Short Description About Quasi-Continuous Method -LITERATURE REVIEW

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FIRDAUS BIN MAHAMAD | UNIVERSITI TEKNOLOGI MALAYSIA

2 LITERATURE REVIEW

2.1 Characteristics of Quasi-Continuous Method

Detailed description of QCM was already made by Lan for planar wigs. However, for

convenience, a brief explanation of the essential feature of QCM is repeated here. One of the

characteristics of QCM is in the treatment of the continuous loading in chordwise direction.

Since the spanwise loading distribution is assumed stepwise constant, we may explain at first the

numerical procedure of the chordwise integration by the case of two-dimensional wing. The

downwash integral is expressed as follow,

� �� = 12 ��

� − ��

… … … … … … … … . �1�

where γ (ξ) is vortex strength and ξ is non-dimensional chordwise coordinate. Transforming the ξ

-coordinate into α-coordinate through the relation

� = 12 �1 − �� … … … . . . . … … … … … … . . �2�

Eq. (1) can be written as

� �� = 12 ��

�� − �� … … … . . �3��

Denoting further

�� = �� … … … … … … … … … … . . �4�

and eliminating the Cauchy singularity in the integrand, Eq. (3) is rewritten as

�� = − 12 � �� − ��

�� − ��

�� − ��

2 �� − ��

��

= − 12 � �� − ��

�� − ��

��, … … . . �5�

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Since g(α) does not contain square root singularity at leading edge, the integrand in Eq. (5) is

infinite everywhere. Thus the integral can be reduced to a finite sum through midpoint

trapezoidal rule as follows.

�� ≅ − 12! " ��#� − ��

�� − �� #

$

#%�

= − 12! " ��#�

�� − �� #

$

#%�+ ��

2! " 1�� − �� #

$

#%�… . . �6�

where

N = number of chordwise integration points,

�# = �2# − 1�2! , �( = 1, 2, … , !� … . … . �7�

Control points, where flow tangency condition is to satisfied, are chosen so as to

eliminate the second term of Eq. (6) by using a characteristic of Chebychev polynomials as

" 1�� * − �� #

$

#%�= 0 … … … … … … … … . . �8�

where

�* = -! , �- = 1, 2, … … , !�, . . … . . … … … … �9�

Then the following equation is obtained

�/�*0 = − 12! " ��#�

�� * − ��# = − 12! " ��#� �� #�� * − �� #

$

#%�

$

#%�… … … … … … … �10�

As shown in the above, chordwise vortex distribution is treated as continuous in QCM,

but the resulting numerical formula Eq. (10) looks like that of VLM in which discrete and

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concentrated vortex distribution is used. In case of two-dimensional wing, vortex distribution

γ(α) obtained by QCM agree very well with analytical solution.

As for three-dimensional wing, the characteristics of QCM can be summarized as

follows;

1) Continuous vortex distribution over a wing is replaced with a quasi-continuous one, i.e.

continuous in chordwise direction and stepwise constant in spanwise direction. In the

surface integral over a chordwise vortex strip, spanwise integration is carried out

analytically, while chordwise integration is carried out numerically by the midpoint

trapezoidal rule. The square root singularity at leading edge and the Cauchy singularity

are properly accounted for as mention above.

2) Loading points (integration points) and control points over a wing are arranged according

to the semi-circle method as shown in Fig. 1. Chordwise arrangement is chosen so as to

eliminate the Cauchy singularity. Since control points are placed on trailing edge, Kutta

condition can be satisfied automatically, i.e. the flow is tangent to wing at the trailing

edge. In the spanwise direction, finer vortex strips are used toward wing tip where

sectional properties vary rapidly. Outermost loading sections do not coincide with wing

tips and it contributes to the quick convergence of calculation.

Fig. 1 Arrangement of loading and control points over a wing

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3) Accuracy in circulation is satisfactory. As an example, the results calculated for a planar

circular wing are shown in comparison with analytical solution in Fig. 2. Spanwise lift

distribution and chordwise loading distributions at center (η=0.0) and at quarter-span

(η=0.50) sections obtained by QCM agree very well with analytical solution. Sufficient

accuracy is retained also in the neighborhood of wing tip (η=0.98). Geometry of the

propeller blade in tip region is similar to that of the circular wing. Then accurate solution

can be expected in the neighborhood of the propeller blade tip.

4) Quick convergence is attained. Fig. 3 shows the result calculated for a planar rectangular

wing of aspect ratio 2.0. The numbers of spanwise control points M and chordwise

control points N were varied systematically. Converged solution is obtained with a small

number of control points.

Fig. 2 Spanwise lift and chordwise loading distributions for a planar circular wing

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Fig. 3 Lift coefficient for a planar rectangular wing of aspect ratio 2.0

2.2 Formulation of Numerical Model of Propeller Lifting Surface

Propeller blades are represented by the distributions of vortices and sources on the mean

camber surface of blades together with the associated distribution of vortices shed into the wake.

The vortex system, i.e. the distribution of horseshoe vortices which consist of bound vortices and

free vortices shed from both edges of bound vortices, represents blade loading and wake, and the

source distribution represents blade thickness.

Continuous distributions of bound vortices and sources are replaced with quasi-

continuous ones according to QCM. Thus the blades are covered with a number or vortex strips

with associated free vortices and source strips.

Loading points and control points are arranged on the mean chamber surface of blade

according to QCM. In the radial direction, loading stations (radial coordinates of the edges of

each vortex/ source strip) and control stations (radial coordinates of control points) are chosen at

the radii defined as follows. For loading stations

123 = 12 �1� + 14� − 1

2 �1� − 15� �� 63 ,

7 = 1, 2, … … , 8 + 1

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and for control stations

19: = 12 �1� + 14� − 1

2 �1� − 15� �� 6: , � = 1, 2, … … , 8,

where

ro = propeller radius,

rB = boss radius,

M = number of radial control points,

63 = 27 − 12�8 + 1� , �7 = 1, 2, … … , 8 + 1�

6: = �8 + 1 , �� = 1, 2, … … , 8 + 1�

At the loading and control stations, chordwise loading and control points are arranged according

to QCM, i.e. for loading points

�3; = �2/1230 + 12 �/1230�1 − �� ;�,

< = 1, 2, … … , !,

and for control points

�:* = �2�19:� + 12 ��19:�/1 − �� *0,

- = 1, 2, … … , !, where

N = number of chordwise control points, �; = �2< − 1�

2! , �< = 1, 2, … … , !�,

�* = -! , �- = 1, 2, … … , !�,

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Then, loading points and control points are arranged on the same fraction-chord lines

respectively. An example of arrangement of loading lines and loading stations with control

points is shown in Fig. 4. Free vortices are shed only from the loading stations due to stepwise

distribution of vortices. The trailing vortex sheet (free vortex sheet in the wake) is, therefore

composed of (M+1) helical vortices.

Fig. 4 Arrangement of loading lines, and

loading stations and control points

Fig. 5 Vortex segment and horseshoe

vortex

2.3 Calculation of Induced Velocities

Induced velocity at a control point Pij due to a vortex segment of unit strength place

between loading point Qµν and Qµ + 1ν is given by Biot-Savart law as

(=:*3;> = 14 1�???= × 1�A?????=

|1�???= × 1�A?????=|A C 1A???=|1A???=| − 1�???=

|1�???=|D . 1�A?????= . … … … … … … �11�

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Consider a horseshoe vortex of unit strength, which consists of bound vortex placed at

Qµν and Qµ + 1ν and two free vortices shed from Qµν and Qµ + 1ν in the chordwise direction along

loading stations (see Fig.5). Induced velocity at a control point Pij due to this horseshoe vortex is

expressed by

(=:*3;E = (=:*3;4 + " F(=:*3G�;HI − (=:*3;HI J + (=:*3G�K − (=:*3K$

;H % ;

where

(=:*3;4 = induced velocity at Pij due to vortex segment (bound vortex) of unit strength

placed between Qµν and Qµ +1ν,

(=:*3;I = induced velocity at Pij due to vortex segment (free vortex) of unit strength

placed between Qµν and Qµ +1ν,

(=:*3 K = induced velocity at Pij due to trailing vortex of unit strength shed from

trailing edge at loading station r = r Lµ,

(=:*3;4 and (=:*3;I can be calculated by Eq.11 and (=:*3K also can be calculated with Eq. 11 by

replacing helical vortex line with vortex segments of appropriate length.

Let the strength of bound vortices in the µ- th vortex strip be γµ(s), then induced velocity

at Pij due to horseshoe vortices on the vortex strip is expressed based on the procedure described

in the previous section, as

L?=:*3E = � �3MK

M2��(=:*3E ��

= �32 � �3�

��(=:*3E ��

≅ �32! " �3$

;%��;�(=:*3E ��;� �� ;

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= �32! " �3;$

;%�(=:*3;E �� ; ,

where

�3 = �/�930, �3; = �3��;�, (=:*3;E = (=:*3E ��;�. The induces velocity a control point Pij due to whole vortex system is obtained by the next equation

L?=:*E = " " L?=:*3NEO

3%�

P

N%�

= 2! " " �3

O

3%�

P

N%�" �3;

$

;%�(=:*;NE �� ; ,

where the suffix k shows the contribution from the k-th blade.

Next, consider the induced velocity due to the source distribution. Induced velocity at a

control point Pij due to a source segment of unit strength placed between loading point Qµν and

Qµ + 1ν (see Fig. 5) is given in a similar manner to that of vortex segment by

(=:*3;M = 14 1�???= × 1�A?????=

|1�???= × 1�A?????=|A × C 1A???=|1A???=| − 1�???=

|1�???=|D . |1�A?????=|.

Denoting the source strength at the ν-th loading point in the µ-th source strip by σµν , the induced

velocity at a control point Pij due to whole source distribution is expressed as

L?=:*M = 2! " " �3

O

3%�

P

N%�" �3;

$

;%�(=:*3;NM �� .

2.4 Calculation of Thrust, Torque and Pressure Distribution

2.4.1 Potential Component

Force acting on a vortex/source segment placed between Qµν and Qµ+1ν can be obtained from Kutta-Joukowski theorem and Lagally’s therorm as

Q=3;= R L?=3; × �=3;− R L?=3; �3;ST=3;S

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where

L?=3;= L?=3;E + L?=3;M + L?=3;U = resultant flow velocity at the midpoint of segment

�=3;= �3;T=3;

= vortex segment vector

T=3; = segment vector V3;V??????????=3G�;

ρ = fluid density

Then, thrust and torque acting on whole propeller are obtained by following equations,

W� = −X � �1�YZ

Y[� Q\

M]

M^��

= −X " �3∗2!O

3%�"�Q\�3;

$

;%�� ; ,

V� = X � �1�YZ

Y[� /−Q̀ . a + Qb . c0��M]

M^

= X " �3∗2!O

3%�" d−/Q̀ 03;a3; + �Qb�3;c3; e

$

;%�� ; ,

where

(Fx)µν, (Fy)µν, (Fz)µν=x-, y- and z- components of Q=3;, xµν, yµν, zµν = x-, y- and z- coordinate at the

midpoint of the segment, �3∗ = cord length at r= ½ (rLµ + rLµ+ 1).

In addition to the forces acting on vortex and source distributions mentioned in the above,

leading edge suction force must be taken into consideration. The leading edge suction force

results from leading edge singularity of vortex distribution under thin wing assumption. The

suction force per unit span is given by

QM = 14 R�fMA

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where

fM = g�hM→M2 ��j� − �2�

and the coefficient Cs can be obtained directly in QCM then, trust and torque resulting from

leading edge suction force are expressed as

WM = X � QM�1�� k�YZ

Y[1��1�

= X �1� − 14�28 " QM�19:� �� k�19:� �� 6:

O

:%�

VM = −X � QM�1�� k�YZ

Y[1��1��1�

= −X �1� − 14�28 " QM�19:� �� k�19:� 19:�� 6:

O

:%�

2.4.2 Viscous Component

Let the viscous drag coefficient of blade element at the radius r be CD (r) , then trust and torque

due to viscous drag are expressed as

Wl = −X 12 R � fl�1��mnYZ

Y[�1��Lo\�1��1��1�

= −XR �1� − 14�48 " fl�19:� mn �19:�. Lo\

O

:%��19:��19:� �� 6:

Vl = X 12 R � fl�1��mnYZ

Y[�1��Lop�1��1��1��1�

= XR �1� − 14�48 " fl�19:� mn �19:�. Lop

O

:%��19:��19:� �� 6:

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where

mn �1� = qLo\�1�A + Lop�1�A,

Lo\�1� = x-component of resultant flow velocity averaged in the chorwise direction.

Lop�1� = θ-component of resultant flow velocity averaged in the chordwise direction.

The total thrust and torque are given by

T= T0+Ts+TD,

Q= Q0+Qs+QD,

The trust coefficient KT, torque coefficient KQ and propeller efficiency eP are defined by

XK = Krstlu , Xv = v

rstlw, xR = yA� . P]

Pz

where

D = propeller diameter,

n = propeller revolutions per unit time,

J = VA/nD : advance distribution

The result flow velocities on back and face at the control points Pij on the mean camber surface are surface are expressed as

L?=:*± = L?=:*E + L?=:*| + L?=:*U ± 12 /�:* . }=:* + ~:*. �o:*0,

where

γij = vortex strength at Pij ,

σij = source strength at Pij,

}=:* = unit chordwise tangent vector at Pij.

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Subscripts ± denote back (suction side) and face (pressure side) respectively. The pressure coefficients at point Pij on back and face are expressed by Bernoulli’s theorem as

f�:*± = �:*± − ��12 Rm�A

= 1 − SL?=:*±SAm�A ,

where

P0 = ambient pressure in uniform flow,

W0 = qL�A + �19:��A .

Short Description About Quasi-Continuous Method -LITERATURE REVIEW

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