Numerical simulations of incompressible aerodynamic flows using viscous/inviscid interaction procedures

Steady two-dimensional laminar incompressible flows over airfoils are simulated using a Helmholtz velocity decomposition where the velocity vector is split into a gradient of a potential and a correction representing the rotational components. For most aerodynamic flows, the rotational components of the velocity vanish outside the boundary layer and the wake region. Therefore, the near and far velocity fields can be represented by a potential function and the pressure can be obtained there from Bernoulli’s law. In the viscous flow region, the momentum equations are integrated to calculate the rotational velocity components. Conservation of mass leads to a Poisson’s equation for the potential function, where the right hand side, the forcing function, is given in terms of the divergence of the rotational velocity component. The latter represents a distribution of sources in the viscous layer and results in a displacement effect on the outside potential field. In this formulation, the pressure in the momentum equations does not play an essential role since the conservation of mass is imposed through the augmented potential equation. After the velocity field is obtained, the correct pressure distribution in the viscous flow region is calculated by integrating the normal momentum equation. Numerical results are presented to confirm the validity and the merits of the present formulation. Solving a Poisson’s equation for the potential function, rather than the pressure as in standard methods for solving Navier–Stokes equations, is the main advantage since the calculations of the rotational velocity components are restricted to the viscous flow region only. Standard numerical methods are applicable to the present formulation including multigrid acceleration techniques for the augmented potential equation.