The numerical solution of the Navier-Stokes equations for incompressible turbulent flow over airfoils

Numerical solutions are obtained for two-dimensional incompressible turbulent viscous flow over airfoils of arbitrary geometry. An algebraic eddy viscosity turbulence model based on Prandtl's mixing length theory is modified for separated adverse pressure gradient flows. Finite difference methods for solving the inviscid stream function equation and the incompressible laminar Navier-Stokes equations are used. A finite difference method for solving the Reynolds averaged incompressible turbulent two-dimensional Navier-Stokes equations is employed. The inviscid stream function equation and the Navier-Stokes equations are transformed using a curvilinear transformation. A body-fitted coordinate system with a constant coordinate line defining the airfoil section surface is transformed to a rectangular coordinate system in the transformed or computational plane. An elliptic partial differential Poisson equation for each coordinate is used to generate the coordinate system in the physical plane for arbitrary airfoils. The two-dimensional time dependent Reynolds averaged incompressible Navier-Stokes equations in the primitive variables of velocity and pressure and a Poisson pressure equation are numerically solved. Turbulence is modelled with an adverse pressure gradient eddy viscosity technique. An implicit finite difference method is used to solve the set of transformed partial differential equations. The system of linearized simultaneous difference equations, at each time step, is solved using successive-over-relaxation iteration.