Introduction

Having established the tools for discretization and methods of analysis of the discretized equation, here we explain methods for solving Navier–Stokes equations of steady and unsteady incompressible viscous flows. Versions of Navier–Stokes equation in primitive and derived variables, expressed in inertial and non-inertial frames have been given in Chapter 2. Various versions of pressure Poisson equation are also derived in Chapter 2, which is to be solved for accurate evaluation of loads and detailed pressure distribution.

Navier–Stokes equation is an evolution equation for vorticity, a primary physical quantity of interest for unsteady laminar and turbulent flows. Primarily, vorticity is generated at physical boundaries for wall-bounded flows, as a consequence of no-slip condition. In free shear layers, vorticity is generated by flow instabilities at interfaces in mixing layers and jets; primary instability mechanisms are attributed to Rayleigh–Taylor, Kelvin–Helmholtz, Görtler mechanisms. Hence, VTE is central for analysis and solution of Navier–Stokes equation. Attendant velocity field can be obtained from the solution of SFE for 2D flows. Poisson equations relating velocity and vorticity field can also be solved for 2D and 3D flows in vorticity–velocity formulation. As stated in Chapter 2, vorticity–stream function formulation is preferred over vorticity–velocity formulation, whenever accurate solution is desired.