Patent Publication Number: US-2023150610-A1

Title: Vessel with stern positioned foil to reduce wave resistance

Description:
TECHNICAL FIELD OF THE INVENTION 
     The present invention relates to the design of seagoing vessels and can be applied to a majority of hull types, from slow-moving ships, rigs and barges to high-speed ships and boats that are operated to over planing speed, including sailing boats and multi hull vessels. 
     In particular, the invention relates to the configuration of a vessel where the vessel&#39;s aft part comprises a device that reduces the wave and turbulence resistance of the vessel. 
     BACKGROUND OF THE INVENTION 
     When a vessel moves at the surface of a water mass, a number of different resistance factors act against the vessel&#39;s motion. The total resistance R t  in Newton [N] for a displacement vessel and a planing hull are illustrated in  FIG.  4   . As can be seen, the total resistance R t  consist of frictional resistance R f  and residual resistance R r . The vessel&#39;s speed is indicated along the x-axis as Froude&#39;s number [F N ]. 
     
       
         
           
             
               
                 
                   
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     As shown in  FIG.  4   , the total resistance R t  for both displacement and planing hulls increases rapidly with increasing speed due to a significant rise in residual resistance R r . 
     For this reason, the speed corresponding to F N  of 0.4 is often referred to as the maximum hull speed for a displacement hull. Also planing hulls, optimized and designed to be operated at a speed above the maximum hull speed for displacement hulls, experience significant residual resistance R r  before it reaches planing speed. 
     The difference between R t  and R f  at a given speed represent the residual resistance, R r , which mainly consist of wave resistance R w . 
         R   t   −R   f   =R   r , and for practical use: R   r   ≈R   w . 
     In the speed range from around F N =0.30 and up to around F N =1.0, the resistance caused by wave making from the hull, R w , usually is the most dominant resistance factor for most hull types. As shown in  FIG.  4   , a typical displacement hull performs better than a planning hull and has lower total resistance R t  at low speed in the range up to F N =0.4. In the speed range F N =0.4−0.8 the semi displacement or planing hull is most efficient even though the total resistance is increasing rapidly as the speed increases also for these vessel types. 
     For this reason, the transition speed range from F N =0.4 to 0.8 has previously been very challenging in view of achieving cost efficient operation. To overcome some of the considerable rise in wave resistance, and thereby to achieve reasonable god fuel economy (resistance/speed), designers have had to design a more slender hull. For a planing hull that needs to be ‘lifted’ out of the water to obtain less resistance, the key objective has been to keep the weight down. This effectively limits the field of application for a planing hull. Consequently, planing hulls are primarily used for smaller and lighter vessels. 
     A typical prior art displacement hull optimized for speed range below F N =0.4 has a streamlined aft hull section with reduced cross sections towards the stern of the vessel, as shown in  FIG.  2 A . When in forward motion, the upward curved shape of the underside of the hull towards the stern gives a water flow an upward direction, thereby contributing to the formation of stern wave and a stern down trim of the vessel, as shown in  FIG.  2 B . The upward momentum in the water flow increases as the speed of the vessel increases. 
     A typical prior art planing hull optimised for higher speed has a rectilinear hull shape from center towards the stern, ending in a flat transom underneath the water surface when floating motionless in a mass of water, as shown in  FIG.  3 A . At low speed the water pressure in the water mass underneath and at the sides of the hull forces the water to rise from the bottom of the hull and fall inward from the hull sides directly behind the transom, resulting in a turbulent water flow behind the transom as shown in  FIG.  3 B . As speed increases, the residual resistance from the aft hull section of the hull gradually changes from turbulence resistance to wave resistance. Due to pressure drop under the hull close to the transom and dynamic lift of the bow area, the sinkage aft and the trim angle of the vessel increases, as shown in  FIG.  3 C . As shown in  FIG.  3 C &amp;D, a water flow separates from the hull at the transom and rises in a stern wave behind the hull. A water flow directly downstream the transom is essential horizontal or slightly downward due to positive trim of the vessel. But since the water flow surface level directly downstream the transom is below the surrounding water surface there is a state of non-equilibrium in the water mass in this region. With other words, there is a lack of equilibrium due to a difference in the water pressure directly behind the transom and the water pressure in the surrounding water masses at the same depth. This lack of equilibrium forces the water downstream the transom to rise, together with water falling inward from the hull sides. This again results in the formation of a stern wave at some distance downstream the transom, contributing to an increasing stern wave until the hull reaches a speed where the dynamic planing forces reduces the draft and rises the hull out of the water, as shown in  FIG.  3 D . 
     The waves generated by the hull, together with turbulence at the stern, represent lost energy. Depending on vessel type and speed, the residual resistance typical contributes to 30-80% of the total resistance to forward motion, as shown in  FIG.  4   . For a typical prior art displacement hull the stern alone typical contributes to 20-50% of the hull&#39;s total residual resistance. For a prior art hull optimized for a higher speed above F N =0.5 having a significant wetted transom, the aft part of the hull often contributes to more than 50% of the total residual resistance for the hull. 
     It is therefore crucial to minimize the wave resistance and turbulence resistance caused by the aft hull section of the vessel to reduce the total resistance to forward motion. 
     PRIOR ART 
     Lifting Foil 
     To reduce total resistance, some high-speed vessels are equipped with lifting foils. The purpose of lifting foils is to reduce the draft of the hull during high speed forward motion of the vessel, and often to lift the entire hull out of the water, thereby reducing the wave resistance for the hull and the wetted area that contributes to frictional resistance. 
     Trim Flaps 
     Trim flaps are widely used to limit a stern down trim for semi displacement and planing hulls. These flaps are usually hinged to the transom flush with the hulls underside. By adjusting the back of these flaps down at operational speed, and thereby forcing a water flow passing under the transom further down, a lift force is applied to the aft part of the hull. 
     Forward Propulsion from Aft Foil 
     Patent publications U.S. Pat. No. 7,617,793 B2 and WO 2016/010423 A1 both describe an aft foil mounted under the water surface at the stern of a displacement vessel, wherein the aft foil develops a continuous forwardly directed propulsion force exerted onto the vessel during forward motion of the vessel and thereby reducing the total resistance of the vessel. 
     To achieve said continuous forwardly directed propulsion force, the aft foil must be located in an upwardly directed water flow. Furthermore, the chord line of the aft foil must be tilted sufficiently downward in respect to the horizontal. 
     Small Concave Aft Foil 
     Patent publication KR200440081 (Y1) describes a small concave foil having a negative chord angel located under the aft part of a vessel having a vertical transom. The transom is located under the water surface during forward motion of the vessel. The small aft foil is claimed to develop a forwardly directed propulsion force exerted onto the vessel and to reduce the turbulence and wave formation behind the wet transom, thereby reducing the propulsion resistance for the vessel. 
     Guiding Fins for Displacement Vessel 
     Patent publication DE 2814260 A1 describes a displacement vessel having fins located under the water surface at the bow and/or stern to suppress bow and/or stern waves. 
     Negative Lift from Foil 
     Patent publication U.S. Pat. No. 4,915,048 A describes a planing vessel. The vessel has a deep draft bow with a fine entrance to prevent dynamic lift from the bow. At the stern the vessel has a foil to generate a downward force to counteract the planing lift from the underside of the aft hull during forward motion of the vessel. The foil is designed to prevent the vessel from being lifted out of the water caused by the planning forces acting on the aft hull and to keep the trim angle for the vessel neutral. 
     GENERAL DESCRIPTION OF THE INVENTION 
     Extended Description of Claims 
     The present invention is set forth and characterized in the independent claim, while the dependent claims describe other characteristics of the invention. 
     In one aspect, the invention concerns a vessel for floating in a body of water. 
     The vessel comprises a longitudinal hull having an aft hull section and an aft body arranged at a distance from the aft hull section, thereby forming a passage between the aft body and a separation line of the aft hull section. 
     The term ‘longitudinal hull’ is hereinafter defined as a hull having a length larger than the width. Further, the separation line is hereinafter defined as a line extending in a transverse direction of the hull at which a water flow originally flowing along the hull is separated from the aft hull section above a minimum forward propulsion of the vessel, for example at maximum forward propulsion of the vessel. The separation line may for example be a step in the aft hull section upstream/in front of the hull&#39;s termination point. 
     Note that the term ‘a line extending in a transverse direction’ shall be understood to include any line having endpoints separated in the transverse direction of the hull. Hence, the separation line may be of any form such as straight, curved, zigzagged, or a combination thereof. The two endpoints of the separation line may be at the same longitudinal position of the hull and/or may be at the same height relative to a common reference level such as the water surface when the vessel is floating in the body of water. 
     The separation line is further defined by the hull having an abrupt change of direction in a longitudinal vertical plane of the hull. In one embodiment, said abrupt change of direction constitute a sharp edge or almost sharp edge. In another embodiment, said abrupt change of direction has a small radius, for example a radius of 50 mm or smaller. 
     Said aft body is defined by a maximum width measured in a horizontal plane in the transverse direction of the hull, a leading edge, a trailing edge and a chord line. 
     The chord line is defined as a straight line extending from the leading edge to the trailing edge in a longitudinal vertical plane of the hull. The length of the chord line may further be defined as the arithmetic mean chord line length calculated along the entire width of the aft body. In case the transverse width of the aft body is centred relative to the transverse width of the vessel&#39;s hull, said longitudinal vertical plane will be at the transverse centre line of the aft body. 
     The vessel further comprises a leading edge area and a trailing edge area. 
     The leading edge area is defined by the smaller of:
         a first area equaling a minimum distance measured in a longitudinal vertical plane of the hull between the leading edge and the aft hull section integrated across the maximum width along the leading edge and   a second area equaling a minimum distance measured in a longitudinal vertical plane of the hull between two parallel lines integrated across the maximum width along the leading edge. Wherein the first line is defined as a tangent line of the aft hull section immediately upstream/in front of the separation line and the second line is defined as the line intersecting the leading edge. Each of the said two lines are oriented along the same longitudinal vertical plane of the hull. The measurement of the minimum distance for calculating the first area is preferably performed immediately upstream/in front of the separation line.       

     The definition of the first area is valid in those cases where the separation line is arranged at or downstream/aft of the leading edge. Likewise, the definition of the second area is valid in those cases where the separation line is arranged upstream/in front of the leading edge. 
     Note that the term ‘longitudinal vertical plane’ refers to a plane oriented perpendicular to a water surface when the vessel is floating motionless in the body of water and parallel to a bow-to-aft longitudinal orientation of the hull. 
     Alternatively, the second area of the leading edge area may be achieved by
         measuring a minimum distance in a longitudinal vertical plane of the hull between the first line and a point on the leading edge within the longitudinal vertical plane,   repeating this measurement over the entire maximum width along the leading edge and   integrating across the maximum width along the leading edge.       

     Of course, in practice, the integration over the maximum width of the leading edge is achieved based on an approximation in which a finite set of minimum distances along the leading edge is acquired, for example at least 3 minimum distances which include the two outermost points and the midpoint of the leading edge relative to the transverse direction. 
     The trailing edge area is defined by the area as seen from astern constrained by the trailing edge, a water surface when the vessel is floating motionless in the body of water at a predetermined load condition and two longitudinal vertical planes intersecting the two points on the surface of the aft body defining the maximum width. As an example, the trailing edge area may be measured when the vessel has no payload or more preferably also without ballast, for example without payload and ballast and with empty fuel tanks and lubricant tanks, i.e. at the lightweight waterline. 
     Note that the term ‘the area as seen from astern’ signifies a vertical cross-sectional ±area of the vessel at the trailing edge of the aft body. 
     Said aft body and said aft hull section is preferably mutually configured so that the leading edge area is at least 0.8 times the trailing edge area, more preferably at least 0.9 times, even more preferably at least 0.95 times, even more preferably at least 1.0 times, for example 1.1 times the trailing edge area. If average values of the leading edge area and the trailing edge area A te  across the maximum widths of the aft body are considered, the leading edge area A le  corresponds to a leading edge distance H 1 , and the trailing edge area A te  corresponds to a trailing edge distance H 2 . 
     In this particular case, H 1  is preferably at least 0.8 times H 2 , even more preferably at least 0.9 times H 2 , even more preferably at least 0.95 times H 2 , even more preferably at least 1.0 times H 2 , for example 1.1 times H 2 . 
     With the above ratio criteria between the leading edge area and the trailing edge area, a sufficient water flow is allowed to flow above the aft body&#39;s top surface to avoid, or at least significantly reduce, deviations from equilibrium in the water masses downstream the aft body during forward propulsion of the vessel. Deviation from equilibrium in the water masses immediately downstream the aft body will cause the formation of a stern wave, and thereby increase the vessel&#39;s total resistance during operation due to wave making. An insufficient amount of water over the aft body will form a depression of the water surface downstream the vessel compared to the level of the surrounding water surface. Any depression will be balanced by the surrounding water consequently contributing to the formation of the stern wave. 
     Another preferred criterion for reducing the vessel&#39;s total resistance is to designing the aft hull section with a double curvature in a longitudinal vertical plane of the vessel and/or such that the angel between tangent lines of the aft hull section immediately upstream/in front of the separation line in the longitudinal direction of the vessel and the water surface is kept small, preferably less than 20 degrees, more preferably less than 15 degrees, even more preferably less than 10 degrees, even more preferably less than 5 degrees, for example 0 degrees (i.e. parallel with the water surface). Such an aft hull section will ensure a minimum upward direction for a water flow in front of the aft body. 
     Please note that the expression “ . . . the separation line is located at or above the water surface” should be interpreted from the point of view of a person skilled in the art, taking into account the measurable technical effect of such a location. Hence the expression “at the water surface” should not be interpreted in a strict mathematical way. 
     Alternatively, or in addition, at least a part of the underside of the aft body, for example the entire underside, may be arranged below the water surface at or below a depth corresponding to 60% of the draft of the hull, for example 80%, when the vessel is floating motionless in the mass of water. As an example, the draft of the hull may be measured when the vessel has no payload or more preferably also without ballast, for example without payload and ballast and with empty fuel tanks and lubricant tanks. 
     The aft body and the aft hull section is preferably configured such that, during forward propulsion of the vessel, the net force component exerted onto the vessel from the aft body in the direction of travel of the vessel is zero or negative in at least a part of the speed range the vessel is operating in, for example in more than 10% of the vessels speed range or more preferably in more than 30% of the vessels speed range, or even more preferably more than 50% of the vessels speed range, or even more preferably more than 70% of the vessels speed range, for example in the full speed range the vessel is operated in. By “the vessels speed range” is meant from 0 knots and up to the vessels maximum speed at full power. The particular design fulfilling such criteria may for example be achieved by performing model tests or full-scale tests while measuring the forces acting on the supports for the aft body to the hull. Such tests can be performed with payload or more preferably without payload. 
     Note that a negative net force component in the direction of travel exerted onto the vessel from the aft body as described herein means that the aft body is adding drag force to the vessel through its supports. 
     Examples of relevant parameters that may be adjusted to achieve a zero or negative net horizontal force component in the longitudinal direction of the vessel are:
         the depth position of the separation line when the vessel is floating motionless in a mass of water and/or   the orientation of the chord line relative to the water surface and/or   the design of the aft hull section, for example the tangent angle of the aft hull section in the longitudinal direction immediately upstream/in front of the separation line relative to the water surface, and/or   the depth of the aft body relative to the vessel.       

     In yet another advantageous configuration, the aft body is designed to give a positive lifting force during forward propulsion of the vessel. Again, the particular design ensuring such an upward direction of the lifting force may be achieved by model tests or full-scale tests of a vessel in accordance with the invention described above. 
     In another advantageous configuration, the design and orientation of the aft body may be chosen such that, during forward propulsion of the vessel, the arithmetic mean direction of a resulting water flow immediately downstream of the trailing edge is orientated in the horizontal plane, i.e. parallel to the water surface, or substantially in the horizontal plane. The resulting water flow is set up by superposing a water flow passing the top surface of the aft body and a water flow passing the underside of the aft body. By such minimization of the upward or downward directed component of the resulting water flow downstream the aft body, said deviation from water flow equilibrium behind the vessel may be further reduced, which again causes a further reduction in the formation of stern waves. A horizontal water flow may be accomplished by for example orienting the chord line parallel or near parallel with said water surface when the vessel is floating motionless in a mass of water. 
     In an alternative configuration, the aft body may also be oriented with a chord line having a positive angle of attack relative to the water surface during forward propulsion of the vessel, for example an angle between 0° and 5° relative to the water surface, more preferable between 0° and 3°, even more preferable between 0° and 2°, for example between 0° and 1,5°. 
     In another alternative configuration, the chord line angle may even be slightly negative, for example −2° or −1°, as long as the result of the configuration yields a net force component exerted onto the aft body in the direction of travel of the vessel that is zero or negative as described above. 
     In yet another advantageous configuration, the chord line is orientated parallel with the water surface when the vessel is floating motionless in a body of water at the lightweight waterline. The term ‘parallel’ shall not be interpreted in its strict mathematical sense. Depending on various parameters such as the vessel&#39;s load conditions, the term ‘parallel’ can be interpreted as an orientation within a range ±2° relative to the water surface, or even within ±1° if the vessel conditions so allows. For example, if the different load conditions of the vessel results in an unchanged trim or near unchanged trim, the term ‘parallel’ may be interpreted narrower, even within ±0.5°. 
     Note that a positive angle is herein defined as an angle pointing upward in the direction of travel relative to the water surface. 
     In yet another advantageous configuration, the leading edge of the aft body is situated less than 20% of the length of the chord line aft of the separation line. This particular embodiment may contribute to reduce turbulence at low speed of the vessel. More favourably the leading edge is situated less than 15% of the length of the chord line aft of the separation or even more favourably less than 10%, even more favourably less the 5%, for example at or upstream the separation line. 
     In yet another advantageous configuration, at least a part of the trailing edge, for example the entire trailing edge, is located deeper than 35% of the maximum draft of the hull without ballast and payload when the vessel is floating motionless in a mass of water, more preferably deeper than 50% of the maximum draft, even more preferably deeper than 60% of the maximum draft, for example 80% of the maximum draft. 
     In yet another advantageous configuration, the length of the chord line is at least equal to the draft of the hull without ballast and payload when the vessel is floating motionless in a mass of water. The cord line length is more preferably 1.2 times the draft, even more preferably 1.5 times the draft, for example 2 times the draft. By exceeding a minimum length of the chord line, turbulence on the top surface and downstream the aft body is prevented or at least significantly reduced. 
     In yet another advantageous configuration, at least a part of the aft body, for example the entire aft body, is located upstream/in front of the vertical projection of a rearmost point of the hull. 
     In yet another advantageous configuration, the leading edge, for example the entire leading edge, is situated half the length of the chord line or more upstream/in front of the separation line, more preferably 60% of the length of the chord line or more, or even more preferably 70% of the length of the chord line or more, for example 80% of the length of the chord line or more, upstream/in front of the separation line. Further, the top surface and position may alternatively, or in addition, be designed such that a minimum distance in a longitudinal vertical plane of the hull between said top surface and the aft hull section upstream/in front of the separation line remains constant or near constant. 
     In yet another advantageous configuration, the aft body constitutes an integrated part of the vessel. 
     In yet another advantageous configuration, at least part of the leading edge, for example the entire leading edge, is located a horizontal length of ½ chord line or less downstream/aft of the separation line, more preferably less than ⅓ chord line, even more preferably less than ¼ chord line, even more preferably less than ⅕ chord line, for example at, or immediately downstream, the separation line. 
     In yet another advantageous configuration at least a part of the aft hull section located downstream/aft of the separation line is situated over said water surface when the vessel is laying still and floating in a mass of water. For example, the transom of the longitudinal hull may be located at or above the water surface. 
     In yet another advantageous configuration, the aft body and the aft hull section is configured so that the aft body during forward propulsion will not contribute to a significant change in draft of the aft hull section. This is in clear contrast to a typical lifting foil having a shape optimized for creating such a lift and contribute to a significant decrease in draft of the hull. 
     In yet another advantageous configuration, the aft body is designed such that a part of a water flow flowing over the top surface of the aft body is lifted above the water surface during forward propulsion of the vessel. 
     In yet another advantageous configuration, the separation line is located at or above the water surface, when the vessel is laying still and floating in a mass of water in a particular load condition such as without ballast and without payload. Another possible load condition may be with maximum ballast or with maximum payload. 
     In yet another advantageous configuration, the vessel further comprises a bow body located at or upstream/in front of a bow area. The bow body is configured to lead the water mass passing the upper surface of the bow body away from the bow area, or essentially parallel to the bow area, or a combination thereof. The design of the bow body and the bow area may be identical or similar to the bow body described in patent publication EP3247620B1, the contents of which are incorporated herein by reference. Particular reference is made to FIGS. 10-12 in EP3247620B1 and its related text. The proprietor of EP3247620B1 is the applicant in this application. 
     In yet another advantageous configuration, the aft body and the aft hull section is configured so that the draft of the hull during forward propulsion of the vessel will be at least 60% of the draft of the hull when the vessel is floating motionless in the body of water, or more preferably at least 70%, or more preferably at least 80%, or more preferably at least 90%, for example 100%. 
     In yet another advantageous configuration the maximum width of the aft body measured in a horizontal plane in the transverse direction of the hull is at least 60% of the maximum width of the hull measured at the water surface in the transverse direction of the hull when the vessel is floating motionless in the body of water, or more preferably at least 70%, or even more preferably at least 80%, or even more preferably at least 90%, for example at least 100%. 
     In yet another advantageous configuration, the longitudinal hull is a displacement hull or a planing hull. 
     In yet another advantageous configuration, the aft body is located between the water surface and 100% of the draft of the hull when the vessel is floating motionless in a mass of water. 
     In yet another advantageous configuration, the length of the chord line of the aft body is at least 5% of the length between perpendiculars of the vessel (L.P.P), more preferably at least 7%, or even more preferably at least 8%, or even more preferably at least 9%, for example at least 10% of the length between perpendiculars of the vessel. 
     The Invention—General Mode of Operation 
       FIG.  1    and  FIG.  5 B  shows the general mode of operation for one embodiment of a vessel according to the invention, where a water flow is indicated by arrows when the vessel is traveling at operational speed. The invention comprises a separation line at the aft hull section where a water flow will separate from the aft hull section during forward propulsion of the vessel. When the vessel is laying still and floating in a body of water, the separation line is located at the water surface (as seen in  FIG.  5 A ) and vertically above the leading edge of the aft body. Further, the chord line of the aft body is orientated parallel to the water surface. 
     During forward travel of the vessel, the upward tapered aft hull section upstream the aft body will give a water flow upstream the aft body a partly upward direction. The underside of the aft body will deflect a partly upwardly directed water flow in front of the aft body, causing a water flow under the aft body to flow in a primarily horizontal direction. The top surface of the aft body has a shape that redirects a water flow passing the top surface of the aft body from a partly upward to a horizontal or slightly downward directed water flow. The combined direction of the water flow downstream the trailing edge of the aft body, i.e. from the water flow passing over and under the aft body, then obtain an essentially horizontal direction. Hence, creation of stern wave due to the upwardly directed water flow at the aft hull section continuing in an upward direction behind the vessel is counteracted. 
     Further, as can be seen in  FIG.  1   , since the height of the water flow over the leading edge of the aft body H 1  is equal to the height from the trailing edge of the aft body to the water surface H 2 , the water mass downstream the aft body achieves a state of equilibrium. This counteracts the creation of a stern wave due to differences in water flow surface level for the water mass immediately downstream the aft body compared to the surrounding water surface. 
     The aft body for a vessel according to the invention will also generate a lifting force that will prevent a stern down trim of the vessel during forward motion. The aft body of such an inventive vessel will however not provide a continuous forwardly directed propulsion force. 
     The aforementioned objects are thus achieved, namely to reduce the vessel resistance to forward motion over a wide speed range due to:
         1) reduced wave resistance, and/or   2) reduced resistance due to turbulence.       

     The inventive vessel can be adopted to different hull types and speed ranges; from typical rounded displacement hulls operated at speeds up to around F N =0.4 as shown in  FIG.  2 A , typical semi displacement hulls with less rounded underside and some wetted transom operating in the speed range of F N , =0.4 to around F N =0.9, and planing hulls with a chine, a straight lined underside and a transom extending under the water surface as shown in  FIG.  3 A , typically operated in the speed range above F N =0.9. 
     The working principal of the inventive vessel is in general the same regardless of speed range and type of vessel. However, the type of vessel and operational cruising speed should be taken into consideration when designing and optimising the geometry of the inventive vessel to a specific hull and to a specific speed range as described later. 
     When applied to a traditional prior art displacement hull, the inventive vessel counteracts the upward directed water flow and the generation of a stern wave downstream the hull and to reduce turbulence under and behind the aft hull section, as shown in  FIG.  5 B . 
     When applied to a vessel with wetted transom below the water surface, like a semi displacement or a planing hull, the invention prevents turbulent flow behind the transom at low speed. At higher speed, when the water starts to separate from the hull behind the flat transom, the inventive vessel will effectively prevent the rise of water behind the hull and thereby counteract creation of stern wave, as shown in  FIG.  6 B . 
     For a typical prior art displacement hull with a streamlined aft hull section, the inventive vessel will usually contribute to reduced propulsion resistance from a speed corresponding to approximately F N =0.17-0.20 and up. For a typical prior art semi displacement or a planing hull, the inventive vessel will usually contribute to reduced propulsion resistance from stand still and all the way up to a speed exceeding F N =1.0. 
     Differences from Prior Art 
     With reference to the description above, the inventive vessel differs from the above described prior art vessels in the following ways: 
     Lifting Foil: 
     Prior art vessels with lifting foils reduce the propulsion resistance at high speed by lifting the prior art vessel partly or fully out of the water during operation. A prior art vessel will typical have two lifting foils, one at the front of the vessel and one towards the stern. Both foils will be located deep under the baseline of the hull to avoid that the low pressure on the top side of the foil has a negative impact on the hull (i.e. “sucking” the hull down). Furthermore, the lifting foils must stay submerged when the prior art vessel is lifted out of the water. If the lifting foils during high speed operation is located close to the water surface they will generate waves and also generate less lift. At low speed the lifting foils will increase the propulsion resistance of the prior art vessel considerably. 
     In contrast, the inventive vessel is designed to maintain the same draft whether it is laying still and floating in a body of water or traveling at operational speed. The inventive vessel lowers the resistance over a broad speed range, starting from low speed. Furthermore, the inventive vessel will have the aft body located between the base line of the hull and the water surface. 
     Trim Flaps: 
     The use of trim flaps is common in prior art vessels to limit the change in trim of semi-displacement and planing hulls due to forward motion of the vessel. By orienting the aft part of these flaps downward at speed, and forcing a water flow passing under the transom further down, a lift force is applied to the aft hull section. The downward directed trim flap effectively lowers the water flow surface level downstream the trim flaps, thereby increasing the distance to equilibrium between a water flow downstream the trim flap and the surrounding water surface. The trim flaps hence contribute to increased stern wave behind the prior art vessel, resulting in increased wave resistance. 
     The inventive vessel also has the ability to counteract an aft down trim of the vessel, but the wave formation is considerably less compared to prior art vessel with trim flaps. A trim flap does not have a water flow over the top side of the trim flaps during forward motion of the vessel, nor a leading edge, a passage or a leading edge area A le  as defined herein. 
     Forward Propulsion from Aft Foil 
     Both patent publication U.S. Pat. No. 7,617,793 B2 and patent publication WO 2016/010423 A1 discloses a displacement hull having an aft foil fixed to the aft part of the prior art vessel which is configured to generate a continuous forwardly directed propulsion force actin on the vessel during forward motion of the vessel. 
     To achieve said propulsion force, the aft foil has to be mounted in a sufficiently upwardly directed water flow during forward motion of the prior art vessel (as documented by model tests later in this document). An upwardly directed water flow can be achieved:
         1) if the aft hull section has steep tangent lines in the longitudinal direction of the hull upstream/in front of the separation line, and/or   2) if the separation line where a water flow separates from the aft hull section during forward motion of the vessel is located sufficiently deep under the water surface, as this will cause a water flow to “shoot” upwards downstream the separation line during forward travel of the vessel to achieve a state of equilibrium in the water masses behind the separation line.       

     In addition, the chord line of the aft foil must be tilted sufficiently downward in the upwardly directed water flow for the aft foil to be able to generate a continuous forward propulsion force. 
     The inventive vessel is not designed with an aft body being configured to generate a continuously forwardly directed propulsion force. (Also this is documented by model tests later in this document.) 
     Prior art vessels disclosed in both patent publication U.S. Pat. No. 7,617,793 B2 and patent publication WO2016/010423 A1 are not designed to lead a sufficient amount of water over the aft foil during forward motion of the vessel. I.e., the leading edge area A le  (as herein defined) of the prior art vessel is smaller than 0.8 times the trailing edge area A te  (as herein defined). Accordingly, a water flow passing over the trailing edge of the aft foil is too small to achieve equilibrium in the water mass downstream the aft foil. In fact, it is an objective when designing the prior art vessels to locate the separation line below the water surface when the vessel is lying motionless in a mass of water to achieve the upwardly directed water flow downstream the separation line during forward motion of the vessel. Accordingly, it is an objective when designing the prior art vessels to keep the A le  divided by A te  ratio small. This is in clear contrast to the inventive vessel where the objective is to have the A le  divided by A te  ratio close to 1 to achieve equilibrium in the water mass downstream the aft body. 
     Aft hull sections of the prior art vessels according to U.S. Pat. No. 7,617,793 B2 are designed with a large angel β, being the angel between the tangent line of the aft hull section immediately upstream/in front of the separation line in the longitudinal direction of the vessel and the water surface, to achieve a sufficiently upwardly directed water flow upstream the aft foil. This is in clear contrast to the inventive vessel where the objective is to direct a water flow horizontally downstream the aft body. 
     Aft foils of the prior art vessels disclosed in both patent publication U.S. Pat. No. 7,617,793 A1 and patent publication WO 2016/010423 A1 are configured with a downward pointing chord line angle in relation to the water surface to generate a continuously forwardly directed propulsion force. The downward pointing chord line contributes to give the water flow passing the aft body&#39;s underside and top surface an upward direction. This is in clear contrast to the inventive vessel, which has an aft body with a chord line essentially parallel to the water surface to counteract such an upward directed water flow, thereby counteracting the formation of a stern wave. 
     The prior art vessels for some of the embodiments in patent publication U.S. Pat. No. 7,617,793 A1 are designed such that the minimum distance, in the longitudinal vertical plane of the vessel, between the top surface of the aft foil and the aft hull section upstream/in front of the separation line is changing (i.e. not constant). This results in a retardation of a water flow from the leading edge of the aft foil over a part of the top surface of the aft foil during forward motion of the prior art vessel. This reduction in the velocity of the water flow passing over a part of the aft foil is adding drag to the vessel. In contrast, an embodiment of the inventive vessel has a geometry of the aft hull section and the aft body&#39;s top surface that is designed to prevent such a retardation of the water flow by having a constant minimum distance (in the longitudinal vertical plane of the vessel, between the top surface of the aft foil and the aft hull section upstream/in front of the separation line). 
     The vessels disclosed in both patent publication U.S. Pat. No. 7,617,793 A1 and patent publication WO2016/010423 A1 are adapted only to displacement hulls, while the inventive vessel is adapted to both displacement hulls and planing hulls. 
     Small Concave Foil 
     Patent publication KR200440081 (Y1) describes a small concave foil located under the aft hull section of a vessel, where a vertical transom extent under the water surface during forward travel of the vessel. The objective of this solution is to reduce the wave formation and the turbulence behind a wet transom and to generate a forward thrust force acting on the small aft concave foil during forward motion of the vessel. Furthermore, the small concave foil is claimed to increase the pressure at the periphery of the leading edge of the small concave foil, thereby reducing stern down trim of the vessel. 
     In contrast to the inventive vessel, this prior art vessel does not achieve equilibrium for the water mass downstream the small concave foil during forward motion of the vessel. The leading edge area A le  of the small concave foil is shown to be about 0.5 times the trailing edge area A te  of the small concave foil. 
     The prior art vessel is designed such that the minimum distance, in the longitudinal vertical plane of the vessel, between the top surface of the small concave foil and the aft hull section upstream/in front of the separation line (i.e. the transom) is changing (i.e. it is not constant), in contrast to the inventive vessel. 
     The inventive vessel does not include a wetted transom where the aft foil&#39;s trailing edge is situated underneath the transom in contrast to the prior art vessel as shown in KR200440081 (Y1). 
     The inventive vessel is not designed to increase the pressure at the periphery of the leading edge of the aft concave foil, thereby reducing the resistance on the vessel, in contrast to the prior art vessel having a small concave foil. 
     The prior art vessel according to KR200440081 (Y1) is designed to generate a forward component Lx of the lift forces L, generating a horizontal forwardly directed thrust force acting on the small concave foil. The inventive vessel is not designed to generate a forwardly directed thrust force acting on the aft body. 
     The maximum width of the small concave foil measured in a horizontal plane in the transverse direction of the hull is only about 15% of the maximum width of the hull, in contrast to the width of the aft body which in one embodiment is at least 50%, preferably close to 100%, of the width of the hull. 
     Guiding Fins for Displacement Vessel 
     Patent publication DE 2814260 A1 describes a displacement vessel having fins located under the water surface at the bow and/or of the vessel stern to suppress bow and/or stern waves, thereby reducing the wave resistance. 
     As can be seen from the figures in DE 2814260 A1, the prior art vessel does not have a separation line as defined herein. I.e. that the separation line has an abrupt change of direction in a longitudinal vertical plane of the hull. Nor does the description of DE 2814260 A1 mention anything about a separation line. 
     In contrary, the inventive vessel includes a defined separation line, which is of vital importance to control a water flow behind the hull at different speeds for the vessel, and to avoid that a water flow will try to follow the shape of the hull giving the water flow an upward direction (i.e. the Coanda effect). 
     Furthermore, the inventive vessel has a superior speed range and is designed to minimize vortexes and turbulence created by the aft foil in the water flow. 
     Negative Lift from Foil 
     Patent publication U.S. Pat. No. 4,915,048 A describes a planing vessel. The vessel has a deep draft bow with a fine entrance. At the stern the vessel has a foil to generate a downwardly directed force to counteract the planing lift from the underside of the hull during forward motion of the vessel. In contrary, the inventive vessel includes an aft body where the force from the aft body acts in the opposite direction to the publication U.S. Pat. No. 4,915,048 A. The aft body of the inventive vessel imposes a lifting force to the aft hull section that counteracts a stern down trim of the vessel as speed increases. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a schematic side view illustration of the aft hull section of the vessel in  FIG.  5    showing the general mode of operation for the invention when the vessel is traveling at operational speed. 
         FIG.  2 A &amp;B show a typical prior art displacement hull, wherein  FIG.  2 A  is a longitudinal vertical plane of the displacement hull at rest and  FIG.  2 B  is a longitudinal vertical plane illustration of the displacement hull in motion, further illustrating the upward direction of a water flow at the aft hull section and the formation of a stern wave. 
         FIG.  3 A-D  are longitudinal vertical plane illustrations of a typical planing hull according to the prior art, at increasing Froude numbers (F N ), wherein  FIG.  3 A  shows a submerged planing hull situated motionless in a body of water,  FIG.  3 B  shows the formation of a stern wave and turbulent reversed water flow behind the planing hull at low (displacement mode) speed (F N =0−0.5),  FIG.  3 C  shows the formation of the stern wave at medium (transition mode) speed (F N =0.5−0.9) and  FIG.  3 D  shows the formation of the stern wave at high (planing mode) speed (F N &gt;0.9). 
         FIG.  4    show a graphic illustration of typical frictional resistance R f  and total resistance R t  as function of the Froude number (F N ) for a prior art displacement hull and planing hull. 
         FIG.  5 A &amp;B show the behaviour of a displacement hull according to the invention, wherein  FIG.  5 A  is an illustration in a longitudinal vertical plane of the displacement hull at rest and  FIG.  5 B  is an illustration in a longitudinal vertical plane of the displacement hull in forward motion, further illustrating the direction of a water flow at the aft hull section and the formation of a reduced stern wave. 
         FIG.  6 A &amp;B are illustrations in longitudinal vertical planes of a planing hull with an aft body according to the invention, wherein  FIG.  6 A  shows the planing hull floating motionless in a body of water and  FIG.  6 B  shows the formation of a reduced stern wave behind the planing hull at speed (F N &gt;0). 
         FIG.  7 A-C  are illustrations in a longitudinal vertical plane of an aft hull section of a vessel in accordance with the invention, submerged in a body of water, wherein  FIG.  7 A  shows the aft hull section designed for low speed (F N &lt;0.4) where the aft body is located closer to the water surface,  FIG.  7 B  shows the aft hull section designed for medium speed (F N =0.4−0.6) where the aft body is located at medium depth and  FIG.  7 C  shows the aft hull section designed for high speed (F N &gt;0.6) where the aft body is located at same depth as the base line of the hull. 
         FIG.  8 A-C  are illustrations in longitudinal vertical planes of aft hull sections of vessels in accordance with the invention, submerged in a body of water, wherein  FIG.  8 A  shows a separation line in the aft hull section arranged upstream/in front of the aft body,  FIG.  8 B  shows the separation line arranged above the aft body and  FIG.  8 C  shows the separation line arranged at the transom of the hull and above the trailing edge of the aft body. 
         FIG.  9 A-D  are perspective illustrations seen obliquely from behind of aft hull sections of vessels in accordance with the invention, wherein  FIG.  9 A  shows an aft body arranged with its trailing edge below the transom of the hull,  FIG.  9 B  shows an aft body arranged with its leading edge below the transom of the hull,  FIG.  9 C  shows a vessel with the hull sides continuing in a straight line all the way back to the trailing edge of the aft body and  FIG.  9 D  shows a vessel where the hull sides below a water surface is sloping towards the longitudinal centre line of the vessel and continuing all the way back to the trailing edge of the aft body. 
         FIG.  10    is a perspective illustration seen obliquely from behind of an aft hull section in accordance with the invention where the leading edge of the aft body is located straight under the transom, showing a leading edge area (A le ) and a trailing edge area (A te ) as herein defined. 
         FIG.  11    is an illustration in a longitudinal vertical plane of a hull in accordance with the invention submerged in a body of water wherein the position and alignment of an aft body and a bow body, according to the applicant&#39;s patent EP3247620B1, and their effect on a water flow during forward propulsion of the vessel. 
         FIG.  12    show a graphic illustration of typical total resistance R t  as function of Froude number (F N ) derived from numerous model tests performed on models, wherein the upper left figure (L-A) is an illustration in a longitudinal vertical plane of a typical displacement vessel according to prior art moving at low speed (F N &lt;0.25), the middle left figure (L-B) is an illustration in a longitudinal vertical plane of a displacement vessel with an aft body according to the invention moving at low speed (F N &lt;0.25), the lower left figure (L-C) is an illustration in a longitudinal vertical plane of a displacement vessel with an aft body according to the invention and a bow body moving at low speed (F N &lt;0.25). The upper right figure (R-A) is an illustration of the prior art displacement vessel in the upper left figure (L-A) moving at higher speed (F N &gt;0.25), the middle right figure (R-B) is an illustration of the displacement vessel in the middle left figure (L-B) moving at higher speed (F N &gt;0.25), the lower right figure (R-C) is an illustration of the displacement vessel in the lower left figure (L-C) moving at higher speed (F N &gt;0.25). The graph indicates the total resistance R t  in Newton as function of the Froude number (F N ) for the three vessels where the prior art displacement vessel (L-A and R-A) is marked with a solid line and with reference Rt(A), the inventive displacement vessel with the aft body (L-B and R-B) is marked with a stippled line and with reference Rt(B) and the inventive displacement vessel with an aft body and a bow body (L-C and R-C) is marked with a dotted line and with reference Rt(C). 
         FIG.  13    show a graphic illustration of typical total resistance R t  as function of Froude number (F N ) derived from numerous model tests performed on models, wherein the upper left figure (L-A) is an illustration in a longitudinal vertical plane of a typical planing vessel according to the prior art moving at a displacement mode speed (F N &lt;0.4), the middle left figure (L-B) is an illustration in a longitudinal vertical plane of a planing vessel with an aft body according to the invention moving at a displacement mode speed (F N &lt;0.4), the lower left figure (L-C) is an illustration in a longitudinal vertical plane of a planing vessel with an aft body according to the invention and a bow body moving at a displacement mode speed (F N &lt;0.4). The upper centre figure (M-A) is an illustration of the prior art planing vessel in the upper left figure (L-A) moving at a transition mode speed (F N =0.4−0.9), the middle centre figure (M-B) is an illustration of the planing vessel in the middle left figure (L-B) moving at a transition mode speed (F N =0.4−0.9), the lower centre figure (M-C) is an illustration of the planing vessel in the lower left figure (L-C) moving at a transition mode speed (F N =0.4−0.9). The upper right figure (R-A) is an illustration of the prior art planing vessel in the upper left figure (L-A) moving at a planing mode speed (F N &gt;0.9), the middle right figure (R-B) is an illustration of the planing vessel in the middle left figure (L-B) moving at a planing mode speed (F N &gt;0.9), the lower right figure (R-C) is an illustration of the planing vessel in the lower left figure (L-C) moving at a planing mode speed (F N &gt;0.9). The graph indicates the total resistance R t  in Newton as function of the Froude number (F N ) for all the three vessels where the prior art planing vessel (L-A, M-A and R-A) is marked with a solid line and with reference Rt(A), the inventive planing vessel with the aft body (L-B, M-B and R-B) is marked with a stippled line and with reference RI(B) and the inventive planing vessel with the aft body and the bow body (L-C, M-C and R-C) is marked with a dotted line and with reference Rt(C). 
         FIG.  14 A  shows a schematic illustrations of an aft hull section of the vessel in accordance with the invention, wherein drawing (a) shows a longitudinal vertical plane of the aft hull section and the position and alignment of an aft body arranged with its leading edge upstream/in front of the separation line and drawing (b) shows drawing (a) seen from behind. 
         FIG.  14 B  drawing (c) shows the aft hull section shown in  FIG.  14 A  seen from below and drawing (d) shows the aft hull section of  FIG.  14 A  drawing (a) illustrating a water flow during forward propulsion of the vessel. 
         FIG.  15 A  shows a schematic illustrations of an aft hull section of the vessel in accordance with the invention, wherein drawing (a) shows a longitudinal vertical plane of the aft hull section and the position and alignment of an aft body arranged with its leading edge downstream/aft of the separation line and drawing (b) shows the aft hull section shown in drawing (a) seen from behind. 
         FIG.  15 B  drawing (c) shows the aft hull section shown in  FIG.  15 A  seen from below and drawing (d) shows the aft hull section of  FIG.  15 A  drawing (a) illustrating a water flow during forward propulsion of the vessel. 
         FIG.  16    are showing upside down perspective illustrations of a model vessel with a slender displacement hull, wherein Model  16 A shows a prior art model vessel and Model  16 B is the same model as Model  16 A but fitted with an aft body according to the invention. 
         FIG.  17    is a perspective illustration of a propulsion system arranged on model vessels to measure propulsion thrust in Newton [N]. 
         FIG.  18 A  are two pictures from model tests showing the formation of stern waves of Model  16 A and Model  16 B at speed corresponding to Froude number (F N ) 0.30. 
         FIG.  18 B  are two pictures from model tests showing the formation of stern waves of Model  16 A and Model  16 B at speed corresponding to Froude number (F N ) 0.36. 
         FIG.  19    shows the total resistance R t  as function of Froude number (F N ) derived from model testing of the prior art Model  16 A and the inventive Model  16 B. 
         FIG.  20    shows upside down perspective illustrations of three different model vessels being compared in model tests, where Model  20 A shows a prior art model vessel with a displacement hull, Model  20 B shows a prior art model vessel with a displacement hull and a bow body, the model having the same length and displacement as the Model  20 A. Model  20 C shows an inventive model vessel which is the same as Model  20 B except for the separation line and the aft body. 
         FIG.  21    shows the total resistance Rt as function of Froude number (F N ) derived from model tests of the prior art Model  20 A, the prior art Model  20 B and the inventive Model  20 C. 
         FIG.  22    shows upside down perspective illustrations of two model vessels, wherein Model  22 A shows is a prior art model vessel with planning hull and Model  22 B is an inventive model vessel with same length, width and displacement as Model  22 A but having a bow body and an aft body. 
         FIG.  23 A  are two pictures from model tests showing the formation of stern waves for the prior art Model  22 A and the inventive Model  22 B at speed corresponding to Froude number (F N ) 0.40. 
         FIG.  23 B  are two pictures from model tests showing the formation of stern waves for the prior art Model  22 A and the inventive Modell  22 B at speed corresponding to Froude number (F N ) 0.50. 
         FIG.  23 C  are two pictures from model tests showing the formation of stern waves for the prior art Model  22 A and the inventive Model  22 B at speed corresponding to Froude number (F N ) 0.65. 
         FIG.  24    shows the power consumption in watt [W] of the electrical propulsion engine as function of Froude number (F N ) from model tests of the prior art Model  22 A and the inventive Model  22 B. 
         FIG.  25    is an upside down perspective illustration of a model vessel having a test set up to measure the horizontal forces in the longitudinal direction of the vessel from the aft body acting on the vessel. 
         FIG.  26    is a side view illustration of the aft hull section of the model vessel shown in  FIG.  25    with a test set up to measure the horizontal forces in the longitudinal direction of the vessel from the aft body acting on the vessel. 
         FIG.  27    is a side view illustration of the aft hull section of the model vessel shown in  FIG.  26    where the chord angel γ is: 0 degrees (i.e. the cord line of the aft body and the water surface is parallel) marked (A), −2 degrees marked (B) and −3 degrees marked (C). (The support to fix the aft body to the hull, incl. the ball bearing, load cell, propeller and rudder is the same as in  FIG.  26    but is for convenience not shown in this figure). 
         FIG.  28    is a side view illustration of the aft hull section of the model vessel shown in  FIG.  26    where the chord angel γ is 0 degrees and the draft of the hull is shown at 80 mm DV(A), at 90 mm DV(B) and at 100 mm DV(C). (The support to fix the aft body to the hull, incl. the ball bearing, load cell, propeller and rudder is the same as in  FIG.  26    but is for convenience not shown in this figure). 
         FIG.  29    is a side view illustration of the aft hull section of the model vessel shown in  FIG.  26    where the chord angel γ is 0 degrees, and where the geometry of the aft hull section is altered to obtain an angle β between a tangent line TH of the aft hull section and the horizontal of 4.5 degrees marked β(A) and 11 degrees marked β(B). (The support to fix the aft body to the hull, incl. the ball bearing, load cell, propeller and rudder is the same as in  FIG.  26    but for convenience is not shown in this figure). 
         FIG.  30    is a side view illustration of the aft hull section of the model vessel shown in  FIG.  26    where the chord angel γ is 0 degrees, and where the aft body is arranged 30 mm above the base line marked (A) and 50 mm above the base line marked (B). (The support to fix the aft body to the hull, incl. the ball bearing, load cell, propeller and rudder is the same as in  FIG.  26    but for is convenience not shown in this figure). 
         FIG.  31    is a side view illustration of the aft hull section of the model vessel shown in  FIG.  26    where the chord angel γ is 0 degrees, and where the cord length of the aft body is 105 mm marked (A) and 145 mm marked (B). (The support to fix the aft body to the hull, incl. the ball bearing, load cell, propeller and rudder is the same as in  FIG.  26    but is for convenience not shown in this figure). 
         FIG.  32    shows a side view illustration of the aft hull section of the model vessel shown in  FIG.  26   :
         where a model vessel according to the invention (A) has a draft DV(A) of 80 mm (which entails A le =1,0*A te ) and an angle β(A) for the tangent line TH of 4.5 degrees and a chord angel γ(A) of 0 degree, and   a model vessel according to prior art (B) has a draft DV(B) of 100 mm (which entails A le =0.71*A te ) and an angle β(B) for the tangent line TH of 11.0 degrees and a chord angel γ(B) of −2 degrees.       

       (The support to fix the aft body to the hull, incl. the ball bearing, load cell, propeller and rudder is the same as in  FIG.  26    but is for convenience not shown in this figure). 
         FIG.  33    shows a graphic presentation of the horizontal forces in Newton acting in the longitudinal direction of the vessel from an aft body onto the aft hull section as function of Froude number (F N ). The graphs are derived from model tests performed on a model with an aft hull section as shown in  FIG.  27    having a chord angle γ of 0 degrees marked (A), a chord angle γ of −2 degrees marked (B) and a chord angle γ of −3 degrees marked (C). A positive force reading equals a backward directed force (i.e. resistance to forward motion) while a negative force reading equals a forward directed force (i.e. propulsion). 
         FIG.  34    shows a graphic presentation of the horizontal forces in Newton acting in the longitudinal direction of the vessel from an aft body onto the aft hull section as function of Froude number (F N ). The graphs are derived from model tests performed on a model with an aft hull section as shown in  FIG.  28    having a leading edge area (A le )=1.0*trailing edge area (A te ) marked (A), (A le )=0.83*(A te ) marked (B), and (A le )=0.71*(A te ) marked (C). A positive force reading equals a backward directed force (i.e. resistance to forward motion) while a negative force reading equals a forward directed force (i.e. propulsion). 
         FIG.  35    shows a graphic presentation of the horizontal forces in Newton acting in the longitudinal direction of the vessel from an aft body onto the aft hull section as function of Froude number (F N ). The graphs are derived from model tests performed on a model with an aft hull section as shown in  FIG.  29    when the geometry of the aft hull section is altered to obtain an angle β between a tangent line TH of the aft hull section and the horizontal of 4.5 degrees marked (A), and 11.0 degrees marked (B). A positive force reading equals a backward directed force (i.e. resistance to forward motion) while a negative force reading equals a forward directed force (i.e. propulsion). 
         FIG.  36    shows a graphic presentation of the horizontal forces in Newton acting in the longitudinal direction of the vessel from an aft body onto the aft hull section as function of Froude number (F N ). The graphs are derived from model tests performed on a model with an aft hull section as shown in  FIG.  30    having an aft body located 30 mm above base line marked (A), and 50 mm above base line marked (B). A positive force reading equals a backward directed force (i.e. resistance to forward motion). 
         FIG.  37    shows a graphic presentation of the horizontal forces in Newton acting in the longitudinal direction of the vessel from the aft body onto the aft hull section as function of Froude number (F N ). The graphs are derived from model tests performed on a model with an aft hull section as shown in  FIG.  31    having a chord line length of the aft body of 105 mm marked (A), and 145 mm marked (B). A positive force reading equals a backward directed force (i.e. resistance to forward motion). 
         FIG.  38    shows a graphic presentation of the horizontal forces in Newton acting in the longitudinal direction of the vessel from the aft body onto the aft hull section as function of Froude number (F N ). The graphs are derived from model tests performed on a model with aft hull sections as shown in  FIG.  32    for a configuration according to the inventive model vessel marked (A) and for a configuration according to the prior art model vessel marked (B). A positive force reading equals a backward directed force (i.e. resistance to forward motion) while a negative force reading indicates a forward directed force (i.e. propulsion). As seen, the aft foil of the prior art model vessel marked (B) is providing a continuously forwardly directed propulsion force. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the following, embodiments of the invention will be described in more detail with reference to the drawings and definitions. However, it is specifically intended that the invention is not limited to the embodiments and illustrations contained herein but includes modified forms of the embodiments including portions of the embodiments and combinations of elements from different embodiments as come within the scope of the claims. 
     Definitions and Reference Numerals 
     Throughout this application, the following definitions, numerals and letters in drawings, shall apply:
     Vessel  1 :
       All vessels that are operated from low displacement speed to above planing speed in excess of F N =1.0.   
       Hull  2 :
       The watertight body of a vessel  1  that makes the vessel  1  seaworthy, but excluding components such as superstructure, the aft body  4 , the bow body  10 , the propeller  12 , the rudder, the keel, the deck, etc.   
       Hull side  2 ′:   

     The hull sides of the vessel  1 . I.e. not including the bow area  21  and the transom  7 .
     Aft hull section  3 :
       For a displacement vessel  1 , the part of the hull  2  which is aft of the cross section of the hull  2  below a waterline  5  with the greatest cross section area, and for a planning vessel  1 , the part of the hull  2  which is aft of mid ship.   
       Aft body  4 :
       The body that is arranged at a distance from the aft hull section  3 .   
       Water surface  5 :
       A strait horizontal surface formed by still sea/water.   
       Separation line  6 :
       A defined line extending primarily in the transverse direction of the vessel  1  in the aft hull section  3  where a water flow  51  passing the hull  2  separates from the hull  2  when the vessel  1  is in forward motion above a minimum speed, for example at operational speed. Furthermore, the separation line  6  is defined by the aft hull section  3  having an abrupt change of direction in a longitudinal vertical plane of the hull  2 .   
       Transom  7 :
       The flat or almost flat part of a hull  2  that forms the stern of a square ended vessel  1 .   
       Support  8 :
       Support to fix the aft body  4  to the vessel  1 .   
       Stern wave  9 :
       A wave behind or at the stern of the vessel  1  created during forward motion of the vessel  1 , for example at operational speed.   
       Bow body  10 :
       The bow body that is arranged at the bow area  21  according to patent EP3247620B1.   
       Propeller shaft  11     Propeller  12     Propeller sleeve (with no thrust bearing)  13     Electric motor  14     Motor housing  15     Motor suspension system  16     Load cell  17     Mounting bracket  18     Base plate  19     Ball bearing slide  20     Bow area  21 :
       The area of the hull  2  seen from in front (over and under the water surface  5  when the vessel  1  is floating in a mass of water), but excluding the bow body  10  if any.   
       Bow wave  22 :
       A wave crest formed ahead of the bow area  21  due to the hull&#39;s  2  deceleration of the oncoming water flow  51 .   
       Leading edge  41 :
       The foremost edge of the aft body  4 , equivalent to “the leading edge” of an airplane wing.   
       Trailing edge  42 :
       The rearmost edge of the aft body  4 , equivalent to “the trailing edge” of an airplane wing.   
       Chord line  43 :
       A straight line in a longitudinal direction of the vessel  1  extending from the leading edge  41  to the trailing edge  42 .   
       Top surface  45 :
       The top surface area of the aft body  4  extending from the leading edge  41  to the trailing edge  42 .   
       Underside  46 :
       The underside area of the aft body  4  extending from the leading edge  41  to the trailing edge  42 .   
       Two vertical planes  49 :
       Two vertical planes in the longitudinal direction of the vessel  1 , each intersecting the point defining the maximum width (W) of the aft body  4  in the transverse direction of the vessel  1 .   
       Passage  50 :
       The area between the aft hull section  3  and the top surface  45  where water can flow through during forward motion of the vessel  1 .   
       Water flow  51 :
       A flow of water relative to the vessel  1  due to vessel&#39;s  1  forward motion, for example at operational speed. Such water flow  51  is also shown as arrows in the figures.   
       Water flow surface level  53 :
       The top surface of a water flow  51  bordering air around a vessel  1  that can be lower or elevated relative to the water surface  5 .   
       Bow passage  56 :
       The area between the bow area  21  and the top surface of the bow body  10 .   
       Base line  58 :
       A horizontal line drawn in the longitudinal direction of the vessel  1  through the draft (DV) of the vessel  1 .   
       LWWL:
       The lightweight waterline is the waterline of the vessel  1  complete in all respect when it is floating motionless in a body of water but without consumables, stores, cargo, crew and effects, and without any liquids on board except that machinery and piping fluids, such as lubricants and hydraulics, are at operating levels. The vessel  1  thereby also have a fixed trim.   
       DV: The draft of the vessel  1  equalling the vertical distance between the water surface  5  and the deepest part of the hull  2  when the vessel  1  is floating motionless in the body of water.   TH: Tangent line of the aft hull section  3  in a longitudinal direction of the vessel  1  immediately upstream/in front of the separation line  6 .   TF: Line in the same longitudinal vertical plane of the vessel  1  as TH above, intersecting the leading edge  41  and parallel to line TH above.   β: The angle in a longitudinal vertical plane of the vessel  1  between TH and the water surface  5  when the vessel  1  is floating motionless in a body of water.   Γ: The angel between the chord line  43  and the water surface  5  in a longitudinal vertical plane of the vessel  1  when the vessel  1  is floating motionless in a body of water. A positive chord angel means that the chord line  43  is pointing upwards in the vessel&#39;s  1  direction of travel, 0 degree angel means the chord line  43  is parallel to the water surface  5  and a negative angel means that the chord line  43  is pointing downwards in the vessels  1  direction of travel.   H 1 : Is the smaller of:
       i) a first minimum distance, measured in a longitudinal vertical plane of the vessel  1 , between the leading edge  41  and the aft hull section  3  at or upstream/in front of the separation line  6  and   ii) a second minimum distance, measured in the same longitudinal vertical plane as i) above, between the two parallel lines TH and TF.   
       H 1 ( w ): The minimum distance H 1  of i) or ii) at point w along the width of the leading edge  41 .   H 2 : The vertical distance between the trailing edge  42  and the water surface  5  measured when the vessel  1  is floating motionless in a body of water at the lightweight waterline (LWWL).   A le : The leading edge area is derived from integrating H 1 ( w ) above from WL at one side of the leading edge  41  to WH at the opposite side of the leading edge  41 . The leading edge area is equal to or substantially equal to the cross sectional area of a water flow  51  above the leading edge  41  at operational speed of the vessel  1 .   A te : The trailing edge area is the area as seen from astern constrained by the trailing edge  42 , the two vertical planes  49  and the water surface  5  when the vessel  1  is floating motionless in a body of water at the lightweight waterline (LWWL).   

     Displacement speed:
         The speed range of a displacement vessel  1 , usually limited by a “hull speed” of about F N =0.4.       

     Transition speed:
         The speed range of a planing hull  2  when it is in transition from displacement speed, usually at around F N , =0.4, until it reaches fully planing speed, usually at around F N =0.9.       

     Planing speed:
         The speed where dynamic lift contributes to a major part of the buoyancy for a planing hull  2 , usually above F N =0.9. Since the invention, especially when operated in combination with a bow body  10  as described in patent publication EP3247620B1, does not rely on lifting the hull  2  out of the water even at speeds above F N =0.4, the reference to transition or planing speed for a hull  2  according to the invention only refers to the speed itself. Hence, it does not refer to planing of the hull  2 .       

     Operational speed:
         The speed interval the vessel  1  is operating at during transit when there are no speed restrictions.       

     General Design Criteria 
     The working principle and the main objective for the invention is the same for both slow or fast vessels  1 . However, certain design issues should be taken in consideration when optimizing the inventive vessel  1 . 
     Leading Edge Area/Trailing Edge Area 
     The common principle for all embodiments of the invention is to allow a sufficient water flow  51  to flow over the top surface  45  of the aft body  4  through the passage  50  during forward propulsion of the vessel  1 . The cross sectional area of a water flow  51  passing the leading edge  41  of the aft body  4  should be equal to, or almost equal to, the area from the trailing edge  42  of the aft body  4  and up to the water surface  5  in order to achieve equilibrium in the water mass downstream the vessel  1  and thereby prevent the formation of a stern wave  9  when the vessel  1  is in forward motion. By equilibrium is meant that the water flow surface level  53  above the trailing edge  42  is at the same level as the (surrounding) water surface  5  during forward motion of the vessel  1 , for example at operational speed. In  FIG.  8 A-C  this is achieved by having H 1 =H 2 . In  FIG.  10    this is achieved by having A le =A te . 
     In general, the width of the aft body  4  in the transverse direction of the vessel  1  will be equal to, or almost equal to, the width of the transom  7  of the vessel  1 . 
     Adaption to Different Speed Ranges 
     Advantageous embodiment for adapting the vessel for different speed ranges are shown and explained by aid of  FIG.  7 A-C . 
     When operating a vessel  1  according to the invention having a displacement hull at a low speed (F N &lt;0.3), the aft body  4  may be positioned closer to the water surface  5 . When the aft body  4  is placed closer to the water surface  5 , the length of the chord line  43  can be reduced compared to a deeper positioning of the aft body  4 . When operating the vessel at higher speed (F N ≥0.3), there might be advantageous to position the aft body  4  deeper and to increase the length of the chord line  43 . 
       FIG.  7 A  show one embodiment of the inventive vessel  1  having a typical displacement hull  2 . The particular arrangement and design of the aft body  4  is optimized for operation at a speed below an F N , of about 0.3. The aft body  4  can in this particular case have a relatively short chord line  43  and be placed at a depth within the upper half of the draft DV of the hull  2 , for example at 30% of the draft DV. 
     As the speed increases, the upward momentum in a water flow  51  passing under the aft hull section  3  upstream the aft body  4  increases. The impact from the aft body  4  onto a water flow  51  should then be enhanced. This is achieved by increasing the length of the chord line  43  and to locate the aft body  4  closer to the base line  58 , as the top surface  45  of the aft body  4  will redirect the water flow  51  more effectively then the underside  46  of the aft body  4 . The increased cord line  43  and deeper located aft body  4  will then effectively counteract the increasing upward momentum of the water flow  51 , thereby achieving an essential horizontal direction of a water flow  51  downstream the aft body  4 . 
       FIG.  7 B  shows the same hull  2  as shown in  FIG.  7 A  but with an aft body  4  designed for a higher speed range; typically, within the speed range F N =0.4−0.6. In order to minimize the propulsion resistance, the aft body  4  is in this particular case located at a depth corresponding to about 50-60% of the draft DV of the hull  2  and the length of the cord line  43  is extended relative to the length shown in  FIG.  7 A . 
       FIG.  7 C  shows the same hull  2  as in  FIG.  7 A  but with an aft body  4  designed for operational speed above F N =0.7. Here the length of the chord line  43  is extended further relative to the length shown in  FIG.  7 B , and the underside  46  of the aft body  4  is located at the base line  58  of the hull  2 . 
     As a rule of thumb, the cord line  43  should be greater than the draft of the aft body  4 , typical by a factor of around 2.0 or greater. 
     At low speed an aft body  4  with a long chord line  43 , and placed relatively deep, only contributes to a minor increase in resistance compared to a smaller and higher placed aft body  4 . If the vessel  1  is supposed to be operated over a wide speed range, it might be advantageous to choose an aft body  4  with long chord line  43  placed at a greater depth optimized for the highest operational speed of the vessel  1 . 
     The optimal depth and optimal length of the chord line  43  for minimizing the total resistance R t  for the vessel  1  may for example be determined by model tests and/or computational fluid dynamics (CFD) analyses. 
     Location of the Separation Line 
     The invention includes a separation line  6  at the aft hull section  3  controlling the separation of a water flow  51  from the aft hull section  3  at a defined line in the transverse direction (w) of the vessel  1  during forward motion of the vessel  1 . The separation line  6  is preferably located close to the water surface  5  when the vessel  1  without payload is laying still and floating in a mass of water. 
     The separation line  6  can either be placed upstream/in front of, vertically above, or downstream/aft of the leading edge  41  as shown in  FIG.  8 A-C . For practical reason the separation line  6  will normally be within the length of one chord line  43  of the leading edge  41 , and preferably within the length of half a chord line  43 . 
       FIG.  8 B-C  shows the geometry of the aft hull section  3  in relation to the top surface  45  of the aft body  4  where the leading edge  41  is located upstream/in front of the separation line  6 . In such design, it is preferable that the minimum distance, in a vertical longitudinal plane of the vessel  1 , between the top surface  45  and the aft hull section  3  upstream/in front of the separation line  6  is held constant to avoid a change in the velocity of a water flow  51  in the passage  50  as such a change in velocity will result in increased resistance for the vessel  1 , especially if the velocity of the water flow  51  is reduced in the passage  50 . It should also be noted that the wetted surface, and accordingly the frictional resistance R f , will increase in the design shown in  FIG.  8    B-C. 
     Turbulence—Design of the Aft Body and its Supports 
     When designing the aft body  4 , including the supports  8  to fix the aft body  4  to the hull  2 , it is advantageous to avoid creation of turbulence and vortexes. 
     If the outer ends of the aft body  4  in the transverse direction of the vessel  1  extends freely in the water during operation, it might be advantageous to reduce the thickness of the aft body  4  in a vertical plane towards the outer ends and/or to make the aft body  4  elliptical when seen from below (as shown in  FIG.  14 B  drawing (c)) and thereby limit creation of tip vortexes. 
     Also winglets, as used in aviation, can be used to reduce the tip vortexes. It would then be natural to also make use of the winglets as supports  8  for the aft body  4 . 
     The aft body  4  should preferably also be shaped according to shape of the aft hull section  3  upstream/in front of the separation line  6  and the resulting angle of attack of the water flow  51  (i.e. the angel between the water flow  51  upstream leading edge  41  and the chord line  43 ). Higher angel of attack requires increased length of the chord line  43 . Furthermore, in order to obtain laminar water flow  51  without turbulence, and to prevent cavitation on the top surface  45 , especially at higher velocity of the water flow  51 , a thicker aft body  4  profile and/or more curved top surface  45 , especially toward the leading edge  41 , would be beneficiary. Alternatively, a high angle of attack for the front part of the aft body  4  can be avoided by keeping the angle γ of the tangent line TH low. 
     When attaching the aft body  4  to the hull  2  some care should be taken when designing the support  8 . Besides ensuring sufficient structural integrity, the support  8  should preferably be made with a streamlined design. In addition, the support  8  should be oriented according to the direction of a water flow  51  where the supports  8  are located to avoid unnecessary propulsion resistance. It should be noted that under the aft hull section  3  of a displacement hull  2 , a water flow  51  can become partly inwardly directed towards the longitudinal center line of the vessel  1 . 
     If the aft body  4  is placed between the hull sides  2 ′, 2 ″, as shown in  FIG.  9 C ,  FIG.  9 D ,  FIG.  15 A  drawing (b) and  FIG.  15 B  drawing (c), it might be advantageous to round the lowest part of the inward facing hull sides  2 ′, 2 ″ in order to make a streamlined inlet at the sides for a water flow  51  to enter the passage  50 . 
       FIG.  9 A  shows the aft body  4  fixed to the hull  2  by two vertical support plates  8 . The number of support plates  8  are decided according to the demand for structural integrity. Each support plate  8  is preferably orientated according to the local direction of water flow  51 . 
     Alternatively, the aft body  4  may be fixed to the transom  7  of the hull  2 . In  FIG.  9 B  such a configuration is exemplified by two triangular support  8  plates, where the curved horizontal edge is fixed to the top surface  45  of the aft body  4 , and the vertical edge is fixed to the transom  7 . 
     In order to prevent a rise of the water flow surface level  53  at the outer side of the hull sides  2 ′, 2 ″ at the aft hull section  3  that might accrue during forward motion, and further to prevent this rise of the water flow surface level  53  to be deflected outward as stern wave  9  from the hull sides  2 ′, 2 ″, it might be advantageous to taper the hull sides  2 ′, 2 ″ of the aft hull section  3  inward towards the longitudinal center line of the vessel  1 . An example of such a tapering of the hull sides  2 ′,  2 ″ is shown in  FIG.  9 D . 
     Adaption to Variation in Draft 
     Some vessels  1  experience a significant variation in draft DV when being operated due to different load conditions. To optimise the vessel  1  for such draft variations it would be advantageous to be able to adjust the amount of water passing over the aft body  4  (i.e. altering H 1 ) according to the vessel&#39;s  1  draft DV, as well as the height H 2  from the trailing edge  42  of the aft body  4  to the water surface  5 . 
     By making the aft body  4  adjustable in a horizontal longitudinal direction of the vessel  1 , an optimal water flow  51  can be led over the aft body  4  at different drafts DV of the hull  2 . At shallow draft DV of the hull  2 , the leading edge  41  can be arranged close to the hull  2 , for example vertically below the separation line  6 . As the vessel  1  is operated at a deeper draft DV of the hull  2 , the leading edge area A le  can be increased by moving the aft body  4  horizontally further downstream the separation line  6 . 
     Alternatively, or in addition, the front part of the aft body  4 , or the entire aft body  4 , can be made tiltable with a rotational axis parallel to the transverse direction of the vessel  1  and parallel to the water surface  5 . When the leading edge  41  is tilted down, a larger water flow  51  is allowed to pass over the top surface  45 . If the entire aft body  4  is tilted around said rotational axis close to aft body&#39;s  4  centre line, the trailing edge  42  will approach the water surface  5  while the leading edge  41  will become deeper as the chord line  43  of the aft body  4  is tilted downward (i.e. a smaller or more negative chord angel γ). This will contribute to a larger leading edge area A le  and a reduced trailing edge area A te . However, to tilt the aft body  4  downward has the disadvantage of creating a non-desired upward direction of a water flow  51  downstream the trailing edge  42 . 
     If the aft body  4  is fixed and the hull  2  is to be operated at different drafts DV, a compromise has to be found. An advantageous compromise could be to adapt the leading edge area A le  in view of the trailing edge area A te  for a draft DV corresponding to the deepest draft DV of the hull  2 , or at least deeper than minimum operational draft DV of the hull  2 . 
     Although the aft body  4  counteracts a stern down trim, the vessel  1  might experience some increased stern down trim as the speed rises, thereby increasing the distance from the trailing edge  42  to the water surface  5  at high speed. With this in mind, it might be advantageous to allow the leading edge area A le  to be greater than the trailing edge area A te  when the vessel  1  is floating motionless in a body of water. Alternatively, the leading edge area A le  can be increased as mentioned above as the speed of the vessel  1  increases. Also the geometry of the aft hull section  3  can be made with a flap or similar to make the leading edge area A le  adjustable. 
     Location of Propeller 
     The vessel&#39;s  1  propeller  12  can be located upstream/in front of the aft body  4  as shown in one embodiment in  FIG.  26   . Furthermore, the propeller  12  can be located vertically under the aft body  4  or vertically above the aft body  4 . Note that positions vertically under or above the aft body  4  include any positions along a horizontal plane. 
     Initial testing performed indicates that a location of the propeller  12  under the aft body  4  can be advantageous as the same thrust force [N] is generated from the propeller with a smaller power consumption [W] from the propulsion engine. 
     A Vessel Having Both an Aft Body and a Bow Body 
     The arrangement of an aft body  4  according to the invention as described will reduce the total resistance R t  for a vessel  1  above a certain speed of the vessel  1 . Further, the inventive vessel  1  will counteract a stern down trim of the vessel  1  if the vessel  1  is being operated at higher speeds, for example above F N =0.3. As a negative consequence, the inventive vessel  1  can experience a larger bow down trim than a prior art vessel  1  not fitted with an aft body  4 . Even if the total resistance R t  of the inventive vessel  1  is lower, the bow down trim of the inventive vessel  1  will result in an increased bow wave  22  and thereby an increased wave resistance R w  from the bow area  21 . 
     The invention disclosed in patent publication EP3247620B1 concerns a bow design with a bow body  10  that counteracts creation of a bow wave  22 , thereby reducing the wave resistance R w  from the bow area  21  and the total resistance R t  for the vessel  1 . However, this particular bow design suffers the disadvantage that such a vessel  1  can experiences an increased stern down trim during forward propulsion, thereby creating greater wave resistance R w  from the stern. In other words, by reducing the formation of waves at one end of the vessel  1 , the formation of waves at the other end of the vessel  1  is often increased. 
     By combining these two inventions (i.e. an aft body  4  and a bow body  10 ) on the same vessel  1  as shown in  FIG.  11   , model tests have shown that a significant synergy effect is achieved. I.e. the reduction in total resistance R t  becomes greater than adding the individual contribution from each invention one at the time on the same vessel  1 . 
     Numerous model tests measuring the total resistance R t  has been performed and an overview picture of these tests are shown in  FIG.  12   . The solid line marked Rt(A) is the total resistance R t  for a prior art displacement vessel  1 . The stippled line Rt(B) is the total resistance R t  for an inventive displacement vessel  1  having an aft body  4  and the dotted line marked Rt(C) is the total resistance R t  for an inventive displacement vessel  1  having both an aft body  4  and a bow body  10 . As can be seen from the graph the displacement vessel  1  having both an aft body  4  and a bow body  10  has superior performance above F N  of about 0.26, while the vessel  1  has somewhat higher total resistance R t  under that speed. 
     A prior art planing hull  2 , as shown in  FIG.  3   , relies upon reaching planing speed to obtain reasonable good resistance/speed ratio. The speed needed to obtain sufficient dynamic lift depends to a large degree on the weight of the vessel  1 . This highly limits the load capacity of a prior planing vessel  1 . In contrast, an inventive vessel  1  as disclosed in  FIG.  11    does not rely on lifting the hull  2  out of the water to achieve a reasonable good resistance/speed ratio. Since there is no need to lift the inventive vessel  1  out of the water, the resistance of the vessel  1  is far less depended on the weight of the vessel  1 . The inventive vessel  1  described herein combined with a bow body  10  thus makes it possible to operate a vessel  1  even at heavy load conditions throughout a wide speed range with far better fuel economy than a prior art vessel  1 . 
     Also for planning vessels  1  numerous model tests has been performed.  FIG.  13    gives an overview based on these model tests, where the solid line marked Rt(A) is the total resistance R t  for a prior art planing vessel  1 . The stippled line Rt(B) is the total resistance R t  for an inventive planing vessel  1  having an aft body  4  and the dotted line marked Rt(C) is the total resistance R t  for an inventive planing vessel  1  having both an aft body  4  and a bow body  10 . From this can be seen that the inventive vessel  1  having both an aft body  4  and a bow body  10  has better performance than a prior art planning vessel  1  above F N  of about 0.25. 
     Detailed Description of a First Embodiment 
       FIG.  14 A  and  FIG.  14 B  shows an aft hull section  3  of a vessel  1  according to a first embodiment. 
       FIG.  14 A  drawing (a) shows a vertical longitudinal plane of the aft hull section  3  when the vessel  1  is floating motionless in a body of water. The vessel  1  comprising a hull  2 , hull sides  2 ′, 2 ″ and a transom  7 . The separation line  6  is located slightly above the water surface  5 . The vessel  1  comprises an aft body  4  having an underside  46  oriented parallel with the water surface  5  and located at approximately 50% of the draft DV of the hull  2 . The aft hull section  3  has decreasing cross sectional area towards the stern of the vessel  1 . The angel between the tangent line TH of the aft hull section  3  immediately upstream/in front of the separation line  6  and the water surface  5  is marked β. The aft body  4  is located at a distance to the hull  2  making a passage  50  between the aft hull section  3  and the top surface  45  of the aft body  4 . The minimum distance between the top surface  45  and the aft hull section  3  is kept constant upstream/in front of the separation line  6  in order to achieve a constant velocity of a water flow  51  in the passage  50  when the vessel  1  is at operational speed. The minimum distance H 1  between the leading edge  41  and the aft hull section  3  is equal to the distance H 2  being the distance from the trailing edge  42  to the water surface  5 . 
       FIG.  14 A  drawing (b) is the aft hull section  3  shown in  FIG.  14 A  drawing (a) seen from behind.  FIG.  14 A  drawing (b) is showing the hull  2  having draft DV and a transom  7 . The separation line  6  is located slightly above the water surface  5 . The aft body  4  is attached to the vessel  1  by two supports  8 . The supports  8  are placed with equal offsets to the longitudinal centre axis of the vessel  1 . The outer ends of the aft body  4  in transverse direction of the vessel  1  have downward tapered top surface  45  to reduce turbulence. The two imaginary vertical planes  49  in the longitudinal direction of the vessel  1  intersecting the two points defining the maximum width (W) of the aft body  4  are marked with stippled lines. 
       FIG.  14 B  drawing (c) shows the aft hull section  3  of  FIG.  14 A  seen from below. The hull  2 , hull sides  2 ′, 2 ″ and the underside  46  of the aft body  4  is shown with solid lines. The two streamlined supports  8  fixing the aft body  4  to the hull  2  are shown with stippled lines. The supports  8  are orientated in the direction of travel of the vessel  1 . The separation line  6  is shown in the transverse direction of the vessel  1  with a stippled line. Also the two imaginary vertical planes  49  are shown with stippled lines. The outer ends in the transverse direction of the aft body  4  have rounded shape when seen from below to reduce turbulence. The leading edge  41  and the trailing edge  42  is extending all the way out to the two points defining the maximum width (W) of the aft body  4  (i.e. all the way out to the hull sides  2 ′ and  2 ″. 
       FIG.  14 B  drawing (d) is the same vertical longitudinal plane of the aft hull section  3  of the inventive vessel  1  as shown in  FIG.  14 A  drawing (a) and having the same draft DV. But here the working principle and the water flow  51  indicated by arrows at the aft hull section  3  and around the aft body  4  at operational speed is illustrated. The water flow  51  direction upstream the leading edge  41  has a partly upward direction due to the tapered design of the aft hull section  3 . This partly upwardly directed water flow  51  entering over the leading edge  41  of the aft body  4  is redirected by the shape of the top surface  45  from partly upward to a horizontal (or slightly downward) direction immediately downstream the trailing edge  42 . The horizontally orientated underside  46  of the aft body  4  will redirect the upwardly directed water flow  51  upstream the aft body  4  that passes under the leading edge  41  to a horizontal (or almost horizontal) direction immediately downstream the trailing edge  42 , thus achieving a resulting horizontal direction of the combined water flow  51  passing over and under the aft body  4  merging downstream the trailing edge  42 . (As a water flow  51  passing under the trailing edge  42  may still have a slightly upwardly direction it can be advantageous that a water flow  51  passing over the trailing edge  42  has a slightly downwardly direction to ensure that the merged water flow  51  downstream the trailing edge  42  has a horizontal direction). The constant minimum distance between the aft hull section  3  and the top surface  45  upstream the separation line  6  contributes to a constant speed of the water flow  51  through the passage  50 . The water flow surface level  53  is elevated slightly above the surrounding water surface  5  over a part of the top surface  45  as indicated. The height of the water flow  51  over the leading edge  41  is marked H 1  and is equal to the height of the water flow  51  over the trailing edge  42  marked H 2 . Since the water flow surface level  53  passing over the trailing edge  42  is at the same level as the surrounding water surface  5 , a state of equilibrium in the water mass downstream the trailing edge  42  is obtained. The creation of a stern wave  9  is thereby greatly reduced. 
     Detailed Description of a Second Embodiment 
       FIG.  15 A  and  FIG.  15 B  shows an aft hull section  3  of a vessel  1  according to a second embodiment. This vessel  1  is similar to the vessel  1  of the first embodiment, with the following main exceptions: 
     As best shown in  FIG.  15 A  drawing (a) (showing a longitudinal vertical plane of an aft hull section  3 ), the separation line  6  is arranged upstream/in front of the leading edge  41 . The separation line  6  is further located slightly below the water surface  5  when the vessel  1  is floating motionless in a body of water having a draft DV. The angel β between the tangent line TH and the water surface  5  is shown. As the separation line  6  is located upstream/in front of the leading edge  41 , the distance H 1  is here defined by the minimum distance between the tangent line TH and a line parallel with TH intersecting the leading edge  41  marked TF. 
     As best shown in  FIG.  15 A  drawing (b) (being the aft hull section  3  shown in  FIG.  15 A  drawing (a) seen from behind), the hull sides  2 ′, 2 ″ are used as supports  8  for the aft body  4 . 
     As best shown in  FIG.  15 B  drawing (c) (showing the aft hull section  3  of  FIG.  15 A  seen from below) the transverse ends of the aft body  4  are straight and oriented along the longitudinal direction of the vessel  1  and are fixed directly to the hull  2 . Each hull side  2 ′, 2 ″ at the aft hull section  3  below the water surface  5  are tapered towards the longitudinal centre axis of the vessel  1  shown with stippled lines. 
       FIG.  15 B  drawing (d) (showing the same vertical longitudinal plane of the aft hull section  3  of the vessel  1  as shown in  FIG.  15 A  drawing (a)) has the same draft DV, but here the working principle and a water flow  51  indicated by arrows at the aft hull section  3  and around the aft body  4  at operational speed is illustrated. The working principle for this second embodiment is the same as for the first embodiment, and the text explaining the working principle of  FIG.  14 B  drawing (d) could be duplicated except for: “The constant minimum distance between the aft hull section  3  and the top surface  45  upstream/in front of the separation line  6  contributes to a constant speed of a water flow  51  through the passage  50 .” which is not relevant for this second embodiment. 
     Detailed Description of a Third Embodiment 
     In a third embodiment, the inventive vessel  1  comprises both an aft body  4  as described herein and a bow body  10  as described in patent publication EP3247620B1, the contents of which are incorporated herein by reference, in particular the  FIGS.  10 - 15    and its related text in EP3247620B1. 
     As a specific example of the third embodiment, reference is made to  FIG.  11    showing a bow body  10  arranged at the bow area  21  of the vessel  1  and an aft body  4 . A water flow  51  around the bow body  10  and a water flow  51  around the aft body  4  are shown with arrows. 
     Model Tests—Total Resistance of Model Vessels 
     To document the mode of operation of the inventive vessel  1  and to verify a reduction in total resistance R t , the inventor has carried out model tests on the model vessels shown in  FIG.  16   ,  FIG.  20    and  FIG.  22   . 
     To be able to monitor thrust from a propeller  12 , all model vessels  1  (except for the planning model vessels  1  shown in  FIG.  22   ) are equipped with powertrain set up as shown in  FIG.  17   . The propeller  12  is connected to an electrical motor  14  by a propeller shaft  11 . Around the propeller shaft  11  there is a propeller sleeve  13  with brass bearings to support the propeller shaft  11 . The propeller sleeve  13  does not absorb any thrust from the propeller  12 . The electrical motor  14  is directly attached to the motor housing  15 . The motor housing  15  is attached to four mounting brackets  18  through four motor suspension systems  16  which are configured to absorb torsional moment but not thrust from the propeller. The four mounting brackets  18  are directly connected to the base plate  19 . The motor suspension system  16  makes the motor housing  15  hover over the base plate  19  without restricting movement for the motor housing  15  in longitudinal direction of the propeller shaft  11 . The base plate  19  and the propeller sleeve  13  are attached to the model vessel  1 . In front of the motor housing  15  there is a high precision load cell  17  attached to the base plate  19  that limits the forward movement of the propeller  12 , the propeller shaft  11 , the electrical motor  14  and the motor housing  15 . The propeller shaft  11  is mounted close to horizontal when the model vessel  1  is laying still and floating in a body of water. 
     During operation, the motor housing  15  applies pressure onto the load cell  17  and all thrust from the propeller  12  is transferred to the load cell  17 . Consequently, the propeller thrust in Newton [N] is monitored and logged during operation of the model vessel  1 . When the model vessels  1  is operated at constant speed the propeller thrust is equal to the total resistance Rt for the model vessel  1 . 
     The speed of all model vessels  1  is measured with high accuracy Doppler GPS. The speed in meters per second [m/s] is converted to Froude number (F N ) for each model vessel  1 . 
     The measured results of the total resistance R t  for the model vessels  1  shown in  FIG.  16    and  FIG.  20    measured in Newton [N] as function of speed (F N ) are logged and plotted in  FIG.  19    and  FIG.  21    respectively, using the test setup as described above. 
     In  FIG.  19   , and in all other graphs, Poly. ( . . . ) means interpolation of measurement points. 
     For the planing model vessels  1  shown in  FIG.  22   , the power consumption [W] of an electric brushless motor attached to a Z-drive and corresponding speed (F N ) is logged using the same high accuracy Doppler GPS. The result is and plotted in  FIG.  24   . 
     All model vessels  1  are radio-controlled. 
     Model Test 1—Double Propelled Slender Displacement Hull 
       FIG.  16    shows an upside down perspective illustrations of a prior art model vessel  1  with a slender displacement hull marked as Model  16 A. Model  16 B is the same model vessel  1  as Model  16 A but fitted with an aft body  4  according to the invention. The model vessel  1  is equipped with two propulsion systems for measuring thrust as described above. 
     The length, width and draft DV of both model hulls  2  are 270 cm, 42 cm and 11 cm, respectively. The full-scale vessel  1  of this model vessel  1  has the separation line  6  located at the water line  5  and so has the model vessel  1 . Consequently, in order to apply an aft body  4  as described above, there is no need to do any cut-out of the aft hull section  3  in order to make A le  equal to A te . 
     With reference to the Model  16 B, the length of the chord line  43  of the aft body  4  is 10 cm. The aft body  4  is attached to the aft hull section  3  such that the leading edge  41  is located 1 cm upstream/in front of the separation line  6 . Further, the maximum (W) of the aft body  4  in the transvers direction (w) of the hull  2  is 42 cm, which is equal to the width of the model vessel  1 . The underside  43  of the aft body  4  is placed 2.7 cm below the water surface  5  and the cord angel γ is orientated parallel to the water surface  5  when the model vessel  1  is floating motionless in a mass of water. The maximum vertical thickness of the aft body  4  is 1.0 cm. The aft hull section  3  has a double curvature in the longitudinal vertical plane of the model vessel  1  and the angel β between the tangent line TH and the water surface  5  is 0 degrees (i.e. parallel with the water surface  5 ). 
     During test runs of the prior art Model  16 A maintained close to neutral trim throughout the entire speed range of the test. However, the model vessel  1  experiences some degree of increasing draft DV as speed increased. 
     Model  16 B according to the invention obtained some bow down trim and increased draft for the bow area  21  as speed increased, leading to an increased bow wave  22  compared to the prior art model vessel  1  at comparable speeds. 
     As clearly seen from the pictures in  FIG.  18 A  and  FIG.  18 B  there is a significant reduction of the stern wave  9  for the inventive Model  16 B compared to the prior art Model  16 A at both speeds (i.e. F N =0.30 and 0.36). 
       FIG.  19    shows the results from the model testing, logged as total resistance R t  [N] as a function of speed (F N ). As clearly seen, even if Model  16 B generates a larger bow wave  22  than Model  16 A at comparable speeds, Model  16 B experiences a reduction in the total resistance R t  compared to Model  16 A in the full speed range from about F N =0.19 and up to about F N =0.37. Above F N =0.3 Model  16 B experiences about 8% lower total resistance R t  than the prior art Model  16 A. (For a full-scale vessel  1 , the reduction in total resistance R t  will be even greater). 
     Model Test 2—Single Propelled Displacement Hull 
       FIG.  20    shows upside down perspective illustrations of three displacement model vessels  1 , where Model  20 A shows a prior art model vessel  1 , Model  20 B shows a prior art model vessel  1  with a bow body  10  and Model  20 C shows an inventive model vessel  1  with both a bow body  10  and an aft body  4 . 
     The bow body  10  on Model  20 B is in accordance with the patent publication EP3247620B1. Further, the configuration of the bow body  10  and bow area  21  is similar to the bow configuration shown in  FIG.  11   . 
     Model  20 C has the same bow body  10  and bow area  21  as Model  20 B, but with an aft hull section  3  similar to the aft hull section  3  shown in  FIG.  11   . 
     All three model vessels  1  have a hull length of 184 cm. The width for Model  20 A is 36 cm and 34 cm for both Model  20 B and Model  20 C. Further, all three model vessels  1  have the same weight and thereby the same displacement volume, resulting in a draft DV of 14 cm for Model  20 A, 15 cm for Model  20 B and 15.2 cm for Model  20 C. 
     The separation line  6  for Model  20 A and Model  20 B is located respectively 2.0 cm and 1.5 cm under the water surface  5  and for Model  20 C at the water surface  5  when the model vessels  1  are floating motionless in a body of water. 
     Moreover, the length of the chord line  43  of the aft body  4  of Model  20 C is 11 cm, and the maximum width (W) of the aft body  4  in the transvers direction (w) of the hull  2  is 33 cm. The chord angel γ is oriented parallel to the water surface  5  and the underside  46  of the aft body  4  is located 7 cm under the water surface  5  when the model vessel  1  is floating motionless in a body of water. The separation line  6  is located 5 cm upstream/in front of the transom  7  and the leading edge  41  of the aft body  4  is located vertically below the separation line  6 . The maximum vertical thickness of the aft body  4  is 1.1 cm. The angel β between the tangent line TH and the water surface  5  is 8.5 degrees. 
     During model tests, all three model vessels  1  have a neutral trim when floating motionless in a body of water. 
     In order to compare the inventive Model  20 C having an aft body  4  with a prior art Model  20 B (without an aft body  4 ), and to exclude the tendency of the inventive Model  20 C to generate a bow down trim of the vessel  1  due to the aft body  4 , thereby preventing a stern down trim, the angel of attach of the bow body  10  for Model  20 B and Model  20 C are adjusted separately to obtain close to neutral trim and unchanged draft for their bow area  21  when they are in motion throughout the testing speed range. The wave making from the bow area  21  is then similar for the Model  20 B and Model  20 C. The bow body  10  contributes to a great reduction of wave resistance R w  from the bow area  21 . The main differences in total resistance R t  between the Model  20 B and the inventive Model  20 C is thus isolated to be the difference between an aft hull section  3  without and with an aft body  4 . 
       FIG.  21    shows the results from the model testing, logged as total resistance R t [N] as a function of speed (F N ). As clearly shown in the graphs, the total resistance R t  of the prior art Model  20 A is significantly higher at a speed above F N =0.3. In the speed range from F N =0.3 to F N =0.4 the total resistance R t  of Model  20 B and Model  20 C are almost the same and the aft body  4  does not have a significant effect for this embodiment in this speed range. Above F N =0.4, wave resistance R w  from the aft hull section  3  becomes more crucial. Model  20 C is seen to have a significantly lower total resistance R t  compared to the prior art Model  20 B at a speed above F N =0.4. The reason for the lower total resistance R t  is the decrease in wave resistance R w  from the aft hull section  3  of the inventive Model  20 C. Hence, it is possible to design an inventive model vessel  1  (Model  20 C) having the same total resistance R t  as a prior art displacement model vessel  1  (Model  20 A) at F N =0.3 but with 46% lower total resistance R t  at F N =0.45. (For a full-scale vessel  1  the reduction in total resistance R t  will be even greater). 
     Model Test 3—Planing Hull 
       FIG.  22    shows upside down perspective illustrations of two model vessels  1 , where Model  22 A is a “flat bottomed” prior art planing hull and Model  22 B is an inventive model vessel  1  with a bow body  10  and an aft body  4 . The bow area  21  of Model  22 B is modified with a bow body  10  according to patent publication EP3247620B1 and the aft hull section  3  is modified with an aft body  4  as described herein. 
     Both model vessels  1  have a length of 120 cm and a width of 40 cm. Further, the weight, and accordingly the displacement volume, is the same for the two model vessels  1 , giving a corresponding draft DV of 5.5 cm for Model  22 A and 6 cm for Model  22 B when the model vessels  1  are floating motionless in a body of water. 
     The separation line  6  of Model  22 A is located 5.5 cm under the water surface  5  and for Model  22 B the separation line  6  is located at the water surface  5  when the model vessels  1  are floating motionless in a body of water. 
     The angel β between the tangent line TH and the water surface  5  is 20 degrees for Model  22 B. The aft hull section  3  of Model  22 B has a similar layout as shown in  FIG.  7 C . The length of the cord line  43  is 14 cm and the chord angle γ is parallel to the water surface  5 . The underside  46  of the aft body  4  is located at the base line  58 . The trailing edge  42  is arranged 3 cm downstream/aft of the separation line  6 . The maximum vertical thickness of the aft body  4  is 1.8 cm. 
     During testing, both prior art Model  22 A and inventive Model  22 B are trimmed to neutral when floating motionless in a body of water. 
       FIG.  23 A ,  FIG.  23 B  and  FIG.  23 C  show pictures of prior art Model  22 A and inventive Model  22 B at speed corresponding to F N =0.4, F N =0.5 and F N =0.65 respectively. At low speed, F N =0.4, the trim and wave making appears to be almost the same for both model vessels  1 . At higher speed, F N =0.5, the prior art Model  22 A gains an increasing positive trim with a significant increase in wave making. At a speed of F N =0.65, the bow area  21  of prior art Model  22 A is lifted out of the water and the aft hull section  3  of the hull  2  causes a significant stern wave compared to the inventive Model  22 B. The pictures hence demonstrate that the inventive model vessel  1  with a bow body  10  and an aft body  4  counteracts increasing sinkage of the aft hull section  3  as speed increases, while the trim and wave making stays almost the same regardless of speed. 
       FIG.  24    shows the results from the model testing logged as required power [W] for the brushless electrical propulsion engine versus speed (F N ) for Model  22 A and Model  22 B. The required power [W] for the inventive Model  22 B is lower than for the prior art Model  22 A throughout the entire speed range tested. Particularly in the speed range from F N =0.4 to F N , =0.7. At F N , =0.6 the inventive Model  22 B requires 40% less power [W] than the prior art Modell  22 A. (For a full-scale vessel  1  the reduction in power [W] will be even greater). 
     Model Tests—Horizontal Forces Provided by an Aft Body 
     To document how the configuration of an aft hull section  3  and an aft body  4  effects the horizontal forces provided by an aft body  4  in a longitudinal direction of a model vessel  1 , a series of model tests have been conducted. The model tests are performed on a model vessel  1  with configurations according to the invention and configuration according to prior art, where the aft body  4  is providing a continuous propulsion force on the model vessel  1  (i.e. a continuous forwardly directed horizontal forces in the longitudinal direction of the model vessel  1 ). 
     Model Vessel—Testing Set Up 
       FIG.  25    shows an upside-down perspective illustration of the model vessel  1  having a test setup to measure the horizontal forces in the longitudinal direction of the model vessel  1  from the aft body  4  acting on the model vessel  1 . The same model vessel  1  is used for all the model tests. The model vessel  1  is configured with a hard chine bow area  21  to prevent a bow down trim of the model vessel  1  during testing (i.e. increased sinkage for the bow area  21 ). 
     Dimensions of the model vessel  1 :
         Length: 185 cm   Width: 34 cm   Draft DV: 8-10 cm   Maximum width (W) of all aft bodies  4 : 34 cm       

     Maximum vertical thickness of all aft bodies  4 : 1.1 cm, except for aft body  4  marked (B) in  FIG.  31    which is 1.4 cm thick. 
       FIG.  26    shows a side view of the aft hull section  3  of the model vessel  1  in  FIG.  25    where the aft body  4  and the two supports  8  are attached to the vessel  1  via two ball bearing slides  20  oriented in a horizontal longitudinal direction of the model vessel  1  when the model vessel  1  is floating motionless in a mass of water. The ball bearing slides  20  are separated by 18 cm in the transvers direction of the hull  2 . The setup enables the aft body  4  to move freely in the horizontal longitudinal direction of the model vessel  1 . A high precision load cell  17  is mounted to the vessel  1  and is further attached to the support  8  arrangement in order to measure the horizontal forces generated by the aft body  4  in the longitudinal direction of the model vessel  1 . A compression force measured in the load cell  17  gives a positive value reading corresponding to the aft body  4  providing a resistance force on the model vessel  1  (i.e. a backwardly directed horizontal forces in the longitudinal direction of the model vessel  1 ). While a tension/stretch force measured in the load cell  17  gives a negative value reading corresponding to the aft body  4  providing a propulsion force on the model vessel  1  (i.e. a forwardly directed horizontal forces in the longitudinal direction of the model vessel  1 ). 
     The model vessel  1  shown in  FIG.  25    and  FIG.  26    is fitted with an interchangeable aft hull section  3  below the water surface  5  in order to compare different geometries of the aft hull section  3  and accordingly different angels β between TH and the water surface  5 . It is further possible to alter the chord angel γ, the draft DV of the hull  2 , the depth of the aft body  4  in relation to the base line  58  and changing to an aft body  4  with a longer chord line  43 . The different configurations tested are shown in  FIGS.  27 - 32   . 
     During all model tests the leading edge  41  of the aft body  4  was located 10 mm downstream/aft of the separation line  6 , the chord angel γ is 0 degree unless otherwise stated and the trim angel of the model vessel  1  was kept neutral when floating motionless in a body of water. 
     Results from Model Tests 
       FIG.  33    is showing the test results for a model vessel  1  having a configuration as shown in  FIG.  27    with a chord angle γ of 0 degrees marked (A), a chord angle γ of −2 degrees marked (B) and a chord angle γ of −3 degrees marked (C). As seen from  FIG.  33    the aft body  4  having a chord angel γ of 0 degrees (graph (A)) is providing a backwardly directed force (i.e. resistance) throughout the entire speed range. Also, a chord angel γ of −2 degrees (graph (B)) is providing a backwardly directed force throughout the entire speed range. As seen for graph (B) the resistance is decreasing from about F N =0.23 to F N =0.44 and increasing thereafter. At a chord angel γ of −3 degrees (graph (C)) there is a forwardly directed force (i.e. propulsion) in the speed range from about F N =0.36 to F N =0.51. Outside this speed range there is a backwardly directed force also for a chord angel of −3 degrees. It should be noted that none of the graphs (A), (B) or (C) in  FIG.  33    provides a continuous forwardly directed propulsion. 
       FIG.  34    shows the effect of altering the ratio of the leading-edge area divided by the trailing edge area (i.e. A le /A te  ratio). This is achieved by varying the draft DV of the model vessel  1  from 80 mm to 90 mm and to 100 mm as shown in  FIG.  28   . Since the width in the transverse direction of the hull  2  of the leading edge  41  is the same as the width of the trailing edge  42 , the A le /A te  ratio equals the H 1 /H 2  ratio. As seen from graph (A) showing A le =1.0×A te , the aft body  4  is providing a backwardly directed force (i.e. resistance) throughout the entire speed range. Also for A le =0.83×A te  (graph (B)) the aft body  4  is providing a backwardly directed force throughout the entire speed range. At A le =0.71×A te  there is a very small forwardly directed force (i.e. propulsion) present in the speed range from about F N =0.22 to F N =0.34. Outside this speed range there is a backwardly directed force also for A le =0.71×A te . As can be seen from  FIG.  34   , a reduction of the A le /A te  ratios (from graph (A) to graph (C)) leads to a reduction in resistance from the aft body  4  throughout the entire speed range and especially at the lower end of the speed range. Such a low ratio is hence important for achieving a propulsion from the aft body  4 . It should however be noted, that none of the graphs (A) or (B) in  FIG.  34    provides a continuous forwardly directed propulsion. 
       FIG.  35    shows the effect of altering the configuration of the aft hull section  3  by changing the angle β of the tangent line TH immediately upstream/in front of the separation line  6  as shown in  FIG.  29   . As seen from graph (A), showing a TH angel β of 4.5 degrees, the aft body  4  is providing a backwardly directed force (i.e. resistance) throughout the entire speed range. From graph (B), showing a TH angel β of 11.0 degrees, there is a very small forwardly directed force (i.e. propulsion) present in the speed range from about F N =0.34 to F N =0.45. Outside this speed range there is a backwardly directed force also for a TH angel β of 11.0 degrees. It is hence concluded that an increasing angle β of the aft hull tangent TH reduces the resistance from the aft body  4  and may contribute to forward propulsion, while a lower angle β will increase the resistance from the aft body  4 . It should be noted that none of the graphs (A), (B) or (C) in  FIG.  35    provides a continuous forwardly directed propulsion. 
       FIG.  36    shows the effect of altering the depth of the aft body  4  in relation to the base line  58  as shown in  FIG.  30   , where graph (A) is showing the resistance for a deep aft body  4  located 30 mm above base line  58  and where graph (B) is showing the resistance for a shallow aft body  4  located 50 mm above the base line  58 . As seen in  FIG.  36   , both the deep (graph (A)) and the shallow (graph (B)) aft body  4  is providing a backwardly directed force (i.e. resistance) throughout the entire speed range, but the shallow aft body  4  (graph (B)) has a lower resistance than the deep aft body  4  (graph (A)) at a speed above F N =0.18. 
       FIG.  37    shows the effect of altering the length of the chord line  43  of the aft body  4  as shown in  FIG.  31   , where graph (A) is showing the resistance for a chord length of 105 mm (having a maximum vertical thickness of 1.1 cm) and graph (B) is showing the resistance for a chord length of 145 mm (having a maximum vertical thickness of 1.4 cm). As seen in  FIG.  37   , both the smaller (graph (A)) and the larger (graph (B)) aft body  4  is providing a backwardly directed force (i.e. resistance) throughout the entire speed range, but the smaller aft body  4  (graph (A)) has a lower resistance than the larger aft body  4  (graph (B)) throughout the speed range. 
       FIG.  38    is showing the test results for a model vessel  1  with aft hull sections  3  as shown in  FIG.  32    for a configuration according to an inventive model vessel  1  marked (A) and for a configuration according to a prior art model vessel  1  marked (B). 
     The geometry of the inventive model vessel  1  (A) is configured to minimize the wave resistance R w  from the aft hull section  3 . Hence, the inventive model vessel  1  (A) has a draft DV(A) of 80 mm, which entails A le =1.0*A te , an angle β(A) for the tangent line TH of 4.5 degrees and a chord angel γ(A) of 0 degree. 
     In order to obtain a continuous forward propulsion from the aft body  4  the configuration of the prior art model vessel  1  (B) is based upon a combination of the configurations found through the model testing to contribute to a forward propulsion. The prior art model vessel  1  (B) hence has a draft DV(B) of 100 mm, which entails A le =0.71*A te , an angle β(B) for the tangent line TH of 11.0 degrees and a chord angel γ(B) of −2 degrees. 
     From  FIG.  38    graph (A) it can be seen that the aft body  4  of the inventive model vessel  1  (A) is providing a continuous backwardly directed force (i.e. resistance) throughout the speed range. The resistance graph (A) is kept relatively steady from F N =0.2 to F N =0.4. Above F N =0.4 the resistance from the aft body  4  of the inventive model vessel  1  (A) is increasing with increasing speed. This is in clear contrast to graph (B), showing a continuous forwardly directed force (i.e. propulsion) throughout the speed range of the prior art model vessel  1  (B) and increasing as the speed increases. 
     Conclusion from Model Testing 
     When measuring the horizontal forces from the aft body  4  of a model vessel  1  according to the invention, it was revealed that the aft body  4  itself applied a backwardly directed force (i.e. resistance) on the model vessel  1 . When the configuration of the aft body  4  and the geometry of the aft hull section  3  was altered, creating a vessel  1  beyond the scope of the invention, a forwardly directed force (i.e. propulsion) from the aft body  4  occurred under certain conditions. 
     A low A le /A te  ratio, a high angel β of the tangent line TH and a downward tilted chord angel γ are important parameters to achieve a propulsion force from the aft body  4 . Furthermore, a reduction of the chord length of the aft body  4  and an arrangement of the aft body  4  closer to the water surface  5  will also contribute to possibly achieve forward propulsion from the aft body  4 . 
     The model tests demonstrate that a configuration seeking to achieve forward propulsion from the aft body  4  are contrary to a configuration seeking to achieve a reduction of the stern wave  9 . 
     Hence a prior art vessel  1  will benefit from a low A le /A te  ratio to achieve forward propulsion from the aft body  4 . This is in clear contrast to the inventive vessel  1  which will benefit of a ratio of A le /A te ≈1.0 to achieve equilibrium in the water mass downstream the aft body  4 . 
     A prior art vessel  1  will benefit from a larger angel β of the tangent line TH to achieve forward propulsion from the aft body  4 . This is also in clear contrast to the inventive vessel  1  which will benefit of small angel β of the tangent line TH to achieve a horizontal direction of a water flow  51  downstream the aft body  4 . 
     A prior art vessel  1  will benefit from a negative chord angel γ to achieve forward propulsion from the aft body  4 . Again, this is in clear contrast to the inventive vessel  1  which will benefit of horizontal, or near horizontal, chord angel γ to achieve a horizontal direction of a water flow  51  downstream the aft body  4 . 
     A prior art vessel  1  having a larger angel β of the tangent line TH would also benefit from an even more negative chord angel γ. However, both a higher angel β of the tangent line TH and an increased negative chord angel γ will contribute to an increasing stern wave  9 . 
     A prior art vessel  1  will benefit from a shorter chord length of the aft body  4  as a shorter chord length results in larger forward propulsion. In contrast, an inventive vessel  1  would need a longer chord length of the aft body  4  to be able redirect the upwardly directed water flow  51  upstream/in front of the aft body  4  to a horizontal water flow  51  downstream the aft body  4  without causing turbulence. 
     A visual comparison of models tested of the inventive vessel  1  described above and the prior art vessels  1  providing forward propulsion from an aft body  4  shows that the inventive vessel  1  generates a smaller stern wave  9  and has less sinkage at the stern relative to the prior art vessel  1 . It is further observed through model tests, not included in this paper, that an inventive vessel  1  seeking to obtain a reduced stern wave  9  contributes to a larger reduction in total resistance R t  for the vessel  1  than a design according to the prior art seeking to obtain forward propulsion from the aft body  4 . 
     In the preceding description, various aspects of the vessel  1  according to the invention have been described with reference to the illustrative embodiment. For purposes of explanation, specific numbers, systems and configurations were set forth in order to provide a thorough understanding of the vessel  1  and its workings. However, this description is not intended to be construed in a limiting sense. Various modifications and variations of the illustrative embodiments, as well as other embodiments of the vessel  1 , which are apparent to persons skilled in the art to which the disclosed subject matter pertains, are deemed to lie within the scope of the present invention.