COMUNICAÇÃO TÉCNICA ______________________________________________________________________________________________________________________________________________________________________________________________________

Nº176546

On the use of blade-section data in the propeller lifting-line theory José Rodolfo Chreim Marcos de Mattos Pimenta Fillipe Rocha Esteves GOEMS, Gustavo de Goes Gustavo Roque da Silva ASSI João Lucas Dozzi Dantas

Palestra apresentada no : ABCM INTERNATIONAL CONGRESS OF MECHANICAL ENGINEERING, 25., 2019, Uberlândia. 25 slides

A série “Comunicação Técnica” compreende trabalhos elaborados por técnicos do IPT, apresentados em eventos, publicados em revistas especializadas ou quando seu conteúdo apresentar relevância pública. ___________________________________________________________________________________________________

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On the Use of Blade-Section Data in the Propeller Lifting-Line Theory

Jose Rodolfo Chreim Fillipe Rocha Esteves Gustavo de Goes Gomes Gustavo Roque da Silva Assi Joao Lucas Dozzi Dantas Marcos de Mattos Pimenta October, 2019

2

Overview

Introduction

Mathematical Formulation

Results

Conclusion & Future Work

3

Overview

Introduction

Mathematical Formulation

Results

Conclusion & Future Work

4

Introduction

Knowledge in the emission estimates for greenhouse gases;

+ Efficient ships (propellers);

+ Compromise KT, KQ, 𝜎;

Tools for design and analysis; + Systematic series;

+ Lifting-line theory;

+ Lifting-surface theory;

+ CFD (Eulerian formulations);

Development of a novel Propeller Lifting-Line formulation;

+ Originally from Wing lifting-line;

+ Adapted to the propeller case; Complexity

Fid

elit

y

Not yet possible

5

Overview

Introduction

Mathematical Formulation

Results

Conclusion & Future Work

6

Mathematical Formulation I General

Novel Propeller LL; + Adaptations from modern

wing LL;

+ Close to Helical HSV;

Numerical expressions using superposition;

+ Force Equivalence + No flux (Pistolesi) Boundary Condition (PBC);

More general geometries;

Nonlinear (viscous) effects on KT, KQ, 𝛽𝑤;

7

Mathematical Formulation II Horseshoe vortex

No analytical expression;

+ Series of straight segments (of velocity

𝑉𝑆𝑆,𝑖,𝑗

𝑏𝑖𝑏𝑗);

+ Velocity of each

horseshoe (𝑉𝐻𝑆,𝑖,𝑗

𝑏𝑖𝑏𝑗,)

𝑉𝐻𝑆,𝑖,𝑗

𝑏𝑖𝑏𝑗 = 𝑉𝑆𝑆,𝑖,𝑗

𝑏𝑖𝑏𝑗

8

Mathematical Formulation III Pitch angle

HSV shed with pitch angle

𝛽𝐵𝑉,#,𝑖𝑏𝑖 .

Linear:

𝛽𝑖 𝑏𝑖 = tan−1

𝑉𝑃,𝑖 𝑏𝑖 ⋅ 𝑒 𝑎,𝑖

𝑏𝑖

𝑉𝑃,𝑖 𝑏𝑖 ⋅ 𝑒 𝑡,𝑖

𝑏𝑖

Nonlinear:

𝛽𝑖 𝑏𝑖 = 𝛼𝑒𝑓𝑓,𝑖

𝑏𝑖 − 𝜃𝑖

𝑅𝑖 𝑏𝑖tan 𝛽𝑖

𝑏𝑖 = 𝑅𝐵𝑉,#,𝑖𝑏𝑖 tan 𝛽𝐵𝑉,#,𝑖

𝑏𝑖

9

Mathematical Formulation IV Hub Model

Hub Influences Γ;

Image vortices;

𝑅𝐼𝑀,𝑖,𝑏𝑖=

𝑅ℎ2

𝑅𝑖,𝑏𝑖

Pressure Drag;

𝐷ℎ =𝜌

16𝜋log

𝑅ℎ

𝑅0+ 3 𝑁𝐵Γℎ

2

10

Mathematical Formulation V Linear Scheme

No flux at each Control Point;

𝑉𝑃,𝑖𝑏𝑖 = 𝑉∞,𝑖

𝑏𝑖 + 𝑉𝑡,𝑖𝑏𝑖 + 𝑉

𝐻𝑆,𝑖,𝑗

𝑏𝑖𝑏𝑗𝑁

𝑗=1

𝑁𝐵

𝑏𝑗=1

u𝑛,𝑖 ⋅ 𝑉𝑃,𝑖𝑏𝑖 = 0 →

𝑛 𝑖 ⋅ 𝑉𝐻𝑆,𝑖,𝑗

𝑏𝑖𝑏𝑗𝑁

𝑗=1

𝑍

𝑏𝑗=1

= −𝑛 𝑖 ⋅ 𝑉∞,𝑖𝑏𝑖 + 𝑉𝑡,𝑖

𝑏𝑖

N × 𝑍 equations for ΓP;

𝕄PΓP = −𝑊∞𝑃

𝕄P =M11 ⋯ M1𝑁𝐵

⋮ M𝑏𝑖𝑏𝐽 ⋮M𝑁𝐵1 ⋯ M𝑁𝐵𝑁𝐵

ΓP =

Γ11

⋮Γ𝑁1

Γ12

⋮

Γ𝑁 𝑁𝐵

, 𝑊∞𝑃 =

W∞11

⋮W∞𝑁

1

W∞12

⋮

W∞𝑁 NB

Hydrodynamic Coefficients

𝐶𝑛,𝑝𝑜𝑡𝑏𝑖 =

𝜌Γi𝑏𝑖𝛿𝑙𝑖sin𝜃𝑖

𝑏𝑖

12 𝜌𝑉

𝑃,𝑖

𝑏𝑖𝛿𝐴𝑖

𝑏𝑖

11

• CP moves to adjust 𝜕𝐶𝑛

𝜕𝛼= 𝐶𝑛𝛼

;

𝑥𝐶𝑃𝑖 =1

2

𝐶𝑛𝛼𝑖

2𝜋+

1

4𝑐𝑖

• 𝐶𝑛𝑃𝑜𝑡, 𝛽𝑠, 𝛼𝑒𝑓𝑓 updated;

𝛼𝑒𝑓𝑓𝑖 =𝐶𝑛𝑃𝑜𝑡𝑖

𝐶𝑛𝛼𝑖+ 𝛼𝐿0𝑖

• 𝐶𝑛𝑉𝑖𝑠from 2-D data;

𝐶𝑛𝑉𝑖𝑠𝑖 = 𝐶𝑛𝑉𝑖𝑠𝛼𝑒𝑓𝑓𝑖 , 𝑅𝑒𝑖

• 𝐶𝑛𝛼

updates;

𝐶𝑛𝛼𝑖 =𝐶𝑛𝑉𝑖𝑠𝑖

𝛼𝑒𝑓𝑓𝑖 − 𝛼𝐿0𝑖

𝐶𝑛𝛼𝑖 = Ω𝐶𝑛𝛼𝑖 + (1 − Ω)𝐶𝑛𝛼𝑉𝑖𝑠

𝑖

Mathematical Formulation VI Nonlinear Scheme

𝑪𝒏𝑷𝒐𝒕𝟏

𝑪𝒏𝑷𝒐𝒕𝟐

𝑪𝒏𝑽𝒊𝒔

𝜶𝒆𝒇𝒇𝟏 𝜶𝑳𝟎 𝜶𝒆𝒇𝒇𝟐

𝑪𝒏

𝜶

Change in 𝒙𝑪𝑷𝒊 (closer to the LE)

12

Mathematical Formulation VII Flowchart

Begin 𝑀𝑃,𝑊𝑃 Lin ΓP , 𝛽 𝐶𝑛𝑃𝑜𝑡

𝐾𝑇 , 𝐾𝑄

𝐶𝑛𝛼 𝐶𝑛𝛼𝑉𝑖𝑠

𝑥 𝐶𝑃

𝛼𝑒𝑓𝑓 𝐶𝑛𝑉𝑖𝑠

End

tol NO

YES

YES

NO

13

Overview

Introduction

Mathematical Formulation

Results

Conclusion & Future Work

14

Convergence of 𝑉 – Single loop vortex;

Results I Previous work

⋯

15

Convergence of 𝑉- number of helices;

Results II Previous work

⋯

⋮

16

Results III Previous work

17

Results IV – Previous work

18

Results V Present work: influence of 2-D data

Assess the influence of the 2-D on the PLL; + Several sources

Theoretical: Thin Foil Theory

𝐶𝑛 = 2𝜋 𝛼 + 2 𝑓0𝑐

𝐶𝑐 = 0

Experimental: Brockett (1966)

𝐶𝑛 = 2𝜋 1 − 0.83𝑡0𝑐

𝛼 + 2 𝑓0𝑐

𝐶𝑐 = 0.0085

Numerical: CFD Simulations using Star-CCM+

Results VI numerical 2-D data

20

Results VII Comparison of 2-D data

21

Results VIII Comparison of 2-D data

22

Results IX – MOD5 Propeller

23

Overview

Introduction

Mathematical Formulation

Results

Conclusion & Future Work

24

Conclusion

Novel propeller LL formulation; + Strictly imposes PBC;

general geometries (rake and skew);

Viscosity (hydrofoil) and 𝛽𝑤;

Results; + Previous (OMAE 2018): Code and solution verification showed

satisfactory p (cosine);

+ Previous (PRADS 2019): Validation showed adequate adherence for a series of geometries and range of advance coefficients;

+ Present: The choice of 2-D data influences the results obtained, specially in terms of 𝐊𝐓; however, one must evaluate the costs for generating the 2-D data.

THANK YOU

Questions?

jrchreim@ipt.br

jrchreim@outlook.com