% Gianni Schena July 2005, schena@units.it
% Lattice Boltzmann LBE, geometry: D2Q9, model: BGK
% Application to permeability in porous media
Restart=false % to restart from an earlier convergence
logical(Restart);
if Restart==false;
close all, clear all % start from scratch and clean ...
Restart=false;
% type of channel geometry ;
% one of the flollowing flags == true
Pois_test=true, % no obstacles in the 2D channel
% porous systems
obs_regolare=false %
obs_irregolare=false %
tic
% IN
% |vvvv| + y
% |vvvv| ^
% |vvvv| | -> + x
% OUT
% Pores in 2D : Wet and Dry locations (Wet ==1 , Dry ==0 )
wXh_Dry=[3,1];wXh_Wet=[3,4];
if obs_regolare, % with internal obstacles
A=repmat([zeros(wXh_Dry),ones(wXh_Wet)],[1,3]);A=[A,zeros(wXh_Dry)];
B=ones(size(A));
C=[A;B] ; D=repmat(C,4,1);
D=[B;D]
end
if obs_irregolare, % with int obstacles
A1=repmat([zeros(wXh_Dry),ones(wXh_Wet)],[1,3]);
A1=[A1,zeros(wXh_Dry)] ;
B=ones(size(A1));
C1=repmat([ones(wXh_Wet),zeros(wXh_Dry)],[1,3]); C1=[C1,ones(wXh_Dry)];
E=[A1;B;C1;B];
D=repmat(E,2,1);
D=[B;D]
end
if ~Pois_test
figure,imshow(D,[])
Channel2D=D;
Len_Channel_2D=size(Channel2D,1); % Length
Width=size(Channel2D,2); % should not be hod
Channel_2D_half_Width=Width/2,
end
% test without obstacles (i.e. 2D channel & no obstacles)
if Pois_test
%over-writes the definition of the pore space
clear Channel2D
Len_Channel_2D=36, % lunghezza canale 2d
Channel_2D_half_Width=8; Width=Channel_2D_half_Width*2;
Channel2D=ones(Len_Channel_2D,Width); % define wet area
%Channel2D(6:12,6:8)=0; % put fluid obstacle
imshow(Channel2D,[]);
end
[Nr Mc]=size(Channel2D); % Number rows and Munber columns
% porosity
porosity=nnz(Channel2D==1)/(Nr*Mc)
% FLUID PROPERTIES
% physical properties
cs2=1/3; %
cP_visco=0.5; % [cP] 1 CP Dinamic water viscosity 20 C
density=1.; % fluid density
Lky_visco=cP_visco/density; % lattice kinematic viscosity
omega=(Lky_visco/cs2+0.5).^-1; % omega: relaxation frequency
%Lky_visco=cs2*(1/omega - 0.5) , % lattice kinematic viscosity
%dPdL= Pressure / dL;% External pressure gradient [atm/cm]
uy_fin_max=-0.2;
%dPdL = abs( 2*Lky_visco*uy_fin_max/(Channel_2D_half_Width.^2) );
dPdL=-0.0125;
uy_fin_max=dPdL*(Channel_2D_half_Width.^2)/(2*Lky_visco); % Poiseuille Gradient;
% max poiseuille final velocity on the flow profile
uy0=-0.001; ux0=0.0001; % linear vel .. inizialization
%
% uy_fin_max=-0.2; % max poiseuille final velocity on the flow profile
% omega=0.5, cs2=1/3; % omega: relaxation frequency
% Lky_visco=cs2*(1/omega - 0.5) , % lattice kinematic viscosity
% dPdL = abs( 2*Lky_visco*uy_fin_max/(Channel_2D_half_Width.^2) ); % Poiseuille Gradient;
%
uyf_av=uy_fin_max*(2/3);; % average fluid velocity on the profile
x_profile=([-Channel_2D_half_Width:+Channel_2D_half_Width-1]+0.5);
uy_analy_profile=uy_fin_max.*(1- ( x_profile /Channel_2D_half_Width).^2 ); % analytical velocity profile
av_vel_t=1.e+10; % inizialization (t=0)
%PixelSize= 5; % [Microns]
%dL=(Nr*PixelSize*1.0E-4); % sample hight [cm]
%
% EXPERIMENTAL SET-UP
% inlet and outlet buffers
inb=2, oub=2; % inlet and outlet buffers thickness
% add fluid at the inlet (top) and outlet (down)
inlet=ones(inb,Mc); outlet=ones(oub,Mc);
Channel2D=[ [inlet]; Channel2D ;[outlet] ] ; % add flux in and down (E to W)
[Nr Mc]=size(Channel2D); % update size
% boundaries related to the experimental set up
wb=2; % wall thickness
Channel2D=[zeros(Nr,wb), Channel2D , zeros(Nr,wb)]; % add walls (no fluid leak)
[Nr Mc]=size(Channel2D); % update size
uy_analy_profile=[zeros(1,wb), uy_analy_profile, zeros(1,wb) ] ; % take into account walls
x_pro_fig=[[x_profile(1)-[wb:-1:1]], [x_profile, [1:wb]+x_profile(end)] ];
% Figure plots analytical parabolic profile
figure(20), plot(x_pro_fig,uy_analy_profile,'-'), grid on,
title('Analytical parab. profile for Poiseuille planar flow in a channel')
% VISUALIZE PORE SPACE & FLUID OSTACLES & MEDIAL AXIS
figure, imshow(Channel2D); title('Vassel geometry');
Channel2D=logical(Channel2D);
% obstacles for Bounce Back ( in front of the grain)
Obstacles=bwperim(Channel2D,8); % perimeter of the grains for bounce back Bound.Cond.
border=logical(ones(Nr,Mc));
border([1:inb,Nr-oub:Nr],[wb+2:Mc-wb-1])=0;
Obstacles=Obstacles.*(border);
figure, imshow(Obstacles); title(' Fluid obstacles (in the fluid)' );
%
Medial_axis=bwmorph(Channel2D,'thin',Inf); %
figure, imshow(Medial_axis); title('Medial axis');
figure(10) % used to visualize evolution of rho
figure(11) % used to visualize ux
figure(12) % used to visualize uy (i.e. top -> down)
% INDICES
% Wet locations etc.
[iabw1 jabw1]=find(Channel2D==1); % indices i,j, of active lattice locations i.e. pore
lena=length(iabw1); % number of active location i.e. of pore space lattice cells
ija= (jabw1-1)*Nr+iabw1; % equivalent single index (i,j)->> ija for active locations
% absolute (single index) position of the obstacles in for bounce back in Channel2D
% Obstacles
[iobs jobs]=find(Obstacles);lenobs=length(iobs); ijobs= (jobs-1)*Nr+iobs; % as above
% Medial axis of the pore space
[ima jma]=find(Medial_axis); lenma=length(ima); ijma= (jma-1)*Nr+ima; % as above
% Internal wet locations : wet & ~obstables
% (i.e. internal wet lattice location non in contact with dray locations)
[iawint jawint]=find(( Channel2D==1 & ~Obstacles)); % indices i,j, of active lattice locations
lenwint=length(iawint); % number of internal (i.e. not border) wet locations
ijaint= (jawint-1)*Nr+iawint; % equivalent singl
NxM=Nr*Mc;
% DIRECTIONS: E N W S NE NW SW SE ZERO (ZERO:Rest Particle)
% y^
% 6 2 5 ^ NW N NE
% 3 9 1 ... +x-> +y W RP E
% 7 4 8 SW S SE
% -y
% x & y components of velocities , +x is to est , +y is to nord
East=1; North=2; West=3; South=4; NE=5; NW=6; SW=7; SE=8; RP=9;
N_c=9 ; % number of directions
% versors D2Q9
C_x=[1 0 -1 0 1 -1 -1 1 0];
C_y=[0 1 0 -1 1 1 -1 -1 0]; C=[C_x;C_y]
% BOUNCE BACK SCHEME
% after collision the fluid elements densities f are sent back to the
% lattice node they come from with opposite direction
% indices opposite to 1:8 for fast inversion after bounce
ic_op = [3 4 1 2 7 8 5 6]; % i.e. 4 is opposite to 2 etc.
% PERIODIC BOUNDARY CONDITIONS - reinjection rules
yi2=[Nr , 1:Nr , 1]; % this definition allows implemening Period Bound Cond
%yi2=[1, Nr , 2:Nr-1 , 1,Nr]; % re-inj the second last to as first
% directional weights (density weights)
w0=16/36. ; w1=4/36. ; w2=1/36.;
W=[ w1 w1 w1 w1 w2 w2 w2 w2 w0];
%c constants (sound speed related)
cs2=1/3; cs2x2=2*cs2; cs4x2=2*cs2.^2;
f1=1/cs2; f2=1/cs2x2; f3=1/cs4x2;
f1=3., f2=4.5; f3=1.5; % coef. of the f equil.
% declarative statemets
f=zeros(Nr,Mc,N_c); % array of fluid density distribution
feq=zeros(Nr,Mc,N_c); % f at equilibrium
rho=ones(Nr,Mc); % macro-scopic density
temp1=zeros(Nr,Mc);
ux=zeros(Nr,Mc); uy=zeros(Nr,Mc); uyout=zeros(Nr,Mc); % dimensionless velocities
uxsq=zeros(Nr,Mc); uysq=zeros(Nr,Mc); usq=zeros(Nr,Mc); % higher degree velocities
% initialization arrays : start values in the wet area
for ia=1:lena % stat values in the active cells only ; 0 outside
i=iabw1(ia); j=jabw1(ia);
f(i,j,:)=1/9; % uniform density distribution for a start
end
uy(ija)=uy0; ux(ija)=ux0; % initialize fluid velocities
rho(ija)=density;
% EXTERNAL (Body) FORCES e.g. inlet pressure or inlet-outlet gradient
% directions: E N W S NE NW SW SE ZERO
force = -dPdL*(1/6)*1*[0 -1 0 1 -1 -1 1 1 0]'; %;
%... E N E S NE NW SW SE RP ...
% the pressure pushes the fluid down i.e. N to S
% While .. MAIN TIME EVOLUTION LOOP
StopFlag=false; % i.e. logical(0)
Max_Iter=3000; % max allowed number of iteration
Check_Iter=1; Output_Every=20; % frequency of check & output
Cur_Iter=0; % current iteration counter inizialization
toler=1.0e-8; % tollerance to declare convegence
Cond_path=[]; % recording values of the convergence criterium
density_path=[]; % recording aver. density values for convergence
end % ends if restart
if(Restart==true)
StopFlag=false; Max_Iter=Max_Iter+3000; toler=1.0e-12;
end
while(~StopFlag)
Cur_Iter=Cur_Iter+1 % iteration counter update
% density and moments
rho=sum(f,3); % density
if Cur_Iter >1 % use inizialization ux uy to start
% Moments ... Note:C_x(9)=C_y(9)=0
ux=zeros(Nr,Mc); uy=zeros(Nr,Mc);
for ic=1:N_c-1;
ux = ux + C_x(ic).*f(:,:,ic) ; uy = uy + C_y(ic).*f(:,:,ic) ;
end
% uy=f(:,:,2) +f(:,:,5)+f(:,:,6)-f(:,:,4)-f(:,:,7)-f(:,:,8); % in short !
% ux=f(:,:,1) +f(:,:,5)+f(:,:,8)-f(:,:,3)-f(:,:,6)-f(:,:,7); % in short !
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ux(ija)=ux(ija)./rho(ija); uy(ija)=uy(ija)./rho(ija);
uxsq(ija)=ux(ija).^2; uysq(ija)=uy(ija).^2;
usq(ija)=uxsq(ija)+uysq(ija); %
% weighted densities : rest particle, principal axis, diagonals
rt0 = w0.*rho; rt1 = w1.*rho; rt2 = w2.*rho;
% Equilibrium distribution
% main directions ( + cross)
feq(ija)= rt1(ija) .*(1 +f1*ux(ija) +f2*uxsq(ija) -f3*usq(ija));
feq(ija+NxM*(2-1))= rt1(ija) .*(1 +f1*uy(ija) +f2*uysq(ija) -f3*usq(ija));
feq(ija+NxM*(3-1))= rt1(ija) .*(1 -f1*ux(ija) +f2*uxsq(ija) -f3*usq(ija));
%feq(ija+NxM*(3)=f(ija)-2*rt1(ija)*f1.*ux(ija); % much faster... !!
feq(ija+NxM*(4-1))= rt1(ija) .*(1 -f1*uy(ija) +f2*uysq(ija) -f3*usq(ija));
% diagonals (X diagonals) (ic-1)
feq(ija+NxM*(5-1))= rt2(ija) .*(1 +f1*(+ux(ija)+uy(ija)) +f2*(+ux(ija)+uy(ija)).^2 -f3.*usq(ija));
feq(ija+NxM*(6-1))= rt2(ija) .*(1 +f1*(-ux(ija)+uy(ija)) +f2*(-ux(ija)+uy(ija)).^2 -f3.*usq(ija));
feq(ija+NxM*(7-1))= rt2(ija) .*(1 +f1*(-ux(ija)-uy(ija)) +f2*(-ux(ija)-uy(ija)).^2 -f3.*usq(ija));
feq(ija+NxM*(8-1))= rt2(ija) .*(1 +f1*(+ux(ija)-uy(ija)) +f2*(+ux(ija)-uy(ija)).^2 -f3.*usq(ija));
% rest particle (.) ic=9
feq(ija+NxM*(9-1))= rt0(ija) .*(1 - f3*usq(ija));
%Collision (between fluid elements)omega=relaxation frequency
f=(1.-omega).*f + omega.*feq;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%add external body force due to the pressure gradient prop. to dPdL
for ic=1:N_c;%-1
for ia=1:lena
i=iabw1(ia); j=jabw1(ia);
% if Obstacles(i,j)==0 % the i,j is not aderent to the boundaries
% if ( f(i,j,ic) + force(ic) ) >0; %! avoid negative distributions
%i=1 ;% force only on the first row !
f(i,j,ic)= f(i,j,ic) + force(ic);
% end
% end
end
end
% % STREAM
% Forward Propagation step & % Bounce Back (collision fluid with obstacles)
%f(:,:,9) = f(:,:,9); % Rest element do not move
feq = f; % temp storage of f in feq
for ic=1:1:N_c-1, % select velocity layer
ic2=ic_op(ic); % selects the layer of the velocity opposite to ic for BB
temp1=feq(:,:,ic); %
% from wet location that are NOT on the border to other wet locations
for ia=1:1:lenwint % number of internal (i.e. not border) wet locations
i=iawint(ia); j=jawint(ia); % so that we care for the wet space only !
i2 = i+C_y(ic); j2 = j+C_x(ic); % Expected final locations to move
i2=yi2(i2+1); % i2 corrected for PBC when necessary (flow out re-fed to inlet)
% i.e the new position (i2,j2) is sure another wet location
% therefore normal propagation from (i,j) to (i2,j2) on layer ic
f(i2,j2,ic)=temp1(i,j); % see circshift(..) fnct for circularly shifts
end ; % i and j single loop
% from wet locations that ARE on the border of obstacles
for ia=1:1:lenobs % wet border locations
i=iobs(ia); j=jobs(ia); % so that we care for the wet space only !
i2 = i+C_y(ic); j2 = j+C_x(ic); % Expected final locations to move
i2=yi2(i2+1); % i2 corrected for PBC
if( Channel2D(i2,j2) ==0 ) % i.e the new position (i2,j2) is dry
f(i,j,ic2) =temp1(i,j); % invert direction: bounce-back in the opposite direction ic2
else % otherwise, normal propagation from (i,j) to (i2,j2) on layer ic
f(i2,j2,ic)=temp1(i,j); % see circshift(..) fnct for circularly shifts
end ; % b.b. and propagations
end ; % i and j single loop
% special treatment for Corners
% f(1,wb+1,ic)=temp1(Nr,Mc-wb); f(1,Mc-wb,ic)=temp1(Nr,wb+1);
% f(Nr,wb+1,ic)=temp1(1,Mc-wb); f(Nr,Mc-wb,ic)=temp1(1,wb+1);
end ; % for ic direction
% ends of Forward Propagation step & Bounce Back Sections
% re-calculate uy as uyout for convergence
rho=sum(f,3); % density
% check velocity
uyout= zeros(Nr,Mc);
for ic=1:N_c-1;
uyout= uyout + C_y(ic).*f(:,:,ic) ; % flow dim.less velocity out
end
% uyout(ija)=uyout(ija)./rho(ija); % from momentum to velocity
% Convergence check on velocity values
if (mod(Cur_Iter,Check_Iter)==0) ; % check for convergence every 'Check_Iter' iterations
% variables monitored
% mean density and
vect=rho(ija); vect=vect(:);
cur_density=mean(vect);
% mean 'interstitial' velocity
% uy(ija)=uy(ija)/rho(ija); ?
vect=uy(ija); av_vel_int= mean(vect) ; % seepage velocity (in the wet area)
% on the whole cross-sectional area of flow (wet + dry)
av_vel_int=av_vel_int*porosity, % av. vel. on the wet + dry area
%av_vel_int=mean2(uy),
av_vel_tp1 = av_vel_int;
Condition=abs( abs(av_vel_t/av_vel_tp1 )-1), % should --> 0
Cond_path=[Cond_path, Condition]; % records the convergence path (value)
density_path=[density_path, cur_density];
%
av_vel_t=av_vel_tp1; % time t & t+1
if (Condition < toler) | (Cur_Iter > Max_Iter)
StopFlag=true;
display( 'Stop iteration: Convergence met or iteration exeeding the max allowed' )
display( ['Current iteration: ',num2str(Cur_Iter),...
' Max Number of iter: ',num2str(Max_Iter)] )
break % Terminate execution of WHILE .. exit the time evolution loop.
end % if(Condition < toler
end
if (mod(Cur_Iter,Output_Every)==0) ; % Output from loop every ...
%if (Cur_Iter>60) ; % Output from loop every ...
rho=sum(f,3); % density
figure(10); imshow(rho,[0.1 0.9]); title(' rho'); % visualize density evolution
figure(11); imshow(ux,[ ]); title(' ux' ); % visualize fluid velocity horizontal
figure(12); imshow(-uy,[ ]); title(' uy' ); % visualize fluid velocity down
figure(14), imshow(-uyout,[]), title('uyout'); % vis vel flow out
up=2; % linear section to visualize up from the lower row
figure(15), hold off, feather(ux(Nr-up,:),uy(Nr-up,:)),
figure(15), hold on , plot(uy_analy_profile,'r-')
title('Analytical vs LB calculated, fluid velocity parabolic profile')
pause(3); % time given to visualize properly
end % every
% pause(1);
end % End main time Evolution Loop
% Output & Draw after the end of the time evolution
figure, plot(Cond_path(2:end)); title('convergence path')
%figure, plot(density_path(2:end)); title('density convergence path')
figure, plot( [uy(Nr-up,:)-uy_analy_profile] ); title('difference : LB - Analytical solution')
toc
% Permeability K
K_Darcy_Porous_Sys= (av_vel_int*porosity)/dPdL*Lky_visco ,
K_Analy_2D_Channel=(Width^2)/12
