ABSTRACT
Several sets of equations that fall between the full Navier–Stokes and boundary-layer equations are of importance in applications in fluid mechanics and heat transfer. This chapter focuses on these approximate equations and begins with the thin-layer equations, particularly for unsteady flow. This reduced set permits approximate solutions to flow fields including separation to be computed in an efficient manner. The parabolized Navier–Stokes (PNS) equations are discussed both for steady and unsteady flows although the major application is in the steady case. Downstream influence is limited in the steady equations, and solutions may be obtained using a marching procedure requiring particular attention to pressure gradient. A number of solution techniques for solving the parabolized equations are presented and include the Beam–Warming and Roe schemes among others. Fully parabolic procedures are considered in both two-dimensional and in three-dimensional flows. Partially parabolized schemes for space-marching solution schemes for subsonic flow are considered and include pressure-correction methods and fully coupled approaches. Methods for solving the viscous shock-layer equations are also included. Throughout the chapter, examples show the application of the reduced sets of Navier–Stokes equations to problems spanning a range from thin-shear-layer flows to those encountered in hypersonic flight.
