Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014
two ways: graphically and mathematically. Graphical interpolation, which uses the test results presented in a graphical
form, is not practical. The emergence of
computers has made it possible to represent the test results of propeller series in the form of mathematical expressions, i.e. it enables mathematical modeling. As in the case described in Sections 1 to 4, the modeling can be done in two ways: by standard mathematical functions and by spline functions. However, in this case, the problem is more complex because the modeling does not relate only to one graphic but to a complete family of graphs.
6.1 MATHEMATICAL MODELING BY STANDARD FUNCTIONS
The diagrams presented in Figures 3 and 4 show what the graphical presentation of dependence of the quantity KT on the quantity J, and the dependence of the quantity KQ on the quantity J, look like. Results of the model testing of a series are presented by the family of those graphical presentations. The task of mathematical modeling by standard functions is the selection of one function which will describe all graphics, i.e. one function which will describe all dependences of
the quantity KT on the
quantity J, and all dependences of the quantity KQ on the quantity J. The mathematical models that are the results of experimental testing of propeller series are functions of a number of variables and they have the following general form
KT = fT (J, Ae /A0, P/D, Z, ) KQ = fQ (J, Ae /A0, P/D, Z, )
(12a) (12b)
The coefficients KT and KQ depend on the quantities J, Ae/Ao, P/D, Z and, i.e. the functions fT and fQ are curves in multidimensional space. By their mathematical nature, the functions fT and fQ can be linear or nonlinear. The polynomials are mathematical expressions that are used in practice for both types of functions.
6.1 (a) Gawn-Burrill KCA Series The
Gawn-Burrill KCA
All mathematical models created are in the form of polynomials. With the authors who used linear functions, the general form of polynomials is
KC JTi() (Ae / Ao)
i 1
n KC JQi() (Ae / Ao)
i 1
m s tui ii v (P / D) ( ( )) i f s tui ii v (P / D) ( ( )) i f (13a) (13b)
and nonlinear functions are in Koushan's model, see expression (14) [5].
Linear functions for f() were used by Blount and Hubble [2], Kozhukharov [3] and Radojcic [4]. They computed the coefficients Ci, si, ti, ui and vi using multiple regression analysis.
Koushan [5] used the nonlinear functions in his
mathematical model, and for computing the coefficients he used artificial neural networks.
In their paper [6], Radojcic et al. compared all mentioned mathematical models of the KCA series with each other and with the data obtained from in experiments.
Each of the above-mentioned authors of mathematical models had their own choice of a function for description of the impact of cavitation. All functions are mutually different. Polynomials have different numbers of terms. The digitization of original graphics was different, i.e. each of them made their own database by means of which they computed the coefficients of polynomials. Their perceptions of the solution of the same problem were different.
Consequently, four mathematical models were obtained giving results that differ from the original data. It can be said that all models are approximate and do not represent the graphics as well as the "natural" dependence between the quantities KT and J, and KQ and J.
6.2 SPLINE FUNCTIONS AND DATABASES
As mentioned above, the results of experimental tests of model
propellers can be described by a family series was tested under
operational conditions both with and with out cavitation. Results of the tests were published and showed in the form of graphics drawn by means of mechanical splines [1]. A number of authors made their own mathematical models of the series. A more difficult part of the problem has been how to make a mathematical model of the graphics representing the behaviour of the propeller under operational conditions when cavitation occurs, see Figure 4. This behaviour is designated as nonlinear. It is known that a mathematical function can be nonlinear, but it is a little strange that a behaviour of a physical phenomenon may be nonlinear.
of
graphics that represent the dependence of KT on J and of KQ on J. These graphics are a database of curves on paper. They look good, but they are not good for practical use. An improvement is to describe this family in electronic form. In this case each graphic from the family must be represented by a spline function. For this, the graphic must be digitized, i.e. it must be represented as a series of pairs of points (KT, J) and (KQ, J) entered in a table, as shown in Table 6. In Sub-section 4.2 it was demonstrated how to make a spline function on the basis of data from a table. One propeller series has as many tables as there are graphics. How one propeller series is created, i.e. how the quantities Ae and P were changed, is also described by а table (see Table 2). Also the change
B-48 ©2014: The Royal Institution of Naval Architects
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