8 Nonlinear and Time-Dependent Flow
8.1 What changes beyond Stokes flow
Navier–Stokes adds the convective term \((\mathbf u\cdot\nabla)\mathbf u\) and possibly a time derivative. Nonlinearity means that a linear solve may be only one step of an iterative method.
8.2 Time discretization
A backward-Euler approximation uses
\[\frac{\partial\mathbf u}{\partial t}\bigg\rvert_{t^{n+1}}\approx \frac{\mathbf u^{n+1}-\mathbf u^n}{\Delta t}.\]
It is first-order accurate in time for smooth solutions under standard assumptions. Higher-order schemes may offer better accuracy but introduce different stability and implementation considerations.
8.3 Linearization
Picard-type iterations lag selected velocity factors in the convection term. Newton-type methods linearize the full nonlinear residual around the current iterate. Both require sensible stopping criteria and residual definitions.
8.4 Conservation is a diagnostic
A small solver residual does not prove acceptable discrete mass balance. Monitor divergence, boundary fluxes, and quantities relevant to the physical problem, particularly as the mesh and time step are refined.
Question: What happens to an apparent steady solution when both mesh size and time step are reduced?
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.