**1.1 Coordinate systems of the vortex cylinder**

The properties of a cylindrical vortex surface are considered in detail by Shaidakov [1]. He analytically investigated the properties of a vortex surface that completely covers a beveled vortex surface. Shaidakov studied a vortex surface that starts from a disk plane, has an arbitrary shape in section, and pointed with its one end to infinity. Vortices on the surface are parallel to the base of the cylinder, which lies in the plane of the beginning of the vortex cylinder. When applied to the rotor disk of a helicopter, it is more appropriate to consider not an arbitrary shape of the vortex surface, but a very specific shape, the cross section of which is shown in **Figure 1**. The disk plane is filled with closed contours formed on two sides by arcs of circles and on the other two sides by radial segments. The number of closed contours depends on the number of calculated points along the blade and the number of points along the azimuth. However, the shape of closed contours remains the same.

To study the velocity field caused by a discrete vortex cylinder, we will select two typical sections of it. One section is located at the beginning plane of the vortex cylinder (lies in the plane of the screw disk); the other is parallel to it and intersects the cylinder at an infinite distance from the first plane. The sections of the cylinder with these planes are conventionally designated 1-1 and 2-2, respectively.

It is convenient to calculate induced velocities in the coordinate systems *Oxyz* (*Ox*1*y*1*z*1) shown in **Figure 2**. The vortex cylinder is tilted from the axis *Oz*(*Oz*1) by the inclination angle of the vortex cylinder *δ*. It is easy to see that the inducedvelocity calculated point is always located in a plane parallel to the disk plane, and the projection of the vortex cylinders on this plane is a disk with a radius equal to the radius of the rotor. The origin of the coordinate system *Oxyz* (*Ox*1*y*1*z*1) is always located in the center of this disk.

The right rectangular coordinate system *Oxyz* is used to record the components of the induced velocity *vx*, *vy*, *vz*. The right-linked coordinate system *Ox*1*y*1*z*<sup>1</sup> with guide orts *e*1,*e*2,*e*<sup>3</sup> is used to calculate the components of the induced velocity *vx*<sup>1</sup> , *vy*<sup>1</sup> , *vz*<sup>1</sup> . The axis *Ox*<sup>1</sup> and *Oz*<sup>1</sup> coincides with the axes *Ox* and *Oz*. The axis *Oy*<sup>1</sup> is

the axis of the cylinder and is inclined to the axis *Ox*<sup>1</sup> at an angle *δ*. In both systems, the base plane *Oxz* (*Ox*1*z*1) is parallel to the plane of the vortex cylinder and contains the induced-velocity calculated point *A*. The left skew coordinate *Oryψ* system is derived from the *Ox*1*y*1*z*1. The left oblique cylindrical coordinate system *Oryψ* is obtained by moving the origin to a point *A*. It is used to bring contour

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

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

The last two axis systems are most convenient for deriving equations of the vortex cylinder surface. It is enough to know the distance *h* along the axis *Oy*<sup>1</sup> from the base plane of the vortex cylinder and the equation of the projection of the

The ratio of induced-velocity's components in systems *Oxyz* and *Ox*1*y*1*z*<sup>1</sup> is

*vx* ¼ *vx*<sup>1</sup> þ *vy*<sup>1</sup> cos *δ*; *vy* ¼ *vy*<sup>1</sup> sin *δ*; *vz* ¼ *vz*<sup>1</sup> *:*

In the accepted coordinate system *Ox*1*y*1*z*<sup>1</sup> for an arbitrary pair of vectors *a*, *b* we

� � � � �

þ cos *δ*

!

� � � �

*ax*<sup>1</sup> *az*<sup>1</sup> *bx*<sup>1</sup> *bz*<sup>1</sup> � � � �

; (2)

These component interdependences are true for any vector [3].

� � � � �

*ay*<sup>1</sup> *az*<sup>1</sup> *by*<sup>1</sup> *bz*<sup>1</sup> (1)

integrals to a form that is convenient for integration.

*Coordinate system of the vortex cylinder with arbitrary form [2].*

cylinder base on the base plane.

**Figure 2.**

**101**

determined by the following dependencies

find a vector product *с* and a scalar product *ab*

*cx*<sup>1</sup> <sup>¼</sup> <sup>1</sup> sin *δ*

**Figure 1.** *Scheme for splitting the disk into discrete vortex cylinders.*

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

#### **Figure 2.** *Coordinate system of the vortex cylinder with arbitrary form [2].*

the axis of the cylinder and is inclined to the axis *Ox*<sup>1</sup> at an angle *δ*. In both systems, the base plane *Oxz* (*Ox*1*z*1) is parallel to the plane of the vortex cylinder and contains the induced-velocity calculated point *A*. The left skew coordinate *Oryψ* system is derived from the *Ox*1*y*1*z*1. The left oblique cylindrical coordinate system *Oryψ* is obtained by moving the origin to a point *A*. It is used to bring contour integrals to a form that is convenient for integration.

The last two axis systems are most convenient for deriving equations of the vortex cylinder surface. It is enough to know the distance *h* along the axis *Oy*<sup>1</sup> from the base plane of the vortex cylinder and the equation of the projection of the cylinder base on the base plane.

The ratio of induced-velocity's components in systems *Oxyz* and *Ox*1*y*1*z*<sup>1</sup> is determined by the following dependencies

$$\begin{aligned} \boldsymbol{\upsilon}\_{\boldsymbol{x}} &= \boldsymbol{\upsilon}\_{\boldsymbol{x}\_{1}} + \boldsymbol{\upsilon}\_{\boldsymbol{\mathcal{y}}\_{1}} \cos \delta; \\ \boldsymbol{\upsilon}\_{\boldsymbol{\mathcal{y}}} &= \boldsymbol{\upsilon}\_{\boldsymbol{\mathcal{y}}\_{1}} \sin \delta; \\ \boldsymbol{\upsilon}\_{\boldsymbol{x}} &= \boldsymbol{\upsilon}\_{\boldsymbol{x}\_{1}}. \end{aligned} \tag{1}$$

These component interdependences are true for any vector [3].

In the accepted coordinate system *Ox*1*y*1*z*<sup>1</sup> for an arbitrary pair of vectors *a*, *b* we find a vector product *с* and a scalar product *ab*

$$c\_{\mathbf{x}\_1} = \frac{\mathbf{1}}{\sin \delta} \left( \begin{vmatrix} a\_{\mathcal{V}\_1} & a\_{x\_1} \\ b\_{\mathcal{V}\_1} & b\_{x\_1} \end{vmatrix} + \cos \delta \begin{vmatrix} a\_{\mathcal{X}\_1} & a\_{x\_1} \\ b\_{\mathcal{X}\_1} & b\_{x\_1} \end{vmatrix} \right);\tag{2}$$

**1.1 Coordinate systems of the vortex cylinder**

*Vortex Dynamics Theories and Applications*

located in the center of this disk.

*vx*<sup>1</sup> , *vy*<sup>1</sup>

**Figure 1.**

**100**

*Scheme for splitting the disk into discrete vortex cylinders.*

The properties of a cylindrical vortex surface are considered in detail by Shaidakov [1]. He analytically investigated the properties of a vortex surface that completely covers a beveled vortex surface. Shaidakov studied a vortex surface that starts from a disk plane, has an arbitrary shape in section, and pointed with its one end to infinity. Vortices on the surface are parallel to the base of the cylinder, which lies in the plane of the beginning of the vortex cylinder. When applied to the rotor disk of a helicopter, it is more appropriate to consider not an arbitrary shape of the vortex surface, but a very specific shape, the cross section of which is shown in **Figure 1**. The disk plane is filled with closed contours formed on two sides by arcs of circles and on the other two sides by radial segments. The number of closed contours depends on the number of calculated points along the blade and the number of points along the azimuth. However, the shape of closed contours remains the same. To study the velocity field caused by a discrete vortex cylinder, we will select two typical sections of it. One section is located at the beginning plane of the vortex cylinder (lies in the plane of the screw disk); the other is parallel to it and intersects the cylinder at an infinite distance from the first plane. The sections of the cylinder

with these planes are conventionally designated 1-1 and 2-2, respectively.

It is convenient to calculate induced velocities in the coordinate systems *Oxyz* (*Ox*1*y*1*z*1) shown in **Figure 2**. The vortex cylinder is tilted from the axis *Oz*(*Oz*1) by the inclination angle of the vortex cylinder *δ*. It is easy to see that the inducedvelocity calculated point is always located in a plane parallel to the disk plane, and the projection of the vortex cylinders on this plane is a disk with a radius equal to the radius of the rotor. The origin of the coordinate system *Oxyz* (*Ox*1*y*1*z*1) is always

The right rectangular coordinate system *Oxyz* is used to record the components of the induced velocity *vx*, *vy*, *vz*. The right-linked coordinate system *Ox*1*y*1*z*<sup>1</sup> with guide orts *e*1,*e*2,*e*<sup>3</sup> is used to calculate the components of the induced velocity

, *vz*<sup>1</sup> . The axis *Ox*<sup>1</sup> and *Oz*<sup>1</sup> coincides with the axes *Ox* and *Oz*. The axis *Oy*<sup>1</sup> is

*Vortex Dynamics Theories and Applications*

$$c\_{\mathcal{Y}\_1} = \frac{1}{\sin \delta} \left( \begin{vmatrix} a\_{x\_1} & a\_{x\_1} \\ b\_{x\_1} & b\_{x\_1} \end{vmatrix} + \cos \delta \begin{vmatrix} a\_{\mathcal{Y}\_1} & a\_{x\_1} \\ b\_{\mathcal{Y}\_1} & b\_{x\_1} \end{vmatrix} \right);\tag{3}$$

$$a\_{\mathfrak{x}\_1} = \sin \delta \begin{vmatrix} a\_{\mathfrak{x}\_1} & a\_{\mathfrak{y}\_1} \\ b\_{\mathfrak{x}\_1} & b\_{\mathfrak{y}\_1} \end{vmatrix};\tag{4}$$

Calculate the components of the inductive velocity vectors from the vortex path

þ cos *δ*

� � � �

� � � �

After describing the determinants, we will have the following expressions for

� � � �

þ cos *δ*

*dξ dζ ξ* � *x*<sup>1</sup> *ζ* � *z*<sup>1</sup>

� ��

� � � �

� ��

� ��

*dξ dζ ξ* � *x*<sup>1</sup> *ζ* � *z*<sup>1</sup>

> 0 *dζ η ζ* � *z*<sup>1</sup>

> > � � � �

½ � cos *δ ζ*ð Þ � *z*<sup>1</sup> *dξ* � cos *δ ξ*ð Þ � *x*<sup>1</sup> *dζ* � *ηdζ* ; (16)

j j*<sup>l</sup>* <sup>3</sup> ½ � cos *δ ζ*ð Þ � *<sup>z</sup>*<sup>1</sup> *<sup>d</sup><sup>ξ</sup>* � cos *δ ξ*ð Þ � *<sup>x</sup>*<sup>1</sup> *<sup>d</sup><sup>ζ</sup>* � *<sup>η</sup>d<sup>ζ</sup>* ; (19)

j j*<sup>l</sup>* <sup>3</sup> ½ � ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> *<sup>d</sup><sup>ξ</sup>* � ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>d</sup><sup>ζ</sup>* � cos *δηd<sup>ζ</sup>* ; (20)

½ � ð Þ *ζ* � *z*<sup>1</sup> *dξ* � ð Þ *ξ* � *x*<sup>1</sup> *dζ* � cos *δηdζ* ; (17)

j j*<sup>l</sup>* <sup>3</sup> *<sup>η</sup>* sin *<sup>δ</sup>dξ:* (18)

j j*<sup>l</sup>* <sup>3</sup> *<sup>η</sup>dξ:* (21)

∞ð

*ηdη* j j*l* 3

3

3

5 (22)

5 (23)

*h*

*ηdη* j j*l* 3

∞ð

*h*

j j*<sup>l</sup>* <sup>3</sup> (24)

ð ∞

*dη* j j*<sup>l</sup>* <sup>3</sup> � *<sup>d</sup><sup>ζ</sup>*

*h*

j j*<sup>l</sup>* <sup>3</sup> � cos *<sup>δ</sup>d<sup>ζ</sup>*

� � � *z* � � � �

� � � � � � � *x*

� � � *y* ; (13)

; (14)

*:* (15)

*ds*. For this purpose, it is necessary to define expressions for projections of the

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

� � � �

0 *dζ η ζ* � *z*<sup>1</sup>

*dξ dζ ξ* � *x*<sup>1</sup> *ζ* � *z*<sup>1</sup>

*<sup>z</sup>* ¼ sin *δ*

� � � �

� � � �

*ds* � *l* � �

1 sin *δ*

> 1 sin *δ*

> > *d*2

*vz*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π dη*

Let us perform integration of formulas (Eqs. (16)–(18)) along the cylinder

<sup>4</sup>*<sup>π</sup>* sin *<sup>δ</sup>*

∞ð

*dη*

*h*

j j*<sup>l</sup>* <sup>3</sup> � cos *δ ξ*ð Þ � *<sup>x</sup>*<sup>1</sup> *<sup>d</sup><sup>ζ</sup>*

∞ð

*dη*

*h*

*ηdη*

∞ð

*h*

vector product *ds* � *l* on the coordinate axis.

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

*ds* � *l* � � *<sup>x</sup>* <sup>¼</sup> <sup>1</sup> sin *δ*

*ds* � *l* � � *<sup>y</sup>* <sup>¼</sup> <sup>1</sup> sin *δ*

induced velocities

*d*2

creators

*dvx*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

**103**

*dvy*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

*vx*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π dη* j j*l* 3

*d*2

*dvx*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

> *dvy*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

> > 1 sin *δ*

*vy*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π dη* j j*l* 3

> 1 sin *δ*

ð ∞

*dη*

∞ð

*dη*

*dvz*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>*

∞ð

*dη*

j j*<sup>l</sup>* <sup>3</sup> � ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>d</sup><sup>ζ</sup>*

<sup>4</sup>*<sup>π</sup>* sin *<sup>δ</sup>d<sup>ξ</sup>*

*h*

*dη*

*dvz*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>*

∞ð

*h*

*h*

cos *δ ζ*ð Þ � *z*<sup>1</sup> *dξ*

sin *<sup>δ</sup>* ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> *<sup>d</sup><sup>ξ</sup>*

2 4 *h*

1 sin *δ*

After the conversion, we will have

2 4

1

$$ab = a\_{\mathbf{x}\_1}b\_{\mathbf{x}\_1} + a\_{\mathbf{y}\_1}b\_{\mathbf{y}\_1} + a\_{\mathbf{z}\_1}b\_{\mathbf{z}\_1} + \left(a\_{\mathbf{x}\_1}b\_{\mathbf{y}\_1} + a\_{\mathbf{y}\_1}b\_{\mathbf{z}\_1}\right)\cos\delta.\tag{5}$$

The modulus of the vector *a* is calculated by the formula (Eq. (5))

$$\mathfrak{a} = \sqrt{\mathfrak{a}\mathfrak{a}} = \sqrt{a\_{\mathfrak{x}\_1}^2 + a\_{\mathfrak{y}\_1}^2 + a\_{\mathfrak{x}\_1}^2 + 2a\_{\mathfrak{x}\_1}a\_{\mathfrak{y}\_1}\cos\delta}. \tag{6}$$

The projection of the vector *c* on the axis of the cylinder is denoted *c*0. Then, using (Eq. (6)), we will have

$$
\mathcal{L}\_0 = \overline{\mathcal{c}} \cdot \overline{\mathcal{e}}\_2 = \mathcal{c}\_{\mathcal{Y}\_1} + \mathcal{c}\_{\mathfrak{x}\_1} \cos \delta. \tag{7}
$$

## **1.2 Components of the induced-velocity vector at any point in area around the rotor**

Let us consider a certain part of a cylindrical vortex surface, with the beginning at the plane of the disk, bounded on two sides by two generatrices and leaving the other side to infinity (**Figure 2**). The beginning of the generatrices is denoted by the point's m and n. Select a vortex element *ds* with circulation *d*Γ at any point *M*ð Þ *ξ*, *η*, *ζ* in the vortex surface:

$$d\Gamma = \gamma d\eta,\tag{8}$$

where *γ* is the running circulation in the direction of the cylinder generatrices. We will calculate the induced velocities from this element at the point *A x*ð Þ 1, 0, *z*<sup>1</sup> using the formula of Biot-Savart

$$d^2\overline{v} = \frac{d\Gamma}{4\pi} \frac{d\overline{s} \times \overline{l}}{|l|^3},\tag{9}$$

where *l* is connecting the point *A* to the point *M* vector; *ds* is a vortex element represented as a vector. The sign of circulation *d*Γ (or *γ*) is determined by the direction of the vector *ds*. A positive value corresponds to a positive direction in the accepted coordinate system. When calculating the inductive effect from the vortex surface, the contour integral is calculated along the vortex lines. The positive direction of the contour traversal in the right coordinate system corresponds to the right rotation, in the left coordinate system—to the left.

We express the vectors included in the formula (Eq. (8)) in terms of affine coordinates

$$d\overline{\mathfrak{s}} = d\xi \overline{e}\_1 + d\zeta \overline{e}\_3;\tag{10}$$

$$l = (\xi - \varkappa\_1)\overline{e}\_1 + \eta \overline{e}\_2 + (\zeta - z\_1)\overline{e}\_3. \tag{11}$$

The modulus of the vector *l* is defined by the formula (Eq. (6))

$$l = \sqrt{(\xi - \varkappa\_1^2) + \eta^2 + (\zeta - \varkappa\_1)^2 + 2(\xi - \varkappa\_1)\eta \cos \delta}. \tag{12}$$

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

Calculate the components of the inductive velocity vectors from the vortex path *ds*. For this purpose, it is necessary to define expressions for projections of the vector product *ds* � *l* on the coordinate axis.

$$\left.d\vec{\sigma} \times \vec{l}\right|\_{\mathfrak{x}} = \frac{1}{\sin\delta} \left( \begin{vmatrix} \mathbf{0} & d\zeta \\ \eta & \zeta - \mathbf{z}\_1 \end{vmatrix} + \cos\delta \begin{vmatrix} d\xi & d\zeta \\ \xi - \mathbf{x}\_1 & \zeta - \mathbf{z}\_1 \end{vmatrix} \right)\bigg|\_{\mathfrak{x}};\tag{13}$$

$$\left.d\vec{\sigma} \times \vec{l}\right|\_{\mathfrak{y}} = \frac{1}{\sin\delta} \left( \begin{vmatrix} d\xi & d\zeta \\ \xi - \varkappa\_1 & \zeta - \varkappa\_1 \end{vmatrix} + \begin{vmatrix} \mathbf{0} & d\zeta \\ \eta & \zeta - \varkappa\_1 \end{vmatrix} \right) \Big|\_{\mathfrak{y}};\tag{14}$$

$$\left.d\overline{\mathfrak{s}} \times \overline{l}\right|\_{x} = \sin\delta\left(\begin{vmatrix} d\xi & d\zeta\\ \xi - \varkappa\_{1} & \zeta - \varkappa\_{1} \end{vmatrix}\right)\Big|\_{x}.\tag{15}$$

After describing the determinants, we will have the following expressions for induced velocities

$$d^2 v\_{\mathbf{x}\_1} = \frac{\chi}{4\pi} \frac{d\eta}{|l|^3} \frac{1}{\sin \delta} [\cos \delta(\zeta - \mathbf{z}\_1) d\xi - \cos \delta(\xi - \mathbf{x}\_1) d\zeta - \eta d\zeta];\tag{16}$$

$$d^2 v\_{\mathcal{Y}\_1} = \frac{\chi}{4\pi} \frac{d\eta}{|l|^3} \frac{1}{\sin\delta} [(\zeta - z\_1)d\xi - (\xi - \varkappa\_1)d\zeta - \cos\delta\eta d\zeta];\tag{17}$$

$$d^2 \upsilon\_{x\_1} = \frac{\chi}{4\pi} \frac{d\eta}{\left| l \right|^3} \eta \sin \delta d\xi. \tag{18}$$

Let us perform integration of formulas (Eqs. (16)–(18)) along the cylinder creators

$$dv\_{\chi\_1} = \frac{\gamma}{4\pi} \frac{1}{\sin \delta} \int\_{\frac{1}{h}}^{\infty} \frac{d\eta}{|l|^3} [\cos \delta(\zeta - z\_1) d\xi - \cos \delta(\xi - \varkappa\_1) d\zeta - \eta d\zeta];\tag{19}$$

$$dv\_{\mathcal{Y}\_1} = \frac{\gamma}{4\pi} \frac{1}{\sin \delta} \left[ \frac{d\eta}{|l|^3} [(\zeta - x\_1)d\xi - (\xi - x\_1)d\zeta - \cos \delta \eta d\zeta];\tag{20}$$

$$dv\_{x\_1} = \frac{\gamma}{4\pi} \sin \delta \int\_{\frac{1}{h}}^{\infty} \frac{d\eta}{|I|^3} \eta d\xi. \tag{21}$$

After the conversion, we will have

$$dv\_{\infty\_1} = \frac{\gamma}{4\pi} \frac{1}{\sin \delta} \left[ \cos \delta (\zeta - z\_1) d\zeta \int\_{\hbar}^{\infty} \frac{d\eta}{|l|^3} - \cos \delta (\xi - \varkappa\_1) d\zeta \int\_{\hbar}^{\infty} \frac{d\eta}{|l|^3} - d\zeta \int\_{\hbar}^{\infty} \frac{\eta d\eta}{|l|^3} \right] \tag{22}$$

$$dv\_{\mathcal{Y}\_1} = \frac{\chi}{4\pi} \frac{1}{\sin \delta} \left[ (\zeta - x\_1) d\xi \int\_{\frac{1}{h}}^{\infty} \frac{d\eta}{|l|^3} - (\xi - x\_1) d\zeta \int\_{\frac{1}{h}}^{\infty} \frac{d\eta}{|l|^3} - \cos \delta d\zeta \int\_{\frac{1}{h}}^{\infty} \frac{\eta d\eta}{|l|^3} \right] \tag{23}$$

$$dv\_{x\_1} = \frac{\gamma}{4\pi} \sin \delta d\xi \int\_h^\infty \frac{\eta d\eta}{|l|^3} \tag{24}$$

*cy*<sup>1</sup> <sup>¼</sup> <sup>1</sup> sin *δ*

*Vortex Dynamics Theories and Applications*

*ab* ¼ *ax*1*bx*<sup>1</sup> þ *ay*<sup>1</sup>

*<sup>a</sup>* <sup>¼</sup> ffiffiffiffiffi *aa* <sup>p</sup> <sup>¼</sup>

using (Eq. (6)), we will have

*M*ð Þ *ξ*, *η*, *ζ* in the vortex surface:

*A x*ð Þ 1, 0, *z*<sup>1</sup> using the formula of Biot-Savart

rotation, in the left coordinate system—to the left.

*ξ* � *x*<sup>1</sup>

*l* ¼

q

**rotor**

coordinates

**102**

*ax*<sup>1</sup> *az*<sup>1</sup> *bx*<sup>1</sup> *bz*<sup>1</sup>

*cz*<sup>1</sup> ¼ sin *δ*

The modulus of the vector *a* is calculated by the formula (Eq. (5))

*a*2 *<sup>x</sup>*<sup>1</sup> þ *a*<sup>2</sup>

q

� � � �

� � � � � þ cos *δ*

*ax*<sup>1</sup> *ay*<sup>1</sup> *bx*<sup>1</sup> *by*<sup>1</sup>

*by*<sup>1</sup> þ *az*1*bz*<sup>1</sup> þ *ax*1*by*<sup>1</sup> þ *ay*<sup>1</sup>

*<sup>y</sup>*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

The projection of the vector *c* on the axis of the cylinder is denoted *c*0. Then,

**1.2 Components of the induced-velocity vector at any point in area around the**

point's m and n. Select a vortex element *ds* with circulation *d*Γ at any point

We will calculate the induced velocities from this element at the point

*d*2 *<sup>v</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>Γ</sup> 4*π*

The modulus of the vector *l* is defined by the formula (Eq. (6))

<sup>2</sup> ð Þþ *η*<sup>2</sup> þ ð Þ *ζ* � *z*<sup>1</sup>

Let us consider a certain part of a cylindrical vortex surface, with the beginning at the plane of the disk, bounded on two sides by two generatrices and leaving the other side to infinity (**Figure 2**). The beginning of the generatrices is denoted by the

where *γ* is the running circulation in the direction of the cylinder generatrices.

where *l* is connecting the point *A* to the point *M* vector; *ds* is a vortex element represented as a vector. The sign of circulation *d*Γ (or *γ*) is determined by the direction of the vector *ds*. A positive value corresponds to a positive direction in the accepted coordinate system. When calculating the inductive effect from the vortex surface, the contour integral is calculated along the vortex lines. The positive direction of the contour traversal in the right coordinate system corresponds to the right

We express the vectors included in the formula (Eq. (8)) in terms of affine

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*ds* � *l*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>z</sup>*<sup>1</sup> þ 2*ax*1*ay*<sup>1</sup> cos *δ*

*c*<sup>0</sup> ¼ *c* � *e*<sup>2</sup> ¼ *cy*<sup>1</sup> þ *cx*<sup>1</sup> cos *δ:* (7)

*d*Γ ¼ *γdη*, (8)

j j*<sup>l</sup>* <sup>3</sup> , (9)

*ds* ¼ *dξ e*<sup>1</sup> þ *dζ e*3; (10)

<sup>2</sup> <sup>þ</sup> <sup>2</sup>ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>η</sup>* cos *<sup>δ</sup>*

*:* (12)

*l* ¼ ð Þ *ξ* � *x*<sup>1</sup> *e*<sup>1</sup> þ *ηe*<sup>2</sup> þ ð Þ *ζ* � *z*<sup>1</sup> *e*3*:* (11)

!

� � � � �

> � � � � �

*ay*<sup>1</sup> *az*<sup>1</sup> *by*<sup>1</sup> *bz*<sup>1</sup> � � � � �

*bz*1 � � cos *δ:* (5)

; (3)

*:* (6)

; (4)

� � � � In formulas (Eq. (22)–(24)) we will denote by

$$\begin{aligned} J\_1 &= \int\_h^\infty \frac{d\eta}{|l|^3} = \int\_h^\infty \frac{d\eta}{\left| (\xi - \mathbf{x}\_1 \mathbf{1}^2) + \eta^2 + (\zeta - \mathbf{z}\_1)^2 + 2(\xi - \mathbf{x}\_1)\eta \cos\delta \right|^3}; \\ J\_2 &= \int\_h^\infty \frac{\eta d\eta}{|l|^3} = \int\_h^\infty \frac{\eta d\eta}{\left| (\xi - \mathbf{x}\_1 \mathbf{1}^2) + \eta^2 + (\zeta - \mathbf{z}\_1)^2 + 2(\xi - \mathbf{x}\_1)\eta \cos\delta \right|^2} \end{aligned} \tag{25}$$

*dvz*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>*

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

sin *<sup>δ</sup>* ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> *<sup>J</sup>*1*d<sup>ξ</sup>* � ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>J</sup>*1*d<sup>ζ</sup>* � cos *<sup>δ</sup><sup>J</sup>* <sup>½</sup> <sup>2</sup>*dζ*�þ

To calculate the velocity from a limited width vortex cylindrical surface, we need to take the contour integral from the projection of this surface on the base

The method of discrete vortex cylinders is based on the linear disk theory of Shaidakov described above. It allows you to calculate the induced velocity from the rotor at any point in the space around the rotor. Consider a main rotor with an infinite number of blades [2]. Imagine a vortex system that descends from the rotor in the form of a vortex column, starting at the plane of the disk and going to infinity. The vortex column is supported by a circle with a radius equal to the radius of the rotor. The angle of vortex column inclination to the disk plane depends on the helicopter forward flight speed and the thrust of the rotor. It is calculated using the

Each partial volume of the vortex column can be considered as an elementary column of dipoles with a constant density of circulation. Alternatively consider it as an elementary vortex cylinder of arbitrary shape with a linear circulation of closed

In case of the beveled cylinder filling with dipoles, to calculate the inductive velocity from the entire vortex column, it is necessary to make integral sums from the *n* number final volumes at *n* ! ∞ for the limit case. In this case, the area of the base of the cylinder is divided into *n* number areas *dσ*1, *dσ*2, … , *dσn*. The area of the cylinder base is divided into finite regions, and the entire vortex column is divided

When filling a vortex column with vortex cylinders, the area of the base is filled with closed contours of a specific shape (**Figure 1**), and the column is entirely filled with vortex cylinders of linear circulation *γ* along the generatrix. The generatrices of the vortex cylinders are parallel to the axis of the vortex column and inclined at an

We propose to consider a vortex column as a collection of a finite number of vortex cylinders resting on the plane of the rotor disk. The plane of the disk is filled with closed vortex contours on two sides by arcs of circles and on the other two sides by radial segments (**Figure 4**). When the disk is split in this way the size of the contours will depend on the number of calculated points along the blade radius and

; *vz*<sup>1</sup> ¼

ð

*s*

ð

*s dvy*<sup>1</sup>

Shaidakov formula for the angle of inclination of the vortex column.

vortex cylinder generatrices. We use the expression (Eq. (7))

<sup>1</sup> � cos <sup>2</sup>*<sup>δ</sup>*

*dvx*<sup>1</sup> ; *vy*<sup>1</sup> ¼

*dv*<sup>0</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

> þ *γ* 4*π*

1

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

1 sin *δ*

After simple transformations we get

*dv*<sup>0</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

plane (the integral along the length of the arc *s*).

*s*

*vx*<sup>1</sup> ¼ ð

**2. Discrete vortex cylinders method**

vortices *γ* along the generatrix.

angle *δ* to the plane of the disk (**Figure 3**).

into infinite volumes.

**105**

We get the formula for the component of the induced velocity directed along the

cos *δ ζ*ð Þ � *z*<sup>1</sup> *J*1*dξ* � cos *δ ξ*ð Þ � *x*<sup>1</sup> *J*1*dζ* � *J* ½ � <sup>2</sup>*dζ* cos *δ:*

<sup>4</sup>*<sup>π</sup>* sin *<sup>δ</sup>J*1*d<sup>ξ</sup>* (33)

sin *<sup>δ</sup>* ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>J</sup>*1*d<sup>ζ</sup>* � ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> *<sup>J</sup>* ½ � <sup>1</sup>*d<sup>ξ</sup> :* (35)

*dvz*<sup>1</sup> ; *v*<sup>0</sup> ¼

ð

*dv*0*:* (36)

*s*

(34)

Using the following formulas *ρ*<sup>2</sup> *<sup>h</sup>* ¼ ð Þ *ξ* � *x*<sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> <sup>2</sup> and *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup><sup>h</sup>* cos *<sup>ϑ</sup>* converting integrals (Eq. (25))

$$\begin{split} J\_1 &= \int\_h^\infty \frac{d\eta}{|l|^3} = \int\_h^\infty \frac{d\eta}{\left|\rho\_h^2 + \eta^2 + 2\rho\_h \cos\vartheta \cos\delta\eta\right|^{\frac{3}{2}}}; \\ J\_2 &= \int\_h^\infty \frac{\eta d\eta}{|l|^3} = \int\_h^\infty \frac{\eta d\eta}{\left|\rho\_h^2 + \eta^2 + 2\rho\_h \cos\vartheta \cos\delta\eta\right|^{\frac{3}{2}}}. \end{split} \tag{26}$$

Integrals (Eq. (26)) are table integrals and can be denoted by the following:

$$f\_1 = \frac{1}{\rho\_h^2 (1 - \cos^2 \delta \cos^2 \theta)} \left( 1 - \frac{h + \rho\_h \cos \delta \cos \theta}{l\_h} \right);\tag{27}$$

$$J\_2 = \frac{1}{\rho\_h (1 - \cos^2 \delta \cos^2 \theta)} \left( \frac{\rho\_h + h \cos \delta \cos \theta}{l\_h} - \cos \delta \cos \theta \right), \tag{28}$$

where *lh* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ρ*2 *<sup>h</sup>* <sup>þ</sup> *<sup>h</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ρhh* cos *<sup>δ</sup>* q .

Using the already known relationships cos *ϑ* ¼ ð Þ *ξ* � *x*<sup>1</sup> *=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *ξ* � *x*<sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> 2 q converting the received formulas

$$J\_1 = \frac{1}{\left(\zeta - \mathbf{z}\_1\right)^2 + \left(1 - \cos^2\delta\right)\left(\xi - \mathbf{x}\_1\right)^2} \left(1 - \frac{h + \cos\delta(\xi - \mathbf{x}\_1)}{l\_h}\right);\tag{29}$$

$$\begin{split} f\_{2} &= \frac{1}{\left(\zeta - \mathbf{z}\_{1}\right)^{2} + \left(1 - \cos^{2}\delta\right)\left(\xi - \mathbf{x}\_{1}\right)^{2}}\\ &\left(\frac{\left(\xi - \mathbf{x}\_{1}\right)^{2} + \left(\zeta - \mathbf{z}\_{1}\right)^{2} + h\cos\delta(\xi - \mathbf{x}\_{1})}{l\_{h}} - \cos\delta(\xi - \mathbf{x}\_{1})\right),\end{split} \tag{30}$$

where *lh* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *ξ* � *x*<sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> <sup>2</sup> <sup>þ</sup> *<sup>h</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>h</sup>* cos *<sup>δ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *ξ* � *x*<sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> 2 r q .

Integrals were obtained for the first time by Shaidakov [3] and are referred to in this chapter as Shaidakov's integrals.

Now it is possible to get expressions for differentials of induced velocity's components on the axis of a coordinate system *Ox*1*y*1*z*<sup>1</sup>

$$dv\_{x\_1} = \frac{\gamma}{4\pi} \frac{1}{\sin \delta} \left\{ (\zeta - z\_1) I\_1 \cos \delta d\xi - [(\xi - x\_1) I\_1 \cos \delta + I\_2] d\zeta \right\} \tag{31}$$

$$dv\_{\mathcal{V}\_1} = \frac{\chi}{4\pi} \frac{1}{\sin \delta} \{ (\zeta - z\_1) J\_1 d\xi - [(\xi - \varkappa\_1) J\_1 + \cos \delta l\_2] d\zeta \} \tag{32}$$

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

$$dv\_{x\_1} = \frac{\gamma}{4\pi} \sin \delta l\_1 d\xi \tag{33}$$

We get the formula for the component of the induced velocity directed along the vortex cylinder generatrices. We use the expression (Eq. (7))

$$\begin{split} d\nu\_{0} &= \frac{\gamma}{4\pi} \frac{1}{\sin\delta} [(\zeta - x\_{1})I\_{1}d\xi - (\xi - x\_{1})I\_{1}d\zeta - \cos\delta I\_{2}d\zeta] + \\ &+ \frac{\gamma}{4\pi} \frac{1}{\sin\delta} [\cos\delta(\zeta - x\_{1})I\_{1}d\xi - \cos\delta(\xi - x\_{1})I\_{1}d\zeta - I\_{2}d\zeta] \cos\delta. \end{split} \tag{34}$$

After simple transformations we get

In formulas (Eq. (22)–(24)) we will denote by

*ξ* � *x*<sup>1</sup>

*ξ* � *x*<sup>1</sup>

<sup>2</sup> ð Þþ *η*<sup>2</sup> þ ð Þ *ζ* � *z*<sup>1</sup>

<sup>2</sup> ð Þþ *η*<sup>2</sup> þ ð Þ *ζ* � *z*<sup>1</sup>

*<sup>h</sup>* ¼ ð Þ *ξ* � *x*<sup>1</sup>

*dη*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>η</sup>* cos *<sup>δ</sup>* � � �

*ηdη*

<sup>2</sup> <sup>þ</sup> <sup>2</sup>ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>η</sup>* cos *<sup>δ</sup>* � � �

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup>

*dη*

*<sup>h</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ρ<sup>h</sup>* cos *<sup>ϑ</sup>* cos *δη* � � �

*ηdη*

*<sup>h</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ρ<sup>h</sup>* cos *<sup>ϑ</sup>* cos *δη* � � �

> <sup>1</sup> � *<sup>h</sup>* <sup>þ</sup> *<sup>ρ</sup><sup>h</sup>* cos *<sup>δ</sup>* cos *<sup>ϑ</sup> lh* � �

<sup>2</sup> <sup>1</sup> � *<sup>h</sup>* <sup>þ</sup> cos *δ ξ*ð Þ � *<sup>x</sup>*<sup>1</sup> *lh* � �

� cos *δ ξ*ð Þ � *x*<sup>1</sup>

ð Þ *ξ* � *x*<sup>1</sup>

� �

� 3 2 ;

(25)

(26)

; (27)

, (28)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup>

; (29)

2

.

(30)

2

ð Þ *ξ* � *x*<sup>1</sup>

,

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� 3 2

� 3 2 ;

> � 3 2 *:*

� cos *δ* cos *ϑ*

q

<sup>2</sup> and *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup><sup>h</sup>* cos *<sup>ϑ</sup>*

*J*<sup>1</sup> ¼ ð ∞

*J*<sup>2</sup> ¼ ∞ð

*h*

*h*

converting integrals (Eq. (25))

*dη* j j*<sup>l</sup>* <sup>3</sup> <sup>¼</sup>

*ηdη* j j*<sup>l</sup>* <sup>3</sup> <sup>¼</sup>

∞ð

*Vortex Dynamics Theories and Applications*

*h*

∞ð

*h*

*J*<sup>1</sup> ¼ ð ∞

*J*<sup>2</sup> ¼ ∞ð

*h*

*h*

*<sup>J</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *ρ*2

*ρ<sup>h</sup>* 1 � cos <sup>2</sup>*δ* cos <sup>2</sup> ð Þ *ϑ*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>h</sup>* <sup>þ</sup> *<sup>h</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*ρhh* cos *<sup>δ</sup>*

*<sup>J</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

q

converting the received formulas

*<sup>J</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> ð Þ *ζ* � *z*<sup>1</sup>

*<sup>J</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> ð Þ *ζ* � *z*<sup>1</sup>

ð Þ *ξ* � *x*<sup>1</sup>

ð Þ *ξ* � *x*<sup>1</sup>

1

*dvy*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

this chapter as Shaidakov's integrals.

*dvx*<sup>1</sup> <sup>¼</sup> *<sup>γ</sup>* 4*π*

*ρ*2

where *lh* ¼

where *lh* ¼

**104**

*dη* j j*<sup>l</sup>* <sup>3</sup> <sup>¼</sup>

*ηdη* j j*<sup>l</sup>* <sup>3</sup> <sup>¼</sup>

ð ∞

*h*

∞ð

*h*

*<sup>h</sup>* 1 � cos <sup>2</sup>*δ* cos <sup>2</sup> ð Þ *ϑ*

Using the already known relationships cos *ϑ* ¼ ð Þ *ξ* � *x*<sup>1</sup> *=*

<sup>2</sup> <sup>þ</sup> <sup>1</sup> � cos <sup>2</sup> ð Þ*<sup>δ</sup>* ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>1</sup> � cos <sup>2</sup> ð Þ*<sup>δ</sup>* ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup>

*lh*

r q

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup>

<sup>2</sup> <sup>þ</sup> ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup>

ponents on the axis of a coordinate system *Ox*1*y*1*z*<sup>1</sup>

1

*ρ*2

*ρ*2

.

Integrals (Eq. (26)) are table integrals and can be denoted by the following:

*ρ<sup>h</sup>* þ *h* cos *δ* cos *ϑ lh*

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin *<sup>δ</sup>* ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> *<sup>J</sup>*<sup>1</sup> cos *<sup>δ</sup>d<sup>ξ</sup>* � ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>J</sup>*<sup>1</sup> cos *<sup>δ</sup>* <sup>þ</sup> *<sup>J</sup>*<sup>2</sup> f g ½ �*d<sup>ζ</sup>* (31)

sin *<sup>δ</sup>* ð Þ *<sup>ζ</sup>* � *<sup>z</sup>*<sup>1</sup> *<sup>J</sup>*1*d<sup>ξ</sup>* � ð Þ *<sup>ξ</sup>* � *<sup>x</sup>*<sup>1</sup> *<sup>J</sup>*<sup>1</sup> <sup>þ</sup> cos *<sup>δ</sup>J*<sup>2</sup> f g ½ �*d<sup>ζ</sup>* (32)

<sup>2</sup> <sup>þ</sup> *<sup>h</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>h</sup>* cos *<sup>δ</sup>*

Integrals were obtained for the first time by Shaidakov [3] and are referred to in

Now it is possible to get expressions for differentials of induced velocity's com-

<sup>2</sup> <sup>þ</sup> *<sup>h</sup>* cos *δ ξ*ð Þ � *<sup>x</sup>*<sup>1</sup>

!

Using the following formulas *ρ*<sup>2</sup>

$$dv\_0 = \frac{\chi}{4\pi} \frac{1 - \cos^2 \delta}{\sin \delta} [(\xi - \chi\_1)I\_1 d\zeta - (\zeta - \mathbf{z}\_1)I\_1 d\xi]. \tag{35}$$

To calculate the velocity from a limited width vortex cylindrical surface, we need to take the contour integral from the projection of this surface on the base plane (the integral along the length of the arc *s*).

$$\boldsymbol{\upsilon}\_{\mathbf{x}\_{1}} = \int\_{\mathbf{s}} d\boldsymbol{\upsilon}\_{\mathbf{x}\_{1}}; \boldsymbol{\upsilon}\_{\mathbf{y}\_{1}} = \int\_{\mathbf{s}} d\boldsymbol{\upsilon}\_{\mathbf{y}\_{1}}; \boldsymbol{\upsilon}\_{\mathbf{x}\_{1}} = \int\_{\mathbf{s}} d\boldsymbol{\upsilon}\_{\mathbf{x}\_{1}}; \boldsymbol{\upsilon}\_{0} = \int\_{\mathbf{s}} d\boldsymbol{\upsilon}\_{0}.\tag{36}$$
