Hydrodynamics of Biomimetic Marine Propulsion and Trends in Computational Simulations

Abstract

The aim of the present paper is to provide the state of the works in the field of hydrodynamics and computational simulations to analyze biomimetic marine propulsors. Over the last years, many researchers postulated that some fish movements are more efficient and maneuverable than traditional rotary propellers, and the most relevant marine propulsors which mimic fishes are shown in the present work. Taking into account the complexity and cost of some experimental setups, numerical models offer an efficient, cheap, and fast alternative tool to analyze biomimetic marine propulsors. Besides, numerical models provide information that cannot be obtained using experimental techniques. Since the literature about trends in computational simulations is still scarce, this paper also recalls the hydrodynamics of the swimming modes occurring in fish and summarizes the more relevant lines of investigation of computational models.

1. Introduction

Biomimetics, also called biomimicry or bionics, involves the imitation of nature in man-made systems. An area of biomimetics that is gaining an increasing interest is marine propulsion. Fish swimming is the result of millions of years of evolutionary optimization to develop a very efficient movement. For this reason, several fish movements inspire technology for marine propulsion. Marine propulsion is a multi-disciplinary area and involves numerous uses. Biomimetic marine propulsors are currently employed in oceanographic observation, military, underwater mine detection, surveillance, pipeline inspection, gas drilling, environmental awareness, rescue, etc. Fish have evolved to achieve different capabilities. Some fish are maneuverable, some are efficient, some are fast swimmers, etc. For engineering purposes, it is necessary to design the most appropriate swimming form depending on the application. For instance, oceanographic observation and pipeline inspection require high maneuvering. Other applications, such as excavation of sunken wreckage or gas drilling, require robustness and maneuvering in narrow spaces. Biological monitoring throughout large regions requires fast vehicles with the capability to travel long distances. In the military area, where spying and avoidance of detect are the two most important fields, noiseless and inconspicuous wakes are very important. In merchant ships, the most important parameter is efficiency in order to overcome the traditional propellers.

In 1936, a decisive paper was published by Gray [1], who initiated the interest in the hydrodynamics of fish swimming. Gray studied the movement of dolphins and postulated that these mammals can obtain high speeds and accelerations with a small muscle mass. He estimated the power required by a rigid model of a dolphin as about a sevenfold shortage in muscle mass to attain such speeds. He concluded that dolphin skin has special drag reduction properties. His research, known as Gray’s paradox, inspired several works about the drag reduction capabilities of dolphins [2,3,4,5,6,7,8,9]. Gray’s paradox remained controversial for the following sixty years, mainly because it is extremely difficult to obtain reliable force measurements from live fish. In the 1990s, several investigations from Massachusetts Institute of Technology (MIT) reported that Gray´s calculations were wrong [10,11,12,13]. Gray had employed a rigid model of a dolphin, and these works showed that drag is lower for an undulating body than that corresponding to the rigid equivalent. Nevertheless, these authors agreed that fish employ techniques to reduce drag forces while swimming. Basically, these techniques are based on the utilization of the energy that exists in the eddies of an oncoming flow by repositioning the vorticities [14].

Great progress was made since the 1990s. Due to the increasing number of investigations about biomimetic marine propulsion, several reviews can be found in the literature. The first and most relevant one was developed by Sfakiotakis et al. [15] in 1999. This paper presented an overview of fish swimming modes, providing a classification and discussion about the relevance to underwater vehicle design. They analyzed the advantages, disadvantages, kinematics, and recommendable applications of several swimming modes. Although many mechanisms were developed after the publication of this paper, the work of Sfakiotakis et al. is still highly referenced due to its excellent explanation and classification of fish swimming modes for aquatic locomotion. Another review is the one of Cheng and Chahine [16], who presented the state of the art of swimming hydrodynamics and their biological relevance. Triantafyllou et al. [17] focused their review on the hydrodynamics of flapping foils and analyzed the implementation of such propulsion systems in marine propulsors. Colgate et al. [18] reviewed the control of robot fish, analyzing the main problems and solutions. Mittal et al. [19] reviewed the works in bio-hydrodynamics, from flapping foils to fin flexion mechanisms. Bandyopadhyay [20] reviewed aspects related to hydrodynamics, artificial muscle technology, and neuroscience-based control. Kato [21] reviewed the studies about kinematics, hydrodynamics, and performance of median and paired fin (MPF) locomotion in fish and biomimetic mechanical systems from the viewpoint of enhancing the propulsive and maneuvering performance of marine vehicles at low speeds. Chu et al. [22] centered on smart actuators used on biomimetic robots, particularly shape memory alloys, ionic polymer metal composites, and lead zirconate titanates. Raj and Thakur [23] discussed several design features like sensing, actuation, autonomy, waterproofing, and morphological structure. Other researchers [24,25,26,27,28] focused on flapping foils.

These reviews constitute a comprehensive state of the art of biomimetic marine propulsion. Nevertheless, the literature about trends in computational techniques is still scarce. According to this, this paper aims to supplement the literature providing the state of the works in the field of hydrodynamics and computational simulations to analyze biomimetic marine propulsors. After a description of fish swimming modes and biomimetic marine propulsors which mimic fishes, aspects related to computational simulations, such as hydrodynamics, non-dimensional parameters, turbulence, grid strategies, and experimental validations, are analyzed.

2. Fish Swimming Modes and Biomimetic Marine Propulsors

A comprehensive classification and description of the fish swimming modes was developed by Breder [29] in 1926 and expanded by Lindsey [30] in 1978. Basically, they differenced two propulsion modes: swimming and non-swimming locomotion. These are explained below.

2.1. Swimming Locomotion

The most common fish swimming modes are summarized in Figure 1. Basically, there are two factors to classify them: the extent to which the propulsion is undulatory or oscillatory and the body structures or fins that generate thrust. Most fish generate thrust using their body and/or caudal fin (BCF), for instance anguilliforms, caranguiforms, thunniforms, and ostraciiforms. Other fish, however, employ their median and/or paired fins (MPF) while the body remains practically rigid. Examples of MPF movement are amiiforms, gymnotiforms, rajiforms, labriforms, and tetraodontiforms. Several reviews [15,16,17,18,19,20,21,22] indicated that BCF modes, especially thunniform and, to a lesser extent, caranguiform, are more efficient than MPF modes. Nevertheless, MPF modes are characterized by a high maneuverability in terms of fast start, short-radius turning, and other factors.

The force actuating on a fish is the result of a viscous (friction) force due to the viscous shear stresses acting on the body and a pressure force due to pressure gradients. Hydrodynamically, fish locomotion can be generated by three procedures: drag-based, added mass, and momentum injection (Figure 2). Most swimmers use a combination of more than one of these procedures.

In the drag-based procedure (Figure 2a) a force due to the friction between the fin and water acts opposite to the direction of motion. In the added mass procedure (Figure 2b), a reaction force is produced by the inertia of the water pushed backward by the body [31]. This propulsive force is approximately the product of the water mass accelerated and its acceleration. The momentum injection procedure (Figure 2c) is based on impulsively injecting momentum into the surrounding fluid. A force is produced by jetting fluid from the body. The momentum, which is approximately the product of the mass flow rate and the jet velocity, creates a reaction force.

Besides these locomotion procedures, most fish evolved vortex control capabilities [10,11,12,13]. This is typical of thunniform swimming mode and is presented in most fishes. As mentioned previously, dolphins have drag reduction capabilities because they employ the energy that exists in the eddies of an oncoming flow by repositioning the vortices. Despite some superficial similarities between thunniform and ostraciiform swimmers, the hydrodynamics of the latter are less efficient and based on a rigid body which achieves less vorticity control. Oscillatory movements also obtain a poorer vorticity control.

A detailed analysis of each movement and repercussion on marine propulsion is shown below.

2.1.1. BCF Swimming Modes

As indicated in Figure 1, the most common BCF swimming modes are anguilliform, caranguiform, thunniform, and ostraciiform. In anguilliform swimming mode, an undulation with constant amplitude takes place along the whole body, from head to tail. It is typical of eels, congers, muraenas, lampreys, lancelets, etc. One advantage of this movement is the turning and accelerating abilities due to the flexibility of the body. Another advantage is the possibility to swim backward, simply passing the undulation from tail to head. Due to its lower efficiency compared with other BCF modes, anguilliform swimming movement has only inspired a few robots that require high maneuverability, mimicking lampreys [32,33,34,35,36,37,38,39,40,41], snakes [42], salamanders [43,44,45], eels [46,47,48,49,50,51], etc.

The caranguiform mode is similar to the anguilliform one, but with the amplitude undulations growing toward the tail. Since the caranguiform swimming movement is more efficient than the anguilliform one, it inspired many more mechanisms mimicking pikes [52], salmons [53,54,55], sea breams [56], mackerels [57,58], etc.

As in the case of caranguiform, in thunniform swimming mode, the amplitude undulations grow toward the tail. Nevertheless, in the latter, the undulations practically take place only in the posterior half of the fish. It is typical of tunas, sharks, whales, dolphins, etc. Many researchers agree that thunniform is the most efficient swimming mode [15,16,17,18,19,20,21,22]. Among thunniform swimmers, tuna is considered the fastest fish in nature. For these reasons, most studies deal with tuna-like mechanisms [59,60,61,62,63,64,65,66,67]. The pioneer tuna-like mechanism was RoboTuna [11,59], made at MIT. This mechanism is a replica of a real tuna. Results from the RoboTuna project did indeed suggest a reduction in the drag force, agreeing with Gray’s paradox. Following the success of the RoboTuna project, Anderson and Chapra at MIT improved this robot providing more vorticity control propulsion and maneuvering [60]. Another important animal that inspired thunniform mechanisms is dolphin [68,69,70,71,72,73,74,75,76,77,78,79,80]. The popularity of Gray’s paradox made dolphins an important source of bioinspiration, and the kinematics of dolphins are among the best understood in nature. At Istanbul Technical University, Dogangil et al. also constructed a robotic dolphin [78] and analyzed the hydrodynamics and kinematics [79]. To a lesser extent, shark also inspired biomimetic propulsion mechanisms [81,82].

Ostraciiform locomotion is characterized by the pendulum-like oscillation of a rather stiff caudal fin, while the body remains essentially rigid. It is typical of cowfishes, boxfishes, coffer fishes, trunkfishes, etc. Due to its low efficiency [15,18,21], this movement has received limited attention in engineering, although an oscillating movement is easier to implement than an undulating one. Most of the investigations found in the literature are inspired by boxfish [83,84,85,86]. Other authors employed flapping foils for oscillating marine propulsors [87,88,89,90,91,92,93,94,95,96,97,98,99,100,101].

Table 1. Summary of body and/or caudal fin (BCF) swimming mechanisms.

Author	Institution	Propulsion Mode	Source of Inspiration
Cohen [32]	Cornell University, USA	Anguilliform	Lamprey
Grillner and Matsushima [33]; Grillner et al. [34]	Karolinska Institute, Sweden	Anguilliform	Lamprey
Ayers et al. [35,36,37]; Wilbur et al. [38]; Westphal et al. [39]	Northeastern University, USA	Anguilliform	Lamprey
Liu et al. [40]	Zhejiang University, China	Anguilliform	Lamprey
Xu et al. [41]	National University of Singapore, Singapore	Anguilliform	Eel/lamprey
Crespi et al. [42]	Swiss Federal Institute of Technology, Switzerland	Anguilliform	Sea snake
Ijspeert et al. [43]; Ijspeert et al. [44]; Crespi et al. [45]	Swiss Federal Institute of Technology, Switzerland	Anguilliform	Salamander
McIsaac and Ostrowski [46,47,48]	University of Western Ontario, Canada	Anguilliform	Eel
Lamas et al. [49,50,51]	University of A Coruña, Spain	Anguilliform	Eel
Kumph [52]	Massachussets Institute of Technology, USA	Caranguiform	Pike
Watts et al. [53,54,55]	University of Glasgow, United Kingdom	Caranguiform	Salmon
Hirata et al. [56]	Ship Research Institute Shinkawa, Japan	Caranguiform	Sea bream
Heo et al. [57] Li et al. [58]	Konkuk University, Korea Beihang University, China	Caranguiform Hybrid	Mackerel Mackerel
Barrett [11]; Barrett et al. [59]; Anderson and Chabra [60]	Massachussets Institute of Technology, USA	Thunniform	Tuna
Liang et al. [61]; Liang et al. [62]; Liang et al. [63]; Wang et al. [64]; Liu et al. [65]	Beihang University, China	Thunniform	Tuna
Suleman and Crawford [66] Kruusmaa et al. [67]	University of Victoria, Canada Tallinn University of Technology, Estonia	Thunniform Thunniform	Tuna Tuna
Nakashima et al. [68]; Nakashima et al. [69]; Nakashima et al. [70]; Nakashima et al. [71]	Tokyo Institute of Technology, Japan	Thunniform	Dolphin
Yu et al. [72]; Yu et al. [73]; Yu et al. [74]; Shen et al. [75]; Shen et al. [76]; Liu et al. [77]	Chinese Academy of Sciences, China	Thunniform	Dolphin
Dogangil et al. [78]; Dogangil et al. [79]	Istanbul Technical University, Turkey	Thunniform	Dolphin
Ho and Lee [80]	Konkuk University, South Korea	Thunniform	Dolphin
Aghbali et al. [81]	College of Engineering, Iran	Thunniform	Shark
Long et al. [82]	Vassar College, USA	Thunniform	Shark
Kodati et al. [83]; Kodati et al. [84]	University of Delaware, USA	Ostraciiform	Boxfish
Gordon et al. [85]	University of California, USA	Ostraciiform	Boxfish
Hu et al. [86]	Peking University, China	Ostraciiform	Boxfish
Anderson et al. [87]	Massachussets Institute of Technology, USA	Ostraciiform	Flapping foil
Read et al. [88]	Massachussets Institute of Technology, USA	Ostraciiform	Flapping foil
Yamamoto et al. [89]	Mitsubishi Heavy Industries, Japan	Ostraciiform	Flapping foil
Streitlien and Triantafyllou [90]	City College of New York, USA	Ostraciiform	Flapping foil
Paterson and Stern [91,92]	University of Lowa, USA	Ostraciiform	Flapping foil
Karpouzian et al. [93]	Univ of Southern California, USA	Ostraciiform	Flapping foil
Yamaguchi and Bose [94]	University of Tokyo, Japan	Ostraciiform	Flapping foil
Saimek and Li [95]	University of Minnesota, USA	Ostraciiform	Flapping foil
Herr and Dennis [96]	MA Institute of Technology, USA	Ostraciiform	Flapping foil
Guo [97]	Kagawa University, Japan	Ostraciiform	Flapping foil
Belibassakis and Politis [98]; Belibassakis and Filippas [99]; Filippas et al. [100]; Koutsogiannakis et al. [101]	National Technical University of Athens, Greece	Ostraciiform	Flapping foil

summarizes the BCF swimming mechanisms mentioned, including the authors, institution, propulsion mode, and source of inspiration.

2.1.2. MPF Swimming Modes

As indicated in Figure 1, the most common MPF swimming modes are rajiform, amiiform, gymnotiform, labriform, and tetraodontiform. Rajiform (rays, skates, mantas, etc.), amiiform (bowfins, etc.), and gymnotiform (electrical eels, American knifefishes, etc.) modes are based on the undulation of the pectoral, dorsal, and anal fins, respectively. Nevertheless, labriform (wrasses, angelfish, surfperches, etc.) is based on the oscillation of the pectoral fin and tetraodontiform (blowfishes, etc.) on the oscillation of both anal and dorsal fins. Speed, performance, and maneuverability are key aspects of pectoral fin locomotion which provide specific application ideas for underwater vehicle technology. MPF locomotion mode is less efficient than BCF, but more maneuverable, and it is easy to reverse thrust [15,16,18]. By controlling the movement of the fins, it is possible to obtain maneuvering forces in the full six degrees of freedom and both rowing and flapping.

Due to its low efficiency, much fewer MPF mechanisms were developed by comparison with BCF mechanisms. Both oscillatory and undulatory MPF mechanisms were developed in the literature, with a preference towards pectoral fin labriform-type swimming. Concerning oscillatory-type MPF movement, labriform swimming has received special attention inspiring mechanisms that mimics black basses [102], sea turtles [103,104,105,106,107], sunfishes [108,109], flapping foils [110,111,112], rays [113,114,115,116,117,118,119,120], knifefishes [121,122,123,124,125,126,127,128], etc. Concerning undulatory-type MPF locomotion, little interest has been showed by comparison with oscillatory MPF movement.

Table 2. Summary of median and paired fin (MPF) swimming mechanisms.

Author	Institution	Propulsion Mode	Source of Inspiration
Kato et al. [102]	Tokai University, Japan	Labriform	Black bass
Kato et al. [103]	Tokai University, Japan	Labriform	Sea turtle
Litch et al. [104,105]	Massachussets Institute of Technology, USA	Labriform	Sea turtle
Kim et al. [106] Salumae et al. [107]	Seoul National University, South Korea Tallinn University of Technology, Estonia	Labriform Labriform	Sea turtle Sea turtle
Bozkurttas et al. [108,109]	Franklin W. Olin College of Engineering, USA	Labriform	Sunfish
Georgiades et al. [110]	McGill University, Canada	Labriform	Flapping foil
Long et al. [111]	Vassar College, USA	Labriform	Flapping foil
Sitorus et al. [112]	Bandung Institute of Technology, Indonesia	Labriform	Flapping foil
Cai et al. [113]	Beihang University, China	Rajiform	Ray
Punning et al. [114]	Tartu Univeristy, Estonia	Rajiform	Ray
Chen et al. [115]	University of Virginia, USA	Rajiform	Ray
Clark and Smiths [116]	Princeton University, USA	Rajiform	Ray
Low, K.H. [117]; Zhang et al. [118,119]	Nanyang Technological University, Singapore	Rajiform	Ray
Low and Willy [120]	Nanyang Technological University, Singapore	Rajiform/gymnotiform	Ray/knifefish
Low [121]; Low [122]; Hu et al. [123]; Low [124]	Nanyang Technological University, Singapore	Gymnotiform	Knifefish
Epstein et al. [125]	Northwestern University, USA	Gymnotiform	Knifefish
McIver et al. [126,127]	Northwestern University, USA	Gymnotiform	Knifefish
Siahmansouri et al. [128]	University of Tabriz, Iran	Gymnotiform	Knifefish

summarizes the MPF swimming mechanisms mentioned, including the authors, institution, propulsion mode, and source of inspiration.

In order to analyze propulsive effectiveness, most researchers use the so-called Froude efficiency. William Froude (1810–1879) was the first who used the propulsive efficiency, defined as the ratio of power output to the rate of energy input. Lamas et al. [49] developed an undulating propulsor actuated by several ribs. After that, Lamas et al. [50,51] employed CFD (Computational Fluid Dynamics) to study several amplitudes, wavelengths, and oscillation frequencies in anguilliform, carangiform, thunniform, ostraciiform, and gymnotiform movements. They concluded that thunniform is the most efficient swimming mode in terms of the Froude efficiency, followed by caranguiform, ostraciiform, anguilliform, and finally, gymnotiform. Caranguiform, ostraciiform, and anguilliform swimming modes are less efficient than thunniform because the latter produce thrust only at the tail. Nevertheless, the large undulation amplitudes along the entire body of caranguiform, anguilliform, ostraciiform, and gymnotiform swimming modes produce wasted power at the head.

It is worth mentioning that a single most appropriate swimming form does not exist. Efficiency is only one among other important assets of bioinspired aquatic propulsion such as low-speed maneuverability, fast start, and short-radius turning [129,130]. These are the most important advantages of employing a bioinspired design. Efficiency is a controversial aspect of aquatic bioinspired propulsion since bioinspired vehicles are not merely built for their energetic efficiency, but for their suitability at dealing with tasks for which common vehicles are unfit. Fish have grown to become specialists; they excel at a range of movements their habitat and respective ecosystems dictate, but at the expense of not being well adapted to other ranges of motion. For instance, while thunniform swimming does provide superior efficiency for high cruise speed in calm water, it performs poorly at lower speeds, and provides only low levels of agility in turning maneuvers [15].

2.2. Non-Swimming Locomotion

Some aquatic animals use other mechanisms than swimming. The most important is jet propulsion (cephalopods, shellfishes, jellyfishes, etc.), based on the ejection of water behind the fish. The phenomenon is shown in Figure 3, which illustrates a vortex ring generated during contraction and another one generated during expansion. Pulsed jetting seems to have gained a growing attention in the scientific community in recent years.

California Institute of Technology, through the works [131,132,133,134,135,136], was the first institution which studied jet propulsion. Particularly, the hydrodynamics of jet propulsion were explained in the review of Dabiri [121]. Briefly, it is based on the momentum injection procedure, i.e., water behind the fish produces a jet which generates thrust. Besides, these researchers found an important vortex optimization capability by analyzing a piston-cylinder mechanism. In their configuration, a boundary layer of vorticity forms on the inner cylinder wall as the piston moves downstream inside the cylinder, and through oscillatory jets they found that the boundary layer is less susceptible to separation; this fact mitigates turbulence. Their measurements of the impulse associated with vortex ring formation showed to be much larger than that expected from the jet velocity alone.

Table 3. Summary of non-swimming mechanisms.

Author	Institution	Propulsion Mode	Source of Inspiration
Gharib et al. [132]; Krueger and Gharib [133]; Krueger [134], Krueger [135], Ruiz et al. [136]	California Institute of Technology, USA	Jet	Piston-cylinder
Guo et al. [137]; Yang et al. [138]; Shi et al. [139]	Kagawa University, Japan	Jet	Jellyfish
Yeom and Oh [140]	Chonnam National University, South Korea	Jet	Jellyfish
Villanueva et al. [141]; Najem et al. [142]	Virginia Tech, USA	Jet	Jellyfish
Krieg and Mohseni [143]; Krieg and Mohseni [144]	University of Colorado, USA	Jet	Squid
Serchi et al. [145]; Serchi et al. [146]	Scuola Superiore Sant’Anna, Italy	Jet	Octopus

summarizes the main non-swimming mechanisms, including the authors, institution, propulsion mode, and source of inspiration.

3. Numerical Models

3.1. Hydrodynamics

A proper design of a mechanism requires a comprehensive characterization of the fluid pattern. In this regard, numerical models can be used as an important tool to analyze the fluid flow in detail. Between them, CFD is highly employed. This is a branch of fluid mechanics based on the split of the domain into small elements called grid or mesh. For each grid element, the governing equations are solved using numerical techniques. In a biologically inspired mechanism, the complex hydrodynamics may be computed by solution of the Navier–Stokes equations using numerical analysis. CFD provides important information that cannot be measured experimentally, such as details of the pressure field, velocity field, hydrodynamic forces, power, and efficiency. It is very useful to complement experimental works using less time and lower cost.

During the movement of the mechanism, water is pushed backward and produces a wake of alternating sign vortices known as a reverse Karman street [50,64,78,110]. This phenomenon is shown in Figure 4a,b, which represents the velocity field overlaid with the pressure field for BCF and MPF swimming modes, respectively [50]. As can be seen in these figures, several vortices are formed behind the fin. Van Buren et al. [147] manipulated the vortex structure in the wake on a pitching panel and verified important effects on thrust and efficiency. As indicated previously, it is possible to control the thrust by manipulating the flow vortex, i.e., by a mechanism of vorticity control [10,12,64]. A fish consumes much less energy to displace than a rigid body because the motion of the fish is associated with a reduction of the drag force and, thus, an increment of the propulsive efficiency by reducing separation and suppressing turbulence [148]. Recent works have shown that undulating fish movement is very efficient compared to rigid bodies [14,59,149].

The pressure differences promote a pressure force, given by:

$$\overrightarrow{F_p}=-{\displaystyle \underset{S}{\mathsf{\int }}p\widehat{n}dS}$$

(1)

where p is the pressure, S the surface, and $\widehat{n}$ a unit vector normal to the surface. On the other hand, a viscous (friction) force is produced due to the viscosity of water, and is given by the following expression:

$$\overrightarrow{F_v}=-{\displaystyle \underset{S}{\mathsf{\int }}{\tau }_{ij}\widehat{n}dS}$$

(2)

where τ_ij is the viscous stress tensor. The total force is the sum of the pressure and viscous forces. A typical distribution of the components of these forces in the direction of the movement as well as the average total force are shown in Figure 5 for anguilliform swimming mode [50]. As can be seen in this figure, the pressure and viscosity forces exhibit two peaks per cycle, corresponding to the forward and backward tail strokes. Under the conditions indicated in the figure, the propulsor accelerates since the total force is positive. If the body is accelerated, the inertia of the system contributes as an additional force, and the additional inertia of the system also promotes an added mass effect [131,150,151].

The average forces per cycle against the cruising velocity for anguilliform swimming are shown in Figure 6 [50]. As can be seen, when the velocity is zero, the pressure force is the maximum, and the viscous force is the minimum. As the velocity is increased, the pressure force is reduced, and the viscous force is increased until the velocity is such that the pressure force equals the viscous force, and thus, the total force is zero. Under this condition, the propulsor displaces at constant velocity, i.e., there is neither acceleration nor deceleration. In Figure 6, this velocity corresponds to 5.5 m/s. If the velocity exceeds this value, the propulsor decelerates.

3.2. Non-Dimensional Parameters

In biomimetic systems, it is useful to employ non-dimensional numbers to generalize the involving parameters [49]. The governing equations corresponding to a flow around a surface are the Navier–Stokes equations of conservation of mass and momentum, given by:

$$\nabla \cdot \overrightarrow{u}=0$$

(3)

$$\frac{\partial \overrightarrow{u}}{\partial t}+\nabla \cdot (\overrightarrow{u}\overrightarrow{u})=-\frac{\nabla p}{\rho }+\nu {\nabla }^2\overrightarrow{u}+\overrightarrow{g}$$

(4)

where u represents the velocity, p the pressure, ρ the density, ν the kinematic viscosity, and g the gravitational acceleration. The parameters in the equations above can be converted to non-dimensional quantities by using reference values L_ref, u_ref, p_ref, t_ref, and g_ref for length, velocity, pressure, time, and gravitational acceleration, respectively. The non-dimensional length, velocity, pressure, time, and gravitational acceleration are given by L* = L/L_ref, u* = u/u_ref, p* = p/p_ref, t* = t/t_ref, and g* = g/g_ref, respectively, and the resulting equations yield:

$$\frac{1}{L_{ref}}{\nabla }^{*}\cdot ({\overrightarrow{u}}^{*}{u}_{ref})=0$$

(5)

$$\frac{1}{t_{ref}}\frac{\partial ({\overrightarrow{u}}^{*}{u}_{ref})}{\partial t^{*}}+\frac{1}{L_{ref}}\nabla \cdot ({\overrightarrow{u}}^{*}{u}_{ref}{\overrightarrow{u}}^{*}{u}_{ref})=-\frac{1}{\rho }\frac{1}{L_{ref}}{\nabla }^{*}(p^{*}{p}_{pref})+\nu \frac{1}{L_{ref}^2}{\nabla }^{*2}({\overrightarrow{u}}^{*}{u}_{ref})+{\overrightarrow{g}}^{*}{g}_{ref}$$

(6)

Employing the reference parameters given in

Table 4. Reference and non-dimensional parameters.

Dimension	Reference Parameter	Non-Dimensional Parameter
Length	$L_{ref}=L$	$L^{*}=L/L_{ref}$
Velocity	$u_{ref}=U$	$u^{*}=u/u_{ref}$
Pressure	$p_{ref}=\rho u_{ref}^2$	$p^{*}=p/p_{ref}$
Time	$t_{ref}=L_{ref}/u_{ref}=L/U$	$t^{*}=t/t_{ref}$
Gravity	$g_{ref}=g$	${\overrightarrow{g}}^{*}=\overrightarrow{g}/g_{ref}$

, where L is the fin length and U the cruising velocity, the resulting equations of conservation of mass and momentum are given by Equations (7) and (8).

$$\frac{1}{L}{\nabla }^{*}\cdot ({\overrightarrow{u}}^{*}U)=0$$

(7)

$$\frac{U}{L}\frac{\partial ({\overrightarrow{u}}^{*}U)}{\partial t^{*}}+\frac{1}{L}\nabla \cdot ({\overrightarrow{u}}^{*}U{\overrightarrow{u}}^{*}U)=-\frac{1}{\rho }\frac{1}{L}{\nabla }^{*}(p^{*}\rho U^2)+\nu \frac{1}{L_{}^2}{\nabla }^{*2}({\overrightarrow{u}}^{*}U)+{\overrightarrow{g}}^{*}\overrightarrow{g}$$

(8)

which yields:

$${\nabla }^{*}\cdot {\overrightarrow{u}}^{*}=0$$

(9)

$$\frac{\partial {\overrightarrow{u}}^{*}}{\partial t^{*}}+\nabla \cdot ({\overrightarrow{u}}^{*}{\overrightarrow{u}}^{*})=-{\nabla }^{*}{p}^{*}+\frac{\nu }{UL}{\nabla }^{*2}{\overrightarrow{u}}^{*}+\frac{Lg}{U^2}{\overrightarrow{g}}^{*}$$

(10)

Expressing the Reynolds number as Re = UL/ν and the Froude number as $Fr=U/\sqrt{Lg}$, the resulting governing equations in non-dimensional form are:

$${\nabla }^{*}\cdot {\overrightarrow{u}}^{*}=0$$

(11)

$$\frac{\partial {\overrightarrow{u}}^{*}}{\partial t^{*}}+{\nabla }^{*}\cdot ({\overrightarrow{u}}^{*}{\overrightarrow{u}}^{*})=-{\nabla }^{*}{p}^{*}+\frac{1}{\mathrm{Re}}{\nabla }^{*}{}^2{\overrightarrow{u}}^{*}+\frac{1}{F{r}^2}{\overrightarrow{g}}^{*}$$

(12)

As can be seen, the Reynolds number, which represents the relation between inertial and viscous effects, and the Froude number, which represents the relation between inertial and gravity effects, are decisive parameters in the hydrodynamics of marine propulsors.

The frequency is usually expressed as the non-dimensional Strouhal number, St, given by St = fA/U, where f is the frequency and A the amplitude. It was highly reported that the Strouhal number is related to the vortex formation. In fact, typical values of the Strouhal number in nature are in a narrow range, approximately 0.2–0.4. Several researchers verified that the Strouhal number is related to the optimal vortex formation, which is produced in this narrow range observed in nature [59,87,149,152,153,154,155].

3.3. Turbulence

A few studies [93,156,157,158] employed inviscid models to analyze the hydrodynamics of biomimetic propulsion. The most important advantage of inviscid simulations is the cheap computational cost, but their accuracy is questionable. An important dimensionless parameter that involves viscosity is the Reynolds number. This parameter constitutes an indication of turbulence. The simulations that can be found in the literature span a wide range of Reynolds numbers due to the variety of values found in nature. Figure 7 shows the Reynolds number for different fish (from zebrafish larvae to stingrays and sharks), amphibians (tadpoles), reptiles (alligators), marine birds (penguins), and large mammals (from manatees and dolphins to belugas and blue whales) according to Gazzola et al. [151].

Another important parameter is the swimming number, Sw = fAL/ν. Although most numerical studies employ the Strouhal number, the swimming number provides the advantage of accounting for two length scales: amplitude and length. The swimming number is a transverse Reynolds number which characterizes the undulating movement. Gazzola et al. [152] analyzed more than 1000 measurements of fish varying in size from millimeters to meters and found that Re ̴ Sw^α, with α = 4/3 for laminar flows and α = 1 for turbulent flows. They verified that around Re 10³–10⁴, the transition from the laminar to the turbulent regime takes place, as indicated in Figure 8.

Some organisms swim at low Reynolds numbers and thus under laminar flow [149,159]. Nevertheless, most engineering biomimetic mechanisms operate at high Reynolds numbers and thus under turbulent flow. According to this, the turbulent regime was the most employed one in the numerical simulations of biomimetic propulsors.

When a body moves along a fluid, a boundary layer exists due to the “no-slip” condition and viscosity. The fluid in contact with the surface has zero velocity and, close to the surface, the fluid is slowed by frictional forces associated to the viscosity of the fluid. The result is a thin layer where the tangential velocity of the fluid increases from zero at the surface to the cruising velocity far away from the surface. The so-called boundary layer is the normal distance between the surface, where the velocity is zero, and the position where the tangential velocity is 0.95–0.99U, depending on the author. Flow over a wavy surface experiences adverse (flow against an increasing pressure) and favorable pressure gradients induced by the wave motion. When a boundary layer undergoes an adverse pressure gradient, the flow near the wall decelerates. If this boundary layer has travelled far enough in the adverse pressure gradient that the velocity becomes negative, i.e., reverses the direction, separation occurs. Figure 9 shows the velocity field in the boundary layer, the last profile corresponds to reverse flow which leads to flow separation.

Flow separation is the result of momentum losses, or decelerations, that eventually prevent the continued streamwise progress of the boundary layer fluid along the body surface. Drag is increased, and thrust is reduced. For this reason, much research has gone in the study of surfaces which delay flow separation and keep the flow attached for as long as possible. Some wave motions reduce the drag force and increase the efficiency by restraining separation and suppressing turbulence [148,160,161]. The travelling wave laminarizes the flow, and the fluid motion in the wave direction is accelerated. The turbulence intensity is increased when separation takes place since separation is the major mechanism for turbulence production in undulating surfaces. Several authors [57,162] found that approximately when the wave phase velocity is larger than the external flow velocity, the wavy surface pushes the fluid so strongly that separation occurs.

RANS (Reynolds-averaged Navier–Stokes), LES (Large-Eddy Simulation), and DNS (Direct Numerical Simulation) approaches were employed in the literature in order to simulate turbulence. RANS procedure computes the mean flow quantities. It includes only the largest flow structures, while the smaller scales are not included. It is robust, computationally cheap, and reasonably accurate for a wide range of engineering problems. Nevertheless, the main disadvantage is that the information about the flow is limited, including the simulation of flow separation [19,163]. In biomimetic marine propulsion, the effect of separation may be important, and thus, the RANS approach does not provide an accurate prediction of the flow in the separated region. Kim et al. [164] compared six two-equation models for detachment flow and determined that the most accurate in their simulations was k-ω and k-ω_SST models. Another disadvantage of RANS models is the simulation of transitional laminar-turbulent flows.

Regarding DNS, this computes all the turbulent scales in the grid and temporal resolution. In DNS, the Reynolds stresses are thus not modeled. The velocity field is obtained by integrating the three-dimensional time-dependent Navier–Stokes equations. The main advantage is that this model provides accurate results, but the disadvantage is the computational cost. In the recent years, computational resources have improved noticeably and made possible several works about direct numerical simulations [165,166,167,168,169]. The spatial resolution required increases as the Reynolds number increases. For this reason, DNS is mostly used to study turbulent flows at low or moderate Reynolds numbers [148,168,169].

Finally, the LES approach lies between RANS and DNS. LES resolves directly large eddies, while small eddies are modeled. Resolving only the large eddies allows to use much coarser meshes and larger time-step sizes than those employed in DNS. However, LES requires substantially finer meshes than RANS. The mesh is related to the turbulence scale. The small turbulence scale becomes smaller at high Reynolds flows, and thus, it needs finer meshes, especially in the near-wall region. The high computational cost has thus become a limitation of LES for high Reynolds simulations and the main advantage is that LES provides flow separation using less computational cost than DNS [170,171,172].

3.4. Grid Strategies

Depending on the mathematical representation of the moving boundary, two approaches to simulate bioinspired marine propulsors can be found in the literature. The first one, known as the interface-tracking or Lagrangian method, consists in employing a moving mesh which follows the movement of the boundary. Therefore, the edges of the mesh cells are aligned to the moving boundary, as indicated in Figure 10a. The second approach, known as interface-capturing or Eulerian method, consists in employing a fixed mesh and implementing the movement of the boundary in the governing equations (Figure 10b). In this approach, the governing equations for both solid and fluid are solved simultaneously and the fluid–solid interaction in computed by source terms in the governing equations.

The habitual approach in the literature was to employ a grid that moves with the surface. Moving meshes provide accuracy but the main disadvantages are the difficulty to handle complex moving boundaries and the high computational cost. Moving meshes must be reconstructed at each time step, and the remeshing techniques increment the computational cost noticeably; in some cases, the convergence may result difficult. Fixed meshes eliminate the computational cost of remeshing procedures during the simulations. Nevertheless, the main disadvantage of fixed meshes is the accuracy. In biomimetic marine propulsion fixed meshes have been employed since the 1980s [173] and have been continued in the recent years [170,171,172,173,174,175,176,177,178,179,180,181,182,183,184]. These are appropriate to represent sharp movements such as free-surface boundary on the incident waves [185,186,187,188].

4. Experimental Validations

The validation process constitutes a crucial part of computational simulations. In order to validate the models and visualize the wake structures behind the propulsor, it is very common to employ PIV (Particle Image Velocimetry) [87,108,109,166,175,189,190]. This is an optical fluid measurement technique that provides instantaneous fluid velocity fields using image processing and statistics. By seeding particles in the fluid and shooting a laser sheet behind the moving surface, the flow pattern is analyzed.

During PIV, the particle concentration is such that it is possible to identify individual particles in an image, but not with certainty to track it between images. When the particle concentration is so low that it is possible to follow an individual particle, it is called PTV (Particle Tracking Velocimetry). This is a velocimetry method, i.e., a technique to measure the velocity of particles that are resident in a fluid. There are several works about PIV analysis of fishes [191,192,193,194,195,196,197] and robotic fishes [198,199]. Anderson et al. [200] compared PIV with PTV and found that both procedures solved the outermost regions of the boundary layer but failed to resolve the flow very close to a moving surface. In the outer regions of the boundary layer, PIV and PTV provided the same velocities; nevertheless, they recommended PTV to analyze the velocity near the surface although the errors in the results also depend on camera pixel resolution, field of view dimensions, particle shape, size, centroid analysis, and image quality. The development of high-speed digital video technology in the recent years and the availability of lower cost continuous wave lasers has facilitated these studies, and PIV and PTV are becoming a common technique among engineers and biologists. Nevertheless, these techniques must be carefully employed in order to analyze the region near the surface. An accurate determination of the velocities in this region is too important to analyze friction.

Other procedures to measure flows are LDV (Laser Doppler Velocimetry) [170], also known as LDA (Laser Doppler Anemometry), and hot-wire anemometry. The main difference between PIV and these techniques is that the former produces two-dimensional or even three-dimensional vector fields, while the other techniques measure the velocity at a point. LDV is the technique of using the doppler shift in a laser beam to measure velocities. Buckles et al. [201] and Kuzan et al. [202] employed LDV to measure velocities working with separating flows and identified the turbulent layer and separated region. Hudson et al. [203,204] used LDV to analyze a wave surface and also analyzed the separated flow and Cherukat et al. [168] developed a computational model to analyze Hudson’s results.

5. Conclusions

This work provides a state about the works in the fields of hydrodynamics and computational simulations to analyze biomimetic marine propulsion. This is an emerging field due to the advantages of biomimetic propulsion over traditional rotary propellers. The advancement of numerical methods in the recent years provides a cheap and efficient tool to analyze the hydrodynamics. The tendency of many current works is to mimic the exact shape of a real fish. One of the reasons to develop a system as a copy is the development of mechanisms capable of capturing energy from a flow. There is a great body of literature describing this phenomenon, and consequently, the hydrodynamics are now better understood. Nevertheless, numerical models provide information that cannot be obtained using experimental techniques such as details of the pressure field, velocity field, hydrodynamic forces, power, and efficiency; analyze in detail all the aspects of the problem; and isolate the effect of each parameter. The characterization of vortex formation is a complex three-dimensional problem. Given the limitations to measure these flows experimentally, numerical models constitute an important tool to characterize the vortex formation.

In the upcoming years, it is expected that the number of numerical analysis grow as computation evolve and improve some deficiencies of current works. For instance, it is very important to improve the computational resources in order to make LES and DNS more affordable. Besides, it is necessary to develop robust moving meshes algorithms which allow abrupt movements with an appropriate convergence.

In order to validate these numerical models and visualize the wake structures behind the propulsor, it is important to develop experimental works such as PIV, PTV, LDV, and other procedures. Besides, it is necessary to improve the current technology in order to obtain more precise data. Future works will focus on developing different numerical models and validate their results using experimental techniques. Once validated, these numerical models will be useful to develop new marine propulsors.