4.2 Velocity potential

Velocity potential simplifies fluid flow analysis by describing irrotational flows with a scalar function. It reduces variables needed and satisfies Laplace's equation for incompressible flows. This powerful tool allows for easier calculation of velocity fields and other flow properties.

Understanding velocity potential is crucial for analyzing various flow scenarios. It enables the study of uniform flows, sources, sinks, and flow around objects like cylinders and spheres. Complex potential theory extends this concept to two-dimensional flows using complex analysis.

Definition of velocity potential

• Velocity potential is a scalar function that describes the velocity field of an irrotational flow
• It simplifies the analysis of fluid flow by reducing the number of variables needed to describe the flow
• The velocity potential is denoted by the Greek letter $\phi$ and has units of $m^2/s$

Irrotational flow and velocity potential

Top images from around the web for Irrotational flow and velocity potential

• 

  Bernoulli’s Equation – University Physics Volume 1 View original

  Is this image relevant?

• 

  Fluid Dynamics – University Physics Volume 1 View original

  Is this image relevant?

• 

  Fluid Dynamics – University Physics Volume 1 View original

  Is this image relevant?

• 

  Bernoulli’s Equation – University Physics Volume 1 View original

  Is this image relevant?

• 

  Fluid Dynamics – University Physics Volume 1 View original

  Is this image relevant?

1 of 3

Top images from around the web for Irrotational flow and velocity potential

• 

  Bernoulli’s Equation – University Physics Volume 1 View original

  Is this image relevant?

• 

  Fluid Dynamics – University Physics Volume 1 View original

  Is this image relevant?

• 

  Fluid Dynamics – University Physics Volume 1 View original

  Is this image relevant?

• 

  Bernoulli’s Equation – University Physics Volume 1 View original

  Is this image relevant?

• 

  Fluid Dynamics – University Physics Volume 1 View original

  Is this image relevant?

1 of 3

• Irrotational flow is a type of fluid flow where the fluid particles do not rotate about their own axis
• In irrotational flow, the curl of the velocity field is zero $(\nabla \times \vec{V} = 0)$
• For irrotational flows, a velocity potential exists such that the velocity field can be expressed as the gradient of the velocity potential $(\vec{V} = \nabla \phi)$

Laplace's equation for velocity potential

• For incompressible and irrotational flows, the velocity potential satisfies Laplace's equation $(\nabla^2 \phi = 0)$
• Laplace's equation is a second-order partial differential equation that describes the spatial distribution of the velocity potential
• Solving Laplace's equation with appropriate boundary conditions allows for the determination of the velocity potential and, consequently, the velocity field

Properties of velocity potential

• The velocity potential is a powerful tool for analyzing irrotational flows due to its unique properties
• These properties simplify the analysis and provide insights into the behavior of the flow

Relationship between velocity potential and velocity field

• The velocity field can be obtained from the velocity potential by taking its gradient $(\vec{V} = \nabla \phi)$
• The x-component of velocity is given by $u = \frac{\partial \phi}{\partial x}$, the y-component by $v = \frac{\partial \phi}{\partial y}$, and the z-component by $w = \frac{\partial \phi}{\partial z}$
• This relationship allows for the determination of the velocity field once the velocity potential is known

Uniqueness of velocity potential

• For a given irrotational flow, the velocity potential is unique up to an additive constant
• This means that if two velocity potentials satisfy the same boundary conditions, they will differ only by a constant value
• The uniqueness property ensures that the solution for the velocity potential is well-defined and consistent

Superposition principle for velocity potential

• The velocity potential satisfies the superposition principle due to the linearity of Laplace's equation
• If $\phi_1$ and $\phi_2$ are velocity potentials for two irrotational flows, then their sum $(\phi_1 + \phi_2)$ is also a valid velocity potential
• The superposition principle allows for the construction of complex flow fields by combining simpler flow solutions

Boundary conditions for velocity potential

• To solve for the velocity potential, appropriate boundary conditions must be specified
• The boundary conditions describe the behavior of the flow at the boundaries of the domain and ensure a unique solution

Solid boundary conditions

• At a solid boundary, the fluid velocity normal to the surface must be zero to satisfy the no-penetration condition
• This condition is expressed as $\frac{\partial \phi}{\partial n} = 0$, where $n$ is the normal direction to the solid surface
• For moving solid boundaries, the normal velocity of the fluid must match the normal velocity of the boundary

Free surface boundary conditions

• At a free surface (e.g., the interface between a liquid and air), two conditions must be satisfied:
  1. Kinematic condition: The fluid particles at the free surface must remain on the surface
  2. Dynamic condition: The pressure at the free surface must be equal to the atmospheric pressure
• These conditions lead to the free surface boundary condition $\frac{\partial \phi}{\partial n} = \frac{\partial \eta}{\partial t}$, where $\eta$ is the free surface elevation

Far-field boundary conditions

• Far away from the region of interest, the flow is assumed to be undisturbed and uniform
• The velocity potential should approach the uniform flow potential $(\phi_\infty)$ as the distance from the origin tends to infinity
• This condition is expressed as $\lim_{r \to \infty} \phi = \phi_\infty$, where $r$ is the radial distance from the origin

Applications of velocity potential

• Velocity potential is a powerful tool for analyzing various types of irrotational flows
• By solving for the velocity potential, the velocity field and other flow properties can be determined

Uniform flow and velocity potential

• For a uniform flow with velocity $U$ in the x-direction, the velocity potential is given by $\phi = Ux$
• The velocity field is constant and equal to $\vec{V} = (U, 0, 0)$
• Uniform flow is a fundamental building block for more complex flows and is often used as a far-field boundary condition

Source, sink, and doublet flow

• A source is a point from which fluid emanates uniformly in all directions, while a sink is a point where fluid is uniformly absorbed
• The velocity potential for a source or sink with strength $m$ is given by $\phi = \frac{m}{4\pi r}$, where $r$ is the radial distance from the source or sink
• A doublet is formed by placing a source and a sink of equal strength infinitesimally close to each other
• The velocity potential for a doublet with strength $\mu$ oriented along the x-axis is given by $\phi = \frac{\mu \cos \theta}{4\pi r^2}$, where $\theta$ is the angle between the radial direction and the x-axis

Flow around a cylinder using velocity potential

• The flow around a circular cylinder can be modeled using a doublet and a uniform flow
• The velocity potential for the flow around a cylinder of radius $a$ with a uniform flow $U$ in the x-direction is given by $\phi = U(r + \frac{a^2}{r})\cos \theta$
• By taking the gradient of the velocity potential, the velocity field and streamlines can be obtained

Flow around a sphere using velocity potential

• The flow around a sphere can be modeled using a doublet and a uniform flow, similar to the flow around a cylinder
• The velocity potential for the flow around a sphere of radius $a$ with a uniform flow $U$ in the x-direction is given by $\phi = U(r + \frac{a^3}{2r^2})\cos \theta$
• The velocity field and streamlines can be obtained by taking the gradient of the velocity potential

Complex potential theory

• Complex potential theory is an extension of velocity potential theory that uses complex analysis to study two-dimensional irrotational flows
• It provides a powerful framework for solving flow problems and visualizing flow patterns

Complex velocity potential definition

• The complex velocity potential $w(z)$ is a complex-valued function that combines the velocity potential $\phi$ and the stream function $\psi$
• It is defined as $w(z) = \phi(x, y) + i\psi(x, y)$, where $z = x + iy$ is the complex coordinate
• The real part of the complex velocity potential represents the velocity potential, while the imaginary part represents the stream function

Relationship between complex potential and velocity field

• The velocity field can be obtained from the complex velocity potential by taking its derivative with respect to the complex coordinate $z$
• The complex velocity is given by $V(z) = \frac{dw}{dz} = u - iv$, where $u$ and $v$ are the x and y components of the velocity, respectively
• This relationship allows for the determination of the velocity field from the complex velocity potential

Conformal mapping using complex potential

• Conformal mapping is a technique that uses complex analysis to transform one flow domain into another while preserving local angles
• By finding a suitable complex velocity potential, a flow in a complicated domain can be mapped to a simpler domain where the solution is known
• Common conformal mappings include the Joukowski transformation, which maps the flow around a cylinder to the flow around an airfoil

Numerical methods for velocity potential

• Analytical solutions for velocity potential are not always possible, especially for flows with complex geometries or boundary conditions
• Numerical methods provide a way to approximate the solution of the velocity potential and the associated flow field

Finite difference method for velocity potential

• The finite difference method discretizes the flow domain into a grid and approximates the derivatives in Laplace's equation using finite differences
• The resulting system of linear equations is solved to obtain the velocity potential at the grid points
• The velocity field can then be calculated from the velocity potential using finite difference approximations of the gradient

Boundary element method for velocity potential

• The boundary element method (BEM) is a numerical technique that discretizes only the boundaries of the flow domain
• It is based on the integral formulation of Laplace's equation and uses fundamental solutions (e.g., sources, sinks, and doublets) to represent the velocity potential
• BEM reduces the dimensionality of the problem and is particularly useful for exterior flow problems with unbounded domains

Comparison of numerical methods for velocity potential

• The choice of numerical method depends on the specific flow problem, the geometry of the domain, and the desired accuracy
• Finite difference methods are straightforward to implement but may require a large number of grid points for accurate solutions
• BEM is more efficient for exterior flow problems and can handle complex geometries, but it requires the evaluation of singular integrals and the solution of dense linear systems
• Other numerical methods, such as finite element methods and spectral methods, can also be used to solve for the velocity potential

Limitations of velocity potential

• While velocity potential theory is a powerful tool for analyzing irrotational flows, it has certain limitations that should be considered

Applicability to rotational flows

• Velocity potential theory is strictly applicable only to irrotational flows, where the curl of the velocity field is zero
• For rotational flows, such as those encountered in turbulence or flows with vorticity, velocity potential theory cannot be directly applied
• In such cases, alternative formulations, such as the stream function or the vorticity-velocity formulation, may be more appropriate

Nonlinear effects and velocity potential

• Velocity potential theory is based on the assumption of inviscid and irrotational flow, which leads to a linear governing equation (Laplace's equation)
• However, many real-world flows exhibit nonlinear effects, such as flow separation, vortex shedding, and turbulence
• These nonlinear effects cannot be captured by the linear velocity potential formulation, and more advanced models, such as the Navier-Stokes equations, are required

Compressibility effects and velocity potential

• Velocity potential theory assumes that the fluid is incompressible, meaning that the density of the fluid remains constant
• For flows with significant compressibility effects, such as high-speed gas flows or flows with large pressure variations, the incompressibility assumption breaks down
• In such cases, the velocity potential formulation must be modified to account for the compressibility of the fluid, leading to more complex governing equations and boundary conditions