Fig 1.
Small cetacean species selected for this study.
1 –Phocoena phocoena, 2 –Delphinus delphis, 3 –Tursiops truncatus, 4 –Lagenorhynchus acutus, 5 –Lagenorhynchus albirostris, 6 –Globicephala melas, 7 –Balaenoptera acutorostrata.
Fig 2.
Dorsal fin and tail fluke’s outline.
1 –Phocoena phocoena, 2 –Delphinus delphis, 3 –Tursiops truncatus, 4 –Lagenorhynchus acutus, 5 –Lagenorhynchus albirostris, 6 –Globicephala melas, 7 –Balaenoptera acutorostrata.
Fig 3.
Basic measurements of fins.
Fig 4.
Scheme of measurement of sweep angle Λ on fins.
Fig 5.
Scheme of the airfoil parameters measured on the cross-sections of fins.
CL–chord length, MT–maximal thickness, PMT–position of maximal thickness, LER–leading edge radius.
Table 1.
Predicted and observed speeds of swimming for the selected species.
Fig 6.
Species-specific differences in the body length L cm, span of the dorsal fin S cm and area of the dorsal fin A cm2, means ± SD.
Table 2.
Dimensional and dimensionless parameters of the dorsal fins, means ± SD.
Table 3.
Correlation matrix of the dimensional and dimensionless parameters of the dorsal fins in the Odontoceti species.
Fig 7.
Comparison of the chord-normalized profile coordinates of cross-sections taken at the base and top of the dorsal fin of the selected species with the conventional airfoils.
Fig 8.
Species-specific differences in the length of the body L cm, span of the fluke S cm and area of the fluke A cm2, means ± SD.
Table 4.
Dimensional and dimensionless parameters of the tail flukes, means ± SD.
Table 5.
Correlation matrix of the dimensional and dimensionless parameters of the tail flukes in the Odontoceti species.
Fig 9.
Comparison of the chord-normalized profile coordinates of cross-sections taken at the base and top of tail flukes of the selected species with the conventional airfoils.
Table 6.
ANOVA table for the dimensionless parameter MT%CL of the dorsal fin and tail fluke cross-sections with the independent factors fin type (Factor 1), section # (Factor 2), species (Factor 3) and body length (Factor 4).
Table 7.
ANOVA table for the dimensionless parameter PMT%CL of the dorsal fin and tail fluke cross-sections with the independent factors fin type (Factor 1), section # (Factor 2), species (Factor 3) and body length (Factor 4).
Table 8.
ANOVA table for the dimensionless parameter LER%CL of the dorsal fin and tail fluke cross-sections with the independent factors fin type (Factor 1), section # (Factor 2), species (Factor 3) and body length (Factor 4).
Fig 10.
Principal Component Analysis of the dimensionless parameters of the dorsal fin and tail fluke cross-sections of all species.
A–A negative relationship between LER%CL and PMT%CL is shown, while MT%CL is unrelated with the LER%CL and has a slight positive correlation with the PMT%CL. B–Separation between the dorsal fin and fluke’s cross-sections based on the difference in LER%CL and PMT%CL. C–Cross-sections located at the base of the fins (blue dots) indicate distinctive hydrofoil design with extreme values for LER%CL and PMT%CL.
Fig 11.
Comparison of the chord-normalized profile coordinates of cross-sections taken at the base and top of the dorsal fin and tail flukes.
A–Cross-sections taken at the base of the dorsal fin. B—Cross-sections taken at the base of the tail fluke. C—Cross-sections taken at the top of the dorsal fin. D—Cross-sections taken at the top of the tail fluke.
Table 9.
Comparison of the hydrodynamic characteristics of the dorsal fin cross-sections with the appropriate airfoils.
Fig 12.
Drag polar diagram of lift CL vs drag Cd.
Calculated for the cross-sections taken at the base of the dorsal fin of the P. phocoena, D. delphis, L. acutus and Eppler 297 airfoil at the averaged Re 5.76E+05 for the fin cross-sections (Table 9).
Fig 13.
Drag polar diagram of lift CL vs drag Cd.
Calculated for the cross-sections taken at the base of the dorsal fin of the L. albirostris, T. truncatus, G. melas and Eppler 297 airfoil at the averaged Re 5.76E+05 for the fin cross-sections (Table 9).
Fig 14.
Drag polar diagram of lift CL vs drag Cd.
Calculated for the cross-sections taken at the base of the dorsal fin of the B. acutorostrata and Eppler 297 airfoil at the averaged Re 5.76E+05 for the fin cross-sections (Table 9).
Table 10.
Comparison of the hydrodynamic characteristics of the tail fluke cross-sections with the appropriate airfoils.
Fig 15.
Drag polar diagram of lift CL vs drag Cd.
Calculated for the cross-sections taken at the base of the tail flukes of the P. phocoena, D. delphis, L. acutus and SD8020 airfoil at the averaged Re 4.77E+05 for the fluke’s cross-sections (Table 10).
Fig 16.
Drag polar diagram of lift CL vs drag Cd.
Calculated for the cross-sections taken at the base of the tail flukes of the L. albirostris, T. truncatus, G. melas and SD8020 airfoil at the averaged Re 4.77E+05 for the fluke’s cross-sections (Table 10).
Fig 17.
Drag polar diagram of lift CL vs drag Cd.
Calculated for the cross-sections taken at the base of the tail flukes of the B. acutorostrata and SD8020 airfoil at the averaged Re 4.77E+05 for the fluke’s cross-sections (Table 10).
Fig 18.
The Λ, in radians vs AR scatterplot shows the fin planform variation in the dorsal fin (orange) and tail flukes (blue).
Different variation along the Λ and AR axes indicates two distinctive patterns of the fin planform. A–G. melas, B–P. phocoena, C–B. acutorostrata.
Fig 19.
Constraints in variation of cross-sections of the dorsal fin (orange) and tail flukes (blue) in the trait space of normalized non-dimensional parameters.
Different variation along the LER%CL, PMT%CL and MT%CL axes indicates two distinctive patterns of the cross-sectional geometry.
Fig 20.
Calculated for the cross-sections of the fluke (blue) and dorsal fin (green) of the D. delphis at simulated swimming speed 2 and 8 m/sec.