2.5 Potential flow

Basics of potential flow

Potential flow is a simplified model of fluid motion built on three assumptions: the flow is irrotational, inviscid, and incompressible. Because viscosity and compressibility are excluded, the governing equations become linear, which means you can find closed-form analytical solutions for many geometries. That's what makes potential flow so useful: it gives you real physical insight into pressure distributions, streamline shapes, and lift without needing a full numerical simulation.

The two central tools are the velocity potential and the stream function. Both are scalar fields that encode the entire velocity field, and both satisfy Laplace's equation. The specific flow you get depends on the boundary conditions you impose.

Velocity potential and stream function

Relationship between velocity potential and velocity field

The velocity potential $\phi$ is a scalar function whose gradient gives the velocity field:

$\mathbf{V} = \nabla \phi$

A velocity potential can only exist when the flow is irrotational, meaning:

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

This is easy to verify: the curl of any gradient is identically zero. The practical benefit is that instead of tracking two or three velocity components, you solve for a single scalar $\phi$ and then differentiate to recover the full velocity field.

Relationship between stream function and velocity field

The stream function $\psi$ is defined for two-dimensional incompressible flows. It relates to the velocity components as:

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

Any $\psi$ defined this way automatically satisfies the 2D continuity equation $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$, so mass conservation is built in by construction.

Contours of constant $\psi$ are streamlines (lines everywhere tangent to the velocity vector). The difference in $\psi$ between two streamlines equals the volume flow rate per unit depth between them, which makes $\psi$ especially handy for flow visualization and flux calculations.

Laplace's equation in potential flow

Derivation of Laplace's equation

Both $\phi$ and $\psi$ satisfy Laplace's equation, but for different reasons:

• For $\phi$: Start with the incompressibility (continuity) condition $\nabla \cdot \mathbf{V} = 0$. Substitute $\mathbf{V} = \nabla \phi$ to get:

$\nabla^2 \phi = 0$

• For $\psi$: Start with the irrotationality condition $\nabla \times \mathbf{V} = 0$. In 2D this reduces to $\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0$. Substituting the stream function definitions yields:

$\nabla^2 \psi = 0$

So continuity drives the equation for $\phi$, and irrotationality drives the equation for $\psi$. Both give you Laplace's equation, but through complementary physical requirements.

Boundary conditions for Laplace's equation

Laplace's equation alone doesn't pick out a unique flow. You need boundary conditions:

• Solid boundaries (no penetration): The velocity component normal to the surface must vanish. In terms of $\phi$, this is $\frac{\partial \phi}{\partial n} = 0$ on the surface. In terms of $\psi$, the surface is a streamline, so $\psi = \text{constant}$ there.
• Far-field (free-stream) conditions: Far from any body, the flow should approach the undisturbed free-stream values. For example, $\phi \to U x$ and $\psi \to U y$ as $r \to \infty$ for a uniform stream of speed $U$.

The geometry of the problem determines which boundary conditions apply and, ultimately, which solution you get.

Elementary flows in potential flow theory

Because Laplace's equation is linear, you can build complex flows by adding together simple "building block" solutions. The four elementary flows below are the most important.

Uniform flow

The simplest potential flow has a constant velocity $\mathbf{V} = (U, 0)$ everywhere:

$\phi = Ux, \quad \psi = Uy$

Streamlines are horizontal straight lines. Uniform flow serves as the background onto which you superpose other elements.

Source and sink flow

A source of strength $Q$ (volume flow rate) emits fluid radially outward from a point; a sink absorbs fluid inward. In 2D:

$\phi = \frac{Q}{2\pi} \ln r, \quad \psi = \frac{Q}{2\pi} \theta$

Here $r$ is the radial distance from the source/sink and $\theta$ is the polar angle. Use $+Q$ for a source and $-Q$ for a sink. The radial velocity decays as $1/r$, so the flow is strongest near the origin.

In 3D the expressions change to $\phi = -\frac{Q}{4\pi r}$ for a source, reflecting the different geometric spreading.

Doublet flow

A doublet (or dipole) is the limiting case of a source and sink brought infinitely close together while their product of strength and separation stays finite. The resulting doublet strength is $\kappa$. In 2D:

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

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

Vortex flow

A free vortex (also called a point vortex) produces purely tangential flow with velocity proportional to $1/r$:

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

Here $\Gamma$ is the circulation, defined as the line integral of velocity around any closed curve enclosing the vortex center:

$\Gamma = \oint_C \mathbf{V} \cdot d\mathbf{l}$

The flow is irrotational everywhere except at the singular point $r = 0$. Vortex elements are essential for modeling lift on airfoils and rotating flows.

Superposition of elementary flows

Linear combination of elementary flows

Because Laplace's equation is linear, any sum of solutions is also a solution. If you have $n$ elementary flows, the combined velocity potential and stream function are:

$\phi = \phi_1 + \phi_2 + \cdots + \phi_n$ $\psi = \psi_1 + \psi_2 + \cdots + \psi_n$

This superposition principle is the single most powerful feature of potential flow theory. It lets you assemble realistic flow fields from simple pieces.

Constructing complex flow patterns

Some classic combinations:

• Rankine half-body: Uniform flow + a single source. The dividing streamline forms a semi-infinite body shape.
• Rankine oval: Uniform flow + a source + a sink (equal and opposite, separated by a distance). The dividing streamline forms a closed oval.
• Non-lifting flow past a cylinder: Uniform flow + a doublet. Produces symmetric streamlines around a circle.
• Lifting flow past a cylinder: Uniform flow + a doublet + a vortex. Breaks the symmetry and generates lift.

Each combination satisfies Laplace's equation automatically. You just need to verify that the boundary conditions (no penetration on the body, correct far-field behavior) are met.

Flow past a circular cylinder

Non-lifting cylinder (uniform flow + doublet)

For a uniform stream of speed $U$ flowing past a cylinder of radius $a$, superpose a uniform flow and a doublet of strength $\kappa = 2\pi U a^2$ (or equivalently, set the doublet to cancel the normal velocity at $r = a$). In polar coordinates centered on the cylinder:

$\phi = U \cos\theta \left(r + \frac{a^2}{r}\right)$ $\psi = U \sin\theta \left(r - \frac{a^2}{r}\right)$

You can verify that $\psi = 0$ at $r = a$ for all $\theta$, confirming the cylinder surface is a streamline.

The flow is perfectly symmetric fore-and-aft, so there is no net drag or lift. This symmetric, drag-free result is known as d'Alembert's paradox and highlights a key limitation of inviscid theory.

Adding circulation for lift

Superposing a vortex of circulation $\Gamma$ shifts the stagnation points and breaks the up-down symmetry. The stream function becomes:

$\psi = U \sin\theta \left(r - \frac{a^2}{r}\right) - \frac{\Gamma}{2\pi} \ln r$

Higher velocity on one side of the cylinder means lower pressure (by Bernoulli's equation), producing a net lift force perpendicular to the free stream.

Flow past a sphere

The 3D analog uses a three-dimensional doublet superposed on a uniform flow. The doublet strength is chosen as $\mu = 2\pi U a^3 / 2$ (again, to enforce no penetration at $r = a$). The total velocity potential in spherical coordinates is:

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

The Stokes stream function for this axisymmetric flow is:

$\psi = \frac{1}{2} U \sin^2\theta \left(r^2 - \frac{a^3}{r}\right)$

As with the cylinder, the flow is symmetric and predicts zero drag. The streamline pattern looks qualitatively similar to the 2D cylinder case, with flow dividing at the front stagnation point and reuniting at the rear.

Lift on a cylinder in potential flow

Kutta-Joukowski theorem

The Kutta-Joukowski theorem states that the lift per unit span on any 2D body in a steady potential flow is:

$L' = \rho U \Gamma$

where $\rho$ is the fluid density, $U$ is the free-stream speed, and $\Gamma$ is the circulation around the body. This result is remarkably general: it applies to cylinders, airfoils, and any 2D shape in potential flow.

Circulation and lift generation

A few key points about how circulation produces lift:

• Without circulation ($\Gamma = 0$), the flow around a cylinder is symmetric and the lift is zero.
• Adding a vortex introduces circulation, which speeds up the flow on one side and slows it on the other.
• By Bernoulli's equation, the faster side has lower pressure. The resulting pressure imbalance is the lift force.
• For airfoils, the Kutta condition determines the correct value of $\Gamma$: the flow must leave the sharp trailing edge smoothly, with finite velocity. This physical requirement selects a unique circulation and therefore a unique lift.

Potential flow vs real fluid flow

Assumptions and limitations

Potential flow rests on three assumptions that real flows violate to varying degrees:

• Inviscid: No viscosity, so no shear stress and no energy dissipation.
• Irrotational: No vorticity anywhere in the flow.
• Incompressible: Constant density (valid for low Mach numbers, roughly $M < 0.3$).

Because of these assumptions, potential flow cannot predict:

• Boundary layer development along surfaces
• Flow separation and wake formation behind bluff bodies
• Drag forces (this is d'Alembert's paradox)
• Turbulence or any rotational flow structures

Where potential flow still works well

Despite its limitations, potential flow gives accurate results in regions away from solid surfaces where viscous effects are confined to thin boundary layers. It predicts pressure distributions and lift on streamlined bodies (like airfoils at small angles of attack) quite well. The outer flow around an aircraft wing, for example, is closely approximated by potential flow.

For situations where viscous effects dominate (separated flows, bluff bodies, high angles of attack), you need the full Navier-Stokes equations or computational fluid dynamics (CFD).

Numerical methods for potential flow

Finite difference method

The finite difference method solves Laplace's equation by discretizing the domain on a grid and replacing derivatives with algebraic approximations. For example, the 2D Laplacian on a uniform grid with spacing $h$ becomes:

$\nabla^2 \phi \approx \frac{\phi_{i+1,j} + \phi_{i-1,j} + \phi_{i,j+1} + \phi_{i,j-1} - 4\phi_{i,j}}{h^2} = 0$

This converts the PDE into a system of linear equations that you solve iteratively (e.g., Gauss-Seidel) or directly. The method is straightforward to implement but can require very fine grids near boundaries to capture steep gradients accurately.

Boundary element method

The boundary element method (BEM) takes a fundamentally different approach: instead of discretizing the entire flow domain, it only discretizes the boundaries (body surfaces and, if needed, far-field boundaries).

BEM exploits Green's theorem to express $\phi$ at any interior point as an integral over boundary values of $\phi$ and $\frac{\partial \phi}{\partial n}$. This reduces a 3D problem to a 2D surface problem (or a 2D problem to a 1D curve problem), which is a significant computational saving.

BEM is particularly well-suited for:

• External flows with unbounded domains (the far-field condition is handled naturally)
• Complex body geometries where volume meshing would be difficult
• Problems where you only need surface quantities (pressure, velocity) rather than the full interior field

Applications of potential flow theory

Aerodynamics and hydrodynamics

• Airfoil design: Panel methods (a form of BEM) based on potential flow are still used in preliminary airfoil design to compute lift coefficients and pressure distributions quickly.
• Submarine and ship hull design: Potential flow models predict the pressure field and wave-making resistance around hull shapes, guiding early-stage design before expensive CFD or tank testing.
• Wind turbine blades: Blade element momentum theory, combined with potential flow models for the induced velocity field, is a standard tool for rotor design.

Groundwater flow and seepage

Groundwater moving slowly through porous media obeys Darcy's law, which has the same mathematical form as potential flow (the hydraulic head plays the role of $\phi$). This means all the tools of potential flow theory carry over directly:

• Seepage under dams: Engineers use flow nets (graphical solutions of Laplace's equation) to estimate seepage rates and pore pressures, which are critical for dam stability.
• Well hydraulics: The drawdown around a pumping well is modeled as a sink in potential flow, giving the classic Thiem and Theis solutions.
• Contaminant transport: Potential flow solutions provide the velocity field needed to track how pollutants move through an aquifer, informing remediation strategies like pump-and-treat systems.