💨Fluid Dynamics Unit 4 – Incompressible inviscid flows

Key Concepts and Definitions

• Incompressible flow assumes fluid density remains constant throughout the flow field
• Inviscid flow neglects the effects of viscosity, assuming no friction between fluid particles or between the fluid and solid boundaries
• Potential flow is a type of inviscid flow where the velocity field can be expressed as the gradient of a scalar potential function
• Irrotational flow has zero vorticity, meaning fluid particles do not rotate about their own axes as they move along streamlines
  • Potential flows are always irrotational, but not all irrotational flows are potential flows
• Streamlines are curves tangent to the velocity vector at every point in a steady flow field
• Streamfunction $\psi$ is a scalar function whose contours represent streamlines in a 2D incompressible flow
• Velocity potential $\phi$ is a scalar function whose gradient gives the velocity field in a potential flow

Governing Equations

• Conservation of mass for incompressible flow: $\nabla \cdot \mathbf{u} = 0$, where $\mathbf{u}$ is the velocity vector
  • This equation is also known as the continuity equation
• Euler's equation for inviscid flow: $\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \rho \mathbf{g}$, where $\rho$ is density, $p$ is pressure, and $\mathbf{g}$ is the gravitational acceleration vector
• For steady, irrotational, inviscid flow, Euler's equation simplifies to the Bernoulli equation: $\frac{1}{2}\rho |\mathbf{u}|^2 + p + \rho g z = \text{constant}$, where $z$ is the vertical coordinate
• Vorticity equation for inviscid flow: $\frac{D \boldsymbol{\omega}}{Dt} = \boldsymbol{\omega} \cdot \nabla \mathbf{u}$, where $\boldsymbol{\omega} = \nabla \times \mathbf{u}$ is the vorticity vector and $\frac{D}{Dt}$ is the material derivative
  • For irrotational flow, $\boldsymbol{\omega} = 0$ everywhere in the flow field

Potential Flow Theory

• Potential flow assumes the velocity field can be expressed as the gradient of a scalar potential function: $\mathbf{u} = \nabla \phi$
• The continuity equation for incompressible potential flow reduces to Laplace's equation: $\nabla^2 \phi = 0$
• Potential flow theory allows for the superposition of elementary flow solutions (uniform flow, source/sink, doublet, vortex) to construct more complex flow patterns
• The velocity potential $\phi$ and streamfunction $\psi$ are related by the Cauchy-Riemann equations in 2D: $\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$ and $\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$
  • This relationship allows for the use of complex potential $w(z) = \phi + i\psi$, where $z = x + iy$ is the complex coordinate
• Potential flow theory is particularly useful for analyzing flows around simple geometries (cylinders, spheres, airfoils) and for understanding the basic features of more complex flows

Streamlines and Streamfunctions

• Streamlines are curves that are everywhere tangent to the velocity vector at a given instant in time
  • In steady flow, fluid particles follow streamlines, while in unsteady flow, streamlines and particle paths may differ
• The streamfunction $\psi$ is a scalar function whose contours represent streamlines in a 2D incompressible flow
  • The velocity components are related to the streamfunction by $u = \frac{\partial \psi}{\partial y}$ and $v = -\frac{\partial \psi}{\partial x}$
• In 2D polar coordinates $(r, \theta)$, the velocity components are $u_r = \frac{1}{r}\frac{\partial \psi}{\partial \theta}$ and $u_\theta = -\frac{\partial \psi}{\partial r}$
• The difference in streamfunction values between two points gives the volume flow rate per unit depth between the corresponding streamlines
• Streamlines cannot cross each other in steady flow, as this would imply multiple velocity vectors at the point of intersection
• Stagnation points occur where the velocity is zero, and the streamfunction has a local maximum, minimum, or saddle point

Velocity Potential and Laplace's Equation

• The velocity potential $\phi$ is a scalar function whose gradient gives the velocity field in a potential flow: $\mathbf{u} = \nabla \phi$
• For incompressible potential flow, the continuity equation reduces to Laplace's equation: $\nabla^2 \phi = 0$
  • In 2D Cartesian coordinates, Laplace's equation is $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$
  • In 2D polar coordinates, Laplace's equation is $\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \phi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \phi}{\partial \theta^2} = 0$
• Solutions to Laplace's equation are called harmonic functions and have important mathematical properties (e.g., uniqueness, maximum principle)
• Elementary potential flow solutions (uniform flow, source/sink, doublet, vortex) are building blocks for more complex potential flows and can be superposed due to the linearity of Laplace's equation
• The velocity potential is related to the pressure through the Bernoulli equation in steady, irrotational, inviscid flow: $\frac{1}{2}\rho |\nabla \phi|^2 + p + \rho g z = \text{constant}$

Boundary Conditions and Flow Patterns

• To solve for the velocity potential or streamfunction, appropriate boundary conditions must be specified
• The no-penetration condition states that the velocity normal to a solid boundary must be zero: $\mathbf{u} \cdot \mathbf{n} = 0$, where $\mathbf{n}$ is the unit normal vector to the boundary
  • This condition is satisfied by setting the streamfunction to a constant value along the boundary
• The no-slip condition, which states that the tangential velocity at a solid boundary must be zero, is not applicable to inviscid flows
• Far-field boundary conditions specify the behavior of the flow at large distances from the object of interest (e.g., uniform flow, quiescent fluid)
• Kutta condition is applied at sharp trailing edges (airfoils) to ensure smooth flow separation and uniqueness of the solution
• Circulation $\Gamma$ around a closed contour is defined as the line integral of the velocity: $\Gamma = \oint_C \mathbf{u} \cdot d\mathbf{l}$
  • In potential flow, circulation is related to the jump in velocity potential: $\Gamma = \Delta \phi$
• Lift on an airfoil is directly proportional to the circulation around it (Kutta-Joukowski theorem): $L = \rho_\infty V_\infty \Gamma$, where $\rho_\infty$ and $V_\infty$ are the freestream density and velocity, respectively

Applications and Examples

• Flow around a circular cylinder can be modeled using a doublet and a uniform flow, with the doublet strength adjusted to satisfy the no-penetration condition
• Flow around a sphere can be modeled using a point source and a uniform flow, with the source strength determined by the sphere's radius and the freestream velocity
• Rankine oval is a simple model for flow around a bluff body, constructed by superposing a uniform flow and a source-sink pair
• Flow around a Joukowski airfoil can be obtained by conformal mapping of the flow around a cylinder using the Joukowski transformation
  • The Kutta condition is applied at the trailing edge to determine the circulation and lift
• Potential flow theory can be used to analyze the flow in a 90-degree bend by superposing a uniform flow and a doublet
• Wind tunnel contraction and diffuser sections can be designed using potential flow theory to minimize flow non-uniformity and separation
• Potential flow solutions serve as a starting point for more advanced numerical methods (panel methods, vortex methods) that can handle more complex geometries and flow conditions

Problem-Solving Techniques

• Identify the key features of the problem (geometry, boundary conditions, far-field behavior) and select appropriate elementary potential flow solutions
• Use the principle of superposition to combine elementary solutions and satisfy the boundary conditions
• Apply conformal mapping techniques to transform the flow around simple geometries (circles, polygons) into the desired flow around more complex shapes (airfoils, ducts)
• Exploit the symmetry of the problem, if present, to simplify the analysis and reduce the computational domain
• Use the Kutta condition to determine the circulation and lift for flows around airfoils and other lifting bodies
• Employ complex potential and complex velocity to streamline the analysis of 2D potential flows
• Verify that the solution satisfies the governing equations (Laplace's equation) and the specified boundary conditions
• Validate the potential flow solution against experimental data or higher-fidelity numerical simulations, and be aware of the limitations of the potential flow assumptions (inviscid, irrotational)