Analysis of unsteady viscous flow past an airfoil. I - Theoretical development

An analysis is presented for the two-dimensional unsteady flow of a viscous incompressible fluid past a finite thickness airfoil at angle of attack. Using techniques commonly employed in ideal-fluid aerodynamic analyses, the airfoil surface is represented by bound-vortex singularities y distributed over the outline of the airfoil. These coexist with the free vorticity to in the boundary layer and wake. Using standard procedures, the integral equation for y is cast into a Fred holm equation of the second kind. This differs from that found in inviscid analyses because of the contribution of to to the induced velocities at the airfoil surface, thus coupling the two vorticity fields. A further coupling arises when the no-slip condition is enforced at the airfoil surface, the enforcement of which causes free vorticity to be produced. The production of free vorticity is modeled by equating the local instantaneous value of y to the amount of free vorticity produced locally and impulsively at the airfoil surface. This free vorticity enters the fluid stream by diffusion, thus giving rise to the essential boundary condition which must be imposed on the transport equation governing the distribution of free vorticity in the fluid. The formulation is developed in detail for the case where the airfoil is impulsively set into translational motion. The analysis makes explicit use of the requirement that the total vorticity of the fluid must remain zero, thus removing any ambiguities in the solution for y. This also insures that the pressure distribution on the airfoil remains single-valued. The numerical formulation and results for a particular airfoil are presented in Part II of the paper.