Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections

•Three-dimensional incompressible velocity is visualized by a novel approach.•Divergence-free quasi-two-dimensional projections of velocity field are used.•The projections are shown to be unique for a given velocity field.•Divergence-free Galerkin bases are used for calculation of the projections.•Flow in a laterally heated cube and lid-driven flow in a cube are used as examples.

Context: A visualization of three-dimensional incompressible flows by divergence-free quasi-two-dimensional projections of velocity field on three coordinate planes is proposed.Objective: To visualize 3D incompressible flow by 3 two-dimensional plots.Method: It is argued that such divergence-free projections satisfying all the velocity boundary conditions are unique for a given velocity field. It is shown that the projected fields and their vector potentials can be calculated using divergence-free Galerkin bases.Results: Using natural convection flow in a laterally heated cube as an example, it is shown that the projections proposed allow for a better understanding of similarities and differences of three-dimensional flows and their two-dimensional likenesses. An arbitrary choice of projection planes is further illustrated by a lid-driven flow in a cube, where the lid moves parallel either to a sidewall or a diagonal plane.Conclusion: A new method for visualization of 3D incompressible flows is developed and described.