# The Effect of Propeller Scaling Methodology on the Performance Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Scaling Methods for Propellers

- Statistical methods
- Analytical methods
- CFD methods
- Combinations of the above methods

#### 2.1. Statistical Methods

#### 2.1.1. ITTC 1978 Method

#### 2.1.2. Derivatives

#### 2.2. Analytical Methods

#### 2.2.1. Meyne Method

#### 2.2.2. ${\beta}_{i}$-Method

#### 2.3. CFD Methods

#### 2.4. Combined Methods

#### 2.4.1. HSVA’s Strip Method

#### 2.4.2. Numerical Section Drag

## 3. Section Drag

#### 3.1. Viscous Drag of a Flat Plate

- Laminar flow and
- turbulent flow.

**Laminar case:**- The flow is laminar over the entire plate. The viscous drag decreases with increasing Reynolds number.
**Transition governed case:**- The flow starts to become turbulent. The higher the Reynolds number, the earlier this transition occurs. Since the viscous resistance of a turbulent flow is higher than that of a laminar flow, the frictional drag increases with higher Reynolds numbers.
**“Fully” turbulent case:**- If the transition point moves close to the leading edge, the influence of the laminar flow on the overall drag diminishes and the viscous drag decreases again with an increase in the Reynolds number.

#### 3.1.1. Friction Line for Laminar Flow

#### 3.1.2. Friction Line for Fully Turbulent Flow

#### 3.1.3. Friction Lines for Transition Region

#### 3.2. Form Drag

## 4. Section Lift

## 5. Methodology

#### 5.1. Performance Prediction Method

#### 5.2. Analysis

- Mean value of all datasets for each scaling method $\mathsf{\lambda}$:$${\overline{C}}_{P,\lambda}=\frac{1}{N}\sum _{i=1}^{N}{C}_{P,i,\lambda},$$
- Normalized model–ship correlation factors:$${C}_{P,i,\lambda}^{*}=\frac{{C}_{P,i,\lambda}}{{\overline{C}}_{P,\lambda}},$$
- Standard deviation ${S}_{P}$ of all normalized datasets for each scaling method $\mathsf{\lambda}$:$${S}_{P,\lambda}^{*}=\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\left({C}_{P,i,\lambda}^{*}-1\right)}^{2}},$$

## 6. Results

- The mean values of the model–ship power correlation factor is about 1 for most investigated scaling methods.
- The scaling methods which do not scale down to the Reynolds number of the self-propulsion test typically perform better than the same method using the scaled down open-water characteristics to analyse the self-propulsion test (B–A, D–C, F–E, H–G, J–I, O–N, Q–P, U–T and W–V, but not S–R).
- The methods using the original Schlichting friction line f for the full scale propeller tend to perform better (E–A, F–B, G–C, H–D, T–P and W–S, but not U–Q and V–R).
- All methods using the local surface friction i trying to capture the transition from laminar to turbulent flow do not perform very well (K, X and Y).
- The ${\mathsf{\beta}}_{\mathrm{i}}$-methods integrating the friction forces over the whole blade perform better than the same method using only the friction force of a significant profile (R–P, S–Q, and W–U, but not V–T and the notable exception Y–X). For the ITTC 1978 methods, this trend is reversed (C–A, D–B, G–E and H–F).
- The most recent methods perform better than the original ITTC 1978 method A.
- The ${\mathsf{\beta}}_{\mathrm{i}}$-methods R and W integrating the ITTC 1978 and Schlichting friction lines h and f over the whole blade perform best, closely followed by other ${\mathsf{\beta}}_{\mathrm{i}}$-methods using different approaches regarding the handling of the viscous resistance. The next best method is the HSVA strip method in a version O, which does not scale down to the Reynolds number range of the self-propulsion tests.

## 7. Discussion

## 8. Conclusions

## 9. Final Note

## Author Contributions

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

DNS | Direct Numerical Simulation |

FS | full scale |

HSVA | Hamburgische Schiffbau-Versuchsanstalt GmbH |

ITTC | International Towing Tank Conference |

ITTC 1978 | ITTC Performance Prediction Method [1] |

LES | Large Eddy Simulation |

OW | open-water |

SMP | Stone Marine Propulsion Ltd. |

SP | self-propulsion |

RANS | Reynolds-averaged Navier-Stokes [equations] |

## Appendix A Review of ITTC 1978 Scaling Procedure

## References

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1. | Data processing was done solely at HSVA and results on trial predictions related to the various propeller scaling approaches were kept anonymous. The anonymous results on trial prediction were exclusively stored in normalized form, meaning that the quality of power and shaft speed predictions were expressed by the two model–ship correlation factors ${C}_{P}$ and ${C}_{N}$. |

**Figure 2.**Mean values ${\overline{C}}_{P,\lambda}$ of the model–ship power correlation factors ${C}_{P,\lambda}$ for all scaling methods $\mathsf{\lambda}$. If the same scaling method was investigated with and without scaling to the Reynolds number of the self-propulsion test, the results are grouped together; the black bar stands for the no scaling to the self-propulsion test. The capital letters reference Table 5.

**Figure 3.**Standard deviation ${S}_{P,\lambda}^{*}$ of the normalized model–ship power correlation factor ${C}_{P,i}^{*}$ for all scaling methods $\mathsf{\lambda}$. If the same scaling method was investigated with and without scaling to the Reynolds number of the self-propulsion test, the results are grouped together, the black bar stands for no scaling to the self-propulsion test. The capital letters reference Table 5.

**Figure 4.**${C}_{N,i,\lambda}^{*\left(\mathrm{ITTC}\right)}$ of the model–ship shaft speed correlation factors ${C}_{N,i,\lambda}$ normalized with the ITTC 1978 propeller scaling methods. The small scatter around the value of 1 indicates a small influence of the propeller scaling method on the prediction of the shaft speed.

**Table 1.**Exemplary values mentioned in the literature for the viscous drags ${c}_{f}$ and ${c}_{F}$ for one side of a flat plate with fully turbulent flow. The first five relationships assume a hydrodynamically smooth surface, the others take the relative roughness $k/c$ of the surface into account.

Local skin friction coefficients: | |
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a Schlichting [14]: | |

${c}_{f}=0.0592/\sqrt[5]{{\mathrm{Rn}}_{x}}$ | |

b von Kármán [15]: | |

$1/\sqrt{{c}_{f}}=4.15{log}_{10}\left({c}_{f}{\mathrm{Rn}}_{x}\right)+1.7$ | |

Friction coefficients for the whole plate: | |

c Schönherr [16]: | |

$0.242/\sqrt{{c}_{F}}={log}_{10}\left({c}_{F}\mathrm{Rn}\right)$ | |

d ITTC 1957 [17]: | |

${c}_{F}=0.075/{\left({log}_{10}\mathrm{Rn}-2\right)}^{2}$ | |

e Streckwall et al. [9] [a]: | |

${c}_{F}=a\left[d/\left(1+{r}^{2}\right)+\left(1-d\right){e}^{-\frac{1}{2}{r}^{2}}\right]$ | |

with $r=\left({log}_{10}\mathrm{Rn}-b\right)/c$ and suitable constants a, b, c and d | |

f Schlichting [14][b,][c]: | |

$\sqrt{2/{c}_{F}}=\frac{1}{\varkappa}ln\left(\mathrm{Rn}\times {c}_{F}/2\right)+2+\frac{1}{\varkappa}ln3.4-\frac{1}{\varkappa}ln\left(3.4+\mathrm{Rn}\sqrt{{c}_{F}/2}\times k/c\right)$ | |

g Schlichting, according to Schulze [7] [b,][c,][d]: | |

$\sqrt{2/{c}_{F}}=\frac{1}{\varkappa}ln\left(\mathrm{Rn}\times {c}_{F}/2\right)+5-\frac{1}{\varkappa}ln\left(3.4+{k}_{\mathit{tech}}^{+}\right)$ | |

with ${k}_{\mathit{tech}}^{+}\approx 0.01\mathrm{Rn}\frac{k}{c}$ | |

h ITTC 1978 [1]: | |

${c}_{F}={\left[1.89+1.62{log}_{10}\left(c/k\right)\right]}^{-2.5}$ |

**Table 2.**Exemplary values for the viscous drag ${c}_{F}$ of a flat plate mentioned in the literature. ${x}_{t}$ is the position from the leading edge along the plate where the flow trips from laminar to turbulent

^{a.}

i Transition (with ${\mathit{\delta}}_{2}$-continuity): | |
---|---|

${c}_{F}={\int}_{0}^{{x}_{t}}{c}_{f,\mathit{lam}}\mathrm{d}x+{\int}_{{x}_{t}}^{c}{c}_{f,\mathit{turb}}\mathrm{d}x$ | |

$\phantom{{c}_{F}}={\int}_{0}^{{x}_{t}}0.664/\sqrt{{\mathrm{Rn}}_{x}}\mathrm{d}x+{\int}_{{x}_{t}}^{c}0.0592/\sqrt[5]{{\mathrm{Rn}}_{x,\delta}}\mathrm{d}x$ | |

with suitable ${x}_{t}$ and ${\mathrm{Rn}}_{x,\delta}$ | |

j ITTC 1978 [1]: | |

${c}_{F}=0.044/{\mathrm{Rn}}^{1/6}-5/{\mathrm{Rn}}^{2/3}$ | |

k Schulze [7] [b]: | |

${c}_{F}=\left\{\begin{array}{c}0.3/\sqrt[3]{\mathrm{Rn}}\hfill \\ 0.003\hfill \\ 3.913/{\left(ln\mathrm{Rn}\right)}^{2.58}-1700/\mathrm{Rn}\hfill \end{array}\right.$ for $\left\{\begin{array}{c}\mathrm{Rn}<{10}^{6}\hfill \\ {10}^{6}\le \mathrm{Rn}\le 1.7\times {10}^{6}\hfill \\ \mathrm{Rn}>1.7\times {10}^{6}\hfill \end{array}\right.$ |

^{5}(suction side) and 2×10

^{5}(pressure side), and for the behind condition as 3×10

^{5}(suction side) and 1×10

^{5}(pressure side) [9].

**Table 3.**Values for the relative form drag $\Delta {c}_{F}/\left(2{c}_{F}\right)$ for symmetrical section profiles mentioned in the literature. For the NACA 64 and 65 laminar profiles, the transition point was fixed at $0.09c$. Hoerner also emphasised that the given relationship for these sections are only valid for rough surfaces [18].

A ITTC 1978 [1]: | |
---|---|

$2{t}_{\mathit{max}}/c$ | |

B Hoerner, ${t}_{\mathit{max}}$ at $0.3c$ [18]: | |

$2{t}_{\mathit{max}}/c$ for ${10}^{6}<\mathrm{Rn}<{10}^{7}$ | |

C Hoerner, NACA 64 and 65 [18]: | |

$1.2{t}_{\mathit{max}}/c$ | |

D Torenbeek [19]: | |

$2.7{t}_{\mathit{max}}/c$ |

**Table 4.**Values for the relative pressure drag ${c}_{P}/\left(2{c}_{F}\right)$ for symmetrical section profiles without flow separation mentioned in the literature. See also explanations given for Table 3.

A ITTC 1978 [1]: | |
---|---|

0 | |

B Hoerner, ${t}_{\mathit{max}}$ at $0.3c$ [18]: | |

$60{\left({t}_{\mathit{max}}/c\right)}^{4}$ for ${10}^{6}<\mathrm{Rn}<{10}^{7}$ | |

C Hoerner, NACA 64 and 65 [18]: | |

$70{\left({t}_{\mathit{max}}/c\right)}^{4}$ | |

D Torenbeek [19]: | |

$100{\left({t}_{\mathit{max}}/c\right)}^{4}$ |

**Table 5.**Scaling methods $\mathsf{\lambda}$ investigated. A mark in the ∫-columns denotes that the scaling procedure integrates the sectional friction over the whole blade. The capital letters specifying the friction lines, form and pressure drag are references to the Table 1, Table 2, Table 3 and Table 4, respectively (OW, open-water; SP, self-propulsion; and FS, full scale). The finished roughness k of the propeller in full scale was assumed to be 20 $\mathsf{\mu}$m.

Friction Lines | Drag | ||||||
---|---|---|---|---|---|---|---|

Method | ∫ | OW | SP | FS | Form | Pressure | |

A | ITTC | j | j | h | A | A | |

B | ITTC | j | — | h | A | A | |

C | ITTC | × | j | j | h | A | A |

D | ITTC | × | j | — | h | A | A |

E | ITTC | j | j | f | A | A | |

F | ITTC | j | — | f | A | A | |

G | ITTC | × | j | j | f | A | A |

H | ITTC | × | j | — | f | A | A |

I | ITTC | e | e | e | incl. | A | |

J | ITTC | e | — | e | incl. | A | |

K | ITTC | i | i | i | A | A | |

L | ITTC | k | k | g | D | D | |

M | Meyne | — | — | — | — | — | |

N | Strip | × | e | e | e | incl. | A |

O | Strip | × | e | — | e | incl. | A |

P | ${\mathsf{\beta}}_{\mathrm{i}}$ | j | j | h | A | A | |

Q | ${\mathsf{\beta}}_{\mathrm{i}}$ | j | — | h | A | A | |

R | ${\mathsf{\beta}}_{\mathrm{i}}$ | × | j | j | h | A | A |

S | ${\mathsf{\beta}}_{\mathrm{i}}$ | × | j | — | h | A | A |

T | ${\mathsf{\beta}}_{\mathrm{i}}$ | j | j | f | A | A | |

U | ${\mathsf{\beta}}_{\mathrm{i}}$ | j | — | f | A | A | |

V | ${\mathsf{\beta}}_{\mathrm{i}}$ | × | j | j | f | A | A |

W | ${\mathsf{\beta}}_{\mathrm{i}}$ | × | j | — | f | A | A |

X | ${\mathsf{\beta}}_{\mathrm{i}}$ | i | i | i | A | A | |

Y | ${\mathsf{\beta}}_{\mathrm{i}}$ | × | i | i | i | A | A |

Discarded Remaining | ||
---|---|---|

Available datasets | — | 360 |

Open-water data could not be calculated | 183 | 177 |

Errors in reference data | 3 | 174 |

Unique hull–propeller combinations | 38 | |

Outliers according to Tukey | 3 | 35 |

Total | 35 |

Ship type | Mainly bulk carriers and container vessels | |
---|---|---|

Ship length | ${L}_{pp}$ | 140–340 m ^{a} |

Ship speed | ${V}_{s}$ | 14–26 km |

Propeller diameter | ${D}_{P}$ | 4.5–9.1 m ^{b} |

Pitch to diameter ratio | $P/D$ | 0.76–1.11 |

Number of blades | N | 4–6 |

Blade area ratio | ${A}_{e}/{A}_{0}$ | 0.4–1.02 |

Number of propellers | Single screw |

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**MDPI and ACS Style**

Helma, S.; Streckwall, H.; Richter, J. The Effect of Propeller Scaling Methodology on the Performance Prediction. *J. Mar. Sci. Eng.* **2018**, *6*, 60.
https://doi.org/10.3390/jmse6020060

**AMA Style**

Helma S, Streckwall H, Richter J. The Effect of Propeller Scaling Methodology on the Performance Prediction. *Journal of Marine Science and Engineering*. 2018; 6(2):60.
https://doi.org/10.3390/jmse6020060

**Chicago/Turabian Style**

Helma, Stephan, Heinrich Streckwall, and Jan Richter. 2018. "The Effect of Propeller Scaling Methodology on the Performance Prediction" *Journal of Marine Science and Engineering* 6, no. 2: 60.
https://doi.org/10.3390/jmse6020060