Wheeled watercraft running on the surface of water

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.

Claims

1. A wheeled watercraft capable of running on water, which constitutes a novel maritime vehicle for fast transportation of small groups of people and commercial merchandise over the surface of rivers, seas, and oceans, wherein the thrust and part of the lift are generated by hydrodynamic reaction of water to roll-and-slide motion of the wheels, and the rest of the lift required to counterbalance the loaded weight of the cruising vehicle is generated by hydrodynamic forces exerted on four plates with adjustable angle of attack, which are mounted on hydraulic struts below the hull, allowing the hull to be lifted above the water line as high as demanded by sea weather conditions.

2. A principal design of the wheel of claim 1, which allows generating all the thrust required for running on the surface of water entirely from hydrodynamic forces exerted on the flat bases of wheels feet.

Description

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' 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'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' 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'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. [0031] I. Hydrodynamic pressure exerted on the surface element in contact with water is proportional to (vn).sup.2, where v is the velocity vector and n is the unit vector to the surface element. [0032] II. For the part of the surface facing air, there is no hydrodynamic force. [0033] 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.sup.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'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. [0036] The list of notations, to be used in hydrodynamic force derivation, is given below: [0037] r—radius of the base circumference of the wheel; [0038] R—radius of the outer circumference of the wheel; [0039] h—depth of vessel immersion when cruising (h=R−r); [0040] a—width of the wheel; [0041] N—number of feet on each wheel (12≦N≦24); [0042] θ—instant angular disposition of the wheel (0≦θ≦2π/N); [0043] ω—angular velocity of wheel rotation; [0044] α—supporting plates' angle of attack (same as plate struts' declination from the vertical); [0045] b—longitudinal size of supporting plates (i.e. in the direction across the hull); [0046] ρ—water density; [0047] ρ.sub.a—air density; [0048] C.sub.D—drag coefficient; [0049] A—vessel's reference area (i.e. projected frontal area); [0050] u—cruising speed of the vessel; [0051] 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:

F.sup.(w)=−∫(½)ρ(vn).sup.2nC.sub.DdS,

[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):

dS=ar[tg(φ+dφ)−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:

n.sub.x=−sin(θ),

n.sub.y=−cos(θ),

v.sub.x=ωr[−cos(θ)−tg(φ)sin(θ)+η],

v.sub.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:

F.sup.(f).sub.x=(½)ρ(ωr).sup.2arC.sub.D sin(θ).sup.(f),

F.sup.(f).sub.y=(½)ρ(ωr).sup.2arC.sub.D cos(θ)I.sup.(f),

[0059] where factor I.sup.(f) stands for the following integral:

I.sup.(f)=[1/(ωr).sup.2]∫(vn).sup.2dφ.

[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:

I.sup.(f)=[tg(2π/N)−tg(θ/2)−2π/N+θ/2]+2η[ ln(cos(2π/N))−ln(cos(θ/2))] sin(θ)+η.sup.2(2π/N−θ/2)sin.sup.2(θ).

[0061] Similarly, for the flat base of the immersed in water back foot of each wheel we have:

n.sub.x=−sin(θ+2π/N),

n.sub.y=−cos(θ+2π/N),

v.sub.x=ωr[−cos(θ+2π/N)−tg(φ)sin(θ+2π/N)+η],

v.sub.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:

F.sup.(b).sub.x=(½)arpC.sub.D(ωr).sup.2 sin(θ+2π/N)I.sup.(b),

F.sup.(b).sub.y=(½)arpC.sub.D(ωr).sup.2 cos(θ+2π/N)I.sup.(b),

[0063] where factor I.sup.(b) stands for the following integral:

I.sup.(b)=[1/(ωr).sup.2]∫(vn).sup.2dφ.

[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:

I.sup.(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.sup.2(2π/N−θ/2−π/N)sin.sup.2(θ+2π/N).

[0065] Total forward thrust generated by all wheels is given then by:

T.sup.(w)=4(F.sup.(f).sub.x+F.sup.(b).sub.x)=2ρ(ωr).sup.2arC.sub.DK.sub.x(η,θ;N),

where

K.sub.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.sup.2(2π/N−θ/2)sin.sup.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.sup.2(2π/N−θ/2−π/N)sin.sup.2(θ+2π/N)} sin(θ+2π/N).

[0066] And the combined upward lift from the four wheels is given by:

L.sup.(w)==4(F.sup.(f).sub.y+F.sup.(b).sub.y)=2ρ(ωr).sup.2arC.sub.DK.sub.y(η,θ;N),

where

K.sub.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.sup.2(2π/N−θ/2)sin.sup.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.sup.2(2π/N−θ/2−π/N)sin.sup.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:

η≦η*(θ) for all θ in the range [0,4π/N].

[0068] The critical function η′(θ) is, obviously, determined by the condition:

(n.sub.xv.sub.x+n.sub.yv.sub.y)|.sub.φ=θ/2=0,

from which we get:

η*(θ)=tg(θ/2)/sin(θ),0≦θ≦4π/N.

[0069] Therefore, η.sub.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.sub.x(η, θ; N) and K.sub.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:

Avg[K.sub.x(0.5,θ;16)]≈0.00194,

Avg[K.sub.y(0.5,θ;16)]≈0.01121.

[0071] The total thrust, and the total lift generated by all four wheels are given then as:

T.sup.(w)=2ρ(ωr).sup.2arC.sub.DAvg[K.sub.x(0.5,θ;N)],

L.sup.(w)=2ρ(ωr).sup.2arC.sub.DAvg[K.sub.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.sub.y and K.sub.x. In the case of N=16, for example, the wheels generate about six times more lift than thrust:

L.sup.(w)/T.sup.(w)|.sub.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:

F.sup.(p)=−∫(½)ρ(vn).sup.2nC.sub.DdS

[0075] For each of the four plates with adjustable angle of attack, α, we have:

n.sub.x=sin(α),

n.sub.y=−cos(α),

v.sub.x=u,

v.sub.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:

F.sub.x.sup.(p)=−(½)ρu.sup.2bhC.sub.D sin.sup.2(α),

F.sub.y.sup.(p)=(½)ρu.sup.2bhC.sub.D sin(α)cos(α).

[0078] For the total drag and the total lift from all four plates we have:

D.sup.(p)=4F.sub.x.sup.(p)=−2ρu.sup.2bhC.sub.D sin.sup.2(α)=−2ρ(ωrη).sup.2bhC.sub.D sin.sup.2(α),

L.sup.(p)=4F.sub.y.sup.(p)=2ρu.sup.2bhC.sub.D sin(α)cos(α)=2ρ(ωrη).sup.2bhC.sub.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:

|L.sup.(p)/D.sup.(p)|=1/tg(5.sup.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.sub.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:

D.sup.(a)=−(½)ρ.sub.a(ωrη).sup.2C.sub.DA,

[0082] where ρ.sub.a is the air density, A is the reference area (projected frontal area of the vehicle), and C.sub.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:

P.sup.(f)=−∫vdF=(½)ρ(vn).sup.3C.sub.DdS=(½)ρ(ωr).sup.3arC.sub.DJ.sup.(f)(η,θ;N),

[0084] where J.sup.(f)(η, θ; N) is the following integral over the segment [θ/2, 2π/N]:

J.sup.(f)(η,θ;N)=∫[tg(φ)−η sin(θ)].sup.3dφ.

[0085] Carrying out integration, we get the following expression for the function J.sup.(f)(η, θ; N):

J.sup.(f)(η,θ;N)=1/[2 cos.sup.2(2π/N)]+ln [ cos(2π/N)]−1/[2 cos.sup.2(θ/2)]−ln [ cos(θ/2)]

−3η sin(θ)[tg(2π/N)−tg(θ/2)−2π/N+θ/2]

−3η.sup.2 sin.sup.2(θ)[ ln(cos(2π/N))−ln(cos(θ/2))]

−η.sup.3 sin.sup.3(θ)(2π/N−θ/2).

[0086] Similarly, for the power applied to the back foot of each wheel we have:

P.sup.(b)=−∫vdF=(½)ρ(vn).sup.3C.sub.DdS=(½)ρ(ωr).sup.3arC.sub.DJ.sup.(b)(η,θ;N),

[0087] where J.sup.(b)(η, θ; N) is the following integral over the segment [θ/2+π/N, 2π/N]:

J.sup.(b)(η,θ;N)=∫[tg(φ)−η sin(θ)].sup.3dφ.

[0088] Carrying out integration, we get a corresponding analytical expression for the integral:

J.sup.(b)(η,θ;N)=1/[2 cos.sup.2(2π/N)]+ln [ cos(2π/N)]−1/[2 cos.sup.2(θ/2+π/N)]−ln [ cos(θ/2+π/N)]

−3η sin(θ+2π/N)[tg(2π/N)−tg(θ/2+π/N)−2π/N+θ/2+π/N]

−3η.sup.2 sin.sup.2(θ+2π/N)[ ln(cos(2π/N))−ln(cos(θ/2+π/N))]

−η.sup.3 sin.sup.3(θ+2π/N)(2π/N−θ/2−π/N).

[0089] Instant horsepower required for driving all four wheels then is:

P=4(P.sup.(f)+P.sup.(b))=2ρ(ωr).sup.3arC.sub.DJ(η,θ;N),

[0090] where factor J(η, θ; N) is the following long expression

J(η,θ;N)≡J.sup.(f)(η,θ;N)+J.sup.(b)(η,θ;N)=

1/[2 cos.sup.2(2π/N)]+ln [ cos(2π/N)]−1/[2 cos.sup.2(θ/2)]−ln [ cos(θ/2)]

−3η sin(θ)[tg(2π/N)−tg(θ/2)−2π/N+θ/2]

−3η.sup.2 sin.sup.2(θ)[ ln(cos(2π/N))−ln(cos(θ/2))]

−η.sup.3 sin.sup.3(θ)(2π/N−θ/2)

+1/[2 cos.sup.2(2π/N)]+ln [ cos(2π/N)]−1/[2 cos.sup.2(θ/2+π/N)]−ln [ cos(θ/2+π/N)]

−3η sin(θ+2π/N)[tg(2π/N)−tg(θ/2+π/N)−2π/N+θ/2+π/N]

−3η.sup.2 sin.sup.2(θ+2π/N)[ ln(cos(2π/N))−ln(cos(θ/2+π/N))]

−η.sup.3 sin.sup.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:

P=4(P.sup.(f)+P.sup.(b))=2ρ(ωr).sup.3arC.sub.DAvg[J(0.5,θ;N)].

[0092] For example, in the case of N=16, we have:

Avg[J(0.5,θ;16)]≈0.00216,

[0093] with the corresponding engine horsepower at P=0.00432ρ(ωr).sup.3arC.sub.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'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'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.

TABLE-US-00001 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.sup.3 Air density ρ.sub.a 1.23 kg/m.sup.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.sup.2 Drag coefficient for C.sub.D 1 — immersed flat surfaces (foot bases and supporting plates) Aerodynamic drag coefficient C.sub.D 0.3 — for the watercraft

TABLE-US-00002 TABLE 2 Output Symbol Formula Calculated As Value Net thrust from the T.sup.(w) 2ρ(ωr).sup.2arC.sub.DAvg 0.00388ρ(ωr).sup.2ar 4108N wheels [K.sub.x(η, θ; N)] Net lift from the L.sup.(w) 2ρ(ωr).sup.2arC.sub.DAvg 0.0224ρ(ωr).sup.2ar 23716N  wheels [K.sub.y(η, θ; N)] Aerodynamic drag on D.sup.(a) 0.5ρ.sub.a(ωrη).sup.2C.sub.DA 0.15ρ.sub.a(0.5ωr).sup.2A 1674N the cruising vehicle Net drag from D.sup.(p) 2ρ(ωrη).sup.2 T.sup.(w) − D.sup.(a) 2434N supporting plates bhC.sub.Dsin.sup.2(α) Supporting plates α — arcsin(✓(D.sup.(p)/ 4.8° angle of attack (2ρ(0.5ωr).sup.2bh))) Net lift from L.sup.(p) 2ρ(ωrη).sup.2bhC.sub.D 2ρ(0.5ωr).sup.2bh 29260N  supporting plates sin(α)cos(α) sin(α)cos(α) Loaded weight of the W — L.sup.(w) + L.sup.(p) 52976N cruising vehicle (≈11905 lb) Required engine P 2ρ(ωr).sup.3arC.sub.DAvg 0.00432ρ(ωr).sup.3ar 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.