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Modelling physics by usage of a mathematical symbolic solver for transferring analytics theory to numerical “First-Principal-Methods”

Wilfried Abels, Hamburg University of Technology, Hamburg/Germany, w.abels@tu-harburg.de

Abstract

This paper describes an approach, how a mathematical symbolic solver can be used to implement complex mathematical theories to a numerical implementation. This will be explained on hand of an unsteady potential theory for calculation of the lift of propeller blades rotating in a given wake. After a short overview about the analytic theory, it will be explained, how a mathematical model has been developed by the usage of the symbolic solver “Maple”. Afterwards this mathematical model has been used to implement and verify a numerical FORTRAN implementation. By doing this it was possible to implement a small and very fast method, which allows to analyse the influence of a wake to the lift-distribution on a blade.

1. Introduction The handling of physical/numerical models is a main task during the development of a technical product. An engineer needs reliable information to design new innovative products. Resulting from the keen competition within the shipbuilding industry, it is from significant interest, to have tools to analyze new design features with only less empirical know-how. Further, such reliable information has to be available in a very short timescale. The concept “Design in Seven Days” (D7D) needs software tools, which generates the needed information fast and reliable, Krüger (2003). Effects of the wake are such an important field within the design process. The wake depends on the ship hull and triggers many questions round about propulsion, aft body and rudder. Especially the interaction between wake and propeller is very important. Beside the question of efficiency, questions of pressure pulse, ruder forces and vibrations are in the focus. Wrong design decisions can result in difficult and expensive problems. It can be a significant competitive advantage to have tools during the early design, which are able to assure a design decision. Tools which are able to do this have to be adjusted to the relevant physical features. Often it is not necessary to analyze the physic in a global way. The problem of an engineer during the early design is to make a decision between two or more design options. For the used software tools this mean, a small qualitative model, which describes the relevant effects, is much more important than a more complex model, which describes many different physical effects. Beside the argument, that complex models need much more computation effort, which is often not available, it is very difficult to evaluate the reliability of such models. As more effects are modeled, as more parameters has to evaluated. From the view of an engineer, this means more uncertainties. A model with well known simplifications and a known reliability is much more useful, than a more detailed model with a not exactly known behavior. This mean, the main task in the development of software tools for the early design, is to think about the relevant physical features needed for the design process. The modeling of physic and a practical analytic description is important and has to be done with accuracy. In a second step the model has to be translated in a numerical description, which can be implemented in software tools used during the early design. In the past, especially from begin to middle of the last century; there have been done many high sophisticated analytic investigations in different aspects of hydrodynamic theories. It was a time without high computer performance. It was essential to calculate mathematical models by hand without computer support. Consequently there was a strong focus on using analytic technologies to solve problems of mathematical models. A numerical discretization of equations was only the absolute last solution and has been only done in the case of no analytic possibilities. Beside the fact, that such theories need a complex mathematical description, the great advantage is the low need in calculation performance. This feature is nowadays very useful during the early design. Such methods are able to handle many different design variations in a short time. They are fast enough to interact with the short time scale of

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the workflow during the early design. Calculation methods which need more as round about one day are useless, because it is feasible that the actual ship design differs already from the start point of the calculation. The former restriction to methods with low calculation demands is nowadays a great benefit for the aim of analyzing many different design variations within a short time. This paper shows an approach to simulate a rotating propeller within a given wake. The aim is to analyze effects of propeller and an unsteady wake during the early design. Although still now, the used theory does not allow to use a dedicated propeller, qualitative effects of a specific wake are taken into account and can be already used to figure out the influence of wake to the unsteady lift distributed on the blade. A significant aspect is to handle the free vortexes downstream the propeller. This is a difficult task. Classical lifting line approaches can only handle a homogenous wake, Isay (1964). Modern viscous approaches based on a three-dimensional domain, have still now problems to discretize the free vortexes in a good way and secondly they are very time consuming. A further approach is the QCM (Quasi-Continuous Method), which based on a panel description of the blades and wake panels to model the free vortexes, Streckwall (1997) and Abels (2006). But this wake panels have the problem, that they generate singularities in the case of a rudder placed behind a propeller. The wake panels of the propeller and the panel description of the rudder collide with each other. To avoid these problems, an unsteady potential theory has been used, which describes the propeller blades as a bounded circulation. For solving the boundary condition, an integration of the law of Biot-Savat has been done from the bounded circulation to infinite downstream the propeller. Because this integration has been done in an analytic way, the whole endless circulation downstream could be taken into account without usage of discrete wake panels. This analytic handling of free vortexes allows to stay compatible with other panel based potential theories. 2. An Overview about the unsteady potential propulsion theory This paper describes the investigation in an unsteady propulsion method, Zwick (1962). The flow had been modeled by a potential theory and a 3-dimensial distribution of free circulation downstream the propeller. There is an incompressible, source and sink free fluid. The propeller and wake are descript with cylindrical coordinates . The propeller has a constant rotation of and it moves with a velocity in the negative direction of the -Axis. The propeller blades will see a wake

with a period of in . From view of a cylindrical coordinate system fixed on the propeller blades, the propeller will see a local flow . Additionally the propeller has blades, the hub is at and the tip at . The cord length is with

Fig.1: Propeller model, Zwick (1962)

The model of the propeller is from a relative definite character. The blades are modeled as cylinder sections and an a constant pitch, Fig.1. In contrast to this, the wake is modeled with axial and radial components, Fig.2.

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Fig.2: axial and radial components of the used wake, Zwick (1962) In spite of the simple propeller model, this unsteady theory could be used to analyze effects of the wake. This mean. effects resulting from the wake, could be investigated in a qualitative way. The propeller is described as an fixed circulation and free circulations drain off downstream. The mathematical model based on an analytic solution of the law from Biot-Savat for the fixed and free circulations at the ¼ point and the flow constraint is solved for the ¾ point at . If now

the following equation has to be solved:

 

(1)

Now the propeller can be modeled as a circulation in the following form:

(2)

By using such a circulation it is in principle easy to describe the free circulation downstream the propeller. Resulting from the “Helmholtz’s theorems” for fluid mechanics, the free transversal circulation is defined as:

(3)

The direction is . In the same way the free longitudinal circulations are defined:

(4)

The direction is in this case . To describe the induced velocities, the law from “Biot-Savart” has to be used:

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(5) In Principle it is easy to describe the unsteady flow by a combination of the equations (1) to (5). The mathematic becomes a bit complicate because of the cylindrical coordinates and the geometrical description of the law from “Biot-Savart”. To calculate the induced velocities on the blades, it is necessary to sum up the integration about the radial axis of every blade and to do an infinite integration in direction of the screw along the free vortexes:

   (6)

The approach for solving this set of equations had been described in Zwick (1962). The solution has been done by a numerical approximation of the circulation in the following form.

 

 

(7)

The coefficients can be calculated by a set of systems of linear equations:

(8)

In equation (8) the Matrix symbolize the effects of the free vortexes and represents the effects of the wake. If this coefficients has been calculated, it is easy to solve this systems of equations. Modern computers are able to do this very fast, because the relevant parameters and are small values less than 10. In contrast to other methods the numerical effort to solve the linear equations is nearly irrelevant. In appendix 999 the definition of Matrix has been shown exemplary. The derivation can been read in the original paper and is not part of this investigation. Important is how such mathematical systems can be transformed in an useful numerical method. 3. Handling analytic models with symbolic mathematical tools As explained above, many high sophisticated models have been already developed in the past. But often they are not easy to handle. Many know how have been lost over the years and often there are no people any more who can explain the backgrounds of them. Furthermore, the handling of analytic theories have been changed over the years. In the past calculations have been done with support of devices as a slide-rule or an analog integrator. Today it is important to bring the model in a form useable by computers.

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This investigation has shown, that the numerical verification of such a complex analytic theory is an ambitious task. The original paper shows only the input data (Fig.2) and the results in the shape of a graphic, Fig.3. Therefore, an analytic math model has been developed and implemented within the symbolic solver maple. Such a symbolic solver has the advantage to describe the mathematical model directly by the needed equations. Integration and derivation can be handled analytically or by an numerical procedure. Problems of numerical modeling are from less importance. Discretization and definition of useable data structures for a specific computer language are not necessary. It will use an own solving mechanism without many user interaction. Although the numerical performance of such a symbolic solver is slow, the great benefit is to use it for rapid prototyping. After building such a mathematical model of a problem, it can be used as a reference model for further developments and for the purpose of testing. This task is not easy to handle, because the numerical implementation of a complex mathematical model, has many other constrains as to map only the mathematical equations to a computer language. A numerical method, which should be proceeded in an efficient way on a microprocessor, has to take into account constrains of the processing unit and the usable data structures of the preferred computer language. This mean, the implementation in software is from another style as the mathematical model.

Fig.3: Examples of calculated circulation at and , Zwick (1962) Nevertheless, the accuracy of the calculation results has to be guaranteed. A rapid prototype model as mentioned above, is very useful for this purpose. During the process of a software implementation, the engineer needs support to control and debug the software. In principle such an implementation is not very complicate. All needed equations are available, but problems result from the effect, that human beings are tend to make mistakes by copying long mathematical structures. In this step there was the main problem of no interim values. The mathematical description could by defined nearly in the same way as listed in appendix A. Between the input data of the wake and the resulting unsteady circulation there are many mathematical calculations. Consequently there are a lot of options for doing faults. And statistically it is sure, to do some. It has already cost a lot of time to develop the mathematical prototype for reproduction of the published circulations. The next step, the implementation in a real computer language, had been in principle much more complicate. Now numerical problems had become in the focus of the development. But a powerful reference system had been available and could be used for a step wise implementation of the pure FORTRAN method. The prototype model allows to generate all needed interim values. By doing the numerical implementation in continuously comparison to the prototype, the developments could be done in a clear and controllable process. The correctness of the implemented functionality could be guaranteed, during the whole time. 4. Transferring the model from analytic to a numerical description As mentioned above, the numerical implementation of such an analytic theory has aspects, which differs a bit, from a pure analytic view. Two examples should be now shown and explained. In both cases the problem results from the fact, that a computer is a digital device and does not calculate in an analog way. The positioning of supporting points could be problematic, if they match discontinuities. Even if it is clear, that such a discontinuity could be canceled, it has to be done in the correct way. It

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has to be found a correct mathematical procedure to handle them, which could be implemented on a digital computer. 4.1. A case study with no support for the equality of to real values The task of the unsteady method is to calculate the vector which describes the solution of the unsteady circulation on the blades. Therefore the matrix and the vector of equation (8) have to be calculated. The matrix is defined in equation (21) in appendix A. But during the calculation of the sum over .a problem occurs if the parameter and becomes equal. The function is defined for the case and for . But the discrete sum causes, that becomes equal to in the case of . This discontinuity has to be handled before the function could be calculated. To do this the relevant part (9) of equation (21) has to analyzed.

(9)

To cancel this discontinuity at the neighborhood has to be taken into account. The sum is an approximation of an integration over . This mean the discontinuity could be canceled by solving the integration over this discontinuity:

 

(10)

This approach solves the discontinuity at can be canceled in the following way: (11)

Now the function can be evaluated everywhere, by adding two new case to the equation (23):

(12)

Further it has to be pointed out, that for the function .does not have implemented with the real parameter and but with the discrete parameter and . This was necessary, because a computer can’t test two reel values to equality in a good way. By matching the case study of (12) to the integer parameters and the problem does not occur any more. 4.2. Example of a numerical integration with a discontinuity A main task during the calculation of is to determine the integral

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(13)

To do this the following two functions are defined:  

(14)

The integral is consequently:

(15)

The advantage within a framework like maple is, that such a discretization is done automatically. Numerical problems are hidden from the user. To implement such a function by hand means to implement a numerical procedure for all valid parameter combination. In this case the fraction from equation (15) becomes undefined if :

(16)

This both equations become zero in the case of . This mean a direct calculation is impossible. But by the usage of the law from l’Hospital this discontinuity can be canceled, too:

(17)

After solving this derivations:

  (18)

Now the result for is:

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(19)

This special case can be now used for the numerical integration of equation (xx):

(20)

This process of transferring the analytic equation into a form which is able to implement within software is not very difficult from the mathematical point of view. But the pure handling of long mathematical terms is not easy for humans. At this point such an analytic math tool is helpful, too. Especially task as using the law from l’Hospital can be handled in an easy way. Building derivations of complex function can be done a fast and sure way. Such tools can be used in way as a mathematical ”CAD-Tool”. Tasks, which are dangerous for humans, as there are algebraic signs, substitution of variables, and so on, are easily done by the computer without errors. The human is now able to concentrate on the mathematical description of a problem and no longer on writing error free equations. 5. Evaluation of the computed unsteady circulation In the way as descript above, the numeric model has been implemented to a standard FORTRAN method which has become part of the ship design tool E4. The wake of paper Zwick (1962) (Fig.4, page 166) has been digitalized and used as input for the purpose of evaluation. The results are shown in the Fig.6 and Fig. 7. Theoretical the results should be the same, but in practice it is a bit complicate to reproduce calculations done 50 years ago. Many information has been lost over the years. The used information has been taken from small pictures copied out of an old publications and has been digitalized. Beside the fact that this procedure contains an information loss, the data is not complete. As seen in Fig.4 information is only available at special radial points. Everything between has to be interpolated. Consequently the real wake used for the former calculation is unknown. The number of original supporting points is unknown as well as the way of interpolation and the way of integration done by the technical staff.

Fig.4: left side: original wake Zwick (1962); right side: imported E4 wake The evaluation of the implemented method has been done with help of the published circulation. The aim was to reproduce them as good as possible. The results are show in Fig.6 and Fig.7. In principle this is an easy task and the results should be exactly the same. In practice it is a little bit more complicate. The input data is only a small subset of the original wake and after 50 years it is not clear which kind of interpolation and integration mechanism have been used The Fig.5 shows the way of discretization. The original data from Fig.4 has be digitalized at fixed

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supporting points and a linear interpolation has been used to connect them (left side Fig.5). Afterwards a Fourier-Analyses has been used to describe the wake. The Fig5 shows the meaning of discretization for such a wake.

Fig.5: Examples of calculated circulation at and

Especially the hard peak at and has a significant influence on the circulation. If this peak is not discretizied accurate enough, the results are useless. The right side of Fig.5 shows further, that the Fourier-Analyses of such an inhomogeneous wake needs enough higher harmonics. Otherwise the results are difficult. The calculations of the original paper has used six harmonics to describe the wake where as this investigation used eight harmonics.

Fig.6: Examples of calculated circulation at

Fig.7: Examples of calculated circulation at The Fig.6 and Fig.7 shows a good reproduction of the original data. The existing differences can be explained with the uncertainties of the available data. It should be pointed out, that the input wake has

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been published in Fig.4 on the supporting points in contrast to this the unsteady circulation has been published for . The results have been calculated for interpolated supporting points. 6. Summary and further developments This paper has described an example how a classical analytic approach for a hydrodynamic theory can be used within a modern computer dominated environment. The missing computer power has lead in former times to many high sophisticated theories. These theories have still today the great advantage of less requirements in computer power. This feature is nowadays very useful for demands of the early design. But the development and the numerical implementation is often not easy. Problems occur if a complex mathematical theory has to be transferred to an algorithm usable by a computer. Discontinuities has to be handled with the needed accuracy. Questions of supporting points and the mapping of real values on a digital floating point arithmetic has to done correctly. To do such a task symbolic math tools can be used nowadays. They can support an engineer by the handling of complex mathematical terms and equations. These kind of software is very useful to implement mathematical prototypes of complex theories, which can be used afterwards to verify the real software implementation. The shown approach is a first step of a development with the aim to simulate the non-viscous effects of a rotating propeller with in a real wake and to get a model which is compatible with other potential theory methods. Calculations, which have been done in the middle of the last century in a time consuming way by hand, could be reproduced with modern computer technique in a time shorter a second. In a next step it is necessary to extend the analytic model by features of a real propeller. Effects of a given distribution of cordlens, pitch and skew have to be taken into account. Even if such things should increase the numerical effort of the original theory, there is still a great advantage for numerical investigations in hydrodynamic effects between propeller, hull and ruder. Especially the interaction between propeller and rudder could be done in a new way. The analytic approach of solving the equations of the propeller induced free vortexes is compatible with other potential theories. A combination with a panel based description of a rudder has not anymore the problem of singularities, as they are by unsteady methods like the QCM. References ABELS, W. (2006), Zuverlässige Prognose propellererregter Druckschwankungen auf die Außenhaut mittels Korrelation direkter Berechnung, ISBN 3-89220-636-8, Schriftenreihe Schiffbau ISAY, W.-H. (1964), Propellertheorie, Springer Verlag KRÜGER, S. (2003), The Role of IT in Shipbuilding, 2nd Conf. Computer and IT Applications in the Maritime Industries (COMPIT), Hamburg STRECKWALL, H. (1997), Description of a Vortex Lattice Method for Propellers in Steady and Non Steady Flow, Hamburgische Schiffbau-Versuchsanstallt GmbH, Report 18/97 ZWICK, W. (1962), Zur Berechnung der Zirkulation und der Kräfte eines Propellers im Nachstrom, Schiffbauforschung 14/1962, Berlin

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A. appendix Definition of matrix elements from Zwick (1962):

  (21)

(22)

Further there is defined:

 

(23

 

 

 

(2

(25)