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Examination of compressibility effects in the Falkner-Skan equations

Miguel Ramirez
High Performance Computing and Visualization Laboratory (HPCVL), Department of Mechanical Engineering, University of Puerto Rico, Mayaguez, PR 00681, USA

Guillermo Araya
High Performance Computing and Visualization Laboratory (HPCVL), Department of Mechanical Engineering, University of Puerto Rico, Mayaguez, PR 00681, USA

DOI: 10.1615/TFEC2020.cmd.031960
pages 687-706

Abstract

The Navier-Stokes (NS) equations are the cornerstone of fluid mechanics. It is solved for given initial and boundary conditions in order to study the dynamics of fluids. Falkner-Skan (FS) developed a solution for these equations in laminar flow, as a generalization of the Blasius boundary layer. The idea is to represent accelerating (Favorable Pressure Gradient, FPG) or decelerating (Adverse Pressure Gradient, APG) flows, whose freestream velocity varies as a power law of the streamwise coordinate. the primary purpose of this work is to solve this set of equations for the laminar compressible flow case subject to streamwise pressure gradients via the use of the Illingworth (which considers only the effects of the local density) and Howarth (which takes into account the changes of the local density with respect to the free stream density) transformations, by considering supersonic and hypersonic Mach numbers and different wall temperature conditions (cold, adiabatic and hot wall conditions). It will also include a study of different pressure gradient parameters (i.e., β parameter). In conclusion, the algorithm developed for Falkner-Skan incompressible and compressible flow was effectively validated. As the Mach number increases up to hypersonic levels, the temperature ratio significantly increases in the near wall region for adiabatic conditions. It is important to mention that as the Mach number increases, the velocity boundary layer thickness decreases, whereas the shear stress near the wall increases.

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