# Triple Decomposition of Velocity Gradient Tensor in Compressible Turbulence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Numerical Methods

#### Numerical Methods

`WENO-Z`) scheme of Don and Borges [29].

`WENO-Z`scheme is conditioned to a smoothness criterion that involves the local values of the normalized spatial variations of both pressure and density [30]. This method is equivalent to the superposition of the four candidate stencils of the

`WENO-Z`scheme when each stencil is evaluated with its optimal weight. As a result, the introduction of such a hybridization makes the used method minimally dissipative, thanks to the use of a central scheme in the majority of the domain, and it yields a reasonable predicted spectrum up to rather high wavenumbers.

## 3. Dns Database

`WENO-Z`scheme, the total amount of dissipation across a shock is independent of numerical viscosity [39]. Indeed, as shown by [40], as long as the shocklet thickness is small, the total amount of dissipation across the shocks is independent of viscosity, and it depends only on the jump conditions across the shocks, which are preserved by the

`WENO-Z`scheme.

## 4. The Velocity Gradient Tensor and the Invariants of Its Characteristic Equation

- ${\Delta}_{{\U0001d4d0}^{\star}}>0$, ${R}^{\star}>0$: compressing towards an unstable focus region (
`UFC`), - ${\Delta}_{{\U0001d4d0}^{\star}}>0$, ${R}^{\star}<0$: stretching away from a stable focus region (
`SFS`), - ${\Delta}_{{\U0001d4d0}^{\star}}<0$, ${R}^{\star}>0$: two saddles with an unstable node (
`UNSS`), and - ${\Delta}_{{\U0001d4d0}^{\star}}<0$, ${R}^{\star}<0$: two saddles with a stable node (
`SNSS`).

## 5. Triple Decomposition of Velocity Gradient Tensor

- ${\Delta}_{{\U0001d4d0}^{\star}}<0$: Non-rotational geometries (${\U0001d4d0}^{\star}={\U0001d4dd}^{\star}+{\U0001d4d7}^{\star}$), the tensor $\U0001d4d0$ has only real eigenvalues; and,
- ${\Delta}_{{\U0001d4d0}^{\star}}>0$: Rotational geometries ($\U0001d4d0={\U0001d4dd}^{\star}+{\U0001d4e1}^{\star}+{\U0001d4d7}^{\star}$), the tensor $\U0001d4d0$ has complex eigenvalues, the flow is locally rotational, and the three tensors ${\U0001d4dd}^{\star}$, ${\U0001d4e1}^{\star}$, and ${\U0001d4d7}^{\star}$ are in general non-zero.

## 6. Results and Discussion

`SFS`and

`UNSS`quadrants. Within these quadrants, the statistical points are mostly aligned with the right branch that corresponds to ${\Delta}_{{\mathcal{A}}^{\star}}=0$. The compression motion tends to reveal a stronger alignment with the right branch, as indicated by a rather sharp joint PDF with the curve ${\Delta}_{{\mathcal{A}}^{\star}}=0$ [27].

`SFS`and

`UNSS`regions, which are dominated by positive enstrophy and strain productions. Furthermore, the probability of occurrence of each topology in the almost incompressible case with ${\mathrm{Ma}}_{t}=0.12$ is very close to that found in the compressible turbulence. This fact proves that (i) the dilatation has a negligible effect on the occurrence of each topology and (ii) compressibility marginally affects the invariant statistics. The occupancy of the non-rotational region that is characterized by ${\U0001d4e1}^{\star}$$\approx 0$ is close to the sum of the conditional data of the

`UNSS`and

`SNSS`topologies.

## 7. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Classification of local three-dimensional streamlines into non-degenerate topologies in the ${Q}^{\star}-{R}^{\star}$ plane for the incompressible turbulence [41]. The curved solid lines are the discriminant, ${\Delta}_{{\U0001d4d0}^{\star}}$, lines.

**Figure 3.**Two-dimensional sketch of the fluid motion deformation due to (

**a**) normal-strain-rate tensor, (

**b**) shear tensor, and (

**c**) rigid-body-rotation tensor. Adapted from [43].

**Figure 4.**PDF of (

**a**) the number of iterations and (

**b**) evaluations of the Jacobian of the objective function performed by the optimizer.

**Figure 5.**Joint PDF of the invariants ${Q}^{\star}$ and ${R}^{\star}$ as function of ${\mathrm{Ma}}_{t}$.

**Figure 6.**Three-dimensional visualization of isosurfaces of ${\omega}_{{\U0001d4e1}^{\star}}$, ${\omega}_{{\U0001d4d7}^{\star}}$ and ${s}_{{\U0001d4e1}^{\star}}$ for ${\mathrm{Ma}}_{t}=0.5$.

**Figure 7.**Color contour of the normalized (

**a**) ${\omega}_{{\U0001d4d0}^{\star}}$ and (

**d**) ${s}_{{\U0001d4d0}^{\star}}$ and their components (

**b**) ${\omega}_{{\U0001d4d7}^{\star}}$, (

**c**) ${\omega}_{{\U0001d4e1}^{\star}}$, and (

**e**) ${s}_{{\U0001d4dd}^{\star}}$ on the x

_{1}-x

_{2}plane for ${\mathrm{Ma}}_{t}\text{}=\text{}0.12$.

**Figure 8.**Color contour of the normalized (

**a**) ${\omega}_{{\U0001d4d0}^{\star}}$ and (

**d**) ${s}_{{\U0001d4d0}^{\star}}$ and their components (

**b**) ${\omega}_{{\U0001d4d7}^{\star}}$, (

**c**) ${\omega}_{{\U0001d4e1}^{\star}}$, and (

**e**) ${s}_{{\U0001d4dd}^{\star}}$ on the x

_{1}-x

_{2}plane for ${\mathrm{Ma}}_{t}\text{}=\text{}0.50$.

**Figure 9.**PDF of ${\omega}_{{\U0001d4d0}^{\star}}$ and ${s}_{{\U0001d4d0}^{\star}}$ and their components (

**a**) ${\omega}_{{\U0001d4d7}^{\star}}$, ${\omega}_{{\U0001d4e1}^{\star}}$, (

**b**) ${s}_{{\U0001d4d7}^{\star}}$ and ${s}_{{\U0001d4dd}^{\star}}$ for six different ${\mathrm{Ma}}_{t}$. The sampling of the data prior to the PDF computation has considered the clipping of the zero values of ${\omega}_{{\U0001d4e1}^{\star}}$.

**Figure 10.**PDF of (

**a**) ${s}_{{\U0001d4d7}^{\star}}/{\omega}_{{\U0001d4d7}^{\star}}$, ${s}_{{\U0001d4d7}^{\star}}/{\omega}_{{\U0001d4e1}^{\star}}$, and (

**b**) ${s}_{{\U0001d4e1}^{\star}}/{\omega}_{{\U0001d4d7}^{\star}}$, ${s}_{{\U0001d4e1}^{\star}}/{\omega}_{{\U0001d4e1}^{\star}}$ for six different ${\mathrm{Ma}}_{t}$.

**Figure 11.**Joint PDF of vorticity with its components and strain rate for ${\mathrm{Ma}}_{t}\text{}=\text{}0.12$ and ${\mathrm{Ma}}_{t}\text{}=\text{}0.89$.

Resolution | ${\mathbf{Re}}_{\mathit{\lambda}}$ | ${\mathbf{Ma}}_{\mathit{t}}$ | ${\U0001d4e4}^{\prime}$ | $\langle \mathit{\epsilon}/\mathit{\rho}\rangle $ | $\mathit{\eta}/\mathbf{\Delta}\mathit{x}$ | ${\U0001d4db}_{\mathit{t}}/\mathit{\eta}$ | $\mathit{\lambda}/\mathit{\eta}$ | $\U0001d4e2{\mathit{k}}_{3}$ | $\U0001d4d5{\mathit{l}}_{3}$ |
---|---|---|---|---|---|---|---|---|---|

${512}^{3}$ | 100 | 0.12 | 0.54 | 0.11 | 1.18 | 151 | 19.80 | −0.43 | 5.50 |

${512}^{3}$ | 100 | 0.32 | 0.53 | 0.10 | 1.17 | 154 | 19.79 | −0.45 | 5.64 |

${512}^{3}$ | 100 | 0.50 | 0.53 | 0.10 | 1.15 | 154 | 19.59 | −0.50 | 5.53 |

${512}^{3}$ | 100 | 0.59 | 0.46 | 0.11 | 1.29 | 181 | 19.59 | −0.51 | 5.94 |

${512}^{3}$ | 100 | 0.73 | 0.45 | 0.09 | 1.35 | 175 | 19.37 | −0.71 | 6.10 |

${512}^{3}$ | 100 | 0.89 | 0.45 | 0.07 | 1.41 | 172 | 19.05 | −1.18 | 8.81 |

**Table 2.**Occupancy of different regions of Figure 2 for the velocity gradient tensor, ${\U0001d4d0}^{\star}$, expressed as a percentage of realizations. The last column indicates the non-rotational non-degenerate topology, where $\parallel {\U0001d4e1}^{\star}\parallel $ is the Frobenius norm of ${\U0001d4e1}^{\star}$.

${\mathbf{Ma}}_{\mathit{t}}$ | UFC | SFS | SNSS | UNSS | $\parallel {\U0001d4e1}^{\star}\parallel =0$ |
---|---|---|---|---|---|

0.12 | 25.71 | 38.36 | 07.99 | 27.90 | 36.13 |

0.32 | 26.06 | 38.84 | 07.96 | 27.11 | 30.52 |

0.50 | 26.71 | 38.93 | 08.04 | 26.29 | 35.02 |

0.58 | 24.67 | 34.95 | 10.47 | 29.88 | 41.23 |

0.73 | 25.68 | 34.07 | 10.35 | 29.88 | 41.88 |

0.89 | 27.42 | 32.97 | 10.03 | 29.54 | 40.03 |

**Table 3.**Coefficients of correlation between ${\omega}_{{\U0001d4d0}^{\star}}/{s}_{{\U0001d4d0}^{\star}}$, ${\omega}_{{\U0001d4d7}^{\star}}/{s}_{{\U0001d4d0}^{\star}}$, and ${\omega}_{{\U0001d4e1}^{\star}}/{s}_{{\U0001d4d0}^{\star}}$.

${\mathbf{Ma}}_{\mathit{t}}$ | $\mathit{r}\left({\mathit{\omega}}_{{\U0001d4d0}^{\star}}/{\mathit{s}}_{{\U0001d4d0}^{\star}}\right)$ | $\mathit{r}\left({\mathit{\omega}}_{{\U0001d4d7}^{\star}}/{\mathit{s}}_{{\U0001d4d0}^{\star}}\right)$ | $\mathit{r}\left({\mathit{\omega}}_{{\U0001d4e1}^{\star}}/{\mathit{s}}_{{\U0001d4d0}^{\star}}\right)$ |
---|---|---|---|

$0.12$ | 0.605 | 0.769 | −0.011 |

$0.32$ | 0.633 | 0.783 | −0.025 |

$0.50$ | 0.661 | 0.787 | −0.054 |

$0.58$ | 0.598 | 0.740 | −0.036 |

$0.73$ | 0.619 | 0.737 | −0.004 |

$0.89$ | 0.628 | 0.714 | −0.040 |

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Boukharfane, R.; Er-raiy, A.; Alzaben, L.; Parsani, M. Triple Decomposition of Velocity Gradient Tensor in Compressible Turbulence. *Fluids* **2021**, *6*, 98.
https://doi.org/10.3390/fluids6030098

**AMA Style**

Boukharfane R, Er-raiy A, Alzaben L, Parsani M. Triple Decomposition of Velocity Gradient Tensor in Compressible Turbulence. *Fluids*. 2021; 6(3):98.
https://doi.org/10.3390/fluids6030098

**Chicago/Turabian Style**

Boukharfane, Radouan, Aimad Er-raiy, Linda Alzaben, and Matteo Parsani. 2021. "Triple Decomposition of Velocity Gradient Tensor in Compressible Turbulence" *Fluids* 6, no. 3: 98.
https://doi.org/10.3390/fluids6030098