6.4 Vortex Sheet and Vortex Filament Models

Vortex sheets and filaments are powerful tools for modeling complex fluid flows. They represent thin surfaces or lines of concentrated vorticity, simplifying analysis of shear layers, wakes, and other vortical structures.

These models capture key physics of incompressible flows with vorticity. By studying their evolution and instabilities, we gain insights into important phenomena like flow separation, mixing, and turbulence in real-world fluid systems.

Vortex sheet concept

Mathematical representation of vortex sheets

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• Vortex sheets form two-dimensional surfaces of discontinuity in fluid flows with jumps in tangential velocity components across the surface
• Sheet strength characterized by circulation per unit length γ(s,t) where s represents arc length along sheet and t denotes time
• Represented mathematically as curves or surfaces in 2D or 3D space with distributed vorticity along length or area
• Velocity field induced by vortex sheet calculated using Biot-Savart law by integrating contributions from infinitesimal sheet segments
• Approximated as closely packed collections of point vortices (2D) or vortex filaments (3D) in zero thickness limit
• Circulation around closed contours intersecting sheet equals integral of sheet strength along intersection line

Physical properties and applications

• Model shear layers between fluid regions moving at different velocities (jet streams, wakes)
• Occur naturally at interfaces between fluids with density or velocity differences (atmosphere, oceans)
• Used to analyze flow separation and reattachment on aerodynamic surfaces (airfoils, bluff bodies)
• Provide simplified representation of complex vortical structures in computational fluid dynamics
• Help explain formation and evolution of coherent vortex structures in turbulent flows
• Useful for studying instabilities in stratified fluids and geophysical flows (atmospheric fronts)

Vortex sheet evolution

Governing equations for inviscid flows

• Evolution governed by Euler equations for inviscid, incompressible flows with appropriate sheet boundary conditions
• Kinematic condition requires sheet moves with local fluid velocity ensuring fluid particles remain on sheet
• Dynamic condition enforces pressure continuity across sheet leading to Kelvin circulation theorem for circulation conservation
• Vortex sheet strength γ(s,t) evolution equation derived by applying conditions and using Helmholtz velocity field decomposition
• Resulting Birkhoff-Rott equation describes self-induced sheet motion and strength evolution
• Birkhoff-Rott equation typically nonlinear integro-differential equation requiring numerical solution methods
• Special care needed for handling singularities arising in evolution equation particularly at sheet edges or endpoints

Numerical methods for vortex sheet simulations

• Point vortex method approximates sheet as discrete vortices to simulate evolution
• Vortex blob method introduces smoothing kernel to regularize singular behavior
• Contour dynamics technique tracks evolution of sheet boundary in potential flows
• Spectral methods employ Fourier series expansions to solve Birkhoff-Rott equation
• Adaptive mesh refinement algorithms concentrate computational resources in regions of high curvature or rapid change
• Time-stepping schemes (Runge-Kutta, Adams-Bashforth) used for temporal integration
• Boundary element methods solve integral equations for sheet evolution in complex geometries

Vortex sheet dynamics and instabilities

Kelvin-Helmholtz instability and vortex sheet roll-up

• Vortex sheets inherently unstable to small perturbations leading to Kelvin-Helmholtz instability in shear layers
• Instability growth rate depends on perturbation wavenumber and vortex sheet strength
• Sheet begins rolling up into discrete vortices as instability develops (vortex sheet roll-up process)
• Nonlinear evolution leads to complex patterns including spiral formations and smaller vortex shedding
• Roll-up process observed in various natural phenomena (cloud formations, ocean currents)
• Kelvin-Helmholtz instability plays crucial role in mixing and entrainment processes in fluids
• Numerical simulations of sheet roll-up require high resolution and careful treatment of singularities

Role in flow separation and wake dynamics

• Vortex sheets form at sharp edges or flow detachment points representing shear layers between main flow and separated regions
• Separation-induced sheet dynamics crucial for determining overall flow structure and body forces in separated flows
• Multiple sheet interactions (bluff body wakes) lead to complex vortex shedding patterns and oscillatory forces
• Vortex sheet models used to study dynamic stall on airfoils and rotor blades
• Sheet behavior influences drag and lift characteristics of aerodynamic bodies
• Wake vortex sheet evolution important for aircraft safety and airport operations
• Vortex sheet dynamics play role in energy harvesting from fluid flows (wind turbines, tidal generators)

Vortex filament models for 3D flows

Vortex filament representation and dynamics

• Vortex filaments idealize thin tube-like vortical structures in 3D flows characterized by circulation and core radius
• Filament motion described using Biot-Savart law relating induced velocity to geometry and circulation
• Self-induced motion (localized induction approximation) derived for small curvature and core radius
• Vortex stretching and reconnection modeled as important 3D vortex dynamics phenomena
• Multiple filament interactions lead to complex behaviors (leapfrogging, merging, vortex knots and links)
• Filament models applied to aircraft wake vortices, vortex rings, and turbulent flows
• Helical vortex filaments used to study propeller and wind turbine wakes

Numerical methods for vortex filament simulations

• Vortex particle method represents filaments as collections of discrete vortex elements
• Vortex-in-cell method combines Lagrangian vortex evolution with Eulerian velocity field calculations
• Discrete vortex filament method tracks evolution of connected vortex segments
• Adaptive remeshing techniques maintain filament resolution during stretching and deformation
• Fast multipole methods accelerate computation of long-range filament interactions
• Vortex sound generation calculated using filament models for aeroacoustic applications
• Parallelization strategies employed for large-scale vortex filament simulations in complex flows