Q:

Velocity Field / 3D Vector Fields in Tikz or PGFplots

I would like to draw "Velocity field" or "Vector Field" like:

I have tried using PGFplots "quiver" but this only works for vector fields on SURFACES: 
\begin{tikzpicture}
\begin{axis}[
domain=0:1,
xmax=1,
ymax=1,
]
\addplot3[cyan,/pgfplots/quiver,
quiver/u=y,
quiver/v=z,
quiver/w=x,
quiver/scale arrows=0.1,
-stealth,samples=10] ({x},{y},{x+y});
\end{axis}
\end{tikzpicture}

Is there a mechanism for drawing vectors on a 3D LATTICE?  Something which does
for i from 1 to 10
   for j from 1 to 10
      for k from 1 to 10
            draw vector (i,j,k) -- f(i,j,k);
      end do;
   end do;
end do;

in PGFplots or Tikz?  (As done here: 3D Vector Fields in Asymptote )

A:

You could unroll the layers in z direction by hand using \pgfplotsinvokeforeach like in the hedgehog example below.  I could not use your example because the parametric function f = (x,y,x+y) actually is a surface.
\documentclass{article}
\usepackage{pgfplots}
\begin{document}

\begin{tikzpicture}
  \begin{axis}[
    domain=-1:1,
    samples=10,
    xmin=-1,xmax=1,
    ymin=-1,ymax=1,
    zmin=-1,zmax=1,
    ]
    \pgfplotsinvokeforeach{-1,-.5,0,.5,1}{
      \addplot3[cyan,quiver,-stealth,
      point meta={sqrt((x)^2+(y)^2+(z)^2)},
      quiver={
        u={x/sqrt((x)^2+(y)^2+(z)^2)},
        v={y/sqrt((x)^2+(y)^2+(z)^2)},
        w={z/sqrt((x)^2+(y)^2+(z)^2)},
        colored,scale arrows=.1}]
      (x,y,#1);
    }
  \end{axis}
\end{tikzpicture}

\end{document}

