APPLICATION OF ENERGY SAVING FINS ON RUDDERS

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1 Proceedings of ASME 25 34th International Conference on Ocean, Offshore and Arctic Engineering OMAE 25 May 3 - June 5, 25, St. John s, Newfoundland, Canada OMAE APPLICATION OF ENERGY SAVING FINS ON RUDDERS Arne Falkenhorst Institute of Ship Design and Ship Safety Hamburg University of Technology Hamburg, D-273 Germany falkenhorst@tuhh.de Stefan Krüger Institute of Ship Design and Ship Safety Hamburg University of Technology Hamburg, D-273 Germany krueger@tu-harburg.de Christoph Michael Steinbach Van der Velden Marine Systems Glinde, D-259 Germany csteinbach@vdvms.com ABSTRACT Due to rising fuel oil prices in the last decade as well as rising design speeds, it has become common practice to build rudders with twisted leading edges to minimize resistance and cavitation risk. The next step in this development is the application of fins on to the rudder. The aim is to generate a distinct amount of thrust through the fins by retrieving rotational kinetic energy from the propeller slipstream. This paper presents a fast method to design and calculate rudder fins in the propeller slipstream, which has been implemented in the ship design environment E4. Because of his working principle, the propeller induces velocities to its slipstream. In the usual setup, the rudder is placed behind the propeller to generate higher steering forces caused by the higher inflow speed in the slipstream. In this arrangement, propeller and rudder together are forming a rotor stator system. The gains of the stator can be maximized by adding fins to the rudder. The main challenge of a fin design is the maximization and prediction of the regained thrust from the propeller slipstream. In order to do this, a steady, three dimensional, direct panel method is used to calculate the flow around the rudder and fin bodies, from which later the pressures and forces are evaluated. A lifting line method is used to predict the inflow velocities caused by the vortex dominated propeller slipstream on each panel. A special focus is on the treatment of the vortex wake, as crossing wake elements can lead to numerical instabilities and a wrong wake alignment produces bad thrust predictions. For the purpose of rudder design steady computation should Address all correspondence to this author. be preferred over fully unsteady computation, since only time average integral values are of interest and the degrees of freedom are reduced to the relevant ones. For example, it is not necessary to know the fluctuation of the angle of attack for the basic design of the profile respective the leading edge of the foil, only the mean value is needed. In the industrial practice, rudder fins are not often used because the calculation is difficult. Until now it is more expensive to design and build the fins than the savings earned by the ship owner. This phenomenon will change in the next years due to better calculations and rising fuel oil prices. NOMENCLATURE β i Hydrodynamic angle of attack. k Pitch of the free-vortex system. r Actual radius in percentage of R a. R a Outer radius. u Inflow velocity. ω Angular velocity. u q Induced axial velocity. v q Induced radial velocity. δ Blade angle. c a Lift coefficient of the respective section. N Number of blades. κ Goldstein factor. D Propeller diameter. T Propeller thrust. Copyright c 25 by ASME

2 v a A Advanced velocity. Propeller disc area. INTRODUCTION This article illustrates a steady method for the calculation of the regained thrust with a rudder-fin configuration from the propeller slipstream. For the calculation of acting forces on the propeller, a steady lifting line method [] is used. To evaluate the velocities in the propeller slipstream a calculation developed by Lerbs and Goldstein is applied. The main prediction of forces on the rudder-fin configuration is done by a direct panel method [2], which takes the already calculated velocities as a non-uniform inflow. In the design process it is not necessary to know the progression of the induced velocities in the propeller slipstream, because only the integral value is from interest for the design of the rudder-fins. Figure shows a typical arrangement of rudderfins on a twisted leading edge rudder. The working principle of this fins is that their leading edge is deflected into the rotating propeller slipstream. So the rectangular to the chord acting lift of the profile section produces a force in the ships x- and z- direction. The part of the force in the ships x-direction is the regained thrust. First, the mathematical model will be described and later, the adaptations, that are made for the case with rudderfins, will be shown. In the end, a conclousion and an outlook for future developments of this method is given. Furthermore, it has to be said that this work is still in progress and under development. It shows the actual state of the work. MATHEMATICAL MODEL In this calculation the mathematical model is divided into three main parts. This includes the lifting line method which determines the forces on the propeller blade, the slipstream calculation for predicting the propeller induced velocities at the location of the rudder and a direct panel method to calculate the forces on the rudder-fin configuration. All three parts are described in the following subsections. Lifting Line The lifting line method is used here to calculate the circulation strength for the respective propeller blade. Since in the steady method only a spatial-continuous result is necessary, here, a procedure is applied, wich is described by Isay [] and based on Goldstein [3]. The procedure of the method is iterative and starts in the first step with neglected induced velocities (u q (r) =, v q (r) = ). This leads to the initial solution of Eqn. (). Next, the Goldstein factor is determined by the ratio between the pitch of the unbounded vortices and the radius of the propeller disc. Figure FIGURE. APPLICATION OF TWISTED FINNS TO A TWISTED LEADING EDGE RUDDER. κ FIGURE 2. PELLER. r/k = 2 r/k = 3 r/k = 4 r/k = r/r a GOLDSTEIN FACTOR OF A FOUR BLADE PRO- 2 shows exemplarily the Goldstein factors of a four bladed propeller. With the evaluated Goldstein factor, Eqn. (2) can be solved and leads to the circulation on the respective section of the propeller blade. 2 Copyright c 25 by ASME

3 tan(β i ) = k r = u + u q ω r + v q ().8 Γ(r) = 2 c a ω r tan(δ ) u l cos(δ ) + 4πκ N (tan(δ ) + k r ) (2) g a.6.4 Out of the calculated circulation strength the induced velocities are determined with Eqn. (3) and (4). With this result the next iteration for a more precise outcome is started. In nearly all cases not more than 2 iterations per section are necessary..2 r/r a =.3 r/r a =.5 r/r a =.7 r/r a = x/d u q (r) = r N Γ(r) k 4πr κ (3) FIGURE 3. LERBS. AXIAL INDUCTION FUNCTION ACCORDING TO v q (r) = N Γ(r) 4πr κ (4).2 Like many other authors have shown, a good accordance for induced velocities in the propeller disc area is delivered by this method. The lifting line determines forces very well and so the propulsion equilibrium is found with a quiet good accuracy. Like all potential methods the lifting line theory represents no viscous influences. Due to this fact, empirical corrections are used. It has to be mentioned, that this calculation can only be used for attached flows. Slipstream Calculation For the calculation of the velocities in the propeller slipstream an approach by Lerbs [4] is taken into account. At first, the induced velocities in the propeller disc are needed and are provided by the lifting line method as described above. Afterwards, the slipstream-helix angel at the point of interest has to be determined. For this position the derivative of the bounded circulation strength in span wise direction in the propeller is evaluated. With this information and the induction function provided by Lerbs [4] the induced velocities at the relevant point can be calculated. Figure 3 shows an example of the axial induction function and Fig. 4 shows the one for the transversal induction function. Eqn. (5) and (6) give the mathematical relationship for the induced velocities. It is obvious that the induced velocities far behind the propeller, when g a becomes, exceed the double absolute value of the one in the propeller disc. g a r/r a =.3 r/r a =.5 r/r a =.7 r/r a = FIGURE 4. LERBS. x/d RADIAL INDUCTION FUNCTION ACCORDING TO ( ( x u q (r,x) = u q (r,x = ) + g a D, r )) R a ( ( x v q (r,x) = v q (r,x = ) + g a D, r )) R a To include a correction for the viscous influence at the boundary of the propeller slipstream a turbulent mixing correction according to Söding [5] is done. It is based on the momentum theorem and includes empirical factors. At first, the thrust (5) (6) 3 Copyright c 25 by ASME

4 load coefficient is calculated in Eqn. (7) and with it the conditions far behind the propeller are determined in Eqn. (8) and (9). Afterwards, the radius r x of the slipstream at distance x behind the propeller disc and the respective velocity v x is calculated with Eqn. () and (). Then, an additional viscous radius r is determined by Eqn. (2) and the correction velocity v corr can be calculated with Eqn. (3). Later, only the correction velocity is used and added to the before determined velocities in the slipstream. c T H = T ρ (7) 2 v2 aa ( r = r + v ) a 2 v (8) v = v a + ct H (9) r x = r.4 (r /r ) 3 + r /r (x/r ).5.4 (r /r ) 3 + (x/r ).5 () v x = v ( r r x ) 2 () r =.5 x vx v a v x + v a (2) ( ) 2 rx v corr = (v x v a ) + v a (3) r x + r Another correction is done for the hub vortex. Here the circulation strength of the most inner section of the propeller blades are taken to produce a hub vortex. With this method the inflow velocity for each panel of the rudder-fin configuration is calculated. Due to this, no propeller wake panels have to be applied and the numerical stability is increased massively, because singularities are avoided. Some weaknesses are provided by this analysis like the missing interaction between free vortices and the rudder-fin configuration. Moreover, it does not represent upstream induction by the rudder or fins. But from the perspective of a design tool it is not necessary to calculate upstream influences of the rudder or fins or interaction effects between rudder or fins and the propeller slipstream. Also it is not necessary to determine the time depended change, it is only the time average value needed to design the mean or thickness line. It can also be stated that the influence of the last-named factors is marginal. Panel Method This is the main calculation of the program. It is a modified version of a program according to Söding [2]. The main differences are the more precise calculation of the inflow in the propeller slipstream and the more precise wake treatment. The applied direct panel method is based on sources and doublets to calculate the flow. It depends on the second Green s identity, shown in Eqn. (4). The function f and g depend on the vector x, that can be differentiated sufficiently. Here Ω means the field and S the surfaces in the field, while n is the normal vector of the surface pointing outwards. Ω [ f g g f ]dω = [ f g g f ] nds (4) This can be formulated with the potential φ for the function f and the Green function of a source point G( x, x ) = x x for g as: Ω [φ( x) G( x, x ) G( x, x ) φ( x)]dω = S [φ( x) G( x, x G( x, x φ( x)] nds (5) S With the boundary condition (no flow through the body surface) seen in Eqn. (6), the Eqn. (5) and (6) are solved to Eqn. (7). It has to be taken into account that at the trailing edge the differentiabillaty in φ is not given. Due to this, wake panels are needed. In Eqn. (7), S b means surface of the body and S w surface of the wake. φ( x) S b +S w n x x = φ( x) n = U n (6) S b U n x x (7) In this method the source point is located inside the body, it is placed in the opposite direction of n and with the distance of % of the square root of the area of the panel, but not more than 3 of the local body thickness. For the numerical calculation of the formulas the bodies (rudder, skeg and fins) are discretized as quadratic and triangular panels, internal the quadratic panels are divided into triangular panels. On every panel a constant potential φ is assumed. The difference in the potential at the upper and lower side of the trailing edge are taken as the potential of the wake panels. Due to 4 Copyright c 25 by ASME

5 practical reasons, the length of the wake panels is set to m, which is more than 2 times the chord length in most cases. The orientation of the wake panels will be discussed in the next section. Afterwards, the potential φ on the wake panels is determined in a more precise way, than just taking the difference at the trailing edge, is taken into account. For this purpose the potential is assumed to be in the center of the panel and then the potential is extrapolated by the last two panels on each section to the trailing edge. The force on each panel is calculated with the steady Bernoulli equation (Eqn. (8)) and the normal vector of the panel, as it can be seen in Eqn. (9). TABLE. GRID RESOLUTION FOR DIFFERENT BODY TYPES. Body Panels Panels per section in span direction rudder 4 2 skeg 26 7 fin 2 8 p p = 2 ρ[( U + φ) 2 U 2 ] (8) F = (p p ) n (9) Later it is quiet easy to sum up the forces and moments on each body or on the complete system. This is how the regained thrust can be calculated, also the forces on the connection to the rudder are fast determined. APPLICATION TO RUDDER-FINS In this section the adoptions that are made to the method for the case of a rudder-fin configuration are described. A typical grid setup is shown in Fig. 5. Discretization of the Grid As mentioned before the grid is discretized by quadratic and triangular panels. For the evaluation of the velocities the quadratic panels are divided internally into triangles. The here described panels are the quadratic ones which provide the sources and doublets. It has been shown from industrial practice that a resolution of the grid for most design cases pointed out in Tab. is sufficient for calculating forces and leads to quiet good results. The grid is described section wise by profiles along the local z-coordinate of the wing (span wise) and the sections are taken from a faring module for rudders in the ship design environment E4. Due to this, a fixed setup can be easily calculated and determines the changes after a geometry modification. Fig. 5 shows that the system is composed by single wings, even though the fins are attached to the rudder body. This is done because it enables fast changes in geometry and reduces the modeling effort dramatically, as described above. If the fins are placed near by the rudder, the influence of the gap is limited. But it has to be stated that the results nearby the gap have to be viewed critical. The determined forces are just slightly affected FIGURE 5. TION. PANEL GRID OF A RUDDER-FIN CONFIGURA- by this approach, due to the small local expansion of the made error. It has also be stated that a minimum gap is needed, otherwise sources and doublets are placed in the other body. This is physical wrong and even so leads to numerical instabilities. Wake Treatment It has been shown by Betz [6] that the angle at which the wake is attached to the trailing edge should be half the angle of attack. This would lead in a propeller slipstream to a non even wake distribution, because the angle of attack changes over the height and also changes the direction (upper and lower part of the rudder). This is good to apply on a case without fins, but when fins are attached to the rudder the wake panels of the fin 5 Copyright c 25 by ASME

6 and the rudder would cut each other. This penetration of wake panels leads to heavy numerical problems, because it produces singularities. To avoid the singularities the wake of the rudder is kept at an angle of zero degrees. This is not correct for the single section, but taking into account that rudder is nearly symmetrical to the x-,y-plane on the height of the propeller hub, it is a good compromise for numerical stability. Also the error that is made on the upper part has nearly the same size as the one on the lower part, but with a different sign. This results in an quite small error of the integral value. For the fins the wakes can be attached with the half angle of attack and the so calculated forces are precise. This can also be seen in Fig. 5. For the orientation of the wake a mean value of angles of attack is determined from the slipstream calculation at the leading edge. This is done to avoid deformed wakes, which would lead again to numerical instabilities. [3] Goldstein, S., 929. On the vortex theory of screw propellers.. In Proceedings of the Royal Society of London, A, pp [4] Lerbs, H., 955. Über gegenläufige schrauben geringsten energieverlustes in radial ungleichförmigem nachstrom. Schiffstechnik. [5] Söding, H., 993. Manoeuvring Technical Manual. Seehafenverlag, ch.. Active and Passive Control Devices, pp [6] Betz, A., 99. Schraubenpropeller mit geringstem energieverlust. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pp CONCLUSION AND OUTLOOK First, it has to be stated that the developed method is able to determine the forces acting on the rudder-fin configuration very well. This leads to the necessary information for the design process of fins. The main reason for the installation is the regain of energy from the propeller slipstream, this can also be predicted by the presented method. Later, it would be of interest to improve the process by the possibility of calculating rudder angles which are non-zero and also treat the wake panels of the rudder right. This leads automatically to penetrating wake panels. With this more precise wake treatment a statement about the influence on the rudder efficiency can be done and the accurancy of the forces is also improved. For this purpose the upcoming singularities have to be solved. This is mathematically demanding and is so far not developed, but it will probably increase the accuracy of the method. Moreover, a in x-direction divided and self orientating wake would be of interest to calculate the forces even more precise. Due to the steady character of the method only, a risk for shed cavitation can be determined. To improve this, an unsteady panel method is needed. ACKNOWLEDGMENT I thank Professor Krüger for supporting my work at our institute. I also thank Christoph Steinbach for the practical input from the industrial point of view and the professional exchange. REFERENCES [] Isay, W. H., 964. Propellertheorie: Hydrodynamische Probleme. Springer. [2] Söding, H., 997. Ruderkraftberechnung. IfS-Bericht. 6 Copyright c 25 by ASME