2.5 Potential flow

Potential flow simplifies fluid motion by assuming irrotational, inviscid, and incompressible flow. This model helps analyze problems where viscosity and compressibility are negligible, allowing for analytical solutions and insights into fluid behavior.

The velocity potential and stream function are key concepts in potential flow. These scalar functions describe the flow field, simplifying analysis and visualization. Laplace's equation governs both, with boundary conditions determining specific flow solutions.

Basics of potential flow

• Potential flow is a simplified model of fluid motion that assumes the flow is irrotational, inviscid, and incompressible
• This model is useful for analyzing fluid flow problems where the effects of viscosity and compressibility are negligible
• Potential flow theory allows for the derivation of analytical solutions to various flow problems, providing valuable insights into fluid behavior

Velocity potential and stream function

Relationship between velocity potential and velocity field

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• The velocity potential $\phi$ is a scalar function whose gradient gives the velocity field $\mathbf{V}$
  • Mathematically, $\mathbf{V} = \nabla \phi$
• The existence of a velocity potential implies that the flow is irrotational, meaning that the curl of the velocity field is zero ($\nabla \times \mathbf{V} = 0$)
• The velocity potential is a useful tool for simplifying the analysis of potential flows, as it reduces the number of variables needed to describe the flow

Relationship between stream function and velocity field

• The stream function $\psi$ is another scalar function that describes the flow field in two-dimensional potential flows
• The stream function is related to the velocity components by $u = \frac{\partial \psi}{\partial y}$ and $v = -\frac{\partial \psi}{\partial x}$
• Streamlines, which are lines tangent to the velocity vector at every point, coincide with the contours of the stream function
• The stream function is particularly useful for visualizing the flow field and determining the volume flow rate between streamlines

Laplace's equation in potential flow

Derivation of Laplace's equation

• In potential flow, both the velocity potential and the stream function satisfy Laplace's equation
• For the velocity potential, Laplace's equation is derived by applying the continuity equation to the definition of the velocity potential
  • $\nabla^2 \phi = 0$
• Similarly, for the stream function in 2D flows, Laplace's equation is obtained by applying the irrotationality condition
  • $\nabla^2 \psi = 0$

Boundary conditions for Laplace's equation

• To solve Laplace's equation for a specific flow problem, appropriate boundary conditions must be specified
• Common boundary conditions include:
  • Solid boundaries: The velocity normal to the boundary must be zero, which translates to $\frac{\partial \phi}{\partial n} = 0$ or $\psi = constant$ along the boundary
  • Free stream conditions: As the distance from the object tends to infinity, the velocity potential and stream function should approach their respective free-stream values
• The choice of boundary conditions depends on the specific flow problem and the geometry of the domain

Elementary flows in potential flow theory

Uniform flow

• Uniform flow is the simplest type of potential flow, characterized by a constant velocity field $\mathbf{V} = (U, 0, 0)$
• The velocity potential for uniform flow is given by $\phi = Ux$, and the stream function (in 2D) is $\psi = Uy$
• Uniform flow is often used as a building block for more complex flow patterns

Source and sink 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/sink is $\phi = \pm \frac{Q}{4\pi r}$, where $Q$ is the volume flow rate and $r$ is the distance from the source/sink
• The stream function (in 2D) for a source/sink is $\psi = \pm \frac{Q}{2\pi} \theta$, where $\theta$ is the angle measured from the x-axis
• Sources and sinks are used to model flows into or out of a domain, such as in the case of a jet or a drain

Doublet flow

• 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 is $\phi = \frac{\mu \cos \theta}{4\pi r^2}$, where $\mu$ is the doublet strength and $\theta$ is the angle measured from the axis of the doublet
• The stream function (in 2D) for a doublet is $\psi = \frac{\mu \sin \theta}{2\pi r}$
• Doublets are used to model the flow around a circular cylinder or a sphere

Vortex flow

• A vortex is a circular flow pattern where the fluid velocity is inversely proportional to the distance from the center
• The velocity potential for a vortex is $\phi = \frac{\Gamma \theta}{2\pi}$, where $\Gamma$ is the circulation strength and $\theta$ is the angle measured from a reference axis
• The stream function (in 2D) for a vortex is $\psi = -\frac{\Gamma}{2\pi} \ln r$, where $r$ is the distance from the center of the vortex
• Vortex flows are used to model the circulation around a lifting body, such as an airfoil or a propeller blade

Superposition of elementary flows

Linear combination of elementary flows

• One of the key advantages of potential flow theory is the ability to construct complex flow patterns by superimposing elementary flows
• The velocity potential and stream function of the resulting flow are obtained by adding the corresponding functions of the individual elementary flows
  • $\phi = \phi_1 + \phi_2 + ... + \phi_n$
  • $\psi = \psi_1 + \psi_2 + ... + \psi_n$
• This linear combination is possible because Laplace's equation, which governs potential flows, is a linear partial differential equation

Constructing complex flow patterns

• By combining various elementary flows, such as uniform flow, sources, sinks, doublets, and vortices, one can model a wide range of flow patterns
• Examples of complex flow patterns include:
  • Flow around a circular cylinder: Superposition of uniform flow and a doublet
  • Flow around a Rankine oval: Superposition of uniform flow, a source, and a sink
  • Flow around a lifting airfoil: Superposition of uniform flow, a vortex, and a doublet
• The superposition principle allows for the analysis of more realistic flow problems by breaking them down into simpler, elementary components

Flow past a circular cylinder

Uniform flow past a circular cylinder

• Consider a uniform flow with velocity $U$ past a circular cylinder of radius $a$
• The velocity potential for the uniform flow is $\phi_\infty = Ux$, and the stream function is $\psi_\infty = Uy$
• To satisfy the boundary condition of no flow through the cylinder surface, a doublet flow is superimposed on the uniform flow

Doublet flow past a circular cylinder

• The doublet flow is placed at the center of the cylinder with its axis aligned with the uniform flow
• The strength of the doublet is chosen to be $\mu = -Ua^2$ to cancel the normal velocity component of the uniform flow at the cylinder surface
• The velocity potential for the doublet flow is $\phi_d = -\frac{Ua^2 \cos \theta}{r}$, and the stream function is $\psi_d = -\frac{Ua^2 \sin \theta}{r}$

Superposition of uniform and doublet flow

• The total velocity potential for the flow past a circular cylinder is $\phi = \phi_\infty + \phi_d = Ux - \frac{Ua^2 \cos \theta}{r}$
• The total stream function is $\psi = \psi_\infty + \psi_d = Uy - \frac{Ua^2 \sin \theta}{r}$
• The resulting flow pattern shows the streamlines dividing and reuniting behind the cylinder, forming a symmetric pattern
• Although this potential flow solution does not predict drag on the cylinder, it provides valuable insights into the flow field around the object

Flow past a sphere

Uniform flow past a sphere

• Consider a uniform flow with velocity $U$ past a sphere of radius $a$
• The velocity potential for the uniform flow is $\phi_\infty = Ux$
• To satisfy the boundary condition of no flow through the sphere surface, a doublet flow is superimposed on the uniform flow

Derivation of velocity potential and stream function

• The doublet flow is placed at the center of the sphere with its axis aligned with the uniform flow
• The strength of the doublet is chosen to be $\mu = -\frac{1}{2}Ua^3$ to cancel the normal velocity component of the uniform flow at the sphere surface
• The velocity potential for the doublet flow is $\phi_d = -\frac{Ua^3 \cos \theta}{2r^2}$
• The total velocity potential for the flow past a sphere is $\phi = \phi_\infty + \phi_d = Ux - \frac{Ua^3 \cos \theta}{2r^2}$
• The stream function for the flow past a sphere can be derived using the relationship between the velocity potential and the stream function in axisymmetric flows
  • $\psi = \frac{1}{r \sin \theta} \frac{\partial \phi}{\partial \theta}$
• The resulting flow pattern shows the streamlines dividing and reuniting behind the sphere, similar to the case of a circular cylinder

Lift on a cylinder in potential flow

Kutta-Joukowski theorem

• The Kutta-Joukowski theorem relates the lift on a cylinder to the circulation around it
• According to the theorem, the lift per unit length on a cylinder is given by $L' = \rho U \Gamma$, where $\rho$ is the fluid density, $U$ is the free-stream velocity, and $\Gamma$ is the circulation
• The circulation is defined as the line integral of the velocity field around a closed contour enclosing the cylinder
  • $\Gamma = \oint_C \mathbf{V} \cdot d\mathbf{l}$

Circulation and lift generation

• In potential flow, a cylinder experiences no lift unless there is circulation around it
• Circulation can be generated by introducing a vortex flow around the cylinder, which can be achieved by superimposing a vortex flow on the uniform and doublet flows
• The strength of the vortex is determined by the Kutta condition, which states that the flow should leave the trailing edge of the cylinder smoothly
• The presence of circulation around the cylinder results in an asymmetric flow pattern, with higher velocities on the upper surface and lower velocities on the lower surface
• This asymmetry in the velocity field leads to a pressure difference between the upper and lower surfaces, generating lift on the cylinder

Potential flow vs real fluid flow

Assumptions and limitations of potential flow theory

• Potential flow theory is based on several simplifying assumptions, such as:
  • The fluid is inviscid (no viscosity)
  • The flow is irrotational (no vorticity)
  • The fluid is incompressible (constant density)
• These assumptions limit the applicability of potential flow theory to certain flow regimes and geometries
• Potential flow theory does not account for flow separation, boundary layer formation, or wake development behind objects
• Consequently, potential flow theory cannot predict drag forces on objects, as drag is primarily a result of viscous effects

Viscous effects in real fluid flow

• In real fluids, viscosity plays a crucial role in the flow behavior near solid boundaries
• Viscous effects lead to the formation of boundary layers, which are thin regions near the surface where the velocity gradients are high
• Boundary layers can separate from the surface when the fluid encounters an adverse pressure gradient, leading to flow separation and wake formation
• The presence of viscosity also results in the generation of vorticity in the flow, which is not accounted for in potential flow theory
• To accurately model real fluid flows, more advanced techniques such as Navier-Stokes equations and computational fluid dynamics (CFD) are employed

Numerical methods for potential flow

Finite difference method

• The finite difference method is a numerical technique for solving partial differential equations, such as Laplace's equation in potential flow
• The method discretizes the flow domain into a grid of points and approximates the derivatives in Laplace's equation using finite differences
• The resulting system of linear equations is then solved to obtain the values of the velocity potential or stream function at the grid points
• The finite difference method is relatively simple to implement but may require a fine grid resolution to achieve accurate results

Boundary element method

• The boundary element method (BEM) is another numerical technique for solving potential flow problems
• BEM focuses on discretizing the boundaries of the flow domain rather than the entire domain, as in the finite difference method
• The method is based on the integral formulation of Laplace's equation, which relates the velocity potential at any point in the domain to the values of the potential and its normal derivative on the boundaries
• BEM reduces the dimensionality of the problem by one, as only the boundaries need to be discretized
• This method is particularly efficient for problems with complex geometries and unbounded domains, as the far-field conditions can be easily incorporated

Applications of potential flow theory

Aerodynamics and hydrodynamics

• Potential flow theory finds extensive applications in aerodynamics and hydrodynamics, where the flow around objects is of primary interest
• Some examples include:
  • Airfoil design: Potential flow theory can be used to calculate the lift and pressure distribution on airfoils, which is essential for the design of aircraft wings and propeller blades
  • Submarine hydrodynamics: Potential flow theory can model the flow around submarines and other underwater vehicles, helping to optimize their shape for reduced drag and improved performance
  • Wind turbine design: Potential flow theory can be used to analyze the flow around wind turbine blades, aiding in the design of more efficient and cost-effective wind energy systems

Groundwater flow and seepage

• Potential flow theory also has applications in the field of groundwater hydrology and seepage analysis
• Groundwater flow through porous media, such as soil and rock, can often be modeled using potential flow theory, as the flow is typically slow and governed by Darcy's law
• Some examples include:
  • Seepage under dams: Potential flow theory can be used to analyze the seepage flow under dams and to design appropriate filters and drainage systems to ensure the stability of the dam
  • Groundwater remediation: Potential flow theory can help in the design of groundwater remediation systems, such as pump-and-treat or permeable reactive barriers, by providing insights into the flow patterns and contaminant transport in the subsurface
• In these applications, potential flow theory provides a simplified yet valuable framework for understanding and predicting the behavior of groundwater flow and seepage