May 18, 2018

# Optimization of the marine propeller shape in a uniform flow

*Industry: Shipbuilding | Product: pSeven | Company: Krylov State Research Centre*

## Objective

This use case is devoted to engineering investigations in shipbuilding, namely the development of the optimization techniques of the propeller shape. The objective was to increase the propeller’s efficiency at the fixed mode with strictly specified constraints on the values of thrust on the propeller blades, the torque on the propeller shaft and minimum pressure in the flow area.

Let's introduce some formulas used in the propulsors design. Usually, the dimensionless characteristics of the thrust and the moment of the propeller are used:

\(K_{T}=\frac{T}{ρn^2D^4}\),

\(K_{Q}=\frac{Q}{ρn^2D^5}\),

where \(T\) - is the thrust (N), \(Q\) – is the torque on the propeller shaft (Nm), \(ρ\) – is water density (kg/m^{3}).

The efficiency of the propeller has defined as follows:

\(η_{0}=\frac{K_{T}}{K_{Q}}\frac{J}{2π}\),

where \(J\) – is the advance ratio. It is the main dimensionless kinematic characteristic of the propeller, which determines the mode of the work in the fluid.

The shape of the propeller prototype is shown in Figure 1.

*Figure 1. The plan view of the propeller prototype*

## Challenges

- Many parameters, describing the propeller blade (more than 100).
- Impossible to automatically get the solid parametric geometric model of the propeller with the help of standard functions of popular CAD systems.
- The parameters’ influence on the propellers hydrodynamic characteristics is not investigated enough.
- There are a few papers and recommendations about creating simple and effective design processes for optimization of the ship propulsors.

## Solution

To overcome the over-mentioned problem with building parametric propulsor models a dedicated Flypoint Parametrica software was developed. Thanks to this software, it became possible to reduce the number of parameters to 23. Now the geometric model has all necessary associative links. Thus, when you change the parameters, you can correctly rebuild all the geometric details in the automatic mode.

For numerical simulation of the propeller, STAR-CCM+ CFD software was used. An unstructured hexahedral grid with a dimension of 200 000 computation cells was built. This task was solved thanks to the steady Reynolds-Averaged Navier–Stokes (RANS) equations closed by the k-omega SST turbulence model with finite volume method (Firuge 2).

*Figure 2. CFD simulation of the propeller vortices in STAR-CCM+*

For the further optimization of the propeller prototype, the advance ratio with maximum efficiency was chosen as a design mode. At this mode deviation in coefficient of thrust is about 1.8%, deviation in coefficient of torque is 8.3% and deviation in efficiency is about 3.7%.

Despite the small number of computational cells for numerical simulation, these deviations were accepted for the optimization on the coarse mesh because the aim was to optimize the shape of the propeller prototype. However, for this propeller prototype the grid convergence occurred on a grid of 26 000 000 cells, so the obtained results should be checked on this grid.

Moreover, the parameters, which have the most significant influence on the hydrodynamic characteristics, were established before optimization. They are a pitch of the propeller blade and a camber of the propeller blade. The achieved results allowed us to formulate the conditions for further optimization of the propeller shape in a uniform flow at the design mode of maximum efficiency (Table 1):

Table 1. Conditions for the optimization problem of the propeller shape

Objective function | Increase \(η_{0}\) at the fixed mode |

Optimization algorithm | Surrogate-based optimization |

Controlled parameters | A pitch of the blade (3 points), a camber of the blade (2 points) |

Constraints | \(T≥T_{prototype}\), \(Q≤Q_{prototype}\), \(P_{min}≥P_{minprototype}\) |

Where \(P_{min}\) – is minimum pressure in the flow area (Pa).

A one-objective optimization problem with constraints was solved with the help of pSeven. The optimization routine is shown in Figure 3. pSeven workflow engine allowed to integrate the in-house software for building geometry, STAR-CCM+ for CFD simulation and to automate and monitor all the exchange of parameters and files. The green lines indicate the blocks united under the name “Solver” into one composite block.

*Figure 3. Optimization workflow in pSeven*

After the end of one composite block run, there was an automatic transition of the obtained data directly to the Optimizer block. This block solved the optimization problem using the Surrogate-Based Optimization algorithm. This algorithm has an advantage in the speed of finding the solution in comparison to genetic optimization.

This workflow is cyclic and the number of cycles is selected automatically, on the basis of the number of objective functions and parameters. In our case, 188 cycles were selected for one objective function and 5 parameters with the global phase intensity equal to 0.5 (by default).

## Results

Optimization results of the propeller shape are presented in Table 2. The profile blade section on a relative radius r/R=0.7 before and after optimization is shown in Figure 4.

Table 2. Optimization results

Before optimization | After optimization | Comparison, ∆ | |

\(P_{min}\) | -135.6 | -128.03 | +5.9% |

\(K_{T}\) | 0.172 | 0.176 | +2.3% |

\(10K_{Q}\) | 0.4064 | 0.4061 | -0.1% |

\(η_{0}\) | 0.601 | 0.616 | +1.5% |

Where \(∆P_{min}=\frac{P_{minopt}-P_{min}}{P_{min}}100\%\),

\(∆K_{T}=\frac{K_{Topt}-K_{T}}{K_{T}}100\%\),

\(∆10K_{Q}=\frac{10K_{Qopt}-10K_{Q}}{10K_{Q}}100\%\),

\(∆η_{0}=(η_{0opt}-η_{0})100\%\).

\(K_{T}\) and \(K_{Q}\) values were calculated from the propeller blades and fillets prototype.

*Figure 4. Blade's sections comparison (r/R)=0.7*

The optimization results on the coarse mesh (200 000 cells) and the fine mesh (26 000 000 cells) were obtained, according to which the pitch and camber distributions were created (Figures 5, 6). Optimization on the base of CFD simulation models usually is a very resource-intensive process, in comparison to ordinary calculations. That’s why the possibility of saving supercomputer resources was also investigated.

One can see that these optimum curves are very close to each other and have the same distribution pattern. Consequently, now there is a possibility to conduct the optimization on the coarse mesh and then perform an analysis in details on the fine mesh, using only one calculation instead of more than one hundred calculations. Thus, we can save a significant amount of the computer time resources and make the propeller optimization more accessible.

*Figure 5. Pitch parameters distribution along the relative radii r/R*

*Figure 6. Camber parameters distribution along the relative radii r/R*

*By Liubov Lavrishcheva, Engineer, Krylov State Research Centre*