A shortfin mako shark (Isurus oxyrinchus) (number of permit: SY202104051) 1.32 m in total length (L) was scanned using a hand-held Laser Scanner HandySCAN BLACK Elite at a resolution of 0.025 mm. In this study, the pectoral fins, first dorsal fin, and caudal fin were fixed within the scanning frame using threadlets to ensure the correct cruising attitude of the shark in the ocean (Figure 1A). A 3D reconstruction of the shark in STL format was used to characterize the main flow field around the shortfin mako shark (Figure 1B).

The computational domain of the smooth shark model should avoid the negative effects of boundary conditions. To ensure this, the computational domain of smooth shark model was set to 10 L × 2 L × 2 L. The tip of the shark’s nose was set as the origin of the coordinate system and the x, y, and z axes denoted the spanwise, streamwise, and wall-normal, respectively (Figure 1C). The inlet boundary was set 3 L from the tip of the shark’s nose and the outlet boundary was placed 6 L downstream of the posterior end of the caudal fin. The perpendicular distance between the wall and the origin of the coordinate system was L. The inlet boundary condition was specified as having a uniform velocity and the outlet boundary condition was set as pressure. The side walls were prescribed to have symmetrical conditions. Both top and bottom walls were set to a shear stress of 0. The Reynolds number was calculated according to R _e = (ρVL) μ ⁻¹ = 4.5 × 10⁶, where ρ is the density of seawater (1030 kg m⁻³), V is the flow velocity of seawater (5 m s⁻¹), and μ is the dynamic viscosity of seawater (1.52 × 10^–3 kg m⁻¹ s⁻¹).

The simulation was started by running a steady-state simulation with a standard k-ε turbulence model to generate more realistic turbulence statistics for LES (Russo and Basse, 2016). To capture the unsteady flow within the turbulent boundary layer, the Smagorinsky-Lilly subgrid-scale turbulence model was adopted in the LES. The transient calculation results using the LES turbulence model have been shown to be close to experimental pressure distribution measurements (Andrus and Post, 2005). The gradients of the solution variables were computed using the Green-Gauss Node based gradient evaluation. The second order scheme and bounded central differencing scheme were used to deal with the spatial discretization of pressure and momentum. The Pressure-Implicit with Splitting of Operators (PISO) pressure-velocity coupling scheme was employed to derive equations for pressure from the momentum equations and the continuity equation.

Tetrahedral grids were employed to discretize the computational domain. The number of grids affect the computational efficiency and simulation accuracy, so it needs to be considered conducting grid accuracy tests. Systematically refined grid schemes with constant refinement ratios of 2 were applied to estimate the errors, as recommended by Wilson (Wilson et al., 2001). The details of the three grid schemes were assessed, including the total number of elements, the distance to the nearest wall, quality, max aspect ratio, non-orthogonality, and max skewness (Table 1). The non-dimensional wall distance y ⁺ was 4 in this article, which met the requirements of the numerical calculations. A grid independence study was performed on the smooth shark model by comparing the surface pressure coefficients of the three different grid schemes. The surface pressure coefficient (C _p ) of the shark is defined by C P = P − P r e f ( ρ V 2 / 2 ) (1) where p is the static pressure at the point at which surface pressure coefficient is being evaluated and P _ref is the static pressure in the freestream.

Comparisons of surface pressure coefficients from the three different grid schemes are presented in Figures 2A,B (Supplementary Table 1). The surface pressure coefficient calculated using Grid1 was much larger than those when using Grid2 and Grid3, while the surface pressure coefficient of Grid2 was basically coincident with Grid3. Because of the huge computational expense required when using Grid3, Grid2 was adopted to calculate the flow field of the shortfin mako shark in this study. The three grid schemes of the shark model are displayed in Figure 2C.

In order to explore the relationship between the flow field around the shark and the adaptive growth of scales, the physical parameters of scales were measured. The morphology, riblet size, roughness, and composition of shark scales have been previously studied in fine detail (Motta et al., 2012; Popp et al., 2020; de Lima Viliod et al., 2021). A sub-set of 45 skin samples of 2 cm × 2 cm were carefully dissected from the different parts of the shark using pre-made templates consisting of plastic substrates and pins (Figure 3A). Because the pins can damage scales, the samples were collected 1 cm horizontally inside of the pins.

Collected samples were fixed on numbered plastic substrates with pins according to the sequence of markings on the shark. The samples were soaked in formalin to prevent them from spoiling and decaying at room temperature. Furthermore, to maintain the original structure and mechanical strength of the shark skin, the samples were processed as follows. First, an ultrasonic cleaner was used to rinse blood and shredded meat from the shark skin (40 min). Samples were then fixed with 2.5% glutaraldehyde (8 h, 4°C) and rinsed (4 × 15 min) in phosphate buffered saline (0.1 mol/L, pH = 7.2, room temperature). Following this, the samples were dehydrated in gradually increasing concentrations of ethanol (30, 50, 75, 80, 95, and 100%, 20 min each), then immersed in a mixture of isopentyl acetate and ethanol (1:1, 15 min) and dipped in 100% isopentyl acetate for 10 min. Finally, the samples were placed on culture dishes and dried under a vacuum (Supplementary Image 1).

Morphology, riblet size, angle, and density are important parameters for characterizing shark skin scales. In this study, the riblet sizes of scales were quantified by measuring crown length (CL), crown width (CW), riblet depth (RD), and riblet spacing (RS) (Figures 3B,C ). Because the gold-spray treatment increases the surface thickness of samples, the RD and RS were recorded first to minimize measurement error. RD and RS of 50 scales per micrograph were obtained with a digital microscope (KEYENCE VHX-6000). The samples were then coated with gold in an argon atmosphere. The morphologies of the scales were characterized using SEM (Zeiss 710) at an accelerating voltage of 20 kV, working distance of 8.5 mm, and magnification of ×100. The riblet angle of each skin sample relative to the flow direction was obtained from 20 randomly selected scales. Because the riblet angle of a scale is measured in a 2D plane, SEM images do not fully express the spatial three-dimensional state of shark skin scales. Ten regions with areas of 1 mm² were randomly selected in each sample to obtain the mean ± S.D. of the density (number of scales, mm⁻²). Five micrographs were taken per sample and 10 scales from each were measured to calculate CL and CW (total N ≥ 50 scales per sample).

This study clearly illustrated the great differences in the morphology, riblet size, and density of scales among surfaces on the shortfin mako shark. The majority of the scales from the measured samples of the shark had three longitudinal riblets, but the scales on the nose and leading edge of fins were flat and round, as shown in Figure 4. The parameters of the scales on the 45 samples are given in Table 2. The maximum CL (286 ± 25 μm) and CW (255 ± 22 μm) values were found on the leading edge of the pectoral fin (P2), and the minimum CL (105 ± 7 μm) and CW (87 ± 4 μm) values were measured on the trailing edge of the caudal fin (C11). The CL and CW of the scales gradually decreased from the leading edge, through the middle, and to the trailing edge of the fins. The ratio of crown length to width (CL/CW) ranged from 1.02 to 1.45. The CL/CW was generally larger on ventral regions (V1, V2, and V3), which indicated that these regions had longer and narrower scales.

The values of the riblet RD in skin samples ranged from 2 ± 0 to 13 ± 2 μm. The riblet RS ranged from 20 ± 2 μm on the trailing edge of the first dorsal fin (D6) to 43 ± 6 μm on the lateral surface (L3). The range of the ratio of riblet depth to spacing (RD/RS) was 0.05–0.36. The RD/RS on the leading edge of the first dorsal fin and caudal fin was nearly zero (smooth scales), and increased progressively from the middle to the trailing edge. In the pectoral fins, the larger values of RD/RS were found in the middle and trailing edge of the ventral surface. The riblets RD/RS of the shark body increased gradually from the anterior zone, the middle zone, and to the posterior zone, with values of 0.05–0.17, 0.08–0.23, 0.32–0.33, respectively.

The higher densities of scales were found on the back regions (B1, B2, B3, and B4) of the shark. The densities of scales on the trailing edge of the first dorsal fin were higher than the leading edge. Interestingly, the scale densities on the dorsal side of the pectoral fin progressively increased from the leading edge, the middle, and to the trailing edge. However, the highest scale density on the ventral side of the pectoral fin was recorded at the leading edge. On the caudal fin, the highest density was found on the lateral surfaces of the dorsal and ventral lobes, and the scales on the trailing edge were denser than on the leading edge.

The Ansys Fluent software was used to carry out all the simulations in this study. The flow field of the shark at a 0° angle of attack was calculated. Velocity and pressure contours for the whole computational domain of the shark model are shown in Figure 5. It can be seen that the domain size was large enough to avoid wall effects. The morphology and size of shark skin scales must be adapted to the surrounding flow field. However, calculating the flow field around a shark with scales was nearly impossible. The microstructures on the shark skin were small enough relative to the macro body, which should have the quantitative effect on the boundary layer flow field. This relatively small quantitative effect would not remarkably change the qualitative distribution regularity of parameters in the macro main flow field, such as the turbulence intensity, pressure, etc. Based on this assumption, the smooth shark model was chosen to study the relationship between the parameters of the macro main flow field and the distribution law of the collected shark scales. The distributions of pressure, vorticity, and velocity (v component) of the smooth shark model are presented in Figure 6. The regions of the shortfin mako shark experiencing the higher pressure were those facing the water flow (nose, leading edges of dorsal and caudal fins, and ventral sides of the pectoral fins), where the densities of the scales were generally lower and the scales were generally smooth (Figure 6A). There were higher density scales with riblets on the middle and trailing edge of the first dorsal fin and caudal fin, where the pressure was relatively low.

The flow states along the first dorsal fin and caudal fin progressed from laminar, transitional, to turbulent (Figure 6B ). Correspondingly, the RD/RS of the scales gradually increased along the laminar-turbulent zone. The distributions of vortices were orderly and obviously, and the main flow field exhibited a laminar regime at the nose and the leading edges of fins. The vortices of the trailing edge of the first dorsal fin and caudal fin were complicated and disorderly, frequently departing from the surface. The values of RD/RS in the first dorsal fin (D5) and caudal fin (C10) were 0.24 and 0.34, respectively. It can be seen that the maximum values of RD/RS on the first dorsal fin and caudal fin were located in the fully developed turbulent region. On the pectoral fins, the laminar-turbulent transition zone occurred closer to the leading edge and the largest RD/RS value was recorded at the trailing edge on the ventral side. Flow separation was obvious at the trailing edge of the first dorsal fin, caudal fin and the posterior zone of the body (Figure 6C). The scales at these regions had riblet structures, and their RD/RS values and densities were larger. The flow separation on the pectoral fin occurred earlier than first dorsal fin and caudal fin.

Analyzing the relative relationship between scale direction and flow field has a certain significance as a reference for arranging riblet surface according to flow field characteristics in engineering. Generally, the swing range of shark body is relatively small at cruising speed (Porter et al., 2011). In the cruising attitude, streamlines of the flow field around the shortfin mako shark and the scale riblets angle along the flow direction, as shown in Figure 7. The flow direction is referenced according to the longitudinal axis of the shark, passing from the nose to the caudal fin keel. The streamlines of the flow field changed with the curvature of the shark body, and the angles between the scale riblets and the streamlines of the shortfin mako shark were generally oriented with the flow (Figure 7A). The angles (Supplementary Table 3) were slightly variable on the lateral surface of shark body, with the values of angles ranging from 2° (L5) to 6° (L2) (Figure 7B).

The relationship between the turbulence intensity and the riblets RD/RS at y + = 20 is shown in Figure 8 (Supplementary Table 2). The riblet RD/RS and turbulence intensity gradually increased along the leading, middle, and trailing edges of the first dorsal fin (Figure 8A). The scales on the leading edge of the first dorsal fin were almost smooth, and the riblet RD/RS was 0. The turbulence intensity at the leading edge of the first dorsal fin was 0.106–0.121. The RD/RS in the middle area was 0.12, and the turbulence intensity was 0.181. The RD/RS of the trailing edge was 0.13–0.24, and the turbulence intensity was 0.198–0.378. The density of the middle area was larger than in the other locations. The riblets RD/RS and turbulence intensity progressively increased along the leading, middle, and trailing edges of the caudal fin (Figure 8B). The riblet RD/RS on the leading edge of the caudal fin was 0, and the turbulence intensity of the leading edge of the caudal fin was 0.078–0.105. The riblet RD/RS and turbulence intensity in the middle area were 0.22–0.24 and 0.112–0.119, respectively. The riblet RD/RS of the trailing edge was 0.27–0.34, and the turbulence intensity was 0.126–0.151. The density also increased along the leading, middle, and trailing edges, but they were smaller in samples C6 and C10. The turbulence intensity of the pectoral fin increased along the leading, middle, and trailing edges (Figure 8C). The scale riblets of leading edge were not prominent, the RD/RS ratio was 0, and the turbulence intensity was 0.107–0.179. The turbulence intensity at the middle and trailing edges were 0.181–0.189 and 0.265–0.371, respectively. The scale riblets of the middle and trailing edges of the pectoral fin were found, and the ratio of RD/RS was 0.1–0.24. The maximum density of 59 was found in sample P7. It can be seen that the riblet RD/RS and turbulence intensity increased gradually from the anterior zone, middle zone, to the posterior zone (Figure 8D). In the anterior zone, the ranges of RD/RS and turbulence intensity were 0.05–0.17 and 0.019–0.021, respectively. The middle zone of shark included the back side and lateral surface, and it had RD/RS values of 0.08–0.23, and turbulence intensities of 0.025–0.071. The densities of scales in the middle zone of the shark body were smaller than other surfaces. The RD/RS and turbulence intensity of the posterior zone of the shark body were 0.32–0.33 and 0.074–0.082, respectively. Sample B3 had the maximum density of 78.

The most inconspicuous riblet structures of scales were found on the leading edges of the shark fins and nose, which allowed us to speculate that the pressure and wall shear stress at these locations may be the physical reasons underlying this tendency. The scale riblet RD/RS values were close to 0 where the pressure or wall shear stress was large (Table 3). For instance, the pressure was high on the leading edge of the pectoral fin (P8), dorsal fin (D1, D2), caudal fin (C1, C7), and the nose (H1). The wall shear stress at these locations, i.e., C2, C3, D3, P1, P2, and P3, was also very high, although the corresponding pressures were smaller.

The riblet structures were found on the surface of the scales where the pressure and wall shear stress are smaller. The riblet RD/RS values on the lateral side of shark were 0.15–0.23, and the ranges of pressure and wall shear stress were relatively small. The larger the riblet RD/RS, the smaller the wall shear stress was on the back of the shark. The riblet RD/RS progressively increased along the back of the shark, with the ratios of 0.05, 0.08, 0.22, 0.32, and 0.33 from anterior to posterior. Along the same span, the wall shear stress gradually decreased, with values of 18, 17, 15, 13, and 11 Pa, respectively. The riblets RD/RS on the ventral side of the shark (V1, V2, and V3) were large, and their pressures and wall shear stresses were small.