MAXIMAL SPEED OF UNDERWATER LOCOMOTION

Background. An increasing interest in unmanned underwater vehicles continues to draw attention to swimming in aquatic animals. Their high speed continues to surprise researchers. In particular, the high dolphin speed caused a series of attempts to explain its paradox, which continues to this day. Some researchers believe that even rigid bodies, shaped like water animals, provide an attached flow pattern, as opposed to the widespread view of the inevitable separation. The possible explanation may be in the perfect body form, which provides an attached flow pattern (without boundary layer separation). Elongated unseparated shapes can not only reduce the pressure drag but also delay the laminar-to-turbulent transition in the boundary layer, significantly reducing the friction drag. Thus, the highest possible swimming speeds are expected in aquatic animals. Objective. We will try to prove that the low drag and the high speed of aquatic animals can only be ensured by their unseparated shape (as a rigid body), neglecting flexibility and compliance. Methods. We will use: a) shape calculations of special bodies of revolution with negative pressure gradients near the tail similar to fish trunks with the use of the developed before approach; b) the known drag estimations of such shapes for laminar and turbulent cases; c) the swimming power balance and the theory of ideal propeller; d) statistical analysis of available data about the length, the speed and the aspect ratio of aquatic animals. Results. The swimming speed of most aquatic animals is proportional to the length of the body in power 7/9. The exception is whale locomotion that occurs in turbulent mode at supercritical Reynolds numbers. Conclusions. The perfect body shapes of most aquatic animals provide an attached laminar flow pattern. Estimated maximum speeds for laminar and turbulent cases show that the special shaped unseparated hulls can greatly increase the speed of underwater vehicles and SWATH ships. Further increase in speed can be achieved by using supercavitation and greater than animal capacity-efficiency.


Introduction
An increasing interest in unmanned underwater vehicles continues to draw attention to swimming in aquatic animals. At one time, the high speed of the dolphin surprised Gray [1] and caused a series of attempts to explain his paradox, which continues to this day [2][3][4][5][6]. In particular, some researchers believe that even rigid bodies, shaped like water animals, provide an attached flow pattern [7][8][9][10][11][12], as opposed to the widespread view of the inevitable separation [13]. Elongated unseparated shapes can not only reduce the pressure drag but also delay the laminar-to-turbulent transition in the boundary layer, significantly reducing the friction drag [10,11,14]. Thus, the highest possible swimming speeds are expected in aquatic animals. In order to investigate their dependence on the body length, a kind of theoretical and statistical analysis was carried out. Maximal speeds for the attached laminar and turbulent hulls were estimated and compared with supercavitating ones.

Calculations of the special shaped bodies of revolution
Special shaped rigid bodies of revolution similar to some fish shapes were calculated with the use of distributions of the sources and sinks on the axis of symmetry. The stream function of the axisymmetric potential flow of the inviscid incompressible fluid was represented as follows [15,9]

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Here , xr are cylindrical coordinates. The corresponding axisymmetric body radius () Rx , flow velocity components , x v r v and pressure coefficient on the surface were calculated with the use of following equations: Here P and P  are pressures on the body surface and in the ambient flow respectively, U is the speed of steady motion,  is the density of water.
Varying the values of constant parameters *1 , , , x a a c different closed ( ( ) 0) RL  and unclosed ( ( ) 0) RL  shapes can be obtained (L is the body length). The links between these parameters are discussed in [9], but every new shape calculation needs some numerical experiments.
Calculation results for different values of the aspect ratio  = L/D (D is the maximal body diameter) are shown in Fig. 1. Some other examples can be found in [9,10,12]). The pressure distributions on these bodies have a negative pressure gradient near the tail (see dashed lines representing the pressure coefficient P С ).  Shapes with similar pressure distributions near the trailing have been calculated [16] and tested in wind tunnels [9,12,17,18]. In particular, Goldschmied body [17] revealed the attached flow patterns only with the use of boundary layer suction. In comparison, the tests of UA-2 body showed unseparated flows without any boundary layer control methods [9,12]. Thus, we expect that similar slender bodies (e.g., shown in Fig. 1) can also ensure attached flows due to their special rigid shape only. In this paper, we do not consider the drag reduction connected with flexible shapes (as, e.g., in [1]) or animal body compliance (as, e.g., in [19]). In order to use the theoretical drag estimations, presented in the next Section, we will also limit our study to only sufficiently slender bodies (

Drag estimations on slender unseparated bodies of revolution
To estimate laminar frictional drag on a slender unseparated body of revolution, we can use the Mangler -Stepanov transformations [20], which reduce the rotationally symmetric boundary-layer equations to a two-dimensional case. Coordinate x for the rotationally symmetric boundary-layer (calculated along the body contour) and the corresponding two-dimensional coordinate , x flow velocity at the outer edge of the boundary layer, the displacement thickness and the skin-friction coefficient are related as follows [20]: All the values in (1) are dimensionless, based on the body length L , the ambient flow velocity U  and 2 0.5 ; dashed values correspond to 2D boundary layer. These equations are valid for an arbitrary rotationally symmetric body provided that the thickness of the boundary layer is small in comparison with the radius, i.e. the flow has to be unseparated. For a slender body, the coordinate x can be calculated along the body's axis and the velocity out U can be supposed to be equal to unity, neglecting the thickness of the boundary layer and the pressure distribution peculiarities [21]. From the second equation (1) the value of out U will also be equal to unity, i.e., the rotationally symmetric boundary layer on a slender body can be reduced to the flat-plate one [10,14,22]. According to the Blasius solution  [20]. Introducing the variable x and using (1) yields the following formula for the laminar skin-friction drag coefficients of a slender rotationally symmetric body [10,14]: With the use of dimensionless values based on the body volume V, (2) can be rewritten as follows [10,11,14]: Note that the volumetric frictional drag coefficient V C does not depend on the slender body shape provided its volume remains constant. This is valid for laminar attached boundary layer and at limited Re , V only. Eq. (3) was compared with the Hoerner formula [23] for the total laminar drag X on standard bodies of revolution (e.g. ellipsoids): (S is the body surface area). Good agreement occurs for slender shapes when the term 0.11(D/L) 2 (connected with separation) can be neglected (see upper dashed lines in Fig. 2).
For a turbulent boundary layer, Hoerner proposed another empirical formula [23]: Eqs. (4) and (5) were used to estimate the drag on standard bodies of revolution with the values D/L coinciding with ones for the closed special shaped bodies of revolution shown in Fig. 1. Dashed lines in Fig. 2 represent the results of these calculations.

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To find the critical Reynolds number for the laminar-to-turbulent transition in the boundary layer on the slender unseparated bodies of revolution, the Tollmin -Schlichting -Lin theory [13] and Mangler -Stepanov transformations [20] were used in [10,11,14]. The boundary-layer on a flat plate remains laminar for any frequencies of disturbances, if And with the use of (1), the condition for the axisymmetrical boundary-layer to remain laminar can be written as follows [10,11,14]: If the Reynolds number is small enough and inequality (6) holds on the entire body surface (x = 1 in (6)), we can calculate the friction drag according to (2) and obtain formula (3). Otherwise, the boundary layer remains laminar only on the nose part of the body surface (on the interval 0 x    ). For the interval 1 x    we can use eq. (5) without the term 7(D/L) 3 , since our bodies are expected to be unseparated. Solid lines in Fig. 2 represent the results of calculations for special shaped bodies of revolution shown in Fig. 1.

Locomotion with the use of attached flow patterns
The power balance for a steady horizontal motion at speed U of an animal or vehicle of weight Mg can be written as follows: where q is the power per unit weight used for locomotion;  (0 1)    is the propulsion efficiency; the hydrodynamic drag X can be expressed with the use of volumetric drag coefficient : Since values q and  are limited, the maximal speed can be achieved at small drag coefficients, in particular, without any boundary layer separation. The trunks of best swimming animals ensure such attached flow patterns [7,8]. The theoretical estimations of V С for laminar and turbulent cases are shown in Table 1 (line 1). Formulas (9a) and (9b) are in good agreement with Hoerner's formula for laminar drag on slender ellipsoids and for the friction drag in developed turbulent flow respectively (see Fig. 2). With the use of assumption of neutral buoyancy () MV  , eqs. (7), (8), (9a), and (9b) yield estimations of the maximal speed at fixed values of the propulsion efficiency  (see Table 1, line 2). Formulas (10a), (10b) illustrate that the maximal speed weakly increases with the body or hull length. Both relationships differ from the known linear dependence (e.g., burst swimming corresponds to speeds of circa 10 body lengths per second for sub-carangiform fish of between 10 and 20 cm in length [25]). Later we will try to estimate the speed-to-length dependence with the use of statistical information about water animals (see Tables  A, B, and C).
Mechanisms of thrust creating and their efficiency can be very different [26,27]. To take into account the probable dependence () L  , we will use here the theory of ideal propeller [28] and corresponding formula: where T = X is the thrust; disc A is the area of an ideal propeller. At small values of the thrust coefficient , T C 1  and formulas (10a) and (10b) can be used. If 1 For the neutral buoyant case () MV  putting eq. (12) into (7), we obtain: Then with the use of (8) and T = X : . Finally, taking into account (9a) and (9b) and assuming 2 disc AL and 3 VL we obtain estimations of the maximal speed for the case of an ideal propeller with 1 T C  (see Table 1, line 3). In comparison with (10a), dependence (13a) is much closer to the linear one. For the turbulent flow pattern, formulae (10b) and (13b) yield the same weak increase of the speed versus length.

Statistical analysis of the speed-to-length dependence for aquatic animals
If we assume that (in such dependences we will measure the swimming speed in m/s and body length in m), then ln u k pl  ( ln , uU  ln ). lL  We can treat l and u as random variables and use linear regression, [29], to estimate constant parameters k and p for the aquatic animal locomotion. A similar statistical approach was used in [30] to link animal speeds with tail beat amplitude and frequency. The length and speed values available in the literature and internet are shown in Tables A, B, and C and in Fig. 3. The results of the statistical analysis are presented in Table 2. We have not used the information about flying animals, human swimming, animals with non-slender trunks (D/L > 0.25) and with the anguilliform propulsion, in order to check our assumptions about unseparated body shapes and estimations presented in Table 1.
Application of a multiple regression for animals represented in Table B (without whales) Maximal speed at 1 T C   Table A; dark blue markers for B; magentafor C; redfor whales from tables B and C. "Stars" correspond to the cases L/D < 4 or animals with the anguilliform propulsion; "triangles" -to flying animals, "circles"to human swimming; all these three cases do not participate in the statistical analysis. Black markers represent vehicles ("stars"underwater or SWATH, "squares" -floating, "circles" -human rowing shells, "triangle" -Hydrofoil craft PTS 150 MK III). Regression results are represented by straight lines (green for Table A; dark blue for B (without whales); magenta for C (without whales); red short linefor whales from tables B and C). Blue line represents the linear regression based on all the animal information from Tables A, B, and C (without whales). Bold black line represents the maximum speed for the laminar flow pattern (eq. (16)); bold red linefor the turbulent one (eq. (21b)). Commercial efficient speeds are shown by dashed yellow and red lines for laminar and turbulent unseparated hulls respectively. Critical speeds for the laminar flow (19) are shown by the black dashed lines for L/D = 4, 10, and 30. Brown line represents the critical velocity 4.72 UL    Table B (see Table 2, line 2), total data (line 4) and formula (15). The corresponding values of the power coefficient p are very close to the theoretical value 7/9 from eq. (13a). Unfortunately, we do not have enough information about whales to check dependencies (10b), (13b), and (14) (see Table 2, line 5). It must be noted that standard assumption about the drag proportionality 2 XU (e.g., it corresponds to the constant volumetric drag coefficient, see eq. (8)) leads to a very weak speed versus length dependence 1/3 UL (see (10b) and (13b)), which is not supported by animals data.

Estimations of the maximal speed for the laminar flow pattern
Since the dependence 7/9 U kL  is substantiated both theoretically and statistically, we can use it to estimate the maximal speed in the laminar flow. For this purpose it is enough to find the maximal value of the coefficient k, i.e. the maximum of the ratio 7/9 / UL . The calculations are presented in Tables A, B, and C. It can be seen that the maximal value of this ratio -28.6 2/9 / mscorresponds to the juvenile Blue shark (C, line 28). Thus, the maximal speed for laminar flow pattern can be estimated from the equation: The black bold line represents dependence (16) in Fig. 3.

Estimations of the commercial efficient speed
Formulae (16) allow us to estimate the maximal speed of animals or well shaped vehicles with effective locomotion, which can be achieved only for short periods of time (in particular, this velocity corresponds for un-aerobic processes in animals). To achieve the maximal range at fixed amount of energy accumulated on board or in body, we need to maximize the weight-to-drag ratio W/X [11,31]. For neutral buoyancy () W Mg Vg    , formula (8) Then with the use of (9a), (9b) and the volume estimation for the special shaped closed unseparated bodies of revolution [10] (with the average value 0.285),  we can calculate the commercial efficient speed for the attached laminar and turbulent cases at different values W/X. The corresponding lines are shown in Fig. 3 by dashed yellow (laminar case) and red (turbulent case) lines for W/X = 50; D/L = 0.1; 6 10   m 2 /s. It can be seen that laminar commercial effective velocities are much lower than the maximal ones, but whale speeds are close to the commercial efficient one. Very small speed of underwater glider (see Fig. 3, Table C, 47) ensures its high commercial efficiency. In Fig. 3, the brown line represents the critical velocity 4.72 UL  corresponding to the drastic increase of the wave drag for ships (vehicles floating on the water surface) [32].

Speed limit for the laminar flow pattern
Formulae (16) demonstrate that speed can increase rather fast with the length of proper shaped laminar unseparated vehicle with effective propulsion. Nevertheless, the use of (16) is limited, since fast and large vehicles (and also animals, e.g., whales) cannot ensure a laminar boundary layer on the entire surface. The corresponding critical Reynolds number was estimated in [10,11,14]: Eqs. (17) and (18) show that slenderer bodies ensure higher values of the critical Reynolds number (see also [10,11,14] and points of bifurcation in Fig. 2). Taking into account (17) and (18) (for 0.285).


At lower speeds a slender special shaped unseparated body of revolution ensures the laminar boundary layer. Dependences (19) are shown in Fig. 3 Relationship (20) demonstrates that the maximal possible velocity with the pure laminar flow pattern is independent on length and that the increase 159 159 of the aspect ratio L/D allows increasing the vehicle speed. In particular, for L/D > 28, a vehicle with a special shaped unseparated hull could exceed the speed 100 m/s. Eq. (20) was used to calculate the maximum possible speed for the pure laminar flow pattern at 6 1. 3

Maximal speed for the turbulent flow pattern
For the developed turbulent flow, we can estimate the maximal speed with the use of (9b), (10b) and the highest value the capacity-efficiency E Cq  [10,11]. According to the data presented in Table C, the maximum 8.5 E Cq    m/s corresponds to the juvenile Blue shark (C-28), see also [11]. Then  Table B and estimation (20).

High-speed underwater options: attached flow or supercavitation
High speeds can cause cavitation in water, since the local pressures and cavitation number  Tables A, B and C do not exceed the corresponding values for cavitation inception on their trunks. This issue, as well as the cavitation inception on fish fins, was discussed in [35]. Decreasing the cavitation number leads to the formation of large enough supercavities, which can cover hulls and thereby reduce the friction drag (see Fig. 4 and [36][37][38]). For the disc cavitator (a part of the hull wetted by water) and when the entire cavity volume is used to locate the hull (as shown in lower pictogram in Fig. 4), the volumetric pressure drag coefficient can be estimated as follows [ (8)). It means that drag on supercavitating vehicles reduces with the increase of speed (when the hull volume is fixed, but its shape changes to be located in the corresponding cavity). This much unexpected conclusion was also confirmed by the results given in [38] for sub-and supersonic speeds.
The supercavitating hulls using the cavity volume completely (see lower pictogram in Fig. 4) must have the aspect ratio L/D coinciding with the  for the cavity, which increases with the diminishing of the cavitation number approximately as 1/  [39].
Since the strength and stability of rigid hulls cannot be ensured at very large L/D values, the applications of the supercavitating flow pattern shown in lower pictogram and formula (23) is limited. It is possible to use only the nose part of the cavity in order to locate the hull with a fixed value of  as shown in upper pictogram. The corresponding values of V C are larger than (23) but small enough to ensure high speeds of supercavitating vehicles (see details in [37]). In particular,


, the advantages of the laminar unseparated hulls over supercavitating ones are limited by two circumstances. First, the length and the volume of the laminar unseparated hulls are limited (see (19) and black dashed lines in Fig. 3). Second, corresponding speeds exceed * C U and the problem of cavitation inception must be solved. Probably, unseparated shapes can delay also the cavitation inception. Some theoretical considerations are available in [33,40]. Experiments presented in [41,42] demonstrate that cavitation started at the region of the boundary layer separation.

Speed limitations for supercavitation
Supercavitation requires compensation of the vehicle weight since its hull moves in gas (as in the case of aircraft, see Fig. 4). It can be implemented by using wings or by hull planning on the cavity surface (with the lift force YW =W). Corresponding additional drag XW limits the advantages of supercavitation [37]. In particular, the additional capacityefficiency [11]: increases linearly with the speed and exceeds the estimation 8.5 m/s respectively. It can be seen that inequality (25) does not hold at higher speed 1000 m/s, but it is possible to achieve this speed for large vehicles with V > 1500 m 3 . Thus, the idea of a superfast supercavitating submarine does not look utopian [38].

Conclusions
As shown by theoretical and statistical analysis, the perfect body shapes of most aquatic animals provide an attached laminar flow pattern. The swimming speed of these animals is proportional to the length of the body in power 7/9. The exception is whale locomotion that occurs in turbulent mode at supercritical Reynolds numbers.
Estimated maximum speeds for laminar and turbulent flow patterns show that the special shaped unseparated hulls can greatly increase the speed of underwater vehicles and SWATH ships. Further increase in speed of underwater motion can be achieved with the use of supercavitating hulls and greater than animal capacity-efficiency. Body lengths were calculated with the use of speed and U/L data presented in [44]. The data corresponding to jumping are eliminated. "Non-slender" animals (with D/L > 0.25) are labeled by "star" and are not used in statistical analysis

Primal information Calculations
Speed U (m/s) Body length L (m) 7  1.3 10     m 2 /s, corresponding to the water temperature approximately 10 C. Unfortunately, there is no information in [7] which value of viscosity was used to calculate the Reynolds number. The whales are marked in red; their statistical analysis was carried out separately. Flying animals are marked in blue, animals with the anguilliform propulsionin yellow. Their data and "Non-slender" animals (with D/L > 0.25) are not used in statistical analysis. Corresponding species are labeled by "star". Speeds exceeding the estimation (20)     For animals the speed and length values are mostly extracted from [45]; D/Lfrom [7]; for Swordfish and Black marline their rostrum is taken into account to calculate D/L (neglected in [7]). The same information was used in [11]. The whales are marked in red; their statistical analysis was carried out separately. Flying fish is marked in blue, eel (with the anguilliform propulsion) -in yellow, human sport activityin magenta. Their data and "non-slender" animals (with D/L > 0. 25