1.5 Potential flow theory

Potential flow theory simplifies fluid dynamics by assuming the flow is irrotational, incompressible, and inviscid. These three assumptions let you describe the entire velocity field using scalar functions (the velocity potential and stream function), which makes complex aerodynamic problems far more tractable than solving the full Navier-Stokes equations.

By combining simple building-block flows (uniform flow, sources, sinks, doublets, and vortices), you can model flow around cylinders, spheres, and airfoils. The theory has real limitations since it ignores viscosity and compressibility, but it remains a core tool for understanding lift generation and pressure distributions in low-speed aerodynamics.

Potential flow assumptions

Potential flow theory rests on three simplifying assumptions that together make the governing equations linear and solvable. Each assumption has a specific mathematical consequence and a specific regime where it holds well.

Irrotational flow

Fluid particles translate through the flow field without spinning. Mathematically, this means the curl of the velocity vector is zero:

$\nabla \times \vec{V} = 0$

This condition guarantees that a scalar velocity potential exists. Irrotational flow is a reasonable approximation in regions away from boundary layers and wakes, where viscous effects are small.

Incompressible flow

The fluid density stays constant everywhere in the flow field. Mathematically, the divergence of the velocity vector is zero:

$\nabla \cdot \vec{V} = 0$

This means the volume of fluid entering any control volume equals the volume leaving it. The incompressibility assumption holds well for low-speed aerodynamics, typically when the Mach number is below about 0.3.

Inviscid flow

The fluid has zero viscosity, so no shear stresses act between fluid layers or between the fluid and solid surfaces. Dropping the viscous terms from the Navier-Stokes equations gives you the Euler equations.

This is a reasonable approximation for high-Reynolds-number flows, where inertial forces dominate and boundary layers are thin relative to the body size.

Velocity potential

Because the flow is irrotational, the velocity field can be written as the gradient of a single scalar function $\phi(x, y, z)$, called the velocity potential:

$\vec{V} = \nabla \phi = \left(\frac{\partial \phi}{\partial x},\; \frac{\partial \phi}{\partial y},\; \frac{\partial \phi}{\partial z}\right)$

The velocity potential has units of $\text{m}^2/\text{s}$. Once you know $\phi$, you can recover the entire velocity field by differentiation.

Laplace's equation

Substituting $\vec{V} = \nabla \phi$ into the incompressible continuity equation $\nabla \cdot \vec{V} = 0$ gives Laplace's equation:

$\nabla^2 \phi = 0$

This is a linear partial differential equation, and that linearity is what makes superposition possible. Solving Laplace's equation with the right boundary conditions gives you the velocity potential for any particular flow problem.

Boundary conditions

Two types of boundary conditions are commonly applied:

• Neumann condition: specifies the normal derivative of $\phi$ at a surface. For a solid wall with no flow through it: $\frac{\partial \phi}{\partial n} = 0$
• Dirichlet condition: specifies the value of $\phi$ itself. For example, matching the freestream far from the body: $\phi = U_\infty x$

Together, these conditions ensure the solution is unique and physically meaningful.

Stream function

In two-dimensional flows, the stream function $\psi(x, y)$ provides an alternative way to describe the velocity field. It also has units of $\text{m}^2/\text{s}$.

Definition of stream function

The stream function is defined so that the continuity equation is automatically satisfied:

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

You can verify this yourself: substituting these into $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$ gives zero identically, regardless of what $\psi$ looks like.

Relationship to velocity potential

In 2D potential flow, $\phi$ and $\psi$ are connected through the Cauchy-Riemann equations:

$\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \qquad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$

These relations mean $\phi$ and $\psi$ are harmonic conjugates: both satisfy Laplace's equation, and knowing one lets you find the other. Lines of constant $\phi$ (equipotential lines) are always perpendicular to lines of constant $\psi$ (streamlines).

Streamlines

Streamlines are curves that are everywhere tangent to the local velocity vector. In 2D potential flow, they correspond to contours where $\psi = \text{constant}$.

Two useful properties of streamlines:

• Fluid never crosses a streamline (the velocity is always tangent to it), so any streamline can be replaced by a solid wall without changing the flow.
• Streamlines that are closer together indicate higher velocity; wider spacing means slower flow.

Elementary flows

Elementary flows are the simplest solutions to Laplace's equation. Each one is fully described by its velocity potential and stream function, and they serve as building blocks for more complex flow fields.

Uniform flow

A uniform flow has constant velocity $U_\infty$ everywhere, directed along the $x$-axis:

• $\phi = U_\infty x$
• $\psi = U_\infty y$

This represents the undisturbed freestream far from any body.

Source and sink flow

A source emits fluid equally in all radial directions from a point; a sink draws fluid inward. For a 2D source/sink of strength $m$ at the origin ($m > 0$ for a source, $m < 0$ for a sink):

• $\phi = \frac{m}{2\pi} \ln r$
• $\psi = \frac{m}{2\pi} \theta$

Here $r$ is the radial distance and $\theta$ is the angle from the positive $x$-axis. The volume flow rate emanating from a 2D source equals $m$ per unit depth.

Doublet flow

A doublet is the limiting case of a source and sink of equal strength brought infinitesimally close together while keeping the product of strength and separation constant. For a 2D doublet of strength $\kappa$ at the origin, oriented along the $x$-axis:

• $\phi = -\frac{\kappa \cos \theta}{2\pi r}$
• $\psi = -\frac{\kappa \sin \theta}{2\pi r}$

Doublets are the key ingredient for modeling flow around cylinders and spheres.

Vortex flow

A point vortex produces circular streamlines with velocity that decays as $1/r$. For a vortex of circulation $\Gamma$ at the origin (positive $\Gamma$ = counterclockwise):

• $\phi = \frac{\Gamma}{2\pi} \theta$
• $\psi = -\frac{\Gamma}{2\pi} \ln r$

Vortex flows are used to model circulation around lifting bodies like airfoils.

Superposition principle

Because Laplace's equation is linear, you can add solutions together and the result is still a valid solution. This is the superposition principle, and it's what makes potential flow theory so powerful.

Linear combination of flows

To build a complex flow, simply add the velocity potentials (or stream functions) of individual elementary flows. For example, combining uniform flow with a doublet:

$\phi = U_\infty x - \frac{\kappa \cos \theta}{2\pi r}, \qquad \psi = U_\infty y - \frac{\kappa \sin \theta}{2\pi r}$

Constructing complex flows

The strategy for modeling flow around a body is:

1. Choose a combination of elementary flows (uniform flow, doublets, vortices, sources/sinks).
2. Write the total $\phi$ or $\psi$ as the sum of the individual contributions.
3. Enforce boundary conditions (typically no flow through the body surface) to determine the unknown strengths and positions.
4. Solve for the velocity and pressure fields from the resulting potential.

For example, uniform flow + doublet gives flow around a cylinder. Adding a vortex to that combination gives a lifting cylinder.

Flow around simple geometries

Flow around a cylinder

Superimposing a uniform flow and a doublet (with $\kappa = 2\pi U_\infty a^2$) produces flow around a circular cylinder of radius $a$:

$\phi = U_\infty \left(r + \frac{a^2}{r}\right) \cos \theta$

$\psi = U_\infty \left(r - \frac{a^2}{r}\right) \sin \theta$

At $r = a$, the radial velocity is zero (no flow through the surface), confirming the boundary condition is satisfied. The streamline pattern is symmetric fore-and-aft, which means the potential flow solution predicts zero drag on the cylinder. This is d'Alembert's paradox: inviscid theory predicts no drag on a body in steady flow.

Flow around a sphere

The 3D analog uses a three-dimensional doublet superimposed on uniform flow:

$\phi = U_\infty \left(r + \frac{a^3}{2r^2}\right) \cos \theta$

The streamline pattern is similar to the cylinder case but extends in three dimensions. Again, the potential flow solution predicts zero drag.

Flow around an airfoil

Modeling an airfoil requires combining:

• Uniform flow for the freestream
• A source-sink pair (or distribution) to create the airfoil's thickness
• A vortex to produce circulation and therefore lift

The vortex strength is not arbitrary. It's fixed by the Kutta condition, which requires the flow to leave the trailing edge smoothly (finite velocity at the sharp trailing edge). Without the Kutta condition, the solution would have an unphysical infinite velocity at the trailing edge.

The resulting flow field gives a reasonable prediction of the pressure distribution and lift, though it cannot capture viscous drag or flow separation.

Kutta-Joukowski theorem

The Kutta-Joukowski theorem connects circulation to lift. For a 2D body in a potential flow, the lift per unit span is:

$L' = \rho U_\infty \Gamma$

where $\rho$ is the fluid density, $U_\infty$ is the freestream velocity, and $\Gamma$ is the circulation around the body.

Circulation and lift

Circulation is defined as the line integral of velocity around a closed curve enclosing the body:

$\Gamma = \oint \vec{V} \cdot d\vec{l}$

For a non-lifting body (like a symmetric cylinder with no vortex), $\Gamma = 0$ and there's no lift. Adding a vortex introduces circulation, and the Kutta-Joukowski theorem tells you exactly how much lift that circulation produces. The relationship is linear: double the circulation, double the lift.

Conformal mapping

Conformal mapping is a mathematical technique that transforms one 2D shape into another while preserving local angles. In potential flow, you can:

1. Solve the (easy) problem of flow around a cylinder with circulation.
2. Apply a conformal mapping to transform the cylinder into an airfoil shape.
3. The flow solution transforms along with the geometry, giving you the flow around the airfoil.

This approach works because Laplace's equation is invariant under conformal transformations.

Joukowski airfoils

The Joukowski transformation is a specific conformal mapping that converts a circle into an airfoil-like shape:

$z = \zeta + \frac{c^2}{\zeta}$

where $\zeta$ is the circle plane and $z$ is the airfoil plane. By shifting the center of the circle slightly off the origin, you get airfoils with:

• A rounded leading edge
• A sharp, cusped trailing edge (which naturally enforces the Kutta condition)
• Controllable camber and thickness depending on the offset

Joukowski airfoils are not used in real aircraft design (they have a cusp at the trailing edge rather than a finite-angle wedge), but they're extremely useful for building intuition about how circulation, camber, and lift are related.

Limitations of potential flow theory

Neglecting viscous effects

Real fluids have viscosity, which creates boundary layers along surfaces and wakes behind bodies. Potential flow theory ignores all of this, so it cannot predict:

• Skin friction drag
• Boundary layer growth
• The viscous contribution to total drag

Inability to predict flow separation

Flow separation happens when the boundary layer detaches from a surface, typically under an adverse pressure gradient. Since potential flow has no boundary layer, it cannot predict where or whether separation occurs. This leads to:

• Overestimation of lift at high angles of attack (where real airfoils stall)
• Underestimation of drag, especially for bluff bodies where large separated regions dominate the drag

Compressibility effects

The incompressibility assumption breaks down as the Mach number increases above about 0.3. At transonic and supersonic speeds, compressibility produces shock waves, wave drag, and density changes that potential flow theory cannot capture. Extensions like the Prandtl-Glauert correction can partially account for compressibility at subsonic speeds, but the basic theory is fundamentally limited to low-speed flows.

Numerical methods for potential flow

Analytical solutions exist only for simple geometries. For realistic shapes (multi-element airfoils, full aircraft configurations), numerical methods are needed.

Panel methods

Panel methods discretize the surface of a body into flat or curved panels, each carrying a distribution of singularities (sources, doublets, or vortices). The solution procedure is:

1. Divide the body surface into $N$ panels.
2. Place a singularity distribution (e.g., source + vortex) on each panel.
3. Write the no-penetration boundary condition at the control point (typically the midpoint) of each panel.
4. This produces a system of $N$ linear equations for the $N$ unknown singularity strengths.
5. Solve the linear system and compute velocities and pressures from the resulting singularity strengths.

Panel methods are widely used in preliminary aerodynamic design. Increasing the number of panels or using higher-order singularity distributions (e.g., linearly varying rather than constant strength) improves accuracy.

Boundary element methods

Boundary element methods (BEM) are closely related to panel methods but use more rigorous mathematical formulations based on Green's theorems. The key advantage is that only the boundary (surface) needs to be discretized, not the entire flow volume. This makes BEM computationally efficient for external flow problems, though the resulting matrices are dense rather than sparse.