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FKS

FKS is a CFD code developed primarily from the work of Dr. Noblesse from NSWC-CD. FKS is called a “free-surface code” which describes its goal of predicting wave making resistance. The free-surface is the boundary between two fluids. For ships, this boundary is between the seawater and the air and mathematically is modeled as a continuous surface. In real life at the free-surface, non-continuum effects can take place. For example, a bow wave from a ship can often roll over on itself, or spray can form and break off free from the rest of the surface. Furthermore, free-surface codes are difficult to construct because little is known about the viscous relationship of wavemaking. FKS is an inviscid code, and this viscous interference is ignored, leading to some error in the solution. Determining this error was a major focus of the testing.

Theory

As waves are formed near the surface of the hull, they radiate towards infinity. However, as they radiate, the waves interfere with each other. As an example of this interference at Fn=0.40, the bow wave has constructive interference with the stern wave which creates larger waves. Since waves are created from the energy of the boat driving through the water, it requires more force for the boat to create the larger bow and stern waves. Therefore, total resistance increases at Froude number 0.40. For ships, Fn=0.40 is considered the "hull speed." Without extra powering, most ships will not be able to travel at faster Froude numbers.

Farfield waves are the summation of waves at a great distance away from the ship. Individual waves created by the hull merge a short distance away from the ship, creating a uniform wave pattern that propagates outward. In the absence of viscosity, these would travel to infinity. By inspecting the farfield waves, the total resistance from all of the wave interference can be calculated.[2]

The Havelock formula (Eqn(1)) calculates the resistance caused by waves through a calculation of the farfield wave-spectrum (S). In Eqn (1), the wave-spectrum is broken into its real (Sr) and imaginary parts (Si).[3]

(1)

The Havelock formula is a function of Froude number, wavenumber (κ), the Fourier variable (β), and the wave-spectrum. Froude number is incorporated into the equation via Eqn(2) which acts as a description for velocity.

(2)

Wavenumber is a description of the wavelength (λ) of the radiated waves (Eqn(3)) and therefore acts as a function of wave celerity (c), or the speed of the waves. [1]

(3)

Wave celerity (Eqn(4)) is also a function of the frequency of the waves (w). The wave frequency, along with wavelength can be used to calculate the energy in each wave.[1]

(4)

Moreover, wavenumber is a function of both velocity and the Fourier variable in Eqn(5).[3]

(5)

The farfield wave-spectrum can be calculated as a function of disturbance velocity, or the velocity of water particles due to the ship’s velocity, along the hull. This calculation, the Fourier-Kochin representation of farfield waves, was used by Dr. Noblesse to develop FKS.[3]

FKS calculates the disturbance velocity at the hull using a slender-ship approximation. This approach treats a hull as a set of cylinders without end conditions. The benefit of this approach is that calculations can occur on limited resources. The negative aspect of this method is that the approximation breaks down once the ship’s beam is too large for its length. Slender-ship theory’s assumptions will fail when the beam is much greater than the total wavelength.[1]

FKS calculates the velocity at the hull of the ship via the slender-ship approximation. Using these velocities, the farfield waves can be calculated using the Fourier-Kochin method. Finally, the Havelock formula calculates the wave resistance from the farfield waves.[3]

Process

In order to calculate the disturbance velocity at the hull, FKS needed to be able to make calculations along a numerical representation of the hull. FKS used a three-dimensional surface model divided into panels to form a structured grid or mesh, Figure 1. The model itself had to be scaled to a non-dimensional length of one unit.

Fig. 1 : Nondimensional Mk II Navy 44 STC with trapezoidal mesh

Care had to be included in the formation of the meshed hull. FKS used only the underwater-body of the hull as the input mesh. Therefore, the mesh was constructed of the body below the waterplane, or z=0. Since the initial free-surface at z=0 moves to both above (+z) and below (-z) the meshed waterplane, FKS calculates the hull at z>0 as a linear continuation of the hull from below the surface of the water. If the meshed hull had protrusions at the waterplane which were not characteristic of the continued hull, the calculated free-surface would be inaccurate. An example would be if the boat was at a maximum heel where the deck touched the free-surface. At this point, FKS would perform its calculations as if there was still more hull for the free-surface to attach.

The CFD meshing program, "Gridgen" was used to create a grid of panels over the underwater surface of the hull. After exporting the domains which represent the hull from Gridgen, a preprocessor was used to cut the trapezoidal panels in half to create triangular panels. The preprocessor, included with FKS, called "g2FKS" created the final model ready for computations.

Besides the hull itself, the only other input to FKS was the Froude number. By running a set of Froude numbers, an upright resistance curve could be completed.

References