# Numerical Investigation of Asymmetric Mach 2.5 Turbulent Shock Wave Boundary Layer Interaction

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## Abstract

**:**

## 1. Introduction

## 2. Experiment

## 3. Computational Methods

#### 3.1. Overview of Simulations

#### 3.2. Governing Equations and Discretization

#### 3.3. Non-Dimensionalization

#### 3.4. Computational Grids

#### 3.5. Boundary Conditions

#### 3.6. Proper Orthogonal Decomposition

## 4. Results

#### 4.1. Turbulent Approach Boundary Layer

#### 4.2. Instantaneous Flow Fields

#### 4.3. Mean Flow Analysis

#### 4.4. Turbulent Statistics

#### 4.5. Unsteady Flow Analysis

#### 4.5.1. Fourier Transforms of Wall-Pressure Coefficient

#### 4.5.2. Probability Density Function of Axial Skin-Friction Coefficient

#### 4.5.3. Proper Orthogonal Decomposition of Pressure Coefficient

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a_{i} | Time coefficient |

C_{f} | Skin-friction coefficient |

C_{p} | Pressure coefficient |

D | Test section diameter |

E | Amplitude |

f | Frequency |

H | Shape factor |

k | Turbulent kinetic energy |

M | Mach number |

n | Fourier mode number |

p | Pressure |

Pr | Prandtl number |

Re | Reynolds number |

r_{SG} | Shock generator centerbody radius |

St | Strouhal number |

t | Time |

T | Temperature |

v_{x},v_{r},v_{t} | Velocities |

r,θ,x | Cylindrical coordinates |

Greek Symbols | |

α | Shock generator cone half-angle |

γ | Ratio of specific heats |

δ | Boundary layer thickness |

δ∗ | Displacement thickness |

ε | turbulent dissipation rate |

ϑ | momentum thickness |

λ | Eigenvalue |

μ | Dynamic viscosity |

v | Kinematic viscosity |

ρ | Density |

Subscripts | |

i | POD mode number |

incomp | Incompressible |

rms | Root-mean-square |

∞ | Approach flow freestream conditions |

Superscripts | |

+ | In wall units |

* | Dimensional quantity |

’ | Fluctuation |

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**Figure 2.**Cross−sectional view of computational grid (outflow boundary) for asymmetric RANS calculation. The axes are made dimensionless by the test section diameter, D. The computational domain boundaries for the ILES are delineated in red.

**Figure 3.**Computational grid for ILES of asymmetric interaction. To enhance clarity, every other grid line in the wall-normal direction was excluded, while every seventh grid line was included in the streamwise direction.

**Figure 4.**Cross—sectional view of computational grid for asymmetric ILES. To improve clarity, every other grid line was excluded in both the wall−normal and spanwise directions. The r and $\theta $ directions of the cylindrical coordinate system are included for future reference (marked in red).

**Figure 5.**Numerical schlieren image obtained from precursor asymmetric RANS calculation ($z=0$ plane). The dashed lines delineate the start and end of the expansion fans.

**Figure 6.**Wall-pressure distributions obtained from temporal averages. Symbols are measurements by Davis et al. [28] for axisymmetric interaction at $R{e}_{D}=4\times {10}^{6}$. Solid and dashed black lines are for ILES and RANS of axisymmetric interaction. Solid and dashed red ($\theta =0$) and blue ($\theta =\pi $) lines are for ILES and RANS of asymmetric interaction.

**Figure 7.**Instantaneous iso-surfaces of (

**a**) $|\nabla \rho |=25$ and (

**b**) $Q=100$ flooded by axial velocity for axisymmetric (

**left**) and asymmetric (

**right**) interaction.

**Figure 8.**Profiles of axial (${v}_{x}$) and tangential (${v}_{t}$) velocity for (

**a**) axisymmetric and (

**b**–

**d**) asymmetric case for (

**b**) $\theta =0$, (

**c**) $\theta =\pi /2$, and (

**d**) $\theta =\pi $. The dotted lines indicate the incident and reflected shock waves.

**Figure 9.**Wall-pressure coefficient contours and inviscid shock impingement line (solid black line) for (

**a**) axisymmetric and (

**b**) asymmetric interaction.

**Figure 10.**Iso-contours of (

**a**) axial, (

**b**) radial, and (

**c**) tangential velocity for axisymmetric interaction [37] (

**left**) and asymmetric interaction at $\theta =1.4$ (

**right**).

**Figure 11.**Skin-friction magnitude iso-contours and skin-friction lines for (

**a**) axisymmetric and (

**b**) asymmetric case.

**Figure 12.**Axial skin-friction coefficient iso-contours and skin-friction lines for (

**a**) axisymmetric and (

**b**) asymmetric case.

**Figure 13.**Axial separation length for the asymmetric (solid lines) and axisymmetric (dashed line) case [37]. The blue line represents the unfiltered data for the asymmetric case.

**Figure 14.**Azimuthal skin-friction iso-contours and skin-friction lines for (

**a**) axisymmetric and (

**b**) asymmetric case.

**Figure 15.**Minimum ${C}_{fx}$ and ${C}_{ft}$ and maximum ${C}_{ft}$ (in axial direction) for axisymmetric (dashed lines) and asymmetric (solid lines) case.

**Figure 17.**Iso-contours of ${v}_{x,rms}^{\prime}$ (

**left**column) and ${v}_{r,rms}^{\prime}$ (

**right**column) for (

**a**) axisymmetric and asymmetric case at (

**b**) $\theta =0$, (

**c**) $\theta =1.4$ and (

**d**) $\theta =\pi $.

**Figure 18.**Iso-contours of ${v}_{t,rms}^{\prime}$ (

**left**column) and k (

**right**column) for (

**a**) axisymmetric and asymmetric case at (

**b**) $\theta =0$, (

**c**) $\theta =1.4$ and (

**d**) $\theta =\pi $.

**Figure 19.**Iso-contours of $\overline{{v}_{x}^{\prime}{v}_{r}^{\prime}}$ (

**left**column) and $\overline{{v}_{x}^{\prime}{v}_{t}^{\prime}}$ (

**right**column) for (

**a**) axisymmetric and asymmetric case at (

**b**) $\theta =0$, (

**c**) $\theta =1.4$, and (

**d**) $\theta =\pi $.

**Figure 20.**Iso-contours of $\overline{{v}_{r}^{\prime}{v}_{t}^{\prime}}$ for (

**a**) axisymmetric and asymmetric case at (

**b**) $\theta =0$, (

**c**) $\theta =1.4$, and (

**d**) $\theta =\pi $.

**Figure 21.**Wall-pressure coefficient Fourier amplitudes for axisymmetric (

**left**) and asymmetric (

**right**) case. Modes (

**a**) $n=2$ ($S{t}_{\delta}=0.003$), (

**b**) $n=6$ ($S{t}_{\delta}=0.013$), (

**c**) $n=137$ ($S{t}_{\delta}=0.22$), and (

**d**) $n=40$ ($S{t}_{\delta}=0.09$). The shock impingement line is marked by the black line.

**Figure 22.**Probability density function of flow separation for axisymmetric (dashed line) and asymmetric (solid lines) case.

**Figure 23.**Wall-pressure coefficient POD eigenvalues for axisymmetric (dashed line) and asymmetric (solid line) case.

**Figure 24.**Fourier transforms of POD time coefficients for axisymmetric (

**left**) and asymmetric (

**right**) case.

**Figure 25.**POD modes (

**a**) $i=1$, (

**b**) $i=2$, (

**c**) $i=3$, and (

**d**) $i=4$ for axisymmetric (

**left**) and asymmetric (

**right**) case.

Property | Axisymmetric [37] | Asymmetric |
---|---|---|

$\Delta {x}^{+}\times \Delta {r}^{+}\times {(r\Delta \theta )}^{+}$ | $10\times 0.13\times 5$ | $10\times 0.13\times 5$ |

${L}_{x}\times {L}_{r}\times {L}_{\theta}$ | $1.4\times 0.25\times 0.17$ | $1.55\times 0.15\times 1.57$ |

${N}_{x}\times {N}_{r}\times {N}_{\theta}$ | $801\times 161\times 203$ | $901\times 126\times 1822$ |

${N}_{total}$ | $2.62\times {10}^{7}$ | $2.07\times {10}^{8}$ |

**Table 2.**Computational grid parameters for earlier grid resolution study [37]. Inner-scale variables calculated at $x=2.65$.

Case | $\Delta {\mathit{x}}^{+}\times \Delta {\mathit{r}}^{+}\times {(\mathit{r}\Delta \mathit{\theta})}^{+}$ | ${\mathit{N}}_{\mathit{x}}\times {\mathit{N}}_{\mathit{r}}\times {\mathit{N}}_{\mathit{\theta}}$ |
---|---|---|

Baseline | $10\times 0.13\times 5$ | $801\times 161\times 203$ |

x-refine | $6.4\times 0.13\times 5$ | $1201\times 161\times 203$ |

r-refine | $10\times 0.09\times 5$ | $801\times 241\times 203$ |

$\theta $-refine | $10\times 0.13\times 3.2$ | $801\times 161\times 304$ |

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**MDPI and ACS Style**

Mosele, J.-P.; Gross, A.; Slater, J.
Numerical Investigation of Asymmetric Mach 2.5 Turbulent Shock Wave Boundary Layer Interaction. *Aerospace* **2023**, *10*, 417.
https://doi.org/10.3390/aerospace10050417

**AMA Style**

Mosele J-P, Gross A, Slater J.
Numerical Investigation of Asymmetric Mach 2.5 Turbulent Shock Wave Boundary Layer Interaction. *Aerospace*. 2023; 10(5):417.
https://doi.org/10.3390/aerospace10050417

**Chicago/Turabian Style**

Mosele, John-Paul, Andreas Gross, and John Slater.
2023. "Numerical Investigation of Asymmetric Mach 2.5 Turbulent Shock Wave Boundary Layer Interaction" *Aerospace* 10, no. 5: 417.
https://doi.org/10.3390/aerospace10050417