Fundamentals of Aerodynamics 2024, 48: Numerical Techniques for Nonlinear Supersonic Flow

This chapter goes into the practical engineering of it all, which means numerical approximation for the problems like nozzle and wing design. While this isn't strictly speaking a topic confined to aerodynamics, I'll still want to read these and might as well get an overview for the different options when it comes to numerical methods, so perhaps I can pick them up in more detail at a later time.

The Method of Characteristics can be employed for steady, inviscid, supersonic flows in 2D. There may be curves along which the derivatives of the flow-field variables are indeterminate or discontinuous. These are the "characteristic lines". One will be able to discern that a point A, two characteristic lines intersect, with slopes of tan(θ ± μ). In fact, the characteristic lines note the Mach waves running through any intersection point, so they are Mach lines. Mathematically, they're described by the compatibility equations.

The points of intersections can either occur within the flow, or at the walls of the flow. In the latter case, it suffices for a single line to impact the wall. In either case, the coefficients for the solutions of the resulting PDE will give a matrix of angles and airspeeds, which can be solved through a Gaussian.

The Finite-Difference Method spans a grid across the outline of the flow, with equidistant grid points along physical planes and create a system of PDEs.

Deriving the flux variables F, G, H and K by x, creates a new set of PDEs for which to solve. Solving it requires more or less the Taylor series and perturbation theory, which may as well be done numerically. The Time-Dependent Technique is in essence a similar approach, though using the momentum, energy and continuity equations as a starting point.