# 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

- Schuster, H.G.; Just, W. Deterministic Chaos: An Introduction; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2006. [Google Scholar]
- Itzykson, C.; Drouffe, J.M. Statistical Field Theory: Volume 2, Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems; Cambridge University Press: Cambridge, UK, 1991; Volume 2. [Google Scholar]
- Egolf, P.W.; Kutter, K. Nonlinear, Nonlocal and Fractional Turbulence; Springer International Publishing: Berlin, Germany, 2020. [Google Scholar]
- Goldenfeld, N.; Shih, H.Y. Turbulence as a problem in non-equilibrium statistical mechanics. J. Stat. Phys.
**2017**, 167, 575–594. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer Science & Business Media: Berlin, Germany, 2009. [Google Scholar]
- L’vov, V.S. Universality of turbulence. Nature
**1998**, 396, 519–521. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Kolmogorov, A.N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech.
**1962**, 13, 82–85. [Google Scholar] [CrossRef][Green Version] - Frisch, U.; Kolmogorov, A.N. Turbulence: The legacy of AN Kolmogorov; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Xu, W.; Wang, Y.; Gao, Y.S.; Liu, J.; Dou, H.; Liu, C. Liutex similarity in turbulent boundary layer. J. Hydrodyn.
**2019**, 31, 1259–1262. [Google Scholar] [CrossRef] - Wilczek, M.; Meneveau, C. Pressure Hessian and viscous contributions to velocity gradient statistics based on Gaussian random fields. J. Fluid Mech.
**2014**, 756, 191–225. [Google Scholar] [CrossRef][Green Version] - Blackburn, H.M.; Mansour, N.N.; Cantwell, B.J. Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech.
**1996**, 310, 269–292. [Google Scholar] [CrossRef][Green Version] - Voelkl, T.; Pullin, D.I.; Chan, D.C. A physical-space version of the stretched-vortex subgrid-stress model for large-eddy simulation. Phys. Fluids
**2000**, 12, 1810–1825. [Google Scholar] [CrossRef][Green Version] - Yang, Z.; Wang, B.C. On the topology of the eigenframe of the subgrid-scale stress tensor. J. Fluid Mech.
**2016**, 798, 598–627. [Google Scholar] [CrossRef] - Kolář, V. Vortex identification: New requirements and limitations. Int. J. Heat Fluid Flow
**2007**, 28, 638–652. [Google Scholar] [CrossRef] - Baysal, K.; Rist, U. Identification and quantification of shear layer influences on the generation of vortex structures. In New Results in Numerical and Experimental Fluid Mechanics VII; Springer: Berlin/Heidelberg, Germany, 2010; pp. 241–248. [Google Scholar]
- Dong, X.; Gao, Y.; Liu, C. New normalized Rortex/vortex identification method. Phys. Fluids
**2019**, 31, 011701. [Google Scholar] [CrossRef] - Jeong, J.; Hussain, F. On the identification of a vortex. J. Fluid Mech.
**1995**, 285, 69–94. [Google Scholar] [CrossRef] - Liu, C.; Gao, Y.; Dong, X.; Wang, Y.; Liu, J.; Zhang, Y.; Cai, X.; Gui, N. Third generation of vortex identification methods: Omega and Liutex/Rortex based systems. J. Hydrodyn.
**2019**, 31, 205–223. [Google Scholar] [CrossRef] - Gao, Y.; Liu, C. Rortex based velocity gradient tensor decomposition. Phys. Fluids
**2019**, 31, 011704. [Google Scholar] [CrossRef] - Keylock, C.J. The Schur decomposition of the velocity gradient tensor for turbulent flows. J. Fluid Mech.
**2018**, 848, 876–905. [Google Scholar] [CrossRef][Green Version] - Maciel, Y.; Robitaille, M.; Rahgozar, S. A method for characterizing cross-sections of vortices in turbulent flows. Int. J. Heat Fluid Flow
**2012**, 37, 177–188. [Google Scholar] [CrossRef] - Eisma, J.; Westerweel, J.; Ooms, G.; Elsinga, G.E. Interfaces and internal layers in a turbulent boundary layer. Phys. Fluids
**2015**, 27, 055103. [Google Scholar] [CrossRef][Green Version] - Nagata, R.; Watanabe, T.; Nagata, K.; da Silva, C. Triple decomposition of velocity gradient tensor in homogeneous isotropic turbulence. Comput. Fluids
**2020**, 198, 104389. [Google Scholar] [CrossRef][Green Version] - Samtaney, R.; Pullin, D.I.; Kosović, B. Direct numerical simulation of decaying compressible turbulence and shocklet statistics. Phys. Fluids
**2001**, 13, 1415–1430. [Google Scholar] [CrossRef][Green Version] - Pirozzoli, S.; Grasso, F. Direct numerical simulations of isotropic compressible turbulence: Influence of compressibility on dynamics and structures. Phys. Fluids
**2004**, 16, 4386–4407. [Google Scholar] [CrossRef] - Wang, J.; Shi, Y.; Wang, L.P.; Xiao, Z.; He, X.; Chen, S. Effect of compressibility on the small-scale structures in isotropic turbulence. J. Fluid Mech.
**2012**, 713, 588–631. [Google Scholar] [CrossRef][Green Version] - Jagannathan, S.; Donzis, D.A. Reynolds and Mach number scaling in solenoidally-forced compressible turbulence using high-resolution direct numerical simulations. J. Fluid Mech.
**2016**, 789, 669–707. [Google Scholar] [CrossRef] - Don, W.S.; Borges, R. Accuracy of the weighted essentially non–oscillatory conservative finite difference schemes. J. Comput. Phys.
**2013**, 250, 347–372. [Google Scholar] [CrossRef] - Adams, N.A.; Shariff, K. A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. J. Comput. Phys.
**1996**, 127, 27–51. [Google Scholar] [CrossRef] - Gottlieb, S.; Shu, C.W. Total variation diminishing Runge–Kutta schemes. Math. Comput.
**1998**, 67, 73–85. [Google Scholar] [CrossRef][Green Version] - Boukharfane, R.; Ferrer, P.J.M.; Mura, A.; Giovangigli, V. On the role of bulk viscosity in compressible reactive shear layer developments. Eur. J. Mech. B/Fluids
**2019**, 77, 32–47. [Google Scholar] [CrossRef][Green Version] - Boukharfane, R.; Er-Raiy, A.; Parsani, M. Compressibility Effects on Homogeneous Isotropic Turbulence Using Schur Decomposition of the Velocity Gradient Tensor. AIAA Scitech 2021 Forum. 2021, p. 1446. Available online: https://research.kaust.edu.sa/en/publications/compressibility-effects-on-homogeneous-isotropic-turbulence-using (accessed on 2 March 2021).
- Martín, M.P.; Candler, G.V. Effect of chemical reactions on decaying isotropic turbulence. Phys. Fluids
**1998**, 10, 1715–1724. [Google Scholar] [CrossRef] - Ristorcelli, J.R.; Blaisdell, G.A. Consistent initial conditions for the DNS of compressible turbulence. Phys. Fluids
**1997**, 9, 4–6. [Google Scholar] [CrossRef][Green Version] - Mansour, N.N.; Wray, A.A. Decay of isotropic turbulence at low Reynolds number. Phys. Fluids
**1994**, 6, 808–814. [Google Scholar] [CrossRef] - Jiménez, J.; Wray, A.A.; Saffman, P.G.; Rogallo, R.S. The structure of intense vorticity in isotropic turbulence. J. Fluid Mech.
**1993**, 255, 65–90. [Google Scholar] [CrossRef][Green Version] - Wang, J.; Wan, M.; Chen, S.; Chen, S. Kinetic energy transfer in compressible isotropic turbulence. J. Fluid Mech.
**2018**, 841, 581–613. [Google Scholar] [CrossRef] - Zhang, Y.T.; Shi, J.; Shu, C.W.; Zhou, Y. Numerical viscosity and resolution of high-order Weighted Essentially Non-Oscillatory schemes for compressible flows with high Reynolds numbers. Phys. Rev. E
**2003**, 68, 046709. [Google Scholar] [CrossRef][Green Version] - Wang, J.; Wang, L.P.; Xiao, Z.; Shi, Y.; Chen, S. A hybrid numerical simulation of isotropic compressible turbulence. J. Comput. Phys.
**2010**, 229, 5257–5279. [Google Scholar] [CrossRef] - Chong, M.S.; Perry, A.E.; Cantwell, B.J. A general classification of three-dimensional flow fields. Phys. Fluids Fluid Dyn.
**1990**, 2, 765–777. [Google Scholar] [CrossRef] - Ooi, A.; Martin, J.; Soria, J.; Chong, M.S. A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J. Fluid Mech.
**1999**, 381, 141–174. [Google Scholar] [CrossRef] - Das, R.; Girimaji, S.S. Revisiting turbulence small-scale behavior using velocity gradient triple decomposition. New J. Phys.
**2020**. [Google Scholar] [CrossRef] - Williams, J. Modern Fortran Edition of the SLSQP Optimizer. GitHub Repository. 2019. Available online: https://github.com/jacobwilliams/slsqp (accessed on 2 March 2021).
- Pátỳ, M.; Lavagnoli, S. A novel vortex identification technique applied to the 3D flow field of a high-pressure turbine. In Proceedings of the ASME Turbo Expo 2019: Turbomachinery Technical Conference and Exposition, Volume 2B: Turbomachinery, Phoenix, AZ, USA, 17–21 July 2019. [Google Scholar]
- Brachet, M.E. Direct simulation of three-dimensional turbulence in the Taylor–Green vortex. Fluid Dyn. Res.
**1991**, 8, 1. [Google Scholar] [CrossRef] - Yeung, P.K.; Pope, S.B. Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech.
**1989**, 207, 531–586. [Google Scholar] [CrossRef][Green Version] - Yeung, P.K.; Pope, S.B.; Lamorgese, A.G.; Donzis, D.A. Acceleration and dissipation statistics of numerically simulated isotropic turbulence. Phys. Fluids
**2006**, 18, 065103. [Google Scholar] [CrossRef][Green Version]

**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 ${\omega}_{{\U0001d4e1}^{\star}}/{s}_{{\U0001d4d0}^{\star}}$.

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

$0.12$ | 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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**MDPI and ACS Style**

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