Validation

This page describes validation test cases and comparisons with analytical solutions, following Stock (2006), §3.1.2, §3.4.4–3.4.5, §3.7.2.

Element Discretization Tests

Stretch Parallel to Vorticity

A flat vortex sheet with $\boldsymbol{\gamma}(t=0) = 1.0\hat{\mathbf{j}}$ is subjected to an artificial velocity field causing stretch parallel to the vorticity direction.

Expected result: $\partial\boldsymbol{\gamma}/\partial t = 0$ (constant vortex sheet strength)

This test validates that the circulation-based discretization correctly handles the interplay between vortex stretching and the in-sheet dilatation term.

Stretch Transverse to Vorticity

A flat vortex sheet with $\boldsymbol{\gamma}(t=0) = 1.0\hat{\mathbf{i}}$ experiences strain transverse to the vorticity direction.

Expected behavior:

• Peak extensional strain (y=0.25): $\gamma_x = e^{-2\pi t}$ (exponential decay)
• Peak compressional strain (y=0.75): $\gamma_x = e^{+2\pi t}$ (exponential growth)

This test verifies that the method handles transverse strain—something neither standard vortex filament nor vortex particle methods can easily account for.

Vortex Ring Circulation Conservation

Setup

A unit-radius sphere centered at the origin with initial vortex sheet strength satisfying the no-through-flow condition in potential flow:

\[\boldsymbol{\gamma} = -\frac{3}{2}r\,(\hat{\mathbf{n}} \times \mathbf{U}_\infty)\]

For $\mathbf{U}_\infty = -\hat{\mathbf{k}}$, this gives $\Gamma_{\text{ring}} = 3$.

Circulation Measurement

The vortex ring circulation is computed as:

\[\Gamma_{\text{ring}} = \sum_{p=1}^{N} \frac{1}{2\pi\|\mathbf{x}_p\|^2} \sum_{i=1}^{3} \Gamma_{p,i}\,(\Delta\mathbf{l}_{p,i} \times \mathbf{x}_p) \cdot \hat{\mathbf{k}}\]

Expected Results

• Initial discretization error: second-order in mean triangle edge length
• Circulation should remain constant (within discretization error) throughout the simulation
• Different midpoint methods (geometric, spline, cylindrical) should all conserve circulation

Baroclinic Validation (Rayleigh–Taylor)

Stable Oscillation

For a sinusoidally-perturbed interface with lighter fluid above ($A > 0$):

Unstable Growth

For heavier fluid above ($A < 0$):

\[\text{Magnification} = \frac{f_{\max}}{F_0} = \cosh(t\sqrt{2\pi})\]

Convergence

• M4' kernel: lowest errors, approximately second-order convergence for stable period
• Area-weighting and Peskin: approximately first-order convergence
• Regularization effect: longer periods (stable) and smaller growth rates (unstable) than theory

Kelvin–Helmholtz Instability

2D Periodic Shear Layer

Setup following Krasny (1986) and Tryggvason (1989):

\[z'(x,t=0) = 0.01\sin(2\pi x), \quad \gamma_y(x,t=0) = 1.0\]

Dimensionless parameters:

• $\delta^* = \delta/\lambda$: regularization length / wavelength
• Typical values: $\delta^* = 0.05$ to $0.2$

Expected behavior:

• Roll-up time scales with $\delta^*$
• Higher resolution (smaller $\delta^*$) → sharper roll-up
• Regularization prevents finite-time singularity

3D Doubly-Periodic Shear Layer

Including spanwise perturbation at the most unstable 3D mode (Pierrehumbert & Widnall, 1982):

\[x' = x + 0.01\sin(2\pi x), \quad z' = z + 0.01\sin(2\pi x) + 0.01\sin(4\pi y)\]

Expected phenomena:

• Initial 2D roll-up
• Development of streamwise ribs
• 3D instability growth

KH with Stabilizing Stratification

At the critical Richardson number ($Ri_c = 1/4$), the shear layer is marginally stable. Linear theory gives the growth rate:

\[n = \frac{ik\,\Delta u\,(\rho_2 - \rho_1)}{2(\rho_1 + \rho_2)} + \sqrt{\frac{k^2(\Delta u)^2\rho_1\rho_2}{(\rho_1+\rho_2)^2} - \frac{gk(\rho_2-\rho_1)}{\rho_1+\rho_2}}\]

Running Validation Tests

Kelvin–Helmholtz (KH) run

Run the advanced example with MPI to produce a time-series JLD2 file with snapshots and (optionally) kinetic energy in the params:

mpirun -n 4 julia --project examples/advanced_kh3d.jl

By default, it writes to checkpoints/advanced_series.jld2. You can change the output path in the example script if desired.

Kinetic energy plot

Use the provided plotting script to generate a KE vs time graphic into the docs assets. Set SERIES_FILE and OUTPUT_PNG:

SERIES_FILE=checkpoints/advanced_series.jld2 \
OUTPUT_PNG=docs/src/assets/ke_series.png \
julia --project examples/plot_series_ke.jl

Expected asset path: docs/src/assets/ke_series.png.

Extracting figures from the thesis (optional)

If the thesis figures are available in mstock_dissertation.pdf, extract selected images using pdfimages (Poppler) or pdftoppm:

• macOS (Homebrew): brew install poppler
• Ubuntu/Debian: sudo apt-get install poppler-utils

Extract embedded images:

pdfimages -png mstock_dissertation.pdf docs/src/assets/fig

or rasterize specific pages to PNGs (e.g., pages 10–12):

pdftoppm -png -f 10 -l 12 mstock_dissertation.pdf docs/src/assets/page

After saving, include figures in the appropriate pages (Theory/Remeshing/Parallelization). For the following thesis figure numbers, save with these suggested filenames:

• 1.23 → docs/src/assets/fig123.png (Theory)
• 3.19 → docs/src/assets/fig319.png (Theory/Parallelization)
• 3.26 → docs/src/assets/fig326.png (Remeshing)
• 3.52 → docs/src/assets/fig352.png (Remeshing)

Direct page extraction (user-provided pages)

You shared the page numbers for these figures: 3.19, 1.23, 3.26, 3.52 → pages 90, 98, 118, 140.

Extract at 300 dpi and save with the suggested filenames:

# Figure 3.19 on page 90
pdftoppm -png -r 300 -f 90 -l 90 mstock_dissertation.pdf docs/src/assets/fig_3_19
mv docs/src/assets/fig_3_19-090.png docs/src/assets/fig_3_19.png || \
  mv docs/src/assets/fig_3_19-90.png docs/src/assets/fig_3_19.png

# Figure 1.23 on page 98
pdftoppm -png -r 300 -f 98 -l 98 mstock_dissertation.pdf docs/src/assets/fig_1_23
mv docs/src/assets/fig_1_23-098.png docs/src/assets/fig_1_23.png || \
  mv docs/src/assets/fig_1_23-98.png docs/src/assets/fig_1_23.png

# Figure 3.26 on page 118
pdftoppm -png -r 300 -f 118 -l 118 mstock_dissertation.pdf docs/src/assets/fig_3_26
mv docs/src/assets/fig_3_26-118.png docs/src/assets/fig_3_26.png

# Figure 3.52 on page 140
pdftoppm -png -r 300 -f 140 -l 140 mstock_dissertation.pdf docs/src/assets/fig_3_52
mv docs/src/assets/fig_3_52-140.png docs/src/assets/fig_3_52.png

Notes

• Ensure that any figure usage aligns with the thesis’ distribution rights.
• For reproducibility, note the exact example parameters used to generate the series file.

Results (example figures)

• Gamma magnitude snapshot:

Generate and embed a snapshot plot of |γ| at a chosen time (snapshot index):

SERIES_FILE=checkpoints/advanced_series.jld2 \
SNAP_INDEX=10 \
OUTPUT_PNG=docs/src/assets/snapshot_gamma.png \
julia --project examples/plot_snapshot_gamma.jl

Expected asset path: docs/src/assets/snapshot_gamma.png.

If you share the thesis figure numbers/pages to include (and captions), we’ll add them directly to the relevant sections with proper references.