# CFD Investigation for Sonar Dome with Bulbous Bow Effect

^{*}

## Abstract

**:**

## 1. Introduction

_{AW}) by using the axe bow. The axe bow effect was verified in the sea trial as well. Sadat-Hosseini et al. [19] analyzed the distribution of R

_{AW}using CFDShip-Iowa for KVLCC2 (KRISO, i.e., Korea Research Institute of Ships and Ocean, Very Large Crude oil Carrier 2) tanker. The R

_{AW}was concentrated in the upper part of the blunt bulbous bow during piercing of the free surface due to the larger ship’s vertical motions in long waves. It also generated an unsteady wave component. Yang and Kim [20] used the Cartesian grid method to investigate the KVLCC2 with an axe bow and Leadge-bow [18] in short waves. The Leadge-bow decreased the R

_{AW}by about 30%, slightly less than the axe bow. Le et al. [21] applied ANSYS-Fluent to study a Non Ballast Water Ship (NBS) tanker with and without a bulbous bow. With the bulbous bow, about a 6% calm water resistance reduction was achieved for Fn = 0.08–0.18. In the wave length of 0.2–0.6 ship length (2 m long model) under Fn = 0.163, the R

_{AW}and total resistance were 48% and 13% lower, respectively.

## 2. Geometry and Test Conditions

_{PP}). The ship’s beam is the maximum beam of the waterline (B

_{WL}). The ship’s draft (t) is the same at the ship’s FP (front perpendicular) and AP (after perpendicular), which is the distance between the water and the ship’s base line. The immersed depth of the sonar dome bottom is deeper than t. In the table, the information about the block coefficient (C

_{B}), wetted area (A

_{W}), and displacement ($\nabla $) are included. The cruise speed of the ship is assigned by the specific Froude number (Fn). Based on the Fn, the ship model was towed under the speed U

_{0}. The water temperature (T) was recorded on-site during the model test. According to L, U

_{0}, and the corresponding water density (ρ) and viscosity ($\mu $) at the T, the model scale Reynolds number (Rn) is calculated.

## 3. CFD Methods

#### 3.1. Numerical Methods and Models

#### 3.2. Computational Domain and Grid Topology

^{+}) targets 150 to be within the logarithmic layer range (y

^{+}between 30 and 200) of the boundary layer. Consequently, the three grid layers are generated with about 0.01 m thickness of the first layer. The layers and unstructured grid can be seen clearly on the ship’s bow part in Figure 5d for three different grid sizes: coarse, medium, and fine grid. They are required for the next section analysis to check grid independency and uncertainty of our CFD method.

#### 3.3. Verification and Validation (V&V) Method

#### 3.3.1. Verification

_{1}), medium (S

_{2}), and coarse grids (S

_{3}), are built and simulated for grid independence check. The refinement ratio $\sqrt{2}$ between S

_{3}and S

_{2}, S

_{2}and S

_{1}grid, is compiled in each spatial direction of boundaries to generate the initial grid. However, the grid number of unstructured grids involves the generation of different element types. For a Cartesian grid, it depends on refinement levels. Therefore, the final total grid number would not be exactly ${N}_{x}\times {N}_{y}\times {N}_{z}$ for the grid number ${N}_{x}$, ${N}_{y}$, ${N}_{z}$ on boundaries in x, y, z direction, respectively. Thus, the ratio of total grid number is managed carefully to be as close to $\sqrt{8}=\sqrt{2}\times \sqrt{2}\times \sqrt{2}$ as possible. Total grid number for different grid sizes is listed in Table 2.

_{2}and S

_{1}grid is less than the difference between S

_{2}and S

_{3}grid, and is so-called monotonic convergence. By increasing grid number, the resistance difference between two different grid sizes is reduced. As a result, the grid independence existing in our CFD method is proven. In other words, our CFD method is verified.

#### 3.3.2. Validation

_{G}) is computed by the factor of safety method with correction factor [32]. The simulation error (E%D) against experimental data (D) is defined as:

_{G}| > |E%D| of S

_{1}represents validation is achieved or our CFD method is validated. The comparison between CFD and experimental results is less uncertain, i.e., more confident, than the comparison only for CFD results of different grid sizes.

#### 3.4. Boundary Conditions

#### 3.4.1. Upstream, Side, and Bottom Boundary

**u**= (U

_{0}, 0, 0) m/s, turbulence dissipation rate ω = 2 s

^{−1}, turbulence viscosity $\nu $

_{t}= ${\mu}_{t}/\rho $ = 0.0000005 m

^{2}/s, turbulence kinetic energy k = 0.00015 m

^{2}/s

^{2}. In VOF, the volume fraction α = 0.5 on the z = 0 plane is defined as the initial location of the free surface. For totally filling with air or water, α = 0 or 1, respectively. As mentioned before (Section 3.2), only starboard flow field is modelled, so the symmetric condition, i.e., ${\nabla}_{n}=0$, is imposed on the middle plane y = 0.

#### 3.4.2. Top Boundary

**u**= 0 with an inverse flow treatment makes sure only outward velocity is solved. The total pressure is zero, but once inverse flow occurs, zero dynamic pressure is forced. For turbulence viscosity, $\nabla \nu $

_{t}= 0. The so-called Inlet Outlet condition is applied: $\nabla $(ω, k, α) = 0, but in case of inverse flow, the constant values of Section 3.4.1 are prescribed.

#### 3.4.3. Downstream Boundary

**u**= 0, but

**u**is automatically adjusted by the average flux of air and water phase if inverse flow happens. For pressure and turbulence viscosity, $\nabla $(p, $\nu $

_{t}) = 0. The Inlet Outlet condition is specified for ω and k. $\nabla $α = 0 for 0 < α < 1. However, to secure 0 < α < 1, α = 0 is forced if α < 0, and α = 1 is limited if α > 1.

#### 3.4.4. Solid Surface Boundary

^{+}in Section 3.2. It is implemented through the following ω and $\nu $

_{t}equation in consideration of surface roughness [33,34]. ${K}_{S}^{+}$ is the non-dimensional roughness height determined by the sand-grain roughness height K

_{S}= 100

^{−6}m.

## 4. Results

#### 4.1. V&V Analysis

_{1}, S

_{2}, and S

_{3}, and their errors E%D computed by Equation (11). As the grid number increases, the overpredicted error decreases to be slightly underpredicted. All absolute errors under 4% are quite low. The lowest error is even less than 0.4% (S

_{1}value in Table 4).

_{G}| larger than |E%D| of S

_{1}in Table 4 indicates the validation is accomplished. The U

_{G}is the percentage of D. To estimate U

_{G}, ${\epsilon}_{12}$ and ${p}_{G}$/${p}_{th}$ are listed in the table as well. ${\epsilon}_{12}$ is defined in Equation (10), and ${p}_{G}$ is the order of accuracy [32] computed by Equation (14) below. The theoretical accuracy order ${p}_{th}$ is 2 here, since the highest order of the numerical methods in the presented work is 2 (Table 2).

_{2}error is only overpredicted by 1%; the medium grid (S

_{2}, total grid number 1.44 M) is our preferred option to study the sonar dome’s length in the next section. F0.225 is the optimal sonar dome with the lowest resistance. For verification, RG < 1 for F0.225 proves the grid is independent. For validation, although F0.225’s experiment is unavailable, U

_{G}can be estimated by the percentage of the S

_{1}value with ${\epsilon}_{12}$ and ${p}_{G}$/${p}_{th}$ in Table 4. Those values are similar between F0 and F0.225, e.g., U

_{G}close to 4%, ${\epsilon}_{12}$ is around 2% of S

_{1}value, and ${p}_{G}$/${p}_{th}$ is in the range of 0.6–0.8 for both sonar domes. In conclusion, the results of F0.225 can be regarded as validated.

^{+}result is included in Table 4 for F0 and F0.225. As the logarithmic layer is targeted in Section 3.2, the resultant y

^{+}= 30–200 on average is confirmed for both F0 and F0.225. All y

^{+}is within 154–169.

#### 4.2. Resistance for Different Sonar Dome Length

_{T}) and its components: pressure resistance (R

_{p}) and frictional resistance (R

_{f}). In the table, the ratio $\Delta X$/L

_{PP}of the sonar dome’s length $\Delta X$ protruding forward and the ship’s length L

_{PP}is also listed as a percentage to give a sense of how long $\Delta X$ is compared to the whole ship. The trend of R

_{T}, R

_{p}, and R

_{p}to $\Delta X$/L

_{PP}, is drawn in Figure 6, respectively. The resistance reduction R

_{d}of the sonar dome with a different length from F0 is defined as:

_{T}, R

_{p}, R

_{p}), the total resistance reduction R

_{d}(R

_{T}), pressure resistance reduction R

_{d}(R

_{p}), and frictional resistance reduction R

_{d}(R

_{f}) is calculated, respectively. The trend R

_{d}(R

_{T}), R

_{d}(R

_{p}), R

_{d}(R

_{f}) to $\Delta X$/L

_{PP}, respectively, is drawn in Figure 7.

_{d}(R

_{T}) is clear up to F0.21 in Figure 7 (R

_{T}decreasing in Figure 6). By further increasing $\Delta X$ (from F0.215 to F0.24), R

_{T}oscillates around 18.5–18.6 N, and R

_{d}(R

_{T}) oscillates around 16.7–17%. The optimal dome F0.225, i.e., $\Delta X$ = 0.225, with lowest R and largest R

_{d}(R

_{T}) is found within the range of $\Delta X$ = 0–0.24 m. Protruding the sonar dome forward 7.5% of the ship’s length can reduce the maximal 17% of the total ship resistance.

_{p}is larger than R

_{f}. Just being slightly longer, such as F0.02, R

_{p}immediately turns out to be smaller than R

_{f}. As $\Delta X$ keeps increasing, R

_{p}drops significantly from around 11 N to 7 N, and R

_{d}(R

_{p}) increases even more dramatically to more than 36%. For the optimal F0.225, R

_{d}(R

_{p}) reaches 37%. Thus, R

_{p}is the major reason for the resistance reduction. In contrast, only F0.02–0.06 have a lower R

_{f}than F0 has. All the other domes produce a larger R

_{f}, i.e., negative R

_{d}(R

_{f}). In the end, R

_{f}rises slightly less than 0.5 N with around −4% of R

_{d}(R

_{p}) because the wetted area increases for longer sonar domes. The influence of R

_{f}on the resistance reduction is minor. Note that the ship’s length L

_{PP}remained the same for all $\Delta X$.

#### 4.3. Ship-Making Wave Pattern

_{PP}in this section. The ship-making wave patterns around the ship’s hull with the F0, F0.14, F0.225, and F0.24 domes are described in Figure 8. For the Kelvin wave system [35] of a ship, the diverging waves and wave cusps are clearer than transverse waves. A comparison of Figure 8a–d indicates all wave patterns are very similar, except that the second bow wave crest of F0.14, F0.225, and F0.24 shows discontinuity laterally. For a longer dome length, the wave amplitude is generally smaller. It is more obvious in Figure 9 shown by the zoom-in and larger contour range bounds. The first and second bow crest (trough) occur with lighter red (blue) as the dome length increases. Since F0.225 is the optimal case, its second crest height is lowest, i.e., smallest brown contour area. By measuring the first bow wave crest, z/L

_{PP}is larger than 0.01 for F0, but z/L

_{PP}is only 0.004–0.006 for F0.225. The wave cancellation related to the ship’s wave-making resistance reduction, i.e., the bulbous bow effect, by F0.225 is confirmed.

_{PP}is:

_{PP}= 0, and the second crest appears around x/L

_{PP}= 0.3. In Table 5, F0.225 corresponds to $\Delta X$/L

_{PP}= 7.5%. Thus, in Figure 8d, the first bow crest emerges around x/L

_{PP}= −0.75, and the second crest happens around x/L

_{PP}= −0.2. Therefore, the wave length of both sonar domes is around 0.3, which agrees with the theoretical value. Furthermore, the 90-degree phase lag is one-quarter wave length. For the wave length 0.3, the one-quarter wave length is 0.075 corresponding to F0.225’s $\Delta X$/L

_{PP}= 7.5%. It means the bow wave crest is generated above the F0.225 tip, and when propagating to the ship’s FP, the wave is at zero height. The wave after FP turns into a trough and cancels the bow wave crest as F0 has.

_{PP}= 1.96 for F0, 1.95 for F0.14, and 1.92 for F0.225 and F0.45. The angle between the wave envelope and middle plane can be calculated as around 19 degrees, close to the theoretical value [36]. The theoretical value for the wave length and envelope angle is based on a single point of the moving pressure source. Since the pressure is a distribution along the ship’s surface in our case, the ship-making waves are more complicated.

#### 4.4. Velocity Field around Ship’s Hull and Sonar Dome

_{PP}. The axial velocity u is non-dimensionalized by the ship’s speed U

_{0}.

_{0}> 1 area of F0.14, F0.225, and F0.24 is larger under the ship. Especially for F0.225 and F0.24, the u/U

_{0}> 1 area is continuous under the sonar dome and through the whole ship’s bottom. In the ship’s wake, the u/U

_{0}> 1 area of F0.14, F0.225, and F0.24 develops very long downstream up to around x/L

_{pp}= 1.9. F0.14’s area is smaller and scattered. Instead, the u/U

_{0}> 1 area of F0 is extremely short, and it is just two small fragments around x/L

_{pp}= 1.2. The higher flow acceleration outside the boundary layer (identified by u/U

_{0}= 0.99 contour line) around the ship and higher wake velocity behind the ship are the evidence for F0.225 performing much lower total resistance. From F0 to F0.24, the u/U

_{0}> 1 area becomes larger. The u/U

_{0}> 1 areas of F0.225 and F0.24 are similar. This trend is consistent with the decreasing resistance trend to nearly a constant discussed in Section 4.2 and in Figure 6.

_{0}distribution and vector field (u/U

_{0}, w/U

_{0}) in Figure 12 illustrates the flow separation behind the sonar domes. In comparison with the other sonar domes, the separation area of F0 is very large with an obvious reverse flow, i.e., u/U

_{0}< 0, and the vectors point opposite to the inflow direction. For F0.225 and F0.24, the separation area is remarkably smaller and thinner, and the reverse flow either does not form or is extremely vague. On F0.14, an area with u/U

_{0}< 0 (dark blue) is still observable but much smaller than F0’s. The boundary layer thickness of F0.14, F0.225, and F0.24 is relatively thin as well. Using the u/U

_{0}= 0.99 contour line as the indicator, check the location it intersects at x/L

_{pp}= 0.17 line, i.e., the vertical axis in the left of Figure 12. It is around z/L

_{pp}= −0.076 for F0. The intersection is −0.07 for F0.14, and −0.066 for F0.225 and F0.45. This local flow field improvement benefits from the less steep back slope of F0.14, F0.225, and F0.24 since the sonar dome is longer under the same depth. It supports the trend of pressure resistance reduction discussed in Section 4.2. The smaller separation implies the smaller pressure difference between the front and back surface of the sonar dome. In conclusion, F0.225 is not only functional in the same way as a bulbous bow to cancel waves, but the viscous flow behavior around the sonar dome is also much improved.

#### 4.5. Pressure Distribution around Ship’s Bow

#### 4.6. Distribution of Wall Shear Stress on Ship’s Bow

#### 4.7. Vortical Structures around Sonar Dome

_{x}, ω

_{y}, and ω

_{z}is illustrated, respectively, in Figure 15a–c to indicate the flow rotational direction. The ω

_{x}, ω

_{y}, and ω

_{z}are the non-dimensional vorticities in the x, y, and z direction. As pointed out in Figure 15a, several kinds of vortex can be observed and categorized: SDV = sonar dome (side or tip) vortex, SDTEV = sonar dome trailing edge vortex, and FSV = free surface vortex. To examine the vortex phenomena, the axial velocity (u/U

_{0}) distribution and vector field (v/U

_{0}, w/U

_{0}) around the F0 sonar dome are plotted on the x/L

_{pp}= 0.08 plane in Figure 16a and the x/L

_{pp}= 0.14 plane in Figure 16b. SDV and SDTEV are the major phenomena that were also explored in [38,39] for the DTMB 5415 hull, and the terminology of [38,39] is adopted here.

_{y}= 0–60 on the SDTEV in Figure 15b agrees with the vector direction of the reverse flow in Figure 12a. In Figure 15c, the high −ω

_{z}on the SDTEV indicates the cross-flow rotation inside the flow separation exists, which is induced by the SDV. As shown in Figure 16a, the SDV rotates counter-clockwise, and correspondingly, the SDTEV rotates clockwise into the separation area (in dark blue). The SDTEV does not last long. It is much shorter than the SDV as presented in Figure 15. Moreover, in Figure 16b, only the sonar dome wake is left beneath the ship’s bottom.

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Ship geometry (side view) and control points (red hollow points) on the sonar dome (red surface). The lines are the surface outline.

**Figure 2.**The details of the distribution of the control points around the original sonar dome, i.e., F0. The white and red lines are the surface outlines of the ship hull and sonar dome, respectively. The red hollow points are the control points on the sonar dome. The yellow hollow points are selected to protrude the dome geometry.

**Figure 3.**Protruding sonar dome geometry. (

**a**) Protruding distance $\Delta X$ of control point; (

**b**) sonar dome shape variations. White lines are the geometry of the ship’s hull and original sonar dome. Red lines are the different sonar dome configurations in this study.

**Figure 4.**The details of the distribution of the control points around the optimal sonar dome, i.e., F0.225. The white and red lines are the surface outlines of the ship’s hull and sonar dome, respectively. The red hollow points are the control points on the sonar dome. The yellow hollow points are selected to protrude the dome geometry.

**Figure 5.**Grid generation for the ship with the optimal sonar dome (F0.225): (

**a**) y = 0 plane (medium grid); (

**b**) z = 0 plane (medium grid); (

**c**) near ship’s hull (medium grid); (

**d**) near ship’s bow and sonar dome (left to right: coarse, medium, fine grid explained in Section 3.3.1).

**Figure 10.**Overlapping comparison for wave pattern between F0 (flooded color contour) and (

**a**) F0.14 (black contour lines); (

**b**) F0.225 (black contour lines); (

**c**) F0.24 (black contour lines).

**Figure 11.**Axial velocity distribution around the ship’s hull (on y = 0 plane): (

**a**) F0; (

**b**) F0.14; (

**c**) F0.225; (

**d**) F0.24. U = U

_{0}.

**Figure 12.**Velocity flow field around the sonar dome (on y = 0 plane): (

**a**) F0; (

**b**) F0.14; (

**c**) F0.225; (

**d**) F0.24. U = U

_{0}.

**Figure 13.**Pressure coefficient (Cp) distribution around the ship’s bow and sonar dome: (

**a**) F0; (

**b**) F0.14; (

**c**) F0.225; (

**d**) F0.24. The black line is the free surface profile on middle plane and ship hull surface.

**Figure 14.**Distribution of wall shear stress ($C{\tau}_{x}$) on ship’s bow and sonar dome surface: (

**a**) F0; (

**b**) F0.14; (

**c**) F0.225; (

**d**) F0.24.

**Figure 15.**Vortical structures around the original sonar dome (isosurface of Q = 500) with the contour flooded by the color of: (

**a**) x-vorticity (ω

_{x}); (

**b**) y-vorticity (ω

_{y}); (

**c**) z-vorticity (ω

_{z}).

**Figure 16.**Axial velocity (u/U

_{0}) distribution and vector field (v/U

_{0}, w/U

_{0}) around F0 sonar dome on: (

**a**) x/L

_{pp}= 0.08 plane; (

**b**) x/L

_{pp}= 0.14 plane. The solid block line around z/L

_{PP}= 0 is the free surface.

**Figure 17.**Vortical structure around different lengths of sonar dome (isosurface of Q = 500) with the contour flooded by the color of x-vorticity ω

_{x}; (

**a**) F0.14; (

**b**) F0.225; (

**c**) F0.24.

Ship’s Particulars | Test Conditions | ||
---|---|---|---|

L = L_{PP} (m) | 3 | U_{0} (m/s) | 1.248 |

B_{WL} (m) | 0.39 | Fn | 0.23 |

t (m) | 0.115 | Rn | 4.307 × 10^{6} |

C_{B} | 0.52 | T (°C) | 26.2 |

A_{W} (m^{2}) | 1.335 | ρ (kg/m^{3}) | 996.7 |

$\nabla $ (m^{3}) | 0.06856 | $\mu $ (m^{2}/s) | 8.663 × 10^{−4} |

Term | Symbol | Method | Order |
---|---|---|---|

Time | $\partial /\partial t$ | Implicit Euler with local time stepping | 1st |

Gradient | $\nabla $ | Central difference | 1st |

Divergence | $\nabla $ | Upwind method | 2nd |

Laplacian | ${\nabla}^{2}$ | Linear interpolation | 1st |

Gradient in normal direction n on surface | ${\nabla}_{n}$ | Explicit with non-orthogonal correction | 2nd |

Fine Grid (S_{1}) | Medium Grid (S_{2}) | Coarse Grid (S_{3}) | |
---|---|---|---|

F0 | 4,258,466 | 1,441,917 | 457,872 |

F0.225 | 4,254,563 | 1,441,124 | 457,871 |

Grid difference | 0.0917% | 0.0550% | 0.0002% |

Sonar Dome | S_{1} (N) | S_{2} (N) | S_{3} (N) | RG | D | ${\mathit{\epsilon}}_{12}$ | ${\mathit{p}}_{\mathit{G}}$$/{\mathit{p}}_{\mathit{t}\mathit{h}}$ | U_{G} | |
---|---|---|---|---|---|---|---|---|---|

F0 | R | 21.937 | 22.289 | 22.829 | 0.653 < 1 → verified | 22.01 | 1.61 | 0.614 | 3.77%D > 0.33% → validated |

E%D | 0.33% | −1.27% | −3.72% | ||||||

average y^{+} | 154.7 | 159.1 | 162.3 | ||||||

F0.225 | R | 18.061 | 18.468 | 19.162 | 0.586 < 1 → verified | - | 2.25 | 0.770 | 3.99%S_{1} (validated) |

average y^{+} | 163.9 | 164.5 | 168.9 |

Geometry | $\Delta \mathit{X}$/L_{PP} (%)
| R_{T} (N) | R_{p} (N) | R_{f} (N) |
---|---|---|---|---|

F0 | 0 | 22.289 | 11.398 | 10.891 |

F0.02 | 0.667 | 21.501 | 10.623 | 10.878 |

F0.04 | 1.333 | 20.713 | 9.833 | 10.881 |

F0.06 | 2.000 | 20.248 | 9.428 | 10.820 |

F0.08 | 2.667 | 19.846 | 8.910 | 10.937 |

F0.10 | 3.333 | 19.387 | 8.397 | 10.990 |

F0.12 | 4.000 | 19.099 | 8.100 | 10.999 |

F0.14 | 4.667 | 18.904 | 7.820 | 11.084 |

F0.16 | 5.333 | 18.885 | 7.724 | 11.161 |

F0.18 | 6.000 | 18.677 | 7.491 | 11.186 |

F0.20 | 6.667 | 18.594 | 7.350 | 11.244 |

F0.21 | 7.000 | 18.530 | 7.261 | 11.269 |

F0.215 | 7.167 | 18.568 | 7.306 | 11.262 |

F0.22 | 7.333 | 18.570 | 7.304 | 11.266 |

F0.2225 | 7.417 | 18.493 | 7.270 | 11.223 |

F0.225 | 7.500 | 18.468 | 7.157 | 11.312 |

F0.2275 | 7.583 | 18.555 | 7.250 | 11.305 |

F0.23 | 7.667 | 18.553 | 7.233 | 11.320 |

F0.235 | 7.833 | 18.517 | 7.190 | 11.327 |

F0.24 | 8.000 | 18.535 | 7.213 | 11.322 |

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## Share and Cite

**MDPI and ACS Style**

Wu, P.-C.; Chen, J.-Y.; Wu, C.-I.; Lin, J.-T. CFD Investigation for Sonar Dome with Bulbous Bow Effect. *Inventions* **2023**, *8*, 58.
https://doi.org/10.3390/inventions8020058

**AMA Style**

Wu P-C, Chen J-Y, Wu C-I, Lin J-T. CFD Investigation for Sonar Dome with Bulbous Bow Effect. *Inventions*. 2023; 8(2):58.
https://doi.org/10.3390/inventions8020058

**Chicago/Turabian Style**

Wu, Ping-Chen, Jiun-Yu Chen, Chen-I Wu, and Jiun-Ting Lin. 2023. "CFD Investigation for Sonar Dome with Bulbous Bow Effect" *Inventions* 8, no. 2: 58.
https://doi.org/10.3390/inventions8020058