Relationship Between the Stream Function and Velocity Potential

In Section 2.15, we demonstrated that for an irrotational flow Y = Уф. At this stage, take a moment and review some of the nomenclature introduced in Section 2.2.5 for the gradient of a scalar field. We see that a line of constant ф is an isoline of ф; since ф is the velocity potential, we give this isoline a specific name, equipotential line. In addition, a line drawn in space such that Уф is tangent at every point is defined as a gradient line; however, since Уф = V, this gradient line is a streamline. In turn, from Section 2.14, a streamline is a line of constant Ф (for a two-dimensional flow). Because gradient lines and isolines are perpendicular (see Section 2.2.5, Gradient of a Scalar Field), then equipotential lines (ф = constant) and streamlines (ф = constant) are mutually perpendicular.

To illustrate this result more clearly, consider a two-dimensional, irrotational, incompressible flow in cartesian coordinates. For a streamline, ф(х, у) = constant. Hence, the differential of ф along the streamline is zero; i. e.,

image181[2.159]

From Equation (2.150a and b), Equation (2.159) can be written as

Подпись:йф = —v dx + и dy = 0

2 ф (or тД) can be defined for axisymmetric flows, such as the flow over a cone at zero degrees angle of attack. However, for such flows, only two spatial coordinates are needed to describe the flow field (see Chapter 6).

Relationship Between the Stream Function and Velocity Potential Relationship Between the Stream Function and Velocity Potential

Solve Equation (2.160) for dy/dx, which is the slope of the ф = constant line, i. e., the slope of the streamline:

Подпись: [2.163]dф = и dx + v dy = 0

Relationship Between the Stream Function and Velocity Potential Relationship Between the Stream Function and Velocity Potential

Solving Equation (2.163) for dy /dx, which is the slope of the ф — constant line, i. e., the slope of the equipotential line, we obtain

Equation (2.165) shows that the slope of а ф = constant line is the negative reciprocal of the slope of а ф = constant line, i. e., streamlines and equipotential lines are mutually perpendicular.

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