**4. Results**

To illustrate the comparison of result, we distribute the induced velocity in the plane of the rotor disk and at a far distance from it (**Figure 5**). This confirms the results of the Momentum theory: the inductive velocities in the plane of the rotor disk are two times less than the inductive velocities at an infinite distance from it [5].

The tip vortices structure of the rotor shown in **Figure 6**. Comparison of the structure under the main rotor with the blade theory results shows an adequate behavior of the vortex surface in modeling.

Introducing the air flow configuration, we can see which areas of the helicopter are influenced by the induced flow and used in the analysis of information corresponding loads.

**Figure 7** shows a comparison of the position of blade vortex theory (disorderly line) and disk theory (black line) for horizontal flight. The tip vortex is shown only from one blade, but the influence of all blades is taken into account.

The results of calculating the normal component of induced velocity were compared with experimental data for forward flight from 75 to 180 km/h. Comparison with the experiment gave good results (**Figure 8**). Experimental and calculated data have got an adequate correlation.

**Figure 5.**

*Normal component of induced velocity on hovering α* ¼ 0°, *μ* ¼ 0,*Ct* ¼ 0*:*01*.*

For a single arbitrary point, this will be a matrix-string for the main rotor and a matrix-string for the tail rotor. Multiplying the row by the column of known air circulations on the disk, we calculate the induced velocities at the

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity*

At different flight speeds, induced velocities have different effects on the tail rotor and stabilizer. In the figures, you can see that induced velocity have a special influence on the low flight speeds mode of *V* ¼ 60 km/h reach 4 m/s (**Figures 7** and **9**). This is 24% of the flight speed. Therefore it is very

important to know and take into account the field of induced velocity's at low

The horizontal and vertical components of induced velocity at a characteristic point in the tail rotor and stabilizer area significantly depend on the flight speed

By drawing the configuration of the air flow, we can see which areas of the helicopter are affected by the induced flow. This allows us to correctly configure the

**Figure 10** shows the position of the tip vortex according to the blade theory

(disordered line) and the disk theory (black line) for forward flight.

The values of the influence matrix elements depend on only the rotor geometric characteristics and the position of the selected point relative to the rotor and the angle of inclination *δ* of the vortex cylinder. In the process of calculations, the values of linear circulations and their corresponding induced velocities are refined. Influence matrices must be built in advance for a possible range of angles. During the calculation process, interpolate between the calculated matrices by the neces-

selected point.

*Induced velocity component in the tail rotor area.*

*DOI: http://dx.doi.org/10.5772/intechopen.93186*

**Figure 7.**

sary values *δ*.

flight speeds.

**111**

(**Figures 7** and **9**).

loads calculating program.

**Figure 6.** *Tip vortex in hovering.*

The application of the described method consists in forming a matrix of influence of the main and tail rotor for any group of points around the rotor. For example, a matrix of influence of the rotor on the fuselage, the matrix of influence of the main rotor on the tail rotor, the matrix of influence of the main rotor on the stabilizer, the matrix effect of the tail rotor on the main rotor, etc., the dimensions of the matrices depend on the selected number of design points on the stabilizer, fuselage, and tail rotor. In this case, these matrices can be formed taking into account the mutual influence of the main and tail rotors.

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity DOI: http://dx.doi.org/10.5772/intechopen.93186*

**Figure 7.** *Induced velocity component in the tail rotor area.*

For a single arbitrary point, this will be a matrix-string for the main rotor and a matrix-string for the tail rotor. Multiplying the row by the column of known air circulations on the disk, we calculate the induced velocities at the selected point.

The values of the influence matrix elements depend on only the rotor geometric characteristics and the position of the selected point relative to the rotor and the angle of inclination *δ* of the vortex cylinder. In the process of calculations, the values of linear circulations and their corresponding induced velocities are refined. Influence matrices must be built in advance for a possible range of angles. During the calculation process, interpolate between the calculated matrices by the necessary values *δ*.

At different flight speeds, induced velocities have different effects on the tail rotor and stabilizer. In the figures, you can see that induced velocity have a special influence on the low flight speeds mode of *V* ¼ 60 km/h reach 4 m/s (**Figures 7** and **9**). This is 24% of the flight speed. Therefore it is very important to know and take into account the field of induced velocity's at low flight speeds.

The horizontal and vertical components of induced velocity at a characteristic point in the tail rotor and stabilizer area significantly depend on the flight speed (**Figures 7** and **9**).

By drawing the configuration of the air flow, we can see which areas of the helicopter are affected by the induced flow. This allows us to correctly configure the loads calculating program.

**Figure 10** shows the position of the tip vortex according to the blade theory (disordered line) and the disk theory (black line) for forward flight.

The application of the described method consists in forming a matrix of influ-

ence of the main and tail rotor for any group of points around the rotor. For example, a matrix of influence of the rotor on the fuselage, the matrix of influence of the main rotor on the tail rotor, the matrix of influence of the main rotor on the stabilizer, the matrix effect of the tail rotor on the main rotor, etc., the dimensions of the matrices depend on the selected number of design points on the stabilizer, fuselage, and tail rotor. In this case, these matrices can be formed taking into

account the mutual influence of the main and tail rotors.

*Normal component of induced velocity on hovering α* ¼ 0°, *μ* ¼ 0,*Ct* ¼ 0*:*01*.*

*Vortex Dynamics Theories and Applications*

**Figure 5.**

**Figure 6.**

**110**

*Tip vortex in hovering.*

#### **Figure 8.**

*The normal component of induced velocity in forward flight. (a) α* ¼ �9*:*2,*Ct* ¼ 0*:*01, *μ* ¼ 0*:*095, *x* ¼ 0, *y* ¼ 0*:*07*, V = 75 km/h. (b) α* ¼ �10*:*1,*Ct* ¼ 0*:*01, *μ* ¼ 0*:*14, *x* ¼ 0, *y* ¼ 0*:*07*, V = 110 km/h. (c) α* ¼ �9*:*5,*Ct* ¼ 0*:*01, *μ* ¼ 0*:*232, *x* ¼ 0, *y* ¼ 0*:*07*, V = 185 km/h.*

Comparison with the blade theory for calculating the position of the tip vortex shows that the method of discrete vortex cylinders gives satisfactory results. The angle of vortex cylinder *δ* calculated from the disk theory inclination coincides with

*Tip vortex at low speed (60 km/h, μ* ¼ 0*:*076*, α* ¼ �1*:*23°*, Ct* ¼ 0*:*01*). (a) side view and (b) overhead view.*

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity*

*DOI: http://dx.doi.org/10.5772/intechopen.93186*

As the graphs show, calculating the position of the tip vortex by the method of

From these results, we can conclude that the greatest convergence with experimental method of discrete vortex cylinder has got in a cross section *X* ¼ 0. In some

.

**Figure 11** shows the results of comparison of the normal component of the average induced velocity calculated in the cross section along the vane theory [3, 6] and the discrete vortex cylinder method with experiment [2]. Error experiment for

the angle *δ* calculated from the blade theory.

**Figure 10.**

**113**

discrete vortex cylinders gives satisfactory results.

flight speed *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*095 is �15% and the angle of attack *<sup>α</sup>* ¼ �9*:*2<sup>∘</sup>

**Figure 9.** *Components of the induced velocity in the stabilizer area.*

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity DOI: http://dx.doi.org/10.5772/intechopen.93186*

**Figure 10.** *Tip vortex at low speed (60 km/h, μ* ¼ 0*:*076*, α* ¼ �1*:*23°*, Ct* ¼ 0*:*01*). (a) side view and (b) overhead view.*

Comparison with the blade theory for calculating the position of the tip vortex shows that the method of discrete vortex cylinders gives satisfactory results. The angle of vortex cylinder *δ* calculated from the disk theory inclination coincides with the angle *δ* calculated from the blade theory.

As the graphs show, calculating the position of the tip vortex by the method of discrete vortex cylinders gives satisfactory results.

**Figure 11** shows the results of comparison of the normal component of the average induced velocity calculated in the cross section along the vane theory [3, 6] and the discrete vortex cylinder method with experiment [2]. Error experiment for flight speed *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*095 is �15% and the angle of attack *<sup>α</sup>* ¼ �9*:*2<sup>∘</sup> .

From these results, we can conclude that the greatest convergence with experimental method of discrete vortex cylinder has got in a cross section *X* ¼ 0. In some

**Figure 8.**

**Figure 9.**

**112**

*The normal component of induced velocity in forward flight. (a) α* ¼ �9*:*2,*Ct* ¼ 0*:*01, *μ* ¼ 0*:*095, *x* ¼ 0, *y* ¼ 0*:*07*, V = 75 km/h. (b) α* ¼ �10*:*1,*Ct* ¼ 0*:*01, *μ* ¼ 0*:*14, *x* ¼ 0, *y* ¼ 0*:*07*, V = 110 km/h.*

*(c) α* ¼ �9*:*5,*Ct* ¼ 0*:*01, *μ* ¼ 0*:*232, *x* ¼ 0, *y* ¼ 0*:*07*, V = 185 km/h.*

*Vortex Dynamics Theories and Applications*

*Components of the induced velocity in the stabilizer area.*

cases, we have strong disagreement with experiment, but similar to the results obtained by the blade theory.

**5. Conclusion**

**Figure 11.**

circulations.

circulations.

**115**

have been conducted. We found the following:

In this chapter, the authors have developed a method of discrete vortex cylinders based on the Shaydakov's disk vortex theory. The capabilities of the discrete vortex cylinder method are demonstrated using a helicopter balancing program based on a "semi-rigid" model of the main and tail rotors. Data from numerical calculations are proposed with experimental data from actual flights. Simulations

*The normal component of the induced velocity at a distance Y/R under and over of the main rotor (μ* ¼ 0*:*095*).*

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity*

*DOI: http://dx.doi.org/10.5772/intechopen.93186*

*(a) X* ¼ 0*, (b) X* ¼ 0*, (c), X* ¼ �0*:*5*, (d) X* ¼ �0*:*5*, (e) X* ¼ 1*:*7*, and (f) X* ¼ �0*:*5*.*

1.The complex procedure for calculating inductive velocities at any point in the space around the rotor was reduced to the procedure of multiplying the row by

coefficients is multiplied by the column of the corresponding vortex cylinders

2.The use of pre-calculated influence matrices makes it much easier to calculate

column. In this case, the row of influence of discrete vortex cylinders

*Discrete Vortex Cylinders Method for Calculating the Helicopter Rotor-Induced Velocity DOI: http://dx.doi.org/10.5772/intechopen.93186*

**Figure 11.**

cases, we have strong disagreement with experiment, but similar to the results

obtained by the blade theory.

*Vortex Dynamics Theories and Applications*

**114**

*The normal component of the induced velocity at a distance Y/R under and over of the main rotor (μ* ¼ 0*:*095*). (a) X* ¼ 0*, (b) X* ¼ 0*, (c), X* ¼ �0*:*5*, (d) X* ¼ �0*:*5*, (e) X* ¼ 1*:*7*, and (f) X* ¼ �0*:*5*.*
