ABSTRACT

The uncalibrated mathematical model resulting from the dimensional analysis procedure (π 1 = ϕ[nothing] = constant for a theoretical model, or π 1 = ϕ[π 2, π 3, …] for an empirical model) (see Chapter 7) may be used to develop criteria/laws that govern dynamic similarity/similitude between two flow situations that are geometrically similar but different in size. Analogous to dimensional analysis, dynamic similitude (i.e., the laws governing dynamic similarity) is also an essential and invaluable tool in the deterministic mathematical modeling of fluid flow problems (or any problems other than fluid flow). Deterministic mathematical models are used in order to predict the performance of physical fluid flow situations. As such, in the deterministic mathematical modeling of physical fluid flow situations (both internal and external flow), while some fluid flow situations may be modeled by the application of pure theory (for e.g., creeping, laminar, sonic, or critical flow), some fluid flow situations (actually most, due to the assumption of real turbulent flow) are too complex for theoretical modeling, such as turbulent, subsonic, supersonic, hypersonic, subcritical, or supercritical flow. However, regardless of the complexity of the fluid flow situation, the analysis phase of mathematical modeling involves the formulation, calibration, and verification of the mathematical model, while the subsequent synthesis or design phase of mathematical modeling involves application of the mathematical model in order to predict the performance of the fluid flow situation. Analysis, by definition, is “to break apart” or “to separate into its fundamental constituents,” where synthesis, by definition, is “to put together” or “to combine separate elements to form a whole.” As such, dynamic similitude is an essential and invaluable tool used in both the analysis and synthesis phases of the deterministic mathematical modeling of fluid flow problems.