Knowlpedia

Definition

Navier–Stokes momentum residual

The defect obtained by evaluating the momentum equation on a trial velocity and pressure.

For sufficiently differentiable trial fields u,pu,p, fixed viscosity ν>0\nu>0, and prescribed force ff, the momentum residual is

Rν(u,p;f)=∂tu+(u⋅∇)u+∇p−νΔu−f.\mathcal R_\nu(u,p;f) =\partial_tu+(u\cdot\nabla)u+\nabla p-\nu\Delta u-f.

It vanishes precisely when the momentum equation in holds. The additional condition ∇⋅u=0\nabla\cdot u=0, initial data, and boundary conditions are separate constraints.

Exact correction identity

For increments w,πw,\pi,

Rν(u+w,p+π;f)=Rν(u,p;f)+∂tw+(u⋅∇)w+(w⋅∇)u+∇π−νΔw+(w⋅∇)w.\mathcal R_\nu(u+w,p+\pi;f)=\mathcal R_\nu(u,p;f) +\partial_tw+(u\cdot\nabla)w+(w\cdot\nabla)u +\nabla\pi-\nu\Delta w+(w\cdot\nabla)w.

The last term is the quadratic error after a linear correction. This sign convention includes −f-f; defining an effective force instead requires stating the corresponding sign explicitly.