Computational Analysis
4.1 Thin Ship Theory
Thin ship theory is a potential flow model and a direct application of Michell's integral equation which was derived by him in 1898. One of the advantages of catamarans, when it comes to computational modelling is that their narrow demihulls allow for the use of thin ship theory. As described in [Tu87], the only requirement is stated that the hulls rate of change of thickness must be small. In other words, so long as the hull has a high length to beam ratio (L/B) and the entry angle on the bow is narrow, this theory is accurate. No means of quantifying this is given but an example hull is presented. Since the L/B and B/D ratios of this example hull are comparable to that of RH1, it was considered a suitable method for determining the resistance of the hull.
4.2 Procedure
- between the
Since section 3.2.4 concluded that the interference between the hull and lifting foils is expected to be small, the computational analysis of the vessel is made much simpler by analysing the hull and foils separately.
Since the computer modelling of RH1 may be computed using thin ship theory which is computationally and in terms of setup, far less time consuming than a complete CFD (RANSE) model. A package using thin ship theory, known as MICHLET [Laz97], which has been used in the past for such research, was therefore selected for computing the resistance on the hull. This does however require several inputs which can only be determined experimentally. Such inputs are running trim, sinkage and WSA across the speed range being calculated. Since towing tank tests are being conducted, these inputs may be measured during towing tank tests and then placed in the input file.
Unfortunately, the option to model heel and leeway has not been incorporated into this program. After searching on the internet and corresponding with the author of MICHLET (Mr L. Lazauskas), it became apparent that only complete potential flow programs at present, allow for such inclusions. One such program that was recommended by Mr Lazauskas, called SPLASH, involved a detailed modelling of both the aerodynamic and hydrodynamic aspects of the boat. The problem with this program is that it is complex
(beyond the scope of this research), designed specifically for monohulls and is expensive. It was therefore decided that the results for tests without heel or leeway could be modelled in MICHLET and so long as those tests corresponded, the model testing methodology can be proven valid and the results achieved for the leeway and heel tests can be presumed correct. It is, after all, desired that a trend regarding the effect of these parameters be found rather than the exact values.
As in past research conducted at the University of Stellenbosch [Mig01], the foils will be modelled using a program known as AUTOWING [KT05]. This will provide the lift and drag on the foil system, including the effect that the foils have on one another for a given configuration. (See section 3.2.4)
The depth at which the foils run at various speeds may be determined by the sinkage and trim results taken from the towing tank experimentation.
Both AUTOWING and MICHLET are potential flow based programs and will therefore not pick up viscous scaling effects. The viscous drag is therefore added on empirically in the calculations. Both programs also do not account for spray drag, but this is expected to be small as sailing catamarans are observed to produce little m spray in most sailing conditions.
4.3 MICHLET
As mentioned before, MICHLET is a computer package that utilises thin ship theory The hull configuration is fed to the program via an input file (Appendix I) which in turn refers to a text file containing points that define the shape of the hull/s. Included in this input file is the speed range and the corresponding running data (WSA, trim and sinkage taken from experimentation)
Some sources of error are derived from the assumptions made by the program. MICHLET assumes that the sides of the boat are vertical from the points defined at the water surface. This may cause error particularly in the region of the bow wave as it is seen in experimentation to be very large with respect to draft. Fortunately there is no flare or tumblehome near the bow so error won't be very large. Similarly, errors may be large for large sinkages however this is not a problem in our case as seen in results - sinkages even without foils are very low.
The form factor of the hull is not known but since the demi-hulls are slender, it is expected to be close to unity. [CMAP97] gives some examples of form factors of slender boats and demonstrates that they are usually significantly higher than unity (note form factors reduce with speed). Also noted that the form factor acquired
from towing tank, wind tunnel and CFD analysis provided large variation in results, towing tank test being the highest. [CMAP97] concludes that the most important factor for determining form factor of high speed, round bilge, transom stern vessels is the L/disp1/3 ratio and that it is reasonably independent of speed and demihulls separation. From the table provided a linear interpolation yields a form factor of about 1.25.
The MICHLET calculations are set for a particular sinkage and trim, which affects the underwater shape (slenderness) and in turn the form factor. For boats without much rocker, the hull becomes very slender as it rises out of the water, thus the form factor tends to 1. RH1 does however have substantial rocker and thus the slenderness doesn't necessarily tend to infinity as the boat emerges, thus the form factor doesn't quite tend to 1. There has apparently been much debate regarding the calculation of form factor. As a result, the form factor was first set to 1 in the calculations and the results compared to determine accuracy.
The convergence of the MICHLET solutions was tested by varying the number of n-theta values (angular division in the hull) which are used to calculate the wave drag on the hull. The value was set initially according to the recommendations of the MICHLET user manual and the number increased until a variation of less than 10% was achieved.
No lifting foils - Comparing Experimental and Computational Results
0.000
0.500
1.000
1.500 Fr_Disp
■ □- - Experimental (hull + rudders + daggerboards)
- -5K- ■ Computational (hull + rudders + daggerboards)
0.500
1.000
1.500 Fr_Disp
2.000
2.500
3.000
Figure 4.1 - Comparing the experimental and computational (MICHLET) resistance curves of RH1
The experimental results of RH1 tested across its speed range are compared to the computational results in figure 4.1. The viscous drag of the rudders and daggerboards (as calculated in appendix H) have been added on to the result of MICHLET and a strong correlation between the results is observed. The correlation between the experimental results and those predicted by MICHLET is excellent and the only major deviation occurs at FrV
where MICHLET under-predicts the resistance by 15%. This may be attributed to assumptions of MICHLET, as mentioned earlier.
4.4 AUTOWING
This is a vortex panel method program, designed specifically for calculating the lift and drag on a given hydrofoil configuration. The cross-sections of each foil are entered first and then a series of points are used to describe the configuration of those foils. The effects of compressibility and viscosity are considered negligible [Dev98]. Since the viscosity is assumed negligible, the residual drag generated by the foils is calculated only and the viscous drag is calculated using the ITTC equation 5.1.
Planing theory may also be applied in this program, thus making it possible to model planing hulls in this program. Unfortunately this does not apply to RH1 as over almost all of the speed range, buoyancy forces dominate over planing forces. This is why the hulls are modelled separately in MICHLET.
Since the foils and hull will be modelled independently, the interference between the two will not be taken into account. This will result in a slight under prediction in-tfie tota± drag and a slight over prediction on the lift. The lifting foils increase the mass flow rate above their jpper surfaces. This will increase velocity of the water in between the foils and the hull, thus increasing the effective form factor. No means of quantifying this effect was found in the literature study, but since it affects only a small portion of the boat, it is not expected to influence the resistance greatly. Where the foils join onto the hull, the hull would act as a turbulent stimulator, thus the transition to turbulent flow may occur sooner. Another effect of the hull's boundary layer would be to reduce the speed of the flow over that section of the hull. This would then reduce the Reynolds number of the flow over the foil. These effects are assumed to cancel.
[KT99] provides a test case where the results of AUTOWING are in good agreement with the experimental results, which verifies the accuracy of the program. A comprehensive analysis of the use of AUTOWING was conducted by Migeotte [Mig01], where the accuracy of AUTOWING was validated and criteria for insuring convergence of the solution were provided. These criteria were then applied to the computational model of the foils modelled in AUTOWING and are summed up in table 4.1.
Large Froude number methodology (often used in hydrofoil calculations) assumes the wave pattern above the foils has a negligible effect on the lift of the foils. This saves vast amounts of computation as the vortex sheet at the surface does not then need to be calculated. According to the AUTOWING manual, this approximation is reasonable only when the Froude number based on the chord of the foils is greater than 4.5. This is often the case for high speed power catamarans but unfortunately the bulk of speed range used for this testing procedure results in Froude numbers lower than this.
For this reason, a linearised free surface calculation was used in AUTOWING, which includes in its calculations the free surface deformation. Since there will be interference from the upstream foil, this free surface condition will best capture these effects.
|
General Convergence |
1. Domain - distance in front of foil 2. Domain - distance behind foil 3. Iterations |
2 chords 10 chord lengths 70 iterations |
|
Hydrofoil vortex lattice |
Spanwise density Chordwise density (Including free surface effects) |
12 32 |
|
Free surface panel density |
Per chord length |
40 |
Table 4.1 - Summary of criteria for convergence of AUTOWING taken from [Mig01]
Table 4.1 - Summary of criteria for convergence of AUTOWING taken from [Mig01]
Normalised lift and drag coefficients as function of iteration number
Iteration number
Figure 4.2 - The normalised coefficients of lift, drag and trim moment plotted against iteration number
Iteration number
Figure 4.2 - The normalised coefficients of lift, drag and trim moment plotted against iteration number
In order to check the convergence of the solution, the coefficients of lift, drag and trim moment on the foil system were plotted against the iteration number. Since lift and drag (and the resulting trim moment) are the outputs which will be used to verify experimental results, it is important that these coefficients are converged. Figure 4.2 below shows the convergence of these coefficients occurs at 40 iterations but since the trim and sinkage were left at zero for this test case, a larger number of iterations (65) was chosen to ensure convergence. For each speed, the convergence of these coefficients was monitored and it was found that 65 iterations was sufficient throughout the speed range.
The centreline wave pattern was also plotted after various iterations (see figure 4.3). It was found that convergence of the centreline wave pattern was only achieved above 75 iterations, but since this was not required for this research, a great deal of computational time was saved by using only 65 iterations.
Graphing Centreline Wave Pattern for Various Number of Iterations
- 0.04
x-position along centreline (m)
Figure 4.3 - The centreline wave pattern for various iteration numbers (Same input values, used for speed, trim, sinkage)
x-position along centreline (m)
Figure 4.3 - The centreline wave pattern for various iteration numbers (Same input values, used for speed, trim, sinkage)
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