Simulation of the Taylor-Couette flow

Coded by Paul Valcke and Luis Bernardos. This codes shows the simulation of a flow around a cylinder. We solved this problem using the steady Navier-Stokes equations. Velocity-pressure non-linear coupling is solved by means of a Newton method, based on the Jacobian matrix. In this code we use a first-order march in time for integrating through time.

There exist an analytical solution for this flow, then we compare it with theory in the last section of this page.

The Taylor-Couette flow consists on two concentrical cylinders. The outter-most cylinder rotates at constant speed, adding a rotational velocity to the interior flow by means of the viscosity. This phenomena is used for measuring the viscosity of fluids.

Through this code we want to perform a calculation of the airflow taking into consieration that we want to use a “deformed” circular mesh, and we want to solve the cartesian-Steady Navier-Stokes equations:

\displaystyle u_j\partial_ju_i = -\partial_ip + \frac{1}{Re}\partial_{jj}^2u_i

Ultimately, we want to compare the velocity profile obtained with the theoretical one.

clear all; clf;


Parameters

Here the user can change the geometry and physics of the flow.

%%%% parameters
Re=10; % reynolds number
Rmax = 1; % Outter cylinder radius
Rmin = 0.5; % Inner cylinde radius
Ntheta=101; % number of grid nodes in theta
Nrho=30; %number of grid nodes in rho
pts=5; % number of points in finite difference stencils
alpha=1; % Under-relaxation
maxIter = 200;
resSTOP = 1e-5;

Lx = 2*pi*Rmax; Lx_vector = linspace(0,2*pi,Ntheta);
Ly = Rmax-Rmin;

% differentiation
[d.x,d.xx,d.wx,x]=dif1D('fd',0,1,Ntheta,pts);
[d.y,d.yy,d.wy,y]=dif1D('fd',0,1,Nrho,pts);
[D,l,~,~,Z,I,NN]=dif2D(d,x,y);


Mesh

We make use of Lspacing for shrinking the mesh near the walls. Also, we make use of map2D for transforming the differentation matrices.

% Meshing
[Theta, Rho] = meshgrid(linspace(0,2*pi,Ntheta),Rmin + Lspacing(Nrho,1,Rmax-Rmin));
[X, Y] = pol2cart(Theta,Rho);
D=map2D(X,Y,D);
D.lap=D.yy+D.xx;

u=(1:NN)'; v=u+NN; p=v+NN;
II=speye(3*NN);
DD=blkdiag(D.y,D.y,D.y);


Boundary Conditions

The boundary conditions are all-dirichlet. No input nor output flow for this case. However, as the mesh joins itself at the overlapping boundary, a link condition has to be established, in both the continuity of u, v and p as well as the derivatives normal to the overlapping boundary (tangency of the unknowns).

% Boundary Conditions
dir=[l.top;l.bot]; % where to put Dirichlet on u and v
lnk1=[u(l.left); v(l.left); p(l.left)];
lnk2=[u(l.right); v(l.right); p(l.right)];
p0loc=1;
loc=[u(dir); v(dir); p(p0loc); ... % Dirichlet

C=[II([u(dir);v(dir);p(p0loc)],:);% Dirichlet on u,v at walls
II(lnk1,:)-II(lnk2,:); % Continuity of the value
DD(lnk1,:)-DD(lnk2,:)]; % Continuity of the normal derivative


We compute a suitably initial guess that will easen the convergence and reduce the number of iterations.

% initial guess
Ur=Rho-Rmin; % Modulus of the tangent velocity imposed on the outter cylinder.
U=-sin(Theta).*Ur;
V=cos(Theta).*Ur;
P=0*X(:);

q0=[U(:);V(:);P(:)];


Calculation

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

% the present solution and its derivatives
U=q(u);
V=q(v);
P=q(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; ...
D.x*U+D.y*V];

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

% Boundary conditions
f(loc)=C*(q-q0);
A(loc,:)=C;
The following commented lines are used for determining whether the matrix A is 

inversible or not. If the solution of Agives a NaN, then it can be beause A is not inversible.

%     % checking the null space of A
%     [UU,S]=eig(full(A));
%    s=diag(S);  [t,o]=sort(-real(s));
%    s=s(o); UU=UU(:,o);
%    rem=abs(s)>1000; s(rem)=[]; UU(:,rem)=[];
%
%    sel=abs(s)<1e-5;
%    s=s(sel), UU=UU(:,sel);
%
%    for num=1:length(s);
%    subplot(3,3,num)
%        U=UU(u,num);
%    V=UU(v,num);
%    P=UU(p,num);
%
%    nu=norm(U)
%    nv=norm(V)
%
%    surf(X,Y,reshape(abs(P),Nrho,Ntheta)); view(2); hold on
%      quiver(X,Y,reshape(U,Nrho,Ntheta),reshape(V,Nrho,Ntheta));
%    end
%
%
%     break

% plotting
Module = sqrt((U.^2+V.^2));
hold on
uu=reshape(U,Nrho,Ntheta); vv=reshape(V,Nrho,Ntheta);
quiver(X(1:5:end,1:5:end),Y(1:5:end,1:5:end),uu(1:5:end,1:5:end),vv(1:5:end,1:5:end));
xlabel('x'); ylabel('y'); title(sprintf('Taylor-Couette Pressure coloured field\nand velocity vector field')); grid off;hold off
axis([-1,1,-1,1]*Rmax)
hold off
drawnow

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

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

colorbar;
set(gcf,'paperpositionmode','auto')
print('-dpng','-r80',sprintf('TaylorCouette.png'));


Results and Validation

The Taylor-Couette flow phisically has a linear velocity profile between the inner cylinder and the outter cylinder. Therefore, the velocity profile will be V_{th} = -R_{min} + \frac{U_{rot}}{R_{max}-R_{min}}\rho. Excellent agreement between theory and calculations is reached:

x_plot = X(l.left);
V_theory = -Rmin + (max(max(Ur))/(Rmax-Rmin)).*x_plot;
figure;
plot(x_plot,V_theory,'-r','DisplayName','Theoretical'); hold on
plot(x_plot,V(l.left),'ob','DisplayName','Calculated'); hold off
grid on;
legend('show','location','NorthWest');
set(gcf,'paperpositionmode','auto')
print('-dpng','-r80',sprintf('TaylorCouette_Val.png'));