Nonprimitive variable formulations in 2D

Abstract

In the study of two-dimensional flows, the incompressible Navier-Stokes equations can be formulated in a convenient alternative manner, by introducing two scalar variables, the vorticity ζ and stream function ψ, in place of the primitive variables, the velocity u and pressure P. The “curl” of ψ is nothing but u, whereas the variable ζ itself is the only vector component of ∇ x u which is different from zero in the particular situation of a plane motion. Such a fully scalar representation of the governing equations is by far the most frequently used for studying both steady and unsteady problems, because it offers the two advantages of reducing the number of the unknowns and of eliminating the in-compressibility condition whose fulfillment may be very difficult to be achieved, numerically.