# On Energy Redistribution for the Nonlinear Parabolized Stability Equations Method

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## Abstract

**:**

## 1. Introduction

## 2. Basic States

## 3. Stability Analysis

#### 3.1. Jokher

#### 3.1.1. Linear Stability Theory

#### 3.1.2. Parabolized Stability Equations

## 4. Linear Results

## 5. Nonlinear Energy Transfer Mechanism

## 6. NPSE Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Mach contours for Case B: A 1 m long, 7° half opening angle, 1 mm circular nose bluntness, straight cone at flow conditions consistent with those expected in AFOSR-Notre Dame Large Mach 6 Quiet Tunnel. Distance along the axis of the cone is given in meters.

**Figure 2.**LST results—

**Upper**: Notre Dame tunnel conditions 4 m flared cone.

**Middle**: Notre Dame tunnel conditions straight cone.

**Lower**: HIFiRE flight conditions straight cone.

**Figure 3.**LPSE results—

**Upper**: Notre Dame tunnel conditions 4 m flared cone.

**Middle**: Notre Dame tunnel conditions straight cone.

**Lower**: HIFiRE flight conditions straight cone.

**Figure 6.**At Notre Dame Tunnel conditions-

**Upper**: NPSE results for a straight cone.

**Lower**: N-factor comparison between traditional discrete (delta-function) and wave packet (WP) methodologies.

**Figure 7.**At HIFiRE flight conditions-

**Upper**: NPSE results for a straight cone.

**Lower**: N-factor comparison between traditional discrete (delta-function) and wave packet (WP) methodologies.

Mach | Re/m | ${\mathit{\rho}}_{\mathit{\infty}}$ [kg/m${}^{3}$] | T${}_{\mathit{\infty}}$ [K] | u${}_{\mathit{\infty}}$ [m/s] | T${}_{\mathbf{wall}}$ [K] | |
---|---|---|---|---|---|---|

Tunnel | 6 | 11.0 × 10${}^{6}$ | 0.0432 | 53.0488 | 875.9795 | 300 |

Flight | 5.3 | 13.42 × 10${}^{6}$ | 0.1190 | 201.4 | 1509.2075 | 393.4 |

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**MDPI and ACS Style**

Khan, A.A.; Liang, T.; Batista, A.; Kuehl, J.
On Energy Redistribution for the Nonlinear Parabolized Stability Equations Method. *Fluids* **2022**, *7*, 264.
https://doi.org/10.3390/fluids7080264

**AMA Style**

Khan AA, Liang T, Batista A, Kuehl J.
On Energy Redistribution for the Nonlinear Parabolized Stability Equations Method. *Fluids*. 2022; 7(8):264.
https://doi.org/10.3390/fluids7080264

**Chicago/Turabian Style**

Khan, Arham Amin, Tony Liang, Armani Batista, and Joseph Kuehl.
2022. "On Energy Redistribution for the Nonlinear Parabolized Stability Equations Method" *Fluids* 7, no. 8: 264.
https://doi.org/10.3390/fluids7080264