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Section PageIntroduction 2Research 7

Foil Design and Construction 9

Instrumentation and Data Analysis 9

Testing 10

Schedule 11

Conclusion 12

List of FiguresFigure Page1 Reynolds Number vs. Arm Speed 42

CDfor cylinder vs. Reynolds Number 53a

CDvs. Arm Speed 53b Drag per Span vs. Arm Speed 5

4 Flow Around Circular Cylinders 7

5 Current Arm Design Concept 8

6 Basic Foil Concept (Cross-Section) 9

7 Test Matrix 10

The techniques to reduce the drag that will be investigated are foils that are free to rotate around the arm segment and low-drag coatings that will be placed on the arm and foils. Through measuring the effort required to move the arm through the water in different configurations of coatings and foils, the effect of each will be determined, and recommendations as to the feasibility of improving underwater space simulation will be made.

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(1)

where *U* is the velocity of the fluid, *L* some characteristic
length, and *v* the kinematic viscosity of the fluid. Figure 1 shows
the Reynolds Number versus arm speed for three different arm diameters.

Thus it is clear that this project will operate under conditions where the Reynolds Number is between 10 and 200, since human arm speeds are approximately between zero and four ft/sec. This regime of Reynolds Number is a part of fluid mechanics where very little research or experimentation is done. Traditionally, research efforts have concentrated on very-low Reynolds number applications, usually in the range of zero to approximately twenty; low Reynolds Number applications, such as boats, sails, and underwater cables, with Reynolds numbers on the order of 10^4 and 10^5; and high Reynolds Number applications, such as airplanes, with Reynolds numbers of the order of 10^6 and above. The region where this project resides is basically omitted because there are very few real applications.

Figure 1 -- Reynolds Number vs. Arm Speed

Luckily, one of the exceptions to this lack of research is in the study of drag on circular cylinders. In general, drag on a body is due to a combination of skin friction forces and pressure forces. In an ideal fluid with no skin friction, by D'Alembert's Paradox, there would be no drag, since the flow would remain laminar and attached to the body along the entire contour, and the pressure forces would exactly balance each other out.[2] However, in reality, a fluid must have a velocity of zero at the surface of any solid, and so a boundary layer is created.[3] This boundary layer serves as a transition between zero velocity at the solid and free-stream velocity some small distance away from the solid. If this boundary layer was to remain attached to the cylinder through the whole contour, then again, there would be no drag due to pressure forces, and the only drag would be the relatively small skin-friction drag. However, this is almost never the case. When the boundary layer separates from the body, a large region of low pressure appears on the trailing edge of the cylinder, which can no longer balance the increase of pressure on the leading edge, so a net drag is produced.[4]

There is no exact analytical solution to the Navier-Stokes equation, which governs incompressible viscous flow, for the case of a cylindrical section.[5] However, many experiments have been performed, and the coefficient of drag

(2)

which is a pure function of Reynolds number and for incompressible flows is
well tabulated. A plot of it is reproduced in Figure 2. For the regions that
the experiment will operate in (10<=*Re* <=1000) *CD* is quite
linear in the log-log plot which corresponds to an approximation of[6]

(3)

Using this approximation, we can plot *CD* and Drag as a function of arm
speed (Figure 3a and 3b)

Figure 2 -- CD for a cylinder vs. Reynolds Number

Figure 3a -- CD vs. Arm Speed Figure 3b -- Drag per span vs. Arm Speed

The reason for the extremely high

It is also worth noting the sudden drop in *CD* at
.
This corresponds to the transition from a laminar boundary layer to a turbulent
one.[7][8] Though it is customary
to associate turbulence with high drag, in this case it actually helps the
situation, because turbulent boundary layers tend to separate later in the flow
than laminar ones. Therefore the counter-intuitive method of making a surface
rougher will actually decrease drag for some situations by forcing the laminar
to turbulent transition to occur at lower Reynolds Numbers. However, it is very
difficult (if not impossible) to conceive that such a transition could be made
to occur in the regime of Reynolds Numbers that this project focuses on.

Finally, the concept of vortex shedding must be addressed. As was stated earlier, for extremely low Reynolds Numbers, the flow around a cylinder is laminar, and skin friction is the only source of drag. As the Reynolds Number increases, separation begins, and the pressure forces become increasingly important. Up to , the overall flow is basically laminar, and a pair of stable vortices form symmetrically off the trailing edge. (Figure 4a) However, at , the two vortices begin to form alternately in time, one above the centerline and then one below. (Figure 4b) This, known as a Von Karman Vortex Sheet,[9][10] leaves a large wake of alternating votrices behind it and adds significantly to the drag. An additional method of overcoming some drag is to attempt to prevent the beginning the vortex shedding, by placing either some ribbons or a fixed flat plate called a Splitter Plate along the centerline facing away from the flow (Figures 4c and 4d).

Figure 4 -- Flow around circular cylinders (gray).[11]

Dark circles represent vortices

The motors and controls will be placed above the main column, so as to remain out of the water. This will greatly reduce the complexity of the water-proofing process. The joint will as frictionless as possible, so as to allow the lower segment (representing the arm) to "float" once it is made neutrally-buoyant. It is currently envisioned that the lower segment will be constructed out of plastic tubing. A possible candidate is ABS pipe often used in drainage and sewer applications. Both ends will be capped, and enough buoyant material, such as Styrofoam, will be inserted to make the segment neutrally-buoyant in water. The actual join will most likely be open to the water, as the water should help in alleviating some of the friction. The biggest concern at the moment is the actual design of the joint and actuator system. It is not that difficult to design a almost-frictionless joint; however, making the connection to the actuator frictionless as well will be very difficult. Designing the joint and the control system will be the primary focus of the arm design for the remainder of the term.Figure 5 -- Current Arm Design Concept

Figure 6 -- Basic foil concept (cross-section)

There will be a brief break in testing while this coating is applied to each of the prospective foils. (Again between two and four) Once this is complete, testing will resume, and each foil (as well as the arm by itself) will be subjected to another series of motions. In this case, some of the motions will attempt to simulate possible EVA tasks. Once again torque and position data will be taken. When all the data has been collected, the final stage of data analysis will begin. The test matrix is pictured in Figure 7.

Figure 7 -- Test Matrix

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Figure 8 -- Project Schedule Spring 1994

Figure 9 -- Project Schedule Fall 1994

A very approximate and conservative budget is as follows[14]:

Motors and control systems $250

Materials for arm $100

Coatings $100

Total $450