# Energy Budget Characterisation of the Optimal Disturbance in Stratified Shear Flow

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Perturbation Energy Budget

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivations of Optimal Perturbation Energy Budget Equation

## References

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**Figure 1.**The optimal perturbation energy (temperature) budget quantities for the configuration ${C}_{1}$; see Table 1. The blue curve corresponds to Gr = 2500, whilst the red curve corresponds to Gr = 3500. The curves are presented at the time when the amplification factor is optimal.

**Figure 2.**Contours of the perturbations to the base state temperature of the optimal perturbation in the r–z plane for $\theta =0.68$. Configurations ${C}_{1}$, ${C}_{2}$, ${C}_{3}$, and ${C}_{4}$ are organized in rows from top to bottom, for selected Gr values, as indicated above each contour plot.

**Figure 3.**Contours of the perturbations to the base state radial velocity of the optimal perturbation in the r–z plane for $\theta =0.68$. Configurations ${C}_{1}$, ${C}_{2}$, ${C}_{3}$, and ${C}_{4}$ are organized in rows from top to bottom, for selected Gr values, as indicated above each contour plot.

**Table 1.**This table displays the 4 configurations examined in [28]. Here, ratio refers to ${\mathrm{Re}}_{i}/{\mathrm{Re}}_{o}$.

Configuration | Re_{i} | Re_{o} | Ratio | n | k | Gr_{c} |
---|---|---|---|---|---|---|

${C}_{1}$ | 591 | $-2588$ | 1:4 | 10 | 1.9940 | 3244.5 |

${C}_{2}$ | 523 | $-2975$ | 1:6 | 11 | 1.9960 | 4804.2 |

${C}_{3}$ | 473 | $-3213$ | 1:8 | 11 | 1.9200 | 5728.5 |

${C}_{4}$ | 405 | $-3510$ | 1:9 | 11 | 1.8390 | 7148.2 |

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**MDPI and ACS Style**

Godwin, L.E.; Trevelyan, P.M.J.; Akinaga, T.; Generalis, S.C.
Energy Budget Characterisation of the Optimal Disturbance in Stratified Shear Flow. *Fluids* **2024**, *9*, 106.
https://doi.org/10.3390/fluids9050106

**AMA Style**

Godwin LE, Trevelyan PMJ, Akinaga T, Generalis SC.
Energy Budget Characterisation of the Optimal Disturbance in Stratified Shear Flow. *Fluids*. 2024; 9(5):106.
https://doi.org/10.3390/fluids9050106

**Chicago/Turabian Style**

Godwin, Larry E., Philip M. J. Trevelyan, Takeshi Akinaga, and Sotos C. Generalis.
2024. "Energy Budget Characterisation of the Optimal Disturbance in Stratified Shear Flow" *Fluids* 9, no. 5: 106.
https://doi.org/10.3390/fluids9050106