# Computational Design Optimization for S-Ducts

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## Abstract

**:**

## 1. Introduction

## 2. State of the Art

## 3. Methods

#### 3.1. Baseline Geometry Configuration

- at the inlet, a cylindrical duct eight times longer than the inlet radius. Its purpose is to ensure uniform inlet conditions;
- at the outlet, a cylindrical duct six times longer than the outlet radius. Its purpose is to guarantee that the outlet conditions do not have any influence on the upstream flow.

#### 3.2. Geometry Parameterization

- since our S-Duct is symmetric with respect to the $x-y$ plane, we decided to design and simulate only half of the duct in order to reduce the computational cost;
- we consider the cylindrical ducts added after and before the S-Duct of fixed geometry, as manufacturing constraints. This means that the only part that have to be parameterized is the S-Duct itself.

- ${X}_{ffd}$ is a vector containing the Cartesian coordinates of the displaced point;
- l, m are the number of control point in S and T direction respectively;
- ${B}_{k}\left(u\right)$ are the degree 3 Bernstein polynomials;
- s, t are the generic point coordinate in the $S-T$ system of reference ($0\le s\le 1$, $0\le t\le 1$);
- ${P}_{ij}$ is a vector containing the Cartesian coordinates of the control point.

- In every cross-section, the deformed geometry is described by a Bezier curve, that is a 1D formulation of the 2D initial FFD problem:$${X}_{ffd}=\sum _{i=0}^{m}{B}_{j}\left(t\right){P}_{i}$$Fixed $m=6$, we inverted this equation in order to find the control points position that correctly interpolate a semicircle. To do this, we imposed the following constraints:
- ${y}_{{P}_{1}}^{\prime}=-r$
- ${y}_{{P}_{1}}^{\prime}={y}_{{P}_{2}}^{\prime}$: tangency condition
- ${z}_{{P}_{3}}={z}_{{P}_{4}}$: symmetry condition
- ${z}_{{P}_{2}}={z}_{{P}_{5}}$: symmetry condition
- ${y}_{{P}_{3}}^{\prime}=-{y}_{{P}_{4}}^{\prime}$: symmetry condition
- ${y}_{{P}_{5}}^{\prime}={y}_{{P}_{6}}^{\prime}$: tangency condition
- ${y}_{{P}_{6}}^{\prime}=r$

where r is the semicircle radius in the particular cross-section. After some calculations we obtained:$$\begin{array}{cc}\hfill {z}_{{P}_{2}}={z}_{{P}_{5}}=& r\frac{4(8\sqrt{2}-9)}{15}\hfill \end{array}$$$$\begin{array}{cc}\hfill {z}_{{P}_{3}}={z}_{{P}_{4}}=& r\frac{2(21-8\sqrt{2})}{15}\hfill \end{array}$$$$\begin{array}{cc}\hfill {y}_{{P}_{3}}^{\prime}=-{y}_{{P}_{4}}^{\prime}=& r\frac{2(64\sqrt{2}-79)}{45}\hfill \end{array}$$ - In order to guarantee tangential condition at the inlet and at the outlet, the control points in the inlet section are copied and translated shortly after. The control points in the outlet section are copied and translated shortly before.

- The control point in the first two cross-section from the S-Duct inlet and the last two before the outlet are fixed. This is due to manufacturing constraints.
- Referring to Figure 4, in every other cross-section we have:
- -
- Point on the symmetry plane (${P}_{1}$, ${P}_{6}$) can only move on the symmetry plane ($do{f}_{{P}_{1}}=2$, $do{f}_{{P}_{2}}=2$).
- -
- To maintain tangency condition, point ${P}_{2}$ and ${P}_{5}$ have the same x and y coordinates as ${P}_{1}$ and ${P}_{6}$ respectively. They can move in z-$direction$ ($do{f}_{{P}_{2}}=1$, $do{f}_{{P}_{5}}=1$).
- -
- point ${P}_{4}$ and ${P}_{5}$ can move in the space ($do{f}_{{P}_{3}}=3$, $do{f}_{{P}_{4}}=3$).

#### 3.3. S-Duct Performance Metrics for the Optimization

- $${f}_{1}=1-\overline{PR}$$$$PR=\frac{{p}_{0,AIP}}{{p}_{0,inlet}}$$
- $${f}_{2}=\left|\overline{\alpha}\right|$$$$\alpha =arctan\left(\frac{{V}_{\theta ,AIP}}{{V}_{x,AIP}}\right)$$

#### 3.4. Optimization Method

- x-direction: between S-Duct inlet and outlet;
- y-direction: $[-10.5{r}_{1},$$9{r}_{1}]$;
- z-direction: $[-4.5{r}_{1},$$9{r}_{1}]$.

- for line upper (UP) and lower (DW) curves in the symmetry plane:$${y}_{UP}\left(x\right)>{y}_{DW}\left(x\right)$$
- Referring for simplicity to the generic cross-section in Figure 4, if ${y}_{{P}_{4}}<{y}_{{P}_{3}}$:$${y}_{{P}_{4}}-{y}_{{P}_{3}}<r1$$
- with ${X}_{{P}_{j}}\left[i\right]$ we indicate the j control point x-range in the generic i cross-section:$${X}_{{P}_{j}}[i-2]\le {X}_{{P}_{j}}\left[i\right]\le {X}_{{P}_{j}}[i+2]$$

#### 3.5. Computational Method

#### 3.5.1. Flow Simulation

#### 3.5.2. Mesh Generation

#### 3.5.3. Boundary Conditions

## 4. Results

#### 4.1. Baseline Analysis

#### 4.2. Results From the Optimization Process

#### 4.3. Multidimensional Data Analysis of the Optimization Process

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AIP | Aerodynamic Interface Plane |

FFD | Free-Form Deformation |

dof | Degree of freedom |

MOTS | Multi-Objective Tabu Search |

## References

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**Figure 3.**Parallelepipedic lattice defined on a plane surface in the symmetry plane (plane $x-y$) of the S-Duct. Dotted lines represent the duct projection on this plane: curves define the main lattice dimensions.

**Figure 4.**Generic cross-section. The black semicircular line represents the baseline geometry. The blue line represents the deformed geometry when the control points ${P}_{1}-{P}_{6}$ are equally spaced on the baseline geometry section.

**Figure 5.**(

**a**) PR and (

**b**) $\alpha $ as a function of the number of mesh elements. The red solid line in (

**a**) represent the experimental result form [16].

**Figure 7.**Baseline pressure recovery distribution comparison: (

**a**) CFD simulation with same geometry as in [16]; (

**b**) CFD simulation with geometry obtained from our new parameterization.

**Figure 9.**Total pressure distribution at AIP. (

**a**) Comparison between Baseline and $op{t}_{CP}$. (

**b**) Comparison between Baseline and $op{t}_{\alpha}$.

**Figure 10.**Total pressure distribution in different cross-section: comparison between $op{t}_{CP}$ (

**left**) and $op{t}_{\alpha}$ (

**right**). Every cross-section is perpendicular to x-direction and situated at $x=2.5{r}_{1}$, $x=4{r}_{1}$, $x=5.5{r}_{1}$, $x=7{r}_{1}$, $x=8.5{r}_{1}$, $x=10{r}_{1}$ (S-Duct outlet) and $x=11{r}_{1}$ (AIP) from S-Duct inlet.

**Figure 11.**S-Duct cross-sections area in optimized solutions compared with Baseline. y-axis represent the ration between the area of the generic cross-section perpendicular to x-direction and inlet area.

**Figure 12.**Axial velocity distribution in different cross-section: comparison between $op{t}_{CP}$ (

**left**) and $op{t}_{\alpha}$ (

**right**). Every cross-section is perpendicular to x-direction and situated at $x=2.5{r}_{1}$, $x=4{r}_{1}$, $x=5.5{r}_{1}$, $x=7{r}_{1}$, $x=8.5{r}_{1}$, $x=10{r}_{1}$ (S-Duct outlet) and $x=11{r}_{1}$ (AIP) from S-Duct inlet.

**Figure 13.**Axial velocity distribution on symmetry plane in optimized solutions compared with Baseline.

**Figure 14.**Recirculation region in optimized solutions compared with Baseline. Blue line represent the total S-Duct axial length. Black lines represent the axial length of recirculation region. Numbers over each lines states the minimum values of x-wall shear stress. Black dot in correspondence of $op{t}_{1}$ indicates the position of the minimum (positive) value of x-wall shear stress for that geometry.

**Figure 16.**Static pressure distribution in different cross-section: comparison between $op{t}_{CP}$ (

**left**) and $op{t}_{\alpha}$ (

**right**). Every cross-section is perpendicular to x-direction and situated at $x=2.5{r}_{1}$, $x=4{r}_{1}$, $x=5.5{r}_{1}$, $x=7{r}_{1}$, $x=8.5{r}_{1}$, $x=10{r}_{1}$ (S-Duct outlet) and $x=11{r}_{1}$ (AIP) from S-Duct inlet.

**Figure 17.**Swirl angle distribution at AIP. (

**a**) Comparison between Baseline and $op{t}_{CP}$. (

**b**) Comparison between Baseline and $op{t}_{\alpha}$.

**Figure 18.**Absolute value of tangential velocity distribution in different cross-section: comparison between $op{t}_{CP}$ (

**left**) and $op{t}_{\alpha}$ (

**right**). Every cross-section is perpendicular to x-direction and situated at $x=2.5{r}_{1}$, $x=4{r}_{1}$, $x=5.5{r}_{1}$, $x=7{r}_{1}$, $x=8.5{r}_{1}$, $x=10{r}_{1}$ (S-Duct outlet) and $x=11{r}_{1}$ (AIP) from S-Duct inlet.

**Figure 19.**The complete dataset is represented in Parallel Coordinates and the two objective functions in the Scatter plot. A selection of the Pareto front is highlighted in blue.

**Figure 20.**Three interval selections expressed for design parameter ×23 and the reflection to the regions of optimality close to the Pareto front.

**Figure 21.**Comparison between the groups of solutions in the compromise region and extreme optimality for the swirl objective function (highlighted in blue).

**Figure 22.**Comparison between the groups of solutions in the compromise region (highlighted in blue) and extreme optimality for the pressure loss objective function.

Individual | $\mathit{CP}$ | $\mathit{\alpha}$$\left[\mathit{deg}\right]$ |
---|---|---|

Baseline | $0.0310$ | $3.3978$ |

Best CP | $0.0251$ | $3.3657$ |

Best swirl | $0.0267$ | $2.9764$ |

Trade-off | $0.0251$ | $3.2827$ |

Individual | $\mathit{CP}$ | $\mathit{\alpha}$$\left[\mathit{deg}\right]$ |
---|---|---|

Baseline | $0.0315$ | $3.4100$ |

Best swirl | $0.0302$ | $2.7500$ |

Trade-off | $0.0239$ | $2.8200$ |

Trade-off | $0.0288$ | $2.7700$ |

Best CP | $0.0237$ | $3.4200$ |

Parameter | Value |
---|---|

${\theta}_{max}$ | ${60}^{\circ}$ |

R | $0.6650m$ |

${r}_{1}$ | $0.0665m$ |

${r}_{2}$ | $0.0820m$ |

Parameter | Values |
---|---|

$Offset$ | $2R(1-cos({\theta}_{max}/2))$ |

${L}_{S-Duct}$ | R |

${L}_{inlet}$ | $8{r}_{1}$ |

${L}_{outlet}$ | $6{r}_{1}$ |

${L}_{AIP}={L}_{inlet}+{L}_{S-Duct}+r1$ | $9{r}_{1}+R$ |

${L}_{TOT}={L}_{inlet}+{L}_{S-Duct}+{L}_{outlet}$ | $14{r}_{1}+R$ |

Parameter | Value |
---|---|

Inlet total pressure | 88,744 Pa |

Inlet static pressure | 69,575 Pa |

Outlet static pressure | 7898 2Pa |

Total temperature | 286.2 K |

**Table 6.**S-Duct performance in baselines geometry: (A) Delot Experiment [16], (a) CFD simulation with same geometry, (b) CFD simulation with geometry obtained from our new parameterization.

Individual | $\mathit{PR}$ | $\mathit{\alpha}$$\left[\mathit{deg}\right]$ |
---|---|---|

Baseline (A) | $0.9711$ | − |

Baseline (a) | $0.9706$ | $4.8511$ |

Baseline (b) | $0.9715$ | $4.3540$ |

**Table 7.**Objective functions comparison between the Baseline geometry, the extreme point and three trade-off solutions in the Pareto front.

Individual | $\mathit{CP}$ | Improvement | $\mathit{\alpha}$$\left[\mathit{deg}\right]$ | Improvement |
---|---|---|---|---|

Baseline (a) | $0.0294$ | − | $4.8511$ | − |

$op{t}_{CP}$ | $0.0252$ | $14.3\%$ | $3.2560$ | $32.9\%$ |

$op{t}_{1}$ | $0.0261$ | $11.2\%$ | $2.5216$ | $48.0\%$ |

$op{t}_{2}$ | $0.0262$ | $10.9\%$ | $1.9972$ | $58.8\%$ |

$op{t}_{3}$ | $0.0264$ | $10.2\%$ | $1.9713$ | $59.4\%$ |

$op{t}_{\alpha}$ | $0.0275$ | $6.5\%$ | $1.4109$ | $70.9\%$ |

© 2018 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) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

D’Ambros, A.; Kipouros, T.; Zachos, P.; Savill, M.; Benini, E. Computational Design Optimization for S-Ducts. *Designs* **2018**, *2*, 36.
https://doi.org/10.3390/designs2040036

**AMA Style**

D’Ambros A, Kipouros T, Zachos P, Savill M, Benini E. Computational Design Optimization for S-Ducts. *Designs*. 2018; 2(4):36.
https://doi.org/10.3390/designs2040036

**Chicago/Turabian Style**

D’Ambros, Alessio, Timoleon Kipouros, Pavlos Zachos, Mark Savill, and Ernesto Benini. 2018. "Computational Design Optimization for S-Ducts" *Designs* 2, no. 4: 36.
https://doi.org/10.3390/designs2040036