├── Bubble rising.m ├── Curvature flow.m ├── FinalProjectdroplet.m ├── LICENSE ├── Numerical simulation of Multiphase flow using level set method.pdf └── README.md /Bubble rising.m: -------------------------------------------------------------------------------- 1 | 2 | % level set method to capture the interface 3 | clear all; clc; close all; 4 | Lx = 1.0; Ly = 1.0; % domain size 5 | gx = 0.0; gy = -100.0; rho1 = 1; rho2 = 2; mu = 0.01; % parameters 6 | unorth = 0; usouth = 0; veast = 0; vwest = 0; % boundary conditions 7 | rad = 0.15; xc = 0.5; yc = 0.5; % initial drop size and location 8 | D=1; % 9 | time = 0.0; plot_freq = 10; 10 | 11 | nx = 256; ny = 256; dx = Lx/nx; dy = Ly/ny; dt = 0.0001; 12 | 13 | nstep = 2000; maxit = 200; maxError = 0.001; omg = 1.5; Nf = 100; 14 | 15 | u=zeros(nx+1,ny+2); ut = u ; uplot = zeros(nx+1,ny+1); 16 | v=zeros(nx+2,ny+1); vt = u ; vplot = zeros(nx+1,ny+1); 17 | 18 | 19 | p=zeros(nx+2,ny+2); tmp1 = p ; tmp2 = p; r = p; chi = p; 20 | C=zeros(nx+2,ny+2); gamma=0; %zeros(nx+2,ny+2); 21 | Cn=C; % C at n time step 22 | phi=zeros(nx+2, ny+2); % signed distance function 23 | phin=zeros(nx+2, ny+2);% phi at n timestep 24 | sphi=zeros(nx+2,ny+2); % signed phi in the reinitialization process 25 | psi=zeros(nx+2,ny+2); % psi to satisfy the steady state solution 26 | psin=zeros(nx+2,ny+2); % fi at n timestep 27 | eps= 1.5*dx; % used for smooth out chi 28 | epsilon=0.0000001; % used for calculating sign of phi 29 | Ddelta=zeros(nx+2,ny+2); % Dirac delta 30 | % 31 | 32 | % stargerred grid 33 | xh = linspace(0,Lx,nx+1) ; yh = linspace(0,Ly,ny+1); % velocity points 34 | x = linspace(-dx/2,Lx+dx/2,nx+2); y = linspace(-dy/2,Ly+dy/2,ny+2); % pressure points 35 | 36 | r = zeros(nx+2,ny+2) + rho2; % initial density 37 | fgx=zeros(nx+2,ny+2); fgy=zeros(nx+2,ny+2); % initial surface tension 38 | 39 | % initialization 40 | % the initial shape is a circle 41 | % initialize front 42 | for i=1:nx+2;for j=1:ny+2 43 | if((x(i)-xc)^2+(y(j)-yc)^2 < rad^2); 44 | phi(i,j)= -( rad-sqrt((x(i)-xc)^2 + (y(j)-yc)^2)); 45 | %chi(i,j)= 1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps);%1.0; 46 | r(i,j) = rho1; 47 | elseif((x(i)-xc)^2+(y(j)-yc)^2 > rad^2); 48 | phi(i,j)= sqrt((x(i)-xc)^2 + (y(j)-yc)^2)-rad; 49 | %chi(i,j)=0.0; 50 | else 51 | phi(i,j)=0.0; 52 | %chi(i,j)=0.0; 53 | end; 54 | if(phi(i,j) <-eps) 55 | chi(i,j)=0.0; 56 | Ddelta(i,j)=0; 57 | elseif(phi(i,j)>eps) 58 | chi(i,j)=1.; 59 | Ddelta(i,j)=0; 60 | else 61 | chi(i,j)=1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps); 62 | Ddelta(i,j)=1/(2*eps)*(1+cos(pi*phi(i,j)/eps)); 63 | end; 64 | end;end; 65 | %r = rho1*chi + rho2*(1-chi); % initial density 66 | figure(1); contourf(x,y,phi'); axis equal; axis([0 Lx 0 Ly]); 67 | hold on;axis equal; axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 68 | ylabel('y', 'Fontsize', 20); 69 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 70 | colorbar;%caxis([-1 1]) 71 | drawnow; hold off 72 | 73 | figure(2); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; 74 | axis equal; axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 75 | ylabel('y', 'Fontsize', 20); 76 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 77 | colorbar;%caxis([-1 1]) 78 | drawnow; hold off 79 | % for i = 2:nx+1; for j = 2:ny+1 80 | % if((x(i)-xc)^2+(y(j)-yc)^2 < rad^2); r(i,j) = rho1; chi(i,j)=1.0; end; 81 | % end; end 82 | 83 | 84 | 85 | 86 | for is=1:nstep 87 | 88 | 89 | 90 | 91 | %%%figure(4); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 92 | %figure(3); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 93 | 94 | ro = r; 95 | r = rho2*chi + rho1*(1-chi); % obtain density from charact func 96 | 97 | figure(5); contourf(x,y,r'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 98 | 99 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 100 | %Calculate the surface tension 101 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 102 | 103 | for i=2:nx+1; for j=2:ny+1 104 | fgx(i,j)=gamma * ((phi(i+1,j)-2.0 * phi(i,j)+phi(i-1,j))/(dx^2) + (phi(i,j+1)-2*phi(i,j)+phi(i,j-1))/(dy^2)) * Ddelta(i,j) * ((phi(i+1,j)-phi(i,j))/dx); % using central 105 | fgy(i,j)=gamma * ((phi(i+1,j)-2.0 * phi(i,j)+phi(i-1,j))/(dx^2) + (phi(i,j+1)-2*phi(i,j)+phi(i,j-1))/(dy^2)) * Ddelta(i,j) * ((phi(i,j+1)-phi(i,j))/dy); 106 | end; 107 | end; 108 | fgx(1:nx+2,1)=fgx(1:nx+2,2);fgx(1:nx+2,ny+2)=fgx(1:nx+2,ny+1); 109 | fgx(1,1:ny+2)=fgx(2,1:ny+2);fgx(ny+2,1:ny+2)=fgx(ny+1,1:ny+2); 110 | fgy(1:nx+2,1)=fgy(1:nx+2,2);fgy(1:nx+2,ny+2)=fgy(1:nx+2,ny+1); 111 | fgy(1,1:ny+2)=fgy(2,1:ny+2);fgy(ny+2,1:ny+2)=fgy(ny+1,1:ny+2); 112 | 113 | %fgx(1:nx+2,2) = fgx(1:nx+2,2) + fgx(1:nx+2,1); fgx(1:nx+2,ny+1) = fgx(1:nx+2,ny+1) + fgx(1:nx+2,ny+2); % bring all forces to interior 114 | % fgy(2,1:ny+2) = fgy(2,1:ny+2) + fgy(1,1:ny+2); fgy(nx+1,1:ny+2) = fgy(nx+1,1:ny+2) + fgy(nx+2,1:ny+2); % boundary condition for surface tension 115 | 116 | 117 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 118 | % Solving N-S equations using projection method 119 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 120 | u(1:nx+1,1) = 2*usouth-u(1:nx+1,2); u(1:nx+1,ny+2) = 2*unorth-u(1:nx+1,ny+1); % tangential vel BC 121 | v(1,1:ny+1) = 2*vwest -v(2,1:ny+1); v(nx+2,1:ny+1) = 2*veast -v(nx+1,1:ny+1); % tangential vel BC 122 | 123 | for i=2:nx; for j=2:ny+1 % temporary u-velocity (boundary values are not touched) 124 | ut(i,j) = (2.0/(r(i+1,j)+r(i,j)))*(0.5*(ro(i+1,j)+ro(i,j))*u(i,j)+ dt* (... 125 | - (0.25/dx)*(ro(i+1,j)*(u(i+1,j)+u(i,j))^2-ro(i,j)*(u(i,j)+u(i-1,j))^2)... 126 | - (0.0625/dy)*( (ro(i,j)+ro(i+1,j)+ro(i,j+1)+ro(i+1,j+1))*(u(i,j+1)+u(i,j))*(v(i+1,j)+v(i,j)) ... 127 | - (ro(i,j)+ro(i+1,j)+ro(i+1,j-1)+ro(i,j-1))*(u(i,j)+u(i,j-1))*(v(i+1,j-1)+v(i,j-1)))... 128 | + mu*((u(i+1,j)-2*u(i,j)+u(i-1,j))/dx^2+ (u(i,j+1)-2*u(i,j)+u(i,j-1))/dy^2)... 129 | + 0.5*(ro(i+1,j)+ro(i,j))*gx + (fgx(i+1,j)+fgx(i,j))/2.0 )); 130 | end; end 131 | 132 | for i=2:nx+1; for j=2:ny % temporary v-velocity (boundary values are not touched) 133 | vt(i,j) = (2.0/(r(i,j+1)+r(i,j)))*(0.5*(ro(i,j+1)+ro(i,j))*v(i,j)+ dt* (... 134 | - (0.0625/dx)*( (ro(i,j)+ro(i+1,j)+ro(i+1,j+1)+ro(i,j+1))*(u(i,j)+u(i,j+1))*(v(i,j)+v(i+1,j)) ... 135 | - (ro(i,j)+ro(i,j+1)+ro(i-1,j+1)+ro(i-1,j))*(u(i-1,j+1)+u(i-1,j))*(v(i,j)+v(i-1,j)) )... 136 | - (0.25/dy)*(ro(i,j+1)*(v(i,j+1)+v(i,j))^2-ro(i,j)*(v(i,j)+v(i,j-1))^2 )... 137 | + mu*((v(i+1,j)-2*v(i,j)+v(i-1,j))/dx^2+(v(i,j+1)-2*v(i,j)+v(i,j-1))/dy^2)... 138 | + 0.5*(ro(i,j+1)+ro(i,j))*gy + (fgy(i,j+1)+fgy(i,j))/2.0 ) ); 139 | end; end 140 | 141 | for i = 2:nx+1; for j = 2:ny+1 142 | tmp1(i,j) = (0.5/dt)*( (ut(i,j)-ut(i-1,j))/dx+(vt(i,j)-vt(i,j-1))/dy ); 143 | tmp2(i,j) =1/( (1/dx)*(1/(dx*(r(i+1,j)+r(i,j)))+ 1/(dx*(r(i-1,j)+r(i,j))) )+ ... 144 | (1/dy)*(1/(dy*(r(i,j+1)+r(i,j)))+ 1/(dy*(r(i,j-1)+r(i,j))) ) ); 145 | end; end 146 | 147 | for it = 1:maxit % solve for pressure by SOR 148 | pold = p; 149 | p(1,:) = p(2,:); p(nx+2,:) = p(nx+1,:); p(:,1) = p(:,2); p(:,ny+2) = p(:,ny+1); % set gosht values 150 | for i=2:nx+1; for j=2:ny+1 151 | p(i,j) = (1.0-omg)*p(i,j) + omg*tmp2(i,j)*( ... 152 | (1/dx)*( p(i+1,j)/(dx*(r(i+1,j)+r(i,j)))+ p(i-1,j)/(dx*(r(i-1,j)+r(i,j))) )+ ... 153 | (1/dy)*( p(i,j+1)/(dy*(r(i,j+1)+r(i,j)))+ p(i,j-1)/(dy*(r(i,j-1)+r(i,j))) ) - tmp1(i,j)); 154 | end; end 155 | if max(max(abs(pold-p))) < maxError; break; end 156 | end 157 | 158 | for i=2:nx; for j=2:ny+1 % correct the u-velocity 159 | u(i,j)=ut(i,j)-dt*(2.0/dx)*(p(i+1,j)-p(i,j))/(r(i+1,j)+r(i,j)); 160 | end; end 161 | 162 | for i=2:nx+1; for j=2:ny % correct the v-velocity 163 | v(i,j)=vt(i,j)-dt*(2.0/dy)*(p(i,j+1)-p(i,j))/(r(i,j+1)+r(i,j)); 164 | end; end 165 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 166 | 167 | 168 | %%%%%%%%%%%%%%%%%%%%%%% 169 | % interface propagation 170 | %%%%%%%%%%%%%%%%%%%%%%%% 171 | phin=phi; 172 | 173 | for i=2:nx+1; 174 | for j=2:ny+1; 175 | if(((u(i,j)+u(i-1,j))/2.0)>=0); 176 | dphi_x=(phin(i,j)-phin(i-1,j))/dx; % upwind scheme 177 | else; 178 | dphi_x=(phin(i+1,j)-phin(i,j))/dx; 179 | end; 180 | if(((v(i,j)+v(i,j-1))/2.0)>=0); 181 | dphi_y=(phin(i,j)-phin(i,j-1))/dy; 182 | else 183 | dphi_y=(phin(i,j+1)-phin(i,j))/dy; 184 | end; 185 | % phi(i,j)= phin(i,j)+ dt * D* ( (phin(i+1,j)-2*phin(i,j)+phin(i-1,j))/(dx^2) + (phin(i,j+1) - 2*phin(i,j) +phin(i,j-1))/(dy^2) ); %curvature flow 186 | phi(i,j)= phin(i,j)- dt * ((u(i,j)+u(i-1,j))/2.0*dphi_x+(v(i,j)+v(i,j-1))/2.0*dphi_y); % drop falling flow 187 | % phi(i,j) =phin(i,j) - u(i,j) 188 | end; 189 | end; 190 | % figure(4); contourf(x,y,phi'); axis equal; axis([0 Lx 0 Ly]); hold on; 191 | % axis equal; axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 192 | % ylabel('y', 'Fontsize', 20); 193 | % title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 194 | % colorbar;%caxis([-1 1]) 195 | % drawnow; hold off 196 | % update phi to n+1 timestep 197 | 198 | %%%%%%%%%%%%%%%%%%%%%%%%%%%% 199 | % interface reinitialization 200 | %%%%%%%%%%%%%%%%%%%%%%%%%%%% 201 | for i=2:nx+1;for j=2:ny+1; 202 | 203 | sphi(i,j)=phi(i,j)/(sqrt(phi(i,j)^2 +epsilon^2)); 204 | end;end; 205 | figure(3); contourf(x,y,sphi'); axis equal; axis([0 Lx 0 Ly]); hold on; 206 | axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 207 | ylabel('y', 'Fontsize', 20); 208 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 209 | colorbar; 210 | drawnow; hold off 211 | psi=phi; 212 | 213 | for k=1:20; % iteration for 20 times 214 | psin=psi; 215 | for i=2:nx+1; 216 | for j=2:ny+1; 217 | dphix1=(psin(i,j)-psin(i-1,j))/dx; 218 | dphix2=(psin(i+1,j)-psin(i,j))/dx; 219 | dphiy1=(psin(i,j)-psin(i,j-1))/dy; 220 | dphiy2=(psin(i,j+1)-psin(i,j))/dy; 221 | 222 | % reinitialization near the interface 223 | if(psi(i,j)>0.); 224 | gphi=1-sqrt(max(max(dphix1,0.)^2, min(dphix2,0.)^2) +max(max(dphiy1, 0.)^2, min(dphiy2, 0.)^2)); 225 | elseif(psi(i,j)<0.); 226 | gphi=1-sqrt(max(min(dphix1,0.)^2, max(dphix2,0.)^2)+max(min(dphiy1, 0.)^2, max(dphiy2, 0.)^2)); 227 | else 228 | gphi=0.; 229 | end; 230 | % gphi=((psin(i+1,j)-psin(i-1,j))/(2*dx))^2 +((psin(i,j+1)-psin(i,j-1))/(2*dy))^2; 231 | psi(i,j)=psin(i,j)+dt*sphi(i,j)*gphi; 232 | end; 233 | end; 234 | 235 | 236 | end; 237 | 238 | phi=psi; 239 | 240 | %%%%%%%%%%get the characteristic function at new time step%%%%%%% 241 | for i=2:nx+1;for j=2:ny+1 242 | if(phi(i,j) <-eps) % inside the drop chi=0 243 | chi(i,j)=0.; 244 | Ddelta(i,j)=0.; 245 | elseif(phi(i,j)>eps) % outside the drop chi=1 246 | chi(i,j)=1.; 247 | Ddelta(i,j)=0.; 248 | else 249 | chi(i,j)=1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps); 250 | Ddelta(i,j)=1/(2*eps)*(1+cos(pi*phi(i,j)/eps)); 251 | end; 252 | end;end; 253 | 254 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 255 | 256 | 257 | time = time+dt 258 | if (mod(is,plot_freq)==0) | (is==1); % plot solution 259 | uu(1:nx+1,1:ny+1)=0.5*(u(1:nx+1,2:ny+2)+u(1:nx+1,1:ny+1)); 260 | vv(1:nx+1,1:ny+1)=0.5*(v(2:nx+2,1:ny+1)+v(1:nx+1,1:ny+1)); 261 | %figure(5); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off %contour(x,y,r'); 262 | figure(6); contourf(x,y,phi'); 263 | axis equal; %hold on;drawnow; hold off 264 | quiver(xh,yh,uu',vv','r'); axis([0 Lx 0 Ly]);hold on; 265 | axis equal; axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 266 | ylabel('y', 'Fontsize', 20); 267 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 268 | colorbar;%caxis([-1 1]) 269 | drawnow; hold off 270 | drawnow; hold off 271 | end 272 | 273 | end 274 | -------------------------------------------------------------------------------- /Curvature flow.m: -------------------------------------------------------------------------------- 1 | 2 | % level set method to capture the interface 3 | 4 | clear all; clc; close all; 5 | Lx = 1.0; Ly = 1.0; % domain size 6 | gx = 0.0; gy = -100.0; rho1 = 2; rho2 = 1; mu = 0.01; % parameters 7 | unorth = 0; usouth = 0; veast = 0; vwest = 0; % boundary conditions 8 | rad = 0.15; xc = 0.5; yc = 0.5; % initial drop size and location 9 | D=1; % diffusion coefficient 10 | time = 0.0; plot_freq = 30; pi=3.415926; 11 | 12 | nx = 256; ny = 256; dx = Lx/nx; dy = Ly/ny; dt = 0.00001; 13 | 14 | nstep = 1200; maxit = 200; maxError = 0.001; omg = 1.5; 15 | 16 | u=zeros(nx+1,ny+2); ut = u ; uplot = zeros(nx+1,ny+1); 17 | v=zeros(nx+2,ny+1); vt = v ; vplot = zeros(nx+1,ny+1); % vt=u??? 18 | 19 | 20 | p=zeros(nx+2,ny+2); tmp1 = p ; tmp2 = p; r = p; chi = p; 21 | C=zeros(nx+2,ny+2); gamma=1; %zeros(nx+2,ny+2); 22 | Cn=C; % C at n time step 23 | phi=zeros(nx+2, ny+2); % signed distance function 24 | phin=zeros(nx+2, ny+2);% phi at n timestep 25 | sphi=zeros(nx+2,ny+2); % signed phi in the reinitialization process 26 | psi=zeros(nx+2,ny+2); % psi to satisfy the steady state solution 27 | psin=zeros(nx+2,ny+2); % psi at n timestep 28 | eps= 1.5*dx; % used for smooth out chi 29 | epsilon=0.00001; % used for calculating sign of phi 30 | Ddelta=zeros(nx+2,ny+2); % Dirac delta 31 | % 32 | R1=zeros(1,nstep+1); R1(1)=rad; % Radius of the circle 33 | %xf=zeros(1,Nf+2); yf=zeros(1,Nf+2); 34 | %un=zeros(1,Nf+2); vn=zeros(1,Nf+2); 35 | 36 | % stargerred grid 37 | xh = linspace(0,Lx,nx+1) ; yh = linspace(0,Ly,ny+1); % velocity points 38 | x = linspace(-dx/2,Lx+dx/2,nx+2); y = linspace(-dy/2,Ly+dy/2,ny+2); % pressure points 39 | 40 | r = zeros(nx+2,ny+2) + rho2; % initial density 41 | fgx=zeros(nx+2,ny+2); fgy=zeros(nx+2,ny+2); % initial surface tension 42 | 43 | % initialization 44 | 45 | 46 | for i=2:nx+1; 47 | for j=2:ny+1; 48 | u(i,j)=1; 49 | end; 50 | end; 51 | 52 | 53 | % the initial shape is a circle 54 | 55 | for i=1:nx+2;for j=1:ny+2 56 | if((x(i)-xc)^2+(y(j)-yc)^2 < rad^2); 57 | phi(i,j)= -( rad-sqrt((x(i)-xc)^2 + (y(j)-yc)^2)); 58 | %chi(i,j)= 1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps);%1.0; 59 | r(i,j) = rho1; 60 | elseif((x(i)-xc)^2+(y(j)-yc)^2 > rad^2); 61 | phi(i,j)= sqrt((x(i)-xc)^2 + (y(j)-yc)^2)-rad; 62 | %chi(i,j)=0.0; 63 | else 64 | phi(i,j)=0.0; 65 | %chi(i,j)=0.0; 66 | end; 67 | if(phi(i,j) <-eps) 68 | chi(i,j)=0; 69 | Ddelta(i,j)=0; 70 | elseif(phi(i,j)>eps) 71 | chi(i,j)=1.; 72 | Ddelta(i,j)=0; 73 | else 74 | chi(i,j)=1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps); 75 | Ddelta(i,j)=1/(2*eps)*(1+cos(pi*phi(i,j)/eps)); 76 | end; 77 | end;end; 78 | 79 | figure(1); contourf(x,y,phi'); axis equal; axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 80 | ylabel('y', 'Fontsize', 20); 81 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 82 | colorbar;%caxis([-1 1]) 83 | drawnow; hold off 84 | figure(2); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; xlabel('x', 'Fontsize', 20); 85 | ylabel('y', 'Fontsize', 20); 86 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 87 | colorbar;%caxis([-1 1]) 88 | drawnow; hold off 89 | % for i = 2:nx+1; for j = 2:ny+1 90 | % if((x(i)-xc)^2+(y(j)-yc)^2 < rad^2); r(i,j) = rho1; chi(i,j)=1.0; end; 91 | % end; end 92 | 93 | 94 | 95 | 96 | for is=1:nstep 97 | 98 | 99 | %%%%%%%%%%%%%%%%%%%%%%% 100 | % interface propagation 101 | %%%%%%%%%%%%%%%%%%%%%%%% 102 | phin=phi; 103 | 104 | %%%% curvature flow %%%%%%%%%%%%%%%%%%%%%%%%%%% 105 | %%%%%%%%validation 106 | for it = 1:maxit 107 | 108 | phi_old=phi; 109 | 110 | for i=2:nx+1; 111 | for j=2: ny+1; 112 | 113 | phi(i,j)= (omg)*1.0/(1.0 + 2.0*dt/dx^2*D+2*dt/dy^2*D)* ... 114 | (((phi(i+1,j)+phi(i-1,j))*dt/dx^2*D +(phi(i,j+1)+phi(i,j-1))*dt/dy^2*D) +phin(i,j) )+ ... 115 | (1-omg)*phi(i,j); 116 | end; 117 | end; 118 | phi(1,:) = phi(2,:); phi(nx+2,:) =phi(nx+1, :); phi(:,1)=phi(:,2); phi(:,ny+2) =phi(:,ny+1) ; % no flux B.C, set ghost values 119 | if max(max(abs(phi_old-phi))) < maxError; break; end 120 | end 121 | 122 | 123 | % figure(4); contourf(x,y,phi'); axis equal; axis([0 Lx 0 Ly]); hold on; 124 | % xlabel('x', 'Fontsize', 20); 125 | % ylabel('y', 'Fontsize', 20); 126 | % title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 127 | % colorbar;%caxis([-1 1]) 128 | % drawnow; hold off 129 | % update phi to n+1 timestep 130 | 131 | %%%%%%%%%%%%%%%%%%%%%%%%%%%% 132 | % interface reinitialization 133 | %%%%%%%%%%%%%%%%%%%%%%%%%%%% 134 | for i=2:nx+1;for j=2:ny+1; 135 | 136 | sphi(i,j)= phi(i,j)/(sqrt(phi(i,j)^2 +epsilon^2)); 137 | end;end; 138 | 139 | figure(3); contourf(x,y,sphi'); axis equal; axis([0 Lx 0 Ly]); hold on; 140 | xlabel('x', 'Fontsize', 20); 141 | ylabel('y', 'Fontsize', 20); 142 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 143 | colorbar;%caxis([-1 1]) 144 | drawnow; hold off 145 | 146 | psi=phi; % initial value of the reinitialization process 147 | 148 | 149 | % if 150 | 151 | for k=1:5; % iteration for 5 time steps 152 | psin=psi; 153 | for i=2:nx+1; 154 | for j=2:ny+1; 155 | dphix1=(psi(i,j)-psi(i-1,j))/dx; 156 | dphix2=(psi(i+1,j)-psi(i,j))/dx; 157 | dphiy1=(psi(i,j)-psi(i,j-1))/dy; 158 | dphiy2=(psi(i,j+1)-psi(i,j))/dy; 159 | 160 | % upwind scheme 161 | if(psi(i,j)>0.); 162 | gphi=1.0-sqrt(max(max(dphix1,0.)^2, min(dphix2,0.)^2) +max(max(dphiy1, 0.)^2, min(dphiy2, 0.)^2)); 163 | elseif(psi(i,j) < 0.); 164 | gphi=1.0-sqrt(max(min(dphix1,0.)^2, max(dphix2,0.)^2)+max(min(dphiy1, 0.)^2, max(dphiy2, 0.)^2)); 165 | else 166 | gphi=0.; 167 | end; 168 | psi(i,j)=psin(i,j)+dt*sphi(i,j)*gphi; 169 | end; 170 | end; 171 | end; 172 | % 173 | % phi=psi; 174 | 175 | 176 | %calculate the characteristic function 177 | for i=2:nx+1;for j=2:ny+1 178 | if(phi(i,j) <-eps) 179 | chi(i,j)=0.; 180 | Ddelta(i,j)=0.; 181 | elseif(phi(i,j)>eps) 182 | chi(i,j)=1.; 183 | Ddelta(i,j)=0.; 184 | else 185 | chi(i,j)=1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps); 186 | Ddelta(i,j)=1/(2.0*eps)*(1+cos(pi*phi(i,j)/eps)); 187 | end; 188 | end;end; 189 | 190 | %%%figure(4); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 191 | %figure(3); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 192 | 193 | ro = r; 194 | r = rho1*chi + rho2*(1-chi); % obtain density from charact func 195 | 196 | 197 | % interpolate the values of the concentration to the front 198 | % for l=1:Nf+1 199 | % ip = floor(xf(l)/dx)+1; jp = floor((yf(l))/dy)+1; % find the closest point 200 | % ax = xf(l)/dx-ip+1; ay = (yf(l))/dy-jp+1; 201 | % cf(l) = (1.0-ax)*(1.0-ay)*C(ip,jp) + ax*(1.0-ay)*C(ip+1,jp) + (1.0-ax)*ay*C(ip,jp+1) + ax*ay*C(ip+1,jp+1); 202 | % end 203 | % cf(Nf+2)=cf(2); 204 | % gammaCf=10.0 -3.0.*cf; % surface tension at the front 205 | 206 | 207 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 208 | %Calculate the surface tension 209 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 210 | 211 | for i=2:nx+1; for j=2:ny+1 212 | fgx(i,j)=gamma * ((phi(i+1,j)-2.0 * phi(i,j)+phi(i-1,j))/(dx^2) + (phi(i,j+1)-2*phi(i,j)+phi(i,j-1))/(dy^2)) * Ddelta(i,j) * ((phi(i+1,j)-phi(i,j))/dx); 213 | fgy(i,j)=gamma * ((phi(i+1,j)-2.0 * phi(i,j)+phi(i-1,j))/(dx^2) + (phi(i,j+1)-2*phi(i,j)+phi(i,j-1))/(dy^2)) * Ddelta(i,j) * ((phi(i,j+1)-phi(i,j))/dy); 214 | end; 215 | end; 216 | 217 | %fgx(1:nx+2,2) = fgx(1:nx+2,2) + fgx(1:nx+2,1); fgx(1:nx+2,ny+1) = fgx(1:nx+2,ny+1) + fgx(1:nx+2,ny+2); % bring all forces to interior 218 | %fgy(2,1:ny+2) = fgy(2,1:ny+2) + fgy(1,1:ny+2); fgy(nx+1,1:ny+2) = fgy(nx+1,1:ny+2) + fgy(nx+2,1:ny+2); % boundary condition for surface tension 219 | 220 | % for l=2:Nf+1 % distribute of Fr (surface tension in a volume)to the fixed grid 221 | % fglx = (gammaCf(l)*tx(l)-gammaCf(l-1)*tx(l-1)); fgly = (gammaCf(l)*ty(l)-gammaCf(l-1)*ty(l-1)); %fglx = gamma*(tx(l)-tx(l-1)); fgly = gamma*(ty(l)-ty(l-1)); % calculate Fr surface tension per volume 222 | % ip = floor(xf(l)/dx)+1; jp = floor((yf(l)+0.5*dy)/dy)+1; 223 | % % closest point as integration point 224 | % 225 | % ax = xf(l)/dx-ip+1; ay = (yf(l)+0.5*dy)/dy-jp+1; 226 | % fgx(ip,jp) = fgx(ip,jp) + (1.0-ax)*(1.0-ay)*fglx/dx/dy; 227 | % fgx(ip+1,jp) = fgx(ip+1,jp) + ax*(1.0-ay)*fglx/dx/dy; 228 | % fgx(ip,jp+1) = fgx(ip,jp+1) + (1.0-ax)*ay*fglx/dx/dy; 229 | % fgx(ip+1,jp+1) = fgx(ip+1,jp+1) + ax*ay*fglx/dx/dy; 230 | % 231 | % ip = floor((xf(l)+0.5*dx)/dx)+1; jp = floor(yf(l)/dy)+1; 232 | % ax = (xf(l)+0.5*dx)/dx-ip+1; ay = yf(l)/dy-jp+1; 233 | % fgy(ip,jp) = fgy(ip,jp) + (1.0-ax)*(1.0-ay)*fgly/dx/dy; 234 | % fgy(ip+1,jp) = fgy(ip+1,jp) + ax*(1.0-ay)*fgly/dx/dy; 235 | % fgy(ip,jp+1) = fgy(ip,jp+1) + (1.0-ax)*ay*fgly/dx/dy; 236 | % fgy(ip+1,jp+1) = fgy(ip+1,jp+1) + ax*ay*fgly/dx/dy; 237 | % end 238 | % 239 | % fgx(1:nx+2,2) = fgx(1:nx+2,2) + fgx(1:nx+2,1); fgx(1:nx+2,ny+1) = fgx(1:nx+2,ny+1) + fgx(1:nx+2,ny+2); % bring all forces to interior 240 | % fgy(2,1:ny+2) = fgy(2,1:ny+2) + fgy(1,1:ny+2); fgy(nx+1,1:ny+2) = fgy(nx+1,1:ny+2) + fgy(nx+2,1:ny+2); % boundary condition for surface tension 241 | 242 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 243 | % Solving N-S equations using projection method 244 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 245 | % u(1:nx+1,1) = 2*usouth-u(1:nx+1,2); u(1:nx+1,ny+2) = 2*unorth-u(1:nx+1,ny+1); % tangential vel BC 246 | % v(1,1:ny+1) = 2*vwest -v(2,1:ny+1); v(nx+2,1:ny+1) = 2*veast -v(nx+1,1:ny+1); % tangential vel BC 247 | % 248 | % for i=2:nx; for j=2:ny+1 % temporary u-velocity (boundary values are not touched) 249 | % ut(i,j) = (2.0/(r(i+1,j)+r(i,j)))*(0.5*(ro(i+1,j)+ro(i,j))*u(i,j)+ dt* (... 250 | % - (0.25/dx)*(ro(i+1,j)*(u(i+1,j)+u(i,j))^2-ro(i,j)*(u(i,j)+u(i-1,j))^2)... 251 | % - (0.0625/dy)*( (ro(i,j)+ro(i+1,j)+ro(i,j+1)+ro(i+1,j+1))*(u(i,j+1)+u(i,j))*(v(i+1,j)+v(i,j)) ... 252 | % - (ro(i,j)+ro(i+1,j)+ro(i+1,j-1)+ro(i,j-1))*(u(i,j)+u(i,j-1))*(v(i+1,j-1)+v(i,j-1)))... 253 | % + mu*((u(i+1,j)-2*u(i,j)+u(i-1,j))/dx^2+ (u(i,j+1)-2*u(i,j)+u(i,j-1))/dy^2)... 254 | % + 0.5*(ro(i+1,j)+ro(i,j))*gx + fgx(i,j) ) ); 255 | % end; end 256 | % 257 | % for i=2:nx+1; for j=2:ny % temporary v-velocity (boundary values are not touched) 258 | % vt(i,j) = (2.0/(r(i,j+1)+r(i,j)))*(0.5*(ro(i,j+1)+ro(i,j))*v(i,j)+ dt* (... 259 | % - (0.0625/dx)*( (ro(i,j)+ro(i+1,j)+ro(i+1,j+1)+ro(i,j+1))*(u(i,j)+u(i,j+1))*(v(i,j)+v(i+1,j)) ... 260 | % - (ro(i,j)+ro(i,j+1)+ro(i-1,j+1)+ro(i-1,j))*(u(i-1,j+1)+u(i-1,j))*(v(i,j)+v(i-1,j)) )... 261 | % - (0.25/dy)*(ro(i,j+1)*(v(i,j+1)+v(i,j))^2-ro(i,j)*(v(i,j)+v(i,j-1))^2 )... 262 | % + mu*((v(i+1,j)-2*v(i,j)+v(i-1,j))/dx^2+(v(i,j+1)-2*v(i,j)+v(i,j-1))/dy^2)... 263 | % + 0.5*(ro(i,j+1)+ro(i,j))*gy + fgy(i,j) ) ); 264 | % end; end 265 | % 266 | % for i = 2:nx+1; for j = 2:ny+1 267 | % tmp1(i,j) = (0.5/dt)*( (ut(i,j)-ut(i-1,j))/dx+(vt(i,j)-vt(i,j-1))/dy ); 268 | % tmp2(i,j) =1/( (1/dx)*(1/(dx*(r(i+1,j)+r(i,j)))+ 1/(dx*(r(i-1,j)+r(i,j))) )+ ... 269 | % (1/dy)*(1/(dy*(r(i,j+1)+r(i,j)))+ 1/(dy*(r(i,j-1)+r(i,j))) ) ); 270 | % end; end 271 | % 272 | % for it = 1:maxit % solve for pressure by SOR 273 | % pold = p; 274 | % p(1,:) = p(2,:); p(nx+2,:) = p(nx+1,:); p(:,1) = p(:,2); p(:,ny+2) = p(:,ny+1); % set gosht values 275 | % for i=2:nx+1; for j=2:ny+1 276 | % p(i,j) = (1.0-omg)*p(i,j) + omg*tmp2(i,j)*( ... 277 | % (1/dx)*( p(i+1,j)/(dx*(r(i+1,j)+r(i,j)))+ p(i-1,j)/(dx*(r(i-1,j)+r(i,j))) )+ ... 278 | % (1/dy)*( p(i,j+1)/(dy*(r(i,j+1)+r(i,j)))+ p(i,j-1)/(dy*(r(i,j-1)+r(i,j))) ) - tmp1(i,j)); 279 | % end; end 280 | % if max(max(abs(pold-p))) < maxError; break; end 281 | % end 282 | % 283 | % for i=2:nx; for j=2:ny+1 % correct the u-velocity 284 | % u(i,j)=ut(i,j)-dt*(2.0/dx)*(p(i+1,j)-p(i,j))/(r(i+1,j)+r(i,j)); 285 | % end; end 286 | % 287 | % for i=2:nx+1; for j=2:ny % correct the v-velocity 288 | % v(i,j)=vt(i,j)-dt*(2.0/dy)*(p(i,j+1)-p(i,j))/(r(i,j+1)+r(i,j)); 289 | % end; end 290 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 291 | 292 | % for i=2:nx+1; 293 | % for j=2:ny+1; 294 | % phi(i,j)=phin(i,j) 295 | % end; 296 | % end; 297 | 298 | 299 | % calculate the radius of the circle; 300 | 301 | for i=(round((nx+3)/2)):nx+1; 302 | %for j=ny/2+1:ny+2; 303 | if phi(i,round((ny+3)/2)) < 0 && phi(i+1, round((ny+3)/2))>0; 304 | x1 =x(i); 305 | x2=x(i+1); 306 | c1= phi(i, round((ny+3)/2)); 307 | c2= phi(i+1, round((ny+3)/2)); 308 | xr= x1-c1/(c2-c1)*(x2-x1); 309 | R1(is+1)=(xr-0.5); 310 | %R(is+1)=sqrt((x(i)-xc)^2 + (y(i)-yc)^2); 311 | break; 312 | end; 313 | %end; 314 | end; 315 | 316 | time = time+dt 317 | if (mod(is,plot_freq)==0) | (is==1); % plot solution 318 | % uu(1:nx+1,1:ny+1)=0.5*(u(1:nx+1,2:ny+2)+u(1:nx+1,1:ny+1)); 319 | % vv(1:nx+1,1:ny+1)=0.5*(v(2:nx+2,1:ny+1)+v(1:nx+1,1:ny+1)); 320 | %figure(5); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off %contour(x,y,r'); 321 | figure(6); contourf(x,y,phi'); 322 | axis equal; axis([0 Lx 0 Ly]); hold on; 323 | xlabel('x', 'Fontsize', 20); 324 | ylabel('y', 'Fontsize', 20); 325 | title([sprintf('time t=%0.3f', time)], 'Fontsize',20); 326 | colorbar; 327 | %caxis([-1 1]); 328 | drawnow; hold off 329 | % quiver(xh,yh,uu',vv','r'); hold on; drawnow; hold off 330 | end 331 | 332 | end 333 | t1=linspace(0,0.012,nstep+1); 334 | r_exact=sqrt(rad^2-2*t1); 335 | figure(7); 336 | plot(t1,r_exact); hold on; 337 | plot(t1,R1,'--r'); 338 | legend('Exact solution', 'level set') 339 | xlabel('t', 'Fontsize', 14); 340 | ylabel('r', 'Fontsize', 14); 341 | drawnow; hold off; 342 | -------------------------------------------------------------------------------- /FinalProjectdroplet.m: -------------------------------------------------------------------------------- 1 | 2 | % level set method to capture the interface 3 | clear all; clc; close all; 4 | Lx = 1.0; Ly = 1.0; % domain size 5 | gx = 0.0; gy = -100.0; rho1 = 2; rho2 = 1; mu = 0.01; % parameters 6 | unorth = 0; usouth = 0; veast = 0; vwest = 0; % boundary conditions 7 | rad = 0.15; xc = 0.5; yc = 0.5; % initial drop size and location 8 | D=1; % 9 | time = 0.0; plot_freq = 10; 10 | 11 | nx = 256; ny = 256; dx = Lx/nx; dy = Ly/ny; dt = 0.0001; 12 | 13 | nstep = 2000; maxit = 200; maxError = 0.001; omg = 1.5; Nf = 100; 14 | 15 | u=zeros(nx+1,ny+2); ut = u ; uplot = zeros(nx+1,ny+1); 16 | v=zeros(nx+2,ny+1); vt = u ; vplot = zeros(nx+1,ny+1); 17 | 18 | 19 | p=zeros(nx+2,ny+2); tmp1 = p ; tmp2 = p; r = p; chi = p; 20 | C=zeros(nx+2,ny+2); gamma=0; %zeros(nx+2,ny+2); 21 | Cn=C; % C at n time step 22 | phi=zeros(nx+2, ny+2); % signed distance function 23 | phin=zeros(nx+2, ny+2);% phi at n timestep 24 | sphi=zeros(nx+2,ny+2); % signed phi in the reinitialization process 25 | psi=zeros(nx+2,ny+2); % psi to satisfy the steady state solution 26 | psin=zeros(nx+2,ny+2); % fi at n timestep 27 | eps= 1.5*dx; % used for smooth out chi 28 | epsilon=0.0000001; % used for calculating sign of phi 29 | Ddelta=zeros(nx+2,ny+2); % Dirac delta 30 | % 31 | 32 | % stargerred grid 33 | xh = linspace(0,Lx,nx+1) ; yh = linspace(0,Ly,ny+1); % velocity points 34 | x = linspace(-dx/2,Lx+dx/2,nx+2); y = linspace(-dy/2,Ly+dy/2,ny+2); % pressure points 35 | 36 | r = zeros(nx+2,ny+2) + rho2; % initial density 37 | fgx=zeros(nx+2,ny+2); fgy=zeros(nx+2,ny+2); % initial surface tension 38 | 39 | % initialization 40 | % the initial shape is a circle 41 | % initialize front 42 | for i=1:nx+2;for j=1:ny+2 43 | if((x(i)-xc)^2+(y(j)-yc)^2 < rad^2); 44 | phi(i,j)= -( rad-sqrt((x(i)-xc)^2 + (y(j)-yc)^2)); 45 | %chi(i,j)= 1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps);%1.0; 46 | r(i,j) = rho1; 47 | elseif((x(i)-xc)^2+(y(j)-yc)^2 > rad^2); 48 | phi(i,j)= sqrt((x(i)-xc)^2 + (y(j)-yc)^2)-rad; 49 | %chi(i,j)=0.0; 50 | else 51 | phi(i,j)=0.0; 52 | %chi(i,j)=0.0; 53 | end; 54 | if(phi(i,j) <-eps) 55 | chi(i,j)=0.0; 56 | Ddelta(i,j)=0; 57 | elseif(phi(i,j)>eps) 58 | chi(i,j)=1.; 59 | Ddelta(i,j)=0; 60 | else 61 | chi(i,j)=1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps); 62 | Ddelta(i,j)=1/(2*eps)*(1+cos(pi*phi(i,j)/eps)); 63 | end; 64 | end;end; 65 | %r = rho1*chi + rho2*(1-chi); % initial density 66 | figure(1); contourf(x,y,phi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 67 | figure(2); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 68 | % for i = 2:nx+1; for j = 2:ny+1 69 | % if((x(i)-xc)^2+(y(j)-yc)^2 < rad^2); r(i,j) = rho1; chi(i,j)=1.0; end; 70 | % end; end 71 | 72 | 73 | 74 | 75 | for is=1:nstep 76 | 77 | 78 | 79 | 80 | %%%figure(4); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 81 | %figure(3); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 82 | 83 | ro = r; 84 | r = rho2*chi + rho1*(1-chi); % obtain density from charact func 85 | 86 | figure(5); contourf(x,y,r'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 87 | 88 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 89 | %Calculate the surface tension 90 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 91 | 92 | for i=2:nx+1; for j=2:ny+1 93 | fgx(i,j)=gamma * ((phi(i+1,j)-2.0 * phi(i,j)+phi(i-1,j))/(dx^2) + (phi(i,j+1)-2*phi(i,j)+phi(i,j-1))/(dy^2)) * Ddelta(i,j) * ((phi(i+1,j)-phi(i,j))/dx); % using central 94 | fgy(i,j)=gamma * ((phi(i+1,j)-2.0 * phi(i,j)+phi(i-1,j))/(dx^2) + (phi(i,j+1)-2*phi(i,j)+phi(i,j-1))/(dy^2)) * Ddelta(i,j) * ((phi(i,j+1)-phi(i,j))/dy); 95 | end; 96 | end; 97 | fgx(1:nx+2,1)=fgx(1:nx+2,2);fgx(1:nx+2,ny+2)=fgx(1:nx+2,ny+1); 98 | fgx(1,1:ny+2)=fgx(2,1:ny+2);fgx(ny+2,1:ny+2)=fgx(ny+1,1:ny+2); 99 | fgy(1:nx+2,1)=fgy(1:nx+2,2);fgy(1:nx+2,ny+2)=fgy(1:nx+2,ny+1); 100 | fgy(1,1:ny+2)=fgy(2,1:ny+2);fgy(ny+2,1:ny+2)=fgy(ny+1,1:ny+2); 101 | 102 | %fgx(1:nx+2,2) = fgx(1:nx+2,2) + fgx(1:nx+2,1); fgx(1:nx+2,ny+1) = fgx(1:nx+2,ny+1) + fgx(1:nx+2,ny+2); % bring all forces to interior 103 | % fgy(2,1:ny+2) = fgy(2,1:ny+2) + fgy(1,1:ny+2); fgy(nx+1,1:ny+2) = fgy(nx+1,1:ny+2) + fgy(nx+2,1:ny+2); % boundary condition for surface tension 104 | 105 | 106 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 107 | % Solving N-S equations using projection method 108 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 109 | u(1:nx+1,1) = 2*usouth-u(1:nx+1,2); u(1:nx+1,ny+2) = 2*unorth-u(1:nx+1,ny+1); % tangential vel BC 110 | v(1,1:ny+1) = 2*vwest -v(2,1:ny+1); v(nx+2,1:ny+1) = 2*veast -v(nx+1,1:ny+1); % tangential vel BC 111 | 112 | for i=2:nx; for j=2:ny+1 % temporary u-velocity (boundary values are not touched) 113 | ut(i,j) = (2.0/(r(i+1,j)+r(i,j)))*(0.5*(ro(i+1,j)+ro(i,j))*u(i,j)+ dt* (... 114 | - (0.25/dx)*(ro(i+1,j)*(u(i+1,j)+u(i,j))^2-ro(i,j)*(u(i,j)+u(i-1,j))^2)... 115 | - (0.0625/dy)*( (ro(i,j)+ro(i+1,j)+ro(i,j+1)+ro(i+1,j+1))*(u(i,j+1)+u(i,j))*(v(i+1,j)+v(i,j)) ... 116 | - (ro(i,j)+ro(i+1,j)+ro(i+1,j-1)+ro(i,j-1))*(u(i,j)+u(i,j-1))*(v(i+1,j-1)+v(i,j-1)))... 117 | + mu*((u(i+1,j)-2*u(i,j)+u(i-1,j))/dx^2+ (u(i,j+1)-2*u(i,j)+u(i,j-1))/dy^2)... 118 | + 0.5*(ro(i+1,j)+ro(i,j))*gx + (fgx(i,j)+fgx(i-1,j))/2.0 )); 119 | end; end 120 | 121 | for i=2:nx+1; for j=2:ny % temporary v-velocity (boundary values are not touched) 122 | vt(i,j) = (2.0/(r(i,j+1)+r(i,j)))*(0.5*(ro(i,j+1)+ro(i,j))*v(i,j)+ dt* (... 123 | - (0.0625/dx)*( (ro(i,j)+ro(i+1,j)+ro(i+1,j+1)+ro(i,j+1))*(u(i,j)+u(i,j+1))*(v(i,j)+v(i+1,j)) ... 124 | - (ro(i,j)+ro(i,j+1)+ro(i-1,j+1)+ro(i-1,j))*(u(i-1,j+1)+u(i-1,j))*(v(i,j)+v(i-1,j)) )... 125 | - (0.25/dy)*(ro(i,j+1)*(v(i,j+1)+v(i,j))^2-ro(i,j)*(v(i,j)+v(i,j-1))^2 )... 126 | + mu*((v(i+1,j)-2*v(i,j)+v(i-1,j))/dx^2+(v(i,j+1)-2*v(i,j)+v(i,j-1))/dy^2)... 127 | + 0.5*(ro(i,j+1)+ro(i,j))*gy + (fgy(i,j)+fgy(i,j-1))/2.0 ) ); 128 | end; end 129 | 130 | for i = 2:nx+1; for j = 2:ny+1 131 | tmp1(i,j) = (0.5/dt)*( (ut(i,j)-ut(i-1,j))/dx+(vt(i,j)-vt(i,j-1))/dy ); 132 | tmp2(i,j) =1/( (1/dx)*(1/(dx*(r(i+1,j)+r(i,j)))+ 1/(dx*(r(i-1,j)+r(i,j))) )+ ... 133 | (1/dy)*(1/(dy*(r(i,j+1)+r(i,j)))+ 1/(dy*(r(i,j-1)+r(i,j))) ) ); 134 | end; end 135 | 136 | for it = 1:maxit % solve for pressure by SOR 137 | pold = p; 138 | p(1,:) = p(2,:); p(nx+2,:) = p(nx+1,:); p(:,1) = p(:,2); p(:,ny+2) = p(:,ny+1); % set gosht values 139 | for i=2:nx+1; for j=2:ny+1 140 | p(i,j) = (1.0-omg)*p(i,j) + omg*tmp2(i,j)*( ... 141 | (1/dx)*( p(i+1,j)/(dx*(r(i+1,j)+r(i,j)))+ p(i-1,j)/(dx*(r(i-1,j)+r(i,j))) )+ ... 142 | (1/dy)*( p(i,j+1)/(dy*(r(i,j+1)+r(i,j)))+ p(i,j-1)/(dy*(r(i,j-1)+r(i,j))) ) - tmp1(i,j)); 143 | end; end 144 | if max(max(abs(pold-p))) < maxError; break; end 145 | end 146 | 147 | for i=2:nx; for j=2:ny+1 % correct the u-velocity 148 | u(i,j)=ut(i,j)-dt*(2.0/dx)*(p(i+1,j)-p(i,j))/(r(i+1,j)+r(i,j)); 149 | end; end 150 | 151 | for i=2:nx+1; for j=2:ny % correct the v-velocity 152 | v(i,j)=vt(i,j)-dt*(2.0/dy)*(p(i,j+1)-p(i,j))/(r(i,j+1)+r(i,j)); 153 | end; end 154 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 155 | 156 | 157 | %%%%%%%%%%%%%%%%%%%%%%% 158 | % interface propagation 159 | %%%%%%%%%%%%%%%%%%%%%%%% 160 | phin=phi; 161 | 162 | for i=2:nx+1; 163 | for j=2:ny+1; 164 | if(((u(i,j)+u(i-1,j))/2.0)>=0); 165 | dphi_x=(phin(i,j)-phin(i-1,j))/dx; % upwind scheme 166 | else; 167 | dphi_x=(phin(i+1,j)-phin(i,j))/dx; 168 | end; 169 | if(((v(i,j)+v(i,j-1))/2.0)>=0); 170 | dphi_y=(phin(i,j)-phin(i,j-1))/dy; 171 | else 172 | dphi_y=(phin(i,j+1)-phin(i,j))/dy; 173 | end; 174 | % phi(i,j)= phin(i,j)+ dt * D* ( (phin(i+1,j)-2*phin(i,j)+phin(i-1,j))/(dx^2) + (phin(i,j+1) - 2*phin(i,j) +phin(i,j-1))/(dy^2) ); %curvature flow 175 | phi(i,j)= phin(i,j)+ dt * ((u(i,j)+u(i-1,j))/2.0*dphi_x+(v(i,j)+v(i,j-1))/2.0*dphi_y); % drop falling flow 176 | % phi(i,j) =phin(i,j) - u(i,j) 177 | end; 178 | end; 179 | figure(4); contourf(x,y,phi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 180 | % update phi to n+1 timestep 181 | 182 | %%%%%%%%%%%%%%%%%%%%%%%%%%%% 183 | % interface reinitialization 184 | %%%%%%%%%%%%%%%%%%%%%%%%%%%% 185 | for i=2:nx+1;for j=2:ny+1; 186 | 187 | sphi(i,j)=phi(i,j)/(sqrt(phi(i,j)^2 +epsilon^2)); 188 | end;end; 189 | figure(3); contourf(x,y,sphi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off 190 | psi=phi; 191 | 192 | for k=1:20; % iteration for 5 times 193 | psin=psi; 194 | for i=2:nx+1; 195 | for j=2:ny+1; 196 | dphix1=(psin(i,j)-psin(i-1,j))/dx; 197 | dphix2=(psin(i+1,j)-psin(i,j))/dx; 198 | dphiy1=(psin(i,j)-psin(i,j-1))/dy; 199 | dphiy2=(psin(i,j+1)-psin(i,j))/dy; 200 | 201 | % reinitialization near the interface 202 | if(psi(i,j)>0.); 203 | gphi=1-sqrt(max(max(dphix1,0.)^2, min(dphix2,0.)^2) +max(max(dphiy1, 0.)^2, min(dphiy2, 0.)^2)); 204 | elseif(psi(i,j)<0.); 205 | gphi=1-sqrt(max(min(dphix1,0.)^2, max(dphix2,0.)^2)+max(min(dphiy1, 0.)^2, max(dphiy2, 0.)^2)); 206 | else 207 | gphi=0.; 208 | end; 209 | % gphi=((psin(i+1,j)-psin(i-1,j))/(2*dx))^2 +((psin(i,j+1)-psin(i,j-1))/(2*dy))^2; 210 | psi(i,j)=psin(i,j)+dt*sphi(i,j)*gphi; 211 | end; 212 | end; 213 | 214 | 215 | end; 216 | 217 | phi=psi; 218 | 219 | %%%%%%%%%%get the characteristic function at new time step%%%%%%% 220 | for i=2:nx+1;for j=2:ny+1 221 | if(phi(i,j) <-eps) % inside the drop chi=0 222 | chi(i,j)=0.; 223 | Ddelta(i,j)=0.; 224 | elseif(phi(i,j)>eps) % outside the drop chi=1 225 | chi(i,j)=1.; 226 | Ddelta(i,j)=0.; 227 | else 228 | chi(i,j)=1/2.0 + phi(i,j)/(2*eps)+1/(2*pi)*sin(pi*phi(i,j)/eps); 229 | Ddelta(i,j)=1/(2*eps)*(1+cos(pi*phi(i,j)/eps)); 230 | end; 231 | end;end; 232 | 233 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 234 | 235 | 236 | time = time+dt 237 | if (mod(is,plot_freq)==0) | (is==1); % plot solution 238 | uu(1:nx+1,1:ny+1)=0.5*(u(1:nx+1,2:ny+2)+u(1:nx+1,1:ny+1)); 239 | vv(1:nx+1,1:ny+1)=0.5*(v(2:nx+2,1:ny+1)+v(1:nx+1,1:ny+1)); 240 | %figure(5); contourf(x,y,chi'); axis equal; axis([0 Lx 0 Ly]); hold on; drawnow; hold off %contour(x,y,r'); 241 | figure(6); contourf(x,y,phi'); 242 | axis equal; axis([0 Lx 0 Ly]); %hold on;drawnow; hold off 243 | quiver(xh,yh,uu',vv','r'); hold on; drawnow; hold off 244 | end 245 | 246 | end 247 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /Numerical simulation of Multiphase flow using level set method.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Dingding-Han/Numerical-simulation-of-multiphase-flow/cdfb92caea2e4f41d844cccf4c0758ea4b187cb8/Numerical simulation of Multiphase flow using level set method.pdf -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | #Level set method for capturing interface of multiphase flows: 2 | (1) bubble rising 3 | (2) curvature flow 4 | (3) droplet impact 5 | --------------------------------------------------------------------------------