Stream Function & Velocity Potential

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In two-dimensional, incompressible flow, two scalar functions are widely used to describe the velocity field: the stream function and the velocity potential. These mathematical constructs simplify analysis, enable visualization of flows, and provide powerful tools for solving fluid flow problems analytically.

1️⃣ Stream Function (ψ)

1.1 Definition

The stream function, denoted by ψ\psi, is a scalar function defined so that its contours represent streamlines of the flow. In two-dimensional, incompressible flow (x–y plane), it is defined such that:

u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}

where:

  • uu is the velocity component in the x-direction,
  • vv is the velocity component in the y-direction.

1.2 Incompressibility Condition

For incompressible, two-dimensional flow, the continuity equation is:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

Substituting the definitions from the stream function automatically satisfies this equation:

\frac{\partial}{\partial x} \left(\frac{\partial \psi}{\partial y}\right) + \frac{\partial}{\partial y} \left(-\frac{\partial \psi}{\partial x}\right) = 0

This property makes the stream function especially useful for incompressible flows.

1.3 Physical Meaning

  • Lines of constant ψ\psi are streamlines—paths tangent to the local velocity vector everywhere.
  • The difference in ψ\psi values between two streamlines equals the volume flow rate per unit depth between them:

\Delta \psi = q

where qq is the flow rate per unit width perpendicular to the plane.

1.4 Applications

  • Visualization of flow fields in two dimensions.
  • Simplification of solving the continuity equation for incompressible flows.
  • Useful in potential flow theory to construct complex flows via superposition.

2️⃣ Velocity Potential (φ)

2.1 Definition

The velocity potential, denoted by ϕ\phi, is a scalar function defined such that the velocity field is the gradient of ϕ\phi:

\mathbf{V} = \nabla \phi

In two-dimensional Cartesian coordinates:

u = \frac{\partial \phi}{\partial x}, \quad v = \frac{\partial \phi}{\partial y}

2.2 Irrotationality Condition

For the existence of a velocity potential, the flow must be irrotational:

\boldsymbol{\omega} = \nabla \times \mathbf{V} = 0

In two dimensions:

\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0

which ensures that the mixed second derivatives of ϕ\phi are equal (a necessary condition for ϕ\phi to be well-defined).

2.3 Laplace’s Equation

If the flow is incompressible and irrotational, ϕ\phi satisfies Laplace’s equation:

\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0

Laplace’s equation is linear, enabling solutions to be superposed to build complex flow fields from simpler ones.

2.4 Physical Meaning

  • Contours of constant ϕ\phi are equipotential lines—lines along which the fluid velocity has no component.
  • Fluid flows from regions of high ϕ\phi to low ϕ\phi.

2.5 Applications

  • Solving potential flow problems analytically.
  • Constructing flow fields around bodies without accounting for viscosity.
  • Superposing elementary solutions (sources, sinks, vortices, uniform flow) to model more complex flows.

3️⃣ Relationship Between ψ and φ

3.1 Orthogonality

In two-dimensional, incompressible, irrotational flow, the stream function and velocity potential are harmonic conjugates. Their contours are everywhere orthogonal:

\frac{\partial \phi}{\partial x} = u = \frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y} = v = -\frac{\partial \psi}{\partial x}

This satisfies the Cauchy–Riemann equations, which confirm the orthogonality of ϕ\phi and ψ\psi lines.

3.2 Complex Potential

For two-dimensional potential flow, a complex potential can be defined:

F(z) = \phi + i \psi

where z=x+iyz = x + i y. This formulation enables the use of complex analysis to solve flow problems efficiently via conformal mapping techniques.

4️⃣ Importance in Aerodynamics

  • Stream function provides a straightforward way to visualize flow patterns and ensure continuity is satisfied automatically.
  • Velocity potential enables solving for irrotational flow fields analytically using Laplace’s equation.
  • Together, they allow aerodynamicists to model idealized, inviscid, incompressible flows around airfoils and bodies, forming the foundation for classic theories of lift and circulation.

These tools are essential in aerodynamics for understanding and predicting flow behavior in regimes where viscosity can be neglected outside thin boundary layers.

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