Cosine Spacing

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KIVANÇ ALİ ANIL April 17, 2007508052003

COSINE SPACING

-c/2 ---- c/2 0 ---- c

( )cos2

c x= −

( )sin2

cdx x dx=

( )( )1 cos2

c x x= −

( )sin2

cdx x dx=

Vortex weight factor

n

dx x x

dxδ δ =

( )sin2

n

c x x

N

π δ =

( )2sin2

n

c x x

N

π δ =

( )21 cos2

n

c x x

N

π δ = −

( )cos2

c x= −

( )2

cos x xc

= −

( )2

cos xc

= −

22

12

n

c x x

c N

π δ

= − −

2

2

41

2n

c x x

c N

π δ = −

22

4n

c x x

N

π δ = −

Vortex weight factor

n

dx x x

dxδ δ =

( )sin2

n

c x x

N

π δ =

( )2sin2

n

c x x

N

π δ =

( )21 cos2

n

c x x

N

π δ = −

( )( )1 cos2

c x x= −

( )2

1 cos xc

= −

( )2

cos 1 xc

= −

22

1 12

n

c x x

c N

π δ

= − −

2

2

4 41 1

2n

c x x x

c c N

π δ

= − − +

2

2

4 41 1

2n

c x x x

c c N

π δ = − + −

( )2

2

4

2n

c

x cx xc N

π δ

= −

( )2

2

4

2n

c x cx x

c N

π δ = −

( )n

x c x x

N

π δ

−=

(Equation 72, Justin E. Kerwin)




-c/2 ---- c/2 0 ---- c

The equation of a parabolic mean line with

maximum camber 0 f is

( )2

021 x f x f c

= −

The equation of a parabolic mean line with

maximum camber 0 f is

( )

2

0

22

1

c x

f x f c

− = −

( )2

0

21

x c f x f

c

− = −

( )2

0

21 1

x f x f

c

= − −

( )2

0 2

4 41 1

x x f x f

c c

= − − +

( )2

0 2

4 4 x x f x f

c c

= −

( ) 0

41

x f x f

c c

= −

(Equation 5.80, Katz&Plotkin)The Slope is:

0

2

8df f x

dx c= −

The Slope is:

0 2

4 8df x f

dx c c

= −

0 24 1

df f x

dx c c

= −

(Equation 5.81, Katz&Plotkin)

We can see from the following figures that both formulations for cosine spacing give thesame results.




Figure 1. Input Window for The Parabolic Mean Line (Cosine Spacing -c/2 - c/2).

Figure 2. Input Window for The flow angle of attack (alpha in degrees) - Thedefault value is for CL = 1.

Figure 3. The Geometry & Results.




Figure 4. Input Window for The Parabolic Mean Line (Cosine Spacing 0 - c).






Figure 7. Input Window for The Parabolic Mean Line (Cosine Spacing 0- c).

Figure 8. Input Window for The flow angle of attack (alpha in degrees) - The

default value is for CL = 1.





Figure 12. Input Window for The Parabolic Mean Line (Cosine Spacing -c/2 - c/2).







Figure 15. Input Window for The Parabolic Mean Line (Cosine Spacing 0 - c).







Figure 18. Input Window for The Parabolic Mean Line (Cosine Spacing 0- c).






Figure 23. Input Window for The FLAT PLATE (Cosine Spacing -c/2 - c/2).






Figure 26. Input Window for The FLAT PLATE (Cosine Spacing -c/2 - c/2).







Figure 29. Input Window for The FLAT PLATE (Cosine Spacing 0 - c).







2-D Vortex Lattice Method with Cosine Spacing

(13.04 LECTURE NOTES HYDROFOILS AND PROPELLERS, Justin E. Kerwin)

pages 53-54

We first define an auxiliary angular variable x such that

( )( )1 cos2

c x x= − (69)

so that 0 x = corresponds to the leading edge and x π = corresponds to the trailing edge

x c= . This is the same as the change of variables introduced by Glauert, except that the x

coordinate has been shifted by c/2 to place the leading edge at 0 x = . We next divide the

chord into N equal intervals of x with common interval N δ π = . Point vortices nΓ are

located at the mid-points of each x interval, and control points are located at the

downstream boundary of each x interval,

(70)

( ) 1 cos2

c

c n x n

N

π = −

Note that with this spacing algorithm the last control point is at the trailing edge, c= .

The velocity induced at the n’th control point is simply,

(71)

where the last equality in 71 is a statement of the boundary condition developed earlier.

Equation 71, written for each of the N control points, represents a set of simultaneous

equations for the unknown point vortex strengths nΓ .

( )( )1/ 2

1 cos2

v

nc x n

N

π − = −

( ) ( )1

1

2

N m

n

m v c

df v U

x m x m dxα

π =

Γ = − = − −

∑




Figure 32. Input Window for VLM-2D for The Parabolic Mean Line



Figure 34. The Geometry & Results (VLM-2D for The Parabolic Mean Line)




Figure 36. Input Window for VLM-2D for The Parabolic Mean Line







Figure 40. Input Window for VLM-2D for The FLAT PLATE



Figure 42. The Geometry & Results (VLM-2D for The FLAT PLATE)




Figure 43. The Geometry & Results (VLM-2D for The FLAT PLATE)



C:\Documents and Settings\xp\My Documents\ZOVAN...\parabolic1.m PageApril 15, 2007 10:06:38

% Kivanc Ali ANIL (2007)

% 508052003%

% The Parabolic Mean Line% Cosine Spacing% -c/2 - c/2

clc, clear, close all,format compact

% ---------------------------------------------def0 = {'1','0.4','8','1'};

dlgTitle = 'The Parabolic Mean Line (Cosine Spacing -c/2 - c/2)';prompt = {'Chord Length (c)',...

'Maximum Chamber (fo) ',...

'Number of Panels (N)',...'Free stream velocity (U)'};

data = inputdlg(prompt,dlgTitle,1,def0);if isempty(data)==1

clearreturn

endc = str2num(char(data(1)));f0 = str2num(char(data(2)));

N = str2num(char(data(3)));U = str2num(char(data(4)));

defalpha = ((1-4*pi*f0/c)/(2*pi))*180/pi;def1 = {num2str(defalpha)};prompt = {'The flow angle of attack (alpha in degrees) - The default value is fo

CL = 1'};data = inputdlg(prompt,dlgTitle,1,def1);

if isempty(data)==1clear

return

endalpha = str2num(char(data(1)))*pi/180;% ---------------------------------------------% Exact solution

xtilda = 0:pi/36:pi;x = -(c/2)*cos(xtilda); % -c/2 - c/2

f = f0*(1-(2*x/c).^2); % -c/2 - c/2CL = 2*pi*alpha+4*pi*f0/c;

gamma = (-2*U*alpha*(1+cos(xtilda))./sin(xtilda))-8*U*sin(xtilda)*f0/c;warning off MATLAB:divideByZero% ---------------------------------------------

% The global coordinates for panelsGCO = [ ];

for k = 1:N+1xgc = -(c/2)*cos((k-1)*pi/N); % -c/2 - c/2

GCO = [GCO; xgc f0*(1-(2*xgc/c)^2)]; % -c/2 - c/2end% ---------------------------------------------

GVP = [ ]; % The global lump vortex points of the panelsGCP = [ ]; % The global collocation points of the panels

for k = 1:Ncp(k) = ((GCO(k,1)-GCO(k+1,1))^2+(GCO(k,2)-GCO(k+1,2))^2)^.5;

theta(k)= atan((GCO(k,2)-GCO(k+1,2))/(GCO(k,1)-GCO(k+1,1)));xgvp = GCO(k,1)+(cp(k)/4)*cos(theta(k));ygvp = GCO(k,2)+(cp(k)/4)*sin(theta(k));




GVP = [GVP; xgvp ygvp];

xgcp = GCO(k,1)+(3*cp(k)/4)*cos(theta(k));ygcp = GCO(k,2)+(3*cp(k)/4)*sin(theta(k));

GCP = [GCP; xgcp ygcp];dx(k) = sqrt((c^2/4)-GVP(k,1)^2)*pi/N; % Vortex weight factors -c/2 - c/2

end

% ---------------------------------------------for i = 1:N

for j = 1:Nr(i,j) = ((GVP(j,1)-GCP(i,1))^2+(GVP(j,2)-GCP(i,2))^2)^.5;

beta(i,j) = theta(i)- atan((GVP(j,2)-GCP(i,2))/(GVP(j,1)-GCP(i,1)));A(i,j) = (1/(2*pi*r(i,j)))*cos(beta(i,j));if j > i

A(i,j) = -A(i,j);end

endend

% [A] {X} = {b}b = -(U*(alpha-theta)); %sin(alpha-theta) = (alpha-theta) since (alpha-theta) is

allb = b'; % TransposeX = A\b;

Xplot = X./dx';%-U*c*pi*alpha % Lumped Vortex Element for Flat plate (f0 = 0, N = 1)

% ---------------------------------------------figure(1)subplot(2,1,1),plot(x,f,'k')

hold onplot(GCO(:,1),GCO(:,2),'r','linewidth',2)

plot(GCO(:,1),GCO(:,2),'g+','linewidth',2)plot(GVP(:,1),GVP(:,2),'o','linewidth',2)

plot(GCP(:,1),GCP(:,2),'x','linewidth',2)

axis equaltitle(['\fontsize{20}\bf{GEOMETRY}']);

grid% ---------------------------------------------

subplot(2,1,2),plot(x(2:length(x)),gamma(2:length(x)),'linewidth',2)hold on

plot(GVP(:,1),Xplot,'or','linewidth',2)legend('\fontsize{15}\bf{Exact Solution}',[num2str(N) , ' Panels'])

xlabel('\fontsize{15}\bf{x}');ylabel('\fontsize{15}\bf{\gamma (x)}');set(figure(1),'Position',[1,1,1400,930])

title(['\fontsize{20}\bf{\gamma (x) (for C_L = }',num2str(CL),')']);grid

% ---------------------------------------------





% 508052003%

% The Parabolic Mean Line% Cosine Spacing% 0 - c


% ---------------------------------------------def0 = {'1','0.4','8','1'};

dlgTitle = 'The Parabolic Mean Line (Cosine Spacing 0 - c)';prompt = {'Chord Length (c)',...




clearreturn






return


xtilda = 0:pi/36:pi;x = (c/2)*(1-cos(xtilda)); % 0 - c

f = f0*(1-(2*(x-c/2)/c).^2); % 0 - cCL = 2*pi*alpha+4*pi*f0/c;

gamma = (-2*U*alpha*(1+cos(xtilda))./sin(xtilda))-8*U*sin(xtilda)*f0/c;warning off MATLAB:divideByZero% ---------------------------------------------

% The global coordinates for panelsGCO = [ ];

for k = 1:N+1xgc = (c/2)*(1-cos((k-1)*pi/N)); % 0 - c

GCO = [GCO; xgc f0*(1-(2*(xgc-c/2)/c)^2)]; % 0 - cend% ---------------------------------------------

GVP = [ ]; % The global lump vortex points of the panelsGCP = [ ]; % The global collocation points of the panels

for k = 1:Ncp(k) = ((GCO(k,1)-GCO(k+1,1))^2+(GCO(k,2)-GCO(k+1,2))^2)^.5;

theta(k)= atan((GCO(k,2)-GCO(k+1,2))/(GCO(k,1)-GCO(k+1,1)));xgvp = GCO(k,1)+(cp(k)/4)*cos(theta(k));ygvp = GCO(k,2)+(cp(k)/4)*sin(theta(k));




GVP = [GVP; xgvp ygvp];

xgcp = GCO(k,1)+(3*cp(k)/4)*cos(theta(k));ygcp = GCO(k,2)+(3*cp(k)/4)*sin(theta(k));

GCP = [GCP; xgcp ygcp];dx(k) = sqrt((c-GVP(k,1))*GVP(k,1))*pi/N;% Vortex weight factors 0 - c

end

% ---------------------------------------------for i = 1:N

for j = 1:Nr(i,j) = ((GVP(j,1)-GCP(i,1))^2+(GVP(j,2)-GCP(i,2))^2)^.5;

beta(i,j) = theta(i)- atan((GVP(j,2)-GCP(i,2))/(GVP(j,1)-GCP(i,1)));A(i,j) = (1/(2*pi*r(i,j)))*cos(beta(i,j));if j > i

A(i,j) = -A(i,j);end

endend

% [A] {X} = {b}b = -(U*(alpha-theta)); %sin(alpha-theta) = (alpha-theta) since (alpha-theta) is

allb = b'; % TransposeX = A\b;

Xplot = X./dx';%-U*c*pi*alpha % Lumped Vortex Element for Flat plate (f0 = 0, N = 1)

% ---------------------------------------------figure(1)subplot(2,1,1),plot(x,f,'k')

hold onplot(GCO(:,1),GCO(:,2),'r','linewidth',2)


plot(GCP(:,1),GCP(:,2),'x','linewidth',2)

axis equaltitle(['\fontsize{20}\bf{GEOMETRY}']);

grid% ---------------------------------------------


plot(GVP(:,1),Xplot,'or','linewidth',2)legend('\fontsize{15}\bf{Exact Solution}',[num2str(N) , ' Panels'])

xlabel('\fontsize{15}\bf{x}');ylabel('\fontsize{15}\bf{\gamma (x)}');set(figure(1),'Position',[1,1,1400,930])


% ---------------------------------------------



C:\Documents and Settings\xp\My Documents\ZOVANC\Tur...\VLM2D.m PageApril 15, 2007 10:11:19


% 508052003%

% VLM-2D for The Parabolic Mean Line% Cosine Spacing% 0 - c


% ---------------------------------------------def0 = {'1','0.4','8','1'};

dlgTitle = 'VLM-2D for The Parabolic Mean Line';prompt = {'Chord Length (c)',...




clearreturn






return


xtilda = 0:pi/36:pi;x = (c/2)*(1-cos(xtilda)); % 0 - c

f = f0*(1-(2*(x-c/2)/c).^2);

CL = 2*pi*alpha+4*pi*f0/c;gamma = (-2*U*alpha*(1+cos(xtilda))./sin(xtilda))-8*U*sin(xtilda)*f0/c;warning off MATLAB:divideByZero

% ---------------------------------------------% The global coordinates for panels

GCO = [ ];for k = 1:N+1

xgc = (c/2)*(1-cos((k-1)*pi/N));GCO = [GCO; xgc 0];

end

% ---------------------------------------------GVP = [ ]; % The global lump vortex points of the panels

GCP = [ ]; % The global collocation points of the panelsfor k = 1:N

xgvp = (c/2)*(1-cos((k-1/2)*pi/N));GVP = [GVP; xgvp 0];xgcp = (c/2)*(1-cos(k*pi/N));



C:\Documents and Settings\xp\My Documents\ZOVANC\Tur...\VLM2D.m PageApril 15, 2007 10:11:19

GCP = [GCP; xgcp 0];

dx(k) = sqrt((c-GVP(k,1))*GVP(k,1))*pi/N;% Vortex weight factorsend

% ---------------------------------------------% Solution of Equation 71 for the Parabolic Mean Line% (13.04 LECTURE NOTES HYDROFOILS AND PROPELLERS, Justin E. Kerwin)

for i = 1:Nb(i,1) = U*((4*f0*(1-2*GCP(i,1)/c)/c)-alpha); %sin(alpha) = (alpha) since (alpha)

smallfor j = 1:N

r(i,j) = GVP(j,1)-GCP(i,1);A(i,j) = -(1/(2*pi*r(i,j)));

end

end% [A] {X} = {b}

X = A\b;Xplot = X./dx';

% ---------------------------------------------figure(1)

subplot(2,1,1),plot(x,f,'k')hold onplot(GCO(:,1),GCO(:,2),'r','linewidth',2)


plot(GCP(:,1),GCP(:,2),'x','linewidth',2)axis equaltitle(['\fontsize{20}\bf{GEOMETRY}']);

grid% ---------------------------------------------


plot(GVP(:,1),Xplot,'or','linewidth',2)

legend('\fontsize{15}\bf{Exact Solution}',[num2str(N) , ' Panels'])xlabel('\fontsize{15}\bf{x}');

ylabel('\fontsize{15}\bf{\gamma (x)}');set(figure(1),'Position',[1,1,1400,930])


% ---------------------------------------------

Cosine Spacing

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