1.5 Potential flow theory

Potential flow theory simplifies fluid dynamics by assuming irrotational, incompressible, and inviscid flow. This approach allows for the use of velocity potential and stream functions to describe flow fields, making complex aerodynamic problems more manageable.

By combining elementary flows like uniform flow, sources, sinks, doublets, and vortices, engineers can model flow around various shapes. While limited by neglecting viscosity and compressibility, potential flow theory remains a valuable tool for understanding basic aerodynamic principles and estimating lift forces.

Potential flow assumptions

• Potential flow theory is a simplified approach to modeling fluid dynamics that relies on several key assumptions to make the governing equations more tractable
• These assumptions allow for the use of potential functions to describe the flow field, which greatly simplifies the analysis of fluid motion around objects

Irrotational flow

Top images from around the web for Irrotational flow

• 

  Aerodynamics/Inviscid and compressible flows - Wikiversity View original

  Is this image relevant?

• 

  fluid dynamics - How do the flow equations relate to the actual situation? - Physics Stack Exchange View original

  Is this image relevant?

• 

  HartleyMath - Vector Fields View original

  Is this image relevant?

• 

  Aerodynamics/Inviscid and compressible flows - Wikiversity View original

  Is this image relevant?

• 

  fluid dynamics - How do the flow equations relate to the actual situation? - Physics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Irrotational flow

• 

  Aerodynamics/Inviscid and compressible flows - Wikiversity View original

  Is this image relevant?

• 

  fluid dynamics - How do the flow equations relate to the actual situation? - Physics Stack Exchange View original

  Is this image relevant?

• 

  HartleyMath - Vector Fields View original

  Is this image relevant?

• 

  Aerodynamics/Inviscid and compressible flows - Wikiversity View original

  Is this image relevant?

• 

  fluid dynamics - How do the flow equations relate to the actual situation? - Physics Stack Exchange View original

  Is this image relevant?

1 of 3

• Assumes that the fluid particles do not rotate or have angular velocity as they move through the flow field
• Mathematically, the curl of the velocity vector is zero ($\nabla \times \vec{V} = 0$), which implies that the velocity field can be described by a scalar potential function
• Irrotational flow is a reasonable approximation for many aerodynamic applications, particularly in regions away from boundary layers and wakes

Incompressible flow

• Assumes that the density of the fluid remains constant throughout the flow field
• Mathematically, the divergence of the velocity vector is zero ($\nabla \cdot \vec{V} = 0$), which implies that the volume of fluid entering a control volume is equal to the volume leaving it
• Incompressible flow is a good approximation for low-speed aerodynamics, where the Mach number is typically less than 0.3

Inviscid flow

• Assumes that the fluid has no viscosity, meaning that there are no shear stresses acting between fluid particles or between the fluid and solid surfaces
• This assumption simplifies the governing equations by removing the viscous terms, resulting in the Euler equations
• Inviscid flow is a reasonable approximation for high-Reynolds-number flows, where the inertial forces dominate over the viscous forces (thin boundary layers)

Velocity potential

• Velocity potential is a scalar function that describes the velocity field of a potential flow
• It is denoted by $\phi(x, y, z)$ and has units of velocity times length
• The velocity components can be obtained by taking the gradient of the velocity potential: $\vec{V} = \nabla \phi = (\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z})$

Definition of velocity potential

• For an irrotational flow, the velocity field can be expressed as the gradient of a scalar function called the velocity potential: $\vec{V} = \nabla \phi$
• The existence of a velocity potential is a consequence of the irrotationality condition, which ensures that the velocity field is conservative

Laplace's equation

• Substituting the definition of velocity potential into the continuity equation for incompressible flow ($\nabla \cdot \vec{V} = 0$) leads to Laplace's equation: $\nabla^2 \phi = 0$
• Laplace's equation is a linear partial differential equation that governs the behavior of the velocity potential in a potential flow
• Solutions to Laplace's equation, subject to appropriate boundary conditions, provide the velocity potential for a given flow problem

Boundary conditions

• To solve Laplace's equation and obtain the velocity potential, appropriate boundary conditions must be specified
• These boundary conditions ensure that the flow satisfies the physical requirements of the problem, such as no flow through solid surfaces and matching the freestream velocity at infinity
• Common boundary conditions include:
  • Neumann boundary condition: specifies the normal derivative of the velocity potential at a surface (e.g., no flow through a solid surface: $\frac{\partial \phi}{\partial n} = 0$)
  • Dirichlet boundary condition: specifies the value of the velocity potential at a surface (e.g., matching the freestream velocity at infinity: $\phi = U_\infty x$)

Stream function

• Stream function is another scalar function used to describe potential flows, particularly in two-dimensional problems
• It is denoted by $\psi(x, y)$ and has units of velocity times length
• The stream function is defined such that its contours, called streamlines, are everywhere tangent to the velocity vector

Definition of stream function

• In two-dimensional potential flow, the stream function is defined by the relations: $u = \frac{\partial \psi}{\partial y}$ and $v = -\frac{\partial \psi}{\partial x}$, where $u$ and $v$ are the velocity components in the $x$ and $y$ directions, respectively
• These definitions ensure that the continuity equation is automatically satisfied, as $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$

Relationship to velocity potential

• The stream function and velocity potential are related by the Cauchy-Riemann equations in two-dimensional flow:
  • $\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$
  • $\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$
• These relations imply that the velocity potential and stream function are harmonic conjugates, which means that they both satisfy Laplace's equation and can be used interchangeably to describe the flow field

Streamlines

• Streamlines are curves in the flow field that are everywhere tangent to the velocity vector
• They represent the paths that fluid particles would follow if released into the flow
• In two-dimensional potential flow, streamlines are defined by the contours of the stream function, $\psi(x, y) = constant$
• Streamlines have important properties:
  • Fluid particles do not cross streamlines, as the velocity vector is always tangent to the streamline
  • The spacing between streamlines is inversely proportional to the velocity magnitude (closer spacing indicates higher velocity)

Elementary flows

• Elementary flows are simple potential flow solutions that can be combined using the superposition principle to construct more complex flow fields
• These flows serve as building blocks for analyzing flow around various geometries and are characterized by their velocity potential and stream function

Uniform flow

• Uniform flow represents a flow field with constant velocity magnitude and direction throughout the domain
• The velocity potential for a uniform flow in the $x$-direction with velocity $U_\infty$ is given by $\phi = U_\infty x$
• The stream function for a uniform flow in the $x$-direction is given by $\psi = U_\infty y$
• Uniform flow is often used to represent the freestream velocity in aerodynamic problems

Source and sink flow

• A source is a point from which fluid emanates uniformly in all directions, while a sink is a point into which fluid converges uniformly from all directions
• The velocity potential for a source or sink with strength $m$ (positive for a source, negative for a sink) at the origin is given by $\phi = \frac{m}{4\pi r}$, where $r$ is the radial distance from the origin
• The stream function for a source or sink with strength $m$ at the origin is given by $\psi = \frac{m}{2\pi} \theta$, where $\theta$ is the angle measured counterclockwise from the $x$-axis
• Sources and sinks can be used to model the flow into or out of a small opening, such as a hole in a wall or a propeller

Doublet flow

• A doublet is formed by placing a source and a sink of equal strength infinitesimally close together
• The velocity potential for a doublet with strength $\mu$ at the origin, oriented along the $x$-axis, is given by $\phi = \frac{\mu \cos \theta}{2\pi r^2}$
• The stream function for a doublet with strength $\mu$ at the origin, oriented along the $x$-axis, is given by $\psi = \frac{\mu \sin \theta}{2\pi r}$
• Doublets can be used to model the flow around a circular cylinder or a sphere

Vortex flow

• A vortex is a circular flow pattern characterized by concentric streamlines and a velocity that varies inversely with the radial distance from the center
• The velocity potential for a vortex with strength $\Gamma$ (positive for counterclockwise rotation) at the origin is given by $\phi = \frac{\Gamma \theta}{2\pi}$
• The stream function for a vortex with strength $\Gamma$ at the origin is given by $\psi = -\frac{\Gamma}{2\pi} \ln r$
• Vortex flows are used to model the circulation around lifting bodies, such as airfoils, and to represent the vortices shed from the trailing edges of wings

Superposition principle

• The superposition principle states that, for linear systems, the net response caused by multiple stimuli is the sum of the responses that would have been caused by each stimulus individually
• In potential flow theory, the superposition principle allows for the combination of elementary flows to construct more complex flow fields
• This principle is applicable because the governing equation (Laplace's equation) and the boundary conditions are linear

Linear combination of flows

• To construct a complex flow field using the superposition principle, the velocity potentials or stream functions of the individual elementary flows are added together
• For example, the velocity potential for a combination of a uniform flow and a doublet can be written as $\phi = U_\infty x + \frac{\mu \cos \theta}{2\pi r^2}$
• Similarly, the stream function for the same combination would be $\psi = U_\infty y + \frac{\mu \sin \theta}{2\pi r}$

Constructing complex flows

• By strategically placing and combining elementary flows, it is possible to construct flow fields that satisfy the boundary conditions for various geometries
• For example, the flow around a circular cylinder can be modeled by superimposing a uniform flow and a doublet at the origin
• The flow around an airfoil can be approximated by combining a uniform flow, a vortex, and a source-sink pair (to represent the thickness of the airfoil)
• The strengths and positions of the elementary flows are determined by enforcing the appropriate boundary conditions (e.g., no flow through the surface of the cylinder or airfoil)

Flow around simple geometries

• Potential flow theory can be used to analyze the flow around simple geometries, such as cylinders, spheres, and airfoils
• By constructing the appropriate combination of elementary flows and enforcing the necessary boundary conditions, the velocity potential and stream function for these flows can be determined

Flow around a cylinder

• The flow around a circular cylinder can be modeled by superimposing a uniform flow and a doublet at the origin
• The velocity potential for this combination is given by $\phi = U_\infty (r + \frac{a^2}{r}) \cos \theta$, where $U_\infty$ is the freestream velocity, $a$ is the radius of the cylinder, and $(r, \theta)$ are the polar coordinates
• The stream function for the flow around a cylinder is given by $\psi = U_\infty (r - \frac{a^2}{r}) \sin \theta$
• The streamlines for this flow show a symmetric pattern, with the fluid dividing smoothly around the cylinder and reuniting behind it

Flow around a sphere

• The flow around a sphere can be modeled using a similar approach to the flow around a cylinder, but with a three-dimensional doublet
• The velocity potential for the flow around a sphere is given by $\phi = U_\infty (r + \frac{a^3}{2r^2}) \cos \theta$, where $a$ is the radius of the sphere and $(r, \theta)$ are the spherical coordinates
• The streamlines for the flow around a sphere show a similar symmetric pattern to the flow around a cylinder, but with the fluid moving in three dimensions

Flow around an airfoil

• The flow around an airfoil can be approximated by combining a uniform flow, a vortex, and a source-sink pair
• The vortex is used to model the circulation around the airfoil, which is responsible for generating lift
• The source-sink pair is used to model the thickness of the airfoil, with the source representing the leading edge and the sink representing the trailing edge
• The strengths and positions of these elementary flows are determined by enforcing the Kutta condition, which requires the flow to leave the trailing edge smoothly
• The resulting flow field provides a reasonable approximation of the pressure distribution and lift generated by the airfoil, although it does not account for viscous effects or flow separation

Kutta-Joukowski theorem

• The Kutta-Joukowski theorem is a fundamental result in aerodynamics that relates the lift generated by an airfoil to the circulation around it
• It states that the lift per unit span, $L'$, is equal to the product of the fluid density, $\rho$, the freestream velocity, $U_\infty$, and the circulation, $\Gamma$: $L' = \rho U_\infty \Gamma$
• This theorem provides a link between the potential flow solution for an airfoil and the actual lift force experienced by the airfoil

Circulation and lift

• Circulation is a measure of the rotational motion of a fluid and is defined as the line integral of the velocity vector along a closed curve: $\Gamma = \oint \vec{V} \cdot d\vec{l}$
• In potential flow, the circulation around an airfoil is related to the strength of the vortex used to model the flow
• The Kutta-Joukowski theorem shows that the lift generated by an airfoil is directly proportional to the circulation around it
• This relationship highlights the importance of circulation in the generation of lift and provides a basis for designing airfoils with desired lift characteristics

Conformal mapping

• Conformal mapping is a mathematical technique used to transform complex geometries into simpler shapes while preserving the local angles between curves
• In the context of potential flow, conformal mapping can be used to transform the flow around an airfoil into the flow around a cylinder or a flat plate
• This transformation simplifies the analysis of the flow and allows for the application of the Kutta-Joukowski theorem to determine the lift generated by the airfoil
• The Joukowski transformation is a specific conformal mapping that transforms a cylinder into an airfoil shape

Joukowski airfoils

• Joukowski airfoils are a family of airfoil shapes obtained by applying the Joukowski transformation to a cylinder with a superimposed vortex
• The shape of the resulting airfoil depends on the strength and position of the vortex, as well as the radius of the cylinder
• Joukowski airfoils have a rounded leading edge and a sharp trailing edge, which helps to enforce the Kutta condition and ensure smooth flow separation
• These airfoils are useful for theoretical studies and provide a simple way to generate airfoil shapes with desired lift characteristics

Limitations of potential flow theory

• While potential flow theory provides a useful framework for analyzing fluid motion, it has several limitations that arise from the assumptions made in its derivation
• These limitations should be considered when applying potential flow theory to real-world aerodynamic problems

Neglecting viscous effects

• Potential flow theory assumes that the fluid is inviscid, meaning that there are no shear stresses acting between fluid particles or between the fluid and solid surfaces
• In reality, all fluids have some degree of viscosity, which leads to the formation of boundary layers and wakes near solid surfaces
• Neglecting viscous effects can lead to inaccurate predictions of drag forces and flow separation, particularly at high angles of attack or in regions of adverse pressure gradients

Inability to predict flow separation

• Flow separation occurs when the boundary layer detaches from the surface of a body, creating a region of recirculating flow
• Potential flow theory, being inviscid, cannot directly predict flow separation, as it does not account for the viscous effects that cause the boundary layer to separate
• This limitation can lead to overestimations of lift and underestimations of drag, particularly for airfoils at high angles of attack or for bluff bodies with large regions of separated flow

Compressibility effects

• Potential flow theory assumes that the fluid is incompressible, meaning that the density remains constant throughout the flow field
• This assumption is valid for low-speed flows, where the Mach number is typically less than 0.3
• However, for high-speed flows, such as those encountered by aircraft at transonic or supersonic speeds, compressibility effects become significant
• Compressibility can lead to the formation of shock waves and other phenomena that cannot be captured by potential flow theory

Numerical methods for potential flow

• While analytical solutions to potential flow problems can be obtained for simple geometries, more complex flows often require the use of numerical methods
• Numerical methods discretize the flow domain into smaller elements and solve the governing equations (Laplace's equation) subject to the appropriate boundary conditions
• Several numerical methods have been developed for solving potential flow problems, each with its own strengths and weaknesses

Panel methods

• Panel methods are a class of numerical techniques that discretize the surface of a body into a series of panels, each with an associated distribution of singularities (sources, sinks, doublets, or vortices)
• The strengths of these singularities are determined by enforcing the boundary conditions at the control points of each panel, resulting in a system of linear equations
• Panel methods are particularly well-suited for analyzing the flow around complex geometries, such as multi-element airfoils or aircraft configurations
• The accuracy of panel methods can be improved by increasing the number of panels or by using higher-order singularity distributions

Boundary element methods

• Boundary element methods (BEM) are similar to panel methods but use more advanced mathematical techniques to solve the