07_Ship_Resistance

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NE224 Spring 2023 Instructor: An Wang 8 Ship Resistance 8.1 Components of resistance — Frictional resistance: — Wave-making resistance — Eddy-making resistance — Air resistance 8.2 Approaches to study ship resistance — Theoretical study — Numerical simulations — Experimental study : full scale experiments, model tests 8.3 Dimensional analysis In order to determine the experimental parameters which represent the true physics of the ship (prototype), proper scaling laws (law of comparison) need to be determined. We first need to determine the dimensional parameters that are relevant to the physics we are trying to study. Relevant parameters to ship resistance ( R , dimension: MLT − 2 ): Ship length: L . (dimension: L) Ship speed: V . (dimension: LT − 1 ) Mass density of water: ρ . (dimension: ML − 3 ) Kinematic viscosity of water: ν . (dimension: L 2 T − 1 ) gravitational acceleration: g . (dimension: LT − 2 ) Hull form coefficients ( C B , C M , C P , etc.). For a model test, the model is scaled down from the prototype of the same hull form coefficients. Thus, hull form coefficients can be removed from the relationship. Assume the resistance can be written as the following function: R = f ( L, V, ρ, ν, g ) (8.1) We don’t know the mathematical form of the function. Buckingham PI theorem: 1

NE224 Spring 2023 Instructor: An Wang If the total number of dimensions of a problem is m and a total number of variables of the problem is n , the relationship among these variables can then be written as relationship among m − n dimensionless parameters, π 1 , π 2 , ..., π n − m . In the current problem, in equation 8.1, R has the dimension of force, or MLT − 2 , L has the dimension of L, V has the dimension of LT − 1 , ρ has the dimension of ML − 3 , ν has the dimension of L 2 T − 1 , g has the dimension of LT − 2 . There are a total of 3 dimensions involved, i.e., mass M, length L and time T. There are a total of 6 dimensional parameters in equation 8.1, i.e., R , L , V , ρ , ν and g . Therefore, according to Bucking PI theorem, the physics represented by equation 8.1 can be reduced to a relationship among 6 − 3 = 3 dimensionless variables. We call them π 1 , π 2 and π 3 . The new dimensionless relationship can be written as π 1 = f ′ ( π 2 , π 3 ) (8.2) Next, we need to determine what is the form of π 1 , π 2 and π 3 . Step 1: Determine the primary dimensions involved in all these variables by observation. As mentioned previously, in the current problem, they are: mass M, length L and time T. Step 2: Determine the repeating (primary) variables. The repeating variables must be dimensionally independent from each other (the dimension of any one of the three variables cannot be formed by the other two variables) and the set of the three primary variable must cover all the three primary dimensions. In the current problem, we can choose L , ρ and V as primary variables. Note: There may be more than one set of repeating variables. But as long as they are dimensionally independent and covers all the primary dimensions, the result of the dimen- sional analysis will be equivalent. Step 3: Multiply each of the remaining variables by exponential form of the primary variables of certain power to form a dimensionless parameter, π i . For example, in the present problem, L , ρ and V are the primary variables, for the first of the remaining variable, R , to form a dimensionless parameter, R should be multiplied by ρ a L b V c to form a dimensionless parameter π 1 = Rρ a V b L c The dimension of the above equation should be one (dimensionless): ( MLT − 2 ) · ( ML − 3 ) a ( LT − 1 ) b ( L ) c = M 0 L 0 T 0 2

NE224 Spring 2023 Instructor: An Wang so 1 + a = 0 1 − 3 a + b + c = 0 − 2 − b = 0 We get a = − 1, b = − 2 and c = − 2. Therefore, the first dimensionless parameter is π 1 = R ρV 2 L 2 . Similarly, dimensionless parameters associated with the other two remaining variables, ν and g , can be found: π 2 = ν V L and π 3 = gL V 2 Note: One can multiply each dimensionless parameter by a constant, or multiply one dimensionless parameter with another one, or give the dimensionless parameter a different power. The physics represented by equation 8.2 will remain the same. By convention, we use π 2 = V L ν and π 3 = V √ gL The resulting relationship in dimensionless parameters is: R ρV 2 L 2 = f ′ V L ν , V √ gL (8.3) In equation 8.3, the L 2 has the dimension of area. We can replace L 2 with a characteristic area, S , then equation 8.3 becomes: π 1 = C = R 0 . 5 ρV 2 S (8.4) , where 0 . 5 ρV 2 is a characteristic pressure and 0 . 5 ρV 2 S is a characteristic force. Therefore, the dimensionless parameter C in the above equation is essentially the ratio of resistance to a characteristic force. This parameter C is called resistance coefficient . The second parameter is called the Reynolds number . Re = V L ν (8.5) Reynolds number Re is the ratio between inertial force and viscous force. 3

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NE224 Spring 2023 Instructor: An Wang The third parameter is called the Froude number . Fr = V √ gL (8.6) Froude number Fr is the ratio between inertial force and gravity force. Therefore, the function of these dimensionless parameter can be rewritten as: C = f ′ ( Re , Fr ) (8.7) The advantage of the above dimensionless equation is obvious. The number of parameters is reduced from 6 to 3. No matter what is the scale ratio between the prototype and the model (unless the scale ratio is too large and results in different physics), the phenomenon will be governed by the same mathematical function described by equation 8.7. In other words, for two models with different scale ratios, as long as Re and Fr are the same for the two models, R will also be the same for the two models. However, the mathematical form of equation 8.7 is still unknown and cannot be de- termined by dimensional analysis. Experiments or physical models are needed to provide further details of equation 8.7. 8.4 Similitude Assume the linear scaling ratio between the ship and model is λ and λ = L s L m = B s B m = T s T m (8.8) , where subscript ‘ s ’ denote “ship” and subscript ‘ m ’ denote “model” and L , B , T denote the length, beam and draft, respectively. 8.4.1 Dynamic similitude Dynamic similitude: as long as both Reynolds number ( Re ) and Froude number ( Fr ) can be matched between the ship and model, the resistance coefficient C of ship and model will be equal (see equation 8.7), i.e., R s 0 . 5 ρ s V 2 s S s = R m 0 . 5 ρ m V 2 m S m or R s R m = ρ s ρ m V 2 s V 2 m S s S m 4

NE224 Spring 2023 Instructor: An Wang Since S s /S m = λ 2 , R s R m = ρ s ρ m V 2 s V 2 m λ 2 (8.9) In a model test with complete dynamic similitude satisfied, in order to find the resistance of the ship, R s , from the resistance measured in the corresponding model test, R m , we also need to find out what is the ratio between the ship speed and the model speed, V s /V m . In the following, the two requirements for complete dynamic similitude ( Re s = Re m and Fr s = Fr m ) will be used to determine the speed ratio. 8.4.2 Matching Froude number In order to match the Froude number between model and ship, Fr s = Fr m , or V s √ gL s = V m √ gL m or V s V m = r L s L m = √ λ (8.10) or V m = V s √ λ (8.11) The speeds of ship and model, V s and V m , respectively, defined by equation 8.10 and 8.11 are called corresponding speeds. 8.4.3 Matching Reynolds number In order to match the Reynolds number between model and ship, Re s = Re m , or V s L s √ ν s = V m L m √ ν m or V m = L s L m ν m ν s V s = λ ν m ν s V s Since the water used in model test and the water where the ship operates have nearly the same viscosity, ν m ≈ ν s , the equation above can be reduced to V m = λV s The above equation for matching Reynolds number is in conflict with equation 8.11, which is needed for matching Froude number between model and ship. This conflict is called 5

NE224 Spring 2023 Instructor: An Wang model speed paradox . This paradox is caused by the fact that it is not feasible to use a liquid media with a specific viscosity so that both Re and Fr can be matched simultaneously. In other words, it is impossible to achieve complete dynamic similitude between ship and model. Note: — The complete dynamic similitude basically says if the two dimensionless parameters ( Re and Fr ) can be matched simultaneously, the equivalency of the physics behind each parameter can be established between the model and ship. Reynolds number: governing parameter for the friction resistance Froude number: governing parameter for the wave-making resistance. Therefore, the paradox indicates that the similitude of friction resistance and wave- making resistance cannot be achieved simultaneously. — In practice, matching Re requires model speed V m = λV s , which is too fast and unrealistic for a model test. On the other hand, matching Fr only requires model speed V m = V s / √ λ , which can be easily achieved in a towing tank. Therefore, model test usually only matches Froude number and set the model speed according to equation 8.11. By matching Froude number, model speed is set to V m = V s / √ λ . Then equation 8.9 can be written as R s R m = ρ s ρ m V 2 s V 2 m λ 2 = ρ s ρ m λ 3 Since the volume of displacement ratio ∇ s / ∇ m = λ 3 , the above equation can be written as R s R m = ρ s ρ m ∇ s ∇ m or R s R m = ∆ s ∆ m (Complete dynamic similitude, cannot be achieved in practice) 8.4.4 Froude’s hypothesis Because Reynolds number and Froude number cannot be matched simultaneously, one solution to this paradox is to divide the total resistance into two parts. It is assumed that one part is governed by Reynolds number and the other part is governed by Froude number. R T = R F + R R (8.12) 6

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NE224 Spring 2023 Instructor: An Wang Divide the above equation by 0 . 5 ρV 2 S , we get C T = C F + C R (8.13) where C T = R T 0 . 5 ρV 2 S and C F = R F 0 . 5 ρV 2 S and C R = R R 0 . 5 ρV 2 S (8.14) . According to the assumption, C F is only a function of Re and C R is only a function of Fr , therefore, C T = C F + C R = f 1 ( Re ) + f 2 ( Fr ) (8.15) R F is the frictional resistance and C F is the frictional resistance coefficient . R R is the residuary resistance and C R is the residuary resistance coefficient . Note: The residuary resistance actually includes both wave-making resistance and eddy- making resistance. The wave-making resistance is dominated by Froude number but eddy- making resistance is not. Therefore, writing C R = f 2 ( Fr ) in equation 8.15 is not exact. However, wave-making resistance is much larger than eddy-making resistance. Therefore, writing residuary resistance as a function of only Fr is approximately correct. Some correc- tion can be added to account for the missing physics in the assumption. The above approach was first introduced by Froude and is called Froude’s hypothesis . 8.5 Procedures to determine the resistance of a ship from model test 1. Model speed: V m = V s / √ λ . Measure total resistance (equal to the towing force) on the model R Tm . 2. Calculate the total resistance coefficient of the model: C Tm = R Tm 0 . 5 ρ m V 2 m S m 3. Calculate the model Reynolds number: Re m = V m L m ν m Use skin friction formula to calculate the model frictional resistance coefficient: C Fm = f 1 ( Re m ). See 8.6 for these skin friction formulas. 7

NE224 Spring 2023 Instructor: An Wang 4. Calculate the model residuary resistance coefficient: C Rm = C Tm − C Fm 5. The ship residuary resistance coefficient equal to the model residuary resistance coefficient: C Rs = C Rm 6. Calculate the ship Reynolds number Re s = V s L s ν s Use the same skin friction formula in step 3 to calculate the frictional resistance coefficient: C Fs = f 1 ( Re s ). 7. Calculate the ship’s total resistance coefficient C Ts = C Fs + C Rs + C A , where C A is a correlation allowance to adjust for the inexact nature of the assumption described above. 8. Calculate the ship resistance: R Ts = C Ts ( 0 . 5 ρ s V 2 s S s ) 8.6 Frictional resistance The calculation of frictional resistance of a ship is usually based on the measurements of the resistance of thin, flat planes or planks. 8.6.1 Frictional resistance determined by Plank tests Froude’s friction formula (original form) : R F = fSV n (8.16) , where R F is the fictional resistance in pounds S is the wetted surface area of the plank, in square feet V is the velocity of the plank, in knots. 8

NE224 Spring 2023 Instructor: An Wang f is called friction coefficient. f decreases as length increases. f generally increases as surface roughness increases. n increases with surface roughness, but depends little on length. Experimental data suggests the values for f and n and the Froude’s friction formula can be written as: R F = fSV 1 . 825 (8.17) where f = 0 . 00871 + 0 . 0530 L + 8 . 8 (seawater) (8.18) f = 0 . 00849 + 0 . 0516 L + 8 . 8 (fresh water) (8.19) and formulas 8.18 and 8.19 for f apply to a water temperature of 15 ◦ C (59 ◦ F) (ITTC has adopted as a standard). For other temperatures, f should be decreased by 0.43% per +1 ◦ C, or by 0.24% per +1 ◦ F. 8.6.2 Modern formulations for frictional resistance The 1947 ATTC Line 0 . 242 √ C F = log 10 ( Re · C F ) (8.20) The ITTC 1957 Line C F = 0 . 075 (log 10 Re − 2) 2 (8.21) 8.7 Correlation allowance Correlation allowance: when calculating the total resistance coefficient of a ship ( C Ts ), an additional coefficient added to the frictional resistance coefficient ( C Fs ) and residuary resistance coefficient ( C Rs ): C Ts = C Fs + C Rs + C A (8.22) Note: the total resistance coefficient of the model ( C Tm ) still consists of two parts, based on Froude’s hypothesis, i.e., the model’s frictional resistance coefficient ( C Fm ) and residuary resistance coefficient ( C Rm ): C Tm = C Fm + C Rm (8.23) 9

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NE224 Spring 2023 Instructor: An Wang In a model test with Froude number matched with prototype, C Rm = C Rs . The purpose of C A is for adjusting ship’s resistance coefficients obtained from the analysis of model test results, in order to account for inaccuracy of the Froude hypothesis and other factors that cannot be accurately scaled (for example, surface roughness of a ship). C A typically decreases as ship’s length increases. Ship length on waterline Correlation allowance meters feet C A 50-150 160-490 +0 . 40 × 10 − 3 150-210 490-690 +0 . 20 × 10 − 3 210-260 690-850 +0 . 10 × 10 − 3 260-300 850-980 0 300-350 980-1,150 − 0 . 10 × 10 − 3 350-450 1,150-1,480 − 0 . 25 × 10 − 3 Table 1: Correlation allowance with ITTC Line (Keller, J. A., 1973.). The value of C A are obtained by comparing the ship’s total resistance coefficient from the analysis of model test data and that from a full-scale test. 8.8 Effective power Effective power: the power needed to tow a ship through perfectly smooth water a given speed or the power required to overcome the total ship resistance at that speed: P E = R T V (8.24) , where P E is the effective power R T is the total ship resistance V is the ship speed. Typically English horsepower (550 ft · lb/s) is used as the unit for effective power, called the effective horsepower (EHP) : EHP = R T V 550 ( R T in lb, V in ft/s) (8.25) or EHP = R T V k 325 . 6 ( R T in lb, V k in knots) (8.26) 10

NE224 Spring 2023 Instructor: An Wang In SI units, the ship effective power ( P E ) is usually expressed in kW and total resistance R T is usually in kN. Example: (Zubaly example 8-1) A resistance test is conducted on a model of a con- tainership whose characteristics are: L WL = 780 . 0 ft B = 101 . 8 ft T = 33 . 7 ft C B = 0 . 580 S = 93 , 590 ft 2 . The model has a length on waterline of 21.97 ft. It is tested in the bare hull (no appendages) condition in 66 ◦ F fresh water. When towed at the model speed corresponding to the design ship speed of 24.5 knots, the towing force is measured at 11.54 pounds. Determine the bare hull effective horsepower of the ship in standard seawater (59 ◦ F) at design speed. Calculate also the power to be added to overcome appendage resistance (estimated to be 5% of the bare hull resistance) and still air resistance if A pt = 54 , 000 ft 2 . Solutions: The scaling ratio λ = L WLs L WLm = 780 ft 21 . 91 ft = 35 . 50 The ratio between characteristic area: S s S m = λ 2 = 35 . 50 2 = 1260 . 25 Model characteristic area: S m = S s λ 2 = 93 , 590 ft 2 35 . 50 2 = 74 . 263 ft 2 Model in fresh water at 66 ◦ F (p. 336-337, Zubaly): — Density: ρ m = 1 . 9371 lb · s 2 / ft 4 — Kinematic viscosity: ν m = 1 . 1103 × 10 − 5 ft 2 / s. Model in sea water at 59 ◦ F: — Density: ρ s = 1 . 9905 lb · s 2 / ft 4 — Kinematic viscosity: ν s = 1 . 2791 × 10 − 5 ft 2 / s. Ship speed: V s = 24 . 5 knot = 24 . 5 × 1 . 68781 ft / s = 41 . 35 ft / s 11

NE224 Spring 2023 Instructor: An Wang Model corresponding speed (matching Froude number): V m = V s √ λ = 41 . 35 ft / s √ 35 . 50 = 6 . 94 ft / s Total resistance of model: R Tm = 11 . 54 lb. Model’s total resistance coefficient C Tm = R Tm 0 . 5 ρ m V 2 m S m = 11 . 54 lb 0 . 5 × 1 . 9371 lb · s 2 / ft 4 × (6 . 94 ft / s) 2 × 74 . 263 ft 2 = 3 . 331 × 10 − 3 Model Reynolds number: Re m = V m L m ν m = 6 . 94 ft / s × 21 . 97 ft 1 . 1103 × 10 − 5 ft 2 / s = 1 . 3732 × 10 7 Use ITTC 1957 skin friction formula (equation 8.21) to calculate the frictional resistance coefficient for model: C Fm = 0 . 075 (log 10 Re m − 2) 2 = 2 . 841 × 10 − 3 Therefore, the model’s residuary resistance coefficient: C Rm = C Tm − C Fm = 3 . 331 × 10 − 3 − 2 . 841 × 10 − 3 = 0 . 49 × 10 − 3 The ships residuary resistance coefficient equals to that of the model because Fr has been matched: C Rs = C Rm = 0 . 49 × 10 − 3 Ship Reynolds number: Re s = V m L s ν m = 41 . 35 ft / s × 780ft 1 . 2791 × 10 − 5 ft 2 / s = 2 . 5215 × 10 9 Use ITTC 1957 skin friction formula (equation 8.21) to calculate the frictional resistance coefficient for ship: C Fs = 0 . 075 (log 10 Re s − 2) 2 = 1 . 369 × 10 − 3 The correlation allowance for a 780 ft long ship is (according to table 1): C A = +0 . 1 × 10 − 3 The ship’s total resistance coefficient: C Ts = C Fs + C Rs + C A = 1 . 369 × 10 − 3 + 0 . 49 × 10 − 3 + 0 . 1 × 10 − 3 = 1 . 959 × 10 − 3 12

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NE224 Spring 2023 Instructor: An Wang The total resistance for the ship: R s = C Ts (0 . 5 ρ s V 2 s S s ) = 1 . 959 × 10 − 3 × 0 . 5 × 1 . 9905lb · s 2 / ft 4 × (41 . 35 ft / s) 2 × 93 , 590 ft 2 = 311995 lb Effective horsepower of the ship: EHP = R T V k 325 . 6 = 311995 × 41 . 35 550 = 23456 hp 13