Table 14 Derived feature fields available in the dataset.
From: A curated dataset for data-driven turbulence modelling
Quantity | Units | Symbol | Field name | Expression |
|---|---|---|---|---|
Features from RANS | ||||
Mean velocity gradient tensor | s−1 | ∇U | gradU | \(\frac{\partial {U}_{i}}{\partial {x}_{j}}\) |
Mean strain rate tensor | s−1 | S | S | \(\frac{1}{2}\left(\nabla U+\nabla {U}^{T}\right)\) |
Mean rotation rate tensor | s−1 | R | R | \(\frac{1}{2}\left(\nabla U-\nabla {U}^{T}\right)\) |
Non-dimensional strain rate tensor | — | \(\widehat{S}\) | Shat | \({T}_{t}S\) |
Non-dimensional rotation rate tensor | — | \(\widehat{R}\) | Rhat | \({T}_{t}R\) |
TKE gradient vector | m/s2 | ∇k | gradk | \(\frac{\partial k}{\partial {x}_{j}}\) |
Pressure gradient vector | m/s2 | ∇p | gradp | \(\frac{\partial p}{\partial {x}_{j}}\) |
Antisymmetric tensor associated with ∇k | m/s2 | Ak | Ak | \(\left[\begin{array}{ccc}0 & -{\partial }_{z}k & {\partial }_{y}k\\ {\partial }_{z}k & 0 & -{\partial }_{x}k\\ -{\partial }_{y}k & {\partial }_{x}k & 0\end{array}\right]\) |
Antisymmetric tensor associated with ∇p | m/s2 | Ap | Ap | See Ak, replacing k with p |
Non-dimensional Ak | — | \({\widehat{A}}_{k}\) | Akhat | \(\frac{\sqrt{k}{A}_{k}}{\varepsilon }\) |
Non-dimensional Ap | — | \({\widehat{A}}_{p}\) | Aphat | \(\frac{{A}_{p}}{| {\rm{D}}U/{\rm{D}}t| }\) |
Turbulent time scale | s | Tt | T_t | \(k/\varepsilon \) |
Kolmogorov time scale | s | Tk | T_k | \(\sqrt{\frac{\nu }{\varepsilon }}\) |
Pope’s 10 basis tensors | — | \({\widehat{{\mathscr{T}}}}_{n}\) | Tensors | See Pope4 |
Pope’s 5 invariants of S and R | — | λi | Lambda | See Pope4 |
47 invariants of \(\left\{\widehat{S},\widehat{R},{\widehat{A}}_{k},{\widehat{S}}_{p}\right\}\), as used by Wu et al.7 | — | \({\mathscr{I}}\) | I | See Wu et al.7 |
Ratio of excess rotation to strain rate | — | — | q[:,0] | \(\frac{{\left\Vert \widehat{R}\right\Vert }^{2}-{\left\Vert \widehat{S}\right\Vert }^{2}}{2{\left\Vert \widehat{S}\right\Vert }^{2}}\) |
Wall-distance based Reynolds number | — | — | q[:,1] | \({\rm{\min }}\left(\frac{\sqrt{k}{y}_{w}}{50\nu },2\right)\) |
Ratio of turbulent time scale to mean strain time scale | — | — | q[:,2] | \(\frac{k}{\varepsilon }\left\Vert S\right\Vert \) |
Ratio of total Reynolds stress to TKE | — | — | q[:,3] | \(\frac{\left\Vert \bar{{u}_{i}^{{\prime} }{u}_{j}^{{\prime} }}\right\Vert }{k}\) |
Wall distance | m | yw | wallDistance | — |
Material derivative of velocity field (equal to convective derivative) | m/s2 | DU/Dt | DUDt | U·∇U |
