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Int. J. Nav. Archit. Ocean Eng. (2015) 7:626~639 http://dx.doi.org/10.1515/ijnaoe-2015-0044 pISSN: 2092-6782, eISSN: 2092-6790
Experimental and numerical study on ice resistance for icebreaking vessels
Jian Hu1 and Li Zhou2
1College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China 2Aker Solutions, Fornebu, Norway
Received 29 October 2014; Revised 11 March 2015; Accepted 30 March 2015
ABSTRACT: Ice resistance is defined as the time average of all longitudinal forces due to ice acting on the ship. Estimation of ship's resistance in ice-covered waters is very important to both designers and shipbuilders since it is closely related to propulsion of a ship and it determines the engine power of the ship. Good ice performance requires ice resistance should be as low as possible to allow different manoeuvres. In this paper, different numerical methods are presented to calculate ice resistance, including semi-analytical method and empirical methods. A model test of an icebreaking vessel that was done in an ice basin has been introduced for going straight ahead in level ice at low speed. Then the comparison between model test results and numerical results are made. Some discussions and suggestions are presented as well to provide an insight into icebreaking vessel design at early stage.
KEYWORDS: Ice resistance; Ice resistance formulas; Model test data; Icebreaking vessels. INTRODUCTION
Growing interest in shipping and drilling operations in ice-covered waters has triggered more investigation on ice loads estimation. Understanding and calculating ice loads form the basis of ship design for ice. For icebreakers and icebreaking vessels, ice resistance estimation is a key mission in preliminary stage of design process and highly related to ship's global performance. Once a certain hull is determined, it is important for designers to have insight into ice loads in order to select propeller and propulsion system which meet power requirement of the ship. The hull designed is expected to undertake low ice resistance and have good performance in ice under certain ice conditions.
Many researchers has done research on ice resistance and thus developed a lot of empirical and analytical formulas since 1970. Enkvist et al. (1979) discussed the main phenomena in the level ice-breaking process. Keinonen et al. (1996) presented equations to calculate ice resistance for a ship traveling at low velocity in level ice based on massive full scale measurements on icebreakers. Riska et al. (1997) proposed a formula to calculate ice resistance. In particular, some researchers divided level ice-hull interaction process into several phases, including ice breaking, rotating, sliding and clearing (Lewis and Edwards, 1970; Kotras et al., 1983; Lindqvist, 1989; Spencer et al., 2001; Valanto, 2001; Jeong, 2010).
More recently, some researchers began to develop numerical simulation in time domain. Wang (2001) initialized a method
Corresponding author: Li Zhou, e-mail: zhouli209@hotmail.com
This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
for simulating the interaction between moving level ice and a fixed conical structure. Based on the ice failure model derived by Wang (2001), Su et al. (2010) refined the ice-ship contact procedure to simulate ship manoeuvres in level ice. The numerical analysis was validated by comparing simulations with ship performance data of icebreaker Tor Viking II.
However, no single and exact method to calculate ice resistance exists. In this paper, a numerical method in time domain and several typical ice resistance formulas often used are described. The calculated results are compared with those from the model tests. Some discussions and suggestions on selections of formulas and how to make a reasonable prediction on ice resistance are made. This might provide help to icebreaker designers to some extent.
METHOD OF NUMERICAL SIMULATION IN TIME DOMAIN In the present simulation, there are two reference frames used, see Fig. 1.
Fig. 1 Earth-fixed (XEYEZE) and body-fixed (XYZ) reference frames in the horizontal plane.
The Earth-fixed frame, denoted as XEYEZE, is placed so that the XEYE plane coincides with the water surface, and the ZE axis is positive downwards.
The body-fixed frame, denoted as XYZ, is fixed to the vessel in such a way that the origin coincides with the centre of gravity, the X-axis is directed from aft to fore along the longitudinal axis of the hull, and the Y-axis is directed to the starboard.
The horizontal position and orientation of the vessel in the Earth-fixed coordinate system are defined by n = [x, y,ty], where the first two variables describe the position and the last variable describes the heading angle. Correspondingly, the translational and rotational body-fixed velocities are defined by V = [u, v, r]. The body-fixed general velocities are transformed to the Earth-fixed frame by
n = J(n)v (1)
cty -Sty 0 "
J(n) = Sty cty 0 (2)
where c, s are compact notations for cosine and sine, respectively.
The equation of motion is first expressed in the Earth-fixed coordinate system and then converted to the body-fixed coordinate system. Based on Newton's second law, the linear coupled differential equations of motion in the body-fixed coordinate can be written in the following form:
(M + A)r(t) + Br (t) + Cr(t) = Fe (t)
Fe(t)=
Rbi Ri Rri Fm1 F owl ' 0 " mvr
Rb2 + R* + Rr2 + Fm2 + 'Fow2 + 0 + -mur
Rb6 R.66 Rr6 F 1 m6 F ow6 0
where r = V , the added mass matrix A in open water is calculated from a boundary element method routine, the damping term B is assumed to be zero in the stationkeeping mode, the hydrostatic restoring coefficient C is zero. The subscripts 1, 2 and 6 refer to the directions of surge, sway, and yaw. The notation Rb is the ice-breaking force, which will be described in Section 3.1.The notation R is the ice submersion force,while the notation Rr denotes the ice force due to ice rubble accumulation. Fm is the restoring force due to the mooring system translated from the earth-fixed coordinate system to body-fixed coordinate system by the rotation matrix. The notation Fow is the drag force due to the motion of the ship relative to the water. The notation represents the moment produced by the heading controller, which is zero if not used in the simulation. The last term in Eq. (4) is due to the translation from the earth-fixed coordinate system into the body-fixed coordinate system. Newmark's method is used to solve the resulting equations of motion.
The ice loads acting on a ship depends significantly on the interaction process by which the hull breaks and displaces the ice. Once the ice contacts the hull, ice is being crushed. The crushing force then increases with increasing contact area until its vertical force component gets large enough to cause bending failure of the ice, after which the broken ice floes start to turn along the ship's hull until they are parallel to the hull. Finally, the floes submerge and slide along the hull as they are pushed by the next broken ice floes. With this concept in mind, an ice force model composed of an ice-breaking model and an ice submersion model is briefly described.
Ice breaking model
The ice-breaking model introduced by Su et al. (2010), is used herein. The basic geometric model for ice-hull interaction includes the full-size waterline of the ship and the edge of the ice. As shown in Fig. 2, the waterline of the ship is discretized into a closed polygon and the edge of the ice is discretized into a poly line in the established simulation program. At each time step, the simulation program is set to detect the ice nodes which are inside the hull polygon. Then, each contact zone can be found. To check whether the ice node is inside the hull polygon, geometric tools from computer graphics are adopted. It is assumed that the contact surface between ice and hull is flat, and the contact area is simply determined by contact length and indentation depth. The contact length is calculated from the distance between adjacent hull nodes, and the indentation depth is calculated from the perpendicular distance from the cusp of ice nodes to the contact surface. More details are referred to Su et al. (2010).
The ice wedges formed in the ice breaking process were determined by bending cracks, which were idealized and described by a very important parameter, namely the icebreaking radius. The icebreaking radius R was derived from the expression given in Wang (2001):
R = Cll(1.0 + Cvvreln)
where v£e1 is the relative normal velocity between the ice and the hull node, Q and Cv are two empirical parameters, Q having a positive value and Cv is a negative value, l is the characteristic length of the ice:
12(1 - v2)pwg
Last Contact Node
Discretized Contact Zone Fig. 3 Idealized ice wedge and discretized contact zone.
Fig. 3 shows that the ice wedge is determined by the interpolation of the icebreaking radius at the first and last contact nodes (i.e., Rf and Ri). The opening angle of the ice wedge is denoted as 0. In order to calculate the contact area Ac, the contact zone was discretized by a number of triangles (the triangles in Fig. 3) based on the hull nodes that were in contact with ice sheet. Then Ac can be approximated by the sum of area of the triangles.
As mentioned above, in the first phase of contact only crushing takes place on the contact surface. The resultant crushing force Fcr is normal to the contact surface and is calculated as the product of the effective crushing strength cc and the contact area Ac, where the ice pressure on the contact surface is assumed to be uniform and equal to the effective crushing strength.
The frictional force is also taken into account in this model by using a coefficient of friction. When the vertical component of the crushing and frictional forces FV exceeds the bending failure load Pf given in Eq. (7), the ice wedge will be formed by a bending crack and break off from the edge of the ice:
Pf = Cf (-fa h
where 0 is the opening angle of the idealized ice wedge shown in Fig. 3, cf is the flexural strength of the ice, h is the ice thickness, and Cf is an empirical parameter.
Ice submersion model
When an ice wedge is formed and broken from intact ice sheet, it will submerge and slide down the surface of the hull. In this case, ice submersion loads introduced during the process has to be included. In this paper, the method by Croasdale (1980) is used. It is assumed that broken ice floe from intact ice sheet is continuously cleared around the structure by some other mechanism and doesn't contribute to ice pile up in front of the structure. The interaction between an ice sheet and a structure sloping at angle a from the horizon is shown in Fig. 4.
Structure
Fig. 4 Geometry for two-dimensional analysis of forces on a sloping structure.
Then the horizontal ice force arouse during ice rotation and sliding process on the sloped hull element per unit width H is written as
, , , . wsma + «cosa cosa
H = lsophig(sma + «cosa)(--1--)
cos a-,« sin a sin a
where ls is the vertical distance that ice is pushed down the slope, 5p is the density difference between water and ice, hi is the ice thickness, g is the acceleration of gravity, ^ is the friction coefficient between ice and the structure.
EMPIRICAL AND ANALYTICAL FORMULAS
As mentioned above, there are many empirical and analytical formulas available to estimate ice resistance. Some of them are presented as follows.
Lindqvist formula
The Lindqvist formula was developed from research done on full scale tests in the Bay of Bothnia (Lindqvist, 1989). It is a rather simple way of estimating the ice resistance. In this model, the resistance is divided into crushing, bending-induced breaking and submergence. The formula gives resistance as a function of main dimensions, hull form, ice thickness, ice friction and strength. The formula is expressed as:
Rice = (R + R )(1 +1.4^=)+ R (1 + 9.4^=l)
2 tan0 + ¿u cos 0 /cos|
Rc = 0.5<bnj -
1 - ¿sin0 / cos|
Rb = 27 <bB h1 tan| + ucos0(1 + _L_) (9)
64 I E cos|sina cos|
\ 12(1 - v2) g Pw
R = (Pw-P) ghB(TB+^+k)
B + ¿1
T B I 1 1
k = «(0.7 L-----h T cos^costy ——- +--- )
tan^ 4 tana ysin^ tana
ty = arctan(-)
where Rjce, Rc and Rb and Rs are total andice resistance, crushing resistance, bending resistance and resistance due to submersion; ab and hj are respectively ice strength in bending and ice thickness; / , ^, a and f are respectively the friction coefficient, stem angle, waterline entrance angle and flare angle; g is the gravity acceleration; pw and pt are water and ice density; E, and u are Young's modulus and Poisson ratio of sea ice respectively. B, T, and L are ship's breadth, draught and waterline length; V is ship speed.
Keinonen formulas
Based on results of a study of escort operations involving five icebreaking vessels, Keinonen et al. (1996) did research on resistance of icebreaking vessels in level ice. In order to investigate low-velocity ice resistance, specific formulas were derived from full scale trials of icebreaker performance in ice at 1 m/s speeds. These prediction formulas include parametric influences for different vessel dimensions, hull forms, hull surface conditions, ice strengths and ambient temperatures. The resistance formula is written as:
R = 0.08 + 0.0HCsChBai L02T 01H125 klk2
k = (1 - 0.0083(t + 30))(0.63 + 0.00074<rf) (10)
k2 = (1 + 0.0018(90 - f )14 )(1 + 0.04(< - 5)15)
where R is total ice resistance in MN; Cs is water salinity coefficient (0 fresh, 1 saline); Ch is hull condition coefficient (1 inerta, 1.33 bare steel); B, T and L are ship beam at waterline, draft and waterline length in meter ; f and <p are average flare angle and buttock angle in degree; t is air temperature; uf is fexural strength; H is ice thickness.
Riska formulas
Riska et al. (1997) proposed a level ice resistance formula by modifying the formulations of Lindqvist (1989). The formulation is based on a set of empirical coefficients, derived from full- scale tests of a number of ships in ice conditions in the Baltic Sea. The main resistance formula is given in Eq. (11), while constants are found in Table 1.
Rlce = C + c2V
C = f BLparh + (1 + 0.021mBh + ULJh) + f4 BLbwh) (11)
— +1 B
C2 = (1 + 0.063« gihi15 + g2 Bh) + g3 h (1 + 1.2T / B)j=
where V, B, T and L are vessel speed, breadth, draught and length, ht is ice thickness, ^ is the stem angle in degrees and Lbow and Lpar are the length of bow and parallel sides section, respectively.
Table 1 Constants in Riska formulation for ice resistance in level ice.
Symbol Value Unit
f: 0.23 [kN/m3]
f2 4.58 [kN/m3]
f3 1.47 [kN/m3]
f4 0.29 [kN/m3]
g1 18.9 [kN/(m/s*m15)]
g2 0.67 [kN/(m/s*m)]
g3 1.55 [kN/(m/s*m25)]
Jeong formulas
Jeong (2010) proposed new ice resistance prediction formula for standard icebreaker model using component method of ice resistance and also predicted the model test results to full-scale using calculated non-dimensional coefficients. The formulas are presented as follow:
R, - 13.14V2 + C^pghfiT + CcFhapiBhy2 + CBRSNP PiBh,V2
where Ri total ice resistance; CB, CC and CBr are coefficient of ice buoyancy resistance, coefficient of ice clearing resistance and coefficient of ice breaking resistance; Fh and SN are Froude number and strength number; a is index of Froude number; p is index of Strength number; pj and pw are ice and water density; Ap is water density minus ice density; g is gravitational constant; h is ice thickness; B and T are beam and draft of the ship; V is ship speed; cf = flexural strength of ice. The constants used are shown in Table 2.
Table 2 Constants in Jeong formulation for ice resistance in level ice.
Symbol Value
Cb 0.5
Cc 1.11
CBR 2.73
a 1.157
P 1.54
Model test description
Zhou et al. (2013) carried out a series of model tests in the ice basin of the Marine Technology Group in the Aalto University, where the icebreaking tanker MT Uikku was deployed. A scaling factor of X=31.56 was used for all tests. The
model was trimmed to even keel without heel angle, so that the center of buoyancy and the center of gravity were in the same longitudinal and transverse location. The particulars of the model and full scale vessel are given in Table 3.
Table 3 Full-scale primary dimensions of MT Uikku.
Item Notation Unit Value
Length L m 150.0
Length of bow Lbow m 39
Length of parallel Lpar m 65
Breadth moulded B m 21.3
Tested draft T m 9.5
Bow waterline angle а deg 21
Bow stem angle Ф deg 30
average flare angle W deg 58
The instrumentations, including a compact 6-component force transducer, one-directional load cell, a dynamic measurement unit and two cameras, were installed to the ship model. The configuration is shown in Fig. 5. The ship model was constrained so that the six force components could be measured. An upper frame with a stiff tube and long beam was used to connect the towing carriage with the load measurement units rigidly, which were attached to the ship model. The measurement data was sampled at a rate of 107 Hz.
Fig. 6 Icebreaking at bow (Zhou et al., 2013). Fig. 7 Open channel after test (Zhou et al., 2013).
As shown in Fig. 6, the ship model was towed straight against unbroken ice at constant predefined speeds by the carriage above the ship model. The open channel created by travelling of the ship model is shown in Fig. 7. Test matrix and measured ice properties are presented in Table 4.
Table 4 Test matrix with head on cases and measured ice properties.
Ice sheet Test No. hi [m] Ob [kPa] Oc [kPa] Ei [MPa] Vi [m/s] V [deg]
I 103 0.77 724 1748 929 0.2 0
104 0.76 844 2192 984 0.5 0
II 205 0.96 920 1840 1685 0.2 0
206 0.95 912 1862 1701 0.5 0
COMPARISONS AND DISCUSSIONS
Time domain simulations are performed in order to compare model test results. In addition, ice resistances by empirical and analytical formulas are calculated and compared with model test results. Some parameters could be derived from Tables 1-4. The other parameters used in the calculation are shown in Table 5.
Table 5 The other parameters in the calculation.
Items Unit Value
Poisson's ratio v - 0.33
Ice-hull friction coefficient / - 0.04
Water density pw kg/m3 989
Ice density p kg/m3 906
Cs - 1.0
Ch - 1.0
Cb - 0.5
Cc - 1.11
CBR - 2.73
Numerical simulation and model test results
Typical time histories of ice forces in the longitudinal direction for all head-on tests from both model test and numerical simulation are presented in Fig. 8. The presented time signals are filtered with an extreme low cut-off frequency in order to show the general evolution of ice forces in time domain. A comparison of results in a quantitative way is made, where the mean values, standard deviations and steady value of the ice forces at the end stage of model test are presented. The differences between simulated data and model test data are explained and identified to some extent.
a) Comparison of test 103 in time domain
400 600 800 1000 1200 1400 1600 1800 2000
Time [s]
b) Comparison of test 104 in time domain
■1800--1-1
400 600 800 1000 1200 1400 1600 1800
Time [s]
c) Comparison of test 205 in time domain Experiment Simulation -Filtered experiment-----Filtered simulation
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400
Time [s]
d) Comparison of test 206 in time domain
200 300 400 500 600 700 800 900 1000
Time [s]
Fig. 8 Comparisons between numerical simulation and model test in time domain.
Table 6 Experimental and simulated level ice forces for zero heading cases.
Test No. Model test (kN) Numerical simulation (kN)
Mean Std. Steady Mean Std. Steady
103 470 120 500 440 190 440
104 560 120 640 530 230 530
205 670 220 770 630 290 640
206 720 200 800 690 320 700
From Fig. 8, it is clear that the numerical simulations capture main trend that was obtained from model test. The smoothed ice forces from simulation approaches to the measured force from the experiment. From Table 6, it is seen that a good agreement is achieved for the mean values of ice forces in the longitudinal direction (ice resistance).
The simulated standard deviation and peak forces are a little higher than the measured value. This is attributed to inherent limitation of the numerical model. In the present numerical model, it is assumed that the broken ice is cleared away immediately after it is broken away from the intact level ice. This is not the case in reality. The broken ice cusp will rotate against the hull, especially at shoulder area to the mid-hull area. After the cusp is broken away from the intact ice, it might be crushed by the hull side in the rotation process until the cusp is parallel to the hull side, which brings in a large transverse force. This phenomenon is hard to reproduce since this process is random in nature. It is very difficult to predict exactly when and where the crushing of broken cusp occurs.
In addition, there exist some deviations between the simulated and measured forces at the steady stage (see Table 6). It should be noted that simulated ice resistances in steady stage almost keep constant while the resistances are increasing for all cases in the model test. The simulated steady ice forces are 15% lower approximately. The main reason is that the broken ice floes accumulated around the hull steadily in the model test, but the ice force calculated due to broken ice floes is assumed to be constant when using the present numerical model. As time goes, more and more broken ice floes may submerge, slide and accumulate at the bow area and even go under the bottom of the hull. However, this mechanism is not considered in the present numerical model.
Empirical and model test results
The empirically calculated results and model test measurements are shown in Table 7. The corresponding results are plotted in Fig. 9 for comparison.
Comparing ice resistance by Lindqvist formulas and model test, it is found the Lindqvist formulas give the lowest prediction among these methods. The highest deviation is around 50% for test 103. The Lindqvist formulas were well evaluated against full scale tests with different ships, but the comparison was done with ice thickness less than 0.65 m. It might be not applicable to estimate level ice resistance of ship in thick ice (>0.65 m), as used in the model test. Lindqvist formulas were evaluated against full scale measurements of several icebreakers in the Baltic, where the ice conditions are identical. How changes in mechanical properties of the ice affect the resistance is still the greatest uncertainty to use the formulas. The full scale tests were done in ice with bending strength up to 660 kPa, but this value has increased to 920 kpa in the model test 205. The ship model might need much more energy to break ice. Lindqvist formulas do not take the effect of crushing strength into account. This might be reasonable for ice in the Baltic with low crushing strength, but the crushing force could be under estimate when high crushing are very large in the model test. The resistance dependency on vessel speed seems to be reasonable.
The ice resistance calculated with Riska formulas is slightly higher (17%~30%) than the model test results. Similar to Lindqvist formula, the Riska formulation assumes a linear relationship between vessel speed and ice thickness. An important difference is however that Riska does not normalize the velocity, which may influence the result to some degree.
Ice thickness, flexural strength and friction coefficient are important factors in prediction of total ice resistance by Jeong's theoretical method. Only one case was predicted well. They seem to be sensitive to vessel speed, which might bring some trouble to ice resistance estimation. Ice buoyancy resistance is calculated from pre-sawn ice test with creeping speed test. The di-mensionless ice buoyancy resistance coefficient, CB was found to be 0.50 for friction coefficient ^=0.02 according to Jeong (2010). However, this parameter is 0.04 in the model tests. It is unclear that how the friction coefficient affects the buoyancy resistance coefficient.
Table 7 Experimental and simulated level ice forces for head-on cases (in kN).
Case No. Model test Numerical simulation Empirical and analytical formulas
Lindqvist Riska Jeong Keinonen1 Keinonen2 Average
1 (Test 103) 470 440 320 610 330 650 510 440
2 (Test 104) 560 530 380 630 520 690 600 530
3 (Test 205) 670 630 510 800 560 930 740 650
4 (Test 206) 720 690 560 840 800 920 810 750
Fig. 9 Comparison of experimental and simulated ice resistance.
It should be noted that the lowest vessel speed for using Keinonen formula is 1 m/s. The calculated results at this speed are shown in Table 7, noted by Keinonen1. Apparently, it is very conservative compared to model test results. But in the model tests, all vessel speeds are below 1 m/s. Correction must be made to consider the effect of vessel speeds at the range below 1 m/s. Herein, this effect presented in Lindqvist formulas is suggested. The ice crushing and breaking forces are dominant. So the correction factor Cf is a function of vessel speed, ice thickness and gravity acceleration, which is express as (Vi= 1 m/s):
C, (13)
The corrected ice resistance is equal to the ice resistance at 1 m/s with Keinonen formula multiplied by correction factor. The correction factors for each case are shown in Table 8. Corrected ice resistances Keinonen2 are lower than those uncorrected, but still over measured ice resistance.
638 Int. J. Nav. Archit. Ocean Eng. (2015) 7:626~639
Table 8 Correction factors for Keinonen formula.
Case No. Correction factor
1 (Test103) 0.78
2 (Test104) 0.87
3 (Test 205) 0.80
4 (Test 206) 0.88
Based on comparisons made above, it is found that every empirical and analytical method has its own advantages and disadvantages. Care should be taken when designers use these formulas. The conditions under which the formulas are suitable should be well evaluated before using them. Under the condition of unknown ice conditions, a general suggestion is to use all those formulas to do a comprehensive analysis to balance pros and cons inherently in those formulas. As shown in Table 7, ice resistances calculated by all empirical formulas described in the paper (except Keinonen1) are averaged. The averaged value agrees well with measured ones (See Fig. 9).
CONCLUSIONS
This paper aims to study ice resistance in both numerical and experimental ways. A numerical simulation method and some empirical and analytical methods to calculate ice resistance are presented in details. The calculated results are compared against model test results.
All empirical methods mentioned in the present paper could predict ice resistance at different accuracy, but any of them could give a good estimation for all cases. A more practical suggestion is to use all empirical formulas and calculate the average value. However, they could be only used in the conceptual design phase and cannot take the details of hull lines into consideration. The numerical simulation method is a relatively complex method considering the bow shape, but could provide reasonable prediction of ice resistance in time domain.
ACKNOWLEDGEMENTS
Project supported by the National Natural Science Foundation of China (Grant Nos. 11102048, 11302057) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20132304120028) .
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