free_surface_gravity_vector_field

Free-surface: displaying the vector field of the surface
The animation
Velocity fields

Free-surface: displaying the vector field of the surface

In this code, we take the code that computes the eigenmodes for free surface waves free_surface_gravity_particles.m and we plot the vector field to confirm the motion of the particles on the surface that we observe in the animation.

clear all; clf; n=100; % number of gridpoints alpha=1; % wavenumber in x L=2; % Fluid height in y rho=1; % fluid density mu=0.0001; % fuid viscosity g=1; % gravity % differentiation matrices scale=-2/L; [y,DM] = chebdif(n,2); D=DM(:,:,1)scale;
DD=DM(:,:,2)scale^2;
y=(y-1)/scale; I=eye(n); Z=zeros(n,n); % renaming the matrices dy=D; dyy=DD; dx=ialphaI; dxx=-alpha^2I; Delta=dxx+dyy; % System matrices A=[muDelta, Z, -dx, Z(:,1); … Z, muDelta, -dy, Z(:,1); … dx, dy, Z, Z(:,1); … Z(1,:),I(n,:),Z(1,:),0]; E=blkdiag(rhoI,rho*I,Z,1); % boundary conditions loc=[1,n,n+1,2*n]; C=[I(1,:),Z(1,:),Z(1,:),0; … Z(1,:),I(1,:),Z(1,:),0; … Z(1,:),Z(1,:),-I(n,:),rho*g; … dy(n,:),dx(n,:),Z(1,:),0]; E(loc,:)=0;
A(loc,:)=C; % compute eigenmodes [U,S]=eig(A,E); s=diag(S); [t,o]=sort(-real(s)); s=s(o); U=U(:,o); rem=abs(s)>1000; s(rem)=[]; U(:,rem)=[];

The animation

We show the position of the free surface stored in q(eta), and the particles to visualise the motion of the particles when the gravity wave propagates on free surface.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% particle animation of the eigenmodes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% parameters
nper=2      % number of periods of oscillation
nt=100       % number of time steps per period
nx=40       % number of points in x
modesel=2;  % which more do animate

% select the eigenmode
u=1:n; v=u+n; p=v+n; eta=3*n+1;
q=U(:,modesel); 
lambda=s(modesel);

% The time and x extent
tvec=linspace(0,nper*2*pi/abs(lambda),nper*nt); 
dt=tvec(2)-tvec(1);
Lx=2*pi/alpha; 
x=linspace(0,Lx,nx);

% scale mode amplitude 
q=0.05*q/abs(q(eta)); 

% initialize tracer particles
[px,py]=meshgrid(linspace(0,Lx,60),linspace(0,L,30)); 
py=py.*(1+2*real(exp(lambda*tvec(1))*q(eta)*exp(i*alpha*px))/L);
px=px(:);py=py(:);  

% time loop
for ind=1:nper*nt

    % expand mode to physical space 
    qq=2*real(exp(lambda*tvec(ind))*q*exp(i*alpha*x));
    
    figure(1)
    % plot pressure
    surf(x,y,qq(p,:)-10,'facealpha',0.3); view(2); shading interp; hold on
    
    % plot free surface
    plot(x,L+qq(eta,:),'k-',x,0*x+L,'k--'); hold on
    
    % plot the particles
    plot(mod(px,Lx),py,'k.');
    
    % compute velocity at position of particles
    pu=interp1(y,q(u),py);
    pv=interp1(y,q(v),py);

    % For particles above L, use Taylor expansion for velocity
    pu(py>L)=q(u(n))+q(eta)*dy(n,:)*q(u);
    pv(py>L)=q(v(n))+q(eta)*dy(n,:)*q(v);
    
    % expand to physical space
    puu=2*real(exp(lambda*tvec(ind))*pu.*exp(i*alpha*px));
    pvv=2*real(exp(lambda*tvec(ind))*pv.*exp(i*alpha*px));
 
    % advect particles
    px=px+puu*dt;
    py=py+pvv*dt;
       
    xlabel('x');    
    ylabel('y'); 
    axis equal; axis([0,Lx,0,1.3*L]); 
    grid off
    hold off
    drawnow
    end
    
    set(gcf,'paperpositionmode','auto');
    print('-dpng','-r100','free_surface_gravity_particles.png');

Velocity fields

We show the position of the free surface stored in q(eta), and velocity vectors for the top particles of the free surface.

    figure(2)
    quiver(x',(L+qq(eta,:))',real(qq(u(n),:)),real(qq(v(n),:)),'b'); hold on
    quiver(x,L+qq(eta,:),imag(qq(u(n),:)),imag(qq(v(n),:)),'r'); hold on
    plot(x,L+qq(eta,:),'k-',x,0*x+L,'k--');

    set(gcf,'paperpositionmode','auto');
    print('-dpng','-r100','free_surface_gravity_velocity_field.png');

The figure at final iteration of the time loop, showing the free surface at the top, the particles in black, and the disturbance pressure field in color

Velocity vectors for the top particles of the free surface

From the figure of the velocity vector, we can confirm the motion of the particles. The velocity vectors show that a particle on the free surface will trace out a counter-clockwise circle as the wave travelling from right to left. This is exactly what we observe in the animation of the free surface gravity waves.