# Entropy Minimization Design Approach of Supersonic Internal Passages

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**Figure 1.**(

**a**) Mach number iso-contour in a supersonic passage; (

**b**) Mach number iso-contour in a subsonic passage; (

**c**) Isentropic Mach number distribution along the two passages.

**Figure 2.**(

**a**) Normalized specific mass flow rate at different inlet Mach number; (

**b**) Flow pattern in a supersonic bladed passage.

## 2. Allowable Turning through Supersonic Passages

_{2}), as defined by Equation (1), enter the passage.

_{2}). Equation (2) defines this isentropic area limitation, which is an exclusive function of the downstream subsonic Mach number (M

_{2}) and the fluid properties.

_{1}) and inlet angle. Let us consider a fully inlet axial flow (α

_{inlet}= 0) when M

_{ref}= M

_{inlet}= 2.0. The maximum turning through this particular passage would be limited to 34 (deg.). In case the inlet Mach number would be augmented by 50%, a much higher turning could be achieved (α

_{throat}could be 44 deg.). Conversely, for the same inlet Mach number, an increase of inlet flow angle allows an increase of the turning.

**Figure 3.**Maximum blade turning in function of the inlet angle (αinlet) and Mach number normalized by M

_{ref}according to the Kantrowitz limit.

## 3. Numerical Flow Analysis Method

#### 3.1. Leading Edge Shock Modeling

_{1}). The detached shock approximates to an asymptotic line for large distances from the leading edge. Hence, Moeckel proposed the use of a simple curve with the characteristics of a hyperbola represented by Equation (4) to define the wave shape.

**Figure 5.**(

**a**) Numerical Schlieren at different inlet Mach numbers, red line represents the predicted/modeled leading edge shock wave, (

**b**) leading edge shock losses prediction along the tangential direction at three inlet Mach numbers and comparison with CFD results

#### 3.2. Method of Characteristics

_{xx}Φ

_{xy}Φ

_{yy}(Φ being the velocity potential).

_{xy}is undetermined. These directions represent the lines where the derivative of the velocity is discontinuous, which are known as characteristic lines. The slope of this characteristic’s line at a certain point in the flow field can be determined with Equation (8).

^{−1}(1/M) and θ is the flow direction at the evaluated location. Equation (8) stipulates that there are two characteristic lines passing through any point in the flow field. A left running characteristic that passes above the streamline (θ + μ known as C

_{+}) and a right running one that passes below the streamline (θ − μ known as C

_{−}). Along each of the characteristic line the flow properties are defined by the algebraic compatibility equations [12].

**Figure 6.**Different unit processes within the characteristic net of a 2D steady and irrotational flow.

_{−})

_{3}= (K

_{−})

_{1}and (K

_{+})

_{3}= (K

_{+})

_{2}. When a boundary condition such as a solid wall is intersected (Point 5) the wall curvature should be equivalent to the flow direction (θ

_{wall}= θ

_{5}). Since, (K

_{+})

_{4}= (K

_{+})

_{5}point 5 can be easily computed. Another possible boundary condition is the downstream intersection with a shock wave, as depicted by point 7. The post shock conditions are estimated in an iterative manner by imposing (K

_{−})

_{6}= (K

_{−})

_{7}and using the oblique shock equations to find β

_{shock}for a certain M

_{inlet}that respects this equality [12].

_{shock}), the upstream flow angle (θ

_{1}) and upstream Mach number (M

_{1}) are known, the computations of the temperature and pressure ratio across the shock can be obtained using Equations (11) and (12).

_{1}and M

_{1}are given by the upstream intersecting characteristic. At both crossings, α

_{shock}is given by the Moeckel method. By contrast, at the remaining intersections, the angle of the shock wave is calculated by the shock-reflection prediction at the wall. Hence, the local entropy production at each intersection can then be calculated with Equation (13) and afterward average in the pitch-wise direction with Equation (14). The outlet entropy value is the addition of all the average values (a to b and b to c in Figure 2b).

#### 3.3. Assessment of the Method of Characteristics

**Figure 8.**(

**a**) Illustration of the characteristic net and the empirically predicted shock waves. (

**b**) Numerical Schlieren visualization obtained with 3D RANS simulations.

_{ax}= 0.75 the leading edge shock wave from the neighboring airfoil impacts with the boundary layer and therefore a separation bubble appeared, which results in a smoother variation of the Mach number, as described by the RANS simulations. By contrast, both Euler and MOC were unable to capture such a shock-boundary layer interaction and therefore they present a very steep variation.

**Figure 9.**Comparison of the isentropic Mach number distribution using the 2D Method of Characteristics (MOC), 3D non-viscous simulations (Euler), and 3D Reynolds Averaged Navier-Stokes simulations (RANS) at mid span.

**Table 1.**Comparison of the turbine outlet flow conditions computed with RANS and MOC simulations and the respective computational time.

P_{0out}/P_{0in} [-] | M_{out} [-] | Θ_{out} [deg.] | Time [s] | |
---|---|---|---|---|

Error | +4.7% | +0.6% | 2[deg.] | −99.5% |

#### 3.4. Numerical Tool Validation

_{01}/P

_{s}

_{3}= 3.8) that lead to a supersonic vane outlet Mach number (M

_{2is}= 1.24). Static pressure measurements were performed along the stator pressure and suction side. In this assessment our computational 3D mesh has four million cells, which ensured a y

^{+}below 1 in all the walls, necessary to resolve the laminar sublayer. Turbulent simulations were performed with the Spalart-Allmaras (SA) turbulence model. Figure 10a shows a 2D cut that sketches the impingent of the trailing edge shock wave on the suction side. Figure 10b shows a good agreement between the experimental isentropic Mach number and the numerical result obtained with CFD++. The solver adequately predicted the trailing edge shock impingement and its impact on the velocity field.

**Figure 10.**(

**a**) 2D cut of the geometry used for validation purposes with representation of the trailing edge shock system. (

**b**) Comparison between the experimental [13] and numerical isentropic Mach number across the passage at mid-span.

## 4. Supersonic Design Approach

#### 4.1. Geometrical Parameterization

- -
- the leading edge thicknesses (T
_{1,le}) - -
- the second leading edge thickness (T
_{2,le}) - -
- and the wedge leading edge angle (α
_{le})

- -
- the first trailing edge thickness (T
_{1,te}) - -
- the wedge trailing edge angle (α
_{Te}) - -
- the second trailing edge thickness (T
_{2,te})

#### 4.2. Optimization Approach

Parameter | Lower–Upper _{SS} | Upper–Lower _{PS} |
---|---|---|

α_{le} [deg.] | 22.5–27.5 | 22.5–27.5 |

P_{3} [mm] | 1.04–2.4 | −1.9–6.1 |

P_{4} [mm] | 1.29–4.9 | −4.4–3.6 |

P_{5} [mm] | 1.6–8.6 | −3.1–4.9 |

P_{6} [mm] | −1.8–6.2 | −3.2–4.8 |

## 5. Results of the Optimization Procedure

#### 5.1. Minimum Losses

_{1,le}) was not altered to ensure a minimum leading edge thickness. Instead, the wedge angle (α

_{le,ss}) was slightly reduced by 2.5 deg., and the second leading edge thickness (T

_{2,le}) was lowered 16%. Consequently, the mechanical properties of the airfoil geometry were slightly modified. Table 3 lists the 2D cross section area (A), the minimum and the maximum momentum of inertia (I

_{min}, I

_{max}) and the angle between the axis of minimum momentum of inertia and the axial direction (α

_{Imin}). Figure 16b shows the thickness distribution for both geometries along the axial direction. A maximum thickness difference occurred at X/C

_{ax}= 50% where it was reduced by 10% in the optimized design. The resulting optimized profile allowed a reduction of the shock detachment distance, as well as a decrease of the shock angle relative to the flow, which consequently minimized the shock losses.

**Figure 16.**(

**a**) Illustration of the initial profile shape (solid line) and the optimized one (dashed line) for minimum shock losses with detailed illustration of the leading edge modification. (

**b**) Thickness variation along the axial direction for both geometries.

**Table 3.**Geometrical features of both geometries: 2D Area, minimum and maximum momentum of inertia and angle of the minimum momentum of inertia.

A/g [m] | I_{min} [m^{4}] | I_{max} [m^{4}] | α_{Imin} [deg.] | |
---|---|---|---|---|

Initial | 9.57 × 10^{−3} | 3.02 × 10^{−9} | 4.37 × 10^{−7} | −12.9 |

Optimized | 8.34 × 10^{−3} | 1.02 × 10^{−9} | 3.84 × 10^{−7} | −13.0 |

_{ax}= 50% is notably reduced. The main differences between the initial and optimized designs are two-fold:

- A higher loading at the leading edge on the optimized geometry due to the greater flow acceleration on the suction side. The smoother acceleration along the pressure side helped to reduce the flow deceleration prior to the shock impact.
- Due to the smaller wedge angle of the leading edge, the shock waves have a higher inclination angle, which directly implies lower losses. As a consequence, the shock impact on the suction and pressure side occurs further downstream (10% of the suction side). Because the Mach number before the shock impingement is detrimental to the intensity of the reflected shock, the optimizer tried to reduce the upstream Mach number by modifying the suction side shape.

**Figure 17.**Isentropic Mach number distribution for the initial and the optimized geometry for minimum shock losses.

#### 5.2. Imposed Mach Number Distribution

**Figure 18.**Representation of the target Mach number distribution and respective initial and the optimized geometry.

**Figure 19.**Illustration of the initial profile shape and the optimized one that better fits the imposed Mach number distribution.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

A- | Flow area (m^{2}) | cp- | Specific heat at constant pressure (J/kgK) |

C- | Characteristic line | g- | Pitch (m) |

H- | Height of the channel (m^{2}) | u- | Velocity in the x direction (m/s) |

K- | Compatibility equation | v- | Velocity in the y direction (m/s) |

M- | Mach number (-) | s- | Entropy (J/kg K) |

P0- | Total pressure (Pa) | m- | mass flow rate (kg/s) |

P- | Bezier control point | ν(M)- | Prandtl Meyer expansion (deg.) |

R- | Specific gas constant (J/kgK) | θ- | Local flow angle (deg.) |

a- | Sound velocity (m/s) | Φ- | Velocity potential (m/s) |

Subscript | |||

+ | Right running characteristic | le- | Leading edge |

− | Left running characteristic | min | Minimum value |

char- | Characteristics line | max | Maximum value |

ref- | Inlet Mach number of the test cases | te- | Trailing edge |

ax | axial direction | out- | Outlet |

in- | Inlet | ss- | Suction side |

ps- | Pressure Side | SB- | Sonic point |

is- | Isentropic value | ||

Greek Symbols | |||

α- | Metal angle (deg.) | ||

α_{shock}- | Shock angle (deg.) | ||

γ- | Specific heat ratio (-) | ||

μ- | Mach angle sin^{−1}(1/M) | ||

Acronyms | |||

MOC- | Method Of Characteristics | ||

RANS- | Reynolds Averaged Navier Stokes |

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Sousa, J.; Paniagua, G.
Entropy Minimization Design Approach of Supersonic Internal Passages. *Entropy* **2015**, *17*, 5593-5610.
https://doi.org/10.3390/e17085593

**AMA Style**

Sousa J, Paniagua G.
Entropy Minimization Design Approach of Supersonic Internal Passages. *Entropy*. 2015; 17(8):5593-5610.
https://doi.org/10.3390/e17085593

**Chicago/Turabian Style**

Sousa, Jorge, and Guillermo Paniagua.
2015. "Entropy Minimization Design Approach of Supersonic Internal Passages" *Entropy* 17, no. 8: 5593-5610.
https://doi.org/10.3390/e17085593