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Reynolds Sensitivity of the Wake Passing Effect on a LPT Cascade Using Spectral/hp Element Methods^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Numerical Approach

#### 2.2. LPT Setup with Wake Passing

#### 2.3. Modelling the Bar Passing Effect

#### 2.4. Low-Speed Experimental Testing of LPTs

## 3. Results

#### 3.1. Evidence of the Transition Mechanism

#### 3.2. Space-Time Boundary Layer Behaviour

#### 3.3. Blade Wall Distributions

#### 3.4. Wake Traverses and Experimental Comparison

#### 3.5. Mixed-Out Measurements

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Nomenclature

## Abbreviations

BF | Body Forcing |

BL | Boundary Layer |

CFD | Computational Fluid Dynamics |

DNS | Direct Numerical Simulation |

IW | Inflow Wakes |

IT | Inflow Turbulence |

LDA | Laser Doppler Anemometry |

LES | Large-Eddy Simulation |

LPT | Low Pressure Turbine |

RANS | Reynolds-Averaged Navier–Stokes |

SVV | Spectral Vanishing Viscosity |

TI | Turbulence Intensity |

URANS | Unsteady RANS |

## Nomenclature

$\alpha $ | Flow angle |

$\omega $ | Total pressure loss coefficient |

$\Phi $ | Flow coefficient |

$\phi $ | Wake passing phase |

$C,\left({C}_{ax}\right)$ | Blade (axial) chord length |

${C}_{f}$ | Skin friction coefficient |

${C}_{p}$ | Static pressure coefficient |

${F}_{\mathrm{red}}$ | Reduced frequency |

H | Boundary layer shape factor |

k | Turbulence kinetic energy |

${L}_{t}$ | Turbulence length scale |

${L}_{z}$ | Spanwise domain size |

n | Blade wall-normal distance |

${N}_{z}$ | Number of Fourier planes |

p | Pressure |

P | Polynomial order |

${P}_{b}$ | Bars pitch |

${P}_{y}$ | Blade pitch |

$Re$ | Reynolds number |

${S}_{0}$ | Suction surface perimeter |

$T,t$ | Time |

${\mathcal{T}}_{b}$ | Wake passing period |

${U}_{b}$ | Bar speed |

${U}_{2}$ | Mixed-out exit velocity |

${U}_{\infty}$ | Reference inlet speed |

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**Figure 2.**Iso-surfaces of Q-Criterion ($Q=200$) contoured by velocity magnitude in case $R{e}_{2}=\mathrm{297,000}$. The computational domain is duplicated for graphical purposes.

**Figure 3.**Instantaneous time–space skin friction on the suction surface at eight consecutive phases, $R{e}_{2}=\mathrm{86,000}$. The dashed boxes identify the regions shown in detail in Figure 4.

**Figure 4.**Instantaneous suction surface statistics for $R{e}_{2}=\mathrm{86,000}$ in the regions identified by the black dashed boxes of Figure 3. The top subfigure shows spanwise vorticity in the blade-normal plane denoted with dash-dotted lines of Figure 3, superimposed with fluctuating velocity vectors. The middle and bottom figures show respectively wall-parallel and wall-normal fluctuating velocity $n/C=0.01$ away from the wall. The solid and dashed lines in the bottom subfigures are iso-lines of ${w}^{\prime}=\pm 0.15$. (

**Left**): $\phi =0.25$; (

**right**): $\phi =0.625$.

**Figure 5.**Instantaneous space–time contour of suction surface BL shape factor, superimposed with iso-lines of wall-shear stress at two levels: ${C}_{f}=0$ (continuous line), ${C}_{f}=-0.024$ (dashed line). (

**Left**): $R{e}_{2}=\mathrm{86,000}$; (

**middle**): $R{e}_{2}=\mathrm{157,000}$; (

**right**): $R{e}_{2}=\mathrm{297,000}$.

**Figure 6.**Blade wall distributions with increasing $R{e}_{2}$, compared with momentum forcing cases at the same flow regime. (

**Left**): pressure distribution; (

**right**): skin-friction coefficient. Y-axis tick labels are omitted due to data sensitivity.

**Figure 7.**Wake traverses at $\widehat{x}=0.513$ with increasing Reynolds numbers. (

**Left**): $R{e}_{2}=\mathrm{86,000}$ (in yellow); (

**middle**): $R{e}_{2}=\mathrm{157,000}$ (in red); (

**right**): $R{e}_{2}=\mathrm{297,000}$ (in burgundy). Three different wake profiles are shown. (

**Top**): velocity magnitude; (

**middle**): total pressure loss coefficient; (

**bottom**): turbulence kinetic energy. The solid lines indicate IW simulations, and the dashed line indicates IW+IT, for $R{e}_{2}=\mathrm{86,000}$ only. The two experimental traverses $S1$ and $S2$ are represented with squares and circles. Y-axis tick labels are omitted due to data sensitivity.

**Figure 8.**Mixed-out wake traverse measurements: (

**left**): total pressure loss coefficient; (

**right**): exit angle, with associated streamtube correction shown with red markers. The orange area indicates the uncertainty associated with the measurement chain, of respectively $2.5\%$ and $\pm 0.{2}^{\circ}$. Y-axis tick labels are omitted due to data sensitivity.

**Table 1.**LPT bar passing setup with cylinder parameters in the upper portion of the table. $R{e}_{2}=\mathrm{86,000}$ was simulated both with inflow wakes (IW) and inflow turbulence (IT), while other regimes analyse IW alone. The compute time is estimated on 1000 cores on the Archer supercomputer, including runtime post-processing and thus providing a conservative estimate.

Re_{2} | $\mathrm{86,000}$ (IW, IW+IT) | $\mathrm{157,000}$ (IW) | $\mathrm{297,000}$ (IW) |
---|---|---|---|

${F}_{\mathrm{red}}$ (${S}_{0}$-based) | 0.624132 | 0.627675 | 0.633188 |

${U}_{b}^{\mathrm{sim}}/{U}_{\infty}$ | 0.705339 | 0.706414 | 0.708116 |

${\Phi}^{\mathrm{sim}}$ | 1.17731 | 1.17414 | 1.16966 |

${\alpha}_{1}\phantom{\rule{0.277778em}{0ex}}{[}^{\circ}]$ | 33.86 | 33.96 | 34.08 |

$\Delta t$ | $2.5\times {10}^{-5}$ | $2.5\times {10}^{-5}$ | $2\times {10}^{-5}$ |

Compute time for $T=1C/{U}_{\infty}$ | 8 h 40 min | 8 h 40 min | 10 h 45 min |

**Table 2.**Summary of percentage relative error between experimental and computational mixed-out quantity, as well as summary of the streamtube contraction factors.

Parameter | ${\mathit{Re}}_{2}=\mathrm{86,000}$ | ${\mathit{Re}}_{2}=\mathrm{157,000}$ | ${\mathit{Re}}_{2}=\mathrm{297,000}$ |
---|---|---|---|

$\frac{\left|\right|{\omega}_{\mathrm{IW}}^{M}-{\omega}_{\mathrm{Exp}}^{M}\left|\right|}{{\omega}_{\mathrm{Exp}}^{M}}\phantom{\rule{0.277778em}{0ex}}[\%]$ | 2.919 | 5.388 | 2.516 |

$\frac{\left|\right|{\alpha}_{2,\mathrm{IW}}^{M}-{\alpha}_{2,\mathrm{Exp}}^{M}\left|\right|}{{\alpha}_{2,\mathrm{Exp}}^{M}}\phantom{\rule{0.277778em}{0ex}}[\%]$ | 0.563 | 0.757 | 0.841 |

$\frac{\left|\right|{\alpha}_{2,\mathrm{IW}}^{M,\mathrm{mod}}-{\alpha}_{2,\mathrm{Exp}}^{M}\left|\right|}{{\alpha}_{2,\mathrm{Exp}}^{M}}\phantom{\rule{0.277778em}{0ex}}[\%]$ | 0.129 | 0.171 | 0.187 |

$\frac{{L}_{z}^{\prime}-{L}_{z}}{{L}_{z}}\phantom{\rule{0.277778em}{0ex}}[\%]$ | −1.164 | −1.564 | −1.734 |

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## Share and Cite

**MDPI and ACS Style**

Cassinelli, A.; Mateo Gabín, A.; Montomoli, F.; Adami, P.; Vázquez Díaz, R.; Sherwin, S.J.
Reynolds Sensitivity of the Wake Passing Effect on a LPT Cascade Using Spectral/*hp* Element Methods. *Int. J. Turbomach. Propuls. Power* **2022**, *7*, 8.
https://doi.org/10.3390/ijtpp7010008

**AMA Style**

Cassinelli A, Mateo Gabín A, Montomoli F, Adami P, Vázquez Díaz R, Sherwin SJ.
Reynolds Sensitivity of the Wake Passing Effect on a LPT Cascade Using Spectral/*hp* Element Methods. *International Journal of Turbomachinery, Propulsion and Power*. 2022; 7(1):8.
https://doi.org/10.3390/ijtpp7010008

**Chicago/Turabian Style**

Cassinelli, Andrea, Andrés Mateo Gabín, Francesco Montomoli, Paolo Adami, Raul Vázquez Díaz, and Spencer J. Sherwin.
2022. "Reynolds Sensitivity of the Wake Passing Effect on a LPT Cascade Using Spectral/*hp* Element Methods" *International Journal of Turbomachinery, Propulsion and Power* 7, no. 1: 8.
https://doi.org/10.3390/ijtpp7010008