# Transient Dynamics in Counter-Rotating Stratified Taylor–Couette Flow

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Model

#### 2.2. Boussinesq Approximation

#### 2.3. Nondimensionalization

#### 2.4. Basic Equations

## 3. Linearization

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Energy Norm

## Appendix B. Definitions of Matrices A and B

## References

- Strogatz, S. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Richtmyer, R.D. Nonlinear Problems: Fluid Dynamics. In Principles of Advanced Mathematical Physics; Springer: Berlin/Heidelberg, Germany, 1978; pp. 364–408. [Google Scholar] [CrossRef]
- Drazin, P.G. Introduction to Hydrodynamic Stability; Cambridge University Press: Cambridge, UK, 2002; Volume 32. [Google Scholar]
- Solomon, T.H. Non-Linear Fluid Flow, Pattern Formation, Mixing and Turbulence. In Encyclopedia of Complexity and Systems Science; Meyers, R.A., Ed.; Springer: New York, NY, USA, 2009; pp. 6195–6206. [Google Scholar] [CrossRef]
- Trefethen, L.N.; Trefethen, A.E.; Reddy, S.C.; Driscoll, T.A. Hydrodynamic stability without eigenvalues. Science
**1993**, 261, 578–584. [Google Scholar] [CrossRef] [Green Version] - Schmid, P.J.; Brandt, L. Analysis of Fluid Systems: Stability, Receptivity, Sensitivity: Lecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013. Appl. Mech. Rev.
**2014**, 66, 024803. [Google Scholar] [CrossRef] - Schmid, P.J.; Springer, D.S.H. Stability and Transition in Shear Flows, 2001. 556 pp. ISBN 0-387-98985-4. J. Fluid Mech.
**2003**, 487, 377–379. [Google Scholar] [CrossRef] - Karp, M.; Cohen, J. Tracking stages of transition in Couette flow analytically. J. Fluid Mech.
**2014**, 748, 896–931. [Google Scholar] [CrossRef] [Green Version] - Nayak, A.; Das, D. Transient growth of optimal perturbation in a decaying channel flow. Phys. Fluids
**2017**, 29, 064104. [Google Scholar] [CrossRef] - Romanov, V.A. Stability of plane-parallel Couette flow. Funct. Anal. Its Appl.
**1973**, 7, 137–146. [Google Scholar] [CrossRef] - Davey, A. On the Stability of Plane Couette flow to Infinitesimal Disturbances. J. Fluid Mech.
**1973**, 57, 369–380. [Google Scholar] [CrossRef] - Drazin, P.G.; Reid, W.H. Hydrodynamic Stability. J. Fluid Mech.
**1982**, 124, 529–532. [Google Scholar] [CrossRef] [Green Version] - Sano, M.; Tamai, K. A universal transition to turbulence in channel flow. Nat. Phys.
**2016**, 12, 249–253. [Google Scholar] [CrossRef] [Green Version] - Eckhardt, B. Transition to Turbulence in Shear Flows. Phys. A
**2018**, 504, 121–129. [Google Scholar] [CrossRef] [Green Version] - Orszag, S.A. Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech.
**1971**, 50, 689–703. [Google Scholar] [CrossRef] [Green Version] - Baines, P.G.; Majumdar, S.J.; Mitsudera, H. The mechanics of the Tollmien-Schlichting wave. J. Fluid Mech.
**1996**, 312, 107–124. [Google Scholar] [CrossRef] [Green Version] - Andersson, P.; Berggren, M.; Henningson, D.S. Optimal disturbances and bypass transition in boundary layers. Phys. Fluids
**1999**, 11, 134–150. [Google Scholar] [CrossRef] - Kreilos, T.; Khapko, T.; Schlatter, P.; Duguet, Y.; Henningson, D.S.; Eckhardt, B. Bypass transition and spot nucleation in boundary layers. Phys. Rev. Fluids
**2016**, 1, 043602. [Google Scholar] [CrossRef] [Green Version] - Meseguer, A.; Mellibovsky, F.; Avila, M.; Marques, F. Instability mechanisms and transition scenarios of spiral turbulence in Taylor-Couette flow. Phys. Rev. E
**2009**, 80, 046315. [Google Scholar] [CrossRef] [Green Version] - Eckert, M. The troublesome birth of hydrodynamic stability theory: Sommerfeld and the turbulence problem. Eur. Phys. J. H
**2010**, 35, 29–51. [Google Scholar] [CrossRef] - Reddy, S.C.; Henningson, D.S. Energy growth in viscous channel flows. J. Fluid Mech.
**1993**, 252, 209–238. [Google Scholar] [CrossRef] - Heaton, C.J.; Peake, N. Transient growth in vortices with axial flow. J. Fluid Mech.
**2007**, 587, 271–301. [Google Scholar] [CrossRef] - Ha, K.; Chomaz, J.M.; Ortiz, S. Transient growth, edge states, and repeller in rotating solid and fluid. Phys. Rev. E
**2021**, 103, 033102. [Google Scholar] [CrossRef] - Quintanilha, H.; Paredes, P.; Hanifi, A.; Theofilis, V. Transient growth analysis of hypersonic flow over an elliptic cone. J. Fluid Mech.
**2022**, 935, A40. [Google Scholar] [CrossRef] - Andereck, C.D.; Liu, S.; Swinney, H.L. Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech.
**1986**, 164, 155–183. [Google Scholar] [CrossRef] [Green Version] - Gebhardt, T.; Grossmann, S. The Taylor–Couette eigenvalue problem with independently rotating cylinders. Z. Phys. B Condens. Matter
**1993**, 90, 475–490. [Google Scholar] [CrossRef] - Bai, Y.; Latrache, N.; Kelai, F.; Crumeyrolle, O.; Mutabazi, I. Viscoelastic instabilities of Taylor-Couette flows with different rotation regimes. Proc. Math. Phys. Eng. Sci.
**2023**, 381, 20220133. [Google Scholar] [CrossRef] - Lopez, J.M.; Lopez, J.M.; Marques, F. Stably stratified Taylor-Couette flows. Proc. Math. Phys. Eng. Sci.
**2023**, 381, 20220115. [Google Scholar] [CrossRef] - Merbold, S.; Hamede, M.H.; Froitzheim, A.; Egbers, C. Flow regimes in a very wide-gap Taylor-Couette flow with counter-rotating cylinders. Proc. Math. Phys. Eng. Sci.
**2023**, 381, 20220113. [Google Scholar] [CrossRef] - Taylor, G.I., VIII. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. A
**1923**, 223, 289–343. [Google Scholar] [CrossRef] [Green Version] - Coles, D. Transition in circular Couette flow. J. Fluid Mech.
**1965**, 21, 385–425. [Google Scholar] [CrossRef] [Green Version] - Van Atta, C. Exploratory measurements in spiral turbulence. J. Fluid Mech.
**1966**, 25, 495–512. [Google Scholar] [CrossRef] [Green Version] - Hegseth, J.J.; Andereck, C.D.; Hayot, F.; Pomeau, Y. Spiral Turbulence and Phase Dynamics. Phys. Rev. Lett.
**1989**, 62, 257–260. [Google Scholar] [CrossRef] - Prigent, A.; Goire, G.G.; Chaté, H.; Dauchot, O.; van Saarloos, W. Large-Scale Finite-Wavelength Modulation within Turbulent Shear Flows. Phys. Rev. Lett.
**2002**, 89, 014501. [Google Scholar] [CrossRef] [Green Version] - Hristova, H.; Roch, S.; Schmid, P.J.; Tuckerman, L.S. Transient growth in Taylor-Couette flow. Phys. Fluids
**2002**, 14, 3475–3484. [Google Scholar] [CrossRef] [Green Version] - Meseguer, A. Energy transient growth in the Taylor-Couette problem. Phys. Fluids
**2002**, 14, 1655–1660. [Google Scholar] [CrossRef] [Green Version] - Maretzke, S.; Hof, B.; Avila, M. Transient growth in linearly stable Taylor–Couette flows. J. Fluid Mech.
**2014**, 742, 254–290. [Google Scholar] [CrossRef] [Green Version] - Lopez, J.M.; Marques, F.; Avila, M. The Boussinesq approximation in rapidly rotating flows. J. Fluid Mech.
**2013**, 737, 56–77. [Google Scholar] [CrossRef] [Green Version] - Akinaga, T.; Generalis, S.; Busse, F. Tertiary and Quaternary States in the Taylor-Couette System. Chaos Solit. Fractals
**2018**, 109, 107–117. [Google Scholar] [CrossRef] [Green Version] - Henningson, D.S.; Reddy, S.C. On the role of linear mechanisms in transition to turbulence. Phys. Fluids
**1994**, 6, 1396–1398. [Google Scholar] [CrossRef] - Schmid, P.J. Nonmodal stability theory. Annu. Rev. Fluid Mech.
**2007**, 39, 129–162. [Google Scholar] [CrossRef] [Green Version] - Mosedale, A.; Drikakis, D. Assessment of very high order of accuracy in Implicit LES models. ASME J. Fl. Eng.
**2007**, 129, 1497–1503. [Google Scholar] [CrossRef]

**Figure 1.**The geometrical configuration of the system under investigation Here, ${\Omega}_{i}>0$, and ${\Omega}_{o}<0$.

**Figure 2.**Plots of the amplification factor for the same parameter values used by Meseguer [36] and Maretzke [37]. Here, ${\mathrm{Re}}_{i}=240$, ${\mathrm{Re}}_{o}=-271.42$, $\eta =0.881$, and $k=\pi $ with (

**a**) $n=1$ and (

**b**) $n=0$. The corresponding results in Hristova et al. [35] are for when Re $=120$ and $\beta =\pi /2$ using the definitions in their paper.

**Figure 3.**Transient growth for different Gr for each configuration C${}_{1}$, C${}_{2}$, C${}_{3}$, and C${}_{4}$ shown in Table 2, for a range of $\mathrm{Gr}\in [1000,\phantom{\rule{3.33333pt}{0ex}}10,000]$.

**Figure 4.**(

**a**) Growth rate with respect to Gr for each configuration. (

**b**) Optimal amplification factor ${G}_{0}$ of the configurations ${C}_{1}$ (solid), ${C}_{2}$ (dotted), ${C}_{3}$ (dot-dashed), and ${C}_{4}$ (dashed) shown in Table 2, for a range of Gr $\in [0,\phantom{\rule{3.33333pt}{0ex}}10,000]$.

**Figure 5.**The transient growth factor G(t) for increasing Gr. The plots from the first to the last row represent ${C}_{1}$, ${C}_{2}$, ${C}_{3}$, and ${C}_{4}$, respectively. The numbers above each plot are the corresponding numerical values for Gr.

**Figure 6.**Contour plots of the amplified perturbations in the radial velocity $\stackrel{\u02d8}{u}$ at the value of Gr when $G={G}_{0}$. The plots from the first to the last row represent ${C}_{1}$, ${C}_{2}$, ${C}_{3}$, and ${C}_{4}$, respectively.

Parameter | Expression |
---|---|

Grashof number | $\mathrm{Gr}=\beta g\Delta T{d}^{3}/{\nu}^{2}$ |

Inner Reynolds number | ${\mathrm{Re}}_{i}={\Omega}_{i}{r}_{i}d/\nu $ |

Outer Reynolds number | ${\mathrm{Re}}_{o}={\Omega}_{o}{r}_{o}d/\nu $ |

Prandtl number | $\mathrm{Pr}=\nu /\kappa $ |

Radius ratio | $\eta ={r}_{i}/{r}_{o}$ |

Relative density | $\u03f5=\beta \Delta T$ |

Config | ${\mathbf{Re}}_{\mathit{i}}$ | ${\mathbf{Re}}_{\mathit{o}}$ | Ratio | n | k | TCF (${\mathit{G}}_{0}$) | STCF (${\mathit{G}}_{0}$) |
---|---|---|---|---|---|---|---|

${C}_{1}$ | 591 | $-2588$ | 1:4 | 10 | 1.9940 | 71.36 | $1.207191\times {10}^{4}$ |

${C}_{2}$ | 523 | $-2975$ | 1:6 | 11 | 1.9960 | 71.58 | $1.855346\times {10}^{4}$ |

${C}_{3}$ | 473 | $-3213$ | 1:8 | 11 | 1.9200 | 71.64 | $2.342350\times {10}^{4}$ |

${C}_{4}$ | 405 | $-3510$ | 1:9 | 11 | 1.8390 | 71.75 | $1.845845\times {10}^{4}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Godwin, L.E.; Trevelyan, P.M.J.; Akinaga, T.; Generalis, S.C.
Transient Dynamics in Counter-Rotating Stratified Taylor–Couette Flow. *Mathematics* **2023**, *11*, 3250.
https://doi.org/10.3390/math11143250

**AMA Style**

Godwin LE, Trevelyan PMJ, Akinaga T, Generalis SC.
Transient Dynamics in Counter-Rotating Stratified Taylor–Couette Flow. *Mathematics*. 2023; 11(14):3250.
https://doi.org/10.3390/math11143250

**Chicago/Turabian Style**

Godwin, Larry E., Philip M. J. Trevelyan, Takeshi Akinaga, and Sotos C. Generalis.
2023. "Transient Dynamics in Counter-Rotating Stratified Taylor–Couette Flow" *Mathematics* 11, no. 14: 3250.
https://doi.org/10.3390/math11143250