sandbox/acastillo/filaments/readme

Vortex Filament Simulations

This module provides comprehensive tools and simulations for vortex filament dynamics using the Biot-Savart law and related frameworks. The implementation includes steady solutions, validation tests, and Navier-Stokes extensions for studying complex vortex phenomena.

Slender Vortex Framework

biot-savart.h

Location: biot-savart.h

Description: Implementation of the vortex filament framework based on the Biot-Savart law for three-dimensional flows. Vortex filaments evolve according to: \begin{aligned} \frac{\partial\vec{x}_c}{\partial t}\bigg\vert_\xi +\frac{\partial\vec{x}_c}{\partial \xi}\bigg\vert_t \frac{\text{d}\xi_0}{\text{d}t} = \vec{u}(\vec{x}_c,t) + \vec{u}^\infty \end{aligned}

The induced velocity is computed using the Biot-Savart law: \begin{aligned} \vec{u}_\text{BS}(\vec{x}) = \frac{1}{2} \kappa \ln\left(\frac{s}{\delta a}\right) \vec{e}_b + \frac{1}{2} \oint_{\mathcal{C}}^{'} \frac{(\vec{x}_c-\vec{x}) \times {\rm d}\vec{T}}{\lVert \vec{x}_c-\vec{x} \rVert^2} \end{aligned} Here, the first term is the local contribution from a curved-line element of length s acting on itself, which is derived using the expression for a vortex ring, while the second term, the non-local contribution, is obtained by integrating along the rest of the filament. The parameter \delta is a dimensionless constant that depends on the vorticity distribution within the vortex core, taken here as \delta \approx 0.8736.

Key Functions:

• local_induced_velocity_segment(): Computes local self-induced velocity using the cutoff formula with arc of circle approximation for three consecutive points
• local_induced_velocity(): Evaluates the complete local auto-induced velocity field for an entire filament
• nonlocal_induced_velocity(): Computes velocity contributions from distant filament segments using discretized Biot-Savart law
• write_filament_state(): Outputs filament configuration and velocity fields to file for analysis and visualization

Implementation Details:

• Frenet-Serret frame computation (tangent, normal, binormal vectors)
• Curvature calculation using three-point arc approximation

Steady Solutions using Biot-Savart Law

steady_solutions_biot_savart/

Location: steady_solutions_biot_savart/

Description: Contains simulations of quasi-steady vortex filament solutions using the Biot-Savart law. Focuses on vortex filaments in the shape of torus unknots, which can be viewed as vortex rings superimposed with Kelvin waves of different azimuthal wavenumbers and amplitudes.

Key Simulations:

• helical_rings_p2.c: Torus unknot with fixed wavenumber p=2 and varying amplitudes (A = 0.1R, 0.2R, 0.3R).
• helical_rings_A015.c: Multiple torus unknots with fixed amplitude A = 0.15R and varying wavenumbers (p = 2, 4, 8)

Associated Data: Each simulation includes filament initialization files (.filament1.txt, .filament2.txt, .filament3.txt) containing segment coordinates, core sizes, and geometric parameters.

Biot-Savart Validation Tests

tests_biot_savart/

Location: tests_biot_savart/

Description: Comprehensive validation suite for the vortex filament framework using analytical solutions and known theoretical results. These tests verify the accuracy of the Biot-Savart implementation across different geometric configurations and discretizations.

Test Cases:

• test_vortex_ring0.c: Test convergence of numerical approach
• test_vortex_ring1.c: Advection of a thin vortex ring
• test_2vortex_rings1.c: Two interacting vortex rings
• test_3vortex_rings1.c: Three interacting vortex rings
• test_vortex_ellipse1.c: Elliptical vortex ring

References