Potential Flow Theory

J. Gordon Leishman

36 Potential Flow Theory

Introduction^[1]

Potential flow methods employ the mathematical principle that the gradient of a scalar field yields a vector field, such as a velocity field. In its simplest incompressible form, a potential flow assumes irrotational and inviscid fluid motion with constant density. More generally, potential-flow methods require the velocity field to be expressible as the gradient of a scalar potential. In contrast, compressible potential flow retains density variations through the continuity equation and an appropriate thermodynamic relation, i.e., an “ideal” flow. These assumptions could be considered significant, if not sweeping, approximations to an actual or “real” flow, in general. However, many interesting and practical flows can be regarded as potential flows for analysis and prediction. More importantly, such assumptions are also used to develop various aspects of airfoil and wing theories, which will be considered in the following chapters, and studying the fundamental elements of potential flows is a necessary prerequisite.

The use of the term “irrotational” in the previous paragraph requires some elaboration. An irrotational flow is mathematically stated as $\nabla \times \vec{V} = 0$ , i.e., the curl of the velocity field is zero. The curl of the velocity field is the vorticity, which is twice the local angular velocity of a fluid element. Therefore, when the vorticity is zero, fluid elements have no local rigid-body rotation. However, irrotationality alone does not eliminate viscous shear stresses or preclude turbulence; these exclusions follow from the additional assumption of inviscid flow.

Additionally, within the inviscid-flow model, viscous shear stresses are identically zero. These assumptions significantly simplify flow analysis, and the inviscid assumption is particularly valid for external flows far from solid boundaries, i.e., outside boundary layers or wakes that may be generated. For example, the flow around an airfoil, as shown in Figure 1, is predominantly potential away from the airfoil’s immediate surface. For clarity, the extent of the viscous flow region is exaggerated for illustration purposes, and the boundary layer is very thin relative to the thickness of the airfoil.

The flow around an airfoil is predominantly potential away from the surface.

Consequently, studying potential flows provides an insightful and tractable framework for predicting and understanding a range of flow problems. For such problems, exact analytical solutions are often possible, yielding valuable insights into the fundamentals of fluid dynamics. Potential flows are foundational to many classical aerodynamic theories, including thin-airfoil theory, lifting-line theory, and lifting-surface theory, which are widely used to model and analyze flow around bodies, airfoils, and wings. Numerical methods, known as surface-singularity or panel methods, can further extend the applicability of potential flow theory, including to complete airplane shapes, making it a valuable approach in flight vehicle design and numerous other engineering applications.

Learning Objectives

Understand the principles of incompressible, irrotational, and inviscid fluid motion, i.e., potential flows.
Become familiar with elementary potential flow solutions, including uniform flow, source/sink flows, doublets, and vortices.
Learn how to combine elementary solutions to construct more complex potential flow fields through linear superposition.
Use complex variable methods to represent and analyze two-dimensional potential flows.
Understand the method of conformal mapping and how it is used to transform simple potential flows onto more useful geometries, such as airfoil shapes.
Know how to use the Prandtl-Glauert compressibility correction and its application in approximating subsonic compressible flow effects using incompressible potential flow solutions.

History

The origins of the potential flow theory can be traced back to the 18th century, with contributions from Daniel Bernoulli and Leonhard Euler. Bernoulli introduced an equation relating velocity and pressure in inviscid, incompressible fluids; the Bernoulli principle, also known as the Bernoulli equation, is a cornerstone of fluid mechanics. Euler expanded on this work by deriving the equations of motion for inviscid flows and introducing the velocity potential, a scalar function whose gradient gives the fluid velocity. Pierre-Simon Laplace further formalized these ideas and contributed to the development of superposition methods for combining elementary flow solutions, such as sources, sinks, and vortices, to model more complex and interesting flows.

In the 19th century, significant advancements in potential flow theory included the mathematical formalization of streamlines. George Stokes published the first and most complete exposition^[2] in 1842. William Rankine analyzed the flow patterns around simple geometries^[3] using a potential flow method, while Hermann von Helmholtz formulated the conservation of vorticity in inviscid flows, a result critical to understanding circulation and lift generation. Rankine was interested in the flows around ship hulls, for which a potential-flow solution provided substantial insight. Figure 2 illustrates his calculations for an oval “half-body,” which Rankine called an “Oögenous Neoïd.”

Rankine’s potential-flow solution about an oval “half-body” or Oögenous Neoïd circa 1864.

However, the limitations of potential flow were also exposed, most notably by Jean le Rond d’Alembert‘s so-called paradox, which showed that the assumption of a steady, inviscid, incompressible potential flow predicts zero drag on closed bodies. This paradox highlighted that additional flow characteristics, specifically the action of viscosity, were necessary to reconcile the discrepancies between predictions and experiments. Despite these limitations, potential flow quickly became a central tool for analyzing fluid motion.

The early 20th century saw the application of potential flow theory to airfoils and wings. Frederick Lanchester played a pivotal role in integrating the concept of circulation into the understanding of lift generation. Martin Kutta and Nikolay (Nikolai) Joukowsky independently developed theories that mathematically linked circulation to lift. These contributions introduced the concept now known as the Kutta condition, which models the smooth flow observed at an airfoil’s sharp trailing edge. Ludwig Prandtl built on this work by developing lifting-line theory for finite wings, while Munk and Glauert helped formulate and systematize thin-airfoil theory for two-dimensional airfoils. Together, these developments provided the first practical models for predicting the lift characteristics of airfoils and the lift and induced drag of finite wings. These theories enabled engineers to design and analyze airfoils and wings with reasonable predictive accuracy despite the assumptions of incompressible, inviscid flow.

Later in the 20th century, potential flow theory was extended to compressible flows, resulting in techniques such as the Prandtl-Glauert transformation for subsonic flows. Numerical methods, such as panel methods, were introduced in the 1970s to solve potential flow problems for more complex geometries, including aircraft fuselages and wings. While potential flow cannot predict viscous effects, flow separation, or turbulence, it remains foundational for understanding basic fluid behavior. It also serves as a starting point for learning about more advanced models, making it an indispensable part of fluid mechanics.

Definition of a Potential Flow

Potential flow methods use the mathematical principle that the gradient of a scalar field is a vector field. For example, if $\phi = \phi\left(x, y, z\right)$ is a scalar field, then the gradient ${\nabla \phi}$ (or grad $\phi$ ) is given by

(1) $\begin{equation*} \mbox{grad } \phi \equiv \nabla \phi = \frac{\partial \phi}{\partial x}\, \vec{i}+ \frac{\partial \phi}{\partial y}\, \vec{j}+\frac{\partial \phi}{\partial z}\, \vec{k} \end{equation*}$

The vector operator $\nabla$ (called nabla or del) is defined as

(2) $\begin{equation*} \nabla = \frac{\partial }{\partial x}\, \, \vec{i}+ \frac{\partial }{\partial y}\, \, \vec{j}+\frac{\partial }{\partial z}\, \, \vec{k} \end{equation*}$

Physically, ${\nabla \phi}$ is the normal (perpendicular) to the surface defined by $\phi\left(x, y, z\right)$ = constant, which is called an equipotential surface, as illustrated in Figure 3.

A gradient of a scalar field is a vector, i.e., a vector that points in the direction of the slope of the scalar field.

Potential Flow Equations

Potential flow methods in aerodynamics simplify the governing equations of fluid motion by assuming inviscid, irrotational flow, thereby allowing the velocity field to be expressed as the gradient of a scalar potential function. In incompressible flow, this approach yields Laplace’s equation, a linear, analytically tractable form of the governing equations of fluid motion. For compressible flow, however, the same assumptions lead to a nonlinear governing equation known as the full potential equation. Derived from the Euler and continuity equations, the full potential equation accounts for variations in velocity, pressure, and density, and forms the basis for more advanced models of subsonic, transonic, and supersonic flow.

Full Potential Equation

For a compressible, inviscid flow, the governing equations are

(3) $\begin{equation*} { \frac{\partial \varrho}{\partial t} + \nabla \bigcdot (\varrho \vec{V}) = 0 \text{~~~(Continuity)} } \end{equation*}$

and

(4) $\begin{equation*} \frac{\partial \vec{V}}{\partial t} + (\vec{V} \bigcdot \nabla)\vec{V} = -\frac{1}{\varrho} \nabla p \text{~~~(Momentum)} \end{equation*}$

If it is assumed that the flow is irrotational, the flow velocity can be written as the gradient of a scalar potential function, i.e.,

(5) $\begin{equation*} \vec{V} = \nabla \phi \end{equation*}$

Substituting this into the momentum equation gives

(6) $\begin{equation*} \frac{\partial \nabla \phi}{\partial t} + (\nabla \phi \bigcdot \nabla) \nabla \phi = -\frac{1}{\varrho} \nabla p \end{equation*}$

Using the identity

(7) $\begin{equation*} (\nabla \phi \bigcdot \nabla) \nabla \phi = \nabla \left( \frac{1}{2} |\nabla \phi|^2 \right) \end{equation*}$

then the unsteady form of the Bernoulli equation in differential form is obtained, i.e.,

(8) $\begin{equation*} \nabla \left( \frac{\partial \phi}{\partial t} + \frac{1}{2} |\nabla \phi|^2 + \int \frac{1}{\varrho} \, dp \right) = 0 \end{equation*}$

Assuming isentropic flow, the pressure and density are related by $dp = a^2 d\varrho$ , where $a$ is the local speed of sound. Therefore,

(9) $\begin{equation*} \int \frac{1}{\varrho} \, dp = \int \frac{a^2}{\varrho} \, d\varrho \end{equation*}$

which defines a compressible enthalpy function ${\cal{H}}(\varrho)$ , leading to

(10) $\begin{equation*} \frac{\partial \phi}{\partial t} + \frac{1}{2} |\nabla \phi|^2 + \mathcal{H}(\varrho) = \text{constant} \end{equation*}$

To eliminate $\varrho$ , the implicit dependence of density on the potential function through the Bernoulli relation can be invoked. For unsteady potential flow, the density is obtained from the isentropic Bernoulli relation and depends on both $\partial \phi/\partial t$ and the local velocity magnitude $V = |\nabla \phi|$ . Substitution into the continuity equation gives the conservative form of the full potential equation. For unsteady compressible potential flow, this equation is

(11) $\begin{equation*} \frac{\partial \varrho}{\partial t} + \nabla \bigcdot \left( \varrho \nabla \phi \right) = 0 \end{equation*}$

For steady compressible potential flow, this reduces to

(12) $\begin{equation*} \nabla \bigcdot \left( \varrho \nabla \phi \right) = 0 \end{equation*}$

where $\varrho$ is obtained from the isentropic Bernoulli relation and therefore depends on the local velocity magnitude $V = |\nabla \phi|$ . This equation retains the essential compressibility effects for steady irrotational, inviscid flow and is the usual starting point for simplified potential-flow models, such as the transonic small-disturbance (TSD) equation and the linearized potential-flow equations.

Small-Disturbance & Wave Equations

The full-potential equation can be further simplified when the disturbance velocities are small relative to the freestream velocity. For steady, compressible, small-disturbance flow about a body in a uniform freestream, the perturbation potential $\phi$ satisfies the linearized small-disturbance potential equation, i.e.,

(13) $\begin{equation*} \left(1 - M_\infty^2 \right) \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0 \end{equation*}$

where $M_\infty$ is the freestream Mach number. This equation is linear for subsonic flow, i.e., $M_\infty < 1$ , and forms the usual basis for the Prandtl-Glauert compressibility correction.
A more general unsteady linearized potential equation includes time-dependent terms. In the absence of a mean flow, the velocity potential satisfies the acoustic wave equation

(14) $\begin{equation*} \nabla^2 \phi - \frac{1}{a_\infty^2} \frac{\partial^2 \phi}{\partial t^2} = 0 \end{equation*}$

or equivalently,

(15) $\begin{equation*} \frac{1}{a_\infty^2} \frac{\partial^2 \phi}{\partial t^2} = \nabla^2 \phi \end{equation*}$

where $a_\infty$ is the speed of sound in the undisturbed medium. For flight through a uniform mean flow, however, the corresponding linearized potential equation involves a convective time derivative rather than a simple stationary-medium wave equation. Therefore, the equation above should be interpreted as the acoustic wave equation for small disturbances in a medium with no mean flow.

For transonic flows, further approximations to the full-potential equation lead to the transonic small-disturbance (TSD) equation. The TSD equation retains selected nonlinear terms that become important when the local Mach number approaches unity. It is therefore useful in transonic aerodynamics, but its precise form depends on the particular nondimensionalization and small-disturbance scaling used.

Laplace’s Equation

For steady, incompressible flows, the velocity potential satisfies Laplace’s equation, i.e.,

(16) $\begin{equation*} \nabla^2 \phi = 0 \end{equation*}$

This result follows directly from the continuity equation for an incompressible flow, i.e.,

(17) $\begin{equation*} \nabla \bigcdot \vec{V} = 0 \end{equation*}$

together with the definition of the velocity potential,

(18) $\begin{equation*} \vec{V} = \nabla \phi \end{equation*}$

so that

(19) $\begin{equation*} \nabla \bigcdot \vec{V} = \nabla \bigcdot \left( \nabla \phi \right) = \nabla^2 \phi = 0 \end{equation*}$

Laplace’s equation is linear, unlike the full-potential and TSD equations. Therefore, elementary potential-flow solutions may be combined by linear superposition to form more complex solutions, i.e.,

(20) $\begin{equation*} \phi_{\text{total}} = c_1 \, \phi_1 + c_2 \, \phi_2 + \cdots + c_N \, \phi_N \end{equation*}$

where each elementary solution $\phi_i$ , for $i = 1 \cdots N$ , satisfies Laplace’s equation and each $c_i$ is a constant. Because

(21) $\begin{equation*} \nabla^2 \left(c_i \, \phi_i\right) = c_i \, \nabla^2 \phi_i = 0 \end{equation*}$

the composite potential $\phi_{\text{total}}$ also satisfies Laplace’s equation. This linearity is one of the main reasons potential-flow theory is so useful in aerodynamics.

Velocity Potential & Stream Function

The velocity potential and stream function are essential in describing and analyzing fluid flows; both are scalar fields from which the flow velocities can be obtained. As previously mentioned, the velocity potential is a scalar function that characterizes irrotational flows, and the fluid’s velocity at any point can be expressed as the gradient of the potential function. The stream function also provides a means of visualizing incompressible two-dimensional flows. Streamlines, which are lines of constant stream function, can reveal flow patterns, i.e., the direction of flow and its relative velocity.

Velocity Potential

The velocity potential, $\phi$ , can now be formally defined as a scalar field whose gradient gives the velocity field of a potential flow, i.e.,

(22) $\begin{equation*} \vec{V} = \nabla \phi \end{equation*}$

The velocity components can then be derived from the gradients of $\phi$ , i.e.,

(23) $\begin{equation*} u = \frac{\partial \phi}{\partial x}, \quad v = \frac{\partial \phi}{\partial y}, \quad \text{and} \quad w = \frac{\partial \phi}{\partial z} \end{equation*}$

For an incompressible flow, continuity requires

(24) $\begin{equation*} \nabla \bigcdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = \frac{\partial}{\partial x} \left(\frac{\partial \phi}{\partial x} \right) + \frac{\partial}{\partial y} \left(\frac{\partial \phi}{\partial y} \right) + \frac{\partial}{\partial z} \left(\frac{\partial \phi}{\partial z} \right) \end{equation*}$

so that

(25) $\begin{equation*} \nabla \bigcdot \vec{V} = \frac{\partial ^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} + \frac{\partial ^2 \phi}{\partial z^2} = \nabla^2 \phi = 0 \end{equation*}$

Stream Function

While the velocity potential is defined for irrotational flows, the scalar stream function, $\psi$ , is most commonly used for two-dimensional incompressible flows. With the usual sign convention, the velocity components are

(26) $\begin{equation*} u = \frac{\partial \psi}{\partial y} \quad \text{and} \quad v = -\frac{\partial \psi}{\partial x} \end{equation*}$

With this definition, the continuity equation is automatically satisfied, i.e.,

(27) $\begin{equation*} \nabla \bigcdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial}{\partial x} \left(\frac{\partial \psi}{\partial y} \right) + \frac{\partial}{\partial y} \left(-\frac{\partial \psi}{\partial x} \right) = \frac{\partial^2 \psi}{\partial x \partial y} - \frac{\partial^2 \psi}{\partial y \partial x} = 0 \end{equation*}$

provided the mixed partial derivatives are continuous.
Lines of constant $\psi$ represent streamlines, and the difference between two adjacent values of $\psi$ gives the volumetric flow rate per unit depth between them. In other words, closer streamlines indicate higher flow velocities, as shown in Figure 4. In a two-dimensional flow, the differential volumetric flow rate per unit depth crossing an elemental line segment is

(28) $\begin{equation*} dQ = u \, dy - v \, dx \end{equation*}$

Substituting the definitions of the velocity components in terms of the stream function gives

(29) $\begin{equation*} dQ = \left( \frac{\partial \psi}{\partial y} \right) dy - \left( -\frac{\partial \psi}{\partial x} \right) dx = \frac{\partial \psi}{\partial y} \, dy + \frac{\partial \psi}{\partial x} \, dx \end{equation*}$

Streamlines passing through a streamtube are used to define the volume flow rate.

The total differential of the stream function $\psi(x, y)$ is

(30) $\begin{equation*} d\psi = \frac{\partial \psi}{\partial x} \, dx + \frac{\partial \psi}{\partial y} \, dy \end{equation*}$

so that comparing Eqs. 29 and 30 gives

(31) $\begin{equation*} dQ = d\psi \end{equation*}$

For example, the flow rate per unit depth between two streamlines, $\psi_1$ and $\psi_2$ , is

(32) $\begin{equation*} Q = \int_{\psi_1}^{\psi_2} dQ = \int_{\psi_1}^{\psi_2} d\psi = \psi_2 - \psi_1 \end{equation*}$

Relationship between Velocity Potential and Stream Function

For a two-dimensional incompressible and irrotational flow, both $\phi$ and $\psi$ exist. The velocity components may then be written as

(33) $\begin{equation*} u = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} \quad \text{and} \quad v = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} \end{equation*}$

or

(34) $\begin{equation*} \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} \quad \text{and} \quad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} \end{equation*}$

These are the Cauchy-Riemann equations for the velocity potential and stream function. They imply that lines of constant $\phi$ and lines of constant $\psi$ are orthogonal, as shown in Figure 5.

Equipotential lines are perpendicular to the streamlines.

The orthogonality follows because

(35) $\begin{equation*} \nabla \phi \bigcdot \nabla \psi = \frac{\partial \phi}{\partial x} \frac{\partial \psi}{\partial x} + \frac{\partial \phi}{\partial y} \frac{\partial \psi}{\partial y} = u(-v) + v u = 0 \end{equation*}$

Therefore, the flow velocity is tangent to the streamlines and normal to the equipotential lines.

Cartesian and Polar Coordinates

The relationship between Cartesian $(x, y)$ and polar $(r, \theta)$ coordinates is

$x = r \cos \theta , \quad y = r \sin \theta , \quad r = \sqrt{x^2 + y^2} , \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$

as shown in the figure below.
The velocity components in polar coordinates are expressed in terms of the radial and angular directions. If $(v_x, v_y)$ are the Cartesian components of velocity, the polar components are given by

$v_r = v_x \cos \theta + v_y \sin \theta, \quad \text{and} \quad v_\theta = -v_x \sin \theta + v_y \cos \theta$

Conversely, the Cartesian velocity components can be expressed in terms of the polar components as

$v_x = v_r \cos \theta - v_\theta \sin \theta , \quad v_y = v_r \sin \theta + v_\theta \cos \theta$

In terms of the polar components, the total velocity can be written as

$\vec{V} = v_r \, \vec{e}_r + v_\theta \, \vec{e}_{\theta}$

where $\vec{e}_r$ and $\vec{e}_{\theta}$ are the unit vectors in the radial and angular directions, respectively.

Finally, it should be noted for reference that the Laplacian operator in polar coordinates is given by

$\nabla^2 \phi = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2}$

Check Your Understanding #1 – Particular solution to Laplace’s equation

Consider a steady, incompressible, irrotational flow in two dimensions where the velocity potential $\phi(x, y)$ satisfies Laplace’s equation, i.e., $\nabla^2 \phi = 0$ . The region of interest is the square $0 \leq x \leq 1$ , $0 \leq y \leq 1$ , with the following boundary conditions: $\phi = 0$ on $x = 0$ and $y = 0$ , $\phi = y$ on $x = 1$ , and $\phi = x$ on ${y = 1}$ .

Assuming a general solution of the form $\phi(x, y) = a x + b y + c x y$ , determine $\phi(x, y)$ .
Determine the velocity field in terms of $u$ and $v$ .
Verify that this velocity field is incompressible and irrotational.
Determine the corresponding stream function for this flow.
Sketch the streamlines and equipotential lines for this flow.

Show solution/hide solution.

1. The velocity potential $\phi(x, y)$ satisfies Laplace’s equation, i.e.,

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

Assume a velocity potential of the form

$\phi(x, y) = a x + b y + c x y$

Substituting this expression into Laplace’s equation shows that it satisfies $\nabla^2 \phi = 0$ for any values of $a$ , $b$ , and $c$ . Applying the boundary conditions gives

$\begin{align*} \phi(0, y) = b y = 0 & \implies b = 0 \\ \phi(x, 0) = a x = 0 & \implies a = 0 \\ \phi(1, y) = c y = y & \implies c = 1 \\ \phi(x, 1) = c x = x & \implies c = 1 \end{align*}$

Therefore, the velocity potential is

$\phi(x, y) = x \, y$

2. The velocity field is given by

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

From $\phi(x, y) = x \, y$ , then

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

3. The out-of-plane vorticity is

$\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$

For $u = y$ and $v = x$ , then

$\frac{\partial v}{\partial x} = 1 \quad \text{and} \quad \frac{\partial u}{\partial y} = 1$

so

$\omega_z = 1 - 1 = 0$

Therefore, the velocity field is irrotational. The continuity equation is also satisfied because

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

Therefore, the velocity field is incompressible.

4. The stream function $\psi(x, y)$ satisfies

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

From $u = y$ , then

$\frac{\partial \psi}{\partial y} = y$

and so

$\psi = \frac{y^2}{2} + f(x)$

From $v = x$ , then

$-\frac{\partial \psi}{\partial x} = x$

or

$\frac{\partial \psi}{\partial x} = -x$

Therefore,

$\psi = -\frac{x^2}{2} + g(y)$

Combining these two results gives

$\psi(x, y) = \frac{y^2}{2} - \frac{x^2}{2} + C$

The arbitrary constant $C$ may be set to zero, so

$\psi(x, y) = \frac{y^2}{2} - \frac{x^2}{2}$

which can be confirmed by differentiation, i.e.,

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

5. The streamlines correspond to constant values of $\psi(x, y)$ , i.e.,

$\frac{y^2}{2} - \frac{x^2}{2} = \text{constant}$

The equipotential lines correspond to constant values of $\phi(x, y)$ , i.e.,

$x \, y = \text{constant}$

These two families of curves are orthogonal hyperbolas, as shown in the figure below.

Elementary Flows

Elementary flows are the building blocks of potential flow theory, each elementary flow satisfying Laplace’s equation. Superimposing elementary flows allows the modeling of complex flow fields. Key elementary flows include:

Uniform flow – A constant velocity field where the flow direction and magnitude remain the same everywhere.
Source and sink flows – Radial flow where fluid emanates from a point (source) or converges to a point (sink).
Vortex flow – A swirling flow around a central axis characterized by circulation and rotational motion.
Doublet flow – A limiting combination of a source and a sink of equal and opposite strength placed infinitesimally close to each other.

Combining these elementary flows enables the analysis and visualization of more intricate flow problems, such as flow over specific classes of body shapes, including circular cylinders and ovals.

Uniform Flow

Figure 6 shows that a uniform flow moves at a constant velocity $U$ in the $x$ direction. The equipotential lines, i.e., $\phi = \text{constant}$ , are vertical lines defined by $x = \text{constant}$ , and the streamlines, i.e., $\psi = \text{constant}$ , are horizontal lines defined by $y = \text{constant}$ , i.e.,

(36) $\begin{equation*} \phi = U \, x \quad \text{and} \quad \psi = U \, y \end{equation*}$

Streamlines and equipotential lines for a uniform flow in the ${x}$ direction.

Notice that

(37) $\begin{equation*} u = \frac{\partial \phi}{\partial x} = U \quad \text{and} \quad v = \frac{\partial \phi}{\partial y} = 0 \end{equation*}$

and, equivalently, from the stream function, then

(38) $\begin{equation*} u = \frac{\partial \psi}{\partial y} = U \quad \text{and} \quad v = -\frac{\partial \psi}{\partial x} = 0 \end{equation*}$

If the flow is moving at a constant velocity $V$ in the ${y}$ -direction, then

(39) $\begin{equation*} \phi = V y \quad \text{and} \quad \psi = -V \, x \end{equation*}$

and for a flow ${V_{\infty}}$ at an angle ${\alpha}$ to the ${x}$ axis then

(40) $\begin{equation*} \phi = V_{\infty} \left( x \cos \alpha + y \sin \alpha\right) \quad \text{and} \quad \psi = V_{\infty} \left( y \cos \alpha - x \sin \alpha\right) \end{equation*}$

These yield the correct velocity components, i.e., from the potential function, then

(41) $\begin{equation*} u = \frac{\partial \phi}{\partial x} = V_{\infty} \cos \alpha \quad \text{and} \quad v = \frac{\partial \phi}{\partial y} = V_{\infty} \sin \alpha \end{equation*}$

and from the stream function, then

(42) $\begin{equation*} u = \frac{\partial \psi}{\partial y} = V_{\infty} \cos \alpha \quad \text{and} \quad v = -\frac{\partial \psi}{\partial x} = V_{\infty} \sin \alpha \end{equation*}$

Therefore, the velocity field is

(43) $\begin{equation*} \vec{V} = u \, \vec{i} + v \, \vec{j} = V_{\infty} \cos \alpha\, \vec{i} + V_{\infty} \sin \alpha\, \vec{j} \end{equation*}$

In polar coordinates $(r, \theta)$ , the potential function $\phi$ and stream function $\psi$ for a uniform flow in the ${x}$ direction are expressed as

(44) $\begin{equation*} \phi = U r \cos \theta \quad \text{and} \quad \psi = U r \sin \theta \end{equation*}$

and the corresponding velocity components in polar coordinates are

(45) $\begin{equation*} v_r = \frac{\partial \phi}{\partial r} = U \cos \theta \quad \text{and} \quad v_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta} = -U \sin \theta \end{equation*}$

This elementary flow forms the basis for modeling freestream flows in nearly all potential-flow problems. The polar representation is beneficial for analyzing flow patterns around circular objects or in cylindrical domains.

Source Flow

To derive the velocity potential and stream function for a point source, consider a steady, two-dimensional, incompressible, and irrotational flow field where fluid is emitted uniformly in all directions from a point at the origin. As shown in Figure 7, a source is a hypothetical flow in which fluid “exits” from a point. The total volume flow rate per unit depth is denoted by $Q$ , which defines the source strength.

Because of the radial symmetry of the flow, the velocity vector has only a radial component $v_r$ , with no tangential component $v_\theta$ , i.e.,

(46) $\begin{equation*} v_r = \frac{Q}{2\pi r} \qquad \text{and} \qquad v_\theta = 0 \end{equation*}$

which satisfies conservation of mass for radial outflow in two dimensions.

The velocity potential $\phi$ satisfies

(47) $\begin{equation*} v_r = \frac{\partial \phi}{\partial r} \end{equation*}$

Substituting for $v_r$ and integrating with respect to $r$ gives

(48) $\begin{equation*} \phi(r) = \int \frac{Q}{2\pi r} \, dr = \frac{Q}{2\pi} \ln r + \text{constant} \end{equation*}$

The constant may be taken as zero without loss of generality, so the velocity potential becomes

(49) $\begin{equation*} \phi = \frac{Q}{2\pi} \ln r \end{equation*}$

To determine the stream function $\psi$ , use the definitions in polar coordinates, i.e.,

(50) $\begin{equation*} v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta} \quad \text{and} \quad v_\theta = -\frac{\partial \psi}{\partial r} \end{equation*}$

Because $v_\theta = 0$ , it follows that $\partial \psi / \partial r = 0$ , so $\psi = \psi(\theta)$ . Substituting into the expression for $v_r$ gives

(51) $\begin{equation*} \frac{1}{r} \frac{d\psi}{d\theta} = \frac{Q}{2\pi r} \quad \Rightarrow \quad \frac{d\psi}{d\theta} = \frac{Q}{2\pi} \end{equation*}$

and integrating with respect to $\theta$ gives

(52) $\begin{equation*} \psi(\theta) = \frac{Q}{2\pi} \theta + \text{constant} \end{equation*}$

Again, setting the constant to zero gives the stream function as

(53) $\begin{equation*} \psi = \frac{Q}{2\pi} \theta \end{equation*}$

The final expressions for the source flow are therefore:

(54) $\begin{equation*} \phi = \frac{Q}{2\pi} \ln r \quad \text{and} \quad \psi = \frac{Q}{2\pi} \theta \end{equation*}$

where $Q$ is the strength of the source, $r$ is the radial distance from the source, and $\theta$ is the polar angle. As shown in Figure 7, the equipotential lines ( $\phi =$ constant) are circles centered at the source, i.e., lines of constant $r$ , and are spaced in a geometric progression. The streamlines ( $\psi =$ constant) are straight lines radiating from the origin, i.e., lines of constant $\theta$ , indicating purely radial flow.

The velocity field is

(55) $\begin{equation*} v_r = \frac{Q}{2\pi r} \quad \text{and} \quad v_\theta = 0 \end{equation*}$

As $r \to 0$ , the radial velocity becomes unbounded, i.e., $v_r \to \infty$ , which marks a singularity in the flow. At the origin, the assumptions of continuity and finite velocity break down. This type of behavior is characteristic of idealized elementary flows, such as sources, sinks, and vortices, which are often referred to as singularities in potential flow theory.

Sink Flow

A sink is the reverse of a source, where fluid is “removed” at a point, as shown in Figure 8. The velocity potential and stream function are given by

(56) $\begin{equation*} \phi = -\frac{Q}{2\pi} \ln r \quad \text{and} \quad \psi = -\frac{Q}{2\pi} \theta \end{equation*}$

where $Q$ is the (positive) strength of the sink, $r$ is the radial distance from the sink, and $\theta$ is the polar angle. The equipotential lines ( $\phi =$ constant) are concentric circles centered at the origin, and the streamlines ( $\psi =$ constant) are radial lines, as in the case of a source, but with flow directed inward.

The radial velocity is

(57) $\begin{equation*} v_r = -\frac{Q}{2\pi r} \quad \text{and} \quad v_\theta = 0 \end{equation*}$

So the flow is purely radial and directed toward the sink at the origin. As $r \to 0$ , the velocity again becomes unbounded, indicating a singularity at the origin. Like sources, sinks are idealized representations that violate the physical assumptions of continuity at the singular point.

Streamlines and equipotential lines for an elementary sink (i.e., negative source) flow.

Vortex Flow (Irrotational Vortex)

An irrotational vortex represents purely tangential motion around the origin, as shown in Figure 9, where the tangential velocity is inversely proportional to the radial distance. The equipotential lines ( $\phi$ = constant) are straight radial lines ( $\theta$ = constant), and the streamlines ( $\psi$ = constant) are concentric circles ( $r$ = constant). The strength of the vortex is defined by the circulation $\Gamma$ , which is the line integral of tangential velocity around any closed loop enclosing the vortex, i.e.,

(58) $\begin{equation*} \Gamma = \oint \vec{V} \bigcdot d\vec{s} \end{equation*}$

By convention, the closed path is taken in the counterclockwise direction, so a positive value of $\Gamma$ corresponds to counterclockwise circulation. By symmetry, the flow has no radial component, i.e.,

(59) $\begin{equation*} v_r = 0 \quad \text{and} \quad v_\theta = \frac{\Gamma}{2\pi r} \end{equation*}$

where $v_\theta$ is the tangential velocity and $r$ is the radial distance from the vortex center. The direction of rotation is determined by the sign of $\Gamma$ , i.e., counterclockwise for positive $\Gamma$ , and clockwise for negative $\Gamma$ .

Streamlines and equipotential lines represent a vortex flow, in this case, with a clockwise direction and, therefore, a negative strength.

To derive the stream function $\psi$ , use the polar coordinate relations, i.e.,

(60) $\begin{equation*} v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta} \quad \text{and} \quad v_\theta = -\frac{\partial \psi}{\partial r} \end{equation*}$

Because $v_r = 0$ , it follows that $\partial \psi / \partial \theta = 0$ , so $\psi = \psi(r)$ . From the second equation,

(61) $\begin{equation*} -\frac{d\psi}{dr} = \frac{\Gamma}{2\pi r} \quad \Rightarrow \quad \frac{d\psi}{dr} = -\frac{\Gamma}{2\pi r} \end{equation*}$

Integrating with respect to $r$ gives

(62) $\begin{equation*} \psi(r) = -\frac{\Gamma}{2\pi} \ln r + \text{constant} \end{equation*}$

Choosing the arbitrary constant to be zero gives

(63) $\begin{equation*} \psi = -\frac{\Gamma}{2\pi} \ln r \end{equation*}$

To derive the velocity potential $\phi$ , use the relationships

(64) $\begin{equation*} v_r = \frac{\partial \phi}{\partial r} \quad \text{and} \quad v_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta} \end{equation*}$

Again, $v_r = 0$ implies $\partial \phi / \partial r = 0$ , so $\phi = \phi(\theta)$ . Therefore,

(65) $\begin{equation*} \frac{1}{r} \frac{d\phi}{d\theta} = \frac{\Gamma}{2\pi r} \quad \Rightarrow \quad \frac{d\phi}{d\theta} = \frac{\Gamma}{2\pi} \end{equation*}$

Integrating with respect to $\theta$ gives

(66) $\begin{equation*} \phi(\theta) = \frac{\Gamma}{2\pi} \theta + \text{constant} \end{equation*}$

Choosing the arbitrary constant to be zero gives

(67) $\begin{equation*} \phi = \frac{\Gamma}{2\pi} \theta \end{equation*}$

Therefore, the final expressions for the vortex flow are

(68) $\begin{equation*} \phi = \frac{\Gamma}{2\pi} \theta \quad \text{and} \quad \psi = -\frac{\Gamma}{2\pi} \ln r \end{equation*}$

where $\Gamma$ is the circulation, $r$ is the radial distance from the vortex center, and $\theta$ is the polar angle. A positive circulation $\Gamma$ produces a counterclockwise vortex, while a negative circulation produces a clockwise vortex. The equipotential lines ( $\phi =$ constant) are radial lines ( $\theta =$ constant), and the streamlines ( $\psi =$ constant) are circles ( $r =$ constant), indicating purely tangential flow.

The velocity field is

(69) $\begin{equation*} v_r = 0 \quad \text{and} \quad v_\theta = \frac{\Gamma}{2\pi r} \end{equation*}$

As $r \to 0$ , the tangential velocity becomes unbounded, i.e., $v_\theta \to \infty$ , again indicating a singularity in the flow.

Understanding Circulation and Vorticity

Circulation provides a global measure of the rotational characteristics of the flow around a closed path. It is an integral quantity of the effect of the velocity field over a finite region. On the other hand, vorticity is a local property of the flow, describing the instantaneous rotation rate of fluid elements at each point in space. For example, in a point vortex, vorticity is concentrated at the singular core, whereas circulation quantifies the net rotational effect along a closed path around the vortex. Understanding the interplay between circulation and vorticity is crucial in aerodynamics, where they play key roles in lift generation.

The property identified as $\Gamma$ is called the circulation, defined mathematically as

(70) $\begin{equation*} \Gamma = \oint_{C} \vec{V} \bigcdot d\vec{s} \end{equation*}$

where $d\vec{s}$ is an infinitesimal arc length element along a closed circuit $C$ , and ${\vec{V}}$ is the local velocity vector of the flow. Circulation represents the net “swirling” or “rotational” behavior of the velocity field around the circuit $C$ , as illustrated in Figure 10.

The property of circulation is established from an integral of the velocity field around a closed path.

The circulation is calculated by integrating the velocity field along a closed path, i.e.,

(71) $\begin{equation*} \Gamma = \oint_C \vec{V} \bigcdot d\vec{s} = \oint_C u \, dx + v \, dy \end{equation*}$

By convention, the contour $C$ is taken in the counterclockwise direction, in which case a positive value of $\Gamma$ corresponds to counterclockwise circulation. In aerodynamics, however, lifting flows are typically associated with a clockwise rotation of the flow around the airfoil, which therefore corresponds to a negative value of $\Gamma$ under this convention.

Vorticity, denoted by $\vec{\omega}$ , is a fundamental property of a fluid flow that is equal to twice the local angular velocity of a fluid element. It is mathematically defined as the curl of the velocity field, i.e.,

(72) $\begin{equation*} \vec{\omega} = \nabla \times \vec{V} \end{equation*}$

For two-dimensional flows in the ${x}$ – ${y}$ plane, the vorticity reduces to a scalar quantity

(73) $\begin{equation*} \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \end{equation*}$

where ${\omega_z}$ is the out-of-plane component of the vorticity vector.

The relationship between circulation and vorticity is given by Stokes’ theorem, which connects the line integral of the velocity field around a closed curve to the surface integral of the vorticity over the enclosed area, i.e.,

(74) $\begin{equation*} \Gamma = \iint_S \vec{\omega} \bigcdot \vec{n} \, dA \end{equation*}$

where $S$ is the surface bounded by the closed curve $C$ , $\vec{n}$ is the unit normal vector to the surface, and $dA$ is the infinitesimal area element. This relationship reveals that the circulation around a closed path is the total vorticity flux through the surface enclosed by that path. If the vorticity within the region is zero and the path encloses no singularity, then the circulation is also zero.

Notice that for potential flow, the vorticity is zero everywhere in the regular part of the flow. However, circulation can result if the path $C$ encloses a singularity such as a point vortex. For a point vortex, the vorticity is zero everywhere except at the singularity. At $r=0$ , the velocity field is not defined in the ordinary sense; mathematically, the vorticity is represented by a concentrated delta-function singularity whose area integral gives the circulation $\Gamma$ .

Check Your Understanding #2 – Pressure-centrifugal balance in vortex flow

Demonstrate that the existence of a steady, incompressible vortex flow requires a radial pressure gradient to balance the centrifugal forces. Use an infinitesimal control volume in the form of a fluid ring element bounded by two adjacent streamlines.

Show solution/hide solution.

Consider a steady, two-dimensional vortex flow in which the fluid moves in concentric circles about a central point. There is no radial motion, so the velocity components are

$v_r = 0 \quad \text{and} \quad v_\theta = v_\theta(r)$

Examine an elemental ring of fluid bounded between circular streamlines at $r$ and $r + dr$ , subtending an angular width $d\theta$ . The area of the element per unit depth is

$dA = r \, dr \, d\theta$

so its mass per unit depth is

$dm = \varrho \, dA = \varrho \, r \, dr \, d\theta$

where $\varrho$ is the fluid density.

This element moves around the origin with tangential velocity $v_\theta(r)$ , so it requires an inward centripetal acceleration of

$a_r = \frac{v_\theta^2}{r}$

The corresponding inward force required to maintain the circular motion is therefore

$dF_r = dm \, \frac{v_\theta^2}{r}$

or

$dF_r = \varrho \, r \, dr \, d\theta \, \frac{v_\theta^2}{r} = \varrho \, v_\theta^2 \, dr \, d\theta$

This inward force must be supplied by the pressure gradient. In a vortex, the pressure is lower near the center and higher farther from the center, so $dp/dr > 0$ . The resulting pressure force acts inward, from the higher pressure at the outer radius toward the lower pressure at the inner radius.

The pressure difference across the element is

$dp = \frac{dp}{dr} \, dr$

and the area of the curved face per unit depth is approximately

$dS = r \, d\theta$

Therefore, the net inward pressure force on the element is

$dF_p = \frac{dp}{dr} \, dr \, r \, d\theta$

For radial equilibrium of the rotating fluid element, this inward pressure force must supply the required centripetal force, so

$\frac{dp}{dr} \, dr \, r \, d\theta = \varrho \, v_\theta^2 \, dr \, d\theta$

Canceling the common factors $dr \, d\theta$ gives

$r \frac{dp}{dr} = \varrho \, v_\theta^2$

or

$\frac{1}{\varrho} \frac{dp}{dr} = \frac{v_\theta^2}{r}$

The result shows that a radial pressure gradient is required to maintain equilibrium in a steady vortex flow. Specifically, the pressure must increase with radial distance so that the higher pressure outside the fluid element supplies the inward force needed to maintain the circular motion. This pressure gradient is a fundamental requirement for the existence of any inviscid vortex flow.

Doublet (Dipole) Flow

A doublet is created by superimposing a source and sink of equal strength placed infinitesimally close together, as shown in Figure 11. A source of strength $+Q$ is placed at $(-d, 0)$ , and a sink of strength $-Q$ at $(+d, 0)$ . The separation tends to zero while the product of strength and separation remains finite.

The velocity potential at a point $(x, y)$ is the superposition of the potentials from the source and sink, i.e.,

(75) $\begin{equation*} \phi(x, y) = \frac{Q}{2\pi} \ln{\sqrt{(x + d)^2 + y^2}} - \frac{Q}{2\pi} \ln{\sqrt{(x - d)^2 + y^2}} \end{equation*}$

Combining logarithmic terms gives

(76) $\begin{equation*} \phi(x, y) = \frac{Q}{4\pi} \ln \left( \frac{(x + d)^2 + y^2}{(x - d)^2 + y^2} \right) \end{equation*}$

Now take the limit as $d \to 0$ and $Q \to \infty$ , such that the product $\mu = 2dQ$ remains finite. This limit defines a doublet of strength $\mu$ , and the velocity potential becomes

(77) $\begin{equation*} \phi(x, y) = \lim_{d \to 0} \frac{Q}{4\pi} \ln \left( \frac{(x + d)^2 + y^2}{(x - d)^2 + y^2} \right) = \frac{\mu}{2\pi} \left( \frac{x}{x^2 + y^2} \right) \end{equation*}$

In polar coordinates, where $x = r \cos \theta$ and $y = r \sin \theta$ , this becomes

(78) $\begin{equation*} \phi(r, \theta) = \frac{\mu}{2\pi} \frac{\cos \theta}{r} \end{equation*}$

where $\mu = 2 \, d \, Q$ is the doublet strength. Note that a doublet has an orientation, and reversing the sign of $\mu$ reverses the direction of the flow.

The corresponding stream function for the doublet is

(79) $\begin{equation*} \psi(x, y) = -\frac{\mu}{2\pi} \left( \frac{y}{x^2 + y^2} \right) \end{equation*}$

which in polar coordinates becomes

(80) $\begin{equation*} \psi(r, \theta) = -\frac{\mu}{2\pi} \frac{\sin \theta}{r} \end{equation*}$

The velocity components in polar coordinates are obtained by differentiation of the velocity potential:

(81) $\begin{equation*} v_r = \frac{\partial \phi}{\partial r} = -\frac{\mu}{2\pi} \frac{\cos \theta}{r^2}, \qquad v_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta} = -\frac{\mu}{2\pi} \frac{\sin \theta}{r^2} \end{equation*}$

As shown in Figure 11, the streamlines of a doublet are symmetric about the $x$ -axis and form circular loops tangent to the origin. The equipotential lines are orthogonal to the streamlines. When superimposed with a uniform flow, a doublet provides the classical potential-flow model of flow around a circular cylinder. As with other elementary flows, the doublet exhibits a singularity at the origin where the velocity becomes unbounded.

Table of Elementary Flows

A handy table of the velocity potentials $\phi(x, y)$ and stream functions $\psi(x, y)$ for the standard elementary flows is given below. Remember that the corresponding velocity fields are obtained using partial differentiation.

Velocity potentials $\phi(x, y)$ and stream functions $\psi(x, y)$ for elementary flows.
Flow Type	Potential Function $\phi(x, y)$	Stream Function $\psi(x, y)$
Uniform flow of speed $U$ at angle of attack $\alpha$	$\phi = U x \cos \alpha + U y \sin \alpha$	$\psi = U y \cos \alpha - U x \sin \alpha$
Uniform flow of speed $U$ at angle of attack $\alpha$	$\phi = U r \cos(\theta - \alpha)$	$\psi = U r \sin(\theta - \alpha)$
Source of strength $Q$ at origin	$\phi = \dfrac{Q}{2\pi} \ln r$	$\psi = \dfrac{Q}{2\pi} \theta$
Sink of strength $Q$ at origin	$\phi = -\dfrac{Q}{2\pi} \ln r$	${ \psi = -\dfrac{Q}{2\pi} \theta}$
Vortex of strength $\Gamma$ at origin	$\phi = \dfrac{\Gamma}{2\pi} \theta$	$\psi = -\dfrac{\Gamma}{2\pi} \ln r$
Doublet of strength $\mu$ aligned with the $x$ -axis	$\phi = \dfrac{\mu}{2\pi} \left( \dfrac{x}{x^2 + y^2} \right)$	$\psi = -\dfrac{\mu}{2\pi} \left( \dfrac{y}{x^2 + y^2} \right)$

Superposition of Elementary Flows

The linearity of Laplace’s equation allows for the principle of superposition, which states that the solution to the equation can be expressed as the sum of individual solutions. The superposition principle is a powerful tool for understanding and analyzing potential flows. It enables the construction of more intricate flow fields from simple, well-understood flow-building blocks. Mathematically, this process can be written as

(82) $\begin{equation*} \phi_{\text{total}} = c_1 \, \phi_1 + c_2 \, \phi_2 + \cdots + c_N \, \phi_N \end{equation*}$

where $c_1 \, \phi_1, c_2 \, \phi_2, \cdots, c_N \, \phi_N$ are the velocity potential functions corresponding to up to $N$ individual elementary flows, the $c$ values being scaling constants. More complex flow fields can be constructed to approximate or model real-world scenarios by superimposing these elementary solutions.

For example, the combination of a uniform flow and a doublet can represent the flow around a cylinder. In contrast, adding vortex and source flows can model flow around rotating bodies or specific streamline patterns. This principle is foundational to panel methods, in which an airfoil or other shape is modeled as a collection of sources, sinks, and vortices distributed along its surfaces. By determining the appropriate strengths of these elements, the flow around the airfoil can be accurately represented, including essential characteristics such as the pressure distribution and lift.

Simulating the Flow Around a Solid Body

A solid body can replace the shape of the dividing streamline between two elementary flows, i.e., the streamline that divides one flow from another; the flow interior to the dividing streamline is irrelevant. In these cases, the value of potential methods for simulating flows around solid bodies will become increasingly apparent.

Rankine Half-Body

A Rankine half-body^[4] is formed by the superposition of a uniform flow $U$ in the ${x}$ -direction and a source of strength $Q$ located at the origin, as shown in Figure 12. Rankine called such shapes “Oögenous Neoïds.”

The velocity potential is

(83) $\begin{equation*} \phi(x, y) = Ux + \frac{Q}{4 \pi} \ln \left( x^2 + y^2 \right) \end{equation*}$

and the corresponding stream function is

(84) $\begin{equation*} \psi(x, y) = U y + \frac{Q}{2 \pi} \tan^{-1} \left( \frac{y}{x} \right) \end{equation*}$

The dividing streamline represents the boundary of the Rankine half-body. For a uniform flow in the positive ${x}$ -direction with a source at the origin, the stagnation point occurs upstream of the source at

(85) $\begin{equation*} (x_s, y_s) = \left( -\frac{Q}{2 \pi U}, \, 0 \right) \end{equation*}$

The upper boundary streamline has $\psi = Q/2$ , so that

(86) $\begin{equation*} U y + \frac{Q}{2\pi}\theta = \frac{Q}{2} \end{equation*}$

or

(87) $\begin{equation*} y = \frac{Q}{2\pi U}\left(\pi - \theta\right) \end{equation*}$

with a corresponding lower boundary streamline by symmetry. Note that $\psi = 0$ does not necessarily indicate a stagnation streamline.

Rankine Oval

A Rankine oval is formed by the superposition of uniform velocity $U$ in the ${x}$ -direction, with a point source $+Q$ at $(-a, 0)$ and a point sink $-Q$ at $(+a, 0)$ , as shown in Figure 13, another Rankine neoïd. He explains, “An oval neoïd differs from an ellipse in being fuller towards the ends and flatter at the sides, and that difference is greater the more elongated the oval is.”

The velocity potential is

(88) $\begin{equation*} \phi(x, y) = Ux + \frac{Q}{4 \pi} \ln \bigg( (x + a)^2 + y^2 \bigg) - \frac{Q}{4 \pi} \ln \bigg( (x - a)^2 + y^2 \bigg) \end{equation*}$

and the stream function is

(89) $\begin{equation*} \psi(x, y) = Uy + \frac{Q}{2 \pi} \theta_1 - \frac{Q}{2 \pi} \theta_2 \end{equation*}$

where

(90) $\begin{equation*} \theta_1 = \tan^{-1}\left(\frac{y}{x+a}\right) \quad \text{and} \quad \theta_2 = \tan^{-1}\left(\frac{y}{x-a}\right) \end{equation*}$

with the angles interpreted using the correct quadrant, i.e., using the two-argument arctangent.

The equation of the dividing streamline may be written as

(91) $\begin{equation*} U y + \frac{Q}{2 \pi} \theta_1 - \frac{Q}{2 \pi} \theta_2 = 0 \end{equation*}$

which forms the oval boundary, i.e., $\psi = 0$ . The stagnation points, where $u = v = 0$ , lie on the ${x}$ -axis at

(92) $\begin{equation*} x = \pm \sqrt{a^2 + \frac{Q a}{\pi U}}, \quad y = 0 \end{equation*}$

Therefore, there are two stagnation points, one upstream and another downstream.

Flow Around a Circular Cylinder

The flow around a circular cylinder is a classic problem in aerodynamics. A potential flow is obtained from the superposition of the stream function from the combination of a uniform flow and a doublet, as shown in Figure 14.

The composite stream function is given by

(93) $\begin{equation*} \psi_{\text{total}} = \psi_{\text{uniform}} + \psi_{\text{doublet}} \end{equation*}$

For uniform flow, then

(94) $\begin{equation*} \psi_{\text{uniform}} = U y \end{equation*}$

and for a doublet at the origin, then

(95) $\begin{equation*} \psi_{\text{doublet}} = -\left( \frac{\mu}{2\pi} \right) \frac{y}{x^2 + y^2} \end{equation*}$

The total stream function is

(96) $\begin{equation*} \psi_{\text{total}} = U y - \frac{\mu}{2\pi} \frac{y}{x^2 + y^2} = U r \sin \theta - \frac{\mu}{2\pi r} \sin \theta \end{equation*}$

The surface of the cylinder corresponds to the streamline $\psi_{\text{total}}=0$ at $r=R$ . Therefore,

(97) $\begin{equation*} U R \sin \theta - \frac{\mu}{2\pi R} \sin \theta = 0 \end{equation*}$

which gives

(98) $\begin{equation*} \mu = 2\pi U R^2 \end{equation*}$

Velocity on the Cylinder’s Surface

The tangential velocity $v_\theta$ is

(99) $\begin{equation*} v_\theta = -\frac{\partial \psi_{\text{total}}}{\partial r} \end{equation*}$

so

(100) $\begin{equation*} v_\theta = -\frac{\partial}{\partial r} \left( U r \sin \theta - \frac{\mu}{2\pi r} \sin \theta \right) \end{equation*}$

Taking the derivative with respect to $r$ gives

(101) $\begin{equation*} v_\theta = - \left( U \sin \theta + \frac{\mu}{2\pi r^2} \sin \theta \right) \end{equation*}$

At the surface of the cylinder, where $r = R$ , the tangential velocity is

(102) $\begin{equation*} v_\theta = - \left( U + \frac{\mu}{2\pi R^2} \right) \sin \theta \end{equation*}$

Using $\mu = 2\pi U R^2$ gives

(103) $\begin{equation*} v_\theta = -2U \sin \theta \end{equation*}$

Pressure on the Cylinder’s Surface

The pressure coefficient, ${C_p}$ , is obtained using Bernoulli’s equation, i.e.,

(104) $\begin{equation*} { C_p = \frac{p - p_\infty}{\frac{1}{2} \, \varrho \, U^2 } } \end{equation*}$

Therefore, the pressure coefficient at a point on the surface of the cylinder is

(105) $\begin{equation*} C_p = 1 - \frac{v_\theta^2}{U^2} \end{equation*}$

Substituting the surface velocity gives

(106) $\begin{equation*} C_p = 1 - 4 \sin^2 \theta \end{equation*}$

This is the well-known expression for the pressure distribution around a circular cylinder in potential flow. Notice that:

At $\theta = 0$ and $\theta = \pi$ (stagnation points), ${\sin^2 \theta = 0}$ , so $C_p = 1$ , indicating maximum pressure.
At $\theta = \pi/2$ and $\theta = 3\pi/2$ (points of maximum velocity), ${\sin^2 \theta = 1}$ , so $C_p = -3$ , indicating minimum pressure.
The top/bottom symmetry of the cylinder will result in zero lift, and the fore/aft symmetry will result in zero drag.

Lifting Circular Cylinder

The flow around a lifting circular cylinder is obtained by combining a uniform flow, a doublet, and a vortex, as shown in Figure 15. The composite stream function is given by

(107) $\begin{equation*} \psi_{\text{total}} = \psi_{\text{uniform}} + \psi_{\text{doublet}} + \psi_{\text{vortex}} \end{equation*}$

The flow around a lifting cylinder can be obtained by adding a vortex flow.

For uniform flow, then

(108) $\begin{equation*} \psi_{\text{uniform}} = U y \end{equation*}$

and for a doublet at the origin, then

(109) $\begin{equation*} \psi_{\text{doublet}} = -\frac{\mu}{2\pi} \frac{y}{x^2 + y^2} \end{equation*}$

Finally, for a vortex

(110) $\begin{equation*} \psi_{\text{vortex}} = -\frac{\Gamma}{2\pi} \ln \left( \frac{r}{R} \right) \end{equation*}$

Therefore, the total stream function becomes

(111) $\begin{equation*} \psi_{\text{total}} = U y - \frac{\mu}{2\pi} \frac{y}{x^2 + y^2} - \frac{\Gamma}{2\pi} \ln \left( \frac{r}{R} \right) = U r \sin \theta - \frac{\mu}{2\pi r} \sin \theta - \frac{\Gamma}{2\pi} \ln \left( \frac{r}{R} \right) \end{equation*}$

The surface of the cylinder corresponds to the streamline $\psi_{\text{total}} = 0$ at $r = R$ . This gives

(112) $\begin{equation*} U R \sin \theta - \frac{\mu}{2\pi R} \sin \theta = 0 \end{equation*}$

so that

(113) $\begin{equation*} \mu = 2\pi U R^2 \end{equation*}$

Velocity on the Cylinder’s Surface

The velocity components in polar coordinates are

(114) $\begin{equation*} v_r = U \left(1 - \frac{R^2}{r^2} \right)\cos\theta \end{equation*}$

(115) $\begin{equation*} v_\theta = - U \left(1 + \frac{R^2}{r^2} \right)\sin\theta + \frac{\Gamma}{2\pi r} \end{equation*}$

On the surface of the cylinder ( $r = R$ ), the radial velocity is zero, i.e.,

(116) $\begin{equation*} v_r = 0 \end{equation*}$

and the tangential velocity becomes

(117) $\begin{equation*} v_\theta = -2U \sin\theta + \frac{\Gamma}{2\pi R} \end{equation*}$

The stagnation points occur where $v_\theta = 0$ , so that

(118) $\begin{equation*} -2U \sin \theta + \frac{\Gamma}{2\pi R} = 0 \quad \Rightarrow \quad \sin \theta = \frac{\Gamma}{4\pi R U} \end{equation*}$

For the stagnation points to remain on the surface of the cylinder, it is necessary that

(119) $\begin{equation*} \left| \frac{\Gamma}{4\pi R U} \right| \leq 1 \end{equation*}$

When this condition is satisfied, the stagnation points are located at

(120) $\begin{equation*} \theta = \sin^{-1} \left( \frac{\Gamma}{4\pi R U} \right) \quad \text{and} \quad \theta = \pi - \sin^{-1} \left( \frac{\Gamma}{4\pi R U} \right) \end{equation*}$

Pressure on the Surface

Using Bernoulli’s equation, the pressure coefficient is given by

(121) $\begin{equation*} C_p = \frac{p - p_\infty}{\frac{1}{2} \, \varrho \, U^2} \end{equation*}$

so that

(122) $\begin{equation*} C_p = 1 - \frac{V^2}{U^2} \end{equation*}$

where the local velocity magnitude satisfies

(123) $\begin{equation*} V^2 = v_r^2 + v_\theta^2 \end{equation*}$

On the surface of the cylinder ( $r = R$ ), $v_r = 0$ , so

(124) $\begin{equation*} V^2 = v_\theta^2 \end{equation*}$

and therefore

(125) $\begin{equation*} C_p = 1 - \frac{v_\theta^2}{U^2} \end{equation*}$

Substituting the expression for $v_\theta$ gives

(126) $\begin{equation*} C_p = 1 - \frac{\left( -2U \sin \theta + \dfrac{\Gamma}{2\pi R} \right)^2}{U^2} = 1 - 4\sin^2\theta + \frac{4 \sin\theta \, \Gamma}{2\pi R U} - \left( \frac{\Gamma}{2\pi R U} \right)^2 \end{equation*}$

Figure 16 shows representative results for two circulation values. With the sign convention used here, upward lift on the cylinder corresponds to negative circulation, for which the speed increases and the pressure decreases over the upper surface. The two initially distinct stagnation points then approach each other on the lower surface and eventually merge. For larger circulation magnitudes, known as super-circulation, the stagnation points move away from the cylinder surface.

Pressure distribution around a lifting circular cylinder for increasing values of the circulation.

Force Components

The pressure distribution results in a force on the cylinder. Decomposing the force into $x$ – and $y$ -directions gives

(127) $\begin{equation*} F_x' = - \int_0^{2\pi} \left( \frac{1}{2} \, \varrho U^2 \, C_p \, R \cos\theta \right) d\theta, \quad F_y' = - \int_0^{2\pi} \left( \frac{1}{2} \, \varrho U^2 \, C_p \, R \sin\theta \right) d\theta \end{equation*}$

After integration, it is found that

(128) $\begin{equation*} F_x' = 0 = D' \end{equation*}$

because of the fore-and-aft symmetry of the pressure distribution, even in lifting flow. The result that the drag is zero in a potential flow is known as D’Alembert’s paradox. The lift force is given by

(129) $\begin{equation*} F_y' = -\varrho \, U \, \Gamma = L' \end{equation*}$

With the sign convention used here, positive circulation is counterclockwise. Therefore, positive upward lift corresponds to clockwise circulation, i.e., $\Gamma < 0$ . In magnitude,

(130) $\begin{equation*} |L'| = \varrho \, U \, |\Gamma| \end{equation*}$

which is the Kutta–Joukowsky theorem.

Note on D’Alembert’s Paradox

D’Alembert’s paradox highlights an apparent contradiction in fluid dynamics, specifically in an ideal, non-viscous, incompressible fluid with steady, irrotational flow: a solid body (such as a cylinder) experiences no drag force. This conclusion is at odds with everyday experience, where drag is consistently observed. This historical paradox applies to drag; lift may be zero for noncirculatory potential flow, but it can be nonzero when circulation is present. Whether this outcome is indeed a paradox depends on one’s perspective. From one viewpoint, the theoretical prediction of zero drag clashes with practical observations, making it seem paradoxical and providing fodder for mathematicians to contribute to ongoing, albeit somewhat scholarly, discussions. However, for those who understand the idealized assumptions of the potential flow model, the absence of drag aligns with the lack of mechanisms for energy dissipation, such as viscosity or turbulence, thereby rendering it a limitation of this idealized solution framework rather than a true paradox.

Method of Images

The method of solid boundary substitution, also known as the method of images, simplifies boundary-condition problems in potential flow by replacing solid boundaries with equivalent image sources, sinks, and vortices. It is based on the principle of superposition and Laplace’s equation. Examples include flows near flat walls (e.g., ground effect), symmetric flows in channels or pipes, and flows around airfoils or lifting surfaces in ground effect. This method ensures uniform enforcement of boundary conditions on walls and provides solutions for linear potential flows, subject to accompanying restrictions.

Source Flow Near a Solid Wall

Consider a point source of strength $Q$ at $(x_0, y_0)$ near an impermeable wall ( $y=0$ ), as shown in Figure 17. To ensure no flow normal to the wall ( $v = 0$ at $y = 0$ ), place an image source of equal strength $+Q$ at $(x_0, -y_0)$ . This ensures that the normal velocity is zero at $y = 0$ . In this case the total velocity potential $\phi(x, y)$ is

(131) $\begin{equation*} \phi(x, y) = \frac{Q}{2\pi} \ln \sqrt{(x - x_0)^2 + (y - y_0)^2} + \frac{Q}{2\pi} \ln \sqrt{(x - x_0)^2 + (y + y_0)^2} \end{equation*}$

from which the velocity field can be found using differentiation.

Two sources separated by a finite distance will give a dividing streamline that can be replaced by a solid wall.

Source Flow in a Corner

Corner flow requires multiple image sources to enforce boundary conditions. For example, a corner flow problem involving a point source requires placing three image sources to satisfy the no-penetration conditions at both walls, as shown in Figure 18. If a point source of strength $Q$ is placed at $(x_0,y_0)$ in a right-angle corner formed by the walls $x=0$ and $y=0$ , then three image sources of the same strength are placed at $(-x_0,y_0)$ , $(x_0,-y_0)$ , and $(-x_0,-y_0)$ . This arrangement makes both walls streamlines and satisfies the no-penetration boundary condition on each wall.

The flow from a source in a corner is simulated using three image sources to create an orthogonal pair of dividing streamlines.

Vortex Flow Near a Solid Wall

The stream function for a vortex of signed circulation $\Gamma_v$ is

(132) $\begin{equation*} \psi = -\frac{\Gamma_v}{2\pi} \ln r \end{equation*}$

where $r$ is the radial distance from the vortex. Consider a real vortex of circulation $-\Gamma$ located at $(0,h)$ , where $\Gamma$ is its positive magnitude. To make the wall at $y=0$ a streamline, an image vortex of circulation $+\Gamma$ is placed at $(0,-h)$ .

The combined stream function for the real vortex and the image vortex at a point ${P(x, y)}$ is

(133) $\begin{equation*} \psi = \frac{\Gamma}{2\pi} \ln r_1 - \frac{\Gamma}{2\pi} \ln r_2 \end{equation*}$

where

(134) $\begin{equation*} r_1 = \sqrt{x^2 + (y - h)^2} \end{equation*}$

and

(135) $\begin{equation*} r_2 = \sqrt{x^2 + (y + h)^2} \end{equation*}$

Substituting $r_1$ and $r_2$ , the combined stream function becomes

(136) $\begin{equation*} \psi = \frac{\Gamma}{2\pi} \left[ \ln \sqrt{x^2 + (y - h)^2} - \ln \sqrt{x^2 + (y + h)^2} \right] \end{equation*}$

The velocity component in the $x$ -direction induced by a vortex of signed circulation $\Gamma_v$ located at $(0,y_0)$ is

(137) $\begin{equation*} u = -\frac{\Gamma_v}{2\pi} \frac{y-y_0}{x^2+(y-y_0)^2} \end{equation*}$

For the real vortex of circulation $-\Gamma$ located at $(0,h)$ ,

(138) $\begin{equation*} u_{\text{real}} = \frac{\Gamma}{2\pi} \frac{y-h}{x^2+(y-h)^2} \end{equation*}$

and for the image vortex of circulation $+\Gamma$ located at $(0,-h)$ ,

(139) $\begin{equation*} u_{\text{image}} = -\frac{\Gamma}{2\pi} \frac{y+h}{x^2+(y+h)^2} \end{equation*}$

The total velocity in the $x$ -direction is

(140) $\begin{equation*} u = u_{\text{real}} + u_{\text{image}} = \frac{\Gamma}{2\pi} \left[ \frac{y-h}{x^2+(y-h)^2} - \frac{y+h}{x^2+(y+h)^2} \right] \end{equation*}$

At the wall where $y=0$ ,

(141) $\begin{equation*} u = \frac{\Gamma}{2\pi} \left[ \frac{-h}{x^2+h^2} - \frac{h}{x^2+h^2} \right] = -\frac{\Gamma}{\pi} \frac{h}{x^2+h^2} \end{equation*}$

Check Your Understanding #3 – Using the method of images

A stationary vortex with negative circulation is located at a height ${h}$ above a solid horizontal wall. 1. Find an expression for the stream function for this flow and sketch the resulting streamlines. 2. Find an expression for the flow velocity induced along the wall.

Show solution/hide solution.

This problem is easily solved using the method of solid boundary substitution, also known as the “method of images.” The first step is to define a convenient coordinate system. In this case, there is a horizontal wall that can be assumed to lie along the ${x}$ axis. The actual or “real” vortex can then be placed at height ${h}$ along the ${y}$ axis. The ${x}$ axis (wall) needs to be a streamline to the flow, so the “image” vortex must be placed below the ${x}$ axis, as shown in the figure below. The real vortex has negative strength (or circulation), so for the image vortex, it must have the correct sign, i.e., $\Gamma$ for the image vortex will be positive.

The stream function for an individual vortex of positive strength (written in polar coordinates) is
$\psi = -\frac{\Gamma}{2\pi} \ln r$

where $r$ is the radial distance. To get the combined stream function for the real vortex and the image vortex, the radial distances of each of the vortices from the arbitrary point P must be determined. Therefore, for the combined flow

$\psi = \frac{\Gamma}{2\pi} \ln r_1 - \frac{\Gamma}{2\pi} \ln r_2$

where $r_1$ and $r_2$ are the radial distances from the real and image vortices, respectively. The solution is incomplete because the combined flow depends on calculating these distances $r_1$ and $r_2$ in terms of ${x}$ and ${y}$ . For the real vortex, then

$r_1 = \sqrt{x^2 + (y-h)^2}$

and for the image vortex, then

$r_2 = \sqrt{x^2 + (y+h)^2}$

Using these distances and remembering the signs of the circulation on the image pair, the combined stream function for the flow is

$\psi = \frac{\Gamma}{2\pi} \ln \sqrt{x^2 + (y-h)^2} - \frac{\Gamma}{2\pi} \ln \sqrt{x^2 + (y+h)^2}$

or

$\psi = \frac{\Gamma}{2\pi} \left( \ln \sqrt{x^2 + (y-h)^2} - \ln \sqrt{x^2 + (y+h)^2} \right)$
The velocity field can be obtained from the stream function by differentiation. Still, the velocity components can also be added in potential flows, which is usually more convenient than differentiating a stream function for which the individual elementary velocity components are already known. Remember that the tangential component $V_{\theta}$ of the induced velocity from a vortex of positive counterclockwise circulation is
$V_{\theta} = \frac{\Gamma}{2 \pi r}$

where $r$ is the distance to the arbitrary point P. Notice that a potential vortex does not induce a $V_r$ component, so the $V_{\theta}$ component can be resolved in any direction. In the ${x}$ direction, i.e., parallel to the wall, the $\sin \theta$ part is used so that

$u = -\frac{\Gamma}{2 \pi r} \sin \theta$

For a Cartesian to polar conversion $\sin \theta = y/r$ , so that

$u = -\frac{\Gamma}{2 \pi r} \left( \frac{y}{r} \right) = -\frac{\Gamma y}{2 \pi r^2}$

Therefore, for the real vortex of circulation $-\Gamma$ at $(0,h)$ and the image vortex of circulation $+\Gamma$ at $(0,-h)$ , the total velocity in the ${x}$ direction is

$u = \frac{\Gamma (y-h)}{2 \pi \left[x^2+(y-h)^2\right]} - \frac{\Gamma (y+h)}{2 \pi \left[x^2+(y+h)^2\right]}$

or

$u = \frac{\Gamma}{2 \pi} \left[ \frac{y-h}{x^2+(y-h)^2} - \frac{y+h}{x^2+(y+h)^2} \right]$

By setting ${y}=0$ , the velocity induced along the wall is

$u = \frac{\Gamma}{2 \pi} \left[ \frac{-h}{x^2+h^2} - \frac{h}{x^2+h^2} \right] = -\frac{\Gamma}{\pi} \frac{h}{x^2+h^2}$

This result clearly shows that the peak velocity on the wall is directly below the vortex and drops quickly on either side as one moves along the wall.

Three-Dimensional Potential Flow Around a Sphere

A uniform flow in the ${x}$ -direction has the velocity potential

(142) $\begin{equation*} \phi_{\text{uniform}} = U \, x \end{equation*}$

where $U$ is the freestream velocity, and $x = r \cos\theta$ in axisymmetric spherical coordinates.

The velocity potential of a doublet aligned along the ${x}$ -axis in spherical coordinates is

(143) $\begin{equation*} \phi_{\text{doublet}} = -\frac{\kappa \cos\theta}{4\pi r^2} \end{equation*}$

where $\kappa$ is the signed doublet strength, $r = \sqrt{x^2 + y^2 + z^2}$ is the radial distance, and $\cos\theta = x / r$ , noting that the flow is axisymmetric about the ${x}$ -axis.

The total velocity potential is the sum of the uniform flow and doublet potentials, i.e.,

(144) $\begin{equation*} \phi_{\text{total}} = \phi_{\text{uniform}} + \phi_{\text{doublet}} = Ux - \frac{\kappa \cos\theta}{4\pi r^2} \end{equation*}$

The no-penetration condition at the surface of a sphere of radius $R$ requires

(145) $\begin{equation*} \kappa = -2\pi U R^3 \end{equation*}$

Therefore, substituting $x = r \cos\theta$ gives

(146) $\begin{equation*} \phi_{\text{total}} = U \, r \cos\theta + \frac{U R^3 \cos\theta}{2 r^2} \end{equation*}$

The velocity components are obtained by taking derivatives of $\phi_{\text{total}}$ , i.e.,

(147) $\begin{equation*} V_r = \frac{\partial \phi_{\text{total}}}{\partial r} = U\cos\theta \left(1 - \frac{R^3}{r^3}\right) \end{equation*}$

and

(148) $\begin{equation*} V_\theta = \frac{1}{r} \frac{\partial \phi_{\text{total}}}{\partial \theta} = -U\sin\theta \left(1 + \frac{R^3}{2r^3}\right) \quad \text{and} \quad V_\phi = 0 \end{equation*}$

because the flow is axisymmetric about the ${x}$ -axis.

The flow tangency (no-penetration) condition at the sphere’s surface at $r = R$ ensures that $V_r = 0$ at $r = R$ .
This latter result is automatically satisfied because

(149) $\begin{equation*} V_r = U\cos\theta \left(1 - \frac{R^3}{R^3}\right) = 0 \end{equation*}$

The corresponding tangential component is

(150) $\begin{equation*} V_\theta = -U\sin\theta \left(1 + \frac{R^3}{2r^3}\right) \end{equation*}$

which for $r = R$ gives

(151) $\begin{equation*} V_\theta = -\frac{3}{2} U\sin\theta \end{equation*}$

Therefore, the pressure coefficient on the surface of the sphere is

(152) $\begin{equation*} C_p = 1 - \frac{v_\theta^2}{U^2} = 1 - \dfrac{9}{4} \sin^2 \theta \end{equation*}$

The flow around a sphere exhibits spherical axisymmetry, meaning the flow field remains invariant about the ${x}$ -axis. The maximum velocity occurs at the sphere’s equator ( $\theta$ = $\pm \pi/2$ ), is 1.5 ${V_{\infty}}$ , with a minimum pressure coefficient of $-5/4$ . In contrast, the flow around a cylinder is two-dimensional, being symmetric only in the plane perpendicular to its axis, with a maximum velocity of $2V_\infty$ and a minimum pressure coefficient of $-3$ . This difference arises from the three-dimensional “relieving effect” in that, in addition to moving over and under the sphere, the flow can also move spanwise. As shown in Figure 19, this extra “freedom” in the flow’s motion reduces the velocity and pressure gradients on the sphere compared to those on the cylinder.

The difference in the pressure distribution between the flow over a circular cylinder and a sphere.

Complex Variable Methods in Potential Flow

Complex variable theory provides a powerful yet elegant framework for solving two-dimensional potential flow problems. A key idea is that instead of working with separate ${x}$ – and ${y}$ -coordinates, these can be combined into a single complex number, i.e.,

(153) $\begin{equation*} z = x + i \, y = r \,\cos \theta + i \, r \, \sin \theta = r \left( \cos \theta + i \, \sin \theta \right) = r \, e^{i \, \theta} \end{equation*}$

where $i$ is the imaginary unit. Notice that

(154) $\begin{equation*} r = \sqrt{x^2 + y^2}, \quad \text{where} \quad \theta = \tan^{-1} \left( \frac{y}{x} \right) \end{equation*}$

This formulation enables both the geometry and the flow to be described as functions of a single complex variable. Notice also that

(155) $\begin{equation*} z^n = r^n \left( \cos n \theta + i \, \sin n \theta \right) \quad \text{and} \quad z^{-n} = r^{-n} \left( \cos n \theta - i \, \sin n \theta \right) \end{equation*}$

In potential flow theory, the flow is represented by a complex potential function given by

(156) $\begin{equation*} w(z) = \phi(x, y) + i \, \psi(x, y) \end{equation*}$

where $\phi$ is the velocity potential and $\psi$ is the stream function. These two quantities describe the fluid motion: streamlines are contours of constant $\psi$ , and equipotential lines are contours of constant $\phi$ .

The complex function $w(z)$ is assumed to be analytic, meaning it has a well-defined complex derivative with respect to ${z}$ . The condition for analyticity is that the real and imaginary parts of $w(z)$ , namely $\phi$ and $\psi$ , satisfy the Cauchy-Riemann equations given previously in Eq. 34. Satisfying these equations ensures that the flow is both irrotational and incompressible. The gradient of $\phi$ gives the velocity vector, and the stream function $\psi$ is orthogonal to it at every point. Hence, the streamlines and equipotential lines are always perpendicular, a fundamental property of potential flows that has already been explained.

Velocity Field

The velocity vector is $V(z) = u + i v$ , so differentiating the complex potential with respect to ${z}$ gives

(157) $\begin{equation*} \frac{dw}{dz} = \frac{\partial \phi}{\partial x} + i \, \frac{\partial \psi}{\partial x} \end{equation*}$

Because $\phi$ and $\psi$ satisfy the Cauchy-Riemann conditions, where

(158) $\begin{equation*} \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} = u \quad \text{and} \quad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x} = v \end{equation*}$

then $\partial \psi / \partial x = -v$ , and it follows that

(159) $\begin{equation*} \frac{dw}{dz} = u - i v \end{equation*}$

which is the complex conjugate of the velocity vector. This expression is central to complex potential theory, as it allows the velocity field to be obtained directly from the derivative of the complex potential.

Satisfying Laplace’s Equation

Because $w(z) = \phi(x, y) + i \psi(x, y)$ is analytic, differentiate the first Cauchy-Riemann equation with respect to $x$ to obtain

(160) $\begin{equation*} \frac{\partial^2 \phi}{\partial x^2} = \frac{\partial^2 \psi}{\partial x \, \partial y} \end{equation*}$

and differentiate the second Cauchy-Riemann equation with respect to $y$ to obtain

(161) $\begin{equation*} \frac{\partial^2 \phi}{\partial y^2} = -\frac{\partial^2 \psi}{\partial y \, \partial x} \end{equation*}$

Adding the latter two results gives

(162) $\begin{equation*} \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = \frac{\partial^2 \psi}{\partial x \, \partial y} - \frac{\partial^2 \psi}{\partial y \, \partial x} = 0 \end{equation*}$

because mixed partials are equal. Therefore,

(163) $\begin{equation*} \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \end{equation*}$

which is Laplace’s equation. A similar argument shows that $\psi$ is harmonic, i.e.,

(164) $\begin{equation*} \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = 0 \end{equation*}$

These results confirm that complex potential theory is fully consistent with the requirements of two-dimensional potential flow.

Because both the $\psi$ and $\phi$ functions satisfy Laplace’s equation, their combination into a single function $w(z)$ inherits the powerful tools of complex analysis, i.e.,

(165) $\begin{equation*} w = w_1 + w_2 + w_3 + \cdots + w_n \end{equation*}$

which is the principle of superposition because

(166) $\begin{equation*} \nabla^2 w = \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} = 0 \end{equation*}$

This complex-variable framework simplifies the construction and manipulation of potential-flow solutions. Simple flow patterns, such as uniform flow, point sources and sinks, doublets (dipoles), and vortices, can now be represented by a corresponding complex velocity potential. These building blocks can be combined through superposition to form more complex flows, as already explained earlier in this chapter.

Another significant advantage of the complex-variable approach is that it provides a direct method for computing the forces and moments acting on bodies from the flow field. These concepts form the theoretical foundation of classical airfoil theory, illustrating the intrinsic relationship between circulation and lift using the Blasius theorem. While the use of complex variables may initially feel unfamiliar to many students, the structure they provide leads to significant analytical clarity and efficiency.

Working with Complex Potentials

The use of complex potentials in potential flow theory is best understood by examining a few examples. Consider a uniform flow of speed $U$ in the $x$ -direction. The complex potential for this flow is

(167) $\begin{equation*} w(z) = U \, z \end{equation*}$

and the complex velocity is

(168) $\begin{equation*} { \frac{dw}{dz} = U } \end{equation*}$

which is constant and purely real. Now, introduce a point source of strength $Q$ at the origin. Its complex potential is

(169) $\begin{equation*} w(z) = \frac{Q}{2\pi} \, \ln z \end{equation*}$

and the corresponding velocity is

(170) $\begin{equation*} \frac{dw}{dz} = \frac{Q}{2\pi \, z} \end{equation*}$

These two flows can be superimposed to give

(171) $\begin{equation*} { w(z) = U \, z + \frac{Q}{2\pi} \, \ln z } \end{equation*}$

The resulting flow describes uniform flow disturbed by a point source, known as a Rankine half-body, which has previously been examined in some detail. The flow field can be visualized as discussed previously by examining the streamlines, which correspond to lines of constant $\psi$ .

Table of Complex Potentials for Elementary Flows

A handy table of the complex potentials for the complete set of standard elementary flows is given below.

Complex potentials for elementary potential flows.
Flow Type	Complex Potential $w(z)$	Velocity $\dfrac{dw}{dz}$
Uniform flow at angle $\alpha$ to the $x$ -axis	$U \, z \, e^{-i \alpha}$	$U \, e^{-i \alpha}$
Source of strength $Q$ at the origin	$\dfrac{Q}{2\pi} \log z$	$\dfrac{Q}{2\pi \, z}$
Sink of strength $-Q$ at the origin	$-\dfrac{Q}{2\pi} \log z$	$-\dfrac{Q}{2\pi \, z}$
Vortex of strength $\Gamma$ at the origin	$-\dfrac{i \,\Gamma}{2\pi} \log \,z$	$-\dfrac{i \Gamma}{2\pi \, z}$
Doublet of strength $\kappa$ at the origin	$\dfrac{\kappa}{2\pi z}$	$-\dfrac{\kappa}{2\pi \, z^2}$

Check Your Understanding #4 – Complex potential for a circular cylinder in a uniform flow

Construct the complex potential for two-dimensional potential flow around a stationary circular cylinder of radius $R$ , centered at the origin, placed in a uniform stream of velocity $U$ directed along the $x$ -axis. Identify the locations of the stagnation points and verify that the cylinder surface corresponds to a streamline. Derive an expression for the stream function in polar coordinates and confirm that the streamline $\psi = 0$ corresponds to the cylinder boundary.

Show solution/hide solution.

The flow can be modeled as a superposition of a uniform flow and a doublet located at the origin. A uniform flow of speed $U$ in the $x$ -direction has the complex potential

$w_{\text{uniform}}(z) = Uz$

A doublet of strength $\kappa$ at the origin has the complex potential

$w_{\text{doublet}}(z) = \frac{\kappa}{2\pi z}$

By linear superposition, the total complex potential is

$w(z) = Uz + \frac{\kappa}{2\pi z}$

To determine the appropriate doublet strength $\kappa$ , consider the location of stagnation points, where the complex velocity vanishes, i.e.,

$\frac{dw}{dz} = U - \frac{\kappa}{2\pi z^2}$

Setting $dw/dz = 0$ gives

$z^2 = \frac{\kappa}{2\pi U} \quad \text{so that} \quad z = \pm \sqrt{\frac{\kappa}{2\pi U}}$

These stagnation points define the outermost streamline enclosing the cylinder. Define the cylinder radius as

$R = \sqrt{\frac{\kappa}{2\pi U}} \quad \text{so that} \quad \kappa = 2\pi U R^2$

Substituting this into the complex potential gives the final expression, i.e.,

$w(z) = Uz + \frac{UR^2}{z}$

The complex velocity field is obtained by differentiation, i.e.,

$V(z) = \frac{dw}{dz} = U - \frac{UR^2}{z^2}$

Setting $V(z)$ = 0 again confirms the stagnation points at $z = \pm R$ , which lie on the surface of the cylinder.

To find the streamlines, use polar coordinates $z = r e^{i\theta}$ so that

$w(z) = U\left( re^{i\theta} + \frac{R^2}{r}e^{-i\theta} \right)$

Taking the imaginary part gives the stream function as

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

The streamline defined by $\psi$ = 0 occurs either when $\sin\theta = 0$ , which corresponds to the axis of symmetry, or when $r = R$ , which defines the surface of the cylinder. This result confirms that the cylinder surface coincides with a streamline and that the potential flow model satisfies the flow-tangency boundary condition.

Conformal Transformations

Before studying conformal transformations, it is advisable to revise the principles of complex mathematics. Within the framework of two-dimensional potential flows in complex variables, conformal transformations (or mappings) are complex analytic functions that locally preserve angles. ^[5] enabling the transfer of closed-form solutions for canonical-shaped boundaries, while rigorously satisfying Laplace’s equation and the non-penetration boundary condition. The mathematical foundations of conformal transformations were laid down in the nineteenth century by Carl Gauss and further developed by William Thomson (Lord Kelvin), who applied these transformation techniques to generate new potential flows from known ones. Bernhard Riemann subsequently completed the theoretical framework that underpins the modern aerodynamic applications of conformal transformations.

A conformal mapping is an analytic transformation, say $z = f(\zeta)$ , which transforms each point given by $\zeta = \xi + i\,\eta$ in one complex plane to new points $z = x + i\,y$ in another plane, while preserving the local angle between intersecting functions and curves. Consequently, streamlines and equipotential lines, which are orthogonal in a potential flow field, remain orthogonal after the transformation, although their shapes will generally deform, as shown in Figure 20.

An example of a conformal mapping, which preserves angles but not values.

Because conformal mappings preserve harmonicity, any velocity potential or stream function satisfying Laplace’s equation in the $\zeta$ -plane also satisfies it in the ${z}$ -plane. This property allows the construction of an aerodynamic flow solution in two steps: first, solve the potential flow around a known geometry in the $\zeta$ -plane; second, apply an appropriate conformal transformation $z=f(\zeta)$ to map that solution onto a boundary in the ${z}$ -plane, thereby obtaining the new potential flow field.

The complex potential in the $\zeta$ -plane is given by

(172) $\begin{equation*} \tilde w(\zeta)=\phi(\xi,\eta)+i\,\psi(\xi,\eta) \end{equation*}$

where

(173) $\begin{equation*} \phi(\xi,\eta)=\Re\{\tilde w(\zeta)\} \quad\text{and}\quad \psi(\xi,\eta)=\Im\{\tilde w(\zeta)\} \end{equation*}$

After mapping $z=f(\zeta)$ , the complex potential in the physical plane becomes

(174) $\begin{equation*} w(z)=\phi(x,y)+i\,\psi(x,y) \end{equation*}$

with $x+i\,y=z$ . The velocity components then follow from

(175) $\begin{equation*} \frac{dw}{dz} =\frac{\dfrac{d\tilde w}{d\zeta}}{\dfrac{dz}{d\zeta}} = u - i\,v \end{equation*}$

Because $\tilde w$ is analytic in the $\zeta$ -plane, its real and imaginary parts satisfy the Cauchy–Riemann relations, i.e.,

(176) $\begin{equation*} \frac{\partial \phi}{\partial \xi} =\frac{\partial \psi}{\partial \eta} \quad\text{and}\quad \frac{\partial \phi}{\partial \eta} =-\frac{\partial \psi}{\partial \xi} \end{equation*}$

so that both $\phi$ and $\psi$ are harmonic and satisfy the Laplace equation, i.e.,

(177) $\begin{equation*} \nabla^2\phi=\phi_{\xi\xi}+\phi_{\eta\eta}=0 \quad \text{and} \quad \nabla^2\psi=\psi_{\xi\xi}+\psi_{\eta\eta}=0 \end{equation*}$

Basic Conformal Transformations in Aerodynamics

In two-dimensional potential flow theory, conformal transformations map simple geometric domains, such as circles, into more complex shapes, such as airfoils. These transformations preserve angles locally and ensure that the governing Laplace equation remains valid under the mapping. Several elementary conformal transformations form the foundation for constructing more complex aerodynamic mappings. The most important of these are translation, rotation, scaling, and inversion.

A translation moves every point in the flow field by a fixed vector

(178) $\begin{equation*} z = \zeta + \zeta_0 \end{equation*}$

where $\zeta_0$ is a complex constant. This operation shifts the geometry and the entire flow field without altering its structure or properties. A rotation about the origin is given by

(179) $\begin{equation*} z = \zeta \, e^{i\theta} \end{equation*}$

which rotates the geometry counterclockwise by an angle $\theta$ . This transformation preserves distances and angles and is commonly used to orient flow fields. Uniform stretching or shrinking of the domain is achieved by

(180) $\begin{equation*} z = a \, \zeta \ , \quad \text{with} \quad a > 0 \end{equation*}$

This transformation scales all lengths by a factor $a$ , modifying the size of the flow domain while preserving shape. The velocity field scales inversely with $a$ , affecting the dynamic pressure. Combined rotation and scaling can be combined into a single transformation, i.e.,

(181) $\begin{equation*} z = A \, \zeta \ , \quad\text{where} \quad A = a \,e^{i\theta} \end{equation*}$

which simultaneously rotates and stretches the geometry. An inversion transforms points near the origin to far away and vice versa, i.e.,

(182) $\begin{equation*} z = \frac{1}{\zeta} \end{equation*}$

This is a nonlinear transformation that maps points near the origin to points far away and vice versa, and it is singular at $\zeta = 0$ . It plays a key role in more advanced mappings such as the Joukowsky transformation. All of these transformations are summarized in Figure 21.

The most important conformal transformations or mappings are translation, rotation, scaling, and inversion, which can be generalized in terms of a Möbius transformation.

Finally, the most general linear-fractional conformal mapping is

(183) $\begin{equation*} z = \frac{a \, \zeta + b}{c \, \zeta + d} \ , \quad \text{with } \quad a \, d - b \, c \neq 0 \end{equation*}$

where ${a}$ , ${b}$ , ${c}$ and ${d}$ are any complex numbers. This is a class of transformations called Möbius transformations, which includes translation, rotation, scaling, and inversion as special cases. Möbius transformations are especially useful in constructing mappings between circular domains or for handling complex boundary conditions. These basic transformations can be composed to generate more elaborate transformations, such as the Joukowsky map. While the underlying mathematics can become intricate, many useful aerodynamic mappings rely only on simple combinations of these fundamental operations.

Flow in a Corner

Consider an example of a conformal transformation of a uniform flow using the mapping $z = \zeta^{1/n}$ . Under a conformal transformation, every analytic solution of Laplace’s equation in the $\zeta$ -plane can be carried over to the ${z}$ -plane simply by substituting $\zeta = z^n$ . The flow in a right-angled corner is obtained for $n = 2$ , which is treated as a special case.

In the computational (or “ $\zeta$ ”) plane, there is a uniform flow of velocity $U$ in the positive $\xi$ direction, as shown in Figure 22. In that plane

(184) $\begin{equation*} w(\zeta)=\phi(\xi,\eta)+i\,\psi(\xi,\eta)=U\,\zeta \end{equation*}$

so that

(185) $\begin{equation*} \phi(\xi,\eta)=U\,\xi \quad\text{and}\quad \psi(\xi,\eta)=U\,\eta \end{equation*}$

Notice that the equipotential lines $\phi=\mathrm{constant}$ are vertical lines $\xi=\mathrm{constant}$ , and the streamlines $\psi=\mathrm{constant}$ are horizontal lines $\eta=\mathrm{constant}$ .

A conformal mapping of a uniform flow with the transfer function with $n$ = 2, i.e., $z = \zeta^{1/2} = \sqrt{\zeta}$ . Notice that the conformal mapping maintains the orthogonal relationship between the streamlines and equipotential lines.

Mapping into the physical (or “ ${z}$ ”) plane using ${\zeta = z^2}$ gives

(186) $\begin{equation*} w(z)=\phi(x,y)+i\,\psi(x,y)=U\,z^2 \end{equation*}$

so that

(187) $\begin{equation*} \phi(x,y)=U\,(x^2 - y^2) \quad\text{and}\quad \psi(x,y)=2\,U\,x\,y \end{equation*}$

Notice that equipotential lines $\phi=\mathrm{constant}$ now become rectangular hyperbolae defined by $x^2 - y^2=\mathrm{constant}$ , while streamlines $\psi=\mathrm{constant}$ become rectangular hyperbolae with $x\,y=\mathrm{constant}$ ; angle preservation ensures they remain orthogonal after the mapping.

In general, substituting $z = \zeta^{1/n}$ (or $\zeta = z^n$ ) transforms a uniform flow in the $\zeta$ -plane into flow in a wedge in the ${z}$ -plane. To see this, let the complex potential in the $\zeta$ -plane be

(188) $\begin{equation*} w(\zeta) = U \, \zeta = U \, r^n \, e^{i \, n \, \theta} \end{equation*}$

so that

(189) $\begin{equation*} \phi = U \, r^n \, \cos(n \theta) \quad \text{and} \quad \psi = U \, r^n \, \sin(n \theta) \end{equation*}$

The streamlines $\psi = \text{constant}$ and equipotential lines $\phi = \text{constant}$ repeat every $2\pi/n$ radians. However, the uniform flow in the $\zeta$ -plane occupies a half-plane with an angular extent of $\pi$ . Therefore, under the transformation $z=\zeta^{1/n}$ , the corresponding wedge angle in the ${z}$ -plane is

(190) $\begin{equation*} \Theta = \frac{\pi}{n} \end{equation*}$

Different values of $n$ produce different wedge geometries. For example, $n=2$ gives

(191) $\begin{equation*} \Theta = \frac{\pi}{2} = 90^\circ \end{equation*}$

which is the right-angled corner shown in Figure 22. Likewise, $n=3$ gives a wedge angle of $60^\circ$ . Values of $n<1$ produce re-entrant wedges with angles greater than $180^\circ$ , although the appropriate branch of the transformation must then be selected carefully.

Transformation of a Circular Cylinder to an Airfoil

This specific conformal transformation is named after Nikolai Zhukovsky (Joukowsky), who generalized the conformal mapping technique^[6] to convert a suitably placed circle into an airfoil profile with a cusped trailing edge, now known as the classic Joukowsky airfoil.

In the $\zeta$ -plane, start with a circular cylinder of radius $R$ centered at

(192) $\begin{equation*} \zeta_0 = \xi_0 + i\,\eta_0 \end{equation*}$

The points on the circle are

(193) $\begin{equation*} \zeta(\theta) = \zeta_0 + R\, e^{i\theta},\quad \text{where} \quad 0 \le \theta < 2\pi \end{equation*}$

Recall that a conformal transformation maps points in the $\zeta$ $(\xi+i\,\eta)$ plane to points in the $z$ $(x+i\,y)$ plane. In this case, the Joukowsky conformal mapping into the $z$ -plane is

(194) $\begin{equation*} z = f(\zeta) = \zeta + \frac{a^2}{\zeta} \end{equation*}$

where $a$ is the mapping parameter and is not, in general, equal to the radius $R$ of the offset circle. Substituting $\zeta(\theta)$ gives

(195) $\begin{equation*} z(\theta) = \bigl(\zeta_0 + R\,e^{i\theta}\bigr) + \frac{a^2}{\,\zeta_0 + R\,e^{i\theta}} \end{equation*}$

Separating the real and imaginary parts gives

(196) $\begin{equation*} \begin{aligned} x(\theta) &= \xi_0 + R\cos\theta + \frac{a^2\,\bigl(\xi_0 + R\cos\theta\bigr)} {\bigl(\xi_0 + R\cos\theta\bigr)^2 + \bigl(\eta_0 + R\sin\theta\bigr)^2}\\[8pt] y(\theta) &= \eta_0 + R\sin\theta - \frac{a^2\,\bigl(\eta_0 + R\sin\theta\bigr)} {\bigl(\xi_0 + R\cos\theta\bigr)^2 + \bigl(\eta_0 + R\sin\theta\bigr)^2} \end{aligned} \end{equation*}$

For a non-lifting flow ( $\Gamma=0$ ) about the offset circle, the complex potential in the $\zeta$ -plane is

(197) $\begin{equation*} \widetilde{w}(\zeta) = U\left[ (\zeta-\zeta_0) + \frac{R^2}{\zeta-\zeta_0} \right] \end{equation*}$

where $R$ is the radius of the circle centered at $\zeta_0$ . Under the mapping $z=f(\zeta)$ , the corresponding complex potential in the physical $z$ -plane is

(198) $\begin{equation*} w(z) = \widetilde{w}\bigl(f^{-1}(z)\bigr) = U\left[ \zeta(z)-\zeta_0 + \frac{R^2}{\zeta(z)-\zeta_0} \right] \end{equation*}$

For a zero offset $\zeta_0$ = 0 with $\xi_0$ = 0 and $\eta_0$ = 0, the transformation produces a flat plate with zero thickness. Because the circle is described by $\zeta(\theta) = \zeta_0 + a\,e^{i\theta}$ , where $0 \le \theta < 2\pi$ , then under the transformation

(199) $\begin{equation*} z = \zeta + \frac{a^2}{\zeta} = a\,e^{i\theta} + \dfrac{a^2}{a\,e^{i\theta}} = a \left(e^{i\theta} + e^{-i\theta} \right) = 2 a \cos \theta \end{equation*}$

which is real and represents a horizontal flat line in the ${z}$ plane. Notice that for the zero-offset circle, the transformation gives a flat plate extending from $x=-2a$ to $x=2a$ , so its chord is $c = 4a$ . For an offset circle, however, the chord is not generally equal to $4a$ ; it must be determined from the mapped leading- and trailing-edge locations.

However, by choosing the offset $\zeta_0 = \xi_0 + i\,\eta_0$ with $\lvert\zeta_0\rvert<a$ , the camber and trailing‐edge shape of the resulting Joukowsky airfoil can be controlled, all while preserving the analytic potential flow solution using a single substitution. As shown in Figure 23, a $\xi_0$ -offset introduces thickness into the mapped airfoil profile without altering its symmetrical shape about the chord line.

Joukowsky transform of the flow around a non-lifting circular cylinder, resulting in a non-cambered airfoil shape with a cusped trailing edge. The center of the cylinder is offset by $-\xi_0$ along the $\xi$ -axis.

A combined ${\xi}$ -axis and ${\eta}$ -axis offset, i.e., $(-\xi_0, -\eta_0)$ , skews the airfoil contour and streamlines, creating both camber and thickness asymmetry, as shown in Figure 24. The stagnation points shift chordwise and/or vertically, demonstrating the complete control of the airfoil shape and its flow that results from displacements $\zeta_0 = \xi_0 + i\,\eta_0$ of the cylinder center in the $\zeta$ -plane.

Joukowsky transform of the flow around a non-lifting circular cylinder, resulting in a cambered airfoil shape with a cusped trailing edge. The center of the cylinder is offset by $\zeta_0 = \xi_0 + i\,\eta_0$ in the $\zeta$ -plane.

Joukowsky transform of the flow around a non-lifting circular cylinder, resulting in a cambered airfoil shape with a cusped trailing edge. The center of the cylinder is offset by $\zeta_0 = \xi_0 + i\,\eta_0$ in the $\zeta$ -plane.

The local complex velocity on the airfoil surface is

(200) $\begin{equation*} \frac{dw}{dz} = \frac{\displaystyle U\left(1-\frac{R^2}{(\zeta(\theta)-\zeta_0)^2}\right)} {\displaystyle 1-\frac{a^2}{\zeta(\theta)^2}} \end{equation*}$

Therefore, the tangential velocity is

(201) $\begin{equation*} V(\theta) = \left|\frac{dw}{dz}\right| = \frac{\displaystyle U\left|1-\frac{R^2}{(\zeta(\theta)-\zeta_0)^2}\right|} {\displaystyle \left|1-\frac{a^2}{\zeta(\theta)^2}\right|} \end{equation*}$

and the pressure coefficient follows as

(202) $\begin{equation*} C_p(\theta) = 1-\left(\frac{V(\theta)}{U}\right)^2 = 1- \frac{\displaystyle \left|1-\frac{R^2}{(\zeta(\theta)-\zeta_0)^2}\right|^2} {\displaystyle \left|1-\frac{a^2}{\zeta(\theta)^2}\right|^2} \end{equation*}$

Figure 25 shows the nominally non-lifting pressure coefficient distribution ${C_p}$ versus chordwise position $x/c$ for the upper and lower surfaces of a Joukowsky airfoil with offsets $\xi_0 = -0.1$ , $\eta_0 = 0.05$ , and $a = 1$ . A slight leading-edge suction peak occurs on the lower surface immediately downstream of the leading edge. Moving aft, the suction gradually weakens, with ${C_p}$ rising toward zero, while the upper surface experiences milder suction and remains closer to ambient pressure. At the trailing edge, the upper- and lower-surface values of ${C_p}$ become equal at $x/c=1$ , reflecting the zero pressure difference at the cusp. This separation of the upper- and lower-surface pressure distributions shows how thickness, from the $\xi_0$ offset, and camber or asymmetry, from the $\eta_0$ offset, produce a nonuniform pressure distribution even when the prescribed circulation is zero. The lifting cases are considered in the next chapter.

Non-lifting pressure coefficient distribution ${C_p}$ versus chordwise position $x/c$ for a Joukowsky airfoil with offsets $\xi_0 = -0.1$ and ${\eta_0 = 0.05}$ .

Forces & Moments Using Complex Potentials

Now consider the lifting cases, which require accounting for circulation. In two-dimensional potential flow, the velocity field around a body can be expressed using a complex potential $w(z) = \phi + i\psi$ , where $\phi$ is the velocity potential and $\psi$ is the stream function. The complex conjugate velocity is given by

(203) $\begin{equation*} \overline{V}(z) = \frac{d w}{dz} \end{equation*}$

This formulation allows the total force on a closed contour enclosing the body to be evaluated using complex-variable methods. The two key results that are central to the process are Blasius’ theorem, which gives the force and moment directly in terms of the complex velocity, and Cauchy’s integral theorem, which supports the derivation and application.

Blasius Theorem

Let $C$ be a positively oriented, counterclockwise contour around a closed body in two-dimensional incompressible, irrotational, inviscid flow, as shown in Figure 26. The complex force exerted by the fluid on the body is defined as

(204) $\begin{equation*} F_x - i F_y = \frac{i \, \varrho}{2} \oint_{\rm C} \left(\frac{dw}{dz}\right)^2 dz \end{equation*}$

where $F_x$ and $F_y$ are the force components in the positive $x$ – and $y$ -directions, respectively. Here, $\dfrac{dw}{dz}=u-iv$ is the complex velocity, and $dz$ is the differential along the contour in the complex plane. Drag and lift are subsequently obtained by resolving $F_x$ and $F_y$ parallel and perpendicular to the freestream direction.

The Blasius theorem eliminates the need for pressure integration to determine the forces on a body in potential flow.

To understand its origin, recall that the force on the body arises solely from the pressure acting normally to the surface. For a smooth surface in two dimensions, the force components are

(205) $\begin{equation*} F_x = -\oint_{\rm C} p \, n_x \, ds \quad \text{and} \quad F_y = -\oint_{\rm C} p \, n_y \, ds \end{equation*}$

For a counterclockwise contour around the body,

(206) $\begin{equation*} n_x \, ds = dy \quad \text{and} \quad n_y \, ds = -dx \end{equation*}$

so that

(207) $\begin{equation*} F_x = -\oint_{\rm C} p \, dy \quad \text{and} \quad F_y = \oint_{\rm C} p \, dx \end{equation*}$

Therefore, the complex force is

(208) $\begin{equation*} F_x - i F_y = -\oint_{\rm C} p \left(dy+i\,dx\right) = -i\oint_{\rm C} p \, d\overline{z} \end{equation*}$

where

(209) $\begin{equation*} d\overline{z}=dx-i\,dy \end{equation*}$

Bernoulli’s equation for steady, incompressible, irrotational, inviscid flow is

(210) $\begin{equation*} p + \frac{1}{2}\varrho\left(u^2+v^2\right)=p_T \end{equation*}$

where $p_T$ is constant. Because

(211) $\begin{equation*} \frac{dw}{dz}=u-iv \end{equation*}$

then

(212) $\begin{equation*} u^2+v^2 = \frac{dw}{dz} \overline{\left(\frac{dw}{dz}\right)} \end{equation*}$

Substituting Bernoulli’s equation into the complex-force expression and noting that the constant total-pressure term integrates to zero gives

(213) $\begin{equation*} F_x-iF_y = \frac{i\varrho}{2} \oint_{\rm C} \frac{dw}{dz} \overline{\left(\frac{dw}{dz}\right)} \,d\overline{z} \end{equation*}$

The body surface is a streamline, so the no-penetration condition requires

(214) $\begin{equation*} \frac{dw}{dz}\,dz = \overline{\left(\frac{dw}{dz}\right)} \,d\overline{z} \end{equation*}$

Therefore,

(215) $\begin{equation*} \frac{dw}{dz} \overline{\left(\frac{dw}{dz}\right)} \,d\overline{z} = \left(\frac{dw}{dz}\right)^2dz \end{equation*}$

and the Blasius force theorem becomes

(216) $\begin{equation*} F_x-iF_y = \frac{i\varrho}{2} \oint_{\rm C} \left(\frac{dw}{dz}\right)^2dz \end{equation*}$

The corresponding moment about the origin (positive clockwise) is similarly given by

(217) $\begin{equation*} M = \frac{\varrho}{2} \oint_{\rm C} z \left( \frac{d w}{dz} \right)^2 dz \end{equation*}$

These expressions apply to flows composed of elementary singularities, such as sources, vortices, and doublets, to be superimposed analytically.

Cauchy’s Integral Theorem

If $f(z)$ is analytic inside and on a simple closed contour C, then by Cauchy’s theorem,

(218) $\begin{equation*} \oint_{\rm C} f(z)\, dz = 0 \end{equation*}$

Consider the integrand $f(z) = \dfrac{1}{z^n}$ and let the contour C represent the boundary of a body in the complex plane. This curve C can now be enclosed by a larger, positively oriented circle of radius $R$ centered at the origin, as shown in Figure 27.

Cauchy’s integral theorem shows how the nonzero value of the integral arises exclusively from the singularity structure of the integrand within the contour.

If $n = 1$ , then the integral over any contour enclosing the origin evaluates to

(219) $\begin{equation*} \oint_{\rm C} \frac{1}{z} \, dz = 2\pi \, i \end{equation*}$

because $z = 0$ is a simple pole enclosed by the contour. This reflects the residue of the singularity at the origin. For all other integer values $n \neq 1$ , the contour integral is zero, i.e.,

(220) $\begin{equation*} \oint_{\rm C} \frac{1}{z^n} \, dz = 0 \quad \text{for } n \neq 1 \end{equation*}$

However, for positive integers $n > 1$ , the function $1/z^n$ is still singular at $z=0$ ; the integral is zero because the residue of a higher-order pole is zero, not because the function is analytic there.

This example illustrates how the nonzero value of the integral arises exclusively from the singularity structure of the integrand within the contour. In aerodynamic applications, such singularities correspond physically to elements like point vortices or sources within the flow domain. The enclosing circular contour of radius $R$ provides a convenient analytic boundary for invoking Cauchy’s theorem or evaluating residues, while the body contour C may lie anywhere inside this outer domain.

These results are beneficial when combined with conformal mapping techniques. By constructing the potential flow in a simpler geometry and mapping it to a physical airfoil shape, the Blasius and Cauchy theorems enable direct calculation of aerodynamic forces from the transformed velocity field, without requiring pressure integration over the physical surface. This framework underpins classic airfoil theory, including the conformal transformation method and thin airfoil theory, which are considered in the next chapter.

Lift on Joukowsky Airfoils using Blasius’ Theorem

Consider again a Joukowsky airfoil generated by the mapping of a circular cylinder (or circle) in the $\zeta$ -plane to the ${z}$ -plane using the transformation

(221) $\begin{equation*} { z = \zeta + \frac{a^2}{\zeta} } \end{equation*}$

The mapping parameter $a$ is not, in general, the radius of the offset circle. If the circle center is $\zeta_0=\xi_0+i\eta_0$ , then the circle radius must be chosen so that the circle passes through the critical point of the Joukowsky transformation, usually $\zeta=a$ , giving

(222) $\begin{equation*} R = \left| a - \zeta_0 \right| \end{equation*}$

for a cusped Joukowsky airfoil. The flow in the $\zeta$ -plane consists of a uniform freestream $U$ , inclined at angle $\alpha$ , and a vortex of circulation $\Gamma$ . The complex potential in the $\zeta$ -plane is

(223) $\begin{equation*} \widetilde{w}(\zeta) = U(\zeta-\zeta_0)e^{-i\alpha} + \frac{U R^2 e^{i\alpha}}{\zeta-\zeta_0} - \frac{i\Gamma}{2\pi}\ln(\zeta-\zeta_0) \end{equation*}$

The complex velocity is

(224) $\begin{equation*} \frac{d\widetilde{w}}{d\zeta} = Ue^{-i\alpha} - \frac{U R^2 e^{i\alpha}}{(\zeta-\zeta_0)^2} - \frac{i\Gamma}{2\pi(\zeta-\zeta_0)} \end{equation*}$

To compute the lift, apply the Blasius theorem in the mapped ${z}$ -plane, i.e.,

(225) $\begin{equation*} F = \frac{i\varrho}{2} \oint_{\rm C} \left( \frac{dw}{dz} \right)^2 dz \end{equation*}$

However, because $w(z)=\widetilde{w}(\zeta)$ and $z=f(\zeta)$ , the chain rule gives

(226) $\begin{equation*} \frac{dw}{dz} = \frac{\dfrac{d\widetilde{w}}{d\zeta}} {\dfrac{dz}{d\zeta}} \end{equation*}$

Also,

(227) $\begin{equation*} dz = \frac{dz}{d\zeta}\,d\zeta \end{equation*}$

Therefore,

(228) $\begin{equation*} \left(\frac{dw}{dz}\right)^2 dz = \frac{\left(\dfrac{d\widetilde{w}}{d\zeta}\right)^2} {\left(\dfrac{dz}{d\zeta}\right)^2} \frac{dz}{d\zeta}\,d\zeta = \frac{\left(\dfrac{d\widetilde{w}}{d\zeta}\right)^2} {\dfrac{dz}{d\zeta}} \,d\zeta \end{equation*}$

Therefore, the Blasius integral transformed into the $\zeta$ -plane becomes

(229) $\begin{equation*} F = \frac{i\varrho}{2} \oint_{\rm C_\zeta} \frac{\left(\dfrac{d\widetilde{w}}{d\zeta}\right)^2} {\dfrac{dz}{d\zeta}} \,d\zeta \end{equation*}$

where

(230) $\begin{equation*} \frac{dz}{d\zeta} = 1-\frac{a^2}{\zeta^2} \end{equation*}$

for the Joukowsky transformation. The transformed integrand must be evaluated in its complete form, including the mapping derivative.

The circulation is selected by imposing the Kutta condition at the trailing-edge critical point, where

(231) $\begin{equation*} \frac{dz}{d\zeta}=0 \end{equation*}$

The corresponding numerator must also vanish there so that the mapped velocity remains finite.

Evaluation of the resulting contour integral gives zero force in the freestream direction, i.e.,

(232) $\begin{equation*} D' = 0 \end{equation*}$

and a lift per unit span of

(233) $\begin{equation*} L' = -\varrho \, U \, \Gamma \end{equation*}$

With the sign convention used in this chapter, positive circulation is counterclockwise. Therefore, positive upward lift requires clockwise circulation, for which $\Gamma<0$ . In magnitude,

(234) $\begin{equation*} |L'| = \varrho \, U \, |\Gamma| \end{equation*}$

This is the Kutta-Joukowsky theorem. The conformal mapping changes the body geometry but does not change the resultant lift associated with the circulation.

General Method of Singularities

Consider now a general method of singularities for modeling two-dimensional potential flows. An understanding of this method is important because it provides a foundation for more advanced numerical techniques for flow analyses, particularly panel methods, which are considered in a later chapter of this eBook.

In the method of singularities, the flow is modeled as a superposition of idealized point singularities such as sources, sinks, vortices, and doublets, each with known analytical potential and velocity fields. Hence, panel methods may be viewed as a numerical implementation of the method of singularities when it is applied to arbitrary body shapes with specified boundary conditions. Panel methods retain the advantages of potential flow theory while enabling the accurate modeling of the flow about realistic aircraft geometries.

Let the flow be composed of a uniform freestream $U$ at an angle of attack $\alpha$ to the body axis, with an arbitrary number of singularities, including sources, sinks, vortices, and doublets enclosed within a closed boundary, as shown in Figure 28. The total complex potential at a point $z = x + i \, y$ can be written as

(235) $\begin{equation*} w(z) = U \, e^{-i \alpha} \, z + \sum_{k=1}^{N_s} \frac{Q_k}{2\pi} \ln(z - z_k) + \sum_{k=1}^{N_v} \frac{-i \, \Gamma_k}{2\pi} \ln(z - z_k) + \sum_{k=1}^{N_d} \frac{\mu_k \, e^{-i\theta_k}}{2\pi(z - z_k)} \end{equation*}$

where ${Q_k}$ is the strength of the $k$ -th source or sink (positive for source), $\Gamma_k$ is the strength of the $k$ -th vortex (positive counterclockwise), $\mu_k$ is the strength of the $k$ -th doublet, $\theta_k$ is its orientation angle, and $z_k$ is the location of the $k$ -th singularity in the complex plane. Notice that a physical flow constraint for a closed body is

(236) $\begin{equation*} \sum_{k=1}^{N_s} Q_k = 0 \end{equation*}$

which ensures mass conservation.

The method of singularities is generalized to an arbitrary number of sources, sinks, vortices, and doublets.

Differentiating the complex potential for this flow gives the complex velocity, i.e.,

(237) $\begin{equation*} { \frac{dw}{dz} = U \, e^{-i\alpha} + \sum_{k=1}^{N_s} \frac{Q_k}{2\pi (z - z_k)} + \sum_{k=1}^{N_v} \frac{-i \, \Gamma_k}{2\pi (z - z_k)} - \sum_{k=1}^{N_d} \frac{\mu_k \, e^{-i \, \theta_k}}{2\pi (z - z_k)^2} } \end{equation*}$

At large distances where $|z| \to \infty$ , each term can be expanded in powers of $1/z$ , i.e.,

(238) $\begin{equation*} \frac{1}{z - z_k} = \frac{1}{z} + \frac{z_k}{z^2} + \frac{z_k^2}{z^3} + \cdots \quad \text{and} \quad \frac{1}{(z - z_k)^2} = \frac{1}{z^2} + \frac{2z_k}{z^3} + \cdots \end{equation*}$

Substituting these into $\dfrac{dw}{dz}$ , the expansion becomes

(239) $\begin{equation*} \frac{dw}{dz} = U \, e^{-i\alpha} + \frac{A_1}{z} + \frac{A_2}{z^2} + \cdots \end{equation*}$

with

(240) $\begin{equation*} A_1 = \sum_{k=1}^{N_s} \frac{Q_k}{2\pi} + \sum_{k=1}^{N_v} \frac{-i \, \Gamma_k }{2\pi} \end{equation*}$

which represents the effects from the sources and vortices. The next coefficient is

(241) $\begin{equation*} A_2 = \sum_{k=1}^{N_s} \frac{Q_k \, z_k}{2\pi} + \sum_{k=1}^{N_v} \frac{-i \, \Gamma_k \, z_k }{2\pi} - \sum_{k=1}^{N_d} \frac{\mu_k \, e^{-i\theta_k}}{2\pi} \end{equation*}$

which includes the effects of the doublets.

Aerodynamic Force

The net complex force per unit span can be obtained using the Blasius theorem, i.e.,

(242) $\begin{equation*} X - i \, Y = \frac{i \varrho}{2} \oint_C \left( \frac{dw}{dz} \right)^2 dz \end{equation*}$

where $C$ is a closed contour enclosing all singularities. Using the previous expansion with some algebra, then

(243) $\begin{equation*} \left( \frac{dw}{dz} \right)^2 = U^2 \, e^{-2 i \, \alpha} + \frac{2 U \, A_1 e^{-i\alpha}}{z} + \frac{A_1^2 + 2 U \, A_2 e^{-i\alpha}}{z^2} + \cdots \end{equation*}$

Only the coefficient of $1/z$ contributes to the contour integral, so

(244) $\begin{equation*} \oint_C \left( \frac{dw}{dz} \right)^2 dz = 2\pi i \left(2 U \, A_1 e^{-i\alpha}\right) \end{equation*}$

and therefore

(245) $\begin{equation*} X - i \, Y = \frac{i\varrho}{2} \left[ 2\pi i \left(2 U \, A_1 e^{-i\alpha}\right) \right] = -2\pi \varrho \, U \, e^{-i\alpha} \, A_1 \end{equation*}$

noting that only the $A_1$ coefficient contributes to the net force. Therefore, only the net source strength and net circulation affect the resultant force; doublets and higher-order terms do not contribute to the net force.

Using

(246) $\begin{equation*} A_1 = \sum_{k=1}^{N_s} \frac{Q_k}{2\pi} - i \sum_{k=1}^{N_v} \frac{\Gamma_k}{2\pi} \end{equation*}$

gives

(247) $\begin{align*} X &= -\varrho U \sum_{k=1}^{N_s} Q_k \cos\alpha + \varrho U \sum_{k=1}^{N_v} \Gamma_k \sin\alpha \\[8pt] Y &= -\varrho U \sum_{k=1}^{N_s} Q_k \sin\alpha - \varrho U \sum_{k=1}^{N_v} \Gamma_k \cos\alpha \end{align*}$

For a closed body, the net source strength must be zero, i.e.,

(248) $\begin{equation*} \sum_{k=1}^{N_s} Q_k = 0 \end{equation*}$

so the source contributions are eliminated. Therefore,

(249) $\begin{equation*} X = \varrho U \Gamma_{\text{tot}} \sin\alpha \quad \text{and} \quad Y = -\varrho U \Gamma_{\text{tot}} \cos\alpha \end{equation*}$

where

(250) $\begin{equation*} \Gamma_{\text{tot}} = \sum_{k=1}^{N_v} \Gamma_k \end{equation*}$

Lift

The lift per unit span is the component of the total aerodynamic force perpendicular to the freestream direction. If the freestream velocity makes an angle $\alpha$ with the $x$ -axis, then

(251) $\begin{equation*} L' = -X \sin\alpha + Y \cos\alpha \end{equation*}$

Substituting the expressions for ${ X }$ and ${Y}$ for a closed body gives

(252) $\begin{align*} L' &= -\left(\varrho U \Gamma_{\text{tot}} \sin\alpha\right)\sin\alpha + \left(-\varrho U \Gamma_{\text{tot}} \cos\alpha\right)\cos\alpha \\[8pt] &= -\varrho U \Gamma_{\text{tot}} \left(\sin^2\alpha+\cos^2\alpha\right) \end{align*}$

which simplifies to

(253) $\begin{equation*} L' = -\varrho U \Gamma_{\text{tot}} \end{equation*}$

This is the classical Kutta-Joukowsky result with the sign convention used here, namely that positive circulation is counterclockwise. Under this convention, positive lift corresponds to clockwise circulation, i.e., $\Gamma_{\text{tot}} < 0$ . In magnitude,

(254) $\begin{equation*} |L'| = \varrho U |\Gamma_{\text{tot}}| \end{equation*}$

Drag

The drag per unit span is the component of the total aerodynamic force in the direction of the freestream velocity. If the freestream velocity makes an angle $\alpha$ with the $x$ -axis, then

(255) $\begin{equation*} D' = X \cos\alpha + Y \sin\alpha \end{equation*}$

Substituting the expressions for ${X }$ and ${Y}$ for a closed body gives

(256) $\begin{align*} D' &= \left(\varrho U \Gamma_{\text{tot}} \sin\alpha\right)\cos\alpha + \left(-\varrho U \Gamma_{\text{tot}} \cos\alpha\right)\sin\alpha \\[8pt] &= \varrho U \Gamma_{\text{tot}} \left(\sin\alpha\cos\alpha-\cos\alpha\sin\alpha\right) = 0 \end{align*}$

Therefore, the drag force is identically zero in potential flow, i.e.,

(257) $\begin{equation*} D' = 0 \end{equation*}$

This result is consistent with d’Alembert’s paradox: the drag on a closed body in steady, inviscid, incompressible potential flow is zero even though the lift may be nonzero when circulation is present.

Prandtl-Glauert Compressibility Rule

At the beginning of the 20th century, advancements in analytical methods in aerodynamics and fluid mechanics assumed incompressible flow. However, this assumption breaks down as the freestream Mach number exceeds about 0.3. During the 1920s, Ludwig Prandtl and Hermann Glauert independently discovered that, for subsonic flows around relatively thin airfoils with small perturbations, compressibility effects can be accounted for by a simple transformation of the flow field.^[7] The procedure, now known as the Prandtl-Glauert transformation, enables the use of an incompressible flow solution to predict the corresponding solution for a linearized, subsonic compressible flow.

For incompressible, irrotational flow, the velocity potential, denoted by $\phi_0$ , satisfies Laplace’s equation, i.e.,

(258) $\begin{equation*} \nabla^{2} \phi_0 = 0 \end{equation*}$

Many classical solutions in aerodynamics, such as those for point sources, doublets, thin-airfoil theory, and lifting-line theory, are derived from this equation. It is therefore attractive to retain these incompressible solutions and apply a correction for compressibility effects, rather than rederive new solutions directly from the compressible form of the governing equations.

Consider steady, inviscid, subsonic flow with small perturbations about a uniform freestream of velocity ${V_{\infty}}$ and freestream Mach number $M_{\infty}$ . Linearising the small-disturbance potential equation and imposing the irrotationality condition gives

(259) $\begin{equation*} \left(1 - M_{\infty}^{2}\right) \frac{\partial^{2} \phi}{\partial x^{2}} + \frac{\partial^{2} \phi}{\partial y^{2}} + \frac{\partial^{2} \phi}{\partial z^{2}} = 0 \end{equation*}$

Notice that this equation would reduce to Laplace’s equation if the first term were not multiplied by the factor $1 - M_{\infty}^{2}$ .

Define the compressibility factor as

(260) $\begin{equation*} \beta = \sqrt{1 - M_{\infty}^{2}} \end{equation*}$

so that Eq. 259 can be written as

(261) $\begin{equation*} \beta^2 \frac{\partial^{2} \phi}{\partial x^{2}} + \frac{\partial^{2} \phi}{\partial y^{2}} + \frac{\partial^{2} \phi}{\partial z^{2}} = 0 \end{equation*}$

Now introduce the transformed coordinates

(262) $\begin{equation*} x' = x, \qquad y' = \beta y, \quad \text{and} \quad z' = \beta z \end{equation*}$

Using the chain rule, the derivatives transform as

(263) $\begin{equation*} \frac{\partial \phi}{\partial y} = \beta \frac{\partial \phi}{\partial y'} \quad \text{and} \quad \frac{\partial^2 \phi}{\partial y^2} = \beta^2 \frac{\partial^2 \phi}{\partial y'^2} \end{equation*}$

and similarly for the ${z}$ component. Substituting into Eq. 259 gives

(264) $\begin{equation*} \beta^2 \left( \frac{\partial^{2} \phi}{\partial x'^{2}} + \frac{\partial^{2} \phi}{\partial y'^{2}} + \frac{\partial^{2} \phi}{\partial z'^{2}} \right) = 0 \end{equation*}$

so that

(265) $\begin{equation*} \frac{\partial^{2} \phi}{\partial x'^{2}} + \frac{\partial^{2} \phi}{\partial y'^{2}} + \frac{\partial^{2} \phi}{\partial z'^{2}} = 0 \end{equation*}$

which is Laplace’s equation expressed in the transformed coordinates.

Therefore, under the assumptions of linearised subsonic compressible flow, the compressible perturbation potential can be related to an incompressible perturbation potential by

(266) $\begin{equation*} \phi(x,y,z) = \frac{1}{\beta}\,\phi_0(x',y',z') = \frac{1}{\beta}\,\phi_0(x,\beta y,\beta z) \end{equation*}$

where ${x' = x}$ , i.e., there is no scaling in the chordwise dimension. The factor $1/\beta$ in the potential is needed so that the linearized surface boundary condition is preserved while the transformed potential satisfies Laplace’s equation. This approach forms the basis of the Prandtl-Glauert transformation, as summarized in Figure 29 for a two-dimensional airfoil section.

A geometric transformation is the essence of the Prandtl-Glauert compressibility correction.

For incompressible flow, the linearised Bernoulli equation gives the pressure coefficient as

(267) $\begin{equation*} C_{p,0} \approx -\frac{2}{V_\infty}\frac{\partial\phi_0}{\partial x} \end{equation*}$

For the corresponding linearized compressible flow, then

(268) $\begin{equation*} C_p \approx -\frac{2}{V_\infty}\frac{\partial\phi}{\partial x} \end{equation*}$

Using the Prandtl-Glauert relation

(269) $\begin{equation*} \phi(x,z) = \frac{1}{\beta}\,\phi_0(x,z'), \qquad z'=\beta z \end{equation*}$

gives

(270) $\begin{equation*} \frac{\partial\phi}{\partial x} = \frac{1}{\beta}\frac{\partial\phi_0}{\partial x} \end{equation*}$

and therefore

(271) $\begin{equation*} C_p \approx -\frac{2}{\beta V_\infty}\frac{\partial\phi_0}{\partial x} \end{equation*}$

Using Eq. 267, this gives the Prandtl-Glauert pressure correction

(272) $\begin{equation*} C_p = \frac{C_{p,0}}{\beta} \end{equation*}$

The pressure values in the compressible solution are therefore greater in magnitude than for the corresponding incompressible flow solution by the factor $1/\beta$ . The assumption here is that the airfoil is sufficiently thin, and that the effects of thickness and camber on the pressure distribution can be treated within the limits of the linearized theory. The additional geometric scaling needed for airfoils with moderate thickness and camber is addressed next in the Göthert transformation.

Finally, integrating the pressure distribution over the surface leads directly to the commonly-used Prandtl-Glauert correction for the force and moment coefficients, i.e.,

(273) $\begin{equation*} C_L = \frac{C_{L,0}}{\beta},\quad C_M = \frac{C_{M,0}}{\beta},\quad \text{and} \quad C_Y = \frac{C_{Y,0}}{\beta} \end{equation*}$

which means they all increase by the factor $1/{\beta}$ . The correction, therefore, represents the additional aerodynamic loading that arises on an airfoil or a wing because of compressibility effects in linearized subsonic flow. For example, consider a thin symmetric airfoil that has an incompressible lift-curve slope of $2\pi$ ( $\approx$ 6.28) per radian. According to the Prandtl-Glauert rule, the compressible value of the slope is

(274) $\begin{equation*} a = \frac{a_0}{\beta} = \frac{2\pi}{\sqrt{1 - M_{\infty}^{2}}} \end{equation*}$

If $M_{\infty} = 0.6$ , then $\beta \approx 0.8$ and $a \approx 7.85$ , so about a 25% increase compared to the incompressible value.

Göthert Transformation

When an airfoil is not reasonably thin, its thickness and camber will alter its pressure field in a way that the simple Prandtl-Glauert rule cannot model. Göthert demonstrated that the problem can still be reduced to the incompressible Laplace equation if both the coordinates and the geometry are scaled.^[8] After applying the linear coordinate transformation

(275) $\begin{equation*} x' = x, \qquad y' = \beta y, \quad \text{and} \quad z' = \beta z \end{equation*}$

where

(276) $\begin{equation*} \beta = \sqrt{1 - M_\infty^2} \end{equation*}$

the transverse ordinates of the airfoil are also transformed. Therefore, the corresponding incompressible problem is solved for a geometrically related airfoil whose thickness and camber ordinates are reduced according to

(277) $\begin{equation*} t'(x) = \beta\, t(x) \quad \text{and} \quad z_c'(x) = \beta\, z_c(x) \end{equation*}$

as shown in Figure 29. This geometry scaling is the essential distinction between the Göthert transformation and the simpler Prandtl-Glauert pressure correction. The result is that incompressible potential flow methods can still be used, even for airfoils with moderate thickness and camber, provided the incompressible solution is obtained for the transformed airfoil geometry.

In the Göthert transformation, the pressure coefficient is then transformed back to the compressible flow using the Prandtl-Glauert pressure scaling, i.e.,

(278) $\begin{equation*} C_p = \frac{C_{p,0}}{\beta} \end{equation*}$

where $C_{p,0}$ is the incompressible pressure coefficient evaluated on the Göthert-scaled geometry. The corresponding lift and pitching moment coefficients are written as

(279) $\begin{equation*} C_L = \frac{C_{L,0}}{\beta} \quad \text{and} \quad C_M = \frac{C_{M,0}}{\beta} \end{equation*}$

where $C_{L,0}$ and $C_{M,0}$ are also evaluated for the transformed incompressible geometry. These expressions retain the familiar Prandtl-Glauert coefficient scaling, but the incompressible reference solution is no longer the solution for the original airfoil shape; it is the solution for the Göthert-scaled airfoil. This is why the Göthert transformation gives a better approximation than applying the simple $1/\beta$ correction directly to the original airfoil when the thickness and camber are not negligible. Yet, it still falls within the scope of the linearized theory.

Summary & Closure

Potential flow theory, which is rooted in the Laplace equation, provides a foundational framework for understanding and analyzing many aerodynamic flows under the idealized assumptions of inviscid, incompressible, and irrotational conditions. By utilizing the velocity potential and stream function, potential theory enables a mathematical representation of flow fields, offering a simplified yet powerful model for predicting aerodynamic behavior. The ability to superimpose elementary flows, such as uniform flow, sources, sinks, and vortices, makes potential flow theory a versatile tool for analyzing composite flow patterns and understanding the fundamental mechanisms of fluid dynamics.

Despite its advantages, potential flow theory has significant limitations that restrict its applicability to practical problems. However, potential flow theory remains an invaluable baseline model, and today it is often integrated with other methods that account for viscous and turbulent effects via various forms of viscous-inviscid coupling. These hybrid approaches can be compelling for analyzing and predicting flow behavior, bridging the gap between theoretical insights and practical engineering solutions. The Prandtl-Glauert transformation also provides a link between incompressible flow theory and subsonic compressible potential flow using a simple geometric transformation. It provides a convenient Mach-number correction for lift, moment, and pressure coefficients and is a valuable predictive approach, provided the flow remains below the critical Mach number and the geometric and kinematic assumptions are satisfied.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

What are stagnation points, and why are they important?
What are the key limitations of potential flow theory in capturing aerodynamic phenomena?
How has potential flow theory evolved historically within the field of aerodynamics?
How do predictions from potential flow theory compare with experimental aerodynamic results?
How might potential flow theory be integrated with other models, such as a boundary layer, to improve predictions?

Other Useful Online Resources

To understand more about the aerodynamics of potential flows, explore some of these online resources:

A good description of potential flow theory on Wikipedia.
Potential flow theory as presented by MIT.
Potential Flow: what it is and why it’s useful.
Potential flow and Laplace’s equation – Engineering Mathematics at the University of Washington.
Potential flow, stream functions, and examples – Engineering Mathematics at the University of Washington.

This chapter is mainly taken from the author's fluid mechanics course notes as a student, the lectures being delivered by Professor Roderick Galbraith. Professor Galbraith is an Honorary Senior Research Fellow in the Division of Aerospace Sciences at the University of Glasgow, renowned for his work in low-speed unsteady aerodynamics. The former Shoda Professor of Aerospace Systems and a Fellow of the Royal Society of Edinburgh, Professor Galbraith has made significant contributions to the understanding of dynamic stall, rotorcraft aerodynamics, and wind turbine performance. He obtained his Ph.D. from the University of Cambridge under the advisement of Prof. M. R. Head. His career has focused on the physics of separated and reattaching flows, with applications in both aerospace and wind energy systems. ↵
Stokes, G., “On the Steady Motion of an Incompressible Fluid,” Transactions of the Cambridge Philosophical Society, 1842. ↵
Rankine, W. J. M., "On Plane Water-Lines in Two Dimensions, Philosophical Transactions of the Royal Society of London, Vol. 154, 1864, pp. 369–391. ↵
Rankine, W. J. M., "Principles Relating to Stream Lines," The Engineer, October 16, 1868. ↵
Conformality is a local property, which means that it applies to small, infinitesimally sized shapes. A conformal mapping might distort the overall shape of a larger area, but it will preserve the angles within that area. ↵
Zhukovsky, N. E., "Über die Konturen der Tragflächen der Drachenflieger," Trudy TsAGI, 1906–1910 (various parts. ↵
Prandtl, L., "General Considerations on the Flow of Compressible Fluids," NACA Technical Memorandum 805, 1936. English translation of: Prandtl, L. Allgemeine Überlegungen über die Strömung zusammendrückbarer Flüssigkeiten, Convegno di Scienze Fisiche, Matematiche e Naturali, Tema: Le Alte Velocità in Aviazione, Reale Accademia d’Italia, Fondazione Alessandro Volta, Rome, Italy, Sept. 30–Oct. 6, 1935. ↵
Göthert, B. H., “Plane and Three-Dimensional Flow at High Speeds,” NACA Technical Memorandum 1105, 1946. English translation of: Göthert, B. H., Ebene und räumliche Strömung bei hohen Unterschallgeschwindigkeiten, Lilienthal-Gesellschaft für Luftfahrtforschung, Bericht 127, 1940. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Introduction[1]

History

Definition of a Potential Flow

Potential Flow Equations

Full Potential Equation

Small-Disturbance & Wave Equations

Laplace’s Equation

Velocity Potential & Stream Function

Velocity Potential

Stream Function

Relationship between Velocity Potential and Stream Function

Elementary Flows

Uniform Flow

Source Flow

Sink Flow

Vortex Flow (Irrotational Vortex)

Understanding Circulation and Vorticity

Doublet (Dipole) Flow

Table of Elementary Flows

Superposition of Elementary Flows

Simulating the Flow Around a Solid Body

Rankine Half-Body

Rankine Oval

Flow Around a Circular Cylinder

Velocity on the Cylinder’s Surface

Pressure on the Cylinder’s Surface

Lifting Circular Cylinder

Velocity on the Cylinder’s Surface

Pressure on the Surface

Force Components

Method of Images

Source Flow Near a Solid Wall

Source Flow in a Corner

Vortex Flow Near a Solid Wall

Three-Dimensional Potential Flow Around a Sphere

Complex Variable Methods in Potential Flow

Velocity Field

Satisfying Laplace’s Equation

Working with Complex Potentials

Table of Complex Potentials for Elementary Flows

Conformal Transformations

Basic Conformal Transformations in Aerodynamics

Flow in a Corner

Transformation of a Circular Cylinder to an Airfoil

Forces & Moments Using Complex Potentials

Blasius Theorem

Cauchy’s Integral Theorem

Lift on Joukowsky Airfoils using Blasius’ Theorem

General Method of Singularities

Aerodynamic Force

Lift

Drag

Prandtl-Glauert Compressibility Rule

Göthert Transformation

Summary & Closure

License

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Introduction^[1]