Abstract:
The subject of the invention is a watercraft equipped with wheels of special design, which is capable of running on the surface of water. The vessel makes a novel maritime vehicle for fast transportation of small groups of people and commercial merchandise over the surface of rivers, seas, and oceans. The thrust, and share of the lift, is generated by hydrodynamic reaction of water to roll-and-slide motion of the wheels; the rest of the lift, which is necessary to counterbalance the loaded weight of the cruising vehicle, is generated by hydrodynamic forces acting on four plates with adjustable angle of attack and mounted on hydraulic struts with adjustable length. The wheeled watercraft, running on the surface of water at cruising speed, can surpass significantly planing vessels and hydrofoils in speed and fuel efficiency, and lower engine power required to achieve high speeds. The invention has the potential for a number of military applications.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
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       STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
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       INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC OR AS A TEXT FILE VIA THE OFFICE ELECTRONIC FILING SYSTEM 
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       STATEMENT REGARDING PRIOR DISCLOSURES BY THE INVENTOR OR A JOINT INVENTOR 
       [0005]    Not Applicable 
       BACKGROUND OF THE INVENTION 
       [0006]    The invention pertains to marine technologies for fast maritime transportation of people and commercial goods. Nearly all sea transportation methods known today use floating principle: boats, ships, aircraft carriers use Archimedes lift force to counterbalance the weight of the vessel. These vessels cannot achieve high cruising speeds, comparable to that of ground transportation vehicles like automobiles and passenger trains, because the water resistance to the motion of vessels grows rapidly with both displacement and speed. Planing waterborne crafts—vessels gliding over the surface of water—are a notable exception. Sports boats of this type can achieve speeds up to 70 knots. In planing mode of operation, the weight of the vessel is supported predominantly by hydrodynamic lift rather than hydrostatic lift (buoyancy). 
         [0007]    The first attempt to depart from the floating principle was made by a famous English shipbuilder pastor Charles Ramus, who proposed in 1870 a project for a 2,500-ton displacement destroyer sliding on the water surface. The Admiralty, impressed by the idea, asked William Froude to immediately test the unusual ship model in the pool. Test results were not too pleasing to the pastor. Froude has concluded that, while sliding on water is possible in principle, Ramus ship will sink under the weight of the engine alone, without which it cannot develop the speed necessary for moving in sliding regime. 
         [0008]    The priest was of persistence character, and he soon offered another model of sliding vessel of 200-ton displacement, but it was also rejected because of want of suitable light engines. Ramos died before his idea came to practical fruition. 
         [0009]    Thirteen years passed, and one of the pioneers of aviation in France, engineer Charles D&#39;Alembert, turned to the idea of Ramus. D&#39;Alembert built a ship of unusual design that rested on four barrels, united by a common frame. Under the barrels—across the vessel obliquely to the surface of the water—four boards were placed, one after the other, on which the vessel was to be supported while moving fast. The unusual design was connected to a team of horses by a sling and had a tow. The experiment was a success: the ship floated up and started gliding over the surface of the water. 
         [0010]    In 1897, D&#39;Alembert tested the first self-propelled sliding vessel, which consisted of two canoes connected by four transverse frames. Under the bottom of the canoes, were fastened four pairs of boards with adjustable angles to the surface of the water. The platform housed the boiler and steam engine, while screw-propeller served as propulsion device. The result has exceeded all expectations: the ship glided over the water at about 20 knots. Building on his success, D&#39;Alembert builds in 1905 a gliding boat with a gasoline engine. Vessels of this type are called “glissers” (from the French word glisser- to slide). 
         [0011]    The glisser principle is based on the inertia of liquid particles; if used properly, it can reduce the resistance of water significantly. Recall how a flat stone, thrown at a small angle to water surface, ricochets. The force, which prevents the pebble from sinking immediately, is but the reaction of water due to the inertia of water particles. 
         [0012]    A similar phenomenon occurs when driving a glisser with a flat bottom at high speed. Under the hull, a force is formed lifting it out of the water which results in dramatic reduction of the water drag. The higher the speed, the more the hull will raise out of the water. 
         [0013]    When standing still, or moving at a very slow rate, the glisser is held at the surface of water by buoyancy alone. With the increase in speed, the moment comes when the inertia of the water particles comes into play, and, in addition to buoyancy, a significant hydrodynamic force builds up at the bottom of the hull. This force, which is directed at almost right angle to the bottom, can be decomposed into two components: one acting vertically upwards, and the other—horizontally in the direction opposite to the vessel&#39;s direction of motion. The first component is known as hydrodynamic lift; combined with the buoyancy force, it counterbalances the weight of the vessel and keeps it on the surface. The other component is the water drag. 
         [0014]    When the speed becomes large, buoyancy lift will be negligible compared to the hydrodynamic lift; water displacement volume goes down accordingly, and the vessel lifts up. Planing begins when the lifting force, which holds the vessel on the surface, becomes almost entirely hydrodynamic. 
         [0015]    The lack (or rather, a small quantity) of wave resistance to the—as if lifted above the water—glisser allows it to achieve speeds much higher than that of a boat of equal displacement and engine power. However, the high speeds are achieved by glissers at high expenditure of power which, in turn, requires powerful yet compact and lightweight engines. Since the ship propeller is fully submerged, a significant share of engine power is wasted stirring up the water, which could be avoided if the propulsion device operated on the surface of water, rather than under the water, yet deriving the required thrust from water reaction alone. This can be achieved by putting the vessel on wheels designed specifically to that end, basically transforming the vessel into a car or heavy truck running on water—a novel type of maritime transport. 
         [0016]    “Give an estimate of the magnitude of the speed, at which a person has to run on water so as not to sink.” This was one of the famous problems P. L. Kapitsa (a Russian physicist, who discovered in 1937 the effect of superfluidity of the liquid helium for which he was later awarded the Nobel Prize in Physics) came up with in the 1960-s for testing his postgraduate students ability for creative thinking. 
         [0017]    Is it indeed feasible to run on water? Short of miracle, obviously no man can do this. Most animals that locomote on water are small insects, whose long limbs deform the water surface to generate surface tension forces to support their body weight. Vertebrates are too dense to support their bodies above the water surface by surface tension alone, and must instead generate sizable hydrodynamic forces to run on water. Under these circumstances, body mass plays a primary role in determining the difficulty of this feat. 
         [0018]    Apart from insects, very few living creatures can run on water. Basilisk lizard, known also as Jesus Christ lizard, is one of them. Basilisk lizards are rather lightweight creatures: females are generally 135 to 194 g, and weigh half as much as males. Younger basilisks can run 10 to 20 meters on water, while adults cross only a few meters before sinking. It shall be noted also that basilisks have to gather sufficient momentum on the ground, before jumping into the water, to pull the trick. 
         [0019]    Few other vertebrates can run on water. The largest animals to accomplish this feat are western and Clark&#39;s grebes. Grebes weigh by an order of magnitude more than the next largest water runners, basilisk lizards, and therefore face a greater challenge to support their body weight. How do these birds produce the hydrodynamic forces necessary to overcome gravity and sustain rushing? The results of the first quantitative study of water running by grebes were presented recently in an article “Western and Clark&#39;s grebes use novel strategies for running on water” ( The Journal of Experimental Biology  (2015) 218, pp. 1235-1243) by Glenna T. Clifton, Tyson L. Hedrick and Andrew A. Biewener. 
         [0020]    If the heavy grebes can run on water, there is no good reason why the same cannot be achieved by technical devices specifically designed for that purpose. This invention is a proof that water skimming vessels on wheels are feasible. Most importantly, wheeled watercrafts running on the surface of water can surpass planing vessels in both speed and fuel efficiency. A watercraft, running on wheels over the rivers, seas and oceans, has the potential of becoming a highly competitive maritime vehicle for fast transportation of small groups of people and commercial merchandise. 
       BRIEF SUMMARY OF THE INVENTION 
       [0021]    The invention presents a watercraft on wheels for running on the surface of water as a novel type of maritime transport. Thrust and lift-generating technique employed by the wheeled watercraft, the invention of which was inspired by western and Clark&#39;s grebes, follows partially the mechanism used by these large birds to run on water. Unlike basilisk lizards, which have to gather sufficient momentum on the ground before jumping into the water and running at the surface, grebes achieve this feat starting right from the water. Besides, grebes are by order of magnitude heavier basilisk lizards. These two factors make it necessary for grebes to use qualitatively different water running technique. 
         [0022]    As shown in the quoted above paper by Clifton et al., grebes use three novel tactics to successfully run on water. First, rushing grebes use exceptionally high stride rates, reaching 10 Hz. Second, grebe foot size and high water impact speed allow grebes to generate up to 30-55% of the required weight support through water slap alone. Finally, flattened foot bones reduce downward drag, permitting grebes to retract each foot from the water laterally. 
         [0023]    The key part of this invention is the wheel ( FIG. 3 ) that models grebe&#39;s foot movement to generate both forward thrust and weight-balancing lift. Since the wheels can rotate at rates much higher than the stride rate of grebes, water skimming vessel on wheels can generate both the required lift and thrust, with only small areas of the wheels—or any other parts of the vessel—immersed. This important technical advantage over the grebes allows improving and simplifying enormously on the mechanism they use to rush on water, for it makes the final, and the most complicated, part of the grebe&#39;s water running tactics—i.e. retraction of the immersed feet from the water laterally in order to reduce the downward draft—simply irrelevant to the design of the wheel. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0024]      FIG. 1  depicts side view of the wheeled watercraft. Supporting plates,  4 , are mounted on struts,  5 , under the hull,  3 , with long sides of the plates oriented across the hull. At the lower end, the struts are connected solidly to the plates at right angles, and hinged to the hull at the upper end. These struts are making forward angle, α, to the vertical line, thus defining plates&#39; angle of attack. The wheels,  1 , are also mounted on struts,  2 , which are forming backward angle, −α, to the vertical line. 
           [0025]      FIG. 2  depicts front view of the wheeled watercraft. Each of the four wheels,  1 , is mounted on a strut,  2 , under the hull,  3 , while each of the four supporting plates,  4 , is mounted on a couple of struts,  5 . The struts are hydraulic, which can be lengthened and shortened as needed. When the vessel is parked, or traveling through calm seas, the struts are completely collapsed; in rough seas, they are extended to lift the hull into the air, eliminating the jarring impact of waves. 
           [0026]      FIG. 3  depicts the principal design of the wheel and its geometrical characteristics. On the base circumference,  6 , of the wheel there is a number of tooth-like projections so the wheel looks like a solid unit comprised of N (12≦N≦24) “feet”. In cruising mode, strut lengths and their orientations are controlled in such a way that the lower points on the outer circumference,  7 , of each wheel and the lower edges of the plates are on the same horizontal level below the water line,  10 , making immersion depth equal to the tooth height (R−r). All forward thrust, as well as some share of upward lift, is generated by hydrodynamic reaction of water to roll-and-slide motion of the wheels; the rest of the lift that is necessary for counterbalancing the weight of the vessel is generated by hydrodynamic forces exerted on supporting plates, which are lifted out of the water and gliding at the surface. 
           [0027]      FIG. 4  depicts instant angular disposition,  6 , of the wheel with respect to the water line,  10 . Net hydrodynamic force, exerted on each wheel at each instant of time, varies with the angle θ, which determines the instant water-wheel contact configuration. When cruising at top speed, the depth of wheel&#39;s immersion does not exceed h=R−r, so only the lower couple of feet ( 8 —front foot, and  9 —back foot) of each wheel fall below the water line,  10 . 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0028]    The forward thrust and upward lift, required for running the watercraft on the surface of water, are generated by four wheels of special design, which are rotating fast, and four plates with adjustable angle of attack. Plates are mounted on struts—under and across the hull—one after the other. At the lower end, the struts are connected solidly to the plates at right angles and hinged to the hull at the upper end, making forward angle, α, to the vertical line, and thus determining plates&#39; angle of attack ( FIG. 1 ). Each wheel is also mounted on a strut that makes backward angle, −α, to the vertical line. The struts are hydraulic and can be lengthened and shortened as needed. When the vessel is parked, or traveling through calm seas, the struts are completely collapsed; in rough seas, they are extended to lift the hull into the air, eliminating the jarring impact of waves. The orientation and overall disposition of plate and wheel struts, with respect to each other and to the position of center of gravity of the hull ( FIG. 1  and  FIG. 2 ), are arranged keeping two primary goals in mind: to confine the strut stresses to compression, minimizing bending component of stress as much as possible, and to gain self-stabilizing interplay of all forces acting on the watercraft. 
         [0029]    On the base circumference of each wheel there is a number of tooth-like projections so the wheel looks like a rigid unit comprised of N (12≦N≦24) “feet” ( FIG. 3 ). In cruising mode, strut lengths and their orientations are controlled in such a way that the lower points on the outer circumference of each wheel and the lower edges of the plates are on the same horizontal level below the water line, making immersion depth equal to the tooth height. While at rest, buoyancy keeps the vessel afloat. When cruising at top speed, the depth of wheel&#39;s immersion does not exceed the tooth height, thus, only the lower couple of feet ( FIG. 4 ) of each wheel fall below the water line. All thrust, and some share of upward lift, is generated by hydrodynamic reaction of water to roll-and-slide motion of the wheels; the rest of the lift, which is necessary to counterbalance the total weight of the vessel, is generated by hydrodynamic forces exerted on the supporting plates, which are lifted out of the water and gliding at the surface. That is, buoyancy plays no role in keeping the vessel above the water when cruising. 
         [0030]    The difficult part of analyzing the motion of the wheeled vessel is, of course, to model the reaction force, which results from the flow around the immersed parts of the wheels and supporting plates. Here we follow the model that was introduced in Ricochet off Water, AMP Memo No. 42.4M (1944) by G. Birkhoff, G. D. Birkhoff, W. E. Bleick, E. H. Handler, F. D. Murnaghan, and T. L. Smith. The model is based on the following assumptions.
       I. Hydrodynamic pressure exerted on the surface element in contact with water is proportional to (vn) 2 , where v is the velocity vector and n is the unit vector to the surface element.   II. For the part of the surface facing air, there is no hydrodynamic force.   III. During the whole process, deformation of water surface is negligible, and the boundary between immersed and non-immersed areas is simply given as the cross section to a horizontal plane at water level.       
 
         [0034]    Reynolds number associated with running on water is rather large (Re˜10 7 ), which makes the validity of these assumptions more credible. 
       Hydrodynamic Forces Exerted on the Wheel 
       [0035]    First, we consider hydrodynamic forces exerted on the wheels. When the watercraft is in cruising mode, only the lower couple of feet of each wheel are immersed and interacting with water. Instant hydrodynamic force, exerted on each wheel at each moment of time, varies with the angle θ that determines the instant water-wheel contact configuration ( FIG. 4 ). We&#39;ll derive below an analytical expression for each component of this instant reaction force, which factors out the dependence of the force on the angle θ, the cruising speed u, (or rather dimensionless speed: η≡u/(ωr)), and the count N of feet on one wheel. Since the wheel rotation rate is quite high (dozens of Hz), it is, evidently, justified to average the factor for each force component over the variable θ and consider it as a function of one variable only, η, when the value of the parameter N is fixed.
       The list of notations, to be used in hydrodynamic force derivation, is given below:   r—radius of the base circumference of the wheel;   R—radius of the outer circumference of the wheel;   h—depth of vessel immersion when cruising (h=R−r);   a—width of the wheel;   N—number of feet on each wheel (12≦N≦24);       θ—instant angular disposition of the wheel (0≦θ≦2π/N);
       ω—angular velocity of wheel rotation;   α—supporting plates&#39; angle of attack (same as plate struts&#39; declination from the vertical);   b—longitudinal size of supporting plates (i.e. in the direction across the hull);   ρ—water density;   ρ a —air density;   C D —drag coefficient;   A—vessel&#39;s reference area (i.e. projected frontal area);   u—cruising speed of the vessel;   η—dimensionless version of the cruising speed (η≡u/(ωr).   
       
 
         [0052]    For the hydrodynamic reaction to the roll-and-slide motion of each of the 4 wheels we have: 
         [0000]        F   (w) =−∫(½)ρ( vn ) 2   nC   D   dS,  
 
         [0053]    where n is the normal vector to the contact surface, v is the velocity vector of base points of the foot (sole points) hitting the water, and dS is the area element of the foot base ( FIG. 3 ): 
         [0000]        dS=ar[tg (φ+ d φ)− tg (φ)]= ardφ.  
 
         [0054]    Integration is to be done over those areas of the base of each immersed foot, which are not only above the water line but also pushing water particles, rather than retreating from them. 
         [0055]    In this simplified hydrodynamic force model, we neglect the skin friction because viscous component of the force is small compared to the normal hydrodynamic pressure exerted at the contact surface. 
         [0056]    For the flat base of immersed in water front foot of each wheel we have: 
         [0000]        n   x =−sin(θ),
 
         [0000]        n   y =−cos(θ),
 
         [0000]        v   x   =ωr [−cos(θ)− tg (φ)sin(θ)+η],
 
         [0000]        v   y   =ωr [sin(θ)− tg (φ)cos(θ)],
 
         [0057]    where η≡u/(ωr) is a dimensionless cruising velocity. 
         [0058]    Therefore, for the components of hydrodynamic force exerted on the front-foot base of the wheel we get: 
         [0000]        F   (f)   x =(½)ρ(ω r ) 2   arC   D  sin(θ) (f) ,
 
         [0000]        F   (f)   y =(½)ρ(ω r ) 2   arC   D  cos(θ) I   (f) ,
 
         [0059]    where factor I (f)  stands for the following integral: 
         [0000]        I   (f) =[1/(ω r ) 2 ]∫( vn ) 2   dφ.  
 
         [0060]    Carrying out integration over the areas of the front-foot base that fall under the water line (segment [θ/2, 2π/N]), we get the following analytical expression for the integral: 
         [0000]        I   (f)   =[tg (2π/ N )− tg (θ/2)−2π/ N+θ/ 2]+2η[ ln(cos(2π/ N ))−ln(cos(θ/2))] sin(θ)+η 2 (2π/ N−θ/ 2)sin 2 (θ).
 
         [0061]    Similarly, for the flat base of the immersed in water back foot of each wheel we have: 
         [0000]        n   x =−sin(θ+2π/ N ),
 
         [0000]        n   y =−cos(θ+2π/ N ),
 
         [0000]        v   x   =ωr [−cos(θ+2π/ N )− tg (φ)sin(θ+2π/ N )+η],
 
         [0000]        v   y   =ωr [sin(θ+2π/ N )− tg (φ)cos(θ+2π/ N )].
 
         [0062]    Therefore, for the components of the force exerted on the base of each back foot we get: 
         [0000]        F   (b)   x =(½) arpC   D (ω r ) 2  sin(θ+2π/ N ) I   (b) ,
 
         [0000]        F   (b)   y =(½) arpC   D (ω r ) 2  cos(θ+2π/ N ) I   (b) ,
 
         [0063]    where factor I (b)  stands for the following integral: 
         [0000]        I   (b) =[1/(ω r ) 2 ]∫( vn ) 2   dφ.  
 
         [0064]    Carrying out integration over the areas of the back-foot base that fall under the water line (segment [π/2+π/N, 2π/N]), we get the following analytical expression for the integral: 
         [0000]        I   (b)   =tg (2π/ N )− tg (θ/2+π/ N )−2π/ N+θ/ 2+π/ N+ 2η[ ln(cos(2π/ N ))−ln(cos(θ/2+π/ N ))] sin(θ+2π/ N )+η 2 (2π/ N−θ/ 2−π/ N )sin 2 (θ+2π/ N ).
 
         [0065]    Total forward thrust generated by all wheels is given then by: 
         [0000]        T   (w) =4( F   (f)   x   +F   (b)   x )=2ρ(ω r ) 2   arC   D   K   x (η,θ; N ),
 
         [0000]      where 
         [0000]        K   x (η,θ; N )={ tg (2π/ N )− tg (θ/2)−2π/ N+ θ/2+2η[ ln(cos(2π/ N ))−ln(cos(θ/2))] sin(θ)+η 2 (2π/ N−θ/ 2)sin 2 (θ)} sin(θ)+{ tg (2π/ N )− tg (θ/2+π/ N )−2π/ N+θ/ 2+π/ N+ 2η[ ln(cos(2π/ N ))−ln(cos(θ/2+π/ N ))] sin(θ+2π/ N )+η 2 (2π/ N−θ/ 2−π/ N )sin 2 (θ+2π/ N )} sin(θ+2π/ N ).
 
         [0066]    And the combined upward lift from the four wheels is given by: 
         [0000]        L   (w) ==4( F   (f)   y   +F   (b)   y )=2ρ(ω r ) 2   arC   D   K   y (η,θ; N ),
 
         [0000]      where 
         [0000]        K   y (η,θ; N )={ tg (2π/ N )− tg (θ/2)−2π/ N+ θ/2+2η[ ln(cos(2π/ N ))−ln(cos(θ/2))] sin(θ)+η 2 (2π/ N−θ/ 2)sin 2 (θ)} cos(θ)+{ tg (2π/ N )− tg (θ/2+π/ N )−2π/ N+ θ/2+π/ N+ 2η[ ln(cos(2π/ N ))−ln(cos(θ/2+π/ N ))] sin(θ+2π/ N )+η 2 (2π/ N−θ/ 2−π/ N )sin 2 (θ+2π/ N )} cos(θ+2π/ N ).
 
         [0067]    It is important to recall now that the above calculations are based on the premise that those areas of each foot base that fall below the water line are pushing water particles, rather than retreating from them. This puts a restriction on the possible values of the dimensionless cruising speed q: 
         [0000]      η≦η*(θ) for all θ in the range [0,4π/ N].  
 
         [0068]    The critical function η′(θ) is, obviously, determined by the condition: 
         [0000]      ( n   x   v   x   +n   y   v   y )| φ=θ/2 =0, 
         [0000]    from which we get: 
         [0000]      η*(θ)= tg (θ/2)/sin(θ),0≦θ≦4π/ N.  
 
         [0069]    Therefore, η max =min(tg(θ/2)/sin(δ))=0.5, i.e. u≦0.5ωr: Translational velocity of the vessel cannot exceed half the linear velocity of points on the base circumference of the wheel. 
         [0070]    Next, using one of the numerical integration utilities available online (for instance, the one at http://www.zweigmedia.com/RealWorld/integral/integral.htnml), we average functions K x (η, θ; N) and K y (η, θ; N) over θ in the range [0, 2π/N], while fixing the cruising speed at its maximum possible value η=0.5. For instance, in the case of N=16, we get: 
         [0000]      Avg[ K   x (0.5,θ;16)]≈0.00194,
 
         [0000]      Avg[ K   y (0.5,θ;16)]≈0.01121.
 
         [0071]    The total thrust, and the total lift generated by all four wheels are given then as: 
         [0000]        T   (w) =2ρ(ω r ) 2   arC   D Avg[ K   x (0.5,θ; N )],
 
         [0000]        L   (w) =2ρ(ω r ) 2   arC   D Avg[ K   y (0.5,θ; N )].
 
         [0072]    To put it differently, the lift to thrust ratio, generated by the wheels when the watercraft is cruising at top speed, is determined entirely by the ratio of averaged over θ values of the functions K y  and K x . In the case of N=16, for example, the wheels generate about six times more lift than thrust: 
         [0000]        L   (w)   /T   (w) | η=0.5;N=16 ≈5.8.
 
         [0073]    In general, this ratio is too small to support the loaded weight of the vessel. The rest of the lift, required to counterbalance the weight of the cruising vessel, without counting on buoyancy, is generated by hydrodynamic forces exerted on the four supporting plates. 
       Hydrodynamic Forces Exerted on Supporting Plates 
       [0074]    Let us now calculate the lift and drag associated with supporting plates: 
         [0000]        F   (p) =−∫(½)ρ( vn ) 2   nC   D   dS  
 
         [0075]    For each of the four plates with adjustable angle of attack, α, we have: 
         [0000]        n   x =sin(α),
 
         [0000]        n   y =−cos(α),
 
         [0000]    
       
      
       v 
       x 
       =u,  
      
     
         [0000]        v   y =0. 
         [0076]    In cruising regime, when all plates are lifted out of the water and sliding at the surface, integration is to be done over the line segment [0, h/sin(α)], where h=r[1/cos(2π/N)−1] is the depth of immersion of the lower edge of the plate. 
         [0077]    Integration is straightforward, and for the components of the hydrodynamic force exerted on each supporting plate we get: 
         [0000]        F   x   (p) =−(½)ρ u   2   bhC   D  sin 2 (α),
 
         [0000]        F   y   (p) =(½)ρ u   2   bhC   D  sin(α)cos(α).
 
         [0078]    For the total drag and the total lift from all four plates we have: 
         [0000]        D   (p) =4 F   x   (p) =−2ρ u   2   bhC   D  sin 2 (α)=−2ρ(ω r η) 2   bhC   D  sin 2 (α),
 
         [0000]        L   (p) =4 F   y   (p) =2ρ u   2   bhC   D  sin(α)cos(α)=2ρ(ω r η) 2   bhC   D  sin(α)cos(α).
 
         [0079]    From this it follows that the drag/lift ratio for supporting plates is determined by the angle of attack, α, alone and, therefore, it can be controlled easily by controlling this angle. For example, with angle of attack at 5°, for each unit of drag on supporting plates we get over 11 units of lift: 
         [0000]      | L   (p)   /D   (p) |=1/ tg (5 0 )≈11.4.
 
         [0080]    Since the foot bases and supporting plate surfaces are all flat, it is reasonable to assume unity for the numerical value of the drag coefficient in all calculations above: C D ≈1. 
       Aerodynamic Drag on the Watercraft 
       [0081]    Finally, to complete the force balance analysis for the vehicle in cruising mode, all is left is to consider the aerodynamic drag on the fast running vehicle: 
         [0000]        D   (a) =−(½)ρ a (ω r η) 2   C   D   A,  
 
         [0082]    where ρ a  is the air density, A is the reference area (projected frontal area of the vehicle), and C D  is the aerodynamic drag coefficient, which is typically around 0.3 for ground transportation vehicles like automobiles. 
       Engine Horsepower Required for Driving the Wheeled Watercraft 
       [0083]    Calculating the power applied by the engine to the front foot of each wheel, we have: 
         [0000]        P   (f)   =−∫vdF =(½)ρ( vn ) 3   C   D   dS =(½)ρ(ω r ) 3   arC   D   J   (f) (η,θ; N ),
 
         [0084]    where J (f) (η, θ; N) is the following integral over the segment [θ/2, 2π/N]: 
         [0000]        J   (f) (η,θ; N )=∫[ tg (φ)−η sin(θ)] 3   dφ.  
 
         [0085]    Carrying out integration, we get the following expression for the function J (f) (η, θ; N): 
         [0000]        J   (f) (η,θ; N )=1/[2 cos 2 (2π/ N )]+ln [ cos(2π/ N )]−1/[2 cos 2 (θ/2)]−ln [ cos(θ/2)]
 
         [0000]      −3η sin(θ)[ tg (2π/ N )− tg (θ/2)−2π/ N+ θ/2]
 
         [0000]      −3η 2  sin 2 (θ)[ ln(cos(2π/ N ))−ln(cos(θ/2))]
 
         [0000]      −η 3  sin 3 (θ)(2π/ N−θ/ 2).
 
         [0086]    Similarly, for the power applied to the back foot of each wheel we have: 
         [0000]        P   (b)   =−∫vdF =(½)ρ( vn ) 3   C   D   dS =(½)ρ(ω r ) 3   arC   D   J   (b) (η,θ; N ),
 
         [0087]    where J (b) (η, θ; N) is the following integral over the segment [θ/2+π/N, 2π/N]: 
         [0000]        J   (b) (η,θ; N )=∫[ tg (φ)−η sin(θ)] 3   dφ.  
 
         [0088]    Carrying out integration, we get a corresponding analytical expression for the integral: 
         [0000]        J   (b) (η,θ; N )=1/[2 cos 2 (2π/ N )]+ln [ cos(2π/ N )]−1/[2 cos 2 (θ/2+π/ N )]−ln [ cos(θ/2+π/ N )]
 
         [0000]      −3η sin(θ+2π/ N )[ tg (2π/ N )− tg (θ/2+π/ N )−2π/ N+θ/ 2+π/ N] 
 
         [0000]      −3η 2  sin 2 (θ+2π/ N )[ ln(cos(2π/ N ))−ln(cos(θ/2+π/ N ))]
 
         [0000]      −η 3  sin 3 (θ+2π/ N )(2π/ N−θ/ 2−π/ N ).
 
         [0089]    Instant horsepower required for driving all four wheels then is: 
         [0000]        P= 4( P   (f)   +P   (b) )=2ρ(ω r ) 3   arC   D   J (η,θ; N ),
 
         [0090]    where factor J(η, θ; N) is the following long expression 
         [0000]        J (η,θ; N )≡ J   (f) (η,θ; N )+ J   (b) (η,θ; N )=
 
         [0000]      1/[2 cos 2 (2π/ N )]+ln [ cos(2π/ N )]−1/[2 cos 2 (θ/2)]−ln [ cos(θ/2)]
 
         [0000]      −3η sin(θ)[ tg (2π/ N )− tg (θ/2)−2π/ N+ θ/2]
 
         [0000]      −3η 2  sin 2 (θ)[ ln(cos(2π/ N ))−ln(cos(θ/2))]
 
         [0000]      −η 3  sin 3 (θ)(2π/ N− θ/2)
 
         [0000]      +1/[2 cos 2 (2π/ N )]+ln [ cos(2π/ N )]−1/[2 cos 2 (θ/2+π/ N )]−ln [ cos(θ/2+π/ N )]
 
         [0000]      −3η sin(θ+2π/ N )[ tg (2π/ N )− tg (θ/2+π/ N )−2π/ N+ θ/2+π/ N] 
 
         [0000]      −3η 2  sin 2 (θ+2π/ N )[ ln(cos(2π/ N ))−ln(cos(θ/2+π/ N ))]
 
         [0000]      −η 3  sin 3 (θ+2π/ N )(2π/ N− θ/2−π/ N ).
 
         [0091]    Averaging the function J(0.5, θ; N) over θ in the range [0, 2π/N], we finally find the engine horsepower required for driving the wheeled watercraft at top cruising speed η=0.5: 
         [0000]        P= 4( P   (f)   +P   (b) )=2ρ(ω r ) 3   arC   D Avg[ J (0.5,θ; N )].
 
         [0092]    For example, in the case of N=16, we have: 
         [0000]      Avg[ J (0.5,θ;16)]≈0.00216,
 
         [0093]    with the corresponding engine horsepower at P=0.00432ρ(ωr) 3 arC D . 
       A Note on Cavitation 
       [0094]    Using hydrofoils (i.e. lifting surfaces or foils, similar in appearance and purpose to airfoils, but operating in water) is a technology that is used to reduce the drag. Boats that use this technology are simply called hydrofoils. As a hydrofoil craft gains speed, the hydrofoils lift the boat&#39;s hull out of the water, decreasing drag and allowing greater speeds. However, boat designers had faced an engineering phenomenon that puts a cap on speed for hydrofoils—cavitation. Hydrodynamic cavitation describes the process of vaporization, bubble generation and bubble implosion that occurs in a flowing liquid as a result of a decrease, and subsequent increase, in local pressure. In addition to the damaging effect that cavitation has on ship propellers, due to the bubble implosion on the surfaces of propeller blades, it disturbs the lift created by the foils as they move through the water at speeds above 60 knots (110 km/h), bending the lifting foil. 
         [0095]    Cavitation will only occur if the local pressure declines to some point below the saturated vapor pressure of the liquid and subsequent recovery above the vapor pressure. Since the watercraft running on the surface of water is using neither ship propellers for generating thrust, nor fully submerged hydrofoils for generating lift, cavitation simply does not occur when the wheeled watercraft is cruising at top speed, therefore, cavitation is not a speed limiting factor for the advanced here technology. 
       Potential for Military Application 
       [0096]    Lightweight watercraft on wheels, running fast on water, is an ideal offshore border patrol vehicle. As will be shown in the case study below, a wheeled watercraft weighing about 12,000 pounds and equipped with a regular diesel or gasoline engine of 675 horsepower only, can cruise at 106 knots. Since the water-running vehicle does not use expensive jet-propulsion system in order to reach high cruising speeds, or powerful thousands horsepower engines to drive submerged ship propellers, it can serve as a fast, reliable, fuel efficient and inexpensive torpedo boat. In addition to that, a low profile torpedo boat on wheels, with disturbance of water confined mostly to the surface layer, will be next to impossible for on-time detection by sonar before it strikes, and—if made of non-magnetic materials—it will be of little or no radar signature. 
         [0097]    By proper selection of design parameters, carrying capacity of the wheeled watercraft can be increased to the point of making it a vessel platform that will assure naval force protection through stealth attack capabilities along with integrated situation awareness. These vessels would create a protective fleet perimeter around aircraft carriers, providing sensor and weapons platforms capable of preventing surface or subsurface attacks. 
         [0098]    These are the same military uses for which the Ghost—a proprietary technology vessel developed by Juliet Marine Systems—is touted. One of the main ideas behind this sophisticated technology is using the supercavitation effect to reduce the skin friction drag, and thereby achieve high speeds. But skin friction is not the main component of the overall drag—the form component of the drag is, by far, the most significant part of hydrodynamic resistance to a watercraft cruising at high speed. The fact that the Ghost achieved only 29 knots so far—way below the 60 knots speed limit imposed by cavitation phenomenon on hydrofoils—is, perhaps, indirect proof of this. Even the 50 knots—the projected speed assumed to be within the reach of the Ghost—is still below the speed limit that can be achieved by ordinary hydrofoils. 
         [0099]    For the wheeled watercraft, skin friction between water and the wheels is not an adverse factor to be avoided. What is more, just like the grip of a tire on a road or a wheel on a rail, the friction of the wheel on water amounts to improved traction that should be enhanced—if possible—not avoided. The skin friction, which takes place on the surface of supporting plates in contact with water, serves no good purpose, of course, but trying to reduce it is not worth the trouble. 
         [0100]    Reliability and cost considerations are of the primary importance when evaluating any new technology. The more complex the design of the craft, the less reliable and more costly it usually is. The Ghost is estimated roughly at $10 million apiece. Due to the sheer simplicity of the main idea behind the water-running vessel on wheels—which is generating the entire thrust and part of the lift from wheels, rather than from ship propellers—and the matching simplicity of the principal design for the wheel that makes water running feasible, the wheeled watercraft can achieve the same military ends for a small fraction of the Ghost&#39;s price tag. 
       Design Case Study 
       [0101]    To get a clear idea of the magnitude of hydrodynamic and other force components (thrust, drag, lift, aerodynamic drag, vehicle weight) exerted on the watercraft when cruising at top speed, we consider next a case study with a characteristic set of values for the involved parameters. The list of input parameters is given in Table 1, and the results of calculations are presented in Table 2. 
         [0000]    
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Design Input Parameter 
                 Symbol 
                 Value 
                 Dimension 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Number of feet on one wheel 
                 N 
                 16 
                 — 
               
               
                 Base radius of the wheel 
                 r 
                 0.350 
                 m 
               
               
                 Outer radius of the wheel 
                 R 
                 0.379 
                 m 
               
               
                 Width of the wheel 
                 a 
                 0.250 
                 m 
               
               
                 Water density 
                 ρ 
                 1000 
                 kg/m 3   
               
               
                 Air density 
                 ρ a   
                 1.23 
                 kg/m 3   
               
               
                 Immersion depth at cruising speed 
                 h 
                 0.029 
                 m 
               
               
                 Wheel rotation rate 
                 — 
                 50 
                 Hz 
               
               
                 Angular velocity of wheel rotation 
                 ω 
                 314 
                 rad/s 
               
               
                 Watercraft cruising speed 
                 u 
                 55 
                 m/s 
               
               
                 Dimensionless cruising speed 
                 η 
                 0.5 
                 — 
               
               
                 Longitudinal size of supporting plate 
                 b 
                 2 
                 m 
               
               
                 Projected frontal area of the watercraft 
                 A 
                 3 
                 m 2   
               
               
                 Drag coefficient for 
                 C D   
                 1 
                 — 
               
               
                 immersed flat surfaces 
               
               
                 (foot bases and supporting plates) 
               
               
                 Aerodynamic drag coefficient 
                 C D   
                 0.3 
                 — 
               
               
                 for the watercraft 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
               
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Output 
                 Symbol 
                 Formula 
                 Calculated As 
                 Value 
               
               
                   
               
             
             
               
                 Net thrust from the 
                 T (w)   
                 2ρ(ωr) 2 arC D Avg 
                 0.00388ρ(ωr) 2 ar 
                 4108N 
               
               
                 wheels 
                   
                 [K x (η, θ; N)] 
               
               
                 Net lift from the 
                 L (w)   
                 2ρ(ωr) 2 arC D Avg 
                 0.0224ρ(ωr) 2 ar 
                 23716N  
               
               
                 wheels 
                   
                 [K y (η, θ; N)] 
               
               
                 Aerodynamic drag on 
                 D (a)   
                 0.5ρ a (ωrη) 2 C D A 
                 0.15ρ a (0.5ωr) 2 A 
                 1674N 
               
               
                 the cruising vehicle 
               
               
                 Net drag from 
                 D (p)   
                 2ρ(ωrη) 2   
                 T (w)  − D (a)   
                 2434N 
               
               
                 supporting plates 
                   
                 bhC D sin 2 (α) 
               
               
                 Supporting plates 
                 α 
                 — 
                 arcsin(✓(D (p) / 
                 4.8° 
               
               
                 angle of attack 
                   
                   
                 (2ρ(0.5ωr) 2 bh))) 
               
               
                 Net lift from 
                 L (p)   
                 2ρ(ωrη) 2 bhC D   
                 2ρ(0.5ωr) 2 bh 
                 29260N  
               
               
                 supporting plates 
                   
                 sin(α)cos(α) 
                 sin(α)cos(α) 
               
               
                 Loaded weight of the 
                 W 
                 — 
                 L (w)  + L (p)   
                 52976N 
               
               
                 cruising vehicle 
                   
                   
                   
                 (≈11905 lb) 
               
               
                 Required engine 
                 P 
                 2ρ(ωr) 3 arC D Avg 
                 0.00432ρ(ωr) 3 ar 
                 503 KW 
               
               
                 horsepower 
                   
                 [J(η, θ; N)] 
                   
                 (≈675 hp) 
               
               
                   
               
             
          
         
       
     
         [0102]    Thus, a wheeled watercraft weighing 11,905 pounds and equipped with a 675-horsepower conventional diesel or gasoline engine, is capable of running on water at cruising speed of 198 km/h (106 knots). This is both a significant gain in speed and a marked reduction in engine horsepower required to drive a vessel of that size at that speed, which is achieved by propulsion technology hitherto unknown for waterborne crafts: using hydrodynamic reaction of water to roll-and-slide motion of wheels that are specifically designed to allow the wheeled watercraft to run on the surface of water, like a regular automobile runs on the ground.