_n} + \frac{\tilde{\Gamma}\left(t + \Delta t\right) - \tilde{\Gamma}\left(t\right)}{\Delta t} = 0$$

where *n n l* is intensity of the distributed vorticity at the trailing edge panel *<sup>n</sup>* , and *<sup>n</sup> l* is the trailing edge panel cord length. By adding the Kutta conditions to the equation set, an over-determined equation set is obtained. It can be reduced to the determined system by the

By analyzing the drawing of the blade wakes, it can be concluded that model applied in this paper gives reasonable simulation of actual wake behavior, specially in the domain of wake boundaries, where wake roll-up occurs (although it is slightly underestimated compared with existing experimental data). In addition, larger wake distortion in the domains of the

Fig. 19. Circulation distribution on the rotor disk; =0.35 (different points of view)

The program results (Figure. 19), show the difference in circulation distributions at different azimuths, as well as the disturbances caused when blades are passing the wakes of other

Obtained results can define suggestions for the future solution improvement. Firstly, by incorporating the transonic flow calculations, the advancing blade tip simulation would be more appropriate. Secondly, viscous interaction should be included as well, which would improve the wake roll-up simulation. The viscous vortex core simulation in time would improve the results concerning the wake vanishing effects far enough from the blade. Finally, introduction of the curvilinear vortex elements would give better results from the aspect of wake self-induction. With these enhancements, this solution should prove to be a

The flight dynamics mathematical model of an aircraft, which would be strictly determined, would comprise of a system of non-linear, non-stationary, partial differential equations. To simplify these equations we introduce a number of assumptions. Ignored are the elastic characteristics of the aircraft so the aircraft can be thought as a rigid body and, in that way, the dispersal of parameters is eliminated. Also, fuel consumption is disregarded and so the

The helicopter is specific in regards to other traffic-transportation means, not just by its structure but also by its motion possibilities. The helicopter can move vertically, float in the air,

forerunning blades or their wakes is noticeable.

blades, and the characteristic reversal flow domains.

useful and efficient tool to the rotorcraft performance evaluation.

non-stationarity due to temporal change in helicopter mass is eliminated.

**3.1 Mathematical models of helicopter flight dynamics** 

**2.3.6 Conclusion** 

**3. Flight dynamics and control** 

method of least squares. After that, it can be solved by some of the usual approaches, by which the unknown values of circulations *i* at the time *t* are obtained.
