Review of Fundamental Equations

Abstract

In this chapter we (re-)familiarize the reader with concepts from mathematics, thermodynamics, and aerodynamics that are fundamental to the methods presented in the remainder of the book. This includes a short review of vector algebra and partial-differential equations to provide the mathematical insight into the governing equations of fluid flow. Examples include the one-dimensional wave equation and the one-dimensional heat equation. A basic review of thermodynamics is given including the equation of state, the first law and the second law of thermodynamics. Also the isentropic relations between pressure, density and temperature are derived. In the aerodynamics section, the Navier-Stokes equations are derived in integral and derivative form. To simulate transonic flow, often simplifications of the Navier-Stokes equations are used. Therefore, it is shown how the Reynolds-Averaged Navier-Stokes (RANS) equations can be obtained and how the k-epsilon turbulence model can be used to close the RANS equations. For inviscid flow the Euler equations, and the full-potential equation are derived. Using examples from the literature, it is shown how well each of these models can predict the outcome of aerodynamic experiments in transonic conditions. This chapter contains 11 examples and concludes with 29 practice problems.