Prediction of Wing Section Lift and Drag from Numerical Solutions of the Navier-Stokes Equations

Doctoral thesis, 1994

A Navier-Stokes finite difference method is developed and applied to two-dimensional, incompressible, viscous flow around wing sections at medium to high Reynolds numbers. An important feature of such flows is laminar boundary layers with spatially amplified Tollmien-Schlichting waves initiating transition to turbulence. Often the transition takes place in a laminar separation bubble. The governing equations in strong conservative form are discretized with second order upwind differences for the convective terms and second order central differences for the other terms. A colocated grid is used and to supress chequerboard oscillations in the pressure, a special form of the Laplace operator is used. Two temporal integration schemes are tried, a four stage explicit Runge-Kutta formula, and a SIMPLER-like implicit formula. With the need to resolve the thin, laminar boundary layer in the present application, the explicit scheme turned out to be prohibitively computer-time consuming, even when using local time-stepping to accelerate convergence. Far-field boundary conditions derived from potential flow around a vortex, source and dipole with the same lift, drag and thickness as the actual wing section are tried but in their present form gives no consitent improvement. Transition is predicted with the en-method. The Baldwin-Lomax turbulence model is used with an empirical formula to model transition. Three wing sections are studied, NACA 0013, NACA (663-018) and FX 67-K-170/17, at Reynolds numbers around 3*106. The lift coefficients are close to experimental values up to moderate angles of attack except for the NACA (663-018) which has a jog in the lift curve slope. This is due to a turbulent separation bubble that is underprediced by the computations. A maximum lift is predicted, but with varying success regarding its value and at what angle of attack it occurs. The drag coefficient is rather close to experimental values up to moderate angles of attack. The low-drag region of the laminar wing sections is accurately predicted. Transition location is accurately predicted but the transitional bubbles are too small or nonexistent.