Helmholtz-Hodge Decomposition

A fundamental theorem of Vector Calculus relevant to Fluid Dynamics that states that any sufficiently smooth Vector Field $w$ can be decomposed into two components orthogonal in the $L^2$ Hilbert Space

$$w=u+\nabla q$$

• The Divergence-Free part $u$
  • $\text{div }u=0$
  • The incompressible part of the flow
  • The part of the flow that forms closed loops/vortices
• The Gradient part $\nabla q$
  • $\text{curl }(\nabla q)=0$
  • The irrotational part of the flow
  • The “straight-line” motion of the flow driver by the scalar potential field $q$ (e.g. pressure or temperature), responsible for the expansion or contraction of the field