FASTEST FISH SHAPES AND OPTIMAL SUPERCAVITATING AND HYPERSONIC BODIES OF REVOLUTION

Background. The best swimmers have a streamlined shape that provides a flow pattern without boundary layer separation and delays the laminar-to-turbulent transition. Their shape itself could be the reason of small drag and high locomotion velocity. The fastest fish, e.g., sailfish, swordfish, black marlin, etc. have another feature of their shape – a very sharp nose – rostrum, the purpose of which remains unclear. Popular belief that the rostrum is used by these predators to pierce their prey is often disputed. Objective. In this study, we analyze the hydrodynamic aspects of the rostrum presence and the possible use of similar hulls for supercavitating underwater vehicles and fast penetration into water. We illustrate that shapes with the very sharp nose could be useful for hypersonic motion in order to eliminate overheating of the vehicle fuselage. Methods. We use the known exact solutions of the Euler equations for the incompressible fluid to simulate the pressure distribution on the bodies of revolution with a sharp nose. The slender body theory is used to simulate the supercavitation and the axisymmetric air flows at high Mach numbers. Results. Bodies of revolution with a rostrum similar to trunks of the fastest fish (sailfish, swordfish, black marlin) and corresponding pressure ant temperature coefficients were calculated. The proposed shapes ensure no stagnation points and no high pressures and temperatures on their noses at suband supersonic speeds both in water and air. The drag on such bodies of revolution was estimated for attached, supercavitating and supersonic flow patterns. Conclusions. A method of calculation of axisymmetric bodies without stagnation points on their surface was proposed. This peculiarity of the shape allows diminishing the maximum pressure and temperature on the nose without a significant increase in drag. Such shapes with the sharp concave nose could be recommended for high-speed attached and supercavitating bodies of revolution and for the hypersonic motion.


Introduction
The high swimming speed of aquatic animals continues to draw attention to the shape of their bodies. For example, the best swimmers have a streamlined shape that provides a flow pattern without boundary layer separation [1][2][3]. This delays the laminar-to-turbulent transition and reduces drag. The estimates in [4] show that the critical Reynolds number can be quite high for special shaped unseparated bodies of revolution, similar to the bodies of aquatic animals. This fact allowed us to conclude that the shape itself can provide low drag inherent in laminar flow [4], and very simply solve the well-known Gray paradox concerning the dolphin swimming [5][6][7][8][9].
The fastest fish, e.g., sailfish Istiophorus platypterus Show and Nodder, swordfish Xiphias gladius L., black marlin Makaira indica Cuv et Val., etc. [1,10,11], have another feature of their shapea very sharp noserostrum, the purpose of which remains unclear. Popular belief that the rostrum is used by these predators to pierce their prey is often disputed [12]. In this study, we will analyze the hydrodynamic aspects of the rostrum presence and the possible use of similar hulls for supercavitating underwater vehicles and fast penetration into water. We will illustrate that shapes with the very sharp nose could be useful for hypersonic motion in order to eliminate overheating of the vehicle fuselage.

Simulation of steady axisymmetric flows by sources and sinks
We will use the standard expressions for Reynolds and Mach numbers: where L and V are the body length and volume;  is the kinematic viscosity; U  and a  are the flow and sound speeds at infinity. We assume the Reynolds numbers (first two equations (1)) to be large enough (for example, Re 50000) L  such that the boundary-layer thickness can be neglected and fluid outside a body and a thin layer on its surface can be treated as ideal.
the potential flow of a source with intensity Q i located at the point ( , 0) i  can be expressed by the streamline function [13]: where x, r are cylindrical coordinates; C i is a constant value. Then the dimensionless (based on ) U  components of the flow velocity x ix vv  and r ir vv  can be calculated from (2) as follows: The rigid body shape R(x) can be obtained from (2) with the use of simple equation [16]: The nose of the body is located at the axis of symmetry r = 0 and its coordinate x n can be calculated with the use of (3) as follows: The velocity at this point is equal to zero (see (3), (4)); the pressure p and pressure coefficient reach their maximum values ( p  and   are the pressure and the fluid density in the ambient flow). Thus the nose of the body has a stagnation point with the maximum pressure.
With the use of n sources and thinks (Q i < 0) located at the axis of symmetry, different bodies of revolution can be simulated by adding corresponding functions , and : Similar to (5), their radii R(x) may be calculated from 1 ( , ( )) 0.  [3,4]). Equations (8) and (9) can be treated as an exact solution of Euler equations for inviscid incompressible fluid.

Slender body theories
The method presented in previous Subsection allows calculating the flow around a body of a given shape or with a given pressure distribution on its surface, provided the use of appropriate algorithms for selecting the distribution of intensities of sources and sinks (in some cases also dipoles or other singularities are necessary, see, e.g., [14]). If the body of revolution is elongated (L/D >>1, D is the maximum diameter, see Fig. 1) and moves along the axis of symmetry, it is possible to use slender body theories which allow calculating the sources distribution q(x) with the use of R(x) at different Mach numbers.
In particular, instead of (8) we can use the following expression for the streamline function [13]: Here we use dimensionless coordinates based on the body length L. The velocity components and the pressure coefficient can be obtained from (10) with the use (3), (4) and (7). In particular, for pressure coefficient the following formula can be used [13]:  (15), black linesdistribution (16). Exact solution (8), (9) was used to calculate body shapes (R(x)10, solid lines); dashed lines show corresponding pressure coefficient (7); dotted lines illustrate the application of the linear theory to calculate the pressure coefficient (12) with the use of exact solution for the body radius The following formulas: can be used to calculate the temperature coefficient for supersonic flows in air, [13]. Here T and T  are temperatures (local and in the ambient flow respectively), R g is the gas constant. Some approximate formulas relate the local pressure on the slender body surface with its local radius [15,16]: It was shown in [16], that equation (14) is valid for both sub-and supersonic flows. If the integral in formula (10) can be replaced by sums similar to (8). In particular, at M = 0 such replacement yields an exact solution of Euler equations coinciding with (8), (9). In our MATLAB codes we have used sums instead of integral (10). Examples of calculations are shown in Fig. 1 for two different sources intensities distributions: where a, b, c, d, a 1 and * x are constant parameters. Fig. 1 illustrates that distribution (16) yields concave nose of the body and much smaller values of pressure on it. The pressure distributions corresponding to the exact solution and the slender body theory are rather close (except for the midline of the body). It follows from (3) and (10) that: near the point (0,0). It follows from (15) and (17) that axial velocity is still infinite (like in the case of single source (3)

Results
The maximal swimming speed of the fastest fish (e.g., sailfish Istiophorus platypterus Show and Nodder, swordfish Xiphias gladius L., black marlin Makaira indica Cuv et Val.) is around 30 m/s [1,10,11]. At such high velocities, the pressure at a stagnation point exceeds the ambient one by 4.5 atm. (see eq. (7)). Probably, rostrums allow these animals to remove stagnation points and high pressures on the body surface. To support this hypothesis, a series of calculations has been carried out with the use of exact solution (8), (9) and the sources/sinks intensity distribution (16). The results are shown in Figs. 2 and 3.
Shapes with sharp concave rostrums and without stagnation points were obtained. The values of the parameters a, b, c, d, a 1 and * x in (16) were chosen to obtain a rather good similarity with the trunks of sailfish, black marlin and swordfish. To investigate what happens when the body nose is not concave, the distribution (15) was used without changing the values of parameters a 1 and * .
x The results of similar calculations can be found also in [4]. The corresponding body radius (red dotted line) and the pressure distribution (red dashed line) are also shown in Figs. 2 and 3. It can be seen that rather small changes of shape (compare green solid line and red dotted one in Fig. 2) cause significant difference in pressure distributions (compare green and red dashed lines). In particular, there is a stagnation point on the shape with convex nose (red dotted line) located very close to the point (0,0), which is not shown in Figs. 2 and 3. The absence of a stagnation point is very important at high velocities when the corresponding pressure increases drastically 2 (pU  according to equation (7)) and the fluid compressibility has to be taken into account. To simulate the influence of Mach number, a new series of calculations has been performed with the use of (8), (9), (12) and the sources/sinks intensity distribution (16). At M > 0 this solution in no more exact, but it accuracy can be rather good as Fig. 1 demonstrates. The results are shown in Fig. 4. The body shape (black lines) was calculated with the use of exact solution at M = 0. The pressure distributions for M = 0, 0.5 and 0.9 were calculated with the use of approximate formula (12) (green, blue and red lines respectively). It can be seen that the body with a sharp concave rostrum can ensure small values of the pressure coefficient and absence of the stagnation point on its surface in a subsonic flow.

Drag on bodies of revolution with rostrum
The absence of the stagnation point does not remove the question of the hydrodynamic drag on bodies with a rostrum. At small Mach numbers the minimum possible value of the drag can be achieved by eliminating the boundary-layer separation. In this case, not only the pressure drag decreases, but also the frictional one due to the delay of the laminar-to-turbulent transition [4]. On the bodies of revolution with the rostrum shown in Figs. 1-4, the pressure increases on their noses (in comparison with the bodies without rostrum shown in Figs. 1-3, calculated and tested in [3,4,[17][18][19]). These positive pressure gradients can cause separation, the presence or absence of which requires further research. But it is worth noting that experiments with rigid bodies similar to the body shape of sailfish and swordfish revealed an attached flow pattern [1]. Unfortunately, the book [1] does not specify in what way the lack of the boundary-layer separation was proved.
If there is no separation on the surface of the body with the rostrum, then its drag X can be estimated by the formula [4]: for Reynolds numbers less than the critical value [4]: It is seen from (18) that the value of drag does not depend on the shape and taking into account also (1) we can obtain: (similar to other slender bodies of revolution without boundary-layer separation [4]).
The maximal speed max U (in m/s) of an animal or a vehicle providing the laminar attached flow pattern was estimated in [4] as follows: The body (hull) length in (19) must be taken in meters. The ratio 7/9 max / UL calculated in [4] is 13-20.8 for sailfish, 11.5-15.1 for swordfish and 8.4-11.8 for black marlin. These values approach the maximal one (see (19)) and are higher or comparable than for some other good swimmers which have no rostrsharks, dolphins, tunas). The 7/9 max / UL ratios for torpedoes Mark 48 and Spearfish (having no rostrum) are 7.23 and 9.17 respectively [4]. Thus, we can conclude that the presence of rostrum does not increase the drag.

Supercavitating bodies of revolution with concave cavitatiors
At high speed motion in water, the local pressures and the cavitation number 22 2( ) 2 ( 10) .  (20) decrease and cavitation occurs [20][21][22][23]. In formula (20) we have neglected the pressure inside the cavity in comparison with the atmospheric pressure, corresponding to the water column of 10 m; h m is the depth of steady motion in meters. Nevertheless, pressures at the stagnation points can be very high according to the formula (7). E.g., at the nose of a slender cone entering the water at speed 1000m/s, the local pressure can reach 5000 atm. and can cause the destruction of the entering body.
To avoid this huge pressure increase, the special shapes without stagnation points can be used. In particular, special shaped concave cavitators (parts of the hulls wetted by water) were proposed in [24]: Here and further in this Subsection, all dimensionless lengths (marked with a "wave") are based on the value of the cavitator radius R 0 at the point of cavity separation 0.

x 
The conical cavitator corresponds to b 1 = 0, and the parameter  is equal to the derivative of the radius of the cavitator at 0. x  At  = 0.1; b 1 = 0.0025, eqs. (11) and (21) allow obtaining the sources distribution (16) with d = 0. It means that corresponding concave cavitator has no stagnation point (see (17)).
The friction drag can be diminished, when some part of the hull is located inside the cavity to avoid a contact with the water flow. To calculate the cavity shape, eq. (14) can be integrated with the use of the conditions of continuity of the radius and its derivative at 0: Here we use the condition .    The accuracy of the first approximation equation (22) can be increased with the use of next approximations which are dependent on the Mach number [25].
The examples of cavitators (eq. (21)), cavities (eq. (22)) and parts of the hulls located inside the corresponding cavities at 0.1     are shown in Fig. 5. Green and black lines represent concave cavitators, but only green one (corresponding b 1 = 0.0025) has no stagnation point. For comparison the conical cavitator is shown by the red line. Each of these 3 cavitators can be combined with the parts of the hull located inside the corresponding cavities (blue, magenta and brown lines). The pressure drag X p on cavitator and corresponding volumetric drag coefficient Vp C was calculated in [24] with the use of exact solution (similar to proposed in [16]) based on (21), (22) and corresponding sources/sinks distributions (11) at M  = 0. The volume of the hull V was assumed to be equal to the total volume of cavitator and the corresponding cavity (the volume the gap between the hull and the cavity surface was neglected). The results of calculations demonstrated that the pressure drag coefficient is minimal for the concave cavitator without stagnation point (b 1 = 0.0025 [24]). The friction drag on concave cavitator could be larger in comparison with the conical one. But for concave cavitator with b 1 = 0.0025, the surfaceto-volume ratio is only 11% higher then for conical one (b 1 = 0). Therefore, at fixed volumes, the bodies of revolution without stagnation points remove high pressures on their surface without increasing the drag.

Hypersonic bodies of revolution with rostrums
For a vehicle moving in the air at high Mach numbers, it is critical to reduce the heating of its surface (see, e.g., [26][27][28][29][30][31]). Many different approaches have been proposed to optimize the shape of hypersonic hulls [32][33][34][35][36][37][38][39], but the von Karman ogive [40] remains the most common shape of the axisymmetric hypersonic forebody [41]. Even nonblunt conical noses proposed in [34,41] cause a stagnation point and very high temperatures. According to (13) the local temperature of air can be estimated as: 2 42 ( 1) 4.97 10 2 in the vicinity of the stagnation point. The velocity U  in (23) must be taken in m/s. For example, at U  = 3000 m/s, the local temperature exceeds the ambient one by 4476 K. If the shape of the body provides a flow without the stagnation point, the heating of its nose surface can be significantly reduced. The results of calculations are shown in Fig. 6 for different values of the Mach number. Since the pressure and temperature coefficient are very close for slender bodies (see (13)), one curve represent the distribution of the pressure and the temperature versus longitudinal coordinate x/L. At M  = 0 the body shape (solid black line) and pressure distribution (dashed black line) were calculated with the use of exact solution (8), (9) and the sources/sinks distribution (16). Pressure and temperature coefficients on the surface of this body were calculated with the use of (12) and (13) at M  = 3; 5; 10 and 20 (dashed blue, magenta, brown and green lines respectively).
A significant pressure drag X p occurs at supersonic speeds. The corresponding drag coefficient was calculated for the slender body of revolution presented in Fig. 6 Fig. 6), but at M = 20 this body has much lower pressure drag coefficient. This fact requires further research. Fig. 6 shows that the temperature coefficient on the entire body surface does not exceed 0.02. This means that special body shapes with rostrums can reduce the maximum temperature on their surface by more than 50 times. In particular, at a speed of 3000 m/s, this maximum temperature exceeds the ambient one by no more than 90 K on the entire surface of the body shown in Fig. 6.

Conclusions
Bodies of revolution with a rostrum similar to trunks of the fastest fish (sailfish, swordfish and black marlin) were calculated with the use of exact solution of Euler equations. The corresponding flow patterns have no stagnation point. This fact allows diminishing the maximum pressure on the surface. Similar shapes with the sharp concave nose could be recommended for cavitators in order to reduce significant local loads on the high-speed supercavitating bodies of revolution without loses in volumetric drag coefficient. Proposed axisymmetric bodies also demonstrate significant reduces in air temperature on the surface at high Mach numbers.