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Key Concepts

Potential Flow

Potential flow theory deals with fluid flows that are both incompressible and irrotational. It simplifies the governing equations by utilizing the existence of a velocity potential and a corresponding stream function, facilitating the analysis of fluid behavior through the use of harmonic functions.

Polar Coordinates

Polar coordinates provide a system for expressing problems with radial symmetry. When working with potential flows, the functions representing the velocity potential and stream function often need to be expressed in polar (or plane polar) coordinates, which require re-formulating the Laplacian and the Cauchy-Riemann equations accordingly.

Harmonic Conjugates

Harmonic conjugates are pairs of functions that together form the real and imaginary parts of an analytic function. In the context of potential flow, if one of these functions (the velocity potential) is harmonic, then there exists a unique (up to a constant) harmonic conjugate (the stream function) that satisfies the Cauchy-Riemann equations in the chosen coordinate system.

Stream Function

A stream function is a scalar function used to represent two?dimensional, incompressible flow. Its contours represent streamlines of the flow, and together with the velocity potential, it forms a pair of harmonic conjugate functions. The stream function ensures that the continuity equation for incompressible fluids is automatically satisfied.

Velocity Potential

A velocity potential is a scalar function whose gradient gives the velocity field in a flow that is irrotational. In potential flow theory, the existence of a velocity potential indicates that the flow can be derived from a potential function, and its validity is contingent upon the function satisfying Laplace's equation, ensuring the field is harmonic.

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