4.2 Velocity potential

Definition of velocity potential

A velocity potential is a scalar function $\phi$ (units of $m^2/s$) that encodes the entire velocity field of an irrotational flow in a single quantity. Instead of tracking three separate velocity components, you only need to find $\phi$ and then differentiate it to recover the full velocity vector.

Irrotational flow and velocity potential

A flow is irrotational when fluid elements translate without spinning about their own axes. Mathematically, this means the vorticity (curl of velocity) vanishes everywhere:

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

A standard vector calculus identity guarantees that any curl-free vector field can be written as the gradient of some scalar. That scalar is the velocity potential:

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

So the existence of $\phi$ is not an assumption you impose; it's a direct consequence of irrotationality.

Laplace's equation for velocity potential

When the flow is both incompressible and irrotational, you can combine the two conditions:

1. Incompressibility requires $\nabla \cdot \vec{V} = 0$.
2. Irrotationality gives $\vec{V} = \nabla \phi$.

Substituting the second into the first yields Laplace's equation:

$\nabla^2 \phi = 0$

This is a linear, second-order PDE. Solving it with the right boundary conditions gives you $\phi$, and from $\phi$ you get the velocity field everywhere in the domain.

Properties of velocity potential

Relationship between velocity potential and velocity field

Taking the gradient of $\phi$ in Cartesian coordinates gives each velocity component directly:

• $u = \frac{\partial \phi}{\partial x}$
• $v = \frac{\partial \phi}{\partial y}$
• $w = \frac{\partial \phi}{\partial z}$

Once you have $\phi$, the velocity field follows immediately from differentiation.

Uniqueness of velocity potential

For a given set of boundary conditions, the velocity potential is unique up to an additive constant. Two solutions $\phi_A$ and $\phi_B$ satisfying the same boundary conditions can differ only by a constant, and since velocity depends on the derivatives of $\phi$, that constant has no physical effect. This guarantees a well-defined velocity field.

Superposition principle

Because Laplace's equation is linear, you can add solutions together and get another valid solution. If $\phi_1$ and $\phi_2$ each satisfy $\nabla^2 \phi = 0$, then:

$\phi = \phi_1 + \phi_2$

also satisfies Laplace's equation. This is extremely useful: you can build up complex flow patterns by superposing simple elementary solutions (uniform flow + source + doublet, etc.).

Boundary conditions for velocity potential

Laplace's equation alone has infinitely many solutions. Boundary conditions pin down the one that matches your physical problem.

Solid boundary conditions

At a solid wall, fluid cannot pass through the surface. This no-penetration condition requires the velocity component normal to the wall to vanish:

$\frac{\partial \phi}{\partial n} = 0$

where $n$ is the outward normal direction. If the solid boundary is moving, the normal fluid velocity must instead match the normal velocity of the boundary.

Free surface boundary conditions

At a free surface (e.g., a water-air interface), two conditions apply simultaneously:

1. Kinematic condition: fluid particles on the surface stay on the surface, expressed as $\frac{\partial \phi}{\partial n} = \frac{\partial \eta}{\partial t}$, where $\eta$ is the free surface elevation.
2. Dynamic condition: the pressure at the free surface equals atmospheric pressure (typically enforced through Bernoulli's equation applied at the surface).

Far-field boundary conditions

Far from the body or region of interest, the flow should return to its undisturbed state. This is written as:

$\lim_{r \to \infty} \phi = \phi_\infty$

where $\phi_\infty$ is the potential for the freestream (usually uniform flow) and $r$ is the distance from the origin. Without this condition, solutions could include spurious disturbances that extend to infinity.

Applications of velocity potential

Uniform flow

The simplest building block. For a uniform stream of speed $U$ in the x-direction:

$\phi = Ux$

The velocity field is constant: $\vec{V} = (U, 0, 0)$. Uniform flow serves as the far-field condition in nearly every external flow problem.

Source, sink, and doublet flow

A source (strength $m > 0$) emits fluid radially outward from a point; a sink ($m < 0$) absorbs fluid radially inward. In three dimensions the potential is:

$\phi = \frac{m}{4\pi r}$

where $r$ is the distance from the source/sink. The sign of $m$ distinguishes source from sink.

A doublet is the limiting case of a source and sink of equal magnitude brought infinitesimally close together while their product of strength and separation stays finite. For a doublet of strength $\mu$ oriented along the x-axis:

$\phi = \frac{\mu \cos\theta}{4\pi r^2}$

Here $\theta$ is measured from the x-axis. Doublets are the key ingredient for modeling closed bodies in a freestream.

Flow around a cylinder

Superposing a uniform flow and a two-dimensional doublet produces the classic potential flow around a circular cylinder of radius $a$:

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

Taking the gradient in polar coordinates gives the radial and tangential velocity components. You can verify that $v_r = 0$ at $r = a$, confirming the no-penetration condition on the cylinder surface.

Flow around a sphere

The three-dimensional analog uses a 3-D doublet superposed with uniform flow. For a sphere of radius $a$:

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

The extra factor of $\frac{1}{2}$ and the $r^{-2}$ dependence (instead of $r^{-1}$) reflect the three-dimensional geometry. Again, differentiating gives the full velocity field around the sphere.

Complex potential theory

In two dimensions, complex analysis provides an elegant framework that packages both the velocity potential and the stream function into a single analytic function.

Complex velocity potential definition

The complex potential is defined as:

$w(z) = \phi(x,y) + i\,\psi(x,y)$

where $z = x + iy$ is the complex coordinate, $\phi$ is the velocity potential, and $\psi$ is the stream function. Because $\phi$ and $\psi$ both satisfy Laplace's equation and are related by the Cauchy-Riemann equations, $w(z)$ is an analytic (holomorphic) function of $z$.

Relationship between complex potential and velocity field

Differentiating $w$ with respect to $z$ gives the complex velocity:

$\frac{dw}{dz} = u - iv$

Note the minus sign on the $v$ component. From this single complex derivative you can read off both velocity components at any point in the flow.

Conformal mapping

Because $w(z)$ is analytic, you can apply conformal mappings to transform a flow in one geometry into a flow in another while preserving local angles between streamlines and equipotential lines. The practical payoff: solve the easy problem (e.g., flow around a cylinder), then map the solution onto a harder geometry.

The Joukowski transformation is the most well-known example. It maps the circle in the $z$-plane to an airfoil-shaped contour in the transformed plane, giving an analytical model for lift generation around airfoil profiles.

Numerical methods for velocity potential

When the geometry or boundary conditions are too complex for analytical solutions, numerical methods approximate $\phi$ on a discretized domain.

Finite difference method

1. Overlay a grid on the flow domain.
2. Replace the partial derivatives in $\nabla^2\phi = 0$ with finite difference approximations (e.g., central differences).
3. Solve the resulting system of linear algebraic equations for $\phi$ at each grid point.
4. Compute velocities from finite difference approximations of $\nabla\phi$.

Finite differences are straightforward to code but can require very fine grids near curved boundaries to maintain accuracy.

Boundary element method

The boundary element method (BEM) discretizes only the boundary of the domain, not the interior. It reformulates Laplace's equation as an integral equation over the boundary using fundamental solutions (sources, sinks, doublets).

This reduces the problem dimension by one (a 3-D problem becomes a 2-D surface problem), which is a major advantage for exterior flows over unbounded domains. The trade-off is that BEM produces dense (not sparse) linear systems and requires careful treatment of singular integrals.

Choosing a method

Other approaches (finite element methods, spectral methods) also apply to Laplace's equation and may be preferred depending on geometry and accuracy requirements.

Limitations of velocity potential

Rotational flows

Velocity potential exists only when $\nabla \times \vec{V} = 0$. Flows with significant vorticity, such as turbulent flows, boundary layers, or wakes, cannot be described by a velocity potential. For those situations, you need the stream function, vorticity transport equations, or the full Navier-Stokes equations.

Nonlinear effects

The linearity of Laplace's equation is both the strength and the limitation of potential flow theory. Real flows exhibit separation, vortex shedding, and turbulence, none of which appear in potential flow solutions. The most famous consequence is d'Alembert's paradox: potential flow predicts zero drag on a body in steady flow, contradicting everyday experience. Viscous and rotational effects must be included to capture drag.

Compressibility effects

The derivation assumes incompressible flow ($\nabla \cdot \vec{V} = 0$). At higher Mach numbers, density changes become significant and the governing equation for $\phi$ is no longer Laplace's equation but a more complex, often nonlinear PDE. Linearized compressible potential flow (e.g., the Prandtl-Glauert equation) extends the theory to small perturbations in subsonic flow, but fully compressible or transonic flows require different formulations entirely.