Chapter 2: CATAMARAN FLOW PHYSICS
2.4 Sectional flow physics
In catamaran case, the chine-unwetted phase may be depicted in Fig. 2.4a and the
chine-wetted phase may be depicted in Fig. 2.4b.
z y ) (x V Chine Large Z C D B E + c z zb+ − c z − b z 0 + j V − j V Large Vs small Vs
z y ) (x V Chine Large Z C D B E CH z zb+ − c z − b z 0 + j V − j V β(x,z)
Figure 2.4b Chine wetted phase of a conventional catamaran
In Fig. 2.4, the body geometry of the catamaran consists of two symmetrical
single hulls with the assumption of small deadrise angle β(x,z). The bottom contour starts from a knuckle, or the keel, denoted as zk (it is denoted as
−
c
z in Fig. 2.4). A hard
chine exists on the outside, at ZCH. V(x) is the downward impact velocity of the section.
−
b
z and zb+ are the inner and outer jet spray-roots.
−
j
V and Vj+ are the jet velocity
respectively.
Analogous to Vorus's description of the planing mono-hull (Vorus, 1996), for
conventional catamarans, in the CUW impact phase, the water surface is forced to turn
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inner jet with a jet velocity Vj−, separated at the keel due to the sharp angle of the keel,
and part of the flow forms the outer jet attached to the contour.
Point B, in Figure 2.4, with coordinate zb+(t), is called the outer jet-head offset,
where the jet is truncated. Point C, with the coordinate zc+(t), called the jet separation
point offset, is the zero dynamic pressure point on the body contour. The inner jet
separates at the keel zk, which denoted as zc−, the inner jet-head is truncated at zb−(t).
Point D, E and Zlarge are reference points.
The jet head point zb+(t) separates the outer flow into branches. The upper branch
is bounded by lines C−D and B−E, and the lower branch is bounded by the line Large
Z
B− located on the free-surface contour bounding the lower flow domain.
Let )Vs(z,t be the cylinder and free surface contour tangential velocity. In the
chine-unwetted flow phase (Fig. 2.4a), the flow velocity Vs(z,t) in the jet head region
(zb+ −zc+) and on the upper branch is much higher than the impact velocity:
) , ( ) , (z t V x t
Vs >> on the upper branch (2.10)
Conversely, on the lower branch, the flow velocity is much lower than the impact
velocity, due to the large volume of the lower flow domain relative to the jet dimensions:
) , ( ) , (z t V x t
In the chine-wetted flow phase (Fig. 2.4b), the separation point has reached the
hard chine, zc+(t)=ZCH. The line C−D is now on the water surface contour, and the flow velocity in the upper branch drops to a lower order. The jet-head moves out across
the free surface.
The description of the outer flow in CW phase is applicable to the inner flow of
catamarans. However, there is a difference for the inner flow of the catamaran. The inner
flow does not exist as a chine-unwetted flow, only as a chine-wetted flow. The inner
separation point zc−(t) is the keel point zk(x).
The flow characteristic that, the CUW flow can only be developed on one side,
the other side always in CW flow phase, described above for catamarans is the same as
the "type B" flow in the asymmetric impact model for planing mono-hulls (Xu et al,
1998). For the catamaran, the outer jet flow is attached to the outside, but the inner flow
separates from the keel. The characteristic of large asymmetry, manifest in steep deadrise
to the inside, clearly make the catamaran flow a “type B” flow (refer to Fig. 1 in Xu et al,
1998).
Following Vorus(1996), the flatness of the bottom contour of the catamaran is
exploited by collapsing the bottom contour and the free surface to the z− axis for the purpose of satisfying boundary conditions.
The z− axis of Figs. 2.4a and 2.4b shows the boundary segments where different boundary conditions are satisfied (refer to Fig. 2.6). The boundary switches from the
upper branch at point B to the lower branch with a discontinuity in jet tangential velocity
) , (z t
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Fig. 2.4c shows a conventional catamaran with a transverse step. In modern
catamaran design, the downstream shoulder of the step will generally return to
approximately the original hull lines, thereby separating the hull into different regions, as
depicted in Fig. 2.4c. The concept of the step is to eliminate the relatively ineffective
chine-wetted region of the hull. Since that part of the hull, aft of the chine wetting point,
has a very small contribution to the useful dynamic lift, but a substantial contribution to
unwanted frictional resistance, its operational efficiency is very low. In today's new
design concepts, a step produces a trip which changes the low-pressure chine-wetted
region to a high-pressure chine-unwetted region, thereby allowing the after part of the
craft to increase its operational efficiency. In essence, the chine-wetted flow is interrupted
and detached by the step and then starts over on re-attachment as a chine-unwetted flow.
The step shown on Figure 2.4c is exaggerated in size for conceptual clarity.
Step
CU W region CW region
Chine K eel