**2.2.1 Introduction**

The basic aim is to determine the lift of helicopter rotor blades by using singularity method, that is to develop an optimal model of aerodynamic rotor planning by simulating unsteady flow and viscous effects. The model is considered corresponding to rotor behavior in real conditions and with high quality from the aspect of engineer use. The idea of unsteady lift modeling and viscous effects simulation by using singularity method is based on the need for avoidance of very expensive experiments in the first stage of rotor planning by using contemporary aerodynamic analysis applied to available computer technique.

It is necessary to modulate vortex wake, unsteady 2D flow field characteristics and blade dynamic characteristics for determination of aerodynamic forces acting upon helicopter rotor blade.

In the first part airfoil is approximated by vortex and source panels, and boundary layer by layer of vortices changing their position during time. Vortex trail and trail of separation are modulated by free vortices. The separation is modulated by free vortices. The dependence of coefficient of lift on angle of attack is determined for known motion of an airfoil. The behavior of one separated vortex in an article time model is spread into a series of positions at a certain number at different vortices. The separated flow velocity profile is approximated by superposition of displacement thickness of these vortices coupled with the potential solution model. After every time step the position of free vortices is changed, for what it is required generation of new vortices that would all together satisfy the airfoil contour boundary conditions.

In addition to a complex problem of analytical modeling these phenomena there is a problem of modeling interactions between effects of these phenomena. This approach to

Vortex wake shape of airfoil oscillating through same angle of attack, with same amplitude, but with different reduced frequency obtained by this method is shown at Fig. 4. It is similar

Numerical method presented in this paper leads to inviscid flow field calculation over airfoil moving arbitrary in time if it is supposed that flow stays attached and that it is separated art trailing edge. Method is completely general although only results of oscillating

**2.2 An optimal main helicopter rotor projection model obtained by viscous effects and** 

This subchapter presents a model of airfoil which allows obtaining a method for optimal main helicopter rotor projection by viscous effect and unsteady lift simulation through algorithm and set of program entireties, applicable to ideological and main project of helicopter rotor. Numerical analysis considerations in this paper can be applied, on basis of a real rotors theoretical consideration, with sufficient accuracy in analysis and constructive realizations of helicopter rotor in real conditions. The method for unsteady viscous flow simulations by inviscid techniques is developed. This allows the to define such optimal conception model of aerodynamic rotor projection which is corresponding to rotor behavior

The basic aim is to determine the lift of helicopter rotor blades by using singularity method, that is to develop an optimal model of aerodynamic rotor planning by simulating unsteady flow and viscous effects. The model is considered corresponding to rotor behavior in real conditions and with high quality from the aspect of engineer use. The idea of unsteady lift modeling and viscous effects simulation by using singularity method is based on the need for avoidance of very expensive experiments in the first stage of rotor planning by using

It is necessary to modulate vortex wake, unsteady 2D flow field characteristics and blade dynamic characteristics for determination of aerodynamic forces acting upon helicopter

In the first part airfoil is approximated by vortex and source panels, and boundary layer by layer of vortices changing their position during time. Vortex trail and trail of separation are modulated by free vortices. The separation is modulated by free vortices. The dependence of coefficient of lift on angle of attack is determined for known motion of an airfoil. The behavior of one separated vortex in an article time model is spread into a series of positions at a certain number at different vortices. The separated flow velocity profile is approximated by superposition of displacement thickness of these vortices coupled with the potential solution model. After every time step the position of free vortices is changed, for what it is required generation of new vortices that would all together satisfy the airfoil contour

In addition to a complex problem of analytical modeling these phenomena there is a problem of modeling interactions between effects of these phenomena. This approach to

in real condition and with sufficiently quality from the aspect of engineer use.

contemporary aerodynamic analysis applied to available computer technique.

with one obtained by methods of visualization in tunnels.

at high frequencies are presented.

**unsteady lift simulation** 

**2.2.1 Introduction** 

rotor blade.

boundary conditions.

helicopter rotor planning allows experimental investigations in tunnels and in flight to be final examinations. In that way, a very useful interaction between numerical calculation and experimental results is achieved.

The finale result is obtaining of unsteady lift dependence of angle of attack. Model established in such a way is characteristic for the helicopter rotor blade airflow and it is based on the influence of the previous lifting surface's wake influence on the next coming blade.

#### **2.2.2 Foundations of the irrotational 2-D flow**

The planar potential flow of incompressible fluid can be treated in Cartesian coordinates *x* and *y*. Two dimensional potential incompressible flow is completely defined by the speed potential and stream function and is presented by Cauchy-Reimann equations:

$$\frac{\partial}{\partial \mathbf{x}} \frac{\partial}{\partial \mathbf{x}} + \frac{\partial \,\boldsymbol{\varphi}}{\partial \mathbf{y}} = \mathbf{0} \qquad \text{i} \qquad \frac{\partial \,\boldsymbol{\varphi}}{\partial \mathbf{x}} - \frac{\partial \,\boldsymbol{\varphi}}{\partial \mathbf{y}} = \mathbf{0}$$

where *u* and *v* are velocities in *x* and *y* directions respectively.

The Laplace partial differential equations can also be introduced <sup>2</sup> 0 and <sup>2</sup> 0 . The fulfillment of Cauchy-Reimann conditions enables combining of the velocity potential and stream function *wz x* (,) (,) *y i x y* . This complex function entirely defines the planar potential flow of incompressible fluid as a function of a complex coordinate. The complex analytical function *w* (*z*), called the complex flow potential, always has a unique value for the first derivative. This derivative of the complex potential is equal to the complex velocity at that point, i.e.:

$$\frac{dw}{dz} = u - iv = \overline{V}$$

where *u* is its real part (velocity component in *x*-direction) and *v* is the imaginary part (velocity component in *y*-direction).

Circulation and flow are equal to zero for any closed curve in complex plane. Complex potential has no singularities except at the stagnation point.

#### **2.2.3 Simulation of the moving vortex**

If we assume that the moving vortex of intensity 0 is at the distance *z*0, than the perturbance potential of such a flow is:

$$\ln w(z) = V\_{\alpha} z e^{-i\alpha} + V\_{\alpha} \frac{a^2}{z} e^{+i\alpha} + i2a \cdot V\_{\alpha} \cdot \sin\left(\alpha - \beta\right) \ln z + \frac{i\Gamma\_0}{2\pi} \ln(z - z\_0) - \frac{i\Gamma\_0}{2\pi} \ln\left(\frac{a^2}{z} - \overline{z}\_0\right)$$

and it's complex velocity:

$$\overline{V} = V\_{\alpha}e^{-i\alpha} - V\_{\alpha}\frac{a^2}{z^2}e^{+i\alpha} + \frac{i}{z}2a \cdot V\_{\alpha} \cdot \sin(\alpha - \beta) + \frac{i\Gamma\_0}{2\pi} \frac{1}{(z - z\_0)} + \frac{i\Gamma\_0(a^2/z^2)}{2\pi\left(\frac{a^2}{z} - \overline{z}\_0\right)}$$

generation of a new vortex 2 which is equal do the difference of vortex intensity before

 

*i i a a i i w z V ze V e i aV sin z z z <sup>z</sup>*

 

*i i a i <sup>i</sup> i az V V e V e a V sin*

11 2 2

( ) ln ln( ) ln

*i ia i i a*

22 2 2

1 2 1 2

*i i i az i az z z a a z z*

*dx dy dt x dt y*

( ) ( ) ( ) ( )

1 1

2 2

 

1 2

2 2

 

Vortices travel down the flowfield by its velocity so that their position is determined by

, . *<sup>i</sup> <sup>i</sup> i i*

Position of the vortex at a new moment can be determined by *dz V dt* m m and its

Now a new distance of each vortex as well as the trajectory of the vortex or any fluid particle

2

*m*

*z z a*

2

*m*

<sup>2</sup>

*aV sin*

 

2 2

*z z*

2 2

*z z*

2

 

2 2

 

*z z zz z*

2 2

 

1 12 2

 2 2 2

 

<sup>1</sup> <sup>2</sup>

*z z z z a*

2 2

2 2 2 2

1 2

*z z z z*

2 2

*z z z zz z*

 

*aV sin*

 

0 0

2 2

 

0 0

0

0 0

2

2 0

(9)

*z z a*

0

1

( )

( )

2

0 0

0

*z z*

( )

(7)

(8)

 

and after displacing of the free moving vortex from time t1 to time t2.

2

 

ln( ) ln ln( ) ln

Now the total complex potential has the value:

 

elementary displacement n s *z z z tV* mmm .

1

2

<sup>n</sup> <sup>m</sup>

1

( )

m

*i i*

 

2

 

*z z z*

*i i m*

 

*<sup>a</sup> <sup>i</sup> i i z z t Ve V e*

solving the system of equations:

is given by n s *z z tV* m m , or:

n s m m

Complex velocity is:

 <sup>2</sup> 4 4 

In this case displacement of the stagnation point appears, which is now at the distance *<sup>i</sup> z ae* . According to the request that complex velocity at the stagnation point must be zero:

$$\frac{dw}{dz} = \overline{V} = 0$$

we can determine position *z* and according to that ' i.e. ( ) 0 *<sup>i</sup> V ae* . The angle ' defines new position of stagnation point, while the difference of angles is = - ' . In order to keep the stagnation point at the steady position, a vortex of intensity 1 must be released.Intensity of the vortices that are released can be determined according to the Kelvin's theorem *d dt* 0 .

So every change in circulation must be compensated by vortex inside of cylinder/airfoil of the opposite sign, whose intensity is a function of the variation of circulation around the cylinder *<sup>i</sup> d dt t* and it is equal do the difference of the intensities before and after the free moving vortex is introduced 1 1 <sup>1</sup> 4 *a V sin sin* .

Now the total complex potential has the value:

$$w(z) = w\_0(z) + w\_{\Gamma\_0}(z) + w\_{\Gamma\_1}(z)$$

where:

$$w\_0(z) = V\_\infty z e^{-i\alpha} + V\_\infty \frac{a^2}{z} e^{+i\alpha} + i2aV\_\infty \sin(\alpha - \beta) \ln z$$

$$w\_{\Gamma\_0}(z) = \frac{i\Gamma\_0}{2\pi} \ln(z - z\_0) - \frac{i\Gamma\_0}{2\pi} \ln\left(\frac{a^2}{z} - \overline{z}\_0\right) \quad , \quad w\_{\Gamma\_1}(z) = \frac{i\Gamma\_1}{2\pi} \ln(z - z\_1) - \frac{i\Gamma\_1}{2\pi} \ln\left(\frac{a^2}{z} - \overline{z}\_1\right)$$

Complex velocity is:

$$\begin{split} \overline{V} &= V\_{\alpha} e^{-i\alpha} - V\_{\alpha} \frac{a^2}{z^2} e^{+i\alpha} + \frac{i}{z} 2a \cdot V\_{\alpha} \cdot \sin\left(\alpha - \beta\right) \\ &+ \frac{i\Gamma\_0}{2\pi} \frac{1}{\left(z - z\_0\right)} + \frac{i\Gamma\_0 \left(a^2 \Big/ z^2\right)}{2\pi \left(\frac{a^2}{z} - \overline{z}\_0\right)} + \frac{i\Gamma\_1}{2\pi} \frac{1}{\left(z - z\_1\right)} + \frac{i\Gamma\_1 \left(a^2 \Big/ z^2\right)}{2\pi \left(\frac{a^2}{z} - \overline{z}\_1\right)} \end{split} \tag{6}$$

After a period of time t induced velocity at the trailing edge is a consequence of the disposition of both moving and released vortex.

Setting again the condition that a point at the circle is stagnation point and complex velocity at that point equal to zero, we determine a new position of the stagnation point by new angle ' . This new angle ' defines a new difference = - '. So the new position of the stagnation point is defined by distance *<sup>i</sup> z ae* . This again causes

In this case displacement of the stagnation point appears, which is now at the

<sup>0</sup> *dw <sup>V</sup> dz*

keep the stagnation point at the steady position, a vortex of intensity 1 must be released.Intensity of the vortices that are released can be determined according to the

So every change in circulation must be compensated by vortex inside of cylinder/airfoil of the opposite sign, whose intensity is a function of the variation of circulation around the cylinder *<sup>i</sup> d dt t* and it is equal do the difference of the intensities before and after

0 1 <sup>0</sup> *wz w z w z w z* () () () ()

 

2 <sup>0</sup>( ) <sup>2</sup> ln *i i <sup>a</sup> w z V ze V e i aV sin z z*

, <sup>1</sup>

2

1 1

 

0 0 1 1

*i i az i i az z z a a z z*

( ) ( ) ( ) ( )

0 1

After a period of time t induced velocity at the trailing edge is a consequence of the

Setting again the condition that a point at the circle is stagnation point and complex velocity at that point equal to zero, we determine a new position of the stagnation point by new angle ' . This new angle ' defines a new difference = - '. So the new

2 2

 

2

the free moving vortex is introduced 1 1 <sup>1</sup> 4

. According to the request that complex velocity at the stagnation point

' i.e. ( ) 0 *<sup>i</sup> V ae* 

 *a V sin sin* 

1 1 1 1 2 2 ( ) ln( ) ln *i ia w z zz z*

 *z*

 

 

2 2 2 2

 

0 1

 

. This again causes

*z z z z*

2 2

. The angle

 

2

(6)

.

 = -  ' . In order to

' defines

distance *<sup>i</sup> z ae*

must be zero:

where:

0

Complex velocity is:

Kelvin's theorem *d dt* 0 .

we can determine position *z* and according to that

Now the total complex potential has the value:

0 0 0 0 2 2 ( ) ln( ) ln *i i <sup>a</sup> w z zz z*

 *z*

> 2 2

position of the stagnation point is defined by distance *<sup>i</sup> z ae*

*i i a i V V e V e a V sin z z*

2 2

 

disposition of both moving and released vortex.

new position of stagnation point, while the difference of angles is

generation of a new vortex 2 which is equal do the difference of vortex intensity before and after displacing of the free moving vortex from time t1 to time t2.

$$\Gamma\_2 = \Delta \Gamma = 4\pi a V\_\infty \sin\left(\alpha - \beta'\right) - 4\pi a V\_\infty \sin\left(\alpha - \beta\right)$$

Now the total complex potential has the value:

$$\begin{split} w(z) &= V\_{\alpha} z e^{-i\alpha} + V\_{\alpha} \frac{a^2}{z} e^{+i\alpha} + i2a V\_{\alpha} \sin(\alpha - \beta) \ln z + \frac{i\Gamma\_0}{2\pi} \ln(z - z\_0) - \frac{i\Gamma\_0}{2\pi} \ln\left(\frac{a^2}{z} - \overline{z}\_0\right) \\ &+ \frac{i\Gamma\_1}{2\pi} \ln(z - z\_1) - \frac{i\Gamma\_1}{2\pi} \ln\left(\frac{a^2}{z} - \overline{z}\_1\right) + \frac{i\Gamma\_2}{2\pi} \ln(z - z\_2) - \frac{i\Gamma\_2}{2\pi} \ln\left(\frac{a^2}{z} - \overline{z}\_2\right) \end{split} \tag{7}$$

Complex velocity is:

$$\begin{aligned} \overline{V} &= V\_{\infty} e^{-i\alpha} - V\_{\infty} \frac{a^2}{z^2} e^{+i\alpha} + \frac{i}{z} 2a \cdot V\_{\infty} \sin(\alpha - \beta) + \frac{i\Gamma\_0}{2\pi} \frac{1}{(z - z\_0)} + \frac{i\Gamma\_0 (a^2/z^2)}{2\pi \left(\frac{a^2}{z} - \overline{z}\_0\right)} \\\\ &+ \frac{i\Gamma\_1}{2\pi} \frac{1}{(z - z\_1)} + \frac{i\Gamma\_1 (a^2/z^2)}{2\pi \left(\frac{a^2}{z} - \overline{z}\_1\right)} + \frac{i\Gamma\_2}{2\pi} \frac{1}{(z - z\_2)} + \frac{i\Gamma\_2 (a^2/z^2)}{2\pi \left(\frac{a^2}{z} - \overline{z}\_2\right)} \end{aligned} \tag{8}$$

Vortices travel down the flowfield by its velocity so that their position is determined by solving the system of equations:

$$\frac{d\mathbf{x}\_i}{dt} = \frac{\partial \wp}{\partial \mathbf{x}}\bigg|\_{i} \qquad \frac{d\mathbf{y}\_i}{dt} = \frac{\partial \wp}{\partial \mathbf{y}}\bigg|\_{i} \dots$$

Position of the vortex at a new moment can be determined by *dz V dt* m m and its elementary displacement n s *z z z tV* mmm .

Now a new distance of each vortex as well as the trajectory of the vortex or any fluid particle is given by n s *z z tV* m m , or:

$$\begin{aligned} \Xi\_{\rm m}^{\rm m} &= \Xi\_{\rm m}^{\rm s} + \Delta t \cdot \left\{ V\_{\rm ce} e^{-i\alpha} - V\_{\rm ce} \frac{a^2}{z^2} e^{+i\alpha} + \frac{i(\Gamma - \Gamma\_m)}{2\pi \, z} + \frac{i\Gamma\_0}{2\pi} \frac{1}{\left(z - z\_0\right)} + \frac{i\Gamma\_0}{2\pi \left(z - \frac{z^2}{a^2} \overline{z}\_0\right)} + \cdots \right\} \\\\ &\quad \sum\_{1}^{n} \left[ \frac{i\Gamma\_m}{2\pi} \frac{1}{\left(z - z\_m\right)} + \frac{i\Gamma\_m}{2\pi \left(z - \frac{z^2}{a^2} \overline{z}\_m\right)} \right] \end{aligned} \tag{9}$$

Analysis of results achieved by unsteady lift modeling and viscous effects simulation method shows that they can be used with sufficient accuracy in rotor analysis and

The aim of this particular simulation is to use advantages of vortex methods. For example, vortex methods use the description of flow field of the smallest range; aerodynamic forces can be obtained with small number of vortices. On the other hand, singular vortex distribution can be accurately determined by using data obtained in small time range. Vortex methods, also, permit boundary layer simulation at large Reynolds's number by

Achieved results imply the direction for the further development of this program

the model should be expanded for transonic flow in the aim of blade tip analysis

 a computational dispersion and gradual disappearing of vortex wake are possible by using viscous vortex shell, as more elegant solution than violent elimination of vortex

an inclusion of curved vortex elements in aim of achieving better results from vortex

On the basis of analysis of presented model and program package for viscous effects and unsteady lift simulation it can be concluded that this subchapter presents an original scientific contribution, applicable in aerodynamic analyses of practical problems in

Fig. 9. Fig. 10.

Fig. 11. Fig. 12.

local concentration of computational points.

wake used in this program

wake self-induction aspect

helicopter rotor projecting.

**2.2.6 Conclusion remarks** 

construction.
