In the previous chapter, the forces acting on a moving fluid element were exhaustively studied. Using Newtons second law of motion, the Navier–Stokes equations for both compressible and incompressible flows were obtained. This chapter uses an alternative approach to developing the Navier–Stokes equations. Namely, by starting from a Eulerian description (as opposed to a Lagrangian description), the integral form and conservation form of the Navier–Stokes equations are developed. The continuity and Navier–Stokes equations in its various forms are tabulated and reviewed in this chapter. This chapter ends by solving some very simple, yet common, problems involving the incompressible Navier–Stokes equations.

The aerodynamics of airplanes designed before and during World War II springs from linear potential theory (discussed in Chapter 3) together with empirical data and lessons learned from previous airplane designs. Jet engines and rocket propulsion enabled vehicles to fly much faster. This uncovered high-speed aerodynamic phenomena that must be understood for the successful design of airplanes capable of trans- and supersonic flight and of space vehicles. This chapter presents the numerical methods employed in computational fluid dynamics (CFD) to treat shock waves. A complete recipe for inviscid nozzle flow is given, with accompanying tutorial software. Mature tools are now standard in the form of industrial-strength CFD codes. A perusal of the user manual for any one of them shows many options and functions. A basic understanding of the theory is needed for the user to set up the code properly for the intended case. While not focusing on constructing such a CFD code, this chapter lays the foundations for training the reader to become an "informed user" of these codes by learning CFD "due diligence." It spells out CFD fundamentals such as constructing the numerical flux, artificial dissipation, approximate Riemann, high-resolution schemes. explicit and implicit time integration, and convergence to steady state.