A Numerical Study of Flow Past a Wall-Mounted Dolphin Dorsal Fin at Low Reynolds Numbers

Abstract

Dolphin swimming has been a captivating subject, yet the dorsal fin’s hydrodynamics remain underexplored. In this study, we conducted three-dimensional simulations of flow around a wall-mounted dolphin dorsal fin derived from a real dolphin scan. The NEK5000 (spectral element method) was employed with a second-order hex20 mesh to ensure high simulation accuracy and efficiency. A total of 13 cases were simulated, covering angles of attack (AoAs) ranging from $0^{°}$ to ${60}^{°}$ and Reynolds numbers (Re) between 691 and 2000. Our results show that both drag and lift increase significantly with the AoA. Almost no vortex was observed at $\mathrm{AoA}=0^{°}$, whereas complex vortex structures emerged for $\mathrm{AoA}\ge {30}^{°}$, including half-horseshoe, hairpin, arch, and wake vortices. This study offers insights that can inform the design of next-generation underwater robots, heat exchangers, and submarine sails.

1. Introduction

Dolphin hydrodynamics has long been a topic of interest for fluid dynamists, robot designers, and biologists [1,2,3,4,5,6,7,8,9]. The dolphin’s dorsal fin plays a crucial role in its underwater propulsion and manoeuvring, contributing to its adaptability to various aquatic environments. Studying the hydrodynamic mechanisms of the dorsal fin can provide insights into both biological understanding of cetaceans [10] and innovative engineering designs for biomimetic underwater robotics [11], heat exchangers [12], and submarine sails [13].

Hydrodynamic studies focusing specifically on the dolphin dorsal fin are relatively rare, and a comprehensive study of the near-fin flow structure is yet to be conducted. While previous research has focused on dolphin swimming, with particular emphasis on dynamic propulsion [7,14] and leaping behaviour above water [15], the current study is more relevant to gliding movements in dolphin-like swimmers [16,17,18]. Pavlov and Rashad [19] conducted numerical simulations of flow past a static dolphin body with a dorsal fin, demonstrating the minimal hydrodynamic impact of a device attached to the dolphin. Nino-Torres et al. [10] discovered high similarities in the dorsal fin shapes of common bottlenose dolphins in the Caribbean Sea, closely resembling the fin geometry used in this study. Okamura et al. [20] found that the dorsal fin, along with the pectoral fins, enhances stability during straight-line locomotion in small cetaceans, such as dolphins. Inspired by the streamlined shape of the dolphin dorsal fin, Bashtani et al. [12] designed a double-tube heat exchanger with reduced friction loss. Han et al. [7] applied the immersed boundary method to study a forward-swimming dolphin with an attached dorsal fin and pectoral fins, analysing the surface pressure distribution. Tanaka et al. [21] measured the time-varying kinematics of a dolphin during burst acceleration, providing a scanned geometry of the dolphin, where the dorsal fin was adapted and applied in the present study. Finally, Rahmani et al. [13] utilised the dolphin dorsal fin shape as the sail of a submarine.

Although the dynamic whole-body swimming dolphin has already been studied, e.g., [14], it remains valuable to study the flow past a static wall-mounted dolphin dorsal fin. The static condition is more relevant for certain applications, such as bionic heat exchangers [12] and submarine sails [13], where the detailed 3-D vortex structure has yet to be fully explored. Moreover, in addition to continuous locomotion, dolphins in nature use gliding as an energy-saving strategy during swimming, making the flow past a static wall-mounted dorsal fin particularly relevant. For instance, adult bottlenose dolphins (Tursiops truncatus) can glide up to 16 metres during a dive [16]. Similarly, newborn dolphin calves (0–1 month old) of the same species spend about one-third of their time gliding [22] rather than actively propelling themselves. It is hypothesised that during gliding over such distances, the flow structure around the dolphin is markedly different from that of an actively propelling dolphin. Additionally, several studies on dolphin-inspired robots have focused specifically on gliding [17,18]. To the best of our knowledge, the unique vortex structures identified in this study have not been previously documented. Therefore, despite the pioneering research on dolphin swimming, the authors believe that investigating this static problem remains valuable.

In summary, no comprehensive study has yet focused on the 3-D flow structures near a dolphin’s dorsal fin, despite its potential applications [11,12,13,23]. Thus, in this paper, we investigate the hydrodynamics of flow past a wall-mounted dorsal fin, with an emphasis on its vortical structures at Reynolds numbers ranging from 691 to 2000, based on the dorsal fin’s base chord length, and angles of attack (AoAs) from $0^{°}$ to ${60}^{°}$. As live dolphins, dolphin robots, or submarine sails resembling a dorsal fin all glide during forward propulsion, they frequently encounter underwater currents from various angles of attack. Dolphins have evolved to be highly adapted to the dynamic underwater environment, so it should be reasonable to postulate that their dorsal fins have evolved to provide advantages in such situations, making them worthy of investigation. Dolphins can glide at an average speed of 1.47 m/s [16], corresponding to $Re={10}^5$ in the present study. However, because of constraints in computational resources, the maximum $Re$ simulated in this study is 2000. Nevertheless, it is not uncommon to use low-Reynolds number (${10}^3$) simulations to discuss moderate-Reynolds number (${10}^5$) experimental results [24].

2. Computational Methods

This section outlines the methodology used in the study. We employ the open-source software NEK5000 [25] to generate accurate spectral element solutions for the flow around a wall-mounted dolphin dorsal fin subjected to steady currents, with varying Reynolds numbers ($Re$) and angles of attack (AoAs). Additionally, a mesh independence study is conducted to ensure the accuracy of the results. Further validation is provided in Appendix A. In this study, the Reynolds number is defined based on the fin’s chord length C as $Re=UC/\nu$, where $\nu$ is the kinematic viscosity and U is the free-stream velocity.

2.1. NEK5000

NEK5000 [25], which is based on a spectral element method, is an open-source package developed at Argonne National Laboratory. It is known for its scalability, efficiency, and capacity to handle unstructured mesh, which is essential for the curved surface of a dolphin dorsal fin. In NEK5000, the governing equation of incompressible Navier–Stokes equations [26] with constant fluid properties can written as

$${\displaystyle \frac{\partial \mathbf{u}}{\partial t}}=-\mathbf{u}·\nabla \mathbf{u}-\nabla p+{\displaystyle \frac{1}{Re}}{\nabla }^2\mathbf{u}+\mathbf{f},\nabla ·\mathbf{u}=0,$$

(1)

where $\mathbf{u}={\{u,v,w\}}^{\mathrm{T}}$ is the velocity vector, p is the pressure, and $\mathbf{f}$ is the body-force term. NEK5000 is particularly advantageous for handling complex vortex structures and flow fields while maintaining high accuracy. It has also been successfully applied to studies involving accelerating swimmers [27]. In the spectral element method (SEM) [28,29], the physical domain is divided into spectral elements. The local approximation of the flow field is then represented as a sum of Lagrange interpolants, which are defined by an orthogonal basis of Legendre polynomials up to degree N. The same polynomial order N is applied in all spatial directions, with a $P_N$–$P_N$ spatial discretisation employed [25]. In most simulations, $N=3$ is used. The equations are advanced in time with a second-order conditionally stable backward differentiation and extrapolation scheme (BDF2/EXT2) [30,31], using an implicit approach for the diffusion term and an explicit approach for the advection term. NEK5000 is highly parallelised and scalable, capable of running on thousands of threads [32]. The results presented in this study were obtained using 64 to 128 processors.

2.2. Problem Setup

This section outlines the setup of the representative problem. As shown in Figure 1, within a rectangular computational domain, a dolphin dorsal fin is mounted on the bottom wall and is subjected to a steady current of incompressible fluid. Both the dorsal fin and the bottom wall are treated as no-slip boundaries. The inlet velocity is uniform at U. Symmetry boundary conditions are applied to the side walls and top wall, effectively acting as slip boundaries. The geometry of the dolphin dorsal fin (Figure 2) is adapted from the scan of a frozen dolphin (Lagenorhynchus obliquidens) by Tanaka et al. [21]. For boundary conditions, at the inlet, Dirichlet conditions are prescribed as ${\{u,v,w\}}^{\mathrm{T}}={\{U,0,0\}}^{\mathrm{T}}$. For non-stress formulation, the outlet boundary condition can be written as $[-p\mathbf{I}+\nu (\nabla \mathbf{u})]·{\widehat{\mathbf{e}}}_n=0$. The symmetric boundary (“SYM” in Figure 1) is prescribed as $\mathbf{u}·{\widehat{\mathbf{e}}}_n=0$, $(\nabla \mathbf{u}·{\widehat{\mathbf{e}}}_t)·{\widehat{\mathbf{e}}}_n=0$, $(\nabla \mathbf{u}·{\widehat{\mathbf{e}}}_b)·{\widehat{\mathbf{e}}}_n=0$. Here, ${\widehat{\mathbf{e}}}_n$ is the unit normal vector, ${\widehat{\mathbf{e}}}_t$ is the unit tangent vector, and ${\widehat{\mathbf{e}}}_b$ is the unit bitangent vector. The wall boundary condition is $\mathbf{u}=0$.

A grid independence study was performed to ensure the selection of an appropriate mesh, as shown in Figure 3. The mesh with 71,612 elements was chosen for the current study.

The drag coefficient is defined as $C_d=2F_x/\left(\rho U^{2}A\right)$, where $F_x$ is the force in the streamwise direction, $\rho$ is the fluid density, and A is the reference area of the fin at $\mathrm{AoA}=0^{°}$ projected onto the side plane. Global and close-up views of the mesh can be found in Figure 4 and Figure 5, respectively.

3. Discussion of Numerical Results

Simulations were carried out at $Re$ = 691–2000 and AoA = 0°–60°. We analysed the results by examining the drag coefficient and the isosurfaces of the Q-criterion, pressure, and velocity magnitude. The force coefficient and flow structure show minimal variation with $Re$ but significant changes with AoA. Therefore, our discussion focuses primarily on the variation with AoA.

At low $Re$ = 691–2000, the drag and lift coefficients both increase with the AoA. As shown in Figure 6a, increasing the AoA from $0^{°}$ to ${60}^{°}$ results in more than a fivefold increase in drag ($C_d$). In Figure 6b, the lift coefficient ($C_L$) rises from 0 at $\mathrm{AoA}=0^{°}$ to approximately 1.5 at $\mathrm{AoA}={30}^{°}$ before reaching a plateau between $\mathrm{AoA}={30}^{°}$ and ${60}^{°}$. Regarding the effect of the Reynolds number, drag decreases slightly as the Reynolds number increases, while the lift coefficient remains largely unaffected by changes in the Reynolds number.

Flow Structure by Q-Criterion Isosurface

The Q-criterion is used to visualise the 3-D vortical structures, with a dimensionless value of $Q^{*}=Q{c}^2{U}^2=7.7$, as seen in Figure 7, Figure 8, Figure 9 and Figure A5. In the following discussion, the naming of vortex types follows conventions established in studies involving flow past wall-mounted objects [33,34,35,36]. As depicted in Figure 7 and Figure 9, the flow structure becomes significantly more complicated as AoA increases from $0^{°}$ to ${60}^{°}$. Generally, the geometry of the dolphin fin avoids the complex flow structures typically seen at $\mathrm{AoA}\le {30}^{°}$; in contrast, a square cylinder in a similar Reynolds number range produces much more intricate flow patterns [33].

At $\mathrm{AoA}=0^{°}$, the flow pattern is symmetric, with no significant flow separation. In this configuration, the dorsal fin is aligned with the incoming flow, and its streamlined geometry effectively minimises the formation of surrounding vortices. This starkly contrasts the flow patterns observed at higher angles of attack ($\mathrm{AoA}\ge {30}^{°}$) and the behaviour of bluff bodies at low Reynolds numbers [33,34].

At $\mathrm{AoA}\ge {30}^{°}$, abundant flow structures emerge, which correspond to the plateauing of the lift coefficient ($C_L$) observed in Figure 6b. Two halves of the horseshoe vortices form on either side of the dorsal fin at $\mathrm{AoA}\ge {30}^{°}$, as shown in Figure 7 and Figure 8. One half detaches from the dorsal fin, resembling the horseshoe vortex structure seen around a square cylinder [33], while the other half remains attached to the fin, merging with the arch vortex, as depicted in Figure 9.

The separated half of the horseshoe vortex can be observed on the right side of the dorsal fin when viewed from above, as shown in Figure 7. As AoA increases, the separated half-horseshoe extends around the right side of the dorsal fin and into the wake. At $\mathrm{AoA}={15}^{°}$, this vortex is barely noticeable, but at $\mathrm{AoA}={30}^{°}$, its length increases more than fourfold. From $\mathrm{AoA}={30}^{°}$ to ${45}^{°}$, its size doubles, with its length rising by another 1.5 times. By the time AoA reaches ${60}^{°}$, the half-horseshoe vortex shows only minor size expansion and slight further lengthening. Thus, the separated half of the horseshoe vortex undergoes significant changes only within the range of AoA = 15°–45°.

Similarly, the attached half-horseshoe vortex can be seen on the left side of the dorsal fin when viewed from above, as illustrated in Figure 7. For a clearer view of the attached vortex, a bottom perspective, as shown in Figure 8, better reveals its extension with increasing AoA. At $\mathrm{AoA}={15}^{°}$, the attached vortex is barely visible, but by $\mathrm{AoA}={30}^{°}$, both its length and size double. However, from $\mathrm{AoA}={45}^{°}$ and beyond, the attached vortex maintains its size and length. Therefore, the attached half of the horseshoe vortex exhibits significant changes only in the range of AoA = 15°–30°, which does not align precisely with the changes observed in the separated vortex.

A layer of the arch vortex forms closely above the dorsal fin, as shown in Figure 9. As AoA increases, the arch vortex tilts upwards and extends further downstream from the dorsal fin, eventually merging with the hairpin-shaped vortex. At AoA = 0°–15°, the arch vortex exhibits only a slight tilt, while at $\mathrm{AoA}={30}^{°}$, it extends approximately three times further downstream. At $\mathrm{AoA}={45}^{°}$, the arch vortex on the back of the fin extends about twice as much, connecting with the base vortex to form a hairpin-shaped structure. The shape of this arch vortex is similar to those observed in the flow around a square cylinder [33,36].

The “hairpin-shaped vortex” is particularly noteworthy. In this case, the U-shaped vortex emerges immediately from the crescent-shaped curve on the back of the dolphin dorsal fin, which differs from square-cylinder scenarios where hairpin vortices form further downstream because of a different mechanism. A view of the hairpin vortex from another angle is shown in Figure A5.

The wake vortex is also an interesting aspect to discuss. It forms at AoA = 30°–60°, as seen in Figure 7 and Figure 8, and its size increases with AoA. The single wake vortex at $\mathrm{AoA}={30}^{°}$ extends directly downstream, which contrasts sharply with the more complex flow structures observed in the wake of a finite square cylinder at similarly low $Re$ [33,35].

In addition, further observations of the flow structure were conducted by reviewing the distribution of pressure and velocity magnitude, as shown in Appendix A.1.

4. Conclusions

In this study, we presented a numerical investigation of the flow past a wall-mounted dolphin dorsal fin at angles of attack (AoAs) ranging from $0^{°}$ to ${60}^{°}$ and low Reynolds numbers between 691 and 2000. These simulations offer insights into the fundamental flow structures that arise when a dolphin glides through oblique currents or performs turning manoeuvres in water. Both drag and lift increase significantly with AoA, while drag shows only a slight decrease with increasing Reynolds number. The vortex structures, pressure, and velocity magnitude distributions vary substantially with AoA, and these variations generally align with changes in the lift coefficient. At $\mathrm{AoA}=0^{°}$, the streamlined shape of the dorsal fin suppresses almost all vortex formation. At $\mathrm{AoA}\ge {30}^{°}$, several distinct vortical structures emerge, which are generally simpler than those observed around a square cylinder at similar Reynolds numbers. Notably, two half-horseshoe vortices are observed: one upstream, which separates from the fin, and another downstream, which remains attached to the fin. A hairpin-shaped vortex appears near the trailing edge, while an arch vortex envelops the downstream fin surface, connecting the attached half-horseshoe vortex with the hairpin vortex. A clear single wake vortex is found at $\mathrm{AoA}={30}^{°}$ and ${45}^{°}$. The pressure magnitude generally increases with AoA, and the velocity magnitude on the upstream side of the fin also rises as AoA increases.

This study is limited by the assumption of a rigid fin, as it does not account for fluid–structure interactions of the flexible dorsal fin. Future work will incorporate these interactions to understand the dorsal fin’s function better. Additionally, we plan to explore higher Reynolds numbers in future studies.