# Theoretical Model and Numerical Analysis for Asymmetry of Shock Train in Supersonic Flows

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## Abstract

**:**

## 1. Introduction

_{u}/h increases. Their experiments at three different confinement levels showed that the shock train is more asymmetric at δ

_{u}/h = 0.26 than those at 0.15 and 0.35. Sugiyama et al. [6,7] observed an asymmetric oblique shock train in Mach 4.0 experiments. Gawehn et al. [2] also observed the flipping unsteadiness of a shock train in a Laval nozzle with a small divergence angle.

## 2. Theoretical Considerations

#### 2.1. Aerodynamic Scheme

- (1)
- Based on the features of the Coanda effect introduced in Section 1, it is assumed that the entrainment of mixing layer only activates on one side, while the separation bubble on the other side stays fixed.
- (2)
- After the shedding of eddies, the mass and pressure are reduced inside the bubble but its size stays constant.
- (3)
- If the final flow is still symmetric, a new amount of reversed flow from downstream enters the bubble at reattachment point R to ensure the balance [19].

_{2}(Figure 2).

_{e}. The rate of mass entrainment can be obtained from Piponniau’s model (will be deduced later). According to the third assumption, the distance between mass escaping from the bubble and mass returning into the bubble is L

_{2}, and then the time that can be used for entrainment is given by:

_{2}is the sonic speed on the low-velocity side of the mixing layer.

#### 2.2. Theoretical Model for Flow Deflection with Entrainment for Multiple SBLIs

_{1}and height h (Figure 2) with an average density of ρ

_{m}. Then, the air mass in the bubble by unit span is given by:

_{2}(x) is the vertical coordinate of the edge of the mixing layer on the low-velocity side and y

_{0}(x) is the vertical coordinate of the centerline of the mixing layer, and x

_{0}= L

_{3}= L

_{1}− L

_{2}denotes the position where large eddies shed, η is the similarity variable, the constant C ≅ 0.14, δ

_{w}and δ

_{w}’ are respectively the local thickness and the spreading rate of the mixing layer, and the latter can be expressed by the following relation according to Papamoschou and Roshko [21]:

_{ref}’ ≅ 0.16 is the spreading rate for subsonic half jet [22]. r = u

_{2}/u

_{1}and s = ρ

_{2}/ρ

_{1}are velocity ratio and density ratio across the mixing layer, respectively. The function Φ(M

_{c}) is the normalized spreading rate of mixing layer and is dependent on the convective Mach number M

_{c}:

_{c}) must be determined by experiment [23] and Figure 5 gives its empirical value depending on the convective Mach number. Piponniau et al. [18] introduced a function g(r,s) as:

_{m}is the initial average pressure in the bubble. Then, according to the first assumption, we can obtain the pressure drop in the bubble:

_{m}= p

_{2}is applied in Equation (12) where p

_{2}denotes the static pressure near the point 2 in Figure 4.

_{1i}is the length of each divided separation bubble, p

_{e}is the pressure at the end of control volume, h

_{e}is the duct height at the end of control volume, $\dot{m}$ is the mass flow rate of the main stream, u

_{e}is the main stream velocity at the exit, and β is the flow deflection angle from horizontal direction (Figure 3). Then from Equation (13), we obtain:

## 3. Model Application Based on Numerical Results of An Isolator

#### 3.1. Isolator Geometry and Numerical Methods

_{0}= 2.45, total pressure p

_{t0}= 310 kPa, total temperature T

_{t0}= 295 K, and the inlet boundary layer velocity profile was given by a 1/7 power-law with a boundary layer thickness of 5.4 mm. The simulations for both full and forced symmetric ducts were conducted.

^{+}< 1. An adiabatic, no slip wall boundary condition was specified for all the walls in these simulations. Back pressure with a ratio of 5.85 relative to the inlet static pressure was imposed on the outlet boundary to produce the shock train in the ducts.

#### 3.2. Numerical Verification of Asymmetric SBLI in Full Duct

#### 3.3. The Overall Deflection Angle Calculation for Asymmetric Shock Train

_{1}CD

_{1}covers the region of the first SBLI and the streamtube AB

_{3}CD

_{3}covers all the three SBLIs. The deflection angles of the upper side and the lower side of a streamtube were averaged to be taken as the overall deflection angle. Taking the streamtube AB

_{3}CD

_{3}as an example, the angle between its upper side AB

_{3}and the horizontal direction is θ

_{3}(taking the counterclockwise as the positive), and the angle between its lower side CD

_{3}and the horizontal direction is δ

_{3}, as shown in Figure 9. We suppose that for the symmetric shock train, the angle between its lower side and the horizontal direction is η (here, η is an absolute value), and the angle between its upper side and the horizontal direction is −η. Then, the deflection angle of the upper side AB

_{3}from the imaginary symmetric flow to the actual asymmetric flow is:

_{3}is:

#### 3.4. Model Application Based on Numerical Results of Half Duct with Forced Symmetry at Ma2.45

_{1}is 469 m/s for the first SBLI. The low-velocity side velocity is taken from the maximum reversed velocity in the separation bubble, i.e., u

_{2}is −76 m/s in Figure 13. After obtaining all the necessary aerodynamic parameters of the first separation (Table 1), Equation (14) was applied to calculate the theoretical flow deflection angle downstream the first SBLI (now n = 1 in Equation (14)).

## 4. Conclusions

- (1)
- The flowfield of an asymmetric shock train in a full duct can be simulated satisfactorily numerically, which agrees well with the experimental result.
- (2)
- The neutral stability of shock train asymmetry was observed in the steady simulations, i.e., the shock train attached randomly to the lower wall or to the upper wall of the duct for the different simulation runs.
- (3)
- The theoretical model for the shock train asymmetry produces a larger range estimation and a close averaged value of the deflection angle compared to the actual full duct flow.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Asymmetric shock train in a nozzle with a small divergence angle [2].

**Figure 2.**Sketch of the entrainment of mixing layer on the separation bubble [18].

**Figure 5.**Normalized spreading rate as a function of the convective Mach number [18].

**Figure 6.**Computational domain and boundary conditions for isolator flow simulations: (

**a**) Full isolator; (

**b**) forced symmetric half isolator.

**Figure 7.**Schlieren picture of shock train from simulation and experiment: (

**a**) Simulation with the coarse mesh; (

**b**) simulation with the fine mesh; (

**c**) experiment [5].

**Figure 8.**Wall pressure distributions from simulation and experiment (x

_{0}denotes the onset of the SBLI): (

**a**) bottom wall; (

**b**) top wall.

**Figure 10.**Streamtube deflection angles for three different SBLI regions from numerical and theoretical results.

**Figure 11.**Numerical schlieren of shock train with forced symmetry (three separation bubbles caused by the first three SBLIs are labeled).

**Figure 12.**Wall pressure distributions and wall shear stresses for the full duct and the half duct (x

_{0}denotes the onset of the SBLI): (

**a**) Wall static pressure distribution; (

**b**) wall shear stress.

**Table 1.**Aerodynamic parameters of the separation and asymmetry for the first SBLI of shock train with forced symmetry.

Part | T_{1}/T_{2} | u_{1} (m/s) | r | s | g(r,s) × 10^{2} | M_{c} | Φ(M_{c}) | h (mm) | L_{1} (mm) | L_{2} (mm) | Δp (Pa) | p_{e}/p_{t0} | h_{e} (mm) | $\dot{\mathit{m}}{\mathit{u}}_{\mathit{e}}\text{}\left(\mathbf{N}\right)$ | β_{1} (°) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

I | 0.64 | 469 | −0.162 | 0.391 | 1.35 | 0.88 | 0.32 | 7.69 | 87.2 | 35.0 | 1218 | 0.097 | 21 | 3927 | −1.4 |

**Table 2.**Aerodynamic parameters of the separation and asymmetry for the three SBLIs with forced symmetry.

Part | T_{1}/T_{2} | u_{1} (m/s) | r | s | g(r,s) × 10^{2} | M_{c} | Φ(M_{c}) | h (mm) | L_{1} (mm) | L_{2} (mm) | Δp (Pa) | p_{e}/p_{t0} | h_{e} (mm) | $\dot{\mathit{m}}{\mathit{u}}_{\mathit{e}}\text{}\left(\mathbf{N}\right)$ | β_{3}(°) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

I | 0.64 | 469 | −0.162 | 0.391 | 1.35 | 0.88 | 0.32 | 7.69 | 87.2 | 35.0 | 1218 | 0.176 | 20 | 3387 | −3.2 |

II | 0.76 | 381 | −0.125 | 0.418 | 1.53 | 0.67 | 0.46 | 6.32 | 44.0 | 17.5 | 1539 | ||||

III | 0.84 | 331 | −0.037 | 0.432 | 1.78 | 0.52 | 0.60 | 2.28 | 21.3 | 8.9 | 3402 |

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**MDPI and ACS Style**

Sun, B.; Wang, C.; Zhuo, C.
Theoretical Model and Numerical Analysis for Asymmetry of Shock Train in Supersonic Flows. *Symmetry* **2020**, *12*, 518.
https://doi.org/10.3390/sym12040518

**AMA Style**

Sun B, Wang C, Zhuo C.
Theoretical Model and Numerical Analysis for Asymmetry of Shock Train in Supersonic Flows. *Symmetry*. 2020; 12(4):518.
https://doi.org/10.3390/sym12040518

**Chicago/Turabian Style**

Sun, Bo, Chengpeng Wang, and Changfei Zhuo.
2020. "Theoretical Model and Numerical Analysis for Asymmetry of Shock Train in Supersonic Flows" *Symmetry* 12, no. 4: 518.
https://doi.org/10.3390/sym12040518