# sandbox/easystab/boundary_layer_2D.m

A code that looks very much like jet_2D.m except for the fact that the top boundary is not a slip boundary. We build the inflow profile with a Blasius boundary layer profile to see how it behaves in 2D with the Navier-Stokes equations. The Blasius profile is computed in blasiusf.m, a function version of the code blasius.m.

See jet_2D.m and then venturi.m to learn how this works.

Dependency:

clear all; clf; format compact

% parameters
Re=2; % Inflow reynolds number
Nx=100; % number of grid nodes in z
Ny=50; %number of grid nodes in r
Lx=100; % domain length
Ly=30; % domain height

% differentiation
[d.x,d.xx,d.wx,x]=dif1D('fd',0,Lx,Nx,5);
[d.y,d.yy,d.wy,y]=dif1D('cheb',0,Ly,Ny);
[D,l,X,Y,Z,I,NN]=dif2D(d,x,y);

% preparing boundary conditions
NN=Nx*Ny;
l.u=(1:NN)'; l.v=l.u+NN; l.p=l.v+NN;
II=speye(3*NN);
D.lap=D.yy+D.xx;

neuploc=[l.ctl;l.ctr;l.ctr-Ny];  % where to impose the neumann condition on the pressure
p0loc=2*Ny; % where to impose zero pressure
dir=[l.left;l.top;l.bot]; % where to put Dirichley on u and w

loc=[l.u(dir); l.v(dir); l.p(p0loc); ...
l.u(l.right); ...
l.v(l.right); ...
l.p(neuploc)];

C=[II([l.u([l.left;l.bot]);l.v(dir);l.p(p0loc)],:); ...     % Dirichlet on u,v,and p
D.y(l.top,:)*II(l.u,:); ... % slip at top for u
D.x(l.right,:)*II(l.u,:); ...   % Neuman on v at outflow
D.x(l.right,:)*II(l.v,:); ...   % Neumann on u at outflow
D.lap(neuploc,:)/Re*II(l.v,:)-D.x(neuploc,:)*II(l.p,:)]; % neuman constraint on pressure

% Compute Blasius velocity profile
[ybla,ubla]=blasiusf(Ly,100); plot(ubla,ybla)
uprof=interp1(ybla,ubla,y);

% initial guess
V=zeros(NN,1);
U=uprof*ones(1,Nx);
P=-X/Re; P=P-P(p0loc); % pressure zero at p0loc
q0=[U(:);V(:);P(:)];

% Newton iterations
disp('Newton loop')
q=q0;
quit=0;count=0;
while ~quit

% the present solution and its derivatives
U=q(l.u); V=q(l.v); P=q(l.p);
Ux=D.x*U; Uy=D.y*U;
Vx=D.x*V; Vy=D.y*V;
Px=D.x*P; Py=D.y*P;

% nonlinear function
f=[-U.*Ux-V.*Uy+D.lap*U/Re-Px; ...
-U.*Vx-V.*Vy+D.lap*V/Re-Py; ...
Ux+Vy];

% Jacobian
A=[-(spd(U)*D.x+spd(Ux)+spd(V)*D.y)+D.lap/Re, -spd(Uy), -D.x; ...
-spd(Vx),-(spd(U)*D.x+spd(V)*D.y+spd(Vy))+D.lap/Re, -D.y; ...
D.x, D.y, Z];

% Boundary conditions
f(loc)=C*(q-q0);
A(loc,:)=C;

% plotting
subplot 311;
surf(X,Y,reshape(P-1,Ny,Nx)); view(2); shading interp; hold on

sely=1:Ny; selx=1:6:Nx;
ww=reshape(U,Ny,Nx); vv=reshape(V,Ny,Nx);
quiver(X(sely,selx),Y(sely,selx),ww(sely,selx),vv(sely,selx),'k');
axis([0,Lx,0,Ly]);
xlabel('z'); ylabel('r'); title('Pressure P'); grid off;hold off

subplot 312;
xlabel('x'); ylabel('y'); title('horizontal velocity U'); grid off

subplot 313;
xlabel('x'); ylabel('y'); title('vertical velocity V'); grid off
drawnow

% convergence test
res=norm(f);
disp([num2str(count) '  ' num2str(res)]);
if count>50|res>1e5; disp('no convergence');break; end
if res<1e-5; quit=1; disp('converged'); continue; end

% Newton step
q=q-A\f;
count=count+1;
end

set(gcf,'paperpositionmode','auto')
print('-dpng','-r80','boundary_layer_2D.png')


The screen output for the convergence:

Blasius: converged
Newton loop
0  33.8846
1  0.067715
2  0.0011648
3  2.9002e-07
converged

And the figure, where we see how the boundary layer diffuses in the vertical direction

%}