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Copy pathmit18086_navierstokes.m
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mit18086_navierstokes.m
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function mit18086_navierstokes
%MIT18086_NAVIERSTOKES
% Solves the incompressible Navier-Stokes equations in a
% rectangular domain with prescribed velocities along the
% boundary. The solution method is finite differencing on
% a staggered grid with implicit diffusion and a Chorin
% projection method for the pressure.
% Visualization is done by a colormap-isoline plot for
% pressure and normalized quiver and streamline plot for
% the velocity field.
% The standard setup solves a lid driven cavity problem.
% 07/2007 by Benjamin Seibold
% http://www-math.mit.edu/~seibold/
% Feel free to modify for teaching and learning.
%-----------------------------------------------------------------------
Re = 1e2; % Reynolds number
dt = 1e-2; % time step
tf = 4e-0; % final time
lx = 1; % width of box
ly = 1; % height of box
nx = 90; % number of x-gridpoints
ny = 90; % number of y-gridpoints
nsteps = 10; % number of steps with graphic output
%-----------------------------------------------------------------------
nt = ceil(tf/dt); dt = tf/nt;
x = linspace(0,lx,nx+1); hx = lx/nx;
y = linspace(0,ly,ny+1); hy = ly/ny;
[X,Y] = meshgrid(y,x);
%-----------------------------------------------------------------------
% initial conditions
U = zeros(nx-1,ny); V = zeros(nx,ny-1);
% boundary conditions
uN = x*0+1; vN = avg(x)*0;
uS = x*0; vS = avg(x)*0;
uW = avg(y)*0; vW = y*0;
uE = avg(y)*0; vE = y*0;
%-----------------------------------------------------------------------
Ubc = dt/Re*([2*uS(2:end-1)' zeros(nx-1,ny-2) 2*uN(2:end-1)']/hx^2+...
[uW;zeros(nx-3,ny);uE]/hy^2);
Vbc = dt/Re*([vS' zeros(nx,ny-3) vN']/hx^2+...
[2*vW(2:end-1);zeros(nx-2,ny-1);2*vE(2:end-1)]/hy^2);
fprintf('initialization')
Lp = kron(speye(ny),K1(nx,hx,1))+kron(K1(ny,hy,1),speye(nx));
Lp(1,1) = 3/2*Lp(1,1);
perp = symamd(Lp); Rp = chol(Lp(perp,perp)); Rpt = Rp';
Lu = speye((nx-1)*ny)+dt/Re*(kron(speye(ny),K1(nx-1,hx,2))+...
kron(K1(ny,hy,3),speye(nx-1)));
peru = symamd(Lu); Ru = chol(Lu(peru,peru)); Rut = Ru';
Lv = speye(nx*(ny-1))+dt/Re*(kron(speye(ny-1),K1(nx,hx,3))+...
kron(K1(ny-1,hy,2),speye(nx)));
perv = symamd(Lv); Rv = chol(Lv(perv,perv)); Rvt = Rv';
Lq = kron(speye(ny-1),K1(nx-1,hx,2))+kron(K1(ny-1,hy,2),speye(nx-1));
perq = symamd(Lq); Rq = chol(Lq(perq,perq)); Rqt = Rq';
fprintf(', time loop\n--20%%--40%%--60%%--80%%-100%%\n')
for k = 1:nt
% treat nonlinear terms
gamma = min(1.2*dt*max(max(max(abs(U)))/hx,max(max(abs(V)))/hy),1);
Ue = [uW;U;uE]; Ue = [2*uS'-Ue(:,1) Ue 2*uN'-Ue(:,end)];
Ve = [vS' V vN']; Ve = [2*vW-Ve(1,:);Ve;2*vE-Ve(end,:)];
Ua = avg(Ue')'; Ud = diff(Ue')'/2;
Va = avg(Ve); Vd = diff(Ve)/2;
UVx = diff(Ua.*Va-gamma*abs(Ua).*Vd)/hx;
UVy = diff((Ua.*Va-gamma*Ud.*abs(Va))')'/hy;
Ua = avg(Ue(:,2:end-1)); Ud = diff(Ue(:,2:end-1))/2;
Va = avg(Ve(2:end-1,:)')'; Vd = diff(Ve(2:end-1,:)')'/2;
U2x = diff(Ua.^2-gamma*abs(Ua).*Ud)/hx;
V2y = diff((Va.^2-gamma*abs(Va).*Vd)')'/hy;
U = U-dt*(UVy(2:end-1,:)+U2x);
V = V-dt*(UVx(:,2:end-1)+V2y);
% implicit viscosity
rhs = reshape(U+Ubc,[],1);
u(peru) = Ru\(Rut\rhs(peru));
U = reshape(u,nx-1,ny);
rhs = reshape(V+Vbc,[],1);
v(perv) = Rv\(Rvt\rhs(perv));
V = reshape(v,nx,ny-1);
% pressure correction
rhs = reshape(diff([uW;U;uE])/hx+diff([vS' V vN']')'/hy,[],1);
p(perp) = -Rp\(Rpt\rhs(perp));
P = reshape(p,nx,ny);
U = U-diff(P)/hx;
V = V-diff(P')'/hy;
% visualization
if floor(25*k/nt)>floor(25*(k-1)/nt), fprintf('.'), end
if k==1|floor(nsteps*k/nt)>floor(nsteps*(k-1)/nt)
% stream function
rhs = reshape(diff(U')'/hy-diff(V)/hx,[],1);
q(perq) = Rq\(Rqt\rhs(perq));
Q = zeros(nx+1,ny+1);
Q(2:end-1,2:end-1) = reshape(q,nx-1,ny-1);
clf, contourf(avg(x),avg(y),P',20,'w-'), hold on
contour(x,y,Q',20,'k-');
Ue = [uS' avg([uW;U;uE]')' uN'];
Ve = [vW;avg([vS' V vN']);vE];
Len = sqrt(Ue.^2+Ve.^2+eps);
quiver(x,y,(Ue./Len)',(Ve./Len)',.4,'k-')
hold off, axis equal, axis([0 lx 0 ly])
p = sort(p); caxis(p([8 end-7]))
title(sprintf('Re = %0.1g t = %0.2g',Re,k*dt))
drawnow
end
end
fprintf('\n')
%=======================================================================
function B = avg(A,k)
if nargin<2, k = 1; end
if size(A,1)==1, A = A'; end
if k<2, B = (A(2:end,:)+A(1:end-1,:))/2; else, B = avg(A,k-1); end
if size(A,2)==1, B = B'; end
function A = K1(n,h,a11)
% a11: Neumann=1, Dirichlet=2, Dirichlet mid=3;
A = spdiags([-1 a11 0;ones(n-2,1)*[-1 2 -1];0 a11 -1],-1:1,n,n)'/h^2;