7.3 Potential Flow Theory

Potential Flow Fundamentals

Potential flow theory simplifies fluid mechanics by assuming the flow is inviscid, incompressible, and irrotational. Under these conditions, the velocity field can be described entirely by two scalar functions: the velocity potential and the stream function. Both satisfy the Laplace equation, which means you can leverage a huge library of known solutions and combine them through superposition to model surprisingly complex flow patterns.

While potential flow can't capture viscous effects like boundary layers or turbulence, it remains a core tool in aerodynamics and hydrodynamics for getting useful first approximations.

Velocity potential and stream function

The velocity potential $\phi$ is a scalar function whose gradient gives the velocity field. It exists whenever the flow is irrotational ($\nabla \times \vec{V} = 0$). The velocity components in 2D Cartesian coordinates are:

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

The stream function $\psi$ is a scalar function whose contour lines (constant $\psi$) are streamlines. It exists whenever the flow is incompressible ($\nabla \cdot \vec{V} = 0$). The velocity components are:

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

Both $\phi$ and $\psi$ satisfy the Laplace equation:

$\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0, \qquad \nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = 0$

This is the key mathematical fact that makes potential flow so tractable: the Laplace equation is linear, well-studied, and has many known solutions.

Elementary flow building blocks:

• Uniform flow (freestream velocity $U_\infty$ in the $x$-direction): $\phi = U_\infty x, \qquad \psi = U_\infty y$

• Source/sink (strength $m$; positive $m$ is a source, negative is a sink): $\phi = \frac{m}{2\pi} \ln(r), \qquad \psi = \frac{m}{2\pi} \theta$ where $(r, \theta)$ are polar coordinates centered on the source/sink. The volume flow rate emanating from a source equals $m$ per unit depth.

• Doublet (strength $\kappa$, oriented along the $x$-axis): $\phi = -\frac{\kappa}{2\pi} \frac{\cos\theta}{r}, \qquad \psi = -\frac{\kappa}{2\pi} \frac{\sin\theta}{r}$ A doublet is the limiting case of a source-sink pair brought infinitely close together while their product of strength and separation stays constant.

• Vortex (circulation $\Gamma$): $\phi = \frac{\Gamma}{2\pi} \theta, \qquad \psi = -\frac{\Gamma}{2\pi} \ln(r)$ This element adds circulation to the flow, which is essential for generating lift on an airfoil.

Flow field analysis techniques

Streamlines are curves everywhere tangent to the local velocity vector. In potential flow, they correspond to lines of constant $\psi$. Equipotential lines are curves of constant $\phi$. A useful geometric property: streamlines and equipotential lines are always mutually perpendicular (they form an orthogonal grid).

The velocity magnitude at any point is:

$|\vec{V}| = \sqrt{u^2 + v^2} = \sqrt{\left(\frac{\partial \phi}{\partial x}\right)^2 + \left(\frac{\partial \phi}{\partial y}\right)^2}$

Stagnation points are locations where $u = 0$ and $v = 0$ simultaneously. These are physically important because pressure reaches a local maximum there (by Bernoulli's equation). Examples include the front of a cylinder in cross-flow or the leading edge of an airfoil.

To find stagnation points in a combined flow:

1. Write out the total $u$ and $v$ components from the superposed solution.
2. Set both equal to zero.
3. Solve the resulting system for the coordinates $(x, y)$ or $(r, \theta)$.

Once you know the velocity field, you can find the pressure distribution using the Bernoulli equation for irrotational flow:

$p + \frac{1}{2}\rho V^2 = p_\infty + \frac{1}{2}\rho U_\infty^2$

This is how potential flow connects velocity results to forces on a body.

Superposition in potential flow

Because the Laplace equation is linear, any sum of solutions is also a solution. This is the principle of superposition, and it's the most powerful feature of potential flow theory.

$\phi_{total} = \phi_1 + \phi_2 + \cdots, \qquad \psi_{total} = \psi_1 + \psi_2 + \cdots$

By combining the elementary building blocks, you can model flow around various body shapes:

• Rankine half-body: Uniform flow + single source. The dividing streamline ($\psi = \frac{m}{2}$) forms a semi-infinite body shape. This is a good model for flow over a blunt nose.

• Rankine oval: Uniform flow + source + sink (equal and opposite, separated by a distance). The dividing streamline forms a closed oval whose shape depends on the ratio of source strength to freestream velocity and the source-sink spacing.

• Flow around a circular cylinder (no lift): Uniform flow + doublet. Setting the doublet strength to $\kappa = U_\infty a^2 \cdot 2\pi$ (where $a$ is the cylinder radius) produces a streamline $\psi = 0$ that coincides with a circle of radius $a$. The resulting stream function is: $\psi = U_\infty r \sin\theta \left(1 - \frac{a^2}{r^2}\right)$

• Lifting cylinder: Uniform flow + doublet + vortex. Adding a free vortex of circulation $\Gamma$ breaks the top-bottom symmetry and produces a net lift force per unit span given by the Kutta-Joukowski theorem: $L' = \rho U_\infty \Gamma$

Steps to build a superposed solution:

1. Identify the body shape or flow pattern you want to model.
2. Select the elementary flows that, when combined, could produce the right streamline pattern.
3. Sum the individual $\phi$ or $\psi$ expressions.
4. Locate stagnation points by setting $u = v = 0$.
5. Identify the dividing streamline (the $\psi$ value that passes through the stagnation points) to find the body contour.
6. Apply Bernoulli's equation to get the pressure distribution and integrate to find forces.

Limitations of potential flow theory

Potential flow assumes the fluid is inviscid, incompressible, and irrotational. Each of these assumptions breaks down in specific situations:

• Inviscid assumption: Real fluids have viscosity. Potential flow cannot predict boundary layers, flow separation, or wake regions. This leads to d'Alembert's paradox: potential flow predicts zero drag on a body in steady flow, which contradicts physical observation.
• Incompressibility assumption: Valid only when the Mach number is below roughly 0.3. Above that, compressibility effects become significant and you need compressible flow theory.
• Irrotationality assumption: Vorticity is generated at solid boundaries due to the no-slip condition. Potential flow ignores this, so it cannot model flows where vorticity has diffused into the bulk of the fluid (e.g., turbulent pipe flow, separated wakes behind bluff bodies).

Despite these limitations, potential flow remains valuable because:

• It gives accurate pressure distributions over the forward portions of streamlined bodies (before separation occurs).
• It provides the outer "inviscid" solution that pairs with boundary layer theory in matched asymptotic approaches.
• Airfoil design, ship hull hydrodynamics, and preliminary aerodynamic analysis all rely heavily on potential flow methods.
• It serves as the foundation for panel methods and vortex lattice methods used in computational aerodynamics.