Kinetic-energy inner product ($⟨v,u⟩=\int v^{\ast }\cdot udV$)

For a constant reference density, fluid perturbation kinetic energy is proportional to $\Vert u{\Vert }^2/2$ under the displayed $L^2$ inner product. An adjoint operator for perturbation evolution must be defined relative to this inner product, including boundary conditions and the projection onto incompressible flow.