(vc_tgv_3d)=

# 3-D Taylor-Green Vortex

The Taylor-Green vortex (TGV) is a canonical benchmark used to assess a solver's ability to reproduce unsteady, spatially periodic flows with known reference data. The initial conditions take the form of smooth trigonometric functions, making the TGV an ideal test for equation-based initialization: the solver must evaluate $\sin$ and $\cos$ expressions at every lattice node at startup and reproduce them to floating-point precision before the time-marching begins.

Two complementary sub-cases are used here. Case 00a is a 2-D TGV, which admits a closed-form analytical solution valid for all time and therefore provides a stringent quantitative accuracy and convergence test. Case 00b is a 3-D TGV, for which no analytical solution exists once the flow has transitioned to turbulence; the dissipation rate is compared against the pseudo-spectral DNS of {footcite:t}`vanRees2011`.

The 3-D TGV starts from a smooth, low-amplitude initial condition that is linearly unstable. At moderate Reynolds numbers the flow develops a cascade of smaller vortical structures and eventually reaches a regime of decaying turbulence. Because an exact solution does not exist for $t > 0$, validation is performed by comparing the kinetic energy dissipation rate $\varepsilon(t) = -\mathrm{d}E/\mathrm{d}t$ against the spectral DNS results of {footcite:t}`vanRees2011`.

(vc_tgv_3d_ic)=

## Initial conditions

The velocity and pressure fields at $t = 0$ are the standard TGV initial condition on the periodic cube $[0,\, 2\pi]^3$:

```{math}
---
label: vc_tgv3d_ic
---
\begin{aligned}
u_x(x, y, z, 0) &= V_0 \sin(k x) \cos(k y) \cos(k z) \\
u_y(x, y, z, 0) &= -V_0 \cos(k x) \sin(k y) \cos(k z) \\
u_z(x, y, z, 0) &= 0 \\
\rho(x, y, z, 0) &= \rho_0 + \frac{\rho_0 V_0^2}{16 c_s^2}
\left[\cos(2kz) + 2\right]
\left[\cos(2kx) + \cos(2ky)\right]
\end{aligned}
```

where $k = 2\pi / L$ and $V_0$ is the peak velocity. The density expression in {eq}`vc_tgv3d_ic` is the weakly-compressible approximation of the incompressible pressure distribution. In the current simulation $\rho_0 = 1$ is used uniformly (i.e. the density correction is omitted) because at $\mathrm{Ma} \approx 0.07$ the maximum perturbation is of order $10^{-3}$ and does not affect the velocity dynamics.

As with Case 00a, these fields are set via `models.initialization.equations`, evaluating the trigonometric expressions on the GPU at startup.

(vc_tgv_3d_validation)=

## Validation approach

The volume-averaged kinetic energy is:

```{math}
---
label: vc_tgv3d_energy
---
E(t) = \frac{1}{N^3} \sum_{i,j,k} \frac{1}{2} \rho(i,j,k,t)
\left(u_x^2 + u_y^2 + u_z^2\right)
```

The dissipation rate is estimated by finite difference:

```{math}
---
label: vc_tgv3d_dissipation
---
\varepsilon(t) = -\frac{\mathrm{d}E}{\mathrm{d}t} \approx
-\frac{E(t + \Delta T) - E(t - \Delta T)}{2\,\Delta T}
```

where $\Delta T$ is the interval between saved snapshots.

The DNS reference of {footcite:t}`vanRees2011` was computed with a pseudospectral code at $Re_\Gamma = 1/\nu = 1600$ on grids up to $768^3$. The reference dissipation curve is digitised from Figure 8a and 8b of the paper (768^3^ PS results, the most converged resolution). The Reynolds number is defined with characteristic length $L^* = 1/k_0 = 1$ (the inverse wavenumber), so in lattice units $Re = V_0 N / (2\pi\nu)$. The peak dissipation time and magnitude are well established and serve as the primary quantitative targets.

(vc_tgv_3d_params)=

## Simulation parameters

| Parameter                  | Value                                                |
| -------------------------- | ---------------------------------------------------- |
| Velocity set               | D3Q27                                                |
| Collision operator         | RRBGK                                                |
| Relaxation time $\tau$     | 0.503056                                             |
| Peak velocity $V_0$        | 0.04 (lattice units)                                 |
| Reference density $\rho_0$ | 1.0                                                  |
| Domain                     | $[0,\, 2\pi]^3$, periodic in all three directions    |
| Grid size $N$              | 256 nodes per side                                   |
| Boundary conditions        | Fully periodic (`periodic_dims: [true, true, true]`) |
| Initialization             | Equation-based (`use_equation_init: true`)           |
| Reference DNS              | {footcite:t}`vanRees2011`, $Re_\Gamma = 1600$        |

```{note}
The Reynolds number follows the van Rees convention $Re_\Gamma = 1/\nu$ with characteristic length $L^* = 1$ (inverse wavenumber) and $V_0 = 1$. In lattice units this becomes $Re = V_0 N / (2\pi\nu)$, giving $\nu \approx 0.001019$ and $\tau = 3\nu + 0.5 \approx 0.503056$.
```

```{toctree}
---
hidden:
---
01_taylor_green_vortex_3d.ipynb
```

```{footbibliography}
```