Three-dimensional viscous flow solutions with a vorticity-stream function formulation

Abstract

A three-dimensional streamlike function/vorticity transport procedure has been developed for the analysis of two-and three-dimensional steady, inviscid and viscous external flows. Special care has been taken to incorporate interacting boundary-layer concepts into this Navier-Stokes procedure in order to take advantage of the features that interacting boundary-layer techniques offer but retain the generality of Navier-Stokes methods. Numerical techniques have been applied to the present formulation that honorthe elliptic nature of the flow in the inviscid region and the parabolic nature in the viscous part of the flow. An implicit, successive line relaxation technique is used to solve two sets of streamlike function/vorticity transport equations for the streamwise and crossflow components of the flow. Solutions are presented for a series of model problems that demonstrate that this Navier-Stokes technique is capable of yielding accurate solutions that simultaneously capture the details of the inviscid and viscous flow regions. Nomenclature Cf x = axial component of skin friction Cf z = spanwise component of skin friction L -reference length p = static pressure Re = freestream reference Reynolds number u = axial (x) component of velocity v = normal (y) component of velocity V = velocity vector Fa, = freestream speed w = spanwise (z) component of velocity x = axial Cartesian coordinate y = normal Cartesian coordinate z -spanwise Cartesian coordinate v = kinematic viscosity coefficient 6 = stream function in z ,y plane \l/ = stream function in x,y plane a? = vorticity vector u x = vorticity component in the axial (x) direction u y = vorticity component in the normal (y) direction co z = vorticity component in the spanwise (z) direction

Vorticity-Velocity Formulation for Three-Dimensional Steady Compressible Flows

“…Some recent successful implementations of the vorticity-vector potential formulation include natural convection problems [2] and inlet and indraft wind tunnel simulations [13]. The use of a streamfunction-vorticity transport procedure for three-dimensional external flows has also been reported [14].…”