Consider the flow field formed by combining a uniform flow...

Consider the flow field formed by combining a uniform flow in the positive $x$ direction and a source located at the origin. Obtain expressions for the stream function, velocity potential, and velocity field for the combined flow. If $U=25 \mathrm{m} / \mathrm{s}$, determine the source strength if the stagnation point is located at $x=-1 \mathrm{m} .$ Plot the streamlines and potential lines.

Key Concepts

Uniform Flow

In potential flow theory, a uniform flow represents a constant velocity field where the speed and direction of the flow are the same at every point in the domain. This concept forms one of the elementary solutions to the potential flow equations and serves as a basis for modeling more complex flows when combined with other singular solutions like sources or vortices.

Source Flow

A source in potential flow mimics the effect of fluid emanating radially from a point in two-dimensional flow. Its mathematical formulation is characterized by a radially symmetric velocity field and is one of the fundamental singularities used to construct complex flow patterns through superposition.

Superposition Principle

The superposition principle allows for the construction of complex flow fields by linearly combining simple, elementary flows such as uniform flows, sources, and vortices. This principle leverages the linearity of Laplace's equation, which governs potential flow, to form resultant fields that satisfy the same governing equations.

Stagnation Point

A stagnation point in a flow field is a point where the local velocity of the fluid is zero. Identifying stagnation points is useful in analyzing flow behavior around objects or in composite flows, as they play a critical role in determining pressure distributions and ensuring the physical consistency of the flow solution.

Stream Function

The stream function is a scalar function whose contours represent streamlines—a visualization of the flow direction. For incompressible two-dimensional flows, the stream function is defined such that its partial derivatives yield the velocity components, ensuring that the continuity equation is automatically satisfied.

Velocity Potential

The velocity potential is a scalar function whose gradient equals the velocity field in irrotational flows. It serves as a convenient tool for solving potential flow problems and is closely connected to the stream function, with both functions representing orthogonal families of curves that describe the flow pattern.

Velocity Field

The velocity field in potential flow is derived from either the gradient of the velocity potential or the derivatives of the stream function. It provides a complete description of the flow's magnitude and direction at every point in the domain and is fundamental for analyzing dynamic properties, pressure distributions, and body forces within the fluid.

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