💨Mathematical Fluid Dynamics Unit 4 – Inviscid and Potential Flow Theory

Key Concepts and Definitions

• Inviscid flow assumes fluid has no viscosity, allowing for simplified analysis of fluid behavior in certain scenarios
• Potential flow theory describes irrotational, inviscid, and incompressible flows using velocity potential and stream functions
• Irrotational flow has zero vorticity $(\nabla \times \vec{V} = 0)$, meaning fluid particles do not rotate as they move
• Incompressible flow assumes constant fluid density, valid for low-speed flows (Mach number < 0.3)
• Velocity potential $(\phi)$ is a scalar function whose gradient gives the velocity field $(\vec{V} = \nabla \phi)$
  • Exists only for irrotational flows
• Stream function $(\psi)$ is another scalar function that describes streamlines, lines tangent to velocity vectors at every point
  • Relationship between velocity potential and stream function: $u = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$, $v = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$
• Elementary flows are simple potential flow solutions (uniform flow, source/sink, vortex, doublet) that can be combined to model more complex flows

Governing Equations

• Continuity equation for incompressible flow: $\nabla \cdot \vec{V} = 0$
  • Expresses conservation of mass
• Euler equations for inviscid flow: $\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{g}$
  • Momentum conservation without viscous terms
• Bernoulli's equation for steady, inviscid, incompressible flow along a streamline: $\frac{1}{2}\rho V^2 + \rho gz + p = \text{constant}$
  • Relates velocity, pressure, and elevation
• Laplace's equation for velocity potential: $\nabla^2 \phi = 0$
  • Governs potential flow, derived from continuity equation and irrotationality condition
• Poisson's equation for stream function: $\nabla^2 \psi = -\omega$, where $\omega$ is vorticity
  • Relates stream function to vorticity in 2D flows
• Boundary conditions:
  • No-penetration: $\vec{V} \cdot \vec{n} = 0$ at solid boundaries
  • Far-field: $\vec{V} \rightarrow \vec{V}_\infty$ as distance from disturbance $\rightarrow \infty$

Assumptions and Limitations

• Inviscid flow assumes no viscosity, neglecting boundary layers and viscous drag
  • Reasonable approximation for high-Reynolds-number flows away from boundaries
• Potential flow theory assumes irrotational, inviscid, and incompressible flow
  • Irrotationality breaks down in regions with strong vorticity (wakes, separated flows)
• Incompressibility assumption valid for low-speed flows (Mach < 0.3), but fails for high-speed or gas flows with significant density changes
• Potential flow theory cannot capture flow separation, stall, or turbulence directly
  • Modifications like Kutta condition and vortex panels can partially address these limitations
• Steady flow assumption neglects time-dependent effects, limiting applicability to unsteady flows
• Two-dimensional simplifications (using complex potential) do not capture 3D effects like wingtip vortices
• Inviscid and potential flow theories provide valuable insights but may require corrections or empirical adjustments for real-world applications

Potential Flow Theory Basics

• Velocity potential $(\phi)$ is a scalar function that fully describes irrotational flow
  • Velocity field is the gradient of velocity potential: $\vec{V} = \nabla \phi$
• Stream function $(\psi)$ is another scalar function that describes 2D incompressible flows
  • Streamlines are lines of constant $\psi$
• Relationship between velocity potential and stream function in 2D: $\phi + i\psi = f(z)$, where $f(z)$ is a complex potential and $z = x + iy$
  • Real part of $f(z)$ is velocity potential, imaginary part is stream function
• Laplace's equation $(\nabla^2 \phi = 0)$ governs velocity potential in potential flow
  • Solutions to Laplace's equation are harmonic functions
• Elementary flows (uniform flow, source/sink, vortex, doublet) are basic solutions to Laplace's equation
  • Can be superposed to model more complex flows
• Circulation $(\Gamma)$ is the line integral of velocity around a closed curve: $\Gamma = \oint_C \vec{V} \cdot d\vec{l}$
  • Related to vorticity and lift generation in potential flow theory

Stream Functions and Velocity Potential

• Stream function $(\psi)$ is a scalar function that describes streamlines in 2D incompressible flow
  • Defined such that $u = \frac{\partial \psi}{\partial y}$ and $v = -\frac{\partial \psi}{\partial x}$
• Streamlines are lines of constant $\psi$, tangent to velocity vectors at every point
  • No flow crosses streamlines in steady flow
• Velocity potential $(\phi)$ is a scalar function whose gradient gives the velocity field: $\vec{V} = \nabla \phi$
  • Exists only for irrotational flows
• In 2D, velocity potential and stream function are related by Cauchy-Riemann equations:
  • $\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$ and $\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$
• Complex potential $f(z) = \phi + i\psi$ combines velocity potential and stream function
  • Powerful tool for analyzing 2D potential flows using complex analysis
• Equipotential lines (constant $\phi$) are perpendicular to streamlines (constant $\psi$)
  • Form an orthogonal grid in 2D potential flow
• Stream function and velocity potential provide a compact description of flow field
  • Useful for flow visualization and calculating velocity, pressure, and forces

Elementary Flows and Superposition

• Elementary flows are simple potential flow solutions that satisfy Laplace's equation
  • Building blocks for more complex flows through superposition
• Uniform flow: $\phi = U_\infty x$, $\psi = U_\infty y$
  • Constant velocity $U_\infty$ in x-direction
• Source/sink: $\phi = \frac{m}{2\pi} \ln r$, $\psi = \frac{m}{2\pi} \theta$
  • Radial flow with strength $m$ (positive for source, negative for sink)
• Vortex: $\phi = \frac{\Gamma}{2\pi} \theta$, $\psi = -\frac{\Gamma}{2\pi} \ln r$
  • Circular flow with circulation $\Gamma$
• Doublet: $\phi = -\frac{\mu}{2\pi} \frac{x}{x^2 + y^2}$, $\psi = -\frac{\mu}{2\pi} \frac{y}{x^2 + y^2}$
  • Formed by bringing source and sink infinitely close with strength $\mu$
• Superposition principle: Solutions to Laplace's equation can be added to obtain new solutions
  • Allows modeling complex flows by combining elementary flows
• Example: Flow around a cylinder can be modeled by superposing uniform flow and doublet
  • Doublet strength chosen to satisfy no-penetration boundary condition
• Conformal mapping techniques use complex analysis to transform elementary flows
  • Enables solving potential flow problems around arbitrary 2D shapes

Boundary Conditions and Uniqueness

• Boundary conditions specify flow behavior at domain boundaries
  • Essential for obtaining unique solutions to potential flow problems
• No-penetration condition: $\vec{V} \cdot \vec{n} = 0$ at solid boundaries
  • Ensures flow does not pass through solid surfaces
• Far-field condition: $\vec{V} \rightarrow \vec{V}_\infty$ as distance from disturbance $\rightarrow \infty$
  • Specifies uniform flow velocity far from the region of interest
• Kutta condition: Smooth flow separation at sharp trailing edges
  • Determines circulation around lifting bodies to ensure finite velocity at trailing edge
• Kelvin's circulation theorem: Circulation around a closed contour remains constant in inviscid, barotropic flow
  • Helps determine circulation and lift in potential flow theory
• Uniqueness theorem: Solution to Laplace's equation with given boundary conditions is unique
  • Guarantees a single, physically meaningful solution for well-posed potential flow problems
• Proper specification of boundary conditions is crucial for obtaining realistic potential flow solutions
  • Incorrect or insufficient boundary conditions can lead to non-unique or unphysical results
• Techniques like conformal mapping and panel methods help enforce boundary conditions
  • Enable solving potential flow problems around complex geometries

Applications and Examples

• Aerodynamics: Potential flow theory is used to analyze lift, drag, and pressure distribution on airfoils and wings
  • Kutta-Joukowski theorem relates circulation to lift: $L = \rho_\infty V_\infty \Gamma$
• Hydrodynamics: Potential flow models are applied to study flow around ships, submarines, and underwater vehicles
  • Helps optimize hull designs for reduced drag and improved performance
• Wind turbines: Potential flow theory is used to design and analyze wind turbine blades
  • Betz limit defines maximum theoretical efficiency of a wind turbine based on potential flow considerations
• Groundwater flow: Potential flow theory is applied to model groundwater flow in porous media
  • Helps predict contaminant transport and design remediation strategies
• Heat transfer: Analogies between potential flow and heat conduction enable solving thermal problems using potential flow techniques
  • Example: Insulated walls in heat conduction are analogous to streamlines in potential flow
• Conformal mapping: Technique for transforming complex geometries into simpler domains using complex analysis
  • Enables solving potential flow problems around arbitrary 2D shapes
• Panel methods: Numerical technique for solving potential flow problems by discretizing surfaces into panels with elementary flow solutions
  • Allows modeling complex 3D geometries and enforcing boundary conditions
• Vortex methods: Lagrangian approach to potential flow that tracks vorticity evolution using discrete vortex elements
  • Captures unsteady and separated flows better than traditional potential flow methods

Advanced Topics and Extensions

• Unsteady potential flow: Extends potential flow theory to time-dependent problems
  • Governed by the unsteady Bernoulli equation: $\frac{\partial \phi}{\partial t} + \frac{1}{2}|\nabla \phi|^2 + \frac{p}{\rho} + gz = \text{constant}$
• Three-dimensional potential flow: Applies potential flow theory to 3D problems
  • Velocity potential satisfies Laplace's equation in 3D: $\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$
• Lifting line theory: Extends potential flow theory to analyze finite-span wings
  • Models wingtip vortices and induced drag
• Vortex panel methods: Combine vortex methods with panel methods to model unsteady and separated flows
  • Captures leading-edge vortices and dynamic stall on airfoils
• Boundary layer corrections: Incorporate viscous effects by coupling potential flow solutions with boundary layer equations
  • Example: Thwaites' method for laminar boundary layers
• Free-streamline theory: Models flows with free surfaces or cavities using potential flow theory
  • Applies to problems like water entry, cavitation, and jet flows
• Vortex sound theory: Relates sound generation to unsteady vorticity in potential flows
  • Helps predict noise from turbomachinery and wind turbines
• Biomechanics applications: Potential flow theory is used to model blood flow in large arteries and aquatic animal propulsion
  • Provides insights into cardiovascular health and bio-inspired robot design
• Coupling with other methods: Potential flow solutions can be used as initial or boundary conditions for more advanced CFD simulations
  • Example: Using potential flow to initialize a RANS simulation for faster convergence