# Shock Capturing in Large Eddy Simulations by Adaptive Filtering

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Spatial Discretization

#### 2.2. Treatment of Regions with Shocks

## 3. Basic Tests

#### 3.1. Riemann Problems

#### 3.2. Interaction of Plane Waves with Shocks

#### 3.3. 3D Evolution

## 4. Jet LES

#### 4.1. Free, Underexpanded Round Jet

_{i}and phase ${\varphi}_{i}$ varied from 0.1 to 0.6 and 0 to $5\pi /6$, respectively. The quantity within square brackets ensures that fluctuations are added within the shear layer only.

#### 4.2. Impinging Round Jet

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Modified wavenumber characteristics of Hixon–Turkel split derivative formulas. (

**b**) Effective orders of truncation error. (

**a**) Modified wavenumber, (

**b**) Order behavior.

**Figure 2.**Scheme for changing filter order near isolated and proximate shocks: (

**a**) isolated shock; and (

**b**) proximate shocks.

**Figure 3.**Response and damping functions of filters and scheme. F2–F10 are filters of various orders, and S6 is the sixth-order difference scheme. Filter parameters: ${\alpha}_{f}\left(F2\right)=0.47,{\alpha}_{f}\left(F4\right)=0.48$ and ${\alpha}_{f}(F6,F8,F10)=0.498$. (

**a**) Filter response functions; and (

**b**) filter damping functions.

**Figure 4.**Solution of shock tube problems. exact (——), numerical (– – –), filter order (•); ${\alpha}_{f}\left(F2\right)=0.47,{\alpha}_{f}\left(F4\right)=0.48$ and ${\alpha}_{f}(F6,F8,F10)=0.498$. (

**a**) Sod’s problem at t = 0.2; and (

**b**) Lax’s problem at t = 0.13.

**Figure 5.**Solution at $t=0.3$ for Case 13 by Lax and Liu [27], and filter adaptation: (

**a**) Density; and (

**b**) Filter adaptation ((3) (

**cyan**) adapted along x and y; (2) (

**red**) along y only; and (1) (

**blue**) along x only).

**Figure 6.**Shu–Osher problem at t = 1.8 (red dotted curve: $\Delta x=0.05$; black curve: $\Delta x=6.25\times {10}^{-3}$).

**Figure 7.**Shock-vorticity/entropy wave interaction. $\Delta y=\pi /32$; Red curve: $\Delta x=\pi /50$; black curve (finer grid): $\Delta x=\pi /100$. (

**a**) Vorticity ${\omega}_{z}$ contours at t = 25 for $\psi $ = 45°, ${k}_{y}$ = 1; (

**b**) ${\omega}_{z}$ along $y=0$ for $\psi $ = 45°, ${k}_{y}$ = 1 at t = 25; and (

**c**) ${\omega}_{z}$ along $y=0$ for $\psi $ = 75°, ${k}_{y}$ = 1 at t = 32.

**Figure 8.**Shock-vorticity/entropy wave interaction. Black dotted line: linear solution; black: grid as in Johnsen et al. [10]; red: medium grid, $\Delta x,\Delta y$ halved; blue: fine grid $\Delta x,\Delta y$ halved again. (

**a**) $\psi $ = 45°, ${k}_{y}$ = 1; (

**b**) $\psi $ = 45°, ${k}_{y}$ = 2; (

**c**) $\psi $ = 75°, ${k}_{y}$ = 1; and (

**d**) $\psi $ = 75°, ${k}_{y}$ = 2.

**Figure 10.**LES of underexpanded jet for ${M}_{j}=2.103$, ${p}_{0}/{p}_{a}=9.187$. Experiment by Norum and Seiner [32] (case on page 27 of their report). (

**a**) Mean pressure on longitudinal plane. (

**b**) Pressure along jet axis. Curves for simulation data, symbol for experiment. (

**c**) Instantaneous pressure on longitudinal plane (Medium grid). (

**d**) Energy spectra (at green dot in the left figure).

**Table 1.**Initial condition for 2D Riemann problem—Case 13 by Lax and Liu [27].

Quadrant | $\mathit{\rho}$ | p | v |
---|---|---|---|

I | 1.0 | 1.0 | −0.3 |

II | 2.0 | 1.0 | 0.3 |

III | 1.0625 | 0.4 | 0.8145 |

IV | 0.5313 | 0.4 | 0.4276 |

${64}^{3}$ | ${128}^{3}$ | ${512}^{3}$ | Brachet et al. [30] | |
---|---|---|---|---|

Inviscid | ||||

Energy (t = 5) | 0.9846 | 0.9943 | 0.9992 | 1.00 |

Enstrophy (t = 3.5) | 3.276 | 3.402 | 3.458 | 3.459 |

Viscous | ||||

Energy (t = 5) | 0.9423 | 0.9468 | 0.9476 | |

Enstrophy (t = 3.5) | 3.080 | 3.145 | 3.154 |

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**MDPI and ACS Style**

Patel, S.K.; Mathew, J.
Shock Capturing in Large Eddy Simulations by Adaptive Filtering. *Fluids* **2019**, *4*, 132.
https://doi.org/10.3390/fluids4030132

**AMA Style**

Patel SK, Mathew J.
Shock Capturing in Large Eddy Simulations by Adaptive Filtering. *Fluids*. 2019; 4(3):132.
https://doi.org/10.3390/fluids4030132

**Chicago/Turabian Style**

Patel, Sumit Kumar, and Joseph Mathew.
2019. "Shock Capturing in Large Eddy Simulations by Adaptive Filtering" *Fluids* 4, no. 3: 132.
https://doi.org/10.3390/fluids4030132