💨Mathematical Fluid Dynamics Unit 6 – Vorticity in Incompressible Flow

Key Concepts and Definitions

• Vorticity ($\omega$) measures the local rotation of a fluid element, defined as the curl of the velocity field ($\nabla \times \mathbf{u}$)
  • In Cartesian coordinates, $\omega = (\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}, \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}, \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$
• Incompressible flow assumes constant fluid density ($\rho$) and satisfies the continuity equation ($\nabla \cdot \mathbf{u} = 0$)
• Vortex lines are curves tangent to the vorticity vector at every point, representing the local axis of fluid rotation
• Vortex tubes are surfaces formed by vortex lines passing through a closed curve, enclosing a region of concentrated vorticity
• Irrotational flow has zero vorticity ($\omega = 0$) everywhere, implying the existence of a velocity potential ($\phi$) such that $\mathbf{u} = \nabla \phi$
• Circulation ($\Gamma$) quantifies the net rotation of a fluid along a closed curve, defined as the line integral of velocity ($\oint_C \mathbf{u} \cdot d\mathbf{l}$)

Mathematical Foundations

• Vector calculus concepts, such as gradient ($\nabla$), divergence ($\nabla \cdot$), and curl ($\nabla \times$), are essential for understanding vorticity
• The Navier-Stokes equations govern the motion of incompressible fluids, relating velocity, pressure, and external forces
  • For constant density and viscosity, the equations are $\rho (\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$
• The vorticity equation is derived from the curl of the Navier-Stokes equations, describing the evolution of vorticity in a fluid
• Vector identities, such as $\nabla \times (\nabla \times \mathbf{u}) = \nabla(\nabla \cdot \mathbf{u}) - \nabla^2 \mathbf{u}$, are used in the derivation of the vorticity equation
• Stokes' theorem relates the circulation along a closed curve to the surface integral of vorticity through the enclosed area ($\Gamma = \iint_S \omega \cdot d\mathbf{S}$)
• Green's theorem connects the line integral of a vector field along a closed curve to the double integral of its curl over the enclosed area

Vorticity Equation Derivation

• Begin with the Navier-Stokes equations for incompressible flow, $\rho (\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$
• Take the curl ($\nabla \times$) of both sides, using the vector identity $\nabla \times (\mathbf{u} \cdot \nabla \mathbf{u}) = (\mathbf{u} \cdot \nabla) (\nabla \times \mathbf{u}) + (\nabla \times \mathbf{u}) \cdot \nabla \mathbf{u}$
  • The curl of the pressure gradient term vanishes, $\nabla \times (\nabla p) = 0$, as the curl of a gradient is always zero
• Simplify using the definition of vorticity ($\omega = \nabla \times \mathbf{u}$) and the vector Laplacian ($\nabla^2 \mathbf{u} = \nabla(\nabla \cdot \mathbf{u}) - \nabla \times (\nabla \times \mathbf{u})$)
• The resulting vorticity equation is $\frac{\partial \omega}{\partial t} + (\mathbf{u} \cdot \nabla) \omega = (\omega \cdot \nabla) \mathbf{u} + \nu \nabla^2 \omega + \frac{1}{\rho} \nabla \times \mathbf{f}$, where $\nu = \mu / \rho$ is the kinematic viscosity
• The terms in the vorticity equation represent the rate of change of vorticity, advection of vorticity, vortex stretching, diffusion of vorticity, and vorticity generation due to external forces, respectively

Physical Interpretation of Vorticity

• Vorticity represents the local spinning motion of fluid elements, with its magnitude indicating the rate of rotation and its direction pointing along the axis of rotation
• In inviscid flows (zero viscosity), vortex lines move with the fluid, behaving like elastic strings that can stretch, twist, and bend
• Vortex stretching occurs when fluid elements are elongated along the vorticity direction, intensifying the vorticity magnitude (ice skater effect)
  • Vortex stretching is absent in 2D flows, as the vorticity vector is always perpendicular to the plane of motion
• Diffusion of vorticity is caused by viscous effects, which smooth out vorticity gradients and dissipate vortical structures over time
• Vorticity can be generated at solid boundaries due to the no-slip condition, which creates a thin boundary layer with high velocity gradients
• The interaction of vortices leads to complex flow phenomena, such as vortex shedding (von Kármán vortex street), vortex merging, and turbulence

Vorticity in 2D vs 3D Flows

• In 2D flows, the vorticity vector is always perpendicular to the plane of motion, simplifying the vorticity equation to $\frac{\partial \omega}{\partial t} + (\mathbf{u} \cdot \nabla) \omega = \nu \nabla^2 \omega$
  • The absence of vortex stretching in 2D flows leads to fundamentally different behavior compared to 3D flows
• 2D vortices interact through merging and mutual advection, forming larger coherent structures over time (inverse energy cascade)
• In 3D flows, the vorticity vector has three components, allowing for more complex vortex interactions and structures
• Vortex stretching in 3D flows can lead to the intensification of vorticity and the formation of smaller-scale structures (direct energy cascade)
  • This process is essential for the energy transfer from large to small scales in turbulent flows
• 3D vortex tubes can undergo stretching, twisting, and folding, creating intricate vortical structures and enhancing mixing
• The study of 3D vorticity is crucial for understanding phenomena such as turbulence, aerodynamic drag, and heat transfer in fluid systems

Circulation and Kelvin's Theorem

• Circulation ($\Gamma$) measures the net rotation of a fluid along a closed curve, defined as the line integral of velocity ($\oint_C \mathbf{u} \cdot d\mathbf{l}$)
• Stokes' theorem relates circulation to the surface integral of vorticity through the enclosed area ($\Gamma = \iint_S \omega \cdot d\mathbf{S}$)
• Kelvin's circulation theorem states that in an inviscid, barotropic fluid with conservative body forces, the circulation around a closed curve moving with the fluid remains constant over time
  • Mathematically, $\frac{D\Gamma}{Dt} = 0$, where $\frac{D}{Dt}$ is the material derivative
• Kelvin's theorem implies the conservation of vorticity in inviscid flows, as vortex lines move with the fluid and maintain their strength
• In viscous flows, circulation can change due to diffusion of vorticity and the presence of non-conservative forces
• The conservation of circulation has important consequences for the formation and stability of vortical structures, such as aircraft wakes and atmospheric vortices

Numerical Methods for Vorticity Calculations

• Numerical simulations of fluid flows often involve solving the vorticity equation instead of the primitive variable formulation (velocity-pressure)
• Vorticity-based methods, such as the vortex-in-cell (VIC) and vortex particle methods, discretize the vorticity field using particles or grid points
  • These methods can efficiently handle complex geometries and capture the evolution of vortical structures
• Finite difference schemes are used to approximate the spatial and temporal derivatives in the vorticity equation
  • Central differences, upwind schemes, and compact schemes are common choices for spatial discretization
• Time integration techniques, such as Runge-Kutta methods and Adams-Bashforth schemes, are employed to advance the vorticity field in time
• Poisson solvers, such as the fast Fourier transform (FFT) or multigrid methods, are used to compute the velocity field from the vorticity distribution
• Boundary conditions for vorticity must be carefully implemented to ensure consistency with the physical problem and numerical stability
  • Common approaches include the creation of vorticity sheets at solid boundaries and the use of vorticity flux boundary conditions
• Adaptive mesh refinement (AMR) techniques can be used to dynamically adjust the grid resolution based on the local vorticity magnitude, improving computational efficiency

Applications in Real-World Fluid Systems

• Understanding vorticity is crucial for the design and analysis of various engineering systems, such as aircraft wings, wind turbines, and combustion engines
• In aerodynamics, the generation and shedding of vortices from airfoils determine the lift and drag forces experienced by the aircraft
  • Vortex generators can be used to control flow separation and enhance lift performance
• Vorticity plays a key role in the formation of wingtip vortices, which pose safety risks for trailing aircraft and contribute to induced drag
• In turbomachinery, such as pumps and turbines, the interaction of vortices with rotating blades affects the efficiency and noise generation of the system
• Vorticity is essential for understanding the mixing and transport processes in environmental flows, such as rivers, oceans, and the atmosphere
  • Coherent vortical structures, like eddies and gyres, control the dispersion of pollutants, nutrients, and heat in these systems
• In combustion systems, vorticity influences the mixing of fuel and oxidizer, affecting the flame stability and pollutant formation
  • Swirl-stabilized combustors rely on the generation of vorticity to anchor the flame and improve combustion efficiency
• The study of vorticity in biological flows, such as blood flow in the cardiovascular system, helps in understanding the development of vascular diseases and designing artificial heart valves