Procedure to calculate a double integral

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PEF
PEF 2012-11-27
Good Morning everybody,
I'd like to perform what follows:
where
The parameters shown in the latter figure can be obtained as follows:
%%Inizialization
time = 614.4; % Analysis Time
Uhub = 11;
HubHt = 90;
alpha = 0.14;
TI = 'A'; % Turbulent Intensity (A,B,C as in the IEC or Specific value)
N1 = 4096;
N2 = 32;
N3 = 32;
N = N1*N2*N3; % Total Number of Point
t = 0:(time/(N1-1)):time; % Sampled Time Vector
L1 = Uhub*time; % Box length along X
L2 = 150; % Box length along Y
L3 = 220; % Box length along Z
dx = L1/N1; % Grid Resolution along X-axis
dy = L2/N2; % Grid Resolution along Y-axis
dz = L3/N3; % Grid Resolution along Z-axis
V = L1*L2*L3; % Analysis Box Volume
gamma = 3.9; % Turbulent Eddies Distorsion Factor
c = 1.476;
b = 5.6;
if HubHt < 60
lambda1 = 0.7*HubHt;
else
lambda1 = 42;
end
L = 0.8*lambda1;
if isequal(TI,'A')
Iref = 0.16;
sigma1 = Iref*(0.75*Uhub + b);
elseif isequal(TI,'B')
Iref = 0.14;
sigma1 = Iref*(0.75*Uhub + b);
elseif isequal(TI,'C')
Iref = 0.12;
sigma1 = Iref*(0.75*Uhub + b);
else
sigma1 = str2num(TI)*Uhub/100;
end
sigma_iso = 0.55*sigma1;
sigma2 = 0.7*sigma1;
sigma3 = 0.5*sigma1;
%%Wave number vectors
ik1 = cat(2,(-N1/2:-1/2),(1/2:N1/2));
ik2 = -N2/2:N2/2-1;
ik3 = -N3/2:N3/2-1;
[x y z] = ndgrid(ik1,ik2,ik3);
k1 = reshape((2*pi*L/L1)*x,N1*N2*N3,1);
k2 = reshape((2*pi*L/L2)*y,N1*N2*N3,1);
k3 = reshape((2*pi*L/L3)*z,N1*N2*N3,1);
k = sqrt(k1.^2 + k2.^2 + k3.^2);
%%Calculation of beta by means of the Energy Spectrum Integration
E = @(k) (1.453*k.^4)./((1 + k.^2).^(17/6));
%//Independent integration on segments
Local_int = arrayfun(@(i)quad(E,i-1,i), 2:(N1*N2*N3));
%//integral additivity + cumulative removal of queues
E_int = 1.5 - [0 fliplr(cumsum(fliplr(Local_int)))]; %//To remove queues
E_int = reshape(E_int,N,1);
S = k.*sqrt(E_int);
beta = (c*gamma)./S;
%%Help Parameters
k30 = k3 + beta.*k1;
k0 = sqrt(k1.^2 + k2.^2 + k30.^2);
C1 = (beta.*k1.^2.*(k1.^2 + k2.^2 - k3.*k30))./(k.^2.*(k1.^2 + k2.^2));
C2 = (k2.*k0.^2./((k1.^2 + k2.^2).^(3/2))).*atan2((beta.*k1.*sqrt(k1.^2 + k2.^2)),(k0.^2 - k30.*k1.*beta));
xhsi1 = C1 - (k2./k1).*C2;
xhsi2 = (k2./k1).*C1 + C2;
E_k0 = (1.453*k0.^4)./((1 + k0.^2).^(17/6));
Could you give me some hints or do you have a clue how to perform what shown in the first figure?
I thank you all in advance.
WKR, Francesco

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