(vc_tgv_2d)=

# 2-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 2-D TGV is a doubly periodic flow on the square domain $[0,\, 2\pi] \times [0,\, 2\pi]$ initialised with a single Fourier mode. Because the nonlinear advection terms are exactly zero for this initial condition, the flow decays purely through viscous diffusion and the velocity field remains a rescaled copy of the initial condition at all times. This property yields an exact analytical solution that the LBM result can be compared against directly.

(vc_tgv_2d_ic)=

## Initial conditions

The velocity and density fields at $t = 0$ are:

```{math}
---
label: vc_tgv2d_ic
---
\begin{aligned}
u_x(x, y, 0) &= U_0 \cos(k x) \sin(k y) \\
u_y(x, y, 0) &= -U_0 \sin(k x) \cos(k y) \\
\rho(x, y, 0) &= \rho_0
\end{aligned}
```

where $k = 2\pi / L$ is the wave number, $L$ is the domain length, and $U_0$ is the peak velocity amplitude. For a lattice of $N$ nodes per side with lattice spacing $\Delta x = 1$, the wave number in lattice units is $k = 2\pi / N$.

These fields are prescribed via the `models.initialization.equations` block introduced in PR #443. Setting `use_equation_init: true` and providing symbolic expressions for $u_x$, $u_y$, and $\rho$ causes the solver to evaluate them on the GPU before the first time step, placing the initial condition to machine precision without any approximation from interpolation or file I/O.

(vc_tgv_2d_analytical)=

## Analytical solution

The exact solution for $t > 0$ is an exponential decay:

```{math}
---
label: vc_tgv2d_analytical
---
\begin{aligned}
u_x(x, y, t) &= U_0 \exp\!\left(-2\nu k^2 t\right) \cos(k x) \sin(k y) \\
u_y(x, y, t) &= -U_0 \exp\!\left(-2\nu k^2 t\right) \sin(k x) \cos(k y)
\end{aligned}
```

The spatially integrated kinetic energy per unit area is:

```{math}
---
label: vc_tgv2d_energy
---
E(t) = \frac{U_0^2}{4} \exp\!\left(-4\nu k^2 t\right) = E_0 \exp\!\left(-4\nu k^2 t\right)
```

The characteristic decay time is $t_\nu = (4\nu k^2)^{-1}$. For a well-resolved simulation the solver result should track {eq}`vc_tgv2d_analytical` with a velocity error that decreases as $O(\Delta x^2)$ when the grid is refined, consistent with the second-order accuracy of the LBM streaming step.

(vc_tgv_2d_params)=

## Simulation parameters

Four grid resolutions are run to assess convergence. All cases use the D2Q9 velocity set with the RR-BGK collision operator (the production 3rd-order Hermite operator).

| Parameter                  | Value                                                         |
| -------------------------- | ------------------------------------------------------------- |
| Velocity set               | D2Q9                                                          |
| Collision operator         | RRBGK                                                         |
| Relaxation time $\tau$     | 0.501 - 0.508 (varies with grid size)                         |
| Peak velocity $U_0$        | 0.01 (lattice units)                                          |
| Reference density $\rho_0$ | 1.0                                                           |
| Reynolds number            | 240 ($Re = U_0 N / \nu$, constant across grids)               |
| Domain                     | $[0,\, 2\pi] \times [0,\, 2\pi]$, periodic in both directions |
| Grid sizes $N$             | 8, 16, 32, 64 nodes per side                                  |
| Boundary conditions        | Fully periodic (`periodic_dims: [true, true]`)                |
| Initialization             | Equation-based (`use_equation_init: true`)                    |

The Mach number $\mathrm{Ma} = U_0 / c_s = U_0 \sqrt{3} \approx 0.017$ is well below the weakly-compressible limit of 0.1, so compressibility errors are negligible.

```{note}
The convergence study runs four independent simulations (N = 8, 16, 32, 64). The expected slope on a log-log plot of the $L_2$ velocity error versus $\Delta x$ is 2, confirming second-order spatial accuracy.
```

```{toctree}
---
hidden:
---
04_taylor_green_vortex_2d.ipynb
```

```{footbibliography}
```