VI. Differential Fluid Flow
Learning Objectives
- Develop most of the common mathematical tools required for modeling potential flows
- Summarize vector calculus definitions (gradient, divergence, curl, etc.) and translate them into tensor notation
- Appreciate importance of vector identities
- Develop a definition of the material derivative in Cartesian and Cylindrical coordinates
- Explore potential flow methods and solutions
- Establish formulation of the Continuity Equation visually and in equation form
- Define the streamfunction visually and in equation form
- Define the equipotential function and its relationship to the streamfunction
- Gain proficiency in understanding the basic potential flows, including uniform flow, source/sink flow, and vortex flow
- Expand upon basic flows to develop superpositions — Rankine oval, doublet, uniform flow over a cylinder
This chapter was revised and remixed from Intermediate Fluid Mechanics by James Liburdy which is licensed under a CC BY NC SA 4.0 International License., except where otherwise noted.
In this chapter, we embark on an exploration of the mathematical foundations of potential flow theory, a fundamental area of fluid dynamics that models flows where viscosity can be neglected. Potential flow theory provides insight into a variety of flow patterns in both natural and engineered systems, making it an essential tool for fields ranging from aerodynamics to hydrodynamics.
Our journey begins by equipping you with the mathematical tools necessary to model potential flows effectively. We will start by revisiting core concepts of vector calculus—such as gradient, divergence, and curl—and translating these into tensor notation for broader application and clarity in more complex fluid systems. Understanding vector calculus also necessitates an appreciation for the importance of vector identities, which provide relationships and simplifications critical for flow analysis.
An essential concept in fluid dynamics, the material derivative—capturing how quantities change with the movement of fluid elements—will be developed in both Cartesian and cylindrical coordinate systems. This concept lays the groundwork for the Continuity Equation, which represents the conservation of mass in a flow field. Through visual representations and mathematical formulations, we will derive and understand this foundational equation.
The chapter also introduces two key functions in potential flow theory: the streamfunction and the equipotential function. These functions describe flow characteristics visually and mathematically, providing a way to visualize and quantify the flow’s behavior. We will explore the interplay between these functions and their critical role in understanding and manipulating potential flows.
From here, we delve into some basic potential flows, including uniform flow, source/sink flow, and vortex flow. Mastery of these flows sets the stage for constructing more complex flow patterns using superposition methods. Through combining elementary flows, we will study configurations such as the Rankine oval, doublet, and uniform flow over a cylinder—configurations essential for understanding flow behavior around bodies.
By the end of this chapter, you will be able to construct, analyze, and interpret a wide range of potential flows. This foundation not only enhances your fluency in fluid dynamics but also prepares you for using potential flow modeling for describing the ocean and its interaction with structures.
