Rayleigh-Bénard convection

One line description of the data: 2D horizontally-periodic Rayleigh-Benard convection.

Longer description of the data: Rayleigh-Bénard convection involves fluid dynamics and thermodynamics, seen in a horizontal fluid layer heated from below, forming convective cells due to a temperature gradient. With the lower plate heated and the upper cooled, thermal energy creates density variations, initiating fluid motion. This results in Bénard cells, showcasing warm fluid rising and cool fluid descending. The interplay of buoyancy, conduction, and viscosity leads to complex fluid motion, including vortices and boundary layers.

Associated paper: Paper 1, Paper 2.

Data generated by: Rudy Morel, Center for Computational Mathematics, Flatiron Institute.

Code or software used to generate the data: Github repository, based on the software Dedalus.

Equation:

While we solve equations in the frequency domain, the original time-domain problem is

\[ \begin{align*} \frac{\partial b}{\partial t} - \kappa \Delta b & = -u\nabla b, \\ \frac{\partial u}{\partial t} - \nu \Delta u + \nabla p - b \vec{e}_z & = -u \nabla u, \end{align*} \]

where \(\Delta = \nabla \cdot \nabla\) is the spatial Laplacian, \(b\) is the buoyancy, \(u = (u_x,u_y)\) the (horizontal and vertical) velocity, and \(p\) is the pressure, \(\vec{e}_z\) is the unit vector in the vertical direction, with the additional constraints \(\int p = 0\) (pressure gauge).

The boundary conditions vertically are as follows:

\[ \begin{align*} b(z=0) = Lz ~~~,~~~ b(z=Lz) = 0 \\ u(z=0) = u(z=Lz) = 0 \end{align*} \]

These PDE are parameterized by the Rayleigh and Prandtl numbers through \(\kappa\) and \(\nu\).

\[ \begin{align*} \text{(thermal diffusivity)} ~~~~~~~ \kappa & = \big(\text{Rayleigh} * \text{Prandtl}\big)^{-\frac12} \\ \text{(viscosity)} ~~~~~~~ \nu & = \bigg(\frac{\text{Rayleigh}}{\text{Prandtl}}\bigg)^{-\frac12}. \end{align*} \]

Dataset	FNO	TFNO	Unet	CNextU-net
rayleigh_benard	0.8395	\(\mathbf{0.6566}\)	1.4860	0.6699

Table: VRMSE metrics on test sets (lower is better). Best results are shown in bold. VRMSE is scaled such that predicting the mean value of the target field results in a score of 1.

About the data

Dimension of discretized data: 200 timesteps of 512 \(\times\) 128 images.

Fields are available in the data: buoyancy (scalar field), pressure (scalar field), velocity (vector field).

Number of simulations: 1750 (35 PDE parameters \(\times\) 50 initial conditions).

Size of the ensemble of all simulations: 342 GB.

Grid type: uniform, cartesian coordinates.

Initial conditions: the buoyancy is composed of a dumped noise added to a linear background \(b(t=0) = (Lz-z)\times\delta b_0 + z(Lz-z) \times\epsilon\) where \(\epsilon\) is a Gaussian white noise of scale \(10^{-3}\). The other fields \(u\) and \(p\) are initialized to \(0\).

Boundary conditions: periodic on the horizontal direction, Dirichlet conditions on the vertical direction.

Simulation time-step: 0.25.

Total time range (\(t_{min}\) to \(t_{max}\)): \(t_{min} = 0\), \(t_{max} = 50\).

Spatial domain size: \(0 \leq x \leq 4\) horizontally, and \(0 \leq z \leq 1\) vertically.

Set of coefficients or non-dimensional parameters evaluated: \(\text{Rayleigh}\in[1e6,1e7,1e8,1e9,1e10], \text{Prandtl}\in[0.1,0.2,0.5,1.0,2.0,5.0,10.0]\). For initial conditions \(\delta b_0\in[0.2,0.4,0.6,0.8,1.0]\), the seed used to generate the initial Gaussian white noise are \(40,\ldots,49\).

Approximate time to generate the data: per input parameter from \(\sim6\,000s\) to \(\sim 50\,000s\) (high Rayleigh numbers take longer), total: \(\sim 60\) hours.

Hardware used to generate the data and precision used for generating the data: 12 nodes of 64 CPU cores with 8 processes per node, in single precision.

What is interesting and challenging about the data:

Rayleigh-Bénard convection datasets offer valuable insights into fluid dynamics under thermal gradients, revealing phenomena like turbulent eddies and convection cells. The position of such cells are highly sensitive to small variations in the initial conditions. Understanding these dynamics is crucial for applications in engineering and environmental science.

Please cite the associated paper if you use this data in your research:

@article{burns2020dedalus,
  title={Dedalus: A flexible framework for numerical simulations with spectral methods},
  author={Burns, Keaton J and Vasil, Geoffrey M and Oishi, Jeffrey S and Lecoanet, Daniel and Brown, Benjamin P},
  journal={Physical Review Research},
  volume={2},
  number={2},
  pages={023068},
  year={2020},
  publisher={APS}
}