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Table 1. Four shapes studied.

Each shape/material combination (with the exception of shape S1 of Polyethylene) was flapped harmonically at frequencies ranging (in 0.1 Hz intervals) from 0.3Hz to 0.8 Hz, while being towed at a speed of 0.08 m/s. The Strouhal number is a non-dimensional number that relates the forward velocity, U to the flapping frequency,f as:

where w is the wake width. In our case, w was treated as the width of a single fin stroke. We define the non-dimensional thrust and drag coefficients as

where Fx and Fy are forward thrust and side way components of the force vector in the horizontal plane, p is the density of the fluid and A is the fin area. r| is defined as a measure of forward propulsive efficiency n =kFxU/(Mzffl) (4)

The denominator in Equation 4 is taken as a measure of the torsional power required to drive the fin where Mz is the moment measured by the force sensor and a (= 2nf) is the circular frequency of flapping. k is a non-dimensional scaling constant.

Fig. 5 shows the variation of CT over the four fin shapes of different materials (Delrin, Polyimide, and Polyethylene). CT increased with frequency except for Shape S1 of Polyimide and Shape S2 of Polyethylene fins. These two fins have a drastic drop in CT after a certain frequency and then it begins to increase again. One possible explanation for this is that the force generation at lower frequencies for these two fins is viscous drag dominant. This effect drops after the Reynolds number, contribution of added mass, and pressure drag start to increase. Fig. 6 shows the variation of n (measure of efficiency) with Strouhal number for all the fins used. The relatively more rigid Delrin fins have peak efficiencies in the range of 0.81. Polyimide fins do not have much variation, for most part, with Strouhal number. The low efficiency of the Polyethylene fins can once again be explained by high viscous drag.

Fig. 5. Variation of thrust coefficient with frequency.
Fig. 6. Variation of efficiency with frequency.

The criteria for fin selection should be high values for CT and r| over a range of frequencies. Caudal fin flapping frequency is one of the key parameters that has to be changed to control the speed of the MUV. Thus, a fin with good thrust production and high efficiency over a range of frequencies is desirable. Shape S3 of Polyimide clearly is the best choice as indicated by the CT and r| values.

3. Body stability

This study seeks to understand the role of different morphological parameters that could be responsible for boxfish stability and transform such findings into engineering design guidelines for the body shape design of a micro underwater vehicle. The analysis was performed using Computer Aided Design and Engineering (CAD/CAE) tools such as solid modeling software and fluid flow simulation software. First approximate models of boxfish were built in 3D modeling software and fluid flow was simulated over such models to analyze the vorticity patterns around the body. The relative role of key morphological features like the dorsal and ventral keels, concavity and convexity of the shape, and changes in the cross section along the length of the body were determined. Different MUV body shape designs were built along these lines and tested for required vortex strength and overall drag to arrive at the best design.

Fig. 7 shows the counter rotating vortex shedding by a bluff body. The vortex shedding should result in moments that can correct disturbances in the pitch and yaw(not shown in the Figure) directions. The shape that can demonstrate this 'selfcorrecting vortex shedding' for pitch and yaw disturbances will be the shape suitable for the body of the MUV. The morphological features that contribute to the unique vortex shedding patterns as reported in (Bartol, et. al, 2005) had to be identified to be incorporated into the design. Approximate 3D models capturing essential features of the boxfish were made and flow at different angles of attack was simulated over such models to study the role of the various features on stability. SolidWorksR , a 3D CAD modeling software was used to build body shapes and fluid flow simulation was carried out using GAMBIT ™ (for defining boundaries of the flow and laying computational grid around the body) and FIDAP™ (for solving the flow and post processing) of FLUENT Inc.

Fig. 7. Cross sections along the body of the buffalo trunk fish.

3D models of boxfish were built in SolidWorks. Essential features like dorsal and ventral keels and variation in cross sections along the length of the body were reproduced in the models. Variation of cross section along the body plays a very significant role in the vorticity

Fig. 7. Self-correcting vortex shedding in boxfish body.

,Vortices

Fig. 7. Self-correcting vortex shedding in boxfish body.

generation: therefore, cross sections near mouth, eye ridge, peduncle, and any other distinct plane were drawn on 2D planes and joined to obtain the desired shape. Fig.7 shows the planes containing important cross sections of the buffalo trunk fish. Only two, the Spotted boxfish and the Buffalo trunkfish, of the four varieties of boxfish studied in (Bartol, et. al, 2005) were considered here. The other two shapes are not significantly different from the buffalo trunkfish. Fig. 8(a) and 9(a) show the planar views of the actual boxfish used in the study by Bartol et. al. 3D models of the fish used in this study are shown in Fig. 8(b) and 9(b). It has to be noted that the 3D models were developed only based on subjective 'resemblance', capturing the key features and not accurate measurements of the boxfish morphology.

Fig. 8. Spotted boxfish

Fig. 9. Buffalo trunkfish

Once created, the 3D models were imported into GAMBIT in IGES format. The imported models were then 'cleaned' by merging unwanted edges and surfaces that may have hindered mesh generation. To take advantage of the lateral symmetry of the boxfish shape, only (left/right) half of the body was considered for simulation. An external brick volume was used for the fluid flow. The symmetry plane of the boxfish was made coincident to one of the brick side walls and the body was placed midway along the height and length of the brick (see Fig. 15). In future simulations, the size of the brick could be increased further to minimize the wall effects. Fig. 15 shows the 3D mesh generated in GAMBIT. Shape functions have been defined such that the mesh is finer near the body surfaces and gets coarser away from the body. The mesh size was kept fairly coarse in order to keep the convergence time reasonable. The course mesh was not detrimental to the simulation results because the Reynolds number was only about 300, and thus did not require a very dense mesh. It has to be noted that the Reynolds number corresponds to the case where a water wave disturbs an otherwise stationary boxfish and therefore is lower than the average Reynolds number of boxfish swimming.

Here the main results and conclusions are summarized and compared to those reported in (Bartol, et. al, 2005). First, it was reported in (Bartol, et. al, 2005)that the ventral keels of all the models produced Leading Edge Vortices (LEVs) that grew in circulation along the bodies, and this was verified in the present study. Vorticity concentration was found at the keel edges for all the models at various cross sections. Vorticity contours at different cross sections for both the fish can be found in (Kodati, 2006). This is expected, as the keels form the sharp corners of the body and hence induce circulation into the oncoming water. However, for a given angle of attack an increase in circulation was found only when there is a sudden increase in the cross section. This is true for both the boxfish shapes. For the spotted boxfish in Fig. 10, the maximum concentrated vorticity is located at the eye ridge, where the cross section increases suddenly. Similar behavior was observed for the Buffalo trunkfish. The concentrated vorticity near the keels increased along the length until it reached a maximum at the mid body plane, where the body curve attains a peak (Fig. 11). Second, the present study also verified that vortices formed "above the keels and increased in circulation as pitch angle became more positive, and formed below the keels and increased in circulation as pitch angle became more negative", verbatim from (Bartol, et. al, 2005). Fig. 11 shows the net vertical lift on the spotted boxfish model varying with pitch angle. In (Bartol, et. al, 2005) the lift coefficients of all the boxfish studied were very near the origin (that is, almost no lift at zero angle of attack). The difference might be error due to the fact that the actual boxfish dimensions were not reproduced in the solid model. Third, it was found that vortices formed along the eye ridges of all the boxfish: this was clear from the simulation - the eye ridge regions for both the models have shown concentrated vorticity, which once again can be attributed to the sudden increase in the cross section profile. Finally, when both boxfish were positioned at various yaw angles, regions of stronger concentrated vorticity formed in far-field locations of the carapace when compared with nearfield areas, and vortex circulation was greatest in the posterior of center of mass, just as described in (Bartol, et. al, 2005). The vorticity contours for the yaw case are not presented here for the sake of brevity and can be found in (Kodati, 2006).

Q = 20° a = -20°

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