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34 Potential Flows

Introduction

A potential flow assumes the aggregate of an incompressible, irrotational, and inviscid fluid motion, i.e., an “ideal” flow. These assumptions are sweeping approximations to an actual flow, in general. However, many interesting and practically valuable flows can be considered as potential flows. More importantly, such assumptions are used in developing various aspects of airfoil and wing theories, which will be considered in the following chapters, for which studying potential flows is a necessary prerequisite.

The term “irrotational” 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. This condition implies that the fluid elements do not undergo spin or rotation,  thereby eliminating viscous shear forces and precluding the generation of turbulence; besides, the absence of viscosity (an inviscid flow) makes the creation of shear stresses impossible anyway. These assumptions simplify the analysis significantly, and the inviscid flow assumption is particularly valid for analyzing external flows away from proximity to solid boundaries, i.e., outside of the boundary layers. For example, the flow about an airfoil, as shown in the figure below, is predominantly a potential flow away from the airfoil’s immediate surface.

The flow about an airfoil is predominantly potential away from the surface. For clarity, the extent of the viscous flow region is exaggerated.

Consequently, studying potential flows provides an insightful and tractable framework for predicting and understanding various flow problems. For these problems, exact analytical solutions are often achievable, leading to valuable results that expose the fundamental fluid dynamics. Potential flows are foundational in many classical aerodynamic theories, including thin airfoil theory, lifting-line theory, and lifting-surface theory, which are widely used to model and analyze airflow behavior around bodies, airfoils, and wings. Numerical methods, called panel methods, can further extend the applicability of potential flow theory, including to complete airplane shapes, making potential flow theory a valuable tool in flight vehicle design and other engineering applications.

Learning Objectives

  • Understand the principles of incompressible, irrotational, and inviscid fluid motion, i.e., potential flows.
  • Become familiar with elementary flow solutions such as uniform flow, sources and sinks, doublets, and vortex flows.
  • Know how to combine elementary potential flow solutions to model composite flow fields.

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 to relate velocity and pressure in inviscid, incompressible fluids, the famous Bernoulli equation being 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 understanding the superposition of elementary flow solutions, such as sources, sinks, and vortices, to model more complex flows.

In the 19th century, significant advancements in potential flow theory included the formalization of streamlines. The first and most complete exposition was published by George Stokes [1] in 1842.  William Rankine analyzed the flow patterns around simple geometries[2] using a potential flow method, while Hermann von Helmholtz formulated the conservation of vorticity in inviscid flows, a concept critical to understanding circulation and lift. Rankine was interested in the flows about ship hulls, for which a potential flow solution provided much insight, as shown in the figure by his calculations about a “half-body,” which Rankine called such shapes “Oögenous Neoïds.”

Rankine’s potential flow solution about an oval 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 potential flow predicts zero drag on symmetric bodies. This paradox emphasized that additional characterisitcs of the flow were needed, specifically viscosity, to rectify the differences between predictions and experiments. Despite these limitations, potential flow became a central tool for analyzing idealized fluid motion.

The early 20th century saw the application of potential flow theory to aerodynamics. Frederick Lanchester played a pivotal role in integrating the concepts of circulation, vortices, and lift into understanding aerodynamics. Martin Kutta and Nikolay (Nikolai) Joukowski independently developed a hypothesis that linked circulation to lift. They introduced the idea of what is now known as the Kutta condition, which models the smooth flow observed at an airfoil’s trailing edge. Ludwig Prandtl built on this work by developing thin airfoil and lifting line theories, providing the first models to predict lift, pitching moments, and induced drag for two-dimensional airfoils and three-dimensional wings. These theories allowed engineers to design and analyze airfoils and wings with good predictive accuracy despite the incompressible and inviscid assumptions.

In the 20th century, potential flow theory was extended to compressible flows, leading to techniques like the Prandtl-Glauert rule for subsonic flows. Numerical methods like panel methods were introduced to solve potential flow problems for complex geometries like 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 in engineering education.

Definition of a Potential Flow

Potential flow methods use the mathematical principle that the gradient of a scalar field is a vector. 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 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 to the surface defined by \phi\left(x, y, z\right) = constant, as illustrated in the figure below.

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.

Development of the Governing Equation

The development of the potential flow method in aerodynamics starts with a governing equation known as the full-potential equation, which is derived from the Euler (momentum) equation. This governing equation applies to compressible flows and is written as

(3)   \begin{equation*} \nabla^2 \phi - \frac{1}{a^2} \Bigg( \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial}{\partial t}(V^2) + \vec{V} \bigcdot \nabla \left( \frac{V^2}{2} \right) \Bigg) = 0 \end{equation*}

where \phi is the potential function, V = |\vec{V}|, and a is the local speed of sound. This equation retains all terms from the Euler and continuity equations and accounts for variations with the Mach number.

The transonic small disturbance (TSD) equation is a variation of the full-potential equation for small disturbances, i.e.,

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

For steady flows, this reduces to

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

The TSD equation is nonlinear and foundational in computational fluid dynamics (CFD). Further, assuming small disturbances reduces the TSD equation to a wave equation, i.e., it becomes

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

where a_\infty is the free-stream speed of sound. The wave equation describes the propagation of the velocity potential as waves pass through the medium. This simplification makes the equation both analytically and computationally tractable, particularly in the study of compressible flows. This reduced form is particularly useful in analyzing acoustic waves and small perturbations in transonic flow fields, serving as a simplified model for studying wave phenomena such as sound waves or minor disturbances in fluid systems.

For steady, incompressible flows, i.e., a_\infty \to \infty, then the TSD equation reduces to Laplace’s equation, i.e.,

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

In this equation, the velocity field can be expressed in terms of the velocity potential \phi as

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

Laplace’s equation is also linear, allowing the superposition of potential flow solutions. This means that multiple more elementary potential flows can be combined to form a composite solution, i.e.,

(9)   \begin{equation*} \phi_{\text{total}} = \phi_1 + \phi_2 + \cdots + \phi_N \end{equation*}

where each elementary solution \phi_i, i = 1 \cdots N satisfies Laplace’s equation, i.e., \nabla^2 \phi_i = 0. This is one of the principles of potential flows that makes it so useful.

Velocity Potential & Stream Function

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 way to visualize incompressible two-dimensional flows. Streamlines, which are lines of constant values of the stream function, can reveal the flow patterns, i.e., where the flow is going and its relative velocity.

Velocity Potential

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

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

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

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

The existence of a velocity potential means that irrotational flow is ensured. Because the condition for irrotationality is \nabla \times \vec{V} = 0, then

(12)   \begin{equation*} \nabla \times (\nabla \phi) = \nabla^2 \phi = 0 \end{equation*}

and so the existence of a velocity potential automatically satisfies the Laplace equation. Notice also that continuity is satisfied, i.e.,

(13)   \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

(14)   \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 valid for up to three-dimensional incompressible flows, the stream function, \psi, is valid only in two dimensions. The velocity components are

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

Again, the continuity equation is automatically satisfied by the existence of  a \psi, i.e.,

(16)   \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\phi}{\partial x \partial y } - \frac{\partial^2\phi}{\partial y \partial x } = 0 \end{equation*}

Lines of constant \psi represent flow streamlines, and the difference in adjacent \psi values gives the volumetric flow rate per unit depth. In other words, closer streamlines indicate higher flow velocities, as shown in the figure below. In a two-dimensional flow, the volumetric flow rate  between two adjacent streamlines is given by

(17)   \begin{equation*} dQ = \int_A^C dy - \int_C^B dy = u \, dy - v \, dx \end{equation*}

Notice that the reason there is a negative sign on the second term is that in determining the flow rate between points C and B, the integration is in the negative x direction. This result then gives

(18)   \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*}

The volumetric flow between adjacent streamlines is constant.

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

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

so that comparing Eqs. 18 and 19 gives

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

For example, the flow rate between two adjacent streamlines, \psi_1 and \psi_2, is

(21)   \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

In two-dimensional incompressible flow, then

(22)   \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*}

This result defines the orthogonality between streamlines (\psi = \text{constant}) and equipotential lines (\phi = \text{constant}), as shown in the figure below.

Equipotential lines run perpendicular to stream functions.

For irrotational and incompressible flows, both \phi and \psi exist, and they are related through the Cauchy-Riemann equations, i.e.,

(23)   \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*}

The velocity potential and stream function are orthogonal, meaning the gradients of \phi and \psi are perpendicular.

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 related to 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

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

Worked Example #1

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, y = 0, \phi = y on x = 1, and \phi = x on y = 1.

  1. Assuming a general solution of the form \phi(x, y) = ax + by + c x y, determine \phi(x, y).
  2. Determine the velocity field in terms of u and v.
  3. Verify that this velocity field is physical, incompressible, and irrotational.
  4. Determine the corresponding stream function for this flow.
  5. 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.,

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

A general solution for the velocity potential of the form is to be assumed, i.e.,

    \[ \phi(x, y) = ax + by + cxy \]

Substituting into Laplace’s equation, it is satisfied for any a, b, c. Using the given boundary conditions leads to

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

Therefore, the solution for 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

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

3. The vorticity is given by

    \[ \omega_z = \omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \]

For u = y and v = x, then

    \[ \frac{\partial v}{\partial x} = 1 \quad \text{and} \quad \frac{\partial u}{\partial y} = 1 \implies \omega = 1 - 1 = 0 \]

Therefore, the velocity field is irrotational. The existence of a velocity potential means that the flow automatically satisfies continuity, i.e.,

    \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial}{\partial x} \left( \frac{\partial \phi }{\partial x} \right) + \frac{\partial}{\partial y} \left( \frac{\partial \phi}{\partial y} \right) = \nabla^2 \phi = 0 \]

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 and v = x, then

    \begin{align*} \frac{\partial \psi}{\partial y} = y & \implies \psi = \int y \, dy = \frac{y^2}{2} + f(x)\\ -\frac{\partial \psi}{\partial x} = x & \implies \psi =\int x \, dx = -\frac{x^2}{2} + f(y) \end{align*}

It will be apparent, therefore, that

    \[ f(x) = -\frac{x^2}{2} \quad \text{and} \quad f(y) = \frac{y^2}{2} \]

So, the stream function is

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

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 \psi(x, y), i.e.,

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

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

    \[ x \, y = \text{constant} \]

which represent 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 combination of a source and a sink placed infinitesimally close to each other, representing the flow around slender bodies like cylinders.

Combining these elementary flows allows for analyzing and visualizing more intricate flow problems, such as flow over certain classes of bodies, such as circular cylinders and ovals.

Uniform Flow

As shown in the figure below, a uniform flow moves at a constant velocity, say 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.,

(24)   \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

(25)   \begin{equation*} u = \frac{\partial \phi}{\partial x} = \frac{\partial (U \, x)}{\partial x} = U \quad \text{and} \quad u = \frac{\partial \psi}{\partial y} = \frac{\partial (U \, y)} {\partial y} = U \end{equation*}

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

(26)   \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

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

(28)   \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

(29)   \begin{equation*} v_r = \frac{\partial \phi}{\partial r} = U \cos \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 free-stream 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

As shown in the figure below, a source is a hypothetical flow where fluid “exits” at a point. The velocity potential \phi and stream function \psi for a sources are given by

(30)   \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 sink, and \theta is the polar angle. Equipotential lines (\phi = \text{constant}) are circles centered at the source (r = \text{constant}) in a geomtrical progression, and streamlines (\psi = \text{constant}) are straight radial lines (\theta = \text{constant}).

Streamlines and equipotential lines for a source flow.

The radial velocity for a source, v_r, varies with the inverse of the distance, r, from the source, i.e.,

(31)   \begin{equation*} v_r = \frac{Q}{2 \pi r} \end{equation*}

and the tangential velocity component for a source is zero, i.e., v_\theta = 0. Notice that v_r \rightarrow \infty as r \rightarrow 0 produces a singular behavior. This is why sources, sinks, vortices, etc., are often called singularities.

Sink Flow

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

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

and the radial velocity is

(33)   \begin{equation*} u_r = -\frac{Q}{2 \pi r} \end{equation*}

with V_r = 0 as in the case of a source.

Streamlines and equipotential lines for a sink flow.

Vortex Flow

A vortex flow represents purely rotational motion around the origin, as shown in the figure below, where the tangential velocity is inversely proportional to the radial distance, as shown in the figure below. The equipotential lines (\phi = \text{constant}) are straight radial lines (\theta = \text{constant}), and streamlines (\psi = \text{constant}) are concentric circles (r = \text{constant}).

 

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

The velocity potential \phi and stream function \psi for a vortex flow are given by

(34)   \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.

The tangential velocity for a vortex flow, v_\theta, is

(35)   \begin{equation*} v_\theta = \frac{\partial \phi}{\partial r} = -\frac{\partial \psi}{\partial \theta} = \frac{\Gamma}{2 \pi r} \end{equation*}

Notice that a vortex flow has no radial velocity component, i.e., v_r = 0. However, a vortex has a sign, a counter-clockwise direction indicating positive \Gamma.

Doublet (Dipole) Flow

A doublet is created by superimposing a source and sink of equal strength placed infinitesimally close together, as shown in the figure below. 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 maintaining a finite product of strength and separation.

Streamlines and equipotential lines for a doublet(dipole) flow.

The velocity potential for a source/sink pair located at (-d, 0) and (d, 0) is

(36)   \begin{equation*} \phi = \frac{Q}{2\pi} \ln{\sqrt{(x+d)^2 + y^2}} - \frac{Q}{2\pi} \ln{\sqrt{(x-d)^2 + y^2}} = -\frac{\mu}{2 \pi r^2} \, \sin \theta \end{equation*}

In the limit as d \to 0 and Q \to \infty while product d \, Q remains finite, then the velocity potential for a doublet is obtained, i.e.,

(37)   \begin{equation*} \phi = -\frac{\mu}{2 \pi r^2} \, \sin \theta \end{equation*}

where the doublet strength is  \mu = d \, Q. Notice that a doublet has a direction, and by changing the sign of the strength, the flow direction will change, too.

The corresponding stream function is

(38)   \begin{equation*} \psi = \frac{\mu}{2\pi} \left( \frac{y}{x^2 + y^2} \right) = \frac{\mu}{2\pi} \, \cos \theta \end{equation*}

The  velocity components for a doublet expressed in polar coordinates are

(39)   \begin{equation*} u_r = \frac{\partial \phi}{\partial r} = \frac{\mu \cos \theta}{r^2} \quad \text{and} \quad u_\theta = \frac{1}{r} \frac{\partial \phi}{\partial \theta} = -\frac{\mu \sin \theta}{r^2} \end{equation*}

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 allows the construction of more intricate flow fields from simple, well-understood flow building blocks. Mathematically, this process can be written as

(40)   \begin{equation*} \phi_{\text{total}} = \phi_1 + \phi_2 + \cdots + \phi_N \end{equation*}

where \phi_1, \phi_2, \cdots, \phi_N are the velocity potential functions corresponding to up to N individual elementary flows. 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 in computational fluid dynamics and analytical techniques. It forms the basis for panel methods, where an airfoil is modeled as a collection of sources, sinks, and vortices distributed along its surface. By solving for the appropriate strengths of these elements, the flow around an airfoil can be accurately represented, including essential characteristics such as the pressure distribution and the lift.

Simulating the Flow About 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 this case, potential methods for simulating flows about solid bodies become apparent.

Rankine Half-Body

A Rankine half-body[3] 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 the figure below. Rankine called such shapes “Oögenous Neoïds.”

 

The superposition of a source in a uniform flow gives what is known as a Rankine half-body.

The velocity potential is

(41)   \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

(42)   \begin{equation*} \psi(x, y) = Uy - \frac{Q}{4 \pi} \tan^{-1} \left( \frac{y}{x} \right) \end{equation*}

The stagnation streamline (\psi = 0) represents the boundary of the Rankine half-body, i.e.,

(43)   \begin{equation*} y = \pm \sqrt{\frac{Q}{2 \pi U} - x}, \quad 0 \leq x \leq \frac{Q}{2 \pi U} \end{equation*}

and the stagnation point (u = v = 0) occurs  on  the x axis at

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

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 the figure below, another Rankine neoid. He explains that: “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 superposition of a source/sink pair in a uniform flow gives what is known as a Rankine oval.

The velocity potential is

(45)   \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

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

The equation of the dividing or stagnation streamline is

(47)   \begin{equation*} \left( x^2 - a^2 + \frac{Q}{2\pi U} \right)^2 + y^2 = \left( \frac{Q}{2\pi U} \right)^2 \end{equation*}

which forms the oval boundary (\psi = 0). Stagnation of the flow (u = v = 0) lies on the x-axis at

(48)   \begin{equation*} x = \pm \sqrt{a^2 - \left( \frac{Q}{2 \pi U} \right)^2 }, \quad y = 0 \end{equation*}

provided Q / (2 \pi U) < a. Therefore, there are two stagnation points, one upstream and another downstream.

Flow About a Circular Cylinder

The flow around a circular cylinder is a classic aerodynamics problem. 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 the figure below.

 

The flow about a circular cylinder can be obtained from the superposition of a doublet in a uniform flow.

The composite stream function is given by

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

For uniform flow, then

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

and for a doublet at the origin, then

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

The total stream function is

(52)   \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*}

Velocity on the Cylinder’s Surface

The tangential velocity v_\theta is

(53)   \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

(54)   \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 (r = R) and v_r = 0, the tangential velocity is

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

Pressure on the Cylinder’s Surface

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

(56)   \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

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

Substituting the expression for v_\theta^2 gives

(58)   \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.

Lifting Circular Cylinder

The flow about a lifting circular cylinder is obtained from the stream function combining a uniform flow, a doublet, and a vortex, as shown in the figure below. The composite stream function is given by

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

 

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

For uniform flow, then

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

and for a doublet at the origin, then

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

Finally, for a vortex

(62)   \begin{equation*} \psi_{\text{vortex}} = \frac{\Gamma}{2\pi} \ln r \end{equation*}

Therefore, the  total stream function becomes

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

Velocity on the Cylinder’s Surface

On the surface of the cylinder (r = R), the velocity components are

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

The stagnation points occur where u_r = 0 and u_\theta = 0 on the cylinder surface (r = R)

(65)   \begin{equation*} u_r = U \left( 1 - \frac{R^2}{R^2} \right) \cos \theta = 0 \end{equation*}

For u_\theta = 0, then

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

The stagnation points are at

(67)   \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 Cylinder’s Surface

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

(68)   \begin{equation*} C_p = 1 - \frac{u_{\theta}^2}{U^2} \end{equation*}

At the surface of the cylinder (r = R), then

(69)   \begin{equation*} u_{\theta}^2 = \left( -2U \sin \theta + \frac{\Gamma}{2\pi R} \right)^2 \end{equation*}

Substituting into Eq. 68 gives

(70)   \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*}

The figure below shows representative results for two circulation values. Notice that the pressure over the upper surface becomes increasingly lower, which is responsible for the lift generation. Also, notice that the two initially distinct stagnation points come closer together on the lower surface and eventually merge. For increasingly higher values of circulation, so-called super-circulation, the stagnation points will move off the surface of the cylinder.

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

(71)   \begin{equation*} F_x = - \int_0^{2\pi} \bigg( \frac{1}{2} \rho U^2C_p \, R \cos\theta \bigg) d\theta \quad \text{and} \quad F_y = - \int_0^{2\pi} \bigg( \frac{1}{2} \rho U^2C_p \, R \sin\theta \bigg) d\theta \end{equation*}

After integration, then it is found that

(72)   \begin{equation*} F_x = 0 = D \end{equation*}

because of the fore-and-aft axisymmetry of the pressure distribution, even in lifting flow. The result that the drag is zero in a potential flow is called D’Alambert’s paradox because the drag is known to be finite in a real flow. The reason for the “paradox” is that the flow is viscous and will create skin friction drag and eventually separate from the cylinder, creating a wake and significant pressure drag.

The lift force is given by

(73)   \begin{equation*} F_y = \rho \, U \, \Gamma = L \end{equation*}

which is called the Kutta-Joukowski theorem, named after Martin Kutta and Nikolay Zhukovsky (or Joukowski). This latter result is very general and applies to any body shape.

The concept of circulation is inherently tied to the generation of lift, which arises from pressure differences created by the flow’s interaction with a body. Indeed, the Kutta-Joukowski theorem’s significance extends beyond theoretical fluid dynamics to practical applications in aerodynamics. Relating lift directly to circulation provides a simple yet powerful framework for understanding and predicting the performance of airfoils, wings, and rotating objects. The theorem also explains phenomena like the Magnus lift effect on spinning balls.

Note on D’Alembert’s Paradox

D’Alembert’s paradox highlights an apparent contradiction in fluid dynamics: in an ideal, nonviscous, incompressible fluid with steady, irrotational flow, a solid body (like a cylinder) experiences no drag force, a conclusion that seems at odds with everyday experience where drag is always observed. This historical paradox refers only to drag, but the lift is also zero.  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 giving food for mathematicians to contribute to never-ending quasi-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 or “method of images” simplifies boundary condition problems in potential flows by replacing the walls and boundaries with equivalent image singularities (sources, sinks, 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 with 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 the figure below. To ensure no flow normal to the wall (v = 0 at y = 0), place an image source of strength -Q at (x_0, -y_0). This ensures the cancellation of the normal velocity at y = 0. In this case he total velocity potential \phi(x, y) is

(74)   \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 involves placing three image sources to satisfy the conditions at both walls, as shown in the figure below. Again, the principles of superposition apply. A stationary vortex with negative circulation -\Gamma is located at a height h above a solid horizontal wall (y=0). To satisfy the boundary condition of no flow normal to the wall (v=0 at y=0), an image vortex with positive circulation +\Gamma is placed at (x, -h).

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

The stream function for an individual vortex in polar coordinates is given by

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

where r is the radial distance from the vortex. The combined stream function for the real vortex and the image vortex at a point P(x, y) is:

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

where

(77)   \begin{equation*} r_1 = \sqrt{x^2 + (y - h)^2} \text{(~distance to real vortex)} \quad \text{and} \quad r_2 = \sqrt{x^2 + (y + h)^2} \ \text{(~distance to image vortex)} \end{equation*}

Substituting r_1 and r_2, the combined stream function becomes

(78)   \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 tangential velocity induced by a vortex at a point is

(79)   \begin{equation*} V_\theta = -\frac{\Gamma}{2\pi r} \end{equation*}

where r is the distance to the vortex. The velocity component in the x-direction (u) is

(80)   \begin{equation*} u = -\frac{\Gamma}{2\pi r} \sin\theta = -\frac{\Gamma}{2\pi} \frac{y}{r^2} \end{equation*}

For the real vortex

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

and for the image vortex

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

The total velocity in the x-direction is

(83)   \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, then

(84)   \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*}

Worked Example #2

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 or the “method of images.” The first step is to define a convenient coordinate system. In this case, there is a horizontal wall, which can be assumed to be placed 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 a negative strength or circulation, so for the image vortex, it must be ensured that it has the correct sign, i.e., \Gamma for the image vortex will be positive.

  1. 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. So, 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

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

  2. 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 (clockwise) circulation is

        \[ V_{\theta} = -\frac{\Gamma}{2 \pi r} \]

    where r is the distance to the arbitrary point P. A potential vortex does not induce a V_r component. Then V_{\theta} 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 our problem (and continuing to remember the signs on the vortices and their relative offsets from the x axis), then

        \[ U = \frac{-(-\Gamma) (y-h)}{2 \pi (x^2+(y-h)^2)} + \frac{\Gamma (y+h)}{2 \pi (x^2+(y+h)^2)} \]

    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 (i.e., on the wall), the flow velocity can be determined there, which is

        \[ U = \frac{\Gamma}{2 \pi} \left(\frac{h}{(x^2+h^2)} + \frac{h}{(x^2+h^2} \right) = \frac{\Gamma}{\pi} \left(\frac{h}{x^2+h^2} \right) \]

    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

(85)   \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

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

where \kappa is the 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.,

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

In spherical coordinates, substituting x = r \cos\theta gives

(88)   \begin{equation*} \phi_{\text{total}} = Ur \cos\theta - \frac{UR^3 \cos\theta}{r^2} \end{equation*}

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

(89)   \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

(90)   \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

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

The corresponding tangential component is

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

which for r = R gives

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

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

(94)   \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, occurring at the sphere’s equator (\theta = \pm \pi/2), is 1.5 V_\infty, with a minimum pressure coefficient of -5/4. In contrast, flow around a cylinder is two-dimensional, symmetric only in the plane perpendicular to its axis, with a maximum velocity of 2 V_\infty and a minimum pressure coefficient of -3. This difference arises from the three-dimensional “relieving effect” in sphere flow: in addition to moving over and under, the flow can also move spanwise. This extra freedom reduces the velocity and pressure gradients on the sphere compared to the cylinder.

Summary & Closure

Potential flow theory, which is rooted in the Laplace equation, provides a foundational framework for understanding and analyzing 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, the potential flow theory has significant limitations that limit its applicability to practical problems. However, potential flow theory remains an invaluable baseline model, and today, it can often be integrated with other methods that account for viscous and turbulent effects. These hybrid approaches can be compelling for analyzing and predicting flow behavior, bridging the gap between theoretical insights and practical engineering solutions.

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 then, explore some of these online resources:

 

  1. Stokes, G., “On the Steady Motion of an Incompressible Fluid,” Transactions of the Cambridge Philosophical Society, 1842. ↵
  2. 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. ↵
  3. Rankine, W. J.M.,  "Principles Relating to Stream Lines," The Engineer, October 16, 1868. ↵

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