#
A CFD-Based Throughflow Method with Three-Dimensional Flow Features Modelling^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Numerical Scheme

## 3. Blade and Dissipative Body Force Model

## 4. Secondary Flow Model

## 5. Tip Leakage Model

## 6. Applications

#### 6.1. T106 Cascade

#### 6.2. CT3 Transonic Stage

^{®}i7-4770 CPU @3.40 GHz (Dresden, Germany). A comparable convergence level in a steady, 3D, viscous, parallel calculation with the TRAF code requires a computational time of about half an hour on a reasonably coarse mesh.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

## Latin Symbols

A | tip gap area |

b | tangential blockage factor $b=\frac{{\vartheta}_{PS}-{\vartheta}_{SS}}{2\pi /N}$ |

${c}_{d}$ | tip leakage discharge coefficient |

d | dissipating body force |

$\mathcal{F},\mathcal{G}$ | inviscid flux vectors |

f | blade body force |

H | specific total enthalpy |

J | Jacobian |

M | Mach number |

m | meridional coordinate |

$\dot{m}$ | mass flow rate |

N | number of blades |

p | pressure |

R | gas constant |

r | radius |

$\mathcal{S}$ | source term vector |

s | specific entropy |

$s,n,h$ | intrinsic coordinates |

T | absolute temperature, leakage loss distribution function |

t | time |

u | axial velocity |

$\mathcal{U},\mathcal{V}$ | contravariant velocities in $\xi $ and $\eta $ directions |

v | radial velocity |

W | power |

w | tangential velocity |

$x,r,\vartheta $ | cylindrical coordinates |

Z | secondary flow penetration depth |

${z}_{0}$ | vortex characteristic length |

## Greek Symbols

$\Gamma $ | vortex circulation, streamsurface |

$\eta $ | efficiency |

$\zeta $ | normalized spanwise coordinate, loss coefficient, $\zeta =\frac{{p}_{t1}-{p}_{t2}}{{p}_{t2}-{p}_{2}}$ |

$\nu $ | kinematic fluid viscosity |

$\xi ,\eta $ | curvilinear coordinates |

$\rho $ | fluid density |

$\Phi $ | secondary loss distribution function |

$\Omega $ | angular velocity |

$\omega $ | vorticity |

## Subscripts

1, 2 | cascade inlet, outlet |

b | blockage-related |

f | force-related |

$is$ | isentropic |

$PS$ | pressure side |

$ref$ | reference |

$SS$ | pressure side |

t | total quantity |

## Superscripts

T | transposed |

## Abbreviations

CFD | Computational Fluid Dynamics |

EU | European Union |

RMS | Root Mean Square |

TATEF2 | Turbine Aero-Thermal External Flows |

TF | Throughflow |

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**Figure 1.**(

**a**) Sketch of a blade row with the intrinsic coordinate system; and (

**b**) Lamb–Oseen vortex and secondary velocities.

**Figure 2.**Comparisons between 3D computational fluid dynamics (CFD) and throughflow results in terms of spanwise distributions of (

**a**) secondary deviation angle [deg] and (

**b**) loss coefficient for different values of the ${c}_{1}$ constant.

**Figure 4.**Predicted and measured spanwise distributions of (

**a**) secondary deviation angle [deg] and (

**b**) loss coefficient for the T106 cascade with tapered endwalls [10].

**Figure 6.**Predicted and measured spanwise distributions of flow quantities in Plane 3 for the CT3 stage at Nom condition with and without 3D effects (abs. flow angle expressed in [deg]).

**Figure 7.**Predicted and measured spanwise distributions of flow quantities in Plane 3 for the CT3 stage at High and Low conditions (abs. flow angle expressed in [deg]).

${\mathit{M}}_{2,\mathit{i}\mathit{s}}$ | ${\mathit{M}}_{3,\mathit{i}\mathit{s}}$ | ${\mathit{p}}_{\mathit{t}1}/{\mathit{p}}_{\mathit{t}3}$ | |
---|---|---|---|

Low | $1.071$ | $0.65$ | $2.19$ |

Nom | $1.242$ | $0.97$ | $3.19$ |

High | $1.249$ | $1.18$ | $3.85$ |

W (kW) | ${\mathit{\eta}}_{\mathbf{tt}}$ | |
---|---|---|

Low: Exp./ TF w 3D/ TF w/o 3D | $713/737/746$ | $0.933/0.9360/0.946$ |

Nom: Exp./ TF w 3D/ TF w/o 3D | $1003/1033/1047$ | $0.891/0.9130/0.925$ |

High: Exp./ TF w 3D/ TF w/o 3D | $1108/1141/1155$ | $0.889/0.9108/0.921$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

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**MDPI and ACS Style**

Pacciani, R.; Marconcini, M.; Arnone, A. A CFD-Based Throughflow Method with Three-Dimensional Flow Features Modelling. *Int. J. Turbomach. Propuls. Power* **2017**, *2*, 11.
https://doi.org/10.3390/ijtpp2030011

**AMA Style**

Pacciani R, Marconcini M, Arnone A. A CFD-Based Throughflow Method with Three-Dimensional Flow Features Modelling. *International Journal of Turbomachinery, Propulsion and Power*. 2017; 2(3):11.
https://doi.org/10.3390/ijtpp2030011

**Chicago/Turabian Style**

Pacciani, Roberto, Michele Marconcini, and Andrea Arnone. 2017. "A CFD-Based Throughflow Method with Three-Dimensional Flow Features Modelling" *International Journal of Turbomachinery, Propulsion and Power* 2, no. 3: 11.
https://doi.org/10.3390/ijtpp2030011