Numerical Solution of Incompressible Turbulent Flow over Airfoils

Date of Award


Document Type


Degree Name

Doctor of Philosophy (Ph.D.)

Institution Granting Degree

Air Force Institute of Technology

Cedarville University School or Department

Engineering and Computer Science


Aerospace materials


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 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 different equations, at each time step, is solved using successive-over-relaxation iteration. Far field boundary conditions are examined. Solutions for a NACA 0012 airfoil at angles of attack varying from five to 11.5 degrees at a chord Reynolds number of 170,000 are obtained. Velocity profiles near the airfoil surface and surface pressure distributions are presented and compared with experimental data. Lift and drag coefficients agree well with experimental values. The computed lift coefficients near stall are within five percent of the experimental measurements, and the numerical drag coefficients agree within ten drag counts in the region of maximum lift to drag ratio. The short laminar separation bubble near the suction pressure peak is numerically determined. The variation of bubble length and turbulent transition length with angle of attack are similar to experimental trends.