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ThreeDEulerEqn.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3D Eulers-Equation solver Periodic in x,z and chebyshev in y
% Spectral Methods - Fourier Series in x,z and Chebyshev Polynomials in y
% Special Tau methods with Green's Functions to enforce divergence free
%
% Nelson Chen
% University of California, Berkeley
% Computational Fluid Dynamics Lab
% Last revision: 6/25/2016
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Global Variables initialization
global n o e r1 r2 g1 g2 g3 g4 I D D2 zero tau_3D tau_matrix
%% initialize Parameters
%Save parameters
save_count = 2;
save_num = 10;
% Number of modes
Nx = 128;
Ny = 129;
Nz = 4;
%Domain size
Lx = 2*pi;
Lz = 2*pi;
Ly = 2;
%Time
T = 8; %Final time
dt = 0.001; %time step-size
Nt = floor(T/dt)+1; %Number of time points
%% Initialize Variables
%increments
inc_x = Lx/Nx;
inc_z = Lz/Nz;
inc_theta = pi/(Ny-1);
%domain grids
xpts = 0:inc_x:2*pi-inc_x;
zpts = 0:inc_z:2*pi-inc_z;
%Chebyshev domain grids, equally spaced in theta
theta = pi:-inc_theta:0;
ypts = (cos(theta))*Ly/2;
%3D points
[X, Y, Z] = ndgrid(xpts, ypts, zpts);
%fourier modes in x and z
mode_x = [0:Nx/2, -Nx/2+1:-1];
mode_z = [0:Nz/2, -Nz/2+1:-1];
%3D array of modes correspond to x,y,z points
[mode_x_3D, mode_y_3D, mode_z_3D] = fModes(Nx, Ny, Nz);
%global tau-arrays for diagnostics
tau_3D = cell(Nx,Nz);
tau_matrix = cell(Nx,Nz);
%k = 2*pi*n/L, 3D arrays
kx_3D = 2*pi*mode_x_3D/Lx;
kz_3D = 2*pi*mode_z_3D/Lz;
VxFFC_2 = zeros(Nx,Ny,Nz);
VyFFC_2 = zeros(Nx,Ny,Nz);
VzFFC_2 = zeros(Nx,Ny,Nz);
Data = cell(floor((Nt-1)/save_num), 4);
DataFFC = cell(floor((Nt-1)/save_num),4);
%Chebychev derivative operators
D = ChDiffnoBC(Ny-1,Ly/2); %First derivative
D2 = D^2; %Second derivative
%dTN/dy
g1 = [D(1:end-2,end-1);1;(-1)^(Ny-2)];
g2 = [D(1:end-2,end);1;(-1)^(Ny-1)];
%identity matrix and zero array for convenience
I = eye(Ny,Ny);
zero = zeros(Nx,Ny,Nz);
%useful vectors for convenience
g3 = [zeros(Ny-2,1); 1;0];
g4 = [zeros(Ny-1,1); 1];
%Boundary vectors Chebyshev
n = 0:Ny-1;
o = ones(1,Ny);
e = (-o).^n;
r1 = o*D;
r2 = e*D;
%% Green's Functions pre-processing
[green_cell, TH_inv_cell] = Gen_green_euler(Nx, Nz, Lx, Lz, mode_x, mode_z, D);
'green'
%% Initial flows
% case 1: Column Vortex
% case 2: Shear flow for Kelvin-Helmholtz Instability
% case 3: Random analytical functions
int_case = 2;
[VxP0, VyP0, VzP0] = init_cond([Lx,Ly,Lz],X,Y,Z,mode_x_3D,mode_z_3D,int_case);
%Transform velocity field to spectral space
VxFFC0 = FFCT(VxP0);
VyFFC0 = FFCT(VyP0);
VzFFC0 = FFCT(VzP0);
%Save initial condition in data arrays
DataFFC(1,:) = {VxFFC0, VyFFC0, VzFFC0, 0};
Data(1,:) = {VxP0, VyP0, VzP0, 0};
%% First Step Forward Euler
%initialize velocity
VxFFC = DataFFC{1,1};
VyFFC = DataFFC{1,2};
VzFFC = DataFFC{1,3};
%physical space
VxP = Data{1,1};
VyP = Data{1,2};
VzP = Data{1,3};
%Calculate vorticity
[WxFFC, WyFFC, WzFFC] = vorticityFFC(VxFFC, VyFFC, VzFFC, Lx, Lz, D, mode_x_3D, mode_z_3D);
%Remove aliasing from nonlinear projection
BxFFC = twothird(WxFFC); ByFFC = twothird(WyFFC); BzFFC = twothird(WzFFC);
AxFFC = twothird(VxFFC); AyFFC = twothird(VyFFC); AzFFC = twothird(VzFFC);
AxP = iFFCT(AxFFC); AyP = iFFCT(AyFFC); AzP = iFFCT(AzFFC);
BxP = iFFCT(BxFFC); ByP = iFFCT(ByFFC); BzP = iFFCT(BzFFC);
%calculate velocity x vorticity in physical space
[VxWxP, VxWyP, VxWzP] = crossProd(AxP,AyP,AzP,BxP,ByP,BzP);
VxWxFFC_1 = FFCT(VxWxP);
VxWyFFC_1 = FFCT(VxWyP);
VxWzFFC_1 = FFCT(VxWzP);
%RHS of diff eq at current time-step
Rx = VxWxFFC_1;
Ry = VxWyFFC_1;
Rz = VxWzFFC_1;
%n+1/2 step
Vx_1half = VxFFC + dt*Rx;
Vy_1half = VyFFC + dt*Ry;
Vz_1half = VzFFC + dt*Rz;
%divergence of half step
DIV = FFCDiv(Vx_1half, Vy_1half, Vz_1half, Lx, Lz, mode_x_3D, mode_z_3D, D);
%special treatment for 0-0 modes
nx = 1; nz = 1;
VxFFC_2(nx,:,nz) = Vx_1half(nx,:,nz);
VyFFC_2(nx,:,nz) = 0;
VzFFC_2(nx,:,nz) = Vz_1half(nx,:,nz);
%calculating pressure and taus to satisfy boundary conditions and
%divergence-free velocity fields mode by mode
nz = 1;
kz = 2*pi*mode_z(nz)/Lz;
for nx = 2:Nx
kx = 2*pi*mode_x(nx)/Lx;
%Pressure Laplacian Matrix
TH_inv = TH_inv_cell{nx,nz};
vx = Vx_1half(nx,:,nz).';
vy = Vy_1half(nx,:,nz).';
vz = Vz_1half(nx,:,nz).';
div = DIV(nx,:,nz).';
[vx_new, vy_new, vz_new] = divFreeFFC(D,kx,kz,nx,nz,...
vx,vy,vz,div,green_cell,TH_inv);
VxFFC_2(nx,:,nz) = vx_new.';
VyFFC_2(nx,:,nz) = vy_new.';
VzFFC_2(nx,:,nz) = vz_new.';
end
for nx = 1:Nx
kx = 2*pi*mode_x(nx)/Lx;
for nz = 2:Nz
kz = 2*pi*mode_z(nz)/Lz;
TH_inv = TH_inv_cell{nx,nz};
vx = Vx_1half(nx,:,nz).';
vy = Vy_1half(nx,:,nz).';
vz = Vz_1half(nx,:,nz).';
div = DIV(nx,:,nz).';
[vx_new, vy_new, vz_new] = divFreeFFC(D,kx,kz,nx,nz,...
vx,vy,vz,div,green_cell,TH_inv);
VxFFC_2(nx,:,nz) = vx_new.';
VyFFC_2(nx,:,nz) = vy_new.';
VzFFC_2(nx,:,nz) = vz_new.';
end
end
%Convert to real velocities
VxP_2 = iFFCT(VxFFC_2);
VyP_2 = iFFCT(VyFFC_2);
VzP_2 = iFFCT(VzFFC_2);
if (save_num == 1)
%Save first step
DataFFC(2,:) = {VxFFC_2, VyFFC_2, VzFFC_2, dt};
Data(2,:) = {VxP_2, VyP_2, VzP_2, dt};
save_count = save_count + 1;
end
%% Adam-BashForth2 for remaining steps
for t = 3:Nt
%initialize velocity
VxFFC = VxFFC_2;
VyFFC = VyFFC_2;
VzFFC = VzFFC_2;
%physical space
VxP = VxP_2;
VyP = VyP_2;
VzP = VzP_2;
%Calculate vorticity
[WxFFC_2, WyFFC_2, WzFFC_2] = vorticityFFC(VxFFC, VyFFC, VzFFC, Lx, Lz, D, mode_x_3D, mode_z_3D);
%2/3 rule to de-alias nonlinear part
BxFFC = twothird(WxFFC_2); ByFFC = twothird(WyFFC_2); BzFFC = twothird(WzFFC_2);
AxFFC = twothird(VxFFC); AyFFC = twothird(VyFFC); AzFFC = twothird(VzFFC);
AxP = iFFCT(AxFFC); AyP = iFFCT(AyFFC); AzP = iFFCT(AzFFC);
BxP = iFFCT(BxFFC); ByP = iFFCT(ByFFC); BzP = iFFCT(BzFFC);
%calculate velocity x vorticity in physical space
[VxWxP_2, VxWyP_2, VxWzP_2] = crossProd(AxP,AyP,AzP,BxP,ByP,BzP);
VxWxFFC_2 = FFCT(VxWxP_2);
VxWyFFC_2 = FFCT(VxWyP_2);
VxWzFFC_2 = FFCT(VxWzP_2);
Rx = 3*VxWxFFC_2 - VxWxFFC_1;
Ry = 3*VxWyFFC_2 - VxWyFFC_1;
Rz = 3*VxWzFFC_2 - VxWzFFC_1;
%n+1/3 step
Vx_1half = VxFFC + 0.5*dt*Rx;
Vy_1half = VyFFC + 0.5*dt*Ry;
Vz_1half = VzFFC + 0.5*dt*Rz;
DIV = FFCDiv(Vx_1half, Vy_1half, Vz_1half, Lx, Lz, mode_x_3D, mode_z_3D, D);
VxFFC_2(1,:,1) = Vx_1half(1,:,1);
VyFFC_2(1,:,1) = 0;
VzFFC_2(1,:,1) = Vz_1half(1,:,1);
for nx = 2:Nx
nz = 1;
kx = 2*pi*mode_x(nx)/Lx;
kz = 2*pi*mode_z(nz)/Lz;
TH_inv = TH_inv_cell{nx,nz};
vx = Vx_1half(nx,:,nz).';
vy = Vy_1half(nx,:,nz).';
vz = Vz_1half(nx,:,nz).';
div = DIV(nx,:,nz).';
[vx_new, vy_new, vz_new] = divFreeFFC(D,kx,kz,nx,nz,...
vx,vy,vz,div,green_cell,TH_inv);
VxFFC_2(nx,:,nz) = vx_new.';
VyFFC_2(nx,:,nz) = vy_new.';
VzFFC_2(nx,:,nz) = vz_new.';
end
for nx = 1:Nx
kx = 2*pi*mode_x(nx)/Lx;
for nz = 2:Nz
kz = 2*pi*mode_z(nz)/Lz;
TH_inv = TH_inv_cell{nx,nz};
vx = Vx_1half(nx,:,nz).';
vy = Vy_1half(nx,:,nz).';
vz = Vz_1half(nx,:,nz).';
div = DIV(nx,:,nz).';
[vx_new, vy_new, vz_new] = divFreeFFC(D,kx,kz,nx,nz,...
vx,vy,vz,div,green_cell,TH_inv);
VxFFC_2(nx,:,nz) = vx_new.';
VyFFC_2(nx,:,nz) = vy_new.';
VzFFC_2(nx,:,nz) = vz_new.';
end
end
VxP_2 = iFFCT(VxFFC_2);
VyP_2 = iFFCT(VyFFC_2);
VzP_2 = iFFCT(VzFFC_2);
%Saving old advection step for Adam-Bashforth
VxWxFFC_1 = VxWxFFC_2;
VxWyFFC_1 = VxWyFFC_2;
VxWzFFC_1 = VxWzFFC_2;
%Print out time step
if (mod(t-1,save_num) == 0)
t
%saving velocities into cell array
DataFFC(save_count,:) = {VxFFC_2, VyFFC_2, VzFFC_2, (t-1)*dt};
Data(save_count,:) = {VxP_2, VyP_2, VzP_2, (t-1)*dt};
save_count = save_count + 1;
DIV_diagFFC = FFCDiv(VxFFC_2, VyFFC_2, VzFFC_2, Lx, Lz, mode_x_3D, mode_z_3D, D);
DIV_diagPPP = iFFCT(DIV_diagFFC);
max(max(max(abs(DIV_diagPPP))))
end
end