5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 1/26
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)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 2/26
KIVANÇ ALİ ANIL April 17, 2007508052003
-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.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 3/26
KIVANÇ ALİ ANIL April 17, 2007508052003
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.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 4/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 4. Input Window for The Parabolic Mean Line (Cosine Spacing 0 - c).
Figure 5. Input Window for The flow angle of attack (alpha in degrees) - Thedefault value is for CL = 1.
Figure 6. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 5/26
KIVANÇ ALİ ANIL April 17, 2007508052003
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 9. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 6/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 10. The Geometry & Results.
Figure 11. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 7/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 12. Input Window for The Parabolic Mean Line (Cosine Spacing -c/2 - c/2).
Figure 13. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 14. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 8/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 15. Input Window for The Parabolic Mean Line (Cosine Spacing 0 - c).
Figure 16. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 17. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 9/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 18. Input Window for The Parabolic Mean Line (Cosine Spacing 0- c).
Figure 19. Input Window for The flow angle of attack (alpha in degrees) - Thedefault value is for CL = 1.
Figure 20. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 10/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 21. The Geometry & Results.
Figure 22. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 11/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 23. Input Window for The FLAT PLATE (Cosine Spacing -c/2 - c/2).
Figure 24. Input Window for The flow angle of attack (alpha in degrees) - Thedefault value is for CL = 1.
Figure 25. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 12/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 26. Input Window for The FLAT PLATE (Cosine Spacing -c/2 - c/2).
Figure 27. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 28. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 13/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 29. Input Window for The FLAT PLATE (Cosine Spacing 0 - c).
Figure 30. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 31. The Geometry & Results.
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 14/26
KIVANÇ ALİ ANIL April 17, 2007508052003
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 = − = − −
∑
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 15/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 32. Input Window for VLM-2D for The Parabolic Mean Line
Figure 33. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 34. The Geometry & Results (VLM-2D for The Parabolic Mean Line)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 16/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 35. The Geometry & Results (VLM-2D for The Parabolic Mean Line)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 17/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 36. Input Window for VLM-2D for The Parabolic Mean Line
Figure 37. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 38. The Geometry & Results (VLM-2D for The Parabolic Mean Line)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 18/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 39. The Geometry & Results (VLM-2D for The Parabolic Mean Line)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 19/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 40. Input Window for VLM-2D for The FLAT PLATE
Figure 41. Input Window for The flow angle of attack (alpha in degrees) - The
default value is for CL = 1.
Figure 42. The Geometry & Results (VLM-2D for The FLAT PLATE)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 20/26
KIVANÇ ALİ ANIL April 17, 2007508052003
Figure 43. The Geometry & Results (VLM-2D for The FLAT PLATE)
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 21/26
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));
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 22/26
C:\Documents and Settings\xp\My Documents\ZOVAN...\parabolic1.m PageApril 15, 2007 10:06:38
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
% ---------------------------------------------
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 23/26
C:\Documents and Settings\xp\My Documents\ZOVAN...\parabolic2.m PageApril 15, 2007 10:10:34
% Kivanc Ali ANIL (2007)
% 508052003%
% The Parabolic Mean Line% Cosine Spacing% 0 - c
clc, clear, close all,format compact
% ---------------------------------------------def0 = {'1','0.4','8','1'};
dlgTitle = 'The Parabolic Mean Line (Cosine Spacing 0 - c)';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)*(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));
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 24/26
C:\Documents and Settings\xp\My Documents\ZOVAN...\parabolic2.m PageApril 15, 2007 10:10:34
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(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
% ---------------------------------------------
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 25/26
C:\Documents and Settings\xp\My Documents\ZOVANC\Tur...\VLM2D.m PageApril 15, 2007 10:11:19
% Kivanc Ali ANIL (2007)
% 508052003%
% VLM-2D for The Parabolic Mean Line% Cosine Spacing% 0 - c
clc, clear, close all,format compact
% ---------------------------------------------def0 = {'1','0.4','8','1'};
dlgTitle = 'VLM-2D for The Parabolic Mean Line';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)*(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));
5/14/2018 Cosine Spacing - slidepdf.com
http://slidepdf.com/reader/full/cosine-spacing 26/26
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(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
% ---------------------------------------------
Top Related