**Visualization of 3D incompressible
flows**

see arXiv:1304.1884

Decompose a 3D divergence free velocity field into 3
components:

So that two dimensional divergence of each component
vanishes:

Now, for each component define vector potential, which
contains only one non-zero component:

**
**
,

* **
**
**,
*

* **
**
**,
*

Therefore, non-zero components of vector potentials are
extensions of two-dimensional stream function: vectors of are tangent to isosurfaces of
, respectively.

__Visualization of lid-driven flow in a
cube__

lid moves parallel to a wall, Re=1000;

lid moves parallel to the
diagonal, Re=1000: in (x,y,z) coordinates; main circulation perpendicular to
diagonal plane

__Visualization of natural convection in a
laterally heated cube__

Adiabatic horizontal and spanwise
boundaries: Ra=10^{3} Ra=10^{4} Ra=10^{5} Ra=10^{6} Ra=10^{7} Ra=10^{8}

Conducting horizontal boundaries: Gr=2.9ª10^{6}

Isothermal
surfaces at different boundary conditions

Animation of slightly
supercritical oscillatory flows. Thermally conducting horizontal boundaries,
Gr=3ª10^{6}.

Thermally conducting spanwise boundaries.

Multiple solutions for thermally insulated
spanwise boundaries: Regime 1, Regime 2