Towing point influence in model tests for high-speed vessels
Dr. Ir. P. de Jong & Dr.-Ing. F.G.J. Kremer
A4ARIN, Wageningen, The Netherlands
ABSTRACT: Model tests are commonly used to determine the behaviour o f ships during the design phase. One o f these tests is a resistance test, in which a towing force is applied to the scaled model in order to replace propulsion. The model is given the desired velocity and the resistance is determined by measuring the applied towing force. However, the location o f this towing force does not always coincide with the working line o f t h e propulsor, causing an additional moment on the ship model. This additional moment influences the measured trim angle, rise and resistance during the model towing test. This specifically applies to high-speed vessels. Therefore, for Froude numbers between 0.6 and 1.2 the towing point may influence the measured resistance signiflcantly.
In this paper the problem o f the location o f the towing point is described and the issues regarding the forces and moments on the vessel are explained. This is quantified for fast mono-hull vessels using both an engineering emperical approach and the potential flow code PANSHIP. The importance o f t h e towing point location is shown and explained for the speed region around Froude number one. In this region the stabilising effect o f t h e buoyancy on the trim angle starts to decrease, while the stabilising effect o f t h e dynamic lift on the trim only is still increasing. Recommendations are given forthe performance determination in model tests for such vessels.
1 INTRODUCTION
For semi-planing and planing hulls, the trim angle and rise (or sinlcage) are a function o f the ship speed (or Froude number, as shown in Figure 1). Theii-progi-ession with speed is related to the development of dynamic lift on the wetted surface o f the hull. As the ship speed increases the weight o f the vessel is increasingly carried by the dynamic lift force, lifting the vessel out o f the water and reducing the buoyancy force. The vessel makes a transition f r o m the displacement regime via the semi-displacement (or semi-planing) to the planing regime, where the majority o f the weight is carried by the dynamic lift. The -equilibrium trim and rise are the result o f the balance resulting f r o m the forces and theii- moments acting on the hull. These forces include the aforementioned (dynamic) lift and (static) buoyancy, together with the gravity, resistance and propulsion forces.
During resistance towing tests performed in towing tanlcs the propulsion force is replaced by a towing force that is measured to obtain the resistance o f the model. This towing force should be applied to the model coinciding with the working line o f the propulsion force it replaces. This can for instance be
achieved by placing the hinge o f a towing post exactly on the working line. Unfortunately this is not always possible, due to the shape and size o f the model and the towing arrangement. Furthermore by definition (ITTC 2016) the towing force is applied horizontally. This may affect the vessel trim and rise, thereby affecting the measured resistance. The question that arises is for which operating conditions this effect becomes significant, and whether correction o f the results or the test setup is necessary.
Displacement Seml-planing Planing
>
Aft Rise
Lift Coefficient
• 1
~0.3 ~1.0 Froude > Figure 1. General trim and rise behaviour as ftinction of the Froude number for high-speed vessels.
XI HSMV - NAPLES OCTOBER 2017
It should be noted that interaction effects between the propulsor and the aft hull by replacing the propulsor with a towing force are not considered in this paper. Additionally, the effect o f applying a horizontal towing force to replace a propulsion force working at an angle is also not detailed. Nevertheless, both effects partly explain why the working lines o f tow force and propulsion force may differ, also causing changes in the trim and rise o f the model and by extension the measured resistance. For water jets these issues are further complicated due to thrust force and its working line being the result o f a rather complex flow through the water jet tunnels and therefore not Imown a priori.
positive moment
Figure 2. The forces acting a on hull during a resistance test.
There are several forces acting on the model during a towing test for a semi- or ftilly planing hull. The forces relevant in horizontal dii"ection are the towing force and the drag force. The drag force is composed o f a contribution due to the pressure distribution on the wetted surface and one due to the viscous drag upon the hull. The forces that act in the vertical direction are the buoyancy (hydrostatic pressure on the hull), the dynamic lift (hydrodynamic pressure on the hull) and the weight o f t h e model itself The trim angle (r) and rise are defined as shown in Figure 2, with positive trim anti-clockwise and positive rise for the centre o f gravity moving upwards. A positive trimming moment is defined as positive bow down. This definition leads to a positive derivative between trim angle and trim moment implying a stable situation, in accordance with usual definitions for transverse stability.
In this paper the stationary solution at a constant velocity in which the model is sailing straight ahead in calm water is o f interest, so all forces and moments should be in equilibrium:
bouymicy Ufl gravity
From these the equilibrium trim and rise are obtained, also referred to as dynamic or running trim and rise.
2 EMPERICAL ANALYSIS M E T H O D
To obtain more insight into the physical background of the vessel behaviour an empirical model was set up based on the often cited model developed by Savitsky (1964). The equations themselves are based on flat plate lift and employ corrections for prismatic shapes, based on analytical considerations combined with empirical formulations. Here the model was solved numerically, while the different contributions to the force and moment balance were kept separate to enable determination o f theii" contribution to the trim balance.
Below the main force contributions are given, mostly in the form as they were given by Savitsky using the following definitions:
L ship length [m]
b beam o f planing prism shape [m] ta draft aft [m]
location o f the centre o f gravity in front o f the transom [m]
u ship speed [m/s]
T trim angle (positive bow up) [rad] (or degrees
where indicated by subscript 'deg')
P deadrise angle (positive bow up) [rad] (or
degrees where indicated by subscript 'deg') A weight o f displacement [N]
The Froude number and the speed coefficient are given by:
U
1
Savitsky derived the following equation for the mean wetted length to beam ratio:
b tan jB
0 _ sinr 27V t a n r
The dynamic lift force is computed combining the lift coefficient for a flat plate with a correction for deadrise angle:
C ; , o = r a e ; ' ( 0 . 0 1 2 0 r - ' )
c,, = q„-o.oo65CX
f
0.0]20r,^;'A'''-0.0065r,,; '°-' (0.0120"'^/l°-'°'')/?,,^
The (static) buoyancy force is based on a speed dependent ' l i f t ' coefficient leading to speed-independent absolute force:
11
0.00551 C
2.5
The viscous drag component is determined by Savitsky by using a friction coefficient based on the Schoenherr friction line, combined with a mean tangential velocity V j over the planing surface:
2 cos yff cos T
In order to simplify the derivations and the empirical character o f the equations here two simplifications were made. First for modest trim angles and regular values for the mean wetted length to beam ratio it was found that the mean velocity was close to the ship velocity:
Next, Savitsky adds both forces and multiplies by the arm to obtain the total moment due to lift and buoyancy. In the text it is noted that the buoyancy force is assumed to act at 33% o f the wetted length from the transom, and the lift force at 75% o f the wetted length and that this leads to the following combined arm for both forces added together:
0.75Ab- Ab
5 . 2 1 ^ + 2.39
This formula leads to an arm o f 33% at zero speed ('buoyancy mode') and an arm o f 75% for Cv values of 4 and higher (depending on the mean wetted length to beam ratio). Although the results for the total moment are similar, in order to better distinguish the separate contributions here the 33% and 75% arms are used:
' i = - OJSAb
The contribution o f the drag force is separated into pressure drag and viscous drag. Both components are assumed to have the same vertical arm:
According to Savitsky and Neidinger (1954) the pressure drag equals the horizontal component of the normal force acting on the planing surface. The vertical component equals the weight o f the vessel, therefore:
= A t a n r
Second, the 1/7 power law was used for the friction line instead o f the mathematically impractical implicit Schoenherr line:
0.027 Re,,^
The magnitude o f the towing force equals the total drag; its vertical arm depends on the height o f the attachment point (usually the hinge o f the towing post) in relation to the centre o f gravity:
Finally, the gravity force acts in the centre o f gravity; its arm is zero.
Substitution o f the above equations into the equilibrium equations and solving the resulting system o f equations yields the equilibrium trim and rise as ftinction o f speed, as well as an estimation o f the resistance of the vessel.
For this study, the equations were used to investigate the characteristics o f the equilibrium trim angle in particular. To simplify the work, the trim and rise as function o f speed obtained from another numerical method (as described in the next section) were used and the moment equilibrium around these equilibrium values was studied by taking the derivative o f the trim moment with respect to the trim angle. Due to the definitions adopted here a positive value o f this derivative implies a stable equilibrium around the equilibrium trim angle. A low value would then imply a less stable situation.
XI H S M V - N A P L E S OCTOBER 2017
while a negative value would mean an unstable behaviour. The derivatives were determined analytically fi-om the above equations by using a symbolic math Python library SymPy. In a similar way the sensitivity o f the resistance to the towing point height was determined.
To estimate the calm water equilibrium trim and rise and the resistance as function o f speed the time-domain panel code was used. This method, termed PANSHIP (Van Walree, 2002, De Jong and Van Walree, 2009 and De Jong, 2011), was originally developed for time domain seakeeping simulations for fast and advanced vessels. The method can be characterised by:
- Three-dimensional transient Green function to account for linearised free surface effects, exact forward speed effects, mean wetted surface, mean radiated and diffracted wave components along the hull and a Kutta condition at the stern;
- Three-dimensional panel method to account for Froude-Id'ylov forces on the instantaneous submerged body;
- Transom flow correction to deal with a dry-transom based on Garme (2005);
- Cross-flow drag method for viscosity effects; - Resistance obtained fi'om pressure integration at
each time step combined with empirical viscous drag;
- Propulsion using propeller open water characteristics or a semi-empirical water jet model;
- Motion control and steering using semi-empirical lifting-surface characteristics, water jet steering, and propeller-rudder interaction coefficients; - Empirical viscous roll damping; and
- Autopilot steering and motion control.
In order to determine the calm water equilibrium condition an iterative approach was followed. First, a mesh was generated on the wetted surface o f the model in an initial position. During simulations at forward speed in calm water the model built up trim and rise. After each simulation the panel mesh was adapted to match the wetted surface again. This was continued iteratively until convergence was achieved
of the trim and the rise o f the model. The total resistance was computed by adding viscous resistance, computed using the ITTC friction line and a form factor, to the resistance obtained fi'om integration o f pressures obtained from the potential flow calculation.
In this study the propulsion was replaced by a horizontal towing force, which was applied in the towing point on the model to simulate the effect o f the towing arrangement on the equilibrium trim and rise o f the model. This process was repeated for a range o f speeds for two vessels (detailed in the next section) and performed for a range o f towing point heights. This allowed the sensitivity o f trim, rise and resistance to the towing height to be studied.
3 RESULTS
Selected results from both the empirical and numerical analysis method are given for two vessels M l and M 2 , both tested at M A R I N . Table I , Figure 3 and Figure 4 show that the two vessels are similar in length and width, but differ in deadrise angle and displacement. Especially vessel M 2 has a pronounced spray rail. Both vessels are small fast craft.
A l l PANSHIP simulations presented in the following were performed for a range o f Froude numbers from 0.5 to 1.5 with a varying number o f intermediate points, as w i l l be addressed further below.
Table I . Particulars o f t h e vessels.
Vessel IVIl M2
Length between perpendiculars, Lpp [m] 20 21.5 Mean chine beam, b [m] 5.1 5.3 Mean deadrise angle, p [deg] 24 18.5 x-position o f centre o f gravity [m] 7.4 8.1 Displacement, A [tonne] 58 76
1
. ^ — • — • — • — — 1
1
STAT. 0 = APP STAT. 20 = FPP
STAT. 0 = APP STAT. 20 = FPP
Figure 4. Lines plan o f model M 2 .
increase o f the confidence interval. Another reason for the large confidence interval is that the curve fit was based on only 5 towing heights to determine the derivative. Finally it is noted to that convergence limits o f the numerical simulations were set to a trim difference o f 0.025 deg and a rise difference o f 0.025 m. Especially decreasing the latter may reduce the scatter in the data.
Figure 5 shows that the equilibrium trim angle computed using the PANSHIP method matches the resistance towing test data reasonably well. For low Froude numbers, however, PANSHIP gives a lower value for the trim angle than the test results for both vessels. This may be caused by the assumption o f dry transom flow in PANSHIP and needs flirther investigation. - z = 0 - z = l -z = 2 - 2 = 3 • t e s t r e s u l t s m a r i n 0.4 0.6 0.8 1 1.2 1.4 1.6 F r o u d e n u r ï i b e r [-]
Figure 5. Trim angle o f model M 2 , based upon PANSHIP simulation (colours) and M A R I N towing test ( z = 1.77 m).
The equilibrium trim angle clearly depends on the towing height, being most sensitive in the region o f maximum trim for both vessels. To quantify this sensitivity, the derivative o f the trim angle to the towing height was determined by means o f a linear curve fit through the PANSHIP simulation data for each speed regarded. Since this fit was not exact, the 95% confidence interval o f this derivative was also determined. This is shown in Figure 6 and Figure 7 for vessels M I and M 2 respectively. The figures also include the trim to towing height derivative determined with the empirical method. As input for the derived empirical equations the equilibrium t r i m and rise resulting from the PANSHIP simulations was used.
Figure 6 and Figure 7 show that the 95% confidence interval o f the derivative is large around Froude number one. Especially spray rails, as were present on vessel M 2 , are difficult to resolve in the numerical method and that may have led to an
- P A N S H I P
- E m p i r i c a l
FroutJe n u m b e r ! - ]
Figure 6. Trim to towing height derivative, M I , PANSHIP (blue, diamonds) and empirical method (red, continuous).
- P A N S H I P
- E m p i r i c a l
4 0,6 0,8 1 1.2 1.4 1.6 F r o u d e n u m b e r [-]
Figure 7. Trim to towing height derivative, M 2 , PANSHIP (blue, diamonds) and empii-ical method (red, continuous).
The numerical results for vessel M I in Figure 6 show a clear minimum in the derivative o f the trim angle to towing height around Froude 0.7-0.8. This implies that the sensitivity o f the trim angle to the towing height is at its maximum in this speed range. The maximum sensitivity for vessel M 2 is visible around Froude 1.1-1.25. This illustrates that the hull lines and displacement have a significant influence on the sensitivity o f trim to towing height.
Compared to the numerical method, the empirical model shows a similar curve; however a 0.15 shift for the minimum value towards a higher Froude number can be observed for both vessels. Nevertheless, the predicted magnitude o f the sensitivity seems close for the empirical model versus the numerical model.
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The progression o f the derivative with speed shows that at Froude numbers below 0.5 the influence o f the towing heigth on the equilibrium trim angle is limited, while at the medium speed range (between Froude numbers 0.6 and 1.3) there is a clear influence, which seems to diminish again towards high speeds. This implies that there w i l l be a significant difference in trim i f the model is not towed with the towing force on the proper working line o f the propulsion force.
Two questions remain. The frst question considers what causes the trim angle to be sensitive to the towing height in the medium speed range determined above. The second is to what extent the model resistance is affected by not towing the model with a tow force on the proper towing height.
The first question is addressed in Figure 8 and Figure 9, both showing the derivative o f the moment contributions acting on the vessel to the trim angle. The more positive the total moment to trim derivative, the more resilient the model is against external disturbances, i.e. an external moment has a smaller influence on the resulting equilibrium trim angle.
- s t a t i c
D y n a m i c
T o t a l
F r o u d e N u m b e r [-]
Figure 8. Moment to trim derivative components and total for model M l ; numerical method PANSHIP.
- B u o y a n c y L i f t - P r e s . d r a g V i s e , cJrag T o t a l f r o u d e N u m b e r [-]
Figure 9. Moment to trim derivative components and total for model M I ; empirical method.
Figure 8 presents the moment contributions derivatives to the trim angle obtained using the numerical method PANSHIP. For each speed the equilibrium trim condition was determined varying the towing height; each variation constituting one single domain solution.To enhance the confidence (and decreasing the 95% confidence intervals shown in the graph) in the numerical fit o f this data this was done for 9 towing heights at each speed. This fit was used to obtain the derivative o f the moment contributions to the trim angle. To save on computatational time the simulations and subsequent analysis were only performed for 5 forward speeds using the numerical method.
Figure 9 shows the same analysis, but now based on the results o f the empirical method. As the moment derivatives o f the empirical method were derived as an analytical function, computational effort was not at issue and the graphs were obtained using a much finer resolution in forward speed compared to the numerical PANSHIP results.
Another difference between the two figures is the way the moment components are split up. PANSHIP only distinguishes a static component (buoyancy and viscous drag) and a dynamic component (dynamic pressure, mainly lift and pressure drag). In the derivation o f the empirical model the components are split up ftuther, disthingsuishing the drag (both presussure and drag) components from the lift and buoyancy components. From Figure 9 it can be concluded that drag does not contribute significantly to the moment derivatives, and that the static component from PANSHIP can be compared to the buoyancy component and the dynamic component fi-om PANSHIP can be compared to the lift component.
For both vessels, the (stabilising) contribution from the static moment decreases with increasing forward speed, while the (stabilising) contribution fi-om the dynamic moment increases. The decrease in static stabilisation moment with speed and the increase o f the dynamic stabilisation moment lead to a clear minimum in stabilising moment in the mid speed range. The speed at which this minimum occurs appears to be close to the speed at which the maximum trim angle difference due to the twoing location occcurs. A n exact match in speed is difficult to discern as the resolution in speed in Figure 8 is lower for reasons explained before. As stated before, the empirical results allow better distinction between the influence o f the different moment contributions.
It becomes apparent that the drag forces do not contribute significantly, and the main contributors are the buoyancy (static) moment and the lift (dynamic) moment. The empirical model for this case is considered only valid for Froude numbers 0.7 and higher, possibly explaining the sharp rise o f the moment contributions towards lower speeds.
OAO 0.60 0.80 1.00 1.20 1.40 1.60
F r o u d e n u m b e r [-]
Figure 10. Normalised towing force to towing height derivative. M l , PANSHIP (blue, diamonds) and empirical method (red, continuous).
0 . 4 0 0 . 6 0 0 , 8 0 1.00 1.20 1.40 1.60 F r o u d e n u m b e r [-]
Figure 11. Normalised towing force to towing height derivative, M 2 , PANSHIP (blue, diamonds) and empirical method (red, continuous).
The second, and main question that this study aims to address is to determine the influence o f towing point height on the measured resistance in a resistance test. To answer this question based on the computations performed here Figure 10 and Figure 11 present the derivative o f towing force with respect to the towing height. The towing force was normalised with the mean towing force (i.e. model resistance) at the same velocity. The numerical results were determined by linear curve fits to the relation o f towing force versus towing height using the PANSHIP simulation data. The empii'ical results were again determined from analytical derivation o f the empirical relations obtained f r o m the Savitsky model. The empirical derivatives were obtained around the equilibrium trim and rise obtained from the PANSHIP simulations. The 95% confidence
intervals o f the curve fits are again included in the graphs.
Similar to the previous results the confidence interval for vessel M 2 compared to vessel M l is larger. This, again, is probably due to the spray rail causing numerical difficulties for the PANSHIP solution, while not being accounted for in the empirical approach. The spray rail caused oscillations in the trim angle resuks, which made it difficuk to determine the exact speed giving the greatest senskivity for the towing point height. This was especially true for model M 2 , which had a pronounced spray rail. The number o f iterations for PANSHIP simulations needed to reach convergence increased significantly from only a few iterations to much more than 10 iterations. For the higher speeds (above Froude number 1.15) reaching f u l l convergence was not feasible within a reasonable number o f computations (that was capped at 30). While the trim (Figure 5) did show some small oscillations for these speeds, the derivatives were significantly more affected.
Disregarding the not very well defined resuks for model M 2 at the higher forward speeds, the resuks shown in Figure 10 and Figure I I indicate that the towing force can be affected between 3% (numerical approach) and 5% (empirical approach) per metre difference in towing point height (at frill scale) for model M l . For model M 2 these percentages become 2.5% for both the numerical and the empirical approach.
Assuming a geometric scale factor o f 1/10 for model tests, this means that a 10 cm difference in towing height with respect to the 'proper' towing height (the towing point at the working line o f the propulsion force) can lead to a resistance difference o f around 3%. As the towing point is often taken higher (due to a lack o f space for the towing arrangement deep in the model) this resuks in a too low a predicted resistance.
This effect is the strongest around the same Froude number range where the trim angle is most sensitive to the towing height, due to the lack o f longkudinal stabilising moment in the same speed range. The buoyancy contribution to this moment has already been reduced due to the vessel being lifted out o f the water, whereas the stabilising moment due to dynamic lift is still building up.
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4 CONCLUSIONS & RECOMMENDATIONS
4.1 Comhisions
This research considered the effect o f the towing point position during resistance tests for fast vessels on the equilibrium trim and measured resistance in calm water. The sensitivity o f equilibrium trim and resistance to the towing point location around the Froude number one was exposed. This was done both by using an empirical model based on the Savitsky equations and by applying a potential f l o w time domain simulation code for fast ships, PANSHIP.
The computations showed that the equilibrium trim angle and the measured calm water resistance are especially sensitive to the towing point height for speeds between Froude 0.6 and 1.2. It was exposed that the hull geometry and displacement affected at which exact speed the resuks were most sensitive. Overall a resistance difference o f 3.2% was predicted to occur due to a towing height error o f 10 cm for a model scale 1:10 for the vessels investigated.
The senskivity was explained by the decrease o f stabilising effect o f the buoyancy on the trim angle with increasing speed, whereas the stabilising effect of the dynamic l i f t only starts to increase towards higher speeds. Theii* overall sum gave a minimum value in the given speed region.
4.2 Recommendations
The senskivky o f the towing height must be considered during model tests for high speed vessels.
The most accurate model test method is to perform a propulsion test to determine the requii'ed thrust o f the ship. Then the remaining towing force (to compensate for the scale effect in the viscous resistance) is much lower and therefore also ks influence on the balance o f moments and trim angle. This w i l l not only reduce the effect o f the towing point height, but w i l l also eliminate other effects, such as a propulsion force working at an angle and the propulsor-hull interaction.
I f resistance tests are performed, the towing height should be varied for crkical speeds around Froude number one. This can be used to extrapolate the resuks towards the towing height where the model is towed on the correct working line. This variation allows for resistance correction afterwards or can be used to give an expected interval o f the resistance i f the exact working line o f the propulsor is not known. 1
Furthermore, this at least makes the error made due to the towing height visible.
Another solution would be to compensate the difference in moment during the towing test, for example by shifting the mass in the model. For this solution, first the drag in a model test without shift is requked. Based on the drag obtained, then an extra moment compensating for the difference in working lines is applied to the model and the correct resistance obtained. This however requires knowing the exact propulsion working line and furthermore increases the required number o f runs.
The senskivky o f the towing point height has been demonstrated theoretically, showing that the largest errors occur when towing a model at an incorrect towing height for speeds close to Froude one. However, these resuks were only compared to resistance towing tests in which a single towing point was used. Akhough this enabled the calculated trim to be compared w k h the measured trim, k did not allow for validating the error that was predicted numerically caused by a change in towing location. The conclusions should therefore be verified by varying the towing height in a resistance towing test for a ship in the region Froude 0.6-1.2.
A C K N O W L E D G E M E N T
The authors wish to acknowledge the contribution o f E. Verboom, A.O. Remola, V. Wieleman, M . Overbeek and T. Hageman in recent years. We wish to thank F. van Wakee for his support on the PANSHIP code.
REFERENCES
I T T C (2016) Recommended Procedures and
guidelines, Resistance test, 7.5-02-02-01, 2011.
Garme, K . (2005) Improved time domain simulation
of planing hulls in M'aves by correction of the near transom lift. International Shipbuilding Progress, 52(3):201-230.
Jong, P. de and Walree, F . van (2009) The
Developmentand Validation of a Time Domain Panel Metiiod for tlie Seaiieeping of High Speed Ships", Proceedings of the 10th International Conference of High Speed Sea Transportation, pp. 141-154, Athens, Greece.
Jong, P. de (2011) Seakeeping behaviour of high
speed ships - An experimental and numerical study, Ph.D. Thesis, Delft University of Technology.
Savitsky, D and Neidinger, J.W. (1954) Wetted
area and center of pressure of planing surfaces at very low speed coejficients, Stevens Institute of Technolog}', Davidson Laboratory Report No. 493, 1954.
Savitslty, D (1964) Hydrodynamic Design of
Planing Hulls, Marine Technolog}', pp. 71-95, 1964.
Walree, F . van, (2002) Development, Validation
and Application of a Time Domain seakeeping method for High Speed Craft M'ith a Ride Control System, Proceedings of the 24th Symposium on Naval Hydrodynamics, Fukuolca, Japan, pp. 475¬ 490.
