Navier-Stokes Solutions for Flow Past a Class of Two-Dimensional Semi-Infinite Bodies

Abstract

The complete Navier-Stokes equations have been used to analyze the symmetric laminar incompressible flow past a class of two-dimension al semi-infinite bodies including the family of parabolas and rectangular slabs as special cases. The problem is formulated in terms of coordinates obtained from the Cartesian coordinates by a conformal transformation. Similarity-type variables are used for the vorticity and stream functions. In these variables, the solution approaches the Blasius solution far downstream and the correct inviscid flow transversely far from the body surface. The formulation also produces the correct starting solution along the stagnation streamline. An alternating direction implicit finite-difference method is used to obtain the numerical solution. Results are presented for the skin-friction function and the surface pressure distributions for various values of the problem parameters. For the rectangular slab with a sharp shoulder, the wall shear is unbounded at the shoulder; however, the vorticity function employed remains bounded. For large Reynolds number, separation and reattachment are observed aft of the region of the shoulder, resulting in a separation bubble of finite, sometimes quite large extent. The flow structure in the separation region is carefully analyzed. Finally, it is shown that a certain boundary-layer-type simplified form of the vorticity equation may be used in separation studies if the displacement effects are correctly accounted for through the complete stream function equation.